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Electronic Theses and Dissertations

1972

An Integrated Circuit Analog Computer

Sthaporn Udomsin

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Recommended Citation Udomsin, Sthaporn, "An Integrated Circuit Analog Computer" (1972). Electronic Theses and Dissertations. 4844. https://openprairie.sdstate.edu/etd/4844

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BY STHAPORN UDOMSIN

A the sis submitted in partial fulfillment of the requirements-for the degree Master of Science, Department of Electrical Engineering, South Dakota State University

1972 ', ,

SOUTH DAKOT� STATE UNIVERSITY LIBRARY AN INTEGRATED CIRCUIT ANALOG COMPUTER

This the sis is approved as a creditable and independent investigation by a candidate for the degree , Master of Science, and is acceptable as meeting the the sis requirements for this degree, but without implying that the conc lusions reached by the candidate are necessarily the conclusions of the major department.

Thesis A�or Date

Head, Electrical Engineering Date Department ACKNOWLEDGMENTS

I would like to take this time to acknowle dge the ge nerous and valuable help give n me by two members of the Elect ric al Engineering

Depart me nt here at S . D . S . Fir st , I wo uld like to express deep U. gr at it ude to R. A. Higgins , advisor, who aided me in both the Dr. my re search andwri t ing of this thesis. Se cond, I would like to thank

R. Mostad for his kind assistance in solvi ng technical Mr. W. proble ms.

s. u.- TABLE OF CONTENTS

Chapter Page

I. INTRODUCTION ...... • • . . • . . 1

A. An application of Computers . . . . . • • . . 1

B. Analog Computer • . . . • . • . • • . • • . . • . 2

Educational Analog Computer • • . • • • . . . 4 c.

D. Purpose of the Study . . • • . . • . • • . • . . 5

- II. OPERATIONAL AMPLIFIER • . . • . • • . • . . • • . . • 6

A. Ideal Operational Amplifier . . . • . . . • . 6

B. A741 Operational Amplifier • • . . 6 J-L • . • . Load Impedance , Input Impedance, Output

c. Impedance and Open-Loop Voltage Gain • . /,; 8

D. Offset, Drift and Noise . . • • ...... 14 ,..--- E. Stability, Bandwidth and Slew Rate . . . . . 23

F. Line rity • . . . • • . . • ...... 25

III. DESIGN ...... • . • • • • . . . . • • • . . • 27

A. General Description ...... • . • . • 27

B. Operational Amplifier • . . • . . . • . • • 29

Power Supply . . • . . • . • . • • . . • 31 c. 10 V egulated Power Supply D. R • . . • . . • 33

E. Panel Meter ...... • 36 m F. A plifier Nulling • . . . • 36

G. Coefficient Potentiometer Setting ...... 39

H. MUltiplier ...... 42 I. Repetitive Operation . . . 42 Ch ap te r Page

APPLICATIONS • • • . . . • . . . • . • • . . • • • 47 IV.

A. Accurac y in Analog Computation • • • • • . • • . . • 47

B. • . Example 1, Si multaneous Algebraic Equations • • 50

Example 2, Second Or de r Differential Equation • • • c. 52 D. Example 3, Simulation of an Electric al Network • . • 53

E. Exampl e 4, Si mul ation of a Nonlinear Control --

System • . • • • . • • • . • • . • . • • . . • 57 ._ . .

F. Example 5, Se cond Degree Differe ntial Equations • 66 .

G. Exa mple 6, Narrowba nd Freque ncy Mo dulation • • . • . 69.

CONCLUSION • . • • ...... • . • . 75 v......

REFERENCES • • . • . • • ...... 79 . . . . . APPENDIX A • • . • . • • . . . • . . . • • • • • 81 . . .

APPENDIX B . • . • • . • • • . • • • . 86 . . . . . LIST OF FIGURES AND TABLES

Figure Page 1. Relative cost ve rsus acc urac y of analog anddi git al

computer for di ffere nt sizes of installations. • • 2. Operational amplifier with inve rting fee dback configuration. • • • • • • • • • • • • • • • ...... 10

3. mum possible load of an operational amplifier. . . . 1 2 Maxi 4. Eq uivalent circuit for st udyi ng effect of bi as curre nt,

offset curre nt andof fset voltage . • • • • • • • •• 16

5. Schematic di agram for measuri ng amplifier dr i :rt. . . . . 18

6. Rec orded results for determining the dr ift. • . . . . ••' 19

?. No ise model of the operational ampli fi er. • • • • . . . . 22

8. Schematic di agram for the analog computer...... 28 9. Ci rcuit di agram for the operational amplifier. JLA ?41 . • • • • • • • • • • • • • • • • • • • 32 10. Ci rcuit diagr for power supply voltage regulator , µA ?23 . • • • • • • • • • • • • • • • • • , • • 34 11. Circuit di agram for 10 V power supply regulator. . . . . 35

12. Panel meter connectio n. • • • • • • • • • • . . . . . J?

13. Simplifie d diagram for amplifier nulling. • • ...... J 8

14. Coefficient-setting pote ntiometer wit h load...... 40

15. fLA?25 connected as a voltage followe r. • • • • • . . . . 41

16. Ci rcuit connection of the AD530K multiplier. . . . . 43 17. Ci rcuit di agram of the multivibrator for repetitive operation. • • • • • • • • • • • • • • • • • • 45 18. The front panel of the integrate d circuit analog computer. • • • • • • • • • • • • • • • • • • • • • • • • 46 Figure Page Component layout on the chassis of the integrated

circuit analog computer. • • • • • • • • • • • • • • • • 46

20. . . Programming diagram for Example 1...... 51

21 . 2. . . Programming diagram for Example ...... 54 22. Computer solution for second order differential equation, Example 2. • • • • • • • • • • • • • . . . . . 55 Computer solution for second order differential equation, Example 2. • • • • • • • • • • • • • '24 . Circuit diagram of the electrical network simulated

in Example • • • • • • • • • • • • • • • • • • • • • • • 3 58 Programming diagram for simulation of an electrical • • • • • • • • • • • • • • . network of Example J. . . 59

26. Computer solution of Example J...... • • • . . 60

27. Block diagram. of general switching controlled plant. • • 62 28. Block diagram of the system to be simulated in Example 4. • • • • • • • • • • • • • • • • • • . . . . . 63 .. 29. Computer setup for simulation of the switching controlled system, Example 4. • • • • • • • • • • • • • • 64

JO. Phase-plane plot of the switching controlled system Example 4. • • • • • • • • • • • • • • • • • • . . .

31 . Programming diagram for Volterra's equation...... 32. Computer solution for Volterra's competition equation, Example 5. • • • • • • • • • • • • • • · • • • • • • • . . 68

JJ. Block diagram for narrowband system of Example 6. 70 FM 34. Computer setup for narrowband system of Example 6. . . 71 FM 35. results for narrowband 6. . . Recorded system, Example 72 FM J 6. Operational amplifiers connected as voltage limiter and frequency discriminator for narrowband frequency modulation detection. • • • • • • • • • • • • • • • • 74 Table Page

1. Characteristics of f.LA?41 in comparison to three other operational amplifiers. • • • • • • • • • • • • • • 30 1

CHAPTER I

INT ROD UCTION

A. An Applicat ion of Computers Tra diti onally, engineers apply both mat he matical analysis and experimentation to the solution of engineering problems . Physical syste ms are frequently expressed in terms of mathe matical models for whi ch the solution may be obtained. ge neral the problem ean be In describe d by a set of differential equations, the soluti on of which gi ve s the dynamic respon se of the system. Without the use of a · com­ put er , solvin g these equations may be tedious and exhaustive. A di gital computer acc omplishe s this task with great ac cur acy; theoreti­ call y the ac curac y of a di gital computer solution is unli mite d. But an analog computer, alth ou gh less accurate , can be used to handle the t.a sk far more rapidly an d easily. Man y engineering pr oblems do not require accurac y be yond three signi ficant fi gu res, and therefore do not require the employment of a di gital computer. Even in the cas es whe re hi gh acc uracy is re quired, it is o�en de sirable to obtain an analog solution before seeking the final solution on a digital computer. The chief advantage of an alog computation is that the operator can gain insight int o the pr oblem. Analog computer r esults are normally presented gra phi cally. The coefficients in the pr oblems c an be varied by corresponding parameter changes and. the block of computing elements can be visuali zed as a mat he matical operation. Considering the cost, an analog computer is less expen sive, but with 2

a specified component accurac y of better than 0.1 percent , its cost ri se s sh arply for even small acc uracy improvement. Figure 1 shows a quantitat ive comparison of the cost between an analog and a digital computer. Besi de s the acc uracy, ot he r di sa dvantages of the an alog computer are it s inabilit y to st ore the results and to make logical decisions. Hybrid computers have been de velopedto reali ze the advantage s of both types of computers and prove d to be the most convenient for engineering de sign applicat ions.

B. Analog Computer Analog computers used toda y may be divide d into two broad categorie s, general purpose and special purpose . The ge neral purpose computers can be further su bdi vide d into two classes: the di rect or ph ysical anal og comp t..er and the indirec t or mat he matical ana log com­ puter.3 Since the mat he matical analog computer simulates the system behavior, it maybe refe rre d to as an an alog si mulat or. but more comm.only as an analog computer. He re an y re fe rence to analog computer will mean in particular the electronic mat he matical analog computer. The electronic analog computer consists of basic element s which perform specific mat he matical ope rat ions.

1. Summation and subtraction of two or more variable s. 2. Multiplicat ion and divisi on of a variab le by a constant.

3� Inte grati on and diffe re nt iat ion of a variable with re spect to time . .,., CIJ 0 {)

.,.,: ...., � &

I I I I I I I lli A I c:rea na.iog \ s:tng comp I Size ute� o:f' in stall ation

10.0 1.o 0.1 Co111po 0 nent .01 acc 0.0 uracy, 01 P82'ce nt o:f' 1'ull. scaie Fig"Ure Re lati V 1. e cos co11tp t Ver uter sus a :f'or di ccur :f':f'e:re acy o:f' nt Si anai zes o og a :f' ·ins nd dig talla i tai tions, 2 -

w 4

4. Multiplication and divisi on of a va ri able by a variable . 5 . Ge nerati on of a function of a va riable or va riables. A ve ry wide range of prob le ms can be solved by application of these computing elements. The first three ope rati on s canbe performed by the use of a high gain de ope rati onal amplifier with suitable feedbac k elements. This is di scussed in va ri ous books of analog computation, such as Refe rences 2 and 6. Man y methods have been employed to do multipli ­ cati on and divisi on of two va riables, discussed in References 2,

6 and 13· Many functions can ge ne rated in the anal og computer as be soluti on s of a diffe rential equation, some ge ne rated by means of nonlinear elements, while some need special circuit ry. Function ge nerating is discussed in detail in Refe re nces 2 and J.

