STATE MODEL APPROACH TO THE SYNTHESIS OF LC NETWORKS AND THE CA-NONIC LC NETWORK TRANSFORMATIONS Thesis for the Degree of Ph. D. MICHIGAN STATE UNIVERSITY 'Radha Krishna Rao Yarlagadda 1964 -w LIBRARY Michigan Sun University This is to certify that the thesis entitled STATE MODEL APPROACH TO THE SYNTHESIS OF LC NETWORKS AND THE CANONIC LC NETWORK TRANSFORMATIONS presented by Radha Krishna Rao Yarlagadda has been accepted towards fulfillment of the requirements for Ph. D. degree in Electrical Engineer’ / Major prtfiessor .Date August 12, 1964 0-169 ROOM USE ONLY w‘ ' "3.,‘y‘ kin; U ru-OU ROOM USE 02355.7! ABSTRACT STATE MODEL APPROACH TO THE SYNTHESIS OF LC NETWORKS AND THE CANONIC LC NETWORK TRANSFORMATIONS by Radha Krishna Rao Yarlagadda In this thesis 1) a state model approach to the synthesis of LC networks and Z) a realization procedure for canonic LC networks using the equivalent network transformations are considered. In Chapter II analysis and formulation of LC networks from the state model point of view is described. In Chapter III a new definition of canonic LC networks is given. This definition is applicable to general n-port LC networks. A general realization procedure of LC networks from the state model equations is described. A state model approach to the realization of reactance or susceptance matrices with the ideal transformers is given. The state model approach is also utilized in the realization of reactance functions without the use of ideal transformers. This procedure is extended to the reactance matrices of order two having dominant resi- due matrices; however, it is also applicable to n-port LC networks of the same class. In Chapter IV a technique for realizing canonic networks is described. This is accomplished by establishing a parametric matrix relation, which relates the parameters of two canonic networks by an orthogonal matrix. Some of the known canonic forms are derived by the application of this technique. STATE MODEL APPROACH TO THE SYNTHESIS OF LC NETWORKS AND THE CANONIC LC NETWORK TRANSFORMATIONS BY Radha Krishna Rao Yarlagadda A THESIS Submitted to Michigan State University in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering 1964 ACKNOWL ED GEMENTS The author is indebted to his thesis advisor, Dr. Yilmaz Tokad, for his guidance and help in the preparation of this thesis. Additional gratitude must be given to Dr. Tokad for his valuable aid in the development of the author's research capabilities. The author wishes to thank his major professor, Dr. Herman E. Koenig, for editing this thesis, and for the help and guidance of the author's graduate program. Acknowledgements are due to Professor J. S. Frame for his valuable discussions with the author. The author wishes to thank the National Science Foundation under whose Sponsorship this research was done. The author wishes to thank his colleagues, friends and relatives for their constant encouragement. Finally, the author would like to eXpress his humble gratitude to his parents for their understanding and patience. ii Chapter I II III INTRODUCTION . . . . , . . TABLE OF CONTENTS ANALYSIS OF LC NETWORKS . . . 2.1 2.2 2.3 2.4 2.5 2.6 SYNTHESIS OF LC NETWORKS . 3.4 3.5 3.6 General........ The State Model . . . . Explicit Expressions for the State Equations of an n-Port LC Network Short-Circuit and Open-Circuit Parameter Matrices from the State Model . . . . . . . . Certain Necessary Conditions for Transformerl-ess Realization An Important Special Case General.......... Canonic Forms of LC Networks Realization of a State Model of an LC n-port without Ideal Transformers . . . Derivation of a State Model from the s-Domain Model - - Realization of Short-Circuit and Open- Circuit Parameters of an n-Port LC Network Using Ideal Transformers . . . . Realization of LC Networks without Ideal Transformers . . . iii 15 23 24 31 31 31 .37 43 -50 58 Chapter IV VI Equivalent Networks ............... 4. 1 General .................. . 4. 2 Equivalent Networks - Transformation Matrix 4. 3 Canonic Transformation for One -Port CanonicNetworks. 4. 4 Realization of One-Port Canonic Networks by the Canonic Transformation. . . . . . . 4. 5 A Realization of Canonic Networks by Tridiagonalization . . . . . . . . . . . . . 4.6 Equivalent Networks - A State Model Approach. ........ EXAMPLES ........... 5.1 General.... ............ 5.2 Example 1 ....... . . . . ...... 5.3 Example 2 ............ . . . . . 5.4 Example 3 . . . ....... . ..... . CONCLUSION . ....... iv Page 88 88 88 91 105 126 131 140 140 140 148 152 163 CHAPTER I INTRODUCTION In classical network theory, the synthesis problem is stated in terms of the impedance or admittance matrices in the s-domain, and its solution is realized by mathematical operations on these matrices which can be interpreted as the interconnection of certain sub-networks. These mathematical operations, in general, neces~ sitate the use of ideal transformers, even though the given matrix is known to be realizable as a network contains only two terminal R, L and C elements. Recently it is recognized by many investigators that the topological approach to the synthesis problem might offer new in— sight. Indeed, the problem of synthesizing n-terminal R networks characterized by the impedance or admittance matrices of order (n-l) is completely solved [PA 1]. Although extension of R-network synthesis to some very Special class of multi-terminal RLC networks is also known [CE 4], the solution of the general problem appears at best, to be very difficult. A natural approach to the RLC network synthesis is by means of state models since, in general, the state model of the network provides more direct information about the network topology than does the network matrices. Recently, some works have been -1- -2- initiated in this direction [RA 1], [DE 2], [KA 1]. Since the properties of the relations between the state models and the tOpology of the corresponding network are not explored fully, in this thesis a natural decision is made and only the synthesis problem of LC networks is considered. In recent literature, new canonic networks have been proposed [RE 1, LE 1, YA 1]. The first set of canonic networks were given by Foster in 1924 [F0 1]; later in 1930 Cauer gave another set of canonic networks. In 1955 Reza [RE 1] stated that there are many more canonic forms which cannot be obtained even if a mixture of Foster and Cauer procedures is used. But he gave neither the properties of such new canonic forms nor the realization procedures. In 1963, Lee [LE 1] gave a lattice canonic form and Yarlagadda and Tokad proposed a different lattice canonic form [YA 1]. None of these known canonic forms are obtained as the result of a general theory. Rather, each is derived by a procedure unique to its form. This thesis presents for the first time two general procedures for deriving one canonic LC network from another. The first proce- dure is based on the equivalent network transformation introduced by Howitt [HO 1], [HO 2] and extended by Cauer, Guillemin [GU l], Schoeffler [SC 1] and others. None of these methods, however, consider major changes in the topology of the network, i.e. , equiva- lent networks are always assumed to have the same number of meshes -3- or node -pairs or one equivalent network is generated from the other by rather obvious network modifications. The second method is based on similarity transformations on the state models of the networks rather than on the mesh or node equations. In Chapter II analysis and formulation of LC networks from the state model point of view is described along with certain similarity transformations. Such an analysis is the backbone of the synthesis procedures considered in the successive Chapters. In Chapter III, a new definition of canonic LC network is given which applies to general n-port networks. A general realiza- tion procedure for the state model equations as an LC network is described in Section 3. 5 and certain conditions for realizability are stated. Although methods exist [KA 1], [G1 1] for derivating the state model from the impedance or admittance matrix, a slightly different approach is given in Section 3. 4. The problem of realizing impedance or admittance matrices with the ideal transformers is solved by state models. Although Kalman [KA 3] in a recent talk has suggested that this result is possible, but so far no published material is available. In Section 3. 6 a realization procedure for the reactance functions is given in terms of state models and the proce- dure extended to a class of 2-port LC networks. -4- In Chapter IV a general procedure for transforming a one- port canonic LC network to another is given. The relation between the element values of one canonic network to another is established and this relation is called parametric matrix relation. For some of the canonic networks given in Table 4. 3. 1, the nonlinear equations obtained from parametric matrix relation are solved analytically. But, in general, an analytical solution for an arbitrary canonic form may not be possible, although computer solution might conceivably give a solution in a particular application. Such numer- ical solutions are not considered in this thesis. Chapter V presents examples, illustrating the procedures described in the thesis. CHAPTER II ANALYSIS OF LC NETWORKS 2.1 General In this chapter the analysis of n-port LC networks in terms of state model formulation is considered. This analysis establishes the conditions on the short circuit and the Open circuit parameters of the n-port LC network which are necessary for its realizability without the ideal transformers. 2. 2 The State Model In classical network theory the analysis, in general, is based on the s-domain 100p or node equations and natural frequencies are defined as Definition 2. 2. l: The finite zeros of the determinantal equation of the loop or the node matrix of an RLC network are called the natural frequencies, and the number of these natural frequencies is called the "order of complexity" of the network. The instanteous behavior of the network can also be described by a set of first order differential equations and a set of algebraic equations of the form d ix = AX+BY+C—-Y dt dt (2.2.1) _ d Y = PX+QY+R——Y dt -6- where X is called the state vector and consists of branch capacitor voltages and chord inductor currents, and Y is a vector which contains the Specified voltages and currents; Y contains the complementary variables of those in Y. The coefficient matrix A is real square and called the operator matrix. All other matrices B, C, P, and Q are real and, in general, they are rectangular. The set of equations in (2. 2. 1) describes the behavior of the network completely and it is called the state model of the network. An alternate definition for the natural frequencies of the network as given in terms of the state model is Definition 2. Z. 2: Natural frequencies: The natural frequencies of an RLC network is defined as the eigenvalues (not necessarily distinct) of the Operator matrix A in the state model (2. 2. 1). In the literature, the order of complexity has been discussed by several authors. Some define it in terms of the topological prOperties of the network [RE 1, GU 1, SE 1 and others], and the others [BR 1, BA 1] define it in terms of state variables. However, the two definitions are equivalent, i.e. , the order of complexity is equal to the order of the state vector. 2. 3 Explicit Expressions for the State Equations of an n--port LC Network A procedure for determining a state vector of minimum order has been described by several authors [BR 1. BA 1. WI 1. KO 1» and others]. -7- The procedure given originally by Wirth is based on a tree of the system graph defined as follows: Let G be the graph of such network. 1. Consider the subgraph GO of G which contains all the elements correSponding to the voltage drivers. Let T0 be a tree (or forest) in G For consistent networks it is necessary that 0° T0 = GO. 2. Let G1 be the subgraph of G which contains GO and all elements corresponding to capacitors. Select a tree T1 in G1 such that T1 contains To. 3. Let G2 be the subgraph of G which contains G1 and all elements correSponding to inductors. Select a tree in G2 such that T2 contains T1. Then T2 is a tree in G and all current drivers are included in its coutree. (For consistent networks current drivers cannot form a cut-set). 4. The voltages of the branch capacitors and the currents of the chord inductors constitute the variables of the state vector. For an LC network the circuit equations correSponding to a tree selected according to the above rules described has the form - _ V0 F l " V B11 B12 0 | U 0 0 be ' bez B21 1322 1323, 0 U o = 0 (2.3.1) I V I CC B31 B32 B33| 0 O U V _. _ cl Vl where Bij are the submatrices of the fundamental circuit equations, U, the unit matrix and the variables are classified as follows: V0 — Voltage drivers vbc - Branch capacitor ”Rages Vb! ' Branch inductor voltages Vcc - Chord capacitor voltages V1 ' Voltages of the current drivers Similarly, the cut-set equations for the same network corres- ponding to the same tree are r _ _ .. Io U 0 0 "BM “1321 'B31 I be T T T .. - - = 0 2.3.2 0 U o 1312 B22 B32 Ibi ( ) I CC 0 0 U o 4323 -B33 I _ __1 cf I1 -9- Let the terminal equations of the branch capacitors and chord inductors be written as L I bc c1 — [1b V _ “J C (2.3.3) and let the terminal equations of the branch inductors and chord capacitors be written as F 1b! V CC Then the state equations can be obtained as _ C b T o + BIZCCBIZ 0 LC + BZ3LbB23 _ T _ o BlZCCBll d at- T 0 B231‘191333 L. _ — 0 Vbc Icf 'Bzz __ L — T q o 1332 + -1321 O (2.3.4) bc Icf (2.3.5) From (2. 3.1) and (2. 3. 2) the currents of the voltage sources and the voltages of the current sources are -10- _ - I CC T T T = + I0 [311 B21 B31] c1 I L 1- _ (2.3.6) V0 V1: ' [B31 B32 B33] Vbc beJ '1 3:: >:< >.~ >1: - The terminal variables V , V , I II are related to the variables 0 l 0' associated with the drivers by >1: >'< I0 = '10 1 V1 ‘ V1 (2.3.7) _ >',< _ -I>< Vo ' Vo I I1 ‘ 1 also (1 d = — = - — + B V Icc C:c dt Vcc Cc dt (B11Vo 12 bc) Substituting this eXpression of Icc and (2. 