PART 7

Circuit Theo Information Theory

Ba IC Sciences Electrostatic Processes

ar

the IEEE International Convention, New York,

21-25, 1

n-PORT SYNTHESIS VIA REACTANCE

~:~

EXTRACTION - PART I

by

D. C. Youla

and

Polytechnic Institute of Brooklyn

Brooklyn, New York

Summary

This paper presents a new approach to the synthesis of a linear, tirne-invariant, lumped n-port N. It is shown that the technique of re act­ance extraction reduces the problem to the syn­thesis of a residual (n +k) - port Na' k being

the n1.1nlber of reactances. Na' containing no

reactances, is a freq1.1ency-insensitive network, and admits always a description in terms of a scattering matrix with constant elements.

Explicit formulas are presented for the minimum number of reactances required for the synthesis of N, and for the minimum numbers of inductors and capacitors, individually, in the reciprocal case.

The properties of the minimal decomposi­tions of rational matrices are fully discussed and are used to derive many properties of equivalent networks.

The LC, RL(RC), and RLC cases are treated; they exhibit different characteristics and varying degrees of difficulty.

':'The work reportedin this paper was supported by the Rome Air Development Center, Air Force Systenls Command, Research and Technology Division, Griffiss Air Force Base, New York, under Contract No. AF 30(602)-3951, and by the Air' Force Office of the Office of Aerospace Research, under Contract No. AF 49(638)-1373.

':":'This work was submitted in partial fulfillment of the requirements for the degree of Doctor of Phi losophy(Electrophysics) at the Polytechnic Institute of Brooklyn.

1. Introduction

The analysis and synthesis of lumped, passive n-ports is now an established discipline but nevertheless the search for structure and more efficient synthesis procedures goes on, as is attested to by the abundance of recent papers

on the subject. 1, Z, 3, 4

In this paper we lay thecornerstones of a new and most promising approach - Reactance Extraction. This method derives its foundations from an algebraic theorem concerning the de­composition of rational matrices first estab­lished in the theory of linear dynamical systems by R. KaIman. In fact at the first Allerton con-

17 ference on network theory KaIman proposed and illustrated, for the first time, the possible application of the theorem as a tool of network synthesis. However, the point of view pro­pounded in this paper i8 considerably different. Instead of attempting to synthesize a network by prescribing its internal topology we seek always the appropriate port description. In this way all realizations are obtained. This attack has been succ.essful enough to enable us to reduce the prob­lem of finding all equivalent (minimal)n-ports to a well-defined one in the theory of matrices.

183

The reader is expected to be farrliliar with n-port synthesis and linear algebra. Neverthe-1ess we have made a serio1.18 attempt to begin at the beginning and great care and thought has gone into the question of motivation. Part H, now in preparation, will, hopefully, ans wer the open questions raised in this paper.

z. Scattering Descriptions of Lumped, Passive n-Ports

It is well known 5 , 6, that an n-port N constructed on a finite graph composed exclu­sively of lumped positive resistors, inductors, capacitors, ideal transforJmers and gyrators may always be described in terms of an nxn scattering matrix S(p) normalized to an arbi­trary set of n positive port numbers. Thus 7, to the voltage-current pair (Vi,Ii ) at port #i is

as signed the incident- reflected "wave" pair (ai' b i ) defined by

Z~a. 1 1

Z..[;; b. 1 1

V.+r.I. 1 1 1

V.-r.I. 1 1 1

( 1)

(Z)

h > 0 . h . th al" b w ere r i lS t e 1- norm lzahon num er, i=l,Z, .. ,no The nxn normalized scattering matrix S(p) relates the co1umn-vector ~ = (bI' b z"'" bn" to the column -vector

::;OO (al' a Z"'" an)' in the linear fashion

b S(p) ~ (3)

Of course, (3) implies that the "ineident" exci­

tation a e P t gives rise to the "reflected"

respons-e 12.ept, p=a+jw being the complex frequency variable. The matrix S(p) is com­pletely characterized by the following attributes:

a. S(p) is nxn, real. foi: real p and rational.

b. S(p) is analytic in Re p.2: O. c. 1 - S':'(jw)S(jIJJ) ;::: 0 for all real IJJ.

n n Alternatively a., b. and c. may be rep1aced by a. and

d. 1 -S':'(p)S(p);z: 0 , Re p > O. n n-

An nxn matrix S(p) possessing the above properties is said to be bounded-rea1 6 and is always realizable as the scattering description (normalized to any set of positive port numbers) of an n-port N composed of a finite number of ideal transformers, gyrators, non-negative resistors, inductors, and capacitors.

A lumped, passive n-port is said to be reactive (or lossless) if it is devoid of resistors in which case c. is replaced by

Cl' 1 -S':'(jIJJ)S(jlV) ::: 0 n n

for all real IJJ· By analytic continuation and re ality this condition may be rewrit­ten as

S '( -p)S(p) = 1 n

(4)

A bounded-real matrix satisfying (4) is regular and~ra-unitary.

184

For reciprocal n.-ports, i. e., structures free of gyrators, S(p) is symmetric:

S(p) = S '(p) . (5)

Conversely6, any symmetrie bounded-real matrix S(p) is synthesizable without gyrators.

3. The Degree of a Rational Matrix and the - Concept of a Minimal Realization

Let W(s) denote an arbitra~y rectan­gular rational matrix function of s.'~ The degree, ö(W), of W(s) may be defined in sev-

eral different but equivalent ways. 3,8,9 One particularly simple and useful version begins by

defining the degree of a pole. Z

De~ s = So (finite or infinite) is a pole of W(s) if at least one entry in W(s) pos­sesses a pole at s = s. The order of s is the

o 0

largest multiplicity it possesses as a pole of any entry in W(s).

DeI. Z: The degree, ö(W;s ), of s=s . -- 0 0

as a pole of W(s) equals the largest multiplicity it possesses as a pole of any minor of W(s).

Dei. 3: The degree, 6(W), of W(s) equals the sum of the degrees of its distinct poles. Hence if sI' 8 Z"'" St are the distinct

poles of W(s) with associated degrees ö(W;si)' i=l,Z, ... ,t,

t 6(W) L: ö(W;s.)

iool

Elementary reasoning suffices lO to establish six important properties of o(W):

1. ö(W) = 0 if and only if W(s) is a constant matrix.

Z. Let W I (s) and W Z( s) be two rational

matrices for which the product W(s) = W 1 (s)WZ(s) is defined. Then,

ö(W) :;;; Ci (W l ) + 6(WZ) .

3. Let W 1 (8) and W Z(s) be two rational

matrices of the same size. If W(s)=W1 (s)+WZ(s),

ö(W):::6(W1)+Ii(WZ) .

4. Let W1(s) and WZ(s) be of the same

size without common poles. Then, if

The reason for changing the variable from p to s will emerge shortly.

W(s) = W 1 (s) + W Z(s),

o (W) = ö( W 1) + 0 ( W Z)

5. Let W( s) be a rational matrix and K any square, nonsingular constant matrix for which the product K W(s) is defined. Then,

o(K W) = 6(W)

6. Let W(s) be a square rational matrix whose determinant is not identically zero.

Then, 1 o(W- ) = Il(W) .

These six properties are chiefly respon­sible for the utility of the concept in netw.prk synthesis and analogue computer design. ','

The significance of ö(W) is brought out

1 dZ 9,1l by lemrnas a.n .

Lemma 1: Any rational matrix W(s) which is finite a.t infinity pos ses ses a decom­position of the form

-1 W(s)=J+H(sIk-F) G (6)

where F, G, Hand J are four constant matrices. Furthermore, in ev~ such breakdown..

1. J = W(oo) z. k.2: 6(W)

If W(s) is real for real s, F, G, Hand J may be chosen to be real matrices. The quadruplet F, G, H, J is said to realize W(s). Symbolieally,

W(s) ... (F, G, H, J).

A realization is said to be minimal if k =: ö( W). That is, if the size of F equals the McMillan degree of W(s). Lemma Z asserts that the dass of minimal realizations is not vacuOUS.

Lemma Z: 9, 11 Any rational matri.x W(s) which is finite at infinity possesses a decomposition (6) with k= 6(W). Let (F G H, J ) and (F, G, H, J) denote any two

0' 0' 0 0

minimal realizations of W(s). Then, there exists a kxk nonsingular constant matrix T such that':":'

H=H T o

G = T- 1 G o

3 "'Il(W) is the MeMillan degree of W(s) .

(7)

(8)

':<>:'Equations (7)-(10) are crucial to the develop­ments in this paper. A constructive proof inde­pendent of references (9) and (11) is given in Appendix 1.

More compactly,

where

and

F = T- l F T

J := J o

(9) (10)

(1Z)

(13 )

Inversely, the representation (6) i6 minimal if and only if

rank =k (14)

and

rank[GIFGI ... \Fk-1GJ =: k. (15)

At this stage we are in a position to out­Ene the centr al network problems studied in this paper.

P l' The prescribed data are an n x n symmetrIe bounded-real matrix S(p) toget~er with npositivenumbers r1,rZ,.··,rn · ItIS

desired to realize S(p) as the scattering des­eriotion (normalized to r. at port # i, i=1, Z, ... , n) of ~ reciproeal n-port cobposed exclusively of a finite number of ideal transforrners, positive resistors, induetors and capacitors.