C. Educ ational Analog·computer Without severe red uct ion in the usage, the cost of an analog computer canbe dramatically reduced by eliminating expen sive units such as function ge nerat ors and using lowerqua lity component s. For educational purposes whe re ec on omic considerations arean import ant factor, many of the proble ms need not be complicated and accurac y is not critical. An educational ana log compute r may consist of a few de operati onal amplifiers, de sources, coefficient potentiomete rs , a panel meter and some computing ele ments. Th is typeof analog computer should be capable of solving ordinary line ar and nonlinear differential equations with re asonable accurac y. 5

Purpose of the Study D. The Heathkit Model EC-1 Educational Analog Computer, priced at $199.95, is the least expensive general· purpose analog computer on the market. This computer contains nine unstabilized, vacuum tube de amplifiers with de open loop voltage gain of about 1000. The low amplifier gain and amplifier drift limit the accuracy of the solution. Results obtained are sometimes confusing. Many integrated circuit operational amplifiers available on the market seem to give much better performance, priced from a few dollars in monolithic form up to hundreds of dollars in hybrid form. The low cost operational amplifiers, designed for various applications, are usually not intended for use in an analog computer. The purpose of this study is to select and evaluate low cost integrated circuit operational amplifiers which are capable of anal.og - computer application. If such operational amplifiers, which meet desired specifications, can be found they are to be incorporated into the analog computer to replace the vacuum tube operational amplifiers. It is expected that these modifications will result in more accurate and more reliable operation with fewer maintenance problems. Another advantage of this modification is its low operating voltage, ± volts, 10 thus enabling the use of lower voltage devices and interconnection to other forms of integrated circuits. In the fast growing field of integrated circuits, digital logic circuits and some function generators are available at reasonable prices. This renders further improvements and hybridization of the analog computer more attractive. 6

CHAPI'ER II

OPERATIONAL AMPLIFIER

A. Ideal Operational Amplifier The operational amplifier was originally conceived to perform linear mathematical operations in analog computation. Ideally, its voltage transfer ratio is solely dependent upon the input and feedback

elements, ZI and ZF. To implement the ideal operational amplifier the following requirements must be met . 1. High open-loop voltage gain. 2. High input impedance.

3. Low out put impedance. 4. 180 Degree phase shift between input and output voltage. 5. Low de voltage and current offset . 6 . Low offset voltage and offset current drift. ?. Low noise. 8. Stable when operat ing at the desired closed-loop gain. 9. Wide bandwidth. 10. High slew rate. 11. Linearity over the desired operating voltage range .

B. fLA?41 Operational Amplifier The J-LA?41 is a monolithic operational amplifier :manufactured by Fairchild Semiconductor. It is intended for a wide range of applica­ tions to replace the popular J.LA 709 which is now obsolete. It is 7

designed for high performance and ease of operation. The fLA741 amplifier is frequency-compensated and short circuit-protected and , in addition, it has a provision for nulling. It is essentially a two­ stage amplifier comprised of a high gain differential amplifier input stage followed by a high gain driver stage with a class AB output. 19 The circuit diagram and electrical characteristic s for the f.L A741 are shown in Appendix A. These data, supplied by the manufacturer, were used to compare it s characteristics with those of the ideal operational. amplifier described in the previous-section. Several other monolithic operational amplifiers were considered, including the fLA709, CAJOJJ and j.LA725, as well as a combination of the fl.A 72 7 and J.LA 741. The f.LA709 and CA303J amplifiers are also suitable for use as a . de amplifier in an analog computer. However, both need an external RC network for frequency compensation and their respective performances are not as good as the f.LA741 . The JLA725 has very good characteristics, including high open-loop voltage gain , low noise, low offset and low drif't. However, when oper­ ating as a unity gain inverter , its slew rate, 0.001 volt per micro­ second , is very poor. Slew rate as measured for an EAIJ80 amplifier, · connected as a unity gain inverter, is about o.1 volt per microsecond. The f.LA727 is a temperature-controlled preamplifier with a gain of 100. Since all components of the circuit are kept at a constant tempera­ ture it has a very low drif't. When used as the input stage of a f.LA741 , the drift due to the fLA741 is suppressed by a fact.tor of 100, the gain 8 of the preamplifier. Thus, the dri:f't of the combination, due largely to the J.lA727, is very low. This combination has low input offset 7 current (2.5 nA), high open loop voltage gain (2x10 ) and very low drift (5 µv /week). Its performance is comparable to a high perfor­ mance discrete component operational. amplifier with chopper stabili- zation. The frequency compensation network connected to the f.LA727 must be selected to obtain stability at the desired closed-loop gain.

As tested, the combination can be used when is limited at 100 kf2 z1 and the closed-loop gain is between 1 and 10. By choosing a proper compensation network, these constraints may be relaxed. This com- · bination was not used in this computer because of its cost, about $23 compared to $5.93 for the Jl- A741. The combination of f.LA727 and flA741 amplifiers would be desirable for a high performance analog computer with high accuracy computing elements.

C. Load Impedance, Input Impedance, Out put Impedance and Open-Loop

Voltage Gain The equivalent circuit of the operational. amplifier with an inverting feedback configuration is shown in Figure 2. The load resistance is assumed to be large enough so its·effect on the transfer characteristic is negligible. An analysis of .the circuit follows.

(2-0 where v. v = in (2-2) e + ZI 9

and

(2-3)

1) Substitute Ve and Vi into Equation (2- and solve for V01

(2-4 ) Voi =

V out expressed in terms of Voi and Vin, assuming RL approaches oo is

Substitute V0i into Equation (2-5)

v z (Z R )-A ( )Z Z = i + R o UJ 1 F out i (2-6) z )(Z ) Z (Z Z 2 ) (C..U)2 Z Vin (ZF+ oi 1+RR + r F+ o1+ i+RR +Ao i r

/ (Z Z ) Assuming z1 >> Zr I F+ 0i z and 01<< ZF

(2-?) -· = Z Z )Z Vin F+ r+Ao (UJ r

(2-8)

Details are given in Reference 8. ZF

Zr Ve lout - --:;. , V out Vin \J � J

+ vi .---+---' RL RR

Figure 2. Operational amplifier with inverting feedback configuration.?

..... 0 11

In de ri ving this princ ip al equation we have made fo ur ass umptions as follows .

1. R1 «> � Loads whi ch be connected to the output of the operational may amplifier include a coefficient potentiometer, the input of the ne xt operational amplifier and the output mete r. Co nsider the worst case whe n all possible loads are connected and each resistance has its

minimum possible value. . The connection is shown in Figure J. Total load resistance is about 4 kn.

= output vo ltage without load. Vout V�ut = output voltage wit h lo ad connected. Th e re lation is gi ven in Reference 7. * v = v (2-9) out out �R of+ L z

= total load resistance � Z0£ = output impedance of the closed-loop amplifier

(2-10)

the percent error of the output voltage * v - v out e = out x 100% (2-11) vout 100 k.Cl

100 krl 10 krl To input of ti' \r----- n e x stage the t * v operational Vin j out amplifier

10 kn< .._ t ) 50-0-50 microammeter -:- -::-

// Total = kD.//10 k.O. 20 kQ R1 10 = k.Q 4 possible Figure ) . load of an operational amplifier. Maximum

...... N 13

2of e = x 100 "' (2-12) 2of+R1

2oi e C: �! x 100% (2 -1 3) Z0i + 1 t A0 [ ( w)] l\ For the J-LA741 operational amplifier using A0(UJ) and Z0i as give n in the specification, Appendix A, 5 Ao (UJ) = 1.5 x 10

zoi = 75 .Q

z1 = ZF = 100 k .0. R1 =4k.Q 75 e, percent error = x1 00 3 75 + [ 1+1.5x105 x4x1o ] � o So the assumption that R1�cois proved to valid. The specification be that R1 2 k.O ensure th at the transfer function will not affected � be by the load re sistance, 4 kf2..

2 . Zj_>>z 1//(Zy+-Z0i ) Sl Zi = 2 M

ZF = z1 = 100 k Q

zoi = ?5fl Zr//(ZF+Zoi ) = 50 k.0.<< 2 Mf2

3. zoi ZF << Z D. oi � ?5

ZF = 100 kQ

269665 SOUTH l1 RJ._qy DAKOTA STATE UNJVERSITY 14

4.

requiring that A (W)Z >>Z • o 1 F+z I For the example note that 5 ) = K.Q>> A0(W Z1 1.5x10 x100 200 kf2.

D. Offset, Drift and Noise For a transistori zed operational amplifier the input connections of the first stage differenti al amp lifier are made directly to the transistor bases, or the gates and grids in case of FET and tube amplifier. Thus , it is unavoidable to have small do bias currents. The average of the two do input currents is termed the input bias current . The difference in the currents into the two inputs with output voltage equal to �e ro is termed the input offset current. The voltage which must be applied between the input terminals to obtain ze ro output voltage is termed the input offset voltage. The tot al offset at the input terminals is equal to the sum of the input offset voltage and the voltage drop across the input impedance due to the input offset current . Although this offset is rather small, its effect maybe significant when the operational ampli­ fier is used as an integrator. 1 7 In mono li thic operational amp lifiers the offset is often compensated by inducing a small difference in the collector current of the two channe ls of the first stage differential ampli fier. A fixed input offset is usually not a problem since it can be compensated, but both input offset voltage and input offset current 15 are not constant . Drift is usually specified as the coefficient of input offset change. Drift,it self, may considere d as a combina­ be tion of thermal noise an d excess noise (1/f noise).9 Three ma.in factors that affect drift are temperature , time and· power supply voltage. Drift in a de amplifier is a serious problem since the offset cannot di stinguished from a chan ge in signal. Due to dri �. be the operational amplifiers use d in analog computers nee d to be balanced after some period of operation. Figure 4 is the equivalent circuit of operational amplifier for studying theef fect of bias current an d offset.

Total offset voltage at the input = VE

Maxi mum total offset vo lt age at the input = VEM ( 2-14)

Set � equal to R1/lRF to cancel the effect of bias current. So (2-1 ) VE = V05tI0s RR 5 ( 2 -1 6) VEM = Vos+IosRR For f.LA741 operational amplifier, from Appendix A

V05 = 1 mV t ypical

I05 = 20 nA typical

RR = 50 krl Expec ted value of the maxi mum total offset at the input of f-LA 741 operational amplifier is - - v = 3 0 9 VEM 10 3+5ox 10 x2ox1 VEM = 2 mV RF

t V V E v:::::�;.�--J•���- • out + -Ao VE

R R

4. Equivalent circuit for studying effect of bi as current � Figure offset current and offset voltage .?

...... °' 1?

For the f.LA741 there was no available data about time drift, so a measurement was ma.de as shown by the sc hema.tic in Fi gure 5 to investigat e this effect.

A 50- 0- 50 mic roammeter in series wit h 490 .(2 resistance was used to de tect the null error. The offset was amplified 8.3/0.101 times. After the null ad jus tm ent was made , the offs et was so small that the time integral of the offset was measured ins tead of the offset itself. Th e recorded curve of integrat ed_ drift is shown in Figure 6.

