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On the synthesis of two-element-kind multiport networks Stein, Richard Adolph 1968

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ON THE SYNTHESIS OF TWO-ELEMENT-KIND MULTIPORT NETWORKS RICHARD ADOLPH STEIN B.Sc, U n i v e r s i t y of A l b e r t a , 1958 M.S., U n i v e r s i t y of I l l i n o i s , 1961 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of E l e c t r i c a l Engineering We accept t h i s t h e s i s as conforming to the required standard Research Supervisor Members of the Committee Head of the Department Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA June, 1968 In presenting t h i s thesis i n p a r t i a l f u l f i l m e n t of the requirements for an advanced degree at the University of B r i t i s h Columbia,, I agree that the Library s h a l l make i t f r e e l y avail able f o r reference and study,, I further agree that permission., f o r extensive copying of t h i s thesis f o r scholarly purposes may be granted by the Head of my Department or by his representatives„ I t i s understood that copying or publication of t h i s thesis for f i n a n c i a l gain s h a l l not be allowed without my written permission. Department of E l e c t r i c a l E n g i n e e r i n g The University of B r i t i s h Columbia Vancouver 8, Canada Date June 24, 1968 ABSTRACT Procedures f o r the synthesis of a c l a s s of two-eli ement-kind mu'ltiport networks are developed and i l l u s t r a t e d by examples. In the RC case, the networks c o n s i s t of the s e r i e s connection of an R network and an RC network. The l a t t e r contains at "least one c a p a c i t o r t r e e so that i t s o p e n - c i r c u i t impedance matrix vanishes at i n f i n i t e frequency. I t i s shown that f o r a network w i t h i n the c l a s s , the o p e n - c i r c u i t impedance matrix and the normal form s t a t e model are equivalent i n t h a t , given one, the other can be w r i t t e n immediately. The. synthesis problem then takes the form of the deter-mination of a normal coordinate transformation such that the t r a n s -formed s t a t e v a r i a b l e s may be i d e n t i f i e d as c a p a c i t o r voltage v a r -i a b l e s i n a passive RC network. Two procedures are described f o r determining a t r a n s f o r -mation (modal) matrix which y i e l d s an i r r e d u c i b l e r e a l i z a t i o n of t h a given kxk, n degree impedance matrix. There are -g-(n-k) (n-k-l) degrees of freedom i n the modal matrix. General a n a l y t i c a l s o l u -t i o n s are p o s s i b l e when n^k+2, one greater than i n e x i s t i n g methods. The main procedure y i e l d s a network w i t h , i n general, n + -gk(k+l) c a p a c i t o r s . A set of necessary c o n d i t i o n s , e a s i l y a p p l i e d to the given impedance matrix, i s derived. Necessary and s u f f i c i e n t c o n d i t i o n s are given f o r the s p e c i a l case k=2, n=3- An a l t e r n a t i v e procedure y i e l d s a network with n c a p a c i t o r s . Using e i t h e r procedure, i t i s p o s s i b l e to simultaneously minimize both the number of elements and the t o t a l capacitance i n the network. By i n t r o d u c i n g a d d i t i o n a l equations i n t o the main proce-dure, numerical s o l u t i o n s f o r the modal matrix may be determined f o r any value of n. With k=2, the procedure y i e l d s a new c l a s s of i i minimal, grounded two-port networks c o n s i s t i n g of one rt-section and n - 2 T-sections connected i n p a r a l l e l . The s e v e r i t y of the r e a l i z a b i l i t y c o n d i t i o n s i s approximately p r o p o r t i o n a l to n. A given 2 x 2 impedance matrix may be r e a l i z e d e x a c t l y , or one d r i v i n g -point impedance f u n c t i o n may be realized, e x a c t l y and the t r a n s f e r impedance f u n c t i o n w i t h a d e s i r e d gain f a c t o r ( w i t h i n the l i m i t s of r e a l i z a b i l i t y ) . A computational procedure i s given which min-imizes the t o t a l capacitance and optimizes the voltage gain f a c t o r . i i i TABLE OP CONTENTS Page LIST" OF ILLUSTRATIONS . . . . . . . v i ACKNOWLEDGEMENT . .". . .". . . . . . . . . . . . . . . v i i i 1. INTRODUCTION . . . . . . . . . . . . . . . . . . " . . . . . . . . . . . . . . . . . . . . . 1 1.1 Statement, of the .Problem . . . ..... . . . . . . . . . . . . . . . . . . . . 1 1.2 Review of Previous Work , 2 1.3 Discussion of the Problem . . . . . . . . . . 6 2. MODELS OF PASSIVE NETWORKS AND DELINEATION AND PROPERTIES OF-A.CLASS OF- TWO-ELEMENT-KIND .-NETWORKS. . . . . 9 2.1 Network Models 9 2.2 Formulation of the State-Model of an RLC Network.. 12 2.3 Models of Two-Element-Kind. Networks and' Delineation of a Special Class o f Networks • 17 2.3.1 LCI and LCV Networks 17 2.3.2 RCI and RLV Networks 22 2.3.3 Summary 25 2.4 Properties of RCI Networks 27 3. SYNTHESIS OF A.CLASS OF RC MULTIPORT NETWORKS 36 3.1 The General Synthesis Procedure .36 3.2 Synthesis Procedure I . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.2.1 Realization of the Matrix Q . . . 44 3.2.2 Determination of the Capacitance Matrix C . . 45 3.2.3 Calculation of the Modal Matrix . 47 3.2.4 Calculation and Realization of the Conductance Matrix J 50 3.2.5 Synthesis of Minimal Networks 52 3.2.6- Factorization of the Residue Matrices 53 3.2.7 Summary of Synthesis Procedure I 56 iv 3-3 Examples 53 3 . 3 . 1 Example One.— k=2, n=3 58 3 . 3 . 2 Example Two — k=2, n=4, Q=0 .. 62 3 . 3 . 3 Example Three — k=l, n=3 69 3 . 4 Synthesis Procedure I I . . . 72 3.5 D i s c u s s i o n 77 4 . SYNTHESIS OF GROUNDED TWO-PORT NETWORKS USING A NUMERICAL TECHNIQUE • 82 . '4.1 E x p l i c i t Determination of the Modal Matrix and Synthesis of the Second Foster Canonical Form 83 4 . 2 A p p l i c a t i o n to the Synthesis of Grounded Two-Port Networks 86 4 . 3 Transmission Gain Optimization 92 4 . 4 Examples and Di s c u s s i o n 98 4 . 4 . 1 Example F i v e 100 4 . 4 . 2 Example Six 107 ' 4 . 4 . 3 D i s c u s s i o n . ... 109 5. SUMMARY AND SUGGESTIONS FOR FURTHER RESEARCH 110 APPENDIX A CEDERBAUM'S DECOMPOSITION ALGORITHM 114 APPENDIX B ON NECESSARY AND SUFFICIENT REALIZABILITY, CONDITIONS 117" B . l Networks Containing a S t a r - L i k e Capacitor Tree 117 B .2 Networks Containing a Lin e a r Capacitor Tree 121 B . 3 Networks w i t h k=2, n=3 • •• 123 APPENDIX C NUMERICAL SOLUTION FOR THE MODAL MATRIX 124 REFERENCES 129 v LIST OF ILLUSTRATIONS Figure Page 2.1 Block diagram of the c l a s s of LCI networks defined by B ^ L ^ ^ = 0... 2 1 2.2 Block diagram of the c l a s s of LCV networks defined by B i 1 c c B 1 2 = 0 2 1 2.3 Summary of the models of a c l a s s of two-element-ki n d networks 26 3.1 Block diagram of RCI networks synthesized by Procedure I 44 3.2 E v o l u t i o n of the network that r e a l i z e s ZRQ(S) of • Example One. Element values are i n ohms, farads and mhos 6 1 3.3(a) Sign p a t t e r n of conductance matrix and c o r r e s -ponding c a p a c i t o r - t r e e c o n f i g u r a t i o n s as func-t i o n s of a f o r Example Two 65 3.3(b) Network corresponding to C o n f i g u r a t i o n l ( a ) and graph of allowed values of a, d^ and d. f o r Example Two .7 66 3.3(c) Network corresponding to C o n f i g u r a t i o n l ( b ) and graph of allowed values of a, d^ and d. f o r Example Two < 7 67 3.3(d) Network corresponding to C o n f i g u r a t i o n 3 and graph of allowed values of d„ and d. f o r a = 70 f o r Example Two < 7.. .. 68 3.4 Complete network and•graph of allowed values of and & f o r r e a l i z a t i o n of Z^^(s) i n Example Three .. 7 1 3«5 R e a l i z a t i o n of Z-n^(s) of Example Four using Proce-dure I I . Element values are i n ohms, farads and mhos 76 4.1 One-pert network formed when J ^ i s a diagonal matrix • 8 5 4.2 Capacitor subnetwork of two-port network formed. when i s diagonal 8 9 t h 4.3 Complete network f o r 5 order problem w i t h one column of J - ^ i n each category 8 9 4.4(a) L i m i t s of transmission gain as fun c t i o n s of a wi t h b=l, c=8 i n Example Five 1 0 2 v i 4.4(b) L i m i t s of transmis s i o n gain as fun c t i o n s of b with a=3, c=8 i n Example Five 103 4 . 4 ( c ) L i m i t s of transmission gain as fun c t i o n s of c ( with a=3, b=l i n Example Five 104 4.5 R e a l i z a t i o n of impedance f u n c t i o n s of Example Five w i t h a=3, b=l and c=8. Element values are i n farads and mhos 105 4 . 6 R e a l i z a t i o n s of the fun c t i o n s In Example S i x . Element values are i n farads and mhos 108 v i i ACKNOWLEDGEMENT The author i s indebted to h i s s u p e r v i s o r , Dr. A. L'. Moore, f o r guidance and a s s i s t a n c e throughout the period of t h i s research. He i s g r a t e f u l to Dr. F. Noakes, Head of the Department o f ! E l e c -t r i c a l Engineering, U.B.C., and Dr. M. Z. Kharadly f o r t h e i r i n t e r -est and support. A p p r e c i a t i o n i s expressed to the author's c o l -leagues f o r h e l p f u l d i s c u s s i o n s and f o r proofreading the manuscript. The work reported i n t h i s t h e s i s was c a r r i e d out under N a t i o n a l Research Council of Canada grants A-68 and A-3357- The author g r a t e f u l l y acknowledges the award of a U.B.C. Graduate Fellow-ship i n 1966-67 and an N.R.C..Studentship i n 1967-68. S p e c i a l thanks are expressed to the author's w i f e , Marie, f o r her patience and s a c r i f i c e . v i i i 1. INTRODUCTION 1.1 Statement of the Problem An outstanding unsolved problem i n e l e c t r i c a l network theory, i s the determination of the co n d i t i o n s under which an th n order immi Ltance matrix may represent the s p e c i f i c a t i o n s of a passive n-port network without transformers . The complete s o l u t i o n i s known only f o r n=l, except i n the case of s i n g l e -element-kind networks where i t i s known f o r n^3. The present work i s concerned with two-element-kind networks. The problem can be sta t e d as f o l l o w s : given an nxn matrix with e n t r i e s which are r a t i o s of polynomials i n the com-plex frequency s, ( i ) what are the necessary and s u f f i c i e n t con-d i t i o n s that i t be the immittance matrix of a passive n-port network c o n t a i n i n g two kinds of elements, and ( i i ) what.is the procedure for. s y n t h e s i z i n g the network? Necessary conditions are determined through a n a l y s i s . S u f f i c i e n c y c o n d i t i o n s take the form of e i t h e r d i r e c t t e s t s on the given matrix or a synthesis procedure which may or may not work out. From the t h e o r e t i c a l viewpoint, the former i s p r e f e r -able. However, f o r the design engineer the l a t t e r may be more . p r a c t i c a l , e s p e c i a l l y i f the d i r e c t t e s t s are d i f f i c u l t to.apply. For t h i s reason, and because of the apparent d i f f i c u l t y i n det-ermining c o n d i t i o n s which can be tested d i r e c t l y , we s h a l l give considerable a t t e n t i o n to the synthesis aspect of the problem. We s h a l l f i r s t review b r i e f l y recent developments i n the synthesis of two-el.ement-kind n-port networks with general t o p o l o g i c a l c o n f i g u r a t i o n s , and then describe the approach 2 taken i n t h i s d i s s e r t a t i o n , g i v i n g some of the important r e s u l t s 1.2 Review of Previous Work . Conventional methods of s y n t h e s i z i n g two-element-kind networks y i e l d c o n f i g u r a t i o n s which are f i x e d by the method chosen. The Foster and Cauer canonical r e a l i z a t i o n s of d r i v i n g -point f u n c t i o n s and Guillemin's method of s y n t h e s i z i n g two-port RC networks i n the form of p a r a l l e l ladders are examples. In recent years, considerable progress has been made i n the synthesis of two-element-kind networks w i t h general top-o l o g i c a l c o n f i g u r a t i o n s . The advances are based f o r the most part on the use of normal coordinate transformations to r e a l i z e o p e n - c i r c u i t impedances. The e s s e n t i a l aspects of t h i s approach (2) f i r s t suggested by G u i l l e m i n i n I960, are given here using RC networks f o r i l l u s t r a t i v e purposes. Consider a k-port, (n+l)-node RC network c h a r a c t e r i z e d on the node b a s i s i n terms of a capacitance matrix C and a con-ductance matrix G. The e q u i l i b r i u m equations are given by Y(s)V(s) = I(s ) 1 where the node admittance matrix i s . Y(s) = sC + G. V(s) i s a column matrix of the n node-pair voltage v a r i a b l e s and l ( s ) i s a column matrix of the corresponding source c u r r e n t s . The normal coordinates of the system are placed i n evidence by a l i n e a r transformation of the v a r i a b l e s . This nor-mal coordinate transformation matrix (by a congruence t r a n s f o r -3 mation) simultaneously diagonalJ.zes the matrices C and G. Let the transformation matrix be designated by M. Then M^Cs^ M = M t(sC + Q)K = sU + L 1 .2 where U i s the i d e n t i t y matrix of order n, and L = diag(\ 1,\ 2,.A n ) -Prom ( l . 2 ) , we have C = (M t)~ 1M" _ 1 and. G = (M t)~ 1LM~ 1. 1 .3 The response of the network i s c h a r a c t e r i z e d by the inverse of Y ( s ) . Y ( s ) " 1 = Z(s) = M(sU + L r V The port v a r i a b l e s are given by a l i n e a r transformation of the node-pair v a r i a b l e s i n ( l . l ) . Without l o s s of g e n e r a l i t y , l e t the node-pairs be chosen so that the f i r s t k node-pairs are i d e n t i c a l .to the k p o r t s . Then the o p e n - c i r c u i t impedance matrix of the k-port network can.be w r i t t e n Z R C ( s ) = M 1(sU + L ) " 1 ] ^ - 1 .4 w h e r e c o n s i s t s of the f i r s t k rows of M. The synthesis problem can be stated as f o l l o w s : given a kxk'matrix ( 1 . 4 ) , determine the l a s t (n-k) rows of M such that the matrices C and G, given by ( 1 . 3 ) , are r e a l i z a b l e by means of single-element-kind networks w i t h nonnegative elements and iden-t i c a l tree c o n f i g u r a t i o n s . The r e a l i z a b i l i t y c o n d i t i o n s are simply stated f o r the important case of common-ground networks where Y(s) i s the usual node-to-datum admittance matrix. They are that C and G must be 4 hyperdominant, that i s , dominant with nonpositive o f f - d i a g o n a l e n t r i e s . This approach leads to common-ground networks with p e r f e c t l y general topologies and a f f o r d s s u f f i c i e n t realizab--i l i t y c r i t e r i a so that no other r e a l i z a b i l i t y c o n d i t i o n s need (3) be considered w / . However, no general method has been given f o r s o l v i n g the many nonlinear i n e q u a l i t i e s that r e s u l t . In reviewing recent developments, we consider f i r s t the a p p l i c a t i o n of normal coordinate transformations to the syn-t h e s i s of one-port networks w i t h general t o p o l o g i c a l c o n f i g u r -a t i o n s . S i g n i f i c a n t c o n t r i b u t i o n s have been made by s e v e r a l workers. Duda ^ \ i n an unpublished d i s s e r t a t i o n , has given a d i r e c t method which y i e l d s a one-parameter f a m i l y of equivalent networks w i t h , i n general, a c a p a c i t o r and a r e s i s t o r i n p a r a l -l e l between every p a i r of nodes. . (5) More r e c e n t l y , Schneider has given a procedure i n which the l i n e a r transformation i s regarded as the product of two transformations, one orthogonal, the other diagonal, which e s t a b l i s h the s i g n p a t t e r n and dominance, r e s p e c t i v e l y , of the parameter matrices. The r e a l i z a t i o n i s always noncanonical w i t h , i n general, a c a p a c i t o r and a r e s i s t o r i n p a r a l l e l between every p a i r of nodes. Either' the number of ca p a c i t o r s or the number of r e s i s t o r s can be reduced to n. In a forthcoming d i s s e r t a t i o n , Mason gives an i n d i r e c t method f o r the synt h e s i s of minimal* one-port networks * A "minimal" network c o n f i g u r a t i o n i s defined here as one i n which the number of elements i s equal to the number of indepen-de n t l y s p e c i f i a b l e parameters. Some authors use "canonical" or "canonic" i n t h i s connection. However, we p r e f e r to reserve " c a n o n i c a l " to describe the minimal c o n f i g u r a t i o n s which are capable of r e a l i z i n g a r b i t r a r y r e a l i z a b l e immittance functions of a given c l a s s (see Lee w) ) . Hence, canonical c o n f i g u r a t i o n s form a subclass of minimal c o n f i g u r a t i o n s . 5 w i t h general t o p o l o g i c a l c o n f i g u r a t i o n s . A f t e r choosing a d e s i r e d c o n f i g u r a t i o n with i n i t i a l element values, an i t e r a t i o n technique i s used to determine the transformation matrix and hence the f i n a l element values. One l i m i t a t i o n of t h i s procedure i s that a s o l u t i o n does not always e x i s t because not a l l minimal c o n f i g u r a t i o n s are capable of r e a l i z i n g a given d r i v i n g - p o i n t f u n c t i o n . Two methods, which are s i m i l a r , f o r s y n t h e s i z i n g • grounded m u l t i p o r t networks i n terms of equivalent network (8) transformations have been given i n an e a r l y r e p o r t by B o x a l l ("5) and i n a recent d i s s e r t a t i o n by Schwab w . In t h e i r methods, an a r b i t r a r y transformation matrix i s chosen y i e l d i n g a network which may have negative element values. Then a transformation i s sought which transforms the network to one w i t h nonnegative element values. Although the r e s u l t i n g nonlinear i n e q u a l i t i e s are simpler i n form than those formed d i r e c t l y from ( 1 . 3 ) , a general s o l u t i o n i s not given. Basson and Halkias have given necessary and suf-f i c i e n t c o n d i t i o n s of a general nature f o r the r e a l i z a t i o n of t h an n order impedance matrix Z ( s ) , where Z(°°) =0, i n the form of a grounded n-port RC network. A procedure i s given f o r det-ermining whether an impedance matrix having no more than (n+l) poles can be r e a l i z e d and, when r e a l i z a b l e , a complete c l a s s of equivalent r e a l i z a t i o n s i s presented. I f a group of r e a l i z a t i o n s e x i s t s , they are a s s o c i a t e d with a r e g i o n bounded by hypersurf-aces s p e c i f i e d by a set of i n e q u a l i t i e s . P o i n t s of i n t e r s e c t i o n of (n+l) surfaces correspond to minimal networks. 6 For grounded n-port networks, a s y n t h e s i s procedure not based on normal coordinate transformations has been pres-ented by C.livares . His method i s a g e n e r a l i z a t i o n ( f o r (n+l)-terminal networks) of the Ozaki-Lucal technique f o r three-t e r m i n a l networks. The basic technique c o n s i s t s of f i n d i n g the p a r t i a l f r a c t i c n expansion of the admittance matrix and separ-a t i n g i t i n t o two matrices, one of which i s r e a l i z a b l e by elem-e n t a l subnetworks i n a mesh c o n f i g u r a t i o n . The remaining matrix i s i n v e r t e d , expanded i n p a r t i a l f r a c t i o n s , and separated i n t o two matrices one of which i s r e a l i z a b l e by elemental subnetworks i n a s t a r c o n f i g u r a t i o n . The remaining matrix i s i n v e r t e d and the procedure repeated. I f the given matrix i s realizable-.by a star-mesh c o n f i g u r a t i o n , the network synthesized by t h i s proc-edure w i l l be minimal with no elements connected between any p a i r of i n t e r n a l nodes. 1.3 D i s c u s s i o n of the Problem C l e a r l y , the problem stated i n Section 1.1 i s f a r from solved. The only known way of s y n t h e s i z i n g networks w i t h general t o p o l o g i c a l c o n f i g u r a t i o n s i s through the use of normal coordinate transformations. While necessary and s u f f i c i e n t c o n d i t i o n s of a general nature are known, p r a c t i c a l s o l u t i o n s have been given only f o r n = k + 1, that i s , f o r k-port networks w i t h k +2 nodes. The synthesis problem i s so d i f f i c u l t because the f r e -quency-domain s p e c i f i c a t i o n s are input-output r e l a t i o n s which give, l i t t l e i n f o r m a t i o n about the i n t e r n a l s t r u c t u r e of the net-work to be synthesized. On the other hand, the s t a t e - v a r i a b l e c h a r a c t e r i z a t i o n (or s t a t e model) i s a time-domain c h a r a c t e r i z -7 a t i o n which i s i n t i m a t e l y r e l a t e d to the i n t e r n a l s t r u c t u r e of the network . Therefore, one would expect the synthesis of a given s t a t e model to be l e s s d i f f i c u l t . In f a c t , the prob-(12) lem has been, solved i n the case of LC networks. v , although not f o r RC or RL networks. A s t a t e model can always be determined, a l b e i t nonun-i q u e l y , from a given frequency-domain model. Therefore, i t i s p o s s i b l e to t r e a t the synthesis problem by f i r s t r e a l i z i n g the given immittance matrix i n the form of a s t a t e model which, i n t u r n , i s r e a l i z e d by a network. We s h a l l adopt t h i s approach he r e i n and we s h a l l show that the synthesis problem can be stated i n terms of a transformation of s t a t e v a r i a b l e s . Accordingly, we s h a l l f i r s t formulate the s t a t e models of two-element-kind networks. In order to have a t r a c t a b l e syn-t h e s i s problem, a broad c l a s s of networks wi t h p a r t i c u l a r l y simple stat e models i s d e l i n e a t e d . In the RC case, a network w i t h i n the c l a s s c o n s i s t s of the s e r i e s connection of two networks w i t h the o p e n - c i r c u i t impedance matrices Z-^(s) and Q, r e s p e c t i v e l y , where Zp^ 0 0) = 0 and Q i s a constant matrix. Thus, the open-c i r c u i t impedance matrix, Z ^ ( s ) , of the s e r i e s connection i s given by Z R C ( s ) = Z^ c(s) + Q . 