• TABLE OF CONTENTS
HIDE
 Title Page
 Dedication
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Abstract
 Introduction
 Calculation of steady-state...
 Frequency-domain aspects of nonlinear...
 Considerations for oscillator...
 Summary and suggestions for future...
 Appendix
 Biographical sketch
 Copyright














Title: Harmonic analysis and design of nonlinear electronic circuits
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Permanent Link: http://ufdc.ufl.edu/UF00082484/00001
 Material Information
Title: Harmonic analysis and design of nonlinear electronic circuits
Alternate Title: Nonlinear electronic circuits
Physical Description: xi, 131 leaves. : illus. ; 28 cm.
Language: English
Creator: Wayne, David Anthony, 1944-
Publication Date: 1971
 Subjects
Subject: Harmonic analysis   ( lcsh )
Electronic circuit design   ( lcsh )
Nonlinear mechanics   ( lcsh )
Electrical Engineering thesis Ph. D   ( lcsh )
Dissertations, Academic -- Electrical Engineering -- UF   ( lcsh )
Genre: bibliography   ( marcgt )
non-fiction   ( marcgt )
 Notes
Thesis: Thesis - University of Florida.
Bibliography: Includes bibliographies.
General Note: Manuscript copy.
General Note: Vita.
 Record Information
Bibliographic ID: UF00082484
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
Rights Management: All rights reserved by the source institution and holding location.
Resource Identifier: aleph - 001035866
oclc - 18271073
notis - AFB8235

Table of Contents
    Title Page
        Page i
    Dedication
        Page ii
    Acknowledgement
        Page iii
    Table of Contents
        Page iv
        Page v
    List of Tables
        Page vi
    List of Figures
        Page vii
        Page viii
        Page ix
    Abstract
        Page x
        Page xi
    Introduction
        Page 1
        Page 2
        Page 3
        Page 4
        Page 5
        Page 6
        Page 7
        Page 8
        Page 9
        Page 10
        Page 11
    Calculation of steady-state response
        Page 12
        Page 13
        Page 14
        Page 15
        Page 16
        Page 17
        Page 18
        Page 19
        Page 20
        Page 21
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        Page 24
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        Page 46
        Page 47
        Page 48
        Page 49
        Page 50
        Page 51
        Page 52
    Frequency-domain aspects of nonlinear circuit design
        Page 53
        Page 54
        Page 55
        Page 56
        Page 57
        Page 58
        Page 59
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        Page 75
        Page 76
        Page 77
        Page 78
        Page 79
        Page 80
        Page 81
    Considerations for oscillator design
        Page 82
        Page 83
        Page 84
        Page 85
        Page 86
        Page 87
        Page 88
        Page 89
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        Page 103
        Page 104
        Page 105
    Summary and suggestions for future research
        Page 106
        Page 107
        Page 108
    Appendix
        Page 109
        Page 110
        Page 111
        Page 112
        Page 113
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        Page 124
        Page 125
        Page 126
        Page 127
        Page 128
        Page 129
        Page 130
    Biographical sketch
        Page 131
        Page 132
        Page 133
    Copyright
        Copyright
Full Text















Harmonic Analysis and Design
Of Nonlinear Electronic Circuits











By



DAVID ANTHONY WAYNE


A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLHl T
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY






UNIVERSITY OF FLORIDA


1971 .


























To

Sharon, my wife
and
Trudy and Ted, my children:

I lock forward to getting to know them again.















ACKNOWLEDGEMENTS


I wish to express sincere appreciation to

Dr. S.W. Director, Chairman of my Supervisory Committee,

for his guidance and enthusiasm during this research.

Thanks are due also to the members of the Supervisory

Committee, Dr. F.A. Lindholm, Dr. E.R. Chenette,

Dr. A.J. Brodersen, and Dr. A.W. Westerberg, for reviewing

and commenting on this work.


Finally, I wish to express loving thanks to my wife,

Sharon, for her patience and understanding over the past

three years and for her excellent typing of this manuscript.


Financial support from an NDEA Title IV Fellowship

and an NSF Traineeship is gratefully acknowledged.


iii
















TABLE OF CONTENTS
Page

ACKNOWLEDGEMENTS . . . . . . . . . iii

LIST OF TABLES . . . . . . . . vi

LIST OF FIGURES . . . . . . .... . . vii

ABSTRACT. . . . . . . . . .. .. x

CHAPTER 1
INTRODUCTION. .... . . . . . . . . 1
1.1 Background . . . . . . . . 1
1.2 Time-Domain Design . . . . . . 4
1.3 References . . . . . . ... . 7

CHAPTER 2
CALCULATION OF STEADY-STATE RESPONSE . . ... 12
2.1 Background . . . . . . . . 12
2.2 The State Variable-Newton Method
of Steady-State Calculation. . . . .. 34
2.3 The Adjoint Network-Newton
Method of Steady-State Calculation .. .. 37
2.4 References . . . . . . . .. 48

CHAPTER 3
FREQUENCY-DOMAIN ASPECTS OF NONLINEAR
CIRCUIT DESIGN. .. . . . . . . . .53
3.1 The Design Algorithm . . . . . .. 55
3.2 Design Examples. . . . . . . .. 64
3.3 Discussion . . . . . . . . 78
3.4 References . . . . ... . . . 80

CHAPTER 4
CONSIDERATIONS FOR OSCTILTLT-OR DESIGN .. 82
4.1 Determination of Frequency of
Oscillation. . . . .... .. . . .. 82
4.2 Calculation of Gradient Components
for Automated Oscillator Design ..... 88
4.3 References . . . . . . . . 105

CHAPTER 5
SUMMARY AND SUGGESTIONS FOR FUTURE RESEARCH . 106











Page


APPENDIX A
DERIVATION OF GRADIENT COMPONENTS . . .. 109
References . . . . . . . . . 122

APPENDIX B
FLOW CHART OF DESIGN ALGORITHM . . . . 123

BIOGRAPHICAL SKETCH. . . . . . . . 131















LIST OF T;.ELE3


Table P age

2-1 Results of Application of Galerkin's
Method to Circuit of Fig. 2-1. . . .. 18

2-2 Results of Application of Adjoint
Network-Optimization Method to
Circuit of Fig. 2-1. . . . ... . 32

2-3 Results of Application of Adjoint
Network-Optimization Method to
Circuit of Fig. 2-6 . . . . .. 35

2-4 Results of Application of Steady-
State Determination Methods to
Circuit of Fig. 2-6. . . . . . ... 50

2-5 Results of Application of Steady-
State Determination Methods "o
Circuit of Fig. 2-7 .. . . ..... 52

3-1 Transistor Parameters ........... 65

3-2 Results of Application of Design
Algorithm to Circuit of Fig. 3-3 . . .. 73

3-3 Results of Application of Design
Algorithm to Circuit of Fig. 3-7 . . .. 79















LIST OF FIGURES
Figure
2- 1 Circuit used in examples of steady-
state determination methods. . . .

2- 2 Flow of operations for implementation
of Galerkin's method ... . . . .

2- 3 Capacitor and inductor initial-
condition models
(a) Linear time-invariant. . . .
(b) Nonlinear. . . . . . .

2- 4 The original and adjoint networks used
ir gradient calculation. . . . .


Page


15


. . 17



. . 20
. . 20


. . 23


2- 5 Flow of operations for adjoint network-
optimization method of steady-state
calculation. . . . . . . . ... .31

2- 6 Voltage doubler circuit of Lhe
second example . . . . . . . . 33

2- 7 Flow of operations for adjoint network-
Newton method of steady-state calculation; . 47

2- 8 Circuit for adjoint network-Newton method
example. . . . . . . . . 51


3- 1 Flow of operations for automated
design algorithm . . . . . . .

3- 2 Arrangement of adjoint network to
obtain gradient components . . . .

3- 3 The shunt-series feedback pair
amplifier of the first example . . .

3- 4 Initial configuration for example
of Fig. 3-3
(a) Output waveform. . . . . .
(b) Frequency-domain components
of output signal . . . . .
(c) Adjoint network excitation. . ..


. 60


S. 62


. 66



. 67

. 67
. 67










Figure


3- 5 Circuit configuration for example
of Fig. 3-3 after one design
iteration
(a) Output waveform. . . . . ... 69
(b) Frequency-domain components
of output signal . . . ... 69
(c) Adjoint network excitation .. ... 69

3- 6 Final circuit configuration for
example of Fig. 3-3
(a) Output waveform. . . . . . 71
(b) Frequency-domain components
of output signal . . . . .. 71
(c) Adjoint network excitation . .. 71

3- 7 The large-signal audio power amplifier
of the second example . . . . ... 74

3- 8 Initial circuit configuration for
example of Fig. 3-7
(a) Output waveform. . . . . ... 75
(b) Frequency-domain components
of output signal . . . ... 75
(c) Adjoint network excitation ..... 75

3- 9 Circuit configuration for example
of Fig. 3-7 after one design iteration
(a) Output waveform. . . . . ... 76
(b) Frequency-domain components
of output signal . . . ... 76
(c) Adjoint network excitation . .. 76

3-10 Final circuit configuration for
example of Fig. 3-7
(a) Output waveform. . . . . ... 77
(b) Frequency-domain components
of output signal . . . ... 77
(c) Adjoint network excitation . . 77

4- 1 Oscillator circuit
(a) Original network . . . . .. 83
(b) Adjoint network. . . . . ... 83

4- 2 Output waveforms of network of Fig. 4-1
(a) Original network output voltage v 84
(b) Adjoint network output voltage 0 84

4- 3 Phase plane diagram for an oscillator . 86


viii


Page










Figure

4- 4 (a) Negative-resistance oscillator.
(b) Characteri'stics of negative-
resistance device . . . .

4- 5 Original and adjoint networks used
to calculate required partial
derivatives. . . . . . . .

4- 6 RC feedback oscillator network . .

4- 7 An oscillator circuit which may not be
partitioned. . . . . . . .

A- 1 Networks used to obtain the summation
terms in the gradient component
expressions. . .' . . . . .

B- 1 Flow chart of main -program . . .

B- 2 Flow chart of SEARCH routine .. ....

B- 3 Flow chart of FUNCT routine . . .

B- 4 Flow chart of DCANAL routine . . .

B- 5 Flow chart of FOURR routine. . . .


