Synthesis of Time-Var y ing Passive Networks

B. D. 0. Anderson

March 1966

Reprinted October 1966

Technical Report No. 6560-7

This work was entirely supported by the Air Force Office of Scientific Research under Grant AF-AFOSR 337-63 except for report publication under Nonr-225I83)

SVSTEIllS THEORV LRBORRTORV

STsRllFORD ELEbTROnlCS ItRBORRTORIES STRIFORO URIUERSITV STRIFORD, CALIFORRIA

SYNTHESIS OF TIME-VARYING PASSIVE NETWORKS

by

B. D. 0. Anderson

March 1966

Reprinted October 1966

Reproduction i n whole o r i n p a r t i s permitted f o r any purpose of t h e United S t a t e s Government

Technical Report No. 6560-7

This work was e n t i r e l y supported by the A i r Force Office of S c i e n t i f i c Research under Grant AF-AFOSR 337-63 except f o r r epor t publ ica t ion under ~onr-225(83)

Systems Theory ~ a b o r a t o r ~ Stanford Elec t ronics Labora tor ies

Stanford Universi ty Stanford, Ca l i fo rn ia

CONTENTS

I . INTRODUCTION

A. A i m s and Philosophy o f t h e Research

B. Review of Previous Work

C. Review of t h e Research Reported

D. Mathematical Techniques Employed

11. REVIEW, DEFINITIONS AND NOTATION

A. Po r t Variables and t h e Sca t t e r ing Matrix

B. P a s s i v i t y and Losslessness

C. Other Network Matr ices

D. Examples of t h e Preceding Concepts

111. PROPERTIES OF PASSIVE SCATTERING MATRICES

A. Antecedal Property

B. Existence of Adjoint

C. X Mapping Property -2

D. Loss less Condition on s -

IV. FINITE NETWORKS

A. Equivalences

B. Ef fec t of F in i t eness on Matrix Descript ion

C. Existence of t h e Antecedal Adjoint

D. The Concept of Degree of a Sca t t e r ing Matrix

E. Cascade Loading

F. Examples of Cascade Loading

Page - 1

1

1

2

4

V. NONDISSIPATIYE NETWORKS 29

A. Loss less and Nondissipat ive Networks 2Y B. Quas i loss less Sca t t e r ing Matr ices 30

C. Sca t t e r ing Matrices of Nondissipat ive 31- Networks--Quasilossless Condition

iii

D. Scattering Matrices of Nondissipative Networks--Mathematical Form

E. Examples of Quasilossless Scattering Matrices

F. Quasilossless Immittance Matrices

VI. FACl'ORIZATION OF SCATTERING MATRICES

A. The Concept of Factorization

B. Conditions for Factorization

C. Conditions for Factorization with Degree Reduction

VII. NONDISSIPATIVE SYNTHESIS

A. Factorization of Quasilossless Scattering Matrices

B. Realization of a Product of Scattering Matrices

C. Realization of Quasilossless Scattering Matrices of Degree Zero and Degree One

D. Synthesis Example

VIII.MSSY SYNTHESIS

A. Statement of Problem and an Outlined Attack

B. Solution Procedure

C. The Multiport Case

D. Example of One Port Synthesis

IX. RELATION TO OTHER WORK

A. Spaulding's Lossless Immittance Synthesis

B. A New Viewpoint of Lossless Time- Invariant Synthesis

C. Relation to Two Problems of Automatic Control

Page - 34

X. CONCLUSIONS

A. Summary of Preceding Work

B. Directions for Future Research

C. Possible Applications

APPENDIX

Verification of Admittance Matrix Formulation

Page - 79 79 79 80

REFERENCES

ILLUSTRATIONS

Figure Page - 1 I l l u s t r a t i o n of S o l v a b i l i t y 6

2 Network wi th Sca t t e r ing Matrix of t h e 18 Form of Equation (3.13)

3 Time-Varying C i r c u i t Equivalences 20

4 I l l u s t r a t i o n of Cascade Loading 25

5 Representat ion of Network a s Cascade Loading 26

6 A Nondissipat ive Network 3

7 Rea l i za t ion of One Por t Sca t t e r ing Matrix 53 Product

8 Rea l i za t ion of Mult iport Sca t t e r ing Matrix Product

54

9 Rea l i za t ion of Mult iple Product 54

10 Proto type Network f o r Degree One S c a t t e r i n g Matrix

58

11 Pass ive Quas i loss less Sca t t e r ing Matr ix 59 Rea l i za t ion

1 2 Reduction i n t h e Gyrators f o r Product Real iza t ion 60

13 Network Dual t o t h a t of Figure 10 61

14 Example I l l u s t r a t i n g Real iza t ion of Quas i loss less 63 Sca t t e r ing Matrix

15 A Lossy Synthesis 71

16 Equivalent Networks with t h e Transformer 77 Turns Rat io ~ ( t ) = cos a t s i n a t

s i n at -cos a t t I

PRINCIPAL SYMBOLS

Symbol f o r a network

Augmented network

Por t vol tage vec to r

Por t current vec to r

Por t inc ident vol tage

Por t r e f l e c t e d vol tage

Exci ta t ion of augmented network.

Unit matr ix (of dimension n )

Zero matr ix (square, of dimension n)

Impedance matr ix

Shunt augmented network impedance matr ix

Admittance matr ix

S e r i e s augmented network admittance matr ix

Sca t t e r ing Matrix

Adjoint of s -

Antecedal ad jo in t of s - +m

Short f o r -m - s( t77)vi (7)d7 - +m

Short f o r J-. zl(t ,A) s 2 ( A , ~ ) d 3

Transpose i n t h e matr ix sense of X - Set membership and i n t e r s e c t i o n symbols

Set of infinitely differentiable vector functions of compact support

Set of infinitely differentiable vector functions with support bounded on the left

Set of vector distributions wlth support bounded on the right

Set of square integrable vector functions.

Unit impulse

Derivative of unit impulse

Unit step function, zero for t< 7, unity for t >T.

Typical vectors

Typical matrices

Typical scalars t

Short for T ( k ) g ( h ) d h -m -

Same as < f, f >t - -

Operator norm.

Energy input up till time t.

Nonnegative definite matrix kernels.

Matrices with elements polynomial in derivative operator.

viii

ACKNOWLEDGMENTS

I t is a most p leasant t a sk t o acknowledge t h e generous Support

provided by t h e Services Canteens Trust, Fund, an agency of t h e

Aust ra l ian Government, over t h e pas t eighteen months, and a l s o t h e

United S t a t e s Educational Eoundation i n Aus t ra l i a f o r a Fulbr ight

Grant. Support by t h e A i r Force Of f i ce of S c i e n t i f i c Research under

Grant A F - A F O S R - ~ ~ T - ~ ~ i s a l s o g r a t e f u l l y acknowledged.

The author would a l s o l i k e t o express i n t h e s t ronges t poss ib le

terms h i s apprec ia t ion and g r a t i t u d e t o Professor Robert W. Newcomb,

who throughout t h e course of t h e research cons tant ly provided ex-

cept ional ly cons t ruc t ive guidance.

The i n t e r e s t displayed by Professors Thomas Cover and John Moll

i n t h e i r reading of t h e manuscript i s a l s o appreciated.

1 I. 1NTIU)DUCTION

A. AINS AND PHILOSOPHY OF THE RESEARCH

Huch of engineering i s concerned with the replacement of physical

e n t i t i e s by models, upon which calculations may be performed; of ten the

aim i s t o interpret physical conditions of the physical en t i ty a s mathe-

matical conditions on some part of a mathematical model, o r vice versa.

This i s precisely the aim here. The physical e n t i t i e s that we consider

are time-varying e l ec t r i ca l networks, which we shal l often assume t o be

imbued with physical properties such a s l inear i ty , passivity, and f i n i t e -

ness, a l l properties well known i n more c lass ica l studies; the mathe-

matical models we consider are pr incipal ly described by network matrices,

such as the s c ~ t t e r i n g matrix o r impedance matrix. We may seek necessary,

suff ic ient , o r necessary and suf f ic ien t conditions on ( for example) a

scat ter ing matrix f o r the associated network to be l inear, o r passive, o r

f i n i t e , o r t o possess any combination of these and other properties; or

we may seek means f o r passing from the network to a matrix description

' (analysis) , and means for passing from a matrix to the network (synthesis) ,

In point of f ac t t h e . l a t t e r i s the principal aim here.

Some of these problems may have well-known answers (e.g. those con-

cerned with l i n e a r i t y ) o r have answers which a re easy to obtain (e.g. those

concerned with reciproci ty) . It is possible t o say for example tha t a net-

work N is l i nea r (and reciprocal i f and only i f i t s scat ter ing matrix s - 1 ' - possesses property a (and p ), where a and @ are two simple mathe-

matical properties [Ref. 11. In t h i s d i sser ta t ion we venturkaway from

the beaten path i n a discussion primarily concerned with networks composed

0% interconnections of possibly time-varying, l inear, nondissipative c i r -

cu i t elements. While section C discusses i n more de ta i l what is attempted,

l e t it be s ta ted a t t h i s stage tha t the object is again to pa i r off physical

and mathematical properties, as much as possible i n a one-to-one fashion.

B. REVIEW OF PREVIOUS WORK

A search of the available l i t e r a t u r e shows a dearth of material con-

sidering time-varying networks from a generalized point of view. Although

Zadeh i n a 1961 review a r t i c l e [Ref. 21 was able t o quote 130 references

concerned wi th time-varying c i r c u i t s i n Some way, only a bare handful of

t h e s e r e fe rences could be described a s applying t o general time-varying

c i r c u i t s , t h a t is, c i r c u i t s where only a minimum number of physical con-

s t r a i n t s were assumed, as opposed t o c i r c u i t s where f o r example s inuso ida l

o r "slow" element v a r i a t i o n s were assumed. Among such we could quote f o r

example two o t h e r a r t i c l e s by Zadeh [Refs. 3>4] and t h a t of Darlington

[ ~ e f . 51, Since 1 9 6 1 t h e s i t u a t i o n has been l i t t l e changed save f o r t h e ap-

pearance of a work by Newcomb [ ~ e f . 61, which l e d on t o f u r t h e r develop-

ments by Newcomb, Spaulding and t h e present author, ac t ing separa te ly and

i n conjunction, e.g. [Ref s. 7-15]. Reference 6 approaches time-varying

network theory axiomatical ly; t h e only p r o p e r t i e s t h a t a r e pos tu la ted

(not necessa r i ly simultaneously) a r e l i n e a r i t y , pas s iv i ty , s o l v a b i l i t y

(equiva lent t o ex is tence of a s c a t t e r i n g matrix, o r t h e p resc r ip t ion of a

unique response t o a c e r t a i n e x c i t a t i o n ) , l o s s l e s sness , rec iproc i ty , and

time-invariance. Prom t h e pos tu l a t ed p rope r t i e s , and these alone, i t is

poss ib l e t o deduce derived p r o p e r t i e s of networks. Of t h e sum t o t a l of

Newcomb and Spaulding 's work, Spaulding's d i s s e r t a t i o n [ ~ e f . 91 probably

is most successfu l i n pursuing a given l i n e of deduction. Af ter an i n i t i a l

cons idera t ion of s c a t t e r i n g and immittance d e s c r i p t i o n s of t h e b a s i c bui ld ing

blocks of networks, i . e . r e s i s t o r s , capac i tors , inductors , gy ra to r s and

mul t ipor t t ransformers, Spaulding deduces t h e mathematical form of t h e im-

mit tance of a l o s s l e s s network, and shows how t o synthes ize a network given

an immittance of t h i s form.

C. REVIEW OF THE RESEARCH REPORTED

The p r i n c i p a l but not t h e s o l e r e s u l t presented he re is concerned

wi th networks composed of nondiss ipa t ive elements. Somewhat paradoxical ly,

such networks a r e not necessa r i ly l o s s l e s s ( i n a sense t o be described) ,

and t h e r e s u l t s on l o s s l e s s s c a t t e r i n g ma t r i ces of Spaulding a r e not d i r -

e c t l y appl icable . A new mathematical condi t ion on t h e s c a t t e r i n g m a t r i x o f such

a network i s described (chapter v), and t h e converse problem i s solved, t h a t of

synthes iz ing a network given a s c a t t e r i n g mat r ix s a t i s f y i n g t h e condition

(chapters V I , V I I ) .

A s a note of caut ion t o t h e reader, we remark t h a t except i n t h e most

important cases f u l l proofs of t h e der iva t ion of these r e s u l t s and o the r s

may not be given here. The d e s i r e f o r brevi ty, t h e a v a i l a b i l i t y of these

proofs i n referenced papers of t h e author, and hopeful ly heightened con-

t i n u i t y of t h e t e x t through t h e use of o u t l i n e proofs a r e our apology f o r

a l l omissions.

I n chapter I11 a s a pre lude t o t h e main mater ia l , we apply methods

of func t iona l analys is t o t h e s tudy of passive s c a t t e r i n g matr ices t o

conclude new and apparently fundamental proper t ies . S imi lar mathematical

techniques y ie ld o t h e r r e s u l t s , most of which a r e d e t a i l e d i n [Ref.l5],

which we draw on t o g ive new i n s i g h t i n t o a e cons t r a in t of loss lessness ,

(chapter 111) and t h e d e s c r i p t i o n of t h e interconnect ion of networks

(termed cascade loading)(chapter IV) . There is a l s o t o be found i n chap-

ter I V a discussion of network equivalences, cu l l ed from Anderson, Spaulding

and Newcomb, [Ref. 81 which i s important i n the sequel.

Following t h e nondiss ipa t ive synthesis , t h e syn thes i s of lo s sy sca t -

t e r i n g matr ices i s considered. While not i n the n i c e form of t h e nondis-

s i p a t i v e synthesis ,

whose so lu t ion , i f it e x i s t s , immediately y ie lds a syn thes i s of a lossy

network s c a t t e r i n g matr ix. (chapter VI I I ) .

Lest system t h e o r i s t s f e e l neglected, we hasten t o mention t h a t a

time-varying equivalent of t h e degree of a matr ix of r a t i o n a l t r a n s f e r

funct ions i s discussed i n chapter IV, while i n chapter I X t h e r e i s mention

of two con t ro l systems r e s u l t s achieved by Newcomb and Anderson [Refs. 16,171 which depend on t h e s o r t of ana lys i s reported here.

Natural ly any time-varying synthes is technique when applied i n t h e

t ime-invariant case should s t i l l go through; the consequences of t h i s a r e

examined i n chapter IX, where r e s u l t s without t h e a s soc ia t ed mathematical

d e t a i l s a r e mentioned. A s t hese r e s u l t s a r e s t r i c t l y pe r iphe ra l t o t h e

main ma te r i a l of t h i s d i s s e r t a t i o n , we do not appologize f o r t h e omission

of purely technica l ma te r i a l .

While it should be c l e a r from t h e t e x t a s t o what ma te r i a l is new, we

poin t ou t here t h a t most of chapter I1 i s not new, t h a t which i s new being

merely r e i n t e r p r e t a t i o n of o l d r e s u l t s . A similar remark can be made con-

cerning sec t ions A and D of chapter I11 and s e c t i o n s B, C, and F of

chapter I V . Other ma te r i a l however was developed by t h e author, though

o f t e n i n conjunction wi th o the r s .

-3-

D. : MATHEMhTICAL TECHNIQUES W U ) Y E D

I t i s probably appropr ia te a t t h i s s t age t o mention t h e s o r t of

mathematics t h a t i s used. Di s t r ibu t ion Theory a s developed by L.

Schwartz [Refs. 18, 1 , 0 i s required t o formulate theorems on net-

works which a r e not necessa r i ly f i n i t e , though f o r f i n i t e networks

heavier demand i s probably placed upon l i n e a r d i f f e r e n t i a l equation

theory.

Functional analys is , i n p a r t i c u l a r r e s u l t s from t h e theory of

Hi lbe r t space, is used t o i l luminate severa l poin ts , a s t h e mathe-

mat ics here enables one t o see "what is r e a l l y going on."

Final ly, t h e no ta t ion and ideas of networks and system theory,

a s developed by many e l e c t r i c a l engineers who found t h e s tandard

mathematics i n s u f f i c i e n t o r unsuitable, is drawn upon from time t o

t i m e .

11. REVIEW, DEFINITIONS AND NOTATION

Most of the notation t o be used is described i n t h i s chapter; a

f u l l l i s t of symbols however appears before chapter I .

A. PORT VARIABLES AND THE SCATTERING MATRIX

For greater de ta i l , Reference 6 should be consulted. A l inear

n-port network N can be described through the allowable port voltage - and current vectors v and , i which are n-nectors i n r9 i . e . , 1 - - -+ ' and i are vector functions which are in f in i te ly dif ferent iable and - zero u n t i l some f i n i t e time. The f u l l reasoning behind the mathematical

r e s t r i c t i ons on v and i is discussed i n Reference 6, but i n brief - - these r e s t r i c t i ons are associated with the fac t that a l l excitations of

a physical system must have commenced a t some f i n i t e time, and only very

unusual physical systems exhibit discontinuit ies i n t he i r responses.

