ISM For Electrical Machinery Fundamentals 4e (Instructor's Manual)

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Instructor’s Manual

to accompany

Chapman

Electric Machinery Fundamentals
Fourth Edition

Stephen J. Chapman
BAE SYSTEMS Australia

i

Instructor’s Manual to accompany Electric Machinery Fundamentals, Fourth Edition
Copyright  2004 McGraw-Hill, Inc.
All rights reserved. Printed in the United States of America. No part of this book may be used or reproduced in
any manner whatsoever without written permission, with the following exception: homework solutions may be
copied for classroom use.
ISBN: ???

ii

CHAPTER 1:

INTRODUCTION TO MACHINERY PRINCIPLES

CHAPTER 2:

TRANSFORMERS

23

CHAPTER 3:

INTRODUCTION TO POWER ELECTRONICS

63

CHAPTER 4:

AC MACHINERY FUNDAMENTALS

103

CHAPTER 5:

SYNCHRONOUS GENERATORS

109

CHAPTER 6:

SYNCHRONOUS MOTORS

149

CHAPTER 7:

INDUCTION MOTORS

171

CHAPTER 8:

DC MACHINERY FUNDAMENTALS

204

CHAPTER 9:

DC MOTORS AND GENERATORS

214

CHAPTER 10:

SINGLE-PHASE AND SPECIAL-PURPOSE MOTORS

1

270

APPENDIX A: REVIEW OF THREE-PHASE CIRCUITS

280

APPENDIX B:

288

COIL PITCH AND DISTRIBUTED WINDINGS

APPENDIX C: SALIENT POLE THEORY OF SYNCHRONOUS MACHINES

295

APPENDIX D: ERRATA FOR ELECTRIC MACHINERY FUNDAMENTALS 4/E

301

iii

PREFACE
TO THE INSTRUCTOR

This Instructor’s Manual is intended to accompany the fourth edition of Electric Machinery Fundamentals. To
make this manual easier to use, it has been made self-contained. Both the original problem statement and the
problem solution are given for each problem in the book. This structure should make it easier to copy pages from
the manual for posting after problems have been assigned.
Many of the problems in Chapters 2, 5, 6, and 9 require that a student read one or more values from a
magnetization curve. The required curves are given within the textbook, but they are shown with relatively few
vertical and horizontal lines so that they will not appear too cluttered. Electronic copies of the corresponding opencircuit characteristics, short-circuit characteristics, and magnetization curves as also supplied with the book. They
are supplied in two forms, as MATLAB MAT-files and as ASCII text files. Students can use these files for
electronic solutions to homework problems. The ASCII files are supplied so that the information can be used with
non-MATLAB software.
Please note that the file extent of the magnetization curves and open-circuit characteristics have changed in this
edition. In the Third Edition, I used the file extent *.mag for magnetization curves. Unfortunately, after the book
was published, Microsoft appropriated that extent for a new Access table type in Office 2000. That made it hard
for users to examine and modify the data in the files. In this edition, all magnetization curves, open-circuit
characteristics, short-circuit characteristics, etc. use the file extent *.dat to avoid this problem.
Each curve is given in ASCII format with comments at the beginning. For example, the magnetization curve in
Figure P9-1 is contained in file p91_mag.dat. Its contents are shown below:
%
%
%
%
%
%
%
%
%
%

This is the magnetization curve shown in Figure
P9-1. The first column is the field current in
amps, and the second column is the internal
generated voltage in volts at a speed of 1200 r/min.
To use this file in MATLAB, type "load p91_mag.dat".
The data will be loaded into an N x 2 array named
"p91_mag", with the first column containing If and
the second column containing the open-circuit voltage.
MATLAB function "interp1" can be used to recover
a value from this curve.
0
0
0.0132
6.67
0.03
13.33
0.033
16
0.067
31.30
0.1
45.46
0.133
60.26
0.167
75.06
0.2
89.74
iv

0.233
0.267
0.3
0.333
0.367
0.4
0.433
0.467
0.5
0.533
0.567
0.6
0.633
0.667
0.7
0.733
0.767
0.8
0.833
0.867
0.9
0.933
0.966
1
1.033
1.067
1.1
1.133
1.167
1.2
1.233
1.267
1.3
1.333
1.367
1.4
1.433
1.466
1.5

104.4
118.86
132.86
146.46
159.78
172.18
183.98
195.04
205.18
214.52
223.06
231.2
238
244.14
249.74
255.08
259.2
263.74
267.6
270.8
273.6
276.14
278
279.74
281.48
282.94
284.28
285.48
286.54
287.3
287.86
288.36
288.82
289.2
289.375
289.567
289.689
289.811
289.950

To use this curve in a MATLAB program, the user would include the following statements in the program:
% Get the magnetization curve. Note that this curve is
% defined for a speed of 1200 r/min.
if_values = p91_mag(:,1);
ea_values = p91_mag(:,2);
n_0 = 1200;
Unfortunately, an error occurred during the production of this book, and the values (resistances, voltages, etc.) in
some end-of-chapter artwork are not the same as the values quoted in the end-of-chapter problem text. I have
attached corrected pages showing each discrepancy in Appendix D of this manual. Please print these pages and
distribute them to your students before assigning homework problems. (Note that this error will be corrected at the
second printing, so it may not be present in your student’s books.)
v

The solutions in this manual have been checked carefully, but inevitably some errors will have slipped through. If
you locate errors which you would like to see corrected, please feel free to contact me at the address shown below,
or at my email address schapman@tpgi.com.au. I greatly appreciate your input! My physical and email
addresses may change from time to time, but my contact details will always be available at the book’s Web site,
which is http://www.mhhe.com/engcs/electrical/chapman/.
I am also contemplating a homework problem refresh, with additional problems added on the book’s Web site midway through the life of this edition. If that feature would be useful to you, please provide me with feedback about
which problems that you actually use, and the areas where you would like to have additional exercises. This
information can be passed to the email address given below, or alternately via you McGraw-Hill representative.
Thank you.

Stephen J. Chapman
Melbourne, Australia
January 4, 2004
Stephen J. Chapman
Caulfield North, VIC 3161
Australia
Phone +61-3-9527-9372

vi

Chapter 1: Introduction to Machinery Principles
1-1.

A motor’s shaft is spinning at a speed of 3000 r/min. What is the shaft speed in radians per second?
SOLUTION The speed in radians per second is
 1 min   2π rad 

 = 314.2 rad/s
 60 s   1 r 

ω = ( 3000 r/min ) 
1-2.

A flywheel with a moment of inertia of 2 kg ⋅ m2 is initially at rest. If a torque of 5 N ⋅ m
(counterclockwise) is suddenly applied to the flywheel, what will be the speed of the flywheel after 5 s?
Express that speed in both radians per second and revolutions per minute.
SOLUTION The speed in radians per second is:
5 N ⋅m
τ 
ω =α t =   t =
( 5 s ) = 12.5 rad/s
J
2
kg ⋅ m 2
 

The speed in revolutions per minute is:
 1 r   60 s 
n = (12.5 rad/s ) 

 = 119.4 r/min
 2π rad   1 min 

1-3.

A force of 5 N is applied to a cylinder, as shown in Figure P1-1. What are the magnitude and direction of
the torque produced on the cylinder? What is the angular acceleration α of the cylinder?

SOLUTION The magnitude and the direction of the torque on this cylinder is:

τ ind = rF sin θ , CCW
τ ind = ( 0.25 m)(10 N ) sin 30° = 1.25 N ⋅ m, CCW
The resulting angular acceleration is:

α=
1-4.

τ
J

=

1.25 N ⋅ m
5 kg ⋅ m 2

A motor is supplying 60 N ⋅ m of torque to its load. If the motor’s shaft is turning at 1800 r/min, what is
the mechanical power supplied to the load in watts? In horsepower?
SOLUTION The mechanical power supplied to the load is
P = τω = ( 60 N ⋅ m )(1800 r/min )

= 11,310 W
1r

1 min
60 s

1

P = (11,310 W )

1-5.

1 hp
= 15.2 hp
746 W

A ferromagnetic core is shown in Figure P1-2. The depth of the core is 5 cm. The other dimensions of the
core are as shown in the figure. Find the value of the current that will produce a flux of 0.005 Wb. With
this current, what is the flux density at the top of the core? What is the flux density at the right side of the
core? Assume that the relative permeability of the core is 1000.

SOLUTION There are three regions in this core. The top and bottom form one region, the left side forms a
second region, and the right side forms a third region. If we assume that the mean path length of the flux is
in the center of each leg of the core, and if we ignore spreading at the corners of the core, then the path
lengths are l1 = 2(27.5 cm) = 55 cm, l 2 = 30 cm, and l3 = 30 cm. The reluctances of these regions are:
R1 =

l
l
0.55 m
=
=
= 58.36 kA ⋅ t/Wb
−7
µ A µr µo A (1000) 4π × 10 H/m ( 0.05 m )(0.15 m )

R2 =

l
l
0.30 m
=
=
= 47.75 kA ⋅ t/Wb
−7
µ A µr µo A (1000 ) 4π × 10 H/m (0.05 m )( 0.10 m )

R3 =

l
l
0.30 m
=
=
= 95.49 kA ⋅ t/Wb
−7
µ A µ r µo A (1000) 4π × 10 H/m ( 0.05 m )( 0.05 m )

(

)

(

)

(

)

The total reluctance is thus

RTOT = R1 + R2 + R3 = 58.36 + 47.75 + 95.49 = 201.6 kA ⋅ t/Wb
and the magnetomotive force required to produce a flux of 0.003 Wb is

F = φ R = ( 0.005 Wb )( 201.6 kA ⋅ t/Wb ) = 1008 A ⋅ t
and the required current is
i=

F 1008 A ⋅ t
=
= 2.52 A
N
400 t

The flux density on the top of the core is
B=

φ
A

=

0.005 Wb
= 0.67 T
0.15
m )( 0.05 m )
(

2

The flux density on the right side of the core is
B=

1-6.

φ
A

=

0.005 Wb
= 2.0 T
(0.05 m )(0.05 m)

A ferromagnetic core with a relative permeability of 1500 is shown in Figure P1-3. The dimensions are as
shown in the diagram, and the depth of the core is 7 cm. The air gaps on the left and right sides of the core
are 0.070 and 0.020 cm, respectively. Because of fringing effects, the effective area of the air gaps is 5
percent larger than their physical size. If there are 4001 turns in the coil wrapped around the center leg of
the core and if the current in the coil is 1.0 A, what is the flux in each of the left, center, and right legs of
the core? What is the flux density in each air gap?

SOLUTION This core can be divided up into five regions. Let R1 be the reluctance of the left-hand portion
of the core, R2 be the reluctance of the left-hand air gap, R3 be the reluctance of the right-hand portion of
the core, R4 be the reluctance of the right-hand air gap, and R5 be the reluctance of the center leg of the
core. Then the total reluctance of the core is
RTOT = R5 +
R1 =
R2 =
R3 =
R4 =
R5 =

l1

µ r µ0 A1

( R1 + R2 ) ( R3 + R4 )
R1 + R2 + R3 + R4

=

1.11 m
= 90.1 kA ⋅ t/Wb
(2000) 4π × 10 H/m (0.07 m )(0.07 m )

(

−7

)

l2
0.0007 m
=
= 108.3 kA ⋅ t/Wb
−7
µ0 A2
4π × 10 H/m ( 0.07 m)(0.07 m )(1.05)

(

l3

µr µ0 A3

=

)

1.11 m
= 90.1 kA ⋅ t/Wb
(2000) 4π × 10 H/m (0.07 m )(0.07 m)

(

−7

)

l4
0.0005 m
=
= 77.3 kA ⋅ t/Wb
−7
µ0 A4
4π × 10 H/m (0.07 m )( 0.07 m )(1.05)

(

l5

µr µ0 A5

=

)

0.37 m
= 30.0 kA ⋅ t/Wb
(2000) 4π × 10 H/m (0.07 m)(0.07 m )

(

−7

)

The total reluctance is
1

In the first printing, this value was given incorrectly as 300.
3

RTOT = R5 +

( R1 + R2 ) ( R3 + R4 ) = 30.0 + (90.1 + 108.3)(90.1 + 77.3) = 120.8 kA ⋅ t/Wb
R1 + R2 + R3 + R4

90.1 + 108.3 + 90.1 + 77.3

The total flux in the core is equal to the flux in the center leg:

φcenter = φTOT =

(400 t )(1.0 A ) = 0.0033 Wb
F
=
RTOT 120.8 kA ⋅ t/Wb

The fluxes in the left and right legs can be found by the “flux divider rule”, which is analogous to the
current divider rule.

φleft =

( R3 + R4 )

R1 + R2 + R3 + R4

( R1 + R2 )

φ right =

φTOT =

R1 + R2 + R3 + R4

(90.1 + 77.3)

90.1 + 108.3 + 90.1 + 77.3

φTOT =

(90.1 + 108.3)

(0.0033 Wb) = 0.00193 Wb

90.1 + 108.3 + 90.1 + 77.3

(0.0033 Wb) = 0.00229 Wb

The flux density in the air gaps can be determined from the equation φ = BA :
Bleft =

φleft

Bright =
1-7.

Aeff

=

φ right
Aeff

0.00193 Wb

(0.07 cm )(0.07 cm )(1.05)
=

= 0.375 T

0.00229 Wb
= 0.445 T
0.07
cm
(
)(0.07 cm )(1.05)

A two-legged core is shown in Figure P1-4. The winding on the left leg of the core (N1) has 400 turns, and
the winding on the right (N2) has 300 turns. The coils are wound in the directions shown in the figure. If
the dimensions are as shown, then what flux would be produced by currents i1 = 0.5 A and i2 = 0.75 A?
Assume µ r = 1000 and constant.

4

SOLUTION The two coils on this core are would so that their magnetomotive forces are additive, so the total
magnetomotive force on this core is
FTOT = N1i1 + N 2i2 = ( 400 t )( 0.5 A ) + ( 300 t )(0.75 A ) = 425 A ⋅ t

The total reluctance in the core is
l
2.60 m
RTOT =
=
= 92.0 kA ⋅ t/Wb
−7
µ r µ0 A (1000 ) 4π × 10 H/m ( 0.15 m)( 0.15 m )

(

)

and the flux in the core is:

φ=
1-8.

FTOT
425 A ⋅ t
=
= 0.00462 Wb
RTOT 92.0 kA ⋅ t/Wb

A core with three legs is shown in Figure P1-5. Its depth is 5 cm, and there are 200 turns on the leftmost
leg. The relative permeability of the core can be assumed to be 1500 and constant. What flux exists in
each of the three legs of the core? What is the flux density in each of the legs? Assume a 4% increase in
the effective area of the air gap due to fringing effects.

SOLUTION This core can be divided up into four regions. Let R1 be the reluctance of the left-hand portion
of the core, R2 be the reluctance of the center leg of the core, R3 be the reluctance of the center air gap,
and R4 be the reluctance of the right-hand portion of the core. Then the total reluctance of the core is
RTOT = R1 +
R1 =
R2 =
R3 =
R4 =

l1

µ r µ0 A1
l2

µ r µ0 A2

( R2 + R3 ) R4

R2 + R3 + R4

=

1.08 m
= 127.3 kA ⋅ t/Wb
(1500) 4π × 10 H/m (0.09 m )(0.05 m)

=

0.34 m
= 24.0 kA ⋅ t/Wb
(1500) 4π × 10 H/m (0.15 m )(0.05 m)

(

−7

(

−7

)

)

l3
0.0004 m
=
= 40.8 kA ⋅ t/Wb
−7
µ0 A3
4π × 10 H/m ( 0.15 m )( 0.05 m)(1.04 )

(

l4

µ r µ0 A4

=

)

1.08 m
= 127.3 kA ⋅ t/Wb
(1500) 4π × 10 H/m (0.09 m)(0.05 m )

(

−7

)

The total reluctance is
5

RTOT = R1 +

( R2 + R3 ) R4

R2 + R3 + R4

= 127.3 +

(24.0 + 40.8)127.3 = 170.2 kA ⋅ t/Wb

24.0 + 40.8 + 127.3

The total flux in the core is equal to the flux in the left leg:

φleft = φTOT =

F
(200 t )( 2.0 A ) = 0.00235 Wb
=
RTOT 170.2 kA ⋅ t/Wb

The fluxes in the center and right legs can be found by the “flux divider rule”, which is analogous to the
current divider rule.
R4
127.3
φ TOT =
(0.00235 Wb) = 0.00156 Wb
24.0 + 40.8 + 127.3
R2 + R3 + R4

φcenter =

R2 + R3
24.0 + 40.8
φTOT =
(0.00235 Wb) = 0.00079 Wb
24.0 + 40.8 + 127.3
R2 + R3 + R4

φ right =

The flux density in the legs can be determined from the equation φ = BA :
Bleft =

φleft

Bcenter =
Bright =

1-9.

=

A

φcenter
A

φleft
A

=

0.00235 Wb

(0.09 cm )(0.05 cm )
=

= 0.522 T

0.00156 Wb

( 0.15 cm )( 0.05 cm )

= 0.208 T

0.00079 Wb
= 0.176 T
( 0.09 cm )( 0.05 cm )

A wire is shown in Figure P1-6 which is carrying 5.0 A in the presence of a magnetic field. Calculate the
magnitude and direction of the force induced on the wire.

SOLUTION The force on this wire can be calculated from the equation

F = i ( l × B ) = ilB = ( 5 A )(1 m )(0.25 T ) = 1.25 N, into the page

6

1-10.

The wire is shown in Figure P1-7 is moving in the presence of a magnetic field. With the information given
in the figure, determine the magnitude and direction of the induced voltage in the wire.

SOLUTION The induced voltage on this wire can be calculated from the equation shown below. The voltage
on the wire is positive downward because the vector quantity v × B points downward.
eind = ( v × B) ⋅ l = vBl cos 45° = (5 m/s)( 0.25 T )( 0.50 m ) cos 45° = 0.442 V, positive down

1-11.

Repeat Problem 1-10 for the wire in Figure P1-8.

SOLUTION The induced voltage on this wire can be calculated from the equation shown below. The total
voltage is zero, because the vector quantity v × B points into the page, while the wire runs in the plane of
the page.
eind = ( v × B) ⋅ l = vBl cos 90° = (1 m/s )( 0.5 T )( 0.5 m ) cos 90° = 0 V

1-12.

The core shown in Figure P1-4 is made of a steel whose magnetization curve is shown in Figure P1-9.
Repeat Problem 1-7, but this time do not assume a constant value of µ r. How much flux is produced in the
core by the currents specified? What is the relative permeability of this core under these conditions? Was
the assumption in Problem 1-7 that the relative permeability was equal to 1000 a good assumption for these
conditions? Is it a good assumption in general?

7

SOLUTION The magnetization curve for this core is shown below:

The two coils on this core are wound so that their magnetomotive forces are additive, so the total
magnetomotive force on this core is
FTOT = N 1i1 + N 2i2 = ( 400 t )( 0.5 A ) + ( 300 t )(0.75 A ) = 425 A ⋅ t

Therefore, the magnetizing intensity H is
8

H=

F 425 A ⋅ t
=
= 163 A ⋅ t/m
lc
2.60 m

From the magnetization curve,
B = 0.15 T
and the total flux in the core is

φTOT = BA = (0.15 T )(0.15 m )( 0.15 m ) = 0.0033 Wb
The relative permeability of the core can be found from the reluctance as follows:

R=

FTOT

φTOT

=

l

µ r µ0 A

Solving for µ r yields

µr =

φTOT l
(0.0033 Wb)(2.6 m )
=
= 714
FTOT µ0 A ( 425 A ⋅ t ) ( 4π × 10-7 H/m ) (0.15 m )( 0.15 m )

The assumption that µ r = 1000 is not very good here. It is not very good in general.

1-13.

A core with three legs is shown in Figure P1-10. Its depth is 8 cm, and there are 400 turns on the center
leg. The remaining dimensions are shown in the figure. The core is composed of a steel having the
(a) What current is required to produce a flux density of 0.5 T in the central leg of the core?
(b) What current is required to produce a flux density of 1.0 T in the central leg of the core? Is it twice the
current in part (a)?
(c) What are the reluctances of the central and right legs of the core under the conditions in part (a)?
(d) What are the reluctances of the central and right legs of the core under the conditions in part (b)?
(e) What conclusion can you make about reluctances in real magnetic cores?

9

SOLUTION The magnetization curve for this core is shown below:

(a)

A flux density of 0.5 T in the central core corresponds to a total flux of

φTOT = BA = ( 0.5 T )( 0.08 m )( 0.08 m ) = 0.0032 Wb
By symmetry, the flux in each of the two outer legs must be φ1 = φ2 = 0.0016 Wb , and the flux density in
the other legs must be
B1 = B2 =

0.0016 Wb
= 0.25 T
(0.08 m)(0.08 m)

The magnetizing intensity H required to produce a flux density of 0.25 T can be found from Figure 1-10c.
It is 50 A·t/m. Similarly, the magnetizing intensity H required to produce a flux density of 0.50 T is 70
A·t/m. Therefore, the total MMF needed is
FTOT = H center lcenter + H outer louter

FTOT = ( 70 A ⋅ t/m )( 0.24 m ) + (50 A ⋅ t/m )( 0.72 m ) = 52.8 A ⋅ t

and the required current is

i=
(b)

FTOT 52.8 A ⋅ t
=
= 0.13 A
N
400 t

A flux density of 1.0 T in the central core corresponds to a total flux of

φTOT = BA = (1.0 T )(0.08 m )( 0.08 m ) = 0.0064 Wb
By symmetry, the flux in each of the two outer legs must be φ1 = φ2 = 0.0032 Wb , and the flux density in
the other legs must be
B1 = B2 =

0.0032 Wb
= 0.50 T
(0.08 m)(0.08 m)

10

The magnetizing intensity H required to produce a flux density of 0.50 T can be found from Figure 1-10c.
It is 70 A·t/m. Similarly, the magnetizing intensity H required to produce a flux density of 1.00 T is about
160 A·t/m. Therefore, the total MMF needed is
FTOT = H center I center + H outer I outer

FTOT = (160 A ⋅ t/m )(0.24 m ) + ( 70 A ⋅ t/m )( 0.72 m ) = 88.8 A ⋅ t

and the required current is

i=

φTOT
N

=

88.8 A ⋅ t
= 0.22 A
400 t

This current is less not twice the current in part (a).

(c)

The reluctance of the central leg of the core under the conditions of part (a) is:

Rcent =

FTOT

φTOT

=

(70 A ⋅ t/m)(0.24 m ) = 5.25 kA ⋅ t/Wb
0.0032 Wb

The reluctance of the right leg of the core under the conditions of part (a) is:

Rright =
(d)

FTOT

φTOT

=

(50 A ⋅ t/m)(0.72 m ) = 22.5 kA ⋅ t/Wb
0.0016 Wb

The reluctance of the central leg of the core under the conditions of part (b) is:

Rcent =

FTOT

φTOT

=

(160 A ⋅ t/m )(0.24 m) = 6.0 kA ⋅ t/Wb
0.0064 Wb

The reluctance of the right leg of the core under the conditions of part (b) is:

Rright =
(e)

1-14.

FTOT

φTOT

=

(70 A ⋅ t/m )(0.72 m) = 15.75 kA ⋅ t/Wb
0.0032 Wb

The reluctances in real magnetic cores are not constant.

A two-legged magnetic core with an air gap is shown in Figure P1-11. The depth of the core is 5 cm, the
length of the air gap in the core is 0.06 cm, and the number of turns on the coil is 1000. The magnetization
curve of the core material is shown in Figure P1-9. Assume a 5 percent increase in effective air-gap area to
account for fringing. How much current is required to produce an air-gap flux density of 0.5 T? What are
the flux densities of the four sides of the core at that current? What is the total flux present in the air gap?

11

SOLUTION The magnetization curve for this core is shown below:

An air-gap flux density of 0.5 T requires a total flux of

φ = BAeff = (0.5 T )( 0.05 m )( 0.05 m )(1.05) = 0.00131 Wb
This flux requires a flux density in the right-hand leg of
Bright =

φ
A

=

0.00131 Wb

(0.05 m)(0.05 m )

= 0.524 T

The flux density in the other three legs of the core is
Btop = Bleft = Bbottom =

φ
A

=

0.00131 Wb
= 0.262 T
(0.10 m)(0.05 m)

12

The magnetizing intensity required to produce a flux density of 0.5 T in the air gap can be found from the
equation Bag = µo H ag :

H ag =

Bag

µ0

=

0.5 T
= 398 kA ⋅ t/m
4π × 10−7 H/m

The magnetizing intensity required to produce a flux density of 0.524 T in the right-hand leg of the core can
be found from Figure P1-9 to be

H right = 410 A ⋅ t/m
The magnetizing intensity required to produce a flux density of 0.262 T in the top, left, and bottom legs of
the core can be found from Figure P1-9 to be

H top = H left = H bottom = 240 A ⋅ t/m
The total MMF required to produce the flux is

FTOT = H ag lag + H right lright + H top ltop + H left lleft + H bottom lbottom
FTOT = ( 398 kA ⋅ t/m )( 0.0006 m ) + ( 410 A ⋅ t/m )( 0.40 m ) + 3 ( 240 A ⋅ t/m )(0.40 m )

FTOT = 278.6 + 164 + 288 = 691 A ⋅ t
and the required current is

i=

FTOT 691 A ⋅ t
=
= 0.691 A
N
1000 t

The flux densities in the four sides of the core and the total flux present in the air gap were calculated
above.

1-15.

A transformer core with an effective mean path length of 10 in has a 300-turn coil wrapped around one leg.
Its cross-sectional area is 0.25 in2, and its magnetization curve is shown in Figure 1-10c. If current of 0.25
A is flowing in the coil, what is the total flux in the core? What is the flux density?

SOLUTION The magnetizing intensity applied to this core is
13

H=

F Ni
(300 t )(0.25 A ) = 295 A ⋅ t/m
=
=
lc
lc (10 in )(0.0254 m/in )

From the magnetization curve, the flux density in the core is

B = 1.27 T
The total flux in the core is

φ = BA = (1.27 T ) ( 0.25 in 2 )
1-16.

0.0254 m
1 in

2

= 0.000205 Wb

The core shown in Figure P1-2 has the flux φ shown in Figure P1-12. Sketch the voltage present at the
terminals of the coil.

SOLUTION By Lenz’ Law, an increasing flux in the direction shown on the core will produce a voltage that
tends to oppose the increase. This voltage will be the same polarity as the direction shown on the core, so it
will be positive. The induced voltage in the core is given by the equation
eind = N

dφ
dt

so the voltage in the windings will be
14

Time
0= 2*pi
wt = wt - 2*pi;
end
while wt < 0
wt = wt + 2*pi;
end
%
a
b
c

Simulate the output of the rectifier.
= sin(wt);
= sin(wt - 2*pi/3);
= sin(wt + 2*pi/3);

volts = max( [ a b c ] );
The function ripple is reproduced below. It is identical to the one in the textbook.

function r = ripple(waveform)
% Function to calculate the ripple on an input waveform.
% Calculate the average value of the waveform
nvals = size(waveform,2);
temp = 0;
for ii = 1:nvals
temp = temp + waveform(ii);
end
average = temp/nvals;
% Calculate rms value of waveform

64

temp = 0;
for ii = 1:nvals
temp = temp + waveform(ii)^2;
end
rms = sqrt(temp/nvals);
% Calculate ripple factor
r = sqrt((rms / average)^2 - 1) * 100;
Finally, the test driver program is shown below.

% M-file: test_halfwave3.m
% M-file to calculate the ripple on the output of a
% three phase half-wave rectifier.
% First, generate the output of a three-phase half-wave
% rectifier
waveform = zeros(1,128);
for ii = 1:128
waveform(ii) = halfwave3(ii*pi/64);
end
% Now calculate the ripple factor
r = ripple(waveform);
% Print out the result
string = ['The ripple is ' num2str(r) '%.'];
disp(string);
When this program is executed, the results are

» test_halfwave3
The ripple is 18.2759%.
This answer agrees with the analytical solution above.

3-2.

Calculate the ripple factor of a three-phase full-wave rectifier circuit, both analytically and using
MATLAB.
SOLUTION A three-phase half-wave rectifier and its output voltage are shown below

65

T/12

v A ( t ) = VM sin ω t

vB (t ) = VM sin (ω t − 2π / 3)
vC (t ) = VM sin (ω t + 2π / 3)

SOLUTION By symmetry, the rms voltage over the interval from 0 to T/12 will be the same as the rms
voltage over the whole interval. Over that interval, the output voltage is:
v ( t ) = vC (t ) − vB ( t ) = VM sin ω t +
v ( t ) = VM sin ω t cos

2π
2π
− VM sin ω t −
3
3

2π
2π
2π
2π
+ cos ω t sin
− VM sin ω t cos
− cos ω t sin
3
3
3
3

v ( t ) = VM 2 cos ω t sin

2π
= 3 cos ω t
3

Note that the period of the waveform is T = 2π / ω , so T/12 is π / 6ω . The average voltage over the
interval from 0 to T/12 is
VDC =
VDC =

6ω

π / 6ω

1
T

v (t ) dt =

3 3

VM = 1.6540 VM

π

π

3VM cos ω t dt =

0

6 3

π

The rms voltage is
Vrms =

1
T

v (t ) 2 dt =

6ω

π

π / 6ω

3VM 2 cos 2 ω t dt

0

66

π / 6ω

VM sin ω t 0

Vrms =

18ω

π

Vrms = VM

VM 2

1
1
t+
sin 2ω t
2 4ω

18ω

π

π

12ω

+

π / 6ω

0

1
π
3 9 3
sin
= VM
+
= 1.6554 VM
4ω
3
2 4π

The resulting ripple factor is
r=

Vrms
VDC

2

− 1 × 100% =

1.6554 VM
1.6540 VM

2

− 1 × 100% = 4.2%

The ripple can be calculated with MATLAB using the ripple function developed in the text. We must
right a new function fullwave3 to simulate the output of a three-phase half-wave rectifier. This output
is just the largest voltage of v A (t ) , v B (t ) , and vC (t ) at any particular time. The function is shown below:

function volts = fullwave3(wt)
% Function to simulate the output of a three-phase
% full-wave rectifier.
%
wt = Phase in radians (=omega x time)
% Convert input to the range 0 <= wt < 2*pi
while wt >= 2*pi
wt = wt - 2*pi;
end
while wt < 0
wt = wt + 2*pi;
end
%
a
b
c

Simulate the output of the rectifier.
= sin(wt);
= sin(wt - 2*pi/3);
= sin(wt + 2*pi/3);

volts = max( [ a b c ] ) - min( [ a b c ] );
The test driver program is shown below.

% M-file: test_fullwave3.m
% M-file to calculate the ripple on the output of a
% three phase full-wave rectifier.
% First, generate the output of a three-phase full-wave
% rectifier
waveform = zeros(1,128);
for ii = 1:128
waveform(ii) = fullwave3(ii*pi/64);
end
% Now calculate the ripple factor
r = ripple(waveform);
% Print out the result
string = ['The ripple is ' num2str(r) '%.'];
disp(string);
67

When this program is executed, the results are

» test_fullwave3
The ripple is 4.2017%.
This answer agrees with the analytical solution above.

3-3.

Explain the operation of the circuit shown in Figure P3-1. What would happen in this circuit if switch S1
were closed?

SOLUTION Diode D1 and D2 together with the transformer form a full-wave rectifier. Therefore, a voltage
oriented positive-to-negative as shown will be applied to the SCR and the control circuit on each half cycle.
(1) Initially, the SCR is an open circuit, since v1 < VBO for the SCR. Therefore, no current flows to the
(2)

Voltage v1 is applied to the control circuit, charging capacitor C1 with time constant RC1.

(3)

When vC > VBO for the DIAC, it conducts, supplying a gate current to the SCR.

(4) The gate current in the SCR lowers its breakover voltage, and the SCR fires. When the SCR fires,
current flows through the SCR and the load.
(5) The current flow continues until iD falls below IH for the SCR (at the end of the half cycle). The
process starts over in the next half cycle.

68

If switch S1 is shut, the charging time constant is increased, and the DIAC fires later in each half cycle.
Therefore, less power is supplied to the load.

3-4.

What would the rms voltage on the load in the circuit in Figure P3-1 be if the firing angle of the SCR were
(a) 0°, (b) 30°, (c) 90°?
SOLUTION The input voltage to the circuit of Figure P3-1 is

vac (t ) = 339 sin ωt , where ω = 377 rad/s
Therefore, the voltage on the secondary of the transformer will be

vac (t ) = 169.5 sin ωt
(a) The average voltage applied to the load will be the integral over the conducting portion of the half
cycle divided by π/ω, the period of a half cycle. For a firing angle of 0°, the average voltage will be
1
ω
v (t ) dt =
T 0
π
T

Vave =

π /ω

1
VM sin ω t dt = − VM cos ω t

0

π

π /ω

0

1
2
Vave = − VM [ −1 − 1] = VM = (0.637 )(169.5 V ) = 108 V

π

(b)

π

For a firing angle of 30°, the average voltage will be
1
ω
=
v (t ) dt =
T π /6
π
T

Vave

π /ω
π /6

1

π /ω

VM sin ω t dt = − VM cos ω t

π

π /6

1
3
2+ 3
=
Vave = − VM −1 −
VM = (0.594 )(169.5 V ) = 101 V
2
2π
π
(c)

For a firing angle of 90°, the average voltage will be
1
ω
=
v (t ) dt =
T π /2
π
T

Vave

π /ω
π /2

1

π /ω

VM sin ω t dt = − VM cos ω t

π

69

π /2

1
1
Vave = − VM [ −1] = VM = ( 0.318)(169.5 V ) = 54 V

π

3-5.

π

For the circuit in Figure P3-1, assume that VBO for the DIAC is 30 V, C1 is 1 µF, R is adjustable in the
range 1-20 kΩ, and that switch S1 is open. What is the firing angle of the circuit when R is 10 kΩ? What
is the rms voltage on the load under these conditions?

Note:

Problem 3-5 is significantly harder for many students, since it involves solving
a differential equation with a forcing function. This problem should only be
assigned if the class has the mathematical sophistication to handle it.

SOLUTION At the beginning of each half cycle, the voltages across the DIAC and the SCR will both be
smaller then their respective breakover voltages, so no current will flow to the load (except for the very tiny
current charging capacitor C), and vload(t) will be 0 volts. However, capacitor C charges up through
resistor R, and when the voltage vC(t) builds up to the breakover voltage of D1, the DIAC will start to
conduct. This current flows through the gate of SCR1, turning the SCR ON. When it turns ON, the
voltage across the SCR will drop to 0, and the full source voltage vS(t) will be applied to the load,
producing a current flow through the load. The SCR continues to conduct until the current through it falls
below IH, which happens at the very end of the half cycle.
Note that after D1 turns on, capacitor C discharges through it and the gate of the SCR. At the end of the
half cycle, the voltage on the capacitor is again essentially 0 volts, and the whole process is ready to start
over again at the beginning of the next half cycle.
To determine when the DIAC and the SCR fire in this circuit, we must determine when vC(t) exceeds VBO
for D1. This calculation is much harder than in the examples in the book, because in the previous problems
the source was a simple DC voltage source, while here the voltage source is sinusoidal. However, the
principles are identical.
(a)

To determine when the SCR will turn ON, we must calculate the voltage vC(t), and then solve for the time
at which vC(t) exceeds VBO for D1. At the beginning of the half cycle, D1 and SCR1 are OFF, and the
voltage across the load is essentially 0, so the entire source voltage vS(t) is applied to the series RC circuit.
To determine the voltage vC(t) on the capacitor, we can write a Kirchhoff's Current Law equation at the
node above the capacitor and solve the resulting equation for vC(t).

i1 + i2 = 0

(since the DIAC is an open circuit at this time)

vC − v1
d
+ C vC = 0
R
dt
70

1
1
d
vC +
vC =
v1
dt
RC
RC
1
d
V
vC +
vC = M sin ωt
dt
RC
RC
The solution can be divided into two parts, a natural response and a forced response. The natural response
is the solution to the differential equation
d
1
vC +
vC = 0
dt
RC
The solution to the natural response differential equation is
vC ,n ( t ) = A e

−

t
RC

where the constant A must be determined from the initial conditions in the system. The forced response is
the steady-state solution to the equation
d
1
V
vC +
vC = M sin ωt
dt
RC
RC
It must have a form similar to the forcing function, so the solution will be of the form
vC , f ( t ) = B1 sin ω t + B2 cos ω t

where the constants B1 and B2 must be determined by substitution into the original equation. Solving for
B1 and B2 yields:
d
1
V
( B1 sin ωt + B2 cos ω t ) + ( B1 sin ω t + B2 cos ωt ) = M sin ω t
dt
RC
RC

(ω B1cos ω t − ω B2 sin ω t ) +

1
V
( B1 sin ω t + B2 cos ωt ) = M sin ω t
RC
RC

cosine equation:

ω B1 +

1
B2 = 0 ⇒
RC

B2 = −ω RC B1

sine equation:
−ω B2 +

1
V
B1 = M
RC
RC

ω 2 RC B1 +
ω 2 RC +

1
V
B1 = M
RC
RC

1
V
B1 = M
RC
RC

1 + ω 2 R 2C 2
V
B1 = M
RC
RC
Finally,

71

B1 =

VM
and
1 + ω 2 R 2C 2

B2 =

−ω RC VM
1 + ω 2 R 2C 2

Therefore, the forced solution to the equation is
vC , f ( t ) =

VM
ω RC VM
sin ωt −
cos ωt
1 + ω 2 R 2C 2
1 + ω 2 R 2C 2

and the total solution is
vC (t ) = vC ,n ( t ) + vC , f ( t )
vC (t ) = Ae

−

t
RC

+

VM
ω RC VM
sin ω t −
cos ω t
1 + ω 2 R 2C 2
1 + ω 2 R 2C 2

The initial condition for this problem is that the voltage on the capacitor is zero at the beginning of the halfcycle:
vC (0 ) = Ae

−

0
RC

+

VM
ω RC VM
sin 0 −
cos 0 = 0
2 2 2
1+ω R C
1 + ω 2 R 2C 2

A−

ω RC VM
=0
1 + ω 2 R 2C 2

A=

ω RC VM
1 + ω 2 R 2C 2

Therefore, the voltage across the capacitor as a function of time before the DIAC fires is
vC (t ) =

ω RC VM −
e
1 + ω 2 R 2C 2

t
RC

+

VM
ω RC VM
sin ω t −
cos ω t
1 + ω 2 R 2C 2
1 + ω 2 R 2C 2

If we substitute the known values for R, C, ω, and VM, this equation becomes
vC (t ) = 42 e −

100t

+ 11.14 sin ωt − 42 cos ωt

This equation is plotted below:

It reaches a voltage of 30 V at a time of 3.50 ms. Since the frequency of the waveform is 60 Hz, the
waveform there are 360° in 1/60 s, and the firing angle α is
72

α = ( 3.50 ms )

360°
= 75.6° or 1.319 radians
1/ 60 s

Note: This problem could also have been solved using Laplace Transforms, if desired.
(b)

The rms voltage applied to the load is
Vrms =

Vrms =

Vrms =

1
T

v (t ) 2 dt =

ω
π

π /ω

VM 2 sin 2 ω t dt

α

VM 2 1
1
ωt − sin 2ω t
4
π 2

π /ω

α

VM 2 1
1
(π − α ) − (sin 2π − sin 2α )
4
π 2

Vrms = VM 0.3284 = 0.573 VM = 97.1 V

3-6.

One problem with the circuit shown in Figure P3-1 is that it is very sensitive to variations in the input
voltage v ac ( t ) . For example, suppose the peak value of the input voltage were to decrease. Then the time
that it takes capacitor C1 to charge up to the breakover voltage of the DIAC will increase, and the SCR will
be triggered later in each half cycle. Therefore, the rms voltage supplied to the load will be reduced both by
the lower peak voltage and by the later firing. This same effect happens in the opposite direction if v ac (t )
increases. How could this circuit be modified to reduce its sensitivity to variations in input voltage?
SOLUTION If the voltage charging the capacitor could be made constant or nearly so, then the feedback
effect would be stopped and the circuit would be less sensitive to voltage variations. A common way to do
this is to use a zener diode that fires at a voltage greater than VBO for the DIAC across the RC charging
circuit. This diode holds the voltage across the RC circuit constant, so that the capacitor charging time is
not much affected by changes in the power supply voltage.

R
vC

73

3-7.

Explain the operation of the circuit shown in Figure P3-2, and sketch the output voltage from the circuit.

SOLUTION This circuit is a single-phase voltage source inverter.
(1) Initially, suppose that both SCRs are OFF. Then the voltage on the transformer T3 will be 0, and
voltage VDC will be dropped across SCR 1 and SCR2 as shown.
(2)

Now, apply a pulse to transformer T1 that turns on SCR1. When that happens, the circuit looks like:

Since the top of the transformer is now grounded, a voltage VDC appears across the upper winding as
shown. This voltage induces a corresponding voltage on the lower half of the winding, charging capacitor
C1 up to a voltage of 2VDC, as shown.
Now, suppose that a pulse is applied to transformer T2. When that occurs, SCR2 becomes a short circuit,
and SCR1 is turned OFF by the reverse voltage applied to it by capacitor C1 (forced commutation). At that
time, the circuit looks like:

Now the voltages on the transformer are reversed, charging capacitor C1 up to a voltage of 2VDC in the
opposite direction. When SCR1 is triggered again, the voltage on C1 will turn SCR2 OFF.
The output voltage from this circuit would be roughly a square wave, except that capacitor C2 filters it
somewhat.

74

(Note: The above discussion assumes that transformer T3 is never in either state long enough for it to
saturate.)

3-8.

Figure P3-3 shows a relaxation oscillator with the following parameters:

R1 = variable
C = 1.0 µ F

R2 = 1500 Ω
VDC = 100 V

VBO = 30 V

I H = 0.5 mA

(a) Sketch the voltages vC (t ) , v D (t ) , and v o (t ) for this circuit.
(b) If R1 is currently set to 500 kΩ, calculate the period of this relaxation oscillator.

SOLUTION
(a) The voltages vC(t), vD(t) and vo(t) are shown below. Note that vC(t) and vD(t) look the same during
the rising portion of the cycle. After the PNPN Diode triggers, the voltage across the capacitor decays with
R1R2
time constant τ2 =
C, while the voltage across the diode drops immediately.
R1 + R2

75

(b) When voltage is first applied to the circuit, the capacitor C charges with a time constant τ1 = R1 C =
(500 kΩ)(1.00 µF) = 0.50 s. The equation for the voltage on the capacitor as a function of time during the
charging portion of the cycle is

vC (t ) = A + B e

−

t
R1C

where A and B are constants depending upon the initial conditions in the circuit. Since vC(0) = 0 V and
vC(∞) = 100 V, it is possible to solve for A and B.
A = vC(∞) = 100 V
A + B = vC(0) = 0 V ⇒ B = -100 V
Therefore,
vC (t ) = 100 − 100 e

−

t
0.50

V

The time at which the capacitor will reach breakover voltage is found by setting vC(t) = VBO and solving for
time t1:
76

t1 = −0.50 ln

100 V − 30 V
= 178 ms
100 V

Once the PNPN Diode fires, the capacitor discharges through the parallel combination of R1 and R2, so the
time constant for the discharge is

τ2 =

(500 kΩ)(1.5 kΩ) 1.0 µ F = 0.0015 s
R1R2
C=
(
)
R1 + R2
500 kΩ + 1.5 kΩ

The equation for the voltage on the capacitor during the discharge portion of the cycle is
vC (t ) = A + B e
vC ( t ) = VBO e

−

−

t

τ2

t

τ2

The current through the PNPN diode is given by
iD (t ) =

t

VBO − τ 2
e
R2

If we ignore the continuing trickle of current from R1, the time at which iD(t) reaches IH is
t2 = − R2C ln

I H R2
(0.0005 A )(1500 Ω ) = 5.5 ms
= − ( 0.0015) ln
30 V
VBO

Therefore, the period of the relaxation oscillator is T = 178 ms + 5.5 ms = 183.5 ms, and the frequency of
the relaxation oscillator is f = 1/T = 5.45 Hz.

3-9.

In the circuit in Figure P3-4, T1 is an autotransformer with the tap exactly in the center of its winding.
Explain the operation of this circuit. Assuming that the load is inductive, sketch the voltage and current
applied to the load. What is the purpose of SCR 2 ? What is the purpose of D2? (This chopper circuit
arrangement is known as a Jones circuit.)

SOLUTION First, assume that SCR1 is triggered. When that happens, current will flow from the power
supply through SCR1 and the bottom portion of transformer T1 to the load. At that time, a voltage will be
applied to the bottom part of the transformer which is positive at the top of the winding with respect to the
bottom of the winding. This voltage will induce an equal voltage in the upper part of the autotransformer

77

winding, forward biasing diode D1 and causing the current to flow up through capacitor C. This current
causes C to be charged with a voltage that is positive at its bottom with respect to its top. (This condition
is shown in the figure above.)
Now, assume that SCR2 is triggered. When SCR2 turns ON, capacitor C applies a reverse-biased voltage
to SCR1, shutting it off. Current then flow through the capacitor, SCR2, and the load as shown below.
This current charges C with a voltage of the opposite polarity, as shown.

SCR2 will cut off when the capacitor is fully charged. Alternately, it will be cut off by the voltage across
the capacitor if SCR1 is triggered before it would otherwise cut off.
In this circuit, SCR1 controls the power supplied to the load, while SCR2 controls when SCR1 will be
turned off. Diode D2 in this circuit is a free-wheeling diode, which allows the current in the load to
continue flowing for a short time after SCR1 turns off.

78

3-10.

A series-capacitor forced commutation chopper circuit supplying a purely resistive load is shown in Figure
P3-5.
VDC = 120 V
R = 20 kΩ
2
I H = 8 mA
RLOAD = 250 Ω
VBO = 200 V
C = 150 µ F
(a) When SCR 1 is turned on, how long will it remain on? What causes it to turn off?
(b) When SCR 1 turns off, how long will it be until the SCR can be turned on again? (Assume that three
time constants must pass before the capacitor is discharged.)
(c) What problem or problems do these calculations reveal about this simple series-capacitor forced
commutation chopper circuit?
(d) How can the problem(s) described in part (c) be eliminated?

Solution
(a) When the SCR is turned on, it will remain on until the current flowing through it drops below IH.
This happens when the capacitor charges up to a high enough voltage to decrease the current below IH. If
we ignore resistor R (because it is so much larger than RLOAD), the capacitor charges through resistor
RLOAD with a time constant τLOAD = RLOADC = (250 Ω)(150 µF) = 0.0375 s. The equation for the voltage
on the capacitor as a function of time during the charging portion of the cycle is
vC (t ) = A + B e

−

t

where A and B are constants depending upon the initial conditions in the circuit. Since vC(0) = 0 V and
vC(∞) = VDC, it is possible to solve for A and B.
A = vC(∞) = VDC
A + B = vC(0) = VDC ⇒ B = -VDC
Therefore,

2

The first printing of this book incorrectly stated that IH is 6 mA.
79

vC (t ) = VDC − VDC e

−

t

V

The current through the capacitor is
iC (t ) = C

d
vC (t )
dt

iC (t ) = C

−
d
VDC − VDC e
dt

iC (t ) =

t

VDC −
e

t

A

Solving for time yields
t = − RLOADC ln

iC ( t ) R2
i (t ) R2
= −0.0375 ln C
VDC
VDC

The current through the SCR consists of the current through resistor R plus the current through the
capacitor. The current through resistor R is 120 V / 20 kΩ = 6 mA, and the holding current of the SCR is
8 mA, so the SCR will turn off when the current through the capacitor drops to 2 mA. This occurs at time
t = −0.0375 ln

(2 mA )(250 Ω ) = 0.206 s
120 V

(b) The SCR can be turned on again once the capacitor has discharged. The capacitor discharges
through resistor R. It can be considered to be completely discharged after three time constants. Since τ =
RC = (20 kΩ)(150 µF) = 3 s, the SCR will be ready to fire again after 9 s.
(c) In this circuit, the ON time of the SCR is much shorter than the reset time for the SCR, so power can
flow to the load only a very small fraction of the time. (This effect would be less exaggerated if the ratio of
R to RLOAD were smaller.)
(d) This problem can be eliminated by using one of the more complex series commutation circuits
described in Section 3-5. These more complex circuits provide special paths to quickly discharge the
capacitor so that the circuit can be fired again soon.

3-11.

A parallel-capacitor forced commutation chopper circuit supplying a purely resistive load is shown in
Figure P3-6.

VDC = 120 V
I H = 5 mA

R1 = 20 kΩ
Rload = 250 Ω
80

VBO = 250 V

C = 15 µ F

(a) When SCR 1 is turned on, how long will it remain on? What causes it to turn off?
(b) What is the earliest time that SCR 1 can be turned off after it is turned on? (Assume that three time
constants must pass before the capacitor is charged.)
(c) When SCR 1 turns off, how long will it be until the SCR can be turned on again?
(d) What problem or problems do these calculations reveal about this simple parallel-capacitor forced
commutation chopper circuit?
(e) How can the problem(s) describe in part (d) be eliminated?

SOLUTION
(a)

When SCR1 is turned on, it will remain on indefinitely until it is forced to turn off. When SCR1 is turned
on, capacitor C charges up to VDC volts with the polarity shown in the figure above. Once it is charged,
SCR1 can be turned off at any time by triggering SCR2. When SCR2 is triggered, the voltage across it
drops instantaneously to about 0 V, which forces the voltage at the anode of SCR1 to be -VDC volts, turning
SCR1 off. (Note that SCR2 will spontaneously turn off after the capacitor discharges, since VDC / R1 < IH
for SCR2.)

(b)

If we assume that the capacitor must be fully charged before SCR1 can be forced to turn off, then the time
required would be the time to charge the capacitor. The capacitor charges through resistor R1, and the time
constant for the charging is τ = R1C = (20 kΩ)(15 µF) = 0.3 s. If we assume that it takes 3 time constants
to fully charge the capacitor, then the time until SCR1 can be turned off is 0.9 s.
(Note that this is not a very realistic assumption. In real life, it is possible to turn off SCR1 with less than a
full VDC volts across the capacitor.)

(c)

SCR1 can be turned on again after the capacitor charges up and SCR2 turns off. The capacitor charges
through RLOAD, so the time constant for charging is

τ = RLOADC = (250 Ω)(15 µF) = 0.00375 s
and SCR2 will turn off in a few milliseconds.
(d)

In this circuit, once SCR1 fires, a substantial period of time must pass before the power to the load can be
turned off. If the power to the load must be turned on and off rapidly, this circuit could not do the job.

81

(e)

This problem can be eliminated by using one of the more complex parallel commutation circuits described
in Section 3-5. These more complex circuits provide special paths to quickly charge the capacitor so that
the circuit can be turned off quickly after it is turned on.

3-12.

Figure P3-7 shows a single-phase rectifier-inverter circuit. Explain how this circuit functions. What are
the purposes of C1 and C2? What controls the output frequency of the inverter?

SOLUTION The last element in the filter of this rectifier circuit is an inductor, which keeps the current flow
out of the rectifier almost constant. Therefore, this circuit is a current source inverter. The rectifier and
filter together produce an approximately constant dc voltage and current across the two SCRs and diodes at
the right of the figure. The applied voltage is positive at the top of the figure with respect to the bottom of
the figure. To understand the behavior of the inverter portion of this circuit, we will step through its
operation.
(1) First, assume that SCR1 and SCR4 are triggered. Then the voltage V will appear across the load
positive-to-negative as shown in Figure (a). At the same time, capacitor C1 will charge to V volts through
diode D3, and capacitor C2 will charge to V volts through diode D2.

(a)
(2) Now, assume that SCR2 and SCR3 are triggered. At the instant they are triggered, the voltage across
capacitors C1 and C2 will reverse bias SCR1 and SCR4, turning them OFF. Then a voltage of V volts will
appear across the load positive-to-negative as shown in Figure (b). At the same time, capacitor C1 will
charge to V volts with the opposite polarity from before, and capacitor C2 will charge to V volts with the
opposite polarity from before.
82

Figure (b)
(3) If SCR1 and SCR4 are now triggered again, the voltages across capacitors C1 and C2 will force
SCR2 and SCR3 to turn OFF. The cycle continues in this fashion.
Capacitors C1 and C2 are called commutating capacitors. Their purpose is to force one set of SCRs to turn
OFF when the other set turns ON.
The output frequency of this rectifier-inverter circuit is controlled by the rates at which the SCRs are
triggered. The resulting voltage and current waveforms (assuming a resistive load) are shown below.