1.5 I t i s shown t h a t , f o r the c l a s s of f u n c t i o n s under c o n s i d e r a t i o n , the normal form of the s t a t e model can be w r i t t e n by i n s p e c t i o n from Z R C ( s ) . A method i s given f o r the r e a l i z a t i o n of a f u n c t i o n (1.5) by means of a network which may or may not have a common 8 ground. When the f u n c t i o n i s r e a l i z a b l e , a complete f a m i l y of equivalent networks can be synthesized. By s u i t a b l y choosing a number of parameters, the network can be made minimal and the t o t a l capacitance minimized at the same time. While the method i s not l i m i t e d i n theory, i n p r a c t i c e general s o l u t i o n s can be found only when n^k+2 (n here i s the degree of Z-^Q(S)). This, however, represents an increase i n order by one over known methods. For n=k+l,solutions are found by a s t r a i g h t f o r w a r d procedure, and f o r n=k+2, by searching over a one-dimensional space. When n^k+2, i t i s p o s s i b l e to choose some parameters a r b i t r a r i l y but i n a manner which has a good chance of l e a d i n g to a passive r e a l i z a t i o n , should one e x i s t : A procedure of t h i s type i s given which leads to a new c l a s s of minimal, grounded two-port networks. S p e c i f i c a l l y , the networks c o n s i s t of (n-2) T-sections,. each w i t h two r e s i s t o r arms and one c a p a c i t o r arm, connected i n p a r a l l e l w i t h one re-section. Each arm of the re-s e c t i o n i s made up of a r e s i s t o r and a c a p a c i t o r connected i n p a r a l l e l . Furthermore, the transmission voltage gain f a c t o r can be optimized and the t o t a l capacitance minimized. The e n t i r e procedure i s e a s i l y programmed f o r d i g i t a l computer execution. 2. MODELS OP PASSIVE NETWORKS AND . DELINEATION AND PROPERTIES OP A CLASS OP TWO-ELEMENT-KIND NETWORKS 2.1 Network Models T r a d i t i o n a l l y , l i n e a r networks have been described mathematically i n terms of the complex frequency v a r i a b l e s. For example, the nxn impedance matrix Z(s) represents r e l a t i o n s between the input currents and the output voltages of an n-port network. Recently, however, network t h e o r i s t s have been g i v i n g i n c r e a s i n g a t t e n t i o n to the s t a t e - v a r i a b l e c h a r a c t e r i z a t i o n of networks. The s t a t e model i s an " i n t e r n a l " d e s c r i p t i o n i n terms of dynamical or s t a t e v a r i a b l e s . As such, i t i s c l o s e l y r e l a t e d to the i n t e r n a l ' s t r u c t u r e of the network. The frequency-domain model, on the other hand, i s an " e x t e r n a l " d e s c r i p t i o n which does not d i s p l a y d i r e c t l y i n f o r m a t i o n about the i n t e r n a l s t r u c t u r e . D e f i n i t i o n s of some of the terms associated w i t h the s t a t e - v a r i a b l e c h a r a c t e r i z a t i o n are i n order. • The s t a t e v a r i a b l e s of a network c o n s i s t of the minimal set of ca p a c i t o r voltage v a r i a b l e s and inductor current v a r i a b l e s (or a nonsingular t r a n s -formation of them) that the network can support. The s t a t e of the network at any time Is determined by the values of the s t a t e v a r i a b l e s at that time. Knowledge of the st a t e at time t plus the e x c i t a t i o n from t onward I s s u f f i c i e n t to completely deter-mine the response from t onward. The state model i s the set of f i r s t - o r d e r d i f f e r e n t i a l equations (Equations (2.1-2) below) which c h a r a c t e r i z e the network. The general form of the s t a t e model of a lumped, l i n e a r , f i n i t e , passive, t i m e - i n v a r i a n t RLC network i s given . ( 1 2 ) hy i. }.X(t) - X(t) = AX(t) + B E ( t ) + CE(t) 2.1 dt E(t) = P X ( t ) + QE(t') + RE(t) . 2.2 where X(t) i s c a l l e d the state v e c t o r , E(t) the input v e c t o r , and E ( t ) the output ve c t o r . The e n t r i e s of X(t) are the state v a r i a b l e s . E ( t ) i s the complement of E ( t ) , that i s , the v a r i -ables i n E(t) are the. complementary currents and vol t a g e s , r e s -p e c t i v e l y , of.those i n E ( t ) . The matrices A, Q and R are r e a l and square; B, C and P are r e a l and, i n general, r e c t a n g u l a r . Equations ( 2 . 1 ) are c a l l e d the s t a t e equations, Equa-t i o n s ( 2 . 2 ) the output equations. The order of complexity of a network i s defined to be the number of s t a t e v a r i a b l e s or the number of f i r s t - o r d e r d i f f e r e n t i a l equations that c h a r a c t e r i z e the network. E q u i v a l e n t l y , the order of complexity may be defined to be the number of f i n i t e ( 1 3 ) n a t u r a l frequencies or the dimension of the A matrix . The 'natural frequencies of the network are simply the eigenvalues of A. The frequency-domain model can be found from the s t a t e model by Laplace transforming ( 2 . 1 - 2 ) and e l i m i n a t i n g the s t a t e v a r i a b l e s . Assuming the network i s i n the zero s t a t e i n i t i a l l y that i s , X ( 0 ) = 0, we f i n d £(s) = [ P ( S U - A ) _ 1 ( B + sC) + Q + sR] £(S) 2.3 where £(s)' =X[ E("t)] , £(s) = £[E(t)] a n d u i s t h e i d e n t i t y matrix*. The t r a n s f e r f u n c t i o n matrix i s given by H(s) = P(sU - A ) _ 1 ( B + sC) + Q + sR and i s , i n general, a hyb r i d matrix. Thus, i t i s apparent that H(s) i s determined from the s t a t e model i n a s t r a i g h t f o r w a r d procedure. On the other hand, determination of a state model from a given H(s) i s a much l e s s t r i v i a l problem. A nonunique set of matrices A, B, C, P, Q and R s a t i s f y i n g (2.3) always e x i s t s (14-16)^ S y S - k e m (2.1-2) given i n numerical form has been c a l l e d a r e a l i z a t i o n * * of H(s) . By t h i s i t i s meant that the system (2.1-2) may be i n t e r p r e t e d as the program f o r an analogue computer which simulates the given H(s). Each r e a l i z a t i o n corresponds to a s p e c i f i c choice of coordinate system f o r the s t a t e v e c t o r . Since H(s) i s determined only by the completely con-t r o l l a b l e and completely observable part of the network ^ ^ ) ^ (17) the order of complexity ( u s u a l l y r e f e r r e d to as the degree ) of H(s)'cannot be greater than the order of complexity of the network. This property f o l l o w s from (2.3) a l s o . On the other hand, the order of the r e a l i z a t i o n of a given H(s) cannot be l e s s than the degree of H(s). R e a l i z a t i o n s of order greater than the degree of H(s) are sa i d to be r e d u c i b l those of order equal to the degree of H(s) are i r r e d u c i b l e . I t i s the o b j e c t i v e of the present work to f i n d the * The l e t t e r U w i l l be used e x c l u s i v e l y f o r the i d e n t i t y matrix w i t h order u s u a l l y obvious from the context. -** However ';• we s h a l l u s u a l l y take "a r e a l i z a t i o n " to mean "a net work r e a l i z a t i o n " . 12 subclass of i r r e d u c i b l e r e a l i z a t i o n s which can be i d e n t i f i e d w i t h passive two-element-kind networks. 2.2 Formulation of the State Model of an RLC Network The modelling of RLC networks i n the form of a set of f i r s t - o r d e r d i f f e r e n t i a l equations was proposed by Bashkow i n 1957. Bryant's (-^ ) m e t h o d of expressing the A matrix i n e x p l i c i t form has been modified by Wilson and Massena to t r e a t source elements i n the same manner as passive c i r c u i t elements. We s h a l l use the l a t t e r method w i t h .a s l i g h t modif-(.21) i c a t i o n * of the n o t a t i o n of Bacon i n reviewing the formul-a t i o n of the st a t e model of a- passive RLC network d r i v e n by i d e a l voltage and current sources. i t i s assumed that the sources form an independent set, that i s , there are no c i r c u i t s of voltage sources and no cut - s e t s of current sources. -;The f i r s t step i n s t a t e - v a r i a b l e f o r m u l a t i o n i s to choose a tree of the network-source graph by s e l e c t i n g i n order:. I) a l l voltage sources, - i i ) as many c a p a c i t o r s as p o s s i b l e , i i i ) as many r e s i s t o r s as p o s s i b l e , and i v ) as many inductors as req u i r e d to complete the t r e e . Thus, the tree contains as many c a p a c i t o r s as p o s s i b l e and the cotree as many inductors as p o s s i b l e . The branches of the tree w i l l be r e f e r r e d to as tree branches, those of the cotree as * In order to conform to normal p r a c t i c e i n the choice of r e f e r -ence d i r e c t i o n of voltages and currents of sources, our output vector w i l l be the negative of Bacon's. 13 chords. C l e a r l y , there are as many chord c a p a c i t o r s as there are c a p a c i t o r and ca p a c i t o r - v o l t a g e source c i r c u i t s . S i m i l a r l y , there are as many tree-branch inductors as there are inductor and i n d u c t 0 1 - c u r r e n t source c u t - s e t s . On the basis of t h i s t r e e , the fundamental c i r c u i t equations take the form u 0 0 0 B l l B 1 2 0 0 0 u 0 0 B 2 1 B 2 2 B 2 3 . 0 0 0 u 0 B 3 1 B 3 2 B 3 3 . B 0 0 0 u B 4 1 B 4 2 B 4 3 B 34 44 V cc V cr V c l V c i V. bv V. be V. br V. b l = 0 2 . 4 or, more c o n c i s e l y , U B. V = 0 where V = c V cc V cr V c l V. . c i V. bv V. > -V K = be br V. b l and. B l l B 1 2 0 0 B 2 1 B 2 2 B 2 3 0 B 3 1 B 3 2 B 3 3 B 3 4 B 4 1 B 4 2 B 4 3 B44 S i m i l a r l y , , the fundamental cut-set equations f o r the same tree are given by I b 0 2 .5 where cc c r c l I . c i bv be br b l In (?.A) and (2.5) the f i r s t s u b s c r i p t on the voltage and current v a r i a b l e s d i s t i n g u i s h e s between tr e e branches (b) and chords ( c ) . The second s u b s c r i p t denotes the type of elem-ent as f o l l o w s : c-- c a p a c i t o r , r - r e s i s t o r , 1 - in d u c t o r , i - current source and v - voltage source. The matrix expresses the t o p o l o g i c a l r e l a t i o n bet-ween the chords and the tree-branches. I t has the property that i t i s unimodular, that i s , a l l of i t s minors are equal to e i t h e r 4 l or 0 (2*0. i j ^ b entry i n B^ i s +1 i f the fundamental th t h c i r c u i t defined by the i chord contains the j tr e e branch and the voltage reference c o i n c i d e s w i t h that of the chord, and th -1 i f the voltage reference i s opposite; i t i s . 0 i f the j tre e branch i s not i n the c i r c u i t . The component te r m i n a l equations can be w r i t t e n i n the form 0 = I b c " 0 c c. V cc_ I cc_ ~ Lb o •" hi' V 0 L c_ V 0 I, " br _ br 0 G c_ V cr. I cr. 2. F i n a l l y , the s t a t e model i s e s t a b l i s h e d by e l i m i n a t i n g 15 a l l v a r i a b l e s except the s t a t e v a r i a b l e s (the tree-branch.cap-a c i t o r voltages and chord i n d u c t o r . c u r r e n t s ) , the source var-i a b l e s and the output v a r i a b l e s from equations ( 2 . 4 - 6 ) . A con-venient form of. the s t a t e model i s D X = AX + B E + C-jE E = P-^ X + C^X + Q E + R^E 2 . 7 where V. X = be ' c l E I . c i - I E bv -V c i 1" p b + B 1 2 C c B 1 2 0 0 L c + B 3 4 L b B 3 4 A l = 0 B^o 32 -0 B 2 2 U - R b B 2 3 -1 0 . R ~ b .^3 + • ; B 3 2 0 : B 3 3 0 _ c 2 ; u -G sB 0 0 0 C 22 B l •= " ° - B 4 2 _ Q B 2 2 U _ R b B 2 3 0 -R bB| 3 + - B 3 1 0 _ ; B 3 3 0 G B„_ c 23 U - G c B 2 1 0 _ '1 1 - B l 2 C c B l l 0 0 0 - B " B 3 4 L b B 4 4 t i r 31 B 4 2 0 + Q l -0 -B t - i 41 41 0 + 0 43 0 -B. t i '21 B 0 -B 21 B 4 3 0 -1 U -R>B" b 23 G B Q 7 U c 23 0 R b B 3 3 n -1 G B o v U c 23 " B o B 2 2 0 0 V « -G B o n 0 c 21 *1 = B l l ^ c E l l 0 0 B44 Lb B44 • (12) The matrix i s square and nonsingular , so ( 2 . 7 ) can be w r i t t e n i n the form of ( 2 . 1 ) , where A = D^" 1 ^, B = d_ 1 b_» c = Di l ci» P = Pi + C I D I ~ 1 A I » Q = Q 1 + C*D~ 1B 1 > R =R1 + C^D^ 1 C R I t i s evident from ( 2 . 7 ) that the order of the s t a t e vector (and the order of complexity of the network) i s equal to the number of tree-branch c a p a c i t o r s plus the number of 'chord i n d u c t o r s . In terms of the c o e f f i c i e n t matrices of ( 2 . 7 ) the frequency-domain model ( 2 . 3 ) becomes ( P 1 + sC^)(sD 1 - A 1) 1 ( B 1 + sC^) + Q 1 + sR 1 £(s) 2. £ ( s ) H l l ( s ) H l 2 ( s ) H 2 1 ( s ) H 2 2 ( s ) where _.(s) = oC and £_(s). = £ , H-^(s) i s a s h o r t -c i r c u i t admittance matrix, R~ 2 2(s) an o p e n - c i r c u i t impedance matrix, H-^2(s) a c u r r e n t - r a t i o t r a n s f e r matrix and H 2^(s) = -H-^2(s) (from r e c i p r o c i t y ) a v o l t a g e - r a t i o t r a n s f e r matrix. In conclusion, formulation of the s t a t e model of an RLC network i s a w e l l - d e f i n e d , s t r a i g h t f o r w a r d procedure. Sub-sequent determination of the frequency-domain model i s e a s i l y 17 accomplished. In the f o l l o w i n g s e c t i o n , we s h a l l examine i n some d e t a i l the models of two-element-kind networks. 2 . 3 Models of Two-Element-Kind Networks and D e l i n e a t i o n of a S p e c i a l Class of Networks Since two-element-kind networks belong to the general c l a s s of RLC networks, t h e i r s t a t e models'and frequency-domain models are given by ( 2 . 7 ) and ( 2.8), r e s p e c t i v e l y . However, because one kind of element i s missing, considerable s i m p l i f i -c a t i o n occurs. In t h i s s e c t i o n , models of the v a r i o u s two-element-kind networks w i l l be given. Because our i n t e r e s t l i e s i n the synthesis of e i t h e r impedance or admittance matrices, we s h a l l consider only networks d r i v e n by one kind of source. S p e c i f i c a l l y four types of net-works w i l l be considered: i ) LC networks d r i v e n by current sources (LCI networks), i i ) LC'networks d r i v e n by voltage sources (LCV networks), i i i ) RC networks d r i v e n by current sources (RCI networks), and i v ) RL networks d r i v e n by voltage sources (RLV networks). With the synthesis problem i n mind, we s h a l l define a c l a s s of two-element-kind networks f o r which the models take on p a r t i c u l a r l y convenient forms. 2 . 3 . 1 LCI and LCV Networks For general LC networks the s t a t e model ( 2 . 7 ) reduces to "c o" o B J : ' 32 + T ' ° B 4 2 bv + 0 L ^ 1 . r B 3 2 0 . f B 3 1 0 . I . c i - B 1 2 C c B l l 0 0 - B 3 4 L b B 4 4 . V bv I . c i .2.9a 18 -V c i 0 42 " B 31 0 V be " c l 3 i l ° o B 1 2 0 0 B 4 4 L b B 3 4 Bll°c Bll 0 0 B 4 4 L b B 4 4 V. bv c i be " c l + 0 41 - B 41 0 v.. bv I . c i + 2 . 9 b where C = C, + B* 0C Bn _ and L = L + B, . L ^ B J b 12 c 12 c 34 b 34 For L C I networks the t r a n s f e r f u n c t i o n matrix, H(s), becomes the o p e n - c i r c u i t impedance matrix, ^ - ^ ( s ) , given by Z ( B ) = B 4 2 ( B C + ^ I . - 1 ^ ) - ^ ^ 3 B 4 4 L b B j 4 B.„(sC + - B ^ L 1 1 4 2 . s 32 ! 5 2 ) 1 B ^ L 1 B 3 4 L B B ^ 4 - B 4 4 L b B 3 4 L _ l B 3 2 ( s C + . i B 3 2 L " l B 3 2 r l B 4 2 ' " s 2 B 4 4 L b B 3 4 ( s L + s B 3 2 C ~ l B 3 2 ) _ l B 3 4 L b B 4 4 ' 2 . 1 0 In the synthesis of a given matrix Z ^ ( s ) , we wish to determine a s t a t e model which i s r e a l i z a b l e as a passive L C network. A f i r s t step i n t h i s process must be the decomposition of Zj£i(s) i n t o the form of ( 2 . 1 0 ) . However, because of the complexity of ( 2 . 1 0 ) , i t i s not c l e a r how one would proceed. Hence, we seek a c l a s s of L C I networks f o r which the decompo-s i t i o n of Z-j-^(s) and, u l t i m a t e l y , the determination of a r e a l -i z a b l e s t a t e model i s more s t r a i g h t f o r w a r d . At the same time we do not wish to r e s t r i c t the s t r u c t u r e of the network too severely. •" Such a c l a s s of networks i s defined by r e q u i r i n g the matrix B ^ L ^ E ^ to vanish. Then the l a s t three terms i n ( 2 . 1 0 ) vanish, l e a v i n g Z L 0 ( s ) =• 3 4 2 ( s C + H 2 L _ 1 b 3 2 ) " 1 b 4 2 + s B 4 4 L b B 4 4 The s t a t e model ( 2 . 9 ) reduces to 2.11 C 0 0 L be • e l l 0 32 " B 32 0 V. be " c l + 3* " 42 0 ci 2.12 - V c i ] = [ B 4 2 °] V. be • c l The decomposition of a given Z ^ ( s ) i n t o the two terms i n ( 2 . 1 l ' now proceeds by i n s p e c t i o n since i n the second term B 4 4 L b B 4 4 =S1}Z s Z L C ( s ) ' The c o n d i t i o n B^^L^B^ = 0 i m p l i e s t h a t , w i t h appro-p r i a t e ordering of tree-branch i n d u c t o r s , the component matrix L, , and the c i r c u i t submatrices B„. and B.. have the forms b' 34 44 L b = L b l 0 0 L. b2 B 34 341 0 and B 44 0 B 442 S t r u c t u r a l l y , t h i s means that no tree-branch inductor i s common to a c i r c u i t defined by a chord inductor and to one defined by a current source. Figure 2 . 1 shows the block diagram of t h i s c l a s s of networks. Included i n the above c l a s s of networks as s p e c i a l cases are those networks f o r which i s empty, i s empty, or they are both empty, that i s , there are no tree-branch induc-t o r s at a l l . I t i s r e c a l l e d that the number of tree-branch inductors equals the number of inductor and inductor-current source cut-sets i n the network. I t f o l l o w s from the method of choosing the tr e e and from the c o n d i t i o n B V.L,B^. t h a t : 34 b 44 i ) chord inductors define c i r c u i t s with elements in. C, b and L b l only, i i ) current sources define c i r c u i t s with elements i n C and only, and i l l ) chord c a p a c i t o r s define c i r c u i t s with elements i n C only. F o l l o w i n g a s i m i l a r procedure, a c l a s s of LCV net-works can be defined having the s t a t e model b b C 0 0 L V. be " c l B, 32 t " 32 0 V. be c l + 0 -B, 31 Vbv] • - I bv ] " c l + Bn\C 11 c 11 V bv 2.13 and s h o r t - c i r c u i t admittance matrix Y L C ( s ) = B ^ ( s L + ^ C T ^ ) ~ 1 B 3 1 + s B ^ C c B i r 2.14 The above formula t i o n i m p l i e s that the matrix B l l ^ c B 1 2 = ^* Hence, wi t h appropriate ordering of chord c a p a c i t o r s , the component, matrix C c and the c i r c u i t submatrices B l l an'^ Bl° ^ a v e "the forms c = C , 0 c l 0 ' c 2 B. 11 B 111 0 and B 12 0 B 122 21 current sources chord capac i tors chord inductors tree-brar.ch inductors tree -branch capac i tors C b tree -branch inductors hi • • • tree Figure 2.1 Block "diagram of the c l a s s of ,LCI networks defined ^ B 3 4 L b B 4 4 = ° -tree -branch inductors • • • chord inductors L c cotree chord capac i tors C c2 voltage eources V chord capac i tors c l t ree -branch capac i tors Figure 2.2 Block diagram of the c l a s s ' o f LCV networks defined by B l l C c B 1 2 = °' 22 S t r u c t u r a l l y , no chord c a p a c i t o r i s common to a cut-set defined by a tree-branch c a p a c i t o r and to one defined by a voltage source. The form of t h i s c l a s s of networks i s shown i n the block diagram Figure 2 . 2 . I t f o l l o w s from the method of choosing the tree and from the c o n d i t i o n .B n^ C B, n = 0 t h a t : 11 c 12 i ) tree-branch inductors define cut-sets w i t h elements i n L c only, i i ) tree-branch c a p a c i t o r s define c u t - s e t s w i t h elements i n L c and C ^ only, and i i i ) voltage sources define cut-sets w i t h elements i n L c and C „ only. c2 0 C l e a r l y , t h i s c l a s s of LCV networks i s the dual of the c l a s s of LCI networks defined vabove. 2 . 3 . 2 RCI and RLV Networks Consider f i r s t RCI networks. With no inductors and v o l -tage sources the fundamental c i r c u i t equations ( 2 . 4 ) reduce to 0 0 U B, V cc 'l2 0 V cr *22 B 2 3 V . Cl = 0 42 B 4 3 _ br S i m i l a r s i m p l i f i c a t i o n s occur i n the fundamental cut-set equations. The s t a t e model of a general RCI network i s found to be CV. be - B 2 2 G B 2 2 V b c + (B 1 - B L G B ^ R , B ^ ) I . 42 22 23 b 4 3 ' c i -V c i ( B 4 2 - B 4 3 H B 2t3 e C B 2 2 ) V B O + B H B j 3 I c l 2.15 where C C b + B 1 2 C c B 1 2 ' G - + B 2 5 R b B 2 3 ; ( R ^ + B ^ G c B 2 3 ) t 1; - i - l and R = The matrices C, G- and R are symmetric and p o s i t i v e d e f i n i t e . The o p e n - c i r c u i t impedance matrix can be found by i n s p e c t i o n from ( 2 . 1 5 ) as Z R C ( s ) ^ B 4 2 ~ B 4 3 R B 2 3 & c B 2 2 ^ ^ s C + B 2 2 G B 2 2 ^ 1^ B42 + B 4 3 R B ^ . B 2 2 G B 2 3 R b B 4 3 } 2.16 Since Z R Q ( S ) i s symmetric f o r r e c i p r o c a l networks, i t i s evident that R B 2 3 G c = ^ G B 2 3 R b ^ ' Fo l l o w i n g the procedure of the l a s t s e c t i o n , we define a c l a s s of RCI networks by r e q u i r i n g the matrix B^jR-^B^ to van-i s h . The state model ( 2 . 1 5 ) and the impedance matrix ( 2 . 1 6 ) reduce t o , r e s p e c t i v e l y , GKo = - B 2 2 G B 2 2 V b c + B 4 2 I c i -V c i B,0V, + B.JRB* I . 42 be 43 43 c i and Z R C ( s ) .= B 4 2 ( s C + B ^ G B 2 2 ) - 1 B j 2 + B 4 3RB^, 2.17 •2.18 The c o n d i t i o n B ^ R ^ B ^ = 0 i m p l i e s t h a t , w i t h appropriate order-i n g of tree-branch r e s i s t o r s , R^, B 2 3 , and B^ 3 are expressed as R, R b l 0 0 R b2 ' B 2 3 231 0 and B 43 0 B 432 We also have G = (G 1 + B0„R,.,B* ) 1 and R = c 231 b l 231 ( R b ! + B 2 3 1 G c B 2 3 i r l ° 0 R. b2 C l e a r l y , with inductors replaced by r e s i s t o r s , the s t r u c t u r e of t h i s c l a s s of RCI networks i s the same as that of the c l a s s of LCI networks defined i n the preceding s e c t i o n and shown i n Figure 2.1. The form of ZJ^Q(S) a n d ~ ^ _ Q ( S ) are sim-i l a r except f o r a frequency transformation. The d i f f e r e n c e s i n form between the state models (2.12) and (2.17) r e f l e c t the d i f f e r e n t dynamical p r o p e r t i e s of r e s i s t o r s and in d u c t o r s . A c l a s s of RLV networks, which i s the dual, of the c l a s of RCI networks above, can be defined having the s t a t e model LI -, = —B„„RB^^I — B„-. V, c l 33 33 c l 31 bv - I , = - B j , I , + B* GB Q 1 V bv 31 c l 21 21 DV 2.19 and admittance matrix Y R L ( s ) = B j n ( s L + B^RB^) 1B, n + B ^ G B o v 31 33 33 31 21 21' 2 . 2 0 This f o r m u l a t i o n i m p l i e s - t h a t the matrix B„^G B„n = 0 . Hence, ^ 23 c 21 ' with appropriate ordering of chord r e s i s t o r s , the matrices•G , B„„- and B0-, have the forms 23 2± G = c G . 0 c l 0 G c2 B 23 B 231 0 and B 21 0 212 I t f o l l o w s that 25 G = ( G c l + B 2 3 1 R b B 2 3 l ' ) 0 -1 0 G c2 and R = + B 2 3 1 G c l B 2 3 1 ^ -1 Obviously, t h i s c l a s s of RLV networks i s r e l a t e d to the c l a s s of LCV networks i n the l a s t s e c t i o n i n the same manner that the c l a s s of RCI networks i s r e l a t e d to the c l a s s of LCI networks. I t should be pointed out t h a t RC (RL) networks d r i v e n by voltage (current) sources only do not a f f o r d the same type of s i m p l i f i c a t i o n of the models as do RCI (RLV) networks. 2 . 