Lessons In Electric Circuits, Volume III Semiconductors Circuits 3

User Manual: Lessons in Electric Circuits Volume 3

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Fifth Edition, last update April 05, 2009

2

Lessons In Electric Circuits, Volume III –
Semiconductors
By Tony R. Kuphaldt
Fifth Edition, last update April 05, 2009

i
c 2000-2010, Tony R. Kuphaldt
This book is published under the terms and conditions of the Design Science License. These
terms and conditions allow for free copying, distribution, and/or modification of this document
by the general public. The full Design Science License text is included in the last chapter.
As an open and collaboratively developed text, this book is distributed in the hope that
it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the Design Science
License for more details.
Available in its entirety as part of the Open Book Project collection at:
www.ibiblio.org/obp/electricCircuits

PRINTING HISTORY
• First Edition: Printed in June of 2000. Plain-ASCII illustrations for universal computer
readability.
• Second Edition: Printed in September of 2000. Illustrations reworked in standard graphic
(eps and jpeg) format. Source files translated to Texinfo format for easy online and printed
publication.
• Third Edition: Printed in January 2002. Source files translated to SubML format. SubML
is a simple markup language designed to easily convert to other markups like LATEX,
HTML, or DocBook using nothing but search-and-replace substitutions.
• Fourth Edition: Printed in December 2002. New sections added, and error corrections
made, since third edition.
• Fith Edition: Printed in July 2007. New sections added, and error corrections made,
format change.

ii

Contents
1 AMPLIFIERS AND ACTIVE DEVICES
1.1 From electric to electronic . . . . . . .
1.2 Active versus passive devices . . . . .
1.3 Amplifiers . . . . . . . . . . . . . . . .
1.4 Amplifier gain . . . . . . . . . . . . . .
1.5 Decibels . . . . . . . . . . . . . . . . .
1.6 Absolute dB scales . . . . . . . . . . .
1.7 Attenuators . . . . . . . . . . . . . . .

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1
1
3
3
6
8
14
16

2 SOLID-STATE DEVICE THEORY
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . .
2.2 Quantum physics . . . . . . . . . . . . . . . . . . .
2.3 Valence and Crystal structure . . . . . . . . . . .
2.4 Band theory of solids . . . . . . . . . . . . . . . . .
2.5 Electrons and “holes” . . . . . . . . . . . . . . . . .
2.6 The P-N junction . . . . . . . . . . . . . . . . . . .
2.7 Junction diodes . . . . . . . . . . . . . . . . . . . .
2.8 Bipolar junction transistors . . . . . . . . . . . . .
2.9 Junction field-effect transistors . . . . . . . . . . .
2.10 Insulated-gate field-effect transistors (MOSFET)
2.11 Thyristors . . . . . . . . . . . . . . . . . . . . . . .
2.12 Semiconductor manufacturing techniques . . . .
2.13 Superconducting devices . . . . . . . . . . . . . . .
2.14 Quantum devices . . . . . . . . . . . . . . . . . . .
2.15 Semiconductor devices in SPICE . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . .

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27
27
28
41
47
50
55
58
60
65
70
73
75
80
83
91
93

3 DIODES AND RECTIFIERS
3.1 Introduction . . . . . . . .
3.2 Meter check of a diode . .
3.3 Diode ratings . . . . . . .
3.4 Rectifier circuits . . . . .
3.5 Peak detector . . . . . . .
3.6 Clipper circuits . . . . . .

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97
98
103
107
108
115
117

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iii

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CONTENTS

iv
3.7 Clamper circuits . . . . . . . .
3.8 Voltage multipliers . . . . . . .
3.9 Inductor commutating circuits
3.10 Diode switching circuits . . . .
3.11 Zener diodes . . . . . . . . . . .
3.12 Special-purpose diodes . . . . .
3.13 Other diode technologies . . . .
3.14 SPICE models . . . . . . . . . .
Bibliography . . . . . . . . . . . . . .

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121
123
130
132
135
143
163
163
171

4 BIPOLAR JUNCTION TRANSISTORS
4.1 Introduction . . . . . . . . . . . . . . .
4.2 The transistor as a switch . . . . . . .
4.3 Meter check of a transistor . . . . . .
4.4 Active mode operation . . . . . . . . .
4.5 The common-emitter amplifier . . . .
4.6 The common-collector amplifier . . . .
4.7 The common-base amplifier . . . . . .
4.8 The cascode amplifier . . . . . . . . .
4.9 Biasing techniques . . . . . . . . . . .
4.10 Biasing calculations . . . . . . . . . .
4.11 Input and output coupling . . . . . . .
4.12 Feedback . . . . . . . . . . . . . . . . .
4.13 Amplifier impedances . . . . . . . . .
4.14 Current mirrors . . . . . . . . . . . . .
4.15 Transistor ratings and packages . . .
4.16 BJT quirks . . . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . .

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173
174
176
179
183
189
202
210
218
222
235
247
256
263
264
269
271
278

5 JUNCTION FIELD-EFFECT TRANSISTORS
5.1 Introduction . . . . . . . . . . . . . . . . . . . .
5.2 The transistor as a switch . . . . . . . . . . . .
5.3 Meter check of a transistor . . . . . . . . . . .
5.4 Active-mode operation . . . . . . . . . . . . . .
5.5 The common-source amplifier – PENDING . .
5.6 The common-drain amplifier – PENDING . .
5.7 The common-gate amplifier – PENDING . . .
5.8 Biasing techniques – PENDING . . . . . . . .
5.9 Transistor ratings and packages – PENDING
5.10 JFET quirks – PENDING . . . . . . . . . . . .

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281
281
283
286
288
297
298
298
298
299
299

6 INSULATED-GATE FIELD-EFFECT TRANSISTORS
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . .
6.2 Depletion-type IGFETs . . . . . . . . . . . . . . . . .
6.3 Enhancement-type IGFETs – PENDING . . . . . . .
6.4 Active-mode operation – PENDING . . . . . . . . . .

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301
301
302
311
311

CONTENTS
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12

v

The common-source amplifier – PENDING . .
The common-drain amplifier – PENDING . .
The common-gate amplifier – PENDING . . .
Biasing techniques – PENDING . . . . . . . .
Transistor ratings and packages – PENDING
IGFET quirks – PENDING . . . . . . . . . . .
MESFETs – PENDING . . . . . . . . . . . . .
IGBTs . . . . . . . . . . . . . . . . . . . . . . .

7 THYRISTORS
7.1 Hysteresis . . . . . . . . . . . . . . . .
7.2 Gas discharge tubes . . . . . . . . . .
7.3 The Shockley Diode . . . . . . . . . . .
7.4 The DIAC . . . . . . . . . . . . . . . .
7.5 The Silicon-Controlled Rectifier (SCR)
7.6 The TRIAC . . . . . . . . . . . . . . .
7.7 Optothyristors . . . . . . . . . . . . . .
7.8 The Unijunction Transistor (UJT) . .
7.9 The Silicon-Controlled Switch (SCS) .
7.10 Field-effect-controlled thyristors . . .
Bibliography . . . . . . . . . . . . . . . . . .

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312
312
312
312
312
313
313
313

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317
317
318
322
329
329
341
344
344
350
352
354

8 OPERATIONAL AMPLIFIERS
8.1 Introduction . . . . . . . . . . . . . . . .
8.2 Single-ended and differential amplifiers
8.3 The ”operational” amplifier . . . . . . .
8.4 Negative feedback . . . . . . . . . . . .
8.5 Divided feedback . . . . . . . . . . . . .
8.6 An analogy for divided feedback . . . .
8.7 Voltage-to-current signal conversion . .
8.8 Averager and summer circuits . . . . .
8.9 Building a differential amplifier . . . .
8.10 The instrumentation amplifier . . . . .
8.11 Differentiator and integrator circuits .
8.12 Positive feedback . . . . . . . . . . . . .
8.13 Practical considerations . . . . . . . . .
8.14 Operational amplifier models . . . . . .
8.15 Data . . . . . . . . . . . . . . . . . . . .

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355
355
356
360
366
369
372
378
380
382
384
385
388
392
408
413

9 PRACTICAL ANALOG SEMICONDUCTOR CIRCUITS
9.1 ElectroStatic Discharge . . . . . . . . . . . . . . . . . .
9.2 Power supply circuits – INCOMPLETE . . . . . . . . .
9.3 Amplifier circuits – PENDING . . . . . . . . . . . . . .
9.4 Oscillator circuits – INCOMPLETE . . . . . . . . . . .
9.5 Phase-locked loops – PENDING . . . . . . . . . . . . .
9.6 Radio circuits – INCOMPLETE . . . . . . . . . . . . . .

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415
415
420
422
422
424
424

CONTENTS

vi
9.7 Computational circuits . . . . . . . . . .
9.8 Measurement circuits – INCOMPLETE
9.9 Control circuits – PENDING . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . .

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10 ACTIVE FILTERS

433
455
456
456
459

11 DC MOTOR DRIVES
461
11.1 Pulse Width Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
12 INVERTERS AND AC MOTOR DRIVES
13 ELECTRON TUBES
13.1 Introduction . . . . . . . . . .
13.2 Early tube history . . . . . .
13.3 The triode . . . . . . . . . . .
13.4 The tetrode . . . . . . . . . .
13.5 Beam power tubes . . . . . .
13.6 The pentode . . . . . . . . . .
13.7 Combination tubes . . . . . .
13.8 Tube parameters . . . . . . .
13.9 Ionization (gas-filled) tubes .
13.10Display tubes . . . . . . . . .
13.11Microwave tubes . . . . . . .
13.12Tubes versus Semiconductors

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467
467
468
471
473
474
476
476
479
481
485
488
491

A-1 ABOUT THIS BOOK

495

A-2 CONTRIBUTOR LIST

499

A-3 DESIGN SCIENCE LICENSE

505

INDEX

509

Chapter 1

AMPLIFIERS AND ACTIVE
DEVICES
Contents
1.1
1.2
1.3
1.4
1.5
1.6
1.7

1.1

From electric to electronic . .
Active versus passive devices
Amplifiers . . . . . . . . . . . . .
Amplifier gain . . . . . . . . . .
Decibels . . . . . . . . . . . . . .
Absolute dB scales . . . . . . .
Attenuators . . . . . . . . . . .
1.7.1 Decibels . . . . . . . . . .
1.7.2 T-section attenuator . . . .
1.7.3 PI-section attenuator . . .
1.7.4 L-section attenuator . . .
1.7.5 Bridged T attenuator . . .
1.7.6 Cascaded sections . . . .
1.7.7 RF attenuators . . . . . .

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1
3
3
6
8
14
16
17
19
20
21
21
23
23

From electric to electronic

This third volume of the book series Lessons In Electric Circuits makes a departure from the
former two in that the transition between electric circuits and electronic circuits is formally
crossed. Electric circuits are connections of conductive wires and other devices whereby the
uniform flow of electrons occurs. Electronic circuits add a new dimension to electric circuits
in that some means of control is exerted over the flow of electrons by another electrical signal,
either a voltage or a current.
1

CHAPTER 1. AMPLIFIERS AND ACTIVE DEVICES

2

In and of itself, the control of electron flow is nothing new to the student of electric circuits. Switches control the flow of electrons, as do potentiometers, especially when connected
as variable resistors (rheostats). Neither the switch nor the potentiometer should be new to
your experience by this point in your study. The threshold marking the transition from electric
to electronic, then, is defined by how the flow of electrons is controlled rather than whether or
not any form of control exists in a circuit. Switches and rheostats control the flow of electrons
according to the positioning of a mechanical device, which is actuated by some physical force
external to the circuit. In electronics, however, we are dealing with special devices able to control the flow of electrons according to another flow of electrons, or by the application of a static
voltage. In other words, in an electronic circuit, electricity is able to control electricity.
The historic precursor to the modern electronics era was invented by Thomas Edison in
1880 while developing the electric incandescent lamp. Edison found that a small current
passed from the heated lamp filament to a metal plate mounted inside the vacuum envelop.
(Figure 1.1 (a)) Today this is known as the “Edison effect”. Note that the battery is only necessary to heat the filament. Electrons would still flow if a non-electrical heat source was used.
control
e-1

e-1

(a)

(b)

e-1

+

- +
(c)

Figure 1.1: (a) Edison effect, (b) Fleming valve or vacuum diode, (c) DeForest audion triode
vacuum tube amplifier.
By 1904 Marconi Wireless Company adviser John Flemming found that an externally applied current (plate battery) only passed in one direction from filament to plate (Figure 1.1 (b)),
but not the reverse direction (not shown). This invention was the vacuum diode, used to convert alternating currents to DC. The addition of a third electrode by Lee DeForest (Figure 1.1
(c)) allowed a small signal to control the larger electron flow from filament to plate.
Historically, the era of electronics began with the invention of the Audion tube, a device
controlling the flow of an electron stream through a vacuum by the application of a small
voltage between two metal structures within the tube. A more detailed summary of so-called
electron tube or vacuum tube technology is available in the last chapter of this volume for those
who are interested.
Electronics technology experienced a revolution in 1948 with the invention of the transistor. This tiny device achieved approximately the same effect as the Audion tube, but in
a vastly smaller amount of space and with less material. Transistors control the flow of elec-

1.2. ACTIVE VERSUS PASSIVE DEVICES

3

trons through solid semiconductor substances rather than through a vacuum, and so transistor
technology is often referred to as solid-state electronics.

1.2

Active versus passive devices

An active device is any type of circuit component with the ability to electrically control electron
flow (electricity controlling electricity). In order for a circuit to be properly called electronic,
it must contain at least one active device. Components incapable of controlling current by
means of another electrical signal are called passive devices. Resistors, capacitors, inductors,
transformers, and even diodes are all considered passive devices. Active devices include, but
are not limited to, vacuum tubes, transistors, silicon-controlled rectifiers (SCRs), and TRIACs.
A case might be made for the saturable reactor to be defined as an active device, since it is able
to control an AC current with a DC current, but I’ve never heard it referred to as such. The
operation of each of these active devices will be explored in later chapters of this volume.
All active devices control the flow of electrons through them. Some active devices allow a
voltage to control this current while other active devices allow another current to do the job.
Devices utilizing a static voltage as the controlling signal are, not surprisingly, called voltagecontrolled devices. Devices working on the principle of one current controlling another current
are known as current-controlled devices. For the record, vacuum tubes are voltage-controlled
devices while transistors are made as either voltage-controlled or current controlled types. The
first type of transistor successfully demonstrated was a current-controlled device.

1.3

Amplifiers

The practical benefit of active devices is their amplifying ability. Whether the device in question be voltage-controlled or current-controlled, the amount of power required of the controlling signal is typically far less than the amount of power available in the controlled current.
In other words, an active device doesn’t just allow electricity to control electricity; it allows a
small amount of electricity to control a large amount of electricity.
Because of this disparity between controlling and controlled powers, active devices may be
employed to govern a large amount of power (controlled) by the application of a small amount
of power (controlling). This behavior is known as amplification.
It is a fundamental rule of physics that energy can neither be created nor destroyed. Stated
formally, this rule is known as the Law of Conservation of Energy, and no exceptions to it have
been discovered to date. If this Law is true – and an overwhelming mass of experimental data
suggests that it is – then it is impossible to build a device capable of taking a small amount of
energy and magically transforming it into a large amount of energy. All machines, electric and
electronic circuits included, have an upper efficiency limit of 100 percent. At best, power out
equals power in as in Figure 1.2.
Usually, machines fail even to meet this limit, losing some of their input energy in the form
of heat which is radiated into surrounding space and therefore not part of the output energy
stream. (Figure 1.3)
Many people have attempted, without success, to design and build machines that output
more power than they take in. Not only would such a perpetual motion machine prove that the

CHAPTER 1. AMPLIFIERS AND ACTIVE DEVICES

4

Pinput

Perfect machine

Efficiency =

Poutput
Pinput

Poutput

= 1 = 100%

Figure 1.2: The power output of a machine can approach, but never exceed, the power input
for 100% efficiency as an upper limit.

Pinput

Realistic machine

Poutput

Plost (usually waste heat)

Efficiency =

Poutput
Pinput

< 1 = less than 100%

Figure 1.3: A realistic machine most often loses some of its input energy as heat in transforming it into the output energy stream.

1.3. AMPLIFIERS

5

Law of Conservation of Energy was not a Law after all, but it would usher in a technological
revolution such as the world has never seen, for it could power itself in a circular loop and
generate excess power for “free”. (Figure 1.4)

Pinput

Perpetual-motion
machine

Efficiency =

Pinput

Poutput
Pinput

Poutput

> 1 = more than 100%

Perpetual-motion
machine

P"free"
Poutput

Figure 1.4: Hypothetical “perpetual motion machine” powers itself?
Despite much effort and many unscrupulous claims of “free energy” or over-unity machines,
not one has ever passed the simple test of powering itself with its own energy output and
generating energy to spare.
There does exist, however, a class of machines known as amplifiers, which are able to take in
small-power signals and output signals of much greater power. The key to understanding how
amplifiers can exist without violating the Law of Conservation of Energy lies in the behavior
of active devices.
Because active devices have the ability to control a large amount of electrical power with a
small amount of electrical power, they may be arranged in circuit so as to duplicate the form
of the input signal power from a larger amount of power supplied by an external power source.
The result is a device that appears to magically magnify the power of a small electrical signal
(usually an AC voltage waveform) into an identically-shaped waveform of larger magnitude.
The Law of Conservation of Energy is not violated because the additional power is supplied
by an external source, usually a DC battery or equivalent. The amplifier neither creates nor
destroys energy, but merely reshapes it into the waveform desired as shown in Figure 1.5.
In other words, the current-controlling behavior of active devices is employed to shape DC
power from the external power source into the same waveform as the input signal, producing
an output signal of like shape but different (greater) power magnitude. The transistor or other
active device within an amplifier merely forms a larger copy of the input signal waveform out
of the “raw” DC power provided by a battery or other power source.
Amplifiers, like all machines, are limited in efficiency to a maximum of 100 percent. Usually, electronic amplifiers are far less efficient than that, dissipating considerable amounts of
energy in the form of waste heat. Because the efficiency of an amplifier is always 100 percent

CHAPTER 1. AMPLIFIERS AND ACTIVE DEVICES

6

External
power source

Pinput

Amplifier

Poutput

Figure 1.5: While an amplifier can scale a small input signal to large output, its energy source
is an external power supply.
or less, one can never be made to function as a “perpetual motion” device.
The requirement of an external source of power is common to all types of amplifiers, electrical and non-electrical. A common example of a non-electrical amplification system would
be power steering in an automobile, amplifying the power of the driver’s arms in turning the
steering wheel to move the front wheels of the car. The source of power necessary for the amplification comes from the engine. The active device controlling the driver’s “input signal” is a
hydraulic valve shuttling fluid power from a pump attached to the engine to a hydraulic piston
assisting wheel motion. If the engine stops running, the amplification system fails to amplify
the driver’s arm power and the car becomes very difficult to turn.

1.4

Amplifier gain

Because amplifiers have the ability to increase the magnitude of an input signal, it is useful to
be able to rate an amplifier’s amplifying ability in terms of an output/input ratio. The technical
term for an amplifier’s output/input magnitude ratio is gain. As a ratio of equal units (power
out / power in, voltage out / voltage in, or current out / current in), gain is naturally a unitless
measurement. Mathematically, gain is symbolized by the capital letter “A”.
For example, if an amplifier takes in an AC voltage signal measuring 2 volts RMS and
outputs an AC voltage of 30 volts RMS, it has an AC voltage gain of 30 divided by 2, or 15:

AV =
AV =

Voutput
Vinput
30 V
2V

AV = 15
Correspondingly, if we know the gain of an amplifier and the magnitude of the input signal,
we can calculate the magnitude of the output. For example, if an amplifier with an AC current

1.4. AMPLIFIER GAIN

7

gain of 3.5 is given an AC input signal of 28 mA RMS, the output will be 3.5 times 28 mA, or
98 mA:

Ioutput = (AI)(Iinput)
Ioutput = (3.5)(28 mA)
Ioutput = 98 mA
In the last two examples I specifically identified the gains and signal magnitudes in terms
of “AC.” This was intentional, and illustrates an important concept: electronic amplifiers often
respond differently to AC and DC input signals, and may amplify them to different extents.
Another way of saying this is that amplifiers often amplify changes or variations in input
signal magnitude (AC) at a different ratio than steady input signal magnitudes (DC). The
specific reasons for this are too complex to explain at this time, but the fact of the matter is
worth mentioning. If gain calculations are to be carried out, it must first be understood what
type of signals and gains are being dealt with, AC or DC.
Electrical amplifier gains may be expressed in terms of voltage, current, and/or power, in
both AC and DC. A summary of gain definitions is as follows. The triangle-shaped “delta”
symbol (∆) represents change in mathematics, so “∆Voutput / ∆Vinput ” means “change in output
voltage divided by change in input voltage,” or more simply, “AC output voltage divided by AC
input voltage”:

DC gains
Voltage

AV =

Current

AI =

AC gains

Voutput

AV =

Vinput
Ioutput

AI =

Iinput
Poutput

Power

AP =

Pinput

AP =

∆Voutput
∆Vinput
∆Ioutput
∆Iinput

(∆Voutput)(∆Ioutput)
(∆Vinput)(∆Iinput)

AP = (AV)(AI)

∆ = "change in . . ."
If multiple amplifiers are staged, their respective gains form an overall gain equal to the
product (multiplication) of the individual gains. (Figure 1.6) If a 1 V signal were applied to the
input of the gain of 3 amplifier in Figure 1.6 a 3 V signal out of the first amplifier would be
further amplified by a gain of 5 at the second stage yielding 15 V at the final output.

CHAPTER 1. AMPLIFIERS AND ACTIVE DEVICES

8

Input signal

Amplifier
gain = 3

Amplifier
gain = 5

Output signal

Overall gain = (3)(5) = 15

Figure 1.6: The gain of a chain of cascaded amplifiers is the product of the individual gains.

1.5

Decibels

In its simplest form, an amplifier’s gain is a ratio of output over input. Like all ratios, this
form of gain is unitless. However, there is an actual unit intended to represent gain, and it is
called the bel.
As a unit, the bel was actually devised as a convenient way to represent power loss in telephone system wiring rather than gain in amplifiers. The unit’s name is derived from Alexander Graham Bell, the famous Scottish inventor whose work was instrumental in developing
telephone systems. Originally, the bel represented the amount of signal power loss due to resistance over a standard length of electrical cable. Now, it is defined in terms of the common
(base 10) logarithm of a power ratio (output power divided by input power):

AP(ratio) =

Poutput
Pinput

AP(Bel) = log

Poutput
Pinput

Because the bel is a logarithmic unit, it is nonlinear. To give you an idea of how this works,
consider the following table of figures, comparing power losses and gains in bels versus simple
ratios:

Table: Gain / loss in bels
Loss/gain as
a ratio
Poutput
Pinput

Loss/gain
in bels
Poutput
log
Pinput

Loss/gain as
a ratio
Poutput
Pinput

Loss/gain
in bels
Poutput
log
Pinput

1000

3B

0.1

-1 B

100

2B

0.01

-2 B

10

1B

0.001

-3 B

0B

0.0001

-4 B

1
(no loss or gain)

It was later decided that the bel was too large of a unit to be used directly, and so it became

1.5. DECIBELS

9

customary to apply the metric prefix deci (meaning 1/10) to it, making it decibels, or dB. Now,
the expression “dB” is so common that many people do not realize it is a combination of “deci-”
and “-bel,” or that there even is such a unit as the “bel.” To put this into perspective, here is
another table contrasting power gain/loss ratios against decibels:

Table: Gain / loss in decibels
Loss/gain as
a ratio
Poutput
Pinput

Loss/gain
in decibels
Poutput
10 log
Pinput

Loss/gain as
a ratio
Poutput
Pinput

Loss/gain
in decibels
Poutput
10 log
Pinput

1000

30 dB

0.1

-10 dB

100

20 dB

0.01

-20 dB

10

10 dB

0.001

-30 dB

0 dB

0.0001

-40 dB

1
(no loss or gain)

As a logarithmic unit, this mode of power gain expression covers a wide range of ratios with
a minimal span in figures. It is reasonable to ask, “why did anyone feel the need to invent a
logarithmic unit for electrical signal power loss in a telephone system?” The answer is related
to the dynamics of human hearing, the perceptive intensity of which is logarithmic in nature.
Human hearing is highly nonlinear: in order to double the perceived intensity of a sound,
the actual sound power must be multiplied by a factor of ten. Relating telephone signal power
loss in terms of the logarithmic “bel” scale makes perfect sense in this context: a power loss of
1 bel translates to a perceived sound loss of 50 percent, or 1/2. A power gain of 1 bel translates
to a doubling in the perceived intensity of the sound.
An almost perfect analogy to the bel scale is the Richter scale used to describe earthquake
intensity: a 6.0 Richter earthquake is 10 times more powerful than a 5.0 Richter earthquake; a
7.0 Richter earthquake 100 times more powerful than a 5.0 Richter earthquake; a 4.0 Richter
earthquake is 1/10 as powerful as a 5.0 Richter earthquake, and so on. The measurement
scale for chemical pH is likewise logarithmic, a difference of 1 on the scale is equivalent to
a tenfold difference in hydrogen ion concentration of a chemical solution. An advantage of
using a logarithmic measurement scale is the tremendous range of expression afforded by a
relatively small span of numerical values, and it is this advantage which secures the use of
Richter numbers for earthquakes and pH for hydrogen ion activity.
Another reason for the adoption of the bel as a unit for gain is for simple expression of system gains and losses. Consider the last system example (Figure 1.6) where two amplifiers were
connected tandem to amplify a signal. The respective gain for each amplifier was expressed as
a ratio, and the overall gain for the system was the product (multiplication) of those two ratios:
Overall gain = (3)(5) = 15
If these figures represented power gains, we could directly apply the unit of bels to the task

CHAPTER 1. AMPLIFIERS AND ACTIVE DEVICES

10

of representing the gain of each amplifier, and of the system altogether. (Figure 1.7)
AP(Bel) = log AP(ratio)

Input signal

AP(Bel) = log 3

AP(Bel) = log 5

Amplifier
gain = 3
gain = 0.477 B

Amplifier
gain = 5
gain = 0.699 B

Output signal

Overall gain = (3)(5) = 15
Overall gain(Bel) = log 15 = 1.176 B

Figure 1.7: Power gain in bels is additive: 0.477 B + 0.699 B = 1.176 B.
Close inspection of these gain figures in the unit of “bel” yields a discovery: they’re additive.
Ratio gain figures are multiplicative for staged amplifiers, but gains expressed in bels add
rather than multiply to equal the overall system gain. The first amplifier with its power gain
of 0.477 B adds to the second amplifier’s power gain of 0.699 B to make a system with an overall
power gain of 1.176 B.
Recalculating for decibels rather than bels, we notice the same phenomenon. (Figure 1.8)
AP(dB) = 10 log AP(ratio)

Input signal

AP(dB) = 10 log 3

AP(dB) = 10 log 5

Amplifier
gain = 3
gain = 4.77 dB

Amplifier
gain = 5
gain = 6.99 dB

Output signal

Overall gain = (3)(5) = 15
Overall gain(dB) = 10 log 15 = 11.76 dB

Figure 1.8: Gain of amplifier stages in decibels is additive: 4.77 dB + 6.99 dB = 11.76 dB.
To those already familiar with the arithmetic properties of logarithms, this is no surprise.
It is an elementary rule of algebra that the antilogarithm of the sum of two numbers’ logarithm
values equals the product of the two original numbers. In other words, if we take two numbers
and determine the logarithm of each, then add those two logarithm figures together, then
determine the “antilogarithm” of that sum (elevate the base number of the logarithm – in this
case, 10 – to the power of that sum), the result will be the same as if we had simply multiplied
the two original numbers together. This algebraic rule forms the heart of a device called a
slide rule, an analog computer which could, among other things, determine the products and
quotients of numbers by addition (adding together physical lengths marked on sliding wood,
metal, or plastic scales). Given a table of logarithm figures, the same mathematical trick
could be used to perform otherwise complex multiplications and divisions by only having to
do additions and subtractions, respectively. With the advent of high-speed, handheld, digital
calculator devices, this elegant calculation technique virtually disappeared from popular use.
However, it is still important to understand when working with measurement scales that are

1.5. DECIBELS

11

logarithmic in nature, such as the bel (decibel) and Richter scales.
When converting a power gain from units of bels or decibels to a unitless ratio, the mathematical inverse function of common logarithms is used: powers of 10, or the antilog.

If:
AP(Bel) = log AP(ratio)
Then:
AP(ratio) = 10AP(Bel)
Converting decibels into unitless ratios for power gain is much the same, only a division
factor of 10 is included in the exponent term:

If:
AP(dB) = 10 log AP(ratio)
Then:

AP(dB)

AP(ratio) = 10

10

Example: Power into an amplifier is 1 Watt, the power out is 10 Watts. Find the power
gain in dB.
AP (dB) = 10 log10 (PO / PI ) = 10 log10 (10 /1) = 10 log10 (10) = 10 (1) = 10 dB
Example: Find the power gain ratio AP (ratio) = (PO / PI ) for a 20 dB Power gain.
AP (dB) = 20 = 10 log10 AP (ratio)
20/10 = log10 AP (ratio)
1020/10 = 10log10 (AP (ratio) )
100 = AP (ratio) = (PO / PI )
Because the bel is fundamentally a unit of power gain or loss in a system, voltage or current
gains and losses don’t convert to bels or dB in quite the same way. When using bels or decibels
to express a gain other than power, be it voltage or current, we must perform the calculation
in terms of how much power gain there would be for that amount of voltage or current gain.
For a constant load impedance, a voltage or current gain of 2 equates to a power gain of 4 (22 );
a voltage or current gain of 3 equates to a power gain of 9 (32 ). If we multiply either voltage
or current by a given factor, then the power gain incurred by that multiplication will be the
square of that factor. This relates back to the forms of Joule’s Law where power was calculated
from either voltage or current, and resistance:

CHAPTER 1. AMPLIFIERS AND ACTIVE DEVICES

12

P=

E2
R

P = I2R
Power is proportional to the square
of either voltage or current
Thus, when translating a voltage or current gain ratio into a respective gain in terms of the
bel unit, we must include this exponent in the equation(s):

AP(Bel) = log AP(ratio)
AV(Bel) = log AV(ratio)2

Exponent required

AI(Bel) = log AI(ratio)2
The same exponent requirement holds true when expressing voltage or current gains in
terms of decibels:

AP(dB) = 10 log AP(ratio)
AV(dB) = 10 log AV(ratio)2
AI(dB) = 10 log AI(ratio)

Exponent required

2

However, thanks to another interesting property of logarithms, we can simplify these equations to eliminate the exponent by including the “2” as a multiplying factor for the logarithm
function. In other words, instead of taking the logarithm of the square of the voltage or current
gain, we just multiply the voltage or current gain’s logarithm figure by 2 and the final result
in bels or decibels will be the same:

For bels:
AV(Bel) = log AV(ratio)2
. . . is the same as . . .
AV(Bel) = 2 log AV(ratio)

AI(Bel) = log AI(ratio)2
. . . is the same as . . .
AI(Bel) = 2 log AI(ratio)

For decibels:
AV(dB) = 10 log AV(ratio)2
. . . is the same as . . .
AV(dB) = 20 log AV(ratio)

AI(dB) = 10 log AI(ratio)2
. . . is the same as . . .
AI(dB) = 20 log AI(ratio)

The process of converting voltage or current gains from bels or decibels into unitless ratios
is much the same as it is for power gains:

1.5. DECIBELS

13

If:
AV(Bel) = 2 log AV(ratio)
Then:

AI(Bel) = 2 log AI(ratio)

AV(Bel)

AV(ratio) = 10

2

AI(Bel)

AI(ratio) = 10

2

Here are the equations used for converting voltage or current gains in decibels into unitless
ratios:

If:
AV(dB) = 20 log AV(ratio)
Then:

AI(dB) = 20 log AI(ratio)

AV(dB)

AV(ratio) = 10

20

AI(dB)

AI(ratio) = 10 20

While the bel is a unit naturally scaled for power, another logarithmic unit has been invented to directly express voltage or current gains/losses, and it is based on the natural logarithm rather than the common logarithm as bels and decibels are. Called the neper, its unit
symbol is a lower-case “n.”

AV(ratio) =

Voutput
Vinput

AV(neper) = ln AV(ratio)

AI(ratio) =

Ioutput
Iinput

AI(neper) = ln AI(ratio)

For better or for worse, neither the neper nor its attenuated cousin, the decineper, is popularly used as a unit in American engineering applications.
Example: The voltage into a 600 Ω audio line amplifier is 10 mV, the voltage across a 600
Ω load is 1 V. Find the power gain in dB.
A(dB) = 20 log10 (VO / VI ) = 20 log10 (1 /0.01) = 20 log10 (100) = 20 (2) = 40 dB
Example: Find the voltage gain ratio AV (ratio) = (VO / VI ) for a 20 dB gain amplifier
having a 50 Ω input and out impedance.
AV (dB) = 20 log10 AV (ratio)
20 = 20 log10 AV (ratio)
20/20 = log10 AP (ratio)
1020/20 = 10log10 (AV (ratio) )
10 = AV (ratio) = (VO / VI )
• REVIEW:
• Gains and losses may be expressed in terms of a unitless ratio, or in the unit of bels (B)
or decibels (dB). A decibel is literally a deci-bel: one-tenth of a bel.

CHAPTER 1. AMPLIFIERS AND ACTIVE DEVICES

14

• The bel is fundamentally a unit for expressing power gain or loss. To convert a power
ratio to either bels or decibels, use one of these equations:
•

AP(Bel) = log AP(ratio)

AP(db) = 10 log AP(ratio)

• When using the unit of the bel or decibel to express a voltage or current ratio, it must be
cast in terms of an equivalent power ratio. Practically, this means the use of different
equations, with a multiplication factor of 2 for the logarithm value corresponding to an
exponent of 2 for the voltage or current gain ratio:

•

AV(Bel) = 2 log AV(ratio)

AV(dB) = 20 log AV(ratio)

AI(Bel) = 2 log AI(ratio)

AI(dB) = 20 log AI(ratio)

• To convert a decibel gain into a unitless ratio gain, use one of these equations:
AV(dB)

AV(ratio) = 10 20
AI(dB)
20

AI(ratio) = 10

AP(dB)

•

AP(ratio) = 10

10

• A gain (amplification) is expressed as a positive bel or decibel figure. A loss (attenuation)
is expressed as a negative bel or decibel figure. Unity gain (no gain or loss; ratio = 1) is
expressed as zero bels or zero decibels.
• When calculating overall gain for an amplifier system composed of multiple amplifier
stages, individual gain ratios are multiplied to find the overall gain ratio. Bel or decibel figures for each amplifier stage, on the other hand, are added together to determine
overall gain.

1.6

Absolute dB scales

It is also possible to use the decibel as a unit of absolute power, in addition to using it as an
expression of power gain or loss. A common example of this is the use of decibels as a measurement of sound pressure intensity. In cases like these, the measurement is made in reference to
some standardized power level defined as 0 dB. For measurements of sound pressure, 0 dB is
loosely defined as the lower threshold of human hearing, objectively quantified as 1 picowatt
of sound power per square meter of area.
A sound measuring 40 dB on the decibel sound scale would be 104 times greater than the
threshold of hearing. A 100 dB sound would be 1010 (ten billion) times greater than the threshold of hearing.
Because the human ear is not equally sensitive to all frequencies of sound, variations of the
decibel sound-power scale have been developed to represent physiologically equivalent sound
intensities at different frequencies. Some sound intensity instruments were equipped with
filter networks to give disproportionate indications across the frequency scale, the intent of

1.6. ABSOLUTE DB SCALES

15

which to better represent the effects of sound on the human body. Three filtered scales became
commonly known as the “A,” “B,” and “C” weighted scales. Decibel sound intensity indications
measured through these respective filtering networks were given in units of dBA, dBB, and
dBC. Today, the “A-weighted scale” is most commonly used for expressing the equivalent physiological impact on the human body, and is especially useful for rating dangerously loud noise
sources.
Another standard-referenced system of power measurement in the unit of decibels has been
established for use in telecommunications systems. This is called the dBm scale. (Figure 1.9)
The reference point, 0 dBm, is defined as 1 milliwatt of electrical power dissipated by a 600 Ω
load. According to this scale, 10 dBm is equal to 10 times the reference power, or 10 milliwatts;
20 dBm is equal to 100 times the reference power, or 100 milliwatts. Some AC voltmeters come
equipped with a dBm range or scale (sometimes labeled “DB”) intended for use in measuring
AC signal power across a 600 Ω load. 0 dBm on this scale is, of course, elevated above zero
because it represents something greater than 0 (actually, it represents 0.7746 volts across a
600 Ω load, voltage being equal to the square root of power times resistance; the square root
of 0.001 multiplied by 600). When viewed on the face of an analog meter movement, this dBm
scale appears compressed on the left side and expanded on the right in a manner not unlike a
resistance scale, owing to its logarithmic nature.
Radio frequency power measurements for low level signals encountered in radio receivers
use dBm measurements referenced to a 50 Ω load. Signal generators for the evaluation of radio
receivers may output an adjustable dBm rated signal. The signal level is selected by a device
called an attenuator, described in the next section.

Table: Absolute power levels in dBm (decibel milliwatt)
Power in
watts

Power in
milliwatts

Power in
dBm

Power in
milliwatts

Power in
dBm

1

1000

30 dB

1

0 dB

0.1

100

20 dB

0.1

-10 dB

0.01

10

10 dB

0.01

-20 dB

0.004

4

6 dB

0.001

-30 dB

0.002

2

3 dB

0.0001

-40 dB

Figure 1.9: Absolute power levels in dBm (decibels referenced to 1 milliwatt).
An adaptation of the dBm scale for audio signal strength is used in studio recording and
broadcast engineering for standardizing volume levels, and is called the VU scale. VU meters
are frequently seen on electronic recording instruments to indicate whether or not the recorded
signal exceeds the maximum signal level limit of the device, where significant distortion will

CHAPTER 1. AMPLIFIERS AND ACTIVE DEVICES

16

occur. This “volume indicator” scale is calibrated in according to the dBm scale, but does not
directly indicate dBm for any signal other than steady sine-wave tones. The proper unit of
measurement for a VU meter is volume units.
When relatively large signals are dealt with, and an absolute dB scale would be useful for
representing signal level, specialized decibel scales are sometimes used with reference points
greater than the 1 mW used in dBm. Such is the case for the dBW scale, with a reference
point of 0 dBW established at 1 Watt. Another absolute measure of power called the dBk scale
references 0 dBk at 1 kW, or 1000 Watts.
• REVIEW:
• The unit of the bel or decibel may also be used to represent an absolute measurement of
power rather than just a relative gain or loss. For sound power measurements, 0 dB is
defined as a standardized reference point of power equal to 1 picowatt per square meter.
Another dB scale suited for sound intensity measurements is normalized to the same
physiological effects as a 1000 Hz tone, and is called the dBA scale. In this system, 0
dBA is defined as any frequency sound having the same physiological equivalence as a 1
picowatt-per-square-meter tone at 1000 Hz.
• An electrical dB scale with an absolute reference point has been made for use in telecommunications systems. Called the dBm scale, its reference point of 0 dBm is defined as 1
milliwatt of AC signal power dissipated by a 600 Ω load.
• A VU meter reads audio signal level according to the dBm for sine-wave signals. Because
its response to signals other than steady sine waves is not the same as true dBm, its unit
of measurement is volume units.
• dB scales with greater absolute reference points than the dBm scale have been invented
for high-power signals. The dBW scale has its reference point of 0 dBW defined as 1 Watt
of power. The dBk scale sets 1 kW (1000 Watts) as the zero-point reference.

1.7

Attenuators

Attenuators are passive devices. It is convenient to discuss them along with decibels. Attenuators weaken or attenuate the high level output of a signal generator, for example, to provide
a lower level signal for something like the antenna input of a sensitive radio receiver. (Figure 1.10) The attenuator could be built into the signal generator, or be a stand-alone device.
It could provide a fixed or adjustable amount of attenuation. An attenuator section can also
provide isolation between a source and a troublesome load.
In the case of a stand-alone attenuator, it must be placed in series between the signal
source and the load by breaking open the signal path as shown in Figure 1.10. In addition,
it must match both the source impedance ZI and the load impedance ZO , while providing a
specified amount of attenuation. In this section we will only consider the special, and most
common, case where the source and load impedances are equal. Not considered in this section,
unequal source and load impedances may be matched by an attenuator section. However, the
formulation is more complex.

1.7. ATTENUATORS

17

ZI

ZI

Attenuator

ZO

ZO

Figure 1.10: Constant impedance attenuator is matched to source impedance ZI and load
impedance ZO . For radio frequency equipment Z is 50 Ω.

T attenuator

Π attenuator

Figure 1.11: T section and Π section attenuators are common forms.
Common configurations are the T and Π networks shown in Figure 1.11 Multiple attenuator
sections may be cascaded when even weaker signals are needed as in Figure 1.19.

1.7.1

Decibels

Voltage ratios, as used in the design of attenuators are often expressed in terms of decibels.
The voltage ratio (K below) must be derived from the attenuation in decibels. Power ratios expressed as decibels are additive. For example, a 10 dB attenuator followed by a 6 dB attenuator
provides 16dB of attenuation overall.
10 dB + 6 db = 16 dB
Changing sound levels are perceptible roughly proportional to the logarithm of the power
ratio (PI / PO ).
sound level = log10 (PI / PO )
A change of 1 dB in sound level is barely perceptible to a listener, while 2 db is readily
perceptible. An attenuation of 3 dB corresponds to cutting power in half, while a gain of 3 db
corresponds to a doubling of the power level. A gain of -3 dB is the same as an attenuation of
+3 dB, corresponding to half the original power level.
The power change in decibels in terms of power ratio is:
dB = 10 log10 (PI / PO )
Assuming that the load RI at PI is the same as the load resistor RO at PO (RI = RO ), the
decibels may be derived from the voltage ratio (VI / VO ) or current ratio (II / IO ):

CHAPTER 1. AMPLIFIERS AND ACTIVE DEVICES

18
PO = V O IO = VO 2 / R = IO 2 R
PI = VI II = VI 2 / R = II 2 R

dB = 10 log10 (PI / PO ) = 10 log10 (VI 2 / VO 2 ) = 20 log10 (VI /VO )
dB = 10 log10 (PI / PO ) = 10 log10 (II 2 / IO 2 ) = 20 log10 (II /IO )
The two most often used forms of the decibel equation are:
dB = 10 log10 (PI / PO )

or

dB = 20 log10 (VI / VO )

We will use the latter form, since we need the voltage ratio. Once again, the voltage ratio
form of equation is only applicable where the two corresponding resistors are equal. That is,
the source and load resistance need to be equal.
Example: Power into an attenuator is 10 Watts, the power out is 1 Watt. Find the
attenuation in dB.
dB = 10 log10 (PI / PO ) = 10 log10 (10 /1) = 10 log10 (10) = 10 (1) = 10 dB
Example: Find the voltage attenuation ratio (K= (VI / VO )) for a 10 dB attenuator.
dB = 10= 20 log10 (VI / VO )
10/20 = log10 (VI / VO )
1010/20 = 10log10 (VI /VO )
3.16 = (VI / VO ) = AP (ratio)
Example: Power into an attenuator is 100 milliwatts, the power out is 1 milliwatt. Find
the attenuation in dB.
dB = 10 log10 (PI / PO ) = 10 log10 (100 /1) = 10 log10 (100) = 10 (2) = 20 dB
Example: Find the voltage attenuation ratio (K= (VI / VO )) for a 20 dB attenuator.
dB = 20= 20 log10 (VI / VO )
1020/20 = 10log10 (VI /VO )
10 = (VI / VO ) = K

1.7. ATTENUATORS

dB = attenuation in decibels
Z = source/load impedance (resistive)
K>1
R1
R1
VI
K =
= 10 dB/20
VO
VI
VO
K- 1
R2
R1 = Z
⇐ Ζ⇒
⇐
Ζ⇒
K+ 1
2K
R2 = Z
T attenuator
K2- 1

19

Resistors for
Z = 50
Attenuation
dB
K=Vi/Vo
1.0
1.12
2.0
1.26
3.0
1.41
4.0
1.58
6.0
2.00
10.0
3.16
20.0
10.00

T-section
R1
2.88
5.73
8.55
11.31
16.61
25.97
40.91

R2
433.34
215.24
141.93
104.83
66.93
35.14
10.10

Figure 1.12: Formulas for T-section attenuator resistors, given K, the voltage attenuation ratio,
and ZI = ZO = 50 Ω.

1.7.2

T-section attenuator

The T and Π attenuators must be connected to a Z source and Z load impedance. The Z(arrows) pointing away from the attenuator in the figure below indicate this. The Z-(arrows)
pointing toward the attenuator indicates that the impedance seen looking into the attenuator
with a load Z on the opposite end is Z, Z=50 Ω for our case. This impedance is a constant (50
Ω) with respect to attenuation– impedance does not change when attenuation is changed.
The table in Figure 1.12 lists resistor values for the T and Π attenuators to match a 50 Ω
source/ load, as is the usual requirement in radio frequency work.
Telephone utility and other audio work often requires matching to 600 Ω. Multiply all R
values by the ratio (600/50) to correct for 600 Ω matching. Multiplying by 75/50 would convert
table values to match a 75 Ω source and load.
The amount of attenuation is customarily specified in dB (decibels). Though, we need the
voltage (or current) ratio K to find the resistor values from equations. See the dB/20 term in
the power of 10 term for computing the voltage ratio K from dB, above.
The T (and below Π) configurations are most commonly used as they provide bidirectional
matching. That is, the attenuator input and output may be swapped end for end and still
match the source and load impedances while supplying the same attenuation.
Disconnecting the source and looking in to the right at VI , we need to see a series parallel
combination of R1 , R2 , R1 , and Z looking like an equivalent resistance of ZIN , the same as the
ource/load impedance Z: (a load of Z is connected to the output.)
ZIN = R1 + (R2 ||(R1 + Z))
For example, substitute the 10 dB values from the 50 Ω attenuator table for R1 and R2 as
shown in Figure 1.13.
ZIN = 25.97 + (35.14 ||(25.97 + 50))
ZIN = 25.97 + (35.14 || 75.97 )
ZIN = 25.97 + 24.03 = 50

CHAPTER 1. AMPLIFIERS AND ACTIVE DEVICES

20

This shows us that we see 50 Ω looking right into the example attenuator (Figure 1.13) with
a 50 Ω load.
Replacing the source generator, disconnecting load Z at VO , and looking in to the left, should
give us the same equation as above for the impedance at VO , due to symmetry. Moreover, the
three resistors must be values which supply the required attenuation from input to output.
This is accomplished by the equations for R1 and R2 above as applied to the T-attenuator
below.

R1=26.0

Z

VI
⇐ Ζ⇒
=50

R1
R2= VO
35.1 ⇐ Ζ⇒
=50

Z

T attenuator
10 dB attenuators for matching input/output to Z= 50 Ω.
Figure 1.13: 10 dB T-section attenuator for insertion between a 50 Ω source and load.

1.7.3

PI-section attenuator

The table in Figure 1.14 lists resistor values for the Π attenuator matching a 50 Ω source/ load
at some common attenuation levels. The resistors corresponding to other attenuation levels
may be calculated from the equations.
dB = attenuation in decibels
Z = source/load impedance (resistive)
K>1
R3
VI
K =
= 10 dB/20
VO
2
R3 = Z K - 1
2K
R4 = Z K+ 1
K- 1

VI
⇐ Ζ⇒

R4

R4

VO
⇐ Ζ⇒

Π attenuator

Resistors for Π-section
Z=50.00
Attenuation
dB
K=Vi/Vo
R3
R4
1.0
1.12
5.77 869.55
2.0
1.26
11.61 436.21
3.0
1.41
17.61 292.40
4.0
1.58
23.85 220.97
6.0
2.00
37.35 150.48
10.0
3.16
71.15
96.25
20.0 10.00
247.50
61.11

Figure 1.14: Formulas for Π-section attenuator resistors, given K, the voltage attenuation
ratio, and ZI = ZO = 50 Ω.
The above apply to the π-attenuator below.

1.7. ATTENUATORS

21

R3=71.2

Z
VI
⇐ Ζ⇒
=50

VO
⇐ Ζ⇒
=50

R4=
96.2 R4

Z

Π attenuator
Figure 1.15: 10 dB Π-section attenuator example for matching a 50 Ω source and load.

What resistor values would be required for both the Π attenuators for 10 dB of attenuation
matching a 50 Ω source and load?
The 10 dB corresponds to a voltage attenuation ratio of K=3.16 in the next to last line of the
above table. Transfer the resistor values in that line to the resistors on the schematic diagram
in Figure 1.15.

1.7.4

L-section attenuator

The table in Figure 1.16 lists resistor values for the L attenuators to match a 50 Ω source/
load. The table in Figure 1.17 lists resistor values for an alternate form. Note that the resistor
values are not the same.
dB = attenuation in decibels
Z = source/load impedance (resistive)
K>1
R5
VI
= 10 dB/20
VO
VI
VO
⇐ Ζ⇒
R6
R5 = Z K- 1
Ζ⇒
K
Z
R6 =
L attenuator
(K-1)

K =

Resistors for L-section
Z=,50.00
Attenuation L
dB
K=Vi/Vo
R5
R6
1.0
1.12
5.44 409.77
2.0
1.26 10.28 193.11
3.0
1.41 14.60 121.20
4.0
1.58 18.45
85.49
6.0
2.00 24.94
50.24
10.0
3.16 34.19
23.12
20.0
10.00 45.00
5.56

Figure 1.16: L-section attenuator table for 50 Ω source and load impedance.
The above apply to the L attenuator below.

1.7.5

Bridged T attenuator

The table in Figure 1.18 lists resistor values for the bridged T attenuators to match a 50 Ω
source and load. The bridged-T attenuator is not often used. Why not?

CHAPTER 1. AMPLIFIERS AND ACTIVE DEVICES

22

dB = attenuation in decibels
Z = source/load impedance (resistive)
K>1
R7
V
K = VI = 10 dB/20
O
VO
VI
R7 = Z(K-1)
R8
⇐ Ζ⇒
Ζ⇒
K
R8 = Z K- 1
L attenuator

Resistors for L-section
Z=50.00
Attenuation
dB K=Vi/Vo
R7
R8
1.0
1.12
6.10 459.77
2.0
1.26 12.95 243.11
3.0
1.41 20.63 171.20
4.0
1.58 29.24 135.49
6.0
2.00 49.76 100.24
10.0
3.16 108.11
73.12
20.0
10.00 450.00
55.56

Figure 1.17: Alternate form L-section attenuator table for 50 Ω source and load impedance.

dB = attenuation in decibels
Z = source/load impedance (resistive)
K>1
VI
= 10 dB/20
VO
R6 = Z
(K-1)
R7 = Z(K-1)
K =

Ζ
VI
⇐ Ζ⇒

R7

Ζ
VO
R6 ⇐ Ζ⇒

Bridged T attenuator

Resistors for bridged T
Z=50.00
Attenuation
dB K=Vi/Vo
R7
R6
1.0
1.12
6.10 409.77
2.0
1.26 12.95 193.11
3.0
1.41 20.63 121.20
4.0
1.58 29.24
85.49
6.0
2.00 49.76
50.24
10.0
3.16 108.11
23.12
20.0
10.00 450.00
5.56

Figure 1.18: Formulas and abbreviated table for bridged-T attenuator section, Z = 50 Ω.

1.7. ATTENUATORS

1.7.6

23

Cascaded sections

Attenuator sections can be cascaded as in Figure 1.19 for more attenuation than may be available from a single section. For example two 10 db attenuators may be cascaded to provide 20
dB of attenuation, the dB values being additive. The voltage attenuation ratio K or VI /VO for
a 10 dB attenuator section is 3.16. The voltage attenuation ratio for the two cascaded sections
is the product of the two Ks or 3.16x3.16=10 for the two cascaded sections.

section 1

section 2

Figure 1.19: Cascaded attenuator sections: dB attenuation is additive.
Variable attenuation can be provided in discrete steps by a switched attenuator. The example Figure 1.20, shown in the 0 dB position, is capable of 0 through 7 dB of attenuation by
additive switching of none, one or more sections.
S1

4 dB

S2

2 dB

S3

1 dB

Figure 1.20: Switched attenuator: attenuation is variable in discrete steps.
The typical multi section attenuator has more sections than the above figure shows. The
addition of a 3 or 8 dB section above enables the unit to cover to 10 dB and beyond. Lower
signal levels are achieved by the addition of 10 dB and 20 dB sections, or a binary multiple 16
dB section.

1.7.7

RF attenuators

For radio frequency (RF) work (<1000 Mhz), the individual sections must be mounted in
shielded compartments to thwart capacitive coupling if lower signal levels are to be achieved
at the highest frequencies. The individual sections of the switched attenuators in the previous
section are mounted in shielded sections. Additional measures may be taken to extend the
frequency range to beyond 1000 Mhz. This involves construction from special shaped lead-less
resistive elements.
A coaxial T-section attenuator consisting of resistive rods and a resistive disk is shown in
Figure 1.21. This construction is usable to a few gigahertz. The coaxial Π version would have
one resistive rod between two resistive disks in the coaxial line as in Figure 1.22.

24

CHAPTER 1. AMPLIFIERS AND ACTIVE DEVICES

metalic conductor

resistive disc
resistive rod
Coaxial T-attenuator for radio frequency work
Figure 1.21: Coaxial T-attenuator for radio frequency work.

metalic conductor

resistive rod
resistive disc
Coaxial Π-attenuator for radio frequency work
Figure 1.22: Coaxial Π-attenuator for radio frequency work.

1.7. ATTENUATORS

25

RF connectors, not shown, are attached to the ends of the above T and Π attenuators.
The connectors allow individual attenuators to be cascaded, in addition to connecting between
a source and load. For example, a 10 dB attenuator may be placed between a troublesome
signal source and an expensive spectrum analyzer input. Even though we may not need the
attenuation, the expensive test equipment is protected from the source by attenuating any
overvoltage.
Summary: Attenuators
• An attenuator reduces an input signal to a lower level.
• The amount of attenuation is specified in decibels (dB). Decibel values are additive for
cascaded attenuator sections.
• dB from power ratio:

dB = 10 log10 (PI / PO )

• dB from voltage ratio:

dB = 20 log10 (VI / VO )

• T and Π section attenuators are the most common circuit configurations.

Contributors
Contributors to this chapter are listed in chronological order of their contributions, from most
recent to first. See Appendix 2 (Contributor List) for dates and contact information.
Colin Barnard (November 2003): Correction regarding Alexander Graham Bell’s country
of origin (Scotland, not the United States).

26

CHAPTER 1. AMPLIFIERS AND ACTIVE DEVICES

Chapter 2

SOLID-STATE DEVICE THEORY
Contents
2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.2

Quantum physics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

2.3

Valence and Crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.4

Band theory of solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

2.5

Electrons and “holes” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.6

The P-N junction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2.7

Junction diodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.8

Bipolar junction transistors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

2.9

Junction field-effect transistors . . . . . . . . . . . . . . . . . . . . . . . . . . 65

2.10 Insulated-gate field-effect transistors (MOSFET) . . . . . . . . . . . . . . . 70
2.11 Thyristors

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2.12 Semiconductor manufacturing techniques . . . . . . . . . . . . . . . . . . . 75
2.13 Superconducting devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.14 Quantum devices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
2.15 Semiconductor devices in SPICE . . . . . . . . . . . . . . . . . . . . . . . . . 91
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

2.1

Introduction

This chapter will cover the physics behind the operation of semiconductor devices and show
how these principles are applied in several different types of semiconductor devices. Subsequent chapters will deal primarily with the practical aspects of these devices in circuits and
omit theory as much as possible.
27

CHAPTER 2. SOLID-STATE DEVICE THEORY

28

2.2

Quantum physics
“I think it is safe to say that no one understands quantum mechanics.”
Physicist Richard P. Feynman

To say that the invention of semiconductor devices was a revolution would not be an exaggeration. Not only was this an impressive technological accomplishment, but it paved the
way for developments that would indelibly alter modern society. Semiconductor devices made
possible miniaturized electronics, including computers, certain types of medical diagnostic and
treatment equipment, and popular telecommunication devices, to name a few applications of
this technology.
But behind this revolution in technology stands an even greater revolution in general science: the field of quantum physics. Without this leap in understanding the natural world, the
development of semiconductor devices (and more advanced electronic devices still under development) would never have been possible. Quantum physics is an incredibly complicated realm
of science. This chapter is but a brief overview. When scientists of Feynman’s caliber say that
“no one understands [it],” you can be sure it is a complex subject. Without a basic understanding of quantum physics, or at least an understanding of the scientific discoveries that led to its
formulation, though, it is impossible to understand how and why semiconductor electronic devices function. Most introductory electronics textbooks I’ve read try to explain semiconductors
in terms of “classical” physics, resulting in more confusion than comprehension.
Many of us have seen diagrams of atoms that look something like Figure 2.1.
e

e

= electron

P = proton
N = neutron
e

e

N
P P
N P
N P
P
N P
N N

e

e

e

Figure 2.1: Rutherford atom: negative electrons orbit a small positive nucleus.
Tiny particles of matter called protons and neutrons make up the center of the atom; electrons orbit like planets around a star. The nucleus carries a positive electrical charge, owing to

2.2. QUANTUM PHYSICS

29

a
pa lp
rt ha
ic
l

es

the presence of protons (the neutrons have no electrical charge whatsoever), while the atom’s
balancing negative charge resides in the orbiting electrons. The negative electrons are attracted to the positive protons just as planets are gravitationally attracted by the Sun, yet the
orbits are stable because of the electrons’ motion. We owe this popular model of the atom to the
work of Ernest Rutherford, who around the year 1911 experimentally determined that atoms’
positive charges were concentrated in a tiny, dense core rather than being spread evenly about
the diameter as was proposed by an earlier researcher, J.J. Thompson.
Rutherford’s scattering experiment involved bombarding a thin gold foil with positively
charged alpha particles as in Figure 2.2. Young graduate students H. Geiger and E. Marsden
experienced unexpected results. A few Alpha particles were deflected at large angles. A few
Alpha particles were back-scattering, recoiling at nearly 180o . Most of the particles passed
through the gold foil undeflected, indicating that the foil was mostly empty space. The fact
that a few alpha particles experienced large deflections indicated the presence of a minuscule
positively charged nucleus.

Gol

d fo

il

Figure 2.2: Rutherford scattering: a beam of alpha particles is scattered by a thin gold foil.
Although Rutherford’s atomic model accounted for experimental data better than Thompson’s, it still wasn’t perfect. Further attempts at defining atomic structure were undertaken,
and these efforts helped pave the way for the bizarre discoveries of quantum physics. Today our
understanding of the atom is quite a bit more complex. Nevertheless, despite the revolution of
quantum physics and its contribution to our understanding of atomic structure, Rutherford’s
solar-system picture of the atom embedded itself in the popular consciousness to such a degree
that it persists in some areas of study even when inappropriate.
Consider this short description of electrons in an atom, taken from a popular electronics
textbook:
Orbiting negative electrons are therefore attracted toward the positive nucleus,
which leads us to the question of why the electrons do not fly into the atom’s nucleus.
The answer is that the orbiting electrons remain in their stable orbit because of two
equal but opposite forces. The centrifugal outward force exerted on the electrons
because of the orbit counteracts the attractive inward force (centripetal) trying to
pull the electrons toward the nucleus because of the unlike charges.

30

CHAPTER 2. SOLID-STATE DEVICE THEORY

In keeping with the Rutherford model, this author casts the electrons as solid chunks of
matter engaged in circular orbits, their inward attraction to the oppositely charged nucleus
balanced by their motion. The reference to “centrifugal force” is technically incorrect (even
for orbiting planets), but is easily forgiven because of its popular acceptance: in reality, there
is no such thing as a force pushing any orbiting body away from its center of orbit. It seems
that way because a body’s inertia tends to keep it traveling in a straight line, and since an
orbit is a constant deviation (acceleration) from straight-line travel, there is constant inertial
opposition to whatever force is attracting the body toward the orbit center (centripetal), be it
gravity, electrostatic attraction, or even the tension of a mechanical link.
The real problem with this explanation, however, is the idea of electrons traveling in circular orbits in the first place. It is a verifiable fact that accelerating electric charges emit
electromagnetic radiation, and this fact was known even in Rutherford’s time. Since orbiting
motion is a form of acceleration (the orbiting object in constant acceleration away from normal,
straight-line motion), electrons in an orbiting state should be throwing off radiation like mud
from a spinning tire. Electrons accelerated around circular paths in particle accelerators called
synchrotrons are known to do this, and the result is called synchrotron radiation. If electrons
were losing energy in this way, their orbits would eventually decay, resulting in collisions with
the positively charged nucleus. Nevertheless, this doesn’t ordinarily happen within atoms.
Indeed, electron “orbits” are remarkably stable over a wide range of conditions.
Furthermore, experiments with “excited” atoms demonstrated that electromagnetic energy
emitted by an atom only occurs at certain, definite frequencies. Atoms that are “excited” by
outside influences such as light are known to absorb that energy and return it as electromagnetic waves of specific frequencies, like a tuning fork that rings at a fixed pitch no matter how
it is struck. When the light emitted by an excited atom is divided into its constituent frequencies (colors) by a prism, distinct lines of color appear in the spectrum, the pattern of spectral
lines being unique to that element. This phenomenon is commonly used to identify atomic elements, and even measure the proportions of each element in a compound or chemical mixture.
According to Rutherford’s solar-system atomic model (regarding electrons as chunks of matter
free to orbit at any radius) and the laws of classical physics, excited atoms should return energy over a virtually limitless range of frequencies rather than a select few. In other words, if
Rutherford’s model were correct, there would be no “tuning fork” effect, and the light spectrum
emitted by any atom would appear as a continuous band of colors rather than as a few distinct
lines.
A pioneering researcher by the name of Niels Bohr attempted to improve upon Rutherford’s model after studying in Rutherford’s laboratory for several months in 1912. Trying to
harmonize the findings of other physicists (most notably, Max Planck and Albert Einstein),
Bohr suggested that each electron had a certain, specific amount of energy, and that their orbits were quantized such that each may occupy certain places around the nucleus, as marbles
fixed in circular tracks around the nucleus rather than the free-ranging satellites each were
formerly imagined to be. (Figure 2.3) In deference to the laws of electromagnetics and accelerating charges, Bohr alluded to these “orbits” as stationary states to escape the implication that
they were in motion.
Although Bohr’s ambitious attempt at re-framing the structure of the atom in terms that
agreed closer to experimental results was a milestone in physics, it was not complete. His
mathematical analysis produced better predictions of experimental events than analyses belonging to previous models, but there were still some unanswered questions about why elec-

an
Lymies
ser

N

Bra
seri cket
es

O

Pasche
series n

P

Balmer
series

2.2. QUANTUM PHYSICS

31

n=6

n=5

discharge lamp
slit

n=4

4102 A
4340
4861

Hδ
Hγ
Hβ
Hα

M

n=3
n=2 (L)
n=1 (K)
nucleus

6563

Balmer series

Figure 2.3: Bohr hydrogen atom (with orbits drawn to scale) only allows electrons to inhabit
discrete orbitals. Electrons falling from n=3,4,5, or 6 to n=2 accounts for Balmer series of
spectral lines.
trons should behave in such strange ways. The assertion that electrons existed in stationary,
quantized states around the nucleus accounted for experimental data better than Rutherford’s
model, but he had no idea what would force electrons to manifest those particular states. The
answer to that question had to come from another physicist, Louis de Broglie, about a decade
later.
De Broglie proposed that electrons, as photons (particles of light) manifested both particlelike and wave-like properties. Building on this proposal, he suggested that an analysis of
orbiting electrons from a wave perspective rather than a particle perspective might make more
sense of their quantized nature. Indeed, another breakthrough in understanding was reached.

node

node

node

antinode

antinode

Figure 2.4: String vibrating at resonant frequency between two fixed points forms standing
wave.
The atom according to de Broglie consisted of electrons existing as standing waves, a phenomenon well known to physicists in a variety of forms. As the plucked string of a musical
instrument (Figure 2.4) vibrating at a resonant frequency, with “nodes” and “antinodes” at stable positions along its length. De Broglie envisioned electrons around atoms standing as waves
bent around a circle as in Figure 2.5.
Electrons only could exist in certain, definite “orbits” around the nucleus because those
were the only distances where the wave ends would match. In any other radius, the wave

CHAPTER 2. SOLID-STATE DEVICE THEORY

32

ti

an

n
nti

tin

no

tin

od

a

od

node
node
nucleus

node

nucleus
node

node

e

node
node

od

an
tin
e

od

(a)

antinode

antinode

node

e

node

e

node

an

e

od

an

de

an

in
nt

de

tin

a

(b)

od

e

no
nti

a

Figure 2.5: “Orbiting” electron as standing wave around the nucleus, (a) two cycles per orbit,
(b) three cycles per orbit.
should destructively interfere with itself and thus cease to exist.
De Broglie’s hypothesis gave both mathematical support and a convenient physical analogy
to account for the quantized states of electrons within an atom, but his atomic model was still
incomplete. Within a few years, though, physicists Werner Heisenberg and Erwin Schrodinger,
working independently of each other, built upon de Broglie’s concept of a matter-wave duality
to create more mathematically rigorous models of subatomic particles.
This theoretical advance from de Broglie’s primitive standing wave model to Heisenberg’s
matrix and Schrodinger’s differential equation models was given the name quantum mechanics, and it introduced a rather shocking characteristic to the world of subatomic particles: the
trait of probability, or uncertainty. According to the new quantum theory, it was impossible
to determine the exact position and exact momentum of a particle at the same time. The
popular explanation of this “uncertainty principle” was that it was a measurement error (i.e.
by attempting to precisely measure the position of an electron, you interfere with its momentum and thus cannot know what it was before the position measurement was taken, and vice
versa). The startling implication of quantum mechanics is that particles do not actually have
precise positions and momenta, but rather balance the two quantities in a such way that their
combined uncertainties never diminish below a certain minimum value.
This form of “uncertainty” relationship exists in areas other than quantum mechanics. As
discussed in the “Mixed-Frequency AC Signals” chapter in volume II of this book series, there
is a mutually exclusive relationship between the certainty of a waveform’s time-domain data
and its frequency-domain data. In simple terms, the more precisely we know its constituent
frequency(ies), the less precisely we know its amplitude in time, and vice versa. To quote
myself:
A waveform of infinite duration (infinite number of cycles) can be analyzed with
absolute precision, but the less cycles available to the computer for analysis, the less
precise the analysis. . . The fewer times that a wave cycles, the less certain its

2.2. QUANTUM PHYSICS

33

frequency is. Taking this concept to its logical extreme, a short pulse – a waveform
that doesn’t even complete a cycle – actually has no frequency, but rather acts as an
infinite range of frequencies. This principle is common to all wave-based phenomena,
not just AC voltages and currents.
In order to precisely determine the amplitude of a varying signal, we must sample it over
a very narrow span of time. However, doing this limits our view of the wave’s frequency.
Conversely, to determine a wave’s frequency with great precision, we must sample it over
many cycles, which means we lose view of its amplitude at any given moment. Thus, we cannot
simultaneously know the instantaneous amplitude and the overall frequency of any wave with
unlimited precision. Stranger yet, this uncertainty is much more than observer imprecision; it
resides in the very nature of the wave. It is not as though it would be possible, given the proper
technology, to obtain precise measurements of both instantaneous amplitude and frequency at
once. Quite literally, a wave cannot have both a precise, instantaneous amplitude, and a precise
frequency at the same time.
The minimum uncertainty of a particle’s position and momentum expressed by Heisenberg
and Schrodinger has nothing to do with limitation in measurement; rather it is an intrinsic
property of the particle’s matter-wave dual nature. Electrons, therefore, do not really exist in
their “orbits” as precisely defined bits of matter, or even as precisely defined waveshapes, but
rather as “clouds” – the technical term is wavefunction – of probability distribution, as if each
electron were “spread” or “smeared” over a range of positions and momenta.
This radical view of electrons as imprecise clouds at first seems to contradict the original
principle of quantized electron states: that electrons exist in discrete, defined “orbits” around
atomic nuclei. It was, after all, this discovery that led to the formation of quantum theory
to explain it. How odd it seems that a theory developed to explain the discrete behavior of
electrons ends up declaring that electrons exist as “clouds” rather than as discrete pieces of
matter. However, the quantized behavior of electrons does not depend on electrons having definite position and momentum values, but rather on other properties called quantum numbers.
In essence, quantum mechanics dispenses with commonly held notions of absolute position and
absolute momentum, and replaces them with absolute notions of a sort having no analogue in
common experience.
Even though electrons are known to exist in ethereal, “cloud-like” forms of distributed probability rather than as discrete chunks of matter, those “clouds” have other characteristics that
are discrete. Any electron in an atom can be described by four numerical measures (the previously mentioned quantum numbers), called the Principal, Angular Momentum, Magnetic,
and Spin numbers. The following is a synopsis of each of these numbers’ meanings:
Principal Quantum Number: Symbolized by the letter n, this number describes the shell
that an electron resides in. An electron “shell” is a region of space around an atom’s nucleus
that electrons are allowed to exist in, corresponding to the stable “standing wave” patterns of
de Broglie and Bohr. Electrons may “leap” from shell to shell, but cannot exist between the
shell regions.
The principle quantum number must be a positive integer (a whole number, greater than
or equal to 1). In other words, principle quantum number for an electron cannot be 1/2 or
-3. These integer values were not arrived at arbitrarily, but rather through experimental evidence of light spectra: the differing frequencies (colors) of light emitted by excited hydrogen
atoms follow a sequence mathematically dependent on specific, integer values as illustrated in

CHAPTER 2. SOLID-STATE DEVICE THEORY

34

Figure 2.3.
Each shell has the capacity to hold multiple electrons. An analogy for electron shells is the
concentric rows of seats of an amphitheater. Just as a person seated in an amphitheater must
choose a row to sit in (one cannot sit between rows), electrons must “choose” a particular shell
to “sit” in. As in amphitheater rows, the outermost shells hold more electrons than the inner
shells. Also, electrons tend to seek the lowest available shell, as people in an amphitheater
seek the closest seat to the center stage. The higher the shell number, the greater the energy
of the electrons in it.
The maximum number of electrons that any shell may hold is described by the equation 2n2 ,
where “n” is the principle quantum number. Thus, the first shell (n=1) can hold 2 electrons;
the second shell (n=2) 8 electrons, and the third shell (n=3) 18 electrons. (Figure 2.6)
K

n= 1
2n2 = 2
observed fill = 2

L

2
8
8

M

N

O

P

Q

3
18
18

4
32
32

18

18

2

Figure 2.6: Principal quantum number n and maximum number of electrons per shell both
predicted by 2(n2 ), and observed. Orbitals not to scale.
Electron shells in an atom were formerly designated by letter rather than by number. The
first shell (n=1) was labeled K, the second shell (n=2) L, the third shell (n=3) M, the fourth
shell (n=4) N, the fifth shell (n=5) O, the sixth shell (n=6) P, and the seventh shell (n=7) Q.
Angular Momentum Quantum Number: A shell, is composed of subshells. One might
be inclined to think of subshells as simple subdivisions of shells, as lanes dividing a road.
The subshells are much stranger. Subshells are regions of space where electron “clouds” are
allowed to exist, and different subshells actually have different shapes. The first subshell
is shaped like a sphere, (Figure 2.7(s) ) which makes sense when visualized as a cloud of
electrons surrounding the atomic nucleus in three dimensions. The second subshell, however,
resembles a dumbbell, comprised of two “lobes” joined together at a single point near the atom’s
center. (Figure 2.7(p) ) The third subshell typically resembles a set of four “lobes” clustered
around the atom’s nucleus. These subshell shapes are reminiscent of graphical depictions of
radio antenna signal strength, with bulbous lobe-shaped regions extending from the antenna
in various directions. (Figure 2.7(d) )
Valid angular momentum quantum numbers are positive integers like principal quantum
numbers, but also include zero. These quantum numbers for electrons are symbolized by the
letter l. The number of subshells in a shell is equal to the shell’s principal quantum number. Thus, the first shell (n=1) has one subshell, numbered 0; the second shell (n=2) has two
subshells, numbered 0 and 1; the third shell (n=3) has three subshells, numbered 0, 1, and 2.

2.2. QUANTUM PHYSICS

35

y
x
z

x
1 of 3

1 of 1

1 of 5
dz2 shown

dx2 - y2 shown
dxy , dyz , dxz similar

px shown
py, pz similar

(s)

1 of 5

(dx2 - y2)

(p)

(dz2)

Figure 2.7: Orbitals: (s) Three fold symmetry. (p) Shown: px , one of three possible orientations
(px , py , pz ), about their respective axes. (d) Shown: dx 2 -y 2 similar to dxy , dyz , dxz . Shown: dz 2 .
Possible d-orbital orientations: five.
An older convention for subshell description used letters rather than numbers. In this notation, the first subshell (l=0) was designated s, the second subshell (l=1) designated p, the third
subshell (l=2) designated d, and the fourth subshell (l=3) designated f. The letters come from
the words sharp, principal (not to be confused with the principal quantum number, n), diffuse,
and fundamental. You will still see this notational convention in many periodic tables, used to
designate the electron configuration of the atoms’ outermost, or valence, shells. (Figure 2.8)
SL

SM
PL

PM

SN
DM

PN

DN SO

SK

2

8

18

18

2

1

2

2

6

6

10 2

6

10 1

electrons

n=

K
1

L
2

M
3
(a)

N
4

O
5

l=
n=
spectroscopic
notation

0
1

0, 1
2

0, 1, 2
3

0, 1, 2
4

0
5

1s2 2s22p6 3s23p63d10 4s24p64d10 5s1

(b)

Figure 2.8: (a) Bohr representation of Silver atom, (b) Subshell representation of Ag with
division of shells into subshells (angular quantum number l). This diagram implies nothing
about the actual position of electrons, but represents energy levels.
Magnetic Quantum Number: The magnetic quantum number for an electron classifies
which orientation its subshell shape is pointed. The “lobes” for subshells point in multiple
directions. These different orientations are called orbitals. For the first subshell (s; l=0), which
resembles a sphere pointing in no “direction”, so there is only one orbital. For the second
(p; l=1) subshell in each shell, which resembles dumbbells point in three possible directions.

CHAPTER 2. SOLID-STATE DEVICE THEORY

36

Think of three dumbbells intersecting at the origin, each oriented along a different axis in a
three-axis coordinate space.
Valid numerical values for this quantum number consist of integers ranging from -l to l, and
are symbolized as ml in atomic physics and lz in nuclear physics. To calculate the number of
orbitals in any given subshell, double the subshell number and add 1, (2·l + 1). For example, the
first subshell (l=0) in any shell contains a single orbital, numbered 0; the second subshell (l=1)
in any shell contains three orbitals, numbered -1, 0, and 1; the third subshell (l=2) contains
five orbitals, numbered -2, -1, 0, 1, and 2; and so on.
Like principal quantum numbers, the magnetic quantum number arose directly from experimental evidence: The Zeeman effect, the division of spectral lines by exposing an ionized
gas to a magnetic field, hence the name “magnetic” quantum number.
Spin Quantum Number: Like the magnetic quantum number, this property of atomic
electrons was discovered through experimentation. Close observation of spectral lines revealed
that each line was actually a pair of very closely-spaced lines, and this so-called fine structure
was hypothesized to result from each electron “spinning” on an axis as if a planet. Electrons
with different “spins” would give off slightly different frequencies of light when excited. The
name “spin” was assigned to this quantum number. The concept of a spinning electron is now
obsolete, being better suited to the (incorrect) view of electrons as discrete chunks of matter
rather than as “clouds”; but, the name remains.
Spin quantum numbers are symbolized as ms in atomic physics and sz in nuclear physics.
For each orbital in each subshell in each shell, there may be two electrons, one with a spin of
+1/2 and the other with a spin of -1/2.
The physicist Wolfgang Pauli developed a principle explaining the ordering of electrons
in an atom according to these quantum numbers. His principle, called the Pauli exclusion
principle, states that no two electrons in the same atom may occupy the exact same quantum
states. That is, each electron in an atom has a unique set of quantum numbers. This limits the
number of electrons that may occupy any given orbital, subshell, and shell.
Shown here is the electron arrangement for a hydrogen atom:

K shell
(n = 1)

subshell orbital
(l)
(ml)
0

0

spin
(ms)
1

/2

One electron

Hydrogen
Atomic number (Z) = 1
(one proton in nucleus)
Spectroscopic notation: 1s1
With one proton in the nucleus, it takes one electron to electrostatically balance the atom
(the proton’s positive electric charge exactly balanced by the electron’s negative electric charge).
This one electron resides in the lowest shell (n=1), the first subshell (l=0), in the only orbital
(spatial orientation) of that subshell (ml =0), with a spin value of 1/2. A common method of
describing this organization is by listing the electrons according to their shells and subshells

2.2. QUANTUM PHYSICS

37

in a convention called spectroscopic notation. In this notation, the shell number is shown as an
integer, the subshell as a letter (s,p,d,f), and the total number of electrons in the subshell (all
orbitals, all spins) as a superscript. Thus, hydrogen, with its lone electron residing in the base
level, is described as 1s1 .
Proceeding to the next atom (in order of atomic number), we have the element helium:

subshell orbital
(l)
(ml)
K shell
(n = 1)

0
0

spin
(ms)

0

-1/2

electron

0

1

electron

/2

Helium
Atomic number (Z) = 2
(two protons in nucleus)
Spectroscopic notation: 1s2
A helium atom has two protons in the nucleus, and this necessitates two electrons to balance the double-positive electric charge. Since two electrons – one with spin=1/2 and the other
with spin=-1/2 – fit into one orbital, the electron configuration of helium requires no additional
subshells or shells to hold the second electron.
However, an atom requiring three or more electrons will require additional subshells to
hold all electrons, since only two electrons will fit into the lowest shell (n=1). Consider the next
atom in the sequence of increasing atomic numbers, lithium:

subshell orbital
(l)
(ml)

spin
(ms)

L shell
(n = 2)

0

0

1

/2

electron

K shell
(n = 1)

0

0

-1/2

electron

0

1

electron

0

/2

Lithium
Atomic number (Z) = 3
Spectroscopic notation: 1s22s1
An atom of lithium uses a fraction of the L shell’s (n=2) capacity. This shell actually has a
total capacity of eight electrons (maximum shell capacity = 2n2 electrons). If we examine the
organization of the atom with a completely filled L shell, we will see how all combinations of
subshells, orbitals, and spins are occupied by electrons:

CHAPTER 2. SOLID-STATE DEVICE THEORY

38

subshell orbital
(l)
(ml)
1

1

-1/2

1

1

1

0

1

- /2

0

1

1

-1

1

- /2

1

-1

1

0

1

- /2

0

0

1

0

0

-1/2

0

1

1
L shell
(n = 2)

1

0

K shell
(n = 1)

spin
(ms)

0

/2
/2

p subshell
(l = 1)
6 electrons

/2
/2

/2

s subshell
(l = 0)
2 electrons
s subshell
(l = 0)
2 electrons

Neon
Atomic number (Z) = 10
Spectroscopic notation: 1s22s22p6
Often, when the spectroscopic notation is given for an atom, any shells that are completely
filled are omitted, and the unfilled, or the highest-level filled shell, is denoted. For example,
the element neon (shown in the previous illustration), which has two completely filled shells,
may be spectroscopically described simply as 2p6 rather than 1s2 2s2 2p6 . Lithium, with its K
shell completely filled and a solitary electron in the L shell, may be described simply as 2s1
rather than 1s2 2s1 .
The omission of completely filled, lower-level shells is not just a notational convenience. It
also illustrates a basic principle of chemistry: that the chemical behavior of an element is primarily determined by its unfilled shells. Both hydrogen and lithium have a single electron in
their outermost shells (1s1 and 2s1 , respectively), giving the two elements some similar properties. Both are highly reactive, and reactive in much the same way (bonding to similar elements
in similar modes). It matters little that lithium has a completely filled K shell underneath its
almost-vacant L shell: the unfilled L shell is the shell that determines its chemical behavior.
Elements having completely filled outer shells are classified as noble, and are distinguished
by almost complete non-reactivity with other elements. These elements used to be classified as
inert, when it was thought that these were completely unreactive, but are now known to form
compounds with other elements under specific conditions.
Since elements with identical electron configurations in their outermost shell(s) exhibit
similar chemical properties, Dimitri Mendeleev organized the different elements in a table
accordingly. Such a table is known as a periodic table of the elements, and modern tables follow
this general form in Figure 2.9.

13

IA

H
1
Hydrogen
1.00794
1s1
Li

Periodic Table of the Elements

Figure 2.9: Periodic table of chemical elements.

IIA

Group new

4
3 Be
Beryllium
Lithium
6.941
9.012182

Symbol

2s1

2

1

K
19
Potassium
39.0983

Name

2s2

Electron
configuration

12
Na
11 Mg
Sodium Magnesium
22.989768 24.3050

IA

4s1

Metalloids
Nonmetals
IVA 15
VA 16
VIA 17 VIIA
13
IIIA 14
9
8 F
7 O
B
5 C
6 N
Fluorine
Oxygen
Nitrogen
Boron
Carbon
18.9984
15.9994
14.0067
10.81
12.011

Group old
Atomic number
Atomic mass
(averaged according to
occurence on earth)

2p1

2p2

2p3

2p4

2p5

VIIIA

He
2
Helium
4.00260
1s2
Ne
10
Neon
20.179
2p6

18
17 Ar
14 P
16 Cl
Al
13 Si
15 S
Argon
Chlorine
Silicon Phosphorus Sulfur
Aluminum
39.948
35.453
28.0855
32.06
26.9815
30.9738

2.2. QUANTUM PHYSICS

1

Metals
IB 12
3
IIIB 4
IVB 5
VB 6
VIB 7
VIIB 8
IIB
3p6
3s2
3p5
3p2
3p4
3s1
3p1
3p3
VIIIB 9 VIIIB 10 VIIIB 11
31 Ge
30 Ga
29 Zn
28 Cu
27 Ni
26 Co
25 Fe
24 Mn
23 Cr
32 As
34 Br
36
22 V
33 Se
35 Kr
21 Ti
K
19 Ca
20 Sc
Gallium Germanium Arsenic
Zinc
Copper
Nickel
Cobalt
Iron
Vanadium Chromium Manganese
Selenium Bromine
Krypton
Scandium Titanium
Potassium Calcium
63.546
55.847
50.9415
83.80
78.96
74.92159
79.904
40.078
44.955910
47.88
51.9961
54.93805
58.93320
58.69
65.39
69.723
72.61
39.0983
4s1

4s2

3d14s2

3d24s2

3d34s2

3d54s1

3d54s2

3d64s2

3d74s2

3d84s2

3d104s1

3d104s2

4p1

4p2

4p3

4p4

4p5

4p6

52 I
Rb
51 Te
50 Sb
49 Sn
48 In
47 Cd
46 Ag
45 Pd
44 Rh
43 Ru
54
42 Tc
41 Mo
53 Xe
40 Nb
39 Zr
38 Y
37 Sr
Antimony Tellurium
Tin
Indium
Cadmium
Silver
Palladium
Xenon
Iodine
Zirconium Niobium Molybdenum Technetium Ruthenium Rhodium
Yttrium
Rubidium Strontium
127.60
121.75
88.90585
102.90550
118.710
114.82
112.411
107.8682
106.42
101.07
(98)
131.30
95.94
92.90638
91.224
126.905
87.62
85.4678
5s1

5s2

4d15s2

4d25s2

4d45s1

4d55s1

4d55s2

4d75s1

4d85s1

4d105s0

4d105s1

4d105s2

5p1

5p2

5p3

5p4

5p5

5p6

83 Po
82 Bi
81 Pb
80 Tl
79 Hg
78 Au
77 Pt
76 Ir
75 Os
74 Re
73 W
Hf
72 Ta
85 Rn
57 - 71
84 At
56
Cs
55 Ba
86
Bismuth
Lead
Thallium
Mercury
Gold
Platinum
Iridium
Osmium
Tantalum Tungsten Rhenium
Astatine
Lanthanide Hafnium
Polonium
Barium
Cesium
Radon
series
208.98037
207.2
204.3833
200.59
196.96654
195.08
192.22
190.2
186.207
183.85
180.9479
178.49
(210)
(209)
137.327
132.90543
(222)
6s1

1

7s

5d26s2

6s2

88
Fr
87 Ra
Radium
Francium
(226)
(223)

89 - 103
Actinide
series

(261)
2

6d 7s

7s

Lanthanide
series

2

6d 7s

Unnilhexium

(263)
4

5d56s2

106 Uns

5d66s2

5d76s2

108

107

5d96s1

5d106s1

5d106s2

6p1

6p2

6p3

6p4

6p5

6p6

109

Unnilseptium

(262)

2

6d 7s

71
70 Lu
69 Yb
68 Tm
67 Er
66 Ho
65 Dy
64 Tb
63 Gd
62 Eu
61 Sm
60 Pm
59 Nd
58 Pr
La
57 Ce
Lutetium
Ytterbium
Thulium
Erbium
Lanthanum Cerium Praseodymium Neodymium Promethium Samarium Europium Gadolinium Terbium Dysprosium Holmium
174.967
173.04
168.93421
167.26
164.93032
162.50
158.92534
157.25
151.965
150.36
(145)
144.24
140.115 140.90765
138.9055
5d16s2

Actinide
series

(262)
3

5d46s2

105 Unh

Unnilquadium Unnilpentium
2

2

5d36s2

104 Unp

Unq

4f15d16s2

4f36s2

4f46s2

4f56s2

4f66s2

4f76s2

4f75d16s2

4f96s2

4f106s2

4f116s2

4f126s2

4f136s2

4f146s2

4f145d16s2

103
102 Lr
101 No
100 Md
99 Fm
98 Es
97 Cf
96 Bk
95 Cm
94 Am
93 Pu
92 Np
91 U
90 Pa
Ac
89 Th
Berkelium Californium Einsteinium Fermium Mendelevium Nobelium Lawrencium
Curium
Thorium Protactinium Uranium Neptunium Plutonium Americium
Actinium
(260)
(259)
(258)
(257)
(252)
(251)
(247)
(247)
(243)
(244)
(237)
232.0381 231.03588 238.0289
(227)
6d17s2

6d27s2

5f26d17s2

5f36d17s2

5f46d17s2

5f66d07s2

5f76d07s2

5f76d17s2

5f96d07s2

5f106d07s2

5f116d07s2

5f126d07s2

5f136d07s2

6d07s2

6d17s2

39

CHAPTER 2. SOLID-STATE DEVICE THEORY

40

Dmitri Mendeleev, a Russian chemist, was the first to develop a periodic table of the elements. Although Mendeleev organized his table according to atomic mass rather than atomic
number, and produced a table that was not quite as useful as modern periodic tables, his development stands as an excellent example of scientific proof. Seeing the patterns of periodicity
(similar chemical properties according to atomic mass), Mendeleev hypothesized that all elements should fit into this ordered scheme. When he discovered “empty” spots in the table,
he followed the logic of the existing order and hypothesized the existence of heretofore undiscovered elements. The subsequent discovery of those elements granted scientific legitimacy to
Mendeleev’s hypothesis, furthering future discoveries, and leading to the form of the periodic
table we use today.
This is how science should work: hypotheses followed to their logical conclusions, and accepted, modified, or rejected as determined by the agreement of experimental data to those
conclusions. Any fool may formulate a hypothesis after-the-fact to explain existing experimental data, and many do. What sets a scientific hypothesis apart from post hoc speculation is
the prediction of future experimental data yet uncollected, and the possibility of disproof as a
result of that data. To boldly follow a hypothesis to its logical conclusion(s) and dare to predict
the results of future experiments is not a dogmatic leap of faith, but rather a public test of that
hypothesis, open to challenge from anyone able to produce contradictory data. In other words,
scientific hypotheses are always“risky” due to the claim to predict the results of experiments
not yet conducted, and are therefore susceptible to disproof if the experiments do not turn out
as predicted. Thus, if a hypothesis successfully predicts the results of repeated experiments,
its falsehood is disproven.
Quantum mechanics, first as a hypothesis and later as a theory, has proven to be extremely
successful in predicting experimental results, hence the high degree of scientific confidence
placed in it. Many scientists have reason to believe that it is an incomplete theory, though,
as its predictions hold true more at micro physical scales than at macroscopic dimensions, but
nevertheless it is a tremendously useful theory in explaining and predicting the interactions
of particles and atoms.
As you have already seen in this chapter, quantum physics is essential in describing and
predicting many different phenomena. In the next section, we will see its significance in the
electrical conductivity of solid substances, including semiconductors. Simply put, nothing in
chemistry or solid-state physics makes sense within the popular theoretical framework of electrons existing as discrete chunks of matter, whirling around atomic nuclei like miniature satellites. It is when electrons are viewed as “wavefunctions” existing in definite, discrete states
that the regular and periodic behavior of matter can be explained.
• REVIEW:
• Electrons in atoms exist in “clouds” of distributed probability, not as discrete chunks of
matter orbiting the nucleus like tiny satellites, as common illustrations of atoms show.
• Individual electrons around an atomic nucleus seek unique “states,” described by four
quantum numbers: the Principal Quantum Number, known as the shell; the Angular
Momentum Quantum Number, known as the subshell; the Magnetic Quantum Number,
describing the orbital (subshell orientation); and the Spin Quantum Number, or simply
spin. These states are quantized, meaning that no “in-between” conditions exist for an
electron other than those states that fit into the quantum numbering scheme.

2.3. VALENCE AND CRYSTAL STRUCTURE

41

• The Principal Quantum Number (n) describes the basic level or shell that an electron
resides in. The larger this number, the greater radius the electron cloud has from the
atom’s nucleus, and the greater that electron’s energy. Principal quantum numbers are
whole numbers (positive integers).
• The Angular Momentum Quantum Number (l) describes the shape of the electron cloud
within a particular shell or level, and is often known as the “subshell.” There are as many
subshells (electron cloud shapes) in any given shell as that shell’s principal quantum
number. Angular momentum quantum numbers are positive integers beginning at zero
and ending at one less than the principal quantum number (n-1).
• The Magnetic Quantum Number (ml ) describes which orientation a subshell (electron
cloud shape) has. Subshells may assume as many different orientations as 2-times the
subshell number (l) plus 1, (2l+1) (E.g. for l=1, ml= -1, 0, 1) and each unique orientation
is called an orbital. These numbers are integers ranging from the negative value of the
subshell number (l) through 0 to the positive value of the subshell number.
• The Spin Quantum Number (ms ) describes another property of an electron, and may be
a value of +1/2 or -1/2.
• Pauli’s Exclusion Principle says that no two electrons in an atom may share the exact
same set of quantum numbers. Therefore, no more than two electrons may occupy each
orbital (spin=1/2 and spin=-1/2), 2l+1 orbitals in every subshell, and n subshells in every
shell, and no more.
• Spectroscopic notation is a convention for denoting the electron configuration of an atom.
Shells are shown as whole numbers, followed by subshell letters (s,p,d,f), with superscripted numbers totaling the number of electrons residing in each respective subshell.
• An atom’s chemical behavior is solely determined by the electrons in the unfilled shells.
Low-level shells that are completely filled have little or no effect on the chemical bonding
characteristics of elements.
• Elements with completely filled electron shells are almost entirely unreactive, and are
called noble (formerly known as inert).

2.3

Valence and Crystal structure

Valence: The electrons in the outer most shell, or valence shell, are known as valence electrons. These valence electrons are responsible for the chemical properties of the chemical
elements. It is these electrons which participate in chemical reactions with other elements. An
over simplified chemistry rule applicable to simple reactions is that atoms try to form a complete outer shell of 8 electrons (two for the L shell). Atoms may give away a few electrons to
expose an underlying complete shell. Atoms may accept a few electrons to complete the shell.
These two processes form ions from atoms. Atoms may even share electrons among atoms in
an attempt to complete the outer shell. This process forms molecular bonds. That is, atoms
associate to form a molecule.

CHAPTER 2. SOLID-STATE DEVICE THEORY

42

For example group I elements: Li, Na, K, Cu, Ag, and Au have a single valence electron.
(Figure 2.10) These elements all have similar chemical properties. These atoms readily give
away one electron to react with other elements. The ability to easily give away an electron
makes these elements excellent conductors.

Li

Na

K

Cu

Ag

Au

Figure 2.10: Periodic table group IA elements: Li, Na, and K, and group IB elements: Cu, Ag,
and Au have one electron in the outer, or valence, shell, which is readily donated. Inner shell
electrons: For n= 1, 2, 3, 4; 2n2 = 2, 8, 18, 32.
Group VIIA elements: Fl, Cl, Br, and I all have 7 electrons in the outer shell. These elements readily accept an electron to fill up the outer shell with a full 8 electrons. (Figure 2.11)
If these elements do accept an electron, a negative ion is formed from the neutral atom. These
elements which do not give up electrons are insulators.

F

Cl

Br

I

Figure 2.11: Periodic table group VIIA elements: F, Cl, Br, and I with 7 valence electrons
readily accept an electron in reactions with other elements.
For example, a Cl atom accepts an electron from an Na atom to become a Cl− ion as shown in
Figure 2.12. An ion is a charged particle formed from an atom by either donating or accepting
an electron. As the Na atom donates an electron, it becomes a Na+ ion. This is how Na and Cl
atoms combine to form NaCl, table salt, which is actually Na+ Cl− , a pair of ions. The Na+ and
Cl− carrying opposite charges, attract one other.
Sodium chloride crystallizes in the cubic structure shown in Figure 2.16. This model is not
to scale to show the three dimensional structure. The Na+ Cl− ions are actually packed similar
to layers of stacked marbles. The easily drawn cubic crystal structure illustrates that a solid
crystal may contain charged particles.
Group VIIIA elements: He, Ne, Ar, Kr, Xe all have 8 electrons in the valence shell. (Figure

2.3. VALENCE AND CRYSTAL STRUCTURE

43
+

-

=
Na

Cl

Na+

Cl-

Figure 2.12: Neutral Sodium atom donates an electron to neutral Chlorine atom forming Na+
and Cl− ions.
below) That is, the valence shell is complete meaning these elements neither donate nor accept
electrons. Nor do they readily participate in chemical reactions since group VIIIA elements
do not easily combine with other elements. In recent years chemists have forced Xe and Kr to
form a few compounds, however for the purposes of our discussion this is not applicable. These
elements are good electrical insulators and are gases at room temperature.

He

Ne

Ar

Kr

Xe

Figure 2.13: Group VIIIA elements: He, Ne, Ar, Kr, Xe are largely unreactive since the valence
shell is complete..
Group IVA elements: C, Si, Ge, having 4 electrons in the valence shell as shown in Figure 2.14 form compounds by sharing electrons with other elements without forming ions. This
shared electron bonding is known as covalent bonding. Note that the center atom (and the
others by extension) has completed its valence shell by sharing electrons. Note that the figure
is a 2-d representation of bonding, which is actually 3-d. It is this group, IVA, that we are
interested in for its semiconducting properties.
Crystal structure: Most inorganic substances form their atoms (or ions) into an ordered
array known as a crystal. The outer electron clouds of atoms interact in an orderly manner.
Even metals are composed of crystals at the microscopic level. If a metal sample is given
an optical polish, then acid etched, the microscopic microcrystalline structure shows as in
Figure 2.15. It is also possible to purchase, at considerable expense, metallic single crystal
specimens from specialized suppliers. Polishing and etching such a specimen discloses no microcrystalline structure. Practically all industrial metals are polycrystalline. Most modern
semiconductors, on the other hand, are single crystal devices. We are primarily interested in
monocrystalline structures.
Many metals are soft and easily deformed by the various metal working techniques. The
microcrystals are deformed in metal working. Also, the valence electrons are free to move
about the crystal lattice, and from crystal to crystal. The valence electrons do not belong to

CHAPTER 2. SOLID-STATE DEVICE THEORY

44

C

Si
(a)

Ge
(b)

Figure 2.14: (a) Group IVA elements: C, Si, Ge having 4 electrons in the valence shell, (b)
complete the valence shell by sharing electrons with other elements.

(a)

(b)

(c)

Figure 2.15: (a) Metal sample, (b) polished, (c) acid etched to show microcrystalline structure.
any particular atom, but to all atoms.
The rigid crystal structure in Figure 2.16 is composed of a regular repeating pattern of
positive Na ions and negative Cl ions. The Na and Cl atoms form Na+ and Cl− ions by transferring an electron from Na to Cl, with no free electrons. Electrons are not free to move about
the crystal lattice, a difference compared with a metal. Nor are the ions free. Ions are fixed in
place within the crystal structure. Though, the ions are free to move about if the NaCl crystal
is dissolved in water. However, the crystal no longer exists. The regular, repeating structure is
gone. Evaporation of the water deposits the Na+ and Cl− ions in the form of new crystals as
the oppositely charged ions attract each other. Ionic materials form crystal structures due to
the strong electrostatic attraction of the oppositely charged ions.
Semiconductors in Group IV also form crystals because of the tetrahedral bonding pattern
of the s2 p2 electrons about the atom, sharing electron-pair bonds to four adjacent atoms. (Figure 2.18(a) ) More correctly the four outer electrons: two in the s-orbital, (sz ) offset along the
z-axis, and two in the p-orbital (px and py ) hybridize to form four sp3 molecular orbitals. These
four electron clouds repell one another to equidistant tetrahedral spacing about the Si atom,
attracted by the positive nucleus as shown in Figure 2.17.
Every semiconductor atom, Si, Ge, or C (diamond) is chemically bonded to four other atoms
by covalent bonds, shared electron bonds. Two electrons may share an orbital if each have
opposite spin quantum numbers. Thus, an unpaired electron may share an orbital with an
electron from another atom. This corresponds to overlapping Figure 2.18(a) of the electron
clouds, or bonding. Figure 2.18 (b) is one fourth of the volume of the diamond crystal structure

2.3. VALENCE AND CRYSTAL STRUCTURE

45

Na+
Cl -

Figure 2.16: NaCl crystal having a cubic structure.

y

+

+

=

x
z
sz 2

px

py

sp3

Figure 2.17: Two s-orbital (sz ) electrons and two p-orbital (sx and sy ) electrons hybridize, (c)
forming four sp3 molecular orbitals.

CHAPTER 2. SOLID-STATE DEVICE THEORY

46

unit cell shown in Figure 2.19 at the origin. The bonds are particularly strong in diamond,
decreasing in strength going down group IV to silicon, and germanium. Silicon and germanium
both form crystals with a diamond structure.

(a)

(b)

Figure 2.18: (a) Tetrahedral bonding of Si atom. (b) leads to 1/4 of the cubic unit cell
The diamond unit cell is the basic crystal building block. Figure 2.19 shows four atoms
(dark) bonded to four others within the volume of the cell. This is equivalent to placing one
of Figure 2.18(b) at the origin in Figure 2.19, then placing three more on adjacent faces to fill
the full cube. Six atoms fall on the middle of each of the six cube faces, showing two bonds.
The other two bonds to adjacent cubes were omitted for clarity. Out of eight cube corners, four
atoms bond to an atom within the cube. Where are the other four atoms bonded? The other
four bond to adjacent cubes of the crystal. Keep in mind that even though four corner atoms
show no bonds in the cube, all atoms within the crystal are bonded in one giant molecule. A
semiconductor crystal is built up from copies of this unit cell.
The crystal is effectively one molecule. An atom covalent bonds to four others, which in turn
bond to four others, and so on. The crystal lattice is relatively stiff resisting deformation. Few
electrons free themselves for conduction about the crystal. A property of semiconductors is that
once an electron is freed, a positively charged empty space develops which also contributes to
conduction.
• REVIEW
• Atoms try to form a complete outer, valence, shell of 8-electrons (2-electrons for the innermost shell). Atoms may donate a few electrons to expose an underlying shell of 8, accept
a few electrons to complete a shell, or share electrons to complete a shell.
• Atoms often form ordered arrays of ions or atoms in a rigid structure known as a crystal.
• A neutral atom may form a positive ion by donating an electron.
• A neutral atom may form a negative ion by accepting an electron
• The group IVA semiconductors: C, Si, Ge crystallize into a diamond structure. Each atom
in the crystal is part of a giant molecule, bonding to four other atoms.
• Most semiconductor devices are manufactured from single crystals.

2.4. BAND THEORY OF SOLIDS

47

Face centered atoms

Atom bonded to 4 others
Other atoms bonded to
chain in cube
Atoms bonded outside of
cube

Figure 2.19: Si, Ge, and C (diamond) form interleaved face centered cube.

2.4

Band theory of solids

Quantum physics describes the states of electrons in an atom according to the four-fold scheme
of quantum numbers. The quantum numbers describe the allowable states electrons may assume in an atom. To use the analogy of an amphitheater, quantum numbers describe how
many rows and seats are available. Individual electrons may be described by the combination
of quantum numbers, like a spectator in an amphitheater assigned to a particular row and
seat.
Like spectators in an amphitheater moving between seats and rows, electrons may change
their statuses, given the presence of available spaces for them to fit, and available energy.
Since shell level is closely related to the amount of energy that an electron possesses, “leaps”
between shell (and even subshell) levels requires transfers of energy. If an electron is to move
into a higher-order shell, it requires that additional energy be given to the electron from an
external source. Using the amphitheater analogy, it takes an increase in energy for a person
to move into a higher row of seats, because that person must climb to a greater height against
the force of gravity. Conversely, an electron “leaping” into a lower shell gives up some of its
energy, like a person jumping down into a lower row of seats, the expended energy manifesting
as heat and sound.
Not all “leaps” are equal. Leaps between different shells require a substantial exchange of
energy, but leaps between subshells or between orbitals require lesser exchanges.
When atoms combine to form substances, the outermost shells, subshells, and orbitals
merge, providing a greater number of available energy levels for electrons to assume. When
large numbers of atoms are close to each other, these available energy levels form a nearly

CHAPTER 2. SOLID-STATE DEVICE THEORY

48

continuous band wherein electrons may move as illustrated in Figure 2.20

Significant leap required
for an electron to move
to the next higher level
3p

Shorter leap
required
3p

Overlap permits
electrons to freely
drift between bands
3p
Overlap

3s
Single atom

3s
Five atoms
in close proximity

3s
Multitudes of atoms
in close proximity

Figure 2.20: Electron band overlap in metallic elements.

It is the width of these bands and their proximity to existing electrons that determines
how mobile those electrons will be when exposed to an electric field. In metallic substances,
empty bands overlap with bands containing electrons, meaning that electrons of a single atom
may move to what would normally be a higher-level state with little or no additional energy
imparted. Thus, the outer electrons are said to be “free,” and ready to move at the beckoning
of an electric field.
Band overlap will not occur in all substances, no matter how many atoms are close to each
other. In some substances, a substantial gap remains between the highest band containing
electrons (the so-called valence band) and the next band, which is empty (the so-called conduction band). See Figure 2.21. As a result, valence electrons are “bound” to their constituent
atoms and cannot become mobile within the substance without a significant amount of imparted energy. These substances are electrical insulators.
Materials that fall within the category of semiconductors have a narrow gap between the
valence and conduction bands. Thus, the amount of energy required to motivate a valence
electron into the conduction band where it becomes mobile is quite modest. (Figure 2.22)
At low temperatures, little thermal energy is available to push valence electrons across
this gap, and the semiconducting material acts more as an insulator. At higher temperatures,
though, the ambient thermal energy becomes enough to force electrons across the gap, and the
material will increase conduction of electricity.
It is difficult to predict the conductive properties of a substance by examining the electron
configurations of its constituent atoms. Although the best metallic conductors of electricity
(silver, copper, and gold) all have outer s subshells with a single electron, the relationship
between conductivity and valence electron count is not necessarily consistent:

2.4. BAND THEORY OF SOLIDS

49

Conduction band
Significant leap required
for an electron to enter
the conduction band and
travel through the material

"Energy gap"
Valence band

Multitudes of atoms
in close proximity
Figure 2.21: Electron band separation in insulating substances.

semiconducting substance
metalic substance for reference

Conduction band
Small leap required
for an electron to enter
the conduction band and
travel through the material

Insignificant leap
for electron to
enter conduction
band

"Energy gap"
Valence band

(a)

(b)

Figure 2.22: Electron band separation in semiconducting substances, (a) multitudes of semiconducting close atoms still results in a significant band gap, (b) multitudes of close metal
atoms for reference.

CHAPTER 2. SOLID-STATE DEVICE THEORY

50

Element

Specific resistance Electron
(ρ) at 20o Celsius configuration

Element

Specific resistance Electron
(ρ) at 20o Celsius configuration

Silver (Ag)

9.546 Ω⋅cmil/ft

4d105s1

Copper (Cu)

10.09 Ω⋅cmil/ft

3d104s1

Zinc (Zn)

35.49 Ω⋅cmil/ft

3d104s2

Gold (Au)

13.32 Ω⋅cmil/ft

5d106s1

Nickel (Ni)

41.69 Ω⋅cmil/ft

3d84s2

Aluminum (Al)

15.94 Ω⋅cmil/ft

3p1

Iron (Fe)

57.81 Ω⋅cmil/ft

3d64s2

Tungsten (W)

31.76 Ω⋅cmil/ft

5d46s2

Platinum (Pt)

63.16 Ω⋅cmil/ft

5d96s1

Molybdenum (Mo) 32.12 Ω⋅cmil/ft

4d55s1

The electron band configurations produced by compounds of different elements defies easy
association with the electron configurations of its constituent elements.
• REVIEW:
• Energy is required to remove an electron from the valence band to a higher unoccupied
band, a conduction band. More energy is required to move between shells, less between
subshells.
• Since the valence and conduction bands overlap in metals, little energy removes an electron. Metals are excellent conductors.
• The large gap between the valence and conduction bands of an insulator requires high
energy to remove an electron. Thus, insulators do not conduct.
• Semiconductors have a small non-overlapping gap between the valence and conduction
bands. Pure semiconductors are neither good insulators nor conductors. Semiconductors
are semi-conductive.

2.5

Electrons and “holes”

Pure semiconductors are relatively good insulators as compared with metals, though not nearly
as good as a true insulator like glass. To be useful in semiconductor applications, the intrinsic
semiconductor (pure undoped semiconductor) must have no more than one impurity atom in
10 billion semiconductor atoms. This is analogous to a grain of salt impurity in a railroad
boxcar of sugar. Impure, or dirty semiconductors are considerably more conductive, though not
as good as metals. Why might this be? To answer that question, we must look at the electron
structure of such materials in Figure 2.23.
Figure 2.23 (a) shows four electrons in the valence shell of a semiconductor forming covalent
bonds to four other atoms. This is a flattened, easier to draw, version of Figure 2.19. All
electrons of an atom are tied up in four covalent bonds, pairs of shared electrons. Electrons are
not free to move about the crystal lattice. Thus, intrinsic, pure, semiconductors are relatively
good insulators as compared to metals.
Thermal energy may occasionally free an electron from the crystal lattice as in Figure 2.23
(b). This electron is free for conduction about the crystal lattice. When the electron was freed,
it left an empty spot with a positive charge in the crystal lattice known as a hole. This hole is
not fixed to the lattice; but, is free to move about. The free electron and hole both contribute

2.5. ELECTRONS AND “HOLES”

(a)

51

hole

(b)

electron

Figure 2.23: (a) Intrinsic semiconductor is an insulator having a complete electron shell. (b)
However, thermal energy can create few electron hole pairs resulting in weak conduction.

to conduction about the crystal lattice. That is, the electron is free until it falls into a hole.
This is called recombination. If an external electric field is applied to the semiconductor, the
electrons and holes will conduct in opposite directions. Increasing temperature will increase
the number of electrons and holes, decreasing the resistance. This is opposite of metals, where
resistance increases with temperature by increasing the collisions of electrons with the crystal
lattice. The number of electrons and holes in an intrinsic semiconductor are equal. However,
both carriers do not necessarily move with the same velocity with the application of an external
field. Another way of stating this is that the mobility is not the same for electrons and holes.
Pure semiconductors, by themselves, are not particularly useful. Though, semiconductors
must be refined to a high level of purity as a starting point prior the addition of specific impurities.
Semiconductor material pure to 1 part in 10 billion, may have specific impurities added
at approximately 1 part per 10 million to increase the number of carriers. The addition of a
desired impurity to a semiconductor is known as doping. Doping increases the conductivity of
a semiconductor so that it is more comparable to a metal than an insulator.
It is possible to increase the number of negative charge carriers within the semiconductor
crystal lattice by doping with an electron donor like Phosphorus. Electron donors, also known
as N-type dopants include elements from group VA of the periodic table: nitrogen, phosphorus,
arsenic, and antimony. Nitrogen and phosphorus are N-type dopants for diamond. Phosphorus,
arsenic, and antimony are used with silicon.
The crystal lattice in Figure 2.24 (b) contains atoms having four electrons in the outer
shell, forming four covalent bonds to adjacent atoms. This is the anticipated crystal lattice.
The addition of a phosphorus atom with five electrons in the outer shell introduces an extra
electron into the lattice as compared with the silicon atom. The pentavalent impurity forms
four covalent bonds to four silicon atoms with four of the five electrons, fitting into the lattice
with one electron left over. Note that this spare electron is not strongly bonded to the lattice as
the electrons of normal Si atoms are. It is free to move about the crystal lattice, not being bound
to the Phosphorus lattice site. Since we have doped at one part phosphorus in 10 million silicon
atoms, few free electrons were created compared with the numerous silicon atoms. However,
many electrons were created compared with the fewer electron-hole pairs in intrinsic silicon.
Application of an external electric field produces strong conduction in the doped semiconductor
in the conduction band (above the valence band). A heavier doping level produces stronger

CHAPTER 2. SOLID-STATE DEVICE THEORY

52

conduction. Thus, a poorly conducting intrinsic semiconductor has been converted into a good
electrical conductor.