3-13.

A simple full-wave ac phase angle voltage controller is shown in Figure P3-8. The component values in
this circuit are:
R = 20 to 300 kΩ, currently set to 80 kΩ
C = 0.15 µF
83

VBO = 40 V (for PNPN Diode D1)
VBO = 250 V (for SCR1)
vs (t ) = VM sin ω t

volts

where VM = 169.7 V and ω = 377 rad/s
(a) At what phase angle do the PNPN diode and the SCR turn on?
(b) What is the rms voltage supplied to the load under these circumstances?

Note:

Problem 3-13 is significantly harder for many students, since it involves
solving a differential equation with a forcing function. This problem should
only be assigned if the class has the mathematical sophistication to handle it.

SOLUTION At the beginning of each half cycle, the voltages across the PNPN diode and the SCR will both be
smaller then their respective breakover voltages, so no current will flow to the load (except for the very tiny
current charging capacitor C), and vload(t) will be 0 volts. However, capacitor C charges up through
resistor R, and when the voltage vC(t) builds up to the breakover voltage of D1, the PNPN diode will start to
conduct. This current flows through the gate of SCR1, turning the SCR ON. When it turns ON, the
voltage across the SCR will drop to 0, and the full source voltage vS(t) will be applied to the load,
producing a current flow through the load. The SCR continues to conduct until the current through it falls
below IH, which happens at the very end of the half cycle.
Note that after D1 turns on, capacitor C discharges through it and the gate of the SCR. At the end of the
half cycle, the voltage on the capacitor is again essentially 0 volts, and the whole process is ready to start
over again at the beginning of the next half cycle.
To determine when the PNPN diode and the SCR fire in this circuit, we must determine when vC(t) exceeds
VBO for D1. This calculation is much harder than in the examples in the book, because in the previous
problems the source was a simple DC voltage source, while here the voltage source is sinusoidal. However,
the principles are identical.
(a)

To determine when the SCR will turn ON, we must calculate the voltage vC(t), and then solve for the time
at which vC(t) exceeds VBO for D1. At the beginning of the half cycle, D1 and SCR1 are OFF, and the
voltage across the load is essentially 0, so the entire source voltage vS(t) is applied to the series RC circuit.
To determine the voltage vC(t) on the capacitor, we can write a Kirchhoff's Current Law equation at the
node above the capacitor and solve the resulting equation for vC(t).
i1 + i2 = 0

(since the PNPN diode is an open circuit at this time)
84

vC − v1
d
+ C vC = 0
R
dt
d
1
1
vC +
vC =
v1
dt
RC
RC

d
1
V
vC +
vC = M sin ωt
dt
RC
RC
The solution can be divided into two parts, a natural response and a forced response. The natural response
is the solution to the equation
d
1
vC +
vC = 0
dt
RC
The solution to the natural response equation is
vC ,n ( t ) = A e

−

t
RC

where the constant A must be determined from the initial conditions in the system. The forced response is
the steady-state solution to the equation
d
1
V
vC +
vC = M sin ωt
dt
RC
RC
It must have a form similar to the forcing function, so the solution will be of the form
vC , f ( t ) = B1 sin ω t + B2 cos ω t

where the constants B1 and B2 must be determined by substitution into the original equation. Solving for
B1 and B2 yields:
d
1
V
( B1 sin ωt + B2 cos ω t ) + ( B1 sin ω t + B2 cos ωt ) = M sin ω t
dt
RC
RC

(ω B1cos ω t − ω B2 sin ω t ) +

1
V
( B1 sin ω t + B2 cos ωt ) = M sin ω t
RC
RC

cosine equation:

ω B1 +

1
B2 = 0 ⇒
RC

B2 = −ω RC B1

sine equation:
−ω B2 +

1
V
B1 = M
RC
RC

ω 2 RC B1 +
ω 2 RC +

1
V
B1 = M
RC
RC

1
V
B1 = M
RC
RC

1 + ω 2 R 2C 2
V
B1 = M
RC
RC
Finally,
85

B1 =

VM
and
1 + ω 2 R 2C 2

B2 =

−ω RC VM
1 + ω 2 R 2C 2

Therefore, the forced solution to the equation is
vC , f ( t ) =

VM
ω RC VM
sin ωt −
cos ωt
1 + ω 2 R 2C 2
1 + ω 2 R 2C 2

and the total solution is
vC (t ) = vC ,n ( t ) + vC , f ( t )
vC (t ) = Ae

−

t
RC

+

VM
ω RC VM
sin ω t −
cos ω t
1 + ω 2 R 2C 2
1 + ω 2 R 2C 2

The initial condition for this problem is that the voltage on the capacitor is zero at the beginning of the halfcycle:
vC (0 ) = Ae

−

0
RC

+

VM
ω RC VM
sin 0 −
cos 0 = 0
2 2 2
1+ω R C
1 + ω 2 R 2C 2

A−

ω RC VM
=0
1 + ω 2 R 2C 2

A=

ω RC VM
1 + ω 2 R 2C 2

Therefore, the voltage across the capacitor as a function of time before the PNPN diode fires is
vC (t ) =

ω RC VM −
e
1 + ω 2 R 2C 2

t
RC

+

VM
ω RC VM
sin ω t −
cos ω t
1 + ω 2 R 2C 2
1 + ω 2 R 2C 2

If we substitute the known values for R, C, ω, and VM, this equation becomes
vC (t ) = 35.76e −

83.3t

+ 7.91 sin ω t − 35.76 cos ωt

This equation is plotted below:

86

It reaches a voltage of 40 V at a time of 4.8 ms. Since the frequency of the waveform is 60 Hz, the
waveform there are 360° in 1/60 s, and the firing angle α is

α = ( 4.8 ms )

360°
= 103.7° or 1.810 radians
1/ 60 s

Note: This problem could also have been solved using Laplace Transforms, if desired.
(b)

The rms voltage applied to the load is
Vrms =

Vrms =

1
T

v (t ) 2 dt =

ω
π

π /ω

VM 2 sin 2 ω t dt

α

VM 2 1
1
ωt − sin 2ω t
4
π 2

π /ω

α

2

Vrms =

VM  1
(π − α ) − 1 (sin 2π − sin 2α )

π 2
4


Since α = 1.180 radians, the rms voltage is
Vrms = VM 0.1753 = 0.419 VM = 71.0 V

3-14.

Figure P3-9 shows a three-phase full-wave rectifier circuit supplying power to a dc load. The circuit uses
SCRs instead of diodes as the rectifying elements.
(a) What will the load voltage and ripple be if each SCR is triggered as soon as it becomes forward biased?
At what phase angle should the SCRs be triggered in order to operate this way? Sketch or plot the
output voltage for this case.
(b) What will the rms load voltage and ripple be if each SCR is triggered at a phase angle of 90° (that is,
half way through the half-cycle in which it is forward biased)? Sketch or plot the output voltage for
this case.
87

SOLUTION Assume that the three voltages applied to this circuit are:
v A ( t ) = VM sin ωt
vB (t ) = VM sin (ω t − 2π / 3)

vC (t ) = VM sin (ωt + 2π / 3)
The period of the input waveforms is T, where T = 2π / ω . For the purpose of the calculations in this
problem, we will assume that ω is 377 rad/s (60 Hz).
(a) The when the SCRs start to conduct as soon as they are forward biased, this circuit is just a threephase full-wave bridge, and the output voltage is identical to that in Problem 3-2. The sketch of output
voltage is reproduced below, and the ripple is 4.2%. The following table shows which SCRs must conduct
in what order to create the output voltage shown below. The times are expressed as multiples of the period
T of the input waveforms, and the firing angle is in degrees relative to time zero.

Start Time
(ω t)

Stop Time
(ω t)

Positive
Phase

Negative
Phase

− T / 12
T / 12
3T / 12
5T / 12
7T / 12
9T / 12
11T / 12

T / 12
3T / 12
5T / 12
7T / 12
9T / 12
11T / 12
T / 12

c
a
a
b
b
c
c

b
b
c
c
a
a
b

88

Conducting
SCR
(Positive)
SCR3
SCR1
SCR1
SCR2
SCR2
SCR3
SCR3

Conducting
SCR
(Negative)
SCR5
SCR5
SCR6
SCR6
SCR4
SCR4
SCR5

Triggered
SCR

Firing
Angle

SCR5
SCR1
SCR6
SCR2
SCR4
SCR3
SCR5

-30°
30°
90°
150°
210°
270°
330°

T/12

(b) If each SCR is triggered halfway through the half-cycle during which it is forward biased, the
resulting phase a, b, and c voltages will be zero before the first half of each half-cycle, and the full
sinusoidal value for the second half of each half-cycle. These waveforms are shown below. (These plots
were created by the MATLAB program that appears later in this answer.)

and the resulting output voltage will be:

89

A MATLAB program to generate these waveforms and to calculate the ripple on the output waveform is
shown below. The first function biphase_controller.m generates a switched ac waveform. The
inputs to this function are the current phase angle in degrees, the offset angle of the waveform in degrees,
and the firing angle in degrees.

function volts = biphase_controller(wt,theta0,fire)
% Function to simulate the output of an ac phase
% angle controller that operates symmetrically on
% positive and negative half cycles. Assume a peak
% voltage VM = 120 * SQRT(2) = 170 V for convenience.
%
%
wt
= Current phase in degrees
%
theta0 = Starting phase angle in degrees
%
fire
= Firing angle in degrees
% Degrees to radians conversion factor
deg2rad = pi / 180;
% Remove phase ambiguities: 0 <= wt < 360 deg
ang = wt + theta0;
while ang >= 360
ang = ang - 360;
end
while ang < 0
ang = ang + 360;
end
% Simulate the output of the phase angle controller.
if (ang >= fire & ang <= 180)
volts = 170 * sin(ang * deg2rad);
elseif (ang >= (fire+180) & ang <= 360)
volts = 170 * sin(ang * deg2rad);
else
90

volts = 0;
end
The main program below creates and plots the three-phase waveforms, calculates and plots the output
waveform, and determines the ripple in the output waveform.

%
%
%
%

M-file: prob3_14b.m
M-file to calculate and plot the three phase voltages
when each SCR in a three-phase full-wave rectifier
triggers at a phase angle of 90 degrees.

% Calculate the waveforms for times from 0 to 1/30 s
t = (0:1/21600:1/30);
deg = zeros(size(t));
rms = zeros(size(t));
va = zeros(size(t));
vb = zeros(size(t));
vc = zeros(size(t));
out = zeros(size(t));
for ii = 1:length(t)
% Get equivalent angle in degrees. Note that
% 1/60 s = 360 degrees for a 60 Hz waveform!
theta = 21600 * t(ii);
% Calculate the voltage in each phase at each
% angle.
va(ii) = biphase_controller(theta,0,90);
vb(ii) = biphase_controller(theta,-120,90);
vc(ii) = biphase_controller(theta,120,90);
end
% Calculate the output voltage of the rectifier
for ii = 1:length(t)
vals = [ va(ii) vb(ii) vc(ii) ];
out(ii) = max( vals ) - min( vals );
end
% Calculate and display the ripple
disp( ['The ripple is ' num2str(ripple(out))] );
% Plot the voltages versus time
figure(1)
plot(t,va,'b','Linewidth',2.0);
hold on;
plot(t,vb,'r:','Linewidth',2.0);
plot(t,vc,'k--','Linewidth',2.0);
title('\bfPhase Voltages');
xlabel('\bfTime (s)');
ylabel('\bfVoltage (V)');
grid on;
legend('Phase a','Phase b','Phase c');
hold off;
91

% Plot the output voltages versus time
figure(2)
plot(t,out,'b','Linewidth',2.0);
title('\bfOutput Voltage');
xlabel('\bfTime (s)');
ylabel('\bfVoltage (V)');
axis( [0 1/30 0 260]);
grid on;
hold off;
When this program is executed, the results are:

» prob_3_14b
The ripple is 30.9547

3-15.

Write a MATLAB program that imitates the operation of the Pulse-Width Modulation circuit shown in
Figure 3-55, and answer the following questions.
(a) Assume that the comparison voltages v x ( t ) and v y ( t ) have peak amplitudes of 10 V and a frequency
of 500 Hz. Plot the output voltage when the input voltage is vin ( t ) = 10 sin 2πft V, and f = 50 Hz.
(b) What does the spectrum of the output voltage look like? What could be done to reduce the harmonic
content of the output voltage?
(c) Now assume that the frequency of the comparison voltages is increased to 1000 Hz. Plot the output
voltage when the input voltage is vin ( t ) = 10 sin 2πft V, and f = 50 Hz.
(d) What does the spectrum of the output voltage in (c) look like?
(e) What is the advantage of using a higher comparison frequency and more rapid switching in a PWM
modulator?
SOLUTION The PWM circuit is shown below:

92

(a)

To write a MATLAB simulator of this circuit, note that if vin > v x , then vu = VDC , and if vin < v x ,

then v u = 0. Similarly, if v in > v y , then v v = 0, and if v in < v y , then v v = VDC . The output voltage is
then v out = v v − vu . A MATLAB function that performs these calculations is shown below. (Note that
this function arbitrarily assumes that VDC = 100 V. It would be easy to modify the function to use any
arbitrary dc voltage, if desired.)

function [vout,vu,vv] = vout(vin, vx, vy)
% Function to calculate the output voltage of a
% PWM modulator from the values of vin and the
% reference voltages vx and vy. This function
% arbitrarily assumes that VDC = 100 V.
%
%
vin
= Input voltage
%
vx
= x reference
%
vy
= y reference
%
vout
= Ouput voltage
%
vu, vv = Components of output voltage
93

%

fire

= Firing angle in degrees

% vu
if ( vin > vx )
vu = 100;
else
vu = 0;
end
% vv
if ( vin >= vy )
vv = 0;
else
vv = 100;
end
% Caclulate vout
vout = vv - vu;
Now we need a MATLAB program to generate the input voltage vin (t ) and the reference voltages v x (t )

and v y (t ) . After the voltages are generated, function vout will be used to calculate vout (t ) and the

frequency spectrum of v out (t ) . Finally, the program will plot v in (t ) , v x (t ) and v y (t ) , v out (t ) , and the

spectrum of v out (t ) . (Note that in order to have a valid spectrum, we need to create several cycles of the
60 Hz output waveform, and we need to sample the data at a fairly high frequency. This problem creates 4
cycles of vout (t ) and samples all data at a 20,000 Hz rate.)

%
%
%
%
%

M-file: prob3_15a.m
M-file to calculate the output voltage from a PWM
modulator with a 500 Hz reference frequency. Note
that the only change between this program and that
of part b is the frequency of the reference "fr".

% Sample the data at 20000 Hz to get enough information
% for spectral analysis. Declare arrays.
fs = 20000;
% Sampling frequency (Hz)
t = (0:1/fs:4/15);
% Time in seconds
vx = zeros(size(t));
% vx
vy = zeros(size(t));
% vy
vin = zeros(size(t));
% Driving signal
vu = zeros(size(t));
% vx
vv = zeros(size(t));
% vy
vout = zeros(size(t));
% Output signal
fr = 500;
% Frequency of reference signal
T = 1/fr;
% Period of refernce signal
% Calculate vx at fr = 500 Hz.
for ii = 1:length(t)
vx(ii) = vref(t(ii),T);
vy(ii) = - vx(ii);
end
% Calculate vin as a 50 Hz sine wave with a peak voltage of
94

% 10 V.
for ii = 1:length(t)
vin(ii) = 10 * sin(2*pi*50*t(ii));
end
% Now calculate vout
for ii = 1:length(t)
[vout(ii) vu(ii) vv(ii)] = vout(vin(ii), vx(ii), vy(ii));
end
% Plot the reference voltages vs time
figure(1)
plot(t,vx,'b','Linewidth',1.0);
hold on;
plot(t,vy,'k--','Linewidth',1.0);
title('\bfReference Voltages for fr = 500 Hz');
xlabel('\bfTime (s)');
ylabel('\bfVoltage (V)');
legend('vx','vy');
axis( [0 1/30 -10 10]);
hold off;
% Plot the input voltage vs time
figure(2)
plot(t,vin,'b','Linewidth',1.0);
title('\bfInput Voltage');
xlabel('\bfTime (s)');
ylabel('\bfVoltage (V)');
axis( [0 1/30 -10 10]);
% Plot the output voltages versus time
figure(3)
plot(t,vout,'b','Linewidth',1.0);
title('\bfOutput Voltage for fr = 500 Hz');
xlabel('\bfTime (s)');
ylabel('\bfVoltage (V)');
axis( [0 1/30 -120 120]);
% Now calculate the spectrum of the output voltage
spec = fft(vout);
% Calculate sampling frequency labels
len = length(t);
df = fs / len;
fstep = zeros(size(t));
for ii = 2:len/2
fstep(ii) = df * (ii-1);
fstep(len-ii+2) = -fstep(ii);
end
% Plot the spectrum
figure(4);
plot(fftshift(fstep),fftshift(abs(spec)));
title('\bfSpectrum of Output Voltage for fr = 500 Hz');
xlabel('\bfFrequency (Hz)');
ylabel('\bfAmplitude');
95

% Plot a closeup of the near spectrum
% (positive side only)
figure(5);
plot(fftshift(fstep),fftshift(abs(spec)));
title('\bfSpectrum of Output Voltage for fr = 500 Hz');
xlabel('\bfFrequency (Hz)');
ylabel('\bfAmplitude');
set(gca,'Xlim',[0 1000]);
When this program is executed, the input, reference, and output voltages are:

96

(b) The output spectrum of this PWM modulator is shown below. There are two plots here, one showing
the entire spectrum, and the other one showing the close-in frequencies (those under 1000 Hz), which will
have the most effect on machinery. Note that there is a sharp peak at 50 Hz, which is there desired
frequency, but there are also strong contaminating signals at about 850 Hz and 950 Hz. If necessary, these
components could be filtered out using a low-pass filter.

97

(c)

%
%
%
%
%

A version of the program with 1000 Hz reference functions is shown below:

M-file: prob3_15b.m
M-file to calculate the output voltage from a PWM
modulator with a 1000 Hz reference frequency. Note
that the only change between this program and that
of part a is the frequency of the reference "fr".

% Sample the data at 20000 Hz to get enough information
% for spectral analysis. Declare arrays.
fs = 20000;
% Sampling frequency (Hz)
t = (0:1/fs:4/15);
% Time in seconds
vx = zeros(size(t));
% vx
vy = zeros(size(t));
% vy
vin = zeros(size(t));
% Driving signal
vu = zeros(size(t));
% vx
vv = zeros(size(t));
% vy
vout = zeros(size(t));
% Output signal
fr = 1000;
% Frequency of reference signal
T = 1/fr;
% Period of refernce signal
% Calculate vx at 1000 Hz.
for ii = 1:length(t)
vx(ii) = vref(t(ii),T);
vy(ii) = - vx(ii);
end
% Calculate vin as a 50 Hz sine wave with a peak voltage of
% 10 V.
for ii = 1:length(t)
vin(ii) = 10 * sin(2*pi*50*t(ii));
end
98

% Now calculate vout
for ii = 1:length(t)
[vout(ii) vu(ii) vv(ii)] = vout(vin(ii), vx(ii), vy(ii));
end
% Plot the reference voltages vs time
figure(1)
plot(t,vx,'b','Linewidth',1.0);
hold on;
plot(t,vy,'k--','Linewidth',1.0);
title('\bfReference Voltages for fr = 1000 Hz');
xlabel('\bfTime (s)');
ylabel('\bfVoltage (V)');
legend('vx','vy');
axis( [0 1/30 -10 10]);
hold off;
% Plot the input voltage vs time
figure(2)
plot(t,vin,'b','Linewidth',1.0);
title('\bfInput Voltage');
xlabel('\bfTime (s)');
ylabel('\bfVoltage (V)');
axis( [0 1/30 -10 10]);
% Plot the output voltages versus time
figure(3)
plot(t,vout,'b','Linewidth',1.0);
title('\bfOutput Voltage for fr = 1000 Hz');
xlabel('\bfTime (s)');
ylabel('\bfVoltage (V)');
axis( [0 1/30 -120 120]);
% Now calculate the spectrum of the output voltage
spec = fft(vout);
% Calculate sampling frequency labels
len = length(t);
df = fs / len;
fstep = zeros(size(t));
for ii = 2:len/2
fstep(ii) = df * (ii-1);
fstep(len-ii+2) = -fstep(ii);
end
% Plot the spectrum
figure(4);
plot(fftshift(fstep),fftshift(abs(spec)));
title('\bfSpectrum of Output Voltage for fr = 1000 Hz');
xlabel('\bfFrequency (Hz)');
ylabel('\bfAmplitude');
% Plot a closeup of the near spectrum
% (positive side only)
figure(5);
plot(fftshift(fstep),fftshift(abs(spec)));
99

title('\bfSpectrum of Output Voltage for fr = 1000 Hz');
xlabel('\bfFrequency (Hz)');
ylabel('\bfAmplitude');
set(gca,'Xlim',[0 1000]);
When this program is executed, the input, reference, and output voltages are:

100

(d)

The output spectrum of this PWM modulator is shown below.

101

(e) Comparing the spectra in (b) and (d), we can see that the frequencies of the first large sidelobes
doubled from about 900 Hz to about 1800 Hz when the reference frequency was doubled. This increase in
sidelobe frequency has two major advantages: it makes the harmonics easier to filter, and it also makes it
less necessary to filter them at all. Since large machines have their own internal inductances, they form
natural low-pass filters. If the contaminating sidelobes are at high enough frequencies, they will never
affect the operation of the machine. Thus, it is a good idea to design PWM modulators with a high
frequency reference signal and rapid switching.

102

Chapter 4: AC Machinery Fundamentals
4-1.

The simple loop is rotating in a uniform magnetic field shown in Figure 4-1 has the following
characteristics:

B = 0.5 T to the right

r = 01
. m

l = 0.5 m

ω = 103 rad/s

(a) Calculate the voltage etot ( t ) induced in this rotating loop.
(b) Suppose that a 5 Ω resistor is connected as a load across the terminals of the loop. Calculate the
current that would flow through the resistor.
(c) Calculate the magnitude and direction of the induced torque on the loop for the conditions in (b).
(d) Calculate the electric power being generated by the loop for the conditions in (b).
(e) Calculate the mechanical power being consumed by the loop for the conditions in (b). How does
this number compare to the amount of electric power being generated by the loop?
ωm

c

N

r

d
vab

vcd

S

b
a

B
B is a uniform magnetic
field, aligned as shown.

SOLUTION
(a)

The induced voltage on a simple rotating loop is given by
eind ( t ) = 2 rωBl sin ω t

(4-8)

eind ( t ) = 2 ( 0.1 m )(103 rad/s)(0.5 T )( 0.5 m ) sin103t

eind ( t ) = 5.15 sin103t V

(b)

If a 5 Ω resistor is connected as a load across the terminals of the loop, the current flow would be:
i (t ) =

(c)

eind 5.15 sin 103t V
=
= 1.03 sin 103t A
R
5Ω

The induced torque would be:

τ ind ( t ) = 2 rilΒ sin θ

τ ind ( t ) = 2 ( 0.1 m )(1.03 sin ω t A )( 0.5 m )( 0.5 T ) sin ω t

τ ind ( t ) = 0.0515 sin 2 ωt N ⋅ m, counterclockwise
(d)

The instantaneous power generated by the loop is:
103

(4-17)

P ( t ) = eindi = (5.15 sin ω t V )(1.03 sin ω t A ) = 5.30 sin 2ωt W
The average power generated by the loop is
Pave =

(e)

1
T

T

5.30 sin 2ω t dt = 2.65 W

The mechanical power being consumed by the loop is:

(

)

P = τ indω = 0.0515 sin 2 ω t V (103 rad/s ) = 5.30 sin 2ω t W

Note that the amount of mechanical power consumed by the loop is equal to the amount of electrical power
created by the loop. This machine is acting as a generator, converting mechanical power into electrical
power.

4-2.

Develop a table showing the speed of magnetic field rotation in ac machines of 2, 4, 6, 8, 10, 12, and 14
poles operating at frequencies of 50, 60, and 400 Hz.
SOLUTION The equation relating the speed of magnetic field rotation to the number of poles and electrical
frequency is
nm =

120 f e
P

The resulting table is
Number of Poles
2
4
6
8
10
12
14

4-3.

f e = 50 Hz

f e = 60 Hz

f e = 400 Hz

3000 r/min
1500 r/min
1000 r/min
750 r/min
600 r/min
500 r/min
428.6 r/min

3600 r/min
1800 r/min
1200 r/min
900 r/min
720 r/min
600 r/min
514.3 r/min

24000 r/min
12000 r/min
8000 r/min
6000 r/min
4800 r/min
4000 r/min
3429 r/min

A three-phase four-pole winding is installed in 12 slots on a stator. There are 40 turns of wire in each slot
of the windings. All coils in each phase are connected in series, and the three phases are connected in ∆.
The flux per pole in the machine is 0.060 Wb, and the speed of rotation of the magnetic field is 1800 r/min.
(a) What is the frequency of the voltage produced in this winding?
(b) What are the resulting phase and terminal voltages of this stator?
SOLUTION
(a)

The frequency of the voltage produced in this winding is
fe =

nm P (1800 r/min )( 4 poles )
=
= 60 Hz
120
120

(b) There are 12 slots on this stator, with 40 turns of wire per slot. Since this is a four-pole machine,
there are two sets of coils (4 slots) associated with each phase. The voltage in the coils in one pair of slots
is
E A = 2π N C φ f = 2π ( 40 t )( 0.060 Wb )( 60 Hz ) = 640 V
There are two sets of coils per phase, since this is a four-pole machine, and they are connected in series, so
the total phase voltage is
104

Vφ = 2 ( 640 V ) = 1280 V
Since the machine is ∆-connected, VL = Vφ = 1280 V .

4-4.

A three-phase Y-connected 50-Hz two-pole synchronous machine has a stator with 2000 turns of wire per
phase. What rotor flux would be required to produce a terminal (line-to-line) voltage of 6 kV?
SOLUTION The phase voltage of this machine should be Vφ = VL / 3 = 3464 V . The induced voltage per
phase in this machine (which is equal to Vφ at no-load conditions) is given by the equation
E A = 2π N C φ f

so

φ=
4-5.

EA
=
2π N C f

3464 V
= 0.0078 Wb
2π ( 2000 t )(50 Hz )

Modify the MATLAB program in Example 4-1 by swapping the currents flowing in any two phases. What
happens to the resulting net magnetic field?
SOLUTION This modification is very simple—just swap the currents supplied to two of the three phases.

% M-file: mag_field2.m
% M-file to calculate the net magetic field produced
% by a three-phase stator.
% Set up the basic conditions
bmax = 1;
% Normalize bmax to 1
freq = 60;
% 60 Hz
w = 2*pi*freq;
% angluar velocity (rad/s)
% First, generate the three component magnetic fields
t = 0:1/6000:1/60;
Baa = sin(w*t) .* (cos(0) + j*sin(0));
Bbb = sin(w*t+2*pi/3) .* (cos(2*pi/3) + j*sin(2*pi/3));
Bcc = sin(w*t-2*pi/3) .* (cos(-2*pi/3) + j*sin(-2*pi/3));
% Calculate Bnet
Bnet = Baa + Bbb + Bcc;
% Calculate a circle representing the expected maximum
% value of Bnet
circle = 1.5 * (cos(w*t) + j*sin(w*t));
% Plot the magnitude and direction of the resulting magnetic
% fields. Note that Baa is black, Bbb is blue, Bcc is
% magneta, and Bnet is red.
for ii = 1:length(t)
% Plot the reference circle
plot(circle,'k');
hold on;
% Plot the four magnetic fields
plot([0 real(Baa(ii))],[0 imag(Baa(ii))],'k','LineWidth',2);
plot([0 real(Bbb(ii))],[0 imag(Bbb(ii))],'b','LineWidth',2);
105

plot([0 real(Bcc(ii))],[0 imag(Bcc(ii))],'m','LineWidth',2);
plot([0 real(Bnet(ii))],[0 imag(Bnet(ii))],'r','LineWidth',3);
axis square;
axis([-2 2 -2 2]);
drawnow;
hold off;
end
When this program executes, the net magnetic field rotates clockwise, instead of counterclockwise.

4-6.

If an ac machine has the rotor and stator magnetic fields shown in Figure P4-1, what is the direction of the
induced torque in the machine? Is the machine acting as a motor or generator?

SOLUTION Since τind = kB R × B net , the induced torque is clockwise, opposite the direction of motion. The
machine is acting as a generator.

4-7.

The flux density distribution over the surface of a two-pole stator of radius r and length l is given by
B = BM cos ( ω m t − α )
(4-37b)
Prove that the total flux under each pole face is

φ = 2 rlBM

106

SOLUTION The total flux under a pole face is given by the equation

φ = B ⋅ dA
Under a pole face, the flux density B is always parallel to the vector dA, since the flux density is always
perpendicular to the surface of the rotor and stator in the air gap. Therefore,

φ = B dA
A differential area on the surface of a cylinder is given by the differential length along the cylinder (dl)
times the differential width around the radius of the cylinder ( rdθ ).
dA = ( dl )( rdθ )

where r is the radius of the cylinder

Therefore, the flux under the pole face is

φ=

B dl r d θ

Since r is constant and B is constant with respect to l, this equation reduces to

φ = rl B dθ
Now, B = BM cos (ωt − α ) = BM cos θ (when we substitute θ = ω t − α ), so

φ = rl B dθ
φ = rl

π /2
−π / 2

π /2

BM cos θ d θ = rlBM [sin θ ]

−π / 2

φ = 2 rlBM

107

= rlBM 1

( 1)

4-8.

In the early days of ac motor development, machine designers had great difficulty controlling the core losses
(hysteresis and eddy currents) in machines. They had not yet developed steels with low hysteresis, and
were not making laminations as thin as the ones used today. To help control these losses, early ac motors
in the USA were run from a 25 Hz ac power supply, while lighting systems were run from a separate 60 Hz
ac power supply.
(a) Develop a table showing the speed of magnetic field rotation in ac machines of 2, 4, 6, 8, 10, 12, and
14 poles operating at 25 Hz. What was the fastest rotational speed available to these early motors?
(b) For a given motor operating at a constant flux density B, how would the core losses of the motor
running at 25 Hz compare to the core losses of the motor running at 60 Hz?
(c) Why did the early engineers provide a separate 60 Hz power system for lighting?
SOLUTION
(a) The equation relating the speed of magnetic field rotation to the number of poles and electrical
frequency is
nm =

120 f e
P

The resulting table is

Number of Poles

f e = 25 Hz

2
4
6
8
10
12
14

1500 r/min
750 r/min
500 r/min
375 r/min
300 r/min
250 r/min
214.3 r/min

The highest possible rotational speed was 1500 r/min.
(b) Core losses scale according to the 1.5th power of the speed of rotation, so the ratio of the core losses
at 25 Hz to the core losses at 60 Hz (for a given machine) would be:
ratio =
(c)

1500
3600

1.5

= 0.269 or 26.9%

At 25 Hz, the light from incandescent lamps would visibly flicker in a very annoying way.

108

Chapter 5: Synchronous Generators
5-1.

At a location in Europe, it is necessary to supply 300 kW of 60-Hz power. The only power sources
available operate at 50 Hz. It is decided to generate the power by means of a motor-generator set
consisting of a synchronous motor driving a synchronous generator. How many poles should each of the
two machines have in order to convert 50-Hz power to 60-Hz power?
SOLUTION The speed of a synchronous machine is related to its frequency by the equation
nm =

120 f e
P

To make a 50 Hz and a 60 Hz machine have the same mechanical speed so that they can be coupled
together, we see that
nsync =

120 (50 Hz ) 120 (60 Hz )
=
P1
P2

P2 6 12
= =
P1 5 10

Therefore, a 10-pole synchronous motor must be coupled to a 12-pole synchronous generator to accomplish
this frequency conversion.

5-2.

A 2300-V 1000-kVA 0.8-PF-lagging 60-Hz two-pole Y-connected synchronous generator has a
synchronous reactance of 1.1 Ω and an armature resistance of 0.15 Ω. At 60 Hz, its friction and windage
losses are 24 kW, and its core losses are 18 kW. The field circuit has a dc voltage of 200 V, and the
maximum I F is 10 A. The resistance of the field circuit is adjustable over the range from 20 to 200 Ω.
The OCC of this generator is shown in Figure P5-1.
(a) How much field current is required to make VT equal to 2300 V when the generator is running at no
(b) What is the internal generated voltage of this machine at rated conditions?
(c) How much field current is required to make VT equal to 2300 V when the generator is running at rated
conditions?
(d) How much power and torque must the generator’s prime mover be capable of supplying?
(e) Construct a capability curve for this generator.

Note:

An electronic version of this open circuit characteristic can be found in file
p51_occ.dat, which can be used with MATLAB programs. Column 1
contains field current in amps, and column 2 contains open-circuit terminal
voltage in volts.

109

SOLUTION
(a) If the no-load terminal voltage is 2300 V, the required field current can be read directly from the
open-circuit characteristic. It is 4.25 A.
(b) This generator is Y-connected, so I L = I A . At rated conditions, the line and phase current in this
generator is
IA = IL =

P
1000 kVA
=
= 251 A at an angle of –36.87°
3 VL
3 ( 2300 V )

The phase voltage of this machine is Vφ = VT / 3 = 1328 V . The internal generated voltage of the machine
is
E A = Vφ + RAI A + jX S I A

E A = 1328∠0° + ( 0.15 Ω )( 251∠ − 36.87° A ) + j (1.1 Ω )( 251∠ − 36.87° A )
E A = 1537∠7.4° V
(c)

The equivalent open-circuit terminal voltage corresponding to an E A of 1537 volts is
VT ,oc = 3 (1527 V ) = 2662 V

From the OCC, the required field current is 5.9 A.
(d)

The input power to this generator is equal to the output power plus losses. The rated output power is
POUT = (1000 kVA )( 0.8) = 800 kW
PCU = 3 I A2 RA = 3 ( 251 A ) ( 0.15 Ω ) = 28.4 kW
2

PF&W = 24 kW
110

Pcore = 18 kW
Pstray = (assumed 0)
PIN = POUT + PCU + PF&W + Pcore + Pstray = 870.4 kW

Therefore the prime mover must be capable of supplying 175 kW. Since the generator is a two-pole 60 Hz
machine, to must be turning at 3600 r/min. The required torque is

τ APP =

(e)

PIN

=

ωm

175.2 kW
= 465 N ⋅ m
 1 min  2π rad 
(3600 r/min )


 60 s  1 r 

The rotor current limit of the capability curve would be drawn from an origin of
Q=−

3Vφ 2
XS

3 (1328 V )
= −4810 kVAR
1.1 Ω
2

=−

The radius of the rotor current limit is
DE =

3Vφ E A
XS

=

3 (1328 V )(1537 V )
= 5567 kVA
1.1 Ω

The stator current limit is a circle at the origin of radius
S = 3Vφ I A = 3 (1328 V )( 251 A ) = 1000 kVA
A MATLAB program that plots this capability diagram is shown below:

% M-file: prob5_2.m
% M-file to display a capability curve for a
% synchronous generator.
% Calculate the waveforms for times from 0 to 1/30 s
Q = -4810;
DE = 5567;
S = 1000;
% Get points for stator current limit
theta = -95:1:95;
% Angle in degrees
rad = theta * pi / 180;
% Angle in radians
s_curve = S .* ( cos(rad) + j*sin(rad) );
% Get points for rotor current limit
orig = j*Q;
theta = 75:1:105;
% Angle in degrees
rad = theta * pi / 180;
% Angle in radians
r_curve = orig + DE .* ( cos(rad) + j*sin(rad) );
% Plot the capability diagram
figure(1);
plot(real(s_curve),imag(s_curve),'b','LineWidth',2.0);
hold on;
plot(real(r_curve),imag(r_curve),'r--','LineWidth',2.0);
% Add x and y axes
111

plot( [-1500 1500],[0 0],'k');
plot( [0,0],[-1500 1500],'k');
% Set titles and axes
title ('\bfSynchronous Generator Capability Diagram');
xlabel('\bfPower (kW)');
ylabel('\bfReactive Power (kVAR)');
axis( [ -1500 1500 -1500 1500] );
axis square;
hold off;
The resulting capability diagram is shown below:

5-3.

Assume that the field current of the generator in Problem 5-2 has been adjusted to a value of 4.5 A.
(a) What will the terminal voltage of this generator be if it is connected to a ∆-connected load with an
impedance of 20∠30° Ω ?
(b) Sketch the phasor diagram of this generator.
(c) What is the efficiency of the generator at these conditions?
(d) Now assume that another identical ∆-connected load is to be paralleled with the first one. What
happens to the phasor diagram for the generator?
(e) What is the new terminal voltage after the load has been added?
(f) What must be done to restore the terminal voltage to its original value?
SOLUTION
(a) If the field current is 4.5 A, the open-circuit terminal voltage will be about 2385 V, and the phase
voltage in the generator will be 2385 / 3 = 1377 V .
The load is ∆-connected with three impedances of 20∠30° Ω . From the Y-∆ transform, this load is
equivalent to a Y-connected load with three impedances of 6.667∠30° Ω . The resulting per-phase
equivalent circuit is shown below:

112

0.15 Ω

j1.1 Ω

IA
+

EA

+
-

Vφ

Z

6.667∠30°

-

The magnitude of the phase current flowing in this generator is
IA =

EA
1377 V
1377 V
=
=
= 186 A
RA + jX S + Z
0.15 + j1.1 + 6.667∠30° 1.829 Ω

Therefore, the magnitude of the phase voltage is
Vφ = I A Z = (186 A )(6.667 Ω ) = 1240 V

and the terminal voltage is
VT = 3 Vφ = 3 (1240 V ) = 2148 V

(b)

Armature current is I A = 186∠ − 30° A , and the phase voltage is Vφ = 1240∠0° V . Therefore, the

internal generated voltage is
E A = Vφ + RAI A + jX S I A

E A = 1240∠0° + ( 0.15 Ω )(186∠ − 30° A ) + j (1.1 Ω)(186∠ − 30° A )
E A = 1377∠6.8° V
The resulting phasor diagram is shown below (not to scale):

E = 1377∠6.8° V
A

θ

Vφ = 1240∠0° V
IA = 186∠-30°
(c)

The efficiency of the generator under these conditions 3can be found as follows:
POUT = 3 Vφ I A cos θ = 3 (1240 V )(186 A )( 0.8) = 554 kW
PCU = 3 I A2 RA = 3 (186 A ) ( 0.15 Ω ) = 15.6 kW
2

PF&W = 24 kW
Pcore = 18 kW
Pstray = (assumed 0)
PIN = POUT + PCU + PF&W + Pcore + Pstray = 612 kW
113

η=
(d)

POUT
554 kW
× 100% =
× 100% = 90.5%
PIN
612 kW

When the new load is added, the total current flow increases at the same phase angle. Therefore,

jX S I S increases in length at the same angle, while the magnitude of E A must remain constant. Therefore,

E A “swings” out along the arc of constant magnitude until the new jX S I S fits exactly between Vφ and
EA .
E′A
E = 1377∠6.8° V
A

θ

V′φ

Vφ = 1240∠0° V

I = 186∠-30°
A

I′

A

(e) The new impedance per phase will be half of the old value, so Z = 3.333∠30° Ω . The magnitude of
the phase current flowing in this generator is

IA =

EA
1377 V
1377 V
=
=
= 335 A
R A + jX S + Z
0.15 + j1.1 + 3.333∠30° 1.829 Ω

Therefore, the magnitude of the phase voltage is
Vφ = I A Z = ( 335 A )(3.333 Ω ) = 1117 V

and the terminal voltage is
VT = 3 Vφ = 3 (1117 V ) = 1934 V

(f)

5-4.

To restore the terminal voltage to its original value, increase the field current I F .

Assume that the field current of the generator in Problem 5-2 is adjusted to achieve rated voltage (2300 V)
at full load conditions in each of the questions below.
(a) What is the efficiency of the generator at rated load?
(b) What is the voltage regulation of the generator if it is loaded to rated kilovoltamperes with 0.8-PFlagging loads?
(c) What is the voltage regulation of the generator if it is loaded to rated kilovoltamperes with 0.8-PFleading loads?
(d) What is the voltage regulation of the generator if it is loaded to rated kilovoltamperes with unity-powerfactor loads?
(e) Use MATLAB to plot the terminal voltage of the generator as a function of load for all three power
factors.
SOLUTION
114

(a) This generator is Y-connected, so I L = I A . At rated conditions, the line and phase current in this
generator is
IA = IL =

P
1000 kVA
=
= 251 A at an angle of –36.87°
3 VL
3 (2300 V)

The phase voltage of this machine is Vφ = VT / 3 = 1328 V . The internal generated voltage of the machine
is
E A = Vφ + RAI A + jX S I A

E A = 1328∠0° + ( 0.15 Ω )( 251∠ − 36.87° A ) + j (1.1 Ω )( 251∠ − 36.87° A )
E A = 1537∠7.4° V
The input power to this generator is equal to the output power plus losses. The rated output power is
POUT = (1000 kVA )( 0.8) = 800 kW
PCU = 3 I A2 RA = 3 ( 251 A ) ( 0.15 Ω ) = 28.4 kW
2

PF&W = 24 kW
Pcore = 18 kW
Pstray = (assumed 0)
PIN = POUT + PCU + PF&W + Pcore + Pstray = 870.4 kW

η=

(b)

POUT
800 kW
×100% =
×100% = 91.9%
PIN
870.4 kW

If the generator is loaded to rated kVA with lagging loads, the phase voltage is Vφ = 1328∠0° V and

the internal generated voltage is E A = 1537∠7.4° V . Therefore, the phase voltage at no-load would be
Vφ = 1537∠0° V . The voltage regulation would be:
VR =
(c)

1537 − 1328
× 100% = 15.7%
1328

If the generator is loaded to rated kVA with leading loads, the phase voltage is Vφ = 1328∠0° V and

the internal generated voltage is
E A = Vφ + RAI A + jX S I A

E A = 1328∠0° + ( 0.15 Ω )( 251∠36.87° A ) + j (1.1 Ω )( 251∠36.87° A )
E A = 1217∠11.5° V
The voltage regulation would be:
VR =

1217 − 1328
× 100% = −8.4%
1328

(d) If the generator is loaded to rated kVA at unity power factor, the phase voltage is
Vφ = 1328∠0° V and the internal generated voltage is
E A = Vφ + RAI A + jX S I A

115

E A = 1328∠0° + ( 0.15 Ω )( 251∠0° A ) + j (1.1 Ω )( 251∠0° A )
E A = 1393∠11.4° V
The voltage regulation would be:
VR =

1393 − 1328
× 100% = 4.9%
1328

(e) For this problem, we will assume that the terminal voltage is adjusted to 2300 V at no load
conditions, and see what happens to the voltage as load increases at 0.8 lagging, unity, and 0.8 leading
power factors. Note that the maximum current will be 251 A in any case. A phasor diagram representing
the situation at lagging power factor is shown below:

EA
θ
δ
θ

Vφ
I

θ

jX I

S A

R I

A A

A

By the Pythagorean Theorem,

(

E A2 = Vφ + R A I A cosθ + X S I A sin θ

) +(X
2

I cosθ − R A I S sin θ )

2

S A

Vφ = E A2 − ( X S I A cosθ − R A I S sin θ ) − RA I A cosθ − X S I A sin θ
2

A phasor diagram representing the situation at leading power factor is shown below:

EA

jX I

S A

I

A

δ

θ

R I

A A

θ

θ

Vφ
By the Pythagorean Theorem,

(

E A2 = Vφ + R A I A cosθ − X S I A sin θ

) +(X
2

I cosθ + R A I S sin θ )

2

S A

Vφ = E A2 − ( X S I A cosθ + R A I S sin θ ) − RA I A cosθ + X S I A sin θ
2

A phasor diagram representing the situation at unity power factor is shown below:

EA
jX I

S A

δ

I

Vφ R AI A

A

116

By the Pythagorean Theorem,

E A2 = Vφ 2 + ( X S I A )
Vφ = E A2 − ( X S I A )

2

2

The MATLAB program is shown below takes advantage of this fact.

%
%
%
%

M-file: prob5_4e.m
M-file to calculate and plot the terminal voltage
of a synchronous generator as a function of load
for power factors of 0.8 lagging, 1.0, and 0.8 leading.

% Define values for this generator
EA = 1328;
% Internal gen voltage
I = 0:2.51:251;
% Current values (A)
R = 0.15;
% R (ohms)
X = 1.10;
% XS (ohms)
% Calculate the voltage for the lagging PF case
VP_lag = sqrt( EA^2 - (X.*I.*0.8 - R.*I.*0.6).^2 ) ...
- R.*I.*0.8 - X.*I.*0.6;
VT_lag = VP_lag .* sqrt(3);
% Calculate the voltage for the leading PF case
VP_lead = sqrt( EA^2 - (X.*I.*0.8 + R.*I.*0.6).^2 ) ...
- R.*I.*0.8 + X.*I.*0.6;
% Calculate the voltage for the unity PF case
VP_unity = sqrt( EA^2 - (X.*I).^2 );
VT_unity = VP_unity .* sqrt(3);
% Plot the terminal voltage versus load
plot(I,abs(VT_lag),'b-','LineWidth',2.0);
hold on;
plot(I,abs(VT_unity),'k--','LineWidth',2.0);
title ('\bfTerminal Voltage Versus Load');
ylabel ('\bfTerminal Voltage (V)');
legend('0.8 PF lagging','1.0 PF','0.8 PF leading');
axis([0 260 1500 2500]);
grid on;
hold off;
The resulting plot is shown below:

117

5-5.

Assume that the field current of the generator in Problem 5-2 has been adjusted so that it supplies rated
voltage when loaded with rated current at unity power factor. (You may ignore the effects of R A when

(a) What is the torque angle δ of the generator when supplying rated current at unity power factor?
(b) When this generator is running at full load with unity power factor, how close is it to the static stability
limit of the machine?
SOLUTION

(a)

The torque δ angle can be found by calculating E A :

E A = Vφ + RAI A + jX S I A
E A = 1328∠0° + ( 0.15 Ω )( 251∠0° A ) + j (1.1 Ω )( 251∠0° A )

E A = 1393∠11.4° V
Thus the torque angle δ = 11.4°.

(b) The static stability limit occurs at δ = 90° . This generator is a very long way from that limit. If we
ignore the internal resistance of the generator, the output power will be given by
P=

3Vφ E A

XS

sin δ

and the output power is proportional to sin δ . Since sin 11.4° = 0.198 , and sin 90° = 1.00 , the static
stability limit is about 5 times the current output power of the generator.

5-6.

A 480-V 400-kVA 0.85-PF-lagging 50-Hz four-pole ∆-connected generator is driven by a 500-hp diesel
engine and is used as a standby or emergency generator. This machine can also be paralleled with the
normal power supply (a very large power system) if desired.
(a) What are the conditions required for paralleling the emergency generator with the existing power
system? What is the generator’s rate of shaft rotation after paralleling occurs?
118

(b) If the generator is connected to the power system and is initially floating on the line, sketch the resulting
magnetic fields and phasor diagram.
(c) The governor setting on the diesel is now increased. Show both by means of house diagrams and by
means of phasor diagrams what happens to the generator. How much reactive power does the generator
supply now?
(d) With the diesel generator now supplying real power to the power system, what happens to the generator
as its field current is increased and decreased? Show this behavior both with phasor diagrams and with
house diagrams.
SOLUTION

(a)

To parallel this generator to the large power system, the required conditions are:
1. The generator must have the same voltage as the power system.
2. The phase sequence of the oncoming generator must be the same as the phase sequence of the
power system.
3. The frequency of the oncoming generator should be slightly higher than the frequency of the
running system.
4. The circuit breaker connecting the two systems together should be shut when the above conditions
are met and the generator is in phase with the power system.

After paralleling, the generator’s shaft will be rotating at

nm =
(b)

120 f e 120 (50 Hz )
=
= 1500 r/min
4
P

The magnetic field and phasor diagrams immediately after paralleling are shown below:

BS

BR

EA

jX I

S A

I

B net

A

Vφ

(c) When the governor setpoints on the generator are increased, the emergency generator begins to supply
more power to the loads, as shown below:

f

e

EA
jX I

S A

PG

P2 P1

I

Psys

A

Vφ

Note that as the load increased with E A constant, the generator began to consume a small amount of
reactive power.

(d) With the generator now supplying power to the system, an increase in field current increases the
reactive power supplied to the loads, and a decrease in field current decreases the reactive power supplied
119

V

T

E A1 E A2

E A3

I A1
Q

G

Q

Q

3

2

Q

I A2

Qsys

1

I A3

Vφ

jX I

S A

V

T

E A2 E A1
I A2

jX I

S A

Q

Q Q

G

5-7.

1

2

I A1

Qsys

Vφ

A 13.8-kV 10-MVA 0.8-PF-lagging 60-Hz two-pole Y-connected steam-turbine generator has a
synchronous reactance of 12 Ω per phase and an armature resistance of 1.5 Ω per phase. This generator is
operating in parallel with a large power system (infinite bus).

(a) What is the magnitude of E A at rated conditions?
(b) What is the torque angle of the generator at rated conditions?
(c) If the field current is constant, what is the maximum power possible out of this generator? How much
reserve power or torque does this generator have at full load?
(d) At the absolute maximum power possible, how much reactive power will this generator be supplying or
consuming? Sketch the corresponding phasor diagram. (Assume I F is still unchanged.)
SOLUTION

(a)

The phase voltage of this generator at rated conditions is

Vφ =

13,800 V
= 7967 V
3

The armature current per phase at rated conditions is

IA =

S
10,000,000 VA
=
= 418 A
3 VT
3 (13,800 V )

Therefore, the internal generated voltage at rated conditions is
E A = Vφ + RAI A + jX S I A

E A = 7967∠0° + (1.5 Ω )( 418∠ − 36.87° A ) + j (12.0 Ω )( 418∠ − 36.87° A )
E A = 12,040∠17.6° V
120

The magnitude of E A is 12,040 V.

(b)

The torque angle of the generator at rated conditions is δ = 17.6°.

(c)

Ignoring R A , the maximum output power of the generator is given by

PMAX =

3 Vφ E A

XS

=

3 ( 7967 V )(12,040 V )
= 24.0 MW
12 Ω

The power at maximum load is 8 MW, so the maximum output power is three times the full load output
power.

(d)

The phasor diagram at these conditions is shown below:
EA

jX I

S A

I

A

R I
Vφ

A A

Under these conditions, the armature current is

IA =

E A − Vφ
RA + jX S

=

12,040∠90° V - 7967∠0° V
= 1194 ∠40.6° A
1.5 + j12.0 Ω

The reactive power produced by the generator at this point is
Q = 3 Vφ I A sin θ = 3 ( 7967 V )(1194 A ) sin (0° − 40.6°) = −18.6 MVAR

The generator is actually consuming reactive power at this time.

5-8.

A 480-V, 100-kW, two-pole, three-phase, 60-Hz synchronous generator’s prime mover has a no-load speed
of 3630 r/min and a full-load speed of 3570 r/min. It is operating in parallel with a 480-V, 75-kW, fourpole, 60-Hz synchronous generator whose prime mover has a no-load speed of 1800 r/min and a full-load
speed of 1785 r/min. The loads supplied by the two generators consist of 100 kW at 0.85 PF lagging.
(a) Calculate the speed droops of generator 1 and generator 2.

