factorization of scattering matrices with … bound... · factorization of scattering matrices and...

VOL. 18 No. 4 AUGUST 1963

BOlTED BY THE RESEARCH LABORATORYOF N.V. PHILIPS' GLOEll..AMPBNFABRIBKEN, EINDHOVEN. NETHERLANDS

R 481 Philips Res. Repts 18, 275-317, 1963

FACTORIZATION OF SCATTERING MATRICES WITHAPPLICATIONS TO PASSIVE-NETWORK SYNTHESIS

by V. BELEVITCH

AbstractIt is proved that the product of two passive scattering matrices 'is apassive scattering matrix, and that the corresponding n-port is realizableby interconnecting the component /I-ports by gyrators. Conversely, thefactorization of scattering matrices (or reflection coefficients) is used asa synthesis technique. In particular, this simplifies the derivation ofDarlington's cascade synthesis for one-ports. The factorization techniquealso yields new results in synthesis. First, a cascade decomposition ofall-pass 2/1-ports into sections of degrees one and two is obtained. Thisresult is then used to establish a synthesis of the Darlington type for/I-ports.

1. Introduction

In the synthesis of one-ports there is a considerable difference in difficultybetween the non-dissipative and the dissipative case. For pure reaetanees theclassical synthesis procedures are based on successive series or shunt extractions,and the canonical realizations have a simple mathematical counterpart sincethey correspond to the expansion of a positive real function into partial, orcontinued, fractions. The elementary techniques generally fail for dissipativeone-ports, and synthesis is then achieved by various more complicated proce-dures due to Brune, Darlington, Bott and Duffin. These procedures are noteasily described, in pure algebraic language, as some expansion theorem fora positive real function.

For n-ports the situation is similar. In the non-dissipative case relativelystraightforward extensions of the techniques used for one-ports are sufficient.On the contrary, the difficulties increase considerably in the dissipative case.The analogue of Brune's synthesis has been achieved independently by severalmethods *), and it has been known for a long time that any passive n-port

*) See ref. 1 for a recent review of the subject.

276 V. BELEVITCH

(even non-reciprocal) is realizable as a non-dissipative 2n-port closed. on'n' re-sistances. Thetrue extension of Darlington's synthesis to rz-ports, i.e, a r~cursive'procedure by which the non-dissipative 2n-port could be realized as a cascadeof simple sections of some restricted types is, however, not yet available. Theproblem of cascade decomposition has not even been solved for the simplestclass of non-dissipative 2n-ports, the all-pass 2n-ports. In other words, theclassical realization, due to Bode, of an all-pass 2-port as a cascade of two typesof section only, has not yet been extended to all-pass 2n-ports.In this paper the synthesis of all-pass 2n-ports and the Darlington synthesis

(in the above sense) for non-reciprocal n-ports are achieved. Moreover, thenew method by which the problems are treated can be described in purelyalgebraic terms as a succession of operations on the network matrices. Themethod also permits a simplification ofthe proofs ofvarious knownprocedures,and seems promising for future progress. The method is essentiaIly based onthe fact that the product of two passive scattering matrices is a passive scatteringmatrix. This multiplicative property is proved in sec. 2 and given a networkinterpretation in sec. 3. These results are immediately translated into a synthesisprocedure if a factorization can be found for an arbitrary passive scatteringmatrix. The factorization is established in sec. 4 for sufficiently wide classes ofmatrices.Due to the non-commutativity ofmatrix multiplication, however, the product

of two symmetric matrices is in general asymmetrie; therefore, the mentionedsynthesis procedure is only simple in the field of non-reciprocal networks. More-over, since complete algebraic factorization into first-degree factors is onlypossible if functions or matrices with complex coefficients are accepted, 'simplealgebraic factorization theorems are only obtained for bounded matrices withcomplex coefficients. Therefore, the" factorization theorems can be imme-diately translated into synthesis procedures only if passive networks with'non-reciprocal and complex elements are accepted. Procedures for real networksare then deduced by temporarily accepting complex elements (imaginaryresistances) and then cancelling imaginaries by combining complex-conjugateelements in pairs. Similarly, procedures for reciprocal networks are obtainedby temporarily accepting gyrators and then cancelling in pairs gyrators ofopposite signs. The process is particularly clear when applied to the cascadesynthesis of all-pass 2-ports, as treated in sec. 5.A complex passive element (imaginary resistance, or frequency-independent

reactance, "definedby v .jXi) has first been introduced by Baum 2). The ideaof accepting imaginary resistances as intermediate tools was then used 3) toobtain a simple proof of Brune's synthesis for one-ports. It was shown that asimple continued-fraction expansion is sufficient for complex positive functions,and that the cancellation of complex-conjugate elements yielding Brune's net-work is simply obtained by a star-delta transformation in a ladder structure.with

FACTORIZATION OF SCATTERING MATRICES AND PASSIVE-NETWORK SYNTHESIS 277

complex parameters. This proves that the imaginary resistance is the, onlyadditional element required for the synthesis of complex passive one-ports.The same idea was used 1) to obtain a Brune synthesis for non-reciprocalcomplex passive n-ports, and the possibility of cancelling imaginaries wasestablished both for reciprocal and non-reciprocal n-ports. The complex idealtransformer, however, was used in ref. las a supplementary intermediate toolwithout duly explaining that such a device is a combination of imaginaryresistances (this is now established in Appendix I); as a consequence, also forn-ports no other complex element is needed.

The new approach to Darlington's synthesis is essentially based on the ideaof all-pass extraction, as will now be explained on the one-port case, treatedin sec. 6. If an impedance Z(P) takes the value Zo = Ro + jXo at po = ao + jwowith ao ~ 0, the impedance Z(P) - jXo (which is still passive, although notreal) takes the positive value Ro at po, so that its reflection coefficient sCp)withrespect to Ro vanishes, and an all-pass factor can be extracted from s(P) withoutaltering its passive character. The basic step in the procedure is thus the seriesextraction of an imaginary resistance, followed by the extraction of an all-pass.This can be followed by the series extraction of a second imaginary resistanceof arbitrary value. It is shown that if the second resistanceis chosen as -jXo,the combined extracted 2-port behaves as an all-pass between the complexconjugate terminations Zo" and Zo. One. thus obtains a procedure of theDarlington- type where an arbitrary complex passive one-port is realized as acascade ofmismatched complex all-pass 2-ports closed on a complex resistance.Darlington's synthesis for real one-ports is discussed in sec. 7. The successiveapplication of the complex procedure at two conjugate values of p yields thebasic real non-reciprocal section called "type E" by Youla 4). Finally, a combi-nation of two such sections with opposite gyrators yields the reciprocalDarlington section, as already established by Youla.

The approach followed for I-ports is extended to n-ports in the remainingsections. The cascade synthesis of all-pass 2n-ports is discussed in sec. 8.A Darlington-type synthesis is achieved in sec. 9 for complex n-ports and insec.IO for real n-ports. The case of real reciprocal n-ports is discussed in sec. 11.The, present paper is, in many respects, a continuation of ref. 1.

Similar notations and terminology are adopted; in particular, M' denotes thetranspose of a matrix M, whereas if is an abbreviation for M'* (conjugatetranspose). Matrices are systematically represented by capitals and vectors bysmall letters.

2. Positive matrices and bounded matricesThe impedance matrix Z(P) of a passive n-port is positive rèal, i.e. real for

real.values of p, and such that Z(p) +Z(p) is (not necessarily strictly) positivedefinite' (p.d.) for Re p ~ 0. The same property holds for admittance matrices.

S = (Z + In)-I(Z - In) = (Z - In)(Z + In)-1 ,and, conversely, one has

(1)

278 V. BELEVITCH

Furthermore, the impedance and admittance matrices of reciprocal n-ports aresymmetric.The scattering matrix of an n-port with respect to unit terminations is defined

by

Z = ,(In- S)-I(In + S) = (In + S) (In- S)-1 . (2)~ ~

For a passive n-port, Sis bounded*), i.e. such that In-SS (and also In-SS)is p.d. for Re p ~ 0, and real. For reciprocal n-ports, S is also symmetric.Independently of any physical interpretation, relations (1) and (2) transform

a positive matrix Z into a bounded matrix S, and vice versa. This also holdstrue for matrices with complex coefficients, provided the notation M be inter-preted as [M'(P)]*. The following Theorem has been proved by Oono andYasuura 6):, A matrix S is bounded if and only if it is (a) strictly Hurwitzian (i.e.analytic in Re p ~ ° including p = (0), (b) such that In - SS is (notnecessarily strictly) positive definite on the imaginary axis., Although the theorem has originally been stated for real bounded matrices,reality is not involved in the proof **).For a non-dissipative n-port one has Z(P) + Z'(-p) = ° and consequently

SCP)S'(-p) = In, for allp. With the notationM(-p) = M*(p)andM'(-p) ==M*'(p) = 4!(P), these conditions are written

Z +?' = 0,

S § = In.

(3)

(4)~

A matrix satisfying (4) is called para-unitary. Since S § and SS coincide onthe imaginary axis, condition (b) of the Oono-Yasuura theorem is satisfied,and a para-unitary matrix is bounded if it is strictly Hurwitzian. By analogy,a positive matrix satisfying (3) should be termed skew para-Hermitian. Whenthese concepts are extended to matrices with complex coefficients, the changeof pinto -p must be accompanied by conjugation of the coefficients; thedefinitions are thus M*(p) = [M(-p*)]* and M(P) = [M'(-p*)]*.

, ~

*) This is the term adopted in ref. 5. The fact that In-SS and In-SS are simultaneouslyp.d. is proved by expressing that Z+Z is p.d. and using the two forms of the secondmember of (2). The result is also physically obvious since one transforms In-SS intoIn-SS by reversing all gyrators, by replacing all imaginary resistances by their conjugatesand by considering the resulting expression at po* rather than at po.

**) The proof is summarized in sec. 6.4 of ref. 7. Th'e theorem is essentially Theorem 1 ofref. 6 When freed from its network interpretation.


It is well known that the sum of positive matrices is positive. Also the inverseof a non-singular positive matrix is positive. Similarly, we have the Theorem:

The product S = SIS~ oftwo bound~d matrices is bounded.Proof: Since In - S1S1 and In - S2S2 are p.d., so are both terms of

~ ~SI(ln - S2S2)SI + (In - SISI) = In - SS,

since the first one is a Hermitian transform of a positive definite matrix.To summarize: positive matrices can be combined by addition, and bounded

matrices by multiplication. The realor non-dissipative character is also pre-served in the combination. On the contrary, reciprocity is preserved in addition(or inversion) of impedance matrices but not in the multiplication of scatteringmatrices, since the product of two symmetric matrices is in general not sym-metric.