3 • 3 Summary Classes of two-element-kind networks d r i v e n by one kind of source have been d e l i n e a t e d . The models of these networks, expressed i n terms of c i r c u i t and component matrices, take on p a r t i c u l a r l y simple forms. The four types of networks, together wi t h t h e i r models, are summarized i n Figure 2 . 3 . With only one kind of source, the t r a n s f e r f u n c t i o n matrix i s symmetric f o r r e c i p r o c a l networks, so that = i n ( 2 . 7 ) . Using t h i s r e l a t i o n s h i p and the property that C^ and e i t h e r Q 1 ( f o r LCI and LCV networks) or R± ( f o r RCI and RLV networks) vanish f o r the networks under c o n s i d e r a t i o n , the st a t e model ( 2 . 7 ) reduces to D X = A-jX + B-j^ E E = B*X + (Q E or R E ) . Thus, the output vector i s made up of two terms; the f i r s t term i s a l i n e a r transformation of the state vector, the second a Type of network Defining condition Transfer function matrix RCI B 2 3 Rb B43 = 0 Z j ^ s ) = B^ 2(sC + B 2 2 G B 2 2 ^ B42 + ^kj^kj RLV B 2 3 G c B 2 1 " 0 YRL ( s )•'" B 3 1 ( s L + B 3 3 R B 3 3 ) " 1 b 3 1 + B 2 l G B 2 l • LCI W 3 ^ • 0 . Z L C ( s ) = B^ 2(sC + ^ B | 2 L "1 B 3 2 ) " 1 B J 2 + s B ^ I ^ B j ^ LCV B n c c B i 2 = 0 V S > " B 5 l ( s L + i - B 3 2 C " l B 3 2 ) " l B 3 1 + S B l l C c B l l State equations Output equations ° V = - B 2 2 G B 2 2 Vbc + : - V c i - \ z \ c + h j ^ c i L I c l = - B 3 3 R B 3 3 I c l " B 3 l V b v - h , = - ^ c l + B 2 1 G B 2 1 Vbv C V = B 3 2 I c l + B 4 2 I c i L * c l " " B 32 Vbc " V c i " V b c + V ^ c i C Vbc = 3 3 2 I c l « LI = -3,„V_ - B_,V, cl j.c be 31 ov -hv- - 4 l X c l + B l V c B l A v Figure 2.3 Summary of the models of a c l a s s of two-element-kind networks. ro 27 l i n e a r transformation of the input vector or i t s time d e r i -v a t i v e . The transformation matrix, B^, i n the f i r s t term has the. important property of being a c i r c u i t matrix and, hence, i t s en-t r i e s are e i t h e r -1: or 0. Therefore, each output v a r i a b l e con-t a i n s a weighted sum of the state v a r i a b l e s where the weighting f a c t o r s a r e , - l or.0. This i s a, most s i g n i f i c a n t property and forms.the bas i s of synthesis procedures to be developed l a t e r . -I t should be pointed out that the necessary and s u f f i c -i e n t c o n d i t i o n s f o r . the- r e a l i z a b i l i t y . of a given s t a t e model as (12) a passive LC network, without transformers are known .. .. As men-... tioned p r e v i o u s l y , a s t a t e model corresponding to a given t r a n s -f e r f u n c t i o n matrix, H(s), can always be found. However, t h i s does not solve the problem of the r e a l i z a b i l i t y of H(s). The non-real-i z a b i l i t y of a s p e c i f i c s t a t e model does not imply that H(s) i s not r e a l i z a b l e since-an equivalent s t a t e model obtained by a l i n e a r t ransformation of the s t a t e v e c t o r may be r e a l i z a b l e . 2.4 P r o p e r t i e s of RCI Networks P r e l i m i n a r y to d i s c u s s i n g synthesis of the c l a s s of two-element-kind networks defined- i n the l a s t s e c t i o n , we s h a l l examine some of t h e i r p r o p e r t i e s . Since the four types of networks are c l o s e l y related, e i t h e r by a frequency transformation or by d u a l i t y i t i s necessary to consider only one. type i n d e t a i l . The r e s u l t s w i l l apply, w i t h appropriate m o d i f i c a t i o n s , to the other three types. Consequently, we choose the RCI network as. the v e h i c l e f o r subsequent development. Throughout the remainder of the t h e s i s we s h a l l be d e a l i n g p r i m a r i l y with the c l a s s of two-element-kind networks defined i n Section 2 . 3 - Hence, except where nec-essary to avoid ambiguity, we s h a l l drop the awkward des-c r i p t i v e phrase w i t h the understanding that i t i s i m p l i e d . Some of the equations i n the previous s e c t i o n are repeated below f o r reference. The fundamental c i r c u i t equati ons f o r R C I networks are given by V U 0 0 B. 12 0 0 0 U 0 B 2 2 ^ 0 .0 0 U B 4 2 0 B 4 3 2 cc V . c r V . c i V be V. b r l V. b r 2 0 2 . 2 1 The s t a t e model i s given by CV be -Ba„GB 0„V., + B ^ 0 I . 22 22 be 42 c i —V . = B.„V, + B. ^ 0R-, 0 B . „ „ I . c i . . 4 2 be 432 b2 432 c i which can be w r i t t e n more c o n c i s e l y as 2 . 2 2 CV, = -JV, + B . 0 O I . be be 42 c i -V . = B , N V ' + Q I . c i 42 be . c i 2 . 2 3 where C = C, + B * C B 1 0 , G = (G 1 + B 0 „ R , , B * ) X ' b 12 c 12 c 2^1 b l . 2 3 1 J = B 2 2 G B 2 2 , 29 Q = B 4 3 2 R b 2 B 4 3 2 ' R = R b l 0  0 Rb2 The o p e n - c i r c u i t impedance matrix becomes Z R C ( s ) = B 4 2 ( s C + J ) _ 1 B 4 2 + Q. 2 .24 Z R Q ( S ) c o n s i s t s of two terms; the f i r s t , . B 4 2 ( s C + J ) ^BJJ^J vanishes at s =oo, and the second, Q, i s a constant. Hence, ZR^ (°°) = Q. Q i s given by a congru-ence transformation of a p o s i t i v e - d e f i n i t e diagonal matrix, R, 0 , where the transformation matrix, B . _ 0 , i s a c i r c u i t dz • ' 452 matrix. Thus, Q i s p o s i t i v e s e m i - d e f i n i t e and, by the (23) Fialkow-G-erst c o e f f i c i e n t c o n d i t i o n s , the magnitude of each o f f - d i a g o n a l entry i s not greater than the diagonal e n t r i e s . o f the same row and the same column. At t h i s point i t i s necessary to give s p e c i f i c dimensions to the network. Let the number of current sources be k. The two terminals to which each source i s connected form a t e r m i n a l - p a i r or port. Hence, the network i s a k-port s t r u c t u r e w i t h input-output r e l a t i o n s given by the kxk impedance matrix Z ^ ^ ( s ) . We; s h a l l r e f e r to the ports defined here as e x t e r n a l ports and to Z R (~,(s) as the e x t e r n a l impedance matrix. Since some terminal s may be common to more than one port, the network may have from (k + l ) to 2k e x t e r n a l t e r m i n a l s . Let the order•of complexity and, e q u i v a l e n t l y , the order of the s t a t e model be n. Hence, the order of C, i s • - . b a l s o n because the s t a t e v a r i a b l e s are the tree-branch c a p a c i t o r v o l t a g e s . Consider the network formed by removing the current sources (or s e t t i n g them to zero) and define a set of i n t e r n a l ports such that the terminals of the i ^ * 1 t h port are coincident w i t h the terminals of the i t r e e -branch c a p a c i t o r . In other words, the port voltages are equal to the tree-branch c a p a c i t o r v o l t a g e s . This i s equi-v a l e n t to s e t t i n g = U. The o p e n - c i r c u i t impedance matrix, Z ( s ) , of the n-port network thus formed i s given by Z n ( s ) = (sC + J ) - 1 . ' 2.25 The matrix Q does not appear i n Z (s) because the r e s i s t o r s i n are i n s e r i e s w i t h the o p e n - c i r c u i t e d e x t e r n a l p o r t s . The i n t e r n a l admittance matrix, defined by Y n ( s ) = Z n ( s ) ~ 1 = sC + J places i n evidence the s h o r t - c i r c u i t capacitance matrix, C, and the s h o r t - c i r c u i t conductance matrix, J , of the n-port network. I t should be pointed out that Y n ( s ) i s equi-v a l e n t to the complete admittance matrix defined by Basson (9) and Halkias as the s h o r t - c i r c u i t admittance matrix of an n-port network c o n t a i n i n g no i n t e r n a l nodes. We assume that the r e s i s t o r subnetwork c o n s i s t i n g of R, and G contains ° b l c no i n t e r n a l nodes. Any i n t e r n a l nodes can be e l i m i n a t e d by the use of g e n e r a l i z e d star-mesh transformations. The matrices C and G are symmetric and p o s i t i v e d e f i n i t e ; J i s symmetric and p o s i t i v e s e m i - d e f i n i t e . There-f o r e , i t i s always p o s s i b l e to f i n d a matrix M which simu l -taneously d i a g o n a l i z e s C and J by a congruence t r a n s f o r -mation., as f o l l o w s : MtCM = U. 2.26 and M tJM = L 2.2? where L - d i a g (X^, X^, •-> X ) . For passive RC networks M, c a l l e d the modal matrix, i s r e a l and the diagonal e n t r i e s i n L are nonnegative. S u b s t i t u t i n g (2.26-27) gives Y n ( s ) = ( M t ) ~ 1 ( s U + DM" 1 and Z (s) = M(sU + D ' V . 2.28 n P a r t i t i o n i n g M i n t o column's M = 1 2 n r? ( \ V- M. M"? v- M. n o n gives ~ V S ) = Z_ 1 1 = 2_ - i - • 2-29 i = l s+X^ 1=1 s+X-^ Equation (2.29) i s i n the form of a p a r t i a l f r a c t i o n expansion although i t i s not, i n general, the usual type of p a r t i a l f r a c t i o n expansion. An expansion .of t h i s type has been r e f e r r e d to as the normal form p a r t i a l f r a c t i o n (3) expansion The p a r t i a l f r a c t i o n expansion (2.29) contains n terms, one term associated w i t h each denominator f a c t o r (s + X^). The X^ are, of course, the eigenvalues of the matrix C "'"J (= -A i n the s t a t e model (2.1)). Now i t i s q u i t e p o s s i b l e that some of the eigen-values are n u m e r i c a l l y equal. Suppose that the eigenvalue are ordered so that n u m e r i c a l l y equal values are grouped together. Let r ^ i n d i c a t e the number of eigenvalues having the value and l e t p be the number of d i s t i n c t v alues. Then P Y~ r . = n 1=1 1 ^ l U r l * 2 U r2 0 0 X u p rp where U . i s the i d e n t i t y matrix of order r. r i J l Let M be p a r t i t i o n e d as f o l l o w s M m 77?2 . . .7/1 where 77?^  i s nxr^, i = 1, 2 , '.. Then z (s) = y n J—. t p 7n±rn± i=i s+x± = E i ~ i B + x ± 2 . 3 0 Equation ( 2 . 3 0 ) i s the usual p a r t i a l f r a c t i o n expansion of Z (s) where each denominator f a c t o r occurs j u s t once. The matrices???^, i = 1 , 2 , ., p are the matrices of residues at the poles s = ~^±> i = 1> 2 , ...,p, r e s p e c t i v e l y . Comparing ( 2 . 2 9 ) and ( 2 . 3 0 ) , i t i s evident that 77) . i s the sum of matrices- M. associated with a l l the numeric a l l y equal denominator f a c t o r s (s + X i ) . 33 Consider now the degree of Z ( s ) . As i s w e l l to n known, the degree of a matrix whose e n t r i e s are r a t i o s of polynomials and whose poles are simple i s equal to the sum of the ranks of i t s residue matrices Thus, i f r'." i s the rank of 77? }^ then degree of Z n ( s ) A S ( Z n ) = £ r ! Prom ( 2 . 3 0 ) , i t i s evident that _P i=Tl 1 s r ! £ r . , i = 1 , 2, ... . ,p Therefore, p < £(Z ) ^ n n That i s , the degree of Z (s) cannot be greater than the order of the s t a t e model or l e s s than the number of d i s t i n c t eigenvalues of the matrix.A. Networks w i t h degree l e s s than n are s a i d to be degenerate. The p h y s i c a l i n t e r p r e t a t i o n of repeated e i g e n v a l -ues i s that there i s more than one way of e x c i t i n g the as-s o c i a t e d n a t u r a l frequency. That i s , there are r ^ indepen-dent i n i t i a l s t a t e vectors that e x c i t e the n a t u r a l frequen-cy -X±. Consider next the e x t e r n a l impedance matrix which, from ( 2 . 2 4 ) and ( 2 . 2 5 ) , can be w r i t t e n Z R C ( s ) = B 4 2 Z n ( s ) B 4 2 + Q ' S u b s t i t u t i n g (2.28) gives Z R C ( s ) = B 4 2M(sU + L)~ 1M tBj 2-,+ Q = K(sU + L - r V + Q 2 . 3 1 where K = B^M. . 2 . 3 2 34 Equation (2.31) i s of the same form as (2.28) except that the matrix K i s kxn whereas M i s nxn and the constant matrix Q. i s added. P a r i t i o n i n g K i n t o columns K L L . . . K 1 2 n w e have, .analogous to ( 2 . 2 9 ) , n Z R C ( s ) i = l K. l i s+\^ n Q = i = l K. i _ •s+X, + 2 . 3 3 The rank of K. i s , i n general, u n i t y . However, i t i s p o s s i b l e that , f o r some i , B.0M. = 0, i n which case K. i s a n u l l matrix. ' ' 42 l - I As a r e s u l t , the corresponding pole f a c t o r (s+\^) from Z (s) does not appear i n Z R (-,(s). S i m i l a r to (2.30), we can w r i t e P Z R C ( s ) ~x±xl s+X. P + Q i = l i = l X i S+X: 1 + Q 2 . 3 4 where = ^42^1' Pr om Y. = B._7??-B.„, the rank o f Y . , rV , i s not greater than the -^ 1 42 1 4 2 ' A i ' 1' to B42 Z R C ( s ) , rank of ^ or Jul^, whichever i s smal l e r . Hence, the degree of 5 ( z R C ) r' i = l P r! s= n. 1 i = l .This proves the statement made e a r l i e r that the degree of the exter-n a l impedance matrix cannot be greater than the order, of the s t a t e model. I t should be pointed out that the above d i s c u s s i o n i s v a l i d f o r general RCI networks as w e l l , w i t h replaced by the matrix (from (2 .16) ' ) B, 0 - B. 7RB^ G B 0 0 . 4-2 43 23 c 22 F i n a l l y , consider the e f f e c t of the simultaneous - i i a g -o n a l i z a t i o n of C and J on the s t a t e model ( 2 . 2 3 ) . S u b s t i t u t i r . g (2.26-27) i n t o ( 2 . 2 3 ) gives (M t)~ 1M" 1V, = -(M t)~ 1LM~ 1¥, + I . v be . be 42 c i c i 42 D C c i P r e m u l t i p l y i n g the s t a t e equations by. and s e t t i n g X = M_1V, 2 . 3 5 be we have X = -LX + E^I . c i -V . = KX + QI .." - 2 . 3 6 c i c i That i s , the transformation of s t a t e v a r i a b l e s ( 2 . 3 5 ) transforms the state equations i n t o the normal form Comparing ( 2 . 3 1 ) and ( 2 . 3 6 ) , we see that the e x t e r n a l impedance matrix and the nor mal form s t a t e model are equivalent i n t h a t , given one, the other can be w r i t t e n immediately. As a r e s u l t , the synthesis problem can be r e s t a t e d as f o l l o w s : given an impedance matrix i n the form ( 2 . 3 1 ) or, equiv-a l e n t l y , a normal form s t a t e model'. ( 2 . 3 6 ) , f i n d a transformation matrix M that w i l l allow the s t a t e v a r i a b l e s to be i d e n t i f i e d as' ca p a c i t o r voltages of a passive RC network. .3. SYNTHESIS OF A CLASS OF RC MULTIPORT NETWORKS We s h a l l now turn to the synthesis problem. In par-t i c u l a r , we s h a l l consider the synthesis of i r r e d u c i b l e RC net-works from s p e c i f i e d open-circuit•impedance matrices i n the form of. ( 2.31). Accordingly, we s h a l l f i r s t o u t l i n e the general syn-t h e s i s procedure. Necessary c o n d i t i o n s w i l l . b e stated and the d i f f i c u l t i e s pointed out and discussed. Then the main synthesis procedure w i l l be given and i l l u s t r a t e d by s e v e r a l examples. I t w i l l be shown that general s o l u t i o n s can be found f o r functions of degree (k+ 2 ) , an increase of one over e x i s t i n g methods. F i n a l l y , we s h a l l give an a l t e r n a t i v e procedure which leads to networks w i t h the minimum number of c a p a c i t o r s . 3•1 The General Synthesis Procedure In c l o s i n g the l a s t chapter, the synthesis problem was r e s t a t e d i n concrete terms. However, Z Rp(s) would normally be given i n the form of the p a r t i a l f r a c t i o n expansion ( 2 . 3 4 ) rather than i n that of ( 2 . 3 3 ) . Equations ( 2 . 3 4 ) and ( 2 . 3 3 ) are repeated here as (3-1) and ( 3 . 2 ) , r e s p e c t i v e l y , Q 3-1 Q 3 . 2 - Z R C ( s ) =Y. i t x " + i = l 1 n E_ ti where the degree of ZR„(s) i s 37 S ( z R C ) = y_ P _ n •i=l For an i r r e d u c i b l e r e a l i z a t i o n ' t h e order, n, of the s t a t e model i s equal to the degree of the impedance matrix. Therefore, we take n = C S ( z R Q ) • In the above, i t i s assumed that <5(Z R^)2rk, which i s the case of most i n t e r e s t . When cS(Z R (0<k, there are at l e a s t k - §(Z RQ) n u l l residue matrices. This c o n d i t i o n represents a degenerate case which we s h a l l not consider. As a p r e l i m i n a r y step, i t i s necessary to w r i t e Z R (^(s) i n the form ( 3 . 2 ) . In order to do t h i s , every residue matrix V. must be factored i n t o the sum of r'.' u n i t rank matrices. I t i s e a s i l y shown that such a f a c t o r i z a t i o n i s always p o s s i b l e , but not unique. We s h a l l show i n Section 3 - 2 . 6 that the r e a l i z a b i l i t y of ZR£,(s) and any r e s u l t i n g network are independent of the f a c t o r - -i z a t i o n chosen. Therefore, without l o s s of g e n e r a l i t y , we s h a l l assume Z R^(s) i s given i n the form ( 3 . 2 ) . With reference to ( 2 . 2 3 ) , the l o g i c a l f i r s t step i n the synthesis process i s seen to be the decomposition of the constant matrix Q i n t o the product Q = B 4 3 2 R b 2 B 4 3 2 3 " where R ^ i s a p o s i t i v e - d e f i n i t e diagonal matrix and B^.^ i s r e a l -i z a b l e as a c i r c u i t matrix of a l i n e a r graph c o n t a i n i n g k indepen-dent c i r c u i t s . I f such a decomposition i s not p o s s i b l e , we can say immediately that Z R ^ ( s ) i s not r e a l i z a b l e as a network within the c l a s s under c o n s i d e r a t i o n . Techniques f o r decomposing Q and f o r t e s t i n g whether B^.^ i s a r e a l i z a b l e c i r c u i t matrix are w e l l (22) known . D e t a i l s are given i n Appendix A. Under the assumption that Q can be decomposed i n t o the form (3.3), we consider now synthesis of the other terms'in Z R Q ( S ) which we w r i t e as Z R 0 ( s ) = X n K. 1 J R C v o / - s+X. i = l 1 As the matrices are of u n i t rank, we have, by i n s p e c t i o n , Z R C ( s ) = K(sU + L ) " 1 ^ 3-4 where L = diag(X 1,\ 2,-. . . , \ n ) ' K = . . . K. 2 * • • Xi-n and K. = K. , i = 1, 2 n . 1 1 1 ' ' ' ' Since Z R (^(s) vanishes at s =00, any r e a l i z a t i o n must have a c a p a c i t o r path between the two terminals of each port. Prom (3-4), l i m s Z R 0 ( s ) = KK t. Thus, i n the l i m i t as s approaches i n f i n i t y , ZR(~,(s) becomes the o p e n - c i r c u i t impedance matrix of a c a p a c i t o r network. Therefore, the synthesis of ZR(~,(s) can' begin from the necessary c o n d i t i o n that ~KK^ he r e a l i z a b l e as the o p e n - c i r c u i t impedance matrix of a k-port c a p a c i t o r network. Discussion ox , the d e t a i l s of s y n t h e s i z i n g the network w i l l be given l a t e r . 1 t For now, we assume that —KK can be r e a l i z e d by a c a p a c i t o r net-work with nonnegative elements. From (2.24), we have the r e l a t i o n Z R C ( s ) = B 4 2 ( s C + J ) _ 1 B J j 2 . 3-5 39 Comparing ( 3 - 5 ) w i t h ( 3 . 4 ) gives. B 4 2 C _ l B 4 2 = ^ • 3.6 where C = C, + B^0C Bn . b 12 c 12 By a n a l y s i n g the c a p a c i t o r network j u s t synthesized, we can determine the component matrices C^ ai±d and the c i r c u i t mat-r i c e s B.^ 2 and B^. At t h i s p o i n t , we observe that the s t r u c t u r e of the cap a c i t o r network must be con s i s t e n t w i t h that of the r e s i s t o r network already synthesized from the Q matrix. The necessary must be a co n d i t i o n i s that the composite matrix r e a l i z a b l e c i r c u i t matrix. B12 0 J 42 "432 .We tu r n next to the modal matrix. Having e s t a b l i s h e d C and B ^ j we can use (2.32) and (2.26), which are given below as (3.7) and (3.8), i n determining M. K = B 4 2M 3.7 MtCM = U 3.8 Equation (3-7) can a l s o be w r i t t e n K ± = B 4 2M i, i = 1, 2, . . ., n. 3.9 Assuming the rank of B ^ i s k, (3-9) gives a set of k l i n e a r eq-uations i n the n e n t r i e s of each column of M. Hence, a t o t a l of 2 kn of the n e n t r i e s can be elim i n a t e d . Equation (3-8) gives a set of n equations which are, i n general, n o n l i n e a r i n the unknown e n t r i e s of M. Because of symmetry, there are only -5-n(n+l) d i s t i n c t equations. However, i t w i l l be shown l a t e r that'(3.7) and (3-8) do not represent 40 an independent set of equations i n general. Thus, we have fewer equations than unknowns and M i s not uniquely determined. However, proceeding on the b a s i s that M has been det-ermined, we can c a l c u l a t e the n x n s h o r t - c i r c u i t conductance matrix, J . From ( 2 . 2 7 ) , J = ( M ^ ^ L M - 1 or s u b s t i t u t i n g ( 3 . 8 ) , t, J = CMLMUC. 3 . 1 0 J must be r e a l i z a b l e as an n-port r e s i s t o r network with nonneg-a t i v e elements. Assuming the r e s i s t o r network can be synthes-i z e d , we can i d e n t i f y the matrices GQ, R ^ i ' B 2 2 a n ( ^ B 2 3 1 i n J = B^ 2(G- 1 + B 2 3 1 R b l B ^ 3 1 ) 1 B 2 2 . F i n a l l y , the r e s i s t o r network and -the c a p a c i t o r network must have been synthesized on i d e n t i c a l port c o n f i g u r a t i o n s (the ports r e f e r r e d to here are t h e . i n t e r n a l ports defined i n Section 2 . 4 ) . The s u f f i c i e n t c o n d i t i o n i s that the composite matrix B 12 0 B 2 2 B 2 3 1 B 4 2 0 B 0 0 432 must be a r e a l i z a b l e c i r c u i t matrix. We observe that C and J must n e c e s a r i l y have the same s i g n patterns. In the approach o u t l i n e d above, the synthesis problem becomes e s s e n t i a l l y that of s y n t h e s i z i n g two single-elemerit-kind networks on i d e n t i c a l port c o n f i g u r a t i o n s . I t remains to be shown 41 whether anything has been gained. There are some seemingly insurmountable problems i n t h i s approach. Foremost among them i s the unsolved problem of the synthesis of single-element-kind n-port networks -which can be s t a t e d b r i e f l y as f o l l o w s : given an nxn symmetric matrix of r e a l e n t r i e s , what are the necessary and s u f f i c i e n t c o nditions f o r t h i s , matrix to be the o p e n - c i r c u i t impedance matrix or s h o r t -c i r c u i t admittance matrix of an n-port r e s i s t o r network? This problem i s unsolved f o r n^3, although some necessary and some s u f f i c i e n t c o n d i t i o n s are known . However, the s o l u t i o n i s known f o r networks w i t h exac-t l y n+l terminals ^ 2 2^. A necessary c o n d i t i o n f o r the n^*1 order symmetric matrix W w i t h constant e n t r i e s to be the immittance mat-r i x of an n-port r e s i s t o r network w i t h n+l terminals i s that i t can be decomposed i n t o the product ¥ = VDVt . - 3 . 11 where D i s a diagonal matrix w i t h p o s i t i v e diagonal e n t r i e s and every entry of V i s e i t h e r —1 or 0 . A necessary and s u f f i c i e n t c o n d i t i o n f o r the r e a l i z a b i l i t y of W as an o p e n - c i r c u i t impedance matrix i s that V be r e a l i z a b l e as a c i r c u i t matrix of a l i n e a r graph c o n t a i n i n g n independent c i r c u i t s , and f o r r e a l i z a b i l i t y as a s h o r t - c i r c u i t admittance matrix,V must be r e a l i z a b l e as a cut-set matrix of a l i n e a r graph w i t h n+l nodes. Hence, we are of n e c e s s i t y constrained to s y n t h e s i z i n g networks i n which the matrix —KK^ i s r e a l i z a b l e as a (k+l)-terminal s c a p a c i t o r network and J i s r e a l i z a b l e as an (n+l)-terminal r e s i s t o r network. The decomposition ( 3 . 