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On the impedance realizing ability of minimal two-element-kind networks Tarnai, Ernest John 1973

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O N T H E I M P E D A N C E R E A L I Z I N G A B I L I T Y O F M I N I M A L T W O - E L E M E N T - K I N D N E T W O R K S by ERNEST JOHN TARNAI B.A.Sc, University of B r i t i s h Columbia, 1968 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENT FOR THE DEGREE OF MASTER OF APPLIED SCIENCE i n the Department of E l e c t r i c a l Engineering We accept t h i s thesis as conforming to the required standards UNIVERSITY OF BRITISH COLUMBIA A p r i l 1973 I n p r e s e n t i n g t h i s t h e s i s i n p a r t i a l f u l f i l m e n t o f the r e q u i r e m e n t s f o r an advanced degree a t t h e U n i v e r s i t y o f B r i t i s h Columbia, I agree t h a t the L i b r a r y s h a l l make i t f r e e l y a v a i l a b l e f o r r e f e r e n c e and s t u d y . I f u r t h e r agree t h a t p e r m i s s i o n f o r e x t e n s i v e c o p y i n g o f t h i s t h e s i s f o r s c h o l a r l y p urposes may be g r a n t e d by the Head o f my Department o r by h i s r e p r e s e n t a t i v e s . I t i s u n d e r s t o o d t h a t c o p y i n g o r p u b l i c a t i o n o f t h i s t h e s i s f o r f i n a n c i a l g a i n s h a l l n o t be a l l o w e d w i t h o u t my w r i t t e n p e r m i s s i o n . Department The U n i v e r s i t y o f B r i t i s h Columbia Vancouver 8, Canada ABSTRACT The Impedance realizing a b i l i t y of minimal two-element kind networks i s considered. As a preamble, a comprehensive survey of relevant mathematics and existing results i s presented. An argument based on group theory i s used to demonstrate the complex nature of the solution for non-canonic networks. The modal matrix of normal co-ordinate transformation on the cut set admittance matrix i s interpreted geometrically as a set of vectors satisfying certain conditions, imposed by the topology and the parameters of the input function of the network, in two Euclidean vector spaces. The existence of the modal matrix, hence the existence of these vectors, i s the necessary and sufficient condition for physical r e a l i z a b i l i t y . Explicit formulas are developed for third order networks and numerical algorithms for the fourth order networks. A necessary condition i s given on the parameters of Z(s) for r e a l i z a b i l i t y for networks containing a linear tree of one kind of element. 1 TABLE OF CONTENTS Page ABSTRACT i TABLE OF CONTENTS i i LIST OF ILLUSTRATIONS i v LIST OF TABLES v ACKNOWLEDGEMENTS . v i LEXICON v i i I. INTRODUCTION 1.1 Historical Background . . . . . . 1 1.2 The Problem 2 II. MATHEMATICAL BACKGROUND AND PREVIOUS RESULTS 2.1 Physically Realizable Two-Element-Kind Driving Point Functions 5 2.2 Minimal -Networks 7 2.3 Normal Co-ordinate Transformation . . 11 2.4 Review of Previous Work 15 III. HOWITT TRANSFORMATIONS AND COMPLEX REALIZATION 3.1 Group Concept 22 3.2 Set of Equivalent Networks 23 3.3 Second Order Networks 24 3.4 Higher Order Networks 28 3.5 Complex Network 30 IV. GEOMETRIC APPROACH 4.1 Mathematical Formulation 36 4.2 Geometric Picture 39 4.3 Third Order Irreducible Network 43 4.4 Fourth Order Networks 51 4.4.1 Polar Angle Approach 57 4.4.2 K-Vector Approach 60 4.5 n t h Order Networks 67 i i Page V. CONCLUSION 70 REFERENCES 71 APPENDIX A: Algebraic Solution for the Third Order Irreducible Network 73 APPENDIX B: Cut-Set Analysis 77 APPENDIX C: Generalized Thales' Theorem 78 APPENDIX D: Formulas for K-Vector Approach . 81 i i i LIST OF ILLUSTRATIONS Page CHAPTER II. 2.4.1 RC Tie-Set . 18 2.4.2 Third Order Irreducible Network 21 CHAPTER III. 3.3.1 Second Order Example 24 3.3.2 Range of Transformation Matrix Coefficients 26 3.5.1 Example 3.5.1 32 3.5.2 Example 3.5.2 33 3.5.3 Example 3.5.3 34 CHAPTER IV. 4.2.1 Foster Type Network 42 4.3.1 Third Order Irreducible Network . . 44 4.3.2 P Plane in Transformed Co-ordinates 48 4.4.1 Fourth Order Irreducible Networks 53 4.4.1.1 0 - i|i Plot 59 4.4.1.2 Refined 6 - ip 60 4.4.2.1 space 63 4.4.2..2 The plane .of.#1 -and £.4 .. . • ... . . 64 4.4.2.3 k 2 - k 3 Plot 65 4.4.2.4 k 2 - k 3 Contour Plot . . . . . . . . . . . . . . . 66 4.4.2.5 Refined k 2 - k 3 Contour plot 66 APPENDIX A A.l Orthogonal Chords of an Ellipse 80 iv LIST OF TABLES Page CHAPTER II 2.1.1 Properties of Two-Element-Kind Driving Point Impedances 6 2.2.1 Properties of Minimal RC Driving Point Im-pedances 10 v ACKNOWLEDGEMENTS The author gratefully acknowledges the award of a U.B.C. Graduate Fellowship i n 1968-69 and the Killam Predoctoral Fellowship in 1969-72. Various members of the Department of E l e c t r i c a l Engineering, headed by the author's supervisor, Dr. A. D. Moore, were most co-operative when advice or encouragement was sought, and the author would li k e to thank them for their help. This work was supported in part by National Research Council grant A-3357. v i L E X I C 0 N Group transformation matrix Incidence matrix Admittance matrix of capacitor subnetwork Member of the set of capacitors i n N R Set of component values of N Domain of impedances realizable by N Member of D Basic vector of -S> space Admittance matrix of resistor subnetwork Vector -of port currents Modal matrix Row vector of M RLC Network Topological configuration of N Minimal two-element-kind network Order of N Space of u_. vectors Resistive path connecting nodes of c^ Set of real numbers Member of the set of resistors i n N K Space of nu vectors Known vector in space v i i s Complex frequency variable Capacitive path connecting nodes of r^ T Time constant matrix, ] |T.. | T1''2 Matrix of mapping $ to ^ , ||/r\ S | | T Rotation in s T Rotation in IR K U Unit matrix V Vector of port voltages x i th i — running variable i n 5 space Short c i r c u i t admittance matrix Y„ Node-to-datum admittance matrix N Yj Input admittance function Z Open ci r c u i t impedance matrix Z Physically realizable RLC impedance Z Physically realizable RC impedance RC Zj. Input impedance function Kronecker delta A Determinant e^ . Basis vector of 3? space X^ Natural frequency Mapping of nu onto £^ i — running variable i n & space E Set of group transformations yielding Z RLC Residue corresponding to X^ a Mapping of s onto T. Time constant, X. 1 i i v i i i CHAPTER I.  INTRODUCTION 1.1 Historical Background In 1924 Foster ^ published his classic paper "A Reactance Theorem" which marks the beginning of the modern theory of network synthesis. In i t , he presents the f i r s t exact synthesis of any posi-* tive real rational functions. His method is well known and need not be considered here. But, one notes that the paramount, characteristic feature of his 2 method, as well as of those who followed, notably Cauer's , Yarlagadda 3 4 5 and Tokad's , and f i n a l l y Lee's ' , i s the manipulation of the immit-tance function in order to arrange i t into a suitable form such that the element values can be determined from the parameters by inspection. These methods w i l l always produce a satisfactory realization provided the specified imittance function i s physically realizable by the particular two element kind network. A l l the above-mentioned procedures have one common, major limitation, namely, the resulting network configuration i s implicit i n the method and, therefore, the designer i s faced with a rather modest set of obtainable networks. To overcome this limitation Mason ^ proposed a numerical realization procedure which yields networks of arbitrary but specified * Some of the terms used throughout this thesis w i l l be defined v ia the discussion of the mathematical background in Chapter II. 1 2 topology. Mason's method, however, i s not a synthesis according to Guillemin's ^ definition: "....synthesis differs from analysis in two major respects: (a) a solution to a stated problem does'not nec-essarily exist, and (b) when i t does exist i t is not unique, there being theoretically an i n f i n i t e number of circuits a l l having the same response function or functions at specified ports of access. For these reasons the approach to a synthe-sis problem must begin by establishing necessary conditions for the existence of a solution and effective means for deter-mining whether these conditions are met in a. given situation." Whereas Mason's method usually results in the required net-work, the procedure does not provide a means for establishing neces-sary conditions. . This shortcoming i s rather important from a practical point of view. For i f the realization procedure f a i l s i t may be caused by ill-chosen i n i t i a l values or by the non-existence of a solution. In the f i r s t case, the process should be restarted with a new' set of i n i t i a l values. In the second case, however, the process should be abandoned. Thus, in case of a fa i l u r e , the designer has to make the choice: which of the above alternatives should he follow. Regretably, his decision must be based on intuition or conjecture, since at present l i t t l e i s known about impedance realizing a b i l i t y of networks with complexity greater than two.. Clearly, the need exists to study the impedance realizing a b i l i t i e s of networks with arbitrary but specified topologies. 