ABSTRACT CAUCHY TYPE INTEGRAL REIRESENTATIONS FOR NETWORK FUNCTIONS By K. S. Bajaj Dumped and distributed network theory is approached from system theory point, which is a very basic and unifying standpoint. An algorithm.developed for reduced state descriptions of lumped linear time invariant systems, has been used for "semi-state reduction" of distributed parameter systems. State descriptions for one and two port distributed networks result in Cauchy type integral representations for network functions and "Cauchy Integral Matrix" for immittance matrices. Necessary and sufficient conditions for network functions, to have RC, RL and LC realizable Cauchy integrals have been given. Realization of Cauchy type of Integrals and "Cauchy matrix integral" has been achieved by dis- tributed tapered networks. An analog between potential theory and Cauchy type of Integrals, has simplified the approximation of ideal characteristics of filters and has resulted into fast converging distributed tapered networks. Approximation of filter characteristics varying as function of w, w%, w"!5 has been achieved by Cauchy type of integrals, resulting in tapered distributed networks. A general procedure for approximation of filter characteristics, analytic in w is being suggested. A question unanswered by this work: is it possible to develop an algorithm for state reduction in distributed parameter systems, as it can be done in lumped linear time invariant systems. CAUCHY TYPE INTEGRAL REPRESENTATIONS FOR.NETWORK FUNCTIONS BY ; Kuljit SJ'Bajaj A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering and System Science 1971 ACKNOWLEDGMENTS The author wishes to express his gratitude to Dr. J.A. Resh, for his supervision, constant encouragement and many helpful comments and suggestions. He also expresses his sincere appreciation to Drs. Robert O. Barr, John B. Kreer, C.E. Weil and J.S. Frame for their interest in the work and useful suggestions. Thanks are also due to Dr. H.E. Koeing. Finally the author is grateful to his friends Artice M. Davis, K. Paryani and Sandy Lenchner for their encouragement. III Chapter I. II. III. IV. TABLE OF CONTENTS INTRODUCTION 1.1 Literature Survey ............... ............ . 1.2 Objectives of Thesis ......... ......... ....... 1.3 smary Of Chapters 0....00000.000.000.0000... DERIVATION OF CANONICAL STATE EQUATIONS FOR n-PORT DISTRIBUTED PARAMETERS NNN o WNH Basis for State Representation ............... Derivation of State Equations ................ Equivalent Reduced State Descriptions for lumped Linear Time Invariant Systems ......... Equivalent Reduced State Description for a Class of Two Port Distributed Systems ........ CAUCHY TYPE INTEGRALS 3.1 3.2 3.3 3.4 3 5 Motivation for Cauchy Type Integrals ..... .... Cauchy Integral Representations for Two Pbrt Network Functions ........ ...... .... ..... Conditions Resulting in RC and RL Realizable Cauchy Type of Integrals ....... ...... ........ Realizability Conditions for RLC Networks .... Realizability Conditions for Two Port RC and Rl.Networks .................................. SYNTHESIS PROCEDURES FOR CAUCHY TYPE OF INTEGRALS 4.1 4.2 4.3 4.4 4.5 4.6 Ladder Networks .............................. An Algorithm for Getting Ladder Networks From Cauchy Type Integral Representations .... Foster Type Networks for Cauchy Integral Representations .............................. Synthesis Procedures for Integral Representa- tions of Two Port Networks ................... Synthesis Methods for Cases where Density Function Matrix is Real ...................... Ladder Networks Representing Some Trans- cendental Functions and Numbers .... ..... ..... [V Page LDNH 21 3O 36 38 43 44 47 52 56 56 68 79 VI. VII. PROPERTIES OF CAUCHY TYPE INTEGRALS FOR NETWORK FUNCTIONS 5.1 5.2 5.3 5.4 Generalization of the Results Established for Lumped Linear Time Invariant‘Networks ........ Generation of Distributed RC Impedence Functions 0.0.IOOOOOOCOOOOOOOIOOOOOOO0.0.0...O EqUivalent Networks OOOOOOOOOOOOOIOOOO00...... PrOperties of Cauchy Integral Matrix Representations 00.000.000.00.0.00.00.00.00... CAUCHY TYPE OF INTEGRALS APPLIED TO APPROXIMATION OF NETWORK FUNCTIONS Basis for the Approximation ............. Determination of Charge Distribution ......... Approximation Procedure ...................... Approximation of a Constant .................. Approximation of Functions Varying by Frequency OO...OOOOOOOOOOOOCOOOOOOOOOOO0.0.... Approximation of Functions Varying as we CONCUISIONS OOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOOO APMDICES C.O...0.000000000000000000000000.0.0.... B IB LIOGRAPHY 82 87 89 90 93 102 104 105 133 140 147 150 156 Figure 4.1 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15s 4.15b 4.16s 4.16b 4.17 4.18s 4.18b LIST OF FIGURES Laplace inversion contour for F(s) RC Ladder networks for { Ejéggfi- RL Ladder networks for { EESERSE. $2 +-x2 (s-hy) f (x ) dx (s+a)2 +x2 Ladder network for example 4.1 LC Ladder networks for {1 M RLC Ladder networks for £ Ladder network for example 4.2 Ladder network for example 4.3 RC distributed networks A representation for RC distributed networks LC distributed networks A representation for LC distributed networks RLC distributed networks RLC distributed network Symmetrical lattice networks Realization of 2a Realization of Zb Foster networks for Za Foster networks for Zb Cauer's network Realization of Za in example 4.4 Realization of Z in example 4.4 b V! Page 31 49 49 51 51 55 55 55 57 57 57 57 58 58 60 6O 60 62 62 62 65 65 Wu. ‘;.J I-‘u ll. ' h HA. I “C i.‘ A Ll Figure 4.19 4.20 4.21 4.22 4.23 4.24a 4.24b 4.24c 4.24d 4.25 4.26 4.27 4.28s 4.28b 6.1a 6.1b 6.2a 6.2b 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 Realization of example 4.4 Realization of example 4.4 by Foster type networks Realization of example 4.5 by Cauer's method Network for example 4.6 Realization of example 4.6 by Cauer's method Network for 2 Network for Z Network for Z Network for Z Cauer's realization for example 4.7 Network for e-SE1(-s) Network for log 2% Network for log 2 Reza's network for log 2 Logarithmic potential for charge at the origin Logarithmic potential for charge at Zm Continuous charge distribution Continuous charge distribution along an arc Electrical transducer Charge distribution along a circle Charge distribution along a semi circle Approximation of a constant a for ‘w‘ s 1 Response of Z(s) outside the approximation band Realization of Z(s) Approximation of constant by Z(p) Realization of Z(P) V11 Page 65 66 66 71 71 77 77 77 78 78 80 80 80 80 93 93 96 96 99 106 106 107 108 110 111 112 Figure 6.1la 6.11b 6.12a 6.12b' 6.12c 6.13 6.14a 6.14b 6.14e 6.14d 6.14e 6.15 6.16 6.17s 6.17b 6.17c 6.17d 6.18 6.19a 6.19b 6.19c 6.19d 6.20 6.21 Parallel plates carrying same charges Parallel plates carrying opposite charges Approximation of a constant by F(s) Response of F(S) outside the desired frequency range Comparison of Butterworth, Chebyschev and Cauchy type integral responses 1.8 2. Approximation of a constant by —_29 f §S+uzgx 2 TI' 0 (8+0) +x Network for F1(s) Network for F2(s) Network for F3(s) Network for F4(s) Network for F5(s) Placement of plates for approximation in the middle band Approximation of a constant in the middle band Network for F1(s) Network for F2(s) Network for F3(s) Network for F4(s) Approximation in the high band Network for F1(p) Network for F2(p) Network for F3(p) Network for F4(p) Placement of conductors for approximation of constant by RC functions Approximation of a constant by F(S) \/III Page 113 113 115 116 117 118 119 119 119 122 122 120 121 122 123 123 123 125 126 126 126 127 128 129 Figure Page 5 6.22a Network for ii & 131 n s+x 4 S xdx 6.22b Network for EA ST; 131 5 -x 6.22c Network for 3; e dx 131 n s + x 5 6.22d Network for 3; 919511 132 n s +-x 5 2 6.23 Network for &-£ x dx 132 n s+x 6.24 Approximation of (1 +-w4) 134 6.25 Approximation of (1 + w2) 135 6.26 Reaponse outside the range of approximation of (1 4-w2) 136 6.27 Placement of conductors for approximation of functions varying linearly as w 137 6.28 Approximation of functions varying linearly with w 138 6.29s Network for ?1(s) 139 6.2% Network for F2(s) 139 6.29c Network for F3(s) 139 6.30 Approximation of w-a 141 10 -% 6.31 Ladder network for F(s) = l;££.£ x dx 144 n s +~x 10 -% _ 1.22 x dx 6.32 Foster type network for F(s) — " £ 8 +'x 144 6.33 Approximation of wa 142 10 -% 6.34 Ladder network for F(s) = lLZZ- §§__fl§, 144 n S +'x 10 -% 1.22 sx dx 0 = — —— 1 6 35 Foster type network for F(s) n g s +'x 44 6.36 Variation of resistance and capacitance along the line 145 1X LIST OF APPENDICES Appendix Page 1 Proof of Lemma (1.1) 150 2 State description for functions having conjugate arcs of discontinuity 152 o'u.--o| ., _;. . I.. .- A-o u a u- ‘ v . .«i ,~ ‘_ ~-. ,-. ‘V'-L CHAPTER I INTRODUCTION Since the advent of new integrated circuit technology, distributed parameter networks have occupied an important place in network theory. Distributed networks are generally characterized by partial differential equations. In general, the solution of partial differential equations is inherently more difficult than the solution of ordinary differential equations, which characterizenetworks with lumped parameters. This is one of the reasons a compact com- prehensive theory is not available for distributed systems as it is for lumped linear time invariant Systems. 1.1 Literature Survey. In classical network theory, realizability of uniformly distributed RC lines (URC's) have been achieved by certain trans- formations of the frequency variable to a form suitable for lumped synthesis procedures. Wyndrum [1] has used positive real trans-- formations to obtain realizability conditions for driving point synthesis of URC networks with identical RC products. O‘Shea's [2] transformations are more general in that they yield realiza- tions consisting of arbitrary connections of URC networks with constant RC products. Rao, Schaffer and Newcomb [3] have treated the realizability of arbitrary n-Port connections of URC networks .,. not 'a | .a.- "p no. O‘D" .. ”I. u ..o __, . I v ... ‘- ‘ Sun I A I I, I v 4 ..... . . t . V' t, -" o. '4'.- ‘ n '0'... . “v-. -‘Ap. ‘ab 5 "Ai‘ 0. 5“ with rationally related (/rcl products. Heizer [4] showed how a class of immittances in the rational form can be realized using a single tapered distributed network. Networks consisting of both distributed lines and lumped elements are treated in references 5-7. Daryanani and Resh [8] showed that, in the frequency domain, nonrational functions (which characterize distributed parameter networks) can be represented by Cauchy type of integrals. Only RC and RL networks were considered. For such networks all singularities lie on the negative real axis of the complex s-plane. The functions considered are sectionally holomorphic ones, with a discontinuity across a line (or a union of line segments). The Russian mathematician, Muskhelishvili, in his monograph on singular integral equations, discusses sufficient conditions a sectionally holo- morphic function should satisfy in order to have Cauchy type of integral representations. Realizations of Cauchy type of integrals have been achieved by distributed networks with Foster type of topologies. Cauchy type integral representations for distributed net- works, give a very basic and unifying approach to distributed net- work theory and hold a promise for a comprehensive theory for dis- tributed networks. 1.2 Objectives of the Thesis. In this thesis state equations for n-Port distributed para- meter networks are developed. A method for reduced state descrip- tion of lumped linear time invariant systems is introduced and applied to a class of distributed parameter systems. State des- cription for one port and two port distributed parameter systems have resulted in Cauchy type of integral representations for driving point immittances. A broad class of network functions, having congugate arcs of discontinuities in the left half plane, are represented by Cauchy type of integrals. Necessary and suf- ficient conditions for realizable Cauchy type of integrals are given. Conditions for integral representation of two ports are obtained. Synthesis of Canchy type of integrals by networks having Cauer and Foster topologies is being suggested. "Cauchy integral matrices" are realized by symmetrical lattice networks and Cauer type of networks. General properties of network functions re- presented by Cauchy type of integrals are discussed and some P.R. transformations suggested. Cauchy type of integrals are introduced in the theory of approximation of network functions using integral representations, Network functions approximating ideal filter characteristics are discussed and realized by RC and RLC ladder networks. A general procedure for approximation of functions is suggested. Summary of Chapters. The second chapter deals with state equations for N-Port distributed parameter networks and their reduced form. Necessary and sufficient conditions for realizable Cauchy Integral repre- sentations for one ports and two ports are discussed in the third chapter. In Chapter four Foster and Cauer type networks from Cauchy integral representations for one port, and lattice and Cauer i ”.“v ' -. P. '5‘ '4..'». . .-“" l: “ "1;" v5 . are C" _...-~ a n“ 1‘ P;' .nv “ "F. u E "-v‘ E 8..“ type networks from Cauchy integral.matrices are obtained. General properties of network functions having Cauchy Integral representa- tions are discussed in Chapter five. In Chapter six, Cauchy integral representations are introduced in the theory of approxima- tion of network functions. Conclusion and certain extensions of this work are given in Chapter seven. 