THE UNIVERSITY OF MICHIGAN INDUSTRY PROGRAM OF THE COLLEGE OF ENGINEERING AN ANALYSIS OF CERTAIN ERRORS IN ELECTRONIC DIFFERENTIAL ANALYZERS by Paul C. Dow, Jr. July, 1957 IP-231

ACKNOWLEDGEMENTS The author wishes to express his sincere appreciation to the chairman of his doctoral committee, Professor R. M. Howe, for his continued advice and counsel during this investigation. He is also grateful to the other members of the committee, Dr. E. O. Gilbert and Professors W. Kaplan, L. L. Rauch, and N. R. Scott, for their guidance and suggestions. The author is indebted to the United States Air Force for providing him with the opportunity to pursue his graduate study and in particular to Dr. G. R. Graetzer and Professor H. C. Larsen for their cooperation and encouragement while the author was associated with the Air Force Institute of Technology. Thanks are also due the University of Michigan Industry Program of the College of Engineering for printing the dissertation. A special debt of gratitude is due the author's wife for her encouragement and for her assistance in the preparation of the manuscript. ii

TABLE OF CONTENTS Page ACKNOWLEDGEMENTS................................................ i i LIST OF TABLES.................................................. LIST OF ILLUSTRATIONS. vi LIST OF SYMBOLS.. ix I. INTRODUCTION.1............................................. II. THEORY OF THE DIFFERENTIAL ANALYZER....................... 5 A. Amplifiers Used in Computers.......................... 5 B. Tile Amplifier Used as a Summer. 8 C. The Amplifier Used as an Integrator.. 10 D. Solution of Linear Differential Equations.. 14 III. ERRORS IN SUMMERS AND INTEGRATORS......................... 16 A. Error Analysis of a Summer.......................... 16 B. Error Analysis of an Integrator: Initial Conditions............................... 18 C. Error Analysis of an Integrator: Transfer Function............................... 21 D. Summary............................................... 27 IV. ERRORS IN THE SOLUTIONS OF EQUATIONS...................... 30 A. Solution of Linear Differential Equations............. 30 B. Simplification of the Exact Machine Equation.......... 35 C. Evaluation of the Extraneous Roots................... 37 D. Comparison of the Exact and Approximate Equations.. 42 E,. Simplification of the Computer Set-up................. 45 F. Rules for Minimizing the Dynamic Errors............... 47 G. Solution of Simultaneous Linear Differential Equations........................................ 49 H. Approximate Solution for Simultaneous Linear Differential Equations........................... 52 I. Extraneous Roots for Simultaneous Equations........... 57 J. General Computer Set-up.58 K. Effect of Dielectric Absorption on Solutions of Equations........................................60 L. Simultaneous Equations with Dielectric Absorption..... 64 M. Effect of Grid Current and Amplifier Unbalance. 65 N. Effect of Initial Condition Errors on Solutions.... 70 O. Errors Due to Inaccuracy in Components........ 71 P. Using the Computer to Analyze Its Own Errors..... 75 iii

TABLE OF CONTENTS (CONT'D) Page V. CONCLUSIONS AND RECOMMENDATIONS............................ 77 A. Conclusions........................................... 77 B. Re-ommendations....................................... 79 APPENDIX I. EFFECTS OF DIFFERENT AMPLIFIER CHARACTERISTICS AND OF STRAY CAPACITANCE. 81 A. Amplifier Characteristics........................ 81 B. Stray Capacitance................................ 86 APPENDIX II. DIELECTRIC ABSORPTION OF CAPACITORS............... 93 A. Theoretical Considerations....................... 93 B. Experimental Investigations...................... 95 C. Measurement of Charging and Discharging Currents.................................... 96 D. Capacitor Model Based on Experimental Results.... 104 E. Evaluation of the Complex Capacitance............ 107 F. Discussion of the Capacitor Voltage Recovery..... 110 G. Measurement of Capacitor Voltage Recovery........ 115 H. Comparison of Results with Theory................ 116 APPENDIX III. APPLICATIONS TO TYPICAL COMPUTER SET-UPS......... 122 A. Simple Harmonic Motion........................... 122 B. Example of a Servomechanism...................... 125 C. Equations of Motion of an Aircraft............... 128 APPENDIX IV. EXPERIMENTAL VERIFICATION OF THEORY............... 133 A. Nature of the Experiments........................ 133 B. Discussion of Results............................ 137 BIBLIOGRAPHY.................................................... 143 iv

LIST OF TABLES Table Page I Roots of the Equations................................ 44 II Experimental Values for Capacitor Model............... 106 III Capacitor Parameters by Two Methods.. i................ 121

LIST OF ILLUSTRATIONS Figure Page 2.1 Frequency Response of Open Loop Amplifier............. 7 2.2 Amplifier Connected as a Summer.................... 8 2.3 Amplifier Connected as an Integrator.................. 10 2.4 Computer Set-up for Linear Differential Equation...... 15 3.1 Integrator Output Fractional Error for Step Function Input..................................... 24 3.2 Integrator Error Due to Dielectric Absorption for Step Function Input.............................25 3.3 Integrator Phase Shift Without Dielectric Absorption.. 28 3.4 Integrator Phase Shift Due Only to Dielectric Absorption........................................ 28 4.1 Computer Set-up for Linear Differential Equations..... 32 4.2 Computer Set-up to Solve Simultaneous Equations....... 51 4.3 Summer and Integrator Inputs to Represent Grid Current and Unbalance.............................. 68 4.4 Computer Set-up for Solution of Simple Harmonic Motion, Including Effect of Grid Current and Unbalance..... 69 4.5 Computer Set-up for Differential Equation......... 7.. 73 Al.1 Open Loop Amplifier Frequency Response................ 82 A1.2 Amplifier Connections Including Stray Capacitance to Ground........................................ 89 A2.1 Theoretical Model for a Capacitor..................... 95 A2.2 Circuit for Current Measurements...................... 7 vi

LIST OF ILLUSTRATIONS (CO0NT'D) Figure Page A2.3 Charge and Discharge Currents for a Type I Capacitor..................................... 100 A2.4 Comparison of Discharge Currents for Type I and Type II Capacitors................. 101 A2.5 Discharge Current Following Different Intervals of Charge..................... 102 A2.6 Model of Absorptive Portion of Capacitor.............. 105 A2.7 Model of a Capacitor Based on Experimental Results.... 106 A2.8 Comparison of the Experimentally Observed Discharge Current with That of the Approximate Model.......... 108 A2.9 Currents Flowing in Capacitor Model During Voltage Recovery........................................ 111 A2.10 Comparison Between Discharge Current and Rate of Voltage Recovery.................................. 117 A2.11 Phase Angle of the Complex Capacitance vs. Frequency.......................................... 120 A3.1 Computer Set-up for Simple Harmonic Motion............ 122 A3.2 Computer Set up for Servomechanism.................... 126 A3.3 Computer Set-up to Solve the Equations of Aircraft Motion.......................................... 130 A4.1 Frequency Response of the Open Loop Amplifiers (Sterling Computer)......................... 134 vii

LIST OF ILLUSTRATIONS (CONT'D) Figure Page A4.2 Frequency Response of the Open Loop Amplifiers (REAC)................................. 136 A4.3 Phase Angle of Complex Capacitance for REAC Capacitor.......................................... 138 A4.4 Damping Ratio Introduced by the Computer, Configuration I. (Sterling Computer)........................ 139 A4.5 Damping Ratio Introduced by the Computer, Configuration II. (Sterling Computer)....................... 140 A4.6 Damping Ratio Introduced by the Computer, Configuration I. (REAC)............ 141 *..

LIST OF SYMBOLS C Amplifier feedback capacitor. CDk Capacitor used to represent dielectric absorption. Co Zero frequency capacitance. C Complex capacitance. C~ Real part of complex capacitance. ~C Negative of imaginary part of complex capacitance. Coo Infinite frequency capacitance. D(s) Function representing absorptive properties of capacitor. e; Error in the i-th root of the characteristic equation. & Operational amplifier gain, open loop. Go Amplifier gain at zero frequency. i Amplifier grid current. IDk Capacitor dielectric absorption current. Probable error in resistor or capacitor X. RDk Resistor used to represent dielectric absorption. Rf Amplifier feedback resistor. Ri Amplifier input resistor. RL Capacitor leakage resistance. Si Roots of the given equation. S' Extraneous roots of the machine equation. TA Time constant of amplifier (Appendix I). Tr Time constant of amplifier (Appendix I). IDT Relaxation time of capacitor dielectric. Time constant of open loop amplifier. ix

Timre constant of a summer. Tsj Time constant of j-th summer. T. Large time constant of an integrator. T.j Large time constant of j-th integrator. Small time constant of an integrator. V Amplifier grid voltage. tV Amplifier unbalance voltage referred to input. Vi Summer or integrator input voltage. VO Amplifier output voltage. Vaec Capacitor recovery voltage. Empirical constant, a measure of the distribution of dielectric relaxation times. O (o) Phase angle of complex capacitance. A parameter which is 1 or 0, depending on amplifier connections. E Complex dielectric constant. I6 Real part of complex dielectric constant. E Negative of imaginary part of complex dielectric constant. F. nInfinite frequency dielectric constant. Io Zero frequency dielectric constant. Damluping ratio introduced by computer. tO Relaxation time of dielectric material. Frequency at which open loop amplifier has gain of unity. (~O~~~~~~~~

I. INTRODUCTION The present day analog computer or differential analyzer may be said to have had its beginning, at least in theory, with the suggestion by Lord Kelvin [1] in 1876 that mechanical integrators could be used to solve differential equations by employing the now familiar device of forcing the inputs at various points to be related to the output. It was not for many years, however, that a practical application of this method was possible. One of the first successful mechanical differential analyzers was described by Bush [2] in 1931, and one of the most elaborate by Bush and Caldwell [3] in 1945. Meanwhile the electronic art was progressing to the point where the solution of differential equations could be performed electronically. The operational amplifier was first described by Ragazzini, Randall, and Russell [4] in 1947. This was followed rapidly by additional papers on the design of analog computers [5,6,7] and their use in engineering problems [8,9]. Further development was evidenced by the description in 1949 of a repetitive computer by Macnee[10]. In 1950 McDonald [11] described the use of analog computers in servo problems. Still more recently several textbooks have appeared which cover quite comprehensively the design, theory, and application of analog computers [12,13,14,15]. It is not the purpose of this dissertation to discuss the design or applications of analog computers. Rather it is the intention of the author to present an analysis of certain errors existent in electronic analog computers, in particular, those errors occurring in -1 -

-2 -electronic differential analyzers when used to solve ordinary linear differential equations with constant coefficients. Certain difficulties present themselves in the application of analog computers to -those areas where great accuracy is required. Unlike the digital computer in which higher accuracy can be obtained by essentially building a larger machine, the analog computer can achieve higher accuracy only by using better quality, more precise, and far more expensive equipment. If the user of an analog computer knows the problem he wishes to solve to within a certain accuracy (that is, the parameters in the mathematical model are known to within this accuracy) and he wishes a solution of comparable accuracy, it should be possible to determine the accuracy required of the computer. For this information to be of use it must be possible to interpret it in terms of the quality and characteristics required in the various components of the computer. In this thesis the author has attempted to show how this can be done. It should be mentioned here that in some problems the accuracy required in the components of the computer may be much higher than is apparently indicated by the accuracy to which the problem parameters are known. For example, if the equation to be solved is that of simple harmonic motion, the introduction by the computer of even a small amount of damping will give rise to a solution which is markedly different from that desired. Again, in some boundary value problems, the solution depends critically on certain boundary conditions. Very small computer errors in these boundary conditions may result in a solution which differs drastically from the correct solution. These situations are considered in the following chapters.

-3 -The problem of errors in analog computers has been studied by others. Raymond [16] in 1950 and Macnee [17] in 1952 reported on methods of determining certain errors in the solutions of differential equations solved by means of particular types of coniputer set-ups. Miller and Murray [16] have presented a mathematical analysis of errors, without considering their source, with a view toward using the computer to evaluate.(and presumably to correct for) its own errors. Cahn [19] discusses the sources of certain errors and methods of measuring them. It is hoped that the analysis which follows provides a logical and useful extension of the previously mentioned work. Chapter II outlines the basic theory which applies to operational amplifiers used in computers, and discusses the characteristics of the type amplifier considered in the rest of the analysis. Then the transfer functions of summers and integrators are developed. Chapter III is concerned with the errors in individual swmmners and integrators; in particular the effect of dielectric absorption on the initial conditions of integrators and on the dynamic response of integrators is studied. In Chapter IV the effects of these and other errors are analyzed when the integrators and summers are connected together to solve linear differential equations. It is shown that the coefficients of the equations actually being solved by the machine can be expressed in terms of the coefficients of the given equations and -the frequency response characteristics of the amplifier. In addition, a method is developed for determining the errors in the roots of the characteristic equation resulting from the effect of the dielectric properties of the integrator capacitors. In this chapter the effect of amplifier unbalance

-4 -and grid current on the solutions of equations is considered, and finally the precision required in the resistors and capacitors is determined. Chapter V presents the conclusions and reconmmendations. Appendix I shows how the method of analysis can be applied to conmputers using amplifiers having frequency response characteristics different from those assumed in Chapter I, and also indicates the modifications needed to account for the effect of stray capacitance appearing in the computer circuits. Appendix II discusses the nature of dielectric absorption and reports on experiments which were undertaken to develop a model which would be useful and susceptible to the error analysis of Chapters III and IV. Appendix III illustrates the application of the methods presented in the text to typical computer set-ups, and Appendix IV gives the results of experiments performed to verify the theory.

II. THEORY OF THE DIFFERENTIAL ANALYZER One of the fundamental components of the electronic analog computer is the feedback amplifier, used for summing and integrating. As a necessary background to what follows, the theory and operation of these amplifiers will be analyzed in some detail. A. Amplifiers Used in Computers In the region of linear operation of the amplifier, the output voltage for sinusoidal inputs is equal to the input voltage, shifted in phase and changed in amplitude, the phase shift and amplitude change being functions of frequency only. For optimum operation of the amplifier, the phase shift would be zero and the amplitude increase infinite at all frequencies. This, however, is physically impossible, and although the amplifier gain can be kept very high (and the phase shift very small) up to quite high frequencies, the amplitude must ultimately decrease and the phase shift increase as higher and higher frequencies are applied. As a result of the large amounts of negative feedback utilized around these high gain amplifiers, particular attention must be given to the stability of the amplifier with feedback. This is achieved by the usual Bode [20] method of analysis, or by similar methods. Satisfactory results are obtained if the amplifier is totally equalized, and has a log amplitude vs. frequency characteristic with a slope of -6 db per octave over a wide range of frequencies on either side of the frequency corresponding to unity gain. In the remainder of this dissertation, it will be assumed that the open loop amplifier characteristic is such as described above, -5 -

-6 -having high gain of i% up to a frequency of TO radians per second, and then decreasing at the rate of 6 db per octave for all higher frequencies. The frequency corresponding to unity gain (zero decibels) is WJO radians per second. The phase shift is uniquely determined by the amplitude vs. frequency characteristic, and is as shown in Figure 2.1. It should be noted that the amplifier introduces a 1800 phase shift above and beyond that shown in Figure 2.1. This will be indicated in what follows by writing the gain of the amplifier as - G, where & G (j )) and includes both the amplitude and phase shift characteristics shown in Figure 2.1. That is =e jf IG-w - jr IGle G-c o = -tan (too) (2.1) By definition, (GI1 I when C = CO, and if o >> I coTo G.0 (2.2) If the Laplace transform variable S is used in place of j c;, the transfer function of the amplifier becomes 1 Most amplifiers will have frequency characteristics which are more complicated. In many cases, however, they can still be closely approximated by the characteristics described here. See Appendix I for a discussion of the effects of amplifiers having othersi frequency response characteristics.