3. 7) into (2. 3. 6) we have >1: T d >:< T d - T1 +BT 1* I0 = 'Io;B11Cc BllETVo+Bllcc BIZdt Vbc BZI c! 31 1 V* V +B V ) 1 ' '(B31V0 + B32 bc 33 b1 * dBLBTI-iBLBTN ' B31Vo+B32Vbc+EiE 33 ‘b 23 c1 dt 33 b 331 -11- The state model obtained above reduces to the form given in (2. 2. 1) upon taking the inversion of the coefficient matrix on the left of (2. 3. 5) and substituting the resulting eXpressions for the derivatives of the state variables into (2. 3. 8). The required inverse always exists I as established by the following theorem. Theorem 2.3.1: Let C , Cc' b L b' LC be diagonal matrices with positive entries and B be arbitrary matrices (in our case they 12' B23 B and T +B 12 are unimodular matrices), then the matrices C 12 b Cc BT L 23 23 b are posuive definite. L+B c Proof: After Tokad and Kesavan [TO 1] The final eXplicit form of the state model for any n-port LC network without ideal transformers is r ‘ r v o (c be b :1. dt _1 I (L + B L ET) B at ' c 23 b 23 _ T “'1 T (Cb + B12CCB12) BlZCcBll o -(L +B _ C o - + + = -1 -(L + B L ET) B c 23 b 23 21 ' T + BIZCCB 22 0 -l T 23LbB23) T B12CcB12) 12) -1 T Bzz BLBT 23b33] -1 T B32 -12- T T '1 T >2 d s T - - B c B -—V I0 ’ {BllccBll BllccB12(Cb+B12CcB12) 12 c 11} dt 0 —1 T T 'r T - 1 + {BllccB12(cb+BlZCcBlz) B22 B21} c1 T T '1 Vs ET 1* - + BllCcB12(Lc +1323L’sz3) 1321 o 31 1 * T T T ’1 T d >=< — B L B — 1 V1 ’ " [{B33LbB33 +B3szBz3(Lc +BZ3LbB23) 23 b 33 dt 1 -1 T T + {B32 ‘ B33LbB23(Lc + B231131323) B22} Vbc -1 T >3 12) B3211 +1331Vo (2.3.9) 3:: " B33LbB2r3(Cb + BirzccB Other investigators (e. g. , [BR 1, 2]) have given alternate procedures for developing the general form of the state model for LC networks with restricted voltage and current drivers (voltage driver in series with an element and current driver in parallel with an element). For this restricted class of networks the procedure of Bryant and that given here provides a method for investigating the network without drivers. When all voltage drivers are replaced by short circuits and all current drivers are replaced by open circuits then the natural frequencies of the resulting network are invariant. -13- However, in the procedure given here there is no restriction on the location of the drivers in a network and, in general, replacing voltage drivers by short circuits and the current drivers by open cir- cuits modifies the network and hence the natural frequencies. However, by the following procedure the general LC network with arbitrary drivers, can be transformed into an LC network without the drivers, while holding the number of natural frequencies invariant. Procedure: 1. Substitute a capacitor for those voltage drivers which form circuits with capacitors in the network and a short circuit for all other voltage drivers. 2. Substitute an inductor for those current drivers which form cut-sets with inductors and open circuit for all others. In this procedure, the size of the capacitors (inductors) substi- tuted for voltage drivers (current drivers) has no effect on the number of natural frequencies. In the case of an arbitrary n-port network, it is always possible to select the type of excitation at the ports such that the number of natural frequencies of the network will be invariant, i. e. , some of the excitations can be taken as voltage drivers, such that they do not form circuits with other capacitors and the remaining excitations can be taken as current drivers, such that they do not form cut-sets with the other inductors. -14- Theorem 2. 3. 2: For an LC network the natural frequencies are located on the imaginary axis, i. e. , the operator matrix has eigenvalues on the imaginary axis. Proof: The operator matrix in (2. 3. 9) can be written as " T -1 W Jr.“ (Cb+B12CcB12) 0 0 B22 X1 A = = = -1 Bl PB1 T -1 o (LC+BZ3LbBZ3)-J -B22 OJ Y1-J (2.3.10) where c +BT c B - x b 12 c 12 ’ 1 Lc +13231‘131323 " Y1 and [- T— 0 B22 : B], _'Bzz U Since B1 is a skew symmetric matrix eigenvalues of B are all on the imaginary axis (See, e. g. , [PE 1], p. 196)? To prove that A also has eigenvalues on the imaginary axis, consider the characteristic equation of A: -15- ll 0 IA — ml (2.3.11) ll 0 IPB1->.UI X From Theorem 2. 3.1, it is known that [ 1 Y1] is a symmetric and positive definite matrix. Therefore, the positive definite matrix, .1 [Xi/2 1/2 I. Y .1 exists, and is unique [PE 1, p. 203]. Hence (2. 3.11) can be written as l 2 1 2 -l 2 det(PB1->\U)=]P/[]P/ Bl-XP /U[ l 2 l 2 1 2 -1 2 =]P/(IP/B1P/-XU([P/I l 2 l 2 = lp / BlP / -xul . 1/2 l/Z. . . . . Where, Since P BlP is a skew symmetrlc matrix, all its eigen- values lie on the imaginary axis. This implies, A = PB1 has eigenvalues on the imaginary axis. 2. 4 Short-Circuit and Open-Circuit Parameter Matrices from the State Model To obtain short circuit or open circuit parameters from the state model equations, i.e. , from (2. 3. 9) or from (2. 3. 5) and (2. 3. 8), let (2. 3. 5) and (2. 3. 8) be repeated here for ready reference. -16- C +BT c B o — r— - _ T- _ - b 12 C 12 Vbc 0 B22 Vbc i T dt = 0 L + - — C 13231-1sz3. _IC£_ LB22 0.1 :01 .J — — - — — q — .1 T s T :1: 13126.3311 0 V0 0 'Bzz Vo -d— + - dt 0 -B L 13T 1* -13 o 1 23 b 33 1 21 1 _ _ _ L L. _ _ L (2.4.1) and * T d * T d T T >:< = — — - I + B I I0 Bu Cc B11 dt V0 + I"3’11 Cc B12 dt Vbc B21 cf 31 1 * * d T T d >1: V ‘ ' {B31 Vo + B32 Vbc +3 B33 Lb I323 Ici ' I3331131333 dt 11} (2.4.2) Although the derivations of state variables appear explicitly in (2. 4.2) they can be eliminated if desired, by direct substitution from the differential equations. Since the coefficient matrix to the left of (2. 4. l) is positive definite, its inverse exists. In (2. 5. 1) introduce the following trans formation of variable 3 1/2 t T 0 V V (Cb‘I'Blzcc B12) bc bC 1/2 1 B L ET) 1 1 L 0 (LC + 23 b 23 c1 c1 (2.4.3) -17- where the coefficient matrix represents a positive definite root of the matrix to be inverted. This transformation takes (2. 4. l) and (2. 4. 2) into the form V' 0 (C +BT C B -1/2BT (L +B L ET) ”2 V' be b 12 c 12) 22 c 23 b 23 bc i - dt ' " T -l/2 T -1/2 ' Ie1 ”(Le+Bz3Lsz3) B22(Cb+BlZCcBlZ) 0 Ie1 L - L _ .. T -1/2 T j o v (Cb+B12CcB12) B12(3cB11 i LBT -1/2B LBT dt 1* 0 '(Lc+Bz3 b 23’ 23 b 33 _1 _ v 0 (c1) + BIZCCBIZ) B32 0 T -1/2 0 1* (Le +B23LbBZ3) l_] and -1/2 ' s 'r d * T T 51 ._. __ C +B C B ) Io BllCcBlldtVO+BllccB12( b 12 c 12 dt bc -1/2 T T I T * - +B I (Le +1323 Lb 323) B211e1 31 1 -13- v* - B v* +13 0 T -1/2 ' 1 ‘ ' 31 o 32‘ b+BIZCc 312) Vbe -1/2 T T d ' T d * +13 + — - — 33LbBZB(Lc 323113323) dt Ie1 B33LbB33 dt I1} (2. 4. 4) It is evident that the operator matrix is still skew symmetric, i.e. , the skew symmetric property of the operator matrix is invariant under the transformation in (Z. 4. 3). For convenience define the following variables 1/2 T 1/2 (Cb +B12Cc B12) ’ X1 1/2 T 1/2 (Le +1323 Lb 323) ‘ Y1 T B12Ce B11 ' X2 T (2.4.5) B23 Lb B33 — Y2 = X Bll Cc B11 3 T 1333 Lb B33 ’ Y3 and let (2. 4. 4) be written as r- .1 P- — I-v ' " . -1/2 T -1/2 V Vbc 0 X1 B22 Y1 bC i _ dt ' ' 1/2 -1/2 ' . _ " 0 I let-J 1 B22 Xl _ _Cl -1 T -l/2 _ ’- *— 7 -1/2 T 7 7 x x o v ° 1 2 d o 0 X1 B32 Vo - «Te - -1 2 >1: - 0 --Y1 / Y2 I1 - 11/2 21 o I: d =3 T -1/2 (1 ' T -1/2 1 T >< I =x -—v +X x -— - o 3 dt 0 2 1 dt Vbe B21Y1 1c! + 133111 ,1 a: -1/2 ' T -1/2 d 1 d >1: v - B v + - - — 1 { 13 0 B32 1 Vbe + Y2 Y1 dt Iez Y3 dt 11} (2.4.6) 2. 4. 1 Short-Circuit Parameters If an LC network contains only voltage drivers then the s-domain short-circuit parameter matrix is of interest and is obtained from the al: (1 l = O and replacing — by s. This, state model in (2. 4. 6) by setting I dt of course, implies that all initial conditions are taken equal to zero. The result is -1/2 T -1/2 ' -1/2 - X 8U X1 BZZYI Vbe 8X1 2 ,3 = VO -1/2 -1/2 ' -1/2 I Y B Y1 132le sU J “J 1 21J and * >:< T -1/2 ' T -1/2' _ _ 1 2.4.7 10 _ 3x3 v0 + sXZ xl Vbc B21 Y1 d ( ) I Solving the first expression in (Z. 4. 7) for Vbc and substituting into the -20- second and third gives (EU 1' -1/2 --1 T -1/2 I 1 +_ _ e1 8 Y1 B22 X1 B22 Y1 Iez) ’ -1/2 -1/2 -1/2 * -1/2 * Y - 1 132le X1 X2 Vo Y1 B21Vo 01' ' 1 -1/2 1/2 1 1/2 >:< I = s +- c1[UsY1B22Xl12T21BY_-l[ZZZIBXXZ'YIBZIJVO (2.4.8) and I*-s(X XTX-IX)V*+(XTX-1BT Y--1/2 BT Y-1/2 0’ 3 2 1 2 o 2 1 22 1 ' 21 1 ) 1 T 1 -1/2 -1 1/2 T -1 T -1/2 T -1/2 :1: EU +§Y1 Bzzx1 BzzY1 :l [XZXI BzzY1 'leYl V0 (2. 4. 9) using (2. 4. 5) this last expression can be rewritten as -1 *. * T T T T = - c B v 10 3|:B11CC B11 BMCC B12(Cb+B12 cC B12) B12 C 11] o -1 -1/2 + (BTCB (c +BTCB ) BT(L +13 LB) 11e12b_12e12 22e23b23 -1/2 T T - BZl.(LC + BZ3 Lb B23) J x -21- -l 1 T -1/2 T -1 T T -1/2 sU+- L +B L s( c 23 sz3) B22(Cb+B12CcBIZ) B22(LC+BZ3LbB23) x BT c B (c +BT c B -1BT L T '1/2 11 e 12 b 12 e 12) 22( e+Bz3Lsz3) -1/2 . T >1: -B L +B L B -| J 21‘ e 23 b 23) — V0 (2.4.10) 2. 4. Z Open-Circuit Parameters In the case when the LC network contains only current drivers, the open-circuit parameters can be derived through a procedure similar . d . to that considered in Section 2. 4. 1. Indeed replacing a by s and a): setting V0 = O in (2. 4. 6), we have -1/2 T -1/2 ' -1/2 T 8U ”X1 BzzY1 Vbe X1 I332 :1: = - 11 -1/2 -1/2 ' -1/2 -sY Y Y1 Bzzx1 8U lo! 1 2J and * -1/2 ' T -1/2' ==< = - - Y I V1 {1332 X1 Vbc+SYZY1 1c! S 3 1 (2.4.11) I Solving the second expression in (Z. 4. 11) for Id and substituting into the first and third gives -22- -1 ' _ 1 -1/2 T -1 -1/2 -1/2 T -1/2 T -1 * Vbe‘{SU+§X1 B22Y1P’22X1 } {"x1 B32+x1 BzzY1 Y2 I1 (2.4.12) * T -1 >2 T -1 -1/2 —1/2 V1‘8‘Y3'Y2 Y1 Y2’I14"Y2Y1 B22x1 'B32 1 ) X -1 T 1 -1/2 T -1 -1/2 T -1 T -1/2 -1/2 4 {sU+-s-X1 BZZYl B22X1 } ‘Y2 Y1 Bzzx1 'B32X1 ) I1 (2.4.13) Using (2. 4. 5) this last expression can be rewritten as . -1 * T T T T >=< V1 ' sI:I:~"3:3Lb]333 ' B33Lsz3 ‘Le J“ B23LbB23) B23Lb1'333] I1 T T T T +|:B L B (LC+B B23) B C +B ZCeB -B32(Cb+B CCB J 33 b 23 23Lb 22‘ b 1 12) 12 12) 1/2 -1 1/2 “1 U+lc+BTCB .. BT(L+B LBT) B (C+BTCB) XSE‘blzelz) 22e23b23 22b12e12 .. -1 2 xB LBT(L+B LBT)1B (c +BTCB )/ 23 b 23 c 23 b 23 22 b 12 c 12 T '1‘? T * -B32(Cb+B12CC B12) 11 (2.4.14) Equations (2. 4.10) and (2.4.14) are the desired basic equations, and the specifications of the networks are usually given by these forms in network synthesis. In symbolic form (2.4. 10) and (2. 4. 14) can be written as -23- I*-YV*-(As+AT(sU+1A -1A V* 2415 o’ ‘ 1 2 s 3) 2) 0 ('°) and -l V*=ZI*=(A s+AT(s+-1-A) A)1* (2416) l l 4 5 s 6 5 1 °' 2. 5 Certain Necessary Conditions for Transformerless Realization For the realization of RLC networks without the ideal transfor- mers, Cederbaum has given the following theorem [CE 1]. Theorem 2. 5.1: A necessary condition for a matrix to be an impedance or admittance matrix of an RLC n-port is that, it is a par amount matrix. Proof: After Cederbaum [CE 1] The following corollary is immediate. Corollary 2. 5.1: A necessary condition for a matrix to be impedance or admittance matrix of an LC n-port without ideal transfor- mers is that, the matrix is symmetric, positive real and paramount. Proof: Follows from Theorem 2. 5.1 Theorem 2. 5. 2: Let 2(5) or Y(s) be a paramount matrix hav- ing a pole at the origin or at infinity, then the residue matrices corres- ponding to these poles are paramount. Proof: Follows from Theorem 2. 5.1 -24- From the state model the following necessary condition for realization of LC network without ideal transformers is developed. I Theorem 2. 5. 3: A necessary condition for an admittance (or impedance) matrix to have a pole at infinity is that the network contains at least one cut-set consisting of inductors and current drivers only (or at least one circuit consisting of capacitors and voltage drivers only). Proof: From (2.4. 10) and (2.4.15) ( (2.4.14) and (2.4.16) ) the existence of the residue matrix A1 (A4) correSponding to the pole at infinity implies that the submatrices B and CC (B and Lb) are non 11 33 empty and from (2. 3. l) the conclusion of the theorem is evident. Z. 6 An Importantfiecial Case Consider an LC network in which there are no circuits of capacitors with or without drivers, no cut-sets of capacitors, no cir- cuits of inductors and no cut-sets of inductors with or without current drivers. However, cut-sets of capacitors with the voltage drivers only and circuits of inductors with the current drivers only are allowed. It is shown in Chapter III that this Special case correSponds to canonic networks. For this reason the LC network satisfying the above condi- tion will be called a canonic LC network. The circuit and cut-set equations for this Special case take on the form -25- ). __ V0 7 VB“ B12 U o vbC = o B B 0 U V 2 22 _ 1 4 .1 J- V14 and r _ Io T U 0 1311 '321 Ibc T T = 0 O U '312 “322 Icf I .. 1 - (2.6.1) Therefore, the state model is given as _ fl _ — )— l T1 '— _ ’— l T- )— Cb vbc 0 'B12 Vbc O 'Bzz Vo d I l d—t =———l—-- +-———’———- - | o -B ' o _. LE. ICI .1 -Blzl .4 L. Cl - 111 J ._11.) T T Io Eula: +132111 V1: 'BZIVo 'Bzz Vbc (2.6.2) at * * Il‘ Again. terminal variables 10. 11, V0 and V1 are related to the driving variables by I*-I V*-V 0"0" 0‘ 0 and 1* I V* v 1"1' 1‘ 1 a) If there are only voltage drivers present then the state model can be simplified further and has the form “' “ F -1 T_ " )— “T Vbc 0 Gb 12 Fvbc 0 d 3:: IF = 1 + Vo 1c! 'Lc BIZ 0 let "Bu __ __ _ .J __ J _ J * T 1o ‘ 'Bu 1c: (2.6.3) By using the transformation of variables (2. 6. 3) becomes — 1/2 _ _ _ )- ' _ Cb Vbc Vbc 1.1/2 I 1' (2.6.4) c cf J cl J we have _ '_ F -1/2 T -1/2" F.1— F " O Vbc 0 Cb B1?.Lc Vbc 2): dit = + V0 ' -1/2 -1/2 -1/2 Icz 'Lc B 12Gb 0 Ian ”Le B 11 -27- _ T -1/2 0 lch 1d (2.6.5) The short circuit parameters are obtained from (2. 6. 5) as -1 * _ T '1/2 1 -1/2 fr -1 -1/2 -1/2 4 I - B Lc (sU +;Cb BIZLc Blsz ) Lc Bllvo (2. 6. 6) which has no pole at infinity. b) If there are only current drivers present in the network then the state model is again simplified and we have — — _ ' -1 T— — " _ -1 T— Vbc 0 ' Cb B12 Vbc ' '06 B22 d * -_. . =.___ ____ I- _.._ _. + I1 dt I - ' o Icl Lc 312' . 