Realizing Trigonometric Functions With The Multifunction Converter LH0094 AN 0637

User Manual: AN-0637

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National Semiconductor
Application Note 637
Hans Palouda
August 1989

This note discusses how to use the LH0094 to generate
output voltages that are the sine and the cosine of the input
voltage. Circuits are developed and the elements dimensioned. Special emphasis is given to the procedure.
The LH0094 can be used to generate a wide variety of output voltages as a function of three input voltages. One application is to generate trigonometric functions.
The LH0094’s transfer function is dependent on three inputs, Vx, Vy, and Vz:
Eo e Vy c (Vz/Vx)m,
(eq. 1)
with m between 0.1 and 10. All voltages are positive. Two
resistors are used to set the value of m (R1, R2 in Figure 2 ).
Eo is the output voltage of the LH0094. In this application Vx
and Vy are held constant at 10V.
In order to be realized by the LH0094 the trig function needs
to be approximated by a polynomial. This can be done by a
Taylor series:
f(x) e f(xo) a x c fÊ (xo) a (x2/2) c f× (xo) a . . .
where xo is the point where the series is developed. This is
also the point of highest accuracy and therefore chosen
close to the middle of the range of x.
The remainder of the series is truncated, because the
LH0094 can provide only one exponential function. For the
Taylor series this is the x2 term. The constant and linear
terms are added with a summing amplifier (Figure 3) .
A better approximation can be achieved if the highest exponent is made a fraction rather than an integer as is the case
for the Taylor series. The LH0094 can accommodate this
easily. The values of the coefficients and of the exponent
can then be determined in such a way that the error within
the value range of x is minimized.
In algebra the independent variable is normally called x. In
this application the independent variable is the input voltage
Vz, while the dependent variable, normally called y, is the
output voltage VOUT.

TL/K/10497 – 2

The clamp diode is for protection. The potentiometer R4 is used to trim the
output Eo to 10V when all 3 inputs Vx, Vy, and Vz are at a 10V.

FIGURE 2. The Multifunction Generator LH0094
Connected to Realize the Function
Eo e 10 c (Vz/10) 2.827

TL/K/10497 – 3

Q1 is the LH0094 connected as shown in Figure 2 . Its output is proportional
to the exponential part of the transfer function.

FIGURE 3. Circuit to Realize the Sine Function
I. REALIZING THE FUNCTION y e sin x

TL/K/10497 – 1

The independent variable has been named Vz to conform with the convention of the LH0094. Both Vy and Vz are voltages.

FIGURE 1. Sinus Function to be Realized

1. OBJECTIVE
A circuit needs to be designed where the output voltage is
the sine of the input voltage, as shown in Figure 1 . The
basic function is
y e sin x.

Realizing Trigonometric Functions with the Multi-Function Converter LH0094

Realizing Trigonometric
Functions with the MultiFunction Converter LH0094

If substitutions are made: x e c c Vz and y e VOUT/k, the
function becomes
VOUT e k c sin c c Vz
(eq. 2)

AN-637

C1995 National Semiconductor Corporation

TL/K/10497

RRD-B30M75/Printed in U. S. A.

accommodating the linear term of the desired transfer function (eq. 5).

This equation connects input and output voltage by a sine
function. The values of k and c can be determined so that
these boundary conditions are met:

4. TRANSFER FUNCTIONS

at VIN e Vz e 0V . . . VOUT e 0V and the argument
c c Vz e 0 radians, at VIN e Vz e 10V . . . VOUT e 10V
and the argument c c Vz e q/2 rad.

The transfer function of the circuit in Figure 3 is
B
D
AaB
c
VOUT e bEo a Vz c
A
CaD
A
b B/A is the inverting voltage gain, (A a B)/A is the non-inverting voltage gain. Vz c D/(C a D) is the fraction of the
input voltage which is fed into the non-inverting input.
With eq. 6 used to substitute for Eo, the transfer function
becomes

Both k and 1/c have the dimension of a voltage, and therefore the value of the sine (VOUT/k) and its argument (c c
Vz) remain without dimension. In the following equations all
voltages need to be measured in Volts.
2. DEVELOPMENT OF A REALIZABLE FUNCTION
From eq. 2 and its boundary conditions it can be shown that
ce

Vzm B
D
AaB
c a Vz
c
(eq. 7)
10m A
CaD
A
The transfer function to be realized is (from eq. 5):
VOUT e Vz c q/2 b Vzm c 10 c (q/20)m/6.28 (eq. 8)