-7 -20 log Go D~ I\ c,-45 "0 0 I/To Co log w V1 -90 Figure 2.1 Frequency Response of Open Loop Amplifier

-8 -G (Af) (2.5) (2.3) B. The Amplifier Used as a Sumnner The amplifier connected as a suimer is shown in Figure 2.2. Rf RI, R2 V 2 Ri V._ O I Figure 2.2 Amplifier Connected as a Summer (Ground connections omitted for clarity) The design of the high gain amplifier is such that the input current to the grid is very small and will be neglected for the present. The output voltage VO is related to the grid voltage V as discussed above, that is V0 = - G-(s)V. 1 The effect of grid current and amplifier drift is considered in Chapter IV.

-9 -Writing the nodal current equation at point P, and assuming zero grid current, gives, t Go +. ' + L if also L1 = v- V V -V V - \O f Rf Combining these equations to eliminate the current terms and the grid voltage V, gives Ri -2 Rjf R I (5)( Z R fi) (2.4) Substituting for G(S) its value given by Equation (2.3) yields RfV RR + T.s. R.i (2.5) If Rt ) I, Equation (2.5) becomes rs R (2.6) (2.6) where Vo (s5) [Vo(t)]... / F,,,, Pr o]

-10 -& o( Ri ) I 2 Ri) (2.7) C. The Amplifier Used as an Integrator The amplifier connected as an integrator is shown in Figure 2.3. D R ic R2 V The presence of stray capacitance in the circuit of Figure 2.2 will V2 o i2 ii Figure 2.3 Amplifier Connected as an Integrator

-11 -The resistor R L connected in parallel to the feedback capacitor C represents the equivalent leakage resistance of the capacitor. The resistor R and capacitor CD represent an approximation to the absorptive property of the capacitor C. It is shown in Appendix II that several such series resistor-capacitor combinations Riust be considered in parallel with capacitor C for an accurate model of the capacitor, but for the time being only one will be used. The effect of additional resistor-capacitor combinations will be considered later. The following relationships can be written as before: Vo = () l + [z t- + i = (c. ' (D Vi -v Ri LL = RE (v-vo) RDC t D = CD dt(V -) Taking the Laplace transform and eliminating the current and grid voltage teris results in ____ _ L Rc, s _____ = -Z \R —S + VO (o) L [ 0+ ' (s) RCDslI (2.8)

-12 -where Vo(O) = Vo(t) when t -0 t( (0) -when t - o If G (s) from Equation (2.3) is substituted in Equation (2.8) and if Ga >> I;: RiC the Laplace transform of the output voltage becomes (2.9) Equation (2.9) is finally written as ___ Z v; __ T7 Vo (o) Vo(s) (hT)D(5) 1 -(r,~+,)(T5, /()17s *1) T, iD (o) ~+,.1T 1 D + c(S+ + )(rs+ )[(+ (2.10) where T s5 + CD/C

-13 --T, RLC + Z RC To I T~ = -rO- = 0(2.11) If more than one resistor-capacitor combination is required to represent the dielectric absorption, an analysis similar to that above shows that Equation (2.10) is modified so that if CDkCoj << C for all J and k D(s) C Dk + I ' ~ (2.12) and the last term on the right hand side becomes k c( st)( st)[s k(I C )] (2.13) where the subscript k denotes the k -th resistor-capacitor combination. If T. approaches infinity, and Tz,,and CD/C approach zero, Equations (2.6) and (2.10) become, in the limit, Rf V (S) Vocs)= - _2_. +Vi(5) V (o) (2.14) 1 These results are also modified by the presence of stray capacitance as discussed in Appendix I.

-14 -or V0 (t) = -Z RF V (t) and J (2.15) which are the relationships for perfect summers and integrators. Since the summers and integrators used in an analog computer are not perfect, certain errors result from these imperfections. In the following chapter these errors will be analyzed in detail and their effect on the solutions of problems solved by the computer will be investigated. D. Solution of Linear Differential Equations Before proceeding with the analysis, however, it will be helpful to indicate a method of utilizing these summers and integrators in the solution of differential equations. This subject is covered exhaustively in the literature [12,13,14,15] and is included here only for convenient reference. The mn -th order linear differential equation with constant coefficients which will be solved is The int egrator and summer co tions sho (2.16) The integrator and summer connections shown in Figure 2.4 would solve this equation if ideal equipment were used. Resistances are in megohms, capacitances in microfarads. The computer set-up shown would be correct only if Equation (2.16) is of even order and the coefficients all positive. The general case is considered in Chapter IV.

I I am-1 a, I I I I I I I, I I, I I I I,I I I I I I f (t), figure2 C pter -S- -- E a m-2 ao Figure 2.4 Computer Set-up for Linear Differential Equation

III. ERRORS IN SUMMERS AND INTEGRATORS It was shown in Chapter II that the output of the summers and integrators differs from the ideal. The purpose of this chapter is to analyze these differences in detail in order to evaluate the errors in the output and to determine the effect of various parameters on these errors. A. Error Analysis of a Summer It can be seen from Equation (2.4) for the summer output voltage that the summer is in error because of the term containing &G(). A finite value of &(S) causes an error in amplitude and a phase shift for sinusoidal inputs. The static or DC output is l I)Vi ~0, R= The desired output is Rf VR ODL.. (3.2) and the error is E = VoD- Vo. Then, if i + (P4- Z e (( (5.4)

-17 -the fractional error is VOD D0 ( Ri (3 5) For a given allowable fractionable error, the required gain can be determined from {+Z Rf (t+ Ri E/ OD, (3.6) In order to evaluate the dynamic behavior of the summer Equation (2.6) is used. Here the static error has been assumed negligible and z_ Re oR; V~ (s), (S) R, (3.7) That is, the summer acts as a simple lag network with time constant Tq If the input to the summer is a step function of amplitude A, that is i Vj= O t< o = A t > then Vo = -A ( -e (3.8) If the input to the summer is sinusoidal, R V = A cos wt the steady state solution is + V0, =A cos(ot - tan cT,). (39) 4I+Tw2T~i-3

-18 -Thus the summer produces a small decrease in amplitude and a phase shift. B. Error Analysis of an Integrator: Initial Conditions It can be seen from Equation (2.10) that the output of an integrator differs from the ideal in several respects. These errors result from the dielectric absorption of the capacitor, from the time constant T (due to capacitor leakage and non-infinite static gain of the amplifier), and froii the time constant T2 (due to the amplifier gain vs. frequency characteristics). The errors affect both the initial conditions set on the integrator and the transfer function itself. The initial conditions will -be discussed in this section. The integrator output resulting from the initial conditions is Vo (t)= Vo(o)e (.10) (3.10) that is, the initial conditions decay exponentially with time constant T1. In addition to the initial conditions set into the integrator, the term in Equation (2.10) due to the initial absorption current, iD (o), produces an effective change in the initial conditions. The Laplace transform of the output due to this ternm is repeated below. ) i D(o) V ) c(TS+l)(TS4n)[5+ #jI * - )J2 (-1

-19 -Referring to Figure 2.3, the value of iD (O) depends on the length of time the initial conditions are applied to C before the integration is started. If this time is very small compared to the dielectric relaxation time, TD, then jD(O) V( -V R If, on the other hand, the initial conditions are applied for a longer time, iD (0) will become less as CD is charged to the initial condibion voltage Vo(0. The inverse transform of Equation (3.11) gives =lD(O) [ t/TFF -D ft/r7 - L-D.2(3.12) if CTT T C (3.13) The last term in Equation (3.12) is small in magnitude and decays rapidly, hence is of little consequence. Also, since Tr 4< T, Equation (3.12) can be written approximately as T(t)= eC [ -t et/ (.14) If iD(O) = - V()/RD then Equation (3.14) becomes V( (O) CD - [ - e-/r] et/' As shown in Appendix II, Equation (A2.36), except for the factor t-/T, Equation (3.15) is just the voltage recovery experienced by the capacitor when charged to -V0 (o) volts for a time long

-20 -compared to h, then briefly discharged, and left open-circuited. (In this case it is actually a voltage decrease from the initial condition voltage.) Equation (3.15) is rewritten as Vo(t) VP-V)e-*/ ' V(t) - - Rec(t) e~tk (3.16) where Vec (t) is the capacitor recovery- voltage. The nature of VRc (t) will be as described above if CD is initially uncharged. However, if the initial conditions are set in for a longer time or conversely, if CD is still partly charged in a negative sense from a previous operation, V/ec (t) will have a different behavior. Finally, Equations (3.10) and (3.16) are combined to give the total integrator output due to the initial conditions: (t) = () Re (t) (317) From Equation (3.17) it is clearly seen that the effect of dielectric absorption is to alter the effective initial conditions with time. This effect would be of particular significance in high speed repetitive computers where high accuracy is required and the initial conditions must be reset in minimum time. It would also be important for boundary value problems in which the solution depends critically on the initial, or boundary, conditions. If the dielectric absorption of the capacitor is represented by several resistor-capacitor combinations as described in Appendix II, the reasoning used above still applies. It is only necessary to add to Equation (3.14) as many additional terms as there are additional resistor-capacitor combinations. In Equations (3.16) and (3.17),

-21 -VRe (C) becomes the voltage recovery from the array representing the dielectric absorption. C. Error Analysis of an Integrator: Transfer Function The transfer function of an integrator from Equation (2.10) is repeated below: V; (5) ~(s)T- I RC D (s) where _1 j~ ~ CDIC D(s) C De s (3.1-9) As was done with the summer, the errors in the integrator will be analyzed by considering the response to a step function and to a sinusoidal function. 1. Step Function Response z =o t<o RiC A t>O Then V5 A D(s) Vo (5) s- Is.T.sI) and Vo (t) - ^A; ((I5-. e (et/ Ft~~~~AX~ COC.r~~~~~~ r () (5.2o) +AT, -Al<(' -e

-22 -if CDk<< I< C In evaluating the error, the first two terms in Equation (3.20) will be considered, and then the error due to the last term, resulting from the absorptive properties of the capacitor, will be discussed. The first term can be expanded in an infinite series and the second term, since T, is very small, can be considered constant equal to -ATZ after the first fraction of a second, giving V0(t)=A(t- ) -AT2. (3.21) Since the desired output is V =At the error is given by E = VV, -Vo E oD (3.22) and the fractional error is vD 2T t t >> T2. (3.23)

-23 -The fractional error times T is plotted in Figure 3.1 as a function of time with T. T as a parameter. It is seen that the maximum permissible error places a limitation on the computing time. However, in most applications an integrator would not continuously be required to integrate a step function. Furthermore, except for very small amplitude inputs, the integrating amplifier would reach saturation before the error from Equation (3.23) became very large. The last termn in Equation (3.20) is the output due to the effects of dielectric absorption. V( AZI CDk t (I et/T ). From Equation (A2.36) in Appendix II, the voltage recovery from an initial charge of A volts is v,,,ec)- T', C ( / e ) (3.24) (In the text, C refers to the infinite frequency capacitance which is given the symbol Coo in Appendix II.) The integral of the voltage recovery is Vo(t) + AttZ c (3.25) Since from Appendix II, Equation (A2.19),

1000 T, T2= 10 100 '0.1 L,.0I W 10 - 0.J0 z 0 o 1.0 < ~ —0.01,~ 0 0.01 0.01 0.1 1.0 10 100 1000 COMPUTING TIME (seconds) Figure 3.1 Integrator Output Fractional Error for Step Function Input

-25 -Co Cok C (3.26) one has finally V0 (t) f eC(f)dt +At(- 1) (3527) The error due to the last term is eliminated if C is regarded as the DC capacitance Co. The fractional error due to the remaining terrm is E I ~D t e (3.28) where VRec (t) is the voltage recovery per unit applied voltage. Figure 3.2 is a plot of Equation (3.28) for the Type I capacitor of Appendix II. 10 0 z <c r UL W O. I 1.0 10 100 I000 I0,000 COMPUTING TIME (Secs) Figure 3.2 Integrator Error Due to Dielectric Absorption for Step Function Input

-26 -2. Response -to Sinusoidal Function i- V(t) A cos Wt Then AT, sD(s) V~ )- (S+( w T s+/)(Ts +)' (5.29) Here only the steady state solution will be considered. Also, for the moment, only one resistor capacitor combination will be considered to represent the dielectric absorption. Under these conditions the inverse transform of Equation (3.29) becomes: V0/o (t) = A +1 cos(St +f) (33.o) where Y t a'tn tat. -tan wTz. If (( J < and tC ~< I> the term in brackets c. becomes approximately m and the phase angle can be written approximately as C (Eq i+..)) cTt aZ Then Equation (3.30) can be rewritten as

-27 -V(t)jA sin wt + C cT; w TC C, L, ct, CGWT 2 C(IWT " ) T' (3.31) If more than one resistor-capacitor combination is required to represent -the dielectric absorption, the last term in the bracket in Equation (3.31) becomes IoCDkTtk k k C( wt rT ) It is shown in Appendix II, Equation (A2.26), that this is just the phase angle, 8, of the complex capacitance. Making this substitution, Equation (3.31) becomes A = (3.32) Since the desired integrator output is V (t) = sin wt D L(3.33) the integrator introduces an erroneous phase lead A =,/-f(oTr, +iS(w). (3.34) Figure 3.3 is a graph of T7 for () = 0 and Figure 3.4 shows s () for the Type I capacitor of Appendix II. D. Summary In this chapter the errors in the outputs of summers and integrators have been determined and evaluated. The simplifying assumptions which have been made are readily satisfied on even the simplest DC

-28 -100 80 60 40 in 20 T, T2 =I0017 0o 01 0.1 O -80 T T2= 100 \1.0 -100 Figure 3.3 Integrator Phase Shift without Dielectric Absorption 2.0 x 1.0 do 0.5O 0 _ I I X - I 0.01 0.1 1.0 10 100 FREQUENCY (rad /sec) Figure 3.4 Integrator Phase Shift Due Only to Dielectric Absorption

-29 -electronic analog computers provided that capacitors with small dielectric absorption are used. An important result obtained is that as a consequence of the assumptions made, the errors in the integrator caused by the dielectric absorption on the one hand, and the amplifier characteristics on the other hand, can be separated and considered individually. That is, the errors due to dielectric absorption can be determined by assuming perfect amplifiers, and the errors due to the amplifiers can be determined by assuming perfect dielectric properties of the capacitor. The total error then is the sum of these two. This situation obtains because only first order error terms have been retained and the device is assumued to be linear, thus permitting application of the superposition principle. Use will be made of this result in the following chapter.