0 LIcl * V1: "322 Vbc (2.6.7) Using the transformation of variables in (2. 6. 4), the state model transforms into — 1‘ _ -1/2 T —1/2 1 —l/2 T V’ -C B Vbc 0 b BIZLc bc b 22 * d + I — = 1 dt ' /2 ' -1/2 -1 0 . c 0 I LIcl I"c B12 b _ _°£_ (_ - * -1/2 ' -23- and, the open circuit parameters are given by a): V =B 22 -1 C b / 2 (sU + 1 - - _ _L 1/2 1 T 1/2 S C 12 b 12 -l BCBL) c -l C:b /2T* 221 (2.6.9) In the above two cases, it is concluded that the short and open circuit parameters cannot have a pole at infinity for the canonic LC networks. c) The derivation of hybrid parameters for general LC networks from the state model is possible, but the final relation is very complex and are not considered in the thesis. However, for completeness a procedure is given below for canonic LC networks. * . relations I = -I 9 V written as After replacing Using the transformation of variables in (2. 6. 4) and the — dt * e * 0 O=VO.V1=V1,andIl— -1/2 T -1/2— 1— '— — 0 Cb B12 c Vbc + -1/2 -1/2 1 0 - c Blsz lo! 4: 'I‘ -l/2' T = - + I I0 1311 c cl B21 1 e * -1/2 ' V1 = 'le Vo " B22 Cb Vbc by s in (2.6.10), we have -11 (2.6.2) can be (2.6.10) -29- _ -1/2 T -1/7 _ '- _ -1/2 T— ' *- sU -c B L - b 12 c Vbc 0 Cb B22 V0 -1/2 -1/2 ' -1/2 * L B C sU - c 12 b 1c! LLc B11 0 I1 and * _ T -1/2' T ab 10 _ -B11LC 1C! +B2111 =3 _ *‘ -1/2 1 V1 ' B21.VO‘i;' BZZCb Vbc (2.6.11) From the first equation in (2. 6.11) 1”Br 1* (2.6.12) -1/2 T -1/2' L I 221 1 i- V ”Ecb B12c cz'ECb Substituting V into the second equation of (2. 6. 11) and solving for be I 1c! gives -1 ' 1 -1/2 -1 T —1/2 -1/2 C-l T >=< =- _ B I 1c: (SU+s c 12 b 12 c c B12 b 22 1 1 1/2 1 T 71/2 ’1 1/2 >:< - — ' ' ' ' V (“”ch Blzcb B12 c ) Lc B11 0 (2.6.13) 1 By substituting for 16! in (2. 6. 12) we have -1 ' _1 -1/2 T -1/2 1 -1/2 -1 T -1/2 L--1/2B ‘C-lBTft Vbc-_2Cb 131ch (SU+;LC Blzcb l312ch ) c 12b 221 S 1 1/2 T 1/2 1 -1/2 ' -1 T —1/2"1L—1/2B v* -gcb 131ch (sU+-S—LC B16201D BIZLC ) C 110 -30- -1/2 T 1* b 22 1, (2.6.14) 1 - - C 3 Substituting (2. 6.13) and (2.6.14) into the last two equations of (2.6. 10) gives the required hybrid parameters as 1 :1): I * _ T -1/2 1 -1/2 -1 T —1/2' -1/2 >1: Io ‘BnLc (511+ch B12cb B12Lc ) Lc B11Vo -1 T 1 L.1/2 1 -1/2 -1 T -1/2 -1/2 1T +[15”21'3'13111‘(“’UJ’EJLC Blzcb B1211: ) Lc B lZCb B 2211 * 1 1 T 1 T 1/2 1 1/21/2 1 = - ' - -— ' 2;;C1TB L' v0 (8 32201) B22 S—z-Bzzc; chlzL ‘5‘“ch B1 12 ) -1/2 -1 T * Le lZCb E2211 1 1T 1/2 1 1/2 1T 1/2 '1 1/2 >1: — " " — " ' ' ' B V -(B21-SB22Cb BIZLC (sU+SLC B1zcb Blch ) LC 11) 0 (2.6.15) Equation (2. 6. 15) can be written in symbolic form as 10 A B v0 - (2.6.16) :1: v: c D 11 h— — L- —- b -- Note that in the last relation the entries of all the submatrices A, B, C, and D are rational functions of 8. Further, the matrices A and D are T Symmetric and also B = -C . CHAPTER III SYNTHESIS OF LC NETWORKS 3.1 General In this chapter the following problems are considered: 1) canonic forms of LC networks 2) realization of state models with and without ideal transformers 3) realization of state models from the given Specifi— cations in s-domain, and 4) realization of s-domain models with and without ideal trans formers. 3. 2 Canonic Forms of LC Networks Although the properties of canonic one port LC (RL and RC) networks are well defined in network theory a clear definition of canonic LC n-ports has not been given. The following definition is based on the topology of LC n-port networks. Definition 3. 2. 1: If all the capacitors of a connected LC network form both a tree and a co-tree in the system graph, then the network is a canonic network. This definition implies the network N has no drivers. The case where N contains drivers will be considered later. The number of elements in a canonic network has certain proper- ties which will be used later in the synthesis procedure. Some of these properties are discussed in the form of theorems. -31- -32- Theorem 3. 2.1: Let G be the graph of a connected LC network N which has no drivers. If N has no circuits of capacitors, cut-sets of capacitors, circuits of inductors and cut-sets of inductors, then N is canonic and the number of capacitors is equal to the number of inductors . Proof: From the hypothesis of the theorem it follows that all capacitors can form a tree, as well as a co-tree. This property implies, by Definition 3. 2. 1, that the network is canonic. Further, since e - v + 1 = v -1 then the number of capacitors and the number of inductors in N are equal. Theorem 3. Z. Z: In an LC network N containing drivers, if the capacitors and the voltage drivers form a tree as well as a co-tree (or equivalently if all the inductors and the current drivers form a tree as well as a co-tree) then in N n +n =n +n, c v f 1 when number of capacitors, n = number of voltage drivers v :3 II number of inductors and n_ = number of current drivers. I 1 :3 ll Proof: Follows immediately from Theorem 3. 2. 1. Theorem 3. Z. 3: Let G be the graph of a connected LC network N. Also in N let there be -33- k1 voltage drivers and k current drivers, 2 r2 independent circuits of inductors with the current drivers, no circuits of capacitors with or without voltage drivers, no cut-sets of inductors with or without current drivers, no cut-sets of capacitors only, and no circuits of inductors only Then in N nc+(k1-r )=n +(kZ-r 1 1 2" Proof: Replace a voltage driver by a capacitor if the driver does not form a cut-set with the capacitors and by a short circuit if it does. Also replace a current driver by an inductor if the driver does not form a circuit with the inductors and by an open circuit if it does. This procedure reduces N to the network considered in Theorem 3. 2.1, and the proof follows. 3. 2.1 Tests for Canonic LC Networks Consider the graph G of a connected LC network N. Let n terminal pairs on this network be Specified as the ports, i. e. , N is considered as an n—port network. Let (31 be the graph of the drivers (the types of drivers yet to be determined) to be connected at the ports of N. 1. Consider the subgraph Gc of GUC}1 which contains the elements correSponding to the capacitors and the subgraph Gl' -34- In (2‘:C if there is a circuit of capacitors then N cannot be a canonic network. If Gc contains no circuit of capacitors then it. is possible to select a tree in CC such that it contains all the capacitors. Further, this tree may contain elements of G1, in this case the drivers correSponding to these elements are taken as voltage drivers and the remaining drivers in the network are taken as current drivers. 2. Let GL be the subgraph of GUC‘:1 which contains all inductors, and consider GCUG If there is a cut-set of inductors with L. or without current drivers then N cannot be a canonic LC. Otherwise consider the next step. 3. Replace a voltage driver by a capacitor if this driver does not form a cut-set with the capacitors and by a short circuit if it does. Replace a current driver by an inductor if this driver does not form a circuit with the inductors and by an open circuit if it does. If the resulting network satisfies Theorem 3. 2. 1 then the n-port network under consideration is a canonic LC network. An n-port canonic LC network can be generated, by the inverse of the above procedure, from a given arbitrary canonic LC network (Definition 3. 2. 1). If the n—terminal pairs are prescribed as the ports of the network, then the problem is reduced to that of selecting the type of drivers to be connected at the ports. These drivers by neces31ty -35- must be current drivers, otherwise there would be a circuit of capacitors with the voltage driver. This procedure, in general, introduces a cir- cuit of inductors with the current driver. If a prescribed port is generated by simply breaking an element and introducing a port, then this port must be excited with a voltage driver. This procedure, in general, introduces a cut-set of capacitors with the voltage driver. Other canonic LC networks with drivers can be generated from a given canonic LC network by replacing some capacitors by voltage drivers and some inductors with current drivers. 3. Z. 2 Degree of a Rational Matrix The degree of a rational matrix, defined first by McMillan [MC 1] in 1951, recently has become an important concept in network theory [DU 1, KA 2]. The objective here is to derive a relation between the degree of the Y or Z matrix of an LC network and the minimum number of elements necessary to realize this matrix. . Definition 3. 2i [DU 1] Let F(s) be an n x n matrix whose elements are rational functions of the complex variables 8, and let A be an n x n matrix of complex constants. Then PA(S) (8) |F(s) +A| = QA -36- where PA(S) and QA(s) are polynomials in s and they are relatively prime. The degree of F(s) is the maximum degree of PA(S) for all possible choices of the constant matrix A. McMillan [MC 1] defines the degree of Z or Y matrices as follows: Let R , R , . . . , Rn’ Rm be the residue matrices of Z or Y O 1 with ranks r0, r1, . . . , 'rn, rm reSpectively. Then the degree of Z or Y is r0 + 22ri+rm (3.2.1) Kalman [KA 2] has shown that, if Z or Y is regular at on , the degree, i.e. , (r + 22r,) is equal to the number of state variables. 1 0 When Z or Y has a pole at infinity, he transforms this form to the previous form by considering a Mobius transformation. However, it is easy to show by constructing a state model from Z or Y, as discussed in Section 3. 3, that the McMillan's degree is equal to the number of state variables plus the rank of the residue matrix corres- ponding to the pole at infinity. In the case of canonic LC networks, the number of elements is equal to the degree of Z or Y matrices. This is true, Since the impedance matrix Z (or admittance matrix Y) cannot have a pole at infinity, and also the inductor and capacitor variables are present in the state vector . -37- 3. 3 Realization of a State Model of an LC n-port without Ideal “ Transformers The Specification of networks are usually given in the s—domain. However, a state model correSponding to the given S-domain model can always be found and such procedure is presented in Section 3. 4. The state model so obtained can be realized by a network containing ideal transformers. This problem is considered in Section 3. 5. In this present section, however, certain conditions are derived which are necessary to realize a given state model by an LC network without the ideal transformers. Given the Specifications -- .— —- —n 1—- — r -! — *— P'" - — *—) O V 0 K V Vbc 0 K1 Vbc K3 0 5 0 d d " ’ - — + dt - dt * ' 0 I -K O I 101 K?- 0 lei-J K4 1 6 l * d * * = — + + K V + K I I0 K7 dt VO K8 Icl 9 O 10 l >'.< d 9.: y): a): " - - I + K V V1" [Klldt I1 +K12 Vbc+K13 1 14 o (3. 3. l) a necessary condition for the realization of this model by an LC network is that the operator matrix must have eigenvalues on the imaginary axis . Comparing (3. 3. 1) with (2. 4. 4) and identifying the matrices we have 11 12 13 T C + (b 131ch 312) B32 T -l L + ( C B23 Lb B23) B22 T T T B1166 B12 (Cb + B12 Cc B12) 1322 ' B21 T T 'Bll Cc B12 (Lc + B23 Lb 323) B21 T T T '1 I333LbB33 + B33LbBZ3(Lc + 323Lb323) B23LbB33 -1 T T L B (LC +B23 Lb B23) B B b23 32 " B33 22 -1 T T T 'B33 Lb B23 (Cb + B12 Cc B12) B32 -39- where K. is known. The problem is that the matrices Bij' Cb, b' 1 and Lc must be determined from the above equations. Let (3. 3. 2) be C c written in the form, T T (Cb + BIZCCB12)K1 — B22 T T (Cb +B12CCI3BZ)K -B12CCB11 C +BT ch B )K -BT (b 12 5’ 32 T (LC + B23 Lb B23) K2 — B22 l U1 1" U3 (LC+B23 Lb B23)K4 23 b 3 = (3.3.3) (Lc+Bz3 Lsz 3)K6 321 K -B c B BT c B K 7 ‘ 11 c 11 11 c 12 3 8 11 c 12 1 21 =- K K9 B11Cc312 6 - L BT +B L B K K11 B33 b 33 33 23 4 -40- where each of the matrices BTCB T C +B C B 9 9 9 ( b 12 c 12) 11 c 12 11 c 11 33 b 33 (Lc+B LBT B LBT.B 23 b 23)' 33 b 23 21’B 22’ B32 are considered as unknowns. Once these matrices are determined their constituents, can be found. Since the system of equations in (3. 3. 3) is linear, a solution exists only if they are consistent. Assuming that the system is consistent, a solution of the unknowns establishes a model of the form given in (2. 5. 1). In this form the submatricesC +BT C B 12 c b and L + B L BT by necessity are c 12 23 b 23 positive definite. ' d t From (2. 5.1) the submatrices Bij' Lb, LC, Cb an CC mus T be determined. Since from the above solution the submatrices Bll T T d + B C B are known, construct the c:c B11' B11 Cc I312 an Cb 12 c 12 matrix —T C B BT C B 1 B11 c 11 11 c 12 Y : (3.3.4) c T T C B LBlZCc B11 Cb+B12 c 12__ Where Y correSponds to the admittance matrix of a capacitor c network and can be obtained by substituting Vbl = Vcl = V1 = 0 in -41- the node equations of the network. To determ' th ' ' me e matrices B11, B12, Cb and Cc 1n (3. 3.4) apply the decomposition algorithm of Cederbaum [CE 2] to (3. 3. 4). Rearranging the rows and columns gives ' i " ‘1 ' 7 T 0 -B11 Cb O O Y : c LU -312 L Czc L-Bll -1312) (3.3.5) which is essentially unique due to the nature of this alogarithm. If entries in either Cb and C are not positive or if c _o BT—l ' 11 U -B 12 L- .- iS not a cut-set matrix, then, the given state model is not realizable. The above procedure can be repeated for a new matrix Z ' ' T B L BT and L + B L BT L havmg the matrices B33 Lb B33, 23 b 33 c 23 b 23 as its submatrices, where ZL corresponds to an impedance matrix of -42- the inductor network. The matrices B B 239 are again L andL C 33’ b determined by the use of the decomposition algorithm. Here the diagonal matrices Lc and Lb must have positive entries and B23, B33 must be submatrices of an unimodular matrix. From the submatrices determined above and the solution of linear equations in (3. 3. 2) all the submatrices appearing in (2. 5. 1) are determined completely, and the circuit matrix is established as _ 1 L o 0 B11 B12 0 I U 1 1 - I 0 U 0 3.3.6 B ’ I321 B22 B23 . ( ) 1 | o 0 U LB31 B32 B33 I1 The problem is reduced, therefore, to the realization of this circuit. matrix. If it is realizable, the topology of the LC network can be determined through any one of the known techniques [TU 1, GO 1, K1 1, BI 1. GU 2, CE 3]. In summary the state model is realizable without ideal transformers if 1. Solution exists for the system (3. 