B
Si

(a)

P

(b)

Si

Si

Si

Si

Si

Si

Si

Si

Si

P

Si

Si

Si

B

Si

Si

Si

Si

Si

Si
electron

Si
Si
Si
Si
(c)
hole
electron
hole movement
movement

Figure 2.24: (a) Outer shell electron configuration of donor N-type Phosphorus, Silicon (for
reference), and acceptor P-type Boron. (b) N-type donor impurity creates free electron (c) Ptype acceptor impurity creates hole, a positive charge carrier.
It is also possible to introduce an impurity lacking an electron as compared with silicon,
having three electrons in the valence shell as compared with four for silicon. In Figure 2.24
(c), this leaves an empty spot known as a hole, a positive charge carrier. The boron atom tries
to bond to four silicon atoms, but only has three electrons in the valence band. In attempting
to form four covalent bonds the three electrons move around trying to form four bonds. This
makes the hole appear to move. Furthermore, the trivalent atom may borrow an electron from
an adjacent (or more distant) silicon atom to form four covalent bonds. However, this leaves
the silicon atom deficient by one electron. In other words, the hole has moved to an adjacent (or
more distant) silicon atom. Holes reside in the valence band, a level below the conduction band.
Doping with an electron acceptor, an atom which may accept an electron, creates a deficiency of
electrons, the same as an excess of holes. Since holes are positive charge carriers, an electron
acceptor dopant is also known as a P-type dopant. The P-type dopant leaves the semiconductor
with an excess of holes, positive charge carriers. The P-type elements from group IIIA of the
periodic table include: boron, aluminum, gallium, and indium. Boron is used as a P-type
dopant for silicon and diamond semiconductors, while indium is used with germanium.
The “marble in a tube” analogy to electron conduction in Figure 2.25 relates the movement
of holes with the movement of electrons. The marble represent electrons in a conductor, the
tube. The movement of electrons from left to right as in a wire or N-type semiconductor is
explained by an electron entering the tube at the left forcing the exit of an electron at the
right. Conduction of N-type electrons occurrs in the conduction band. Compare that with the
movement of a hole in the valence band.
For a hole to enter at the left of Figure 2.25 (b), an electron must be removed. When moving
a hole left to right, the electron must be moved right to left. The first electron is ejected from
the left end of the tube so that the hole may move to the right into the tube. The electron is
moving in the opposite direction of the positive hole. As the hole moves farther to the right,
electrons must move left to accommodate the hole. The hole is the absence of an electron in
the valence band due to P-type doping. It has a localized positive charge. To move the hole in
a given direction, the valence electrons move in the opposite direction.
Electron flow in an N-type semiconductor is similar to electrons moving in a metallic wire.

2.5. ELECTRONS AND “HOLES”

53

electron movement

(a)

hole movement

(b)

electron movement

Figure 2.25: Marble in a tube analogy: (a) Electrons move right in the conduction band as
electrons enter tube. (b) Hole moves right in the valence band as electrons move left.
The N-type dopant atoms will yield electrons available for conduction. These electrons, due
to the dopant are known as majority carriers, for they are in the majority as compared to the
very few thermal holes. If an electric field is applied across the N-type semiconductor bar in
Figure 2.26 (a), electrons enter the negative (left) end of the bar, traverse the crystal lattice,
and exit at right to the (+) battery terminal.
electron enters

electron exits

crystal lattice

(a) N-type

(b) P-type

Figure 2.26: (a) N-type semiconductor with electrons moving left to right through the crystal
lattice. (b) P-type semiconductor with holes moving left to right, which corresponds to electrons
moving in the opposite direction.
Current flow in a P-type semiconductor is a little more difficult to explain. The P-type
dopant, an electron acceptor, yields localized regions of positive charge known as holes. The
majority carrier in a P-type semiconductor is the hole. While holes form at the trivalent dopant
atom sites, they may move about the semiconductor bar. Note that the battery in Figure 2.26
(b) is reversed from (a). The positive battery terminal is connected to the left end of the P-type
bar. Electron flow is out of the negative battery terminal, through the P-type bar, returning to
the positive battery terminal. An electron leaving the positive (left) end of the semiconductor
bar for the positive battery terminal leaves a hole in the semiconductor, that may move to the
right. Holes traverse the crystal lattice from left to right. At the negative end of the bar an
electron from the battery combines with a hole, neutralizing it. This makes room for another
hole to move in at the positive end of the bar toward the right. Keep in mind that as holes move
left to right, that it is actually electrons moving in the opposite direction that is responsible for
the apparant hole movement.

CHAPTER 2. SOLID-STATE DEVICE THEORY

54

The elements used to produce semiconductors are summarized in Figure 2.27. The oldest group IVA bulk semiconductor material germanium is only used to a limited extent today.
Silicon based semiconductors account for about 90% of commercial production of all semiconductors. Diamond based semiconductors are a research and development activity with considerable potential at this time. Compound semiconductors not listed include silicon germanium
(thin layers on Si wafers), silicon carbide and III-V compounds such as gallium arsenide. IIIVI compound semiconductors include: AlN, GaN, InN, AlP, AlAs, AlSb, GaP, GaAs, GaSb, InP,
InAs, InSb, Alx Ga1−x As and Inx Ga1−x As. Columns II and VI of periodic table, not shown in
the figure, also form compound semiconductors.
Elemental semiconductors
C(diamond), Si, Ge
B
P-type dopant for C

13

IIIA

B

5
Boron
10.81

2p1

14
C

IVA

6
Carbon
12.011

2p2

15
N

VA

7
Nitrogen
14.0067

2p3

B, Al, Ga, In
P-type dopant for Si

Al
13 Si
15
14 P
Aluminum
Silicon Phosphorus
26.9815
30.9738
28.0855

Al, Ga, In
P-type dopant for Ge

Ga
31 Ge
33
32 As
Gallium Germanium Arsenic
69.723
72.61
74.92159

3p1

4p1
In

49
Indium
114.82

5p1

3p2

4p2

N, P
N-type dopant for C

3p3

P, As, Sb
N-type dopant for Si, Ge

4p3
Sb
51
Antimony
121.75
5p3

Figure 2.27: Group IIIA P-type dopants, group IV basic semiconductor materials, and group
VA N-type dopants.
The main reason for the inclusion of the IIIA and VA groups in Figure 2.27 is to show the
dopants used with the group IVA semiconductors. Group IIIA elements are acceptors, P-type
dopants, which accept electrons leaving a hole in the crystal lattice, a positive carrier. Boron
is the P-type dopant for diamond, and the most common dopant for silicon semiconductors.
Indium is the P-type dopant for germanium.
Group VA elements are donors, N-type dopants, yielding a free electron. Nitrogen and
Phosphorus are suitable N-type dopants for diamond. Phosphorus and arsenic are the most
commonly used N-type dopants for silicon; though, antimony can be used.
• REVIEW:
• Intrinsic semiconductor materials, pure to 1 part in 10 billion, are poor conductors.
• N-type semiconductor is doped with a pentavalent impurity to create free electrons. Such
a material is conductive. The electron is the majority carrier.
• P-type semiconductor, doped with a trivalent impurity, has an abundance of free holes.
These are positive charge carriers. The P-type material is conductive. The hole is the
majority carrier.

2.6. THE P-N JUNCTION

55

• Most semiconductors are based on elements from group IVA of the periodic table, silicon
being the most prevalent. Germanium is all but obsolete. Carbon (diamond) is being
developed.
• Compound semiconductors such as silicon carbide (group IVA) and gallium arsenide (group
III-V) are widely used.

2.6

The P-N junction

If a block of P-type semiconductor is placed in contact with a block of N-type semiconductor in
Figure 2.28(a), the result is of no value. We have two conductive blocks in contact with each
other, showing no unique properties. The problem is two separate and distinct crystal bodies.
The number of electrons is balanced by the number of protons in both blocks. Thus, neither
block has any net charge.
However, a single semiconductor crystal manufactured with P-type material at one end and
N-type material at the other in Figure 2.28 (b) has some unique properties. The P-type material
has positive majority charge carriers, holes, which are free to move about the crystal lattice.
The N-type material has mobile negative majority carriers, electrons. Near the junction, the
N-type material electrons diffuse across the junction, combining with holes in P-type material.
The region of the P-type material near the junction takes on a net negative charge because
of the electrons attracted. Since electrons departed the N-type region, it takes on a localized
positive charge. The thin layer of the crystal lattice between these charges has been depleted
of majority carriers, thus, is known as the depletion region. It becomes nonconductive intrinsic
semiconductor material. In effect, we have nearly an insulator separating the conductive P
and N doped regions.

no charge
separation

N

electron
(a)

P

N

crystal lattice
hole
(b)

P

intrinsic
charge
separation

Figure 2.28: (a) Blocks of P and N semiconductor in contact have no exploitable properties. (b)
Single crystal doped with P and N type impurities develops a potential barrier.
This separation of charges at the PN junction constitutes a potential barrier. This potential
barrier must be overcome by an external voltage source to make the junction conduct. The
formation of the junction and potential barrier happens during the manufacturing process.
The magnitude of the potential barrier is a function of the materials used in manufacturing.
Silicon PN junctions have a higher potential barrier than germanium junctions.

CHAPTER 2. SOLID-STATE DEVICE THEORY

56

In Figure 2.29(a) the battery is arranged so that the negative terminal supplies electrons
to the N-type material. These electrons diffuse toward the junction. The positive terminal
removes electrons from the P-type semiconductor, creating holes that diffuse toward the junction. If the battery voltage is great enough to overcome the junction potential (0.6V in Si),
the N-type electrons and P-holes combine annihilating each other. This frees up space within
the lattice for more carriers to flow toward the junction. Thus, currents of N-type and P-type
majority carriers flow toward the junction. The recombination at the junction allows a battery
current to flow through the PN junction diode. Such a junction is said to be forward biased.

electrons
N

(a) Forward

holes

electrons
P

N

depletion region
holes
P

(b) Reverse

Figure 2.29: (a) Forward battery bias repells carriers toward junction, where recombination
results in battery current. (b) Reverse battery bias attracts carriers toward battery terminals,
away from junction. Depletion region thickness increases. No sustained battery current flows.
If the battery polarity is reversed as in Figure 2.29(b) majority carriers are attracted away
from the junction toward the battery terminals. The positive battery terminal attracts N-type
majority carriers, electrons, away from the junction. The negative terminal attracts P-type
majority carriers, holes, away from the junction. This increases the thickness of the nonconducting depletion region. There is no recombination of majority carriers; thus, no conduction.
This arrangement of battery polarity is called reverse bias.
The diode schematic symbol is illustrated in Figure 2.30(b) corresponding to the doped
semiconductor bar at (a). The diode is a unidirectional device. Electron current only flows in
one direction, against the arrow, corresponding to forward bias. The cathode, bar, of the diode
symbol corresponds to N-type semiconductor. The anode, arrow, corresponds to the P-type
semiconductor. To remember this relationship, Not-pointing (bar) on the symbol corresponds
to N-type semiconductor. Pointing (arrow) corresponds to P-type.
If a diode is forward biased as in Figure 2.30(a), current will increase slightly as voltage is
increased from 0 V. In the case of a silicon diode a measurable current flows when the voltage
approaches 0.6 V at (c). As the voltage is increases past 0.6 V, current increases considerably
after the knee. Increasing the voltage well beyond 0.7 V may result in high enough current to
destroy the diode. The forward voltage, VF , is a characteristic of the semiconductor: 0.6 to 0.7
V for silicon, 0.2 V for germanium, a few volts for Light Emitting Diodes (LED). The forward
current ranges from a few mA for point contact diodes to 100 mA for small signal diodes to tens
or thousands of amperes for power diodes.
If the diode is reverse biased, only the leakage current of the intrinsic semiconductor flows.
This is plotted to the left of the origin in Figure 2.30(c). This current will only be as high as
1 µA for the most extreme conditions for silicon small signal diodes. This current does not

2.6. THE P-N JUNCTION
electrons
N

57
holes
P

I
mA
forward
bias V
0.7

reverse bias
(a)
N-type
(not pointing)
(b)

cathode

P-type
(pointing)
anode

breakdown
(c)

µA

Figure 2.30: (a) Forward biased PN junction, (b) Corresponding diode schematic symbol (c)
Silicon Diode I vs V characteristic curve.
increase appreciably with increasing reverse bias until the diode breaks down. At breakdown,
the current increases so greatly that the diode will be destroyed unless a high series resistance
limits current. We normally select a diode with a higher reverse voltage rating than any applied voltage to prevent this. Silicon diodes are typically available with reverse break down
ratings of 50, 100, 200, 400, 800 V and higher. It is possible to fabricate diodes with a lower
rating of a few volts for use as voltage standards.
We previously mentioned that the reverse leakage current of under a µA for silicon diodes
was due to conduction of the intrinsic semiconductor. This is the leakage that can be explained
by theory. Thermal energy produces few electron hole pairs, which conduct leakage current
until recombination. In actual practice this predictable current is only part of the leakage current. Much of the leakage current is due to surface conduction, related to the lack of cleanliness
of the semiconductor surface. Both leakage currents increase with increasing temperature, approaching a µA for small silicon diodes.
For germanium, the leakage current is orders of magnitude higher. Since germanium semiconductors are rarely used today, this is not a problem in practice.
• REVIEW:
• PN junctions are fabricated from a monocrystalline piece of semiconductor with both a
P-type and N-type region in proximity at a junction.
• The transfer of electrons from the N side of the junction to holes annihilated on the P side
of the junction produces a barrier voltage. This is 0.6 to 0.7 V in silicon, and varies with
other semiconductors.
• A forward biased PN junction conducts a current once the barrier voltage is overcome.
The external applied potential forces majority carriers toward the junction where recombination takes place, allowing current flow.

CHAPTER 2. SOLID-STATE DEVICE THEORY

58

• A reverse biased PN junction conducts almost no current. The applied reverse bias attracts majority carriers away from the junction. This increases the thickness of the nonconducting depletion region.
• Reverse biased PN junctions show a temperature dependent reverse leakage current.
This is less than a µA in small silicon diodes.

2.7

Junction diodes

There were some historic crude, but usable semiconductor rectifiers before high purity materials were available. Ferdinand Braun invented a lead sulfide, PbS, based point contact rectifier
in 1874. Cuprous oxide rectifiers were used as power rectifiers in 1924. The forward voltage
drop is 0.2 V. The linear characteristic curve perhaps is why Cu2 O was used as a rectifier for
the AC scale on D’Arsonval based multimeters. This diode is also photosensitive.
Selenium oxide rectifiers were used before modern power diode rectifiers became available.
These and the Cu2 O rectifiers were polycrystalline devices. Photoelectric cells were once made
from Selenium.
Before the modern semiconductor era, an early diode application was as a radio frequency
detector, which recovered audio from a radio signal. The “semiconductor” was a polycrystalline
piece of the mineral galena, lead sulfide, PbS. A pointed metallic wire known as a cat whisker
was brought in contact with a spot on a crystal within the polycrystalline mineral. (Figure 2.31)
The operator labored to find a “sensitive” spot on the galena by moving the cat whisker about.
Presumably there were P and N-type spots randomly distributed throughout the crystal due
to the variability of uncontrolled impurities. Less often the mineral iron pyrites, fools gold,
was used, as was the mineral carborundum, silicon carbide, SiC, another detector, part of a
foxhole radio, consisted of a sharpened pencil lead bound to a bent safety pin, touching a rusty
blue-blade disposable razor blade. These all required searching for a sensitive spot, easily lost
because of vibration.
Replacing the mineral with an N-doped semiconductor (Figure 2.32(a) ) makes the whole
surface sensitive, so that searching for a sensitive spot was no longer required. This device was
perfected by G.W.Pickard in 1906. The pointed metal contact produced a localized P-type region
within the semiconductor. The metal point was fixed in place, and the whole point contact diode
encapsulated in a cylindrical body for mechanical and electrical stability. (Figure 2.32(d) ) Note
that the cathode bar on the schematic corresponds to the bar on the physical package.
Silicon point contact diodes made an important contribution to radar in World War II, detecting giga-hertz radio frequency echo signals in the radar receiver. The concept to be made
clear is that the point contact diode preceded the junction diode and modern semiconductors
by several decades. Even to this day, the point contact diode is a practical means of microwave
frequency detection because of its low capacitance. Germanium point contact diodes were once
more readily available than they are today, being preferred for the lower 0.2 V forward voltage
in some applications like self-powered crystal radios. Point contact diodes, though sensitive to
a wide bandwidth, have a low current capability compared with junction diodes.
Most diodes today are silicon junction diodes. The cross-section in Figure 2.32(b) looks a
bit more complex than a simple PN junction; though, it is still a PN junction. Starting at the
cathode connection, the N+ indicates this region is heavily doped, having nothing to do with

2.7. JUNCTION DIODES

59

Figure 2.31: Crystal detector

Anode

Anode

P+

P+

N

NN+

(a)

(b)
Cathode

Cathode

Anode

Cathode
(c)

(d)

Figure 2.32: Silicon diode cross-section: (a) point contact diode, (b) junction diode, (c) schematic
symbol, (d) small signal diode package.

CHAPTER 2. SOLID-STATE DEVICE THEORY

60

polarity. This reduces the series resistance of the diode. The N− region is lightly doped as
indicated by the (-). Light doping produces a diode with a higher reverse breakdown voltage,
important for high voltage power rectifier diodes. Lower voltage diodes, even low voltage power
rectifiers, would have lower forward losses with heavier doping. The heaviest level of doping
produce zener diodes designed for a low reverse breakdown voltage. However, heavy doping
increases the reverse leakage current. The P+ region at the anode contact is heavily doped
P-type semiconductor, a good contact strategy. Glass encapsulated small signal junction diodes
are capable of 10’s to 100’s of mA of current. Plastic or ceramic encapsulated power rectifier
diodes handle to 1000’s of amperes of current.
• REVIEW:
• Point contact diodes have superb high frequency characteristics, usable well into the microwave frequencies.
• Junction diodes range in size from small signal diodes to power rectifiers capable of 1000’s
of amperes.
• The level of doping near the junction determines the reverse breakdown voltage. Light
doping produces a high voltage diode. Heavy doping produces a lower breakdown voltage,
and increases reverse leakage current. Zener diodes have a lower breakdown voltage
because of heavy doping.

2.8

Bipolar junction transistors

The bipolar junction transistor (BJT) was named because its operation involves conduction by
two carriers: electrons and holes in the same crystal. The first bipolar transistor was invented
at Bell Labs by William Shockley, Walter Brattain, and John Bardeen so late in 1947 that it
was not published until 1948. Thus, many texts differ as to the date of invention. Brattain
fabricated a germanium point contact transistor, bearing some resemblance to a point contact
diode. Within a month, Shockley had a more practical junction transistor, which we describe in
following paragraphs. They were awarded the Nobel Prize in Physics in 1956 for the transistor.
The bipolar junction transistor shown in Figure 2.33(a) is an NPN three layer semiconductor sandwich with an emitter and collector at the ends, and a base in between. It is as if a third
layer were added to a two layer diode. If this were the only requirement, we would have no
more than a pair of back-to-back diodes. In fact, it is far easier to build a pair of back-to-back
diodes. The key to the fabrication of a bipolar junction transistor is to make the middle layer,
the base, as thin as possible without shorting the outside layers, the emitter and collector. We
cannot over emphasize the importance of the thin base region.
The device in Figure 2.33(a) has a pair of junctions, emitter to base and base to collector,
and two depletion regions.
It is customary to reverse bias the base-collector junction of a bipolar junction transistor
as shown in (Figure 2.33(b). Note that this increases the width of the depletion region. The
reverse bias voltage could be a few volts to tens of volts for most transistors. There is no current
flow, except leakage current, in the collector circuit.

2.8. BIPOLAR JUNCTION TRANSISTORS
+- P - +
+- - +
+- - +
+- - +
+- - +
+- - +
+- - +
+- - +

N

emitter

61

N

base

+ - P+- +- +- +- +- +- +- -

N

collector

emitter

+
+
+
+
+
+
+
+

base

N

collector
-

E
(a)

C
B

E
(b)

+

C
B

-

+

Figure 2.33: (a) NPN junction bipolar transistor. (b) Apply reverse bias to collector base junction.
In Figure 2.34(a), a voltage source has been added to the emitter base circuit. Normally we
forward bias the emitter-base junction, overcoming the 0.6 V potential barrier. This is similar
to forward biasing a junction diode. This voltage source needs to exceed 0.6 V for majority
carriers (electrons for NPN) to flow from the emitter into the base becoming minority carriers
in the P-type semiconductor.
If the base region were thick, as in a pair of back-to-back diodes, all the current entering
the base would flow out the base lead. In our NPN transistor example, electrons leaving the
emitter for the base would combine with holes in the base, making room for more holes to be
created at the (+) battery terminal on the base as electrons exit.
However, the base is manufactured thin. A few majority carriers in the emitter, injected as
minority carriers into the base, actually recombine. See Figure 2.34(b). Few electrons injected
by the emitter into the base of an NPN transistor fall into holes. Also, few electrons entering
the base flow directly through the base to the positive battery terminal. Most of the emitter
current of electrons diffuses through the thin base into the collector. Moreover, modulating the
small base current produces a larger change in collector current. If the base voltage falls below
approximately 0.6 V for a silicon transistor, the large emitter-collector current ceases to flow.
In Figure 2.35 we take a closer look at the current amplification mechanism. We have an
enlarged view of an NPN junction transistor with emphasis on the thin base region. Though
not shown, we assume that external voltage sources 1) forward bias the emitter-base junction,
2) reverse bias the base-collector junction. Electrons, majority carriers, enter the emitter from
the (-) battery terminal. The base current flow corresponds to electrons leaving the base terminal for the (+) battery terminal. This is but a small current compared to the emitter current.
Majority carriers within the N-type emitter are electrons, becoming minority carriers when
entering the P-type base. These electrons face four possible fates entering the thin P-type base.
A few at Figure 2.35(a) fall into holes in the base that contributes to base current flow to the
(+) battery terminal. Not shown, holes in the base may diffuse into the emitter and combine
with electrons, contributing to base terminal current. Few at (b) flow on through the base to
the (+) battery terminal as if the base were a resistor. Both (a) and (b) contribute to the very
small base current flow. Base current is typically 1% of emitter or collector current for small
signal transistors. Most of the emitter electrons diffuse right through the thin base (c) into

CHAPTER 2. SOLID-STATE DEVICE THEORY

62

P-

N

-

+
+
+
+
+
+
+
+

N

+

-

P
-

N

-

+

E
(a)

C
+

B

-

+

N

+

E

-

+
+
+
+
+
+
+
+

-

+

C
+

B

-

+

(b)

Figure 2.34: NPN junction bipolar transistor with reverse biased collector-base: (a) Adding
forward bias to base-emitter junction, results in (b) a small base current and large emitter and
collector currents.

(c) (a)
holes
electrons
- + depletion
- + region

(b)
N
emitter

(d)
+
+
+
+
depletion
region
N
P
collector
base

Figure 2.35: Disposition of electrons entering base: (a) Lost due to recombination with base
holes. (b) Flows out base lead. (c) Most diffuse from emitter through thin base into basecollector depletion region, and (d) are rapidly swept by the strong depletion region electric field
into the collector.

2.8. BIPOLAR JUNCTION TRANSISTORS

63

the base-collector depletion region. Note the polarity of the depletion region surrounding the
electron at (d). The strong electric field sweeps the electron rapidly into the collector. The
strength of the field is proportional to the collector battery voltage. Thus 99% of the emitter
current flows into the collector. It is controlled by the base current, which is 1% of the emitter
current. This is a potential current gain of 99, the ratio of IC /IB , also known as beta, β.
This magic, the diffusion of 99% of the emitter carriers through the base, is only possible if
the base is very thin. What would be the fate of the base minority carriers in a base 100 times
thicker? One would expect the recombination rate, electrons falling into holes, to be much
higher. Perhaps 99%, instead of 1%, would fall into holes, never getting to the collector. The
second point to make is that the base current may control 99% of the emitter current, only if
99% of the emitter current diffuses into the collector. If it all flows out the base, no control is
possible.
Another feature accounting for passing 99% of the electrons from emitter to collector is that
real bipolar junction transistors use a small heavily doped emitter. The high concentration of
emitter electrons forces many electrons to diffuse into the base. The lower doping concentration
in the base means fewer holes diffuse into the emitter, which would increase the base current.
Diffusion of carriers from emitter to base is strongly favored.
The thin base and the heavily doped emitter help keep the emitter efficiency high, 99% for
example. This corresponds to 100% emitter current splitting between the base as 1% and the
collector as 99%. The emitter efficiency is known as α = IC /IE .
Bipolar junction transistors are available as PNP as well as NPN devices. We present a
comparison of these two in Figure 2.36. The difference is the polarity of the base emitter diode
junctions, as signified by the direction of the schematic symbol emitter arrow. It points in
the same direction as the anode arrow for a junction diode, against electron current flow. See
diode junction, Figure 2.30. The point of the arrow and bar correspond to P-type and N-type
semiconductors, respectively. For NPN and PNP emitters, the arrow points away and toward
the base respectively. There is no schematic arrow on the collector. However, the base-collector
junction is the same polarity as the base-emitter junction compared to a diode. Note, we speak
of diode, not power supply, polarity.

N

P

-

+

N

-

+

(a)

N

+

-

P

+

-

(b)
E

-

P

C
+

B

-

E
+

+

C
-

B

+

-

Figure 2.36: Compare NPN transistor at (a) with the PNP transistor at (b). Note direction of
emitter arrow and supply polarity.
The voltage sources for PNP transistors are reversed compared with an NPN transistors

CHAPTER 2. SOLID-STATE DEVICE THEORY

64

as shown in Figure 2.36. The base-emitter junction must be forward biased in both cases. The
base on a PNP transistor is biased negative (b) compared with positive (a) for an NPN. In both
cases the base-collector junction is reverse biased. The PNP collector power supply is negative
compared with positive for an NPN transistor.

Collector

Collector

+

N

Base

NP

(b) Emitter
P+

+

N

Base

Emitter
+

N
P base
N collector epitaxial layer

Collector
N+

N+ buried

Base

Emitter (a)

(c)

P substrate

Figure 2.37: Bipolar junction transistor: (a) discrete device cross-section, (b) schematic symbol,
(c) integrated circuit cross-section.
Note that the BJT in Figure 2.37(a) has heavy doping in the emitter as indicated by the
N+ notation. The base has a normal P-dopant level. The base is much thinner than the notto-scale cross-section shows. The collector is lightly doped as indicated by the N− notation.
The collector needs to be lightly doped so that the collector-base junction will have a high
breakdown voltage. This translates into a high allowable collector power supply voltage. Small
signal silicon transistors have a 60-80 V breakdown voltage. Though, it may run to hundreds
of volts for high voltage transistors. The collector also needs to be heavily doped to minimize
ohmic losses if the transistor must handle high current. These contradicting requirements are
met by doping the collector more heavily at the metallic contact area. The collector near the
base is lightly doped as compared with the emitter. The heavy doping in the emitter gives the
emitter-base a low approximate 7 V breakdown voltage in small signal transistors. The heavily
doped emitter makes the emitter-base junction have zener diode like characteristics in reverse
bias.
The BJT die, a piece of a sliced and diced semiconductor wafer, is mounted collector down
to a metal case for power transistors. That is, the metal case is electrically connected to the
collector. A small signal die may be encapsulated in epoxy. In power transistors, aluminum
bonding wires connect the base and emitter to package leads. Small signal transistor dies
may be mounted directly to the lead wires. Multiple transistors may be fabricated on a single
die called an integrated circuit. Even the collector may be bonded out to a lead instead of the
case. The integrated circuit may contain internal wiring of the transistors and other integrated
components. The integrated BJT shown in (Figure ??) is much thinner than the “not to scale”
drawing. The P+ region isolates multiple transistors in a single die. An aluminum metalization
layer (not shown) interconnects multiple transistors and other components. The emitter region
is heavily doped, N+ compared to the base and collector to improve emitter efficiency.
Discrete PNP transistors are almost as high quality as the NPN counterpart. However, in-

2.9. JUNCTION FIELD-EFFECT TRANSISTORS

65

tegrated PNP transistors are not nearly a good as the NPN variety within the same integrated
circuit die. Thus, integrated circuits use the NPN variety as much as possible.
• REVIEW:
• Bipolar transistors conduct current using both electrons and holes in the same device.
• Operation of a bipolar transistor as a current amplifier requires that the collector-base
junction be reverse biased and the emitter-base junction be forward biased.
• A transistor differs from a pair of back to back diodes in that the base, the center layer, is
very thin. This allows majority carriers from the emitter to diffuse as minority carriers
through the base into the depletion region of the base-collector junction, where the strong
electric field collects them.
• Emitter efficiency is improved by heavier doping compared with the collector. Emitter
efficiency: α = IC /IE , 0.99 for small signal devices
• Current gain is β=IC /IB , 100 to 300 for small signal transistors.

2.9

Junction field-effect transistors

The field effect transistor was proposed by Julius Lilienfeld in US patents in 1926 and 1933
(1,900,018). Moreover, Shockley, Brattain, and Bardeen were investigating the field effect
transistor in 1947. Though, the extreme difficulties sidetracked them into inventing the bipolar
transistor instead. Shockley’s field effect transistor theory was published in 1952. However, the
materials processing technology was not mature enough until 1960 when John Atalla produced
a working device.
A field effect transistor (FET) is a unipolar device, conducting a current using only one
kind of charge carrier. If based on an N-type slab of semiconductor, the carriers are electrons.
Conversely, a P-type based device uses only holes.
At the circuit level, field effect transistor operation is simple. A voltage applied to the gate,
input element, controls the resistance of the channel, the unipolar region between the gate
regions. (Figure 2.38) In an N-channel device, this is a lightly doped N-type slab of silicon
with terminals at the ends. The source and drain terminals are analogous to the emitter and
collector, respectively, of a BJT. In an N-channel device, a heavy P-type region on both sides of
the center of the slab serves as a control electrode, the gate. The gate is analogous to the base
of a BJT.
“Cleanliness is next to godliness” applies to the manufacture of field effect transistors.
Though it is possible to make bipolar transistors outside of a clean room, it is a necessity
for field effect transistors. Even in such an environment, manufacture is tricky because of
contamination control issues. The unipolar field effect transistor is conceptually simple, but
difficult to manufacture. Most transistors today are a metal oxide semiconductor variety (later
section) of the field effect transistor contained within integrated circuits. However, discrete
JFET devices are available.
A properly biased N-channel junction field effect transistor (JFET) is shown in Figure 2.38.
The gate constitutes a diode junction to the source to drain semiconductor slab. The gate is

CHAPTER 2. SOLID-STATE DEVICE THEORY

66

Dra

in
N

Ga

te

Channel

P
So

urc

e
N

+

Figure 2.38: Junction field effect transistor cross-section.
reverse biased. If a voltage (or an ohmmeter) were applied between the source and drain, the
N-type bar would conduct in either direction because of the doping. Neither gate nor gate bias
is required for conduction. If a gate junction is formed as shown, conduction can be controlled
by the degree of reverse bias.
Figure 2.39(a) shows the depletion region at the gate junction. This is due to diffusion of
holes from the P-type gate region into the N-type channel, giving the charge separation about
the junction, with a non-conductive depletion region at the junction. The depletion region
extends more deeply into the channel side due to the heavy gate doping and light channel
doping.
The thickness of the depletion region can be increased Figure 2.39(b) by applying moderate
reverse bias. This increases the resistance of the source to drain channel by narrowing the
channel. Increasing the reverse bias at (c) increases the depletion region, decreases the channel width, and increases the channel resistance. Increasing the reverse bias VGS at (d) will
pinch-off the channel current. The channel resistance will be very high. This VGS at which
pinch-off occurs is VP , the pinch-off voltage. It is typically a few volts. In summation, the
channel resistance can be controlled by the degree of reverse biasing on the gate.
The source and drain are interchangeable, and the source to drain current may flow in
either direction for low level drain battery voltage (¡ 0.6 V). That is, the drain battery may
be replaced by a low voltage AC source. For a high drain power supply voltage, to 10’s of
volts for small signal devices, the polarity must be as indicated in Figure 2.40(a). This drain
power supply, not shown in previous figures, distorts the depletion region, enlarging it on the
drain side of the gate. This is a more correct representation for common DC drain supply
voltages, from a few to tens of volts. As drain voltage VDS is increased,the gate depletion
region expands toward the drain. This increases the length of the narrow channel, increasing
its resistance a little. We say ”a little” because large resistance changes are due to changing
gate bias. Figure 2.40(b) shows the schematic symbol for an N-channel field effect transistor

2.9. JUNCTION FIELD-EFFECT TRANSISTORS

S N

S N

D
G

P-type

67

D
G

(a)

S N

D
G

(c)

S N

D
G

(b)

(d)

Figure 2.39: N-channel JFET: (a) Depletion at gate diode. (b) Reverse biased gate diode increases depletion region. (c) Increasing reverse bias enlarges depletion region. (d) Increasing
reverse bias pinches-off the S-D channel.
compared to the silicon cross-section at (a). The gate arrow points in the same direction as
a junction diode. The “pointing” arrow and “non-pointing” bar correspond to P and N-type
semiconductors, respectively.
electron current

S

to G

P-type
N

D
S
G

(a)

D
G

(b)

Figure 2.40: N-channel JFET electron current flow from source to drain in (a) cross-section, (b)
schematic symbol.
Figure 2.40 shows a large electron current flow from (-) battery terminal, to FET source,
out the drain, returning to the (+) battery terminal. This current flow may be controlled by
varying the gate voltage. A load in series with the battery sees an amplified version of the
changing gate voltage.
P-channel field effect transistors are also available. The channel is made of P-type material. The gate is a heavily dopped N-type region. All the voltage sources are reversed in the
P-channel circuit (Figure 2.41) as compared with the more popular N-channel device. Also
note, the arrow points out of the gate of the schematic symbol (b) of the P-channel field effect
transistor.
As the positive gate bias voltage is increased, the resistance of the P-channel increases,
decreasing the current flow in the drain circuit.
Discrete devices are manufactured with the cross-section shown in Figure 2.42. The cross-

CHAPTER 2. SOLID-STATE DEVICE THEORY

68
to G
S

P

D
S
G

D
G

N-type

(a)

(b)

Figure 2.41: P-channel JFET: (a) N-type gate, P-type channel, reversed voltage sources compared with N-channel device. (b) Note reversed gate arrow and voltage sources on schematic.
section, oriented so that it corresponds to the schematic symbol, is upside down with respect
to a semiconductor wafer. That is, the gate connections are on the top of the wafer. The
gate is heavily doped, P+ , to diffuse holes well into the channel for a large depletion region.
The source and drain connections in this N-channel device are heavily doped, N+ to lower
connection resistance. However, the channel surrounding the gate is lightly doped to allow
holes from the gate to diffuse deeply into the channel. That is the N− region.

Drain

Drain
Gate

N+
N
P+

N+

Gate Source

Source

(b)

-

Source
P+
(a)

Gate

Drain

P+
N
(c) P substrate

Figure 2.42: Junction field effect transistor: (a) Discrete device cross-section, (b) schematic
symbol, (c) integrated circuit device cross-section.
All three FET terminals are available on the top of the die for the integrated circuit version
so that a metalization layer (not shown) can interconnect multiple components. (Figure 2.42(c)
) Integrated circuit FET’s are used in analog circuits for the high gate input resistance.. The
N-channel region under the gate must be very thin so that the intrinsic region about the gate
can control and pinch-off the channel. Thus, gate regions on both sides of the channel are not
necessary.
The static induction field effect transistor (SIT) is a short channel device with a buried gate.
(Figure 2.43) It is a power device, as opposed to a small signal device. The low gate resistance
and low gate to source capacitance make for a fast switching device. The SIT is capable of
hundreds of amps and thousands of volts. And, is said to be capable of an incredible frequency
of 10 gHz.[24]

2.9. JUNCTION FIELD-EFFECT TRANSISTORS

69

Junction field-effect transistor
(static induction type)

Cross-section
Drain

Schematic symbol
+

N

Drain

NGate

+

P

+

Gate
+

P

P

+

P

Source

N+
Source

(a)

(b)

Figure 2.43: Junction field effect transistor (static induction type): (a) Cross-section, (b)
schematic symbol.

Drain
Gate
(a)

Source

(b)

Source

Gate

N+

N-

Drain
N+

substrate

Figure 2.44: Metal semiconductor field effect transistor (MESFET): (a) schematic symbol, (b)
cross-section.

CHAPTER 2. SOLID-STATE DEVICE THEORY

70

The Metal semiconductor field effect transistor (MESFET) is similar to a JFET except the
gate is a schottky diode instead of a junction diode. A schottky diode is a metal rectifying
contact to a semiconductor compared with a more common ohmic contact. In Figure 2.44 the
the source and drain are heavily doped (N+ ). The channel is lightly doped (N− ). MESFET’s
are higher speed than JFET’s. The MESET is a depletion mode device, normally on, like a
JFET. They are used as microwave power amplifiers to 30 gHz. MESFET’s can be fabricated
from silicon, gallium arsenide, indium phosphide, silicon carbide, and the diamond allotrope of
carbon.
• REVIEW:
• The unipolar junction field effect transistor (FET or JFET) is so called because conduction
in the channel is due to one type of carrier
• The JFET source, gate, and drain correspond to the BJT’s emitter, base, and collector,
respectively.
• Application of reverse bias to the gate varies the channel resistance by expanding the
gate diode depletion region.

2.10

Insulated-gate field-effect transistors (MOSFET)

The insulated-gate field-effect transistor (IGFET), also known as the metal oxide field effect
transistor (MOSFET), is a derivative of the field effect transistor (FET). Today, most transistors are of the MOSFET type as components of digital integrated circuits. Though discrete
BJT’s are more numerous than discrete MOSFET’s. The MOSFET transistor count within an
integrated circuit may approach hundreds of a million. The dimensions of individual MOSFET
devices are under a micron, decreasing every 18 months. Much larger MOSFET’s are capable
of switching nearly 100 amperes of current at low voltages; some handle nearly 1000 V at lower
currents. These devices occupy a good fraction of a square centimeter of silicon. MOSFET’s
find much wider application than JFET’s. However, MOSFET power devices are not as widely
used as bipolar junction transistors at this time.
The MOSFET has source, gate, and drain terminals like the FET. However, the gate lead
does not make a direct connection to the silicon compared with the case for the FET. The
MOSFET gate is a metallic or polysilicon layer atop a silicon dioxide insulator. The gate bears
a resemblance to a metal oxide semiconductor (MOS) capacitor in Figure 2.45. When charged,
the plates of the capacitor take on the charge polarity of the respective battery terminals.
The lower plate is P-type silicon from which electrons are repelled by the negative (-) battery
terminal toward the oxide, and attracted by the positive (+) top plate. This excess of electrons
near the oxide creates an inverted (excess of electrons) channel under the oxide. This channel
is also accompanied by a depletion region isolating the channel from the bulk silicon substrate.
In Figure 2.46 (a) the MOS capacitor is placed between a pair of N-type diffusions in a Ptype substrate. With no charge on the capacitor, no bias on the gate, the N-type diffusions, the
source and drain, remain electrically isolated.
A positive bias applied to the gate, charges the capacitor (the gate). The gate atop the oxide
takes on a positive charge from the gate bias battery. The P-type substrate below the gate takes

2.10. INSULATED-GATE FIELD-EFFECT TRANSISTORS (MOSFET)

71
inverted
channel

+

++++++

oxide

oxide

- -depletion
- - - -

P

P

(a)

(b)

Figure 2.45: N-channel MOS capacitor: (a) no charge, (b) charged.
Source

Gate

-

Drain

Ν

S
+
N
N+

N+

N+

G

+ + ++++
-- ----

P

P
(a)

+

depletion

(b)

inverted channel

D
N+
depletion

-

+

Figure 2.46: N-channel MOSFET (enhancement type): (a) 0 V gate bias, (b) positive gate bias.
on a negative charge. An inversion region with an excess of electrons forms below the gate
oxide. This region now connects the source and drain N-type regions, forming a continuous
N-region from source to drain. Thus, the MOSFET, like the FET is a unipolar device. One
type of charge carrier is responsible for conduction. This example is an N-channel MOSFET.
Conduction of a large current from source to drain is possible with a voltage applied between
these connections. A practical circuit would have a load in series with the drain battery in
Figure 2.46 (b).
The MOSFET described above in Figure 2.46 is known as an enhancement mode MOSFET.
The non-conducting, off, channel is turned on by enhancing the channel below the gate by
application of a bias. This is the most common kind of device. The other kind of MOSFET will
not be described here. See the Insulated-gate field-effect transistor chapter for the depletion
mode device.
The MOSFET, like the FET, is a voltage controlled device. A voltage input to the gate
controls the flow of current from source to drain. The gate does not draw a continuous current.
Though, the gate draws a surge of current to charge the gate capacitance.
The cross-section of an N-channel discrete MOSFET is shown in Figure 2.47 (a). Discrete
devices are usually optimized for high power switching. The N+ indicates that the source and
drain are heavily N-type doped. This minimizes resistive losses in the high current path from
source to drain. The N− indicates light doping. The P-region under the gate, between source
and drain can be inverted by application of a positive bias voltage. The doping profile is a
cross-section, which may be laid out in a serpentine pattern on the silicon die. This greatly
increases the area, and consequently, the current handling ability.

CHAPTER 2. SOLID-STATE DEVICE THEORY

72

Drain
N+

Drain

N-

inversion
P

(a)

Gate

Gate
Source

N+
Source

= silicon dioxide
insulator

(b)

Figure 2.47: N-channel MOSFET (enhancement type): (a) Cross-section, (b) schematic symbol.
The MOSFET schematic symbol in Figure 2.47 (b) shows a “floating” gate, indicating no
direct connection to the silicon substrate. The broken line from source to drain indicates that
this device is off, not conducting, with zero bias on the gate. A normally “off ” MOSFET is
an enhancement mode device. The channel must be enhanced by application of a bias to the
gate for conduction. The “pointing” end of the substrate arrow corresponds to P-type material,
which points toward an N-type channel, the “non-pointing” end. This is the symbol for an
N-channel MOSFET. The arrow points in the opposite direction for a P-channel device (not
shown). MOSFET’s are four terminal devices: source, gate, drain, and substrate. The substrate
is connected to the source in discrete MOSFET’s, making the packaged part a three terminal
device. MOSFET’s, that are part of an integrated circuit, have the substrate common to all
devices, unless purposely isolated. This common connection may be bonded out of the die for
connection to a ground or power supply bias voltage.

Drain
N+

Drain
inversion

N-

Gate
P
N+

P
N+

(a)
Gate

Source

Source
= silicon dioxide
insulator

(b)

Figure 2.48: N-channel “V-MOS” transistor: (a) Cross-section, (b) schematic symbol.
The V-MOS device in (Figure 2.48) is an improved power MOSFET with the doping profile
arranged for lower on-state source to drain resistance. VMOS takes its name from the V-

2.11. THYRISTORS

73

shaped gate region, which increases the cross-sectional area of the source-drain path. This
minimizes losses and allows switching of higher levels of power. UMOS, a variation using a
U-shaped grove, is more reproducible in manufacture.
• REVIEW:
• MOSFET’s are unipoar conduction devices, conduction with one type of charge carrier,
like a FET, but unlike a BJT.
• A MOSFET is a voltage controlled device like a FET. A gate voltage input controls the
source to drain current.
• The MOSFET gate draws no continuous current, except leakage. However, a considerable
initial surge of current is required to charge the gate capacitance.

2.11

Thyristors

Thyristors are a broad classification of bipolar-conducting semiconductor devices having four
(or more) alternating N-P-N-P layers. Thyristors include: silicon controlled rectifier (SCR),
TRIAC, gate turn off switch (GTO), silicon controlled switch (SCS), AC diode (DIAC), unijunction transistor (UJT), programmable unijunction transistor (PUT). Only the SCR is examined
in this section; though the GTO is mentioned.
Shockley proposed the four layer diode thyristor in 1950. It was not realized until years
later at General Electric. SCR’s are now available to handle power levels spanning watts
to megawatts. The smallest devices, packaged like small-signal transistors, switch 100’s of
milliamps at near 100 VAC. The largest packaged devices are 172 mm in diameter, switching
5600 Amps at 10,000 VAC. The highest power SCR’s may consist of a whole semiconductor
wafer several inches in diameter (100’s of mm).

Anode

Anode

P
N

N

P

P

Gate

N

(a)

Cathode

+
Gate

−
(b)

Cathode

Figure 2.49: Silicon controlled rectifier (SCR): (a) doping profile, (b) BJT equivalent circuit.
The silicon controlled rectifier is a four layer diode with a gate connection as in Figure 2.49
(a). When turned on, it conducts like a diode, for one polarity of current. If not triggered on,
it is nonconducting. Operation is explained in terms of the compound connected transistor

CHAPTER 2. SOLID-STATE DEVICE THEORY

74

equivalent in Figure 2.49 (b). A positive trigger signal is applied between the gate and cathode
terminals. This causes the NPN equivalent transistor to conduct. The collector of the conducting NPN transistor pulls low, moving the PNP base towards its collector voltage, which causes
the PNP to conduct. The collector of the conducting PNP pulls high, moving the NPN base in
the direction of its collector. This positive feedback (regeneration) reinforces the NPN’s already
conducting state. Moreover, the NPN will now conduct even in the absence of a gate signal.
Once an SCR conducts, it continues for as long as a positive anode voltage is present. For the
DC battery shown, this is forever. However, SCR’s are most often used with an alternating
current or pulsating DC supply. Conduction ceases with the expiration of the positive half of
the sinewave at the anode. Moreover, most practical SCR circuits depend on the AC cycle going
to zero to cutoff or commutate the SCR.
Figure 2.50 (a) shows the doping profile of an SCR. Note that the cathode, which corresponds to an equivalent emitter of an NPN transistor is heavily doped as N+ indicates. The
anode is also heavily doped (P+ ). It is the equivalent emitter of a PNP transistor. The two
middle layers, corresponding to base and collector regions of the equivalent transistors, are
less heavily doped: N− and P. This profile in high power SCR’s may be spread across a whole
semiconductor wafer of substantial diameter.

Anode

schematic symbols

+

P

Anode

Anode

-

N

Gate
+

P

P

+

N

(a)
Gate

Gate
Cathode

Cathode

SCR

GTO

(b)

(c)

Cathode

Figure 2.50: Thyristors: (a) Cross-section, (b) silicon controlled rectifier (SCR) symbol, (c) gate
turn-off thyristor (GTO) symbol.
The schematic symbols for an SCR and GTO are shown in Figures 2.50 (b & c). The basic
diode symbol indicates that cathode to anode conduction is unidirectional like a diode. The
addition of a gate lead indicates control of diode conduction. The gate turn off switch (GTO)
has bidirectional arrows about the gate lead, indicating that the conduction can be disabled by
a negative pulse, as well as initiated by a positive pulse.
In addition to the ubiquitous silicon based SCR’s, experimental silicon carbide devices have
been produced. Silicon carbide (SiC) operates at higher temperatures, and is more conductive
of heat than any metal, second to diamond. This should allow for either physically smaller or
higher power capable devices.
• REVIEW:

2.12. SEMICONDUCTOR MANUFACTURING TECHNIQUES

75

• SCR’s are the most prevalent member of the thyristor four layer diode family.
• A positive pulse applied to the gate of an SCR triggers it into conduction. Conduction
continues even if the gate pulse is removed. Conduction only ceases when the anode to
cathode voltage drops to zero.
• SCR’s are most often used with an AC supply (or pulsating DC) because of the continuous
conduction.
• A gate turn off switch (GTO) may be turned off by application of a negative pulse to the
gate.
• SCR’s switch megawatts of power, up to 5600 A and 10,000 V.

2.12

Semiconductor manufacturing techniques

The manufacture of only silicon based semiconductors is described in this section; most semiconductors are silicon. Silicon is particularly suitable for integrated circuits because it readily
forms an oxide coating, useful in patterning integrated components like transistors.
Silicon is the second most common element in the Earth’s crust in the form of silicon dioxide,
SiO2 , otherwise known as silica sand. Silicon is freed from silicon dioxide by reduction with
carbon in an electric arc furnace
SiO2 + C = CO2 + Si
Such metalurgical grade silicon is suitable for use in silicon steel transformer laminations,
but not nearly pure enough for semiconductor applications. Conversion to the chloride SiCl4
(or SiHCl3 ) allows purification by fractional distillation. Reduction by ultrapure zinc or magnesium yields sponge silicon, requiring further purification. Or, thermal decomposition on a
hot polycrystalline silicon rod heater by hydrogen yields ultra pure silicon.
Si + 3HCl
= SiHCl3 + H2
SiHCl3 + H2 = Si + 3HCl2
The polycrystalline silicon is melted in a fused silica crucible heated by an induction heated
graphite suceptor. The graphite heater may alternately be directly driven by a low voltage
at high current. In the Czochralski process, the silicon melt is solidified on to a pencil sized
monocrystal silicon rod of the desired crystal lattice orientation. (Figure 2.51) The rod is rotated and pulled upward at a rate to encourage the diameter to expand to several inches. Once
this diameter is attained, the boule is automatically pulled at a rate to maintain a constant
diameter to a length of a few feet. Dopants may be added to the crucible melt to create, for
example, a P-type semiconductor. The growing apparatus is enclosed within an inert atmosphere.
The finished boule is ground to a precise final diameter, and the ends trimmed. The boule
is sliced into wafers by an inside diameter diamond saw. The wafers are ground flat and
polished. The wafers could have an N-type epitaxial layer grown atop the wafer by thermal
deposition for higher quality. Wafers at this stage of manufacture are delivered by the silicon
wafer manufacturer to the semiconductor manufacturer.
The processing of semiconductors involves photo lithography, a process for making metal
lithographic printing plates by acid etching. The electronics based version of this is the pro-

CHAPTER 2. SOLID-STATE DEVICE THEORY

76

lift rod
Si boule
fused silica crucible
graphite suceptor

RF induction coil
Si melt
Figure 2.51: Czochralski monocrystalline silicon growth.

Si boule
cut wafers
diamond blade
driven edge

Figure 2.52: Silicon boule is diamond sawed into wafers.

2.12. SEMICONDUCTOR MANUFACTURING TECHNIQUES

77

cessing of copper printed circuit boards. This is reviewed in Figure 2.53 as an easy introduction
to the photo lithography involved in semiconductor processing.

(a) copper PCB

(e) remove artwork

(b) apply photoresist (c) place artwork

(f) develop resist

(g) etch copper

(d) expose

(h) strip resist

Figure 2.53: Processing of copper printed circuit boards is similar to the photo lithographic
steps of semiconductor processing.
We start with a copper foil laminated to an epoxy fiberglass board in Figure 2.53 (a). We
also need positive artwork with black lines corresponding to the copper wiring lines and pads
that are to remain on the finished board. Positive artwork is required because positive acting
resist is used. Though, negative resist is available for both circuit boards and semiconductor
processing. At (b) the liquid positive photo resist is applied to the copper face of the printed
circuit board (PCB). It is allowed to dry and may be baked in an oven. The artwork may be
a plastic film positive reproduction of the original artwork scaled to the required size. The
artwork is placed in contact with the circuit board under a glass plate at (c). The board is
exposed to ultraviolet light (d) to form a latent image of softened photo resist. The artwork is
removed (e) and the softened resist washed away by an alkaline solution (f). The rinsed and
dried (baked) circuit board has a hardened resist image atop the copper lines and pads that
are to remain after etching. The board is immersed in the etchant (g) to remove copper not
protected by hardened resist. The etched board is rinsed and the resist removed by a solvent.
The major difference in the patterning of semiconductors is that a silicon dioxide layer atop
the wafer takes the place of the resist during the high temperature processing steps. Though,
the resist is required in low temperature wet processing to pattern the silicon dioxide.
An N-type doped silicon wafer in Figure 2.54 (a) is the starting material in the manufacture
of semiconductor junctions. A silicon dioxide layer (b) is grown atop the wafer in the presence
of oxygen or water vapor at high temperature (over 1000o C in a diffusion furnace. A pool
of resist is applied to the center of the cooled wafer, then spun in a vacuum chuck to evenly
distribute the resist. The baked on resist (c) has a chrome on glass mask applied to the wafer
at (d). This mask contains a pattern of windows which is exposed to ultraviolet light (e).
After the mask is removed in Figure 2.54 (f), the positive resist can be developed (g) in
an alkaline solution, opening windows in the UV softened resist. The purpose of the resist is
to protect the silicon dioxide from the hydrofluoric acid etch (h), leaving only open windows
corresponding to the mask openings. The remaining resist (i) is stripped from the wafer before
returning to the diffusion furnace. The wafer is exposed to a gaseous P-type dopant at high

CHAPTER 2. SOLID-STATE DEVICE THEORY

78

(a) N-type wafer

(b) grow SiO2

(e) expose

(f) remove mask

(c) apply photoresist (d) place mask

(g) develop resist

(h) HF etch

BH3

(i) strip resist

(j) P-type diffusion

Figure 2.54: Manufacture of a silicon diode junction.
temperature in a diffusion furnace (j). The dopant only diffuses into the silicon through the
openings in the silicon dioxide layer. Each P-diffusion through an opening produces a PN
junction. If diodes were the desired product, the wafer would be diamond scribed and broken
into individual diode chips. However, the whole wafer may be processed further into bipolar
junction transistors.
To convert the diodes into transistors, a small N-type diffusion in the middle of the existing P-region is required. Repeating the previous steps with a mask having smaller openings
accomplishes this. Though not shown in Figure 2.54 (j), an oxide layer was probably formed in
that step during the P-diffusion. The oxide layer over the P-diffusion is shown in Figure 2.55
(k). Positive photo resist is applied and dried (l). The chrome on glass emitter mask is applied
(m), and UV exposed (n). The mask is removed (o). The UV softened resist in the emitter
opening is removed with an alkaline solution (p). The exposed silicon dioxide is etched away
with hydrofluoric acid (HF) at (q)
After the unexposed resist is stripped from the wafer (r), it is placed in a diffusion furnace
(Figure 2.55 (s) for high temperature processing. An N-type gaseous dopant, such phosphorus
oxychloride (POCl) diffuses through the small emitter window in the oxide (s). This creates
NPN layers corresponding to the emitter, base, and collector of a BJT. It is important that the
N-type emitter not be driven all the way through the P-type base, shorting the emitter and
collector. The base region between the emitter and collector also needs to be thin so that the
transistor has a useful β. Otherwise, a thick base region could form a pair of diodes rather
than a transistor. At (t) metalization is shown making contact with the transistor regions.
This requires a repeat of the previous steps (not shown here) with a mask for contact openings
through the oxide. Another repeat with another mask defines the metalization pattern atop
the oxide and contacting the transistor regions through the openings.
The metalization could connect numerous transistors and other components into an integrated circuit. Though, only one transistor is shown. The finished wafer is diamond scribed

2.12. SEMICONDUCTOR MANUFACTURING TECHNIQUES

79

(k) grow SiO2

(l) apply photoresist

(m) place mask

(n) expose

(o) remove mask

(p) develop resist

(q) HF etch

(r) strip resist

POCl
C B

(s) N-type diffusion

E

(t) metalization

Figure 2.55: Manufacture of a bipolar junction transistor, continuation of Manufacture of a
silicon diode junction.
and broken into individual dies for packaging. Fine gauge aluminum wire bonds the metalized
contacts on the die to a lead frame, which brings the contacts out of the final package.
• REVIEW:
• Most semiconductor are based on ultra pure silicon because it forms a glass oxide atop the
wafer. This oxide can be patterned with photo lithography, making complex integrated
circuits possible.
• Sausage shaped single crystals of silicon are grown by the Czochralski process, These are
diamond sawed into wafers.
• The patterning of silicon wafers by photo lithography is similar to patterning copper
printed circuit boards. Photo resist is applied to the wafer, which is exposed to UV light
through a mask. The resist is developed, then the wafer is etched.
• hydrofluoric acid etching opens windows in the protective silicon dioxide atop the wafer.
• Exposure to gaseous dopants at high temperature produces semiconductor junctions as
defined by the openings in the silicon dioxide layer.
• The photo lithography is repeated for more diffusions, contacts, and metalization.
• The metalization may interconnect multiple components into an integrated circuit.

80

2.13

CHAPTER 2. SOLID-STATE DEVICE THEORY

Superconducting devices

Superconducting devices, though not widely used, have some unique characteristics not available in standard semiconductor devices. High sensitivity with respect to amplification of electrical signals, detection of magnetic fields, and detection of light are prized applications. High
speed switching is also possible, though not applied to computers at this time. Conventional
superconducting devices must be cooled to within a few degrees of 0 Kelvin (-273 o C). Though,
work is proceeding at this time on high temperature superconductor based devices, usable at 90
K and below. This is significant because inexpensive liquid nitrogen may be used for cooling.
Superconductivity: Heike Onnes discovered superconductivity in mercury (Hg) in 1911,
for which he won a Nobel prize. Most metals decrease electrical resistance with decreasing
temperature. Though, most do not decrease to zero resistance as 0 Kelvin is approached.
Mercury is unique in that its resistance abruptly drops to zero Ω at 4.2 K. Superconductors
lose all resistance abruptly when cooled below their critical temperature, Tc A property of
superconductivity is no power loss in conductors. Current may flow in a loop of superconducting
wire for thousands of years. Super conductors include lead (Pb), aluminum, (Al), tin (Sn) and
niobium (Nb).
Cooper pair: Lossless conduction in superconductors is not by ordinary electron flow. Electron flow in normal conductors encounters opposition as collisions with the rigid ionic metal
crystal lattice. Decreasing vibrations of the crystal lattice with decreasing temperature accounts for decreasing resistance– up to a point. Lattice vibrations cease at absolute zero, but
not the energy dissipating collisions of electrons with the lattice. Thus, normal conductors do
not lose all resistance at absolute zero.
Electrons in superconductors form a pair of electrons called a cooper pair, as temperature
drops below the critical temperature at which superconductivity begins. The cooper pair exists
because it is at a lower energy level than unpaired electrons. The electrons are attracted to
each other due to the exchange of phonons, very low energy particles related to vibrations. This
cooper pair, quantum mechanical entity (particle or wave) is not subject to the normal laws
of physics. This entity propagates through the lattice without encountering the metal ions
comprising the fixed lattice. Thus, it dissipates no energy. The quantum mechanical nature
of the cooper pair only allows it to exchange discrete amounts of energy, not continuously
variable amounts. An absolute minimum quantum of energy is acceptable to the cooper pair.
If the vibrational energy of the crystal lattice is less, (due to the low temperature), the cooper
pair cannot accept it, cannot be scattered by the lattice. Thus, under the critical temperature,
the cooper pairs flow unimpeded through the lattice.
Josephson junctions: Brian Josephson won a Nobel prize for his 1962 prediction of the
Josepheson junction. A Josephson junction is a pair of superconductors bridged by a thin insulator, as in Figure 2.56 (a), through which electrons can tunnel. The first Josephson junctions
were lead superconductors bridged by an insulator. These days a tri-layer of aluminum and
niobium is preferred. Electrons can tunnel through the insulator even with zero voltage applied across the superconductors.
If a voltage is applied across the junction, the current decreases and oscillates at a high
frequency proportional to voltage. The relationship between applied voltage and frequency
is so precise that the standard volt is now defined in terms of Josephson junction oscillation
frequency. The Josephson junction can also serve as a hyper-sensitive detector of low level magnetic fields. It is also very sensitive to electromagnetic radiation from microwaves to gamma

2.13. SUPERCONDUCTING DEVICES

81

rays.

lead (Pb)

ga

te

lead oxide
(a)

(b)

Figure 2.56: (a) Josephson junction, (b) Josephson transistor.
Josephson transistor: An electrode close to the oxide of the Josephson junction can influence the junction by capacitive coupling. Such an assembly in Figure 2.56 (b) is a Josephson
transistor. A major feature of the Josephson transistor is low power dissipation applicable
to high density circuitry, for example, computers. This transistor is generally part of a more
complex superconducting device like a SQUID or RSFQ.
SQUID: A Superconduction quantum interference device or SQUID is an assembly of Josephson junctions within a superconducting ring. Only the DC SQUID is considered in this discussion. This device is highly sensitive to low level magnetic fields.
A constant current bias is forced across the ring in parallel with both Josephson junctions
in Figure 2.57. The current divides equally between the two junctions in the absence of an
applied magnetic field and no voltage is developed across across the ring. [3] While any value
of Magnetic flux (Φ) may be applied to the SQUID, only a quantized value (a multiple of the
flux quanta) can flow through the opening in the superconducting ring.[2] If the applied flux is
not an exact multiple of the flux quanta, the excess flux is cancelled by a circulating current
around the ring which produces a fractional flux quanta. The circulating current will flow in
that direction which cancels any excess flux above a multiple of the flux quanta. It may either
add to, or subtract from the applied flux, up to ±(1/2) a flux quanta. If the circulating current
flows clockwise, the current adds to the top Josepheson junction and subtracts from the lower
one. Changing applied flux linearly causes the circulating current to vary as a sinusoid.[?] This
can be measured as a voltage across the SQUID. As the applied magnetic field is increased, a
voltage pulse may be counted for each increase by a flux quanta.[18]
A SQUID is said to be sensitive to 10−14 Tesla, It can detect the magnetic field of neural
currents in the brain at 10−13 Tesla. Compare this with the 30 x 10−6 Tesla strength of the
Earth’s magnetic field.
Rapid single flux quantum (RSFQ): Rather than mimic silicon semiconductor circuits,
RSFQ circuits rely upon new concepts: magnetic flux quantization within a superconductor,
and movement of the flux quanta produces a picosecond quantized voltage pulse. Magnetic flux
can only exist within a section of superconductor quantized in discrete multiples. The lowest
flux quanta allowed is employed. The pulses are switched by Josephson junctions instead of
conventional transistors. The superconductors are based on a triple layer of aluminum and

CHAPTER 2. SOLID-STATE DEVICE THEORY

82

+
Iconstant
-

JJ

Φ
±∆Ι
JJ

counter
V

Figure 2.57: Superconduction quantum interference device (SQUID): Josephson junction pair
within a superconducting ring. A change in flux produces a voltage variation across the JJ
pair.
niobium with a critical temperature of 9.5 K, cooled to 5 K.
RSQF’s operate at over 100 gHz with very little power dissipation. Manufacture is simple
with existing photolithographic techniques. Though, operation requires refrigeration down
to 5 K . Real world commercial applications include analog-to-digital and digital to analog
converters, toggle flip-flops, shift registers, memory, adders, and multipliers.[4]
High temperature superconductors: High temperature superconductors are compounds
exhibiting superconductivity above the liquid nitrogen boiling point of 77 K. This is significant
because liquid nitrogen is readily available and inexpensive. Most conventional superconductors are metals; widely used high temperature superconductors are cuprates, mixed oxides of
copper (Cu), for example YBa2 Cu3 O7−x , critical temperature, Tc = 90 K . A list of others is
available.[22] Most of the devices described in this section are being developed in high temperature superconductor versions for less critical applications. Though they do not have the
performance of the conventional metal superconductor devices, the liquid nitrogen cooling is
more available.
• REVIEW:
• Most metals decrease resistance as they approach absolute 0; though, the resistance does
not drop to 0. Superconductors experience a rapid drop to zero resistance at their critical
temperature on being cooled. Typically Tc is within 10 K of absolute zero.
• A Cooper pair, electron pair, a quantum mechanical entity, moves unimpeded through the
metal crystal lattice.
• Electrons are able to tunnel through a Josephson junction, an insulating gap across a
pair of superconductors.
• The addition of a third electrode, or gate, near the junction constitutes a Josephson transistor.
• A SQUID, Superconduction quantum interference device, is an highly sensitive detector
of magnetic fields. It counts quantum units of a magnetic field within a superconducting
ring.
• RSFQ, Rapid single flux quantum is a high speed switching device based on switching
the magnetic quanta existing withing a superconducting loop.

2.14. QUANTUM DEVICES

83

• High temperature superconductors, Tc above liquid nitrogen boiling point, may also be
used to build the superconducting devices in this section.

2.14

Quantum devices

Clasical view

Energy

Energy

Energy

Most integrated circuits are digital, based on MOS (CMOS) transistors. Every couple of years
since the late 1960’s a geometry shrink has taken place, increasing the circuit density– more
circuitry at lower cost in the same space. As of this writing (2006), the MOS transistor gate
length is 65-nm for leading edge production, with 45-nm anticipated within a year. At 65nm leakage currents were becoming evident. At 45-nm, heroic innovations were required to
minimize this leakage. The end of shrinkage in MOS transistors is expected at 20- to 30nm. Though some think that 1- to 2-nm is the limit. Photolithography, or other lithographic
techniques, will continue to improve, providing ever smaller geometry. However, conventional
MOS transistors are not expected to be usable at these smaller geometries below 20- to 30-nm.
Improved photolithography will have to be applied to other than the conventional transistors, dimensions (under 20- to 30-nm). The objectional MOS leakage currents are due to
quantum mechanical effects–electron tunneling through gate oxide, and the narrow channel.
In summary, quantum mechanical effects are a hindrance to ever smaller conventional MOS
transistors. The path to ever smaller geometry devices involves unique active devices which
make practical use of quantum mechanical principles. As physical geometry becomes very
small, electrons may be treated as the quantum mechanical equivalent: a wave. Devices
making use of quantum mechanical principles include: resonant tunneling diodes, quantum
tunneling transistors, metal insulator insulator metal diodes, and quantum dot transistors.
Quantum tunneling: is the passing of electrons through an insulating barrier which is
thin compared to the de Broglie (page 31) electron wavelength. If the “electron wave” is large
compared to the barrier, there is a possibility that the wave appears on both sides of the barrier.

Quantum mechanical view

Figure 2.58: Classical view of an electron surmounting a barrier, or not. Quantum mechanical
view allows an electron to tunnel through a barrier. The probability (green) is related to the
barrier thickness. After Figure 1 [21]
In classical physics, an electron must have sufficient energy to surmount a barrier. Otherwise, it recoils from the barrier. (Figure 2.58) Quantum mechanics allows for a probability
of the electron being on the other side of the barrier. If treated as a wave, the electron may
look quite large compared to the thickness of the barrier. Even when treated as a wave, there
is only a small probability that it will be found on the other side of a thick barrier. See green

CHAPTER 2. SOLID-STATE DEVICE THEORY

84

(a)

InGaAs

10

20

30

barrier
energy
level

Energy eV

1

barrier
well

Energy eV

AlAs

InGaAs

GaAs

0

AlAs

GaAs

1

InGaAs

Energy eV

portion of curve, Figure 2.58. Thinning the barrier increases the probability that the electron
is found on the other side of the barrier. [21]
Tunnel diode: The unqualified term tunnel diode refers to the esaki tunnel diode, an
early quantum device. A reverse biased diode forms a depletion region, an insulating region,
between the conductive anode and cathode. This depletion region is only thin as compared to
the electron wavelength when heavily doped– 1000 times the doping of a rectifier diode. With
proper biasing, quantum tunneling is possible. See (page 144)for details.
RTD, resonant tunneling diode: This is a quantum device not to be confused with the
Esaki tunnel diode, (page 144) , a conventional heavily doped bipolar semiconductor. Electrons
tunnel through two barriers separated by a well in flowing source to drain in a resonant tunneling diode. Tunneling is also known as quantum mechanical tunneling. The flow of electrons
is controlled by diode bias. This matches the energy levels of the electrons in the source to the
quantized level in the well so that electrons can tunnel through the barriers. The energy level
in the well is quantized because the well is small. When the energy levels are equal, a resonance occurs, allowing electron flow through the barriers as shown in Figure 2.59 (b). No bias
or too much bias, in Figures 2.59 (a) and (c) respectively, yields an energy mismatch between
the source and the well, and no conduction.

0

1

0
source

40 nm

10

(b)

drain
20

30

40 nm

10

20

30

40 nm

(c)

Figure 2.59: Resonant tunneling diode (RTD): (a) No bias, source and well energy levels not
matched, no conduction. (b) Small bias causes matched energy levels (resonance); conduction
results. (c) Further bias mismatches energy levels, decreasing conduction.
As bias is increased from zero across the RTD, the current increases and then decreases,
corresponding to off, on, and off states. This makes simplification of conventional transistor
circuits possible by substituting a pair of RTD’s for two transistors. For example, two back-toback RTD’s and a transistor form a memory cell, using fewer components, less area and power
compared with a conventional circuit. The potential application of RTD’s is to reduce the
component count, area, and power dissipation of conventional transistor circuits by replacing
some, though not all, transistors. [10] RTD’s have been shown to oscillate up to 712 gHz. [7]
Double-layer tunneling transistor: The Deltt, otherwise known as the Double-layer tunneling transistor is constructed of a pair of conductive wells separated by an insulator or high
band gap semiconductor. (Figure 2.60) The wells are so thin that electrons are confined to two
dimensions. These are known as quantum wells. A pair of these quantum wells are insulated
by a thin GaAlAs, high band gap (does not easily conduct) layer. Electrons can tunnel through
the insulating layer if the electrons in the two quantum wells have the same momentum and
energy. The wells are so thin that the electron may be treated as a wave– the quantum mechanical duality of particles and waves. The top and optional bottom control gates may be
adjusted to equalize the energy levels (resonance) of the electrons to allow conduction from

2.14. QUANTUM DEVICES

85

source to drain. Figure 2.60, barrier diagram red bars show unequal energy levels in the wells,
an “off-state” condition. Proper biasing of the gates equalizes the energy levels of electrons in
the wells, the “on-state” condition. The bars would be at the same level in the energy level
diagram.
bottom quantum well
contact (drain)
top gate

well

top quantum well
tunneling

barrier
well
depletion
barrier
diagram

top quantum well
contact (drain)

top depletion
gate

bottom
depletion gate

GaAs
GaAlAs

depletion
insulator

GaAs
bottom quantum well
bottom gate (optional)

Figure 2.60: Double-layer tunneling transistor (Deltt) is composed of two electron containing
wells separated by a nonconducting barrier. The gate voltages may be adjusted so that the
energy and momentum of the electrons in the wells are equal which permits electrons to tunnel
through the nonconductive barrier. (The energy levels are shown as unequal in the barrier
diagram.)
If gate bias is increased beyond that required for tunneling, the energy levels in the quantum wells no longer match, tunneling is inhibited, source to drain current decreases. To summarize, increasing gate bias from zero results in on, off, on conditions. This allows a pair of
Deltt’s to be stacked in the manner of a CMOS complementary pair; though, different p- and ntype transistors are not required. Power supply voltage is about 100 mV. Experimental Deltt’s
have been produced which operate near 4.2 K, 77 K, and 0o C. Room temperature versions are
expected.[10] [12] [19]
MIIM diode: The metal-insulator-insulator-metal (MIIM) diode is a quantum tunneling device, not based on semiconductors. See “MIIM diode section” Figure 2.61. The insulator layers
must be thin compared to the de Broglie (page 31) electron wavelength, for quantum tunneling
to be possible. For diode action, there must be a prefered tunneling direction, resulting in a
sharp bend in the diode forward characteristic curve. The MIIM diode has a sharper forward
curve than the metal insulator metal (MIM) diode, not considered here.
The energy levels of M1 and M2 are equal in “no bias” Figure 2.61. However, (thermal)
electrons cannot flow due to the high I1 and I2 barriers. Electrons in metal M2 have a higher
energy level in “reverse bias” Figure 2.61, but still cannot overcome the insulator barrier. As
“forward bias” Figure 2.61 is increased, a quantum well, an area where electrons may exist,
is formed between the insulators. Electrons may pass through insulator I1 if M1 is bised at
the same energy level as the quantum well. A simple explanation is that the distance through
the insulators is shorter. A longer explanation is that as bias increases, the probability of
the electron wave overlapping from M1 to the quantum well increases. For a more detailed

CHAPTER 2. SOLID-STATE DEVICE THEORY

I1

Energy

Energy

M1

quantum
well

Energy

86

I2
M2

M1 I1 I2 M2

M1 I2 I1 M2

Distance
MIIM diode
section

M1 I1 I2

Distance

No bias

M2

Distance

Forward bias

Reverse bias

Figure 2.61: Metal insulator insulator metal (MIIM) diode: Cross section of diode. Energy
levels for no bias, forward bias, and reverse bias. After Figure 1 [20].
explanation see Phiar Corp. [20]
MIIM devices operate at higher frequencies (3.7 THz) than microwave transistors. [15] The
addition of a third electrode to a MIIM diode produces a transistor.
Quantum dot transistor: An isolated conductor may take on a charge, measured in
coulombs for large objects. For a nano-scale isolated conductor known as a quantum dot, the
charge is measured in electrons. A quantum dot of 1- to 3-nm may take on an incremental
charge of a single electron. This is the basis of the quantum dot transistor, also known as a
single electron transistor.
A quantum dot placed atop a thin insulator over an electron rich source is known as a single
electron box. (Figure 2.62 (a)) The energy required to transfer an electron is related to the size
of the dot and the number of electrons already on the dot.
A gate electrode above the quantum dot can adjust the energy level of the dot so that quantum mechanical tunneling of an electron (as a wave) from the source through the insulator is
possible. (Figure 2.62 (b)) Thus, a single electron may tunnel to the dot.

+
gate

+++

quantum
dot

tunneling

tunnel
barrier

--+++

source

drain

---

tunnel
barrier

source
(a)

(b)

(c)

Figure 2.62: (a) Single electron box, an isolated quantum dot separated from an electron source
by an insulator. (b) Positive charge on the gate polarizes quantum dot, tunneling an electron
from the source to the dot. (c) Quantum transistor: channel is replaced by quantum dot surrounded by tunneling barrier.
If the quantum dot is surrounded by a tunnel barrier and embedded between the source

2.14. QUANTUM DEVICES

87

and drain of a conventional FET, as in Figure 2.62 (c) , the charge on the dot can modulate the
flow of electrons from source to drain. As gate voltage increases, the source to drain current
increases, up to a point. A further increase in gate voltage decreases drain current. This is
similar to the behavior of the RTD and Deltt resonant devices. Only one kind of transistor is
required to build a complementary logic gate.[10]
Single electron transistor: If a pair of conductors, superconductors, or semiconductors
are separated by a pair of tunnel barriers (insulator), surrounding a tiny conductive island,
like a quantum dot, the flow of a single charge (a Cooper pair for superconductors) may be
controlled by a gate. This is a single electron transistor similar to Figure 2.62 (c). Increasing
the positive charge on the gate, allows an electron to tunnel to the island. If it is sufficiently
small, the low capacitance will cause the dot potential to rise substantially due to the single
electron. No more electrons can tunnel to the island due the electron charge. This is known at
the coulomb blockade. The electron which tunneled to the island, can tunnel to the drain.
Single electron transistors operate at near absolute zero. The exception is the graphene
single electron transistor, having a graphene island. They are all experimental devices.
Graphene transistor: Graphite, an allotrope of carbon, does not have the rigid interlocking crystalline structure of diamond. None the less, it has a crystalline structure– one atom
thick, a so called two-dimensional structure. A graphite is a three-dimensional crystal. However, it cleaves into thin sheets. Experimenters, taking this to the extreme, produce micron
sized specks as thin as a single atom known as graphene. (Figure 2.63 (a)) These membranes
have unique electronic properties. Highly conductive, conduction is by either electrons or holes,
without doping of any kind. [11]
Graphene sheets may be cut into transistor structures by lithographic techniques. The
transistors bear some resemblance to a MOSFET. A gate capacitively coupled to a graphene
channel controls conduction.
As silicon transistors scale to smaller sizes, leakage increases along with power dissipation.
And they get smaller every couple of years. Graphene transistors dissipate little power. And,
they switch at high speed. Graphene might be a replacement for silicon someday.
Graphene can be fashioned into devices as small as sixty atoms wide. Graphene quantum dots within a transistor this small serve as single electron transistors. Previous single
electron transistors fashioned from either superconductors or conventional semiconductors
operate near absolute zero. Graphene single electron transistors uniquely function at room
temperature.[23]
Graphene transistors are laboratory curiosities at this time. If they are to go into production two decades from now, graphene wafers must be produced. The first step, production of
graphene by chemical vapor deposition (CVD) has been accomplished on an experimental scale.
Though, no wafers are available to date.
Carbon nanotube transistor: If a 2-D sheet of graphene is rolled, the resulting 1-D structure is known as a carbon nanotube. (Figure 2.63 (b)) The reason to treat it as 1-dimensional is
that it is highly conductive. Electrons traverse the carbon nanotube without being scattered by
a crystal lattice. Resistance in normal metals is caused by scattering of electrons by the metallic crystalline lattice. If electrons avoid this scattering, conduction is said to be by ballistic
transport. Both metallic (acting) and semiconducting carbon nanotubes have been produced.
[5]
Field effect transistors may be fashioned from a carbon nanotubes by depositing source
and drain contacts on the ends, and capacitively coupling a gate to the nanotube between the

CHAPTER 2. SOLID-STATE DEVICE THEORY

88

(a)

(b)

Figure 2.63: (a) Graphene: A single sheet of the graphite allotrope of carbon. The atoms are
arranged in a hexagonal pattern with a carbon at each intersection. (b) Carbon nanotube: A
rolled-up sheet of graphene.
contacts. Both p- and n-type transistors have been fabricated. Why the interest in carbon
nanotube transistors? Nanotube semiconductors are Smaller, faster, lower power compared
with silicon transistors. [6]
Spintronics: Conventional semiconductors control the flow of electron charge, current.
Digital states are represented by “on” or “off ” flow of current. As semiconductors become more
dense with the move to smaller geometry, the power that must be dissipated as heat increases
to the point that it is difficult to remove. Electrons have properties other than charge such
as spin. A tentative explanation of electron spin is the rotation of distributed electron charge
about the spin axis, analogous to diurnal rotation of the Earth. The loops of current created by
charge movement, form a magnetic field. However, the electron is more like a point charge than
a distributed charge, Thus, the rotating distributed charge analogy is not a correct explanation
of spin. Electron spin may have one of two states: up or down which may represent digital
states. More precisely the spin (ms ) quantum number may be ±1/2 the angular momentum (l)
quantum number. [1]
Controlling electron spin instead of charge flow considerably reduces power dissipation and
increases switching speed. Spintronics, an acronym for SPIN TRansport electrONICS, is not
widely applied because of the difficulty of generating, controlling, and sensing electron spin.
However, high density, non-volatile magnetic spin memory is in production using modified
semiconductor processes. This is related to the spin valve magnetic read head used in computer
harddisk drives, not mentioned further here.
A simple magnetic tunnel junction (MTJ) is shown in Figure 2.64 (a), consisting of a pair of
ferromagnetic, strong magnetic properties like iron (Fe), layers separated by a thin insulator.
Electrons can tunnel through a sufficiently thin insulator due to the quantum mechanical
properties of electrons– the wave nature of electrons. The current flow through the MTJ is a
function of the magnetization, spin polarity, of the ferromagnetic layers. The resistance of the
MTJ is low if the magnetic spin of the top layer is in the same direction (polarity) as the bottom
layer. If the magnetic spins of the two layers oppose, the resistance is higher. [8]
The change in resistance can be enhanced by the addition of an antiferromagnet, material
having spins aligned but opposing, below the bottom layer in Figure 2.64 (b). This bias magnet
pins the lower ferromagnetic layer spin to a single unchanging polarity. The top layer magnetization (spin) may be flipped to represent data by the application of an external magnetic field

2.14. QUANTUM DEVICES

89

contact
ferromagnet
tunneling
insulator
ferromagnet
contact
antiferromagnet
contact
(a)

(b)

Figure 2.64: (a) Magnetic tunnel junction (MTJ): Pair of ferromagnetic layers separated by
a thin insulator. The resistance varies with the magnetization polarity of the top layer (b)
Antiferromagnetic bias magnet and pinned bottom ferromagnetic layer increases resistance
sensitivity to changes in polarity of the top ferromagnetic layer. Adapted from [8] Figure 3.

not shown in the figure. The pinned layer is not affected by external magnetic fields. Again,
the MTJ resistance is lowest when the spin of the top ferromagnetic layer is the same sense as
the bottom pinned ferromagnetic layer. [8]
The MTJ may be improved further by splitting the pinned ferromagnetic layer into two
layers separated by a buffer layer in Figure 2.65 (a). This isolates the top layer. The bottom
ferromagnetic layer is pinned by the antiferromagnet as in the previous figure. The ferromagnetic layer atop the buffer is attracted by the bottom ferromagnetic layer. Opposites attract.
Thus, the spin polarity of the additional layer is opposite of that in the bottom layer due to
attraction. The bottom and middle ferromagnetic layers remain fixed. The top ferromagnetic
layer may be set to either spin polarity by high currents in proximate conductors (not shown).
This is how data are stored. Data are read out by the difference in current flow through the
tunnel junction. Resistance is lowest if the layers on both sides of the insulting layer are of the
same spin. [8]
An array of magnetic tunnel junctions may be embedded in a silicon wafer with conductors
connecting the top and bottom terminals for reading data bits from the MTJ’s with conventional CMOS circuitry. One such MTJ is shown in Figure 2.65 (b) with the read conductors.
Not shown, another crossed array of conductors carrying heavy write currents switch the magnetic spin of the top ferromagnetic layer to store data. A current is applied to one of many “X”
conductors and a “Y” conductor. One MTJ in the array is magnetized under the conductors’
cross-over. Data are read out by sensing the MTJ current with conventional silicon semiconductor circuitry. [9]
The main reason for interest in magnetic tunnel junction memory is that it is nonvolatile. It
does not lose data when powered “off ”. Other types of nonvolatile memory are capable of only
limited storage cycles. MTJ memory is also higher speed than most semiconductor memory
types. It is now (2006) a commercial product. [17]
Not a commercial product, or even a laboratory device, is the theoretical spin transistor

CHAPTER 2. SOLID-STATE DEVICE THEORY

90

data

top contact
ferromagnet

pinned layers

tunneling
insulator
ferromagnet
coupling layer
ferromagnet
antiferromagnet

(a)

bottom contact

(b)

Figure 2.65: (a)Splitting the pinned ferromagnetic layer of (b) by a buffer layer improves stability and isolates the top ferromagnetic unpinned layer. Data are stored in the top ferromagnetic
layer based on spin polarity (b) MTJ cell embedded in read lines of a semiconductor die– one
of many MTJ’s. Adapted from [9]
which might one day make spin logic gates possible. The spin transistor is a derivative of the
theoretical spin diode.
It has been known for some time that electrons flowing through a cobalt-iron ferromagnet
become spin polarized. The ferromagnet acts as a filter passing electrons of one spin preferentially. These electrons may flow into an adjacent nonmagnetic conductor (or semiconductor)
retaining the spin polarization for a short time, nano-seconds. Though, spin polarized electrons may propagate a considerable distance compared with semiconductor dimensions. The
spin polarized electrons may be detected by a nickel-iron ferromagnetic layer adjacent to the
semiconductor. [1] [14]
It has also been shown that electron spin polarization occurs when circularly polarized light
illuminates some semiconductor materials. Thus, it should be possible to inject spin polarized
electrons into a semiconductor diode or transistor. The interest in spin based transistors and
gates is because of the non-dissipative nature of spin propagation, compared with dissipative
charge flow. As conventional semiconductors are scaled down in size, power dissipation increases. At some point the scaling down will no longer be practical. Researchers are looking
for a replacement for the conventional charge flow based transistor. That device may be based
on spintronics. [13]
• REVIEW:
• As MOS gate oxide thins with each generation of smaller transistors, excessive gate leakage causes unacceptable power dissipation and heating. The limit of scaling down conventional semiconductor geometry is within sight.
• Resonant tunneling diode (RTD): Quantum mechanical effects, which degrade conven-

2.15. SEMICONDUCTOR DEVICES IN SPICE

91

tional semiconductors, are employed in the RTD. The flow of electrons through a sufficiently thin insulator, is by the wave nature of the electron– particle wave duality. The
RTD functions as an amplifier.
• Double layer tunneling transistor (Deltt): The Deltt is a transistor version of the RTD.
Gate bias controls the ability of electrons to tunnel through a thin insulator from one
quantum well to another (source to drain).
• Quantum dot transistor: A quantum dot, capable of holding a charge, is surrounded by a
thin tunnel barrier replacing the gate of a conventional FET. The charge on the quantum
dot controls source to drain current flow.
• Spintronics: Electrons have two basic properties: charge and spin. Conventional electronic devices control the flow of charge, dissipating energy. Spintronic devices manipulate electron spin, a propagative, non-dissipative process.

2.15

Semiconductor devices in SPICE

The SPICE (simulation program, integrated circuit emphesis) electronic simulation program
provides circuit elements and models for semiconductors. The SPICE element names begin
with d, q, j, or m correspond to diode, BJT, JFET and MOSFET elements, respectively. These
elements are accompanied by corresponding “models” These models have extensive lists of
parameters describing the device. Though, we do not list them here. In this section we provide
a very brief listing of simple spice models for semiconductors, just enough to get started. For
more details on models and an extensive list of model parameters see Kuphaldt. [16] This
reference also gives instructions on using SPICE.
Diode: The diode statement begins with a diode element name which must begin with “d”
plus optional characters. Some example diode element names include: d1, d2, dtest, da, db,
d101, etc. Two node numbers specify the connection of the anode and cathode, respectively, to
other components. The node numbers are followed by a model name, referring to a “.model”
statement.
The model statement line begins with “.model”, followed by the model name matching one
or more diode statements. Next is a “d” indicating that a diode is being modeled. The remainder of the model statement is a list of optional diode parameters of the form ParameterName=ParameterValue. None are shown in the example below. For a list, see reference,
“diodes”. [16]
General form: d[name] [anode] [cathode] [model]
.model [modelname] d ( [parmtr1=x] [parmtr2=y] . . .)
Example:

d1 1 2 mod1
.model mod1 d

Models for specific diode part numbers are often furnished by the semiconductor diode manufacturer. These models include parameters. Otherwise, the parameters default to so called
“default values”, as in the example.

CHAPTER 2. SOLID-STATE DEVICE THEORY

92

BJT, bipolar junction transistor: The BJT element statement begins with an element
name which must begin with “q” with associated circuit symbol designator characters, example: q1, q2, qa, qgood. The BJT node numbers (connections) identify the wiring of the collector,
base, emitter respectively. A model name following the node numbers is associated with a
model statement.
General form: q[name] [collector] [base] [emitter] [model]
.model [modelname] [npn or pnp] ([parmtr1=x] . . .)
Example:
Example:

q1 2 3 0 mod1
.model mod1 pnp
q2 7 8 9 q2n090
.model q2n090 npn ( bf=75 )

The model statement begins with “.model”, followed by the model name, followed by one
of “npn” or “pnp”. The optional list of parameters follows, and may continue for a few lines
beginning with line continuation symbol “+”, plus. Shown above is the forward β parameter
set to 75 for the hypothetical q2n090 model. Detailed transistor models are often available
from semiconductor manufacturers.
FET, field effect transistor The field effect transistor element statement begins with an
element name beginning with “j” for JFET associated with some unique characters, example:
j101, j2b, jalpha, etc. The node numbers follow for the drain, gate and source terminals, respectively. The node numbers define connectivity to other circuit components. Finally, a model
name indicates the JFET model to use.
General form: j[name] [drain] [gate] [source] [model]
.model [modelname] [njf or pjf] ( [parmtr1=x] . . .)
Example:

j1 2 3
.model
j3 4 5
.model

0 mod1
mod1 pjf
0 mod2
mod2 njf ( vto=-4.0 )

The “.model” in the JFET model statement is followed by the model name to identify this
model to the JFET element statement(s) using it. Following the model name is either pjf or njf
for p-channel or n-channel JFET’s respectively. A long list of JFET parameters may follow. We
only show how to set Vp , pinch off voltage, to -4.0 V for an n-channel JFET model. Otherwise,
this vto parameter defaults to -2.5 V or 2.5V for n-channel or p-channel devices, respectively.
MOSFET, metal oxide field effect transistor The MOSFET element name must begin
with “m”, and is the first word in the element statement. Following are the four node numbers
for the drain, gate, source, and substrate, respectively. Next is the model name. Note that the
source and substrate are both connected to the same node “0” in the example. Discrete MOSFET’s are packaged as three terminal devices, the source and substrate are the same physical
terminal. Integrated MOSFET’s are four terminal devices; the substrate is a fourth terminal.
Integrated MOSFET’s may have numerous devices sharing the same substrate, separate from
the sources. Though, the sources might still be connected to the common substrate.
General form: m[name] [drain] [gate] [source] [substrate] [model]
.model [modelname] [nmos or pmos] ( [parmtr1=x] . . . )

BIBLIOGRAPHY

Example:

93

m1 2 3
m5 5 6
.model
.model

0 0 mod1
0 0 mod4
mod1 pmos
mod4 nmos ( vto=1 )

The MOSFET model statement begins with “.model” followed by the model name followed
by either “pmos” or “nmos”. Optional MOSFET model parameters follow. The list of possible
parameters is long. See Volume 5, “MOSFET” for details. [16] MOSFET manufacturers provide
detailed models. Otherwise, defaults are in effect.
The bare minimum semiconductor SPICE information is provided in this section. The models shown here allow simulation of basic circuits. In particular, these models do not account
for high speed or high frequency operation. Simulations are shown in the Volume 5 Chapter 7,
“Using SPICE ...”.
• REVIEW:
• Semiconductors may be computer simulated with SPICE.
• SPICE provides element statements and models for the diode, BJT, JFET, and MOSFET.

Contributors
Contributors to this chapter are listed in chronological order of their contributions, from most
recent to first. See Appendix 2 (Contributor List) for dates and contact information.
Maciej Noszczyski (December 2003): Corrected spelling of Niels Bohr’s name.
Bill Heath (September 2002): Pointed out error in illustration of carbon atom – the nucleus
was shown with seven protons instead of six.

Bibliography
[1] David D. Awschalom, Michael E. Flatte, Nitin Samarth, “Spintronics”, Scientific American, June 2002 at http://www.sciam.com
[2] John Bland,
“The Fluxoid” in “A Mossbauer Spectroscopy and Magnetometry Study of Magnetic Multilayers and Oxides”,
Oliver Lodge
Laboratory,
Department of Physics,
University of Liverpool,
2002,
at
http://www.cmp.liv.ac.uk/frink/thesis/thesis/node45.html
¡bibitem¿[JBb]John Bland, “Superconducting Quantum Interference Device (SQUID)”
in “A Mssbauer Spectroscopy and Magnetometry Study of Magnetic Multilayers and
Oxides”, Oliver Lodge Laboratory, Department of Physics, University of Liverpool, 2002,
at http://www.cmp.liv.ac.uk/frink/thesis/thesis/node48.html
[3] John Bland, “SQUID Magnetometer” in “A Mossbauer Spectroscopy and
Magnetometry Study of Magnetic Multilayers and Oxides”, Oliver Lodge
Laboratory,
Department of Physics,
University of Liverpool,
2002,
at
http://www.cmp.liv.ac.uk/frink/thesis/thesis/node48.html

94

CHAPTER 2. SOLID-STATE DEVICE THEORY

[4] Darren K. Brock, “RSFQ Technology: Circuits and Systems”, Hypres, Inc., NY, at
http://www.hypres.com/papers/Brock-RSFQ-CirSys-Rev01.pdf
[5] Matthew Broersma , “Nanotubes break semiconducting record”, Cnet News, December
19, 2003, at http://news.com.com/2100-1006-5129761.html
[6] “Carbon Nanotube Transistor”, Physics News Graphics,
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May

13,

1998,

at

[7] E. R. Brown, C. D. Parker, “Resonant Tunnel Diodes as Submillimetre-Wave
Sources”, Philosophical Transactions:
Mathematical, Physical and Engineering Sciences, Vol. 354, No. 1717, The Current Status of Semiconductor Tunnelling Devices (Oct. 15, 1996), pp. 2365-2381 at http://links.jstor.org/
sici?sici=1364-503X(19961015)354%3A1717%3C2365%3ARTDASS%3E2.0.CO%3B2-Q
[8] W. J. Gallagher, S. S. P. Parkin, “Development of the magnetic tunnel junction MRAM at IBM: From first junctions to a 16-Mb MRAM demonstrator chip”,
IBM Research and Development, Spintronics, Volume 50, Number 1, 2006, at
http://www.research.ibm.com/journal/rd/501/gallagher.html
[9] “IBM, Infineon Develop Most Advanced MRAM Technology to Date”, IBM Research, at
http://domino.research.ibm.com/comm/pr.nsf/pages/news.20030610 mram.html
[10] Linda Geppert “Quantum Transistors: toward nanoectronic”, IEEE Spectrum, September
2000, at http://www.ece.osu.edu/˜berger/press/quan0900.pdf
[11] A. K. Geim1 and K. S. Novoselov1 , “The rise of graphene”, Nature Materials, 6, 2007, at
http://www.nature.com/nmat/journal/v6/n3/full/nmat1849.html
[12] Ilan Greenberg, “Transistor Technology Takes a Quantum Leap”, Wired News, December
8, 1997, at http://www.wired.com/news/technology/0,1282,8994,00.html
[13] R. Colin Johnson, “Spintronics approach advances toward live chips,” EE Times,
11/06/2006, at http://www.eetimes.com/showArticle.jhtml?articleID=193500309
[14] R. Colin Johnson “ U. of Delaware researchers edge closer to spintronics,” EE Times,
07/26/2007, at http://www.eetimes.com/news/design/showArticle.jhtml?articleID=201201400
[15] R. Colin Johnson, “Can metal-insulator electronics do it better, sans semiconductors?”
http://www.eetimes.com/showArticle.jhtml?articleID=201200024
[16] Tony R. Kuphaldt, “Lessons in Electricity”, Reference, Vol. 5, Ch 7, 2007 at
http://www.ibiblio.org/obp/electricCircuits/Ref/spice.html
[17] Tom
Lee,
“Is
nonvolatile
MRAM
right
for
your
consumer
bedded
device
application?
”,
Freescale
Semiconductor
http://www.acumeninfo.com/subscriber/article/getArticle.jhtml?
articleId=197006965
[18] HyperPhysics,
“SQUID
Magnetometer”,
HyperPhysics
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[19] Phillip
F.
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Ben
Stein,
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February 4,
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[20] “Why MIIM?”, Phiar Corporation, at http://www.phiar.com/whyMIIM.php4
[21] “What
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http://www.phiar.com/whatQuantum.php4

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[22] Oxford University, “Theory, Superconductor Synthesis”, Oxford University, 1996, at
http://www.chem.ox.ac.uk/vrchemistry/super/theory.htm
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articleID=197700700
[24] Ying-Yu
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96

CHAPTER 2. SOLID-STATE DEVICE THEORY

Chapter 3

DIODES AND RECTIFIERS
Contents
3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9
3.10

Introduction . . . . . . . . . . . .
Meter check of a diode . . . . .
Diode ratings . . . . . . . . . . .
Rectifier circuits . . . . . . . . .
Peak detector . . . . . . . . . . .
Clipper circuits . . . . . . . . . .
Clamper circuits . . . . . . . . .
Voltage multipliers . . . . . . . .
Inductor commutating circuits
Diode switching circuits . . . .
3.10.1 Logic . . . . . . . . . . . . .
3.10.2 Analog switch . . . . . . . .
3.11 Zener diodes . . . . . . . . . . . .
3.12 Special-purpose diodes . . . . .
3.12.1 Schottky diodes . . . . . . .
3.12.2 Tunnel diodes . . . . . . . .
3.12.3 Light-emitting diodes . . . .
3.12.4 Laser diodes . . . . . . . . .
3.12.5 Photodiodes . . . . . . . . .
3.12.6 Solar cells . . . . . . . . . .
3.12.7 Varicap or varactor diodes .
3.12.8 Snap diode . . . . . . . . . .
3.12.9 PIN diodes . . . . . . . . . .
3.12.10 IMPATT diode . . . . . . . .
3.12.11 Gunn diode . . . . . . . . . .
3.12.12 Shockley diode . . . . . . . .
3.12.13 Constant-current diodes . .

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98
103
107
108
115
117
121
123
130
132
132
134
135
143
143
144
146
150
152
154
158
159
159
160
161
162
162

CHAPTER 3. DIODES AND RECTIFIERS

98
3.13 Other diode technologies
3.13.1 SiC diodes . . . . . .
3.13.2 Polymer diode . . . .
3.14 SPICE models . . . . . . .
Bibliography . . . . . . . . . . .

3.1

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163
163
163
163
171

Introduction

A diode is an electrical device allowing current to move through it in one direction with far
greater ease than in the other. The most common kind of diode in modern circuit design is
the semiconductor diode, although other diode technologies exist. Semiconductor diodes are
symbolized in schematic diagrams such as Figure 3.1. The term “diode” is customarily reserved
for small signal devices, I ≤ 1 A. The term rectifier is used for power devices, I > 1 A.

Figure 3.1: Semiconductor diode schematic symbol: Arrows indicate the direction of electron
current flow.
When placed in a simple battery-lamp circuit, the diode will either allow or prevent current
through the lamp, depending on the polarity of the applied voltage. (Figure 3.2)

-

+
(a)

-

(b)

+

Figure 3.2: Diode operation: (a) Current flow is permitted; the diode is forward biased. (b)
Current flow is prohibited; the diode is reversed biased.
When the polarity of the battery is such that electrons are allowed to flow through the
diode, the diode is said to be forward-biased. Conversely, when the battery is “backward” and
the diode blocks current, the diode is said to be reverse-biased. A diode may be thought of as
like a switch: “closed” when forward-biased and “open” when reverse-biased.
Oddly enough, the direction of the diode symbol’s “arrowhead” points against the direction
of electron flow. This is because the diode symbol was invented by engineers, who predominantly use conventional flow notation in their schematics, showing current as a flow of charge
from the positive (+) side of the voltage source to the negative (-). This convention holds true
for all semiconductor symbols possessing “arrowheads:” the arrow points in the permitted direction of conventional flow, and against the permitted direction of electron flow.

3.1. INTRODUCTION

99

+

-

-

+

Hydraulic
check valve

(a)

Flow permitted

(b)

Flow prohibited

Figure 3.3: Hydraulic check valve analogy: (a) Electron current flow permitted. (b) Current
flow prohibited.
Diode behavior is analogous to the behavior of a hydraulic device called a check valve. A
check valve allows fluid flow through it in only one direction as in Figure 3.3.
Check valves are essentially pressure-operated devices: they open and allow flow if the
pressure across them is of the correct “polarity” to open the gate (in the analogy shown, greater
fluid pressure on the right than on the left). If the pressure is of the opposite “polarity,” the
pressure difference across the check valve will close and hold the gate so that no flow occurs.
Like check valves, diodes are essentially “pressure-” operated (voltage-operated) devices.
The essential difference between forward-bias and reverse-bias is the polarity of the voltage
dropped across the diode. Let’s take a closer look at the simple battery-diode-lamp circuit
shown earlier, this time investigating voltage drops across the various components in Figure 3.4.
6.0 V

0.7 V

VΩ
VΩ
A

+

A

COM

COM

+ -

- +

5.3 V

0.0 V

VΩ

6V
(a)

A

-

COM

VΩ

6V
(b)

A

COM

+

Figure 3.4: Diode circuit voltage measurements: (a) Forward biased. (b) Reverse biased.
A forward-biased diode conducts current and drops a small voltage across it, leaving most
of the battery voltage dropped across the lamp. If the battery’s polarity is reversed, the diode
becomes reverse-biased, and drops all of the battery’s voltage leaving none for the lamp. If we
consider the diode to be a self-actuating switch (closed in the forward-bias mode and open in
the reverse-bias mode), this behavior makes sense. The most substantial difference is that the
diode drops a lot more voltage when conducting than the average mechanical switch (0.7 volts
versus tens of millivolts).

CHAPTER 3. DIODES AND RECTIFIERS

100

This forward-bias voltage drop exhibited by the diode is due to the action of the depletion
region formed by the P-N junction under the influence of an applied voltage. If no voltage
applied is across a semiconductor diode, a thin depletion region exists around the region of the
P-N junction, preventing current flow. (Figure 3.5 (a)) The depletion region is almost devoid of
available charge carriers, and acts as an insulator:

P-type
material

N-type
material

P-N junction representation

(a)
Depletion region
Anode

Cathode
Schematic symbol

(b)

Stripe marks cathode

Real component appearance
(c)

Figure 3.5: Diode representations: PN-junction model, schematic symbol, physical part.
The schematic symbol of the diode is shown in Figure 3.5 (b) such that the anode (pointing
end) corresponds to the P-type semiconductor at (a). The cathode bar, non-pointing end, at (b)
corresponds to the N-type material at (a). Also note that the cathode stripe on the physical
part (c) corresponds to the cathode on the symbol.
If a reverse-biasing voltage is applied across the P-N junction, this depletion region expands, further resisting any current through it. (Figure 3.6)

-

P
Reverse-biased

+

N
Depletion region

Figure 3.6: Depletion region expands with reverse bias.
Conversely, if a forward-biasing voltage is applied across the P-N junction, the depletion
region collapses becoming thinner. The diode becomes less resistive to current through it. In

3.1. INTRODUCTION

101

order for a sustained current to go through the diode; though, the depletion region must be
fully collapsed by the applied voltage. This takes a certain minimum voltage to accomplish,
called the forward voltage as illustrated in Figure 3.7.

0.4 V

0.7 V

Partial forward-biased

P
(a)

Forward-biased

N
Depletion region

P
(b)

N

Depletion region fully collapsed

Figure 3.7: Inceasing forward bias from (a) to (b) decreases depletion region thickness.
For silicon diodes, the typical forward voltage is 0.7 volts, nominal. For germanium diodes,
the forward voltage is only 0.3 volts. The chemical constituency of the P-N junction comprising
the diode accounts for its nominal forward voltage figure, which is why silicon and germanium
diodes have such different forward voltages. Forward voltage drop remains approximately
constant for a wide range of diode currents, meaning that diode voltage drop is not like that of a
resistor or even a normal (closed) switch. For most simplified circuit analysis, the voltage drop
across a conducting diode may be considered constant at the nominal figure and not related to
the amount of current.
Actually, forward voltage drop is more complex. An equation describes the exact current
through a diode, given the voltage dropped across the junction, the temperature of the junction,
and several physical constants. It is commonly known as the diode equation:

ID = IS (eqVD/NkT - 1)
Where,
ID = Diode current in amps
IS = Saturation current in amps
(typically 1 x 10-12 amps)
e = Euler’s constant (~ 2.718281828)
q = charge of electron (1.6 x 10-19 coulombs)
VD = Voltage applied across diode in volts
N = "Nonideality" or "emission" coefficient
(typically between 1 and 2)
k = Boltzmann’s constant (1.38 x 10-23)
T = Junction temperature in Kelvins

CHAPTER 3. DIODES AND RECTIFIERS

102

The term kT/q describes the voltage produced within the P-N junction due to the action of
temperature, and is called the thermal voltage, or Vt of the junction. At room temperature,
this is about 26 millivolts. Knowing this, and assuming a “nonideality” coefficient of 1, we may
simplify the diode equation and re-write it as such:

ID = IS (eVD/0.026 -1)
Where,
ID = Diode current in amps
IS = Saturation current in amps
(typically 1 x 10-12 amps)
e = Euler’s constant (~ 2.718281828)
VD = Voltage applied across diode in volts
You need not be familiar with the “diode equation” to analyze simple diode circuits. Just
understand that the voltage dropped across a current-conducting diode does change with the
amount of current going through it, but that this change is fairly small over a wide range of
currents. This is why many textbooks simply say the voltage drop across a conducting, semiconductor diode remains constant at 0.7 volts for silicon and 0.3 volts for germanium. However,
some circuits intentionally make use of the P-N junction’s inherent exponential current/voltage
relationship and thus can only be understood in the context of this equation. Also, since temperature is a factor in the diode equation, a forward-biased P-N junction may also be used as a
temperature-sensing device, and thus can only be understood if one has a conceptual grasp on
this mathematical relationship.
A reverse-biased diode prevents current from going through it, due to the expanded depletion region. In actuality, a very small amount of current can and does go through a reversebiased diode, called the leakage current, but it can be ignored for most purposes. The ability
of a diode to withstand reverse-bias voltages is limited, as it is for any insulator. If the applied reverse-bias voltage becomes too great, the diode will experience a condition known as
breakdown (Figure 3.8), which is usually destructive. A diode’s maximum reverse-bias voltage
rating is known as the Peak Inverse Voltage, or PIV, and may be obtained from the manufacturer. Like forward voltage, the PIV rating of a diode varies with temperature, except that
PIV increases with increased temperature and decreases as the diode becomes cooler – exactly
opposite that of forward voltage.
Typically, the PIV rating of a generic “rectifier” diode is at least 50 volts at room temperature. Diodes with PIV ratings in the many thousands of volts are available for modest prices.
• REVIEW:
• A diode is an electrical component acting as a one-way valve for current.
• When voltage is applied across a diode in such a way that the diode allows current, the
diode is said to be forward-biased.
• When voltage is applied across a diode in such a way that the diode prohibits current, the
diode is said to be reverse-biased.

3.2. METER CHECK OF A DIODE

103

ID
forward

reverse-bias

forward-bias

0.7
V

breakdown!

VD

reverse

Figure 3.8: Diode curve: showing knee at 0.7 V forward bias for Si, and reverse breakdown.
• The voltage dropped across a conducting, forward-biased diode is called the forward voltage. Forward voltage for a diode varies only slightly for changes in forward current and
temperature, and is fixed by the chemical composition of the P-N junction.
• Silicon diodes have a forward voltage of approximately 0.7 volts.
• Germanium diodes have a forward voltage of approximately 0.3 volts.
• The maximum reverse-bias voltage that a diode can withstand without “breaking down”
is called the Peak Inverse Voltage, or PIV rating.