Given port vectors v and i i t i s then possible t o define inci- - - i r

dent and ref lected voltage vectors v and v where - -

A scat ter ing matrix is an n x n matrix of dis t r ibut ions i n two vari- i

ables [Refs. 18, 19, 201, o r kernel, - s(t,'r), mapping arbi t rary - v r

in to v 6 _+ - fi through

o r i n br ie fe r notation,

r i v = s o v - - -

Linear networks possessing a scat ter ing matrix are termed solvable

[Ref. 61. This is because t h e existence of a scattering matrix for - N

i s equivalent t o the series-augmented network N formed from N i n -a

Fig. 1 possessing a unique current response i t o an arbi t rary ex- - c i t a t i on e e &, see Anderson and Newcomb [ ~ e f . 141. Plainly solvabi l i ty - w i l l be a property normally possessed by real-world networks

FIGURE 1

ILLUSTRATION OF SOLVABILITY;

1 denotes n un i t res i s tors ; -n

ports of N a r e not shown separately. -

In addition to the notation of Eq.(2.2b) whichis shorthand f o r an

operator acting on a function, it is convenient to define a notation for

a composite operator defined by l e t t i n g one operator act on another. For

example i f s ( t , ~ ) and s 2 ( t , 7 ) a r e two kernels which are matrices of -1 dis t r ibu t ions i n two variables t and T and which map $ in to g+, then we may define s ( t , ~ ) , the composition of s and s by

-3 -1 -2'

r i v = s . v = s . ( s 2 e x i ) - -3 - -1 (2.3)

i . Because s maps g+ into $I s v is i n the domain of s and in -2 3' -2 - r -1 '

f ac t we w i l l have v - e $ as a resu l t .

For convenience we write

o r (as may be. logical ly ver i f ied from Eq. (2.3))

A s is discussed i n References 15 and 20, the composition of more than

two operators i s not necessari ly associative. In the case where the op-

erators map. &! in to &! however, associat ivi ty does apply. 3 4

We remark tha t when s and s are scat ter ing matrices we w i l l -1 -2

of ten term s the product of s and s and say tha t s possesses -3 -1 -2' -3

the factorization s o s -1 -2.

B. PASSIVITY AND MSSLESSNESS

We introduce the notation f o r two n-vector functions of time f and - g -

and

assuming the in tegra l s always ex is t . The superscript t i l d e denotes matrix

transposition.

Suppose now f and g are taken as the port voltage vector v and - - - simultaneous current vector i of a network N, with 1, 5 assumed to - be i n &&. Thus we form

which has the physical meaning of the energy input t o the network up till

time t. Reference 6 defines N as passive i f t h i s quantity i s non- - negative for a l l possible choices of simultaneous pa i r s v and 2, and - times t . An a l te rna t ive expression for ~q' ,(2.6a), following direct ly

from ~ q . (2.1), i s

i Suppose f o r the moment we r e s t r i c t N t o being passive. I f v can - -

i be taken a s square-integrable, i . e . v e B n g then &(a) i s wel l - 4 -2'

i defined, being nonnegative and bounded above by / v i f . It f u r t h e r -

r fo l lows from Eq. (2.6b) and t h e nonnegativity of &(a) t h a t v - e $2, and thus t h e por t vol tage and current vec tors v and i a r e a l s o - -

i r square integrable, being l i n e a r combinations of v and v by - - ~ q . (2.1).

Lossless N [Ref.6] a r e now defined a s those N which a r e - - pass ive and solvable, and have one addi t ional property, v i z . t h a t f o r

i every v g+ nJ2, &(m) = 0 . The physical motivation f o r t h i s re- - quirement i s t h a t t h e energy de l ivered i n t o the c i r c u i t up t i ll t = m

should be zero, provided t h a t t h e exc i t a t ions f a l l t o zero f a s t enough

a s t approaches i n f i n i t y . We comment t h a t p a s s i v i t y impl ies t h a t i

~ ( m ) is f i n i t e , while s o l v a b i l i t y permits the choice of any v - r € 5

with t h e knowledge t h a t t h e r e w i l l be a corresponding v . - To c lose t h i s sec t ion we remark t h a t some of t h e c o n s t r a i n t s

imposed by p a s s i v i t y and los s l e s sness on s c a t t e r i n g ma t r i ces w i l l be

discussed i n t h e next chapter .

C. OTHER NETWORK MATRICES

We excuse t h e f a c t t h a t our t h e o r e t i c a l d iscuss ions a r e normally

i n terms of the l e s s known s c a t t e r i n g matrix r a t h e r than t h e b e t t e r

known immittance matr ices on t h e grounds t h a t t h e l a t t e r have more

r e s t r i c t e d a p p l i c a b i l i t y . None t h e l e s s the r e l a t i o n of s c a t t e r i n g

t o immittance matr ices should be understood i f f o r no o t h e r reason than

t h a t t h e l a t t e r w i l l be en te r ing i n t o subsequent d iscuss ions on occasions.

The time va r i ab le impedance z( t ,T) and admittance y(t ,T) of a net- - - work N a r e defined i n a completely na tu ra l way. I f v and i a r e - - - p o r t vol tage and current vectors , then

and +m

i ( t ) = J Y ( ~ , T ) v ( T ) ~ T = y . V - - - - - -m

The mere inscr ipt ion of Eq.(2.7) of course does not guarantee the I

existence of t h e relevant matrices; though we cap usually, but not al-

ways, expect a scat ter ing matrix t o exis t , thJs i 6 not the case for

immittance matrices. A transformer f o r example has neither an impedance

o r an admittance matrix.

Equations ( 2 . 1 , (2.2) and (2.7) allow deduction of the inter-

re la t ions between s, ? and y, assuming existence. We have -

Here 61 i s shorthand f o r the un i t matrix multiplying the impulse -n

6(t-?); thus 6 1 @ w = w for any n-vector w. -n - - - Two other network matrices are sometimes of interest , the augmented

admittance matrix y ( t , ~ ) and augmented impedance matrix z' ( t , ~ ) . The -A -A

network of Fig.1, consisting of N with a un i t res i s tor i n se r ies with - each port, has admittance y , and N with a uni t r e s i s to r i n pa ra l l e l

-A - with each por t has impedance z' Reference 14 shows tha t the existence

-A' of an; one of y z ' and s guarantees the existence of the other two,

-A' -A - and tha t t he following equations r e l a t e the matrices:

D. EXAMPLES OF THE PRECEDING CONCEITS

To illustrate some of the preceding concepts we shall consider the

description by network matrices of the time-varying inductor, which has

the differential equation description

Here q t ) is the instantaneous inductance. We observe immediately from

Eq. (2.10) that

(where the prime denotes differentiation) and thus

From Eq.(2.10) it also follows that

t

(where u(t-7) is the unit step function, 0 for tq, 1 for t%)

Thus:

i r By substituting 2v = v+i and 2v = v-i in Eq.(2.10) the dif-

ferential equation relating the incident and reflected voltages is

found to be

(where the superscript dot denotes differentiation)

This f i r s t order equation is readily solvable, and has the impulse re-

sponse. ' [~ef. 131

The augmented admittance could .be formed from s(t, T ) using Eq. (2.9)

o r by writing down the d i f f e r en t i a l equation re la t ing e and i for the

network of Fig.1. One obtains

The impedance z l ( t ,T) i s determined the same way, &uta t ig routandfs, or A

from 2': - 61-y see Eq. (2.9). . A - A '

111. PROPERTIES OF PASSIVE SCATTERING MATRICES

In t h i s chapter we examine t h e cons t ra in t s on a sca t t e r ing matrix

imposed by p a s s i v i t y and loss lessness .

A. ANTECEDAL PROPERTY

Denoting t h e n x n zero matrix by 0 w e can show f o r a passive -n3 network

s ( t , 7 ) = Cn - f o r t < T (3.1)

I n essence t h i s equation s t a t e s t h a t N i s i n some sense causal, which - i s a property t h a t i s f a m i l i a r i n real-world systems. More prec ise ly s - is antecedal [ ~ e f .21]. But although E ~ . (3.1) might indeed be assumed on

physical grounds, it i n f a c t follows, a s i s now shown, from our e a r l i e r

de f in i t ion of pass iv i ty . i

Let to be f ixed and take - v ( t ) = 0 f o r tito. Equation (2.6b) - . shows - v r ( t ) = - 0 f o r t<to s i n c e e(t)zO by assumption. Therefore,

with an obvious extension of t h e < > nota t ion t o cope with double in te -

g r a l s

n n

whenever y(t)=O - f o r t>t and - X ( T ) = ~ f o r ~ < t with 9, if e & the s e t 0 0'

of i n f i n i t e l y d i f f e r e n t i a b l e functions zero ou t s ide some i n t e r v a l I.

A s t h i s equation i s t r u e i r r e s p e c t i v e of and - x f o r tCto<7, equation

(3.1) follows.[Ref. 8 p.26, p.1081.

B. EXISTENCE OF AD3OINT a .

Associated wi th s t h e r e is defined an ad jo in t transformation, 5 , - through

f o r a l l 5, 2 s. A s i s discussed i n Ref. 17, it follows read i ly from

-12-

a t h i s that s is rela ted simply to s through - -

where, it w i l l be recalled, the t i l d e denotes matrix transposition. It a is not t rue tha t s maps A+ in to &9 but merely, see Ref. 20, that -

a 4' s maps 5 - in to A where A i s the s e t of dis t r ibut ions zero

-4 ' 4 af te r some f i n i t e time. ( In fact , i f the direction of time flow is re-

a a versed, s is antecedal) I t follows tha t a product of the form s o s - - - is not a p r i o r i defined, see section A of chapter I1 o r Ref. 15, and

a a certainly tha t one may not a p r io r i wri te s o (s o s) = s o s ) o s . - - (- - -

I t should also be noted that the inner product on the r ight s ide of

(3.2) is well defined, as it is essent ia l ly an inner product of a d i s t r i - a

bution, o cp, and a tes t ing function, 5. [Ref. 18, p.241 This use - - of the inner product notation w i l l appear i n future pages also.

C. - g2 MAPPING PRaPERTY

The most fundamental property of s is associated with the f ac t that - i

for a passive network, a square integrable v is mapped into a square - r

integrable v . From Eqs. (2.6) and the passivi ty of N - -

i Choosing v c n g2 and l e t t i n g t = m shows tha t s is a bounded l i nea r - - continuous transformation on a subset of g v iz &I+ n ; we can there- - 2' fore make an extension [Ref. 22, p.2981 defining - s i n a bounded manner

f o r a l l vi - =J2. The def ini t ion of the adjoint v i a the inner product a

of Eq.(3.2) means tha t s w i l l be the adjoint of s when s i s re- - - - garded as an operator mapping the Hilbert space g2 into i t s e l f ; thus -

a s i s a lso a bounded l i nea r continuous transformation on S Then noting - 2'

tha t s and sa as .& operators have the same operator norm [ ~ e f .22, - - p.2011, E ~ . (3.4) yields

Equation (3.5) is a necessary condition for the passivity of N; - essen t ia l ly it is also suff ic ient i f N i s l inear and solvable: -

Theorem 3.1. A l inear solvable network N is passive i f - and only i f (1) A scat ter ing matrix s ex is t s mapping v

i - - € % in to

vr c 5 and - (2) s maps 22 in to g 2 and

(3) z(t,r) = gn for t < ~ and

(4) Ik114 1.

Proof: The only i f portion has been proven by the reasoning leading

t o Eq. (3.5). To show the i f portion we f i r s t observe that i f conditions

(I), (2) and (3) are not s a t i s f i ed then N must f a i l t o be e i ther l inear - o r solvable o r passive.

A s we are assuming N l inear and solvable we are led t o consider the - existence of an s sat isfying 1 but which is not passive. Then -

i there is some port voltage Y and current i associated with v =

-1 -1 -1 $ (zl + i ) 6 6 and some f i n i t e constant T such that < v i > = 8 ( T ) < 0. -1 4 -1' -1 T 1

i Let a second excitation v 8 be defined as

-2 4

i . f o r a rb i t r a r i l y small C- 0 (v is defined i n an in f in i te ly dif ferent iable

-2 i manner i n T < t < T + E ) . Then 2v2 = v + i . by solvabi l i ty v - v

-2 -2' -2 - -1 and i2 = Al f o r t S T. Then -

where Y ( G ) can be made a rb i t r a r i l y small by properly choosing E since i r

E ( t ) i s evaluated as an integral . But x2eL2 and thus also v2€Z2 by

1 1 1 1 . Equation (3.6b) then shows, by llsll < 1,

Choosing 0 < y(c) < - E1(T) shows that the assumption of E ~ ( T ) < 0 i s

violated, and hence N must be passive. - Q. E. D.

A negative r e s i s t o r of -2 ohms i s act ive under usual definit ions and

should be so under the above definit ions. As may be easi ly found, it has

s = 3S(t-7). Conditions (l), (2) and (3) of the theorem are sat isf ied, but - condition (4) f a i l s as 1k11 = 3.

We remark tha t it i s now possible t o wri te down a well defined operator a

s o 2, provided i ts domain i s res t r ic ted t o S, functions. For i f x E Z2, - -2 - a a s Q x E X2, and (s o s ) 0 x = s 0 ( s a x) i s then well defined. - - - - - - - - -

Any 2 sat isfying the conditions of Theorem 3.1 i s termed passive;

as an easy consequence of the theorem we have

Theorem 3.2. I f and s are passive then s o s i s passive. -2 -1 -2

Proof: (Outline only) The f i r s t three conditions of theorem 3.1 are easi ly shown t o be sa t i s f ied . For the fourth, we have lkl o s211 lklll 1k211 < 1. For detai ls , see Ref. 15.

We sha l l l a t e r show how a network with scat ter ing matrix 2l o s2 can be constructed from networks of scat ter ing matrices s and s2. -1 Scattering matrix products and factor izat ions w i l l be of considerable

importance f o r the nondissipative synthesis of chapter V I I .

A convenient a l ternat ive formulation of the passivi ty c r i te r ion can

now be given. By definit ion, a rea l dis t r ibut ional kernel - k(a,p) w

which i s self-adjoint , ~ ( p , a) = - k (a, p), i s cal led nonnegative def in i te

writ ten k 2 0, i f f o r a l l x E 8 - - -

Considering Eq.(2.6b), w e can write

The in t roduct ion of u ( t ) allows replacement of t h e i n t e g r a l s over

(-a, t$ by i n t e g r a l s over ( -m, a). Manipulations d e t a i l e d i n Ref.15

then show t h a t we can w r i t e

where

Q (a,@) = u ( t - a ) 6 ( m @ ) l - %,a) 0 {u(t-X)s(h,@)I (3.9d) -t -n -

and Q is a s e l f ad jo in t bounded operator , mapping J2 i n t o J2. -t

Theorem 3.3. A l i n e a r so lvable network N is pass ive i f and only i f condit ions (I), ( 2 ) and (3) of thGorem 3.1 a r e s a t i s f i e d and f o r a l l f i n i t e t

Corollary: By (4) of theorem 3 .1 t h e following version of ( 4 ' ) i s s u f f i c i e n t :

One advantage of t h i s formulation i s t h a t it shows t h e na ture of re-

s u l t s when z o r y, Eqs. (2.7) , e x i s t . Thus, using ~ ( t ) = < v , i > - - - - t = < i , v >t, t h e manipulations of Eq.s (3.9), and a r e s u l t s i m i l a r t o Eq.(3.3), -- we s e e t h a t an equivalent statement of condi t ion ( 4 ' ) is: f o r each t t h e

form

I n genera l R t w i l l not be a bounded opera tor on J i n con t ra s t t o Q . -2' -t

For example, i f z i s chosen t o be u(t-T), corresponding t o a 1 fa rad - capaci tor , R ( q p ) = ~ ( t - a ) u ( a - p ) + ~ ( t - p ) u ( p - a ) . Because of t h e i n t e r - t g r a t i o n s involved, it w i l l be impossible f o r ((R ( c I , ~ ) c i (p) l I t o be bounded

t by ~ l l i ( p ) I I f o r a f ixed M i n t h i s example.

D. LOSSLESS CONDITION ON s - Suppose t h a t N i s a l i n e a r , solvable, passive network which i s loss- -

i l e s s . Then &(m) is zero f o r any inc ident vol tage v - E 5 n L ~ , and thus i n t h e language of t h e previous sect ion, from Eq.(3.9c)

from which it read i ly fol lows ( s e e Ref. 15) by applying known r e s u l t s

from t h e theory of Hi lber t spaces [Ref. 23, p.2671 t h a t

Q = o -Q) -

a Observing Eq. (3.9d) we see t h a t C&= 6Ln - o S. Thus w e have -

Theorem 3.4. A pass ive s i s l o s s l e s s i f and only i f -

a s 0 s = 6 1 - - a (3.12)

An important example of a l o s s l e s s s c a t t e r i n g matr ix is

where cp(t) i s square in t eg rab le over [c,co) f o r a l l f i n i t e c. The

network t o which t h i s s c a t t e r i n g matr ix corresponds ac tua l ly c o n s i s t s 1

Of a t ransformer with time-varying t u r n s - r a t i o ~ ( t )

which i s loaded at i ts secondary i n a u n i t capaci tor , see Fig.2.

FIGURE 2

NETWORK WITH SCATTERINGMATRIX OF THE FORM OF EQUATION (3.13)

That ~ q . (3.12) is i n f ac t s a t i s f i ed by the expression of Eq. (3.13) is

not hard t o verify.

Though t h i s network w i l l be loss less for any cp square integrable a on [c,m) for a l l f i n i t e c, i . e . s o s = 61 , it w i l l not necessarily

a be the case that a t the same time s o s = 61. In f ac t d i rec t cal-

culation shows tha t a necessary and suff ic ient additional condition for a

s o s = 6 1 i s that cp not be square integrable on (-m,m). I t should thus - a

be noted that s i n Eq.(3.12) is not necessarily a r ight inverse of - s i n general. -

We also point out that by s ta r t ing with ~ q . ( 3 . 1 0 ) i t i s not possible

t o conclude that %= 0. Hilbert space theory cannot be applied here, - and indeed such a resu l t would clearly be wrong, since R (a,'$) =?(a,@) -a

f o r W. > p, and there are c lear ly non-zero loss less impedances, e.g.

z=u(t-T), a 1 farad capacitor. Although Spaulding [ ~ e f .g] derives the

loss less constraint on s, the importance of the d: mapping properties - -2 i s not made clear ; the preceding remark should therefore emphasize t h i s

importance.