3. Network interpretation of the multiplication of scattering matrices

Most network-synthesis procedures are based on the additivity of impedanceand admittance matrices, whereas the possibility of multiplying scatteringmatrices does not seem to have been fully exploited. It has, however, occurredindirectly in at least two instances which will now be discussed and extended.

For scalars, the combination s = S1S2of two reflection coefficients is trans-lated by (1) and (2)·into the combination

:i

1 + Z1Z2z=---ZI +Z2

(5)

of the corresponding impedances. The combination (5) appears in the Bott-Duffin synthesis, where it is realized as balanced bridge, thus requiring twiceas many elements as the formula would suggest. This interpretation of the

z,

Z2Fig. 1. Realization of 8 = 8182.

essential step of the Bott-Duffin synthesis as a factorization of a reflectioncoefficient has been mentioned by Belevitch 8). On the other hand, a realizationusing each impedance only once is obtained by means of a circulator 3-port(fig. I). A wave of unit amplitude incoming at port 1 is circulated to port 3,where it is reflected by Z2 with the amplitude S2; the reflected wave S2 is cir-

280 V. BELEVITCH

culated to port 2 where it is reflected by ZI, thus multiplied by SI; finally, theamplitude SlS2 is circulated back to port 1; the resulting reflection at port 1is thus S = SlS2, so thatthe impedance seen from port 1 is that of (5). A classical

2

Fig. 2. Detailed schematic of fig. 1.

Z1

2'

realization of the circulator is shown in fig. 2, where the gyrator has the impe-dance matrix

Provided R be chosen as the normalizing resistance with respect to whichreflection coefficients are defined, the network of fig. 2 thus affords the realizationof the impedance (5). This realization is due to Hazony and Schott 9), but thesimple circulator interpretation is new. If Zl is disconnected in fig. 1, theremaining 2-port (circulator closed on Z2) has the scattering matrix

as is obvious from its reflection and transmission properties. We have thus ob-tained the following Theorem:

The one-port of reflection coefficient S = SlS2 is realizable as a 2-port ofscattering matrix (7) closed on a one-port of reflection coefficient SI.

The above result is immediately extended to x-ports: to produce the n-port ofscattering matrix S = SlS2 from two n-ports of matrices SI and S2, it is sufficientto combine separately each port of SI with the corresponding port of S2 by acirculator, thus forming the corresponding port of S. This shows physically thatSlS2 is bounded, or para-unitary, with SI and S2, and explains the non-conservation of reciprocity. It is again convenient to summarize the result inthe form of a Theorem:

The n-port of scattering matrix S= SlS2 is realizable as a 2n-port of scat-tering matrix

(-~ ~). (6)

(7)


1~) (8)

closed on an n-port of scattering matrix SI.Having thus interpreted the multiplicative property of reflection coefficients,

we relate it with the similar property of the transmission coefficients of constant-impedance 2-ports. Such a reciprocal 2-port has the scattering matrix

(~ ~) (9)

between matched terminations. It is well known that the cascade combinationof two 2-ports of this type is again a constant-impedance 2-port, and thetransmission coefficientss combine multiplicatively. If (9) is transformed bythe orthogonal matrix

(10)

it becomes

(11)

and this yields the canonic realization of the constant-impedance 2-port as alattice, or as a Jaumann structure, with dual arms, having the reflectioncoefficients sand +-S, When a lattice of impedances Zl, 1/Z1.iS cascaded witha lattice of impedances Z2, 1/Z2, the equivalent single lattice has arms ofimpedance (5) and its dual. On the other hand, the Bott-Duffin bridge alsorealizes simultaneously (5) and the dual impedance seen from the oppositediagonal *), so that the factorization of reflection coefficients on which theBott-Duffin synthesis is based is identical to the factorization of transmissioncoefficients required in the cascade synthesis of constant-impedance 2-ports.

*) The title of ref. 9 seems to imply that the original Bott-Duffin synthesis cannot be organizedinto a cascade realization without gyrators. If ideal transformers of unit ratio are ..allowed,the algebra of the Bott-Duffin expansion can, however, be interpreted as a true cascaderealization, simultaneously for Zand I/Z, since two dual impedances are both requiredand produced at each step. The unit-ratio transformers between the bridges correspondingto the successive steps are merely necessary to prevent internal parasitic loops. This showsthat the exponential increase of the number of elements in the Bott-Duffin synthesis is onlyrequired in order to dispense with unit-ratio transformers. Conversely, the main difficultyarising from the prohibition of transformers is not the unavailability of voltage or currentgain, but the impossibility of realizing simultaneously certain Kirchhoff constraints atvarious ports if the ports are not insulated from each other.In the extension to reciprocal n-ports, the analogue of (9) is a matrix with submatrices S

and S', and the back transformation by a direct sum of matrices of type (IO) does not leadto a form similar to (11) because the non-main-diagonal submatrices become ;I:(S-S') =p O.This explains why there is no general extension of the Bott-Duffin process to reciprocalu-ports.

282 V. BELEVITCH

4. Factorization theorems

The theorems of sec. 3lead to a synthesis procedure whenever Sis factorizable.In the scalar case, the following factorization Theorem is well known since it issimply a restatement of Bode's extraction of all-pass factors from a non-minimum-phase transfer function:

A rational passive reflection coefficient s(P) having a .zero po with Re po > 0can be factorized into s = SlS2, where

S2 = (p - po)/(P +po*) (12)

and Sl is a passive reflection coefficient; the degree of si is equal to the degreeof s, unless -po* is a pole of s, in which case the degree decreases by one unit.

The theorem is proved by noting that sand Sl have the same modulus on theimaginary axis and are simultaneously strictly Hurwitzian, and applying theOono- Yasuura theorem of sec. 2.

The extension of this theorem to matrices involves the concept of the degreeof a rational matrix, as introduced by Tellegen 10,12),Me Millan 11), and furtherdiscussed by Belevitch 7) and Youla 13). The degree of an n-port is the degreeof its impedance, admittance, or scattering matrix and corresponds to thenumber of independent initial conditions defining the transients in the n-portclosed on general-resistive terminations. When a rational matrix is expandedinto simple fractions with distinct poles, the degree of the matrix is the sumof the degrees of the terms. In the case of simple poles, the terms are of theform A/(P-po) where A is a constant residue matrix; the degree of such a termis the rank of A. In the case of multiple poles the evaluation of the degree ismore difficult but is always possible from the Smith-McMiIlan canonic formofthe matrix; This occurs in a particular instance of Appendix 11. Finally, oneshould note that the degree of a rational matrix is generally not equal to thedegree of its determinant *). It is, however, well known 6,13) that the degreeof a bounded para-unitary matrix is the degree of its determinant.

The natural matrix extension of the scalar factorization theorem based on(12) is a matrix factorization S = SlS2 where S2 is a bounded para-unitary ma-trix of degree one having (12) as determinant, so that the matrix factorizationis related to the scalar factorization of its determinant. It is proved in Appen-dix 11 that, except for a constant unitary multiplier which can be transferredarbitrarily from S2 to Si, a bounded para-unitary matrix of degree one isnecessarily of the form

*) Consider the following simple example of a passive 2-port formed by two separate impe-dances Zl = Lp + RI and Z2 = RI + (R~-R2)/Lp withL, R and Rl positive and Ri > R,so that Z2 contains a positive capacitance, The reflection coefficients with respect to RareSI = [P+(RI-R)/L]/[p+(RI+R)/L] and S2 = (RI-R)/SI(RI+R). The scattering matrixis diagonal, of elements SI and S2, and has a constant determinant, although the networkis obviously of degree 2.

1fS(po).u = o. (15)


where no= Re po > o,·andwhere u is a vector, which can be normalized toI,!r -...

;-.' uu = 1. ~ (14)

It is furthef proved that the residual matrix SI = S S21 remains bounded,and has}~ degree not exceeding the degree of S, if the parameters of (13) areso chtfsen that

. A non-zero solution vector of (15) can always be found if po is a zero of<ietS, but this is not necessary *). The values of Po at which (15) admits a non-trivial solution can be found by the following method: rewriting (15) as Su = vwith v = 0, one transforms the first equation into u = S-IV whence it appearsthat one may have a finite non-zero u for v= 0 and S-1= 00. As a conclusion,(15) requires Po to be a pole of S-I. Such a pole is not necessarily a zero ofdet S because of possible cancellations.The conditions under which a reduction of degree (at most by one unit since

(13) is of degree one) actually occurs will now be discussed. If the degree re-duetion for the matrix could be judged by the degree reduction in the determi-nant (this holds true in the case of a para-unitary matrix), one would expectthe reduction to take place if det S(po) = 0 and det S(-po*) = 00, i.e.det ~-1 (Po)= o. Since the condition det S(po) = 0 is already too strong and mustbe replaced by (15), one similarly expects the condition det §-l(PO) = 0 to beweak-ened into §-I(PO).v = 0 where v is some non-zero vector and, by analogy withthe preceding paragraph, this requires po to be a pole of §. But a degree reduc-tion in the determinant does not necessarily imply a degree reduction in thematrix, so that the true condition must be stronger than (15) and §-I(PO).v = o.It is proved in Appendix II that, if Po is a simple pole of § (this is of course the"general" case) and if §-1 is analytic **) atpo, the onlysupplementaryconditionis that the vector v of the last equation be identical to the vector u of (15); Thecondition is thus

(16)

The conditions of degree reduction at a multiple pole of §, or at a simplepole of ~coincidingwithapole of §-I,havenot been found and require a deeper

.) In the example of the preceding footnote, det S is constant and has no zeros, whereas (15)admits for instance the solution III = 1,112 = 0 at a zero of SI •

•• ) This prevents po from being simultaneously a pole of § and of §-l.and excludes such casesas the example of the footnote, where § and §-l are identical up to a constant factor.

284 V. BELEVITCH

investigation of the structure of the. Smith-McMillan canonic form of thematrix. There are, however, various particular situations (which can often bereached by choosing appropriately the terminating resistances with respect towhich the scattering matrix is defined) where the general conditions of degreereduction can be stated without reference to the Smith-Mclvlillan form, even inthe case of multiple poles. As discussed in Appendix 11, these situations corre-spond to the case where one ofthe conditions (15) or (16) becomes redundant.This occurs if all elements of S or §-1 vanish at po, for one of the conditions isthen satisfied with an arbitrary u; this also occurs if S is para-unitary, since (15)and (16) become identical when S= §-1. The resulting conditions are more pre-cisely stated in the following Theorem, which also summarizes the earlier state-ments *):

A rational bounded matrix Ssatisfying (15) at somepo such that ao = Re po> °witha vector u normalized by (14) can be factorized into S = SIS2, where S2is the para-unitary bounded matrix (13) of degree one and SI is a bounded ma-trix such that deg SI ::::;;;deg S; the degree inequality is strict if one ofthefollow-ing additional sets of conditions is satisfied:

(a) po is a simple pole of §, §-l is analytic atpo, and u satisfies (16),

(b) S-l(PO) = 0,

(c) S(po) = 0, §-l is analytic at po, and u satisfies (16),

(d) S is para-unitary.