3.2

Meter check of a diode

Being able to determine the polarity (cathode versus anode) and basic functionality of a diode
is a very important skill for the electronics hobbyist or technician to have. Since we know
that a diode is essentially nothing more than a one-way valve for electricity, it makes sense
we should be able to verify its one-way nature using a DC (battery-powered) ohmmeter as in
Figure 3.9. Connected one way across the diode, the meter should show a very low resistance
at (a). Connected the other way across the diode, it should show a very high resistance at (b)
(“OL” on some digital meter models).
Of course, to determine which end of the diode is the cathode and which is the anode, you
must know with certainty which test lead of the meter is positive (+) and which is negative (-)
when set to the “resistance” or “Ω” function. With most digital multimeters I’ve seen, the red
lead becomes positive and the black lead negative when set to measure resistance, in accordance with standard electronics color-code convention. However, this is not guaranteed for all
meters. Many analog multimeters, for example, actually make their black leads positive (+)

CHAPTER 3. DIODES AND RECTIFIERS

104

V

A

V

A

V

A

V

A

+
Anode

OFF

+
Cathode

OFF

Cathode
A

Anode

-

COM

(a)

A

-

COM

(b)

Figure 3.9: Determination of diode polarity: (a) Low resistance indicates forward bias, black
lead is cathode and red lead anode (for most meters) (b) Reversing leads shows high resistance
indicating reverse bias.
and their red leads negative (-) when switched to the “resistance” function, because it is easier
to manufacture it that way!
One problem with using an ohmmeter to check a diode is that the readings obtained only
have qualitative value, not quantitative. In other words, an ohmmeter only tells you which way
the diode conducts; the low-value resistance indication obtained while conducting is useless. If
an ohmmeter shows a value of “1.73 ohms” while forward-biasing a diode, that figure of 1.73
Ω doesn’t represent any real-world quantity useful to us as technicians or circuit designers.
It neither represents the forward voltage drop nor any “bulk” resistance in the semiconductor
material of the diode itself, but rather is a figure dependent upon both quantities and will vary
substantially with the particular ohmmeter used to take the reading.
For this reason, some digital multimeter manufacturers equip their meters with a special
“diode check” function which displays the actual forward voltage drop of the diode in volts,
rather than a “resistance” figure in ohms. These meters work by forcing a small current
through the diode and measuring the voltage dropped between the two test leads. (Figure 3.10)
The forward voltage reading obtained with such a meter will typically be less than the
“normal” drop of 0.7 volts for silicon and 0.3 volts for germanium, because the current provided
by the meter is of trivial proportions. If a multimeter with diode-check function isn’t available,
or you would like to measure a diode’s forward voltage drop at some non-trivial current, the
circuit of Figure 3.11 may be constructed using a battery, resistor, and voltmeter
Connecting the diode backwards to this testing circuit will simply result in the voltmeter
indicating the full voltage of the battery.
If this circuit were designed to provide a constant or nearly constant current through the
diode despite changes in forward voltage drop, it could be used as the basis of a temperaturemeasurement instrument, the voltage measured across the diode being inversely proportional
to diode junction temperature. Of course, diode current should be kept to a minimum to avoid
self-heating (the diode dissipating substantial amounts of heat energy), which would interfere
with temperature measurement.
Beware that some digital multimeters equipped with a “diode check” function may output
a very low test voltage (less than 0.3 volts) when set to the regular “resistance” (Ω) function:

3.2. METER CHECK OF A DIODE

V

105

A

V

OFF

A

+

A

Anode

COM

Cathode

Figure 3.10: Meter with a “Diode check” function displays the forward voltage drop of 0.548
volts instead of a low resistance.

V

+

A

V

A
OFF

Resistor

+

-

+

Diode

V
-

A

COM

Battery

(a)

(b)

Figure 3.11: Measuring forward voltage of a diode without“diode check” meter function: (a)
Schematic diagram. (b) Pictorial diagram.

CHAPTER 3. DIODES AND RECTIFIERS

106

too low to fully collapse the depletion region of a PN junction. The philosophy here is that
the “diode check” function is to be used for testing semiconductor devices, and the “resistance”
function for anything else. By using a very low test voltage to measure resistance, it is easier for a technician to measure the resistance of non-semiconductor components connected
to semiconductor components, since the semiconductor component junctions will not become
forward-biased with such low voltages.
Consider the example of a resistor and diode connected in parallel, soldered in place on
a printed circuit board (PCB). Normally, one would have to unsolder the resistor from the
circuit (disconnect it from all other components) before measuring its resistance, otherwise any
parallel-connected components would affect the reading obtained. When using a multimeter
which outputs a very low test voltage to the probes in the “resistance” function mode, the
diode’s PN junction will not have enough voltage impressed across it to become forward-biased,
and will only pass negligible current. Consequently, the meter “sees” the diode as an open (no
continuity), and only registers the resistor’s resistance. (Figure 3.12)

V

OFF

A

D1

A

1 kΩ

V

Printed circuit board

R1

k

A

COM

Figure 3.12: Ohmmeter equipped with a low test voltage (<0.7 V) does not see diodes allowing
it to measure parallel resistors.
If such an ohmmeter were used to test a diode, it would indicate a very high resistance
(many mega-ohms) even if connected to the diode in the “correct” (forward-biased) direction.
(Figure 3.13)
Reverse voltage strength of a diode is not as easily tested, because exceeding a normal
diode’s PIV usually results in destruction of the diode. Special types of diodes, though, which
are designed to “break down” in reverse-bias mode without damage (called zener diodes), which
are tested with the same voltage source / resistor / voltmeter circuit, provided that the voltage
source is of high enough value to force the diode into its breakdown region. More on this subject
in a later section of this chapter.
• REVIEW:
• An ohmmeter may be used to qualitatively check diode function. There should be low
resistance measured one way and very high resistance measured the other way. When
using an ohmmeter for this purpose, be sure you know which test lead is positive and

3.3. DIODE RATINGS

107

M

V

A

V

OFF

A

A

COM

Figure 3.13: Ohmmeter equipped with a low test voltage, too low to forward bias diodes, does
not see diodes.
which is negative! The actual polarity may not follow the colors of the leads as you might
expect, depending on the particular design of meter.
• Some multimeters provide a “diode check” function that displays the actual forward voltage of the diode when its conducting current. Such meters typically indicate a slightly
lower forward voltage than what is “nominal” for a diode, due to the very small amount
of current used during the check.

3.3

Diode ratings

In addition to forward voltage drop (Vf ) and peak inverse voltage (PIV), there are many other
ratings of diodes important to circuit design and component selection. Semiconductor manufacturers provide detailed specifications on their products – diodes included – in publications
known as datasheets. Datasheets for a wide variety of semiconductor components may be
found in reference books and on the internet. I prefer the internet as a source of component
specifications because all the data obtained from manufacturer websites are up-to-date.
A typical diode datasheet will contain figures for the following parameters:
Maximum repetitive reverse voltage = VRRM , the maximum amount of voltage the diode
can withstand in reverse-bias mode, in repeated pulses. Ideally, this figure would be infinite.
Maximum DC reverse voltage = VR or VDC , the maximum amount of voltage the diode can
withstand in reverse-bias mode on a continual basis. Ideally, this figure would be infinite.
Maximum forward voltage = VF , usually specified at the diode’s rated forward current. Ideally, this figure would be zero: the diode providing no opposition whatsoever to forward current.
In reality, the forward voltage is described by the “diode equation.”
Maximum (average) forward current = IF (AV ) , the maximum average amount of current
the diode is able to conduct in forward bias mode. This is fundamentally a thermal limitation:
how much heat can the PN junction handle, given that dissipation power is equal to current (I)
multiplied by voltage (V or E) and forward voltage is dependent upon both current and junction
temperature. Ideally, this figure would be infinite.

CHAPTER 3. DIODES AND RECTIFIERS

108

Maximum (peak or surge) forward current = IF SM or if (surge) , the maximum peak amount
of current the diode is able to conduct in forward bias mode. Again, this rating is limited by the
diode junction’s thermal capacity, and is usually much higher than the average current rating
due to thermal inertia (the fact that it takes a finite amount of time for the diode to reach
maximum temperature for a given current). Ideally, this figure would be infinite.
Maximum total dissipation = PD , the amount of power (in watts) allowable for the diode to
dissipate, given the dissipation (P=IE) of diode current multiplied by diode voltage drop, and
also the dissipation (P=I2 R) of diode current squared multiplied by bulk resistance. Fundamentally limited by the diode’s thermal capacity (ability to tolerate high temperatures).
Operating junction temperature = TJ , the maximum allowable temperature for the diode’s
PN junction, usually given in degrees Celsius (o C). Heat is the “Achilles’ heel” of semiconductor
devices: they must be kept cool to function properly and give long service life.
Storage temperature range = TST G , the range of allowable temperatures for storing a diode
(unpowered). Sometimes given in conjunction with operating junction temperature (TJ ), because the maximum storage temperature and the maximum operating temperature ratings
are often identical. If anything, though, maximum storage temperature rating will be greater
than the maximum operating temperature rating.
Thermal resistance = R(Θ), the temperature difference between junction and outside air
(R(Θ)JA ) or between junction and leads (R(Θ)JL ) for a given power dissipation. Expressed in
units of degrees Celsius per watt (o C/W). Ideally, this figure would be zero, meaning that the
diode package was a perfect thermal conductor and radiator, able to transfer all heat energy
from the junction to the outside air (or to the leads) with no difference in temperature across
the thickness of the diode package. A high thermal resistance means that the diode will build
up excessive temperature at the junction (where its critical) despite best efforts at cooling the
outside of the diode, and thus will limit its maximum power dissipation.
Maximum reverse current = IR , the amount of current through the diode in reverse-bias
operation, with the maximum rated inverse voltage applied (VDC ). Sometimes referred to as
leakage current. Ideally, this figure would be zero, as a perfect diode would block all current
when reverse-biased. In reality, it is very small compared to the maximum forward current.
Typical junction capacitance = CJ , the typical amount of capacitance intrinsic to the junction, due to the depletion region acting as a dielectric separating the anode and cathode connections. This is usually a very small figure, measured in the range of picofarads (pF).
Reverse recovery time = trr , the amount of time it takes for a diode to “turn off ” when the
voltage across it alternates from forward-bias to reverse-bias polarity. Ideally, this figure would
be zero: the diode halting conduction immediately upon polarity reversal. For a typical rectifier
diode, reverse recovery time is in the range of tens of microseconds; for a “fast switching” diode,
it may only be a few nanoseconds.
Most of these parameters vary with temperature or other operating conditions, and so a
single figure fails to fully describe any given rating. Therefore, manufacturers provide graphs
of component ratings plotted against other variables (such as temperature), so that the circuit
designer has a better idea of what the device is capable of.

3.4

Rectifier circuits

3.4. RECTIFIER CIRCUITS

109

Now we come to the most popular application of the diode: rectification. Simply defined,
rectification is the conversion of alternating current (AC) to direct current (DC). This involves
a device that only allows one-way flow of electrons. As we have seen, this is exactly what a
semiconductor diode does. The simplest kind of rectifier circuit is the half-wave rectifier. It
only allows one half of an AC waveform to pass through to the load. (Figure 3.14)

+

AC
voltage
source

Load

-

Figure 3.14: Half-wave rectifier circuit.
For most power applications, half-wave rectification is insufficient for the task. The harmonic content of the rectifier’s output waveform is very large and consequently difficult to
filter. Furthermore, the AC power source only supplies power to the load one half every full
cycle, meaning that half of its capacity is unused. Half-wave rectification is, however, a very
simple way to reduce power to a resistive load. Some two-position lamp dimmer switches apply full AC power to the lamp filament for “full” brightness and then half-wave rectify it for a
lesser light output. (Figure 3.15)
Bright
AC
voltage
source

Dim

Figure 3.15: Half-wave rectifier application: Two level lamp dimmer.
In the “Dim” switch position, the incandescent lamp receives approximately one-half the
power it would normally receive operating on full-wave AC. Because the half-wave rectified
power pulses far more rapidly than the filament has time to heat up and cool down, the lamp
does not blink. Instead, its filament merely operates at a lesser temperature than normal,
providing less light output. This principle of “pulsing” power rapidly to a slow-responding load
device to control the electrical power sent to it is common in the world of industrial electronics.
Since the controlling device (the diode, in this case) is either fully conducting or fully nonconducting at any given time, it dissipates little heat energy while controlling load power, making
this method of power control very energy-efficient. This circuit is perhaps the crudest possible
method of pulsing power to a load, but it suffices as a proof-of-concept application.
If we need to rectify AC power to obtain the full use of both half-cycles of the sine wave,
a different rectifier circuit configuration must be used. Such a circuit is called a full-wave
rectifier. One kind of full-wave rectifier, called the center-tap design, uses a transformer with a
center-tapped secondary winding and two diodes, as in Figure 3.16.

CHAPTER 3. DIODES AND RECTIFIERS

110

AC
voltage
source

+
Load

-

Figure 3.16: Full-wave rectifier, center-tapped design.
This circuit’s operation is easily understood one half-cycle at a time. Consider the first halfcycle, when the source voltage polarity is positive (+) on top and negative (-) on bottom. At this
time, only the top diode is conducting; the bottom diode is blocking current, and the load “sees”
the first half of the sine wave, positive on top and negative on bottom. Only the top half of the
transformer’s secondary winding carries current during this half-cycle as in Figure 3.17.

+

+

-

-

+
-

Figure 3.17: Full-wave center-tap rectifier: Top half of secondary winding conducts during
positive half-cycle of input, delivering positive half-cycle to load..
During the next half-cycle, the AC polarity reverses. Now, the other diode and the other half
of the transformer’s secondary winding carry current while the portions of the circuit formerly
carrying current during the last half-cycle sit idle. The load still “sees” half of a sine wave, of
the same polarity as before: positive on top and negative on bottom. (Figure 3.18)

+

+

+
-

Figure 3.18: Full-wave center-tap rectifier: During negative input half-cycle, bottom half of
secondary winding conducts, delivering a positive half-cycle to the load.
One disadvantage of this full-wave rectifier design is the necessity of a transformer with a

3.4. RECTIFIER CIRCUITS

111

center-tapped secondary winding. If the circuit in question is one of high power, the size and
expense of a suitable transformer is significant. Consequently, the center-tap rectifier design
is only seen in low-power applications.
The full-wave center-tapped rectifier polarity at the load may be reversed by changing the
direction of the diodes. Furthermore, the reversed diodes can be paralleled with an existing
positive-output rectifier. The result is dual-polarity full-wave center-tapped rectifier in Figure 3.19. Note that the connectivity of the diodes themselves is the same configuration as a
bridge.
Loads

+
+
AC voltage source

-

Figure 3.19: Dual polarity full-wave center tap rectifier
Another, more popular full-wave rectifier design exists, and it is built around a four-diode
bridge configuration. For obvious reasons, this design is called a full-wave bridge. (Figure 3.20)

AC
voltage
source

+
Load

Figure 3.20: Full-wave bridge rectifier.
Current directions for the full-wave bridge rectifier circuit are as shown in Figure 3.21 for
positive half-cycle and Figure 3.22 for negative half-cycles of the AC source waveform. Note
that regardless of the polarity of the input, the current flows in the same direction through
the load. That is, the negative half-cycle of source is a positive half-cycle at the load. The
current flow is through two diodes in series for both polarities. Thus, two diode drops of the
source voltage are lost (0.7·2=1.4 V for Si) in the diodes. This is a disadvantage compared with
a full-wave center-tap design. This disadvantage is only a problem in very low voltage power
supplies.
Remembering the proper layout of diodes in a full-wave bridge rectifier circuit can often be
frustrating to the new student of electronics. I’ve found that an alternative representation of
this circuit is easier both to remember and to comprehend. It’s the exact same circuit, except
all diodes are drawn in a horizontal attitude, all “pointing” the same direction. (Figure 3.23)

CHAPTER 3. DIODES AND RECTIFIERS

112

+
-

+
-

Figure 3.21: Full-wave bridge rectifier: Electron flow for positive half-cycles.

+

+
-

Figure 3.22: Full-wave bridge rectifier: Electron flow for negative half=cycles.

AC
voltage
source

+
Load

Figure 3.23: Alternative layout style for Full-wave bridge rectifier.

3.4. RECTIFIER CIRCUITS

113

One advantage of remembering this layout for a bridge rectifier circuit is that it expands
easily into a polyphase version in Figure 3.24.

3-phase
AC source

+
Load

-

Figure 3.24: Three-phase full-wave bridge rectifier circuit.
Each three-phase line connects between a pair of diodes: one to route power to the positive
(+) side of the load, and the other to route power to the negative (-) side of the load. Polyphase
systems with more than three phases are easily accommodated into a bridge rectifier scheme.
Take for instance the six-phase bridge rectifier circuit in Figure 3.25.

6-phase
AC source

+
Load

-

Figure 3.25: Six-phase full-wave bridge rectifier circuit.
When polyphase AC is rectified, the phase-shifted pulses overlap each other to produce a DC
output that is much “smoother” (has less AC content) than that produced by the rectification of
single-phase AC. This is a decided advantage in high-power rectifier circuits, where the sheer
physical size of filtering components would be prohibitive but low-noise DC power must be
obtained. The diagram in Figure 3.26 shows the full-wave rectification of three-phase AC.
In any case of rectification – single-phase or polyphase – the amount of AC voltage mixed
with the rectifier’s DC output is called ripple voltage. In most cases, since “pure” DC is the desired goal, ripple voltage is undesirable. If the power levels are not too great, filtering networks
may be employed to reduce the amount of ripple in the output voltage.

CHAPTER 3. DIODES AND RECTIFIERS

114
1

2

3

TIME
Resultant DC waveform

Figure 3.26: Three-phase AC and 3-phase full-wave rectifier output.

Sometimes, the method of rectification is referred to by counting the number of DC “pulses”
output for every 360o of electrical “rotation.” A single-phase, half-wave rectifier circuit, then,
would be called a 1-pulse rectifier, because it produces a single pulse during the time of one
complete cycle (360o ) of the AC waveform. A single-phase, full-wave rectifier (regardless of
design, center-tap or bridge) would be called a 2-pulse rectifier, because it outputs two pulses
of DC during one AC cycle’s worth of time. A three-phase full-wave rectifier would be called a
6-pulse unit.
Modern electrical engineering convention further describes the function of a rectifier circuit
by using a three-field notation of phases, ways, and number of pulses. A single-phase, halfwave rectifier circuit is given the somewhat cryptic designation of 1Ph1W1P (1 phase, 1 way,
1 pulse), meaning that the AC supply voltage is single-phase, that current on each phase of
the AC supply lines moves in only one direction (way), and that there is a single pulse of DC
produced for every 360o of electrical rotation. A single-phase, full-wave, center-tap rectifier
circuit would be designated as 1Ph1W2P in this notational system: 1 phase, 1 way or direction
of current in each winding half, and 2 pulses or output voltage per cycle. A single-phase, fullwave, bridge rectifier would be designated as 1Ph2W2P: the same as for the center-tap design,
except current can go both ways through the AC lines instead of just one way. The three-phase
bridge rectifier circuit shown earlier would be called a 3Ph2W6P rectifier.
Is it possible to obtain more pulses than twice the number of phases in a rectifier circuit?
The answer to this question is yes: especially in polyphase circuits. Through the creative use
of transformers, sets of full-wave rectifiers may be paralleled in such a way that more than
six pulses of DC are produced for three phases of AC. A 30o phase shift is introduced from
primary to secondary of a three-phase transformer when the winding configurations are not
of the same type. In other words, a transformer connected either Y-∆ or ∆-Y will exhibit
this 30o phase shift, while a transformer connected Y-Y or ∆-∆ will not. This phenomenon
may be exploited by having one transformer connected Y-Y feed a bridge rectifier, and have
another transformer connected Y-∆ feed a second bridge rectifier, then parallel the DC outputs
of both rectifiers. (Figure 3.27) Since the ripple voltage waveforms of the two rectifiers’ outputs
are phase-shifted 30o from one another, their superposition results in less ripple than either
rectifier output considered separately: 12 pulses per 360o instead of just six:

3.5. PEAK DETECTOR

115

3Ph2W12P rectifier circuit
Secondary

Primary
3-phase
AC input

+

DC
output

-

Secondary

Figure 3.27: Polyphase rectifier circuit: 3-phase 2-way 12-pulse (3Ph2W12P)
• REVIEW:
• Rectification is the conversion of alternating current (AC) to direct current (DC).
• A half-wave rectifier is a circuit that allows only one half-cycle of the AC voltage waveform
to be applied to the load, resulting in one non-alternating polarity across it. The resulting
DC delivered to the load “pulsates” significantly.
• A full-wave rectifier is a circuit that converts both half-cycles of the AC voltage waveform
to an unbroken series of voltage pulses of the same polarity. The resulting DC delivered
to the load doesn’t “pulsate” as much.
• Polyphase alternating current, when rectified, gives a much “smoother” DC waveform
(less ripple voltage) than rectified single-phase AC.

3.5

Peak detector

A peak detector is a series connection of a diode and a capacitor outputting a DC voltage equal
to the peak value of the applied AC signal. The circuit is shown in Figure 3.28 with the corresponding SPICE net list. An AC voltage source applied to the peak detector, charges the
capacitor to the peak of the input. The diode conducts positive “half cycles,” charging the capacitor to the waveform peak. When the input waveform falls below the DC “peak” stored
on the capacitor, the diode is reverse biased, blocking current flow from capacitor back to the
source. Thus, the capacitor retains the peak value even as the waveform drops to zero. Another
view of the peak detector is that it is the same as a half-wave rectifier with a filter capacitor
added to the output.
It takes a few cycles for the capacitor to charge to the peak as in Figure 3.29 due to the
series resistance (RC “time constant”). Why does the capacitor not charge all the way to 5 V?

CHAPTER 3. DIODES AND RECTIFIERS

116

1

1.0 kΩ
5 Vp-p
0 Voffset
1 kHz 0

3

2
V(2)
output

0.1 uF

*SPICE 03441.eps
C1 2 0 0.1u
R1 1 3 1.0k
V1 1 0 SIN(0 5 1k)
D1 3 2 diode
.model diode d
.tran 0.01m 50mm
.end

Figure 3.28: Peak detector: Diode conducts on positive half cycles charging capacitor to the
peak voltage (less diode forward drop).

It would charge to 5 V if an “ideal diode” were obtainable. However, the silicon diode has a
forward voltage drop of 0.7 V which subtracts from the 5 V peak of the input.

Figure 3.29: Peak detector: Capacitor charges to peak within a few cycles.

The circuit in Figure 3.28 could represent a DC power supply based on a half-wave rectifier.
The resistance would be a few Ohms instead of 1 kΩ due to a transformer secondary winding
replacing the voltage source and resistor. A larger “filter” capacitor would be used. A power
supply based on a 60 Hz source with a filter of a few hundred µF could supply up to 100 mA.
Half-wave supplies seldom supply more due to the difficulty of filtering a half-wave.
The peak detector may be combined with other components to build a crystal radio ( page
424).

3.6. CLIPPER CIRCUITS

3.6

117

Clipper circuits

A circuit which removes the peak of a waveform is known as a clipper. A negative clipper is
shown in Figure 3.30. This schematic diagram was produced with Xcircuit schematic capture
program. Xcircuit produced the SPICE net list Figure 3.30, except for the second, and next to
last pair of lines which were inserted with a text editor.

1

2
1.0 kΩ

5 Vp
0 Voffset
0 1 kHz

V(2)
output

*SPICE 03437.eps
* A K ModelName
D1 0 2 diode
R1 2 1 1.0k
V1 1 0 SIN(0 5 1k)
.model diode d
.tran .05m 3m
.end

Figure 3.30: Clipper: clips negative peak at -0.7 V.
During the positive half cycle of the 5 V peak input, the diode is reversed biased. The diode
does not conduct. It is as if the diode were not there. The positive half cycle is unchanged
at the output V(2) in Figure 3.31. Since the output positive peaks actually overlays the input
sinewave V(1), the input has been shifted upward in the plot for clarity. In Nutmeg, the SPICE
display module, the command “plot v(1)+1)” accomplishes this.

Figure 3.31: V(1)+1 is actually V(1), a 5 Vptp sinewave, offset by 1 V for display clarity. V(2)
output is clipped at -0.7 V, by diode D1.
During the negative half cycle of sinewave input of Figure 3.31, the diode is forward biased,

CHAPTER 3. DIODES AND RECTIFIERS

118

that is, conducting. The negative half cycle of the sinewave is shorted out. The negative half
cycle of V(2) would be clipped at 0 V for an ideal diode. The waveform is clipped at -0.7 V
due to the forward voltage drop of the silicon diode. The spice model defaults to 0.7 V unless
parameters in the model statement specify otherwise. Germanium or Schottky diodes clip at
lower voltages.
Closer examination of the negative clipped peak (Figure 3.31) reveals that it follows the
input for a slight period of time while the sinewave is moving toward -0.7 V. The clipping
action is only effective after the input sinewave exceeds -0.7 V. The diode is not conducting for
the complete half cycle, though, during most of it.
The addition of an anti-parallel diode to the existing diode in Figure 3.30 yields the symmetrical clipper in Figure 3.32.

1
1.0 kΩ

5 Vp
0 Voffset
0 1 kHz

2
D2

D1

*SPICE 03438.eps
D1 0 2 diode
D2 2 0 diode
R1 2 1 1.0k
V1 1 0 SIN(0 5 1k)
.model diode d
.tran 0.05m 3m
.end

Figure 3.32: Symmetrical clipper: Anti-parallel diodes clip both positive and negative peak,
leaving a ± 0.7 V output.
Diode D1 clips the negative peak at -0.7 V as before. The additional diode D2 conducts for
positive half cycles of the sine wave as it exceeds 0.7 V, the forward diode drop. The remainder
of the voltage drops across the series resistor. Thus, both peaks of the input sinewave are
clipped in Figure 3.33. The net list is in Figure 3.32
The most general form of the diode clipper is shown in Figure 3.34. For an ideal diode, the
clipping occurs at the level of the clipping voltage, V1 and V2. However, the voltage sources
have been adjusted to account for the 0.7 V forward drop of the real silicon diodes. D1 clips
at 1.3V +0.7V=2.0V when the diode begins to conduct. D2 clips at -2.3V -0.7V=-3.0V when D2
conducts.
The clipper in Figure 3.34 does not have to clip both levels. To clip at one level with one
diode and one voltage source, remove the other diode and source.
The net list is in Figure 3.34. The waveforms in Figure 3.35 show the clipping of v(1) at
output v(2).
There is also a zener diode clipper circuit in the “Zener diode” section. A zener diode replaces
both the diode and the DC voltage source.
A practical application of a clipper is to prevent an amplified speech signal from overdriving
a radio transmitter in Figure 3.36. Over driving the transmitter generates spurious radio
signals which causes interference with other stations. The clipper is a protective measure.
A sinewave may be squared up by overdriving a clipper. Another clipper application is the
protection of exposed inputs of integrated circuits. The input of the IC is connected to a pair of
diodes as at node “2” of Figure ??. The voltage sources are replaced by the power supply rails

3.6. CLIPPER CIRCUITS

119

Figure 3.33: Diode D1 clips at -0.7 V as it conducts during negative peaks. D2 conducts for
positive peaks, clipping at 0.7V.

1

2

0

1.0 kΩ

V3
5 Vp
0 Voffset
1 kHz

D1
3
V1
+
1.3
V
−

V2
D2
4
V2
+
-2.3
V
−

*SPICE 03439.eps
V1 3 0 1.3
V2 4 0 -2.3
D1 2 3 diode
D2 4 2 diode
R1 2 1 1.0k
V3 1 0 SIN(0 5 1k)
.model diode d
.tran 0.05m 3m
.end

Figure 3.34: D1 clips the input sinewave at 2V. D2 clips at -3V.

CHAPTER 3. DIODES AND RECTIFIERS

120

Figure 3.35: D1 clips the sinewave at 2V. D2 clips at -3V.

clipper
preamp

transimtter

microphone

Figure 3.36: Clipper prevents over driving radio transmitter by voice peaks.

3.7. CLAMPER CIRCUITS

121

of the IC. For example, CMOS IC’s use 0V and +5 V. Analog amplifiers might use ±12V for the
V1 and V2 sources.
• REVIEW
• A resistor and diode driven by an AC voltage source clips the signal observed across the
diode.
• A pair of anti-parallel Si diodes clip symmetrically at ±0.7V
• The grounded end of a clipper diode(s) can be disconnected and wired to a DC voltage to
clip at an arbitrary level.
• A clipper can serve as a protective measure, preventing a signal from exceeding the clip
limits.

3.7

Clamper circuits

The circuits in Figure 3.37 are known as clampers or DC restorers. The corresponding netlist
is in Figure 3.38. These circuits clamp a peak of a waveform to a specific DC level compared
with a capacitively coupled signal which swings about its average DC level (usually 0V). If the
diode is removed from the clamper, it defaults to a simple coupling capacitor– no clamping.
What is the clamp voltage? And, which peak gets clamped? In Figure 3.37 (a) the clamp
voltage is 0 V ignoring diode drop, (more exactly 0.7 V with Si diode drop). In Figure 3.38, the
positive peak of V(1) is clamped to the 0 V (0.7 V) clamp level. Why is this? On the first positive
half cycle, the diode conducts charging the capacitor left end to +5 V (4.3 V). This is -5 V (-4.3
V) on the right end at V(1,4). Note the polarity marked on the capacitor in Figure 3.37 (a). The
right end of the capacitor is -5 V DC (-4.3 V) with respect to ground. It also has an AC 5 V peak
sinewave coupled across it from source V(4) to node 1. The sum of the two is a 5 V peak sine
riding on a - 5 V DC (-4.3 V) level. The diode only conducts on successive positive excursions
of source V(4) if the peak V(4) exceeds the charge on the capacitor. This only happens if the
charge on the capacitor drained off due to a load, not shown. The charge on the capacitor is
equal to the positive peak of V(4) (less 0.7 diode drop). The AC riding on the negative end,
right end, is shifted down. The positive peak of the waveform is clamped to 0 V (0.7 V) because
the diode conducts on the positive peak.
Suppose the polarity of the diode is reversed as in Figure 3.37 (b)? The diode conducts on
the negative peak of source V(4). The negative peak is clamped to 0 V (-0.7 V). See V(2) in
Figure 3.38.
The most general realization of the clamper is shown in Figure 3.37 (c) with the diode
connected to a DC reference. The capacitor still charges during the negative peak of the source.
Note that the polarities of the AC source and the DC reference are series aiding. Thus, the
capacitor charges to the sum to the two, 10 V DC (9.3 V). Coupling the 5 V peak sinewave
across the capacitor yields Figure 3.38 V(3), the sum of the charge on the capacitor and the
sinewave. The negative peak appears to be clamped to 5 V DC (4.3V), the value of the DC
clamp reference (less diode drop).
Describe the waveform if the DC clamp reference is changed from 5 V to 10 V. The clamped
waveform will shift up. The negative peak will be clamped to 10 V (9.3). Suppose that the

CHAPTER 3. DIODES AND RECTIFIERS

122

4

1000 pF
+ - 1

+ 5 Vpeak
0 Voffset
0
- 1 kHz
(a)

-4.3 VDC

-

1000 pF
4
2
- +

4.3 VDC

+0

-

1000 pF
4
3
- +

+ 5V +
−
(b)

(c)

9.3 VDC

0

Figure 3.37: Clampers: (a) Positive peak clamped to 0 V. (b) Negative peak clamped to 0 V. (c)
Negative peak clamped to 5 V.

*SPICE 03443.eps
V1 6 0 5
D1 6 3 diode
C1 4 3 1000p
D2 0 2 diode
C2 4 2 1000p
C3 4 1 1000p
D3 1 0 diode
V2 4 0 SIN(0 5 1k)
.model diode d
.tran 0.01m 5m
.end
Figure 3.38: V(4) source voltage 5 V peak used in all clampers. V(1) clamper output from
Figure 3.37 (a). V(1,4) DC voltage on capacitor in Figure (a). V(2) clamper output from Figure
(b). V(3) clamper output from Figure (c).

3.8. VOLTAGE MULTIPLIERS

123

amplitude of the sine wave source is increased from 5 V to 7 V? The negative peak clamp level
will remain unchanged. Though, the amplitude of the sinewave output will increase.
An application of the clamper circuit is as a “DC restorer” in “composite video” circuitry in
both television transmitters and receivers. An NTSC (US video standard) video signal “white
level” corresponds to minimum (12.5%) transmitted power. The video “black level” corresponds
to a high level (75% of transmitter power. There is a “blacker than black level” corresponding to
100% transmitted power assigned to synchronization signals. The NTSC signal contains both
video and synchronization pulses. The problem with the composite video is that its average
DC level varies with the scene, dark vs light. The video itself is supposed to vary. However,
the sync must always peak at 100%. To prevent the sync signals from drifting with changing
scenes, a “DC restorer” clamps the top of the sync pulses to a voltage corresponding to 100%
transmitter modulation. [2]
• REVIEW:
• A capacitively coupled signal alternates about its average DC level (0 V).
• The signal out of a clamper appears the have one peak clamped to a DC voltage. Example:
The negative peak is clamped to 0 VDC, the waveform appears to be shifted upward. The
polarity of the diode determines which peak is clamped.
• An application of a clamper, or DC restorer, is in clamping the sync pulses of composite
video to a voltage corresponding to 100% of transmitter power.

3.8

Voltage multipliers

A voltage multiplier is a specialized rectifier circuit producing an output which is theoretically
an integer times the AC peak input, for example, 2, 3, or 4 times the AC peak input. Thus,
it is possible to get 200 VDC from a 100 Vpeak AC source using a doubler, 400 VDC from a
quadrupler. Any load in a practical circuit will lower these voltages.
A voltage doubler application is a DC power supply capable of using either a 240 VAC or 120
VAC source. The supply uses a switch selected full-wave bridge to produce about 300 VDC from
a 240 VAC source. The 120 V position of the switch rewires the bridge as a doubler producing
about 300 VDC from the 120 VAC. In both cases, 300 VDC is produced. This is the input to a
switching regulator producing lower voltages for powering, say, a personal computer.
The half-wave voltage doubler in Figure 3.39 (a) is composed of two circuits: a clamper at
(b) and peak detector (half-wave rectifier) in Figure 3.28, which is shown in modified form in
Figure 3.39 (c). C2 has been added to a peak detector (half-wave rectifier).
Referring to Figure 3.39 (b), C2 charges to 5 V (4.3 V considering the diode drop) on the
negative half cycle of AC input. The right end is grounded by the conducting D2. The left end
is charged at the negative peak of the AC input. This is the operation of the clamper.
During the positive half cycle, the half-wave rectifier comes into play at Figure 3.39 (c).
Diode D2 is out of the circuit since it is reverse biased. C2 is now in series with the voltage
source. Note the polarities of the generator and C2, series aiding. Thus, rectifier D1 sees a total
of 10 V at the peak of the sinewave, 5 V from generator and 5 V from C2. D1 conducts waveform
v(1) (Figure 3.40), charging C1 to the peak of the sine wave riding on 5 V DC (Figure 3.40 v(2)).

CHAPTER 3. DIODES AND RECTIFIERS

124

4

1000 pF

1

D1

C2

D2
5 Vp-p
1000 pF
0 Voffset
1 kHz 0

(a)

5V

2
C1

-

-

C2
D2

+
(b)

5V

+

+

+

D1

C1

+

C2
5V

-

10V

-

(c)

Figure 3.39: Half-wave voltage doubler (a) is composed of (b) a clamper and (c) a half-wave
rectifier.
Waveform v(2) is the output of the doubler, which stabilizes at 10 V (8.6 V with diode drops)
after a few cycles of sinewave input.

*SPICE 03255.eps
C1 2 0 1000p
D1 1 2 diode
C2 4 1 1000p
D2 0 1 diode
V1 4 0 SIN(0 5 1k)
.model diode d
.tran 0.01m 5m
.end
Figure 3.40: Voltage doubler: v(4) input. v(1) clamper stage. v(2) half-wave rectifier stage,
which is the doubler output.
The full-wave voltage doubler is composed of a pair of series stacked half-wave rectifiers.
(Figure 3.41) The corresponding netlist is in Figure 3.41. The bottom rectifier charges C1 on
the negative half cycle of input. The top rectifier charges C2 on the positive halfcycle. Each
capacitor takes on a charge of 5 V (4.3 V considering diode drop). The output at node 5 is the
series total of C1 + C2 or 10 V (8.6 V with diode drops).
Note that the output v(5) Figure 3.42 reaches full value within one cycle of the input v(2)
excursion.
Figure 3.43 illustrates the derivation of the full-wave doubler from a pair of opposite polarity half-wave rectifiers (a). The negative rectifier of the pair is redrawn for clarity (b). Both
are combined at (c) sharing the same ground. At (d) the negative rectifier is re-wired to share

3.8. VOLTAGE MULTIPLIERS

2
5 Vp-p
0 Voffset
1 kHz

125

D2

5
1000 pF

C2

3
1000 pF

C1

0
D1

*SPICE 03273.eps
*R1 3 0 100k
*R2 5 3 100k
D1 0 2 diode
D2 2 5 diode
C1 3 0 1000p
C2 5 3 1000p
V1 2 3 SIN(0 5 1k)
.model diode d
.tran 0.01m 5m
.end

Figure 3.41: Full-wave voltage doubler consists of two half-wave rectifiers operating on alternating polarities.

Figure 3.42: Full-wave voltage doubler: v(2) input, v(3)voltage at mid point, v(5) voltage at
output

CHAPTER 3. DIODES AND RECTIFIERS

126

one voltage source with the positive rectifier. This yields a ±5 V (4.3 V with diode drop) power
supply; though, 10 V is measurable between the two outputs. The ground reference point is
moved so that +10 V is available with respect to ground.
+5V

+5V

+5V

+10V

+5V

+
+5V

+
-

-5V
-5V

-5V

-5V

(a)

(b)

(c)

(d)

(e)

Figure 3.43: Full-wave doubler: (a) Pair of doublers, (b) redrawn, (c) sharing the ground, (d)
share the same voltage source. (e) move the ground point.
A voltage tripler (Figure 3.44) is built from a combination of a doubler and a half wave
rectifier (C3, D3). The half-wave rectifier produces 5 V (4.3 V) at node 3. The doubler provides
another 10 V (8.4 V) between nodes 2 and 3. for a total of 15 V (12.9 V) at the output node 2
with respect to ground. The netlist is in Figure 3.45.

1000 pF

4
5 Vp-p
0 Voffset
1 kHz
(a)

C2

Doubler

1

2
D1

D2

3

C1
10V

1000 pF

15V
C3

Single stage rectifier

1000 pF
D3

0

5V

Figure 3.44: Voltage tripler composed of doubler stacked atop a single stage rectifier.
Note that V(3) in Figure 3.45 rises to 5 V (4.3 V) on the first negative half cycle. Input v(4)
is shifted upward by 5 V (4.3 V) due to 5 V from the half-wave rectifier. And 5 V more at v(1)
due to the clamper (C2, D2). D1 charges C1 (waveform v(2)) to the peak value of v(1).
A voltage quadrupler is a stacked combination of two doublers shown in Figure 3.46. Each
doubler provides 10 V (8.6 V) for a series total at node 2 with respect to ground of 20 V (17.2
V). The netlist is in Figure 3.47.
The waveforms of the quadrupler are shown in Figure 3.47. Two DC outputs are available:
v(3), the doubler output, and v(2) the quadrupler output. Some of the intermediate voltages

3.8. VOLTAGE MULTIPLIERS

127

*SPICE 03283.eps
C3 3 0 1000p
D3 0 4 diode
C1 2 3 1000p
D1 1 2 diode
C2 4 1 1000p
D2 3 1 diode
V1 4 3 SIN(0 5 1k)
.model diode d
.tran 0.01m 5m
.end
Figure 3.45: Voltage tripler: v(3) half-wave rectifier, v(4) input+ 5 V, v(1) clamper, v(2) final
output.

1000 pF

4
5 Vp-p
0 Voffset
1 kHz

D1

1

2

C2
C1
D2

doubler 1
doubler 2
1000 pF
C22

1000 pF
3

10V
20V

C11
D22

5

1000 pF
D11

0

10V

Figure 3.46: Voltage quadrupler, composed of two doublers stacked in series, with output at
node 2.

128

CHAPTER 3. DIODES AND RECTIFIERS

at clampers illustrate that the input sinewave (not shown), which swings by ¡plusminus)¿5 V,
is successively clamped at higher levels: at v(5), v(4) and v(1). Strictly v(4) is not a clamper
output. It is simply the AC voltage source in series with the v(3) the doubler output. None the
less, v(1) is a clamped version of v(4)

*SPICE 03441.eps
*SPICE 03286.eps
C22 4 5 1000p
C11 3 0 1000p
D11 0 5 diode
D22 5 3 diode
C1 2 3 1000p
D1 1 2 diode
C2 4 1 1000p
D2 3 1 diode
V1 4 3 SIN(0 5 1k)
.model diode d
.tran 0.01m 5m
.end
Figure 3.47: Voltage quadrupler: DC voltage available at v(3) and v(2). Intermediate waveforms: Clampers: v(5), v(4), v(1).
Some notes on voltage multipliers are in order at this point. The circuit parameters used
in the examples (V= 5 V 1 kHz, C=1000 pf) do not provide much current, microamps. Furthermore, load resistors have been omitted. Loading reduces the voltages from those shown. If the
circuits are to be driven by a kHz source at low voltage, as in the examples, the capacitors are
usually 0.1 to 1.0 µF so that milliamps of current are available at the output. If the multipliers
are driven from 50/60 Hz, the capacitor are a few hundred to a few thousand microfarads to
provide hundreds of milliamps of output current. If driven from line voltage, pay attention to
the polarity and voltage ratings of the capacitors.
Finally, any direct line driven power supply (no transformer) is dangerous to the experimenter and line operated test equipment. Commercial direct driven supplies are safe because
the hazardous circuitry is in an enclosure to protect the user. When breadboarding these circuits with electrolytic capacitors of any voltage, the capacitors will explode if the polarity is
reversed. Such circuits should be powered up behind a safety shield.
A voltage multiplier of cascaded half-wave doublers of arbitrary length is known as a
Cockcroft-Walton multiplier as shown in Figure 3.48. This multiplier is used when a high
voltage at low current is required. The advantage over a conventional supply is that an expensive high voltage transformer is not required– at least not as high as the output.
The pair of diodes and capacitors to the left of nodes 1 and 2 in Figure 3.48 constitute a
half-wave doubler. Rotating the diodes by 45o counterclockwise, and the bottom capacitor by
90o makes it look like Figure 3.39 (a). Four of the doubler sections are cascaded to the right for

3.8. VOLTAGE MULTIPLIERS

1000 pF
99
1
5 Vp-p
0 Voffset
1 kHz
1000 pF

129

1000 pF

3

1000 pF

5

4

2
1000 pF

1000 pF

7

6
1000 pF

8
1000 pF

Figure 3.48: Cockcroft-Walton x8 voltage multiplier; output at v(8).
a theoretical x8 multiplication factor. Node 1 has a clamper waveform (not shown), a sinewave
shifted up by 1x (5 V). The other odd numbered nodes are sinewaves clamped to successively
higher voltages. Node 2, the output of the first doubler, is a 2x DC voltage v(2) in Figure 3.49.
Successive even numbered nodes charge to successively higher voltages: v(4), v(6), v(8)
D1 7 8 diode
C1 8 6 1000p
D2 6 7 diode
C2 5 7 1000p
D3 5 6 diode
C3 4 6 1000p
D4 4 5 diode
C4 3 5 1000p
D5 3 4 diode
C5 2 4 1000p
D6 2 3 diode
D7 1 2 diode
C6 1 3 1000p
C7 2 0 1000p
C8 99 1 1000p
D8 0 1 diode
V1 99 0 SIN(0 5
1k)
.model diode d
.tran 0.01m 50m
.end
Figure 3.49: Cockcroft-Walton (x8) waveforms. Output is v(8).
Without diode drops, each doubler yields 2Vin or 10 V, considering two diode drops (101.4)=8.6 V is realistic. For a total of 4 doublers one expects 4·8.6=34.4 V out of 40 V. Consulting
Figure 3.49, v(2) is about right;however, v(8) is <30 V instead of the anticipated 34.4 V. The
bane of the Cockcroft-Walton multiplier is that each additional stage adds less than the previous stage. Thus, a practical limit to the number of stages exist. It is possible to overcome this
limitation with a modification to the basic circuit. [3] Also note the time scale of 40 msec com-

CHAPTER 3. DIODES AND RECTIFIERS

130

pared with 5 ms for previous circuits. It required 40 msec for the voltages to rise to a terminal
value for this circuit. The netlist in Figure 3.49 has a “.tran 0.010m 50m” command to extend
the simulation time to 50 msec; though, only 40 msec is plotted.
The Cockcroft-Walton multiplier serves as a more efficient high voltage source for photomultiplier tubes requiring up to 2000 V. [3] Moreover, the tube has numerous dynodes, terminals requiring connection to the lower voltage “even numbered” nodes. The series string of
multiplier taps replaces a heat generating resistive voltage divider of previous designs.
An AC line operated Cockcroft-Walton multiplier provides high voltage to “ion generators”
for neutralizing electrostatic charge and for air purifiers.
• REVIEW:
• A voltage multiplier produces a DC multiple (2,3,4, etc) of the AC peak input voltage.
• The most basic multiplier is a half-wave doubler.
• The full-wave double is a superior circuit as a doubler.
• A tripler is a half-wave doubler and a conventional rectifier stage (peak detector).
• A quadrupler is a pair of half-wave doublers
• A long string of half-wave doublers is known as a Cockcroft-Walton multiplier.

3.9

Inductor commutating circuits

A popular use of diodes is for the mitigation of inductive “kickback:” the pulses of high voltage
produced when direct current through an inductor is interrupted. Take, for example, this
simple circuit in Figure 3.50 with no protection against inductive kickback.
V

+

+

+

+

−

−

−

−

−

+

(a)

(b)

(c)

off
(d)

on
off

Figure 3.50: Inductive kickback: (a) Switch open. (b) Switch closed, electron current flows
from battery through coil which has polarity matching battery. Magnetic field stores energy.
(c) Switch open, Current still flows in coil due to collapsing magnetic field. Note polarity change
on coil. (d) Coil voltage vs time.
When the pushbutton switch is actuated, current goes through the inductor, producing
a magnetic field around it. When the switch is de-actuated, its contacts open, interrupting
current through the inductor, and causing the magnetic field to rapidly collapse. Because the
voltage induced in a coil of wire is directly proportional to the rate of change over time of
magnetic flux (Faraday’s Law: e = NdΦ/dt), this rapid collapse of magnetism around the coil
produces a high voltage “spike”.

3.9. INDUCTOR COMMUTATING CIRCUITS

131

If the inductor in question is an electromagnet coil, such as in a solenoid or relay (constructed for the purpose of creating a physical force via its magnetic field when energized), the
effect of inductive “kickback” serves no useful purpose at all. In fact, it is quite detrimental
to the switch, as it causes excessive arcing at the contacts, greatly reducing their service life.
Of the practical methods for mitigating the high voltage transient created when the switch is
opened, none so simple as the so-called commutating diode in Figure 3.51.

+

+

+

-

-

-

(a)

(b)

+
-

+

-

-

+

+

(c)

Figure 3.51: Inductive kickback with protection: (a) Switch open. (b)Switch closed, storing
energy in magnetic field. (c) Switch open, inductive kickback is shorted by diode.
In this circuit, the diode is placed in parallel with the coil, such that it will be reverse-biased
when DC voltage is applied to the coil through the switch. Thus, when the coil is energized,
the diode conducts no current in Figure 3.51 (b).
However, when the switch is opened, the coil’s inductance responds to the decrease in current by inducing a voltage of reverse polarity, in an effort to maintain current at the same
magnitude and in the same direction. This sudden reversal of voltage polarity across the coil
forward-biases the diode, and the diode provides a current path for the inductor’s current, so
that its stored energy is dissipated slowly rather than suddenly in Figure 3.51 (c).
As a result, the voltage induced in the coil by its collapsing magnetic field is quite low:
merely the forward voltage drop of the diode, rather than hundreds of volts as before. Thus,
the switch contacts experience a voltage drop equal to the battery voltage plus about 0.7 volts
(if the diode is silicon) during this discharge time.
In electronics parlance, commutation refers to the reversal of voltage polarity or current direction. Thus, the purpose of a commutating diode is to act whenever voltage reverses polarity,
for example, on an inductor coil when current through it is interrupted. A less formal term for
a commutating diode is snubber, because it “snubs” or “squelches” the inductive kickback.
A noteworthy disadvantage of this method is the extra time it imparts to the coil’s discharge.
Because the induced voltage is clamped to a very low value, its rate of magnetic flux change
over time is comparatively slow. Remember that Faraday’s Law describes the magnetic flux
rate-of-change (dΦ/dt) as being proportional to the induced, instantaneous voltage (e or v). If
the instantaneous voltage is limited to some low figure, then the rate of change of magnetic
flux over time will likewise be limited to a low (slow) figure.
If an electromagnet coil is “snubbed” with a commutating diode, the magnetic field will dissipate at a relatively slow rate compared to the original scenario (no diode) where the field
disappeared almost instantly upon switch release. The amount of time in question will most
likely be less than one second, but it will be measurably slower than without a commutating diode in place. This may be an intolerable consequence if the coil is used to actuate an

CHAPTER 3. DIODES AND RECTIFIERS

132

electromechanical relay, because the relay will possess a natural “time delay” upon coil deenergization, and an unwanted delay of even a fraction of a second may wreak havoc in some
circuits.
Unfortunately, one cannot eliminate the high-voltage transient of inductive kickback and
maintain fast de-magnetization of the coil: Faraday’s Law will not be violated. However, if slow
de-magnetization is unacceptable, a compromise may be struck between transient voltage and
time by allowing the coil’s voltage to rise to some higher level (but not so high as without a
commutating diode in place). The schematic in Figure 3.52 shows how this can be done.
V
off

(d)
(e)

on

(c)
(a)

(b)

off

Figure 3.52: (a) Commutating diode with series resistor. (b) Voltage waveform. (c) Level with
no diode. (d) Level with diode, no resistor. (e) Compromise level with diode and resistor.
A resistor placed in series with the commutating diode allows the coil’s induced voltage to
rise to a level greater than the diode’s forward voltage drop, thus hastening the process of demagnetization. This, of course, will place the switch contacts under greater stress, and so the
resistor must be sized to limit that transient voltage at an acceptable maximum level.

3.10

Diode switching circuits

Diodes can perform switching and digital logic operations. Forward and reverse bias switch a
diode between the low and high impedance states, respectively. Thus, it serves as a switch.

3.10.1

Logic

Diodes can perform digital logic functions: AND, and OR. Diode logic was used in early digital
computers. It only finds limited application today. Sometimes it is convenient to fashion a
single logic gate from a few diodes.
An AND gate is shown in Figure 3.53. Logic gates have inputs and an output (Y) which is
a function of the inputs. The inputs to the gate are high (logic 1), say 10 V, or low, 0 V (logic
0). In the figure, the logic levels are generated by switches. If a switch is up, the input is
effectively high (1). If the switch is down, it connects a diode cathode to ground, which is low
(0). The output depends on the combination of inputs at A and B. The inputs and output are
customarily recorded in a “truth table” at (c) to describe the logic of a gate. At (a) all inputs
are high (1). This is recorded in the last line of the truth table at (c). The output, Y, is high
(1) due to the V+ on the top of the resistor. It is unaffected by open switches. At (b) switch A
pulls the cathode of the connected diode low, pulling output Y low (0.7 V). This is recorded in

3.10. DIODE SWITCHING CIRCUITS

133

V+

1

A

1

B

Y=1

V+

0

A

1

B

(a)

A B Y
0 0 0
0 1 0
1 0 0
1 1 1

Y=0

(b)

(c)

Figure 3.53: Diode AND gate
the third line of the truth table. The second line of the truth table describes the output with
the switches reversed from (b). Switch B pulls the diode and output low. The first line of the
truth table recordes the Output=0 for both input low (0). The truth table describes a logical
AND function. Summary: both inputs A and B high yields a high (1) out.
A two input OR gate composed of a pair of diodes is shown in Figure ??. If both inputs are
logic low at (a) as simulated by both switches “downward,” the output Y is pulled low by the
resistor. This logic zero is recorded in the first line of the truth table at (c). If one of the inputs
is high as at (b), or the other input is high, or both inputs high, the diode(s) conduct(s), pulling
the output Y high. These results are reordered in the second through fourth lines of the truth
table. Summary: any input “high” is a high out at Y.
V+
0

A

0

B

V+
Y=0

1

A

0

B

Y=1

line
operated
power
supply

A B Y
0 0 0
0 1 1
1 0 1
1 1 1

+
backup
battery

(a)

(b)

+

load

(d)

(c)
+

Figure 3.54: OR gate: (a) First line, truth table (TT). (b) Third line TT. (d) Logical OR of power
line supply and back-up battery.
A backup battery may be OR-wired with a line operated DC power supply in Figure 3.54 (d)
to power a load, even during a power failure. With AC power present, the line supply powers
the load, assuming that it is a higher voltage than the battery. In the event of a power failure,
the line supply voltage drops to 0 V; the battery powers the load. The diodes must be in series
with the power sources to prevent a failed line supply from draining the battery, and to prevent
it from over charging the battery when line power is available. Does your PC computer retain
its BIOS setting when powered off? Does your VCR (video cassette recorder) retain the clock
setting after a power failure? (PC Yes, old VCR no, new VCR yes.)

CHAPTER 3. DIODES AND RECTIFIERS

134

3.10.2

Analog switch

Diodes can switch analog signals. A reverse biased diode appears to be an open circuit. A
forward biased diode is a low resistance conductor. The only problem is isolating the AC signal
being switched from the DC control signal. The circuit in Figure 3.55 is a parallel resonant
network: resonant tuning inductor paralleled by one (or more) of the switched resonator capacitors. This parallel LC resonant circuit could be a preselector filter for a radio receiver. It
could be the frequency determining network of an oscillator (not shown). The digital control
lines may be driven by a microprocessor interface.

decoupling
capacitor

RFC

RFC

RFC

switched
resonator
capacitor

RFC

switching
diode

resonant
tuning
inductor
large value
DC blocking
capacitor

+5V
digital control

Figure 3.55: Diode switch: A digital control signal (low) selects a resonator capacitor by forward
biasing the switching diode.

The large value DC blocking capacitor grounds the resonant tuning inductor for AC while
blocking DC. It would have a low reactance compared to the parallel LC reactances. This
prevents the anode DC voltage from being shorted to ground by the resonant tuning inductor.
A switched resonator capacitor is selected by pulling the corresponding digital control low. This
forward biases the switching diode. The DC current path is from +5 V through an RF choke
(RFC), a switching diode, and an RFC to ground via the digital control. The purpose of the RFC
at the +5 V is to keep AC out of the +5 V supply. The RFC in series with the digital control is to
keep AC out of the external control line. The decoupling capacitor shorts the little AC leaking
through the RFC to ground, bypassing the external digital control line.
With all three digital control lines high (≥+5 V), no switched resonator capacitors are selected due to diode reverse bias. Pulling one or more lines low, selects one or more switched
resonator capacitors, respectively. As more capacitors are switched in parallel with the resonant tuning inductor, the resonant frequency decreases.
The reverse biased diode capacitance may be substantial compared with very high frequency or ultra high frequency circuits. PIN diodes may be used as switches for lower capacitance.

3.11. ZENER DIODES

3.11

135

Zener diodes

If we connect a diode and resistor in series with a DC voltage source so that the diode is
forward-biased, the voltage drop across the diode will remain fairly constant over a wide range
of power supply voltages as in Figure 3.56 (a).
According to the “diode equation” (page 101), the current through a forward-biased PN
junction is proportional to e raised to the power of the forward voltage drop. Because this is an
exponential function, current rises quite rapidly for modest increases in voltage drop. Another
way of considering this is to say that voltage dropped across a forward-biased diode changes
little for large variations in diode current. In the circuit shown in Figure 3.56 (a), diode current
is limited by the voltage of the power supply, the series resistor, and the diode’s voltage drop,
which as we know doesn’t vary much from 0.7 volts. If the power supply voltage were to be
increased, the resistor’s voltage drop would increase almost the same amount, and the diode’s
voltage drop just a little. Conversely, a decrease in power supply voltage would result in an
almost equal decrease in resistor voltage drop, with just a little decrease in diode voltage drop.
In a word, we could summarize this behavior by saying that the diode is regulating the voltage
drop at approximately 0.7 volts.
Voltage regulation is a useful diode property to exploit. Suppose we were building some
kind of circuit which could not tolerate variations in power supply voltage, but needed to be
powered by a chemical battery, whose voltage changes over its lifetime. We could form a circuit
as shown and connect the circuit requiring steady voltage across the diode, where it would
receive an unchanging 0.7 volts.
This would certainly work, but most practical circuits of any kind require a power supply
voltage in excess of 0.7 volts to properly function. One way we could increase our voltage
regulation point would be to connect multiple diodes in series, so that their individual forward
voltage drops of 0.7 volts each would add to create a larger total. For instance, if we had ten
diodes in series, the regulated voltage would be ten times 0.7, or 7 volts in Figure 3.56 (b).

≈ 0.7 V

(a)

≈≈ 7.0 V

(b)

Figure 3.56: Forward biased Si reference: (a) single diode, 0.7V, (b) 10-diodes in series 7.0V.
So long as the battery voltage never sagged below 7 volts, there would always be about 7
volts dropped across the ten-diode “stack.”
If larger regulated voltages are required, we could either use more diodes in series (an inelegant option, in my opinion), or try a fundamentally different approach. We know that diode
forward voltage is a fairly constant figure under a wide range of conditions, but so is reverse
breakdown voltage, and breakdown voltage is typically much, much greater than forward voltage. If we reversed the polarity of the diode in our single-diode regulator circuit and increased
the power supply voltage to the point where the diode “broke down” (could no longer withstand

CHAPTER 3. DIODES AND RECTIFIERS

136

the reverse-bias voltage impressed across it), the diode would similarly regulate the voltage at
that breakdown point, not allowing it to increase further as in Figure 3.57 (a).
≈ 50 V

Zener diode

+

Cathode

150 V
(a)

≈ 100 V

-

(b)

Anode

Figure 3.57: (a) Reverse biased Si small-signal diode breaks down at about 100V. (b) Symbol
for Zener diode.
Unfortunately, when normal rectifying diodes “break down,” they usually do so destructively. However, it is possible to build a special type of diode that can handle breakdown
without failing completely. This type of diode is called a zener diode, and its symbol looks like
Figure 3.57 (b).
When forward-biased, zener diodes behave much the same as standard rectifying diodes:
they have a forward voltage drop which follows the “diode equation” and is about 0.7 volts.
In reverse-bias mode, they do not conduct until the applied voltage reaches or exceeds the socalled zener voltage, at which point the diode is able to conduct substantial current, and in
doing so will try to limit the voltage dropped across it to that zener voltage point. So long as
the power dissipated by this reverse current does not exceed the diode’s thermal limits, the
diode will not be harmed.
Zener diodes are manufactured with zener voltages ranging anywhere from a few volts to
hundreds of volts. This zener voltage changes slightly with temperature, and like common
carbon-composition resistor values, may be anywhere from 5 percent to 10 percent in error
from the manufacturer’s specifications. However, this stability and accuracy is generally good
enough for the zener diode to be used as a voltage regulator device in common power supply
circuit in Figure 3.58.

+
≈ 12.6 V

Figure 3.58: Zener diode regulator circuit, Zener voltage = 12.6V).
Please take note of the zener diode’s orientation in the above circuit: the diode is reversebiased, and intentionally so. If we had oriented the diode in the “normal” way, so as to be
forward-biased, it would only drop 0.7 volts, just like a regular rectifying diode. If we want
to exploit this diode’s reverse breakdown properties, we must operate it in its reverse-bias
mode. So long as the power supply voltage remains above the zener voltage (12.6 volts, in this
example), the voltage dropped across the zener diode will remain at approximately 12.6 volts.

3.11. ZENER DIODES

137

Like any semiconductor device, the zener diode is sensitive to temperature. Excessive temperature will destroy a zener diode, and because it both drops voltage and conducts current,
it produces its own heat in accordance with Joule’s Law (P=IE). Therefore, one must be careful to design the regulator circuit in such a way that the diode’s power dissipation rating is
not exceeded. Interestingly enough, when zener diodes fail due to excessive power dissipation,
they usually fail shorted rather than open. A diode failed in this manner is readily detected: it
drops almost zero voltage when biased either way, like a piece of wire.
Let’s examine a zener diode regulating circuit mathematically, determining all voltages,
currents, and power dissipations. Taking the same form of circuit shown earlier, we’ll perform
calculations assuming a zener voltage of 12.6 volts, a power supply voltage of 45 volts, and a
series resistor value of 1000 Ω (we’ll regard the zener voltage to be exactly 12.6 volts so as to
avoid having to qualify all figures as “approximate” in Figure 3.59 (a)
If the zener diode’s voltage is 12.6 volts and the power supply’s voltage is 45 volts, there will
be 32.4 volts dropped across the resistor (45 volts - 12.6 volts = 32.4 volts). 32.4 volts dropped
across 1000 Ω gives 32.4 mA of current in the circuit. (Figure 3.59 (b))
32.4 V
1 kΩ

1 kΩ

+
45 V

+
12.6 V

(a)

32.4 mA

45 V

12.6 V

(b)

32.4 mA

Figure 3.59: (a) Zener Voltage regulator with 1000 Ω resistor. (b) Calculation of voltage drops
and current.
Power is calculated by multiplying current by voltage (P=IE), so we can calculate power
dissipations for both the resistor and the zener diode quite easily:

Presistor = (32.4 mA)(32.4 V)
Presistor = 1.0498 W
Pdiode = (32.4 mA)(12.6 V)
Pdiode = 408.24 mW
A zener diode with a power rating of 0.5 watt would be adequate, as would a resistor rated
for 1.5 or 2 watts of dissipation.
If excessive power dissipation is detrimental, then why not design the circuit for the least
amount of dissipation possible? Why not just size the resistor for a very high value of resistance, thus severely limiting current and keeping power dissipation figures very low? Take this
circuit, for example, with a 100 kΩ resistor instead of a 1 kΩ resistor. Note that both the power
supply voltage and the diode’s zener voltage in Figure 3.60 are identical to the last example:

CHAPTER 3. DIODES AND RECTIFIERS

138

32.4 V
100 kΩ

+

324 µA

-

324 µA

12.6 V

45 V

Figure 3.60: Zener regulator with 100 kΩ resistor.
With only 1/100 of the current we had before (324 µA instead of 32.4 mA), both power
dissipation figures should be 100 times smaller:

Presistor = (324 µA)(32.4 V)
Presistor = 10.498 mW
Pdiode = (324 µA)(12.6 V)
Pdiode = 4.0824 mW
Seems ideal, doesn’t it? Less power dissipation means lower operating temperatures for
both the diode and the resistor, and also less wasted energy in the system, right? A higher
resistance value does reduce power dissipation levels in the circuit, but it unfortunately introduces another problem. Remember that the purpose of a regulator circuit is to provide a stable
voltage for another circuit. In other words, we’re eventually going to power something with
12.6 volts, and this something will have a current draw of its own. Consider our first regulator
circuit, this time with a 500 Ω load connected in parallel with the zener diode in Figure 3.61.
32.4 V
1 kΩ

+

32.4 mA

45 V

-

25.2 mA

7.2 mA
32.4 mA

12.6 V

Rload
500 Ω

25.2 mA

Figure 3.61: Zener regulator with 1000 Ω series resistor and 500 Ω load.
If 12.6 volts is maintained across a 500 Ω load, the load will draw 25.2 mA of current. In
order for the 1 kΩ series “dropping” resistor to drop 32.4 volts (reducing the power supply’s
voltage of 45 volts down to 12.6 across the zener), it still must conduct 32.4 mA of current.
This leaves 7.2 mA of current through the zener diode.
Now consider our “power-saving” regulator circuit with the 100 kΩ dropping resistor, deliv-

3.11. ZENER DIODES

139

ering power to the same 500 Ω load. What it is supposed to do is maintain 12.6 volts across
the load, just like the last circuit. However, as we will see, it cannot accomplish this task.
(Figure 3.62)
44.776 V
447.76 µA

100 kΩ
447.76 µA

0 µA
224 mV

45 V
447.76 µA

Rload
500 Ω

447.76 µA

Figure 3.62: Zener non-regulator with 100 KΩ series resistor with 500 Ω load.¿
With the larger value of dropping resistor in place, there will only be about 224 mV of
voltage across the 500 Ω load, far less than the expected value of 12.6 volts! Why is this? If we
actually had 12.6 volts across the load, it would draw 25.2 mA of current, as before. This load
current would have to go through the series dropping resistor as it did before, but with a new
(much larger!) dropping resistor in place, the voltage dropped across that resistor with 25.2
mA of current going through it would be 2,520 volts! Since we obviously don’t have that much
voltage supplied by the battery, this cannot happen.
The situation is easier to comprehend if we temporarily remove the zener diode from the
circuit and analyze the behavior of the two resistors alone in Figure 3.63.
44.776 V
100 kΩ

447.76 µA

447.76 µA
224 mV

45 V
447.76 µA

Rload
500 Ω

447.76 µA

Figure 3.63: Non-regulator with Zener removed.
Both the 100 kΩ dropping resistor and the 500 Ω load resistance are in series with each
other, giving a total circuit resistance of 100.5 kΩ. With a total voltage of 45 volts and a total
resistance of 100.5 kΩ, Ohm’s Law (I=E/R) tells us that the current will be 447.76 µA. Figuring
voltage drops across both resistors (E=IR), we arrive at 44.776 volts and 224 mV, respectively.
If we were to re-install the zener diode at this point, it would “see” 224 mV across it as well,
being in parallel with the load resistance. This is far below the zener breakdown voltage of the
diode and so it will not “break down” and conduct current. For that matter, at this low voltage
the diode wouldn’t conduct even if it were forward-biased! Thus, the diode ceases to regulate
voltage. At least 12.6 volts must be dropped across to “activate” it.

CHAPTER 3. DIODES AND RECTIFIERS

140

The analytical technique of removing a zener diode from a circuit and seeing whether or
not enough voltage is present to make it conduct is a sound one. Just because a zener diode
happens to be connected in a circuit doesn’t guarantee that the full zener voltage will always
be dropped across it! Remember that zener diodes work by limiting voltage to some maximum
level; they cannot make up for a lack of voltage.
In summary, any zener diode regulating circuit will function so long as the load’s resistance
is equal to or greater than some minimum value. If the load resistance is too low, it will
draw too much current, dropping too much voltage across the series dropping resistor, leaving
insufficient voltage across the zener diode to make it conduct. When the zener diode stops
conducting current, it can no longer regulate voltage, and the load voltage will fall below the
regulation point.
Our regulator circuit with the 100 kΩ dropping resistor must be good for some value of
load resistance, though. To find this acceptable load resistance value, we can use a table to
calculate resistance in the two-resistor series circuit (no diode), inserting the known values of
total voltage and dropping resistor resistance, and calculating for an expected load voltage of
12.6 volts:

Rdropping
E

Rload

Total

12.6

45

I
R

Volts
Amps
Ohms

100 k

With 45 volts of total voltage and 12.6 volts across the load, we should have 32.4 volts across
Rdropping :

E

Rdropping

Rload

Total

32.4

12.6

45

I
R

Volts
Amps
Ohms

100 k

With 32.4 volts across the dropping resistor, and 100 kΩ worth of resistance in it, the current
through it will be 324 µA:

Rdropping

Rload

Total

E

32.4

12.6

45

I

324 µ

Amps

R

100 k

Ohms

Ohm’s Law
I= E
R

Volts

3.11. ZENER DIODES

141

Being a series circuit, the current is equal through all components at any given time:

Rdropping

Rload

Total

E

32.4

12.6

45

Volts

I

324 µ

324 µ

324 µ

Amps

R

100 k

Ohms

Rule of series circuits:
ITotal = I1 = I2 = . . . In
Calculating load resistance is now a simple matter of Ohm’s Law (R = E/I), giving us 38.889
kΩ:

Rdropping

Rload

Total

E

32.4

12.6

45

Volts

I

324 µ

324 µ

324 µ

Amps

R

100 k

38.889 k

Ohms

Ohm’s Law
R= E
I
Thus, if the load resistance is exactly 38.889 kΩ, there will be 12.6 volts across it, diode or
no diode. Any load resistance smaller than 38.889 kΩ will result in a load voltage less than
12.6 volts, diode or no diode. With the diode in place, the load voltage will be regulated to a
maximum of 12.6 volts for any load resistance greater than 38.889 kΩ.
With the original value of 1 kΩ for the dropping resistor, our regulator circuit was able
to adequately regulate voltage even for a load resistance as low as 500 Ω. What we see is a
tradeoff between power dissipation and acceptable load resistance. The higher-value dropping
resistor gave us less power dissipation, at the expense of raising the acceptable minimum load
resistance value. If we wish to regulate voltage for low-value load resistances, the circuit must
be prepared to handle higher power dissipation.
Zener diodes regulate voltage by acting as complementary loads, drawing more or less current as necessary to ensure a constant voltage drop across the load. This is analogous to
regulating the speed of an automobile by braking rather than by varying the throttle position:
not only is it wasteful, but the brakes must be built to handle all the engine’s power when
the driving conditions don’t demand it. Despite this fundamental inefficiency of design, zener
diode regulator circuits are widely employed due to their sheer simplicity. In high-power applications where the inefficiencies would be unacceptable, other voltage-regulating techniques
are applied. But even then, small zener-based circuits are often used to provide a “reference”
voltage to drive a more efficient amplifier circuit controlling the main power.
Zener diodes are manufactured in standard voltage ratings listed in Table 3.1. The table
“Common zener diode voltages” lists common voltages for 0.3W and 1.3W parts. The wattage

CHAPTER 3. DIODES AND RECTIFIERS

142

corresponds to die and package size, and is the power that the diode may dissipate without
damage.
Table 3.1:
0.5W
2.7V 3.0V
5.1V 5.6V
10V
11V
20V
24V
1.3W
4.7V 5.1V
9.1V 10V
18V
20V
36V
39V
68V
75V

Common zener diode voltages
3.3V
6.2V
12V
27V

3.6V
6.8V
13V
30V

3.9V
7.5V
15V

4.3V
8.2V
16V

4.7V
9.1V
18V

5.6V
11V
22V
43V
100V

6.2V
12V
24V
47V
200V

6.8V
13V
27V
51V

7.5V
15V
30V
56V

8.2V
16V
33V
62V

Zener diode clipper: A clipping circuit which clips the peaks of waveform a approximately
the zener voltage of the diodes. The circuit of Figure 3.64 has two zeners connected series
opposing to symmetrically clip a waveform at nearly the Zener voltage. The resistor limits
current drawn by the zeners to a safe value.

1
1.0 kΩ

20 Vp
0 Voffset
0 1 kHz

2
V(2)
output

*SPICE 03445.eps
D1 4 0 diode
D2 4 2 diode
R1 2 1 1.0k
V1 1 0 SIN(0 20
1k)
.model diode d
bv=10
.tran 0.001m 2m
.end

Figure 3.64: Zener diode clipper:
The zener breakdown voltage for the diodes is set at 10 V by the diode model parameter
“bv=10” in the spice net list in Figure 3.64. This causes the zeners to clip at about 10 V. The
back-to-back diodes clip both peaks. For a positive half-cycle, the top zener is reverse biased,
breaking down at the zener voltage of 10 V. The lower zener drops approximately 0.7 V since
it is forward biased. Thus, a more accurate clipping level is 10+0.7=10.7V. Similar negative
half-cycle clipping occurs a -10.7 V. (Figure 3.65) shows the clipping level at a little over ±10 V.
• REVIEW:
• Zener diodes are designed to be operated in reverse-bias mode, providing a relatively low,
stable breakdown, or zener voltage at which they begin to conduct substantial reverse

3.12. SPECIAL-PURPOSE DIODES

143

Figure 3.65: Zener diode clipper: v(1) input is clipped at waveform v(2).
current.
• A zener diode may function as a voltage regulator by acting as an accessory load, drawing
more current from the source if the voltage is too high, and less if it is too low.

3.12

Special-purpose diodes

3.12.1

Schottky diodes

Schottky diodes are constructed of a metal-to-N junction rather than a P-N semiconductor
junction. Also known as hot-carrier diodes, Schottky diodes are characterized by fast switching
times (low reverse-recovery time), low forward voltage drop (typically 0.25 to 0.4 volts for a
metal-silicon junction), and low junction capacitance.
The schematic symbol for a schottky diode is shown in Figure 3.66.
Anode
Cathode

Figure 3.66: Schottky diode schematic symbol.
The forward voltage drop (VF ), reverse-recovery time (trr ), and junction capacitance (CJ ) of
Schottky diodes are closer to ideal than the average “rectifying” diode. This makes them well
suited for high-frequency applications. Unfortunately, though, Schottky diodes typically have
lower forward current (IF ) and reverse voltage (VRRM and VDC ) ratings than rectifying diodes

CHAPTER 3. DIODES AND RECTIFIERS

144

and are thus unsuitable for applications involving substantial amounts of power. Though they
are used in low voltage switching regulator power supplies.
Schottky diode technology finds broad application in high-speed computer circuits, where
the fast switching time equates to high speed capability, and the low forward voltage drop
equates to less power dissipation when conducting.
Switching regulator power supplies operating at 100’s of kHz cannot use conventional silicon diodes as rectifiers because of their slow switching speed . When the signal applied to a
diode changes from forward to reverse bias, conduction continues for a short time, while carriers are being swept out of the depletion region. Conduction only ceases after this tr reverse
recovery time has expired. Schottky diodes have a shorter reverse recovery time.
Regardless of switching speed, the 0.7 V forward voltage drop of silicon diodes causes poor
efficiency in low voltage supplies. This is not a problem in, say, a 10 V supply. In a 1 V supply
the 0.7 V drop is a substantial portion of the output. One solution is to use a schottky power
diode which has a lower forward drop.

3.12.2

Tunnel diodes

Tunnel diodes exploit a strange quantum phenomenon called resonant tunneling to provide a
negative resistance forward-bias characteristics. When a small forward-bias voltage is applied
across a tunnel diode, it begins to conduct current. (Figure 3.67(b)) As the voltage is increased,
the current increases and reaches a peak value called the peak current (IP ). If the voltage
is increased a little more, the current actually begins to decrease until it reaches a low point
called the valley current (IV ). If the voltage is increased further yet, the current begins to
increase again, this time without decreasing into another “valley.” The schematic symbol for
the tunnel diode shown in Figure 3.67(a).
Tunnel diode
Anode

Forward
current

+
−

IP

Cathode
IV

(a)

(b)

VP

VV

Forward voltage

(c)

Figure 3.67: Tunnel diode (a) Schematic symbol. (b) Current vs voltage plot (c) Oscillator.
The forward voltages necessary to drive a tunnel diode to its peak and valley currents are
known as peak voltage (VP ) and valley voltage (VV ), respectively. The region on the graph
where current is decreasing while applied voltage is increasing (between VP and VV on the
horizontal scale) is known as the region of negative resistance.
Tunnel diodes, also known as Esaki diodes in honor of their Japanese inventor Leo Esaki,
are able to transition between peak and valley current levels very quickly, “switching” between
high and low states of conduction much faster than even Schottky diodes. Tunnel diode characteristics are also relatively unaffected by changes in temperature.

3.12. SPECIAL-PURPOSE DIODES

145

Breakdown voltage (V)

1000

Ge
100

Si

Ga

As

Ga

P

10
ng

eli

n
tun

1
1014

1015
1016
Doping concentration (cm-3)

1017

1018

Figure 3.68: Reverse breakdown voltage versus doping level. After Sze [22]

Tunnel diodes are heavily doped in both the P and N regions, 1000 times the level in a
rectifier. This can be seen in Figure 3.68. Standard diodes are to the far left, zener diodes near
to the left, and tunnel diodes to the right of the dashed line. The heavy doping produces an
unusually thin depletion region. This produces an unusually low reverse breakdown voltage
with high leakage. The thin depletion region causes high capacitance. To overcome this, the
tunnel diode junction area must be tiny. The forward diode characteristic consists of two regions: a normal forward diode characteristic with current rising exponentially beyond VF , 0.3
V for Ge, 0.7 V for Si. Between 0 V and VF is an additional “negative resistance” characteristic
peak. This is due to quantum mechanical tunneling involving the dual particle-wave nature
of electrons. The depletion region is thin enough compared with the equivalent wavelength of
the electron that they can tunnel through. They do not have to overcome the normal forward
diode voltage VF . The energy level of the conduction band of the N-type material overlaps
the level of the valence band in the P-type region. With increasing voltage, tunneling begins;
the levels overlap; current increases, up to a point. As current increases further, the energy
levels overlap less; current decreases with increasing voltage. This is the “negative resistance”
portion of the curve.
Tunnel diodes are not good rectifiers, as they have relatively high “leakage” current when
reverse-biased. Consequently, they find application only in special circuits where their unique
tunnel effect has value. To exploit the tunnel effect, these diodes are maintained at a bias
voltage somewhere between the peak and valley voltage levels, always in a forward-biased
polarity (anode positive, and cathode negative).
Perhaps the most common application of a tunnel diode is in simple high-frequency oscillator circuits as in Figure 3.67(c), where it allows a DC voltage source to contribute power to an
LC “tank” circuit, the diode conducting when the voltage across it reaches the peak (tunnel)
level and effectively insulating at all other voltages. The resistors bias the tunnel diode at a

CHAPTER 3. DIODES AND RECTIFIERS

146

few tenths of a volt centered on the negative resistance portion of the characteristic curve. The
L-C resonant circuit may be a section of waveguide for microwave operation. Oscillation to 5
GHz is possible.
At one time the tunnel diode was the only solid-state microwave amplifier available. Tunnel
diodes were popular starting in the 1960’s. They were longer lived than traveling wave tube
amplifiers, an important consideration in satellite transmitters. Tunnel diodes are also resistant to radiation because of the heavy doping. Today various transistors operate at microwave
frequencies. Even small signal tunnel diodes are expensive and difficult to find today. There is
one remaining manufacturer of germanium tunnel diodes, and none for silicon devices. They
are sometimes used in military equipment because they are insensitive to radiation and large
temperature changes.
There has been some research involving possible integration of silicon tunnel diodes into
CMOS integrated circuits. They are thought to be capable of switching at 100 GHz in digital
circuits. The sole manufacturer of germanium devices produces them one at a time. A batch
process for silicon tunnel diodes must be developed, then integrated with conventional CMOS
processes. [21]
The Esaki tunnel diode should not be confused with the resonant tunneling diode (page
84), of more complex construction from compound semiconductors. The RTD is a more recent
development capable of higher speed.

3.12.3

Light-emitting diodes

Diodes, like all semiconductor devices, are governed by the principles described in quantum
physics. One of these principles is the emission of specific-frequency radiant energy whenever
electrons fall from a higher energy level to a lower energy level. This is the same principle
at work in a neon lamp, the characteristic pink-orange glow of ionized neon due to the specific
energy transitions of its electrons in the midst of an electric current. The unique color of a neon
lamp’s glow is due to the fact that its neon gas inside the tube, and not due to the particular
amount of current through the tube or voltage between the two electrodes. Neon gas glows
pinkish-orange over a wide range of ionizing voltages and currents. Each chemical element
has its own “signature” emission of radiant energy when its electrons “jump” between different,
quantized energy levels. Hydrogen gas, for example, glows red when ionized; mercury vapor
glows blue. This is what makes spectrographic identification of elements possible.
Electrons flowing through a PN junction experience similar transitions in energy level, and
emit radiant energy as they do so. The frequency of this radiant energy is determined by the
crystal structure of the semiconductor material, and the elements comprising it. Some semiconductor junctions, composed of special chemical combinations, emit radiant energy within
the spectrum of visible light as the electrons change energy levels. Simply put, these junctions glow when forward biased. A diode intentionally designed to glow like a lamp is called a
light-emitting diode, or LED.
Forward biased silicon diodes give off heat as electron and holes from the N-type and P-type
regions, respectively, recombine at the junction. In a forward biased LED, the recombination of
electrons and holes in the active region in Figure 3.69 (c) yields photons. This process is known
as electroluminescence. To give off photons, the potential barrier through which the electrons
fall must be higher than for a silicon diode. The forward diode drop can range to a few volts for
some color LEDs.

3.12. SPECIAL-PURPOSE DIODES

147

Diodes made from a combination of the elements gallium, arsenic, and phosphorus (called
gallium-arsenide-phosphide) glow bright red, and are some of the most common LEDs manufactured. By altering the chemical constituency of the PN junction, different colors may be
obtained. Early generations of LEDs were red, green, yellow, orange, and infra-red, later generations included blue and ultraviolet, with violet being the latest color added to the selection.
Other colors may be obtained by combining two or more primary-color (red, green, and blue)
LEDs together in the same package, sharing the same optical lens. This allowed for multicolor
LEDs, such as tricolor LEDs (commercially available in the 1980’s) using red and green (which
can create yellow) and later RGB LEDs (red, green, and blue), which cover the entire color
spectrum.
The schematic symbol for an LED is a regular diode shape inside of a circle, with two small
arrows pointing away (indicating emitted light), shown in Figure 3.69.

+
Anode

long

Cathode
(a)

short
(b)

flat
(c)

p-type
active region
n-type
substrate
electron
hole

−

Figure 3.69: LED, Light Emitting Diode: (a) schematic symbol. (b) Flat side and short lead of
device correspond to cathode. (c) Cross section of Led die.
This notation of having two small arrows pointing away from the device is common to the
schematic symbols of all light-emitting semiconductor devices. Conversely, if a device is lightactivated (meaning that incoming light stimulates it), then the symbol will have two small
arrows pointing toward it. LEDs can sense light. They generate a small voltage when exposed
to light, much like a solar cell on a small scale. This property can be gainfully applied in a
variety of light-sensing circuits.
Because LEDs are made of different chemical substances than silicon diodes, their forward
voltage drops will be different. Typically, LEDs have much larger forward voltage drops than
rectifying diodes, anywhere from about 1.6 volts to over 3 volts, depending on the color. Typical
operating current for a standard-sized LED is around 20 mA. When operating an LED from
a DC voltage source greater than the LED’s forward voltage, a series-connected “dropping”
resistor must be included to prevent full source voltage from damaging the LED. Consider the
example circuit in Figure 3.70 (a) using a 6 V source.
With the LED dropping 1.6 volts, there will be 4.4 volts dropped across the resistor. Sizing
the resistor for an LED current of 20 mA is as simple as taking its voltage drop (4.4 volts) and
dividing by circuit current (20 mA), in accordance with Ohm’s Law (R=E/I). This gives us a
figure of 220 Ω. Calculating power dissipation for this resistor, we take its voltage drop and
multiply by its current (P=IE), and end up with 88 mW, well within the rating of a 1/8 watt
resistor. Higher battery voltages will require larger-value dropping resistors, and possibly
higher-power rating resistors as well. Consider the example in Figure ?? (b) for a supply

CHAPTER 3. DIODES AND RECTIFIERS

148
Rdropping

+

220 Ω

Rdropping
Red LED,

+
24 V

VF = 1.6 V typical

6V

−

IF = 20 mA typical
(a)

1.12 kΩ

−
(b)

Figure 3.70: Setting LED current at 20 ma. (a) for a 6 V source, (b) for a 24 V source.
voltage of 24 volts:
Here, the dropping resistor must be increased to a size of 1.12 kΩ to drop 22.4 volts at 20
mA so that the LED still receives only 1.6 volts. This also makes for a higher resistor power
dissipation: 448 mW, nearly one-half a watt of power! Obviously, a resistor rated for 1/8 watt
power dissipation or even 1/4 watt dissipation will overheat if used here.
Dropping resistor values need not be precise for LED circuits. Suppose we were to use a
1 kΩ resistor instead of a 1.12 kΩ resistor in the circuit shown above. The result would be a
slightly greater circuit current and LED voltage drop, resulting in a brighter light from the
LED and slightly reduced service life. A dropping resistor with too much resistance (say, 1.5
kΩ instead of 1.12 kΩ) will result in less circuit current, less LED voltage, and a dimmer light.
LEDs are quite tolerant of variation in applied power, so you need not strive for perfection in
sizing the dropping resistor.
Multiple LEDs are sometimes required, say in lighting. If LEDs are operated in parallel,
each must have its own current limiting resistor as in Figure ?? (a) to ensure currents dividing
more equally. However, it is more efficient to operate LEDs in series (Figure 3.71 (b)) with a
single dropping resistor. As the number of series LEDs increases the series resistor value must
decrease to maintain current, to a point. The number of LEDs in series (Vf ) cannot exceed the
capability of the power supply. Multiple series strings may be employed as in Figure 3.71 (c).
In spite of equalizing the currents in multiple LEDs, the brightness of the devices may not
match due to variations in the individual parts. Parts can be selected for brightness matching
for critical applications.

+

220 Ω

+

6V

−

140 Ω

(a)

+

6V

−

140 Ω
6V

−
(b)

(c)

Figure 3.71: Multiple LEDs: (a) In parallel, (b) in series, (c) series-parallel
Also because of their unique chemical makeup, LEDs have much, much lower peak-inverse

3.12. SPECIAL-PURPOSE DIODES

149

voltage (PIV) ratings than ordinary rectifying diodes. A typical LED might only be rated at
5 volts in reverse-bias mode. Therefore, when using alternating current to power an LED,
connect a protective rectifying diode anti-parallel with the LED to prevent reverse breakdown
every other half-cycle as in Figure 3.72 (a).
Rdropping

+

Red LED,
1.12 kΩ
24 V

(a)

−

rectifying diode

VF = 1.6 V typical
IF = 20 mA typical
VR = 5 V maximum

Figure 3.72: Driving an LED with AC
The anti-parallel diode in Figure 3.72 can be replaced with an anti-parallel LED. The resulting pair of anti-parallel LED’s illuminate on alternating half-cycles of the AC sinewave.
This configuration draws 20 ma, splitting it equally between the LED’s on alternating AC half
cycles. Each LED only receives 10 mA due to this sharing. The same is true of the LED antiparallel combination with a rectifier. The LED only receives 10 ma. If 20 mA was required for
the LED(s), The resistor value could be halved.
The forward voltage drop of LED’s is inversely proportional to the wavelength (λ). As wavelength decreases going from infrared to visible colors to ultraviolet, Vf increases. While this
trend is most obvious in the various devices from a single manufacturer, The voltage range for
a particular color LED from various manufacturers varies. This range of voltages is shown in
Table 3.2.
Table 3.2: Optical and electrical properties of LED’s
LED
λ nm (= 10 −9 m) Vf (from) Vf (to)
infrared
940
1.2
1.7
red
660
1.5
2.4
orange
602-620
2.1
2.2
yellow, green
560-595
1.7
2.8
white, blue, violet 3
4
ultraviolet
370
4.2
4.8
As lamps, LEDs are superior to incandescent bulbs in many ways. First and foremost is
efficiency: LEDs output far more light power per watt of electrical input than an incandescent
lamp. This is a significant advantage if the circuit in question is battery-powered, efficiency
translating to longer battery life. Second is the fact that LEDs are far more reliable, having
a much greater service life than incandescent lamps. This is because LEDs are “cold” devices:
they operate at much cooler temperatures than an incandescent lamp with a white-hot metal
filament, susceptible to breakage from mechanical and thermal shock. Third is the high speed
at which LEDs may be turned on and off. This advantage is also due to the “cold” operation
of LEDs: they don’t have to overcome thermal inertia in transitioning from off to on or vice

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150

versa. For this reason, LEDs are used to transmit digital (on/off) information as pulses of light,
conducted in empty space or through fiber-optic cable, at very high rates of speed (millions of
pulses per second).
LEDs excel in monochromatic lighting applications like traffic signals and automotive tail
lights. Incandescents are abysmal in this application since they require filtering, decreasing
efficiency. LEDs do not require filtering.
One major disadvantage of using LEDs as sources of illumination is their monochromatic
(single-color) emission. No one wants to read a book under the light of a red, green, or blue
LED. However, if used in combination, LED colors may be mixed for a more broad-spectrum
glow. A new broad spectrum light source is the white LED. While small white panel indicators
have been available for many years, illumination grade devices are still in development.

Lamp type
White LED
White LED, future
Incandescent
Halogen
Compact fluorescent
Sodium vapor, lp
Mercury vapor

Table 3.3: Efficiency of lighting
Efficiency lumen/watt Life hrs
35
100,000
100
100,000
12
1000
15-17
2000
50-100
10,000
70-200
20,000
13-48
18,000

notes
costly
R&D target
inexpensive
high quality light
cost effective
outdoor
outdoor

A white LED is a blue LED exciting a phosphor which emits yellow light. The blue plus
yellow approximates white light. The nature of the phosphor determines the characteristics of
the light. A red phosphor may be added to improve the quality of the yellow plus blue mixture
at the expense of efficiency. Table 3.3 compares white illumination LEDs to expected future
devices and other conventional lamps. Efficiency is measured in lumens of light output per
watt of input power. If the 50 lumens/watt device can be improved to 100 lumens/watt, white
LEDs will be comparable to compact fluorescent lamps in efficiency.
LEDs in general have been a major subject of R&D since the 1960’s. Because of this it
is impractical to cover all geometries, chemistries, and characteristics that have been created
over the decades. The early devices were relatively dim and took moderate currents. The efficiencies have been improved in later generations to the point it is hazardous to look closely
and directly into an illuminated LED. This can result in eye damage, and the LEDs only required a minor increase in dropping voltage (Vf) and current. Modern high intensity devices
have reached 180 lumens using 0.7 Amps (82 lumens/watt, Luxeon Rebel series cool white),
and even higher intensity models can use even higher currents with a corresponding increase
in brightness. Other developments, such as quantum dots, are the subject of current research,
so expect to see new things for these devices in the future

3.12.4

Laser diodes

The laser diode is a further development upon the regular light-emitting diode, or LED. The
term “laser” itself is actually an acronym, despite the fact its often written in lower-case letters.
“Laser” stands for Light Amplification by Stimulated Emission of Radiation, and refers to

3.12. SPECIAL-PURPOSE DIODES

151

another strange quantum process whereby characteristic light emitted by electrons falling
from high-level to low-level energy states in a material stimulate other electrons in a substance
to make similar “jumps,” the result being a synchronized output of light from the material. This
synchronization extends to the actual phase of the emitted light, so that all light waves emitted
from a “lasing” material are not just the same frequency (color), but also the same phase as
each other, so that they reinforce one another and are able to travel in a very tightly-confined,
nondispersing beam. This is why laser light stays so remarkably focused over long distances:
each and every light wave coming from the laser is in step with each other.

"white"
light
source
(a)
monochromatic
light
source
(b)
laser
light
source
(c)

Figure 3.73: (a) White light of many wavelengths. (b) Mono-chromatic LED light, a single
wavelength. (c) Phase coherent laser light.
Incandescent lamps produce “white” (mixed-frequency, or mixed-color) light as in Figure 3.73
(a). Regular LEDs produce monochromatic light: same frequency (color), but different phases,
resulting in similar beam dispersion in Figure 3.73 (b). Laser LEDs produce coherent light:
light that is both monochromatic (single-color) and monophasic (single-phase), resulting in
precise beam confinement as in Figure 3.73 (c).
Laser light finds wide application in the modern world: everything from surveying, where
a straight and nondispersing light beam is very useful for precise sighting of measurement
markers, to the reading and writing of optical disks, where only the narrowness of a focused
laser beam is able to resolve the microscopic “pits” in the disk’s surface comprising the binary
1’s and 0’s of digital information.
Some laser diodes require special high-power “pulsing” circuits to deliver large quantities of
voltage and current in short bursts. Other laser diodes may be operated continuously at lower
power. In the continuous laser, laser action occurs only within a certain range of diode current,
necessitating some form of current-regulator circuit. As laser diodes age, their power requirements may change (more current required for less output power), but it should be remembered
that low-power laser diodes, like LEDs, are fairly long-lived devices, with typical service lives
in the tens of thousands of hours.

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152

3.12.5

Photodiodes

A photodiode is a diode optimized to produce an electron current flow in response to irradiation by ultraviolet, visible, or infrared light. Silicon is most often used to fabricate photodiodes;
though, germanium and gallium arsenide can be used. The junction through which light enters the semiconductor must be thin enough to pass most of the light on to the active region
(depletion region) where light is converted to electron hole pairs.
In Figure 3.74 a shallow P-type diffusion into an N-type wafer produces a PN junction near
the surface of the wafer. The P-type layer needs to be thin to pass as much light as possible.
A heavy N+ diffusion on the back of the wafer makes contact with metalization. The top
metalization may be a fine grid of metallic fingers on the top of the wafer for large cells. In
small photodiodes, the top contact might be a sole bond wire contacting the bare P-type silicon
top.

+
+

top metal contact
p diffusion
depletion region
n type

-

n+ contact region
bottom metal contact

-

Figure 3.74: Photodiode: Schematic symbol and cross section.
Light entering the top of the photodiode stack fall off exponentially in with depth of the
silicon. A thin top P-type layer allows most photons to pass into the depletion region where
electron-hole pairs are formed. The electric field across the depletion region due to the built in
diode potential causes electrons to be swept into the N-layer, holes into the P-layer. Actually
electron-hole pairs may be formed in any of the semiconductor regions. However, those formed
in the depletion region are most likely to be separated into the respective N and P-regions.
Many of the electron-hole pairs formed in the P and N-regions recombine. Only a few do so
in the depletion region. Thus, a few electron-hole pairs in the N and P-regions, and most in
the depletion region contribute to photocurrent, that current resulting from light falling on the
photodiode.
The voltage out of a photodiode may be observed. Operation in this photovoltaic (PV) mode
is not linear over a large dynamic range, though it is sensitive and has low noise at frequencies
less than 100 kHz. The preferred mode of operation is often photocurrent (PC) mode because
the current is linearly proportional to light flux over several decades of intensity, and higher
frequency response can be achieved. PC mode is achieved with reverse bias or zero bias on the
photodiode. A current amplifier (transimpedance amplifier) should be used with a photodiode

3.12. SPECIAL-PURPOSE DIODES

153

in PC mode. Linearity and PC mode are achieved as long as the diode does not become forward
biased.
High speed operation is often required of photodiodes, as opposed to solar cells. Speed
is a function of diode capacitance, which can be minimized by decreasing cell area. Thus, a
sensor for a high speed fiber optic link will use an area no larger than necessary, say 1 mm2 .
Capacitance may also be decreased by increasing the thickness of the depletion region, in the
manufacturing process or by increasing the reverse bias on the diode.
PIN diode The p-i-n diode or PIN diode is a photodiode with an intrinsic layer between
the P and N-regions as in Figure 3.75. The P-Intrinsic-N structure increases the distance
between the P and N conductive layers, decreasing capacitance, increasing speed. The volume
of the photo sensitive region also increases, enhancing conversion efficiency. The bandwidth
can extend to 10’s of gHz. PIN photodiodes are the preferred for high sensitivity, and high
speed at moderate cost.

+
top metal contact
p diffusion
intrinsic region
(larger depletion
region)
n type
n+ contact region
bottom metal contact

-

Figure 3.75: PIN photodiode: The intrinsic region increases the thickness of the depletion
region.
Avalanche photo diode:An avalanche photodiode (APD)designed to operate at high reverse bias exhibits an electron multiplier effect analogous to a photomultiplier tube. The reverse bias can run from 10’s of volts to nearly 2000 V. The high level of reverse bias accelerates
photon created electron-hole pairs in the intrinsic region to a high enough velocity to free additional carriers from collisions with the crystal lattice. Thus, many electrons per photon result.
The motivation for the APD is to achieve amplification within the photodiode to overcome noise
in external amplifiers. This works to some extent. However, the APD creates noise of its own.
At high speed the APD is superior to a PIN diode amplifier combination, though not for low
speed applications. APD’s are expensive, roughly the price of a photomultiplier tube. So, they
are only competitive with PIN photodiodes for niche applications. One such application is
single photon counting as applied to nuclear physics.

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154

3.12.6

Solar cells

A photodiode optimized for efficiently delivering power to a load is the solar cell. It operates in
photovoltaic mode (PV) because it is forward biased by the voltage developed across the load
resistance.
Monocrystalline solar cells are manufactured in a process similar to semiconductor processing. This involves growing a single crystal boule from molten high purity silicon (P-type),
though, not as high purity as for semiconductors. The boule is diamond sawed or wire sawed
into wafers. The ends of the boule must be discarded or recycled, and silicon is lost in the saw
kerf. Since modern cells are nearly square, silicon is lost in squaring the boule. Cells may be
etched to texture (roughen) the surface to help trap light within the cell. Considerable silicon
is lost in producing the 10 or 15 cm square wafers. These days (2007) it is common for solar
cell manufacturer to purchase the wafers at this stage from a supplier to the semiconductor
industry.
P-type Wafers are loaded back-to-back into fused silica boats exposing only the outer surface
to the N-type dopant in the diffusion furnace. The diffusion process forms a thin n-type layer on
the top of the cell. The diffusion also shorts the edges of the cell front to back. The periphery
must be removed by plasma etching to unshort the cell. Silver and or aluminum paste is
screened on the back of the cell, and a silver grid on the front. These are sintered in a furnace
for good electrical contact. (Figure 3.76)
The cells are wired in series with metal ribbons. For charging 12 V batteries, 36 cells at
approximately 0.5 V are vacuum laminated between glass, and a polymer metal back. The
glass may have a textured surface to help trap light.

top metal contact
N diffusion
depletion region
P type wafer
bottom metal
contact

+
Figure 3.76: Silicon Solar cell
The ultimate commercial high efficiency (21.5%) single crystal silicon solar cells have all
contacts on the back of the cell. The active area of the cell is increased by moving the top (-)
contact conductors to the back of the cell. The top (-) contacts are normally made to the N-type
silicon on top of the cell. In Figure 3.77 the (-) contacts are made to N+ diffusions on the bottom
interleaved with (+) contacts. The top surface is textured to aid in trapping light within the
cell.. [18]
Multicyrstalline silicon cells start out as molten silicon cast into a rectangular mold. As

3.12. SPECIAL-PURPOSE DIODES

155

Antireflectrive coating
Silicon dioxide passivation
N-type diffusion
P-type wafer
N+ diffusion
- contact
P+ diffusion
+ contact
N+ diffusion
- contact

Figure 3.77: High efficiency solar cell with all contacts on the back. Adapted from Figure 1 [18]
the silicon cools, it crystallizes into a few large (mm to cm sized) randomly oriented crystals
instead of a single one. The remainder of the process is the same as for single crystal cells. The
finished cells show lines dividing the individual crystals, as if the cells were cracked. The high
efficiency is not quite as high as single crystal cells due to losses at crystal grain boundaries.
The cell surface cannot be roughened by etching due to the random orientation of the crystals.
However, an antireflectrive coating improves efficiency. These cells are competitive for all but
space applications.
Three layer cell: The highest efficiency solar cell is a stack of three cells tuned to absorb
different portions of the solar spectrum. Though three cells can be stacked atop one another,
a monolithic single crystal structure of 20 semiconductor layers is more compact. At 32 %
efficiency, it is now (2007) favored over silicon for space application. The high cost prevents
it from finding many earth bound applications other than concentrators based on lenses or
mirrors.
Intensive research has recently produced a version enhanced for terrestrial concentrators
at 400 - 1000 suns and 40.7% efficiency. This requires either a big inexpensive Fresnel lens or
reflector and a small area of the expensive semiconductor. This combination is thought to be
competitive with inexpensive silicon cells for solar power plants. [9] [23]
Metal organic chemical vapor deposition (MOCVD) deposits the layers atop a P-type germanium substrate. The top layers of N and P-type gallium indium phosphide (GaInP) having
a band gap of 1.85 eV, absorbs ultraviolet and visible light. These wavelengths have enough
energy to exceed the band gap. Longer wavelengths (lower energy) do not have enough energy
to create electron-hole pairs, and pass on through to the next layer. A gallium arsenide layers
having a band gap of 1.42 eV, absorbs near infrared light. Finally the germanium layer and
substrate absorb far infrared. The series of three cells produce a voltage which is the sum of
the voltages of the three cells. The voltage developed by each material is 0.4 V less than the
band gap energy listed in Table 3.4. For example, for GaInP: 1.8 eV/e - 0.4 V = 1.4 V. For all
three the voltage is 1.4 V + 1.0 V + 0.3 V = 2.7 V. [4]
Crystalline solar cell arrays have a long usable life. Many arrays are guaranteed for 25

156

CHAPTER 3. DIODES AND RECTIFIERS
Table 3.4: High efficiency triple layer solar cell.
Layer
Band gap Light absorbed
Gallium indium phosphide 1.8 eV
UV, visible
Gallium arsenide
1.4 eV
near infrared
Germanium
0.7 eV
far infrared

years, and believed to be good for 40 years. They do not suffer initial degradation compared
with amorphous silicon.
Both single and multicrystalline solar cells are based on silicon wafers. The silicon is both
the substrate and the active device layers. Much silicon is consumed. This kind of cell has
been around for decades, and takes approximately 86% of the solar electric market. For further
information about crystalline solar cells see Honsberg. [8]
Amorphous silicon thin film solar cells use tiny amounts of the active raw material, silicon. Approximately half the cost of conventional crystalline solar cells is the solar cell grade
silicon. The thin film deposition process reduces this cost. The downside is that efficiency is
about half that of conventional crystalline cells. Moreover, efficiency degrades by 15-35% upon
exposure to sunlight. A 7% efficient cell soon ages to 5% efficiency. Thin film amorphous silicon
cells work better than crystalline cells in dim light. They are put to good use in solar powered
calculators.
Non-silicon based solar cells make up about 7% of the market. These are thin-film polycrystalline products. Various compound semiconductors are the subject of research and development. Some non-silicon products are in production. Generally, the efficiency is better than
amorphous silicon, but not nearly as good as crystalline silicon.
Cadmium telluride as a polycrystalline thin film on metal or glass can have a higher
efficiency than amorphous silicon thin films. If deposited on metal, that layer is the negative
contact to the cadmium telluride thin film. The transparent P-type cadmium sulfide atop the
cadmium telluride serves as a buffer layer. The positive top contact is transparent, electrically
conductive fluorine doped tin oxide. These layers may be laid down on a sacrificial foil in place
of the glass in the process in the following pargraph. The sacrificial foil is removed after the
cell is mounted to a permanent substrate.
A process for depositing cadmium telluride on glass begins with the deposition of N-type
transparent, electrically conducive, tin oxide on a glass substrate. The next layer is P-type
cadmium telluride; though, N-type or intrinsic may be used. These two layers constitute the
NP junction. A P+ (heavy P-type) layer of lead telluride aids in establishing a low resistance
contact. A metal layer makes the final contact to the lead telluride. These layers may be laid
down by vacuum deposition, chemical vapor deposition (CVD), screen printing, electrodeposition, or atmospheric pressure chemical vapor deposition (APCVD) in helium. [10]
A variation of cadmium telluride is mercury cadmium telluride. Having lower bulk resistance and lower contact resistance improves efficiency over cadmium telluride.
Cadmium Indium Gallium diSelenide: A most promising thin film solar cell at this
time (2007) is manufactured on a ten inch wide roll of flexible polyimide– Cadmium Indium
Gallium diSelenide (CIGS). It has a spectacular efficiency of 10%. Though, commercial grade
crystalline silicon cells surpassed this decades ago, CIGS should be cost competitive. The
deposition processes are at a low enough temperature to use a polyimide polymer as a substrate

3.12. SPECIAL-PURPOSE DIODES

157

−
glass substrate
n Tin oxide
cadmium suflide
p cadmium telluride
(phosphorus doped)
p+ lead telluride
metal substrate
metal contact

+

+

Figure 3.78: Cadmium telluride solar cell on glass or metal.

top contact
N-type transparent
conductor
buffer layer

Tin oxide
Zinc oxide
Cadmium suflide
CIGS Cadmium Indium
Gallium diSelenide
Molybdenum

P type
bottom contact

Polyimide substrate

-

Figure 3.79: Cadmium Indium Gallium diSelenide solar cell (CIGS)

CHAPTER 3. DIODES AND RECTIFIERS

158

instead of metal or glass. (Figure 3.79) The CIGS is manufactured in a roll to roll process,
which should drive down costs. GIGS cells may also be produced by an inherently low cost
electrochemical process. [7]
• REVIEW:
• Most solar cells are silicon single crystal or multicrystal because of their good efficiency
and moderate cost.
• Less efficient thin films of various amorphous or polycrystalline materials comprise the
rest of the market.
• Table 3.5 compares selected solar cells.
Table 3.5: Solar cell properties
Maximum Practical Notes
effiefficiency
ciency
Selenium, polycrystalline
0.7%
1883, Charles Fritts
Silicon, single crystal
4%
1950’s, first silicon solar cell
Silicon, single crystal PERL, 25%
solar cars, cost=100x commerterrestrial, space
cial
Silicon, single crystal, commer- 24%
14-17%
$5-$10/peak watt
cial terrestrial
Cypress Semiconductor, Sun- 21.5%
19%
all contacts on cell back
power, silicon single crystal
Gallium Indium Phosphide/ 32%
Preferred for space.
Gallium Arsenide/ Germanium, single crystal, multilayer
Advanced terrestrial version of 40.7%
Uses optical concentrator.
above.
Silicon, multicrystalline
18.5%
15.5%
Thin films,
Silicon, amorphous
13%
5-7%
Degrades in sun light. Good indoors for calculators or cloudy
outdoors.
Cadmium telluride, polycrys- 16%
glass or metal substrate
talline
Copper indium arsenide dise- 18%
10%
10 inch flexible polymer web.
lenide, polycrystalline
[17]
Organic polymer, 100% plastic
4.5%
R&D project
Solar cell type

3.12.7

Varicap or varactor diodes

A variable capacitance diode is known as a varicap diode or as a varactor. If a diode is reverse
biased, an insulating depletion region forms between the two semiconductive layers. In many

3.12. SPECIAL-PURPOSE DIODES

159

capacitance

diodes the width of the depletion region may be changed by varying the reverse bias. This
varies the capacitance. This effect is accentuated in varicap diodes. The schematic symbols is
shown in Figure 3.80, one of which is packaged as common cathode dual diode.

symbol

−

Vcontrol

Clarge

L

+
voltage

Coptional

varicap diode

Figure 3.80: Varicap diode: Capacitance varies with reverse bias. This varies the frequency of
a resonant network.
If a varicap diode is part of a resonant circuit as in Figure 3.80, the frequency my be varied
with a control voltage, Vcontrol . A large capacitance, low Xc , in series with the varicap prevents
Vcontrol from being shorted out by inductor L. As long as the series capacitor is large, it has
minimal effect on the frequency of resonant circuit. Coptional may be used to set the center
resonant frequency. Vcontrol can then vary the frequency about this point. Note that the required active circuitry to make the resonant network oscillate is not shown. For an example of
a varicap diode tuned AM radio receiver see “electronic varicap diode tuning,” (page 428)
Some varicap diodes may be referred to as abrupt, hyperabrupt, or super hyper abrupt.
These refer to the change in junction capacitance with changing reverse bias as being abrupt
or hyper-abrupt, or super hyperabrupt. These diodes offer a relatively large change in capacitance. This is useful when oscillators or filters are swept over a large frequency range.
Varying the bias of abrupt varicaps over the rated limits, changes capacitance by a 4:1 ratio,
hyperabrupt by 10:1, super hyperabrupt by 20:1.
Varactor diodes may be used in frequency multiplier circuits. See “Practical analog semiconductor circuits,” Varactor multiplier

3.12.8

Snap diode

The snap diode, also known as the step recovery diode is designed for use in high ratio frequency multipliers up to 20 gHz. When the diode is forward biased, charge is stored in the PN
junction. This charge is drawn out as the diode is reverse biased. The diode looks like a low
impedance current source during forward bias. When reverse bias is applied it still looks like
a low impedance source until all the charge is withdrawn. It then “snaps” to a high impedance
state causing a voltage impulse, rich in harmonics. An applications is a comb generator, a
generator of many harmonics. Moderate power 2x and 4x multipliers are another application.

3.12.9

PIN diodes

A PIN diode is a fast low capacitance switching diode. Do not confuse a PIN switching diode
with a PIN photo diode (page 153). A PIN diode is manufactured like a silicon switching diode

CHAPTER 3. DIODES AND RECTIFIERS

160

with an intrinsic region added between the PN junction layers. This yields a thicker depletion
region, the insulating layer at the junction of a reverse biased diode. This results in lower
capacitance than a reverse biased switching diode.

top metal contact
p+ contact region
p diffusion
intrinsic region
(larger depletion
region)
n type
n+ contact region
bottom metal contact

Figure 3.81: Pin diode: Cross section aligned with schematic symbol.
PIN diodes are used in place of switching diodes in radio frequency (RF) applications, for
example, a T/R switch (page 431). The 1n4007 1000 V, 1 A general purpose power diode is
reported to be usable as a PIN switching diode. The high voltage rating of this diode is achieved
by the inclusion of an intrinsic layer dividing the PN junction. This intrinsic layer makes the
1n4007 a PIN diode. Another PIN diode application is a the antenna switch (page 431) for a
direction finder receiver.
PIN diodes serve as variable resistors when the forward bias is varied. One such application
is the voltage variable attenuator (page 431). The low capacitance characteristic of PIN diodes,
extends the frequency flat response of the attenuator to microwave frequencies.