IV. ERRORS IN THE SOLUTIONS OF EQUATIONS In the previous chapter the errors in summers and integrators were discussed. The purpose of this chapter is to determine the errors in the solution of the differential equations where summers and integrators are combined to solve an equation, for example, as shown in Figure 2.4. More precisely, if the equation to be solved, called hereafter the "given equation", is set up on the computer, the errors discussed previously cause the computer to solve a different equation, called the "machine equation". In the following sections, methods will be presented for determining the machine equation when the given equation and computer set-up are known. This problem has been studied by Raymond [16] and Macnee [17] for particular computer set-ups. This investigation uses rather a different method of approach and can -be applied to any computer set-up solving linear differential equations with constant coefficients. In addition the method reported here gives the machine equation directly rather than the errors in the roots of the equation in terms of the true roots, which may not be known. This facilitates direct comparison between the given equation and the machine equation. A. Solution of Linear Differential Equations To begin the analysis, consider the solution to Equation (2.16) which is rewritten here X x+a".x. +' + a,o x + =V f (t) XA.i.'+a 1 (4.1) There is, of course, no limit to the number of different ways in which the analog comiputer may be set up to solve this problem. However, at -3o

-31 -the outset, consider the method shown in Figure 2.4 which is redrawn as Figure 4.1. To determine the equation which the set-up of Figure 4.1 is solving, the voltage at y is evaluated in terms of y then y is set equal to ye by connecting the two points together. If the integrators and summers are perfect, one has,( ++,) 4 (t) (4.2) which becomes (sexa",S"m+-.a, s ao)y =s"'f(t). (4.3) Finally, since the relation between X and y' (or y ) is one has (-ea-t,- asea x =f(.t (4.4) Equation (4.4) is the machine equation if ideal equipment is used, and is seen to be identical to Equation (4.1).2 However, since the equipment actually used is not ideal, the computer must be examined from the point of view of the transfer functions given by Equations (2.6) and (2.10). For the present it will be assumed that there are no errors due to dielectric absorption; these errors will be discussed later. Therefore D(s) = I in Equation (2.10). 1 When the operator S appears in the equations in this chapter, it is understood to mean 4/dt 2 The effect of initial conditions will be considered later in the chapter.

I I om-l <a IIII I am-2 a0 Figure 4.1 Computer Set-up for Linear Differential Equations

-33 -By using different values for the resistors and capacitors in Figure 4.1 the same equation can be set up but the time constants of the summers and integrators will be different. Therefore these time constants are left in general terms. The relationship between y and y then becomes ~ emILao, 1T,1 Y Y ( [ (5 TS +. 5 )(TS + 1)(S; st,( 5te l) + ' ' a,", Ta, - ~ + IX TS +X Istl)(, s I Sit) (4-5) wof the 2nd ser 2 = T; of the achst integrator (from left to right) T, T; of the 2nd integrator, etc. rs, = rT of the 1st summer IS2 T rT of the 2nd summer r2 = T. of each integrator (They are all equal if the amplifiers are all identical.) Si I ) if the coefficient ak of the term in which it appears is determined by a resistor connected to the input of the first summer 8 = O if the resistor is connected to the second suAmmer; = I or O in the last term of Equation (4.5) if f(t) is an input to the first or second summer, respectively

-34 -Equation (4.5) may be written more compactly as Y-l [rn-k T j E ko (mSTI 5 + )(T S,st)(T25s~-i - TT (j f(t) (sT,,s+)(rs+-I) ' (4.6) Multiplying both sides of Equation (4.6) by j T,, j and noting that IT N Equation (4.6) becomes 1 (osl s+l)( s is+)(TstI, V"T T: ) +=-o r J=Wm-k+f 'J f(t) - (STs, S,+1XY stI)(T Stl)m (4.7) Equation (4.7) is the exact equation solved by the analog computer setup shown in Figure 4.1. It is seen that the characteristic equation is of order 2mn + 2 in S. The given equation, Equation (4.1), has a characteristic equation of order m in S. Thus m + 2 extraneous

-35 -roots are introduced by the m integrators and two smuimers. The remaining m roots, if the errors introduced by the imperfect components are small, are approximately equal to the m roots of Equation (4.1). B. Simplification of the Exact Machine Equation If it is stipulated that the errors be small, it is possible to make certain approximations which enable the characteristic equation of the machine equation, Equation (4.7), to be reduced to an equation of order m in S, the coefficients of which are approximately equal to those of the given equation. The difference between the coefficients in the Inachine equation and the given equation will be an indication of the error in the solution. In addition, these approximations permit evaluation of the m + 2 extraneous roots. The approximations which follow are fundamental to the rest of the analysis. Let 5i (i = I) 0' ) ) be the roots of the characteristic equation of the given equation of order m. Then if ~Si 4 () I, T. 7-T 7; T S T 2 (4.8) the following approximations will be valid in those regions of the complex plane where S - 5i I I-Ts, i - Tst S 1 It is shown in Appendix I that stray capacitance in the computer circuit causes changes in the values of r1, TS, and T2 It will also produce additional extraneous roots.

T2s ' I (r, s + I)(T2 s I) I + Trs, $ 'T2 s (Ts +I)m~ I + TS _ _ _ _I I, Tsjs etc. (4.9) If these approximations are made, Equation (4.7) becomes ~T, s 1 - ',j -ks)T5 (-s Ts -T2s -mT s)f(t), This can be further simplified to 1kS [k { k -ak, ({ T, 'TS T +)(-ki)T2( X =f (t)-(S T +T,s + MT2) d f (t) -;i!re it is understood that a ) m i0,+, = o ) -I = Since a feedback resistor in Figure 4.1 will be connected to.i-r first summer or second sunmer depending on the sign of ak and

-37 -on the order of the equation, the following general rule will apply: When aLK >.k S = aK if (k+m) is odd 0 if (k+m) is even. When l (o 0 a. S = 0 if (k+m) is odd aK if (k+m) is even. Equation (4.10) is an m-th order equation in S as is the given equation. Furthermore, in the limit as T Tsz T IJ and T; approach zero, Equation (4.10) becomes identical to the given equation. The coefficients of Equation (4.10) can be computed and compared with those of the given equation. Methods of determining the effects of these differences on the computer solution are discussed in a later section. C. Evaluation of the Extraneous Roots It was shown previously that the exact machine equation contains m+2 extraneous roots. It is important to determine the nature of these roots in order to evaluate their effect on the computer solution. The exact machine characteristic equation, from Equation (4.7) is (,st.)(Ts s +IT )(tr2sl)m (s e+ ) * I,(iJsI T j -I- o k=O ja ( ) m-k '+i (4.11) where it is assumed that each 8Si. The terms which would otherwise be

-38 -present would drop out later in the analysis and not affect the results. Equation (4.11) is of order 2m + 2, and if T T T- = Equation (4.11) degenerates to the given equation of order m, and the m + 2 extraneous roots are lost. In order to locate the extraneous roots on the complex plane when TrS ) T' ) approach zero, the parameter 6 is introduced, such that e = aT, =bT cTd51 _ T, (4.12) where a, b, C and d are positive constants. In order to keep the resulting expressions from becoming unwieldly, the simplification is made that these constants are all equal to unity. It will be seen as the development proceeds that this simplification will not affect the result. Substituting Equations (4.12) in Equation (4.11) and letting a= b= c = d, yields it2 m 'MI -' k '(5+ 1 (5t) + k = 0. (4.13) Expressed in powers of S Equation (4.13) becomes S -(m-6 s + - 5Z fja k ( o. kto (4.14) The omitted terms are all factoris of S raised to powers less than 2m + 1 and of C and, as will be seen, do not enter into the development. Since Equation (4.14) still degenerates to lower order when 6 - O, a change in variable is introduced. Let = 2m+Z Then after multiplying through by 0T Equation (4.14) becomes

-39 -E6f + (IE +)~ O' ' +t T k ~ O. ko (r.j_) Now when 6 approaches zero, Equation (4.15) becomes T*2 Zk C- =, (4.16) There are m + 2 roots Ur = O corresponding to the reciprocals of the extraneous roots and m additional roots which are the reciprocals of the roots of the given equation. Now that the extraneous roots have been located at the origin of the complex O -plane for 6 = O, it is necessary to determine their behavior as 6 takes on values other than zero. This is done by taking the partial derivative of Equation (4.15) with respect to E. This yields h ere t h e C) tted (I t;)erm s a (4.17) where the omitted terms are factors of. From Equation (4.17), can be evaluated at the origin of the complex O -plane by letting 0 = 0. This gives C) = _ + 2 C) 37= En_ + _ _ _ < 2 (4.18) Since, the extraneous roots in the complex T -plane A start at the origin for 6 = 0 and move to the left along the negative real axis as 6 takes on non-zero values. It follows then that in the complex S -plane the extraneous roots start on the negative real axis at — O~ for 6 = O and move along the negative real

-40 -axis toward the origin as 6 takes on non-zero values. (The roots may, of course, leave the real axis as 6 increases; only their initial direction has been determined here.) If 6 is made small enough, all the extraneous roots will be far to the left of the S5, the roots of the given equation. That is, 151 > I Si I where the S are the extraneous roots. The part of the computer solution due to the i. will consist of terms such as e and since 5 is large and negative these terms will damp out rapidly compared to the terms resulting from the correct roots and will have negligible effect on the solution. The question still remains: How small must 6 be so that I sl > > I S I? To answer this question it is stipulated that I S IX > I Sjl, and the conditions which must be met by E for this inequality -to be valid are determined. Letting S = S in Equation (4.11) and substituting 6 for T1, T and TZ yields ( M ~2i '+ -) K 5-, + ) k ak(Es t) n. (5'. t ) = k1o jjm k+I (4.19) Since it is stipulated that I S >> Sia and since 5i >> froml inequality (4.8), Equation (4.19) becomes approximately [s (ES+ I) +l + as k '+l) = o. [5~K~~o~~ynz] Zjaks(ES*I) = (4.20) For every term aK S (E 5 I)K in the summation it is readily seen that there is a corresponding term S (E S + I) in the

-41 -brackets. The assertion is made that The ter,- in the sumination is always very much less in magnitude -than the corresponding terni in the brackets if I SI >> M / Sj 5. This will be proved by showing that a,< I < Is< l -k Let Sj be the largest of the roots Si of the given equation. Then from the binomial expansion of (S-5j)m If sjl '< < S'l/m rn: m-k | l k|<~ (m-k)! k! k)'k (4.21) But (m-k)! k! nmak Therefore.akI < < S. The proof is complete and Equation (4.20) can be written approximately $51 6(51 $' r (4.22) From Equation (4.22) it is seen that E.= - " Since I Sl >> )l I Si, the condition which 6 must satisfy is X <K lsil. From the definition of 6 it follows that if r i <A ) _and - I the extraneous roots will be mis1) I and ___ very much larger in magnitude than the desired roots; in fact, they will be very much larger than m times the magnitude of the desired roots.

-42 -It is possible now to evaluate -he extraneous roots approximately by returning to Equation (4.11). If pWiSil << rI r and T it follows from the previous proof that Equation (4.11) can be writ en approximately as s'm(S r T,, s', S/Il)( t I S- o. s~t8Es s Tzs+ = ~(4.23) Equation (4.23) has iii roots 1 1 -T2 (4.24) and two additional roots S' I TI S = (4.25) It is clear from Equations (4.24) and (4.25) that if I I But 1 1 Isil << T and that Il. But in the previous discussion it was found necessary to have Inl[ S j << T ). in order to evaluate the extraneous roots. -fSI I The question then occurs, if Isil << ) '' but msil SI is not very much less than TS' I, is I si (< s' So It would appear that the answer is affirmative, but that in such a case the extraneous roots would be given less accurately by Equations (4.24) and (4.25). However, this has not been proved. D. Comparison of the Exact and Approximate Equations To see more clearly the nature of the approximations which have been made, and to gain a better insight into the effect of the extraneous roots, a numerical example will be considered.

-43 -Let the given equation be (s' + 3 S - 2) X - (t). (4.26) Equation (4.26) is to be solved by the computer set-up of Figure 4.1. The computer time constants are as follows: T,- o.oos T, T- =T 0Oo. The exact machine equation is obtained by substituting the numerical values in Equation (4.7) and this becomes (o.oo5s+,)(o.o001o + I)(S *. oo) +3(0o. 0oo01)(o. oo s + 2(o. ooss+) x = f(t). (4.27) The approximate machine equation is obtained by substituting in Equation (4.10) and this yields (0.96705 s -t-2.988s +.o03)X = (-0.01o2s)f(t). (4.28) The errors in the coefficients, as given by the approximate method, can be determined at once by comparing Equations (4.28) and (4.26). Further comparison can be made by evaluating the characteristic roots of the given equation, the exact machine equation, and the approximate machine equation. This was done by solving the equations numerically with the aid of a desk calculator. The results are tabulated in Table I. 1 Time constants of this magnitude might be found in an inexpensive computer of relatively low accuracy. Larger machines would typically have much smaller T I T$1, T and miuch larger Til and 7.5

-44 -TABLE I. ROOTS OF THE EQUATIONS Given Exact Machine Approximate Equation Equation Machine Equation -1.000 -o.9850 -0.9831 -2.000 -2.1069 -2.1069 -999.9 + ill -1000* -999.9 - ill -1000* -226.12 - 200* -170. 95 - 200* * Extraneous roots given by Equations (4.24) and (4.25). The extraneous roots do not appear in the approximate equation, of course, and their values as listed in the column for -the approxi.late equation are those given by the method of the previous section. It is seen from Table I that the approximation predicts the true roots quite accurately. The order of magnitude of the extraneous roots is correctly given by Equations (4.24) and (4.25), although their location is not exactly correct. However, since the order of miagnitude is usually all that one would wish to know about the extraneous roots, the results may be considered adequate. It is interesting to note that the approximations predict large negative real values for the extraneous roots, while actually one pair of roots is complex, although very close on the complex $-plane to its predicted location. The comparison is carried one step further by determining the comlplete solution, X (t), for a given input function. If f(t) is

a unit impulse at time t = O, its Laplace transfb rm is unity and X (t) will be the impulse response or weighting function. Define U (t) = x (t) when f (t) is a unit impulse. The solution of Equations (4.26), (4.27), and (4.28) is straightforward and the results are listed below. Desired solution: -Zt U(t). 1.0000 e-t - I. oooo e (4.29) Exact machine solution: 0o.98 3t -2,1o8 9t a(t) - 0. 93032 e - o.9340 e -170.9 St -226.12t t O. 03676 e -0 02400 e -999.9t + 0.00S69e sin (lt +2,8372 ). (4.30) Approximate machine solution: -0o9831t -2.1069t u (t) = 0.93 106 e - o.9434 7 e (4.31) Thus it is seen that the approximate method accurately gives not only the roots of the equation but also the coefficients of the terms in the solution corresponding to the true roots. It is also seen that not only do the terms corresponding to the extraneous roots damp out very rapidly but in addition their contribution to the solution, even initially, is quite small. E. Simplification of the Comiputer Set-up The question which immediately arises is, can these changes, or errors, be reduced by using fewer amplifiers? It is certainly true

-46 -that fewer anplifiers can be used than in the circuit of Figure 4.1. The first summing amplifier can, in general, be removed and the connections made to the integrator. The number of integrators, of course, must always equal the order of the equation. It will now be determined whether the elimination of the first summer has any effect on the inaccuracy introduced by the imperfect components. If the first summer is removed, and all connections to its input are made instead to the input of the first integrator, the exact machine equation is the same as Equation (4.)) except that r = ~ and S- 1 refers now to the only sumner remaining in the circuit. In approximate form, the machine equation becomes 5ko Lak kA- -A {k-T (m-4-I)Ta}] x =f(t) -(s-rT2 f. d3t2 (4.32) where Te = time constant of the summer T2 and T = time constants of the j-th integrator ZJ (counting from left) When ak > 0 adks = o if (k+m) is odd ak if (k+rr0 is even. When ak < aR s — i k if (k+m) is odd = O if (k+m) is even.