3. 3) 2. The circuit matrix B in (3. 3.6), is realizable, and -43- 3. Cb, CC, Lb and Lc are diagonal matrices with positive diagonal entries. It is interesting to note that if the conditions given in the above are not satisfied, this does not imply that the network is not realizable without the ideal transformers. Indeed, another state model obtained from that given by a similarity transformation might be realized as an LC network without the ideal transformers. 3. 4 Derivation of a State Model from the s-domain Equations The derivation of a state model from the s—domain model has been considered by Gilbert and Kalman [CI 1, KA l, ZA 1] and they have described certain procedures for this derivation. In this thesis, since our main interest is LC networks, a Slightly different method for deriving the state model is given. The procedures are the same for the Open and short circuit parameters. Let Y be eXpanded into partial fractions to obtain 2 +Rms (3.4.1) Y=R -l—+Z‘R, S 1 5 +631 0 Where R R, and R00 are the residue matrices whose properties are 1 0' well known [CA 1]. The operator matrix in (2. 4. 6) correSponding to an LC network 0 a ‘ 1 ' w ° ' ' . s is Ske symmetric and consequently it has pure imag1nary eigenvaiue till: tjkz, . . . . Note that some of the eigenvalues may be zero. -44- Theorem 3.4.1: Let A be a skew symmetric matrix of the form 1- T— o A 1 A : -A1 0 _ _1 Then there exists an orthogonal matrix P such that T P AP = A. where the matrix J\_ has one of the two forms F 1, — 1 l | x2 0 ' 1 | k . F IO 0 o I: 1° I0 0 0 JL: ______ _.__.__l _______ :— 1 1 -1 0.01 1-x 1 1 | O ._ , 1 ")‘k 0. 01 _) (3.4.2) -45- 01‘ T :11 owe—1 x O ' Z 1 _ 1 1 F ' ' 1 k 0 0 O'D' O _ 1 1 1 A2- ———————— '— _________ : -_‘__.- 1 -D' _ 1 1. _, ; o 2 l 0) 1 — ' J -xkl O 1 9 1 I l 0 01 _ ._J 1' 1 ." O | DT : ___1___ , 1 -D | O __ , J (3.4.3) The expression for JLl applies when the number of branch capacitors is greater than the number of chord inductors, and _A_2 applies when the number of branch capacitors is less than the number of chord inductors. In-/\_1 and_/\. 2 , N's are taken as the absolute values of the eigenvalues 1 of A. Proof: Follows from the application of Theorem 5311 in Perilis [PE 1] -46- To apply the above theorem to the operator matrix in the State * model of (2. 4.6), (II = 0), let P be a transformation matrix such that 1—,-1 -— V V bc 2 P = (3.4.4) Llcl 3_] The tranSposed state model is then — I- )— —- — — ._ —-( o D'El v W x'l/ZX 0 V2 2 1 2 :1: T ' -d— = -PT i V - P v dt dt 0 0 ' o ”fl/2B * d 4 T -1/2 d T -1/2 T f — ._ — - B Y 9 3.4.5 Io’x3dtVoJ"X2 X1 PdtVZ 21 1 3 ‘ ) and the short circuit equations are >1< T —1 >1: T --1 T —l/2 _ T Y~1/1Z p 10 = s(x3 - X2 X1 x2) V0 + (x2 x1 BZZYl B21 1 ) x -1 2T 5': 1 1 'T T T —1 T 4.1/2 _ T -1/ v' [Us +-S- D D ] P [X2 X1 BZZYI B21 Yl ] 0 (3.4.6) which is of the form * 1 S * 3 4 7) _ _ V ( . . IO-[Rms+ROS+Z 2 2Ri10 -47- The fundamental problem is to derive (3. 4. 5), or an equivalent state model from (3. 4. 7). Since the short circuit and the open circuit matrices are positive real, the residue matrices R , Rm and R1 are all positive 0 definite or semi-definite [CA 1]. Decomposition of each residue matrix Rj into the form Rj = K? K_ is always possible as stated in J the following theorem. Even though this theorem is well known, the proof given here presents a procedure for decomposing the residue matrices . Theorem 3. 4. 2: The necessary and sufficient condition for the decomposition of any real symmetric matrix R, into the products of the form KTK, is that R is positive definite or positive semi- definite . Proof: Necessity is evident [HN 1]. To prove the sufficiency, consider an orthogonal matrix V which transforms R into diagonal matrix D, i. e. , VT R V : D1; where D1 is a diagonal matrix consisting of the eigenvalues of R. Since R is positive definite or semidefinite, all the elements in D1 are either positive or zero. Let the positive 1/2 , 11 th Square root of the matrix of D1 be denoted by D1 , i.e. , a-- . e elements in D1/2 are all non-negative. Then R can be written as T 1/2 1/2 1 )(v D1 1 w I (V D (3.4.8) KTK -48- which proves the Sufficiency. In (3. 4. 8) K is a square matrix, consisting of zero rows corresponding to the number of zero eigenvalues of R. These rows 1 in K can be deleted, to give a rectangular matrix K for which '1" 1 1 R=K K=KTK From the above theorem, (3. 4. 7) can be written as * 'T ' 'T ' l K K V* (3 4 9) IO— [Km KcoS+KO KO's—Z 7—: Ii 1.] 0 . . S +61)i From the information contained in (3. 4. 5) and (3. 4. 6), a state model corresponding to this equation can be constructed as —— q 1- “IT. r- —1 r—1 — V2 0 D V2 0 i = + J dt , . I -D 0 I K 3 _ 3_1 - - L - _ - >3 1T 1 d :lt. 'T ' = _ + I (3.4.10) Io Km Km dt V0 K 3 where F- , -‘ K1 1 " _ (3.4.11) K - Kn I K0 -49- I and Ki i = 1, Z, . . . n are the submatrices obtained from the decomposi- | tion of Ri' and D has the form given in (3. 4. Z) or (3. 4. 3). The form of D in (3.4. 2) or (3.4. 3) is D = 2 (3.4.12) , 2 2 where Di 18 a scalar matrix associated with d, = a), . Note that the 1 1 order of D, is equal to the rank of Ri' and each D, is arranged in the 1 1 I same order as of Ki' An interesting result in the decomposition is that, the procedure gives a Spectral decomposition of the residue matrices, indeed R. = VDIVT r—d — 1 =v - vT ’d n _ ._ 0 II am <1 9.. < r-l ll .Mb <', 0. <1 " r-i H 31M: HQ: H< < H »—3 Where each of the matrices (V, V?) have the same order as R and each 1 represents a constituent matrix of R. -50.. 3. 5 Realization of Short Circuit and Open Circuit Parameters of an n-port LC Network Using Ideal Transformers In the literature, the technique for realizing the short circuit and open circuit parameters of an LC network by using the ideal transformers is well established [CA 1]. The procedure described here for the realization of LC n-port networks is slightly different from that of Cauer and utilizes a state model for realizing these parameters. This method also realizes the network with minimum number of reactive elements, this number being equal to the degree of Z or Y. The method is identical for short circuit and open circuit parameters. Let the given short circuit parameters be * ’5‘ = (3.5.1) IO Y V0 where Y is an n x n positive real matrix. Expanding Y into partial fr actions give 3 1 ._.—L— 35.?» Y=RmS+R0§+ZRi Z Z ( I s +00i From Section 3. 4, (3. 5.1) can be written in terms of the decomposed residue matrices as. IT 1 l S 'T °° 0 O s s +wi 1 :1: IT 10 = [Km I >}< " 3.5.3 Ki] V0 ( ) -51- and the state model given in (3. 4.10) is _ ' _ _ ._. Fv ro D T Fv To d 2 _ 2 ,5 a - 1 l V0 I3 -D O I K _ 4 _ J -33 _ _ * _ 'T ' d * 'T ‘ IO—Km Kmd—tVO+K 13 (3.5.4, In order to realize the state model in (3. 5.4), consider an ideal transformer network with the following terminal equations. ' _ I 1 _ I _ (n) F10 F0 0 : K T Fvo (rbc) 1'2 = 0 o : D'T V; (3.5.5) -- — ————— I — -— ——- I - t - I I 0 II (rcfl) _V3J _ K D l a _ 3J Let the terminal graph of the transformer be selected as shown in Fig. 3. 5. 1. Further, let two-terminal inductors and capacitors be connected to the external terminals of the transformer network, as shown in Fig. 3. 5. Z. -52.- Fig. 3.5.1 Fig. 3.5.2 Note that the number of ports of the ideal transformer, as indicated in (3. 5. 5), is equal to the number of state variables in (3. 5. 4) plus the order of the admittance matrix in (3. 5. 2). It will be shown that, the ideal transformer network which is loaded by the two-terminal inductors and capacitors has the state model given in (3. 5. 4) except the first term in the algebraic part of (3. 5. 4). The transformer network is then modified as required to give a complete solution. From this point the synthesis procedure reduces to the determination of the submatrices in the coefficient matrix of (3. 5. 5). If the element values of the inductors and the capacitors are taken equal as unity, then the terminal equations of these components are -53- d v2 127 a; I == (3.5.6) __ 3- LV3_ ‘— T — 1‘ — HIT“ P .- _ '3 12. -I2 0 -D V2 0 I V = v = D o ' + ' V0 - I -K _. 'T_ __ _, _ _ O D V2 0 ' = + V I I O -D O I -K _ _ L 3__ L _ (3.5.7) Therefore, the state model is of the form _ _. _ “1,1 ._ ._ _ _ 0 d V2 0 D VZ — = + V dt ' I O I -D 0 I +K 3 IT I IT _._. ._. .. (3. 5. 8) I0 K I3 K I3 In the above model, the residue matrix correSponding to infinity is not included. This matrix is derived separately. Indeed, let the terminal equations of another transformer network, correSponding to the terminal graph in Fig. 3. 5. 3, be -54- F— H— F- 1 - _ - T II 10 0 Km v0 = (3.5.9) II I II v3 -Km 0 13 II Connecting capacitors with unit values across the ports indicated by 3 , the terminal representation of the resulting network with reference to Fig. 3. 5. 4, can be derived as follows: The terminal equations of the capacitors are 5; V?) = 12” (3.5.10) 11 11 11 3 0 3 0 0(3) 3 3( ) Fig. 3.5.3 Fig. 3.5.4 From Fig. 3. 5.4 and (3. 5. 9), we have -55- (3) _ H - 1T 11- 1T (3) I0 — Io .. -Km I3—Km 13 'T d (3) 'T d 11 Ken 31! V3 ’ Koo 'd-t V3 IT I d 11 -Km Km 8? V0 'T ' d (3) Km K” dt Vo (3.5.11) Hence the first term in the algebraic part of (3.5.4) is realized. To combine the transformer networks correSponding to (3. 5. 5) and (3. 5. 9) as indicated in Fig. 3. 5. 5 we have * (3) * (3) = - = = 305.12 I0 (I0 + IO) and V0 V0 V0 ( ) Substituting (3. 5.8) and (3.5.11) in (3. 5.12) gives F." — F- IT_ -— —- )—— -— 0 D V 0 d V2 2 ,1 — = + V' dt 1 K1 0 I3 -D 0 I3 _ _ _. d _ .1 .- .J * .. 'T ' d * 'T (3.5.13) 10-K,, KmdtVOJrK 13 Which is the desired state model. The network correSponding to this state model is shown in Fig. 3. 5. 6. The state model given in (3. 5. 13) is not unique. (31 0 Fig. 3. 5. 5 ~56- But any such derived state model can be transformed into the state model (3. 5. 13) by means of a similarity transformation P. Therefore, the general form for the realization of the given short circuit parameters is given in Fig. 3.5. 7. Transformer 6:9 Network CorreSponding to (3. 5. 5) ‘ ‘11th -.J Transformer Network Corresponding to (3. 5. 9) Fig. 3.5.6 '__I_. +3" -57- Fig. 3.5.7 Transformer Network Corresponding toP"1 "—_‘L J h— -58- The above realization is based on element values of unity magnitude. These values can theoretically be altered to any desired values by including a transformer of the desired ratio. Further, multi- terminal inductance and capacitance networks as well can be used since the realization of one -kind of n-port networks can be reduced to the form discussed above [CA 1]. The realization of the transformer networks correSponding to the terminal equations of the forms given in (3. 5. 5) and (3. 5. 9) is classical and can be found elsewhere [BE 1]. 3. 6 Realization of LC Networks without Ideal Transformers Realization of LC n-port networks without ideal transformers has been an outstanding problem in Network Theory. Only certain sufficient conditions are known to realize a given reactance matrix [SL 1, F0 1, SO 1]. Although general necessary and sufficient conditions are stated by Cederbaum [CE 1], the application of these conditions to the synthesis procedure seem impossible. A new approach to the synthesis problem, not utilizing the reactance matrices directly, is considered here. First attempts in this direction were made by Kalman [KA 3] who gave a procedure for realizing LC driving point admittance function using the state models. The approach given here for one-port LC network synthesis differs from his. Also a procedure for realizing 2-port LC networks is given with the restriction that the residue matrices are dominant. -59- 1. One -Port Realization Procedure The procedure developed here is applicable to any driving point admittance or impedance, which does not have a pole at infinity. A consideration of the functions Y1(s), 22(5), 23(5) and I Y4(s) is sufficient to cover all possible LC driving point functions (see Section 4. 2). The partial fraction eXpansion of 2 (s + a2) (3 + a ) 1 I I O Zn-l Y1(s) = 2 2 2 2 (3.6.1) s(s + a2) . . . (s + aZn) gives u r s 1 K ‘ 213. “:1. S + a7: where ri i = l, . . . , n are the residues of Y1. Therefore, a state model correSponding to (3. 6. 2) can be constructed, through the procedure given in Section 3.4. This resulting model is of the form F— : 0 a2 _ . F O I a4 __ rv; : . °. sz l Q d l 0 a'2n a; = -—-— —--— -— -— -1 ———————— + —--—- I o o o 1 1 «fTO’ __ 3d 1 L 3) -a, I 2 NE?— '34 : O 7' .' I an (_ '8'2n I __( L- J -60- and I=['\/—r7)...'\}r2n]13 Premultiplying both sides of (3. 6. 3) by the diagonal matrix (- 1 a I 1 (3.6.4) -61- I O a — I 2 I I | _ I 1—- I l 0 a2n V2 0 | — ————————— l—__.__ ______ + ._.-._. 0 O l l I a I 3 - 2 1 __ _) 1 .[T | 1 I l 1 I 1__ I l I a I 2n I - I «('5— Zn I 1 (3.6.5) and I=[\/—;B» '“anI3 To retain the skew symmetric property of the operator matrix in (3- 6- 5). apply the similarity transformation (3.6.6) -62,- Z 1 3 V .I- V T.- r _ F L 7 _ _ d3 _ .m- _ 1 W _ _ _ _ _ n .l- 2 _ _ r _ _ _ 1 F10 _ _ m _ 2 _ rm _ l—r " _ U _ m u 1_r0 _ _ m. 1111111111111111111 6 _ __ 3. U _ - J m _ V7- .13 1w _ _ - m V _ w -63- — I a 1 Z l0 I II; 0 ' ' 1 a _ '1 I 2n (0 J"— l r2n V2 ———————— '—-—-—————— + —-- V I 1 o .o ' I I 3 1 a l - 2 | r l “2. . 0 L1- I a 1 ... 2nI ‘Vrz' n _( (3.6.7) and To reduce the operator matrix in (3. 6. 7) to the desired form, apply the Similarity transformation in (3. 6. 7). 7 I. T 8.2 I ,r—r | Z | ° 1 F _ aZn I I— V' ' V2 2 “/an 1 _____ I— —— — — _ 1 ' ‘ I I I L13 ' U _ 3J - I (3.6.8) (3.6.9) -64- and -65- Fig. 3.6.1 -66- The circuit matrix as obtained is (see Chapter II), ._. P .— -1 I 0 I ‘1 V I I I I 1 -l I 1. 1 U v2 = 0 (3.6.10) : I . I -1 I 1 I I; I I _ ._ L _ The circuit matrix in (3. 6. 10) correSponds to second Foster form indicated in Fig. 3. 6. l. The element values obtained from the coefficient matrix in (3. 6. 9) are indicated in Fig. 3. 