3.12.10

IMPATT diode

¡The IMPact Avalanche Transit Time diode is a high power radio frequency (RF) generator
operating from 3 to 100 gHz. IMPATT diodes are fabricated from silicon, gallium arsenide, or
silicon carbide.
An IMPATT diode is reverse biased above the breakdown voltage. The high doping levels
produce a thin depletion region. The resulting high electric field rapidly accelerates carriers
which free other carriers in collisions with the crystal lattice. Holes are swept into the P+
region. Electrons drift toward the N regions. The cascading effect creates an avalanche current
which increases even as voltage across the junction decreases. The pulses of current lag the
voltage peak across the junction. A “negative resistance” effect in conjunction with a resonant
circuit produces oscillations at high power levels (high for semiconductors).
The resonant circuit in the schematic diagram of Figure 3.82 is the lumped circuit equivalent of a waveguide section, where the IMPATT diode is mounted. DC reverse bias is applied

3.12. SPECIAL-PURPOSE DIODES

+
−

161

N+
N-

choke
drift

N+
avalanche

P+

resonant circuit

Figure 3.82: IMPATT diode: Oscillator circuit and heavily doped P and N layers.
through a choke which keeps RF from being lost in the bias supply. This may be a section of
waveguide known as a bias Tee. Low power RADAR transmitters may use an IMPATT diode
as a power source. They are too noisy for use in the receiver. [20]

3.12.11

Gunn diode

Diode, gunn Gunn diode
A gunn diode is solely composed of N-type semiconductor. As such, it is not a true diode.
Figure 3.83 shows a lightly doped N− layer surrounded by heavily doped N+ layers. A voltage
applied across the N-type gallium arsenide gunn diode creates a strong electric field across the
lightly doped N− layer.

I

+
−

choke

N+

threshold

-

N

N+
resonant circuit

1

V

Figure 3.83: Gunn diode: Oscillator circuit and cross section of only N-type semiconductor
diode.
As voltage is increased, conduction increases due to electrons in a low energy conduction
band. As voltage is increased beyond the threshold of approximately 1 V, electrons move from
the lower conduction band to the higher energy conduction band where they no longer contribute to conduction. In other words, as voltage increases, current decreases, a negative resistance condition. The oscillation frequency is determined by the transit time of the conduction
electrons, which is inversely related to the thickness of the N− layer.
The frequency may be controlled to some extent by embedding the gunn diode into a resonant circuit. The lumped circuit equivalent shown in Figure 3.83 is actually a coaxial trans-

CHAPTER 3. DIODES AND RECTIFIERS

162

mission line or waveguide. Gallium arsenide gunn diodes are available for operation from 10
to 200 gHz at 5 to 65 mw power. Gunn diodes may also serve as amplifiers. [19] [14]

3.12.12

Shockley diode

The Shockley diodeis a 4-layer thyristor used to trigger larger thyristors. It only conducts in
one direction when triggered by a voltage exceeding the breakover voltage, about 20 V. See
“Thyristors,” The Shockley Diode. The bidirectional version is called a diac. See “Thyristors,”
The DIAC.

3.12.13

Constant-current diodes

A constant-current diode, also known as a current-limiting diode, or current-regulating diode,
does exactly what its name implies: it regulates current through it to some maximum level.
The constant current diode is a two terminal version of a JFET. If we try to force more current
through a constant-current diode than its current-regulation point, it simply “fights back” by
dropping more voltage. If we were to build the circuit in Figure 3.84(a) and plot diode current
against diode voltage, we’d get a graph that rises at first and then levels off at the current
regulation point as in Figure 3.84(b).
Rdropping

Idiode

constant-current
diode
(a)

(b)
Vdiode

Figure 3.84: Constant current diode: (a) Test circuit, (b) current vs voltage characteristic.
One application for a constant-current diode is to automatically limit current through an
LED or laser diode over a wide range of power supply voltages as in Figure ??.
Of course, the constant-current diode’s regulation point should be chosen to match the LED
or laser diode’s optimum forward current. This is especially important for the laser diode, not
so much for the LED, as regular LEDs tend to be more tolerant of forward current variations.
Another application is in the charging of small secondary-cell batteries, where a constant
charging current leads to predictable charging times. Of course, large secondary-cell battery
banks might also benefit from constant-current charging, but constant-current diodes tend to
be very small devices, limited to regulating currents in the milliamp range.

3.13. OTHER DIODE TECHNOLOGIES

3.13

Other diode technologies

3.13.1

SiC diodes

163

Diodes manufactured from silicon carbide are capable of high temperature operation to 400o C.
This could be in a high temperature environment: down hole oil well logging, gas turbine
engines, auto engines. Or, operation in a moderate environment at high power dissipation.
Nuclear and space applications are promising as SiC is 100 times more resistant to radiation
compared with silicon. SiC is a better conductor of heat than any metal. Thus, SiC is better
than silicon at conducting away heat. Breakdown voltage is several kV. SiC power devices are
expected to reduce electrical energy losses in the power industry by a factor of 100.

3.13.2

Polymer diode

Diodes based on organic chemicals have been produce using low temperature processes. Hole
rich and electron rich conductive polymers may be ink jet printed in layers. Most of the research and development is of the organic LED (OLED). However, development of inexpensive
printable organic RFID (radio frequency identification) tags is on going. In this effort, a pentacene organic rectifier has been operated at 50 MHz. Rectification to 800 MHz is a development goal. An inexpensive metal insulator metal (MIM) diode acting like a back-to-back zener
diode clipper has been delveloped. Also, a tunnel diode like device has been fabricated.

3.14

SPICE models

The SPICE circuit simulation program provides for modeling diodes in circuit simulations.
The diode model is based on characterization of individual devices as described in a product
data sheet and manufacturing process characteristics not listed. Some information has been
extracted from a 1N4004 data sheet in Figure 3.85.
The diode statement begins with a diode element name which must begin with “d” plus
optional characters. Example diode element names include: d1, d2, dtest, da, db, d101. Two
node numbers specify the connection of the anode and cathode, respectively, to other components. The node numbers are followed by a model name, referring to a subsequent “.model”
statement.
The model statement line begins with “.model,” followed by the model name matching
one or more diode statements. Next, a “d” indicates a diode is being modeled. The remainder of the model statement is a list of optional diode parameters of the form ParameterName=ParameterValue. None are used in Example below. Example2 has some parameters
defined. For a list of diode parameters, see Table 3.6.
General form:

d[name] [anode] [cathode] [modelname]
.model ([modelname] d [parmtr1=x] [parmtr2=y] . . .)

Example:

d1 1 2 mod1
.model mod1 d

CHAPTER 3. DIODES AND RECTIFIERS
100
CJ junction capacitance (pF)

10

1.0

0.1
0.925

IF instaneous forward current (I)

164

0.01
0.6

0.8

1.0

1.2

1.4

30

10

1
1

1.6

10

100

VR reverse voltage (V)

VF instaneous forward voltage (V)
1

Forward voltage drop VF (V)

Peak repetitive reverse voltage VRRM (V) 400

@ IF (A)

1

Max reverse current IR (µΑ)

5

Max avg rectified current IO (A)
Peak forward surge current IFSM (A)
Total capacitance CT (pF)

30
15

1

@ VR (V) 400

Figure 3.85: Data sheet 1N4004 excerpt, after [6].

Example2:

D2 1 2 Da1N4004
.model Da1N4004 D (IS=18.8n RS=0 BV=400 IBV=5.00u CJO=30

M=0.333 N=2)
The easiest approach to take for a SPICE model is the same as for a data sheet: consult
the manufacturer’s web site. Table 3.7 lists the model parameters for some selected diodes. A
fallback strategy is to build a SPICE model from those parameters listed on the data sheet.
A third strategy, not considered here, is to take measurements of an actual device. Then,
calculate, compare and adjust the SPICE parameters to the measurements.
If diode parameters are not specified as in “Example” model above, the parameters take on
the default values listed in Table 3.6 and Table 3.7. These defaults model integrated circuit
diodes. These are certainly adequate for preliminary work with discrete devices For more
critical work, use SPICE models supplied by the manufacturer [5], SPICE vendors, and other
sources. [16]
Otherwise, derive some of the parameters from the data sheet. First select a value for spice
parameter N between 1 and 2. It is required for the diode equation (n). Massobrio [1] pp 9,
recommends ”.. n, the emission coefficient is usually about 2.” In Table 3.7, we see that power
rectifiers 1N3891 (12 A), and 10A04 (10 A) both use about 2. The first four in the table are not
relevant because they are schottky, schottky, germanium, and silicon small signal, respectively.
The saturation current, IS, is derived from the diode equation, a value of (VD , ID ) on the graph
in Figure 3.85, and N=2 (n in the diode equation).

3.14. SPICE MODELS

Symbol
IS
RS
n
τD
CD (0)
φ0
m
Eg
pi
kf
af
FC
BV
IBV

Table 3.6: Diode SPICE parameters
Name Parameter
IS
Saturation current (diode equation)
RS
Parsitic resistance (series resistance)
N
Emission coefficient, 1 to 2
TT
Transit time
CJO
Zero-bias junction capacitance
VJ
Junction potential
M
Junction grading coefficient
0.33 for linearly graded junction
0.5 for abrupt junction
EG
Activation energy:
Si: 1.11
Ge: 0.67
Schottky: 0.69
XTI
IS temperature exponent
pn junction: 3.0
Schottky: 2.0
KF
Flicker noise coefficient
AF
Flicker noise exponent
FC
Forward bias depletion capacitance coefficient
BV
Reverse breakdown voltage
IBV
Reverse breakdown current

165

Units
A

Default
1E-14

Ω

0

s
F
V
-

1
0
0
1
0.5
-

eV
-

1.11
3.0
0
1
0.5

V
A

∞
1E-3

Table 3.7: SPICE parameters for selected diodes; sk=schottky Ge=germanium; else silicon.
Part
IS
RS
N
TT
CJO M
VJ EG
XTI BV IBV
Default
1E-14 0
1
0
0
0.5
1
1.11 3
∞
1m
1N5711 sk 315n
2.8
2.03
1.44n 2.00p 0.333 0.69 2
70
10u
1N5712 sk 680p
12
1.003 50p
1.0p
0.5
0.6 0.69 2
20
1N34 Ge
200p
84m
2.19
144n 4.82p 0.333 0.75 0.67 60
15u
1N4148
35p
64m
1.24
5.0n
4.0p
0.285 0.6 75
1N3891
63n
9.6m
2
110n 114p 0.255 0.6 250 10A04 10A 844n
2.06m 2.06
4.32u 277p 0.333 400 10u
1N4004
76.9n 42.2m 1.45
4.32u 39.8p 0.333 400 5u
1A
1N4004
18.8n 2
30p
0.333 400 5u
data sheet

CHAPTER 3. DIODES AND RECTIFIERS

166
ID = IS (eVD /nVT − 1)
VT = 26 mV at 25o C

n = 2.0

VD = 0.925 V at 1 A from graph

1 A = IS (e(0.925V )/(2)(26mV ) − 1)
IS =

18.8E-9

The numerical values of IS=18.8n and N=2 are entered in last line of Table 3.7 for comparison to the manufacturers model for 1N4004, which is considerably different. RS defaults to 0
for now. It will be estimated later. The important DC static parameters are N, IS, and RS.
Rashid [15] suggests that TT, τ D , the transit time, be approximated from the reverse recovery stored charge QRR , a data sheet parameter (not available on our data sheet) and IF ,
forward current.
ID = IS (eVD /nVT − 1)
τD =

QRR /IF

We take the TT=0 default for lack of QRR . Though it would be reasonable to take TT for a
similar rectifier like the 10A04 at 4.32u. The 1N3891 TT is not a valid choice because it is a
fast recovery rectifier. CJO, the zero bias junction capacitance is estimated from the VR vs CJ
graph in Figure 3.85. The capacitance at the nearest to zero voltage on the graph is 30 pF at
1 V. If simulating high speed transient response, as in switching regulator power supplies, TT
and CJO parameters must be provided.
The junction grading coefficient M is related to the doping profile of the junction. This is not
a data sheet item. The default is 0.5 for an abrupt junction. We opt for M=0.333 corresponding
to a linearly graded junction. The power rectifiers in Table 3.7 use lower values for M than 0.5.
We take the default values for VJ and EG. Many more diodes use VJ=0.6 than shown in
Table 3.7. However the 10A04 rectifier uses the default, which we use for our 1N4004 model
(Da1N4001 in Table 3.6). Use the default EG=1.11 for silicon diodes and rectifiers. Table 3.6
lists values for schottky and germanium diodes. Take the XTI=3, the default IS temperature
coefficient for silicon devices. See Table 3.6 for XTI for schottky diodes.
The abbreviated data sheet, Figure 3.85, lists IR = 5 µA @ VR = 400 V, corresponding to
IBV=5u and BV=400 respectively. The 1n4004 SPICE parameters derived from the data sheet
are listed in the last line of Table 3.7 for comparison to the manufacturer’s model listed above
it. BV is only necessary if the simulation exceeds the reverse breakdown voltage of the diode,
as is the case for zener diodes. IBV, reverse breakdown current, is frequently omitted, but may
be entered if provided with BV.
Figure 3.86 shows a circuit to compare the manufacturers model, the model derived from
the datasheet, and the default model using default parameters. The three dummy 0 V sources
are necessary for diode current measurement. The 1 V source is swept from 0 to 1.4 V in 0.2
mV steps. See .DC statement in the netlist in Table 3.8. DI1N4004 is the manufacturer’s diode
model, Da1N4004 is our derived diode model.

3.14. SPICE MODELS

167

1

D1

+
− 1V
+
0V −
0

D2

+
0V −

D3

+
0V −

Figure 3.86: SPICE circuit for comparison of manufacturer model (D1), calculated datasheet
model (D2), and default model (D3).

Table 3.8: SPICE netlist parameters: (D1) DI1N4004 manufacturer’s model, (D2) Da1N40004
datasheet derived, (D3) default diode model.
*SPICE circuit <03468.eps> from XCircuit v3.20
D1 1 5 DI1N4004
V1 5 0 0
D2 1 3 Da1N4004
V2 3 0 0
D3 1 4 Default
V3 4 0 0
V4 1 0 1
.DC V4 0 1400mV 0.2m
.model Da1N4004 D (IS=18.8n RS=0
BV=400 IBV=5.00u CJO=30
+M=0.333
N=2.0 TT=0)
.MODEL DI1N4004 D (IS=76.9n RS=42.0m BV=400 IBV=5.00u CJO=39.8p
+M=0.333 N=1.45 TT=4.32u)
.MODEL Default D
.end

168

CHAPTER 3. DIODES AND RECTIFIERS

We compare the three models in Figure 3.87. and to the datasheet graph data in Table 3.9.
VD is the diode voltage versus the diode currents for the manufacturer’s model, our calculated
datasheet model and the default diode model. The last column “1N4004 graph” is from the
datasheet voltage versus current curve in Figure 3.85 which we attempt to match. Comparison
of the currents for the three model to the last column shows that the default model is good at
low currents, the manufacturer’s model is good at high currents, and our calculated datasheet
model is best of all up to 1 A. Agreement is almost perfect at 1 A because the IS calculation is
based on diode voltage at 1 A. Our model grossly over states current above 1 A.

Figure 3.87: First trial of manufacturer model, calculated datasheet model, and default model.
The solution is to increase RS from the default RS=0. Changing RS from 0 to 8m in the
datasheet model causes the curve to intersect 10 A (not shown) at the same voltage as the
manufacturer’s model. Increasing RS to 28.6m shifts the curve further to the right as shown in
Figure 3.88. This has the effect of more closely matching our datasheet model to the datasheet
graph (Figure 3.85). Table 3.10 shows that the current 1.224470e+01 A at 1.4 V matches the
graph at 12 A. However, the current at 0.925 V has degraded from 1.096870e+00 above to
7.318536e-01.
Suggested reader exercise: decrease N so that the current at VD=0.925 V is restored to 1
A. This may increase the current (12.2 A) at VD=1.4 V requiring an increase of RS to decrease
current to 12 A.
Zener diode: There are two approaches to modeling a zener diode: set the BV parameter
to the zener voltage in the model statement, or model the zener with a subcircuit containing a
diode clamper set to the zener voltage. An example of the first approach sets the breakdown
voltage BV to 15 for the 1n4469 15 V zener diode model (IBV optional):
.model D1N4469 D ( BV=15 IBV=17m )

3.14. SPICE MODELS

169

Table 3.9: Comparison of manufacturer model, calculated datasheet model, and default model
to 1N4004 datasheet graph of V vs I.

1N4004
index
graph
3500
0.01
4001
0.13
4500
0.7
4625
1.0
5000
2.0
5500
3.3
6000
5.3
6500
8.0
7000

model

model

model

manufacturer

datasheet

default

7.000000e-01

1.612924e+00

1.416211e-02

5.674683e-03

8.002000e-01

3.346832e+00

9.825960e-02

2.731709e-01

9.000000e-01

5.310740e+00

6.764928e-01

1.294824e+01

9.250000e-01

5.823654e+00

1.096870e+00

3.404037e+01

1.000000e-00

7.395953e+00

4.675526e+00

6.185078e+02

1.100000e+00

9.548779e+00

3.231452e+01

2.954471e+04

1.200000e+00

1.174489e+01

2.233392e+02

1.411283e+06

1.300000e+00

1.397087e+01

1.543591e+03

6.741379e+07

1.400000e+00

1.621861e+01

1.066840e+04

3.220203e+09 12.

VD

Figure 3.88: Second trial to improve calculated datasheet model compared with manufacturer
model and default model.

CHAPTER 3. DIODES AND RECTIFIERS

170

Table 3.10: Changing Da1N4004 model statement RS=0 to RS=28.6m decreases the current at
VD=1.4 V to 12.2 A.
.model Da1N4004 D (IS=18.8n RS=28.6m BV=400 IBV=5.00u CJO=30
+M=0.333
N=2.0 TT=0)
model
model
1N4001
index
VD
manufacturer
datasheet
graph
3505
7.010000e-01
1.628276e+00
1.432463e-02
0.01
4000
8.000000e-01
3.343072e+00
9.297594e-02
0.13
4500
9.000000e-01
5.310740e+00
5.102139e-01
0.7
4625
9.250000e-01
5.823654e+00
7.318536e-01
1.0
5000
1.000000e-00
7.395953e+00
1.763520e+00
2.0
5500
1.100000e+00
9.548779e+00
3.848553e+00
3.3
6000
1.200000e+00
1.174489e+01
6.419621e+00
5.3
6500
1.300000e+00
1.397087e+01
9.254581e+00
8.0
7000
1.400000e+00
1.621861e+01
1.224470e+01
12.
The second approach models the zener with a subcircuit. Clamper D1 and VZ in Figure ??
models the 15 V reverse breakdown voltage of a 1N4477A zener diode. Diode DR accounts for
the forward conduction of the zener in the subcircuit.

A

1

D1

−
+ 13.7 V
3

K

2

.SUBCKT DI-1N4744A
1 2
* Terminals A K
D1 1 2 DF
DZ 3 1 DR
VZ 2 3 13.7
.MODEL DF D (
IS=27.5p RS=0.620
N=1.10
+ CJO=78.3p
VJ=1.00 M=0.330
TT=50.1n )
.MODEL DR D (
IS=5.49f RS=0.804
N=1.77 )
.ENDS

Figure 3.89: Zener diode subcircuit uses clamper (D1 and VZ) to model zener.
Tunnel diode: A tunnel diode may be modeled by a pair of field effect transistors (JFET)
in a SPICE subcircuit. [11] An oscillator circuit is also shown in this reference.
Gunn diode: A Gunn diode may also be modeled by a pair of JFET’s. [12] This reference
shows a microwave relaxation oscillator.
• REVIEW:

BIBLIOGRAPHY

171

• Diodes are described in SPICE by a diode component statement referring to .model statement. The .model statement contains parameters describing the diode. If parameters are
not provided, the model takes on default values.
• Static DC parameters include N, IS, and RS. Reverse breakdown parameters: BV, IBV.
• Accurate dynamic timing requires TT and CJO parameters
• Models provided by the manufacturer are highly recommended.

Contributors
Contributors to this chapter are listed in chronological order of their contributions, from most
recent to first. See Appendix 2 (Contributor List) for dates and contact information.
Jered Wierzbicki (December 2002): Pointed out error in diode equation – Boltzmann’s
constant shown incorrectly.

Bibliography
[1] Paolo Antognetti, Giuseppe Massobrio “Semiconductor Device Modeling with SPICE,”
ISBN 0-07-002107-4, 1988
[2] ATCO
Newsletter,
Volume
14
No.
http://www.atco.tv/homepage/vol14 1.pdf

1,

January

1997

at

[3] D.A. Brunner, et al,, “A Cockcroft-Walton Base for the FEU84-3 Photomultiplier Tube,”
Department of Physics, Indiana University, Bloomington, Indiana 47405 January 1998,
at http://dustbunny.physics.indiana.edu/˜paul/cwbase/
[4] Brenton Burnet, “The Basic Physics and Design of III-V Multijunction Solar,” NREL, at
photochemistry.epfl.ch/EDEY/NREL.pdf
[5] Diodes Incorporated http://www.diodes.com/products/spicemodels/index.php
[6] Diodes
Incorporated,
“1N4001/L
1N4007/l,
http://www.diodes.com/datasheets/ds28002.pdf

1.0A

rectifier,”

at

[7] “Solar firm gains $30 million in funding,”
EE Times,
07/12/2007 at
http://www.eetimes.com/news/latest/showArticle.jhtml?articleID=201001129
[8] Christiana
Honsberg,
Stuart
Bowden,
http://www.udel.edu/igert/pvcdrom/

“Photovoltaics

CDROM,”

at

[9] R. R. King, et. al., “40% efficient metamorphic GaInP/GaInAs/Ge multijunction solar cells”, Applied Physics Letters, 90, 183516 (2007) , at
http://scitation.aip.org/getabs/servlet/GetabsServlet?prog=normal&
id=APPLAB000090000018183516000001&idtype=cvips&gifs=yes

172

CHAPTER 3. DIODES AND RECTIFIERS

[10] Kim W Mitchell, “Method of making a thin film cadmium telluride solar cell,” United
States Patent 4734381,http://www.freepatentsonline.com/4734381.html
[11] Karl H. Muller “RF/Microwave Analysis” Intusoft Newsletter #51, November 1997, at
http://www.intusoft.com/nlhtm/nl51.htm
[12] “A Gunn Diode Relaxation Oscillator,” Intusoft Newsletter #52, February 1998, at
http://www.intusoft.com/nlhtm/nl52.htm
[13] OAK
Solar.,
“Technical
LED’s
LED
http://www.oksolar.com/led/led color chart.htm

color

chart,”

at

[14] Ian Poole, “Summary of the Gunn Diode,” at http://www.radio-electronics.com/
nfo/data/semicond/gunndiode/gunndiode.php
[15] Muhammad H. Rashid, “SPICE for Power Electronics and Electric Power,” ISBN 0-13030420-4, 1993
[16] “SPICE model index,” V2.16 30-Nov-05, at http://homepages.which.net/
˜paul.hills/Circuits/Spice/ModelIndex.html
[17] Neil Thomas, “Advancing CIGS Solar Cell Manufacturing Technology,” April 6, 2007 at
http://www.renewableenergyaccess.com/rea/news/story?id=48033&src=rss
[18] P.J. Verlinden, Sinton, K. Wickham, R.M. Swanson Crane, “BACKSIDECONTACT SILICON SOLAR CELLS WITH IMPROVED EFFICIENCY.” at
http://www.sunpowercorp.com/techpapers/EPSEC97.pdf
[19] Christian Wolff, “Radar Principles,” Radar components, Gunn diodes at at
http://www.radartutorial.eu/17.bauteile/bt12.en.htm
[20] L. Yuan, M. R. Melloch, J. A. Cooper, K. J. Webb,“Silicon Carbide IMPATT Oscillators for High-Power Microwave and Millimeter-Wave Generation,” IEEE/Cornell Conference on Advanced Concepts in High Speed
Semiconductor Devices and Circuits, Ithaca, NY, August 7-9, 2000. at
http://www.ecn.purdue.edu/WBG/Device Research/IMPATT Diodes/Index.html
[21] Alan
Seabaugh,
Zhaoming
HU,
Qingmin
LIU,
David
Wang,
“Silicon
Based
Tunnel
Diodes
and
Integrated
http://www.nd.edu/˜nano/0a1003QFDpaper v1.pdf

Rink,
Jinli
Circuits,”
at

[22] S. M. Sze, G. Gibbons, “Avalanche breakdown voltages of abrupt and linearly graded p-n
junctions in Ge, Si, GaAs, and Ga P,” Appl. Phys. Lett., 8, 111 (1966).
[23] Lisa Zyga, “40% efficient solar cells to be used for solar electricity”, PhysOrgForum, at
http://www.physorg.com/news99904887.html

Chapter 4

BIPOLAR JUNCTION
TRANSISTORS
Contents
4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9
4.10

4.11
4.12
4.13
4.14
4.15
4.16

Introduction . . . . . . . . . . . . .
The transistor as a switch . . . .
Meter check of a transistor . . . .
Active mode operation . . . . . .
The common-emitter amplifier .
The common-collector amplifier
The common-base amplifier . . .
The cascode amplifier . . . . . . .
Biasing techniques . . . . . . . . .
Biasing calculations . . . . . . . .
4.10.1 Base Bias . . . . . . . . . . . .
4.10.2 Collector-feedback bias . . . .
4.10.3 Emitter-bias . . . . . . . . . .
4.10.4 Voltage divider bias . . . . .
Input and output coupling . . . .
Feedback . . . . . . . . . . . . . . .
Amplifier impedances . . . . . . .
Current mirrors . . . . . . . . . . .
Transistor ratings and packages
BJT quirks . . . . . . . . . . . . . .
4.16.1 Nonlinearity . . . . . . . . . .
4.16.2 Temperature drift . . . . . . .
4.16.3 Thermal runaway . . . . . . .
4.16.4 Junction capacitance . . . . .
4.16.5 Noise . . . . . . . . . . . . . .

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174
176
179
183
189
202
210
218
222
235
235
236
237
243
247
256
263
264
269
271
272
272
274
274
275

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

174

4.16.6 Thermal mismatch (problem with paralleling transistors) . . . . . . . . . 275
4.16.7 High frequency effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 278

4.1

Introduction

The invention of the bipolar transistor in 1948 ushered in a revolution in electronics. Technical
feats previously requiring relatively large, mechanically fragile, power-hungry vacuum tubes
were suddenly achievable with tiny, mechanically rugged, power-thrifty specks of crystalline
silicon. This revolution made possible the design and manufacture of lightweight, inexpensive
electronic devices that we now take for granted. Understanding how transistors function is of
paramount importance to anyone interested in understanding modern electronics.
My intent here is to focus as exclusively as possible on the practical function and application
of bipolar transistors, rather than to explore the quantum world of semiconductor theory. Discussions of holes and electrons are better left to another chapter in my opinion. Here I want
to explore how to use these components, not analyze their intimate internal details. I don’t
mean to downplay the importance of understanding semiconductor physics, but sometimes
an intense focus on solid-state physics detracts from understanding these devices’ functions
on a component level. In taking this approach, however, I assume that the reader possesses
a certain minimum knowledge of semiconductors: the difference between “P” and “N” doped
semiconductors, the functional characteristics of a PN (diode) junction, and the meanings of
the terms “reverse biased” and “forward biased.” If these concepts are unclear to you, it is best
to refer to earlier chapters in this book before proceeding with this one.
A bipolar transistor consists of a three-layer “sandwich” of doped (extrinsic) semiconductor
materials, either P-N-P in Figure 4.1(b) or N-P-N at (d). Each layer forming the transistor has
a specific name, and each layer is provided with a wire contact for connection to a circuit. The
schematic symbols are shown in Figure 4.1(a) and (d).
collector

collector

collector

collector
P
N

base

base

base

emitter
(a)

base

P

N
P
N

emitter
emitter

(b)

(c)

emitter

(d)

Figure 4.1: BJT transistor: (a) PNP schematic symbol, (b) physical layout (c) NPN symbol, (d)
layout.
The functional difference between a PNP transistor and an NPN transistor is the proper
biasing (polarity) of the junctions when operating. For any given state of operation, the current
directions and voltage polarities for each kind of transistor are exactly opposite each other.

4.1. INTRODUCTION

175

Bipolar transistors work as current-controlled current regulators. In other words, transistors restrict the amount of current passed according to a smaller, controlling current. The main
current that is controlled goes from collector to emitter, or from emitter to collector, depending
on the type of transistor it is (PNP or NPN, respectively). The small current that controls the
main current goes from base to emitter, or from emitter to base, once again depending on the
kind of transistor it is (PNP or NPN, respectively). According to the standards of semiconductor
symbology, the arrow always points against the direction of electron flow. (Figure 4.2)

B

C
E

= small, controlling current

B

C
E

= large, controlled current

Figure 4.2: Small electron base current controls large collector electron current flowing against
emitter arrow.
Bipolar transistors are called bipolar because the main flow of electrons through them takes
place in two types of semiconductor material: P and N, as the main current goes from emitter
to collector (or vice versa). In other words, two types of charge carriers – electrons and holes –
comprise this main current through the transistor.
As you can see, the controlling current and the controlled current always mesh together
through the emitter wire, and their electrons always flow against the direction of the transistor’s arrow. This is the first and foremost rule in the use of transistors: all currents must be
going in the proper directions for the device to work as a current regulator. The small, controlling current is usually referred to simply as the base current because it is the only current
that goes through the base wire of the transistor. Conversely, the large, controlled current is
referred to as the collector current because it is the only current that goes through the collector
wire. The emitter current is the sum of the base and collector currents, in compliance with
Kirchhoff ’s Current Law.
No current through the base of the transistor, shuts it off like an open switch and prevents
current through the collector. A base current, turns the transistor on like a closed switch and
allows a proportional amount of current through the collector. Collector current is primarily
limited by the base current, regardless of the amount of voltage available to push it. The next
section will explore in more detail the use of bipolar transistors as switching elements.
• REVIEW:
• Bipolar transistors are so named because the controlled current must go through two
types of semiconductor material: P and N. The current consists of both electron and hole

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

176

flow, in different parts of the transistor.
• Bipolar transistors consist of either a P-N-P or an N-P-N semiconductor “sandwich” structure.
• The three leads of a bipolar transistor are called the Emitter, Base, and Collector.
• Transistors function as current regulators by allowing a small current to control a larger
current. The amount of current allowed between collector and emitter is primarily determined by the amount of current moving between base and emitter.
• In order for a transistor to properly function as a current regulator, the controlling (base)
current and the controlled (collector) currents must be going in the proper directions:
meshing additively at the emitter and going against the emitter arrow symbol.

4.2

The transistor as a switch

Because a transistor’s collector current is proportionally limited by its base current, it can be
used as a sort of current-controlled switch. A relatively small flow of electrons sent through
the base of the transistor has the ability to exert control over a much larger flow of electrons
through the collector.
Suppose we had a lamp that we wanted to turn on and off with a switch. Such a circuit
would be extremely simple as in Figure 4.3(a).
For the sake of illustration, let’s insert a transistor in place of the switch to show how it can
control the flow of electrons through the lamp. Remember that the controlled current through
a transistor must go between collector and emitter. Since it is the current through the lamp
that we want to control, we must position the collector and emitter of our transistor where the
two contacts of the switch were. We must also make sure that the lamp’s current will move
against the direction of the emitter arrow symbol to ensure that the transistor’s junction bias
will be correct as in Figure 4.3(b).

+
switch

+
NPN
transistor

PNP
transistor
+

(a)

(b)

(c)

Figure 4.3: (a) mechanical switch, (b) NPN transistor switch, (c) PNP transistor switch.
A PNP transistor could also have been chosen for the job. Its application is shown in Figure 4.3(c).
The choice between NPN and PNP is really arbitrary. All that matters is that the proper
current directions are maintained for the sake of correct junction biasing (electron flow going
against the transistor symbol’s arrow).

4.2. THE TRANSISTOR AS A SWITCH

177

Going back to the NPN transistor in our example circuit, we are faced with the need to
add something more so that we can have base current. Without a connection to the base wire
of the transistor, base current will be zero, and the transistor cannot turn on, resulting in a
lamp that is always off. Remember that for an NPN transistor, base current must consist of
electrons flowing from emitter to base (against the emitter arrow symbol, just like the lamp
current). Perhaps the simplest thing to do would be to connect a switch between the base and
collector wires of the transistor as in Figure 4.4 (a).

+
switch
(a)

+
switch
(b)

Figure 4.4: Transistor: (a) cutoff, lamp off; (b) saturated, lamp on.
If the switch is open as in (Figure 4.4 (a), the base wire of the transistor will be left “floating”
(not connected to anything) and there will be no current through it. In this state, the transistor
is said to be cutoff. If the switch is closed as in (Figure 4.4 (b), however, electrons will be able
to flow from the emitter through to the base of the transistor, through the switch and up to
the left side of the lamp, back to the positive side of the battery. This base current will enable
a much larger flow of electrons from the emitter through to the collector, thus lighting up the
lamp. In this state of maximum circuit current, the transistor is said to be saturated.
Of course, it may seem pointless to use a transistor in this capacity to control the lamp.
After all, we’re still using a switch in the circuit, aren’t we? If we’re still using a switch to
control the lamp – if only indirectly – then what’s the point of having a transistor to control
the current? Why not just go back to our original circuit and use the switch directly to control
the lamp current?
Two points can be made here, actually. First is the fact that when used in this manner, the
switch contacts need only handle what little base current is necessary to turn the transistor on;
the transistor itself handles most of the lamp’s current. This may be an important advantage
if the switch has a low current rating: a small switch may be used to control a relatively highcurrent load. More important, the current-controlling behavior of the transistor enables us to
use something completely different to turn the lamp on or off. Consider Figure 4.5, where a
pair of solar cells provides 1 V to overcome the 0.7 VBE of the transistor to cause base current
flow, which in turn controls the lamp.
Or, we could use a thermocouple (many connected in series) to provide the necessary base
current to turn the transistor on in Figure 4.6.
Even a microphone (Figure 4.7) with enough voltage and current (from an amplifier) output
could turn the transistor on, provided its output is rectified from AC to DC so that the emitterbase PN junction within the transistor will always be forward-biased:
The point should be quite apparent by now: any sufficient source of DC current may be
used to turn the transistor on, and that source of current only need be a fraction of the current

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

178

solar
cell
Figure 4.5: Solar cell serves as light sensor.

thermocouple
+
source of
heat
Figure 4.6: A single thermocouple provides 10s of mV. Many in series could produce in excess
of the 0.7 V transistor VBE to cause base current flow and consequent collector current to the
lamp.

microphone

source of
sound

Figure 4.7: Amplified microphone signal is rectified to DC bias the base of the transistor providing a larger collector current.

4.3. METER CHECK OF A TRANSISTOR

179

needed to energize the lamp. Here we see the transistor functioning not only as a switch, but
as a true amplifier: using a relatively low-power signal to control a relatively large amount
of power. Please note that the actual power for lighting up the lamp comes from the battery
to the right of the schematic. It is not as though the small signal current from the solar cell,
thermocouple, or microphone is being magically transformed into a greater amount of power.
Rather, those small power sources are simply controlling the battery’s power to light up the
lamp.
• REVIEW:
• Transistors may be used as switching elements to control DC power to a load. The
switched (controlled) current goes between emitter and collector; the controlling current
goes between emitter and base.
• When a transistor has zero current through it, it is said to be in a state of cutoff (fully
nonconducting).
• When a transistor has maximum current through it, it is said to be in a state of saturation
(fully conducting).

4.3

Meter check of a transistor

Bipolar transistors are constructed of a three-layer semiconductor “sandwich,” either PNP or
NPN. As such, transistors register as two diodes connected back-to-back when tested with a
multimeter’s “resistance” or “diode check” function as illustrated in Figure 4.8. Low resistance
readings on the base with the black negative (-) leads correspond to an N-type base in a PNP
transistor. On the symbol, the N-type material corresponds to the “non-pointing” end of the
base-emitter junction, the base. The P-type emitter corresponds to “pointing” end of the baseemitter junction the emitter.

A

V

A

V

A

V

A

V

A

ecto

V

A
OFF

V

A

OFF

r

OFF

r

V

ecto

coll

A

coll

V

OFF

base
A

COM

A

base
A

COM

A

COM

r

itte

em

COM

r

itte

em

Figure 4.8: PNP transistor meter check: (a) forward B-E, B-C, resistance is low; (b) reverse
B-E, B-C, resistance is ∞.

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

180

Here I’m assuming the use of a multimeter with only a single continuity range (resistance)
function to check the PN junctions. Some multimeters are equipped with two separate continuity check functions: resistance and “diode check,” each with its own purpose. If your meter
has a designated “diode check” function, use that rather than the “resistance” range, and the
meter will display the actual forward voltage of the PN junction and not just whether or not it
conducts current.
Meter readings will be exactly opposite, of course, for an NPN transistor, with both PN
junctions facing the other way. Low resistance readings with the red (+) lead on the base is the
“opposite” condition for the NPN transistor.
If a multimeter with a “diode check” function is used in this test, it will be found that
the emitter-base junction possesses a slightly greater forward voltage drop than the collectorbase junction. This forward voltage difference is due to the disparity in doping concentration
between the emitter and collector regions of the transistor: the emitter is a much more heavily
doped piece of semiconductor material than the collector, causing its junction with the base to
produce a higher forward voltage drop.
Knowing this, it becomes possible to determine which wire is which on an unmarked transistor. This is important because transistor packaging, unfortunately, is not standardized. All
bipolar transistors have three wires, of course, but the positions of the three wires on the actual
physical package are not arranged in any universal, standardized order.
Suppose a technician finds a bipolar transistor and proceeds to measure continuity with a
multimeter set in the “diode check” mode. Measuring between pairs of wires and recording the
values displayed by the meter, the technician obtains the data in Figure 4.9.

• Meter touching wire 1 (+) and 2 (-): “OL”
• Meter touching wire 1 (-) and 2 (+): “OL”
• Meter touching wire 1 (+) and 3 (-): 0.655 V
• Meter touching wire 1 (-) and 3 (+): “OL”

1

• Meter touching wire 2 (+) and 3 (-): 0.621 V

2

3

• Meter touching wire 2 (-) and 3 (+): “OL”

Figure 4.9: Unknown bipolar transistor. Which terminals are emitter, base, and collector?
Ω-meter readings between terminals.
The only combinations of test points giving conducting meter readings are wires 1 and 3
(red test lead on 1 and black test lead on 3), and wires 2 and 3 (red test lead on 2 and black test
lead on 3). These two readings must indicate forward biasing of the emitter-to-base junction
(0.655 volts) and the collector-to-base junction (0.621 volts).
Now we look for the one wire common to both sets of conductive readings. It must be the
base connection of the transistor, because the base is the only layer of the three-layer device
common to both sets of PN junctions (emitter-base and collector-base). In this example, that
wire is number 3, being common to both the 1-3 and the 2-3 test point combinations. In both

4.3. METER CHECK OF A TRANSISTOR

181

those sets of meter readings, the black (-) meter test lead was touching wire 3, which tells us
that the base of this transistor is made of N-type semiconductor material (black = negative).
Thus, the transistor is a PNP with base on wire 3, emitter on wire 1 and collector on wire 2 as
described in Figure 4.10.

• E and C high R: 1 (+) and 2 (-): “OL”
• C and E high R: 1 (-) and 2 (+): “OL”
• E and B forward: 1 (+) and 3 (-): 0.655 V

1
Emitter

• E and B reverse: 1 (-) and 3 (+): “OL”

2
3
Collector Base

• C and B forward: 2 (+) and 3 (-): 0.621 V
• C and B reverse: 2 (-) and 3 (+): “OL”

Figure 4.10: BJT terminals identified by Ω-meter.
Please note that the base wire in this example is not the middle lead of the transistor, as one
might expect from the three-layer “sandwich” model of a bipolar transistor. This is quite often
the case, and tends to confuse new students of electronics. The only way to be sure which lead
is which is by a meter check, or by referencing the manufacturer’s “data sheet” documentation
on that particular part number of transistor.
Knowing that a bipolar transistor behaves as two back-to-back diodes when tested with a
conductivity meter is helpful for identifying an unknown transistor purely by meter readings.
It is also helpful for a quick functional check of the transistor. If the technician were to measure continuity in any more than two or any less than two of the six test lead combinations,
he or she would immediately know that the transistor was defective (or else that it wasn’t a
bipolar transistor but rather something else – a distinct possibility if no part numbers can be
referenced for sure identification!). However, the “two diode” model of the transistor fails to
explain how or why it acts as an amplifying device.
To better illustrate this paradox, let’s examine one of the transistor switch circuits using the
physical diagram in Figure 4.11 rather than the schematic symbol to represent the transistor.
This way the two PN junctions will be easier to see.
A grey-colored diagonal arrow shows the direction of electron flow through the emitter-base
junction. This part makes sense, since the electrons are flowing from the N-type emitter to the
P-type base: the junction is obviously forward-biased. However, the base-collector junction is
another matter entirely. Notice how the grey-colored thick arrow is pointing in the direction
of electron flow (up-wards) from base to collector. With the base made of P-type material and
the collector of N-type material, this direction of electron flow is clearly backwards to the direction normally associated with a PN junction! A normal PN junction wouldn’t permit this
“backward” direction of flow, at least not without offering significant opposition. However, a
saturated transistor shows very little opposition to electrons, all the way from emitter to collector, as evidenced by the lamp’s illumination!

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

182

solar
cell

N
P
N

collector
base
emitter

+

Figure 4.11: A small base current flowing in the forward biased base-emitter junction allows a
large current flow through the reverse biased base-collector junction.
Clearly then, something is going on here that defies the simple “two-diode” explanatory
model of the bipolar transistor. When I was first learning about transistor operation, I tried to
construct my own transistor from two back-to-back diodes, as in Figure 4.12.

no light!

solar
cell

no current!

Figure 4.12: A pair of back-to-back diodes don’t act like a transistor!
My circuit didn’t work, and I was mystified. However useful the “two diode” description
of a transistor might be for testing purposes, it doesn’t explain how a transistor behaves as a
controlled switch.
What happens in a transistor is this: the reverse bias of the base-collector junction prevents
collector current when the transistor is in cutoff mode (that is, when there is no base current).
If the base-emitter junction is forward biased by the controlling signal, the normally-blocking
action of the base-collector junction is overridden and current is permitted through the collector, despite the fact that electrons are going the “wrong way” through that PN junction. This
action is dependent on the quantum physics of semiconductor junctions, and can only take
place when the two junctions are properly spaced and the doping concentrations of the three
layers are properly proportioned. Two diodes wired in series fail to meet these criteria; the
top diode can never “turn on” when it is reversed biased, no matter how much current goes
through the bottom diode in the base wire loop. See (page ??) for more details.
That doping concentrations play a crucial part in the special abilities of the transistor is
further evidenced by the fact that collector and emitter are not interchangeable. If the transistor is merely viewed as two back-to-back PN junctions, or merely as a plain N-P-N or P-N-P
sandwich of materials, it may seem as though either end of the transistor could serve as collec-

4.4. ACTIVE MODE OPERATION

183

tor or emitter. This, however, is not true. If connected “backwards” in a circuit, a base-collector
current will fail to control current between collector and emitter. Despite the fact that both the
emitter and collector layers of a bipolar transistor are of the same doping type (either N or P),
collector and emitter are definitely not identical!
Current through the emitter-base junction allows current through the reverse-biased basecollector junction. The action of base current can be thought of as “opening a gate” for current
through the collector. More specifically, any given amount of emitter-to-base current permits a
limited amount of base-to-collector current. For every electron that passes through the emitterbase junction and on through the base wire, a certain, number of electrons pass through the
base-collector junction and no more.
In the next section, this current-limiting of the transistor will be investigated in more detail.
• REVIEW:
• Tested with a multimeter in the “resistance” or “diode check” modes, a transistor behaves
like two back-to-back PN (diode) junctions.
• The emitter-base PN junction has a slightly greater forward voltage drop than the collectorbase PN junction, because of heavier doping of the emitter semiconductor layer.
• The reverse-biased base-collector junction normally blocks any current from going through
the transistor between emitter and collector. However, that junction begins to conduct if
current is drawn through the base wire. Base current may be thought of as “opening a
gate” for a certain, limited amount of current through the collector.

4.4

Active mode operation

When a transistor is in the fully-off state (like an open switch), it is said to be cutoff. Conversely, when it is fully conductive between emitter and collector (passing as much current
through the collector as the collector power supply and load will allow), it is said to be saturated. These are the two modes of operation explored thus far in using the transistor as a
switch.
However, bipolar transistors don’t have to be restricted to these two extreme modes of operation. As we learned in the previous section, base current “opens a gate” for a limited amount
of current through the collector. If this limit for the controlled current is greater than zero
but less than the maximum allowed by the power supply and load circuit, the transistor will
“throttle” the collector current in a mode somewhere between cutoff and saturation. This mode
of operation is called the active mode.
An automotive analogy for transistor operation is as follows: cutoff is the condition of no
motive force generated by the mechanical parts of the car to make it move. In cutoff mode, the
brake is engaged (zero base current), preventing motion (collector current). Active mode is the
automobile cruising at a constant, controlled speed (constant, controlled collector current) as
dictated by the driver. Saturation the automobile driving up a steep hill that prevents it from
going as fast as the driver wishes. In other words, a “saturated” automobile is one with the
accelerator pedal pushed all the way down (base current calling for more collector current than
can be provided by the power supply/load circuit).

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

184

Let’s set up a circuit for SPICE simulation to demonstrate what happens when a transistor
is in its active mode of operation. (Figure 4.13)

Vammeter
2

3
0V

1

V1

Q1

Current I1
source
0

0

0

bipolar transistor
simulation
i1 0 1 dc 20u
q1 2 1 0 mod1
vammeter 3 2 dc 0
v1 3 0 dc
.model mod1 npn
.dc v1 0 2 0.05
.plot dc
i(vammeter)
.end

Figure 4.13: Circuit for “active mode” SPICE simulation, and netlist.
“Q” is the standard letter designation for a transistor in a schematic diagram, just as “R”
is for resistor and “C” is for capacitor. In this circuit, we have an NPN transistor powered
by a battery (V1 ) and controlled by current through a current source (I1 ). A current source
is a device that outputs a specific amount of current, generating as much or as little voltage
across its terminals to ensure that exact amount of current through it. Current sources are
notoriously difficult to find in nature (unlike voltage sources, which by contrast attempt to
maintain a constant voltage, outputting as much or as little current in the fulfillment of that
task), but can be simulated with a small collection of electronic components. As we are about
to see, transistors themselves tend to mimic the constant-current behavior of a current source
in their ability to regulate current at a fixed value.
In the SPICE simulation, we’ll set the current source at a constant value of 20 µA, then
vary the voltage source (V1 ) over a range of 0 to 2 volts and monitor how much current goes
through it. The “dummy” battery (Vammeter ) in Figure 4.13 with its output of 0 volts serves
merely to provide SPICE with a circuit element for current measurement.

The constant base current of 20 µA sets a collector current limit of 2 mA, exactly 100 times
as much. Notice how flat the curve is in (Figure 4.15 for collector current over the range of
battery voltage from 0 to 2 volts. The only exception to this featureless plot is at the very
beginning, where the battery increases from 0 volts to 0.25 volts. There, the collector current
increases rapidly from 0 amps to its limit of 2 mA.
Let’s see what happens if we vary the battery voltage over a wider range, this time from 0
to 50 volts. We’ll keep the base current steady at 20 µA. (Figure 4.15)
Same result! The collector current in Figure 4.15 holds absolutely steady at 2 mA, although
the battery (v1) voltage varies all the way from 0 to 50 volts. It would appear from our simulation that collector-to-emitter voltage has little effect over collector current, except at very low
levels (just above 0 volts). The transistor is acting as a current regulator, allowing exactly 2
mA through the collector and no more.

4.4. ACTIVE MODE OPERATION

185

Figure 4.14: A Sweeping collector voltage 0 to 2 V with base current constant at 20 µA yields
constant 2 mA collector current in the saturation region.

bipolar transistor
simulation
i1 0 1 dc 20u
q1 2 1 0 mod1
vammeter 3 2 dc 0
v1 3 0 dc
.model mod1 npn
.dc v1 0 50 2
.plot dc
i(vammeter)
.end
Figure 4.15: Sweeping collector voltage 0 to 50 V with base current constant at 20 µA yields
constant 2 mA collector current.

186

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

Now let’s see what happens if we increase the controlling (I1 ) current from 20 µA to 75
µA, once again sweeping the battery (V1 ) voltage from 0 to 50 volts and graphing the collector
current in Figure 4.16.

bipolar transistor
simulation
i1 0 1 dc 75u
q1 2 1 0 mod1
vammeter 3 2 dc 0
v1 3 0 dc
.model mod1 npn
.dc v1 0 50 2 i1
15u 75u 15u
.plot dc
i(vammeter)
.end
Figure 4.16: Sweeping collector voltage 0 to 50 V (.dc v1 0 50 2) with base current constant at
75 µA yields constant 7.5 mA collector current. Other curves are generated by current sweep
(i1 15u 75u 15u) in DC analysis statement (.dc v1 0 50 2 i1 15u 75u 15u).
Not surprisingly, SPICE gives us a similar plot: a flat line, holding steady this time at 7.5
mA – exactly 100 times the base current – over the range of battery voltages from just above
0 volts to 50 volts. It appears that the base current is the deciding factor for collector current,
the V1 battery voltage being irrelevant as long as it is above a certain minimum level.
This voltage/current relationship is entirely different from what we’re used to seeing across
a resistor. With a resistor, current increases linearly as the voltage across it increases. Here,
with a transistor, current from emitter to collector stays limited at a fixed, maximum value no
matter how high the voltage across emitter and collector increases.
Often it is useful to superimpose several collector current/voltage graphs for different base
currents on the same graph as in Figure 4.17. A collection of curves like this – one curve plotted
for each distinct level of base current – for a particular transistor is called the transistor’s
characteristic curves:
Each curve on the graph reflects the collector current of the transistor, plotted over a range
of collector-to-emitter voltages, for a given amount of base current. Since a transistor tends to
act as a current regulator, limiting collector current to a proportion set by the base current, it is
useful to express this proportion as a standard transistor performance measure. Specifically,
the ratio of collector current to base current is known as the Beta ratio (symbolized by the
Greek letter β):

4.4. ACTIVE MODE OPERATION

187

9

Ibase = 75 µA

8
7

Icollector
(mA)

6
5

Ibase = 40 µA

4
3

Ibase = 20 µA

2

Ibase = 5 µA

1
0
0

1

2

3

4

5

6

7

8

9

10

11

12

13

14

Ecollector-to-emitter (V)
Figure 4.17: Voltage collector to emitter vs collector current for various base currents.

β=

Icollector
Ibase

β is also known as hfe
Sometimes the β ratio is designated as “hf e ,” a label used in a branch of mathematical semiconductor analysis known as “hybrid parameters” which strives to achieve precise predictions
of transistor performance with detailed equations. Hybrid parameter variables are many, but
each is labeled with the general letter “h” and a specific subscript. The variable “hf e ” is just
another (standardized) way of expressing the ratio of collector current to base current, and is
interchangeable with “β.” The β ratio is unitless.
β for any transistor is determined by its design: it cannot be altered after manufacture. It is
rare to have two transistors of the same design exactly match because of the physical variables
afecting β . If a circuit design relies on equal β ratios between multiple transistors, “matched
sets” of transistors may be purchased at extra cost. However, it is generally considered bad
design practice to engineer circuits with such dependencies.
The β of a transistor does not remain stable for all operating conditions. For an actual
transistor, the β ratio may vary by a factor of over 3 within its operating current limits. For
example, a transistor with advertised β of 50 may actually test with Ic /Ib ratios as low as 30
and as high as 100, depending on the amount of collector current, the transistor’s temperature,
and frequency of amplified signal, among other factors. For tutorial purposes it is adequate to
assume a constant β for any given transistor; realize that real life is not that simple!
Sometimes it is helpful for comprehension to “model” complex electronic components with a
collection of simpler, better-understood components. The model in Figure 4.18 is used in many

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

188
introductory electronics texts.

C

C
B

B
E
NPN
diode-rheostat
model

E

Figure 4.18: Elementary diode resistor transistor model.
This model casts the transistor as a combination of diode and rheostat (variable resistor).
Current through the base-emitter diode controls the resistance of the collector-emitter rheostat (as implied by the dashed line connecting the two components), thus controlling collector
current. An NPN transistor is modeled in the figure shown, but a PNP transistor would be
only slightly different (only the base-emitter diode would be reversed). This model succeeds in
illustrating the basic concept of transistor amplification: how the base current signal can exert
control over the collector current. However, I don’t like this model because it miscommunicates
the notion of a set amount of collector-emitter resistance for a given amount of base current.
If this were true, the transistor wouldn’t regulate collector current at all like the characteristic curves show. Instead of the collector current curves flattening out after their brief rise as
the collector-emitter voltage increases, the collector current would be directly proportional to
collector-emitter voltage, rising steadily in a straight line on the graph.
A better transistor model, often seen in more advanced textbooks, is shown in Figure 4.19.

C
C
B

B
E

NPN
diode-current source
model

E

Figure 4.19: Current source model of transistor.
It casts the transistor as a combination of diode and current source, the output of the current source being set at a multiple (β ratio) of the base current. This model is far more accurate

4.5. THE COMMON-EMITTER AMPLIFIER

189

in depicting the true input/output characteristics of a transistor: base current establishes a certain amount of collector current, rather than a certain amount of collector-emitter resistance
as the first model implies. Also, this model is favored when performing network analysis on
transistor circuits, the current source being a well-understood theoretical component. Unfortunately, using a current source to model the transistor’s current-controlling behavior can be
misleading: in no way will the transistor ever act as a source of electrical energy. The current
source does not model the fact that its source of energy is a external power supply, similar to
an amplifier.
• REVIEW:
• A transistor is said to be in its active mode if it is operating somewhere between fully on
(saturated) and fully off (cutoff).
• Base current regulates collector current. By regulate, we mean that no more collector
current can exist than what is allowed by the base current.
• The ratio between collector current and base current is called “Beta” (β) or “hf e ”.
• β ratios are different for every transistor, and
• β changes for different operating conditions.

4.5

The common-emitter amplifier

At the beginning of this chapter we saw how transistors could be used as switches, operating in
either their “saturation” or “cutoff ” modes. In the last section we saw how transistors behave
within their “active” modes, between the far limits of saturation and cutoff. Because transistors
are able to control current in an analog (infinitely divisible) fashion, they find use as amplifiers
for analog signals.
One of the simpler transistor amplifier circuits to study previously illustrated the transistor’s switching ability. (Figure 4.20)

solar
cell
Figure 4.20: NPN transistor as a simple switch.
It is called the common-emitter configuration because (ignoring the power supply battery)
both the signal source and the load share the emitter lead as a common connection point shown
in Figure 4.21. This is not the only way in which a transistor may be used as an amplifier, as
we will see in later sections of this chapter.

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

190

Vout

B
solar
cell

Vin

C

Load

E

Figure 4.21: Common-emitter amplifier: The input and output signals both share a connection
to the emitter.
Before, a small solar cell current saturated a transistor, illuminating a lamp. Knowing now
that transistors are able to “throttle” their collector currents according to the amount of base
current supplied by an input signal source, we should see that the brightness of the lamp in
this circuit is controllable by the solar cell’s light exposure. When there is just a little light
shone on the solar cell, the lamp will glow dimly. The lamp’s brightness will steadily increase
as more light falls on the solar cell.
Suppose that we were interested in using the solar cell as a light intensity instrument. We
want to measure the intensity of incident light with the solar cell by using its output current
to drive a meter movement. It is possible to directly connect a meter movement to a solar
cell (Figure 4.22) for this purpose. In fact, the simplest light-exposure meters for photography
work are designed like this.

+

-

solar
cell
Figure 4.22: High intensity light directly drives light meter.
Although this approach might work for moderate light intensity measurements, it would
not work as well for low light intensity measurements. Because the solar cell has to supply the
meter movement’s power needs, the system is necessarily limited in its sensitivity. Supposing
that our need here is to measure very low-level light intensities, we are pressed to find another
solution.
Perhaps the most direct solution to this measurement problem is to use a transistor (Figure 4.23) to amplify the solar cell’s current so that more meter deflection may be obtained for
less incident light.
Current through the meter movement in this circuit will be β times the solar cell current.
With a transistor β of 100, this represents a substantial increase in measurement sensitivity.
It is prudent to point out that the additional power to move the meter needle comes from the
battery on the far right of the circuit, not the solar cell itself. All the solar cell’s current does

4.5. THE COMMON-EMITTER AMPLIFIER

191

-

+
+

+
solar cell

-

Figure 4.23: Cell current must be amplified for low intensity light.
is control battery current to the meter to provide a greater meter reading than the solar cell
could provide unaided.
Because the transistor is a current-regulating device, and because meter movement indications are based on the current through the movement coil, meter indication in this circuit
should depend only on the current from the solar cell, not on the amount of voltage provided by
the battery. This means the accuracy of the circuit will be independent of battery condition, a
significant feature! All that is required of the battery is a certain minimum voltage and current
output ability to drive the meter full-scale.
Another way in which the common-emitter configuration may be used is to produce an
output voltage derived from the input signal, rather than a specific output current. Let’s replace
the meter movement with a plain resistor and measure voltage between collector and emitter
in Figure 4.24

R
+
+
solar cell

Voutput

-

Figure 4.24: Common emitter amplifier develops voltage output due to current through load
resistor.
With the solar cell darkened (no current), the transistor will be in cutoff mode and behave
as an open switch between collector and emitter. This will produce maximum voltage drop
between collector and emitter for maximum Voutput , equal to the full voltage of the battery.
At full power (maximum light exposure), the solar cell will drive the transistor into saturation mode, making it behave like a closed switch between collector and emitter. The result
will be minimum voltage drop between collector and emitter, or almost zero output voltage.
In actuality, a saturated transistor can never achieve zero voltage drop between collector and
emitter because of the two PN junctions through which collector current must travel. However, this “collector-emitter saturation voltage” will be fairly low, around several tenths of a

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

192

volt, depending on the specific transistor used.
For light exposure levels somewhere between zero and maximum solar cell output, the transistor will be in its active mode, and the output voltage will be somewhere between zero and
full battery voltage. An important quality to note here about the common-emitter configuration
is that the output voltage is inversely proportional to the input signal strength. That is, the
output voltage decreases as the input signal increases. For this reason, the common-emitter
amplifier configuration is referred to as an inverting amplifier.
A quick SPICE simulation (Figure 4.26) of the circuit in Figure 4.25 will verify our qualitative conclusions about this amplifier circuit.

R

2

3

5 kΩ
1

Q1

I1
0

0

Voutput

+
V1

15 V
0

*common-emitter
amplifier
i1 0 1 dc
q1 2 1 0 mod1
r 3 2 5000
v1 3 0 dc 15
.model mod1 npn
.dc i1 0 50u 2u
.plot dc v(2,0)
.end

Figure 4.25: Common emitter schematic with node numbers and corresponding SPICE netlist.

Figure 4.26: Common emitter: collector voltage output vs base current input.
At the beginning of the simulation in Figure 4.26 where the current source (solar cell) is
outputting zero current, the transistor is in cutoff mode and the full 15 volts from the battery

4.5. THE COMMON-EMITTER AMPLIFIER

193

is shown at the amplifier output (between nodes 2 and 0). As the solar cell’s current begins to
increase, the output voltage proportionally decreases, until the transistor reaches saturation
at 30 µA of base current (3 mA of collector current). Notice how the output voltage trace on
the graph is perfectly linear (1 volt steps from 15 volts to 1 volt) until the point of saturation,
where it never quite reaches zero. This is the effect mentioned earlier, where a saturated
transistor can never achieve exactly zero voltage drop between collector and emitter due to
internal junction effects. What we do see is a sharp output voltage decrease from 1 volt to
0.2261 volts as the input current increases from 28 µA to 30 µA, and then a continuing decrease
in output voltage from then on (albeit in progressively smaller steps). The lowest the output
voltage ever gets in this simulation is 0.1299 volts, asymptotically approaching zero.
So far, we’ve seen the transistor used as an amplifier for DC signals. In the solar cell light
meter example, we were interested in amplifying the DC output of the solar cell to drive a
DC meter movement, or to produce a DC output voltage. However, this is not the only way
in which a transistor may be employed as an amplifier. Often an AC amplifier for amplifying
alternating current and voltage signals is desired. One common application of this is in audio
electronics (radios, televisions, and public-address systems). Earlier, we saw an example of the
audio output of a tuning fork activating a transistor switch. (Figure 4.27) Let’s see if we can
modify that circuit to send power to a speaker rather than to a lamp in Figure 4.28.

microphone

source of
sound

Figure 4.27: Transistor switch activated by audio.
In the original circuit, a full-wave bridge rectifier was used to convert the microphone’s AC
output signal into a DC voltage to drive the input of the transistor. All we cared about here was
turning the lamp on with a sound signal from the microphone, and this arrangement sufficed
for that purpose. But now we want to actually reproduce the AC signal and drive a speaker.
This means we cannot rectify the microphone’s output anymore, because we need undistorted
AC signal to drive the transistor! Let’s remove the bridge rectifier and replace the lamp with a
speaker:
Since the microphone may produce voltages exceeding the forward voltage drop of the baseemitter PN (diode) junction, I’ve placed a resistor in series with the microphone. Let’s simulate
the circuit in Figure 4.29 with SPICE. The netlist is included in (Figure 4.30)
The simulation plots (Figure 4.30) both the input voltage (an AC signal of 1.5 volt peak
amplitude and 2000 Hz frequency) and the current through the 15 volt battery, which is the
same as the current through the speaker. What we see here is a full AC sine wave alternating
in both positive and negative directions, and a half-wave output current waveform that only

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

194

speaker
microphone
source of
sound

Figure 4.28: Common emitter amplifier drives speaker with audio frequency signal.

8Ω

3
1
Vinput
1.5 V
2 kHz

R1

4

speaker

2

Q1

V1

15 V

1 kΩ
0

0

0

Figure 4.29: SPICE version of common emitter audio amplifier.

common-emitter
amplifier
vinput 1 0 sin (0
1.5 2000 0 0)
r1 1 2 1k
q1 3 2 0 mod1
rspkr 3 4 8
v1 4 0 dc 15
.model mod1 npn
.tran 0.02m 0.74m
.plot tran v(1,0)
i(v1)
.end
Figure 4.30: Signal clipped at collector due to lack of DC base bias.

4.5. THE COMMON-EMITTER AMPLIFIER

195

pulses in one direction. If we were actually driving a speaker with this waveform, the sound
produced would be horribly distorted.
What’s wrong with the circuit? Why won’t it faithfully reproduce the entire AC waveform
from the microphone? The answer to this question is found by close inspection of the transistor
diode current source model in Figure 4.31.

C
C
B

B
E

NPN
diode-current source
model

E

Figure 4.31: The model shows that base current flow in on direction.
Collector current is controlled, or regulated, through the constant-current mechanism according to the pace set by the current through the base-emitter diode. Note that both current
paths through the transistor are monodirectional: one way only! Despite our intent to use the
transistor to amplify an AC signal, it is essentially a DC device, capable of handling currents
in a single direction. We may apply an AC voltage input signal between the base and emitter,
but electrons cannot flow in that circuit during the part of the cycle that reverse-biases the
base-emitter diode junction. Therefore, the transistor will remain in cutoff mode throughout
that portion of the cycle. It will “turn on” in its active mode only when the input voltage is
of the correct polarity to forward-bias the base-emitter diode, and only when that voltage is
sufficiently high to overcome the diode’s forward voltage drop. Remember that bipolar transistors are current-controlled devices: they regulate collector current based on the existence of
base-to-emitter current, not base-to-emitter voltage.
The only way we can get the transistor to reproduce the entire waveform as current through
the speaker is to keep the transistor in its active mode the entire time. This means we must
maintain current through the base during the entire input waveform cycle. Consequently, the
base-emitter diode junction must be kept forward-biased at all times. Fortunately, this can
be accomplished with a DC bias voltage added to the input signal. By connecting a sufficient
DC voltage in series with the AC signal source, forward-bias can be maintained at all points
throughout the wave cycle. (Figure 4.32)
With the bias voltage source of 2.3 volts in place, the transistor remains in its active mode
throughout the entire cycle of the wave, faithfully reproducing the waveform at the speaker.
(Figure 4.33) Notice that the input voltage (measured between nodes 1 and 0) fluctuates between about 0.8 volts and 3.8 volts, a peak-to-peak voltage of 3 volts just as expected (source
voltage = 1.5 volts peak). The output (speaker) current varies between zero and almost 300
mA, 180o out of phase with the input (microphone) signal.
The illustration in Figure 4.34 is another view of the same circuit, this time with a few

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

196

speaker
8Ω

3
1
Vinput
1.5 V
2 kHz

R1

2

Q1

4

V1

15 V

1 kΩ
+
5
Vbias
2.3 V

0

0

Figure 4.32: Vbias keeps transistor in the active region.

common-emitter
amplifier
vinput 1 5 sin (0
1.5 2000 0 0)
vbias 5 0 dc 2.3
r1 1 2 1k
q1 3 2 0 mod1
rspkr 3 4 8
v1 4 0 dc 15
.model mod1 npn
.tran 0.02m 0.78m
.plot tran v(1,0)
i(v1)
.end
Figure 4.33: Undistorted output current I(v(1) due to Vbias

4.5. THE COMMON-EMITTER AMPLIFIER

197

oscilloscopes (“scopemeters”) connected at crucial points to display all the pertinent signals.

speaker
8Ω
V1

R1
Vinput
1.5 V
2 kHz

Q1

1 kΩ
+

+

15 V
-

Vbias
Figure 4.34: Input is biased upward at base. Output is inverted.
The need for biasing a transistor amplifier circuit to obtain full waveform reproduction is
an important consideration. A separate section of this chapter will be devoted entirely to the
subject biasing and biasing techniques. For now, it is enough to understand that biasing may
be necessary for proper voltage and current output from the amplifier.
Now that we have a functioning amplifier circuit, we can investigate its voltage, current,
and power gains. The generic transistor used in these SPICE analyses has a β of 100, as
indicated by the short transistor statistics printout included in the text output in Table 4.1
(these statistics were cut from the last two analyses for brevity’s sake).
Table 4.1: BJT SPICE model parameters.
type
is
bf
nf
br
nr

npn
1.00E-16
100.000
1.000
1.000
1.000

β is listed under the abbreviation “bf,” which actually stands for “beta, forward”. If we
wanted to insert our own β ratio for an analysis, we could have done so on the .model line of
the SPICE netlist.
Since β is the ratio of collector current to base current, and we have our load connected in
series with the collector terminal of the transistor and our source connected in series with the
base, the ratio of output current to input current is equal to beta. Thus, our current gain for
this example amplifier is 100, or 40 dB.
Voltage gain is a little more complicated to figure than current gain for this circuit. As
always, voltage gain is defined as the ratio of output voltage divided by input voltage. In order
to experimentally determine this, we modify our last SPICE analysis to plot output voltage
rather than output current so we have two voltage plots to compare in Figure 4.35.

198

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

common-emitter
amplifier
vinput 1 5 sin (0
1.5 2000 0 0)
vbias 5 0 dc 2.3
r1 1 2 1k
q1 3 2 0 mod1
rspkr 3 4 8
v1 4 0 dc 15
.model mod1 npn
.tran 0.02m 0.78m
.plot tran v(1,0)
v(3)
.end
Figure 4.35: V(3), the output voltage across rspkr , compared to the input.

Plotted on the same scale (from 0 to 4 volts), we see that the output waveform in Figure 4.35
To be honest, this low voltage gain is not characteristic to all common-emitter amplifiers.
It is a consequence of the great disparity between the input and load resistances. Our input
resistance (R1 ) here is 1000 Ω, while the load (speaker) is only 8 Ω. Because the current gain
of this amplifier is determined solely by the β of the transistor, and because that β figure
is fixed, the current gain for this amplifier won’t change with variations in either of these
resistances. However, voltage gain is dependent on these resistances. If we alter the load
resistance, making it a larger value, it will drop a proportionately greater voltage for its range
of load currents, resulting in a larger output waveform. Let’s try another simulation, only this
time with a 30 Ω in Figure 4.36 load instead of an 8 Ω load.
This time the output voltage waveform in Figure 4.36 is significantly greater in amplitude
than the input waveform. Looking closely, we can see that the output waveform crests between
0 and about 9 volts: approximately 3 times the amplitude of the input voltage.
We can do another computer analysis of this circuit, this time instructing SPICE to analyze
it from an AC point of view, giving us peak voltage figures for input and output instead of a
time-based plot of the waveforms. (Table 4.2)
Peak voltage measurements of input and output show an input of 1.5 volts and an output
of 4.418 volts. This gives us a voltage gain ratio of 2.9453 (4.418 V / 1.5 V), or 9.3827 dB.

4.5. THE COMMON-EMITTER AMPLIFIER

199

common-emitter
amplifier
vinput 1 5 sin (0
1.5 2000 0 0)
vbias 5 0 dc 2.3
r1 1 2 1k
q1 3 2 0 mod1
rspkr 3 4 30
v1 4 0 dc 15
.model mod1 npn
.tran 0.02m 0.78m
.plot tran v(1,0)
v(3)
.end
Figure 4.36: Increasing rspkr to 30 Ω increases the output voltage.

Table 4.2: SPICE netlist for printing AC input and output voltages.
common-emitter amplifier
vinput 1 5 ac 1.5
vbias 5 0 dc 2.3
r1 1 2 1k
q1 3 2 0 mod1
rspkr 3 4 30
v1 4 0 dc 15
.model mod1 npn
.ac lin 1 2000 2000
.print ac v(1,0) v(4,3)
.end
freq
v(1)
v(4,3)
2.000E+03
1.500E+00
4.418E+00

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

200

AV =

Vout
Vin

AV = 4.418 V
1.5 V
AV = 2.9453
AV(dB) = 20 log AV(ratio)
AV(dB) = 20 log 2.9453
AV(dB) = 9.3827 dB
Because the current gain of the common-emitter amplifier is fixed by β, and since the input and output voltages will be equal to the input and output currents multiplied by their
respective resistors, we can derive an equation for approximate voltage gain:

AV = β

Rout
Rin

AV = (100)

30 Ω
1000 Ω

AV = 3
AV(dB) = 20 log AV(ratio)
AV(dB) = 20 log 3
AV(dB) = 9.5424 dB
As you can see, the predicted results for voltage gain are quite close to the simulated results.
With perfectly linear transistor behavior, the two sets of figures would exactly match. SPICE
does a reasonable job of accounting for the many “quirks” of bipolar transistor function in its
analysis, hence the slight mismatch in voltage gain based on SPICE’s output.
These voltage gains remain the same regardless of where we measure output voltage in
the circuit: across collector and emitter, or across the series load resistor as we did in the
last analysis. The amount of output voltage change for any given amount of input voltage
will remain the same. Consider the two following SPICE analyses as proof of this. The first
simulation in Figure 4.37 is time-based, to provide a plot of input and output voltages. You
will notice that the two signals are 180o out of phase with each other. The second simulation
in Table 4.3 is an AC analysis, to provide simple, peak voltage readings for input and output.
We still have a peak output voltage of 4.418 volts with a peak input voltage of 1.5 volts. The
only difference from the last set of simulations is the phase of the output voltage.
So far, the example circuits shown in this section have all used NPN transistors. PNP tran-

4.5. THE COMMON-EMITTER AMPLIFIER

201

common-emitter
amplifier
vinput 1 5 sin (0
1.5 2000 0 0)
vbias 5 0 dc 2.3
r1 1 2 1k
q1 3 2 0 mod1
rspkr 3 4 30
v1 4 0 dc 15
.model mod1 npn
.tran 0.02m 0.74m
.plot tran v(1,0)
v(3,0)
.end
Figure 4.37: Common-emitter amplifier shows a voltage gain with Rspkr =30Ω

Table 4.3: SPICE netlist for AC analysis
common-emitter amplifier
vinput 1 5 ac 1.5
vbias 5 0 dc 2.3
r1 1 2 1k
q1 3 2 0 mod1
rspkr 3 4 30
v1 4 0 dc 15
.model mod1 npn
.ac lin 1 2000 2000
.print ac v(1,0) v(3,0)
.end
freq
v(1)
v(3)
2.000E+03
1.500E+00
4.418E+00

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

202

sistors are just as valid to use as NPN in any amplifier configuration, as long as the proper
polarity and current directions are maintained, and the common-emitter amplifier is no exception. The output invertion and gain of a PNP transistor amplifier are the same as its NPN
counterpart, just the battery polarities are different. (Figure 4.38)

+
Vinput

-

+

Vbias
Figure 4.38: PNP version of common emitter amplifier.

• REVIEW:
• Common-emitter transistor amplifiers are so-called because the input and output voltage
points share the emitter lead of the transistor in common with each other, not considering
any power supplies.
• Transistors are essentially DC devices: they cannot directly handle voltages or currents
that reverse direction. To make them work for amplifying AC signals, the input signal
must be offset with a DC voltage to keep the transistor in its active mode throughout the
entire cycle of the wave. This is called biasing.
• If the output voltage is measured between emitter and collector on a common-emitter
amplifier, it will be 180o out of phase with the input voltage waveform. Thus, the commonemitter amplifier is called an inverting amplifier circuit.
• The current gain of a common-emitter transistor amplifier with the load connected in
series with the collector is equal to β. The voltage gain of a common-emitter transistor
amplifier is approximately given here:

•

AV = β

Rout
Rin

• Where “Rout ” is the resistor connected in series with the collector and “Rin ” is the resistor
connected in series with the base.

4.6

The common-collector amplifier

Our next transistor configuration to study is a bit simpler for gain calculations. Called the
common-collector configuration, its schematic diagram is shown in Figure 4.39.

4.6. THE COMMON-COLLECTOR AMPLIFIER

203

+
+
-

Vin
-

Vout

Rload

Figure 4.39: Common collector amplifier has collector common to both input and output.
It is called the common-collector configuration because (ignoring the power supply battery)
both the signal source and the load share the collector lead as a common connection point as
in Figure 4.40.

C
B
E
Vin

Vout

Rload

Figure 4.40: Common collector: Input is applied to base and collector. Output is from emittercollector circuit.
It should be apparent that the load resistor in the common-collector amplifier circuit receives both the base and collector currents, being placed in series with the emitter. Since the
emitter lead of a transistor is the one handling the most current (the sum of base and collector
currents, since base and collector currents always mesh together to form the emitter current),
it would be reasonable to presume that this amplifier will have a very large current gain. This
presumption is indeed correct: the current gain for a common-collector amplifier is quite large,
larger than any other transistor amplifier configuration. However, this is not necessarily what
sets it apart from other amplifier designs.
Let’s proceed immediately to a SPICE analysis of this amplifier circuit, and you will be able
to immediately see what is unique about this amplifier. The circuit is in Figure 4.41. The
netlist is in Figure 4.42.
Unlike the common-emitter amplifier from the previous section, the common-collector produces an output voltage in direct rather than inverse proportion to the rising input voltage.
See Figure 4.42. As the input voltage increases, so does the output voltage. Moreover, a close
examination reveals that the output voltage is nearly identical to the input voltage, lagging

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

204

2
1

V1

3

Vin

2

Q1

1

15 V

Rload
5 kΩ
0

0

0

Figure 4.41: Common collector amplifier for SPICE.

common-collector
amplifier
vin 1 0
q1 2 1 3 mod1
v1 2 0 dc 15
rload 3 0 5k
.model mod1 npn
.dc vin 0 5 0.2
.plot dc v(3,0)
.end
Figure 4.42: Common collector: Output equals input less a 0.7 V VBE drop.

4.6. THE COMMON-COLLECTOR AMPLIFIER

205

behind by about 0.7 volts.
This is the unique quality of the common-collector amplifier: an output voltage that is
nearly equal to the input voltage. Examined from the perspective of output voltage change
for a given amount of input voltage change, this amplifier has a voltage gain of almost exactly
unity (1), or 0 dB. This holds true for transistors of any β value, and for load resistors of any
resistance value.
It is simple to understand why the output voltage of a common-collector amplifier is always
nearly equal to the input voltage. Referring to the diode current source transistor model in Figure 4.43, we see that the base current must go through the base-emitter PN junction, which
is equivalent to a normal rectifying diode. If this junction is forward-biased (the transistor
conducting current in either its active or saturated modes), it will have a voltage drop of approximately 0.7 volts, assuming silicon construction. This 0.7 volt drop is largely irrespective
of the actual magnitude of base current; thus, we can regard it as being constant:

C
B

Vin

+
0.7 V
-

E
+
Rload
-

Figure 4.43: Emitter follower: Emitter voltage follows base voltage (less a 0.7 V VBE drop.)
Given the voltage polarities across the base-emitter PN junction and the load resistor, we
see that these must add together to equal the input voltage, in accordance with Kirchhoff ’s
Voltage Law. In other words, the load voltage will always be about 0.7 volts less than the input
voltage for all conditions where the transistor is conducting. Cutoff occurs at input voltages
below 0.7 volts, and saturation at input voltages in excess of battery (supply) voltage plus 0.7
volts.
Because of this behavior, the common-collector amplifier circuit is also known as the voltagefollower or emitter-follower amplifier, because the emitter load voltages follow the input so
closely.
Applying the common-collector circuit to the amplification of AC signals requires the same
input “biasing” used in the common-emitter circuit: a DC voltage must be added to the AC
input signal to keep the transistor in its active mode during the entire cycle. When this is
done, the result is the non-inverting amplifier in Figure 4.44.
The results of the SPICE simulation in Figure 4.45 show that the output follows the input.
The output is the same peak-to-peak amplitude as the input. Though, the DC level is shifted

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

206

2
1

+

V1

3

1.5 V
2 kHz

Vin

2

Q1

1

15 V

Rload
5 kΩ
-

4
Vbias
2.3 V

0

0

common-collector
amplifier
vin 1 4 sin(0 1.5
2000 0 0)
vbias 4 0 dc 2.3
q1 2 1 3 mod1
v1 2 0 dc 15
rload 3 0 5k
.model mod1 npn
.tran .02m .78m
.plot tran v(1,0)
v(3,0)
.end

Figure 4.44: Common collector (emitter-follower) amplifier.
downward by one VBE diode drop.

Figure 4.45: Common collector (emitter-follower): Output V3 follows input V1 less a 0.7 V VBE
drop.
Here’s another view of the circuit (Figure 4.46) with oscilloscopes connected to several
points of interest.
Since this amplifier configuration doesn’t provide any voltage gain (in fact, in practice it
actually has a voltage gain of slightly less than 1), its only amplifying factor is current. The
common-emitter amplifier configuration examined in the previous section had a current gain
equal to the β of the transistor, being that the input current went through the base and the

4.6. THE COMMON-COLLECTOR AMPLIFIER

207

+
V1
Vin

1.5 V
2 kHz
+

Rload
5 kΩ

15 V

-

-

Figure 4.46: Common collector non-inverting voltage gain is 1.
output (load) current went through the collector, and β by definition is the ratio between the
collector and base currents. In the common-collector configuration, though, the load is situated
in series with the emitter, and thus its current is equal to the emitter current. With the emitter
carrying collector current and base current, the load in this type of amplifier has all the current
of the collector running through it plus the input current of the base. This yields a current gain
of β plus 1:

AI =

Iemitter
Ibase

AI =

Icollector+ Ibase
Ibase

AI =

Icollector
+1
Ibase

AI = β + 1
Once again, PNP transistors are just as valid to use in the common-collector configuration
as NPN transistors. The gain calculations are all the same, as is the non-inverting of the
amplified signal. The only difference is in voltage polarities and current directions shown in
Figure 4.47.
A popular application of the common-collector amplifier is for regulated DC power supplies,
where an unregulated (varying) source of DC voltage is clipped at a specified level to supply
regulated (steady) voltage to a load. Of course, zener diodes already provide this function of
voltage regulation shown in Figure 4.48.
However, when used in this direct fashion, the amount of current that may be supplied to
the load is usually quite limited. In essence, this circuit regulates voltage across the load by
keeping current through the series resistor at a high enough level to drop all the excess power
source voltage across it, the zener diode drawing more or less current as necessary to keep the
voltage across itself steady. For high-current loads, a plain zener diode voltage regulator would
have to shunt a heavy current through the diode to be effective at regulating load voltage in
the event of large load resistance or voltage source changes.

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

208

+

Vin
Rload
-

+

Figure 4.47: PNP version of the common-collector amplifier.

R
Unregulated
DC voltage
source

Zener
diode

Rload

Regulated voltage
across load

Figure 4.48: Zener diode voltage regulator.
One popular way to increase the current-handling ability of a regulator circuit like this is
to use a common-collector transistor to amplify current to the load, so that the zener diode
circuit only has to handle the amount of current necessary to drive the base of the transistor.
(Figure 4.49)

R
Unregulated
DC voltage
source

Zener
diode

Rload

Figure 4.49: Common collector application: voltage regulator.
There’s really only one caveat to this approach: the load voltage will be approximately 0.7
volts less than the zener diode voltage, due to the transistor’s 0.7 volt base-emitter drop. Since
this 0.7 volt difference is fairly constant over a wide range of load currents, a zener diode with

4.6. THE COMMON-COLLECTOR AMPLIFIER

209

a 0.7 volt higher rating can be chosen for the application.
Sometimes the high current gain of a single-transistor, common-collector configuration isn’t
enough for a particular application. If this is the case, multiple transistors may be staged together in a popular configuration known as a Darlington pair, just an extension of the commoncollector concept shown in Figure 4.50.

C

B

E
Figure 4.50: An NPN darlington pair.
Darlington pairs essentially place one transistor as the common-collector load for another
transistor, thus multiplying their individual current gains. Base current through the upperleft transistor is amplified through that transistor’s emitter, which is directly connected to the
base of the lower-right transistor, where the current is again amplified. The overall current
gain is as follows:

Darlington pair current gain
AI = (β1 + 1)(β2 + 1)
Where,
β1 = Beta of first transistor
β2 = Beta of second transistor
Voltage gain is still nearly equal to 1 if the entire assembly is connected to a load in commoncollector fashion, although the load voltage will be a full 1.4 volts less than the input voltage
shown in Figure 4.51.
Darlington pairs may be purchased as discrete units (two transistors in the same package),
or may be built up from a pair of individual transistors. Of course, if even more current gain
is desired than what may be obtained with a pair, Darlington triplet or quadruplet assemblies
may be constructed.
• REVIEW:
• Common-collector transistor amplifiers are so-called because the input and output voltage points share the collector lead of the transistor in common with each other, not con-

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

210

+
0.7 V +

+
0.7 V -

-

Rload

Vin

Vout

Vout = Vin - 1.4
Figure 4.51: Darlington pair based common-collector amplifier loses two VBE diode drops.
sidering any power supplies.
• The common-collector amplifier is also known as an emitter-follower.
• The output voltage on a common-collector amplifier will be in phase with the input voltage, making the common-collector a non-inverting amplifier circuit.
• The current gain of a common-collector amplifier is equal to β plus 1. The voltage gain is
approximately equal to 1 (in practice, just a little bit less).
• A Darlington pair is a pair of transistors “piggybacked” on one another so that the emitter
of one feeds current to the base of the other in common-collector form. The result is an
overall current gain equal to the product (multiplication) of their individual commoncollector current gains (β plus 1).

4.7

The common-base amplifier

The final transistor amplifier configuration (Figure 4.52) we need to study is the common-base.
This configuration is more complex than the other two, and is less common due to its strange
operating characteristics.
It is called the common-base configuration because (DC power source aside), the signal
source and the load share the base of the transistor as a common connection point shown in
Figure 4.53.
Perhaps the most striking characteristic of this configuration is that the input signal source
must carry the full emitter current of the transistor, as indicated by the heavy arrows in the
first illustration. As we know, the emitter current is greater than any other current in the
transistor, being the sum of base and collector currents. In the last two amplifier configurations, the signal source was connected to the base lead of the transistor, thus handling the least
current possible.

4.7. THE COMMON-BASE AMPLIFIER

211

Rload

Vin
−
+

−

+

Figure 4.52: Common-base amplifier

E

C
B

Rload V
out

Vin
−
+

−

+

Figure 4.53: Common-base amplifier: Input between emitter and base, output between collector and base.
Because the input current exceeds all other currents in the circuit, including the output
current, the current gain of this amplifier is actually less than 1 (notice how Rload is connected
to the collector, thus carrying slightly less current than the signal source). In other words,
it attenuates current rather than amplifying it. With common-emitter and common-collector
amplifier configurations, the transistor parameter most closely associated with gain was β.
In the common-base circuit, we follow another basic transistor parameter: the ratio between
collector current and emitter current, which is a fraction always less than 1. This fractional
value for any transistor is called the alpha ratio, or α ratio.
Since it obviously can’t boost signal current, it only seems reasonable to expect it to boost
signal voltage. A SPICE simulation of the circuit in Figure 4.54 will vindicate that assumption.

Q1

2

4
Rload

R1
100 Ω
Vin
+
1 −

0 −

V1

5.0 kΩ
+ 3

15 V
Figure 4.54: Common-base circuit for DC SPICE analysis.
Notice in Figure 4.55 that the output voltage goes from practically nothing (cutoff) to 15.75
volts (saturation) with the input voltage being swept over a range of 0.6 volts to 1.2 volts. In
fact, the output voltage plot doesn’t show a rise until about 0.7 volts at the input, and cuts off

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

212

common-base
amplifier
vin 0 1
r1 1 2 100
q1 4 0 2 mod1
v1 3 0 dc 15
rload 3 4 5k
.model mod1 npn
.dc vin 0.6 1.2
.02
.plot dc v(3,4)
.end
Figure 4.55: Common-base amplifier DC transfer function.
(flattens) at about 1.12 volts input. This represents a rather large voltage gain with an output
voltage span of 15.75 volts and an input voltage span of only 0.42 volts: a gain ratio of 37.5,
or 31.48 dB. Notice also how the output voltage (measured across Rload ) actually exceeds the
power supply (15 volts) at saturation, due to the series-aiding effect of the input voltage source.
A second set of SPICE analyses (circuit in Figure 4.56) with an AC signal source (and DC
bias voltage) tells the same story: a high voltage gain

Vin
2
100 Ω

1 −

4

Q1
Rload
5.0 kΩ

0.12 Vp-p
0 Voffset
2 kHz

R1

5

+

Vbias 0.95 V

0 −

V1

Vout

+ 3

15 V

Figure 4.56: Common-base circuit for SPICE AC analysis.
As you can see, the input and output waveforms in Figure 4.57 are in phase with each other.
This tells us that the common-base amplifier is non-inverting.
The AC SPICE analysis in Table 4.4 at a single frequency of 2 kHz provides input and
output voltages for gain calculation.

4.7. THE COMMON-BASE AMPLIFIER

213

common-base
amplifier
vin 5 2 sin (0
0.12 2000 0 0)
vbias 0 1 dc 0.95
r1 2 1 100
q1 4 0 5 mod1
v1 3 0 dc 15
rload 3 4 5k
.model mod1 npn
.tran 0.02m 0.78m
.plot tran v(5,2)
v(4)
.end
Figure 4.57:

Table 4.4: Common-base AC analysis at 2 kHz– netlist followed by output.
common-base amplifier
vin 5 2 ac 0.1 sin
vbias 0 1 dc 0.95
r1 2 1 100
q1 4 0 5 mod1
v1 3 0 dc 15
rload 3 4 5k
.model mod1 npn
.ac dec 1 2000 2000
.print ac vm(5,2) vm(4,3)
.end
frequency
mag(v(5,2))
mag(v(4,3))
-------------------------------------------0.000000e+00
1.000000e-01
4.273864e+00

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

214

Voltage figures from the second analysis (Table 4.4) show a voltage gain of 42.74 (4.274 V /
0.1 V), or 32.617 dB:

AV =

Vout
Vin

AV = 4.274 V
0.10 V
AV = 42.74
AV(dB) = 20 log AV(ratio)
AV(dB) = 20 log 42.74
AV(dB) = 32.62 dB
Here’s another view of the circuit in Figure 4.58, summarizing the phase relations and DC
offsets of various signals in the circuit just simulated.

Vin
Q1
Rload

−

+

−

V1

+

Vbias
Figure 4.58: Phase relationships and offsets for NPN common base amplifier.
. . . and for a PNP transistor: Figure 4.59.
Predicting voltage gain for the common-base amplifier configuration is quite difficult, and
involves approximations of transistor behavior that are difficult to measure directly. Unlike the
other amplifier configurations, where voltage gain was either set by the ratio of two resistors
(common-emitter), or fixed at an unchangeable value (common-collector), the voltage gain of
the common-base amplifier depends largely on the amount of DC bias on the input signal. As
it turns out, the internal transistor resistance between emitter and base plays a major role in
determining voltage gain, and this resistance changes with different levels of current through
the emitter.
While this phenomenon is difficult to explain, it is rather easy to demonstrate through the
use of computer simulation. What I’m going to do here is run several SPICE simulations on
a common-base amplifier circuit (Figure 4.56), changing the DC bias voltage slightly (vbias
in Figure 4.60 ) while keeping the AC signal amplitude and all other circuit parameters constant. As the voltage gain changes from one simulation to another, different output voltage

4.7. THE COMMON-BASE AMPLIFIER

215

Vin
Q1
Rload

Vbias
+
−

+

V1

−

Figure 4.59: Phase relationships and offsets for PNP common base amplifier.
amplitudes will be noted.
Although these analyses will all be conducted in the “transfer function” mode, each was
first “proofed” in the transient analysis mode (voltage plotted over time) to ensure that the
entire wave was being faithfully reproduced and not “clipped” due to improper biasing. See
”*.tran 0.02m 0.78m” in Figure 4.60, the “commented out” transient analysis statement. Gain
calculations cannot be based on waveforms that are distorted. SPICE can calculate the small
signal DC gain for us with the “.tf v(4) vin” statement. The output is v(4) and the input as vin.
At the command line, spice -b filename.cir produces a printed output due to the .tf statement: transfer function, output impedance, and input impedance. The abbreviated output
listing is from runs with vbias at 0.85, 0.90, 0.95, 1.00 V as recorded in Table 4.5.
A trend should be evident in Table 4.5. With increases in DC bias voltage, voltage gain
(transfer function) increases as well. We can see that the voltage gain is increasing because
each subsequent simulation (vbias= 0.85, 0.8753, 0.90, 0.95, 1.00 V) produces greater gain
(transfer function= 37.6, 39.4 40.8, 42.7, 44.0), respectively. The changes are largely due to
minuscule variations in bias voltage.
The last three lines of Table ??(right) show the I(v1)/Iin current gain of 0.99. (The last
two lines look invalid.) This makes sense for β=100; α= β/(β+1), α=0.99=100/(100-1). The
combination of low current gain (always less than 1) and somewhat unpredictable voltage gain
conspire against the common-base design, relegating it to few practical applications.
Those few applications include radio frequency amplifiers. The grounded base helps shield
the input at the emitter from the collector output, preventing instability in RF amplifiers. The
common base configuration is usable at higher frequencies than common emitter or common
collector. See “Class C common-base 750 mW RF power amplifier” (page 431). For a more
elaborate circuit see “Class A common-base small-signal high gain amplifier” (page 431).
• REVIEW:
• Common-base transistor amplifiers are so-called because the input and output voltage
points share the base lead of the transistor in common with each other, not considering
any power supplies.
• The current gain of a common-base amplifier is always less than 1. The voltage gain is a
function of input and output resistances, and also the internal resistance of the emitter-

216

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

common-base amp
vbias=0.85V
vin 5 2 sin (0 0.12 2000
0 0)
vbias 0 1 dc 0.85
r1 2 1 100
q1 4 0 5 mod1
v1 3 0 dc 15
rload 3 4 5k
.model mod1 npn
*.tran 0.02m 0.78m
.tf v(4) vin
.end

common-base amp current gain
Iin 55 5 0A
vin 55 2 sin (0 0.12 2000 0
0)
vbias 0 1 dc 0.8753
r1 2 1 100
q1 4 0 5 mod1
v1 3 0 dc 15
rload 3 4 5k
.model mod1 npn
*.tran 0.02m 0.78m
.tf I(v1) Iin
.end
Transfer function
information:
transfer function =
9.900990e-01
iin input impedance =
9.900923e+11
v1 output impedance =
1.000000e+20

Figure 4.60: SPICE net list: Common-base, transfer function (voltage gain) for various DC
bias voltages. SPICE net list: Common-base amp current gain; Note .tf v(4) vin statement.
Transfer function for DC current gain I(vin)/Iin; Note .tf I(vin) Iin statement.

4.7. THE COMMON-BASE AMPLIFIER

Table 4.5: SPICE output: Common-base transfer function.
Circuit: common-base amp vbias=0.85V
transfer function = 3.756565e+01
output impedance at v(4) = 5.000000e+03
vin#input impedance = 1.317825e+02
Circuit: common-base amp vbias=0.8753V Ic=1 mA
Transfer function information:
transfer function = 3.942567e+01
output impedance at v(4) = 5.000000e+03
vin#input impedance = 1.255653e+02
Circuit: common-base amp vbias=0.9V
transfer function = 4.079542e+01
output impedance at v(4) = 5.000000e+03
vin#input impedance = 1.213493e+02
Circuit: common-base amp vbias=0.95V
transfer function = 4.273864e+01
output impedance at v(4) = 5.000000e+03
vin#input impedance = 1.158318e+02
Circuit: common-base amp vbias=1.00V
transfer function = 4.401137e+01
output impedance at v(4) = 5.000000e+03
vin#input impedance = 1.124822e+02

217

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

218

base junction, which is subject to change with variations in DC bias voltage. Suffice to
say that the voltage gain of a common-base amplifier can be very high.
• The ratio of a transistor’s collector current to emitter current is called α. The α value for
any transistor is always less than unity, or in other words, less than 1.

4.8

The cascode amplifier

While the C-B (common-base) amplifier is known for wider bandwidth than the C-E (commonemitter) configuration, the low input impedance (10s of Ω) of C-B is a limitation for many
applications. The solution is to precede the C-B stage by a low gain C-E stage which has moderately high input impedance (kΩs). See Figure 4.61. The stages are in a cascode configuration,
stacked in series, as opposed to cascaded for a standard amplifier chain. See “Capacitor coupled three stage common-emitter amplifier” (page 253) for a cascade example. The cascode
amplifier configuration has both wide bandwidth and a moderately high input impedance.

Vo
Vi

Vo
RL

Common
base

Vo

Vi

Vi
RL

RL

Common
emitter

Common-base

Common-emitter

Cascode

Figure 4.61: The cascode amplifier is combined common-emitter and common-base. This is an
AC circuit equivalent with batteries and capacitors replaced by short circuits.
The key to understanding the wide bandwidth of the cascode configuration is the Miller
effect. The (page 277) is the multiplication of the bandwidth robbing collector-base capacitance
by voltage gain Av . This C-B capacitance is smaller than the E-B capacitance. Thus, one would
think that the C-B capacitance would have little effect. However, in the C-E configuration, the
collector output signal is out of phase with the input at the base. The collector signal capacitively coupled back opposes the base signal. Moreover, the collector feedback is (1-Av ) times
larger than the base signal. Thus, the small C-B capacitance appears (1-Av ) times larger than
its actual value. This capacitive gain reducing feedback increases with frequency, reducing the
high frequency response of a C-E amplifier.
The approximage voltage gain of the C-E amplifier in Figure 4.62b is -RL /REE . The emitter
current is set to 1.0 mA by biasing. REE = 26mV/IE = 26mV/1.0ma = 26 Ω. Thus, Av = -RL /REE =
-4700/26 = -181. The pn2222 datasheet list Ccbo = 8 pF.[5] The miller capacitance is Ccbo (1-Av ).
Gain Av = -181, negative since it is inverting gain. Cmiller = Ccbo (1-Av ) = 8pF(1-(-181)=1456pF

4.8. THE CASCODE AMPLIFIER

219

A common-base configuration is not subject to the Miller effect because the grounded base
shields the collector signal from being fed back to the emitter input. Thus, a C-B amplifier has
better high frequency response. To have a moderately high input impedance, the C-E stage
is still desirable. The key is to reduce the gain (to about 1) of the C-E stage which reduces
the Miller effect C-B feedback to 1·CCBO . The total C-B feedback is the feedback capacitance
1·CCB plus the actual capacitance CCB for a total of 2·CCBO . This is a considerable reduction
from 181·CCBO . The miller capacitance for a gain of -2 C-E stage is Cmiller = Ccbo (1-Av )= Cmiller
= Ccbo (1-(-1)) = Ccbo ·2.
The way to reduce the common-emitter gain is to reduce the load resistance. The gain of a
C-E amplifier is approximately RC /RE . The internal emitter resistance REE at 1mA emitter
current is 26Ω. For details on the 26Ω, see “Derivation of REE ”, see (page 241). The collector
load RC is the resistance of the emitter of the C-B stage loading the C-E stage, 26Ω again. CE
gain amplifier gain is approximately Av = RC /RE =26/26=1. This Miller capacitance is Cmiller
= Ccbo (1-Av ) = 8pF(1-(-1)=16pF. We now have a moderately high input impedance C-E stage
without suffering the Miller effect, but no C-E dB voltage gain. The C-B stage provides a
high voltage gain, AV = -181. Current gain of cascode is β of the C-E stage, 1 for the C-B, β
overall. Thus, the cascode has moderately high input impedance of the C-E, good gain, and
good bandwidth of the C-B.

1
V4

V3

3

R3
80 kΩ

2

+
− 11.5 V
C3
10n F
4
0.1 Vp-p
0 Voffset
1 kHz

C2
10n F

Α
V5

5
R5
80 kΩ
1.5 V
6
+−
V6

Cascode

Q2

Q3

19
R1
4.7 kΩ

R4
4.7 kΩ

13

9
VCC

+
− 20 V
V3

10 nF
4
C1

15
Q1
R2
80 kΩ
V1 +
− 10 V
1.5 V
16 −
+
V2
Common-base

Figure 4.62: SPICE: Cascode and common-base for comparison.
The SPICE version of both a cascode amplifier, and for comparison, a common-emitter amplifier is shown in Figure 4.62. The netlist is in Table 4.6. The AC source V3 drives both
amplifiers via node 4. The bias resistors for this circuit are calculated in an example problem
(page 246).
The waveforms in Figure 4.63 show the operation of the cascode stage. The input signal
is displayed multiplied by 10 so that it may be shown with the outputs. Note that both the
Cascode, Common-emitter, and Va (intermediate point) outputs are inverted from the input.
Both the Cascode and Common emitter have large amplitude outputs. The Va point has a DC
level of about 10V, about half way between 20V and ground. The signal is larger than can be
accounted for by a C-E gain of 1, It is three times larger than expected.
Figure 4.64 shows the frequency response to both the cascode and common-emitter ampli-

220

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

Figure 4.63: SPICE waveforms. Note that Input is multiplied by 10 for visibility.

Figure 4.64: Cascode vs common-emitter banwidth.

4.8. THE CASCODE AMPLIFIER

Table 4.6: SPICE netlist for printing AC input and output voltages.
SPICE
circuit
<03502.eps> from XCircuit v3.20
*
V1 19 0 10
Q1 13 15 0 q2n2222
Q2 3 2 A q2n2222
R1 19 13 4.7k
V2 16 0 1.5
C1 4 15 10n
R2 15 16 80k
Q3 A 5 0 q2n2222
V3 4 6 SIN(0 0.1 1k) ac 1
R3 1 2 80k
R4 3 9 4.7k
C2 2 0 10n
C3 4 5 10n
R5 5 6 80k
V4 1 0 11.5
V5 9 0 20
V6 6 0 1.5
.model q2n2222 npn (is=19f bf=150
+ vaf=100 ikf=0.18 ise=50p ne=2.5 br=7.5
+ var=6.4 ikr=12m isc=8.7p nc=1.2 rb=50
+ re=0.4 rc=0.3 cje=26p tf=0.5n
+ cjc=11p tr=7n xtb=1.5 kf=0.032f af=1)
.tran 1u 5m
.AC DEC 10 1k 100Meg
.end

221

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

222

fiers. The SPICE statements responsible for the AC analysis, extracted from the listing:
V3 4 6 SIN(0 0.1 1k)
.AC DEC 10 1k 100Meg

ac 1

Note the “ac 1” is necessary at the end of the V3 statement. The cascode has marginally
better mid-band gain. However, we are primarily looking for the bandwidth measured at the
-3dB points, down from the midband gain for each amplifier. This is shown by the vertical solid
lines in Figure 4.64. It is also possible to print the data of interest from nutmeg to the screen,
the SPICE graphical viewer (command, first line):
nutmeg 6 -> print frequency db(vm(3)) db(vm(13))
Index
frequency
db(vm(3)) db(vm(13))
22
0.158MHz
47.54
45.41
33
1.995MHz
46.95
42.06
37
5.012MHz
44.63
36.17
Index 22 gives the midband dB gain for Cascode vm(3)=47.5dB and Common-emitter vm(13)=45.4dB.
Out of many printed lines, Index 33 was the closest to being 3dB down from 45.4dB at 42.0dB
for the Common-emitter circuit. The corresponding Index 33 frequency is approximately 2Mhz,
the common-emitter bandwidth. Index 37 vm(3)=44.6db is approximately 3db down from
47.5db. The corresponding Index37 frequency is 5Mhz, the cascode bandwidth. Thus, the
cascode amplifier has a wider bandwidth. We are not concerned with the low frequency degradation of gain. It is due to the capacitors, which could be remedied with larger ones.
The 5MHz bandwith of our cascode example, while better than the common-emitter example, is not exemplary for an RF (radio frequency) amplifier. A pair of RF or microwave
transistors with lower interelectrode capacitances should be used for higher bandwidth. Before the invention of the RF dual gate MOSFET, the BJT cascode amplifier could have been
found in UHF (ultra high frequency) TV tuners.
• REVIEW
• A cascode amplifier consists of a common-emitter stage loaded by the emitter of a commonbase stage.
• The heavily loaded C-E stage has a low gain of 1, overcoming the Miller effect
• A cascode amplifier has a high gain, moderately high input impedance, a high output
impedance, and a high bandwidth.

4.9

Biasing techniques

In the common-emitter section of this chapter, we saw a SPICE analysis where the output
waveform resembled a half-wave rectified shape: only half of the input waveform was reproduced, with the other half being completely cut off. Since our purpose at that time was to
reproduce the entire waveshape, this constituted a problem. The solution to this problem was
to add a small bias voltage to the amplifier input so that the transistor stayed in active mode
throughout the entire wave cycle. This addition was called a bias voltage.

4.9. BIASING TECHNIQUES

223

A half-wave output is not problematic for some applications. In fact, some applications may
necessitate this very kind of amplification. Because it is possible to operate an amplifier in
modes other than full-wave reproduction and specific applications require different ranges of
reproduction, it is useful to describe the degree to which an amplifier reproduces the input
waveform by designating it according to class. Amplifier class operation is categorized with
alphabetical letters: A, B, C, and AB.
For Class A operation, the entire input waveform is faithfully reproduced. Although I didn’t
introduce this concept back in the common-emitter section, this is what we were hoping to attain in our simulations. Class A operation can only be obtained when the transistor spends
its entire time in the active mode, never reaching either cutoff or saturation. To achieve this,
sufficient DC bias voltage is usually set at the level necessary to drive the transistor exactly
halfway between cutoff and saturation. This way, the AC input signal will be perfectly “centered” between the amplifier’s high and low signal limit levels.

Vinput
+

Class A
Amplifier

Vbias
Figure 4.65: Class A: The amplifier output is a faithful reproduction of the input.
Class B operation is what we had the first time an AC signal was applied to the commonemitter amplifier with no DC bias voltage. The transistor spent half its time in active mode
and the other half in cutoff with the input voltage too low (or even of the wrong polarity!) to
forward-bias its base-emitter junction.

Vinput

Little or no

Class B
Amplifier

DC bias voltage

Figure 4.66: Class B: Bias is such that half (180o ) of the waveform is reproduced.
By itself, an amplifier operating in class B mode is not very useful. In most circumstances,
the severe distortion introduced into the waveshape by eliminating half of it would be unacceptable. However, class B operation is a useful mode of biasing if two amplifiers are operated

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

224

as a push-pull pair, each amplifier handling only half of the waveform at a time:

Q1
+

Input components
omitted for simplicity

−

Power
supply

Voutput
Q2

Figure 4.67: Class B push pull amplifier: Each transistor reproduces half of the waveform.
Combining the halves produces a faithful reproduction of the whole wave.
Transistor Q1 “pushes” (drives the output voltage in a positive direction with respect to
ground), while transistor Q2 “pulls” the output voltage (in a negative direction, toward 0 volts
with respect to ground). Individually, each of these transistors is operating in class B mode,
active only for one-half of the input waveform cycle. Together, however, both function as a team
to produce an output waveform identical in shape to the input waveform.
A decided advantage of the class B (push-pull) amplifier design over the class A design is
greater output power capability. With a class A design, the transistor dissipates considerable
energy in the form of heat because it never stops conducting current. At all points in the wave
cycle it is in the active (conducting) mode, conducting substantial current and dropping substantial voltage. There is substantial power dissipated by the transistor throughout the cycle.
In a class B design, each transistor spends half the time in cutoff mode, where it dissipates
zero power (zero current = zero power dissipation). This gives each transistor a time to “rest”
and cool while the other transistor carries the burden of the load. Class A amplifiers are simpler in design, but tend to be limited to low-power signal applications for the simple reason of
transistor heat dissipation.
Another class of amplifier operation known as class AB, is somewhere between class A
and class B: the transistor spends more than 50% but less than 100% of the time conducting
current.
If the input signal bias for an amplifier is slightly negative (opposite of the bias polarity
for class A operation), the output waveform will be further “clipped” than it was with class B
biasing, resulting in an operation where the transistor spends most of the time in cutoff mode:
At first, this scheme may seem utterly pointless. After all, how useful could an amplifier be
if it clips the waveform as badly as this? If the output is used directly with no conditioning of
any kind, it would indeed be of questionable utility. However, with the application of a tank

4.9. BIASING TECHNIQUES

225

Vinput
Vbias

Class C
Amplifier

+

Figure 4.68: Class C: Conduction is for less than a half cycle (< 180o ).
circuit (parallel resonant inductor-capacitor combination) to the output, the occasional output
surge produced by the amplifier can set in motion a higher-frequency oscillation maintained
by the tank circuit. This may be likened to a machine where a heavy flywheel is given an
occasional “kick” to keep it spinning:

Vinput

Vbias

Class C
Amplifier
with resonant
output

+

Figure 4.69: Class C amplifier driving a resonant circuit.
Called class C operation, this scheme also enjoys high power efficiency due to the fact that
the transistor(s) spend the vast majority of time in the cutoff mode, where they dissipate zero
power. The rate of output waveform decay (decreasing oscillation amplitude between “kicks”
from the amplifier) is exaggerated here for the benefit of illustration. Because of the tuned
tank circuit on the output, this circuit is usable only for amplifying signals of definite, fixed
amplitude. A class C amplifier may used in an FM (frequency modulation) radio transmitter.
However, the class C amplifier may not directly amplify an AM (amplitude modulated) signal
due to distortion.
Another kind of amplifier operation, significantly different from Class A, B, AB, or C, is
called Class D. It is not obtained by applying a specific measure of bias voltage as are the other
classes of operation, but requires a radical re-design of the amplifier circuit itself. It is a little
too early in this chapter to investigate exactly how a class D amplifier is built, but not too early
to discuss its basic principle of operation.
A class D amplifier reproduces the profile of the input voltage waveform by generating a
rapidly-pulsing squarewave output. The duty cycle of this output waveform (time “on” versus
total cycle time) varies with the instantaneous amplitude of the input signal. The plots in
(Figure 4.70 demonstrate this principle.
The greater the instantaneous voltage of the input signal, the greater the duty cycle of

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

226

Input

Output

Figure 4.70: Class D amplifier: Input signal and unfiltered output.
the output squarewave pulse. If there can be any goal stated of the class D design, it is to
avoid active-mode transistor operation. Since the output transistor of a class D amplifier is
never in the active mode, only cutoff or saturated, there will be little heat energy dissipated
by it. This results in very high power efficiency for the amplifier. Of course, the disadvantage
of this strategy is the overwhelming presence of harmonics on the output. Fortunately, since
these harmonic frequencies are typically much greater than the frequency of the input signal,
these can be filtered out by a low-pass filter with relative ease, resulting in an output more
closely resembling the original input signal waveform. Class D technology is typically seen
where extremely high power levels and relatively low frequencies are encountered, such as
in industrial inverters (devices converting DC into AC power to run motors and other large
devices) and high-performance audio amplifiers.
A term you will likely come across in your studies of electronics is something called quiescent, which is a modifier designating the zero input condition of a circuit. Quiescent current,
for example, is the amount of current in a circuit with zero input signal voltage applied. Bias
voltage in a transistor circuit forces the transistor to operate at a different level of collector
current with zero input signal voltage than it would without that bias voltage. Therefore, the
amount of bias in an amplifier circuit determines its quiescent values.
In a class A amplifier, the quiescent current should be exactly half of its saturation value
(halfway between saturation and cutoff, cutoff by definition being zero). Class B and class C
amplifiers have quiescent current values of zero, since these are supposed to be cutoff with no
signal applied. Class AB amplifiers have very low quiescent current values, just above cutoff.
To illustrate this graphically, a “load line” is sometimes plotted over a transistor’s characteristic
curves to illustrate its range of operation while connected to a load resistance of specific value
shown in Figure 4.71.
A load line is a plot of collector-to-emitter voltage over a range of collector currents. At the
lower-right corner of the load line, voltage is at maximum and current is at zero, representing
a condition of cutoff. At the upper-left corner of the line, voltage is at zero while current is at a
maximum, representing a condition of saturation. Dots marking where the load line intersects
the various transistor curves represent realistic operating conditions for those base currents

4.9. BIASING TECHNIQUES

227

saturation
Ibase = 75 µA
"Load line"
Icollector

Ibase = 40 µA
Ibase = 20 µA
Ibase = 5 µA
0

Ecollector-to-emitter

Vsupply cutoff

Figure 4.71: Example load line drawn over transistor characteristic curves from Vsupply to
saturation current.
given.
Quiescent operating conditions may be shown on this graph in the form of a single dot along
the load line. For a class A amplifier, the quiescent point will be in the middle of the load line
as in (Figure 4.72.