E =-1 e t (2-'1?) o RC OS whe re E =ou tput voltage at the end of ti me t 0 eos =off se t voltage at the input

1 = 1 =10 s conds RC 0.1x1

t = integrating time =1 minute E e = o OS 10x60 The record was ma.d e during th e period July 23 to July 26, 19?1. The maximumE obtained was 0.1 volt on July 24 , at 3:00 p.m. 0 9

e = 0.1 9 =0.3 2 mV OS 10x60

Thus, within a four-day period, th e offse t dri.fted to a maxim um va lue of 0.32 mV. The approximat e time drift is 0. 1 mV/day. fl 8. 3 M

Mrl v 0.101 +15 2 To strip chart recorder

50 kD. 50-0-50 mic roammeter

-15 v

Figure 5. Sc he ma.t ic diagram for mea surin g ampl ifier drift.

.­ CX> 0 .3 1 ,--... en +l rl 0 > l � p. 0.2 ·r-1 July 24, 3:00 m. July 25, 1:00 p.m. ��

t10 ...... July 1:QO I 23, 0.1 p.m. July 26, 9:00 a.m.

July 24, 11:00 a.m.

- T • u 1 · - v u��,.... ".) � � - J , a.m. 1 I 9:00 0 30 6b5 ., - ; , 9:00 p.m • (Integration time seconds) Julv 2'i _ in 6. Figure Recorded results determining for the dri�. 20

In the j.LA 741 the offse t is compensated by a 10 k fl. potentiometer connec ted between pins 1 and 5. This te chnique is capable of compen­ sating up to 15 mV of the offset at the input and avoids the use of a 1 MO resistance for �· ZI and ZF are selec ted to per form the mathe matical operation required. Thus the equivalent resistance connec ted to the inverting input, Rr//�, is not fixed at 50 kn as assumed, and so the idea that th e effec t of the bias current is cancelled by setting Ra = �/ /� can not always be achieved because of va rying gain requi rements. R1//RF can take a minimum. value of 4. 76 k0. (when both inputs are connec ted · to 10 k fl and ZF is 100 kfl) to a maxim um value of 100 k.Q (when one input . is connec te d to 100 k.0. and ZF is a capacitance ). The error voltage at th e input depends upon the equivalent resistance connec ted to th e inve rti ng input and could as high as 5 was chosen to be be m.V. Ra (100 50 kfland the nulling c:lrcuit se t as (Rr//�) = kll//8.J Mfl). This is proposed to minimize th e effec t or bias current when the operati onal amplifier is used as an integrator. The operational amplifier with FET input stage has lower input bias current and lower input offset current. The error due to input offse t current may be reduced by th e use of th e operationalamp lifier with FET input stage. However, th e input offset of the input F.ET operational amplifier is more sensiti ve to te mperature change. It may not be very satisfac tory unless operating in a temperatu re- controlle d environmen t. 21

Higher frequency changes in the input current and voltage are referred to as noise, which is a combination of thermal noise and shot noise.9 Noise model of' the operational amplifier for studying the effect of noise is shown in Figure ?.

Esig = signal voltage

Ens = thermal noise of source resistance (2-18) = 4KTRs6f

En = equivalent noise voltage generator

�2 (2-19) = ) � en df J f1

In = equivalent noise current generator

(2-20)

Eni = total equivalent noise voltage at the input

(2-21)

Note that the last term approachesrzero because C, correlation coefficient, is zero over most of the frequency range.9

Zs = source impedance

= Zr//� = 100 let = � kn 50 = kQ 20 4 6 1.61x10- x.5x10 x10 V = 4KTRs �f = -10 = B-.o.sxio v 22

�o

°' . "4 a> � ;q �

';1s::: 0 .,...

�,...

s::: H 8.0

�..., � 0 ,..... Q) a Q) Cl) .,... 0 z

• � Q)

Cl) bD """' � Cl) � pf � 23

Eni is given in the data sheet, Appendix A, for the frequency k range 10 Hz - 100 Hz. Curve 15 on Page 84 indicates 12 µ..v rms for Zs = 50 kfl. 2 2 be From. the curves 13 and 14, Page 84, en and in appear to constant in the region 100 k Hz to 1 M Hz. 1 e = - 6 � 4x10 v2/Hz 2 2 in = 3x1o-25 A /Hz For the total noise at the input in the frequency range 10 Hz _ 1 MHz, taken as the bandwidth of the operational amplifier, w obtain:

�1=

12 = �(144+360+6?5)xto-

0 = Ju. ?9xto-� • = 34.3 JLVrms I Total noise at the input is 34.3 at a temperature of 25°c. ILVrms r- E. Stability, Bandwidth and Slew Rate The gain of an ideal operational amplifier is constant, independent of frequency. But due to stray capacitance in the circuit and high frequency limitation of transistors the gain of an operational amplifier decreases and the phase shift between output and input voltages increases as the frequency increases. The attenuation of the amplifier 24 gain and the increase of negative phase shift with frequency creates the possibility of self-oscillation and increases the output error. The voltage transfer function of an inverting feedback amplifier may be written:

(W) -A0 - 1 - 1:. (2- 22) 1+A 0 (W)p f3 1 1 + -Ao-( W;;;..._) ,,- where A0(UJ) is the open-loop voltage gain, a function of frequency, and f> is the voltage transfer function of the feedback path. As the open-loop voltage gain, decreases, A0(UJ), 1/A0 (UJ)fa increases and so increases the output error. Consider the situation when R = -1, = - , and the system is unstable causing (W) r OUt CO A0 V For self-oscillation. stable operation the loop gain, A0 (W) fo , of an amplifier must not greater than unity at the frequency where be phase shirt is 180°. A conventional design criterion for unconditional stability is the requirement that the attenuation rate be at least -6 dB/octave at the closed loop gain crossing. For analog computer application the smallest required closed loop gain is usually unity so the operational amplifier must be frequency-compensated to obtain stability at unity closed loop gain. Frequency compensation in a monolithic operational amplifier is usually achieved by the method of pole cancellation. An network RC is connected to the output stage to introduce a pole at a lower frequency and a zero to cancel the original pole. 25

This compensation network significantly reduces the slew rate. Slew rate is defined as the time rate of change of closed loop ampli­ fier output voltage under large signal oonditions.7 It appears that the slew rate is usually related to the bandwidth of the open-loop amplifier but not the bandwidth of the closed-loop amplifier nor the closed .J.oop gain of the amplifier. Bandwidth is defined as the frequency range over which the gain is approximately constant. The negative feedback extends the higher cutoff frequency by the factor 1+A0(UJ)f' The bandwidth of an amplifier is a function of its closed-loop gain. The product of band- width and the closed-loop gain of an amplifier with a fixed compen- sation network is nearly a constant, usually referred to as the gain- bandwidth product.

The f.LA741 operation � amplifier is internally compensated. Its gain-bandwidth product is 106, that is, the bandwidth is 1 MHz when the closed-loop gain is unity and 100 k Hz when the closed-loop gain is ten. Its slew rate is 0.5 V/fLsec.

F. Linearity Output and input of an operational amplifier should be linearly voltage of any related, that is Vout/Vin = _zF/z1• But the output realizable amplifier is limited, and the relationship between input and output voltage is not linear throughout the entire r�ge of out­ put voltage. At the higher limits of output voltage the amplifier is operating in the saturation region. To ensure that the closed-loop gain is linear, operating the amplifier with the output voltage near 26 the higher limits should be avoided. To avoid operating the operational amplifier in its saturation region which leads to inaccurate results and possible damage to the amplifier, overvoltage indicators are used to warn the operator whenever the output voltage is beyond the desired range. When the fLA 741 operational amplifier is supplied by !15 V power supply its ·output voltage range is ±14 This assures that it will V. ·operate linearly in the required range of ±10 Although operating V. in the saturation region will not damage the f.LA741 operational ampli­ fier, it leads to incorrect results. The overvoltage indicators were not included in this computer. The approximate maximum value for each amplifier should be determined during magnitude scaling. When a problem is run on the computer, any critical value should be checked by the output readout device to ensure that the maxim um voltage at the output of any amplifier does not exceed ±12 V. 2 7

CHA PTER III

DESIGN

A. General Description The schematic diagram of the analog computer is shown in Figure 8. It consists of: Nine operational amplifiers manually balanced by screwdriver adjustments on the front panel. One four quadrant analog multiplier. !1 5 Volts regulated power supply for all active components, capable of supplying current higher than 150 mA. 10 Volts ungrounded, regulated power supply for coefficient potenti meter setting and initial condition voltage supply.

50-0-50 Microammeter, 2 percent scale accuracy, full calibrated to read ±1 V, ±3 V, and ±1 0 V. Meter is also used for output voltage reading, coefficient potentio­ meter setting and offset nulling. One four contact relay for manual operation, repetitive operation and reset. In the repetitive operation mode the rate of repetition can be varied from 1/15 Hz up to 35 Hz.

Five 10 k fl coefficient potentiometers. - iO kI1 t.'f14t4 ------+1.5 v voltage :regulator Opention&l. a111pli f'1er #2 through 18 AD5)0K

- - - _ fo � , � J 111 �·�I 1 �1� : � : ------J ' 101-L -IT,�- 11 ' ... - ,.. ,I --- :'M-0 • - I I 1 -- L,I l-..h>� \ _ I . I To - ::: Mllo .. ::: ' -[- •- -: I l� I Mlohi 0 t .� ,ur;;, i-�I.., 1--h>I I I -J_Mo . ' tod l� •oollo" switch '"'10 ___ __ '.'.______115 '/'41! L- ...t 1N2}5 �: 60 ep= Hol:i.y - i ....I � 11'.a.\n I I 1 fl - - � P..oFAtit\ve control - - ! -- - ljl "' - :i - L U -- �JF-- - -� - H - - - _ I_ - - --q:::::::::: - � - -i, - � 1'25t - I 1i 1- - Voltaee follow!' I �� -: 1N2.51 10 v c: -., Voltage I N I regulator I.. ..,, >;:. <1--rO I

.. .:. .J

All switches viewed from th• !'t!ar

Figure Schema.tic diagram fer 8. the analog computer

N co 2 9

These components are enc lo sed in the case for the EC- 1 He athkit Analog Computer wit h ori ginal vacuumtub e components removed. Computer programmin g is done on the front panel using plug-in computing elements and pat ch cords.