The re la t ion (3.12) can be likened t o a similar resu l t of time-invar- - i an t theory, vie _S(-p)S(p) - = _1 (where S(p) i s the Laplace-transformed

n - s ( t ) ) . For time-invariant systems, the composition becomes convolution, - which i n turn becomes multiplication on taking Laplace transforms.

IV. FINITE NETWORKS

A. EQUIVaENCES

Fini te networks a r e those networks consisting of an inter-

connection of a f i n i t e number of res is tors , capacitors, inductors,

gyrators and (multiport) transformers, any of which may be time-

variable. For l a t e r calculations, it i s convenient t o replace

certain time-variable elements by equivalent c i r c u i t s where the

only time-variable elements appearing i n these equivalent c i rcu i t s

are time-variable transformers.

A time-varying r e s i s to r r ( t ) has an equivalence of the form

[ ~ e f . 81 shown i n Fig. 3(a).

The gyrator, inductor and capacitor have the equivalences shown i n

Pig, 3(b), (c) , and (d), provided the i r time var ia t ion i s "sufficiently"

smooth.

Several features a r e common to these real izat ions . F i r s t , a l l time-

variation i s i n the transformers, Second, the transformer turns r a t i o s

are functions of r( t) , c ( t ) , etc, as the case may be. For example, i f

r ( t) 2 0,then tl = (r(t), t2 = 0, while i f r ( t ) < 0, tl = 0 and

t2 = J-r(t) . The precise formulae for the t . a re i n Ref. 8. Third, 1 -

i f the time-variable element i s passive, then no ac t ive elements w i l l

appear i n i ts equivalent c i rcu i t [ ~ e f , 81. This means f o r example tha t

i n the case of the inductor, t and t w i l l be zero. 2 4

Consequently we may assume for any f i n i t e network a l l time var ia t ion

c m by the use of c i r cu i t equivalences be included i n transforaners, and

i f the time-varying elements are passive, a l l elements i n the c i rcu i t

equivalence a re passive.

In the passive case, element scattering matrices are known t o

exis t . We have already noted tha t of the inductor, see ~ q . (2.14). That

of the capacitor may be obtained by replacing &(t) by @(t) i n Eq.(2.14),

and changing the sign of so Eqa(2.8b) yie lds quick calculation of S

for the r e s i s t o r and gyrator.

To simulate a time-varying ci rcui t , it is evident tha t the time-

variable transformer is going to require simulation. This problem is

discussed i n Ref. 12, where it is pointed out t ha t variable autotrans-

FIGURE 3

TIME-VARYING CIRCUIT EQUIVALENCES

formers can approximate t h e d e s i r e d behavior. The cascade of two gyra to r s

a l s o behaves l i k e a transformer, and thus i f a t ime-variable gyra to r can be

constructed, a simulation is a v i l a b l e . The idea l behavior is hardest t o

approximate, whatever means a r e used, i n the v i c i n i t y of zero tu rns - ra t io .

This seems t o be because, while leakage inductance i s t o be kept minimal, - t h e mutual inductance must be a s large+%s p o s s i b l 9 yet change s ign a s t h e

t u r n s - r a t i o goes through zero. A t some s t age the mutual inductance must

cease t o be f a r l a r g e r than t h e leakage inductance, and i d e a l behavior is

l o s t . I f t h e time i n t e r v a l over which t h i s occurs is small however, i ts

e f f e c t on t h e c i r c u i t may be n e g l i g i b l e i n some s i t u a t i o n s .

Since t h e e f f e c t i v e list of c i r c u i t elements includes; t ime-invariant

(posi t ive-valued) r e s i s t o r s , capac i to r s and inductors, t ime-invariant

gy ra to r s and time-variable t ransformers, we choose t o descr ibe such net-

works with no r e s i s t o r s a s nondiss ipa t ive . I t is of course known t h a t

t ime-invariant inductors, capac i to r s and gyra tors a r e l o s s l e s s . The time-

v a r i a b l e transformer is a l s o l o s s l e s s [ ~ e f . 123; i f zl, x2, 51, i2 a r e

t h e primary voltage, secondary voltage, primary current , secondary cur- - r e n t , it i s known t h a t vl = 2 v2, A2 = -2 il where N is t h e turns- - r a t i o matrix. Then '; i + i t h e net power i n t o t h e transformer, i s

-1-1 -S2' i d e n t i c a l l y zero.

The reason f o r naming networks containing no r e s i s t o r s nondiss ipa t ive

r a t h e r than l o s s l e s s w i l l become c l e a r a t t h e s t a r t of t h e next chapter .

B. EFFECT OF FINITENESS ON MATRIX DESCRIPTIONS i r

For a f i n i t e n-port network N v and v a r e r e l a t e d by an ordinary -7 - -

d i f f e r e n t i a l equation.

where C and D a r e n x n ma t r i ces whose elements a r e polynomial i n - - t h e d e r i v a t i v e operator p d/dt wi th time-varying c o e f f i c i e n t s . I f

s i s passive, then i t can be shown [Refs. 9, 131 t h a t -

Here A ( t ) is an n x n matrix, and @ and Y a r e n x r matrices, - - - depending on the order of t h e d i f f e r e n t i a l equation (4.1) .

Without explicit mention, we shall henceforth assume that the

element values of N are always infinitely differentiable. This - i

means that an infinitely differentiable v will yield an infinitely - r

differentiable v , and that the entries of A(t), g(t) and y(t) are - - - all infinitely differentiable.

Passivity, as applied to the case of a finite network, requires

inter alia the following:

1. The eigenvalues of A(t) are bounded in modulus by unity. - 2 . The elements of @(t) are functions square integrable on -

LT,~) for all finite T.

3. The elements of y(t) are functions square integrable on - (--,TI for all finite T.

Statement (1) can be shown to follow from Eq.(3.5) while (2) and

(3) are consequences of Eq.(3.1) and the L mapping property of s. -2 - Similar analysis applied to passive immittances of finite net-

works shows they have the form [~efs. 9, 111

The matrix entries are again all infinitely differentiable, with L - and M n x p matrices, (p depending on the order of the defining - differential equation). The matrix T is q x n, q s n. -

One consequence of passivity is that Z f is nonnegative de- -0 -0

finite.

C. EXISTENCE OF THE ANTECEDAL ADJOINT

Given a scattering matrix s(t,'T) we have described the existence - of an adjoint - sa(t,~) in Eq. (3.3):

sa(t,7) = Z(7,t) - -

For s of the form of ~q.(4.2), i.e. -

s(~,T) = a(t)s(t-T) + 2(t)z(~)uft-~) - -

The presence of U(T-t) means that t h i s sa is not antecedal, and the - d i f f i c u l t i e s associate& with including sa i n a composition product - have been discussed. To eliminate t h i s problem and for other reasons,

we define an antecedal adjoint:

a a We note tha t s i s formed from s by replacing U(T-t) with

-a - a a

-u(t-7); t h i s means tha t s and s sa t i s fy the same d i f fe ren t ia l -a -

equation. I f s however i s not determined from a d i f fe ren t ia l - a a

equation it may not be possible to define s (though s can always -a -

be defined) . a

Naturally a f i n i t e z defines a z . For the z of (4.3) one - -a - has the antecedal solution of the d i f fe ren t ia l equation sa t i s f ied by a

Z -

D. THE CONCEPT O F DEGREE O F A SCATTERING MATRIX

The matrices @ and \Y i n the equation f o r 2, (4.2), are n x r ; - n is the number of por ts of the network N, while r is a measure of

the complexity of the network, termed i ts degree. (S t r i c t l y speaking,

the columns of both - @ and 2 are required t o be l inear ly independent on

(-a, m ) . I f t h i s i s not the case they may readi ly be made so. I f rl

< r, and r columns only of @ are l inear ly independent, m ( t ) = ml(t)c 1 - -

where - @ i s n x r C is constant and r x r ; then m ( t ) Y ( T ) = m,(t)Yl(7) 1: - 1 where lyl(t) = l y ( t ) ~ and i s also n x r ).

1

The concept of t h e degree of a t ime-invariant network o r a matrix

of r a t i o n a l t r a n s f e r funct ions has been known i n network theory f o r

some time, e .g. [Refs. a4, 251. We r e c a l l t h a t t h e degree of a passive

s c a t t e r i n g matr ix - s ( ~ ) is t h e minimal number of energy s torage elements

i n a network synthesizing t h e matrix. I f such a s c a t t e r i n g matrix i s

w r i t t e n i n time-domain notat ion, a s i n Eq.(4.2), t h e elements of t h e

@ and matr ices become exponentials, and t h e number of l i n e a r l y - - independent columns over (-a, a) equals t h e degree of t h e matrix.

I t i s the re fo re p leas ing t o note t h a t t h e syn thes i s procedures

given f o r s c a t t e r i n g matr ices of (passiv4,non-dissipative and then

lossy networks l a t e r i n t h i s work a r e minimal degree ones, i n the

sense t h a t t h e number of r e a c t i v e elements used i n t h e synthes is i s

equal t o t h e number of l i n e a r l y independent columns of @ and Y. - - Abstract arguments contained i n ~ e f . 2 6 and discussed i n Ref.27 show

t h a t synthes is wi th fewer r eac t ive elements i s not poss ib le . Ref. -

should a l s o be consulted f o r a f u l l e r discussion of t h e degree concept

i n t h e present context, including its extension t o immittances.

Denoting t h e degree of s by deg [?I, f u r t h e r p a r a l l e l s with t h e - t ime-invariant theory a r e exhib i ted by such r e s u l t s a s deg [s] =

deg [s + ~ ? 5 ] where K i s a time varying matrix, and deg ~ s - l ] = deg [?I, - - - - provided the inverse e x i s t s [ ~ e f . 281.

E. CASCADE LOADING

In order t o o b t a i n somewhat more s p e c i f i c r e s u l t s f o r networks of

considerable importance we t u r n t o a usefu l method of combining networks,

t h a t of cascade loading. I n p a r t i c u l a r w e c a l c u l a t e t h e input s c a t t e r i n g

matr ix s when a c e r t a i n inverse e x i s t s and show t h a t under such a con- - d i t i o n s i s pass ive when t h e subnetworks are . -

Consider an ( n + m)-port N whose v a r i a b l e s a r e p a r t i t i o n e d a s t h e - -i %r -r

por ts , t h a t is, v = [ yi 1, v = [!I, v2] with t h e subsc r ip t s 1 and -c -1' -2 -z 2 respect ive ly denoting n- and m-vectors. Then an m-port 3 i s s a i d

t o cascade load N i f -z i r r i 2 = x2 and 3 =

i r where 3 and 3 a r e incident and r e f l e c t e d vol tages f o r N This 4 connection def ines a new n-port 2, a s i l l u s t r a t e d i n Fig. 4.

- 2k-

FIGURE 4 ILLUSTRATION OF CASCADE LOADING

The incident and reflected voltages of N are -

i i r r v = v a n d v = v - -1 - -1 (4.7b)

subject to the constraints placed on the coupling network N by loading -z it with Partitioning the scattering matrix z of N according to - -C its port partition and defining and s as the scattering matrices - of and N lead to -

r i where Eqs.(4.7) have been used in v = z o . In order to gain

. -z some insight into various of the following manipulations we write these

out fully as

i We wish t o so lve f o r vr i n terms of 1 , t o equate t o ~ ~ . ( 4 . 8 c ) by -

i el iminat ing v Doing t h i s when t h e ind ica ted inverse e x i s t s y i e lds &'

Unfortunately t h e inverse requi red i n (4.10) may not always e x i s t

even though N and 3 a r e pass ive . I n con t ra s t t o t h e time-invar- -x i a n t case [Ref. 291 t h e r e is not always a way around t h i s d i f f i c u l t y .

A transformer with tu rns - ra t io ~ ( t ) with secondary open-circuited does

not have a wel l defined s c a t t e r i n g matr ix i f T ( ~ ) = o f o r some t CRef.151, and exemplif ies t h e d i f f i c u l t y . I n f u t u r e appl ica t ions of these r e s u l t s

we s h a l l normally be assuming exis tence; f o r a discussion of t h e d i f f i -

c u l t i e s involved, see Ref. 15.

One app l i ca t ion of t h e idea of cascade loading i s t o represent any

network N a s a combination of N cons i s t ing of t h e c i r c u i t elements - 4' of N unconnected t o one another, and N a network cons is t ing of con- - -2 necting wires alone, a s shown i n Fig. 5 . The interconnect ion of - any two

networks N and 2 2

can a l s o be viewed a s a cascade loading, [Ref .29]. -1

I n Fig. 5 %would cons is t of N and N2 uncoupled t o one another. -1 -

and Shor ts

FIGURE 5 REPRESENTATION OF NETWORK AS CASCADE LOADING

This technique has been f r u i t f u l l y applied i n t h e examination of

t ime-invariant network problems, [ ~ e f s . 29, 301, and it w i l l be used i n

t h e next chapter i n a considerat ion of nondiss ipa t ive networks.

F. EXAMPLES OF CASCADE MADING

We examine here the e f fec t of connecting an orthogonal transformer

a t the ports of a network 3 with scattering matrix s ( t , ~ ) . An 4 orthogonal transformer i s defined by

where T is an orthogonal matrix, and the subscripts r e f e r to primary - and secondary .ports.

Ref. 6 out l ines the calculation of the scat ter ing matrix of a trans-

former with the turns-ratio matrix ~ ( t ) being an arbi t rary m x n matrix. - The resu l t i s

~ ( t , 7 ) = 6(t-7) -

- -1 (r, +E) (Am - *) L

(4.12)

I f now T is r e s t r i c t ed t o being orthogonal (with m=n), considerable - simplification ensues with

Equation (4.10) now allows calculation of the scat ter ing matrix s ( t , ~ ) - of the transformer loaded a t i t s secondary with the network N The re-

4' su l t is

and i s of some importance. I f T i s not orthogonal then the cascade - loading calculation using (4.10) w i l l c lear ly not be as straightforward.

A s a second example, w e t a k e N t o be an a r b i t r a r y network, and -C -

t o cons is t of m u n i t r e s i s t o r s , one f o r each por t , s o t h a t 2.=

Then i f

d i r e c t app l i ca t ion of t h e cascade loading formula (4.10) y i e l d s

Effec t ive ly , we d e l e t e rows and columns of 2 by matching a t some of - i ts por t s . This technique w i l l be used i n t h e sequel, chapters V I I

and V I I I .

V. NONDISSIPATIVE NETWORKS

A. IDSSLESS AND NONDISSIPATIVE NETWORKS

In t h i s section we draw a d i s t inc t ion between loss less networks a

( i . e . those networks with scat ter ing matrices satisfying s o s = - - 6 1 ) and f i n i t e nondissipative networks ( i . e . those composed of f i n i t e -n

interconnections of loss less elements). That there is a difference i s

seen by exhibiting a non-lossless nondissipative network, which is i n

f ac t an interconnection of loss less subnetworks.

Specifically we consider the s i tua t ion shown i n Fig.6

FIGURE 6 A NONDISSIPATIVE NETWORK

Calculations outlined i n Ref. 15 show tha t

(5.1)

Here the rea l f i n i t e constant a is arbi t rary.

The adjoint is found i n the usual fashion: f 7

a (5.2)

The composition product s o s can now be formed; it consists of the - - sum of a term 6(t-7), a term Y ( ~ , t ) u ( ~ - t ) and a term F ( t , ~ ) u ( t - T ) , where

F ( ~ , T ) i s given by

a If ( P(A)d), i s not divergenl, then i t i s impossible t o have s o s

= 6; on t h e o t h e r hand i f T (~ )d ) , is divergent , then d i r e c t cal- a Jcx '

cu la t ion shows t h a t s o s = 6 . Thus t h e network of Fig. 6 is loss-

less i f and only i f T (h)dh = m f o r any a. This conclusion is a l s o L 2 reached bv an a l t e r n a t i v e anaLysis elsewhere [ ~ e f . 3]: where the pro - p e r t i e s of s i m i l a r networks a r e more f u l l y described. The physical

reasoning suggesting t h i s r e s u l t is a l s o given i n t h i s reference; i n

b r i e f we may say t h a t i f ~ ( t ) vanishes f a s t enough as t -t m, it i s

poss ib le f o r t h e capaci tor t o t r a p charge and r e t a i n it t h e r e a f t e r . Then

&(m) f o r t h e network is nonzero.

The aim of t h i s chapter i s t o f i n d condi t ions on t h e s c a t t e r i n g

matr ix of a network which correspond t o t h e network being nondiss ipa t ive .

I t i s p l a i n t h a t we must appeal t o some more involved concept than t h e

e ( m ) = O property, which i s evident ly not preserved under interconnect ion.

W e in t roduce t h i s concept i n t h e next sec t ion .

B. QUASILOSSLESS SCATTERING MATRICES

Suppose w e a r e given a s c a t t e r i n g matr ix of t h e form of Eq. (4 .2) :

s ( ~ , T ) = A(t)6(t-?) + $ ( t ) y ( ? ) u ( t - r ) - - -

Direct ca l cu la t ions then y i e l d

a - -., s o s = ~ ( t ) ~ ( t ) b ( t - r ) + F ( ~ , T ) u ( ~ - T ) + ! ( ~ , t ) u ( ~ - t ) (5.4) \ - - - - -

where

t a

We may also form s o s, which we reca l l is w e l l defined as s and -a - -

a s both map bQ+ in to XI+ . Again by d i rec t calculation -a

By analogy with the loss less condition on scat ter ing matrices, we cal l

a scat ter ing matrix quasilossless i f

We note tha t i f s is lossless, then it i s automatically quasilossless. - - a

For i f s i s lossless, so tha t s o s = 6 1 , Eq.(5.4) immediately yields - - - - -n ~ ( t ) ~ ( t ) = Ln and F ( ~ , T ) = s. When these are subst i tuted i n ~ ~ . ( 5 . 6 ) , - - - equation (5.7) follows. The converse resu l t does not hold however.