5. All-pass 2-ports

A real reciprocal all-pass 2-port has a scattering matrix (normalized to a realresistance R) of the form (9) and the transmission coefficient sCP) is a para-unitary strictly Hurwitzian real function, thus of the form

(17)

where g(P) is a Hurwitz polynomial and where g* denotes g(-p); the factoriza-tion of g(P) into first- and second-degree factors then corresponds to thecanonical realization due to Bode 16). On the other hand, the all-pass can be.realized as a whole in the form of a Jaumann structure with dual reactances.

*) Since the scalar factorization theorem corresponds to (5) when interpreted in terms ofimpedances, and is thus related with Richard's theorem, it is interesting to see the relationof the matrix factorization theorem with Bayard's extension 14) of Richard'stheorem toreciprocal n-ports ; if R = Z(ao), the matrix Zl = R{aoR- pZ)-l(aoZ - pR) is positivereal and symmetric with Z. Just as in the one-port case, this theorem is equivalent to thefactorization S = S1S2, where Sand S1 are the scattering matrices of Sand Z1 withrespect to R, while S2 is the scalar all-pass factor (ao-p)/(ao+p). Such a: factorization,however, requires ao to be a zero of all elements of S, thus a zero of order /I of det S,and extracts an all-pass of degree 11 without. generally lowering by /I the degree of theresidual matrix. The statement of Hazony and Nain 16) is simply a rewording of Bayard'stheorem.

(18)


A real transmission coefficient (17) satisfies the condition s(P) s(-p) = 1,expressing that the matrix (9) is para-unitary. For complex reciprocal all-passesthis condition IS still written

provided s* be interpreted as [s(-p*)]* in accordance with the convention ofsec. 2. The general form of a complex transmission coefficient is still (17) butg(P) is now a polynomial with complex coefficients having zeros in the lefthalf-plane. The Jaumann realization still holds true, but the dual impedancescontain imaginary resistances.

A non-reciprocal all-pass has a scattering matrix of the form

S~) (19)

where the transmission coefficients SI and S2 in either direction are different,but both of the form (17). The realization with the help of a 4-port circulator(fig. 3) is immediate: by elementary signal-flow analysis on the wave vectors,the transmission coefficient from 1 to 2 is the reflection, coefficient S2 at port 3 ;similarly, the transmission from 2 to 1 is related to the reflection at port 4.Since a circulator 4-port is realizable with one gyrator 17), any non-reciprocalall-pass 2-port is realizable. with at most one gyrator, in accordance with ageneral theorem of Oono-Yasuura 6) for non-dissipative 2-ports. In the parti-cular case of the scattering matrix (7) where the transmission coefficient in onedirection is unity, ZI Of fig. 3 is a short-circuit, and the circulator reducesto a 3-port circulator, as used in figs 1 and 2.

A cascade decomposition is obtained by factorizing the denominators of gl(P)and g2(P) of SI and S2. In the real reciprocal case, g(P) is factorized into first- andsecond-degree polynomials, this corresponding to a factorization of s(P) intofactors I

2

Z2Fig. 3. Realization of a non-reciprocal all-pass 2-port.

286 V. BELEVITCH

ao-pao +p (20)

or(ao - p)2 + W02

(ao + p)2 + W02(21)

with ao > 0 in either case. The corresponding elementary sections in Jaumannform are shown in figs 4 and 5, respectively.

1 2

2ROCOIl'o1----......-------o2'Fig. 4. Real reciprocal all-pass of degree 2.

I R<OC02+(02)1'O-------~-...-------02'

Fig. 5. Real reciprocal all-pass of degree 4.

In the complex reciprocal case, the elementary factors are of the first degree, _of the form (17), thus

ao - jwo - pao +jwo + p

(22)

with ao > O. The realization is shown in fig. 6, which reduces to fig. 4 forwo = O.Finally, the real section of fig. 5 is equivalent to a cascade connectionof two complex-conjugate sections of the type of fig, 6..In the non-reciprocal case, it is sufficient to consider-sections with a matrix

of the form (7), for (19) is immediately realizable as a 2-port with SI = 1

PACfORIZATION OF SCATTERING MATRICES AND PASSIVE-NETWORK SYNTHESIS 287

l' 2'Fig. 6. Complex reciprocal all-pass of degree 2. The circle is used as a symbol for an imaginaryresistance.

-1

2

followed by a 2-port with S2 = 1. In the real case, the elementary factors areof the form (20) or (21), and the realization by means of a circulator 3-portyields the sections of figs 7 and 8. In the complex case, the elementary factorsare of the form (22), and the realization is shown in fig. 9. The particular caseWo = 0 corresponds to the previous realization of fig. 7, whereas in the caseao = 0, corresponding to s = -1, the network of fig. 9 reduces to the gyratoralone (giving no phase shift in one direction and 1800 in the other).

0--- ---.0 2

1 0---1"---022OCo

l' 2'Fig. 7. Real non-reciprocalall-pass of degree 1.

Fig. 8. Real non-reciprocalall-pass of degree 2.

As a result of the above discussion, the section of fig. 4 is equivalent to thecascade of fig. 7 and of its transpose (direction of the gyrator reversed), whereasthe section of fig. 5 is equivalent to a cascade of four sections according tofig. 9 (the section, its transpose, and their two conjugates, in an arbitrary order).It is only in the domain of complex non-reciprocal networks that the fullfactorization into first-degree 2-ports of a single type is reached.Numerous variants are available for the section of fig. 9 and its particular

cases. For instance, by changing S2 into -S2 (which changes Z2 into its dual)one obtains the all-pass of fig. 10. .

288 V. BELEVITCH

Consider an arbitrary point po = aO+jwo of the right half-plane (ao > 0)and the corresponding value of a positive function Z(P), which is not a purereactance:

Fig. 9. Complex non-reciprocalall-pass of degree 1.

6. Darlington synthesis for complex one-ports

Fig. 10. Dual of fig. 9.

The positive function .

Z(po) = Zo = Ro + jXo , Ro > O. (23)

Za(P) = Z(P)- jXo (24)

takes the value Ro af Po, so that the corresponding reflection coefficient sCP)vanishes at this point. One may thus factorize s into SlS2: and extract the complexall-pass function (12). The first theorem of sec. 4 guarantees that Sl is also thereflection coefficient of a positive function, to be called"Zl(p): As explained insec. 3, Za is the input impedance of a non-reciprocal all-pass of transmission. . .coefficients S2(P) and 1, closed on Zi. Since (12) is the negative of (22) with woreplaced by -wo, the all-pass is the, one of fig.. 10, with the same substitution.Finally, Z.is realized by insertingjXo in series at the input. The result is shownin 'fig. 11.

We have thus proved that the non-dissipative 2-port of fig. 11can be extractedfrom any positive function Z(P) at an arbitrary point po = ao;+ jwo of theright half-plane. The degree of the residual impedance Zl after the extraction.is identical to the degree of Z unless the degree of si is reduced by one unit,compared with the degree of s. This only occurs if s has a pole at p =-po* == -ao +jwo. By the definition of the reflection coefficient this means

and, by (24),Za (-ao + jwo) = -Ro (25)

Z(-ao + jwo) =-Ro + jXo. (26)

2'Fig. 11. Illustrating the synthesis process for complex one-ports.

Comparison of (23) and the conjugate of (26) finally shows that a reductionof degree by one unit occurs if Po is a root of . .

Equation (27) then also has a root at -po*. Only half the number of rootsof (27) are, however, acceptable, for one must have Re Po > 0.After the extraction of fig. 11, the remaining roots of (27) are those of the

similar expression in Zl, i.e. Zl(P) + Zl*(P) =,0, as established in Appendix Ill.By applying the process of fig. 11 successively to all roots of (27) one obtainsa residual impedance whose degree is reduced by one unit at each step, and whichstill satisfies (27) at the remaining zeros ofthe right half-plane. After all zeros inthe right half-plane of Z + Z* have been used, the residual impedance Z; issuch that Z; + Zr* is a constant. Such an impedance is a constant (complex)resistance in series with a real reactance. One thus has realized an arbitrarycomplex impedance Z(P) as the input impedance of a (complex) non-dissipative2-port closed on one real resistance.

In the beginning of this section we have assumed ao> 0. For ao = ° theprocess breaks down, since (12) reduces to unity, and there is no all-pass

. extraction. The limiting case of ao and.Ro vanishing together is, however, ofinterest, for the element values in fig. 11 only involve Ro/ao. Since, for ao small,Z(po) =jXo + aoLo, where J;.oand jXo are the values' of dZ/dp and Z at jwo,and since I/Lois real as the residue at jwo of the positive function (Z-jXO)-l,Ro/ao tends to Lo. On the other hand, for Ro :_ ° the gyrator disappears andandthe 2-port is' reciprocal but complex. .In the network of fig. 11 one is free to insert an arbitrary constant imaginary

resistancejàr at the output ofthe all-pass and substract it from thetermination,thus changing Zl(p) into

jXo

Z(p)"~ Z 1 (p)

(27)

(28)

290 V. BELEVITCH

which is still a positive function ofthe same degree as Zl(p) .We will now showthat the particular choice Xl = -Xo is most appropriate for the simplificationof further computations and has an interesting theoreticai justification.

The impedance matrix of the complete extracted section is

p- jwo + ao )p - jw'o + jaOXI/ Ro

Ro (p - jwo ~ jaoXo/Ro,ao \ p- JWo- ao

and its chain matrix (A,B,C,D) is

1 (p-jwo +jaoXo/Ro j(p- jwo) (Xo +Xl) + ao(Ro- XOXl/RO)). (30)p-ao-jwo\ ao/Ro p- jwo +jaoXI/Ro

(29)

It is obvious that a simplification can only occur in the element B of (30),which falls to degree zero for Xo +Xl = O.The simplified chain matrix, cor-responding to the 2-port of fig. 12, is

1 (P - jwo + jaoXo/Ro ao(Ro + Xo2/Ro) ) (31)p- aojwo ao/Ro p- jwo- jaoXo/Ro .