Page

. . 90

. . 90



. . 96

. . 103


. . 104



. . 119

. . 124

. . 125

. . 126

* . 129

* . 130










Abstract of Dissertation Presented to the Graduate Council
of the University of Florida in Partial Fulfillment
of the Requirements for the Degree of Doctor of Philosophy



HARMONIC ANALYSIS AND DESIGN
OF NONLINEAR ELECTRONIC CIRCUITS


By


David Anthony Wayne

December, 1971


Chairman: Dr. Stephen W. Director
Major Department: Electrical Engineering


Methods are developed for the automated analysis and

design of nonlinear networks whose performance is charac-

terized in frequency-domain terms. Such circuits are

generally found in communications systems and include large-

signal amplifiers, mixers, frequency multipliers, oscillators,

and others. These design procedures utilize the efficient

"adjoint network" method, sparse matrix techniques and the

fast Fourier transform.


The problem of determination of steady-state response of

a nonlinear network to a periodic input is also treated. Two

methods are developed which calculate this response efficiently.


Special considerations for oscillator design are

discussed. The adjoint network corresponding to an

oscillator network may exhibit an unstable response: one










which grows without bound. Methods for dealing with this

difficulty are presented. Techniques are also developed

for calculating frequency of oscillation.


Several computer programs have been written to utilize

the algorithms presented. A number of problems using these

programs are presented to demonstrate the effectiveness of

the procedures.














CHAPTER 1

INTRODUCTION


The purpose of this work is the development of methods

for the automated design of nonlinear electronic networks

whose response is characterized in frequency-domain terms.

These circuits are generally found in communications systems

and include large-signal amplifiers, mixers, frequency

multipliers and dividers, oscillators and others.


1.1 Background


The analysis and design of nonlinear networks by

conventional analytic or graphical means are difficult and

tedious. In addition, these methods are generally applicable

only in situations in which one or two nonlinearities dominate

the behavior of the circuit. The resulting solution may only

be a first-order approximation and detailed information about

the circuit found only through the construction of prototypes,

an expensive luxury in_';:t -j sted circuit production.


High-speed digital computers have had a profound impact

on the methods of circuit analysis and design. Computer

methods make practical the numerical simulation of large

nonlinear circuits without unrealistically limiting the

complexity of the nonlinearities. Development of automated





2


circuit analysis. algorithms began in the early 1960's. Programs

such as ECAP [1] and SCEPTRE [2] allowed the circuit design

engineer to accurately analyze a network. The designer then

modified the circuit, using his experience and intuition.

Again the computer was used to perform the analysis task. This

procedure was repeated iteratively until the design goals were

met (or until the designer gave up).

Increasing circuit complexity has been the incentive for

the development of fully automated design methods. In these

procedures, the computer is given an active role in the design

cycle. Moreover, in addition to analyzing the network, the

computer determines how the design parameters must be adjusted

in order to achieve the design goals. Techniques that may lead

to fully automated design have evolved through the efforts of

many researchers [3,4,5]. Hachtel and Rohrer [6,7] first

presented in 1967 a highly efficient scheme for automated

design which used parameter optimization. In their approach,

variational calculus methods were applied to the state-variable

formulation of the network equations to obtain the required

parameter-space error gradient. Director and Rohrer [8]

developed even more practical algorithms based on the "adjoint

network" concept. The adjoint network technique has been

successfully applied to dc [91 and small-signal ac [10] net-

work design. Simultaneous design for dc and small-signal dc

performance has also been investigated [11,12]. Time-domain

design of switching circuits has been described by several



References will appear at the end of each chapter.









authors [13,14,15]. Time-domain automated design provides

the basis for the present work.


For a time-domain design scheme to be practical, the

required numerical integration must be done efficiently.

Computational effort required for numerical integration is

roughly proportional to the number of increments, or "steps,"

into which the time interval is divided. The "order" of a

numerical integration method may be defined as the highest

order of a polynomial in the time variable to which the method

would give an exact solution. Higher order methods may be

seen to provide a better approximation where the responses

are rapidly varying. Gear [16,17] has proposed a scheme in

which order of integration and time step are both adjusted to

minimize computational effort. Calahan [18] has developed

techniques for the application of Gear's algorithm to time-

domain network analysis.


The system of equations which describes a network is

frequently "sparse"; that is, the coefficient matrix contains

relatively few non-zero entries. Exploitation of this sparse-

ness has two advantages: First, computer memory requirements

are reduced by saving only the non-zero coefficients. Second,

only the non-zero entries need to be processed during the

solution--resulting in a time savings. Techniques which take

advantage of sparseness: in-network equations have been

developed by Gustavson et al. [19], Tinney and Walker [20],









and others. A novel design technique based on sparse matrix

manipulation methods--called the "tableau" approach--has been

presented by Hachtel et al. [21,22].


1.2 Time-Domain Design


Time-domain automated design begins with specification of

desired time responses v(t) and i(t), as well as an initial

network configuration. A performance function which reflects

the design specifications is then formulated. A suitable

performance function is the least-squares function:
tf 1 2
(x) = W/ {v(t)[v(t,x) v(t)
o0 (1.1)
+ wi(t)[i(t,x) i(t)]2}dt,

where x represents the vector of designable parameters and

w and wi are non-negative weighting functions which allow for

design flexibility. At its minimum, E(x) yields the optimum

set of network parameters x. This minimum is found by a

technique such as the method of steepest descent, the Fletcher-

Powell method, or another function minimization scheme [23,24,25].

These procedures minimize the performance function iteratively

by adjusting the parameter vector in a direction dependent

on the gradient

SxT
~= -i'xl' "x2 X (1.2)


(where superscript T indicates transposition).









For these optimization schemes to be useful, the

performance function e and the gradient Vc must be found in

an efficient manner. A successful technique for calculation

of the gradient components is the adjoint network approach

mentioned previously. This method evolves from application

of Tellegen's theorem [26] to the network of interest and a

topologically equivalent adjoint network.

A brief description of the method to calculate the

gradient components follows:

1. Analyze the original network. Retain branch
voltages and currents of designable elements
(vB(t) and iB(t)).

2. Analyze the adjoint network in reverse time
(T = t + tf t). Retain branch voltages
and currents of designable elements (B (T)
and (BT))

3. Compute the gradient components:

t
S iR(t) R(t + tf t)]dt (1.3)
to
for resistor branches,

s tf dvC(t)
c / dt )c(t0 + tf t)]dt (1.4)
t
for capacitor branches, and


t di (t)
= / dt L (t0 + t t)Idt (1.5)
to
for inductor branches.

S(These expressions are derived in Appendix A.)










Notice that only two complete network analyses are

required to obtain all gradient components. There are,

however, drawbacks in the method. Due to the time reversal

in the adjoint network analysis, values of branch voltage

and current must be stored for all time points during the

analysis of the original circuit. The large number of

values to be saved may cause storage problems. Also, since

the adjoint network excitation may not be a smooth time

function, the adjoint network must be analyzed with care.


In some nonlinear transient applications, design specifi-

cations may be more meaningful if expressed in terms of

frequency rather than time. In the following chapters, we

present techniques for dealing with several aspects of non-

linear network design for frequency-domain specifications.

The fast Fourier transform is used to obtain harmonic infor-

mation about the network response. A performance function

is then formulated which reflects the frequency-domain aspects

of the design requirements. The parameter-space gradient of

this performance function is found by use of the adjoint

network method. The performance function is minimized to

realize the design requirements. In Chapter 2, we discuss

techniques for calculation of the steady-state response of a

nonlinear network driven by a periodic source. Use of the

fast Fourier transform to obtain harmonic information requires

that the network be in the steady-state condition. Network

examples illustrate the effectiveness of the steady-state










calculation. The algorithm for harmonic automated design is

developed in Chapter 3. Two practical design examples are

included. In Chapter 4, we consider special techniques

required for automated oscillator design. The determination

of oscillation frequency is one of the problems discussed.

Chapter 5 contains a summary and recommendations for future

research.


1. 3 References


1. "IBM 1620 Electronic circuit analysis program," IBM

Corporation, White Plains, New York, Application Program

1620--EE-02X, 1965.


2. H.W. Mathers, S.R. Sedore, and J.R. Sents, "Automated

digital computer program for determining responses of

electronic circuits to transient nuclear radiation

(SCEPTRE)," IBM Corporation, Oswego, New York, Technical

Report No. AXWL-TR-66-126, Sceptre Users Manual, February,

1967.


3. D.A. Calahan, "Computer solution of the network reali-

zation problem," Proceedings of the Second Annual Allerton

Conference on Circuit and System Theory, pp. 175-193, 1964.


4. P.O. Scheibe and E.A. Huber, "The application of Carroll's

optimization technique:.to:,.network synthesis," Proceedings

of the Third Annual Allerton Conference on Circuit and

System Theory, pp. 182-191, 1965.










5. D.A. Calahan, "Computer design of linear frequency-

selective networks," Proceedings of the IEEE, Vol. 53,

No. 11, pp. 1701-1706, November, 1965.


6. R.A. Rohrer, "Fully automated network design by digital

computer: preliminary considerations," Proceedings of

the IEEE, Vol. 55, No. 11, pp. 1929-1939, November, 1967.


7. G.D. Hachtel and R.A. Rohrer, "Methods for the optimal

design and synthesis of switching circuits and other

nonlinear networks," Proceedings of the IEEE, Vol. 55,

No. 1 pp. 1864-1877, November, 1967.


8. S.W. Director and R.A. Rohrer, "The generalized adjoint

network and network sensitivities," IEEE Transactions

on Circuit Theory, Vol. CT-16, No. 3, pp. 318-322,

August, 1969.


9. S.W. Director and R.A. Rohrer, "On the design of resistance

n-port networks by digital computer," IEEE Transactions

on Circuit Theory, Vol. CT-16, No. 3, pp. 337-346,

August, 1969.


10. S.W. Director and R.A. Rohrer, "Automated network design--

the frequency-domain case," IEEE Transactions on Circuit

Theory, Vol. CT-16, No. 3, pp. 330-337, August, 1969.


11. A.J. Brodersen, S.W. Director, and W.A. Bristol, "Simul-

taneous automated ac and dc design of linear integrated

circuit amplifiers," IEEE Transactions on Circuit Theory,

Vol. CT-18, No. 1, pp. 50-58, January, 1971.










12. B.A. Wooley, "Automated design of dc-coupled monolithic

broadband amplifiers," IEEE Journal of Solid-State

Circuits, Vol. SC-6, No. 1, pp. 24-34, February, 1971.


13. D.A. Calahan, "Optimization of switching circuits,"

Proceedings of the Second Biennial Cornell Electrical

Engineering Conference, pp. 282-288, October, 1969.


14. P.M. Russo and R.A. Rohrer, "Computer optimization of

the transient response of an ECL gate," IEEE Transactions

on Circuit Theory, Vol. CT-18, No. 1, pp. 197-199,

January, 1971.