B. Operat ional Amplifier Det ailed characterist ics of the operational ampl ifier are given in Chapter 2. Table 1 shows it s characteristics in comparison to three other operat ional ampli fiers used in EC- 1, EASE and EAI380. These characteristics are as given in specifications released by the manufac-· turer. Some parameters, ho weve r, are not ava ilable . It may not be convenient or ne ce ssary in all cases to make the measurements, so est imations will be ma.d e instead from the circ uit diagrams and available data. EC-1 He ath kit 's amplifier is the most simple operational ampli fier, cons ist in g of a pentode cascaded to a triode. This gives an unstabil ized ampl ifier with open- lo op voltage ga in of 103 . EASE•s amplifier consists of a high ga in de amplifier of ga in 105 wit h stabil izing ampli fier of ga in 103. This gives a hi gh performance 8 operat ional ampl ifier of ga in 10 and ve ry low drift. It s defic ienc ie s in clude poor characterist ics of the tube; la ck of ru ggedness, short life-time, large phys ical size, and le ngthy warm-up period . The EAI380 amplifier consists of a hi gh ga in de ampli fi er and a chopper stabilize r. This trans istor operat ional ampl ifier gives de 7 10 The comput in g gain of x 10 , operat ing in the range of � 3 V. elements are conne ct ed inte rnally. JO

Table 1. Characteristics of A741 in comparison to three other operational amplifiers.f.L

Mod. 1480 Mod. 6.614-2 EC-1 op-amp dual de amp Heathkit EASE EAIJ80 fLA741 Electronic element Tubes Tubes Transistors Integrated circuit

Power supply +JOO V +250 v !15 v !15 v -150 v -465 v +JO v - Power consumption 1.5 4.8 W de 50 (stand by) w 11.97 ac mW W Operating range !60 v !100 v ±10 v ±10 v voltage gain 103 108 3x1o7 2x105 DC * * * 2in High 5x106 D. 104 n 2xto6 D. * * 2out 400 n 0.01 n 50 n 75 D. ** * ** Input offset a:rter 75 mV 100 µv 20 µv 0.1 mV - balancing .. * ** Drift !'5 mV 100 V/day Very low 1ooµv/day short term µ 5 . V Input noise 4 mVrms mVrms 1.06 µvrms J4 3fL rms 1 MHz Bandwidth of unity 00 Hz 10 kHz 125 kHz gain inverter 5 ** ** ** 5 5 Slew rate (V/sec) 5xto4 2xt-o 6xto

* Estimated from available data

** Measured value and otherwise obtained from References 13, 14 15. 31

The Jl-A741 integrated circuit operational amplifier which was

selected to replace the :&::-1 Heathkit's operational amplifier is

extremely small and has very low power consumption. The ma.in advan- tages of an integrated circuit operational amplifier· are its circuit

simplicity and its low cost. Its disadvantages are due to limited

performance caused by price versus performance considerations in the

integrated circuit manufacturing. The fLA741 , although not considered

as a high perform.:i.nce computing amplifier in terms of drift and voltage

gain, satisfactorily fulfills the requirements of a computing amplifier.

The circuit diagram of the operational amplifier is shown in Figure 9.

C. Power Supply

Each operational amplifier requires a power supply of ! 15 volts with supply currents of 1. 7 up to a maximum of 2.8 The power mA mA. supply must provide ! 15 lts with sufficient current for these nine � operational amplifiers, the voltage follower and the multiplier.

Total required current is about 35 The power supply must be line­ mA. regulated to prevent the variation of output voltages due to line vol­

tage changes and load current-regulated to prevent interference between various operational amplifiers. The specifications for regulated power

supplies depend on the size and precision of the computer, generally

regulated to within 0.1 to 1.0 percent of their nominal values for 1 anticipated variations in the line voltage and load current. The (0. 01 / power supply should also have a low temperature coefficient percent 18 o 0. 05 / (JmV to 100 mV ). For an c to percent OC) and low ripple PP PP +15 v

To front panel amplifier connection 2 6 To front panel amplifier connection

3

-15 v

Figure 9. Circuit diagram for the operational amplifier, µ.A741. ·

w l\) JJ

integrated circuit operational amplifier the power supply sensitivity

is always specified, usually in terms of the ratio of the change in

input offset voltage to the change in the supply voltage. Specifi­

cations of the power supply may be based on thi s characteristic of

the operational amplifier. f.LA723 integrated circuit voltage regulators are used . The circuit diagram is shown in Figure 10. Thi s connection is as given by the manufac turer, Fairchild Semiconductor. The temperature coefficient is 0.002 -percent/0c, ripple reduction is 74 dB , line re gulation is 0.02 percent and load regulation is 0.03 percent.

Both positive and negative output voltages change less than

0.5 V when the input voltage changes from to 30 V and the ! 18 ! V load current changes from zero to 150 Ripple cannot be detected mA. on an oscilloscope with a scale sensitivity of 50 mv/cm.

D. 10 V Regulated Power Supply

The computer is equipped with a. 10 V regulated power supply to

used as a reference voltage in coefficient potentiometer setting be and as an initial condition voltage supply. The circuit diagram of the power supply is shown in Figure 11. The Darlington pair � Qi, f element. performs as a comparator, unctions as a series control Q) comparing a sample of the output voltage to the reference voltage drop ac ross the Zener diode.

The out put voltage remains constant at 10 V as the input voltage to changes from v to 30 and the load current changes from 0 50 18 mA. v h a scale sensitivity Ripple cannot detec ted on an oscilloscope wit be of mV cm. .50 / +23 v -23 v

�

_. . V N C � v V · v __ v+V V . .,___ . ref O ref outc U' 2N2.564 V . . ..---- +15 v z v -15 v µA723 c:� µA723 Cz . 3.01 CL. ...____ L kn (""'\

Cs. s c . �0 N.I. · N.I. Inv. • � . � v- Com�. ·I C!� v-: (""'\ 500 pF 500 pF

-:- '7

Figure 10. Circuit diagram for power supply voltage regulator, µ.A?2J.

\...> � 22 .6rl ------+10 +25 v v

20 kfl 20000. \ \. / ' I s 2140f).

1 2ooofl

2 kfl 20 I I � � I 25 I 30100. v .a lN'/51 ">

Q , 2N65?A 1 � 2N956 �

Figure 11. Circuit diagram for 10 V power supply regulator.

\..u \J\ 36

E. Panel Meter

A 50-0-50 microa.mmeter is the only readout device furnished in

the computer. An recorder, oscilloscope , or strip chart recorder XY may used to display the output by connec ting to the amplifier out­ be put terminals. The meter is also used to read the voltage applied to

the meter input terminals to detect the amplifier nulling. Instead and of using exact resistance , resistances in series with potentiometers are used to provide meter calibration. The meter connection is shown in Figure 12. Current through the meter is limited at about 140 A· fL . The voltage drop across the 5 k f2 series resistance is limited by the breakdown voltage of the diode 1N251 , 0.7 V.

F. Amplifier Nulling

Offset of the operational amplifiers can be compensated without n­ removing the problem setup: A nulling switch on the front panel co nects the amplifier to ground through 0.101 M flinput re sistance and M :teedback resistance. The input offset is thus amplified 83 8.3 fl times and then detected by the microammeter in series with 490fl resistance. This gives a very sensitive detector . The input offset m Two of the operational ampli fier can be reduced to le ss than 0 .1 V • ect it 1N251 diodes are connected in parallel with the meter to prot ssing the from transient high voltage that may occur while depre any nulling is shown nulling switch. The simplified diagram for amplifier in Figure 13. From meter function switch

.1 Output no • - "' Meter switch 2 range Output no .

C!� 0 '4) �

\r\ c: � 0 \r\

4900. 1N251

1N251 50-0-50 J-LA

-:- 7 7 - -

12. Figure Panel meter connection.

'v.> """' +15 v

-15 v

� To problem board , I

Nulling switch

0.101 Mfl

-:- Meter function switch

50-0-50 microa.mmeter 1N251

-. -. -.

Figure Simplified diagram for amplifier nulling. 13 . \..tJ co 39

G. Coeffic ie nt Potentiometer Sett ing

A 10 V regulated power supply incorporated with the meter is used for setting coefficient potentiometers. The main error in coefficient potent iometer setting is due to the add it ional voltage drop in (1-Cl) � caused by the current flowing to the load res istance, 11_· Figure 14 shows coefficient-setting potent iometer with connecting load. The effect is described by an expression for the effect ive potent iometer sett ing. E 0 k - - - Er a Note that k, the volt age attenuation, is a function of the potentiometer sett ing,Q. , and the rat io of the potentiometer resis­ tance to the load res ist ance, Rpfl): Error correction by us ing the given equation is tediou '!. But with a high input res istance meter, accurate potent iometer setting is easily ac compl ished by a direct measurement of the volt age with the load res ist ance connected. The meter in put binding posts are connected to the meter through a volt age follower which provides a high input resistance of 400 Mf1. The µ A 725 integrated circuit operational ampl if ie r is connected as a volt age follower. The frequency compensated network shown in Figure 15 is connected to prevent osc illat ion. The µ A725 was selected because of it s low drift and low offset . With the volt age follower configura­ tion it s offset and drift are negl igible . The in put offset is 0.6 mV and the drift is 1.5 µv/day. With no connection to the meter input binding post , the output of the volt age follower is -13.0 V. E1

Eo = kE 10 I v ftp Hi l � -:-

Figure 14. Coefficient-setting potentiometer with load.

g +15 v

2 To meter function switch

From meter input binding post 3

-15 v C2 I = 40S1. = 10. 5 Rt R2 fl c 1 = 0.02 fLF c2 = 0.05 fLF

Figure 15. µA?25 connected as a voltage follower.

� ..... 42

H. Multiplier

Among various types of electronic multipliers on the market , the time division multiplier gives the best accuracy and the best perfor­ mance attainable , while the transconductance multiplier gives moderate accuracy at lowest cost . The AD530K is a monolithic transconductance multiplier manufactured by Analog Devices Inc., priced at $45.00. The

scale factor is externally adjustable . The input and output offsets are externally trimmed. Its total error is about one percent .

Detailed specificat ions and the pin configuration as given by the manufacturer are in Appendix B.

Figure 16 shows circuit connection of the multiplier. When used as a multiplier the Zin terminal is connected to the output nnected terminal , and when used as a divider the Yin terminal is co to the output terminal . Squaring and square root operations are

. - ected accomplished by connecting Xin to Yin when the AD530K is conn as a multiplier and a divider, re spectively. When connected as a di� vider a negative value of X, the divisor , is forbidden.

I. Repetitive Operation

Sometimes it is desirable to investigate the different behaviors resulting from a change in the parameters and /or initial conditions of the system. aut omatically switching the computer between OPERATE By or ial and RESET modes while simultaneously changing the parameters init served on an conditions, the effects of the change can be immediately ob 1 ulation to oscilloscope . This is very convenient when using sim optimize the design of a system. -15 +1 5 v v 20 k.0.

20 kfl

20 kSl

2 10 7 9

6 3 • in '. ., zin. x. 1 I AD530K y in · Output 1 41 I 5 kfl � 5 8 750 D. '-� j -

Figure 16. Circuit connection of the AD530K multiplier.

� \.iJ 44

A multivibrator is used to operate the relay for repetitive operation. The rate of repetition can be varied from 1/15 Hz up to

35 Hz . The duratdton of the period is about one millisecond RESET which is long enough to let the capacitor discharge through the closed switch of the relay.

The circuit diagram of the multi vibrator is shown in Figure 17.

is a unijunction transistor operating as an astable multivibrator Q1 whose period of oscillation can be adjusted by a 40 krlvariable re sistor. The output from the multivibrator is fed to a Darlington pair �, � which handles the high current required in operating the relay.

Photographs of the computer front panel and the computer chassis are shown in Figures 18 and 19. +30 v

Relay

+15 v -----.------t------, , To OPERATION switch

klL

Qi 150 500 Jl-F kD.