1.2 The Problem Let N denote a linear, passive, time invariant e l e c t r i c a l network without mutual inductance. The element or component values of N are not defined, i.e., the physical characteristics of the branches 3 of N are known qualitatively. If a numerical value d\ Is assigned to th ~ the i element of N and i assumes a l l the integer values from one to m, where m i s the number of elements i n N, a network N results for which the port variables have the following well-defined mathematical relationship: N = {v(s), I(s): V(s) = Z(s) T(s)} (1.2.1) Clearly N i s a function which maps a set of element values D = {d^: i = 1, ..., m} into Z(s), i.e. N: D -> Z(s) (1.2.2) The range of this function for physically real networks i s A (N) =' {d : 6 < d ±eR #} (1.2.3) the m-tuple of positive real numbers. The domain of N however J& (N) = UZ(s) (1.2.4) depends on the function N. The ultimate goal of any study of impedance realizing a b i l i -ty of e l e c t r i c a l networks is knowing the exact nature of (N) or alternatively, determining whether a specified input impedance Z(s) is contained in 25 (N) for some prescribed N. The mathematical d i f f i c u l t y involved in tackling so general a problem is forbidding. Therefore, to obtain any results whatever, 4 N should be severely restricted. In this thesis the following r e s t r i c -tions are used: a) N i s accessible via one port. R b) N contains two types of elements. R c) N contains the minimum number of elements. R Unfortunately, even with these restrictions, the problem was found to be too ambitious to obtain a general solution. In the following pages, techniques based on co-ordinate trans-formations are presented. These are used to gain some insight into the nature of £> ( N ) , to obtain the analytic solution for networks of com-plexity three, and to construct numerical procedures to discover whether or not a fourth order prescribed impedance i s contained i n (N R). x" R' ' Before proceeding with our results, in the next chapter we w i l l present some of the mathematical background required, and summar-ize previous results. CHAPTER II MATHEMATICAL BACKGROUND AND PREVIOUS RESULTS 2.1 Physically Realizable Two-Element-Kind Driving Point Functions The material presented in this chapter is intended to f u l f i l several functions: a) laying the mathematical foundation for new developments in subsequent chapters; b) defining terms used throughout the thesis; and c) presenting a summary of a l l rele^vent materials in a uni-fied fashion, The intersection of the domains [)2> (N), as N assumes a l l pos-sible configurations, is the set of a l l physically realizable RLC (res-istance, inductance, capacitance) networks, {Z^_}. In his celebrated g .b.ut .challenging .paper. MacMillan derived the necessary and sufficient conditions for Z(s), an arbitrary matrix of rational functions, to be a member of {Z }. His results are stated i n Theorem 2.1.1. JxLL. Theorem 2.1.1 (MacMillan*s Theorem) Let Z(s) = 1 z„(s)||" be a matrix, where (i) each z..(s) is a rational function ( i i ) Z i j ( s ) = z...(s) ( i i i ) z...(s) = z (s> (iv) For each .set of real constants {k^: i = l,...,n} the function n <|>(s) - I z. . (s) k.k. . . . . 13 i 2 i j = l J J has a non-negative real part where (s) > 0. Then there exists a f i n i t e passive n port network which has the 6 impedance matrix Z(s). Conversely, i f a passive n-port network has an impedance matrix Z(s), i t must have properties (i) to (iv) . The formally indentical dual results hold for open-circuit admittance matrix Y(s). Matrices having properties (i) to (iv) in the above theorem are said to be positive real or PR. Theorem 2.1.1 i s the generalization of Brune's classic result, which i s rephrased in the next theorem. Theorem 2.1.2 (Brune's Theorem) The necessary and sufficient condition that a rational func-tion Z(s) is the driving point immittance function of a one port passive network is that Z^(s) is PR. In addition to positive realness, required by Brune's theorem, two-element-kind networks -are characterized by pole zero patterns of marked simplicity. The specific properties of these patterns are summa-rized in Table 2.1.1. TYPE OF NETWORK LC RC RL ALTERNATING SIMPLE POLES AND ZEROS ON j - AXIS NEGATIVE REAL AXIS CRITICAL FREQUENCY NEAREST TO ORIGIN IS - A POLE A ZERO Table 2.1.1 Properties of Two-Element-Kind Driving Point Impedances 7 This simplicity facilitates transformations, known as Cauer transformations, of one type of two-element-kind network into others. For example, ZRL ( S ) = ZRC ( b (2'1-1) he ( S ) = 7 Z R L ( s 2 ) ( 2 - 1 ' 2 ) The subscripts in these equations denote the type of elements required to realize the appropriate impedance function. By virtue of the Cauer transformation, results obtained for one type of two-element-kind network are directly applicable to a l l other types. Thus no generality is lost by studying one type only. Hence, this research is executed using RC impedances. A convenient expression for an RC driving point impedance function is i t s p a r t i a l fraction form: Z R C < S ) - "-I + I 7TT7 + o~ (2-1'3) 1=1 i where s denotes the complex frequency variable and and denote the th i natural frequency and the corresponding residue respectively. 2.2 Minimal Networks The minimum number of elements or components required to real-ize a prescribed impedance function is the number of independent para-meters in this function. Networks containing this number of components are called minimal networks. The physical and mathematical characteristics of minimal net-works are discussed by Seshu and Reed1''' and are summarized in the follow-ing two theorems: 8 Theorem 2.2.1 A one-terminal-pair network without mutual inductance i s minimal in reactive elements i f and only i f (i) there are no all-inductor or all-capacitor loops, ( i i ) there are no all-inductor or all-capacitor cut-sets in the network obtained by shorting the driving terminals together. Theorem 2.2.2 The driving-point impedance of a passive network without mutual inductance has (i) a pole at s = 0 i f and only i f there i s an a l l -capacitor cut-set such that i t s removal from the network places the driving terminals in different connected parts, ( i i ) a pole at s = K i f and only i f there i s an a l l -inductor cut-set such that i t s removal from the network . places the driving terminals in different connected parts, ( i i i ) a zero at s = 0 i f and only i f there is an a l l -inductor path between the driving terminals, (iv) a zero at s = 0 0 i f and only i f there i s an a l l -capacitor path between the driving terminals. A cut-set of a connected network i s defined as a minimum set Unfortunately, as Ponstein points out, "terminology in Graph Theory is somewhat chaotic. Several names are known for the same object and, much worse, the same name i s sometimes used for different objects". Therefore a l l terms borrowed from Graph Theory should be defined precisely. However, to improve readability, we depart from this ideal path and i n -stead refer the interested reader to Ponstein for the definitions of terms we deem familiar to most readers such as network, graph, connectedness, path, loop and consequently do not define here. 9 of branches of the network such that the removal of these branches d i s -connects the network. Both theorems above apply to a r b i t r a r y RLC networks. Conse-quently, Theorem 2.2.1 applies to LC networks and through the Cauer transformations to a l l two-element-kind networks. In terms of RC net-works t h i s theorem states that the component networks, i . e . the sub-networks composed of the set of r e s i s t o r s and the set of capacitors, are e i t h e r trees or networks containing two separate parts with one of the nodes of the d r i v i n g ports i n each of the parts i f the network i s minimal. The tree i s a subnetwork which contains a l l nodes, no loops and i s connected. Theorem 2.2.2 states the dependence of the d r i v i n g point im-pedance function on the nature of the subnetwork of the component net-works connecting the input ports. The properties of minimal RC networks are summarized i n table 2.2.1. The binary designation of the networks i n the above table, introduced by Mason^, denotes the presence (1) or absence (o) of pm and p Q r e s p e c t i v e l y i n equation (2.1.3). Minimal networks other than type 00 may be generated from t h i s type by considering a network of type 00 with the appropriate e l e -ment connecting the input nodes, and then removing t h i s element. Thus any properties of minimal two element kind networks may be deduced by studying RC networks of type 00. Both component networks of t h i s c o n f iguration are trees, hence they are c a l l e d complementary tree s t r u c t u r e s . NETWORK 00 10 01 11 subnetwork connecting input ports RC R C Behaviour at s = 0 cons tan t cons tant pole p o l e Behaviour at g = CO zero constant zero constant i s p =0? o yes yes no no i s p =0? oo yes no yes no Table 2.2.1 P r o p e r t i e s o f Minimal RC D r i v i n g P o i n t Impedances 11 The input impedance function of an n order complementary tree RC network in pa r t i a l fraction form i s n p. z i ( s ) = E T T i r ( 2 - 2 > 1 ) i=l s f 2.3 Normal Co-ordinate Transformation In the early nineteen-sixties much activity in network theory centered around the mathematical method known as normal co-ordinate transformation. The quest in this direction was encouraged by several 13 eminent experts in the f i e l d , notably by Guillemin . They predicted the development of completely general synthesis of any linear device based on co-ordinate transformation of this type. The consequent investigation proved normal co-ordinate trans-formation to be a useful tool,.yet, due to the mathematical d i f f i c u l t i e s involved, no results as general as were envisaged have been obtained. A significant portion of this research i s based on normal co-ordinate transformation, therefore an outline of the