(b) Find the operating frequency of the power system.
(c) Find the power being supplied by each of the generators in this system.
(d) If VT is 460 V, what must the generator’s operators do to correct for the low terminal voltage?
SOLUTION The no-load frequency of generator 1 corresponds to a frequency of
f nl1 =

nm P (3630 r/min )( 2 )
=
= 60.5 Hz
120
120

The full-load frequency of generator 1 corresponds to a frequency of
121

f fl1 =

nm P ( 3570 r/min )( 2 )
=
= 59.5 Hz
120
120

The no-load frequency of generator 2 corresponds to a frequency of
nm P (1800 r/min )( 4 )
=
= 60.00 Hz
120
120

f nl2 =

The full-load frequency of generator 2 corresponds to a frequency of
f fl2 =

(a)

nm P (1785 r/min )( 4 )
=
= 59.50 Hz
120
120

The speed droop of generator 1 is given by
SD1 =

nnl − nfl
3630 r/min − 3570 r/min
× 100% =
× 100% = 1.68%
nfl
3570 r/min

The speed droop of generator 2 is given by
SD2 =

(b)

nnl − nfl
1800 r/min − 1785 r/min
× 100% =
× 100% = 0.84%
nfl
1785 r/min

The power supplied by generator 1 is given by

(

P1 = sP1 f nl1 − f sys

)

and the power supplied by generator 1 is given by

(

P2 = sP 2 f nl2 − fsys

)

The power curve’s slope for generator 1 is
sP 1 =

P
0.1 MW
=
= 0.1 MW/Hz
f nl − f fl 60.5 Hz − 59.5 Hz

The power curve’s slope for generator 1 is
sP 2 =

P
0.075 MW
=
= 0.150 MW/Hz
f nl − f fl 60.00 Hz − 59.50 Hz

The no-load frequency of generator 1 is 60.5 Hz and the no-load frequency of generator 2 is 60 Hz. The
total power that they must supply is 100 kW, so the system frequency can be found from the equations

PLOAD = P1 + P2

(

)

(

PLOAD = sP1 f nl1 − fsys + sP 2 f nl2 − f sys

(

)
)

(

100 kW = ( 0.1 MW/Hz ) 60.5 Hz − f sys + (0.15 MW/Hz ) 60.0 Hz − fsys

)

100 kW = 6050 kW − ( 0.10 MW/Hz ) fsys + 9000 kW − (0.15 MW/Hz ) fsys

(0.25 MW/Hz ) f sys = 6050 kW + 9000 kW − 100 kW
f sys =
(c)

14,950 kW
= 59.8 Hz
0.25 MW/Hz

The power supplied by generator 1 is
122

(

)

P1 = sP1 f nl1 − fsys = ( 0.1 MW/Hz )( 60.5 Hz − 59.8 Hz ) = 70 kW
The power supplied by generator 2 is

(

)

P2 = sP 2 f nl2 − fsys = ( 0.15 MW/Hz )( 60.0 Hz − 59.8 Hz ) = 30 kW
(d) If the terminal voltage is 460 V, the operators of the generators must increase the field currents on
both generators simultaneously. That action will increase the terminal voltages of the system without
changing the power sharing between the generators.

5-9.

Three physically identical synchronous generators are operating in parallel. They are all rated for a full
load of 3 MW at 0.8 PF lagging. The no-load frequency of generator A is 61 Hz, and its speed droop is 3.4
percent. The no-load frequency of generator B is 61.5 Hz, and its speed droop is 3 percent. The no-load
frequency of generator C is 60.5 Hz, and its speed droop is 2.6 percent.
(a) If a total load consisting of 7 MW is being supplied by this power system, what will the system
frequency be and how will the power be shared among the three generators?

(b) Create a plot showing the power supplied by each generator as a function of the total power supplied to
all loads (you may use MATLAB to create this plot). At what load does one of the generators exceed
its ratings? Which generator exceeds its ratings first?
(c) Is this power sharing in (a) acceptable? Why or why not?
(d) What actions could an operator take to improve the real power sharing among these generators?
SOLUTION

(a)

Speed droop is defined as
SD =

so

f fl =

nnl − nfl
f − f fl
× 100% = nl
× 100%
nfl
f fl

f nl
SD
+1
100

Thus, the full-load frequencies of generators A, B, and C are
f nl,A
61 Hz
=
= 59.0 Hz
SD A
3.4
+1
+1
100
100
f nl,B
61.5 Hz
f fl,B =
=
= 59.71 Hz
SD B
3.0
+1
+1
100
100
f nl,C
60.5 Hz
f fl,C =
=
= 58.97 Hz
SDC
2.6
+1
+1
100
100
and the slopes of the power-frequency curves are:
3 MW
S PA =
= 1.5 MW/Hz
2 Hz
3 MW
S PB =
= 1.676 MW/Hz
1.79 Hz
3 MW
S PC =
= 1.961 MW/Hz
1.53 Hz
f fl,A =

123

(a)

The total load is 7 MW, so the system frequency is
PLOAD = sPA f nlA − f sys + sPB f nlB − f sys + sPC f nlC − f sys

(

)

(

(

)

)

(

(

)

)

(

7 MW = (1.5) 61.0 − f sys + (1.676) 61.5 − f sys + (1.961) 60.5 − f sys

)

7 MW = 91.5 − 1.5f sys + 103.07 − 1.676f sys + 118.64 − 1.961f sys
5.137 fsys = 306.2

f sys = 59.61 Hz

The power supplied by each generator will be

(
(f
(f

)
) = (1.676 MW/Hz )(61.5 Hz − 59.61 Hz ) = 3.17 MW
) = (1.961 MW/Hz )(60.5 Hz − 59.61 Hz ) = 1.74 MW

PA = s PA f nlA − fsys = (1.5 MW/Hz )(61.0 Hz − 59.61 Hz ) = 2.09 MW

PB = sPB
PC = sPC
(b)

nlB

− f sys

nlC

− fsys

The equation in part (a) can be re-written slightly to express system frequency as a function of load.

(

)

(

)

(

PLOAD = (1.5) 61.0 − f sys + (1.676) 61.5 − f sys + (1.961) 60.5 − fsys

)

PLOAD = 91.5 − 1.5f sys + 103.07 − 1.676fsys + 118.64 − 1.961fsys
5.137 fsys = 313.2 − PLOAD

f sys =

5.137

A MATLAB program that uses this equation to determine the power sharing among the generators as a
function of load is shown below:

% M-file: prob5_9b.m
% M-file to calculate and plot the power sharing among
% three generators as a function of load.
% Define values for this generator
fnlA = 61.0;
fnlB = 61.5;
fnlC = 60.5;
spA = 1.5;
% Slope of Gen
spB = 1.676;
% Slope of Gen
spC = 1.961;
% Slope of Gen
% Load in MW
% Calculate the system frequency
fsys = (313.2 - Pload) ./ 5.137;
% Calculate
PA = spA .*
PB = spB .*
PC = spC .*

the power of each generator
( fnlA - fsys);
( fnlB - fsys);
( fnlC - fsys);

% Plot the power sharing versus load
124

of Gen A
of Gen B
of Gen C
A (MW/Hz)
B (MW/Hz)
C (MW/Hz)

hold on;
plot([0 10],[3 3],'k','LineWidth',1.0);
plot([0 10],[0 0],'k:');
title ('\bfPower Sharing Versus Total Load');
xlabel ('\bfTotal Load (MW)');
ylabel ('\bfGenerator Power (MW)');
legend('Generator A','Generator B','Generator C','Power Limit');
grid on;
hold off;
The resulting plot is shown below:

This plot reveals that there are power sharing problems both for high loads and for low loads. Generator B
is the first to exceed its ratings as load increases. Its rated power is reached at a total load of 6.45 MW.
On the other hand, Generator C gets into trouble as the total load is reduced. When the total load drops to
2.4 MW, the direction of power flow reverses in Generator C.
(c)

The power sharing in (a) is not acceptable, because Generator 2 has exceeded its power limits.

(d) To improve the power sharing among the three generators in (a) without affecting the operating
frequency of the system, the operator should decrease the governor setpoints on Generator B while
simultaneously increasing them in Generators A and C.

5-10.

A paper mill has installed three steam generators (boilers) to provide process steam and also to use some its
waste products as an energy source. Since there is extra capacity, the mill has installed three 5-MW
turbine generators to take advantage of the situation. Each generator is a 4160-V 6250-kVA 0.85-PFlagging two-pole Y-connected synchronous generator with a synchronous reactance of 0.75 Ω and an
armature resistance of 0.04 Ω. Generators 1 and 2 have a characteristic power-frequency slope sP of 2.5
MW/Hz, and generators 2 and 3 have a slope of 3 MW/Hz.
(a) If the no-load frequency of each of the three generators is adjusted to 61 Hz, how much power will the
three machines be supplying when actual system frequency is 60 Hz?

125

(b) What is the maximum power the three generators can supply in this condition without the ratings of one
of them being exceeded? At what frequency does this limit occur? How much power does each
generator supply at that point?
(c) What would have to be done to get all three generators to supply their rated real and reactive powers at
an overall operating frequency of 60 Hz?
(d) What would the internal generated voltages of the three generators be under this condition?
SOLUTION
(a) If the system frequency is 60 Hz and the no-load frequencies of the generators are 61 Hz, then the
power supplied by the generators will be

(
(f
(f

)
) = (2.5 MW/Hz )(61 Hz − 60 Hz ) = 2.5 MW
) = (3.0 MW/Hz )(61 Hz − 60 Hz ) = 3.0 MW

P1 = sP1 f nl1 − fsys = ( 2.5 MW/Hz )(61 Hz − 60 Hz ) = 2.5 MW
P2 = s P 2

nl2

− fsys

P3 = s P 3

nl3

− fsys

Therefore the total power supplied by the generators is 8 MW.
(b) The maximum power supplied by any one generator is (6250 kVA)(0.85) = 5.31 MW. Generator 3
will be the first machine to reach that limit. Generator 3 will supply this power at a frequency of

(

5.31 MW = ( 3.0 MW/Hz ) 61 Hz − f sys

)

f sys = 59.23 Hz
At this point the power supplied by Generators 1 and 2 is

(

)

P1 = P2 = sP1 f nl1 − f sys = ( 2.5 MW/Hz )( 61 Hz − 59.23 Hz ) = 4.425 MW
The total power supplied by the generators at this condition is 14.16 MW.
(c) To get each of the generators to supply 5.31 MW at 60 Hz, the no-load frequencies of Generator 1
and Generator 2 would have to be adjusted to 62.12 Hz, and the no-load frequency of Generator 3 would
have to be adjusted to 61.77 Hz. The field currents of the three generators must then be adjusted to get
them supplying a power factor of 0.85 lagging. At that point, each generator will be supplying its rated
real and reactive power.
(d) Under the conditions of part (c), which are the rated conditions of the generators, the internal
generated voltage would be given by
E A = Vφ + RAI A + jX S I A

The phase voltage of the generators is 4160 V /
their rated current is
IA = IL =

3 = 2402 V, and since the generators are Y-connected,

S
6250 kVA
=
= 867 A
3 VT
3 ( 4160 V )

The power factor is 0.85 lagging, so I A = 867∠ − 31.8° A . Therefore,

E A = Vφ + RAI A + jX S I A
E A = 2402∠0° + ( 0.04 Ω )(867 ∠ − 31.8° A ) + j ( 0.75 Ω )(867∠ − 31.8° A )

126

E A = 2825∠10.9° V
Problems 5-11 to 5-21 refer to a two-pole Y-connected synchronous generator rated at 470 kVA, 480 V, 60 Hz,
and 0.85 PF lagging. Its armature resistance RA is 0.016 Ω. The core losses of this generator at rated conditions
are 7 kW, and the friction and windage losses are 8 kW. The open-circuit and short-circuit characteristics are
shown in Figure P5-2.

127

Note:

5-11.

An electronic version of the saturated open circuit characteristic can be found
in file p52_occ.dat, and an electronic version of the air-gap characteristic
can be found in file p52_ag_occ.dat. These files can be used with
MATLAB programs. Column 1 contains field current in amps, and column 2
contains open-circuit terminal voltage in volts. An electronic version of the
short circuit characteristic can be found in file p52_scc.dat. Column 1
contains field current in amps, and column 2 contains short-circuit terminal
current in amps.

(a) What is the saturated synchronous reactance of this generator at the rated conditions? (b) What is the
unsaturated synchronous reactance of this generator? (c) Plot the saturated synchronous reactance of this
generator as a function of load.
SOLUTION
(a)

The rated armature current for this generator is
IA = IL =

S
470 kVA
=
= 565 A
3 VT
3 ( 480 V )

The field current required to produce this much short-circuit current may be read from the SCC. It is 0.534
A3. The open circuit voltage at 0.532 A is 880 V4, so the open-circuit phase voltage (= E A ) is 880/ 3 =
508 V. The approximate saturated synchronous reactance X S is
3

If you have MATLAB available, you can use the file p52_scc.dat and the interp1 function to look up this
value as shown below. Note that column 1 of p52_scc contains field current, and column 2 contains short-circuit
terminal current.
if = interp1(p52_scc(:,2),p52_scc(:,1),565)
if =

128

XS =

508 V
= 0.899 Ω
565 A

(b) The unsaturated synchronous reactance X Su is the ratio of the air-gap line to the SCC. This is a
straight line, so we can determine its value by comparing the ratio of the air-gap voltage to the short-circuit
current at any given field current. For example, at I F = 0.50 A, the air-gap line voltage is 1040 V, and the
SCC is 532 A.
X Su =

(1040 V ) /
532 A

3

= 1.13 Ω

(c) This task can best be performed with MATLAB. The open-circuit characteristic is available in a file
called p52_occ.dat, and the short-circuit characteristic is available in a file called p52_scc.dat.
Each of these files are organized in two columns, where the first column is field current and the second
column is either open-circuit terminal voltage or short-circuit current. A program to read these files and
calculate and plot X S is shown below.

%
%
%
%

M-file: prob5_11c.m
M-file to calculate and plot the saturated
synchronous reactance of a synchronous
generator.

% Load the open-circuit characteristic. It is in
% two columns, with the first column being field
% current and the second column being terminal
% voltage.
if_occ = p52(:,1);
vt_occ = p52(:,2);
% Load the short-circuit characteristic. It is in
% two columns, with the first column being field
% current and the second column being line current
% (= armature current)
if_scc = p52(:,1);
ia_scc = p52(:,2);
% Calculate Xs
if = 0.001:0.02:1;
vt = interp1(if_occ,vt_occ,If);
ia = interp1(if_scc,ia_scc,If);
Xs = (vt ./ sqrt(3)) ./ ia;

% Current steps
% Terminal voltage
% Current

0.534
4

If you have MATLAB available, you can use the file p52_occ.dat and the interp1 function to look up this
value as shown below. Note that column 1 of p52_occ contains field current, and column 2 contains open-circuit
terminal voltage.
vt = interp1(p52_occ(:,1),p52_occ(:,2),0.534)
vt =
880.400

129

% Plot the synchronous reactance
figure(1)
plot(If,Xs,'LineWidth',2.0);
title ('\bfSaturated Synchronous Reactance \itX_{s} \rm');
xlabel ('\bfField Current (A)');
ylabel ('\bf\itX_{s} \rm\bf(\Omega)');
grid on;
The resulting plot is:

5-12.

(a) What are the rated current and internal generated voltage of this generator? (b) What field current
does this generator require to operate at the rated voltage, current, and power factor?
SOLUTION
(a)

The rated line and armature current for this generator is
IA = IL =

S
470 kVA
=
= 565 A
3 VT
3 ( 480 V )

The power factor is 0.85 lagging, so I A = 565.3∠ − 31.8° A . The rated phase voltage is Vφ = 480 V /

3

= 277 V. The saturated synchronous reactance at rated conditions was found to be 0.450 Ω in the previous
problem. Therefore, the internal generated voltage is
E A = Vφ + RAI A + jX S I A

E A = 277∠0° + ( 0.016 Ω )(565.3∠ − 31.8° A ) + j ( 0.899 Ω)( 565.3∠ − 31.8° A )
E A = 509 ∠30.5° V
(b) This internal generated voltage corresponds to a no-load terminal voltage of
From the open-circuit characteristic, the required field current would be 0.535 A.

5-13.

What is the voltage regulation of this generator at the rated current and power factor?
SOLUTION The voltage regulation is
130

3 (509 ) = 881 V.

VR =

5-14.

VT ,nl − VT ,fl
881 − 480
× 100% =
× 100% = 83.5%
480
VT ,fl

If this generator is operating at the rated conditions and the load is suddenly removed, what will the
terminal voltage be?
SOLUTION From the above calculations, VT will be 881 V.

5-15.

What are the electrical losses in this generator at rated conditions?
SOLUTION The electrical losses are
PCU = 3 I A2 RA = 3 ( 565 A ) ( 0.016 Ω ) = 15.3 kW
2

5-16.

If this machine is operating at rated conditions, what input torque must be applied to the shaft of this
generator? Express your answer both in newton-meters and in pound-feet.
SOLUTION To get the applied torque, we must know the input power. The input power to this generator is
equal to the output power plus losses. The rated output power and the losses are
POUT = ( 470 kVA )( 0.85) = 400 kW
PCU = 3 I A2 RA = 3 ( 565 A ) ( 0.016 Ω ) = 15.3 kW
2

PF&W = 8 kW
Pcore = 7 kW
Pstray = (assumed 0)
PIN = POUT + PCU + PF&W + Pcore + Pstray = 430.3 kW

Therefore, the applied torque is

τ APP =

or

5-17.

τ APP =

PIN

ωm

=

430.3 kW
(1800 r/min ) 2π rad
1r

1 min
60 s

= 2280 N ⋅ m

7.04 P 7.04 (430.3 kW )
=
= 1680 lb ⋅ ft
1800 r/min
nm

What is the torque angle δ of this generator at rated conditions?
SOLUTION From the calculations in Problem 5-12, δ = 30.5°.

5-18.

Assume that the generator field current is adjusted to supply 480 V under rated conditions. What is the
static stability limit of this generator? (Note: You may ignore R A to make this calculation easier.) How
close is the full-load condition of this generator to the static stability limit?
SOLUTION At rated conditions, E A = 509 ∠30.5° V . Therefore, the static stability limit is
PMAX =

3 Vφ E A
XS

=

3 ( 277 V )(509 V )
= 471 kW
0.899 Ω

The full-load rated power of this generator is reasonably close to the static stability limit.
generators would have more margin than this.
131

Normal

5-19.

Assume that the generator field current is adjusted to supply 480 V under rated conditions. Plot the power
supplied by the generator as a function of the torque angle δ. (Note: You may ignore R A to make this
calculation easier.)
SOLUTION We will again ignore R A to make this calculation easier. The power supplied by the generator is
PG =

3 Vφ E A
XS

sin δ =

3 ( 277 V )(509 V )
sin δ = ( 471 kW ) sin δ
0.899 Ω

The power supplied as a function of the torque angle δ may be plotted using a simple MATLAB program:

%
%
%
%

M-file: prob5_19.m
M-file to plot the power output of a
synchronous generator as a function of
the torque angle.

% Calculate Xs
delta = (0:1:90);
% Torque angle (deg)
Pout = 561 .* sin(delta * pi/180); % Pout
% Plot the output power
figure(1)
plot(delta,Pout,'LineWidth',2.0);
title ('\bfOutput power vs torque angle \delta');
xlabel ('\bfTorque angle \delta (deg)');
ylabel ('\bf\itP_{OUT} \rm\bf(kW)');
grid on;
The resulting plot is:

5-20.

Assume that the generator’s field current is adjusted so that the generator supplies rated voltage at the rated
load current and power factor. If the field current and the magnitude of the load current are held constant,
how will the terminal voltage change as the load power factor varies from 0.85 PF lagging to 0.85 PF

132

leading? Make a plot of the terminal voltage versus the impedance angle of the load being supplied by this
generator.
SOLUTION If the field current is held constant, then the magnitude of E A will be constant, although its
angle δ will vary. Also, the magnitude of the armature current is constant. Since we also know R A , X S ,
and the current angle θ, we know enough to find the phase voltage Vφ , and therefore the terminal voltage

VT . At lagging power factors, Vφ can be found from the following relationships:
EA
θ
δ
θ

Vφ
I

θ

jX I

S A

R I

A A

A

By the Pythagorean Theorem,

E A = (Vφ + R A I A cos θ + X S I A sin θ ) + ( X S I A cosθ − R A I S sin θ )
2

2

2

Vφ = E A − ( X S I A cos θ − RA I S sin θ ) − R A I A cos θ − X S I A sin θ
2

2

At unity power factor, Vφ can be found from the following relationships:

EA
jX I

S A

δ

I

Vφ R AI A

A

By the Pythagorean Theorem,

E A = Vφ + ( X S I A )
2

2

2

Vφ = E A − ( X S I A )
2

2

At leading power factors, Vφ can be found from the following relationships:

EA

jX I

S A

I

A

δ

θ

R I

A A

θ

Vφ
By the Pythagorean Theorem,
133

θ

E A = (Vφ + R A I A cos θ − X S I A sin θ ) + ( X S I A cos θ + R A I S sin θ )
2

2

2

Vφ = E A − ( X S I A cos θ + R A I S sin θ ) − R A I A cos θ + X S I A sin θ
2

2

If we examine these three cases, we can see that the only difference among them is the sign of the term
sin θ . If θ is taken as positive for lagging power factors and negative for leading power factors (in other
words, if θ is the impedance angle), then all three cases can be represented by the single equation:

Vφ = E A − ( X S I A cos θ − RA I S sin θ ) − R A I A cos θ − X S I A sin θ
2

2

A MATLAB program that calculates terminal voltage as function of impedance angle is shown below:

%
%
%
%
%

M-file: prob5_20.m
M-file to calculate and plot the terminal voltage
of a synchronous generator as a function of impedance
angle as PF changes from 0.85 lagging to 0.85

% Define values for this generator
EA = 509;
% Internal gen voltage
I = 361;
% Current (A)
R = 0.04;
% R (ohms)
X = 0.695;
% XS (ohms)
% Calculate impedance angle theta in degrees
theta = -31.8:0.318:31.8;
th = theta * pi/180;
% Calculate the phase voltage and terminal voltage
VP = sqrt( EA^2 - (X.*I.*cos(th) - R.*I.*sin(th)).^2 ) ...
- R.*I.*cos(th) - X.*I.*sin(th);
VT = VP .* sqrt(3);
% Plot the terminal voltage versus power factor
figure(1);
plot(theta,abs(VT),'b-','LineWidth',2.0);
title ('\bfTerminal Voltage Versus Impedance Angle');
xlabel ('\bfImpedance Angle (deg)');
ylabel ('\bfTerminal Voltage (V)');
%axis([0 260 300 540]);
grid on;
hold off;
The resulting plot of terminal voltage versus impedance angle (with field and armature currents held
constant) is shown below:

134

5-21.

Assume that the generator is connected to a 480-V infinite bus, and that its field current has been adjusted
so that it is supplying rated power and power factor to the bus. You may ignore the armature resistance
R A when answering the following questions.
(a) What would happen to the real and reactive power supplied by this generator if the field flux (and
therefore E A ) is reduced by 5%.
(b) Plot the real power supplied by this generator as a function of the flux φ as the flux is varied from 75%
to 100% of the flux at rated conditions.
(c) Plot the reactive power supplied by this generator as a function of the flux φ as the flux is varied from
75% to 100% of the flux at rated conditions.
(d) Plot the line current supplied by this generator as a function of the flux φ as the flux is varied from 75%
to 100% of the flux at rated conditions.
SOLUTION
(a) If the field flux in increase by 5%, nothing would happen to the real power. The reactive power
supplied would increase as shown below.
V
T

E A1 E A2

Q

G

Q2

Q1

I A1

Qsys

I A2

The reactive power
135

Vφ
Q ∝ I sin θ
A

jX I

S A

(b) If armature resistance is ignored, the power supplied to the bus will not change as flux is varied.
Therefore, the plot of real power versus flux is

(c)

If armature resistance is ignored, the internal generated voltage E A will increase as flux increases,

but the quantity E Asin δ will remain constant. Therefore, the voltage for any flux can be found from the
expression

φ
E A = 
 φr


 E Ar


and the angle δ for any E A can be found from the expression

 E Ar

sin δ r 
 EA


δ = sin −1 

where φ is the flux in the machine, φ r is the flux at rated conditions, E Ar is the magnitude of the internal
generated voltage at rated conditions, and δ r is the angle of the internal generated voltage at rated
conditions. From this information, we can calculate I A for any given load from equation

IA =

E A − Vφ
jX S

and the resulting reactive power from the equation

Q = 3Vφ I A sin θ
where θ is the impedance angle, which is the negative of the current angle. Ignoring R A , the internal
generated voltage at rated conditions is

E A = Vφ + jX S I A
E A = 277∠0° + j ( 0.899 Ω )(565.3∠ − 31.8° A )
136

E A = 695∠38.4° V
so E Ar = 461 V and δ r = 27.5° . A MATLAB program that calculates the reactive power supplied
voltage as a function of flux is shown below:

%
%
%
%

M-file: prob5_21c.m
M-file to calculate and plot the reactive power
supplied to an infinite bus as flux is varied from
75% to 100% of the flux at rated conditions.

% Define values for this generator
flux_ratio = 0.90:0.01:1.00; % Flux ratio
Ear = 695;
% Ea at full flux
dr = 38.4 * pi/180;
% Torque ang at full flux
Vp = 277;
% Phase voltage
Xs = 0.899;
% Xs (ohms)
% Calculate Ea for each flux
Ea = flux_ratio * Ear;
% Calculate delta for each flux
d = asin( Ear ./ Ea .* sin(dr));
% Calculate Ia for each flux
Ea = Ea .* ( cos(d) + j.*sin(d) );
Ia = ( Ea - Vp ) ./ (j*Xs);
% Calculate reactive power for each flux
theta = -atan2(imag(Ia),real(Ia));
Q = 3 .* Vp .* abs(Ia) .* sin(theta);
% Plot the power supplied versus flux
figure(1);
plot(flux_ratio,Q/1000,'b-','LineWidth',2.0);
title ('\bfReactive power versus flux');
xlabel ('\bfFlux (% of full-load flux)');
ylabel ('\bf\itQ\rm\bf (kVAR)');
grid on;
hold off;

137

When this program is executed, the plot of reactive power versus flux is

(d) The program in part (c) of this program calculated I A as a function of flux. A MATLAB program
that plots the magnitude of this current as a function of flux is shown below:

%
%
%
%

M-file: prob5_21d.m
M-file to calculate and plot the armature current
supplied to an infinite bus as flux is varied from
75% to 100% of the flux at rated conditions.

% Define values for this generator
flux_ratio = 0.75:0.01:1.00; % Flux ratio
Ear = 695;
% Ea at full flux
dr = 38.4 * pi/180;
% Torque ang at full flux
Vp = 277;
% Phase voltage
Xs = 0.899;
% Xs (ohms)
% Calculate Ea for each flux
Ea = flux_ratio * Ear;
% Calculate delta for each flux
d = asin( Ear ./ Ea .* sin(dr));
% Calculate Ia for each flux
Ea = Ea .* ( cos(d) + j.*sin(d) );
Ia = ( Ea - Vp ) ./ (j*Xs);
% Plot the armature current versus flux
figure(1);
plot(flux_ratio,abs(Ia),'b-','LineWidth',2.0);
title ('\bfArmature current versus flux');
xlabel ('\bfFlux (% of full-load flux)');
ylabel ('\bf\itI_{A}\rm\bf (A)');
grid on;
138

hold off;
When this program is executed, the plot of armature current versus flux is

5-22.

A 100-MVA 12.5-kV 0.85-PF-lagging 50-Hz two-pole Y-connected synchronous generator has a per-unit
synchronous reactance of 1.1 and a per-unit armature resistance of 0.012.
(a) What are its synchronous reactance and armature resistance in ohms?
(b) What is the magnitude of the internal generated voltage E A at the rated conditions? What is its torque
angle δ at these conditions?
(c) Ignoring losses in this generator, what torque must be applied to its shaft by the prime mover at full
SOLUTION The base phase voltage of this generator is Vφ ,base = 12,500 / 3 = 7217 V . Therefore, the base
impedance of the generator is
Z base =
(a)

3 Vφ ,base 2
S base

3 (7217 V )
=
= 1.56 Ω
100,000,000 VA
2

The generator impedance in ohms are:
R A = (0.012 )(1.56 Ω ) = 0.0187 Ω
X S = (1.1)(1.56 Ω ) = 1.716 Ω

(b)

The rated armature current is
IA = IL =

S
100 MVA
=
= 4619 A
3 VT
3 (12.5 kV )

The power factor is 0.8 lagging, so I A = 4619∠ − 36.87° A . Therefore, the internal generated voltage is

E A = Vφ + RAI A + jX S I A
139

E A = 7217∠0° + (0.0187 Ω )( 4619∠ − 36.87° A ) + j (1.716 Ω )( 4619∠ − 36.87° A )
E A = 13,590∠27.6° V
Therefore, the magnitude of the internal generated voltage E A = 13,590 V, and the torque angle δ = 23°.
(c)

Ignoring losses, the input power would equal the output power. Since
POUT = (0.85)(100 MVA ) = 85 MW

and
120 f e 120 (50 Hz )
=
= 3000 r/min
P
2

nsync =

the applied torque would be

τ app = τ ind =
5-23.

85,000,000 W
= 270, 000 N ⋅ m
(3000 r/min )(2π rad/r )(1 min/60 s)

A three-phase Y-connected synchronous generator is rated 120 MVA, 13.2 kV, 0.8 PF lagging, and 60 Hz.
Its synchronous reactance is 0.9 Ω, and its resistance may be ignored.
(a) What is its voltage regulation?
(b) What would the voltage and apparent power rating of this generator be if it were operated at 50 Hz
with the same armature and field losses as it had at 60 Hz?
(c) What would the voltage regulation of the generator be at 50 Hz?
SOLUTION
(a)

The rated armature current is
IA = IL =

S
120 MVA
=
= 5249 A
3 VT
3 (13.2 kV )

The power factor is 0.8 lagging, so I A = 5249∠ − 36.87° A . The phase voltage is 13.2 kV /
V. Therefore, the internal generated voltage is

3 = 7621

E A = Vφ + RAI A + jX S I A
E A = 7621∠0° + j ( 0.9 Ω)(5249 ∠ − 36.87° A )

E A = 11,120∠19.9° V
The resulting voltage regulation is
VR =

11,120 − 7621
× 100% = 45.9%
7621

(b) If the generator is to be operated at 50 Hz with the same armature and field losses as at 60 Hz (so
that the windings do not overheat), then its armature and field currents must not change. Since the voltage
of the generator is directly proportional to the speed of the generator, the voltage rating (and hence the
apparent power rating) of the generator will be reduced by a factor of 5/6.
VT ,rated =
S rated =

5
(13.2 kV) = 11.0 kV
6

5
(120 MVA ) = 100 MVA
6
140

Also, the synchronous reactance will be reduced by a factor of 5/6.
XS =

(c)

5
(0.9 Ω) = 0.75 Ω
6

At 50 Hz rated conditions, the armature current would be
IA = IL =

S
100 MVA
=
= 5247 A
3 VT
3 (11.0 kV )

The power factor is 0.8 lagging, so I A = 5247∠ − 36.87° A . The phase voltage is 11.0 kV /
V. Therefore, the internal generated voltage is

3 = 6351

E A = Vφ + RAI A + jX S I A

E A = 6351∠0° + j ( 0.75 Ω )(5247 ∠ − 36.87° A )
E A = 9264∠19.9° V
The resulting voltage regulation is
VR =

9264 − 6351
× 100% = 45.9%
6351

Because voltage, apparent power, and synchronous reactance all scale linearly with frequency, the voltage
regulation at 50 Hz is the same as that at 60 Hz. Note that this is not quite true, if the armature resistance
R A is included, since R A does not scale with frequency in the same fashion as the other terms.

5-24.

Two identical 600-kVA 480-V synchronous generators are connected in parallel to supply a load. The
prime movers of the two generators happen to have different speed droop characteristics. When the field
currents of the two generators are equal, one delivers 400 A at 0.9 PF lagging, while the other delivers 300
A at 0.72 PF lagging.
(a) What are the real power and the reactive power supplied by each generator to the load?
(b) What is the overall power factor of the load?
(c) In what direction must the field current on each generator be adjusted in order for them to operate at the
same power factor?
SOLUTION
(a)

The real and reactive powers are
P1 = 3 VT I L cos θ = 3 ( 480 V )( 400 A )( 0.9 ) = 299 kW
Q1 = 3 VT I L sin θ = 3 ( 480 V )( 400 A ) sin cos −1 ( 0.9 ) = 145 kVAR
P2 = 3 VT I L cos θ = 3 ( 480 V )( 200 A )( 0.72 ) = 120 kW
Q2 = 3 VT I L sin θ = 3 ( 480 V )( 200 A ) sin cos −1 ( 0.72 ) = 115 kVAR

(b)

The overall power factor can be found from the total real and reactive power supplied to the load.
PTOT = P1 + P2 = 299 kW + 120 kW = 419 kW
QTOT = Q1 + Q2 = 145 kVAR + 115 kVAR = 260 kVAR

The overall power factor is
141

QTOT
= 0.850 lagging
PTOT

PF = cos tan −1

(c) The field current of generator 1 should be increased, and the field current of generator 2 should be
simultaneously decreased.

5-25.

A generating station for a power system consists of four 120-MVA 15-kV 0.85-PF-lagging synchronous
generators with identical speed droop characteristics operating in parallel. The governors on the
generators’ prime movers are adjusted to produce a 3-Hz drop from no load to full load. Three of these
generators are each supplying a steady 75 MW at a frequency of 60 Hz, while the fourth generator (called
the swing generator) handles all incremental load changes on the system while maintaining the system's
frequency at 60 Hz.
(a) At a given instant, the total system loads are 260 MW at a frequency of 60 Hz. What are the no-load
frequencies of each of the system’s generators?
(b) If the system load rises to 290 MW and the generator’s governor set points do not change, what will the
new system frequency be?
(c) To what frequency must the no-load frequency of the swing generator be adjusted in order to restore the
system frequency to 60 Hz?
(d) If the system is operating at the conditions described in part (c), what would happen if the swing
generator were tripped off the line (disconnected from the power line)?
SOLUTION
(a) The full-load power of these generators is (120 MVA )(0.85) = 102 MW and the droop from noload to full-load is 3 Hz. Therefore, the slope of the power-frequency curve for these four generators is
sP =

102 MW
= 34 MW/Hz
3 Hz

If generators 1, 2, and 3 are supplying 75 MW each, then generator 4 must be supplying 35 MW. The noload frequency of the first three generators is

(

P1 = sP1 f nl1 − f sys

)

75 MW = ( 34 MW/Hz ) ( f nl1 − 60 Hz )

f nl1 = 62.21 Hz
The no-load frequency of the fourth generator is

(

P4 = sP 4 f nl4 − fsys

)

35 MW = ( 34 MW/Hz ) ( f nl1 − 60 Hz )

f nl1 = 61.03 Hz
(b)

The setpoints of generators 1, 2, 3, and 4 do not change, so the new system frequency will be

(

)

(
) (
) (
)
) + (34) (62.21 − f ) + (34) (62.21 − f ) + (34) (61.03 − f )

PLOAD = sP1 f nl1 − fsys + sP 2 f nl2 − f sys + sP 3 f nl3 − fsys + sP 4 f nl4 − f sys

(

290 MW = ( 34 ) 62.21 − fsys

sys

8.529 = 247.66 − 4 f sys
142

sys

sys

f sys = 59.78 Hz
(c) The governor setpoints of the swing generator must be increased until the system frequency rises back
to 60 Hz. At 60 Hz, the other three generators will be supplying 75 MW each, so the swing generator must
supply 290 MW – 3(75 MW) = 65 MW at 60 Hz. Therefore, the swing generator’s setpoints must be set
to

(

P4 = sP 4 f nl4 − fsys

)

65 MW = ( 34 MW/Hz ) ( f nl1 − 60 Hz )

f nl1 = 61.91 Hz
(d) If the swing generator trips off the line, the other three generators would have to supply all 290 MW
of the load. Therefore, the system frequency will become

(

)

(
) (
)
) + (34) (62.21 − f ) + (34 ) (62.21 − f )

PLOAD = sP1 f nl1 − fsys + sP 2 f nl2 − f sys + sP 3 f nl3 − f sys

(

290 MW = ( 34 ) 62.21 − f sys

sys

sys

8.529 = 186.63 − 3 f sys

f sys = 59.37 Hz

Each generator will supply 96.7 MW to the loads.

5-26.

Suppose that you were an engineer planning a new electric co-generation facility for a plant with excess
process steam. You have a choice of either two 10 MW turbine-generators or a single 20 MW turbine
generator. What would be the advantages and disadvantages of each choice?
SOLUTION A single 20 MW generator will probably be cheaper and more efficient than two 10 MW
generators, but if the 20 MW generator goes down all 20 MW of generation would be lost at once. If two
10 MW generators are chosen, one of them could go down for maintenance and some power could still be
generated.

5-27.

A 25-MVA three-phase 13.8-kV two-pole 60-Hz synchronous generator was tested by the open-circuit test,
and its air-gap voltage was extrapolated with the following results:

Open-circuit test
Field current, A
Line voltage, kV
Extrapolated air-gap voltage, kV

320
13.0
15.4

365
13.8
17.5

380
14.1
18.3

475
15.2
22.8

570
16.0
27.4

475
1550

570
1885

The short-circuit test was then performed with the following results:

Short-circuit test
Field current, A
Armature current, A

320
1040

365
1190

380
1240

The armature resistance is 0.24 Ω per phase.
(a) Find the unsaturated synchronous reactance of this generator in ohms per phase and in per-unit.
(b) Find the approximate saturated synchronous reactance X S at a field current of 380 A. Express the
answer both in ohms per phase and in per-unit.
143

(c) Find the approximate saturated synchronous reactance at a field current of 475 A. Express the answer
both in ohms per phase and in per-unit.
(d) Find the short-circuit ratio for this generator.
SOLUTION
(a) The unsaturated synchronous reactance of this generator is the same at any field current, so we will
look at it at a field current of 380 A. The extrapolated air-gap voltage at this point is 18.3 kV, and the
short-circuit current is 1240 A.
Since this generator is Y-connected, the phase voltage is
Vφ = 18.3 kV/ 3 = 10,566 V and the armature current is I A = 1240 A . Therefore, the unsaturated
synchronous reactance is
X Su =

10,566 V
= 8.52 Ω
1240 A

The base impedance of this generator is
Z base =

3 Vφ ,base 2
S base

3 ( 7967 V )
= 7.62 Ω
25,000,000 VA
2

=

Therefore, the per-unit unsaturated synchronous reactance is
X Su ,pu =

8.52 Ω
= 1.12
7.62 Ω

(b) The saturated synchronous reactance at a field current of 380 A can be found from the OCC and the
SCC. The OCC voltage at I F = 380 A is 14.1 kV, and the short-circuit current is 1240 A. Since this
generator is Y-connected, the corresponding phase voltage is Vφ = 14.1 kV/ 3 = 8141 V and the armature
current is I A = 1240 A . Therefore, the saturated synchronous reactance is
X Su =

8141 V
= 6.57 Ω
1240 A

and the per-unit unsaturated synchronous reactance is
X Su ,pu =

6.57 Ω
= 0.862
7.62 Ω

(c) The saturated synchronous reactance at a field current of 475 A can be found from the OCC and the
SCC. The OCC voltage at I F = 475 A is 15.2 kV, and the short-circuit current is 1550 A. Since this
generator is Y-connected, the corresponding phase voltage is Vφ = 15.2 kV/ 3 = 8776 V and the armature
current is I A = 1550 A . Therefore, the saturated synchronous reactance is
X Su =

8776 V
= 5.66 Ω
1550 A

and the per-unit unsaturated synchronous reactance is
X Su ,pu =

5.66 Ω
= 0.743
7.62 Ω

(d) The rated voltage of this generator is 13.8 kV, which requires a field current of 365 A. The rated line
and armature current of this generator is

144

IL =

25 MVA
= 1046 A
3 (13.8 kV )

The field current required to produce a short-circuit current of 10465 A is about 320 A. Therefore, the
short-circuit ratio of this generator is
SCR =

5-28.

365 A
= 1.14
320 A

A 20-MVA 12.2-kV 0.8-PF-lagging Y-connected synchronous generator has a negligible armature
resistance and a synchronous reactance of 1.1 per-unit. The generator is connected in parallel with a 60-Hz
12.2-kV infinite bus that is capable of supplying or consuming any amount of real or reactive power with
no change in frequency or terminal voltage.
(a) What is the synchronous reactance of the generator in ohms?
(b) What is the internal generated voltage E A of this generator under rated conditions?
(c) What is the armature current I A in this machine at rated conditions?
(d) Suppose that the generator is initially operating at rated conditions. If the internal generated voltage
E A is decreased by 5 percent, what will the new armature current I A be?
(e) Repeat part (d) for 10, 15, 20, and 25 percent reductions in E A .
(f) Plot the magnitude of the armature current I A as a function of E A . (You may wish to use MATLAB
to create this plot.)
SOLUTION
(a) The rated phase voltage of this generator is 12.2 kV /
generator is
Z base =

3 Vφ ,base 2
S base

3 = 7044 V. The base impedance of this

3 (7044 V )
= 7.44 Ω
20,000,000 VA
2

=

Therefore,
R A ≈ 0 Ω (negligible)
X S = (1.1)( 7.44 Ω ) = 8.18 Ω
(b)

The rated armature current is
IA = IL =

S
20 MVA
=
= 946 A
3 VT
3 (12.2 kV )

The power factor is 0.8 lagging, so I A = 946∠ − 36.87° A . Therefore, the internal generated voltage is

E A = Vφ + RAI A + jX S I A
E A = 7044 ∠0° + j (8.18 Ω )( 946∠ − 36.87° A )

E A = 13,230∠27.9° V
(c)

From the above calculations, I A = 946∠ − 36.87° A .

145

(d)

If E A is decreased by 5%, the armature current will change as shown below. Note that the infinite

bus will keep Vφ and ω m constant. Also, since the prime mover hasn’t changed, the power supplied by the
generator will be constant.

E A2 E A1

I A2
I A1

Vφ

jX I

S A

Q ∝ I sin θ
A

P=

3Vφ E A
XS

sin δ = constant , so E A1 sin δ1 = E A2 sin δ 2

With a 5% decrease, E A2 = 12,570 V , and

δ 2 = sin −1

E A1
13,230 V
sin δ 2 = sin −1
sin 27.9° = 29.5°
E A2
12,570 V

Therefore, the new armature current is

IA =
(e)

E A2 − Vφ
jX S

=

12,570∠29.5° − 7044∠0°
= 894∠ − 32.2° A
j8.18

Repeating part (d):

With a 10% decrease, E A2 = 11,907 V , and

δ 2 = sin −1

E A1
13,230 V
sin δ 2 = sin −1
sin 27.9° = 31.3°
E A2
11,907 V

Therefore, the new armature current is

IA =

E A2 − Vφ
jX S

=

11,907∠31.3° − 7044∠0°
= 848∠ − 26.8° A
j8.18

With a 15% decrease, E A2 = 11,246 V , and

δ 2 = sin −1

E A1
13,230 V
sin δ 2 = sin −1
sin 27.9° = 33.4°
E A2
11,246 V

Therefore, the new armature current is

IA =

E A2 − Vφ
jX S

=

11,246∠33.4° − 7044∠0°
= 809 ∠ − 20.7° A
j8.18

With a 20% decrease, E A2 = 10,584 V , and

δ 2 = sin −1

E A1
13,230 V
sin δ 2 = sin −1
sin 27.9° = 35.8°
E A2
10,584 V

Therefore, the new armature current is
146

IA =

E A2 − Vφ
jX S

=

10,584∠35.8° − 7044 ∠0°
= 780∠ − 14.0° A
j8.18

With a 25% decrease, E A2 = 9,923 V , and

δ 2 = sin −1

E A1
13,230 V
sin δ 2 = sin −1
sin 27.9° = 38.6°
E A2
9,923 V

Therefore, the new armature current is

IA =

E A2 − Vφ
jX S

=

9,923∠38.6° − 7044∠0°
= 762 ∠ − 6.6° A
j8.18

(f) A MATLAB program to plot the magnitude of the armature current I A as a function of E A is shown
below.

% M-file: prob5_28f.m
% M-file to calculate and plot the armature current
% supplied to an infinite bus as Ea is varied.
% Define values for this generator
Ea = (0.65:0.01:1.00)*13230; % Ea
Vp = 7044;
% Phase voltage
d1 = 27.9*pi/180;
% torque angle at full Ea
Xs = 8.18;
% Xs (ohms)
% Calculate delta for each Ea
d = asin( 13230 ./ Ea .* sin(d1));
% Calculate Ia for each flux
Ea = Ea .* ( cos(d) + j.*sin(d) );
Ia = ( Ea - Vp ) ./ (j*Xs);
% Plot the armature current versus Ea
figure(1);
plot(abs(Ea)/1000,abs(Ia),'b-','LineWidth',2.0);
title ('\bfArmature current versus \itE_{A}\rm');
xlabel ('\bf\itE_{A}\rm\bf (kV)');
ylabel ('\bf\itI_{A}\rm\bf (A)');
grid on;
hold off;

147

The resulting plot is shown below:

148

Chapter 6: Synchronous Motors
6-1.

A 480-V, 60 Hz, four-pole synchronous motor draws 50 A from the line at unity power factor and full
load. Assuming that the motor is lossless, answer the following questions:
(a) What is the output torque of this motor? Express the answer both in newton-meters and in pound-feet.
(b) What must be done to change the power factor to 0.8 leading? Explain your answer, using phasor
diagrams.
(c) What will the magnitude of the line current be if the power factor is adjusted to 0.8 leading?
SOLUTION
(a) If this motor is assumed lossless, then the input power is equal to the output power. The input power
to this motor is
PIN = 3VT I Lcos θ = 3 ( 480 V )( 50 A )(1.0) = 41.6 kW

The output torque would be

POUT

ωm

=

41.6 kW
1 min
(1800 r/min )
60 s

1r

= 221 N ⋅ m

In English units,

7.04 POUT (7.04 )( 41.6 kW )
=
= 163 lb ⋅ ft
nm
(1800 r/min )

(b) To change the motor’s power factor to 0.8 leading, its field current must be increased. Since the
power supplied to the load is independent of the field current level, an increase in field current increases
E A while keeping the distance E A sin δ constant. This increase in E A changes the angle of the current

I A , eventually causing it to reach a power factor of 0.8 leading.

}

∝P

I A2
Vφ

I A1

jX I

S A

E A1

Q ∝ I sin θ

}

∝P

E A2

A

(c)

The magnitude of the line current will be
IL =

6-2.

P
=
3 VT PF

41.6 kW
= 62.5 A
3 ( 480 V )( 0.8)

A 480-V, 60 Hz, 400-hp 0.8-PF-leading six-pole ∆-connected synchronous motor has a synchronous
reactance of 1.1 Ω and negligible armature resistance. Ignore its friction, windage, and core losses for the
purposes of this problem.

149

(a) If this motor is initially supplying 400 hp at 0.8 PF lagging, what are the magnitudes and angles of E A
and I A ?
(b) How much torque is this motor producing? What is the torque angle δ? How near is this value to the
maximum possible induced torque of the motor for this field current setting?
(c) If E A is increased by 15 percent, what is the new magnitude of the armature current? What is the
motor’s new power factor?
(d) Calculate and plot the motor’s V-curve for this load condition.
SOLUTION
(a)

If losses are being ignored, the output power is equal to the input power, so the input power will be
PIN = ( 400 hp )(746 W/hp ) = 298.4 kW

This situation is shown in the phasor diagram below:
Vφ
IA

jX I

S A

EA

The line current flow under these circumstances is
IL =

P
298.4 kW
=
= 449 A
3 VT PF
3 ( 480 V )( 0.8)

Because the motor is ∆-connected, the corresponding phase current is I A = 449 / 3 = 259 A . The angle of
the current is − cos−1 ( 0.80) = −36.87° , so I A = 259 ∠ − 36.87° A . The internal generated voltage E A is

E A = Vφ − jX S I A
E A = ( 480∠0° V ) − j (1.1 Ω )( 259∠ − 36.87° A ) = 384 ∠ − 36.4° V
(b) This motor has 6 poles and an electrical frequency of 60 Hz, so its rotation speed is nm = 1200 r/min.
The induced torque is

τ ind =

POUT

ωm

=

298.4 kW
= 2375 N ⋅ m
1 min 2π rad
(1200 r/min )
60 s
1r

The maximum possible induced torque for the motor at this field setting is

τ ind,max =

(c)

3 Vφ E A

ωm X S

=

3 ( 480 V )( 384 V )
= 4000 N ⋅ m
1 min 2π rad
Ω
1200
r/min
1.1
(
)
(
)
60 s
1r

If the magnitude of the internal generated voltage E A is increased by 15%, the new torque angle can

be found from the fact that E A sin δ ∝ P = constant .
E A2 = 1.15 E A1 = 1.15 ( 384 V ) = 441.6 V

150

δ 2 = sin −1

E A1
384 V
sin δ1 = sin −1
sin ( −36.4° ) = −31.1°
E A2
441.6 V

The new armature current is

I A2 =

Vφ − E A2
jX S

=

480∠0° V − 441.6∠ − 31.1° V
= 227∠ − 24.1° A
j1.1 Ω

The magnitude of the armature current is 227 A, and the power factor is cos (-24.1°) = 0.913 lagging.
(d)

A MATLAB program to calculate and plot the motor’s V-curve is shown below:

% M-file: prob6_2d.m
% M-file create a plot of armature current versus Ea
%
for the synchronous motor of Problem 6-2.
% Initialize values
Ea = (1:0.01:1.70)*384;
% Magnitude of Ea volts
Ear = 384;
% Reference Ea
deltar = -36.4 * pi/180;
% Reference torque angle
Xs = 1.1;
% Synchronous reactance
Vp = 480;
% Phase voltage at 0 degrees
Ear = Ear * (cos(deltar) + j * sin(deltar));
% Calculate delta2
delta2 = asin ( abs(Ear) ./ abs(Ea) .* sin(deltar) );
% Calculate the phasor Ea
Ea = Ea .* (cos(delta2) + j .* sin(delta2));
% Calculate Ia
Ia = ( Vp - Ea ) / ( j * Xs);
% Plot the v-curve
figure(1);
plot(abs(Ea),abs(Ia),'b','Linewidth',2.0);
xlabel('\bf\itE_{A}\rm\bf (V)');
ylabel('\bf\itI_{A}\rm\bf (A)');
title ('\bfSynchronous Motor V-Curve');
grid on;

151

The resulting plot is shown below
Synchronous Motor V-Curve
260

250

A

I (A)

240

230

220

210

200
350

400

450

500

550

600

650

700

E (V)
A

6-3.

A 2300-V 1000-hp 0.8-PF leading 60-Hz two-pole Y-connected synchronous motor has a synchronous
reactance of 2.8 Ω and an armature resistance of 0.4 Ω. At 60 Hz, its friction and windage losses are 24
kW, and its core losses are 18 kW. The field circuit has a dc voltage of 200 V, and the maximum I F is 10
A. The open-circuit characteristic of this motor is shown in Figure P6-1. Answer the following questions
about the motor, assuming that it is being supplied by an infinite bus.
(a) How much field current would be required to make this machine operate at unity power factor when
(b) What is the motor’s efficiency at full load and unity power factor?
(c) If the field current were increased by 5 percent, what would the new value of the armature current be?
What would the new power factor be? How much reactive power is being consumed or supplied by the
motor?
(d) What is the maximum torque this machine is theoretically capable of supplying at unity power factor?
At 0.8 PF leading?

Note:

An electronic version of this open circuit characteristic can be found in file
p61_occ.dat, which can be used with MATLAB programs. Column 1
contains field current in amps, and column 2 contains open-circuit terminal
voltage in volts.