Q - 2N489IA 1

Q2 _; 2N171? Q - 2N3 1 J ? ?

Figure 17. Circuit diagram of the multivibrator for repetitive operation.

� 46

e • e . . 0 1 T frcn pa.n f th t t i comput er .

a o..,. e • A 1 er. 47

CHAPI'ER IV

APPLICATIONS

The complexity of the problems that can be solved is limited because this analog computer contains only nine operational ampli- fiers and each amplifier has only two inputs. These restrictions are due to the front panel layout on the EC-1 Analog Computer and are not restrictions imposed by the u�e of integrated circuits. The analog computer will be used to solve a few simple equations that are often encountered in the study of electrical circuits and control system engineering. The solutions obtained from the analog computer will then be compared to the solutions obtained from hand calculation. A discussion follows concerning the accuracy of analog computations in general and an investigat.�on of the sources of error which may arise .

A. Accuracy in Analog Computation The accuracy of a solution obtained from analog computation depends upon the nature of the problem and the ability of the operator as well as the accuracy of the instruments. 3 It is not possible to ,. specify the accuracy of an analog computer without considering the precision of the computing components. Even if the precision of the computing components are known , it still may not be possible to specify the ac curacy of the analog computer in a given problem. _ The precision of these components, such as input and feedback elements, limits the accuracy of solutions obtainable from an analog computer. 48

Instrume ntal error is due to individual errors in each of the instrument s combined in some unspecified fashion . The instrumental error can be minimized by selecting suitable magnitude and time scales which make optimum use of an accurate readout device which has suffi­ cient frequency response . Magnitude of the signal should be maximum without overloading the amplifiers while the computing time should be minimum. provided that the results can be followed by the readout device.

Finite input impedance , finite open-loop voltage gain and nonzero output impedance of the operational amplifier introduce static error.

But in this case the error is dominantly that due to inaccurate values of the computing elements, potentiometer settings and initial condition settings . All carbon film. and deposited carbon resistors used for the computation were matched to within one percent. For economic reasons , five percent mylar capaci"'tors (0.1 µ F and 1.0 f1.. F) and polystyrene capacitors (10 f.L F) were used. The coefficient potentiometers were inexpensive one -turn potentiometers without dial calibration . This necessitated setting potentiometers by means of the reference voltage and panel meter. This method gave accurate results to within two percent provided that the load remained connected in the circuit while setting the potentiometers .

Drift in the operational amplifiers is the major error when computation extends over a long period of time. The operational amplifiers should be nulled before each computer setup and the null verified whenever a hi gh preci sion result is desired . 49

Frequency response and slew rate establish an upper frequency limit fo the signal. The gain-bandwidth product of the µA741 is 106 • Thus the operational amplifier is capable of operating at a frequency above 10 k Hz. Its slew rate is 0.5 V/µ_,sec. The voltage transfer function of an analog integrator is approximately equal to -1/RCS. Due to R1, leakage resistance associated with the feedback capacitance, the voltage transfer -Rl 1 function of an integrator is __ __ The low frequency gain R R1cs + 1 of an integrator is limited by either open-loop voltage gain of the operational amplifier or the ratio of leakage resistance and input resistance, R1/R. This nonideal frequency response of the integrator may have significant effect upon some problems. Sources of noise in the computer, including amplifier noise, pickup noise and noise in the readout device establish a threshold of usable signal at about 0. 5 mV. The linear operating region of the operational amplifier establishes a maximum amplitude for the signal so the amplifier output voltage must not exceed !12 V. Accuracy of the solution is often limited by the readability and the accuracy of the readout device. Typical accuracies for these readout devices are : panel meter; 2.0%, XY plotter; 0.1% - 0.25%, strip chart recorder; 0. 5% _ 1. 0� , and oscilloscope; 5.0% - 10.0%, when they are properly calibrated and operating within their specified frequency ranges. 50

Accuracy of the computing components in this analog computer is limited by tolerances of the computing elements used; 1� in summation and 5% in integration. In general the accuracy of the solutions

obtained from this computer can not be better than these limits.

B. Example 1, Simultaneous Algebraic Equations

ax x = o 1 + � + 3 (4-1)

X1 + 2X2 + X3 = .7 (4-2) + x1 + 12 2x3 = 5 (4-3) To ensure stability of the computer setup the machine equations must be chosen so that the self-loop gain for each variable is less than unity.4 Solving for the variable in Equations (4-1), (4-2) and (4-3)

= -) 2X1 0 - x2 - x3 (4-4

2 = - (4-5) 12 7 x1 - x3

2x = 5 - x - x (4-6) 3 1 2 The programming diagram for this problem is shown in Figure 20 with the loop gain for each variable equal to 1/12.

Computer results are x1 = -2.9 4.o x2 = x3 = 1.9

X = - J . Calculated results are 1 O 4.o x2 =

x3 = 2.0 Accuracy of the computed results is about 5 percent • -2x1

) -2 x2

(x (5-x -Xz) 1+xz) 1 2 - x3

Fi gure 20. Programming diagram for Example 1 .

\J\ ..... 52

C. Example 2 , Second Order Diffe rential Equation

The second order differential equation can be written in

standard form as follows.

d2x 2 + 2 + x = 0 n dx w (4-7) dt2 �t:w dt n where

W = undamped natural frequency in radians per second n �- = damping ratio Solution to this equation is

- W t w F-i t W t n n - n x = e K e S K e (4-8) [ 1 + 2 Fi 12 Wave shape of x depends on the damping ratio , (; S • J

= 0, x COS t Undamped � X = 0 ll\.i - fWt x = x e cosW Underda.mped �<1, o n

-Wt X n � = , x = o ( 1 +t) e Critically damped S 1 2

- t x x e cosh W t Qverdamped �>1, = 0 �C4i n R 2 1 d x ) The machine equation is + � + x = 0 (4-9 d 10 dt nm t2° dx

= 2 The correspondin = 10 radians per second and g wn ( f3/

The initial conditions are 0 : (0) = and x (0) = 8 53

The programming diagram is shown in Figure 21. Waveforms of x ,

1 and -X versus for various values of are shown in dx L dx 10 dt 10 dt R. r Figure s 22 and 23. From the recorded re sults, frequencies of oscil- lation and percent overshoot of x were measured and compared to the calculated results.

.. o.o e= �/2 0.10 0.25 0.50 1 .0 2.5 me asured 9 .8 9. 9 9. 7 7.0 0 0 Frequency _ of oscillation calculated 10.00 9. 95 9. 70 8.66 0 o ·

measured 100 81 20 0 0 Percent 50 overshoot of x calculated 100 72 15 0 0 44

The accuracy of this solution is about 20 percent . Resolution of the oscilloscope trace severely limited the accuracy for this problem. Hi gher accuracy can be obtained by scaling down the problem time and using a more accurate recording device , such as recorder. X-Y Note that noise seen in the pictures was noise in the oscilloscope rather than noise from the computer. This could be eliminated by using another oscilloscope .

Example 3, Simulat ion of an Electrical Network. D. Kirchhoff 's voltage equations of the circuit diagram shown in

Fi gure 24 can be written as follows .

1 i dt (4-10 ) :� i dt - 2 e1 = + � t � f Li f . N

I _.0 1 �

0

0.2

1 x - - vs . 110 dxdt _,.,.,,. 10 dt

scale , 500 \J\ msec /c \J\ scale , 5 V/cm Voltage

Filltlre 22 .. Computer solution f

ime scale , 500 msec/c Volta�e scale , 5 V/c

ple , 23.. Comput olution for equatio�:. second order differential Fi.. 2 .. � 57

0 = Ri L di2 1 2 + 2 + i2dt - ! i1dt (4-1 1 ) dt c f c f

let q = i dt and = 1 f 1 ciz f i2dt (4-12)

- - q" 1 q 2 - -R 2 (4-13) 12 12c For the values given we can write the ma.chine equations qt = -100 qt + 100 q2 + (0.5 sin 20t) x 20 (4-1/f)

= + q (4-15) q2 -10 q2 - 100 q2 100 1 Programming diagram is shown in Figure 25. Waveforms of 10e 1. , 1 - 0i1 , -10i2 , 10ei versus -10i1 and 10ei versus -10i2 are shown on Figure 26. The factors of ten were necessary for magnitude scaling of the equations. It st be noted that photographing reversed the pictures from left to right . Time scale must be read toward the lef't hand . Results obtained from the pictures are o i1 = 1.45 t?90 ei o.40 i2 = /120° ei Results from hand calculation are

= .9 e i1 1.44 j-86 ° i

= 26.8° e i2 0.40 /1 i The solution to this problem has an accuracy within 5 percent.

E. Example 4, Simulation of a Nonlinear Control System. A block diagram for general switching controlled plant is shown in Fi gure 27. R 11

1 2 11 c = ) e1 0.5 sin 20t ' R = 0.5 fl = 11 0.05 henry 1 = 0.05 henry 2 = 0. 20 farad C

Figure 24. Circuit diagram of the electrical network simulated in Example ). \.)'\ CD 10ei = sin 20t 5 -100q -10i1 100q1 1 10 1

1 1

-10i2 100q2 -100� 10 0 10 1 10

1 1

Figure 25. Programming diagram. for simulation of an electrical network of Example 3.

\J\ '° 1 'J ei 0 ei

f"'.!

- ! r'li l 2

f;.,: ·�/S o vs e

10e:': 1 (l.l .. -· . ' 1011 2

scale , 5 V/c Voltaf!e msec/cm Time $cale , 200 °' 0

Fi�e Com.puter solutton .pla 61

Specially consider the case of an ideal relay having control

effe ct N with the governing function as F(t) = e(t) + de (t) /dt . k1 k2 2 ) The plant transfer function is 1/(s + 100 . The block diagram of the system to be simulated is shown in Figure 28. Let the driving function r(t) be equal to zero and investiga�e the characteristic re sponse to a set of initial conditions. As r(t) is equal to zero, e(t) is equal to -c (t) , the computer setup can be simplified as shown in Figure 29. The control equation of the system written as may be

e(t ) + 2 e(t) = -N sgn F (t) lt.) 0 (4-16)

Amplifiers and #7, used for changing signs of -e and -e , f6 were disconnected for no sign change . All resistances are 100 kfl and capac itances are 1 J.LF. Amplifier #1 with two zener diodes was used to simulate the relay with two possible outputs of +4.5 V ide l and -4. 5 V.

The phase-plane plots of the system with various governing functions are shown in Figure 30 . The re sponse of the system depends upon the governing function F(t ). sponse is and = the system re For = O, k = 1 k , k 0 ki 2 - 1 2 divergent. 0 the system re sonse is F = d k = 1, = or k = -1 k 0 an k 1 ' 2 1 2 oscillatory.

e has a For the response is undesirable for k = o, ki = 1 1 finite value when input is zero. system seems to be satisfactory. For the case k = k = 1 the 1 2 c (t r . - e (t ) ( ) ' ... - F t - Relay Plant j-- . ll

Figure 27. Block diagram of general switching controlled plant .

