R 895 Philips Res. Repts 30, 106*-121*, 1975Issue in honour of c. J. Bouwkamp

RECIPROCITY INVARIANTS INEQUIV ALENT NETWORKS

by V. BELEVITCH and Y. GENIN

MBLE Research LaboratoryBrussels, Belgium

(Received November 29, 1974)

AbstractThe invariance ofthe difference between the instantaneous magnetic andelectric energies in equivalent passive reciprocal networks leads to acharacterization of any real transient (even secular) as dominantlymagnetic or electric. Algebraic expressions for the associated invariantsare deduced both from the state-space and the scattering descrip-tions of the network. This yields bounds on the number of independentresistances in any realization. By contrast, a network having only com-plex eigenmodes is continuously transformable into its dual and alwaysadmits an antimetrie equivalent.

1. Introduetion

The equivalence problem for closed passive reciprocal networks has beentreated in a previous paper 1) and various canonic forms were obtained. Twoequivalent networks have the same determinant (the same eigenvalues) and thesame elementary divisors (imposing identical structures to eigenspaces associatedwith multiple eigenvalues). Moreover, reciprocity preservation forbids changingRL-circuits into RC-circuits, so that simple real eigenvalues are distinctlycharacterized as magnetic or electric.In this paper we establish a similar distinction for real secular modes, although

these necessarily involve all three kinds of elements (R, L, C). More precisely,for a fully defective real mode of order s, where every state is of the formf(t) exp (-cxo t), withf a polynomial of degree-s; s - 1, the Lagrangian (dif-ference between the instantaneous magnetic and electric energies) is ofthe form

L = T,1I - Te = get) exp (-2 CXo t)

where g is a polynomial of degree ::;;;s - 1 whose leading coefficient has aconstant sign, for all initial conditions exciting that secular mode alone, sothat the asymptotic decay of L is either dominantly magnetic or dominantlyelectric. Every real mode (secular or not) is thus characterized by an invariantreactivity index Cr = +1for a magnetic, 7: = -1 for an electric mode) whichwill be computed algebraically from the network description.

In secs 2 to 6 the state-space description is used. The relation between ree-

F= T-lHT (1)

RECIPROCITY !NVARIANTS IN EQUIVALENT NETWORKS 107*

iprocity preservation and Lagrangian invariance is discussed in sec. 2, supple-menting our fust treatment 1). This invariance essentially prevents a networkhaving real eigenvalues to be equivalent to its dual; by contrast, dual equiv-alenee is allowed for networks with only complex eigenvalues, as proved in sec. 3.Real secular modes are analyzed in sec. 4, and their energy behaviour in sec. 6.The reactivity indices are computed in sec. 5 from the state-space descriptionand in sec. 7 from the scattering description. In sec. 8 a lower bound is deducedfor the number of resistances in any realization of a prescribed transientbehaviour.

2. Lagrangian invariance

In state-space synthesis, a minimal realization of an n-port of degree m isobtained as a frequency-independent (n + m)-port closed on m unit reactances.If H is the hybrid matrix of the (n + m)-port, all equivalent minimal realiza-tions are deduced by the similarity transformation replacing H by

where T is an arbitrary non-singular matrix leaving invariant the n free ports.If, in addition, the initial realization is reciprocal, the matrix H is B-sym-

metric, i.e. satisfies

BH= H'B (2)

where B is diagonal of ± 1 entries. Reciprocity is preserved in (1) if one alsohas

(3)

for some matrix BI of the same type as B. From (1) to (3) one deduces

HU=UH (4)where

U= TBl T'B. (5)

Youla and Tissi 2) have proved algebraically that for minimal realizations theonly solution U of (4) is the unit matrix. This produces

(6)and

TB T' = T' BT = B. (7)

If x denotes the vector of independent variables at the ports of the (n + m)-port, (7) expresses the invariance of the quadratic form

L = x' ex (8)

in the transformation from x to T x. In (8), the terms corresponding to the

v. BELEVITCH AND Y. GENIN

n free ports are invariant anyway in case of identical excitation, and the remain-ing terms represent 2 (T", - Te). The invariance of Till - Te has been provedpreviously by Tellegen 3) and implies, in particular, the invariance of thereactive power in steady state.