PZ' The given data are an nxn boml.~e.d­real non-symmetric matrix S(p) and n posltlVe nunlbers r ,rZ"'" r. It is desired to realize

I n S(p) as the scattering description (normalized to r, at port #i, i=: I, Z, ... ,n) of a non-reciproeal n~port composed exclusively of a finite nun:~er of ideal tl'ansformers, ideal gyrators, pOSitIve resistors, inductors and capacitors.

P 3' In addition to. the propertie s . enu­merated in P1(P Z), S(p) 16 also para-umtary and

rnust be realized as the normalized scattering matrix of a lossles s, reciproc al (non- reciproe aJ) n-port devoid of resistors. These two problems are labeled P + P and P Z+ P ~. respectively.

1 3 J

Focusing attention on PI first, let N be any

185

reciprocal n-port network realization of the nxn s~mmetric, bounded-real matrix S(p) emploY1ng k reactances of which kare

1 inductors and k Z are capacitors (k=kl +kZ)'

Denote the corresponding (positive) inductances by LI' L Z' ... , L k and the corresponding

1 (positive) capacitance values by C C C l' Z'···, k'

Z Upon extracting these k reactances a residual (n+k)-port Na is created (Fig. 1a) composed solely of ideal transformers and non-negative resistors. Hence, Na possesses an (n+k)x(n+k) scattering matrix description S . Clearly, Sa is real, constant and symmetric a (Sa= S~). Because of the dissipative character of Na'

1 -S'S >0 n+k a a- n+k (16)

By prescription Sa is normalized to r. at port # i, i = 1, Z, ... ,n. We are of course at 1 liberty to choose any set of positive normal­ization numbers at the remaining k ports. The following choice proves useful: .

T] L t' t = 1, Z, ... ,k1 '

( C )-1 T] .f, ,.e=kl +l,k1+Z •...• k,

where T] is an arbitrary positive constant of proportionality whoseprecise value is, for the moment. immaterial.

The reactance p L{ possesses the reflection coefficient

(17)

,f=1,Z, ...• k 1• while that associated with C- l / is given by { P

+--d--p t T] t

_1_ +_1_ pC T] C

t t

- p • ( 18)

t = 1, Z •...• k Z'

S(p) in terms of partition S :

To obtain an expression for

S a and p it is expedient to

a n k

["11 5 12 ] n

S = (19) a S{Z SZZ k

Since S = S '. S = S ' and S = S' a a 11 11 ZZ ZZ· By the very

definition of Sa (refer to Fig. 1a).

~l (ZO)

186

Set

I: = [~l 01-1:~1

J

(ZZ)

Evidently. the ensemble of k uncoupled reactmlces conshtutes a k-port with scattering matrix

B(p) = p I: (Z3)

By c10sing the output k ports of N on their respective reactances (Fig. Ib) we il'ecover N and at the same time impose the constraint

~Z = B(p) ~Z

Substituting (Z4) into (Zl),

(Z4)

::Z = (B-l_SZZ)-lS{Z~l (Z5)

and* inserting (Z5) into (ZO),

But ~ 1 = S(p):: 1 and the identification

S(p) S +S (B -S )-lS' = 11 lZ ,~ZZ lZ (Z6)

is therefore justified. Introductiop of the new variable

s = p+T] _ -1 P-T] - p (Z7)

:,-nd some slight rearrangement throws Eqs. (Z6) 1nto the standard form

S(p) = Sll+SIZ(slk -I:SZZ)-lI:S{Z'

(Z8)

Naturally, S(p) mayaiso be regarded as a func­tion of s:

( s + 1 W s) == S(T] s::-r) = S(p) . (Z9)

W(s) == Sll+SIZ(s l k - I:SZZ)-IL:S{Z

(30)

Thus. to solve PI' S(p) must be written in the

*In general A*(p) == A '( -p). In the case of B( ) B () p'+ T] -1 ( -1 -1

p. ':' p = p_ T] L: = p L:= pI:) = B (p).

Z Note that L: = 1k .

form (Z8). Sa satisfying (16) and the symmetry

requirement S = S'. To solve P I + P the a a 3

decomposition (Z8) must be subjected to the requirements

S'S = I +k • a a n

S = S' a a

(31)

(3Z)

In problem P Z gyrators are admissible and all reactances may be assumed inductive. Instead of (Z8) we riow obtain

S(p) =Sll+SlZ(slk-SZz)-ISZ1' (33)

the real constant matrix

(34)

being arbitrary except for the passivity con­straint (16). Of course for P"+P3' (16) goes over into (31). "

It follows, from either (Z8) or (33) and the properties of degree that

5(S) ~ k (35)

In words. any realization of S(p) employs a number of non-trivial reactances at least equal to the degree of S(p). In this paper we concen­trate primarily on the case k= II(S); i. e .• the case in which S(p) is synthesized with the mini­mum number of reactances. Such a realization is called minimal. In the next section the break­down (6). with k=&(W). is investigated in some detail. The results are then applied to (Z8) and (33).

4. Properties of Minimal Decompositions of Rational Matrices

Let W(s} be any real. rational matrix finite at infinity and consider the decomposition

W(s) = J+H(slk-F)-lG (36)

where F, G. H, J are real constant matrices and k= 5(W). We begin by identifying the minimal and characteristic polynomials of F.

Lemma 3: Consider any representation (36) in which k- Il(W). Denote the distinct poles of W(s) by SI' sZ •...• s~. Let their respective orders and degrees be wntten r .• v .• i= 1. Z .... ,t.

:::<: 1 1 Let t r.

g(s) = TI (s-s.) 1 • (37) i=l 1

*If all entries in W( s) have relatively prime numerators and denominators. g(s) is the monic least common multiple of all denominators. Of course. vi?r i • i = 1. Z ..... t .

187

t v· *(s) == lT (s-s.) 1. (38)

i==l 1

Then. g(s) and *(s) are the minimal and char­acteristic polynomials of F. respectively.

Proof: Evidently. J = W(oo). Hence, if W(s) = W(s) - J. W(oo) = O. Moreover the poles of W( s) are precisely those of W( s) and all ors:!ers and degrees are preserved. Specifically. ~(W) = 5(W). Write

- -1 W(s)=H(slk-F) G (39)

and denote the minimum and characteristic polynomials of F by h(s) and cp(s). respec­tively. We must show that h(s) =g(s) and cp(s) = *(s).

lZ From a weIl known result.

-1 (s lk- F) ..

1. J

f .. ( s)

= -hlsr-. i,j = L z ....• k . (40)

The fij'S are polynomials in sand at least one numerator is prime to h(s). If Al' AZ' ...• Am are the distinct eigenvalues of

F. m q.

h(s) TT (s -·A.) 1 (41 ) i=l 1

and m n.

cp(s) TT(s-A.) 1 i==l 1

(4Z)

where n i ~ qi' i = 1. Z, ...• m. Obviously, the

poles of W(s) must be zeros of h(s); i. e .• with the proper ordering.

si=\' i==l.Z •... ,t

Again. the order of any pole of W(s) cannot exceed its multiplicity as a zero of h(s) whence,

r i ::: qi' i=l,Z ..... t (43)

Thus 5(g) S. ö(h) and g(s) divides into h(~) without remainder. To prove that h(s) divides g(s) is more challenging.

Let r==rl+rZ+ .. +r t andset

r r-l g( s) = s + a l s + ... + a r

(44)

Since W(oo) =0. W(s) may be written as

r-I r-Z BIS +BZs + ... +Br

g(s) (45) W(s)

where BI' B Z,···. B r are constant matrices.

In the neighborhood of s == 00.

I i=O

A. 1

i+T s

Combining (44), (45) and (46),

Identifying eoefficients of like powers of s,

A o BI'

A l +a1 A o =B Z '

Ar_1+a1 A r _ Z+'" +ar _ 1 A o =: B r

Ai+a1 A i _ 1 + ... +arAi _r '" 0

i= r,r+l, ... ,,,

( 46)

(47)

However, expanding the right-hand side of (39),

A=HFiG, i=O,l,.,.,.. (48) 1

Substituting (48) into the last equation in (47) ':~

yields

Hg(F)Fi-rcr = 0 , (49)

i=r, r+1, ... , ..

In partieular,

From lemma Z

Thus (50) implies

Hg(F)= 0 .

It follow8 immediately from (51) that for any power of F, Fi say,

Henee

H HF

--k-=T HF

g(F) ::: 0

and invoking lemma Z onee more,

~:~

F eommutes with g(F).

( 50)

(51 )

( 52)

188

g(F) '" 0 (53)

In other words g(s) is an annihilatingpoly­nomial for Fand therefore divisible without remainder by the minimal polynomial h(s), But it has already been established that g(s) divides

h(s) and sinee both are monie,

g( s) = h( s) . (54)

All that remains i8 the proof of the equality ilJ (s) = cp(s), This is aeeomplished by going baek to (39) and observing that any minor of (s lk- Ft 1

lZ is of the form

f.l( s) "iPTsT ' (55)

where f.l(s) is a polynomial and cp(s) is the eharaeteristie polynomial of F, It iollows that the distinet pole s of the minors of W( s) are all eigenvalues of Fand in view of the polynomial nature of f.l(s),

v. < n., 1- 1

i=1,2, ... ,t (56)

t t '\ :::; In. < k (57) L vi 1-

,

i=l i=1

k being the size of F. By hypothesis (mini­mality)

,f z: v·

i= 1 1

k

t :8 n, ::: k i=l 1

(58)

whieh, after taking (56) into aeeount, gives t=n and

Thatis, W(s)=cp(s), Q.E.D.