152

SOLUTION
(a)

At full load, the input power to the motor is

PIN = POUT + Pmech + Pcore + PCU
We can’t know the copper losses until the armature current is known, so we will find the input power and
armature current ignoring that term, and then correct the input power after we know it.
PIN = (1000 hp )(746 W/hp ) + 24 kW + 18 kW = 788 kW

Therefore, the line and phase current at unity power factor is
IA = IL =

P
788 kW
=
= 198 A
3 VT PF
3 ( 2300 V )(1.0 )

The copper losses due to a current of 198 A are
PCU = 3 I A2 RA = 3(198 A ) ( 0.4 Ω) = 47.0 kW
2

Therefore, a better estimate of the input power at full load is
PIN = (1000 hp )( 746 W/hp ) + 24 kW + 18 kW + 47 kW = 835 kW
and a better estimate of the line and phase current at unity power factor is
153

IA = IL =

P
835 kW
=
= 210 A
3 VT PF
3 ( 2300 V )(1.0 )

The phasor diagram of this motor operating a unity power factor is shown below:

I

Vφ

A

jX I

S A

EA
The phase voltage of this motor is 2300 /

R I

A A

3 = 1328 V. The required internal generated voltage is

E A = Vφ − RAI A − jX S I A
E A = 1328∠0° V − (0.4 Ω )( 210∠0° A ) − j ( 2.8 Ω )( 210∠0° A )

E A = 1376∠ − 25.3° V
This internal generated voltage corresponds to a terminal voltage of

3 (1376) = 2383 V . This voltage

would require a field current of 4.6 A.
(b)

The motor’s efficiency at full load and unity power factor is

η=
(c)

POUT
746 kW
× 100% =
× 100% = 89.3%
PIN
835 kW

To solve this problem, we will temporarily ignore the effects of the armature resistance R A . If R A is

ignored, then E A sin δ is directly proportional to the power supplied by the motor. Since the power
supplied by the motor does not change when I F is changed, this quantity will be a constant.
If the field current is increased by 5%, then the new field current will be 4.83 A, and the new value of
the open-circuit terminal voltage will be 2450 V. The new value of E A will be 2450 V /
Therefore, the new torque angle δ will be

δ 2 = sin −1

3 = 1415 V.

E A1
1376 V
sin δ1 = sin −1
sin ( −25.3°) = −24.6°
E A2
1415 V

Therefore, the new armature current will be

IA =

Vφ − E A
RA + jX S

=

1328∠0° V − 1415∠-25.3° V
= 214.5∠3.5° A
0.4 + j 2.8 Ω

The new current is about the same as before, but the phase angle has become positive. The new power
factor is cos 3.5° = 0.998 leading, and the reactive power supplied by the motor is
Q = 3 VT I L sin θ = 3 ( 2300 V )( 214.5 A ) sin ( 3.5°) = 52.2 kVAR
(d)

The maximum torque possible at unity power factor (ignoring the effects of R A ) is:

τ ind,max =

3 Vφ E A

ωm X S

=

3 (1328 V )(1376 V )
1 min 2π rad
(3600 r/min )
60 s
1r
154

( 2.8 Ω)

= 5193 N ⋅ m

If we are ignoring the resistance of the motor, then the input power would be 788 kW (note that copper
losses are ignored!). At a power factor of 0.8 leading, the current flow will be
IA = IL =

P
=
3 VT PF

788 kW
= 247 A
3 ( 2300 V )(0.8)

so I A = 247∠36.87° A . The internal generated voltage at 0.8 PF leading (ignoring copper losses) is

E A = Vφ − RAI A − jX S I A
E A = 1328∠0° V − j ( 2.8 Ω )( 247∠36.87° A )

E A = 1829∠ − 17.6° V
Therefore, the maximum torque at a power factor of 0.8 leading is

τ ind,max =

6-4.

3 Vφ E A

ωm X S

=

3 (1328 V )(1829 V )
1 min 2π rad
(3600 r/min )
60 s
1r

( 2.8 Ω )

= 6093 N ⋅ m

Plot the V-curves ( I A versus I F ) for the synchronous motor of Problem 6-3 at no-load, half-load, and fullload conditions. (Note that an electronic version of the open-circuit characteristics in Figure P6-1 is
available at the book’s Web site. It may simplify the calculations required by this problem. Also, you may
assume that R A is negligible for this calculation.)
SOLUTION The input power at no-load, half-load and full-load conditions is given below. Note that we are
assuming that R A is negligible in each case.
PIN,nl = 24 kW + 18 kW = 42 kW
PIN,half = (500 hp )( 746 W/hp ) + 24 kW + 18 kW = 373 kW
PIN,full = (1000 hp )( 746 W/hp ) + 24 kW + 18 kW = 788 kW

If the power factor is adjusted to unity, then armature currents will be
I A,nl =

P
42 kW
=
= 10.5 A
3 VT PF
3 ( 2300 V )(1.0 )

I A,fl =

P
=
3 VT PF

I A,fl =

P
788 kW
=
= 198 A
3 VT PF
3 ( 2300 V )(1.0 )

373 kW
= 93.6 A
3 ( 2300 V )(1.0 )

The corresponding internal generated voltages at unity power factor are:
E A = Vφ − jX S I A

E A,nl = 1328∠0° V − j ( 2.8 Ω )(10.5∠0° A ) = 1328.3∠ − 1.27° V
E A,half = 1328∠0° V − j (1.5 Ω )( 93.6∠0° A ) = 1354∠ − 11.2° V

E A,full = 1328∠0° V − j ( 2.8 Ω )(198∠0° A ) = 1439∠ − 22.7° V

These values of E A and δ at unity power factor can serve as reference points in calculating the
synchronous motor V-curves. The MATLAB program to solve this problem is shown below:
155

% M-file: prob6_4.m
% M-file create a plot of armature current versus field
%
current for the synchronous motor of Problem 6-4 at
%
% First, initialize the field current values (21 values
% in the range 3.8-5.8 A)
If = 2.5:0.1:8;
% Get the OCC
if_values = p61_occ(:,1);
vt_values = p61_occ(:,2);
% Now initialize all other values
Xs = 1.5;
% Synchronous reactance
Vp = 1328;
% Phase voltage
% The following values of Ea and
% power factor. They will serve
% when calculating the V-curves.
d_nl = -1.27 * pi/180;
d_half = -11.2 * pi/180;
d_full = -22.7 * pi/180;
Ea_nl = 1328.3;
Ea_half = 1354;
Ea_full = 1439;

delta are for unity
as reference values
%
%
%
%
%
%

delta
delta
delta
Ea at
Ea at
Ea at

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Calculate the actual Ea corresponding to each level
% of field current
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
Ea = interp1(if_values,vt_values,If) / sqrt(3);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Calculate the armature currents associated with
% each value of Ea for the no-load case.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% First, calculate delta.
delta = asin ( Ea_nl ./ Ea .* sin(d_nl) );
% Calculate the phasor Ea
Ea2 = Ea .* (cos(delta) + j .* sin(delta));
% Now calculate Ia
Ia_nl = ( Vp - Ea2 ) / (j * Xs);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Calculate the armature currents associated with
% each value of Ea for the half-load case.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% First, calculate delta.
delta = asin ( Ea_half ./ Ea .* sin(d_half) );
% Calculate the phasor Ea
Ea2 = Ea .* (cos(delta) + j .* sin(delta));
156

% Now calculate Ia
Ia_half = ( Vp - Ea2 ) / (j * Xs);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Calculate the armature currents associated with
% each value of Ea for the full-load case.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% First, calculate delta.
delta = asin ( Ea_full ./ Ea .* sin(d_full) );
% Calculate the phasor Ea
Ea2 = Ea .* (cos(delta) + j .* sin(delta));
% Now calculate Ia
Ia_full = ( Vp - Ea2 ) / (j * Xs);
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
% Plot the v-curves
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
plot(If,abs(Ia_nl),'k-','Linewidth',2.0);
hold on;
plot(If,abs(Ia_half),'b--','Linewidth',2.0);
plot(If,abs(Ia_full),'r:','Linewidth',2.0);
xlabel('\bfField Current (A)');
ylabel('\bfArmature Current (A)');
title ('\bfSynchronous Motor V-Curve');
grid on;
The resulting plot is shown below. The flattening visible to the right of the V-curves is due to magnetic
saturation in the machine.

6-5.

If a 60-Hz synchronous motor is to be operated at 50 Hz, will its synchronous reactance be the same as at
60 Hz, or will it change? (Hint: Think about the derivation of X S .)
157

SOLUTION The synchronous reactance represents the effects of the armature reaction voltage Estat and the
armature self-inductance. The armature reaction voltage is caused by the armature magnetic field B S , and
the amount of voltage is directly proportional to the speed with which the magnetic field sweeps over the
stator surface. The higher the frequency, the faster B S sweeps over the stator, and the higher the armature
reaction voltage Estat is. Therefore, the armature reaction voltage is directly proportional to frequency.
Similarly, the reactance of the armature self-inductance is directly proportional to frequency, so the total
synchronous reactance X S is directly proportional to frequency. If the frequency is changed from 60 Hz to
50 Hz, the synchronous reactance will be decreased by a factor of 5/6.

6-6.

A 480-V 100-kW 0.85-PF leading 50-Hz six-pole Y-connected synchronous motor has a synchronous
reactance of 1.5 Ω and a negligible armature resistance. The rotational losses are also to be ignored. This
motor is to be operated over a continuous range of speeds from 300 to 1000 r/min, where the speed changes
are to be accomplished by controlling the system frequency with a solid-state drive.
(a) Over what range must the input frequency be varied to provide this speed control range?
(b) How large is E A at the motor’s rated conditions?
(c) What is the maximum power the motor can produce at the rated conditions?
(d) What is the largest E A could be at 300 r/min?
(e) Assuming that the applied voltage Vφ is derated by the same amount as E A , what is the maximum
power the motor could supply at 300 r/min?
(f) How does the power capability of a synchronous motor relate to its speed?
SOLUTION
(a)

A speed of 300 r/min corresponds to a frequency of
fe =

nm P (300 r/min ) ( 6)
=
= 15 Hz
120
120

A speed of 1000 r/min corresponds to a frequency of
fe =

nm P (1000 r/min ) ( 6)
=
= 50 Hz
120
120

The frequency must be controlled in the range 15 to 50 Hz.
(b)

The armature current at rated conditions is
IA = IL =

P
100 kW
=
= 141.5 A
3 VT PF
3 ( 480 V )( 0.85)

so I A = 141.5∠31.8° A . This machine is Y-connected, so the phase voltage is Vφ = 480 /

3 = 277 V.

The internal generated voltage is

E A = Vφ − RAI A − jX S I A
E A = 277∠0° V − j (1.5 Ω )(141.5∠31.8° A )

E A = 429∠ − 24.9° V
So E A = 429 V at rated conditions.
(c)

The maximum power that the motor can produce at rated speed with the value of E A from part (b) is
158

Pmax =
(d)

3 Vφ E A
XS

3 ( 277 V )( 429 V )
= 238 kW
1.5 Ω

Since E A must be decreased linearly with frequency, the maximum value at 300 r/min would be
E A,300 =

(e)

=

15 Hz
50 Hz

(429 V ) = 129 V

If the applied voltage Vφ is derated by the same amount as E A , then Vφ = (15/50)(277) = 83.1 V.

Also, note that X S = (15/50)(1.5 Ω) = 0.45 Ω. The maximum power that the motor could supply would
be
Pmax =

3 Vφ E A
XS

=

3 (83.1 V )(129 V )
= 71.5 kW
0.45 Ω

(f) As we can see by comparing the results of (c) and (e), the power-handling capability of the
synchronous motor varies linearly with the speed of the motor.
A 208-V Y-connected synchronous motor is drawing 40 A at unity power factor from a 208-V power
system. The field current flowing under these conditions is 2.7 A. Its synchronous reactance is 0.8 Ω.
Assume a linear open-circuit characteristic.
(a) Find the torque angle δ.
(b) How much field current would be required to make the motor operate at 0.8 PF leading?
(c) What is the new torque angle in part (b)?
SOLUTION
(a)

The phase voltage of this motor is Vφ = 120 V, and the armature current is I A = 40∠0° A .

Therefore, the internal generated voltage is

E A = Vφ − RAI A − jX S I A
E A = 120∠0° V − j (0.8 Ω )( 40∠0° A )

E A = 124 ∠ − 14.9° V
The torque angle δ of this machine is –14.9°.
(b)

A phasor diagram of the motor operating at a power factor of 0.78 leading is shown below.
∝P

}

6-7.

I A2
Vφ

I A1

jX I

S A

E A1

}

∝P

E A2

Since the power supplied by the motor is constant, the quantity I A cos θ , which is directly proportional to
power, must be constant. Therefore,
I A2 (0.8) = ( 40 A )(1.00)
159

I A2 = 50∠36.87° A
The internal generated voltage required to produce this current would be

E A2 = Vφ − R AI A2 − jX S I A2
E A2 = 120∠0° V − j (0.8 Ω )(50∠36.87° A )

E A2 = 147.5∠ − 12.5° V
The internal generated voltage E A is directly proportional to the field flux, and we have assumed in this
problem that the flux is directly proportional to the field current. Therefore, the required field current is
IF2 =

(c)

6-8.

E A2
147 V
I F1 =
(2.7 A ) = 3.20 A
124 V
E A1

The new torque angle δ of this machine is –12.5°.

A synchronous machine has a synchronous reactance of 2.0 Ω per phase and an armature resistance of 0.4
Ω per phase. If E A =460∠-8° V and Vφ = 480∠0° V, is this machine a motor or a generator? How
much power P is this machine consuming from or supplying to the electrical system? How much reactive
power Q is this machine consuming from or supplying to the electrical system?
SOLUTION This machine is a motor, consuming power from the power system, because E A is lagging Vφ .
It is also consuming reactive power, because E A cos δ < Vφ . The current flowing in this machine is

IA =

Vφ − E A
R A + jX S

=

480∠0° V − 460∠ − 8° V
= 33.6∠ − 9.6° A
0.4 + j 2.0 Ω

Therefore the real power consumed by this motor is
P = 3Vφ I A cos θ = 3 ( 480 V )( 33.6 A ) cos ( 9.6°) = 47.7 kW

and the reactive power consumed by this motor is
Q = 3Vφ I A sin θ = 3 ( 480 V )( 33.6 A ) sin ( 9.6°) = 8.07 kVAR

6-9.

Figure P6-2 shows a synchronous motor phasor diagram for a motor operating at a leading power factor
with no RA . For this motor, the torque angle is given by
X S I A cos θ
tan δ =
Vφ + X S I A sin θ

δ =tan -1

X S I A cos θ
Vφ + X S I A sin θ

Derive an equation for the torque angle of the synchronous motor if the armature resistance is included.

160

SOLUTION The phasor diagram with the armature resistance considered is shown below.
A

X I sin θ
Vφ

θ

δ

S A

}

I

jX I

θ

S A

θ

X I cos θ
S A

R I

A A

}

EA

}

R I cos θ
A A

Therefore,
tan δ =

X S I A cosθ + R A I A sin θ
Vφ + X S I A sin θ − RA I A cos θ

δ = tan −1
6-10.

X S I A cosθ + RA I A sin θ
Vφ + X S I A sin θ − RA I A cosθ

A 480-V 375-kVA 0.8-PF-lagging Y-connected synchronous generator has a synchronous reactance of 0.4
Ω and a negligible armature resistance. This generator is supplying power to a 480-V 80-kW 0.8-PFleading Y-connected synchronous motor with a synchronous reactance of 1.1 Ω and a negligible armature
resistance. The synchronous generator is adjusted to have a terminal voltage of 480 V when the motor is
drawing the rated power at unity power factor.
(a) Calculate the magnitudes and angles of E A for both machines.
(b) If the flux of the motor is increased by 10 percent, what happens to the terminal voltage of the power
system? What is its new value?
(c) What is the power factor of the motor after the increase in motor flux?
SOLUTION
(a)

The motor is operating at rated power and unity power factor, so the current flowing in the motor is
I A,m = I L,m =

P
80 kW
=
= 96.2 A
3 VT PF
3 ( 480 V )(1.0)

so I A,m = 96.2 ∠0° A . This machine is Y-connected, so the phase voltage is Vφ = 480 /
internal generated voltage of the motor is
161

3 = 277 V. The

E A,m = Vφ ,m − jX S I A,m
E A,m = 277∠0° V − j (1.1 Ω )( 96.2∠0° A )

E A,m = 297∠ − 20.9° V
This same current comes from the generator, so the internal generated voltage of the generator is
E A,g = Vφ ,g + jX S I A,g

E A,g = 277∠0° V + j ( 0.4 Ω )( 96.2∠0° A )
E A,g = 280∠7.9° V
IA,g

j0.4 Ω

EA,g

+
-

IA,m

+

+

Vφ,g

Vφ,m

-

j1.1 Ω

+
-

EA,m

-

E A,g
I

jX

A

I

S,g A

I

Vφ

A

Vφ

jX

I

S,m A

E A,m
Generator

Motor

(b) The power supplied by the generator to the motor will be constant as the field current of the motor is
varied. The 10% increase in flux will raise the internal generated voltage of the motor to (1.1)(297 V) =
327 V.
To make finding the new conditions easier, we will make the angle of the phasor E A, g the reference during
the following calculations. The resulting phasor diagram is shown below.

I

E A,g

A

δg

jX I

δm

S,g A

Vφ
jX
E A,m
Then by Kirchhoff’s Voltage Law,

E A, g = E A,m + j ( X S , g + X S ,m ) I A

162

I

S,m A

IA =

or

E A, g − E A,m
j ( X S , g + X S ,m )

Note that this combined phasor diagram looks just like the diagram of a synchronous motor, so we can
apply the power equation for synchronous motors to this system.
P=

3E A, g E A,m
X S , g + X S ,m

sin γ

where γ = δ g + δ m . From this equation,

γ = sin −1

(X

S ,g

)

+ X S ,m P

3E A, g E A,m

= sin −1

(1.5 Ω )(80 kW ) = 25.9°
3 ( 280V )( 327 V )

Therefore,

IA =

E A, g − E A,m
j ( X S , g + X S ,m )

=

280∠0° V − 327∠ − 25.9° V
= 95.7∠5.7° A
j1.5 Ω

The phase voltage of the system would be
Vφ = E A, g − jX S , g I A = 280∠0° V − j ( 0.4 Ω )( 95.7∠5.7° A ) = 286∠ − 7.6° V

If we make Vφ the reference (as we usually do), these voltages and currents become:
E A, g = 280∠7.6° V

Vφ = 286∠0° V
E A,m = 327∠ − 18.3° V

I A = 95.7∠13.3° A
The new terminal voltage is VT = 3 ( 286 V ) = 495 V , so the system voltage has increased.
(c)

The power factor of the motor is now PF = cos ( −13.3° ) = 0.973 leading , since a current angle of

-18.3° implies an impedance angle of 18.3°.

Note: The reactive power in the motor is now
Qmotor = 3Vφ I A sin θ = 3 ( 286 V )( 95.7 A ) sin ( −13.3°) = −18.9 kVAR

The motor is now supplying 18.9 kVAR to the system. Note that an increase in machine flux has
increased the reactive power supplied by the motor and also raised the terminal voltage of the system.
This is consistent with what we learned about reactive power sharing in Chapter 5.

6-11.

A 480-V, 100-kW, 50-Hz, four-pole, Y-connected synchronous motor has a rated power factor of 0.85
leading. At full load, the efficiency is 91 percent. The armature resistance is 0.08 Ω, and the synchronous
reactance is 1.0 Ω. Find the following quantities for this machine when it is operating at full load:
(a) Output torque
(b) Input power
(c) nm
(d) E A
163

(e) I A
(f) Pconv
(g) Pmech + Pcore + Pstray
SOLUTION
(a)

Since this machine has 8 poles, it rotates at a speed of

nm =

120 f e 120 (50 Hz )
=
= 1500 r/min
P
4

If the output power is 100 kW, the output torque is

(b)

ωm

=

(100,000 W )
(1500 r/min )

1r

1 min
60 s

= 637 N ⋅ m

The input power is
PIN =

(c)

Pout

POUT 100 kW
=
= 110 kW
0.91
η

The mechanical speed is
nm = 1500 r/min

(d)

The armature current is
IA = IL =

P
110 kW
=
= 156 A
3 VT PF
3 ( 480 V )(0.85)

I A = 156∠31.8° A
Therefore, E A is
E A = Vφ − RAI A − jX S I A

E A = ( 277 ∠0° V ) − ( 0.08 Ω )(156∠31.8° A ) − j (1.0 Ω )(156∠31.8° A )
E A = 375∠ − 21.8° V
(e)

The magnitude of the armature current is 375 A.

(f)

The power converted from electrical to mechanical form is given by the equation Pconv = PIN − PCU
PCU = 3 I A2 RA = 3 (156 A ) ( 0.08 Ω ) = 5.8 kW
2

Pconv = PIN − PCU = 110 kW − 5.8 kW = 104.2 kW
(g)

The mechanical, core, and stray losses are given by the equation
Pmech + Pcore + Pstray = Pconv − POUT = 104.2 kW − 100 kW = 4.2 kW

6-12.

The Y-connected synchronous motor whose nameplate is shown in Figure 6-21 has a per-unit synchronous
reactance of 0.90 and a per-unit resistance of 0.02.
(a) What is the rated input power of this motor?
(b) What is the magnitude of E A at rated conditions?
164

(c) If the input power of this motor is 10 MW, what is the maximum reactive power the motor can
simultaneously supply? Is it the armature current or the field current that limits the reactive power
output?
(d) How much power does the field circuit consume at the rated conditions?
(e) What is the efficiency of this motor at full load?
(f) What is the output torque of the motor at the rated conditions? Express the answer both in newtonmeters and in pound-feet.

SOLUTION The base quantities for this motor are:
VT ,base = 6600 V
6600 V
= 3811 V
3
= I L ,base = 1404 A

Vφ ,base =

I A,base

S base = Prated = 3 VT I L PF = 3 ( 6600 V )(1404 A )(1.0 ) = 16.05 MW

(a)

The rated input power of this motor is
PIN = 3 VT I L PF = 3 ( 6600 V )(1404 A )(1.0) = 16.05 MW

(b)

At rated conditions, Vφ = 1.0∠0° pu and Iφ = 1.0∠0° pu , so E A is given in per-unit quantities as
E A = Vφ − RAI A − jX S I A

E A = (1∠0° ) − (0.02 )(1.0∠0°) − j (0.90 )(1∠0° )
E A = 1.33∠ − 42.6° pu
The base phase voltage of this motor is 6600 /

3 = 3810 V, so E A is

E A = (1.33∠ − 42.6°)( 3810 V ) = 5067 ∠ − 42.6° V

(c) From the capability diagram, we know that there are two possible constraints on the maximum
reactive power—the maximum stator current and the maximum rotor current. We will have to check each
one separately, and limit the reactive power to the lesser of the two limits.
The stator apparent power limit defines a maximum safe stator current. This limit is the same as the rated
input power for this motor, since the motor is rated at unity power factor. Therefore, the stator apparent
165

power limit is 16.05 MVA. If the input power is 10 MW, then the maximum reactive power that still
protects the stator current is
Q = S 2 − P2 =

(16.05 MVA )2 − (10 MW )2

= 12.6 MVAR

Now we must determine the rotor current limit. The per-unit power supplied to the motor is 10 MW /
16.05 MW = 0.623. The maximum E A is 5067 V or 1.33 pu, so with E A set to maximum and the motor
consuming 10 MW, the torque angle (ignoring armature resistance) is

δ = sin −1

XSP
= sin −1
3Vφ E A

(0.90)(0.623)
(1.0)(1.33)

= 24.9°

At rated voltage and 10 MW of power supplied, the armature current will be

IA =

Vφ − E A
R A + jX S

=

1∠0° − 1.33∠ − 24.9°
= 0.663∠20.2° pu
j0.90

In actual amps, this current is
I A = (1404 A )( 0.663∠20.2° ) = 931∠20.2° A

The reactive power supplied at the conditions of maximum E A and 10 MW power is
Q = 3Vφ I A sin θ = 3 ( 3811 V )( 931 A ) sin ( 20.2°) = 3.68 MVAR

Therefore, the field current limit occurs before the stator current limit for these conditions, and the
maximum reactive power that the motor can supply is 3.68 MVAR under these conditions.
(d)

At rated conditions, the field circuit consumes
Pfield = VF I F = (125 V )(5.2 A ) = 650 W

(e)

The efficiency of this motor at full load is

η=
(f)

(21000 hp )(746 W/hp) × 100% = 97.6%
POUT
× 100% =
PIN
16.05 MW

The output torque in SI and English units is

POUT

ωm

=

( 21000 hp)(746 W/hp)
(1200 r/min ) 1 min 2π rad
60 s

6-13.

= 124,700 N ⋅ m

1r

5252 P 5252 ( 21000 hp )
=
= 91,910 lb ⋅ ft
nm
(1200 r/min )

A 440-V three-phase Y-connected synchronous motor has a synchronous reactance of 1.5 Ω per phase.
The field current has been adjusted so that the torque angle δ is 28° when the power supplied by the
generator is 90 kW.
(a) What is the magnitude of the internal generated voltage E A in this machine?
(b) What are the magnitude and angle of the armature current in the machine? What is the motor’s power
factor?
(c) If the field current remains constant, what is the absolute maximum power this motor could supply?
166

SOLUTION
(a)

The power supplied to the motor is 90 kW. This power is give by the equation
3Vφ E A
P=
sin δ
XS

so the magnitude of E A is
EA =
(b)

XS P
(1.5 Ω )(90 kW ) = 377 V
=
3Vφ sin δ 3 ( 254 V ) sin 28°

The armature current in this machine is given by

IA =

Vφ − E A
jX S

=

254∠0° V − 377∠ − 28°
= 129∠ 24° A
j1.5

The power factor of the motor is PF = cos 24º = 0.914 leading.
(c)

The maximum power that the motor could supply at this field current
Pmax =

6-14.

3Vφ E A
XS

=

3 ( 254 V )( 377 V )
= 191.5 kW
1.5 Ω

A 460-V, 200-kVA, 0.80-PF-leading, 400-Hz, six-pole, Y-connected synchronous motor has negligible
armature resistance and a synchronous reactance of 0.50 per unit. Ignore all losses.
(a) What is the speed of rotation of this motor?
(b) What is the output torque of this motor at the rated conditions?
(c) What is the internal generated voltage of this motor at the rated conditions?
(d) With the field current remaining at the value present in the motor in part (c), what is the maximum
possible output power from the machine?
SOLUTION
(a)

The speed of rotation of this motor is
120 f e 120 ( 400 Hz )
nsync =
=
= 8000 r/min
6
P

(b) Since all losses are ignored, PIN ,rated = POUT ,rated = S rated × PF = ( 200 kVA )( 0.8) = 160 kW . The output
torque of this motor is
P
160 kW
τ load = OUT =
= 191 N ⋅ m
ωm
(8000 r/min ) 1 min 2π rad
60 s
1r
(c)

The phase voltage of this motor is 460 V /
IA = IL =

3 = 266 V. The rated armature current of this motor is

P
160 kW
=
= 251 A
3 VT PF
3 ( 460 V )(0.80 )

Therefore, I A = 251∠36.87° A . The base impedance of this motor is
Z base =

3Vφ ,base 2
S base

3 ( 266 V )
= 1.06 Ω
200,000 VA
2

=

167

so the actual synchronous reactance is X S = ( 0.50 pu )(1.06 Ω ) = 0.53 Ω . The internal generated voltage
of this machine at rated conditions is given by

E A = Vφ − jX S I A
E A = 266∠0° V − j ( 0.53 Ω )( 251∠36.87° A ) = 362∠ − 17.1° V

(d)

The maximum power that the motor could supply at these conditions is
PMAX =

6-15.

3Vφ E A
XS

=

3 ( 266 V )( 362 V )
= 545 kW
0.53 Ω

A 100-hp 440-V 0.8-PF-leading ∆-connected synchronous motor has an armature resistance of 0.22 Ω and
a synchronous reactance of 3.0 Ω. Its efficiency at full load is 89 percent.
(a) What is the input power to the motor at rated conditions?
(b) What is the line current of the motor at rated conditions? What is the phase current of the motor at
rated conditions?
(c) What is the reactive power consumed by or supplied by the motor at rated conditions?
(d) What is the internal generated voltage E A of this motor at rated conditions?
(e) What are the stator copper losses in the motor at rated conditions?
(f) What is Pconv at rated conditions?
(g) If E A is decreased by 10 percent, how much reactive power will be consumed by or supplied by the
motor?
SOLUTION
(a)

The input power to the motor at rated conditions is
P
(100 hp )(746 W/hp ) = 83.8 kW
PIN = OUT =
0.89
η

(b)

The line current to the motor at rated conditions is
IL =

P
83.8 kW
=
= 137 A
3 VT PF
3 ( 440 V )( 0.8)

The phase current to the motor at rated conditions is
Iφ =

(c)

I L 137 A
=
= 79.4 A
3
3

The reactive power supplied by this motor to the power system at rated conditions is
Qrated = 3Vφ I A sin θ = 3 ( 440 V )( 79.4 A ) sin 36.87° = 62.9 kVAR

(d)

The internal generated voltage at rated conditions is
E A = Vφ − RAI A − jX S I A

E A = 440∠0° V − ( 0.22 Ω )( 79.4∠36.87° A ) − j (3.0 Ω )( 79.4∠36.87° A )
E A = 603∠ − 19.5° V
(e)

The stator copper losses at rated conditions are
168

PCU = 3 I A2 RA = 3 ( 79.4 A ) ( 0.22 Ω ) = 4.16 kW
2

(f)

Pconv at rated conditions is
Pconv = PIN − PCU = 83.8 kW − 4.16 kW = 79.6 kW

(g) If E A is decreased by 10%, the new value if E A = (0.9)(603 V) = 543 V. To simplify this part of the
problem, we will ignore R A . Then the quantity E A sin δ will be constant as E A changes. Therefore,

δ 2 = sin −1

E A1
603 V
sin δ1 = sin −1
sin ( −19.5° ) = −21.8°
E A2
543 V

Therefore,

IA =

Vφ − E A
jX S

=

440∠0° V − 543∠ − 21.8°
= 70.5∠17.7° A
j3.0

and the reactive power supplied by the motor to the power system will be
Q = 3Vφ I A sin θ = 3 ( 440 V )( 70.5 A ) sin (17.7° ) = 28.3 kVAR

6-16.

Answer the following questions about the machine of Problem 6-15.
(a) If E A = 430∠13.5° V and Vφ = 440∠0° V, is this machine consuming real power from or supplying
real power to the power system? Is it consuming reactive power from or supplying reactive power to
the power system?
(b) Calculate the real power P and reactive power Q supplied or consumed by the machine under the
conditions in part (a). Is the machine operating within its ratings under these circumstances?
(c) If E A = 470∠-12° V and Vφ = 440∠0° V, is this machine consuming real power from or supplying
real power to the power system? Is it consuming reactive power from or supplying reactive power to
the power system?
(d) Calculate the real power P and reactive power Q supplied or consumed by the machine under the
conditions in part (c). Is the machine operating within its ratings under these circumstances?

SOLUTION
(a)

This machine is a generator supplying real power to the power system, because E A is ahead of Vφ .

It is consuming reactive power because E A cos δ < Vφ .
(b)

This machine is acting as a generator, and the current flow in these conditions is

IA =

E A − Vφ
RA + jX S

=

430∠13.5° − 440∠0° V
= 34.2∠16.5° A
0.22 + j 3.0

The real power supplied by this machine is
P = 3Vφ I A cos θ = 3 ( 440 V )( 34.2 A ) cos ( −16.5° ) = 43.3 kW

The reactive power supplied by this machine is
Q = 3Vφ I A sin θ = 3 ( 440 V )( 34.2 A ) sin ( −16.5°) = −12.8 kVAR

169

(c)

This machine is a motor consuming real power from the power system, because E A is behind Vφ . It

is supplying reactive power because E A cos δ > Vφ .
(d)

This machine is acting as a motor, and the current flow in these conditions is

IA =

Vφ − E A
RA + jX S

=

440∠0° V − 470∠ − 12°
= 33.1∠15.6° A
0.22 + j 3.0

The real power consumed by this machine is
P = 3Vφ I A cos θ = 3 ( 440 V )( 33.1 A ) cos (15.6°) = 42.1 kW

The reactive power supplied by this machine is
Q = 3Vφ I A sin θ = 3 ( 440 V )( 33.1 A ) sin (15.6°) = +11.7 kVAR

170

Chapter 7: Induction Motors
7-1.

A dc test is performed on a 460-V ∆-connected 100-hp induction motor. If VDC = 24 V and I DC = 80 A,
what is the stator resistance R1 ? Why is this so?
SOLUTION If this motor’s armature is connected in delta, then there will be two phases in parallel with one
phase between the lines tested.
VDC

R1

R1

R1

Therefore, the stator resistance R1 will be
R ( R + R1 )
VDC
2
= 1 1
= R1
I DC R1 + ( R1 + R1 ) 3
R1 =

7-2.

3 VDC 3 24 V
=
= 0.45 Ω
2 I DC 2 80 A

A 220-V, three-phase, two-pole, 50-Hz induction motor is running at a slip of 5 percent. Find:
(a) The speed of the magnetic fields in revolutions per minute
(b) The speed of the rotor in revolutions per minute
(c) The slip speed of the rotor
(d) The rotor frequency in hertz
SOLUTION
(a)

The speed of the magnetic fields is
nsync =

(b)

120 f e 120 (50 Hz )
=
= 3000 r/min
2
P

The speed of the rotor is
nm = (1 − s ) nsync = (1 − 0.05)( 3000 r/min ) = 2850 r/min

(c)

The slip speed of the rotor is
nslip = snsync = (0.05)(3000 r/min ) = 150 r/min

(d)

The rotor frequency is
fr =

7-3.

nslip P
120

=

(150 r/min )( 2 ) = 2.5 Hz
120

Answer the questions in Problem 7-2 for a 480-V, three-phase, four-pole, 60-Hz induction motor running at
a slip of 0.035.
SOLUTION
(a)

The speed of the magnetic fields is
171

nsync =
(b)

120 f e 120 (60 Hz )
=
= 1800 r/min
4
P

The speed of the rotor is
nm = (1 − s ) nsync = (1 − 0.035)(1800 r/min ) = 1737 r/min

(c)

The slip speed of the rotor is
nslip = snsync = (0.035)(1800 r/min ) = 63 r/min

(d)

The rotor frequency is
fr =

7-4.

nslip P
120

=

(63 r/min )(4 ) = 2.1 Hz
120

A three-phase, 60-Hz induction motor runs at 890 r/min at no load and at 840 r/min at full load.
(a) How many poles does this motor have?
(b) What is the slip at rated load?
(c) What is the speed at one-quarter of the rated load?
(d) What is the rotor’s electrical frequency at one-quarter of the rated load?

SOLUTION
(a)

This machine has 8 poles, which produces a synchronous speed of
nsync =

(b)

The slip at rated load is
s=

(c)

120 f e 120 (60 Hz )
=
= 900 r/min
8
P

nsync − nm
nsync

× 100% =

900 − 840
× 100% = 6.67%
900

The motor is operating in the linear region of its torque-speed curve, so the slip at ¼ load will be
s = 0.25(0.0667) = 0.0167

The resulting speed is
nm = (1 − s ) nsync = (1 − 0.0167 )(900 r/min ) = 885 r/min

(d)

The electrical frequency at ¼ load is
f r = sf e = ( 0.0167 )( 60 Hz ) = 1.00 Hz

7-5.

A 50-kW, 440-V, 50-Hz, six-pole induction motor has a slip of 6 percent when operating at full-load
conditions. At full-load conditions, the friction and windage losses are 300 W, and the core losses are 600
W. Find the following values for full-load conditions:
(a) The shaft speed nm
(b) The output power in watts
(c) The load torque τ load in newton-meters
(d) The induced torque τ ind in newton-meters
172

(e) The rotor frequency in hertz
SOLUTION
(a)

The synchronous speed of this machine is
nsync =

120 f e 120 (50 Hz )
=
= 1000 r/min
6
P

Therefore, the shaft speed is
nm = (1 − s ) nsync = (1 − 0.06 )(1000 r/min ) = 940 r/min
(b)

The output power in watts is 50 kW (stated in the problem).

(c)

The load torque is

(d)

POUT

=

ωm

50 kW
(940 r/min )
1r

1 min
60 s

= 508 N ⋅ m

The induced torque can be found as follows:
Pconv = POUT + PF&W + Pcore + Pmisc = 50 kW + 300 W + 600 W + 0 W = 50.9 kW

τ ind =

(e)

Pconv

ωm

=

50.9 kW
(940 r/min ) 2π rad
1r

1 min
60 s

= 517 N ⋅ m

The rotor frequency is
f r = sf e = ( 0.06 )(50 Hz ) = 3.00 Hz

7-6.

A three-phase, 60-Hz, four-pole induction motor runs at a no-load speed of 1790 r/min and a full-load
speed of 1720 r/min. Calculate the slip and the electrical frequency of the rotor at no-load and full-load
conditions. What is the speed regulation of this motor [Equation (4-68)]?
SOLUTION The synchronous speed of this machine is 1800 r/min. The slip and electrical frequency at noload conditions is
nsync − nnl

snl =

nsync

× 100% =

1800 − 1790
× 100% = 0.56%
1800

f r ,nl = sf e = ( 0.0056)( 60 Hz ) = 0.33 Hz

The slip and electrical frequency at full load conditions is
sfl =

nsync − nnl
nsync

× 100% =

1800 − 1720
× 100% = 4.44%
1800

f r ,fl = sf e = ( 0.0444 )( 60 Hz ) = 2.67 Hz

The speed regulation is
SR =

7-7.

nnl − nfl
1790 − 1720
× 100% =
× 100% = 4.1%
1720
nfl

A 208-V, two-pole, 60-Hz Y-connected wound-rotor induction motor is rated at 15 hp. Its equivalent
circuit components are
173

R1 = 0.200 Ω

R2 = 0.120 Ω

X 1 = 0.410 Ω

X 2 = 0.410 Ω

Pmech = 250 W

Pmisc ≈ 0

X M = 15.0 Ω

Pcore = 180 W

For a slip of 0.05, find
(a) The line current I L
(b) The stator copper losses PSCL
(c) The air-gap power PAG
(d) The power converted from electrical to mechanical form Pconv
(e) The induced torque τind
(g) The overall machine efficiency
(h) The motor speed in revolutions per minute and radians per second
SOLUTION The equivalent circuit of this induction motor is shown below:
IA
+

R1

jX1

0.20 Ω

j0.41 Ω
j15 Ω

Vφ

jX2

R2

j0.41 Ω

0.120 Ω

1− s 
R2 

 s 

jXM

2.28 Ω

-

(a) The easiest way to find the line current (or armature current) is to get the equivalent impedance Z F
of the rotor circuit in parallel with jX M , and then calculate the current as the phase voltage divided by the
sum of the series impedances, as shown below.
IA
+

R1

jX1

0.20 Ω

j0.41 Ω

jXF

RF

Vφ
-

The equivalent impedance of the rotor circuit in parallel with jX M is:
1
1
ZF =
=
= 2.220 + j 0.745 = 2.34 ∠18.5° Ω
1
1
1
1
+
+
jX M Z 2
j15 Ω 2.40 + j 0.41
The phase voltage is 208/ 3 = 120 V, so line current I L is
174

IL = IA =

Vφ

=

R1 + jX 1 + RF + jX F

120∠0° V
0.20 Ω + j 0.41 Ω + 2.22 Ω + j 0.745 Ω

I L = I A = 44.8∠ − 25.5° A
(b)

The stator copper losses are
PSCL = 3I A2 R1 = 3 ( 44.8 A ) ( 0.20 Ω ) = 1205 W
2

(c)

The air gap power is PAG = 3 I 2 2

R2
= 3I A2 RF
s

R2
, since the only resistance in the original rotor circuit was R2 / s , and
s
the resistance in the Thevenin equivalent circuit is RF . The power consumed by the Thevenin equivalent
circuit must be the same as the power consumed by the original circuit.)

(Note that 3 I A2 RF is equal to 3 I 2 2

PAG = 3 I 2 2
(d)

R2
2
= 3I A 2 RF = 3 ( 44.8 A ) ( 2.220 Ω ) = 13.4 kW
s

The power converted from electrical to mechanical form is
Pconv = (1 − s ) PAG = (1 − 0.05)(13.4 kW ) = 12.73 kW

(e)

The induced torque in the motor is

τ ind =

(f)

PAG

ω sync

=

13.4 kW
(3600 r/min )
1r

1 min
60 s

= 35.5 N ⋅ m

The output power of this motor is
POUT = Pconv − Pmech − Pcore − Pmisc = 12.73 kW − 250 W − 180 W − 0 W = 12.3 kW

The output speed is
nm = (1 − s ) nsync = (1 − 0.05)( 3600 r/min ) = 3420 r/min

Therefore the load torque is

(g)

(h)

POUT

ωm

=

12.3 kW
(3420 r/min )
1r

= 34.3 N ⋅ m

The overall efficiency is

η=

POUT
POUT
× 100% =
× 100%
PIN
3Vφ I A cosθ

η=

12.3 kW
× 100% = 84.5%
3 (120 V )( 44.8 A ) cos 25.5°

The motor speed in revolutions per minute is 3420 r/min. The motor speed in radians per second is

ω m = ( 3420 r/min )
7-8.

1 min
60 s

1r

1 min
60 s

For the motor in Problem 7-7, what is the slip at the pullout torque? What is the pullout torque of this
motor?
175

SOLUTION The slip at pullout torque is found by calculating the Thevenin equivalent of the input circuit
from the rotor back to the power supply, and then using that with the rotor circuit model.

Z TH =

jX M ( R1 + jX 1 ) ( j15 Ω )( 0.20 Ω + j 0.41 Ω )
=
= 0.1895 + j0.4016 Ω = 0.444∠64.7° Ω
R1 + j ( X 1 + X M ) 0.20 Ω + j ( 0.41 Ω + 15 Ω )

VTH =

jX M
( j15 Ω )
Vφ =
(120∠0° V ) = 116.8∠0.7° V
R1 + j ( X 1 + X M )
0.22 Ω + j ( 0.43 Ω + 15 Ω )

The slip at pullout torque is
smax =
smax =

R2
RTH + ( X TH + X 2 )
2

2

0.120 Ω

(0.1895 Ω ) + (0.4016 Ω
2

+ 0.410 Ω )

2

= 0.144

The pullout torque of the motor is

τ max =

τ max =

2
3VTH
2
2ω sync RTH + RTH
+ ( X TH + X 2 )

2

3 (116.8 V )
2 ( 377 rad/s) 0.1895 Ω +

2

(0.1895 Ω)2 + (0.4016 Ω
176

+ 0.410 Ω )

2

τ max = 53.1 N ⋅ m
7-9.

(a) Calculate and plot the torque-speed characteristic of the motor in Problem 7-7. (b) Calculate and plot
the output power versus speed curve of the motor in Problem 7-7.
SOLUTION
(a)

A MATLAB program to calculate the torque-speed characteristic is shown below.

% M-file: prob7_9a.m
% M-file create a plot of the torque-speed curve of the
%
induction motor of Problem 7-7.
% First, initialize the values needed in this program.
r1 = 0.200;
% Stator resistance
x1 = 0.410;
% Stator reactance
r2 = 0.120;
% Rotor resistance
x2 = 0.410;
% Rotor reactance
xm = 15.0;
% Magnetization branch reactance
v_phase = 208 / sqrt(3);
% Phase voltage
n_sync = 3600;
% Synchronous speed (r/min)
w_sync = 377;
% Synchronous speed (rad/s)
% Calculate the Thevenin voltage and impedance from Equations
% 7-41a and 7-43.
v_th = v_phase * ( xm / sqrt(r1^2 + (x1 + xm)^2) );
z_th = ((j*xm) * (r1 + j*x1)) / (r1 + j*(x1 + xm));
r_th = real(z_th);
x_th = imag(z_th);
% Now calculate the torque-speed characteristic for many
% slips between 0 and 1. Note that the first slip value
% is set to 0.001 instead of exactly 0 to avoid divide% by-zero problems.
s = (0:1:50) / 50;
% Slip
s(1) = 0.001;
nm = (1 - s) * n_sync;
% Mechanical speed
% Calculate torque versus speed
for ii = 1:51
t_ind(ii) = (3 * v_th^2 * r2 / s(ii)) / ...
(w_sync * ((r_th + r2/s(ii))^2 + (x_th + x2)^2) );
end
% Plot the torque-speed curve
figure(1);
plot(nm,t_ind,'k-','LineWidth',2.0);
xlabel('\bf\itn_{m}');
ylabel('\bf\tau_{ind}');
title ('\bfInduction Motor Torque-Speed Characteristic');
grid on;

The resulting plot is shown below:

177

Induction Motor Torque-Speed Characteristic
60

50

τ

ind

40

30

20

10

0

0

500

1000

1500

2000
n

2500

3000

3500

4000

m

(b)

A MATLAB program to calculate the output-power versus speed curve is shown below.

% M-file: prob7_9b.m
% M-file create a plot of the output pwer versus speed
%
curve of the induction motor of Problem 7-7.
% First, initialize the values needed in this program.
r1 = 0.200;
% Stator resistance
x1 = 0.410;
% Stator reactance
r2 = 0.120;
% Rotor resistance
x2 = 0.410;
% Rotor reactance
xm = 15.0;
% Magnetization branch reactance
v_phase = 208 / sqrt(3);
% Phase voltage
n_sync = 3600;
% Synchronous speed (r/min)
w_sync = 377;
% Synchronous speed (rad/s)
% Calculate the Thevenin voltage and impedance from Equations
% 7-41a and 7-43.
v_th = v_phase * ( xm / sqrt(r1^2 + (x1 + xm)^2) );
z_th = ((j*xm) * (r1 + j*x1)) / (r1 + j*(x1 + xm));
r_th = real(z_th);
x_th = imag(z_th);
% Now calculate the torque-speed characteristic for many
% slips between 0 and 1. Note that the first slip value
% is set to 0.001 instead of exactly 0 to avoid divide% by-zero problems.
s = (0:1:50) / 50;
% Slip
s(1) = 0.001;
nm = (1 - s) * n_sync;
% Mechanical speed (r/min)
wm = (1 - s) * w_sync;
% Mechanical speed (rad/s)
% Calculate torque and output power versus speed
for ii = 1:51

178

t_ind(ii) = (3 * v_th^2 * r2 / s(ii)) / ...
(w_sync * ((r_th + r2/s(ii))^2 + (x_th + x2)^2) );
p_out(ii) = t_ind(ii) * wm(ii);
end
% Plot the torque-speed curve
figure(1);
plot(nm,p_out/1000,'k-','LineWidth',2.0);
xlabel('\bf\itn_{m} \rm\bf(r/min)');
ylabel('\bf\itP_{OUT} \rm\bf(kW)');
title ('\bfInduction Motor Ouput Power versus Speed');
grid on;

The resulting plot is shown below:

7-10.

For the motor of Problem 7-7, how much additional resistance (referred to the stator circuit) would it be
necessary to add to the rotor circuit to make the maximum torque occur at starting conditions (when the
shaft is not moving)? Plot the torque-speed characteristic of this motor with the additional resistance
inserted.
SOLUTION To get the maximum torque at starting, the smax must be 1.00. Therefore,
smax =
1.00 =

R2
RTH + ( X TH + X 2 )
2

2

R2

(0.1895 Ω ) + (0.4016 Ω
2

+ 0.410 Ω )

2

R2 = 0.833 Ω
Since the existing resistance is 0.120 Ω, an additional 0.713 Ω must be added to the rotor circuit. The
resulting torque-speed characteristic is:

179

7-11.

If the motor in Problem 7-7 is to be operated on a 50-Hz power system, what must be done to its supply
voltage? Why? What will the equivalent circuit component values be at 50 Hz? Answer the questions in
Problem 7-7 for operation at 50 Hz with a slip of 0.05 and the proper voltage for this machine.
SOLUTION If the input frequency is decreased to 50 Hz, then the applied voltage must be decreased by 5/6
also. If this were not done, the flux in the motor would go into saturation, since

φ=

1
N

∫ v dt
T

and the period T would be increased. At 50 Hz, the resistances will be unchanged, but the reactances will
be reduced to 5/6 of their previous values. The equivalent circuit of the induction motor at 50 Hz is shown
below:
IA
+

Vφ

R1

jX1

0.20 Ω

j0.342 Ω
j12.5 Ω

R2

j0.342 Ω

0.120 Ω

jXM

1− s 
R2 

 s 
2.28 Ω

-

(a)

jX2

The easiest way to find the line current (or armature current) is to get the equivalent impedance Z F

of the rotor circuit in parallel with jX M , and then calculate the current as the phase voltage divided by the
sum of the series impedances, as shown below.

180

IA
+

R1

jX1

0.20 Ω

j0.342 Ω

jXF

RF

Vφ
-

The equivalent impedance of the rotor circuit in parallel with jX M is:
1
1
ZF =
=
= 2.197 + j 0.744 = 2.32∠18.7° Ω
1
1
1
1
+
+
jX M Z 2
j12.5 Ω 2.40 + j 0.342
The line voltage must be derated by 5/6, so the new line voltage is VT = 173.3 V . The phase voltage is
173.3 / 3 = 100 V, so line current I L is
Vφ
100∠0° V
=
IL = IA =
R1 + jX 1 + RF + jX F 0.20 Ω + j 0.342 Ω + 2.197 Ω + j 0.744 Ω
I L = I A = 38.0∠ − 24.4° A
(b)

The stator copper losses are
PSCL = 3I A 2 R1 = 3 ( 38 A ) (0.20 Ω) = 866 W
2

(c)

The air gap power is PAG = 3 I 2 2

R2
= 3I A2 RF
s

R2
, since the only resistance in the original rotor circuit was R2 / s , and
s
the resistance in the Thevenin equivalent circuit is RF . The power consumed by the Thevenin equivalent
circuit must be the same as the power consumed by the original circuit.)

(Note that 3 I A2 RF is equal to 3 I 2 2

PAG = 3 I 2 2
(d)

R2
2
= 3I A 2 RF = 3 ( 38 A ) ( 2.197 Ω ) = 9.52 kW
s

The power converted from electrical to mechanical form is
Pconv = (1 − s ) PAG = (1 − 0.05)( 9.52 kW ) = 9.04 kW

(e)

The induced torque in the motor is

τ ind =

PAG

ω sync

=

9.52 kW
(3000 r/min )
1r

1 min
60 s

= 30.3 N ⋅ m

(f) In the absence of better information, we will treat the mechanical and core losses as constant despite
the change in speed. This is not true, but we don’t have reason for a better guess. Therefore, the output
power of this motor is
POUT = Pconv − Pmech − Pcore − Pmisc = 9.04 kW − 250 W − 180 W − 0 W = 8.61 kW

The output speed is
nm = (1 − s ) nsync = (1 − 0.05)( 3000 r/min ) = 2850 r/min

181

Therefore the load torque is

(g)

(h)

POUT

ωm

=

8.61 kW
(2850 r/min )
1r

= 28.8 N ⋅ m

The overall efficiency is

η=

POUT
POUT
× 100% =
× 100%
PIN
3Vφ I A cosθ

η=

8.61 kW
× 100% = 82.9%
3 (100 V )( 38.0 A ) cos 24.4°

The motor speed in revolutions per minute is 2850 r/min. The motor speed in radians per second is

ω m = ( 2850 r/min )
7-12.

1 min
60 s

1r

1 min
60 s

Figure 7-18a shows a simple circuit consisting of a voltage source, a resistor, and two reactances. Find the
Thevenin equivalent voltage and impedance of this circuit at the terminals. Then derive the expressions for
the magnitude of VTH and for RTH given in Equations (7-41b) and (7-44).

SOLUTION The Thevenin voltage of this circuit is

VTH =

jX M
Vφ
R1 + j ( X 1 + X M )

The magnitude of this voltage is
VTH =

XM
R + ( X1 + X M )
2
1

2

Vφ

If X M >> X 1 , then R12 + ( X 1 + X M ) ≈ ( X 1 + X M ) , so
2

VTH ≈

2

XM
Vφ
X1 + X M

The Thevenin impedance of this circuit is
Z TH =

jX M ( R1 + jX 1 )
R1 + j ( X 1 + X M )

182

Z TH =

Z TH =

jX M ( R1 + jX 1 ) R1 − j ( X 1 + X M )

R1 + j ( X 1 + X M )

R1 − j ( X 1 + X M )

− R1 X 1 X M + R1 X 1 X M + R1 X M 2 + j R12 X M + X 12 X M + X 1 X M 2
R12 + ( X 1 + X M )

2

Z TH = RTH + jX TH =

R1 X M 2

R12 + ( X 1 + X M )

The Thevenin resistance is RTH =

2

R12 X M + X 12 X M + X 1 X M 2

+j

R12 + ( X 1 + X M )

R1 X M 2

R + ( X1 + X M )
2
1

2

. If X M >> R1 , then R12 + ( X 1 + X M ) ≈ ( X 1 + X M ) ,
2

2

2

so
RTH ≈ R1

XM
X1 + X M

2

The Thevenin reactance is X TH =

R12 X M + X 12 X M + X 1 X M 2
R12 + ( X 1 + X M )

2

.