� --- t ) -- 1 c( de - · - � e(t)_ � - ..., -- kie k2 .. + dt ·· s2+100 ±-N F J.

I I

Figure 28. Block diagram of the system to simulated in Example 4. be

°' \.A) Relay Plant ------I ------1 I I I I I o I I x I I I . I I I I - I l _c I I (t) I I I I 1N747 I I : ' I I -=- ,------J - - - I r------

______· L _I 1 1 , ------' I - -- 1

I ,__..(\A A - I I I I I I I I I � I I · · I · 1 All resistances are 100 k Q

______, ______Capacitances are F F t J 1.0 f.L ( )

Figure 29. Computer setup for simulation of the switching controlled system, Example 4.

°' � ::: :::: :::: k2 -1 ki k2 1

:::: - t - kz o ,·-1... - 1 k_· -z 0

'!:: = ::: � 1 1 k2 - k1 0,

B

) X � � 4 V/c sea.le Y scale B) = = V/cm X scale Y scale 2 Jv. 4. Fi 1lot of t °' tchin :mple \J\ 66

Details may be found in Reference 10. Mathematical analysis of

this system is rather involved so the accuracy of the computed result

was not determined.

· similar switching control system with various nonlinear

parameters can be simulated on this analog computer. The analog

computer is useful in nonlinear system analysis because mathematical

methods may be rather complicated and may not give very accurate

result s due to approximations made in the solutions.

F. Example 5 , Second Degree Differential Fiquations.

The example to be simulated was Volterra's competition equation

with the general form

=ax - R xy (4-1 7) dtdx � = + (4-18) dt� -YY C X1 .. the equilibrium condition is at x = Y/8 and y = a/13 . The phase-plane plot of the system can be obtained from the equation

_ 8x-Y � = i (4-19) x y- 0 dx fo the equations for computer setup are

= 2x (4-20) dtdx - xy (4-21 ) = -5Y + 2xy dt� -p the programming diagram is shown in Figure 31 and the phase lane plot plotter is as obtained from the analog computer, recorded by an XY eq the shown in Figure 32. From the recorded result the uilibrium. of 1

2 :!l 1 >------10

Yo

1 y 2

1 1 xy 10 -xy

Figure Programming diagram for Volterra's equation. 31.

°' ....:> r I

___.,.._ ti:'>_ tI t ' 1- - '' �' �(��t -·-=>' ·:·� � ttt � -:� 'i. \ ' ' . t - ·- : : ·· -1 .. -. _ - . ' . \; \ .• ! "::,i:L �- ' ..,...... ;.._ ..(...... ;. : t 'j '"'-;- r·.�· ..., __, __, '"'· . . ..:... "'" . --��· I; I s I -+ i ,- : l. �� ... .• l : · � �,.I .: , i : - ·--- . ;-· @j t� ·--�··· � 1J I � \

'I I . ·\ t --- l ····--- l- ··--· -,.-·--1· ...... ·--·-· '\ / � ·�---� -·- $ -.1� '·· \f i ' - !: .. , � ?t���:�r "�': ...... ? -� I • '·f n• '!\\''•. ' �·�· --·--·-·--· - •• t -�-... j ;;;;ji:b.'!!it• ., J · . 1 -"" I '·· � -�... I 1 + f ', �, - i'"-<:.:t.- .... ,. -;ctt"-�•+�-+.�---r...... !"i t_: . '"k,

V/in. Seale , 1 )2. competition equation , Example 5. °' Figure Computet' solution for Voltarra1s Ct> system is at x = 2.4, y = 1.9. From hand calc ulation the equilibrium

of the system is at x= 2 . 5 , y = 2.0. The accuracy of the computed

re sult is about 5 percent .

G. Example 6, Narrowband Frequency Modulation .

The equat ion for phase modulation of a sinusoidal signal may

written as be ( ) = ( ) f t cos Wt sin W t · (4-22) c c + f3 m_ (�-23)

where

f ( t ) is the narrowband out put signal. c FM W is the angular frequency of the carrier signal . c Wm is the angular frequency of the modulating signal . (3, is the m� phaseshift of the output . maxi FM for f><<· or fo 0 . 2 radian {[ < (4-24)

From this approximation , a narrowband frequency modulated signal

can be generated by the system whose block diagram is shown in

Figure 33. Diagram of the computer setup is shown in 34. Figure Waveforms of the modulating si gnal, carrier, output of the multiplier and the narrowband output are shown 35. FM in Figure Because the phaseshift, f3,is very smal , the frequency maximum � deviation of the output can not be observed. Also there was some amplitude modulation in the output . The output signal looks more like Integrator Balanced modulator Adder

f(t) +

+

Carrier

90° Phase shi�er

Figure 33. Block diagram for narrowband system of Example 6. FM

� 1600 pF

1Mf2 100 kfl.

Signal output FM 2 2'1fx1 .25x10 t · Em sin

Carrier

E0 sin 27Tx1 0--rt

16o pF

Figure J4. Computer setup for narrowband system of Example 6. FM -...J ... Carri.er

ModulatJ n.r:: signal

Out put from bala:ncecl modula.tol'"

FM Narrowband

V/c scale .

Y scale , C/C

10 35 .. Fi corded re sults , E�pl � 73

low modulation index AM signal than signal . In order to show that FM the output really contains frequency change , the output was fed to a

voltage limiter and a frequency discriminator . The voltage limiter

constrained the amplitude of the signal to ! 4.7 volts, thus eliminating the amplitude modulation . The output from the voltage

· limiter was fed to a frequency discriminator which converted the

frequency deviation to amplitude deviation . The output �f the fre­ "' quency discriminator contained a measurable but small amplitude

change . The result was not recorded.

The computer setup for the voltage limiter and the frequency

discriminator is shown in Figure J6. The voltage transfer function

of the frequency discriminator was similar to an tank circuit with LC a resonant frequency at 0.81 x 103 Hz. - Modulation and demodulation are nonlinear processes, often

requiring multiplication . This is usually accomplished by using devices whose characteristics are nonlinear or linear time-varying.

The analog computer, designed for this research, contains one analog

multiplier and nine operational ampli fiers which can perform a wide

variety of operat.ions. This computer is thus a convenient instrument

to simulate many systems of modulation and demodulation. pF 1N750 I I 130 I kQ kfl 100 I I 100 'i · I I1 130 pF

Voltage limiter Frequency discriminator

Figure 36. Operational amplifiers connected as voltage limiter and frequency discriminator for narrowband frequency modulation detection .

� 75

CHAPI'ER V

CONCLUSION

The purpose of this study has been fulfilled. The vacuum-tube

amplifiers in the Heathkit EC-1 Analog Computer have been replaced

by the integrated circuit operational amplifiers , f..L A741 . All com­ puting fac ilities and control circuits which were contained in the

EC-1 Analog Computer were redesigned to operate with the f.L A?41 . One analog multiplier was included to extend the use of the computer to second degree equations. The computer was tested and used to solve several representative problems . Every unit f"unctioned per­ fectly and the results obtained were satisfactory. Over-all performance of the analog computer is considerably better than the

EC-1 Analog Computer.

Chapter I serves as an introduction to the analog computer in general and specifies the problem of the research.

Chapter II is a study of the operational amplifier, the most essential part in an analog computer. In . this chapter it is shown that general .purpose integrated circuit operational amplifiers are suitable for the use as a de amplifier in an analog computer. As a result , an analog computer can be built much less sophisticated and conventional method of using at muc h lower expense than by the discrete-component amplifiers.

Characteristics of the nonideal operational amplifier and their ga . Chapter II effects upon computed results were investi ted also . 76

discusses which parameters are considered to be important in

selecting an operational amplifier. Manufacturer's technical data

are of'ten sufficient in evaluation of an operational amplifier.

Table 1 shows characteristic s of the f.!A741 operational amplifier

in comparison with three other operational amplifiers. One may

consult these data as an aid in selecting an integrated circ uit

operational amplifier.

- Chapter III contains circuit diagrams and circuit descriptions

of every unit in the analog computer. Design concepts and speci-

fications for each unit are discussed . This chapter answers some

questions that may arise in designing an analog computer.

Chapter IV presents typical examples of linear and nonlinear equations solvable on this computer. By this means some measure of the accuracy of the solution from this computer is obtained. This chapter may also be useful in the application of the analog computer to the solut ion of specific problems .

For a low cost analog computer with component acc uracy of 1 per- cent , one may follow the design in Chapter III, but instead making use of higher precision computing elements , since it has been shown that the precision of this computer is limited by the precision of these com­ puting components. The complexity of the problem solvable on an analog computer is usually limited by the number o� operational ampli­ fiers it contains . The front panel of the EC-1 Analog Computer has allowed space for only nine operational amplifiers and each amplifier can have only two inputs. For a more flexible computer, one should 77

discard the EC-1 case and design a better front panel. The use of

internally connected input and feedback elements would result in a

compact front panel. A removable patch panel allows the preservation

of a computer setup for later reuse and increases the utility of a

computer appreciably. A well-designed front panel layout is very

important in a large analog and hybrid computer.

The author suggests that the present work be extended by

designing a high precision and high performance analog or hybrid

computer, using integrated circuits. It has been shown in this wo�k

that the major error of the operational amplifier is due to drift and

offset current of the operational amplifier. This error can be sup­

pressed by using a low dri:f't and low input offset current operational

amplifier. The operational amplifier with FET input has very low offset current . Although it is more sensitive to temper�ture change , it may be suitable if used in an air conditioned room. The use of a temperature-controlled preamplifier is advisable . The combination of

�A727, temperature-controlled preamplifier, and f.LA741 , general· purpose operational amplifier, was tested and. found to be very satisfactory.

Error due to computing elements such as input and :feedback impedances, potentiometers and initial conditions can be overcome by using high precision components . A higher accuracy panel meter is necessary because the accuracy of potentiometer and initial condition . setting is limited by the accuracy of the panel meter A digital rac of voltmeter is expensive but necessary if an accu y better 78 than 0.1 percent is desired. Otherwise potentiometer and initial condition settings must be done by means of a reference voltage , reference potentiometer and a null-meter, which is less reliable .

Overload indicators and some operating control systems may be necessary. 79

REFERENCES

1. We yrick, R. C. Fundamentals of Analog Computer. Englewood Cliffs , New Jersey: Prentice-Hall , Inc. , T969.

2. Ko rn , G. A. and Korn , T. M. Electronic Analog Computers . Sec ond edition , New York : McGraw- Hill Book Company , Inc. , 1956 .

Johnson 3. , C. L. Analog Computer Techniques. New York : �Graw-Hill Book Company , Inc . , 1956. ·

4. Warfield , J . · N . Electronic Analog Computers . Englewood Cli ffs , New Jersey: Prentice-HalI , Inc ., 1959.

5. Ashley, R. J. Introduction to Analog Computation . New York : John Wiley and Sons , Inc., 1963. 6. Zdenek Nenodal and Bohum.il Mirtes. Analog and Hybrid Comgut ers. Trans . Grew, R. J. M. , London : ILIFFE Books Lt d. , 19B.