If the network of fig. 12 is closed on Re + jXo at its output, the all-pass partis terminated on Ro, which is reproduced at the input; the impedance seen from

jXo -jXo

OC'o

Fig. 12. Complex all-pass extracted in the synthesis process.

the input of the complete section is thus Ro +jXo. Conversely, if the networkis closed on Ro - jXo at the input, the same impedance is seen from the output.The impedances Zo and Zo" are thus the iterative impedances of the completesection. If the section is closed on the conjugate impedances (Zo* at the inputand Zo at the output) conjugate matching occurs at both ends and the trans-mission coefficients are the ones of the all-pass part. As a consequence, thescattering matrix of the complete section operating between Zo* and Zo is (7)with s given by (12). This assumes that the scattering matrix is normalized with

woRo + aoXon= .

woRo - aoXo

The chain matrix of the network of the second extraction is obtained byreplacing wo, Xo and Ro in (31) by -wo, -nXo and nRo, respectively. Theproduct of the two chain matrices turns out to be real. After a few simplifica-tions resulting from (34), the product matrix becomes

(

P2 + (a02+ w02)/n [w02(n- 1)2 J)-:----:--::-1 pao(n+ l)Ro a02(n+ 1)2+ 1 • (35)(p-ao)2 + w02

aop(n+1)/Ron p2 + (a02 + w02)n

(34)

FACTORIZATION OF SCATIERING,MATRICES,AND PASSIVE-NETWORK SYNTHESIS 291

respect to complex terminations in accordance with Youla's definition 18),which is further discussed in Appendix IV. Anyway, the complete sectionappears as an all-pass with complex-conjugate reference impedances.A last remark on the extraction process may be of some interest. If the 2-port

of fig. 12 is closed not on Zf(P) but on a constant complex impedanceequalto Zf(PO), the input impedance will not be Z, but an approximate impedanceZapp (certainly equal to Zo at po). 'The 2-port of fig. 12 closed on a constantcomplex impedance gives a complex impedance of the first degree. Such animpedance is ofthe form (ap+b)/(cp+d) and contains 3 complex, that is 6 realparameters. It is easily shown that the impedance,Zapp is completely definedby the 3 complex conditions expressing that Zapp and dZapp/dp coincide withZand dZ/dp at Po, and that Zapp*(-po*) is -Z(po).

7. Darlington synthesis for real one-portsFor real one-ports, a Darlington-type. synthesis yielding a non-reciprocal

non-dissipative real 2-port closed on a resistance is easily deduced from thecomplex process of sec. 6. First, if po is real (= ao) in the process of sec. 6,one has Xo= 0 for a real one-port, and the network of fig. 12 reduces to areal all-pass. We now show that if Po is complex the process can be repeatedat the conjugate point, and that the combination of the two complex sectionsextracted according to the process of sec. 6 is, a real section of degree two.If Z(P) is real, at aojwo it takes the value Ro- jXo, so that Za is Ro - 2jXoand its reflection coefficient is s = - jXo/(Ro - jXo). Since (12) takes the value-jwo/(aojwo), one has successively, at aojwo,

(ao - jwo)XoSl = (Ro _ jXo)wo ' (32)

1+ Sl Rowo + aoXo 2XowoRoZl = Ro-- = Ro - j ------ , (33)

1 - Sl Rowo - aoXo Rowo - aoXo

Zf = Zl +jXo = neRo - jXo),with

292 V. BELEVITCH

The.corresponding impedance matrix, decomposed into partial fractions, is

2n~o. ( 0n + 1-1

1) nRop (1'o + (n.+ l)ao 1

1) + nRo(ao2 + wo2)'(1/n1 (n + 1) aop 1

:), (36)

and yields the canonic 2-port of fig. 13 (E-section) with the positive elementvalues

nRo woRo + aoXoL= =-----

(n + l)ao 2aowo

c = en + l)ao(ao2 + wo2)Ro

(37)

2aowo(38)

(ao2 + wo2) (woRo - aoXo) ,

2nRo woRo + aoXoR= --=----n + 1 wo'

(39)

and n given by (34). By' incorporating an ideal transformer of ratio n/1 incascade at the output, one obtains the E-section of fig. 14, already derivedby Youla 4) and Hazony 19) by different methods *).

Fig. 13. Type-E section. Fig. 14. Equivalent form of fig. 13.

For Ro = 0 (this can only occur for ao = 0 for a positive function), thegyrator disappears in fig. 14 and the network reduces to Brune's section. Theelement values (34, 37, 38) then become apparently undeterminate, but thedifficulty disappears by replacing Ro/ao by Lo as in sec. 6.

For Ro =1= 0, the section of fig. 14is non-reciprocal. The process can, however,be iterated at the same pair of conjugate points ao ± jwo but with reversedgyrators (interchange of s 'and 1 in the extracted all-passes); the resultingcombined structure is then reciprocal and has the form of Darlington's section.

*) The positiveness of the elements (37-39) constitutes an independent proof of the inequalityRo/ao > IXo/wol for ao > 0, an inequality usually deduced from Schwarz's lemma. Itsextension to /I-ports is discussed in ref. 20.


This has been proved by Youla 4) but is more easily established by the followingargument. First, the determinant of the chain matrix (31) is found to be thereciprocal of (12), so that the determinant of (35) is the reciprocal of (21).Next, formulas (34, 37-39) show that the reversal of the gyrator correspondsto the change of ao and Ro into -ao and -Ro; the change of Ro does notaffect the determinant whereas the change of ao transforms it into its reciprocal.The determinant of the combined 4-section structure is thus unity, so that thesection is reciprocal. Each component section (fig. 12) is of degree one: it iseasily checked that its impedance, admittance and scattering matrices havedeterminants of degree one. The combined structure is thus of degree 4. Finally,Darlington's section is simply the lossless reciprocal 2-port of degree 4 havingthe maximum number of independent parameters, as shown by Tellegen *).As a result of the above discussion, a Darlington-type structure can be

extracted from anyone-port at an arbitrary pair of conjugate points ao ± jwo,but in general this gives no reduction of degree in the residual one-port. Areduction of degree by 2 (one at each point) only occurs if condition (27) issatisfied. For a real one-port this condition becomes Z(P) + Z(-p) = O.Complex zeros ofthe even part of Z are in fact associated in tetrads ± ao ± jwo,but only two of them are in the right half-plane, and have to be used twice,while the reduction of degree occurs only once.The case of a pair of imaginary zeros, at ± jwo, necessarily corresponds to

Ro = 0 for a real one-port; this is the case of Brune's section discussed above.The case of a real zero at ao corresponds to an all-pass in the process of sec. 6.The reduction of degree is by one unit, but the section extracted in sec. 6 innon-reciprocal. To arrive to a reciprocal section (the so-called modified Brune-type-C section), one must repeat the process at the same point (ao) with areversed gyrator; the second step gives no reduction of degree. Generally, thenon-reciprocal all-passes at the two steps have not the same image impedance,so that their combination is not an all-pass, but a generallossIess reciprocal2-port of degree 2. Here again, Brune's section is merely the general form ofsuch a 2-port. For a classical treatment of the subject we refer to ref. 21.

Sa)o ' (40)

8. All-pass 2n-ports

In this section we extend to 2n-ports some of the results of sec. 5 for 2-ports.An all-pass 2n-port has a scattering matrix of the form

where Sa and Sb are both bounded para-unitary. Moreover, one has Sb = Sa'if the 2n-port is reciprocal, but even then Sa and Sb need not be symmetric,so that the corresponding impedance matrices Za and Zb are generally not*) Ref. 10, p. 180.

294 v. BELEVITCH

symmetric and there is no canonic realization by a generalized lattice or aJaumann structure. On the contrary, in the non-reciprocal case the extensionof the realization of fig. 3 (set of n 4-port circ~lators closed on the n-ports ofmatrices Za and Zb) is immediate, but wasteful in gyrators, for Za and Zbrequire themselves gyrators in their realization.

Following the second part of sec. 5, we therefore investigate the possibilityof a cascade synthesis by sections of degree one for complex all-pass 2n-ports,and then discuss the possibility of cancelling imaginaries in the case of real2n-ports, and of cancelling gyrators in the case of reciprocal 2n-ports. In orderto obtain decomposability theorems it is sufficient to deal with mathematicalproperties of various scattering matrices, so that existence theorems for varioussections of degrees 1, 2 and 4 will first be obtained, whereas the discussion dealingwith their most economical realization (for real n-ports only) is postponed tillthe end.

In the non-reciprocal case it is sufficient to deal with scattering matrices (40)with Sa = In, since a cascade combination with an all-pass having a matrixof the transpose form yields the general form (40). One is thus led to considermatrices of the form (8), and a factorization of the sub matrix Sb (replacing S2)is interpreted as a cascade decomposition of the all-pass 2n-port in two 2n-portsof the same type. Since the submatrix is bounded para-unitary, the matrixfactorization theorem of sec. 4, c~se (d) applies to Sb at each zero of its deter-minant and involves a reduction of degree by one unit at each step. Afactorization of Sb into bounded para-unitary matrix factors of degreeone is thus achieved, except for a constant unitary matrix multiplier. As aconsequence, a complex all-pass 2n-port is a cascade of first-degree sectionshaving scattering matrices of the form (8) or its transpose, with S2 given by (13),followed by a constant lossless 2n-port.

In the case of a real all-pass 2n-port, det Sb may have real zeros and pairsof complex-conjugate zeros. In the case of a real zero ao, the vector u satisfying(15) is also real if Sb is a real matrix, and (13) reduces to

2aouu'S2 = In- ---,

p + ao(41)

which is a real matrix, so that the residue factor is also a real matrix and theprocess can be iterated. Consider now the case of a pair of complex-conjugatezeros po and po*. Having extracted a factor S2 having a zero at po by writingSb = S1S2, we further extract a factor S3 at po* by writing S1 = S3S4. Theresulting factorization is Sb = S3(S4S2). It is proved in Appendix V that if Sis real, so is S4S2, and, by consequence, S3. The proof is based on the effectivecomputation of

(42)

4aoM = [G- (p+ao)F] ,

a02 + w02- a02 t t* .(43)

FAl-'TORIZATION OF SCATTERING MATRICES AND PASSIVE-NETWORK SYNTHESIS 295

where the real matrix M is obtained as

with the notations

F= (a02 + W02)A- a02 C + aOwO D,

G = wo (a02 + W02) B + aO (woG + aoD),

t = u'u,

(44)

(45)

(46)

where A, B, C, D are defined by the real and imaginary parts of

u u = A +jB, t* u u' = C+jD. (47)

As a consequence, every real all-pass 2n-port is realizable as a cascade of realnon-reciprocal sections of degree one or two, of scattering matrix (8) or itstranspose, where S2 is (41) or (42), respectively, followed by a constant reallossless 2n-port. . '

In the case of real reciprocal all-pass 2n-ports, a cascade realization by sectionsof degrees 2 and 4 is immediately deduced by applying step by step the factoriza-tion process ofthe non-reciprocal case to Sb of(40), the transposed factorizationto Sa = Sb', and by combining the factors at every step. The cascade connectionof a section having the scattering matrix (8), where S2 is (41) or (42), with asection having the transposed matrix, yields a reciprocal all-pass of degree 2 or 4since its scattering matrix is para-unitary and symmetric. After each step ofsymmetric extraction the residual matrix is again symmetric so that the processcan be iterated.