Although there appear to be fewer erroneous terms in the coefficient of Sk in Equation (4.52) than in Equation (4.10), it is not readily apparent which terms are smaller since the timle constants Tj and TS Ts, Ts are not the same. In fact, since the erroneous termsrmay be of opposite sign, it is quite possible that even though one or both of them is made smaller by the elimination of the extra sumnmer, their sum may well be larger than before, so that the removal of the summer may eliminate an error which previously was nearly cancelled by another error, with the net result that a larger error exists. This can ornly be determined by examining the problem when specific values of the parameters are known. Thus one cannot say that the elimination of the first summer, per se, from Figure 4.1 will result in any improvement in accuracy. The advantage of the method presented in this chapter is the ease with which it may be applied to any computer set-up which describes a differential equation which is a function of one dependent variable and in which the independent variable is the time. Examples of more complex problems are given in Appendix III. The use of this method in the case of several dependent variables (simultaneous linear differential equations) is treated later in the chapter. F. Rules for Minimizing the Dynamic Errors In returning again to the general equations solved by the machine, Equation (4.10) and Equation (4.32), it is worthwhile to see whether any general rules can be established which, if applied when setting up a problem on the computer, will minimize the errors resulting from the imperfect components. The erroneous terms introduced into the coefficient of Sk in Equation (4.10) are

-48 -Gak, T i and -axk, +B] Tsz +(M-k{i)T Ideally, the sum of these two terms should be zero, and this can be achieved by proper selection of the values of the various resistors and capacitors. In most cases, the labor involved in making the sum of these terms total zero would be comparable to the labor required to solve the original problem directly, and would therefore be unjustified. More reasonable would be an attempt to minimize the individual terms; this would be accomplished by making Z1, T p-m-k 7j TS, and as small as possible. With given amplifiers, Ti a) and is a constant. T. is inversely proportional to the product of the input resistor and the feedback capacitor. T51 and T5S are inversely proportional to the input resistors and proportional to the feedback resistor. This means that the sum of the (Ri C) 's for the integrators should be a maximum and Rf /R for each sulimer should be a minimum. That is, each integrator and suiimer should multiply by a constant which is as small as possible. However, the multiplication performed by all the integrators, that is, the product of all the (Ri C)s of the integrators, or rather the reciprocal of this product, must have a constant value determined by the coefficients of the differential equation. It can be shown by the calculus of variations that if the sum of several variables is to be a maximum while their product is constant, then the variables should

-49 -be equal. Thus one concludes that the (Ri C) 's of the several integrators should be made equal. The general rule can be stated, then, that in order to minimize the individual error terms introduced into -the coefficients of the differential equation, multiplication by a constant should be divided among several amplifiers whenever possible, each amplifier multiplying by the same amount. Furthermore, the multiplication performed by any integrator or summer should be miiade as small as possible. It should be emphasized that these rules are in a sense incompatible, since the problem coefficients determine the amount of multiplication which must be performed within a given feedback loop. But they do indicate that although the nultiplication cannot be avoided it can still be judiciously distributed among the amplifiers. There are other factors which may require modifications of these rules. One is the scaling problem. The constants of multiplication may be largely determined by the magnitudes of the input quantities. That is, it may be necessary to multiply by large numbers to avoid output voltages which are inconveniently low, or conversely, to multiply by small numbers to avoid amplifier saturation. Another factor is stray capacitance. As shown in Appendix I stray capacitance in the input and feedback resistors causes changes in the values of the time constants which would affect the magnitude of the errors. Still another consideration is the effect of integrator drift; this is discussed later in the chapter. G. Solution of Simultaneous Linear Differential Equations The method of analysis of the preceding sections will be used to evaluate the dynamic errors in systems of simultaneous linear

-50 -differential equations. As before, only one method of setting up the problem on the computer will be considered at the outset, but the extension of the method of analysis to other configurations will be made clear later on. The general expression for a set of n first order linear differential equations can be written a, -a,,x, +I * VI ft) ~n a, 1I+ ' a. n fnt) (433) This can be written more compactly in matrix notation as;]= [ai4'x ] ~ffij (4.34) or, X =AX + F where represents the coluurm matrix [X;] A represents the square matrix [ij] of order n X represents the column matrix Xj] F represents the column matrix LfiR The computer set up to solve Equation (4.34) is shown in Figure 4.2, where it is assumed that all the coefficients aij are positive. If unit resistors and capacitors are used throughout, the potentiometers should be set, as indicated, to the value of the coefficient. If other than unit resistors and capacitors are used, the potentiolaeter values will be different, and the time constants will also

-51 -1 2 f2o ----'- v — 1 2X2 fn n- vv- n Xn Figure 4.2 Computer Set-up to Solve Simultaneous Equations

-52 -be different. Using Equations (2.6) and (2.10) it is possible, as before, to write the exact equation being solved by the machine. (Note that if any of the 'ij 's is negative, the corresponding potentioiimeter would be connected to the input of the integrator instead of to that of the summer. The symbol a is used as before, C = I when the input is to the summer, a = O when it is connected to the integrator.) Then the voltage )i is given by Exl ' ( ETs l Ts )(i +zz l eTs )] [i s s+ Sei)( +Tli S,)(1+ e (4.1+ where; i = T for the i-th (from the top) integrator =Ts B T for the i-th summer T2 = T for each integrator Equation (4.35) is the exact equation solved by the computer and is seen to be at most of order 3n if all n summers are used. That is, it is of order 2n plus the number of summers. H. Approximate Solution for Simultaneous Linear Differential Equations Using the same approximations that were made previously, given by Equations (4.9), Equation (4.35p can be written

-53 -+ '(rsT s)(i +Tis)(l*Ts)j or [aj (I- Ts T5 s T2 s) -Iij + [T [fI (,- s TS-T2 5 Of. Finally, this becomes sai,r i+T] + I=E] - ' T., 'X j (4.36) Let B =[a,j (siT) +, Iij] G = [f1 - (STfi jT2)f.] (4.37)

-54 -Then Equation (4.36) can be written more conmpactly as BX =CX +G. (4.38) Thus by using the above approximations, the exact equation solved by the machine has been reduced to the approximate Equation (4.38) which is a set of n first order equations, as was the given equation, Equation (4.34). To facilitate comparison of the two equations, both sides of Equation (4.38) are pre-multiplied by the inverse of B. Then * = BiC + B'G. (4.59) It is possible to make use of the previous assumptions to evaluate B quite simiply. Let A j= [s ] = [aij (~TSXir Tji (4.40) Then or [bL~J3= at ( 4.41) The inverse of B is given by Bj' = [de (b-t) (4.42) where Bij is the co-factor of bij and det (bj) is the determinant formed by the elements of the matrix B de t(bi)= - (-)h(b,ii, b2 " bn.J) (4.43)

-55 -where the sunimation extends over all possible arrangements (il, i2,., in) of the n second subscripts (1, 2,..., n) and h is the total number of inversions in the sequences of the second subscripts, that is, the total number of times any of the numbers il, i2,...' i precedes a smaller number [21]. n Since all of the sij are small numbers, -their products will be neglected in evaluating the determinant and co-factors. Therefore the only terms in the summation which will contribute to the result are those in which all but one of the elements bkiK are of the form (I+ kk), that is, all but one must be bKK But then the remaining term must also be of that form. Therefore, et(bj) b,, b22, bBn = + SI, + S2a2.2 (4.44) By similar reasoning the co-factors are BLj = - Sji 'j Bkk: I+$11t 22 1nn kk ( Then F b-T'j 1 J (4.46) and in general [b,]' - Iij -AJ] -'= - ~ I (4.47)

Making use of this approximation for the inverse of B yields B-'C = -I, - 4 (S8rf 7 ) tNj[a~r i -Tj x — a - a,.k aj(STs+Ti (4.48) where products of small terms have been neglected and the subscript k appearing in each term of a product implies summation over k. Finally, Li=~ ~~~~~~ 6[~j T.,(4.49) Similarly the last term in Equation (4.39) becomes B-'s = [Ii -at(STj( TS l)JfI-(STsi ~T2)fe7 Equation (4.39), the machine equation, can be written as X = A'X F' (4.51) where A' = B-'C F= B-'G(4.52) and can be compared term by term with the given equation X =AX i F. (455)

-57 -I. Extraneous Roots for Simultaneous Equations As in the case of the single differential equation, extraneous roots are introduced in the solution of simultaneous equations. If these roots are S and are very much larger in magnitude than the roots of the given equation, they can be evaluated from the exact machine equation, Equation (4.,).1 Substituting S/ for S in Equation (4.3p), and setting the forcing functions equal to zero yields Tr a j l HO *. S S 5') + TSi TS')(I r2 5)1 i After rearranging this becomes L-/ )(+ a ES I')(I I sX = iU I/ > > this simplif to(4. If s >'> r, this simplifies to bai - 4j s5(' rS + )(+T S )3X,S[x j] -o0. (4.56) The characteristic equation of the system of equations given -by Equation (4.56) is det,aj -Ij s'(, +8 iT')(lC )] - O. T. The conditions for the extraneous roots to be much larger in magnitude than the roots of the given equation will be -the sale as those which applied in the case of a single differential equation. The proof is entirely analogous to the proof given in the previous section on extraneous roots.

-58 -The characteristic equation of the given equation is (4.58) where the Sk are the roots of the given equation. If S'I>> n Iski, (k = 1,..., n) then Equation (4.57) becomes approximately det [I(j S'( SI* T, 5'XI-F s')j - o. l (4.59) or S'n(lS sis (Iv T..s X l s - ~ (4.60) This equation has n roots / I ~~T2;~~~ ~(4.61) and n additional roots Ss, Fsn (4.62) Thus the number of extraneous roots, as in the case of the single equation, is equal to the number of summers and integrators. As before, they are negative and very large, hence damp out rapidly. J. General Computer\ Set-up The method of analysis for evaluating the errors in the coefficients of the equations has so far been applied to specific computer set-ups. In this section the application of the method to an arbitrary set-up will be described. 1 The proof is analogous to that given in the section on extraneous roots for a single equation.

-59 -Let the computer set-up have m integrators having outputs X;, i = 1 to m, and n summers having outputs Xi, i = m+l to m+n. Let aij be the coefficient of the input to the i-th amplifier from the output of the j-th amplifier and let f, be the forcing function input to the i-th aamiplifier. Then the output of the integrators can be written at = (T'st)(rse)(j=,atji +(4.63) for i = 1 to m. The output of the summers can be written )(=,Thsjl)Xj ti) (4.64) for i = m + 1 to m + n. If T S and T S ~ I, Equations (4.63) and (4.64) become approximately, AII (S + ) (ali += to tn XL - (IChsa +j tal ) h- tb?M+n. (4.65) Use would be made of the fact that S > T when Equations (4.65) are combined to eliminate some of the variables. The manipulation of Equations (4.65) would be determined by the circumstances. One would normally wish to put the equations in a form similar to the given equations used to set up the computer. Examination of these equations would indicate which of the XL should be

-60 -eliminated from Equations (4.65), to reduce them to a. single equation or to a set of equations having a particular form. Frequently it is possible to by-pass the intermediate steps indicated and write the machine equation directly from the computer circuit in a form which can conveniently be compared to the given equation. Just how this would be done must be determined by examining the computer set-up and the given equation. The method of procedure would vary froim one situation to another, but the technique is illustrated by examples in Appendix III and it is apparent from these examples how the method could be applied to other computer circuits. K. Effect of Dielectric Absorption on Solutions of Equations It was stated at the beginning of the chapter that the errors in the solutions of differential equations resulting from amplifier gain characteristics could be considered separately from those resulting from dielectric absorption and other effects as long as these errors are small and their products can be neglected. In this section the summers and integrators will be assumed ideal except for the effect of dielectric absorption. Then from Equation (2.10) for an integrator V, (s) Vo (s) - - RC D(s (4.66) where I ~ i CDk D(5) C k TDK5+I (4.67)

The relation between y and y' in Figure 4.1 becomes, if D () is the same for each integrator, 4s)- yD(s (4.68) Since y = y, the characteristic equation becomes -s) D(s) I = - Am_-I S ]- 0 C0 S — (4.69) Multiplying through by D(s) yields m L D(s) D(s))I J+, + + a = (470) Equation (4.70) is the exact machine equation. The sinmplifying assumptions of the type made in Section B are not available here. Due to the wide range of values of the time constants TDk it will generally not be possible to assume that Dk 5 <( or TDk S >> for all rDR. However, for any particular root S, these assumptions will be valid for most of the rD, but not all. Even to make these assumptions requires prior knowledge of the roots. Thus it appears that -the errors can only be determined in terms of the true roots. Equation (4.70) can be written akD(S (4 I7)

-62 -The given characteristic equation is, from Equation (4.1), 71k 6a S i 0 k a k;o (4.72) where S; (i = I) a e) V )) are the roots of the' given equation. Comparison of Equations (4.71) and (4.72) shows that Si = D (s) (4.73) The roots of the machine equation can be expressed in terms of the roots of the given equation and the errors in these roots 5 =s i + ei (4.74) where e' is the error in the root Si. Making this substitution in Equation (4.73) gives si + ee D (Cs + et) (4.75) From Equation (4.67) - = I+ I Ck Dcs) C k TDk S (4.76) and Equation (4.75) becomes (5s +e;) CDk St = s-+et CT L-TDk (s5e ) k (+r~kC~ik L (4.7()

-63 -Solving Equation (4.77) for e, and recalling that if the errors are to be small e ~ SL, one has approximately; CDk C k + ks (4.78) and S CDk S= Sil ( Z I TrOS Si (4.79) Equations (4.78) and (4.79) permit one to find the roots and the errors in the roots of the machine equation in terms of the roots of the given equation. Although these results are not as convenient as in the previous sections in which the coefficients of the machine equation could be compared directly with those of the given equation, they nevertheless have utility. For example, by observing the computer solution the machine roots can be obtained, at least approximately, and since they are nearly equal to Si they can be substituted in Equation (4.78) to obtain e. (Actually, this will be the exact solution for e[, as can be seen from Equation (4.77), if the machine roots are exactly determined.) Then knowing S and eL, St can be obtained from Equation (4.74).