If X M >> R1 and X M >> X 1 then X 1 X M 2 >> R12 X M + X 12 X M and ( X 1 + X M ) ≈ X M 2 >> R12 , so
2

X TH ≈

7-13.

X1 X M 2
= X1
XM2

Figure P7-1 shows a simple circuit consisting of a voltage source, two resistors, and two reactances in
parallel with each other. If the resistor RL is allowed to vary but all the other components are constant, at
what value of RL will the maximum possible power be supplied to it? Prove your answer. (Hint: Derive
an expression for load power in terms of V, RS , X S , RL and X L and take the partial derivative of that
expression with respect to RL .) Use this result to derive the expression for the pullout torque [Equation (754)].

SOLUTION The current flowing in this circuit is given by the equation
IL =

IL =

V
RS + jX S + RL + jX L

V

( RS + RL ) + ( X S + X L )2
2

The power supplied to the load is

183

V 2 RL

P = I L 2 RL =
∂P
=
∂RL

( RS + RL )2 + ( X S + X L )2
( RS + RL )2 + ( X S + X L )2 V 2 − V 2 RL 2 ( RS + RL )
( RS + RL )2 + ( X S + X L )2

2

To find the point of maximum power supplied to the load, set ∂P / ∂RL = 0 and solve for RL .

( RS + RL )2 + ( X S + X L )2

V 2 − V 2 RL 2 ( RS + RL ) = 0

( RS + RL )2 + ( X S + X L )2

= 2 RL ( RS + RL )

RS 2 + 2 RS RL + RL 2 + ( X S + X L ) = 2 RS RL + 2 RL 2
2

RS 2 + RL 2 + ( X S + X L ) = 2 RL 2
2

RS 2 + ( X S + X L ) = RL 2
2

Therefore, for maximum power transfer, the load resistor should be
RL = RS 2 + ( X S + X L )

7-14.

2

A 440-V 50-Hz two-pole Y-connected induction motor is rated at 75 kW.
parameters are
R1 = 0.075 Ω

R2 = 0.065 Ω

X 1 = 0.17 Ω

X 2 = 0.17 Ω

PF&W = 1.0 kW

Pmisc = 150 W

X M = 7.2 Ω

Pcore = 1.1 kW

For a slip of 0.04, find
(a) The line current I L
(b) The stator power factor
(c) The rotor power factor
(d) The stator copper losses PSCL
(e) The air-gap power PAG
(f) The power converted from electrical to mechanical form Pconv
(g) The induced torque τ ind
(i) The overall machine efficiency η
(j) The motor speed in revolutions per minute and radians per second
SOLUTION The equivalent circuit of this induction motor is shown below:

184

The equivalent circuit

IA
+

R1

jX1

0.075 Ω

j0.17 Ω

R2

j0.17 Ω

0.065 Ω

1− s 
R2 

 s 

jXM

j7.2 Ω

Vφ

jX2

1.56 Ω

-

(a)

The easiest way to find the line current (or armature current) is to get the equivalent impedance Z F

of the rotor circuit in parallel with jX M , and then calculate the current as the phase voltage divided by the
sum of the series impedances, as shown below.
IA
+

R1

jX1

0.075 Ω

j0.17 Ω

jXF

RF

Vφ
-

The equivalent impedance of the rotor circuit in parallel with jX M is:
1
1
ZF =
=
= 1.539 + j 0.364 = 1.58∠13.2° Ω
1
1
1
1
+
+
jX M Z 2
j 7.2 Ω 1.625 + j0.17
The phase voltage is 440/ 3 = 254 V, so line current I L is
Vφ
254∠0° V
=
IL = IA =
R1 + jX 1 + RF + jX F 0.075 Ω + j0.17 Ω + 1.539 Ω + j 0.364 Ω
I L = I A = 149.4∠ − 18.3° A
(b)

The stator power factor is
PF = cos (18.3° ) = 0.949 lagging

(c)

To find the rotor power factor, we must find the impedance angle of the rotor

θ R = tan −1

X2
0.17
= tan −1
= 5.97°
R2 / s
1.625

Therefore the rotor power factor is
PFR = cos5.97° = 0.995 lagging
(d)

The stator copper losses are
PSCL = 3I A2 R1 = 3 (149.4 A ) ( 0.075 Ω ) = 1675 W
2

(e)

The air gap power is PAG = 3 I 2 2

R2
= 3I A2 RF
s
185

R2
, since the only resistance in the original rotor circuit was R2 / s , and
s
the resistance in the Thevenin equivalent circuit is RF . The power consumed by the Thevenin equivalent
circuit must be the same as the power consumed by the original circuit.)
(Note that 3 I A2 RF is equal to 3 I 2 2

PAG = 3 I 2 2
(f)

R2
2
= 3I A 2 RF = 3 (149.4 A ) (1.539 Ω ) = 103 kW
s

The power converted from electrical to mechanical form is
Pconv = (1 − s ) PAG = (1 − 0.04 ) (103 kW ) = 98.9 kW

(g)

The synchronous speed of this motor is
120 f e 120 (50 Hz )
=
= 3000 r/min
2
P
2π rad 1 min
= ( 3000 r/min )
1r
60 s

nsync =

ω sync

Therefore the induced torque in the motor is

τ ind =

(h)

PAG

ω sync

=

103 kW
(3000 r/min ) 2π rad
1r

1 min
60 s

= 327.9 N ⋅ m

The output power of this motor is
POUT = Pconv − Pmech − Pcore − Pmisc = 98.8 kW − 1.0 kW − 1.1 kW − 150 W = 96.6 kW

The output speed is
nm = (1 − s ) nsync = (1 − 0.04) (3000 r/min ) = 2880 r/min
Therefore the load torque is

(i)

(j)

POUT

ωm

=

98.8 kW
(2880 r/min ) 2π rad
1r

= 327.6 N ⋅ m

The overall efficiency is

η=

POUT
POUT
× 100% =
× 100%
PIN
3Vφ I A cosθ

η=

96.6 kW
× 100% = 89.4%
3 ( 254 V )(149.4 A ) cos (18.3°)

The motor speed in revolutions per minute is 2880 r/min. The motor speed in radians per second is

ω m = ( 2880 r/min )
7-15.

1 min
60 s

1r

1 min
60 s

For the motor in Problem 7-14, what is the pullout torque? What is the slip at the pullout torque? What is
the rotor speed at the pullout torque?
SOLUTION The slip at pullout torque is found by calculating the Thevenin equivalent of the input circuit
from the rotor back to the power supply, and then using that with the rotor circuit model.
186

Z TH =

jX M ( R1 + jX 1 ) ( j 7.2 Ω )( 0.075 Ω + j 0.17 Ω )
=
= 0.0731 + j 0.1662 Ω = 0.182 ∠66.3° Ω
R1 + j ( X 1 + X M ) 0.075 Ω + j (0.17 Ω + 7.2 Ω )

VTH =

jX M
( j7.2 Ω )
Vφ =
(254∠0° V ) = 248∠0.06° V
R1 + j ( X 1 + X M )
0.075 Ω + j ( 0.17 Ω + 7.2 Ω )

The slip at pullout torque is
smax =
smax =

R2
RTH + ( X TH + X 2 )
2

2

0.065 Ω

(0.0731 Ω ) + (0.1662 Ω
2

+ 0.17 Ω )

2

= 0.189

The pullout torque of the motor is

τ max =
τ max =

2
3VTH
2
2
2ω sync  RTH + RTH
+ ( X TH + X 2 ) 



3 ( 248 V )
2 ( 314.2 rad/s) 0.0731 Ω +

2

(0.0731 Ω )2 + (0.1662 Ω

+ 0.17 Ω )

2

τ max = 704 N ⋅ m
7-16.

If the motor in Problem 7-14 is to be driven from a 440-V 60-Hz power supply, what will the pullout
torque be? What will the slip be at pullout?
SOLUTION If this motor is driven from a 60 Hz source, the resistances will be unchanged and the reactances
will be increased by a ratio of 6/5. The resulting equivalent circuit is shown below.
IA
+

R1

jX1

0.075 Ω

j0.204 Ω
j8.64 Ω

Vφ

jX2

R2

j0.204 Ω

0.065 Ω

jXM

1− s 
R2 

 s 
1.56 Ω

-

The slip at pullout torque is found by calculating the Thevenin equivalent of the input circuit from the rotor
back to the power supply, and then using that with the rotor circuit model.
Z TH =

jX M ( R1 + jX 1 ) ( j8.64 Ω )( 0.075 Ω + j 0.204 Ω )
=
= 0.0731 + j 0.1994 Ω = 0.212 ∠69.9° Ω
R1 + j ( X 1 + X M ) 0.075 Ω + j ( 0.204 Ω + 8.64 Ω)

VTH =

jX M
( j8.64 Ω )
Vφ =
(254∠0° V ) = 248∠0.05° V
R1 + j ( X 1 + X M )
0.075 Ω + j ( 0.204 Ω + 8.64 Ω )

The slip at pullout torque is
smax =

R2
RTH 2 + ( X TH + X 2 )

2

187

smax =

0.065 Ω

(0.0731 Ω ) + (0.1994 Ω

+ 0.204 Ω )

2

2

= 0.159

The synchronous speed of this motor is
120 f e 120 ( 60 Hz )
=
= 3600 r/min
2
P
2π rad 1 min
= ( 3600 r/min )
1r
60 s

nsync =

ω sync

Therefore the pullout torque of the motor is

τ max =

τ max =

2
3VTH
2
2ω sync RTH + RTH
+ ( X TH + X 2 )

2

3 ( 248 V )
2 ( 377 rad/s) 0.0731 Ω +

2

(0.0731 Ω )2 + (0.1994 Ω

+ 0.204 Ω )

2

τ max = 507 N ⋅ m
7-17.

Plot the following quantities for the motor in Problem 7-14 as slip varies from 0% to 10%: (a) τ ind (b)
Pconv (c) Pout (d) Efficiency η. At what slip does Pout equal the rated power of the machine?
SOLUTION This problem is ideally suited to solution with a MATLAB program. An appropriate program is
shown below. It follows the calculations performed for Problem 7-14, but repeats them at many values of
slip, and then plots the results. Note that it plots all the specified values versus nm , which varies from
2700 to 3000 r/min, corresponding to a range of 0 to 10% slip.
% M-file: prob7_17.m
% M-file create a plot of the induced torque, power
%
converted, power out, and efficiency of the induction
%
motor of Problem 7-14 as a function of slip.
% First, initialize the values needed in this program.
r1 = 0.075;
% Stator resistance
x1 = 0.170;
% Stator reactance
r2 = 0.065;
% Rotor resistance
x2 = 0.170;
% Rotor reactance
xm = 7.2;
% Magnetization branch reactance
v_phase = 440 / sqrt(3);
% Phase voltage
n_sync = 3000;
% Synchronous speed (r/min)
w_sync = 314.2;
% Synchronous speed (rad/s)
p_mech = 1000;
% Mechanical losses (W)
p_core = 1100;
% Core losses (W)
p_misc = 150;
% Miscellaneous losses (W)
% Calculate the Thevenin voltage and impedance from Equations
% 7-41a and 7-43.
v_th = v_phase * ( xm / sqrt(r1^2 + (x1 + xm)^2) );
z_th = ((j*xm) * (r1 + j*x1)) / (r1 + j*(x1 + xm));
r_th = real(z_th);
x_th = imag(z_th);

188

% Now calculate the torque-speed characteristic for many
% slips between 0 and 0.1. Note that the first slip value
% is set to 0.001 instead of exactly 0 to avoid divide% by-zero problems.
s = (0:0.001:0.1);
% Slip
s(1) = 0.001;
nm = (1 - s) * n_sync;
% Mechanical speed
wm = nm * 2*pi/60;
% Mechanical speed
% Calculate torque, P_conv, P_out, and efficiency
% versus speed
for ii = 1:length(s)
% Induced torque
t_ind(ii) = (3 * v_th^2 * r2 / s(ii)) / ...
(w_sync * ((r_th + r2/s(ii))^2 + (x_th + x2)^2) );
% Power converted
p_conv(ii) = t_ind(ii) * wm(ii);
% Power output
p_out(ii) = p_conv(ii) - p_mech - p_core - p_misc;
% Power input
zf = 1 / ( 1/(j*xm) + 1/(r2/s(ii)+j*x2) );
ia = v_phase / ( r1 + j*x1 + zf );
p_in(ii) = 3 * v_phase * abs(ia) * cos(atan(imag(ia)/real(ia)));
% Efficiency
eff(ii) = p_out(ii) / p_in(ii) * 100;
end
% Plot the torque-speed curve
figure(1);
plot(nm,t_ind,'b-','LineWidth',2.0);
xlabel('\bf\itn_{m} \rm\bf(r/min)');
ylabel('\bf\tau_{ind} \rm\bf(N-m)');
title ('\bfInduced Torque versus Speed');
grid on;
% Plot power converted versus speed
figure(2);
plot(nm,p_conv/1000,'b-','LineWidth',2.0);
xlabel('\bf\itn_{m} \rm\bf(r/min)');
ylabel('\bf\itP\rm\bf_{conv} (kW)');
title ('\bfPower Converted versus Speed');
grid on;
% Plot output power versus speed
figure(3);
plot(nm,p_out/1000,'b-','LineWidth',2.0);
xlabel('\bf\itn_{m} \rm\bf(r/min)');
ylabel('\bf\itP\rm\bf_{out} (kW)');
title ('\bfOutput Power versus Speed');
axis([2700 3000 0 180]);

189

grid on;
% Plot the efficiency
figure(4);
plot(nm,eff,'b-','LineWidth',2.0);
xlabel('\bf\itn_{m} \rm\bf(r/min)');
ylabel('\bf\eta (%)');
title ('\bfEfficiency versus Speed');
grid on;

The four plots are shown below:
Induced Torque versus Speed
700

600

400

τ

ind

(N-m)

500

300

200

100

0
2700

2750

2800

n

m

190

2850
(r/min)

2900

2950

3000

This machine is rated at 75 kW. It produces an output power of 75 kW at 3.1% slip, or a speed of 2907
r/min.

7-18.

A 208-V, 60 Hz, six-pole Y-connected 25-hp design class B induction motor is tested in the laboratory,
with the following results:

208 V, 22.0 A, 1200 W, 60 Hz

Locked rotor:

24.6 V, 64.5 A, 2200 W, 15 Hz

DC test:

13.5 V, 64 A

Find the equivalent circuit of this motor, and plot its torque-speed characteristic curve.
191

SOLUTION From the DC test,
2 R1 =

13.5 V
64 A

⇒

R1 = 0.105 Ω
IDC
+
R1
VDC
R1

R1

-

In the no-load test, the line voltage is 208 V, so the phase voltage is 120 V. Therefore,
X1 + X M =

Vφ
I A,nl

=

120 V
= 5.455 Ω @ 60 Hz
22.0 A

In the locked-rotor test, the line voltage is 24.6 V, so the phase voltage is 14.2 V. From the locked-rotor
test at 15 Hz,
Z LR
′ = RLR + jX LR
′ =

θ LR
′ = cos−1

Vφ
I A,LR

PLR
= cos−1
S LR

=

14.2 V
= 0.2202 Ω
64.5 A

2200 W
= 36.82°
3 ( 24.6 V )(64.5 A )

Therefore,
RLR = Z LR
′ cos θ LR = ( 0.2202 Ω ) cos ( 36.82° ) = 0.176 Ω

⇒
⇒

R1 + R2 = 0.176 Ω
R2 = 0.071 Ω

X LR
′ = Z LR
′ sinθ LR = ( 0.2202 Ω ) sin ( 36.82°) = 0.132 Ω

At a frequency of 60 Hz,
X LR =

60 Hz
15 Hz

X LR
′ = 0.528 Ω

For a Design Class B motor, the split is X 1 = 0.211 Ω and X 2 = 0.317 Ω . Therefore,
X M = 5.455 Ω − 0.211 Ω = 5.244 Ω
The resulting equivalent circuit is shown below:

192

IA
+

Vφ

R1

jX1

0.105 Ω

j0.211 Ω
j5.244 Ω

jX2

R2

j0.317 Ω

0.071 Ω

jXM

I2
1 − s 
R2 

 s 

-

A MATLAB program to calculate the torque-speed characteristic of this motor is shown below:
% M-file: prob7_18.m
% M-file create a plot of the torque-speed curve of the
%
induction motor of Problem 7-18.
% First, initialize the values needed in this program.
r1 = 0.105;
% Stator resistance
x1 = 0.211;
% Stator reactance
r2 = 0.071;
% Rotor resistance
x2 = 0.317;
% Rotor reactance
xm = 5.244;
% Magnetization branch reactance
v_phase = 208 / sqrt(3);
% Phase voltage
n_sync = 1200;
% Synchronous speed (r/min)
w_sync = 125.7;
% Synchronous speed (rad/s)
% Calculate the Thevenin voltage and impedance from Equations
% 7-41a and 7-43.
v_th = v_phase * ( xm / sqrt(r1^2 + (x1 + xm)^2) );
z_th = ((j*xm) * (r1 + j*x1)) / (r1 + j*(x1 + xm));
r_th = real(z_th);
x_th = imag(z_th);
% Now calculate the torque-speed characteristic for many
% slips between 0 and 1. Note that the first slip value
% is set to 0.001 instead of exactly 0 to avoid divide% by-zero problems.
s = (0:1:50) / 50;
% Slip
s(1) = 0.001;
nm = (1 - s) * n_sync;
% Mechanical speed
% Calculate torque versus speed
for ii = 1:51
t_ind(ii) = (3 * v_th^2 * r2 / s(ii)) / ...
(w_sync * ((r_th + r2/s(ii))^2 + (x_th + x2)^2) );
end
% Plot the torque-speed curve
figure(1);
plot(nm,t_ind,'b-','LineWidth',2.0);
xlabel('\bf\itn_{m}');
ylabel('\bf\tau_{ind}');
title ('\bfInduction Motor Torque-Speed Characteristic');
grid on;

193

The resulting plot is shown below:

7-19.

A 460-V, four-pole, 50-hp, 60-Hz, Y-connected three-phase induction motor develops its full-load induced
torque at 3.8 percent slip when operating at 60 Hz and 460 V. The per-phase circuit model impedances of
the motor are
R1 = 0.33 Ω

X M = 30 Ω

X 1 = 0.42 Ω

X 2 = 0.42 Ω

Mechanical, core, and stray losses may be neglected in this problem.
(a) Find the value of the rotor resistance R2 .
(b) Find τ max , smax , and the rotor speed at maximum torque for this motor.
(c) Find the starting torque of this motor.
(d) What code letter factor should be assigned to this motor?
SOLUTION The equivalent circuit for this motor is
IA
+

Vφ

R1

jX1

0.33 Ω

j0.42 Ω
j30 Ω

jX2

R2

j0.42 Ω

??? Ω

jXM

I2

1− s 
R2 

 s 

-

The Thevenin equivalent of the input circuit is:
Z TH =

jX M ( R1 + jX 1 ) ( j 30 Ω )( 0.33 Ω + j 0.42 Ω )
=
= 0.321 + j 0.418 Ω = 0.527∠52.5° Ω
R1 + j ( X 1 + X M ) 0.33 Ω + j ( 0.42 Ω + 30 Ω )
194

VTH =

jX M
( j30 Ω )
Vφ =
(265.6∠0° V ) = 262∠0.6° V
R1 + j ( X 1 + X M )
0.33 Ω + j ( 0.42 Ω + 30 Ω)

(a) If losses are neglected, the induced torque in a motor is equal to its load torque. At full load, the
output power of this motor is 50 hp and its slip is 3.8%, so the induced torque is
nm = (1 − 0.038)(1800 r/min ) = 1732 r/min

τ ind = τ load =

(50 hp)(746 W/hp)
(1732 r/min ) 2π rad 1min
1r

= 205.7 N ⋅ m

60 s

The induced torque is given by the equation

τ ind =

2
3VTH
R2 / s

ω sync ( RTH + R2 / s ) + ( X TH + X 2 )
2

2

Substituting known values and solving for R2 / s yields
3 ( 262 V ) R2 / s
2

205.7 N ⋅ m =

38,774 =

(188.5 rad/s) (0.321 + R2 / s ) + (0.418 + 0.42 )2
2

205,932 R2 / s

(0.321 + R2 / s )2 + 0.702

(0.321 + R2 / s )2 + 0.702

= 5.311 R2 / s

0.103 + 0.642 R2 / s + ( R2 / s ) + 0.702 = 5.311 R2 / s
2

R2
s

2

− 4.669

R2
+ 0.702 = 0
s

R2
= 0.156, 4.513
s

R2 = 0.0059 Ω, 0.172 Ω
These two solutions represent two situations in which the torque-speed curve would go through this specific
torque-speed point. The two curves are plotted below. As you can see, only the 0.172 Ω solution is
realistic, since the 0.0059 Ω solution passes through this torque-speed point at an unstable location on the
back side of the torque-speed curve.

195

Induction Motor Torque-Speed Characteristic
450
R2 = 0.0059 ohms
R2 = 0.172 ohms

400

350

300

τ

ind

250

200

150

100

50

0
1600

1620

1640

1660

1680

1700
n

1720

1740

1760

1780

1800

m

(b) The slip at pullout torque can be found by calculating the Thevenin equivalent of the input circuit
from the rotor back to the power supply, and then using that with the rotor circuit model. The Thevenin
equivalent of the input circuit was calculate in part (a). The slip at pullout torque is
smax =
smax =

R2
RTH + ( X TH + X 2 )
2

2

0.172 Ω

(0.321 Ω) + (0.418 Ω
2

+ 0.420 Ω )

2

= 0.192

The rotor speed a maximum torque is
npullout = (1 − s ) nsync = (1 − 0.192 )(1800 r/min ) = 1454 r/min

and the pullout torque of the motor is

τ max =

τ max =

2
3VTH
2
2ω sync RTH + RTH
+ ( X TH + X 2 )

2

3 ( 262 V )
2 (188.5 rad/s) 0.321 Ω +

2

(0.321 Ω )2 + (0.418 Ω

+ 0.420 Ω )

2

τ max = 448 N ⋅ m
(c)

The starting torque of this motor is the torque at slip s = 1. It is

τ ind =

2
3VTH
R2 / s

ω sync ( RTH + R2 / s ) + ( X TH + X 2 )
2

2

3 ( 262 V ) (0.172 Ω )
2

τ ind =

(188.5 rad/s) (0.321 + 0.172 Ω )2 + (0.418 + 0.420)2
196

= 199 N ⋅ m

(d) To determine the starting code letter, we must find the locked-rotor kVA per horsepower, which is
equivalent to finding the starting kVA per horsepower. The easiest way to find the line current (or
armature current) at starting is to get the equivalent impedance Z F of the rotor circuit in parallel with
jX M at starting conditions, and then calculate the starting current as the phase voltage divided by the sum
of the series impedances, as shown below.
IA,start
+

R1

jX1

0.33 Ω

j0.42 Ω

jXF

RF

Vφ
-

The equivalent impedance of the rotor circuit in parallel with jX M at starting conditions (s = 1.0) is:
1
1
Z F ,start =
=
= 0.167 + j 0.415 = 0.448∠68.1° Ω
1
1
1
1
+
+
jX M Z 2
j 30 Ω 0.172 + j 0.42
The phase voltage is 460/ 3 = 266 V, so line current I L,start is

I L,start = I A =

Vφ
R1 + jX 1 + RF + jX F

=

266∠0° V
0.33 Ω + j 0.42 Ω + 0.167 Ω + j 0.415 Ω

I L ,start = I A = 274 ∠ − 59.2° A

Therefore, the locked-rotor kVA of this motor is
S = 3 VT I L ,rated = 3 ( 460 V )( 274 A ) = 218 kVA

and the kVA per horsepower is
kVA/hp =

218 kVA
= 4.36 kVA/hp
50 hp

This motor would have starting code letter D, since letter D covers the range 4.00-4.50.

7-20.

Answer the following questions about the motor in Problem 7-19.
(a) If this motor is started from a 460-V infinite bus, how much current will flow in the motor at starting?
(b) If transmission line with an impedance of 0.35 + j0.25 Ω per phase is used to connect the induction
motor to the infinite bus, what will the starting current of the motor be? What will the motor’s terminal
voltage be on starting?
(c) If an ideal 1.4:1 step-down autotransformer is connected between the transmission line and the motor,
what will the current be in the transmission line during starting? What will the voltage be at the motor
end of the transmission line during starting?
SOLUTION
(a)

The equivalent circuit of this induction motor is shown below:

197

IA
+

R1

jX1

0.33 Ω

j0.42 Ω
j30 Ω

Vφ

jX2

R2

j0.42 Ω

0.172 Ω

I2

1− s 
R2 

 s 

jXM

-

The easiest way to find the line current (or armature current) at starting is to get the equivalent impedance
Z F of the rotor circuit in parallel with jX M at starting conditions, and then calculate the starting current
as the phase voltage divided by the sum of the series impedances, as shown below.
IA
+

R1

jX1

0.33 Ω

j0.42 Ω

jXF

RF

Vφ
-

The equivalent impedance of the rotor circuit in parallel with jX M at starting conditions (s = 1.0) is:
1
1
ZF =
=
= 0.167 + j0.415 = 0.448∠68.0° Ω
1
1
1
1
+
+
jX M Z 2
j 30 Ω 0.172 + j 0.42
The phase voltage is 460/ 3 = 266 V, so line current I L is
Vφ
266∠0° V
=
IL = I A =
R1 + jX 1 + RF + jX F 0.33 Ω + j0.42 Ω + 0.167 Ω + j 0.415 Ω

I L = I A = 273∠ − 59.2° A
(b) If a transmission line with an impedance of 0.35 + j0.25 Ω per phase is used to connect the induction
motor to the infinite bus, its impedance will be in series with the motor’s impedances, and the starting
current will be
Vφ ,bus
IL = I A =
Rline + jX line + R1 + jX 1 + RF + jX F
266∠0° V
0.35 Ω + j 0.25 Ω + 0.33 Ω + j 0.42 Ω + 0.167 Ω + j0.415 Ω
I L = I A = 193.2∠ − 52.0° A

IL = I A =

The voltage at the terminals of the motor will be

Vφ = I A ( R1 + jX 1 + RF + jX F )

Vφ = (194.1∠ − 52.3° A )(0.33 Ω + j 0.42 Ω + 0.167 Ω + j 0.415 Ω )
Vφ = 187.7∠7.2° V

Therefore, the terminal voltage will be

3 (187.7 V ) = 325 V . Note that the terminal voltage sagged by

about 30% during motor starting, which would be unacceptable.
198

(c) If an ideal 1.4:1 step-down autotransformer is connected between the transmission line and the motor,
the motor’s impedances will be referred across the transformer by the square of the turns ratio a = 1.4. The
referred impedances are
R1′ = a 2 R1 = 1.96 ( 0.33 Ω ) = 0.647 Ω

X 1′ = a 2 X 1 = 1.96 (0.42 Ω ) = 0.823 Ω

RF′ = a 2 RF = 1.96 ( 0.167 Ω) = 0.327 Ω

X F′ = a 2 X F = 1.96 (0.415 Ω ) = 0.813 Ω

Therefore, the starting current referred to the primary side of the transformer will be
Vφ ,bus
I′L = I′A =
Rline + jX line + R1′ + jX 1′ + RF′ + jX F′
266∠ 0° V
I ′L = I ′A =
0.35 Ω + j0.25 Ω + 0.647 Ω + j 0.823 Ω + 0.327 Ω + j 0.813 Ω
I′L = I′A = 115.4∠ − 54.9° A
The voltage at the motor end of the transmission line would be the same as the referred voltage at the
terminals of the motor
Vφ′ = I ′A ( R1′ + jX 1′ + RF′ + jX F′ )

Vφ = (115.4∠ − 54.9° A )( 0.647 Ω + j0.823 Ω + 0.327 Ω + j0.813 Ω )

Vφ = 219.7 ∠4.3° V
Therefore, the line voltage at the motor end of the transmission line will be

3 ( 219.7 V ) = 380.5 V . Note

that this voltage sagged by 17.3% during motor starting, which is less than the 30% sag with case of
across-the-line starting.

7-21.

In this chapter, we learned that a step-down autotransformer could be used to reduce the starting current
drawn by an induction motor. While this technique works, an autotransformer is relatively expensive. A
much less expensive way to reduce the starting current is to use a device called Y-∆ starter. If an induction
motor is normally ∆-connected, it is possible to reduce its phase voltage Vφ (and hence its starting current)
by simply re-connecting the stator windings in Y during starting, and then restoring the connections to ∆
when the motor comes up to speed. Answer the following questions about this type of starter.
(a) How would the phase voltage at starting compare with the phase voltage under normal running
conditions?
(b) How would the starting current of the Y-connected motor compare to the starting current if the motor
remained in a ∆-connection during starting?
SOLUTION
(a) The phase voltage at starting would be 1 /
conditions.

3 = 57.7% of the phase voltage under normal running

(b) Since the phase voltage decreases to 1 / 3 = 57.7% of the normal voltage, the starting phase current
will also decrease to 57.7% of the normal starting current. However, since the line current for the original
delta connection was 3 times the phase current, while the line current for the Y starter connection is
equal to its phase current, the line current is reduced by a factor of 3 in a Y-∆ starter.
For the ∆-connection:

I L ,∆ = 3 I φ ,∆

199

I L,Y = I φ ,Y

For the Y-connection:
But I φ ,∆ =

7-22.

3I φ ,Y , so I L ,∆ = 3I L ,Y

A 460-V 100-hp four-pole ∆-connected 60-Hz three-phase induction motor has a full-load slip of 5 percent,
an efficiency of 92 percent, and a power factor of 0.87 lagging. At start-up, the motor develops 1.9 times
the full-load torque but draws 7.5 times the rated current at the rated voltage. This motor is to be started
with an autotransformer reduced voltage starter.
(a) What should the output voltage of the starter circuit be to reduce the starting torque until it equals the
rated torque of the motor?
(b) What will the motor starting current and the current drawn from the supply be at this voltage?
SOLUTION
(a)

The starting torque of an induction motor is proportional to the square of VTH ,
2

τ start2
V
= TH2
τ start1
VTH1

V
= T2
VT 1

2

If a torque of 1.9 τ rated is produced by a voltage of 460 V, then a torque of 1.00 τ rated would be produced
by a voltage of
1.00 τ rated
VT 2
=
1.90 τ rated
460 V

(460 V )

2

2

VT 2 =
(b)

1.90

= 334 V

The motor starting current is directly proportional to the starting voltage, so
334 V
460 V

I L2 =

I L1 = ( 0.726 ) I L1 = ( 0.726 ) (7.5I rated ) = 5.445 I rated

The input power to this motor is
PIN =

POUT

η

=

(100 hp )(746 W/hp ) = 81.1 kW
0.92

The rated current is equal to
I rated =

PIN
(81.1 kW ) = 117 A
=
3 VT PF
3 ( 460 V )( 0.87 )

Therefore, the motor starting current is
I L 2 = 5.445 I rated = (5.445)(117 A ) = 637 A

The turns ratio of the autotransformer that produces this starting voltage is
N SE + N C 460 V
=
= 1.377
NC
334 V

so the current drawn from the supply will be

200

I line =

7-23.

I start
637 A
=
= 463 A
1.377 1.377

A wound-rotor induction motor is operating at rated voltage and frequency with its slip rings shorted and
with a load of about 25 percent of the rated value for the machine. If the rotor resistance of this machine is
doubled by inserting external resistors into the rotor circuit, explain what happens to the following:
(a) Slip s
(b) Motor speed nm
(c) The induced voltage in the rotor
(d) The rotor current
(e) τ ind
(f) Pout
(g) PRCL
(h) Overall efficiency η
SOLUTION
(a)

The slip s will increase.

(b)

The motor speed nm will decrease.

(c)

The induced voltage in the rotor will increase.

(d)

The rotor current will increase.

(e) The induced torque will adjust to supply the load’s torque requirements at the new speed. This will
depend on the shape of the load’s torque-speed characteristic. For most loads, the induced torque will
decrease.

(f)

The output power will generally decrease: POUT = τ ind ↓ ω m ↓

(g)

The rotor copper losses (including the external resistor) will increase.
201

(h)

7-24.

The overall efficiency η will decrease.

Answer the following questions about a 460-V ∆-connected two-pole 75-hp 60-Hz starting code letter E
induction motor:
(a) What is the maximum current starting current that this machine’s controller must be designed to
handle?
(b) If the controller is designed to switch the stator windings from a ∆ connection to a Y connection during
starting, what is the maximum starting current that the controller must be designed to handle?
(c) If a 1.25:1 step-down autotransformer starter is used during starting, what is the maximum starting
current that will be drawn from the line?
SOLUTION
(a) Starting code letter E corresponds to a 4.50 – 5.00 kVA/hp, so the maximum starting kVA of this
motor is
Sstart = ( 75 hp )( 5.00 ) = 375 kVA

Therefore,
I start =

S
375 kVA
=
= 471 A
3 VT
3 ( 460 V )

(b) The line voltage will still be 460 V when the motor is switched to the Y-connection, but now the
phase voltage will be 460 / 3 = 266 V.

Before (in ∆):
I φ ,∆ =

Vφ ,∆

( RTH + R2 ) + j ( X TH + X 2 )

But the line current in a ∆ connection is
I L ,∆ = 3I φ ,∆ =

=

460 V
( RTH + R2 ) + j ( X TH + X 2 )

3 times the phase current, so

3Vφ ,∆

=

797 V

( RTH + R2 ) + j ( X TH + X 2 ) ( RTH + R2 ) + j ( X TH + X 2 )

After (in Y):
I L,Y = I φ ,Y =

Vφ ,Y

( RTH + R2 ) + j ( X TH + X 2 )

=

265.6 V
( RTH + R2 ) + j ( X TH + X 2 )

Therefore the line current will decrease by a factor of 3 when using this starter. The starting current with a
∆-Y starter is
I start =

471 A
= 157 A
3

(c) A 1.25:1 step-down autotransformer reduces the phase voltage on the motor by a factor 0.8. This
reduces the phase current and line current in the motor (and on the secondary side of the transformer) by a
factor of 0.8. However, the current on the primary of the autotransformer will be reduced by another factor
of 0.8, so the total starting current drawn from the line will be 64% of its original value. Therefore, the
maximum starting current drawn from the line will be
I start = ( 0.64 )( 471 A ) = 301 A
202

7-25.

When it is necessary to stop an induction motor very rapidly, many induction motor controllers reverse the
direction of rotation of the magnetic fields by switching any two stator leads. When the direction of
rotation of the magnetic fields is reversed, the motor develops an induced torque opposite to the current
direction of rotation, so it quickly stops and tries to start turning in the opposite direction. If power is
removed from the stator circuit at the moment when the rotor speed goes through zero, then the motor has
been stopped very rapidly. This technique for rapidly stopping an induction motor is called plugging. The
motor of Problem 7-19 is running at rated conditions and is to be stopped by plugging.
(a) What is the slip s before plugging?
(b) What is the frequency of the rotor before plugging?
(c) What is the induced torque τ ind before plugging?
(d) What is the slip s immediately after switching the stator leads?
(e) What is the frequency of the rotor immediately after switching the stator leads?
(f) What is the induced torque τ ind immediately after switching the stator leads?
SOLUTION
(a)

The slip before plugging is 0.038 (see Problem 7-19).

(b)

The frequency of the rotor before plugging is f r = sf e = ( 0.038)( 60 Hz ) = 2.28 Hz

(c)

The induced torque before plugging is 205.7 N⋅m in the direction of motion (see Problem 7-19).

(d) After switching stator leads, the synchronous speed becomes –1800 r/min, while the mechanical speed
initially remains 1732 r/min. Therefore, the slip becomes
s=

nsync − nm
nsync

=

−1800 − 1732
= 1.962
−1800

(e)

The frequency of the rotor after plugging is f r = sf e = (1.962 )( 60 Hz ) = 117.72 Hz

(f)

The induced torque immediately after switching the stator leads is

τ ind =

2
3VTH
R2 / s

ω sync ( RTH + R2 / s ) + ( X TH + X 2 )
2

2

3 ( 262 V ) (0.172 Ω /1.962 )
2

τ ind =

(188.5 rad/s) (0.321 + 0.172 Ω /1.962)2 + (0.418 + 0.420)2
3 ( 262 V ) (0.0877 )
2

τ ind =

(188.5 rad/s) (0.321 + 0.0877)2 + ( 0.418 + 0.420)2

τ ind = 110 N ⋅ m, opposite the direction of motion

203

Chapter 8: DC Machinery Fundamentals
8-1.

The following information is given about the simple rotating loop shown in Figure 8-6:
B = 0.8 T

VB = 24 V

l = 0.5 m

R = 0.4 Ω

ω = 250 rad/s
(a) Is this machine operating as a motor or a generator? Explain.
r = 0.125 m

(b) What is the current i flowing into or out of the machine? What is the power flowing into or out of the
machine?
(c) If the speed of the rotor were changed to 275 rad/s, what would happen to the current flow into or out
of the machine?
(d) If the speed of the rotor were changed to 225 rad/s, what would happen to the current flow into or out
of the machine?

204

(a)

If the speed of rotation ω of the shaft is 500 rad/s, then the voltage induced in the rotating loop will be
eind = 2 rlBω

eind = 2 ( 0.125 m )( 0.5 m )(0.8 T )( 250 rad/s) = 25 V
Since the external battery voltage is only 24 V, this machine is operating as a generator, charging the
battery.
(b)

The current flowing out of the machine is approximately
i=

eind − VB
25 V − 24 V
=
= 2.5 A
R
0.4 Ω

(Note that this value is the current flowing while the loop is under the pole faces. When the loop goes
beyond the pole faces, eind will momentarily fall to 0 V, and the current flow will momentarily reverse.
Therefore, the average current flow over a complete cycle will be somewhat less than 2.5 A.)
(c)

If the speed of the rotor were increased to 275 rad/s, the induced voltage of the loop would increase to
eind = 2 rlBω

eind = 2 ( 0.125 m )( 0.5 m)( 0.8 T )( 275 rad/s) = 27.5 V
and the current flow out of the machine will increase to
i=

(d)

eind − VB
27.5 V − 24 V
=
= 8.75 A
R
0.4 Ω

If the speed of the rotor were decreased to 450 rad/s, the induced voltage of the loop would fall to
eind = 2 rlBω

eind = 2 ( 0.125 m )( 0.5 m )( 0.8 T )( 225 rad/s ) = 22.5 V

Here, eind is less than VB , so current flows into the loop and the machine is acting as a motor. The current
flow into the machine would be
i=

8-2.

VB − eind
24 V - 22.5 V
=
= 3.75 A
R
0.4 Ω

Refer to the simple two-pole eight-coil machine shown in Figure P8-1. The following information is given

B = 10
. T in air gap
l = 0.3 m (length of coil sides)
r = 0.08 m (radius of coils)
n = 1700 r/min CCW
The resistance of each rotor coil is 0.04 Ω.
(a) Is the armature winding shown a progressive or retrogressive winding?
(b) How many current paths are there through the armature of this machine?
(c) What are the magnitude and the polarity of the voltage at the brushes in this machine?
(d) What is the armature resistance RA of this machine?
205

(e) If a 10 Ω resistor is connected to the terminals of this machine, how much current flows in the
machine? Consider the internal resistance of the machine in determining the current flow.
(f) What are the magnitude and the direction of the resulting induced torque?
(g) Assuming that the speed of rotation and magnetic flux density are constant, plot the terminal voltage of
this machine as a function of the current drawn from it.

SOLUTION

(a) This winding is progressive, since the ends of each coil are connected to the commutator segments
ahead of the segments that the beginnings of the coils are connected to.
(b) There are two current paths in parallel through the armature of this machine (this is a simplex lap
winding).
(c) The voltage is positive at brush x with respect to brush y, since the voltage in the conductors is
positive out of the page under the North pole face and positive into the page under the South pole face.
(d)

There are 8 coils on this machine in two parallel paths, with each coil having a resistance of 0.04 Ω.
Therefore, the total resistance R A is
RA =

(0.04 Ω

+ 0.04 Ω + 0.04 Ω + 0.04 Ω )(0.04 Ω + 0.04 Ω + 0.04 Ω + 0.04 Ω )
0.04 Ω + 0.04 Ω + 0.04 Ω + 0.04 Ω + 0.04 Ω + 0.04 Ω + 0.04 Ω + 0.04 Ω

206

R A = 0.08 Ω
(e)

The voltage produced by this machine can be found from Equations 8-32 and 8-33:
ZvBl Zrω Bl
=
a
a

EA =

where Z is the number of conductors under the pole faces, since the ones between the poles have no voltage
in them. There are 16 conductors in this machine, and about 12 of them are under the pole faces at any
given time.

ω = (1700 r/min )

1r

1 min
60 s

Zrω Bl (12 cond )( 0.08 m )(178 rad/s )(1.0 T )( 0.3 m )
=
= 25.6 V
2 current paths
a

EA =

Therefore, the current flowing in the machine will be
IA =

(f)

EA
25.6 V
=
= 2.54 A
RA + Rload 0.08 Ω + 10 Ω

The induced torque is given by Equation 8-46:

τ ind =

ZrlBI A (12 cond )( 0.08 m )( 0.3 m )(1.0 T )( 2.54 A )
=
a
2 current paths

τ ind = 0.366 N ⋅ m, CW (opposite to the direction of rotation)
8-3.

Prove that the equation for the induced voltage of a single simple rotating loop
2
(8-6)
eind = φ ω
π
is just a special case of the general equation for induced voltage in a dc machine
EA = K φ ω

(8-38)

SOLUTION From Equation 8-38,

where

EA = K φ ω
ZP
K=
2π a

For the simple rotation loop,
Z = 2 (There are 2 conductors)
P = 2 (There are 2 poles)
a = 1 (There is one current path through the machine)
Therefore,
K=

ZP ( 2 ) ( 2 ) 2
=
2π a 2π (1) π

and Equation 8-38 reduces to Equation 8-6.

8-4.

A dc machine has 8 poles and a rated current of 100 A. How much current will flow in each path at rated
conditions if the armature is (a) simplex lap-wound, (b) duplex lap-wound, (c) simplex wave-wound?
SOLUTION
207

(a)

Simplex lap-wound:
a = mP = (1)(8) = 8 paths

Therefore, the current per path is
I=
(b)

I A 100 A
=
= 12.5 A
a
8

Duplex lap-wound:
a = mP = ( 2 )(8) = 16 paths
Therefore, the current per path is
I=

(c)

I A 100 A
=
= 6.25 A
a
16

Simplex wave-wound:
a = 2 m = ( 2 )(1) = 2 paths

Therefore, the current per path is
I=

8-5.

I A 100 A
=
= 50 A
a
2

How many parallel current paths will there be in the armature of a 12-pole machine if the armature is (a)
simplex lap-wound, (b) duplex wave-wound, (c) triplex lap-wound, (d) quadruplex wave-wound?
SOLUTION
(a)

Simplex lap-wound:
a = mP = (1)(12) = 12 paths

(b)

Duplex wave-wound:

a = 2m = (2)(2) = 4 paths
(c)

Triplex lap-wound:

a = mP = (3)(12) = 36 paths
(d)

a = 2m = (2)(4) = 8 paths
8-6.

The power converted from one form to another within a dc motor was given by
Pconv = E A I A = τ indω m
Use the equations for E A and τ ind [Equations (8-38) and (8-49)] to prove that E A I A = τ ind ω m ; that is,
prove that the electric power disappearing at the point of power conversion is exactly equal to the
mechanical power appearing at that point.
SOLUTION
Pconv = E A I A
Substituting Equation (8-38) for E A
208

Pconv = ( K φ ω ) I A
Pconv = ( K φ I A ) ω
But from Equation (8-49), τ ind = K φ I A , so
Pconv = τ indω

8-7.

An eight-pole, 25-kW, 120-V DC generator has a duplex lap-wound armature, which has 64 coils with 16
turns per coil. Its rated speed is 2400 r/min.
(a) How much flux per pole is required to produce the rated voltage in this generator at no-load conditions?
(b) What is the current per path in the armature of this generator at the rated load?
(c) What is the induced torque in this machine at the rated load?
(d) How many brushes must this motor have? How wide must each one be?
(e) If the resistance of this winding is 0.011 Ω per turn, what is the armature resistance R A of this
machine?
SOLUTION
(a)

E A = Kφω =

ZP
φω
2π a

In this machine, the number of current paths is
a = mP = ( 2 )(8) = 16
The number of conductor is
Z = (64 coils )(16 turns/coil )( 2 conductors/turn ) = 2048

The equation for induced voltage is
ZP
φω
2π a
so the required flux is
(2048 cond )(8 poles ) φ 2400 r/min 2π rad
120 V =
(
)
2π (16 paths)
1r
120 V = 40,960 φ
φ = 0.00293 Wb
EA =

(b)

1 min
60 s

At rated load, the current flow in the generator would be
IA =

25 kW
= 208 A
120 V

There are a = m P = (2)(8) = 16 parallel current paths through the machine, so the current per path is
I=

(c)

I A 208 A
=
= 13 A
a
16

The induced torque in this machine at rated load is

τ ind =

ZP
φI A
2π a
209

τ ind =

(2048 cond )(8 poles) 0.00293 Wb 208 A
(
)(
)
2π (16 paths)

τ ind = 99.3 N ⋅ m

(d) This motor must have 8 brushes, since it is lap-wound and has 8 poles. Since it is duplex-wound,
each brush must be wide enough to stretch across 2 complete commutator segments.
(e)

There are a total of 1024 turns on the armature of this machine, so the number of turns per path is
NP =

1024 turns
= 64 turns/path
16 paths

The total resistance per path is RP = ( 64 )( 0.011 Ω) = 0.704 Ω . Since there are 16 parallel paths through
the machine, the armature resistance of the generator is
RA =

8-8.

0.704 Ω
= 0.044 Ω
16 paths

Figure P8-2 shows a small two-pole dc motor with eight rotor coils and four turns per coil. The flux per
pole in this machine is 0.0125 Wb.
(a) If this motor is connected to a 12-V dc car battery, what will the no-load speed of the motor be?
(b) If the positive terminal of the battery is connected to the rightmost brush on the motor, which way will
it rotate?
(c) If this motor is loaded down so that it consumes 50 W from the battery, what will the induced torque of
the motor be? (Ignore any internal resistance in the motor.)

SOLUTION
(a) At no load, VT = E A = Kφω . If K is known, then the speed of the motor can be found. The constant
K is given by
ZP
K=
2π a
On the average, about 6 of the 8 coils are under the pole faces at any given time, so the average number of
active conductors is
Z = (6 coils)(4 turns/coil)(2 conductors/turn) = 48 conductors
210

There are two poles and two current paths, so
ZP ( 48 cond )( 2 poles )
K=
=
= 7.64
2π a
2π ( 2 paths)
The speed is given by

ω=

EA
12 V
=
K φ ( 7.64 )( 0.0125 Wb )

nm = (125.6 rad/s )

1r

60 s
= 1200 r/min
1 min

(b) If the positive terminal of the battery is connected to the rightmost brush, current will flow into the
page under the South pole face, producing a CW torque ⇒ CW rotation.
(c)

If the motor consumes 50 W from the battery, the current flow is
I=

P 50 W
=
= 4.17 A
VB 12 V

Therefore, the induced torque will be

τ ind = Kφ I A = ( 7.64 )( 0.0125 Wb )(4.17 A ) = 0.40 N ⋅ m, CW
8-9.

Refer to the machine winding shown in Figure P8-3.
(a) How many parallel current paths are there through this armature winding?
(b) Where should the brushes be located on this machine for proper commutation? How wide should they
be?
(c) What is the plex of this machine?
(d) If the voltage on any single conductor under the pole faces in this machine is e, what is the voltage at
the terminals of this machine?

211

SOLUTION
(a)

This is a duplex, two-pole, lap winding, so there are 4 parallel current paths through the rotor.

(b) The brushes should be shorting out those windings lying between the two poles. At the time shown,
those windings are 1, 2, 9, and 10. Therefore, the brushes should be connected to short out commutator
segments b-c-d and j-k-l at the instant shown in the figure. Each brush should be two commutator
segments wide, since this is a duplex winding.
(c)

Duplex (see above)

(d) There are 16 coils on the armature of this machine. Of that number, an average of 14 of them would
be under the pole faces at any one time. Therefore, there are 28 conductors divided among 4 parallel paths,
which produces 7 conductors per path. Therefore, E A = 7e = VT for no-load conditions.
212

8-10.

Describe in detail the winding of the machine shown in Figure P8-4. If a positive voltage is applied to the
brush under the North pole face, which way will this motor rotate?

SOLUTION This is a 2-pole, retrogressive, lap winding. If a positive voltage is applied to the brush under
the North pole face, the rotor will rotate in a counterclockwise direction.

213

Chapter 9: DC Motors and Generators
Problems 9-1 to 9-12 refer to the following dc motor:
Prated = 15 hp
I L ,rated = 55 A

VT = 240 V
nrated = 1200 r/min
RA = 0.40 Ω
RS = 0.04 Ω

N F = 2700 turns per pole
N SE = 27 turns per pole
RF = 100 Ω
Radj = 100 to 400 Ω

Rotational losses = 1800 W at full load. Magnetization curve as shown in Figure P9-1.

Note:

An electronic version of this magnetization curve can be found in file
p91_mag.dat, which can be used with MATLAB programs. Column 1
contains field current in amps, and column 2 contains the internal generated
voltage EA in volts.
In Problems 9-1 through 9-7, assume that the motor described above can be connected in shunt. The equivalent
circuit of the shunt motor is shown in Figure P9-2.
214

Note:

9-1.

Figure P9-2 shows incorrect values for RA and RF in the first printing of this
book. The correct values are given in the text, but shown incorrectly on the
figure. This will be corrected at the second printing.

If the resistor Radj is adjusted to 175 Ω what is the rotational speed of the motor at no-load conditions?
SOLUTION At no-load conditions, E A = VT = 240 V . The field current is given by
IF =

VT
240 V
240 V
=
=
= 0.873 A
Radj + RF 175 Ω + 100 Ω 250 Ω

From Figure P9-1, this field current would produce an internal generated voltage E Ao of 271 V at a speed

no of 1200 r/min. Therefore, the speed n with a voltage E A of 240 V would be
EA
n
=
E Ao no

n=

9-2.

EA
E Ao

no =

240 V
(1200 r/min ) = 1063 r/min
271 V

Assuming no armature reaction, what is the speed of the motor at full load? What is the speed regulation of
the motor?
SOLUTION At full load, the armature current is
IA = IL − IF = IL −

VT
= 55 A − 0.87 A = 54.13 A

The internal generated voltage E A is
E A = VT − I A R A = 240 V − (54.13 A )( 0.40 Ω ) = 218.3 V

The field current is the same as before, and there is no armature reaction, so E Ao is still 271 V at a speed

no of 1200 r/min. Therefore,
n=

EA
E Ao

no =

218.3 V
(1200 r/min ) = 967 r/min
271 V

The speed regulation is
SR =

1063 r/min − 967 r/min
nnl − nfl
× 100% =
× 100% = 9.9%
nfl
967 r/min

215

9-3.