Thus the time-invariant capacitor, inductor and gyrator, and the

time-varying transformer a r e a l l quasilossless. So is any interconnecting

network, a s N of Fig.5; t h i s i s because such networks are loss less -E

[Ref. 291. It is the quasilossless property which w i l l be shown i n succeeding

sections t o be retained on interconnecting quasilossless networks.

Henceforth, unless exp l i c i t assumption i s otherwise made, quasiloss-

l e s s scat ter ing matrices w i l l be taken as being also passive.

C. SCATTERING MATRICES OF NONDISSIPATIVE NETWORKS -- THE QUASIIDSSLESS CONDITION

We apply the cascade loading methods of chapter I V t o demonstrate t ha t

the scat ter ing matrix ofanondiss ipat ive network is quasilossless.

More specifically, we represent a nondissipative network N a s a - nondissipative network 3 of uncoupled c i r cu i t elements loading a net-

work N consisting en t i re ly of interconnections, as i n Fig. 5. Then -C'

though N need not be lossless, both N and are, the former be- - -z cause it contains no c i r cu i t elements a t a l l , t he l a t t e r because the

c i r c u i t elements t h a t c o n s t i t u t e N a r e los s l e s s , and a r e disconnected 4 from one another. Put another way, % i s t h e " d i r e c t sum" of a number

of l o s s l e s s networks, one corresponding t o each c i r c u i t element of 3 Following t h e n o t a t i o n of sec t ion C, t he s c a t t e r i n g matr ix s of -

N i s given i n terms of t h e s c a t t e r i n g matr ices C and 3 of N and - - -z a s ( ~ ~ . 4 . 1 0 ) ,

where we assume t h e ind ica ted inverse e x i s t s . Then, using t h e eas i ly a

derived r e s u l t ( h o k )a = ka o h f o r two opera tors h, k mapping - - a -a -a -t

i n t o 8+

a The product s o s can then be formed; a l l composition products w i l l

-a - be well defined and as soc ia t ive , s ince a l l f a c t o r s map i n t o s.

4 a

I n t h i s r e su l t ing expression f o r s o 2, it is poss ib le t o make -a

s impl i f i ca t ion using

These equations a r e consequences of t h e l o s s l e s s cha rac te r of N and -z % (and the r e s u l t i n g quas i los s l e s s charac ter of t h e i r s c a t t e r i n g matr ices) .

When a l l s impl i f i ca t ions a r e made, one f inds

That is, s is quas i los s l e s s . The d e t a i l e d ca lcu la t ions , though not - hard, a re avai lable i n Ref. 27. This reference a l s o e s t a b l i s h e s t h e

r e s u l t f o r t h e case when [gl -1 - o ] does not necessar i ly e x i s t .

I n f u l l , we have

Theorem 7.1. Let N be a nondissipat ive n-port network with - s c a t t e r i n g matr ix s. Then s is quas i los s l e s s , t h a t is, - -

We comment t h a t N must be solvable f o r t h e theorem t o hold; - t h e theorem does not e s t a b l i s h t h e exis tence of 2, but a condition

-1 on s when i t i s known t o e x i s t . The assumption t h a t [g& - Zp2 o 3,] - e x i s t s i s s u f f i c i e n t (but not necessary) f o r t h e exis tence of 2.

a In con t ra s t t o t h e l o s s l e s s case, where s o s = 6 1 does - - -

a a not imply s o s = 61, it is t r u e t h a t s o s = Q implies - - -

a -a - s o s = 61. One way o f e s t ab l i sh ing t h i s i s t o observe t h a t f o r a - -a s i n g l e element of t h e network % ca lcu la t ions r e a d i l y e s t a b l i s h t h a t a

s f o r t h i s element i s a r i g h t and l e f t inverse. I t follows t h a t the -a a a same is t r u e of s being a d i r e c t sum of t h e s e elemental s . I t can +' -a a l s o be e a s i l y seen t h a t i s a r i g h t and l e f t i nve r se f o r x; t h i s

-a - is because i s a constant matr ix mult iplying 6(t-7) . Using these -

a f a c t s , t h e c a l c u l a t i o n s used t o e s t a b l i s h s o s = 6 1 may be mod-

-a - - a

i f i e d only slightly t o show s o sa = 61. I n s e c t i o n F of t h i s chapter, - we indica te an a l t e r n a t i v e der iva t ion r e ly ing on knowledge of the quasi-

l o s s l e s s condit ion f o r immittances. Thus

Theorem 5.2. With t h e same condit ions a s requi red i n Theorem 5.1,

a a s 0 s = s o s = 6Ln -a - - -a

One of t h e p r i n c i p a l t a sks s t i l l ahead of u s ( t o be covered i n

Chapter VII ) i s t o demonstrate t h a t a s c a t t e r i n g matr ix s a t i s f y i n g Eq.

(5.10) possesses a syn thes i s a s a nondiss ipa t ive network. This r e s u l t

i s t h e converse of t h a t e s t ab l i shed i n t h i s sec t ion .

D. SCATTERING MATRICES OF NONDISSIPATIVE NETWORKS -- MATHEMATICAL FORM

In t h i s section we derive more specif ic re la t ions between the

matrices A, 9, 2 of ~ q . (4.2) -

which are consequences of the quasilossless character of s . We recal l , - see section B, Eqs. (5.5) and (5.6),

where

e ~ t , ~ ) = y t ) ~ c t ) ~ ( ~ ) + get) [Jrl(~i~(i)a] I(T) - t

If - s i s quasilossless, sa t isfying ( 5 . l O ) , we obtain immediately

w , - A ( t ) ~ ( t ) - = 1

-n (5.11)

F ( ~ , T ) - F ( T , ~ ) = o - - -n (5.12)

For convenience se t

m

o ( t ) = y t ) i ( t ) + y( t ) [ J z(A)J(A)~A] - t (5.13)

Then (5.12) becomes

Using the l inear independence of the columns of y, which as explained - ea r l i e r we are f ree t o assume, i t follows from (5.14), see Ref. 27, that

there ex is t s a constant matrix C such that -

with

This reference also establishes that passivity is ensured if and only if

C is nonnegative definite, written -

The method used to establish (5.17) is to show that corresponding to an i

incident voltage v , the associated &(a) is -

i The independence of the columns of y means that v can be selected - - to yield any value for the integral terms. Then for e ( m ) 2 0, (5.17) is necessary and sufficient.

The preceding manipulations have thus established

Theorem 5.3. The scattering matrix

is quasilossless if and only if [see E~S. (5.11), (5.13)) (5.15) (5.16)) and (5.17)1

w ~(t)~(t) = 1 - - -n (5.19.)

where C is a symmetric nonnegative definite matrix. -

An alternative formulation of the quasilossless condition can be

derived from either (5.19) or the second half of Eq. (5.10)--hitherto a

unused--vie. s o s = 61.The symmetry between Theorem 5.3 and the - -a -n

following theorem should be clear.

Theorem 5.4. The scattering matrix

s(t,~) = ;(t)6(t-~) + m(t)Y(~)u(t-~) -

is quasilossless if and only if

- ~(t)~(t) = 1 - - -n (5.20a)

where D is a symmetric nonnegative definite matrix. -

The two equations (5.19) and (5.20) can be written in more com- pact notation, as may be directly verified. Respectively, they are

and

To conclude this section, we point out modifications of these

results which apply to a subclass of quasilossless scattering matrices,

namely the lossless ones.

From the equation (5.4) of section B,

it follows that an additional condition to be fulfilled for losslessness

is

F(~,T) = 0 - -n (5.22)

In place of ~ ~ . ( 5 . 1 4 ) we obtain @ ( t ) Y ( T ) = 0, where @(t) is given by - - - - ~ q . (5 .13) .~ne l i nea r independence of the columns of y then requires - tha t @(t) =g, which i s ~ ~ . ( 5 . 1 5 ) with C = 0. That C = 0 also follows - - - - - from E ~ . (5.18), since f o r loss less s we require ~ ( m ) = 0 for a l l -

i square integrable v . Accordingly we have -

Theorem 5.5. The Scattering matrix

i s loss less i f

and

o r equivalently,

Contrary to what might be expected, it i s not necessarily t rue that

s e y = 2. This is i n f ac t because i t is not necessari ly t rue that - - a

5 0 s = & A n . - -

This concludes our discussion of the detai led mathematical con-

di t ions on quasilossless scat ter ing matrices. In summary, the ortho-

gonality condition on the coefficient matrix of the 6( t -T) term i n 2, a

and the resul t s a 3 = -y C (with C = 0 i f s i s lossless, and - - - - -- - symmetric nonnegative de f in i t e 0therwise)are the conclusions of the

analysis.

E. EXAMF'LES OF QUASIIOSSLESS SCATTERING MATRICES

A s a f i r s t example, w e consider

where A is orthogonal; l a t e r (chapter VII) it w i l l be shown t h a t t h i s - matrix can be synthesized from transformers and gyra to r s alone. We

have a t once

and a a

S 0 S = S 0 S = 6( t -7) - - -a -

a a by t h e orthogonali ty. I n t h i s instance, not merely s but a l s o s i s

-a - a r i g h t inverse f o r s ; t h e l a t t e r need not always be t h e case, a s seen by - t h e example of sec t ion D, chapter 111.

A s a more complicated example we consider t h e capacitor-loaded

transformer of Fig. 6. We found i n sec t ion A t h a t with ~ ( t ) t h e tu rns

r a t i o

s ( t , ~ ) = -6(t-7) + . T ( T ) ~ X P

(5.1) - Thus here the coe f f i c i en t of 6 ( t -T) is c e r t a i n l y orthogonal, being -1.

The matr ices @(t) and y ( ~ ) have become s c a l a r funct ions cp and $. - - We ca lcu la t e

m a

s ocp = - 2 ~ ( t )

where

He observe tha t c i s nonnegative, and i s zero i f P ( h ) d k

diverges; t h i s has already been established i n section A as the loss-

l e s s condition.

It may happen tha t s is not given i n the form above where ~ ( t )

appears expl ic i t ly , i . e . we may be given s purely as

Then we see using Eq.(5.19) that s w i l l be quasilossless i f $ i s

related t o by

where c i s a nonnegative number. Then the s of Eq.(5.26) is the

scattering matrix of a capacitor loaded transformer with turns r a t i o

2 This follows by comparing (5.1) with (5.261, (observe that T ( t ) =

v( t )q ( t ) ) .

F. QUASILOSSLESS IMMITTANCE MATRICES

There i s no a p r io r i reason to suppose tha t the existence of a

quasilossless condition on scattering matrices requires the existence

of one f o r immittance matrices. Indeed by analogy with the loss less

case, where t h e r e appears t o he no meaningfully simple l o s s l e s s con-

d i t i o n on immittance matr ices , one might expect t h e same f o r the quasi-

l o s s l e s s case. Spaulding [Ref. 91 has given a condit ion which is neces-

sary f o r lo s s l e s sness of f i n i t e networks i n terms of an impedance matr ix

z which i s -7

The network of Fig. 6 which has

is such t h a t Eq.(5.29) i s s a t i s f i e d i r r e s p e c t i v e of whether o r not - m

J a ~ ~ ( h ) d ? , diverges. Thus ~ q . (7.29) i s c e r t a i n l y not a s u f f i c i e n t con-

d i t i o n f o r lo s s l e s sness .

Spaulding g ives a syn thes i s procedure f o r impedances s a t i s f y i n g

(5.29) which always y ie lds a nondiss ipa t ive network. Thus we may wel l

ask, i s Eq.(5.29) both a necessary and s u f f i c i e n t condit ion f o r t h e

immittance of a nondiss ipa t ive network? The answer i s yes.

Equation (5.29) can be shown t o follow from t h e condit ion a

s 0 s = i3An i n t h e following way. From Eq.(2.8b), -a -

w e have

and thus

with the composition products a s soc ia t ive . Multiplying on t h e r i g h t by a

( z - + 6Ln), on t h e l e f t by ( z + 61 ), and evalua t ing t h e r e s u l t i n g pro- -a -n ducts leads a t once t o

W e have thus concluded t h a t a solvable nondiss ipa t ive network N - with an impedance matr ix z has z sa t i s fy ing (5.29). The r e s u l t can - - be extended t o nonsolvable networks; any nonsolvable f i n i t e network N - wi th an impedance matr ix z can be s p l i t i n t o t h e s e r i e s combination - of transformer-coupled inductors , and a solvable network N [Ref. 111. -1 Now N i s such t h a t (5.29) is s a t i s f i e d ; the transformer coupled

-1 inductors have an impedance s a t i s f y i n g Eq. (5.29) a lso . Then t h e im-

pedance of N s a t i s f i e s (5.29), being t h e sum of two such impedances. - Consequently w e have t h e fol lowing theorem, i n p a r t due t o Spaulding:

Theorem 5.6. Let N be a network with impedance matr ix z. - Then i f - N is non&ssipative,

and i f z s a t i s f i e s t h i s quaBilossless condition, t h e r e is a nondissiFat ive network synthes iz ing N. -

A dual theorem holds f o r admittances.

A s remarked i n sec t ion C, w e can now suggest new reasons why

sa i s a r i g h t inverse of s i f it i s a l e f t inverse. I f t h e network -a -

a t o which s corresponds possesses an impedance matrix, then z + z = 0 - - -a -7

a and by t h e symmetry of t h i s equat ion e i t h e r of s o s = 6 1 and - -a -

a s o s = 61 implies t h e o the r . I n t h e case where no impedance matr ix -a - - e x i s t s , minor adjustments d e t a i l e d i n Ref. 27 reduce t h e problem t o t h e

case where t h e impedance matr ix does e x i s t .

VI. FACTORIZATION OF SCATTERING MATRICES

A. THE CONCEPT OF FACTORIZATION

We have already mentioned the idea of factorization in section

C of Chapter 111. In this chapter we are concerned with some of the

conditions under which a passive scattering matrix s can be repre- - sented as the composition product of two passive scattering matrices

s and s a -1 -2'

The motivation for this is strong. The matrix s can be syn- - thesized, using an algorithm to be described in the next chapter, if

the matrices s and s can be synthesized. While the synthesis -1 -2

of an arbitrary s may not be immediately possible, syntheses of s - -1 and s may be immediately possible, since these matrices may have a

-2 simpler form than s Again, taking as a yardstick of the simplicity -1' of a scattering matrix its degree, a particularly pertinent problem

for investigation is the determination of conditions guaranteeing s -1

in ~q~(6.1) to have degree one, with s to have degree one less than -2

s. Then although a synthesis of s is not generally immediately - -2 possible, a synthesis for s is perhaps immediately possible, especially

-1 if we can ensure that s is quasilossless. Thus although with this -1 factorization a synthesis of s is not immediate, the problem of syn- - thesizing s has been reduced to a simpler problem, namely that of - synthesizing s of degree less than s.

-2 - The next section will consider the problem of obtaining a factor-

ization as in (6.1) with s quasilossless of degree one and s not -1 -2 -

necessarily of degree less than 2, while the following section will con- sider the requirements necessary to achieve degree reduction (i.e., s -2 of degree less than s). The results obtained will be used in the next - chapter to give a complete synthesis of any quasilossless s, -

B. CONDITIONS FOR FAClVRIZATION

The pr incipal resu l t i s as follows:

Theorem 6.1. Let s be a passive scat ter ing matrix of the - form of Eq. (4-2) :

s ( ~ , T ) = A ( t ) 6 ( t - ~ ) + ~ ( ~ ) Y ( T ) U ( ~ - T ) - -

and l e t s be a degree one scat ter ing matrix of the form -1

Then a decomposition

is possible f o r a l l c > 0 i f is a square integrable on [T,-) f o r a l l f i n i t e T, and s a t i s f i e s

where 1 i s a constant r-vector ( r being the degree of 5, or the number of columns i n y, assumed l inear ly independent).

Before outl ining the proof of t h i s theorem, we comment that

s i s a quasilossless scattering matrix, t h i s fac t following from an -1 application of theorem 5.3 with the simplifications A ( t ) = I.., 2 - and

y vectors ra ther than general matrices, and C a scalar . There are - - certainly quasilossless degree one s without A ( t ) = In, (e.g. -s ), - - - -1 but it is convenient t o assume the form of Eq. (6.2).

The theorem above precludes the case c=O, corresponding to s -1

lossless, a s d i s t i n c t from quasilossless. This is deal t with i n

Theorem 6.2.

I t should be noted that the theorem makes no claims as to the

passivity o r otherwise of s . s however is passive, since y i s -2' -1

square integrable on [T,m), fo r a l l f i n i t e T, and c > 0 (see Theorem

5.3). The passivi ty of s2 w i l l be discussed l a t e r .

