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 LIFE SUPPORT POLICY NATIONAL’S PRODUCTS ARE NOT AUTHORIZED FOR USE AS CRITICAL COMPONENTS IN LIFE SUPPORT DEVICES OR SYSTEMS WITHOUT THE EXPRESS WRITTEN APPROVAL OF THE PRESIDENT OF NATIONAL SEMICONDUCTOR CORPORATION. As used herein: 1. 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