If the motor is operating at full load and if its variable resistance Radj is increased to 250 Ω, what is the
new speed of the motor? Compare the full-load speed of the motor with Radj = 175 Ω to the full-load speed
with Radj = 250 Ω. (Assume no armature reaction, as in the previous problem.)
SOLUTION If Radj is set to 250 Ω, the field current is now
IF =

VT
240 V
240 V
=
=
= 0.686 A
Radj + RF 250 Ω + 100 Ω 325 Ω

Since the motor is still at full load, E A is still 218.3 V. From the magnetization curve (Figure P9-1), the
new field current I F would produce a voltage E Ao of 247 V at a speed no of 1200 r/min. Therefore,
n=

EA
E Ao

no =

218.3 V
(1200 r/min ) = 1061 r/min
247 V

Note that Radj has increased, and as a result the speed of the motor n increased.

9-4.

Assume that the motor is operating at full load and that the variable resistor Radj is again 175 Ω. If the
armature reaction is 1200 A⋅turns at full load, what is the speed of the motor? How does it compare to the
result for Problem 9-2?
SOLUTION The field current is again 0.87 A, and the motor is again at full load conditions. However, this
time there is an armature reaction of 1200 A⋅turns, and the effective field current is
I F* = I F −

AR
1200 A ⋅ turns
= 0.87 A −
= 0.426 A
2700 turns
NF

From Figure P9-1, this field current would produce an internal generated voltage E Ao of 181 V at a speed

no of 1200 r/min. The actual internal generated voltage E A at these conditions is
E A = VT − I A R A = 240 V − (54.13 A )( 0.40 Ω ) = 218.3 V

Therefore, the speed n with a voltage of 240 V would be
EA
218.3 V
n=
no =
(1200 r/min ) = 1447 r/min
E Ao
181 V
If all other conditions are the same, the motor with armature reaction runs at a higher speed than the motor
without armature reaction.

9-5.

If Radj can be adjusted from 100 to 400 Ω, what are the maximum and minimum no-load speeds possible
with this motor?
SOLUTION The minimum speed will occur when Radj = 100 Ω, and the maximum speed will occur when
Radj = 400 Ω. The field current when Radj = 100 Ω is:
IF =

VT
240 V
240 V
=
=
= 1.20 A
Radj + RF 100 Ω + 100 Ω 200 Ω

From Figure P9-1, this field current would produce an internal generated voltage E Ao of 287 V at a speed
no of 1200 r/min. Therefore, the speed n with a voltage of 240 V would be

216

EA
n
=
E Ao no

n=

EA
E Ao

no =

240 V
(1200 r/min ) = 1004 r/min
287 V

The field current when Radj = 400 Ω is:
IF =

VT
240 V
240 V
=
=
= 0.480 A
Radj + RF 400 Ω + 100 Ω 500 Ω

From Figure P9-1, this field current would produce an internal generated voltage E Ao of 199 V at a speed
no of 1200 r/min. Therefore, the speed n with a voltage of 240 V would be
EA
n
=
E Ao no

n=

9-6.

EA
E Ao

no =

240 V
(1200 r/min ) = 1447 r/min
199 V

What is the starting current of this machine if it is started by connecting it directly to the power supply VT ?
How does this starting current compare to the full-load current of the motor?
SOLUTION The starting current of this machine (ignoring the small field current) is
I L,start =

VT
240 V
=
= 600 A
RA 0.40 Ω

The rated current is 55 A, so the starting current is 10.9 times greater than the full-load current. This much
current is extremely likely to damage the motor.

9-7.

Plot the torque-speed characteristic of this motor assuming no armature reaction, and again assuming a
full-load armature reaction of 1200 A⋅turns.
SOLUTION This problem is best solved with MATLAB, since it involves calculating the torque-speed values
at many points. A MATLAB program to calculate and display both torque-speed characteristics is shown
below.
% M-file: prob9_7.m
% M-file to create a plot of the torque-speed curve of the
%
the shunt dc motor with and without armature reaction.
% Get the magnetization curve. Note that this curve is
% defined for a speed of 1200 r/min.
if_values = p91_mag(:,1);
ea_values = p91_mag(:,2);
n_0 = 1200;
% First, initialize the values needed in this program.
v_t = 240;
% Terminal voltage (V)
r_f = 100;
% Field resistance (ohms)
% Adjustable resistance (ohms)
r_a = 0.40;
% Armature resistance (ohms)
i_l = 0:1:55;
% Line currents (A)
n_f = 2700;
% Number of turns on field

217

f_ar0 = 1200;

% Armature reaction @ 55 A (A-t/m)

% Calculate the armature current for each load.
i_a = i_l - v_t / (r_f + r_adj);
% Now calculate the internal generated voltage for
% each armature current.
e_a = v_t - i_a * r_a;
% Calculate the armature reaction MMF for each armature
% current.
f_ar = (i_a / 55) * f_ar0;
% Calculate the effective field current with and without
% armature reaction. Ther term i_f_ar is the field current
% with armature reaction, and the term i_f_noar is the
% field current without armature reaction.
i_f_ar = v_t / (r_f + r_adj) - f_ar / n_f;
i_f_noar = v_t / (r_f + r_adj);
% Calculate the resulting internal generated voltage at
% 1200 r/min by interpolating the motor's magnetization
% curve.
e_a0_ar = interp1(if_values,ea_values,i_f_ar);
e_a0_noar = interp1(if_values,ea_values,i_f_noar);
% Calculate the resulting speed from Equation (9-13).
n_ar = ( e_a ./ e_a0_ar ) * n_0;
n_noar = ( e_a ./ e_a0_noar ) * n_0;
% Calculate the induced torque corresponding to each
% speed from Equations (8-55) and (8-56).
t_ind_ar = e_a .* i_a ./ (n_ar * 2 * pi / 60);
t_ind_noar = e_a .* i_a ./ (n_noar * 2 * pi / 60);
% Plot the torque-speed curves
figure(1);
plot(t_ind_noar,n_noar,'b-','LineWidth',2.0);
hold on;
plot(t_ind_ar,n_ar,'k--','LineWidth',2.0);
xlabel('\bf\tau_{ind} (N-m)');
ylabel('\bf\itn_{m} \rm\bf(r/min)');
title ('\bfShunt DC Motor Torque-Speed Characteristic');
legend('No armature reaction','With armature reaction');
axis([ 0 125 800 1250]);
grid on;
hold off;

218

The resulting plot is shown below:
Shunt DC Motor Torque-Speed Characteristic
1250

1200
No armature reaction
With armature reaction
1150

nm (r/min)

1100

1050

1000

950

900

850

800

0

20

40

60
τ

ind

80

100

120

(N-m)

For Problems 9-8 and 9-9, the shunt dc motor is reconnected separately excited, as shown in Figure P9-3. It has a
fixed field voltage V F of 240 V and an armature voltage V A that can be varied from 120 to 240 V.

Note:

9-8.

Figure P9-3 shows incorrect values for RA and RF in the first printing of this
book. The correct values are given in the text, but shown incorrectly on the
figure. This will be corrected at the second printing.

What is the no-load speed of this separately excited motor when Radj = 175 Ω and (a) V A = 120 V, (b) V A
= 180 V, (c) V A = 240 V?
SOLUTION At no-load conditions, E A = VA . The field current is given by
IF =

VF
240 V
240 V
=
=
= 0.873 A
Radj + RF 175 Ω + 100 Ω 275 Ω

From Figure P9-1, this field current would produce an internal generated voltage E Ao of 271 V at a speed
no of 1200 r/min. Therefore, the speed n with a voltage of 240 V would be
219

EA
n
=
E Ao no

n=
(a)

(1200 r/min ) = 531 r/min

180 V
271 V

(1200 r/min ) = 797 r/min

If V A = 240 V, then E A = 240 V, and
n=

9-9.

120 V
271 V

If V A = 180 V, then E A = 180 V, and
n=

(a)

no

If V A = 120 V, then E A = 120 V, and
n=

(a)

EA
E Ao

240 V
271 V

(1200 r/min ) = 1063 r/min

For the separately excited motor of Problem 9-8:
(a) What is the maximum no-load speed attainable by varying both V A and Radj ?
(b) What is the minimum no-load speed attainable by varying both V A and Radj ?
SOLUTION
(a)

The maximum speed will occur with the maximum V A and the maximum Radj . The field current

when Radj = 400 Ω is:
IF =

VT
240 V
240 V
=
=
= 0.48 A
Radj + RF 400 Ω + 100 Ω 500 Ω

From Figure P9-1, this field current would produce an internal generated voltage E Ao of 199 V at a speed
no of 1200 r/min. At no-load conditions, the maximum internal generated voltage E A = V A = 240 V.
Therefore, the speed n with a voltage of 240 V would be
EA
n
=
E Ao no

n=
(b)

EA
E Ao

no =

240 V
(1200 r/min ) = 1447 r/min
199 V

The minimum speed will occur with the minimum V A and the minimum Radj . The field current when

Radj = 100 Ω is:

IF =

VT
240 V
240 V
=
=
= 1.2 A
Radj + RF 100 Ω + 100 Ω 200 Ω

From Figure P9-1, this field current would produce an internal generated voltage E Ao of 287 V at a speed
no of 1200 r/min. At no-load conditions, the minimum internal generated voltage E A = V A = 120 V.
Therefore, the speed n with a voltage of 120 V would be
220

EA
n
=
E Ao no

n=

9-10.

EA
E Ao

no =

120 V
(1200 r/min ) = 502 r/min
287 V

If the motor is connected cumulatively compounded as shown in Figure P9-4 and if Radj = 175 Ω, what is
its no-load speed? What is its full-load speed? What is its speed regulation? Calculate and plot the torquespeed characteristic for this motor. (Neglect armature effects in this problem.)

Note:

Figure P9-4 shows incorrect values for RA + RS and RF in the first printing of
this book. The correct values are given in the text, but shown incorrectly on
the figure. This will be corrected at the second printing.

SOLUTION At no-load conditions, E A = VT = 240 V . The field current is given by
IF =

VF
240 V
240 V
=
=
= 0.873 A
Radj + RF 175 Ω + 100 Ω 275 Ω

From Figure P9-1, this field current would produce an internal generated voltage E Ao of 271 V at a speed
no of 1200 r/min. Therefore, the speed n with a voltage of 240 V would be
EA
n
=
E Ao no

n=

EA
E Ao

no =

240 V
(1200 r/min ) = 1063 r/min
271 V

At full load conditions, the armature current is
IA = IL − IF = IL −

VT
= 55 A − 0.87 A = 54.13 A

The internal generated voltage E A is
E A = VT − I A ( RA + RS ) = 240 V − (54.13 A )(0.44 Ω) = 216.2 V
The equivalent field current is
I F* = I F +

N SE
27 turns
I A = 0.873 A +
(54.13 A ) = 1.41 A
NF
2700 turns

221

From Figure P9-1, this field current would produce an internal generated voltage E Ao of 290 V at a speed
no of 1200 r/min. Therefore,
n=

EA
E Ao

no =

216.2 V
(1200 r/min ) = 895 r/min
290 V

The speed regulation is
SR =

1063 r/min − 895 r/min
nnl − nfl
× 100% =
× 100% = 18.8%
nfl
895 r/min

The torque-speed characteristic can best be plotted with a MATLAB program. An appropriate program is
shown below.
% M-file: prob9_10.m
% M-file to create a plot of the torque-speed curve of the
%
a cumulatively compounded dc motor without
%
armature reaction.
% Get the magnetization curve. Note that this curve is
% defined for a speed of 1200 r/min.
if_values = p91_mag(:,1);
ea_values = p91_mag(:,2);
n_0 = 1200;
% First, initialize the values needed in this program.
v_t = 240;
% Terminal voltage (V)
r_f = 100;
% Field resistance (ohms)
% Adjustable resistance (ohms)
r_a = 0.44;
% Armature + series resistance (ohms)
i_l = 0:55;
% Line currents (A)
n_f = 2700;
% Number of turns on shunt field
n_se = 27;
% Number of turns on series field
% Calculate the armature current for each load.
i_a = i_l - v_t / (r_f + r_adj);
% Now calculate the internal generated voltage for
% each armature current.
e_a = v_t - i_a * r_a;
% Calculate the effective field current for each armature
% current.
i_f = v_t / (r_f + r_adj) + (n_se / n_f) * i_a;
% Calculate the resulting internal generated voltage at
% 1200 r/min by interpolating the motor's magnetization
% curve.
e_a0 = interp1(if_values,ea_values,i_f);
% Calculate the resulting speed from Equation (9-13).
n = ( e_a ./ e_a0 ) * n_0;
% Calculate the induced torque corresponding to each
% speed from Equations (8-55) and (8-56).

222

t_ind = e_a .* i_a ./ (n * 2 * pi / 60);
% Plot the torque-speed curves
figure(1);
plot(t_ind,n,'b-','LineWidth',2.0);
xlabel('\bf\tau_{ind} (N-m)');
ylabel('\bf\itn_{m} \rm\bf(r/min)');
title ('\bfCumulatively-Compounded DC Motor Torque-Speed
Characteristic');
axis([0 125 800 1250]);
grid on;

The resulting plot is shown below:

Compare this torque-speed curve to that of the shunt motor in Problem 9-7. (Both curves are plotted on the
same scale to facilitate comparison.)

9-11.

The motor is connected cumulatively compounded and is operating at full load. What will the new speed of
the motor be if Radj is increased to 250 Ω? How does the new speed compared to the full-load speed
calculated in Problem 9-10?
SOLUTION If Radj is increased to 250 Ω, the field current is given by
IF =

VT
240 V
240 V
=
=
= 0.686 A
Radj + RF 250 Ω + 100 Ω 350 Ω

At full load conditions, the armature current is
I A = I L − I F = 55 A − 0.686 A = 54.3 A
The internal generated voltage E A is
E A = VT − I A ( RA + RS ) = 240 V − (54.3 A )(0.44 Ω ) = 216.1 V

223

The equivalent field current is
I F* = I F +

N SE
27 turns
I A = 0.686 A +
(54.3 A ) = 1.23 A
NF
2700 turns

From Figure P9-1, this field current would produce an internal generated voltage E Ao of 288 V at a speed
no of 1200 r/min. Therefore,
n=

EA
E Ao

no =

216.1 V
(1200 r/min ) = 900 r/min
288 V

The new full-load speed is higher than the full-load speed in Problem 9-10.

9-12.

The motor is now connected differentially compounded.
(a) If Radj = 175 Ω, what is the no-load speed of the motor?
(b) What is the motor’s speed when the armature current reaches 20 A? 40 A? 60 A?
(c) Calculate and plot the torque-speed characteristic curve of this motor.
SOLUTION
(a)

At no-load conditions, E A = VT = 240 V . The field current is given by
IF =

VF
240 V
240 V
=
=
= 0.873 A
Radj + RF 175 Ω + 100 Ω 275 Ω

From Figure P9-1, this field current would produce an internal generated voltage E Ao of 271 V at a speed
no of 1200 r/min. Therefore, the speed n with a voltage of 240 V would be
EA
n
=
E Ao no

n=
(b)

EA
E Ao

no =

240 V
(1200 r/min ) = 1063 r/min
271 V

At I A = 20A, the internal generated voltage E A is
E A = VT − I A ( RA + RS ) = 240 V − ( 20 A )(0.44 Ω ) = 231.2 V

The equivalent field current is
I F* = I F −

N SE
27 turns
I A = 0.873 A −
(20 A ) = 0.673 A
NF
2700 turns

From Figure P9-1, this field current would produce an internal generated voltage E Ao of 245 V at a speed
no of 1200 r/min. Therefore,
n=

EA
E Ao

no =

231.2 V
(1200 r/min ) = 1132 r/min
245 V

At I A = 40A, the internal generated voltage E A is
E A = VT − I A ( RA + RS ) = 240 V − ( 40 A )(0.44 Ω ) = 222.4 V

The equivalent field current is
224

I F* = I F −

N SE
27 turns
I A = 0.873 A −
(40 A ) = 0.473 A
NF
2700 turns

From Figure P9-1, this field current would produce an internal generated voltage E Ao of 197 V at a speed
no of 1200 r/min. Therefore,
n=

EA
E Ao

no =

227.4 V
(1200 r/min ) = 1385 r/min
197 V

At I A = 60A, the internal generated voltage E A is
E A = VT − I A ( RA + RS ) = 240 V − ( 60 A )( 0.44 Ω ) = 213.6 V
The equivalent field current is
I F* = I F −

N SE
27 turns
I A = 0.873 A −
(60 A ) = 0.273 A
NF
2700 turns

From Figure P9-1, this field current would produce an internal generated voltage E Ao of 121 V at a speed
no of 1200 r/min. Therefore,
n=

EA
E Ao

no =

213.6 V
(1200 r/min ) = 2118 r/min
121 V

(c) The torque-speed characteristic can best be plotted with a MATLAB program. An appropriate
program is shown below.
% M-file: prob9_12.m
% M-file to create a plot of the torque-speed curve of the
%
a differentially compounded dc motor withwithout
%
armature reaction.
% Get the magnetization curve. Note that this curve is
% defined for a speed of 1200 r/min.
if_values = p91_mag(:,1);
ea_values = p91_mag(:,2);
n_0 = 1200;
% First, initialize the values needed in this program.
v_t = 240;
% Terminal voltage (V)
r_f = 100;
% Field resistance (ohms)
% Adjustable resistance (ohms)
r_a = 0.44;
% Armature + series resistance (ohms)
i_l = 0:50;
% Line currents (A)
n_f = 2700;
% Number of turns on shunt field
n_se = 27;
% Number of turns on series field
% Calculate the armature current for each load.
i_a = i_l - v_t / (r_f + r_adj);
% Now calculate the internal generated voltage for
% each armature current.
e_a = v_t - i_a * r_a;
% Calculate the effective field current for each armature

225

% current.
i_f = v_t / (r_f + r_adj) - (n_se / n_f) * i_a;
% Calculate the resulting internal generated voltage at
% 1200 r/min by interpolating the motor's magnetization
% curve.
e_a0 = interp1(if_values,ea_values,i_f);
% Calculate the resulting speed from Equation (9-13).
n = ( e_a ./ e_a0 ) * n_0;
% Calculate the induced torque corresponding to each
% speed from Equations (8-55) and (8-56).
t_ind = e_a .* i_a ./ (n * 2 * pi / 60);
% Plot the torque-speed curves
figure(1);
plot(t_ind,n,'b-','LineWidth',2.0);
xlabel('\bf\tau_{ind} (N-m)');
ylabel('\bf\itn_{m} \rm\bf(r/min)');
title ('\bfDifferentially-Compounded DC Motor Torque-Speed
Characteristic');
axis([0 100 800 1600]);
grid on;

The resulting plot is shown below:

Compare this torque-speed curve to that of the shunt motor in Problem 9-7 and the cumulativelycompounded motor in Problem 9-10. (Note that this plot has a larger vertical scale to accommodate the
speed runaway of the differentially-compounded motor.)

9-13.

A 7.5-hp 120-V series dc motor has an armature resistance of 0.2 Ω and a series field resistance of 0.16 Ω.
At full load, the current input is 58 A, and the rated speed is 1050 r/min. Its magnetization curve is shown
226

in Figure P9-5. The core losses are 200 W, and the mechanical losses are 240 W at full load. Assume that
the mechanical losses vary as the cube of the speed of the motor and that the core losses are constant.

Note:

An electronic version of this magnetization curve can be found in file
p95_mag.dat, which can be used with MATLAB programs. Column 1
contains field current in amps, and column 2 contains the internal generated
voltage EA in volts.

(a) What is the efficiency of the motor at full load?
(b) What are the speed and efficiency of the motor if it is operating at an armature current of 35 A?
(c) Plot the torque-speed characteristic for this motor.
SOLUTION
(a)

The output power of this motor at full load is
POUT = ( 7.5 hp )( 746 W/hp ) = 5595 W

The input power is
227

PIN = VT I L = (120 V )(58 A ) = 6960 W
Therefore the efficiency is

η=
(b)

5595 W
POUT
× 100% =
× 100% = 80.4%
PIN
6960 W

If the armature current is 35 A, then the input power to the motor will be
PIN = VT I L = (120 V )( 35 A ) = 4200 W

The internal generated voltage at this condition is
E A2 = VT − I A ( RA + RS ) = 120 V − ( 35 A )(0.20 Ω + 0.16 Ω ) = 107.4 V

and the internal generated voltage at rated conditions is
E A1 = VT − I A ( RA + RS ) = 120 V − (58 A )(0.20 Ω + 0.16 Ω ) = 99.1 V

The final speed is given by the equation
E A2 K φ2 ω 2 E Ao ,2 n2
=
=
E A1 K φ2 ω 2 E Ao ,1 n1
since the ratio E Ao ,2 / E Ao ,1 is the same as the ratio φ 2 / φ1 . Therefore, the final speed is
E A 2 E Ao ,1
n1
E A1 E Ao ,2

n2 =

From Figure P9-5, the internal generated voltage E Ao,2 for a current of 35 A and a speed of no = 1200
r/min is E Ao,2 = 115 V, and the internal generated voltage E Ao,1 for a current of 58 A and a speed of no =
1200 r/min is E Ao ,1 = 134 V.
E A2 E Ao ,1
107.4 V
n1 =
E A1 E Ao ,2
99.1 V

n2 =

134 V
(1050 r/min ) = 1326 r/min
115 V

The power converted from electrical to mechanical form is
Pconv = E A I A = (107.4 V )( 35 A ) = 3759 W
The core losses in the motor are 200 W, and the mechanical losses in the motor are 240 W at a speed of
1050 r/min. The mechanical losses in the motor scale proportionally to the cube of the rotational speedm
so the mechanical losses at 1326 r/min are
Pmech

n
= 2
n1

3

1326 r/min
(240 W ) =
1050 r/min

3

(240 W ) = 483 W

Therefore, the output power is
POUT = Pconv − Pmech − Pcore = 3759 W − 483 W − 200 W = 3076 W
and the efficiency is

η=
(c)

3076 W
POUT
× 100% =
× 100% = 73.2%
PIN
4200 W

A MATLAB program to plot the torque-speed characteristic of this motor is shown below:
228

% M-file: prob9_13.m
% M-file to create a plot of the torque-speed curve of the
%
the series dc motor in Problem 9-13.
% Get the magnetization curve. Note that this curve is
% defined for a speed of 1200 r/min.
if_values = p95_mag(:,1);
ea_values = p95_mag(:,2);
n_0 = 1200;
% First, initialize the values needed in this program.
v_t = 120;
% Terminal voltage (V)
r_a = 0.36;
% Armature + field resistance (ohms)
i_a = 9:1:58;
% Armature (line) currents (A)
% Calculate the internal generate voltage e_a.
e_a = v_t - i_a * r_a;
% Calculate the resulting internal generated voltage at
% 1200 r/min by interpolating the motor's magnetization
% curve. Note that the field current is the same as the
% armature current for this motor.
e_a0 = interp1(if_values,ea_values,i_a,'spline');
% Calculate the motor's speed, using the known fact that
% the motor runs at 1050 r/min at a current of 58 A. We
% know that
%
%
Ea2
K' phi2 n2
Eao2 n2
% ----- = ------------ = ---------%
Ea1
K' phi1 n1
Eao1 n1
%
%
Ea2
Eao1
%
==> n2 = ----- ------ n1
%
Ea1
Eao2
%
% where Ea0 is the internal generated voltage at 1200 r/min
% for a given field current.
%
% Speed will be calculated by reference to full load speed
% and current.
n1 = 1050;
% 1050 r/min at full load
Eao1 = interp1(if_values,ea_values,58,'spline');
Ea1 = v_t - 58 * r_a;
% Get speed
Eao2 = interp1(if_values,ea_values,i_a,'spline');
n = (e_a./Ea1) .* (Eao1 ./ Eao2) * n1;
% Calculate the induced torque corresponding to each
% speed from Equations (8-55) and (8-56).
t_ind = e_a .* i_a ./ (n * 2 * pi / 60);
% Plot the torque-speed curve

229

figure(1);
plot(t_ind,n,'b-','LineWidth',2.0);
hold on;
xlabel('\bf\tau_{ind} (N-m)');
ylabel('\bf\itn_{m} \rm\bf(r/min)');
title ('\bfSeries DC Motor Torque-Speed Characteristic');
grid on;
hold off;

The resulting torque-speed characteristic is shown below:

9-14.

A 20-hp 240-V 76-A 900 r/min series motor has a field winding of 33 turns per pole. Its armature
resistance is 0.09 Ω, and its field resistance is 0.06 Ω. The magnetization curve expressed in terms of
magnetomotive force versus EA at 900 r/min is given by the following table:

Note:

EA , V

95

150

188

212

229

243

F, A ⋅ turns

500

1000

1500

2000

2500

3000

An electronic version of this magnetization curve can be found in file
prob9_14_mag.dat, which can be used with MATLAB programs. Column
1 contains magnetomotive force in ampere-turns, and column 2 contains the
internal generated voltage EA in volts.

Armature reaction is negligible in this machine.
(a) Compute the motor’s torque, speed, and output power at 33, 67, 100, and 133 percent of full-load
armature current. (Neglect rotational losses.)
(b) Plot the terminal characteristic of this machine.
SOLUTION Note that this magnetization curve has been stored in a file called prob9_14_mag.dat. The
first column of the file is an array of mmf_values, and the second column is an array of ea_values.
These values are valid at a speed no = 900 r/min. Because the data in the file is relatively sparse, it is
230

important that interpolation be done using smooth curves, so be sure to specify the 'spline' option in
the MATLAB interp1 function:
mmf_values = prob9_14_mag(:,1);
ea_values = prob9_14_mag(:,2);
...
Eao = interp1(mmf_values,ea_values,mmf,'spline')

(a) Since full load corresponds to 76 A, this calculation must be performed for armature currents of 25.3
A, 50.7 A, 76 A, and 101.3 A.
If I A = 23.3 A, then
E A = VT − I A ( RA + RS ) = 240 V − ( 25.3 A )(0.09 Ω + 0.06 Ω ) = 236.2 V
The magnetomotive force is F = NI A = ( 33 turns)( 25.3 A ) = 835 A ⋅ turns , which produces a voltage E Ao
of 134 V at no = 900 r/min. Therefore the speed of the motor at these conditions is
n=

EA
236.2 V
no =
(900 r/min ) = 1586 r/min
E Ao
134 V

The power converted from electrical to mechanical form is
Pconv = E A I A = ( 236.2 V )( 25.3 A ) = 5976 W

Since the rotational losses are ignored, this is also the output power of the motor. The induced torque is

τ ind =

Pconv

ωm

=

5976 W
(1586 r/min ) 2π rad
1r

1 min
60 s

= 36 N ⋅ m

If I A = 50.7 A, then
E A = VT − I A ( RA + RS ) = 240 V − (50.7 A )( 0.09 Ω + 0.06 Ω ) = 232.4 V

The magnetomotive force is F = NI A = ( 33 turns )(50.7 A ) = 1672 A ⋅ turns , which produces a voltage E Ao
of 197 V at no = 900 r/min. Therefore the speed of the motor at these conditions is
n=

EA
232.4 V
no =
E Ao
197 V

(900 r/min ) = 1062 r/min

The power converted from electrical to mechanical form is
Pconv = E A I A = ( 232.4 V )(50.7 A ) = 11, 780 W

Since the rotational losses are ignored, this is also the output power of the motor. The induced torque is

τ ind =

Pconv

ωm

=

11,780 W
(1062 r/min )
1r

1 min
60 s

= 106 N ⋅ m

If I A = 76 A, then
E A = VT − I A ( RA + RS ) = 240 V − (76 A )(0.09 Ω + 0.06 Ω ) = 228.6 V
231

The magnetomotive force is F = NI A = (33 turns)(76 A ) = 2508 A ⋅ turns , which produces a voltage E Ao
of 229 V at no = 900 r/min. Therefore the speed of the motor at these conditions is
n=

EA
228.6 V
no =
E Ao
229 V

(900 r/min ) = 899 r/min

The power converted from electrical to mechanical form is
Pconv = E A I A = ( 228.6 V )( 76 A ) = 17,370 W
Since the rotational losses are ignored, this is also the output power of the motor. The induced torque is

τ ind =

Pconv

ωm

=

17,370 W
(899 r/min ) 2π rad
1r

1 min
60 s

= 185 N ⋅ m

If I A = 101.3 A, then
E A = VT − I A ( RA + RS ) = 240 V − (101.3 A )( 0.09 Ω + 0.06 Ω ) = 224.8 V
The magnetomotive force is F = NI A = (33 turns)(101.3 A ) = 3343 A ⋅ turns , which produces a voltage
E Ao of 252 V at no = 900 r/min. Therefore the speed of the motor at these conditions is
n=

EA
224.8 V
no =
E Ao
252 V

(900 r/min ) = 803 r/min

The power converted from electrical to mechanical form is
Pconv = E A I A = ( 224.8 V )(101.3 A ) = 22,770 W

Since the rotational losses are ignored, this is also the output power of the motor. The induced torque is

τ ind =

(b)

Pconv

ωm

=

22,770 W
(803 r/min ) 2π rad
1r

1 min
60 s

= 271 N ⋅ m

A MATLAB program to plot the torque-speed characteristic of this motor is shown below:

% M-file: series_ts_curve.m
% M-file to create a plot of the torque-speed curve of the
%
the series dc motor in Problem 9-14.
% Get the magnetization curve. Note that this curve is
% defined for a speed of 900 r/min.
mmf_values = prob9_14_mag(:,1);
ea_values = prob9_14_mag(:,2);
n_0 = 900;
% First, initialize the values needed in this program.
v_t = 240;
% Terminal voltage (V)
r_a = 0.15;
% Armature + field resistance (ohms)
i_a = 15:1:76;
% Armature (line) currents (A)
n_s = 33;
% Number of series turns on field
% Calculate the MMF for each load

232

f = n_s * i_a;
% Calculate the internal generate voltage e_a.
e_a = v_t - i_a * r_a;
% Calculate the resulting internal generated voltage at
% 900 r/min by interpolating the motor's magnetization
% curve. Specify cubic spline interpolation to provide
% good results with this sparse magnetization curve.
e_a0 = interp1(mmf_values,ea_values,f,'spline');
% Calculate the motor's speed from Equation (9-13).
n = (e_a ./ e_a0) * n_0;
% Calculate the induced torque corresponding to each
% speed from Equations (8-55) and (8-56).
t_ind = e_a .* i_a ./ (n * 2 * pi / 60);
% Plot the torque-speed curve
figure(1);
plot(t_ind,n,'b-','LineWidth',2.0);
hold on;
xlabel('\bf\tau_{ind} (N-m)');
ylabel('\bf\itn_{m} \rm\bf(r/min)');
title ('\bfSeries DC Motor Torque-Speed Characteristic');
%axis([ 0 700 0 5000]);
grid on;
hold off;

The resulting torque-speed characteristic is shown below:

9-15.

A 300-hp 440-V 560-A, 863 r/min shunt dc motor has been tested, and the following data were taken:
Blocked-rotor test:
233

V A = 16.3 V exclusive of brushes

VF = 440 V

I A = 500 A

I F = 8.86 A

V A = 16.3 V including brushes

I F = 8.76 A

I A = 231
. A

n = 863 r/min

What is this motor’s efficiency at the rated conditions? [Note: Assume that (1) the brush voltage drop is 2
V; (2) the core loss is to be determined at an armature voltage equal to the armature voltage under full load;
and (3) stray load losses are 1 percent of full load.]
SOLUTION The armature resistance of this motor is
RA =

VA,br 16.3 V
=
= 0.0326 Ω
I A,br 500 A

Under no-load conditions, the core and mechanical losses taken together (that is, the rotational losses) of
this motor are equal to the product of the internal generated voltage E A and the armature current I A , since
this is no output power from the motor at no-load conditions. Therefore, the rotational losses at rated speed
can be found as
E A = VA − Vbrush − I A RA = 442 V − 2 V − ( 23.1 A )(0.0326 Ω ) = 439.2 V
Prot = Pconv = E A I A = ( 439.2 V )( 23.1 A ) = 10.15 kW

The input power to the motor at full load is
PIN = VT I L = ( 440 V )(560 A ) = 246.4 kW

The output power from the motor at full load is
POUT = PIN − PCU − Prot − Pbrush − Pstray
The copper losses are
PCU = I A2 RA + VF I F = (560 A ) ( 0.0326 Ω ) + ( 440 V )(8.86 A ) = 14.1 kW
2

The brush losses are
Pbrush = Vbrush I A = ( 2 V )(560 A ) = 1120 W

Therefore,
POUT = PIN − PCU − Prot − Pbrush − Pstray
POUT = 246.4 kW − 14.1 kW − 10.15 kW − 1.12 kW − 2.46 kW = 218.6 kW
The motor’s efficiency at full load is
218.6 kW
POUT
× 100% =
× 100% = 88.7%
PIN
246.4 kW
Problems 9-16 to 9-19 refer to a 240-V 100-A dc motor which has both shunt and series windings.
characteristics are
RA = 0.14 Ω
N F = 1500 turns

η=

RS = 0.04 Ω

N SE = 12 turns
234

Its

RF = 200 Ω
nm = 1200 r/min
Radj = 0 to 300 Ω, currently set to 120 Ω
This motor has compensating windings and interpoles. The magnetization curve for this motor at 1200 r/min is
shown in Figure P9-6.

Note:

9-16.

An electronic version of this magnetization curve can be found in file
p96_mag.dat, which can be used with MATLAB programs. Column 1
contains field current in amps, and column 2 contains the internal generated
voltage EA in volts.

The motor described above is connected in shunt.
(a) What is the no-load speed of this motor when Radj = 120 Ω?
(b) What is its full-load speed?
(c) Under no-load conditions, what range of possible speeds can be achieved by adjusting Radj ?
SOLUTION Note that this magnetization curve has been stored in a file called p96_mag.dat. The first
column of the file is an array of ia_values, and the second column is an array of ea_values. These
values are valid at a speed no = 1200 r/min. These values can be used with the MATLAB interp1
function to look up an internal generated voltage as follows:

235

if_values = p96_mag(:,1);
ea_values = p96_mag(:,2);
...
Ea = interp1(if_values,ea_values,if,'spline')

(a)

If Radj = 120 Ω, the total field resistance is 320 Ω, and the resulting field current is
IF =

VT
240 V
=
= 0.75 A
RF + Radj 200 Ω + 120 Ω

This field current would produce a voltage E Ao of 256 V at a speed of no = 1200 r/min. The actual E A is
240 V, so the actual speed will be
n=

(b)

EA
240 V
no =
(1200 r/min ) = 1125 r/min
E Ao
256 V

At full load, I A = I L − I F = 100 A − 0.75 A = 99.25 A , and
E A = VT − I A R A = 240 V − (99.25 A )( 0.14 Ω ) = 226.1 V

Therefore, the speed at full load will be
n=

(c)

EA
226.1 V
no =
(1200 r/min ) = 1060 r/min
E Ao
256 V

If Radj is maximum at no-load conditions, the total resistance is 500 Ω, and
IF =

VT
240 V
=
= 0.48 A
RF + Radj 200 Ω + 300 Ω

This field current would produce a voltage E Ao of 200 V at a speed of no = 1200 r/min. The actual E A is
240 V, so the actual speed will be
n=

EA
240 V
no =
(1200 r/min ) = 1440 r/min
E Ao
200 V

If Radj is minimum at no-load conditions, the total resistance is 200 Ω, and
IF =

VT
240 V
=
= 1.2 A
RF + Radj 200 Ω + 0 Ω

This field current would produce a voltage E Ao of 287 V at a speed of no = 1200 r/min. The actual E A is
240 V, so the actual speed will be
n=

9-17.

EA
240 V
no =
(1200 r/min ) = 1004 r/min
E Ao
287 V

This machine is now connected as a cumulatively compounded dc motor with Radj = 120 Ω.
(a) What is the full-load speed of this motor?
(b) Plot the torque-speed characteristic for this motor.
(c) What is its speed regulation?
SOLUTION
236

(a)

At full load, I A = I L − I F = 100 A − 0.75 A = 99.25 A , and
E A = VT − I A ( RA + RS ) = 240 V − (99.25 A )(0.14 Ω + 0.05 Ω ) = 221.1 V

The actual field current will be
IF =

VT
240 V
=
= 0.75 A
RF + Radj 200 Ω + 120 Ω

and the effective field current will be
I F* = I F +

N SE
12 turns
I A = 0.75 A +
(99.25 A ) = 1.54 A
NF
1500 turns

This field current would produce a voltage E Ao of 290 V at a speed of no = 1200 r/min. The actual E A
is 240 V, so the actual speed at full load will be
n=

(b)

EA
221.1 V
no =
(1200 r/min ) = 915 r/min
E Ao
290 V

A MATLAB program to calculate the torque-speed characteristic of this motor is shown below:

% M-file: prob9_17.m
% M-file to create a plot of the torque-speed curve of the
%
a cumulatively compounded dc motor.
% Get the magnetization curve.
if_values = p96_mag(:,1);
ea_values = p96_mag(:,2);
n_0 = 1200;
% First, initialize the values needed in this program.
v_t = 240;
% Terminal voltage (V)
r_f = 200;
% Field resistance (ohms)
% Adjustable resistance (ohms)
r_a = 0.19;
% Armature + series resistance (ohms)
i_l = 0:2:100;
% Line currents (A)
n_f = 1500;
% Number of turns on shunt field
n_se = 12;
% Number of turns on series field
% Calculate the armature current for each load.
i_a = i_l - v_t / (r_f + r_adj);
% Now calculate the internal generated voltage for
% each armature current.
e_a = v_t - i_a * r_a;
% Calculate the effective field current for each armature
% current.
i_f = v_t / (r_f + r_adj) + (n_se / n_f) * i_a;
% Calculate the resulting internal generated voltage at
% 1800 r/min by interpolating the motor's magnetization
% curve.
e_a0 = interp1(if_values,ea_values,i_f);

237

% Calculate the resulting speed from Equation (9-13).
n = ( e_a ./ e_a0 ) * n_0;
% Calculate the induced torque corresponding to each
% speed from Equations (8-55) and (8-56).
t_ind = e_a .* i_a ./ (n * 2 * pi / 60);
% Plot the torque-speed curves
figure(1);
plot(t_ind,n,'b-','LineWidth',2.0);
xlabel('\bf\tau_{ind} (N-m)');
ylabel('\bf\itn_{m} \rm\bf(r/min)');
title ('\bfCumulatively-Compounded DC Motor Torque-Speed
Characteristic');
axis([0 200 900 1600]);
grid on;

The resulting torque-speed characteristic is shown below:

(c) The no-load speed of this machine is the same as the no-load speed of the corresponding shunt dc
motor with Radj = 120 Ω, which is 1125 r/min. The speed regulation of this motor is thus
SR =

9-18.

1125 r/min - 915 r/min
nnl − nfl
× 100% =
× 100% = 23.0%
nfl
915 r/min

The motor is reconnected differentially compounded with Radj = 120 Ω. Derive the shape of its torquespeed characteristic.
SOLUTION A MATLAB program to calculate the torque-speed characteristic of this motor is shown below:
% M-file: prob9_18.m
% M-file to create a plot of the torque-speed curve of the

238

%

a differentially compounded dc motor.

% Get the magnetization curve.
if_values = p96_mag(:,1);
ea_values = p96_mag(:,2);
n_0 = 1200;
% First, initialize the values needed in this program.
v_t = 240;
% Terminal voltage (V)
r_f = 200;
% Field resistance (ohms)
% Adjustable resistance (ohms)
r_a = 0.19;
% Armature + series resistance (ohms)
i_l = 0:2:40;
% Line currents (A)
n_f = 1500;
% Number of turns on shunt field
n_se = 12;
% Number of turns on series field
% Calculate the armature current for each load.
i_a = i_l - v_t / (r_f + r_adj);
% Now calculate the internal generated voltage for
% each armature current.
e_a = v_t - i_a * r_a;
% Calculate the effective field current for each armature
% current.
i_f = v_t / (r_f + r_adj) - (n_se / n_f) * i_a;
% Calculate the resulting internal generated voltage at
% 1800 r/min by interpolating the motor's magnetization
% curve.
e_a0 = interp1(if_values,ea_values,i_f);
% Calculate the resulting speed from Equation (9-13).
n = ( e_a ./ e_a0 ) * n_0;
% Calculate the induced torque corresponding to each
% speed from Equations (8-55) and (8-56).
t_ind = e_a .* i_a ./ (n * 2 * pi / 60);
% Plot the torque-speed curves
figure(1);
plot(t_ind,n,'b-','LineWidth',2.0);
xlabel('\bf\tau_{ind} (N-m)');
ylabel('\bf\itn_{m} \rm\bf(r/min)');
title ('\bfDifferentially-Compounded DC Motor Torque-Speed
Characteristic');
axis([0 200 900 1600]);
grid on;

239

The resulting torque-speed characteristic is shown below:

This curve is plotted on the same scale as the torque-speed curve in Problem 6-17. Compare the two
curves.

9-19.

A series motor is now constructed from this machine by leaving the shunt field out entirely. Derive the
torque-speed characteristic of the resulting motor.
SOLUTION This motor will have extremely high speeds, since there are only a few series turns, and the flux
in the motor will be very small. A MATLAB program to calculate the torque-speed characteristic of this
motor is shown below:
% M-file: prob9_19.m
% M-file to create a plot of the torque-speed curve of the
%
a series dc motor. This motor was formed by removing
%
the shunt field from the cumulatively-compounded machine
%
if Problem 9-17.
% Get the magnetization curve.
if_values = p96_mag(:,1);
ea_values = p96_mag(:,2);
n_0 = 1200;
% First, initialize the values needed in this program.
v_t = 240;
% Terminal voltage (V)
r_a = 0.19;
% Armature + series resistance (ohms)
i_l = 20:1:45;
% Line currents (A)
n_f = 1500;
% Number of turns on shunt field
n_se = 12;
% Number of turns on series field
% Calculate the armature current for each load.
i_a = i_l;

240

% Now calculate the internal generated voltage for
% each armature current.
e_a = v_t - i_a * r_a;
% Calculate the effective field current for each armature
% current. (Note that the magnetization curve is defined
% in terms of shunt field current, so we will have to
% translate the series field current into an equivalent
% shunt field current.
i_f = (n_se / n_f) * i_a;
% Calculate the resulting internal generated voltage at
% 1800 r/min by interpolating the motor's magnetization
% curve.
e_a0 = interp1(if_values,ea_values,i_f);
% Calculate the resulting speed from Equation (9-13).
n = ( e_a ./ e_a0 ) * n_0;
% Calculate the induced torque corresponding to each
% speed from Equations (8-55) and (8-56).
t_ind = e_a .* i_a ./ (n * 2 * pi / 60);
% Plot the torque-speed curves
figure(1);
plot(t_ind,n,'b-','LineWidth',2.0);
xlabel('\bf\tau_{ind} (N-m)');
ylabel('\bf\itn_{m} \rm\bf(r/min)');
title ('\bfSeries DC Motor Torque-Speed Characteristic');
grid on;

The resulting torque-speed characteristic is shown below:

241

The extreme speeds in this characteristic are due to the very light flux in the machine. To make a practical
series motor out of this machine, it would be necessary to include 20 to 30 series turns instead of 12.

9-20.

An automatic starter circuit is to be designed for a shunt motor rated at 15 hp, 240 V, and 60 A. The
armature resistance of the motor is 0.15 Ω, and the shunt field resistance is 40 Ω. The motor is to start
with no more than 250 percent of its rated armature current, and as soon as the current falls to rated value,
a starting resistor stage is to be cut out. How many stages of starting resistance are needed, and how big
should each one be?
SOLUTION The rated line current of this motor is 60 A, and the rated armature current is I A = I L − I F = 60
A – 6 A = 54 A. The maximum desired starting current is (2.5)(54 A) = 135 A. Therefore, the total initial
starting resistance must be
240 V
= 1.778 Ω
135 A
Rstart,1 = 1.778 Ω − 0.15 Ω = 1.628 Ω

R A + Rstart,1 =

The current will fall to rated value when E A rises to
E A = 240 V − (1.778 Ω )(54 A ) = 144 V

At that time, we want to cut out enough resistance to get the current back up to 135 A. Therefore,
240 V − 144 V
= 0.711 Ω
135 A
= 0.711 Ω − 0.15 Ω = 0.561 Ω

R A + Rstart,2 =

Rstart,2

With this resistance in the circuit, the current will fall to rated value when E A rises to
E A = 240 V − ( 0.711 Ω )(54 A ) = 201.6 V

At that time, we want to cut out enough resistance to get the current back up to 185 A. Therefore,
240 V − 201.6 V
= 0.284 Ω
135 A
= 0.284 Ω − 0.15 Ω = 0.134 Ω

R A + Rstart,3 =
Rstart,3

With this resistance in the circuit, the current will fall to rated value when E A rises to
E A = 240 V − ( 0.284 Ω )(54 A ) = 224.7 V

If the resistance is cut out when E A reaches 228,6 V, the resulting current is
IA =

240 V − 224.7 V
= 102 A < 135 A ,
0.15 Ω

so there are only three stages of starting resistance. The three stages of starting resistance can be found
from the resistance in the circuit at each state during starting.
Rstart,1 = R1 + R2 + R3 = 1.628 Ω
Rstart,2 = R2 + R3 = 0.561 Ω

Rstart,3 = R3 = 0.134 Ω
Therefore, the starting resistances are
R1 = 1.067 Ω
R2 = 0.427 Ω
R3 = 0.134 Ω
242

9-21.

A 15-hp 120-V 1800 r/min shunt dc motor has a full-load armature current of 60 A when operating at rated
conditions. The armature resistance of the motor is RA = 0.15 Ω, and the field resistance RF is 80 Ω.
The adjustable resistance in the field circuit Radj may be varied over the range from 0 to 200 Ω and is
currently set to 90 Ω. Armature reaction may be ignored in this machine. The magnetization curve for this
motor, taken at a speed of 1800 r/min, is given in tabular form below:

Note:

EA , V

5

78

95

112

118

126

IF , A

0.00

0.80

1.00

1.28

1.44

2.88

An electronic version of this magnetization curve can be found in file
prob9_21_mag.dat, which can be used with MATLAB programs. Column
1 contains field current in amps, and column 2 contains the internal generated
voltage EA in volts.

(a) What is the speed of this motor when it is running at the rated conditions specified above?
(b) The output power from the motor is 7.5 hp at rated conditions. What is the output torque of the motor?
(c) What are the copper losses and rotational losses in the motor at full load (ignore stray losses)?
(d) What is the efficiency of the motor at full load?
(e) If the motor is now unloaded with no changes in terminal voltage or Radj , what is the no-load speed of
the motor?
(f) Suppose that the motor is running at the no-load conditions described in part (e). What would happen
to the motor if its field circuit were to open? Ignoring armature reaction, what would the final steadystate speed of the motor be under those conditions?
(g) What range of no-load speeds is possible in this motor, given the range of field resistance adjustments
available with Radj ?
SOLUTION
(a)

If Radj = 90 Ω, the total field resistance is 170 Ω, and the resulting field current is
IF =

VT
230 V
=
= 1.35 A
RF + Radj 90 Ω + 80 Ω

This field current would produce a voltage E Ao of 221 V at a speed of no = 1800 r/min. The actual E A is
E A = VT − I A RA = 230 V − ( 60 A )( 0.15 Ω ) = 221 V

so the actual speed will be
n=

EA
221 V
no =
(1800 r/min ) = 1800 r/min
E Ao
221 V

(b) The output power is 7.5 hp and the output speed is 1800 r/min at rated conditions, therefore, the
torque is

τ out =

Pout

ωm

=

(15 hp)(746 W/hp)
(1800 r/min ) 2π rad 1 min
1r

(c)

60 s

The copper losses are
243

= 59.4 N ⋅ m

PCU = I A2 RA + VF I F = ( 60 A ) ( 0.15 Ω ) + ( 230 V )(1.35 A ) = 851 W
2

The power converted from electrical to mechanical form is
Pconv = E A I A = ( 221 V )( 60 A ) = 13,260 W
The output power is
POUT = (15 hp )( 746 W/hp ) = 11,190 W
Therefore, the rotational losses are
Prot = Pconv − POUT = 13,260 W − 11,190 W = 2070 W
(d)

The input power to this motor is
PIN = VT ( I A + I F ) = ( 230 V )(60 A + 1.35 A ) = 14,100 W

Therefore, the efficiency is

η=
(e)

The no-load E A will be 230 V, so the no-load speed will be
n=

(f)

11,190 W
POUT
× 100% =
× 100% = 79.4%
PIN
14,100 W

EA
230 V
no =
(1800 r/min ) = 1873 r/min
E Ao
221 V

If the field circuit opens, the field current would go to zero ⇒ φ drops to φ res ⇒ E A ↓ ⇒ I A ↑⇒

τ ind ↑ ⇒ n↑ to a very high speed. If I F = 0 A, E Ao = 8.5 V at 1800 r/min, so
n=

EA
230 V
no =
(1800 r/min ) = 48,700 r/min
E Ao
8.5 V

(In reality, the motor speed would be limited by rotational losses, or else the motor will destroy itself first.)
(g)

The maximum value of Radj = 200 Ω, so
IF =

VT
230 V
=
= 0.821 A
RF + Radj 200 Ω + 80 Ω

This field current would produce a voltage E Ao of 153 V at a speed of no = 1800 r/min. The actual E A is
230 V, so the actual speed will be
n=

EA
230 V
no =
(1800 r/min ) = 2706 r/min
E Ao
153 V

The minimum value of Radj = 0 Ω, so
IF =

VT
230 V
=
= 2.875 A
RF + Radj 0 Ω + 80 Ω

This field current would produce a voltage E Ao of about 242 V at a speed of no = 1800 r/min. The actual
E A is 230 V, so the actual speed will be

244

n=

9-22.

EA
230 V
no =
(1800 r/min ) = 1711 r/min
E Ao
242 V

The magnetization curve for a separately excited dc generator is shown in Figure P9-7. The generator is
rated at 6 kW, 120 V, 50 A, and 1800 r/min and is shown in Figure P9-8. Its field circuit is rated at 5A.
The following data are known about the machine:

Note:

An electronic version of this magnetization curve can be found in file
p97_mag.dat, which can be used with MATLAB programs. Column 1
contains field current in amps, and column 2 contains the internal generated
voltage EA in volts.

245

R A = 0.18 Ω
Radj = 0 to 30 Ω

VF = 120 V
RF = 24 Ω

N F = 1000 turns per pole
(a) If this generator is operating at no load, what is the range of voltage adjustments that can be achieved
by changing Radj ?
(b) If the field rheostat is allowed to vary from 0 to 30 Ω and the generator’s speed is allowed to vary from
1500 to 2000 r/min, what are the maximum and minimum no-load voltages in the generator?
SOLUTION
(a) If the generator is operating with no load at 1800 r/min, then the terminal voltage will equal the
internal generated voltage E A . The maximum possible field current occurs when Radj = 0 Ω. The current
is
I F ,max =

VF
120 V
=
=5A
RF + Radj 24 Ω + 0 Ω

From the magnetization curve, the voltage E Ao at 1800 r/min is 129 V. Since the actual speed is 1800
r/min, the maximum no-load voltage is 129 V.
The minimum possible field current occurs when Radj = 30 Ω. The current is
I F ,max =

VF
120 V
=
= 2.22 A
RF + Radj 24 Ω + 30 Ω

From the magnetization curve, the voltage E Ao at 1800 r/min is 87.4 V. Since the actual speed is 1800
r/min, the minimum no-load voltage is 87 V.
(b) The maximum voltage will occur at the highest current and speed, and the minimum voltage will
occur at the lowest current and speed. The maximum possible field current occurs when Radj = 0 Ω. The
current is
I F ,max =

VF
120 V
=
=5A
RF + Radj 24 Ω + 0 Ω

From the magnetization curve, the voltage E Ao at 1800 r/min is 129 V. Since the actual speed is 2000
r/min, the maximum no-load voltage is
246

EA
n
=
E Ao no
EA =

n
2000 r/min
E Ao =
(129 V ) = 143 V
no
1800 r/min

The minimum possible field current occurs when Radj = 30 Ω. The current is
I F ,max =

VF
120 V
=
= 2.22 A
RF + Radj 24 Ω + 30 Ω

From the magnetization curve, the voltage E Ao at 1800 r/min is 87.4 V. Since the actual speed is 1500
r/min, the maximum no-load voltage is
EA
n
=
E Ao no
EA =

9-23.

n
1500 r/min
E Ao =
(87.4 V ) = 72.8 V
no
1800 r/min

If the armature current of the generator in Problem 9-22 is 50 A, the speed of the generator is 1700 r/min,
and the terminal voltage is 106 V, how much field current must be flowing in the generator?
SOLUTION The internal generated voltage of this generator is
E A = VT + I A RA = 106 V + (50 A )( 0.18 Ω ) = 115 V

at a speed of 1700 r/min. This corresponds to an E Ao at 1800 r/min of
EA
n
=
E Ao no

E Ao =

no
1800 r/min
EA =
(115 V ) = 121.8 V
n
1700 r/min

From the magnetization curve, this value of E Ao requires a field current of 4.2 A.