An immediate eorollary of this result 18 that in any minimal representation of W(s) every eigenvalue of F i8 a pole of W(s).

Lemma 4: The eigenvalues of areal k xk matrix Aare all strietly less than one in modulus if and only if the equation

S-ASA'" Q (59)

possesses a unique real, symmetrie, positive definite k x k matrix solution S for every ehoiee of kxk real, symmetrie, positive definite matrix Q.

~:' Proof: Appendix Z,

~:' If Q is semi-positive definite S is again

unique and either semi or positive definite,

Lemma 5: Let the real, rational nxn matrix W(s) possess an orthogonal minimal realization; i. e. ,

in whieh

MM' " 1 (61) n+k

M denoting the real (n+k) x (n+k) matrix

M (62)

Consider any other minimal realization

M:= (1 +-T-1jM(1 +T) (63) n n'

T real. Then, M is orthogonal if and only if T is orthogonal.

Proof: Sufficieney i8 obvious. Neeess~sing (61) and (63). M M':::: In +k reads

M(l +K) n

(1 -I- K)M, n

K TT'.

Letting

and substituting into (64) we obtain

H6 0

6G 0

F6 6F

From (68) and (69),

l:.(FG) F(6G) = 0

Suppose 6(F t C) :::: O. Then, with the (69) •

6 (FU 1 G) 6 F(FtG)

n(Ft G )

Consequently,

Invoking lemma 2, Eq, (15), 6= 0 and

K=TT' = lk

Henee T 1S orthogonal, Q. E, D.

(64)

(65)

(66)

(67)

( 68)

( 69)

(70)

aid of

0 ( 71)

(72)

Lemma 6: Let W(s) be a real, rational n x n matrix üfdegree k, Let

189

where

and k l , k Z

that W(s) sueh that

are non-negative integer8, Suppose

possesses a minimal realization M

(In + 2:) M " symmetrie matrix.

Consider any other minimal realization

M:::: (1 + T-1)M(1 + Tl . (75) n n

Then, there exists a kxk diagonal matrix

sueh that

(1 -I- I:) M" symmetrie matrix n

if and onl y if

(76)

1. 2:

2. 2:

2:; i.e., k1 "k l ,k2 "kZ and

T 2: T' .

Proof: Sufficien~

(l + 2:) M = (l + Z T- 1) M(I +- T} n n n

(1 +T'Z)M(l + Tl n n

(I + T 1)( 1 -I- z:) M (l +- Tl n n n

symmetrie matrix,

beeause (ln + 2:)M 1S symmetrie (by hypothesis).

In reaehing this eonclusion we have used the

relations z: = Z', 2:2 = lk and the faet that

Tl: T':::: 2: implies T'Z::::: :>:T- 1 .

Neeessity: Frorn (75),

(l +~)M= (I +~T-l)M(l +T). n n n

Sinee (I + l:) M i8 symmetrie, n

(1 + l:) M = M' (l +- 2:) n n

(77)

M' == (l -i- l:) M (l + 2:) (78) n n

and (1 -+ f)Ivl " M' (l +~) goes over into n n

M(l + K):::: (l + K)M , (79) n n

K = T~T/l: (80)

Exploiting the minimality of M (see lemma 5),

K = lk = T ~ T I l:

T~T' = l: . (81)

12 From a theorem of Sylvester ,(81) and the nonsingularity of T imply that E and l: possess the same number of plus on$s and minus ones along the diagonal. Thus, l: = l: and

Tl: T I = l:, Q. E. D. (82)

Lemma 7: Let W(s) be areal nxn rational matrix possessing the following prop-

* erties

I s I ~ 1

1. W(s) is finite at s = 00 •

2. W(s) is analytic in the region and at s = O.

1 3. W/(s)W(s) = In forall s.

Then, W(s) admits a real, orthogonal, minimal realization

M [b I ~] ; MM ' = ln+k' (83)

k= ö(W) •

Proof: According to lemma 1 W(s) pos­ses ses a real, minimal realization

-1 W(s) = J +H(s lk - F) G, k= 6 (W).

(84)

We shall establish (constructively) the existence of a unique real, symmetric kxk positive definite matrix L such that

M = (1 + i-I)M(1 + L) (85) n n

- - I isorthogonal (MM = ln+k)' Fromlemma3

and the minimalÜy of (84) it is known that every eigenvalue of F is a pole of W(s). Since W(s) is analytic in I s I ~ I all eigenvalues of F must be strictly less than one in modulus. In addi­tion, the analyticity of W(s) at s = 0 demands the nonsingularity of F. Clearly, J:: W(oo). We maintain that J is also nonsingular because

det J = 0 taken together with W '( ~) W(s) = In + k

implies the non- analyticity of W( s) at s = 0, a c ontr adic tion.

*Properties 1,2 ilnd 3 provide the translation of the regular para-unitary condition in the p-plane into the s-plane via the substitution

s ; P+T] ,T]> O. The analyticity of W(s) at P-T]

s = 0 corresponds to the analyti'city of S(p) at P=-T] .

In view of 3,

or, from (84),

W-l(s) = J/+G/(.!.1 _F/)-IH ' s k

After a little algebraic juggling,

W-l(S)=J~/(F,,-lH/_G/(F/) -1 [slk _(F/ )-l r l(F/ ) -IR'.

(86)

However it is easy to verif/' directly that W-l(s) may also be written as

W-l(s) = J +H (s l k - F )-1 G a a a a

(87)

where J- I J , (88)

a

H _J- 1 H (89) a

G = G J- I , (90) a

F = F - G J- 1 H (91 ) a

We invite the reader to supply the proof. Equa­tions (86) and (87) constitute two real, minimal

-1 ** realizations of W (s) and from lemma 2 we may assert the existence of areal kxk non­singular matriJlS T such that

J/_G /(F / )-I H, = J- l (92)

G/(F/)-l = J- 1 HT (93)

(F/)-IH I

(F 1)-1 =

T- 1 GJ- 1, (94)

T-I(F_GJ-IH) T.

(95)

To condense these equations we employ a weIl known formula for the inverse of a partitioned

matrix: 13

'~Consider the dynamical system

x Fx + Gu

r Hx + J u

where x is the state vector, u is the input and y is the output. If J is square and nonsingular, - -1 -1 ~ = J r- J Hx. Hence,

-1 1 x (F - G J H) ~ + G J- r

-1 -1 ~ -J H~ + J r

** -1 Remember that 5( W ) = Ii( W) = k.

190

rA IB]-l =[ (A_BD-IC)-l -A-lB(D-CA-1Bfll,

[C D _(D_CA- l B)-l CA-1 (D_CA-IB)-l J (96)

provided of course that det A f. 0, det D f. O. With the aid of Eqs. (92)-(96) it is possible to express the inverse of M in terms of M' and T:

[ .-!.I H] -1 = [---+--J I ---,-G 'K- l ] GTF KH ' KF' K- l '

(97)

K=-T. The verification is simple. For exam­pIe, from (92)

Identifying A with J, B with H, C with G and D with F, the (1,1) block in (96) is precisely J: etcetera. A more compact form of (97) is

M- I = (1 + K) M' (l + K- 1). (99) n n

Hence,

M(l + K)M ' n

(1 + K) n

(100)

Equating blocks in the lower right-hand corners of both sides of (100) yields

K-FKF' = GG ' . (101)

Now G G' is non-negative definite and the eigenvalues of ,F are all strict1y less than one in modulus. Calling upon lemma 4 we conc1ude that K is uniquely determined by (101) as a real, symmetric non-negative definite matrix. But since it is nonsingular (K = - T) it is actu­ally positive definite. As such it admits a unique real, symmetric, positive definite

square _ root L 12:

(102)

Write

M = (1 + L- I )M(1 + L) n n

( 103)

From (99) and (102),

MM ' = In+k' Q. E. D.

Lemma 7 is a major result and solves (section 5) the problem of finding all passive lossless n-ports which realize a prescribed nxn regular para-unitary matrix S(p) and employ the minimum number of reactances.

The next lemma complements lemma 6.,

Lemma 8: Let W(s) be any nxn real, symmetric rational matrix function of s of degree k which is finite at infinity. Let

be any real minimal realization. Then,

1. there exists areal, constant, non­singular kxk matrix T such that

M / = (1 +T-l)M(l +T) n n

2. moreover T is symmetric; i. e. ,

T=T' .

Proof: Transposing the expression

-1 W( s) = J + H( s lk - F) G

and using W = W I we obtain

W(s) =J'+G/(slk-F)-lH/.

Hence, M' is also a minimal realization of W(s) and there exists areal, constant non­singular kxk matrix T such that

M' = (1 + T- 1)M(1 + T) n n

(103a)

Iterating (103a) once,

191

(1 + K) M = M(1 + K) n n (103b)

where

K =T(T,)-l (103c)

Invoking the minimality of M (see lemma 5), K= lk' Thus,

T = T " Q. E. D.