1 l ) i s performed using an algorithm 42 due to Cederbaum . As t h i s decomposition plays an important r o l e i n the synthesis procedures', developed h e r e i n , it.was f e l t appropriate to o u t l i n e the procedure of the , algorithm (see Appendix A). '•""'. The decomposition of W, t e s t i n g f o r the r e a l i z s . b i l i t y of V and determining.the s t r u c t u r e ,of the network, while s t r a i g h t -forward, n e c e s s i t a t e a great many computations. A t o p o l o g i c a l method given by G u i l l e m i n (24) anr\f independently, by B i o r c i and (25) ' ' C i v a l l e r i • i s often more .suited to p r a c t i c a l a p p l i c a t i o n . We s h a l l o u t l i n e the procedure f o r the r e a l i z a t i o n of an admit-tance matrix. F i r s t , a network tr e e i s constructed. One begins w i t h any branch and s u c c e s s i v e l y adds the other branches, b e i n g . guided as to t h e i r r e l a t i v e p o s i t i o n s by the signs of the e n t r i e s i n the r e s p e c t i v e rows and columns of W. Construction of the tree i s s t r a i g h t f o r w a r d and always p o s s i b l e unless one encounters a c o n t r a d i c t i o n , i n which case no tre e e x i s t s and W has no r e a l -ization i n an (n+l)-node network. I f a tree i s found to e x i s t , W i s transformed i n t o another matrix r e f e r r e d to a set of node-to-datum independent v o l t a g e s . A r e a l i z a t i o n w i t h nonnegative element values e x i s t s i f and only i f the transformed matrix i s hyperdominant. The other problems encountered i n v o l v e the l a c k of un-iqueness of some of the steps, and the nonlinear character of equations i n the e n t r i e s of M. However, non-uniqueness may be more of an asset than a problem because i t can be used to advan-tage i n searching f o r a s o l u t i o n or i n f i n d i n g an optimal s o l u -t i o n . On the other hand, the non l i n e a r character of the equations w i l l prove to be the major stumbling block i n applying our 43 procedure. We s h a l l give two synthesis procedures which d i f f e r -mainly i n the r e a l i z a t i o n of the c a p a c i t o r network. In Proced-ure I, developed i n the f o l l o w i n g s e c t i o n , part of the capa c i t o r network i s synthesized by decomposing the matrix (KK^) i n the manner of (3.11)- This leads to a network which, i n general, contains c a p a c i t o r c i r c u i t s . On the other hand, i n Procedure I I , developed i n Section 3-4, part of the ca p a c i t o r network i s synthesized by decomposing the matrix KK^. This leads to a net-work wi t h the minimum number of c a p a c i t o r s , that i s , with no cap-a c i t o r c i r c u i t s . Procedure I i s the p r e f e r r e d method because i t i s more s t r a i g h t f o r w a r d computationally and because Procedure I I cannot be used i n .every case. As a r e s u l t , we s h a l l give considerably more a t t e n t i o n to the former. 3.2 Synthesis Procedure I In t h i s section,, we s h a l l give a step-by-step procedure f o r the synthesis of a given kxk o p e n - c i r c u i t impedance matrix of degree n. We assume the form Z R C ( s ) ='K(sU + L - r V + Q 3.12 where, as p r e v i o u s l y , we l e t Z R C ( s ) = K(sU + I O ' V . We are constrained to the synthesis of the c l a s s of networks shown i n the block diagram i n Figure 3-1 because of our l i m i t e d knowledge of the synthesis of single-element-kind net-works. As a r e s u l t , the p o r t i o n of the network with impedance 44-matrix Z R Q ( S ) must contain at l e a s t one c a p a c i t o r tree,' that i s there are no tree-branch r e s i s t o r s ( p r e v i o u s l y designated by chord capacitors C current sources ^ i k+1 tree-branch r e s i s t o r s k+1 tree-branch capacitors n+l chord r e s i s t o r s G. 5— ZHC(S) Z : c ( s ) Figure 3 . 1 Block diagram of RCI networks synthesized by Procedure I In terms of the component and c i r c u i t m a t r i c e s * Z R C ( s ) = B 4 2 ( s C + J ) ^ + B 4 3 R b B j 3 ' 3 . 1 3 where C = C, + B^0C Bn _ and J = B^0G Bnn . b 12 c 12 22 c 22 The o b j e c t i v e of the 'synthesis procedure Is to decompose ( 3 . 1 2 ) i n t o the form ( 3 . 1 3 ) which gives us the s t a t e model. Further-more, the decomposition must lead to a r e a l i z a b l e RC network with nonnegative element values. 3 . 2 . 1 R e a l i z a t i o n of the Matrix Q Synthesis of the p o r t i o n of the network designated by R^ i n Figure 3 . 1 r e q u i r e s the decomposition of the symmetric, constant matrix Q i n t o the product * For s i m p l i c i t y , we s h a l l new drop the s u b s c r i p t " 2 " i n B.„„ and i n R^* 45 t Q = B 4 3 R b B 4 3 where R, i s a diagonal matrix and the e n t r i e s of B.., are e i t h e r b fa 43 +1 or 0. For a passive r e a l i z a t i o n the diagonal e n t r i e s of R^ must be p o s i t i v e and B.„ must be a r e a l i z a b l e c i r c u i t matrix.* 45 The above i s e x a c t l y the decomposition given by (3.11)• Hence, a necessary c o n d i t i o n f o r the synthesis of Z ^ ( s ) i s that Q be r e a l i z a b l e as a r e s i s t o r network with k independent c i r c u i t s . 3.2.2 Determination of the Capacitance Matrix C Next, we synthesize an (n+l)-node c a p a c i t o r network, given the kxk o p e n - c i r c u i t impedance matrix —KK^. C l e a r l y , . when n>k, the c a p a c i t o r network w i l l not be unique. Thus, the given matrix must be augmented i n some way to the nxn matrix associated w i t h an n-port network. •We must a l s o bear i n mind that the network c o n f i g u r a t i o n chosen here w i l l a f f e c t the synthesis of the conductance network because C and J must be r e a l i z a b l e by networks w i t h i d e n t i c a l ( i n t e r n a l ) port c o n f i g u r a t i o n s . Hence, i t i s d e s i r a b l e to synthe-s i z e the c a p a c i t o r network i n a manner that c o n s t r a i n s the port c o n f i g u r a t i o n as l i t t l e as p o s s i b l e . With t h i s objective''in mind, we take the kxn c i r c u i t mat-r i x B ^ to have the form B 42 [ u 0 3.14 That i s , Z Rp(s) i s taken to be the kxk p r i n c i p a l submatrix of the pxn i n t e r n a l impedance matrix Z ( s ) . The t o p o l o g i c a l meaning of th t h i s choice i s that the i (external) current source defines a c i r c u i t c o n t a i n i n g but one, the i " ^ , tree-branch c a p a c i t o r . *- For b r e v i t y , we s h a l l use the c o n d i t i o n that a p-row matrix B "must be a r e a l i z a b l e c i r c u i t matrix" to imply that B "must be realiza.ble as a c i r c u i t matrix of a l i n e a r graph c o n t a i n i n g p independent c i r c u i t s " . 46 Let the nxn capacitance matrix be p a r t i t i o n e d as f o l l o w s C = '11 '12 C12 C22 I t i s evident that minimum c o n s t r a i n t w i l l be placed upon the i n t e r n a l port c o n f i g u r a t i o n by t a k i n g and Then °12 - 0 C 2 2 = diag ( c k + 1 , c k + 2 , . . ., c n) C X 1 = ( K K t ) ~ 1 . For r e a l i z a t i o n as an (n+l)-node network, C must be decomposable i n t o ,t C• = |B^2 U C 0 c 0 C, B. 12 U °b + B 1 2 C c B 1 2 which i s equivalent to (3.11). In p a r t i t i o n e d form, the decom-p o s i t i o n becomes (KK1)"1 0 0 '22 °bl + B121°c B121 0 0 '22 °bl 0 0 c 22 + B. 121 0 B121 0 1 • ] The usual conditions must be s a t i s f i e d , namely, the diagonal mat-r i c e s C, and C must be p o s i t i v e d e f i n i t e and B, „, must be a r e a l -b c r 121 i z a b l e c i r c u i t matrix. 47 The above decomposition gives enough information to construct the p o r t i o n of the network containing ( i ) the current sources, ( i i ) the tree-branch r e s i s t o r s ( i n R^), ( i i i ) the chord c a p a c i t o r s , and ( i v ) k of the tree-branch c a p a c i t o r s . For top-o l o g i c a l r e a l i z a b i l i t y of t h i s subnetwork, the composite matrix B. 12 0 B 42 B 43 B 121 U 0 0 0 B 43 must be a r e a l i z a b l e c i r c u i t matrix. The diagonal e n t r i e s i n C 2 2 are not s p e c i f i e d here and are f r e e to be chosen at some l a t e r stage, with the r e s t r i c t i o n that they be nonnegative. For l a t e r convenience, we define where D 2 = d i a g ( d k + 1 , d k + 2 , d . = . / c . d j n The p o s i t i v e square root i s i m p l i e d . In summary, we have taken the inverse of KK U as the kxk p r i n c i p a l submatrix of the capacitance matrix C. By t a k i n g a l l other o f f - d i a g o n a l e n t r i e s of C to be zero, we synthesize at t h i s stage a k-port c a p a c i t o r network only, l e a v i n g the c o n f i g u r a t i o n of the remaining n-k c a p a c i t o r s f o r a l a t e r step. In t h i s way, the minimum c o n s t r a i n t i s placed upon the i n t e r n a l port c o n f i g -u r a t i o n . 3.2.3 C a l c u l a t i o n of the Modal Matrix P a r t i t i o n i n g the modal matrix M. 1 M = M. 48 whe re i s kxn and i s (n-k;xn, and s u b s t i t u t i n g B^ 2 =[u o j i n t o K = B^M, we obtain K = M. 1' Hence, the f i r s t k rows of the modal matrix are given d i r e c t l y by the s p e c i f i e d impedance matrix ( 3 - 1 ? ) • From MtCM = U, we have C"1 = MM13. which, i n the p a r t i t i o n e d form, becomes KK 0 t 0" U 2 2 M 2 KK M 2K l KM.2 3.15 t h I f we t h i n k of the e n t r i e s of the i row of M as measure numbers, — th on an orthonormal b a s i s , of a vector on^, then the i j entry i n i s the s c a l a r product . .'From ( 3 . 1 5 ) , KM t 2 M 2 M 2 0 °22 - D2 or, i n terms of s c a l a r products, . . . ., k; j = k+1, k+2, [0, 3 £ i ana in. .m . = 0, i = l , 2 , I D 1 j 1,2 > , i , j = k+1, k+2, , n, d. i = i 1 d s t t h Therefore, the (k+l) to n vectors are orthogonal to each other'and to the l s ^ to vectors.- Thus, ( 3 . 1 5 ) w i l l be r e f e r r e d to as the o r t h o g o n a l i t y c o n d i t i o n s . I t i s convenient to normalize vectors w , i = k+1, k+2, . 49 . ., n by setting where Thus, N = DM "U 0 0 D D = 2 N = 1 N 2 " K D' M 2 • 2_ C l e a r l y , N 2 i s independent of. the parameters d^, The o r t h o g o n a l i t y c o n d i t i o n s become and KN^ = .0 N 2N 2 = U 3.16 3.17 Equation (3.16) gives a set of k(.n-k) l i n e a r equations and (3.17) a set of -Hn-k+l) (n-k) nonlinear equations i n the n(n-k) unknown e n t r i e s of . Let f equal the number of unknowns l e s s the num-ber of equations. Thus, . f = i ( n - k ) ( n - k - 1 ) . We can t h i n k of f as the number of degrees of freedom i n the ( k + l ) s ^ to 'vectors. They are, i n f a c t , r o t a t i o n a l degrees of freedom. For example, f o r n-k = 2, there i s one degree of freedom which i s the angle of r o t a t i o n of the orthogonal vectors rrn and w ..about an a x i s normal to the plane of the two vectors, n—JL n Because of the nonlinear nature of (3.17), i t i s not p o s s i b l e , i n general, to determine a n a l y t i c a l l y i n terms of f of i t s e n t r i e s . Of course, i t i s always p o s s i b l e to choose f e n t r i e s a r b i t r a r i l y , but then we cannot be sure of f i n d i n g a pas-s i v e r e a l i z a t i o n should one e x i s t . 50 3.2.4- C a l c u l a t i o n and R e a l i z a t i o n of the Conductance Matrix J Assuming that the modal matrix has been determined i n some way, the s h o r t - c i r c u i t conductance matrix can be c a l c u l a t e d using (3-10). In. the p a r t i t i o n e d form, we have J = J l l J12 J21 J22 ( K K B ) • 0 • U 0 t N - l 0 0 D 2 K L ( K K 1 ) - 1 0 ( K K * ) 1 K L K T ( K K T ) ~ 1 - ( K K ^ ) " ^ K L N ^ N 2 L N 2 0 4 U 0 0 D, DJ D. We note that the matrix J 1 ( K K T ) ~ 1 K I J K T ( K K T ) ~ 1 N 2 L K ^ ( K K ^ ) 1 ( K K * ) - 1 K L N | N 2 L N ^ is independent of the unknown diagonal e n t r i e s i n D. I t i s also evident t h a t , because D i s a p o s i t i v e - d e f i n i t e diagonal matrix, J and have i d e n t i c a l s i g n p atterns. To be r e a l i z a b l e as an n-port r e s i s t o r network with n+l nodes, J must be decomposable i n t o J =. B^ 2G cB 2 2 where 0 i s a p o s i t i v e - d e f i n i t e diagonal matrix and B„„ i s a r e a l -c • * to 22 i z a b l e c i r c u i t matrix. The r e a l i z a t i o n of J y i e l d s the (n+l)-terminal r e s i s t o r network designated by &q i n Figure 3.1. The ports of G are the 51 p r e v i o u s l y defined i n t e r n a l ports. I t i s r e c a l l e d that the t e r -th minals of the i i n t e r n a l port are c o i n c i d e n t with the terminals Th of the i tree-branch c a p a c i t o r . Hence, by s y n t h e s i z i n g G-^, we i n e f f e c t determine the c o n f i g u r a t i o n of the tree-branch capac-i t o r s . However, t h i s c o n f i g u r a t i o n must be t o p o l o g i c a l l y c o n s i s -tent with the network synthesized from Q and i n e a r l i e r steps. This requirement i s embodied i n the necessary c o n d i t i o n that the composite matrix B. 12 B 22 0 0 B 4 2 B 4 3 where B. 12 B 1 2 1 0 and B 42 U 0 , must be a r e a l i z a b l e . c i r -d can often be c u i t matrix. The parameters c^+l, ^k+2' ' ' n chosen to advantage f o r t o p o l o g i c a l r e a l i z a b i l i t y of the network as w e l l as f o r nonnegative element values. A necessary c o n d i t i o n which can be a p p l i e d d i r e c t l y to the s p e c i f i e d impedance matrix w i l l now be formulated. R e a l i z a -b i l i t y of B^ i m p l i e s that C and J must be r e a l i z a b l e as n-port networks with i d e n t i c a l port c o n f i g u r a t i o n s from.which i t . f o l l o w s that C-^ and J - ^ must be r e a l i z a b l e as k-port networks w i t h iden-t i c a l port c o n f i g u r a t i o n s . The r e a l i z a b i l i t y of was c o n s i -dered i n Section 3 . 2 . 2 . J - ^ i s r e a l i z a b l e i f , i n the decomposi-t i o n , J l l = = 2 2 ^ 2 2 G^  i s a diagonal matrix with p o s i t i v e diagonal e n t r i e s and. B^^ i s a r e a l i z a b l e c i r c u i t matrix. F i n a l l v , C, -, and J n , are r e a l -" 11 11 i z a b l e with i d e n t i c a l port c o n f i g u r a t i o n s i f the composite matrix 52 B 1 2 1 i B 2 2 i s a r e a l i z a b l e c i r c u i t matrix.. 3 . 2 . 5 Synthesis of Minimal Networks As discussed, e a r l i e r , an nxn symmetric constant mat-r i x W can be r e a l i z e d as an (n+l)-node, n-port network through the use of Cederbaum's decomposition algorithm. The decompos-th i t i o n of W y i e l d s an m order diagonal matrix whose diagonal e n t r i e s are the element values. I t i s shown i n Appendix A that l ^ m ^ ^ n ( n + l ) . C l e a r l y , m = ^ ( n + l ) unless there are special, r e l a t i o n s h i p s among the e n t r i e s of W, such as some of the o f f - d i a g o n a l e n t r i e s equal to zero. In other words, ^ ( n + l ) elements are required to r e a l -i z e W i n general. We note that an (n+l)-node network with a f u l l graph contains -sn(n+l) elements where one element i s connected between every p a i r of nodes. A minimal network has been defined p r e v i o u s l y as one i n which the number of elements i s equal to the number of indep-endently s p e c i f i a b l e . p a r a m e t e r s . Hence, the r e a l i z a t i o n of W above y i e l d s a minimal network because there are -g-n(n+l) indep-endent e n t r i e s i n an n^*1 order symmetric matrix. In the kxk impedance matrix Z R C ( s ) = K(sU + L ) " 1 ! ^ + Q there are -5k(k+l) independent parameters i n Q, kn i n K and n i n L. The r e a l i z a t i o n of Q r e q u i r e s i>"k(k+l) r e s i s t o r s i n general. The number of c a p a c i t o r s r e q u i r e d c o n s i s t s of -§-k(k-l) i n C c iand n 53 i n C^. F i n a l l y , the r e a l i z a t i o n of J req u i r e s ^n(n+l) r e s i s t o r ' s . Hence, there are, i n genera.l, a t o t a l of n - k + -Hn-k) (n-k-l) more elements than independently s p e c i f i a b l e parameters. However, an equal number of u n s p e c i f i e d parameters i s introduced by the synthesis procedure. These c o n s i s t of n-k d i a -gonal e n t r i e s i n and -g-(n-k) (n-k-l) degrees of freedom i n the . modal matrix. Therefore, a minimal network r e s u l t s i f the unspec-i f i e d parameters are chosen such that an equal number of conduc-tances ( i n G ) vanish. The procedure w i l l be i l l u s t r a t e d by ex-amples i n Section 3 - 3 . P r a c t i c a l s ynthesis of minimal networks i s l i m i t e d to cases where n-k i s small because of the nonlinear i t y of the o r t h o g o n a l i t y c o n d i t i o n s . In the synthesis procedure o u t l i n e d i n previous sec-t i o n s , the decompositions of Q, C.^ and J could lead to fewer than the normal number of network elements. Although such an occurrence appears to r e s u l t i n fewer elements than independently s p e c i f i a b l e . parameters, the number of t h e ' l a t t e r i s , i n e f f e c t , decreased by an amount equal to the number of s p e c i a l r e l a t i o n s h i p s among the s p e c i f i e d parameters. 3 . 2 . 6 F a c t o r i z a t i o n of the Residue Matrices In t h i s s e c t i o n , we s h a l l show that n e i t h e r the r e a l -i z a b i l i t y of Z R Q ( S ) nor the r e s u l t i n g network are a f f e c t e d by the p a r t i c u l a r f a c t o r i z a t i o n of residue matrices chosen. C l e a r l y , i t i s s u f f i c i e n t to show that the matrices C and J are independent of the f a c t o r i z a t i o n . Suppose, f o r s i m p l i c i t y , that every residue matrix but 54 one has u n i t rank. Then we can w r i t e n K. Z R C ( s ) = " s i \ + 1 i=r+l s+A.^  where the rank of i s r and the rank of i s un i t y , Some f a c t o r i z a t i o n x i = l 1 leads to a matrix K K l l \2 where K 1 1 K 1 1 ~ ^ 1" Equation (3.18) becomes Z R C ( s ) = K L 1 K 1 2 sU + L ' 0 0 -1 K K. t • 11 t 12 where and U 2 a r e i d e n t i t y . matrices of order r and h-r, respec-t i v e l y , L x = \ 1^ and L 2 = d i a g ( \ 1,\ r + 2',•• . .., X f i). • We s h a l l show next that any other f a c t o r i z a t i o n of leads to a matrix K which i s an orthogonal transformation of K. Since K^ 2 corresponds to u n i t rank residue matrices, i t i s not changed by the transformation. Hence, K = KT = where But T = "11 0 K T K 11 11 12 0 U A A 4-K 1 1 K 1 1 < K l l T l i > < T i l K l l > = x 1 Therefore, ^ l ^ l l = ^ a n <^ "^l] ( a n d hence T) i s an orthogonal mat-r ix . 55 From K = B^M, the nodal matrix i s given by M = K l l K12 M ¥ 21 22 and transformed modal matrix by M = M T = 11 11 12 M T M 21 11 22 a t i o n i s The inverse capacitance matrix f o r the f i r s t f a c t o r i z -,-1 and f o r the transformed f a c t o r i z a t i o n C"1 = MM^  = (MT).' (T V) = MM13. Thus, the capacitance matrix i s independent of the f a c t o r i z a t i o n . In order to show that the conductance matrix i s indep-endent of the f a c t o r i z a t i o n , i t i s necessary only to examine the matrix MLM^. For the transformed f a c t o r i z a t i o n , we have MIM^ = M T L T V . In p a r t i t i o n e d form TLT^  T 0 11 0 U \ 1 u 0 0 L rpt Q 11 0 U \ 1 u 1 0 0 .Lv L . Therefore, J = CMLM C i s a l s o independent of the f a c t o r i z a t i o n . C l e a r l y , t h i s r e s u l t i s e a s i l y extended to the case where any number of residue matrices must be factored. Hence, we have shown that the synthesis procedure i s independent of the p a r t i c -u l a r f a c t o r i z a t i o n and one need only choose the most convenient method f o r f a c t o r i n g the residue matrices. 56 3.2.7•_Summary of Synthesis Procedure I We s h a l l now summarize the synthesis procedure dev-eloped i n preceding s e c t i o n s . I t i s assumed.that the kxk open-c i r c u i t impedance matrix Z R (-,(s), which i s of degree n, i s given i n the form Z R C ( s ) . X(sU + D ' V + Q. Necessary conditions are stated i n each step of the procedure. Step 1: Decompose: Q = B^^R^B^^. Conditions: R^ must have p o s i t i v e diagonal e n t r i e s , and B ^ must be a r e a l i z a b l e c i r c u i t matrix. Step 2: Take B 42 [u o ] , C = ( K K V 1 0 Decompose:. (KK1)'1 = C_ + B^ nC.B n b l 1 2 1 c 1 2 1 0. D, Conditions: C,, and C must have p o s i t i v e diagonal b l c r to t e n t r i e s , . B 1 2 1 mus^ ^ e a r e a - l i Z 8 - h l e c i r c u i t matrix, and Step 3^ Decompose: Conditions B. 1 2 1 U B 0 4 3 must be a r e a l i z a b l e c i r c u i t matrix. (KK*) 1 K L K t ( K K t ) 1 B ' G' B' ^ 2 2 a 2 2 ' G'Q must have p o s i t i v e diagonal e n t r i e s , B ^ must be a r e a l i z a b l e c i r c u i t matrix, and 1 2 1 •B 0 22 0 U B 43 must be a r e a l i z a b l e c i r c u i t matrix. 57 Step 4- Determine from the o r t h o g o n a l i t y c o n d i t i o n s , KN"2 = 0 and .NgN^  = U, where l ( n - k ) ( n - k - l ) e n t r i e s i n must be a r b i t r a r i l y chosen. Then the modal matrix K M D-%2 Step 5 Ca l c u l a t e : Decompose: Conditions J = CMLMtC. J = B^ 2G CB 2 2-G must have p o s i t i v e diagonal e n t r i e s , c B 22 must be a r e a l i z a b l e c i r c u i t matrix, and B, B 12 B 22 0 0 B42 B 4 3 must be a r e a l i z a b l e c i r -c u i t matrix. Determine D^  to s a t i s f y the above necessary c o n d i t i o n s , i f p o s s i b l e . Step 6: Construct the network from c i r c u i t matrix B^. Steps 1, 2 and 3 above are s t r a i g h t f o r w a r d . The modal matrix i s determined i n Step 4- For n = k+l, there are zero deg-rees of freedom i n M. Therefore, t h i s case i s s t r a i g h t f o r w a r d . For n = k+2, there i s one degree of freedom, but i t i s s t i l l pos-s i b l e to w r i t e a n a l y t i c a l expressions f o r the e n t r i e s of i n terms of a parameter. Thus, t h i s case, too, can be handled a n a l -y t i c a l l y , although i t i n v o l v e s searching over the range of the parameter f o r s o l u t i o n s . When n = k+3, the number of degrees of freedom i n M i s large and i t i s v i r t u a l l y Impossible to f i n d an a n a l y t i c a l s o l -5c u t i o n . One i s forced to choose a r b i t r a r i l y a number of e n t r i e s i n M. A l t e r n a t i v e l y , as w i l l be shown l a t e r , i t i s p o s s i b l e to generate a d d i t i o n a l equations which allow M to be c a l c u l a t e d ex-p l i c i t l y . 