relevant theory i s warranted. The essence of normal co-ordinate tranformation i s the simul-taneous diagonalization of two real, symmetric, square matrices. The problem was f i r s t studied in Mechanics1"*'^ in connection with small oscillations. In this context the matrices are associated with quad-ra t i c forms representing the potential and kinetic energies of the sys-tem; the second of these matrices i s obviously positive definite. The equations of motion for such vibrating systems can always be expressed in a set of generalized co-ordinates, {n }, each of which 12 executes one simple oscillation corresponding to one of the natural fre-quencies. In this set of co-ordinates the quadratic forms are sums of squares; i.e., the relevant matrices are diagonal. The simple periodic vibrations are called normal oscillations or normal modes and the transformation matrix the modal matrix. This heuristic reasoning indicates the existence of the con-gruent transformation, which simultaneously diagonalizes two real, sym-metric matrices one of which i s positive definite and the other positive semi-definite. The existence of this transformation can be proven rigorously. An elegant proof i s provided by Gantmacher^ in his book: Theory of Matrices. The condition requiring one of the matrices to be positive definite may, as B o x a l l ^ showed, be replaced by a weaker condition more appropriate to el e c t r i c networks, by . requiring the sum of the matrices to be positive definite. For i f C and G are matrices such that A = C + G (2.3.1) i s positive definite, then A and B = C - G (2.3.2) .form a pencil of quadratic forms which can be diagonalized by congruent transformation M. But clearly, the sum and difference of diagonal matrices are also diagonal therefore MT (A + B)M = 2MT C M (2.3.3) MT (A - B)M = 2MT G M (2.3.4) 13 are both diagonal. This r e s u l t i s stated as Theorem 2.3.1. Theorem 2.3.1 I f C and G are r e a l , symmetric p o s i t i v e semi-definite then the s u f f i c i e n t condition for the existence of a congruent transformation which simultaneously diagonalizes both of these matrices i s that t h e i r sum i s p o s i t i v e d e f i n i t e . For any set of independent voltage v a r i a b l e s the open c i r c u i t admittance matrix of an RC network i s Y(s) = sC + G (2.3.5) where C and G are the admittance matrices of the component networks. According to MacMillan's theorem ( i ) C and G are r e a l and symmetric, ( i i ) t h e i r sum i s p o s i t i v e d e f i n i t e , * ( i i i ) C and G are hyperdominant . Hence, the admittance matrix (2.3.5) i s p h y s i c a l l y r e a l i z a b l e i f and only i f C and G may simultaneously be diagonalized as M l Y M = s C, + G, * (2.3.6) a a The diagonal matrices C, and G, have the structure a d A matrix A = | I a - j j I 1 ^ byperdominant i f and only i f ( i ) a ± J $ 0 ; i * j n ( i i ) I a * 0 ; i = 1, .. ., n 3=1 2 14 C d = U 0 c 0 0 (2.3.7) and G d = 0 0 0 A (2.3.8) where U is a unit matrix of rank r and A i s a diagonal matrix of rank c c r . Thus 8 T M YM = sU 0 0 ng 0 Y d(s) 0 0 U nc = Y (2.3.9) where U and U are unit matrices of ranks ng = n - r and nc = n - r ng nc 8 c respectively and the submatrix Y^ i s Y d ( s ) - (s + X.)6..| * ; A = r + r - n (2.3.10) • i n " l c g When the admittance matrix i s diagonalized as i n equation (2.3.9) an explicit expression for the impedance matrix can be obtained i n terms of the elements of the modal matrix, as Z ( s ) = | | Z . . ( s ) l | ^ - Y _ 1 (s) " - I T M(Y) A M (2.3.11) or 15 ng m.,m., ng+£ m., m. . n . . (S) = I I _^LJJ + l 1 2 k=l S k=ng+l S ng+k-2 k=ng+ +1 l k 2 * (2.3.12) Comparison of equations (2.3.12) and (2.1.3) suggests using normal co-ordinate transformations to f a c i l i t a t e solution of either a synthesis or an analysis problem. In analyzing a network, the modal matrix i s easily found by considering C and G in equation (2.3.5) as a pencil of quadratic forms'^. The modal matrix i s the characteristic matrix of this pencil and can be determined by conventional algorithms. But i n synthesizing an impedance function, no general systematic method exists to find the modal matrix. The information i n a specified input function i s mathematically sufficient to construct this matrix, 6 13 17 18 but, although several * ' ' attempts have been made, the solution to this problem i s s t i l l elusive. 2.4 Review of Previous Work Circuits obtained by conventional two-element-kind synthesis procedures, such as Foster or Cauer, are canonic. A network i s said to be canonic i f i t is minimal and able to realize any PR driving point function. The only significant results relevant to impedance realizing 19 a b i l i t y are published by Lee. He states some necessary conditions 20 for the network to be canonic and some bounds on the natural frequen-cies i f these conditions are not met. This concluding section of the present chapter i s devoted to 16 the summary of the essence of Lee's work. The node-pairs associated with r e s i s t o r r ^ of the complementary tree RC structure N „ are connected by a c a p a c i t i v e path, S.. S i m i l a r l y , K 1 each capacitor, c^,possesses a unique r e s i s t i v e path, R^, connecting i t s nodes. The fundamental set of loops {r., S. i = 1, n} e N„ de-r l l 1 R fines the impedance matrix Z (s) = R + - S (2.4.1) s ttl til where the i diagonal element of R i s r ^ and the i diagonal element til of S i s the sum of elastances i n the i loop, S^. Considering Z(s) as a p e n c i l of quadratic forms in d i c a t e s that the set of eigenvalues of N possesses the extremal properties of the R c h a r a c t e r i s t i c values of a regular p e n c i l of forms'^: I i m m s = -=-1 m a x 1 v and l max -T -v Rv s m m 1 v -T -v Sv -T -v Rv (2.4.2) -T -v Sv (2.4.3) where v i s an a r b i t r a r y , r e a l vector of dimension n. The above equations _T _ _T — show that the r a t i o of quadratic forms u Ru and u Su l i e s between the smallest and larges t eigenvalues of the system for any vector vi, i n c l u d i n g th the one which has unity as i t s i component and zero f o r a l l other components. An argument, s i m i l a r to the foregoing, holds for the system i n which the c a p a c i t i v e and r e s i s t i v e behaviours are inter-changed. 17 The results are stated i n Lemma 2.4.1 If r^ is any one of the resistors of the complementary tree RC structure, N^, and i s the capacitive path associated with r^; furthermore, i f c^ i s any one of the capacitors of N„ and the associated resistive path R.; then R i r. |X | >: - r 1 >. |x . | (2.4.4) 1 max1 S. 1 nan' l R |X | > \>. \\ . | (2.4.5) 1 max1 -1 1 mxn1 c i where X and X . are the largest and smallest natural fre-max min quencies of N , respectively. A consequence of this lemma, as shown below, i s that the natural frequencies of N are severely restricted i f each fundamental loop of contains three or more branches. Let .n ^  denote the number of elastances, the smallest of rwhich i s c^ \ i n the capacitive. path, S^ , .associated with the smallest resistor r^ of N R. Then, obviously, Sk>. n ^ c f (2.4.6) But according to lemma 2.4.1 therefore r, >, S, X . | (2.4.7) k k 1 min1 ' k ^ c k ^ >minl ( 2 ' 4 - 8 ) 18 Similarly -1 , nr£ Tl cz * ] x — r 1 max' (2.4.9) where n^ ,^ i s the number of resistors in the resistive path, R^ , associ-ated with c^, the smallest of which i s r^. Combining equations (2.4.8) and (2.4.9) yields max A . m m ^ nr£ nck r, (2.4.10) But r k is smaller than or equal to r^ and according to the basic assump-tion both n . and n , are greater than or equal to two, therefore r)6 etc max m m 5 4 (2.4.11) the corollary of which is stated as Lemma 2.4.2 If the natural frequencies of N R are free from constraint i t must contain a parallel RC branch (or tie-set) such as shown in f i gure 2.4.1. £ c, Figure 2.4.1 RC Tie-Set The node-to-datum short circuit admittance matrix Y.T = sC + G N (2.4.12) of N R may be arranged so that the (n,n) element is the admittance of an RC tie-set provided N R contains such a branch. Short circuiting this 19 RC tie-set reduces N,, to L 1 a complementary tree RC structure with com-plexity (n-1). The admittance matrix of N , K Y ' = s C' + G' N (2.A.13) may be obtained from Y^ in equation (2.4.12) by deleting the last row and column. The extremal properties of the characteristic values of the pen-c i l of forms shows that max |x |-m a x N R V _T _ v Cv _T -v Gv (2.4.14) and max1 max v _T v C'v v T G'v (2.4.15) th If u i s an arbitrary vector, except for the requirement that the n com-ponent equals zero, then max v _T -v Cv _T -v Gv max u _T -u Cu _T _ u Gu (2.4.16) But the left-hand side of this inequality i s the largest eigenvalue of N and the right side is the maximum eigenvalue of N'. An analogous line of reasoning holds for the smallest charac-t e r i s t i c values of N and Nl; hence K . R Lemma 2.4.3 R If Nl i s the reduced network obtained from N by short circuiting an RC tie-set, then for any choice of component values X X max max X . mm $ N : X . mm (2.4.17) N R 20 The foregoing lemmas may be combined i n Theorem 2.4.1 A necessary c o n d i t i o n f o r a complementary tree s t r u c -ture to be able to r e a l i z e any set of n a t u r a l frequencies i s a) That i t can be reduced to a s i n g l e node by succes-s i v e l y short c i r c u i t i n g RC t i e - s e t s ; or b) That i t can be reduced to (n+1) i s o l a t e d nodes by s u c c e s s i v e l y open c i r c u i t i n g RC cut s e t s . Networks of t h i s type are s a i d to be r e d u c i b l e . The proof of b) i s the formal dual of the proof of a) given above. I f N i s r e d u c i b l