15. C.-W. Ho, "Time-domain sensitivity computation for networks

containing transmission lines," IEEE Transactions on

Circuit Theory, Vol. CT-18, No. 1, pp. 114-122, January,

1971.


16. C.W. Gear, "Simultaneous numerical solution of differential-

algebraic equations," IEEE Transactions on Circuit Theory,

Vol. CT-18, No. 1, pp. 89-94, January, 1971.


17. C.W. Gear, "The automatic integration of ordinary

differential equations," Communications of the ACM, Vol.14,

No. 3, pp. 176-179, March, 1971.


18. D.A. Calahan, "Numerical considerations for implementation

of a nonlinear transient circuit analysis program,"

IEEE Transactions on Circuit Theory, Vol. CT-18, No. 1,

pp. 66-73, January, 1971.










19. F.G. Gustavson, W. Liniger, and R. Willoughby, "Symbolic

generation of an optimal Crout algorithm for sparse

systems of linear equations," Journal of the ACM, Vol. 17,

No. 1, pp. 87-109, January, 1970.


20. W.F. Tinney and J.W. Walker, "Direct solutions of sparse

network equations by optimally ordered triangular factori-

zation," Proceedings of the IEEE, Vol. 55, No. 11, pp. 1801-

1809, November, 1967.


21. G.D. Hachtel, F.G. Gustavson, R. Brayton, and T. Grapes,

"A sparse matrix approach to network analysis," Proceedings

of the Second Biennial Cornell Electrical Engineering

Conference, pp. 68-82, October, 1969.


22. G.D. Hachtel, R.K. Brayton, and F.G. Gustavson, "The sparse

tableau approach to network analysis and design," IEEE

Transactions on Circuit Theory, Vol. CT-18, No. 1, pp. 101-

113, January, 1971.


23. R. Fletcher and C.M. Reeves, "Function minimization by

conjugate gradients," Computer Journal, Vol. 7, No. 2,

pp. 149-154, July, 1964.


24. R. Fletcher and M.J.D. Powell, "A rapidly convergent

descent method for minimization," Computer Journal,

Vol. 6, No. 2, pp. 163-168, July, 1963.






11



25. D.G. Wilde and C.S. Beightler, Foundations of Optimi-

zation, Englewood Cliffs, New Jersey: Prentice-IIall,

1967.


26. C.A. Desoer and E.S. Kuh, Basic Circuit Theory, New York:

McGraw-Hill, 1969.















CHAPTER 2

CALCULATION OF PERIODIC
STEEADY-STATE RESPONSE


A rapid method for periodic steady-state analysis is

of utmost importance in the harmonic design of nonlinear

networks. A conventional technique is to merely

numerically integrate the system equations until the response

becomes periodic. In lightly damped systems the initial

transient may be significant for a length of time much

greater than the time of one period. In these cases, the

method of continuously integrating the differential

equations until the transient part of the solution becomes

negligible is unacceptable, since a considerable amount of

computer time is wasted on calculation of the initial

transient. In this chapter, a method is presented which

uses a Newton iteration to obtain the periodic steady-state

response of a nonlinear network.


2.1 Background


Several methods have been described for the determin-

ation of the periodic steady-state response. One such

method proposes the use of a "residual" which indicates

how well an assumed solution satisfies the system

differential equations. This technique is referred to as









Galerkin's method [1], and was used by Baily [2,3] for

steady-state analysis of nonlinear circuits. The solution

of the nonlinear differential equation


x(t) f[x(t), t] = 0 (2.1)


is assumed to be a linear combination of suitable functions.

Since a periodic solution is sought, the solution is

expressed as a Fourier series:

L
x(t) t) Xt) = a + a a sin(nt) + b cos(nt), (2.2)
n=l


where L, the number of harmonics, is chosen to yield suitable

accuracy. Since the solution X(t) is not exact, if it is

substituted into the original differential equation a non-

zero residual results:


e(t) = X(t) f[X(t), t] (2.3)


This residual is a measure of how well the assumed solution

satisfies the original equation. The square of the residual

integrated over the time interval of interest is then

minimized by suitable adjustment of the coefficients a
n
and b Any of the mult-idimensional search routines such
n
as Fletcher-Powell [4] or conjugate gradients [5] may be

used for this purpose. These algorithms require calculation

of the gradient of the residual with respect to the

coefficients a and b
n n









In Baily's formulation of Galerkin's method, the state

equations of the network are written and the above approxi-

mation is made for each state variable. An alternate

method would be to use nodal analysis. The above approxi-

mation would then be used for each node voltage.


Galerkin's method is easily illustrated in terms of

an example. Consider the network of Fig. 2-1. The equation

describing this system ip.


Gv(t) + Cv(t) + f(v) A sin t = 0, (2.4)

where

f(v) = I [exp(Xv) 1]. (2.5)


The assumed solution is

L
V(t) = a0 + [an sin(nt) + b cos(nt)], (2.6)
n=l


and the residual is


e(t) = GV(t) + CV(t) + f(V) A sin t. (2.7)


Since we wish to minimize

t +T
E = f/ [e(t)]2 dt, (2.8)
to


the gradient components -, c ... are required. These
a 1
gradient components are easily calculated:

t 0+T
t0+T
=- = e[G + XI exp(XV)] dt, (2.9)
0 t, s
























id=Is [exp (Xv) -1]


Fig. 2-1
Circuit used in examples
of steady-state determination methods


A sin t









t0+T
C / e[{G + XI exp(XV)}sin(nt) + n cos(nt)]dt,
n t0 (2.10)



t T
t0+T
= / e[{G + XI exp(XV)}cos(nt) n sin(nt)]dt,
n tO (2.11)


for n = 1, ***, L, where L is the number of harmonics to be

included. A FORTRAN computer program was written to test

Galerkin's method. A "package" conjugate gradients search

routine [6] was chosen. The algorithm is illustrated in the

flow chart of Fig. 2-2. Results of the analysis for several

values of L are given in Table 2-1.


A drawback of Galerkin's method is that if L harmonics

are required for suitable accuracy and there are m variables,

then m(2L+l) parameters must be determined. In practical

examples this number can be quite large. Moreover, optimi-

zation methods tend to use excessive computer time,

especially if the number of optimization variables is large.

As may be seen from Table 2-1, even a very simple network

may require an excessive amount of computation.


A more efficient scheme emerges upon consideration

of what initial values of capacitor voltage and inductor

current yield periodic steady-state operation. First

we recognize that if a network operates in the periodic

steady-state, all voltages and currents in the network are

periodic. In particular, if the period is T then for all








SStart.


Spe ci fy
initial
estimate of
solution


Calculate error:
Approximate
t +T
S= /0 1 e2dt
t
by
= (h.1 2 1 2
2 2 n 1 2 n
= ( 1 2- en
(Trapezoidal Rule)


Small yesStop


Calculate gradients:
use Trapezoidal Rule
to numerically find
t +T
f --=f / e[G+XI exp(Xv)]dt
o t
0


t +T
/ e[{G+XI exp(Av)}sin(nt)+n
t
0
t 0+T
f e[{G+Xs exp(Av) }cos(nt)-n
t
0


cos (nt) ]dt



sin (nt) ] dt


for harmonics i = 1, N


Adjust parameters


Fig. 2-2
Flow of operations for implementation
of Galerkin's method


Da.
1


b=
I












Table 2-1
Results of Application of Galerkin's Method
to Circuit of Fig. 2-1

L -1 2 3 4
do -1.432 1.096 -1.102 -1.049
sin 1 1.688 1.817 1.836 1.833
2 --- 0.5987 0.6016 0.6224
3 --- --- 7.052E-2 5.163E-2
4 --- --- --- 8.217E-2

cos 1 -1.219 1.342 1.298 1.308
2 --- 0.1687 0.1018 0.1170
3 --- --- 2.229E-2 0.1439
4 --- --- --- 6.112E-2
Final error 6.765 3.911 3.506 2.029
Iterations 43. 121. 158. 240
CPU time 4.2 Sec. 34.3 sec. 62.6 sec. 177.2 sec.









capacitor voltages


vC(t) = vC(t + T), (2.12)


and for all inductor currents


iL (t) = i (t + T). (2.13)


At the initial time tO, we wish to determine a capacitance

voltage vC(t0) and inductance current iL(t ) such that


vC(t0 = V 0(t + T) (2.14)

and


iL(tO) = iL(t0 + T). (2.15)


It is convenient to model the capacitor with initial

voltage vC(t0) by a series combination of a capacitor with

zero initial conditions and a dc voltage source of value

ve = V(t 0). Inductors with initial current iL(t0) are

modeled by an inductor with zero initial conditions and a

parallel dc current source i = iL (t0) as shown in Fig. 2-3.

Then

vC(t) = vC' (t) + ve (2.16)

and


iL(t) = i '(t) + ij, (2.17)

where


V (t0) = 0


(2.18)











+ Lict)
vc' (t)

()ve = VC(tO)


+ C (t)
v (t) c





t+ i,(t)
vL (t-)
J'Lit L


+ C(t)

vC (t

0f


qc(t) = f (vc(t),PC't)
d
iC(t) dt qC(t)


+ i (t)

vL (t

T


XL(t) = fL(iL(t),PLt)

vL(t) dt L(t)


+ T (t)
v,' (t) y





C(t) f (v (t)pt)
Ct) dt


+
vL (t)


= iL(t)


L(t) = fL' (i L'(t),PL't)
d
vL(t) dt (t)


Fig. 2-3
Capacitor and inductor initial-condition models
(a) Linear time-invariant
(b) Nonlinear


+
vL (t)


ij = i(t0)









and

iL (t0) = 0. (2.19)


Thus if initial condition source values are chosen to cause

steady-state operation,


vC' (t0 + T) = 0 -(2.20)


and

iL (t0 + T) = 0. (2.21)

The determination of these 'initial condition source values

may be accomplished in two ways: an optimization procedure

or a Newton iteration. In the optimization approach, an

error criterion is formulated, namely


E[= [ 1 [v(to + T)]2 + .1 [i' (t + T)]2. (2.22)
C L


This error function is zero if all initial capacitor voltages

and inductor currents are correct and greater than zero

otherwise. The object is then to minimize e by adjusting

the initial condition source values v and i.. All that is
e 3
required to use a .funct-io .miinimization algorithm to

perform this adjustment is the gradient of E with respect

to all the initial condition source values. The number of

optimization variables has thus been reduced to one per

energy-storage element. The undesirable convergence

properties of optimization procedures (e.g., converging to









an incorrect local minimum) still remain. For this reason

a Newton-Raphson technique is preferred. However, in the

interest of completeness, the optimization method is

described in the following.