1
q
c
and k e 10V
10V
2

VOUT e b10

The desired function (eq. 2) becomes
q
V
VOUT
e sin c Z , or, in a shorter form,
10V
2
10V
VOUT e 10 sin (Vz c q/20),
(eq. 3)
where Vz is the input voltage.
Because the LH0094 can only create an exponential function, the sine function in eq. 3 needs to be substituted by a
polynomial. The approximation
sin x e x b xm/6.28,
(eq. 4)

5. DIMENSIONING THE CIRCUIT PARAMETERS
For the two transfer functions, eq. 7 and eq. 8, to be identical, in both functions the coefficients of Vzm must be equal,
and also the coefficients of Vz must be equal to each other.
Vzm B
c e b 10 c (q/20)m/6.28
10m A
With m e 2.827 (eq. 4) this results in
B/A e 0.5708.
The linear coefficients set equal give
AaB
D
c
e q/2
CaD
A
b 10

with x e Vz c q/20 and m e 2.827
lets the desired function be written as
VOUT e 10 [Vz c q/20 b (Vz c q/20)m/6.28]. (eq. 5)
This approximation is accurate for Vz from 0V to a 10V with
a theoretical accuracy of 0.23%.

With the known value for B/A (eq. 9) this yields
C/D e 0
or D e infinite, or open, C having a finite value.

3. DEVELOPMENT OF THE CIRCUIT
3a. Realization of the Exponential Term
The first task is to realize a function proportional to Vzm.
Then it can be scaled and added to the other terms to yield
the required transfer function (eq. 5).

(eq. 9)

(eq. 10)

6. DIMENSIONING OF THE RESISTORS
10 kX is a good first choice for both load and source impedances. Load impedances need to be larger than 2 kX.
Sources driving Q2 can be quite high, if the LF411 is chosen, because of its FET input. However, the impedances
should not be too high, because of the time constants
formed together with the circuit capacitances. Below 1 MX
this concern will not arise.
If A is chosen as A e 10 kX,
B becomes (eq. 9) . . . B e 0.5708 A e 5.708 kX.
D is infinite,
C is chosen so it is approximately equal to
A c B/(A a B) e 3.6 kX.
In this case both inputs of the op-amp Q2 see the same
impedance.

Figure 2 shows how to connect the LH0094 to perform the
exponential portion of the desired function (eq. 5). From eq.
1 and with the input voltages Vx e Vy e a 10V the transfer
function of the LH0094 becomes
Eo e 10 c (Vz/10)m.
(eq. 6)
From eq. 4 it is known that m e 2.827
The value of m is set by R1 and R2, with
m e (R1 a R2)/R1
(from the datasheet) and the recommendation that
R1 a R2 e 200X
(approximately). Suitable values are
R1 e 127.89X,
R2 e 70X.
R3 and the potentiometer R4 in Figure 2 serve to trim the
maximum output Eo to a 10V when the input voltage Vz e
a 10V.
Both Vx and Vy should be connected to a regulated a 10V,
since the accuracy of this voltage directly influences the
accuracy of the output voltage.

7. LIST OF RESISTORS
R1 e 127.89
A e 10k
B e 5.708k
R2 e 70
R3 e 2 M
C e 3.6k, 5%
R4 e 10k POT
D e open
II. REALIZING THE FUNCTION y e cos x
1. OBJECTIVE
A circuit needs to be designed where the output voltage is
the cosine of the input voltage, as shown in Figure 1 . The
basic function is
y e cos x.

3b. Circuit to Realize the Sine Function

Figure 3 shows the schematic for the complete circuit. Q1 is
the LH0094 connected as shown in Figure 2 to perform the
exponential term of the transfer function (eq. 5). The output
voltage Eo is amplified by Q2, which also adds some fraction of the input voltage Vz coming through C and D, thus
2

Figure 5 shows how to connect the LH0094 to perform the
exponential portion of the desired transfer function (eq. 14).
The resistors R1 and R2 set the value of m (from the datasheet):
m e (R1 a R2)/R1.
For m e 1.504 (eq. 13), and the recommendation that R1
a R2 be approximately 200X, suitable values are R1 e
65.52X and R2 e 130X.
R3 and the potentiometer R4 in Figure 2 serve to trim the
maximum output Eo to 10V when the input voltage Vz e
10V.
Both Vx and Vy should be connected to a regulated a 10V,
since the accuracy of this voltage directly influences the
accuracy of the circuit.