Corollary: W(s) possesses a realization M such that

(ln + l:)M = symmetric matrix,

l: being an ordered k x k diagonal matrix of plus and minus ones, if and only if W(s) = W'(s). In addition, l:: is unique.

Proof: Necessity: Suppose that for some 2

M and l::(l: = lk)

(1 + l:) M = M I (1 + l:) n n

Then, J = J I, G I = H l:, F I = ~ F l:: and

W/(s) J/+G/(sl _F/,-lH' k

J + H l:: (s 1 k - 2: FI;) - 1 EG

J +H(s lk- F)-1 G

W(s), Q. E. D.

Sufficiency: Suppose W(s)= W/(s) and let M be any real minimal realization. From lemma 8,

M' = (l + T- l ) M(l + Tl ,(103d) n n

for some real, symmetrie T. Write

T = LL: L'

where L is kxk, real and nonsingular and

Set M= (1 tL-1)M(1 tL) .

n n (103e)

From (1 03d),

(1 +L:)M = M'(l +L:) n n

(103f)

and !Vi is the desired realization. The unique­ness assertion follows fron, lemma 6.

Lemma 8 and its eorollary justify ealling L: the "reaetanee signature matrix" of the real, symmetrie rational matrix W(s). Further justifieation is provided in the next seetion.

Lemma 9: Let W(s) be areal nxn rational matrix possessing the following prop­

:::~

erties: 1. W( s) is finite at s = CD.

Z. In I si> 1, 1 - W( sl W':'( s) ~ ° . - n n

3. W(s) isa.nalytieat 8=0.

Then, W(s) realization

M=

adnlits a real, minimal

n k

whieh is bounded; i. e. ,

1 -MM/>O n+k - n+k'

k = 6(Wl .

(103g)

Proof: Aceording to a weH known

resultZ, 6 in the theory of lumped, passive net­works, there exist three rational matriees WlZ(s), WZ1(s) and WZZ(s) analytie in \sl~l and of sizes nxt, txn and .tx~", respeetively, such that

O"Property 2. translates boundedness in Re p;;::: 0-into boundedne s s in I s l ;;; 1 via the change of

variable s = P+r], Ti> O. Again, 3. is equiva-p-r]

lent to assuming S(p) analytie at p = -r] .

(103h)

is para-unitary and analytie in \s I > 1 and at s = O. Thus Wa (s) satisfies 1. ,Z. -and 3. of lemma 7. Moreover, and this is most important,

1. ö(Wa ) = Ii(W).

z. If W(s) = W'(s), the bordering may always be earried out so as to guarantee the syrn­metry ofW (s); i. e., W (8) = W/(s).

a a a

3. t ~ normal rank [I - W(s)W'(.!)J n s

Invoking lemma 7 we may assert the existenee of a minimal realization

W (s) = J + H (s 1 k- F ) -1 G a a a a a

wherein k=Ii(W )=6(W) and a

M a " [*J is real-orthogonal (Ma M~ ln+t)'

Clearly,

W(s) = 11. W (s) a

Ic I ,

where 11. = [lnl°of)

Henee, from (103i)

( 103i)

(103j)

( 103k)

(l031)

(103m)

is aminimal realization of W(s). In other words, the (n+k) x (n+k) matrix

M = (At lk)Ma (11. / + lk) (103n)

eonstitutes a minimal realization of W(s).

Let

M= [W] be any other real, minimal realization of W(s). Then, there exists areal, eonstant nonsingular kxk matrix T suchthat

192

M = (l + T) 1:1 (1 + T- 1 ) n n

= (A~' Tl M (AI-+, T- 1) a

(1030)

( 103p)

Evidently, beeause of the orthogonality of M a and the form of A,

Consequently any realization M may be con­verted into a bounded one by means of the trans­formation

(1 +T-l)M(l tT)=M, n n

where T is the matrix appearing in (103p). Given M, how do we find T?

Observe first that 11. + T pos ses ses the right-inverse Alt T-l:

(A + T)(A I + T- 1) = In+k

Let E ::: M( 11. + T) - (A + T) M

a (103q)

From (103p), ( 103r)

Put n .f. k

E ::: [EI I E Z IE 3 J (n+k) .

Using (l03r) we find easily

E =0 3 n +k, k

and

Thus, E(A / + T) = 0n+k= E(A/t TI) . (1038)

From (103q) and M MI::: 1 + +k' a a n L

[M(At Tl-EJ[M(A + T)-EJ I = 1 + T T' n

Or, simplifying with the help of (103s),

(l .~ T T') - M( 1 + T T') M'::: E E' . n n

(103t)

Unfortunately, E depends on T as weH as M a . In fact writing

J a

n t

Ga = [ G all G aZ ] k

n

it i6 easily seen from 103q that

E, " -[TJ~~J and (l03t) goes over into

(1 + T TI) _ M( 1 + T TI) M'::: [""J,..,a;;:;.Z;;;...J-;a;;-1 ~:;--I-=J-:=a;-z_G-:=~'Fz.,..T='l n n TGaZJ aZ TG azG:2TJ

(103u)

Equating blocks in the lower right-hand corners we obtain

193

T(l -G G' )T/-F(TT/)F'=GG ' . (103v) k aZ aZ

The kxk matrix lk-GaZG~2 18 sym­

metrie positive definite. This i8 established by invoking the orthogonality of M. From MaM~:::ln+t+k we get a

G G'+F F'=l . a a a a k' or,

Henee lk -GaZG~Z i8 non-negative definite. Sup­

pose it is singular. Then for some real non­trivial k-veetor x, FiX = 0k' G/1x" 0k' But

- a- - a--beeause of the a8sumed analyticity of W(s) and therefore of W (s) at s = 0, F is nonsingular.

a a Thus ;;" ~k' a eontradietion. Note,that if W(s)

is already para-unitary no bordering is required, GaZG~2 = 0k and (103v) eollapses into (l0I) with

K" TT /.

In brief, given W(s) we find a minimal realization M (Appendix Il. M may or may not be bounded. If not, W( s) is borde red into the regular para-unitary matrix W (s) and G Z

a a determined. Employing (103v) we solve for T. With T we eonstruet the bounded realization M=(l + T-I)M(l t T)" (A+1 k )M (A /+ lk)' One n n a last point 1s worth mentioning. If W(s) is sym­metrie we border it into asymmetrie para­unitary matrix W a ( s) and employ Appendix I and

the method deseribed in the next seehon to eon­struet an orthogo/nal M a . satisfying (l n + t + I;)Ma:::Ma(ln+L + ~), L: the associated

reaetanee signature matrix. Thus,

M" (A -+ L:) MI (A ' +- L:) a

Henee (1 +I:)M=(A+-lk)M/(A/+-~)

n a and

M/(l +~) = (A+lk)M/.(A ' +- r) n a

A thorough study of Eq. (1 03v) will be presented in Part 11.

5. Synthesis of Reciproeal, Lumped, Passive n-ports

Our starting point i6 Eq. (30):

-1 I W(s) " S11 + SlZ(s lk-tsZZ) L:S 12 = S(p) ,

s " ~. Tl> 0 , P - 11' 'I

(104)

(l05)

k= 5(S) .

To make eontaet with seetion 4 it suffiees to put

J = S11

H= 5 1Z

G= r S{Z'

F = 2:5 22 ,

Henee,

M (1 + 1:) 5 (106) n a

where

M= [~J [ S1l ; S = ( ... I

a "'12 S12] 5 Z2

(107)

Bee'luse of minimality every eigenvalue of F =2: S i8 a pole of W(s) (lemma 3). Let 8

Z2 0

be any eigenvalue of F. Then, S(p) possesses a pole at the point

(l08)

Repo (109)

and 1801:::: 1 implies Repo:::: O. Beeause oi its

bounded- re al nature 5(p) is free of pole 8 in Re p ~ 0, infinity included. Consequently, in any minimal realization (55) all eigenvalues .~f 2:S Z2 are strietly less than one in modulus .. ,.

If SZ2 is singular So = 0 is an eigen­

va1ue of 2:5 22 and S(p) possesses a pole at

p = -Yi. Thus if P = -Yi is not a pole of S(p), SZ2

is nonsingular. 'FrmTI now on it is understood that Yi is so chosen.

Let S(p) be n x n, symmetrie, bounde d­real and of degree k. A pair (2:, Sa) satisfying (l06) is said to eonstitute a minimal reeiproeal network realization of S(p) if

1. k 1+k2=k=I\(S) (110)

2. S = S' ( 111) a a

3. 1 -S'S >0 n + k a a - n+ k

( llZ)

For short we s ay that Ci, S) is an MRN re a1i­

zation of S(p). A pair \2::, S) satisfying (l06)

and (110) but not necessarily (111) or (11Z) is

>:~ This is valid irrespective of whether or not

satisfies the passivity requirement, 5

a

I n + k - S~Sa ~ 0n+k

called a realization. If it satisfies (l06), (lIO) and (111) it is a symmetrie re alization.