3.3 Examples Synthesis Procedure I i s i l l u s t r a t e d i n t h i s s e c t i o n by t r e a t i n g s e v e r a l examples. The r e s i s t o r s and c a p a c i t o r s i n the network diagrams are l a b e l l e d i n accordance w i t h the f o l l o w i n g matrices: C b = d i a g ( c 1 , c 2 , . , ., c n) C = diag(c , c n> ... *.) c fa n+1' n+2 R b = d i a g ( r 1 , r 2 , . . . ) G Q = d i a g ( g 1 , g 2 , . . . ) . The examples are chosen to emphasize c e r t a i n aspects of the synthesis procedure as ibllows: i ) Example One i s worked i n d e t a i l to i l l u s t r a t e the step-by-step procedure, i i ) Example Two i s given to show how a l l p o s s i b l e s o l u -t i o n s can be found by a search method when n-.-= k+2, and i i i ) i n Example Three an a d d i t i o n a l equation i n the ent-r i e s of N 2 i s generated by s e t t i n g one o f f - d i a g o n a l entry i n J to zero. 3.3-1 Example One — k = 2, n = 3 The given impedance matrix i s 59 Z R C ( s ) 1 3 s 5 + 2 4 s 2 + 5 4 s + 3 5 s 5+8s 2+22sfl7 s 5+8s 2 +22s+17 2s 5+23s 2+66s+47 T&TT) (s+2)(s+3) Performing the p a r t i a l f r a c t i o n expansion gives Z R C ( s ) = sTl Because the rank of each residue matrix i s u n i t y , no f a c t o r i z a t i o n i s necessary. Z Rp(s) can be w r i t t e n i n the form (3.12) by inspec-t i o n . 1 1 1 3 4 -2 "3 l " 1 + 1 1 1 + s+2 _3 9_ s+3 -2 1 + 1 2 s+1 0 0 -1 ~1 1 _ 1 1 2 Z R C ( s ) = 1 3 -1 0 s+2 0 1 3 _0 0 s+3 2 -1 + 3 1 1 .2 Step 1: Q = ~3 l " 1 1 0 1 2_ _1 0 1_ — 1 0 0 1 1 0 2 0 1 0 0 0 1 0 1 B. „R, B . „ 43 b 43 By s y n t h e s i z i n g the network i n Figure 3.2(a), i t f o l l o w s that the necessary co n d i t i o n s are s a t i s f i e d . 1 0 0' Step 2: B 42 ( K K V 1 0 1 0 = 1/62 " l l -2 9/62 0 " 1 -2 6 0 . 4/62_ + -1 [2/62] [ l - l ] c 11/62 -2/62 0 •2/62 6/62 0 0-0 4 C b l + B 1 2 1 C c E 1 2 1 C c = diag(2/62) b diag(9/62,4/62,dJ) By s y n t h e s i z i n g the network i n Figure 3.2(b). i t f o l l o w s that the necessary conditions are s a t i s f i e d . 1849 -524 Step 3: J-Q 1 62 2 -524 828 1 1 0 - 1 0 1 1 62 2 . diag(524, 1325, 304) 1 -1 1 0 0 1 - B'^G'B' ~ B22 c 22' By s y n t h e s i z i n g the network i n Figure 3.2(c), i t f o l l o w s that the necessary conditions are s a t i s f i e d . 1 1 2 1 . 3 -1 •Step 4: N = K n 31 n 32 n 33 Because n=k+l i n t h i s example, I\T can be completely determined from t t the o r t h o g o n a l i t y c o n d i t i o n s K l ^ = 0 and N^N^ = U. 1 Step 5: J 62< N 2 = l/s/62 1849 -524 - I I I N / 6 2 d 3 -3 -2 -524 828 -52v/62 d -111762 d 5 ! 3 -52762 d 3 4898 d Decomposing J gives B 22 1 1 0 1 0 0 - 1 0 1 0 1 0 0 - 1 - 1 0 0 1 61 . 0 r-^  <= 1, r 2 •=• 2, r-j ra 1 . (a) The network at Step 1. L. c 1 9/62, c 2 = k/62, . « 2/62 (b) The network at Step 2, g[-52k/62z, ^ = 1325/62 2, g^-30V62 z (c) The network at Step 3« r 2 A/~ Of A * 1 A / ~ ^2 A T g 3 A> g 6 *3 g x - $2k/622, g 2 * lllx / 6 2 ,d 3 / 6 2 2 , g^ «* ZzJoHd^/SZ2, g,+ «= (1325 - Ulyf6?^)/622 e 5 - (30^- 52s^62,d3)/622, g 6= ('+898d 2» l63>/621d3)/622, cy= a 2 (d) The f i n a l network where . 163\/62/4898sd^304/52\(62, Figure 3-2 Ev o l u t i o n of the network that r e a l i z e s Z„„(s) of •KG Example One. Element values are i n ohms, farads and mhos. r <^ G 62 2 diag(524, 111J.62 d 7 , 52-/62 d , 1325-lllv/62 d 3 : 304-52J62 A y 4898d 2 -a63J62 d^). The complete network i s shown i n Figure 3 - 2 ( d ) where, f o r nonneg-a t i v e conductances, d^ i s constrained to the i n t e r v a l 163^62/4898<d^ s 304/52^62. At the lower l i m i t gg = 0 and at the upper l i m i t g^ = 0, giving i n each case a minimal network. 3 . 3 - 2 Example Two — k=2, n=4, Q=0 The given impedance matrix i s Z R C ( s ) = K(sU + L ) - 1 ^ where L = d i a g ( l , 2, 3, 4) 1 1 1 r K 1 2 - 3 2 Since Q=0, Step 1 does not apply. Completing Steps 2 and 3 gives "4 2 2 18 ' B121 = [ 1 -1 C b = diag(8/34, 1/34, d 2 , d 2) , Cc'= diag(l/34) 1 11 34 790 -118 -118 ' 202 B 22 1 -1 1 0 0 1 G^  .= 3 4 2 d i a g ( H 8 , 672, 84.). A l l necessary co n d i t i o n s to t h i s point are s a t i s f i e d . With n=k+2, there i s one degree of freedom i n the modal matrix. The or t h o g o n a l i t y c o n d i t i o n s are used to determine i n 63 terms of a s i n g l e parameter, which, from Section 3.2.3, i s most conveniently taken as an angle a. N = 1 1 1 cosa 1 2 1 . — cosa- + — since ^34 ' N/2 5 • . 1 2 . since . - — cosa - — s m a J2 34 1 — cosa /34 1 . — s i n a 34 1 2 1 . ^_cosa - — sma 34 \[2 1 2 . - — cosa - — s m a 42 v/34 The conductance matrix i s then expressed i n terms of the unknowns a, d^ and d.. 3 4 0.6834 -0.1021 -0.1021 0.1747 0.4976d,sin(a-125.81°) 3 0.4976d4sin(a+144.19°) 0.4976d^sin(a-125.81°) 3 0.1346d5sin(a-157.99°) 0.1346d5sin(a-157.99°) 0.1346d4sin(a+112.01°) 0.4976d4sin(a+144.19°) 0.1346d4sin(a+112.01°) d 2(2.2647+0.8809sin(2a-56.59°)) 0.8809d 5d 4sin(2a»146.59°) . . 0.8809d^d.sin(2a-146.59°) ' d 2(2.2647+0.8809sin(2a+123.41°)) 3 4 4 While independent of d^ and d^, the s i g n p a t t e r n of J i s a f u n c t i o n of a. Each d i f f e r e n t s i g n p a t t e r n leads to one or more port (or capacitor) tree c o n f i g u r a t i o n s . In order to f i n d a l l pos-s i b l e s o l u t i o n s , each c o n f i g u r a t i o n must be examined f o r passive r e a l i z a t i o n s . C l e a r l y , only t r i v i a l v a r i a t i o n s occur i n J i f a i s va-r i e d over more than 90°. Three d i s t i n c t s i g n patterns are found i n the quadrant -16 .71°-=;a<:73• 29°• Figure 3 . 3(a) shows the s i g n 64 patterns and the range of o; over which each occurs.. Also shown are the p o s s i b l e c a p a c i t o r - t r e e c o n f i g u r a t i o n s corresponding to each s i g n p a t t e r n . Of the four p o s s i b l e c o n f i g u r a t i o n s , only Configurations 1(a) and 3 are found to lead to r e a l i z a t i o n s w i t h nonnegative e l -ement values. The complete network corresponding to l ( a ) and a graph of allowed values ofcX, d^ and d^ are shown i n Figure 3 . 3 ( b ) . Eight d i f f e r e n t minimal networks can be constructed by t a k i n g d^ and a at i n t e r s e c t i o n p o i n t s of the curves d • vs <* and t a k i n g d^ from e i t h e r of the curves d^/d^ vs <x. Three conductances vanish i n each case, as f o l l o w s : i ) g y and g^, i i ) ' g y gy a n d g 1 0 ' i i i ) grj, g 8 and g^, i v ) grj, gg and g 1 Q , v) g2» §3 a n d S5» v i ) g 2 , g^ and g 1 Q , v i i ) g 2 , gg and g^, and v i i i ) g 2 , gg and g 1 Q . The network corresponding to C o n f i g u r a t i o n l ( b ) i s shown i n Figure 3 - 3 ( c ) . I t i s seen from the accompanying graph that a r e a l i z a t i o n w i t h nonnegative element values does not e x i s t . In a s i m i l a r manner, i t can be shown that C o n f i g u r a t i o n 2 does not lead to a r e a l i z a t i o n w i t h nonnegative element values. The network corresponding to C o n f i g u r a t i o n 3 and a graph of allowed values of d^ and d^ f o r one value of cx(.70°) are shown i n Figure 3 . 3 ( d ) . R e a l i z a t i o n s with nonnegative element values Range 1: -16.7K cX<35-81 + - - + - + + + + - + Range 2: 35.810<cX<67.99° - + - + - - + -- + - + Range 3: 67-99 «X<73.29° - + - -- - + -(a) c l ( b ) c2 Figure 3»3(a) Sign p a t t e r n of conductance matrix and corresponding c a p a c i t o r - t r e e c o n f i g u r a t i o n s as func t i o n s of a f o r Example Two. V J l 66 \°3 II ^ Vr 8 f 1 0 8 16 CX i n degrees — Figure I^Kb) Network corresponding to C o n f i g u r a t i o n 1(a) and graph, of allowed values of a, d^ and d 4 f o r Example Two. 67 -16 -8 0 8 16 2k 32 (X i n degrees >~ Figure 3•3(c) Network corresponding to C o n f i g u r a t i o n 1(b) and graph of allowed values of a, d^ and d^ f o r Example Two. 68 0 0.5 l.o 1.5 d 3 ^ Figure 3.5(d) Network corresponding to C o n f i g u r a t i o n 3 and graph of allowed values of d^ and d^ f o r a = 70° f o r Example Two. 69 e x i s t over the e n t i r e range of <x w i t h only small changes i n allowed values of d-, and d. and the conductances as cA v a r i e s . In t h i s case, 3 4 too, eight d i f f e r e n t minimal networks can he constructed by t a i l -i n g a equal to one of the l i m i t values and .taking d^ and d^ at i n t e r s e c t i o n p o i n t s on the graph. Thus, f o r a. = 67-99°, g^ = 0 and f o r cK = 7 3 - 2 9 ° , gg = 0 and e i t h e r - g ^ and g Q , g^ and gg, g g and g 1 Q , or gg and g ^ vanish. Whenever the c a p a c i t o r tree i s s t a r - l i k e , as i s the case i n C o n f i g u r a t i o n 3 , "the allowed values of d. , d, ' , :. . '.., d b . ' K+1' k+2 • ' n w i l l be bounded by hyperplanes i n (n-k)-dimensional space. The hyperplanes are defined by s e t t i n g to zero the conductances of the " r e s i s t o r s i n p a r a l l e l with the tree-branch capacitor's. Thus, i n Figure 3 - 3 ( d ) , the allowed region i s bounded by s t r a i g h t l i n e s corresponding to g^ = 0 , gg = 0 , g^ = 0 and g ^ = 0 . 3 . 3 - 3 Example Three 1, n = 3 The given f u n c t i o n i s Z R C ( s ) - s+1 + s+2 + s+3 The f i r s t three steps are t r i v i a l . The matrix N i s 1 1 1 N = n 2 1 n 2 2 n 2 3 n n„ 0 n „ 31 32 33_ Since n=k+2, there i s one degree of freedom i n N. Instead of f o l -lowing the procedure i n Example Two, consider s e t t i n g one of the o f f - d i a g o n a l e n t r i e s i n J to zero. In t h i s way, an a d d i t i o n a l equation i s formed, al l o w i n g N to be completely determined. I f the term set to zero i s i - ^ o r J-j^' a d d i t i o n a l equation, i s l i n -70 ear i n the unknowns. S e t t i n g = 0 , we f i n d 1 1 1 N -l/s/6 2/46 -l/v/6" -l/s / 2 0 •1/42 and J = 2/3 0 d 3 / 3 / 2 0 d3/3v/2 2 d 2 - d ^ / v f j -d„dv/^/3 2d!; 2 3 3 The complete network i s shown i n Figure 3-4 where the conductances are given by G d i a g ( d /3V2, d ? d , / J J , 2/3 - d 2 & 1 - d 0d , / ^ 3 2 d | - d 5/3Nf2 - d 2 d A / J ) . The allowed Values of d 0 and d.-, are determined as f o l l o w s : 2 3 f o r g , 2 0 , d^ ^ 2 N / ^ 3 3 f o r g ^ O , d 2 ^ d /2V3~ f o r g^^O, d^ ^ c 2 / 2 / j + l/6>/2. These r e l a t i o n s are shown i n g r a p h i c a l form i n Figure 3 . 4 Minimal networks are formed by choosing d 2 and d^ at points of i n t e r s e c t i o n making two conductances vanish. By s e t t i n g ^13 o r ^23 ^° z e r o » a d d i t i o n a l minimal networks can be found, a l -though some may be only t r i v i a l v a r i a t i o n s of others. The method used i n t h i s example i s c e r t a i n l y applicable to d r i v i n g - p o i n t impedances of l a r g e r n, although i t may be nec-essary to r e s o r t to numerical methods to solve f o r the modal mat-r i x . Thus, we can construct new minimal networks not p o s s i b l e using c l a s s i c a l methods. I t should be pointed out that the net-works so formed are complernentarjr t r e e s t r u c t u r e s which have been studied e x t e n s i v e l y by Lee and Mason 71 72 3 . 4 Synthesis Procedure I I In Synthesis Procedure I, the inverse of KE^ i s r e a l -i z e d as a network c o n t a i n i n g the chord c a p a c i t o r s and the f i r s t k tree-branch c a p a c i t o r s . In the present s e c t i o n , we consider the r e a l i z a t i o n of the elastance matrix KK^ d i r e c t l y . The r e s t r i c t i o n to networks w i t h k independent c i r c u i t s i m p l i e s that KK^ must be decomposable i n t o the form ( 3 . 1 1 ) . Prom ( 3 . 6 ) , i t i s apparent that the decomposition must take the form ^ = B 4 2 C b l B 4 2 ' Thus, the p o r t i o n of the network represented by Z ^ ( s ) (see P i g - . ure 3-1) w i l l have a s i n g l e c a p a c i t o r t r e e , that i s , there w i l l be no chord c a p a c i t o r s and hence no c a p a c i t o r c i r c u i t s . With the above approach, Step 2 of Procedure I i s mod-i f i e d as f o l l o w s . P i r s t , KK° i s decomposed i n t o m ± = B 4 2 1 C b l B 4 2 1 3 .19 where i s a diagonal matrix. I t i s necessary that the d i a g -onal e n t r i e s of C,., be p o s i t i v e and that B. 0, be a r e a l i z a b l e c i r -b l r 421 , • B 4 2 1 B 4 3 must c u i t matrix. Furthermore, the composite matrix be a r e a l i z a b l e c i r c u i t matrix. Let m equal the order of and define the diagonal ma-t r i x by 2 2 2 2 D l = C b l = d i a g ( d 1 , d 2 , . . ., d m ) . As shown i n Appendix A, 1 ^ m ^ Jrk(k+l). For an i r r e d u c i b l e r e a l i z a t i o n , m cannot be greater than n. I f <=n, i t i s necessary to augment a r i d ^° ^ x n a n d n x n > res-m p e c t i v e l y . The most convenient augmentation i s to add n-m c o l -umns of zeros to a n d as many u n s p e c i f i e d diagonal e n t r i e s to D^ , as f o l l o w s where B 42 B421 0 and D 0 0 D 2 = diag(d r r i^ n , d m+1' m+2' , d ). n In general, the o r i g i n a l Step 3 does not apply here because the f i r s t k rows of the modal matrix cannot be determined e x p l i c i t l y from K = B^M which, w i t h column p a r t i t i o n i n g , becomes Ki = B 4 2 M i ' 1 = 1 ' 2 ' V •• ' ' n * This merely gives a set of k l i n e a r equations i n the f i r s t m ent-r i e s of each column of M. An exception occurs when m ^ k and the rank of B ^ equals m. In t h i s case, the f i r s t m rows of M and hence the mxm p r i n c i -p a l submatrix of J can be determined. Thus, wi t h some m o d i f i c a -t i o n s , Step 3 can be c a r r i e d out. The o r t h o g o n a l i t y c o n d i t i o n s of Step 4 now take the form MM^  = CT 1 = D~2 or, i f we define N = DM, NN* = U. The previous d i s c u s s i o n ( i n Section 3-2.3) regarding the number of independent equations and the number of degrees of freedom i n N i s v a l i d here w i t h the exception that k i s replaced by the rank, k', of B^2- Hence, the number of degrees of freedom i n N i s f = •Hn-k' ) (n-k 1-!). 74 The c a l c u l a t i o n and r e a l i z a t i o n of the s h o r t - c i r c u i t conduc-tance matrix J of Step 5 proceeds as before. As i n Synthesis Procedure I , minimal networks can be constructed by choosing s u i t a b l e values f o r the u n s p e c i f i e d par-ameters. Normally, m = -g-k(k+l) and k' = k. Then there are n—§-k(k+l) u n s p e c i f i e d e n t r i e s i n and -g-(n-k) (n-k-l) degrees of freedom i n the modal matrix which can be chosen to make an equal number of conductances vanish. The minimal network then c o n s i s t s of 4"k(k+l) r e s i s t o r s i n R^, n ca p a c i t o r s i n and nk r e s i s t o r s i n G . The cases m'^yk(k+l) and k' ^  k represent s p e c i a l r e l a t i o n -ships, among the e n t r i e s of K, with the r e s u l t that fewer network elements than normal are requi r e d . In order to i l l u s t r a t e the procedure, developed above, consider the f o l l o w i n g example. Example Four The given impedance matrix i s the same as that f o r Ex-ample One. Z R C ( s ) = 1 1 2 1 3 -1 s+1 0 0 0 s+2 0 0 0 s+3 -1 r 1 1 1 3 2 -1 + 3 1 1 2 Q i s r e a l i z e d as before. KK i s decomposed i n t o " l 1 1 0 KK t 6 2 " l 1 0 2 11 1 0 1 diag(2, 4, 9) 0 1 = B 4 2 D 2 ] 342* Since m = n, no augmentation i s necessary. By s y n t h e s i z i n g the network i n Figure 3-5(a), i t f o l l o w s -that the necessary conditions 75 are s a t i s f i e d . Using K =r B^M, two e n t r i e s i n each column of M can be elimi n a t e d . Thus, M m l l 1-m mn 11 12 1-m m 13 12 2-m 13 1 _ m l l . 5 _ m 1 2 ~ 1 _ m 1 3 The orthogonality' c o n d i t i o n s give three equations, a l l o w i n g M to be completely determined as M = _1 31 13 21 14 18 • 10 48 18 72 -45 ± 31 -7 3 2 7 - 3 - 2 7 - 3 - 2 The ambiguity w i t h respect to s i g n i s resolved when i t i s found that only the minus s i g n leads to a passive r e a l i z a t i o n . R e a l i z a t i o n of the conductance matrix, which i s found to be ' _ ' • (3061-714 727/4 (1248-309 J2)/8 (-54+24972/18 J =' 31' (1248 - 3 0 9 7 2 ) / 8 (8858-133272 )./l6 (-3294+77472)/36 ( - 5 4 + 2 4 9 7 2 ) / l 8 (-3294+77472)/36 (18189+21672)/81 completes the r e a l i z a t i o n of Z^^(s). The f i n a l network, which i s minimal, i s shown i n Figure 3 . 5 ( h ) . As noted p r e v i o u s l y , Procedure I I cannot be used to f i n d i r r e d u c i b l e r e a l i z a t i o n s when m>n. Unless there are s p e c i a l r e l a -t i o n s h i p s among the e n t r i e s of KK^, m = ,-§-k(k+l) so that we would normally r e q u i r e n s-g-k(k+1) . For example, when k=3 we would r e -quire n>6. On the other hand, i n Procedure I we re q u i r e only 76 © r 2 <=2 °3 r 3 © < 1 © r2 A T r l o 1 ' r 2 r = 2 ' r 3 t = 1 > (a) The network at Step 2 g 3 A T *6 A/~ © g i = ^ i 2 ' S 2 = J i 3 ' g 3 = " ^ 2 3 ' h\~ hz~ ^5 1 3 ^22- J l 2 + ^23' ^6" J 3 3 - J 1 3 + J23 • (b) The f i n a l network. Figure 3.5 R e a l i z a t i o n of Z R Q ( S ) of Example Four vising Procedure I I . Element values are i n ohms, farads and mhos. 77 risk.- However, f o r the important p r a c t i c a l case of two-port net-works, Procedure I I i s a p p l i c a b l e when n^3 which includ.es most problems of i n t e r e s t . I t should be pointed out that r e d u c i b l e r e a l i z a t i o n s may be found by Procedure I I when m>n by augmenting Z R (^(s) w i t h Y —— / s+A.. i : r:-i 1 I where = 0, i = n+1, n+2,. . • , m and the \_ are u n s p e c i f i e d . Then the r e s u l t i n g network w i l l have m-n n a t u r a l frequencies which are unobservable at the e x t e r n a l p o r t s . The disadvantage of t h i s , approach i s that i t increases the order of the modal matrix and hence the complexity of the o r t h o g o n a l i t y . c o n d i t i o n s . In summary, Procedure I I d i f f e r s from Procedure I mainly i n the way i n which the capacitance matrix i s formed. C l e a r l y , the two methods are i d e n t i c a l f o r impedance matrices i n which the matrix KK^ i s diagonal. In general, the method here i s s l i g h t l y more cumbersome to use and does not afford, the necessary-condition check of Step 3- There may be i n s t a n c e s , however, when one method leads to a passive r e a l i z a t i o n while the other does not, and v i c e versa. 3.5 D i s c u s s i o n Two synthesis procedures have been given i n t h i s chapter, Procedure I i n Section 3.2 and Procedure I I i n Section 3.4- For i r r e d u c i b l e r e a l i z a t i o n s , the l a t t e r i s i n general r e s t r i c t e d to cases where nsyk(k+l) because the decomposition (3-19) normally leads to a c a p a c i t o r matrix of t h i s order. The former i s a p p l i -78 cable f o r any n^k, and i s somewhat more convenient to use. Hence, f o r the most p a r t , the d i s c u s s i o n to f o l l o w i s confined to Proce-dure I , although many of the statements apply to Procedure J l as w e l l . Consider f i r s t the question of necessary and s u f f i c i e n t W t h c o n d i t i o n s . We are given a kxk, n degree matrix, Z R P ( s ) , which can be w r i t t e n Z R C ( s ) = K(sU + L ) - 1 ^ + Q. The f o l l o w i n g question i s posed: what are the necessary and suf-f i c i e n t c o n d i t i o n s that ZR(~,(s) can be r e a l i z e d (by Procedure I) as a ( k + l ) - t e r m i n a l , k-port passive RC network i n the form of Figure 3.1? A set of necessary and s u f f i c i e n t c o n d i t i o n s has.not been found f o r k^2. As i s w e l l known, i n the one-port case, the neces-sary and s u f f i c i e n t c o n d i t i o n i s that ZR(-,(s) can be w r i t t e n i n the above form. Although a number of necessary co n d i t i o n s have been given, the only general s u f f i c i e n t c o n d i t i o n i s the s u c c e s s f u l com-p l e t i o n of the synthesis procedure. The necessary co n d i t i o n s given i n Steps 1, 2 and 3 of the synthesis procedure (Section 3.2.7) can be stated more co n c i s e l y , F i r s t , define the s h o r t - c i r c u i t admittance matrix Y-^(s) by Y±1(s) A s C ^ + J X 1 = (KK t)~ 1K(sU + L ) K t ( K K t ) _ 1 where KK^ i s n e c e s s a r i l y nonsingular. The necessary.conditions are that i ) " ^ i i ^ s ) must be r e a l i z a b l e as a k-port, passive RC network with k+l nodes, i i ) Q must be r e a l i z a b l e as a k-port, passive r e s i s t o r 79 network wi t h k independent c i r c u i t s , and i i i ) the o p e n - c i r c u i t impedance matrix of the s e r i e s connec-t i o n of the two networks must "be given by Z R C ( S ) = Y ~ i ( s ) + Q. The Fialkow-Gerst ' c o e f f i c i e n t c onditions are a l s o nec-essary conditions- which, c l e a r l y , are not e n t i r e l y independent, of the above c o n d i t i o n s . For completeness, we s h a l l state the Fialkow-G-erst c o n d i t i o n s . W r i t i n g the e n t r i e s of Z^Q(S) as r a t i o s of polynomials with a common denominator, we have f o r ports- p and q, n-1 z (s) = • ) . a . s 1 pp P(s) Z i i=0 n-1 z (s) = -nV^ s / b . s 1 qq P(s) / . i 1=0 n-1 and z (s) = -D/1\ / c . s 1 pq P i s ) Z i 1=0 n where P(s) (s + \, ). i = l Then f o r Z R Q ( S ) to be r e a l i z a b l e by means of an RC network, i t i s necessary that 0 — lc. I a. , b., i = 0, 1, . . . , n-1 f o r a l l p, q. When ports p and q share a common t e r m i n a l , we have the f u r t h e r c o n d i t i o n that c^z. 0, i = 0, 1, . . ., n-1 80 where the standard reference d i r e c t i o n s of voltage and current are assumed. Our c o n d i t i o n s ( i ) , ( i i ) and ( i i i ) plus the F-G condi-t i o n s do not form a set of s u f f i c i e n t c o nditions i n general. How-ever, i t i s shown i n Appendix B that they are s u f f i c i e n t f o r the s p e c i a l case k=2, n=3, when a common-ground•network i s sought. Another i n t e r e s t i n g problem occurs when Q=0 and KE^ i s a diagonal matrix. Then every o p e n - c i r c u i t t r a n s f e r impedance f u n c t i o n has a double zero at s = 0 0 , a not .unusual occurrence. The necessary c o n d i t i o n s then reduce to our c o n d i t i o n ( i ) plus the F-G c o n d i t i o n s . In t h i s case, i t i s only necessary to determine M such that J i s r e a l i z a b l e as an n-port r e s i s t o r network (pos-s i b l y with more than n+l t e r m i n a l s ) , since no c o n s t r a i n t s whatso-ever are placed on the i n t e r n a l port c o n f i g u r a t i o n i n the previous steps. For t h i s reason, i t i s conjectured that the above condi-ti o n s , may be necessary and sufficient."' To support the above conjecture, a problem was considered i n which some of the necessary c o n d i t i o n s were s a t i s f i e d with the e q u a l i t y s i g n . For the given matrix Z R^,(s) with L = d i a g ( l . 