6.1. II. Two-Port LC Network Realization Procedure This part is devoted to the realization of a Special class of 2 x 2 reactance or susceptance matrices having dominant residue matrices. Slepian and Weinberg [SL 1] have described a general procedure of realizing n-port Z(Y) matrices having dominant residue matrices. Using state model approach, we shall show that Z x 2 Z or Y matrices can be realized yielding an identical solution to that obtained by Slepian and Weinberg. Since the realization procedure is identical to both Z and Y matrices, only Y matrices are considered here. Consider first the susceptance matrix Y, which does not have a pole at infinity. The general form of the network is shown in Fig. 3.6. 2. -67- -68- In this network, it can be observed that all the capacitors and the voltage drivers can be included in a tree T, which also contains an inductor. This inductor has to be selected from a series or a cross arm of the lattice. Select an inductor between any one of the terminal pairs (A, B), (A, D), (B, C) or (C, D). If such an inductor does not exist, select an inductor from the resonators between any one of these terminal pairs. Let the graph of this network be Shown in Fig. 3. 6. 3 with the arbitrary orientations of edges. Because of the above selections of the inductors we have two possible cases. 1. An inductor exists between any of the terminal pairs. Then the circuit equations can be written as V I311 U: B13 : U Vbc = 0 3.6.11 1 1 U Vb1 ( ) B21 0 1 B14 : V " I — cl V12 where V = voltage drivers 4 11 ca acitor volta es bc P g voltage of the branch inductor < 11 bl -69- Vcl V 12 Then the state model can be constructed as follows. _ fl __ _., o o C:b Vbc T T d 0 I"Jolt-”b11313Bl3 [blBl3Bl4 d—t I61 T T L0 £b£B14B13 L2+1b1B14B14 L 112 — I 1 '3 — _ _ 7 O I U: Q Vbc O _. __ _ _L _ _ _ _ _ __ - + - - 1 -13 U l Icf 11 1 O 1 O 112 'Bz1 __ _ __ _ u _ and 1" 1 I * T T c1 1 _ -[B11 1321] In >i< >1< where I and V represen to the driving function as and voltages of the inductors which are in the resonators voltages of the inductors which are not mentioned above. (3.6.12) ts the terminal variables which are related -70- 2. There is no inductor between any of the terminal pairs (A, B), (A, D), (B, C), or (C, D) in Fig. 3.6.2. In this case the synthesis procedure can be reduced to the previous case. Indeed, in this case, since Y matrix does not have a pole at the origin, then we introduce such a pole by simply adding a term correSponding to the pole at the origin. After the realization for this augmented matrix the inductances correSponding to the pole at the origin are removed to yield the network for Y. In the following we shall discuss the procedure for realizing 2 x Z susceptance matrices which have dominant residue matrices. Consider first the susceptance matrix Yl which does not have a pole at infinity. The partial fraction exPansion gives 1 S = _ E:_____ R 3.6.13 Yl ROS+ 2 2 i ( ) S +wi where R0 and R, are the residue matrices which are assumed to be dominant. The procedure for decomposing the residue matrices differ from the procedure described in Section 3. 4. Consider the following two p03 8 ibilitie s: a) Let (3.6.14) -71- represent any one of the residue matrices in (3.6.13), such that a >b , c >b andb >0, i.e., R.is dominant. ThenR in(3.6.l4) l- 1 1— 1 1- J j can be decomposed into the form —b b T F b 0— _ ‘1 1 1 a1 - 1 O 0 R. = + + J b - L _ L. _ _. _- — — —- --1 — —1 lb]. 3.1-bl 0 = (dbl \Ibl] + [Val-.131 0] + [o «FEET-Tl] «Ibl O cl-b1 L. _. _. ._. ._ -_. 1—— ——q I_- — Vbl dbl «(bl al-b1 0 a -b 0 = K K = 1 1 J N/bl 0 cl-b1 0 cl-b1 ._ _ I. (3.6.15) b) If the residue matrix is of the form a -b — 2 2 (3 6 16) Rk — -bZ c2 L_. .4 > . h R ' 3.6.16 canbe where azibz, c :bzandb2_0 T en k1n( ) decomposed as follows -b _ _ 2 I-az b2 0 To + + b O O 0 c 24 _ _) _ 3 F _ “az'bz [4 b2 " \I ha] 'I' [‘V aZ-bz O L J «I aZ-b2 0 b2 NI aa-b2 O c -b 2 2 0 ._J _ 0 - sz I- “1 O] + [O ch-bz] NIc -b2 _. _I -N/‘b_2 0 (3.6.17) The correSponding state model can be constructed by the procedure already described in Section 3. 4. The result is I—I ‘_' 1— V 0 g 2 - _-_ dt ’ -D I3 I = KTI where v Fo— (3.6.18) -73- _ K01 and D has the same form as in (3. 4. Z) or (3. 4. 3). Applying a proce- dure identical to that applied for one ports, i. e. , premultiplying both sides of (3. 6. 18) by a diagonal matrix and applying the similarity transformation, the equivalent state model is r— I 1 o— I 1 O I F .1 o . -. —,- C2 V2 : 1 0 V2 1 :: —————— I ——————— + dt I I I L I _1 13 ._ 3- _ 3.3 ' _ J -1 I I 1 O -1 I | o o. o , and I I : BT13 (3.6.19) where B contains t 1, 0 as its entries and C2 and L3 are diagonal matrices . -74- To determine the network corresponding to (3. 6. 19) the only requirement is that B' must be a submatrix of a circuit matrix. If it is, then the realization follows immediately. If B' is not a sub- matrix of a circuit matrix, then the state model obtained above must be modified by using a proper similarity transformation on the state vector so that the new model is realizable. This is always possible I and is shown next. The first column of B contains only the entries I +1, or 0 but not -1 and the second column of each Bi (the submatrix I of B ) has the following possible forms J.“- 1 (a) O l I. d 0 (b) (3.6.20) 1 + (C) I 1] In (3. 6. 20a), the plus sign is taken if the off-diagonal entry in the residue matrix is positive, and the negative sign is taken if otherwise. Equation (3. 6. 20b) appears if the residue matrix is diagonal and (3. 6. 20c) appears if the entries in the residue matrix are equal in absolute value. Again the plus sign is taken if the off- -75.. diagonal entry of the residue matrix is positive and the negative sign is taken if otherwise. In the following,three cases are considered separately. In Case 1 it is assumed that the off-diagonal entries in the residue matrices are all positive, in Case 2 all are negative and in Case 3 they are arbitrary. I Case 1: If each submatrix Bi of B contains one of the following matrices _ '1 l l 1 0 l 1 l 0 or . . [1 1]: [1 0] O 1 1 O ._0 1 (3.6.21) then B is a. submatrix of a circuit matrix. Then the circuit equations as obtained ar e _ 7 _ 1 I 1 I V I | l 1 I -B' | 1 ' U v2 = o I I | I ' 0 I V3 (3.6.22) These equations correSpond to the network shown in Fig. 3. 6. 4. -76.. I 3 *6 I-er—ll—UU— “I T I Fig. 3.6.4 The element values in this network are dictated by the diagonal matrices Czand L3 in the state model. Case 2: If each B, represents any one of the following matrices 1 1 -1 1 o 1 -1 1 o . . , [1 -1], [1 0] o 1 1 o _o 1—I (3.6.23) I Then B is a submatrix of a circuit matrix of the form _ _ _ 1 II I v I I " 1 lU V —0 -Bl 1 I 2 _ I ', I L I .0: V3 and the correSponding network for the circuit matrix is very similar to Fig. 3.6. 4 and is shown in Fig. 3.6.5. -77- re .3533..— ~I I Case 3: If some Bi have the form in (3.6. 21) and some Bi Fig. 3.6.5 I have the form in (3. 6. 23), then the resultant B may not be a sub- I matrix of circuit matrix. A typical form of B in this case is I— l OJ (3.6.24) In this case, although B may not be a realizable submatrix, the state model can be transformed to a realizable state model by first augmenting the state model and then using a similarity transformation on the state vector. -78- The augmentation of the state model can be achieved by decomposing the residue matrices as follows: «[6172 NHEITZ 46177 dbl/'2 —~l bz/Z «162/2 -d13272 -NFE;TZ .- l-b1 0 O cl-b1 .1 Z-bz 0 0 cz-b2 Vin W al-b1 O 0 cl-b1 (3.6.28) «IbZ/Z -'\I bZ/z (3.6.9) In order to illustrate the procedure,the following example is considered. Example: Y(S) = be the given susceptanc Let s(Zsz+5) (s2+l)(sz+4) 3s (sz+l)(sz+4) _ Adding the term 3s (sz+l) (sz+4) 2 s(ZS +5) (sz+1)(sz+4) (3.6.30) e matrix which has no pole at the origin. tn].— to Y(s) we have Note that R0 matrices as and :II— 6.11 éIII-- §I|~ UJII—a -l 1 1 1 l 1 +R2 is arbitrary but it is dominant. EII~ ill“ §|~ §I~ (nII—a +4 Em w 31F :II“ -l s 2 l 6 +4 (3. 6. 31) Decomposing the residue -80- -1] [l 1 d 2 av VL A state model can be constructed: a . 2 2 2 2 a o _ . _ . 21 Z 2 2 —1 1 1 1 _ _ L + 3 4 5 6 7 8 9 1 l v v v V 1 .1 1 1 1 _ _ _0 0 0 0 _ A _ 2 _ 2 _ O l _ _ 1 _ _ _ 7.- o _ Z 0 _ — O _ 1 o - _1 0 — _ _ __ _ 0 .l. 3 4 5 6 7 co 9 1 l V V V V 1 .1 1 1 1 _ d...m (3.6.32) L 1“- and -81.. Applying the transformation of variables we have _ -— r.— 1 V 3 3’ 1 v 7 i‘ 1 v - 5 8 I l d v .. _ 6 = 8 dt , Z i 2 .7 2 i 2 f3 l 19 I ‘10 I 111 I—- — _ —I I—' — I O ' l v ' .3 I 1 o v 4 O : 1 0 ' Q _ _ V I 1 o 5 v1 ___-I _______ 4,---- ____ I V6 -1 ' ' i 1 1 -1 l "7 V2 I . 1 1 - .- -1 18 I ‘1' O 1' 1 -1 I 9 I o o 0 0| . 1 _1 — - 110 I i 1 -1 L 1L L _ (3.6.33) -82- The submatr ix r—- —I 1 l l l I B : l -l l -1 l -1 I_ .3 is not a submatrix of a circuit matrix, therefore, a similarity trans- I formation T must be used to reduce B , to the desired form. General description of the transformation matrix is given later. Let the transformation matrix be taken as of the form ,— _ l — — — (3.6.34) Applying the transformation to the inductor currents in (3. 6. 33) and then using another set of transformation of variables similar to that used in one ~port case, the final state model is I. 2 TV v _ _ 3 0 1 $V3 $V4- $ V5 $V6 $.17 $.18 $.19 $.11 $.11.- _ . .I. 0 .I- .I- II— 7 _ _ . . . . ._.. o _ J _ — 0 .l. .l. .I. 1 w.- l r _ _ T + \III/ _ O _ 1 *2.*4.*5*6*7*8*9*1*1 .1 .1 .1 01 o _ _ _ 1 v v v v 1 _ _I _ 2 2.- _ 2 _ O 0 0 0 _ I. 8 _ . Z _ _ l _ _ z 1 _ 0 {III\ _ _ I l _ T 1 _ _ IMIIIIIH " 1 o .1—8 _ . _ _ l 0 1—8 0 _ - _ _ .I. 0 Z . _ _ l O _ _ and where Hence NIH NIH NII-Il NIH NIH -84.. Ho *xl-X- Ho H*H*\O*m ._.; Ho Ho Ho (3.6.35) (3.6.36) (3.6.37) -35- Then the circuit matrix can be constructed from (3. 6. 35) and (3. 6. 37) as o +1I1 l-lll q I I I -1 ol 1 I 1| 1 I I I B: -1 1| 1 lol 1 I I I -1 1| 1| 0| 1 I I I -1 1Io o 0 0| 1| 1 (3.6.38) The network can be realized from (3. 6. 38) and the element values are determined from (3. 6. 36) and (3. 6. 37), and the result is shown in Fig. 3. 6. 6. 1-1 1"?- 11 *— If 2 1 8 1 — 2 _1_ I2 2 l 2 8 J! L/UUD'OYIU‘ Ir _1_ 7- Fig. 3.6.6 The network corresponding to the given matrix Y can be obtained by removing the two inductors 11 and 12 in this figure. -86.. _S_e_lection of the Transformation Matrix T Consider B1 B2 ' O B = - Bk BO where Bi is obtained from the residue matrices as discussed earlier. The leading two rows of Bi (i 16 O) are one of the forms: [1 l] , [1 -1] , [1 O] , [0 1]. The last row of B can be taken as [1 1] , 0 or [1 -l]. Then T must have the form —— I '— U , T T ._- -———|——-— (3.6.39) 0 I 1 — I —.I where -37- contains t 1, O as entries. If the last row of B0 is [1 1], then considering all the submatrices, Bj’ whose first two rows contain [1 -1], we see that it is necessary to take T, =-1 T, = 1 J1 or J1 sz= Z sz : -1 and for all the other entries Ti 2 0. If the last row of B0 is [1 -l], consider all the submatr1ces B3, whose first two rows contain [1 1], then we shall select either I T : -1 T -‘ I] ' ‘1 J1 or , 8 3'2: 1 T32: 1I In the above discussion we assumed that Y has no pole at T infinity. If it has a pole at infinity, the residue matrix can be realized without considering the state model approach and the corresponding network can be connected to that obtained from the realization of the other residue matrices. The above procedure can be extended to n-port LC networks for which the residue matrices are dominant [SL 1]. If the dominancy condition is not imposed on the residue matrices, the realization by the procedure given here, in general, may not be possible. CHAP TER IV EQUIVALENT NETWORKS 4.1 General This chapter is concerned with the parameter transformations on one port canonic networks. The theory of equivalent networks goes back to 1930's. In the classical theory one equivalent network is obtained from another by applying a non-singular transformation to the mesh impedance or node admittance matrices such that certain parameters of the network are invariant. In the following sections, these classical transformation techniques are summarized along with the development of certain new contributions to equivalent networks. In the last section state model equations are used to generate one equivalent network from the other. 4. 2 Equivalent Networks - Transformation Matrix Two n-port networks are said to be equivalent if they have identical terminal equations correSponding to a given terminal graph. The important point is that the two n-port networks have the same terminal characteristics. Their internal construction need not be identical. In general, the transformation of a network into another equivalent network is possible only if the internal constructions of .88... -89.. these two networks are known. If only the terminal equations are given such transformation is possible if the network is of one element-kind. This type of transformation, called the congruent transformation, is discussed by Cauer [CA 1] and has very limited applications. Howitt in his doctoral thesis, later in his papers [HO 1, HO 2] discussed the equivalence transformation of RLC networks, and showed that the equivalent electrical networks form a group. He also considered the necessary minimum number of elements to realize the given driving point impedance function. The equivalent networks are also considered by Guillemin [GU 1], [GU 3], who attempted to simplify the problem by using the normal coordinate transformations. Recently Schoeffler [SC 1], [SC 2] considered a transformation of equivalent networks, keeping the same network configuration but with different element values. Guillemin applies the transformation on the mesh impedance or node admittance matrices, while Schoeffler applies such transformations to the parameter matrices, thereby introducing additional constraints on the transformation matrix. In all of these existing techniques the topolOgy of the equivalent networks is held rigid, i. e. , in general, equivalent networks derived have identical topologies (networks having either the same number of meshes or the same number of nodes). In the following, the principles of Howitt transformation is considered briefly with some additional remarks. -90- Let the mesh equations of an n-port RLC network in the 8- domain be written as l Vzm+Ls+EIQI (41” where V = [v v , v ]T (mesh voltages) 1' Z ’ n I = [i , i , ..., i ]T (mesh currents) 1 Z n and R, L, K are square real matrices, i.e. , mesh resistance, mesh inductance and me sh elastance matrices. Let the vector I and V be I I transformed into vectors I and V by I = CI (4.2.2.) I V = CTV where C is a non-singular matrix C11 c:12. ' ° ' C1n C21 C22 '° ‘ C2n C = . . (4.2. 