9-24.

Assuming that the generator in Problem 9-22 has an armature reaction at full load equivalent to 400
A⋅turns of magnetomotive force, what will the terminal voltage of the generator be when I F = 5 A, nm =
1700 r/min, and I A = 50 A?
SOLUTION When I F is 5 A and the armature current is 50 A, the magnetomotive force in the generator is
Fnet = NI F − FAR = (1000 turns )(5 A ) − 400 A ⋅ turns = 4600 A ⋅ turns

or

I F * = Fnet / N F = 4600 A ⋅ turns / 1000 turns = 4.6 A

The equivalent internal generated voltage E Ao of the generator at 1800 r/min would be 126 V. The actual
voltage at 1700 r/min would be
EA =

n
1700 r/min
E Ao =
(126 V ) = 119 V
no
1800 r/min

Therefore, the terminal voltage would be
VT = E A − I A RA = 119 V − (50 A )(0.18 Ω ) = 110 V

247

9-25.

The machine in Problem 9-22 is reconnected as a shunt generator and is shown in Figure P9-9. The shunt
field resistor Radj is adjusted to 10 Ω, and the generator’s speed is 1800 r/min.

(a) What is the no-load terminal voltage of the generator?
(b) Assuming no armature reaction, what is the terminal voltage of the generator with an armature current
of 20 A? 40 A?
(c) Assuming an armature reaction equal to 200 A⋅turns at full load, what is the terminal voltage of the
generator with an armature current of 20 A? 40 A?
(d) Calculate and plot the terminal characteristics of this generator with and without armature reaction.
SOLUTION
(a) The total field resistance of this generator is 34 Ω, and the no-load terminal voltage can be found
from the intersection of the resistance line with the magnetization curve for this generator. The
magnetization curve and the field resistance line are plotted below. As you can see, they intersect at a
terminal voltage of 112 V.

248

(b)

At an armature current of 20 A, the internal voltage drop in the armature resistance is
As shown in the figure below, there is a difference of 3.6 V between E A and

(20 A )(0.18 Ω) = 3.6 V .

VT at a terminal voltage of about 106 V.

A MATLAB program to locate the position where the triangle exactly fits between the E A and VT lines is
shown below. This program created the plot shown above. Note that there are actually two places where
the difference between the E A and VT lines is 3.6 volts, but the low-voltage one of them is unstable. The
code shown in bold face below prevents the program from reporting that first (unstable) point.
% M-file: prob9_25b.m
% M-file to create a plot of the magnetization curve and the
%
field current curve of a shunt dc generator, determining
%
the point where the difference between them is 3.6 V.
% Get the magnetization curve. This file contains the
% three variables if_values, ea_values, and n_0.
clear all
if_values = p97_mag(:,1);
ea_values = p97_mag(:,2);
n_0 = 1800;
% First, initialize the values needed in this program.
r_f = 24;
% Field resistance (ohms)
% Adjustable resistance (ohms)
r_a = 0.19;
% Armature + series resistance (ohms)
i_f = 0:0.02:6;
% Field current (A)
n = 1800;
% Generator speed (r/min)
% Calculate Ea versus If
Ea = interp1(if_values,ea_values,i_f);

249

% Calculate Vt versus If
Vt = (r_f + r_adj) * i_f;
% Find the point where the difference between the two
% lines is 3.6 V. This will be the point where the line
% line "Ea - Vt - 3.6" goes negative. That will be a
% close enough estimate of Vt.
diff = Ea - Vt - 3.6;
% This code prevents us from reporting the first (unstable)
% location satisfying the criterion.
was_pos = 0;
for ii = 1:length(i_f);
if diff(ii) > 0
was_pos = 1;
end
if ( diff(ii) < 0 & was_pos == 1 )
break;
end;
end;
% We
disp
disp
disp

have the intersection. Tell
(['Ea = ' num2str(Ea(ii)) '
(['Vt = ' num2str(Vt(ii)) '
(['If = ' num2str(i_f(ii)) '

user.
V']);
V']);
A']);

% Plot the curves
figure(1);
plot(i_f,Ea,'b-','LineWidth',2.0);
hold on;
plot(i_f,Vt,'k--','LineWidth',2.0);
% Plot intersections
plot([i_f(ii) i_f(ii)], [0 Ea(ii)], 'k-');
plot([0 i_f(ii)], [Vt(ii) Vt(ii)],'k-');
plot([0 i_f(ii)], [Ea(ii) Ea(ii)],'k-');
xlabel('\bf\itI_{F} \rm\bf(A)');
ylabel('\bf\itE_{A} \rm\bf or \itV_{T}');
title ('\bfPlot of \itE_{A} \rm\bf and \itV_{T} \rm\bf vs field
current');
axis ([0 5 0 150]);
set(gca,'YTick',[0 10 20 30 40 50 60 70 80 90 100 110 120 130 140
150]')
set(gca,'XTick',[0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0]')
legend ('Ea line','Vt line',4);
hold off;
grid on;

At an armature current of 40 A, the internal voltage drop in the armature resistance is
(40 A )(0.18 Ω) = 7.2 V . As shown in the figure below, there is a difference of 7.2 V between E A and

VT at a terminal voltage of about 98 V.

250

(c) The rated current of this generated is 50 A, so 20 A is 40% of full load. If the full load armature
reaction is 200 A⋅turns, and if the armature reaction is assumed to change linearly with armature current,
then the armature reaction will be 80 A⋅turns. The figure below shows that a triangle consisting of 3.6 V
and (80 A⋅turns)/(1000 turns) = 0.08 A fits exactly between the E A and VT lines at a terminal voltage of
103 V.

251

The rated current of this generated is 50 A, so 40 A is 80% of full load. If the full load armature reaction
is 200 A⋅turns, and if the armature reaction is assumed to change linearly with armature current, then the
armature reaction will be 160 A⋅turns. There is no point where a triangle consisting of 3.6 V and (80
A⋅turns)/(1000 turns) = 0.16 A fits exactly between the E A and VT lines, so this is not a stable operating
condition.
(c) A MATLAB program to calculate the terminal characteristic of this generator without armature
reaction is shown below:
% M-file: prob9_25d.m
% M-file to calculate the terminal characteristic of a shunt
%
dc generator without armature reaction.
% Get the magnetization curve. This file contains the
% three variables if_values, ea_values, and n_0.
if_values = p97_mag(:,1);
ea_values = p97_mag(:,2);
n_0 = 1800;
% First, initialize the values needed in this program.
r_f = 24;
% Field resistance (ohms)
% Adjustable resistance (ohms)
r_a = 0.18;
% Armature + series resistance (ohms)
i_f = 0:0.005:6;
% Field current (A)
n = 1800;
% Generator speed (r/min)

252

% Calculate Ea versus If
Ea = interp1(if_values,ea_values,i_f);
% Calculate Vt versus If
Vt = (r_f + r_adj) * i_f;
% Find the point where the difference between the two
% lines is exactly equal to i_a*r_a. This will be the
% point where the line line "Ea - Vt - i_a*r_a" goes
% negative.
i_a = 0:1:50;
for jj = 1:length(i_a)
% Get the voltage difference
diff = Ea - Vt - i_a(jj)*r_a;
% This code prevents us from reporting the first (unstable)
% location satisfying the criterion.
was_pos = 0;
for ii = 1:length(i_f);
if diff(ii) > 0
was_pos = 1;
end
if ( diff(ii) < 0 & was_pos == 1 )
break;
end;
end;
% Save terminal voltage at this point
v_t(jj) = Vt(ii);
i_l(jj) = i_a(jj) - v_t(jj) / ( r_f + r_adj);
end;
% Plot the terminal characteristic
figure(1);
plot(i_l,v_t,'b-','LineWidth',2.0);
xlabel('\bf\itI_{L} \rm\bf(A)');
ylabel('\bf\itV_{T} \rm\bf(V)');
title ('\bfTerminal Characteristic of a Shunt DC Generator');
hold off;
axis( [ 0 50 0 120]);
grid on;

253

The resulting terminal characteristic is shown below:

A MATLAB program to calculate the terminal characteristic of this generator with armature reaction is
shown below:
% M-file: prob9_25d2.m
% M-file to calculate the terminal characteristic of a shunt
%
dc generator with armature reaction.
% Get the magnetization curve. This file contains the
% three variables if_values, ea_values, and n_0.
clear all
if_values = p97_mag(:,1);
ea_values = p97_mag(:,2);
n_0 = 1800;
% First, initialize the values needed in this program.
r_f = 24;
% Field resistance (ohms)
% Adjustable resistance (ohms)
r_a = 0.18;
% Armature + series resistance (ohms)
i_f = 0:0.005:6;
% Field current (A)
n = 1800;
% Generator speed (r/min)
n_f = 1000;
% Number of field turns
% Calculate Ea versus If
Ea = interp1(if_values,ea_values,i_f);
% Calculate Vt versus If
Vt = (r_f + r_adj) * i_f;
% Find the point where the difference between the Ea
% armature reaction line and the Vt line is exactly
% equal to i_a*r_a. This will be the point where

254

% the line "Ea_ar - Vt - i_a*r_a" goes negative.
i_a = 0:1:37;
for jj = 1:length(i_a)
% Calculate the equivalent field current due to armature
% reaction.
i_ar = (i_a(jj) / 50) * 200 / n_f;
% Calculate the Ea values modified by armature reaction
Ea_ar = interp1(if_values,ea_values,i_f - i_ar);
% Get the voltage difference
diff = Ea_ar - Vt - i_a(jj)*r_a;
% This code prevents us from reporting the first (unstable)
% location satisfying the criterion.
was_pos = 0;
for ii = 1:length(i_f);
if diff(ii) > 0
was_pos = 1;
end
if ( diff(ii) < 0 & was_pos == 1 )
break;
end;
end;
% Save terminal voltage at this point
v_t(jj) = Vt(ii);
i_l(jj) = i_a(jj) - v_t(jj) / ( r_f + r_adj);
end;
% Plot the terminal characteristic
figure(1);
plot(i_l,v_t,'b-','LineWidth',2.0);
xlabel('\bf\itI_{L} \rm\bf(A)');
ylabel('\bf\itV_{T} \rm\bf(V)');
title ('\bfTerminal Characteristic of a Shunt DC Generator w/AR');
hold off;
axis([ 0 50 0 120]);
grid on;

255

The resulting terminal characteristic is shown below:

9-26.

If the machine in Problem 9-25 is running at 1800 r/min with a field resistance Radj = 10 Ω and an
armature current of 25 A, what will the resulting terminal voltage be? If the field resistor decreases to 5 Ω
while the armature current remains 25 A, what will the new terminal voltage be? (Assume no armature
reaction.)
SOLUTION If I A = 25 A, then I A RA = ( 25 A )( 0.18 Ω ) = 4.5 V. The point where the distance between the

E A and VT curves is exactly 4.5 V corresponds to a terminal voltage of 104 V, as shown below.

256

If Radj decreases to 5 Ω, the total field resistance becomes 29 Ω, and the terminal voltage line gets
shallower. The new point where the distance between the E A and VT curves is exactly 4.5 V corresponds
to a terminal voltage of 115 V, as shown below.

Note that decreasing the field resistance of the shunt generator increases the terminal voltage.

9-27.

A 120-V 50-A cumulatively compounded dc generator has the following characteristics:
R A + RS = 0.21 Ω
N F = 1000 turns
RF = 20 Ω
N SE = 20 turns
Radj = 0 to 30 Ω, set to 10 Ω
nm = 1800 r/min
The machine has the magnetization curve shown in Figure P9-7. Its equivalent circuit is shown in Figure

(a) If the generator is operating at no load, what is its terminal voltage?
(b) If the generator has an armature current of 20 A, what is its terminal voltage?
257

(c) If the generator has an armature current of 40 A, what is its terminal voltage'?
(d) Calculate and plot the terminal characteristic of this machine.
SOLUTION
(a) The total field resistance of this generator is 30 Ω, and the no-load terminal voltage can be found
from the intersection of the resistance line with the magnetization curve for this generator. The
magnetization curve and the field resistance line are plotted below. As you can see, they intersect at a
terminal voltage of 121 V.

(b)

If the armature current is 20 A, then the effective field current contribution from the armature current
N SE
20
IA =
(20 A ) = 0.4 A
NF
1000

and the I A ( R A + RS ) voltage drop is I A ( R A + RS ) = ( 20 A ) ( 0.21 Ω ) = 4.2 V . The location where the
N SE
I A and I A RA exactly fits between the E A and VT lines corresponds to a terminal
NF
voltage of 120 V, as shown below.

triangle formed by

258

(c)

If the armature current is 40 A, then the effective field current contribution from the armature current

N SE
15
IA =
(40 A ) = 0.6 A
NF
1000
and the I A (R A + RS ) voltage drop is I A (RA + RS ) = (80 A )(0.20 Ω ) = 8 V . The location where the
triangle formed by

N SE
I A and I A R A exactly fits between the E A and VT lines corresponds to a terminal
NF

voltage of 116 V, as shown below.

259

A MATLAB program to locate the position where the triangle exactly fits between the E A and VT lines is
shown below. This program created the plot shown above.
% M-file: prob9_27b.m
% M-file to create a plot of the magnetization curve and the
%
field current curve of a cumulatively-compounded dc generator
%
when the armature current is 20 A.
% Get the magnetization curve. This file contains the
% three variables if_values, ea_values, and n_0.
clear all
if_values = p97_mag(:,1);
ea_values = p97_mag(:,2);
n_0 = 1800;
% First, initialize the values needed in this program.
r_f = 20;
% Field resistance (ohms)
% Adjustable resistance (ohms)
r_a = 0.21;
% Armature + series resistance (ohms)
i_f = 0:0.02:6;
% Field current (A)
n = 1800;
% Generator speed (r/min)
n_f = 1000;
% Shunt field turns
n_se = 20;
% Series field turns
% Calculate Ea versus If
Ea = interp1(if_values,ea_values,i_f);
% Calculate Vt versus If
Vt = (r_f + r_adj) * i_f;
% Calculate the Ea values modified by mmf due to the
% armature current
260

i_a = 20;
Ea_a = interp1(if_values,ea_values,i_f + i_a * n_se/n_f);
% Find the point where the difference between the
% enhanced Ea line and the Vt line is 4 V. This will
% be the point where the line "Ea_a - Vt - 4" goes
% negative.
diff = Ea_a - Vt - 4;
% This code prevents us from reporting the first (unstable)
% location satisfying the criterion.
was_pos = 0;
for ii = 1:length(i_f);
if diff(ii) > 0
was_pos = 1;
end
if ( diff(ii) < 0 & was_pos == 1 )
break;
end;
end;
% We
disp
disp
disp
disp
disp

have the
(['Ea_a
(['Ea
(['Vt
(['If
(['If_a

intersection. Tell user.
= ' num2str(Ea_a(ii)) ' V']);
= ' num2str(Ea(ii)) ' V']);
= ' num2str(Vt(ii)) ' V']);
= ' num2str(i_f(ii)) ' A']);
= ' num2str(i_f(ii)+ i_a * n_se/n_f) ' A']);

% Plot the curves
figure(1);
plot(i_f,Ea,'b-','LineWidth',2.0);
hold on;
plot(i_f,Vt,'k--','LineWidth',2.0);
% Plot intersections
plot([i_f(ii) i_f(ii)], [0 Vt(ii)], 'k-');
plot([0 i_f(ii)], [Vt(ii) Vt(ii)],'k-');
plot([0 i_f(ii)+i_a*n_se/n_f], [Ea_a(ii) Ea_a(ii)],'k-');
% Plot compounding triangle
plot([i_f(ii) i_f(ii)+i_a*n_se/n_f],[Vt(ii) Vt(ii)],'b-');
plot([i_f(ii) i_f(ii)+i_a*n_se/n_f],[Vt(ii) Ea_a(ii)],'b-');
plot([i_f(ii)+i_a*n_se/n_f i_f(ii)+i_a*n_se/n_f],[Vt(ii)
Ea_a(ii)],'b-');
xlabel('\bf\itI_{F} \rm\bf(A)');
ylabel('\bf\itE_{A} \rm\bf or \itE_{A} \rm\bf(V)');
title ('\bfPlot of \itE_{A} \rm\bf and \itV_{T} \rm\bf vs field
current');
axis ([0 5 0 150]);
set(gca,'YTick',[0 10 20 30 40 50 60 70 80 90 100 110 120 130 140
150]')
set(gca,'XTick',[0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0]')
legend ('Ea line','Vt line',4);
hold off;
grid on;

261

(d) A MATLAB program to calculate and plot the terminal characteristic of this generator is shown
below.
% M-file: prob9_27d.m
% M-file to calculate the terminal characteristic of a
%
cumulatively compounded dc generator without armature
%
reaction.
% Get the magnetization curve. This file contains the
% three variables if_values, ea_values, and n_0.
clear all
if_values = p97_mag(:,1);
ea_values = p97_mag(:,2);
n_0 = 1800;
% First, initialize the values needed in this program.
r_f = 20;
% Field resistance (ohms)
% Adjustable resistance (ohms)
r_a = 0.21;
% Armature + series resistance (ohms)
i_f = 0:0.02:6;
% Field current (A)
n = 1800;
% Generator speed (r/min)
n_f = 1000;
% Shunt field turns
n_se = 20;
% Series field turns
% Calculate Ea versus If
Ea = interp1(if_values,ea_values,i_f);
% Calculate Vt versus If
Vt = (r_f + r_adj) * i_f;
% Find the point where the difference between the two
% lines is exactly equal to i_a*r_a. This will be the
% point where the line line "Ea - Vt - i_a*r_a" goes
% negative.
i_a = 0:1:50;
for jj = 1:length(i_a)
% Calculate the Ea values modified by mmf due to the
% armature current
Ea_a = interp1(if_values,ea_values,i_f + i_a(jj)*n_se/n_f);
% Get the voltage difference
diff = Ea_a - Vt - i_a(jj)*r_a;
% This code prevents us from reporting the first (unstable)
% location satisfying the criterion.
was_pos = 0;
for ii = 1:length(i_f);
if diff(ii) > 0
was_pos = 1;
end
if ( diff(ii) < 0 & was_pos == 1 )
break;
end;
end;

262

% Save terminal voltage at this point
v_t(jj) = Vt(ii);
i_l(jj) = i_a(jj) - v_t(jj) / ( r_f + r_adj);
end;
% Plot the terminal characteristic
figure(1);
plot(i_l,v_t,'b-','LineWidth',2.0);
xlabel('\bf\itI_{L} \rm\bf(A)');
ylabel('\bf\itV_{T} \rm\bf(V)');
string = ['\bfTerminal Characteristic of a Cumulatively ' ...
'Compounded DC Generator'];
title (string);
hold off;
axis([ 0 50 0 130]);
grid on;

The resulting terminal characteristic is shown below. Compare it to the terminal characteristics of the
shunt dc generators in Problem 9-25 (d).

9-28.

If the machine described in Problem 9-27 is reconnected as a differentially compounded dc generator, what
will its terminal characteristic look like? Derive it in the same fashion as in Problem 9-27.
SOLUTION A MATLAB program to calculate and plot the terminal characteristic of this generator is shown
below.
% M-file: prob9_28.m
% M-file to calculate the terminal characteristic of a
%
differentially compounded dc generator without armature
%
reaction.
% Get the magnetization curve. This file contains the
263

% three variables if_values, ea_values, and n_0.
clear all
if_values = p97_mag(:,1);
ea_values = p97_mag(:,2);
n_0 = 1800;
% First, initialize the values needed in this program.
r_f = 20;
% Field resistance (ohms)
% Adjustable resistance (ohms)
r_a = 0.21;
% Armature + series resistance (ohms)
i_f = 0:0.02:6;
% Field current (A)
n = 1800;
% Generator speed (r/min)
n_f = 1000;
% Shunt field turns
n_se = 20;
% Series field turns
% Calculate Ea versus If
Ea = interp1(if_values,ea_values,i_f);
% Calculate Vt versus If
Vt = (r_f + r_adj) * i_f;
% Find the point where the difference between the two
% lines is exactly equal to i_a*r_a. This will be the
% point where the line line "Ea - Vt - i_a*r_a" goes
% negative.
i_a = 0:1:26;
for jj = 1:length(i_a)
% Calculate the Ea values modified by mmf due to the
% armature current
Ea_a = interp1(if_values,ea_values,i_f - i_a(jj)*n_se/n_f);
% Get the voltage difference
diff = Ea_a - Vt - i_a(jj)*r_a;
% This code prevents us from reporting the first (unstable)
% location satisfying the criterion.
was_pos = 0;
for ii = 1:length(i_f);
if diff(ii) > 0
was_pos = 1;
end
if ( diff(ii) < 0 & was_pos == 1 )
break;
end;
end;
% Save terminal voltage at this point
v_t(jj) = Vt(ii);
i_l(jj) = i_a(jj) - v_t(jj) / ( r_f + r_adj);
end;
% Plot the terminal characteristic
figure(1);

264

plot(i_l,v_t,'b-','LineWidth',2.0);
xlabel('\bf\itI_{L} \rm\bf(A)');
ylabel('\bf\itV_{T} \rm\bf(V)');
string = ['\bfTerminal Characteristic of a Cumulatively ' ...
'Compounded DC Generator'];
title (string);
hold off;
axis([ 0 50 0 120]);
grid on;

The resulting terminal characteristic is shown below. Compare it to the terminal characteristics of the
cumulatively compounded dc generator in Problem 9-28 and the shunt dc generators in Problem 9-25 (d).

9-29.

A cumulatively compounded dc generator is operating properly as a flat-compounded dc generator. The
machine is then shut down, and its shunt field connections are reversed.
(a) If this generator is turned in the same direction as before, will an output voltage be built up at its
terminals? Why or why not?
(b) Will the voltage build up for rotation in the opposite direction? Why or why not?
(c) For the direction of rotation in which a voltage builds up, will the generator be cumulatively or
differentially compounded?
SOLUTION
(a) The output voltage will not build up, because the residual flux now induces a voltage in the opposite
direction, which causes a field current to flow that tends to further reduce the residual flux.
(b) If the motor rotates in the opposite direction, the voltage will build up, because the reversal in voltage
due to the change in direction of rotation causes the voltage to produce a field current that increases the
residual flux, starting a positive feedback chain.
(c)

The generator will now be differentially compounded.

265

9-30.

A three-phase synchronous machine is mechanically connected to a shunt dc machine, forming a motorgenerator set, as shown in Figure P9-11. The dc machine is connected to a dc power system supplying a
constant 240 V, and the ac machine is connected to a 480-V 60-Hz infinite bus.

The dc machine has four poles and is rated at 50 kW and 240 V. It has a per-unit armature resistance of
0.04. The ac machine has four poles and is Y-connected. It is rated at 50 kVA, 480 V, and 0.8 PF, and its
saturated synchronous reactance is 2.0 Ω per phase.
All losses except the dc machine’s armature resistance may be neglected in this problem. Assume that the
magnetization curves of both machines are linear.
(a) Initially, the ac machine is supplying 50 kVA at 0.8 PF lagging to the ac power system.
1. How much power is being supplied to the dc motor from the dc power system?
2. How large is the internal generated voltage E A of the dc machine?
3. How large is the internal generated voltage E A of the ac machine?
(b) The field current in the ac machine is now increased by 5 percent. What effect does this change have
on the real power supplied by the motor-generator set? On the reactive power supplied by the motorgenerator set? Calculate the real and reactive power supplied or consumed by the ac machine under
these conditions. Sketch the ac machine’s phasor diagram before and after the change in field current.
(c) Starting from the conditions in part (b), the field current in the dc machine is now decreased by 1
percent. What effect does this change have on the real power supplied by the motor-generator set? On
the reactive power supplied by the motor-generator set? Calculate the real and reactive power supplied
or consumed by the ac machine under these conditions. Sketch the ac machine’s phasor diagram before
and after the change in the dc machine’s field current.
(d) From the above results, answer the following questions:
1. How can the real power flow through an ac-dc motor-generator set be controlled?
2. How can the reactive power supplied or consumed by the ac machine be controlled without
affecting the real power flow?
SOLUTION
(a)

The power supplied by the ac machine to the ac power system is
PAC = S cos θ = (50 kVA )( 0.8) = 40 kW

266

and the reactive power supplied by the ac machine to the ac power system is
QAC = S sin θ = (50 kVA ) sin cos −1 (0.8) = 30 kvar

The power out of the dc motor is thus 40 kW. This is also the power converted from electrical to
mechanical form in the dc machine, since all other losses are neglected. Therefore,
Pconv = E A I A = (VT − I A R A ) I A = 40 kW
VT I A − I A 2 R A − 40 kW = 0
The base resistance of the dc machine is
VT ,base 2 ( 230 V )
=
= 1.058 Ω
Pbase
50 kW
2

Rbase,dc =

Therefore, the actual armature resistance is
R A = (0.04)(1.058 Ω ) = 0.0423 Ω
Continuing to solve the equation for Pconv , we get
0.0423 I A 2 − 230 I A + 40, 000 = 0

I A2 − 5434.8 I A + 945180 = 0
I A = 179.9 A
and E A = 222.4 V.
Therefore, the power into the dc machine is VT I A = 41.38 kW , while the power converted from electrical

to mechanical form (which is equal to the output power) is E A I A = ( 222.4 V )(179.9 A ) = 40 kW . The
internal generated voltage E A of the dc machine is 222.4 V.

The armature current in the ac machine is
IA =

S
50 kVA
=
= 60.1 A
3 Vφ
3 ( 480 V )

I A = 60.1∠ − 36.87° A
Therefore, the internal generated voltage E A of the ac machine is
E A = Vφ + jX S I A

E A = 277∠0° V + j ( 2.0 Ω )( 60.1∠ − 36.87° A ) = 362∠15.4° V
(b) When the field current of the ac machine is increased by 5%, it has no effect on the real power
supplied by the motor-generator set. This fact is true because P = τω , and the speed is constant (since the
MG set is tied to an infinite bus). With the speed unchanged, the dc machine’s torque is unchanged, so the
total power supplied to the ac machine’s shaft is unchanged.
If the field current is increased by 5% and the OCC of the ac machine is linear, E A increases to
E A′ = (1.05)( 262 V ) = 380 V

The new torque angle δ can be found from the fact that since the terminal voltage and power of the ac
machine are constant, the quantity E A sinδ must be constant.
267

E A sin δ = E A′ sin δ ′

δ ′ = sin −1

EA
362 V
sin δ = sin −1
sin15.4° = 14.7°
E A′
380 V

Therefore, the armature current will be

IA =

E A − Vφ
jX S

=

380∠14.7° V − 277∠0° V
= 66.1∠ − 43.2° A
j 2.0 Ω

The resulting reactive power is
Q = 3 VT I L sin θ = 3 ( 480 V )( 66.1 A ) sin 43.2° = 37.6 kvar

The reactive power supplied to the ac power system will be 37.6 kvar, compared to 30 kvar before the ac
machine field current was increased. The phasor diagram illustrating this change is shown below.

E A1 E A2

I A1
I A2

Vφ

jX I

S A

(c) If the dc field current is decreased by 1%, the dc machine’s flux will decrease by 1%. The internal
generated voltage in the dc machine is given by the equation E A = K φ ω , and ω is held constant by the
infinite bus attached to the ac machine. Therefore, E A on the dc machine will decrease to (0.99)(222.4 V)
= 220.2 V. The resulting armature current is
I A,dc =

VT − E A 230 V − 220.2 V
=
= 231.7 A
RA
0.0423 Ω

The power into the dc motor is now (230 V)(231.7 A) = 53.3 kW, and the power converted from electrical
to mechanical form in the dc machine is (220.2 V)(231.7 A) = 51 kW. This is also the output power of the
dc machine, the input power of the ac machine, and the output power of the ac machine, since losses are
being neglected.
The torque angle of the ac machine now can be found from the equation
Pac =

3Vφ E A
XS

δ = sin −1

sin δ

Pac X S
(51 kW )( 2.0 Ω ) = 18.9°
= sin −1
3Vφ E A
3 ( 277 V )( 380 V )

The new E A of this machine is thus 380∠18.9° V , and the resulting armature current is

IA =

E A − Vφ
jX S

=

380∠18.9° V − 277∠0° V
= 74.0∠ − 33.8° A
j 2.0 Ω

The real and reactive powers are now
P = 3 VT I L cos θ = 3 ( 480 V )(74.0 A ) cos 33.8° = 51 kW

Q = 3 VT I L sin θ = 3 ( 480 V )( 74.0 A ) sin 33.8° = 34.2 kvar

268

The phasor diagram of the ac machine before and after the change in dc machine field current is shown
below.

E A2
E A1
jX I

S A

I A1

Vφ

I A2

(d) The real power flow through an ac-dc motor-generator set can be controlled by adjusting the field
current of the dc machine. (Note that changes in power flow also have some effect on the reactive power of
the ac machine: in this problem, Q dropped from 35 kvar to 30 kvar when the real power flow was
The reactive power flow in the ac machine of the MG set can be adjusted by adjusting the ac machine’s
field current. This adjustment has basically no effect on the real power flow through the MG set.

269

Chapter 10: Single-Phase and Special-Purpose Motors
A 120-V, 1/3-hp 60-Hz, four-pole, split-phase induction motor has the following impedances:
R1 = 1.80 Ω
X 1 = 2.40 Ω
X M = 60 Ω
R2 = 2.50 Ω
X 2 = 2.40 Ω
At a slip of 0.05, the motor’s rotational losses are 51 W. The rotational losses may be assumed constant
over the normal operating range of the motor. If the slip is 0.05, find the following quantities for this
motor:
(a)
(b)
(c)
(d)

Input power
Air-gap power

Pconv
Pout
(e) τ ind

(g) Overall motor efficiency
(h) Stator power factor
SOLUTION The equivalent circuit of the motor is shown below
R1
+

jX1

1.8 Ω

j0.5X2

{
j2.4 Ω

j0.5XM

0.5ZF

-

ZB =

R2
s

0.5

R2
2−s

Forward

jX2

{

0.5ZB

ZF =

0.5

j30 Ω

V

ZF =

j1.20 Ω

j1.20 Ω

j0.5XM
j30 Ω

( R2 / s + jX 2 )( jX M )
R2 / s + jX 2 + jX M

(50 +

j 2.40)( j 60 )
= 28.15 + j 24.87 Ω
50 + j 2.40 + j 60

R2 / ( 2 s )

jX 2

( jX M )

R2 / ( 2 − s ) + jX 2 + jX M

270

{

I1

{

10-1.

Reverse

ZB =

(a)

(1.282 +

j 2.40)( j 60)
= 1.185 + j 2.332 Ω
1.282 + j 2.40 + j 60

The input current is
I1 =

V
R1 + jX 1 + 0.5Z F + 0.5Z B

I1 =

120∠ 0° V
= 5.23∠ − 44.2° A
1.80
j
2.40
0.5
28.15
+
+
+ j 24.87 ) + 0.5 (1.185 + j 2.332 )
(
)
(

PIN = VI cos θ = (120 V )(5.23 A ) cos 44.2° = 450 W

(b)

The air-gap power is
PAG,F = I12 ( 0.5RF ) = (5.23 A ) (14.1 Ω ) = 386 W
2

PAG,B = I12 ( 0.5 RB ) = (5.23 A ) (0.592 Ω ) = 16.2 W
2

PAG = PAG,F − PAG,B = 386 W − 14.8 W = 371 W
(c)

The power converted from electrical to mechanical form is
Pconv,F = (1 − s ) PAG,F = (1 − 0.05)(386 W ) = 367 W

Pconv,B = (1 − s ) PAG,B = (1 − 0.05)(16.2 W ) = 15.4 W

Pconv = Pconv,F − Pconv,B = 367 W − 15.4 W = 352 W
(d)

The output power is
POUT = Pconv − Prot = 352 W − 51 W = 301 W

(e)

The induced torque is
PAG

τ ind =

(f)

ω sync

1 min
60 s

POUT

ωm

=

301 W
(0.95)(1800 r/min ) 2π rad
1r

The overall efficiency is

η=
(h)

371 W
(1800 r/min ) 2π rad
1r

301 W
POUT
× 100% =
× 100% = 66.9%
PIN
450 W

The stator power factor is
PF = cos 44.2° = 0.713 lagging

10-2.

= 1.97 N ⋅ m

The load torque is

(g)

=

Repeat Problem 10-1 for a rotor slip of 0.025.
ZF =
ZF =

( R2 / s + jX 2 )( jX M )
R2 / s + jX 2 + jX M

(100 + j 2.40 )( j 60 ) = 28.91 +
100 + j 2.40 + j 60

j 43.83 Ω

271

1 min
60 s

= 1.68 N ⋅ m

R2 / ( 2 s )

ZB =
ZB =

(a)

jX 2

( jX M )

R2 / ( 2 − s ) + jX 2 + jX M

(1.282 +

j 2.40)( j 60)
= 1.170 + j 2.331 Ω
1.282 + j 2.40 + j 60

The input current is
I1 =

V
R1 + jX 1 + 0.5Z F + 0.5Z B

I1 =

120∠ 0° V
= 4.03∠ − 59.0° A
(1.80 + j 2.40 ) + 0.5 (25.91 + j 43.83) + 0.5 (1.170 + j 2.331)

PIN = VI cos θ = (120 V )( 4.03 A ) cos 59.0° = 249 W

(b)

The air-gap power is
PAG,F = I12 ( 0.5RF ) = ( 4.03 A ) (12.96 Ω ) = 210.5 W
2

PAG,B = I12 (0.5RB ) = ( 4.03 A ) ( 0.585 Ω ) = 9.5 W
2

PAG = PAG,F − PAG,B = 210.5 W − 9.5 W = 201 W
(c)

The power converted from electrical to mechanical form is
Pconv,F = (1 − s ) PAG,F = (1 − 0.025)( 210.5 W ) = 205 W

Pconv,B = (1 − s ) PAG,B = (1 − 0.025)( 9.5 W ) = 9.3 W
Pconv = Pconv,F − Pconv,B = 205 W − 9.3 W = 196 W

(d)

The output power is
POUT = Pconv − Prot = 205 W − 51 W = 154 W

(e)

The induced torque is

τ ind =

(f)

210.5 W
(1800 r/min )
1r

1 min
60 s

= 1.12 N ⋅ m

POUT

ωm

=

154 W
(0.975)(1800 r/min ) 2π rad
1r

1 min
60 s

= 0.84 N ⋅ m

The overall efficiency is

η=
(h)

=

The load torque is

(g)

PAG

ω sync

154 W
POUT
× 100% =
× 100% = 61.8%
PIN
249 W

The stator power factor is
PF = cos 59.0° = 0.515 lagging

10-3.

Suppose that the motor in Problem 10-1 is started and the auxiliary winding fails open while the rotor is
accelerating through 400 r/min. How much induced torque will the motor be able to produce on its main

272

winding alone? Assuming that the rotational losses are still 51 W, will this motor continue accelerating or
will it slow down again? Prove your answer.
SOLUTION At a speed of 400 r/min, the slip is
s=

1800 r/min − 400 r/min
= 0.778
1800 r/min

ZF =
ZF =

ZB =
ZB =

( R2 / s + jX 2 )( jX M )
R2 / s + jX 2 + jX M

(100 + j 2.40)( j 60 ) = 2.96 +
100 + j 2.40 + j 60

R2 / ( 2 s )

jX 2

j 2.46 Ω

( jX M )

R2 / ( 2 − s ) + jX 2 + jX M

(1.282 + j 2.40 )( j 60 ) = 1.90 + j 2.37 Ω
1.282 + j 2.40 + j 60

The input current is
I1 =

V
R1 + jX 1 + 0.5Z F + 0.5Z B

I1 =

120∠ 0° V
= 18.73∠ − 48.7° A
(1.80 + j 2.40) + 0.5 ( 2.96 + j 2.46 ) + 0.5 (1.90 + j 2.37 )

The air-gap power is
PAG,F = I12 ( 0.5RF ) = (18.73 A ) (1.48 Ω ) = 519.2 W
2

PAG,B = I12 ( 0.5RB ) = (18.73 A ) ( 0.945 Ω ) = 331.5 W
2

PAG = PAG,F − PAG,B = 519.2 W − 331.5 W = 188 W
The power converted from electrical to mechanical form is
Pconv,F = (1 − s ) PAG,F = (1 − 0.778 )(519.2 W ) = 115.2 W

Pconv,B = (1 − s ) PAG,B = (1 − 0.778)( 331.5 W ) = 73.6 W
Pconv = Pconv,F − Pconv,B = 115.2 W − 73.6 W = 41.6 W
The induced torque is

τ ind =

PAG

ω sync

=

188 W
(1800 r/min ) 2π rad
1r

1 min
60 s

= 1.00 N ⋅ m

Assuming that the rotational losses are still 51 W, this motor is not producing enough torque to keep
accelerating. Pconv is 41.6 W, while the rotational losses are 51 W, so there is not enough power to make
up the rotational losses. The motor will slow down5.

10-4.

5

Use MATLAB to calculate and plot the torque-speed characteristic of the motor in Problem 10-1, ignoring
the starting winding.

Note that in the real world, rotational losses decrease with decreased shaft speed. Therefore, the losses will really
be less than 51 W, and this motor might just be able to keep on accelerating slowly—it is a close thing either way.
273

SOLUTION This problem is best solved with MATLAB, since it involves calculating the torque-speed values
at many points. A MATLAB program to calculate and display both torque-speed characteristics is shown
below. Note that this program shows the torque-speed curve for both positive and negative directions of
rotation. Also, note that we had to avoid calculating the slip at exactly 0 or 2, since those numbers would
produce divide-by-zero errors in Z F and Z B respectively.
% M-file: torque_speed_curve3.m
% M-file create a plot of the torque-speed curve of the
%
single-phase induction motor of Problem 10-4.
% First, initialize the values needed in this program.
r1 = 1.80;
% Stator resistance
x1 = 2.40;
% Stator reactance
r2 = 2.50;
% Rotor resistance
x2 = 2.40;
% Rotor reactance
xm = 60;
% Magnetization branch reactance
v = 120;
% Single-Phase voltage
n_sync = 1800;
% Synchronous speed (r/min)
w_sync = 188.5;
% Synchronous speed (rad/s)
% Specify slip ranges to plot
s = 0:0.01:2.0;
% Offset slips at 0 and 2 slightly to avoid divide by zero errors
s(1)
= 0.0001;
s(201) = 1.9999;
% Get the corresponding speeds in rpm
nm = ( 1 - s) * n_sync;
% Caclulate Zf and Zb as a function of slip
zf = (r2 ./ s + j*x2) * (j*xm) ./ (r2 ./ s + j*x2 + j*xm);
zb = (r2 ./(2-s) + j*x2) * (j*xm) ./ (r2 ./(2-s) + j*x2 + j*xm);
% Calculate the current flowing at each slip
i1 = v ./ ( r1 + j*x1 + zf + zb);
% Calculate the air-gap power
p_ag_f = abs(i1).^2 .* 0.5 .* real(zf);
p_ag_b = abs(i1).^2 .* 0.5 .* real(zb);
p_ag = p_ag_f - p_ag_b;
% Calculate torque in N-m.
t_ind = p_ag ./ w_sync;
% Plot the torque-speed curve
figure(1)
plot(nm,t_ind,'Color','b','LineWidth',2.0);
xlabel('\itn_{m} \rm(r/min)');
ylabel('\tau_{ind} \rm(N-m)');
title ('Single Phase Induction motor torque-speed
characteristic','FontSize',12);
grid on;
hold off;

274

The resulting torque-speed characteristic is shown below:

10-5.

A 220-V, 1.5-hp 50-Hz, two-pole, capacitor-start induction motor has the following main-winding
impedances:
R1 = 1.40 Ω
X 1 = 2.01 Ω
X M = 105 Ω
R2 = 1.50 Ω
X 2 = 2.01 Ω
At a slip of 0.05, the motor’s rotational losses are 291 W. The rotational losses may be assumed constant
over the normal operating range of the motor. Find the following quantities for this motor at 5 percent slip:
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)

Stator current
Stator power factor
Input power
PAG

Pconv
Pout

τ ind
Efficiency

SOLUTION The equivalent circuit of the motor is shown below

275

R1
1.4 Ω

j0.5X2

{
j1.9 Ω

j1.90 Ω

j0.5XM

0.5ZF

-

ZB =
ZB =

(a)

(b)

{

j0.5XM
j100 Ω

Reverse

( R2 / s + jX 2 )( jX M )
R2 / s + jX 2 + jX M

(30 + j1.90)( j100) = 26.59 +
30 + j1.90 + j100

R2 / ( 2 s )

jX 2

j9.69 Ω

( jX M )

R2 / ( 2 − s ) + jX 2 + jX M

(0.769 +

j1.90 )( j100 )
= 0.741 + j1.870 Ω
0.769 + j1.90 + j100

The input stator current is
I1 =

V
R1 + jX 1 + 0.5Z F + 0.5Z B

I1 =

220∠ 0° V
= 13.0∠ − 27.0° A
(1.40 + j1.90) + 0.5 (26.59 + j 9.69 ) + 0.5 (0.741 + j1.870 )

The stator power factor is

The input power is
PIN = VI cos θ = ( 220 V )(13.0 A ) cos 27° = 2548 W

(d)

R2
2−s

j1.90 Ω

PF = cos 27° = 0.891 lagging
(c)

0.5

Forward

jX2

0.5ZB

ZF =

R2
s

j30 Ω

V = 220∠0° V

ZF =

0.5

{

+

jX1

{

I1

The air-gap power is
PAG,F = I12 ( 0.5RF ) = (13.0 A ) (13.29 Ω ) = 2246 W
2

PAG,B = I12 ( 0.5 RB ) = (13.0 A ) (0.370 Ω ) = 62.5 W
2

PAG = PAG,F − PAG,B = 2246 W − 62.5 W = 2184 W
276

(e)

The power converted from electrical to mechanical form is
Pconv,F = (1 − s ) PAG,F = (1 − 0.05)( 2246 W ) = 2134 W

Pconv,B = (1 − s ) PAG,B = (1 − 0.05)(62.5 W ) = 59 W

Pconv = Pconv,F − Pconv,B = 2134 W − 59 W = 2075 W

(f)

The output power is
POUT = Pconv − Prot = 2134 W − 291 W = 1843 W

(g)

The induced torque is
PAG

τ ind =

(h)

ω sync

1 min
60 s

= 6.95 N ⋅ m

POUT

ωm

=

1843 W
(0.95)(3000 r/min ) 2π rad
1r

1 min
60 s

= 6.18 N ⋅ m

The overall efficiency is

η=
10-6.

2184 W
(3000 r/min )
1r

The load torque is

(i)

=

1843 W
POUT
× 100% =
× 100% = 72.3%
PIN
2548 W

Find the induced torque in the motor in Problem 10-5 if it is operating at 5 percent slip and its terminal
voltage is (a) 190 V, (b) 208 V, (c) 230 V.
ZF =
ZF =

ZB =
ZB =

(a)

( R2 / s + jX 2 )( jX M )
R2 / s + jX 2 + jX M

(30 + j1.90)( j100) = 26.59 +
30 + j1.90 + j100

R2 / ( 2 s )

jX 2

j9.69 Ω

( jX M )

R2 / ( 2 − s ) + jX 2 + jX M

(0.769 +

j1.90 )( j100 )
= 0.741 + j1.870 Ω
0.769 + j1.90 + j100

If VT = 190∠0° V,
I1 =

V
R1 + jX 1 + 0.5Z F + 0.5Z B

I1 =

190∠ 0° V
= 11.2∠ − 27.0° A
(1.40 + j1.90) + 0.5 (26.59 + j 9.69 ) + 0.5 (0.741 + j1.870 )

PAG,F = I12 ( 0.5RF ) = (11.2 A ) (13.29 Ω ) = 1667 W
2

PAG,B = I12 ( 0.5 RB ) = (11.2 A ) ( 0.370 Ω ) = 46.4 W
2

PAG = PAG,F − PAG,B = 1667 W − 46.4 W = 1621 W

277

τ ind =

(b)

PAG

ω sync

=

1621 W
(3000 r/min ) 2π rad
1r

1 min
60 s

= 5.16 N ⋅ m

If VT = 208∠0° V,
I1 =
I1 =

V
R1 + jX 1 + 0.5Z F + 0.5Z B

(1.40 +

208∠0° V
= 12.3∠ − 27.0° A
j1.90) + 0.5 ( 26.59 + j 9.69 ) + 0.5 (0.741 + j1.870 )

PAG,F = I12 ( 0.5RF ) = (12.3 A ) (13.29 Ω ) = 2010 W
2

PAG,B = I12 ( 0.5 RB ) = (12.3 A ) ( 0.370 Ω ) = 56 W
2

PAG = PAG,F − PAG,B = 2010 W − 56 W = 1954 W

τ ind =

(c)

PAG

ω sync

=

1954 W
(3000 r/min ) 2π rad
1r

1 min
60 s

= 6.22 N ⋅ m

If VT = 230∠0° V,
I1 =

V
R1 + jX 1 + 0.5Z F + 0.5Z B

I1 =

230∠ 0° V
= 13.6∠ − 27.0° A
(1.40 + j1.90) + 0.5 (26.59 + j 9.69 ) + 0.5 (0.741 + j1.870 )

PAG,F = I12 ( 0.5RF ) = (13.6 A ) (13.29 Ω ) = 2458 W
2

PAG,B = I12 ( 0.5RB ) = (13.6 A ) ( 0.370 Ω ) = 68 W
2

PAG = PAG,F − PAG,B = 2458 W − 68 W = 2390 W

τ ind =

PAG

ω sync

=

2390 W
(3000 r/min ) 2π rad
1r

1 min
60 s

= 7.61 N ⋅ m

Note that the induced torque is proportional to the square of the terminal voltage.

10-7.

What type of motor would you select to perform each of the following jobs? Why?
(a) Vacuum cleaner

(b) Refrigerator

(c) Air conditioner compressor

(d) Air conditioner fan

(e) Variable-speed sewing machine

(f) Clock

(g) Electric drill
SOLUTION
(a)

Universal motor—for its high torque

(b) Capacitor start or Capacitor start and run—For its high starting torque and relatively constant
speed at a wide variety of loads
(c)

Same as (b) above
278

(d)

Split-phase—Fans are low-starting-torque applications, and a split-phase motor is appropriate

(e)

Universal Motor—Direction and speed are easy to control with solid-state drives

(f)

Hysteresis motor—for its easy starting and operation at nsync . A reluctance motor would also do

nicely.
(g) Universal Motor—for easy speed control with solid-state drives, plus high torque under loaded
conditions.

10-8.

For a particular application, a three-phase stepper motor must be capable of stepping in 10° increments.
How many poles must it have?
SOLUTION From Equation (10-18), the relationship between mechanical angle and electrical angle in a
three-phase stepper motor is

θm =
so

10-9.

2
θe
P

P=2

θe
60°
=2
= 12 poles
θm
10°

How many pulses per second must be supplied to the control unit of the motor in Problem 10-7 to achieve a
rotational speed of 600 r/min?
SOLUTION From Equation (10-20),
nm =
so

1
npulses
3P

npulses = 3 P nm = 3 (12 poles )( 600 r/min ) = 21, 600 pulses/min = 360 pulses/s

10-10. Construct a table showing step size versus number of poles for three-phase and four-phase stepper motors.
SOLUTION For 3-phase stepper motors, θ e = 60° , and for 4-phase stepper motors, θ e = 45° . Therefore,

Number of poles
2
4
6
8
10
12

Mechanical Step Size
3-phase ( θ e = 60° )
4-phase ( θ e = 45° )
60°
30°
20°
15°
12°
10°

279

45°
22.5°
15°
11.25°
9°
7.5°

Appendix A: Review of Three-Phase Circuits
A-1.

Three impedances of 4 + j3 Ω are ∆-connected and tied to a three-phase 208-V power line. Find I φ , I L ,
P, Q, S, and the power factor of this load.
SOLUTION
IL
+
Iφ

240 V

Zφ

-

Zφ = 3 + j4 Ω

Zφ

Zφ

Here, VL = Vφ = 208 V , and Zφ = 4 + j 3 Ω = 5∠36.87° Ω , so
Iφ =

Vφ

=

Zφ

208 V
= 41.6 A
5Ω

I L = 3 I φ = 3 ( 41.6 A ) = 72.05 A

P=3

Vφ 2

Q=3

Z
Vφ 2
Z

2
208 V )
(
cosθ = 3
cos 36.87° = 20.77 kW

5Ω

sin θ = 3

(208 V )2 sin 36.87° = 15.58 kvar
5Ω

S = P + Q = 25.96 kVA
PF = cos θ = 0.8 lagging
2

A-2.

2

Figure PA-1 shows a three-phase power system with two loads. The ∆-connected generator is producing a
line voltage of 480 V, and the line impedance is 0.09 + j0.16 Ω. Load 1 is Y-connected, with a phase
impedance of 2.5∠36.87° Ω and load 2 is ∆-connected, with a phase impedance of 5∠-20° Ω.

280

(a) What is the line voltage of the two loads?
(b) What is the voltage drop on the transmission lines?
(c) Find the real and reactive powers supplied to each load.
(d) Find the real and reactive power losses in the transmission line.
(e) Find the real power, reactive power, and power factor supplied by the generator.
SOLUTION To solve this problem, first convert the two deltas to equivalent wyes, and get the per-phase
equivalent circuit.
0.090 Ω

j0.16 Ω

+
Line

277∠0° V

+
-

Zφ 2

Zφ 1

Z φ1 = 2.5∠36.87° Ω
Zφ 2 = 1.67∠ − 20° Ω

(a)

The phase voltage of the equivalent Y-loads can be found by nodal analysis.
Vφ ,load − 277∠0° V
0.09 + j0.16 Ω

+

2.5∠36.87° Ω

+

1.67∠ − 20° Ω

=0

(5.443∠ − 60.6°) ( Vφ ,load − 277∠0° V ) + (0.4∠ − 36.87°) Vφ ,load + (0.6∠20°) Vφ ,load = 0
(5.955∠ − 53.34°) Vφ ,load = 1508∠ − 60.6°
Vφ ,load = 253.2∠ − 7.3° V
281

Therefore, the line voltage at the loads is VL 3 Vφ = 439 V .
(b)

The voltage drop in the transmission lines is
∆Vline = Vφ ,gen − Vφ ,load = 277∠0° V − 253.2∠-7.3° = 41.3∠52° V

(c)

The real and reactive power of each load is
P1 = 3

Vφ 2

Q1 = 3
P2 = 3

Z
Vφ 2

Z
Vφ 2

Q2 = 3
(d)

Z
Vφ 2
Z

cosθ = 3

(253.2 V)2 cos 36.87° = 61.6 kW
2.5 Ω

2
253.2 V )
(
sin θ = 3
sin 36.87° = 46.2 kvar

2.5 Ω

cosθ = 3

( 253.2 V )2 cos

(-20°) = 108.4 kW

sin θ = 3

(253.2 V )2 sin

(-20°) = −39.5 kvar

1.67 Ω

1.67 Ω

The line current is
I line =

41.3∠52° V
∆Vline
=
= 225∠ − 8.6°A
Z line
0.09 + j0.16 Ω

Therefore, the loses in the transmission line are
Pline = 3I line 2 Rline = 3 ( 225 A ) ( 0.09 Ω ) = 13.7 kW
2

Qline = 3I line 2 X line = 3 ( 225 A ) (0.16 Ω ) = 24.3 kvar
2

(e)

The real and reactive power supplied by the generator is
Pgen = Pline + P1 + P2 = 13.7 kW + 61.6 kW + 108.4 kW = 183.7 kW

Qgen = Qline + Q1 + Q2 = 24.3 kvar + 46.2 kvar − 39.5 kvar = 31 kvar
The power factor of the generator is
PF = cos tan -1

A-3.