For two non-minimal realizations of an n-port, even (1) does not hold ingeneral. In the case of reciprocal n-ports, however, the additional freedomproduced by non-minimality corresponds to internal states that are both inob-servable and uncontrollable from the ports, so that the invariance of Tm - Testill holds for zero-state equivalence, as mentioned by Tellegen on one example.In this paper we only deal with the equivalence problem for closed reciprocal

networks (without free ports). If the network is the m-port of hybrid matrix Hclosed on m reactances, the transients are ruled by the equation

dxHx+-=O.

dt(9)

If x is changed into T x in (9), premultiplication by T-1 of the result yieldsF x + dxjdt = 0 where F is (1), so that all equivalent networks are obtain-able by similarity. On the other hand, reciprocity preservation leads to (4) and(5), but it is no longer true that the only solution of (5) is U = I",. For instance,for T = I"" any diagonal matrix F = H satisfies (2) and (3) with 8 and 81separately arbitrary, so that (5) reduces to 818, an arbitrary matrix of ± entries.Consequently the invariance of (8) for equivalent closed networks, whereequivalence is defined as the preservation of the form (9) of the state equation,is not an algebraic consequence of reciprocity conservation. Indeed, an RL-network and an RC-network of identical degrees obey the same equation (9)but with different physical meanings of H and x, whereas (8) is positive in thefirst case and negative in the second. In conclusion, one must either accept thatan RL-circuit is equivalent to an RC-network, which seems violently un-physical, or strengthen the very concept of equivalence with respect to themere preservation of the form of (9).At this stage it is important to analyze the origin of the difference between

the case of equivalent minimal n-ports, where (7) is a consequence ofreciprocity,and the case of closed networks, where it is not. For n-ports, the possibility ofapplying forced states x exp (p t) for all p allows one to explore the internalbehaviour from the ports, whereas only free states at discrete frequencies existin closed networks and there is no possibility of analytical continuation. Physicalintuition thus suggests the possibility of proving (7), also for closed networks,provided some form of continuous perturbation is allowed. In other words, theconcept of a frozen network without interaction with its environment is anexcessive idealization in which some essential physical properties are lost.

Continuous perturbations can be introduced in a number of ways, and the

REÇIPROCITY INVARIANTS IN EQUIVALENT NETWORKS

choice is a matter of taste. One may couple very weakly all reactances of a closednetwork to some common port and apply continuous excitations from the port.Alternatively one may allow tolerances on the components and require reciproc-ity to be preserved for all reciprocal perturbations. The second approach isfollowed in the appendix, where (7) is thus established for closed networks.

3. Networks without real modes

If a matrix H has no real eigenvalues, its characteristic polynomial

is a product of quadratic factors that are sums of two squares of polynomials,and it is itself of the formf2(A) + g2(A). Since H satisfies its own characteristicequation, one has J2(H) + g2(H) = 0 for some polynomials f and g in H,with scalar coefficients. All powers of Hare 8-orthogonal with H and com-mute, so that the rational function r = fig is defined and the matrix Q = r(H)is also 8-orthogonal and commutes with H. One thus has

Q2 =-1".; 8Q = Q' 8; Q-l HQ = H. (10)Since the order of a matrix without real eigenvalues is even, we write m = 2n.

Moreover 4), one has tr 8 = O. We thus define the matrices

and one has

'P' 'P= 1".; 'P2 = -I".; 'P' 8'P = -8.

The partitions on the hybrid matrices induced by (ll) are

[RN]H- .- -N' G ' [ G N']lJI-1 H 'P =

-N R

and the second network is the twisted dual of the first one, since the transfor-mation of matrix lJI also interchanges inductances and capacitances owing tothe last equation (12). Since 'P is thus not 8-orthogonal, (8) is not invariant inthe transformation (13). We now prove that, if H has no real eigenvalues, thereexists a 8-orthogonal similarity transformation relating both matrices of (13).In fact, (7) and

T-l HT= lJI-l H'P

become identities for T = Q 'P, by (10) and (12).A network satisfying

lJI-1 HlJI = H,

109*

(ll)

(12)

(13)

(14)

110* V. BELEVITCH AND Y. GENIN

i.e. G = R, N = N', is antimetrie. Having proved that a network without realeigenvalues is reciprocally equivalent to its own twisted dual, i.e. that it ispotentially antimetrie, we now show that it can be made actually antimetrie,i.e. that there exists a 8-orthogonal transformation Tfrom H into (1), such that(14) holds for F. One must thus find a matrix T such that