7 . Fitchen , F. C. Electronic Integrated Circuits and Systems . New York : Van Nostrand Reinhold Company , 1970 .

8. RCA Linear Integrated Circuits . Harri son , New Jersey: Radio Corporation of America, f 96 7.

Amplifier Handbooi Hopkins , 9. Motchenbacher, C. J. Low Noi se . Minnesota : Honeywell Re searc h Center.

10. Flugge-Lot z, Di scontinuous and Optimal Control. New York : McGIrmgraw-ardHill. Book Company , 1968.

11. Gibson , J. E. Nonlinear Automatic Control. New York : McGraw­ Hill Book Company , Inc • , 1963 .

12 . Van Valkenburg, M. E. Network Analysis. . Second edition . , , . Englewood Cliffs , New Jersey : Prentice-Hall Inc. 1964

Data Book. Second 13. "AN-489" and "AN-490 ," The Microelectronic ola Inc. , 1969. edition. Phoenix , Arizona : Motor

at ional Analog Comp ut er 14. Operat ional for the Heath Educ gan : Heath Company , 1959 . Model ECManual-1 . Ben ton Harbor, Michi

Richmond , California: 15. EASE Analog Comput er Instruction . Beckman Instruments, Inc. , 19Manual56.

/ /

• 80

6 1 . EAI 380 Analog/Hybrid Computing Systems Maintenance Series. We st Long Branch, New Jersey: Electronic Associates Inc., 1 969.

1 7. Schick, L. L. "Linear Circuit Application of Operational Amplifiers," IEEE Spectrum, Vol . 8, No. 4, April , 1 971 .

18 . Hickey, J. "Powering your IC •s," Electronic Engineer, Vol. 30 , No. 4 , April, 1971 . 19. A New High Performance Monolithic Operational Amplifier, J..LA741 . Mountain View, California: Fairchild Semiconductor, 1971 . 81

·- APPENDIX A

TECHNICAL DATA FOR fl-A7 41,

· OPERATIONAL AMPLIFIER 82

•

"Tl :0 fTI -0 c l"T'1 z () -< 0 0 � -0 fTI z tJI NO r------• FREQUENCY COMPr�SilTION kf.(ttrr;cp � l"T'1 • SHORl-CIRCIJIT PlilHLCHON 1 PHYS���;.,�����;IONS 0 • OFFS£[ VOllAG� M'.lll (,J\!'�i>rurv J£0EC E /\NO011 1 ERH�W, 1"1'. ·�.u ·rr.NG>. ·; ,.., :0 • LOW POWrR l'ONSUMl'JINl )> -i • NO LATCH UI' 0 z GENERAL DfSCRll'TION �

,, ...... > 1 smsle s1ilcon cl"'' th� 1 t t.' t'• · . �1 ,• ·1 i ' 1v• • r;lJ.1 r · ·' ·,!r 1".1·':1 ·;,i ,c!i:: i_.i.1�:1' 1 • d'l.51-)f! '" , . � applications High co:1i;:.,.c •:1 , o!I i�,. i r'}.•! ,.J ',. 't, ,r ·1:· h i,� l :•:fo'l(H· . r. J�f.' �r.,: Ai·.:. l 1CJ�a1 " � for use as a voltoki 1. ..,,, r ...:1· · 1;,.·r�.: J, 'f.it n• :. • f'V"�'· :.i··p,�;: .r �t'.'l(,ii'l =;; ': .. " ;. ance 1n integ ·u�<.:r, 1t'm• r. � 1; · 11 1,., . 1 �,.,.,, '"'k ·, � 1 �· .A : l t,h: •! •. ··u.f t1ffc�n1 ;Ti , .. . :0 has the Sclme pin t. r.t1, •.1;, " ' • r 1 . 1, .1J, .. I tor +rl' \1,. •( • •.::1 11, : .. 11,. , , component:; lj "Tl applicdt1on; > ii () :I: ABSOLUTfMAXIMIJM RATINGS I F Supply Voltag� 0 '" I r lnterndlPow,:1 Di:i.:.·ri�t 11n11 rf\ ,·JO TW z D•tfe1ent1at 1npu1 V<' '""-' + 'O V I � ; � I ... ,. Input Vo!tage iN•JIC ··1 I I 'l'• • • W :0 ... , \loildge belw<·-" i:'ts•' �'- "· " i·0.5 v l • • .• ... ••�.. • .•. 2 I r,,··�·'"�(N -i Storage femp,1•t ·ie E�·.::' rri G) Op11rating Tt11µ»r.,',:•c fl ow· )"1"r l• -� lb'(; :0 .. > Lead fernpera!uri_ '',\. J,·! r:�� ti.' 't' �fJO"C OROER PART NO. U587741312 rri-i Output Short t;;ir,qf (1u•.d,,,-. 1\.,, ' l•l•n1te I · - --·- ______, 0 L ()

ii() c: EQUIVALENT CIR\Ulf

· ·t . I• r � NC

�". ' '. ·1 .. � • *

1.- .. I . I ' ' .i .. ; , l Olf;E • �"" J, " � 0- • ; . f .:

• P:.lnarts a patented f'a1rch1fd process. NOTES: .... · "' r-t!airf, a• '-. ,!'ifJ'�C fl':r amb1.a11t !e1Poua1u PS a001e -/5� (1) R1t1ne appltl'!s for \.ii'- iemrer.:Jlt • v _ �·,,.. je:e't' :. Its:. tl':.?rr :!: 11· ;:;:Ji ·r.:t;e equal 'o 'ti� s<.:1,p!1 \"11tato·• (2) For supply volt•gt'!!'. .JVthe a-bsc;.•1.;te rr• -,.t:1'11.1rn ; .. lcm�raiunt .ilpp!.cs lo Jf':C. ·( n10:.e trmperatUre o; 1s•c imbtl!nt (3) Short circwt may be to �. •

A. 49 30M COPYRlf;KT FAIRo..HILD SfMICONDUCTOR !969 • PRINTED IN U.S 04-BR 0024 3025589. 3054161. 3108359, 3117260; OTH(R PATENTS PENDING .... u f''IH:N1S 2981877 101 5048. MANUFACTURED UNDER ONt: or1 •..i OFl! Dr Ti-•L FOLLO\ l'H• BJ

FAIRCHILD LIN EAR . INTEGRATED CIRCUITS µA741

£LICTRICAL CHARACTERISTICS (V5 = ±15 V, TA = 25°C unless otherwise specified)

PARAMETERS (see definitions) CONDITIONS MIN. TYP. MAX. UNITS

Input Offset Voltage Rs :S 10 11.n 1.0 5.0 rnV Input Offset Current 20 200 nA

f, put Bias Current 80 500 nA lnp1Jt Resistance 0.3 2.0 Mn Input Capacitance 1.4 pf Offset Voltage Adjustment Range ±15 mV Large-Signal Voltage Gain RL � 2 kf!, voot = ±lOV 50,000 200,000

O:itput Resistance 75 n Output Short·Circuit Current 25 mA

Supply Current 1 .7 2.8 rnA Power Consumption 50 85 mW

Transient Response (unity gain) V;n = 20 rnV, RL = 2 kn. CL � 100 pF Risetime 0.3 µS Overshoot 5.0 % Siew Rate RL � 2 kf! 0.5 V/µ.S

The following specifications apply for - 55°C s T,., :S +125°C: Input Offset Voltage Rs s 10 tm 1.0 6.0 mV Input Offset Current TA = +12s•c 7.0 200 nA TA = -ss•c 85 500 nA Input Bias Current TA = +125°c 0.03 0. 5 µA TA = -ss•c 0.3 1.5 µA Input Voltage Range :t:l2 ±13 v

Common Mode Rejection Ratio Rs s 10 IC.!! 70 90 dB Supply Voltage Rejection Rcitio Rs s 10 kn 30 150 µV/V 25,000 Large-Signal Voltage Gain RL � 2 kf!, v""1= ±lOV Output Voltage Swing RL � 10 kf! ±12 ±14 v �L � 2 kf! ±10 ± 13 v 2.5 Supply Current TA = +1z5·c 1.5 mA T,., = -55"C 2.0 3.3 mA 45 75 mW Power Consumption TA = +125•c 60 100 mW TA = -5s0c

TYPICAL PERFORMANCE CURVES

OUTPUT VOLTAGE SWING J INPUT COMMON MODE llPEN LOOP VOLTAG E GAIN AS FUNCTION OF VOLTAGE RANGE AS A 1 AS A FUNCTION OF A SUPPLY VOLTAGE FUNCTION OF SUPPLY VOLTAGE SUPPLY VOLTAGE 2 ' -ss·t;-;T i •12S"C I -ttt•r... 1�'t � I \�lka 0_r" . z�c i,...- lZ 1...... - I I [...;i,...- - It 1.-/1I .... /- / � ...v. ' v / ...__-- I I I I ...v v /// l �� / I / v 1I l/ 17 e I/ • I 0 l I ' � • m �o 1 • • 1 m � w

llJl'f'lY Yt'lfA<;(•!V

2 84

A C ������ F IR H I L D UNEAR l NTE RA C C A �-______�___�___G___T_ED IR U lTS_:_� 1 �______7������� TYPICAL PERFORMANCE CURVES 4 =oJ

4 POWER CONSUMPTION 6 5 INPUT BIAS CURRENT INPUT RESISTAr•cE AS A FUNCTION OF AS A FUNCTION OF AS A FUNCTION OF VOLTAGE SUPPLY AMBIENT TEMPERATURE AMBIENT TEMPERATURE Ull IO.O TA v, • t1'ol/ •ZS't V, • tlSV s.o � _J.-+ I l.O / I "' / i // �LO // / \' � � 11.S f 11.l / 1111 '\ .. /v ...... '..,_ � 0 v 0 11.l J -611 -20 20 !Cl -«I -20 20 UX) lCl SUPPl.YYCl.T AG[·tY n:MPERATURl • "C TlMPERATUR£ ··c

7 INPUT OFFSET CURRENT 8 INPUT OFFSET CURRENT 9 POWER CONSUMPl"ION AS A FUNCTION OF AS FUNCTION OF A AS A FUNCTION OF • SUPPLY VOLTAGE AMBIENT TEMPERATURE AMBIENT TEMPERATURE

V1 •t l5V

� )II Mi f--<-+--+--'-ii -+-+---1---1---11--'--� I � " ! "'\ � "'...... _ � 1.) _,,,,...-- i\. ' i � f--+-+--+-"'+-=-.....cl-r---+'-----1--1-f-----I ·,__....._....._....._._-I-+-+-+--1--l 1--+--+--+---t-�--i 01 � IO i � \. a \. '-...._ JC 1---1-+-+-+-+-+--+-+--+--I

e -fQ ·ID IOO 20 lCl -- C SUPR.Y Y

' " > I• 1---+--+-I-+-·�­ \. n 1---+-+-'-¥--+---+-'--+-t- ---1 r-...... �i. 'i-.... - � 20 - B LI \... 16 1--+--J-++- �-�-+-+--j ...... " \...... , "\ 1---t--l+-++--+--t-+-rl---1 ' · � IA � u 1-...... +.1-+-++--�--+--+-H---I 15 '