e , i t can r e a l i z e any a r b i t r a r y s e t of n a t u r a l frequencies w i t h i n any s p e c i f i e d e r r o r bound by " b u i l d i n g - u p " the network v i a a process which i s the reverse of reducing N .' S t a r t i n g from a s i n g l e node, N i s constructed by s p l i t t i n g t h i s node i n t o two K and i n s e r t i n g a t i e - s e t , between the h a l v e s , which has one of the s p e c i f i e d n a t u r a l frequencies. The next n a t u r a l frequency i s r e a l i z e d by c u t t i n g i n t o a s h o r t c i r c u i t and i n s e r t i n g a t i e - s e t w i t h the appro-p r i a t e n a t u r a l frequency. In general t h i s new frequency i n t e r a c t s w i t h the o l d causing an e r r o r i n both. However, i f the impedance-level of the t i e - s e t to be i n s e r t e d i s s u f f i c i e n t l y d i f f e r e n t from that of the network, the r e s u l t i n g e r r o r can be kept w i t h i n the s p e c i f i e d e r r o r bound. The process i s r e p e a t e d . u n t i l N_ i s complete. Thus the c o n d i -K t i o n s t a t e d i n the theorem above i s not only necessary f o r N to r e a -ix. l i z e an u n r e s t r i c t e d set of n a t u r a l f r e q u e n c i e s , but i s a l s o s u f f i c e n t . Before concluding t h i s chapter we note that the i n e q u a l i t y 21 (2.4.11) i s not a greatest lower bound ( g . l . b . ) , and i n fact the r a t i o of extreme n a t u r a l frequencies i s much higher. For the simplest i r r e -20 ducible network, shown i n figure 2.4.2 the value of the g.l.b. f o r this r a t i o i s max A . mm = 17 + 12v^" (2.4.18) Figure 2.4.2 Third Order I r r e d u c i b l e Network In t h i s chapter, i n addition to a summary of e x i s t i n g r e s u l t s , the mathematical foundation required to the material i n t h i s thesis i s l a i d . The d e s c r i p t i o n of new r e s u l t s commences a f t e r the introductory s e c t i o n of the next chapter. 22 CHAPTER I I I HOWITT TRANSFORMATIONS AND COMPLEX REALIZATION 3.1 Group Concept A s p e c i f i e d p o s i t i v e r e a l RC d r i v i n g point impedance function, Z ( s ) , can always be r e a l i z e d by any of the w e l l known canonic network configurations. If another network of a r b i t r a r y but s p e c i f i e d t o p o l o g i c a l configuration has t h i s same impedance f u n c t i o n , Z j ( s ) , f o r a set of element values, D = {d^} then the two networks are said to be equivalent. Hence, impedance r e a l i z i n g a b i l i t y can be investigated by studying c i r c u i t s equivalent to canonic networks. 21 In 1931, Howitt proposed applying group theory, the abstract mathematic notation then enjoying a peak of p o p u l a r i t y amongst quantum p h y s i c i s t s , to the study of equivalent networks. He showed that e l e c t r i c networks form a continuous group wi t h the impedance func-t i o n as an absolute i n v a r i a n t and a l i n e a r tranformation of the i n s t a n -taneous mesh currents and charges as the group operation. 22 Howitt's work was extended by MacFarlane , who applied group transformations to the state vectors of l i n e a r systems, i n order to obtain more general sets of equivalent systems. The group transformation i s u s e f u l to generate an i n f i n i t e set of equivalent networks of any given network. Unfortunately, how-ever, no method e x i s t s to determine whether or not a network with a p a r t i c u l a r t o p o l o g i c a l configuration i s a member of such a s e t . Nevertheless, some i n s i g h t into the nature of non-canonic 23 networks may be gained by considering a group consisting of the set of equivalent admittances and the group operation of linear transformation of node voltages. (Transformation of flux-linkages need not be consi-dered, since the networks of present interest contain no inductances.) This group is the dual of the one considered by Howitt. 3.2 Set of Equivalent Networks If the nodes of a network are numbered so that the terminals of the input port are the f i r s t and the reference node, then the input admittance function i s YjCs) = A/A11 (3.2.1) where A and A n are the determinant and the (1,1) minor, respectively, of the short ci r c u i t node-to-datum admittance matrix, = sC + G (3.2.2) N Consequently the congruent transformation, A, does not alter the value of Y j i f Y.'N = A TYj^. = sC' + G' (3.2.3) where a l i = 6 l i (3.2.4) i Y^ is the admittance matrix of a physically real network i f , according to MacMillan's theorem, the transformed component matrices i t C and G are hyperdominant. The element values of the transformation 24 matrices which ensure this hyperdominance form a closed set, H. The networks defined by the elements of 5 and the transformation i s a group. This group is the dual of the one introduced by Howitt. The advantage of this formulation, as shown in the consequent example, i s that one or more of the branches of the maximally connected network vanish i f the elements of the transformation matrix define a boundary point of H. 3.3 Second Order Networks The second order admittance function Y = s + 4s + 3 s + 2 (3.3.1) may be used as a specific example to demonstrate some of the foregoing concepts. One possible realization i s shown in figure 3.3.1. 1 A 1 A Figure 3.3.1 Second Order Example 25 The transformation of the component matrices C = 2 -1 -1 1 and G = 2 L - l -1 2. according to equation (3.2.3) yields C = ( a 2 1 - 2a 2 1 + 2) _ a 2 2 ( a 2 1 - 1) and G = 2(a 2 1 - a 2 1 + 1) a 2 2 ( 2 a 2 i - 1) a 2 2 ( a 2 i - 1) 2 a 2 2 a 2 2 ( 2 a 2 i - 1) 2 2 a 2 2 (3.3.2) (3.3.3) (3.3.4) (3.3.5) To ensure hyperdominance of these matrices a 2 j . and a 2 2 must satisfy inequalities: a 2 2 ( 2 a 2 i - 1) j: 0 (3.3.6) 2 a 2 2 ( 2 a 2 1 - 1) + 2 a 2 2 * 0 (3.3.7) a 2 2 ( 2 a 2 1 - 1) + 2 ( a 2 2 - a 2 1 + 1) ^ 0 (3.3.8) S 0 (3.3.9) * 0 (3.3.10) a 2 2 ( a 2 1 - 1) 2 a 2 2 a 2 2 ^ a 2 1 - 1) a 2 2 ( a 2 j - 1) + ( a 2 i + 2a 2i +2) 5 0 (3.3.11) Figure 3.3.2 is a graphic representation of these inequalities. Figure 3.3.2 Range of Transformation M a t r i x Coef-f i c i e n t s The dotted l i n e s correspond to the bounds when (3.3.6) to (3.3.8) are s a t i s f i e d with the equality signs. The closed i n t e r i o r of these l i n e s defines the set of transformation matrix elements which y i e l d hyper-dominant C matrices. S i m i l a r l y , the dashed l i n e s define the bounds i on (3.3.9) to (3.3.11) ensuring the p h y s i c a l r e a l i z a b i l i t y of G . The i n t e r s e c t i o n of these regions (shown cross-hatched on f i g u r e 3.3.2), defines the closed set of transformations, 5, y i e l d i n g p h y s i c a l l y r e a l equivalent networks. The i n t e r i o r points of 5 y i e l d maximally connected networks, i . e . networks wherein both kinds of elements connect a l l p o s s i b l e com-binations of node-pairs. The boundary l i n e s of E define transformations which y i e l d networks i n which one of the elements i s absent. The i n t e r -s e c t i o n of two of these l i n e s y i e l d s networks wherein two of the elements are missing, i . e . minimal networks which r e a l i z e admittance (3.3.1). The points i d e n t i f i e d on f i g u r e 3.3.2 are the c o e f f i c i e n t s of transformations r e s u l t i n g i n w e l l known canonic networks. These are A u n i t transformation Foster Networks Cauer Networks This example i n d i c a t e s that group transformations, at l e a s t f o r the second order case, are u s e f u l to generate a l l equivalent networks of a l l p o s s i b l e configurations v i a the construction of 5. 2 8 3 . 4 H i g h e r O r d e r N e t w o r k s t h T h e m a t r i x o f t r a n s f o r m a t i o n o f a n n o r d e r n e t w o r k 2 c o n t a i n s ( n - n ) a r b i t r a r y e l e m e n t s . T h u s , 5, t h e r e g i o n o f p r o p e r t r a n s f o r m a t i o n s o f n e t w o r k s o f o r d e r t h r e e ( o r m o r e ) a n a l o g o u s t o f i g u r e 3 . 3 . 2 , a r e h y p e r v o l u m e s i n s i x ( o r m o r e ) d i m e n s i o n a l s p a c e s . Due t o t h e u n i v e r s a l i n a b i l i t y t o v i s u a l i z e s u c h s p a c e s o n l y some g e n e r a l c o n c l u s i o n s c a n b e d r a w n a b o u t t h e n a t u r e o f t h e s e t s o f e q u i v a l e n t n e t w o r k s . T h e h y p e r d o m i n a n c e o f t h e t r a n s f o r m e d c o m p o n e n t m a t r i c e s i m p o s e s t w o t y p e s o f c o n s t r a i n t s o n t h e e l e m e n t s o f t h e s e m a t r i c e s a n d h e n c e o n t h e e l e m e n t s o f t h e t r a n s f o r m a t i o n m a t r i c e s . T h e s e c o n s t r a i n t s may b e g i v e n g e o m e t r i c i n t e r p r e t a t i o n i n t h e n ( n - 1) d i m e n s i o n a l h y p e r s p a c e o f t h e e l e m e n t s o f t h e t r a n s f o r m a t i o n m a t r i c e s ( a ^ : i £ j ) s i m i l a r t o t h e t w o d i m e n s i o n a l c a s e . a ) T h e n o n - p o s i t i v e n e s s o f t h e o f f - d i a g o n a l e l e m e n t s o f C a n d G , ° i j * 0 ; i * j ( 3 . 4 . 1 ) and * 0 ; i * j ( 3 . 4 . 2 ) i m p l y t h a t t h e h y p e r s p a c e i s d i v i d e d i n t o r e g i o n s b y t h e p l a n e s r g^- = 0 a n d c ' i j = 0 * I n t n e r e g i o n o n o n e s i d e o f g ^ = 0 , i n e q u a -l i t y ( 3 . 5 . 1 ) i s s a t i s f i e d a n d n o t o n t h e o t h e r ; s i m i l a r l y , i n e q u a l i t y ( 3 . 5 . 