IJl-I T-1 HTIJI = T-1 HT

and (7) hold. From (10) one deduces

(Q - j lm) (Q +j lm) = 0

(15)

(16)

so that Q has n eigenvalues +l and n eigenvalues -jo Moreover, the nullity ofeach factor of (16) is at least n, so that Q has exactly neigenvectors for eacheigenvalue and is non-defective. As a consequence 5), Q is 8-orthogonallysimilar to '1', so that

Q = TIJIT-1 (17)

holds for some T satisfying (7). This allows to replace T IJl by Q T in (15),which is thus established. In fact, antimetrie canonic forms have been obtainedin a previous paper 6).

4. Jordan canonic forms

The Jordan canonic formJ= T-1 HT (18)

of a general 8-orthogonal matrix is a direct sum of blocks Jk + Àk Ik, oneblock per independent eigenvector of H. Since J is not 8-symmetric, T is not8-orthogonal, but (2) imposes

V J = J' V (19)

whereV= T' 8T (20)

is non-singular symmetric. On the other hand, T is generally not unique sinceany transformation matrix A leaving J invariant, i.e. satisfying

JA=AJ (21)

can be incorporated into T, which becomes T J, and this changes (20) intoA' V A. With Ek denoting a matrix of order k of unit entries on the seconddiagonal, and calling E the direct sum of blocks Ek conformal to the partitionof J, one has

EJ =.T' E (22)

so that, if A satisfies (21),

RECIPROCITY INVARIANTS IN EQUIVALENT NETWORKS

V=EAsatisfies (19).The general solution of (21) is well known from the theory of commuting

matrices 7). If each Jordan block corresponds to a distinct eigenvalue, A is adirect sum of blocks Ab and each block is of the form

Correspondingly, (23) separates into symmetric blocks of the form

If two blocks J1 and Jk (possibly of different orders) have the same eigenvalue,the corresponding superblock of A (corresponding to the direct sum of J1 and Jk)

is of the form

where each submatrix is of the form (24), the off-diagonal submatrices beingcompleted by additional rows and columns of zeros when they are rectangular.When (26) is used to generate the corresponding superblock of V, by (23), thearbitrary off-diagonal parameters of (26) are further restricted by the sym-metry of V. The resulting form of the superblocks of V is illustrated by theexample

ao !ao al l bo

..~~ ~.~ ~~..L~? ~.~..bo ! Co

s; i, ~Co Cl

which occurs if a 5-tuple eigenvalue has a Jordan structure of orders 3 + 2.If V is (27), the off-diagonal matrices corresponding to Vb are reduced to

L.._ ~ ~ __ ~ _

111*

(23)

(24)

(25)

(26)

(27)

112* v. BELEVITCH AND Y. GENIN

zeros in the transform A' V A, where A is (26) with unit diagonal submatrices,with Aki = 0 and with A'k determined by

(28)

This diagonalization process for submatrices is exactly isomorphic to the clas-sical Gauss algorithm reducing scalar off-diagonal entries to zeros in a sym-metric matrix. The pivot subrnatrix, Va in (27), must be non-singular, and thisis automatically true if Va is the submatrix of largest order: since (27) is non-singular, one has ao =1= O. In the case of a partition into submatrices of iden-tical orders, both diagonal submatrices of V can be singular, but the off-diagonalsubrnatrix is then non-singular; as in the Gauss algorithm one then generatesnon-singular diagonal submatrices by an orthogonal transformation of matrix

1 [Ik Ik ]V2 Ik -Ik .

One thus obtains for Va direct sum of blocks (25), to be called Vk, of the sameorders as the Jordan blocks Jk, in all cases.The blocks Vk corresponding to real eigenvalues are real. The corresponding

subrnatrix Ak = Ek Vk of the form (24) has a k-tuple eigenvalue ao. Forao > 0, Ak has thus a unique real square root of the same form, and thetransform (A' k)-1/2 Vk Ak-1/2 is (A' k)-1/2 s, Ak 1/2 which is s, because everymatrix M of the form (24) satisfies M' Ek = Ek M. Consequently one canreduce Vk to Ok Ek, where Ok is the sign of the entry ao in Vk. For blocksassociated to complex eigenvalues Ak -1/2 can be complex, and one can alwaysachieve Vk = Ek· .In conclusion, we have proved that a B-symmetric matrix is similar to its

Jordan form by a transformation T satisfying (20), where V is a direct sum ofblocks Vk = Ek for complex eigenvalues and of blocks Vk = Ok Ek withOk = ± 1 for real eigenvalues*).