Ibase = 75 µA

Icollector

Ibase = 40 µA

Quiescent point
for class A
operation

Ibase = 20 µA
Ibase = 5 µA
0

Ecollector-to-emitter

Vsupply

Figure 4.72: Quiescent point (dot) for class A.
In this illustration, the quiescent point happens to fall on the curve representing a base
current of 40 µA. If we were to change the load resistance in this circuit to a greater value, it
would affect the slope of the load line, since a greater load resistance would limit the maximum
collector current at saturation, but would not change the collector-emitter voltage at cutoff.
Graphically, the result is a load line with a different upper-left point and the same lower-right

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

228
point as in (Figure 4.73)

Ibase = 75 µA

Icollector

The nonhorizontal
portion of
the curve
represents
transistor
saturation

Ibase = 40 µA
Ibase = 20 µA
Ibase = 5 µA
0

Ecollector-to-emitter

Vsupply

Figure 4.73: Load line resulting from increased load resistance.
Note how the new load line doesn’t intercept the 75 µA curve along its flat portion as before.
This is very important to realize because the non-horizontal portion of a characteristic curve
represents a condition of saturation. Having the load line intercept the 75 µA curve outside
of the curve’s horizontal range means that the amplifier will be saturated at that amount of
base current. Increasing the load resistor value is what caused the load line to intercept the
75 µA curve at this new point, and it indicates that saturation will occur at a lesser value of
base current than before.
With the old, lower-value load resistor in the circuit, a base current of 75 µA would yield
a proportional collector current (base current multiplied by β). In the first load line graph, a
base current of 75 µA gave a collector current almost twice what was obtained at 40 µA, as the
β ratio would predict. However, collector current increases marginally between base currents
75 µA and 40 µA, because the transistor begins to lose sufficient collector-emitter voltage to
continue to regulate collector current.
To maintain linear (no-distortion) operation, transistor amplifiers shouldn’t be operated at
points where the transistor will saturate; that is, where the load line will not potentially fall
on the horizontal portion of a collector current curve. We’d have to add a few more curves to
the graph in Figure 4.74 before we could tell just how far we could “push” this transistor with
increased base currents before it saturates.
It appears in this graph that the highest-current point on the load line falling on the
straight portion of a curve is the point on the 50 µA curve. This new point should be considered
the maximum allowable input signal level for class A operation. Also for class A operation, the
bias should be set so that the quiescent point is halfway between this new maximum point and
cutoff shown in Figure 4.75.
Now that we know a little more about the consequences of different DC bias voltage levels,
it is time to investigate practical biasing techniques. So far, I’ve shown a small DC voltage
source (battery) connected in series with the AC input signal to bias the amplifier for whatever
desired class of operation. In real life, the connection of a precisely-calibrated battery to the

4.9. BIASING TECHNIQUES

229

Ibase = 75 µA
Ibase = 60 µA
Ibase = 50 µA

Icollector

Ibase = 40 µA
Ibase = 20 µA
Ibase = 5 µA
0

Ecollector-to-emitter

Vsupply

Figure 4.74: More base current curves shows saturation detail.

Ibase = 75 µA
Ibase = 60 µA
Ibase = 50 µA

Icollector

Ibase = 40 µA
Ibase = 20 µA

New quiescent point

Ibase = 5 µA
0

Ecollector-to-emitter

Vsupply

Figure 4.75: New quiescent point avoids saturation region.

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

230

input of an amplifier is simply not practical. Even if it were possible to customize a battery to
produce just the right amount of voltage for any given bias requirement, that battery would
not remain at its manufactured voltage indefinitely. Once it started to discharge and its output
voltage drooped, the amplifier would begin to drift toward class B operation.
Take this circuit, illustrated in the common-emitter section for a SPICE simulation, for
instance, in Figure 4.76.

speaker
8Ω

3
R1

1
Vinput
1.5 V
2 kHz

2

Q1

4

V1

15 V

1 kΩ
+

-

5

0

0

Vbias
2.3 V

Figure 4.76: Impractical base battery bias.
That 2.3 volt “Vbias ” battery would not be practical to include in a real amplifier circuit. A
far more practical method of obtaining bias voltage for this amplifier would be to develop the
necessary 2.3 volts using a voltage divider network connected across the 15 volt battery. After
all, the 15 volt battery is already there by necessity, and voltage divider circuits are easy to
design and build. Let’s see how this might look in Figure 4.77.

speaker
4
R2

R1

1
Vinput
1.5 V
2 kHz

1 kΩ
Vbias

R3
0

3

0

2

8Ω

Q1

0

4

V1

15 V

0

Figure 4.77: Voltage divider bias.
If we choose a pair of resistor values for R2 and R3 that will produce 2.3 volts across R3 from
a total of 15 volts (such as 8466 Ω for R2 and 1533 Ω for R3 ), we should have our desired value
of 2.3 volts between base and emitter for biasing with no signal input. The only problem is,
this circuit configuration places the AC input signal source directly in parallel with R3 of our

4.9. BIASING TECHNIQUES

231

voltage divider. This is not acceptable, as the AC source will tend to overpower any DC voltage
dropped across R3 . Parallel components must have the same voltage, so if an AC voltage source
is directly connected across one resistor of a DC voltage divider, the AC source will “win” and
there will be no DC bias voltage added to the signal.
One way to make this scheme work, although it may not be obvious why it will work, is
to place a coupling capacitor between the AC voltage source and the voltage divider as in
Figure 4.78.

speaker
4
C
1
Vinput
1.5 V
2 kHz
0

R2

8.466 kΩ
5 R1 2

R3

1 kΩ
1.533 kΩ

5

0

3
Q1

0

8Ω

4

V1

15 V

0

Figure 4.78: Coupling capacitor prevents voltage divider bias from flowing into signal generator.
The capacitor forms a high-pass filter between the AC source and the DC voltage divider,
passing almost all of the AC signal voltage on to the transistor while blocking all DC voltage
from being shorted through the AC signal source. This makes much more sense if you understand the superposition theorem and how it works. According to superposition, any linear,
bilateral circuit can be analyzed in a piecemeal fashion by only considering one power source
at a time, then algebraically adding the effects of all power sources to find the final result.
If we were to separate the capacitor and R2 −−R3 voltage divider circuit from the rest of the
amplifier, it might be easier to understand how this superposition of AC and DC would work.
With only the AC signal source in effect, and a capacitor with an arbitrarily low impedance
at signal frequency, almost all the AC voltage appears across R3 :
With only the DC source in effect, the capacitor appears to be an open circuit, and thus
neither it nor the shorted AC signal source will have any effect on the operation of the R2 −−R3
voltage divider in Figure 4.80.
Combining these two separate analyses in Figure 4.81, we get a superposition of (almost)
1.5 volts AC and 2.3 volts DC, ready to be connected to the base of the transistor.
Enough talk – its about time for a SPICE simulation of the whole amplifier circuit in
Figure 4.82. We will use a capacitor value of 100 µF to obtain an arbitrarily low (0.796 Ω)
impedance at 2000 Hz:
Note the substantial distortion in the output waveform in Figure 4.82. The sine wave is
being clipped during most of the input signal’s negative half-cycle. This tells us the transistor
is entering into cutoff mode when it shouldn’t (I’m assuming a goal of class A operation as
before). Why is this? This new biasing technique should give us exactly the same amount of
DC bias voltage as before, right?

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

232

≈ 1.5 V
2 kHz

Vinput
1.5 V
2 kHz

R2

8.466
kΩ

R3

1.533
kΩ

Figure 4.79: Due to the coupling capacitor’s very low impedance at the signal frequency, it
behaves much like a piece of wire, thus can be omitted for this step in superposition analysis.

2.3 V
R2

8.466
kΩ
V1

R3

15 V

1.533
kΩ

Figure 4.80: The capacitor appears to be an open circuit as far at the DC analysis is concerned

C
Vinput
1.5 V
2 kHz

R2

8.466
kΩ
V1

R3

1.533
kΩ

Figure 4.81: Combined AC and DC circuit.

15 V

4.9. BIASING TECHNIQUES

233

voltage divider
biasing
vinput 1 0 sin (0
1.5 2000 0 0)
c1 1 5 100u
r1 5 2 1k
r2 4 5 8466
r3 5 0 1533
q1 3 2 0 mod1
rspkr 3 4 8
v1 4 0 dc 15
.model mod1 npn
.tran 0.02m 0.78m
.plot tran v(1,0)
i(v1)
.end
Figure 4.82: SPICE simulation of voltage divider bias.
With the capacitor and R2 −−R3 resistor network unloaded, it will provide exactly 2.3 volts
worth of DC bias. However, once we connect this network to the transistor, it is no longer
unloaded. Current drawn through the base of the transistor will load the voltage divider,
thus reducing the DC bias voltage available for the transistor. Using the diode current source
transistor model in Figure 4.83 to illustrate, the bias problem becomes evident.

speaker
R2
C
Vinput

IR3 + Ibias
R1

R3

Ibias
IR3

Q1

V1

IR3

Figure 4.83: Diode transistor model shows loading of voltage divider.
A voltage divider’s output depends not only on the size of its constituent resistors, but also
on how much current is being divided away from it through a load. The base-emitter PN
junction of the transistor is a load that decreases the DC voltage dropped across R3 , due to the
fact that the bias current joins with R3 ’s current to go through R2 , upsetting the divider ratio
formerly set by the resistance values of R2 and R3 . To obtain a DC bias voltage of 2.3 volts, the

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

234

values of R2 and/or R3 must be adjusted to compensate for the effect of base current loading.
To increase the DC voltage dropped across R3 , lower the value of R2 , raise the value of R3 , or
both.
voltage divider
biasing
vinput 1 0 sin (0
1.5 2000 0 0)
c1 1 5 100u
r1 5 2 1k
r2 4 5 6k <--- R2
decreased to 6 k
r3 5 0 4k <--- R3
increased to 4 k
q1 3 2 0 mod1
rspkr 3 4 8
v1 4 0 dc 15
.model mod1 npn
.tran 0.02m 0.78m
.plot tran v(1,0)
i(v1)
.end
Figure 4.84: No distortion of the output after adjusting R2 and R3.
The new resistor values of 6 kΩ and 4 kΩ (R2 and R3 , respectively) in Figure ?? results in
class A waveform reproduction, just the way we wanted.
• REVIEW:
• Class A operation is an amplifier biased to be in the active mode throughout the entire
waveform cycle, thus faithfully reproducing the whole waveform.
• Class B operation is an amplifier biased so that only half of the input waveform gets
reproduced: either the positive half or the negative half. The transistor spends half its
time in the active mode and half its time cutoff. Complementary pairs of transistors
running in class B operation are often used to deliver high power amplification in audio
signal systems, each transistor of the pair handling a separate half of the waveform cycle.
Class B operation delivers better power efficiency than a class A amplifier of similar
output power.
• Class AB operation is an amplifier is biased at a point somewhere between class A and
class B.
• Class C is an amplifier biased to amplify only a small portion of the waveform. Most of the
transistor’s time is spent in cutoff mode. In order for there to be a complete waveform at
the output, a resonant tank circuit is often used as a “flywheel” to maintain oscillations for

4.10. BIASING CALCULATIONS

235

a few cycles after each “kick” from the amplifier. Because the transistor is not conducting
most of the time, power efficiencies are high for a class C amplifier.
• Class D operation requires an advanced circuit design, and functions on the principle of
representing instantaneous input signal amplitude by the duty cycle of a high-frequency
squarewave. The output transistor(s) never operate in active mode, only cutoff and saturation. Little heat energy dissipated makes energy efficiency high.
• DC bias voltage on the input signal, necessary for certain classes of operation (especially
class A and class C), may be obtained through the use of a voltage divider and coupling
capacitor rather than a battery connected in series with the AC signal source.

4.10

Biasing calculations

Although transistor switching circuits operate without bias, it is unusual for analog circuits to
operate without bias. One of the few examples is “TR One, one transistor radio” (page 425) with
an amplified AM (amplitude modulation) detector. Note the lack of a bias resistor at the base in
that circuit. In this section we look at a few basic bias circuits which can set a selected emitter
current IE . Given a desired emitter current IE , what values of bias resistors are required, RB ,
RE , etc?

4.10.1

Base Bias

The simplest biasing applies a base-bias resistor between the base and a base battery VBB . It
is convenient to use the existing VCC supply instead of a new bias supply. An example of an
audio amplifier stage using base-biasing is “Crystal radio with one transistor . . . ” (page 425).
Note the resistor from the base to the battery terminal. A similar circuit is shown in Figure ??.
Write a KVL (Krichhoff ’s voltage law) equation about the loop containing the battery, RB ,
and the VBE diode drop on the transistor in Figure 4.85. Note that we use VBB for the base
supply, even though it is actually VCC . If β is large we can make the approximation that IC
=IE . For silicon transistors VBE ∼
=0.7V.

+

RB

_
+
V _
0.7BE =
V

RC

VBB - IΒRB - VBE = 0
VCC
+
_

VBB - VBE = IBRB
IB =

(KVL)

VBB - VBE
RB

IE = (β+1)ΙΒ ≈ βIB
IE =

VBB - VBE
RB / β

Figure 4.85: Base-bias

(IE base-bias)

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

236

Silicon small signal transistors typically have a β in the range of 100-300. Assuming that
we have a β=100 transistor, what value of base-bias resistor is required to yield an emitter
current of 1mA?
Solving the IE base-bias equation for RB and substituting β, VBB , VBE , and IE yields
930kΩ. The closest standard value is 910kΩ.

β = 100
RB =

VBB = 10V

VBB - VBE
IE / β

=

IC ≈ IE = 1ma
10 - 0.7
= 930k
1mA / 100

What is the the emitter current with a 910kΩ resistor? What is the emitter current if we
randomly get a β=300 transistor?

β = 100
VBB = 10V
RB = 910k VBE = 0.7V
VBB - VBE
10 - 0.7
IE =
=
= 1.02mA
910k / 100
RB / β
β = 300
10 - 0.7
3.07mA
IE =
=
910k / 300
The emitter current is little changed in using the standard value 910kΩ resistor. However,
with a change in β from 100 to 300, the emitter current has tripled. This is not acceptable in
a power amplifier if we expect the collector voltage to swing from near VCC to near ground.
However, for low level signals from micro-volts to a about a volt, the bias point can be centered
for a β of square root of (100·300)=173. The bias point will still drift by a considerable amount
. However, low level signals will not be clipped.
Base-bias by its self is not suitable for high emitter currents, as used in power amplifiers.
The base-biased emitter current is not temperature stable. Thermal run away is the result
of high emitter current causing a temperature increase which causes an increase in emitter
current, which further increases temperature.

4.10.2

Collector-feedback bias

Variations in bias due to temperature and beta may be reduced by moving the VBB end of the
base-bias resistor to the collector as in Figure 4.86. If the emitter current were to increase, the
voltage drop across RC increases, decreasing VC , decreasing IB fed back to the base. This, in
turn, decreases the emitter current, correcting the original increase.
Write a KVL equation about the loop containing the battery, RC , RB , and the VBE drop.
Substitute IC ∼
=IE /β. Solving for IE yields the IE CFB-bias equation. Solving for IB
=IE and IB ∼
yields the IB CFB-bias equation.
Find the required collector feedback bias resistor for an emitter current of 1 mA, a 4.7K
collector load resistor, and a transistor with β=100 . Find the collector voltage VC . It should be
approximately midway between VCC and ground.

4.10. BIASING CALCULATIONS

+
RC
_
+

VC

RB
_

VCC +

+
V _
BE =
0.7
V

_

237

IC = βIB
IC ≈ IE
IE ≈ βIB
VCC - ICRC - IBRB - VBE = 0
VCC - IERC - (IE / β)RB - VBE = 0
VCC - VBE = IERC + (IE / β)RB
VCC - VBE = IE((RB / β) + RC)
VCC - VBE
IE =
RB / β + RC
RB = β

VCC - VBE
IE

- RC

(KVL)

(IE CFB-bias)

(RB CFB-bias)

Figure 4.86: Collector-feedback bias.

β = 100
RB = β

IC ≈ IE = 1ma

VCC = 10V

VCC - VBE
IE

- RC = 100

RC = 4.7k

10 - 0.7 -4.7k
1mA

= 460k

VC = VCC - ICRC = 10 - (1mA)⋅(4.7k) = 5.3V
The closest standard value to the 460k collector feedback bias resistor is 470k. Find the
emitter current IE with the 470 K resistor. Recalculate the emitter current for a transistor
with β=100 and β=300.

β = 100
IE =

VCC = 10V

VCC - VBE
RB / β + RC

β = 300
VCC - VBE
IE =
RB / β + RC

RC = 4.7k

=

=

RB = 470k

10 - 0.7
470k / 100 + 4.7k
10 - 0.7
470k / 300 + 4.7k

= 0.989mA

= 1.48mA

We see that as beta changes from 100 to 300, the emitter current increases from 0.989mA
to 1.48mA. This is an improvement over the previous base-bias circuit which had an increase
from 1.02mA to 3.07mA. Collector feedback bias is twice as stable as base-bias with respect to
beta variation.

4.10.3

Emitter-bias

Inserting a resistor RE in the emitter circuit as in Figure 4.87 causes degeneration, also known
as negative feedback. This opposes a change in emitter current IE due to temperature changes,
resistor tolerances, beta variation, or power supply tolerance. Typical tolerances are as follows:
resistor— 5%, beta— 100-300, power supply— 5%. Why might the emitter resistor stabilize a

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

238

change in current? The polarity of the voltage drop across RE is due to the collector battery
VCC . The end of the resistor closest to the (-) battery terminal is (-), the end closest to the
(+) terminal it (+). Note that the (-) end of RE is connected via VBB battery and RB to the
base. Any increase in current flow through RE will increase the magnitude of negative voltage
applied to the base circuit, decreasing the base current, decreasing the emitter current. This
decreasing emitter current partially compensates the original increase.

VBB -IΒRB - VBE - IERE = 0
IE = (b+1)IB ≈ βIB
VBB -(IΕ/β)RB - VBE - IERE = 0
VBB - VBE = IE ( (RB / β) + RE )

RC
V VCC
0.7BE =
V

_+

_

_

VBB - VBE
RB / β + RE
VBB - VBE
RB/β + RE =
IE

IE =

RB = β

VBB - VBE
IE

- RE

(IE emitter-bias)

(RB emitter-bias)

KVL

_

loop

RB +
+
VBB RE
+
_

+

Figure 4.87: Emitter-bias
Note that base-bias battery VBB is used instead of VCC to bias the base in Figure 4.87.
Later we will show that the emitter-bias is more effective with a lower base bias battery.
Meanwhile, we write the KVL equation for the loop through the base-emitter circuit, paying attention to the polarities on the components. We substitute IB ∼
=IE /β and solve for emitter
current IE . This equation can be solved for RB , equation: RB emitter-bias, Figure 4.87.
Before applying the equations: RB emitter-bias and IE emitter-bias, Figure 4.87, we need
to choose values for RC and RE . RC is related to the collector supply VCC and the desired
collector current IC which we assume is approximately the emitter current IE . Normally the
bias point for VC is set to half of VCC . Though, it could be set higher to compensate for the
voltage drop across the emitter resistor RE . The collector current is whatever we require or
choose. It could range from micro-Amps to Amps depending on the application and transistor
rating. We choose IC = 1mA, typical of a small-signal transistor circuit. We calculate a value
for RC and choose a close standard value. An emitter resistor which is 10-50% of the collector
load resistor usually works well.

VC = VCC / 2 = 10 / 2 = 5V
RC = Vc / IC = 5/1mA = 5k (4.7k standard value)
RE = 0.10RC = 0.10(4.7K) = 470Ω

4.10. BIASING CALCULATIONS

239

Our first example sets the base-bias supply to high at VBB = VCC = 10V to show why a lower
voltage is desirable. Determine the required value of base-bias resistor RB . Choose a standard
value resistor. Calculate the emitter current for β=100 and β=300. Compare the stabilization
of the current to prior bias circuits.

β = 100

IE ≈ IC = 1ma

RB = β VBB - VBE
IE

- RE

Vcc = VBB= 10V
= 100

RE = 470Ω

10 - 0.7
0.001

= 883k

- 470

An 883k resistor was calculated for RB , an 870k chosen. At β=100, IE is 1.01mA.

β = 100
IE =

RB = 870k

VBB - VBE
RB / β + RE

=

10 - 0.7
870K/100 + 470

= 1.01mA

=

10 - 0.7
870K/300 + 470

= 2.76mA

β = 300
IE =

VBB - VBE
RB / β + RE

For β=300 the emitter currents are shown in Table 4.7.
Table 4.7: Emitter current comparison for β =100, β =300.
Bias circuit
IC β=100 IC β=300
base-bias
1.02mA
3.07mA
collector feedback bias 0.989mA
1.48mA
emitter-bias, VBB =10V 1.01mA
2.76mA
Table 4.7 shows that for VBB = 10V, emitter-bias does not do a very good job of stabilizing
the emitter current. The emitter-bias example is better than the previous base-bias example,
but, not by much. The key to effective emitter bias is lowering the base supply VBB nearer to
the amount of emitter bias.
How much emitter bias do we Have? Rounding, that is emitter current times emitter resistor: IE RE = (1mA)(470) = 0.47V. In addition, we need to overcome the VBE = 0.7V. Thus,
we need a VBB >(0.47 + 0.7)V or >1.17V. If emitter current deviates, this number will change
compared with the fixed base supply VBB ,causing a correction to base current IB and emitter
current IE . A good value for VB >1.17V is 2V.

β = 100
RB = β

IE ≈ IC = 1ma
VBB - VBE
IE

- RE

Vcc = 10V
= 100

VBB = 2V
2 - 0.7
0.001

RE = 470Ω
- 470

= 83k

The calculated base resistor of 83k is much lower than the previous 883k. We choose 82k
from the list of standard values. The emitter currents with the 82k RB for β=100 and β=300
are:

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

240

β = 100 RB = 82k
IE =

VBB - VBE
RB / β + RE

=

2 - 0.7
82K/100 + 470

= 1.01mA

=

2 - 0.7
82K/300 + 470

= 1.75mA

β = 300
IE =

VBB - VBE
RB / β + RE

Comparing the emitter currents for emitter-bias with VBB = 2V at β=100 and β=300 to
the previous bias circuit examples in Table 4.8, we see considerable improvement at 1.75mA,
though, not as good as the 1.48mA of collector feedback.
Table 4.8: Emitter current comparison for β =100, β =300.
Bias circuit
IC β=100 IC β=300
base-bias
1.02mA
3.07mA
collector feedback bias 0.989mA
1.48mA
emitter-bias, VBB =10V 1.01mA
2.76mA
emitter-bias, VBB =2V
1.01mA
1.75mA
How can we improve the performance of emitter-bias? Either increase the emitter resistor
RB or decrease the base-bias supply VBB or both. As an example, we double the emitter resistor
to the nearest standard value of 910Ω.

β = 100

RB = β

IE ≈ IC = 1ma

VBB - VBE
IE

- RE

Vcc = 10V
= 100

VBB = 2V
2 - 0.7
0.001

RE = 910Ω
- 910

= 39k

The calculated RB = 39k is a standard value resistor. No need to recalculate IE for β = 100.
For β = 300, it is:

β = 300 RB = 39k
IE =

VBB - VBE
RB / β + RE

=

2 - 0.7
39K/300 + 910

= 1.25mA

The performance of the emitter-bias circuit with a 910¡Onega¿ emitter resistor is much
improved. See Table 4.9.
As an exercise, rework the emitter-bias example with the base resistor reverted back to
470Ω, and the base-bias supply reduced to 1.5V.

β = 100
RB = β

IE ≈ IC = 1ma
VBB - VBE
IE

- RE

Vcc = 10V
= 100

VBB = 1.5V RE = 470Ω
1.5 - 0.7
0.001

- 470

= 33k

4.10. BIASING CALCULATIONS

241

Table 4.9: Emitter current comparison for β =100, β =300.
Bias circuit
IC β=100 IC β=300
base-bias
1.02mA
3.07mA
collector feedback bias
0.989mA
1.48mA
emitter-bias, VBB =10V
1.01mA
2.76mA
emitter-bias, VBB =2V, RB =470 1.01mA
1.75mA
emitter-bias, VBB =2V, RB =910 1.00mA
1.25mA
The 33k base resistor is a standard value, emitter current at β = 100 is OK. The emitter
current at β = 300 is:

IE =

VBB - VBE
RB / β + RE

=

1.5 - 0.7
33K/300 + 470

= 1.38mA

Table 4.10 below compares the exercise results 1mA and 1.38mA to the previous examples.
Table 4.10: Emitter current comparison for β =100, β =300.
Bias circuit
IC β=100 IC β=300
base-bias
1.02mA
3.07mA
collector feedback bias
0.989mA
1.48mA
emitter-bias, VBB =10V
1.01mA
2.76mA
emitter-bias, VBB =2V, RB =470
1.01mA
1.75mA
emitter-bias, VBB =2V, RB =910
1.00mA
1.25mA
emitter-bias, VBB =1.5V, RB =470 1.00mA
1.38mA
The emitter-bias equations have been repeated in Figure 4.88 with the internal emitter
resistance included for better accuracy. The internal emitter resistance is the resistance in
the emitter circuit contained within the transistor package. This internal resistance REE is
significant when the (external) emitter resistor RE is small, or even zero. The value of internal
resistance RE is a function of emitter current IE , Table 4.11.
Table 4.11: Derivation of REE
REE = KT/IE m
where:
K=1.38×10−23 watt-sec/o C, Boltzman’s constant
T= temperature in Kelvins ∼
=300.
IE = emitter current
m = varies from 1 to 2 for Silicon
0.026V/I
REE ∼
=
E = 26mV/IE
For reference the 26mV approximation is listed as equation REE in Figure 4.88.
The more accurate emitter-bias equations in Figure 4.88 may be derived by writing a KVL
equation. Alternatively, start with equations IE emitter-bias and RB emitter-bias in Figure 4.87, substituting RE with REE +RE . The result is equations IE EB and RB EB, respectively

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

242

RC
VCC +
_

+_ + REE
_

RB 0VBE =
.
+ + 7V
_

VBB

VBB -IΒRB - VBE - IEREE - IERE= 0
IE = (β +1)IB ≈ βIB
VBB -(IE / β)RB - VBE - IEREE - IERE = 0
VBB - VBE = (IE(RB / β) + IEREE + IERE)

(KVL)

VBB - VBE
RB / β + REE + RE
VBB - VBE
RB/β + REE + RE =
IE

IE =
_

RE

RB = β

VBB - VBE
IE

(IE EB)

- REE -RE

(RB EB)

RE E = 26mV/ IE

(REE)

Figure 4.88: Emitter-bias equations with internal emitter resistance REE included..
in Figure 4.88.
Redo the RB calculation in the previous example (page 239) with the inclusion of REE and
compare the results.

β = 100

IE ≈ IC = 1ma

Vcc = 10V

VBB= 2V

RE = 470Ω

RE E = 26mV/1mA = 26Ω
RB = β

Vcc - VBE
IE

- REE - RE

= 100

2.0 - 0.7
- 26 - 470
0.001

= 80.4k

The inclusion of REE in the calculation results in a lower value of the base resistor RB a
shown in Table 4.12. It falls below the standard value 82k resistor instead of above it.
Table 4.12: Effect of inclusion of REE on calculated RB
REE ?
REE Value
Without REE 83k
With REE
80.4k
Bypass Capacitor for RE
One problem with emitter bias is that a considerable part of the output signal is dropped
across the emitter resistor RE (Figure 4.89). This voltage drop across the emitter resistor is in
series with the base and of opposite polarity compared with the input signal. (This is similar to
a common collector configuration having <1 gain.) This degeneration severely reduces the gain
from base to collector. The solution for AC signal amplifiers is to bypass the emitter resistor
with a capacitor. This restores the AC gain since the resistor is a short for AC signals. The DC

4.10. BIASING CALCULATIONS

243

emitter current still experiences degeneration in the emitter resistor, thus, stabilizing the DC
current.

RC
Ccoupling

Rin
Vin

VCC +
RB

+

RE

_

_

RC
4.7k V
CC +

Ccoupling

Rin
Vin

+

RB
33k
RE
470

Cbypass

_

_

Figure 4.89: Cbypass is required to prevent AC gain reduction.
What value should the bypass capacitor be? That depends on the lowest frequency to be
amplified. For radio frequencies Cbpass would be small. For an audio amplifier extending down
to 20Hz it will be large. A “rule of thumb” for the bypass capacitor is that the reactance should
be 1/10 of the emitter resistance or less. The capacitor should be designed to accommodate the
lowest frequency being amplified. The capacitor for an audio amplifier covering 20Hz to 20kHz
would be:

XC =

1
2πfC

C=

1
2πfXC

C=

1
= 169µF
2π20(470/10)

Note that the internal emitter resistance REE is not bypassed by the bypass capacitor.

4.10.4

Voltage divider bias

Stable emitter bias requires a low voltage base bias supply, Figure 4.90. The alternative to a
base supply VBB is a voltage divider based on the collector supply VCC .
The design technique is to first work out an emitter-bias design, Then convert it to the voltage divider bias configuration by using Thevenin’s Theorem. [4] The steps are shown graphically in Figure 4.91. Draw the voltage divider without assigning values. Break the divider
loose from the base. (The base of the transistor is the load.) Apply Thevenin’s Theorem to yield
a single Thevenin equivalent resistance Rth and voltage source Vth.
The Thevenin equivalent resistance is the resistance from load point (arrow) with the battery (VCC ) reduced to 0 (ground). In other words, R1||R2.The Thevenin equivalent voltage is
the open circuit voltage (load removed). This calculation is by the voltage divider ratio method.

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

244

RC

_

VCC +

R1

_

+
R2
_

+_ + REE
_

RB 0VBE =
.
+ + 7V
_

VBB

VCC +

+
RE

RC

+_ + REE
_

V
0.7BE =
V

+
RE

_

Emitter-bias

_

_

Voltage divider bias

Figure 4.90: Voltage Divider bias replaces base battery with voltage divider.

RC

R1
_
R2
+

R1

VCC +

VCC +

_

_

+_ + REE
_

V
0.7BE =
V

+
RE

Rth
Vth +
Vth

+
R2
_

_

_

Figure 4.91: Thevenin’s Theorem converts voltage divider to single supply Vth and resistance
Vth.

4.10. BIASING CALCULATIONS

245

R1 is obtained by eliminating R2 from the pair of equations for Rth and Vth. The equation of
R1 is in terms of known quantities Rth, Vth, Vcc. Note that Rth is RB , the bias resistor from
the emitter-bias design. The equation for R2 is in terms of R1 and Rth.

Rth = R1 || R2
1
Rth
1
Rth

=

1
R1

=

+

R2
R1 +R2

Vth
=
VCC

R2
R1 +R2

1
R2

f=

R2 +R1
R1⋅R2

R1 = Rth
f

Vth = VCC

=

= Rth

1 R2 +R1
= 1 ⋅ 1
R1 R2
R1 f

VCC
Vth

1
=
R2

1
1
Rth
R1

Convert this previous emitter-bias example to voltage divider bias.

RC
VCC +
10V
RB
VBB

33k
+
RE

R1

RC

?

VCC +

_

10V
R2

470

?
RE

_

_

470

Figure 4.92: Emitter-bias example converted to voltage divider bias.

These values were previously selected or calculated for an emitter-bias example

β = 100
RB = β

IE ≈ IC = 1ma
VBB - VBE
IE

- RE

Vcc = 10V
= 100

VBB = 1.5V RE = 470Ω
1.5 - 0.7
0.001

- 470

= 33k

Substituting VCC , VBB , RB yields R1 and R2 for the voltage divider bias configuration.

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

246

VBB = Vth = 1.5V
RB = Rth = 33k
R1 = Rth

VCC
Vth

R1 = 33k 10 = 220k
1.5

1
=
R2

1
1
Rth
R1

1
=
R2

1
1
33k
220k

R2 = 38.8k

R1 is a standard value of 220K. The closest standard value for R2 corresponding to 38.8k is
39k. This does not change IE enough for us to calculate it.
Problem: Calculate the bias resistors for the cascode amplifier in Figure 4.93. VB2 is the
bias voltage for the common emitter stage. VB1 is a fairly high voltage at 11.5 because we
want the common-base stage to hold the emitter at 11.5-0.7=10.8V, about 11V. (It will be 10V
after accounting for the voltage drop across RB1 .) That is, the common-base stage is the load,
substitute for a resistor, for the common-emitter stage’s collector. We desire a 1mA emitter
current.

VCC = 20V
RB1
Q1

+

VBB1 = 11.5V

RL

VB1
A
Vi

Q2
RB2

Vo

VCC
+

IE = 1mA β = 100

IE =

VBB - VBE
RB / β

RB1 =

VBB - VBE
IE / β

RB2 =

VBB2 - VBE
IE / β

VB2
+

VA = 10V

RL = 4.7k

VBB2 = 1.5V
(IE base-bias)

=

(VBB1 - VA) - VBE
(11.5-10) - 0.7
=
= 80k
1mA / 100
IE / β
=

(1.5) - 0.7
1mA / 100

= 80k

Cascode
Figure 4.93: Bias for a cascode amplifier.

Problem: Convert the base bias resistors for the cascode amplifier to voltage divider bias
resistors driven by the VCC of 20V.

4.11. INPUT AND OUTPUT COUPLING
RBB1 = 80k

VCC = Vth = 20V

247
RBB2 = 80k

VBB1 = 11.5V

VBB2 = 1.5V

VBB = Vth = 11.5V

VBB = Vth = 1.5V

RB = Rth = 80k

RB = Rth = 80k

R1 = Rth

VCC
Vth

R3 = Rth

VCC
Vth

R1 = 80k 20 = 139.1k
11.5

R3 = 80k 20 = 1.067Meg
1.5

1
=
R2

1
1
Rth
R1

1
=
R4

1
1
Rth
R3

1
=
R2

1
1
80k 139.1k

1
=
R4

1
1
80k 1067k

R2 = 210k

R4 = 86.5k

The final circuit diagram is shown in the “Practical Analog Circuits” chapter, “Class A
cascode amplifier . . . ” (page 431).
• REVIEW:
• See Figure 4.94.
• Select bias circuit configuration
• Select RC and IE for the intended application. The values for RC and IE should normally
set collector voltage VC to 1/2 of VCC .
• Calculate base resistor RB to achieve desired emitter current.
• Recalculate emitter current IE for standard value resistors if necessary.
• For voltage divider bias, perform emitter-bias calculations first, then determine R1 and
R2.
• For AC amplifiers, a bypass capacitor in parallel with RE improves AC gain. Set XC ≤0.10RE
for lowest frequency.

4.11

Input and output coupling

To overcome the challenge of creating necessary DC bias voltage for an amplifier’s input signal
without resorting to the insertion of a battery in series with the AC signal source, we used a
voltage divider connected across the DC power source. To make this work in conjunction with
an AC input signal, we “coupled” the signal source to the divider through a capacitor, which
acted as a high-pass filter. With that filtering in place, the low impedance of the AC signal
source couldn’t “short out” the DC voltage dropped across the bottom resistor of the voltage
divider. A simple solution, but not without any disadvantages.

248

RC
VCC

RB

RC
VCC +

VBB

VCC +

_

VCC - VBE
RB / β + RC

VBB - VBE
RB / β

IE =

RB =

VBB - VBE
IE / β

RB = β

VCC - VBE
IE

- RC

Collector feedback bias

_

R2

RE

RE

VBB - VBE
RB / β + RE

VBB = Vth

RE ⇐ RE + REE
to include REE

RB = Rth

REE = 26mv/IE

R1 = Rth

IE =

RB = β
Base-bias

RB
+
_

_

IE =

VCC +

_

VC

RB

+

VBB - VBE
IE

Emitter-bias

- RE

1
=
R2

VCC
Vth
1
1
Rth
R1

Voltage divider bias

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

Figure 4.94: Biasing equations summary.

RC

RC

R1

4.11. INPUT AND OUTPUT COUPLING

249

Most obvious is the fact that using a high-pass filter capacitor to couple the signal source
to the amplifier means that the amplifier can only amplify AC signals. A steady, DC voltage
applied to the input would be blocked by the coupling capacitor just as much as the voltage
divider bias voltage is blocked from the input source. Furthermore, since capacitive reactance
is frequency-dependent, lower-frequency AC signals will not be amplified as much as higherfrequency signals. Non-sinusoidal signals will tend to be distorted, as the capacitor responds
differently to each of the signal’s constituent harmonics. An extreme example of this would be
a low-frequency square-wave signal in Figure 4.95.

C

R2
V1

Vinput

R3

Figure 4.95: Capacitively coupled low frequency square-wave shows distortion.
Incidentally, this same problem occurs when oscilloscope inputs are set to the “AC coupling” mode as in Figure 4.97. In this mode, a coupling capacitor is inserted in series with the
measured voltage signal to eliminate any vertical offset of the displayed waveform due to DC
voltage combined with the signal. This works fine when the AC component of the measured
signal is of a fairly high frequency, and the capacitor offers little impedance to the signal.
However, if the signal is of a low frequency, or contains considerable levels of harmonics over
a wide frequency range, the oscilloscope’s display of the waveform will not be accurate. (Figure 4.97) Low frequency signals may be viewed by setting the oscilloscope to “DC coupling” in
Figure 4.96.
In applications where the limitations of capacitive coupling (Figure 4.95) would be intolerable, another solution may be used: direct coupling. Direct coupling avoids the use of capacitors
or any other frequency-dependent coupling component in favor of resistors. A direct-coupled
amplifier circuit is shown in Figure 4.98.
With no capacitor to filter the input signal, this form of coupling exhibits no frequency
dependence. DC and AC signals alike will be amplified by the transistor with the same gain
(the transistor itself may tend to amplify some frequencies better than others, but that is
another subject entirely!).
If direct coupling works for DC as well as for AC signals, then why use capacitive coupling
for any application? One reason might be to avoid any unwanted DC bias voltage naturally
present in the signal to be amplified. Some AC signals may be superimposed on an uncontrolled DC voltage right from the source, and an uncontrolled DC voltage would make reliable
transistor biasing impossible. The high-pass filtering offered by a coupling capacitor would

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

250

FUNCTION GENERATOR

40.00 Hz
coarse

fine

1

10 100 1k 10k 100k 1M

DC

output

OSCILLOSCOPE
vertical
Y
V/div

DC GND AC

trigger

timebase
X
s/div

DC GND AC

Figure 4.96: With DC coupling, the oscilloscope properly indicates the shape of the square
wave coming from the signal generator.
work well here to avoid biasing problems.
Another reason to use capacitive coupling rather than direct is its relative lack of signal
attenuation. Direct coupling through a resistor has the disadvantage of diminishing, or attenuating, the input signal so that only a fraction of it reaches the base of the transistor. In many
applications, some attenuation is necessary anyway to prevent signal levels from “overdriving”
the transistor into cutoff and saturation, so any attenuation inherent to the coupling network
is useful anyway. However, some applications require that there be no signal loss from the input connection to the transistor’s base for maximum voltage gain, and a direct coupling scheme
with a voltage divider for bias simply won’t suffice.
So far, we’ve discussed a couple of methods for coupling an input signal to an amplifier, but
haven’t addressed the issue of coupling an amplifier’s output to a load. The example circuit
used to illustrate input coupling will serve well to illustrate the issues involved with output
coupling.
In our example circuit, the load is a speaker. Most speakers are electromagnetic in design:
that is, they use the force generated by an lightweight electromagnet coil suspended within a
strong permanent-magnet field to move a thin paper or plastic cone, producing vibrations in
the air which our ears interpret as sound. An applied voltage of one polarity moves the cone
outward, while a voltage of the opposite polarity will move the cone inward. To exploit cone’s
full freedom of motion, the speaker must receive true (unbiased) AC voltage. DC bias applied to
the speaker coil offsets the cone from its natural center position, and this limits the back-andforth motion it can sustain from the applied AC voltage without overtraveling. However, our
example circuit (Figure 4.98) applies a varying voltage of only one polarity across the speaker,

4.11. INPUT AND OUTPUT COUPLING

251

FUNCTION GENERATOR

40.00
coarse

Hz
1

10 100 1k 10k 100k 1M

fine

DC

output

OSCILLOSCOPE
vertical
Y
V/div

DC GND AC

trigger

timebase
X
s/div

DC GND AC

Figure 4.97: Low frequency: With AC coupling, the high-pass filtering of the coupling capacitor
distorts the square wave’s shape so that what is seen is not an accurate representation of the
real signal.

4
1
Vinput

Rinput

R2

3
2

4
speaker

Q1

V1

R3
0

0

0

0

Figure 4.98: Direct coupled amplifier: direct coupling to speaker.

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

252

because the speaker is connected in series with the transistor which can only conduct current
one way. This would be unacceptable for any high-power audio amplifier.
Somehow we need to isolate the speaker from the DC bias of the collector current so that
it only receives AC voltage. One way to achieve this goal is to couple the transistor collector
circuit to the speaker through a transformer in Figure 4.99)

speaker

R
Vinput

R
Q1

V1

R

Figure 4.99: Transformer coupling isolates DC from the load (speaker).
Voltage induced in the secondary (speaker-side) of the transformer will be strictly due to
variations in collector current, because the mutual inductance of a transformer only works on
changes in winding current. In other words, only the AC portion of the collector current signal
will be coupled to the secondary side for powering the speaker. The speaker will “see” true
alternating current at its terminals, without any DC bias.
Transformer output coupling works, and has the added benefit of being able to provide
impedance matching between the transistor circuit and the speaker coil with custom winding
ratios. However, transformers tend to be large and heavy, especially for high-power applications. Also, it is difficult to engineer a transformer to handle signals over a wide range of
frequencies, which is almost always required for audio applications. To make matters worse,
DC current through the primary winding adds to the magnetization of the core in one polarity
only, which tends to make the transformer core saturate more easily in one AC polarity cycle
than the other. This problem is reminiscent of having the speaker directly connected in series with the transistor: a DC bias current tends to limit how much output signal amplitude
the system can handle without distortion. Generally, though, a transformer can be designed to
handle a lot more DC bias current than a speaker without running into trouble, so transformer
coupling is still a viable solution in most cases. See the coupling transformer between Q4 and
the speaker, (page 425) as an example of transformer coupling.
Another method to isolate the speaker from DC bias in the output signal is to alter the
circuit a bit and use a coupling capacitor in a manner similar to coupling the input signal
(Figure 4.100) to the amplifier.
This circuit in Figure 4.100 resembles the more conventional form of common-emitter amplifier, with the transistor collector connected to the battery through a resistor. The capacitor
acts as a high-pass filter, passing most of the AC voltage to the speaker while blocking all
DC voltage. Again, the value of this coupling capacitor is chosen so that its impedance at the

4.11. INPUT AND OUTPUT COUPLING

253

R
R
Vinput

C
speaker

R
Q1

V1

R

Figure 4.100: Capacitor coupling isolates DC from the load.
expected signal frequency will be arbitrarily low.
The blocking of DC voltage from an amplifier’s output, be it via a transformer or a capacitor,
is useful not only in coupling an amplifier to a load, but also in coupling one amplifier to another
amplifier. “Staged” amplifiers are often used to achieve higher power gains than what would
be possible using a single transistor as in Figure 4.101.

Voutput

Vinput

First stage

Second stage

Third stage

Figure 4.101: Capacitor coupled three stage common-emitter amplifier.
While it is possible to directly couple each stage to the next (via a resistor rather than a
capacitor), this makes the whole amplifier very sensitive to variations in the DC bias voltage of
the first stage, since that DC voltage will be amplified along with the AC signal until the last
stage. In other words, the biasing of the first stage will affect the biasing of the second stage,
and so on. However, if the stages are capacitively coupled shown in the above illustration, the
biasing of one stage has no effect on the biasing of the next, because DC voltage is blocked from
passing on to the next stage.
Transformer coupling between amplifier stages is also a possibility, but less often seen due
to some of the problems inherent to transformers mentioned previously. One notable exception
to this rule is in radio-frequency amplifiers (Figure 4.102) with small coupling transformers,
having air cores (making them immune to saturation effects), that are part of a resonant circuit
to block unwanted harmonic frequencies from passing on to subsequent stages. The use of

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

254

resonant circuits assumes that the signal frequency remains constant, which is typical of radio
circuitry. Also, the “flywheel” effect of LC tank circuits allows for class C operation for high
efficiency.

Voutput

Vinput

First stage

Second stage

Third stage

Figure 4.102: Three stage tuned RF amplifier illustrates transformer coupling.
Note the transformer coupling between transistors Q1, Q2, Q3, and Q4, (page 425). The
three intermediate frequency (IF) transformers within the dashed boxes couple the IF signal
from collector to base of following transistor IF amplifiers. The intermediate freqency ampliers
are RF amplifiers, though, at a different frequency than the antenna RF input.
Having said all this, it must be mentioned that it is possible to use direct coupling within a
multi-stage transistor amplifier circuit. In cases where the amplifier is expected to handle DC
signals, this is the only alternative.
The trend of electronics to more widespread use of integrated circuits has encouraged the
use of direct coupling over transformer or capacitor coupling. The only easily manufactured
integrated circuit component is the transistor. Moderate quality resistors can also be produced.
Though, transistors are favored. Integrated capacitors to only a few 10’s of pF are possible.
Large capacitors are not integrable. If necessary, these can be external components. The same
is true of transformers. Since integrated transistors are inexpensive, as many transistors as
possible are substituted for the offending capacitors and transformers. As much direct coupled
gain as possible is designed into ICs between the external coupling components. While external
capacitors and transformers are used, these are even being designed out if possible. The result
is that a modern IC radio (See “IC radio”, (page 428)) looks nothing like the original 4-transistor
radio (page 425).
Even discrete transistors are inexpensive compared with transformers. Bulky audio transformers can be replaced by transistors. For example, a common-collector (emitter follower)
configuration can impedance match a low output impedance like a speaker. It is also possible
to replace large coupling capacitors with transistor circuitry.
We still like to illustrate texts with transformer coupled audio amplifiers. The circuits
are simple. The component count is low. And, these are good introductory circuits— easy to
understand.
The circuit in Figure 4.103 (a) is a simplified transformer coupled push-pull audio amplifier.
In push-pull, pair of transistors alternately amplify the positive and negative portions of the
input signal. Neither transistor nor the other conducts for no signal input. A positive input

4.11. INPUT AND OUTPUT COUPLING

255

signal will be positive at the top of the transformer secondary causing the top transistor to
conduct. A negative input will yield a positive signal at the bottom of the secondary, driving the
bottom transistor into conduction. Thus the transistors amplify alternate halves of a signal.
As drawn, neither transistor in Figure 4.103 (a) will conduct for an input below 0.7 Vpeak.
A practical circuit connects the secondary center tap to a 0.7 V (or greater) resistor divider
instead of ground to bias both transistor for true class B.
+Vcc
R3

R1
Q2

input

+Vcc

Q3

Q1
input
Q4
R4

(a)

R5

R2

(b)

Figure 4.103: (a) Transformer coupled push-pull amplifier. (b) Direct coupled complementarypair amplifier replaces transformers with transistors.
The circuit in Figure 4.103 (b) is the modern version which replaces the transformer functions with transistors. Transistors Q1 and Q2 are common emitter amplifiers, inverting the
signal with gain from base to collector. Transistors Q3 and Q4 are known as a complementary
pair because these NPN and PNP transistors amplify alternate halves (positive and negative, respectively) of the waveform. The parallel connection the bases allows phase splitting
without an input transformer at (a). The speaker is the emitter load for Q3 and Q4 . Parallel
connection of the emitters of the NPN and PNP transistors eliminates the center-tapped output transformer at (a) The low output impedance of the emitter follower serves to match the
low 8 Ω impedance of the speaker to the preceding common emitter stage. Thus, inexpensive
transistors replace transformers. For the complete circuit see“ Direct coupled complementary
symmetry 3 w audio amplifier,” (page 423)
• REVIEW:
• Capacitive coupling acts like a high-pass filter on the input of an amplifier. This tends
to make the amplifier’s voltage gain decrease at lower signal frequencies. Capacitivecoupled amplifiers are all but unresponsive to DC input signals.
• Direct coupling with a series resistor instead of a series capacitor avoids the problem of
frequency-dependent gain, but has the disadvantage of reducing amplifier gain for all
signal frequencies by attenuating the input signal.

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

256

• Transformers and capacitors may be used to couple the output of an amplifier to a load,
to eliminate DC voltage from getting to the load.
• Multi-stage amplifiers often make use of capacitive coupling between stages to eliminate
problems with the bias from one stage affecting the bias of another.

4.12

Feedback

If some percentage of an amplifier’s output signal is connected to the input, so that the amplifier amplifies part of its own output signal, we have what is known as feedback. Feedback
comes in two varieties: positive (also called regenerative), and negative (also called degenerative). Positive feedback reinforces the direction of an amplifier’s output voltage change, while
negative feedback does just the opposite.
A familiar example of feedback happens in public-address (“PA”) systems where someone
holds the microphone too close to a speaker: a high-pitched “whine” or “howl” ensues, because
the audio amplifier system is detecting and amplifying its own noise. Specifically, this is an
example of positive or regenerative feedback, as any sound detected by the microphone is amplified and turned into a louder sound by the speaker, which is then detected by the microphone
again, and so on . . . the result being a noise of steadily increasing volume until the system
becomes “saturated” and cannot produce any more volume.
One might wonder what possible benefit feedback is to an amplifier circuit, given such an
annoying example as PA system “howl.” If we introduce positive, or regenerative, feedback into
an amplifier circuit, it has the tendency of creating and sustaining oscillations, the frequency
of which determined by the values of components handling the feedback signal from output to
input. This is one way to make an oscillator circuit to produce AC from a DC power supply.
Oscillators are very useful circuits, and so feedback has a definite, practical application for us.
See “Phase shift oscillator” (page 423) for a practical application of positive feedback.
Negative feedback, on the other hand, has a “dampening” effect on an amplifier: if the
output signal happens to increase in magnitude, the feedback signal introduces a decreasing
influence into the input of the amplifier, thus opposing the change in output signal. While
positive feedback drives an amplifier circuit toward a point of instability (oscillations), negative
feedback drives it the opposite direction: toward a point of stability.
An amplifier circuit equipped with some amount of negative feedback is not only more
stable, but it distorts the input waveform less and is generally capable of amplifying a wider
range of frequencies. The tradeoff for these advantages (there just has to be a disadvantage to
negative feedback, right?) is decreased gain. If a portion of an amplifier’s output signal is “fed
back” to the input to oppose any changes in the output, it will require a greater input signal
amplitude to drive the amplifier’s output to the same amplitude as before. This constitutes a
decreased gain. However, the advantages of stability, lower distortion, and greater bandwidth
are worth the tradeoff in reduced gain for many applications.
Let’s examine a simple amplifier circuit and see how we might introduce negative feedback
into it, starting with Figure 4.104.
The amplifier configuration shown here is a common-emitter, with a resistor bias network
formed by R1 and R2 . The capacitor couples Vinput to the amplifier so that the signal source
doesn’t have a DC voltage imposed on it by the R1 /R2 divider network. Resistor R3 serves

4.12. FEEDBACK

257

Rload

R1

+

R3
Voutput

−

R2

Vinput

Figure 4.104: Common-emitter amplifier without feedback.
the purpose of controlling voltage gain. We could omit it for maximum voltage gain, but since
base resistors like this are common in common-emitter amplifier circuits, we’ll keep it in this
schematic.
Like all common-emitter amplifiers, this one inverts the input signal as it is amplified. In
other words, a positive-going input voltage causes the output voltage to decrease, or move
toward negative, and vice versa. The oscilloscope waveforms are shown in Figure 4.105.

Rload

R1
R3

+
-

Vinput

R2

Figure 4.105: Common-emitter amplifier, no feedback, with reference waveforms for comparison.
Because the output is an inverted, or mirror-image, reproduction of the input signal, any
connection between the output (collector) wire and the input (base) wire of the transistor in
Figure 4.106 will result in negative feedback.
The resistances of R1 , R2 , R3 , and Rf eedback function together as a signal-mixing network
so that the voltage seen at the base of the transistor (with respect to ground) is a weighted
average of the input voltage and the feedback voltage, resulting in signal of reduced amplitude

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

258

R1

Rfeedback

Rload
+

R3

Vinput

R2

Figure 4.106: Negative feedback, collector feedback, decreases the output signal.
going into the transistor. So, the amplifier circuit in Figure 4.106 will have reduced voltage
gain, but improved linearity (reduced distortion) and increased bandwidth.
A resistor connecting collector to base is not the only way to introduce negative feedback
into this amplifier circuit, though. Another method, although more difficult to understand at
first, involves the placement of a resistor between the transistor’s emitter terminal and circuit
ground in Figure 4.107.

Rload

R1

+

R3

Vinput

R2

Rfeedback

Figure 4.107: Emitter feedback: A different method of introducing negative feedback into a
circuit.
This new feedback resistor drops voltage proportional to the emitter current through the
transistor, and it does so in such a way as to oppose the input signal’s influence on the baseemitter junction of the transistor. Let’s take a closer look at the emitter-base junction and see
what difference this new resistor makes in Figure 4.108.
With no feedback resistor connecting the emitter to ground in Figure 4.108 (a) , whatever
level of input signal (Vinput ) makes it through the coupling capacitor and R1 /R2 /R3 resistor

4.12. FEEDBACK

259

network will be impressed directly across the base-emitter junction as the transistor’s input
voltage (VB−E ). In other words, with no feedback resistor, VB−E equals Vinput . Therefore, if
Vinput increases by 100 mV, then VB−E increases by 100 mV: a change in one is the same as a
change in the other, since the two voltages are equal to each other.
Now let’s consider the effects of inserting a resistor (Rf eedback ) between the transistor’s
emitter lead and ground in Figure 4.108 (b).

Icollector

Ibase
+

+
Vinput

VB-E
-

+

+

Iemitter

Icollector

Ibase

Vinput

VB-E
-

Iemitter

+
-

(a)

Rfeedback Vfeedback

(b)

Figure 4.108: (a) No feedback vs (b) emitter feedback. A waveform at the collector is inverted
with respect to the base. At (b) the emitter waveform is in-phase (emitter follower) with base,
out of phase with collector. Therefore, the emitter signal subtracts from the collector output
signal.
Note how the voltage dropped across Rf eedback adds with VB−E to equal Vinput . With
Rf eedback in the Vinput −− VB−E loop, VB−E will no longer be equal to Vinput . We know that
Rf eedback will drop a voltage proportional to emitter current, which is in turn controlled by the
base current, which is in turn controlled by the voltage dropped across the base-emitter junction of the transistor (VB−E ). Thus, if Vinput were to increase in a positive direction, it would
increase VB−E , causing more base current, causing more collector (load) current, causing more
emitter current, and causing more feedback voltage to be dropped across Rf eedback . This increase of voltage drop across the feedback resistor, though, subtracts from Vinput to reduce the
VB−E , so that the actual voltage increase for VB−E will be less than the voltage increase of
Vinput . No longer will a 100 mV increase in Vinput result in a full 100 mV increase for VB−E ,
because the two voltages are not equal to each other.
Consequently, the input voltage has less control over the transistor than before, and the
voltage gain for the amplifier is reduced: just what we expected from negative feedback.
In practical common-emitter circuits, negative feedback isn’t just a luxury; its a necessity
for stable operation. In a perfect world, we could build and operate a common-emitter transistor amplifier with no negative feedback, and have the full amplitude of Vinput impressed across
the transistor’s base-emitter junction. This would give us a large voltage gain. Unfortunately,
though, the relationship between base-emitter voltage and base-emitter current changes with
temperature, as predicted by the “diode equation.” As the transistor heats up, there will be
less of a forward voltage drop across the base-emitter junction for any given current. This
causes a problem for us, as the R1 /R2 voltage divider network is designed to provide the correct
quiescent current through the base of the transistor so that it will operate in whatever class

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

260

of operation we desire (in this example, I’ve shown the amplifier working in class-A mode). If
the transistor’s voltage/current relationship changes with temperature, the amount of DC bias
voltage necessary for the desired class of operation will change. A hot transistor will draw
more bias current for the same amount of bias voltage, making it heat up even more, drawing
even more bias current. The result, if unchecked, is called thermal runaway.
Common-collector amplifiers, (Figure 4.109) however, do not suffer from thermal runaway.
Why is this? The answer has everything to do with negative feedback.

R1
+

R3

Vinput

R2

Rload

Figure 4.109: Common collector (emitter follower) amplifier.
Note that the common-collector amplifier (Figure 4.109) has its load resistor placed in exactly the same spot as we had the Rf eedback resistor in the last circuit in Figure 4.108 (b):
between emitter and ground. This means that the only voltage impressed across the transistor’s base-emitter junction is the difference between Vinput and Voutput , resulting in a very low
voltage gain (usually close to 1 for a common-collector amplifier). Thermal runaway is impossible for this amplifier: if base current happens to increase due to transistor heating, emitter
current will likewise increase, dropping more voltage across the load, which in turn subtracts
from Vinput to reduce the amount of voltage dropped between base and emitter. In other words,
the negative feedback afforded by placement of the load resistor makes the problem of thermal
runaway self-correcting. In exchange for a greatly reduced voltage gain, we get superb stability
and immunity from thermal runaway.
By adding a “feedback” resistor between emitter and ground in a common-emitter amplifier,
we make the amplifier behave a little less like an “ideal” common-emitter and a little more like
a common-collector. The feedback resistor value is typically quite a bit less than the load,
minimizing the amount of negative feedback and keeping the voltage gain fairly high.
Another benefit of negative feedback, seen clearly in the common-collector circuit, is that
it tends to make the voltage gain of the amplifier less dependent on the characteristics of the
transistor. Note that in a common-collector amplifier, voltage gain is nearly equal to unity
(1), regardless of the transistor’s β. This means, among other things, that we could replace
the transistor in a common-collector amplifier with one having a different β and not see any
significant changes in voltage gain. In a common-emitter circuit, the voltage gain is highly dependent on β. If we were to replace the transistor in a common-emitter circuit with another of

4.12. FEEDBACK

261

differing β, the voltage gain for the amplifier would change significantly. In a common-emitter
amplifier equipped with negative feedback, the voltage gain will still be dependent upon transistor β to some degree, but not as much as before, making the circuit more predictable despite
variations in transistor β.
The fact that we have to introduce negative feedback into a common-emitter amplifier to
avoid thermal runaway is an unsatisfying solution. Iis it possibe to avoid thermal runaway
without having to suppress the amplifier’s inherently high voltage gain? A best-of-both-worlds
solution to this dilemma is available to us if we closely examine the problem: the voltage gain
that we have to minimize in order to avoid thermal runaway is the DC voltage gain, not the AC
voltage gain. After all, it isn’t the AC input signal that fuels thermal runaway: its the DC bias
voltage required for a certain class of operation: that quiescent DC signal that we use to “trick”
the transistor (fundamentally a DC device) into amplifying an AC signal. We can suppress DC
voltage gain in a common-emitter amplifier circuit without suppressing AC voltage gain if we
figure out a way to make the negative feedback only function with DC. That is, if we only feed
back an inverted DC signal from output to input, but not an inverted AC signal.
The Rf eedback emitter resistor provides negative feedback by dropping a voltage proportional
to load current. In other words, negative feedback is accomplished by inserting an impedance
into the emitter current path. If we want to feed back DC but not AC, we need an impedance
that is high for DC but low for AC. What kind of circuit presents a high impedance to DC but
a low impedance to AC? A high-pass filter, of course!
By connecting a capacitor in parallel with the feedback resistor in Figure 4.110, we create
the very situation we need: a path from emitter to ground that is easier for AC than it is for
DC.

Rload

R1

+

R3

Vinput

R2

Rfeedback

Cbypass

Figure 4.110: High AC voltage gain reestablished by adding Cbypass in parallel with Rf eedback
The new capacitor “bypasses” AC from the transistor’s emitter to ground, so that no appreciable AC voltage will be dropped from emitter to ground to “feed back” to the input and
suppress voltage gain. Direct current, on the other hand, cannot go through the bypass capacitor, and so must travel through the feedback resistor, dropping a DC voltage between emitter
and ground which lowers the DC voltage gain and stabilizes the amplifier’s DC response, preventing thermal runaway. Because we want the reactance of this capacitor (XC ) to be as low

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

262

as possible, Cbypass should be sized relatively large. Because the polarity across this capacitor
will never change, it is safe to use a polarized (electrolytic) capacitor for the task.
Another approach to the problem of negative feedback reducing voltage gain is to use multistage amplifiers rather than single-transistor amplifiers. If the attenuated gain of a single
transistor is insufficient for the task at hand, we can use more than one transistor to make
up for the reduction caused by feedback. An example circuit showing negative feedback in a
three-stage common-emitter amplifier is Figure 4.111.

Rfeedback

+

Rin
Vinput

Voutput

-

Figure 4.111: Feedback around an “odd” number of direct coupled stages produce negative
feedback.
The feedback path from the final output to the input is through a single resistor, Rf eedback .
Since each stage is a common-emitter amplifier (thus inverting), the odd number of stages
from input to output will invert the output signal; the feedback will be negative (degenerative).
Relatively large amounts of feedback may be used without sacrificing voltage gain, because the
three amplifier stages provide much gain to begin with.
At first, this design philosophy may seem inelegant and perhaps even counter-productive.
Isn’t this a rather crude way to overcome the loss in gain incurred through the use of negative
feedback, to simply recover gain by adding stage after stage? What is the point of creating a
huge voltage gain using three transistor stages if we’re just going to attenuate all that gain
anyway with negative feedback? The point, though perhaps not apparent at first, is increased
predictability and stability from the circuit as a whole. If the three transistor stages are designed to provide an arbitrarily high voltage gain (in the tens of thousands, or greater) with no
feedback, it will be found that the addition of negative feedback causes the overall voltage gain
to become less dependent of the individual stage gains, and approximately equal to the simple
ratio Rf eedback /Rin . The more voltage gain the circuit has (without feedback), the more closely
the voltage gain will approximate Rf eedback /Rin once feedback is established. In other words,
voltage gain in this circuit is fixed by the values of two resistors, and nothing more.
This is an advantage for mass-production of electronic circuitry: if amplifiers of predictable
gain may be constructed using transistors of widely varied β values, it eases the selection and
replacement of components. It also means the amplifier’s gain varies little with changes in
temperature. This principle of stable gain control through a high-gain amplifier “tamed” by
negative feedback is elevated almost to an art form in electronic circuits called operational

4.13. AMPLIFIER IMPEDANCES

263

amplifiers, or op-amps. You may read much more about these circuits in a later chapter of this
book!
• REVIEW:
• Feedback is the coupling of an amplifier’s output to its input.
• Positive, or regenerative feedback has the tendency of making an amplifier circuit unstable, so that it produces oscillations (AC). The frequency of these oscillations is largely
determined by the components in the feedback network.
• Negative, or degenerative feedback has the tendency of making an amplifier circuit more
stable, so that its output changes less for a given input signal than without feedback.
This reduces the gain of the amplifier, but has the advantage of decreasing distortion and
increasing bandwidth (the range of frequencies the amplifier can handle).
• Negative feedback may be introduced into a common-emitter circuit by coupling collector
to base, or by inserting a resistor between emitter and ground.
• An emitter-to-ground “feedback” resistor is usually found in common-emitter circuits as
a preventative measure against thermal runaway.
• Negative feedback also has the advantage of making amplifier voltage gain more dependent on resistor values and less dependent on the transistor’s characteristics.
• Common-collector amplifiers have much negative feedback, due to the placement of the
load resistor between emitter and ground. This feedback accounts for the extremely stable voltage gain of the amplifier, as well as its immunity against thermal runaway.
• Voltage gain for a common-emitter circuit may be re-established without sacrificing immunity to thermal runaway, by connecting a bypass capacitor in parallel with the emitter
“feedback resistor.”
• If the voltage gain of an amplifier is arbitrarily high (tens of thousands, or greater), and
negative feedback is used to reduce the gain to reasonable levels, it will be found that
the gain will approximately equal Rf eedback /Rin . Changes in transistor β or other internal component values will have little effect on voltage gain with feedback in operation,
resulting in an amplifier that is stable and easy to design.

4.13

Amplifier impedances

Input impedance varies considerably with the circuit configuration shown in Figure 4.112. It
also varies with biasing. Not considered here, the input impedance is complex and varies with
frequency. For the common-emitter and common-collector it is base resistance times β. The
base resistance can be both internal and external to the transistor. For the common-collector:
Rin = βRE
It is a bit more complicated for the common-emitter circuit. We need to know the internal
emitter resistance REE . This is given by:

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

264

REE = KT/IE m
where:
K=1.38×10−23 watt-sec/o C, Boltzman’s constant
T= temperature in Kelvins ∼
=300.
IE = emitter current
m = varies from 1 to 2 for Silicon
RE ∼
= 0.026V/IE = 26mV/IE
Thus, for the common-emitter circuit Rin is
Rin = βREE /IE
As an example the input resistance of a, β = 100, CE configuration biased at 1 mA is:
REE = 26mV/1mA = 0.26
Rin = βREE = 100(26) = 2600Ω
Moreover, a more accurate Rin for the common-collector should have included Re’
Rin = β(RE + REE )
This equation (above) is also applicable to a common-emitter configuration with an emitter
resistor.
Input impedance for the common-base configuration is Rin = REE .
The high input impedance of the common-collector configuration matches high impedance
sources. A crystal or ceramic microphone is one such high impedance source. The commonbase arrangement is sometimes used in RF (radio frequency) circuits to match a low impedance
source, for example, a 50 Ω coaxial cable feed. For moderate impedance sources, the commonemitter is a good match. An example is a dynamic microphone.
The output impedances of the three basic configurations are listed in Figure 4.112. The
moderate output impedance of the common-emitter configuration helps make it a popular
choice for general use. The Low output impedance of the common-collector is put to good
use in impedance matching, for example, tranformerless matching to a 4 Ohm speaker. There
do not appear to be any simple formulas for the output impedances. However, R. Victor Jones
develops expressions for output resistance. [3]
• REVIEW:
• See Figure 4.112.

4.14

Current mirrors

An often-used circuit applying the bipolar junction transistor is the so-called current mirror,
which serves as a simple current regulator, supplying nearly constant current to a load over a
wide range of load resistances.
We know that in a transistor operating in its active mode, collector current is equal to base
current multiplied by the ratio β. We also know that the ratio between collector current and
emitter current is called α. Because collector current is equal to base current multiplied by β,
and emitter current is the sum of the base and collector currents, α should be mathematically
derivable from β. If you do the algebra, you’ll find that α = β/(β+1) for any transistor.
We’ve seen already how maintaining a constant base current through an active transistor
results in the regulation of collector current, according to the β ratio. Well, the α ratio works

4.14. CURRENT MIRRORS
Basic circuit

265

Common emitter

Common collector

Common base

Cascode

Vo

+

Vo
Vo

+

Vo

+

- -

+

+

- -

+

-

+-

+

+

Voltage gain

high

less than unity

high, same as CE

high, same as CB

Current gain

high

high

less than unity

high, same as CE

Power gain

high

moderate

moderate

highest

no

no

yes

highest ≈ 300 k

low ≈ 50 Ω

same as CE, ≈ 1 k

low ≈ 300 Ω

highest ≈ 1 Meg

same as CB, ≈ 1 Meg

Phase inversion yes
Input
moderate ≈ 1 k
impedance
Output
moderate ≈ 50 k
impedance

Figure 4.112: Amplifier characteristics, adapted from GE Transistor Manual, Figure 1.21.[2]

similarly: if emitter current is held constant, collector current will remain at a stable, regulated
value so long as the transistor has enough collector-to-emitter voltage drop to maintain it in
its active mode. Therefore, if we have a way of holding emitter current constant through a
transistor, the transistor will work to regulate collector current at a constant value.
Remember that the base-emitter junction of a BJT is nothing more than a PN junction, just
like a diode, and that the “diode equation” specifies how much current will go through a PN
junction given forward voltage drop and junction temperature:

ID = IS (eqVD/NkT - 1)
Where,
ID = Diode current in amps
IS = Saturation current in amps
(typically 1 x 10-12 amps)
e = Euler’s constant (~ 2.718281828)
q = charge of electron (1.6 x 10-19 coulombs)
VD = Voltage applied across diode in volts
N = "Nonideality" or "emission" coefficient
(typically between 1 and 2)
k = Boltzmann’s constant (1.38 x 10-23)
T = Junction temperature in Kelvins
If both junction voltage and temperature are held constant, then the PN junction current
will be constant. Following this rationale, if we were to hold the base-emitter voltage of a

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

266

transistor constant, then its emitter current will be constant, given a constant temperature.
(Figure 4.113)

Icollector
(constant)

Ibase
Vbase
(constant)

Rload

(constant)

β (constant)
α (constant)
Iemitter
(constant)

Figure 4.113: Constant VBE gives constant IB , constant IE , and constant IC .
This constant emitter current, multiplied by a constant α ratio, gives a constant collector
current through Rload , if enough battery voltage is available to keep the transistor in its active
mode for any change in Rload ’s resistance.
To maintain a constant voltage across the transistor’s base-emitter junction use a forwardbiased diode to establish a constant voltage of approximately 0.7 volts, and connect it in parallel
with the base-emitter junction as in Figure 4.114.

Rbias

Icollector
(constant)

Ibase (constant)
0.7 V
(constant)

Idiode
(constant)

Rload
β (constant)
α (constant)
Iemitter
(constant)

Figure 4.114: Diode junction 0.7 V maintains constant base voltage, and constant base current.
The voltage dropped across the diode probably won’t be 0.7 volts exactly. The exact amount
of forward voltage dropped across it depends on the current through the diode, and the diode’s
temperature, all in accordance with the diode equation. If diode current is increased (say, by
reducing the resistance of Rbias ), its voltage drop will increase slightly, increasing the voltage
drop across the transistor’s base-emitter junction, which will increase the emitter current by
the same proportion, assuming the diode’s PN junction and the transistor’s base-emitter junction are well-matched to each other. In other words, transistor emitter current will closely

4.14. CURRENT MIRRORS

267

equal diode current at any given time. If you change the diode current by changing the resistance value of Rbias , then the transistor’s emitter current will follow suit, because the emitter
current is described by the same equation as the diode’s, and both PN junctions experience the
same voltage drop.
Remember, the transistor’s collector current is almost equal to its emitter current, as the
α ratio of a typical transistor is almost unity (1). If we have control over the transistor’s
emitter current by setting diode current with a simple resistor adjustment, then we likewise
have control over the transistor’s collector current. In other words, collector current mimics, or
mirrors, diode current.
Current through resistor Rload is therefore a function of current set by the bias resistor, the
two being nearly equal. This is the function of the current mirror circuit: to regulate current
through the load resistor by conveniently adjusting the value of Rbias . Current through the
diode is described by a simple equation: power supply voltage minus diode voltage (almost a
constant value), divided by the resistance of Rbias .
To better match the characteristics of the two PN junctions (the diode junction and the
transistor base-emitter junction), a transistor may be used in place of a regular diode, as in
Figure 4.115 (a).

Rbias

Rload

+
−

(a) current sinking

+

Rbias

Rload

−

(b) current-sourcing

Figure 4.115: Current mirror circuits.
Because temperature is a factor in the “diode equation,” and we want the two PN junctions
to behave identically under all operating conditions, we should maintain the two transistors at
exactly the same temperature. This is easily done using discrete components by gluing the two
transistor cases back-to-back. If the transistors are manufactured together on a single chip of
silicon (as a so-called integrated circuit, or IC), the designers should locate the two transistors
close to one another to facilitate heat transfer between them.
The current mirror circuit shown with two NPN transistors in Figure 4.115 (a) is sometimes
called a current-sinking type, because the regulating transistor conducts current to the load
from ground (“sinking” current), rather than from the positive side of the battery (“sourcing”
current). If we wish to have a grounded load, and a current sourcing mirror circuit, we may
use PNP transistors like Figure 4.115 (b).
While resistors can be manufactured in ICs, it is easier to fabricate transistors. IC designers
avoid some resistors by replacing load resistors with current sources. A circuit like an operational amplifier built from discrete components will have a few transistors and many resistors.

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

268

An integrated circuit version will have many transistors and a few resistors. In Figure 4.116
One voltage reference, Q1, drives multiple current sources: Q2, Q3, and Q4. If Q2 and Q3 are
equal area transistors the load currents Iload will be equal. If we need a 2·Iload , parallel Q2 and
Q3. Better yet fabricate one transistor, say Q3 with twice the area of Q2. Current I3 will then
be twice I2. In other words, load current scales with transistor area.

Rbias

Rload

+

Iload

Iload

−
Q1

Q2

Q3

Q4

Figure 4.116: Multiple current mirrors may be slaved from a single (Q1 - Rbias ) voltage source.
Note that it is customary to draw the base voltage line right through the transistor symbols
for multiple current mirrors! Or in the case of Q4 in Figure 4.116, two current sources are
associated with a single transistor symbol. The load resistors are drawn almost invisible to
emphasize the fact that these do not exist in most cases. The load is often another (multiple)
transistor circuit, say a pair of emitters of a differential amplifier, for example Q3 and Q4 in
”A simple operational amplifier” (page 410). Often, the collector load of a transistor is not a
resistor but a current mirror. For example the collector load of Q4 collector (page 410) is a
current mirror (Q2).
For an example of a current mirror with multiple collector outputs see Q13 in the model 741
op-amp (page 410). The Q13 current mirror outputs substitute for resistors as collector loads
for Q15 and Q17. We see from these examples that current mirrors are preferred as loads over
resistors in integrated circuitry.
• REVIEW:
• A current mirror is a transistor circuit that regulates current through a load resistance,
the regulation point being set by a simple resistor adjustment.
• Transistors in a current mirror circuit must be maintained at the same temperature for
precise operation. When using discrete transistors, you may glue their cases together to
do this.
• Current mirror circuits may be found in two basic varieties: the current sinking configuration, where the regulating transistor connects the load to ground; and the current
sourcing configuration, where the regulating transistor connects the load to the positive
terminal of the DC power supply.

4.15. TRANSISTOR RATINGS AND PACKAGES

4.15

269

Transistor ratings and packages

Like all electrical and electronic components, transistors are limited in the amounts of voltage and current each one can handle without sustaining damage. Since transistors are more
complex than some of the other components you’re used to seeing at this point, these tend to
have more kinds of ratings. What follows is an itemized description of some typical transistor
ratings.
Power dissipation: When a transistor conducts current between collector and emitter, it also
drops voltage between those two points. At any given time, the power dissipated by a transistor is equal to the product (multiplication) of collector current and collector-emitter voltage.
Just like resistors, transistors are rated for how many watts each can safely dissipate without sustaining damage. High temperature is the mortal enemy of all semiconductor devices,
and bipolar transistors tend to be more susceptible to thermal damage than most. Power ratings are always referenced to the temperature of ambient (surrounding) air. When transistors
are to be used in hotter environments (>25o , their power ratings must be derated to avoid a
shortened service life.
Reverse voltages: As with diodes, bipolar transistors are rated for maximum allowable
reverse-bias voltage across their PN junctions. This includes voltage ratings for the emitterbase junction VEB , collector-base junction VCB , and also from collector to emitter VCE .
VEB , the maximum reverse voltage from emitter to base is approximately 7 V for some
small signal transistors. Some circuit designers use discrete BJTs as 7 V zener diodes with
a series current limiting resistor. Transistor inputs to analog integrated circuits also have a
VEB rating, which if exceeded will cause damage, no zenering of the inputs is allowed.
The rating for maximum collector-emitter voltage VCE can be thought of as the maximum
voltage it can withstand while in full-cutoff mode (no base current). This rating is of particular
importance when using a bipolar transistor as a switch. A typical value for a small signal transistor is 60 to 80 V. In power transistors, this could range to 1000 V, for example, a horizontal
deflection transistor in a cathode ray tube display.
Collector current: A maximum value for collector current IC will be given by the manufacturer in amps. Typical values for small signal transistors are 10s to 100s of mA, 10s of A for
power transistors. Understand that this maximum figure assumes a saturated state (minimum collector-emitter voltage drop). If the transistor is not saturated, and in fact is dropping
substantial voltage between collector and emitter, the maximum power dissipation rating will
probably be exceeded before the maximum collector current rating. Just something to keep in
mind when designing a transistor circuit!
Saturation voltages: Ideally, a saturated transistor acts as a closed switch contact between
collector and emitter, dropping zero voltage at full collector current. In reality this is never
true. Manufacturers will specify the maximum voltage drop of a transistor at saturation, both
between the collector and emitter, and also between base and emitter (forward voltage drop
of that PN junction). Collector-emitter voltage drop at saturation is generally expected to be
0.3 volts or less, but this figure is of course dependent on the specific type of transistor. Low
voltage transistors, low VCE , show lower saturation voltages. The saturation voltage is also
lower for higher base drive current.
Base-emitter forward voltage drop, kVBE , is similar to that of an equivalent diode, ∼
=0.7 V,
which should come as no surprise.
Beta: The ratio of collector current to base current, β is the fundamental parameter char-

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

270

acterizing the amplifying ability of a bipolar transistor. β is usually assumed to be a constant
figure in circuit calculations, but unfortunately this is far from true in practice. As such,
manufacturers provide a set of β (or “hf e ”) figures for a given transistor over a wide range of
operating conditions, usually in the form of maximum/minimum/typical ratings. It may surprise you to see just how widely β can be expected to vary within normal operating limits. One
popular small-signal transistor, the 2N3903, is advertised as having a β ranging from 15 to
150 depending on the amount of collector current. Generally, β is highest for medium collector
currents, decreasing for very low and very high collector currents. hf e is small signal AC gain;
hF E is large AC signal gain or DC gain.
Alpha: the ratio of collector current to emitter current, α=IC /IE . α may be derived from
β, being α=β/(β+1) .
Bipolar transistors come in a wide variety of physical packages. Package type is primarily dependent upon the required power dissipation of the transistor, much like resistors: the
greater the maximum power dissipation, the larger the device has to be to stay cool. Figure 4.117 shows several standardized package types for three-terminal semiconductor devices,
any of which may be used to house a bipolar transistor. There are many other semiconductor
devices other than bipolar transistors which have three connection points. Note that the pinouts of plastic transistors can vary within a single package type, e.g. TO-92 in Figure 4.117. It
is impossible to positively identify a three-terminal semiconductor device without referencing
the part number printed on it, or subjecting it to a set of electrical tests.

5.8
5.3

E ΒC

TO-39
TO-3

16

case, Collector
9.4

E
B
16.89
30.15
39.37

TO-3 (300 w)

10.7

5.2

21
5.3

6.6
15.5

E ΒC

EΒC
ECΒ

TO-92
TO-18

ΒCE

TO-220 (150 w)

B C E
(TO-247 250 w)

Figure 4.117: Transistor packages, dimensions in mm.
Small plastic transistor packages like the TO-92 can dissipate a few hundred milliwatts.
The metal cans, TO-18 and TO-39 can dissipate more power, several hundred milliwatts. Plas-

4.16. BJT QUIRKS

271

tic power transistor packages like the TO-220 and TO-247 dissipate well over 100 watts, approaching the dissipation of the all metal TO-3. The dissipation ratings listed in Figure 4.117
are the maximum ever encountered by the author for high powered devices. Most power transistors are rated at half or less than the listed wattage. Consult specific device datasheets
for actual ratings. The the semiconductor die in the TO-220 and TO-247 plastic packages is
mounted to a heat conductive metal slug which transfers heat from the back of the package to
a metal heatsink, not shown. A thin coating of thermally conductive grease is applied to the
metal before mounting the transistor to the heatsink. Since the TO-220 and TO-247 slugs, and
the TO-3 case are connected to the collector, it is sometimes necessary to electrically isolate the
these from a grounded heatsink by an interposed mica or polymer washer. The datasheet ratings for the power packages are only valid when mounted to a heatsink. Without a heatsink, a
TO-220 dissipates approximately 1 watt safely in free air.
Datasheet maximum power disipation ratings are difficult to acheive in practice. The maximum power dissipation is based on a heatsink maintaining the transistor case at no more than
25o C. This is difficult with an air cooled heatsink. The allowable power dissipation decreases
with increasing temperature. This is known as derating. Many power device datasheets include a dissipation versus case termperaure graph.
• REVIEW:
• Power dissipation: maximum allowable power dissipation on a sustained basis.
• Reverse voltages: maximum allowable VCE , VCB , VEB .
• Collector current: the maximum allowable collector current.
• Saturation voltage is the VCE voltage drop in a saturated (fully conducting) transistor.
• Beta: β=IC /IB
• Alpha: α=IC /IE

α= β/(β+1)

• TransistorPackages are a major factor in power dissipation. Larger packages dissipate
more power.

4.16

BJT quirks

An ideal transistor would show 0% distortion in amplifying a signal. Its gain would extend
to all frequencies. It would control hundreds of amperes of current, at hundreds of degrees C.
In practice, available devices show distortion. Amplification is limited at the high frequency
end of the spectrum. Real parts only handle tens of amperes with precautions. Care must be
taken when paralleling transistors for higher current. Operation at elevated temperatures can
destroy transistors if precautions are not taken.

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

272

Figure 4.118: Distortion in large signal common-emitter amplifier.

4.16.1

Nonlinearity

The class A common-emitter amplifier (similar to Figure 4.34)is driven almost to clipping in
Figure 4.118 . Note that the positive peak is flatter than the negative peaks. This distortion
distortion is unacceptable in many applications like high-fidelity audio.
Small signal amplifiers are relatively linear because they use a small linear section of the
transistor characteristics. Large signal amplifiers are not 100% linear because transistor characteristics like β are not constant, but vary with collector current. β is high at low collector
current, and low at very low current or high current. Though, we primarily encounter decreasing β with increasing collector current.
The SPICE listing in Table 4.119 illustrates how to quantify the amount of distortion. The
”.fourier 2000 v(2)” command tells SPICE to perm a fourier analysis at 2000 Hz on the output
v(2). At the command line ”spice -b circuitname.cir” produces the Fourier analysis output in
Table 4.119. It shows THD (total harmonic distortion) of over 10%, and the contribution of the
individual harmonics.
A partial solution to this distortion is to decrease the collector current or operate the amplifier over a smaller portion of the load line. The ultimate solution is to apply negative feedback.
See (page 256).

4.16.2

Temperature drift

Temperature affects the AC and DC characteristics of transistors. The two aspects to this
problem are environmental temperature variation and self-heating. Some applications, like
military and automotive, require operation over an extended temperature range. Circuits in a
benign environment are subject to self-heating, in particular high power circuits.

4.16. BJT QUIRKS

common-emitter amplifier
Vbias 4 0 0.74
Vsig 5 4 sin (0 125m 2000 0 0)
rbias 6 5 2k
q1 2 6 0 q2n2222
r 3 2 1000
v1 3 0 dc 10
.model q2n2222 npn (is=19f bf=150
+ vaf=100 ikf=0.18 ise=50p ne=2.5
br=7.5
+ var=6.4 ikr=12m isc=8.7p nc=1.2
rb=50
+ re=0.4 rc=0.3 cje=26p tf=0.5n
+ cjc=11p tr=7n xtb=1.5 kf=0.032f
af=1)
.fourier 2000 v(2)
.tran 0.02m 0.74m
.end

273

spice -b ce.cir
Fourier analysis
v(2):
THD: 10.4688
Har Freq
Norm
Mag
--- -----------0
0
0
1
2000
1
2
4000
0.0979929
3
6000
0.0365461
4
8000
0.00438709
5
10000
0.00115878
6
12000
0.00089388
7
14000
0.00021169
8
16000
3.8158e-05
9
18000
3.3726e-05

Figure 4.119: SPICE net list: for transient and fourier analyses. Fourier analysis shows 10%
total harmonic distortion (THD).

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

274

Leakage current ICO and β increase with temperature. The DC β hF E increases exponentially. The AC β hf e increases, but not as rapidly. It doubles over the range of -55o to 85o C. As
temperature increases, the increase in hf e will yield a larger common-emitter output, which
could be clipped in extreme cases. The increase in hF E shifts the bias point, possibly clipping
one peak. The shift in bias point is amplified in multi-stage direct-coupled amplifiers. The
solution is some form of negative feedback to stabilize the bias point. This also stabilizes AC
gain.
Increasing temperature in Figure 4.120 (a) will decrease VBE from the nominal 0.7V for
silicon transistors. Decreasing VBE increases collector current in a common-emitter amplifier,
further shifting the bias point. The cure for shifting VBE is a pair of transistors configured as
a differential amplifier. If both transistors in Figure 4.120 (b) are at the same temperature, the
VBE will track with changing temperature and cancel.
+Vcc

+Vcc

+
VBE -

(a)

+
VBE -

-Vee

(b)

+
- VBE

-Vee

Figure 4.120: (a) single ended CE amplifier vs (b) differential amplifier with VBE cancellation.
The maximum recommended junction temperature for silicon devices is frequently 125o C.
Though, this should be derated for higher reliability. Transistor action ceases beyond 150o C.
Silicon carbide and diamond transistors will operate considerably higher.

4.16.3

Thermal runaway

The problem with increasing temperature causing increasing collector current is that more current increase the power dissipated by the transistor which, in turn, increases its temperature.
This self-reinforcing cycle is known as thermal run away, which may destroy the transistor.
Again, the solution is a bias scheme with some form of negative feedback to stabilize the bias
point.

4.16.4

Junction capacitance

Capacitance exists between the terminals of a transistor. The collector-base capacitance CCB
and emitter-base capacitance CEB decrease the gain of a common emitter circuit at higher
frequencies.

4.16. BJT QUIRKS

275

In a common emitter amplifier, the capacitive feedback from collector to base effectively
multiplies CCB by β. The amount of negative gain-reducing feedback is related to both current
gain, and amount of collector-base capacitance. This is known as the Miller effect, (page 277).

4.16.5

Noise

The ultimate sensitivity of small signal amplifiers is limited by noise due to random variations
in current flow. The two major sources of noise in transistors are shot noise due to current flow
of carriers in the base and thermal noise. The source of thermal noise is device resistance and
increases with temperature:

Vn = 4kTRBn
where
k = boltzman’s conatant (1.38•10−23 watt-sec/K)
T = resistor tempeature in kelvins
R = resistance in Ohms
Bn = noise bandwidth in Hz
Noise in a transistor amplifier is defined in terms of excess noise generated by the amplifier,
not that noise amplified from input to output, but that generated within the amplifier. This is
determined by measuring the signal to noise ratio (S/N) at the amplifier input and output. The
AC voltage output of an amplifier with a small signal input corresponds to S+N, signal plus
noise. The AC voltage with no signal in corresponds to noise N. The noise figure F is defined in
terms of S/N of amplifier input and output:

F=

(S/N)i
(S/N)o

FdB = 10 log F
The noise figure F for RF (radio frequency) transistors is usually listed on transistor data
sheets in decibels, FdB . A good VHF (very high frequency, 30 MHz to 300 Mhz) noise figure
is < 1 dB. The noise figure above VHF increases considerable, 20 dB per decade as shown in
Figure 4.121.
Figure 4.121 also shows that noise at low frequencies increases at 10 dB per decade with
decreasing frequency. This noise is known as 1/f noise.
Noise figure varies with the transistor type (part number). Small signal RF transistors
used at the antenna input of a radio receiver are specifically designed for low noise figure.
Noise figure varies with bias current and impedance matching. The best noise figure for a
transistor is achieved at lower bias current, and possibly with an impedance mismatch.

4.16.6

Thermal mismatch (problem with paralleling transistors)

If two identical power transistors were paralleled for higher current, one would expect them
to share current equally. Because of differences in characteristerics, transistors do not share
current equally.

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

0

1/

fn

dB

oi

/d

ec

shot noise and
thermal noise

se

/de
ca
de

-1

ad

dB

e

white noise region

fLn

20

Noise figure F (decibels)

276

fHn
Log Frequency

Figure 4.121: Small signal transistor noise figure vs Frequency. After Thiele, Figure 11.147 [1]

+V

Incorrect

+V

Correct

Figure 4.122: Transistors paralleled for increased power require emitter ballast resistors

4.16. BJT QUIRKS

277

It is not practical to select identical transistors. The β for small signal transistors typically
has a range of 100-300, power transistors: 20-50. If each one could be matched, one still might
run hotter than the other due to environmental conditions. The hotter transistor draws more
current resulting in thermal runaway. The solution when paralleling bipolar transistors is to
insert emitter resistors known as ballast resistors of less than an ohm. If the hotter transistor
draws more current, the voltage drop across the ballast resistor increases— negative feedback.
This decreases the current. Mounting all transistors on the same heatsink helps equalize
current too.

4.16.7

High frequency effects

The performance of a transistor amplifier is relatively constant, up to a point, as shown by the
small signal common-emitter current gain with increasing frequency in Figure 4.123. Beyond
that point the performance of a transistor degrades as frequency increases.
Beta cutoff frequency, fT is the frequency at which common-emitter small signal current
gain (hf e ) falls to unity. (Figure 4.123) A practical amplifier must have a gain >1. Thus, a
transistor cannot be used in a practical amplifier at fT . A more usable limit for a transistor is
0.1·fT .

100

hfe

10

1
fT
log f
Figure 4.123: Common-emitter small signal current gain (hf e ) vs frequency.
Some RF silicon bipolar transistors are usable as amplifers up to a few GHz. Silicongermanium devices extend the upper range to 10 GHz.
Alpha cutoff frequency, falpha is the frequency at which the α falls to 0.707 of low frequency α,0 α=0.707α0 . Alpha cutoff and beta cutoff are nearly equal: falpha ∼
=fT Beta cutoff fT
is the preferred figure of merit of high frequency performance.
fmax is the highest frequency of oscillation possible under the most favorable conditions of
bias and impedance matching. It is the frequency at which the power gain is unity. All of the
output is fed back to the input to sustain oscillations. fmax is an upper limit for frequency of
operation of a transistor as an active device. Though, a practical amplifier would not be usable
at fmax .
Miller effect: The high frequency limit for a transistor is related to the junction capacitances. For example a PN2222A has an input capacitance Cobo =9pF and an output capacitance

278

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

Cibo =25pF from C-B and E-B respectively. [5] Although the C-E capacitance of 25 pF seems
large, it is less of a factor than the C-B (9pF) capacitance. because of the Miller effect, the C-B
capacitance has an effect on the base equivalent to beta times the capacitance in the commonemitter amplifier. Why might this be? A common-emitter amplifier inverts the signal from
base to collector. The inverted collector signal fed back to the base opposes the input on the
base. The collector signal is beta times larger than the input. For the PN2222A, β=50–300.
Thus, the 9pF C-E capacitance looks like 9·50=450pF to 9·300=2700pF.
The solution to the junction capacitance problem is to select a high frequency transistor for
wide bandwidth applications— RF (radio frequency) or microwave transistor. The bandwidth
can be extended further by using the common-base instead of the common-emitter configuration. The grounded base shields the emitter input from capacitive collector feedback. A
two-transistor cascode arrangement will yield the same bandwidth as the common-base, with
the higher input impedance of the common-emitter.
• REVIEW:
• Transistor amplifiers exhibit distortion because of β variation with collector current.
• Ic , VBE , β and junction capacitance vary with temperature.
• An increase in temperature can cause an increase in IC , causing an increase in temperature, a vicious cycle known as thermal runaway.
• Junction capacitance limits high frequency gain of a transistor. The Miller effect makes
Ccb look β times larger at the base of a CE amplifier.
• Transistor noise limits the ability to amplify small signals. Noise figure is a figure of
merit concerning transistor noise.
• When paralleling power transistors for increased current, insert ballast resistors in series
with the emitters to equalize current.
• FT is the absolute upper frequency limit for a CE amplifier, small signal current gain
falls to unity, hf e =1.
• Fmax is the upper frequency limit for an oscillator under the most ideal conditions.

Bibliography
[1] A. G. Thiele in Loyd P. Hunter, “Handbook of Semiconductor Electronics,” Low Frequency
Amplifiers, ISBN -07-031305-9, 1970
[2] “GE Transistor Manual”, General Electric, 1964.
[3] R. Victor Jones, “Basic BJT Amplifier Configurations”, November 7, 2001. at
http://people.seas.harvard.edu/˜jones/es154/lectures/lecture 3/
bjt amps/bjt amps.html

BIBLIOGRAPHY

279

[4] Tony Kuphaldt,“Lessons in Electric Circuits”, Vol. 1, DC, DC Network Analysis,
Thevenin’s Theorem, at http://www.openbookproject.net/electricCircuits/
DC/DC 10.html#xtocid102679
[5] “PN2222
Datasheet”,Fairchild
Semiconductor
Corporation,
http://www.fairchildsemi.com/ds/PN/PN2222A.pdf

2007

at

280

CHAPTER 4. BIPOLAR JUNCTION TRANSISTORS

Chapter 5

JUNCTION FIELD-EFFECT
TRANSISTORS
Contents
5.1
5.2
5.3
5.4
5.5
5.6
5.7
5.8
5.9
5.10

Introduction . . . . . . . . . . . . . . . . . . . . .
The transistor as a switch . . . . . . . . . . . .
Meter check of a transistor . . . . . . . . . . . .
Active-mode operation . . . . . . . . . . . . . .
The common-source amplifier – PENDING . .
The common-drain amplifier – PENDING . . .
The common-gate amplifier – PENDING . . .
Biasing techniques – PENDING . . . . . . . . .
Transistor ratings and packages – PENDING
JFET quirks – PENDING . . . . . . . . . . . . .

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281
283
286
288
297
298
298
298
299
299

*** INCOMPLETE ***

5.1

Introduction

A transistor is a linear semiconductor device that controls current with the application of a
lower-power electrical signal. Transistors may be roughly grouped into two major divisions:
bipolar and field-effect. In the last chapter we studied bipolar transistors, which utilize a small
current to control a large current. In this chapter, we’ll introduce the general concept of the
field-effect transistor – a device utilizing a small voltage to control current – and then focus
on one particular type: the junction field-effect transistor. In the next chapter we’ll explore
another type of field-effect transistor, the insulated gate variety.
All field-effect transistors are unipolar rather than bipolar devices. That is, the main current through them is comprised either of electrons through an N-type semiconductor or holes
281

CHAPTER 5. JUNCTION FIELD-EFFECT TRANSISTORS

282

through a P-type semiconductor. This becomes more evident when a physical diagram of the
device is seen:

N-channel JFET
drain
drain
gate

gate

P N

source
source
schematic symbol

physical diagram

In a junction field-effect transistor, or JFET, the controlled current passes from source to
drain, or from drain to source as the case may be. The controlling voltage is applied between
the gate and source. Note how the current does not have to cross through a PN junction on
its way between source and drain: the path (called a channel) is an uninterrupted block of
semiconductor material. In the image just shown, this channel is an N-type semiconductor.
P-type channel JFETs are also manufactured:

P-channel JFET
drain
drain
gate

gate

N P

source
source
schematic symbol

physical diagram

Generally, N-channel JFETs are more commonly used than P-channel. The reasons for this
have to do with obscure details of semiconductor theory, which I’d rather not discuss in this
chapter. As with bipolar transistors, I believe the best way to introduce field-effect transistor
usage is to avoid theory whenever possible and concentrate instead on operational characteristics. The only practical difference between N- and P-channel JFETs you need to concern
yourself with now is biasing of the PN junction formed between the gate material and the
channel.

5.2. THE TRANSISTOR AS A SWITCH

283

With no voltage applied between gate and source, the channel is a wide-open path for electrons to flow. However, if a voltage is applied between gate and source of such polarity that it
reverse-biases the PN junction, the flow between source and drain connections becomes limited, or regulated, just as it was for bipolar transistors with a set amount of base current.
Maximum gate-source voltage ”pinches off ” all current through source and drain, thus forcing
the JFET into cutoff mode. This behavior is due to the depletion region of the PN junction
expanding under the influence of a reverse-bias voltage, eventually occupying the entire width
of the channel if the voltage is great enough. This action may be likened to reducing the flow of
a liquid through a flexible hose by squeezing it: with enough force, the hose will be constricted
enough to completely block the flow.

water

hose

nozzle

water

Hose constricted by squeezing,
water flow reduced or stopped
Note how this operational behavior is exactly opposite of the bipolar junction transistor.
Bipolar transistors are normally-off devices: no current through the base, no current through
the collector or the emitter. JFETs, on the other hand, are normally-on devices: no voltage
applied to the gate allows maximum current through the source and drain. Also take note
that the amount of current allowed through a JFET is determined by a voltage signal rather
than a current signal as with bipolar transistors. In fact, with the gate-source PN junction
reverse-biased, there should be nearly zero current through the gate connection. For this
reason, we classify the JFET as a voltage-controlled device, and the bipolar transistor as a
current-controlled device.
If the gate-source PN junction is forward-biased with a small voltage, the JFET channel
will ”open” a little more to allow greater currents through. However, the PN junction of a
JFET is not built to handle any substantial current itself, and thus it is not recommended to
forward-bias the junction under any circumstances.
This is a very condensed overview of JFET operation. In the next section, we’ll explore the
use of the JFET as a switching device.

5.2

The transistor as a switch

Like its bipolar cousin, the field-effect transistor may be used as an on/off switch controlling
electrical power to a load. Let’s begin our investigation of the JFET as a switch with our
familiar switch/lamp circuit:

284

CHAPTER 5. JUNCTION FIELD-EFFECT TRANSISTORS

switch

Remembering that the controlled current in a JFET flows between source and drain, we
substitute the source and drain connections of a JFET for the two ends of the switch in the
above circuit:

If you haven’t noticed by now, the source and drain connections on a JFET look identical
on the schematic symbol. Unlike the bipolar junction transistor where the emitter is clearly
distinguished from the collector by the arrowhead, a JFET’s source and drain lines both run
perpendicular into the bar representing the semiconductor channel. This is no accident, as the
source and drain lines of a JFET are often interchangeable in practice! In other words, JFETs
are usually able to handle channel current in either direction, from source to drain or from
drain to source.
Now all we need in the circuit is a way to control the JFET’s conduction. With zero applied
voltage between gate and source, the JFET’s channel will be ”open,” allowing full current to
the lamp. In order to turn the lamp off, we will need to connect another source of DC voltage
between the gate and source connections of the JFET like this:

switch

Closing this switch will ”pinch off ” the JFET’s channel, thus forcing it into cutoff and turning the lamp off:

switch

5.2. THE TRANSISTOR AS A SWITCH

285

Note that there is no current going through the gate. As a reverse-biased PN junction, it
firmly opposes the flow of any electrons through it. As a voltage-controlled device, the JFET
requires negligible input current. This is an advantageous trait of the JFET over the bipolar
transistor: there is virtually zero power required of the controlling signal.
Opening the control switch again should disconnect the reverse-biasing DC voltage from
the gate, thus allowing the transistor to turn back on. Ideally, anyway, this is how it works. In
practice this may not work at all:

switch

No lamp current after the switch opens!
Why is this? Why doesn’t the JFET’s channel open up again and allow lamp current through
like it did before with no voltage applied between gate and source? The answer lies in the operation of the reverse-biased gate-source junction. The depletion region within that junction acts
as an insulating barrier separating gate from source. As such, it possesses a certain amount of
capacitance capable of storing an electric charge potential. After this junction has been forcibly
reverse-biased by the application of an external voltage, it will tend to hold that reverse-biasing
voltage as a stored charge even after the source of that voltage has been disconnected. What
is needed to turn the JFET on again is to bleed off that stored charge between the gate and
source through a resistor:

switch

Resistor bleeds off stored charge in
PN junction to allow transistor to
turn on once again.
This resistor’s value is not very important. The capacitance of the JFET’s gate-source junction is very small, and so even a rather high-value bleed resistor creates a fast RC time constant, allowing the transistor to resume conduction with little delay once the switch is opened.
Like the bipolar transistor, it matters little where or what the controlling voltage comes
from. We could use a solar cell, thermocouple, or any other sort of voltage-generating device to
supply the voltage controlling the JFET’s conduction. All that is required of a voltage source
for JFET switch operation is sufficient voltage to achieve pinch-off of the JFET channel. This
level is usually in the realm of a few volts DC, and is termed the pinch-off or cutoff voltage.
The exact pinch-off voltage for any given JFET is a function of its unique design, and is not a

CHAPTER 5. JUNCTION FIELD-EFFECT TRANSISTORS

286

universal figure like 0.7 volts is for a silicon BJT’s base-emitter junction voltage.
• REVIEW:
• Field-effect transistors control the current between source and drain connections by a
voltage applied between the gate and source. In a junction field-effect transistor (JFET),
there is a PN junction between the gate and source which is normally reverse-biased for
control of source-drain current.
• JFETs are normally-on (normally-saturated) devices. The application of a reverse-biasing
voltage between gate and source causes the depletion region of that junction to expand,
thereby ”pinching off ” the channel between source and drain through which the controlled
current travels.
• It may be necessary to attach a ”bleed-off ” resistor between gate and source to discharge
the stored charge built up across the junction’s natural capacitance when the controlling
voltage is removed. Otherwise, a charge may remain to keep the JFET in cutoff mode
even after the voltage source has been disconnected.

5.3

Meter check of a transistor

Testing a JFET with a multimeter might seem to be a relatively easy task, seeing as how it
has only one PN junction to test: either measured between gate and source, or between gate
and drain.

V

A

V

A

N-channel transistor

OFF

A

drain
+

COM

drain
gate

gate

-

P N

source
+
source
V

A
physical diagram

V

A
OFF

A

COM

Both meters show non-continuity
(high resistance) through gatechannel junction.

5.3. METER CHECK OF A TRANSISTOR

V

287

A

V

A

N-channel transistor

OFF

A

drain
-

COM

drain
gate

gate

+
+

P N

source
source
V

A

V

A

physical diagram
OFF

A

COM

Both meters show continuity (low
resistance) through gate-channel
junction.

Testing continuity through the drain-source channel is another matter, though. Remember from the last section how a stored charge across the capacitance of the gate-channel PN
junction could hold the JFET in a pinched-off state without any external voltage being applied
across it? This can occur even when you’re holding the JFET in your hand to test it! Consequently, any meter reading of continuity through that channel will be unpredictable, since you
don’t necessarily know if a charge is being stored by the gate-channel junction. Of course, if you
know beforehand which terminals on the device are the gate, source, and drain, you may connect a jumper wire between gate and source to eliminate any stored charge and then proceed
to test source-drain continuity with no problem. However, if you don’t know which terminals
are which, the unpredictability of the source-drain connection may confuse your determination
of terminal identity.
A good strategy to follow when testing a JFET is to insert the pins of the transistor into
anti-static foam (the material used to ship and store static-sensitive electronic components)
just prior to testing. The conductivity of the foam will make a resistive connection between all
terminals of the transistor when it is inserted. This connection will ensure that all residual
voltage built up across the gate-channel PN junction will be neutralized, thus ”opening up” the
channel for an accurate meter test of source-to-drain continuity.
Since the JFET channel is a single, uninterrupted piece of semiconductor material, there is
usually no difference between the source and drain terminals. A resistance check from source
to drain should yield the same value as a check from drain to source. This resistance should be
relatively low (a few hundred ohms at most) when the gate-source PN junction voltage is zero.
By applying a reverse-bias voltage between gate and source, pinch-off of the channel should be
apparent by an increased resistance reading on the meter.

CHAPTER 5. JUNCTION FIELD-EFFECT TRANSISTORS

288

5.4

Active-mode operation

JFETs, like bipolar transistors, are able to ”throttle” current in a mode between cutoff and
saturation called the active mode. To better understand JFET operation, let’s set up a SPICE
simulation similar to the one used to explore basic bipolar transistor function:

Vammeter
2

3
0V

1

V1

Q1

Vin

0

0

0

jfet simulation
vin 0 1 dc 1
j1 2 1 0 mod1
vammeter 3 2 dc 0
v1 3 0 dc
.model mod1 njf
.dc v1 0 2 0.05
.plot dc i(vammeter)
.end
Note that the transistor labeled ”Q1 ” in the schematic is represented in the SPICE netlist as
j1. Although all transistor types are commonly referred to as ”Q” devices in circuit schematics
– just as resistors are referred to by ”R” designations, and capacitors by ”C” – SPICE needs to
be told what type of transistor this is by means of a different letter designation: q for bipolar
junction transistors, and j for junction field-effect transistors.