0 , 2.0, 3-0, 4-677977) 2.0 1.0 3-0 1.0 -1.1433051 0.381017 0.381017 0.762034 and K the numerator c o e f f i c i e n t matrix of the constant term and J ^ ^ are both dominant with the e q u a l i t y s i g n . We s h a l l g i v e only the gen-e r a l r e s u l t here. F o l l o w i n g the same procedure used e a r l i e r i n s o l v i n g Example Two, two passive r e a l i z a t i o n s based on d i f f e r e n t 81 c a p a c i t o r tree c o n f i g u r a t i o n s were found. As might be expected, two of the three parameters (o(, and d^) are l i m i t e d to d i s c r e t e values i n each r e a l i z a t i o n . Minimal networks are formed by t a k i n g the t h i r d parameter to one of the l i m i t s of i t s allowed range. While the above r e s u l t s are i n c o n c l u s i v e , they do i n d i c -ate that the question of necessary and s u f f i c i e n t r e a l i z a b i l i t y c o n d i t i o n s f o r the case of a diagonal KK^ matrix merits f u r t h e r study. An i n t e r e s t i n g property of Procedure I i s that i t appears to l e a d to networks which are economical i n t o t a l capacitance as w e l l as i n the number of c a p a c i t o r s . This property i s advantag-eous i n the design of m i c r o e l e c t r o n i c networks since the area of the c a p a c i t o r i s p r o p o r t i o n a l to i t s capacitance. The t o t a l capacitance i s minimized i n two'ways.' F i r s t , the r e a l i z a t i o n of KK^ i n the admittance form leads to a minimum capacitance r e a l i z a t i o n of t h i s term. I t i s easy to show that any other r e a l i z a t i o n of KK^, f o r example, i n the impedance form i n Procedure I I , must have a l a r g e r t o t a l capacitance. Second, f o r a s p e c i f i c i n t e r n a l port c o n f i g u r a t i o n , the u n s p e c i f i e d diag-onal e n t r i e s i n can be minimized, hence minimizing the capac-2 ita n c e s c^ = d^, i = k+1, k+2,. . . , n. The allowed configur-a t i o n s , I f there are more than one, can be examined to determine which one leads to minimum t o t a l capacitance. 4. SYNTHESIS OF GROUNDED TWO-PORT NETWORKS USING A NUMERICAL TECHNIQUE . r-The major stumbling block i n the p r a c t i c a l implemen-t a t i o n of the synthesis procedures given i n the previous chapter i s that the set of second order a l g e b r a i c equations a r i s i n g from the o r t h o g o n a l i t y c o n d i t i o n s i s not amenable to a n a l y t i c a l s o l u -t i o n . Furthermore, since there are fewer equations than unknowns, i t i s not p o s s i b l e to use numerical techniques to f i n d general s o l -u t i o n s . Our o b j e c t i v e i n t h i s chapter i s to f i n d a method which leads to a s o l u t i o n i n a s t r a i g h t f o r w a r d manner. Obviously, t h i s can only be done at the cost of.some g e n e r a l i t y . I t i s c l e a r that a s u f f i c i e n t number of a d d i t i o n a l equations must be generated to . allow the modal matrix to be determined e x p l i c i t l y . At the same time, we do not want to decrease g r e a t l y the chance of f i n d i n g a r e a l i z a t i o n with nonnegative element values. The synthesis problem most often encountered i n p r a c t i c e i s that of grounded two-port networks. Therefore, our e f f o r t s w i l l be d i r e c t e d p r i m a r i l y towards t h i s problem. For s i m p l i c i t y , we assume Q.= 0 so that the s p e c i f i c a t i o n s take the form Z R C ( s ) = K(sU + L r V . 4.1 The extension to cases where Q / 0 i s s t r a i g h t f o r w a r d and w i l l not be given. In the f o l l o w i n g s e c t i o n s , i t w i l l be shown that -M can be computed e x p l i c i t l y when i s required to be a diagonal matrix. Then, f o r k=l and k=2, a given matrix (4.1) i s always t o p o l o g i c a l l y r e a l i z a b l e . A method f o r the synthesis of a new c l a s s of grounded two-port networks w i t h optimum voltage g a i n . w i l l be given. r-4.1 E x p l i c i t Determination of the Modal Matrix and Synthesis of  the Second Foster Canonical Form In Synthesis Procedure I, i t was shown that there are T-(n-k) (n-k-l) degrees of freedom i n the determination of the modal matrix. This number e x a c t l y equals the number of .off-diagonal ent-r i e s i n ^22' 3 0 ^ e a d d i t i o n a l equations generated by r e q u i r i n g to be a diagonal matrix are s u f f i c i e n t to allow the modal mat-r i x to be determined e x p l i c i t l y . We s h a l l consider next the e f f e c t of f o l l o w i n g t h i s course. Since both a n d ^22 a r e n o w ^-iag°na-'- matrices, the f i r s t e f f e c t we n o t i c e i s that there w i l l be no elements connected between any p a i r o f ' i n t e r n a l nodes. I t w i l l a l s o be shown that, f o r k=l and k=2, J w i l l always be t o p o l o g i c a l l y r e a l i z a b l e , prov-ided that the known necessary c o n d i t i o n s are s a t i s f i e d . Consider f i r s t the one-port case. The capacitance and conductance matrices are given by C ( K K V 1 0 d' 0 n 84 J = h i d 2 j 1 2  d 3 j 1 3 drAn 2 J 1 2 ,2 . 2 J 2 2 0 0 d 3 j 1 3 0 d 3 j 3 3 0 d j n n J In 0 0 a 2 j n°nn where j' = (KK*) 1 KLK t (KK t ) X. C l e a r l y , the admittance matrix Y (s) = sC + J n can be r e a l i z e d i n the form of the network shown i n Figure 4 . 1 where the element -values are given by c l = S i = ( K I ^ ) " 1 n j=2 J c . d . d 2 j . .-d . 3 3 J • 3 J = 2 , 3 , , n. Conductances g., j= 2 , 3 : n are always nonnegative. S p e c i f i c , lower bounds are placed on the parameters d.. by the require-ment that conductances g'.'be nonnegative and upper bounds a r i s e from the requirement that be nonnegative. At the lower bound the primed conductances vanish and the network reduces to the second Foster canonical form. In t h i s 85 case 1 ^ T o y 1 > o Therefore, we conclude that i f and only i f Z j ^ ( s ) can be writven i n t h e form ( 4 . 1 ) there e x i s t s a set of parameters. d . s 3 3 = 2, 3, n 33 such that Zgrj(s) i s r e a l i z a b l e by means of the network i n Figure 4 . 1 with nonnegative element values. ?«2 n Jn g n Figure 4•1 One-port network formed when i s a diagonal matrix. I t has been shown by Duda that the second Foster canonical form i s the minimum capacitance r e a l i z a t i o n of a d r i v i n g -point f u n c t i o n . Therefore, by t a k i n g the values of"d^, 3=2, 3> • . ., n equal to t h e i r lower bounds, the method given above y i e l d s the minimum capacitance r e a l i z a t i o n . D i s c u s s i o n of the two-port case i s postponed u n t i l the f o l l o w i n g s e c t i o n . At t h i s p o i n t , we t u r n our a t t e n t i o n to the method of s o l v i n g f o r the modal matrix. The equations formed by r e q u i r i n g J " 2 2 be diagonal are n o n l i n e a r i n the e n t r i e s of , having a form s i m i l a r to the equations a r i s i n g from the o r t h o g o n a l i t y con-d i t i o n s . To summarize, we 'now have the f o l l o w i n g r e l a t i o n s h i p s : i ) KN^ = 0, g i v i n g k (n-k) l i n e a r equations, t ' i i ) ^2^2 = ^' S i v ^ n S (n-k) (n-k+l) nonlinear equations, and i i i ) N^LN^ i s diagonal, g i v i n g \ ( n - k ) ( n - k - l ) nonlinear equa-t i o n s . Thus, there are a t o t a l of n(n-k) equations i n the same number of unknown e n t r i e s i n N^. Most cases of i n t e r e s t (nsk+2) give r i s e to a l a r g e number of second order a l g e b r a i c equations so that numerical meth-ods are c l e a r l y necessary f o r t h e i r s o l u t i o n . Although an exhaus-t i v e study of a v a i l a b l e numerical methods was not made, a method (26) due to F l e t c h e r and Powell was found to be adequate. I t i s a powerful i t e r a t i v e descent method which converges to a . l o c a l min-imum of the given f u n c t i o n . D e t a i l s of i t s a p p l i c a t i o n to the prob lem at hand are given i n Appendix C. In every one of the l a r g e number of problems solved, con vergence to the g l o b a l minimum was observed. In many cases, the i n i t i a l estimates of the parameter values were not near the f i n a l v a l ues. Thus, i t appears that the f u n c t i o n to be minimized (see Ap-pendix C) contains no l o c a l minima, that i s , i t i s a monotonically decreasing f u n c t i o n . Systems wi t h n=10 (up to 90 equations) are conveniently handled by the IBM 7044 computer a v a i l a b l e . Computation times are reasonable. For example, one problem w i t h 30 equations required 76 i t e r a t i o n s , t a k i n g 141 seconds to compute. 4.2 - A p p l i c a t i o n to the Synthesis of Grounded Two-Port Networks For two-port networks w i t h a diagonal matrix, the capacitance and conductance matrices are 87 °11 0 0 c 22 '11 '12 '12 '22 0 0 0 0 d n J = J l l J 1 2 J 1 2 J 2 2 h i J 1 2 j d i r± I n j 1 2 3 2 2 ! d"3* i23 dn^ 2 n d 3 j 1 3 d 3 ^ 2 3 ! d 2 i u 3 J 33 0 d n h n dn^ 2 n ; 0 n nn where C ^ = 1 and J = (KK13) ^ K ^ K K * ) 1 . From the kn.own necessary conditions, C^^ and J^-^ must be dominant mat-r i c e s w i t h i d e n t i c a l s i g n p a t t e r n s . , The c a p a c i t o r subnetwork i s shown i n Figure 4•2 where.the placement of ca p a c i t o r s c^ to c^ i s determined from the corresponding column of the matrix J . When c-^ a n d 3 n 0 a r e p o s i t i v e (negative), 12 one t e r m i n a l of c a p a c i t o r c., j = 3 , 4 , • • .., n i s connected to a) node 1 when j-, . and j ? . have the same (opposite) signs and 2D b) node 2 when . and • have the same (opposite) signs and 3u '2d •J , and c) node n+l when j - ^ and j 2 j have opposite (the same) signs. The e x t e r n a l ports are i n p a r a l l e l with c a p a c i t o r s c-^  and c 2 , that i s , the terminals of current source 1-^  are connected to 88 nodes 1 and n+1 and the terminals of current source 1^ "to nodes 2 and n+1. V/e r e f e r to nodes 1, 2 and n+1 as e x t e r n a l nodes and nodes 3 "bo n as i n t e r n a l nodes. From the form of J , i t i s c l e a r that there w i l l g e n e r a l l y be a r e s i s t o r connected between every p a i r of nodes where at l e a s t one of the p a i r i s an ex t e r n a l node. .Thus, there w i l l be a t o t a l of 3 ( n - l ) r e s i s t o r s . , F o r " s i m p l i c i t y i n formulating expressions f o r the element th values, l e t us consider a 5 order problem where one column of J - ^ f a l l s i n t o each of the Categories ( a ) , (b) and ( c ) . Let J be w r i t -ten as fo l l o w s J = h i J12 d a j l a < V i b d A c h2 hz d a J 2 a dbJ2b d C J 2 c V i a d a^2a d2 j a J a a 0 0 hhh d b^2b 0 d2 • b J bb 0 d c h o . d c j 2 c 0 0 d2 j c cc ,rd ,th .. where the 3 column f a l l s i n t o Category (aj , the 4 • i n t o Category t h (b) and the 5 i n t o Category ( c ) . The complete network f o r t h i s problem, drawn i n Figure 4 . 3 : i s seen to c o n s i s t of three d i f f e r e n t T-sections connected i n pa r a l -l e l w i t h one i t - s e c t i o n . The values of the capacitances are given by a = d 12 2 "a c l = c l l c, = d-, , c b b' c '12 d 2 '2 '22 '12 By w r i t i n g the conductance matrix f o r the network, expressions f o r the conductances become evident as f o l l o w s : 89 Figure 4.2 Capacitor subnetwork of two-port network formed when i s diagonal, L l W c l th Figure 4-.3 Complete network.for 5 order problem with one • column of J^p in. each category. 90 g = d °a a 32a > g a = d a ( 3 l a - 3 2a g b = d b 3 l b g b = d b ( 32b - 3 l b ), g" = d 2 j -d rr" — rl 2 i — f\ ' ' b b - b Jbb b j = d 5c c 'lc , g' = d >6 J12 d a '2a - d, '2c 'lb g'" = d 2 i ' -d ( ' & c c J c c c 'lc + ' l a '2b '2c ), g l - 311 j 1 2 3 l a - 32a ) - d c 3 l c g2 = 322 j 1 2 - V 32b - 3 l b ) - .1 ' • c 32c C l e a r l y , conductances g , g^, g b > g f e, g Q and g^ are always nonnegative. The c o n d i t i o n that g^, g^ and g^ be nonnegative y i e l d s the i n e q u a l i t i e s below which place lower bounds on the values of d^, d d, and d . b c d J l a a d; '2b aa 'bb d s= c ' l c + '2c 'cc The c o n d i t i o n that g^, g 2 and g^ be nonnegative y i e l d s the f o l l o w i n g i n e q u a l i t i e s which place upper bounds on the values of d , d, and d . a u c J12 d a '2a + d b 3 ' l l - 312 - da< 3 l a 322~ 312 - d b ( 32b 3 l b Jo a l b ) + d. + d 'lc '2c In c o n c l u s i o n, a r e a l i z a t i o n of Z j ^ ( s ) by means of a network of the form Figure 4-3 with nonnegative element values e x i s t s i f and only i f every upper bound i s not l e s s than the corresponding lower bound. I f t h i s c o n d i t i o n I s s a t i s f i e d , a three-parameter f a m i l y of equiva-l e n t networks e x i s t s . The above r e s u l t s are e a s i l y extended to the general prob-lem. One merely groups every column of J-^ 2 i n t o one of the three 91 categories (a), (b) or ( c ) . Then the number of T-sections. of each type i s equal to the number of columns f a l l i n g i n t o the correspon-ding category. The conductances and the lower bound f o r d. a r i s i n g from column j are given as f o l l o w s : when column j f a l l s i n t o Category (a): g-i =• d, 2 J '2j ;'. = d . ( 2 2 J 2 j g" = d.j . . - d . B 2 2^22 J d . > --2 2 when column j f a l l s i n t o Category (b) ;. = d . 2 J ; = d . ( 2 D ' 2 j = d . j . . - d . 2 3 JJ J J 2 J d . >-|J2.il 4 . 2 when column j f a l l s i n t o Category (c) ; = d 2 j . . - d .( 'J rJD J g'. = d. & J J ' 2 j 3 l j + '2j ), d.,-+ _ 2 i _ . J33 The remaining conductances become 3n+l '12 E d . (a) J '2j r d. (b) j 3 i - j n 2 ~ D 2 2 '12 J 12 Z d.( (a) J H d.( (b) 3" J 1 J - j 2 j j 2 j -l j ) ) r d. (c) J £ d (c) d j J l j '2j 4 . 3 E x p l i c i t . u p p e r bounds f o r the values of d ., j = 3» 4 , . • . , n cannot be determined because the c o n d i t i o n that g^, g^, and g 2_ b e n o n n e g a t i v e represents only three equations i n n-2 parameters 92 However, one can determine whether a r e a l i z a t i o n with nonnegative element values e x i s t s by s u b s t i t u t i n g the values of the lower bounds i n ( 4 . 3 ) and checking that g^, and SnijL a r e nonnegative. In summary, a method has been given f o r s y n t h e s i z i n g a f a m i l y of equivalent grounded two-port networks c o n s i s t i n g of (n - 2 ) T-sections connected i n p a r a l l e l with one re-section. A u s e f u l prop-e r t y i s t h a t , by s u i t a b l y choosing n-2 parameters, the network can be made minimal and minimum capacitance.. However, i t remains to be • shown whether the c o n d i t i o n of nonnegative element values places severe r e s t r i c t i o n s on the c l a s s of fu n c t i o n s which are r e a l i z a b l e by t h i s method. This question w i l l be discussed i n Section 4 . 4 where an example with v a r i a b l e parameters i s examined. The i n d i c a t i o n i s that the method i s not unduly r e s t r i c t i v e f o r small n (^10) . 4 . 3 Transmission G-ain Optimization In the synthesis, of two-port networks, i t i s often only necessary to r e a l i z e the impedance fu n c t i o n s to w i t h i n m u l t i p l i c a t i v e constants. In other cases, i t may be d e s i r a b l e to optimize the tra n s m i s s i o n voltage gain parameter. Therefore, we modify here the method given i n the l a s t s e c t i o n to inc l u d e an adjustable gain para-meter, a. In order to do t h i s , we pre-'and p o s t - m u l t i p l y the given f u n c t i o n by the matrix " l 0 A 0 1/c a to form Z R C ( s ) ; = ( l / c x ) z 1 2 ( s ) ( l / a ) z 1 2 ( s ) (1/a ) z 0 , ( s ) 22 AK(SU + L) V A . ' 4 . 4 93 We can consider l/cx as the gain from port 1 to port 2 and a as the gain from port 2 to port 1. The e f f e c t on the modal matrix, which now becomes AK M B 2 > 2 i s to d i v i d e the second row by a. C l e a r l y , the o r t h o g o n a l i t y con-d i t i o n s are unaffected by t h i s . The capacitance and conductance matrices become C = (MM13) 1 A 1 ( K K t ) XA 1 0 0 , and J = CMLMtC = A _ 1 ( K K t ) " 1 K L K t ( K K t ) 1 A ~ 1 D 2W 2LK t(l^K t) XA 1 A 1(KKt) 1KLS^D D 2N 2LN 2D 2 Thus, the second row and second column of the o r i g i n a l C and J mat-r i c e s are m u l t i p l i e d by a. With these changes the capacitances are given by 'n+1 a '12 C l ~ C l l a '12 -2 - a ^ 2 ; a '12 4.5 o . = d. , j = 3 , 4, , n Equations (4.2-3) f o r the conductances and the lower bounds f o r d. are e a s i l y modified to incl u d e the parameter a. I t i s advantageous to take the parameters d., j= 3, 4, • . ., n equal to t h e i r lower bounds because ( i ) i t i s evident that i f , f o r some value of a, a r e a l i z a t i o n with nonnegative element v a l 94 ues. e x i s t s , then one al s o e x i s t s at the lower bound, ( i i ) i t can be shown that the range of a f o r which a r e a l i z a t i o n e x i s t s i s greatest at the lower bound, and ( i i i ) t a k i n g the lower bound val -ues leads to a minimal network with minimum capacitance. Using the lower bounds, we have the f o l l o w i n g : when column j f a l l s i n t o Category (a) : d . = D D Li! J i . i 3 2 . j DD g . = Of 1 1 • • J J0D J l . i l ( i i i a h i 'D D DO and when column j f a l l s i n t o Category (b) D d . = a D 21L • l 3 l . i J 2 . i l , g | 3 2 / a P 2 . i DD D 0 DO 4 . 6 when column j f a l l s i n t o Category (c) d . D J j j l + A|D 2;i P l . i l ( h l . j l + a l j 2 . i l } JD'D D DD a P 2 i ( l 3 l i l + a 3DD 3n+l = a '12 (a),(b) 3 l i 3 2 . i DD *1 ~ 311 a '12 (a) hra\h?h?\ JDD (c) . 2 3 l i + a 3 l i 3 2 , i DD 4 . 7 2 . a J 2 2 - a '12 2 .2 a 3 2 . j - g l 3 l . i 3 2 . i l a 3 2 3 + a (b) J00 2 .2 3 l . i 3 2 . i l (c) DD Assuming that f o r the given func t i o n c-j 2 and j ^ 2 have the same sig n s , a r e a l i z a t i o n e x i s t s i f some value of a can be found 95 such that the element values c^, c^, g1 and are nonneg-a t i v e provided that g ^ ^ O a l s o . From ( 4 . 7 ) , the c o n d i t i o n that gn+2_-^> i s independent of a and i s given by '12 4 . 8 (a),(b) 3 j j A l l other element values are a u t o m a t i c a l l y nonnegative. The con-d i t i o n s C2==0 and g^ 0 place a lower bound on the value of a. Let J 1 2 a . = maximum of mm c 121 3 1 , j 3 2 , j ana J i . i 3 2 . j l .i .i 22 '22 .2 l 2 i 11. • 4 . 9 (b),(c) 3 3 S i m i l a r l y , the. conditions of c-^sO and g^ s-0 place an upper bound on the value of a. Let 3 1 1 > i i a = minimum of max c 1 1 1 2 and (a),(c) J j 3 '12 3 l , i 3 2 , j (a) + 3 i . i 32 , j (c) 4.10 We can now s t a t e a set of necessary and s u f f i c i e n t con-d i t i o n s f o r the r e a l i z a b i l i t y of Z^Q(S) by a grounded network with (n-2) T-sections connected i n p a r a l l e l with one re-section. They are: i ) c-^2 and j - ^ must be both n o n p o s i t i v e or both nonnegative, i i ) I n e q u a l i t y ( 4 . 8 ) must be s a t i s f i e d , and i i i ) a . ^ a mm max Whether Z-^Cs) s a t i s f i e s the above con d i t i o n s i s .determined only by 96 completing a major p o r t i o n of the synthesis procedure. However, there are a number of necessary co n d i t i o n s which can be tested d i r e c t l y . F i r s t , from the Fialkow-Gerst c o e f f i c i e n t c o n d i t i o n s f o r grounded networks, the numerator c o e f f i c i e n t s i n z ^ i s ) must be e i t h e r a l l nonnegative or a l l n o n p o s i t i v e . Second, i t f o l l o w s from c o n d i t i o n ( i ) above that, j - ^ must be nonpositive' (nonnegative) when the numerator c o e f f i c i e n t s i n z-^(s) are nonnegative (non-p o s i t i v e ) . I t i s evident t h a t , while the number of columns of J - ^ f a l l i n g i n t o Category (c) i s independent of a, the d i s t r i b u t i o n of the remaining columns i n t o Categories (a) and (b) i s not. There-f o r e , the value of a (or at l e a s t an approximate value) should be known before proceeding with the grouping of columns. We s h a l l show next how i n i t i a l l i m i t s f o r the value of a can be determined. In the form of a r a t i o of polynomials, we have K( s U ' + D - V = f I i y £ K.s 1 1=0 whence (4.4) becomes n-1 = p f e r £ A i i A s i - 4.11 where n P(s) = J | (s + X±) 1=1 By the F-G c o n d i t i o n s , the matrices AIL A, i = 0, 1, . . ., n-1 must be dominant. Prom these dominance co n d i t i o n s and the domin-ance of J-Q> a lower dominance l i m i t and an upper dominance l i m i t 97 f o r a can be determined. Then f o r a w i t h i n these l i m i t s , a l l of the dominance con d i t i o n s are s a t i s f i e d . The complete procedure f o r the synthesis of a given func-t i o n (4.4) by means of a minimal grounded network w i t h optimun. gain can now be summarized as f o l l o w s : 1. P r e l i m i n a r y steps: i ) Check that the numerator c o e f f i c i e n t s i n z^is) are a l l non-negative (nonpositive) and i-s nonpositive (nonnegative). i i ) Solve f o r the modal matrix n u m e r i c a l l y and compute the con-ductance matrix. Check that I n e q u a l i t y ( 4 . 8 ) i s s a t i s f i e d , i i i ) Compute the dominance l i m i t s f o r a. 2. M i n i m i z a t i o n of a ( f o r maximum voltage gain from port l.toport2): i ) With a equal to the lower dominance l i m i t , group the c o l -umns of i n t o Categories ( a ) , (b) and ( c ) . i i ) C a l c u l a t e a . and 0: using ( 4 - 9-10). I f a . >cx , s k i p mm max to mm max' r step ( i i i ) . . • i i i ) I f , w i t h a = "the grouping of columns i s unchanged from the previous grouping, the minimum value of a sought i s equal to a m j _ n - The synthesis procedure i s then completed by computing the element values using ( 4 . 