(29)

5. Invariance of the reactivity indicesWenowprovethatthenumbers Ok associated to real eigenvalues are invariant,

i.e. that the same set is deduced from any diagonalization process of V intoseparate blocks Vk• From (18) and (20) one deduces the identity

(30)

If (t, is a real eigenvalue of multiplicity s of H associated to a single eigen-vector, the principal value of CH +p Im)-l near p = -(t, is determined bythe partial fraction expansion

*) This lemma was used in the proof of theorem 5 of ref. 1 but the derogatory case was notanalyzed in detail. In the meantime the same lemma appeared in F. Uh l ig, Lin. Alg.Appl. 8, 351-354, 1974.

RECIPROCITY INVARIANTS IN EQUIVALENT NETWORKS 113*

(31)

where all matrices KI of order m are 8-symmetric. Correspondingly, the prin-cipal value of (J +p Im)-1 results from the expansion

1 J. J. 2 . J. .-1[J. ( + )1 ]-1' +' +( 1).-1_' __

s + al P. = P+ al - (p + al)2 (p + al)3 • . • - (p + al)'

(32)

of a single Jordan block. Since (32) is premultiplied in (30) by V. = 7: Es, onehas by comparison with T' 8 (31) 8,

(k = 0, 1, ... , s - 1). (33)

In particular, for k = s - 1, (33) has rank one, and

7: = (_1)'-1 sgn tr (8 K,). (34)

Since the sequence of matrices E. J.k of (33) written in reverse order(k = s - 1, s - 2, ... ) is

[ J, [ i ,]. L '] (35)

the successive ranks (signatures) are 1,2, 3, ... (1,0, 1, ... ) and the remainingrelations (33) fix the ranks and signatures of all symmetric matrices 8Kk fork ~s-2.The case where the eigenvalue al is associated to several eigenveetors will be

discussed on the basis of the example of two eigenveetors corresponding to twoJordan blocks of orders sand t.Assuming s ~ t, the expansion of (H +P Im)-lreplacing (31) thus stops with Kso whereas (32) must be replaced by the directsum of two expansions, so that (33) is replaced by

[

7: E J. k(-I)k 1;' (k = 0, 1, ... , s- 1). (36)

If s = t, (36) has rank two for k = s - 1 and 8K, is congruent to

(_1)'-1 [7:1 0 J.o 7:2

Consequently 7:1 and 7:2 are determined by the signature of 8 Kso within anirrelevant permutation. If s > t, one has J/-l = ° and i'l is determined bythe signature of 8K, in (36) written for k = s - 1. If k is then decreased, one

x=Ty

and (18), the equation (9) reduces to

(37)

114* v. BELEVITCH AND Y. GENIN

still has J,k = 0 in (36) as long as k > t - 1, so that the ranks and signaturesvary as in (35). For k = t - 1, i2 appears for the first time in (36) so that therank suddenly increases by 2 for k changing from t - 2 to t - 1, and thedirection of the deviation in the signature evolution from the pattern 1, 0, 1,0, ... determines i2• It is clear from this example that in all cases the reactivityindices il of all Jordans blocks associated to a multiple real eigenvalue are de-duced uniquely from the ranks and signatures of the matrices e KI arising inthe partial fraction expansion of the resolvent matrix e (H + p lm)-1.

6. The energy of a secular state

With

dyJy+-=O

dt(38)

and splits into separate equations in accordance with the partition of J. In thefollowing we thus only treat an isolated system corresponding to some realeigenvalue cts of order s associated to a single eigenvector, and thus discussthe equation

dy(Js + cts Is) y + - = °

dt(39)

whose solution is

y = exp (-cts t) exp (Js t) a (40)

where a is an arbitrary s-vector. In (40) one has

ts-1exp (Js f) = Is + t Js + ... + J;-1

(s - 1) I(41)

since higher powers of Js vanish.By (37) and (20), the Lagrangian (8) is L = y' V y and reduces to • y' Es y

for the mode (40). By (40) and (22), one has

L = i exp (-2 cts t) a' exp (J.' t) Es exp (Js t) a= i exp (-2 cts t) a' Es exp (2 Js t) a

s-1

L (2t)k= i exp (-2 cts t) Ck--

kl(42)

k=O

RECIPROCITY INVARIANTS IN EQUIVALENT NETWORKS 115*

where

The vector a can be noted

where A is (24) of order s. Since (41) is also the matrix

1 1 12/2

o 1

y = exp (-O(s t)P Au.