'Irl I- " .F ' -‘A , , ._ . ,. n... . Q .. g \ I. ii ’ v u' v a.” v.. v on. 1? _'t. CHAPTER II DERIVATION OF CANONICAL STATE EQUATIONS FOR n-PORT DISTRIBUTED PARAMETER NETWORKS 2.1 38318 for State Representation. An n-Port can be defined as an operator that maps a certain Space of distributions into itself. Some basic properties possessed by an n-Port having an immitance matrix are single valuedness, linearity, time invariance and continuity. According to a theorem of Schwartz [9] these four properties can be replaced by the entirely equivalent condition asserting that the n-port has a convolution representation. Definition 2.1. The convolution of an n X n matrix distribution f with an n X 1 vector distribution g, which is denoted by f * g, is the n X 1 vector distribution obtained by convolving the elements of f with those of g according to ordinary rule for multiplication for matrices. Definition 2.2. An n—Port is said to have a convolution re- presentation if the responding vector V is the convolution of an n X n matrix 2, whose elements are distributions, with the n X 1 driving vector U; and if D(n) [Domain of the operator] contains all g for which Z * Q exists. In symbols l< II N X- 1C: 1‘ ‘.,eq‘1' ‘ D- ‘. . .‘...a~ .- u Iv- " "u .n - ",.:p-~v! \. ':.1.‘..1-H " ; 0a! “.3" , g». a. nu ‘4 Ni .‘1 p r ‘U "V . u. sn- 1.. 1‘ A as i 1 ‘l '.l L..: t , k ‘. \ ‘1. "-..::] 'v ‘u ‘v J‘ . FL- . .4» I . we ‘5 2.2 Derivation of State Equations. In this section we will give the main results in the derivation of state equations for n-Port. All relevant conditions [10] for getting state equations are given in the appendix 1. Lemma 2.1. 1 (z * g): mj_12(S)-[exp(st) * g(t)]ds :£ Proof is given in appendix 1. Theorem 2.1. g1 and g2 in g(_m,tO] are equivalent if t t o o LGXMS (to-fl) 'Ql('r) 'd’T = {meXMS (to-7)) ~g2(¢) 'd'r for all complex 8. t O S 332- j‘ eXP(5(tO'T))'gl(T)dT = - e XUI(tO-x)dx w 3"30 where to-T = x . Let gl(to-x) = §1(x) then SX 21(X)€ d" = £1(-8) 0918 cc Similarly [Ezeki’fi = §2(-s) where g2(x) = 92(tO-x) 0 Now £1(-s) = £2(-s) implies £1(x) = £2(x) V x > O or 91(to-X) = 92(to-x) V'x > 0 or gl(t) = g2(t) V t S to Hence the proof of the theorem. < I .-._ .A'_ a, 1P5 - h“ I ,.~ We can now define H; (go) for each 90 in g(-m,to) to consist of those vector functions U' in U such that NO ~(‘m3t01 t t o o ' = - j expdw j“ exp>gO(T)dT ca) -oo Next we want to find a set Z(to) for the partition H; just defined. We make the obvious choice: the label for H; (U0) will be the pattern of values t O [xs(to) = I exP(S(tO‘T))'QO(T)dT] which characterizes its members. Z(to) itself is then the collec- tion of all such patterns as go ranges over g(_m,to]. Finally we define the input, output state relations by taking (to,w) portion of the reaponse Puck») (00,11) (t) = 211. £_IZ(S)xS(t)ds The development of the state equations for n-Pbrt distributed parameters is completed by noting that xs(t) satisfies the state differential equation given below. %;§(S.t) = S §(S.t) +g(t) 2.1 1 z - , d 2.2 310:) = 21—1-14 (5) 35(5 t) S In particular for two port case the type of the equations which we will be using are x1(s,t) s o x1(s,t) l 0 11(t) a + 2.3 x2(s,t) s x2(s,t) O 1 12(t) d dt V102) 211(3) 212(3) X1(S,t) v2(t> 221(3) 222(8) x2(s.t> if where Il(t) and 12(t) are the input currents. V1(t) and V2(t) are the output voltages. 2.3 EQuivalent Reduced State Descriptions. In order to have minimum number of elements in the realization of a system, it is important to have the equivalent reduced state description of the system. In this section we will develop a general procedure for constructing reduced state description of linear lumped time invariant systems, and in the next section we will apply it to a class of distributed parameter systems. Before we can find the method we have to state some theorems and defini- tions. Theorem 2.2 [11]. A continuous time system described by Ax +-Bu 2.5 x. II where x n vector u = r vector A = n X n matrix B = n X r matrix is completely state controllable a the composite n X nr matrix P where n-l , P = [B: AB: ... (A) B] 18 of rank n. 2.6 Theorem 2.3 [11]. The system described by ""‘.P°,C ,-p.s.- I;- ' , , b. .l'l-r u-n., ...‘o. 1 .’l . uti'y r‘- .v .‘. u ‘g‘ 1‘ I‘ I .o '. n. : . 'I-"IA a..‘ . no. A. ' C "V: 1. a7. 1 f . J (at: v 1" 5'» “" UK; I l \ {a 25).,- I a“ In {In ‘} CM v“ , ’1 J N Na" .. ..lalr I‘: x = Ax y = Cx where x = n vector m vector ~< II A = n X n matrix 0 ll m X n matrix is completely observable a the composite n X mn matrix P where * * * * - * P = [C : A C : ... (A )n 1C ] is of rank n 2.7 Definition 2.3. A realization of Z is irreducible if its state Space has minimal dimension nmin(z)' Theorem 2.4 [12]. A linear dynamical system is an irreducible realization of an impulse reSponse matrix if and only if the system is completely controllable and completely observable. Definition 2.4. State Equivalence. State ai is equivalent to a if and only if, for all inputs U, the reSponse segment of j (7 (abstract oriented object) starting in state ai is identical with the reSponse segment of (7 starting in state aj. Inswmob {a1 “-' 05-} r {Raw = Kojm v u Now if a system is (a) Observable but not controllable (b) Controllable but not observable (c) Neither controllable nor observable, then in all the three cases, we can find a system which is state equivalent to the given system and in which the dimension of the 10 state is minimal. We prOpose a method for obtaining such reduced state descriptions in the following theorems. Theorem 2.5. Let the state equations be given in the form i = Ax +-Bu y=Cx where x = n vector u = scalar A = n X n matrix B = n X 1 matrix y = m vector C=an and assume that the system is not controllable, but observable. Then the equivalent reduced state description given by AA + Bu 2.8 1 Cl Y . is obtained by the following procedure. 1) 11) iii) Compute the controllability matrix (n X n) p = [B: AB: AZB: ...An-IB] If the rank of P is me then, by row operations on P, 0 one can produce a matrix of the form [3,] where rank of P ‘= # rows of P = nc- C811 ‘i’ the reduced controllability matr'ix. NOW define B to be first column of the reduced Po matrix and C:'v I, ..- A an; I. ,3 , A. ..k. I ,. p- r .1 v Q P ’..§ me... 1. Y ., ..‘c .‘ ‘ cit . ‘ n 11 > > n ... (A) °fi][fi: Ki: ... (A) E] 2.10 W) where n-l [B: AB: A213: (A) C B] is the nonsingular matrix formed from the first nC columns of P. iv) C is found from ) A CB = CB CAB=CAB ) A: n -1 n -1 C(A) c 1‘3 =C(A) C B ['1 or E = [CB: CAB: ... CA C B][B: AB: ... (A) C B] 2.11 Proof. (i) Consider the matrix 2C = {8: A8: A28: Ari-1B] if this matrix has rank nC then it is easy to see that the matrix formed from the first nC columns of P is of rank nc. Hence the system X = Ax + Bu is controllable. (ii) Proof of the state equivalence of two systems for i = Ax + Bu y = Cx t x(t) = eAt{x(o) +j‘ e-ATBu(T)dT} o t >'(t) = CeAt[x(o) +-f e’ATBu(T)dT} 2.12 O 12 Zero state output is given by 1: At -A fie) = Ce ye Tamed: o o t A y1(t) = c f e TBU(t--r)d«r. 0 We can express m-l . eA'r = E (311('r)A1 i=0 :11: degree of minimal polynomial of A. o n-l y i y (t)=C 2 2y. AB. 1 i=0 j=1 13 J where t Yij = g ai(T)Uj(t-T)dT = scalar O _ . . n-l y1(t) - C[B. AB. A BJEXij] 2.13 By performing row operations on [B: AB: A B] we have P matrix as 2 .. [13: LAB: LAB: 1A“ 113] 2.14 it is Of the form 9 9 C = 15. P} where K is nc X nC nonsingular matrix or we can write (2.14) as LLB: LAL- 13: (LAL'1)213: (LAL'1)“‘113] 2.15 . o;’v I... ..,.. “IA"- \ Lu” 3..-.- n n ”s a i. "h“ “4“ 5 ‘F 1 1 .0 u - v H a h L- \ ..-_ “ . 5.. o o | a. J 13 where Now we have \A - m = \P - uHA - m which Shows that A satisfies the characteristic polynomial of A and they have nC common eigenvalues. If we expand eAT and A h e T only in terms of An and (A) for n = 0,2,...,(nC-l) then Yij will be same in eqn. (2.13). From equation (2.13) we have T yi(t) = C[LB: LAB: LA B] xi]. n y:(t) = T 1C[B: AB: ... (A) C B] xij y‘im Eifiz A13: (A) C B] 31° 2.16 Comparing (2.13) and (2.16) we have > A A 6(2) C B = CA C B Hence the proof of the theorem. Extension of the above theorem for cases where we have r vector and U B = n X r matrix . P' ‘1‘ r .x i g... o I". rt» o I . I. u . lea-1... fl ...... 3“ n .M . O l. ..' 3.;- \ '. ‘ 1 ' ..~. ._ 3:?“ Lt 14 is quite simple. After row reduction of the controllability matrix. First r columns of the row reduced matrix comprise of B. While for A: we have to pick nC independent columns out of nr columns (fl controllability matrix and A can be found AA A 20 A 'nCA A A nC-IA -1 [ABI: ()Bl: (A) Bl][B1: A81: (A) B] 2.17 where ‘B1 is the first column of B and n -1 [81:A81: ... (A) C B] are the independent columns of row reduced controllability matrix P. We can pick any of the columns of B and find A or in general.we can write 3» — 25." -(A)zfi - (mncfi - [ B . Y . ... Y B : (A)B : ... (A) Y] 2 ][ Y1 Y2 Y C C 11 Next we can find C by Eq. (2.12). Exagple 2.1. Consider the system :21 0 0 -6 x1] 4 £1 :22 = 1 0 -11 x2 + 5 U i3 0 1 -6 x34 1 y = [l 2 3] x1 X2 x3 whiCh is; not controllable but is observable. The controllability 15 4 -6 6 0 J 0 0 -7 5 _. 4§-6 6 1 -1 -1 1 3-1 -1 w: u dbl-D. L___J “E: ii: :3 {‘1 :2] The observability matrix is 1 l 3 2 2 -46 and the system is observable. 3 -46 231 In order to calculate C we use ) ) CB ll 9 AAA CAB CAB E = [-.6 19.4] Heruze we have the equivalent reduced state description i O -6 x 4 A2 1 -5 )2 l y = [-.6 19.4] [A1] "2 (3n comparison, one sees that the transfer functions of 21 and 22 are identical. E523212_j§;§- Consider multi-input-output system 'n e-(~ \ rut) .M x1 2 0 x1 1 2 U1 :1: = + x2 0 2 x2 2 4 U2 y1 1 3 x1 y2 3 2 X2 which is observable, but not controllable. The controllability matrix is a) ll r"'l H N I.._J and from CB = CB Hence we have the reduced system given by 22: i = 2) + [1 2] [U1] U2 3'1 7 yz 7 Th¢°rem 2.6. Let the state equations be given as 17 i = Ax +~BU y = Cx where U = r vector y = scalar C = 1 X n matrix A = n X n matrix B = n X r matrix- 'The system is controllable but not observable with the reduced state description Ai-rfiu i C). 3' is given by the following procedure i) Find the observability matrix Q [C*:A*C*: ... (A*)"Clc*] ii) Find the reduced observability matrix Q a x * iii) Then first column of 9 comprises of (C) and 11 (3)4. = [(A)*C*: (A*)2C*: ... ((A )*) CC*][C*: A*c*: ... (A ) C C*]-1 2.19 A 1V) B is found from CE = CB CAB = CAB n '1. n -1 . ~ ..w' aI-o 3’ «an ‘ .. .44...- 1. I ‘ ' o I. . on J 1. ... a . MI. 2'; -'u~ .' .‘ F " DOAV . 18 Now extension of the above theorem for multi-output system is simple. Lat C = m X n matrix y m vector then after row reduction of 9 which is (n X mn) matrix first * tn columns of Q comprise of C and -l . ..*..* 1* 2.9: .4: n ,9: .* * .* .3: “ -1 (A)*=[AC :(A)C :... (A)CC ][C :(A)C :. (A)C c ] 2.21 Y1 Y2 Yn Y1 Y2 Yn c c where n -l 0* A*A* A* 5* [C :A C : ... (A ) C C ] Y1 Y2 V“ C are the independent columns of reduced 0. B can be found from equation (2.12). Example 2.3. Consider the system i 3 2 x 1 1 U The system is controllable but not observable and the observabi lity matr ix is 1 -1 1 -1 4 -4 0 0 A* 15* C = 1 = -l B=[5 5] 19 Hence the reduced state description is A = -)\ + [5 5] U1 22: U 2 y = [1]).. Theorem 2.7. Let the state equations be given as = AX +-BU )4: y = Cx where x = n vector U = r vector A = n X n matrix B = n X r matrix y = m vector C = m X n matrix and the system is neither controllable nor observable.Then the equi‘valent reduced system is given by the following procedure 1) 2) 3) 4) 5) Find the rank of controllability matrix, call it nc1 Find the rank of observability matrix, call it nc2 If nc1 < nc2 then apply theorem 2.5 to get reduced state description If nc < nc 2 1 then apply theorem 2.6 to get reduced state description If nc2 = nc then either of the theorems 2.5 or 2.6 can be 1 used. EEQQE- For nc1 < nc2 if we use theorem (2.5) we get the minimum POSSIble dimension of A matrix from the reduced controllability 20 matrix and C is being obtained from CAVE = CAvB for minimum possible v. Hence we get the reduced state description in one stroke. Now if we use theorem (2.6) for this case, then once we get C', B', A' we have to check the system y: = C'Z agaixg for controllability this time and reduce it again. Example 2.4. Consider the system x1 1 0 0 x1 5 {:1 x2 = 0 1 0 x2 + 3 U x3 1 -1 -1 x3 1 = 1 l 1 y I 3 X1 x2 x3 whitflm is neither controllable nor observable. 1. Controllability matrix C = 5 5 5 3 3 3 is of rank 1 1 l l 2. Observability matrix Q = 2 1 l 0 1 is of rank 2 ...-l I H H . . IA. 4;. 21 According to theorem 2.7 we will apply theorem 2.5 to get reduced state description for the system. From reduced controllability matrix A = 1 B = 1 From CB = “A C=9 Hence reduced state description is given by A + U )1. ll 2 : 2 9A ‘4 II 12.5 Equivalent Reduced State Description for a Class of Two Port Distributed Systems. The concept of reduced state descriptions for distributed Parameter systems is quite vague. In this section we will Suggest a few steps for the reduction of such state descriptions, but it is ruat clear that the final form of the state equations is irreducible. Statue equations for two port distributed system are x1(s,t) s 0 x1(s,t) 1 0 U1(t) d_ dt x2(s,t) o s x2(S,t) o 1 U2(t) =J 211(8) 212s) x1 222 Hence we see that when Z(s) has a normal rank = l the state description can be further reduced. The case of normal rank = 0 is trivial. Next we will consider further reduction of the state des- cription given in equation (2.3). This can be achieved by noticing that 23 t x1(S,t) = [mCS(t-T)U1(T)dT t x2(s,t) = f es(t-T) '(D are entire functions of s U2(T)dT at each instant of t. Thus both sides of equation (2.3) can be differentiated w.r.t. s arbitrarily many times. Denoting differentiation of x1(s,t) we a]a. FT get Px1(sat) T x{1)(s.t> U1(t) 1120:) 0 0 s 0 l s (n) n times by x1 (s,t) b IF x1(s,t) x§1)(s.t) x§C) X2(S:t) x5“)(s.tvj 2.24 24 As the output is given by y1(t> 211(5) 212(5) x1 £1 221m 222m x2(s,t> Resolving 211(8), 222(3), Z12(s), 221(3) into partial fractions we have nK _ 1 n2 n 211(8) — z [8.5 + 2 + 0.0. K J (8-3) 12 N 0:2 Ci: CnKn 212=21(S>" EC _ + K] (8-8) (8-8) 2 n C22 CnK 222(S)=n§_1:[—E—: +"° n x] +(S'Sn )2 (s-s ) 11 n C1 CnKn I] y1(t)= g n21:[_r.11_§_ x,1(s t) +Erl2—2-x1(s, t) + —-—-K—-x1(s,t) + (9‘ S ) n n (s-sn) 12 12 C12 C N C X (S t) nK £nEIsL n1 x 2,(S t)+— an 22 +....—'n——IE-X2(s,t)] (3 S “)2 (s-s ) II n 12 N as: y2(t)=J‘ 2[(s-s) X,1(S t) +J-_—§+...._—Lf-x1(8,t)] + ()21 n=1 n (s- s n) n (S-S) C2 C2 GEE—‘— n J. 2'. [Eff-Lx2(s,t) +_£Z__2_x2(s,t) +.... x2(s Q] “=1 8n (s-s ) n (s-sn)Kn Now using Cauchy Integral formula we have 25 1 if (1) CnKn (K-l) y1(t)=n21n[:c 1.x1(sn ,t) + _n_2_ x1(sn,t) + le (s l,t) n 12 12 C C12 0 nK (K -1) (S t) + _I_1_2 X(1) n n Cnl X2 a! 0—1. x2 (sn,t) + ....