If substitutions are made: x e c c Vz and y e VOUT/k, the
function becomes
VOUT e k c cos c c Vz
(eq. 11)
This equation connects input and output voltage by a cosine
function (see Figure 4 ). The values of k and c can be determined so that these boundary conditions are met:
at VIN e Vz e 0V . . . VOUT e 10V and the argument
c c Vz e 0 radians, at VIN e Vz e 10V . . . VOUT e 0V
and the argument c c Vz e q/2 rad.
Both k and 1/c have the dimension of a voltage, and therefore the value of the cosine (VOUT/k) and its argument (c c
Vz) remain without dimension. In the following equations all
voltages need to be measured in Volts.

TL/K/10497 – 4

The independent variable has been named Vz to conform with the convention of the LH0094. Both Vy and Vz are voltages.

FIGURE 4. Cosine Function to be Realized
2. DEVELOPMENT OF A REALIZABLE FUNCTION
From eq. 2 and its boundary conditions it can be shown that
ce

1
q
c
10V
2

and

TL/K/10497 – 5

The clamp diode is for protection. R4 trims the output voltage to 10V when
all three inputs Vx, Vy, and Vz, are at a 10V.

k e 10V

FIGURE 5. The LH0094 Connected to
Output the Function Eo e 10 c (Vz/10)1.504

The desired function (eq. 2) becomes
q
Vz
VOUT
e cos c
, or, in a shorter form,
10V
2
10V
(eq. 12)
VOUT e 10 cos (Vz c q/20),
where Vz is the input voltage.
Because the LH0094 can only create an exponential function, the cosine function in eq. 12 needs to be substituted by
a polynomial.
The approximation
cos x e 1 a 0.2325 x b xm/1.445,
(eq. 13)

3b. Circuit to Realize the Cosine Function
Figure 6 shows the schematic for the complete circuit. Q1 is
the LH0094 connected as shown in Figure 5 to perform the
exponential term of the desired transfer function (eq. 14).
The output voltage Eo is amplified by Q2, which also adds
some fraction of the input voltage Vz coming through C, as
well as a constant voltage coming through R.

with x e Vz c q/20 and m e 1.504,
allows the desired function to be written as
VOUT e 10 [ 1 a 0.2325 Vz c q/20 b
(Vz c q/20)m/1.445 ].

(eq. 14)
This approximation is accurate for Vz from 0V to a 10V with
a theoretical accuracy of 0.75%.
3. DEVELOPMENT OF THE CIRCUIT
3a. Realization of the Exponential Term
The LH0094 is used to realize a function proportional to
(Vz)m, which will then be scaled and added to the other
voltages which correspond to the remainder of the terms in
eq. 13.
From eq. 1, with Vx and Vy held constant at a 10V, the
transfer function of the LH0094 becomes
Eo e 10 (Vz/10)m.
(eq. 15)

TL/K/10497 – 6

Q1 is the LH0094 connected as per Figure 5 . Its output is proportional to the
exponential term in the transfer function.

FIGURE 6. Circuit to Realize the Cosine Function

3

4. TRANSFER FUNCTIONS

6. DIMENSIONING OF THE RESISTORS

The transfer function of this circuit (Figure 3) is
R ll D
C ll D
B
AaB
aV
Vz
,
VOUT e bEo a
A
A
C a R ll D
F a C ll D
where R ll D e R c D/(R a D), and C ll D e C c D/(C a D).
The factor (bB/A) is the inverting voltage gain, (A a B)/A
the non-inverting voltage gain. The term z c (R ll D)/(C a R ll
D) describes the fraction of the input voltage fed into the
non-inverting input, and V c (C ll D)/(F a C ll D) is the DC
voltage at the non-inverting input. The op-amp sums and
amplifies all three voltages.
With eq. 15 used to substitute for Eo, the transfer function of
the circuit becomes

10 kX is a good choice for both load and source impedances. Load impedances need to be larger than 2 kX. Sources
driving Q2 can be quite high, if the LF411 is chosen, because of its FET input. However, the impedances should not
be too high, because of the time constants formed together
with the circuit capacitances. Below 1 MX this concern will
not arise.