2 ) i s s a t i s f i e d i n t h e r e g i o n o n o n e s i d e o f t h e p l a n e c ^ = 0 a n d 29 not on the other. The networks corresponding to transformation matrix elements lying on the above planes are characterized by the absence of the appropriate element connecting nodes I and j . b) The other type of restrictions, i.e., I g], « 0 ; j > i (3.4.3) i = 1 J and n i - 1 c $ 0 ; j > i (3.4.4) can also be interpreted as planes, dividing the hyperspace. In this case, the vanishing elements in the networks, corresponding to boun-th dary points, connect the j and the datum node. The intersection of a l l the 2n regions i n which an inequality i s satisfied i s a set of disjoint hypervolumes corresponding to trans-formations between physically realizable networks. The complement of this set transform networks into circuits containing negative elements. Points lying on a straight line connecting the points corres-ponding to two equivalent realizations define an i n f i n i t e set of continuous transformations between the two networks. The intermediate networks may or may not be physically realizable depending on whether or not the end points of the line are i n the interior of the same * This suggests an algorithm to generate a required realization from a canonic network numerically. The advantage of this algorithm over existing ones is that i t does not require an arbitrary set of i n i t i a l conditions. However, the algorithm does not establish r e a l i z a b i l i t y conditions, which was the purpose of this research. Therefore, i t was not implemented. 30 volume. The apex of a hypervolume defined by the i n t e r s e c t i o n of 2n planes, corresponds to a minimal network. The existence of a f i n i t e set of such apexes i s ensured by the existence of canonic networks. However, the i n t e r s e c t i o n of the 2n planes required to transform a network into a non-canonic configuration may be empty, i n which case the s o l u t i o n i s not part of the hyperspace and the r e s u l t i n g network w i l l contain complex, i . e . , non-physical, elements. 3.5 Complex Networks The complex nature of networks may be demonstrated by f o r c i n g a non-canonic network to r e a l i z e an impedance function not i n Z> (N^) For example, i f the required impedance of the networks shown i n f i g u r e 2.4.2 has na t u r a l frequencies v i o l a t i n g i n e q u a l i t y (2.4.18), then the network should have complex elements. For t h i s simple case t h i s can be v e r i f i e d by: a) analyzing the network, b) equating c o e f f i c i e n t s , and c) s o l v i n g the r e s u l t i n g set of equations. Using such a procedure, the parameters of the impedance fu n c t i o n , 2 3 Z(s) = I A i s 1 / I B i s 1 (3.5.1) i = 0 i = 0 can be written i n terms of the component values as: 31 81 83 AO = — - (3,5.2) C1 C2 C3 Al i — { g i ( c 1 + c 2) + g 3 ( c 2 + c 3)} (3.5.3) p l C 2 - 3 A 2 = 7 ^ < cl c3 + c l c 2 + C2 C3> ( 3 ' 5 - 4 ) 818060 BO = X Z J (3.5.5) C1 C2 C3 BI C1 C2 C3 B2 C1 C2 C3 {g 2[g 1(c 1 + c 2) + g 3 f c 2 + c 3)] + g l g 3 ( C l + c 2 + c 3)} (3.5.6) K s 3 ( c 2 +  C3 ) + C 3 g l ( c l + c 2 ) + g 2 ( C l c 2 + C ; L c 3 + c 2c 3)} (3.5.7) B3 = 1 (3.5.8) The above six equations can be solved for the six unknowns c^ and g^ ; i = 1,2,3. The algebra involved, even for this simple case, i s quite complicated. To i l l u s t r a t e this complexity, the details of the calculations are shown in Appendix A. The results of the calculations were used to produce the following examples, shown on figures 3.5.1 to 3.5.3. The f i r s t example shows an impedance which is realizable by the specified topology and the two equivalent realizations. The b i -quadratic equation for c^ (in Appendix A) has four roots. Nevertheless, this impedance has only two realizations since the two values of c 3 1.27924477740 01 1.3475477037D-01 7.27974559340-02 Z(S> = • : • S + 4.03935385650 01 S + 9-17472B83280 00 S + 4.31732602130-01 H-c H CD U> C l NETWORK ffl RE I I 1.C00O0CD 00 -0.0 NETWORK 92 R E IH 2.3865140-01 -0.0 M X ta 3 X) M ro co cz C3 5.000000D-01 0.0 1.0000000-01 -0.0 1.2353640 00 0.0 1.2498490-01 -0.0 Gl « 4.00COOOD 00 0.0 1 . 6 5 0 1 1 9 D 00 0 . 0 G2 » 2.00X3COOD 00 0.0 2.0000000 00 0.0 G3 - 1.0000000 00 . 0.0 1.7878970 00 0 . 0 THE LARGEST ERROR APPEARS IN THE SECOND COEFFICIENT OF THE DENOMINATOR OF THE RATIONAL FRACTION FORK OF Z(S> WHICH IS: w 5.0000000 01 to THF COMPUTED COEFFICIENT IS: 5.000000D 01 0.0 5.0000000 01 0.0 THE PERCENT ERROR IS: 0.0000000000028706* 0.0000000001745875S 8.5786437627D-02 0-00000000000 00 2.91421356240 00 US) » • • S + 1.71572875250-01 S • 1.00000000000 00 S + 5. 8284271247D 00 OP c I-i ro NETWORK *1 RE NETWORK #2 P.E IK W X 3 C l C2 1.OOOOOOOOOOD 00 O.OOOOOOOOOOD 00 l-OOOOOOOOOOD 00 O-OOOOOOOOOCO 00 1.00000000000 00 0.00000000000 00 l.OOOOCOOOOOO 00 0.0C0U3OOOOO0 00 C3 1.OOOOOOOOOOD 00 O.OCOOOOOOOOD 00 1.00000000000 00 0.00000000000 00 Cl 1.OOOOOOOOOOD 00 0.OOOOOOOOOOD 00 1.00000000000 00 0.OOOOOOOOOOD 00 1.00000000000 00 0.00000000000 00 t.OOOOOOOOOOD 00 O.OOOOOOOOOOD 00 G3 1.OOOOOOOOOOD 00 O.OOOOOOOOOOD 00 1.00000000000 00 O.OOOOOOOOOOD 00 THE LARGEST ERROR APPEARS IN THE SECOND COEFFICIENT OF THE DENOMINATOR OF THE RATIONAL FRACTION FORM OF US) WHICH I S : 7.OOOOOOOOOOD 00 THE COMPUTED COEFFICIENT IS: THE PERCENT ERROR IS: 7.000000000CU 00 O.OOOOOOOOOOD 00 o.nooonnooooD oo 7, 7.OOOOOOOOOOD 00 O.OOOOOOOOOOD 00 O.OOOOOOOOOOD 00 X CO LO Z { S) 1.OOOOOOOOOOD OO 1.OOOOOOOOOOD 00 1.OOOOOOOOOOD 00 S + I.OOOOOOOOOOD 00 S • 2.OOOOOOOOOOD 00 S + 3.OOOOOOOOOOD 00 NETWORK ffl NETWORK *2 c l-l ro Ln LO w 3 M ro LO Ln LO C l C2 C3 RE -3.2915740-01 3.966942D-01 3.374219D-01 IM 2.6013T4U-01 -2.0913060-01 -5.1006780-02 RE 3.9669420-01 3.3742190-01 IM -3.2915740-01 -2.601374D-01 2.091306D-01 5- 10C678D-02 GI 2.7284340 00 8.7419140 00 2.7284340 00 -3.7419140 00 G2 5.4545450-01 5.4545450-01 0.0 G3 » 6.495491D-02 4.1572fl2()-02 6.495491D-02 -4.1572320-02 THE LARGEST ERROR APPEARS IM THE SECOND COEFFICIENT OF THE DENOMINATOR OF THE RATIONAL FRACTION FORM OF Z(S) WHICH IS: • 6.000000D 00 THE COMPUTEO COEFFICIENT IS: THE PERCENT ERROR IS: 6.000000D 00 -8.2644070-15 0.000000000000138251 6.0000000 00 8.2644070-15 0.000000O000001382S LO 4> 35 must also be roots of this equation, due to the symmetry of the net-work configuration. The next example (figure 3.5.2) shows an impedance which l i e s 20 x ~ on the boundary of Jj (N ). The two realizations of the impedance in this limiting situation coincide. The impedance given in the third example obviously violates Lee's inequality (2.4.18). In this case the solutions for c^ are com-plex, and the element values associated with the same branch i n the two realizations are complex conjugate pairs. Our approach to group theory shows that non-canonic networks occur in complex conjugate pairs i f the impedance parameters are not included in the domain of r e a l i z a b i l i t y . This conclusion i s verified •in the above examples. 36 CHAPTER IV  GEOMETRIC APPROACH 4.1 Mathematical Formulation The impedance r e a l i z i n g a b i l i t y of a l i n e a r network may be deduced from the exi s t e n c e of a modal m a t r i x . In order to determine t h i s e x i s t e n c e the impedance f u n c t i o n should be expressed i n terms of the d e s c r i p t i o n of network topology. For minimal RC s t r u c t u r e s , the p r e -sence of a c a p a c i t o r t r e e suggests the use of the s t a t e model as the v e h i c l e f o r such a f o r m u l a t i o n . (The same r e s u l t s may be obta i n e d , a l t e r n a t i v e l y , as shown i n Appendix B, using cut set a n a l y s i s . ) The fundamental c i r c u i t equation w r i t t e n f o r the normal t r e e , i . e . , the c a p a c i t o r t r e e , of a RC complementary-tree s t r u c t u r e has the form " U 0 B r c 0 U B. U IC -J The f i r s t s u b s c r i p t s on the components of the v o l t a g e v e c t o r denote tree-branches (b) and chords (c) and the second s u b s c r i p t s denote the type of elements: r e s i s t o r s ( r ) , c a p a c i t o r s (c) and in p u t p o r t ( i ) . The submatrix B^ c expresses the t o p o l o g i c a l r e l a t i o n between chords A ( r e s i s t o r s ) and the tree-branches ( c a p a c i t o r s ) . I t i s an E-matrix w i t h i , j element u n i t y i f the c i r c u i t defined by the i * " * 1 chord contains t l i th the j tree-branch; t h a t i s , the j c a p a c i t o r i s contained i n a * (24) An E-matrix has +1, -1, or 0 f o r the determinant of every f i n i t e square submatrix. V cr V c i be = 0 (4.1.1) 37 simple path formed from tree-branches between the v e r t i c e s f o r which the i ^ * 1 r e s i s t o r i s i n c i d e n t . B . i s an E - m a t r i x w i t h i , k element u n i t y IC -" J til i f the k c a p a c i t o r i s p a r t of an a l l - c a p a c i t o r , simple path between th the nodes of the i input p o r t ; of other elements and B . are zero. B . J v v ' i c xc i s a row ma t r i x f o r one por t networks. The fundamental cut-set equation i s T T [ - B 1 -B7 U ] r c xc cr c i be = 0 (4.1.2) and the branch volt-ampere r e l a t i o n s are given by ^bc "be = I , and G V = 1 c c r c r (4.1.3) where C , and G are diagonal matrices w i t h elements corresponding to b e capacitance and conductance v a l u e s , r e s p e c t i v e l y . The s t a t e model i s e s t a b l i s h e d by e l i m i n a t i n g a l l v a r i a b l e s except the s t a t e v a r i a b l e s ( c a p a c i t o r v o l t a g e s ) , source v a r i a b l e s ( p o r t c u r r e n t s ) , and output v a r i a b l e s (port v o l t a g e s ) . A convenient form of the s t a t e model i s : T T G B V, + B T r c c r c be xc c i °b V b c = G B V, + B T I (4.1.4) V = - B . V. c i i c be (4.1.5) The in p u t impedance Z(s) , i n terms of c i r c u i t matrices i s ob-ta i n e d from the Laplace transform of the s t a t e model as Z(s) = B . [s C, + B T G B ] 1 B T xc b r c c r c xc (4.1.6) Since the matrices i n s i d e the brackets i n the above equation are r e a l , 38 symmetric and p o s i t i v e - d e f i n i t e , they can be considered as generating a p e n c i l of qu a d r a t i c forms which can be reduced simultaneously to sums of squares by a s u i t a b l e non-singular t r a n s f o r m a t i o n . In other words, T there e x i s t s a r e a l m a t r i x M which w i l l d i a g o n a l i z e C, and B G B D r c c r c T con c u r r e n t l y by a congruent t r a n s f o r m a t i o n . The ma t r i x sC, + B G B J3 TC C i s the cut-set admittance matr i x of the network on which M,the modal m a t r i x , performs the normal co-ordinate t r a n s f o r m a t i o n . Since M i s n o n - s i n g u l a r , Z(s) may be w r i t t e n as —1 T —1 T -1 T T Z(s) = B. MM .