———(Kn_2)! x (sn,t)J 2.25 12 u 12 c12 (1) CnKn (Kn'1> y2(t) = 2 [Cnl x1(sn,t) +_ 0! (Sn ’ H) le (Sn’t) n=1 [1' 22 c2 2 Cu x(1) nKn (Kn-1) c:r11 X2(sn,t) +— 0! x2 (sn,t) + ___(Kn-2)! x2 (snug) 2.26 Now we can discard the equations and variables which are not needed in eqns. (2.25) and (2.26), and we get the form of the equ at ions as Fx1(t) 1 x2(t> x§ xi x<2>(t) C3-C1: n 7%}(t) .1 1 x2 x<2) 5x Ho... (t) (2) 1 (t) (t) + + 26 12.1 9 92 123 0 U 13“ g 1 2.27 U Q 121 2 : 132 g : 0 b. L'" n J 11 11 11 12 12 12 r 1 7 yl El .2 0000 oooooo C-rl 531 C-2 0 ..... 0 CE X—l(t) = C2]. C21 C21; C22 C22 C22 1 t y2 .1 2 0000000000 3 {-1- -2- 0090000. B )2.) i xn(t) 2.28 i“) (2) L-Xz (t) where .23 ="1-1 for n = 1,2,... 0 L01 A good look at the equations given in (2.27) and (2.28) shows that we have one complete set of natural frequencies for 27 each port or in other words there seems to be a duplication of the 8 ‘plane 0 To anyone familiar with state description of lumped systmes, this has to seem odd; state vectors do not partition up into subvectors each aligned with a separate port! Or do they? The answer is partly yes, partly no. Let us consider an example of a two port system whose inmfittance matrix is given by Curl ('7’ 2(3) 1 1 rx1 xfim 2 x1 Lx§1 y1 y2 (.23-3 25-3 (s-l)(s-2) (s-1)(s-2) 2.29 0 L (s-z) J State description for this system is r0 1 o 0 0T fixing r.1 o1 O 2 0 O x:(t) +- 1 O U1(t) 2.30 2 O O 1 O x1(t) 0 1 U2(t) 2 Lo 0 0 2J Lx2(t)j L0 lj 1 1 1 1 1 1 Fxlm 0 0 0 1 x:(t) 2.31 2 x1(t) 2 LX2(t)J It is interesting to observe that each port has one complete set of frequencies. two subvectors, each aligned tum observable and using the The state vector is partitioned into with a port. Note that the system is method developed in the previous Section we get the reduced state description as . u A I ' a — | ... ,p U 'U 1 .. .‘.A, i Q d i .. V a \- - I: | . U V . 0,...5 T', - - n‘m. . 1‘. C il“.l‘?1v ...... A h 28 x1 1 O 0 x1 1 1 ['UI] d 1 — = i— 2.32 dt x2 0 2 0 x2 + l 0 U2 x3 0 O 2 x3 0 1 r y1 1 1 1 X1 = x2 2.33 y2 0 O 1 y LX3 In equation (2.32) x is being controlled by both ports 1 while x2 and x3 are being controlled by port 1 and port 2 reSpectively. System represented by equations (2.32) and (2.33) is equivalent to 11 1 o (1 rip] 1 1 U1 511-ng — 1 2 0 12 + 2 1 U2 2.34 x3 -1 0 2 1.13.1 2 2 y1 [O O I] )‘1 [y2]= 0 -1 1 12 2.35 *3 in which both the ports control all the three states. The fact that each port controls a state vector, in our description of Eqn. (2.3) is further justified by, looking into the form of state equations for multi-inputdmulti-output systems [13], which are given by >2 = 3x0) +13 U(t) ale. n Where all" (Ill .— t ldu =72» ' ~ 5 um. h u sste .,' {.A ....ta' L NASHS and m> >> I I I L I I I I I I I I I I I I I ----- --'~-- ---.-- I F. m Am-l ;' f' : : 3 : 9.: E ' I : I boo-:.--..J: : I I I 9: Q; 9 g . L--+---}-- : I I I : I . I I I g I , I 1'01"". : I o: o: 0 :. ~| N I! I '1: i I I * I I -------fl--------- ----------- 9 29 p) O ...: --ucno-boc--o--.oooonooaoo'ucuooonco ---u- L --c> c> J I 'H I 10 -.-.\ 10 >> L IC‘. a 0 0 O 0 Xkl XEJ From equation (2.36) it is clear that any multi-input SyStem can be transformed into a collection of coupled single input systems 0 Another obvious result is: state equations for lumped linear time invariant systems are Special form of the state equa- tions of the distributed systems given by Equation (2.3). CHAPTER III CAUCHY TYPE INTEGRALS 3.1 Motivation for Cauchy Iype Integrals. In the last chapter we developed the State equations for distributed networks. In this section we will show that under some conditions, it is natural to represent the driving point impedance of such networks by Cauchy type integrals. Until now only networks whose line of singularity was composed of arcs on the negative half of the real axis or on the imaginary axis have been considered [8]. Network functions with conjugate arcs of discontinuity in the left half plane will be considered, so that we can cover a very broad class of networks. State equations for one port distributed parameter net- works are given by g;- I(S,t) = sI + i(t) V(t) = ETITj-afef(s)q,(s,t)ds where i(t) input current V(t) = output voltage. Now if F(s), the driving point impedance of the network, has a finite number of conjugate arcs of discontinuity L1.L2, 3,...,Ln in the left half of the plane as shown in Fig. (3-1). Then the modified state equations, derived in Appendix 2, after evaluating the Laplace inversion integral, are 30 31 A 81 A 32 a; I(Sv(p)at) = Sv(p)-W(Sv(p),t) + i(t) 3.1 and f}; I(S"v_(p),t) = svm W(Sv(9),t) + i(t) 3.2 V(t) = nvm (9)) Ms (9») [II/(8 (p) :t)] d9 v=1 3.3 Ms (9) t) where sv(p) is the equation of the vth arc .1 + . f (S (9)) ‘ f (S (9)) = 2111 MS (9)) v v v From (3.1) and (3.2) _. _ -1 ¢(Sv(p),8) — 1[s sv(p>1 and I(Sv(p).8) = i(s>[s - sv(p>]'1 Substituting in (3.3) we have ___ N () MS ()) F(S) E11 HL; : )98) +—-———-—" D ] dp 3.4 p sv(p)-s As a limiting case, equation 3.4 gives a general representa- tion for driving point impedances of distributed parameter networks having conjugate arcs of discontinuity. From equation (3.4) we Can get representations for RC networks as mfg) F(S) =15”: dx 3.5 o For LC networks the conjugate line of discontinuity is contained in the Jw-axis so from equation (3.4) 33 m.) =fcm+m s-jx s+jx m sf1(x) = i -§——§—- where f1(x) 18 real. The general form of RLC networks is given by F(S) = 2': n (8-Sv(p)¢(sv(p)) + @(Sv(p))(S-Sv(p)) dp WA [8+9v(p3+jwv(p))[8fiv(p)~j¢v(p)] V n [(S-fVUD) +§v(9))1d9 3.7 F(S) = 3 . 2 v S +28gv(p)+§v(p)J v=l where fv(p), fv(p), gv(p), §V(p) are real functions of p. The above developments show that it is natural to represent driving point immitances by Cauchy type of integrals. We will next identify the necessary and sufficient conditions a function should satisfy in order to have Cauchy type integral representa- tions. Before we can develop this comprehensive theory, we have to give some definitions. nginition 3.1 Arc: A continuous arc is defined as a set of points (x,y) Such that x = M9) y = M9) and s(p) = @(p) +3 V(D) Where Q(p) and ¢(p) are continuous functions of a real variable 9- If no two distinct values of p correSpond to same point (x,y), the arc is called Jordan arc. QEEInition 3.2: Sectionally Holomorphic Functions: A function F(S) is sectionally holomorphic with arcs Lv of discontinuities, 34 if F(s) is holomorphic in the plane not including L\) and if F(s) is continuous (in the sense defined below) on Lv from left and right with possible exception of the ends near which the follow- ing inequality holds Const. ‘0! USO/<1. \F(s)| < S‘C F(s) is said to be continuous at tv on LV from the left if F(S) tends to a definite limit F+(tv) as s approaches tv along any path remaining on the left of Lv' A similar defini- tion holds for F-(tv) and continuity from the right. Definition 3.3: H61der Condition [14]: A function f(t) is said to satisfy the H31der condition on an are if for any two points t1 and t2 on the arc \U 3.8 |f(t2) - f(t1){ s A‘t2 - t1 where A and U are positive constants. A is called Holder's constant and U is called Holder's index. nginition 3.4: Class H [14]: f(t) belongs to class H on L if it satisfies the H(U) condition on each of the closed arcs Lj of L for some U > O. nginition 3.6: Class H* [14]: If f(t) satisfies the H(U) condition on every closed part of L not containing the ends, and if near any end c it is of the form * f(t)=-f—‘9— OSQ<1 3.9 °’ h * W ere f (t) belongs to class H on L. 35 The following theorem gives the necessary and sufficient conditions a function must satisfy in order to have Cauchy integral representation. Theorem 3.1: If a function F(s) has the following properties b : F(s) a O as s a m b : F(s) is sectionally holomorphic with are of dis- continuity L b : F(S) satisfies the boundary condition F'm - F+(t) = 2nj f(-t> where f(-t) is a function of point t on are L b4: f(-t) belongs to class H* on L. Then Cauchy integral with density function f(-t) is the unique representation for F(s). Conversely if F(s) has a Cauchy integral representation with -f(-t) satisfying H con- dition then it satisfies the first three conditions of the theorem t-S F(S) ={M * 3.10 Proof of the theorem can be found in Reference (8). Now if, in the above theorem, F(s) were sectionally holo- morphic with conjugate arcs of discontinuity L1 and Li, then F(S) will have a representation of the form F(S) = { —§-l' 't +£ 4” 't 3.11 ' f t-s - 1 1 t-s 36 3.2 Cauchy Integral Representations for Two Port Network Functions Two port RC and RL networks have singularities on the negative half of the real axis.state representations for such networks are a special form of Eq. (2.3), viz d x1(-o,t) -o 0 x1(‘o,t) 1 0 U1(t) a; = + 3.12 x2(-o.t> o -o x2(-c.t) o 1 U2(t) = 3.13 y2 f21(o) f22(0) X2(-U)t) L From equations (3.12) and (3.13) we get the impedence matrix of such networks as r- ‘W f11(o) f12(a) F(S) = '1;E;" “313'- do f21(o) f22(o) L L 5+0 8+0 f (o) f (a) 11 12 do 3.14 01' F(S) = f21(°) f22(°) 8+6 L We will refer to integral representations of the form (3.14) as "Cauchy integral matrix representations". Next we have to find the necessary and sufficient conditions for a given (2 x 2) matrix to have a Cauchy integral matrix representation. Again we require Some definitions before giving the conditions. 3Definition 3.7: A matrix F(t) is said to be sectionally holo- uIOrphic if all its elements fij(t) are sectionally holomorphic furundons. Thus a matrix F(t) is sectionally holomorphic with 37 line of discontinuity L if F(t) is holomorphic in the open plane excluding L, and if F(t) is continuous on L from the left and right, with the possible exception of the ends near which the following inequality holds: \F(s)\ s—A-- 3.15 (t-c)“ where ‘F(s)‘ denotes matrix F(s) with modulus of each element and A is a constant matrix. Definition 3.8 Holder's Condition: A matrix F(t) is said to satisfy Holder's condition if each element (fij) of it satisfies the H-condition. Definition 3.9 Class H*: If F(t) satisfies H(U) condition on every closed part of L not containing the ends and if near any end c it is of the form * 1 F(t)=F(t)~—-‘—— 05a<1 (t-c)“ where F*(t) is a matrix which satisfies H condition then F(t) is said to belong to H* on L. Cauchy Integral Matrix: Let F(t) be a matrix defined and bounded on L with the possible exception of points cj where ‘f(t)‘SA'—L— a<1 (t-c)” then 2nj t-s F(s) =—1-{Lf—911dc 3.17 where the matrix 1-1— [f(t)] is called the density function matrix. an 38 Theorem 3.2: If a matrix F(s) has the following prOperties B1: F(S) a 9 (where 9 is a null matrix) as s a m 32: F(s) is sectionally holomorphic with line of dis- continuity L B3: F(S) satisfies the boundary conditions - + . F (t) - F (t) = 2n) f(-t) where F(-t) is a matrix defined on L then F(S) is uniquely determined in the entire complex plane except 8 E L. Theorem 3.3: If a matrix F(s) satisfies B1, B2 and B3 and also B4: The density function matrix belongsto class H*. Then the Cauchy integral matrix with density function matrix -[f(-t)] is the unique representation for F(s). i.e. F(s) ={Mdt 3.18 t-S Conversely if F(s) has a Cauchy integral matrix repre- sentation with density function matrix satisfying H condition, then it satisfies 31’ B2 and B3. 3.3 Conditions Resulting in RC and RL Realizable Cauchy Type of Integrals In this section we give conditions, which will guarantee that the Cauchy type of integrals can be realized by RC, RL and combination of RC and RL networks. Ihgorem 3.4: If a function F(s) satisfies conditions b1, b2, b3 and b4 then F(s) = { Eéfifigi- can be realized by RC networks 39 I F(s) f;(§T—‘< O . if and only if Proof: It is clear from the previous theorem that the first four conditions result in Cauchy type of integrals with f(x) real. Now the sufficiency of the theorem is established by showing that ImF(s) ——< 0 results in f(x) > 0 Vx I (S) m _ _l_ . _ _ . _ _ . f(x) - Zni 11m.[F( x 1y) F( x + iy)] yao as F(-x + iy) = F(-x - iy) f(x) = —l.— 11m [F<-x - iy) - F(-x - iy)] 3.19 2111 y-+O From the condition ImF(a) < 0 Im(8) we have for y > O ImF(S) < O 3.20 and y < 0 ImF(s) > 0 3.21 from eqn. (3.19) and using eqns. (3.20) and (3.21) we have 1 . - Zni [f1(x) +>1 f2(x)] - [f1(x) - i f2(x)] where f2(x), the imaginary part of the limit of the function, is always positive £200 f(x) = 2“ >0 Vx d Hence F(s) = {‘géfii—E' is RC realizable. In order to see the necessary part of the theorem*we separate £§§£2§- into real and imaginary parts: 4O f(xzdx a 4; f(x)(u+x)dx + i {I -vf(x)dx (u+iv+x) (u+x)2+v (u+x)2+v2 I mF(S)= {M where f(x) > 0 (ubc) 2+v2 I mF(s) ' ._i£L____. 3.22 111:5;j ={. (11+x)2-+v2 As the right hand side of Eqn. (3.22) is always positive, it follows that I F(s) m 1m Theorem 3.5: If a function F(s) satisfies conditions b2 and b3 and in addition F1(s) = F(s)/s satisfies conditions b1, b2, b3 and b4 then F(s) has 8 RL realizable integral representation if I F(s) d 'L— 0 an only if Im(S) > 0 Before we can prove the theorem we have to state the follow- ing Lemma. Lemma 3.1: If the singularities of F(s) are all on the negative real axis and if F-(x) - F+(x) = 2n] f(x) then for F1(S) = Eéil F-(x) - F+(x) = 2nj Eiél' 3.23 1 2 -x I mF(S) Proof of Theorem: Condition (6)1m > 0 guarantees f(-x) < 0 ‘Vx in an f(-x) F-(x) - F+(x) thus f1(-x) in - + . . . 