Proof: (Outline only) With sl and s as given, we can - a form the well defined transformation s o s. Direct calculation

-la - ( fo r de t a i l s see Ref. 27) yields

(6.4a)

We now define

Then the quasilossless character of s means tha t s o s = s o a -1 -1 -2 -1

O S a ) O S = 6 A n O S = S. (zla O S) = ( s ~ -la - - Associativity of the products follows from the fact tha t a l l factors map $3 in to --+ %*

For notational convenience, we se t

where

(h )~ (h )dh , f tg(h(IH~)dh+ c 1

Although our def ini t ion of s w i l l not always make it passive, -2

w e sha l l t r y t o make i t so by choice of appropriate Cp; one condition - that must be f u l f i l l e d for passivi ty i s tha t y2(7) have elements

which are square integrable over (-a, TI fo r a l l f i n i t e T (chapter I V ,

section B). Although - y ( ~ ) has t h i s property ( s being passive) we must - i n addition require

t o be square integrable over (-=,TI. In order t ha t a t the same time

x 2 ( ~ ) have no greater a number of l inear ly independent columns than y(7) - ( fo r otherwise s would have greater degree than s), we sha l l suppose

-2 - that

where i s a constant vector. Direct calculation shows t h i s i s the - same as

Assuming ~ q . ( 6 . 3 ) , o r equivalently Eq.(6.5), we then obtain

from Eqs.(6.4) the following equation for s . -2'

where i t w i l l be noticed that the dimensions of the matrices corres-

ponding t o @ and y i n s a re the same as the dimensions of $ and - - - - y, namely n x r. This completes the proof. -

We comment tha t there do ex is t cp which sa t i s fy equations of - t he form (6.3). We sha l l be exhibiting such i n the next chapter.

The passivity of s may be examined with the a id of the fol- -2

lowing relation, which i s established in Ref.27. The re la t ion w i l l be

used i n the next chapter t o es tabl ish passivity i n our quasilossless

synthesis technique.

This must be nonnegative def in i te t o ensure passivity (Theorem 3.3, corollary).

Reference 27 also car r ies out the calculations which es tabl ish

the following theorem closely re la ted to Theorem 6.1; we now deal with

the factoring out of a loss less s -1'

Theorem 6.2. Let s be a passive scattering matrix of the - form of E ~ . (4.2):

and l e t s be a degree one scattering matrix of the form -1

then a passive decomposition

is possible i f i s square integrable on [T,") f o r a l l f i n i t e T, and s a t i s f i e s

We comment tha t i n t h i s case the passivi ty of s follows -2

automatically by v i r tue of the following eas i ly established resu l t

[Ref - 271,

The matrix s is i n fac t given by (6.6), with c and Y -2

both zero.

C. CONDITIONS FOR FACTORIZATION WITH DEGREE REDUCTION

In section A we pointed out the des i r ab i l i t y of achieving a

factor izat ion with factors of l e s s degree than the original scat-

ter ing matrix s, while i n section B we detai led a factorization

where one of the fac tors had the same degree as s. Evidently some - additional condition(s) must be s a t l s f i e d i f degiee redugtion i s t o

occur. These conditions are given i n the following theorem:

Theorem 6 . 3 Let the hypotheses of Theorem 6.1 o r Theorem 6.2 hold, with the conditions for factor izat ion being sa t i s f ied . Then the fac tor s has degree one l e s s than s i f the fol- - lowing additional-gonditions a r e f u l f i l l e d :

9 i s a column of 5 - (6.11)

and when c > 0, c is the i - t h entry of y i f i s the i - t h column of $. - Proof: From Eq. (6.6) we have

* '3)T)uit-T)

(with c=O and y=O i n the loss less case).

For brevity we sha l l write t h i s as

Here @ and y are n x r matrices, and s w i l l have degree r -2 - -2

i f g2 and y have r l inear ly independent columns. Now y - - certainly has t h i s property, but ~q . (6 .11) can now be used t o show

that @ has one column zero (and thus fewer than r l inear ly in- -2

dependent columns).

I f T is the i - th column of @, then immediately the i - th - - column of @ i s -2

This completes the proof.

In summary, Theorem 6.1 has shown tha t factor izat ion of s is - -

a possible i f there ex i s t s a square integrable such tha t s 2 =

-Y Y j for some constant vector y; the more useful factor izat ion with - - degree reduction occurs i f T is also a column of 3, with a pos i t iv i ty - condition on an element of y then being required (Theorem 6.3). - Questions of passivi ty when a quasilossless, a s d i s t i nc t from lossless,

s i s factored out can i n theory be decided by Eq.(6.7). -1 In the next chapter we shal l be applying these resu l t s t o the

case where s is quasilossless. One interest ing feature is that we - shal l be able t o show tha t the result ing s2 i s (passive and) quasi-

lossless.

VII. NONDISSIPATIVE SYNTHESIS

In t h i s chapter we present a synthesis of quasilossless scat-

t e r ing matrices. The synthesis proceeds by factoring a quasilossless

scat ter ing matrix into the product of degree zero and degree one

quasilossless scat ter ing matrices, exhibiting syntheses f o r these in-

dividual matrices, and a synthesis f o r a scattering matrix product.

A. FACTORIZATION OF QUASILOSSLESS SCATTERING MATRICES

We commence with a scat ter ing matrix of degree r, assumed

quasilossless, of the form of Eq. (4.2).

The scat ter ing matrix

w i l l then be quasilossless, i n f ac t lossless, by the orthogonality of

~ ( t ) ; moreover s o s must then be quasilossless for - --O -

This matrix is also passive, f o r as may readily be verified,

a " o ? ) ~ o ( Z O O ) = 6 1 - s o s 6Ln - (s -n - -

Since s = s o (s o s), the problem of factoring s is now a - -0 -0 - - - problem of factoring s o 2, which we have pointed out above i s s t i l l

-0

quasilossless, but, as direct calculation shows, has as the fj(t-7) co-

e f f i c i en t matrix 1 rather than an arbi t rary orthogonal A ( t ) . Ac- -n -

cordingly, consider now quasilossless r-th degree s ( t , ~ ) , given by -

Suppose now the f i r s t column of @ i s designated as Cp. - - Equation (5.21a) which requires f o r any quasilossless s tha t -

f o r some nonnegative def in i te 5, then yields

where Y i s the f i r s t column of C. The conditions of Theorem 6.3 - - are now d i rec t ly applicable t o yield

where

Here c is the f i r s t entry of y and thus the 1, 1 entry of C . 11 -

I t i s therefore nonnegative since C is nonnegative def ini te . The - vector y(t) i s square integrable on [ ~ , m ) , because the elements of

@(t) have t h i s property, (chapter IV, section B) . The matrix 6 - -1 i s thus f i r s t degree, ( ~ a s s i v e ) and quasilossless by Theorem 5.3.

An important resu l t of t h i s factor izat ion i s that the matrix

ol, of degree (r-1), is also (passive and) quasilossless. To observe a

the quasilossless property f i r s t , Eq. (6.4b) yields S - -1 - Sla O 2' a a a a Then S o S = s o s o s o s = s 0 6 & ~ 0 5 = 6 L ~ .

-la -1 -a -1 -la - -a

To demonstrate passivity, we r eca l l tha t from Eq.(6.7), c f 0,

a Since &An - - s o - s = - ~ ( t , r ) u ( t - 7 ) - - ? ( r , t )u ( r - t ) - from Eq. (5 .4) ,

Eqs. (5.5) and (5.19) together yield

a 6Ln - S o - s = - y ( t ) $(r)[u(t-T) + ~ ( r - t ) ] (7.7)

and thus

1 This w i l l be nonnegative def in i te i f and only i f - C - - c - - y 7 i s non-

negative. But recal l ing the def ini t ion of - y as t he f i r s t column 1

of C, we see tha t C - 1 is indeed nonnegative def in i te by - - vi r tue of the eas i ly ver i i i ed re la t ions

and - T C T = c @ D - - - 11 - (7.8b)

where @ denotes the d i rec t sum and

In the case where c ,la= 0, passivity of S follows immediately -1

a - as 6Ln - El 0 g1 - 6Ln - 2 o - s, see ~ q . ( 6 . 1 0 ) .

The coefficient matrix of the &(t-7) term of S i s orthogonal by -1

the quasilosslessness of S I t i s in f ac t Ln a s d i rec t calculation of -1 '

a S = s o s shows. -1 -la -

Accordingly S is, l i k e s, passive, quasilossless and has also i t s -1

6(t-7) coefficient matrix 1 . It must therefore possess a factor i - -n

zation of its own

Here now s i s (passive), quasilossless, of degree one and S i s -2 -2

(passive), quasilossless, of degree (r-2)) and with i ts 6( t -T) co-

e f f ic ien t matrix 1 . This process can then be repeated with S -n -3'

S etc . un t i l S a t which point the process terminates as we reach 4 -r a matrix of degree zero. The number of degree one fac tors is thus r, the degree of s. -

The preceding arguments collected together then es tabl ish the

following theorem, which i s the fundamental theorem of t he thesis:

Theorem 7.1. Let s be a passive quasilossless scat ter ing - matrix of the form

s ( ~ , T ) = ~ ( t ) 6 ( t - ~ ) + m ( t ) x ( ~ ) ~ ( t - ~ ) - -

then there ex i s t s a factorization of s of the form - S = s 0 s 0 s O . . . S - -0 -1 -2 -r (7.10)

where s i s of degree zero, s s -1' -2' ' ' s are of degree one, and a l l 3 h e s . are (passive and) quasilossless. The scat- tering matrice3 sl, s~,...s, are of the form

wllere 'P. i s square integrable on [ ~ , m ) f o r a l l f i n i t e T, -1

ci2 0; the scat ter ing matrix s i s of t he form -0

where A ( t ) i s orthogonal. -

B. REALIZATION OF A PRODUCT OF SCATTERING MATRICES

We consider f i r s t the rea l iza t ion of s o s when s and 1 2 1

s a re scat ter ing matrices of one port networks N and N2. With 2 1

the gyrator i n Fig. 7a possessing the scat ter ing matrix

6 ( t - ~ ) , the network of Fig. 7b i s readily found to have scattering

matrix

FIGURE 7 REALIZATION OF ONE PORT SCATTERING MATRIX PRODUCT

The scattering matrix of the network i n Fig. 7c can be found

using the cascade loading resu l t of ~ ~ . ( 4 . 1 0 ) , and is [ ~ e f . 321

To r ea l i ze the matrix s o s i n the n-port case, we use n un- . -. -1 -2

coupled g y r a t k s (so that the scat ter ing matrix of t h i s network con-

sists of the direct sum of n matrices s ). The i - th ports of El -g

and N are connected to the i-th gyrator, (i = 1, 2, .. .n), as in -2

Fig. 7. Calculations for this case are almost as simple as those for

the one port. Schematically, we represent the realization by Fig. 8 below.

FIGURE 8 RIWLIZATION OF MULTIPORT SCATTERING MATRIX PRODUCT

Extension to the multiplication by more than one scattering

matrix is simple. Figure 9 shows by way of example two realizations for s o s o s

-1 -2 -3'

FIGURE 9 REALIZATION OF MULTIPLE PRODUCT

C. REALIZATION OF QUASILOSSLESS SCATTERING MATRICES OF DEGREE ZERO AND DEGREE ONE

Hitherto we have shown how t o factor an arbi t rary quasilossless

scattering matrix into the prpduct of quasilossless scat ter ing matrices

of degree zero and d~gree-on*; i n the preceding section we have also

shown how t o rea l ize scattering matrix products, given realizations of

the factors. This section s e t s out realizations of the factors, thus

completing the synthesis of arbi t rary quasilossless scattering matrices.

There is more than one nontrivial variant realizing s ( t ,7 ) = A(t) - - ~ ( t - 7 ) where A is orthogonal, see Ref. Zi' for two. We w i l l content - ourselves with presenting one such real izat ion here.

Since - A(t) is orthogonal, there ex is t s an orthogonal T ( t ) Such -0

tha t [Ref. 33, p.2851

where hi is +1 o r -1.

Scattering matrices +&(t- 'r) and -6 ( t -7 ) can b e synthesized by an

open c i r cu i t and a short c i r cu i t respectively.

Now consider i n i t i a l l y the matrix, closely re la ted t o one of the

blocks of (7.14), . . . .

. .

s in e j ( t )

Since t h i s m a t r i x is symmetric, there ex is t s an orthogonal 2 x 2 matrix

I n chapter IV, sec t ion F, i t was es tabl i shed t h a t an orthogonal

transformer of t u r n s r a t i o ~ ( t ) connected a t t h e p o r t s of a network - N of s c a t t e r i n g matr ix s y ie lds a network N of s c a t t e r i n g -2 -2 -1 matrix ( see Eq. 4.14)

We note that .

can be synthesized a s a two por t of t h i s type, with ~ ( t ) = ~ . ( t ) , a n d -J

N an uncoupled s h o r t - c i r c u i t and an open-circui t . -1

By connecting a u n i t gyra tor ac ross t h e f i r s t p o r t only of t h i s

network, we o b t a i n a network of s c a t t e r i n g matr ix

Physical ly, t h i s i s because t h e gyra to r causes a 180 degree phase

s h i f t i n waves t r a v e l l i n g through it i n one d i r e c t i o n only. The

mathematical v e r i f i c a t i o n i s not d i f f i c u l t .

By simple juxtaposit ion, the sca t t e r ing matrix

- s i n B . ( t )

s i n B . ( t )

can then be synthesized, and, f i n a l l y , by connecting a transformer

of turns r a t i o matrix T ( t ) a t the input ports, it follows from -0

Eq.(7.14) and another appl ica t ion of t h e r e s u l t of chapter I V ,

sec t ion F, t h a t we achieve a network of scat ter ing matrix

The r e a l i z a t i o n of

i s physical ly more straightforward. This - s is of t h e form of the

quas i loss less passive degree one sca t t e r ing matrices r e s u l t i n g from

t h e fac to r iza t ion of Theorem 7.1, when we take c 2 0 and - cp t o

be square in tegrable on [T ,~J ) f o r a l l T.

Consider the network of Fig. 10

FIGURE 10

PROTOTYPE NETWORK FOR DEGREE ONE SCATTERING MATRIX

Easy calculations yield that the admittance of t h i s network is

given by

y ( t , ~ ) = n(t)u(t-T);(T) - - - (7.19)

when n is an n-vector whose i - th entry is n. . We take - 1

Th& t h i s network i s precisely a real izat ion of ~q . (7 .18) , as may

be proved by showing tha t ( 6 1 + y)-l 0 (61 - y ) is precisely the -n - -n -

s of ~ ~ ~ ( 7 . 1 8 ) . The actual calculations a r e detailed i n the appendix. -

We note t h a t t h i s r e a l i z a t i o n of a degree'one s c a t t e r i n g matrix

uses only one energy s torage element.

The work of t h i s and t h e preceding two sec t ions then y ie lds the

following theorem.

Theorem 7.2. A passive quas i los s l e s s n-port s c a t t e r i n g matrix may be r e a l i z e d a s a nondissipat ive network. The number of energy s to rage elements used equals t h e degree of t h e sca t t e r ing matrix.

The forms of t h e r e a l i z a t i o n a r e many. One poss ib le one i s shown

i n Figure 11 below.

FIGURE 11

PASSIVE QUASILOSSLESS $CATTERING MATRIX REALIZATION

Here N i s a degree zero network, and N through N a re degree -0 -1 -r

one networks of t h e type of Fig. 10. Considerable v a r i a t i o n s a r e pos-

s i b l e , and many a r e discussed i n Ref. 27. Among t h e more s ign i f i can t ,

w e would mention:

1. An a l t e r n a t i v e r e a l i z a t i o n of N . -0

2. The replacement of t h e banks of n gyra to r s required t o

r e a l i z e any one product with one gyra to r and o the r cir-

c u i t r y . More s p e c i f i c a l l y , t h e r e exists an equivalence

of t h e f o m shown i n Fig. 12, where ~ ( t ) i s a mult iport - transformer.

REDUCTION IN THE

A possible reduction i n the number of gyrators required

for the gyrator bank t o which N i s connected, and -Q

N i t s e l f ; and a possible reduction i n the complexity -0

of the transformer realizing N . -Q

The use of a network dual t o Ni i n place of N . . The -1

dual network has scattering matrix which is the negative

of tha t of Ni, and can always be used by multiplying

fac tors i n Eq. (7.10) by -1; i f N . has the rea l iza t ion -1

of Fig. 10, then its dual network has the rea l iza t ion of

Fig. 13.

FIGURE 13

NETWORK DUAL TO THAT OF FIGURE 10

D. SYNTHESIS EXAMPLE

We consider the scat ter ing matrix of degree 2:

The quasilossless character of t h i s scattering matrix can be

established by d i rec t computation; a l ternat ively we sha l l be able t o

observe that the matrix has a factor izat ion into quasilossless matrices.

The f i r s t column of the - @ matrix is [2=] and t h i s generates

the matrix

Note tha t [e7, 01 comes from [2e-', o]{ rk?2'M)-1. The constant c 11 'T

a 0 of equation (7.>) calculates t o zero, as s 0 [2e , 01 = 0. By - -

a calculating s o s = s we derive the degree one scattering matrix

-la - -2

which i s readily ver i f ied as quasilossless. The matrix - s is now

factored into s o s where s and s are both quasilossless -1 -2 -1 -2

and degree one.

The turns-ratio vector of the transformer i n Fig. 10 which

rea l izes s i s -1

Fig. 14 below real izes s For s the turns r a t i o vector i s -1 ' -2

Figure 14b real izes s In accordance with the formation of -2'

products of scattering matrices, Fig. l 4c real izes - s. Figure l4d

gives a realization using networks dual to those of Fig. 14a and l4b.

1- 2 ..i 1 HENRY

'-

(a)

1 HENRY

1 HENRY

..

1 HENRY

(dl

FIGURE 14

EXAMPLE ILLUSTRATING REALIZATION OF QUASIIBSSLESS SCATTERING MATRIX

V I I I . LOSSY SYNTHESIS

A. STATNENT OF PROBLEM AND AN OUTLINED ATTACK

The problem we consider can be posed simply: given a s c a t t e r i n g

mat r ix s ( t , ~ ) of a f i n i t e network, wi th s assumed passive, how may - - s be synthesized? -

When s i s not quas i lo s s l e s s , we term s lossy. For convenience - - we s h a l l r e s t r i c t cons idera t ion i n i t i a l l y t o t h e lossy one p o r t ; t h e

techniques t o be described a r e extended t o t h e mul t ipor t case i n sec t ion

C.