The adjoint network approach [7] yields an efficient

means for evaluating the necessary gradient components.

Consider the situation depicted in Fig. 2-4. We wish to

calculate the steady-state response of networki2, in which

capacitors and inductors have been modeled as discussed

earlier. The network is the topologically equivalent

adjoint network. Application of Tellegen's theorem [8] to

this situation (over one period) allows us to write

t +T
/ tT [vB(t)B(T) ig(t)PB(T)] dt = 0, (2.23)
t B

where vB and iB are the Bth branch voltage and current in

the original network, ~B and B are the Bth branch voltage

in the adjoint network /, and the indicated summation is

over all network branches. Upon an incremental change in

the parameters of the original network, (2.23) becomes

t +T
t0+T
/ [{vB(t) +Av (t)} B((T)
to B


{iB(t) + AiB(t) }B(T)] dt = 0, (2.24)

so that
t0+T

t B






















ii v Li

VL V
vv (c
v v
used in gradient calculation






I-vI--------I c


-R-.
11.













i L_
v
n --e






i\__L






Fig. 2-4
The original and adjoint networks
used in gradient calculation








This equation may now be written

t +T


/t
0


S [Av (t) (T) Ai (t)i (T)] dt
v


t0
t, i


t0
+ /
to


t0+T
+ /


t0+T
0


[Av.(t).i(T) Ai.(t) .i(T)] dt




[Av (t) j(T) Ai (t)4 (T)] dt




[Av. (t)(.(T) Ai .(t)$.(T)] dt




[ [AvN(t) (N T) AiN(t) N(T)] dt, (2.26)
N


where the summations over v, i, e, j, and N represent,

respectively, independent voltage source branches, independent
current source branches, capacitor initial condition voltage
sources, inductor initial current sources, and non-source
branches. Since the values of independent sources do not

vary,


Av (t) = Aii(t) = 0.


(2.27)


The adjoint network branches which correspond to sources

in the original network are zero-valued sources. Thus









9 (T) = (.(T) = 0,
V 1

and

e (T) = 0.j(T) = 0.


Therefore, (2.26) becomes


tO"+
Ave (t)e) (T) dt +T
e to


t +T
= -
to


SAi (t) j(T) dt
j


S [AVN (t)N (T) AiN(t)N(T)] dt. (2.30)
N


The non-source branches are now considered. Only nonlinear

resistor, capacitor, and inductor branches are assumed to

make up the original network. Other network element types

are considered in Ref. [7]. Equation (2.30) thus becomes
t0+T t+T
/ X Av e (T)dt / I Aijqj(T) dt
tO e to 3


t0+T
= /
t

0+T
tO


t0+T
0



tO
0
-I/
to


S[AvR(t) R(T) AiR(t) R(T)] dt
R



I [Av' (t) C(T) Aic(t)lc(T)] dt
C


S[AvL (t) L(T) Ai' (t)L(T)] dt.
L


(2.28)


t +T

t0


(2.29)


(2.31)








By a proper interpretation of network variables--as in
Appendix A--we may write


t +T

t0+
to


SAv ee (T) dt -
e


t 0+T
j
to


}Ai i (T) dt
j JJ


t +T
= /t0+T
t 0
to


t +T

C to


f R(t)
SP R(T) R dt



S8f '(t)
-C( dt pC aC] dt


Dfc (tf)
+ v (to) AvC '(tf)


8f (0 )
c t 0 (t)) Avy (tO)
C-


+ t0+T
L t


d pf (t)
[L() dt PL AL] dt
L


Df (t )
- aiL (t0) AiL'(tf)
L

DfL (to)
+ i L( Ai L ) ,
L4


(2.32)


where tf = t0 + T and T = T t. Since energy-storage
elements are modeled as described before,

qC(t) = f C(vc' (t),P',t) = fc(vc(tO) + VC'(t),PC't)
(2.33)









and


L(t) L I(iL(t)PL't) = fL(iL (t0) + iL (t),Pct)

(2.34)

Because element parameters pR', PC and pL do not vary,


e(T) dt Av /0
j t0


j (T) dt Aij


af'' (t ) a c '(t
C (t ) v f c c(tf)Avc (to)
C *C c


fL' (tf)
L iL L 0 L f


DfL (t )

iL L (tf) (t 0


(2.35)


Now, by letting


Av' (t0) = AiL' (t) = 0


v '(t )
C(t0) = fc'(t )

VC


iL (tf)
L(t0) = F (tf)

L

and dT = dt ,

we obtain


(2.36)


(2.37)




(2.38)




(2.39)


t +T
e (T) dTAve / +
j to


_.(T-) dT i.
3 3


t0+T

e tO


t0+T
e tO









Sv (tf)v' (tf) + i (t'f) L' (tf)
C L

The variation of the error (2.22) is

AE = vc' (tf)AvC' (tf) + : iL' (tf)AiL' (tf)
C L


= : A v + i Ai. .i
e e j j 3

Upon consideration of (2.40) and (2.41) we may write


(2.40)


(2.41)


DC t +T
8v /
e tO


Th
j


- t0+

t0


8f (t ) Df '(tf)
ge(T) dT = avc e(tf)- vc e(t0)

(2.42)

T 3fL (t0)
j (T) d= 3im -j (tf)


(fL f)
L (t(t )
Di L 0


(2.43)


For linear time-invariant capacitors and inductors, (2.42)

and (2.43) become


e
9v
e


C [ (t ) 9(t )]
e f e 0~)


and


D C
i .
j


Notice that two network analyses suffice to obtain the


required gradients.


(2.44)


(2.45)


and


- L [ j(tf) (to) .









The following algorithm emerges:


1. Estimate initial condition source values v
and i.. e
3
2. Analyze the original network over one period T.

3. Calculate the error c.

4. Analyze the adjoint network over one period in
reverse time (,- = T t), letting initial
capacitor voltages be


vC' (tf)
'C (t0 (tf)

vL

or for linear time-invariant capacitors


vC f
I c(to) = C
C 0 C '

and initial inductor current be



i L (tf)
S= fL (tf)
aVv
V C


or for linear time-invariant inductors

iL' (t f)



5. Determine the gradient components by calculating



e FfC' (tf) fC' (t )
8v v e f ay v e 0
e C C










and


j 8fi (t ) 8f '(t )j(t)
= --(t -) --(t t ) ,
SLL


or for linear time-invariant elements,


-= C[e (tf) e(t0)]
e



j L[j.(tf) j(t]


6. Alter the initial source values according to
some gradient search technique (e.g., Fletcher-
Powell or conjugate gradients).


A FORTRAN program was written to implement this

algorithm. The flow chart of the resulting program is shown

in Fig. 2-5. The results from application of this program

to the network of Fig. 2-1 are given in Table 2-2. Since

this technique yields the steady-state time response of

the network, the fast Fourier transform must be used if

harmonic information is desired. In the program example,

the IBM Scientific Subroutine Package fast Fourier transform

routine HARM [6] was used. Comparison of Tables 2-1 and

2-2 indicate the speed advantage of the adjoint network-

optimization technique. However, when the adjoint network-

optimization approach was applied to the circuit of

Fig. 2-6, a local minimum caused the method to yield an

incorrect solution. Results of this calculation are given











Start



Initial
estimate of
initial condition



Analyze network
over one
period



Calculate error e



c smal Yes




Analyze adjoint
network in reverse
time: T = t + tf t




Calculate gradient
components



Adjust initial condition
source values








Fig. 2-5
Flow of operations for adjoint network-
optimization method of steady-state calculation












Table 2-2
Results of Application of Adjoint
Network-Optimization Method to Circuit of Fig. 2-1


dc term


sin
coefficients


cos
coefficients


-1.086


1.852

0.5405

2.204E-2

-1. 450E-2

-5.226E-2


-1.273

8.922E-2

-1. 591E-1

-5.296E-2

-1. 807E-2


final error 4.5E-13

iterations 3

CPU time 3.52 sec


i
























if







10 sin(27Tt)
1 f






















Fig. 2-6
Voltage doubler circuit of the second example










in Table 2-3.. The fact that the optimization approach was

unable to yield the correct solution for such a simple

network spurred the development of a more reliable method.


2.2 The State Variable Newton Method of Steady-State
Calculation


A Newton method for calculating the steady-state

response of a nonlinear network was described by Aprille

and Trick [9,101. In their approach, state variable

analysis was used to obtain the response over one cycle.

A state transition matri: is calculated for an "auxiliary"

system of linear differential equations with time-varying

coefficients. This state transition matrix is used to

compute the new set of initial condition values. The

periodic steady-state solution is sought for the system of

equations


x= f(x, t). (2.46)


If the backward Euler numerical integration formula is applied

to (2.46), we obtain


x(jh) = x[(j l)h + bh f[x(jh), jh] (2.47)


where h is the step size. Newton's method may be used to

solve (2.47):

i+l i i
x (jh) x (jh) {I- hF[x (jh)]}-l

{x (jh) x[(j l)h] hf[xi(jh)]} (2.48)












Table 2-3
Results of Application of Adjoint
Network-Optimization Method to Circuit of Fig. 2-6

v (to) v (t )
Iteration CPU Time* 1 2 Error* *


24.9
35.4
43.0
52.4
66.7
70.5
77.3


0.394
-3.108
-4.729
-4.415
-4.349
-4.348
-4.34's


*Seconds IBM 360/65 CPU Time
A IC1 (to) + 4.3381
**Error = 33 +
4.338


VC2 (t) 9.8021
9.802


5.184
10.320
10.133
9.994
9.840
9.838
9.838


1.561
0.336
0.124
3. 734E-2
6.412E-3
5.978E-3
5.978E-3










where


F[x (jh)] -= f[x(jh),jh]
~x(jh)


(2.49)


is the Jacobian of f. The "auxiliary" system of equations

is


Z = F[x(t)] Z .


(2.50)


Application of the backward Euler formula to (2.50) yields


Z(h) = Z(0) + hF[x(h) ] Z(h)


= {I hF[x(h)]}-1 Z(0) ,


(2.51)


so that


k
k -1
Z(T) = Z(kh) = {I hF[x(ih)] }- Z(0)
i=l


(2.52)


Thus, the state transition matrix 0 of (2.50) is simply


S= H {I hF[x(ih)]} .
=1-


(2.53)


Define x0 to be the vector of initial condition values.