3) c c . . . c _— n1 n2 nn_J This transformation carries (4. 2. 1) into a new form for which the mesh resistance, mesh inductance and mesh elastance matrices are of the form .91.- I R =CTRC . T L =c LC (41% I K =CTKC To hold the driving point impedance invariant through the transformation, it is necessary that i1 = i1 and v1 = v1. Consequently, if C is taken as T1 o...ofl c21 C22 C2n (13$ _Cnl C111:- In general, it is easy to show that for the invariance of first k mesh voltages and currents, C must have the form ——- MJA) where Uk is a unit matrix of order k. 4. 3 Canonic Transformation for One-Port Canonic Networks The oldest known one -port LC canonic networks were obtained by Foster [F0 1] through the partial fraction expansion of the -92- reactance functions. Later Cauer [CA 1] gave other forms, which are canonic by considering the continued fraction eXpansion of the reactance functions. Recently Lee [LE 1] gave a lattice canonic form for the one- port LC networks which are derived by applying a combination of the Foster and Cauer procedures and modifying the final network by an ideal transformer. Yarlagadda and Tokad [YA 1] considered a rather restricted lattice canonic form which differes from that given by Lee and is derived from two cascaded Brune sections. The known canonic networks introduced by Foster, Cauer and Lee are shown in Table 4. 3. 1. In Section 3. 2 procedures have already been given for deriving one -port canonic networks from a general canonic network. There are four possible classes of one -port canonic networks obtainable by this process. These four classes correspond to the four reactance functions. Class 1: Z 2 Z Z Z () s(s +a2). . .(s +a2n) s = 2 2 2 Z 1 (s + a1). . . (s + aZn-l) °r 2 2 2 ( ) (s + a1) . . . (s + aZn-l) Y s = 1 Z Z Z s(s + a2) . . . (s + aZn) Class 2: 2 (s + a ) . (s + azn_1) N N A m V I N N NNI—‘N Class 3: s(sz + a2) . . (s + a: 1) 23(8) = 2 2 an. (s + a1). . (s + azn) Class 4: Z (s + a?) . (s + aim) Z4(s) _ 2 2 2 (4.3.1) s(s + a2) . (s + aZn-l) or Z 2 2 y (8) : s(s + a2) . . (s + aZn-l) 4 Z Z 2 Each class corresponds to a class of canonic networks as follows. Class 1: Class 1 is obtained by replacing a capacitor by a voltage driver in a canonic network. Therefore, the number of inductros is one more than the number of capacitors. For Y1(s) in (4. 3.1), one -port canonic network contains (n + 1) inductors, n capacitors and a voltage driver. Class 2: This class of canonic networks are obtained by replacing an inductor by a current driver in a canonic network. Therefore, the number of capacitors is one more than the number of inductors. For 22(3) in (4. 3. 1), one -port canonic network of Class 2 contains n + l capacitors, n inductors and a current driver. -94- T ,1 -1 - T EIKPHI I I . I IfkarI/blklt F I.- LIFE] L- THIfi (.I_I ,9 F _J{ Ame? IIiIgp-UI I“ I. .- :3, mas-I @QI awe-I Sam-I x a W a a I. a i Ir ..W- - 2. Jr :33..- a- W w _ , w . . v mdeU . .. m wwfl~0 N mdeU H mdeU W6— H .m .v 0..an III’DDIIu‘ ’l IIII’III, 46m _ WWW. 4 _ -_W m. ,. ._.-W I _wa WIWMMW.) v mdeO m nmdanv . N mdeU u mdeU -96- Class 3: This class of canonic networks are obtained from a canonic network by inserting a current driver between arbitrary nodes. Therefore, the number of capacitors is equal to the number of inductors. For 23(3) in (4.3. 1), one port canonic network of Class 3 contains n inductors, n capacitors and a current driver. Note that the current driver forms a circuit with some of the inductors in the network. Class 4: If in a canonic network, a branch or a chord is opened, and a voltage driver is inserted in series with this branch or chord, the fourth class of canonic networks results. In this class of networks the number of capacitors is equal to the number of inductors. For Y4(s) in (4. 3. 1), one port canonic network of Class 4 contains n inductors, n capacitors and a voltage driver. Note that the voltage driver forms a cut-set with some of the capacitors. The admittance functions are considered for Classes 1 and 4, and impedance functions are for Classes 2 and 3 for the obvious reasons that Y1(s), 22(3), 23(3) and Y4(s) have no poles at infinity. Transformation Matrix C Correspondingto Canonic Networks In considering the equivalent networks, it is generally assumed that one canonic network is given, and the problem is to determine the element values of another canonic network of known topology. Since the topology of both networks are known, certain restrictions are im- posed on the transformation matrix C. -97- Class 1: Let N1 be a one -port canonic network having the cut- set equations [U A1] = 0 (4.3.2) and let N2 be second one -port canonic network of the class having the cut-set equations [U A2] = o (4.3. 3) where N1 is assumed to be known completely (i. e. , topology and element values are known) and for N only the tOpology is known. The problem 2 is solved if the element values of NZ are determined. Because of the nature of this class of canonic forms, the known submatrices Al and A2 in (4. 3. Z) and (4. 3. 3) are nonsingular. The branch equations for network N1, are. 0 I TD 8 70 I U I * I 1 I 3:: 11: ”": A1 1 "“17" V1 U -L A __ ' _I __ s {I _ 1 _I (4.3.4) and for network NZ -98- ’0 I _ F-D s I — I—o: U_I $I I :::I I = —--—' A -—--—-— ——'-—- v 1 I 2 1 T 1 U I I 3L2 A2 _ I .3 _ I _I _ _I, (4.3.5) where the entries in the diagonal matrices D1 and D2 are the element values of the capacitors, the entries in the diagonal matrices 1..1 and L2 are the inverses of the element values of the inductors in these .I I *1 I V=V I=-I andV=Varethe * canonic networks, and I1 = ~11. 1 1: 1 1 1 1 ter minal variable 3 . Expressing (4. 3. 4) and (4. 3. 5) in symbolic form, we have 1* * ~ 4 3 6 1 — Yl V1 ( . . ) and 3:" 3“ 4 7 : .30 1 Y1V1 ( ) v* c v*' 1 ' 1 (4.3.8) *I T >:< I1 — C I1 from which it follows that I“ CT Y c v*' 4 3 Il _ 1 1 ( - -9) Comparing (4. 3.9), (4. 3. 7), (4. 3. 5) and (4. 3.4) gives which is an identity for all values of s. (4. 3.10) gives Equation (4. 3.11) can be rewritten as P O where V is an arbitrary orthogonal matrix. 1 To“ U I. D[0 _ — 0 D [0 U]C= 1/2 -99.. U 0 I D s 0 I U3 = I Z I T C" ----| AZ 1 __;_ I -. 1 U, SL2 A2 _JI ._ _ _.. .—II _. .— (4.3.10) For large values of s = jw. -- -I 0 U] c = D2 [0 U] (4.3.11) U o 2 Dilly VTDi/ [0 U] (4.3.12) U From (4.3.12) — -I o 131/2v (4.3.13) 2 U From (4. 2. 5) the driving point impedance remains invariant if -100- I—l 0...0_ C21 C22 . . . C211 0 = ' ' Cnl an o . o Cnn_] and from (4. 3. 13) it follows that C21: =Cn1=0 The final form of C is, therefore, TI 0— _ 1_I and (4. 3.13) can be rewritten as CTDl/Z = D1/2 1 1 2 IV T _ 1/2 -1/2 (:1 - 132 'VJDI Where it is evident that C1 is a non-singular matrix. For small values of s = ja) , (4.3.10) gives T T T C AlLlAl C - AZLZAZ 01' (4.3.14) (4.3.15) (4.3.16) (4.3.17) (4.3.18) -101- I—l — [_1 _ A L AT - A L AT CT 1 1 1 - 2 Z Z c _. 1J I. 1J (4.3.19) Substituting the expression of C1 from (4. 3. 17) into the above equation gives T1 —I I-l _I F1 _ ['1 _ ['1 — ’1 _I 1/ AILIAT 2 -1/2 -1/2 T 1/2 D V D D V D _ Z J _ J _ I J _ 1 _I _ J _ ZJ T = A2 L2 A2 (4.3.20) 01‘ T1 _ F1 _ I—l — r'1 - I-l _ I? T T A L A = A L A v D.1/2 1 1 1 D.1/2 VT -1/2 2 Z 2 D - J 1 J _. 1 J __ J _ 7- J _ 2 parametric matrix of N1 parametric matrix of N (4.3.21) This result indicates that the parametric equations of network N1 can be transformed to the parametric equations of network N2 by an orthogonal matrix. The unknowns in this transformation are the matrices VI . D2 and L2 -102- Class 2: The procedure for the derivation of the transformation for this class is similar to that given above except for the fact that a current driver replaces voltage drivers and circuit equations rather than cut-set equations are considered. The result then has the same general form as that in (4. 3. 21) with cut-set matrices replaced by the circuit matrices. Class 3: Let the canonic network Nl have the circuit equations _V T u] 1 = 0 (4.3.22) V2 — - [Bl and let the canonic network N2 have the circuit equations _ '... V1 [B U] = 0 (4.3.23) 2 I V 2. L J where again N1 is known completely but only the topology of canonic network N is known. Note that unlike in Class 1 and 2, B1 and B2 2 are rectangular matrices. Deriving the chord equations for network N1, we have * 1 v = B I _____ 12 (4.3.24) and for network N where the entries in the diagonal matrices D1 of the capacitor element values, similarly the entries in L the inductor element values, and V *I I2 :1: I I -103- J; 2 -12 are the terminal variables. Consider the following transformation D i ls 2 "‘=V,1 $ 2 (4.3.25) and D are the inverses =-1 For large values of s = ju), from (4. 3. 27) we have — Which implies 0 U m J 0 L1[ P U]C= O U _ L 2 [O 2 >1: Z’V U] 2 I 1, L2 are I = V2, and (4.3.26) (4.3.27) (4.3.28) -104- where V is an arbitrary orthogonal matrix. C has the form For small values of s = jog, (4. 3. 27) gives _ ,._ _ r II 1 1 -1/2 BlDlBl V 1 J I—l : D B -1/2 B2 2 2 L2 (4.3.29) The transformation matrix (4.3.30) -1/2 L2 (4.3.31) where the relation in (4. 3. 29) has been used in the final form. The unknowns in the transformation are V. L2 and D2. Class 4: The derivation of the transformation for Class 4 canonic forms is identical to that in Class 3 except for obvious interchange of voltage and current drivers and circuit and cut-set matrices . -105- 4. 4 Realization of One -Port Canonic Networks by the Canonic Trans formation Consider first the realization of some of the known one -port canonic networks by using the canonic transformation developed in the last two sections. The parameters for equivalent LC networks are related by (4. 3. 21), where the matrices V, D2 and L2 are yet to be determined. For a canonic network of Class 1, containing n inductors and n-1 capacitors, the number of unknowns in (4. 3. 21) is n +(n-1) + (n-1)Z(n-2) = 3912451) (4.4.1) where (n-1)(n-2) correSponds to the number of unknown entries in the 2 orthogonal matrix V. In (4. 3. 21) the matrix F. 2 R T -1/2 A2 L2 A2 -1/2 is symmetric. Therefore, the number of equations is equal to the number of unknowns. However, the equations in (4. 3. 21) can be Simplified by rewriting 1 A11 A12 1 1 B11 12 = _ IT I -1/2 v A A V 12 22 2 _12 22_ __ _ _ 2 _ L _ _ _ (4. 4. 2) -106- , , 1 2 Combining DZ/ with the matrix V, we have — — F — — _- l V ' A11 A12 1 B11 1/2 T T 1/2 = 'T D __ 2 V A12 A221 V D2 312 _ ._. g. _I _— from which it follows that If we let T12 VDZ/zxT the (4. 4. 5) can be written as A = B 11 11 T I A12X ’ B12 T I XAZZX _ 132.2 These equations are quadratic in the unknowns. (4.4.3) (4. 4. 4) (4.4. 5) (4.4.6) Steepest descent, Gauss-Scidel or Newton's method can perhaps be used to obtain a numerical solution [HH 1, ZU l]. The difficulty, however, is that the approximate solution must be given. This appr oximate solution -107- can perhaps be obtained by using the analog simulation methods discussed by Rybashov [RY 1]. For some of the known canonic forms an analytical solution is possible. In the following we shall discuss the realization of certain known canonic forms of Class 1. In obtaining these forms, parametric equations in (4. 3. 21) are considered. Canonic network N1 is assumed to be known. I. Cauer's First Canonic Form The topology of the first Cauer form is given in Table 4. 3. 1. In (4. 3. 21), A correSponds to the submatrix of the cut-set matrix for 2 the Cauer network. 71 It is evident that A2 LZ A; is in tridiagonal form, for any diagonal matrix L2,, and this property is not altered after pre- and post- multiplication by the diagonal matrix V1 _Fb -108- Fl— T v _J Fall A12 T J A12 Azzj ‘: V A (4.4.7) where A is a tridiagonal matrix and all other matrices are defined by comparison with (4. 3. 21). The known matrix B can be reduced to A by an orthogonal matrix transformation described by Frame [FR 1]. An alternate procedure, and one which has advantage over the tridia- gonal method in that it also provides a proof that element values so obtained are all positive, is presented here. Since B is positive definite, it has real positive eigenvalues where B : ll 21 n1 T PJLP P12 ... Pln P22 . . P211 PnZ Pun—J (4.4.8) (4. 4. 9) -109- andJ\. is a diagonal matrix containing the eigenvalues of A. Substituting (4. 4. 8) in (4. 4. 7) we have l T 1 P_/\_P = A (4.4.10) V VT or 1 l =PT A P=QTAQ (4.4.11) VT V where " 1 T — "‘ ‘1 o a T q11'” qnl T 1 p11 p21 pnl Q = . . =P : . 0 . O T O V . . . q1n qn: _ — _pln pm: _0 V _. (4.4.12) Comparing the terms in the above matrix equation, we obtain : z ..., -_- (4.4.13) q11 p11’ q12 plz’ qln p1n Since Q in (4. 4. 11) is an orthogonal matrix, it transforms A into a diagonal form. The matrices Q and A are obtained, simultaneously, from the above information as demonstrated in the following. Since q, is an ‘eigenvector 1 (A -in)qi=o (i: 1, 2, n) (4.4.14) ~110- 01' all i ‘312 0 0 0 -a12 aZZ-kl -a23 O 0 r—q11_ 0 "’23 a33'x1 ’a34 0 qu : O 0 0 "‘34 8144*. 0 q , _ 111.1 o _ _ (4.4.15) From the first equation of (4. 4.15) (3‘11" x1) q11 : a‘12 q21 (4.4.16) (all ”‘2’ q12 = 2"12 q22 (all " kn) qln = 3‘12 an where b1 and ql, are known. After squaring and adding the 1 1za11 equations in (4. 4. 16) we have 2 2 2 2 2 2 2 2 = = "' .0. + -x a12(q21+”‘ +q2n) 8‘12 (all x1) (111+ (all n) qln (4.4.17) Taking the positive value of a12 as determined from (4. 4. 17), the values of qz, can be obtained from (4. 4.16). Then 1 2 x + + 271 azz‘q21 1 qzn n -111- The remaining coefficients ai, in (4. 4. 