We r e c a l l from s e c t i o n A of chapter I V t h a t every f i n i t e network

can be represented by an equiva lent network containing t ime-invariant

r e s i s t o r s , capaci tors , inductors , and gyrators , and t ime-variable

t ransformers. By s u i t a b l e u s e of t ransformers we can assume t h e time-

inva r i an t elements, i n p a r t i c u l a r t h e r e s i s t o r s , t o be normalized t o

one ohm, so t h a t t h e network may then be regarded a s being composed of

a nondiss ipa t ive network N i n conjuc t ion with a number of u n i t re- -C

s i s t o r s . The p o r t s of t h e nondiss ipa t ive network w i l l be t h e p o r t s of

t h e o v e r a l l network, toge ther wi th terminal p a i r s across which u n i t

r e s i s t o r s a r e connected i n t h e o v e r a l l network.

I f we assume t h a t t h e nond i s s ipa t ive network possesses a sca t -

t e r i n g matr ix C, then we may w r i t e -

The b a s i s f o r tak ing t h e 1, 1 e n t r y of C a s s was e s t ab l i shed - i n sec t ion F of chapter I V ; when t h e p o r t s of

NC corresponding t o

t h e 2, 2 terms i n t h e p a r t i t i o n of C a r e terminated i n u n i t re- - s i s t o r s ,

= s must be t h e s c a t t e r i n g matr ix of t h e r e s u l t i n g

network.

For a one port we make the assumption that s can be synthesized

using one resistor at most, as is possible for the time-invariant case,

see e.g. Ref. 32. Then the C . . are scalars, with I J

By the finiteness, we know that s is of the form

and accordingly we assume the following form for C -

which is of the same degree as s. Since - C is quasilossless (cor-

responding to a nondissipative network) the coefficient matrix of 2

6(t-7) must be orthogonal. Accordingly we suppose (noting a S 1 by

passivity)

The choice of the particular orthogonal matrix multiplying 6(t-7) is not

critical.

At this stage we are now faced with the following problem: determine

the unknowns Cp+ and * in Eq.(8.5) so that C is quasilossless. Ob- - - serve that all other terms in C are known, via ~~.(8.3), the stated -

form f o r s.

Once having found ifx and J* we have a synthes is f o r s, - f o r we can synthesize C by t h e h i the r to e s t ab l i shed methods, and - then simply obta in s by terminating the second p o r t of t h e network

synthesizing C i n a u n i t r e s i s t o r . - One poss ib le way t o go about f inding p and $+ is de ta i l ed -

i n t h e next sec t ion .

B. SOLUTION PROCEDURE

Our aim w i l l be t o der ive equations s a t i s f i e d by t h e unknowns a ifx and $* which a r e p o t e n t i a l l y more use fu l than merely C o C = 6 1

-a - -2' Theorem 5.3 o u t l i n e s equations which must be s a t i s f i e d t o

make the - G of ~ q . ( 8 . 5 ) quas i loss less . When applied t o t h e case i n hand,

we have

Examining j u s t t h e f i r s t column of t h i s matrix equation, we obta in an

equation t h a t is independent of I)*: -

where

From Eq.(8.7) we conclude t h a t knowledge of ~ ( t ) i s s u f f i c i e n t t o - determine <pic and 2, t h e l a t t e r by evaluat ing l i m ~ ( t ) , t h e former - t- - by f i r s t computing ~ ( t ) . From (8.7a) we have -

while from (8.7b)

Consequently we have on equating these two equations

The determination of P i s thus equivalent t o solving t h i s matrix - Riccati d i f fe ren t ia l equation [ ~ e f .34]. We comment that whereas we

require P to be defined over (-m,m) a d i f f e r en t i a l equation of the - form of Eq.(8.9) need not necessarily possess a solution defined over

2 an in f in i t e interval. We note also tha t the [L-a ( t )] denominators

appearing i n Eq.(8.9) indicated that possible s ingular i t i es can occur 2

i n P; a ( t ) can have values lying between -1 or +1, and thus 1-a ( t ) - may be zero everywhere, over an interval, o r merely a t an isolated

point.

Once P has been found, (and with it cpx) the function $* can - - immediately be obtained from Eq. (8.6), as

Accordingly, by assuming a form f o r the network synchesizing s,

we are led t o the nonlinear d i f fe ren t ia l equation (8.9). I f t h i s can

be solved over (-m,m) then a synthesis f o r s is achievable.

C. THE MULTIPORT CASE

By analogy again with t h e t ime-invariant case, we seek a non-

d i s s i p a t i v e 2n-port network N such t h a t , when the l a s t n p o r t s of -C

N a r e terminated i n u n i t r e s i s t o r s , a network N of prescr ibed -C - pass ive s c a t t e r i n g matr ix s i s obtained. -

We suppose

s = ~ ( t ) 6 ( t - 7 ) + @ ( t ) y ( ~ ) u ( t - 7 ) - - - - and

Here t h e A. . ( t ) a r e matr ices chosen t o make t h e coe f f i c i en t matr ix -13 w w

of 6( t -T) orthogonal. This i s p o s s i b l e because 1 -A A and 1 -A A -n - - -n - - a r e nonnegative d e f i n i t e by p a s s i v i t y . h he n x n matrix A ( t ) may be f i r s t

-12 chosen t o make t h e n rows of [A(t)A&t)] - orthon0:rniai; then [A ( t ) _ ~ ~ ~ ( t ) ] i s

-21 s e l e c t e d t o complete these n rows t o a set of 2n orthonormal row vec to r s . )

Theorem 5.3 allows us t o w r i t e down t h e following r e l a t i o n in-

volving t h e unknows @* and t h e constant nonnegative d e f i n i t e matr ix -

where

A s f o r t h e one port , knowledge of P serves t o determine @* and - - C. Again a s f o r t h e one port , it fol lows t h a t the re is a R icca t i -

d i f f e r e n t i a l equat ion f o r g,. provided t h a t w e assume t h e exis tence -1

Of +21 (corresponding t o [l-a']-' i n t h e one p o r t case) . This

equation i s

The remaining unknown i n C i s Y*, which may be determined from - - t h e following equation, r e s u l t i n g from an app l i ca t ion o f Theorem 5.3:

D. EXlUlPLE OF ONE PORT SYNTHESIS

A s an example of t h e preceding theory, consider t h e one por t

s c a t t e r i n g matr ix

We seek a quas i los s l e s s C of t h e form -

+ 2e

(8.171

The matr ix P of Eq.(8.7b) becomes a s c a l a r funct ion; it s a t i s f i e s - t h e d i f f e r e n t i a l equat ion (8.9):

@(e-2t+ e -t 2 @(e-2t+ e-t * + 2 ~ 2L-4. + 2e -2. ] - P e-'Le-.t + 2e -2.3 + 2e-2t =

A so lu t ion of t h i s equat ion is

-2t which y ie lds CEO and cpn = -e . Equation (8.10) then y i e l d s

$* = (-e 2 -2t) and thus

The methods of chapter VII can then he applied to synthesize C. - First we synthesize the matrix

which has the form shown in Fig. l5a. The matrix

is then synthesized, and has the f o m of Fig. l5b. Note that

C = s o s Then it is clear that Fig. l5c synthesizes s. - -0 -1' -

1 ' -5) -1:n 1 HENRY

2 '

FIGURE 13

A LOSSY SYNTHESIS

Several comments can be made here. F i r s t , ca lculat ions of the

type out l ined i n the previous chapter y i e l d

Second, because of t h e shor t c i r c u i t and open c i r c u i t appearing i n

Fig. l 5 a and again i n Fig. l5c, t h e a s soc ia t ed transformer is cap-

a b l e of s impl i f i ca t ion . The transformer may be replaced by a s i n g l e

two-port t ransformer of tu rns r a t i o (1+J2): 1, t h e primary i n s e r i e s

wi th 1 - l', t h e secondary i n s e r i e s with 2 - 2 ' . Third, we observe

t h a t p r e c i s e l y one r e a c t i v e element i s used f o r t h e ove ra l l synthesis ,

corresponding t o t h e degree one na ture of t h e prescr ibed s(t ,r) .

Fina l ly , may be synthesized a s t h e product of two sca t t e r ing matrices,

-s and -s The networks of Figs. 15a and l 5 b a r e then replaced by -0 -1 '

t h e i r duals . I n t h e case of Fig. l5a, t h e shor t c i r c u i t and open c i r -

c u i t a r e simply interchanged, while Fig. 13 i l l u s t r a t e s t h e dual of

Fig. l5b.

IX. RELATION TO OTHER WORK

A. SPAULDING'S MSSLESS IMMITTANCE SYNTHESIS

Spaulding's immittance synthesis of lossless and more generally

quasilossless immittances [~ef .g] can be applied to yield a quasiloss-

less scattering matrix synthesis. Suppose the given quasilossless

scattering matrix is

We can construct s by synthesizing -

S = A(t)6(t-T) - - (9.2a) and

s = 1 6(t-T) + x(t)@(t)Y(-i)u(t-T) -1 -n - - - (9.2b)

As is discussed in Ref.11, it is possible to form (6Ln + S1)-l because

the term multiplied by 6(t-T) in 61 + S is a nonsingular matrix. -n -1

Then the admittance

evidently exists, and is quasilossless because S is. Spaulding's -1 synthesis can then be used on Y -1 '

TWO remarks should be made. First, in the synthesis the number

of reactive elements used is equal to the degree of Y1 (by Spaulding's results) which is in turn equal to the degree of - s and S (by the -1' results of section D of chapter IV). Second, since s = (-S) - o (-El), we can use an impedance rather than admittance quasilossless synthesis

if we wish, with the impedance possessing scattering matrix -S1.

The converse problem of using t h e s c a t t e r i n g matr ix synthes is

given a quas i los s l e s s immittance can a l s o be considered. Suppose

the re fo re we a r e given a pass ive quas i los s l e s s immittance without

l o s s of g e n e r a l i t y assumed an impedance Z. I t i s known [Refs. 9, - 111 t h a t Z must be of the form -

(9.3)

where r i s a skew symmetric matr ix. Physically, t h i s corresponds - t o t h e s e r i e s connection of transformer-coupled inductors, t r ans -

former-coupled gyra tors , and transformer-coupled capaci tors .

Sca t t e r ing matr ices need not e x i s t f o r such Z ( ~ , T ) ; it i s known 3 f o r example t h a t t 6 ' ( t - 7 ) ~ ~ has no s c a t t e r i n g matrix CRef.351.

However it i s a l s o known t h a t Z can be synthesized a s t h e s e r i e s - combination of an inductor-terminated transformer with impedance

matr ix - y ( t ) 6 ' ( t -T)T(T) - and a pass ive impedance Z with Z given -1' -1

by

Moreover [Ref. 111, Z1 has a s c a t t e r i n g matr ix S which i s of course -1'

quas i los s l e s s i f Z is. Consequently, although Z i t s e l f may not -1 -

be suscept ib le t o t h e s c a t t e r i n g matr ix synthesis , we can, by ex t rac t ing

t h e induct ive p a r t of Z, arrange t o synthesize t h e remainder by t h e sca t -

t e r i n g matr ix synthes is .

B. A NEW VIEWPOINT OF LQSSLESS TIME-INVARIANT SYNTHESIS

Since time-invariant networks a r e merely spec ia l cases of time-

varying networks, and l o s s l e s s t ime-invariant networks s p e c i a l cases

of nondissipat lve time-varying networks, both the quas i los s l e s s sca t -

t e r i n g matr ix and immittance matr ix synthes is should work f o r time-

inva r i an t matr ices. Indeed they do, and i t is i n s t r u c t i v e t o consider

t h e r e s u l t s .

The time-invariance p laces a cons t r a in t f i r s t upon t h e mathematical

form of s o r z, and then upon t h e network r e a l i z i n g the matrix. - - I n t h e case of

j - a t ~ ( t ) is a cons tant ; t h e columns of @ a r e of t h e form u t e cos b t - - - -

j - a t . o r u t e s i n b t , where u i s a constant vector, a and b a r e - -

j +a t r e a l with a p o s i t i v e ; t h e columns of Y ( t ) a r e of t h e form v t e , - -

j +a t j +at . v t e cos b t o r v t e s i n b t . I t should be noted t h a t these - - r e s t r i c t i o n s a r e not s u f f i c i e n t , but merely necessary, f o r time-

invarianee.

For a quas i lose le s s impedance of t h e form of ~ q . ( 9 . 3 )

t h e time-invariance causes T and t o be constant , and ~ ( t ) t o - - - be of t h e form N(0)exp Ct where C i s a skew matrix. When t h i s - - - s u b s t i t u t i o n is made one der ives

w N(t)u( t -T)N(T) = F ( o ) ~ @ cos a i ( t -T) - s i n a . ( t - T ) - - - -"[ 1

s i n a ( t - T ) cos a . ( t - T ) i 1 I

where @ denotes d i r e c t sum, P i s a constant orthogonal matrix, and - t h e ai a r e r e a l nonzero constants .

When t h e cascade synthes is is applied t o t h e - s of (9.4) , two

c l a s s e s of f i r s t degree networks r e s u l t . Corresponding t o t h e s e l e c t i o n j - a t

of a column of @ of t h e form t e - t o genera te a f i r s t degree f a c t o r ,

a t ime-invariant network i s ext rac ted . On t h e o t h e r hand a column Of t h e

j - a t j - a t . form t e cos b t o r t e s l n b t leads t o t h e ex t rac t ion of a

time-varying subnetwork. A t t h e next ex t rac t ion however, one may

e x t r a c t a second time-varying degree one network, which i n combination

wi th t h e f i r s t , behaves l i k e a t ime-invariant degree two network. Thus

t h e degree two networks t h a t appear i n t h e c l a s s i c a l cascade synthes is

[ ~ e f . 321 may be regarded a s cascades of time-varying degree one net-

works (with r e a l parameters) .

A s f a r a s t h e impedance syn thes i s goes, we note from Eq. ( 9 . 5 )

t h a t t h e use o f a t r a n s f o r m e r of t u r n s r a t i o P N(0) reduces t h e pro- - - blem t o t h a t of synthesizing

a ( t -T) - s i n a ( t - T ) ~ ( t - T )

a ( t - T ) cos a ( t - T ) 1 -I

When Laplace transformed, t h i s i s

which has the synthes is of Fig. 16a below. But a l s o

a t - - s i n a ( t - T )

a( t- ' r ) cos a ( t - T ) 1 s i n a t -cos a t

u ( t -T) p s a7 s i n aT1

1 s i n a'r -cos aT J and t h e synthes is with time-varying transformers of Fig. 16b app l i e s .

Thus a time-varying transformer terminated i n capaci tors r e a l i z e s a

time-invariant network which, using time-invariant elements only,

normally requires a gyrator i n t he real izat ion.

( a ) (b)

FIGURE 16

EQUIVALENT NETWORKS, WITH THE TRANSFOF&!ER

TURNS RATIO - M( t ) = rS at s in a]

s i n a t -cos a t

A s indicated by Eq.(g.5), time-invariant transformers combined with

blocks of the type shown i n Flg. 16 w i l l r ea l ize any loss less time-

invariant Z, once removal has been made of se r ies inductors, cor- - - responding t o - ~ ( t ) 6 ' ( t - T ) T ( T ) , - gyrators, corresponding t o - T ( t ) ,

and capacitors, corresponding t o ( 0 - - - r - - 1 P ~ ( 0 ) . The time-variable

synthesis procedures lead however t o dif ferent forms f o r the blocks.

W e comment that fo r a reciprocal network, there w i l l be no se r ies

gyrators extracted, while when the blocks of Fig. 16a are connected

together correctly, e i ther 1-1' or 2-2' w i l l remain open-circuited or

short-circuited for each block. This w i l l allow the removal of the

gyrators, and the replacement of some of the capacitors by inductors

[ ~ e f . 361.

C. RELATION TO TWO PROBLEMS OF AUTOMATIC CONTROL

We make brief mention here of how the material tha t has been developed

has shed l igh t on some problems of contvol systems which a re reported i n

Refs. 16, 17.

One of t h e aims of applying feedback around a system is t o i m -

prove t h e system's performance by decreasing i t s s e n s i t i v i t y t o var i -

a t ions i n t h e p lan t parameters. For t ime-invariant systems, a re-

i n t e r p r e t a t i o n of known r e s u l t s es tabl i shed t h a t t h e inverse of t h e ,f return-difference" matr ix of t h e system (which is a matr ix c lose ly

r e l a t e d t o the loop ga in ) must be t h e s c a t t e r i n g matrix of a l i n e a r ,

passive, t ime-invariant network, i f improvement i n t h e system's per-

formance was t o be achieved. Using t h e mathematical c r i t e r i a contained

here in f o r pass ive tine-varying s c a t t e r i n g matrices, t h i s r e s u l t was

extended t o time-varying systems, thus enabling the wr i t ing down of a

s p e c i f i c mathematical c r i t e r i o n f o r improvement i n system performance

given a mathematical desc r ip t ion of t h e system.

The design of l i n e a r r egu la to r s f o r time-varying systems using

optimal cont ro l theory has l e d t o a f u r t h e r usefu l r e s u l t .