#

J

6a. First Round of Dimensioning
If A is chosen, as first try, A1 e 10 kX,
B becomes (eq. 17) B1 e 1.3649 A1 e 13.649 kX.
D is chosen as D1 e 10 kX,
C then becomes (eq. 21) C1 e 2.7373 D1 e 27.373 kX,
and F becomes (eq. 22) F1 e 0.99968 D1 e 9.9968 kX

Vzm B
AaB
c a
10m A
A
(eq. 16)
R ll D
C ll D
aV
Vz
C a R ll D
F a C ll D
The transfer function to be realized is (eq. 14):
VOUT e 10 [1 a 0.2325 Vz c q/20b(Vz c q/20)m/1.445].
VOUT e b10

#

6b. Second Round of Dimensioning
The values of the resistors are in the proper range as far as
loading and input impedance are concerned. However, for
reasons of accuracy the amplifier Q2 needs source impedances on its inverting and non-inverting inputs that are approximately equal.
A1 ll B1 e A1 c B1/(A1 a B1) e 5.772 kX
C1 ll D1 ll F1 e 1/(1/C1 a 1/D1 a 1/F1) e 4.227 kX
These values are relatively close to each other, but it is easy
to make them even closer. The scaling factor is
5.772/4.227 e 1.3653, and with it the new values for C, D
and F become
C2 e 1.365 C1 e (1.365) c (27.373) e 37.373 kX,
D2 e 1.365 D1 e (1.365) c (10) e 13.653 kX
F2 e 1.365 F1 e (1.365) c (9.9968) e 13.649 kX
The resistors A1 and B1 remain unchanged:
A2 e A1 e 10 kX
B2 e B1 e 13.649 kX

J

5. DIMENSIONING OF THE CIRCUIT PARAMETERS
For the two transfer functions (eq. 14, 16) to be identical, in
both functions the coefficients of Vzm must be equal, the
coefficients of Vz must be equal, and the constants must be
equal.
The coefficients of Vzm set equal give
Vzm B
c e b 10 (q/20)m/1.445
10m A
(eq. 17)
This yields B/A e 1.365.
The linear coefficients set equal give
AaB
F ll D
q
c
e 10 (0.2325) c
(eq. 18)
A
C a F ll D
20
This yields
C
B
q
1
c
e 5.475 e a (eq. 19)
e
a1 c
F ll D
A
2
0.2325
b 10

#

6c. Final Dimensions
Three resistors have about the same value: B1, D2, and F2.
If they are all made 13.65 kX the error is 0.02%. This error
will in most cases be negligible. To get standard values the
three resistors can be set 20 kX. The remaining resistors A1
and C2 will be proportionally scaled with a factor of
20/13.65 e 1.4652. Since all resistors are scaled by the
same factor, all the previously established relationships, like
voltage attenuation and impedance matching, remain unchanged.

J

The coefficients of the constants set equal give
B
C ll D
a1 cVc
e 10, or, with V e 10V,
A
F a C ll D

#

F
C

ll D

e

J
B
#A 1J
a

c

10
b 1 e 1.3649 e b
10

(eq. 20)

7. LIST OF RESISTORS
R1 e 65.52
A e 14.652k
R2 e 130
B e 20k
R3 e 2 M
C e 54.759k
R4 e 10k POT
D e 20k
F e 20k

Eq. 19 and eq. 20 are two equations for the three unknowns
C, D, and R. This means one of the unknowns can be chosen. This makes sense, because in voltage dividers it is the
ratio of resistors that counts. The terms a and b are used as
abbreviations for easier use.
The resistor D will be chosen and considered known. With
this premise it can be shown the equations 19 and 20 yield
C e D c (a c b b 1)/(b a 1) e 2.7373 D
(eq. 21)
F e D c (a c b b 1)/(a a 1) e 0.9997 D
(eq. 22)

For good accuracy the voltage V e a 10V needs to be
regulated and is best connected to Vx and Vy, which need
to be regulated as well.

4

TL/K/10497 – 7

TL/K/10497 – 9

FIGURE 7. Input and Output Voltage of the Sine Circuit

FIGURE 9. Input and Output Voltage
of the Cosine Circuit

TL/K/10497 – 8

FIGURE 8. Input and Output Voltage
of the Sine Circuit, Detail

TL/K/10497 – 10

FIGURE 10. Input and Output Voltage
of the Cosine Circuit, Detail

5

Realizing Trigonometric Functions with the Multi-Function Converter LH0094
AN-637

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Title                           : Realizing Trigonometric Functions with the Multifunction Converter LH0094
Subject                         : AN-637
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Keywords                        : Application, Notes
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