L, Simultaneous Equations with Dielectric Absorption The errors in the solutions of simultaneous differential equations when the capacitors exhibit dielectric absorption follow directly from the preceding sections. Referring to Figure 4.2, the equation solved by the computer, considering only the effect of dielectric absorption, is [j) JS-k [[ kS1 (k 8ok where the forcing functions are set equal to zero since only the characteristic equation is of interest here. Equation (4.80) can be rearranged to give Ik-~S ]c~ l 0I (4.81) The characteristic equation is dPetfaj - Ij D(cs)II (4.82) and the given characteristic equation, from Equation (4.34), is det aJk Ijk S] 0 (4.83) where i (i = I) '' n) are the roots of the given equation. Comparison of Equations (4.82) and (4.83) shows that S 5. = a~ (4.84) D(s)

Since this relationship is identical to Equation (4.73), the errors in the roots of the given equation can be found as in the previous section, from Equation (4.78). M. Effect of Grid Current and Amplifier Unbalance It will be assumed in this section that the only errors in summers and integrators are those due to grid current and asimplifier unbalance. The output voltage under these conditions will be derived and the effect on the solution of differential equations will be discussed. Referring to Figure 2.2 for a summer, let the grid current in the first stage of the amplifier from the summing junction P be represented by l. Let the voltage unbalance referred to the amplifier input be represented by t. (That is, a voltage 'V applied to the input of the open loop amplifier would produce the given unbalance at the output.) For simplicity assume only one input resistor, Ri Then the following relationships apply: vo = -G(V +v) vi -V _o + v. Ri R+ (4.85) Eliminating V from Equations (4.85) yields Ri R.()8 (4.e6)

-66 -If the error due to finite ailplifier gain, G, is neglected, Equation (4.86) becomes / = -R; i- R; - + R i (4.87) Referring to Figure 2.3 for an integrator, and assuming no dielectric absorption or leakage resistance, and a single input resistance Ri, let i and V' be defined as for the summer. Then vo =-G (vt v) jRi V- vv s ((4.88) Eliminating V from Equations (4.88) yields sRvC C s SC I G (I+sRiC (4.89) If the errors due to finite amplifier gain are neglected, Equation (4.89) becomes Vo - sRiC (I sRiC) sC (4.90) Examination of Equations (4.87) and (4.90) shows that the magnitude of the output error due to the voltage unbalance depends on the multiplication constant of the amplifier, while that due to grid current depends only on the magnitude of the feedback resistor or capacitor and is independent of the input resistor. Furthermore, the

-67 -integrator error increases w-th timle due to the integration effect and is usually referred to as integrator drift. It was shown earlier in the chapter that in order to minimize the errors due to the amplifier characteristics it was necessary to have each summer or integrator multiply by a constant as small as possible. Now it is seen that to minimize the effect of unbalance the same criterion applies, but to minimize the effect of grid current the magnitude of only the feedback component must be considered. Obviously -to minimize the effect of both unbalance and grid current requires knowledge of their relative magnitudes. If this is not known one can compromise by avoiding unusually large feedback resistors and unusually sirmall input resistors and feedback capacitors, and multiply by constants which are approximately equal to unity whenever possible. The effect of the unbalance and grid current errors on the solution of equations will be considered by noting that the summer or integrator can be represented by ideal devices which have additional inputs to produce the outputs caused by grid current and voltage unbalance or drift. This is illustrated in Figure 4.3. When the sunmers and integrators are connected together for the solution of differential equations, the effect of these spurious inputs would be to act as an effective forcing function to the equation being solved. It would cause an additional output which would be added to the desired solution. As an example, the solution of the equation of simple harmonic motion will be analyzed. The extension of the method to more complex equations offers no difficulties and will not be considered here. The equation to be solved is

Ri Rf V + — Vi- [+ V+-iRf Ri Ri Ri -i - i Ri C Vi (,dv )Ri sC Figure 4.3 Simnmer and Integrator Inputs -to Represent Grid Current and Unbalance x + X 0 (4.91) and the computer set-up is shown in Figure 4.4 where the spurious inputs are included. In Figure 4.4 the input voltages eS, e,, and e2 have the following values: e, (c, t + R) -

-69 -Rf C1 C2 eS ek e2 Figure 4.4 Computer Set-up for Solution of Simple Harmonic Motion, Including Effect of Grid Current and Unbalance z R 2 'I (4.92) 2[ R2 IfRf-R.= s=, 2 and Rj = Ax, the equation I I 1(4.93) Assuming that the grid current and voltage unbalance in each case is constant, the effective forcing function becomes — t)) — +JItE + i, =constant. (4.94) If some other functions had been assumed for the grid current and drift, a similar result would be obtained. That is, the equivalent forcing function -would have the same form as the functions and their derivatives. Thie samre statement applies for higher order equations.

-70 -N. Effect of Initial Condition Errors on Solutions The effect of initial condition errors on the solution of equations has been deferred to this section since they can be treated exactly as the grid current and unbalance errors discussed in the preceding section. From Equation (3.17) the integrator output due to the initial conditions is VOe C) - [Vfo() eVR c (t)]/T e (4.95) where VO (0) is the initial condition voltage, VRec (t) is the capacitor recovery voltage, and T is the large integrator time constant. For t < T1, and neglecting products of small quantities, Equation (4.95) can be written approximately as V, (f )= V, (0t -V0 )(t). \/0 (t)- \4(~)~ T-; t /Rec (t(4.96) It is readily seen that the output given by Equation (4.96) is that of an ideal integrator having initial condition VO (o) and input Voo ) d VRe (t). Thus each integrator in the computer circuit can be represented (for the purpose of the initial condition errors) by an ideal integrator having a forcing function determined by its initial conditions and capacitor recovery characteristics. The analysis is considerably complicated, however, by the fact that the capacitor recovery voltage depends not only on the dielectric properties of the capacitor but also on the previous history of the capacitor; that is, on the previous solutions and the length of time during which the initial conditions have been applied.

-71 -O. Errors Due to Inaccuracy in Coliiponeiits In mosb computer,work, the values of the resistors and capacitors used are assumed to be their nominal values. In the language of statistics, this means that the assumed value is the mean value of the parent population from which the component is selected. The true value of any component differs from the mean value, and the extent of this difference in a large number of samples is determined by the standard deviation or probable error of the parent population [22]. The purpose of this section is to determine the probable error allowable in the parent population if the probable error in the constant coefficients of the differential equation is specified. Tie following definitions are miiade: PR = probable error in the nominal value of resistor R pc = probable error in the nominal value of capacitor C PX = probable error in the nominal value of resistor or capacitor X PX = fractional probable error in the nominal value of X In this section only the errors due to inaccuracies in the components will be considered, and the summers and integrators will otherwise be considered ideal. The desired equation for a slummer is then and foranRi ing(4.97) and for an integrator RC,( 4..8

-72 -where Rf, Ri,and C are the nominal values. The actual values of R /R. and RiC differ from the nominal values, For convenience let a = Rf/Ri and bi = I/RiC. Then to determine the probable error in the ai and the bi, one makes use of the law of the propagation of precision indices. When a quantity, U, cannot be measured directly but must be calculated from the mean values of two or more independently measured quantities, XY ~ the probable error of U -imay be calculated from those of X,Y, ' with the aid of the law of propagation of precision indices. In general, if U = f(X,Y, '") the probable error of U, PU is given by PU >, x +. [22] Applying this to the.ij and bi, one obtains \a PRf / PRj R. and PbC PRi ( 1~~~~~~ (4.99) If the fractional probable error in all the components is the same, and equal to P Pai Pbi Px ai bi X (4.100)

-75 -That is, -the constant of multiplication of a summer or integrator will have a fractional probable error which is 1.414 times that of the individual components. This method can be applied directly to the computer circuit for the solution of an ordinary differential equation with constant coefficients or a set of simulbaneous differential equations to determine the probable error in -the constant coefficients. Figure 4.5 shows a possible computer circuit for the solution of the equation X( )ta,X(')+ -.. t-ax.O (4.101) RA Rc CI cm b RaRm-R Figure 4.5 Computer Set-up for Differential Equation The coefficients a1. m, ), are related to the resistors and capacitors and potentiometer settings by Ra RC a, -b RR Ra._ RR,IC,

-74 -Rc aO= b Ra R1,C, WRC (4.102) If the fractional probable errors of the resistors and capacitors are the same and equal to P /X, then the fractional probable error in a, a is given by Ia - A = /Pbm.,i +6 P( ao bo!x (fa~)= (fbo\2 +(2 m 2) (4.105) In general, the fractional probable error in any constant coefficient is given by ( /Pb Pa ) (P^)(Px ) (4.104) where a = the constant coefficient b = the potentiometer setting in the feedback loop which determines aL

75 — = the number, of resistors and capacitors which determine L.1 An identical result is obtained by a similar method in the case of simultaneous differential equations. As an illustration of the use of this result, consider a differential equation in which the coefficient a is known within a fractional probable error of one percent. Then if the value of a is determined by ) components of equal fractional probable error (assume no potentiometer), the fractional probable error of the n components should be no greater than O0.0 /E P. Using the Computer to Analyze Its Own Errors Having found the errors introduced by the computer from the methods of the previous sections it is natural to inquire whether the computer itself can be used to determine the errors and their effects on the solutions. Two methods of doing so will be suggested here. Let the given equation be k a on k x =f(B. (4.105) Then for a given X (t) one can readily solve Equation (4.105) for f (t). Using the f (t) thus obtained as the input to the computer set-up, the computer solution can be compared with the given X (t) and 1 If the fractional probable error of the resistors is not equal to that of the capacitors, Equation (4.104) is modified to - ()- + n (P) + nc where n R is the nunber of resistors and nC is the number of capacitors which determine.

their differences will be due to the computer errors. Then the computer set-up can be adjusted (for example, by changing potentiometer settings) to cause the conmputer solution to agree with the given solution. A variety of such "test solutions" could be employed; for example, sinusoidal and exponential functions would permit ready evaluation of f(t). One might expect that by using several such functions a procedure could be established for a given problem to correct the computer solution in a logical manner. The method would be useful as a means of periodically checking complicated computer set-ups such as simulators, but in these comiplicated cases it is doubtful if the method would indicate the source of the errors. The second method of using the computer to analyze the effect of its own errors applies to those situations in which the errors appear as changes in the constant coefficients. Then the resistors or potentiometers in the computer circuit which determine the coefficients can be perturbed by an amount corresponding to the errors in the coefficients and the resulting change in the nature of the computer solution can be observed. The difference in the solutions before and after the perturbation will then be an indication of the magnitude and nature of the error. The coefficients could be perturbed individually to determine the sensitivity of the solution to errors in a particular coefficient.

V. CONCLUSIONS AND RECOMMENDATIONS A. Conclusions In the preceding chapters it has been shown that the suimmers and integrators used in differential analyzers differ from the ideal because of many factors. Among these are the amplifier gain vs. frequency characteristics, capacitor dielectric absorption, stray capacitance, amplifier drift, and inaccuracy in resistor and capacitor values. The effects of these differences can be represented by changes in the summer and integrator transfer functions and by additional inputs to the amplifiers. When the summers and integrators so represented are used to solve linear differential equations with constant coefficients, it was seen that the equation being solved (the machine equation) is not the same as the given equation. The machine equation is of higher order than the given equation and contains additional inputs or forcing functions. The machine solution differs front the solution of the given equation. If the errors in the solution are small, it is possible to consider the total error as being made up of the sum of the individual errors. If the error due to dielectric absorption is temporarily neglected, the machine equation can be reduced to an equation of the same order as the given equation. The reduced machine equation and the given equation can then be compared term by term and the errors, will appear as changes in the constant coefficients and as additional forcing functions. (The extraneous roots which were lost in reducing the machine equation can be determined approximately, and their contribution -77 -

-78 -to the error will be quite small initially and will damp out very rapidly.) The effect of dielectric absorption by the integrating capacitors is to cause small changes in the location of the roots of the given characteristic equation. These changes or errors in the roots can be determined from the complex capacitance of the integrating capacitors and the roots of the given equation (or the roots of the machine equation). Thus with a given equation and computer set-up, one can deterniine with a high degree of accuracy the effect of the computer errors on the given equation. Conversely, if one has an equation which must be solved with a given accuracy, it is possible to determine what characteristics are required in the individual components of the computer -to achieve this accuracy. In addition, methods have been described by which the computer itself may be used to determine the influence of the errors on the solution. In the opinion of the author, the chief contributions of this investigation are as follows: 1. Although others have shown [16,17] that the amplifier characteristics cause errors in the locations of the roots of the characteristic equation, it has been shown here that these errors can be represented by changes in the constant coefficients and forcing functions of the given equation, even if the roots of the given characteristic equation are not known. 2. This investigation has included the errors resulting from stray capacitance, amplifier grid current and unbalance, and lack of precision in the values of the resistors and capacitors used in summers and integrators.

-79 -3. The effect of capacitor dielectric absorption has been determined, with respect to the integrator transfer function as well as the integrator initial conditions. These effects have been analyzed not only for individual integrators, but also for integrators connected to solve differential equations. 4. The methods of analysis are not restricted to particular computer set-ups, but can be applied to any computer set-up solving linear differential equations with constant coefficients. 5. Methods have been described by which the significant dielectric properties of the capacitor can be obtained experimentally by using the computer and without the necessity of removing the capacitor from its installation. B. Recommendations This investigation has indicated a number of topics which merit further study. Some of these are discussed below. The method of obtaining the complex capacitance described in Appendix II requires a rather tedious curve fitting procedure, although the experimental data are readily obtained. It would be desirable to develop a method which would permit more direct use of the transient current information without the intervening analysis. It was suggested that by using certain "test functions" as inputs to the computer, it should be possible, by comparing the machine solution with the known correct solution, to adjust the values of the coefficients on the computer to yield the correct solution. The application of this technique to typical problems would be useful.

This investigation has been limited to the solution of linear differential equations. Perhaps the most useful and challenging study would be an analysis of computer errors in the solution of non-linear equations. Although the errors in individual multiplication devices have been analyzed, the author is not aware of a comprehensive investigation of the effect of these errors when the multipliers are used in the solution of non-linear differential equations.