Qgen
Pgen

= cos tan −1

31 kvar
= 0.986 lagging
183.7 kW

Figure PA-2 shows a one-line diagram of a simple power system containing a single 480 V generator and
three loads. Assume that the transmission lines in this power system are lossless, and answer the following
questions.
(a) Assume that Load 1 is Y-connected. What are the phase voltage and currents in that load?
(b) Assume that Load 2 is ∆-connected. What are the phase voltage and currents in that load?
(c) What real, reactive, and apparent power does the generator supply when the switch is open?
(d) What is the total line current I L when the switch is open?
(e) What real, reactive, and apparent power does the generator supply when the switch is closed?
(f) What is the total line current I L when the switch is closed?
(g) How does the total line current I L compare to the sum of the three individual currents I1 + I 2 + I 3 ? If
they are not equal, why not?
282

SOLUTION Since the transmission lines are lossless in this power system, the full voltage generated by G1
will be present at each of the loads.
(a)

Since this load is Y-connected, the phase voltage is
Vφ1 =

480 V
= 277 V
3

The phase current can be derived from the equation P = 3Vφ I φ cos θ as follows:
I φ1 =

(b)

100 kW
P
=
= 133.7 A
3Vφ cos θ 3 ( 277 V )(0.9 )

Since this load is ∆-connected, the phase voltage is
Vφ 2 = 480 V

The phase current can be derived from the equation S = 3Vφ I φ as follows:
Iφ 2 =

S
80 kVA
=
= 55.56 A
3Vφ 3 ( 480 V )

(c) The real and reactive power supplied by the generator when the switch is open is just the sum of the
real and reactive powers of Loads 1 and 2.
P1 = 100 kW

(

)

Q1 = P tan θ = P tan cos −1 PF = (100 kW )( tan 25.84°) = 48.4 kvar

P2 = S cosθ = (80 kVA )( 0.8) = 64 kW

Q2 = S sin θ = (80 kVA )( 0.6) = 48 kvar

PG = P1 + P2 = 100 kW + 64 kW = 164 kW
QG = Q1 + Q2 = 48.4 kvar + 48 kvar = 96.4 kvar
(d)

The line current when the switch is open is given by I L =

θ = tan −1
IL =

QG
96.4 kvar
= tan −1
= 30.45°
PG
164 kW

P
164 kW
=
= 228.8 A
3 VL cosθ
3 ( 480 V ) cos (30.45°)

283

P
Q
, where θ = tan −1 G .
PG
3 VL cos θ

(e) The real and reactive power supplied by the generator when the switch is closed is just the sum of the
real and reactive powers of Loads 1, 2, and 3. The powers of Loads 1 and 2 have already been calculated.
The real and reactive power of Load 3 are:
P3 = 80 kW

(

)

Q3 = P tan θ = P tan cos−1 PF = (80 kW ) tan ( 31.79

)

49.6 kvar

PG = P1 + P2 + P3 = 100 kW + 64 kW + 80 kW = 244 kW
QG = Q1 + Q2 + Q3 = 48.4 kvar + 48 kvar − 49.6 kvar = 46.8 kvar

(f)

The line current when the switch is closed is given by I L =

θ = tan −1
IL =
(g)

P
Q
, where θ = tan −1 G .
PG
3 VL cos θ

QG
46.8 kvar
= tan −1
= 10.86°
PG
244 kW

P
244 kW
=
= 298.8 A
3 VL cosθ
3 ( 480 V ) cos (10.86°)

The total line current from the generator is 298.8 A. The line currents to each individual load are:
P1
100 kW
I L1 =
=
= 133.6 A
3 VL cosθ1
3 ( 480 V )(0.9)
I L2 =

S2
80 kVA
=
= 96.2 A
3 VL
3 ( 480 V )

I L3 =

P3
80 kW
=
= 113.2 A
3 VL cosθ 3
3 ( 480 V )( 0.85)

The sum of the three individual line currents is 343 A, while the current supplied by the generator is 298.8
A. These values are not the same, because the three loads have different impedance angles. Essentially,
Load 3 is supplying some of the reactive power being consumed by Loads 1 and 2, so that it does not have
to come from the generator.
A-4.

Prove that the line voltage of a Y-connected generator with an acb phase sequence lags the corresponding
phase voltage by 30°. Draw a phasor diagram showing the phase and line voltages for this generator.
SOLUTION If the generator has an acb phase sequence, then the three phase voltages will be
Van = Vφ ∠0°
Vbn = Vφ ∠ − 240°
Vcn = Vφ ∠ − 120°

The relationship between line voltage and phase voltage is derived below. By Kirchhoff’s voltage law, the
line-to-line voltage Vab is given by
Vab = Va − Vb
Vab = Vφ ∠0° − Vφ ∠ − 240°

1
3
3
3
V = Vφ − j
V
Vab = Vφ − − Vφ + j
2
2 φ
2
2 φ
Vab = 3Vφ

3
1
−j
2
2

Vab = 3Vφ∠ − 30°

284

Thus the line voltage lags the corresponding phase voltage by 30°. The phasor diagram for this connection
is shown below.
Vbc
Vbn

Van

Vab

Vcn

A-5.

Find the magnitudes and angles of each line and phase voltage and current on the load shown in Figure P23.

SOLUTION Note that because this load is ∆-connected, the line and phase voltages are identical.
Vab = Van − Vbn = 120∠0° V - 120∠ - 120° V = 208∠30° V
Vbc = Vbn − Vcn = 120∠ − 120° V - 120∠ - 240° V = 208∠ - 90° V
Vca = Vcn − Van = 120∠ − 240° V - 120∠0° V = 208∠150° V

285

I ab =

Vab 208∠30° V
=
= 20.8∠10° A
10∠20° Ω
Zφ

I bc =

Vbc 208∠ − 90° V
=
= 20.8∠ − 110° A
Zφ
10∠20° Ω

I ca =

Vca 208∠150° V
=
= 20.8∠130° A
Zφ
10∠20° Ω

Ia = Iab − I ca = 20.8∠10° A - 20.8∠130° A = 36∠ - 20° A
Ib = I bc − I ab = 20.8∠ − 110° A - 20.8∠10° A = 36∠ - 140° A
Ic = Ica − Ibc = 20.8∠130° A - 20.8∠-110° A = 36∠100° A

A-6.

Figure PA-4 shows a small 480-V distribution system. Assume that the lines in the system have zero
impedance.

(a) If the switch shown is open, find the real, reactive, and apparent powers in the system. Find the total
current supplied to the distribution system by the utility.
(b) Repeat part (a) with the switch closed. What happened to the total current supplied? Why?
SOLUTION
(a)

With the switch open, the power supplied to each load is
2
V
(480 V )2 cos 30° = 59.86 kW
P1 = 3 φ cos θ = 3
Z
10 Ω
Vφ 2
(480 V )2 sin 30° = 34.56 kvar
sin θ = 3
Q1 = 3
10 Ω
Z
2
V
(277 V )2 cos 36.87° = 46.04 kW
P2 = 3 φ cos θ = 3
Z
4Ω
2
Vφ
277
( V )2 sin 36.87° = 34.53 kvar
sin θ = 3
Q2 = 3
4Ω
Z
PTOT = P1 + P2 = 59.86 kW + 46.04 kW = 105.9 kW
QTOT = Q1 + Q2 = 34.56 kvar + 34.53 kvar = 69.09 kvar
The apparent power supplied by the utility is
STOT = PTOT + QTOT = 126.4 kVA
The power factor supplied by the utility is
2

2

286

PF = cos tan -1

QTOT
69.09 kvar
= cos tan −1
= 0.838 lagging
PTOT
105.9 kW

The current supplied by the utility is
IL =

(b)

PTOT
105.9 kW
=
= 152 A
3 VT PF
3 ( 480 V ) (0.838)

With the switch closed, P3 is added to the circuit. The real and reactive power of P3 is
P3 = 3

Vφ 2

cos θ = 3

(277 V )2 cos (-90°) = 0 kW

Z
5Ω
2
2
Vφ
277 V )
(
sin θ = 3
sin ( -90°) = −46.06 kvar
P3 = 3
Z
5Ω
PTOT = P1 + P2 + P3 = 59.86 kW + 46.04 kW + 0 kW = 105.9 kW
QTOT = Q1 + Q2 + Q3 = 34.56 kvar + 34.53 kvar − 46.06 kvar = 23.03 kvar
The apparent power supplied by the utility is
STOT = PTOT + QTOT = 108.4 kVA
The power factor supplied by the utility is
2

PF = cos tan -1

2

QTOT
23.03 kVAR
= cos tan −1
= 0.977 lagging
PTOT
105.9 kW

The current supplied by the utility is
IL =

PTOT
=
3 VT PF

105.9 kW
= 130.4 A
3 ( 480 V ) (0.977 )

(c) The total current supplied by the power system drops when the switch is closed because the capacitor
bank is supplying some of the reactive power being consumed by loads 1 and 2.

287

Appendix B: Coil Pitch and Distributed Windings
B-1.

A 2-slot three-phase stator armature is wound for two-pole operation. If fractional-pitch windings are to be
used, what is the best possible choice for winding pitch if it is desired to eliminate the fifth-harmonic
component of voltage?
SOLUTION The pitch factor of a winding is given by Equation (B-19):

k p = sin

υρ
2

To eliminate the fifth harmonic, we want to select ρ so that sin

5ρ
= 0 . This implies that
2

5ρ
= (180°) n , where n = 0, 1, 2, 3, …
2
or

ρ=

2(180°) n
= 72°, 144°, ...
5

These are acceptable pitches to eliminate the fifth harmonic. Expressed as fractions of full pitch, these
pitches are 2/5, 4/5, 6/5, etc. Since the desire is to have the maximum possible fundamental voltage, the
best choice for coil pitch would be 4/5 or 6/5. The closest that we can approach to a 4/5 pitch in a 24-slot
winding is 10/12 pitch, so that is the pitch that we would use.
At 10/12 pitch,

150°
= 0.966
2
(5)(150°) = 0.259
k p = sin
2

k p = sin

for the fundamental frequency
for the fifth harmonic

288

B-2.

Derive the relationship for the winding distribution factor kd in Equation B-22.

SOLUTION The above illustration shows the case of 5 slots per phase, but the results are general. If there
are 5 slots per phase, each with voltage E Ai , where the phase angle of each voltage increases by γ° from
slot to slot, then the total voltage in the phase will be

E A = E A1 + E A 2 + E A3 + E A 4 + E A5 + ... + E An
The resulting voltage E A can be found from geometrical considerations. These “n” phases, when
drawn end-to-end, form equally-spaced chords on a circle of radius R. If a line is drawn from the center of
a chord to the origin of the circle, it forma a right triangle with the radius at the end of the chord (see
voltage E A5 above). The hypotenuse of this right triangle is R, its opposite side is E / 2 , and its smaller
angle is γ / 2 . Therefore,

sin

γ
2

=

E/2
⇒
R

1
E
R= 2
γ
sin
2

(1)

The total voltage E A also forms a chord on the circle, and dropping a line from the center of that chord to
the origin forms a right triangle. For this triangle, the hypotenuse is R, the opposite side is E A / 2 , and the
angle is nγ / 2 . Therefore,

nγ E A / 2
sin
=
R
2

⇒

1
EA
2
R=
nγ
sin
2

Combining (1) and (2) yields

289

(2)

1
1
E
EA
2
= 2
γ
nγ
sin
sin
2
2

nγ
sin
EA
2
=
γ
E
sin
2
Finally,

nγ
sin
EA
2
kd =
=
nE n sin γ
2
since k d is defined as the ratio of the total voltage produced to the sum of the magnitudes of each
component voltage.
B-3.

A three-phase four-pole synchronous machine has 96 stator slots. The slots contain a double-layer winding
(two coils per slot) with four turns per coil. The coil pitch is 19/24.
(a) Find the slot and coil pitch in electrical degrees.
(b) Find the pitch, distribution, and winding factors for this machine.
(c) How well will this winding suppress third, fifth, seventh, ninth, and eleventh harmonics? Be sure to
consider the effects of both coil pitch and winding distribution in your answer.
SOLUTION
(a) The coil pitch is 19/24 or 142.5°. Note that these are electrical degrees. Since this is a 4-pole
machine, the coil pitch would be 71.25 mechanical degrees.
There are 96 slots on this stator, so the slot pitch is 360°/96 = 3.75 mechanical degrees or 7.5 electrical
degrees.
(b)

The pitch factor of this winding is

k p = sin

ρ

= sin

2

142.5°
= 0.947
2

The distribution factor is

kd =

sin

nγ
2

n sin

γ
2

The electrical angle γ between slots is 7.5°, and each phase group occupies 8 adjacent slots. Therefore, the
distribution factor is

290

nγ
(8)(15°)
sin
2 =
2
kd =
= 0.956
15°
γ
8 sin
n sin
2
2
sin

The winding factor is

k w = k p k d = (0.947 )(0.956) = 0.905
B-4.

A three-phase four-pole winding of the double-layer type is to be installed on a 48-slot stator. The pitch of
the stator windings is 5/6, and there are 10 turns per coil in the windings. All coils in each phase are
connected in series, and the three phases are connected in ∆. The flux per pole in the machine is 0.054 Wb,
and the speed of rotation of the magnetic field is 1800 r/min.
(a) What is the pitch factor of this winding?
(b) What is the distribution factor of this winding?
(c) What is the frequency of the voltage produced in this winding?
(d) What are the resulting phase and terminal voltages of this stator?
SOLUTION
(a)

The pitch factor of this winding is

k p = sin

ρ
2

= sin

150°
= 0.966
2

(b) The coils in each phase group of this machine cover 4 slots, and the slot pitch is 360/48 = 7.5
mechanical degrees or 15 electrical degrees. Therefore, the distribution factor is

nγ
(4)(15°)
sin
2 =
2
= 0.958
kd =
15°
γ
4 sin
n sin
2
2
sin

(c)

The frequency of the voltage produces by this winding is

fe =

nm P (1800 r/min )(4 poles )
=
= 60 Hz
120
120

(d) There are 48 slots on this stator, with two coils sides in each slot. Therefore, there are 48 coils on the
machine. They are divided into 12 phase groups, so there are 4 coils per phase. There are 10 turns per
coil, so there are 40 turns per phase group. The voltage in one phase group is

EG = 2πN P k p k d φf e = 2π (40 turns )(0.966)(0.958)(0.054 Wb)(60 Hz ) = 533 V
There are two phase groups per phase, connected in series (this is a 4-pole machine), so the total phase
voltage is Vφ = 2 EG = 1066 V . Since the machine is ∆-connected,

VT = Vφ = 1066 V
B-5.

A three-phase Y-connected six-pole synchronous generator has six slots per pole on its stator winding. The
winding itself is a chorded (fractional-pitch) double-layer winding with eight turns per coil. The
distribution factor k d = 0.956, and the pitch factor k p = 0.981. The flux in the generator is 0.02 Wb per
291

pole, and the speed of rotation is 1200 r/min. What is the line voltage produced by this generator at these
conditions?
SOLUTION There are 6 slots per pole × 6 poles = 36 slots on the stator of this machine. Therefore, there are
36 coils on the machine, or 12 coils per phase. The electrical frequency produced by this winding is

fe =

n m P (1200 r/min )(6 poles)
=
= 60 Hz
120
120

The phase voltage is

Vφ = 2πN P k p k d φf e = 2π (96 turns )(0.981)(0.956)(0.02 Wb )(60 Hz ) = 480 V
Therefore, the line voltage is

VL = 3Vφ = 831 V
B-6.

A three-phase Y-connected 50-Hz two-pole synchronous machine has a stator with 18 slots. Its coils form
a double-layer chorded winding (two coils per slot), and each coil has 60 turns. The pitch of the stator coils
is 8/9.
(a) What rotor flux would be required to produce a terminal (line-to-line) voltage of 6 kV?
(b) How effective are coils of this pitch at reducing the fifth-harmonic component of voltage? The seventhharmonic component of voltage?
SOLUTION
(a)

The pitch of this winding is 8/9 = 160°, so the pitch factor is

k p = sin

160°
= 0.985
2

The phase groups in this machine cover three slots each, and the slot pitch is 20 mechanical or 20 electrical
degrees. Thus the distribution factor is

(3)(20°)
nγ
sin
2
2 =
= 0.960
kd =
γ
20°
n sin
3 sin
2
2
sin

The phase voltage of this machine will be

Vφ = 2πN P k p k d φf e = 2π (6 coils )(60 turns/coil )(0.985)(0.960) φ (50 Hz )

Vφ = 75621φ
The desired phase voltage is 6 kV /

φ=
(b)

3 = 3464 V, so

3464 V
= 0.046 Wb
75621

The fifth harmonic: k p = sin

(5)(160°) = 0.643

The seventh harmonic: k p = sin

2

(7)(160°) = −0.342
2

292

Since the fundamental voltage is reduced by 0.985, the fifth and seventh harmonics are suppressed relative
to the fundamental by the fractions:
5th:

0.643
= 0.653
0.985

7th:

0.342
= 0.347
0.985

In other words, the 5th harmonic is suppressed by 34.7% relative to the fundamental, and the 7th harmonic is
suppressed by 65.3% relative to the fundamental frequency.
B-7.

What coil pitch could be used to completely eliminate the seventh-harmonic component of voltage in ac
machine armature (stator)? What is the minimum number of slots needed on an eight-pole winding to
exactly achieve this pitch? What would this pitch do to the fifth-harmonic component of voltage?
SOLUTION To totally eliminate the seventh harmonic of voltage in an ac machine armature, the pitch factor
for that harmonic must be zero.

k p = 0 = sin
⇒

7ρ
2

7ρ
= (180°) n , n = 0, 1, 2, …
2
2(180°) n
ρ=
7

In order to maximize the fundamental voltage while canceling out the seventh harmonic, we pick the value
of n that makes ρ as nearly 180° as possible. If n = 3, then ρ = 154.3°, and the pitch factor for the
fundamental frequency would be

k p = sin

154.3°
= 0.975
2

This pitch corresponds to a ratio of 6/7. For a two-pole machine, a ratio of 6/7 could be implemented with
a total of 14 slots. If that ratio is desired in an 8-pole machine, then 56 slots would be needed.
The fifth harmonic would be suppressed by this winding as follows:

k p = sin
B-8.

(5)(154.3°) = 0.434
2

A 13.8-kV Y-connected 60-Hz 12-pole three-phase synchronous generator has 180 stator slots with a
double-layer winding and eight turns per coil. The coil pitch on the stator is 12 slots. The conductors from
all phase belts (or groups) in a given phase are connected in series.
(a) What flux per pole would be required to give a no-load terminal (line) voltage of 13.8 kV?
(b) What is this machine’s winding factor k w ?
SOLUTION
(a)

The stator pitch is 12/15 = 4/5, so ρ = 144° , and

k p = sin

144°
= 0.951
2

293

Each phase belt consists of (180 slots)/(12 poles)(6) = 2.5 slots per phase group. The slot pitch is 2
mechanical degrees or 24 electrical degrees. The corresponding distribution factor is

nγ
(2.5)(24°)
sin
2 =
2
= 0.962
kd =
24°
γ
2.5 sin
n sin
2
2
sin

Since there are 60 coils in each phase and 8 turns per coil, all connected in series, there are 480 turns per
phase. The resulting voltage is

Vφ =

2π N P k p k d φ f e =

2π (480 turns

) (0.951 )(0.962 ) φ (60 Hz )

Vφ = 117,061 φ
The phase voltage of this generator must be 13.8 kV / 3 = 7967 V , so the flux must be

φ=
(b)

7967 V
= 0.068 Wb
117,061

The machine’s winding factor is

k w = k p k d = (0.951)(0.962) = 0.915

294

Appendix C: Salient Pole Theory of Synchronous Machines
C-1.

A 480-V 200-kVA 0.8-PF-lagging 60-Hz four-pole Y-connected synchronous generator has a direct-axis
reactance of 0.25 Ω, a quadrature-axis reactance of 0.18 Ω, and an armature resistance of 0.03 Ω.
Friction, windage, and stray losses may be assumed negligible. The generator’s open-circuit characteristic
is given by Figure P5-1.
(a) How much field current is required to make VT equal to 480 V when the generator is running at no
(b) What is the internal generated voltage of this machine when it is operating at rated conditions? How
does this value of E A compare to that of Problem 5-2b?
(c) What fraction of this generator’s full-load power is due to the reluctance torque of the rotor?

SOLUTION
(a) If the no-load terminal voltage is 480 V, the required field current can be read directly from the opencircuit characteristic. It is 4.55 A.
(b)

At rated conditions, the line and phase current in this generator is

IA = IL =

200 kVA
P
=
= 240.6 A at an angle of –36.87°
3 VL
3 (480 V )

295

E ′′A = Vφ + RA I A + jX q I A

E′A′ = 277∠0° + (0.03 Ω )(240.6∠ − 36.87° A ) + j (0.18 Ω )(240.6∠ − 36.87° A )
E′A′ = 310∠5.61° V
Therefore, the torque angle δ is 5.61°. The direct-axis current is

I d = I A sin (θ + δ ) ∠δ − 90°

I d = (240.6 A ) sin (42.48°) ∠ − 84.4°

I d = 162.5 ∠ − 84.4° A
The quadrature-axis current is

I q = I A cos(θ + δ ) ∠δ

I q = (240.6 A ) cos(42.48°) ∠5.61°
I q = 177.4 ∠5.61° A
Therefore, the internal generated voltage of the machine is

E A = Vφ + R A I A + jX d I d + jX q I q

E A = 277∠0° + (0.03)(240.6∠ − 36.87°) + j (0.25)(162.5∠ − 84.4° ) + j (0.18)(177.4∠5.61° )
E A = 322∠5.61° V

E A is approximately the same magnitude here as in Problem 5-2b, but the angle is about 2.2° different.
(c)

The power supplied by this machine is given by the equation

P=

3Vφ E A
Xd

sin δ +

3Vφ2  X d − X q 

 sin 2δ
2  X d X q 

3(277 )(322)
3(277)  0.25 − 0.18 
sin 5.61° +

 sin 11.22°
0.25
2  (0.25)(0.18) 
P = 104.6 kW + 34.8 kW = 139.4 kW
2

P=

The cylindrical rotor term is 104.6 kW, and the reluctance term is 34.8 kW, so the reluctance torque
accounts for about 25% of the power in this generator.
296

C-2.

A 14-pole Y-connected three-phase water-turbine-driven generator is rated at 120 MVA, 13.2 kV, 0.8 PF
lagging, and 60 Hz. Its direct-axis reactance is 0.62 Ω and its quadrature- axis reactance is 0.40 Ω. All
rotational losses may be neglected.
(a) What internal generated voltage would be required for this generator to operate at the rated conditions?
(b) What is the voltage regulation of this generator at the rated conditions?
(c) Sketch the power-versus-torque-angle curve for this generator. At what angle δ is the power of the
generator maximum?
(d) How does the maximum power out of this generator compare to the maximum power available if it
were of cylindrical rotor construction?
SOLUTION
(a)

At rated conditions, the line and phase current in this generator is

IA = IL =

120 MVA
P
=
= 5249 A at an angle of –36.87°
3 VL
3 (13.2 kV )

E′′A = Vφ + RA I A + jX q I A

E′A′ = 7621∠0° + 0 + j (0.40 Ω )(5249∠ − 36.87° A )
E′A′ = 9038∠10.7° V
Therefore, the torque angle δ is 10.7°. The direct-axis current is

I d = I A sin (θ + δ ) ∠δ − 90°

I d = (5249 A ) sin (47.57°) ∠ − 79.3°
I d = 3874 ∠ − 79.3° A
The quadrature-axis current is

I q = I A cos(θ + δ ) ∠δ

I q = (5249 A ) cos(47.57°) ∠10.7°
I q = 3541 ∠10.7° A
Therefore, the internal generated voltage of the machine is

E A = Vφ + RA I A + jX d I d + jX q I q

E A = 7621∠0° + 0 + j (0.62 )(3874∠ − 79.3°) + j (0.40)(3541∠10.7° )
E A = 9890∠10.7° V
(b)

The voltage regulation of this generator is

Vnl − Vfl
9890 − 7621
× 100% =
× 100% = 29.8%
Vfl
7621
(c)

The power supplied by this machine is given by the equation

P=

3Vφ E A
Xd

3Vφ2  X d − X q 

 sin 2δ
sin δ +
2  X d X q 
297

3(7621)(9890)
3(7621)  0.62 − 0.40 
P=
sin δ +

 sin 2δ
0.62
2
 (0.62 )(0.40 ) 
P = 364.7 sin δ + 77.3 sin 2δ MW
2

A plot of power supplied as a function of torque angle is shown below:

The peak power occurs at an angle of 70.6°, and the maximum power that the generator can supply is
392.4 MW.
(d) If this generator were non-salient, PMAX would occur when δ = 90°, and PMAX would be 364.7 MW.
Therefore, the salient-pole generator has a higher maximum power than an equivalent non-salint pole
generator.
C-3.

Suppose that a salient-pole machine is to be used as a motor.
(a) Sketch the phasor diagram of a salient-pole synchronous machine used as a motor.
(b) Write the equations describing the voltages and currents in this motor.
(c) Prove that the torque angle δ between E A and Vφ on this motor is given by

δ = tan -1

I A X q cos θ - I A R A sin θ
Vφ + I A X q sin θ + I A R A cos θ

SOLUTION

298

299

C-4.

If the machine in Problem C-1 is running as a motor at the rated conditions, what is the maximum torque
that can be drawn from its shaft without it slipping poles when the field current is zero?
SOLUTION When the field current is zero, E A = 0, so
2
3Vφ  X d − X q 

 sin 2δ
2  X d X q 
2
3(277)  0.25 − 0.18 
P=

 sin 2δ = 179 sin 2δ kW
2  (0.25)(0.18) 

P=

At δ = 45° , 179 kW can be drawn from the motor.

300

Appendix D: Errata for Electric Machinery Fundamentals 4/e
(Current at 10 January 2004)
Please note that some or all of the following errata may be corrected in future reprints of the
book, so they may not appear in your copy of the text. PDF pages with these corrections are attached to
this appendix; please provide them to your students.
1.

Page 56, Problem 1-6, there are 400 turns of wire on the coil, as shown on Figure P1-3. The body of
the problem incorrectly states that there are 300 turns.

2.

Page 56, Problem 1-7, there are 400 turns of wire on the left-hand coil, and 300 turns on the righthand coil, as shown on Figure P1-4. The body of the problem is incorrect.

3.

Page 62, Problem 1-19, should state: “Figure P1-14 shows a simple single-phase ac power system
with three loads. The voltage source is V = 120∠ 0° V , and the three loads are …”

4.

Page 64, Problem 1-22, should state: “If the bar runs off into a region where the flux density falls to
0.30 T… ”. Also, the load should be 10 N, not 20.

5.

Page 147, Problem 2-10, should state that the transformer bank is Y-∆, not ∆-Y.

6.

Page 226, Problem 3-10, the holding current I H should be 8 mA.

7.

Page 342, Figure p5-2, the generator for Problems 5-11 through 5-21, the OCC and SCC curves are
in error. The correct curves are given below. Note that the voltage scale and current scales were
both off by a factor of 2.

301

8.

Page 344, Problem 5-28, the voltage of the infinite bus is 12.2 kV.

9.

Page 377, Problem 6-11, the armature resistance is 0.08 Ω, and the synchronous reactance is 1.0 Ω.

10. Page 470, Problem 7-20 (a), the holding the infinite bus is 460-V.
302

11. Page 623, Figure P9-2 and Figure P9-3, R A = 0.40 Ω and RF = 100 Ω . Values are stated correctly
in the text but shown incorrectly on the figure.
12. Page 624, Figure P9-4, R A + RS = 0.44 Ω and RF = 100 Ω . Values are stated correctly in the text
but shown incorrectly on the figure.
13. Page 627, Problem 9-21, Radj is currently set to 90 Ω. Also, the magnetization curve is taken at 1800
r/min.
14. Page 627, Problem 9-22, RA is 0.18 Ω.
15. Page 630, Figure P9-10, R A + RS = 0.21 Ω N SE is 20 turns. Values are stated correctly in the text
but shown incorrectly on the figure.
16. Page 680, Problem 10-6, refers to Problem 10-5 instead of Problem 10-4.

303

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1–4. A motor is supplying 60 N • m of torque to its load. If the motor’s shaft is turning at
1800 r/min, what is the mechanical power supplied to the load in watts? In horsepower?
1–5. A ferromagnetic core is shown in Figure P1–2. The depth of the core is 5 cm. The
other dimensions of the core are as shown in the figure. Find the value of the current
that will produce a flux of 0.005 Wb. With this current, what is the flux density at
the top of the core? What is the flux density at the right side of the core? Assume
that the relative permeability of the core is 1000.
10 cm

20 cm

5 cm

15 cm
i

φ

+

15 cm

400 turns
–

φ
15 cm

Core depth  5 cm
FIGURE P1–2
The core of Problems 1–5 and 1–16.

1–6. A ferromagnetic core with a relative permeability of 1500 is shown in Figure P1–3.
The dimensions are as shown in the diagram, and the depth of the core is 7 cm. The
air gaps on the left and right sides of the core are 0.070 and 0.050 cm, respectively.
Because of fringing effects, the effective area of the air gaps is 5 percent larger than
their physical size. If there are 400 turns in the coil wrapped around the center leg
of the core and if the current in the coil is 1.0 A, what is the flux in each of the left,
center, and right legs of the core? What is the flux density in each air gap?
1–7. A two-legged core is shown in Figure P1–4. The winding on the left leg of the core
(N1) has 400 turns, and the winding on the right (N2) has 300 turns. The coils are
wound in the directions shown in the figure. If the dimensions are as shown, then
what flux would be produced by currents i1  0.5 A and i2  0.75 A? Assume r 
1000 and constant.
1–8. A core with three legs is shown in Figure P1–5. Its depth is 5 cm, and there are 200
turns on the leftmost leg. The relative permeability of the core can be assumed to be
1500 and constant. What flux exists in each of the three legs of the core? What is the
flux density in each of the legs? Assume a 4 percent increase in the effective area of
the air gap due to fringing effects.

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0.010

φ (Wb)

0.005

0

1

2

3

4

5

6

7

8

t (ms)

–0.005
– 0.010

FIGURE P1–12
Plot of flux  as a function of time for Problem 1–16.

4 cm
i
N=?
N turns

4 cm
Depth = 4 cm

lr = 4 cm
lg = 0.05 cm
lc = 48 cm

4 cm
FIGURE P1–13
The core of Problem 1–17.

(d) Calculate the reactive power consumed or supplied by this load. Does the load
consume reactive power from the source or supply it to the source?
1–19. Figure P1–14 shows a simple single-phase ac power system with three loads. The
voltage source is V = 120∠0° V, and the impedances of the three loads are

Z1  530° 

Z2  545° 

Z3  590° 

(a) Assume that the switch shown in the figure is open, and calculate the current I,
the power factor, and the real, reactive, and apparent power being supplied by

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ELECTRIC MACHINERY FUNDAMENTALS

(a) If this bar has a load of 10 N attached to it opposite to the direction of motion,
what is the steady-state speed of the bar?
(b) If the bar runs off into a region where the flux density falls to 0.30 T, what happens to the bar? What is its final steady-state speed?
(c) Suppose VB is now decreased to 80 V with everything else remaining as in
part b. What is the new steady-state speed of the bar?
(d) From the results for parts b and c, what are two methods of controlling the
speed of a linear machine (or a real dc motor)?

REFERENCES
1. Alexander, Charles K., and Matthew N. O. Sadiku: Fundamentals of Electric Circuits, McGrawHill, 2000.
2. Beer, F., and E. Johnston, Jr.: Vector Mechanics for Engineers: Dynamics, 6th ed., McGraw-Hill,
New York, 1997.
3. Hayt, William H.: Engineering Electromagnetics, 5th ed., McGraw-Hill, New York, 1989.
4. Mulligan, J. F.: Introductory College Physics, 2nd ed., McGraw-Hill, New York, 1991.
5. Sears, Francis W., Mark W. Zemansky, and Hugh D. Young: University Physics, Addison-Wesley,

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TRANSFORMERS

147

2–8. A 200-MVA, 15/200-kV single-phase power transformer has a per-unit resistance of
1.2 percent and a per-unit reactance of 5 percent (data taken from the transformer’s
nameplate). The magnetizing impedance is j80 per unit.
(a) Find the equivalent circuit referred to the low-voltage side of this transformer.
(b) Calculate the voltage regulation of this transformer for a full-load current at
power factor of 0.8 lagging.
(c) Assume that the primary voltage of this transformer is a constant 15 kV, and
plot the secondary voltage as a function of load current for currents from no
load to full load. Repeat this process for power factors of 0.8 lagging, 1.0, and
2–9. A three-phase transformer bank is to handle 600 kVA and have a 34.5/13.8-kV voltage ratio. Find the rating of each individual transformer in the bank (high voltage,
low voltage, turns ratio, and apparent power) if the transformer bank is connected to
(a) Y–Y, (b) Y–, (c) –Y, (d) –, (e) open , (f) open Y–open .
2–10. A 13,800/480-V three-phase Y--connected transformer bank consists of three
identical 100-kVA 7967/480-V transformers. It is supplied with power directly from
a large constant-voltage bus. In the short-circuit test, the recorded values on the
high-voltage side for one of these transformers are
VSC  560 V

ISC  12.6 A

PSC  3300 W

(a) If this bank delivers a rated load at 0.85 PF lagging and rated voltage, what is
the line-to-line voltage on the high-voltage side of the transformer bank?
(b) What is the voltage regulation under these conditions?
(c) Assume that the primary voltage of this transformer is a constant 13.8 kV, and
plot the secondary voltage as a function of load current for currents from noload to full-load. Repeat this process for power factors of 0.85 lagging, 1.0, and
(d) Plot the voltage regulation of this transformer as a function of load current for
currents from no-load to full-load. Repeat this process for power factors of 0.85
lagging, 1.0, and 0.85 leading.
2–11. A 100,000-kVA, 230/115-kV – three-phase power transformer has a resistance of
0.02 pu and a reactance of 0.055 pu. The excitation branch elements are RC  110 pu
and XM  20 pu.
(a) If this transformer supplies a load of 80 MVA at 0.85 PF lagging, draw the phasor diagram of one phase of the transformer.
(b) What is the voltage regulation of the transformer bank under these conditions?
(c) Sketch the equivalent circuit referred to the low-voltage side of one phase of this
transformer. Calculate all the transformer impedances referred to the low-voltage
side.
2–12. An autotransformer is used to connect a 13.2-kV distribution line to a 13.8-kV distribution line. It must be capable of handling 2000 kVA. There are three phases, connected Y–Y with their neutrals solidly grounded.
(a) What must the NC /NSE turns ratio be to accomplish this connection?
(b) How much apparent power must the windings of each autotransformer handle?
(c) If one of the autotransformers were reconnected as an ordinary transformer,
what would its ratings be?
2–13. Two phases of a 13.8-kV three-phase distribution line serve a remote rural road (the
neutral is also available). A farmer along the road has a 480-V feeder supplying

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3–10. A series-capacitor forced commutation chopper circuit supplying a purely resistive
load is shown in Figure P3–5.
VDC  120 V
IH  8 mA
VBO  200 V

R1  20 k
C  150 F

(a) When SCR1 is turned on, how long will it remain on? What causes it to turn off?
(b) When SCR1 turns off, how long will it be until the SCR can be turned on again?
(Assume that 3 time constants must pass before the capacitor is discharged.)
(c) What problem or problems do these calculations reveal about this simple seriescapacitor forced-commutation chopper circuit?
(d) How can the problem(s) described in part c be eliminated?
+
SCR

+
R1

vc

C

–
+

VDC

D

–












–
FIGURE P3–5
The simple series-capacitor forced-commutation circuit of Problem 3–10.

3–11. A parallel-capacitor forced-commutation chopper circuit supplying a purely resistive load is shown in Figure P3–6.
VDC  120 V
IH  5 mA
VBO  250 V

R1  20 k
C  15 F

(a) When SCR1 is turned on, how long will it remain on? What causes it to turn off?
(b) What is the earliest time that SCR1 can be turned off after it is turned on?
(Assume that 3 time constants must pass before the capacitor is charged.)
(c) When SCR1 turns off, how long will it be until the SCR can be turned on again?
(d) What problem or problems do these calculations reveal about this simple parallelcapacitor forced-commutation chopper circuit?
(e) How can the problem(s) described in part d be eliminated?
3–12. Figure P3–7 shows a single-phase rectifier-inverter circuit. Explain how this circuit
functions. What are the purposes of C1 and C2? What controls the output frequency
of the inverter?

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Page 342

Open Circuit Characteristic
1200
1100
1000

Open-circuit voltage, V

900
800
700
600
500
400
300
200
100
0

0

0.1

0.2

0.3

0.4

0.5

0.6 0.7 0.8 0.9
Field current, A
(a)

1

1.1

1.2

1.3

1.4

Short Circuit Characteristic
1600

1400

Armature current, A

1200

1000

800

600

400

200

0

0.6
0.8
1
1.2
1.4
Field current, A
FIGURE P5–2
(b)
(a) Open-circuit characteristic curve for the generator in Problems 5–11 to 5–21. (b) Short-circuit
characteristic curve for the generator in Problems 5–11 to 5–21.

342

0

0.2

0.4

1.5

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ELECTRIC MACHINERY FUNDAMENTALS

5–27. A 25-MVA, three-phase, 13.8-kV, two-pole, 60-Hz Y-connected synchronous generator was tested by the open-circuit test, and its air-gap voltage was extrapolated
with the following results:
Open-circuit test
Field current, A

320

365

380

475

570

Line voltage, kV

13.0

13.8

14.1

15.2

16.0

Extrapolated air-gap voltage, kV

15.4

17.5

18.3

22.8

27.4

The short-circuit test was then performed with the following results:
Short-circuit test
Field current, A
Armature current, A

320

365

380

475

570

1040

1190

1240

1550

1885

The armature resistance is 0.24 per phase.
(a) Find the unsaturated synchronous reactance of this generator in ohms per phase
and per unit.
(b) Find the approximate saturated synchronous reactance XS at a field current of
380 A. Express the answer both in ohms per phase and per unit.
(c) Find the approximate saturated synchronous reactance at a field current of
475 A. Express the answer both in ohms per phase and in per-unit.
(d) Find the short-circuit ratio for this generator.
5–28. A 20-MVA, 12.2-kV, 0.8-PF-lagging, Y-connected synchronous generator has a negligible armature resistance and a synchronous reactance of 1.1 per unit. The generator is connected in parallel with a 60-Hz, 12.2-kV infinite bus that is capable of supplying or consuming any amount of real or reactive power with no change in
frequency or terminal voltage.
(a) What is the synchronous reactance of the generator in ohms?
(b) What is the internal generated voltage EA of this generator under rated conditions?
(c) What is the armature current IA in this machine at rated conditions?
(d) Suppose that the generator is initially operating at rated conditions. If the internal generated voltage EA is decreased by 5 percent, what will the new armature
current IA be?
(e) Repeat part d for 10, 15, 20, and 25 percent reductions in EA.
(f) Plot the magnitude of the armature current IA as a function of EA. (You may wish
to use MATLAB to create this plot.)

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SYNCHRONOUS MOTORS

377

6–9. Figure P6–2 shows a synchronous motor phasor diagram for a motor operating at a
leading power factor with no RA. For this motor, the torque angle is given by
tan  

XSIA cos
V  XSIA sin

  tan1

(

XSIA cos
V  XSIA sin

)

Derive an equation for the torque angle of the synchronous motor if the armature resistance is included.

IA

XSIA sin

V

jXSIA

(

XSIA cos
= tan–1 ––—————–
V + XSIA sin

(

XSIA cos

EA

FIGURE P6–2
Phasor diagram of a motor at a leading power factor.

6–10. A 480-V, 375-kVA, 0.8-PF-lagging, Y-connected synchronous generator has a synchronous reactance of 0.4 and a negligible armature resistance. This generator is
supplying power to a 480-V, 80-kW, 0.8-PF-leading, Y-connected synchronous motor with a synchronous reactance of 1.1 and a negligible armature resistance. The
synchronous generator is adjusted to have a terminal voltage of 480 V when the motor is drawing the rated power at unity power factor.
(a) Calculate the magnitudes and angles of EA for both machines.
(b) If the flux of the motor is increased by 10 percent, what happens to the terminal voltage of the power system? What is its new value?
(c) What is the power factor of the motor after the increase in motor flux?
6–11. A 480-V, 100-kW, 50-Hz, four-pole, Y-connected synchronous motor has a rated
power factor of 0.85 leading. At full load, the efficiency is 91 percent. The armature
resistance is 0.08 , and the synchronous reactance is 1.0 . Find the following
quantities for this machine when it is operating at full load:
(a) Output torque
(b) Input power
(c) nm
(d) EA
(e) |IA|
(f) Pconv
(g) Pmech  Pcore  Pstray

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7–15.
7–16.
7–17.

7–18.

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ELECTRIC MACHINERY FUNDAMENTALS

(a) The line current IL
(b) The stator power factor
(c) The rotor power factor
(d) The stator copper losses PSCL
(e) The air-gap power PAG
(f) The power converted from electrical to mechanical form Pconv
(g) The induced torque ind
(i) The overall machine efficiency
(j) The motor speed in revolutions per minute and radians per second
For the motor in Problem 7–14, what is the pullout torque? What is the slip at the
pullout torque? What is the rotor speed at the pullout torque?
If the motor in Problem 7–14 is to be driven from a 440-V, 60-Hz power supply,
what will the pullout torque be? What will the slip be at pullout?
Plot the following quantities for the motor in Problem 7–14 as slip varies from 0 to
10 percent: (a) ind; (b) Pconv; (c) Pout; (d) efficiency . At what slip does Pout equal
the rated power of the machine?
A 208-V, 60 Hz six-pole, Y-connected, 25-hp design class B induction motor is
tested in the laboratory, with the following results:

208 V, 22.0 A, 1200 W, 60 Hz

Locked rotor:

24.6 V, 64.5 A, 2200 W, 15 Hz

DC test:

13.5 V, 64 A

Find the equivalent circuit of this motor, and plot its torque–speed characteristic
curve.
7–19. A 460-V, four-pole, 50-hp, 60-Hz, Y-connected, three-phase induction motor develops its full-load induced torque at 3.8 percent slip when operating at 60 Hz and 460
V. The per-phase circuit model impedances of the motor are
R1  0.33 

XM  30 

X1  0.42 

X2  0.42 

Mechanical, core, and stray losses may be neglected in this problem.
(a) Find the value of the rotor resistance R2.
(b) Find max, smax, and the rotor speed at maximum torque for this motor.
(c) Find the starting torque of this motor.
(d) What code letter factor should be assigned to this motor?
7–20. Answer the following questions about the motor in Problem 7–19.
(a) If this motor is started from a 460-V infinite bus, how much current will flow in
the motor at starting?
(b) If transmission line with an impedance of 0.35 j0.25  per phase is used to
connect the induction motor to the infinite bus, what will the starting current of
the motor be? What will the motor’s terminal voltage be on starting?
(c) If an ideal 1.4:1 step-down autotransformer is connected between the transmission line and the motor, what will the current be in the transmission line during
starting? What will the voltage be at the motor end of the transmission line during starting?

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DC MOTORS AND GENERATORS

IA

623

IL

RA
0.40

IF
+
–

RF

100

EA

VT = 240 V

LF

FIGURE P9–2
The equivalent circuit of the shunt motor in Problems 9–1 to 9–7.

IF

IA

RA

IL

+

+
0.40

+

RF = 100

VF = 240 V

–

VA = 120 to 240 V

EA

LF
–

–

FIGURE P9–3
The equivalent circuit of the separately excited motor in Problems 9–8 and 9–9.

9–10. If the motor is connected cumulatively compounded as shown in Figure P9–4 and if
Radj  175 , what is its no-load speed? What is its full-load speed? What is its
speed regulation? Calculate and plot the torque–speed characteristic for this motor.
(Neglect armature effects in this problem.)
9–11. The motor is connected cumulatively compounded and is operating at full load.
What will the new speed of the motor be if Radj is increased to 250 ? How does the
new speed compare to the full-load speed calculated in Problem 9–10?
9–12. The motor is now connected differentially compounded.
(a) If Radj  175 , what is the no-load speed of the motor?
(b) What is the motor’s speed when the armature current reaches 20A? 40 A? 60 A?
(c) Calculate and plot the torque–speed characteristic curve of this motor.
9–13. A 7.5-hp, 120-V series dc motor has an armature resistance of 0.2 and a series field
resistance of 0.16 . At full load, the current input is 58 A, and the rated speed is

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ELECTRIC MACHINERY FUNDAMENTALS

IA

IL

LS

+
0.44

= Cumulatively compounded
= Differentially compounded

= RA + RS

IF
+
–

RF

100

EA

VT = 240 V

LF
–
FIGURE P9–4
The equivalent circuit of the compounded motor in Problems 9–10 to 9–12.

1050 r/min. Its magnetization curve is shown in Figure P9–5. The core losses are 200
W, and the mechanical losses are 240 W at full load. Assume that the mechanical
losses vary as the cube of the speed of the motor and that the core losses are constant.
(a) What is the efficiency of the motor at full load?
(b) What are the speed and efficiency of the motor if it is operating at an armature
current of 35 A?
(c) Plot the torque–speed characteristic for this motor.
9–14. A 20-hp, 240-V, 76-A, 900 r/min series motor has a field winding of 33 turns per
pole. Its armature resistance is 0.09 , and its field resistance is 0.06 . The magnetization curve expressed in terms of magnetomotive force versus EA at 900 r/min
is given by the following table:
EA, V
, A • turns

95

150

188

212

229

243

500

1000

1500

2000

2500

3000

Armature reaction is negligible in this machine.
(a) Compute the motor’s torque, speed, and output power at 33, 67, 100, and 133
percent of full-load armature current. (Neglect rotational losses.)
(b) Plot the torque–speed characteristic of this machine.
9–15. A 300-hp, 440-V, 560-A, 863 r/min shunt dc motor has been tested, and the following data were taken:
Blocked-rotor test:
VA  16.3 V exclusive of brushes

VF  440 V

IA  500 A

IF  8.86 A

VA  16.3 V including brushes

IF  8.76 A

IA  23.1 A

n  863 r/min

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DC MOTORS AND GENERATORS

627

9–19. A series motor is now constructed from this machine by leaving the shunt field out
entirely. Derive the torque–speed characteristic of the resulting motor.
9–20. An automatic starter circuit is to be designed for a shunt motor rated at 15 hp, 240
V, and 60 A. The armature resistance of the motor is 0.15 , and the shunt field resistance is 40 . The motor is to start with no more than 250 percent of its rated armature current, and as soon as the current falls to rated value, a starting resistor
stage is to be cut out. How many stages of starting resistance are needed, and how
big should each one be?
9–21. A 15-hp, 230-V, 1800 r/min shunt dc motor has a full-load armature current of 60 A
when operating at rated conditions. The armature resistance of the motor is RA 
0.15 , and the field resistance RF is 80 .The adjustable resistance in the field circuit Radj may be varied over the range from 0 to 200 and is currently set to 90 .
Armature reaction may be ignored in this machine. The magnetization curve for this
motor, taken at a speed of 1800 r/min, is given in tabular form below:
EA, V

8.5

150

180

215

226

242

IF, A

0.00

0.80

1.00

1.28

1.44

2.88

(a) What is the speed of this motor when it is running at the rated conditions specified above?
(b) The output power from the motor is 7.5 hp at rated conditions. What is the output torque of the motor?
(c) What are the copper losses and rotational losses in the motor at full load (ignore
stray losses)?
(d) What is the efficiency of the motor at full load?
(e) If the motor is now unloaded with no changes in terminal voltage or Radj, what
is the no-load speed of the motor?
(f) Suppose that the motor is running at the no-load conditions described in part e.
What would happen to the motor if its field circuit were to open? Ignoring armature reaction, what would the final steady-state speed of the motor be under
those conditions?
(g) What range of no-load speeds is possible in this motor, given the range of field
9–22. The magnetization curve for a separately excited dc generator is shown in Figure
P9–7. The generator is rated at 6 kW, 120 V, 50 A, and 1800 r/min and is shown in
Figure P9–8. Its field circuit is rated at 5A. The following data are known about the
machine:
RA  0.18
Radj  0 to 30

VF  120 V
RF  24

NF  1000 turns per pole
(a) If this generator is operating at no load, what is the range of voltage adjustments
that can be achieved by changing Radj?
(b) If the field rheostat is allowed to vary from 0 to 30 and the generator’s speed
is allowed to vary from 1500 to 2000 r/min, what are the maximum and minimum no-load voltages in the generator?

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ELECTRIC MACHINERY FUNDAMENTALS

IA

RA + RS

IL

Nse = 20 turns

+
0.21

LS

IF

+
–

EA

RF

20

LF

VT

NF = 1000
turns
–

FIGURE P9–10
The compounded dc generator in Problems 9–27 and 9–28.

The machine has the magnetization curve shown in Figure P9–7. Its equivalent circuit is shown in Figure P9–10. Answer the following questions about this machine,
assuming no armature reaction.
(a) If the generator is operating at no load, what is its terminal voltage?
(b) If the generator has an armature current of 20 A, what is its terminal voltage?
(c) If the generator has an armature current of 40 A, what is its terminal voltage?
(d) Calculate and plot the terminal characteristic of this machine.
9–28. If the machine described in Problem 9–27 is reconnected as a differentially compounded dc generator, what will its terminal characteristic look like? Derive it in the
same fashion as in Problem 9–27.
9–29. A cumulatively compounded dc generator is operating properly as a flatcompounded dc generator. The machine is then shut down, and its shunt field connections are reversed.
(a) If this generator is turned in the same direction as before, will an output voltage
be built up at its terminals? Why or why not?
(b) Will the voltage build up for rotation in the opposite direction? Why or why
not?
(c) For the direction of rotation in which a voltage builds up, will the generator be
cumulatively or differentially compounded?
9–30. A three-phase synchronous machine is mechanically connected to a shunt dc machine, forming a motor–generator set, as shown in Figure P9–11. The dc machine is
connected to a dc power system supplying a constant 240 V, and the ac machine is
connected to a 480-V, 60-Hz infinite bus.
The dc machine has four poles and is rated at 50 kW and 240 V. It has a per-unit
armature resistance of 0.04. The ac machine has four poles and is Y-connected. It is
rated at 50 kVA, 480 V, and 0.8 PF, and its saturated synchronous reactance is 2.0
per phase.
All losses except the dc machine’s armature resistance may be neglected in this
problem. Assume that the magnetization curves of both machines are linear.
(a) Initially, the ac machine is supplying 50 kVA at 0.8 PF lagging to the ac power
system.

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ELECTRIC MACHINERY FUNDAMENTALS

10–6.
10–7.

10–8.
10–9.
10–10.

(f) Pout
(g) ind
(i) Efficiency
Find the induced torque in the motor in Problem 10–5 if it is operating at 5 percent
slip and its terminal voltage is (a) 190 V, (b) 208 V, (c) 230 V.
What type of motor would you select to perform each of the following jobs? Why?
(a) Vacuum cleaner
(b) Refrigerator
(c) Air conditioner compressor
(d) Air conditioner fan
(e) Variable-speed sewing machine
(f) Clock
(g) Electric drill
For a particular application, a three-phase stepper motor must be capable of stepping in 10° increments. How many poles must it have?
How many pulses per second must be supplied to the control unit of the motor in
Problem 10–8 to achieve a rotational speed of 600 r/min?
Construct a table showing step size versus number of poles for three-phase and
four-phase stepper motors.

REFERENCES
1. Fitzgerald, A. E., and C. Kingsley, Jr. Electric Machinery. New York: McGraw-Hill, 1952.
2. National Electrical Manufacturers Association. Motors and Generators, Publication No. MG11993. Washington, D.C.: NEMA, 1993.
3. Veinott, G. C. Fractional and Subfractional Horsepower Electric Motors. New York: McGrawHill, 1970.
4. Werninck, E. H. (ed.). Electric Motor Handbook. London: McGraw-Hill, 1978.



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