II 0 -20 IOO !Cl 2S 6S as 1.0 -Ml 11.2 11.> LO S.0 ![MP£RATUlll• ·c LOADR1SISTANCE -K11

INPUT NOISE CURRENT INPUT NOISE VOLTAGE 14 AS A FUNCTION OF BROADBAND NOISE FOR A FUNCTION OF 13 AS FREQUENCY VARIOUS BANDWIDTHS FREQUENCY 15 JilO 1111 l5V I! Vs ·± lSV rl ,_v5 • .! II r,. - 2s·c -J •2S"C ! 111 r A I I I II t ll ! I - Ir• � � �ili�il:l�� :::;=cF+-i-r--i I 10-IOOOH• N! ia22 1: I {�I· I 1 ioi�1-t- -+#"'"+-� ...._ I [I ! -J..li ,!....r - 10-llll<� g� t::=:-:t-,,,,.,...--4==*=� ::!wzi l I M- -,./ ill � I "-I I I II I I v i!I � ,.... I I I 10-WU ,__.. s: I ' II I �wl• ;� I 111• :-17 1-+-+H--+ --+t+t--t- i . ! I IO sm=Fn�� ,_ _ I IIi I II zs 1 11•! ; �lii 11 1 .l II !ID !00 II lac i2'. IOI) IOO IDk 100 Flll.QlECY•HJ SOUllC[ llESISTAllC£ •a FR(Ql,{Nl'.Y •Hl

3 a;

FAIRCHILD LINETYPICALAR PERFINTEGORMRATEDANCE CURCIRCUVESITS µA741

16 OPEN LOOP VOll.ASE GAIN 1 7 OP'EH LOOP PHASE RESPONSE 18 OUTPUT VOLTAGE SWIHG AS A FUNCTION OF AS A FUNCTION OF AS A FUNCTION OF FREQUENCr FREQUENCY FREQUENCY •' v1 I vi ., 1'V •il5V II I vs ·:.l.5Y 25'C 2''C i--... TA · 25 'C T • TA• I A \•Ullo I I ' ' I !'.... i\ II :1: � I 11·11, I � I , I. "" \ II -� I II � •US \ �:1 II II I ill r"\. ,,

- . I ·JI) 0 1 • • [\ I JO 11111 Ii 10c JOO< IM !OM Im Ii Mil I l\m RlQIJflC'l'-llr FllBIUENCY•Hl flll!QUUICY • IHI r--.Ull

19 INPUT RESISJ: CE AND 20 OUTPUT RESISTANCE 21 COMMON MODE REJECTION INPUT CAPAC.IT CE AS A AS A FUNCTION OF RATIO AS A FUNCTION OF FUNCTION OF FREQU ENCY FREQUENCY FREQUENCY

VI • .t1'V vs •! )JV l5"C T • IA •l5°C A -'\. ITII 1 I1 "'1� !\.

- ,I f{. I\. III I � \. "1-- "� i ' I\. 111 ' I/ - � - Ill I �I ' Q.I 0 111 IM DI 11111 JO Im la lllt m L" lllM FllEQUlNCY•Hz FlllQUOICY-'Hz 2 TRANSIENT RESPONSE 3 VOLTAGE FOLLOWER 22 TRANSIENT llfSPONSE TEST CIRCUIT LARGE-SIGNAL PULSE RESPONSE

V •.!lSV S 1--1-�>--l>--l'--l-+- TA • 15'C

2D ...._ 1001P111 � \ 0 J � ) INPUT·! � ·Z l--1---U-1-/l-+-1---+--1---1--+--.t+--+i-+-\-+--'\-+--l---I--+-�� I § ... �...... ·-+- -+--+--+ -- +-.....l _.+-,__ --+\��-+---+-,__ ·6 1--1--1--1--1--1--1--1--1--�-1 � 1--1--1--1---1---1--1--1--1--� •• ....._,__,__,__..__..__..__..__,__.J.....-J • • 20 � c � c n m • .S 1.D 1,5 Z.O Z.5 TIK -115 TIMC•,.S

FREQUENCY CHARACTERISTICS FREQUENCY CHARACTERISTICS 24 AS A FUNCTION OF A fUNCllON OF VOLTAGE OFFSET 2; AS AMBIENT TEMPERATURE SUPPLY VOITAGE NULL CIRCUIT

· .. ._, _.__..__,__.__2'I__._ _ _.___.___.___,_�C 1111 MD y -«! ·211 !lJUUATUal•'C SUPP\.Y �-1V

4 86

APPENDIX B

TECHNICAL DATA FOR AD530 ,

ANALOG MULTIPLIER 87

PRELIMINARY

GENERAL DESCRIPTION CHARACTERIZATION

ihe Analog Dfv1ces AD530J and A05JOK perlorrn the functions of multiplying, divid·

ing, squa• mg, dnJ <.;qua1e rootin� b,r comb111109 d difforential input transconductance DATA mult1plyin1 ell'mE:'nt, nn opt>rationcJI amplifier, and a unH reguJnted current sourc on a t>ssentia1ly �If sinole cn1:' ot s;lwon fhe devices .m� cr)ntainod, requiring only four performed tr1mmmg p1:it.,nt1orn>!t11'1ri. T"e AD530J and AD530K feature AD530J high 11ccurac:v, .;xcellent temper ature · tabtlity ..il'l

ABSOLUTE MAXIMUM RATI NGS MULTIPLIER

Supply Voll;Jgp [Vsl ±18V DIVIDER Powe� 01ss.p.�t.on 500mW SQUAR E ROOTER XYZ Voit¥jes :tVs SQUARER Storage T ,m1 ">er..iture Range 65°C to + 1 50°C QpP,r

ELECTRICAL CHARACTE RISTICS IV� "' U5V, RL .? 2KH, TA '"' 25QC)

Parametttr AD530J AD530K Uniu

MIN TYP MAX MI N TYP MAX

Multiplier Characterimcs

Out pvt F ..inctwn XY:10 XY/10 fotal Accuracy O.B 2 0.5 % (extemilll•; tr1m111ed) Scale Factor Adju5ldble Adjustable

Diwider Charactariitu:.s

Output Fvnctoor-. lfll // lOZ/X Total Accurn•:v (ex tern.sllv trnf?me(l / x o.i 0.3 % x - l 0 l.5 % Scale Factor /1-d111>table Adjustable

Genera! Characteristtcs

Linearity

X fnp� (X 20V 11 p •o a •0.5 Y" '10Vrlc) t0.2 Y Input(Y 20V pp •0.3 % X � � 10Vdc) Feedthrough (externally trimmed) 0, 100 mV p-p X � Y : 20V o-p, 30 200 30 50Hz f � 30 100 mV p-p Y � 0, X � 20V p p 30 150 f - 50Hz 0 Output Offset 0 mV (ad1ustable w zero) (12·70)

Thrs information is of a preliminary nature and is subject to changewithout notice. 88

AD530 TRIMMING PROCEDURES

A. The suggested procedure for trimming the AD530 for multiplier use is as follows:

1. With X;0 = Y;0 = 0 volts, adjust Z0 for zero output. 2. With Y;0 = 20 volts p-p (at f = 50Hz) and X;n = 0 volts, adjust X0 for minimum feedthrough. 3. With X;n = 20 volts p-p (at f = 50Hz) and Y;n = 0 volts, adjust Y 0 for TO 100 PIN CON FIGURATIC"'N minimum feedthrough. 4. Readjust Z0 • if necessary. 5. With Xin = 10 volts de and Y;0 = 20 volts p-p (at f = 501-fz),adjust Gain for Output = Y;0. NOTE 1: For best accuracy over limited voltage ranges (e.g. ±SV), ga in and feed­ through adjustments shou ld be optimized with the i11puts in the desi red range. When so optimized, the error may be greater over the specified (± TOV) range. However, linearity is considerably hetter over sm aller ranges of input.

B. The suggested procedure for trimming the AD530 for use- as a ciivider is as follows: · 1. Set all pots at mid -sca le. 2. With Z;n = 0, trim Z0 to hold the output constant, as X;0 is varied from r v- -10V th ough -1r V. 3. With Z;0 = 0, t i m Y 0 for zero, X in = -10V. Top View . 4. With 2;0 = X;0 and/or Z;n = - X ;0 , trim X0 for the minirrum worst-case vari­ ation as X in is varied from -1OV to - 1 V. 5. Repeat steps 2 and 3 if step 4 rf'quired a large initial adjustment.

6. With Z;n = X in and/or Z ;0 = - X in , trim the scale factor for the closest average approach to ±10V output as X;., is varied from - 10V to -J.OV .

APPLICATIONS OF TH E AD530

------0 V "15V MULTIPLIER DIVIDER =

(OV T0 -10V)

6 Xin 4 Xin AD530

Z;,, 3 yin 5 GAIN] 5 5k

7500!"2 V -15V = -= '="

V +15V SQUARER SQUARE ROOTER =

7 OUTPUT = 6 OUTPUT = X; AD530 4 x; 3 10- Z;n (OV T0 +10V) 8

V -15V = ':'" V = -15V 89

ELECTRICAL CHARACTE RISTICS (Continued)

Parameter A0530J AD530K Units

MIN TYP MAX MIN TYP MAX

Dynamic Response

Small Signal 1.0 1.0 MHz (-3d8point)

Full Power 750 750 KHz Slew Rate 45 45 V/µsec 1% Amplitude Error 75 75 KHz 1% Vector Error 5 5 KHz (0.5° phasesh ift) Settling Time (to 2% of µsec final va lue)(± 10V pulse) Overload Recovery (to µsec 2% of final value) Output Noise 5Hz to lOKHz 0.6 0.6 mV rms

5Hz to 5MHz 3 3 mV rms Input Resistance

X Input 10 10 Mn Y Input 6 6 Mn Z Input 36 36 Kn Input Bias Current

X Input, Y Input 2 2 µA Z Input 25 25 µA Power S1J pply Variation Multiplier Accuracy 0.2 0.2 %/% Output Offset 70 70 MVN Sca le Factor 0.1 0. 1 %/% Quiescent Cu rrent 3.5 6.0 3.5 6.0 mA

;;;ii: 0°C to ELECTRICAL CHARACTERISTICS (Vs = ±15V, R L 2Kn, TA = +70°C)

Multiplier Characteristics

Total Accuracy 3 2 % (externally trimmed )

Genera I Characteristics

Average Temperature Coefficient of Accuracy 0.06 0.06 %/°C of Output Offset ±0.2 ±0.2 - mV/°C of Scale Factor 0.04 0.04 %/°C Input Bias Current X Input, Y Input 0.6 5 0.6 5 µA Z Input 1.0 25 1.0 25 µA Input Voltage K, Y, Z for rated accuracy ±10 ±10 v Output Voltage Sw ing Rt ;;;;i.: 2Kn, ±10 ±10 v CL � 1000pF