I n t h e l i n e a r r egu la to r problem, t h e ob jec t i s t o design a feed-

back law which w i l l r e t u r n t h e system t o t h e zero s t a t e when i t i s

d isp laced from t h e zero s t a t e , i n an optimal fashion. By " i n an

optimal fashion" w e o f t e n mean by minimizing t h e sum of ( i ) t h e energy

required t o r e t u r n t h e system t o t h e zero s t a t e and (ii) t h e i n t e g r a l

squared e r r o r ( i .e . t h e e r r o r being t h e displacement of t h e s t a t e vec tor

from zero) . Using such a c r i t e r i o n , ( involving a quadra t i c l o s s funct ion) ,

i t tu rns out t h a t a l i n e a r feedback law provides minimization, t h a t is,

t h e instantaneous input vector . Reference 17 e s t a b l i s h e s t h a t t h e

feedback law computed v i a t h e optimal cont ro l theory w i l l automatical ly

y i e l d a system whose performance i s improved with r e spec t t o p l a n t para-

meter s e n s i t i v i t y . The der iva t ion of t h i s r e s u l t depends on t h e e x p l i c i t

formulation mentioned above of t h e c r i t e r i o n f o r improvement i n system

performance; t h e r e s u l t is a use fu l l i n k between modern and c l a s s i c a l

cont ro l concepts.

X. CONCLUSIONS

A. S W R Y OF PRECEDING WORK

I n r e t rospec t it i s poss ib le t o say t h a t much of t h e work s e t

f o r t h has been a f o r m a l i s t i c p resen ta t ion of p roper t i e s of time-

varying networks, where t h e r e has been l i t t l e o r none before. But

it would be wrong t o say t h a t t h i s formalism represented t h e s o l e

content of t h e t h e s i s . Of paramount i n t e r e s t has been t h e iden t i -

f i c a t i o n of t h e concepts of nondiss ipa t ive networks and quasi loss-

less s c a t t e r i n g matr ices. By chapter V, a n a l y t i c a l tebhniques were

a b l e t o e s t a b l i s h t h a t every so lvable nondiss ipa t ive network possesses

a q u a s i l o s s l e s s s c a t t e r i n g matrix. How s a t i s f y i n g it was then t o s e e

t h a t i n chapter V I I , syn the t i ca l techniques could e s t a b l i s h t h a t t h e r e

i s a nondiss ipa t ive network corresponding t o a quas i los s l e s s matrix.

Of less but not minor importance, t h e r e a r e two concepts we would

mention: t h a t of t h e pass ive s c a t t e r i n g matrix, examined v i a Hi lber t

space techniques i n chapter 111, and t h a t of t h e lossy synthesis , pre-

sented i n chapter V I I I . Here we were a b l e t o reduce t h e physical pro-

blem f i r s t t o a mathematical problem, t h a t of solving a fuc t iona l eq-

uat ion, and then t o a more usual mathematical problem, t h a t of solving

a d i f f e r e n t i a l equation.

Perhaps w e could mention t h e cascade loading concept too, though

without decrying i t s importance it i s poss ib ly of more use i n time-

inva r i an t s i t u a t i o n s than time-varying ones.

Throughout t h e web of a l l t h e preceding theory, one golden th read

i s t o be seen: t h e cen t ra l aim i n t h i s s tudy has been, and w i l l cont inue

t o be, t h e t r a n s l a t i o n of physical concepts i n t o mathematical concepts

and v i c e versa . Without exception, a l l t h e ma te r i a l presented i n t h i s

t h e s i s se rves t h i s aim.

B. DIRECTION FOR FUTURE RESEARCH

One outs tanding problem remaining i s a f u l l so lu t ion of t h e l o s s y

syn thes i s problem. Insofar a s we found i n chapter VII an equation whose

so lu t ion y i e l d s a lossy synthesis , one a t t a c k on t h e lossy synthes is

problem would be t o demonstrate t h e exis tence of a s o l u t i o n of t h i s

equation. Hopefully t h i s ex is tence would he a consequence of pas-

s i v i t y .

While t h e r e s u l t s here have been concerned with passive networks,

no doubt many could be applied t o the study of a c t i v e networks. The

s o r t of study which might be rewarding, i n terms of both t h e o r e t i c a l

and p r a c t i c a l r e s u l t s , would be one embracing networks composed of

passive t ime-invariant elements and time-varying transducers (e.g.

i dea l ampl i f i e r s whose ga ins a r e funct ions of time).

A s i s apparent from t h e theory, a l l time v a r i a t i o n s can be

lumped i n t o time-varying transformers. A s t h e p r i n c i p a l time-varying

element therefore , a more de ta i l ed study of such devices, p a r t i c u l a r l y

from a p r a c t i c a l poin t of view, seems warranted. A spec ia l problem is

involved where t u r n s r a t i o s change s ign during t h e passage of time.

The f a c t o r i z a t i o n theorems have presumably wider a p p l i c a b i l i t y

than t h e purpose f o r which they were used here, i . e . t h e quas i los s l e s s

synthesis . Two problems could p r o f i t a b l y be examined, t h e f ac to r ing

of degree one los sy s c a t t e r i n g matrices, and the p o s s i b i l i t y of fac-

to r ing renormalized quas i los s l e s s s c a t t e r i n g matr ices t o achieve a

lossy synthesis , a s may be done i n t h e t ime-invariant case, [ ~ e f . 321. Although many more suggestions could be made, s e e Ref. 37, we

content ourse lves with one more, which c e r t a i n l y has general system

t h e o r e t i c overtones. For obvious reasons we a r e never in t e res t ed i n

behavior a s f a r back a s t = -m; perhaps a shade l e s s obviously, we a r e

never i n t e r e s t e d i n behavior a s f a r ahead a s t = +m. We would not he

doing p o s t e r i t y an i n j u s t i c e by l imi t ing our cons idera t ion of devices 6 t o say 10 years pas t t h e present time. What then a r e t h e consequences

of r e s t r i c t i n g our considerat ions t o a f i n i t e r a t h e r than i n f i n i t e time

in terva l?

C. POSSIBLE APPLICATIONS

Venturing s t i l l f u r t h e r in to the world of speculat ion, we hesi-

t a n t l y o f f e r suggestions as t o the app l i ca t ion of t h i s theory.

Its use i n t h e study of commonly encountered time-varying c i r -

c u i t s seems possible, though more so f o r pass ive devices such a s a

varac to r diode, than f o r a c t i v e ones such a s a frequency converting

heptode. Control systems problems may be a t tacked wi th some of t h e

techniques used i n our study, and have already been successfu l ly so

a t tacked i n two instances, a s repor ted i n chapter I X .

Just a s many t ime-invariant physical systems have been modelled

by e l e c t r i c a l c i r c u i t s because t h e problems as soc ia t ed wi th the l a t t e r

have been b e t t e r understood, so w e can suggest t h e same might happen

with time-varying phys ica l systems. While not f a r removed from t h e

idea of e l e c t r i c a l c i r c u i t s , t h e ionosphere with i t s time-varying

propagation c h a r a c t e r i s t i c s can perhaps be examined t h i s way. Again,

an i n t e r e s t i n g study would be a search f o r phys ica l systems which

could be modelled by nondiss ipa t ive networks and described by quasi-

l o s s l e s s s c a t t e r i n g matr ices.

The problems a r e obviously many, but t h i s is only r i g h t ; i f we

ever th ink t h a t we have run out of problems t o solve, we w i l l have

l o s t t h e a b i l i t y t o so lve any problems a t a l l .

Ah Love! Could you and I but wi th Fate conspire

To grasp t h i s so r ry scheme of th ings e n t i r e ,

Would not we s h a t t e r it t o bits--and then

Remould it nearer t o t h e Hear t ' s Desire .

Omar Khayyam, The Rubaiyat

REFERENCES

1. B. D. Anderson and R. W. Newcomb, "On Reciprocity and Time-Variable ~etworks," Proceedings of the IEEE, vol. 53, No. 10, October 1965, p. 1674.

2. L. Zadeh, "~ipe-Varying Networks, I," Proceedings of the IRE, vol. 49. No. 10, October 1961, pp. 1488-1503.

3. L. Zadeh, "The Determination of the Impulsive Response of Variable Networks, "Journal of Applied Physics, vol. 2 1 July 1950, pp.642-645.

4. L. Zadeh, " c i r cu i t Analysis of Linear Varying-Parameter ~etworks," Journal of Applied Physics, vol. 21, November 1950, pp. 1171-1177.

5. S. Darlington, "~ime-variable Transducers," Proceedings of the Brooklyn Polytechnic Symposium on Active Networks and Feedback Systems, vol. 10, pp. 621-633, 1960.

6. R. W. Newcomb, "The Foundations of Network Theory," The In s t i t u t i on of Engineers Australia, E lec t r ica l and Mechanical Engineering Trans- actions, vol. EM-6, No, 1, May 1964, pp. 7-12.

7 . D. A. Spaulding and R. W. Newcomb, "Synthesis of Lossless Time-Varying ~etworks," ICMCI Summaries of Papers, Part 2, Tokyo, September 1964, PP. 95-96.

8. B. D. Anderson, D. A. Spaulding and R. W. Newcomb, "useful. Time-Variable Circuit-Element Equivalences," Electronics Let ters , vol. 1, No. 3, May 1965, PP. 56-37.

9. D. A. Spaulding, "passive Time-Varying Networks," Ph.D. Dissertation, Stanford University, January 1965.

10. R. W. Newcomb, "on the Definition of a ~etwork," Proceedings of the IEEE, vol. 53, No. 5, May 1965, pp. 547-548.

11. B. D. Anderson, roperti ties of Time-Varying N-Port Impedance Matrices, 11

Proceedings of the Eighth Midwest Symposium on Circuit Theory, June 1965, pp. 8-0 - 8-13.

12. B. D. Anderson, D. A. Spaulding and R . W. Newcomb, "The Time-Variable ~ransformer," Proceedings of the IEEE, vol. 53, No. 6, June 1965, p.634.

9 ,

13. D. A. Spaulding and R. W. Newcomb, ne he Time-Variable Scattering Matrix, Proceedings of the IEEE, vol . 53, No. 6, June 1965, pp. 651-652.

14. B. D. Anderson and R. W. Newcomb, "on Relations Between Series-and Shunt- Augmented ~etworks," Proceedings of the ,-, vol. 53, No. 7, Ju ly 1965, P. 725.

- 15. B. D. Anderson and R. W. Newcomb, "Functional Analysis of Linear Passive Net- works," International Journal of Engineering Science, ( t o ' be published).

B. D. Anderson and R. W. Newcomb, "An Approach to the Time-Varying Sensi t ivi ty problem," Report No. SEL-66-06, (TR No. 6560-l), Stanford Electronics Laboratories, Stanford, California.

B. D. Anderson, " sens i t i v i ty Improvement Using Optimal Design, I,

Proceedings of the IEE, ( t o be published).

L. Schwartz, Thgorie des D i s t r i b u t i o n ~ ~ v o l . I, Hermann, Paris, 1957.

L. Schwartz, ~he*orie des Distributions, vol. 11, Hermann, Paris, 1959.

L. Schwartz, "Thgorie des noyaux, "proceedings of t he International Congress of Mathematicians, Cambridge, Mass., 1950, pp.220-230.

M Bunge, "causali ty, Chance and ~aw," American Scient is t , vol. 49, NO. 4 December 1961, pp. 432-448.

F. Riesz and B. Sz-Nagy, Functional Analysis, Ungar, New York, 1955.

G. F. Simmons, Topology and Modern Analysis, McGrawHill, New York, 1963.

B. Mdil lan, "Introduction t o Formal Realizabil i ty ~heory," Bell System Technical Journal, vol. 31, NO. 2, March 1952, pp. 217-279; and vol. 31, No. 3, May 1952, pp. 541-600.

9 , R. J . Duffin and D. Hazony, h he Degree of a Rational Matrix Function,

Journal of the Society f o r Industrial and Applied Mathematics, vol. 11, No. 3, September 1963, pp. 645-658.

R. E. Kalman, "Algebraic Structure of Linear Dynamical Systems," ( t o be published).

B. D. Anderson, "Cascade Synthesis of Time-Varying Nondissipative N-~orts ," (submitted f o r publication).

R. W. Newcomb and B.D. Anderson, "The Degree of a Matrix of Separable Time- Variable Impulse Response Functions," ( i n preparation).

B. D. Anderson and R. W. Newcomb, "on the Cascade Connection f o r Time- Invariant N-Ports," Proceedings of the IEE, to ,be published.

B. D. Anderson and R. W. Newcomb, "On Reciprocity i n Linear- Time-Invariant Networks," Report No. SEL-65-101 (TR No. 6558-3), Stanford Electronics Laboratories, Stanford, California, November 1965.-

B. D. Anderson and R. W. Newcomb, "Apparently Lossless Time-Varying Net- works," Report No. SEL-65-094 (TR No. 6759-1)) Stanford Electronics Laboratories, Stanford, California, September 1965.

32. V. Belevitch, '"Factorization of Scattering Matrices with Applications t o Passive Network Synthesis, " Phil ips Research Reports, vol. 18, No. 4,. ~ u g u s t 1963, pp. 275-317.

33. F. R. Gantmacher, The Theory of Matrices, Chelsea Publishing Company, New York, 1959.

34. J. J. Levin, "On the Matrix Riccati Equation," Proceedings of the American Mathematical Society, vol. 10, 1 , pp. 519-524.

35. R. W. Newcomb and B. D. Anderson,"Degenerate Networks," Prooeedings of the IEEE, to be published.

36. R. W. Newcomb, Linear Multiport Synthesis, McGraw-Hill, 1966.

37. Proposal to A i r Force Office of Sc ien t i f ic Research f o r "Functional Network Analysis and Synthesis," Stanford Electronics Labra tor ies , SEL 8-65, December 1965.

APPENDIX. VERIFICATION O F ADMITTANCE MATRIX FOWlJLATION

The object i s t o identify t he scattering matrix and admittance

matrix of Eqs. (7.18) and (7.19) as corresponding t o the same network.

By v i r tue of the resu l t of Eq.(2.8a)

i t i s suf f ic ien t to show that

where s and y are given by Eqs.(7.18) and (7.19). We have by - - di rec t calculation:

Now,

and thus,

y o s = fe( t)

u( t -7) 3 7 )

- - j r 1-

= - Y + (6An - z) - 22

That is,

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NOTE: 1 CY-TO EACH ADDRESSEE UNLESS OTHERWISE INDICATED

Page 1 O N R ( S ~ ) 10/66

SSD (SSTRT/L~. s t a r b e c k ) AFETR (ETLLG-1 ) AFUPO STINFO O f f i c e r (For ~ i b r a r y ) Los Angeles, C a l i f o r n i a 90045 P a t r i c k AFB, F l o r i d a 32925

DET #6, OAR (LOOAR) A i r Force Uni t P o s t O f f i c e Los Angeles, C a l i f o r n i a 90045

Systems Engineer ing Group (RTD) Techn ica l I n f o r m a t i o n Refe rence

Branch Attn: SEPIR D i r e c t o r a t e of E n g i n e e r i n g S tandards

& Techn ica l I n f o r m a t i o n Wright-Pat terson AFB, Ohio 45433

ARL (ARIY) Wright-Pat terson AFB, Ohio 45433

AFAL (AVT) Wright-Pat terson AFB, Ohio 45433

AFAL (AWE/R.D. L a r s o n ) Wright-Pat terson AFB, Ohio 45433

O f f i c e of Research A n a l y s i s At tn : Techn ica l L i b r a r y Branch Holloman AFB, New Mexico 88330

Commanding Genera l At tn : STEWS-WS-VT White Sands Missile Range New Mexico 88002 (2 CYS)

RADC (EMLAL-1) Attn: Documents L i b r a r y G r i f f i s s AFB, New York 13442

Academy L i b r a r y (DFSLB) U.S. A i r Force Academy Colorado 80840

FJSRL USAF Academy, Colorado 80840

APGC (PGBPS-12) E g l i n AFB, F l o r i d a 32542

AFETR Techn ica l L i b r a r y (ETV, MU-135) P a t r i c k AFB, F l o r i d a 32925

AFCRL (CRMXLR) AFCRL Research L i b r a r y , S top 29 L.G. Hanscom F i e l d Bedford, Mass. 01731

Department of t h e A i r Force AFSC S c i e n t i f i c and Technical

L i a i s o n O f f i c e 111 E a s t 1 6 t h S t r e e t New York, N e w York 10003

ESD (ESTI) L.G. Hanscom F i e l d Bedford , Mass. 01731 (2 c y s )

AEDC (ARO; 1 n c . ) A t t n : ~ i b r a r ~ / ~ o c u m e n t s Arnold AFS, Tenn. 37389

European O f f i c e of Aerospace Res. S h e l l B u i l d i n g 4 7 Rue C a n t e r s t e e n B r u s s e l s , Belgium ( 2 c y s )

L t . C o l . R. K a l s i c h C h i e f , E l e c t r o n i c s Div. (SREE) D i r e c t o r a t e of E n g i n e e r i n g S c i . A i r F o r c e O f f i c e of S c i . Res. 1400 Wilson Boulevard A r l i n g t o n , V i r g i n i a 22209 ( 5 c y s )

Department of t h e A i r Force AF F l i g h t Dynamics Labora to ry At tn: FDCC Wrigh t -Pa t t e r son AFB, Ohio 45433

DEPARTMENT OF THE ARMY

U.S. Army Research Off i c e At tn : P h y s i c a l S c i e n c e s D i v i s i o n 3045 Columbia P i k e A r l i n g t o n , V i r g i n i a 22204

Research P l a n s O f f i c e U.S. Army Research O f f i c e 3045 Columbia P i k e A r l i n g t o n , V i r g i n i a 22204

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Commanding General U.S. Army Mater ie l Command At tn : AMCRD-RS-PE-E Washington, D.C. 20315

Commanding General U.S. Army S t r a t e g i c Communicatiorls

Command Washington, D.C. 20315

Commanding O f f i c e r U.S. Army Ma te r i a l s Research Watertown Arsenal Watertown, Massachusetts 02172

Commanding O f f i c e r U.S. Army B a l l i s t i c s Research Lab At tn : V.W. Richards Aberdeen Proving Ground Aberdeen, Maryland 21005

Commandant U.S. Army A i r Defense School At tn : Mis s i l e Sciences Div.