5.4. ACTIVE-MODE OPERATION

289

Here, the controlling signal is a steady voltage of 1 volt, applied with negative towards the
JFET gate and positive toward the JFET source, to reverse-bias the PN junction. In the first
BJT simulation of chapter 4, a constant-current source of 20 µA was used for the controlling
signal, but remember that a JFET is a voltage-controlled device, not a current-controlled device
like the bipolar junction transistor.
Like the BJT, the JFET tends to regulate the controlled current at a fixed level above a
certain power supply voltage, no matter how high that voltage may climb. Of course, this
current regulation has limits in real life – no transistor can withstand infinite voltage from
a power source – and with enough drain-to-source voltage the transistor will ”break down”
and drain current will surge. But within normal operating limits the JFET keeps the drain
current at a steady level independent of power supply voltage. To verify this, we’ll run another
computer simulation, this time sweeping the power supply voltage (V1 ) all the way to 50 volts:

jfet simulation
vin 0 1 dc 1
j1 2 1 0 mod1
vammeter 3 2 dc 0
v1 3 0 dc
.model mod1 njf
.dc v1 0 50 2
.plot dc i(vammeter)
.end

290

CHAPTER 5. JUNCTION FIELD-EFFECT TRANSISTORS

Sure enough, the drain current remains steady at a value of 100 µA (1.000E-04 amps) no
matter how high the power supply voltage is adjusted.
Because the input voltage has control over the constriction of the JFET’s channel, it makes
sense that changing this voltage should be the only action capable of altering the current
regulation point for the JFET, just like changing the base current on a BJT is the only action
capable of altering collector current regulation. Let’s decrease the input voltage from 1 volt to
0.5 volts and see what happens:

jfet simulation
vin 0 1 dc 0.5
j1 2 1 0 mod1
vammeter 3 2 dc 0
v1 3 0 dc
.model mod1 njf
.dc v1 0 50 2
.plot dc i(vammeter)
.end

5.4. ACTIVE-MODE OPERATION

291

As expected, the drain current is greater now than it was in the previous simulation. With
less reverse-bias voltage impressed across the gate-source junction, the depletion region is not
as wide as it was before, thus ”opening” the channel for charge carriers and increasing the
drain current figure.
Please note, however, the actual value of this new current figure: 225 µA (2.250E-04 amps).
The last simulation showed a drain current of 100 µA, and that was with a gate-source voltage
of 1 volt. Now that we’ve reduced the controlling voltage by a factor of 2 (from 1 volt down to
0.5 volts), the drain current increased, but not by the same 2:1 proportion! Let’s reduce our
gate-source voltage once more by another factor of 2 (down to 0.25 volts) and see what happens:

jfet simulation
vin 0 1 dc 0.25
j1 2 1 0 mod1
vammeter 3 2 dc 0
v1 3 0 dc
.model mod1 njf
.dc v1 0 50 2
.plot dc i(vammeter)
.end

292

CHAPTER 5. JUNCTION FIELD-EFFECT TRANSISTORS

With the gate-source voltage set to 0.25 volts, one-half what it was before, the drain current
is 306.3 µA. Although this is still an increase over the 225 µA from the prior simulation, it isn’t
proportional to the change of the controlling voltage.
To obtain a better understanding of what is going on here, we should run a different kind
of simulation: one that keeps the power supply voltage constant and instead varies the controlling (voltage) signal. When this kind of simulation was run on a BJT, the result was a
straight-line graph, showing how the input current / output current relationship of a BJT is
linear. Let’s see what kind of relationship a JFET exhibits:

jfet simulation
vin 0 1 dc
j1 2 1 0 mod1
vammeter 3 2 dc 0
v1 3 0 dc 25
.model mod1 njf
.dc vin 0 2 0.1
.plot dc i(vammeter)
.end

5.4. ACTIVE-MODE OPERATION

293

This simulation directly reveals an important characteristic of the junction field-effect transistor: the control effect of gate voltage over drain current is nonlinear. Notice how the drain
current does not decrease linearly as the gate-source voltage is increased. With the bipolar
junction transistor, collector current was directly proportional to base current: output signal
proportionately followed input signal. Not so with the JFET! The controlling signal (gatesource voltage) has less and less effect over the drain current as it approaches cutoff. In this
simulation, most of the controlling action (75 percent of drain current decrease – from 400
µA to 100 µA) takes place within the first volt of gate-source voltage (from 0 to 1 volt), while
the remaining 25 percent of drain current reduction takes another whole volt worth of input
signal. Cutoff occurs at 2 volts input.
Linearity is generally important for a transistor because it allows it to faithfully amplify a
waveform without distorting it. If a transistor is nonlinear in its input/output amplification,
the shape of the input waveform will become corrupted in some way, leading to the production
of harmonics in the output signal. The only time linearity is not important in a transistor
circuit is when its being operated at the extreme limits of cutoff and saturation (off and on,
respectively, like a switch).
A JFET’s characteristic curves display the same current-regulating behavior as for a BJT,
and the nonlinearity between gate-to-source voltage and drain current is evident in the disproportionate vertical spacings between the curves:

CHAPTER 5. JUNCTION FIELD-EFFECT TRANSISTORS

294

|VDS| = |VP| - |VGS|
Below pinch-off
Triode region

Above pinch-off
Saturation region

regio

n

Vgate-to-source = 0 V

Vgate-to-source = 0.5 V

Ohmic

Idrain

Vgate-to-source = 1 V
Vgate-to-source = 2 V = VP

(pinch-off)

Edrain-to-source
To better comprehend the current-regulating behavior of the JFET, it might be helpful to
draw a model made up of simpler, more common components, just as we did for the BJT:

D
G
S
N-channel JFET diode-regulating diode model
D

G

S
In the case of the JFET, it is the voltage across the reverse-biased gate-source diode which
sets the current regulation point for the pair of constant-current diodes. A pair of opposing
constant-current diodes is included in the model to facilitate current in either direction be-

5.4. ACTIVE-MODE OPERATION

295

tween source and drain, a trait made possible by the unipolar nature of the channel. With
no PN junctions for the source-drain current to traverse, there is no polarity sensitivity in the
controlled current. For this reason, JFETs are often referred to as bilateral devices.
A contrast of the JFET’s characteristic curves against the curves for a bipolar transistor
reveals a notable difference: the linear (straight) portion of each curve’s non-horizontal area is
surprisingly long compared to the respective portions of a BJT’s characteristic curves:

Vgate-to-source = 0 V

Idrain

Vgate-to-source = 0.5 V

Vgate-to-source = 1 V
Vgate-to-source = 2 V (pinch-off)
Edrain-to-source
"Ohmic regions"

Ibase = 75 µA

Icollector

Ibase = 40 µA
Ibase = 20 µA
Ibase = 5 µA
Ecollector-to-emitter

A JFET transistor operated in the triode region tends to act very much like a plain resistor
as measured from drain to source. Like all simple resistances, its current/voltage graph is a
straight line. For this reason, the triode region (non-horizontal) portion of a JFET’s characteristic curve is sometimes referred to as the ohmic region. In this mode of operation where there

CHAPTER 5. JUNCTION FIELD-EFFECT TRANSISTORS

296

isn’t enough drain-to-source voltage to bring drain current up to the regulated point, the drain
current is directly proportional to the drain-to-source voltage. In a carefully designed circuit,
this phenomenon can be used to an advantage. Operated in this region of the curve, the JFET
acts like a voltage-controlled resistance rather than a voltage-controlled current regulator, and
the appropriate model for the transistor is different:

D
G
S
N-channel JFET diode-rheostat model
(for saturation, or "ohmic," mode only!)
D

G

S
Here and here alone the rheostat (variable resistor) model of a transistor is accurate. It
must be remembered, however, that this model of the transistor holds true only for a narrow
range of its operation: when it is extremely saturated (far less voltage applied between drain
and source than what is needed to achieve full regulated current through the drain). The
amount of resistance (measured in ohms) between drain and source in this mode is controlled
by how much reverse-bias voltage is applied between gate and source. The less gate-to-source
voltage, the less resistance (steeper line on graph).
Because JFETs are voltage-controlled current regulators (at least when they’re allowed to
operate in their active), their inherent amplification factor cannot be expressed as a unitless
ratio as with BJTs. In other words, there is no β ratio for a JFET. This is true for all voltagecontrolled active devices, including other types of field-effect transistors and even electron
tubes. There is, however, an expression of controlled (drain) current to controlling (gate-source)
voltage, and it is called transconductance. Its unit is Siemens, the same unit for conductance
(formerly known as the mho).
Why this choice of units? Because the equation takes on the general form of current (output
signal) divided by voltage (input signal).

5.5. THE COMMON-SOURCE AMPLIFIER – PENDING

gfs =

297

∆ID
∆VGS

Where,
gfs = Transconductance in Siemens
∆ID = Change in drain current
∆VGS = Change in gate-source voltage
Unfortunately, the transconductance value for any JFET is not a stable quantity: it varies
significantly with the amount of gate-to-source control voltage applied to the transistor. As we
saw in the SPICE simulations, the drain current does not change proportionally with changes
in gate-source voltage. To calculate drain current for any given gate-source voltage, there is
another equation that may be used. It is obviously nonlinear upon inspection (note the power
of 2), reflecting the nonlinear behavior we’ve already experienced in simulation:

ID = IDSS ( 1 -

VGS
VGS(cutoff)

2

)

Where,
ID = Drain current
IDSS = Drain current with gate shorted to source
VGS = Gate-to-source voltage
VGS(cutoff) = Pinch-off gate-to-source voltage
• REVIEW:
• In their active modes, JFETs regulate drain current according to the amount of reversebias voltage applied between gate and source, much like a BJT regulates collector current
according to base current. The mathematical ratio between drain current (output) and
gate-to-source voltage (input) is called transconductance, and it is measured in units of
Siemens.
• The relationship between gate-source (control) voltage and drain (controlled) current is
nonlinear: as gate-source voltage is decreased, drain current increases exponentially.
That is to say, the transconductance of a JFET is not constant over its range of operation.
• In their triode region, JFETs regulate drain-to-source resistance according to the amount
of reverse-bias voltage applied between gate and source. In other words, they act like
voltage-controlled resistances.

5.5

The common-source amplifier – PENDING

*** PENDING ***

CHAPTER 5. JUNCTION FIELD-EFFECT TRANSISTORS

298
• REVIEW:
•
•
•

5.6

The common-drain amplifier – PENDING

*** PENDING ***
• REVIEW:
•
•
•

5.7

The common-gate amplifier – PENDING

*** PENDING ***
• REVIEW:
•
•
•

5.8

Biasing techniques – PENDING

*** PENDING ***
• REVIEW:
•
•
•

5.9. TRANSISTOR RATINGS AND PACKAGES – PENDING

5.9

Transistor ratings and packages – PENDING

*** PENDING ***
• REVIEW:
•
•
•

5.10

JFET quirks – PENDING

*** PENDING ***
• REVIEW:
•
•
•

299

300

CHAPTER 5. JUNCTION FIELD-EFFECT TRANSISTORS

Chapter 6

INSULATED-GATE
FIELD-EFFECT TRANSISTORS
Contents
6.1
6.2
6.3
6.4
6.5
6.6
6.7
6.8
6.9
6.10
6.11
6.12

Introduction . . . . . . . . . . . . . . . . . . . . .
Depletion-type IGFETs . . . . . . . . . . . . . .
Enhancement-type IGFETs – PENDING . . . .
Active-mode operation – PENDING . . . . . .
The common-source amplifier – PENDING . .
The common-drain amplifier – PENDING . . .
The common-gate amplifier – PENDING . . .
Biasing techniques – PENDING . . . . . . . . .
Transistor ratings and packages – PENDING
IGFET quirks – PENDING . . . . . . . . . . . .
MESFETs – PENDING . . . . . . . . . . . . . . .
IGBTs . . . . . . . . . . . . . . . . . . . . . . . . .

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301
302
311
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313
313
313

*** INCOMPLETE ***

6.1

Introduction

As was stated in the last chapter, there is more than one type of field-effect transistor. The
junction field-effect transistor, or JFET, uses voltage applied across a reverse-biased PN junction to control the width of that junction’s depletion region, which then controls the conductivity of a semiconductor channel through which the controlled current moves. Another type
of field-effect device – the insulated gate field-effect transistor, or IGFET – exploits a similar
principle of a depletion region controlling conductivity through a semiconductor channel, but
it differs primarily from the JFET in that there is no direct connection between the gate lead
301

CHAPTER 6. INSULATED-GATE FIELD-EFFECT TRANSISTORS

302

and the semiconductor material itself. Rather, the gate lead is insulated from the transistor
body by a thin barrier, hence the term insulated gate. This insulating barrier acts like the dielectric layer of a capacitor, and allows gate-to-source voltage to influence the depletion region
electrostatically rather than by direct connection.
In addition to a choice of N-channel versus P-channel design, IGFETs come in two major
types: enhancement and depletion. The depletion type is more closely related to the JFET, so
we will begin our study of IGFETs with it.

6.2

Depletion-type IGFETs

Insulated gate field-effect transistors are unipolar devices just like JFETs: that is, the controlled current does not have to cross a PN junction. There is a PN junction inside the transistor, but its only purpose is to provide that nonconducting depletion region which is used to
restrict current through the channel.
Here is a diagram of an N-channel IGFET of the ”depletion” type:
N-channel, D-type IGFET
drain
drain
substrate

gate
source

gate

N P

substrate

insulating
barrier
source

schematic symbol

physical diagram

Notice how the source and drain leads connect to either end of the N channel, and how the
gate lead attaches to a metal plate separated from the channel by a thin insulating barrier.
That barrier is sometimes made from silicon dioxide (the primary chemical compound found in
sand), which is a very good insulator. Due to this Metal (gate) - Oxide (barrier) - Semiconductor
(channel) construction, the IGFET is sometimes referred to as a MOSFET. There are other
types of IGFET construction, though, and so ”IGFET” is the better descriptor for this general
class of transistors.
Notice also how there are four connections to the IGFET. In practice, the substrate lead
is directly connected to the source lead to make the two electrically common. Usually, this
connection is made internally to the IGFET, eliminating the separate substrate connection,
resulting in a three-terminal device with a slightly different schematic symbol:

6.2. DEPLETION-TYPE IGFETS

303

N-channel, D-type IGFET
drain
drain
gate

gate

N P

substrate

insulating
barrier

source

source
schematic symbol

physical diagram

With source and substrate common to each other, the N and P layers of the IGFET end up
being directly connected to each other through the outside wire. This connection prevents any
voltage from being impressed across the PN junction. As a result, a depletion region exists
between the two materials, but it can never be expanded or collapsed. JFET operation is based
on the expansion of the PN junction’s depletion region, but here in the IGFET that cannot
happen, so IGFET operation must be based on a different effect.
Indeed it is, for when a controlling voltage is applied between gate and source, the conductivity of the channel is changed as a result of the depletion region moving closer to or further
away from the gate. In other words, the channel’s effective width changes just as with the
JFET, but this change in channel width is due to depletion region displacement rather than
depletion region expansion.
In an N-channel IGFET, a controlling voltage applied positive (+) to the gate and negative
(-) to the source has the effect of repelling the PN junction’s depletion region, expanding the
N-type channel and increasing conductivity:

Rload
drain

gate
N
controlling
voltage

P

source

Channel expands for greater conductivity
Reversing the controlling voltage’s polarity has the opposite effect, attracting the depletion
region and narrowing the channel, consequently reducing channel conductivity:

304

CHAPTER 6. INSULATED-GATE FIELD-EFFECT TRANSISTORS

Rload
drain

gate

N
P

controlling
voltage

source

Channel narrows for less conductivity

The insulated gate allows for controlling voltages of any polarity without danger of forwardbiasing a junction, as was the concern with JFETs. This type of IGFET, although its called
a ”depletion-type,” actually has the capability of having its channel either depleted (channel
narrowed) or enhanced (channel expanded). Input voltage polarity determines which way the
channel will be influenced.

Understanding which polarity has which effect is not as difficult as it may seem. The key
is to consider the type of semiconductor doping used in the channel (N-channel or P-channel?),
then relate that doping type to the side of the input voltage source connected to the channel by
means of the source lead. If the IGFET is an N-channel and the input voltage is connected so
that the positive (+) side is on the gate while the negative (-) side is on the source, the channel
will be enhanced as extra electrons build up on the channel side of the dielectric barrier. Think,
”negative (-) correlates with N-type, thus enhancing the channel with the right type of charge
carrier (electrons) and making it more conductive.” Conversely, if the input voltage is connected
to an N-channel IGFET the other way, so that negative (-) connects to the gate while positive
(+) connects to the source, free electrons will be ”robbed” from the channel as the gate-channel
capacitor charges, thus depleting the channel of majority charge carriers and making it less
conductive.

For P-channel IGFETs, the input voltage polarity and channel effects follow the same rule.
That is to say, it takes just the opposite polarity as an N-channel IGFET to either deplete or
enhance:

6.2. DEPLETION-TYPE IGFETS

305

Rload
drain

gate
P
controlling
voltage

N

source

Channel expands for greater conductivity

Rload
drain

gate

controlling
voltage

P
N

source

Channel narrows for less conductivity
Illustrating the proper biasing polarities with standard IGFET symbols:

306

CHAPTER 6. INSULATED-GATE FIELD-EFFECT TRANSISTORS

N-channel

P-channel

+

-

Enhanced
(more drain
current)

-

+

Depleted
(less drain
current)

+

+

-

When there is zero voltage applied between gate and source, the IGFET will conduct current between source and drain, but not as much current as it would if it were enhanced by
the proper gate voltage. This places the depletion-type, or simply D-type, IGFET in a category
of its own in the transistor world. Bipolar junction transistors are normally-off devices: with
no base current, they block any current from going through the collector. Junction field-effect
transistors are normally-on devices: with zero applied gate-to-source voltage, they allow maximum drain current (actually, you can coax a JFET into greater drain currents by applying
a very small forward-bias voltage between gate and source, but this should never be done in
practice for risk of damaging its fragile PN junction). D-type IGFETs, however, are normally
half-on devices: with no gate-to-source voltage, their conduction level is somewhere between
cutoff and full saturation. Also, they will tolerate applied gate-source voltages of any polarity,
the PN junction being immune from damage due to the insulating barrier and especially the
direct connection between source and substrate preventing any voltage differential across the
junction.
Ironically, the conduction behavior of a D-type IGFET is strikingly similar to that of an electron tube of the triode/tetrode/pentode variety. These devices were voltage-controlled current
regulators that likewise allowed current through them with zero controlling voltage applied. A
controlling voltage of one polarity (grid negative and cathode positive) would diminish conductivity through the tube while a voltage of the other polarity (grid positive and cathode negative)
would enhance conductivity. I find it curious that one of the later transistor designs invented
exhibits the same basic properties of the very first active (electronic) device.
A few SPICE analyses will demonstrate the current-regulating behavior of D-type IGFETs.
First, a test with zero input voltage (gate shorted to source) and the power supply swept from
0 to 50 volts. The graph shows drain current:

6.2. DEPLETION-TYPE IGFETS

307

Vammeter
1

2
0V
V1

Q1

0
0

0

n-channel igfet characteristic curve
m1 1 0 0 0 mod1
vammeter 2 1 dc 0
v1 2 0
.model mod1 nmos vto=-1
.dc v1 0 50 2
.plot dc i(vammeter)
.end

As expected for any transistor, the controlled current holds steady at a regulated value over
a wide range of power supply voltages. In this case, that regulated point is 10 µA (1.000E-05).
Now let’s see what happens when we apply a negative voltage to the gate (with reference to
the source) and sweep the power supply over the same range of 0 to 50 volts:

308

CHAPTER 6. INSULATED-GATE FIELD-EFFECT TRANSISTORS

Vammeter
1

2
0V
V1

Q1

3

0.5 V

0

0

n-channel igfet characteristic curve
m1 1 3 0 0 mod1
vin 0 3 dc 0.5
vammeter 2 1 dc 0
v1 2 0
.model mod1 nmos vto=-1
.dc v1 0 50 2
.plot dc i(vammeter)
.end

Not surprisingly, the drain current is now regulated at a lower value of 2.5 µA (down from
10 µA with zero input voltage). Now let’s apply an input voltage of the other polarity, to enhance
the IGFET:

6.2. DEPLETION-TYPE IGFETS

309

Vammeter
1

2
0V
V1

Q1

3

0.5 V

0

0

n-channel igfet characteristic curve
m1 1 3 0 0 mod1
vin 3 0 dc 0.5
vammeter 2 1 dc 0
v1 2 0
.model mod1 nmos vto=-1
.dc v1 0 50 2
.plot dc i(vammeter)
.end

With the transistor enhanced by the small controlling voltage, the drain current is now
at an increased value of 22.5 µA (2.250E-05). It should be apparent from these three sets
of voltage and current figures that the relationship of drain current to gate-source voltage is
nonlinear just as it was with the JFET. With 1/2 volt of depleting voltage, the drain current is
2.5 µA; with 0 volts input the drain current goes up to 10 µA; and with 1/2 volt of enhancing
voltage, the current is at 22.5 µA. To obtain a better understanding of this nonlinearity, we

310

CHAPTER 6. INSULATED-GATE FIELD-EFFECT TRANSISTORS

can use SPICE to plot the drain current over a range of input voltage values, sweeping from
a negative (depleting) figure to a positive (enhancing) figure, maintaining the power supply
voltage of V1 at a constant value:

n-channel igfet
m1 1 3 0 0 mod1
vin 3 0
vammeter 2 1 dc 0
v1 2 0 dc 24
.model mod1 nmos vto=-1
.dc vin -1 1 0.1
.plot dc i(vammeter)
.end

Just as it was with JFETs, this inherent nonlinearity of the IGFET has the potential to
cause distortion in an amplifier circuit, as the input signal will not be reproduced with 100
percent accuracy at the output. Also notice that a gate-source voltage of about 1 volt in the
depleting direction is able to pinch off the channel so that there is virtually no drain current.
D-type IGFETs, like JFETs, have a certain pinch-off voltage rating. This rating varies with
the precise unique of the transistor, and may not be the same as in our simulation here.
Plotting a set of characteristic curves for the IGFET, we see a pattern not unlike that of the
JFET:

6.3. ENHANCEMENT-TYPE IGFETS – PENDING

Vgate-to-source = +0.5 V

Idrain
Vgate-to-source = 0 V

Vgate-to-source = -0.5 V
Edrain-to-source
• REVIEW:
•
•
•

6.3

Enhancement-type IGFETs – PENDING

• REVIEW:
•
•
•

6.4

Active-mode operation – PENDING

• REVIEW:
•
•
•

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CHAPTER 6. INSULATED-GATE FIELD-EFFECT TRANSISTORS

312

6.5

The common-source amplifier – PENDING

• REVIEW:
•
•
•

6.6

The common-drain amplifier – PENDING

• REVIEW:
•
•
•

6.7

The common-gate amplifier – PENDING

• REVIEW:
•
•
•

6.8

Biasing techniques – PENDING

• REVIEW:
•
•
•

6.9

Transistor ratings and packages – PENDING

• REVIEW:
•
•
•

6.10. IGFET QUIRKS – PENDING

6.10

313

IGFET quirks – PENDING

• REVIEW:
•
•
•

6.11

MESFETs – PENDING

• REVIEW:
•
•
•

6.12

IGBTs

Because of their insulated gates, IGFETs of all types have extremely high current gain: there
can be no sustained gate current if there is no continuous gate circuit in which electrons may
continually flow. The only current we see through the gate terminal of an IGFET, then, is
whatever transient (brief surge) may be required to charge the gate-channel capacitance and
displace the depletion region as the transistor switches from an ”on” state to an ”off ” state, or
vice versa.
This high current gain would at first seem to place IGFET technology at a decided advantage over bipolar transistors for the control of very large currents. If a bipolar junction
transistor is used to control a large collector current, there must be a substantial base current
sourced or sunk by some control circuitry, in accordance with the β ratio. To give an example,
in order for a power BJT with a β of 20 to conduct a collector current of 100 amps, there must be
at least 5 amps of base current, a substantial amount of current in itself for miniature discrete
or integrated control circuitry to handle:

Rload
100 A
Control
circuitry

5A

β = 20
105 A

314

CHAPTER 6. INSULATED-GATE FIELD-EFFECT TRANSISTORS

It would be nice from the standpoint of control circuitry to have power transistors with high
current gain, so that far less current is needed for control of load current. Of course, we can
use Darlington pair transistors to increase the current gain, but this kind of arrangement still
requires far more controlling current than an equivalent power IGFET:

Rload
0.238 A
100 A

Control
circuitry

β = 20
5A
105 A

Rload
100 A
Control
circuitry

≈0A
100 A

Unfortunately, though, IGFETs have problems of their own controlling high current: they
typically exhibit greater drain-to-source voltage drop while saturated than the collector-toemitter voltage drop of a saturated BJT. This greater voltage drop equates to higher power
dissipation for the same amount of load current, limiting the usefulness of IGFETs as highpower devices. Although some specialized designs such as the so-called VMOS transistor have
been designed to minimize this inherent disadvantage, the bipolar junction transistor is still
superior in its ability to switch high currents.
An interesting solution to this dilemma leverages the best features of IGFETs with the
best of features of BJTs, in one device called an Insulated-Gate Bipolar Transistor, or IGBT.
Also known as an Bipolar-mode MOSFET, a Conductivity-Modulated Field-Effect Transistor
(COMFET), or simply as an Insulated-Gate Transistor (IGT), it is equivalent to a Darlington
pair of IGFET and BJT:

6.12. IGBTS

315

Insulated-Gate Bipolar Transistor (IGBT)
(N-channel)
Schematic symbols
Collector

Equivalent circuit

Collector

Collector

Gate
Gate
Emitter

Emitter

Gate
Emitter

In essence, the IGFET controls the base current of a BJT, which handles the main load
current between collector and emitter. This way, there is extremely high current gain (since
the insulated gate of the IGFET draws practically no current from the control circuitry), but
the collector-to-emitter voltage drop during full conduction is as low as that of an ordinary BJT.
One disadvantage of the IGBT over a standard BJT is its slower turn-off time. For fast
switching and high current-handling capacity, its difficult to beat the bipolar junction transistor. Faster turn-off times for the IGBT may be achieved by certain changes in design, but only
at the expense of a higher saturated voltage drop between collector and emitter. However, the
IGBT provides a good alternative to IGFETs and BJTs for high-power control applications.
• REVIEW:
•
•
•

316

CHAPTER 6. INSULATED-GATE FIELD-EFFECT TRANSISTORS

Chapter 7

THYRISTORS
Contents
7.1 Hysteresis . . . . . . . . . . . . . . . . . .
7.2 Gas discharge tubes . . . . . . . . . . . .
7.3 The Shockley Diode . . . . . . . . . . . .
7.4 The DIAC . . . . . . . . . . . . . . . . . . .
7.5 The Silicon-Controlled Rectifier (SCR)
7.6 The TRIAC . . . . . . . . . . . . . . . . . .
7.7 Optothyristors . . . . . . . . . . . . . . . .
7.8 The Unijunction Transistor (UJT) . . .
7.9 The Silicon-Controlled Switch (SCS) . .
7.10 Field-effect-controlled thyristors . . . .
Bibliography . . . . . . . . . . . . . . . . . . . .

7.1

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317
318
322
329
329
341
344
344
350
352
354

Hysteresis

Thyristors are a class of semiconductor components exhibiting hysteresis, that property whereby
a system fails to return to its original state after some cause of state change has been removed.
A very simple example of hysteresis is the mechanical action of a toggle switch: when the
lever is pushed, it flips to one of two extreme states (positions) and will remain there even
after the source of motion is removed (after you remove your hand from the switch lever). To
illustrate the absence of hysteresis, consider the action of a ”momentary” pushbutton switch,
which returns to its original state after the button is no longer pressed: when the stimulus is
removed (your hand), the system (switch) immediately and fully returns to its prior state with
no ”latching” behavior.
Bipolar, junction field-effect, and insulated gate field-effect transistors are all non-hysteric
devices. That is, these do not inherently ”latch” into a state after being stimulated by a voltage
or current signal. For any given input signal at any given time, a transistor will exhibit a
317

CHAPTER 7. THYRISTORS

318

predictable output response as defined by its characteristic curve. Thyristors, on the other
hand, are semiconductor devices that tend to stay ”on” once turned on, and tend to stay ”off ”
once turned off. A momentary event is able to flip these devices into either their on or off
states where these will remain that way on their own, even after the cause of the state change
is taken away. As such, these are useful only as on/off switching devices – much like a toggle
switch – and cannot be used as analog signal amplifiers.
Thyristors are constructed using the same technology as bipolar junction transistors, and
in fact may be analyzed as circuits comprised of transistor pairs. How then, can a hysteric device (a thyristor) be made from non-hysteric devices (transistors)? The answer to this question
is positive feedback, also known as regenerative feedback. As you should recall, feedback is the
condition where a percentage of the output signal is ”fed back” to the input of an amplifying
device. Negative, or degenerative, feedback results in a diminishing of voltage gain with increases in stability, linearity, and bandwidth. Positive feedback, on the other hand, results in
a kind of instability where the amplifier’s output tends to ”saturate.” In the case of thyristors,
this saturating tendency equates to the device ”wanting” to stay on once turned on, and off
once turned off.
In this chapter we will explore several different kinds of thyristors, most of which stem from
a single, basic two-transistor core circuit. Before we do that, though, it would be beneficial to
study the technological predecessor to thyristors: gas discharge tubes.

7.2

Gas discharge tubes

If you’ve ever witnessed a lightning storm, you’ve seen electrical hysteresis in action (and
probably didn’t realize what you were seeing). The action of strong wind and rain accumulates tremendous static electric charges between cloud and earth, and between clouds as well.
Electric charge imbalances manifest themselves as high voltages, and when the electrical resistance of air can no longer hold these high voltages at bay, huge surges of current travel
between opposing poles of electrical charge which we call ”lightning.”
The buildup of high voltages by wind and rain is a fairly continuous process, the rate of
charge accumulation increasing under the proper atmospheric conditions. However, lightning
bolts are anything but continuous: they exist as relatively brief surges rather than continuous
discharges. Why is this? Why don’t we see soft, glowing lightning arcs instead of violently brief
lightning bolts? The answer lies in the nonlinear (and hysteric) resistance of air.
Under ordinary conditions, air has an extremely high amount of resistance. It is so high, in
fact, that we typically treat its resistance as infinite and electrical conduction through the air
as negligible. The presence of water and dust in air lowers its resistance some, but it is still an
insulator for most practical purposes. When enough high voltage is applied across a distance
of air, though, its electrical properties change: electrons become ”stripped” from their normal
positions around their respective atoms and are liberated to constitute a current. In this state,
air is considered to be ionized and is called a plasma rather than a gas. This usage of the word
”plasma” is not to be confused with the medical term (meaning the fluid portion of blood), but
is a fourth state of matter, the other three being solid, liquid, and vapor (gas). Plasma is a
relatively good conductor of electricity, its specific resistance being much lower than that of the
same substance in its gaseous state.
As an electric current moves through the plasma, there is energy dissipated in the plasma

7.2. GAS DISCHARGE TUBES

319

in the form of heat, just as current through a solid resistor dissipates energy in the form of heat.
In the case of lightning, the temperatures involved are extremely high. High temperatures are
also sufficient to convert gaseous air into a plasma or maintain plasma in that state without
the presence of high voltage. As the voltage between cloud and earth, or between cloud and
cloud, decreases as the charge imbalance is neutralized by the current of the lightning bolt, the
heat dissipated by the bolt maintains the air path in a plasma state, keeping its resistance low.
The lightning bolt remains a plasma until the voltage decreases to too low a level to sustain
enough current to dissipate enough heat. Finally, the air returns to a gaseous state and stops
conducting current, thus allowing voltage to build up once more.
Note how throughout this cycle, the air exhibits hysteresis. When not conducting electricity,
it tends to remain an insulator until voltage builds up past a critical threshold point. Then,
once it changes state and becomes a plasma, it tends to remain a conductor until voltage falls
below a lower critical threshold point. Once ”turned on” it tends to stay ”on,” and once ”turned
off ” it tends to stay ”off.” This hysteresis, combined with a steady buildup of voltage due to the
electrostatic effects of wind and rain, explains the action of lightning as brief bursts.
In electronic terms, what we have here in the action of lightning is a simple relaxation oscillator. Oscillators are electronic circuits that produce an oscillating (AC) voltage from a steady
supply of DC power. A relaxation oscillator is one that works on the principle of a charging
capacitor that is suddenly discharged every time its voltage reaches a critical threshold value.
One of the simplest relaxation oscillators in existence is comprised of three components (not
counting the DC power supply): a resistor, capacitor, and neon lamp in Figure 7.1.

R

C

Neon lamp

Figure 7.1: Simple relaxation oscillator
Neon lamps are nothing more than two metal electrodes inside a sealed glass bulb, separated by the neon gas inside. At room temperatures and with no applied voltage, the lamp
has nearly infinite resistance. However, once a certain threshold voltage is exceeded (this voltage depends on the gas pressure and geometry of the lamp), the neon gas will become ionized
(turned into a plasma) and its resistance dramatically reduced. In effect, the neon lamp exhibits the same characteristics as air in a lightning storm, complete with the emission of light
as a result of the discharge, albeit on a much smaller scale.
The capacitor in the relaxation oscillator circuit shown above charges at an inverse exponential rate determined by the size of the resistor. When its voltage reaches the threshold
voltage of the lamp, the lamp suddenly ”turns on” and quickly discharges the capacitor to a
low voltage value. Once discharged, the lamp ”turns off ” and allows the capacitor to build up a

CHAPTER 7. THYRISTORS

320

charge once more. The result is a series of brief flashes of light from the lamp, the rate of which
is dictated by battery voltage, resistor resistance, capacitor capacitance, and lamp threshold
voltage.
While gas-discharge lamps are more commonly used as sources of illumination, their hysteric properties were leveraged in slightly more sophisticated variants known as thyratron
tubes. Essentially a gas-filled triode tube (a triode being a three-element vacuum electron tube
performing much a similar function to the N-channel, D-type IGFET), the thyratron tube could
be turned on with a small control voltage applied between grid and cathode, and turned off by
reducing the plate-to-cathode voltage.
+
Rload
Thyratron
Tube

high voltage
AC source

control
voltage

Figure 7.2: Simple thyratron control circuit
In essence, thyratron tubes were controlled versions of neon lamps built specifically for
switching current to a load. The dot inside the circle of the schematic symbol indicates a gas
fill, as opposed to the hard vacuum normally seen in other electron tube designs. In the circuit
shown above in Figure 7.2. the thyratron tube allows current through the load in one direction
(note the polarity across the load resistor) when triggered by the small DC control voltage
connected between grid and cathode. Note that the load’s power source is AC, which provides a
clue about how the thyratron turns off after its been triggered on: since AC voltage periodically
passes through a condition of 0 volts between half-cycles, the current through an AC-powered
load must also periodically halt. This brief pause of current between half-cycles gives the
tube’s gas time to cool, letting it return to its normal ”off ” state. Conduction may resume only
if enough voltage is applied by the AC power source (some other time in the wave’s cycle) and
if the DC control voltage allows it.
An oscilloscope display of load voltage in such a circuit would look something like Figure 7.3.
As the AC supply voltage climbs from zero volts to its first peak, the load voltage remains
at zero (no load current) until the threshold voltage is reached. At that point, the tube switches
”on” and begins to conduct, the load voltage now following the AC voltage through the rest of
the half cycle. Load voltage exists (and thus load current) even when the AC voltage waveform
has dropped below the threshold value of the tube. This is hysteresis at work: the tube stays
in its conductive mode past the point where it first turned on, continuing to conduct until
there the supply voltage drops off to almost zero volts. Because thyratron tubes are one-way
(diode) devices, no voltage develops across the load through the negative half-cycle of AC. In

7.2. GAS DISCHARGE TUBES

321

Threshold voltage

Load voltage
AC supply voltage
Figure 7.3: Thyratron waveforms
practical thyratron circuits, multiple tubes arranged in some form of full-wave rectifier circuit
to facilitate full-wave DC power to the load.
The thyratron tube has been applied to a relaxation oscillator circuit. [1] The frequency
is controlled by a small DC voltage between grid and cathode. (See Figure 7.4) This voltagecontrolled oscillator is known as a VCO. Relaxation oscillators produce a very non-sinusoidal
output, and they exist mostly as demonstration circuits (as is the case here) or in applications
where the harmonic rich waveform is desirable. [2]

R

C
Controlling
voltage

Figure 7.4: Voltage controlled thyratron relaxation oscillator
I speak of thyratron tubes in the past tense for good reason: modern semiconductor components have obsoleted thyratron tube technology for all but a few very special applications. It
is no coincidence that the word thyristor bears so much similarity to the word thyratron, for
this class of semiconductor components does much the same thing: use hysteretically switch
current on and off. It is these modern devices that we now turn our attention to.
• REVIEW:
• Electrical hysteresis, the tendency for a component to remain ”on” (conducting) after it
begins to conduct and to remain ”off ” (nonconducting) after it ceases to conduct, helps to
explain why lightning bolts exist as momentary surges of current rather than continuous
discharges through the air.
• Simple gas-discharge tubes such as neon lamps exhibit electrical hysteresis.

CHAPTER 7. THYRISTORS

322

• More advanced gas-discharge tubes have been made with control elements so that their
”turn-on” voltage could be adjusted by an external signal. The most common of these
tubes was called the thyratron.
• Simple oscillator circuits called relaxation oscillators may be created with nothing more
than a resistor-capacitor charging network and a hysteretic device connected across the
capacitor.

7.3

The Shockley Diode

Our exploration of thyristors begins with a device called the four-layer diode, also known as
a PNPN diode, or a Shockley diode after its inventor, William Shockley. This is not to be
confused with a Schottky diode, that two-layer metal-semiconductor device known for its high
switching speed. A crude illustration of the Shockley diode, often seen in textbooks, is a fourlayer sandwich of P-N-P-N semiconductor material, Figure 7.5.

Anode

P
N
P
N

Cathode

Figure 7.5: Shockley or 4-layer diode
Unfortunately, this simple illustration does nothing to enlighten the viewer on how it works
or why. Consider an alternative rendering of the device’s construction in Figure 7.6.

P

Anode

N

N

P

P
N

Cathode

Figure 7.6: Transistor equivalent of Shockley diode

7.3. THE SHOCKLEY DIODE

323

Shown like this, it appears to be a set of interconnected bipolar transistors, one PNP and
the other NPN. Drawn using standard schematic symbols, and respecting the layer doping
concentrations not shown in the last image, the Shockley diode looks like this (Figure 7.7)
Anode

Anode
P
N
P

N
P
N

Cathode
Physical diagram

Cathode
Equivalent schematic

Schematic symbol

Figure 7.7: Shockley diode: physical diagram, equivalent schematic diagram, and schematic
symbol.
Let’s connect one of these devices to a source of variable voltage and see what happens:
(Figure 7.8)

Figure 7.8: Powered Shockley diode equivalent circuit.
With no voltage applied, of course there will be no current. As voltage is initially increased,
there will still be no current because neither transistor is able to turn on: both will be in cutoff
mode. To understand why this is, consider what it takes to turn a bipolar junction transistor
on: current through the base-emitter junction. As you can see in the diagram, base current
through the lower transistor is controlled by the upper transistor, and the base current through
the upper transistor is controlled by the lower transistor. In other words, neither transistor
can turn on until the other transistor turns on. What we have here, in vernacular terms, is
known as a Catch-22.
So how can a Shockley diode ever conduct current, if its constituent transistors stubbornly
maintain themselves in a state of cutoff? The answer lies in the behavior of real transistors
as opposed to ideal transistors. An ideal bipolar transistor will never conduct collector current
if no base current flows, no matter how much or little voltage we apply between collector and

CHAPTER 7. THYRISTORS

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emitter. Real transistors, on the other hand, have definite limits to how much collector-emitter
voltage each can withstand before one breaks down and conduct. If two real transistors are
connected in this fashion to form a Shockley diode, each one will conduct if sufficient voltage is
applied by the battery between anode and cathode to cause one of them to break down. Once
one transistor breaks down and begins to conduct, it will allow base current through the other
transistor, causing it to turn on in a normal fashion, which then allows base current through
the first transistor. The end result is that both transistors will be saturated, now keeping each
other turned on instead of off.
So, we can force a Shockley diode to turn on by applying sufficient voltage between anode
and cathode. As we have seen, this will inevitably cause one of the transistors to turn on, which
then turns the other transistor on, ultimately ”latching” both transistors on where each will
tend to remain. But how do we now get the two transistors to turn off again? Even if the applied
voltage is reduced to a point well below what it took to get the Shockley diode conducting, it
will remain conducting because both transistors now have base current to maintain regular,
controlled conduction. The answer to this is to reduce the applied voltage to a much lower point
where too little current flows to maintain transistor bias, at which point one of the transistors
will cutoff, which then halts base current through the other transistor, sealing both transistors
in the ”off ” state as each one was before any voltage was applied at all.
If we graph this sequence of events and plot the results on an I/V graph, the hysteresis
is evident. First, we will observe the circuit as the DC voltage source (battery) is set to zero
voltage: (Figure 7.9)

Circuit
current

Applied voltage
Figure 7.9: Zero applied voltage; zero current
Next, we will steadily increase the DC voltage. Current through the circuit is at or nearly
at zero, as the breakdown limit has not been reached for either transistor: (Figure 7.10)
When the voltage breakdown limit of one transistor is reached, it will begin to conduct
collector current even though no base current has gone through it yet. Normally, this sort
of treatment would destroy a bipolar junction transistor, but the PNP junctions comprising a
Shockley diode are engineered to take this kind of abuse, similar to the way a Zener diode is
built to handle reverse breakdown without sustaining damage. For the sake of illustration I’ll
assume the lower transistor breaks down first, sending current through the base of the upper
transistor: (Figure 7.11)
As the upper transistor receives base current, it turns on as expected. This action allows
the lower transistor to conduct normally, the two transistors ”sealing” themselves in the ”on”

7.3. THE SHOCKLEY DIODE

325

Circuit
current

Applied voltage
Figure 7.10: Some applied voltage; still no current

Circuit
current

Applied voltage
Figure 7.11: More voltage applied; lower transistor breaks down

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326
state. Full current is quickly seen in the circuit: (Figure 7.12)

Circuit
current

Applied voltage
Figure 7.12: Transistors are now fully conducting.
The positive feedback mentioned earlier in this chapter is clearly evident here. When one
transistor breaks down, it allows current through the device structure. This current may be
viewed as the ”output” signal of the device. Once an output current is established, it works
to hold both transistors in saturation, thus ensuring the continuation of a substantial output
current. In other words, an output current ”feeds back” positively to the input (transistor base
current) to keep both transistors in the ”on” state, thus reinforcing (or regenerating) itself.
With both transistors maintained in a state of saturation with the presence of ample base
current, each will continue to conduct even if the applied voltage is greatly reduced from the
breakdown level. The effect of positive feedback is to keep both transistors in a state of saturation despite the loss of input stimulus (the original, high voltage needed to break down one
transistor and cause a base current through the other transistor): (Figure 7.13)

Circuit
current

Applied voltage
Figure 7.13: Current maintained even when voltage is reduced
If the DC voltage source is turned down too far, though, the circuit will eventually reach a
point where there isn’t enough current to sustain both transistors in saturation. As one transistor passes less and less collector current, it reduces the base current for the other transistor,
thus reducing base current for the first transistor. The vicious cycle continues rapidly until
both transistors fall into cutoff: (Figure 7.14)
Here, positive feedback is again at work: the fact that the cause/effect cycle between both

7.3. THE SHOCKLEY DIODE

327

Circuit
current

Applied voltage
Figure 7.14: If voltage drops too low, both transistors shut off.
transistors is ”vicious” (a decrease in current through one works to decrease current through
the other, further decreasing current through the first transistor) indicates a positive relationship between output (controlled current) and input (controlling current through the transistors’
bases).
The resulting curve on the graph is classically hysteretic: as the input signal (voltage) is
increased and decreased, the output (current) does not follow the same path going down as it
did going up: (Figure 7.15)

Circuit
current

Applied voltage
Figure 7.15: Hysteretic curve
Put in simple terms, the Shockley diode tends to stay on once its turned on, and stay off
once its turned off. No ”in-between” or ”active” mode in its operation: it is a purely on or off
device, as are all thyristors.
A few special terms apply to Shockley diodes and all other thyristor devices built upon the
Shockley diode foundation. First is the term used to describe its ”on” state: latched. The word
”latch” is reminiscent of a door lock mechanism, which tends to keep the door closed once it has
been pushed shut. The term firing refers to the initiation of a latched state. To get a Shockley
diode to latch, the applied voltage must be increased until breakover is attained. Though this
action is best described as transistor breakdown, the term breakover is used instead because
the result is a pair of transistors in mutual saturation rather than destruction of the transistor. A latched Shockley diode is re-set back into its nonconducting state by reducing current
through it until low-current dropout occurs.
Note that Shockley diodes may be fired in a way other than breakover: excessive voltage

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328

rise, or dv/dt. If the applied voltage across the diode increases at a high rate of change, it
may trigger. This is able to cause latching (turning on) of the diode due to inherent junction
capacitances within the transistors. Capacitors, as you may recall, oppose changes in voltage
by drawing or supplying current. If the applied voltage across a Shockley diode rises at too
fast a rate, those tiny capacitances will draw enough current during that time to activate the
transistor pair, turning them both on. Usually, this form of latching is undesirable, and can be
minimized by filtering high-frequency (fast voltage rises) from the diode with series inductors
and parallel resistor-capacitor networks called snubbers: (Figure 7.16)

Series inductor

Shockley
diode

RC "snubber"

Figure 7.16: Both the series inductor and parallel resistor-capacitor “snubber” circuit help
minimize the Shockley diode’s exposure to excessively rising voltage.
The voltage rise limit of a Shockley diode is referred to as the critical rate of voltage rise.
Manufacturers usually provide this specification for the devices they sell.
• REVIEW:
• Shockley diodes are four-layer PNPN semiconductor devices. These behave as a pair of
interconnected PNP and NPN transistors.
• Like all thyristors, Shockley diodes tend to stay on once turned on (latched), and stay off
once turned off.
• To latch a Shockley diode exceed the anode-to-cathode breakover voltage, or exceed the
anode-to-cathode critical rate of voltage rise.
• To cause a Shockley diode to stop conducting, reduce the current going through it to a
level below its low-current dropout threshold.

7.4. THE DIAC

7.4

329

The DIAC

Like all diodes, Shockley diodes are unidirectional devices; that is, these only conduct current
in one direction. If bidirectional (AC) operation is desired, two Shockley diodes may be joined
in parallel facing different directions to form a new kind of thyristor, the DIAC: (Figure 7.17)

DIAC equivalent circuit

DIAC schematic symbol

Figure 7.17: The DIAC
A DIAC operated with a DC voltage across it behaves exactly the same as a Shockley diode.
With AC, however, the behavior is different from what one might expect. Because alternating
current repeatedly reverses direction, DIACs will not stay latched longer than one-half cycle. If
a DIAC becomes latched, it will continue to conduct current only as long as voltage is available
to push enough current in that direction. When the AC polarity reverses, as it must twice
per cycle, the DIAC will drop out due to insufficient current, necessitating another breakover
before it conducts again. The result is the current waveform in Figure 7.18.

Breakover voltage

DIAC current
AC supply voltage

Breakover voltage

Figure 7.18: DIAC waveforms
DIACs are almost never used alone, but in conjunction with other thyristor devices.

7.5

The Silicon-Controlled Rectifier (SCR)

Shockley diodes are curious devices, but rather limited in application. Their usefulness may
be expanded, however, by equipping them with another means of latching. In doing so, each
becomes true amplifying devices (if only in an on/off mode), and we refer to these as siliconcontrolled rectifiers, or SCRs.

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330

The progression from Shockley diode to SCR is achieved with one small addition, actually
nothing more than a third wire connection to the existing PNPN structure: (Figure 7.19)
Anode

Anode
P
N

Gate

P

Anode
N
P
N

Cathode
Physical diagram

Gate

Gate
Cathode

Cathode
Equivalent schematic

Schematic symbol

Figure 7.19: The Silicon-Controlled Rectifier (SCR)
If an SCR’s gate is left floating (disconnected), it behaves exactly as a Shockley diode. It
may be latched by breakover voltage or by exceeding the critical rate of voltage rise between
anode and cathode, just as with the Shockley diode. Dropout is accomplished by reducing
current until one or both internal transistors fall into cutoff mode, also like the Shockley diode.
However, because the gate terminal connects directly to the base of the lower transistor, it
may be used as an alternative means to latch the SCR. By applying a small voltage between
gate and cathode, the lower transistor will be forced on by the resulting base current, which
will cause the upper transistor to conduct, which then supplies the lower transistor’s base
with current so that it no longer needs to be activated by a gate voltage. The necessary gate
current to initiate latch-up, of course, will be much lower than the current through the SCR
from cathode to anode, so the SCR does achieve a measure of amplification.
This method of securing SCR conduction is called triggering, and it is by far the most common way that SCRs are latched in actual practice. In fact, SCRs are usually chosen so that
their breakover voltage is far beyond the greatest voltage expected to be experienced from the
power source, so that it can be turned on only by an intentional voltage pulse applied to the
gate.
It should be mentioned that SCRs may sometimes be turned off by directly shorting their
gate and cathode terminals together, or by ”reverse-triggering” the gate with a negative voltage
(in reference to the cathode), so that the lower transistor is forced into cutoff. I say this is
”sometimes” possible because it involves shunting all of the upper transistor’s collector current
past the lower transistor’s base. This current may be substantial, making triggered shut-off
of an SCR difficult at best. A variation of the SCR, called a Gate-Turn-Off thyristor, or GTO,
makes this task easier. But even with a GTO, the gate current required to turn it off may be
as much as 20% of the anode (load) current! The schematic symbol for a GTO is shown in the
following illustration: (Figure 7.20)
SCRs and GTOs share the same equivalent schematics (two transistors connected in a
positive-feedback fashion), the only differences being details of construction designed to grant
the NPN transistor a greater β than the PNP. This allows a smaller gate current (forward or
reverse) to exert a greater degree of control over conduction from cathode to anode, with the
PNP transistor’s latched state being more dependent upon the NPN’s than vice versa. The

7.5. THE SILICON-CONTROLLED RECTIFIER (SCR)

331

Anode
Gate
Cathode
Figure 7.20: The Gate Turn-Off thyristor (GTO)
Gate-Turn-Off thyristor is also known by the name of Gate-Controlled Switch, or GCS.
A rudimentary test of SCR function, or at least terminal identification, may be performed
with an ohmmeter. Because the internal connection between gate and cathode is a single PN
junction, a meter should indicate continuity between these terminals with the red test lead on
the gate and the black test lead on the cathode like this: (Figure 7.21)

gate

V

cathode

A

V

A
OFF

A

COM

Figure 7.21: Rudimentary test of SCR
All other continuity measurements performed on an SCR will show ”open” (”OL” on some
digital multimeter displays). It must be understood that this test is very crude and does not
constitute a comprehensive assessment of the SCR. It is possible for an SCR to give good
ohmmeter indications and still be defective. Ultimately, the only way to test an SCR is to
subject it to a load current.
If you are using a multimeter with a ”diode check” function, the gate-to-cathode junction
voltage indication you get may or may not correspond to what’s expected of a silicon PN junction (approximately 0.7 volts). In some cases, you will read a much lower junction voltage:
mere hundredths of a volt. This is due to an internal resistor connected between the gate and
cathode incorporated within some SCRs. This resistor is added to make the SCR less susceptible to false triggering by spurious voltage spikes, from circuit ”noise” or from static electric
discharge. In other words, having a resistor connected across the gate-cathode junction requires that a strong triggering signal (substantial current) be applied to latch the SCR. This

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332

feature is often found in larger SCRs, not on small SCRs. Bear in mind that an SCR with an
internal resistor connected between gate and cathode will indicate continuity in both directions
between those two terminals: (Figure 7.22)

Anode

Gate

Gate-to-Cathode
resistor

Cathode

Figure 7.22: Larger SCRs have gate to cathode resistor.
”Normal” SCRs, lacking this internal resistor, are sometimes referred to as sensitive gate
SCRs due to their ability to be triggered by the slightest positive gate signal.
The test circuit for an SCR is both practical as a diagnostic tool for checking suspected SCRs
and also an excellent aid to understanding basic SCR operation. A DC voltage source is used
for powering the circuit, and two pushbutton switches are used to latch and unlatch the SCR,
respectively: (Figure 7.23)

off

on

SCR under
test

Figure 7.23: SCR testing circuit
Actuating the normally-open ”on” pushbutton switch connects the gate to the anode, allowing current from the negative terminal of the battery, through the cathode-gate PN junction,
through the switch, through the load resistor, and back to the battery. This gate current should
force the SCR to latch on, allowing current to go directly from cathode to anode without further
triggering through the gate. When the ”on” pushbutton is released, the load should remain energized.
Pushing the normally-closed ”off ” pushbutton switch breaks the circuit, forcing current
through the SCR to halt, thus forcing it to turn off (low-current dropout).

7.5. THE SILICON-CONTROLLED RECTIFIER (SCR)

333

If the SCR fails to latch, the problem may be with the load and not the SCR. A certain
minimum amount of load current is required to hold the SCR latched in the ”on” state. This
minimum current level is called the holding current. A load with too great a resistance value
may not draw enough current to keep an SCR latched when gate current ceases, thus giving
the false impression of a bad (unlatchable) SCR in the test circuit. Holding current values for
different SCRs should be available from the manufacturers. Typical holding current values
range from 1 milliamp to 50 milliamps or more for larger units.
For the test to be fully comprehensive, more than the triggering action needs to be tested.
The forward breakover voltage limit of the SCR could be tested by increasing the DC voltage
supply (with no pushbuttons actuated) until the SCR latches all on its own. Beware that
a breakover test may require very high voltage: many power SCRs have breakover voltage
ratings of 600 volts or more! Also, if a pulse voltage generator is available, the critical rate of
voltage rise for the SCR could be tested in the same way: subject it to pulsing supply voltages
of different V/time rates with no pushbutton switches actuated and see when it latches.
In this simple form, the SCR test circuit could suffice as a start/stop control circuit for a DC
motor, lamp, or other practical load: (Figure 7.24)

Motor

off

on

SCR under
test

Figure 7.24: DC motor start/stop control circuit
Another practical use for the SCR in a DC circuit is as a crowbar device for overvoltage
protection. A ”crowbar” circuit consists of an SCR placed in parallel with the output of a
DC power supply, for placing a direct short-circuit on the output of that supply to prevent
excessive voltage from reaching the load. Damage to the SCR and power supply is prevented
by the judicious placement of a fuse or substantial series resistance ahead of the SCR to limit
short-circuit current: (Figure 7.25)
Some device or circuit sensing the output voltage will be connected to the gate of the SCR,
so that when an overvoltage condition occurs, voltage will be applied between the gate and
cathode, triggering the SCR and forcing the fuse to blow. The effect will be approximately the
same as dropping a solid steel crowbar directly across the output terminals of the power supply,
hence the name of the circuit.
Most applications of the SCR are for AC power control, despite the fact that SCRs are inherently DC (unidirectional) devices. If bidirectional circuit current is required, multiple SCRs
may be used, with one or more facing each direction to handle current through both half-cycles
of the AC wave. The primary reason SCRs are used at all for AC power control applications
is the unique response of a thyristor to an alternating current. As we saw, the thyratron tube
(the electron tube version of the SCR) and the DIAC, a hysteretic device triggered on during a

CHAPTER 7. THYRISTORS

334

Transformer
Filter

AC
power
source

Fuse

Load

Rectifier
Crowbar
(triggering circuit
omitted for simplicity)

Figure 7.25: Crowbar circuit used in DC power supply
portion of an AC half-cycle will latch and remain on throughout the remainder of the half-cycle
until the AC current decreases to zero, as it must to begin the next half-cycle. Just prior to the
zero-crossover point of the current waveform, the thyristor will turn off due to insufficient current (this behavior is also known as natural commutation) and must be fired again during the
next cycle. The result is a circuit current equivalent to a ”chopped up” sine wave. For review,
here is the graph of a DIAC’s response to an AC voltage whose peak exceeds the breakover
voltage of the DIAC: (Figure 7.26)

Breakover voltage

DIAC current
AC supply voltage

Breakover voltage

Figure 7.26: DIAC bidirectional response
With the DIAC, that breakover voltage limit was a fixed quantity. With the SCR, we have
control over exactly when the device becomes latched by triggering the gate at any point in
time along the waveform. By connecting a suitable control circuit to the gate of an SCR, we
can ”chop” the sine wave at any point to allow for time-proportioned power control to a load.
Take the circuit in Figure 7.27 as an example. Here, an SCR is positioned in a circuit to
control power to a load from an AC source.
Being a unidirectional (one-way) device, at most we can only deliver half-wave power to
the load, in the half-cycle of AC where the supply voltage polarity is positive on the top and
negative on the bottom. However, for demonstrating the basic concept of time-proportional
control, this simple circuit is better than one controlling full-wave power (which would require
two SCRs).

7.5. THE SILICON-CONTROLLED RECTIFIER (SCR)

335

Load
AC
source

SCR

Figure 7.27: SCR control of AC power
With no triggering to the gate, and the AC source voltage well below the SCR’s breakover
voltage rating, the SCR will never turn on. Connecting the SCR gate to the anode through
a standard rectifying diode (to prevent reverse current through the gate in the event of the
SCR containing a built-in gate-cathode resistor), will allow the SCR to be triggered almost
immediately at the beginning of every positive half-cycle: (Figure 7.28)

Load
AC
source

Load current

AC source voltage
Figure 7.28: Gate connected directly to anode through a diode; nearly complete half-wave
current through load.
We can delay the triggering of the SCR, however, by inserting some resistance into the
gate circuit, thus increasing the amount of voltage drop required before enough gate current
triggers the SCR. In other words, if we make it harder for electrons to flow through the gate by
adding a resistance, the AC voltage will have to reach a higher point in its cycle before there
will be enough gate current to turn the SCR on. The result is in Figure 7.29.
With the half-sine wave chopped up to a greater degree by delayed triggering of the SCR,
the load receives less average power (power is delivered for less time throughout a cycle). By
making the series gate resistor variable, we can make adjustments to the time-proportioned
power: (Figure 7.30)
Unfortunately, this control scheme has a significant limitation. In using the AC source
waveform for our SCR triggering signal, we limit control to the first half of the waveform’s
half-cycle. In other words, it is not possible for us to wait until after the wave’s peak to trigger
the SCR. This means we can turn down the power only to the point where the SCR turns on at

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336

Load
AC
source

Load current

AC source voltage
Figure 7.29: Resistance inserted in gate circuit; less than half-wave current through load.

Load
AC
source
trigger
threshold

Figure 7.30: Increasing the resistance raises the threshold level, causing less power to be
delivered to the load. Decreasing the resistance lowers the threshold level, causing more power
to be delivered to the load.

7.5. THE SILICON-CONTROLLED RECTIFIER (SCR)

337

the very peak of the wave: (Figure 7.31)

Load
AC
source
trigger
threshold

Figure 7.31: Circuit at minimum power setting
Raising the trigger threshold any more will cause the circuit to not trigger at all, since not
even the peak of the AC power voltage will be enough to trigger the SCR. The result will be no
power to the load.
An ingenious solution to this control dilemma is found in the addition of a phase-shifting
capacitor to the circuit: (Figure 7.32)

Load
AC
source

Capacitor voltage
Figure 7.32: Addition of a phase-shifting capacitor to the circuit
The smaller waveform shown on the graph is voltage across the capacitor. For the sake of
illustrating the phase shift, I’m assuming a condition of maximum control resistance where the
SCR is not triggering at all with no load current, save for what little current goes through the
control resistor and capacitor. This capacitor voltage will be phase-shifted anywhere from 0o to
90o lagging behind the power source AC waveform. When this phase-shifted voltage reaches a

CHAPTER 7. THYRISTORS

338

high enough level, the SCR will trigger.
With enough voltage across the capacitor to periodically trigger the SCR, the resulting load
current waveform will look something like Figure 7.33)

Load
AC
source
trigger
threshold

load
current

Capacitor voltage
Figure 7.33: Phase-shifted signal triggers SCR into conduction.
Because the capacitor waveform is still rising after the main AC power waveform has
reached its peak, it becomes possible to trigger the SCR at a threshold level beyond that peak,
thus chopping the load current wave further than it was possible with the simpler circuit.
In reality, the capacitor voltage waveform is a bit more complex that what is shown here, its
sinusoidal shape distorted every time the SCR latches on. However, what I’m trying to illustrate here is the delayed triggering action gained with the phase-shifting RC network; thus, a
simplified, undistorted waveform serves the purpose well.
SCRs may also be triggered, or ”fired,” by more complex circuits. While the circuit previously shown is sufficient for a simple application like a lamp control, large industrial motor
controls often rely on more sophisticated triggering methods. Sometimes, pulse transformers
are used to couple a triggering circuit to the gate and cathode of an SCR to provide electrical
isolation between the triggering and power circuits: (Figure 7.34)

...
...
to triggering
circuit
...

pulse
transformer

SCR
to power
circuit
...

Figure 7.34: Transformer coupling of trigger signal provides isolation.
When multiple SCRs are used to control power, their cathodes are often not electrically common, making it difficult to connect a single triggering circuit to all SCRs equally. An example

7.5. THE SILICON-CONTROLLED RECTIFIER (SCR)

339

of this is the controlled bridge rectifier shown in Figure 7.35.

SCR1

SCR4

SCR2

SCR3

Load

Figure 7.35: Controlled bridge rectifier
In any bridge rectifier circuit, the rectifying diodes (in this example, the rectifying SCRs)
must conduct in opposite pairs. SCR1 and SCR3 must be fired simultaneously, and SCR2 and
SCR4 must be fired together as a pair. As you will notice, though, these pairs of SCRs do not
share the same cathode connections, meaning that it would not work to simply parallel their
respective gate connections and connect a single voltage source to trigger both: (Figure 7.36)

SCR1

SCR2

triggering
voltage
(pulse voltage
source)

SCR4

SCR3

Load

Figure 7.36: This strategy will not work for triggering SCR2 and SCR4 as a pair.
Although the triggering voltage source shown will trigger SCR4 , it will not trigger SCR2
properly because the two thyristors do not share a common cathode connection to reference
that triggering voltage. Pulse transformers connecting the two thyristor gates to a common
triggering voltage source will work, however: (Figure 7.37)
Bear in mind that this circuit only shows the gate connections for two out of the four SCRs.
Pulse transformers and triggering sources for SCR1 and SCR3 , as well as the details of the
pulse sources themselves, have been omitted for the sake of simplicity.
Controlled bridge rectifiers are not limited to single-phase designs. In most industrial control systems, AC power is available in three-phase form for maximum efficiency, and solid-state

CHAPTER 7. THYRISTORS

340

SCR1

SCR2

SCR4

SCR3

pulse
voltage
source

Load

Figure 7.37: Transformer coupling of the gates allows triggering of SCR2 and SCR4 .
control circuits are built to take advantage of that. A three-phase controlled rectifier circuit
built with SCRs, without pulse transformers or triggering circuitry shown, would look like
Figure 7.38.

3-phase source
Controlled
rectifier

+
Load
-

Figure 7.38: Three-phase bridge SCR control of load
• REVIEW:
• A Silicon-Controlled Rectifier, or SCR, is essentially a Shockley diode with an extra terminal added. This extra terminal is called the gate, and it is used to trigger the device
into conduction (latch it) by the application of a small voltage.
• To trigger, or fire, an SCR, voltage must be applied between the gate and cathode, positive
to the gate and negative to the cathode. When testing an SCR, a momentary connection
between the gate and anode is sufficient in polarity, intensity, and duration to trigger it.
• SCRs may be fired by intentional triggering of the gate terminal, excessive voltage (breakdown) between anode and cathode, or excessive rate of voltage rise between anode and

7.6. THE TRIAC

341

cathode. SCRs may be turned off by anode current falling below the holding current value
(low-current dropout), or by ”reverse-firing” the gate (applying a negative voltage to the
gate). Reverse-firing is only sometimes effective, and always involves high gate current.
• A variant of the SCR, called a Gate-Turn-Off thyristor (GTO), is specifically designed to
be turned off by means of reverse triggering. Even then, reverse triggering requires fairly
high current: typically 20% of the anode current.
• SCR terminals may be identified by a continuity meter: the only two terminals showing
any continuity between them at all should be the gate and cathode. Gate and cathode
terminals connect to a PN junction inside the SCR, so a continuity meter should obtain
a diode-like reading between these two terminals with the red (+) lead on the gate and
the black (-) lead on the cathode. Beware, though, that some large SCRs have an internal
resistor connected between gate and cathode, which will affect any continuity readings
taken by a meter.
• SCRs are true rectifiers: they only allow current through them in one direction. This
means they cannot be used alone for full-wave AC power control.
• If the diodes in a rectifier circuit are replaced by SCRs, you have the makings of a controlled rectifier circuit, whereby DC power to a load may be time-proportioned by triggering the SCRs at different points along the AC power waveform.

7.6

The TRIAC

SCRs are unidirectional (one-way) current devices, making them useful for controlling DC
only. If two SCRs are joined in back-to-back parallel fashion just like two Shockley diodes were
joined together to form a DIAC, we have a new device known as the TRIAC: (Figure 7.39)

Main Terminal 2
(MT2)

Gate
Main Terminal 1
(MT1)
TRIAC equivalent circuit

Main Terminal 2
(MT2)
Gate
Main Terminal 1
(MT1)

TRIAC schematic symbol

Figure 7.39: The TRIAC SCR equivalent and, TRIAC schematic symbol
Because individual SCRs are more flexible to use in advanced control systems, these are
more commonly seen in circuits like motor drives; TRIACs are usually seen in simple, lowpower applications like household dimmer switches. A simple lamp dimmer circuit is shown in

CHAPTER 7. THYRISTORS

342

Figure 7.40, complete with the phase-shifting resistor-capacitor network necessary for afterpeak firing.

Lamp

AC
source

Figure 7.40: TRIAC phase-control of power
TRIACs are notorious for not firing symmetrically. This means these usually won’t trigger
at the exact same gate voltage level for one polarity as for the other. Generally speaking, this
is undesirable, because unsymmetrical firing results in a current waveform with a greater variety of harmonic frequencies. Waveforms that are symmetrical above and below their average
centerlines are comprised of only odd-numbered harmonics. Unsymmetrical waveforms, on
the other hand, contain even-numbered harmonics (which may or may not be accompanied by
odd-numbered harmonics as well).
In the interest of reducing total harmonic content in power systems, the fewer and less
diverse the harmonics, the better – one more reason individual SCRs are favored over TRIACs
for complex, high-power control circuits. One way to make the TRIAC’s current waveform
more symmetrical is to use a device external to the TRIAC to time the triggering pulse. A
DIAC placed in series with the gate does a fair job of this: (Figure 7.41)

Lamp

AC
source

Figure 7.41: DIAC improves symmetry of control
DIAC breakover voltages tend to be much more symmetrical (the same in one polarity
as the other) than TRIAC triggering voltage thresholds. Since the DIAC prevents any gate
current until the triggering voltage has reached a certain, repeatable level in either direction,
the firing point of the TRIAC from one half-cycle to the next tends to be more consistent, and
the waveform more symmetrical above and below its centerline.
Practically all the characteristics and ratings of SCRs apply equally to TRIACs, except
that TRIACs of course are bidirectional (can handle current in both directions). Not much
more needs to be said about this device except for an important caveat concerning its terminal
designations.

7.6. THE TRIAC

343

From the equivalent circuit diagram shown earlier, one might think that main terminals
1 and 2 were interchangeable. These are not! Although it is helpful to imagine the TRIAC
as being composed of two SCRs joined together, it in fact is constructed from a single piece of
semiconducting material, appropriately doped and layered. The actual operating characteristics may differ slightly from that of the equivalent model.
This is made most evident by contrasting two simple circuit designs, one that works and
one that doesn’t. The following two circuits are a variation of the lamp dimmer circuit shown
earlier, the phase-shifting capacitor and DIAC removed for simplicity’s sake. Although the
resulting circuit lacks the fine control ability of the more complex version (with capacitor and
DIAC), it does function: (Figure 7.42)

Lamp

AC
source

Figure 7.42: This circuit with the gate to MT2 does function.
Suppose we were to swap the two main terminals of the TRIAC around. According to the
equivalent circuit diagram shown earlier in this section, the swap should make no difference.
The circuit ought to work: (Figure 7.43)

Lamp

AC
source

Figure 7.43: With the gate swapped to MT1 , this circuit does not function.
However, if this circuit is built, it will be found that it does not work! The load will receive
no power, the TRIAC refusing to fire at all, no matter how low or high a resistance value the
control resistor is set to. The key to successfully triggering a TRIAC is to make sure the gate
receives its triggering current from the main terminal 2 side of the circuit (the main terminal
on the opposite side of the TRIAC symbol from the gate terminal). Identification of the MT1
and MT2 terminals must be done via the TRIAC’s part number with reference to a data sheet
or book.
• REVIEW:
• A TRIAC acts much like two SCRs connected back-to-back for bidirectional (AC) operation.

CHAPTER 7. THYRISTORS

344

• TRIAC controls are more often seen in simple, low-power circuits than complex, highpower circuits. In large power control circuits, multiple SCRs tend to be favored.
• When used to control AC power to a load, TRIACs are often accompanied by DIACs connected in series with their gate terminals. The DIAC helps the TRIAC fire more symmetrically (more consistently from one polarity to another).
• Main terminals 1 and 2 on a TRIAC are not interchangeable.
• To successfully trigger a TRIAC, gate current must come from the main terminal 2 (MT2 )
side of the circuit!

7.7

Optothyristors

Like bipolar transistors, SCRs and TRIACs are also manufactured as light-sensitive devices,
the action of impinging light replacing the function of triggering voltage.
Optically-controlled SCRs are often known by the acronym LASCR, or Light Activated
SCR. Its symbol, not surprisingly, looks like Figure 7.44.

Light Activated SCR

LASCR
Figure 7.44: Light activated SCR
Optically-controlled TRIACs don’t receive the honor of having their own acronym, but instead are humbly known as opto-TRIACs. Their schematic symbol is shown in Figure 7.45.

Opto-TRIAC

Figure 7.45: Opto-TRIAC
Optothyristors (a general term for either the LASCR or the opto-TRIAC) are commonly
found inside sealed ”optoisolator” modules.

7.8

The Unijunction Transistor (UJT)

Unijunction transistor: Although a unijunction transistor is not a thyristor, this device can
trigger larger thyristors with a pulse at base B1. A unijunction transistor is composed of a bar

7.8. THE UNIJUNCTION TRANSISTOR (UJT)

345

of N-type silicon having a P-type connection in the middle. See Figure 7.46(a). The connections
at the ends of the bar are known as bases B1 and B2; the P-type mid-point is the emitter.
With the emitter disconnected, the total resistance RBBO , a datasheet item, is the sum of RB1
and RB2 as shown in Figure 7.46(b). RBBO ranges from 4-12kΩ for different device types. The
intrinsic standoff ratio η is the ratio of RB1 to RBBO . It varies from 0.4 to 0.8 for different
devices. The schematic symbol is Figure 7.46(c)

B2

B2
RBB0 = RB1 + RB2

N

RB2

E P

η=
RB1

B1
(a)

B1
(b)

η=

RB1
RB1 + RB2

B2
E

B1

RB1
RBB0
(c)

Figure 7.46: Unijunction transistor: (a) Construction, (b) Model, (c) Symbol
The Unijunction emitter current vs voltage characteristic curve (Figure 7.47(a) ) shows
that as VE increases, current IE increases up IP at the peak point. Beyond the peak point,
current increases as voltage decreases in the negative resistance region. The voltage reaches a
minimum at the valley point. The resistance of RB1 , the saturation resistance is lowest at the
valley point.
IP and IV , are datasheet parameters; For a 2n2647, IP and IV are 2µA and 4mA, respectively. [5] VP is the voltage drop across RB1 plus a 0.7V diode drop; see Figure 7.47(b). VV is
estimated to be approximately 10% of VBB .
The relaxation oscillator in Figure 7.48 is an application of the unijunction oscillator. RE
charges CE until the peak point. The unijunction emitter terminal has no effect on the capacitor until this point is reached. Once the capacitor voltage, VE , reaches the peak voltage
point VP , the lowered emitter-base1 E-B1 resistance quickly discharges the capacitor. Once
the capacitor discharges below the valley point VV , the E-RB1 resistance reverts back to high
resistance, and the capacitor is free to charge again.
During capacitor discharge through the E-B1 saturation resistance, a pulse may be seen on
the external B1 and B2 load resistors, Figure 7.48. The load resistor at B1 needs to be low to
not affect the discharge time. The external resistor at B2 is optional. It may be replaced by a
short circuit. The approximate frequency is given by 1/f = T = RC. A more accurate expression
for frequency is given in Figure 7.48.
The charging resistor RE must fall within certain limits. It must be small enough to allow
IP to flow based on the VBB supply less VP . It must be large enough to supply IV based on the
VBB supply less VV . [6] The equations and an example for a 2n2647:

CHAPTER 7. THYRISTORS

346

VE

B2
ne
ga

VP

RE

e
tiv
ta
sis
nc

VP

a

r
satu

e

VV

tion

RB2

0.7V
+ -

re

Peak
point

+
ηVBB
-

IV
(a)

RB1

-

B1

Valley point
IP

+

VBB

VP = 0.7 + ηVBB

IE

VV ≈ 0.10(VBB )

(b)

Figure 7.47: Unijunction transistor: (a) emitter characteristic curve, (b) model for VP.

VBB
RE
100k

VBB

10V
470Ω

VRE

RE
100k

E

B1
47Ω

470Ω
B2

B2
CE
10nF

10V

VCE

E

B1

CE
10nF

VRB1

2n2647 RBBO = 4.7— 9.1k η = 0.68—0.82 IV = 8mA IP = 2µA
f=

1
1
= 1.39kHz
RC ln(1/(1- η)) = (100k)(10nF) ln(1/(1- 0.75))

Figure 7.48: Unijunction transistor relaxation oscillator and waveforms. Oscillator drives SCR.

7.8. THE UNIJUNCTION TRANSISTOR (UJT)

347

2n2647 RBBO = 4.7— 9.1k η = 0.68—0.82 IV = 8mA IP = 2µA
VP = 0.7 + ηVBB

VP = 0.7 + 0.75(10) = 8.2V

VV = 0.10(VBB )

VV = 0.10(10) = 1V

VBB - VV
< RE <
IV

VBB - VP
IP

10 - 1
< RE <
8mA

10 - 8.2
2µA

1.125k < RE <

900k

Programmable Unijunction Transistor (PUT): Although the unijunction transistor is
listed as obsolete (read expensive if obtainable), the programmable unijunction transistor is
alive and well. It is inexpensive and in production. Though it serves a function similar to the
unijunction transistor, the PUT is a three terminal thyristor. The PUT shares the four-layer
structure typical of thyristors shown in Figure 7.49. Note that the gate, an N-type layer near
the anode, is known as an “anode gate”. Moreover, the gate lead on the schematic symbol is
attached to the anode end of the symbol.

VA
VP

A
P
G

VV

N

G

A

P
IA
IP

IV

N

K

K

Figure 7.49: Programmable unijunction transistor: Characteristic curve, internal construction,
schematic symbol.
The characteristic curve for the programmable unijunction transistor in Figure 7.49 is similar to that of the unijunction transistor. This is a plot of anode current IA versus anode
voltage VA . The gate lead voltage sets, programs, the peak anode voltage VP . As anode current inceases, voltage increases up to the peak point. Thereafter, increasing current results in
decreasing voltage, down to the valley point.
The PUT equivalent of the unijunction transistor is shown in Figure 7.50. External PUT
resistors R1 and R2 replace unijunction transistor internal resistors RB1 and RB2 , respectively.
These resistors allow the calculation of the intrinsic standoff ratio η.
Figure 7.51 shows the PUT version of the unijunction relaxation oscillator Figure 7.48.
Resistor R charges the capacitor until the peak point, Figure 7.49, then heavy conduction
moves the operating point down the negative resistance slope to the valley point. A current
spike flows through the cathode during capacitor discharge, developing a voltage spike across
the cathode resistors. After capacitor discharge, the operating point resets back to the slope up

CHAPTER 7. THYRISTORS

348

E

B2
E

A

G

B2

RBB0 = R1 + R2

R2

η=

VS

VS = ηVBB

R1

VP = VT + VS

B1
K

B1
Unijunction

PUT equivalent

R1
R1 + R2

RG =

R1⋅R2
R1 + R2

Figure 7.50: PUT equivalent of unijunction transistor
to the peak point.

VBB
R

10V
R2

A
K

C
47Ω

G

VG

VP
VC

VRK

0V

R1
VRK

Figure 7.51: PUT relaxation oscillator
Problem: What is the range of suitable values for R in Figure 7.51, a relaxation oscillator?
The charging resistor must be small enough to supply enough current to raise the anode to VP
the peak point (Figure 7.49) while charging the capacitor. Once VP is reached, anode voltage
decreases as current increases (negative resistance), which moves the operating point to the
valley. It is the job of the capacitor to supply the valley current IV . Once it is discharged,
the operating point resets back to the upward slope to the peak point. The resistor must be
large enough so that it will never supply the high valley current IP . If the charging resistor
ever could supply that much current, the resistor would supply the valley current after the
capacitor was discharged and the operating point would never reset back to the high resistance
condition to the left of the peak point.
We select the same VBB =10V used for the unijunction transistor example. We select values
of R1 and R2 so that η is about 2/3. We calculate η and VS . The parallel equivalent of R1, R2 is
RG , which is only used to make selections from Table 7.1. Along with VS =10, the closest value

7.8. THE UNIJUNCTION TRANSISTOR (UJT)

349

to our 6.3, we find VT =0.6V, in Table 7.1 and calculate VP .

R1 = 27k R2 = 16k
η=

R1

η=

27

= 0.6279
27 + 16
VS = 0.6279(10) = 6.279V

R1 + R2

VS = ηVBB
RG =

VBB = 10V

R1⋅R2
R1 + R2

RG =

27k⋅16k
= 10k
27k + 16k

For RG=10k and VS=10V, VT = 0.6V
VP = 0.6 + 6.3 = 6.9V

VP = VT + VS

We also find IP and IV , the peak and valley currents, respectively in Table 7.1. We still need
VV , the valley voltage. We used 10% of VBB = 1V, in the previous unijunction example. Consulting the datasheet, we find the forward voltage VF =0.8V at IF =50mA. The valley current
IV =70µA is much less than IF =50mA. Therefore, VV must be less than VF =0.8V. How much
less? To be safe we set VV =0V. This will raise the lower limit on the resistor range a little.

For RG=10k and VS=10V, IP = 4.0µA
For RG=10k and VS=10V, IV = 70µA
VV = 0.10(VBB ) not used

VV = 0V

VBB - VV
< RE <
IV

10 - 0
< RE <
70µA

10 - 6.9
4µA

143k < RE <

755k

VBB - VP
IP

Choosing R > 143k guarantees that the operating point can reset from the valley point after
capacitor discharge. R < 755k allows charging up to VP at the peak point.
Table 7.1: Selected 2n6027 PUT parameters, adapted from 2n6027 datasheet. [4]
Parameter Conditions
min typical max units
VT
V
VS =10V, RG =1Meg 0.2
0.7
1.6
VS =10V, RG =10k
0.2
0.35
0.6
IP
µA
VS =10V, RG =1Meg 1.25
2.0
VS =10V, RG =10k
4.0
5.0
IV
µA
VS =10V, RG =1Meg 18
50
VS =10V, RG =10k
70
150
VS =10V, RG =200Ω 1500 VF
IF =50mA
0.8
1.5
V
Figure 7.52 show the PUT relaxation oscillator with the final resistor values. A practical

CHAPTER 7. THYRISTORS

350

application of a PUT triggering an SCR is also shown. This circuit needs a VBB unfiltered
supply (not shown) divided down from the bridge rectifier to reset the relaxation oscillator
after each power zero crossing. The variable resistor should have a minimum resistor in series
with it to prevent a low pot setting from hanging at the valley point.

VBB
R
270k

A

10V

G

C
3.7nF
47Ω

K

VRK

VBB
R2
16k

R
270k
A

VG

C

R1
27k

33
nF

G

R2
16k
VG

K
PUT

R1
27k

SCR

Figure 7.52: PUT relaxation oscillator with component values. PUT drives SCR lamp dimmer.
PUT timing circuits are said to be usable to 10kHz. If a linear ramp is required instead of
an exponential ramp, replace the charging resistor with a constant current source such as a
FET based constant current diode. A substitute PUT may be built from a PNP and NPN silicon
transistor as shown for the SCS equivalent circuit in Figure 7.53 by omitting the cathode gate
and using the anode gate.
• REVIEW:
• A unijunction transistor consists of two bases (B1, B2) attached to a resistive bar of silicon, and an emitter in the center. The E-B1 junction has negative resistance properties;
it can switch between high and low resistance.
• A PUT (programmable unijunction transistor) is a 3-terminal 4-layer thyristor acting like
a unijunction transistor. An external resistor network “programs” η.
• The intrinsic standoff ratio is η=R1/(R1+R2) for a PUT; substitute RB1 and RB2 , respectively, for a unijunction transistor. The trigger voltage is determined by η.
• Unijunction transistors and programmable unijunction transistors are applied to oscillators, timing circuits, and thyristor triggering.

7.9

The Silicon-Controlled Switch (SCS)

If we take the equivalent circuit for an SCR and add another external terminal, connected to
the base of the top transistor and the collector of the bottom transistor, we have a device known
as a silicon-controlled-switch, or SCS: (Figure 7.53)

7.9. THE SILICON-CONTROLLED SWITCH (SCS)
Anode

351

Anode
P
N

Cathode
Gate

P

N

Anode
Gate

Anode
Gate

P

Cathode
Gate

N
Cathode

Anode
Anode
Gate
Cathode
Gate
Cathode

Cathode

Physical diagram

Equivalent schematic

Schematic symbol

Figure 7.53: The Silicon-Controlled Switch(SCS)
This extra terminal allows more control to be exerted over the device, particularly in the
mode of forced commutation, where an external signal forces it to turn off while the main
current through the device has not yet fallen below the holding current value. Note that the
motor is in the anode gate circuit in Figure 7.54. This is correct, although it doesn’t look right.
The anode lead is required to switch the SCS off. Therefore the motor cannot be in series with
the anode.

Motor

Motor

R1

R1

R2

R2
+

SCS
+

on
SCS

−
off

on

−
off

Figure 7.54: SCS: Motor start/stop circuit, equivalent circuit with two transistors.
When the ”on” pushbutton switch is actuated, the voltage applied between the cathode gate
and the cathode, forward-biases the lower transistor’s base-emitter junction, and turning it on.
The top transistor of the SCS is ready to conduct, having been supplied with a current path
from its emitter terminal (the SCS’s anode terminal) through resistor R2 to the positive side of
the power supply. As in the case of the SCR, both transistors turn on and maintain each other
in the ”on” mode. When the lower transistor turns on, it conducts the motor’s load current, and
the motor starts and runs.
The motor may be stopped by interrupting the power supply, as with an SCR, and this is
called natural commutation. However, the SCS provides us with another means of turning

CHAPTER 7. THYRISTORS

352

off: forced commutation by shorting the anode terminal to the cathode. [3] If this is done
(by actuating the ”off ” pushbutton switch), the upper transistor within the SCS will lose its
emitter current, thus halting current through the base of the lower transistor. When the lower
transistor turns off, it breaks the circuit for base current through the top transistor (securing
its ”off ” state), and the motor (making it stop). The SCS will remain in the off condition until
such time that the ”on” pushbutton switch is re-actuated.
• REVIEW:
• A silicon-controlled switch, or SCS, is essentially an SCR with an extra gate terminal.
• Typically, the load current through an SCS is carried by the anode gate and cathode
terminals, with the cathode gate and anode terminals sufficing as control leads.
• An SCS is turned on by applying a positive voltage between the cathode gate and cathode
terminals. It may be turned off (forced commutation) by applying a negative voltage
between the anode and cathode terminals, or simply by shorting those two terminals
together. The anode terminal must be kept positive with respect to the cathode in order
for the SCS to latch.

7.10

Field-effect-controlled thyristors

Two relatively recent technologies designed to reduce the ”driving” (gate trigger current) requirements of classic thyristor devices are the MOS-gated thyristor and the MOS Controlled
Thyristor, or MCT.
The MOS-gated thyristor uses a MOSFET to initiate conduction through the upper (PNP)
transistor of a standard thyristor structure, thus triggering the device. Since a MOSFET requires negligible current to ”drive” (cause it to saturate), this makes the thyristor as a whole
very easy to trigger: (Figure 7.55)

MOS-gated thyristor
equivalent circuit

Anode

Gate
Cathode
Figure 7.55: MOS-gated thyristor equivalent circuit
Given the fact that ordinary SCRs are quite easy to ”drive” as it is, the practical advantage
of using an even more sensitive device (a MOSFET) to initiate triggering is debatable. Also,

7.10. FIELD-EFFECT-CONTROLLED THYRISTORS

353

placing a MOSFET at the gate input of the thyristor now makes it impossible to turn it off by a
reverse-triggering signal. Only low-current dropout can make this device stop conducting after
it has been latched.
A device of arguably greater value would be a fully-controllable thyristor, whereby a small
gate signal could both trigger the thyristor and force it to turn off. Such a device does exist, and
it is called the MOS Controlled Thyristor, or MCT. It uses a pair of MOSFETs connected to a
common gate terminal, one to trigger the thyristor and the other to ”untrigger” it: (Figure 7.56)

MOS Controlled Thyristor
(MCT) equivalent circuit

Anode

Gate

Cathode
Figure 7.56: MOS-controlled thyristor (MCT) equivalent circuit
A positive gate voltage (with respect to the cathode) turns on the upper (N-channel) MOSFET, allowing base current through the upper (PNP) transistor, which latches the transistor
pair in an ”on” state. Once both transistors are fully latched, there will be little voltage dropped
between anode and cathode, and the thyristor will remain latched as long as the controlled
current exceeds the minimum (holding) current value. However, if a negative gate voltage is
applied (with respect to the anode, which is at nearly the same voltage as the cathode in the
latched state), the lower MOSFET will turn on and ”short” between the lower (NPN) transistor’s base and emitter terminals, thus forcing it into cutoff. Once the NPN transistor cuts off,
the PNP transistor will drop out of conduction, and the whole thyristor turns off. Gate voltage
has full control over conduction through the MCT: to turn it on and to turn it off.
This device is still a thyristor, though. If zero voltage is applied between gate and cathode,
neither MOSFET will turn on. Consequently, the bipolar transistor pair will remain in whatever state it was last in (hysteresis). So, a brief positive pulse to the gate turns the MCT on, a
brief negative pulse forces it off, and no applied gate voltage lets it remain in whatever state
it is already in. In essence, the MCT is a latching version of the IGBT (Insulated Gate Bipolar
Transistor).
• REVIEW:

354

CHAPTER 7. THYRISTORS

• A MOS-gated thyristor uses an N-channel MOSFET to trigger a thyristor, resulting in an
extremely low gate current requirement.
• A MOS Controlled Thyristor, or MCT, uses two MOSFETS to exert full control over the
thyristor. A positive gate voltage triggers the device; a negative gate voltage forces it
to turn off. Zero gate voltage allows the thyristor to remain in whatever state it was
previously in (off, or latched on).

Bibliography
[1] “Phattytron PT-1 Vacuum Tube Synthesizer”, The Audio Playground Synthesizer Museum at http://www.keyboardmuseum.com/ar/m/meta/pt1.html
[2] “At last, a pitch source with tube power”, METASONIX, PMB 109, 881
11th Street, Lakeport CA 95453 USA at http://www.metasonix.com/i
ndex.php?option=com content&task=view&id=14&Itemid=31
[3] “Silicon Contolled Switches”, GE Transistor Manual, The General Electric Company,
1964, Figure 16.19(M).
[4] “2N6027,
2N6028 Programmable Unijunction Transistor ”,
http://www.onsemi.com/pub link/Collateral/2N6027-D.PDF

datasheet

at

[5] “Unijunction
Transistor
”,
American
Microsemiconductor,
http://www.americanmicrosemi.com/tutorials/unijunction.htm

at

[6] Matthew
H.
Williams,
“Unijunction
http://baec.tripod.com/DEC90/uni tran.htm
http://baec.tripod.com/DEC90/uni tran.htm

at
by

Transistor
”,
Unijunction
Transistor

Chapter 8

OPERATIONAL AMPLIFIERS
Contents
8.1
8.2
8.3
8.4
8.5
8.6
8.7
8.8
8.9
8.10
8.11
8.12
8.13

Introduction . . . . . . . . . . . . . . . . . .
Single-ended and differential amplifiers
The ”operational” amplifier . . . . . . . .
Negative feedback . . . . . . . . . . . . . .
Divided feedback . . . . . . . . . . . . . . .
An analogy for divided feedback . . . . .
Voltage-to-current signal conversion . .
Averager and summer circuits . . . . . .
Building a differential amplifier . . . . .
The instrumentation amplifier . . . . . .
Differentiator and integrator circuits . .
Positive feedback . . . . . . . . . . . . . . .
Practical considerations . . . . . . . . . .
8.13.1 Common-mode gain . . . . . . . . . . .
8.13.2 Offset voltage . . . . . . . . . . . . . .
8.13.3 Bias current . . . . . . . . . . . . . . .
8.13.4 Drift . . . . . . . . . . . . . . . . . . .
8.13.5 Frequency response . . . . . . . . . . .
8.13.6 Input to output phase shift . . . . . . .
8.14 Operational amplifier models . . . . . . .
8.15 Data . . . . . . . . . . . . . . . . . . . . . . .

8.1

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355
356
360
366
369
372
378
380
382
384
385
388
392
393
396
398
404
404
405
408
413

Introduction

The operational amplifier is arguably the most useful single device in analog electronic circuitry. With only a handful of external components, it can be made to perform a wide variety
355

CHAPTER 8. OPERATIONAL AMPLIFIERS

356

of analog signal processing tasks. It is also quite affordable, most general-purpose amplifiers
selling for under a dollar apiece. Modern designs have been engineered with durability in
mind as well: several ”op-amps” are manufactured that can sustain direct short-circuits on
their outputs without damage.
One key to the usefulness of these little circuits is in the engineering principle of feedback,
particularly negative feedback, which constitutes the foundation of almost all automatic control
processes. The principles presented here in operational amplifier circuits, therefore, extend
well beyond the immediate scope of electronics. It is well worth the electronics student’s time
to learn these principles and learn them well.

8.2

Single-ended and differential amplifiers

For ease of drawing complex circuit diagrams, electronic amplifiers are often symbolized by a
simple triangle shape, where the internal components are not individually represented. This
symbology is very handy for cases where an amplifier’s construction is irrelevant to the greater
function of the overall circuit, and it is worthy of familiarization:

General amplifier circuit symbol
+Vsupply
Input

Output
-Vsupply

The +V and -V connections denote the positive and negative sides of the DC power supply,
respectively. The input and output voltage connections are shown as single conductors, because
it is assumed that all signal voltages are referenced to a common connection in the circuit called
ground. Often (but not always!), one pole of the DC power supply, either positive or negative,
is that ground reference point. A practical amplifier circuit (showing the input voltage source,
load resistance, and power supply) might look like this:

+V
Input

+
Output

30 V

Vinput
-V

Rload

-

Without having to analyze the actual transistor design of the amplifier, you can readily
discern the whole circuit’s function: to take an input signal (Vin ), amplify it, and drive a load

8.2. SINGLE-ENDED AND DIFFERENTIAL AMPLIFIERS

357

resistance (Rload ). To complete the above schematic, it would be good to specify the gains of
that amplifier (AV , AI , AP ) and the Q (bias) point for any needed mathematical analysis.
If it is necessary for an amplifier to be able to output true AC voltage (reversing polarity)
to the load, a split DC power supply may be used, whereby the ground point is electrically
”centered” between +V and -V. Sometimes the split power supply configuration is referred to
as a dual power supply.

+V

+

Input

Output

15 V

Vinput
Rload
-V

+

15 V
The amplifier is still being supplied with 30 volts overall, but with the split voltage DC
power supply, the output voltage across the load resistor can now swing from a theoretical
maximum of +15 volts to -15 volts, instead of +30 volts to 0 volts. This is an easy way to
get true alternating current (AC) output from an amplifier without resorting to capacitive or
inductive (transformer) coupling on the output. The peak-to-peak amplitude of this amplifier’s
output between cutoff and saturation remains unchanged.
By signifying a transistor amplifier within a larger circuit with a triangle symbol, we ease
the task of studying and analyzing more complex amplifiers and circuits. One of these more
complex amplifier types that we’ll be studying is called the differential amplifier. Unlike normal amplifiers, which amplify a single input signal (often called single-ended amplifiers), differential amplifiers amplify the voltage difference between two input signals. Using the simplified
triangle amplifier symbol, a differential amplifier looks like this:

Differential amplifier
+Vsupply
Input1

−

Input2

+

Output
-Vsupply

The two input leads can be seen on the left-hand side of the triangular amplifier symbol, the
output lead on the right-hand side, and the +V and -V power supply leads on top and bottom.
As with the other example, all voltages are referenced to the circuit’s ground point. Notice that
one input lead is marked with a (-) and the other is marked with a (+). Because a differential
amplifier amplifies the difference in voltage between the two inputs, each input influences the

CHAPTER 8. OPERATIONAL AMPLIFIERS

358

output voltage in opposite ways. Consider the following table of input/output voltages for a
differential amplifier with a voltage gain of 4:
(-) Input1

0

0

0

0

1

2.5

7

3

-3

-2

(+) Input2

0

1

2.5

7

0

0

0

3

3

-7

Output

0

4

10

28

-4

-10

-28

0

24

-20

Voltage output equation: Vout = AV(Input2 - Input1)
or
Vout = AV(Input(+) - Input(-))

An increasingly positive voltage on the (+) input tends to drive the output voltage more
positive, and an increasingly positive voltage on the (-) input tends to drive the output voltage
more negative. Likewise, an increasingly negative voltage on the (+) input tends to drive the
output negative as well, and an increasingly negative voltage on the (-) input does just the
opposite. Because of this relationship between inputs and polarities, the (-) input is commonly
referred to as the inverting input and the (+) as the noninverting input.
It may be helpful to think of a differential amplifier as a variable voltage source controlled
by a sensitive voltmeter, as such:

+V

G
+
+

-V

Bear in mind that the above illustration is only a model to aid in understanding the behavior of a differential amplifier. It is not a realistic schematic of its actual design. The ”G” symbol
represents a galvanometer, a sensitive voltmeter movement. The potentiometer connected between +V and -V provides a variable voltage at the output pin (with reference to one side of
the DC power supply), that variable voltage set by the reading of the galvanometer. It must be
understood that any load powered by the output of a differential amplifier gets its current from
the DC power source (battery), not the input signal. The input signal (to the galvanometer)
merely controls the output.
This concept may at first be confusing to students new to amplifiers. With all these polarities and polarity markings (- and +) around, its easy to get confused and not know what the
output of a differential amplifier will be. To address this potential confusion, here’s a simple
rule to remember:

8.2. SINGLE-ENDED AND DIFFERENTIAL AMPLIFIERS

Differential
input voltage

Differential
input voltage

-

−

+

+

+

−

359

+

Output
- voltage

+

-

+

Output
voltage

When the polarity of the differential voltage matches the markings for inverting and noninverting inputs, the output will be positive. When the polarity of the differential voltage clashes
with the input markings, the output will be negative. This bears some similarity to the mathematical sign displayed by digital voltmeters based on input voltage polarity. The red test lead
of the voltmeter (often called the ”positive” lead because of the color red’s popular association
with the positive side of a power supply in electronic wiring) is more positive than the black,
the meter will display a positive voltage figure, and vice versa:

Differential
input voltage

Differential
input voltage

blk

-

-

6V
+

+

red

+

blk

-

6V
-

red

+

+ 6.00 V
Digital Voltmeter

- 6.00 V
Digital Voltmeter

Just as a voltmeter will only display the voltage between its two test leads, an ideal differential amplifier only amplifies the potential difference between its two input connections, not
the voltage between any one of those connections and ground. The output polarity of a differential amplifier, just like the signed indication of a digital voltmeter, depends on the relative
polarities of the differential voltage between the two input connections.
If the input voltages to this amplifier represented mathematical quantities (as is the case
within analog computer circuitry), or physical process measurements (as is the case within
analog electronic instrumentation circuitry), you can see how a device such as a differential
amplifier could be very useful. We could use it to compare two quantities to see which is
greater (by the polarity of the output voltage), or perhaps we could compare the difference
between two quantities (such as the level of liquid in two tanks) and flag an alarm (based on the
absolute value of the amplifier output) if the difference became too great. In basic automatic
control circuitry, the quantity being controlled (called the process variable) is compared with
a target value (called the setpoint), and decisions are made as to how to act based on the
discrepancy between these two values. The first step in electronically controlling such a scheme

CHAPTER 8. OPERATIONAL AMPLIFIERS

360

is to amplify the difference between the process variable and the setpoint with a differential
amplifier. In simple controller designs, the output of this differential amplifier can be directly
utilized to drive the final control element (such as a valve) and keep the process reasonably
close to setpoint.
• REVIEW:
• A ”shorthand” symbol for an electronic amplifier is a triangle, the wide end signifying
the input side and the narrow end signifying the output. Power supply lines are often
omitted in the drawing for simplicity.
• To facilitate true AC output from an amplifier, we can use what is called a split or dual
power supply, with two DC voltage sources connected in series with the middle point
grounded, giving a positive voltage to ground (+V) and a negative voltage to ground (-V).
Split power supplies like this are frequently used in differential amplifier circuits.
• Most amplifiers have one input and one output. Differential amplifiers have two inputs
and one output, the output signal being proportional to the difference in signals between
the two inputs.
• The voltage output of a differential amplifier is determined by the following equation:
Vout = AV (Vnoninv - Vinv )

8.3

The ”operational” amplifier

Long before the advent of digital electronic technology, computers were built to electronically
perform calculations by employing voltages and currents to represent numerical quantities.
This was especially useful for the simulation of physical processes. A variable voltage, for instance, might represent velocity or force in a physical system. Through the use of resistive
voltage dividers and voltage amplifiers, the mathematical operations of division and multiplication could be easily performed on these signals.
The reactive properties of capacitors and inductors lend themselves well to the simulation
of variables related by calculus functions. Remember how the current through a capacitor
was a function of the voltage’s rate of change, and how that rate of change was designated
in calculus as the derivative? Well, if voltage across a capacitor were made to represent the
velocity of an object, the current through the capacitor would represent the force required to
accelerate or decelerate that object, the capacitor’s capacitance representing the object’s mass:
iC = C dv
F = m dv
dt
dt
Where,
iC = Instantaneous current
through capacitor
C = Capacitance in farads
dv = Rate of change of
dt
voltage over time

Where,
F = Force applied to object
m = Mass of object
dv = Rate of change of
dt
velocity over time

8.3. THE ”OPERATIONAL” AMPLIFIER

361

This analog electronic computation of the calculus derivative function is technically known
as differentiation, and it is a natural function of a capacitor’s current in relation to the voltage
applied across it. Note that this circuit requires no ”programming” to perform this relatively
advanced mathematical function as a digital computer would.

Electronic circuits are very easy and inexpensive to create compared to complex physical
systems, so this kind of analog electronic simulation was widely used in the research and
development of mechanical systems. For realistic simulation, though, amplifier circuits of high
accuracy and easy configurability were needed in these early computers.

It was found in the course of analog computer design that differential amplifiers with extremely high voltage gains met these requirements of accuracy and configurability better than
single-ended amplifiers with custom-designed gains. Using simple components connected to
the inputs and output of the high-gain differential amplifier, virtually any gain and any function could be obtained from the circuit, overall, without adjusting or modifying the internal
circuitry of the amplifier itself. These high-gain differential amplifiers came to be known as
operational amplifiers, or op-amps, because of their application in analog computers’ mathematical operations.

Modern op-amps, like the popular model 741, are high-performance, inexpensive integrated
circuits. Their input impedances are quite high, the inputs drawing currents in the range of
half a microamp (maximum) for the 741, and far less for op-amps utilizing field-effect input
transistors. Output impedance is typically quite low, about 75 Ω for the model 741, and many
models have built-in output short circuit protection, meaning that their outputs can be directly
shorted to ground without causing harm to the internal circuitry. With direct coupling between
op-amps’ internal transistor stages, they can amplify DC signals just as well as AC (up to
certain maximum voltage-risetime limits). It would cost far more in money and time to design
a comparable discrete-transistor amplifier circuit to match that kind of performance, unless
high power capability was required. For these reasons, op-amps have all but obsoleted discretetransistor signal amplifiers in many applications.

The following diagram shows the pin connections for single op-amps (741 included) when
housed in an 8-pin DIP (Dual Inline Package) integrated circuit:

CHAPTER 8. OPERATIONAL AMPLIFIERS

362

Typical 8-pin "DIP" op-amp
integrated circuit

7

1

2

−

8

Offset
Output null
6

5

+

No
+V
connection

3

4

Offset
null

-V

Some models of op-amp come two to a package, including the popular models TL082 and
1458. These are called ”dual” units, and are typically housed in an 8-pin DIP package as well,
with the following pin connections:

Dual op-amp in 8-pin DIP
+V
8

7

6

5

−
+
+
−
1

2

3

4
-V

Operational amplifiers are also available four to a package, usually in 14-pin DIP arrangements. Unfortunately, pin assignments aren’t as standard for these ”quad” op-amps as they
are for the ”dual” or single units. Consult the manufacturer datasheet(s) for details.
Practical operational amplifier voltage gains are in the range of 200,000 or more, which

8.3. THE ”OPERATIONAL” AMPLIFIER

363

makes them almost useless as an analog differential amplifier by themselves. For an op-amp
with a voltage gain (AV ) of 200,000 and a maximum output voltage swing of +15V/-15V, all
it would take is a differential input voltage of 75 µV (microvolts) to drive it to saturation or
cutoff! Before we take a look at how external components are used to bring the gain down to a
reasonable level, let’s investigate applications for the ”bare” op-amp by itself.

One application is called the comparator. For all practical purposes, we can say that the
output of an op-amp will be saturated fully positive if the (+) input is more positive than the (-)
input, and saturated fully negative if the (+) input is less positive than the (-) input. In other
words, an op-amp’s extremely high voltage gain makes it useful as a device to compare two
voltages and change output voltage states when one input exceeds the other in magnitude.

+V
Vin

−

LED

+

-V

In the above circuit, we have an op-amp connected as a comparator, comparing the input
voltage with a reference voltage set by the potentiometer (R1 ). If Vin drops below the voltage
set by R1 , the op-amp’s output will saturate to +V, thereby lighting up the LED. Otherwise, if
Vin is above the reference voltage, the LED will remain off. If Vin is a voltage signal produced
by a measuring instrument, this comparator circuit could function as a ”low” alarm, with the
trip-point set by R1 . Instead of an LED, the op-amp output could drive a relay, a transistor, an
SCR, or any other device capable of switching power to a load such as a solenoid valve, to take
action in the event of a low alarm.

Another application for the comparator circuit shown is a square-wave converter. Suppose
that the input voltage applied to the inverting (-) input was an AC sine wave rather than a
stable DC voltage. In that case, the output voltage would transition between opposing states
of saturation whenever the input voltage was equal to the reference voltage produced by the
potentiometer. The result would be a square wave:

CHAPTER 8. OPERATIONAL AMPLIFIERS

364

+V
−
Vin

Vout

+

-V
Vin

Vout

Adjustments to the potentiometer setting would change the reference voltage applied to
the noninverting (+) input, which would change the points at which the sine wave would cross,
changing the on/off times, or duty cycle of the square wave:

+V
−
Vin

Vout

+

-V
Vin

Vout

It should be evident that the AC input voltage would not have to be a sine wave in particular
for this circuit to perform the same function. The input voltage could be a triangle wave,
sawtooth wave, or any other sort of wave that ramped smoothly from positive to negative to
positive again. This sort of comparator circuit is very useful for creating square waves of
varying duty cycle. This technique is sometimes referred to as pulse-width modulation, or
PWM (varying, or modulating a waveform according to a controlling signal, in this case the
signal produced by the potentiometer).
Another comparator application is that of the bargraph driver. If we had several op-amps

8.3. THE ”OPERATIONAL” AMPLIFIER

365

connected as comparators, each with its own reference voltage connected to the inverting input,
but each one monitoring the same voltage signal on their noninverting inputs, we could build a
bargraph-style meter such as what is commonly seen on the face of stereo tuners and graphic
equalizers. As the signal voltage (representing radio signal strength or audio sound level)
increased, each comparator would ”turn on” in sequence and send power to its respective LED.
With each comparator switching ”on” at a different level of audio sound, the number of LED’s
illuminated would indicate how strong the signal was.

+V

Simple bargraph driver circuit

−

LED4

+
−

LED3

+
−

LED2

+
−

LED1

+
-V

-V
Vin

In the circuit shown above, LED1 would be the first to light up as the input voltage increased
in a positive direction. As the input voltage continued to increase, the other LED’s would
illuminate in succession, until all were lit.
This very same technology is used in some analog-to-digital signal converters, namely the
flash converter, to translate an analog signal quantity into a series of on/off voltages representing a digital number.
• REVIEW:
• A triangle shape is the generic symbol for an amplifier circuit, the wide end signifying
the input and the narrow end signifying the output.
• Unless otherwise specified, all voltages in amplifier circuits are referenced to a common
ground point, usually connected to one terminal of the power supply. This way, we can
speak of a certain amount of voltage being ”on” a single wire, while realizing that voltage
is always measured between two points.

CHAPTER 8. OPERATIONAL AMPLIFIERS

366

• A differential amplifier is one amplifying the voltage difference between two signal inputs.
In such a circuit, one input tends to drive the output voltage to the same polarity of the
input signal, while the other input does just the opposite. Consequently, the first input is
called the noninverting (+) input and the second is called the inverting (-) input.
• An operational amplifier (or op-amp for short) is a differential amplifier with an extremely
high voltage gain (AV = 200,000 or more). Its name hails from its original use in analog
computer circuitry (performing mathematical operations).
• Op-amps typically have very high input impedances and fairly low output impedances.
• Sometimes op-amps are used as signal comparators, operating in full cutoff or saturation
mode depending on which input (inverting or noninverting) has the greatest voltage.
Comparators are useful in detecting ”greater-than” signal conditions (comparing one to
the other).
• One comparator application is called the pulse-width modulator, and is made by comparing a sine-wave AC signal against a DC reference voltage. As the DC reference voltage
is adjusted, the square-wave output of the comparator changes its duty cycle (positive
versus negative times). Thus, the DC reference voltage controls, or modulates the pulse
width of the output voltage.

8.4

Negative feedback

If we connect the output of an op-amp to its inverting input and apply a voltage signal to
the noninverting input, we find that the output voltage of the op-amp closely follows that
input voltage (I’ve neglected to draw in the power supply, +V/-V wires, and ground symbol for
simplicity):

−
Vin

+

Vout

As Vin increases, Vout will increase in accordance with the differential gain. However, as
Vout increases, that output voltage is fed back to the inverting input, thereby acting to decrease
the voltage differential between inputs, which acts to bring the output down. What will happen
for any given voltage input is that the op-amp will output a voltage very nearly equal to Vin ,
but just low enough so that there’s enough voltage difference left between Vin and the (-) input
to be amplified to generate the output voltage.
The circuit will quickly reach a point of stability (known as equilibrium in physics), where
the output voltage is just the right amount to maintain the right amount of differential, which
in turn produces the right amount of output voltage. Taking the op-amp’s output voltage and
coupling it to the inverting input is a technique known as negative feedback, and it is the key
to having a self-stabilizing system (this is true not only of op-amps, but of any dynamic system
in general). This stability gives the op-amp the capacity to work in its linear (active) mode, as

8.4. NEGATIVE FEEDBACK

367

opposed to merely being saturated fully ”on” or ”off ” as it was when used as a comparator, with
no feedback at all.
Because the op-amp’s gain is so high, the voltage on the inverting input can be maintained
almost equal to Vin . Let’s say that our op-amp has a differential voltage gain of 200,000. If Vin
equals 6 volts, the output voltage will be 5.999970000149999 volts. This creates just enough
differential voltage (6 volts - 5.999970000149999 volts = 29.99985 µV) to cause 5.999970000149999
volts to be manifested at the output terminal, and the system holds there in balance. As you
can see, 29.99985 µV is not a lot of differential, so for practical calculations, we can assume
that the differential voltage between the two input wires is held by negative feedback exactly
at 0 volts.

The effects of negative feedback
−
29.99985 µV

+
5.999970000149999 V

6V

The effects of negative feedback
(rounded figures)
−
0V

+

6V

6V

One great advantage to using an op-amp with negative feedback is that the actual voltage
gain of the op-amp doesn’t matter, so long as its very large. If the op-amp’s differential gain
were 250,000 instead of 200,000, all it would mean is that the output voltage would hold just
a little closer to Vin (less differential voltage needed between inputs to generate the required
output). In the circuit just illustrated, the output voltage would still be (for all practical purposes) equal to the non-inverting input voltage. Op-amp gains, therefore, do not have to be
precisely set by the factory in order for the circuit designer to build an amplifier circuit with

CHAPTER 8. OPERATIONAL AMPLIFIERS

368

precise gain. Negative feedback makes the system self-correcting. The above circuit as a whole
will simply follow the input voltage with a stable gain of 1.
Going back to our differential amplifier model, we can think of the operational amplifier
as being a variable voltage source controlled by an extremely sensitive null detector, the kind
of meter movement or other sensitive measurement device used in bridge circuits to detect a
condition of balance (zero volts). The ”potentiometer” inside the op-amp creating the variable
voltage will move to whatever position it must to ”balance” the inverting and noninverting
input voltages so that the ”null detector” has zero voltage across it:

+V

-

6V
0V

null
+

-V

6V

As the ”potentiometer” will move to provide an output voltage necessary to satisfy the ”null
detector” at an ”indication” of zero volts, the output voltage becomes equal to the input voltage:
in this case, 6 volts. If the input voltage changes at all, the ”potentiometer” inside the op-amp
will change position to hold the ”null detector” in balance (indicating zero volts), resulting in
an output voltage approximately equal to the input voltage at all times.
This will hold true within the range of voltages that the op-amp can output. With a power
supply of +15V/-15V, and an ideal amplifier that can swing its output voltage just as far, it
will faithfully ”follow” the input voltage between the limits of +15 volts and -15 volts. For this
reason, the above circuit is known as a voltage follower. Like its one-transistor counterpart,
the common-collector (”emitter-follower”) amplifier, it has a voltage gain of 1, a high input
impedance, a low output impedance, and a high current gain. Voltage followers are also known
as voltage buffers, and are used to boost the current-sourcing ability of voltage signals too weak
(too high of source impedance) to directly drive a load. The op-amp model shown in the last
illustration depicts how the output voltage is essentially isolated from the input voltage, so
that current on the output pin is not supplied by the input voltage source at all, but rather
from the power supply powering the op-amp.
It should be mentioned that many op-amps cannot swing their output voltages exactly to
+V/-V power supply rail voltages. The model 741 is one of those that cannot: when saturated,
its output voltage peaks within about one volt of the +V power supply voltage and within about
2 volts of the -V power supply voltage. Therefore, with a split power supply of +15/-15 volts,
a 741 op-amp’s output may go as high as +14 volts or as low as -13 volts (approximately), but
no further. This is due to its bipolar transistor design. These two voltage limits are known

8.5. DIVIDED FEEDBACK

369

as the positive saturation voltage and negative saturation voltage, respectively. Other op-amps,
such as the model 3130 with field-effect transistors in the final output stage, have the ability to
swing their output voltages within millivolts of either power supply rail voltage. Consequently,
their positive and negative saturation voltages are practically equal to the supply voltages.
• REVIEW:
• Connecting the output of an op-amp to its inverting (-) input is called negative feedback.
This term can be broadly applied to any dynamic system where the output signal is ”fed
back” to the input somehow so as to reach a point of equilibrium (balance).
• When the output of an op-amp is directly connected to its inverting (-) input, a voltage
follower will be created. Whatever signal voltage is impressed upon the noninverting (+)
input will be seen on the output.
• An op-amp with negative feedback will try to drive its output voltage to whatever level
necessary so that the differential voltage between the two inputs is practically zero. The
higher the op-amp differential gain, the closer that differential voltage will be to zero.
• Some op-amps cannot produce an output voltage equal to their supply voltage when saturated. The model 741 is one of these. The upper and lower limits of an op-amp’s output
voltage swing are known as positive saturation voltage and negative saturation voltage,
respectively.