Since P A u is the last column of P A one finally has

a=

of order s, one has

(43)

oo

=Au (44)

o1

P=001

tS-l ts-2ao + al + ...(s- I)! (s- 2)!

(45)

On the other hand (43) is

whereCk = u' A' Es J/ A u = u' Es A2 J/ u = u' Es J/ B u (46)

is a matrix of type (24) with

Finally J,k B is a matrix of type (24) where the subscripts of the entries hl aredecreased by k, the negative subscripts being replaced by zeros. In (46), the

ho = ao2,hl = 2 ao al'h2 = 2ao a2 + a12,

(47)

116* V. BELEVITCH AND Y. GENIN

matrix Es Jsk B is then of the type (25) and the operation u' . . . u selects itslowest rightmost entry bs-k-1' Consequently one has Ck = bS-k-1 and (42)becomes

((2t)"-1 (2t)"- 2 )

L="t"exp(-2ast) bo--+bl--+···+bs-2t+bs-l·(s-I)! (s-2)!

For ao =1= 0, the dominant term of (45) contains ts-1 and so does the dom-inant term of (48) whose sign is "t" because bo is a square. For ao = 0, thedominant term of (45) is tS-2;but one then has bo= bs = ° by (47) and thedominant term in (48) is t":3 and has again the sign of "t" because its coeffi-cient b2 then reduces to a12 in (47). By induction, whenever the dominant termin the state is e:' (i = 1,2, ... , s), the dominant term inL is t,-2'+1 and hasthe sign of "t" for i:::;;; (s + 1)/2, but one has L = ° for i> (s + 1)/2. Con-sequently for any initial conditions exciting only the mode corresponding tothe Jordan block J, associated to a real eigenvalue, either L is identically zero,or L decays asymptotically to zero through positive (negative) values for"t" = +1 (= -1), so that the mode can be characterized as dominantly

(48)

magnetic (electric).

7. Scattering description

We now describe the closed network as a lossless r-port terminated on somenumber r of unit resistances and characterize the r-port by its symmetric para-unitary scattering matrix SCp), of degree m. If a (b) denotes the incident(reflected) wave vector, the network equation b = Sa for a forced stateexp (p t) can also be written a = s=' b or

a = S(-p)b (49)

and reduces to S(-p,) b = ° for a free state exp (p, t), hence in particular to

(50)

for a real mode exp (-ex, t). Moreover, since the scattering description andthe state-space description must yield the same eigenvalues and the same num-bers of independent eigenvectors, the matrices H + p, lm and S(-p,) musthave identical nullities. Since the nullity of S is at most its dimension r (thenumber of resistances), one has the restrietion

(51)

Consider again a forced state exp (p t) with p = ex+ JOJ. The reactivepower 2w (Tm - Te) = Im i v entering the r-port expressed in the vector b is

b [S(-p*) - S(-p)] bTm~Te= .

8jOJ(52)

(55)

RECIPROCITY INVARIANTS IN EQUIVALENT NETWORKS 117*

For a real state (co= 0, b real), (52) reduces by l'Höpital's rule to

For a simple real transient exp (-ex, t), the instantaneous Lagrangian is then

(53)

and its reactivity index is

dS"ti = sgnb'-b

dex,(54)

where b is the only solution of (50).For a network containing only one resistance, S reduces to a scalar all-pass

function scp), and (54) is the sign of ds/dex, at the positive real zero exlof scp).If sgn s(ex) is plotted versus ex,one thus has "t > 0 (r < 0) at a simple zerowhere the plot looks like fig. lA (fig. lB). If multiple real zeros occur, (51)with r = 1 forces each eigenvalue to be fully defective, so that multiple zeroscan be treated as arising from the confluence of simple zeros. The asymptoticbehaviour of the instantaneous Lagrangian corresponding to several distinctsimple modes is determined by the slowest decaying mode, i.e. by the smallest ex"and this remains true by continuity after confluence. Consequently for a zeroof odd order resulting from the confluence of a pattern ABA ... (BAB ... ),"t is postitive (negative) so that the rules of fig. lA (fig.lB) holdafterconfluencefor odd orders. For a zero of even order originating from AB ... (BA ... ) thesignature plot after confluence is fig. 1C (fig. ID) and "t is then positive (nega-tive). If the distinct eigenvalues are ordered such that

and if the multiplicity of ex,is called n" it results from the plot of sgn s(ex)inspected in the reverse direction, i.e. starting from