The Newton iteration to determine these values is


x = x [I T'(x [ T(x)]
~0 0 0 H -0 0


= [I T'(x)]-1 [T(x ) T'(x )x ]


S[{T' (x )}. Ii [{T' (x )}- T(x) -x] (2.54)
0 -e -

where

T(x3) x3(T, x3) (2.55)
0 ( 2 5










and

ax(T,x )
T'(x0) = (2.56)
~ ~ 0 3
x
I0


Aprille and Trick showed in their presentation that

k
T' (x ) = = {I.- hF [x (ih)]}- (2.57)
~ ~ ~ i=1 ~ ~ ~

and
S-1 -
{T'(x3)} = = H {I hF [x (ih)]}. (2.58)
-01


Hence, from (2.54),


j+1 1 -1 xj
1 [1 I] [-1 x(T) x0] (2.59)



gives the new initial state. This "state variable-Newton"

method must be implemented with a state variable analysis

scheme. Moreover, it is dependent upon the numerical

technique used to integrate the systems of equations (2.46)

and (2.50). In the following section, the adjoint network

concept is used to develop a Newton method for computation

of steady-state response which avoids these restrictions.


2.3 The Adjoint Network N- Newton Method of Steady-State
Calculation


Define x0 to be the vector of initial condition
-0
source values:

A T
x = (ve, ij) (2.60)
~0* u ev ~ i










and a function f :


f = f(x0) [V (tf), L' (tf)] (2.61)


which is zero for the proper choice of initial conditions

x0. Newton's method for determining the values of x

which cause f to be zero may be expressed as


F Ax = f (2.62)

where Dv '(t ) 3v '(tf)
Dv Di.
-~e
F ---------- ------ (2.63)

iL (tf) L (tf)
Dv Di.
~e -3


is the Jacobian matrix. The "change vector" Ax is obtained

by solving the simultaneous linear equations (2.62). This

solution may be accomplished by a Gaussian elimination or

L-U factorization procedure [11]. The change vector is
k
then added to the previous iterate x to obtain the new

iterate:

k+l k
x0 = x + Ax


When Ax is sufficiently small, the iteration is terminated.


In order to apply Newton's method to the steady-state

determination, the partial derivatives which make up the

Jacobian matrix must be determined. The adjoint network

concept may be employed to calculate these partial









derivatives efficiently. Once again, capacitors and
inductors are modeled as shown in Fig. 2-3. Upon appli-
cation of Tellegen's theorem to the original networkIZ
and its mutually reciprocal adjoint network Z--as shown
in Fig. 2-4--we may write

t +T
/tT [vB(t)B)(T) iB(t)B(T)] dt = 0 (2.64)
t0 B

where vB and i are the Bth branch voltage and current in
the original networkZ, ,B' and )B are the Bth branch
voltage and current in the adjoin'- network't, and the
indicated summation is over all network branches. Upon an
incremental change in the parameters of the original network,
(2.64) becomes

t +T
t0+T I [{vB(t) + AvB(t)} (B(T)
t B

{iB(t) + Ai(t)} B(T)] dt = 0 (2.65)

Subtraction of (2.64) from (2.65) yields

t +T
0 [AvB(t)cB(T) AiB(t)BB(T)] dt = 0. (2.66)
t B

Equation (2.66) may be rewritten as

t0+T
/ [Av (t)v (T) Ai (t)i (T)] dt
t v
0t+T
+ [Av.(t)4.(T) Ai.(t).i(T)] dt
0to








t +T
+ / [Av (t)A (T) Ai (t)4) (T)] dt
t e e e e
to

+ 0 [Avj(t) j(T) Ai (t)Mj(T) ] dt
t +


= / X [AvI(t)N(T) AiN(t)N(T)] dt ,
to N
0


(2.67)


where the summations over v, i, e, j, and N represent,

respectively, independent voltage source branches, independent

current source branches, capacitor initial condition voltage

sources, inductor initial condition current sources, and

non-source branches. Since the values of the independent

sources are constant,


AV (t) = Ai (t) = 0 .


(2.68)


The adjoint network branches which correspond to sources in

the original network are zero-valued sources, so


v (T) = (T) = 0
V 1


(2.69)


(2.70)


( (T) = i (T) = 0 .
e- J


Thus, (2.67) becomes

to+T t +T
SAV e e(T) dt /
t e t


SAi" j(T) dt
j


= /+T

to


I [AvN(t)N(T) AiN(t)pN(T)] at.
N


and


(2.71)









The non-source branches are now considered. For brevity

in this presentation, only nonlinear resistor, capacitor,

and inductor branches are assumed to make up the original
network. Other elements are considered in Ref. [7].
Equation (2.71) thus becomes


t +T
t0


t +T
Av e) (T) at t
ee to


t+T


t0+T
to

t +T

t


Ai (T) dt
J j


I [A'v (tip(T) Ai (t)R(T)]dt
R


[AvC' (t)(C () Aic(t)lC(T)] at
C



[ [AvL(t) L (T) AiL'(t) L(T)] dt (2.72)
L


By a proper interpretation of the adjoint network
variables--as in Appendix A--we may write


t0+T
tO


t +T
SAv (T) dt /+T
e to


SAi j (T) dt
J


_ -o0 8f (fR(t)
ft R i PR p R(T) Ap dt
t R ap R R


+ +T d fC' (t)
C+ [-t o( Z)- PC Apc] dt








fC' (t )
+ 3vcC c(t0) Av '(tf)


Sfc( (to)
v, c~(tf) Avc' (tO)


t0+T fL(t)
L+ t L Ltp L] dt


I f (tf)
iL L(t0)AiL f)

8f, (t )
+ iaiL L(tf) AiL'(t) (2.73)


where tf = t0 + T and T = T t. Since energy-storage
elements are modeled as an element with zero initial
conditions and a source as shown in Fig. 2-3,

qC(t) = f' (vc'(t),Pc,t) = fC(vc(t0) + vc'(t),PCt)

and

L(t) = fL '(iL' t),PLt) = fL(iL(tO) + iL'(t),PLrt)-

(2.74)

Because element parameters pR, PC' and PL do vary,

t0+T t+T
S/ 4e (T) dt Av x / 0 j(T) dt Ai.
e t e e jt0

v fc' (tf)

c '(t0)
C v C C 0 C f
Cf C (t ) c (t)AV '(t

S v C C f (to)C 0






43

f '(tf)
Lf
L- L L L 0 L f)


f L' (t )
i C c(tf)Ai (Lo) (2.75)


By letting

Avc' (t) = AiL' (t0) = 0 (2.76)

Df (t f)
c (t0) = 1/ vc (2.77)



C. (to) = 0 \j i (2.78)



L (t0) = 0 (2.79)

and

dT = dt (2.80)


we obtain

t t
S/ c (T)dT Av J / t (T)dT Ai
e tO e j t 3


= Av (tf) (2.81)
1

Since
8v' (t ) 8v' (t )
C f CD f
Av' (tf) = 1 Av + ----Ai, (2.82)
i e e 3 3

the partial derivatives are seen to be

Dv (tf) t
S = e(T) dT
ei 0 k






44

CfC (t0) Cf (tf
vck ek(t) (t)
k k (2.83)

Dv (tf t
C. f tf
i. f (T) dT
jm to m
Df (t0) fm (t) t
mL m m
DiL (tf) DiL

m m (2.84)

DiL (tf)
Dvek-= / (e(T) dT
av e
ek t k
k 0

af' (to) Dfk (tf)
t-0--) t -Pe (t )
vCk eek(tf C ek(t0)
Dvk k k k (2.85)
and
iL. (tf) t
1 tF
Di. = / .jm (T) dT
3m t 0
DfL (t0) DfL (tf)
m m
= (t ) i (t) .
m mm (2.86)


For linear time-invariant elements, these partial

derivatives are

v'. (tf)
v = C [ tf) ek(t0)] (2.87)
Sk k k


Dv. (t f)

L [ (t) ) j(t 0)] (2.88)
mm m









i' (t )
L. f
= [ek(tf ek (t0), (2.89)
ek k k

and
iL. f
i
S = L [L j (t, ) (t0)] (2.90)
Di. m (tf 0
3m m m

Thus, all that is required is to set one capacitor or
afC(tf)
inductor initial condition to the value 1/ or
f' (t ) C
-1/ i- respectively (or to 1/C or -1/L for

linear time-invariant elements), in the adjoint network.

The adjoint network analysis is performed and the partial

derivatives of the final condition of that element with

respect to all initial conditions are calculated by

performing the subtractions indicated in (2.83) through

(2.86) or (2.87) through (2.90). These partial derivatives

occupy one row in the Jacobian (2.63). This procedure is

repeated, setting a different initial condition in the

adjoint network until all partial derivatives have been

calculated. A total of M adjoint network analyses are

thus required if there are M energy-storage elements in the

network. The (M + 1) total analyses are the same number

as required to calculate the partial derivaties by

perturbation of the initial conditions. However, the

adjoint network method yields the exact partial derivatives

rather than the approximate value arrived at by the pertur-

bation technique.









Once all the partial derivatives have been determined,

Newton's method (2.62) is applied to obtain the new initial

condition source values. The resulting algorithm is

summarized below:


1. Analyze the original network over one period.

2. Set initial condition in adjoint network and
perform analysis of adjoint network.

3. Repeat step 2 until all Jacobian entries have
been calculated.

4. Solve for change vector.

5. Calculate new initial conditions.

6. If change vector is greater than desired
go to step 1.

7. Stop.


A variation of the Newton-Raphson method may be used

to decrease the computation required. The Jacobian is

evaluated once and used until the error either stops

decreasing or increases. This procedure is known as the

"fixed tangent" method [11]. The algorithm which results

from incorporation of this technique is illustrated in

Fig. 2-7.


To establish the effectiveness of the algorithm, two

practical examples are presented. The first example is

the voltage doubler circuit depicted in Fig. 2-6. The


























































Fig. 2-7
Flow of operations for adjoint network-
Newton method of steady-state calculation










steady-state solution was obtained by two methods:

continuous integration and the adjoint network-Newton

algorithm described earlier. Gear's numerical integration

procedure was used. Both methods were started at

V = VC = 0. The period was one second. Results of
-I 2
this example are summarized in Table 2-4. The speed

advantage of the adjoint network-Newton procedure is

obvious.


The second example 'is the circuit shown in Fig. 2-8.

Again, continuous integration and the adjoint network-

Newton method are compared. Initial conditions were

C = 2 = vC3 = i = 0. The period was 1/60 second.
c 2 3
A considerable saving in computer time is realized by use

of the adjoint network-Newton method as may be seen in

Table 2-5.


2.4 References


1. W.J. Cunningham, Nonlinear Analysis, New York:

McGraw-Hill, 1958.


2. E.M. Baily, "Steady-s.tate harmonic analysis of nonlinear

networks,"-Ph.D. The'sis, Stanford University, Stanford,

California, 1968.