8.5

Divided feedback

If we add a voltage divider to the negative feedback wiring so that only a fraction of the output
voltage is fed back to the inverting input instead of the full amount, the output voltage will be
a multiple of the input voltage (please bear in mind that the power supply connections to the
op-amp have been omitted once again for simplicity’s sake):

The effects of divided negative feedback
6 mA R1
1 kΩ
6V

0V

R2

6V
−
+

6V

6 mA

1 kΩ
12 V

All voltage figures shown in
reference to ground

CHAPTER 8. OPERATIONAL AMPLIFIERS

370

If R1 and R2 are both equal and Vin is 6 volts, the op-amp will output whatever voltage is
needed to drop 6 volts across R1 (to make the inverting input voltage equal to 6 volts, as well,
keeping the voltage difference between the two inputs equal to zero). With the 2:1 voltage
divider of R1 and R2 , this will take 12 volts at the output of the op-amp to accomplish.
Another way of analyzing this circuit is to start by calculating the magnitude and direction
of current through R1 , knowing the voltage on either side (and therefore, by subtraction, the
voltage across R1 ), and R1 ’s resistance. Since the left-hand side of R1 is connected to ground (0
volts) and the right-hand side is at a potential of 6 volts (due to the negative feedback holding
that point equal to Vin ), we can see that we have 6 volts across R1 . This gives us 6 mA of current
through R1 from left to right. Because we know that both inputs of the op-amp have extremely
high impedance, we can safely assume they won’t add or subtract any current through the
divider. In other words, we can treat R1 and R2 as being in series with each other: all of the
electrons flowing through R1 must flow through R2 . Knowing the current through R2 and the
resistance of R2 , we can calculate the voltage across R2 (6 volts), and its polarity. Counting up
voltages from ground (0 volts) to the right-hand side of R2 , we arrive at 12 volts on the output.
Upon examining the last illustration, one might wonder, ”where does that 6 mA of current
go?” The last illustration doesn’t show the entire current path, but in reality it comes from the
negative side of the DC power supply, through ground, through R1 , through R2 , through the
output pin of the op-amp, and then back to the positive side of the DC power supply through the
output transistor(s) of the op-amp. Using the null detector/potentiometer model of the op-amp,
the current path looks like this:

R1

R2

1 kΩ

1 kΩ
+V

null
+

-V

6V

The 6 volt signal source does not have to supply any current for the circuit: it merely
commands the op-amp to balance voltage between the inverting (-) and noninverting (+) input
pins, and in so doing produce an output voltage that is twice the input due to the dividing effect
of the two 1 kΩ resistors.
We can change the voltage gain of this circuit, overall, just by adjusting the values of R1
and R2 (changing the ratio of output voltage that is fed back to the inverting input). Gain can
be calculated by the following formula:

8.5. DIVIDED FEEDBACK

AV =

371

R2
+1
R1

Note that the voltage gain for this design of amplifier circuit can never be less than 1. If
we were to lower R2 to a value of zero ohms, our circuit would be essentially identical to the
voltage follower, with the output directly connected to the inverting input. Since the voltage
follower has a gain of 1, this sets the lower gain limit of the noninverting amplifier. However,
the gain can be increased far beyond 1, by increasing R2 in proportion to R1 .
Also note that the polarity of the output matches that of the input, just as with a voltage
follower. A positive input voltage results in a positive output voltage, and vice versa (with
respect to ground). For this reason, this circuit is referred to as a noninverting amplifier.
Just as with the voltage follower, we see that the differential gain of the op-amp is irrelevant, so long as its very high. The voltages and currents in this circuit would hardly change
at all if the op-amp’s voltage gain were 250,000 instead of 200,000. This stands as a stark contrast to single-transistor amplifier circuit designs, where the Beta of the individual transistor
greatly influenced the overall gains of the amplifier. With negative feedback, we have a selfcorrecting system that amplifies voltage according to the ratios set by the feedback resistors,
not the gains internal to the op-amp.
Let’s see what happens if we retain negative feedback through a voltage divider, but apply
the input voltage at a different location:

6 mA R1
1 kΩ
6V

R2

0V
−
+

0V

6 mA

1 kΩ
-6 V

All voltage figures shown in
reference to ground

By grounding the noninverting input, the negative feedback from the output seeks to hold
the inverting input’s voltage at 0 volts, as well. For this reason, the inverting input is referred
to in this circuit as a virtual ground, being held at ground potential (0 volts) by the feedback,
yet not directly connected to (electrically common with) ground. The input voltage this time
is applied to the left-hand end of the voltage divider (R1 = R2 = 1 kΩ again), so the output
voltage must swing to -6 volts in order to balance the middle at ground potential (0 volts).
Using the same techniques as with the noninverting amplifier, we can analyze this circuit’s
operation by determining current magnitudes and directions, starting with R1 , and continuing
on to determining the output voltage.
We can change the overall voltage gain of this circuit, overall, just by adjusting the values
of R1 and R2 (changing the ratio of output voltage that is fed back to the inverting input). Gain
can be calculated by the following formula:

R
AV = − 2
R1
Note that this circuit’s voltage gain can be less than 1, depending solely on the ratio of R2

CHAPTER 8. OPERATIONAL AMPLIFIERS

372

to R1 . Also note that the output voltage is always the opposite polarity of the input voltage.
A positive input voltage results in a negative output voltage, and vice versa (with respect to
ground). For this reason, this circuit is referred to as an inverting amplifier. Sometimes, the
gain formula contains a negative sign (before the R2 /R1 fraction) to reflect this reversal of
polarities.
These two amplifier circuits we’ve just investigated serve the purpose of multiplying or
dividing the magnitude of the input voltage signal. This is exactly how the mathematical
operations of multiplication and division are typically handled in analog computer circuitry.

• REVIEW:

• By connecting the inverting (-) input of an op-amp directly to the output, we get negative
feedback, which gives us a voltage follower circuit. By connecting that negative feedback
through a resistive voltage divider (feeding back a fraction of the output voltage to the
inverting input), the output voltage becomes a multiple of the input voltage.

• A negative-feedback op-amp circuit with the input signal going to the noninverting (+)
input is called a noninverting amplifier. The output voltage will be the same polarity as
the input. Voltage gain is given by the following equation: AV = (R2 /R1 ) + 1

• A negative-feedback op-amp circuit with the input signal going to the ”bottom” of the
resistive voltage divider, with the noninverting (+) input grounded, is called an inverting
amplifier. Its output voltage will be the opposite polarity of the input. Voltage gain is
given by the following equation: AV = -R2 /R1

8.6

An analogy for divided feedback

A helpful analogy for understanding divided feedback amplifier circuits is that of a mechanical
lever, with relative motion of the lever’s ends representing change in input and output voltages,
and the fulcrum (pivot point) representing the location of the ground point, real or virtual.
Take for example the following noninverting op-amp circuit. We know from the prior section
that the voltage gain of a noninverting amplifier configuration can never be less than unity (1).
If we draw a lever diagram next to the amplifier schematic, with the distance between fulcrum
and lever ends representative of resistor values, the motion of the lever will signify changes in
voltage at the input and output terminals of the amplifier:

8.6. AN ANALOGY FOR DIVIDED FEEDBACK

373

Vout
R2

R1

Vout = 2(Vin)

Vin
R1

R2

1 kΩ

−

0V

+

1 kΩ

Vout

Vin

Physicists call this type of lever, with the input force (effort) applied between the fulcrum
and output (load), a third-class lever. It is characterized by an output displacement (motion) at
least as large than the input displacement – a ”gain” of at least 1 – and in the same direction.
Applying a positive input voltage to this op-amp circuit is analogous to displacing the ”input”
point on the lever upward:

CHAPTER 8. OPERATIONAL AMPLIFIERS

374

Vout

Vin
Vout = 2(Vin)
R1

R2

1 kΩ

−

1 kΩ
+

0V

+

Vout
-

+
Vin
Due to the displacement-amplifying characteristics of the lever, the ”output” point will move
twice as far as the ”input” point, and in the same direction. In the electronic circuit, the output
voltage will equal twice the input, with the same polarity. Applying a negative input voltage is
analogous to moving the lever downward from its level ”zero” position, resulting in an amplified
output displacement that is also negative:

Vin
Vout
R1

R2

1 kΩ

1 kΩ

−

0V

+
Vin

+

Vout = 2(Vin)
Vout
+

8.6. AN ANALOGY FOR DIVIDED FEEDBACK

375

If we alter the resistor ratio R2 /R1 , we change the gain of the op-amp circuit. In lever terms,
this means moving the input point in relation to the fulcrum and lever end, which similarly
changes the displacement ”gain” of the machine:

Vout

R2

R1

Vout = 4(Vin)

Vin
R1

R2

1 kΩ

−

0V

+

3 kΩ

Vout

Vin

Now, any input signal will become amplified by a factor of four instead of by a factor of two:

CHAPTER 8. OPERATIONAL AMPLIFIERS

376

Vout
Vin
Vout = 4(Vin)
R1

R2

1 kΩ

−

0V

3 kΩ
+
Vout
-

+
+
Vin
-

Inverting op-amp circuits may be modeled using the lever analogy as well. With the inverting configuration, the ground point of the feedback voltage divider is the op-amp’s inverting
input with the input to the left and the output to the right. This is mechanically equivalent to
a first-class lever, where the input force (effort) is on the opposite side of the fulcrum from the
output (load):

Vin
Vin

R1

R2

R1

R2

1 kΩ

1 kΩ

−
+

Vout
Vout = -(Vin)

Vout

With equal-value resistors (equal-lengths of lever on each side of the fulcrum), the output
voltage (displacement) will be equal in magnitude to the input voltage (displacement), but of
the opposite polarity (direction). A positive input results in a negative output:

8.6. AN ANALOGY FOR DIVIDED FEEDBACK

Vout

Vin
R1
+
Vin

377

1 kΩ

-

R2
−

Vout = -(Vin)

1 kΩ
Vout
+

+

Changing the resistor ratio R2 /R1 changes the gain of the amplifier circuit, just as changing
the fulcrum position on the lever changes its mechanical displacement ”gain.” Consider the
following example, where R2 is made twice as large as R1 :

Vout

Vin
R1
+
Vin
-

1 kΩ

R2
−
+

Vout = -2(Vin)

2 kΩ
Vout
+

With the inverting amplifier configuration, though, gains of less than 1 are possible, just
as with first-class levers. Reversing R2 and R1 values is analogous to moving the fulcrum to
its complementary position on the lever: one-third of the way from the output end. There, the
output displacement will be one-half the input displacement:

CHAPTER 8. OPERATIONAL AMPLIFIERS

378

Vout
Vin

+
Vin
-

R1

R2

2 kΩ

1 kΩ

−
+

8.7

Vout = -0.5(Vin)
Vout
+

Voltage-to-current signal conversion

In instrumentation circuitry, DC signals are often used as analog representations of physical
measurements such as temperature, pressure, flow, weight, and motion. Most commonly, DC
current signals are used in preference to DC voltage signals, because current signals are exactly equal in magnitude throughout the series circuit loop carrying current from the source
(measuring device) to the load (indicator, recorder, or controller), whereas voltage signals in
a parallel circuit may vary from one end to the other due to resistive wire losses. Furthermore, current-sensing instruments typically have low impedances (while voltage-sensing instruments have high impedances), which gives current-sensing instruments greater electrical
noise immunity.
In order to use current as an analog representation of a physical quantity, we have to have
some way of generating a precise amount of current within the signal circuit. But how do
we generate a precise current signal when we might not know the resistance of the loop?
The answer is to use an amplifier designed to hold current to a prescribed value, applying
as much or as little voltage as necessary to the load circuit to maintain that value. Such an
amplifier performs the function of a current source. An op-amp with negative feedback is a
perfect candidate for such a task:

8.7. VOLTAGE-TO-CURRENT SIGNAL CONVERSION

379

4 to 20 mA

250 Ω

Rload
+

−
+

4 to 20 mA

+
Vin

1 to 5 volt signal range
-

The input voltage to this circuit is assumed to be coming from some type of physical transducer/amplifier arrangement, calibrated to produce 1 volt at 0 percent of physical measurement, and 5 volts at 100 percent of physical measurement. The standard analog current signal
range is 4 mA to 20 mA, signifying 0% to 100% of measurement range, respectively. At 5 volts
input, the 250 Ω (precision) resistor will have 5 volts applied across it, resulting in 20 mA of
current in the large loop circuit (with Rload ). It does not matter what resistance value Rload is,
or how much wire resistance is present in that large loop, so long as the op-amp has a high
enough power supply voltage to output the voltage necessary to get 20 mA flowing through
Rload . The 250 Ω resistor establishes the relationship between input voltage and output current, in this case creating the equivalence of 1-5 V in / 4-20 mA out. If we were converting the
1-5 volt input signal to a 10-50 mA output signal (an older, obsolete instrumentation standard
for industry), we’d use a 100 Ω precision resistor instead.
Another name for this circuit is transconductance amplifier. In electronics, transconductance is the mathematical ratio of current change divided by voltage change (∆I / ∆ V), and
it is measured in the unit of Siemens, the same unit used to express conductance (the mathematical reciprocal of resistance: current/voltage). In this circuit, the transconductance ratio is
fixed by the value of the 250 Ω resistor, giving a linear current-out/voltage-in relationship.
• REVIEW:
• In industry, DC current signals are often used in preference to DC voltage signals as
analog representations of physical quantities. Current in a series circuit is absolutely
equal at all points in that circuit regardless of wiring resistance, whereas voltage in a
parallel-connected circuit may vary from end to end because of wire resistance, making
current-signaling more accurate from the ”transmitting” to the ”receiving” instrument.
• Voltage signals are relatively easy to produce directly from transducer devices, whereas
accurate current signals are not. Op-amps can be used to ”convert” a voltage signal into
a current signal quite easily. In this mode, the op-amp will output whatever voltage is
necessary to maintain current through the signaling circuit at the proper value.

CHAPTER 8. OPERATIONAL AMPLIFIERS

380

8.8

Averager and summer circuits

If we take three equal resistors and connect one end of each to a common point, then apply
three input voltages (one to each of the resistors’ free ends), the voltage seen at the common
point will be the mathematical average of the three.

"Passive averager" circuit
R1
R2
Vout =
R3
V1

V2

V3

V1
V2
V3
R1 + R2 + R3
1
1
1
R1 + R2 + R3

With equal value resistors:
V1 + V2 + V3
Vout =
3

This circuit is really nothing more than a practical application of Millman’s Theorem:

R1

R2

R3
Vout =

V1

V2

V3

V1
V2
V3
R1 + R2 + R3
1
1
1
R1 + R2 + R3

This circuit is commonly known as a passive averager, because it generates an average voltage with non-amplifying components. Passive simply means that it is an unamplified circuit.
The large equation to the right of the averager circuit comes from Millman’s Theorem, which
describes the voltage produced by multiple voltage sources connected together through individual resistances. Since the three resistors in the averager circuit are equal to each other, we
can simplify Millman’s formula by writing R1 , R2 , and R3 simply as R (one, equal resistance
instead of three individual resistances):

8.8. AVERAGER AND SUMMER CIRCUITS

Vout =

V1
V2
V3
+
+
R
R
R
1
1
1
R + R + R

Vout =

V1 + V2 + V3
R
3
R

Vout =

V1 + V2 + V3
3

381

If we take a passive averager and use it to connect three input voltages into an op-amp
amplifier circuit with a gain of 3, we can turn this averaging function into an addition function.
The result is called a noninverting summer circuit:

1 kΩ

V1
V2
V3

2 kΩ

R

−

R

+

R

Vout

With a voltage divider composed of a 2 kΩ / 1 kΩ combination, the noninverting amplifier
circuit will have a voltage gain of 3. By taking the voltage from the passive averager, which
is the sum of V1 , V2 , and V3 divided by 3, and multiplying that average by 3, we arrive at an
output voltage equal to the sum of V1 , V2 , and V3 :

Vout = 3

V1 + V2 + V3
3

Vout = V1 + V2 + V3
Much the same can be done with an inverting op-amp amplifier, using a passive averager
as part of the voltage divider feedback circuit. The result is called an inverting summer circuit:

CHAPTER 8. OPERATIONAL AMPLIFIERS

382

V1
V2
V3

R

I1

R
R

R

0V
I2
I3

I1 + I2 + I3

−
+

Vout

0V
Now, with the right-hand sides of the three averaging resistors connected to the virtual
ground point of the op-amp’s inverting input, Millman’s Theorem no longer directly applies as
it did before. The voltage at the virtual ground is now held at 0 volts by the op-amp’s negative
feedback, whereas before it was free to float to the average value of V1 , V2 , and V3 . However,
with all resistor values equal to each other, the currents through each of the three resistors
will be proportional to their respective input voltages. Since those three currents will add at
the virtual ground node, the algebraic sum of those currents through the feedback resistor will
produce a voltage at Vout equal to V1 + V2 + V3 , except with reversed polarity. The reversal in
polarity is what makes this circuit an inverting summer:

Vout = -(V1 + V2 + V3)
Summer (adder) circuits are quite useful in analog computer design, just as multiplier and
divider circuits would be. Again, it is the extremely high differential gain of the op-amp which
allows us to build these useful circuits with a bare minimum of components.

• REVIEW:

• A summer circuit is one that sums, or adds, multiple analog voltage signals together.
There are two basic varieties of op-amp summer circuits: noninverting and inverting.

8.9

Building a differential amplifier

An op-amp with no feedback is already a differential amplifier, amplifying the voltage difference between the two inputs. However, its gain cannot be controlled, and it is generally too high
to be of any practical use. So far, our application of negative feedback to op-amps has resulting
in the practical loss of one of the inputs, the resulting amplifier only good for amplifying a single voltage signal input. With a little ingenuity, however, we can construct an op-amp circuit
maintaining both voltage inputs, yet with a controlled gain set by external resistors.

8.9. BUILDING A DIFFERENTIAL AMPLIFIER

R

383

R

V1
−
Vout

+
R

R

V2
If all the resistor values are equal, this amplifier will have a differential voltage gain of 1.
The analysis of this circuit is essentially the same as that of an inverting amplifier, except that
the noninverting input (+) of the op-amp is at a voltage equal to a fraction of V2 , rather than
being connected directly to ground. As would stand to reason, V2 functions as the noninverting
input and V1 functions as the inverting input of the final amplifier circuit. Therefore:

Vout = V2 - V1
If we wanted to provide a differential gain of anything other than 1, we would have to
adjust the resistances in both upper and lower voltage dividers, necessitating multiple resistor
changes and balancing between the two dividers for symmetrical operation. This is not always
practical, for obvious reasons.
Another limitation of this amplifier design is the fact that its input impedances are rather
low compared to that of some other op-amp configurations, most notably the noninverting
(single-ended input) amplifier. Each input voltage source has to drive current through a resistance, which constitutes far less impedance than the bare input of an op-amp alone. The
solution to this problem, fortunately, is quite simple. All we need to do is ”buffer” each input
voltage signal through a voltage follower like this:

V1

+

R

R

−
−
Vout

+
−
V2

R

R

+

Now the V1 and V2 input lines are connected straight to the inputs of two voltage-follower
op-amps, giving very high impedance. The two op-amps on the left now handle the driving of
current through the resistors instead of letting the input voltage sources (whatever they may
be) do it. The increased complexity to our circuit is minimal for a substantial benefit.

CHAPTER 8. OPERATIONAL AMPLIFIERS

384

8.10

The instrumentation amplifier

As suggested before, it is beneficial to be able to adjust the gain of the amplifier circuit without
having to change more than one resistor value, as is necessary with the previous design of
differential amplifier. The so-called instrumentation builds on the last version of differential
amplifier to give us that capability:

V1

+

R

3
−

R
1

−

2

+

Rgain

−
V2

+

R

R

R

Vout

R

4

This intimidating circuit is constructed from a buffered differential amplifier stage with
three new resistors linking the two buffer circuits together. Consider all resistors to be of equal
value except for Rgain . The negative feedback of the upper-left op-amp causes the voltage at
point 1 (top of Rgain ) to be equal to V1 . Likewise, the voltage at point 2 (bottom of Rgain ) is
held to a value equal to V2 . This establishes a voltage drop across Rgain equal to the voltage
difference between V1 and V2 . That voltage drop causes a current through Rgain , and since the
feedback loops of the two input op-amps draw no current, that same amount of current through
Rgain must be going through the two ”R” resistors above and below it. This produces a voltage
drop between points 3 and 4 equal to:

V3-4 = (V2 - V1)(1 +

2R )
Rgain

The regular differential amplifier on the right-hand side of the circuit then takes this voltage drop between points 3 and 4, and amplifies it by a gain of 1 (assuming again that all ”R”
resistors are of equal value). Though this looks like a cumbersome way to build a differential
amplifier, it has the distinct advantages of possessing extremely high input impedances on the
V1 and V2 inputs (because they connect straight into the noninverting inputs of their respective op-amps), and adjustable gain that can be set by a single resistor. Manipulating the above
formula a bit, we have a general expression for overall voltage gain in the instrumentation
amplifier:

AV = (1 + 2R )
Rgain
Though it may not be obvious by looking at the schematic, we can change the differential
gain of the instrumentation amplifier simply by changing the value of one resistor: Rgain . Yes,
we could still change the overall gain by changing the values of some of the other resistors,

8.11. DIFFERENTIATOR AND INTEGRATOR CIRCUITS

385

but this would necessitate balanced resistor value changes for the circuit to remain symmetrical. Please note that the lowest gain possible with the above circuit is obtained with Rgain
completely open (infinite resistance), and that gain value is 1.
• REVIEW:
• An instrumentation amplifier is a differential op-amp circuit providing high input impedances
with ease of gain adjustment through the variation of a single resistor.

8.11

Differentiator and integrator circuits

By introducing electrical reactance into the feedback loops of op-amp amplifier circuits, we can
cause the output to respond to changes in the input voltage over time. Drawing their names
from their respective calculus functions, the integrator produces a voltage output proportional
to the product (multiplication) of the input voltage and time; and the differentiator (not to be
confused with differential) produces a voltage output proportional to the input voltage’s rate of
change.
Capacitance can be defined as the measure of a capacitor’s opposition to changes in voltage.
The greater the capacitance, the more the opposition. Capacitors oppose voltage change by
creating current in the circuit: that is, they either charge or discharge in response to a change
in applied voltage. So, the more capacitance a capacitor has, the greater its charge or discharge
current will be for any given rate of voltage change across it. The equation for this is quite
simple:

Changing
DC
voltage

C

i = C dv
dt
The dv/dt fraction is a calculus expression representing the rate of voltage change over
time. If the DC supply in the above circuit were steadily increased from a voltage of 15 volts
to a voltage of 16 volts over a time span of 1 hour, the current through the capacitor would
most likely be very small, because of the very low rate of voltage change (dv/dt = 1 volt / 3600
seconds). However, if we steadily increased the DC supply from 15 volts to 16 volts over a
shorter time span of 1 second, the rate of voltage change would be much higher, and thus the
charging current would be much higher (3600 times higher, to be exact). Same amount of
change in voltage, but vastly different rates of change, resulting in vastly different amounts of
current in the circuit.
To put some definite numbers to this formula, if the voltage across a 47 µF capacitor was
changing at a linear rate of 3 volts per second, the current ”through” the capacitor would be
(47 µF)(3 V/s) = 141 µA.
We can build an op-amp circuit which measures change in voltage by measuring current
through a capacitor, and outputs a voltage proportional to that current:

CHAPTER 8. OPERATIONAL AMPLIFIERS

386

Differentiator
C
Vin

0V
0V
0V

R
−
+

Vout

The right-hand side of the capacitor is held to a voltage of 0 volts, due to the ”virtual ground”
effect. Therefore, current ”through” the capacitor is solely due to change in the input voltage.
A steady input voltage won’t cause a current through C, but a changing input voltage will.
Capacitor current moves through the feedback resistor, producing a drop across it, which
is the same as the output voltage. A linear, positive rate of input voltage change will result
in a steady negative voltage at the output of the op-amp. Conversely, a linear, negative rate
of input voltage change will result in a steady positive voltage at the output of the op-amp.
This polarity inversion from input to output is due to the fact that the input signal is being
sent (essentially) to the inverting input of the op-amp, so it acts like the inverting amplifier
mentioned previously. The faster the rate of voltage change at the input (either positive or
negative), the greater the voltage at the output.
The formula for determining voltage output for the differentiator is as follows:

Vout = -RC

dvin
dt

Applications for this, besides representing the derivative calculus function inside of an analog computer, include rate-of-change indicators for process instrumentation. One such rateof-change signal application might be for monitoring (or controlling) the rate of temperature
change in a furnace, where too high or too low of a temperature rise rate could be detrimental. The DC voltage produced by the differentiator circuit could be used to drive a comparator,
which would signal an alarm or activate a control if the rate of change exceeded a pre-set level.
In process control, the derivative function is used to make control decisions for maintaining
a process at setpoint, by monitoring the rate of process change over time and taking action to
prevent excessive rates of change, which can lead to an unstable condition. Analog electronic
controllers use variations of this circuitry to perform the derivative function.
On the other hand, there are applications where we need precisely the opposite function,
called integration in calculus. Here, the op-amp circuit would generate an output voltage proportional to the magnitude and duration that an input voltage signal has deviated from 0 volts.
Stated differently, a constant input signal would generate a certain rate of change in the output voltage: differentiation in reverse. To do this, all we have to do is swap the capacitor and
resistor in the previous circuit:

8.11. DIFFERENTIATOR AND INTEGRATOR CIRCUITS

387

Integrator

Vin

R

C

0V
0V
0V

−
+

Vout

As before, the negative feedback of the op-amp ensures that the inverting input will be held
at 0 volts (the virtual ground). If the input voltage is exactly 0 volts, there will be no current
through the resistor, therefore no charging of the capacitor, and therefore the output voltage
will not change. We cannot guarantee what voltage will be at the output with respect to ground
in this condition, but we can say that the output voltage will be constant.
However, if we apply a constant, positive voltage to the input, the op-amp output will fall
negative at a linear rate, in an attempt to produce the changing voltage across the capacitor
necessary to maintain the current established by the voltage difference across the resistor.
Conversely, a constant, negative voltage at the input results in a linear, rising (positive) voltage
at the output. The output voltage rate-of-change will be proportional to the value of the input
voltage.
The formula for determining voltage output for the integrator is as follows:

dvout
Vin
=dt
RC
or
t

Vout = ∫ 0

Vin
dt + c
RC

Where,
c = Output voltage at start time (t=0)
One application for this device would be to keep a ”running total” of radiation exposure,
or dosage, if the input voltage was a proportional signal supplied by an electronic radiation
detector. Nuclear radiation can be just as damaging at low intensities for long periods of time
as it is at high intensities for short periods of time. An integrator circuit would take both
the intensity (input voltage magnitude) and time into account, generating an output voltage
representing total radiation dosage.
Another application would be to integrate a signal representing water flow, producing a
signal representing total quantity of water that has passed by the flowmeter. This application
of an integrator is sometimes called a totalizer in the industrial instrumentation trade.

CHAPTER 8. OPERATIONAL AMPLIFIERS

388
• REVIEW:

• A differentiator circuit produces a constant output voltage for a steadily changing input
voltage.
• An integrator circuit produces a steadily changing output voltage for a constant input
voltage.
• Both types of devices are easily constructed, using reactive components (usually capacitors rather than inductors) in the feedback part of the circuit.

8.12

Positive feedback

As we’ve seen, negative feedback is an incredibly useful principle when applied to operational
amplifiers. It is what allows us to create all these practical circuits, being able to precisely
set gains, rates, and other significant parameters with just a few changes of resistor values.
Negative feedback makes all these circuits stable and self-correcting.
The basic principle of negative feedback is that the output tends to drive in a direction
that creates a condition of equilibrium (balance). In an op-amp circuit with no feedback, there
is no corrective mechanism, and the output voltage will saturate with the tiniest amount of
differential voltage applied between the inputs. The result is a comparator:
With negative feedback (the output voltage ”fed back” somehow to the inverting input), the
circuit tends to prevent itself from driving the output to full saturation. Rather, the output
voltage drives only as high or as low as needed to balance the two inputs’ voltages:

Negative feedback

−
0V

Vout

+
Vout = Vin

Vin

Whether the output is directly fed back to the inverting (-) input or coupled through a set of
components, the effect is the same: the extremely high differential voltage gain of the op-amp
will be ”tamed” and the circuit will respond according to the dictates of the feedback ”loop”
connecting output to inverting input.
Another type of feedback, namely positive feedback, also finds application in op-amp circuits. Unlike negative feedback, where the output voltage is ”fed back” to the inverting (-)
input, with positive feedback the output voltage is somehow routed back to the noninverting

8.12. POSITIVE FEEDBACK

389

(+) input. In its simplest form, we could connect a straight piece of wire from output to noninverting input and see what happens:

Positive feedback
+
Vout

−

The inverting input remains disconnected from the feedback loop, and is free to receive an
external voltage. Let’s see what happens if we ground the inverting input:

+
0V

−

Vout

With the inverting input grounded (maintained at zero volts), the output voltage will be
dictated by the magnitude and polarity of the voltage at the noninverting input. If that voltage
happens to be positive, the op-amp will drive its output positive as well, feeding that positive
voltage back to the noninverting input, which will result in full positive output saturation. On
the other hand, if the voltage on the noninverting input happens to start out negative, the opamp’s output will drive in the negative direction, feeding back to the noninverting input and
resulting in full negative saturation.
What we have here is a circuit whose output is bistable: stable in one of two states (saturated positive or saturated negative). Once it has reached one of those saturated states, it
will tend to remain in that state, unchanging. What is necessary to get it to switch states is a
voltage placed upon the inverting (-) input of the same polarity, but of a slightly greater magnitude. For example, if our circuit is saturated at an output voltage of +12 volts, it will take an
input voltage at the inverting input of at least +12 volts to get the output to change. When it
changes, it will saturate fully negative.
So, an op-amp with positive feedback tends to stay in whatever output state its already in.
It ”latches” between one of two states, saturated positive or saturated negative. Technically,
this is known as hysteresis.
Hysteresis can be a useful property for a comparator circuit to have. As we’ve seen before,
comparators can be used to produce a square wave from any sort of ramping waveform (sine
wave, triangle wave, sawtooth wave, etc.) input. If the incoming AC waveform is noise-free
(that is, a ”pure” waveform), a simple comparator will work just fine.

CHAPTER 8. OPERATIONAL AMPLIFIERS

390
+V
−
Vin

Vout

+

-V
Square wave
output voltage
DC reference
voltage

AC input
voltage
A "clean" AC input waveform produces predictable
transition points on the output voltage square wave

However, if there exist any anomalies in the waveform such as harmonics or ”spikes” which
cause the voltage to rise and fall significantly within the timespan of a single cycle, a comparator’s output might switch states unexpectedly:
+V
−
Vin

+

Vout

-V
Square wave
output voltage
DC reference
voltage

AC input
voltage

Any time there is a transition through the reference voltage level, no matter how tiny that
transition may be, the output of the comparator will switch states, producing a square wave
with ”glitches.”
If we add a little positive feedback to the comparator circuit, we will introduce hysteresis
into the output. This hysteresis will cause the output to remain in its current state unless the
AC input voltage undergoes a major change in magnitude.

8.12. POSITIVE FEEDBACK

391

+V
−
Vin

Vout

+

-V

Positive feedback
resistor

What this feedback resistor creates is a dual-reference for the comparator circuit. The
voltage applied to the noninverting (+) input as a reference which to compare with the incoming
AC voltage changes depending on the value of the op-amp’s output voltage. When the opamp output is saturated positive, the reference voltage at the noninverting input will be more
positive than before. Conversely, when the op-amp output is saturated negative, the reference
voltage at the noninverting input will be more negative than before. The result is easier to
understand on a graph:

square wave
output voltage

DC reference voltages
upper
center
lower

AC input
voltage

When the op-amp output is saturated positive, the upper reference voltage is in effect, and
the output won’t drop to a negative saturation level unless the AC input rises above that upper
reference level. Conversely, when the op-amp output is saturated negative, the lower reference
voltage is in effect, and the output won’t rise to a positive saturation level unless the AC input
drops below that lower reference level. The result is a clean square-wave output again, despite
significant amounts of distortion in the AC input signal. In order for a ”glitch” to cause the
comparator to switch from one state to another, it would have to be at least as big (tall) as the
difference between the upper and lower reference voltage levels, and at the right point in time
to cross both those levels.
Another application of positive feedback in op-amp circuits is in the construction of oscillator circuits. An oscillator is a device that produces an alternating (AC), or at least pulsing,
output voltage. Technically, it is known as an astable device: having no stable output state (no
equilibrium whatsoever). Oscillators are very useful devices, and they are easily made with
just an op-amp and a few external components.

CHAPTER 8. OPERATIONAL AMPLIFIERS

392

Oscillator circuit using positive feedback
C

Vramp

R

−
Vout

+
R

R
Vref

Vref
Vramp

Vout is a square wave just like Vref, only taller
When the output is saturated positive, the Vref will be positive, and the capacitor will
charge up in a positive direction. When Vramp exceeds Vref by the tiniest margin, the output
will saturate negative, and the capacitor will charge in the opposite direction (polarity). Oscillation occurs because the positive feedback is instantaneous and the negative feedback is
delayed (by means of an RC time constant). The frequency of this oscillator may be adjusted
by varying the size of any component.
• REVIEW:
• Negative feedback creates a condition of equilibrium (balance). Positive feedback creates
a condition of hysteresis (the tendency to ”latch” in one of two extreme states).
• An oscillator is a device producing an alternating or pulsing output voltage.

8.13

Practical considerations

Real operational have some imperfections compared to an “ideal” model. A real device deviates
from a perfect difference amplifier. One minus one may not be zero. It may have have an offset
like an analog meter which is not zeroed. The inputs may draw current. The characteristics
may drift with age and temperature. Gain may be reduced at high frequencies, and phase
may shift from input to output. These imperfection may cause no noticable errors in some
applications, unacceptable errors in others. In some cases these errors may be compensated
for. Sometimes a higher quality, higher cost device is required.

8.13. PRACTICAL CONSIDERATIONS

8.13.1

393

Common-mode gain

As stated before, an ideal differential amplifier only amplifies the voltage difference between
its two inputs. If the two inputs of a differential amplifier were to be shorted together (thus
ensuring zero potential difference between them), there should be no change in output voltage
for any amount of voltage applied between those two shorted inputs and ground:

−
+

Vcommon-mode

Vout

Vout should remain the same
regardless of Vcommon-mode

Voltage that is common between either of the inputs and ground, as ”Vcommon−mode ” is
in this case, is called common-mode voltage. As we vary this common voltage, the perfect
differential amplifier’s output voltage should hold absolutely steady (no change in output for
any arbitrary change in common-mode input). This translates to a common-mode voltage gain
of zero.

AV =

Change in Vout
Change in Vin

. . . if change in Vout = 0 . . .
0
=0
Change in Vin
AV = 0
The operational amplifier, being a differential amplifier with high differential gain, would
ideally have zero common-mode gain as well. In real life, however, this is not easily attained.
Thus, common-mode voltages will invariably have some effect on the op-amp’s output voltage.
The performance of a real op-amp in this regard is most commonly measured in terms of its
differential voltage gain (how much it amplifies the difference between two input voltages) versus its common-mode voltage gain (how much it amplifies a common-mode voltage). The ratio
of the former to the latter is called the common-mode rejection ratio, abbreviated as CMRR:

CMRR =

Differential AV
Common-mode AV

An ideal op-amp, with zero common-mode gain would have an infinite CMRR. Real op-amps
have high CMRRs, the ubiquitous 741 having something around 70 dB, which works out to a

CHAPTER 8. OPERATIONAL AMPLIFIERS

394

little over 3,000 in terms of a ratio.
Because the common mode rejection ratio in a typical op-amp is so high, common-mode gain
is usually not a great concern in circuits where the op-amp is being used with negative feedback. If the common-mode input voltage of an amplifier circuit were to suddenly change, thus
producing a corresponding change in the output due to common-mode gain, that change in output would be quickly corrected as negative feedback and differential gain (being much greater
than common-mode gain) worked to bring the system back to equilibrium. Sure enough, a
change might be seen at the output, but it would be a lot smaller than what you might expect.
A consideration to keep in mind, though, is common-mode gain in differential op-amp circuits such as instrumentation amplifiers. Outside of the op-amp’s sealed package and extremely high differential gain, we may find common-mode gain introduced by an imbalance of
resistor values. To demonstrate this, we’ll run a SPICE analysis on an instrumentation amplifier with inputs shorted together (no differential voltage), imposing a common-mode voltage
to see what happens. First, we’ll run the analysis showing the output voltage of a perfectly
balanced circuit. We should expect to see no change in output voltage as the common-mode
voltage changes:

1 +
−

V1
0

R3

3

E1

9
R1

2

7

2

−

Rgain
Rjump
(jumper
wire)

5
5

4
instrumentation amplifier
v1 1 0
rin1 1 0 9e12
rjump 1 4 1e-12
rin2 4 0 9e12
e1 3 0 1 2 999k
e2 6 0 4 5 999k
e3 9 0 8 7 999k
rload 9 0 10k
r1 2 3 10k
rgain 2 5 10k
r2 5 6 10k
r3 3 7 10k
r4 7 9 10k
r5 6 8 10k

R2

−
+

R4

7

E2

+

8
R5

6

E3

Vout

9

R6
8

0

8.13. PRACTICAL CONSIDERATIONS

395

r6 8 0 10k
.dc v1 0 10 1
.print dc v(9)
.end
v1
0.000E+00
1.000E+00
2.000E+00
3.000E+00
4.000E+00
5.000E+00
6.000E+00
7.000E+00
8.000E+00
9.000E+00
1.000E+01

v(9)
0.000E+00
1.355E-16
2.710E-16
0.000E+00
5.421E-16
0.000E+00
0.000E+00
0.000E+00
1.084E-15
-1.084E-15
0.000E+00

As you can see, the output voltage v(9)
hardly changes at all for a common-mode
input voltage (v1) that sweeps from 0
to 10 volts.

Aside from very small deviations (actually due to quirks of SPICE rather than real behavior
of the circuit), the output remains stable where it should be: at 0 volts, with zero input voltage
differential. However, let’s introduce a resistor imbalance in the circuit, increasing the value of
R5 from 10,000 Ω to 10,500 Ω, and see what happens (the netlist has been omitted for brevity
– the only thing altered is the value of R5 ):
v1
0.000E+00
1.000E+00
2.000E+00
3.000E+00
4.000E+00
5.000E+00
6.000E+00
7.000E+00
8.000E+00
9.000E+00
1.000E+01

v(9)
0.000E+00
-2.439E-02
-4.878E-02
-7.317E-02
-9.756E-02
-1.220E-01
-1.463E-01
-1.707E-01
-1.951E-01
-2.195E-01
-2.439E-01

This time we see a significant variation
(from 0 to 0.2439 volts) in output voltage
as the common-mode input voltage sweeps
from 0 to 10 volts as it did before.

Our input voltage differential is still zero volts, yet the output voltage changes significantly
as the common-mode voltage is changed. This is indicative of a common-mode gain, something
we’re trying to avoid. More than that, its a common-mode gain of our own making, having
nothing to do with imperfections in the op-amps themselves. With a much-tempered differential gain (actually equal to 3 in this particular circuit) and no negative feedback outside the
circuit, this common-mode gain will go unchecked in an instrument signal application.
There is only one way to correct this common-mode gain, and that is to balance all the resistor values. When designing an instrumentation amplifier from discrete components (rather
than purchasing one in an integrated package), it is wise to provide some means of making

CHAPTER 8. OPERATIONAL AMPLIFIERS

396

fine adjustments to at least one of the four resistors connected to the final op-amp to be able to
”trim away” any such common-mode gain. Providing the means to ”trim” the resistor network
has additional benefits as well. Suppose that all resistor values are exactly as they should
be, but a common-mode gain exists due to an imperfection in one of the op-amps. With the
adjustment provision, the resistance could be trimmed to compensate for this unwanted gain.
One quirk of some op-amp models is that of output latch-up, usually caused by the commonmode input voltage exceeding allowable limits. If the common-mode voltage falls outside of the
manufacturer’s specified limits, the output may suddenly ”latch” in the high mode (saturate at
full output voltage). In JFET-input operational amplifiers, latch-up may occur if the commonmode input voltage approaches too closely to the negative power supply rail voltage. On the
TL082 op-amp, for example, this occurs when the common-mode input voltage comes within
about 0.7 volts of the negative power supply rail voltage. Such a situation may easily occur in
a single-supply circuit, where the negative power supply rail is ground (0 volts), and the input
signal is free to swing to 0 volts.
Latch-up may also be triggered by the common-mode input voltage exceeding power supply
rail voltages, negative or positive. As a rule, you should never allow either input voltage to
rise above the positive power supply rail voltage, or sink below the negative power supply
rail voltage, even if the op-amp in question is protected against latch-up (as are the 741 and
1458 op-amp models). At the very least, the op-amp’s behavior may become unpredictable. At
worst, the kind of latch-up triggered by input voltages exceeding power supply voltages may
be destructive to the op-amp.
While this problem may seem easy to avoid, its possibility is more likely than you might
think. Consider the case of an operational amplifier circuit during power-up. If the circuit
receives full input signal voltage before its own power supply has had time enough to charge
the filter capacitors, the common-mode input voltage may easily exceed the power supply rail
voltages for a short time. If the op-amp receives signal voltage from a circuit supplied by a
different power source, and its own power source fails, the signal voltage(s) may exceed the
power supply rail voltages for an indefinite amount of time!

8.13.2

Offset voltage

Another practical concern for op-amp performance is voltage offset. That is, effect of having
the output voltage something other than zero volts when the two input terminals are shorted
together. Remember that operational amplifiers are differential amplifiers above all: they’re
supposed to amplify the difference in voltage between the two input connections and nothing
more. When that input voltage difference is exactly zero volts, we would (ideally) expect to have
exactly zero volts present on the output. However, in the real world this rarely happens. Even
if the op-amp in question has zero common-mode gain (infinite CMRR), the output voltage may
not be at zero when both inputs are shorted together. This deviation from zero is called offset.

8.13. PRACTICAL CONSIDERATIONS

397

+15 V
−
Vout = +14.7 V (saturated +)

+
-15 V

A perfect op-amp would output exactly zero volts with both its inputs shorted together and
grounded. However, most op-amps off the shelf will drive their outputs to a saturated level,
either negative or positive. In the example shown above, the output voltage is saturated at a
value of positive 14.7 volts, just a bit less than +V (+15 volts) due to the positive saturation
limit of this particular op-amp. Because the offset in this op-amp is driving the output to a
completely saturated point, there’s no way of telling how much voltage offset is present at the
output. If the +V/-V split power supply was of a high enough voltage, who knows, maybe the
output would be several hundred volts one way or the other due to the effects of offset!
For this reason, offset voltage is usually expressed in terms of the equivalent amount of
input voltage differential producing this effect. In other words, we imagine that the op-amp
is perfect (no offset whatsoever), and a small voltage is being applied in series with one of the
inputs to force the output voltage one way or the other away from zero. Being that op-amp
differential gains are so high, the figure for ”input offset voltage” doesn’t have to be much to
account for what we see with shorted inputs:

+15 V
−
+

Vout = +14.7 V (saturated +)

-15 V
Input offset voltage
(internal to the real op-amp,
external to this ideal op-amp)
Offset voltage will tend to introduce slight errors in any op-amp circuit. So how do we
compensate for it? Unlike common-mode gain, there are usually provisions made by the manufacturer to trim the offset of a packaged op-amp. Usually, two extra terminals on the op-amp
package are reserved for connecting an external ”trim” potentiometer. These connection points
are labeled offset null and are used in this general way:

CHAPTER 8. OPERATIONAL AMPLIFIERS

398

+15 V
−
+

Vout

-15 V
Potentiometer adjusted so that
Vout = 0 volts with inputs shorted together
On single op-amps such as the 741 and 3130, the offset null connection points are pins 1
and 5 on the 8-pin DIP package. Other models of op-amp may have the offset null connections
located on different pins, and/or require a slightly difference configuration of trim potentiometer connection. Some op-amps don’t provide offset null pins at all! Consult the manufacturer’s
specifications for details.

8.13.3

Bias current

Inputs on an op-amp have extremely high input impedances. That is, the input currents entering or exiting an op-amp’s two input signal connections are extremely small. For most purposes
of op-amp circuit analysis, we treat them as though they don’t exist at all. We analyze the circuit as though there was absolutely zero current entering or exiting the input connections.
This idyllic picture, however, is not entirely true. Op-amps, especially those op-amps with
bipolar transistor inputs, have to have some amount of current through their input connections in order for their internal circuits to be properly biased. These currents, logically, are
called bias currents. Under certain conditions, op-amp bias currents may be problematic. The
following circuit illustrates one of those problem conditions:

+V
Thermocouple

−
Vout

+
-V

At first glance, we see no apparent problems with this circuit. A thermocouple, generating a
small voltage proportional to temperature (actually, a voltage proportional to the difference in
temperature between the measurement junction and the ”reference” junction formed when the
alloy thermocouple wires connect with the copper wires leading to the op-amp) drives the opamp either positive or negative. In other words, this is a kind of comparator circuit, comparing
the temperature between the end thermocouple junction and the reference junction (near the
op-amp). The problem is this: the wire loop formed by the thermocouple does not provide a

8.13. PRACTICAL CONSIDERATIONS

399

path for both input bias currents, because both bias currents are trying to go the same way
(either into the op-amp or out of it).

+V
I?

−

Thermocouple

Vout

+

I?

-V
This comparator circuit won’t work
In order for this circuit to work properly, we must ground one of the input wires, thus
providing a path to (or from) ground for both currents:

+V

I
−

Thermocouple

Vout

+
I

I

-V

This comparator circuit will work
Not necessarily an obvious problem, but a very real one!
Another way input bias currents may cause trouble is by dropping unwanted voltages across
circuit resistances. Take this circuit for example:

+V
Voltage drop due
to bias current:
- Rin +
Vin

Ibias

−
Vout

+
-V

Voltage at (+) op-amp input
will not be exactly equal to Vin
We expect a voltage follower circuit such as the one above to reproduce the input voltage
precisely at the output. But what about the resistance in series with the input voltage source?
If there is any bias current through the noninverting (+) input at all, it will drop some voltage
across Rin , thus making the voltage at the noninverting input unequal to the actual Vin value.
Bias currents are usually in the microamp range, so the voltage drop across Rin won’t be very
much, unless Rin is very large. One example of an application where the input resistance

CHAPTER 8. OPERATIONAL AMPLIFIERS

400

(Rin ) would be very large is that of pH probe electrodes, where one electrode contains an ionpermeable glass barrier (a very poor conductor, with millions of Ω of resistance).
If we were actually building an op-amp circuit for pH electrode voltage measurement, we’d
probably want to use a FET or MOSFET (IGFET) input op-amp instead of one built with
bipolar transistors (for less input bias current). But even then, what slight bias currents may
remain can cause measurement errors to occur, so we have to find some way to mitigate them
through good design.
One way to do so is based on the assumption that the two input bias currents will be the
same. In reality, they are often close to being the same, the difference between them referred
to as the input offset current. If they are the same, then we should be able to cancel out the
effects of input resistance voltage drop by inserting an equal amount of resistance in series
with the other input, like this:

-

Ibias

Rin(-)

+V
+ −

- Rin(+)+
Ibias

Vin

Vout

+
-V

With the additional resistance added to the circuit, the output voltage will be closer to Vin
than before, even if there is some offset between the two input currents.
For both inverting and noninverting amplifier circuits, the bias current compensating resistor is placed in series with the noninverting (+) input to compensate for bias current voltage
drops in the divider network:

Noninverting amplifier with
compensating resistor
R1

R2
−

Vin

Rcomp

+

Rcomp = R1 // R2

Vout

8.13. PRACTICAL CONSIDERATIONS

401

Inverting amplifier with
compensating resistor
R1

R2
−

Vin
Rcomp

+

Vout

Rcomp = R1 // R2
In either case, the compensating resistor value is determined by calculating the parallel
resistance value of R1 and R2 . Why is the value equal to the parallel equivalent of R1 and R2 ?
When using the Superposition Theorem to figure how much voltage drop will be produced by
the inverting (-) input’s bias current, we treat the bias current as though it were coming from
a current source inside the op-amp and short-circuit all voltage sources (Vin and Vout ). This
gives two parallel paths for bias current (through R1 and through R2 , both to ground). We
want to duplicate the bias current’s effect on the noninverting (+) input, so the resistor value
we choose to insert in series with that input needs to be equal to R1 in parallel with R2 .
A related problem, occasionally experienced by students just learning to build operational
amplifier circuits, is caused by a lack of a common ground connection to the power supply. It is
imperative to proper op-amp function that some terminal of the DC power supply be common
to the ”ground” connection of the input signal(s). This provides a complete path for the bias
currents, feedback current(s), and for the load (output) current. Take this circuit illustration,
for instance, showing a properly grounded power supply:

R1

R2

1 kΩ

1 kΩ
+V

null
+

-V

6V

Here, arrows denote the path of electron flow through the power supply batteries, both for
powering the op-amp’s internal circuitry (the ”potentiometer” inside of it that controls output

CHAPTER 8. OPERATIONAL AMPLIFIERS

402

voltage), and for powering the feedback loop of resistors R1 and R2 . Suppose, however, that the
ground connection for this ”split” DC power supply were to be removed. The effect of doing this
is profound:

A power supply ground is essential to circuit operation!
R1

R2

1 kΩ

1 kΩ
+V

-

broken
connection

null
+

-V

6V

No electrons may flow in or out of the op-amp’s output terminal, because the pathway to the
power supply is a ”dead end.” Thus, no electrons flow through the ground connection to the left
of R1 , neither through the feedback loop. This effectively renders the op-amp useless: it can
neither sustain current through the feedback loop, nor through a grounded load, since there is
no connection from any point of the power supply to ground.

The bias currents are also stopped, because they rely on a path to the power supply and back
to the input source through ground. The following diagram shows the bias currents (only), as
they go through the input terminals of the op-amp, through the base terminals of the input
transistors, and eventually through the power supply terminal(s) and back to ground.

8.13. PRACTICAL CONSIDERATIONS

403

Bias current paths shown, through power supply

-

+V

Ibias
Ibias
+

-V

6V

Without a ground reference on the power supply, the bias currents will have no complete
path for a circuit, and they will halt. Since bipolar junction transistors are current-controlled
devices, this renders the input stage of the op-amp useless as well, as both input transistors
will be forced into cutoff by the complete lack of base current.
• REVIEW:
• Op-amp inputs usually conduct very small currents, called bias currents, needed to properly bias the first transistor amplifier stage internal to the op-amps’ circuitry. Bias currents are small (in the microamp range), but large enough to cause problems in some
applications.
• Bias currents in both inputs must have paths to flow to either one of the power supply
”rails” or to ground. It is not enough to just have a conductive path from one input to the
other.
• To cancel any offset voltages caused by bias current flowing through resistances, just add
an equivalent resistance in series with the other op-amp input (called a compensating
resistor). This corrective measure is based on the assumption that the two input bias
currents will be equal.
• Any inequality between bias currents in an op-amp constitutes what is called an input
offset current.
• It is essential for proper op-amp operation that there be a ground reference on some terminal of the power supply, to form complete paths for bias currents, feedback current(s),
and load current.

CHAPTER 8. OPERATIONAL AMPLIFIERS

404

8.13.4

Drift

Being semiconductor devices, op-amps are subject to slight changes in behavior with changes
in operating temperature. Any changes in op-amp performance with temperature fall under
the category of op-amp drift. Drift parameters can be specified for bias currents, offset voltage,
and the like. Consult the manufacturer’s data sheet for specifics on any particular op-amp.
To minimize op-amp drift, we can select an op-amp made to have minimum drift, and/or we
can do our best to keep the operating temperature as stable as possible. The latter action may
involve providing some form of temperature control for the inside of the equipment housing
the op-amp(s). This is not as strange as it may first seem. Laboratory-standard precision
voltage reference generators, for example, are sometimes known to employ ”ovens” for keeping
their sensitive components (such as zener diodes) at constant temperatures. If extremely high
accuracy is desired over the usual factors of cost and flexibility, this may be an option worth
looking at.
• REVIEW:
• Op-amps, being semiconductor devices, are susceptible to variations in temperature. Any
variations in amplifier performance resulting from changes in temperature is known as
drift. Drift is best minimized with environmental temperature control.

8.13.5

Frequency response

With their incredibly high differential voltage gains, op-amps are prime candidates for a phenomenon known as feedback oscillation. You’ve probably heard the equivalent audio effect
when the volume (gain) on a public-address or other microphone amplifier system is turned
too high: that high pitched squeal resulting from the sound waveform ”feeding back” through
the microphone to be amplified again. An op-amp circuit can manifest this same effect, with
the feedback happening electrically rather than audibly.
A case example of this is seen in the 3130 op-amp, if it is connected as a voltage follower
with the bare minimum of wiring connections (the two inputs, output, and the power supply
connections). The output of this op-amp will self-oscillate due to its high gain, no matter
what the input voltage. To combat this, a small compensation capacitor must be connected
to two specially-provided terminals on the op-amp. The capacitor provides a high-impedance
path for negative feedback to occur within the op-amp’s circuitry, thus decreasing the AC gain
and inhibiting unwanted oscillations. If the op-amp is being used to amplify high-frequency
signals, this compensation capacitor may not be needed, but it is absolutely essential for DC or
low-frequency AC signal operation.
Some op-amps, such as the model 741, have a compensation capacitor built in to minimize
the need for external components. This improved simplicity is not without a cost: due to that
capacitor’s presence inside the op-amp, the negative feedback tends to get stronger as the
operating frequency increases (that capacitor’s reactance decreases with higher frequencies).
As a result, the op-amp’s differential voltage gain decreases as frequency goes up: it becomes
a less effective amplifier at higher frequencies.
Op-amp manufacturers will publish the frequency response curves for their products. Since
a sufficiently high differential gain is absolutely essential to good feedback operation in op-amp

8.13. PRACTICAL CONSIDERATIONS

405

circuits, the gain/frequency response of an op-amp effectively limits its ”bandwidth” of operation. The circuit designer must take this into account if good performance is to be maintained
over the required range of signal frequencies.
• REVIEW:
• Due to capacitances within op-amps, their differential voltage gain tends to decrease as
the input frequency increases. Frequency response curves for op-amps are available from
the manufacturer.

8.13.6

Input to output phase shift

In order to illustrate the phase shift from input to output of an operational amplifier (op-amp),
the OPA227 was tested in our lab. The OPA227 was constructed in a typical non-inverting
configuration (Figure 8.1).

Figure 8.1: OPA227 Non-inverting stage
The circuit configuration calls for a signal gain of ∼
=50 dB. The input excitation
=34 V/V or ∼
at Vsrc was set to 10 mVp, and three frequencies of interest: 2.2 kHz, 22 kHz, and 220 MHz.
The OPA227’s open loop gain and phase curve vs. frequency is shown in Figure 8.2.
To help predict the closed loop phase shift from input to output, we can use the open loop
gain and phase curve. Since the circuit configuration calls for a closed loop gain, or 1/β, of
∼
=50 dB, the closed loop gain curve intersects the open loop gain curve at approximately 22
kHz. After this intersection, the closed loop gain curve rolls off at the typical 20 dB/decade for
voltage feedback amplifiers, and follows the open loop gain curve.
What is actually at work here is the negative feedback from the closed loop modifies the
open loop response. Closing the loop with negative feedback establishes a closed loop pole at
22 kHz. Much like the dominant pole in the open loop phase curve, we will expect phase shift
in the closed loop response. How much phase shift will we see?
Since the new pole is now at 22 kHz, this is also the -3 dB point as the pole starts to roll off
the closed loop again at 20 dB per decade as stated earlier. As with any pole in basic control
theory, phase shift starts to occur one decade in frequency before the pole, and ends at 90o of
phase shift one decade in frequency after the pole. So what does this predict for the closed loop
response in our circuit?

406

CHAPTER 8. OPERATIONAL AMPLIFIERS

Figure 8.2: AV and Φ vs. Frequency plot
This will predict phase shift starting at 2.2 kHz, with 45o of phase shift at the -3 dB point of
22 kHz, and finally ending with 90o of phase shift at 220 kHz. The three Figures shown below
are oscilloscope captures at the frequencies of interest for our OPA227 circuit. Figure 8.3 is
set for 2.2 kHz, and no noticeable phase shift is present. Figure 8.4 is set for 220 kHz, and
∼
=45o of phase shift is recorded. Finally, Figure 8.5 is set for 220 MHz, and the expected ∼
=90o
of phase shift is recorded. The scope plots were captured using a LeCroy 44x Wavesurfer. The
final scope plot used a x1 probe with the trigger set to HF reject.

Figure 8.3: OPA227 Av=50dB @ 2.2 kHz

8.13. PRACTICAL CONSIDERATIONS

Figure 8.4: OPA227 Av=50dB @ 22 kHz

Figure 8.5: OPA227 Av=50dB @ 220 kHz

407

CHAPTER 8. OPERATIONAL AMPLIFIERS

408

8.14

Operational amplifier models

While mention of operational amplifiers typically provokes visions of semiconductor devices
built as integrated circuits on a miniature silicon chip, the first op-amps were actually vacuum
tube circuits. The first commercial, general purpose operational amplifier was manufactured
by the George A. Philbrick Researches, Incorporated, in 1952. Designated the K2-W, it was
built around two twin-triode tubes mounted in an assembly with an octal (8-pin) socket for easy
installation and servicing in electronic equipment chassis of that era. The assembly looked
something like this:

The Philbrick Researches
op-amp, model K2-W

approx.
4 inches
GAP/R
MODEL
K2-W

The schematic diagram shows the two tubes, along with ten resistors and two capacitors, a
fairly simple circuit design even by 1952 standards:
The Philbrick Researches op-amp, model K2-W
+300 V

220 kΩ

680 kΩ

510 kΩ

NE-68
12AX7

12AX7

7.5 pF

1 MΩ

Inverting (-)
input

2.2 MΩ
Noninverting (+)
input

Output
221 kΩ
220 kΩ

500 pF

9.1 kΩ

120 kΩ

4.7 MΩ

-300 V

In case you’re unfamiliar with the operation of vacuum tubes, they operate similarly to N-

8.14. OPERATIONAL AMPLIFIER MODELS

409

channel depletion-type IGFET transistors: that is, they conduct more current when the control
grid (the dashed line) is made more positive with respect to the cathode (the bent line near the
bottom of the tube symbol), and conduct less current when the control grid is made less positive
(or more negative) than the cathode. The twin triode tube on the left functions as a differential
pair, converting the differential inputs (inverting and noninverting input voltage signals) into
a single, amplified voltage signal which is then fed to the control grid of the left triode of
the second triode pair through a voltage divider (1 MΩ −− 2.2 MΩ). That triode amplifies
and inverts the output of the differential pair for a larger voltage gain, then the amplified
signal is coupled to the second triode of the same dual-triode tube in a noninverting amplifier
configuration for a larger current gain. The two neon ”glow tubes” act as voltage regulators,
similar to the behavior of semiconductor zener diodes, to provide a bias voltage in the coupling
between the two single-ended amplifier triodes.
With a dual-supply voltage of +300/-300 volts, this op-amp could only swing its output +/50 volts, which is very poor by today’s standards. It had an open-loop voltage gain of 15,000
to 20,000, a slew rate of +/- 12 volts/µsecond, a maximum output current of 1 mA, a quiescent
power consumption of over 3 watts (not including power for the tubes’ filaments!), and cost
about $24 in 1952 dollars. Better performance could have been attained using a more sophisticated circuit design, but only at the expense of greater power consumption, greater cost, and
decreased reliability.
With the advent of solid-state transistors, op-amps with far less quiescent power consumption and increased reliability became feasible, but many of the other performance parameters
remained about the same. Take for instance Philbrick’s model P55A, a general-purpose solidstate op-amp circa 1966. The P55A sported an open-loop gain of 40,000, a slew rate of 1.5
volt/µsecond and an output swing of +/- 11 volts (at a power supply voltage of +/- 15 volts),
a maximum output current of 2.2 mA, and a cost of $49 (or about $21 for the ”utility grade”
version). The P55A, as well as other op-amps in Philbrick’s lineup of the time, was of discretecomponent construction, its constituent transistors, resistors, and capacitors housed in a solid
”brick” resembling a large integrated circuit package.
It isn’t very difficult to build a crude operational amplifier using discrete components. A
schematic of one such circuit is shown in Figure 8.6.
While its performance is rather dismal by modern standards, it demonstrates that complexity is not necessary to create a minimally functional op-amp. Transistors Q3 and Q4 form
the heart of another differential pair circuit, the semiconductor equivalent of the first triode
tube in the K2-W schematic. As it was in the vacuum tube circuit, the purpose of a differential pair is to amplify and convert a differential voltage between the two input terminals to a
single-ended output voltage.
With the advent of integrated-circuit (IC) technology, op-amp designs experienced a dramatic increase in performance, reliability, density, and economy. Between the years of 1964
and 1968, the Fairchild corporation introduced three models of IC op-amps: the 702, 709, and
the still-popular 741. While the 741 is now considered outdated in terms of performance, it is
still a favorite among hobbyists for its simplicity and fault tolerance (short-circuit protection
on the output, for instance). Personal experience abusing many 741 op-amps has led me to the
conclusion that it is a hard chip to kill . . .
The internal schematic diagram for a model 741 op-amp is shown in Figure 8.7.
By integrated circuit standards, the 741 is a very simple device: an example of smallscale integration, or SSI technology. It would be no small matter to build this circuit using

CHAPTER 8. OPERATIONAL AMPLIFIERS

410

+V

Q1

Q2
Output

Q3

input (+)

Q4

Q5

(-) input

A simple operational
amplifier made from
discrete components

Q6

-V
Figure 8.6: A simple operational amplifier made from discrete components.

+V

Internal schematic of a model 741 operational amplifier

Q8

Q9

Q12

Q14

Q13

(-) input
Q1

(+) input

R5

Q2

Q15

Q18

R6
Output

Q19
Q3

R10

Q4

R7
Q21

C1
Q16

Q7
Q5

Q20

Q22
Q6

Q11

Q10

Q23

Q17

offset null
offset null
R1

R3

R2

R4

R9

R8

Q24

R11

-V

Figure 8.7: Schematic diagram of a model 741 op-amp.

8.14. OPERATIONAL AMPLIFIER MODELS

411

discrete components, so you can see the advantages of even the most primitive integrated
circuit technology over discrete components where high parts counts are involved.
For the hobbyist, student, or engineer desiring greater performance, there are literally hundreds of op-amp models to choose from. Many sell for less than a dollar apiece, even retail!
Special-purpose instrumentation and radio-frequency (RF) op-amps may be quite a bit more
expensive. In this section I will showcase several popular and affordable op-amps, comparing
and contrasting their performance specifications. The venerable 741 is included as a ”benchmark” for comparison, although it is, as I said before, considered an obsolete design.

Model
number
TL082
LM301A
LM318
LM324
LF353
LF356
LF411
741C
LM833
LM1458
CA3130

Table 8.1: Widely used operational amplifiers
Devices/ Power
Bandwidth Bias
Slew
package supply
current
rate
(count)
(V)
(MHz)
(nA)
(V/µS)
2
12 / 36
4
8
13
1
10 / 36
1
250
0.5
1
10 / 40
15
500
70
4
3 / 32
1
45
0.25
2
12 / 36
4
8
13
1
10 / 36
5
8
12
1
10 / 36
4
20
15
1
10 / 36
1
500
0.5
2
10 / 36
15
1050
7
2
6 / 36
1
800
10
1
5 / 16
15
0.05
10

Output
current
(mA)
17
25
20
20
20
25
25
25
40
45
20

Listed in Table 8.1 are but a few of the low-cost operational amplifier models widely available from electronics suppliers. Most of them are available through retail supply stores such as
Radio Shack. All are under $1.00 cost direct from the manufacturer (year 2001 prices). As you
can see, there is substantial variation in performance between some of these units. Take for
instance the parameter of input bias current: the CA3130 wins the prize for lowest, at 0.05 nA
(or 50 pA), and the LM833 has the highest at slightly over 1 µA. The model CA3130 achieves
its incredibly low bias current through the use of MOSFET transistors in its input stage. One
manufacturer advertises the 3130’s input impedance as 1.5 tera-ohms, or 1.5 x 1012 Ω! Other
op-amps shown here with low bias current figures use JFET input transistors, while the high
bias current models use bipolar input transistors.
While the 741 is specified in many electronic project schematics and showcased in many
textbooks, its performance has long been surpassed by other designs in every measure. Even
some designs originally based on the 741 have been improved over the years to far surpass
original design specifications. One such example is the model 1458, two op-amps in an 8-pin
DIP package, which at one time had the exact same performance specifications as the single
741. In its latest incarnation it boasts a wider power supply voltage range, a slew rate 50
times as great, and almost twice the output current capability of a 741, while still retaining
the output short-circuit protection feature of the 741. Op-amps with JFET and MOSFET input
transistors far exceed the 741’s performance in terms of bias current, and generally manage to
beat the 741 in terms of bandwidth and slew rate as well.

CHAPTER 8. OPERATIONAL AMPLIFIERS

412

My own personal recommendations for op-amps are as such: when low bias current is a
priority (such as in low-speed integrator circuits), choose the 3130. For general-purpose DC
amplifier work, the 1458 offers good performance (and you get two op-amps in the space of
one package). For an upgrade in performance, choose the model 353, as it is a pin-compatible
replacement for the 1458. The 353 is designed with JFET input circuitry for very low bias
current, and has a bandwidth 4 times are great as the 1458, although its output current limit
is lower (but still short-circuit protected). It may be more difficult to find on the shelf of your
local electronics supply house, but it is just as reasonably priced as the 1458.
If low power supply voltage is a requirement, I recommend the model 324, as it functions
on as low as 3 volts DC. Its input bias current requirements are also low, and it provides four
op-amps in a single 14-pin chip. Its major weakness is speed, limited to 1 MHz bandwidth and
an output slew rate of only 0.25 volts per µs. For high-frequency AC amplifier circuits, the 318
is a very good ”general purpose” model.
Special-purpose op-amps are available for modest cost which provide better performance
specifications. Many of these are tailored for a specific type of performance advantage, such as
maximum bandwidth or minimum bias current. Take for instance the op-amps, both designed
for high bandwidth in Table 8.2.

Model
number
CLC404
CLC425

Table 8.2: High bandwidth operational amplifiers
Devices/ Power
Bandwidth Bias
Slew
package supply
current
rate
(count)
(V)
(MHz)
(nA)
(V/µS)
1
10 / 14
232
44,000
2600
1
5 / 14
1900
40,000
350

Output
current
(mA)
70
90

The CLC404 lists at $21.80 (almost as much as George Philbrick’s first commercial opamp, albeit without correction for inflation), while the CLC425 is quite a bit less expensive at
$3.23 per unit. In both cases high speed is achieved at the expense of high bias currents and
restrictive power supply voltage ranges. Some op-amps, designed for high power output are
listed in Table 8.3.

Model
number
LM12CL
LM7171

Table 8.3: High current operational amplifiers
Devices/ Power
Bandwidth Bias
Slew
package supply
current
rate
(count)
(V)
(MHz)
(nA)
(V/µS)
1
15 / 80
0.7
1000
9
1
5.5 / 36
200
12,000
4100

Output
current
(mA)
13,000
100

Yes, the LM12CL actually has an output current rating of 13 amps (13,000 milliamps)!
It lists at $14.40, which is not a lot of money, considering the raw power of the device. The
LM7171, on the other hand, trades high current output ability for fast voltage output ability
(a high slew rate). It lists at $1.19, about as low as some ”general purpose” op-amps.
Amplifier packages may also be purchased as complete application circuits as opposed to
bare operational amplifiers. The Burr-Brown and Analog Devices corporations, for example,

8.15. DATA

413

both long known for their precision amplifier product lines, offer instrumentation amplifiers
in pre-designed packages as well as other specialized amplifier devices. In designs where high
precision and repeatability after repair is important, it might be advantageous for the circuit
designer to choose such a pre-engineered amplifier ”block” rather than build the circuit from
individual op-amps. Of course, these units typically cost quite a bit more than individual opamps.

8.15

Data

Parametrical data for all semiconductor op-amp models except the CA3130 comes from National Semiconductor’s online resources, available at this website: (http://www.national.com).
Data for the CA3130 comes from Harris Semiconductor’s CA3130/CA3130A datasheet (file
number 817.4).

Contributors
Contributors to this chapter are listed in chronological order of their contributions, from most
recent to first. See Appendix 2 (Contributor List) for dates and contact information.
Wayne Little (June 2007): Author, “Input to output phase shift” subsection, in “Practical
considerations” section.

414

CHAPTER 8. OPERATIONAL AMPLIFIERS

Chapter 9

PRACTICAL ANALOG
SEMICONDUCTOR CIRCUITS
Contents
9.1

ElectroStatic Discharge . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.1 ESD Damage Prevention . . . . . . . . . . . . . . . . . . . . . . . . .
9.1.2 Storage and Transportation of ESD sensitive component and boards
9.1.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2 Power supply circuits – INCOMPLETE . . . . . . . . . . . . . . . . . .
9.2.1 Unregulated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.2 Linear regulated . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.3 Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.2.4 Ripple regulated . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.3 Amplifier circuits – PENDING . . . . . . . . . . . . . . . . . . . . . . . .
9.4 Oscillator circuits – INCOMPLETE . . . . . . . . . . . . . . . . . . . . .
9.4.1 Varactor multiplier . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.5 Phase-locked loops – PENDING . . . . . . . . . . . . . . . . . . . . . . .
9.6 Radio circuits – INCOMPLETE . . . . . . . . . . . . . . . . . . . . . . .
9.7 Computational circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.8 Measurement circuits – INCOMPLETE . . . . . . . . . . . . . . . . . .
9.9 Control circuits – PENDING . . . . . . . . . . . . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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415
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*** INCOMPLETE ***

9.1

ElectroStatic Discharge

Volume I chapter 1.1 discusses static electricity, and how it is created. This has a lot more
significance than might be first assumed, as control of static electricity plays a large part in
415

416

CHAPTER 9. PRACTICAL ANALOG SEMICONDUCTOR CIRCUITS

modern electronics and other professions. An ElectroStatic Discharge event is when a static
charge is bled off in an uncontrolled fashion, and will be referred to as ESD hereafter.
ESD comes in many forms, it can be as small as 50 volts of electricity being equalized up
to tens of thousands of volts. The actual power is extremely small, so small that no danger
is generally offered to someone who is in the discharge path of ESD. It usually takes several
thousand volts for a person to even notice ESD in the form of a spark and the familiar zap
that accompanies it. The problem with ESD is even a small discharge that can go completely
unnoticed can ruin semiconductors. A static charge of thousands of volts is common, however
the reason it is not a threat is there is no current of any substantial duration behind it. These
extreme voltages do allow ionization of the air and allow other materials to break down, which
is the root of where the damage comes from.
ESD is not a new problem. Black powder manufacturing and other pyrotechnic industries
have always been dangerous if an ESD event occurs in the wrong circumstance. During the
era of tubes (AKA valves) ESD was a nonexistent issue for electronics, but with the advent of
semiconductors, and the increase in miniaturization, it has become much more serious.
Damage to components can, and usually do, occur when the part is in the ESD path. Many
parts, such as power diodes, are very robust and can handle the discharge, but if a part has
a small or thin geometry as part of their physical structure then the voltage can break down
that part of the semiconductor. Currents during these events become quite high, but are in the
nanosecond to microsecond time frame. Part of the component is left permanently damaged
by this, which can cause two types of failure modes. Catastrophic is the easy one, leaving
the part completely nonfunctional. The other can be much more serious. Latent damage may
allow the problem component to work for hours, days or even months after the initial damage
before catastrophic failure. Many times these parts are referred to as ”walking wounded”, since
they are working but bad. Figure 9.1 is shown an example of latent (”walking wounded”) ESD
damage. If these components end up in a life support role, such as medical or military use,
then the consequences can be grim. For most hobbyists it is an inconvenience, but it can be an
expensive one.
Even components that are considered fairly rugged can be damaged by ESD. Bipolar transistors, the earliest of the solid state amplifiers, are not immune, though less susceptible. Some
of the newer high speed components can be ruined with as little as 3 volts. There are components that might not be considered at risk, such as some specialized resistors and capacitors
manufactured using MOS (Metal Oxide Semiconductor) technology, that can be damaged via
ESD.

9.1.1

ESD Damage Prevention

Before ESD can be prevented it is important to understand what causes it. Generally materials
around the workbench can be broken up into 3 categories. These are ESD Generative, ESD
Neutral, and ESD Dissipative (or ESD Conductive). ESD Generative materials are active static
generators, such as most plastics, cat hair, and polyester clothing. ESD Neutral materials are
generally insulative, but don’t tend to generate or hold static charges very well. Examples of
this include wood, paper, and cotton. This is not to say they can not be static generators or an
ESD hazard, but the risk is somewhat minimized by other factors. Wood and wood products,
for example, tend to hold moisture, which can make them slightly conductive. This is true of
a lot of organic materials. A highly polished table would not fall under this category, because

9.1. ELECTROSTATIC DISCHARGE

417

Figure 9.1:

the gloss is usually plastic, or varnish, which are highly efficient insulators. ESD Conductive
materials are pretty obvious, they are the metal tools laying around. Plastic handles can be
a problem, but the metal will bleed a static charge away as fast as it is generated if it is on a
grounded surface. There are a lot of other materials, such as some plastics, that are designed
to be conductive. They would fall under the heading of ESD Dissipative. Dirt and concrete are
also conductive, and fall under the ESD Dissipative heading.
There are a lot of activities that generate static, which you need to be aware of as part of
an ESD control regimen. The simple act of pulling tape off a dispenser can generate extreme
voltage. Rolling around in a chair is another static generator, as is scratching. In fact, any
activity that allows 2 or more surfaces to rub against each other is pretty certain to generate
some static charge. This was mentioned in the beginning of this book, but real world examples
can be subtle. This is why a method for continuously bleeding off this voltage is needed. Things
that generate huge amounts of static should be avoided while working on components.
Plastic is usually associated with the generation of static. This has been gotten around in
the form of conductive plastics. The usual way to make conductive plastic is an additive that
changes the electrical characteristics of the plastic from an insulator to a conductor, although
it will likely still have a resistance of millions of ohms per square inch. Plastics have been
developed that can be used as conductors is in low weight applications, such as those in the
airline industries. These are specialist applications, and are not generally associated with ESD
control.
It is not all bad news for ESD protection. The human body is a pretty decent conductor.
High humidity in the air will also allow a static charge to dissipate harmlessly away, as well
as making ESD Neutral materials more conductive. This is why cold winter days, where the
humidity inside a house can be quite low, can increase the number of sparks on a doorknob.
Summer, or rainy days, you would have to work quite hard to generate a substantial amount of
static. Industry clean rooms and factory floors go the effort to regulate both temperature and
humidity for this reason. Concrete floors are also conductive, so there may be some existing
components in the home that can aid in setting up protections.

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CHAPTER 9. PRACTICAL ANALOG SEMICONDUCTOR CIRCUITS

To establish ESD protection there has to be a standard voltage level that everything is
referenced to. Such a level exists in the form of ground. There are very good safety reasons
that ground is used around the house in outlets. In some ways this relates to static, but not
directly. It does give us a place to dump our excess electrons, or acquire some if we are short,
to neutralize any charges our bodies and tools might acquire. If everything on a workbench is
connected directly or indirectly to ground via a conductor then static will dissipate long before
an ESD event has a chance to occur.
A good grounding point can be made several different ways. In houses with modern wiring
that is up to code the ground pin on the AC plug in can be used, or the screw that holds the
outlets cover plate on. This is because house wiring actually has a wire or spike going into
the earth somewhere where the power is tapped from the main power lines. For people whose
house wiring isn’t quite right a spike driven into the earth at least 3 feet or a simple electrical
connection to metal plumbing (worst option) can be used. The main thing is to establish an
electrical path to the earth outside the house.
Ten megohms is considered a conductor in the world of ESD control. Static electricity is
voltage with no real current, and if a charge is bled off seconds after being generated it is
nullified. Generally a 1 to 10 megohm resistor is used to connect any ESD protection for this
reason. It has the benefit of slowing the discharge rate during an ESD event, which increases
the likelihood of a component surviving undamaged. The faster the discharge, the higher the
current spike going though the component. Another reason such a resistance is considered
desirable is if the user is accidentally shorted to high voltage, such as household current, it
won’t be the ESD protections that kill them.
A large industry has grown up around controlling ESD in the electronics industry. The
staple of any electronics construction is the workbench with a static conductive or dissipative
surface. This surface can be bought commercially, or home made in the form of a sheet of metal
or foil. In the case of a metal surface it might be a good idea to lay thin paper on top, although it
is not necessary if you are not doing any powered tests on the surface. The commercial version
is usually some form of conductive plastic whose resistance is high enough not to be a problem,
which is a better solution. If you are making your own surface for the workbench be sure to
add the 10 megohm resistor to ground, otherwise you have no protection at all.
The other big item that needs ESD grounded is you. People are walking static generators.
Your body being conductive it is relatively easy to ground it though, this is usually done with
a wrist strap. Commercial versions already have the resistor built in, and have a wide strap to
offer a good contact surface with your skin. Disposable versions can be bought for a few dollars.
A metal watchband is also a good ESD protection connection point. Just add a wire (with the
resistor) to your grounding point. Most industries take the issue seriously enough to use real
time monitors that will sound an alarm if the operator is not properly grounded.
Another way of grounding yourself is a heel strap. A conductive plastic part is wrapped
around the heel of your shoe, with a conductive plastic strap going up and under your sock for
good contact with the skin. It only works on floors with conductive wax or concrete. The method
will keep a person from generating large charges that can overwhelm other ESD protections,
and is not considered adequate in and of itself. You can get the same effect by walking barefoot
on a concrete floor.
Yet another ESD protection is to wear ESD conductive smocks. Like the heel strap, this is
a secondary protection, not meant to replace the wrist strap. They are meant to short circuit
any charges that your clothes may generate.

9.1. ELECTROSTATIC DISCHARGE

419

Figure 9.2:
Moving air can also generate substantial static charges. When you blow dust off your electronics their will be static generated. An industrial solution to the problem to this issue is two
fold: Firstly, air guns have a small, well shielded radioactive material implanted within the
air gun to ionize the air. Ionized air is a conductor, and will bleed off static charges quite well.
Secondly, use high voltage electricity to ionize the air coming out of a fan, which has the same
effect as the air gun. This will effectively help a workstation reduce the potential for ESD
generation by a large amount.
Another ESD protection is the simplest of all, distance. Many industries have rules stating
all Neutral and Generative materials will be at least 12 inches or more from any work in
progress.
The user can also reduce the possibility of ESD damage by simply not removing the part
out of its protective packaging until it is time to insert it into the circuit. This will reduce the
likelihood of ESD exposure, and while the circuit will still be vulnerable, the component will
have some minor protection from the rest of the components, as the other components will offer
different discharge paths for ESD.

9.1.2

Storage and Transportation of ESD sensitive component and
boards

It does no good to follow ESD protections on the workbench if the parts are being damaged
while storing or carrying them. The most common method is to use a variation of a Faraday
cage, an ESD bag. An ESD bag surrounds the component with a conductive shield, and usually
has a non static generating insulative layer inside. In permanent Faraday cages this shield is
grounded, as in the case of RFI rooms, but with portable containers this isn’t practical. By

420

CHAPTER 9. PRACTICAL ANALOG SEMICONDUCTOR CIRCUITS

putting a ESD bag on a grounded surface the same thing is accomplished. Faraday cages work
by routing the electric charge around the contents and grounding them immediately. A car
struck by lightning is an extreme example of a Faraday cage.
Static bags are by far the most common method of storing components and boards. They
are made using extremely thin layers of metal, so thin as to be almost transparent. A bag with
a hole, even small ones, or one that is not folded on top to seal the content from outside charges
is ineffective.
Another method of protecting parts in storage is totes or tubes. In these cases the parts are
put into conductive boxes, with a lid of the same material. This effectively forms a Faraday
cage. A tube is meant for ICs and other devices with a lot of pins, and stores the parts in a
molded conductive plastic tube that keeps the parts safe both mechanically and electrically.

Figure 9.3:

9.1.3

Conclusion

ESD can be a minor unfelt event measuring a few volts, or a massive event presenting real
dangers to operators. All ESD protections can be overwhelmed by circumstance, but this can
be circumvented by awareness of what it is and how to prevent it. Many projects have been
built with no ESD protections at all and worked well. Given that protecting these projects is a
minor inconvenience it is better to make the effort.
Industry takes the problem very seriously, as both a potential life threatening issue and a
quality issue. Someone who buys an expensive piece of electronics or high tech hardware is not
going to be happy if they have to return it in 6 months. When a reputation is on the line it is
easier to do the right thing.

9.2

Power supply circuits – INCOMPLETE

There are three major kinds of power supplies: unregulated (also called brute force), linear
regulated, and switching. A fourth type of power supply circuit called the ripple-regulated, is a
hybrid between the ”brute force” and ”switching” designs, and merits a subsection to itself.

9.2. POWER SUPPLY CIRCUITS – INCOMPLETE

9.2.1

421

Unregulated

An unregulated power supply is the most rudimentary type, consisting of a transformer, rectifier, and low-pass filter. These power supplies typically exhibit a lot of ripple voltage (i.e.
rapidly-varying instability) and other AC ”noise” superimposed on the DC power. If the input
voltage varies, the output voltage will vary by a proportional amount. The advantage of an
unregulated supply is that its cheap, simple, and efficient.

9.2.2

Linear regulated

A linear regulated supply is simply a ”brute force” (unregulated) power supply followed by a
transistor circuit operating in its ”active,” or ”linear” mode, hence the name linear regulator.
(Obvious in retrospect, isn’t it?) A typical linear regulator is designed to output a fixed voltage
for a wide range of input voltages, and it simply drops any excess input voltage to allow a
maximum output voltage to the load. This excess voltage drop results in significant power
dissipation in the form of heat. If the input voltage gets too low, the transistor circuit will lose
regulation, meaning that it will fail to keep the voltage steady. It can only drop excess voltage,
not make up for a deficiency in voltage from the brute force section of the circuit. Therefore, you
have to keep the input voltage at least 1 to 3 volts higher than the desired output, depending
on the regulator type. This means the power equivalent of at least 1 to 3 volts multiplied by
the full load current will be dissipated by the regulator circuit, generating a lot of heat. This
makes linear regulated power supplies rather inefficient. Also, to get rid of all that heat they
have to use large heat sinks which makes them large, heavy, and expensive.

9.2.3

Switching

A switching regulated power supply (”switcher”) is an effort to realize the advantages of both
brute force and linear regulated designs (small, efficient, and cheap, but also ”clean,” stable
output voltage). Switching power supplies work on the principle of rectifying the incoming AC
power line voltage into DC, re-converting it into high-frequency square-wave AC through transistors operated as on/off switches, stepping that AC voltage up or down by using a lightweight
transformer, then rectifying the transformer’s AC output into DC and filtering for final output. Voltage regulation is achieved by altering the ”duty cycle” of the DC-to-AC inversion on
the transformer’s primary side. In addition to lighter weight because of a smaller transformer
core, switchers have another tremendous advantage over the prior two designs: this type of
power supply can be made so totally independent of the input voltage that it can work on any
electric power system in the world; these are called ”universal” power supplies.
The downside of switchers is that they are more complex, and due to their operation they
tend to generate a lot of high-frequency AC ”noise” on the power line. Most switchers also have
significant ripple voltage on their outputs. With the cheaper types, this noise and ripple can be
as bad as for an unregulated power supply; such low-end switchers aren’t worthless, because
they still provide a stable average output voltage, and there’s the ”universal” input capability.
Expensive switchers are ripple-free and have noise nearly as low as for some a linear types;
these switchers tend to be as expensive as linear supplies. The reason to use an expensive
switcher instead of a good linear is if you need universal power system compatibility or high

422

CHAPTER 9. PRACTICAL ANALOG SEMICONDUCTOR CIRCUITS

efficiency. High efficiency, light weight, and small size are the reasons switching power supplies
are almost universally used for powering digital computer circuitry.

9.2.4

Ripple regulated

A ripple-regulated power supply is an alternative to the linear regulated design scheme: a
”brute force” power supply (transformer, rectifier, filter) constitutes the ”front end” of the circuit, but a transistor operated strictly in its on/off (saturation/cutoff) modes transfers DC power
to a large capacitor as needed to maintain the output voltage between a high and a low setpoint. As in switchers, the transistor in a ripple regulator never passes current while in its
”active,” or ”linear,” mode for any substantial length of time, meaning that very little energy
will be wasted in the form of heat. However, the biggest drawback to this regulation scheme is
the necessary presence of some ripple voltage on the output, as the DC voltage varies between
the two voltage control setpoints. Also, this ripple voltage varies in frequency depending on
load current, which makes final filtering of the DC power more difficult.
Ripple regulator circuits tend to be quite a bit simpler than switcher circuitry, and they
need not handle the high power line voltages that switcher transistors must handle, making
them safer to work on.

9.3

Amplifier circuits – PENDING

Note, Q3 and Q4 in Figure 9.4 are complementary, NPN and PNP respectively. This circuit
works well for moderate power audio amplifiers. For an explanation of this circuit see “Direct
coupled complementary-pair,”
(page 255).

9.4

Oscillator circuits – INCOMPLETE

The phase shift oscillator of Figure 9.5 produces a sinewave output in the audio frequency
range. Resistive feedback from the collector would be negative feedback due to 180o phasing
(base to collector phase inversion). However, the three 60o RC phase shifters ( R1 C1 , R2 C2 ,
and R3 C3 ) provide an additional 180o for a total of 360o . This in-phase feedback constitutes
positive feedback. Oscillations result if transistor gain exceeds feedback network losses.

9.4.1

Varactor multiplier

A Varactor or variable capacitance diode with a nonlinear capacitance vs frequency characteristic distorts the applied sinewave f1 in Figure 9.6, generating harmonics, f3.
The fundamental filter passes f1, blocking the harmonics from returning to the generator.
The choke passes DC, and blocks radio frequencies (RF) from entering the Vbias supply. The
harmonic filter passes the desired harmonic, say the 3rd, to the output, f3. The capacitor at
the bottom of the inductor is a large value, low reactance, to block DC but ground the inductor

9.4. OSCILLATOR CIRCUITS – INCOMPLETE

423

+22 V
R1
39 kΩ
input
C1

R4
390 Ω

R3
47 kΩ

Q3

Q2

C2
25 µF
Q1

R6 2.2 kΩ

R9
2.2 Ω

C4
3.3 nF

R10
2.2 Ω

C5
4000 µF

220 nF

R2

C3
250 µF

120 kΩ

R7
15 Ω

Q4
R5

R8

22 Ω

560 Ω

Figure 9.4: Direct coupled complementary symmetry 3 w audio amplifier. After Mullard. [2]

Vcc
R5
C1

R1

C2

R2

R6

C3

R3

R4

C4

Figure 9.5: Phase shift oscillator. R1 C1 , R2 C2 , and R3 C3 each provide 60o of phase shift.

CHAPTER 9. PRACTICAL ANALOG SEMICONDUCTOR CIRCUITS

424

Vbias
RF blocking
choke
capacitance

fundamental
filter
varactor
diode

f1

harmonic
filter
Resonant
inductor

f3

DC blocking
capacitor

voltage

Figure 9.6: Varactor diode, having a nonlinear capacitance vs voltage characteristic, serves in
frequency multiplier.
for RF. The varicap diode in parallel with the indctor constitutes a parallel resonant network.
It is tuned to the desired harmonic. Note that the reverse bias, Vbias , is fixed.
The varicap multiplier is primarily used to generate microwave signals which cannot be
directly produced by oscillators. The lumped circuit representation in Figure 9.6 is actually
stripline or waveguide sections. Frequenies up to hundreds of gHz may be produced by varactor
multipliers.

9.5

Phase-locked loops – PENDING

9.6

Radio circuits – INCOMPLETE

(b)
L1
240
µΗ

C1
365
pF

(a)

C2
1000
pf

(c)
(d)

Figure 9.7: (a) Crystal radio. (b) Modulated RF at antenna. (c) Rectified RF at diode cathode,
without C2 filter capacitor. (d) Demodualted audio to headphones.

An antenna ground system, tank circuit, peak detector, and headphones are the the main
components of a crystal radio. See Figure 9.7 (a). The antenna absorbs transimtted radio
signals (b) which flow to ground via the other components. The combination of C1 and L1
comprise a resonant circuit, refered to as a tank circuit. Its purpose is to select one out of many

9.6. RADIO CIRCUITS – INCOMPLETE

425

available radios signals. The variable capacitor C1 allows for tuning to the various signals.
The diode passes the positive half cycles of the RF, removing the negative half cycles (c). C2
is sized to filter the radio frequencies from the RF envelope (c), passing audio frequencies (d)
to the headset. Note that no power supply is required for a crystal radio. A germanium diode,
which has a lower forward voltage drop provides greater sensitvity than a silicon diode.
While 2000Ω magnetic headphones are shown above, a ceramic earphone, sometimes called
a crystal earphone, is more sensitive. The ceramic earphone is desirable for all but the strongest
radio signals
The circuit in Figure 9.8 produces a stronger output than the crystal detector. Since the
transistor is not biased in the linear region (no base bias resistor), it only conducts for positive
half cycles of RF input, detecting the audio modulation. An advantage of a transistor detector
is amplification in addition to detection. This more powerful circuit can readily drive 2000Ω
magnetic headphones. Note the transistor is a germanuim PNP device. This is probably more
sensitive, due to the lower 0.2V VBE , compared with silicon. However, a silicon device should
still work. Reverse battery polarity for NPN silicon devices.

2000Ω double headphones

365
pF
Ge
transistor

5nf
1.5V
+

-

Coil - #34 AWG magnet wire
close wound over 1 in. length on
1 1/4 in. dia. form. Tap 1/4 in.
from bottom.

Figure 9.8: TR One, one transistor radio. No-bias-resistor causes operation as a detector. After
Stoner, Figure 4.4A. [8]
The 2000Ω headphones are no longer a widely available item. However, the low impedance
earbuds commonly used with portable audio equipment may be substituted when paired with
a suitable audio transformer. See Volume 6 Experiments, AC Circuits, Sensitive audio detector
for details.
The circuit in Figure 9.9 adds an audio amplifier to the crystal detector for greater headphone volume. The original circuit used a germanium diode and transistor. [8] A schottky
diode may be substituted for the germanium diode. A silicon transistor may be used if the
base-bias resistor is changed according to the table.
For more crystal radio circuits, simple one-transistor radios, and more advanced low transistor count radios, see Wenzel [9]
The circuit in Figure 9.11 is an integrated circuit AM radio containing all the active radio
frequency circuitry within a single IC. All capacitors and inductors, along with a few resistors,
are external to the IC. The 370 Pf variable capacitor tunes the desired RF signal. The 320
pF variable capacitor tunes the local oscillator 455 KHz above the RF input signal. The RF
signal and local oscillator frequencies mix producing the sun and difference of the two at pin
15. The external 455 KHz ceramic filter between pins 15 and 12, selects the 455 KHz difference
frequency. Most of the amplification is in the intermediate frequency (IF) amplifier between

426

CHAPTER 9. PRACTICAL ANALOG SEMICONDUCTOR CIRCUITS
2000Ω double
headphones

5
nf

365
pF
Ge diode

500
pF

1.5-6V

−
+

Resistor
1.5V 6V
Ge 47k 220k
Si 120k 1Meg
Coil - #34 AWG magnet
wire close wound over
1 in. length on 1 1/4 in.
dia. form. Tap 1/4 in.
from bottom.

Figure 9.9: Crystal radio with one transistor audio amplifer, base-bias. After Stoner, Figure
4.3A. [8]
pins 12 and 7. A diode at pin 7 recovers audio from the IF. Some automatic gain control (AGC)
is recovered and filtered to DC and fed back into pin 9.
Figure 9.12 shows conventional mecahnical tuning (a) of the RF input tuner and the local
oscillator with varactor diode tuning (b). The meshed plates of a dual variable capacitor make
for a bulky component. It is ecconomic to replace it with varicap tuning diodes. Increasing the
reverse bias Vtune decreases capacitance which increases frequency. Vtune could be produced
by a potentiometer.
Figure 9.13 shows an even lower parts count AM radio. Sony engineers have included the
intermediate frequency (IF) bandpass filter within the 8-pin IC. This eliminates external IF
transformers and an IF ceramic filter. L-C tuning components are still required for the radio
frequency (RF) input and the local oscillator. Though, the variable capacitors could be replaced
by varicap tuning diodes.
Figure 9.14 shows a low-parts-count FM radio based on a TDA7021T integrated circuit by
NXP Wireless. The bulky external IF filter transformers have been replaced by R-C filters.
The resistors are integrated, the capacitors external. This circuit has been simplified from
Figure 5 in the NXP Datasheet. See Figure 5 or 8 of the datasheet for the omitted signal
strength circuit. The simple tuning circuit is from the Figure 5 Test Circuit. Figure 8 has a
more elaborate tuner. Datasheet Figure 8 shows a stereo FM radio with an audio amplifier for
driving a speaker. [7]
For a construction project, the simplified FM Radio in Figure 9.14 is recommended. For the
56nH inductor, wind 8 turns of #22 AWG bare wire or magnet wire on a 0.125 inch drill bit or
other mandrel. Remove the mandrel and strech to 0.6 inch length. The tuning capacitor may
be a miniature trimmer capacitor.
Figure 9.15 is an example of a common-base (CB) RF amplifier. It is a good illustration
because it looks like a CB for lack of a bias network. Since there is no bias, this is a class C
amplifier. The transistor conducts for less than 180o of the input signal because at least 0.7 V
bias would be required for 180o class B. The common-base configuration has higher power gain
at high RF frequencies than common-emitter. This is a power amplifier (3/4 W) as opposed to
a small signal amplifier. The input and output π-networks match the emitter and collector to
the 50 Ω input and output coaxial terminations, respectively. The output π-network also helps

9.6. RADIO CIRCUITS – INCOMPLETE

Figure 9.10: Regency TR1: First mass produced transistor radio, 1954.

560 Ω

560 Ω

268
pF
volume
1K
2 µF

20,000 pF
Q2

Q1

− +

Q3

10,000
pF
50,000
pF

1000 pF

50,000
pF

1000 pF

20,000
pF

Q4

1000
pF

3.9 K
1K

10 K
40
µF

1000
pF
2.2 K

2.7 K
560 Ω
2.2 K

2.7 K

+

40 µF

−

33 K

2.2 K

470 K

100 K

+
−

5 µF

+

22.5 V

−

427

CHAPTER 9. PRACTICAL ANALOG SEMICONDUCTOR CIRCUITS

428

370pF
Vcc
16

RF

14

TCA440

IF

2

RF in

AF

7

1

10nF

47pF
6
5
4

330pF

9
Osc.

8

15

11

13

3

10

39K

47pF

Vcc

320pF

12

100nF

5µF

100nF
100nF

1.5nF

8.2K

455 kHz
Ceramic filter

Vcc

25µF

12K

3.3nF

Figure 9.11: IC radio, After Signetics [3]

370pF

TCA440

TCA440

BB113
RF in

2

RF in
10nF

6
5
4

330pF

2
330nF

1 RF

Osc
BB113

Vcc
(a)

+Vtune

1 RF
6
5
4

330pF

8

320pF

10nF

Osc

8

270K

Vcc

(b)

Figure 9.12: IC radio comparison of (a) mechanical tuning to (b) electronic varicap diode
tuning.[3]

9.6. RADIO CIRCUITS – INCOMPLETE

429

4.7
µF 130
µH

160
µF

1500
µF

160
µF

3V
220
µF

560
µH

8

7

RF in

6

OL AGC

-

5
Lo Osc

Vcc

RF Amp

Mixer

Osc

Overload
& AGC

BPF

IF Amp

AGC

+

Detector

CXA1600MP
Audio
Amp

Vol
AGC

Vol

1

Gnd

2
22
µF

3
100K

Audio

4
0.1
µF

220µF
22
µF

Figure 9.13: Compact IC radio eliminates external IF filters. After Sony [4]

CHAPTER 9. PRACTICAL ANALOG SEMICONDUCTOR CIRCUITS

antenna

audio

430

100
nF
16

3.3
nF
15

220
pF
14

13

4.7
nF

220
pF
12

Field
strength
11

10

9

7

8

RF
TDA7021T

Mixer

Demodulator
VCO
1

2

3V
+

10
nF

100
nF

3

4
10
nF

56
nH

IF
5

6
100
nF
40
pF

1.5
nF

820
pF

Figure 9.14: IC FM radio, signal strength circuit not shown. After NXP Wireless Figure 5. [7]

9.6. RADIO CIRCUITS – INCOMPLETE

431

filter harmonics generated by the class C amplifier. Though, more sections would likely be
required by modern radiated emissions standards.
L1
25nH

100pF

45-380
pF

L2
25nH

2N2863

1.2µH
RFC

9-180
pF

8-60
pF

100pF

0.68µH
RFC
100pF

9-180
pF
+10 V

Figure 9.15: Class C common-base 750 mW RF power amplifier. L1 = #10 Cu wire 1/2 turn, 5/8
in. ID by 3/4 in. high. L2 = #14 tinned Cu wire 1 1/2 turns, 1/2 in. ID by 1/3 in. spacing. After
Texas Instruments [5]
An example of a high gain common-base RF amplifier is shown in Figure 9.16. The commonbase circuit can be pushed to a higher frequency than other configurations. This is a common
base configuration because the transistor bases are grounded for AC by 1000 pF capacitors.
The capacitors are necessary (unlike the class C, Figure 9.15) to allow the 1KΩ-4KΩ voltage
divider to bias the transistor base for class A operation. The 500Ω resistors are emitter bias
resistors. They stablize the collector current. The 850Ω resistors are collector DC loads. The
three stage amplifier provides an overall gain of 38 dB at 100 MHz with a 9 MHz bandwidth.
4-30
pF

68
pF
10nH

80nH
2N1141

4-30
pF

500
Ω

1000
pF

80nH
2N1141

500
Ω

1000
pF
1K
4K

4-30
pF
1000
pF

2N1141
100nH
1000
pF

500
Ω
1000
pF

820
Ω

4-30
pF

1K
4K

1000
pF

820
Ω

1K
4K

820
Ω
-25 V

2
nF

100µH RFC

2
nF

100µH RFC

2
nF

Figure 9.16: Class A common-base small-signal high gain amplifier. After Texas Instruments
[6]
A cascode amplifier has a wide bandwdth like a common-base amplifier and a moderately
high input impedance like a common emitter arrangement. The biasing for this cascode amplifier (Figure 9.17) is worked out in an example problem (page 246).
This circuit (Figure 9.17) is simulated in the “Cascode” section of the BJT chapter (page
219). Use RF or microwave transistors for best high frequency response.

CHAPTER 9. PRACTICAL ANALOG SEMICONDUCTOR CIRCUITS

432

VCC
RL
4.7k

R1
150k

C2
10nF

R2
220k
C1
10nF
Vi

20V

R3
1Meg

Vo

Q1

C3
10nF

Q2

R4
87k

VB2

Figure 9.17: Class A cascode small-signal high gain amplifier.

10 V

10 V

1K

1K

Transmitter

Receiver
10 V
Transmit
Receive

1K

Figure 9.18: PIN diode T/R switch disconnects receiver from antenna during transmit.

left antenna

1K

right antenna

right

5V

RFC

1K
Receiver

left

-5 V

1K

Figure 9.19: PIN diode antenna switch for direction finder receiver.

9.7. COMPUTATIONAL CIRCUITS

433
Vcontrol = 0 to 5 V

47 nF

47 nF

330

150

330
1.25 V

47 nF

150
330

47 nF

47 nF

Figure 9.20: PIN diode attenuator: PIN diodes function as voltage variable resistors. After Lin
[1].

The PIN diodes are arranged in a π-attenuator network. The anti-series diodes cancel some
harmonic distortion compared with a single series diode. The fixed 1.25 V supply forward
biases the parallel diodes, which not only conducting DC current from ground via the resistors,
but also, conduct RF to ground through the diodes’ capacitors. The control voltage Vcontrol ,
increases current through the parallel diodes as it increases. This decreases the resistance
and attenuation, passing more RF from input to output. Attenuation is about 3 dB at Vcontrol =
5 V. Attenuation is 40 dB at Vcontrol = 1 V with flat frequency response to 2 gHz. At Vcontrol =
0.5 V, attenuation is 80 dB at 10 MHz. However, the frequency response varies too much to
use. [1]

9.7

Computational circuits

When someone mentions the word ”computer,” a digital device is what usually comes to mind.
Digital circuits represent numerical quantities in binary format: patterns of 1’s and 0’s represented by a multitude of transistor circuits operating in saturated or cutoff states. However,
analog circuitry may also be used to represent numerical quantities and perform mathematical
calculations, by using variable voltage signals instead of discrete on/off states.
Here is a simple example of binary (digital) representation versus analog representation of
the number ”twenty-five:”

434

CHAPTER 9. PRACTICAL ANALOG SEMICONDUCTOR CIRCUITS

A digital circuit representing the number 25:
1
2
4
8
16
32

16 + 8 + 1 = 25

An analog circuit representing the number 25:

Voltmeter
100 V

0

50

100

Digital circuits are very different from circuits built on analog principles. Digital computational circuits can be incredibly complex, and calculations must often be performed in
sequential ”steps” to obtain a final answer, much as a human being would perform arithmetical calculations in steps with pencil and paper. Analog computational circuits, on the other
hand, are quite simple in comparison, and perform their calculations in continuous, real-time
fashion. There is a disadvantage to using analog circuitry to represent numbers, though: imprecision. The digital circuit shown above is representing the number twenty-five, precisely.
The analog circuit shown above may or may not be exactly calibrated to 25.000 volts, but is
subject to ”drift” and error.

9.7. COMPUTATIONAL CIRCUITS

435

In applications where precision is not critical, analog computational circuits are very practical and elegant. Shown here are a few op-amp circuits for performing analog computation:

Analog summer (adder) circuit
1 kΩ

1 kΩ
−

R

Input2

Output

+

Input1
R

Output = Input1 + Input2

Analog subtractor circuit
R

R

Input(-)
−
Output

+
R

R

Input(+)

Output = Input(+) - Input(-)

436

CHAPTER 9. PRACTICAL ANALOG SEMICONDUCTOR CIRCUITS

Analog averager circuit

−
R

+

Input1
Input2

Output
(Buffer optional)

R

Output =

Input1 + Input2
2

Analog inverter (sign reverser) circuit
R

R

Input
−
+

Output

Output = - Input

Analog "multiply-by-constant" circuit
K

−
Input

+

Output = (K)(Input)

Output

9.7. COMPUTATIONAL CIRCUITS

437

Analog "divide-by-constant" circuit

−

Input

+

K

Output
(Buffer optional)

Input
Output =

K

Analog inverting "multiply/divideby-constant" circuit
K
Input
−
+

Output

Output = - (K)(Input)

Each of these circuits may be used in modular fashion to create a circuit capable of multiple
calculations. For instance, suppose that we needed to subtract a certain fraction of one variable
from another variable. By combining a divide-by-constant circuit with a subtractor circuit, we
could obtain the required function:

CHAPTER 9. PRACTICAL ANALOG SEMICONDUCTOR CIRCUITS

438

Divide-by-constant
−

Input2
K

R

Subtractor
R

+
−
Output

+
R

R

Input1
Output = Input1 -

Input2
K

Devices called analog computers used to be common in universities and engineering shops,
where dozens of op-amp circuits could be ”patched” together with removable jumper wires to
model mathematical statements, usually for the purpose of simulating some physical process
whose underlying equations were known. Digital computers have made analog computers all
but obsolete, but analog computational circuitry cannot be beaten by digital in terms of sheer
elegance and economy of necessary components.
Analog computational circuitry excels at performing the calculus operations integration and
differentiation with respect to time, by using capacitors in an op-amp feedback loop. To fully
understand these circuits’ operation and applications, though, we must first grasp the meaning
of these fundamental calculus concepts. Fortunately, the application of op-amp circuits to realworld problems involving calculus serves as an excellent means to teach basic calculus. In
the words of John I. Smith, taken from his outstanding textbook, Modern Operational Circuit
Design:
”A note of encouragement is offered to certain readers: integral calculus is one of
the mathematical disciplines that operational [amplifier] circuitry exploits and, in
the process, rather demolishes as a barrier to understanding.” (pg. 4)
Mr. Smith’s sentiments on the pedagogical value of analog circuitry as a learning tool for
mathematics are not unique. Consider the opinion of engineer George Fox Lang, in an article
he wrote for the August 2000 issue of the journal Sound and Vibration, entitled, ”Analog was
not a Computer Trademark!”:
”Creating a real physical entity (a circuit) governed by a particular set of equations and interacting with it provides unique insight into those mathematical statements. There is no better way to develop a ”gut feel” for the interplay between physics
and mathematics than to experience such an interaction. The analog computer was
a powerful interdisciplinary teaching tool; its obsolescence is mourned by many educators in a variety of fields.” (pg. 23)

9.7. COMPUTATIONAL CIRCUITS

439

Differentiation is the first operation typically learned by beginning calculus students. Simply put, differentiation is determining the instantaneous rate-of-change of one variable as it
relates to another. In analog differentiator circuits, the independent variable is time, and so
the rates of change we’re dealing with are rates of change for an electronic signal (voltage or
current) with respect to time.
Suppose we were to measure the position of a car, traveling in a direct path (no turns), from
its starting point. Let us call this measurement, x. If the car moves at a rate such that its
distance from ”start” increases steadily over time, its position will plot on a graph as a linear
function (straight line):

x
Position
x

Time
If we were to calculate the derivative of the car’s position with respect to time (that is,
determine the rate-of-change of the car’s position with respect to time), we would arrive at
a quantity representing the car’s velocity. The differentiation function is represented by the
fractional notation d/d, so when differentiating position (x) with respect to time (t), we denote
the result (the derivative) as dx/dt:

x
Position

Velocity
dx
dt

x

Time

Time

For a linear graph of x over time, the derivate of position (dx/dt), otherwise and more
commonly known as velocity, will be a flat line, unchanging in value. The derivative of a
mathematical function may be graphically understood as its slope when plotted on a graph,

CHAPTER 9. PRACTICAL ANALOG SEMICONDUCTOR CIRCUITS

440

and here we can see that the position (x) graph has a constant slope, which means that its
derivative (dx/dt) must be constant over time.
Now, suppose the distance traveled by the car increased exponentially over time: that is, it
began its travel in slow movements, but covered more additional distance with each passing
period in time. We would then see that the derivative of position (dx/dt), otherwise known as
velocity (v), would not be constant over time, but would increase:

x
Position

Velocity
dx
dt

x

Time

Time

The height of points on the velocity graph correspond to the rates-of-change, or slope, of
points at corresponding times on the position graph:

Position

Velocity
dx
dt

x

Time

Time

What does this have to do with analog electronic circuits? Well, if we were to have an analog
voltage signal represent the car’s position (think of a huge potentiometer whose wiper was
attached to the car, generating a voltage proportional to the car’s position), we could connect a
differentiator circuit to this signal and have the circuit continuously calculate the car’s velocity,
displaying the result via a voltmeter connected to the differentiator circuit’s output:

9.7. COMPUTATIONAL CIRCUITS

441

Differentiator
−
+
x V
Position

-

+

- dx
V
dt
Velocity
+

x
Recall from the last chapter that a differentiator circuit outputs a voltage proportional to
the input voltage’s rate-of-change over time (d/dt). Thus, if the input voltage is changing over
time at a constant rate, the output voltage will be at a constant value. If the car moves in such
a way that its elapsed distance over time builds up at a steady rate, then that means the car
is traveling at a constant velocity, and the differentiator circuit will output a constant voltage
proportional to that velocity. If the car’s elapsed distance over time changes in a non-steady
manner, the differentiator circuit’s output will likewise be non-steady, but always at a level
representative of the input’s rate-of-change over time.
Note that the voltmeter registering velocity (at the output of the differentiator circuit) is
connected in ”reverse” polarity to the output of the op-amp. This is because the differentiator
circuit shown is inverting: outputting a negative voltage for a positive input voltage rate-ofchange. If we wish to have the voltmeter register a positive value for velocity, it will have to be
connected to the op-amp as shown. As impractical as it may be to connect a giant potentiometer to a moving object such as an automobile, the concept should be clear: by electronically
performing the calculus function of differentiation on a signal representing position, we obtain
a signal representing velocity.
Beginning calculus students learn symbolic techniques for differentiation. However, this
requires that the equation describing the original graph be known. For example, calculus students learn how to take a function such as y = 3x and find its derivative with respect to x (d/dx),
3, simply by manipulating the equation. We may verify the accuracy of this manipulation by
comparing the graphs of the two functions:

CHAPTER 9. PRACTICAL ANALOG SEMICONDUCTOR CIRCUITS

442

d 3x = 3
dx
y = 3x

(y = 3)

y

y
x = 2; y = 3

x = 2.5; slope = 3
x = 2; slope = 3
x = 1; slope = 3
x

x = 1; y = 3

x = 2.5; y = 3
x

Nonlinear functions such as y = 3x2 may also be differentiated by symbolic means. In this
case, the derivative of y = 3x2 with respect to x is 6x:
d 3x2 = 6x
dx

y = 3x2

(y = 6x)

x = 2; slope = 12
y

y

x = 2; y = 12

x = 1; y = 6
x = 1; slope = 6
x
x = 0; slope = 0

x
x = 0; y = 0

In real life, though, we often cannot describe the behavior of any physical event by a simple
equation like y = 3x, and so symbolic differentiation of the type learned by calculus students
may be impossible to apply to a physical measurement. If someone wished to determine the
derivative of our hypothetical car’s position (dx/dt = velocity) by symbolic means, they would
first have to obtain an equation describing the car’s position over time, based on position measurements taken from a real experiment – a nearly impossible task unless the car is operated
under carefully controlled conditions leading to a very simple position graph. However, an
analog differentiator circuit, by exploiting the behavior of a capacitor with respect to voltage,
current, and time i = C(dv/dt), naturally differentiates any real signal in relation to time, and
would be able to output a signal corresponding to instantaneous velocity (dx/dt) at any moment. By logging the car’s position signal along with the differentiator’s output signal using
a chart recorder or other data acquisition device, both graphs would naturally present them-

9.7. COMPUTATIONAL CIRCUITS

443

selves for inspection and analysis.