I+ +

IA B c o

Fig. 1. (A) zero of odd order, 1: > 0; (B) zero of odd order 1: < 0; (C) zero of even order1: > 0; (D) zero of even order 1: < O.

118* v. BELEVITCH AND Y. GENIN

s(oo) = (100 = ± I, (56)

that one has by recurrence

<I = (_l)n1-l (100;

and this is equivalent to

<, = (-I)nl "'-I (i = 2, 3, ... ,4) (57)

(58)

We now return to the general matrix case of (50) and (54) and assume thatS(a,) has some nullity n. Since S(a2) is symmetric it can be diagonalized byaconstant orthogonal transformation to be disregarded since it leaves invariantthe r resistances. After the transformation one has

[0 0 Jcn)

S(a,) = 0 X (r-n) (59)

and the solutions of (50) are the unit vectors along the first n coordinates, sothat only the principal submatrix [dSjda']n of order n of dSjdal is relevantin (53). Since (59) is still invariant by an arbitrary orthogonal transformationon the first n coordinates, one can also diagonalize [dSjda,]n which becomesdiag {Öl> 152, ••• , ön}. If all 15, =1= 0, the 11 solutions of (50) inserted in (54)produce n non-zero indices

<, = sgn 15, (60)

thus characterizing n distinct non-secular modes at a,. If the first k, say, 15,are zero, only n - k non-zero indices are produced in (60) and the first kvectors correspond to secular modes. One can then diagonalize the submatrix[d2Sjda,2]k by an additional orthogonal transformation which does not alterthe previous submatrices [S(a,)h and [dSjda,h since these are zero. Let thus[d2Sjda?h = diag {SI' S2' ..• , sd. If all s, =1= 0, one has separated the prob-lem into k distinct scalar secular problems of order 2, and the first k still missingindices of (60) are given, in accordance with (58) with 11, = 2, by

(61)

If some s, are still zero, one continues the process by considering successivederivatives of higher orders, until a complete set of n non-zero indices <, isobtained.

8. Bounds on the number of resistances

For a prescribed transient behaviour to be realizable by a network containingonly one resistance, it is necessary by (SI) that all multiple eigenvalues (if any)

(63)

RECIPROCITY INVARIANTS IN EQUIVALENT NETWORKS 119*

possess only one eigenvector. In addition, adjacent pairs of reactivity indicesmust comply with the second relation (57). The first relation (57) can alwaysbe satisfied, for it merely determines the sign of (56), i.e. the ambiguous signin the scalar reflectance

( )- ±g(-p)

sp - --g(p)

(62)

where g(P) is the specified network determinant. The conditions are thus alsosufficient and the state-space realization of (62) is known explicitly 8). Theresulting hybrid matrix is in fact a permutation and renormalization of a tri-diagonal matrix discovered by Bückner 9) and interpreted in network termsby Bashkow and Desoer 10). Bückner's matrix has been rediscovered by Puriand Weygandt 11). The recent discussion by Chen 12) is erroneous in the de-rogatory case. A slight variant is the Schwarz form discussed by Barnett andStorey 13). .

When the conditions of the previous paragraph are not met, more than oneresistance is required and we wish to find the minimum. We first consider onlythe set of real modes (55) and call a" b., c., dl the number of modes at ctlhaving the behaviour of fig. 1, A, B, C, D, respectively. The problem is toconstruct a scattering matrix of minimum dimension accepting the prescribedpattern for its real eigenvalues and is solved by following the signature of thematrix on the real p-axis, thus generalizing the similar procedure used in theOono-Yasuura synthesis 14). One must link together the prescribed number offigures ABC D at successive (not necessarily adjacent) ctl in the naturalorderso that the input and output ± signs are matched at every junction, whilekeeping minimum the total number of chains generated in the linking process.Let some number of chains be alreadyformed which have exhausted all modesfrom ctn to al+ 1 and let PI+ 1 (111+1) designate the number of positive (negative)signs appearing at the outputs of the PI+ 1+ 111+1 chains. In crossing a" al + dlpositive outputs sign are generated and at most bi + dl among the PI+l pastpositive output signs can be consumed, if PI+ 1> bi + d.; so that the numberof inherited positive signs is at least I(PI+ 1- bl- di), where