(sC. + B G B ) (M ) M BT IC D r c c r c IC = (B. M)(sM TC,M+ [B M ] T G [B M ] ) _ 1 ( B . M ) T i c b r c c r c i c -1 -1 T = K(sU + T ) K • K Mi-^ h-Mi k T. <4-1-7> or n k k z,, = I — —^TT (4.1.8) J £=1 s + T. 1 Thus f o r every given RC complementary-tree s t r u c t u r e the modal m a t r i x w i t h elements from the f i e l d of r e a l numbers i s dependent upon the t o -p o l o g i c a l c o n f i g u r a t i o n and component values of the network and i s unique T i f the requirement that M C^ M be the u n i t m a t r i x i s met. For e q u i v a l e n t networks w i t h the same topology and same Z(s) but d i f f e r e n t element v a l u e s , the modal matrices are d i f f e r e n t . Conversely, i f Z(s) and the network topology are given, the component values may be determined from a modal ma t r i x . Hence Z(s) i s p h y s i c a l l y r e a l i z a b l e i f and only i f such a m a t r i x e x i s t s , and the 39 number of eq u i v a l e n t r e a l i z a t i o n s i s equal to the number of d i s t i n c t modal matrices. Therefore the problem of s y n t h e s i s of RC complementary-tree s t r u c t u r e s i n the widest sense, i . e . , r e a l i z a b i l i t y and r e a l i z a t i o n of a l l e q u i v a l e n t networks, may be solved by determining the modal matrices. This f o r m u l a t i o n i s more general than those used by G u i l l e m i n and others i n connection w i t h normal co-ordinate t r a n s f o r m a t i o n s . A l l previous methods are c h a r a c t e r i z e d by the d i a g o n a l i z a t i o n of a node-to-datum admittance m a t r i x by some modal m a t r i x M'. The elements of M' are obtained by equating an element of Z (s) = M' | ls(<y^>m| [ M , T (4.1.9) to a p r e s c r i b e d f u n c t i o n . The i m p l i c i t assumption i n t h i s f o r m u l a t i o n i s that the network i s common ground, which i s not true i n ge n e r a l . 4.2 Geometric P i c t u r e The essence of the method f o r determining the modal m a t r i x , proposed i n t h i s t h e s i s , i s i n t e r p r e t i n g the modal matri x g e o m e t r i c a l l y , and f i n d i n g the s o l u t i o n to the geometric problem so obtained. Let the modal ma t r i x , be a set of row v e c t o r s . M = m l m n (4.2.1) From equation (4.1.7), (or from the nature of the c h a r a c t e r i s t i c m a t r i x of p e n c i l s of q u a d r a t i c forms) f o l l o w s that the rows of M form 40 an orthogonal s e t ; i . e . , MT M = U (4.2.3) or C^1 = S b = M M T (4.2.4) tin The el a s t a n c e value of the i tree-branch i s equal to the squared norm of v e c t o r m\. Fu r t h e r , from the same equation, ( B r c M ) T G c (B^M) = T _ 1 (4.2.5) G " 1 = (B _M) T (B C M ) T (4.2.6) R = (B M T 1 / 2 ) ( B M T 1 / 2 ) T (4.2.7) c r c r c 1/2 Hence the rows of the ma t r i x B M T a l s o form an orthogonal set and r c b the squared norm of the j ^ r o w equals the value of the r e s i s t a n c e of .th , , the j chord. Comparing equations (2.2.1) and (4.1.7) e s t a b l i s h e s the r e -l a t i o n s h i p between the residues of the p r e s c r i b e d impedance f u n c t i o n and the elements of M as p. = k..k., = [J [B. ]. m.. ][7 [B. ]. m ,] „ O N p k i i j k l^ 1 i c J j jk J L£ L i c J i j k (4.2.8) For input impedance (4.2.8) reduces to i . e . , i s equal to the k*"*1 component of the v e c t o r obtained by summing the vectors m^  where the index i i s determined by the non-zero elements of B. . i c 41 Summarizing the above, the problem of determining the modal matrix may be stated in geometric terms as follows: $ and are n-dimensional, Euclidean, vector spaces with orthonormal basis sets {e^} and {e^} respectively. The linear operator 1/2 - - 1/2 T maps the vectors of S onto , i.e., i f m, e $ then = nu T e 2 c » or, in matrix notation, y l i nx n lx m . nx (4.2.10) The unknown to be determined is a set of vectors in S space, {nu | 1=1, n} ; which satisfies equations (4.2.3) to (4.2.9). The geometric interpretations of these equations are: i) The set {nu | i=l, n} consists of mutually orthogonal vectors in 3" space. i i ) The set of vectors {v\ | i=l, ..., n} consists of mutually orthogonal vectors in OR. space. i s the vector obtained by summing u, , u. , u , ... i f capacitors c, , c„ , c , ... form a simple all-capacitor k I m k & m path between the vertices for which resistor r. is incident. r x i i i ) The vector s formed by summing the vectors m^ , m^ , m^ , i f capacitors c^, c^, c^, ... form a path connecting the nodes of the input port, i s known: s = ( • • • (4.2.11) The set of vectors {m. i=l, n} contains n unknowns. The orthogonality conditions yield n(n-l)/2 equations each, and the known vector gives n equations. Thus the orthogonality conditions and 42 the known vector provide sufficient information to determine the modal matrix. As an i l l u s t r a t i v e example, consider a canonic network of the Foster form (Figure 4.2.1)-The impedance function and the relevant matrices for this configuration are: Z ( s ) - I — i i=l s + T -1 (4.2.12) B. — [1,1,. ...,1.1 i c rc 1 0 1 (4.2.13) (4.2.14) The known vector s in 5 space i s n s = I m. i=l 1 (4.2.15) r, I—WW j — w -II II II Figure 4.2.1 Foster Type Network Since a l l m^  vectors are mutually orthogonal, they a l l termin-ate on a hypersphere with origin at s/2 and radius |s/2|. The orthogonal set {ir^} maps onto an orthogonal set {y i>. The only complete set of vectors which preserve t h e i r orthogonality under the mapping from -S" to X are p a r a l l e l to the coordinate axes. Therefore the m vectors are the i n t e r s e c t i o n of the coordinate axes and the hypersphere: i = l Z J Jp7 -\ 2 n p . i = l 4 (A.2.16) Hence the modal matrix i s : M = n J (4.2.17) which y i e l d s the well-known r e s u l t s : c. = p. \ and r. = T. p.. And, 1 1 1 1 1 since the rows of M may be numbered a r b i t r a r i l y , there are n! d i s t i n c t M matrices; the number of equivalent networks therefore equals the num-ber of permutations of the sections shown i n Figure 4.2.1. The r e s u l t s of t h i s example are not new and are stated only to i l l u s t r a t e the geometric method which can also be applied to non-canonic networks. 4.3 Third Order Irreducible Network The lowest order of i r r e d u c i b l e (non-canonic) RC complementary tree structures i s the t h i r d order network shown i n Figure 2.4.2 and repeated here f o r convenience as Figure 4.3.1. The submatrices of the fundamental c i r c u i t equation f o r t h i s network are: B. = [1 1 1] i c J (4.3.1) 44 Figure 4.3.1 Third Order Irreducible Network B rc 1 1 0 1 1 1 0 1 1 (4.3.2) The geometric constraints, formulated in the previous section, on the set of vectors {m^ | i=l, 2, 3} to be determined are as follows: i ) I n -S space the vector 3 3 = y m. = y ^p^" e-1=1 1=1 (4.3.3) is known. i i ) In -S space, the three vectors m^ , ra^ and m^  are mutually orthogonal. i i i ) In 7Z. space the three vectors - + P 2 + u3» 45 V 1 = ^1 + P2 a n c* V3 = y2 + V3 a r e m u t u a x x y orthogonal. In 1R. space v 2 is known since i t i s the mapping of s e -S ; - A - - 1/2 i.e., = o = s T = £ /x^ e^. and span a subspace, P, 1,-orthogonal to v 2« The vectors ^ + y2 + y3 3 1 1 ( 1 2~^1 + ^3^ a r e i n ^' where II denotes the class of vectors, £, congruent modulo P to a: £ = a (mod P) (4.3.4) The normal vector of II and the equation of II are (4.3.5) and y /r.p.c. — y T'.P. = o L ±^±^x L x i (4.3.6) respectively, where £^ i s the i ' * 1 running variable. 1/2 The transformation T and i t s inverse map planes onto planes, Thus m^ , m^ , (m^  + and £ m^  terminate on a plane P orthogonal to n = 3 I i=l i=l J J 3 2 T i ^ T g i (4.3.7) The equation of P is 3 3 I T±^T x ± - I Tj.pi = 0 i=l i=l (4.3,8) where x^ denotes the i ^ variable. But in -S space a l l nu vectors terminate on the surface of a sphere with origin at s / 2 and radius | s / 2 J . Hence the vector 46 (m^  + m^)/2 must terminate on a sphere with origin at s/4 and radius |s/4|. Therefore the intersection of P and is the locus of (m^  + m^)/2, The subsequent development is f a c i l i t a t e d by the rotation of the co-ordinate axes of both spaces so that one of the basis vectors, becomes collinear with the normal vectors of P in S and IT in 33. spaces. The matrix of transformation, T^, which rotates the co-ordinate axes of an n-dimensional space so that the basis vector w i l l align with a given vector A = £ a± e± 1 S : (4.3.9) where "jk t.