2nj f1(x) = F1(x) - F1(x) = is always p051t1ve f1(-x) 3 m 'X 41 F1(s) satisfies Cauchy integral conditions hence f1(x) s+x dx F s F1(s) = -SL-)- =4: f1(X) or F(S)={s s-l-x dx which is RL realizable. In order to prove the necessary condition sf (x) — 1 U I O we separate F(S) - { -§;;——-dx into real and imaginary parts (Uz‘l'vz-ZHJx)f1(X)dX Re F(S)= i: (U+x)2 + v2 x f 1(x)vdx Im F(S)= { (U+x)2 +vZ or m = x f1 dx > O VX 1‘“ (S) (U+x)2 + v hence the theorem is proved. In the next theorem we will give conditions under which a function can be realized by a sum of RC and RL functions. Lemma 3.2: If a density function f(x) satisfies Holder's con- dition on [a,b] with Holder's index U = 1, then it can be expressed as f(x) = f1(x> - f2 Where f1(x) and f2(x) are bounded, nonnegative, monotonic increasing functions. meg: f(x) satisfies Hb‘lder's condition with U = 1 i.e. “(1:1) - £(x2)\ SM‘xl xz‘ 42 for ab partitioned into xi intervals n n z ‘f(xi) - f(xi+1)\ SM '13 1x1 ' xi+1‘ - 0 i=0 1: n z taxi) - f(x i=0 i+l)‘ S s 3.24 Thus f(x) is absolutely continuous on [a,b]. According to theorem [15] which states if f(x) is absolutely continuous on [a,b] then it is of bounded variations on [a,b]. Now if f(x) is of bounded variations on [a,b], it can be expressed as the difference between two bounded, nonnegative and monotonic non- decreasing functions. Hence the proof of the lemma. Theorem 3.6: If a function has the following prOperties l) F(s) = F1(s) + F(m), where lim F(s) = F(m) Sam 2) F(s) is sectionally holomorphic with line of dis- continuity L_ = [a,b] 3) F(s) satisfies condition b3. 4) density function f(x) satisfies Holder's condition with U = l f2(X) 5) F(s) ={ x +R1 where f2(s) is as defined in Lemma3.3 OSR10 Vx ii) b(x), c(x), d(x) 2 O 'V x iii) a(x), c(x) 2 b(x) 'V x - We can see that conditions (i) and (ii) are obtained from the requirement that all poles and zeroes be in left half plane. Third condition is obtained from Re(JW) 2 0 44 b(X)[82+d(x)L- szaIX)c(X)dx Re(F(JW)) = EVF(S)‘S=JW = 4: [82 + d(x)]2 - $2[C(X)]2 s=JW w2 (a (x) ' c(x) '13 ()0) + b (x)d (x) :Jx [d(x)-w ] +'W [c(x)] this is 2 0 if a(x)c(x) 2 b(x) . 3.5 Realizability Conditions for Two Pbrt RC and RI.Networks Definition 3.10: An n X n matrix F(s) is called positive real if it satisfies the following conditions 1) F(s) is holomorphic in a > 0 * * 2) F (s) = F(s ) in o > 0 3) FH(s) 2 o in a > o where FH(s) = FT(s*) + F(s) For two port RC networks F(S) = [:f11(x) f12(x)] dx £21“) £22“) 3 +-x L For F(s) to be realizable we should have i) f11(x) and f22(x) >0 Vx on L ii) f12(x) and f21(x) real iii) f11(x) f22(x) 2f12(x) f21(x) Vx on L Conditions i, ii and iii are of the same type as the lumped cases. To compare the similarities between Cauchy integrals and partial fraction expansions we will call f11(x), f12(x), f21(x) and f22(x) as distributed residues. Note 3.1: While trying to find realizability conditions for resistive networks, author struck upon a conjecture, which can be applied to RC networks also. Given a matrix A with all 45 positive entries then a necessary and sufficient condition that it can be realized by resistive networks without transformers is that p(D-1(E+F)) < l where A = Id +§+E and D = Diagonal matrix E is strictly upper triangular matrix F is strictly lower triangular matrix p(M) denotes the Spectral radius of the matrix M Now for two port distributed RC networks - f11(x) f12(x) dx f21(x) f22(x) s +-x F(S) ’- L f22(x) are all positive then f(s) can be realized by RC networks and if f11(x), f12(x) ,f21(x), without transformers if and only if p(D-1(E +-F)) < l.‘V x on L where f11(x) f12(x) f21(x) f22(x> = D(x) +ng) + F(x) while D(x), E(x), E(x) are defined as above. This conjecture may be the solution of an important un- solved problem in network theory i.e. necessary and sufficient conditions for realizability of resistive N-port networks wihout transformers. Theorem 3.7: If a matrix F(s) satisfies conditions 31’ 32, B3, B and if det H%E—SFE>-n > O with diagonal entries negative V s 4 then 46 f11(x) f12(x) dx £21“) £2200 s +'x F(s) =, is realizable by two port RC reciprocal networks. Proof: As in the matrix Im f11(s) Im f12(s) the diagonal Im f21(s) Im f22(s) Im (5) entries are negative, thus from theorem (3.4) f11(x) and f22(x) will be positive. det Hi: :(8 1 > O guarantees f11(x)f22(x) > f12(x)f21(x) ‘V x) thus proving the theorem. CHAPTER IV SYNTHESIS PROCEDURES FOR CAUCHY TYPE OF INTEGRALS 4.1 Ladder Networks In this section we will find methods to realize Cauchy type of integrals by ladder networks. Continued fraction expansion of Cauchy type of integrals forms the basis of ladder networks. Stieltjes[l6] in his classical work used power series for getting continued fractions from Cauchy type of integrals. The following theorems give continued fraction expansion for power series and Cauchy type of integrals. c Theorem 4.1: If 1 +-;l'+'-%'+'-%-+ ... be power series then s 3 this can be expressed equal to continued fraction given by a a a a - 1‘ 2‘ 3‘ 4| = - ¢v+l¢v-l 1+TE-+1-1—+-‘S—+T1—+... where a1 @1 aZv— vav -Q ¢ v+1 v - a = ——————- v = 1,2,3,... 6 = l w - l and 2v+1 ¢v+1§v o 1 c1 c2 .... cV c2 c3 .... cV Qv = c2 c3 .... Cv+1 wv = c3 c4 : CV Cv+1 CZv-l Cv CZv-Z v = 1,2,3, v = ‘2,3,4,... a a2+l +4‘+... Continued fraction given by T—-|+ T—-+ «T——+ -V—— can be converted into a more useful form given by 47 48 where b “ 1‘ = 2 4 2V 4.2 1 a1 2v+l a a a 1 3 . 2v+1 b — 8.18.3 ... azv-l 4 3 2V a2a4 coo azv Theorem 4.2: If ¢(x) is a monotonic increasing function of x for x 2 O and the integral ! xk-1d¢(x) for k = 1,2,3,... exists then the integral Ids+: has correSponding continued L fraction 1 + 1 +- 1L + 1L + ... where b are positive bls b2 \b3s 1b4 v real and when 2 b diverges the continued fraction converges v for all values of 3, except for the values on the negative real axis, to {9195). s+x For proof it suffices to show that [ii-£51 = {Cl/s - :—2+:—§ ...qu;(x) 4.4 As { xk-1d¢(x) exists for k = 1,2,3,... mlo ...: C C 2 3 +'—§'+'_§I+ 000 4.5 S S which according to theorem (4.1) has continued fraction of the given form. The fact that this continued fraction converges to the above integral can be seen by comparing the nth approximant A -E- of the continued fraction with that of Riemann sum approxima- B n tion of the integral. 4.11 RC Ladder Networks From.Theorem 4.2 we see that 49 b2 b4 b6 b8 Jv\ JVx—___/\/\____x\/\___ ......... Fig. (4.1) b b2 b4 b6 8 t 6 0 o I I I l ’“ " " ‘ F(s) 1/bl 1/153 1/b5 1/b7 1/1»9 Fig. (4.2) J In ID- 0‘, .1 - Wt] ' It.‘ - ‘ . .- 17.? C ‘~ an .-..u‘ ' ‘n.\, D“ 5,7,“, (" ‘fi'&: V flu“ int-j 50 f(s) = { g%£§l- where d(x) an increasing function 1 1 bls+b2"'bsl+—l-— 3 154+... where b1, b2,... are all real and positive. Hence we get a ladder network for f(s) = { égéil- given by fig. (4.1) 4.12 RL Ladder Networks: For RL networks the driving point 5 +-x 1 fraction is given by T%I-+'Tgi£— +-T%§-+-T%ZE-+-... . So the ladder networks for RL functions are given by fig. (4.2). d impedance is of the form f(s) = { §—-i£§l- and the continued 4.13 LC Ladder Networks: Driving point impedance for LC networks is given by §§g1£§%- and the following theorem gives continued 8 +~x fraction expansion for it. Theorem 4.3: If i(x) is a monotonic increasing function of x 2(k-l) for x 2 0 and the integral {. x dW(x) for k = 1,2,3,... exists then the integral {é_%1$§l§. has the corresponding con- s +-x tinued fraction given by 1L+ 11 + 1! + 11 + O... \bls |b2s \st 1b45 where bv are positive real and when E b diverges the continued v fraction converges for all values of 5 except for the values on the Jx axis, to {IS d xz . s +~x Proof: f(s) = { §§§11§% let 52 = p then eXpand { 91—%- in s i-x p+x 2 continued fraction. Replace p by s and multiply it by s. 51 b2 b4 b6 b8 1 4290. i JML 47m Jim—~— ----- 1. ..L -1 ..- — b1 b3 T- b5 -r- b7 -- b9 sd x Fig. (4.3) I 43—12 2 L 3 +x b b a b b a b b a b l b 5 7 L 1/l)71y .J l/bso -_ Iii—1 j—‘W #l‘n—l 52 2 i d Hence the ladder networks for {.§§_1£§2. are given by s +-x fig. (4.3). 4.14 Ladder Networks for a class of RLC functions: For distortionless uniformly distributed lines, the line of singularities is a straight line parallel to the imaginary axis, a units to the left of it. The impedence function using 2 (8 +01) <11le Eqn. (3.4) is given by f(s) = 2 2 . Continued fraction (s+e) + x for such functions is given by 1 ll 11 ...—lL___ T—L‘bflsm) + ”2001”) + ”33+!” + ‘b4(s+u) + 4.6 Hence the ladder networks are given by fig. (4.4). 4.2 An Algorithm for Getting Ladder Networks from Cauchy Type Integral Representations In the last section we observed that we can get ladder networks from Cauchy type of integrals. In this section we will give an algorithm [171 for getting ladder networks from the Cauchy type of integrals satisfying theorem (4.2). Given f(s) = { 91351 s+x l x2 X = _'-—+_"ooo d¢(X) {Cs .2 .3 3 C C C .. .2 _1_ __2_ — S + 2+ 3+... 5 8 After getting [c0,c1,c2,...] we proceed as follows 600 = 1 = a C0500 1 610 = 1 53 c6 =aa=h 1 10 1 2 1 - h1 h2 = 3‘ 1 [520 0 522] - [510 0] [E o o - a2[o o 500] 1 0 [C2 0 C11 620 = a182613 = h2 o 622 a = hz 3 ala2 2 3o 1 2 3 4 3 o 5 1 32 1.0_J h a = 3 4 alaza3 F— 'fi [540 0 542 0 644] = [530 o 532 0] 1 o o o o - o 1 o o o o o 1 o 0 L9 0 o 1 oj a4[0 0 620 0 622] [ 0 0 E -‘ = a a a a a = h c4 C3 C2] 40 1 2 3 4 5 4 o 542 o 54 ha a = a a a a l 2 3 4 5 So [35,86,....] can be calculated. Now we can find the values of series resistors in fig. (4.1) as 8183 o. o. 82v_1 azal‘ 00.. 82v b2v = and the value of the capacitors is given by bl = 1“[1 and b2v+l = a a .... a A computer programme using this algorithm will be used, in the next section, to find the elements of the networks. Example 4.1: Consider the function f(s) = [(s +-sz)% + s +-%] using theorem 3.2 we have the Cauchy integral representation for f(s) (X_X2% l f(s) - n x +-s CDC-fiv- Applying the algorithm for getting ladder networks, we have the network given in Fig. (4.5). Example 4.2: 5+2 f(s)=log('s':1- =29; { s+x Using the algorithm we have ladder networks as in Fig. (4.6). 2 Example 4.3: f(s) = 3 log 82+3 8 +2 =S3£1_x__ g sz+x 55 .249 .083 ,041 .024 .0163 .0115 .0084 __/\/\_____/\/\_j__./\/\___J\/\_____/\/\__ ----.. _i. .4L— ‘L' -J- .J_. _L_ -- -J- “8.0 “"3205 7 72.3 T- 129.1 ——- 203.3 296.2 7409.5 ‘7 Fig. (4.5) .666 .025 .81x10'3 .24x10'4 .74x10'6 J\/\____/\/\__ _______ _ F(S) --1.0 7"26.9 ‘F‘844.7 .27x105 .929106 .31x108 Fig. (4.6) .40 .54110'2 .59.».10'4 .621-10'6 eann——T-Jmm——u——nn0————QmI——«--—-— J: .4— J— J— ‘ J- F(S) 1.0 ""74.9 "f7684x104 ‘T’.65~106 -T- Fig. (4.7) 56 LC ladder network is given in fig. (4.7). Foster Type Networks for Cauchy Integral Representations: 4.31 RC Distributed Networks: For RC distributed networks, driving point impedance f(s) = {.Eééigz’ has a.Foster type realization shown in fig. (4.8). Continum of networks as shown in fig.(4.&)will be represented as shown in fig. (4.9). 4.32 LC Distributed Networks: For LC distributed networks the driving point impedence f(s) = { fiiéifilgfi- has foster type realiza- tion given in fig. (4.10) and reprZse:t:d by fig. (4.11). 4.33 RLC Networks: We have seen that f(s) for RLC networks having arc singularities is given by f(s) = 4: [films + b(x)ldx_ s + sc(x) + d(x) Network realizations of f(s) by Foster type of networks are shown in fig. (4.12) and (4.13). Realization of f(s) by Darlington's reciprocal and non- reciprocal networks and by Youla's method can also be achieved easily. 4.4 Synthesis Procedures for Integral Representations of Two Port Networks In the last chapter we obtained Cauchy integral matrix representations for two port distributed parameter networks. In this section we will develop synthesis procedures for them. The given impedance matrix satisfies the realizability conditions given in Section (3.5) and is of the form ----d r--- '—1 = f(x)dx c X 57 ub- ------ -’-.X Fig. (4.8) :f(x) /xdxf H“; l/cx = f(x)dx Fig. (4.9) f1(x)/x dx f1(x)/x dx Farah—... f l/cX = f ;) 1( Fig. (4.11) dx F\- 58 aSXZ dX bSX>dXI d(X) d(X) J l/CX '— F(S) ‘T‘a(x)dx f x dx 3%:gdx b(x) F1g (4.12) 83x2 f(x) dx .../v»... agxz d 1/c d(x) x b F(s) [5W _ f x f .251 b(x) (1X: Fig. (4.13) QVT‘F—v-Il . ’1 N...“ M— “l ..u vsrxvv ‘ fi .“54. '16 l H .\ 59 [f11(x) f12(x)] dx f21(x) f22(x) f(s) = s +-x 4.7 4.41 Symmetrical Lattice Networks: Symmetrical lattice networks as shown in fig. (4.14) have long been used in classical network theory [18]. We will use them for realization of f(s) given in equation (4.7). For symmetrical lattice networks — -_1_ 211 222 — 2 [28 +-zb] 4.8 __1_ 212 2 [2b Za1 or Za = 211 - Z12 4.9 Z = Z +-Z 4.10 As 211 = 222 so f11(x) = f22(x) in eqn. (4.7) [f11(x) - f12(x)]dx = .11 23 s + x 4 Now as [f11(x) - f12(x)] 2 O and if £‘x(k'1)(£11(x) - f12(x))dx exists for k = 1,2,... 1 1 1 1 Z =T—L—+TL-+‘—l—+<|—-L+... 4.12 a bls b2 b3s b4 _ [f11(x) + f12(x)]dx . Z - as f (x) + f (x) 2 O and if b s +-x ll 12 then { xk-1[f11(x) + f12(x)]dx exist then 1 1 l 1 Zb =‘T:£; +'T;:'+'T:§; + —;i+ ... 60 1 2a / 2 1' /------------ 2' Fig. (4.14) b6 b8 ___/\/\ Za ‘L' "b b b ‘ b fir- S 7 '1 Fig. (4.15a) c2 c4 c C8 AT ’\/‘ ’V‘ zb .JL C C JLC JL— C J1— C 1" 1 T 3 1r- 5 " 1r- Fig. (4.15b) OI 61 Hence each arm of the lattice network is of the form shown in fig. (4.15a and b). Instead of having ladder networks for Z3 and Zb we can also find Foster type of networks for them. Now we can state the following theorem for lattice network realization. Theorem 4 .4: [f11(x) f12(x)] dx f21(x) f22(x) s +-x f(s)= where f11(x) = f22(x) and f12(x) = f21(x), can be realized by symmetrical lattice networks containing ladder networks in each arm if i) (f11(x) - f12(x)) > 0 and f11(x) + f12(x) > 0 ii) { xk-1(f11(x) - f12(x)) and { xk-1[f11(x) + f12(x)]dx exist for k = 1,2,3,... Satisfaction of condition (i) is sufficient for Foster type of networks in each arm. 4.42: Cauer's Networks: The network configuration used in Cauer's method of network realization [18] is shown in fig. (4.17). We will use this type of configuration for realization of Eqn. (4.7). For network given in fig. (4.17) 11 - a c Z12 — azc 2 2 Z22 a Zb +-a ZC or Za = Z11 - l/a 212 4.14 2 ’2b — l/a Z22 - l/a 212 4.15 l/c = f(x)dx X int—J 23 6 x dx X Fig. (4.168) 1/6X = fl(x)dx Fan—J Zb £1(x)dx X Fig. (4.16b) Zc 1' l : a Fig. (4.17) 2! 63 2c = l/a 212 For f(s) given in eqn. (4.7) [f11(x) - l/a f12(x)]dx za - s + x [l/a2 f ( - 1/a f d z = 22 x) 12(x)] X b s +-x f (x) _ 1/a{_];.2__dx c s +>x N I Now it is possible to pick a in such a way as to make 1) f(x) = [f11(x) — l/a f12(x)] > 0 ii) f1(x) =[l/a2 f22(x) - l/a f12(x8 > 0 iii) f2(x) 1/a f12(x) > 0 thus we can realize Za’ Zb, 1) f xk'1%(x)dx L ii) £ xk'1 f1(x)dx iii) { xk-1 f2(x)dx exist for k = 1,2,3,... can realize Za’ Zb, ZC by ladder networks. Example 4.4: j 10 3+3 1 8+2 2 g s+2 g 8+3 f(8) = 3+2 s+3 1 .___ ___ L_Og 3+3 2 log 3+2 J ZC by Foster type of networks. 4.16 4.17 4.18 4.19 If 4".