5 - 7 ) . i v ) Should a . >a or should one or more columns, p r e v i o u s l y mxn max ' * J i n Category (a ) , s h i f t to Category ( b ) , . f i n d a new i n i t i a l value by i n c r e a s i n g a from the previous valxie u n t i l one c o l -umn s h i f t s from (a) to (b). With the new value of a and the new grouping, r e t u r n to step ( i i ) . v) I t can be concluded that no r e a l i z a t i o n i s p o s s i b l e a f t e r f i n d i n g that e i t h e r a . -^a. when no columns remain i n Cate-mm max gory (a)or a must be increased beyond the upper dominance 98 l i m i t . •3. Maximization of a ( f o r maximum voltage gain from port 2 to port l ) : i ) With a equal to the upper dominance l i m i t , group the c o l -umns of i n t o Categories ( a ) , (b) and ( c ) . i i ) C a l c u l a t e a . and a usi n g (4.9-10). I f a . >a m o s k i p ' mm max to mm max ^ step ( i i i ) . i i i ) I f , w i t h a = a , the grouping of columns i s unchanged from the previous grouping, the maximum value of a sought i s equal to a . The synthesis procedure i s then'com-max ple t e d by computing the element values using (4-5-7). i v ) Should a . >a or should one or more columns, p r e v i o u s l y mm max . ^ i n Category (b), s h i f t to Category ( a ) , f i n d a new i n i t i a l value by decreasing a from the previous value u n t i l one column s h i f t s from (b) to (a). With the new value of a and the new grouping, r e t u r n to step ( i i ) . v) I t can be concluded that no r e a l i z a t i o n i s p o s s i b l e a f t e r f i n d i n g that e i t h e r a . >-a when no columns remain i n ° mm max Category (b) or a must be decreased below the lower domi-nance l i m i t . Obviously, when both minimum and maximum values of a are sought, and part 2 shows no r e a l i z a t i o n i s p o s s i b l e , i t i s not necessary to complete part 3-A d i g i t a l computer program has been w r i t t e n to carry out the e n t i r e s ynthesis procedure, g i v i n g both the minimum and maximum values of a and the corresponding element values. Several examples w i l l be given i n the next s e c t i o n . 4.4 Examples and D i s c u s s i o n In order to i l l u s t r a t e the synthesis method developed i n preceding s e c t i o n s , we s h a l l present two examples. A f r e q u e n t l y encountered problem i s that of r e a l i s i n g a given voltage, t r a n s f e r f u n c t i o n q 9 ( s ) T ( s ) = ifrsr where q^ and q^ are polynomials i n s. For an unterminated network v" 2(s) zs 1 2(s) T ( s ) = v^TsT = . ^ T s T so that i t i s common to take q 1 ( s ) q 2 ( s ) z' -, (s) = — 7 — r — and z.-,0(s) = —7—r—. . 1 1 p(s) 12 p(s) where the polynomial p(s) may be chosen a r b i t r a r i l y but subject to the r e s t r i c t i o n that z-^(s) must be a p o s i t i v e , r e a l f u n c t i o n . For RC networks, the poles and zeros of z-^(s) must a l t e r n a t e on the negative r e a l a x i s of the s-plane with a pole nearest to the o r i g i n and a-zero nearest to s = - 0 0 . ' In the f i r s t of the examples to f o l l o w , we are given z-^ and z ^ 2 w i t h n=4. The three f i n i t e zeros of z-^2 are v a r i a b l e and z 2 2 i s chosen, so that every residue matrix has u n i t rank. The pur-pose of t h i s example i s to compare the l i m i t s of r e a l i z a b i l i t y , i n terms of the transmission gain f a c t o r a,with the l i m i t s determined from the F-G c o e f f i c i e n t c o n d i t i o n s and w i t h our necessary c o n d i t i o n that Y^^(s) must be r e a l i z a b l e (see Section 3-5). In the second example, we are again given z ^ and z-^2 with. n=4 but o n l y t h e t r a n s m i s s i o n gain f a c t o r i s v a r i a b l e . A comparison i s made between the network synthesized by our method.and the con- . v e n t i o n a l ladder r e a l i z a t i o n . The r e s u l t i s that a much, higher gain 100 i s achieved by our network than by the ladder network. The f o r -mer a l s o contains a much smaller value of t o t a l capacitance. 4.4.1 Example Five The given f u n c t i o n s are Z I : L ( S ) 2(s+2)(s+4)(s+6) TT+TTVs+3) ( s + 5 ) (iT7T and (o\ (s+c) (s2+2as+a 2+b)  Z 1 2 l s j - 4(s+l) ( s + 3 ) ( s + 5 ) . ( s + 7 ) where the r e a l numbers a, b and c a r e v a r i a b l e . Note that when \><0, z 1 2 ^ s ^ ^"as ^bree r e a l zeros and when b^-0, one r e a l zero and a p a i r of complex conjugate zeros. Upon expanding z-^ and z^^ i n p a r t i a l f r a c t i o n s and t a k i n g z^ so that every residue matrix has u n i t rank, we f i n d (c - 1)(1 - 2a + a 2 + b ) / 4 8 j i 0 ( 3 ~ c ) ( 9 - 6 a + a + b ) / l 6 v / 6 (c - 5 ) ( 2 5 - 10a + a 2 + b ) / 16^6 K t 5 / 2 N / 2 3 / 2 s / 2 3 / 2 x f 2 5 / 2 J 2 ( 7 - c)(49 - 14 -a •+ a -1- b ) / 48\/T0 The procedure given at the end of Section 4 - 3 i s used to determine the maximum and minimum values of a. The r e a l i z a b i l i t y l i m i t s are shown in- Figure 4.4 as f u n c t i o n s of a s i n g l e parameter with the other two parameters f i x e d . The c e n t r a l point i s a - 3 , b=l, c=8. Thus, Figure 4 . 4 ( a ) shows the l i m i t s of r e a l i z a b i l i t y as func-t i o n s of a with b=l, c=8, Figure 4 . 4 ( b ) as f u n c t i o n s of b with a= 3 ? c=8 and Figure 4 . 4 ( c ) as f u n c t i o n s of c with a= 3, b=l. The various r e a l i z a b i l i t y l i m i t s and t h e i r r e s p e c t i v e des-i g n a t i o n s i n Figure 4 - 4 are given as f o l l o w s : a) the dominance l i m i t s (F-G conditions) of AK^A i n ( - 4 . i l ) are designated by " s 1 " , i=0, 1, 2, 3 , 101 b) the r e a l i z a b i l i t y of Y-^(s) = sC-^ + J^-^ gives i ) l i m i t s -within which c-^ a r i d j - ^ ^ a v e i d e n t i c a l s i g n s , designated by " s i g n " , and i i ) dominance l i m i t s of J - ^ which are designated by "J-Q"' and c) the a c t u a l r e a l i z a b i l i t y l i m i t s , designated by "c^", "C2" "g^", ' ^ 2 " and "g^", are determined by the cond i t i o n s that c^, c^, g-^, 1 a n-d g^, r e s p e c t i v e l y , must be nonnegative. Note that the l i m i t s i n (a) apply to any RC r e a l i z a t i o n of the g i v -en impedance f u n c t i o n s . The l i m i t s i n (b) are derived from a d d i -t i o n a l c o n d i t i o n s which are r e l a t e d to our method and are a p p l i e d d i r e c t l y to the given matrix. In Figure 4 . 4 , the cross-hatched areas i n d i c a t e that the l i m i t s (b) are more r e s t r i c t i v e than the l i m i t s ( a ) . Shaded areas show d i f f e r e n c e s between the a c t u a l r e a l i z a b i l i t y l i m i t s (c) and the most r e s t r i c t i v e l i m i t s ( i n (a) and (b)) determined by the nec-essary c o n d i t i o n s a p p l i e d d i r e c t l y to the given matrix. Figure 4-5 shows the complete network with element values f o r maximum gain and f o r minimum gain at the c e n t r a l point (a=3, b=l, c=8). . There i s one'T-section corresponding to Category (a) (see Section 4 . 2 ) and one.to Category (c) at both gain l e v e l s . How ever, every, p o s s i b l e combination of T-sections was observed as the parameters a, b and c were v a r i e d over the ranges i n d i c a t e d . Fre-quently, a d i f f e r e n t combination of T-sections occurred at maximum gain than at minimum gain at the same point (a, b, c ) . For example at a=3, b=-4, c=8, there are two Category (a) T-sections at maximum gain, but two Category (b) T-sections at minimum gain. From the graphs shown i n Figure 4 . 4 , we make the f o l l o w i n - I I I L_ vo vr\ J - r> 105 port 1 O— «1 g 5 g3<> A / -port 2 for maximum gain for minimum gain (nec. cond.) 4.80 2.123894-l/of (actual) if. 28177? 2.123894-°1 0.4-31202 0.280810 c2 0.12854-1 0.825616 c 3 0.106051 0.106051 V 0.272880 0.570344 c 5 0.14-8023 0.2984-15 «1 0.988689 0.0 g2 0.0 2.374-802 g 3 0.186305 0.375591 *3 0.320192 0.130906 % 0.390270 0.564219 0.305018 0.888995 . 0.801907 1.61664-8 total capacitance 1.086697 2.081236 Figure 4 . 5 R e a l i z a t i o n of impedance f u n c t i o n s of Example Five •with a = J, b = 1 and c = 8 . Element values are i n farads and mhos. 106 observations: i ) over considerable ranges of the parameters a, b and c the a c t u a l r e a l i z a b i l i t y l i m i t s c o i n c i d e w i t h those deter-mined by the F-G c o n d i t i o n s , i i ) where the a c t u a l l i m i t s are w i t h i n those determined by d i r -e c t l y a p p l i e d c o n d i t i o n s , the d i f f e r e n c e s (shaded areas) between them are g e n e r a l l y small compared to the d i f f e r -ences between maximum and minimum gain, i i i ) only over r e l a t i v e l y small ranges of a and b are t h e . l i m i t s '• determined by the dominance of J - ^ more r e s t r i c t i v e than those determined by the F-G c o n d i t i o n s , and i v ) the c o n d i t i o n g^sO determines four extreme.values, namely, a • , a „ , b and c . , but b . i s determined by the mm' max max mm mm ° F-G c o n d i t i o n s and c = max In view of the f a c t that the F-G c o n d i t i o n s are known to be necessary but not s u f f i c i e n t , i t i s c l e a r that our synthesis procedure i s cap-able of a c h i e v i n g an almost f u l l range of gain values over ranges of a, b and c i n which a r e a l i z a t i o n e x i s t s . While i t i s not p o s s i b l e to g e n e r a l i z e from one example, i t seems reasonable to conclude t h a t , when passive r e a l i z a t i o n s e x i s t , values of ga i n close to the l i m i t s determined by the F-G conditions can u s u a l l y be achieved. The most r e s t r i c t i v e c o n d i t i o n s a r i s i n g i n the procedure appear to be the. s i g n c o n d i t i o n on j - ^ a n-d "the con-d i t i o n g n +]_-0* The former determines the t o p o l o g i c a l r e a l i z a b i l i t y of Y - ^ ( s ) . From ( 4 . 7 ) , the l a t t e r depends upon the number of c o l -umns of f a l l i n g i n t o Categories (a) and (b) . Thus, the e f f e c t of t h i s c o n d i t i o n i s l i k e l y to be q u i t e severe f o r l a r g e n. 3.07 4.4.2 Example Six The given functions are the same as those i n Example Fiv e -with a=3, b=0, c=8. Thus, (s +"l)(s + 3)(s + 5)(s + 7) 2(s +2)(s +4) (s + 6) and z 1 0 ( s ) 4(s + i ) ( s + 3)(s + 5)(s:-+ 7) ' (s + 3 ) 2 ( s + 8) Note that a f a c t o r (s + 3) cancels i n z-^ s o "that z^ has a p r i -vate pole at s = -3. Three r e a l i z a t i o n s of the given f u n c t i o n s are shown i n Figure 4-6, namely, edure f o r n=4, b) a r e a l i z a t i o n determined by us i n g our synthesis procedure f o r n=3 a f t e r f i r s t removing the p r i v a t e pole of z-^, and c) a conventional ladder r e a l i z a t i o n determined by removing the p r i v a t e pole as the f i r s t step. Upon comparing the three r e a l i z a t i o n s , we make the f o l l o w -i n g observations: i ) r e a l i z a t i o n (a)' has the l a r g e s t gain f a c t o r and the smal-l e s t t o t a l capacitance, i i ) r e a l i z a t i o n s (b) and (c) have roughly equal values of t o t a l capacitance but (b) has a much l a r g e r gain f a c t o r , and i i i ) each of the r e a l i z a t i o n s i s minimal. Observation ( i i i ) i s v a l i d even though r e a l i z a t i o n (a) has one more element than the other r e a l i z a t i o n s . A zero residue i n z-^ does not decrease the number of independently s p e c i f i a b l e parameters un-l e s s the p r i v a t e pole i s f i r s t removed to decrease n.' a) a d i r e c t r e a l i z a t i o n determined by us i n g our synthesis proc-108 0.267? 0.1302 1.5525 o ON 0.2942 - A r 0.1700 - / \ r -3 — o CO -O ON O -3" -O l/cX = 4.9847, C t o t a l = 1.1603 (a) D i r e c t r e a l i z a t i o n w i t h n = 4 . 0.8004 A / l/c< - 4.6042, C total = 4.0672 (b) R e a l i z a t i o n with n = 3 a f t e r removing the p r i v a t e pole from z. J l l * 8.0 l/cx = 1.6875, c t o t a l " 3 , 9 5 9 9 .(c) Conventional ladder r e a l i z a t i o n with-removal of the p r i v a t e pole from z-^ as the f i r s t step. Figure 4.. 6 R e a l i z a t i o n s of the fun c t i o n s i n Example S i x . Elem-ent values are i n farads and mhos. 109 4-4.3 D i s c u s s i o n A synthesis procedure which y i e l d s a new c l a s s of minim-a l , grounded two-port RC networks has been presented. The f o l l o w i n g d e s i r a b l e features are claimed: i ) i t i s p o s s i b l e to r e a l i z e a given 2x2 impedance matrix e x a c t l y , : i i ) a complete f a m i l y of equivalent networks i s obtained, i i i ) the t o t a l capacitance i n the network can.be minimized, i v ) i t i s p o s s i b l e to r e a l i z e complex transmission zeros, v) given z-^ and ^ s P o s s i b l e to r e a l i z e z-^ e x a c t l y and z-^ 2 w i t h the optimum gain f a c t o r , and v i ) the procedure i s e a s i l y programmed f o r d i g i t a l computer execution. On the debit s i d e , a set of necessary and s u f f i c i e n t r e a l -i z a b i l i t y c o n d i t i o n s has not been found. .This i s a serious s h o r t -coming which plagues most matrix synthesis methods. However, our method represents an extension of synthesis techniques because the known methods f o r the r e a l i z a t i o n of an impedance matrix by means of a two-element-kind network without transformers are l i m i t e d to cases where n^k+1. With regard to the r e a l i z a t i o n of z-^ e x a c t l y and z-^ "to w i t h i n a m u l t i p l i c a t i v e constant, our method i s somewhat more r e s -t r i c t i v e than some conventional/methods, e s p e c i a l l y f o r l a r g e n.. However, n i s small (^lO) i n most problems encountered i n engineering design, i n which case our method i s not unduly r e s t r i c t i v e . 5. SUMMARY AND SUGGESTIONS FOR FURTHER RESEARCH New procedures have been given f o r the synthesis of a c l a s s of two-element-kind networks from, s p e c i f i e d immittance mat-r i c e s . In terms of RC networks, the given kxk o p e n - c i r c u i t im-pedance matrix Z R C ( s ) = K(sU + + Q which i s of degree n, i s equivalent to the normal form s t a t e model X = -LX + K^I o V = KX + QI o .o th t h where I i s the k order input v e c t o r , V q the k order output vec-th t o r and X the n . order s t a t e v e c t o r . The o b j e c t i v e of the synthe-s i s procedure i s to f i n d a matrix M which, by the transformation V, = MX be allows the transformed s t a t e v a r i a b l e s to be i d e n t i f i e d as a set of independent c a p a c i t o r voltage v a r i a b l e s i n a passive RC network. In the main synthesis procedure (Procedure I i n Chapter 3), Z'.^(s) = Z R^(s) - Q i s taken as the kxk p r i n c i p a l submatrix of the nxn matrix Z (s) = M(sU + I O ' V . n In t h i s way, the f i r s t k rows of M are given by K. The remaining n-k rows of M must be determined such that the s h o r t - c i r c u i t capac-it a n c e matrix C = (M^) 1 M 1 and the s h o r t - c i r c u i t conductance mat-r i x J "(M^ ) "*"LM 1 are r e a l i z a b l e as (n+1)-terminal, single-element-kind networks with i d e n t i c a l port c o n f i g u r a t i o n s . By s e t t i n g as many o f f - d i a g o n a l e n t r i e s as p o s s i b l e . i n C to zero, the r e a l i z a -I l l b i l i t y c o n ditions on C and J are s i m p l i f i e d and a set of orthogon-a l i t y c o n ditions i n v o l v i n g the e n t r i e s of M are e s t a b l i s h e d . The n-k .unspecified diagonal e n t r i e s i n C and the remaining -g-(n-k) (n-k-l) degrees of freedom i n M represent parameters which can be chosen to advantage i n determining a passive r e a l i z a t i o n . When a passive r e a l i z a t i o n i s found, i t i s always p o s s i b l e to take the diagonal e n t r i e s i n C equal to t h e i r smallest allowed values i n order to minimize the t o t a l capacitance i n the network. A set of necessary and s u f f i c i e n t conditions, has not been found except f o r the s p e c i a l case of k=2, n=3. However, a number . of necessary conditions which can be ap p l i e d d i r e c t l y to the given f u n c t i o n are s t a t e d . Using Procedure I , r e a l i z a t i o n s of Z^^(s) can be deter-mined a n a l y t i c a l l y when n=k+l and n=k+2. This represents an increase i n n by one over e x i s t i n g methods. Furthermore, minimal networks are.synthesized by s u i t a b l y choosing the (n-k) + -Hn-k)(n-k-l) parameters. When nak+3, a n a l y t i c a l s o l u t i o n s are precluded by the n o n l i n e a r i t y of the o r t h o g o n a l i t y c o n d i t i o n s . However, i t i s s t i l l p o s s i b l e to f i n d r e a l i z a t i o n s e i t h e r by a r b i t r a r i l y choosing a num-ber of parameters i n M or by generating a d d i t i o n a l equations so that-M i s determined e x p l i c i t l y . The l a t t e r method i s used ( i n Chapter 4) to synthesize a new c l a s s of minimal, grounded two-port networks c o n s i s t i n g of (n-2) T-sections and one J t-section connected i n p a r a l l e l . To the author's knowledge, t h i s i s the f i r s t s t r a i g h t f o r w a r d method of r e a l i z i n g by matrix methods a 2x2 impedance matrix with n^4. The procedure may al s o be used to r e a l i z e z-^ e x a c t l y and z-^ w i t h i n a v a r i a b l e gain f a c t o r which can be optimized. The method i s somewhat more r e s t r i c -t i v e than, some conventional methods (such as the ladder r e a l i z a t i o n " 112 method) but has the advantage of p r o v i d i n g c o n t r o l of the gain f a c -t o r w i t h the r e s u l t that l a r g e r values of gain may be achieved. An a l t e r n a t i v e synthesis procedure (Procedure I I i n Chap-t e r 3) y i e l d s a network w i t h the minimum number (n) of capacitors by t a k i n g C as a diagonal matrix. In the i n i t i a l step, a p r i n c i p a l submatrix of C.is determined by using Cederbaum's algorithm to decom-t pose the matrix KK . From t h i s p o i n t , the procedure i s s i m i l a r to Procedure I and i s a l s o capable of y i e l d i n g minimal networks. How-ever, Procedure I I i s more r e s t r i c t i v e than Procedure I and not as s t r a i g h t f o r w a r d computationally. The above procedures, with a simple frequency transformation, are a p p l i c a b l e to the r e a l i z a t i o n of a given LC impedance matrix. With appropriate m o d i f i c a t i o n s of a dual nature, Procedures I and I I may be used to r e a l i z e LC and RL admittance matrices. However, because the procedure normally leads to nonplanar networks, a given matrix may not be r e a l i z a b l e i n both the impedance and the admittance form; that i s , dual networks may not e x i s t . Thus, the procedure i n Chapter 4 i s a p p l i c a b l e to ZR(-,(s) and Z ^ ( s ) but not to Y ^ ( s ) and T R L ( B ) . Several aspects of the present work merit f u r t h e r i n v e s t i -g a t i o n . Recently, a number of methods have been given f o r the r e a l -i z a t i o n of an nxn constant matrix as an n-port r e s i s t o r network with (27-29) more than n+1 terminals . Incorporation of one of these me-thods may p o s s i b l y widen the c l a s s of functions r e a l i z a b l e by the procedures developed h e r e i n . An i n t e r e s t i n g problem concerns the r e a l i z a t i o n of ZR(-,(s) when every t r a n s f e r impedance f u n c t i o n has at l e a s t a double zero at s = 0 0 , that i s KK^ i s a diagonal matrix and Q = 0. Then C i s a d i a -113 gonal matrix and places no c o n s t r a i n t s upon the i n t e r n a l port con-f i g u r a t i o n . Hence, only the r e a l i z a b i l i t y of J need be considered i n determining the modal matrix. This problem should be i n v e s t i -gated w i t h the o b j e c t i v e of developing a method f o r determining M which always leads to a r e a l i z a b l e J provided a w e l l - d e f i n e d set of c o n d i t i o n s i s s a t i s f i e d . In the two-port case, such a method would be a p p l i c a b l e to the r e a l i z a t i o n of voltage t r a n s f e r f u nctions with at l e a s t one zero at s = 0 0 . Grounded two-port networks c o n s i s t i n g of (n-2) T-sections and one at-section connected i n p a r a l l e l possess d e s i r a b l e p r o p e r t i e s , but f u r t h e r i n v e s t i g a t i o n i s needed to e s t a b l i s h necessary and suf-f i c i e n t r e a l i z a b i l i t y c o n d i t i o n s . I t may a l s o be p o s s i b l e to f i n d other new c l a s s e s of networks by using d i f f e r e n t c r i t e r i a f o r gen-e r a t i n g a d d i t i o n a l equations i n the e n t r i e s of M. APPENDIX A ' CEDERBAUM'S DECOMPOSITION ALGORITHM (22) In 1959, Cederbaum removed the l a s t obstacle to the synthesis of single-element-kind n-port networks with n+1 t e r m i n a l s . By developing a decomposition algorithm, he provided the means f o r answering the question: i s the given symmetric matrix V/ of constant numbers decomposable i n t o the form W = VDY* where D i s a diagonal matrix w i t h p o s i t i v e e n t r i e s on the diagonal, and every entry i n V i s e i t h e r —1 or 0? A negative answer means that W cannot be the o p e n - c i r c u i t impedance matrix of an n-port r e s i s t o r network w i t h n independent c i r c u i t s , or that W cannot be the s h o r t - c i r c u i t admittance matrix of an n-port r e s i s t o r network w i t h n+1 nodes. A p o s i t i v e answer re q u i r e s f u r t h e r i n v e s t i g a t i o n . In the impedance case, V must be r e a l i z a b l e as a c i r c u i t matrix, of a l i n e a r graph c o n t a i n i n g n indep-endent c i r c u i t s and, i n the admittance case, V must be r e a l i z a b l e as a cut-set matrix of a l i n e a r graph w i t h n+1 nodes. A d i r e c t a l -gebraic, method of t e s t i n g f o r the r e a l i z a b i l i t y of V' i s known ^0 ) ^ Cederbaum's decomposition algorithm may be used to decom-pose any nxn symmetric matrix W = jw^JJ. Because W i s symmetric, we s h a l l r e f e r only to e n t r i e s on and above the main diagonal and'it w i l l be understood that the s u b s c r i p t s on w.. are ordered so that j s i . The procedure can be o u t l i n e d as f o l l o w s : i ) Take the nonzero o f f - d i a g o n a l entry of ¥ with the minimum absolute value (or one such e n t r y ) . Let t h i s entry be w '. Let d, be the f i r s t diagonal entry i n D. Then pq 1 115 d l = w pq t h i i ) Let be the f i r s t column i n V and l e t be the i entry, i n V^. Then take v = +1 and set v ^ .= +1 when w >0 and v , = -1 when w <0. p<* .q.i t h i i i ) The r entry i n V^, v ^ , i s determined by examining the 3x3 p r i n c i p a l submatrix of ¥ c o n t a i n i n g rows (and columns) p, q and r , r / p, q. When w^^>0, '+1, i f w >0 and w -^0 ' . pr qr v , = / - l , i f w -<0 and w ^0 r l \ ' pr qr and when w ^0, pq 0, otherwise +1, i f w =>0 and w ^-0 ' pr qr v , = /-1, i f w ^0 and w r l \ ' pr qr s . 0, otherwise Note that v -, = 0 whenever w or w i s zero. A t o t a l of r l pr qr (n-2). 