15) are obtained by an identical procedure. After obtaining the matrix A, the network parameters are determined as follows. Let A be written as _1 _ r1 -1 _ F1 -1 1 1 -1 F1 A '—'-' L2 D-l/Z -1 l 1 -l 2 L. __ (4.4.18) with d2 T 1/2 d3 D- : o . 2 ° (1 n and 11 l 2 L : Z 1 n Forming the matrix products in (4. 4. 18) ~112- O1 (— 1 -1 d o - O - 1 1 2 [ F—all a12 0 ~11d2 (11+12)d: —1 d d o .. . o -a a 2 2 3 12 22 '323 0 A: o ~£2d2d3 (1 +2 )d‘Z . . . = o -a a -a 2 .3 3 23 33 34"' o o o o _J o 0 (4.4.19) Therefore, network parameters are 6111 = ’1 a d2 ==._LE a 11 1 _ a11 a22 _ all 2 2 — 2 -a11 — 7(311322-312) (4.4.20) alz a12 a a d z 23 12 3 a - a 11 22 12 a a a. (a a a —) a 2 1 ._ 33 23 1 33 11 22 12 " 11 a23 ’ -§—"‘___— :'___——_ .3 d3 d2d3 a112‘13 aL12357-3 . 1 . Note that the numerical values of the capacitors 1s -—2- and inductors (1 § 1 . 1 theorem. i The positiveness of the parameters is established by the following -113- Theorem 4.4.1: Let B be a symmetric and positive definite matrix of order n and let N be a real non-singular matrix, then NBN is positive definite . Proof: After Hohn [HN 1] Since the matrix A1 L1 Al 15 p031t1ve def1n1te,the matrix A IS positive definite and the submatrices corresponding to the principal minors of A are positive definite. In general, the eXpression for Ii contains the first principal minor of order i x i, which is positive. Therefore, there exists a first Cauer form for a given canonic network with positive element values. II. Second Foster Form The topology of the second Foster form is given in Table 4. 3.1. In this case the matrix A2 corresponds to the submatrix of the cut—set matrix and has the form _ _. 1 1 1. 1 A = l 0 (4.4.21) 2 1 o Therefore, for any diagonal L2 and D2 matrices we have -ll4- Fa a T . . . a _ _. __ 11 2 1 1 fl 1 In F; A _1 T a a ll 12 A L A = 12 22 O -_— 2. 2 2 D-l/Z -1/2 . AT A _ Z ‘ _ 2 a a 12 22 1n 0 nn —' — (4.4. 22) where A.22 is also a diagonal matrix. Writing the above parametric equation in the form ”1 ‘ i. ‘ ’1 ‘ ' A ‘ 11 1312 311 12 = (4. 4. 23) T T V B12 B22 V A12 A22 _. _ (._. ...( ._. —J ._ .— and equating the submatrices gives a‘11 = b11 T T = 4.4. 24 V B12 A12 ( ) T V B22 V - A22 Since V is an orthogonal matrix which transforms B22 into its diagonal form, the eigenvalues of B22 are the diagonal entries in A22. The matrix V can be obtained by any one of several methods, and the matrix A is obtained as a solution to (4. 4. 24). y writing (4. 4. 22) expli- The parameter values can be obtained‘b citly in terms of L2 and D2 as -115- F-l I F. 1 1 - F. -1 F. '7 . I 1 d2 ii: 1 Z n d all 312 aln . ° 12 I 2 a a ° 2 12 22 d . . d o n ' o _ _j In '1 n - . n " J __ _J _aln a‘nnJ 04.4.25) which gives 2 a.. 2 d = i 1 a , 141 11 a1i Ii = a— (4. 4. 26) 1¢1 1‘ and 1 1 where —2 represents the capacitor element values and T are the d, i 1 inductor element values. Since B22 is positive definite, the eigenvalues of B22 are all positive, implying aii (1 ¢ 1) is positive (all is also positive since b11 > 0). Since A is positive definite, 11 is positive. 2 By the above reasoning all L's and di '5 are positive. Note that 1 aii' i 16 1 correSponds to the resonant frequencies of the resonators in the second Foster form and that they are distinct. -116- 111. Second Cauer Form The topology of the second Cauer form is given in Table 4. 3. 1. The matrix A2 correSponds to the submatrix of the cut-set matrix and has the form _ _1 1 1 1 1 1 1 1 1 1 1 1 1 1 1...1 A = (4.4.27) 2 1 1. 1 '1 -1 . . . . For canonic networks of Class 1, A ex1sts and 1t 15 g1ven as Z l —l l —l -1 A2 = 1 “'1 (4.4. 28) o '1 l In this case, taking the inverse of the correSponding parametric equat1on, i. e. , taking the inverse of each matrix in (4. 3. 21) we have ~117- L'A (A2) 2 2 _ 2 J 2 (4.4.29) Since the right hand side of (4. 4. 29) has the tridiagonal form, the techniques already presented can be used to find the matrix A, and the network parameters. It can be shown that all the element values are positive . IV. First Foster Form The topology of the first Foster form is given in Table 4. 3.1. In this case the matrix A2 corresponding to the submatrix of the cut-set matrix has the form 1 o o o 1 1 A = O (4. 4. 30a) 2 1 1 1 O 1_ The parametric equation in symbolic form is F — — — — a A T T T V B12 322 V A12 A22__ L. ._d -— —‘ '_ - — which can be rewritten as ~118- Tb B — Fl ‘1 Fa A _ —1 _ 11 12 11 12 T _ T B” B27- V A112 A22 V __ A __ _l _ __ ._ 3 (4.4.30) Since B is a symmetric and positive definite matrix, there exists an orthogonal matrix P which transforms B into its diagonal form, i. e. , T 1212T B=PAP =P_/\_/_/\_/P (4.4.31) where P and .A. are calculatable. If Q is an arbitrary orthogonal matrix, then (4. 4. 30) can be written as 12 T A LlleQTL/ AZ 2 2 2 - 2 -1/2 VT D2” D2 L — — a—J — —J h 2 T : (pA1/2)(_,(_1/ p) (4.4.32) It follows from (4. 4. 32) that _ _ __ _ 1 1 1/2 AzLi/ZQ=P_A_ (4.4.33) T -1/2. v D — —a ._. 2 -—J or 22 3 23 Zn 2 PJLl/ 2... 33... -119- —' F 1/2 0 f l 'V n2 1/2 1/2 dzll dzfz ‘Vn3 V' d [I/Z nn n.l '— F 1/2 , 1 P12 pln 1 p22 pZn pnn x1/2 A1/2 12 2 1n n 1/2 1/2 A 22x2 2n n 1/2 X pnn n '7 T- 0 q11 qu SZn q21 qzz Snn L‘an ' d 1/2 1 2 1 / 11n xl/Z (4.4.34) -120- where 8,, is a function of v”, I, and d . 1J 1J 1 i From the first row of (4. 4. 34) 1/2 _ 1/2 21 q11 ‘ p11)‘1 11/2 41/2 (4.4.35) 1 q12 :: p12 2 1/2 1/2 = X £1 qln 1n :1 where Ii/Z = b112, p,, and L's are known from (4. 4. 31). Therefore, U 1 from (4. 4. 35) ql, can be calculated. To obtain the other qij' pre- 1 multiply the matrix equation (4. 4. 33) by its tranSpose, which can be written as 71 QTLl/ZAT 1 AZLléZQzAzQTCQ (4.4.36) 2 2 .D- U Furthermore, because of the Special form of A2 in (4. 4.30a), C has the form F- .— C11 C12 C1n (4.4.37) C '3 C12 C22 0 c1n 0 Gun From (4. 4. 36), for q, an eigenvector of C, 1 -121_ (C ->\. U) q. = 0 1 1 or f—' _— 4. C11 1 C12 C1n _ _ c c ..x o o q” 12 22 i q . = 0 21 o -x . C33 i 9 cl 0 o. ..c -1, qu _ n nn _1_ - (4.4.38) where Cll' ql, are known quantities. From the second row of the 1 matrix equation in (4. 4. 37) we have (C2241) q21 = 'q11 C12 - = - 4.4.39 (C22 12) q22 q12 C12 ( ) (CZZ-Xn) q2n : "q12 C12 for ql. )1 O, and 1 q21 _ (:12 q11 k1"sz C 32—2 ._- 12 (4.4.40) q12 xz'czz an _ C:12 q11'1 n 22 -122.—- Multiplying each one of the equations in (4. 4. 40) by (1:1, qu’ . . . , Z q1n re Spectively, give 3 C111C121 ) -c ' q12 2-2 )..-c (4'4'41) _ C12‘11n q1n an - RY «c?Z In (4. 4.37) c,, (i gt 1) correSponds to the resonant frequencies of 11 the resonators in the first Foster form and they are all distinct. Further- more, cl, is unequal to zero in the matrix C. Therefore, the following 1 theorem can be stated. Theorem 4. 4. 2; Let r‘ C "‘1 C11 C12 ' ' ‘ 111 C12 C22 0 c 2 (4.4.42) C33 Cln Cnn . o ‘ . . o I 3 he a non-Singular symmetric matrix W1th eigenvalues X1. - - . . kn (..,i> 0) Such that the entries c and c,, are all non-zero. Let c” # c,, for 11 11 11 JJ 1% l, i#j. Then c__ #hj for alli?é 1. 11 -123- Proof: The proof of this theorem is established by contradic- tion. Let Q be an orthogonal matrix which transforms C into the diagonal form. Assume that c22 = Xk. Then for )1 -X . c11 1. C12 Cln _ 1 q 1k C12 0 0 q = 0 (4.4.43) -). 2 C33 k . k 0 -1 q C1D. Cnn kl 1..— 111(— From the second row of the matrix equation (4. 4.43), C12 q2k = O, which implies q2k = 0, since c1i ¢ 0 by hypothesis. Then from the third, fourth, . . . , nth row of (4. 4. 43) q3k=q4k:°" =an=0 Also from the first row c‘12 qZk Which implies qZk Therefore, the eigenvector qk corresponding to the eigenvalue Xk is identically zero and a contradiction is established --- by definition an eigenvector is a non-zero vector. An identical proof can be repeated for all c,, (i a)! l) to establish the theorem. 11 -124- Now return to the solution of (4. 4. 41), since Q is an orthogonal matrix q11q21+q12q22+... +q1nq2n=0 (4.4.44) Substituting for qu q2j from (4.4.41) into (4. 4. 51) gives 2 2 c q c q c q 2 2 2 )‘lf—Cll + iL-C—l_+ ...+)\—l-F-l-E: 0 (4.4.45) 1 22 2‘ 22 n 22 But since C12 16 0, it follows that ‘12 q2 q2 2 i—lé-l— + «inl—u + . . . + ‘K—l:—- : O (4.4. 46) 1 22 2'°22 n“ 22 Repeating the procedure given above establishes all c” (i 1! 1) as a 11 solution to the polynomial obtained from q q2 (12 11 + 12 +... + 1n = 0 (4.4.47) X -C.. X -C.. X -C.. l 11 2 11 n 11 The degree of this polynomial in cii is (n - 1). By using the positive pair property, it can be shown that all cii (i )1 l) are positive real and distinct (from the unpublished notes of Dr. Tokad). Therefore, the roots of (4.4. 47) gives C . q __ 11 11 “(111 x -c,, 1 11 C11 in : qlz X -C (4.4.48) . 2 11 c 11 q : in q1n X -c n -125.. After squaring and adding, we have ( 2 2 2 2 2 q11 qln qi1 + ... +q, — l - C i X _c” +... + X -C,, 1 11 n 11 ‘ (4. 4. 49) where c1i is the only unknown. The network parameters are found by applying a procedure similar as described before for the previous cases. The procedures for Class 2 canonic networks are similar to those of Class 1 and are not consider further. The procedures for Class 3 and 4 are similar to 1, except when it needs an inverse of the submatrix B2. But, in general, Class 3 and 4 can be reduced to Class 2 and 1, respectively, by the following procedure; a canonic network of Class 3 is reduced to Class 2 by introducing a capacitor C0 in series with the current driver and of Class 4 is reduced to Class 1 by connecting an inductor parallel to the voltage driver. In considering the solution, N1 and N2 of Class 3 (4) are augmented by the above procedure with an identical capacitor (an identical inductor). In conclusion of this section, realization of canonic networks by using transformation matrix needs a solution of nonlinear algebraic equations. In general, an analytical solution may not be possible. However, in certain Special cases, these equations can be linearized. ~126- 4. 5 A Realization of Canonic Networks by Tridiagonalization In this procedure, the transformation from one canonic network, N1, to the other, N2' is realized in two steps. First, transform the given network N into its equivalent first Cauer form 1 NC; then transform N into N2. The reason for the use of Cauer c network as an intermediate step is obvious --- the parameter matrix for NC is tridiagonal. The transformation matrix T from N to N is then obtained 1 2 as the product of two transformation matrices T1 and T2 where T1 is used to transform N1 into N and T2 transforms N into N2. The c c construction of T is already given in Section 4. 4. To construct T l 2 let the parametric equation be written as T ATT = A T = B (4.5.1) where A is tridiagonal and corresponds to the first Cauer form Nc’ B corresponds to the unknown canonic form N2 and V is an orthogonal matrix. The topology of N2 is assumed to be known. The transformation in (4. 5. 1) can also be written as A = 13 (4.5. 2) The matrices B and A have the following form -lZ7- 7' _ bll b12 ° ' ° b1n blz bzz O O O bZn B = ' ° ' (4.5.3) bln bZn ' ' ' bnn and F— ... all all "a12 azz “a23 A : -.'=1Z3 3.33 (4.5.4) an-1,n-l -an,n-l -a a n,n-1 nnJ The first row in B is reduced to tridiagonal form by the symmetric orthOgonal matrix [FR 1] 2 T X1 X1 Where Fblz + 3.12 bl3 X = - l _bln ._.: ~128- and F1 — {—1 7 F (1) bll -b12 O . . . O (1) (1) (l) (1) -b12 b22 bZ3 . . . b2n = ° (4.5.5) (1) 9 P23 (1) '(1) 0 b2n . . . bnn _ ..J Where b”) f t' s ofb andb =a b =a b =a . ij are unc 10H ija 11 119 12 12: 22 22 The second row of B1 is transformed to tridiagonal form by an ortho- gonal matrix Where -129- _ bm + 23 a 23 (1) b24 '(1) bZn — A and n 2 2. a = E: b 20 23 ”1,2 1 The matrix B2 has the form —l — Fl — _all -a12 0 . . O— Bz = 1 1 1 -a12 a22 ”(225) O 0 v. v. . by: — — _ — 9 _ o o Lg? . . .331: (4. 5. 6) where b(223)___ az3, b33233: a33 and the entries bi?) (i, j )6 2, 3) are obtained (1) in terms of b,. . 1J The number of orthogonal transformations involved in this procedure This procedure is continued until B is tridiagonalized. is n - Z and in the last stage, we have F— + (n-3) (n-3) a'n-1,n-l - an-1,n bn-1,n-l bn-l,n Bn"2 — + : Vn-Z ( 3) ( 3) Vn-Z - a a b n- b n- n-1,n n,n n-l,n nn _ _ _. J (4.5. 7) The equations of equality involved in the successive transformation are 11 11 n 2 3‘12 = Z bli 1=l 8.22 = f(bij) n 2 a : b(21,) 23 i¢1.2 1 _ (1) __ a33 _ {(15ij ) .. f(bi,j) (4.5.8) (n-3) _ an-1,n-l f(bij ) ’ f(bid) _. (n-3) _ an-l,n3 ‘ f(bij ) " f(bi.j) _ (W3) ._ an'n _ f(bij ) _ Mb“) In the above set there are -131- n + n-l = Zn-l equations, correSponding to the Zn-l unknowns. However, these equations are nonlinear, and have no Special advantage over the quadratic equations of Section 4. 4 insofar as solution is concerned, except the number of equations are minimized and are eXpressed in terms of the network parameters. Synthesis equations similar to those in (4. 5. 8) can be obtained by a different procedure, which is as follows. Since the tepology of the canonic network N2 is known, the entries of the admittance matrix Y2 can be obtained as an explicit function of the network parameters. The driving point admittance is the ratio of the principal minor and the determinant of Y2. Due to the topolOgical properties of canonic networks, the principal minor and the determinant of Y do not have any factors in common. 2 In a similar manner the driving point admittance of N1 can be obtained. The desired synthesis equations are obtained by equating the two admittance functions. The resulting equations are also non- linear in network parameters. 4. 6 Equivalent Networks - A State Model Approach In Section 4. 2 the properties of equivalent networks are studied from their ‘mesh or nodal equations. Equivalent networks can also be generated from each other by considering their state models with the restriction, of course, that the operator matrices are related by a similarity transformation. -l32- The state model for canonic networks considered in Chapter 2 is of the form r T r O Cb d Vbc B12 Vbc 0 * a-t- : + V0 Lc Icl ~B12 0 1c! -Bll I— A. _ ...J -_ —.J .