C & S Dept. P .O. Box 9390 F o r t B l i s s , Texas 79916

Commanding General U.S. Army Mis s i l e Command Attn: Technical L ib ra ry Redstone Arsenal , Alabama 35809

Commanding General Frankford Arsenal At tn : SMUFA-L6000 ( ~ r . S idney ROSS)

Ph i l ade lph i a , Pa. 19137

U.S. Army Munitions Command At tn : Technical Informat ion Branch P i ca t i nney Arsenal Dover, New J e r s e y 07801

Commanding O f f i c e r Har ry Diamond Labora tor ies At tn : Microminiatur izat ion Branch Connect icut Ave. and Van Ness S t r e e t

N . W . Washington, D .C . 20438

Commanding O f f i c e r U.S. Army Secu r i t y Agency Ar l ing ton Ha l l Ar l ing ton , V i rg in i a 22212

Commanding O f f i c e r U.S. Army Limited War Laboratory Attn: Technica l D i r e c t o r Aberdeen Proving Ground Aberdeen, Maryland 21005

Commanding O f f i c e r Human Engineer ing Labo ra to r i e s Aberdeen Proving Ground, Maryland 21005

Di r ec to r U.S . Army Engineer Geodesy, I n t e l l i g e n c e and Mapping Research and Development Agency Fo r t Be lvo i r , V i r g i n i a 22060

Commandant U.S. Army Command and General

S t a f f Col lege Attn: S e c r e t a r y Fo r t Leavenworth, Kansas 66270

D r . H . Robl, Deputy Chief S c i e n t i s t U.S. Army Research O f f i c e ( ~ u r h a m ) Box CM, Duke S t a t i o n Durham, North Ca ro l i na 27706

Commanding O f f i c e r U . S . Army Research Off ice ( ~ u r h a m ) Attn: CRD-AA-IP ( ~ i c h a r d 0. ~ l s h ) Box CM, Duke S t a t i o n Durham, North Ca ro l i na 27706

Superintendent U .S . M i l i t a r y Academy West P o i n t , New York 10996

The Walter Reed I n s t i t u t e of Res. Walter Reed Medical Cente r Washington, D . C . 20012

Commanding O f f i c e r U.S. Army Engineer ing R & D Lab. Attn: STINFO Branch Fo r t Be lvo i r , V i r g i n i a 22060

Dr. S . Benedict Levin, D i r ec to r I n s t i t u t e f o r Exp lo ra to ry Research U.S. Army E l e c t r o n i c s Command Fo r t Monmouth, New J e r s e y 07703

Page 3 O N R ( S ~ ) 10/66

D i r e c t o r I n s t i t u t e f o r Explora tory Research U.S. Army E l e c t r o n i c s Command At tn : M r . Robert 0 . Parker

Execut ive Sec re t a ry , JSTAC (AMSEL-M-D)

F o r t Monmouth, New J e r s e y 07703

Commanding General U.S. Army E l e c t r o n i c s Command F o r t Monmouth, New J e r s e y 07703 At tn : AMSEL-SC

RD-D RD-G RD-GF RD-MAF-1 RD-MAT XL-D XL -E XL-C ILL -S HL-D HL-L HL-J HL-P HL-0 HL-R NL-D NL-A NL-P NL-R NL-S KL-D KL-E KL-1 KL-S KL-T VL-D WL-D WL -C WL-N WL-S

NOTE: 1 CY TO EACH SYMBOL LISTED INDIVIDUALLY ADDRESSED

Commanding O f f i c e r U.S. Army Combat Development Command

Communications-Electronics Agency CDC-ECOM Liaison O f f i c e F o r t Monmouth, New J e r s e y 07703

DEPARTMENT OF THE N A W

Chief of Naval Research Attn: Code 427 Washington, D.C. 20360 ( 3 c y s )

Chie f , Bureau of Weapons Department of t h e Navy Washington, D.C. 20360 ( 3 cys )

Commanding O f f i c e r Off ice of Naval Research Branch

Off i c e Box 39, Navy No. 100 F.P.O. New York, New York 09510 (2 c y s )

Of f i ce of Naval Res. Branch Of f i ce At tn : Deputy Chief S c i e n t i s t 1076 Mission S t r e e t San Franc isco , C a l i f o r n i a 94103

Commanding O f f i c e r O f f i c e of Naval Research Branch

O f f i c e 219 South Dearborn S t r e e t Chicago, I l l i n o i s 60604 ( 3 c y s )

Commanding O f f i c e r O f f i c e of Naval Research Branch

Off i c e 1030 E a s t Green S t r e e t Pasadena, C a l i f . 91101

Commanding O f f i c e r Of f i ce of Naval R e s . Branch O f f i c e 207 West 24th S t r e e t New York, New York 10011

Commanding O f f i c e r Of f i ce of Naval R e s . Branch O f f i c e 495 Summer S t r e e t Boston, Massachusetts 02210

D i r e c t o r , Naval Research Laboratory Technical Informat ion O f f i c e r Washington, D . C . Attn: Code 2000 ( 7 c y s )

Code 5204

Commander Naval A i r Development and Mater ia l

Cente r J o h n s v i l l e , Pennsylvania 18974

Page 4 O N R ( S ~ ) 10/66

L i b r a r i a n U.S. Naval E l e c t r o n i c s L a b o r a t o r y San Diego, C a l i f o r n i a 95152 (2 cys)

Commanding O f f i c e r and D i r e c t o r U.S. Naval Underwater Sound Lab. F o r t Trumbull New London, C o n n e c t i c u t 06840

L i b r a r i a n U.S. Navy P o s t Graduate School Monterey, C a l i f o r n i a 93940

Commander U.S. Naval A i r Missile T e s t C e n t e r P o i n t Mugu, C a l i f o r n i a 93041

D i r e c t o r U.S. Naval Observa to ry Washington, D.C. 20390

Chie f of Naval O p e r a t i o n s OP-07 Washington, D.C. 20360 ( 2 c y s )

D i r e c t o r , U.S. Naval S e c u r i t y Group At tn : G43 3801 Nebraska Ave. Washington, D.C. 20390

Commanding O f f i c e r Naval Ordnance L a b o r a t o r y White Oak, Maryland 21502 ( 2 c y s )

Commanding O f f i c e r Naval Ordnance T e s t S t a t i o n China Lake, C a l i f o r n i a 93555

Commanding O f f i c e r Naval Avion ics F a c i l i t y I n d i a n a p o l i s , I n d i a n a 46206

Commanding O f f i c e r Naval T r a i n i n g Device Cen te r Or lando , F l o r i d a 32813

U.S. Naval Weapons Laboratory Dahlgren, V i r g i n i a 22448

Weapons Systems T e s t D i v i s i o n Naval A i r T e s t C e n t e r P a t u x t e n t R i v e r , Maryland 20670 A t t n : L i b r a r y

U.S. Naval Ordnance L a b o r a t o r y D i e l e c t r i c s & Semiconductors Branch Corona, C a l i f o r n i a 91720

Commander Naval S h i p Systems Command Main Navy B u i l d i n g Department o f t h e Navy Washington, D.C. 20360

Department of t h e Navy Naval E l e c t r o n i c Systems Command Semiconductors & M i c r o e l e c t r o n i c s

Uni t B a i l e y s C r o s s r o a d s , V i r g i n i a 22041

OTHER GOVERNMENT AGENCIES

M r . C h a r l e s F. Yost S p e c i a l A s s i s t a n t t o t h e D i r e c t o r

of Resea rch N a t i o n a l Aeronau t i c s and Space Admin. Washington, D.C. 20546

D r . H. H a r r i s o n , Code RRE C h i e f , E l e c t r o p h y s i c s Branch N a t i o n a l Aeronau t i c s and Space Admin. Washington, D.C. 20546

Goddard Space F l i g h t C e n t e r N a t i o n a l Aeronau t i c s and Space Ad. A t t n : L i b r a r y , Documents S e c t i o n

Code 252 G r e e n b e l t , Maryland 20771

NASA Lewis Research C e n t e r At tn : L i b r a r y 21000 Brookpark Road Cleve land , Ohio 44135

N a t i o n a l S c i e n c e Foundat ion At tn : Dr. John R. Lehmann

D i v i s i o n of E n g i n e e r i n g 1800 G S t r e e t , N.W. Washington, D.C. 20550

U.S. Atomic Energy Commission D i v i s i o n of Tech. I n f o r m a t i o n E x t . P.O. Box 62 Oak Ridge, Tenn. 37831

Page 5 oN~(83) 10166

The U n i v e r s i t y of Oklahoma Col lege of Engineer ing School of E l e c t r i c a l Engineer ing Norman, Oklahoma 73069

Hughes A i r c r a f t Company Hughes Research Labora tory 3011 Malibu Canyon Road Malibu, C a l i f o r n i a 90265

The Un ive r s i t y of Wisconsin Laboratory of Neurophysiology Medical School 283 Medical Sc i ences Bui ld ing Madison, Wisconsin 53706

Fede ra l S c i e n t i f i c Corpora t ion 615 West 131s t S t r e e t New York, New York 10027

The Un ive r s i t y of San t a C l a r a The School of Engineer ing Dept. of E l e c t r i c a l Engineer ing San ta C l a r a , C a l i f o r n i a 95053

Sprague E l e c t r i c Company Research and Development Labo ra to r i e s North Adams, Massachuset ts 01247

Worcester Po ly t echn i c I n s t i t u t e Dept. of E l e c t r i c a l Engineer ing Worcester , Massachuset ts 01609

Westinghouse E l e c t r i c Corporat ion Research Labo ra to r i e s Chu rch i l l Borough P i t t s b u r g h , Pennsylvania 15235

I n t e r n a t i o n a l Business Machines Corp. Monterey & C o t t l e Roads San J o s e , C a l i f o r n i a 95114

Un ive r s i t y of Notre Dame Dept . of E l e c t r i c a l Engineer ing Notre Dame, Ind iana 46556

Texas Ins t ruments , I n c . Technica l Reports Se rv i ce P.O. Box 5012 D a l l a s , Texas 75222

Radio Corpora t ion of America RCA Engineer ing L ib r a ry Moorestown, New J e r s e y 08057

Lockheed-Georgia Company Department 72-34, Zone 400 Research Informat ion Mar i e t t a , Georgia

Varian Assoc ia tes Technica l L i b r a r y 611 Hansen Way P a l o A l to , C a l i f o r n i a 94303

P r ince ton Un ive r s i t y School of Engineer ing and

Applied Sc ience

P r ince ton , New J e r s e y 08540

Vice P re s iden t B e l l Telephone Labo ra to r i e s , I nc . Murray H i l l , New J e r s e y 07971

Un ive r s i t y of I l l i n o i s Dept. of E l e c t r i c a l Engineer ing Urbana, I l l i n o i s 61801

M r . John J . S i e , P r e s iden t Micro S t a t e E l e c t r o n i c s Corp. 152 F l o r a l Avenue Murray H i l l , N e w J e r s e y 07971

Un ive r s i t y of Alber ta Department of Phys ics Edmonton, A lbe r t a Canada

S t a f f Vice P r e s i d e n t M a t e r i a l s and Device Research Radio Corpora t ion of America David Sarnoff Research Center P r i nce ton , New J e r s e y 08540

Informat ion Technology L ib r a ry AUERBACH Corpora t ion 121 North Broad S t r e e t P h i l a d e l p h i a , Pennsylvania 19107

Page 10 ONR( 83) 10166

P h i l c o Corpora t ion Module Engineer ing Department Mic roe l ec t ron i c s D iv i s i on 504 South Main S t r e e t Sp r ing C i t y , Pennsylvania 19475

Motorola, I n c . Government E l e c t r o n i c s D iv i s i on 8201 E a s t McDowell Road S c o t t s d a l e , Arizona 85252

L i b r a r i a n Microwave Assoc i a t e s , I n c . Northwest I n d u s t r i a l Pa rk Bur l ing ton , Massachuset ts 01803

P r o f e s s o r A . S . Vander Vors t Labo ra to i r e d l E l e c t r o n i q u e U n i v e r s i t e de Louvain 94, Kardinaal Merc ie r laan Hever lee - Leuven, Belgium

General E l e c t r i c Company Research and Development Center E l e c t r o n i c s Phys i c s Labora tory P.O. Box 8 Schenectady, New York 12301

United A i r c r a f t Corpora t ion Norden Div i s ion Technica l L i b r a r y Norwalk, Connec t icu t 06851

Watkins-Johnson Company 3333 Hi l lv iew Avenue S t an fo rd I n d u s t r i a l Park P a l o A l to , C a l i f o r n i a 94304

The Ohio S t a t e Un ive r s i t y D e p t . of E l e c t r i c a l Engineer ing 2024 Nei l Avenue Columbus, Ohio 43210

Un ive r s i t y of F l o r i d a Col lege of Engineer ing Dept. of E l e c t r i c a l Engineer ing G a i n e s v i l l e , F l o r i d a 32603

Sylvan ia E l e c t r o n i c Systems Western Operat ion Advanced Technology Labora tory P.O. Eox 2.05 Mountain View, C a l i f o r n i a 94042

The Un ive r s i t y of Western A u s t r a l i a Dept. of E l e c t r i c a l Engineer ing Nedlands, Western A u s t r a l i a

Radio Corpora t ion of America Manager of Engineer ing 415 South 5 t h S t r e e t Ha r r i son , New J e r s e y 07029

General Telephone & E l e c t r o n i c s Labo ra to r i e s , I nc .

208-20 Willets P o i n t Boulevard Bayside, New York 11360

The Boeing Company O f f i c e of t h e Vice P r e s i d e n t Research and Development P.O. Box 3707 S e a t t l e , Washington 98124

P ro fe s so r R.E. Vowels, Head School of E l e c t r i c a l Engineer ing The Un ive r s i t y of N.S.W. Box 1 P.O. Kensington, N.S.W. A u s t r a l i a

The George Washington U n i v e r s i t y School of Engineer ing and Applied S c i . Department of E l e c t r i c a l Engineer ing Washington, D.C. 20006

Georgia I n s t i t u t e of Technology P r i c e G i l b e r t Memorial L ib r a ry A t l an t a , Georgia 30332

Sandia Cor9ora t ion Technica l L i b r a r y P.O. Box 5800, Sandia Base Albuquerque, New Mexico 87115

Corne l l Aeronaut ica l Laboratory, I n c . of C o r n e l l U n i v e r s i t y P.O. Box 235 Buf fa lo , New York 14221

The Un ive r s i t y of Michigan E l ec t ron Phys ics Laboratory Dept. of E l e c t r i c a l Engineer ing 3505 Eas t Engineer ing Bui lding Ann Arbor, Michigan 48104

Page 11 O N R ( ~ ~ ) 10/66

AUTONETICS Research and E n g i n e e r i n g D i v i s i o n T e c h n i c a l In fo rmat ion Cen te r P.O. Box 4173 3370 Miraloma Avenue Anaheim, C a l i f o r n i a 92803

Mar t in -Mar ie t t a C o r p o r a t i o n Mart in Company, B a l t i m o r e D i v i s i o n Chief of L i b r a r i e s Ba l t imore , Maryland 21203

S p e r r y Gyroscope Company D i v i s i o n of S p e r r y Rand Corpora t ion Head, E n g i n e e r i n g Department Countermeasures E n g i n e e r i n g Grea t Neck, New York 11020

R e n s s e l a e r P o l y t e c h n i c I n s t i t u t e Dept. of E l e c t r i c a l Eng ineer ing R u s s e l l Sage L a b o r a t o r y Troy, New York 12181

Genera l E l e c t r i c Company L i a i s o n Eng ineer Computer L a b o r a t o r y P.O. Box 1285 Sunnyvale, C a l i f o r n i a 94088

U n i v e r s i t y of I l l i n o i s Dept . of Computer S c i e n c e D i g i t a l Computer Labora to ry Room 230 Urbana, I l l i n o i s 61803

U n i v e r s i t y of Chicago I n s t i t u t e f o r Computer Research 5640 South E l l i s Avenue Chicago, I l l i n o i s 60637

S c i e n t i f i c Advisor Lockheed E l e c t r o n i c s Company M i l i t a r y Systems P l a i n f i e l d , New J e r s e y 07060

C a l i f o r n i a I n s t i t u t e of Technology J e t P r o p u l s i o n L a b o r a t o r y L i b r a r y S u p e r v i s o r 4800 Oak Grove Dr ive Pasadena, C a l i f o r n i a 91103

E l e c t r o - O p t i c a l Systems, I n c . Micro-Data Opera t ions 300 North H a l s t e a d S t r e e t Pasadena, C a l i f o r n i a 91107

Manager Systems Engineer ing Radio Corpora t ion o f America Communications Systems D i v i s i o n F r o n t and Market S t r e e t s , Bldg. 17C-6 Camden, New J e r s e y 08102

E n g i n e e r i n g L i b r a r i a n Var ian A s s o c i a t e s LEL D i v i s i o n Akron S t r e e t Copiague, New York 11726

D r . Alan M. Schne ide r U n i v e r s i t y of C a l i f o r n i a , San Diego Department of t h e Aerospace and

Mechanical Eng ineer ing S c i e n c e s P.O. Box 109 La J o l l a , C a l i f o r n i a 92038

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