3. E.M. Baily, "Steady-state harmonic analysis of nonlinear

networks--a computer-oriented approach," Proceedings

of the First Annual Houston Conference on Circuits,

Systems, and Computers, pp. 390-399, 1969.










4. R. Fletcher and M.J.D. Powell, "A rapidly convergent

descent method for minimization," Computer Journal,

Vol. 6, No. 2, pp. 163-168, July, 1963.


5. R. Fletcher and C.M. Reeves, "Function minimization

by conjugate gradients," Computer Journal, Vol. 7,

No. 2, pp. 149-154, July, 1964.


6. "IBM System/360 scientific subroutine package," IBM

Corporation, White Plains, New York, 1966.


7. S.W. Director and R.A. Rohrer, "The generalized adjoint

network and network sensitivities," IEEE Transactions

on Circuit Theory, Vol. CT-16, No. 3, pp. 318-322,

August, 1969.


8. C.A. Desoer and E.S. Kuh, Basic Circuit Theory,

New York: McGraw-Hill, 1969.


9. T.J. Aprille and T.N. Trick, "Computer-aided steady-state

analysis of nonlinear circuits," Proceedings of the 14th

Midwest Symposium on Circuit Theory, 1971.


10. T.J. Aprille and T.N. Trick, "Steady-state analysis of

nonlinear networks that have a periodic response,"

University of Illinois Coordinated Science Laboratory

Report R-515, June, 1971.


11. C.-E. Fr6berg, Introduction to Numerical Analysis,

Reading, Massachusetts: Addison-Wesley, 1969.














Table 2-4
Results of Application of Steady-State Determination
Methods to Circuit of Fig. 2-6

Continuous Integration Adjoint Network-Newton Method

v (t ) (t ) CP vC (t0) C (t0) CPU
Iteration 1 C2 Error* Time** 1 2 Error* Time**

1 0.3091 2.570 1.809 3.0 0.3091 2.570 1.809 11.8
2 .2518 4.160 1.518 4.2 -4.094 9.067 0.131 12.7
3 .9192 5.301 1.247 5.4 -4.310 9.705 1.6E-2 13.5
4 -1.462 6.155 1.035 6.2 -4.330 9.801 2.1E-3 14.4
5 -1.950 6.821 0.895 7.1 -4.338 9.802 0.0 15.3
6 -2.361 7.360 0.705 7.9
7 -2.703 7.797 0.581 8.0
8 -3.015 8.128 0.476 9.6
9 -3.239 8.425 0.394 10.4
10 -3.424 8.671 0.326 11.2
20 -4.194 9.646 4.91E-2 18.8
40 -4.334 9.792 1.94E-3 33.3
100 -4.335 9.803 7.94E-4 75.2


Ivcl(to) + 4.3381
*Error = 4.338


1vC2(to) 9.802j
9.802


**Seconds IBM 360/65 CPU time
























5Q 0.1 h



-6 L0
10- 6 L

+
+ + v _---
L C v v 1 k
C C
V2 3
-3
10 f LO- 3f

10 sin(21r60t)






















Fig. 2-8
Circuit for adjoint network-Newton method example





0


Table 2-5
Results of Application of Steady-State Determination Methods
to Circuit of Fic. 2-7
Continuous Integration Adjoint Network-Newton Method
VC (to) i ) CPU vC (t) i (t) CPU
Iteration 1 L 0 Error* Time** 1 L 0 Error* Time**
1 -1.970 0.3665 40.40 5.1 -1.970 0.3665 40.40 36.5
2 -3.365 -1.160E-3 1.76 8.4 -7.311 2.054E-2 1.470 38.7
3 -7.635 1.364E-2 0.67 11.6 -8.343 2.863E-2 2.252 41.0
4 -6.222 0.1459 15.48 14.2 -8.695 1.838E-2 1.078 43.2
5 -7.462 -6.471E-2 8.35 16.9 -8.864 1.392E-2 0.5648 48.4
10 -8.287 -9.572E-2 11.69 29.1 -9.053 9.320E-3 3.424E-2 56.2
14 -9.119 2.587E-2 1.87 38.6 -9.066 9.024E-3 0.0 64.8
20 -8.930 2.546E-2 1.84 52.4
30 -9.108 1.231E-2 0.37 75.0
40 -9.078 6.620E-3 0.27 98.0
50 -9.059 8.523E-3 5.63E-2 120.9
60 -9.064 9.392E-3 4.11E-2 143.6
70 -9.068 9.098E-3 8.42E-3 167.2
80 -9.067 8.963E-3 6.87E-3 190.2
90 -9.066 9.009E-3 1.66E-3 213.2
100 -9.066 9.028E-3 4.43E-4 236.5


Error =


I Cl (t0)


+ 9.066I liL(tO) -
+


9.024 10-31


9.024 10-3


**Seconds IBM 360/65 CPU time


9.066















CHAPTER 3

FREQUENCY-DOMAIN ASPECTS
OF NONLINEAR CIRCUIT DESIGN


In some instances, nonlinear networks are designed

to meet specifications in terms of frequency-domain, rather

than time-domain, behavior. The rationale for development

of automated design methods for these network types may be

established by several practical examples.


Large-signal amplifiers are often designed by methods

involving linearization or graphical constructions [1]. The

added complexity of integrated circuits decreases the

desirability of these approaches. Possible design require-

ments for large-signal amplifiers might include specifications

concerning:


1. Power efficiency; i.e., the ratio of average
power delivered to the load to the power
consumed by the amplifier;

2. Harmonic distortion;

3. Maintenance of .. and 2. within some tolerance
over a specified frequency range.


Consider the specification of harmonic distortion.

Design for minimum distortion makes a great deal more sense

if specifications are formulated in frequency-domain terms.









There is no salient feature of a time waveform which reflects

distortion. Measuring distortion by taking the difference

between the response and a pure sinusoid is inadequate

because the two signals must have the same frequency and

phase for the subtraction to indicate waveform distortion.

Moreover, even if the two waves did have the same frequency

and phase, a loss of numerical accuracy results from

subtraction of two similar numbers. In the frequency domain,

distortion is found easily: it is simply the ratio of the

sum of the harmonics to the fundamental component. Thus, a

method which exploits the frequency-domain nature of this

problem is required.


Frequency multiplier circuits make use of the nonlinear

characteristics of some electronic circuit elements to

generate harmonics. Usually only one of these harmonics is

desired. Graphical methods are often used in the design of

frequency multiplier circuits [1]. Here again, an automated

design method based on frequency-domain information--the

desired harmonic-- may be advantageously applied.


Finally, sinusoidal oscillators are generally designed

to meet specifications regarding

1. frequency of oscillation;

2. power output;

3. spectral quality (harmonic content of output);










4. efficiency;

5. frequency and amplitude stability.


In this chapter, general concepts are developed which

lead to an effective algorithm for the automated design of

nonlinear circuits for specified harmonic content. The

nature of oscillator operation dictates a need for special

methods. Consideration of these methods is undertaken in

Chapter 4.


3.1 The Design Algorithm


The algorithm for harmonic design is based on the time-

domain automated design concepts discussed in Chapter 1.

The proposed method can be illustrated in terms of a large

signal amplifier. Let v(t) denote the actual output wave-

form of the amplifier. Under reasonable assumptions (e.g.,

finite energy content [2])the Fourier transform of v(t)

exists:

Vfj -J2nft A
V(f) = v(t) e-jf dt I=[v(t)]. (3.1)
-00

The inverse transform is


v fe j2'itft A -1
v(t) = V(f) e it df =2- [V(f)]. (3.2)
-00

Only a discrete set of sampled values of v(t) is actually

stored in the computer... Therefore, in computer work the

discrete Fourier transform is used. Let v(k) represent the









sampled values of v(t) at the equally spaced times t = k At,

k = 0, 1, -**, N 1, where N is the number of samples.

V(n) denotes the discrete values of V(f) at the equally-

spaced frequency points fn = n Af, n = 0, 1, ".*, N 1.

The discrete Fourier transform is [3,4]:

N-1
V(n) = [ v(k) e-j2 nAf kAt
k=0 (3.3)

n = 0, 1, ***, N 1; and


N-l
v(k) = V(n) ej2" nAf kAt
n=O (3.4)

k = 0, 1, ***, N 1 .


Since At = C/N and Af = 1/J where 3 is the time interval

of interest,

N-l
N- (k) -j2nk/N A
V(n) = 1 v(k) e =aD [V(n)],
(3.5)

n = 0, 1, ***, N 1; and


N-1
v(k) = 1 V(n) ej2nk/N [V(n)],
n=o (3.6)

k = 0, 1, '-, N 1.


The Fourier series coefficients may be shown to be

1
D = V(n)
n N
(3.7)
n = 0, 1, "', N 1.










In practice, the calculation of the discrete Fourier

transform is accomplished by the fast Fourier transform

(FET), which is also called the Cooley-Tukey algorithm [5].


The output of a distortionless amplifier, v(t), is a

sinusoid of appropriate amplitude with period T = 1/fl,

where fl is the input signal frequency. The discrete Fourier

transform of v(t), denoted by V(n), has non-zero components

only at the fundamental-frequency (n = 1) and possibly a dc

component (n = 0).


Consider the performance function


e(x) /T [e(t)]2 dt, (3.8)
2 0

where
A
e(t) = v(t) (t). (3.9)


The performance function is zero when the actual output is

distortionless and non-zero otherwise. At its minimum it

yields the optimum parameter values x. Since v(t) and v(t)

could be about equal in magnitude, the indicated subtraction

could give rise to numerical errors in e(t) and thus in E.

From the relationship [2]'

T
F2 = 1f2(t)dt (3.10)
S0

for the mean squared value of the signal
00
f(t) = D ej2Tnt/T



SD + 2 D ej2 nt/T (3.11)
n=l









we may write

0O
2 2 D2
F2 = D + 2 D (3.12)
n=l


where D = ID Equation (3.12) is a result of the
n -n
orthogonality of the terms of the Fourier series. This

equation and (3.7) may now be used to evaluate the performance

function (3.8):

T
T ^ 2
e = / [v(t) v(t)] dt
0
L
T ^ 2 2
2 J(Do Do + 2 1 (Dn- D) }
n=l
L
{[V(0) V(0)] + 2 1 [V(n) V(n)]2}. (3.13)
n=l

The summation is taken over the fi--st L harmonics. The value

of E is seen to be calculated directly from the results of

the fast Fourier transform: no integration over time is

necessary. For design flexibility, non-negative weights may

be inserted:

L
T {W[V(0) V(0)]2 + 2Y Wn[V(n) V(n)]2}. (3.14)


The gradient of e with respect to all designable parameters

remains to be found. As indicated earlier, the adjoint

network concept is an extremely efficient method for finding

gradients. An additional concern is calculation of the

adjoint network excitation signals. The adjoint network

will be excited by the error signal


Q(t) = w(t) [v(t) v(t)]


(3.15)










or

c(k) = w(k) [v(k) v(k)]. (3.16)


Again, to avoid numerical inaccuracies which may arise from

taking differences of approximately equal numbers, the fast

Fourier transform is used:

-1.
#(k) =D1 [Wn {V(n) V(n)}] (3.17)


The following design algorithm emerges for meeting

harmonic requirements:


1. Analyze the original net.rork. Retain branch
voltages and currents of designable elements
(vB(t) and iB(t)).

2. Use the fast Fourier transform to determine
the performance function and adjoint network
excitations in the frequency domain.