We may take the principle of differentiation one step further by applying it to the velocity
signal using another differentiator circuit. In other words, use it to calculate the rate-of-change
of velocity, which we know is the rate-of-change of position. What practical measure would we
arrive at if we did this? Think of this in terms of the units we use to measure position and
velocity. If we were to measure the car’s position from its starting point in miles, then we would
probably express its velocity in units of miles per hour (dx/dt). If we were to differentiate the
velocity (measured in miles per hour) with respect to time, we would end up with a unit of miles
per hour per hour. Introductory physics classes teach students about the behavior of falling
objects, measuring position in meters, velocity in meters per second, and change in velocity over
time in meters per second, per second. This final measure is called acceleration: the rate of
change of velocity over time:

x
Position

Velocity
d2x

dx
dt

x

Time

Acceleration

dt2
Time

Differentiation

Time
Differentiation

The expression d2 x/dt2 is called the second derivative of position (x) with regard to time (t).
If we were to connect a second differentiator circuit to the output of the first, the last voltmeter
would register acceleration:

CHAPTER 9. PRACTICAL ANALOG SEMICONDUCTOR CIRCUITS

444

Differentiator
−
+
x V
Position

Differentiator

- dx
V
dt
Velocity
+

+

-

−
+

V
Acceleration

+ d2x
2
- dt

x

Deriving velocity from position, and acceleration from velocity, we see the principle of differentiation very clearly illustrated. These are not the only physical measurements related to
each other in this way, but they are, perhaps, the most common. Another example of calculus
in action is the relationship between liquid flow (q) and liquid volume (v) accumulated in a
vessel over time:

Water
supply

dv = flow
dt

Water

LT v = volume
A ”Level Transmitter” device mounted on a water storage tank provides a signal directly
proportional to water level in the tank, which – if the tank is of constant cross-sectional area
throughout its height – directly equates water volume stored. If we were to take this volume
signal and differentiate it with respect to time (dv/dt), we would obtain a signal proportional
to the water flow rate through the pipe carrying water to the tank. A differentiator circuit

9.7. COMPUTATIONAL CIRCUITS

445

connected in such a way as to receive this volume signal would produce an output signal proportional to flow, possibly substituting for a flow-measurement device (”Flow Transmitter”)
installed in the pipe.

Returning to the car experiment, suppose that our hypothetical car were equipped with a
tachogenerator on one of the wheels, producing a voltage signal directly proportional to velocity.
We could differentiate the signal to obtain acceleration with one circuit, like this:

Differentiator
−
+

+

v V
Velocity

-

+

- dv d2x
V
= 2
dt
Acceleration
dt
+

Gen

x

By its very nature, the tachogenerator differentiates the car’s position with respect to time,
generating a voltage proportional to how rapidly the wheel’s angular position changes over
time. This provides us with a raw signal already representative of velocity, with only a single
step of differentiation needed to obtain an acceleration signal. A tachogenerator measuring
velocity, of course, is a far more practical example of automobile instrumentation than a giant potentiometer measuring its physical position, but what we gain in practicality we lose in
position measurement. No matter how many times we differentiate, we can never infer the
car’s position from a velocity signal. If the process of differentiation brought us from position
to velocity to acceleration, then somehow we need to perform the ”reverse” process of differentiation to go from velocity to position. Such a mathematical process does exist, and it is called
integration. The ”integrator” circuit may be used to perform this function of integration with
respect to time:

CHAPTER 9. PRACTICAL ANALOG SEMICONDUCTOR CIRCUITS

446

Integrator
−
+

Position

V ∫ v dt = x
+

Differentiator
−
+

+

v V
Velocity

-

+

- dv d2x
V
= 2
dt
Acceleration
dt
+

Gen

x
Recall from the last chapter that an integrator circuit outputs a voltage whose rate-ofchange over time is proportional to the input voltage’s magnitude. Thus, given a constant
input voltage, the output voltage will change at a constant rate. If the car travels at a constant
velocity (constant voltage input to the integrator circuit from the tachogenerator), then its
distance traveled will increase steadily as time progresses, and the integrator will output a
steadily changing voltage proportional to that distance. If the car’s velocity is not constant,
then neither will the rate-of-change over time be of the integrator circuit’s output, but the
output voltage will faithfully represent the amount of distance traveled by the car at any given
point in time.
R
The symbol for integration looks
something like a very narrow, cursive letter ”S” ( ). The
R
equation utilizing this symbol ( v dt = x) tells us that we are integrating velocity (v) with
respect to time (dt), and obtaining position (x) as a result.
So, we may express three measures of the car’s motion (position, velocity, and acceleration)
in terms of velocity (v) just as easily as we could in terms of position (x):

9.7. COMPUTATIONAL CIRCUITS

447

x
Position
∫ v dt

Velocity
dv
dt

v

Time

Time
Integration

Acceleration

Time
Differentiation

If we had an accelerometer attached to the car, generating a signal proportional to the rate
of acceleration or deceleration, we could (hypothetically) obtain a velocity signal with one step
of integration, and a position signal with a second step of integration:

CHAPTER 9. PRACTICAL ANALOG SEMICONDUCTOR CIRCUITS

448

Integrator
−
+

+
V ∫∫ a dt = x
Position
-

Integrator
−
+
a V
Acceleration

+

Velocity

-

V ∫ a dt = v
+

Accel.

x
Thus, all three measures of the car’s motion (position, velocity, and acceleration) may be
expressed in terms of acceleration:

x
Position
∫∫ a dt

Velocity
∫ a dt

Time

Acceleration
a

Time
Integration

Time
Integration

9.7. COMPUTATIONAL CIRCUITS

449

As you might have suspected, the process of integration may be illustrated in, and applied
to, other physical systems as well. Take for example the water storage tank and flow example
shown earlier. If flow rate is the derivative of tank volume with respect to time (q = dv/dt),
then we could also say that volume is the integral of flow rate with respect to time:

FT f = flow
Water
supply

Water
∫ f dt = volume

If we were to use a ”Flow Transmitter” device to measure water flow, then by time-integration
we could calculate the volume of water accumulated in the tank over time. Although it is theoretically possible to use a capacitive op-amp integrator circuit to derive a volume signal from
a flow signal, mechanical and digital electronic ”integrator” devices are more suitable for integration over long periods of time, and find frequent use in the water treatment and distribution
industries.
Just as there are symbolic techniques for differentiation, there are also symbolic techniques
for integration, although they tend to be more complex and varied. Applying symbolic integration to a real-world problem like the acceleration of a car, though, is still contingent on the
availability of an equation precisely describing the measured signal – often a difficult or impossible thing to derive from measured data. However, electronic integrator circuits perform
this mathematical function continuously, in real time, and for any input signal profile, thus
providing a powerful tool for scientists and engineers.
Having said this, there are caveats to the using calculus techniques to derive one type
of measurement from another. Differentiation has the undesirable tendency of amplifying
”noise” found in the measured variable, since the noise will typically appear as frequencies
much higher than the measured variable, and high frequencies by their very nature possess
high rates-of-change over time.
To illustrate this problem, suppose we were deriving a measurement of car acceleration
from the velocity signal obtained from a tachogenerator with worn brushes or commutator
bars. Points of poor contact between brush and commutator will produce momentary ”dips” in
the tachogenerator’s output voltage, and the differentiator circuit connected to it will interpret
these dips as very rapid changes in velocity. For a car moving at constant speed – neither
accelerating nor decelerating – the acceleration signal should be 0 volts, but ”noise” in the
velocity signal caused by a faulty tachogenerator will cause the differentiated (acceleration)
signal to contain ”spikes,” falsely indicating brief periods of high acceleration and deceleration:

CHAPTER 9. PRACTICAL ANALOG SEMICONDUCTOR CIRCUITS

450

Differentiator
−
+

+

v V
Velocity

-

+

- dv
V
dt
Acceleration
+

Gen

x

Noise voltage present in a signal to be differentiated need not be of significant amplitude to
cause trouble: all that is required is that the noise profile have fast rise or fall times. In other
words, any electrical noise with a high dv/dt component will be problematic when differentiated, even if it is of low amplitude.

It should be noted that this problem is not an artifact (an idiosyncratic error of the measuring/computing instrument) of the analog circuitry; rather, it is inherent to the process of
differentiation. No matter how we might perform the differentiation, ”noise” in the velocity
signal will invariably corrupt the output signal. Of course, if we were differentiating a signal
twice, as we did to obtain both velocity and acceleration from a position signal, the amplified noise signal output by the first differentiator circuit will be amplified again by the next
differentiator, thus compounding the problem:

9.7. COMPUTATIONAL CIRCUITS

451
more noise

little noise

Differentiator
−
+
x V
Position

-

+

even more noise!

Differentiator

- dx
V
dt
Velocity
+

−
+

V
Acceleration

+ d2x
2
- dt

x

Integration does not suffer from this problem, because integrators act as low-pass filters,
attenuating high-frequency input signals. In effect, all the high and low peaks resulting from
noise on the signal become averaged together over time, for a diminished net result. One
might suppose, then, that we could avoid all trouble by measuring acceleration directly and
integrating that signal to obtain velocity; in effect, calculating in ”reverse” from the way shown
previously:

CHAPTER 9. PRACTICAL ANALOG SEMICONDUCTOR CIRCUITS

452

Integrator
−
+
a V
Acceleration

-

+

Velocity

V ∫ a dt = v
+

Accel.

x

Unfortunately, following this methodology might lead us into other difficulties, one being a
common artifact of analog integrator circuits known as drift. All op-amps have some amount of
input bias current, and this current will tend to cause a charge to accumulate on the capacitor
in addition to whatever charge accumulates as a result of the input voltage signal. In other
words, all analog integrator circuits suffer from the tendency of having their output voltage
”drift” or ”creep” even when there is absolutely no voltage input, accumulating error over time
as a result. Also, imperfect capacitors will tend to lose their stored charge over time due
to internal resistance, resulting in ”drift” toward zero output voltage. These problems are
artifacts of the analog circuitry, and may be eliminated through the use of digital computation.

Circuit artifacts notwithstanding, possible errors may result from the integration of one
measurement (such as acceleration) to obtain another (such as velocity) simply because of the
way integration works. If the ”zero” calibration point of the raw signal sensor is not perfect, it
will output a slight positive or negative signal even in conditions when it should output nothing. Consider a car with an imperfectly calibrated accelerometer, or one that is influenced by
gravity to detect a slight acceleration unrelated to car motion. Even with a perfect integrating
computer, this sensor error will cause the integrator to accumulate error, resulting in an output
signal indicating a change of velocity when the car is neither accelerating nor decelerating.

9.7. COMPUTATIONAL CIRCUITS

453

Integrator
(slight positive
voltage)

−
+
a V
Acceleration

-

(calibration error)

+

Velocity

V ∫ a dt = v
+

(small rate
of change)

Accel.
(no motion)

x

As with differentiation, this error will also compound itself if the integrated signal is passed
on to another integrator circuit, since the ”drifting” output of the first integrator will very soon
present a significant positive or negative signal for the next integrator to integrate. Therefore,
care should be taken when integrating sensor signals: if the ”zero” adjustment of the sensor is
not perfect, the integrated result will drift, even if the integrator circuit itself is perfect.

So far, the only integration errors discussed have been artificial in nature: originating from
imperfections in the circuitry and sensors. There also exists a source of error inherent to the
process of integration itself, and that is the unknown constant problem. Beginning calculus students learn that whenever a function is integrated, there exists an unknown constant (usually
represented as the variable C) added to the result. This uncertainty is easiest to understand
by comparing the derivatives of several functions differing only by the addition of a constant
value:

454

CHAPTER 9. PRACTICAL ANALOG SEMICONDUCTOR CIRCUITS

y = 3x2 + 4
y = 3x2

y

d 3x2 + 4 = 6x
dx
d 3x2 = 6x
dx
d 3x2 - 6 = 6x
dx
(y’ = 6x)

y’

y = 3x2 - 6

x

x

Note how each of the parabolic curves (y = 3x2 + C) share the exact same shape, differing
from each other in regard to their vertical offset. However, they all share the exact same
derivative function: y’ = (d/dx)( 3x2 + C) = 6x, because they all share identical rates of change
(slopes) at corresponding points along the x axis. While this seems quite natural and expected
from the perspective of differentiation (different equations sharing a common derivative), it
usually strikes beginning students as odd from the perspective of integration, because there are
multiple correct answers for the integral of a function. Going from an equation to its derivative,
there is only one answer, but going from that derivative back to the original equation leads us
to a range of correct solutions. In honor of this uncertainty, the symbolic function of integration
is called the indefinite integral.
When an integrator performs live signal integration with respect to time, the output is
the sum of the integrated input signal over time and an initial value of arbitrary magnitude,
representing the integrator’s pre-existing output at the time integration began. For example,
if I integrate the velocity of a car driving in a straight line away from a city, calculating
R that a
constant velocity of 50 miles per hour over a time of 2 hours will produce a distance ( v dt) of
100 miles, that does not necessarily mean the car will be 100 miles away from the city after 2
hours. All it tells us is that the car will be 100 miles further away from the city after 2 hours of
driving. The actual distance from the city after 2 hours of driving depends on how far the car
was from the city when integration began. If we do not know this initial value for distance, we
cannot determine the car’s exact distance from the city after 2 hours of driving.
This same problem appears when we integrate acceleration with respect to time to obtain
velocity:

9.8. MEASUREMENT CIRCUITS – INCOMPLETE

455

Integrator
−
+
a V
Acceleration

-

+

V
∫ dt

Velocity

+

∫ a dt = v + v0
Accel.

Where,
v0 = Initial velocity

x
In this integrator system, the calculated velocity of the car will only be valid if the integrator
circuit is initialized to an output value of zero when the car is stationary (v = 0). Otherwise,
the integrator could very well be outputting a non-zero signal for velocity (v0 ) when the car
is stationary, for the accelerometer cannot tell the difference between a stationary state (0
miles per hour) and a state of constant velocity (say, 60 miles per hour, unchanging). This
uncertainty in integrator output is inherent to the process of integration, and not an artifact
of the circuitry or of the sensor.
In summary, if maximum accuracy is desired for any physical measurement, it is best to
measure that variable directly rather than compute it from other measurements. This is not
to say that computation is worthless. Quite to the contrary, often it is the only practical means
of obtaining a desired measurement. However, the limits of computation must be understood
and respected in order that precise measurements be obtained.

9.8

Measurement circuits – INCOMPLETE

Figure 9.21 shows a photodiode amplifier for measuring low levels of light. Best sensitivity
and bandwidth are obtained with a transimpedance amplifier, a current to voltage amplifier,
instead of a conventional operational amplifier. The photodiode remains reverse biased for
lowest diode capacitance, hence wider bandwidth, and lower noise. The feedback resistor sets
the “gain”, the current to voltage amplification factor. Typical values are 1 to 10 Meg Ω. Higher
values yield higher gain. A capacitor of a few pF may be required to compensate for photodiode
capacitance, and prevents instability at the high gain. The wiring at the summing node must
be as compact as possible. This point is sensitive to circuit board contaminants and must be
thoroughly cleaned. The most sensitive amplifiers contain the photodiode and amplifier within
a hybrid microcircuit package or single die.

CHAPTER 9. PRACTICAL ANALOG SEMICONDUCTOR CIRCUITS

456

−
Vo
+

Figure 9.21: Photodiode amplifier.

9.9

Control circuits – PENDING

Contributors
Contributors to this chapter are listed in chronological order of their contributions, from most
recent to first. See Appendix 2 (Contributor List) for dates and contact information.
Warren Young (August 2002): Initial idea and text for ”Power supply circuits” section.
Paragraphs modified by Tony Kuphaldt (changes in vocabulary, plus inclusion of additional
concepts).
Bill Marsden (April 2008) Author of “ElectroStatic Discharge” section.

Bibliography
[1] Chin-Leong Lim, Lim Yeam Ch’ng, Goh Swee Chye, “Diode
Foundation For PIN Diode Attenuator,” Microwaves & RF, May
http://www.mwrf.com/Articles/Index.cfm?Ad=1&ArticleID=12523

Quad
2006,

Is
at

[2] “Transistor Audio and Radio Circuits,” TP1399, 2nd Ed., pp 39-40, Mullard, London, 1972.
[3] “AM Receiver Circuit TCA440,” Analog Data Manual, 2nd Ed., pp 14-20 to 14-26, Signetics, 1982.
[4] Sony “8-pin Single-Chip AM Radio with Builot-in Power Amplifier,” pp 5, at
http://www.datasheetcatalog.com/datasheets pdf/C/X/A/1/CXA1600.shtml
[5] Texas Instruments “Solid State Communications,” pp 318, McGraw-Hill, N.Y., 1966.
[6] Texas Instruments “Transistor Circuit Design,” pp 290, McGraw-Hill, N.Y., 1963.
[7] “Datasheet
TDA7021T”,
STR-NXP
Wireless,
at
acrobat download/datasheets/TDA7021T CNV 2.pdf

http://www.nxp.com/

[8] Donald L. Stoner, L. A. Earnshaw, “The Transistor Radio Handbook,” pp 76, Editors and
Eenineers, Sumerland, CA, 1963.

BIBLIOGRAPHY
[9] ,Charles Wenzel, “Crystal Radio
lectronics/crystal.html.

457
Circuits,”

at

http://www.techlib.com/e

458

CHAPTER 9. PRACTICAL ANALOG SEMICONDUCTOR CIRCUITS

Chapter 10

ACTIVE FILTERS
Contents

*** PENDING ***

459

460

CHAPTER 10. ACTIVE FILTERS

Chapter 11

DC MOTOR DRIVES
Contents
11.1 Pulse Width Modulation

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*** INCOMPLETE ***

11.1

Pulse Width Modulation

Pulse Width Modulation (PWM) uses digital signals to control power applications, as well as
being fairly easy to convert back to analog with a minimum of hardware.
Analog systems, such as linear power supplies, tend to generate a lot of heat since they are
basically variable resistors carrying a lot of current. Digital systems don’t generally generate
as much heat. Almost all the heat generated by a switching device is during the transition
(which is done quickly), while the device is neither on nor off, but in between. This is because
power follows the following formula:
P = E I, or Watts = Voltage X Current
If either voltage or current is near zero then power will be near zero. PWM takes full
advantage of this fact.
PWM can have many of the characteristics of an analog control system, in that the digital
signal can be free wheeling. PWM does not have to capture data, although there are exceptions
to this with higher end controllers.
One of the parameters of any square wave is duty cycle. Most square waves are 50%, this
is the norm when discussing them, but they don’t have to be symmetrical. The ON time can
be varied completely between signal being off to being fully on, 0% to 100%, and all ranges
between.
Shown below are examples of a 10%, 50%, and 90% duty cycle. While the frequency is the
same for each, this is not a requirement.
461

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CHAPTER 11. DC MOTOR DRIVES

The reason PWM is popular is simple. Many loads, such as resistors, integrate the power
into a number matching the percentage. Conversion into its analog equivalent value is straightforward. LEDs are very nonlinear in their response to current, give an LED half its rated
current you you still get more than half the light the LED can produce. With PWM the light
level produced by the LED is very linear. Motors, which will be covered later, are also very
responsive to PWM.
One of several ways PWM can be produced is by using a sawtooth waveform and a comparator. As shown below the sawtooth (or triangle wave) need not be symmetrical, but linearity of
the waveform is important. The frequency of the sawtooth waveform is the sampling rate for
the signal.

If there isn’t any computation involved PWM can be fast. The limiting factor is the comparators frequency response. This may not be an issue since quite a few of the uses are fairly
low speed. Some microcontrollers have PWM built in, and can record or create signals on
demand.
Uses for PWM vary widely. It is the heart of Class D audio amplifiers, by increasing the
voltages you increase the maximum output, and by selecting a frequency beyond human hearing (typically 44Khz) PWM can be used. The speakers do not respond to the high frequency,
but duplicates the low frequency, which is the audio signal. Higher sampling rates can be used
for even better fidelity, and 100Khz or much higher is not unheard of.

Another popular application is motor speed control. Motors as a class require very high
currents to operate. Being able to vary their speed with PWM increases the efficiency of the

11.1. PULSE WIDTH MODULATION

463

total system by quite a bit. PWM is more effective at controlling motor speeds at low RPM
than linear methods.
PWM is often used in conjunction with an H-Bridge. This configuration is so named because it resembles the letter H, and allows the effective voltage across the load to be doubled,
since the power supply can be switched across both sides of the load. In the case of inductive
loads, such as motors, diodes are used to suppress inductive spikes, which may damage the
transistors. The inductance in a motor also tends to reject the high frequency component of
the waveform. This configuration can also be used with speakers for Class D audio amps.
While basically accurate, this schematic of an H-Bridge has one serious flaw, it is possible
while transitioning between the MOSFETs that both transistors on top and bottom will be
on simultaneously, and will take the full brunt of what the power supply can provide. This
condition is referred to as shoot through, and can happen with any type of transistor used in a
H-Bridge. If the power supply is powerful enough the transistors will not survive. It is handled
by using drivers in front of the transistors that allow one to turn off before allowing the other
to turn on.

Switching Mode Power Supplies (SMPS) can also use PWM, although other methods also
exist. Adding topologies that use the stored power in both inductors and capacitors after the
main switching components can boost the efficiencies for these devices quite high, exceeding
90% in some cases. Below is an example of such a configuration.

Efficiency in this case is measured as wattage. If you have a SMPS with 90% efficiency, and
it converts 12VDC to 5VDC at 10 Amps, the 12V side will be pulling approximately 4.6 Amps.
The 10% (5 watts) not accounted for will show up as waste heat. While being slightly noisier,
this type of regulator will run much cooler than its linear counterpart.

Contributors
Contributors to this chapter are listed in chronological order of their contributions, from most
recent to first. See Appendix 2 (Contributor List) for dates and contact information.
Bill Marsden (February 2010) Author of “Pulse Width Modulation” section.

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Chapter 12

INVERTERS AND AC MOTOR
DRIVES
Contents

*** PENDING ***

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CHAPTER 12. INVERTERS AND AC MOTOR DRIVES

Chapter 13

ELECTRON TUBES
Contents
13.1 Introduction . . . . . . . . . . .
13.2 Early tube history . . . . . . .
13.3 The triode . . . . . . . . . . . .
13.4 The tetrode . . . . . . . . . . . .
13.5 Beam power tubes . . . . . . .
13.6 The pentode . . . . . . . . . . .
13.7 Combination tubes . . . . . . .
13.8 Tube parameters . . . . . . . .
13.9 Ionization (gas-filled) tubes .
13.10Display tubes . . . . . . . . . .
13.11Microwave tubes . . . . . . . .
13.12Tubes versus Semiconductors

13.1

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467
468
471
473
474
476
476
479
481
485
488
491

Introduction

An often neglected area of study in modern electronics is that of tubes, more precisely known as
vacuum tubes or electron tubes. Almost completely overshadowed by semiconductor, or ”solidstate” components in most modern applications, tube technology once dominated electronic
circuit design.
In fact, the historical transition from ”electric” to ”electronic” circuits really began with
tubes, for it was with tubes that we entered into a whole new realm of circuit function: a way of
controlling the flow of electrons (current) in a circuit by means of another electric signal (in the
case of most tubes, the controlling signal is a small voltage). The semiconductor counterpart
to the tube, of course, is the transistor. Transistors perform much the same function as tubes:
controlling the flow of electrons in a circuit by means of another flow of electrons in the case of
the bipolar transistor, and controlling the flow of electrons by means of a voltage in the case of
467

CHAPTER 13. ELECTRON TUBES

468

the field-effect transistor. In either case, a relatively small electric signal controls a relatively
large electric current. This is the essence of the word ”electronic,” so as to distinguish it from
”electric,” which has more to do with how electron flow is regulated by Ohm’s Law and the
physical attributes of wire and components.
Though tubes are now obsolete for all but a few specialized applications, they are still a
worthy area of study. If nothing else, it is fascinating to explore ”the way things used to be
done” in order to better appreciate modern technology.

13.2

Early tube history

Thomas Edison, that prolific American inventor, is often credited with the invention of the
incandescent lamp. More accurately, it could be said that Edison was the man who perfected
the incandescent lamp. Edison’s successful design of 1879 was actually preceded by 77 years by
the British scientist Sir Humphry Davy, who first demonstrated the principle of using electric
current to heat a thin strip of metal (called a ”filament”) to the point of incandescence (glowing
white hot).
Edison was able to achieve his success by placing his filament (made of carbonized sewing
thread) inside of a clear glass bulb from which the air had been forcibly removed. In this
vacuum, the filament could glow at white-hot temperatures without being consumed by combustion:

clear, glass bulb
air removed
filament

In the course of his experimentation (sometime around 1883), Edison placed a strip of metal
inside of an evacuated (vacuum) glass bulb along with the filament. Between this metal strip
and one of the filament connections he attached a sensitive ammeter. What he found was that
electrons would flow through the meter whenever the filament was hot, but ceased when the
filament cooled down:

13.2. EARLY TUBE HISTORY

469

metal strip
A
+

The white-hot filament in Edison’s lamp was liberating free electrons into the vacuum of
the lamp, those electrons finding their way to the metal strip, through the galvanometer, and
back to the filament. His curiosity piqued, Edison then connected a fairly high-voltage battery
in the galvanometer circuit to aid the small current:

A
more
current

+

Sure enough, the presence of the battery created a much larger current from the filament to
the metal strip. However, when the battery was turned around, there was little to no current
at all!

no
current ! A

+
-

In effect, what Edison had stumbled upon was a diode! Unfortunately, he saw no practical
use for such a device and proceeded with further refinements in his lamp design.
The one-way electron flow of this device (known as the Edison Effect) remained a curiosity until J. A. Fleming experimented with its use in 1895. Fleming marketed his device as a
”valve,” initiating a whole new area of study in electric circuits. Vacuum tube diodes – Fleming’s ”valves” being no exception – are not able to handle large amounts of current, and so
Fleming’s invention was impractical for any application in AC power, only for small electric
signals.

CHAPTER 13. ELECTRON TUBES

470

Then in 1906, another inventor by the name of Lee De Forest started playing around with
the ”Edison Effect,” seeing what more could be gained from the phenomenon. In doing so, he
made a startling discovery: by placing a metal screen between the glowing filament and the
metal strip (which by now had taken the form of a plate for greater surface area), the stream of
electrons flowing from filament to plate could be regulated by the application of a small voltage
between the metal screen and the filament:

The DeForest "Audion" tube

"plate"
-

"grid"
"filament"

A

control
voltage

+

plate current can be controlled by the
application of a small control voltage
between the grid and filament!
De Forest called this metal screen between filament and plate a grid. It wasn’t just the
amount of voltage between grid and filament that controlled current from filament to plate, it
was the polarity as well. A negative voltage applied to the grid with respect to the filament
would tend to choke off the natural flow of electrons, whereas a positive voltage would tend to
enhance the flow. Although there was some amount of current through the grid, it was very
small; much smaller than the current through the plate.
Perhaps most importantly was his discovery that the small amounts of grid voltage and grid
current were having large effects on the amount of plate voltage (with respect to the filament)
and plate current. In adding the grid to Fleming’s ”valve,” De Forest had made the valve
adjustable: it now functioned as an amplifying device, whereby a small electrical signal could
take control over a larger electrical quantity.
The closest semiconductor equivalent to the Audion tube, and to all of its more modern tube
equivalents, is an n-channel D-type MOSFET. It is a voltage-controlled device with a large
current gain.
Calling his invention the ”Audion,” he vigorously applied it to the development of communications technology. In 1912 he sold the rights to his Audion tube as a telephone signal
amplifier to the American Telephone and Telegraph Company (AT and T), which made longdistance telephone communication practical. In the following year he demonstrated the use of
an Audion tube for generating radio-frequency AC signals. In 1915 he achieved the remarkable feat of broadcasting voice signals via radio from Arlington, Virginia to Paris, and in 1916
inaugurated the first radio news broadcast. Such accomplishments earned De Forest the title
”Father of Radio” in America.

13.3. THE TRIODE

13.3

471

The triode

De Forest’s Audion tube came to be known as the triode tube, because it had three elements:
filament, grid, and plate (just as the ”di” in the name diode refers to two elements, filament
and plate). Later developments in diode tube technology led to the refinement of the electron
emitter: instead of using the filament directly as the emissive element, another metal strip
called the cathode could be heated by the filament.
This refinement was necessary in order to avoid some undesired effects of an incandescent
filament as an electron emitter. First, a filament experiences a voltage drop along its length,
as current overcomes the resistance of the filament material and dissipates heat energy. This
meant that the voltage potential between different points along the length of the filament wire
and other elements in the tube would not be constant. For this and similar reasons, alternating
current used as a power source for heating the filament wire would tend to introduce unwanted
AC ”noise” in the rest of the tube circuit. Furthermore, the surface area of a thin filament was
limited at best, and limited surface area on the electron emitting element tends to place a
corresponding limit on the tube’s current-carrying capacity.
The cathode was a thin metal cylinder fitting snugly over the twisted wire of the filament.
The cathode cylinder would be heated by the filament wire enough to freely emit electrons,
without the undesirable side effects of actually carrying the heating current as the filament
wire had to. The tube symbol for a triode with an indirectly-heated cathode looks like this:

CHAPTER 13. ELECTRON TUBES

472

plate

grid
cathode

filament

Since the filament is necessary for all but a few types of vacuum tubes, it is often omitted in
the symbol for simplicity, or it may be included in the drawing but with no power connections
drawn to it:

...

...

no filament
shown at all

no connections shown
to filament wires
...

A simple triode circuit is shown to illustrate its basic operation as an amplifier:

Triode amplifier circuit

R
output
voltage

"plate supply"
DC power
source

input
voltage

The low-voltage AC signal connected between the grid and cathode alternately suppresses,
then enhances the electron flow between cathode and plate. This causes a change in voltage on
the output of the circuit (between plate and cathode). The AC voltage and current magnitudes
on the tube’s grid are generally quite small compared with the variation of voltage and current
in the plate circuit. Thus, the triode functions as an amplifier of the incoming AC signal

13.4. THE TETRODE

473

(taking high-voltage, high-current DC power supplied from the large DC source on the right
and ”throttling” it by means of the tube’s controlled conductivity).
In the triode, the amount of current from cathode to plate (the ”controlled” current is a
function both of grid-to-cathode voltage (the controlling signal) and the plate-to-cathode voltage (the electromotive force available to push electrons through the vacuum). Unfortunately,
neither of these independent variables have a purely linear effect on the amount of current
through the device (often referred to simply as the ”plate current”). That is, triode current
does not necessarily respond in a direct, proportional manner to the voltages applied.
In this particular amplifier circuit the nonlinearities are compounded, as plate voltage (with
respect to cathode) changes along with the grid voltage (also with respect to cathode) as plate
current is throttled by the tube. The result will be an output voltage waveform that doesn’t
precisely resemble the waveform of the input voltage. In other words, the quirkiness of the
triode tube and the dynamics of this particular circuit will distort the waveshape. If we really wanted to get complex about how we stated this, we could say that the tube introduces
harmonics by failing to exactly reproduce the input waveform.
Another problem with triode behavior is that of stray capacitance. Remember that any
time we have two conductive surfaces separated by an insulating medium, a capacitor will
be formed. Any voltage between those two conductive surfaces will generate an electric field
within that insulating region, potentially storing energy and introducing reactance into a circuit. Such is the case with the triode, most problematically between the grid and the plate. It
is as if there were tiny capacitors connected between the pairs of elements in the tube:

Cgrid-plate
Cplate-cathode
Cgrid-cathode
Now, this stray capacitance is quite small, and the reactive impedances usually high. Usually, that is, unless radio frequencies are being dealt with. As we saw with De Forest’s Audion
tube, radio was probably the prime application for this new technology, so these ”tiny” capacitances became more than just a potential problem. Another refinement in tube technology was
necessary to overcome the limitations of the triode.

13.4

The tetrode

As the name suggests, the tetrode tube contained four elements: cathode (with the implicit
filament, or ”heater”), grid, plate, and a new element called the screen. Similar in construction
to the grid, the screen was a wire mesh or coil positioned between the grid and plate, connected
to a source of positive DC potential (with respect to the cathode, as usual) equal to a fraction
of the plate voltage. When connected to ground through an external capacitor, the screen
had the effect of electrostatically shielding the grid from the plate. Without the screen, the

CHAPTER 13. ELECTRON TUBES

474

capacitive linking between the plate and the grid could cause significant signal feedback at
high frequencies, resulting in unwanted oscillations.
The screen, being of less surface area and lower positive potential than the plate, didn’t
attract many of the electrons passing through the grid from the cathode, so the vast majority
of electrons in the tube still flew by the screen to be collected by the plate:

Tetrode amplifier circuit

R
R
"screen"

R
input
voltage

"plate supply"
DC power
source
C

With a constant DC screen voltage, electron flow from cathode to plate became almost exclusively dependent upon grid voltage, meaning the plate voltage could vary over a wide range
with little effect on plate current. This made for more stable gains in amplifier circuits, and
better linearity for more accurate reproduction of the input signal waveform.
Despite the advantages realized by the addition of a screen, there were some disadvantages
as well. The most significant disadvantage was related to something known as secondary
emission. When electrons from the cathode strike the plate at high velocity, they can cause
free electrons to be jarred loose from atoms in the metal of the plate. These electrons, knocked
off the plate by the impact of the cathode electrons, are said to be ”secondarily emitted.” In a
triode tube, secondary emission is not that great a problem, but in a tetrode with a positivelycharged screen grid in close proximity, these secondary electrons will be attracted to the screen
rather than the plate from which they came, resulting in a loss of plate current. Less plate
current means less gain for the amplifier, which is not good.
Two different strategies were developed to address this problem of the tetrode tube: beam
power tubes and pentodes. Both solutions resulted in new tube designs with approximately the
same electrical characteristics.

13.5

Beam power tubes

In the beam power tube, the basic four-element structure of the tetrode was maintained, but
the grid and screen wires were carefully arranged along with a pair of auxiliary plates to create
an interesting effect: focused beams or ”sheets” of electrons traveling from cathode to plate.
These electron beams formed a stationary ”cloud” of electrons between the screen and plate

13.5. BEAM POWER TUBES

475

(called a ”space charge”) which acted to repel secondary electrons emitted from the plate back
to the plate. A set of ”beam-forming” plates, each connected to the cathode, were added to help
maintain proper electron beam focus. Grid and screen wire coils were arranged in such a way
that each turn or wrap of the screen fell directly behind a wrap of the grid, which placed the
screen wires in the ”shadow” formed by the grid. This precise alignment enabled the screen to
still perform its shielding function with minimal interference to the passage of electrons from
cathode to plate.

grid wires
(cross-sectional view)

beam-forming plates
(2)
"space charge"
- -- - -- -- - - - - ---- -- -- -- - -- -- --- - -

cathode

plate

electron beams

screen wires
(cross-sectional view)
This resulted in lower screen current (and more plate current!) than an ordinary tetrode
tube, with little added expense to the construction of the tube.
Beam power tetrodes were often distinguished from their non-beam counterparts by a different schematic symbol, showing the beam-forming plates:

The "Beam power" tetrode tube
plate

grid
cathode

screen

CHAPTER 13. ELECTRON TUBES

476

13.6

The pentode

Another strategy for addressing the problem of secondary electrons being attracted by the
screen was the addition of a fifth wire element to the tube structure: a suppressor. These
five-element tubes were naturally called pentodes.

The pentode tube
plate
suppressor

screen

grid

cathode
The suppressor was another wire coil or mesh situated between the screen and the plate,
usually connected directly to ground potential. In some pentode tube designs, the suppressor
was internally connected to the cathode so as to minimize the number of connection pins having
to penetrate the tube envelope:

plate
(suppressor internally
connected to cathode)
screen

grid

cathode
The suppressor’s job was to repel any secondarily emitted electrons back to the plate: a
structural equivalent of the beam power tube’s space charge. This, of course, increased plate
current and decreased screen current, resulting in better gain and overall performance. In
some instances it allowed for greater operating plate voltage as well.

13.7

Combination tubes

Similar in thought to the idea of the integrated circuit, tube designers tried integrating different tube functions into single tube envelopes to reduce space requirements in more modern
tube-type electronic equipment. A common combination seen within a single glass shell was
two either diodes or two triodes. The idea of fitting pairs of diodes inside a single envelope
makes a lot of sense in light of power supply full-wave rectifier designs, always requiring multiple diodes.
Of course, it would have been quite impossible to combine thousands of tube elements into
a single tube envelope the way that thousands of transistors can be etched onto a single piece

13.7. COMBINATION TUBES

477

of silicon, but engineers still did their best to push the limits of tube miniaturization and
consolidation. Some of these tubes, whimsically called compactrons, held four or more complete
tube elements within a single envelope.
Sometimes the functions of two different tubes could be integrated into a single, combination tube in a way that simply worked more elegantly than two tubes ever could. An example
of this was the pentagrid converter, more generally called a heptode, used in some superheterodyne radio designs. These tubes contained seven elements: 5 grids plus a cathode and a plate.
Two of the grids were normally reserved for signal input, the other three relegated to screening
and suppression (performance-enhancing) functions. Combining the superheterodyne functions of oscillator and signal mixer together in one tube, the signal coupling between these two
stages was intrinsic. Rather than having separate oscillator and mixer circuits, the oscillator
creating an AC voltage and the mixer ”mixing” that voltage with another signal, the pentagrid
converter’s oscillator section created an electron stream that oscillated in intensity which then
directly passed through another grid for ”mixing” with another signal.
This same tube was sometimes used in a different way: by applying a DC voltage to one of
the control grids, the gain of the tube could be changed for a signal impressed on the other control grid. This was known as variable-mu operation, because the ”mu” (µ) of the tube (its amplification factor, measured as a ratio of plate-to-cathode voltage change over grid-to-cathode
voltage change with a constant plate current) could be altered at will by a DC control voltage
signal.
Enterprising electronics engineers also discovered ways to exploit such multi-variable capabilities of ”lesser” tubes such as tetrodes and pentodes. One such way was the so-called ultralinear audio power amplifier, invented by a pair of engineers named Hafler and Keroes, utilizing a tetrode tube in combination with a ”tapped” output transformer to provide substantial
improvements in amplifier linearity (decreases in distortion levels). Consider a ”single-ended”
triode tube amplifier with an output transformer coupling power to the speaker:

Speaker

input
voltage

If we substitute a tetrode for a triode in this circuit, we will see improvements in circuit gain

CHAPTER 13. ELECTRON TUBES

478

resulting from the electrostatic shielding offered by the screen, preventing unwanted feedback
between the plate and control grid:

Standard
configuration
of tetrode tube
in a single-ended
audio amplifier

Speaker

input
voltage

However, the tetrode’s screen may be used for functions other than merely shielding the
grid from the plate. It can also be used as another control element, like the grid itself. If
a ”tap” is made on the transformer’s primary winding, and this tap connected to the screen,
the screen will receive a voltage that varies with the signal being amplified (feedback). More
specifically, the feedback signal is proportional to the rate-of-change of magnetic flux in the
transformer core (dΦ/dt), thus improving the amplifier’s ability to reproduce the input signal
waveform at the speaker terminals and not just in the primary winding of the transformer:

"Ultralinear"
configuration
of tetrode tube
in a single-ended
audio amplifier

input
voltage

Speaker

13.8. TUBE PARAMETERS

479

This signal feedback results in significant improvements in amplifier linearity (and consequently, distortion), so long as precautions are taken against ”overpowering” the screen with
too great a positive voltage with respect to the cathode. As a concept, the ultralinear (screenfeedback) design demonstrates the flexibility of operation granted by multiple grid-elements
inside a single tube: a capability rarely matched by semiconductor components.
Some tube designs combined multiple tube functions in a most economic way: dual plates
with a single cathode, the currents for each of the plates controlled by separate sets of control
grids. Common examples of these tubes were triode-heptode and triode-hexode tubes (a hexode
tube is a tube with four grids, one cathode, and one plate).
Other tube designs simply incorporated separate tube structures inside a single glass envelope for greater economy. Dual diode (rectifier) tubes were quite common, as were dual triode
tubes, especially when the power dissipation of each tube was relatively low.

Dual triode tube

The 12AX7 and 12AU7 models are common examples of dual-triode tubes, both of low-power
rating. The 12AX7 is especially common as a preamplifier tube in electric guitar amplifier
circuits.

13.8

Tube parameters

For bipolar junction transistors, the fundamental measure of amplification is the Beta ratio
(β), defined as the ratio of collector current to base current (IC /IB ). Other transistor characteristics such as junction resistance, which in some amplifier circuits may impact performance
as much as β, are quantified for the benefit of circuit analysis. Electron tubes are no different,
their performance characteristics having been explored and quantified long ago by electrical
engineers.
Before we can speak meaningfully on these characteristics, we must define several mathematical variables used for expressing common voltage, current, and resistance measurements
as well as some of the more complex quantities:

CHAPTER 13. ELECTRON TUBES

480

µ = amplification factor, pronounced "mu"
(unitless)
gm = mutual conductance, in siemens
Ep = plate-to-cathode voltage
Eg = grid-to-cathode voltage
Ip = plate current
Ik = cathode current
Es = input signal voltage
rp = dynamic plate resistance, in ohms
∆ = delta, the Greek symbol for change

The two most basic measures of an amplifying tube’s characteristics are its amplification
factor (µ) and its mutual conductance (gm ), also known as transconductance. Transconductance is defined here just the same as it is for field-effect transistors, another category of
voltage-controlled devices. Here are the two equations defining each of these performance
characteristics:

µ=

gm =

∆Ep
∆Eg
∆Ip
∆Eg

with constant Ip (plate current)

with constant Ep (plate voltage)

Another important, though more abstract, measure of tube performance is its plate resistance. This is the measurement of plate voltage change over plate current change for a constant
value of grid voltage. In other words, this is an expression of how much the tube acts like a
resistor for any given amount of grid voltage, analogous to the operation of a JFET in its ohmic
mode:

rp =

∆Ep
∆Ip

with constant Eg (grid voltage)

The astute reader will notice that plate resistance may be determined by dividing the amplification factor by the transconductance:

13.9. IONIZATION (GAS-FILLED) TUBES

µ=

∆Ep

gm =

∆Eg

481

∆Ip
∆Eg

. . . dividing µ by gm . . .
∆Ep
rp =

∆Eg
∆Ip
∆Eg

rp =

rp =

∆Ep

∆Eg

∆Eg

∆Ip

∆Ep
∆Ip

These three performance measures of tubes are subject to change from tube to tube (just as
β ratios between two ”identical” bipolar transistors are never precisely the same) and between
different operating conditions. This variability is due partly to the unavoidable nonlinearities
of electron tubes and partly due to how they are defined. Even supposing the existence of a
perfectly linear tube, it will be impossible for all three of these measures to be constant over
the allowable ranges of operation. Consider a tube that perfectly regulates current at any
given amount of grid voltage (like a bipolar transistor with an absolutely constant β): that
tube’s plate resistance must vary with plate voltage, because plate current will not change
even though plate voltage does.
Nevertheless, tubes were (and are) rated by these values at given operating conditions, and
may have their characteristic curves published just like transistors.

13.9

Ionization (gas-filled) tubes

So far, we’ve explored tubes which are totally ”evacuated” of all gas and vapor inside their glass
envelopes, properly known as vacuum tubes. With the addition of certain gases or vapors,
however, tubes take on significantly different characteristics, and are able to fulfill certain
special roles in electronic circuits.
When a high enough voltage is applied across a distance occupied by a gas or vapor, or when
that gas or vapor is heated sufficiently, the electrons of those gas molecules will be stripped
away from their respective nuclei, creating a condition of ionization. Having freed the electrons
from their electrostatic bonds to the atoms’ nuclei, they are free to migrate in the form of a
current, making the ionized gas a relatively good conductor of electricity. In this state, the gas
is more properly referred to as a plasma.
Ionized gas is not a perfect conductor. As such, the flow of electrons through ionized gas
will tend to dissipate energy in the form of heat, thereby helping to keep the gas in a state

CHAPTER 13. ELECTRON TUBES

482

of ionization. The result of this is a tube that will begin to conduct under certain conditions,
then tend to stay in a state of conduction until the applied voltage across the gas and/or the
heat-generating current drops to a minimum level.
The astute observer will note that this is precisely the kind of behavior exhibited by a class
of semiconductor devices called ”thyristors,” which tend to stay ”on” once turned ”on” and tend
to stay ”off ” once turned ”off.” Gas-filled tubes, it can be said, manifest this same property of
hysteresis.
Unlike their vacuum counterparts, ionization tubes were often manufactured with no filament (heater) at all. These were called cold-cathode tubes, with the heated versions designated
as hot-cathode tubes. Whether or not the tube contained a source of heat obviously impacted
the characteristics of a gas-filled tube, but not to the extent that lack of heat would impact the
performance of a hard-vacuum tube.
The simplest type of ionization device is not necessarily a tube at all; rather, it is constructed
of two electrodes separated by a gas-filled gap. Simply called a spark gap, the gap between the
electrodes may be occupied by ambient air, other times a special gas, in which case the device
must have a sealed envelope of some kind.

Spark gap
enclosure (optional)

electrodes
A prime application for spark gaps is in overvoltage protection. Engineered not to ionize,
or ”break down” (begin conducting), with normal system voltage applied across the electrodes,
the spark gap’s function is to conduct in the event of a significant increase in voltage. Once
conducting, it will act as a heavy load, holding the system voltage down through its large
current draw and subsequent voltage drop along conductors and other series impedances. In a
properly engineered system, the spark gap will stop conducting (”extinguish”) when the system
voltage decreases to a normal level, well below the voltage required to initiate conduction.
One major caveat of spark gaps is their significantly finite life. The discharge generated
by such a device can be quite violent, and as such will tend to deteriorate the surfaces of the
electrodes through pitting and/or melting.
Spark gaps can be made to conduct on command by placing a third electrode (usually with
a sharp edge or point) between the other two and applying a high voltage pulse between that
electrode and one of the other electrodes. The pulse will create a small spark between the
two electrodes, ionizing part of the pathway between the two large electrodes, and enabling
conduction between them if the applied voltage is high enough:

13.9. IONIZATION (GAS-FILLED) TUBES

483

Triggered spark gap

main
voltage
source

(high voltage,
high current)

Load

spark gap

third electrode

triggering voltage source
(high voltage, low current)

Spark gaps of both the triggered and untriggered variety can be built to handle huge
amounts of current, some even into the range of mega-amps (millions of amps)! Physical size
is the primary limiting factor to the amount of current a spark gap can safely and reliably
handle.

When the two main electrodes are placed in a sealed tube filled with a special gas, a discharge tube is formed. The most common type of discharge tube is the neon light, used popularly as a source of colorful illumination, the color of the light emitted being dependent on the
type of gas filling the tube.

Construction of neon lamps closely resembles that of spark gaps, but the operational characteristics are quite different:

CHAPTER 13. ELECTRON TUBES

484

high voltage power supply (AC or DC)

NEON LAMP

electrode

current through the tube
causes the neon gas to glow

electrode

glass tube

small neon
indicator lamp
Neon lamp schematic symbol

By controlling the spacing of the electrodes and the type of gas in the tube, neon lights
can be made to conduct without drawing the excessive currents that spark gaps do. They
still exhibit hysteresis in that it takes a higher voltage to initiate conduction than it does to
make them ”extinguish,” and their resistance is definitely nonlinear (the more voltage applied
across the tube, the more current, thus more heat, thus lower resistance). Given this nonlinear
tendency, the voltage across a neon tube must not be allowed to exceed a certain limit, lest the
tube be damaged by excessive temperatures.
This nonlinear tendency gives the neon tube an application other than colorful illumination:
it can act somewhat like a zener diode, ”clamping” the voltage across it by drawing more and
more current if the voltage decreases. When used in this fashion, the tube is known as a glow
tube, or voltage-regulator tube, and was a popular means of voltage regulation in the days of
electron tube circuit design.
Rseries

glow-discharge
voltage regulator
tube

Rload

voltage across load
held relative constant
with variations of voltage
source and load resistance

Please take note of the black dot found in the tube symbol shown above (and in the neon
lamp symbol shown before that). That marker indicates the tube is gas-filled. It is a common
marker used in all gas-filled tube symbols.
One example of a glow tube designed for voltage regulation was the VR-150, with a nominal
regulating voltage of 150 volts. Its resistance throughout the allowable limits of current could

13.10. DISPLAY TUBES

485

vary from 5 kΩ to 30 kΩ, a 6:1 span. Like zener diode regulator circuits of today, glow tube
regulators could be coupled to amplifying tubes for better voltage regulation and higher load
current ranges.
If a regular triode was filled with gas instead of a hard vacuum, it would manifest all the
hysteresis and nonlinearity of other gas tubes with one major advantage: the amount of voltage applied between grid and cathode would determine the minimum plate-to cathode voltage
necessary to initiate conduction. In essence, this tube was the equivalent of the semiconductor
SCR (Silicon-Controlled Rectifier), and was called the thyratron.
+
Rload
Thyratron
Tube

high voltage
AC source

control
voltage

It should be noted that the schematic shown above is greatly simplified for most purposes
and thyratron tube designs. Some thyratrons, for instance, required that the grid voltage
switch polarity between their ”on” and ”off ” states in order to properly work. Also, some thyratrons had more than one grid!
Thyratrons found use in much the same way as SCR’s find use today: controlling rectified
AC to large loads such as motors. Thyratron tubes have been manufactured with different
types of gas fillings for different characteristics: inert (chemically non-reactive) gas, hydrogen
gas, and mercury (vaporized into a gas form when activated). Deuterium, a rare isotope of
hydrogen, was used in some special applications requiring the switching of high voltages.

13.10

Display tubes

In addition to performing tasks of amplification and switching, tubes can be designed to serve
as display devices.
Perhaps the best-known display tube is the cathode ray tube, or CRT. Originally invented
as an instrument to study the behavior of ”cathode rays” (electrons) in a vacuum, these tubes
developed into instruments useful in detecting voltage, then later as video projection devices
with the advent of television. The main difference between CRTs used in oscilloscopes and
CRTs used in televisions is that the oscilloscope variety exclusively use electrostatic (plate)
deflection, while televisions use electromagnetic (coil) deflection. Plates function much better
than coils over a wider range of signal frequencies, which is great for oscilloscopes but irrelevant for televisions, since a television electron beam sweeps vertically and horizontally at fixed
frequencies. Electromagnetic deflection coils are much preferred in television CRT construction because they do not have to penetrate the glass envelope of the tube, thus decreasing the

CHAPTER 13. ELECTRON TUBES

486
production costs and increasing tube reliability.

An interesting ”cousin” to the CRT is the Cat-Eye or Magic-Eye indicator tube. Essentially, this tube is a voltage-measuring device with a display resembling a glowing green ring.
Electrons emitted by the cathode of this tube impinge on a fluorescent screen, causing the
green-colored light to be emitted. The shape of the glow produced by the fluorescent screen
varies as the amount of voltage applied to a grid changes:

"Cat-Eye" indicator tube displays

large shadow

slight shadow

minimal shadow

The width of the shadow is directly determined by the potential difference between the control electrode and the fluorescent screen. The control electrode is a narrow rod placed between
the cathode and the fluorescent screen. If that control electrode (rod) is significantly more negative than the fluorescent screen, it will deflect some electrons away from the that area of the
screen. The area of the screen ”shadowed” by the control electrode will appear darker when
there is a significant voltage difference between the two. When the control electrode and fluorescent screen are at equal potential (zero voltage between them), the shadowing effect will be
minimal and the screen will be equally illuminated.
The schematic symbol for a ”cat-eye” tube looks something like this:

"Cat-Eye" or "Magic-Eye"
indicator tube
plate

fluorescent
screen
control
electrode

amplifier
grid
cathode
Here is a photograph of a cat-eye tube, showing the circular display region as well as the
glass envelope, socket (black, at far end of tube), and some of its internal structure:

13.10. DISPLAY TUBES

487

Normally, only the end of the tube would protrude from a hole in an instrument panel, so
the user could view the circular, fluorescent screen.
In its simplest usage, a ”cat-eye” tube could be operated without the use of the amplifier
grid. However, in order to make it more sensitive, the amplifier grid is used, and it is used like
this:

"Cat-Eye" indicator tube circuit

R

signal

As the signal voltage increases, current through
the tube is choked off. This decreases the voltage
between the plate and the fluorescent screen,
lessening the shadow effect (shadow narrows).
The cathode, amplifier grid, and plate act as a triode to create large changes in plate-tocathode voltage for small changes in grid-to-cathode voltage. Because the control electrode
is internally connected to the plate, it is electrically common to it and therefore possesses the
same amount of voltage with respect to the cathode that the plate does. Thus, the large voltage
changes induced on the plate due to small voltage changes on the amplifier grid end up causing
large changes in the width of the shadow seen by whoever is viewing the tube.

CHAPTER 13. ELECTRON TUBES

488

Control electrode negative with
respect to the fluorescent screen.
This is caused by a positive
amplifier grid voltage (with
respect to the cathode).

No voltage between control
electrode and flourescent screen.
This is caused by a negative
amplifier grid voltage (with
respect to the cathode).

”Cat-eye” tubes were never accurate enough to be equipped with a graduated scale as is
the case with CRT’s and electromechanical meter movements, but they served well as null
detectors in bridge circuits, and as signal strength indicators in radio tuning circuits. An
unfortunate limitation to the ”cat-eye” tube as a null detector was the fact that it was not
directly capable of voltage indication in both polarities.

13.11

Microwave tubes

For extremely high-frequency applications (above 1 GHz), the interelectrode capacitances and
transit-time delays of standard electron tube construction become prohibitive. However, there
seems to be no end to the creative ways in which tubes may be constructed, and several highfrequency electron tube designs have been made to overcome these challenges.
It was discovered in 1939 that a toroidal cavity made of conductive material called a cavity
resonator surrounding an electron beam of oscillating intensity could extract power from the
beam without actually intercepting the beam itself. The oscillating electric and magnetic fields
associated with the beam ”echoed” inside the cavity, in a manner similar to the sounds of
traveling automobiles echoing in a roadside canyon, allowing radio-frequency energy to be
transferred from the beam to a waveguide or coaxial cable connected to the resonator with a
coupling loop. The tube was called an inductive output tube, or IOT:

The inductive output tube (IOT)
coaxial
output
cable

RF power
output

RF
signal
input
electron beam
toroidal
cavity

DC supply

13.11. MICROWAVE TUBES

489

Two of the researchers instrumental in the initial development of the IOT, a pair of brothers
named Sigurd and Russell Varian, added a second cavity resonator for signal input to the
inductive output tube. This input resonator acted as a pair of inductive grids to alternately
”bunch” and release packets of electrons down the drift space of the tube, so the electron beam
would be composed of electrons traveling at different velocities. This ”velocity modulation” of
the beam translated into the same sort of amplitude variation at the output resonator, where
energy was extracted from the beam. The Varian brothers called their invention a klystron.

The klystron tube
coaxial
output
cable

RF
signal
input

RF power
output

Beam
control
electron beam

DC supply
Another invention of the Varian brothers was the reflex klystron tube. In this tube, electrons
emitted from the heated cathode travel through the cavity grids toward the repeller plate, then
are repelled and returned back the way they came (hence the name reflex) through the cavity
grids. Self-sustaining oscillations would develop in this tube, the frequency of which could be
changed by adjusting the repeller voltage. Hence, this tube operated as a voltage-controlled
oscillator.

The reflex klystron tube
cavity
grids

repeller

RF output
cavity
control grid
cathode

490

CHAPTER 13. ELECTRON TUBES

As a voltage-controlled oscillator, reflex klystron tubes served commonly as ”local oscillators” for radar equipment and microwave receivers:

Reflex klystron tube used as
a voltage-controlled oscillator

Initially developed as low-power devices whose output required further amplification for
radio transmitter use, reflex klystron design was refined to the point where the tubes could
serve as power devices in their own right. Reflex klystrons have since been superseded by
semiconductor devices in the application of local oscillators, but amplification klystrons continue to find use in high-power, high-frequency radio transmitters and in scientific research
applications.

One microwave tube performs its task so well and so cost-effectively that it continues to
reign supreme in the competitive realm of consumer electronics: the magnetron tube. This device forms the heart of every microwave oven, generating several hundred watts of microwave
RF energy used to heat food and beverages, and doing so under the most grueling conditions
for a tube: powered on and off at random times and for random durations.

Magnetron tubes are representative of an entirely different kind of tube than the IOT and
klystron. Whereas the latter tubes use a linear electron beam, the magnetron directs its electron beam in a circular pattern by means of a strong magnetic field:

13.12. TUBES VERSUS SEMICONDUCTORS

491

The magnetron tube
cavity
resonators

electron
cathode

RF output

beam

Once again, cavity resonators are used as microwave-frequency ”tank circuits,” extracting
energy from the passing electron beam inductively. Like all microwave-frequency devices using
a cavity resonator, at least one of the resonator cavities is tapped with a coupling loop: a loop
of wire magnetically coupling the coaxial cable to the resonant structure of the cavity, allowing
RF power to be directed out of the tube to a load. In the case of the microwave oven, the output
power is directed through a waveguide to the food or drink to be heated, the water molecules
within acting as tiny load resistors, dissipating the electrical energy in the form of heat.
The magnet required for magnetron operation is not shown in the diagram. Magnetic flux
runs perpendicular to the plane of the circular electron path. In other words, from the view of
the tube shown in the diagram, you are looking straight at one of the magnetic poles.

13.12

Tubes versus Semiconductors

Devoting a whole chapter in a modern electronics text to the design and function of electron
tubes may seem a bit strange, seeing as how semiconductor technology has all but obsoleted
tubes in almost every application. However, there is merit in exploring tubes not just for
historical purposes, but also for those niche applications that necessitate the qualifying phrase
”almost every application” in regard to semiconductor supremacy.
In some applications, electron tubes not only continue to see practical use, but perform their
respective tasks better than any solid-state device yet invented. In some cases the performance
and reliability of electron tube technology is far superior.
In the fields of high-power, high-speed circuit switching, specialized tubes such as hydrogen
thyratrons and krytrons are able to switch far larger amounts of current, far faster than any
semiconductor device designed to date. The thermal and temporal limits of semiconductor
physics place limitations on switching ability that tubes – which do not operate on the same
principles – are exempt from.
In high-power microwave transmitter applications, the excellent thermal tolerance of tubes
alone secures their dominance over semiconductors. Electron conduction through semiconducting materials is greatly impacted by temperature. Electron conduction through a vacuum

492

CHAPTER 13. ELECTRON TUBES

is not. As a consequence, the practical thermal limits of semiconductor devices are rather
low compared to that of tubes. Being able to operate tubes at far greater temperatures than
equivalent semiconductor devices allows tubes to dissipate more thermal energy for a given
amount of dissipation area, which makes them smaller and lighter in continuous high power
applications.
Another decided advantage of tubes over semiconductor components in high-power applications is their rebuildability. When a large tube fails, it may be disassembled and repaired at
far lower cost than the purchase price of a new tube. When a semiconductor component fails,
large or small, there is generally no means of repair.
The following photograph shows the front panel of a 1960’s vintage 5 kW AM radio transmitter. One of two ”Eimac” brand power tubes can be seen in a recessed area, behind the
glass door. According to the station engineer who gave the facility tour, the rebuild cost for
such a tube is only $800: quite inexpensive compared to the cost of a new tube, and still quite
reasonable in contrast to the price of a new, comparable semiconductor component!

Tubes, being less complex in their manufacture than semiconductor components, are potentially cheaper to produce as well, although the huge volume of semiconductor device production in the world greatly offsets this theoretical advantage. Semiconductor manufacture is
quite complex, involving many dangerous chemical substances and necessitating super-clean
assembly environments. Tubes are essentially nothing more than glass and metal, with a vacuum seal. Physical tolerances are ”loose” enough to permit hand-assembly of vacuum tubes,
and the assembly work need not be done in a ”clean room” environment as is necessary for
semiconductor manufacture.
One modern area where electron tubes enjoy supremacy over semiconductor components
is in the professional and high-end audio amplifier markets, although this is partially due to
musical culture. Many professional guitar players, for example, prefer tube amplifiers over
transistor amplifiers because of the specific distortion produced by tube circuits. An electric
guitar amplifier is designed to produce distortion rather than avoid distortion as is the case
with audio-reproduction amplifiers (this is why an electric guitar sounds so much different
than an acoustical guitar), and the type of distortion produced by an amplifier is as much a
matter of personal taste as it is technical measurement. Since rock music in particular was

13.12. TUBES VERSUS SEMICONDUCTORS

493

born with guitarists playing tube-amplifier equipment, there is a significant level of ”tube
appeal” inherent to the genre itself, and this appeal shows itself in the continuing demand for
”tubed” guitar amplifiers among rock guitarists.
As an illustration of the attitude among some guitarists, consider the following quote taken
from the technical glossary page of a tube-amplifier website which will remain nameless:
Solid State: A component that has been specifically designed to make a guitar
amplifier sound bad. Compared to tubes, these devices can have a very long lifespan,
which guarantees that your amplifier will retain its thin, lifeless, and buzzy sound
for a long time to come.
In the area of audio reproduction amplifiers (music studio amplifiers and home entertainment amplifiers), it is best for an amplifier to reproduce the musical signal with as little distortion as possible. Paradoxically, in contrast to the guitar amplifier market where distortion is a
design goal, high-end audio is another area where tube amplifiers enjoy continuing consumer
demand. Though one might suppose the objective, technical requirement of low distortion
would eliminate any subjective bias on the part of audiophiles, one would be very wrong. The
market for high-end ”tubed” amplifier equipment is quite volatile, changing rapidly with trends
and fads, driven by highly subjective claims of ”magical” sound from audio system reviewers
and salespeople. As in the electric guitar world, there is no small measure of cult-like devotion
to tube amplifiers among some quarters of the audiophile world. As an example of this irrationality, consider the design of many ultra-high-end amplifiers, with chassis built to display
the working tubes openly, even though this physical exposure of the tubes obviously enhances
the undesirable effect of microphonics (changes in tube performance as a result of sound waves
vibrating the tube structure).
Having said this, though, there is a wealth of technical literature contrasting tubes against
semiconductors for audio power amplifier use, especially in the area of distortion analysis.
More than a few competent electrical engineers prefer tube amplifier designs over transistors,
and are able to produce experimental evidence in support of their choice. The primary difficulty in quantifying audio system performance is the uncertain response of human hearing.
All amplifiers distort their input signal to some degree, especially when overloaded, so the
question is which type of amplifier design distorts the least. However, since human hearing is
very nonlinear, people do not interpret all types of acoustic distortion equally, and so some amplifiers will sound ”better” than others even if a quantitative distortion analysis with electronic
instruments indicates similar distortion levels. To determine what type of audio amplifier will
distort a musical signal ”the least,” we must regard the human ear and brain as part of the
whole acoustical system. Since no complete model yet exists for human auditory response,
objective assessment is difficult at best. However, some research indicates that the characteristic distortion of tube amplifier circuits (especially when overloaded) is less objectionable than
distortion produced by transistors.
Tubes also possess the distinct advantage of low ”drift” over a wide range of operating
conditions. Unlike semiconductor components, whose barrier voltages, β ratios, bulk resistances, and junction capacitances may change substantially with changes in device temperature and/or other operating conditions, the fundamental characteristics of a vacuum tube
remain nearly constant over a wide range in operating conditions, because those characteristics are determined primarily by the physical dimensions of the tube’s structural elements

494

CHAPTER 13. ELECTRON TUBES

(cathode, grid(s), and plate) rather than the interactions of subatomic particles in a crystalline
lattice.
This is one of the major reasons solid-state amplifier designers typically engineer their circuits to maximize power-efficiency even when it compromises distortion performance, because
a power-inefficient amplifier dissipates a lot of energy in the form of waste heat, and transistor
characteristics tend to change substantially with temperature. Temperature-induced ”drift”
makes it difficult to stabilize ”Q” points and other important performance-related measures in
an amplifier circuit. Unfortunately, power efficiency and low distortion seem to be mutually
exclusive design goals.
For example, class A audio amplifier circuits typically exhibit very low distortion levels, but
are very wasteful of power, meaning that it would be difficult to engineer a solid-state class A
amplifier of any substantial power rating due to the consequent drift of transistor characteristics. Thus, most solid-state audio amplifier designers choose class B circuit configurations for
greater efficiency, even though class B designs are notorious for producing a type of distortion
known as crossover distortion. However, with tubes it is easy to design a stable class A audio
amplifier circuit because tubes are not as adversely affected by the changes in temperature
experienced in a such a power-inefficient circuit configuration.
Tube performance parameters, though, tend to ”drift” more than semiconductor devices
when measured over long periods of time (years). One major mechanism of tube ”aging” appears to be vacuum leaks: when air enters the inside of a vacuum tube, its electrical characteristics become irreversibly altered. This same phenomenon is a major cause of tube mortality,
or why tubes typically do not last as long as their respective solid-state counterparts. When
tube vacuum is maintained at a high level, though, excellent performance and life is possible.
An example of this is a klystron tube (used to produce the high-frequency radio waves used in
a radar system) that lasted for 240,000 hours of operation (cited by Robert S. Symons of Litton
Electron Devices Division in his informative paper, ”Tubes: Still vital after all these years,”
printed in the April 1998 issue of IEEE Spectrum magazine).
If nothing else, the tension between audiophiles over tubes versus semiconductors has
spurred a remarkable degree of experimentation and technical innovation, serving as an excellent resource for those wishing to educate themselves on amplifier theory. Taking a wider
view, the versatility of electron tube technology (different physical configurations, multiple
control grids) hints at the potential for circuit designs of far greater variety than is possible
using semiconductors. For this and other reasons, electron tubes will never be ”obsolete,” but
will continue to serve in niche roles, and to foster innovation for those electronics engineers,
inventors, and hobbyists who are unwilling to let their minds by stifled by convention.

Appendix A-1

ABOUT THIS BOOK

A-1.1

Purpose

They say that necessity is the mother of invention. At least in the case of this book, that adage
is true. As an industrial electronics instructor, I was forced to use a sub-standard textbook
during my first year of teaching. My students were daily frustrated with the many typographical errors and obscure explanations in this book, having spent much time at home struggling
to comprehend the material within. Worse yet were the many incorrect answers in the back of
the book to selected problems. Adding insult to injury was the $100+ price.
Contacting the publisher proved to be an exercise in futility. Even though the particular
text I was using had been in print and in popular use for a couple of years, they claimed my
complaint was the first they’d ever heard. My request to review the draft for the next edition
of their book was met with disinterest on their part, and I resolved to find an alternative text.
Finding a suitable alternative was more difficult than I had imagined. Sure, there were
plenty of texts in print, but the really good books seemed a bit too heavy on the math and the
less intimidating books omitted a lot of information I felt was important. Some of the best
books were out of print, and those that were still being printed were quite expensive.
It was out of frustration that I compiled Lessons in Electric Circuits from notes and ideas I
had been collecting for years. My primary goal was to put readable, high-quality information
into the hands of my students, but a secondary goal was to make the book as affordable as
possible. Over the years, I had experienced the benefit of receiving free instruction and encouragement in my pursuit of learning electronics from many people, including several teachers
of mine in elementary and high school. Their selfless assistance played a key role in my own
studies, paving the way for a rewarding career and fascinating hobby. If only I could extend
the gift of their help by giving to other people what they gave to me . . .
So, I decided to make the book freely available. More than that, I decided to make it ”open,”
following the same development model used in the making of free software (most notably the
various UNIX utilities released by the Free Software Foundation, and the Linux operating
495

APPENDIX A-1.

496

ABOUT THIS BOOK

system, whose fame is growing even as I write). The goal was to copyright the text – so as to
protect my authorship – but expressly allow anyone to distribute and/or modify the text to suit
their own needs with a minimum of legal encumbrance. This willful and formal revoking of
standard distribution limitations under copyright is whimsically termed copyleft. Anyone can
”copyleft” their creative work simply by appending a notice to that effect on their work, but
several Licenses already exist, covering the fine legal points in great detail.
The first such License I applied to my work was the GPL – General Public License – of the
Free Software Foundation (GNU). The GPL, however, is intended to copyleft works of computer
software, and although its introductory language is broad enough to cover works of text, its
wording is not as clear as it could be for that application. When other, less specific copyleft
Licenses began appearing within the free software community, I chose one of them (the Design
Science License, or DSL) as the official notice for my project.
In ”copylefting” this text, I guaranteed that no instructor would be limited by a text insufficient for their needs, as I had been with error-ridden textbooks from major publishers. I’m sure
this book in its initial form will not satisfy everyone, but anyone has the freedom to change it,
leveraging my efforts to suit variant and individual requirements. For the beginning student
of electronics, learn what you can from this book, editing it as you feel necessary if you come
across a useful piece of information. Then, if you pass it on to someone else, you will be giving
them something better than what you received. For the instructor or electronics professional,
feel free to use this as a reference manual, adding or editing to your heart’s content. The
only ”catch” is this: if you plan to distribute your modified version of this text, you must give
credit where credit is due (to me, the original author, and anyone else whose modifications are
contained in your version), and you must ensure that whoever you give the text to is aware of
their freedom to similarly share and edit the text. The next chapter covers this process in more
detail.
It must be mentioned that although I strive to maintain technical accuracy in all of this
book’s content, the subject matter is broad and harbors many potential dangers. Electricity
maims and kills without provocation, and deserves the utmost respect. I strongly encourage
experimentation on the part of the reader, but only with circuits powered by small batteries
where there is no risk of electric shock, fire, explosion, etc. High-power electric circuits should
be left to the care of trained professionals! The Design Science License clearly states that
neither I nor any contributors to this book bear any liability for what is done with its contents.

A-1.2

The use of SPICE

One of the best ways to learn how things work is to follow the inductive approach: to observe
specific instances of things working and derive general conclusions from those observations.
In science education, labwork is the traditionally accepted venue for this type of learning, although in many cases labs are designed by educators to reinforce principles previously learned
through lecture or textbook reading, rather than to allow the student to learn on their own
through a truly exploratory process.
Having taught myself most of the electronics that I know, I appreciate the sense of frustration students may have in teaching themselves from books. Although electronic components
are typically inexpensive, not everyone has the means or opportunity to set up a laboratory
in their own homes, and when things go wrong there’s no one to ask for help. Most textbooks

A-1.3. ACKNOWLEDGEMENTS

497

seem to approach the task of education from a deductive perspective: tell the student how
things are supposed to work, then apply those principles to specific instances that the student
may or may not be able to explore by themselves. The inductive approach, as useful as it is, is
hard to find in the pages of a book.
However, textbooks don’t have to be this way. I discovered this when I started to learn a
computer program called SPICE. It is a text-based piece of software intended to model circuits
and provide analyses of voltage, current, frequency, etc. Although nothing is quite as good as
building real circuits to gain knowledge in electronics, computer simulation is an excellent alternative. In learning how to use this powerful tool, I made a discovery: SPICE could be used
within a textbook to present circuit simulations to allow students to ”observe” the phenomena
for themselves. This way, the readers could learn the concepts inductively (by interpreting
SPICE’s output) as well as deductively (by interpreting my explanations). Furthermore, in
seeing SPICE used over and over again, they should be able to understand how to use it themselves, providing a perfectly safe means of experimentation on their own computers with circuit
simulations of their own design.
Another advantage to including computer analyses in a textbook is the empirical verification it adds to the concepts presented. Without demonstrations, the reader is left to take
the author’s statements on faith, trusting that what has been written is indeed accurate. The
problem with faith, of course, is that it is only as good as the authority in which it is placed and
the accuracy of interpretation through which it is understood. Authors, like all human beings,
are liable to err and/or communicate poorly. With demonstrations, however, the reader can
immediately see for themselves that what the author describes is indeed true. Demonstrations
also serve to clarify the meaning of the text with concrete examples.
SPICE is introduced early in volume I (DC) of this book series, and hopefully in a gentle
enough way that it doesn’t create confusion. For those wishing to learn more, a chapter in
the Reference volume (volume V) contains an overview of SPICE with many example circuits.
There may be more flashy (graphic) circuit simulation programs in existence, but SPICE is
free, a virtue complementing the charitable philosophy of this book very nicely.

A-1.3

Acknowledgements

First, I wish to thank my wife, whose patience during those many and long evenings (and
weekends!) of typing has been extraordinary.
I also wish to thank those whose open-source software development efforts have made this
endeavor all the more affordable and pleasurable. The following is a list of various free computer software used to make this book, and the respective programmers:
• GNU/Linux Operating System – Linus Torvalds, Richard Stallman, and a host of others
too numerous to mention.
• Vim text editor – Bram Moolenaar and others.
• Xcircuit drafting program – Tim Edwards.
• SPICE circuit simulation program – too many contributors to mention.
• TEX text processing system – Donald Knuth and others.

APPENDIX A-1.

498

ABOUT THIS BOOK

• Texinfo document formatting system – Free Software Foundation.
• LATEX document formatting system – Leslie Lamport and others.
• Gimp image manipulation program – too many contributors to mention.
Appreciation is also extended to Robert L. Boylestad, whose first edition of Introductory
Circuit Analysis taught me more about electric circuits than any other book. Other important
texts in my electronics studies include the 1939 edition of The ”Radio” Handbook, Bernard
Grob’s second edition of Introduction to Electronics I, and Forrest Mims’ original Engineer’s
Notebook.
Thanks to the staff of the Bellingham Antique Radio Museum, who were generous enough
to let me terrorize their establishment with my camera and flash unit.
I wish to specifically thank Jeffrey Elkner and all those at Yorktown High School for being
willing to host my book as part of their Open Book Project, and to make the first effort in contributing to its form and content. Thanks also to David Sweet (website: (http://www.andamooka.org))
and Ben Crowell (website: (http://www.lightandmatter.com)) for providing encouragement, constructive criticism, and a wider audience for the online version of this book.
Thanks to Michael Stutz for drafting his Design Science License, and to Richard Stallman
for pioneering the concept of copyleft.
Last but certainly not least, many thanks to my parents and those teachers of mine who
saw in me a desire to learn about electricity, and who kindled that flame into a passion for
discovery and intellectual adventure. I honor you by helping others as you have helped me.
Tony Kuphaldt, July 2001
”A candle loses nothing of its light when lighting another”
Kahlil Gibran

Appendix A-2

CONTRIBUTOR LIST

A-2.1

How to contribute to this book

As a copylefted work, this book is open to revision and expansion by any interested parties.
The only ”catch” is that credit must be given where credit is due. This is a copyrighted work:
it is not in the public domain!
If you wish to cite portions of this book in a work of your own, you must follow the same
guidelines as for any other copyrighted work. Here is a sample from the Design Science License:
The Work is copyright the Author. All rights to the Work are reserved
by the Author, except as specifically described below. This License
describes the terms and conditions under which the Author permits you
to copy, distribute and modify copies of the Work.
In addition, you may refer to the Work, talk about it, and (as
dictated by "fair use") quote from it, just as you would any
copyrighted material under copyright law.
Your right to operate, perform, read or otherwise interpret and/or
execute the Work is unrestricted; however, you do so at your own risk,
because the Work comes WITHOUT ANY WARRANTY -- see Section 7 ("NO
WARRANTY") below.
If you wish to modify this book in any way, you must document the nature of those modifications in the ”Credits” section along with your name, and ideally, information concerning how
you may be contacted. Again, the Design Science License:
Permission is granted to modify or sample from a copy of the Work,
499

APPENDIX A-2.

500

CONTRIBUTOR LIST

producing a derivative work, and to distribute the derivative work
under the terms described in the section for distribution above,
provided that the following terms are met:
(a) The new, derivative work is published under the terms of this
License.
(b) The derivative work is given a new name, so that its name or
title can not be confused with the Work, or with a version of
the Work, in any way.
(c) Appropriate authorship credit is given: for the differences
between the Work and the new derivative work, authorship is
attributed to you, while the material sampled or used from
the Work remains attributed to the original Author; appropriate
notice must be included with the new work indicating the nature
and the dates of any modifications of the Work made by you.
Given the complexities and security issues surrounding the maintenance of files comprising
this book, it is recommended that you submit any revisions or expansions to the original author
(Tony R. Kuphaldt). You are, of course, welcome to modify this book directly by editing your
own personal copy, but we would all stand to benefit from your contributions if your ideas were
incorporated into the online “master copy” where all the world can see it.

A-2.2

Credits

All entries arranged in alphabetical order of surname. Major contributions are listed by individual name with some detail on the nature of the contribution(s), date, contact info, etc. Minor
contributions (typo corrections, etc.) are listed by name only for reasons of brevity. Please understand that when I classify a contribution as “minor,” it is in no way inferior to the effort
or value of a “major” contribution, just smaller in the sense of less text changed. Any and all
contributions are gratefully accepted. I am indebted to all those who have given freely of their
own knowledge, time, and resources to make this a better book!

A-2.2.1

Tony R. Kuphaldt

• Date(s) of contribution(s): 1996 to present
• Nature of contribution: Original author.
• Contact at: liec0@lycos.com

A-2.2. CREDITS

A-2.2.2

501

Dennis Crunkilton

• Date(s) of contribution(s): July 2004 to present
• Nature of contribution: Original author: Karnaugh mapping chapter; 04/2004; Shift
registers chapter, June 2005.
• Nature of contribution: Mini table of contents, all chapters except appendicies; html,
latex, ps, pdf; See Devel/tutorial.html; 01/2006.
• Nature of contribution: Completed Ch4 Bipolar junction transistors, CH7 Thyristors;
Ch9 Practical anlog ckts, a few additions; Ch8 Opamps, minor; 04/2009
• Contact at: dcrunkilton(at)att(dot)net

A-2.2.3

Bill Marsden

• Date(s) of contribution(s): May 2008
• Nature of contribution: Original author: “ElectroStatic Discharge” Section, Chapter
9.
• Nature of contribution: Chapter 3, LED’s update, photodiode update, Feburary 2009.
• Nature of contribution: Chapter 11, Section author: ”Pulse Width Modulation”, Feburary 2010.
• Contact at: bill marsden2(at)hotmail(dot)com

A-2.2.4

Your name here

• Date(s) of contribution(s): Month and year of contribution
• Nature of contribution: Insert text here, describing how you contributed to the book.
• Contact at: my email@provider.net

A-2.2.5

Typo corrections and other “minor” contributions

• line-allaboutcircuits.com (June 2005) Typographical error correction in Volumes 1,2,3,5,
various chapters ,(:s/visa-versa/vice versa/).
• Colin Creitz (May 2007) Chapters: several, s/it’s/its.
• Dennis Crunkilton (October 2005) Typographical capitlization correction to sectiontitles, chapter 9.
• Jeff DeFreitas (March 2006)Improve appearance: replace “/” and ”/” Chapters: A1, A2.
• Paul Stokes, Program Chair, Computer and Electronics Engineering Technology, ITT
Technical Institute, Houston, Tx (October 2004) Change (10012 = -810 + 710 = -110 ) to
(10012 = -810 + 110 = -110 ), CH2, Binary Arithmetic

502

APPENDIX A-2.

CONTRIBUTOR LIST

• Paul Stokes, Program Chair Computer and Electronics Engineering Technology, ITT
Technical Institute, Houston, Tx (October 2004) Near ”Fold up the corners” change Out=B’C’
to Out=B’D’, 14118.eps same change, Karnaugh Mapping
• The students of Bellingham Technical College’s Instrumentation program, .
• Roger Hollingsworth (May 2003) Suggested a way to make the PLC motor control
system fail-safe.
• Jan-Willem Rensman (May 2002) Suggested the inclusion of Schmitt triggers and gate
hysteresis to the ”Logic Gates” chapter.
• Don Stalkowski (June 2002) Technical help with PostScript-to-PDF file format conversion.
• Joseph Teichman (June 2002) Suggestion and technical help regarding use of PNG
images instead of JPEG.
• Unregistered@allaboutcircuits.com (November 2007) “Boolean algebra”, images 14019.pes
14021.eps output of gates incorrect s/0/A s/1/A .
• Dan Simon (February 2008) “Numeration Systems”, After BINARY TO OCTAL CONVERSION, position of decimal point —.
• Timothy Kingman (March 2008) Changed default roman font to newcent.
• Imranullah Syed (March 2008) Suggested centering of uncaptioned schematics.
• Chris01720@allaboutcircuits.com (March 2008) Ch 15, Inaccuracy involving CD-ROM
production.
• studiot@allaboutcircuits.com (March 2008) Ch 15, s/disk/disc/ in CDROM .
• Keith@allaboutcircuits.com (April 2008) Ch 12, s/laralel-out/parallel-out/ .
• Ken Braswell (May 2008) Ch 3, s/drips/drops/.
• Guest@allaboutcircuits.com (Oct 2008) Ch 2, s/are in close/are close/.
• Radoslav@allaboutcircuits.com (Oct 2008) Ch 8, s/that 1 mA of/that 6 mA/.
• Scanman@allaboutcircuits.com (Dec 2008) Ch 2, s/shells are hold/shells hold/.
• dgeorge@allaboutcircuits.com (Dec 2008) Ch 7, image 03320.png, swapped anode and
anode gate. left diagram.
• Unregistered Guest@allaboutcircuits.com (Feb 2009) Ch 2 s/than FET’s/than JFET’s.
• Unregistered Guest@allaboutcircuits.com (March 2009) Ch 8, 13061.png, change
formula for inverting gain to include ”-” .
• dezurtrat@allaboutcircuits.com (March 2009) Ch 3, 03443.png, s/p-p/peak.

A-2.2. CREDITS

503

• Bill Marsden@allaboutcircuits.com (April 2009) Ch 3, s/I would/It would/
• Peter O@allaboutcircuits.com (April 2009) Ch 1, closing parenthesis, above replaced
with reference to figure.
• Nanophotonics@allaboutcircuits.com (April 2009) Ch 9, image 53009.jpg s/courtisy/courtesy.
• Bill Marsden@allaboutcircuits.com (April 2009) Ch 8, images 2001.png, 2002.png
appearance.
• D Crunkilton (April 2009) Ch 4, images 23006.png, 23007.png updated.
• Unregistered Guest@allaboutcircuits.com (June 2009) Ch 7, s/SCR schematic symbol/TRIAC schematic symbol .
• Peter O’Dette (June 2009) Ch 1, s/is 1 watts/is 1 Watt , s/10 watt/10 Watts , s/ watt/ Watt
.
• Unregistered Guest@allaboutcircuits.com (June 2009) Ch 3, s/being/begin , near
”voltage at which they” . s/is/in near ”The diodes must be”.
• regrehan@allaboutcircuits.com (June 2009) Ch 4, s/r1 1 2 1/r1 1 2 1k in commonemitter amplifier SPICE list.
• Unregistered Guest@allaboutcircuits.com (July 2009) Ch 3, s/Note polarity change
on coil changed/Note polarity change on coil.
• Unregistered Guest@allaboutcircuits.com (August 2009) Ch 4, Swap PNP & NPN at
(b) & (c), caption of 03075.png .
• Unregistered Guest@allaboutcircuits.com (August 2009) equation typos 03077.png
03479.png .
• Peter O’Dette@allaboutcircuits.com (August 2009) Ch 2, Numerous changes, and
03409.png .
• Bill Marsden@allaboutcircuits.com (November 2009) Ch 4, Beta formula, ”Transistor
atings and Packages”.
• Unregistered Guest@allaboutcircuits.com (November 2009) Ch 3, Image 03288.eps
changed polarized capacitor to non-polarized.

• Unregistered Guest@allaboutcircuits.com (November 2009) Ch 4 s/hasre/share/ s/common=emitter/co
emitter/ .
• Uisge@allaboutcircuits.com (November 2009) Ch 3, s/once every half-cycle/one half of
every full cycle/ , s/much/half/ .
• Unregistered Guest@allaboutcircuits.com (November 2009) Ch 4 s/To maintaining/To
maintain .

504

APPENDIX A-2.

CONTRIBUTOR LIST

• Unregistered Guest@allaboutcircuits.com (November 2009) Ch 3 s/[model]/[modelname]/
.
• gareththegeek@allaboutcircuits.com (November 2009) Ch 2 numerous typos, omissions .
• Dcrunkilton@allaboutcircuits.com (November 2009) Ch 2 minor chages to text and
image 03392.eps .
• waynerr@allaboutcircuits.com (December 2009) Ch 4 equations 4 and 7 of image
03488.eps .
• jkenny@allaboutcircuits.com (January 2010) Ch 7 s/will will/will/ .
• BHijazi@allaboutcircuits.com (February 2010) Ch 1, Clarification of text between images 03378.png and 03379.png .
• SgtWookiei@allaboutcircuits.com (March 2010) Ch 4, image 03375.png, flipped pnp
and battery .
• Bill Marsden@allaboutcircuits.com (March 2010) Ch 9, Changes to ESD section.
• SgtWookiei@allaboutcircuits.com (April 2010) Ch 4, image 03078.png, added resistors.
• silv3rmOOn@allaboutcircuits.com (April 2010) Ch 4, typo in SPICE listing near image 20004.png.
• optomist1@allaboutcircuits.com (July 2010) Ch 2, typo s/campared/compared/.
• Bill Marsden@allaboutcircuits.com (July 2010) Ch 11, change [I] to italic tags in dcdrive.sml .
• Unregistered guest @allaboutcircuits.com (August 2010) Ch 2, s/The bopolar transistor/The bipolar junction transistor/ .
• Unregistered guest @allaboutcircuits.com (August 2010) Ch 4,
• D Crunkilton (Sept 2010) Ch 2 s/minuscule/minuscule; Ch 3 ,4 ,5, 7, s/useable/usable.
• beenthere@allaboutcircuits.com (Oct 2010) Ch 3, AC line powered LED material removed.

Appendix A-3

DESIGN SCIENCE LICENSE

Copyright c 1999-2000 Michael Stutz stutz@dsl.org
Verbatim copying of this document is permitted, in any medium.

A-3.1

0. Preamble

Copyright law gives certain exclusive rights to the author of a work, including the rights
to copy, modify and distribute the work (the ”reproductive,” ”adaptative,” and ”distribution”
rights).
The idea of ”copyleft” is to willfully revoke the exclusivity of those rights under certain
terms and conditions, so that anyone can copy and distribute the work or properly attributed
derivative works, while all copies remain under the same terms and conditions as the original.
The intent of this license is to be a general ”copyleft” that can be applied to any kind of work
that has protection under copyright. This license states those certain conditions under which
a work published under its terms may be copied, distributed, and modified.
Whereas ”design science” is a strategy for the development of artifacts as a way to reform
the environment (not people) and subsequently improve the universal standard of living, this
Design Science License was written and deployed as a strategy for promoting the progress of
science and art through reform of the environment.

A-3.2

1. Definitions

”License” shall mean this Design Science License. The License applies to any work which
contains a notice placed by the work’s copyright holder stating that it is published under the
terms of this Design Science License.
”Work” shall mean such an aforementioned work. The License also applies to the output of
the Work, only if said output constitutes a ”derivative work” of the licensed Work as defined by
copyright law.
505

506

APPENDIX A-3.

DESIGN SCIENCE LICENSE

”Object Form” shall mean an executable or performable form of the Work, being an embodiment of the Work in some tangible medium.
”Source Data” shall mean the origin of the Object Form, being the entire, machine-readable,
preferred form of the Work for copying and for human modification (usually the language,
encoding or format in which composed or recorded by the Author); plus any accompanying
files, scripts or other data necessary for installation, configuration or compilation of the Work.
(Examples of ”Source Data” include, but are not limited to, the following: if the Work is an
image file composed and edited in ’PNG’ format, then the original PNG source file is the Source
Data; if the Work is an MPEG 1.0 layer 3 digital audio recording made from a ’WAV’ format
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necessary for compilation constitute the Source Data.)
”Author” shall mean the copyright holder(s) of the Work.
The individual licensees are referred to as ”you.”

A-3.3

2. Rights and copyright

The Work is copyright the Author. All rights to the Work are reserved by the Author, except as
specifically described below. This License describes the terms and conditions under which the
Author permits you to copy, distribute and modify copies of the Work.
In addition, you may refer to the Work, talk about it, and (as dictated by ”fair use”) quote
from it, just as you would any copyrighted material under copyright law.
Your right to operate, perform, read or otherwise interpret and/or execute the Work is unrestricted; however, you do so at your own risk, because the Work comes WITHOUT ANY
WARRANTY – see Section 7 (”NO WARRANTY”) below.

A-3.4

3. Copying and distribution

Permission is granted to distribute, publish or otherwise present verbatim copies of the entire
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Permission is granted to distribute, publish or otherwise present copies of the Object Form
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(a) The Source Data is included in the same distribution, distributed under the terms of
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section above; or

A-3.5. 4. MODIFICATION

507

(c) A third party’s written offer for obtaining the Source Data at no cost, as described in
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You may copy and distribute the Work either gratis or for a fee, and if desired, you may
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The aggregation of the Work with other works which are not based on the Work – such as
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the License for the Work.

A-3.5

4. Modification

Permission is granted to modify or sample from a copy of the Work, producing a derivative
work, and to distribute the derivative work under the terms described in the section for distribution above, provided that the following terms are met:
(a) The new, derivative work is published under the terms of this License.
(b) The derivative work is given a new name, so that its name or title can not be confused
with the Work, or with a version of the Work, in any way.
(c) Appropriate authorship credit is given: for the differences between the Work and the
new derivative work, authorship is attributed to you, while the material sampled or used from
the Work remains attributed to the original Author; appropriate notice must be included with
the new work indicating the nature and the dates of any modifications of the Work made by
you.

A-3.6

5. No restrictions

You may not impose any further restrictions on the Work or any of its derivative works beyond
those restrictions described in this License.

A-3.7

6. Acceptance

Copying, distributing or modifying the Work (including but not limited to sampling from the
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License, any rights granted to you by the License are null and void. The copying, distribution or
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If for any reason, conditions are imposed on you that forbid you to fulfill the conditions of
this License, you may not copy, distribute or modify the Work at all.
If any part of this License is found to be in conflict with the law, that part shall be interpreted in its broadest meaning consistent with the law, and no other parts of the License shall
be affected.

APPENDIX A-3.

508

A-3.8

DESIGN SCIENCE LICENSE

7. No warranty

THE WORK IS PROVIDED ”AS IS,” AND COMES WITH ABSOLUTELY NO WARRANTY,
EXPRESS OR IMPLIED, TO THE EXTENT PERMITTED BY APPLICABLE LAW, INCLUDING BUT NOT LIMITED TO THE IMPLIED WARRANTIES OF MERCHANTABILITY OR
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A-3.9

8. Disclaimer of liability

IN NO EVENT SHALL THE AUTHOR OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS
OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION)
HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT,
STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
ANY WAY OUT OF THE USE OF THIS WORK, EVEN IF ADVISED OF THE POSSIBILITY
OF SUCH DAMAGE.
END OF TERMS AND CONDITIONS
[$Id:

dsl.txt,v 1.25 2000/03/14 13:14:14 m Exp m $]

Index
α ratio, 210, 264
β ratio, 186, 264
10-50 milliamp signal, 379
4-20 milliamp signal, 379
4-layer diode, 322
741 operational amplifier, 361
A-weighted dB scale, 14
A/D converter, 365
AC-DC power supply schematic, 333
Active device, 3
Active mode, transistor, 183
Alpha ratio, 210, 264
Amplification, definition, 3
Amplifier, differential, 357
Amplifier, inverting, 371
Amplifier, noninverting, 371
Amplifier, single-ended, 357
Analog-to-digital converter, 365
Angular Momentum quantum number, 33
Anti-static foam, 287
Antilogarithm, 10
Artifact, measurement, 450
Astable, 391
Attenuator, 16
Attenuator, bridged T, 21
Attenuator, coaxial, 23
Attenuator, L, 21
Attenuator, PI, 20
Attenuator, rf, 23
Attenuator, T, 19
Avalanche photodiode, 153
Averager, 380
Band, electron, 47
Bandwidth, amplifier, 256
Bardeen, John, 60, 65

Beam power tube, 474
Bel, 8
Beta ratio, 186, 264
Beta ratio, bipolar transistor, 479
Beta variations, 187
Bias current, op-amp, 398
Bias, BJT, base, 235
Bias, BJT, calculations, 235
Bias, BJT, collector-feedback, 236
Bias, BJT, emitter, 237
Bias, BJT, voltage divider, 243
Bias, diode, 98
Bias, transistor, 195, 222
Bilateral, 294
Bipolar-mode MOSFET, 314
Bistable, 389
Brattain, Walter, 60, 65
Breakdown, diode, 102
Breakdown, transistor, 327
Breakover, thyristor, 327
Bridge rectifier circuit, 111
Bridge rectifier circuit, polyphase, 111
Bypass capacitor, 261
Calculus, 360, 386, 438
Capacitance, diode, 108
Capacitor, bypass, 261
Capacitor, coupling, 231
Capacitor, op-amp compensation, 404
Cascode amplifier, 218
Cat-Eye tube, 486
Cathode, 471
Cathode Ray Tube, 485
Center-tap rectifier circuit, 109
Characteristic curves, transistor, 186, 293
Check valve, 98
Clamper circuit, 121
509

INDEX

510
Class A amplifier operation, 223
Class AB amplifier operation, 224
Class B amplifier operation, 223
Class C amplifier operation, 225
Class D amplifier operation, 225
Class, amplifier operation, 222
Clipper circuit, 117
clipper, zener diode, 142
CMRR, 393
Cockcroft-Walton, voltage multiplier, 128
Coherent light, 151
Cold-cathode tube, 482
COMFET, 314
Common-base amplifier, 210
Common-collector amplifier, 202
Common-emitter amplifier, 189
Common-mode rejection ratio, 393
Common-mode voltage, 393
Commutating diode, 130, 131
Commutation, 131
Commutation time, diode, 108
Commutation, forced, 350, 351
Commutation, natural, 333, 351
Comparator, 363
Compensation capacitor, op-amp, 404
Conduction band, 48
Conductivity-Modulated Field-Effect Transistor, 314
Constant-current diode, 162
Controlled rectifier, 338
Conventional flow, 98
Cooper pair, 80
Coupling capacitor, 231
Coupling loop, resonator, 488, 491
Critical rate of voltage rise, 328, 330
Crossover distortion, 494
Crowbar, 333
CRT, 485
Crystal radio, 424
Current mirror, 264
Current source, 184, 378
Current sourcing vs. sinking, 267
Current, diode leakage, 108
Current-limiting diode, 162
Current-regulating diode, 162
Curve, characteristic, 186, 293

Cutoff voltage, 285
Cutoff, transistor, 177, 183
Czochralski process, silicon , 75
Darlington pair, 209
Datasheet, component, 107
dB, 8
dB, absolute power measurements, 15, 16
dB, sound measurements, 14
dBA, 14
dBk, 16
dBm, 15
dBW, 16
DC restorer circuit, 121
Decibel, 8
Decibels, attenuator, 17
Decineper, 13
Degenerative feedback, 256
Derivative, calculus, 439
DIAC, 329
Differential amplifier, 357
Differential pair, 408, 409
Differentiation, 360
Differentiation, calculus, 386, 438
Diode, 98
Diode check, meter function, 104, 180
Diode equation, the, 101
Diode junction capacitance, 108
Diode leakage current, 108
Diode PIV rating, 102
Diode tube, 471
Diode, 4-layer, 73
Diode, constant-current, 162
Diode, Esaki, 144
Diode, four-layer, 322
Diode, hot carrier, 143, 158
Diode, IMPATT, 160
Diode, laser, 150
Diode, light-activated, 152
Diode, light-emitting, 146
diode, MIIM, 85
diode, MIM, 163
Diode, pin, 159
Diode, PNPN, 322
Diode, schottky, 143
Diode, Shockley, 322

INDEX
Diode, snap, 159
Diode, SPICE, 163
Diode, tunnel, 144
Diode, varactor, 158
Diode, varicap, 158
Diode, zener, 136
DIP, 361
Discharge tube, 483
Distortion, amplifier, 256
Distortion, crossover, 494
dn, 13
Double-layer tunneling transistor, 84
Drift, op-amp, 404
Dropout, thyristor, 327
Dual Inline Package, 361
Dual power supply, 357
Duty cycle, square wave, 364
Duty cycle, squarewave, 225
Edison effect, 469
Effect, Edison, 469
Electrode, cathode, 471
Electrode, grid, 470
Electrode, screen, 473
Electrode, suppressor, 476
Electron, 28
Electron flow, 98
Emitter follower, 205
Emitter-follower amplifier, 202
Equation, diode, 101
Equilibrium, 366
Esaki diode, 144
Exclusion principle, 36
Failure mode, zener diode, 136
Faraday’s Law, 130, 131
Feedback, amplifier, 256
Feedback, negative, 366
Feedback, positive, 388
FET, field effect transistor, 65
Field effect transistor , 65
Firing, thyristor, 327
Flash converter, 365
Floating, 177, 330
Flow, electron vs. conventional, 98
Foam, anti-static, 287

511
Forced commutation, 350, 351
Forward bias, 98
Forward voltage, diode, 100
Four-layer diode, 322
Frequency response, op-amp, 404
Full-wave rectifier circuit, 109, 111
Gain, 6
Gain, AC versus DC, 7
Gate turn off switch, 73
Gate-Controlled Switch, 330
Gate-Turn-Off thyristor, 330
GCS, 330
Glow tube, 484
Grid, 470
Ground, 356
Ground, virtual, 371
GTO, 330
GTO, gate turn off switch, 73
Half-wave rectifier circuit, 108
Harmonic, 342
Harmonic, even vs. odd, 342
Harmonics and waveform symmetry, 342
Heptode, 477
hfe, 187
High temperature superconductors:, 82
Holding current, SCR, 332
hot carrier diode, 143
Hot-cathode tube, 482
Hybrid parameters, 187
Hysteresis, 389, 482
IC, 267
IGBT, 314, 353
IGFET, insulated gate field effect transistor,
70
IGT, 314, 353
IMPATT diode, 160
Inductive output tube, 488
Inert elements, 38
Input, inverting, 358
Input, noninverting, 358
Insulated gate field effect transistor, 70
Insulated-Gate Bipolar Transistor, 314, 353
Insulated-Gate Transistor, 314, 353

INDEX

512
Integrated circuit, 267
Integration, calculus, 386, 438
Inverting amplifier, 192, 371
Inverting summer, 381
Ionization, 318, 481
JFET, junction field effect transistor, 65
Josephson junctions, 80
Josephson transistor, 81
Joule’s Law, 11, 136
Junction capacitance, diode, 108
Kickback, inductive, 130
Kirchhoff ’s Current Law, 175
Kirchhoff ’s Voltage Law, 205
Klystron, 488
Laser diode, 150
Laser light, 151
Latch-up, 396
Latching, thyristor, 327
Leakage current, diode, 102, 108
LED, 146
Light-emitting diode, 146
Lilienfeld, Julius, 65
Load line, 226
Logarithm, 10
Magic-Eye tube, 486
Magnetic quantum number, 33
Magnetic tunnel junction, 88
Mechanics, quantum, 32
MESFET, metal semiconductor field effect transistor, 68
Metal oxide field effect transistor, 70
Mho, 296
Microphonics, electron tube, 493
MIIM, diode, 85
Miller effect, 277
MIM diode, 163
Monochromatic light, 151
MOS Controlled Thyristor, 352
MOS-gated thyristor, 352
MOSFET, metal oxide field effect transistor,
70
MTJ, magnetic tunnel junction, 88

Mu, tube amplification factor, 477
Multiplier circuit, diode, 123
Multiplier, frequency, varactor, 422
Natural commutation, 333, 351
Negative feedback, 256, 366
Negative resistance, 144
Neper, 13
Neutron, 28
Noble elements, 38
Noninverting amplifier, 371
Noninverting summer, 381
Number, quantum, 33
Offset null, op-amp, 397
Offset voltage, op-amp, 396
Ohmic region, JFET, 295
Op-amp, 262, 361
Operational amplifier, 262, 361
Orbital, electron, 35
Oscillator, 256
Oscillator, op-amp, 391
oscillator, phase shift, 422
Oscillator, relaxation, 319
Oscillator, voltage-controlled, 489
Over-unity machine, 5
Passive averager, 380
Passive device, 3
Pauli, exclusion principle, 36
PCB, 106
Peak detector, 115
Pentagrid tube, 477
Pentode tube, 306
Perpetual motion machine, 3
Phase shift, op-amp, 405
Photodiode, 152
Photodiode amplifier, 455
Photodiode, APD, 153
Photodiode, PIN, 153
PI-network , 16
PIN diode, 159
PIN, photodiode, 153
Pinch-off voltage, 285
PIV rating, diode, 102
Plasma, 318, 481

INDEX
PNPN diode, 322
Polyphase bridge rectifier circuit, 111
Positive feedback, 256, 318, 388
Power supply schematic, AC-DC, 333
Principal quantum number, 33
Printed circuit board, 106
Process variable, 359
Programmable unijunction transistor, 347
Proton, 28
Pulse-width modulation, 364
Push-pull amplifier, 223
PWM, 364
Quantum dot, 86
Quantum dot transistor, 86
Quantum mechanics, 32
Quantum number, 33
Quantum physics, 28
quantum tunneling, 83
Quiescent, 226
Radio, crystal, 424
Rail voltage, 368
Rectifier, 98
Rectifier circuit, 108
Rectifier circuit, full-wave, 109, 111
Rectifier circuit, half-wave, 108
Rectifier, controlled, 338
Reference junction, thermocouple, 398
Reflex klystron, 489
Regenerative feedback, 256, 318
Regulator, voltage, 207
Relaxation oscillator, 319
Resistance, negative, 144
Resonant tunneling diode, 84
Restorer circuit, 121
Reverse bias, 98
Reverse recovery time, diode, 108
Reverse voltage rating, diode, 102
Rheostat, 188, 296
Richter scale, 9
Ripple voltage, 113
Runaway, thermal, 259
s,p,d,f subshell notation, 34
Saturable reactor, 3

513
Saturation voltage, 368
Saturation, transistor, 177, 183
Schottky diode, 143
SCR, 329, 485
SCR bridge rectifier, 338
SCR, silicon controlled rectifier, 73
Screen, 473
SCS, 350
Secondary emission, 474
Semiconductor, defined, 48
Sensitive gate, SCR, 332
Setpoint, 359
Shell, electron, 33
Shockley diode, 322
Shockley, William, 60, 65, 73
Siemens, 296, 379
Signal, 10-50 milliamp, 379
Signal, 4-20 milliamp, 379
Silicon controlled rectifier, 73
Silicon-controlled rectifier, 329, 485
Silicon-controlled switch, 350
Single-ended amplifier, 357
Sink, current, 267
Slicer circuit, 117
Slide rule, 10
Small-scale integration, 409
Snap diode, 159
Snubber, 131
Solar cell, 154
Solid-state, 2
Sound intensity measurement, 14
Spark gap, 482
SPICE, diode, 163
Spin quantum number, 33
Spintronics, 88
Split power supply, 357
SQUID:, 81
SSI, 409
Step recovery diode, 159
Subshell notation, 34
Subshell, electron, 34
Superconduction quantum interference device,
81
Superconductivity, 79
Superposition theorem, 231
Suppressor, 476

INDEX

514
Switching time, diode, 108
T-network , 16
Tetrode tube, 306, 473
Theorem, Superposition, 231
Thermal runaway, BJT, 259
Thermal voltage, diode, 101
Thermocouple, 398
Three-phase bridge rectifier circuit, 111
Thyratron, 485
Thyratron tube, 320
Thyristor, 73, 482
Time, diode switching, 108
Totalizer, 387
Transconductance, 296, 379
Transconductance amplifier, 379
Transistor, field effect, 65
Transistor, insulated gate field effect, 70
Transistor, Josephson, 81
Transistor, metal oxide field effect, 70
Transistor, programmable unijunction, 347
Transistor, single electron, 86
Transistor, unijunction, 344
Triode tube, 306, 320, 471
Tube, discharge, 483
Tunnel diode, 144
Tunnel junction, magnetic, 88
tunneling, quantum, 83
Unijunction transistor, 344
Unipolar, conduction, 65
Unit, bel, 8
Unit, decineper, 13
Unit, mho, 296
Unit, neper, 13
Unit, siemens, 296, 379
Valence band, 48
Valence shell, 34
Valve, “check”, 98
Varactor diode, 158
Varicap diode, 158
VCO, 321
Virtual ground, 371
VMOS transistor, 314
Voltage buffer, 368

Voltage doubler circuit, 123
Voltage follower, 205, 368
Voltage multiplier circuit, 123
Voltage multiplier, Cockcroft-Walton, 128
Voltage regulator, 207
Voltage regulator tube, 484
Voltage rise, critical rate of, 328, 330
Voltage, bias, 195, 222
Voltage, common-mode, 393
Voltage, forward, 100
Voltage, op-amp output saturation, 368
Voltage, ripple, 113
Voltage-controlled oscillator, 321, 489
Volume units, 15
VU scale, 15
Waveform symmetry and harmonics, 342
Zener diode, 136
Zener diode failure mode, 136
Zener diode, clipper, 142

INDEX
.

515

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Title                           : "Lessons In Electric Circuits, Volume III -- Semiconductors"
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Lessons In Electric Circuits, Volume III Semiconductors Circuits 3

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