APPENDIX I. EFFECTS OF DIFFERENT AMPLIFIER CHARACTERISTICS AND OF STRAY CAPACITANCE A. Amplifier Characteristics It was assumed in Chapter It that the operational amplifier used in the analog computer had certain anplitude-frequency characteristics, namely that it could be considered as a simple first order system. Since many amplifiers in use actually have more complicated gain characteristics, it is important to provide for the analysis of computers using such amplifiers. Methods of measuring the amplifier gain are described by Johnson [231]. Whatever the characteristics of the amplifier may be, they will have certain features in common. They must all have high gain at zero frequency, and they must fall off at approximately -6db per octave in the vicinity of unity gain to provide stability when used for summing and integrating. They may, however, fall off more rapidly than -6db per octave at intermediate frequencies. Figure Al.1 illustrates such behavior where the DC gain is and the break point frequencies are TA and T The transfer function for the amplifier of Figure Al.1 is (Al.1) A summer using such an amplifier will have a transfer function, from Equation (2.4), of

20 log Go < X - 12 DB/OCTAVE o I\,,I, '~-6 DB/OCTAVE 0 I w '/TA W='/TB LOG FREQUENCY Figure Al.1 Open Loop Amplifier Frequency Response Vo () I - RRf V (s (TA S J) (I (Al.2) After simplification, and if I, Equation (A1.2) becomes Vo(S) -o (r,,s+ ) -Zw v(,) (-z1+2 RjF555(+ WjA~(5+,]+ (A1.3)

-83 -It was shown in Chapters III ~nd IV that the amplifier contribution to errors in the solution of differential equations results primarily from the lack of infinite DC gain and the extraneous phase shift introduced in the output of summers and integrators. Consequently if the amplifier of Figure Al.l can be replaced by one having the same static gain and producing the same phase shift when used in a summer or integrator, the errors produced by the two amplifiers will be the same. The phase shift for sinusoidal frequencies for the summer given by Equation (A1.3) will be e=ani' (wTB) tan` 2(I -[)TA(l +Gi Tj (Al.4) If 42 (< <Go (t+ z Rm);, Ri, (Al.-) and tA) ((o 2(It+Z Ri)rT a (Al.6) then the phase shift becomes approximately 2TA (I Rf ) (A1.7) The equivalent transfer function for a summer is then

Vo(s) 1 -Z-Vj(S)s ss (A.8) where (Al. 9) This is equivalent to replacing the wamplifier of Figure Al.1 by one having a transfer function G( 2G (s) = I (Al.10) The transfer function of an integrator (not considering dielectric absorption) is, from Equation (2.8), Vo(s) I RiC G() < [S, RC + (R R-CF c(S) RLC G-(s)+ RI C (Al.11) If the value of Gr (5) fromi Equation (Al.1) is substituted in Equation (Al.11), and a procedure is followed similar to that for a summer, the result is Vo(s) T, s Vi(S) (T, s + I)(T2s I)

-85 -where I I v I RLC G. G0RiC 2 TA T2=G (Al.13) provided that Go ____ Tr2Z~L ~ 2TAZ&R~c < 1. TAZ ZTTAA+GTF ZA Ric 2 I << I (A1.14) As before, this is equivalent to replacing the amplifier by one having a transfer function given by Equation (Al.10). In summary, if the amplifier has a characteristic similar to that shown in Figure Al.1, it is only necessary in the analysis of Chapters III and IV to substitute for rS and T2 the following value s: -C.Tso= (,+ Ri -rs= 2jj z(.z) (Al.l5)

-86 -B. Stray Capacitance In Chapter II it was assuiied that the input and feedback resistors did not contain stray capacitance. Actual resistors, particularly those that are wire-wound, do have some capacitance in parallel with them. In addition, there will usually be some capacitance between the summing junction, or amplifier input, and ground. In this section the effect of this stray capacitance will be determined. It will first be convenient to develop the amplifier transfer function with arbitrary input and feedback impedances. Let the input impedance be Zi and the feedback impedance be Zf. Then following a procedure similar to that used in Chapter II, one has Vo= -G(s)V Vi - V _V - eO Z; Zf (Al.16) Eliminating V yields Zf V; G S, ( _ I+ vo) (I+ Z (Al.17) For a summer, let the input resistor Rj have in parallel with it a capacitor Cj, and the feedback resistor Rf have in parallel with it a capacitor Cf. Then one has R; Zf I -s RfCf Zi l R C(A1.18) I+sRiCi

-87 -Substituting in Equation (Al.17) the values of Z; and Zi given by Equation (Al.18) and the value of & (5) given by Equation (2.3) yields R i+sRiCi V = i + s R;C.) 1vo l~~~~+ &O5+1 01+ Rf '(Al.l9) If Ci and Cf are stray capacities, it will usually be true that S R Ci and s RfCf << I. Then Equation (Al.l9) becomes approximately -V ~Rjf- 1+ RCiRIC - RfCf V0, { (R + TS + I s(RiCi- RifC (A1.20) Letting RiCi - RfCf = Tsc, Equation (A1.20) can be written R --- Vi Ri 'o0 _____ To r s _<_ (A1.21) If Gand ( S (< O,Equation (A1.21) becomes approximately Rf v0 = Ri (A ~+ Trs s (A1.22)

where TS = Go(1+ Ri T5C (Al.23) Thus stray capacitance in parallel with the resistors causes a change in the summer time constant from -that given by Equation (2.7). Equation (Al.19) will apply to an integrator if R = RL, the capacitor leakage resistance, and C; = C, the feedback capacitor. After making these substitutions and rearranging, Equation (Al.l9) becomes RVLC +(I +s R C) RL+ S [rS(l C;)+ R+C + I + (Al.24) If -o Ri C To <<I C R — << I RL G,>>I (A:l.25) Equation (A1.24) can be written RiC 1Dielectricabr (Tpn(.26) 1 Dielectric absorption has been neglected.

-89 -If — S and RiCi << I, Equation (A1.26) becomes T, i RiC (Ts +i)(Ts +I) (A.27) where I I I 1T - RLC G GORiC T, = - Ri (Al.28) Thus the effect of stray capacitance in parallel with the integrator input resistor is to reduce the small integrator time constant T2 from the value given by Equation (2.11). The large integrator time constant T. is not affected. The effect of stray capacitance between the summing junction and ground will be considered next. Referring to Figure A1.2, CS is the stray capacity to ground and Zf may be either a resistor or capacitor. Ri V C S Figure A1.2 Amplifier Connections Including Stray Capacitance to Ground

The following relationships can be written: V = -V G(s) -V;-,-, vo + S Cs V. R; Z* (A1.29) Eliminating V yields V..CRi Z) G-(s) Ri (A. 30) For a summer Zf = R. Substituting for G(S) from Equation (2.3) yields _ Rf Vl+.(1 )+ RFTo(16G) R.1 GO GS (Al.31) If Go >>(I+ Ri and ROCs << To(l Ri Equation (A1.31) becomes Rf V; where The term in the denominator of Equation (A1.32) involving S8 will be very snmall for the normal spectrum of computing frequencies

and will make a negligible contribution to the amplifier phase shift. Although this term is unimportant from the point of view of the error analysis, it must be considered in the design of the amplifier since it may have a detrimental effect on stability. It will be noted that the summer time constant TS is not affected by the stray capacitance between sumiling junction and ground. For an integrator, Zf in Equation (A1.30) becomes RL and one has I + RLC V; + L + s IS [ \I I RLC GO C RLC RiC (A1.35) If ~ I G R;C R <<l (Al.34) Go >> Equation (Al.33) becomes approximately T, V; vO~ RiC T( s +.i T)( -S I) (Al 35) where IT_ I I T, RLC G oRiC 2 -(Al.6)

-92 -Comparison of Equations (A1.36) and (2.11) shows that the presence of stray capacitance between amplifier input and ground has negligible effect on the transfer function of an integrator.

APPENDIX II. DIELECTRIC ABSORPTION OF CAPACITORS A. Theoretical Considerations A well known phenomenon exhibited by dielectric materials is that of dielectric absorption [24]. When a potential difference is applied to a dielectric, the polarization current or charging current consists of two distinct types. The first is the charging current which occurs practically instantaneously; second is that which occurs more slowly during a measurable period of time. The former is caused by the rapidly-forming or instantaneous polarizations and the latter by slowly-forming or absorptive polarizations. The instantaneous polarizations consist of two types characterized by different relaxation times. (The relaxation time is a quantitative measure of the time required for a polarization to form or disappear.) The electronic and atomic polarizations occur practically instantaneously with relaxation times less than 1010 seconds. The absorptive polarizations also consist of two types; first is the type due to the effect of the applied field on the orientation of molecules with permanent electric moments, referred to as dipole polarization. Second is the interfacial polarization resulting from lack of homogeneity in the material. Only the interfacial polarizations have relaxation times large enough to be of interest in analog computer applications. The effect of the absorptive polarizations is to cause a capacitor containing a dielectric material to depart from the ideal characteristic in which the charging current leads the applied voltage by 90~. Instead, the slowly-forming polarizations cause the current to lead -93 -

-94 -the voltage by less than 900. The theory of dipole polarizations has been developed by Debye [25] and that of interfacial polarizations was developed earlier by Maxwell [26] and Wagner [27]. In both cases the theory shows that the dielectric constant can be considered a complex function defined by E6 =E jE' (A2.1) When expressed in terms of frequency and relaxation time, the complex dielectric constant becomes.o - E~ +., (A2.2) where Eo is the zero frequency or static dielectric constant, E0 is the infinite frequency dielectric constant, and TO is the relaxation time which is a function of temperature. Let a capacitor with air dielectric have capacitance Ca When filled with a material having a dielectric constant given by Equation (A2.2), it will have a capacitance CI. Such a capacitor can be represented by a model consisting of ideal components as shown in Figure A2.1. Unfortunately experimental results using solid dielectrics do not entirely confirm the theory. Yager [28], and Fuoss and Kirkwood [29], have explained the discrepancy by considering the dielectric to have not one but many relaxation times, rather closely spaced. This suggests that the physical model in Figure A2.1 should have more than one capacitor-resistor combination shunted across the ideal capacitor.

-95 -C t a 0 Figure A2.1 Theoretical Model for a Capacitor Adequate experimental data were not available to make use of the theory for application to capacitors of the type used in analog computers. Consequently the experiments described in the following sections were carried out. B. Experimental Investigations These tests were conducted using two types of capacitors manufactured by the Southern Electronics Corporation, Burbank, California. Both types utilize polystyrene as the dielectric and are considered typical of those found in many analog computers. Type I is used in the Sterling Computer discussed in Appendix IV. In order to determine how the model shown in Figure A2.1 should be altered to conform with experimental results, and to determine the values of the parameters needed to describe the model, 1 Type I: Model Po-1-1-200 SB, 200 volt, 1MFD, Type II: Model Po-1-0-200 E, 200 voli. 1MFD.

-96 -essentially two types of measurements were made. The first was a measurement of the charging current when a constant voltage was suddenly applied -to the capacitor terminals, and the discharge current when the teriinals of the charged capacitor were short-circuited. The second method was to measure the build-up of voltage on the opencircuited terminals of the capacitor when the capacitor had been charged for a given time and then briefly discharged by short-circuiting the terminals. The method of measuring charging and discharging current is first described in detail. From the results of these measurements a physical model of the capacitor is determined. Finally, the voltage build-up which would occur with this physical model is determined and compared with the experimentally observed voltage build-up. C. Measurement of Charging and Discharging Currents For the measurement of the charging and discharging currents an electrometer was used.1 The current was measured by observing the rate of charge on the electrometer. The circuit diagram is shown in Figure A2.2. The battery voltage, VS, is constant and equal to the voltage across the capacitor, V, plus the voltage on the electrorneter, Vi Vs= V V. (A2.3) Since Co represents the ideal portion of the capacitor, that is, the portion whose current consists of instantaneous polarizations and dipole polarizations, the rate of change of voltage across CO is proportional to the current through Coo. 1 Vibrating Reed Electrometer, Model 30, Applied Physics Corporation, Pasadena, California.

-97 -__.__._..< — CAPACITOR CI I I, _ __. ELECTROMETER V.lIoCi Ri2 2 AND RECORDER s! i3 i 4 VB = BATTERY VOLTAGE Coo = IDEAL PORTION OF CAPACITOR Z =ELEMENTS WHICH MUST BE ADDED TO Coo TO REPRESENT PHYSICALLY OBSERVED BEHAVIOR C =i EQUIVALENT INPUT CAPACITANCE OF THE ELECTROMETER Ri -EQUIVALENT INPUT RESISTANCE OF THE ELECTROMETER Figure A2.2 Circuit for Current Measurements

-98 -dV i, *t Coo (A2.4) Similar expressions can be written relating the voltage and current for Ci and Ri dt C; dt (A2.5) From this it follows that t I (A(2.6 In the electrometer used, R 106 ohms, C; 10-11 farads. Thus j4 can be kept very much smaller than 13 by occasionally closing switch S2 and reducing the charge on Ci to zero. From Kirchoff's law I, * 12 = 13 + i' (A2.7) If i4 < < i3, Equation (A2.7) becomes + I = In (A2.8) From Equations (A2.4) and (A2.5), j = CoO dV/st and i3 = Ci JVi/t. Also, since V is constant, from Equation (A2.3) one has JV/4t + d /t o Substituting in Equation (A2.8) finally gives i2 (Ci Co) at (A2.9)

-99 -If Ci < C oo (in the tests reported here C 00 = 10-6 f, Ci 10-11 f), the final expression for the current through Z becomes dV; =2 = Coo dt (A2.10) The procedure for measuring the charging current was as follows. The capacitor was short-circuited for a given length of time by placing S in position 1 with S2 closed. Then S1 was moved to position 2, and after the first surge of charging current S2 was opened. V; was recorded and its rate of change was evaluated to give 12. The discharge current was then measured by closing S2, placing S1 in position 1, and then opening S2. It was noted, as would be expected, that the charging current at a given time depended on the length of the previous discharge interval. Consequently, measurements were made following increasing discharge intervals until repeatable results were obtained. Increasing the discharge interval beyond fifteen minutes did not result in a charging current which was significantly greater than that following a fifteen minute discharge. The same observation applies to the discharge current following a fifteen minute charging interval. Battery voltages varying from 23 volts to 110 volts were used, and the charging and discharging currents were, in each case, proportional to the voltage used. Figure A2.3 shows typical current vs. time plots for charge and discharge of a Type I Capacitor. Figure A2.4 is a comparison of Type I and Type II capacitors, and Figure A2.5 shows the discharge current following two different charging intervals. It will be noted that there is no significant difference in the discharge current

-100 -id5.0 0 In 10 40 1 x0 DISCHARGE ---— /x -7r _ I I I I I 0 100 200 300 400 500 TIME (seconds) Figure A2.3 Charge and Discharge Currents for a Type I Capacitor

-101 --5 10 o):, -6 0 0 1 0020030040TYP0E 50 0I 20I I I 40o5I 0 100 200 300 400 500 TIME (seconds) Figure A2.4 Comparison of Discharge Currents for Type I and Type II Capacitors