3. Use the inverse fast Fourier transform to
form the time-domain adjoint excitations.

4. Analyze the adjoint network in reverse time
(T = t0 + tf t), using the error signal
as excitation. Retain branch voltages and
currents of designable elements ( B(T) and
cB(T)).
5. Calculate the gradient components.


The performance function c and gradient Ve are used by a

function minimization algorithm to adjust the parameter

values. The gradient components are calculated as outlined

in Appendix A. A flow chart of the design algorithm is

shown in Fig. 3-1.








Initial
network
configuration
1I


Analyze
original
network


Perform
FFT


Form
error terms
and performance
function


Perform
inverse
FFT


Analyze
adjoint
network


Calculate
gradient
components
,v


Conjugate
gradients routine:
obtain new
parameter vector


Fig. 3-1
Flow of operations for automated design algorithm


-dI


I


L


Done S
---Stopj










For the design algorithm to be practical, efficient

techniques are mandatory for analysis of the original and

adjoint networks. Numerical integration methods of fixed

order are frequently used for transient analysis. Increased

efficiency is obtained by adjusting order of integration as

required to achieve the largest time step consistent with

allowable error at each time point. This is the basic

premise of Gear's integration algorithm [6,7]. Moreover,

an increase in speed and a decrease in the amount of storage

required is realized by using sparse matrix techniques. In

these methods, only the non-zero elements of the appropriate

matrices are stored and operated upon.


A FORTRAN program has been developed which uses the

'Gear integration method and sparse matrix techniques for

automated harmonic design. A detailed flow-chart of this

program appears in Appendix B. In the analysis of the

adjoint network, the gradient components and network

equations are integrated concurrently. This integration is

done in a manner similar to that described by Hachtel et al.

[8], and may be visualized by appending to the network a

parallel combination of a unit-valued capacitor and a

current source. This concept is illustrated in Fig. 3-2.

The current source values are equal to the appropriate

products of branch voltages and currents so that, upon

integration, the node voltages correspond to the time-

integral in the gradient component expressions. For






























SR DC DL




in 1 1 1
nt
:k) I1 2 13





















Fig. 3-2
Arrangement -of, adjoint network
to obtain gradient components









example:


I = iR) R(T) (3.18)

dv (t)
2 = dt C (T) (3.19)

and
diL(t)
13 d= t L(T) (3.20)


are used to obtain gradient components for a resistor,

capacitor, and inductor, respectively.


The integration is carried out by the "companion model"

method described by Calahan [9]. In this approach, energy-

storage elements are replaced--at each time point--by a

model which represents the numerical integration formula

applied to the element. This technique allows a time-domain

analysis problem to be reduced to a repetitive dc analysis

problem.


The sparse matrix techniques used are based upon the

methods presented by Gustavson et al. [10]. Executable

machine code is generated for solution of the specific

circuit under consideration. The equations are ordered so

that the solution code which is generated reflects to best

advantage the original sparsity. The procedure utilized

corresponds to the second method of Tinney and Walker [11].









3.2 Design Examples


Two practical examples are presented to demonstrate

the effectiveness of the design algorithm. In order to

speed the simulation, charge-storage effects were neglected

in the transistor models used in the analysis. That is,

frequencies of interest were assumed to be so low that the

static (dc) Ebers-Moll transistor model was applicable.

This assumption was verified by a trial analysis in which

capacitive elements of the model were included. Comparison

with an analysis of the same network using the static model

revealed a negligible difference in response. Hence, the

original assumption was valid.


In the examples, function minimization was accomplished

by the conjugate gradients method [12]. Transistor parameters

used in the examples are shown in Table 3-1.


A shunt-series feedback pair amplifier, shown in Fig. 3-3,

is to be designed. This circuit is a Class A amplifier in

which the feedback and biasing elements are to be adjusted

so that operation in a linear region occurs. A desired gain

of 30 was specified and harmonic distortion was minimized.

Figures 3-4, 3-5, and 3-6 illustrate the progress of the

design. The initial output waveform, spectrum, and adjoint

excitation signal are shown in Fig. 3-4. Notice the

reflection in the adjoint excitation of the distortion of

the output signal. Figure 3-5 reveals the result of one














Table 3-1

Transistor Parameters



BF = 100.


R = 0.1

-14
I = I = 1.0 x 10 A.
CS ES
















+12v.




R >1K




r--!1----0




600I



+ RR2
0.1 sin(21l000t) 1200
















Fig. 3-3
The shunt-series feedback pair amplifier
of the first example


















v0
volts 4

2

0



10

1

0.1
Vjn 10i-2


-5
10-




4


i
amps


.2 .4 .6 .8


0 1 2 3 4 5 6 7 8 9 10


.2 .4


Fig. 3-4
Initial configuration for example of Fig. 3-3
(a) Output waveform
(b) Frequency-domain components of output signal
(c) Adjoint network excitation


, ms.
















n











t, ms.































0












Fig. 3-5
Circuit configuration for example of Fig. 3-3
after one design iteration
(a) Output waveform
(b) Frequency-domain components
of output signal
(c) Adjoint network excitation









12

10

V0o 8

volts 6
4.

2


0



10 -


.2 .4


.6 .8 1.


0.1.
Vn 10 -2

-3
10
-4
10

-5
10 n
0 1 2 3 4 5 6 7 8 9 10
(b)

0.3



0.2


amps
0.1



0o


t, ms.


.6 .8 1.


ms.









9


Fig. 3-6
Final circuit configuration for example of Fig. 3-3
(a) Output waveform
(b) Frequency-domain components
of output signal
(c) Adjoint network excitation










12

10

8

6

4

2

0 t, ms.
.2 .4 .6 .8 1.
(a)

10

1

0.1

10-2
10-3
10 3

10-

10-5 -I- 1 I 1 >8-n
0 1 2 3 4 5 6 7 8 9.10

(b)

.05
.04
.03
.02

.01

0 I\ --' 'A--- t, ms.


-.01

-.02


.b .0 iL.










design iteration. The output waveform is more nearly

sinusoidal and the reduction in the error between desired

and actual frequency components causes a much smaller

adjoint excitation signal. After four design iterations,

the situation is that shown in Fig. 3-6. Note the change

in the scale of Fig. 3-6(c). In circuits such as this,

gain and distortion are conflicting specifications. That

is, for a given input signal, increased gain results in

increased distortion. The design procedure accounts for

this conflict, yielding a design which compromises between

the two conflicting specification.. The availability of

weighting factors allows one or the other of the specifi-

cations to be emphasized, if desired. The compromise is

seen by consideration of Table 3-2. Notice that although

distortion was quite small after the first iteration, the

fundamental component was too small. After the fourth

iteration, both gain and distortion were acceptable.


The large-signal Class B audio power amplifier of

Fig. 3-7 is to be designed for minimum distortion at a

power output of 20 watts (RMS). Four resistances are

adjusted by the program to achieve the design goals. The

design progress is depicted in Figs. 3-8, 3-9, and 3-10.

The initial output waveform, spectrum, and adjoint network

excitation signal are shown in Fig. 3-8. In this example,

the weighting factors of the harmonics were increased to

emphasize the requirement of minimum distortion. This












Results of Application of


Table 3-2
Design Algorithm


to Circuit of Fig. 3-3


Initial It. 1 It. 2 It. 3 It. 4 Desired

R1 5000 4747 4725 4708 3111
R2 5000 1982 2095 2110 2061 --
Dist.(%) 28.2 1.38E-2 1.46E-2 1.47E-2 1.26E-2 0
v0 (dc) 5.99 5.71 5.70 5.70 5.51 5.50
v1 (fund) 5.81 2.86 3.01 3.03 2.96 3.00
v2 0.719 1.60E-4 1.88E-4 1.92E-4 1.30E-4 0
v3 0.492 6.55E-5 7.41E-5 7.53E-5 7.01E-5 0
v4 : 0.145 3.12E-5 3.17E-5 3.19E-5 3.18E-5 0
v5 2.08E-2 1.82E-5 1.89E-5 1.87E-5 1.87E-5 0
V6 0.110 2.96E-5 2.13E-5 3.14E-5 3.07E-5 0
v7 4.74E-2 2.26E-5 2.38E-5 2.40E-5 2.33E-5 0
v8 3.91E-2 2.42E-5 2.55E-5 2.57E-5 2.50E-5 0
v9 3.94E-2 2.35E-5 2.47E-5 2.49E-5 2.43E-5 0
vl0 2.26E-2 2.07E-5 2.18E-5 2.20E-5 2.13E-5 0

Performance 4.45E-3 3.30E-5 2.04E-5 2.02E-5 6.92E-7 0
function


0













+50v.
R3


R4


1001F=

S_390
20001


R O

)IF 2.7K




!v 10 390 390
'Hz K


100
pF



R1





Fig. 3-7
The large-signal audio power amplifier
of the second example



















.6 .8


0 1 2


3 4 5
(b)


1L .1. I I I _t I -


6 7 8 9 10


Fig. 3-8
Initial circuit configuration for example of Fig. 3-7
(a) Output waveform
(b) Frequency-domain components of output signal
(c) Adjoint network excitation


V0
v0
volts


-10

-20


SVni


t, ms.























n


0.1
-2
10

10-3

10-4


8
6
4
i 2


amps


- 2
- 4
-6
- 8
-10
-12


t, ms.














V0
volts
volts


0

-10

-20





10-


1-

0.1.


10-2
10

-3
10


.6 .8 1.


10- 4
0



1


_________________ L


1 2 3 4 5
(b)


7 8 9 10


.6 .8 1.