— C 3 i ii f(s) = 2dx s+x dx s+x Realization by Lattice Networks: 2dx s+x Referring to fig. (4.14) + 3 9s. g s+x s+x 's. 2dx dx‘J s+x s+x 3 = E 3dx 3 g s+x Realization for Za and Zb are shown in fig. (4.18a) and (4.18b) and symmetrical lattice realization using Foster type of networks is given by fig. (4.19). Realization by Cauer's Method: Referring to equations 4.13, 4.14 and 4.15 and taking a = -1 dx s+x s+x dx s+x E z = g 95- i Ladder network realization for Za = Zb = Zc is shown in fig. (4.18b) while the network using Foster type networks is shown in fig. (4.20). 1.20 .0162 .177x10'3 .186x10'5 )--‘--‘- -- -- a . .333 24.9 -) .22x10 .216x10 ~1- Fig. (4.183) .40 .54x10'2 .59x10'4 .621x10'6 we » ------- Zb .0 74.9 .684x104'I'.65x106 Fig. (4.18b) 1L——~ ----------.-. 2' Fig. (4.19) d x/x dx /x dx/x D 2! F. 1g. (4.20 ) F g 1. Q (4.21 ) 67 Example 4.5: Let S-1/3 -% C11 " f(s) ' -% -1/3 -s azs First we will find Cauchy integral representation for f(s) and then realize it i) f(s) is sectionally holomorphic matrix,with line of discontinuity as negative real axis- ii) A8 S-bao f(s)-99 iii) Using theorem (3.2) we can find the density function matrix f (x) =/3 “V3 11 '2? “1x x-ls f12(x) g ' T. 3: X f21(x) — - n = [2 ‘1/3 f22(x) 2n “zx' Hence the Cauchy integral representation is °° a, x-1/3 fi 2n 1 n _% dx _x QC! 4” n 211 2 f(3)'= s +-x o -1 -2 3 for realizability ‘25'X / 0102 2 x2 4n W By picking values of a1 and a this condition can be 2 met. f(s) can now be realized by Cauer's method as shown in fig. (4.21). 68 By using the turm>ratio for the transformer to be -a Q -l/3 xJ5 Za = 2n 01X - a}; = f1(x) 0/3 '1/3 1 ";5 A Z = -———'x - -——'x = f (X) b 2 a n 2 2a1n 1 _._l_ ‘% _ * Zc - aifi X — f3(x) 4.5 Synthesis Methods for Cases where Density Function Matrix is Real In the last section we considered cases where the density function matrix satisfied the realizability conditions. In the next few theorems we will suggest ways to realize impedance matrices, having density function matrices which do not satisfy realizability conditions. Theorem 4.5: Let f11(x) f12(x) dx f21(x) f22(x) s +rx f(8) = L where f11(x) and f22(x) are nonpositive ‘v x but satisfy the condition f11(x)f22(x) 2 f12(x)f21(x) \7 x, then f(s) can be realized by RL networks if '% f(s) has a Cauchy integral matrix representation. Proof: If '% f(s) has a Cauchy integral matrix representation then according to Lemma (3.7-) f11(x) f12(x) s -x -x dx f12(x) f22(x) f(s) a) ‘x S +x 'x ' 4.20 69 f11(x) and f22(x) where of the density function matrix are positive and satisfy the condition f11(X)f22(x) 2 f12(x)f21(x) hence realizable by RL distributed networks. Theorem 4.6: Let f (X) f (X) 11 12 dx R11 R12 f (x) f (x) _ 21 22 f(s) — s +-x + R21 R22 where R is a constant matrix and f11(x) and f22(x) are non- positive functions but satisfy the condition f11(x)f22(x) 2 f12(x)f22(x) ‘V x then f(s) can be realized by RL networks if f 1) R11 R12 = f11(x) 12(x) Y11 Y12 -x -x dx + R21 R22 f ( ) f (x) Y21 Y22 21 x 22 -x -x f (X) f (X) f (X) f (X) ii) 1: , 1: , 2: , 2: are integrable. Y iii) 11 12 is P.R. Y21 Y22 Proof: f (X) f (X) f (x) f' (x) -x -x 21 22 f(S) = + dX + 'Y 'Y L s +~x a f21(x) f22(x) 21 22 -x -x K V” '1 L f11(x) f12(x) s -x -x dx Y Y f21(x) f22(x) ll 12 f(s) '/ -x -x 70 First matrix in equation (4.21) is realizable by RL networks while the second one is realizable by lumped resistive networks. Example 4.6: F s_+_1_ ___s+3 “ 4 log 3 (8+3) log (334-3) f(8) = 5+3 5+3 log 33+3 S 10g s+lJ r L T F 3 _ dx +_3 dx 3 dx _ 3 Q5. { s+x { x i s x 3 dx _ 3 d_ s 3 dx L_{ s+x { x {hs+x .J 4 3 l/x dx S 3 -1/x de { s+x { s+x -1[x dx 3s 3 dx L. .[ s+x {s-l-x J 3 4/x dx -1/x dx S -1/x dx 3 dx f(s) = s +-x 4.22 1 Which we can realize by RL networks. Referring to fig. (4.17). Taking a = -1 Z = s 3 l/x dx { s+x 3 21) = 8 { SB‘I/Xde s+x 3 3/x dx 2 = s _.__..__.. c fi' 84x Ladder networks for 28, Z are shown in fig. (4.22 a,b and c). b’ Zc 71 1.81 .175 .0134 .997x10"3 do to 1; ,, ------- 3.3 .303 2.56x10"2 .86><10'3 1.36,~<.10‘4 . -2 -3 .603 .0583 .448x10 .332x10 H 44 )1 ,, ------.. .1 .108 :.4x10'3 :.6x10"4 55410”5 2.40 .209 .0164 .123x10‘2 ll 0‘ ‘1 a; -----_- 4.9 .385 3.12x10'2 2.3x10‘3 1.69x10'4 Fig. (4.22) 3/x dx (3-1/x)dx 2 . 3/x dx (3-1/x)/x dx l/xzdx l/x dx r 1 : -1 Fig. (4.23) 72 Network using Foster type of networks for Za’ Zb and z c is shown in fig. (4.23). Next we will consider the case where the density function matrix is real i.e. f11(x) f12(x) dx f21(x) f22(x) f(s) = s +-x L where f11(x), f12(x), f21(x), f22(x) are real separating each function into positive and negative parts we can write 6:100 + £11m f‘1"2(x> + £1200 + _ + _ dx f(s) = f21(x) +-f21(x) f22(x) + f22(x) 4 23 s +' x ' L Case 1: If the Cauchy matrix integrals + + - - 511(x) f12(x) f11(x) f12(x) dx dx + + - - f21(x) f22(x) and f21(x) f22(x) s +-x .. s +-x L L exist then let + + f11(x) f12(x) dx + f” (x) f (x) 21 22 4 24 f = 1(8) L S +-x If the density function of this integral satisfies the realizability conditions then it can be realized by symmetrical lattice or Cauer's method. 73 Also let f11(x) f12(x) _ dx f (8) = f21(x) f22(x) 2 s +'x L Now if l/s f2(s) has Cauchy integral matrix representa— tion then f11(x) £;2(x)7 'X 'X s _ _ dx f21(x) f22(x) -x -x _J f2(s) = i s +-x ' 4.25 Further if the density function matrix in Eqn. (4.25) satisfies the realizability conditions then f2(s) can be realized by RL networks. Case II: f(s) in eqn. (4.23) can be separated into + - - + . f (x) f (x) A f (x) f (x) f1(s) = J 11 12 dx and f2(s) = 11 12 dx - + + - 62100 13220:) £2100 £2200 S +'X s +'x L L and if f1(s) exists and the density function of f1(s) satisfies the realizability conditions then f1(s) can be realized by RC networks. Again if f2(s) exists and 1/s f2(s) has Cauchy integral matrix representation then we can write f‘ (x)/-x £+'(x)/-x f2(s) = s 11 12 dx 4.26 f:i(x)/-x f;2(x)/-x s +-x 74 and if the density function matrix in eqn. (4.26) satisfies the realizability conditions then f2(s) can be realized by RL networks. Now if f(m) = R and the density function matrix is real then the following two theorems give the realizability conditions for f(s), Theorem 4.7: Let f(x) = R and f(s) = f+(s) + f-(s) where dx + + - - f (X) f (X) f (X) f (X) + - f (S) = 11 12 dX and f (S) = 11 12 + + . - - f21(x) f22(x) f21(x) f22(x) s +-x s +-x L L exist and R = £1 +'r where f- (x) f (x) .. 1:. .. 1:. . R1 = - - ' f (x) f (x) (_ 21 d 22 dx _x "X L Y11 Y12 £= Y21 Y22 exist then f(s) has a network realization if i) f+(s) is P.R. ii) [f-(s) + R1] is P.R. iii) 5 is P.R. Theorem 4.8: Let f(m) = R and f(s) = %1(s) + f2(s) where + - - + . f (x) f (x) f (x) f (x) f1(s) = 11 12 and f2(s) = 11 12 dx iglm £225") £2300 £5200 s +-x s +-x 7S exist and R = R1 +'y where - + x _ f (x) f (x) R1‘ 1} 1? Y11 Y12 x x dx + + .- f21(x) f22(x) Y21 Y22 L -x -x exist then f(s) has a network realization if n* 1) Elm) is P.R. ii) [f2(s) + R1] is P.R. iii) y is P.R. .E Example 4.7: f 3+2 2(S+1 s+l s+l [S-Zs log §:E:+-2 log 7)] [log-;:§ - 1 - 5 log 8+2 f(s) = s+l s+l s+l [log 3+2 - l - s lo og— 3+2 [10- -23 log (: %) + 2 log 8+2 2 2 dx dx dx -— - 2+ -—— - --—— - 1 [CE 8+9, MES-13:; + (108 5)] [E- 8+}, SJ; “x ] f(8) = 2 2 2 dx dx dx pix—{fl 29,-; S,“ [E- s+x’23{s+x+10] 2 2 1 2 - d -S—— - —--— .. {(2: $7?) x Eon-81+ fi)dx zj‘a/x Sflmx {(l/x 1/s+x)dx 1 f(s) 2 +- 2 2 {0 -1+ -—)dx i(z-S —-)dx ~£[1/x-1/si-xmx 2£(l/x-1/s+x)dx + 2 -log 2 -log 2 7-2 log 2 r2 2xdx 2 xdx"1 r;.2 2/xdx 8.2 1/xdx1 l s+x i s+x l s+x { s+x f(s) = +' 2 2_ xdx 2 2xdx 8.2 l/xdx s 2/xdx { s+x { s+x { s+x { s+x J a .J L. + 2 - log 2 - log 2 7-2 log 2 = f1(s) +- f2(s) +> f3(s) we can realize f1(s) by RC symmetrical lattice. Refering to fig. (4.14) Z = 2 3xdx a £ s+x 2 z_»_<_d_x. b i s+x f2(s) can be realized by RL Symmetrical lattice networks s+x l l . . . Ladder networks for Za’ Zb’ Za’ Zb are given in fig. (4.24). Finally we can realize f3(S) (approximately) by Cauer's method, taking a = -1 (2) la = (2 - log 2) 2152) = (7 - 4 log 2) 2c = (log 2) Hence the final network is given in fig. (4.25). ~'H 77 2.89 .103 .333x10'2 .101x10'3 MAP—$— -..------- r 6 2a "' .222 ~£6.70 :_J:'_.206><103 ‘ .626x104 .225x10 T T V Fig. (4.24a) -2 -4 .964 .0345 .111x10 .338x10 W—M—W%_ _______ _3_. -S Zb .666 2:: 20.1 AWE-.62x103 E; .202x105 ...... .675x10 Fig. (4.24b) -2 -4 1.44 .056 .178X10 .541x10 i _ _ Z5 2,2.12 .95x10 5 $2.38x10 6 Fig. (4.24c) 78 .480 .0189 .59Sx10'3 ,180x10‘ .795x10'7 . wur‘- Fig. (4.24d) (2 log 2) (7 - 4 log 2) ——v u 2! Fig. (4.25) l! 78 .480 .0189 Fig. (4.24d) .595x10'3 .180x10' 4 .795x10- / Fig. (4.25) 2! 7 79 4.7 Ladder Networks_Representing Some Transcendental Functions and Numbers A riumber of papers (19-22) have been written on getting ladder networks from transcendental numbers. In this section we will use Cauchy integral representations for functions to get ladder networks from transcendental functions and numbers. Example 4.8: 00 ‘X = 'S - = 3.93 f . ram 1“... Using algorithm for expansion of Cauchy type of integrals we have the continued fraction expansion for this as l “Eta—1.1 . s + —-— l ‘5 + gfi; 333'+'—L- 1 . 3+.25+g_l; 1 .20 + Hence the network is given in fig. (4.26). Next we will consider eXpansion of a transcendental number. Example 4.9: f(s) = log 2% 3 dx = % log 2 = % & g;; for S = 1 5+3 - % 10g (m Substituting 3 = 1 gives = % log 2 We will use the algorithm for continued fraction expansion of 3 dx % £‘gz; and in the expansion substitute for s = l 8O 1/2 1/3 1/4 1/5 «4AVA.____...;Avfi_____r___J\/\____T___/\/\________/\¢\________J\/\_____r ...... ‘i 1 ‘: 2’ 1 -- 1 -L 1 ‘b “E W T f " T_ 7' ‘r I -sF _ e 1< s) Fig. (4.26) log 2 -—a- log 2 --f- .5 .0454 .00356 .265><10'3 AVXe --------- .0835 .665x10' 3.64x10‘5 Fig. (4.27) 1 .09 .715x10'2 5.22x10’4 1.4x10'5 1/6 1.33x10’2 1x10"3 7.28x10’5 Fig. (4.28a) 1 4/9 4x16/9x25 #«vk - ‘AVA __-- 4x16 4x16x36 1 4 9 9x25 Fig. (4.28b) 81 1 =— 1 s +-—- l '5 +12.05 + .3... El? 1 .0454 + 1 .00356 +'-l- 151.ZS-+ Substituting for s = l we get a resistive ladder network for log (2)}5 Similarly resistive network for log 2 is given by fig. (4.28a). Reza (19), using the expansion of log 2, got the follow- ing network fig. (4.28b). It is interesting to note that we have got a more rapidly converging resistive structure than that of Reza's. id? | CHAPTER V 5.1 Properties of Cauchy Type Integrals for Network Functions In this section we will give some of the properties of RC, RL and LC functions represented by Cauchy type of integrals. Many of the results in this section are generalization of the results for lumped networks. Lemma 5.1: Let f(s) = Eéfiigz- be a distributed RC functionoThen i) Re f(JW) is a monotonic decreasing function for w > 0 ii) Re {(0) > Re f(m) Proof: f(s) = { £15195- s+x Separating into real and imaginary parts we have Re f(s) =f im):(x)d: 5-1 L (0+x) +-w d Re f(JW) -_— m x2 +w2 Let w < W d Re f(le) - Re F(sz) ei ZLESEL‘LX - £15132}. dx x +»wi x2 + w: 2 2 xf(x)(w2 - wl) = > O for w > 0 2 2 (x +wi) (x Ni) 82 83 Hence Re f(JW) is a monotonic decreasing function for w 2 O. The proof for the second part is quite obvious. Theorem 5.1: Let f(s) = {.figifilgg. be the driving point impedance 8 + x of an LC network then i) §2_£(§l. 2 0 'v S Re (8) f(ol) f(oz) ii) 5 for 61 >»02 > 0 0'1 0'2 Proof: f(s) = §§£$§l§.dx s + x _ (0+JW)f(x)dx — 2 2 (0+JW) + X 3 2 2 Re f(s) = f(§)[° 2+'°" +’°x gag, 5.2 2 (o -w +x)2+4ow Re f(s) g f(x)[c2 + w2 +-x2]dx 2 O V s Re (8) (02 _ w2 +-x2)2+ 402w2 Hence the proof. The proof for part ii) can be obtained as in Lemma 5.1. Next we will look into the Schlicht (23) behavior of impedence functions for RC and RL distributed parameter networks. Definition 5.1: A single valued, analytic function f(s) is called Schlicht in a domain if in that domain the function satisfies the condition f(sl) # f(sz) for s1 # 32 Theorem 5.2: The driving point impedance of a distributed para- meter resistive and capacitive networks, represented by 84 f(s) = { ££§lfl§_ is a Schlicht function of s in the right half s+x plane. Proof: f(s) = i Eliléi. f(x) > 0 -——-—- s+x let us assume f(sl) = f(sz) for s1 # $2 i.e. f(x2dx _ fgx2dx 3 +x { 8 +x 1 2 _ f(x)dx (32 31) { (51+x)(sz+x) 0 which means I f(x)dx 0 L (sl+x)(sz+x) = 5'3 Separating (5.3) into real and imaginary parts we have f(x)[(oz +X)(a 4X) - B B ldx Re f = 1 2 2 2 1 2 = 0 5.4 (31-1-14 ‘sz-Px‘ HRH-.8 (01 +70 + B (d 41)]dx Im f = 22 1 2 1 2 = 0 5.5 \sl+x‘ ‘sz+x\ The following two cases need to be examined i) 31-52 > 0 ii) 51-52 s 0 If Bl and 82 are both positive or both negative, (5.5) is contradicting otherwise, (5.4) is violated. Hence the proof of the theorem. Theorem 5.3: The driving point impedance of a distributed para- meter resistive and inductive networks represented by 85 f(s) = { Egééfil-dx is a Schlicht function of a complex variable 8 in the right half plane. Proof- _—° ‘: d f(s) = i s f({) x s+x f0.) = f—/L:Lf£{dx 5.6 This is RC distributed parameter function and by theorem (5.2) a Schlicht function. Hence the proof of the theorem. Lemma 5.2: Every distributed RC function, represented by f(s) = { Eéfifigi' is an infinitely differentiable function every where in 3 plane except points on L. Proof is given in most of texts on complex analysis. Lemma 5.3 (8): If f1(s) and f2(s) have Cauchy integral re- presentations, which are distributed RC realizable then f1(f2(s)) has a distributed RC realizable Cauchy integral representations. Lemma 5.4: If f1(s) has a distributed RC realizable Cauchy integral representation then f1(l/s) has distributed RL realizable Cauchy integral representation. Proof: f1(s) = [ ESElgE s+x f1(1/s) = s f(x2dx _ s f(x)/x l +-xs — l/x + s which is RC realizable. 86 f (x) f2(x)dx lemma 5.5: If f1(s) = { 8+1: dx and {2(5) = { T have RC realizable Cauchy integral representation then so does [fl(s) + f2(s)]. lemma 5.6: If f1(s) has RC realizable Cauchy integral representa- tion then f2(s) = [k f1(s) + s f1(s)] has RC-R1.realizable Cauchy integral representation for k > 0. g“ Theorem 5.4: If f1(s) = { Eéfifigé' has RC realizable Cauchy in- ’ tegral representation then for 11 E L (not an end point) [)1f1(s) + s f1()1)] is distributed RCL realizable. i Proof: f1(3) + S f1(X1) = X1 {% +S{ :; 1+x under the condition that f(x) satisfies H61der's condition (14) {£932- always exists. 