-102 --5 10 0 x 15 MINUTE CHARGE > | XO o 60MINUTE CHARGE 8 x 6 Z -6 I 9 0 0 0 - 70 10 x 0x 0 I I I I I 0 100 200 300 400 500 TIME (seconds) Figure A2.5 Discharge Current Following Different Intervals of Charge

-103 -during the first 300 seconds of discharge for charging times of 15 minutes and 60 minutes. It will be observed that -the charging current in Figure A2.3 is greater than the discharge current. This is due in part to the leakage resistance of the capacitor which results in a constant difference between the two currents. Since the difference in Figure A2.3 is approximately constant after 300-400 seconds, the higher charging current for observation times greater than 400 seconds is assumned due to leakage resistance. For times less than 300 seconds, the difference between the two curves becomes increasingly greater. This can be explained as follows. When the capacitor is charged by connecting it directly across the battery ternminals, the battery experiences a current drain which causes its terminal voltage to decrease. After the first instant of charge the subsequent current is very small and the battery recovers its original voltage. This rate of change of battery voltage causes a proportional current to flow to the capacitor, in addition to the absorptive current. For example, at the end of ten seconds the difference between charge and discharge currents would be the current flowing in a one microfarad capacitor when the applied voltage is changing at the rate of about 0.6 millivolts per second. This phenomenon was verified by comparing the voltage of two batteries, using a null technique, before and after one of the batteries was used to charge a capacitor. The rate of voltage recovery of the latter battery was found to be the same order of magnitude as that required to explain the difference in capacitor charge and discharge curves. Since there was some variation in the battery behavior depending on its condition, it is felt that the charging current

measurements are of doubtful value and in the remainder of the analysis only the discharge current will be considered (except for the determination of the leakage resistance). If -the difference between charge and discharge current in Figure A2.3 after 400 seconds is due to the leakage resistance, RL, it is seen that RL = 5 x 1012 ohis. The scatter in the points is primarily due to the fact that they were obtained by evaluating the rate of change of the recorded electrometer voltage. These tests were conducted at temperatures ranging from 66~F to 102~F. No significant differences in results were noted at the various temperatures, other factors remaining constant. The charging and discharging current was also measured by connecting a resistor in series with the capacitor and measuring the voltage drop across the resistor with the electrometer. The resistor was short-circuited during the first instant when the instantaneous polarizations occurred. However, it was still necessary to correct for the time constant formed between the capacitor and resistor. Due to the large value of resistance required to achieve a measurable voltage drop (107 ohms or greater) the correction was of the same order of magnitude as the uncorrected current for the first few seconds. Nevertheless, this method showed excellent agreement with the results obtained by the rate of charge method. D. Capacitor Model Based on Experimental Results The discharge curve in Figure A2.3 can be approximated by the function

co v = 21 ake (A2.11) where the number of terms would be determined by the accuracy required in the approximation. In Figure A2.6 the current I, following the application of voltage VB with the capacitors initially discharged, is given by ___i -t/RkCk CoVo k RkCoo (A2.12) I -. V OR' Rk Rn 1 -T TC... Z Ck jcn Figure A2.6 Model of Absorptive Portion of Capacitor The discharge current which would flow if the terminals in Figure A2.6 were shorted after all capacitors are charged to VB volts is, of course, also given by Equation (A2.12). Equations (A2.11) and (A2.12) will be the same if the resistors and capacitors have values given by the following relations: Ck C_ =Th ak (A2.15) co

-106 -It follows directly then that the equivalent circuit for -the capacitor is as shown in Figure A2.7, where the leakage resistance RL is included. Coo R, R, Rk Rn C, Ck Cn Figure A2.7 Model of a Capacitor Based on Experimental Results It is possible, using five resistor-capacitor combinations, to fit the data given in Figure A2.3 with a maximum error of 10 percent if the values used for the components in Figure A2.7 are those given in Table II. The discharge current which would result from -this TABLE II. EXPERIMENTAL VALUES FOR CAPACITOR MODEL k Ck/Coo RkCo (sees) Tk -= RkCk (sees) 1 1.40 x 10-4 3.56 x 106 500 -4 2 2.00 x 104 2.50 x 10 50 3 2.70 x 10- 2.00 x 104 5.4 4 1.93 x 10-4 3.03 x 103 0.585 5 1.20 x 10-4 3.34 x lo2 o.o40 RLC - 5.0 x 106 L 00

-107 -approximation is plotted in Figure A2.8 for comparison with the experimentally observed discharge current. E. Evaluation of the Complex Capacitance If the capacitor represented by the model in Figure A2.7 has an air capacitance Co, it will have a complex capacitance C when filled with a material having a complex dielectric constant E where C = ~C' E*. ~(A2.14) The complex capacitance can readily be evaluated from the admittance of the model in Figure A2.7 since, for a capacitor, Y(ji: jc6CY. (A2.15) Circuit theory gives for the admittance of the model n I 0 cCE Rk + jWCC (A2.16) where the admittance due to the leakage resistance is not included since it is not part of the polarization process. The complex capacitance follows directly from Equation (A2.16) " Ck C*= Co + I+jwTk (A2.17) where Tk is the k-th relaxation time, Rk Ck. From Equation (A2.17) it is clear that the infinite frequency capacitance is Coo = C Jag (A2.18)

4'- EXPERIMENTAL 10 L --- APPROXIMATION THIS INVESTIGAT ION z -6 ' 10 10 10 0.01 0.1 1.0 10 K0 1000 TIME (seconds) Figure A2.8 Comparison of the Experimentally Observed Discharge Current with that of the Approximate Model

-109 -and zero frequency capacitance is 0 CJ c = +coo Ck (A2.19) The complex capacitance can also be expressed in terms of its real and imaginary parts as C * C - j C c* c'-jc" (A2.2o) where Ck k=1 A Tk^2.) (A2.21) n WCkTk (A2.22) If C is expressed in polar coordinates, one has C = IcI ed (A2.23) where IIC C tan 8 C (A2.24) Tan $ is the loss tangent or dissipation factor of the dielectric. If I C ( CO the following approximate expressions can be derived:

-110 -Ic*C= Z 0k kIl I+'TW (A2.25) wn CkTk CO, I + W1'T (A2.26) F. Discussion of the Capacitor Voltage Recovery If a capacitor is charged at a constant voltage for a period of time, then briefly discharged by short-circuiting its terminals, a voltage is observed to build up on the open-circuited terminals. On poor quality capacitors this recovery voltage may even approach the original charging voltage in magnitude. This behavior is, of course, due to the slow forming polarizations of the dielectric material and provides a convenient means of measuring the same properties of the dielectric which were determined by the transient current measurements described above. If the model of the capacitor shown in Figure A2.7 is charged with a voltage VB for a length of time long compared to the longest relaxation time RkCk, then each of the capacitors Ck will be charged to essentially VB volts. The voltage VB is removed and the terminals of the capacitors are short-circuited for a length of time which is short compared to the shortest relaxation time, reducing the voltage on CO0 to zero and leaving the voltage on the other capacitors still essentially at VB. The short-circuit is

-111 -removed and the terminals of the capacitor are left open-circuited. The following analysis permits determination of the voltage which then appears on the terminals. Time is measured from the instant the shortcircuit is removed. VReC 'i 1 t RiT ' iRk i Rn RLC C C Ck IC n Figure A2.9 Currents Flowing in Capacitor Model During Voltage Recovery With reference to Figure A2.9 the following set of differential equations can be written: R~a+ +C =0 R'S-t C, C. dik Ik co+ tjdi. +i io R ~+ C+ - =0 d iL ia -RL it + C - ik -iL I =o O. IC2~~~~~~~ ~(A2.27)

-112 -Combining the last two equations to eliminate iL and taking the Laplace transform of the remaining equations yields (S+ RCi, + R,Clo ~ ( L+ ii(s = k(O) RkCk RkC o (S, ' —,)i. (S)+ (S)) RvhR- C ios) =i Elk (S) s IL) () (A2.28) where ik~s ~olik W ik (~) ik(t]t o (A2.29) Since the voltage on Coo is zero when t = O and the voltage on the other capacitors is VB when t 0 VB k ()= v8 RX iL'o ( 0) ~ (A2.30) After substituting for the initial conditions the solution for io (s) expressed in determinants becomes

-115 -(S +, ~ ) 0V o O (oi\ R, S S S o i (s) = (A2.31) ~+ R ) co o mRat ion yields $ S ( -(+ 32) If (C( < the evaluation of i0(S) with this approximation yields S RRLc

-114 -Taking the inverse Laplace transformation gives = X - e/R, ~C VB Z Ck e/RLC0. t= ku Rk keRI Coo (A2.33) If RLCOo >> RkCk and the time of observation, t, is of the same order of magnitude as the longest relaxation time, Rk Ck, the last term in Equation (A2.33) will be negligible compared to the sum of the other terms. From Figure A2.9 it is seen that dVRe _ 10 dC Co0 (A2.34) Substituting for 10 from Equation(A2.33) gives | VRC C, P I et/RkCk (A2.35) and VRec k- ) (A2.36) The right-hand side of Equation (A2.35) is seen to be identical to that of Equation (A2.12). Therefore '4~~~at. CIt0.VB ~(A2.57)

-115 -where VRec is the recovery voltage and I is the discharge current. Thus as long as Ck/C < I and RkCk << RLCoo the measurement of the capacitor recovery voltage can be used to obtain precisely the same information as can be obtained from current discharge measurements. G. Measurement of Capacitor Voltage Recovery The capacitor voltage recovery was measured in two ways. The first was by means of the electrometer described earlier. After briefly discharging the capacitor, its terminals were connected to the electrometer input. Due to the high input impedance of the electrometer the capacitor was effectively open-circuited. The voltage build up was recorded and its rate of change evaluated. As in the case of current measurements, the charging interval was made long enough so that by increasing it further no appreciable change in voltage recovery was observed. The second method of measuring voltage recovery was by means of the analog computer. The capacitor was used as the feedback for a high gain amplifier. The capacitor was charged to V8 volts by using the initial-condition setting capability with the computer in RESET. With the computer set to OPERATE, the capacitor was discharged briefly by short-circuiting its terminals. The build-up of voltage on the capacitor then appeared as the output voltage of the high gain amplifier. This was amplified to a readable value by connecting the output of the amplifier to two summing amplifiers in series connected to multiply by a constant high enough to give a convenient voltage for measurement.

Results of the two methods of measuring voltage recovery are shown in Figure A2.10, in which the discharge current is also plotted for comparison. The method in which the computer is used is certainly the most convenient for determining the absorption characteristics of the capacitors used in a particular computer, since no additional equipment is required, and the data can be obtained without removing the capacitor from -the computer. One difficulty in using the computer for voltage recovery measurements results from the integrator drift which produces a voltage output increasing linearly with time which may be of the same order of magnitude as the capacitor recovery voltage.l This effect can be minimized, however, by making two tests using the same amplifier, the second test with the charging voltage equal in magnitude but opposite in sign to the first. When the results of the two tests are subtracted, the integrator drift voltage will be eliminated. This was done to obtain the data plotted in Figure A2.10. H. Comparison of Results with Theory It was mentioned earlier that the simple description of a dielectric developed by Debye [25] and others must be modified by considering a particular type of polarization to be characterized not by a single relaxation time but rather by a distribution of relaxation times. One method of doing this was developed in the preceding sections. Another method is described below. An empirical relationship developed by Cole and Cole [30] based on the experimental observation of dielectric materials riodifies 1 It is shown in Chapter IV that the integrator drift will be due only to the grid current if there is no input; that is, if the input resistance is infinitely large.

-117 -m 0 -> U O 0 " -' —z DI SCHA RGE CURRENT x RATE OF VOLTAGE RECOVERY (ELECTROMETER) 0 0 RATE OF VOLTAGE RECOVERY x> |\ (COMPUTER) 0 Ldx 0 0 -I0 o z x 'IJ W: tn -7 0 and Rate o Voltage Recovery and Rate of~ Voltage Recovery

the complex dielectric constant of Equation (A2.2) to give E* =E+ + " -C |@( j W To) (A2.38) where O( is an empirical constant with values from 0 to 1, a measure of the distribution of the relaxation times. Cole and Cole, in a second paper [31], have evaluated the transient current following the application of a constant voltage to a capacitor having a dielectric constant given by Equation (A2.38). These results have been presented by Field [32] in a manner which facilitates comparison with experimental data. The experimental discharge curve for the Type I capacitor (Figure A2.8) was found to agree with the theoretical discharge curves given by Field [32] if the parameters have the following values: oe = o.6 1'w = 7.0 seconds Co C = _10-3 (A2.39) coo Although the method of Cole and Cole permits the complex capacitance to be described by fewer parameters than does the method of this investigation, the latter method lends itself more readily to the error analysis which is the purpose of this thesis. The complex capacitance has been obtained in the time domain by the method of this report and by the method of Cole and Cole. It is informative to compare the complex capacitance obtained by the eqnc two methods by examining them in the frequency domain.

-119 -The phase shift, or loss tangent, of the complex capacitance is given by C _ ta n = C' (A2.40) For the complex capacitance given by the method of Cole and Cole, Smyth [33] has shown that the real and imaginary parts are given by C Coo o - coo s i nh [(i- )X] 2 | cosh[O-o')X] + si l (o(r/2J (CO C.)cos(o T/z) 2{coshl-o)XJ + sin r/)} (A2.41) where X = In co r. The expression for the phase shift obtained by the method of this investigation is given by Equation (A2.26). Figure A2.11 is a plot of the angle S vs. frequency for the two methods. Table III presents a summary of the comparison between the method of Cole and Cole and the method of this investigation for the Type I capacitor. It should be mentioned that an attempt to extrapolate the results of these experiments to predict the capacitor characteristics for frequencies outside the range given by experimental data would be

COLE 8 COLE 'o --- APPROXIMATION THIS INVESTIGATION 2.0 -_ID,._> 1.5 low, C). 0.5- I 0 0.001 0.01 0. I 1.0 10 100 FREQUENCY (rad/sec) Figure A2.11 Phase Angle of the Complex Capacitance vs. Frequency

-121 -TABLE III. CAPACITOR PARAMETERS BY TWO METHODS This Invetigatic- Cole and Cole 6o - CO 0.923 x 10-9 f 1.000 x 10-9 f 6^* (1.82 x 10-4 rad 1.62 x 10-4 rad ~Crn ** 0.158 rad/sec 0.145 rad/sec * Em is the maximum value of. ** W)n is the frequency at which $ has its maximum value. of doubtful value. Particularly at the higher frequencies the dipole polarizations would begin to appear, causing the loss tanjent to increase; this would be evidenced in Figure A2.11 by further peaks in S at higher frequencies [24]. Furthermore, since a finite time was required to discharge C, the determination of the Ck corresponding to the shortest relaxation time is undoubtedly somewhat low. This would cause 8 in Figure A2.11 also to be lower than its true value for frequencies above about 5 rad/sec. 1 The transient current was measured for discharge times between about 0.1 and 600 seconds. The data thus obtained permitted determination of the capacitor phase angle for frequencies of approximately 0.002 to 10 rad/sec.