Before discussing the realization of the various sections, it is convenient, forfurther reference, to compute the chain matrix of the complex section of degreeone. The terminal impedances with respect to which the scattering matrix (8,13)is defined, must, however, first be specified. In order to be as general as possible,while still keeping zero transmission loss, we consider matched conjugate-trans-pose terminations characterized by a gener~l impedance matrix Zo (not neces-sarily real nor symmetric) at the output and Zo at the input. We further introducethe notations -

Ro = (Zo + Zo)/2,

h = aoiR01,

k = Zs h,

(48)

(49)

(50)

where R01 is the Hermitian square root 22) of RO-l which can be computedas explained in Appendix IV. It is proved in Appendix VI that the chainmatrix corresponding to (8,13) is

296 V. BELEVITCH

_1_ ((p- po)l~ + kp- po h h

, kk )(p- po)ln + hk . (51)

It should also be remarked that, with the notations (48-50), the normalizationcondition (14) is replaced by

or

~ ~kh + hk = 2ao,

h Ro hlao = 1.

(52)

(53)

In the case of a real all-pass of degree one (po, thus u, real) working betweenreal (not necessarily reciprocal) terminations (Zo at the output, Zo' at the input),hand k are real by (49-50), and (51-53) reduce to

_1_ ((P - ao)ln+k h'p- ao hh'

kk' )(p - ao) In + h k' ,

(54)

k' h = ao, (55)

(56)h' Ro hlao = 1.

The network is thus specified by a total of 2n independent parameters: ao,the n components of one of the vectors, say k, and the n-l components ofthe other vector h, since one parameter disappears due to the constraint (55),A most economical realization is shown in :fig.IS, where the 2-port representedas a black box is detailed in :fig.16; the proof is given in Appendix VII. Therepresentation of :fig. 15 is drawn for the case n = 3, but its generalization isobvious. The input ports are la, 2a, 3a and the output ports h, 2b, 3b. Thetransformer ratios are related to the components kt, and Iu of the vectors kand h. There is a total of 2n-2 ratios; the additional two elements of :fig. 16make the total of 2n, equal to the number of parameters. The realization of:fig.16 assumes hl =1= 0, but this is not a serious restrietion since another choicecould have been made, and h has at least one non-zero component since it isrelated with u =1= 0 by (49). .

No economic realization has yet been obtairîêd for the non-reciprocal sectionof degree 2, nor for the reciprocal section of degree 4. A most economic realiza-tion for the reciprocal section of degree 2 is obtained later as a particular caseof a more general result (sec. 11 and end of Appendix X).

9. Darlington synthesi~ for complex n':'ports

As mentioned in the introduction, the particular case of Brune's synthesis hasbeen treated in ref. 1 for complex n-ports (reciprocal or not). The furtherdevelopments ofthe present paper are an extension of ref. 1, in the same way asthe contents of secs 6 and 7 extend ref. 3. We assume that the n-port is de:fined

(57)

FAç:TORIZATION OF SCATTERING MATRICES AND PASSIVE-NETWORK SYNTHESIS 297

Fig. 15. Canonic form of various all-pass 211-port Fig. 16.Part Q of fig. 15producing a non-sections. reciprocal all-pass 2n-port of degree 1.

by its impedance matrix Z which is not identically singular, for in degeneratecases preliminary reductions by ideal transformers can be made.We consider an arbitrary point Po = ao +jwo with ao > 0, and separate Z(po)

into its Hermitian and skew-Hermitian parts by (23) with Ro and Xo Hermitian.Moreover, we assume that Rojs strictly p.d., since it cannot be singular in theright half-plane unless Z + Z is identically singular; Z is then resistance-reduced, and the Brune process of ref. I or simple gyrator extraction applies. Wenext define the positive matrix *) Za by (24). In accordance with Youla'sdefinition (see Appendix IV, eq~103), the scattering matrix S of Za, withrespect to the Hermitian termination characterized by the matrix Ro, is given by

S = Ro-t (Za - Ro) (Za + RO)-1 Rot .

By (23) and (24), Za - Ro, thus S, vanishes at po. In accordance with the secondtheorem of sec. 4, we thus factorize S into S_1S2, where SI is bounded ..and whereS2 is still given by (13) with u arbitrary, since S vanishes identically at po.

*) In our previous treatment 1) of Brune synthesis, instead of extracting jXo, a matrix jH, . of rank one was extracted in sucha wayso as to make the skew-Hermitian part of Z(P)- jH

singular at Po. It has now been found that this complication was not necessary, and Hmay be replaced by Xo in all results of ref.I.'

298. V. BELEVITCH

As in sec. 3, the factorization is interpreted as a realization of Za as the inputimpedance matrix of an all-pass 2n-port of scattering matrix (8) closed on ann-port of scattering matrix SI, all scattering matrices being defined with respectto the matrix Rs: Finally, by (24), the basic step in the synthesis procedure isthe series extraction of a complex resistance n-port of impedance matrix jXofollowed by the extraction of a complex all-pass 2n-port of degree one.If a degree reduction in the factorization is desired, the second theorem of

sec.4, case (c) imposes to choose Poso that(16)is satisfied for some vector uandso that §-1 is analytic atpo. By (57) and (23-24), one has

§-1 = RÖ(? - ZO)-l (? + Zo)Rë( . (58)

Ifpo is not a pole of(?...,- ZO)-l, condition (16), with the notation (49), becomes(Z+Zo)h = 0 and shows thatpo must be chosen sothat the equa~ion

(59)

admits a non-zero solution h at po.This happens whenPo is a zero of det (Z+?),a condition generalizing (27), but is not necessary as mentioned in sec. 4: it issufficient that Po be a pole of (Z + Z)-l. Since .-(? - ZO)-l = [In - 2(? + Zo)-lRo]-l t? + ZO)-l

reduces to Ro-1/2 at a pole of (Z+ZO)-l; (58) is automatically analytic at thepoles po of (?+Z)-l. The extraction processbased on the poles of (?"+Z)-lgives a complex iterative realization procedure for a general passive impedancematrix. The process terminates when (?"+Z)-l has no more poles in the righthalf-plane. Since -po* is a pole of (?+Z)-l with po, the only remaiEing poles. are located on the imaginary axis. On this axis ?+Z coincides with Z+Z, andZ is then resistance-reduced. If (-?"+Z)-l has no poles at all, not even on theimaginary axis, it is a constant matrix, and Z is then the sum of a constant re-sistance matrix and a lossless impedance matrix.As in the one-port case, the extracted section, formed by the series n-port

of impedance matrix jXo followed by the all-pass of scattering matrix (8)where S2 is (13), can be completed at the output by an arbitrary series n-portof matrixjX1 withX1Hermitian. Here again, it is convenient to choose Xl =-Xoin order to simplify the results as much as possible. The chain matrix of theoverall section will now be computed by multiplying the individual chainmatrices. The chain matrix of a series n-port of impedance matrix jX is

whereas the chain matrix of the all-pass part is (51) with Zo replaced by Roin (50). By multiplication of the three component matrices, the overall chainmatrix is found as (51) with the original notation (50), so that the overall

z =h' k,

t = (1- zz*/popo*)-l.

(66)

(67)


section is an all-pass 2n-port having the complex reference impedances Zó at,the input and Zo at the output. This generalizes the result obtained in the 2-portcase of sec. 6.The chain matrix (51) involves the parameters po, hand k. As a solution of

(59), h is only determined up to a complex scalar factor, and k, computedby (50), contains the same factor. The modulus of this factor is determinedby the constraint (52), whereas its phase remains arbitrary, but cancels in allelements of (51).

10. Darlington synthesis for real n-portsIf po is real, thus wo = 0, the vector h solution of (59) is real, and so are Zo

and the value (50) of k. The extracted section thus reduces to a real all-passof degree one, as was the case in one-port synthesis.In the case of a complex po, the residual impedance matrix Zr, after the

extraction of the complex 2n-port of chain matrix (51), is computed by

Zf = (Ze- A)-l (B- Z D), (60)

where A, B, e, D are the submatrices of (51) and Z is the initially prescribed :matrix. The extraction process is then repeated on Zf at the conjugate point po*.The chain matrix of the second extracted 2n-port is of the form (51), with po,hand k replaced by po*, ha and ka. The vector ha is the solution of

[Zf(PO*)+ Zf(PO*)] ha = 0,

playing the role of (59), whereas the vector ka is defined by

(61)

ka = Za ha, (62)

similar to (50), where Za denotes Zf(PO*). Final1y, ha must be normalized bya condition similar to (52), i.e. by'

~ ~ha ka + ka ha = 2ao.

It is shown in Appendix VIII that the solution vectors are given by

ha = t(h* - z*h/po*),

ka = t(k* - z*k/po*),

(63)

(64)

(65)where

By combining the two sections, that is multiplying their chain matrices, oneobtains a chain matrix of the form

__ 1_ (AN(P - ao)2 + w02 Cs

(68)

300 V. BELEVITCH

withAN = [(p-ao)2 + wo2]ln + (p-ao)AI + AD,

B.N = (p-aO)BI + BD,

CN = (p-aO)CI + Co,

DN = [(p-ao)2 + wo2]ln + (p-ao)DI + Do,

(69)

where the coefficient matrices are defined by

Al =2tRekha' =2tRekah',

~Dl = Al = 2t Re h ka' = 2t Re ha k',

(70)

AD= -2wo t Im k ha',

Bi = 2t Re ka k" = 2t Re k ka',

BD =- 2wot Im k ka',

Cl = 2t Re h ha' = 2t Re ha h',

Co = -2wo t Im h ha',

Do = -2wo t Im h ka',

as proved in Appendix VIII; all expressions are real.We did not find an economical realization of this section. The total number

of parameters is 4n, which is twice the number of parameters of the sectionof degree one: in addition to aDand wo, each of the complex vectors hand kcontains 2n real parameters, but hand k contain one arbitrary common phasefactor (which does not affect the results) whereas one common real factor isdetermined by (52). An economical realization should use one inductance, onecapacitance and 4n-2 frequency-independent elements. By analogy with the caseof degree one, one expects that a realization with one gyrator and 4n-3 turnratios is possible.We now discuss a number of particular cases of (68). First the case wo= 0

is excluded, for the first section is then real and the second extraction is notneeded; (64) yields, indeed, ha = 0 for h real. Secondly, in the case of a 2-port(n = 1), (68) should reduce to the chain matrix (35) of the E-type section;this is checked in Appendix IX. The cases po = 0 and po = co are trivial, fora series or shunt extraction of a reactance n-port from Z(P) then produces adegree reduction.It remains to discuss the case aD= 0 (with wo =ft. 0 and finite), where Ro

may be singular. For aD= 0, (59) reduces to Roh = 0 and (50) yields

k =jXoh. (71)