{X for x> 0

I(x) = o for x:::;;; o.Finally, one has

and a similar reasoning on the negative signs yields

III ,ç. bi + Cl +f(nl+1 - ai - Cl). (64)

120* V. BELEVITCH AND Y. GENIN

The number of chains produced at the end is Pl + nl' Since the right-handsides of (63) and (64) are monotone non-decreasing functions of PHl (nl+l),all inequalities operate in the same direction, so that any strategy producingequalities in (63) and (64) at every step is optimal. The simplest strategy thenstarts from an + b; + en + d; chains after exn and proceeds by consuming inan arbitrary way the largest possible number of past positive and negative signsat every new exl and by initiating new chains only when strictly necessary. Onethus constructs a diagonal scattering matrix of order r =Pl + nl' Complexpairs of conjugate eigenvalues can then be distributed arbitrarily among theexisting diagonal entries if (51) is satisfied, else the size of the diagonal scatteringmatrix must be further increased and is then determined by (51).

Appendix

We consider perturbations of the equivalent networks of hybrid matrices Hand F, whereas the reactances which are normalized to unit values remainunperturbed. We thus change H into H + (jH and (1) into F + (jF where(jF = T-l (jH T. By requiring F + (jF to remain reciprocal for arbitraryreciprocal perturbations of H one establishes (jH U = U (jH, where U is (5),for all (jH such that (J (jH = (jH' (J. By taking (jH diagonal one proves that Uis diagonal, and by taking (jH proportional to various permutation matricesone establishes the equality of all diagonal entries of U, so that one hasU = a 1m. We now set a = ± b2 and

(A. I)

so that (5) becomes

(A.2)

The transformation (A.l) from H to F thus appears as the product of Tl bythe scalar transformation l/b which cancels in (1) but changes x into xlb, sothat it must be interpreted as a change of the unit (watt)l/2 of measurement.By requiring both equivalent networks to be described in terms of the sameunit, one forces b = 1, hence T = Tl' On the other hand, by Sylvester's lawof inertia, (Jl in (A.2) is a permutation of ± (J, say ± P (J P', so that (A.2)reduces to

TP8P'T'= 8. (A.3)

If one now requires the transformation from H to F to depend continuouslyon some parameters, so that T changes gradually from lm to T, (A.3) must alsohold for T = lm so that one has P (JP' = e, and (A.3) reduces to (7). Wehave thus proved that (6) and (7) hold for continuously equivalent closedreciprocal networks, and the invariance of (8) for corresponding initial con-ditions (x changed into T x) results.

RECIPROCITY INVARIANTS IN EQUIVALENT NETWORKS 121*

REFERENCES1) V. Belevitch, Philips Res. Repts 29, 214-242, 1974.2) D. C. Youla and P. Tissi, IEEE Cony. Rec. 14, Part 7, 183-200, 1966.3) B. D. H. TeIIegen, Philips Res. Repts 7, 259-269, 1952.4) Ref. 1, theorem 6.5) Ref. 1, theorem 1.6) Ref. 1, eq. (50).7) F. R. Gantmacher, Matrizenrechnung, VDE Verlag, Berlin, 1958, vol. I, p. 204.8) Ref. 1, theorem 7.9) H. Bückner, Quart. Math. 10,205-213, 1952.

10) T. R. Bas hko w and C. A. Desoer, Quart. appl. Math. 14, 423-426, 1957.11) N. N. Puri and C. N. Weygandt, J. Frankl. Inst. 276, 365-384, 1963.12) C. F. Chen, Proc. 1974 IEEE Syrnp. Circuits and Systems, pp. 606-610.13) S. Barnett and C. Storey, Matrix methods in stability theory, Nelson, London, 1970.14) Y. Oono and K. Yasuura, Mem. Fac. Eng. Kyushu Univ. 14, 2, 125-177, 1954.