APPENDIX III. APPLICATIONS TO TYPICAL COMPUTER SET-UPS In this appendix several examples of typical computer set-ups are analyzed by the methods developed in Chapter IV. The examples are not all restricted to the computer set-up in Figure 4.1, and thus serve to illustrate the general method of approach. A. Simple Harmonic Motion The equation to be solved is X + C X =O (A3.l) and Figure A3.1 shows the computer set-up. Ri RI R2 Figure A3.1 Computer Set-up for Simple Harmonic Motion (1) Configuration I Let R C, R -C = R I Ri= -;-. (A3.2) Then the set-up is as shown in Figure 4.1 and the machine equation can be found directly from Equation (4.10), which becomes i* Y-,' r,l - (T~ +1 T) s * c X - o. -122 -

If the three tamplifiers have identical gain vs. frequency characteristics, T5=.. TS II T,, - - C GO T., rid RLC &0 (A3.4) and Equation (A3.3) becomes F 2 2 3_ S+4] 0.2 + + 2 + -O. (A3.5) Equation (A3.5) is modified by the effect of dielectric absorption, as discussed in Chapter IV. The characteristic roots fromi Equation (A3.1) are Si = -ij. The roots of the machine equation (considering only dielectric absorption) are from Equation (4.79) [ ~I C 2 l s )OTpH X(A3.6) which simplifies to WCDKTPk F, CDk -s ~ Z W C 2 ] L ~ |W T] CO (A5.7) Referring to Appendix II, Equation (A2.26), one sees that the real part of S is just — W times the phase angle & of the complex capacitance. The imaginary part of S is simply related to the real

-124 -part of the complex capacitance C, given by Equntion (A2.21). (The symbol C refers to the same quantity denoted by COO in Appendix II.) Making these substitutions yields s -wX8 ijC( c) (A3.8) Since C is very nearly equal to C,the characteristic equation can be written approximately as S2 2 $c2, WS cWz2 0. (A5.9) From Equation (A3.9) it is seen that the damnping ratio is just the phase angle 8, which from Figure A2.11 is a function of frequency. Combining Equations (A3.5) and (A3.9) yields for the machine equation 2 2- 2- 3w +* - 2 or 52 f 2505 X2= (A3.10) where + + $. 11 3w+iW3 l RLC GJ 2 w (A3.11) (2) Configuration II Let RIC, R2C2 = ~~Ri~~~~~ i~' ~(A.12)

-125 -The machine equation is still given by Equation (A3.3), but the time constants are now T 2 Ts = -~0 T = ~I I I I 0 T11 RLC r (A 3.13) and the characteristic equation becomes URLC 4 ]O s (A3.14) Since the effect of dielectric absorption is the same as for Configuration I, the damping ratio for Configuration II becomes I + I ZW c aW RLC Gc WO RLC (A3.15), B. Example of a Servomechanism The equation for a servomechanism with position, error rate, and integral control as given by Howe [34] is o - k (E+ Ce 0 CE t O (A3.16) where eo = the servo output E = 0 - ei = the error e& = the servo input

-126 -k = constant, determined by servo inertia and gain Ce and Ci = constants which determine the amount of error rate and integral control respectively. Figure A3.2 shows a computer set-up which, with ideal components, would solve Equation (A3.16). To prevent the equations from becoming unduly long, all the integrators and summers are assumed to have, respectively, the same time constants. Potentiometers are used to represent the constant coefficients, and feedback and input resistors and feedback capacitors are not shown. (This assumption imposes no limitation on the application of the method, however.) -Ce8 eo Figure A3.2 Computer Set-up for Servomechanism Comparing voltages at y gives

-127 -k I 1-c,3 y —k y s,)~(r'-( +)-(r, s,1)3(T, S++) 3(-r (ss) + ~CeT; 1 k i (Tnsl+,)('f s+()( rss.t) T s+ (A3.17) Also, e; - eo kE s s +I Ts s+I (A.18) Multiplying through by + 3 and rearranging yields T, 3 _r_( __ k (Tst) k C; + kkCe(,S) (Ts t. )3 T2 (rTs + I)(Ts tl)l 3' (A3.19) Making use of the assumption that J << $ < eI (A3.2o0)

-128 -Equation (A3.19) becomes 6[S3( +l r3 )+ks(lt i -T - 2T2s) +kCi(l-2Ts -3Ts) + kCeS2(1+ 2 TSs - T2s T, (:=o s3 (1+ T). After collecting terms this becomes eo+ 3 o -k *- 2-Te c;-(3r2-+r} E,(Ce-TS -2T,)E [{ - Ce (Tz + 2 s) (c + —)fE dt] = 0 (A3.21) Equation (A3.21) is seen to be of the same order as Equation (A3.16); the coefficients of the various terms have been changed, an effective viscous damping term has been added, and a higher derivative of the input, resulting from the term containing E, appears. C. Equations of Motion of an Aircraft As an example of the application of the method to simultaneous differential equations, the linearized equations of the longitudinal motion of an aircraft are used. These equations and a computer set-up for solving them are given by Howe [35]; to keep the resulting equations from becoiing inconveniently lengthy it is assumed here that the elevator control surface is locked and the variation in total aircraft

velocity is negligible. Then the equations of motion reduce to Xk =kzD +(l+ k q)q q = koso t k t, c + k,q (A-.22) where Z and are perturbations in angle of attack and pitch rate, and the constants kzO, etc., are the stability derivatives determined by aircraft geometry, equilibrium flight conditions, altitude, etc. The computer set-up to solve Equations (A3.22) is shown in Figure A3.3 which is taken from Howe [35], simplified according to the above assumptions. Since the computer set-up is not the same as in Figure 4.2, the equations developed in Chapter IV are not directly applicable, but the same method of analysis applies. Referring to Figure A3.3, the outputs of summers 3 and 5 are evaluated in terms of -the inputs on the left. Note, however, that the output of integrator 4 is actually not -- but rather q divided by the transfer function of summer 5. Similarly - a is the output of sumnmer 3 divided by the transfer functions of integrator 2 and summer 3. The machine equations are -50(T = -k2 q _. __S~)(rs+I) =t (T;4...... i)S (A.25 )

-130 -Kz /5 q o -Kqo /50 0.1 -5a 0 ~-q q1 -Kqq /4 0.25 Figure A3.3 Computer Set-up to Solve the Equations of Aircraft Motion

-1)1 -From the col:puter diagram it is seen thatTS,14T,;T =Ts T and if I I I I I — ) - and - > S > and, the equations become TS, TS 3 rT + Ts c0(s + # ) = k,,o(1- 7rs) + k,4 (I-STiS) + (I - 72 S) qs+-~ E+) kS o( I- 3T S )+ k q(-T2s)+k~a e rk ) Finally these become (I+ 7T2 kz)( T+ T2(5kz+ i = (ykZk. )kc( +( + k2)q (-k +3T2 k)ck - (( kT') = (Ik + ) k (A3.24> Except for the terms containing TS, Equations (A3.22) and (A3.24) are seen to be identical. Thus the effect of the errors is to change the stability derivatives and add the term containing ~ to the left side of the first equation. (This term can be thought of as causing a further change in the stability derivatives. If is eliminated from the first equation by substituting its value from the second, only those terms remain which can be compared directly with the corresponding terms of Equation (A3.22). This procedure would be analogous to that described in Chapter IV in which the machine equation

-132 -was put into the same form as the given equation by multiplying both sides by an inverse matrix.) The machine equations can be further compared with the given equations, if desired, by evaluating the characteristic equations from Equations (A3.22) and (A3.24).

APPENDIX IV. EXPERIMENTAL VERIFICATION OF THEORY In this appendix experiments are described which were conducted for the purpose of comparing the errors predicted by the theory previously developed with the errors actually observed by experiments on typical differential analyzers. A. Nature of the Experiments It was shown in Appendix III that the damping ratio for the machine equation with the computer set-up shown in Figure A3.1 is given by 5 I Xo ILC Go 2 O I F I +2 3w+c I =wRLC GO _ O (A4.1) where the subscripts on Y refer to Configurations I and II. The analog computers used for the experiments were the Sterling Computerl and the REAC2. To determine the parameters G0 and L for the Sterling Computer, frequency response measurements were made for the open loop amplifiers. These showed some variation in G0 between amplifiers, but W0 was virtually the same for all. On the average they agreed with the same data given in the instruction manual for the Sterling Computer [36]. Figure A4.1 shows the gain vs. frequency characteristics for the three amplifiers used in the experiments described below. The values of the amplifier parameters 1 Sterling Model LM-10 Electronic Differential Analyzer, Sterling Instruments Company, Detroit, Michigan. Reeves Electronic Analogue Computer C-101 Mod 5, Reeves Instrument Corporation, New York, N. Y. -135 -

Go 100 x X * AMPLIFIER I o AMPLIFIER 2 80 X AMPLIFIER 3 60 z~~~~~~~~ 40 20 /2v 0 I I\' 00 I 10 100 IKC 10 KC 01KC 1000KC FREQUENCY (cps) Figure A4.1 Frequency Response of the Open Loop Amplifiers (Sterling Computer)

-125 -are seen to be = 2.0 x 105 and C0 = 5.0 x 106 rad/sec. The values of RLC and S (w) were taken from the measurements described in Appendix II. ( RLC = 5.0 x 106 and S () as oiven by Figure A2.11.) 'he gain characteristics of the REAC were determined by measuring only -the DC gain and comparing this with the data given in the instruction book [37]. The DC gain of the direct coupled amnplifiers was found to be the same as that given in the instruction book but the DC gain of the auxiliary chopping amplifiers was found to be considerably below that given. Figure A4.2 shows the amplifier frequency response as given by the REAC instruction book [37] after being corrected for the lower auxiliary amplifier gain.l It will be noted -that the REAC amplifier frequency response is similar to that shown in Figure Al.l. The significant parameters were determined from Figure A4.2 and were found to be GO = 5.0 x 106 and TA = 5 seconds. From the analysis of Appendix I it follows that the equivalent value of OJO for use in Equations (A4.1) for the REAC will be 2 = 5.0 x 105 rad/sec. 2TA The value of the leakage resistance of the REAC capacitors -was determined by charging the capacitor and measuring the decrease il voltage after several days. It was found that RL 20 x 1012 ohms and its contribution would be insignificant compared to the other terms in Equation (A4.1). 1 The correction was made by adjusting the amplifier DC gain to agree with the measured value and by assuming that the rate of fall-off with frequency was the same as that given in the instruction book.

140- TA 130 \-12 DB/OCTAVE 120 I,10 \\ COMBINED 100\ 1 AMPLIFIERS 100 w=-. 90 80 DIRECT COUPLED 70 AMPLIFIER -6 DB / OCTAVE 50 -a 40 -X AUXILARY CHOPPING 30- AMPLIFIER 20 -I0 -0.01 0.1 1 10 100 IKC IOKC FREQUENCY (cps) Figure A4.2 Frequency Response of the Open Loop Amplifiers (REAC)

-137 -The capacitor phase angle 8 was determined for the REAC capacitors by the voltage recovery method described in Appendix II, using the computer to make the measurements. The results are shown in Figure A4.5. The irregular curve results from using only three resistor-capacitor combinations to represent the dielectric absorption characteristics; -the smoothed curve is an estimate of the result which would be obtained by a more precise approximation. The smoothed curve is taken to be the correct one in the subsequent analysis. Substituting the appropriate values in the expressions for 5I and 51 gives values of damping ratio as a function of frequency. These are plotted in Figures A4.4, A4.5, and A4.6. The Sterling Collputer was set up in Configurations I and II and the REAC in Configuration I. The dmnping ratio was determined by measuring the timie for the oscillations to decay or grow to a given fraction of the initial amplitude. The results are plotted on Figures A4.4, A4.5 and A4.6. (Although not presented here, similar data were measured on combinations involving several other amplifiers in groups of three and when the proper parameters were substituted in the expressions for the agreement with theory and experiment was similar;o that shown here. Also, no differences in results were noted by starting the oscillations at different amplitudes; initial amplitudes varied from 100 volts to a fraction of a volt for some frequencies, and in each case the same damiping ratio was observed.) B. Discussion of Results The comparison of experimental results -with those predicted by theory is quite good in the frequency range tested for Configuration I. The portion of damping contributed by the dielectric properties of

-138 -o -APPROXIMATION 1.0 --- SMOOTHED CURVE O 0.8 0 _.0 < 0.4 J0.2 0. on I / \ 0.001 0.01 O. I 1.0 10 FREQUENCY (rod/sec) Figure A4.3 Phase Angle of Complex Capacitance for REAC Capacitor

4 3 TOTAL DAMPING RATI 0 2C DAMPING RATIO DUI T DIELECTRIC A BSORPTO 0 x 0.1 1.0 10 -nI -It o ~~~~~~(rod/ sec) -2C —THEDRY o1 20 o EXPERIMENT -10 0 x -20 -30 Figure A4.4 Damping Ratio Introduced by the Computer, Configuration I (Sterling Computer)

THEORY 0 EXPERIMENT TO TAL DAMPING RAT IO DAMPING RATIO DUE TO O DIELECTRIC ABSORPTION O O.I I I 0.1 1.0 10 w (rad/sec) Figure A4.5 Dmping Ratio Introduced by the Computer, Configuration II (Sterling Computer)

3 DAMPING RATIO DUE TO 2- DIELECTRIC ABSORPTION 4 - 10~' I~ - o o 0.1 JJ 0 0o.1 1.0 X 10 I x -I (road/sec) ~ TOTAL DAMPING AJ- RATIO THEORY -3C o EXPERIMENT Figure A4.6 Damping Ratio Introduced by the Computer, Configuration I. (REAC)

-142 -the capacitor in this configuration is indicated in Figures A4.4 and A4.6, and for frequencies between about 0.1 and 5.0 rad/sec for the Sterling Computer and 0.1 and 2.0 rad/sec for the REAC it is seen to be the major portion of the damping. Above and below these frequencies the gain properties of the amplifiers are more significant. For Configuration II the damping is due almost entirely to the dielectric properties in the frequency range tested. Agreement is seen to be good up to about 4 rad/sec. Beyond this frequency the capacitor loss angle is apparently greater than was determined from the measurements in Appendix II. However, as was mentioned in that appendix, the results at the higher frequencies were considered to be less reliable than at lower frequencies. In Configuration I this difference became insignificant because of the great increase in the contribution from the amplifier characteristics at the higher frequencies. It was shown in Chapter IV that as a general rule it is better to divide multiplication among as many amplifiers as possible. Figures A4.4 and A4.5 provide striking evidence of the effect of this rule. Although the computer in each configuration is solving the same equation, in Configuration I the necessary multiplication is performed by a single summer while in Configuration II it is evenly divided between the two integrators. The error in Configuration II is less than that in Configuration I for all frequencies except a rather narrow range where the error for Configuration I goes through zero. For frequencies below 0.1 and above 10 rad/sec the advantage of Configuration II is unmistakable.

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UNIVERSITY OF MICHIGAN I3 9I01 03527 i38011111 3 9015 03527 5380