~hLoh = 1, (72)

FACTORIZATION OF SCAITERlNG MATRICES AND PASSIVE-NETWORK SYNTIiESIS 301

On the otlier hand, (53) becomes indeterminate, but the true value is, byl'Hopital's rule,

where Lo is dZ/dp at jwo, a Hermitian p.d. matrix. The only differences withthe general case are thus the replacement of (53) by (72) and the replacementof no by 0 in (68-69). Equation (66) simplifies to z = jh' Xoh and, since h con-tains an arbirtary phase factor, the normalization can be chosen so that h' Xohis real. With the abbreviation

.:\= h' Xeh] wo, (73)(67), (64) and (65) become

t = (1- .:\2)4,

ha = t(h* - M); ka = t(k* - .:\k),

(74)

(75)

but this does not simplify appreciably (70) and (69).An important simplification occurs, however, in the case of a reciprocal

n-port where Ro, and consequently the solution h of Roh = 0, is real; on theother hand, k = jXoh is purely imaginary and will be denoted by jg with

g =Xoh. (76)

By (75), ha = t(I- .:\)h is real and ka = is» imaginary with ga = -t(I + .:\)g.As a result, one has Al = Bs = Co = Dl = 0 in (70), and (68-69) give thechain matrix

1 ((P2 + wo2)In- 2wogh'/(I + .:\) 2p gg'/(I- .:\) ) (. 77)

p2 + wo2 2p hh'/(l + .:\) (P2 + wo2) In+ 2wohg'/(I-.:\)

This is identical with the chain matrix of the reciprocal Brune sectiondescribed in ref. 1,*).A new variant of the realization is obtained in Appendix VIIas a network of the same general form as fig. 15 (with kt replaced by gt) andwith the 2-port having the form of the Brune section of fig. 17, with the elementvalues

1 + .:\ 2 (1 - .:\)hI2L = -- C = ----,----

2hl2 ' wo2(1 + .:\)2 '

2glhln=I----:-:-

wo(I + .:\)

(78)

(79)

*) On the contrary, the chain matrix of the non-reciprocal Brune section obtained in ref. 1is different from (68-69) with ao = 0, even after the modification mentioned in the precedingfootnote. This is due to the fact that a value of Xl different from -Xo was adopted. Alossless non-reciprocal section of a given degree is only defined up to a cascade connectionwith an arbitrary lossless section of degree zero. In the reciprocal case, the lossless networkof degree zero merely contains ideal transformers, whereas gyrators mayalso occur in thenon-reciprocal case.

302 v. BELEVITCH

Fig. 17. Part Q of fig. 15 producing a reciprocal all-pass 211-port of degree 2.

11. Darlington synthesis for real reciprocal n-ports

'It is well known 6) that a real reciprocal n-port is realizable as a real Iosslessreciprocal 2n-port closed on n resistors. The process of sec. 10, when appliedto a reciprocal n-port, yields a realization involving a lossless non-reciprocal2n-port closed on n resistors. It remains to see whether the process can bemodified, by introducing excess extractions without degree reduction, in sucha way that each non-reciprocal section of degree one or two is followed by asimilar section with reversed gyrators, thus producing reciprocal sections ofdegrees two and four, as in the scalar process of sec. 7. We have not been ableto prove the reciprocity for the case of degree four; the proof for the case ofdegree two will now be given.If Z(P) is symmetric and if (59) has a real zero at ao, h is real and Zo = Ro

is real and symmetric. The section of chain matrix (54) having been extracted,we repeat the extraction at the same point on the residual impedance matrix (60)where A, B, C, D are the subrnatrices of (54). Replacing k by its value (50)one obtains

z, = (Z - Zo)hh' _ ln)-l (Zo- Z)hh'Zo' _ z) .p- ao \ p- ao .

For p = ao the resulting value is, by l'Hopital's rule,

Za = Zf(PO) = (In- Lo hh')-l (Zo + Lohh'Zo/).

(80)

(81)

In the particular case under discussion this reduces to

Za = (In- Lohh')-l (In+ Lohh')Ro. (82)

The second extraction at po will generally produce no degree reduction, for (82)does not satisfy det(Za + Za) = 0 at po, so that it can be based on somearbitrary vector ha. It is proved in Appendix X that if one chooses

ha = [(1 - ),,)/(1 + )..)]t h (83)

with)..= h'Lo h (84)

(86)

,.

FACTORIZATION OF SCATIERING MATIUCES AND PASSIVE-NETWORK SYNTHESIS . 303

and factorizes S into S2S1 rather than into S1S2 as was done in sec. 4 (inorder to reverse the gyrator), one obtains à second section which combineswith the first one into a reciprocal section of degree two of chain matrix

1 (P2_ ao2)ln + 2aokh'j(1 + À)p2 _ ao2 2p hh' j(1 + À)

2p kk'j(l- À) ) (. 85)

(p2_ ao2)ln + 2ahk'j(l- À)

This matrix is very similar to (77) and the realization of Appendix VII appliesimmediately. One obtains the network of fig. 15where the 2-port is fig. 17 withthe element values' .

(2k1hl )-1

n = 1 - ao(1+ À) (87)

This result extends the Brune section of type C to n-ports.

12. Concluding remarksSeveral questions raised in this paper have not been solved. First, economical

realizations have not been obtained for the basic 2n-ports of degree two, neitherfor the all-pass of scattering submatrix (42), nor for the non-reciprocal Darling-ton section of chain matrix (68), not even in the Brune case (ao = 0). Secondly,in the Darlington synthesis of reciprocal n-ports the possibility of cancellinggyrators in the combination of two sections of degree two into one reciprocalsection of degree four has not been proved and, consequently, the section ofdegree four has not been realized. In both problems the difficulty is merelydue to the heavy algebraic computations, but no questions of principle areinvolved.Various attempts have been made to use the possibility of extracting a second

section without degree reduction, also in the case of non-reciprocal n-ports, withthe idea that a cancellation of gyrators at every extraction would yield a syn-thesis procedure realizing an arbitrary non-reciprocal n-port as a losslessreciprocal 2n-port closed on a non-reciprocal n-port of degree zero. If thiswere true, a non-~eciprocal n-port would be realizable with nj2 gyrators atmost. Since the minimum number of gyrators seems to be related to the rankof Z- Z', by analogy with the minimum number of resistors which is the rankof Z+Z, as mentioned by Oono 23), such a conjecture appears natural. It seems,however, impossible to obtain a cancellation of gyrators, even in the simplestcase of an extraction at ao.In the case of lossless non-reciprocal n-ports, it is known 6) that the minimum

number of gyrators is at most n-l, and this figure coincides with nj2 for n=2.Although the conjecture thus seemsmore likely in this case, and although much

304 V. BELEVITCH

more freedom is available for the extraction, since (59) is identically satisfied forall po and h, all attempts to attain even partial success have failed.

Acknowledgement

A part of this paper was the subject of a colloquium at the PolytechnicInstitute of Brooklyn, on February 20,1962; the writer is grateful to H. J. Carlinand D. C. Youla for interesting discussions. Further work has been discussedwith B. D. H. Tellegen, D.'C. Youla and R. W. Newcomb, whose criticisms onpreliminary versions of the paper have been of great help.

MBLE Research Laboratory Brussels, June 1963

Appendix J. Complex ideal transformers

A complex ideal transformer, defined by

V2 = n VI, Î! =-n* ie, (88)

can be realized by means of imaginary resistances and real passive elements.To show this we first remark that a complex ratio can be obtained from itsreal and imaginary parts by parallel-connecting the inputs and series-connectingthe outputs, so that it is sufficient to realize the purely imaginary transformer

V2 =jn VI, Î! =jn i2. (89)

Elementary network analysis shows that (89) are the equations of the 2-portof fig. 18 if n = -XjR and if the ideal gyrator has the impedance matrix (6).Another realization of a complex ideal transformer, containing two gyrators.was described by Carlin 17).

jX j X2

l' 2'Fig. 18. Realization of an imaginary transformer.

By combining in series and parallel a number of real and imaginary trans-former 2-ports, one easily realizes a complex ideal transformer n-port defined by

Vb= NVa, ia =-Rib, (90)

where N is an arbitrary rectangular matrix of complex constants and where thevectors Va, ia are the electrical variables at some set of ports, while the vectors Vb,ib are the variables at the remaining ports.


Appendix 11. Proof of the matrix factorization theorem of sec. 4In the factorization S = S1S2 one can incorporate an arbitrary constant

unitary factor B in SI and the inverse matrix B-1 in S2. We first show that,except for this arbitrary factor, a para-unitary matrix of degree one is necessarilyof the form (13). Since the determinant of a para-unitary matrix is para-unitary 6), det S2must be equal to ± (12). Since the matrix S2 is of degree one,the common denominator of its elements is of degree one and thus equal top + po*. Since S2 can have no pole at infinity, its numerator S2(P + po*) isa polynomial matrix of degree one which can be written -2aoA + B(p + po*).Since for p = 00, S2 reduces to B, B must be unitary and one can set B = In,for an arbitrary unitary factor can be incorporated in SI. Finally, one has

2aoAS2 = ln-. (91)

p + po*

Expressing that (91) is para-unitary, one obtains

A(p- aojwo) + A(-p- ao + jwo) + 2ao A A = o. (92)

The cancellation of the coefficient of p forces A to be Hermitian and the con-dition then reduces to A2 = A, so that A is idempotent. Moreover, since S2must be of degree one, A must have rank one. A Hermitian matrix A of rankone is of the form k u U, where k is a real constant and u a constant vector,which may be normalized by (14). Finally, the idempotency requires k = 1,and the form (13) is established. At po, (13) reduces to In - U U and, since(In - U u)u = 0, (15) is satisfied. This proves that u is some solution of (15)_This solution is unique if S has rank n - 1 at po.We next prove that SI = S S2-1 is bounded with S. Since S2 is para-unitary,

one has

(93)

To prove that S is bounded, it is sufficient to check the conditions (a) and (b)of the theorem of sec. 2. Condition (b) is automatically satisfied since one hasS2S2 = In on the imaginary axis, so that In - S105'lcoincides there withIn - ss. As regards condition Cb), the expression

(2ao U~)

SI = S S2-1 = S In + --p-po

(94)

shows that the only possible pol.eof SI in the right half-plane is po. Since Sisstrictly Hurwitzian, thus analytic at Po, its Taylor expansion near po is of theform P + (p- po)Q + ... , where P = S(po) satisfies (15), i.e., Pu = O. Dueto this condition, the coefficient of the only term in (P - PO)-1 of (94) vanishes,and SI is analytic, even at po.Wenowprovetheproperty degS'[ ~ deg S. In the case ofa bounded matrixS

306 v. BELEVlTCH

(without poles at infinity) the degreeis computed in the following way 7.11.13):

let S = Nlg, where g is the least common denominator of the elements of S,.and N the matrix of the numerators; let E be the Smith canonic form of N(E is a diagonal matrix, and one has N =P E Q with Pand Q unimodular, i.e.polynomial matrices with constant determinants; each element e( divides thenext et+1) and reduce the elements of Elg to their lowest terms edg = fiJgtto obtain the so-called McMillan canonic form (where each.fi divides the nextand each gt divides the preceding); the degree of S is the sum of the degreesof the gr.By (94) one has S; = N1/g(P - po) where N1 = N[(P - po)ln + 2aoü~]. By

(15) one has N(po).u = 0, for the denominator g of S is a Hurwitz polynomialand cannot vanish at po, since Re Po> O. One thus has N1(P~) = O. This provesthat all elements of N1 are divi.sibleby p - po, so that all elements ofthe Smithform of N1 contain this factor, which cancels with the common denominatorg(p - po) in the McMillan form of Sl. Since N1 is a right multiple of N, theelements of its Smith form are multiples of the et. Passing from the Smith formof N1 to the McMillan form of Si, every cancellation occurring in S will thusoccur in SI, since the common denominator has become g after cancellationof p - po. The denominators of the McMillan form of Ss. are thus at most thedenominators of the McMillan form of S. This proves deg S: ~ deg S.