— _1 _ J (4.6.1) * T Io ' "1311ch and its transformed state model is 7 u— (— - 2 - 2— F ' _ 7 _ V 0 C 1/ BT L l/ V 0 d bc b 12 c bc dt " 0 1 -1/2 -1/2 ' -1/2 Ion Lc BlZCb 0 let 'Lc B11 and (4.6.2) I T -1/2 ' o ’ 11 c C! or r " — T“ r- “ r a 0 0 V2 A1 V2 d >§< dt _ + V0 I3 -A1 0 I3 -Bl_) _. ._. ... _J _. ... ... (4.6.3) 3:: T IO — ~B1 I3 -133-— Since a similarity transformation does not effect the terminal character- istics of the correSponding network, it is always possible to transform the state model of canonic network N into the state model of canonic 1 network N2. Let the state model of N2 be written in symbolic form as r- . — r‘ _ (— — F — T I d V2 0 A2 V2 0 ,4 _ ._. v' dt . , + 0 I -A 0 I -B _ 34 _ Z _ _ 3‘ _ 2_) 1* — ET 1* 4 6 4 o " ' 2 3 ( ' ° ) >i< * Note that V0 and I0 in (4. 6. 3) and (4.6. 4) are identical. A procedure for constructing the transformation matrix is given by the following theorems. Theorem 4. 6.1: Let the operator matrix correSponding to (4. 6. 2) be _ T... 0 A1 A : (4.6.5) -A1 0 _ _J Then the non-zero eigenvalues of A1 A? and A? A1 are identical. Proof: Consider —l34- . 0 A? 1 s1 .A;r IBU'AI =det([8U]- >._. -A o A s 1 1 z . _ ..J 1 (4.6.6) where S1 and S2 are the scalar matrices of the forms sU1 and sUz where subscripts l and Z are used to distinguish the order of these matrices . Expanding the determinant given in (4.6.6) [GA 1], we have (1) IsU-Al '1‘ -1 lszl |sl +A1 (52) All 1. 'r '52’ (51+ E A1A1| Z T -l ISZI ls U1+A1A1| Isl) (4.6.7) or expanding by the “wand row (1) z 1T -1 )aU-Al = [sllls U2+A1A1l lszl (4.6.8) Therefore, from (4.6. 7) And (4.6.8) we have (1) z T -1 2 T -1 IBU‘AI = 'SZI IIU1+A1AII (81' :3 '51. '8U2+A1A1' 'Szl (4.6.9) T It follow! from (4. 6. 9) that the non-zero eigenvalue. of Al A1 and A1 A? are identical. Theorem 4. 6. 2: Let Am and Am be two operator matrices having the same eigenvalues. exists an orthogonal matrix of the form -l35- which transforms A”) into A( ). Proof: Consider first the lemma Lemma :4. 6.1: Let (1) and (4.6.10) (4.6.11) Then there (4.6.12) where D has one of the forms shown in (3.4.2) or (3.4. 3). Let A”) and A3 have the same eigenvalues. Then there exists an orthogonal matrix of the form p“) = (4.6.13) which transforms Al to A3. Proof: Since (A A?) is a symmetric matrix, there exists an 1 orthogonal matrix Pl [PE 1] such that T T T = DD (4.6.14) Pl (A1A1)Pl To show that this is true we follow a technique similar to that used in l the proof of Theorem 4.6.1. Indeed, since A( ) and A3 have the same eigenvalues we can write -1 PA(1)P = A3 Where P is orthogonal, hence lsU-PA(1)P-ll = ISU'A3 ~137- IPI (su-A<1>) (p4) = IsU-A 3 lsU—A‘”! = lsU-A3I From (4. 6. 9) We have 2 T -1 2 T -1 l51||5U+A1All [szl=|sl| IsU+D DI'SZI _ 2 T -1 2 T -1 —ISZIISU+A1A1||81|=|SZIISU+DD(|SII (4.6.15) . T T . . . Wthh shows that A1 A1 and D D have 1dent1cal eigenvalues and (4.6.14) is established. Also from (4. 6.15) it follows that A1 A? and T . D D have identical eigenvalues. Therefore, there ex1sts an ortho- gonal matrix P2 such that T T T _ T P1(A1(P2PZ)A1)P1— DD 01' PTAP =D M6Jm 2 2 ( ) has the same (3) Continuing with the proof of the main theorem; since A eigenvalues as A”), it follows from the Lemmas that A can be H transformed into A3 using an orthogonal matrix P of the form given in (4. 6.13). Thus (P')T A(1)P' = A3 and ”T (p ) A(2) )1 -138- 01‘ Therefore H l T g H (P )(P) Alp (p )T = A2 where l H P = p (p )T Consider now the similarity transformation from N1 to N2 r ‘ ‘ 1' 1‘ V2 F131 V2 = (4.6.17) I I P I 3 2 3 _ J _ 4 _ .1 From (4.6. 3) we have '- ' r- - r .. r " - "- F- "1 — _ ' T T T V P 0 V2 P1 0 A1 P1 2 1 g: t ' T ' T I P -A 0 P2 I3 PZ -Bl _ 3i 1_ Z__ _ l _J _ .. _ _ _ J ... _ * T ' (4.6.18) I0 — -B1 P213 which is to be identical to (4.6.4), i.e. , we require that P — F- — -— T— 14“ To .1 o A. T -A 0 P7- -A1 _ i _ J (4.6.19) -139- T 13231-132 T T T P1 Ale—AZ and (4.6.20) P 131 =13Z Since N1 is given and the topology of N2 is known, the forms of Az and B2 are known. Hence the problem is to find orthogonal matrices P1 and P2 such that (4.6. 20) is satisfied. The network parameters can be obtained by compariing (4. 6. 2) with (4. 6. 4). The realization of any canonic form by this procedure may not be possible, in general, because of the nonlinear algebraic equations involved in (4. 6. 20). However, some of the known canonic froms can be realized by this procedure. An example is illustrated in Chapter V. The development for n-port canonic networks is identical to * . the above, except V0 is now taken as a vector correSponding to the drivers. CHAPTER V EXAMPLES 5.1 General This chapter is devoted to the examples which illustrate the various techniques presented in the previous chapters. 5. 2 Example 1 Realize the LC network whose Specifications are given by the following state model. I I 1 J T — Q {P 0 0 l. l. v --];- 0 '1' V2 '4 4 2 4 4 P v 0 0 0 l v 0 0 0 v d - -V + a = dt 0 ' 0 O 0 1 17 l O O 0 17 .- 1 _ l l - 0 — 0 18 --2- -2 0 0 18 z L 1 _ _. .. .. - 4 — 1 (5.2.1) and r. _1 1 7 3:: 23 d * 1 3 -1* i = —— v + [ -' Z ] 1 0 4 dt 0 4 i8 v2 * * d .* + — - —- + -l 0 v V1 ‘ 3 dt 11 [ 1 0 V3 -l40- -14l- Solution: Since the eigenvalues of the operator matrix are pure imaginary, one of the necessary conditions is satisfied. Comparing (3. 3. 2) and the state model in (5. 2.1), gives _l 1- 'Z 4' K1 _ 1 0 z — "1 1 0 ...K : 2 1 l '2 2 _1- K _ '4 3 _ o L J (5.2.2) = 0 K4 1 K _ 4 5 _ o — 0— ..K : 6 _1_ 2 (__ ... 23 K7 - 4 1 3 K8 = [4 Z] K K K K 10— ll 12— l3 l4— -142- Let the following matrices be identified as (5.2.3) b T 9 B23 Lb B33 ;' b1o — ._J b5 I321 = b6 B11061311 = 11’7 and (_- 1 _ £11 12 L +13 L BT = c 23 b 23 ‘12 ’22 Therefore, (3. 3. 3) can be written in matrix form: fixv ’1' SF -144- N- 3' V I 0000000000oooooogk-wwwonToo l . J l3£§$$$3z§d—-dnf~n¢r—mmfll LuovngnageIufinmnbngpnn—l I _ _WH 11 _ -— —1+ l | l 1 _ I .. -l+—-Id _ ~H—H-N I 4.1—(«I —|¢\| —IN—1~- AM I 1 I T . T . 1 l I 1 -l+ -N‘ 4* --1+ -Iv —-I+ l 134+ 4+ 4+ '0 Npkfrl. -Nn¢\ntgl\cn0‘QZS’BSTEEESNN‘NNN -145- In (5. l. 4) there are 24 linear equations, in 20 unknowns. However, equations 6, 7, l9 and 24 are redundant, and they can be omitted. By writing the remaining 20 equations in matrix form, it is verified that the coefficient matrix is nonsingular and the solution is given by cll=4, clz=0, C22=4' b3Z=-l, b34=l, b4z=0, b43rl, 111-1, [la-0. 122 - 2, bl=-1, bz=0. b3=.1' 134:0, b5=o’ b6="l, b7=‘g b8=3, 139:0, .. 5.2.5 b1o 0 ( ) From (5. 2. 3) and (5.2.5) .... _ ... I .1 T T 6 -l 0 131166 311 I311 96 B12 _L Y = = -l . ‘ 0 C T T l 0 0 BIZCc Bll Cb + 312Cc B12 I — Applying Cederbaum's algorithm to Y6 and rearranging the rows and columns, we have ("0 o : 1 1 3 0 1 0 l 4 0 O 1 Y = 1 o (-1 o (5.2-6) c l 1 1 -1 o o 1 ' o o . 1.... -- 5 1 0 0‘ Similarly T Lc + B23 Lb B23 2 = L T B33 Lb B23 -l46- T B23 Lst3 T B L B 33 b 321 and after the application of algorithm 22 32 21 B31 r3 —P Therefore, the circuit matrix is (5. 2. 7) (5.2.8) -l47- (— v0 F—l 1 o o 1 v2 -1 o o o 1 v3 0 -1 o o 1 v4 ___ o -l l l O 1 v5 _ 1 -1 o 1 1J v6 V7 V8 ._ VI & Which is realizable and the correSponding network is given in Fig. 5. 2.1, with the element values taken from (5. 2. 6) and (5. 2. 8). CS} 4.5 Fig. 50 201 -148— 5. 3 The following example is given to illustrate the procedure des- cribed in Chapter IV, Section 5. Example 2 Determine the element values of the lattice canonic network ‘ in Fig. 5. 3. 2 which has identical driving point impedance with the Cauer network given in Fig. 5. 3. l. l 2 3 . L 4 WW j—rvmm -;J( A j (_2—E— L Z(3)‘—'—) | (__ F—ermi— L5 Fig. 5.3.1 Fig. 5.3.2 le Z(s) ——> 13‘- ll Solution: Let the linear graphs of these equivalent LC networks be. drawn as in Fig. 5. 3. 3. Fig. 5. 3. 3 -149- The parameter matrix for the Cauer form is O F'all -fl — 12 " “a12 1 — 0 4.1 12 22 23 (5.3.1) The parameter matrix for the lattice section will be of the form F1 " B : -1/2 2 l4-+15 +16 = -(£5-+£6)d2 d (14-+15) 3 L. b11 "b12 = “blz b22 b13 'b23 71 T A2 L2 A2 D-l/2 2 _ _J - d (£5+£6)d2 (24+l5) 3 1 d2 I d d (15+ 6) 2 " 2 3 -15 d2 d3 (14+15)d3 b13 'b23 b33 (5.3.2) -150- Since the first row of B is not in tridiagonal form, an orthogonal matrix defined by V1=U-ZTXXT XX is used to transform B into a tridiagonal form where -. ‘1 ’b12 + a12 2 X = +b "H2+bn = 13 Therefore, Vl can be expressed in terms of b,, as Since and the n u 1 b12 ‘b13 V1 =Zf-3— ‘b -b 13 12 — — - —' _l — b bll : b12 b13 11 __—)————-- = (1) I - V -b 1 ”b12 I b22 b23 1 12 d I — — O b -b b (l) _ b12 " a12 12 (5.3.3) (1) _ “b12 0 (1) (l) b22 b23 (l) (1) b23 b33_ (5.3.4) Although in (5. 3. 5), there are two possibilities for the Sign of b shall take the positive sign. r (1) bZZ + (1) - Z3 01' __ b22 -b -b23 b "b23 b33 _J -151- + (1)— ' 23 hm 33J - 23 In this case 71 4 ._.4l Nod V73 17? l 4 ._ .bpd (5.3.5) (1) 23 (5.3.6) (5.3.7) From (5. 3. 2) and the tridiagonalization procedure we also have the relations 2. 2 3 = b12 + b13 and 2 2 ‘312 b33 + b13 b22 Substituting the expressions of b12' b22’ 1323. 1333 8i 01‘ b12 b13 b23 = 2b 22 b33 (5.3.8) (5.3.9) ,we ven in (5. 3. 7) and -152- (5. 3.8) into (5. 3. 9), we obtain 3/2 1/2 1 4 1 2 1 «l3 2 «1‘3 2 3 54 b13"1"8'b13+8'+‘3'6'(3'b13) b13" 36 (3'b13) b13 ‘ 0 (5. 3. 10) . . . . . Z whlch is equivalent to a fourth degree equation in the unknown b13' One of the solutions of (5. 3. 10) is 3 b13‘2 CorreSponding to this solution from (5. 3. 8) and (5. 3. 7), we have \f— 3 b12= —23' ' b22927 ' b23= _17’ b3: 2 (53°11) 8/‘3 Hence from (5. 3. 2), the network parameters are determined as follows: 1 1 C -3 ’ C ’1 ’ L4- 1 '2"L5 l5). .1; (5. 3. 12) Other possible solutions can be investigated similarly. 5. 4 The procedure for the transformation of the state model described in Chapter IV, Section 6 is illustrated by the following example. Example 3 Determine the element values of the Cauer network, N2, shown in Fig. 5. 4. l, which is equivalent to the Foster network, N1, g1ven in Fig. 5. 4. 2. -153- 14 15 Mmgm .- Z(s)———) C2 .3: C3 #1 £6 Z(s)__)%z 6 l 1 A 2 “Ta 4 , g I J Fig. 5.4.1 Fig. 5.4.2 are given in Solution: The linear graphs for N1 and N2 Fig. 5. 4. 3. Fig. 5. 4. 3 The state model of N2 is and -154- 01-1 J Applying the transformation described in Chapter III, the state model in (5. 4.1) can be brought to the form -155- (— " F — v2 0 I v3 0 ' -1/2 x 14 + [4 V1 I i5 0 I i O _ 6i _ J and T'- i4 -l/2 ' i1 -[14 0 0] 15 I i _ 61 v1 = V1 and 1* - i l - 1 we have 1" r '- l - O F”; , ll1 ll2 I O l v3 I 0 ‘3 "4 .4 - ————— '— ----- t ' - | 14 -a1 0 I | 15 "Z '33 ' O i' o : a 1"," (5.4. 2) (‘3‘) 14 .* [b 0 I 1 = ' 1 1 O] 15 I i6 —. _J or in symbolic form r ,- r- “ - ‘r -' T F: d Vbc 0 A2 Vbc 0 '5? ' = ' + V1 (5.4.4) v -A o I -B cl 2 2 _ J _ - _CIJ _. _ T I I1 ’ 'Bz cl Repeating the same transformation for N its state model takes on the 1 form )— -" F- | -— r- - F" — v* 0 (3 0 v* 0 2 l 2 O l * >I< v I 0 O 1 v 0 d 3 I 3 * i4 0 O I 14 F 3 l 1* - 3 o ' O 1* —1-— 5 I 5 W? 3): I _* 1 i 0 '1 I 16 — _ 6J ._ J _ .1 MIT- and -157- F— *— i4 * l 1 1 >1: «[3 «I? «f3: 35‘ 16-J or symbolically (_- *— (— T.- a}: _ (— _- v o d be Al v‘DC o - = + v 5.4.6 dt 1* A o 1* l ( ) c1 " 1 c1 “Bl _ J _ J _ J _ J and T I1 “ "B1161 Now consider the transformation described in Section 4. 6. To transform the state model of N1 into that of N2, from (4.6. 20), we write PT B " B or B " P B 2 1 ' 2 1 - 2 2 and (5.4.7) T Pl A1 Pl — A2 where P1 and P2 are orthogonal matrices. The first equation gives T _qur" (5.4.8) |~ill~i||~ -158- 01‘ (5.4.9) 0" II squaring and adding these equations, we have 2 2 2 2 2 IC’1(P11+‘-"21+p31) ‘ 1 ‘ b1 01‘ where the sign of b1 is decided from the topology of N2. Therefore, (5.4.9) gives 1 1 p = II— , p = — (5.4. 10) 21 «re? 31 .17 l 75' VAlso from the second equation in (5. 4- 7). it follow3 that P A A: "Zr-SA AT = 3 (5.4.11) where r- a1 'alaz AAT aa a+a2 22= 12 2 3 0 -a3a4 Equation (5. 4. 11) indicates the eigenvalues of A are 0, 3 and l. (5. 4. 10) the first eigenvector is known. a2 _ 2 l _ l = + — . 3 O 6 or = 1 a1 Further from the relation or r 2 x a 31 " i “*1 2 2 .. - X al a2 a2 + a3 1 0 --a3 a4 we have -159- Then 2 2 x pll k1+ p21 )‘2 + p31 3 l + E o 1 - l 0 0 pi1 -a3 a4 pi2 2 - )4 , a4 fl p13 (5.4.12) From 0 (5.4.13) -160— 2 a‘1 p11 z a’1 8‘2 p12 (a2 3) — 1 p21 ' a1 aL21°22 2 (a1 ' 1) 1331 = aL1“‘2 p32 Substituting the values of pi1 and a1, (5. 4. 14) gives = EL1 a2 p12 = a1 212 p22 i)|~ 31F 0 = a1a2p32 from which 01' Therefore, from (5.4. 14), we have 1 _ Z -0 p12 = J31” 922"~/-Z—' p32 Since 2 2 2 2 2 x _ a2+a3=plzx1+P22x2+P32 3 we alsohave a = 1 (5.4.14) (5.4.15) -161- The second row of (5. 4. 13), after substituting the known variables, gives 1 + Z a a .— — -_- p J? «I? 3 4 13 1 + 2 a a '— — - P «1‘6 a 3 4 23 _1_ _ a a p If? 3 4 33 from which we have l+l+i-l-z«izaZ-l 3 6 2 - — 3 4 - or a4 == 1 Therefore, p _ 1 p -_1_ p -._1_. 13 “—3 23 J? 33 (5 Since all the entries are found, (5. 4. 3) is of the form F- .fi !— — '— '-1 _ 1 v2 : 1 -1 0 v2 0 I <:> I ' v I 0 1 ’1 V3 0 d 3 * — = ————|———-- + V dt ' 0 ' .' 1 1 . _ 1 I I ' i5 1 '1 l O 15 0 | t | ,' O i 0 l 1 _ 6_ _ I _J __ 6- L. .. ~162- and Note that, with this procedure, although the transformation matrix P is necessary to be evaluated, it is not needed for the final part of synthesis. Comparing (5.4. 2) and (5. 4.16), we have the element values CHAPTER VI C ONCLUSION The use of state models is shown to be effective in realizing reactance functions and they have served as a basis for extending the realization of Z-port Z and Y matrices having dominant residue matrices. The same procedure can be applied to n-port matrices, having dominant residue matrices. One of the significant features of the state model approach is that the realization procedures do not differ from one- to n-port networks. The similarity transformations given in Chapter IV provide simple and practical ways of relating the parameters of one canonic network to another. Although the solution to the resulting equations is given explicitly for certain classes of canonic networks, their solution for certain other classes may be very difficult either analyti- cally or numerically. One of the difficulties with numerical techniques is an initial guess. 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