11+“ Hence the proof of the theorem. lemma 5.7: Let f(s) be the driving point impedance of distributed b LC networks given by I EEELEL , then there always exists f1(s) a s +-x the driving point impedance of lumped RL networks such that f(fl(s)) has a Cauchy integral representation for RLC networks. Proof: f(s) = $2 513$)— dx a $2 +'x let f1(s) = (qls + B) then b ((21 s + B)f(x) f(f1(s)) =£ 5.7 8(a18 + 8 2)+x Which is a representation for RLC networks and is realizable. 87 5.2 Generation of Distributed chlmpedance Functions Generation of driving point functions for lumped RC networks have often (24) been mentioned in network theory. In the following few theorems we will discuss generation of distributed RC and RL impedance functions. Lemma 5.8: Let Eééfigi- be RC realizable Cauchy integral and let f1(x) be any non negative function then f(x).f1(x)dx s +'x 1) [f(x) + f1(x)]dx 11) i s +-x are RC distributed network realizable if f1(x) satisfies H condition on L. Proof: As [f(x)f1(x)] and [f1(x) + f(x)] are non negative for all x and f1(x) = [f(x).f1(x)] and f2(x) = [f(x) + f1(x)] satisfy H condition. Hence both f(x)f1(x)dx [f(x) +-f1(x)]dx and are RC realizable. s +vx s +-x Theorem 5.5: Let I Eéfil-dx be distributed parameter realizable L Cauchy integral with f(x) satisfying H condition with a = l everywhere on L = [a,b] then f(x)f1(x)dx 1) s +~x (f(x) + fl(x))dx ii) 3 +-x f(x)/f1(x)dx iii) 3 +'x (f1(x) is bounded away from zero) 88 are RC and RL realizable if f1(x) satisfies H condition on L with a = l. .ggggfi: As f(x) and f1(x) are both functions of bounded varia- tions so are [f(x)-$100]. [f(x) +2160] and [f(x)/Elem on L. By theorem (3.6) we can realize functions of bounded variations by RC-RL networks. Lemma 5.9: Let { géil-dx be distributed parameter realizable Cauchy integral with f(x) satisfying H condition with a = l and let f(x) be any function which is monotone on L 6 [a,b]; then all the three cases defined in theorem 5.5 are RC-RL realizable. .Egggflz Any function monotone on a compact set is a function of bounded variations. Now applying theorem (5.5) we have the proof. Theorem 5.6: Let 1 and {‘géil-dx be a basis; then the following operations always generate lumped-distributed RC, RL functions. a) Multiplication by a constant b) Replacement of s by l/s c) Addition of two functions. Proof is clear from the above discussed properties of Cauchy type of integrals. 5.2 Integral Representation of PR Functions Definition 5.2: A function f(s) is called P.R if i) when s is real, f(s) is real ii) Re f(s) 2 0 when Re 3 2 0 Theorem 5.7 (15): Any function f(s) which is regular in right s-half plane and such that Re f(s) 2 0 there and f(s) is real for real 3 can be represented as 89 f(s) = ch +1? $5ng 5.8 0 where c is a nonnegative constant and W is a nondecreasing function. Lemma 5.10: Every positive real function can be realized by LC ladder networks with a lumped inductor in series with ladder net- °° k-l . work if Ix d¢(x) exists for k = 1,2,3,... . 0 Proof of the lemma is clear from theorem 3.3. 5.3 Equivalent Networks Now we state a theorem which gives us the necessary and sufficient conditions under which the distributed parameter RC networks represented by Cauchy type of integrals are equivalent. Theorem 5.8: A necessary and sufficient condition that two RC ladder networks represented by Cauchy type of integrals with density functions f1(x) and f2(x) are equivalent is { xnf1(x)dx = { xnf2(x)dx n = 1,2,... f1(x)dx f2(x)dx Proof: f1(s) = L T f2(s) = { —s-+-1_<—— f (x) - f (x) f (x) - f (x) 2 3 1 2 _ 1 2 §_ §___ §_ Now { s +-x — { s [l - 5 +82 33 + ...:de Now as we can integrate the series term by term hence the nec- essary and sufficient condition for identical vanishing of right hand side is {‘xnf1(x)dx = i xnf2(x)dx 90 5.4 Properties of Cauchy Integral Matrix Representations Theorem 5.9: Let ' f11(x) f1299] dx [£2100 f22(x) F(s) = s +-x L = [a,b] I... be RC realizable. Now if f1(s) be a lumped RC realizable admittance function then F(f1(s)) has a Cauchy integral matrix representation i.e. F[f1(S)] = i11 i12 A 5 dx f21(x) f22(x) 1‘ s +-x which is also RC realizable. Proof: Let f1(s) = %%§%- is an RC realizable network admittance Fif1) = f11 £12 dx f21(x) f22(x) 5 9 L P(S) 1 . “ V1(x) Now -7;T-::; can be written as vo(x) +-1213;;;;E;T where -ai(x) are nonpositive (8). Substituting this into (5.9) we have 91 f11(x) f12 N f11(X)Yi(X) f12(x>vi(x> F(f1(s)) = yo(x) dx +> 2 dx f12(x) £22(x i=1 f21(X)Yi(X) f22(x)vi(x> 1» L S + ai(X) f11(X) f12(x> i11 i12 = yo(x) dx +' A dx 5.10 f21(x) f22(x) E21(X) f22(x) L L s +-x First integral in eqn. 5.7 can be realized by resistive networks while the second one by RC networks. Lemma 5.8: Let f11(x) f12\d§\ 6.3 c where ‘d§| is an element of length on the contour. In general we write w=v+i¢ 95 where V is the potential and W the stream function. We note in (6.2) that W is analytic everywhere in the finite part of the z-plane except at points occupied by the charges. Similarly, W in (6.3) is analytic everywhere except on the contour (c) and at infinity. We may use the theory of analytic functions of a complex variable to obtain various properties of the potential and of the stream function. The derivative of W is unique, and may be written in either of the forms fl=fl+ial=ai-iay., dz ax 5x 5y 5y whence V and W satisfy the Cauchy-Riemann relations, 51:-5l, al=al BX BY BY 8X The components of the electric intensity are obtained from V by the relation E = -grad V. Thus we have the various forms, Ex = - BK_= - AK = - re §H_ 6.4 ax 5y dz by ax dz The stream function W may be interpreted in terms of the flux of the field intensity across a curve in the z-plane. The flux of a vector across a given curve is the line integral of the normal component of the vector, 6 = I Ends 96 (6.23) Fig. (6.2b) Fig. 97 hence the flux of E crossing the curve of fig. (6.2 a) between the points 20 and z is 2 Q = I (-Eydx + Exdy) z 0 all .31 = - (- ax dx ay dy) i(zo) ¢(z) , ll NL—‘a N o in the clockwise direction when viewed from 20. The flux depends only on the values of V at the ends of the curve. For a point charge q at the origin the stream function is W = -q@ + const, and the outward flux through a closed contour surrounding the charge, Fig. (6.2 b) is Q = -q@o +'q(¢0 +'2n) = an 6.6 The flux from a set of charges is additive, so that equa- tion (6.6) is general, when q is interpreted as the total charge inside the contour. Consider now a charge distributed continuously on a contour (c), and let q(z) be the total charge on the are extending from 20 to 2. If we surround this arc by an infinitely narrow closed contour, fig. (6.2 b). The flux leaving the enclosing contour through part 2 is Q2 = ¢2(ZO) - ¢2(Z) 9 6.7 where $2 is the stream function in the region on the correSponding side of (c). Similarly the flux leaving the enclosing contour 98 through part 1 is Q]- = 411(2) -¢1(ZO) 9 608 where $1 is the stream function in the region on that side of (c). Since the total flux, Q + @2, is given by (6.6) we see that l the stream function is discontinuous across the line charge, and the amount of the discontinuity is [w1<2) - w11 - [w2 - ¢23 = 2nq 6.9 On the other hand the potential is continuous across the line charge. To prove this we note that the potential is the real part of the complex potential W in (6.3) and therefore given by v = 'l 90;) log |z-g|\dg| + const. 6.10 (c) The integral depends on the distance \z-g‘ between a typical point g on c and the given point z. For two points 21 and 22 just on opposite sides of (c) the distance is the same, so that V(zz) = V(z,). 6.12 Analog between potential theory and transmission functions Before we can suggest a relation between Cauchy type of integral representations, for network functions, and potential theory, we will discuss in brief an existing analog (25) between transmission functions and potential theory. Consider the transmission function of a typical transducer Fig. (6.3). The absolute value of the ratio of the output voltage to the input voltage represents the gain in transmission through 99 NETWORK h—<—~| F(s) = a + is = log V/E. Fig. (6.3) 100 the network, while the phase of the ratio represents the phase shift. If a is the gain in nepers and B the phase shift in radians we have V/E = ea.ei8 and we define the transmission function as the logarithm of this ratio F(iw) = log(V/E) = a + is 6.11 For a finite network with lumped elements the ratio V/E is a rational fraction and the transmission function may be repre- sented by an expression of the form (s-si)(s-s£) ... F(s) log K (s-s'l') (s-s'z') . . . log K + 2 log (s-sé) - 2 log (s-s") 6.12 n Comparing equations (6.2) and (6.12) we see that the trans- mission function F(s) in the complex 3 plane may be identified with the complex potential W of a system of discrete charges. If we assume that unit positive charges are located at the poles, sg, : of the network, and unit negative charges at the zeros, s; the complex potential in the s-plane is W = - 2 log (s-sg) + 2 log (s-s'm) + const. The gain of the associated network is given by the potential on the imaginary axis, and the phase by the correSponding stream function. 101 In the next theorem, we give a new relationship, between network functions represented by Cauchy type of integrals, and potential theory. Theorem 6.1: W(s) be complex potential for continuous charge dis- tribution -q(-x) on a contour L in the left half plane and F(s) be network function represented by Cauchy type of integral having L_ as the line of discontinuity and -q(-x) as the density function then d x dx EW(S) =F(S) =J‘ ELL L s+x + where L+ is the mirror image of L_ in the right half plane. Proof: Complex potential for a charge q placed at any point 8m is given by W = -q log (s-sm) for a set of point charges the total potential is simply the sum of individual potentials = - 1 - O W Z q 08 (S S ) Now for a continuous distribution of charge -q(-x) over a contour L_, the sum is replaced by an integral W(s) = -{ -q(-x)log (s-x)dx 6.13 where dx is an element of length on the contour. From equation (6.13) 102 dWSs) = -gS-x)dx ds f x-s or M1_{ 1991}: 6.14 ds s+x + 6.2 Determination of Charge Distribution Once the complex potential is known in the analytic form, we have to determine the charge distribution on a contour, whose complex potential is the same as the given one. As the selection of the contour is arbitrary we will pick circular contour for our use, and will use the method described below for finding charge distribution on circular contours (16). Expansion of complex potential for a given set of lumped charges qn located on the circle at points Sn, F(s) = Cons - Z qn log (s-sn) n inside the circle we have ‘s‘ < sn for each of charge points sn therefore each of logarthmic terms may be expanded as convergent series in S/Sn' Hence 5 F1(S) - Const - Z qn log (-sn) - 2 qn log (1 - ;—) n n n s 82 =C0nSt-2q "_._-0000 s 2 n n 25n F1(S) = 80 + 2 a s 6.15 outside the circle we have ‘s‘ > sn so logrithmic terms may be expanded in convergent series of sn/s 103 s n F s = Const - 10 s - lo 1 - —— e() Eqn 8 an g( s) n 2 s s n n =b'-b 108- -—--—-.. g Eqn( 8 2 ) n 28 6.16 Suitable power series expansion for exterior potential is a) = ' _ Fe(s) b0 b0 log 8 + 2 bm s m=1 -m The constant bO represents the total charge on the circle. the circle of radius wO we have so that just inside c the interior potential is Fi(@) = 30 + 53mg m E.‘ m while just outside c the exterior potential is m = I _ i? -m -im@ Fe(®) bo bolog (woe ) +- 2 b w e m o m=1 Separating 6.18 and 6.19 into real and imaginary parts v(¢)=a +za WmCosmtb ¢.(@)=za meian 1 0 n1 0 1 m o Ve(¢) bo bolog1wo\ + z bm wo Cos mo 'm V (9) = 'b a - 2 b w Sin me e O m 0 On 6.17 6.18 6.19 6.20 from the condition that V must be continuous across c we have ' - = b0 b0 log‘wo‘ 30 b = w a m > 0 104 Now the charge distribution is determined by the discon- tinuity in V we have m -m , Zflq (é) -[bo¢ + 2(8tho + bmwo )S in ma] 1 m -m . (1(4)) - 2" [1306 + 2(amwo + bmwO ) Sin mtg 6.21 or b Q 1 co - q(¢)= -9—+- 2: meSian 6.22 2n n m=1 o o For unit circle Eqn. (6.22) reduces to q(¢)=%+T-1Tzamsmmq> 6.23 where e is the total charge on the unit circle 9 = ao/log‘w;‘ 6.3 Approximation Procedure When |F(Jw)‘ is given as an analytic function of w, we will suggest the following steps, for obtaining Cauchy integral representation for a network function F(s) which, on the Jw axis approximates ‘F(Jw)‘ in the desired frequency range, and drOps off sharply outside the desired frequency range. (1) From the given analytic function of w form F(s). (2) Complex potential W(s) is obtained by indefinite integral of F(s). (3) Select a contour and find charge distribution on the contour, which has complex potential W(s). (4) Adjust the value of the charge on the contour such that positive charge is on the left half portion of the contour 105 while negative charge on the right side of it. (5) Flip the half portion of the contour in the right half to the left half plane and double the charge density on the contour in the left half plane. (6) Take the contour in the left half plane as an arc of discon- tinuity and charge distribution as the boundary value. Now apply the theory develOped in Chapter III to get Cauchy type of integrals for such functions. 6.2 Approximation of a Constant Let us consider that we want to approximate a function whose magnitude is constant on the j-axis in the interval -1 s w s l, i.e. \F(Jw)| =01 Oswsl and \F(Jw)\=o for w>l Forming F(s) by analytic continuation we have F(S)=Q/ OSWSI using theorem 6.1 w(s) = as +-ao 6.24 Selecting a circular contour which embraces the required frequency range (fig. 6.3), we can find charge distribution on the surface of the contour, which has same complex potential as W(s). Applying power series method discussed in section 6.11, we have the charge distribution given by 2cm) = figsm e +1?- 6.14 106 ““1 311/2 Fig. (6.4) Fig. (6.5) where e is the total charge on the surface of the contour. With- 2 out any loss of generality we can take 6 = 0. Thus 2q(®) = Fg-Sin Q. From physically realizable conditions we should use only the left half of the contour with double charge density given by (1(9) = 3% Sin (Q - n/Z) 11/2 S 4) S 311/2 -20 q(é) = 7 Cos Q 6.25 Translating back the problem from potential theory to network theory, we want to find a function which has the following properties i) F(s)-+0 s—ooo ii) F(s) has conjugate quadrant of a circle as line of discon- tinuity iii) q(é) is the boundary value of the function iv) q(é) satisfies H condition on the line of discontinuity. Hence the function n/2 . —.§1 Cos Q Cos Q F(S)-.11Jéts+Cos)2 + SinZQ 107 3.8 .me 3 DwH m4. m. m. o. m. c. m. N. H o a. N. m. a. m. o. om. m. 4 JG 4 d a I 1 u 4 d d # d d m. n w 5339330 5509326 .3 scion—“80254 Amy I. o uuvuo :uuoDuouusm .3 so“. uuafldhan< A3 a: a Snags: E 3. IO. Im- a m 3 you a ucwumaoo a mo con-awxouanz S+¢eoo 8+ 5 e N t «in moo + $0 30 2mm I Anvu 7332 108 :..: .wE o umppo >o£ommnm£o mo mmcoammm Amv o poppotnouosuwuusm «0 wmcoammm ANV AmYN mo mwaoamom Adv AH + 0 «00 mm + mmv o .C. 33 moo + a: moo QMd u AmvN pawn zocosvmum pouwmmp wsu opamuso omcoammm a .umsoo m mo GOwumEfixOpam< _§_ 1381 109 Response of F(s) for a = 1 has been shown in fig. (6.6) for 0 s w s 1. As desired, ‘F(Jw)| is constant for O s w s 1 and falls off very quickly outside the desired frequency range. In fig (6.6) we have also compared the approximation of a constant by our method alternative to that of Butterworth order 6 and Chebyshev order 6 with s = .5. It is very interesting to note that, inside the approximation band, results of our method are better than that of Butterworth and Chebyschev, but outside the approximation band Butterworth's and Chebyschev's methods give better results. 6.21 High Band Case Let us consider we want to approximate \F(Jw)| = a for w 2 w . 0 Function F(s) in equation (6.26) approximates a in 0 s w 5 WO for wo = 1 using transformation a = 1/p in equa- tion (6.26) we get the desired approximation. fig F/2(1/p + Cos algos Q dé fi