We now prove that one has a strict inequality for the degrees under the condi-tion (b) of the last theorem of sec. 4, which is equivalent to S-l(-PO*) = O.This makes p + po* a factor of all denominators in the McMillan form of S.Since a cancellation by p +po* occurs in det Sl = det Sldet S2, the cancellation'must take place in some element of the McMillan form of Si, and this reducesthe degree of the corresponding denominator and therefore the degree of thematrix.

Case (c) results from case (b) by replacing Sby s=, and this changes (15) into(16). But S is strictly Hurwitzian, thus analytic at po, and this property was usedin the proof. In order to legitimate the change from case (b) to case (c), one mustexplicitly require §-1 to be analytic atpo.

Case (d) is trivial. Since the determinant of a bounded para-unitary matrixis a bounded para-unitary function 6.13), it has a set of zeros Pt, all hi the openright half-plane, and a corresponding set of poles -Pt*. Since the degree of apara-unitary matrix is the degree of its determinant, a reduction of degreeoccurs whenever po is one of the zeros Pt of det S, and no supplementary con-dition is required.

The last step is to establish the conditions of degree reduction covered bycase (a) of the last theorem of sec. 4. Since in (94), Ss: is analytic at po, Ss. and Shave the same poles and the principal value of Ss. near anypole can be com-puted by (94) from the principal value of S. In the statement of case (a), itis assumed that -po* is a simple pole of S; let K be the residue matrix at this


pole; the residue matrix of SI at -po* is then' K(In-uu), and a degree reductionoccurs if the rank of this matrix is smaller than the rank of K. The conjugatetranspose of the rank inequality is

~ ~rank (In - uu)K < rank K,

which means that a vector x =1= 0 must exist such that

(in-uu)Kx = 0 (95)with

Kx=y=l=O. (96)

This requirement is satisfied if y = u, for (96) premultiplied. by In ~ U~ yields(95), thanks to (14). The requirement of degree reduction is then Kx = u withx =1= O. Since ~ ~

K = [(P*+po)S(p)]P=-P~ = [(Po - p)§(P)]P=Pothe requirement becomes

[(po - p)§(P)]p=Po x = u (97)

with x =1= o. If §-1 is analytic at po, (97) premultiplied by §-I(PO)reduces to (16).

Appendix ID. Proof of the invariance of the roots of (27) in the process of sec. 6

Consider the expression

Z12 Z21Zl = ----- Z22,

Z11-Z

where the Ztj are the elements of the impedance matrix of the extracted 2-portof fig. 11. Thanks to the property Z1.j*=- Zjt which holds by (3) for a non-dissipative network, one obtains

(98)

At Po, Z21 vanishes and Z reduces to Z11 which is non-dissipative, so that bothfactors in the denominator vanish. But since po was also a zero of Z + Z*,this zero is finally cancelled in (98). This also occurs for -po*, where Z12,Z* + Z, Z* + Z11 and Z11 - Z all vanish. As a result two zeros ± ao + jwoof Z + Z* are suppressed in Zi + ZI*. On the other hand, degree considera-tions easily show that no zeros have been added. With the restrietion to zerosin the right half-plane, the zeros of ZI + ZI* are the ones of Z + Z*, exceptthe zero Po at which the extraction has been made.

Appendix IV. Scattering matrix with complex terminations

Youla's definiti0I?- 18) ofthe scattering matrix ofa general n-port of impedance

308 V. BELEVITCH

matrix Z with respect to à strictly passive termination characterized by amatrix Z, (not necessarily diagonal, nor reciprocal, nor real), such thatRi = (Z, + Zt)/2 is strictly p.d. can be justified in the <following way. LetZ, = Rt +JXt be the decomposition of Zi in its Hermitian and skew-Hermitianparts, so that both Rt and Xt are Hermitian. Substracting the lossless part jXtfrom the termination and incorporating it in the n-port, which becomes Z +jX«;does not change the active-power distribution. It is therefore equivalent to finda definition for the scattering matrix of the n-port Z + jXt closed on Rt. If Rtwere diagonal, one would normalize the n-port impedance matrix by the trans-formation Rt-l ... Rt-l and apply (1). Since Ri is generally not diagonal, onemust first apply a diagonalizing transformation. The closed system is againunchanged if an arbitrary complex transformer 2n-port of matrix N, followedby the inverse transformer 2n-port, is inserted between the n-port and the ter-minatien (fig. 19). Since Nis lossless, it is equivalent, as regards active-powerdistribution, to cut the system at point B of fig. 19, rather than at point A,thus considering the n-port N-l(Z + jXt)N-l closed on N-IRtN-l. On theother hand, a linear transformation on Zin (1) only induces the same transfor-mation on S if the transformation preserves the unit matrix, i.e. if it is unitary.

A BRt - N: 1n 1n:N - Z+jXt-

Fig. 19. Illustrating thedefinition ofthe scattering matrix with respect to complex terminations.

As a consequence, the classical normalization and the definition (1) can onlybe used at point B of fig. 19 if N is unitary. Since Ri is strictly p.d., a unitary Nexists which makes N-IRtN-l = Ll diagonal with positive elements. Thenormalization at point B then replaces N-l(Z +jXt)N-l = N(Z + jXt)N by

Ll-l N(Z + jXt)NLl-!,

and the normalized impedance matrix at point A is

N Ll-l N(Z + jXt)N Ll-t N. (99)The matrix

(100)

which is Herrnitian p.d. (for Llt is diagonal with positive elements) is classicallycalled the Hermitian square root of Rt. With this notation (99) becomes

(101)

and this shows that the classical normalization by Rt-l ... Rt-l can be used,even for non-diagonal terminations, provided the Hermitian square root be


adopted. Finally, the definition (1) of the scattering matrix with Z replacedby (101) yields

S - Rt-t(Z + Zt)-I(Z - Zt)Rtt

= Rt-t(Z - Zt) (Z + Zt)-IRtt.

(102)

(103)

An equivalent form of (103) is

(In - Rtt S Rt-!) Z = Zt + Rtt SR-t Zi,

so that the equations of an n-port of scattering matrix S for a termination Z, are

Sb(PO*) . u* = O. (107)

Appendix v. <;ombina~on of two conjugate all-pass matrices of degree one

Consider the factorizations Sb = S1S2 and SI = S3S4 of sec. 8. The secondtheorem of sec. 4 applied to SI at po* replaces (13) by

(105)

where u« satisfies an equation similar to (15), thus

SI(PO*) . u« = O. (106)

But the initial matrix Sb being supposed real, the conjugate of (15) may bewritten as

By Sb = S1S2 and (13), this becomes

- )ao U USI(PO*) (In + . u* = O.

-ao + JWO(108)

From a comparison between (106) and (108), it appears that u« is

(ao U U )In + u*,

-ao + jwo

except for a norrnalization factor to be introduced in order to have UaUa = 1.The norm of (109) is simplified, thanks to (14), into

(109)

(ao2 U U )

U' In - ---- u*.ao2 + wo2

With the notation (46), thus t* = uu*, and by the conjugate of (14), the lastresult becomes

310 V. BELEVITCH

a02 t t*1-----

a02.+ w02'

so that one must take

üa = (1- a02t t* )-t (In+ aou U ) u* .

a02 + w02 -ao + jwoThe combined extracted matrix of degree two is

(2aOUaUa) (In- 2ao u U)S4S2 = ln-p + po p + po* '

(110)

which is (42) with

M =-2ao(P + ao) (UaUa + uu) + 2aojwo (UaUa- uu) + 4a02uaUaUU. (111)

One now replaces Ua in (111) by (110) and obtains, after long but elementarycomputations, the lengthy explicit expression for M defined by (43) to (47).

Appendix VI. Chain matrix of an all-pass 2n-port

We wish to compute the chain matrix of the 2n-port having the scatteringmatrix (8) when it operates between arbitrary passive ~conjugate-transposeterminations, of impedance matrix Zo at the output, and Zo at the input. Thenetwork equation is (104) with

~

z, = (Zoo 0) (112)Zo .

and S replaced by (8). If one denotes by Ro the Hermitian part of Zo, (104)becomes

-Zo In

where the subscripts a and b refer to input and output variables, respectively.The chain equations are obtained by premultiplying (113) by the inverse ofthe left-half submatrix, which is

-Zo Ro-t S2-1 Ro-t)-Ro-t S2-1 Ro-t

~ ~Zo Ro-1Zo = Zo Ro-1Zo, (114)

as can be checked, taking into account the identity

which is easily established by taking the inverse and replacing Roby its definition.


Finally, the chain matrix is

t (ZORO-l + ZoRo-t S2-l Ro-tRo-t S2-lRo-t-Ro-l

_ZORO-lZO + ZORO-tS2-lRO-tio) .RO-lZO + RO-tS2-lRo--t Zo (115)

We now particularize this result for the general (complex) all-pass of degreeone, where S2 is (13), so that S2-l is (93). The matrix (115) becomes

1 ((p- po)ln + aoZoRo--tu U Ro-t aoZoRo-t u U Ro--t Zo )-_ _l ~ i _i ~ _i ~ • (116)p - po aoRo-"J:u U Ro>: (p - po)ln + aoRo>: u uRo -,;ZoWith the notations (49-50), (116) simplifies to (51).

'I

Appendix VII. Realization of 2n-port sections

Consider a chain matrix of the form .