± . / / l + t. ; j < n J = n (4.3.10) 2 7 £ 2 t. = a.,, / ) a. J J + 1 / i=l 1 (4.3.11) kj f \ / a j + l -1 - t , k $ j k = j+1 k > j+1 (4.3.12) The required matrices of rotations T and T in S" and 33. spaces respectively can be readily determined and the mapping from S to 31 , with respect to the new set of coordinate axes can be expressed in terms of these matrices as 1/2 T T R s (4.3.13) 47 With respect to the new bases, the equation of the vector s i n -S" space i s s = (s^, s', s') , . , i i x=l i = l (4.3.14) and the equation of the plane P i s * 3 - s 3 (4.3.15) The equations of spheres and S2 are 3 I i = l X i " 2 4 ^ p i 4 1=1 (4.3.16) s! 2 J i H ^ } ^ J i ' i (4.3.17) The i n t e r s e c t i o n of plane P and spheres S^, S£ becomes s! 2 f r 2 2 = r ^ (4.3.18) 3 ^ 2 2 [ x i " Tl = 16 'I p i " i = U 4 J 1=1 L S i r , 2 - r^j (4.3.19) The vector (m| + mp /2 i s a common point of P and 82^ fore, the i n t e r s e c t i o n of P and S2 i s not empty. There-Thus must be r e a l , which implies that 48 - 1 I S 3 4 * 0 (4.3.20) S u b s t i t u t i n g the value of s^ we o b t a i n a necessary c o n d i t i o n on the parameters of Z(s) which i s r e q u i r e d f o r r e a l i z a b i l i t y ; namely, -3 i 2 . \ V i 3 2 5! T.p . . , 1 1 1=1 J L i = l 3 (4.3.21) To o b t a i n s u f f i c i e n c y c o n d i t i o n s f o r the t h i r d - o r d e r case the geometric problem must be f u l l y s o l v e d . This may be accomplished by f i n d i n g the p o i n t (m| + m'^)/2. Consider the plane P i n the transformed coordinates (Figure 4.3.2). Figure 4.3.2 P plane i n Transformed Coordinates 49 Let the l i n e s a-a' and b-b' map onto the d i r e c t i o n of the minor and major axes of the e l l i p s e s i n 33. space corresponding to the c i r c l e s i n -S space. These p r i n c i p a l axes are p a r a l l e l to the e i g e n -~ -1 T ~ -1 vec t o r s of the matri x T = ( T T ) (T^ ) where T^ i s obtained from T^ by d e l e t i n g the l a s t row and the l a s t column. The e x p l i c i t form of T i s T = t . . 7 i i 1 J ; i 11 V 2 ( T 1 P 1 + T 2 P 2 ) 2 ^ 2 T 1 P1 + T 2 P2 ( T l - T 2 ) V l T 2 P 2 T 3 P 3 ( T l p l + T 2 p 2 ) ( T l p l + T 2 p 2 ^ T i p i T 3 ( T 1 P 1 + T 2 P 2 ^ V i ( T 1 P 1 + T 2 p 2 ^ T i p i 1 2 (4.3.22) "The l i n e p a r a l l e l to a a-a 1 through < T 2 - T l ) P 1 P 2 2 2 L T l p l + T 2 p 2 i [ T 3 ( T 1 P 1 + T 2 P 2 ) " ( T 1 P 1 + P 2 P 2 ) ] i _ 2 ( T 1 P 1 + T 2 P 2 ^ T i P i (4.2.23) i n t e r s e c t s the c i r c l e i n p o i n t c. I t can be shown that the c i r c l e S„, which i s drawn through 0, w i t h o r i g i n 0„ l o c a t e d on the l i n e 0.. - c A l " X2 - -and w i t h r a d i u s r 0 = (-—-—:—)r../2 i s the locus o f (mi + m')/2 which J A , T A „ 1 1 J See Appendix C 50 ensures the orthogonality of (u^ + u^) and (u^ + u^) i n ^ s P a c e « X^ and X 2 are the eigenvalues of T. But (m| + m3)/2 must l i e on the c i r c l e S^J therefore i n order that Z(s) be r e a l i z a b l e S 2 and must i n t e r s e c t . In other words r 2 - r 3 | * |0 2 - 0 3| « | r 2 + r 3 | (4.3.24) Inequality (4.3.24) can be expressed i n terms of the parameters of Z(s) as where V 2 I V i v 2 u 1 1 J x -4 s ,2 (4.3.25) K = 1 -2t t 11 22 ~2 ~2 ~2 ^ 1 1 + Z12 + fc22 (4.3.26) X = ( s 2 2 " s i 2 ) + 2 s i s 2 * 1 + 1l> (4.3.27) 2 t l l fc12 ~2 ~2 ~2 (4.3.28) and t ^ and s!^  are given i n equations 4.3.22 and 4.3.23 r e s p e c t i v e l y . The r e l a t i o n s h i p between the parameters of a given Z(s) ex-pressed i n i n e q u a l i t y 4.3.25 i s the necessary and s u f f i c i e n t condition f o r r e a l i z a b i l i t y by the t o p o l o g i c a l configuration shown i n fig u r e 4.3.1. Inequality 4.3.26 and figure 4.3.2 i n d i c a t e three d i f f e r e n t 51 cases: 1) I n e q u a l i t y i s s a t i s f i e d . C i r c l e s and i n t e r s e c t at po i n t s e and f corresponding to two d i s t i n c t s o l u t i o n s . Hence two e q u i -v a l e n t networks e x i s t which r e a l i z e Z ( s ) . 2) E q u a l i t y i s s a t i s f i e d . C i r c l e s and are tangent to each other. The two s o l u t i o n s , hence the two eq u i v a l e n t networks, be-come c o i n c i d e n t . 3) I n e q u a l i t y 4.3.25 i s not s a t i s f i e d . C i r c l e s and have no: common p o i n t and Z(s) i s not r e a l i z a b l e by the network shown i n f i g u r e 4.3.1. Thus the r e s u l t s of the geometric f o r m u l a t i o n are c o n s i s t e n t w i t h the conclusions derived from group theory. On f i g u r e 4.3.2 e i t h e r p o i n t e or f corresponds to the v e c t o r ~(m^ + m^). The r e a l i z a t i o n of the 'given Z(s) may be accomplished when the l o c a t i o n s of these p o i n t s are known as f o l l o w s : The l i n e i n the plane P passing through e (or f ) and perpendi-c u l a r to the l i n e j o i n i n g 0^ and e (or f ) i n t e r s e c t s the c i r c l e at m^  and m^ . The ve c t o r m^ , i s obtained by s u b t r a c t i n g + m^ from s' . - - T The t r a n s f o r m a t i o n m. = m! (T Q) y i e l d s the set of ve c t o r s {nu| i = 1, 2, 3} hence the rows of the modal ma t r i x . The r e q u i r e d element values can be c a l c u l a t e d from the modal mat r i x . 4.4 Fourth Order Networks The t h i r d order networks can adequately be s t u d i e d by i n v e s -t i g a t i n g the only i r r e d u c i b l e s t r u c t u r e of t h i s complexity. For h i g h e r 52 order networks, however, the number of non-canonic t o p o l o g i c a l c o n f i g u r -a t i o n s increases r a p i d l y w i t h the network complexity and even the enumeration of these c o n f i g u r a t i o n s becomes a formidable task. For f o u r t h order networks, the number of non-canonic networks i s s t i l l low enough to l i s t them as shown i n f i g u r e 4.4.1. I r r e d u c i b l e networks of order greater than three may be grouped i n t o two c l a s s e s : (a) networks which are r e d u c i b l e to lower order con-f i g u r a t i o n s ( f i g u r e 4.4.1 (a) to (e)) and (b) networks which c o n t a i n no RC p a r a l l e l branch or RC cut s e t ( f i g u r e 4.4.1 ( f ) to ( h ) ) . An argument i d e n t i c a l to the one used to prove p a r t (b) of theorem 2.4.1 i n d i c a t e s that the n a t u r a l frequency r e a l i z i n g a b i l i t i e s of networks i n c l a s s (a) are the same as those of the lower order networks to which they may be reduced. No s i m i l a r statement can be made about the residues of these networks. To determine the impedance r e a l i z i n g a b i l i t i e s of networks be-l o n g i n g to c l a s s ( b ), the geometric problem i n n-dimensional space should be solved. Before proceeding w i t h the s o l u t i o n f o r n = 4, some general observations concerning the d i f f e r e n c e between the geometric problems i n 3-dimensional and hyper spaces are worthwhile. For s i m p l i c i t y assume that the c a p a c i t o r s form a l i n e a r t r e e such that the ends are the input nodes to the network. Then the S space o r t h o g o n a l i t y c o n d i t i o n s (m., m . ) = c . ^ 6 . . (4.4.1) imply that the set of vectors { i r u : i = l,...,n} terminate on the su r f a c e of the hypersphere: o S 1 (x) = (x,x) - (s,x) = 0 (4.4.2) Figure 4.4.1 Fourth Order I r r e d u c i b l e Networks 54 where n n s = I m i = I e ± (4.4.3) i = l i = l i s a known v e c t o r . This hypersphere maps onto the h y p e r e l l i p s o i d l^l) = (L T _ 1 l) - (a, O = 0 (4.4.4) i n "K, space where -1/2 -a = T 1 s (4.4.5) Thus the vectors of "& space must ( i ) s a t i s f y the o r t h o g o n a l i t y c o n d i t i o n s i n & space; and ( i i ) terminate on the e l l i p s o i d 1-^ (0 = 0 The second of these c o n d i t i o n s i s a necessary, but f o r n > 3, not a s u f f i c i e n t c o n d i t i o n to ensure 5 space o r t h o g o n a l i t y . This i s the preeminent d i f f e r e n c e between the geometric problem i n three dim-e n s i o n a l and hyper spaces, because lx (p.) =0 ==> _S 1 (m.) = 0 (4.4.6) and i n the three dimensional s i t u a t i o n S 1(m i) = (nu, nu) - (n^ + nu, + m3, nu) (4.4.7) Therefore S 1(m 1) = 0 ==> (m]L,m2) + (mv = 0 (4.4.8) S 1(m 2) = 0 (m 1 5m 2) + (nu,,^) = 0 (4.4.9) 55 S 1(m 3) = 0 (m1,m3) + (nym.^) = 0 (4.4.10) o r 1 0 1 1 1 o - (m^,m2) 1 ^ m l ' m 3 ^ 1 . _ (m2,m3) = 0 (4.4.11) But the determinant of the c o e f f i c i e n t m a t r i x above i s non zero i m p l y i n g that the vec t o r s formed from the i n n e r products (nu,m ) i s i d e n t i c a l l y equal to zero; i . e . , the vectors i n -S space are mutually orthogonal as req u i r e d . The r e s t r i c t i o n (u.^ ) =0: i = 1, 2, 3} forms an indepen-dent or b a s i s set of c o n s t r a i n t s on the v e c t o r s , {vu}. Any other i n d e -pendent set of c o n s t r a i n t s l i k e w i s e ensures the S space o r t h o g o n a l i t y . In f a c t the set of c o n s t r a i n t s used f o r the s o l u t i o n of the t h i r d order problem i s (y^) = 0, l± = 0, l(v± •+ = 0}. For higher order spaces (vu) =0: i = 1, ..., n} no longer ensures the -S space o r t h o g o n a l i t y c o n d i t i o n s . For example, i f n = 4 the above c o n s t r a i n t s provide four equations f o r the s i x i n n e r products. 