The problem of MRN equivalence is essen­tially the following: Given one MRN re alization of 5(p) find all others. The solution is achieved step-by-step. Let (r,5 a ) be any (fixed) MRN

realization of S(p). Then,

M=(l+r:)S n a

where

and

5 = (1 + r)M = asymmetrie bounded-real matrix. a n

Let Ü:,13 ) be any other MRN realization of S(p). Then a

"~ [I~I I.I:J . ~I + '2 c k

and

Ei ,,( 1 + E) M = symmetrie matrix. a n

From lemma 6 we know that

1. ~ = Z and

Z. lV1= (I + T-1jM(1 +- Tl n n

(113)

where T is areal, eonstant, nonsingular kxk matrix s atisfying

3. TI: T ' == 2: . (114)

The equality ~ = 2: reveals a most interesting physieal fact. Namely, all MRN realizations of asymmetrie, bounded-real matrix S(p) employ the same number of inductors and the same nurn-

~:~

ber of eapacitors.

194

Sub stituting (114) into (113) yields

S == (1 + T ')S (1 + Tl , a n a n

(115)

in whieh guise the symmetry of S is obvious. a

However, in order to qualify as an MRN reali­zation Ei must also be bounded; i. e. ,

a

1 -55'=1 -SZa;a:On+k (116) n+k aa n+k

In terms of M,

In+k-M'M;a: 0n+k . (117)

':'In other words, k1 = k 1 and kZ= k Z' All this is

a consequenee of minimality and reciprocity and justifies ealling L: the "reactance signature matrix" associated with S(p).

Thus, the problem reduees to finding all real T's which satisfy (114) and generate, via (115), matriees S with eigenvalues completely eon-

a tained in the interval -1::; x ::; I (by hypothesis _,_

all eigenvalues of S lie in this same intervall. a

Theorem 1 supplies a eomplete answer to prob­lem P l +P3 .

Theorem 1: Any n x n symmetrie regu­lar para-unitary matrix S(p) of degree k pos­sesses an MRN realization (2:, Sa) wherein Sa

is orthogonal. The matrix

is uniquely determined as the reaetance signature matrix of S(p). Moreover, in any other MRN realization (r, Ei) in which S 1S orthogonal. . a a

-a. I: '" Z and

b. Ei == (1 + T ')S (l + T) a n a n

( 118)

where T=T 1+ TZ' Tl and TZ

being two .~lC?itrary real orthogonal matrices of sizes k 1 x k 1 and k 2 x k Z' respectively.

Proof: All we need demonstrate is the necessJty of conditions a and b. In Appendix 1 a method i8 deseribed for finding a realization (r, Sa) in whieh Sa 1S re al- symmetrie but not

neeessarilyorthogonal. Write

M::: (1 + I:)S n a

( ll9)

Then,

(1 + 2:)M= S = symmetrie matrix. n a

Aec:ording to lemma 7, the regular para-unitary eharac:ter of S(p) implies the existenee of a unique real, symmetrie p0sitive definite matrix 'L whieh generates, via the formula

. 1 A

M = (l + L - ) M(l + Lj , n n

(1 ZO)

an orthogonal (n = k) x (n + k) matrix M. If we write

(121 )

L is determined as the unique symmetrie, posi­tive definite square-root of the solution K oi the linear system

( 122)

"~Sinee S 18 real-symmetrie, (116) attains if a

and only if all eigenva1ues of Sa He in -lS;x'::;1.

The matrix

S = (l + I:)M a n

(123)

is obviously orthogonal. We shall show that it is also symmetrie and therefore that (I:, S a) i6 an

MRN realization with 5 real-orthogonal. a

Clearly M orthogonal implies that (refer to (120) )

( 124)

is also orthogonal. But from the symmetry of Set

and (124) reads

• A • -1 (1 -I- L z: )M( 1 +!:L )::: orthogonal matrix. n n ~

. A'_l (1 + L LI: )M( 1 + 2:L 2:)::: orthogonal matrix. n ,. n

But L L z: i6 symmetrie positive definite. By the uniqueness assertion,

-1 2:L2:=L (125)

or, (1Z6)

Going back to (120) and using (125) and (119) we see that

(1 +I:)M =(1 +LjS (l+L) n n a n

" symmetrie matrix, Q. E. D.

Now let S ) be any other MRN reali-zation of S(p) with a S orthogonal. Frorn

a (114) and (115), there exists areal kxk matrix T such that

S (1+T'j50+T) a n a n

and L: =T2:T' (127)

Invoking lemma 5 and the orthogonality of S a' T

is an orthogonal matrix; i. e., T T' := lk' Put

195

k 1 k 2

T [2lJ~] k]

TZ 1 I r 2 k Z

5ubs tituting into (I Z 7), written as

TI: = ZT ,

where Tl and TZ are real-orthogonaL

eompletes the proof of theorem 1.

Henee

This

Corollary: Let Na be any residual (n +k)-port eomposed exclusively of ideal trans­formers whieh when closed on its output k ports in k 1 preseribed induetors LI' L Z"'" L k

1 and k Z preseribed eapaeitors Cl' C Z" .. , C k

Z yields an n-port realization of the preseribed symmetrie, regular nxn para-unitary matrix S(p) of degre~ k. Then, any other such residual (n+k)-port Na is obtained by easeading Na on

its output side in two arbitrary ideal-transformer all-pass networks NI and N Z' The first i8 a

Z kl-port and the seeond a Z kZ-port (refer to

Fig. Z). Setting

and

L diag [LI' L Z"'" L k ] 1

-1 . [-1 -1 -1 C =dlag Cl ,CZ "",Ck ]

Z

the eorresponding turns-ratio matriees PI and P z are arbitrary subjeet 801e1y to the eonstraints

L ,

Proof: From the formula

S (I.} T ') S (I .} T) a n a n

where

( 1Z8)

( 1Z9)

( 130)

TI and TZ orthogonal, it follows that Na is

obtained by easeading the output side of Na with

an all-pass possessing the scattering deseription

However sinee T is a direet sum, this all-pass reduees to two uneoupled an-passes, one in the first output k 1 ports and the other in the remaining output k Z ports of Na' Now these

two all-passes NI and N Z are simply ideal

transformer banks: The first preserves the k i

uneoupled impedanees pLI' P L Z' ... ,p L k . I

while the seeond preserves the kZ uneoupled -1 -1 -1

impedanees pC l ,pC Z , ... ,pCk . Using the Z

well known formula for the transformation indueed by an ideal transformer bank we obtain (lZ8) and (lZ9) for the respeetive turns-ratio matriees PI and P Z' Q. E. D.

An nxn symmetrie bounded-real seat­tering matrix S(p) is said to be of type RL if it possesses a network realization eontaining only ideal transformers, positive resistors and

196

induetors. Similarly, it is of type RC if it pos­ses ses a realization eontaining only ideal trans­formers, positive resistors and eapacitors. Let k 1 and k Z denote the number of plus ones and

minus ones in the reactanee signature matrix of S(p). Clearly, S(p) is RL if and only if k Z= 0

and RC if and only if k 1 = O. In the first ease

Zoo lk while in the seeond Z=-lk(k=k1+k 2 =o(W)=

degree S). These observations lead to theorem 2.

Theorem Z: Let 0::, S ) be any symmetrie realization of the RL(RC) syrJinetrie bounded­real matrix S(p). Then

1 -SS/~O n+k a a n+k

(131 )

That is, the symmetry of Sa automati~ally

guarantees the MRN eharacter of the doublet (>:, S). In any other MRN realization (Z, § ), _ a a :>: :=:>: and

S (1';' T ')S (1 .} T) a n a n

( 13Z)

where T is an.arbitrary real, orthogonal kxk matrix.

Proof: Suppose S(p) i8 RL. Then, :>: = lk' Let (lk' S .) be any realization with S a -- a symmetrie. Aceording to (lI:!) and (115) any other symmetrie realization Sa is of the form

S := (l .~ T ')S (1 .~ T) a n a n

where TZT' OO:>:. But ')= lk whence, TT'= lk'

Since T is orthogonal, the passivity constraint

i,s either satisfied by all symmetrie realizations S a or by none. The existenee of one bounded

symmetrie re alization is as sured by lemma 9. Henee, any symmetrie realization Sa is auto-

matieally bounded. The same argument is applieable to S's of the RC type, Q. E. D.

Corollary: Let Na be any MRN reali­zation of the RL bounded-real nxn matrix S(p) of degree k employing the preseribed induetanee values L l , L Z' ... L k . Then any other such

MRN realization N is obtained by eascading N a a

on its output side in a Zk-port ideal transformer bank with turns-ratio matrix P satisfying

PLP'= L (133) where

( 134)

Proof: Identie al to that given for the corollary to theorem 1. Of course, if S(p) i6 RC, (133) and (134) go over into

PC-1p'=C- 1

and [ -1 -1 -1 1

diag Cl' C z , ... , C k ] = C- .

As we have seen, the problem of finding all minimal passive network equivalents of a bounded-real matrix S(o) reduees to the follow­ing. Given an (n + k) x (n + k) real symmetrie matrix S a whose eigenvalues are eompletely

eontained in -1:;;; x :;;; 1 and a diagonal matrix

" [l~l l_l:J k1+kZoqSj

find al1 real kxk matriees T satisfying

TZT'=:>:

whieh generate, via the formula

s = (1 + T/)S (I + T) a n a n

matriees S with eigenvalues also eomplete1y a

contained in -1::s. x:;;; 1. We return to this question in part H.