(In + a k h'

ehh'b k k' )

In + dhk' , (117)

where a, b, e, d are arbitrary scalar functions, while hand k are vectors whosescalar product is denoted by

k'h=m. (118)

The network equations deduced from (117) are

Va - Vb= k (ah'vb - b k'ib),

ia + ib = h (eh'vb - d k'ib).

(119)

(120)

This suggests that the 2n-port is equivalent to a 2-port combined with two setsof ideal transformers characterized by the vectors k and h. Itwill be shown thatthe realization of fig. 15 is obtained if the impedance matrix of the 2-port11'22' represented as a black box in fig. 15 is suitably related to the para-meters a, b, e, d of (117). The electrical variables at the ports of the 2-port aredenoted VI, V2, ti, iz whereas a notation such as ial designates the first compo-nent of ia, i.e. the input current at port la.

The downward current in the winding lei is io: - h = iz - in, whereas thedownward current in the windings +h; in series is

Ïl + i2 =I«: + ia2. (121)

The current constraints imposed by the transformers hi are the n - 1 equationscontained in the vector relation

. (122)

the first equation (122) reducing to (121). The voltage at 1'2' of the 2-port isVal __:_'Vl = Vbl - V2, so that one has

312 V. BELEVITCH

(123)Val - Vbl = VI - V2.

The voltage drop between the same point and la'2a' is

Vbl- V2= -h2 Vb2/hl - h3 vselh: - ... ,

.which yields the vector relationV2= h'Vb/hl.

The current constraint imposed by the transformers kt is

kl (ial - h) + kz ia2+ k3 ia3+ ... = 0,thus

h = k' ia/kl,

~:~'~

(124)

(125)I

t ,~I

whereas the associated voltage relations are the n -v- I last equations contained in

(126)

multiplied by h. It remains to eliminate á'vs by (124) and k'ib by (125) combinedwith (122) premultiplied by k' /kl. Thanks to (128), this yields

k'ib/kl = i: (m/hlkt - 1) + iz m/hlkl'

After the elimination, (127-128) become

Equations (129) define the 2-port 11'22' offig. 15. The elements ofits impedancematrix are obtained by solving the equations in terms of VI, V2; the results are

Va- Vb= (VI - v2)k/kl,

the first equation (126) reducing to (123).Equation (123) shows that (119) results from the scalar relation

(VI- v2)/kl = ah'vs-« bk'is

multiplied by k, whereas (122) shows that (120) is

(h + i2)/hl = ch'vs-« dk'ib

(vi - v2)/kl = a hsv« - b ii (m/hI - kl) + i2m/hl, ~

(h + i2)/hl .= C hiv» - d t: (m/hlkl) + i2mlhs, ~

Zn = 1+ md + (a-d)klhl + klhl (m- klhl) (ad- bc)/chl2,I

Zl2 = 1+ md + a ksh: + m ksh: (ad- bc)/chl2,

Z2l = (1 + md- d klhl)/chI2,

Z22 = (1 + md)/chI2.

(127)

. (128)

(129)

(130)

The matrix (54) is of the form (117) with a = b = c = d = (p- ao)-l andm = ao by (55). With these values the impedance matrix (130) reduces to


p/hl2 + kl/hl)p/hl2

(131)

and corresponds to an inductance l/hl2 in series with a gyrator kl/hl as shownin fig. 16.

The matrix (77) is of the form (117) with

2wo 2pa=---- ,b= ,

(P2 + w02)(1 + À) (p2 + w02) (1- À)(132)

2p 2woc = d = -------(P2 + w02) (1 + À) , (P2 + w02) (1 - À) ,

and with k replaced by g. Moreover, (76) and (73) give m = g' h = h'Xoh = woÀ.With the values and the notations (78-79) the impedance matrix (130) is

1)+ __2_ (n2 n)1 Cp n 1

(133)

and is realized by fig. 17.

Appendix VIII. Combination of conjugate sections ID the Darlington process forreal n-ports

We. start by determining the value Za of (60) at po*. With the notation

(Zo:- Zo*)/2j = Z, (134)

defining a real matrix, and with

(135)one easily obtains

Za = Zf(PO*) = M (Zo* + Z; hh ZO/wo). (136)

From (60) one also deduces

- -~f(PO*) = [Zo hh(Zo - Zo*)/2po* + ~o*] [In + hh(?o* - Zo)/2pO*]-l, (137)

where ~o* is the value of ?(P) = Z'(-p) atpo*, i.e. Z'(-po*). From (136) and037) one thus deduces

[Zf + ~f]Po* = M(Zo* + ~o*) [In + hh(~o* - Zo)/2pO*]-l. (138)

Since the conjugate of (59) is

(Zo* + ?o*)h* = 0, (139)

314 v. BELEVITCH

- -[In + hh (?"o*- Zo)/2po*]h*,

it appears that (138) vanishes when multiplied on the rig~t by the vector

(140)

and this yields a solution vector ha for (61), after normalization by (63). In (140), one replaces ~o* by -Zo* thanks to (139). Introducing the notation (66), i.e.

z = h'k = !z'Zoh = h' (Zo + Zo') h/2, (141)

where the skew part does notcontributeto asymmetric( even complex) quadraticform, one finally transforms h(Zo* - Zo)!z* of (140) into -2z*. One thus hasobtained (64), where t is a real scalar to be determined by (63). One first verifiesby substitution of (136), (135) and (64) into (62) that

ka = t(Zo*h* - z*Zoh/po*), (142)

which can also be written as (65). By substitution of (64) and (65) into (63),one finally establishes (67).

The expressions (69) as they result from the product of two chain matrices. of the form (51) are

Appendix IX. Reduction of (68) to (35) for n = 1

For Ro scalar, (56) becomes h h = ao/Ro. Since the phase of h is arbitraryone may take h = (ao/Ro)!. Equations (141), (67), (64) and (65) then becomerespectively

,.., ...., ,.." .....

Ao = jwo(k h - kaha) + k(h ka + k ha)ha,

_ ..... N ,..,,..,

Bo =jwo(k k - kaka) + k(h ka + k ha)ka,

,.., ,.., ,.., ,..; "'"

Co = jwo(h h - haha) + h(h ka + k ha)ha,

,.., ,.., ,.., ,..,,..,

Do = jwo(h k - haka) + h(h ka + k ha)ka.

The scalar occurring in several expressions is- -h ka + k ha = -2t z* jwo/po*.

Replacement of ha and ka by (64-65) yields (70).

(143)

(144)

In) (va + Roia).o Vb + Roib

(145)

FAGrORIZATION OF SCATTERING MATRICES AND PASSIVE-NETWORK SYNTHESIS 315

aoXo - woRo woRo + aoXoha =it ,ka =-jt Zo* - .

(ao - jwo)Ro (aojwo)R02

The expressions (70) become

Roao - woXoAl =Dl = 2ao, Ao = 2woao ,

. woRo + aoXo

2aowo(R02 + X02) 2a02wo(R02 + X02)Bl = Bo = --------

Ro(woRo - aoXo)' Ro(Rowo - aoXo) ,

2aowoRo 2woa02RoCl = ; Co =----.

woRo + aoXo woRo + aoXo

aoRo + woXoDo = 2woao -

woRo - aoXo

The numerators (69) of the chain matrix (68) are thus

Roao - w~XoAN = p2_ a02 + w02 + 2woao .,

Rowo + aoXo

2aowopCN= ,

woRo + aoXo

aoRo + woXoDN = p2_ a02 + w02 + 2aowo .

woRo - aoXo

Itcan be checked that these results are identical to (35), taking (34) into account.

Appendix X. Combination of real all-pass 2n-ports of degree one with oppositegyrators into a reciprocal all-pass of degree two .

In sec. 8 it has been established that, the chain matrix of the 2n-port ofscattering matrix (8) with S2 given by (41), operating between terminations ofreal symmetric impedance matrix Ro, is given by (54) with h defined by (49)and k by (50) reducing to k = Ro h. We want to replace (8) by its transpose,and prove that the resulting modification in the chain matrix is simply a changeof sign in ao and Ro (thus in k). This is most easily seen from the equations (104)of the 2n-port which reduce in the case considered to

316 V. BELEVITCH '

Comparison between (41) and the particular case of (93) for Po = ao showsthat S2 is changed into its reciprocal by the change of ao into -ao. Ifthe sub-matrix occurring in (145) is written (aoRo)t S2(aoRo)-t, this remains true ofthe submatrix if Ro is also changed into =R«. Premultiplying (145) by the in-verse of the entire scattering matrix and changing Ro into - Ro in the vectors,one obtains an equation only differing from (145) by the fact that the scatteringmatrix is replaced by its transpose.

Consider now the second extraction of sec. 11. Since Za is generally notsymmetric, the extraction starts by a subtraction of its skew part jXa. Theresulting impedance Zf- jXa has a scattering matrix S, relatively to Ra, whichvanishes at po. If S is factorized into S2S1, Zf- jXa is realized as an all-passhaving as scattering matrix -the transpose of (8) closed on Zi. As shown inthe preceding paragraph, the chain matrix of the all-pass is (54) with ao, kand h replaced by -ao, -Raha and ha, respectively. The entire section of thesecond extraction is the all-pass preceded by the series branch jXa and, as insec. 9, followed by the series branch -jXa• The resulting chain matrix isobtained by multiplying the three component matrices and is

1 ((P + ao)ln- kaha' kaka') (146)p + ao haha' CP + ao)ln - haka'

withka = (Ra - jX a)h = Za' ha. (147)

It remains to show that with a suitable choice of ha, the product of (54) by(146) is realizable as a reciprocal 2n-port, taking into account (82), (147), (55)and the similar constraint

(153)

ka'ha = ao· (148)

We first show that the appropriate choice is

ha = uh, (149)

where f.Lis a scalar to be determined. By (82), (147) and (149), one has

ka = f.LRo(ln + hh'Lo) (ln- hh'Lo)-lh.

With the. notation (84) it is easily checked that

(ln - hh'Lo)-lh = hl(l- À)

(150)

(151)

. so that (150) reduces toka = f.L(1+ À)kl(l- À).

The condition (148), taking into account (55), gives then

f.L= [(1 - À)/(l + À)]t,

(152)

and (152) reduces toka = klf.L. . (154)

REFERENCES

FACTORIZATION OF SCATIERING MATRICES AND PASSIVE-NETWORK SYNTHESIS 317

Comparison of (149) and (153) establishes (83). On the other hand, the productof (54) by (146), táking into account (149, 153,' 154), yields (85). .The reciprocal all-pass 2n-port of degree two mentioned in sec. 8 is easily

obtained as a particular case. Since the first extracted section is an all-passwith terminations specifiedby the real symmetric matrix Ro, the second extractedsection reduces to an all-pass with identical terminations if (82) reduces to Ro,

. i.e. if Loh = 0, thus À = ° by (84) and fL = 1 by (153). The all-pass is thus'figs 15-17 with the element values (86-87) where À = 1.

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