Fig. 3-9
Circuit configuration for example of Fig. 3-7
after one design iteration
(a) Output waveform
(b) Frequency-domain components of output signal
(c) Adjoint network excitation


I n


t, ms.






















n









t, ms.


i
amps






77




20

10
volts
0 -- t, ms.
.2 .4 / .6 .8 1.
-10


-20 (a)




10

1

S 0.1


10-2

10- 3

-4 >I I I I -n
0 1 2 3 4 5 6 7 8 9 10
(b)

0.4 4
0.3
0.2
amps
0.1
0 I t, ims.
-0.1
-0.2
-0.3 (c)

Fig. 3-10
Final circuit configuratich for example of Fig. 3-7
(a) Output waveform
(b) Frequency-domain components of output signal
(c) Adjoint network excitation









emphasis is reflected in the adjoint excitation signal.

Figure 3-9 illustrates the results of the first design

iteration. The final situation, after three design

iterations, is depicted in Fig. 3-10. The final distortion

level is 0.3 per cent: an acceptable level for audio appli-

cations. Results appear in Table 3-3.


3.3 Discussion


A method has been developed which is effective in

designing nonlinear networks for frequency-domain specifi-

cations. This method makes use of three powerful concepts:

the adjoint network approach, Gear's numerical integration

algorithm, and the fast Fourier transform. Practical

examples using the FORTRAN program were presented to

demonstrate the effectiveness of the design procedure.


Several points need to be mentioned, however. The

examples were chosen so that no initial transient was

present in the simulation. In general, the simulation will

contain both transient and steady-state components. The

program--to be applicable in these cases--needs to be

altered to allow for this situation. The steady-state

determination procedure of Chapter 2 should be incorporated.

A method of determining the transversal terms (see

Appendix A) must also be included.










Table 3-3

Results of Application of Design Algorithm
to Circuit of Fig. 3-7


Initial


It. 1


It. 2


Desired


1 50.K 47.7K 44.9K --

R2 130.K 129.K 129.K--

3 1.50K 6.23K 5.81K --

R
4 6.80K 1.14K 1.10K ---

Dist. (%) 8.08 0.299 0.303 0

V0 (dc) 0.145 5.48E-2 5.09E-2 0

V1 (fund) 21.3 20.9 20.0 20.0

V
2 0.379 4.18E-3 3.67E-3 0

V3 0.308 2.29E-2 2.24E-2 0

V
4 0.296 9.47E-4 9.05E-4 0

V5 0.267 1.41E-2 1.36E-2 0

V6 0.194 7.46E-4 7.18E-4 0

V7 0.124 1.02E-2 9.90E-3 0

V8 8.61E-2 7.09E-4 6.85E-4 0

V9 5.47E-2 8.14E-3 7.89E-3 0

V10 1.08E-2 7.31E-4 7.11E-4 0

Performance
Function 1.85E-2 4.63E-4 2.96E-5 0









3.4 References


1. F.K. Manasse, J.A. Ekiss, and C.R. Gray, Modern Transistor

Electronics Analysis and Design, Englewood Cliffs, New

Jersey: Prentice-Hall, 1967.


2. M. Javid and E. Brenner, Analysis Transmission and

Filtering of Signals, New York: McGraw-Hill, 1963.


3. E.O. Brigham and R.E. Morrow, "The fast Fourier

transform" IEEE Spectrum, Vol. 4, No. 11, pp. 63-70,

December, 1967.


4. G.O. Bergeland, "A guided tour of the fast Fourier

transform," IEEE Spectrum, Vol. 6, No. 7, pp. 41-52,

July, 1969.


5. J.W. Cooley and J..W. Tukey, "An algorithm for the

machine calculation of complex Fourier series,"

Mathematics of Computation, Vol. 19, pp. 297-301,

April, 1965.


6. C.W. Gear,"Simultaneous numerical solution of

differential-algebraic equations," IEEE Transactions

on Circuit Theory, Vol. CT-18, No. 1, pp. 89-94,

January, 1971.


7. C.W. Gear, "The automatic integration of ordinary

differential equations," Communications of the ACM,

Vol. 14, No. 3, pp. 176-179, March, 1971.









8. G.D. Hachtel, R.K. Brayton, and F.G. Gustavson, "The

sparse tableau approach to network analysis and design,"

IEEE Transactions on Circuit Theory, Vol. CT-18, No. 1,

pp. 101-113, January, 1971.


9. D.A. Calahan, "Numerical considerations for implemen-

tation of a nonlinear transient circuit analysis

program," IEEE Transactions on Circuit Theory, Vol. CT-18,

No. 1, pp. 66-.73, January, 1971.


10. F.G. Gustavson, W. Liniger, and R.A. Willoughby, "Symbolic

generation of an optimal Crouc algorithm for sparse

systems of linear equations," Journal of the ACM, Vol. 17,

No. 1, pp. 87-109, January, 1970.


11. W.F. Tinney and J.W. Walker, "Direct solutions of sparse

network equations by optimally-ordered triangular

factorization," Proceedings of the IEEE, Vol. 55, No. 11,

pp. 1801-1809, November, 1967.


12. R. Fletcher and C.M. Reeves, "Function minimization by

conjugate gradients," Computer Journal, Vol. 7, No. 2,

pp. 149-154, July, 1964.















CHAPTER 4

CONSIDERATIONS FOR OSCILLATOR DESIGN


Automated design of oscillator circuits requires

several special methods in addition to the techniques

developed in the preceding chapters. Automated network

design is based on iterative analysis of the network under

consideration and of the "adjoint" network. In order to

design oscillator circuits, therefore, it is necessary

to be able to determine the frequency of oscillation as

well as the steady-state waveform. Moreover, the steady-

state determination algorithm described in Chapter 2

requires knowledge of the period of the waveform. Investi-

gation of oscillator design reveals that the unstable

nature of oscillator circuits causes additional difficulties

in the analysis of the adjoint network. For example,

analysis of the oscillator network of Fig. 4-1 yielded the

results shown in Fig. 4-2. The adjoint network response

grows without bound. This chapter is concerned with

possible solutions to these problems.


4.1 Determination of Frequency of Oscillation


The frequency of oscillation must be accurately

determined in order to apply the harmonic design methods
















i=f(v0)


i=f(v0)


gm =-3.5


slope = 1
(1l)


slope = 100
(o .on)


-0 r

r(T)


gm= 3.5


(b)





Fig. 4-1
Oscillator circuit
(a) Original network
(b) Adjoint network












v(t)






-t




(a)






o(T)













(T=T-t)



(b)



Fig. 4-2
Output waveforms of networks of Fig. 4-1
(a) Original network output voltage v0
(b) Adjoint network output voltage 0









described in Chapter 3. A straightforward approach to this

determination may be based on the phase plane concepts

frequently used in control systems analysis [1,2]. The

phase plane is the plane of the variable under consideration

(for example, output voltage) and its time derivative.

Stable oscillations in this variable give rise to closed

loops in the phase plane referred to as "limit cycles."

One revolution around such a limit cycle represents one

period of the steady-state response. In the following

discussion, we refer to Fig. 4-3, which is assumed to be a

phase 'plane diagram representative of an oscillator circuit.

If the initial conditions of the circuit correspond to

point A, the "trajectory" is seen to eventually converge on

the 'limit cycle which represents steady-state operation.

One .revolution around the phase plane from point A to

point B may be taken as an estimate of the period of

oscillation, while the distance vA vB may be thought of

as an error which is zero when the circuit is operating in

the .steady-state condition. This error may be used in a

method which adjusts the initial condition to achieve steady-

state operation.


One such method is the adjoint network-Newton method

of Chapter 2. This technique makes use of derivative

information to calculate a new initial condition which

reduces the error. Another revolution around the phase


































VV

Limit
Cycle
















Fig. 4-3
Phase plane diagram for an oscillator









plane is undertaken using the new initial condition value.

This process is repeated until the error decreases to an

acceptable amount. The period of oscillation is then

known: it is the time required to traverse one revolution

around the limit cycle.,


The application of the phase plane method just

described to a network".anialysis program is quite simple.

A phase plane plot is "constructed" for each capacitor

voltage and its derivative and each inductor current and

its derivative in the circuit. The initial conditions

determine an angle OC for each capacitor and 0L for each

inductor, where


S= ArctanC (t0) (4.1)


and

S= ArctanL i(0) (4.2)



If the initial capacitor voltage vC(t0) is given, the

current iC(t0) is found by replacing the capacitor by a

voltage source of value'.t; (t). A dc analysis is performed

and the current through this voltage source equals i(t0) .

Similarly, an inductor is replaced by a current source of

value iL(t0). A dc analysis is undertaken and the voltage

across this current source is vL(t0). A time-domain analysis

is then performed and the angles 0C and 8L are calculated









for each capacitor and inductor, respectively, at each step

of the analysis. When at least a full revolution has

occurred for each energy-storage element, the analysis is

halted. Interpolation may be necessary to obtain this

angle precisely. The time of the estimated period of

oscillation is calculated and one iteration of the steady-

state algorithm may then be applied. When periodic

steady-state operation has been achieved, the period is

known.


It should be noted that in !.ome instances, vC and VC

or iL and iL must be used because of the presence of a dc

component. The second derivative may be calculated by

differencing; i.e.,

v (t0+h) Vc(t )
v ~ h (4.3)

or
S L(t0+h) 1L(t0)
iL (4.4)


In some numerical integration methods which use higher

derivatives (such as Gear's algorithm [3]) this differencing

may not be necessary.


4.2 Calculation of Gradient Components for Automated
Oscillator Design


Proper operation of practical oscillator circuits

depends upon two factors: instability and nonlinearity.

The unstable nature of oscillator circuits insures the









growth of oscillations, while the nonlinear property limits

the growth and establishes a steady oscillation. Since the

adjoint network is tq be used in oscillator design, several

facts must be established about the adjoint network of an

oscillator. We demonstrated in Appendix A that nonlinear

elements in the original network become linear time-

varying elements in the adjoint network. Thus, the

limiting of the growth of signals may not occur in the

adjoint network which-corresponds to an oscillator. As

an example, consider the circuit shown in Fig. 4-4. In

this network, an increasing sinusoid is generated which

grows until excursions of the output reach the positive

segments of the nonlinear resistance characteristic. The

change in resistance limits these excursions and a steady-

state condition is achieved. In the adjoint network

corresponding to this circuit, the resistor has a negative

value during the time intervals corresponding to time

segments in the original network analysis during which the

slope of the nonlinear resistance was negative. The signal

grows during this time interval. When the slope of the

resistance in the, ori inai>lnetwork was positive, the

adjoint network resistance also has a positive value.

During this time, the signal will decay. Situations thus

exist in which the effect of the growth is greater than

that of the decay.




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