In part II we also undertake the eonstrue­tion of non- minimal re ali zations. N eve rthele s s theorem Z is within our reach.

Theorem Z: Let k 1 and k Z be the num­

ber of inductors and eapacitors employed in any minimal, reeiproeal network realization of the bounded-real matrix S(p). Let t l and t z be

the number employed in any non-minimal recip­roeal realization. Then, t l ~ k i and t z 2: k Z'

Stated otherwi.se, a minimal reciproeal reali­zation not only employs the minimum number of total reaetanees but also the minimum number of induetors and minimum number of capaeitors indi viduall y!

Proof: Lemma 10, Appendix 1.

6. An Example

The one-port N of Fig. 3 possesses the adrnittanee

y(p) z

p +p + 1 Z

Zp + Zp + 1 (135)

Because Il Ly{p)] ::; Z, any realization of y(p) requires at least two reaetanees. Perforrning the extraetion of the two reaetances, we obtain the residual network N , of Fig. 4.

a The scattering matrix of Na' under unit

normalization, is given by

1 1 1 '4 '2 '4

S 1 0

a 2' Z

1 1 1 '4 "2 '4

Sinee k1=kZ= 1, then

I: == ~ _OJ ( 136)

Aceording to the results of section 5, any other Mr::!R of N is described by a seattering matrix Sa' given by

S "(II.} TI)S (1 1+ T) a a

(137)

where T and S are subjeeted to the eonditions a

and

respeetively.

TZ:;T'= I:

-Z 13- Sa;::03

(138)

( 139)

The general solution of Eq. (138), in the special ease where k 1 == k Z' ean be found in refer-

ence (18). With r given by (136), we obtain

T= [0. 0] [eOSh x sinh x] r ~ ° J ' o B sinh x coshx l ° 1'1

(140)

where x is an arbitrary finite real parameter, and the greek symbols may take on the values +1 or-I, independently of eaeh other. Substituting (140) into (137), we get

S UQU (141 ) w.here a

[ 20 co"hx + 8 ,inh x

2a cosh x + ß sinh x Zasinhx + Seoshx

l Q 1 sinhZx + ZaSsinh(Zx) ZaSeosh(Zx) + i sinh(2x) '4

Za sinh x + ß eosh x Zaß cosh( Zx) + i sinh( Zx) cosh Z x + ZaS sinh( Zx}

( HZ)

197

and U = diag [1,8,11 ]

In order that the passivity eondition (139) be satisfied, it is neeessary and suffieient that a11 eigenvalues of Q be less than one in n1ag­nitude. For eonvenience, we impose instead the eondition that the eigenvalues of 4Q be restrieted to the interval (-4,4). The character­istic equation for 4Q is given by:

det(Al 3 -4Q)=A 3_ A 2( l+cosh(2x)+4as sinh(2x)) _

-4A(l+cosh(2x)) = 0 ( 143)

Obviously, one root is always zero, irrespective of the values attributed to the parameters a,s, and x. This was expected because we started from a singular S a'

In orde r to exanline the roots of (143), let us first dispose of the case when as x 2: O. Equation (143) becomes

\.3_ A2 ( l+cosh(2x)+41 sinh(2x) I) -4x( l+cosh(2x) ) = O.

(144)

It is easy to verify that one root is always larger than 4, except when x = O. We must then impose the condition as x S O. The characteristie equa­tion for 4Q then becomes:

\.3-A2(l+cosh(2x)-4Isinh(2x) I )-4A(l+cosh(2x)) = O.

(145)

An investigation of the roots shows that the three roots are contained in the prescribed interval only when

Os lxi S 0.4407 . ( 146)

All MNR are then completely described in tenns of their scattering matrix, Eqs. (141) and (142), to which the eondition as x$.O, and the restric­tion on x, give.n by (146), should be added. Without loss of generality, we may assurne a 13 = -1, x.2: O.

In order to exhibi.t a concrete re ali zation, it is convenient to work with the admittance matrix 1'a . This is related to S via the

. a expresslon

which leads to

( 147)

[

cosh2x - sinh(2x)

a,(coshx- sinhx)

-13 sinh x

Any such l' can be realized as an ideal trans-a

former 3-port containing at nlOst 3 resistors. It is of some interest, however, to investigate the existence of transformer-less equivalents.

A necessary and sufficient condition for the realizability without transformers of an admittance matrix of order 3 is known14, J 5. The matrix must be paramount, i. e., each principal minor must be no less than the absolute value of any minor buHt from the same rows. For Y as

a in (148), the 2x2 princiJ'al minor formed with the corner elements of Y is however smaller

a than" the magnitude of the minor formed with the elements from the same rows, and from the sec­ond and third columns. This i8 true for a11 x' s in the interval (146), except at the end points. The end point x == 0 corresponds to the network we started with, and at the other point l' i8 not

a defined (8ince sinh{2x 0.4407) == 1). We conclude

then that there are no transformerless equivalents for N (except for topological ones), which use the same reactanees as N. lf we allow for ideal transformers at ports 2 and 3, which may be eliminated later by changing the value of the reactances, then transformerless equivalents may be found. Let n 2 , n 3 be the turns-ratio at

ports 2 and 3 respeetively, as in Fig. 5. The admittance matrix of the 3-port ro obtained 1S given by

cosh2x-s inh( 2x)

aI1z( coshx-s inhx}

ClI1z{coshx-sinhx) -B~sinhx

2 2 n Z 11f13cosh( 2x)

(149) where

y :::;

A detailed examination of th!, inequalities to be satisfied by the elements of Yal , in order

to yield a paramount matrix, gives the following bounds for the parameter x and for the turns raUes n 2 and n 3 :

o $.x$. O. Z741 (150)

coshx-sinhx(l+cosh(2x}) < < coshZ,*,sinh(2x) 2 - n Z -

198

l-sinh x coshx-sinhx

a{coshx- sinhx}

2

eosh(2x)

-13 sinhx l 'Oe~{2xl J

(151 )

( 148)

{ 1

i j ! i J i 1

I

1-sinh{2x}

cosh 2x eosh(2x) + sinh2x_. sinh( Zx) cosh(2x) - i sinh(2x) (152)

A nurnerical realization is presented in Fig. 6. The parameters are:

x " O. 15

n = Z O. 8344 (::lower bound of (151))

1. 1200 (=upper bound of (l52)).

For these values (and a= 1, 13" -1), Eq, (149) yields

[

1. 0326

1. 0326

O. 2023

1. 0326

2. 0030

1.1719

O. 2023] 1. 1719

1. 2556

From this admittance, the network of Fig. 6 is obtained by the method described in reference (16), for transformerless 3-ports on four nodes.

7. Conclusion

In order to p1ace in its proper perspective the method of synthesis via reactance extraction, we will recall briefly how one proceeds to solve a synthesis problen1, pointing out S0me of the salient features.

Given an arbitrary rational bounded real scattering matrix, S(p), to be realized as a pas­sive network, the first quantity to be determined i8 the minimum nurnber of reactances, k. This number may be eomputed either via the definition of degree or via the expression (I39 ), of Appendix

1. If we are dealing with a reciprocal realization, then the minimum number of inductors and of

199

capacitors, respectively, are given again by (134) and (135 ), The special choice for norn1al-

ization numbers at the k ports n + 1, n + 2, ... ,n+k permits all reactance va1ues to be arbitrary. One then proceeds to determine a11 MNR's. A complete explicit solution was given for an two-elen1ent networks. Specifically, we found that any n1ini­mal realization of a para-unitary S(p) may be translated into a lossless one, from which all lossless realizations are then derived via an orthogonal direct sum similarity transformation. In the RL, 01' RC, case the situation i8 particu1ar­ly interesting, since any symmetrie realization autornatically satisfies the passivity requirement and any other MN R is obtained via an arbitrary orthogonal transformation. In the RLC case, we have demonstrated the existence oi MN R's in a constructive manner, and we have indicated how a bordering process n1ay be applied, in order to

reduce the problem, again, to the realiza tion of a para-unitary matrix. In the example of section 6 this proeess could be avoided, sinee the low order of complexity allowed the use of direct computation.

Onee an MNR is at hand, it comes in the form of the sc attering matrix of the re act.ance­free network N , to be terminated upon k indue-

a 1 tors and k 2 capaeitors of given values. If we allow

the use of ideal transformers, the physical representation of Na i6 readily obtained via

classical methods. If transformerless realiza­tions of S(p) are desired, then we must look for transformerless representations of Na' The

problem of synthesis without transformers may therefore be restricted to re actance-free net­works. The example illustrates how an appro­priate choice of the reactanee values may lead to a whole class of transformerless realizations.

Appendix I

The purpose of this appendix is to prove that any real, rational matrix W(s) which is finite at infinity possesses a realization of the form

where J, H, Fand G are real, constant matrices of compatible sizes and k = O(W).