- " o (l/p + Cos 02 + Sin2§ n/z 2 15(1)) = EELJ‘ (P + 2 C03 QICOS T dé 6.27 O p2 +'2p Cos Q + 1 ‘F(Jw)\ approximates a constant a in the range w 2 1. Response of F(p) for higher frequencies, with a = l, is shown in fig. (6.9). A network realization of F(p) by dis- tributed RLC networks is given in fig. (6.10). 6.3 Approximation of a Constant Continued Next we will propose a simpler approach to the approximation C>f a constant, in a given frequency range. The basic principle of 110 I ”IN A X X v D: x F .1—fgx2dx b(x) (a) 85X) [ “d“ __J\f\_1 h——JUb——J dx agxz b(x) d(X) ~[J (X) dx f x dx a(x) b(x) I ~u—a (b) Fig. (6.8) where a(x) b(x) c(x) d(X) = n 111 m oaxma Oaxma onoa 1 m d n q n x Ga man 1 Oaxqa 1‘ n ofixmn d m Ao.ov .wwh +||3 Oaxwa Oaxaa oaxoa x x m m noaxmnoaxw MOHXN moaxc mOH n nod c n d 1 4 4 a 1 a T 1 onn d m oaxm d n oaxfi o «00 mm + a + H o : €Z8Q+amwe a Twas: N ~\: mvcmm cwws aw undumnoo m we newuqafixouan< 1 0.." .Asnvu# 112 f(x) d(x) 7“. - d(x Ex %%§%-dx a(x) \ ] a(x) f(x) d" K __J\/\—— L_.fJbfi_.. 9921 dx / J Fig. (6.10) where a(x), b(x), d(x), f(x) are the same as in Fig. (6.8). 113 the method is as follows a "w a' E a' o ' _ + 'l' + ,1 ... 6__a__._ ... . ‘0. ‘.a ' ”—UPJ b I b' b I b' Fig. (6.11s) Fig. (6.11b) Let us suppose we have two parallel plates each having con- stant density distributed charge (fig. 6.11) and placed ata4distance a on each side of the origin. As both the plates have the same charge, the field transverse to the imaginary axis will be zero, while the field parallel to the imaginary axis will be constant i.e. gg- is constant and 53- is zero. Now it is clear from theorem 6.1 that the network function constructed from this potential analog, will have magnitude of its value constant on -Jw axis. As before for the network function to be realizable, we will consider only one plate with double charge density and will accept the phase accompanied by this. Taking the trace of this plate as line of discontinuity L_ for F(s) and the value of the charge density as density function we have Cauchy type integral representation for F(s) i.e. N fit) ; {[Tfigi')‘ 4’ fl] dx + ID (s+u)q(x)dx 6.28 s+e)2 +2:2 F(s) :l 114 Approximation can be improved by changing the value of a in equation (6.10). Fig. (6.12) gives reSponse of F(s) on Jw axis, for different values of a. Again approximation can be improved by changing the length of the plates. It becomes better as the length of the plates extends over the band width. Fig. (6.13) shows the approximation of a constant for 0 s w 5 1 while the length of the plates is 1.8. The approximation of the constant is nearly perfect. Now instead of taking positive charge on both the plates, we can also take positive charge on the plate in the left half of the plane and negative charge on the plate in the right half of the plane. Now the field transverse to the imaginary axis is constant while the field along the imaginary axis is zero. Again we can trans- fer the plate carrying negative charge from the right half of the plane to the left and in doing so we must change the sign of the change to keep phase invariant and we will accept the value of the gain func- tion which accompanies it. Besides the constant charge distribution on the plates, we have also been able to find some other charge distributions, which result in network functions, approximating a constant. A few net- work functions with their realizations are given below l l F1(s) = é-I (8+ujgx 2 a = l \F1(Jw)‘ = 1.0 0 s w 5 1 n 0 (8+3!) + x 6 29 4 1 xgs-lu)dx 2 F (s) ='- a = 1 F (Jw) — .44 6.30 2 n i (sin!)2 +x2 ‘ 2 ‘ 115 A34: .wE All 3 0.4 00 w. No Q. m. Q. no NO #0 - q u - o q a d d 1“ H u .a sums uncommon wouauxouaaz Tb o. u 5 no; mason—a3. wouwsfixouaaa. any a. u a :3: 38¢qu pause—€8.33. ANV «magnum @9539 A3 x + 313 o N N 7. u a: sauna“ c c ~ I/ m 1X /* u- 11:3”! .228 a we 83.55334 “O o..— “WEI lfil6 33.8 .wE o.¢ o.m o.“ o.@ o.m o.¢ o.n w novuo zuuosuouusm an masonmoa Anv od nacho >o£omhnonu an oncoauua Amy Amvm he uncommom AHV x + 3+3 0 c Jammlsa omen» mucosv0uw vouanov ozu ukuuso uncommon o.N o.~ V ow. on. co. oh. ow. N.~ {Wm 117 33.8 .wE lbllllll 3 o.H m. w. w. o. m. d. m. N. H. o P — b p p p P b r p m. u o LuH3 OH popuo >m£ommnofio Amv w pompo Lupozhouusm ANV .éN. HmuwmucH xsoamo .3 H u a so; .omcoo mo coEwEonuaaas AHV 10¢. AM mu w ‘00. wow. AnH.ov .waa 7) lllllllll 1L18 o.H a. m. m. m. m. c. m. N. H. o r ON. : ca MN m k omcoamom poHHme HNV v cc. assesses eseeeaxoeee< AHV OIIIOOOIICCE 1 IIIIII IIIOOCOI IODIIIIln-nIIIDIII! o.H accomcoo a mo coHumEonuaad 119 3.81 3.81 3.96 3.96 3.98 3.98 3.98 3.98 3.99 3.99 Man—W W—W—MW ,____ -7891.— l i l— l f L F1(s) 27 1.02 1.01 1.01 .01 -.. (.009 T— 98 .993 .996T .997T .998 ...— Fig. (6.14a) 1.27 1.27 .636 .636 .424 .424 .318 .318 .254 .254 ¥ WWW—WW ------- i. i f f F2(S> 572‘: .637 .212 .128 092 .671 .058 4.71]; 7. .81I 10.1—— 14.1T 17.2 L ._ _1— Fig. (6.14b) .707 .707 .20 .200 .096 .096 .055 .055 .036 .036 F3(S)2.35£ .425 J:— .081 i— .033 .017 I .011 f .0076 I 12.3 L 30.3 ‘ 3 ‘ 90. 2 132.1]:1 \ Fig. (6.14e) 120 4'1 x2(s+u)dx n; (s-l-a)2 +~x .28 6.31 3 F3(s) 2 \F3(Jw)| } $706241)ch ‘F (Jw)\ O (3+U)2 +'x2 4 an» 4 F4(s) 1.45 6.32 1 2 5 F5(s) é-f (1+x%(s+agdx \F5(Jw)\ o (8+a) +'x 1.25 6.33 :1 In fig. (6.12 c) we have compared the results of approximation 1 of a constant by F(s) = &. [(S+U)gx 2 0(S+U) +-x n for a = l with Butterworth's order 8 and Chebyschev order 10. Inside the approximation band, result of our method is better than Chebyschev's and Butterworth's. But outside the approximation band Chebyschev's and Butterworth's methods have better reSponse. One great advantage of our method is that it results in tapered distributed networks. 6.31 Approximation in the Middle Band Let us suppose we want to approximate ‘F(Jw)‘ = B for W0 5 w s wl. From the theory for approximation of constant in the lower bands it is clear that, we can approximate a constant in the middle band by placing charged plates parallel to the band of desired frequency as shown in the Fig. (6.15). lw q(X)-’1 , """" W : O - €~oe£+j - ...... (we q(Xl/'{ j """ W Fig. (6.15) 121 3?: .3» L r 1‘ :oHuuaonuafiH no mean «I... A33 w: .6 .N «.3..— vuon 02.3! any 3 53303 no 8Hu¢BHuouaa< Khan 4.91 4.91 ——ML—— 122 4.4 4.4 W—j 4.2 4.2 4.18 4.18 W 4.14 4.14 W .04 F (s) -;: 1.91 £— 1.19 ‘L— 1.10 i— 1.06 E 1.05 «[— 4 '523 .84L .91L .94 .951 .96I: Fig. (6.14d) 4.24 4.24 3.97 3.97 3.98 3.98 3.99 3.99 3.99 3.99 WWWWW J IL 1— L f F (s) -L .7 1.03 1.01 «L— 1.01 1.01 I 5 '589 .97 .99I .99I; 4 .99L .99L Fig. (6.14e) .166 .083 . .0184 .0092 my}: and: _—/\/LJ73\.-- 171(3) .392 I: 1.28 ‘l: I 35.8I Fig. (6.17s) 123 .208 .104 .0254 .0127 .0028 .0014 Fig. (6.17b) 2.22 1.11 .272 .136 .032 .016 .0038 .0019 /\/\-1331—-—"-‘\PLJQHf\-— F3(s) f 1.49 l— .152 i— .018 336 I ' 3.32 26. 228.0 Fig. (6.17c) 3.3 1.65 .442 .221 .054 .027 .0062 .0031 V ML" ‘ 'VHZUL‘T W F403) 1:. 2.18 I .242 f .030 003 l— i— .235 2.06 16.5 141.2 I;1 I; Fig. (6. 17d) 124 Let the charge on each plate be q(x). Then the network function approximating a constant in the middle band is given by a.) = - I $333? 6.3. Response of F(s) for a = 2 q(x) = 2 and w = 6 wo = 5 is shown in the firg. (6.16). Following are some of the functions which approximate a constant in the middle band. 6 1 F1(s) = i. 2(s+u%dx 2 a = 2 IF1(Jw)I = 0.65 6.35 (3+0) +'X 2 2 F2(s) = 3- 31§+“%dx 2 a = 2 IF2(Jw)I = 0.64 6.36 (8+a) +-x 3 F (s) -.- 5‘. 2 x2(8+nr)dx 0’ = 2 IF (Jw)I g 99 6 37 3 11 (841102 +x2 3 2 4 F4(s) = fl' £l+x)§S+U)gx a = 2 6-33 n (s+u) +~x Network realizations for these functions are shown in fig. (6.17 a-d). 6.32 Approximation in High Band For approximation in high bands replace 8 by l/p in equa- tion (6.34) i.e. 6.39 an a} amnesia: n w o(l/p-m)2 f-xz Following are a few of the functions which approximate a con- stant in high band i.e. w 2 1 lJZS HwH.ov .wam I3 oooa com com ooh coo con cos com ecu ooH o i w. A Q0 m .. Q In...“ . m. o.a u+ Here\~v : a .Hlelqu. 1.3.— ue +. \H 3 one:— AanH eH 60H uaHMH—nnd 126 813. .262 3. 96 .252 3.98 .252 3.98 .252 3.98 .252 __—/v\_..4 |-——I——’\/¥H-————N\41__—JV\—|L——um- F1(s) 1.27 1. 27 1. 02 1.01 1.01 1.01 1.01 1.02 1. 01 1.01 1.0 1.01 Fig. (6.19a) 1.27 .79 .636 1.57 .424 2.36 .318 3.14 .254 3.94 F2(s) .637 .637 .212 .128 .092 .071 .058 . .212 .128 .092 .071 .058 Fig. (6.19b) .707 1.41 .20 5.0 .096 10.4 .055 18.2 .036 27. 8 AA—n“ ’V\—H————-/\/\-4b—1—/\/\-41-—I—/V\—-| 1——I— F3(s) .424 .424 .081 .033 .017 .011 .0076 .081 .033 .017 .O11.0076 Fig. (6.19c) 127 1.01 4.24 °236 3.97 .252 3.98 .252 3. 99 .251 3.99 .251 F4(s) 1.7 1.7 1.03 1.01 1. 01 1.01 1.03 1.01 1.0101 1.01 Fig. (6.19d) 128 1 1 F1(P) = 5'- ] 9’1““)? 2 01 = 1 6.40 n o (Up-hr) + x 1 4 gjl/pdu)dx 2 F (p) = -' 6.41 2 "I; (1/p+a)2+x2 4 1 xzil/mex 3 F3(p) = 51 2 2 6.42 o (l/p+u) + x 4 1 114.7(2) (10241ny 4 F4(p) = -j' 2 2 6.43 " o (1/p+o/) + x Network realizations for such functions are given in fig. (6.19). Fig. (6.18) gives ‘F(Jw)‘ for w 2 l. 6.4 Approximation of a Constant by RC Functions In the previous sections we approximated a constant in the desired frequency range, while the reSponse outside the desired frequency range fell sharply. Now if we do not care for the reSponse outside the frequency range and want only the reSponse in the desired frequency range, we can approximate a constant by the following method. 129 A34: .wZ +l'3 o.~ co. ow. oh. co. on. cc. on. ON. a. o a d d [1 3 11 1 A! T 1 Tea. VON. d“ m :on.‘nH o>h=o vsuwaoa Amy oppsu counewxouad< Aav 10¢. 1cm. on. foo. ucuunaoo a mo cofiuqaaxouma< 130 Now instead of placing plates as shown in the fig. (6.11), we can place them as in fig. (6.20). The basic principle of the method is the same as that of last section. Again by adjusting the position of the plates approximation can be improved. The correspond- ing network function is given by F(s) =J. m 6.44 8! 8+1: Fig. (6.21) shows the reSponse of magnitude of 5 . 2d F(s) =£ 3:1"? for 0 S w S 1. Following are some of the functions approximating a constant. 5 _ 4_ 2dx = 1 F1(s) - "21—8“ 01 .56 6.45 5 4 xdx _ 2 "i S+x (II-1.27 6.46 5 -x 3 9.4: e d" a = 3.34 x 10'3 6.47 n s+x 4 5 l+x dx n s +'x 4 szd 5 —I X a = 5.71 6.49 'n 4 s+x Network realizations for the above functions are shown in fig. (6.22). 131 .565 .23x10'2 .77x10'5 .248x10"7 AJN¢\» al\/\ j\fL, JNVA_____T ______ —d — _1— 3 .4... 5 -*- 8 F (s) _- .392 _ 95.4 - 28.7x10 ”— 84.8x10 — 28.4x10 1 F i" f Fig. (6.?2a) 1.26 .52x1o'2 .173x10"4 .55x10'6 /\/\ _______ 192(3) .174 :: 42.9 -: 12.9x103 £540.3x105 1.2x1o9 Fig. (6.22b) ,333x10"2 .133x10'4 .443x10'7 .141x1o'9 _M M M _/\/\_____ _______ F3(S) ‘7‘; 67.8 f: 1.66x104 A- 5.0x106 J- 1.57x109 i 4.9x1011 T Fig. (6.22c) 132 1.55 .63x10‘2 .21x10’4 .67x10' _/V\ _M N ;_M__ _____ F ‘“ " 1 5 103 32 9x105 1 0x109 4(5) q- 0142 -... 35.0 1". 0. X . o c Fig. (6.22d) -4 -6 5.70 .0231 .77x10 .24x10 _‘M 4M N.- ;M-— ...... 175(9) 1" .033 9.66 ~-2.83x10 9.0x10 T 2.83x10 Fig. (6.22c) 133 It can be observed that all the functions result in tapered distributed RC lines. 6.5 Approximation of Functions Varying by Frequency The general procedure developed in section 6.1 can be used for functions varying with frequency. Let us consider that we want to design a network for which ‘F(Jw)‘ = a +w4 for lw‘ S 1. By analytic continuation 4 F(s)=a+s Oswsl By theorem 6.1 S W(s) = as +.%_ 6.50 Now again selecting a unit circle as the contour, charge dis- tribution for complex potential W(s) is given by l . (1(9)’ T-T- [0! Sin 0 + Sin 51>] taking the right half portion of the contour to the left hand side and doubling the charge on the half contour Q(§) =-%'[a Cos Q + C08 50] n/2 s 0 s 3n/2 and the correSponding network function is given by F(s) =é_ "E, Zga Cos Q +-Cos SQ)(s +-Cos §)d§ 6.51 82 +-23 Cos Q + 1 Response of F(s) on the Jw axis for a = 1 and 0 s w s 1 is shown in fig. (6.23). It is interesting to note that for 0 s w s l the approximated value of ‘F(JW)‘ and the exact value 1314 omcoamwm vmufimoa any meson-0m vauuswxouaa< adv ~ Am~.ov .wam H + o moo mm + mm seas moo + sussa 600 + e meow m 3 M o a 3.+ a I .Aawvm. O C N\t wo nodudawxouna< u~.~ rn.H r HsN .~.~ .AaaVN, 1L35 OJ >O\ w h 0 oncoamox wouwmoo Amy omcoawmm vmuqawxounad AHV «~31. s Hmawo N Asm.sc .wee 3+ H 1:3,:— N 30... as 681.2 88 .w 1 3m H+oooom~+u w s: No acuunauuouaad ]J36 Am~.sv .wes 3.7080 3+ 3 o {1. N % u Asap owns soo+ -VAo sou + on 6003 ~\: ”Saul—582:; mo amend may commune magaaom oJ w..