3x3 submatrices must be examined i n order to com-p l e t e l y determine V^. i v ) Calculate, the matrix W^  by •w1 = W - d ^ V * I f W^  has some nonzero o f f - d i a g o n a l e n t r i e s , the procedure i s r e -peated to f i n d a new column , a new diagonal entry d 2 and a new matrix W^  given by t W2 = W]_ - d 2V 2V 2.. I f W2 has some nonzero o f f - d i a g o n a l e n t r i e s , the procedure i s again repeated u n t i l , a f t e r m steps, W i s a diagonal matrix. A matrix, 116 V , with rn columns and an rnxm diagonal matrix, D', have been deter-mined. The next step i s to w r i t e W = UW ifk = YUDUYU± m m where U i s the i d e n t i t y matrix of order n. D" i s .formed by c r o s s i n g out the i ^ * 1 r o w and column of ¥ m whenever the i ^ * 1 diagonal entry i s zero, and V" i s formed by c r o s s i n g out the corresponding columns of U. F i n a l l y , we have the de s i r e d matrices D' 0 0 D" Each step decreases the number of nonzero o f f - d i a g o n a l en-t r i e s i n W by one, so the number of steps i s f i n i t e and not greater than the number of o f f - d i a g o n a l e n t r i e s i n W. Thus, m£-g-n(n-l). C l e a r l y , the.' order, r , of D i s given by lir<-jn(n+l). The matrices V a n d D determined by the above procedure are e s s e n t i a l l y unique. V/e say " e s s e n t i a l l y " because a number of t r i v i a l v a r i a t i o n s are p o s s i b l e . For example, any column V may be m u l t i p l i e d by -1, or the columns of V may be reordered- as long as the diagonal e n t r i e s of D are.reordered i i the same way. The above d e s c r i p t i o n i s , of n e c e s s i t y , very b r i e f . For a more d e t a i l e d development of the algorithm, the reader i s r e f e r r e d to the o r i g i n a l paper by Cederbaum. V [V V"] and D APPENDIX B ON NECESSARY AND SUFFICIENT REALIZABILITY CONDITIONS Although general necessary and s u f f i c i e n t c o n d i t i o n s f o r the r e a l i z a b i l i t y of a f u n c t i o n Z R C ( s ) = K(sU + L) 1 K t B . l by Synthesis Procedure I have not been found, we s h a l l formulate here co n d i t i o n s f o r two s p e c i a l cases, namely, ( i ) networks co n t a i n -i n g a s t a r - l i k e c a p a c i t o r t r e e , and ( i i ) networks c o n t a i n i n g a l i n -ear c a p a c i t o r t r e e . Necessary and s u f f i c i e n t r e a l i z a b i l i t y condi-t i o n s w i l l be derived f o r the case of k=2, n=3• B.1 Networks Containing a S t a r - L i k e Capacitor Tree . For l a t e r use, the f o l l o w i n g theorem i s st a t e d and proved. th Theorem B . l : Let A be an n order symmetric, p o s i t i v e - d e f i n i t e matrix with nonpositive o f f - d i a g o n a l e n t r i e s . Then every entry i n i t s inverse i s nonnegative. Proof: The theorem i s c l e a r l y true f o r n=2. Suppose that i t i s al s o true f o r n=m and consider an (m.+l)s^ order matrix, p a r t i t i o n e d as f o l l o w s A m 1 A. 12 A, 12 22 The inverse of A i s given by m 1 A -1 A " + A 1 1 A 1 2 ^ 1 A I ^ A I 1 11 A" 1 ' A 1 2 A 1 1 12 1 -A "'"A i A11 A12 .-1 ,-1 B.2 where - A _ A A 1 A _ A 2 2 A i 2 ^ i l 12' 118 By the s u p p o s i t i o n above, every entry i n i s nonnegative and, because A i s p o s i t i v e d e f i n i t e , ?>0. Therefore, every entry i n A ^ i s nonnegative and the theorem i s proved by i n d u c t i o n . Synthesis Procedure I y i e l d s and where J D, c. . 13 DJ D , .11 0 0 '22 (KK*)"" 1 0 d i a g ( d k + 1 , d k + 2 , D d ) , D = n U 0 0 D, B .3 J l = 3 ± 3 . J. J. 11 t 12 12 J 22 N 2LN^ A necessary c o n d i t i o n f o r the r e a l i z a b i l i t y of a given impedance matrix (B.l) as a network wi t h a s t a r - l i k e c a p a c i t o r tree i s that C and J , given by ( B . 3 ) , have non p o s i t i v e o f f - d i a g o n a l ent-r i e s ( a f t e r m u l t i p l y i n g some rows of M by -1, i f necessary). A nec-essary and s u f f i c i e n t c o n d i t i o n i s that C and J be hyperdominant. Using Procedure I , any r e a l i z a t i o n w i l l have a common ground because the e x t e r n a l ports are i n p a r a l l e l w i t h tree-branch c a p a c i t o r s and thus a l s o form a s t a r - l i k e c o n f i g u r a t i o n . Suppose that C and J have nonpositive o f f - d i a g o n a l ent-r i e s . Whether or not C i s dominant can be determined d i r e c t l y 119 because the dominance of i m p l i e s the dominance of C. In the case of J, the dominance of J - ^ i s a necessary but not s u f f i c i e n t c o n d i t i o n . We must determine under what co n d i t i o n s there e x i s t s a set of paiameters d, -, , d, „, . . . , d such that J i s dominant r k+1' k+2' ' n Fo i : the equations (5) F o l l o w i n g a procedure given by Schneider , we form n Y. 3, ,d = 6, , 1=1, 2, . . . , n. B.4 3=1 ID 3 i . • t h M u l t i p l y i n g the i equation by d^ gives n Yl 3 ' i i d i ^ i = d i A ' 1 = 1 ' 2, . . ., n. B.5 3=1 t h C l e a r l y , d^p\ i s the sum of the e n t r i e s i n the i row of J . R e c a l -l i n g that every o f f - d i a g o n a l entry of i s n o n p o s i t i v e , J w i l l be dominant i f _ d i p i s 0 , 1=1, 2, . . ., n. Equation (B.4) can be w r i t t e n i n the matrix form J 1 c o l ( d i ) = col(B j L) or, as J-^ i s p o s i t i v e d e f i n i t e , col(d.) = J n 1 c o l ( B . ). B.6 l 1 M i Thus, J can be made dominant i f , f o r some choice of nonnegative p\ , the d^ determined by (B.6) are nonnegative. In the problem under c o n s i d e r a t i o n , d^ = 1, i = l , 2, . . ., k, that i s , some of the d^ are known. From J = (M*) 1LM 1 , we have. J " 1 = ML~1Mt = D~ 1NL~ 1N tD~ 1, Thus, J i 1 = NL V . Then, -with the usual p a r t i t i o n i n g , (B.6) becomes 120 c c l ( l ) c o l ( d i ) 2 K l T 1 ^ KL X N 2 N 2L 1 K t N 2L % 2 c o l ( B . i ' l c o l ( p i ) 2 B.7 whence c o l ( B ^ ) ^ (KL V ) 1 c o l ( l ) - (KL V ) XKL ^ c o l C f L ^ B.i and c o l ( d i ) 2 = N 2L 1 K t ( K L V ) 1 c o l ( l ) + -N 2L" 1K t(KL~ 1K t) ^ K l T 1 ^ N 2 L " l N 2 c o l t f L ^ B.9 Using ( B . 2 ) , (B . 7 - 8 ) become, r e s p e c t i v e l y , c o l ( B i ) 1 c o l ( d i ) 2 (KL V j c o l d ) + J 1 2 J 2 2 c o l ( B i ) 2  J 2 2 ~^2°°1^1^ + c o 1 ^ P 2 * B . 10 B . l l For nonnegative B^, every entry i n J ^ 2 c o l ( l ) + col(8 n. ) i'2 is 1 . nonnegative and, by Theorem B.I, every entry i n J 2 2 i s nonnegative. t h Therefore, d^>0 and hence the i row of J s a t i s f i e s the dominance c o n d i t i o n f o r i = k+l, k+2, . . ., n. In order that rows 1 to k of J s a t i s f y the dominance con-d i t i o n , the e n t r i e s of col(S^).-^ must be nonnegative f o r some choice of c o l ( B ^ ) 2 . In (B . 1 0 ) , the e n t r i e s of Jj .2^22 c°l^Pj_^2 a r e n o n _ p o s i t i v e , and vanish when c o l ( B ^ ) 2 = 0 , g i v i n g c o l ( B i ) 1 = ( K L ~ 1 K t ) ~ 1 c o l ( l ) Z R J ( 0 ) c o l ( l ) B .12 But the admittance matrix of a common-ground network has nonpositive. o f f - d i a g o n a l e n t r i e s at zero frequency. Therefore, the c o n d i t i o n Bj-rO, i = l , . 2 , . '. ., k i m p l i e s that Z R J ( o ) must be dominant. 121 In summary, a given impedance matrix (B.l) i s r e a l i z a b l e by Procedure I as a common-ground network w i t h a s t a r - l i k e capaci-t o r t r e e i f and only i f i ) Z R Q ( 0 ) , C - J ^ and <L^ are hyperdominant matrices, and i i ) has nonpositive o f f - d i a g o n a l e n t r i e s . The u t i l i t y of t h i s r e s u l t i s l i m i t e d , however, because f o r n>k+2 i t i s d i f f i c u l t to determine whether there e x i s t s an M such that has the desired s i g n p a t t e r n . . B.2 Networks Containing a Linear Capacitor Tree Consider next the case where every entry i n C and J i s nonnegative ( a f t e r m u l t i p l y i n g some rows of M by -1, i f necessary). Then any r e a l i z a t i o n found by Procedure I must contain a l i n e a r cap-a c i t o r t r e e . One might expect the existence of r e a l i z a b i l i t y con-d i t i o n s s i m i l a r to those found above f o r networks with a s t a r - l i k e c a p a c i t o r t r e e . However, stronger conditions appear to be necessary f o r n> 3 • The necessary and s u f f i c i e n t c o n d i t i o n that the matrix J with nonnegative e n t r i e s be r e a l i z a b l e on a l i n e a r p o r t - t r e e c o n f i g -th u r a t i o n i s that i t be uniformly tapered. An n order symmetric mat-r i x , A = [ a i j ' w ^ e r e aj_-pO> i s uniformly tapered i f the rows and columns can be renumbered ( i f necessary) so that l j i - l '3+1- 1-1, J x ' j + l f o r a l l i - ^ i , w i t h the a d d i t i o n a l n o t a t i o n a. -, = a~ . = 0. d' i,n+l 0,j Consider the s p e c i a l case k=2, n= 3 with c. .2 0 and j . .^0. . I t i s e a s i l y shown that the inverse of a 3 ^ 3 hyperdominant matrix i s n e c e s s a r i l y uniformly tapered ( t h i s i s not true f o r matrices of l a r -ger order). Thus, a s u f f i c i e n t c o n d i t i o n f o r the r e a l i z a b i l i t y of J i s that a nonnegative value of d^ e x i s t such that J -- 1- i s hyperdom-122 i n a n t . we have Follo w i n g the- same procedure used i n the previous s e c t i o n , c o l ( l / d i ) =. J 1 c o l ( 6 i ) or 1 1 1/d 3. 11 t 12 J 12 J 22 whence and J l 1 = J -1 11 l / d 5 = J* 2J - i | ; J i i J i 2 P 3 + ( J 2 2 - J i 2 J l l J 1 2 ) P 3 C l e a r l y , the 2x1 matrix J dominant. Also -1 11 1 1 B.14 has nonnegative e n t r i e s i f J - ^ i s J 22 ~ J 12 J 11 J 12 (N 2L hp 1>0, Therefore, f o r any choice of B^ £0, d„, determined by (B . 14) , i s non-negative. Furthermore, from (B.13) , i t i s evident that B-^ O and B^O. C and J must be r e a l i z a b l e on i d e n t i c a l tree s t r u c t u r e s . In general, c- 2^>0 so that c a p a c i t o r s c-^  and c 2 share a common (ground) node. Thus, f o r the r e a l i z a b i l i t y of J , we requ i r e ^12sd3^13 0 r ^12~d3^23' However, by t a k i n g B^ la r g e enough so that e i t h e r 8^ = 0 or B 2 = 0 i n (B .13), i t i s e a s i l y shown that d^ can always be chosen to s a t i s f y the above c o n d i t i o n . 123 In summary, a given impedance matrix ( B . l ) , where k=2 and n=3j i s r e a l i z a b l e as a 'common-ground network with a l i n e a r capac-i t o r tree ^ f and only i f i ) KK- and KD~-LKl" are hyperdominant matrices, i i ) J-j^  has nonnegative e n t r i e s , and i i i ) J-j^ i s dominant. B.3 Networks with k=2. n=3 Necessary and s u f f i c i e n t c o n d i t i o n s have been e s t a b l i s h e d above f o r the case of n=3 because a tree c o n s i s t i n g of three branches can be only s t a r - l i k e or l i n e a r . The r e s u l t s of Sections B . l and B.2 f o r t h i s case are summarized by the f o l l o w i n g theorem. Theorem B.2: A given 2x2 impedance matrix (B.l) of degree 3 i s r e a l -i z a b l e by Procedure I as a grounded network i f and only i f t - I t i j KK and KL K are dominant wi t h i d e n t i c a l s i g n patterns, and i i ) (KK^) "'"KLK^(KK^) 1 Is dominant wi t h a s i g n p a t t e r n i d e n t i -c a l to that of (KK*)" 1. Condition ( i ) i s equivalent to the Fialkow-Gerst c o e f f i c -i e n t c o n d i t i o n s for. t h r e e - t e r m i n a l networks a p p l i e d to the c o e f f i c -2 0 t i e n t s of s and s . When KK i s a diagonal matrix, that. i s , c^^f i t i s p o s s i b l e to r e a l i z e a p o s i t i v e r e a l t r a n s m i s s i o n zero. The r e a l i z a t i o n , i f one e x i s t s , w i l l not have a common ground and hence must contain a l i n e a r c a p a c i t o r t r e e . APPENDIX C NUMERICAL SOLUTION FOR THE MODAL MATRIX In t h i s appendix, we describe the numerical technique used to solve f o i the modal matrix i n the synthesis, procedure of Chapter (26) 4. The basic method, due to F l e t c h e r and Powell , i s an i t e r a -t i v e descent method' f o r f i n d i n g a l o c a l minimum of a f u n c t i o n of se v e r a l v a r i a b l e s . We s h a l l describe b r i e f l y the F l e t c h e r — P o w e l l method aid then consider i t s a p p l i c a t i o n to the problem at hand. I t i s convenient to use the Dirac bra-ket n o t a t i o n . In t h i s n o t a t i o n the column vector (x-^,x2, . . ., x ) i s w r i t t e n as |x> . The row vector with the same e n t r i e s i s denoted by <x| . Let f denote the f u n c t i o n to be minimized, |x) denote i t s arguments and |g>denote i t s gradient. Let the current point be Ix1)1 with gradient j g 1 ) . The i t e r a t i o n can then be stated as f o l l o w s . i ) Set J s 1 ) - -H 1|g 1)>, where H 1 i s a square matrix determined i n the previous i t e r a t i o n , i i ) Obtain a 1 such that f C J x 1 ) + a 1 j s 1 ) > ) i s a minimum and a 1>0. ( a 1 can always be chosen to be p o s i t i v e . ) i i i ) - Set ICT 1) = a 1 I s 1 ) , i v ) Set |x 1 + 1> = I x 1 ) + I d 1 ) . v) Evaluate f ( | x 1 + ± > ) and |g 1 + 1>. v i ) Set l y 1 ) = |g i + 1> - j g 1 ) . v i i ) Set H 1 + 1 = H 1 + A 1 + B 1 ' ' v, ' A i ' I t r 1 ) < c r i l , p i - H 1 1 y 1)* < y 1 l H 1 . where A = —: and B = -—. ,-y . ,—. ,y — ^ " l y 1 ) <y 1|H 1|y 1) v i i i ) Set i = i + l and repeat. Matrix H may i n i t i a l l y be chosen to be any p o s i t i v e - d e f i n i t e symmet-125 r i c matrix. However, i t i s convenient to take the i d e n t i t y matrix i n i t i a l l y f o r H so that the f i r s t d i r e c t i o n i s down the l i n e of steepest descent. The F l e t c h e r - P o w e l l method always converges, and converges r a p i d l y , and does not re q u i r e i n i t i a l estimates close to the f i n a l values. Furthermore, the method i s r e l a t i v e l y easy to program f o r d i g i t a l computer execution. In our a p p l i c a t i o n , l e t the normalized modal matrix, N n. . , he p a r t i t i o n e d i n t o rows so that N* i d . . . 1 2 n The f i r s t k rows of N are known. From Section 4.1, we have the f o l -lowing equations. i = 1, 2, . . . , n N i N J = &lj>I j = k+1' k+2' ' * " n 3 - i N'.LN. = 0 , i 1 i , j =k+l, k+2, . . ., n The f u n c t i o n to be minimized i s chosen to be n n-1 n f = (N.N* i ) 2 + y ( v ) y n - l n + i=k+l i = l j=k+l 3>i i=k+l j=k+2-(N.LN:)2 n n n - l n n - r G i+k+l m=l 2 - i ) 2 + l m • ( i = l j=k+l m=l n. n . )' lm jm 126 n - l n n + n . \ n . ) 2 . • C l / mi m 3m i=k+l j-k+2 m=l The gradient f u n c t i o n s are computed as n & = 4n (N - 1) +Y" 2n. (N.N*) — 3 n v o r s r r / i s 1 r i = l 3 r s " d~ra 1=1 n + ) 2\ n. (N.LN*), f r = k+l, k+2, . . ., n / s i s 1 r ) ' ' ' i=k+l ) i ?Z r ( s = l , 2, . . . . n At Step ( i i ) of the i t e r a t i o n procedure, i t i s necessary to f i n d the value of cx which minimizes the f u n c t i o n f(|n)> + a|s)> ), where I h> = col(n, -, -,,n, -, 0, . . . , n, -, , n, 0 , , . . . , n ) I / k+1,1' k+l,2 ' k+l,n' k+2,1' ' nn = col(N, ,N. _, . . . , N ) . k+l' k+2' ' n S u b s t i t u t i n g (n. . + as. .) f o r n. . i n ( C l ) gives & 13 I J 13 & ' n n f(|n> + a| s > ) = 2_ ( Z _ < n i m + a s i m ) 2 - ^ i=k+l m=l n - l n n + / / ( / (n. + as . ) (n . + as . ) )' Z Z Z im im jm jnr i = l j=k+l m=l. n - l n n 2 + > > ( > (n: + as. ) \ (n. + as. )) / / / ,im im m jm jm i=k+l j=k+2 m=l 127 A f t e r c o l l e c t i n g terms, we f i n d f ( | xi} + oj s> ) r n f(|n>) + a n n n - l n i=k+l m=l m=l n n n. s . ) im im + Y~~ V ~ 2 ( V ~ n . n . )(^ (n. s . + n . s. )) Z Z Z i m J m Z i m 3 m J m im i = l j=k+l • m-1 n - l . n n m=l n + 2(\ n. \ n . ) ( \ (n. \ s. + n . \ s . )) w im m jm / im m jm jm m im" i=k+l j=k+2 m=l m=l + a ' n n n n V ~ ( 4 ( V " n . s. ) 2 + 2( >~~n2 - l ) (Y™ s 2 )) Z Z im im z ^ l m Z , i m i=k+l m=l m=l m=l + + n - l n_ n ((' n n i = l j=k+l m=l n - l n n [n. s . + n . s. ) ) 2 + 2 ( ) n . n . ) ( ) s. s . )) im jm jm im / im jm V i m D m m=l n m=l ' n . ) y ~ ( (^ ( n - \" s . +n. \ s. ) ) 2 + 2 ( ) n . \ n . ) ( Y ~ s . \ s j ) Z z Z im m jm jm m im i m m J 1 1 Z im m jn i=k+l j=k+2 m=l m=l m=l + a n n n n-l n n n 4(> n . s. ) ( ) s f ) + ' 11 im Z im 2( ) (n s. +n! s.J) () as. '") / am jn . jm im / am ."in-i=k+l m=l n-1 n m=l i = l j=k+l m=l m=l + i=k+l j=k+2 m=l n n 2( ) n. \ s. + n . \ s . ))(> s. \ s . ) ' im m jm jm m im / im m jm m=l 128 + J n n 2 n ~ ^ - n n n - l n n / / im / /_ / urn jn /_ / / .im m jrr; i = k f l m=l i = l j=k+l m=l i=k+l j=k+2 m=l C.2 Upon s u b s t i t u t i n g the current values of | n) and s) , a th 4 order polynomial i n a i s formed. In abbreviated form (C.2) becomes 2 3 4 f ( |n> + cc|s)>) = f ( | n ) ) + a ^ a + + a^a + a^a . The value of a that minimizes f(|n> + a | s ) ) i s determined by. f i n d i n g rd the roots of the 3 order equation f ( I n ) + a Is)) = a n + 2a 0oc + 3 a ^ a 2 + 4a. o? = 0 o a • I 1 2 3 4 and s e l e c t i n g the smallest p o s i t i v e r o o t . The roots are e a s i l y found numeri c a l l y by us i n g the Newton-Raphson method, f o r example. The remaining steps of the i t e r a t i o n procedure are c a r r i e d out In numerical form and re q u i r e no f u r t h e r comment. I t was found convenient to terminate the procedure when every entry in|CP) i s l e s s than a prescribed'value. In t h i s way, the magnitude of nj_-j> which i s . l e s s than u n i t y , i s determined to a pres-cr i b e d number of decimal places. REFERENCES ' "The R e a l i z a t i o n of n-Port Networks Without Transfor-mers—A Panel D i s c u s s i o n " , IRE Trans, on C i r c u i t Theory, v o l . CT-9, Sept., 1962, pp. 202-214. 2. G u i l l e m i n , E. A., "An Approach to the Synthesis of Linear Net-works Through Use of Normal Coordinate Transformations Leading to More General Topological C o n f i g u r a t i o n s " , IRE Trans, on C i r -c u i t Theory, v o l . CT-7, Aug., I960, pp. 40-48. 3. Schwab, W. C , "Synthesis of RC Networks by Normal Coordinate Transformations", Ph.D. d i s s e r t a t i o n , Dept. of E l e c t . Engrg., Mass. I n s t . Tech., Cambridge, Sept., 1962. 4. . Duda, R. 0., "Equivalent and Optimal Equivalent E l e c t r i c a l Net works", Ph.D. d i s s e r t a t i o n , Dept. of E l e c t . Engrg., Mass. I n s t . Tech., Cambridge, June, 1962. 1 5. Schneider, A. J . , "RC D r i v i n g - P o i n t Impedance R e a l i z a t i o n by Linear Transformations", IEEE Trans, on C i r c u i t Theory, v o l . CT-13, Sept., 1966, pp. 265-270.. 6. Mason, L. J . , "Synthesis of Minimal Two-Element-Kind One-Port Networks", Ph.D. d i s s e r t a t i o n , Dept. of E l e c t . Engrg., U. of B r i t . C o l . , V a n c o u v e r f o r t h c o m i n g i n 1968. 7. Lee, H. B., "On the D i f f e r i n g A b i l i t i e s of RL Structures to R e a l i z e N a t u r a l Frequencies", IEEE Trans, on C i r c u i t Theory, v o l . CT-12, Sept., 1965, pp. 365-373-8. B o x a l l , F.S., "Synthesis, of M u l t i t e r m i n a l Two-Element-Kind Net works", E l e c t r o n i c s Research Laboratory, Stanford U., Technical Report No. 95, Nov., 1955. 9. Basson, D., and H a l k i a s , C. C , "The R e a l i z a t i o n of RC n-Ports IEEE Trans, on C i r c u i t Theory, v o l . CT-12, June, 1965, pp. 247-256. 10. 0 1 i v a r e s , J . E., J r . , "Synthesis of Grounded RC n-Ports by Canonical Star-Mesh Networks", Proc. Third Annual A l l e r t o n Conf on C i r c u i t and System Theory, U. of I l l i n o i s , Oct., 1965. 11. Kuh, E. S., and Rohrer, R. A., "The S t a t e - V a r i a b l e Approach to Network A n a l y s i s " , Proc. IEEE, v o l . 53, J u l y , 1965, pp. 672-686 12. Yarlagadda, R., and Tokad, Y., "Synthesis of LC Networks - A State-Model Approach", Proc. IEE, vol.-113, June, 1966, pp. 975-981. 13. Bryant, P. R., "The Order of Complexity of E l e c t r i c a l Networks Proc. IEE, v o l . 106C, June, 1959, pp.' 174-188. 130 14. G i l b e r t , E. G., " C o n t r o l l a b i l i t y and O b s e r v a b i l i t y i n M u l t i -v a r i a b l e C o n t r o l Systems", J. Soc. Indus t . Appl. Math.,. Ser. A, v o l . 1, 1 9 6 3 , pp. 128-151. 15. Kalman, R. E., "On a New C h a r a c t e r i z a t i o n of Linear Passive Systems", Proc. F i r s t Annual A l l e r t o n Conf. on C i r c u i t and Sys-tem Theory, U. of I l l i n o i s , Nov., 1963, pp. 456-470. 16. Kalman, R. E., "Mathematical D e s c r i p t i o n of Linear Dynamical Systems", J . Soc. Indust. Appl. Math., Ser. A, v o l . 1, 1963, pp. 152-192. 17. McMillan, B., " I n t r o d u c t i o n to Formal R e a l i z a b i l i t y Theory", B e l l System Tech. J . , v o l . 31, Pt. 1, March, 1952, pp. 217-279 and Pt. 2, May, 1952, pp. 541-600. 18. Bashkow, T. R., "The A M a t r i x , A New Network D e s c r i p t i o n " , IRE Trans, on C i r c u i t Theory, v o l . CT-4, Sept., 1957, pp. 117-120 f. 1 9 . Bryant, P. R., "The E x p l i c i t Form of Bashkow's A Ma t r i x " , IRE Trans, on C i r c u i t Theory, v o l . CT-9, Sept., 1962, pp. 303-306. 20. Wilson, R. L., and Massena, W. A., "An Extension of Bryant--Bashkow A M a t r i x " , IEEE Trans, on C i r c u i t Theory, v o l . CT-12, Mar., 1965, pp. 120-122. 2 1 . Bacon, C. M., "Time-Domain S o l u t i o n Transformations f o r the n-Port RLC Network", Proc. Ninth Midwest Symposium on C i r c u i t Theory, Oklahoma State U., May, 1966. 22. Cederbaum, I . , " A p p l i c a t i o n s of Matrix Algebra to Network . Theory",' IRE Trans.'on. C i r c u i t Theory, v o l . CT-6, May, 1959, pp. 127 -137 . -23. Fialkow, A., and Gerst, I . , "The Transfer Function of General Two-Terminal-Pair RC Networks", Quart. Appl. Math., v o l . 10, J u l y , 1952, pp. 113-127. 24. G u i l l e m i n , E. A., "On the A n a l y s i s and Synthesis of Single-Elem-ent-Kind Networks", IRE Trans, on C i r c u i t Theory, v o l . CT-7, Sept., I960, pp. 303-312. 25. B i o r c i , G., and C i v a l l e r i , P. P., "Conditions f o r the R e a l i z a - . b i l i t y of a Conductance M a t r i x " , IRE Trans, on C i r c u i t Theory, v o l . CT-8, Sept., 1961, pp. 312-317. 26. F l e t c h e r , R., and Powell, M. J . D., "A Rapidly Convergent Des-cent Method f o r M i n i m i z a t i o n " , Computer J . , v o l . 6, J u l y , 1963, pp. 163-168. 27. Jambotkar, C. G., and Tokad, Y., "Topological Synthesis of n-Port R e s i s t i v e Networks With (n+2) Nodes", Proc. Tenth Midwest :Symposium on C i r c u i t Theory, Purdue U., May, 1967. 131 •28. Lupo, F. J . , "The Synthesis of n-Port Networks on k-Tree P o r t -S t r u c t u r e s " , Proc. Tenth Midwest Symposium on C i r c u i t Theory, Purdue U., May, 1967. 29- Lempel, A., and Cederbaum, I . , "Terminal Configurations of n-Port Networks", IEEE Trans. on C i r c u i t Theory, v o l . CT-15, Mar., 1968, pp. 50-53-30. Gould, R., "Graphs and Vector Spaces", J. Math. Physics, v o l . 37, 1958, pp. 193-214. 

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