Clearly,

J", W(oo) . (1Z)

Hence the matrix W(s) := W(s)-J vanishes at infinity and may be written as

r-l r-Z BIs +BZs +o .. +B r

gIs)

where B l , B Z' ... ,B r are real, constant matri­

ces and

is the monic least common multiple of all denomi­ators in W(s). The integer r is the order, O(W), of W(s). Writing ---

W(s)

it follows from Ego (47) that

i=r, r+l, .. o, ..

A glance at (11) reve als the identific ation

Ai:::HFiG, i=O,l, ... ,.. (17)

Put

P. 1

and

[ ~: T. 1

A. 1

[

H 1

J], Al

Az

Ai+l'

.A.

1 1

. Ai + I

. A Zi

Then, the equalities (17) are subsumed in the 19

compact statement

(19)

200

Since the realization (F, G, H, J) is to be minimal,

~:~

Thus rank Ti::: k, i.2;: k - 1 .

It is an easy conseguence of (I6 ) that

rank Ti=rank T r _. l , i",r-l, r, ... , ..

and therefore

rank Ti" k =ö(W). i = r-l, r, 0 •• , o. (lU)

We shall obtain the components H, Fand G by an appropriate factorization of the matrix T r-I

which is uniquely determined by the expansion of W( s} in the neighborhood of s = 00. Moreove r rank Tr_1=k= 6(W). If we assurne that W(s) =

mxn, all Ai's are mxn and we introduce the

"generalized" companion matrix

0 1 0 .. 0 m m m m

0 0 1 .. °m m m m

o = 0 0 0 ... 1

m m m m -a 1 -a 1 . - a 1 r m r-I m 1m

o possesses r "block" columns and r "block" rows. The polynomial gis) annihilates 0:

g(O) = 0

Using (16) it is a straightfoward matter to ve rify the following identitie s:

A ~.

A t+l

0

A t +r-l

OT l=T 10', r- r-

A t + 1

A t+ 2

-p::---t+r

t= 0,1, ... , ..

(1 17 )

With the aid of (~16) and (1 17 ) it is possible to

exhibit T r-I in two alternative ways. Let

','

Because P.Q." T., rank T. :s. k for i.2;: k - 1. 1 1 1 1

However, rank T i 2=: rank P i + rank Qi-k=

Zk-k=k, i.2;:k- 1 and(IIZ) follows.

Then,

and

A=

A o

T ::: [Ä 10Ä \,l Ä I ... lor-I Ä] (120) r-I

T r-I

Given W(s) we calculate T r-i and O·

Then we factor T into the product r-I

T := MN r-I

where number of columns in M =: number of ro,,::s in N ::: k := rank T l' Hence rank M:= rank N -

r- A A

k and in any other such factorization MN = T r-I'

say,

A

N

T an arbitrary kx.k real, constant nonsingular matrix. We now partition M and N conformably with P and Q l' Thus we write

r-I r-

and

M=

M"-' r-l

N:= [NoIN11 ... \Nr_1J .

Comparing (120) and (IZ2 ) we obtain

Hence

MN. ).

N. ).

MN i

-1 0M)N (Me' i

(M-10M)i N , i:::O,l, ... ,r-l . o

-1 1 f . r M Th' Of course, M i6 the e t-Inverse 0, ,. 16

inverse exists because M has rank equal to the nurnber of its columns.

Again, cornparing (121 ) and (122),

M.N 1

A(O,)i, i= 0, 1, ... , r-l

M.NO' 1

M.(N Cl' N- 1) 1

('1' -li. 1 ]'vi (N" N ) ,1=0,1, ... , r-o

Hence (N- 1 is the right-inverse of N),

H

F

1t i6 easily shown that (IZ9 ) and (132) yield the

same value for F. From (1 15 ) and (IZ2 )'

OMN MNO'

Since g(s) annihilates Cl it also annihi­

lates F == M- 1 Cl M. To see this observe, with the aid of (I I5 ), that

F Z (M-1ClM)(NO'N- 1)

M- 1 ('2 T ('2'N- 1 r-I

M- 1 OZ T N- 1 r-l

M- 1 ci, M .

In general, whence,

F t oo M-10tM for all integers t;:: 1

-1 0 g(F) "M g(olM == .

Now the H, Fand G constructed above obviously

satisfy HFiG=Ai for i=O,l, ... ,r-l. But

then, from (16 ) and (I32a),

201

Ar -alAr_l-aZAr_Z-,···,-arAo

r-l 2 + J)G -H(alF +aZ + ... a r

HF r G

i Continuing in this fashion we find Ai'" HF G for

all integers i;:; 0 and therefore (F, G, H) con­stitutes a solution-triplet of the infinite system (1 7),

A Evideptly if M and N are replaced by M:=MT alld N=T-IN,

H HT,

d ::: T- 1 G

and

a well known result. 11

In summary

J W(oo) H M

o G N

o

F M-IO M = NO 'N- 1

where MN = T l' ('J is given by (I ) and num-r- 14

ber of eolumns in M = number of rows in N == rank T r _ 1 = 0 ( W).

Suppose W(s) is symmetrie; i. e. ,

W(s) := W'(s) .

Then J, the Ai"s and therefore T r _ l are also

symmetrie. Henee one ean write

T r _ 1 = ML:M' (133 )

where M is real, eontains k= 6(W) eolumns and

The matrix L: is uniquely determined by W(s) and is precisely the assoeiated reaetanee­signature matrix. To make eontaet with the above developments we set M = M. N =!: M' and get

and

and

H"M o

G=L:M ' o

.'. L: F " F 'L:

[ J I Mo] ~ S

a

a symmetrie matrix.

For the sake of eompleteness we point out the following formulas for k 1 and k 2 :

_ 1 . k1-Z(rank T r _ 1 +s1gnature T r _ 1),

1 k 2 "Z(rank Tr_1-signature T r _ 1)·

202

Up until now we have restrieted ourselves to minimal realizations of W(s). However sup­pose

W( s) " J + H( s 1 - F) - I G t

where t.2 k:: ö( W). Also suppose that for some

L: ::

(In +~) [i I ; ] = symmetrie matrix.

(I38 )

As we have already seen, mined by W(s) if t = k. denoteL: by l:o:

L: is uniquely deter­In this ease let us

Lemma 10: For t ~ k,

Proof: Again r=O(W), the order of W(s}. It follows from (I38 ) that

Thus PL:P ' T =PL:P ' r r r-l 0 0 0

in whieh

LtJ P r

and

H 0

H F o 0

P 0

H F r - l o 0

The triplet (F • G ,H ) is any minimal reali-000

zation of W( s) s atisfying

11n+ "0) ~:o I ~o] symmetrie matrix.

Beeause of the existenee of p: l , the left­inverse of Po' (I42 ) yields

Q L:Q I ,

Q p- 1 P = k x t real matrix of rank k. o

For any real k-veetor x set

and

From (I43 ),

2 2 2 Z 2 2 xl+x2+"'+~ -~ +l-~ +Z-, .. ·,-~

1 1 1 2 2 2 2 2 2

yl+yZ+···+ytl-ytl+l-ytl+2-' .. ··-yt . (I45 )

Or,

Bearing in mind that we are trying to establish the inequalities t 1 2: k 1 and ~'2 2: k 2 let us

assume for example that -/'1 <klo Consider the linear system

~ + 1 = ~ + 2 =, ... , = x k = 0 1 1

Y 1 = Y 2 =, ... , " Y t l = 0

(147 ) eonstitutes, via (145 ), a homogeneous set of

t l +k2 linear equations in the k unknowns

"1' x 2 • ... ,xk · Sinee t l < k 1, the number of

these equations is less than k 1 +k2= k. Henee we

have fewer equations than unknowns and a non­trivial solution for ~ always exists. Substituting this ehoiee for ~ into (146 ) gives xl = x 2 :: •... =~ =

O. In other words ~ = ~ k' a eontradietion. ,~ The same argument shows that t 2 ':::: k Z' Q. E. D:" ~:<

After eompletion of this paper. we noted that several of the results in Appendix I were also obtained by B. L, Ho and R. E. KaIman.':O

Appendix II

Let A and Q be two preseribed real kxk matriees. The matrix equation

S-ASA ' = Q

for the unknown real kxk matrix S is obviously linear. Let F denote the linear operator defined by

F(S) :: S- ASA ' .

F is defined on the eolleetion of all k x k matri­ees Sand (II I ) reads

F(S) = Q

Lemma 1: Let 1.. 1,1.. 2"'" Ak denote the

k eigenvalues of the matrix A. Then, the eigen­values of F are the k Z numbers

f.l .. 1 J

l-\Aj , i, j=l,Z, ... ,k

Proof: Let

Ax. 1... X. _1 1 _1

Ax. t...x. -J J -J

Sinee x. and x. are not trivial, the dyad x. x.' ~ 1 -J _1 -J

is not the zero matrix. Clearly,

F(x. x .') = X. x.' - 1...1... x . x! -1-J -1-J 1 J-1-J

(1 - 1..,1...) X. x.' . 1 J -1-J

Thus the numbers in (1I4 ) are all eigenvalues of Fand the remaining task is to prove that there are no others.

Suppose that f.l is an eigenvalue of Fand S an assoeiated eigenveetor. Then

S- ASA ' = I-lS or

Assume first that A is nonsingular. Then,

from whieh we deduee immediately

for any polynomial f(x). We maintain that the -1

matriees (l-I-l)A and A' must possess at least

203

one eommon eigenvalue. Suppose this is false.