— o.~ ~.N 137 are very close to each other. Figs. (6.24) and (6.25) show the reSponse of F1(s) which approximates ‘F(Jw)‘ = 1 +-w2 for O s w s 1. Again, the approximated and the exact curve are very close for 0 s w s 1. Network realization for F(s) in eqhation (6.51) is like the one as in fig. (6.10). Realization of equation (6.51) by Darlington's method and Youla's method can be easily achieved. 6.51 Approximation of Functions Varyipg Linearly as w E.S. Kuh (26) found a method for the approximation of functions whose magnitude varies as Kw for 0 S‘w s 1. In this section we will use Cauchy type of integrals for the approximation of such functions. It is interesting to note that our method is much simpler than that of Kuh's and results in a tapered distributed LC line. Consider ‘F(Jw)| = Kw 0 s w s 1. Network function approximating such reSponse can be obtained by placing two charged conductors as shown in fig. (6.27). By adjusting the distance A “a .1 - . - c .-+—._ 0 o +ra' b. Fig. (6.27))' between the conductors, field between 0 S w s 1 can be made to vary as kw. Thus the network function obtained from this potential analog has the desired response and is given by 1L38 .wsa Awm.sv IbIIII 7) e; s. s. s. s. s. p p p p p . spasm vmuwmoa Amy m>u=o wouwewxouaa< hay 3 no“: zuuwocwa maghud> couuocsh mo coHuqaaxoumad 1 ON. 1.00. u.oe. HMDJ‘ .uom. 1. 84 o¢.H 139 .125 .002 .27x10' .345x10'7 __ 3 5 7 F1(s) .392 _ 24.0 1.8x10 1.42x10 5} 1.13x10 Fig. (6.29a) - — -6 .279 .45x10 .60x10 .76x10 Jam 42m 4320 Jim—— -------- 2 4 , 6 F2(s) - .174 10.8 8.18x10 6.4x10 e-5.12\10 r ’ T Fig. (6.29b) 1.25 .020 .268x10‘ .341x10"6 JUL am an ----- _ ... 2 4 6 F3(s) _.0386 2.45 1.84x10 1.44x10 1.15x10 Fig. (6.29c) 140 b F(s) = j ALL-82 x d}; 6.52 a S + X 4 5 25d Fig. (6.28) shows the reSponse of F (s) ='— x 1 71 82-11(2 for 0 s w s 1. Following are some of the functions whose reSponse varies as kw. A 4 5 23dx 1 F (s) = -'j k = .127 for 0 S w s l 6.53 1 rr 22 48+): 4 5 sxdx 2 F (s) - - k = .284 6.54 2 n 2 2 8 +x 3 F(s)=5’-55’5-‘12E k-127 655 3 n 4 82 2 _ . - Network realizations for the above functions are shown in fig. (6.29). 6.6 Approximation of Functions Varying as w-a (0 < a < 1) Very often network theorists (27) have tried to design net- works having ‘F(Jw)| = w'°’ 0 s w s 1 6.56 06 10 J1; 2%?- L95. , fig. (6.30) gives the reSponse For a = 10 i.e. F(s) = s 4-x of F(s) for 0 s w s 1. Network realizations for F(s) are given in fig. (6.31).and (6.32). 6.61 Approximation of Functions Varying as wa Let us consider a function such that ‘F(Jw)‘=wa 0 Amy 1 ON. xuoauoc ecu macaw moauuwwmom mo chMuuwuu> adv roe. Foo. AV vow. T 0 AV 0 u vo~.H 3 o no use Rowan m- u H H 4 oc.~ sprain u; aouanrondvo U u; abusnslsau 146 Cauchy integral representation for F(s) is given by _ 1 m S- xdx F(S) _11j‘s-l-x o where B = (1 - a) for a = 5 1 ° -% F(S) =- i."___d_>£ n £S+x 6.58 The reSponse of ‘F(Jw)} for O s w s l for the function 10 -% - 1.22 S x dx I ' ‘ F(s) _._1;_.£ S +~x is shown in fig. (6.33) and the network realizations for F(s) is given in fig. (6.34) and (6.35). Fig. (6.36)gives an idea about the taper in the distributed line shown in fig. (6.34). CHAPTER VII CONCLUSIONS This thesis has looked into the possibility of building a comprehensive theory for distributed parameter networks. Re- duced state descriptions for distributed parameters, developed in this thesis, have resulted in Cauchy type of integral representa- tion for network functions. A very broad class of network functions, which cannot be handled by the classical theory for distributed parameters, can be easily represented by Cauchy type of Integrals (see Chapter I for summary of the work done in the thesis). Cauchy type of integral representations have resulted in Foster and Cauer type of networks. It is clear from the results of Chapter V that most of the previously established results for lumped networks can be generalized for Cauchy type of integral representations. This gives weight to the idea that a comprehensive theory for dis- tributed networks can be built for such representations. One of the most interesting usesof Cauchy type of Integrals lies in approximation of network functions. The present work has produced an analog between potential theory and Cauchy type of Integrals. This forms the basis for such approximations. Approxima- tion of ideal filter characteristics by Cauchy type of Integrals compares favourably with the results obtained by Butterworth and Chebyschev approximations. Approximation of such functions by 147 148 Cauchy type of Integrals has an added advantage in that the resulting network is a tapered distributed line. In the thesis Cauer type of networks for ”/29 + cos 6)f(6)d6 2 have not been given. It will be interesting 0 (s + cos 6) + sin 5 to know if they exist for such functions. Possible Extensions of the Work: 1. From the reduced state description in Chapter II, the notion of complexity of networks can be expressed in terms of ranks of controllability and observability matrices. In addition, bounds on the complexity of networks connected in i) Series ii) Parallel iii) Cascade iv) Feedback can be found. 2. In the lumped networks, generalizations of the network functions for multivariables have been obtained. It seems possible that generalization of Cauchy type of integrals for multivariables can be obtained. 3. Cauchy type of integral representations for active distributed networks. 4. Correlation between the classical distributed parameter theory and Cauchy type of integral representations for distributed networks. 5. Given a filter characteristics it is possible to develop a computer programme which obtains a Cauchy type of integral approximating the characteristic and find the network. 6. The development of concept of reduced state descriptions in distributed parameter theory and the derivation of an algorithm for obtaining such reduced state descriptions. W°'muflu:nvfi~“‘n' 149 Cauchy type integral representations can be applied to time domain network synthesis. It may be possible to find distributed networks equivalent to lumped networks. APPENDICES APEND IX 1 PROOF OF LEMMA (1.1) State equations for distributed n-Pbrt networks are given as in equation (2.3). To derive these equations we have the follow- ing assumptions. 1) The input Space of the system consists of all real regular dis- tributions based on locally bounded and measurable vector testing functions U(t) such that for some finite T dependent upon U U(t) = O for all t < T. 2) §=s+§ where H is a 2 X 2 matrix and each of its element is of the k form h = z +- z dk6(k). Note that z(t) = 0 t < 0. k=0 3) z(t).exp(-ct) is absolutely integrable over (-m,m) for some finite c and a: 4) 2(8) (defined as I z(t)exp(-st)dt for Re(s) > c by assumption -m 3) is sumable along the path c - ioo to c + ion i.e. co \J z(c +-iw)‘ < w . -co Lemma 1.1: (2 * U)t = 5%: .f 1z(s)[exp(st) * U(t)]ds =£ Proof: (2 * U)t = (UT * zT)Tt where z is a symmetric matrix and UT is the transpose of U 150 151 C+ioo T = CUT“) * 2.11—if z(s) .Exp(st) .dsJ C-iao t on T a 21+1j' UT(t) J[z(c + iw)mcp(c + iw) (t - T)dW] (11' T -oo t 99 T a Eta-951 £[UT(T) jz(c + iw).exp((-c'r 4' 1W) (t " T))dWJ ‘11 \ Since each element of 90-) and exp(-crr) are locally integrable, and each element of z(c + iw) is absolutely integrable, Flt—arr J‘ |UT(T).z(c + iw)exp(-cq- + iw(t-'r))‘dw.d'r t T on a i ‘U (ir)exp(-c'r)‘ I ‘z(c + iw)‘dw d'r t s i |UT<¢).exp<-cs)|d¢.1_i 5 M1 Thus we can change the order of integration to obtain a t T (2 * U)t - Eggs-Elf .[uT(vr)exp(-cir + w(t-ir)).dvr z(c + iw)dw] C+im 1 = 571—1 c-iaz(3)exP(3t) * 1.10:) ".1... APPENDIX 2 STATE DESCRIPTION FOR FUNCTIONS HAVING CONJUGATE ARCS OF DISCONTINUITY Let c be a closed contour within and on which a function f is analytic except for a finite number of area as shown in fig. (1). Fig. (1) Now according to Cauchy-Goursat theorem extended to such regions I f(s)ds - I f(s)ds - j f(s)ds .... f f(s)ds = 0 C C1 C2 Cn f f(s)ds = f f(s)ds +if f(s)ds .... +-f f(s)ds C C1 C2 en New I f(s)ds = I f(s)ds +-f f(s)ds c1 c: c; let f(s) = f+(t) on c:. and f(8) = f-(t) on c; then we have 152 153 j‘ f(s)ds =J‘ f+(t)dt +j f-(t)dt C1 c+I - C 1 1 defining f(t) - f+ = f(t) jf(s)ds = {4“ f(t)dt +£ f2(t)dt + {fn(t)dt] (1) C 1 2 n Now let us consider a function F(s) which has conjugate arcs of discontinuity in the left half plane. The state description for driving point impedance having conjugate arcs of discontinuity is given by d _ . a: V(Sst) - SMSJ) + 1(t) 1 V(t) = I-l 2:3-F(S)¢(s’t)ds where {fl contour is shown in the fig. (3.1). Now we want to evaluate the Laplace inversion contour. f 1F(s)¢(s,t)ds = I ’ B F(s)((s,t)ds +j F(s)¢(s,t)ds + j F(s)¢(s,t)ds i C C l 1 2 +]‘ F(s)¢(s,t)ds (2) C Now we want to evaluate f f(s)¢(s,t)ds C t where ¢(8,t) = I es(t-T)i(T)dT -w assuming i(T) = 0 for t < —T and 1(7) s M for all t > T 2 -T 154 s(t-T) t If(s)¢(s,t)ds = If(s)(§ e i(T)dT)ds C c -1- = f (f£(s)e“ (t flds)i(T)dT 3n/2 Now ‘jf(s)es (t T)ds| s R Sup \f(s)| I eR(t'T)C°SQde (3) ‘s‘=R n/2 we can evaluate in/Z eR(t-T)Cos ede = 2 E eR(t-T)Cos ede n/2 n/Z n/2 . = 2 f e'R(t'T)Sln Qdo where e = (Q +'n/2) On using the fact that Sin 6 2 33* for 0 s Q s n/2 “/2 . 11/2 - 34-29- (t-T) 2 f e'R(t'T>31“ Tdé s 2 i e " di (4) O 11 _ -R(t'T) = R(t-T) T1 e 1 hence ‘If(s)e S‘t T>s31 5 Sup |f| [1 - e'R(t'T)] ls|=R T £f(s)¢(s, t)ds s n Sup |£(s)\ Tr -111-[1 - e-R(t-T)]dT (5) ‘s\=R t-T for higher values of R 3 =R t-T =11 |Sup \f(s)\[T mdrr Under the conditions i(T) satisfies H condition as given in t i d Chapter III f -éE%-I- exists hence -T 155 f f(s)¢(s,t)ds s n Sup ‘f(8)"M1 c |S‘=R \f(s)\ s o as \sl 4 m hence the integral I f(s)¢(s,t)ds = O c Also I f(s)¢(s,t)ds = - I f(s)¢(s,t)ds C1 C2 from equation (2) we have I f(s)¢(s,t)ds = g f(s)¢(s,t)ds {fl 1 Using equation (1) g f(s)¢-¢w,t> + f<"‘s2(p'>>¢(s'2"

,tjdp 2 + [ [f(sn

)w,c) + f(sn(pm,c)] dp n where sY(p) is the equation of yth arc. Hence n = .. 2 f — , V(t) v=1 [f(sY(p) (SY(p))] [Mg/(p) a] (6) W(sy(p):t) BIB LIOGRAPHY 10. BIBLIOGRAPHY R.W. Wyndrum, Jr. "The exact synthesis of distributed RC networks" Lab. for Electroscience Research, New York University, New York, N.Y. Tech. Rept. 400-76, May 1963. R.P.O. Shea "Synthesis using distributed RC networks", IEEE Trans. Circuit Theory, Vol. CT-lZ, pp. 546-554, December 1965. T.N. Rao, C.V. Shaffer, and RJW. Newcomb "Realizability conditions for distributed RC networks" Stanford Electronics Labs., Stanford University, Stanford, Calif., Tech. Rep. 6558-7, May 1965. K.W. Heizer "Distributed RC networks with rational transfer functions" IRE Trans. Circuit Theory, Vol. CT-9, pp. 356-362, December 1962. D.C. Youla "The synthesis of networks containing lumped and distributed elements - Part I" 1966. Proc. Poly- technic Symp. on Generalized Networks. T.N. Rao and R.W. Newcomb "Synthesis of lumped distributed RC n-ports" IEEE Trans. Circuit Theory, Vol. CT-13, pp. 458-459 (CorreSpondence), December 1966. E.N. Protonotarius and 0. Wing "Theory of nonuniform RC lines. Part 1: Analytic prOperties and realizability conditions in the frequency domain" IEEE Trans. Circuit Theory, Vol. CT-14, pp. 2-12, March 1967. G.T. Daryanani and J.A. Resh "Foster-distributed-lumped network synthesis" IEEE Trans. Circuit Theory, Vol. CT-16, pp. 429-434, November 1969. A.H. Zemanian "An N-port realizability theory based on the theory of distributions" IEEE Trans. Circuit Theory, Vol. CT-9, pp. 265-274, June 1963. J.A. Resh and 1.0. Goknar "Derivation of Canonical state equations for a class of distributed systems" Sixth Allerton Conference on Circuit and System Theory, 1968. 156 n.” I (‘1 W.- O " "V‘M" ll. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 157 Katsuhiko Ogata "State Space Analysis of Control Systems" Prentice-Hall Series, 1967. R.E. Kalman "Irreducible realizations and the degree of a rational matrix" J. S.I.AWM., Vol. 13, No. 2, June 1965. D.C. Luenberger "Canonical forms for linear multivariable systems" IEEE Trans. Control Theory, Vol. 12, 1967. N.I. Muskhelishvili "Singular Integral Equations" (Translated from the Russian Edition by J.R.M. Radok). Groningen, Holland: P. Noordhoff N.V., 1953. H.L. Royden "Real Analysis" Macmillan, 1968. O. Perron "Die Lehre Von Den Kettenbrfichen. Band II, 8.6. Teubner Verlagsgesellschaft, Stuttgart, 1957. H.S. Wall "Note on the expansion of a power series into a continued fraction" Bull. Amer. Math. Soc., Vol. 51 (1945) , pp. 97 -1050 Van Valkenburg "Introduction to Modern Network Synthesis" John Wiley & Sons, New York, 1967. F. Reza "Some ladder networks representing certain trans- cedental numbers" IRE Circuit Theory Proceedings, Vol. 39, pp. 1693, 1951. F.F. Kuo "The use of convergents in the synthesis of Mth root‘ladder networks" IEEE Trans. Circuit Theory, Vol. CT‘IZ, pp. 442, 19650 S. Karni "On ladder networks and continued fractions" Proc. IRE, Vol. 50, pp. 334, 1962. S. Karni "More on ladder networks synthesis of irrational numbers" IEEE Trans. Circuit Theory, Vol. CT-14, pp. 343, 1967. F.M. Reza "0n the Schlicht behaviour of certain impedence functions" IRE Transactions on Circuit Theory, Vol. CT-9, pp. 231, 1962. S.K. Mitra "On the generation of RC impedence functions" IEEE Trans., Vol. CT-17, pp. 257-258, May 1970. Darlington, S. "The potential analogue method of network synthesis" Bell System Technical Journal, Vol. 30, pp. 315-165, April 1951. 26. 27. 28. 29. 30. 158 E.S. Kuh "Potential analog network synthesis for arbitrary loss functions" J. Appl. Phys., Vol. 24, No. 7, p. 897, July 1953. R.M. Lerner "The design of a constant-angle or power law magnitude impedence" IEEE Trans. Circuit Theory, Vol. CT’l-O, pp. 98-107, MarCh 19630 J.A. Resh "On Canonical state equations for distributed systems" Conference Record, Eleventh Midwest Symposium on Circuit Theory, University of Notre Dame, Notre Dame, Indiana, May 1968. R.E. Kalman "Mathematical description of linear dynamical systems" J. S.I.A.M. Control, 1 (1963), pp. 152-192. G.T. Daryanani "Foster distributed-lumped network synthesis" Ph.D. Thesis, Michigan State University, 1968. ”9 “5 "1 ms E II. V ”o M ”'3 U "o E m3 A mg "2 “1 m3