Then it is easy':' to eonstruet a polynomial f(x) sueh that

','

We leave the eonstruetion to the reader.

f [( 1-1J.) A -1 ]

f(A ')

which when substituted into (!I7 ) yie1ds S = 0k'

a contradiction. It follows that for some A. and some 1..., 1

J ( -1 1-IJ.)A. = A.

1 J

.'.IJ. = 1-1...1.. .. 1 J

The case A = singular matrix is handled by a continuity argument, Q. E. D.

Corollary: The operator F is nonsingu­lar if and only if

\Aj F I, i, j=l,2, ... ,k.

In particular F is nonsingular if the eigenvalues of Aare all strictly less than one in m~griitude and Eq. (!I l ) ~>ossesses a unique solutIOn S for every matnx Q. An explicit and enlightening infinite series solution is available. Write

and iterate. t,

S = Q+ASA '

Thus, for any non-negative integer

S=Q +AQ A '+A2Q(A ,)2+ ... +A.1.,Q(A 1).1.,+ A.1.,+1S (A').1.,+1.

Since';<

limit A.1., + l S(A / ).1.,+1 .1., .... 00

00. .

S = L: A 1Q(A ,)1 i=O

If Q is non-negative definite, Q = G G' for some real matrix G. Substituting into (!I8),

00 . .

S = L: (A1 G)(A1 G) I

i=O

and being the sum of non-negative definite matri­ces is itself non-negative definite. In a special case we can actual1y conc1ude that S is positive definite. Let

S = limit Q. Q.' i."oo 1 1

Hence if rank Qk 1= k, Q Q' - k-1 k-1

::~

is positive

According to a standard matrix result, limit t .1., .... 00

A = 0k if and on1y if all eigenva1ues of Aare

strictly 1ess than one in modulus.

204

definite and S is nonsingular. In other words S is also posi~ive-definite~ To complete the proof of lemma 4 It must be shown that the eigenvalues of Aare necessarily all strictly less than one in magnitude if the solution S of (!I1) is symmetrie

pos~t~ve def~n~te for every choice of symmetrie posItive defInIte matrix Q. Let A 'x = AX .

~ultip1ying (!I l ) on the 1eft with x~< :nd o~ the nght wIth ~ we find -

(1 .. I 1..1 2) x* S x = x* Q x . - - - -Evidently x';< S x > 0 and x* Q x> 0 imply

I I.. 12< I, Q. E. D.

REFERENCES

1. V. Be1evitch, "Factorization of Scattering Matrices with Applications to Passive­Network Synthesis", Philips Res. Repts., Vol. 18, pp. 275-317; Reprint R 481.

2. D. C. Youla, "Cascade Synthesis of Passive n-Ports", Technical Report No. RADC-TDR-64-332, Rome Air Development Center, Griffiss Air Force Base, New York, August 1964.

3. D. Hazony, "Elements of Network Synthesis", Reinhold Publishing Corp. ,New York, 1963.

4. R. J. Duffin, D. Hazony, N. Morrison, "The Gyration Operator in Network Theory", Scientific Report No. 7, AF 19(628)1699, J anuary 1965, CRS TI, Sills Building, 5285 Port Royal Rd., Springfield, Virginia, 22151.

5. V. Belevitch, "Synthesis of Passive Electric Networks with N Terminal Pairs from a Pr&­scribed Scatte~ing Matrix" (In French), Anna1es des Te1ecommunications, Vol. 6 (August, 1951), pp. 302-312.

6. Y. Oono and K. Yasuura, "Synthesis of Finite Passive 2n-Terminal Networks with Prescribed Scattering Matrices", Mem. Kyushu Univ. (Engin. ), Vol. 14, 2(1954). Also (In French) in Ann. Telecommun., Vol. 9, (March, April, May 1954).

7. H. J. Carlin, "The Scattering Matrix in Net­work Theory", Transactions of the IRE, Vol. CT-3, 2 (June 1956), pp. 88-97.

8. B. Me Millan, "Introduction to Formal Reali­zability Theory", Bell System Technical Journal, Vol. 31 (1952), pp. 217-279,March, pp. 541-600, May.

9. R. E. KaIman, "Irreducible Realizations and the Degree of a Matrix of Rational Functions" Technical Report No. 64-5, March 1964, Research Institute for Advanced Studies, 7212 Bellona Ave., Baltimore, Maryland 21212.

:-:~

This explains why the matrix K in Eq. (10l) is symmetrie positive definite instead of just semi­definite.

10. D. C. Youla, "The Degree of a Rational Matrix", Notes for E.P. 732, 1965 (Spring Term); Po1ytechnic Institute of Brooklyn, E1ectrophysics Dept., Farmingda1e, N. Y.

11. D. C. You1a, "The Synthesis of Linear Dyna­mical Systems from Prescribed Weighting Patterns", PIB Report No. PIBMRI-1271-65, 1 June 1965, Polytechnic Institute of Brook-1yn, Brooklyn 1, N. Y.

12. F. R. Gantmacher, "The Theory of Matrices" Vol. 1, Chelsea Publishing Co., New York.

13. D. K. Faddeev and V. N. Faddeeva, "Compu­tationa1 Methods of Linear Algebra",W. H. Freeman, San Francisco, 1963.

14. B. D. H. Tellegen, "Theorie der Wisselstro­men", (Dee1 II!, "Theorie der E1ectrische Netwerken") P. Noordhoff, N. V. Groningen, Djakarta, pp. 166-168; 1952.

15. P. Slepian and L. Weinberg, "Synthesis Applications of Paramount and Dominant Matrices", Proceedings of the National E1ectronics Conference, Vol. XIV (October 1958), pp. 611-630.

16. F. T. Boesch, "On the Synthesis of Resistor N-Ports", Ph. D. Dissertation, June 1963, Po1ytechnic Institute of Brooklyn, Brook1yn I, N. Y.

17. R. E. KaIman, "On a New Characterization of Line ar Pas s ive Systems", Proceeding s of the First Allerton Conference on Circuit and System Theory, Univ. of Illinois, Nov. 1963.

18. D. C. You1a, "Weissfloch Equivalents for Los sIe s s 2n-Ports", Tr ans actions of the IRE, Vo1, CT-1,3 (Sept. 1960), pp. 193-199.

19. D. C. You1a and P. Tissi, "An Explicit For­mu1a for the Degree of a Rational Matrix", PIB Report No. PIBMRI-1273-65, 3 June 1965, Polytechnic Institute of Brooklyn, Brooklyn 1, N. Y.

20. B. L. Ho and R. E. Kalman, "Effective Con­struction of Linear State- Variable Models from Input-Output Data", presented at the Allerton Conference on Circuit and System Theory, October 20-22 1965, University of Illinois, Monticello, Illinois.

205

~I -

~I -- ( I)

... (2)

Frequency Insensitive

( n + k) - port

-E.2 -02 -

· · · ~

S(p)- [ SIl1'12] - SO" 'I 5 12 522

S(p) -

--- • • I So = So • (n) ... -

No

Fig. la The residual (ntk)-port Na"

,------ --I I

( I ) I I , I

(2) I No

-I :, ( n) ~

I

I 1 _____ -

Fig. 1 b The n-port N.

206

j

~ • • •

~

:+:

I I

L2 I ~N

Lk I I I

C, I .* C2 I • • •

* I

Ck2 I I

__ J

L = diag. [ LI. L2'. • • , L kl ] •

-I . [-I -I -I ] C = dia<;! CI' C2 • 0 •• , Ck2 •

I-I-=- Z- ~~ -, -7N:L'Sa1-i

I I Na ' (L, Sa ) I I I I

(a) (a )

I I I ~ I I Ideal

All - ideal I 2k l - port I ~ (I) : I Transformer I

Transformer • I I •

I (ntk)-port • bank I 1 •

I • • I (2) I I constituting

I I • • a fixe d ~a =PI ~a' I I minimal I I ~

I residual I I I realization

I I of S( p) I I S(p)- I I I I

I I I (b 1 (bl I I I I Ideal

I -* I I

2k2 - port I *' I I transformer I I • bank • I I • I (n) I I •

I • I I I I • • I

I I :tb = P2 :tb' ~ * I I I I

I _______ J I 1 ___________ .J

I PI LPI = L •

-I I -I P2 C P2 = C

Fig. 2 The generation of all residual (ntk)-ports (from a given one) for a prescribed symmetrie regular para-unitary matrix S(p).

207

LI

L2

Lk l

c,

C2

Ck 2

y (p I -:-. ~l-----"I H ""'........--. -----'J I F

\ Fig. 3 The one-port N.

( I )

In In ~

Fig. 4 The residual 3-port N • a

0---

( I )

0.9696mho O. 2023 mho

0.0837 mho ;;:j;:; 1.1451F

0.6965H 0.8303mho

Fig. 6 A transforrnerless one-port equivalent to N.

r----------------- i n : I I

I I

I I

Na I I I

I I

2

~ n3

.~

cL: (2)

I : I I

~ (3)

~ I L _________________ ~

Fig. 5 The 3-port Nal obtained by the insertion of ideal transformers at ports 2 and 3 of Na'

208

(2)

(3 )