1 1 0 0 1 1 0 0 1 0 0 1 0 1 1 0 0 0 1 0 0 1 1 1 (m 1»m 2) (m^,m3) (m-^rn^) (m2,m3) ( m2 , m4^ (m3,m^) = 0 (4.4.12) To guarantee that the v e c t o r of i n n e r products, (nu,m..): i f j vanishes the c o e f f i c i e n t matrix above must be augmented by two independent 56 rows, so that i t has a non zero determinant. In terms of c o n s t r a i n t s , i n addition to the requirement that a l l y_ vectors terminate on the hyper e l l i p s o i d , I^(u^) = 0> two more independent r e s t r i c t i o n s are ne-cessary f o r -S space orthogonality. Such a p a i r of independent constraints i s , f o r example, ^1 ( ^ i + V = ° 5 1 * jI x ( P ± + y k) = 0 ; k i i or j (4.4.13) (4.4.14) For the n-dimensional s i t u a t i o n the number of vanishing inner products i n -5 space i s n ( n - l ) / 2 , and the number of constraints imposed by the requirement \^ (LU) = 0 i s n; therefore the number of a d d i t i o n a l constraints must be (n-3)n/2. In addition to the above d i s p a r i t y , the major d i f f i c u l t y i n obtaining solutions f o r the higher order geometric problems i s the lack of a b i l i t y to v i s u a l i z e hyper spaces. Even f or the fourth order networks, an e x p l i c i t s o l u t i o n ( s i m i l a r to the t h i r d order case) was beyond the ingenuity of the author, and instead two numerical algorithms were de-veloped and implemented to determine r e a l i z a b i l i t y conditions. The network shown i n figure 4.4.1-(f) i s chosen, a r b i t r a r i l y , as the v e h i c l e f o r the following development. This network i s charac-t e r i z e d by the incidence matrix: rc 1 1 0 0 1 1 1 0 1 0 0 0 1 1 1 1 (4.4.15) 57 Hence, the .R. space orthogonality conditions are 3 4 _ 4 I V± _L * I u±» I ~ V±> M2 } (4.4.16) i=l i=l i=2 4 4 I v,. _L t I y,, y-> (4.4.17) i=2 i=l J y 2 - L y 3 (4.4.18) Thus the geometric problem may be stated: find a set of four orthogonal vectors in 5" space, {rru}, such that a) their sum is known (s — (>/p^, ...» Jp~^)) , and, b) the vectors obtained by mapping {nu} onto space satisfy orthogonality conditions (4.4.16) to (4.4.18). The algorithms to solve this problem are the topics of the following two sections. 4.4.1 Polar angle approach In the set of bases vectors obtained by rotating the co-ordin-ate axes of 43 and ?t spaces by transformations T and T respectively, o R. where the transformations have the same role as in the third order case, 4 the f i r s t three components of the vector a' = ][ y! and the last com-3 _ 4 ± = 1_ 1 ponents of the vectors v| = £ y^ and = £ y^ are zero. The latter i=l i=2 two of these vectors span a plane, I I ' , in a three dimensional space. The normal vector, a', of II' is expressible in terms of two polar angles, 6 and (j), as a' = (sinij) sin6, sin<f> cos0, cos<j>) (4.4.1.1) 58 A second p a i r of t r a n s f o r m a t i o n s , T' and T', which render the t h i r d b a s i s v e c t o r i n 1ft space c o l l i n e a r w i t h a' and leave the f o u r t h b a s i s v e c t o r unchanged, are f u n c t i o n s of 0 and cj). These transformations s i m p l i f y the d e s c r i p t i o n of the locus of v'^  and v^1, which i s the i n t e r -ii _ s e c t i o n of e l l i p s o i d ^ ( 5 " ) = 0 and the plane ^ = ^ = 0. Thus, the v e c t o r s can be found using the procedure described i n s e c t i o n 4.3. The planes, orthogonal to manifolds spanned by {m^, m^'} and {u^1, y^} i n -S and 33, spaces r e s p e c t i v e l y , i n t e r s e c t ' and £j i n a c i r c l e and i n an e l l i p s e . These curves are the l o c i of (m^' m 3 ^ a n ^ {u", u"} i n t h e i r r e s p e c t i v e spaces. But both s e t s of these v e c t o r s c o n t a i n orthogonal pairs*, t h e r e f o r e m^' and m'3 must map i n t o the e i g e n -v e c t o r s of the t r a n s f o r m a t i o n matrix. A l l the v e c t o r s are expressed i n terms of parameters 0 and <j>. These q u a n t i t i e s can be determined from the *R. space o r t h o g o n a l i t y con-d i t i o n s not yet used, namely: 1^(0,4) = (y 2', y j + y p = 0 (4.4.1.2) h 2(6 ,<j , ) = (y'3', y3' + = 0 (4.4.1.3) A computer program was w r i t t e n to d i s p l a y the zeros of h^ and h 2 i n the 0 - tf> plane. The user c o n t r o l s the range of the v a r i a b l e s and the step s i z e ; hence, i n an i t e r a t i v e f a s h i o n , he can determine 0 and cj> w i t h i n any d e s i r e d accuracy or conclude that the impedance i s not r e a l i z a b l e i f the l i n e s corresponding to the zeros of h^ and h 2 do not i n t e r s e c t . 59 The program may be modified to eliminate the i n t e r a c t i v e feature of the procedure, but t h i s was not done since i n th i s research the e x i s -tence of the s o l u t i o n , not a numerical r e s u l t , i s of consequence. To i l l u s t r a t e the polar angle approach the synthesis of the impedance (4.4.1.4) nr \ 43.3 , 18.8 , 3.57 0.377 Z(s) = . , „ + :—„ H :—- + s + 15.0 s + 0.032 s + 0.590 s + 0.94 i s considered. The 6 - <j> pl o t i s shown i n figure 4.4.1.1. The approxi-mate lo c a t i o n s of the in t e r s e c t i o n s i o f the curves (shown as the cross hatched area) are determined by ins p e c t i o n . The range of va r i a b l e s i s set to include only t h i s area to improve the accuracy of the s o l u t i o n (figure 4.4.1.2). This process can be repeated u n t i l the required ac-curacy f o r the s o l u t i o n i s obtained. 0 ~l f TT/2 TT i r TT/2 T~\—r~i—I—I 0 Figure 4.4.1.1 0 - ij; P l o t 60 0.7 TT 0.8 TT 0.4 T 03 TT Figure 4.4.1.2 Refined 0 - <J> p l o t The previous f i g u r e s i n d i c a t e the e x i s t e n c e of two s o l u t i o n s , one of which i s {0 = .274rr, ty = . 822TT}. The component values computed, f o r the network shown i n f i g u r e 4.4.1-(f), using t h i s s o l u t i o n are: c 1 = 0.1, c 2 = 0.3, c 3 .= 1.0, c^ = 0.7 g± = 1.0, g 2 = 2.0, g 3 = 3.0, g^ = 4.0 4.4.2 K-Vector approach An a l t e r n a t i v e procedure may be developed i f the a l g e b r a i s c a r -r i e d f u r t h e r than i n the previous s e c t i o n . According to the theory described at the beginning of s e c t i o n 4.4, the s o l u t i o n of the geometric problem i s p o s s i b l e by c o n s i d e r i n g ¥i space only, s i n c e the -S space o r t h o g o n a l i t y c o n d i t i o n s are 6 1 automatically satisfied i f the vectors of H space are, in addition to the requirement that they l i e on the hypexellipsoid l ± C O = (I, T"1 I) - (5, O = 0 (4.4.2.1) where o = { / p i / x i e\: i = 1, 2, 3, 4} (4.4.2.2) constrained by two independent restrictions to be introduced later. According to the orthogonality conditions defined by the i n -cidence matrix (4.4.15), OR. space can be separated into two 2-dimensional subspaces, and 3^2' s o t n a t •^1 + ^ 2 = ^ (4.4.2.3) {v± = y^ -I- y 2' 2^ = ^3 + V £ ^ 1 (4.4.2.4) {v 3 = ]i± + y 2 + Vy = y 2 + y 3 + y^ ,} e ^ 2 (4.4.2.5) The transformation, T , renders the vector, a' collinear with the JK fourth co-ordinate axis of the space. In this new set of co-ordinates an orthonormal set of basis vectors spanning i s t'K » o'/|o'|) where K = \/l - k 2 2 - k 3 2 l± + k 2 e 2 + k 3 e 3 (4.4.2.6) A second transformation of the space, which w i l l leave the fourth co-ordinate axis unchanged but align K with the third axis, can be accomplished by the rotation T : 62 K k_ 2 2 i-k? 2 2 i - k ; 4 l-k^ -k2 k 2 k 3 (4.4.2.7) In t h i s l a t e s t s et of co-ordinates the subspaces, and bave p a r t i c u l a r l y simple forms; i . e . , the l a s t two components of the vectors of ^ 2 a n c* t' i e f i - r s t t w o components of the vec t o r s of tt, arc zero. The equation of the h y p e r e l l i p s o i d i n t h i s s et Of axes i s X'j G") = ( £ " > T 1 ?••) — ± - ( t 4 , 5") = 0 (4.4.2.8) 44 where t ^ i s the (4,4) element and t ^ i s the f o u r t h row of the m a t r i x T = l i t I I = T T T T T 1 1 i j 1 ! 1 K R R K (4.4.2.9) The i n t e r s e c t i o n of QR, ^  and ^ i s shown i n Figure 4.4.2.1 Since the vectors and V2 are orthogonal i n both spaces, they must be p a r a l l e l to the major axes of the e l l i p s e ^ ^ f l Therefore 1 v , = . . == (ce- + [1 + /l e.) 1 V^ t4(i + c2) 3 4 (4.4.2.10) 63 Figure 4.4.2.1 "H.^ space v 9 = ~. (^ + '[1 - A + ? 2 ] e.) (4.4.2.11) 2 Vt 4 4d + ? 2 ) ^ 4 where 2 t 3 4 5 (4.4.2.12) 44 c23 To determine and the procedure described i n section 4.3 i s used. The vectors are part of the class I B d (mod 7t2) (4.4.2.13) shown i n figure 4.4.2.2. 64 A / \ i A \ v \ \ W \ A M x A \ \ \ \ \ \ l A — r v Figure 4.4.2.2 The Plane of v 3 and The e l l i p s e , . i s the locus of 1/2 G'^ + y^) . The e l l i p s e P3 and the p o i n t q are determined according to Thales' theorem to ensure the o r t h o g o n a l i t y of v'^  and v^. These v e c t o r s are then found from the i n t e r s e c t i o n of and the l i n e p a s sing through p o i n t s x and q. The co-ordinates of the p o i n t s and hence the v e c t o r s ( v ^ , I K : i = 1, 2, 3, can be expressed i n terms of the elements of T, which are themselves fu n c t i o n s of and k^. The attempt to express the above vectors e x p l i -c i t l y i n terms of these v a r i a b l e s f a i l e d , due to the complexity and the overwhelming amount of a l g e b r a ' i n v o l v e d . N e v e r t h e l e s s , the f o r m u l a t i o n i s w e l l s u i t e d f o r numerical computation and the components of the vec-tors can be c a l c u l a t e d r e a d i l y as { k ^ ^ ^ } sweeps out the u n i t c i r c l e . See Appendix C A A See Appendix D 65 To determine the required values f o r these v a r i a b l e s , the a d d i t i o n a l constraints, £ (v^) = 0 and jj^  (y^) = 0 are introduced. I f the zeros of these equations are p l o t t e d , the s o l u t i o n which s a t i s f i e s both constraints i s found from the i n t e r s e c t i o n of the two curves. Such 0-6 ~n k, 1111111111 \ j 1 1 1 1 1 1 1 11 i 1 1 1 1 1 1 1 111 1 •1.0 -0/8 . -0.6 -0.4 TTTTTTTTrj I I I I I I I I I | I I I I TTmTI I I I I I I I I I I ITT | 0.2 G.4_ 0.6 0.8 1.0 -0.4 -Figure 4.4.2.3 k 2 - k 3 p l o t p l o t , for the example i n the previous sec t i o n , i s shown on figu r e 4.4.2.3. Occasionally, the i n t e r s e c t i o n s of these l i n e s are obscure. This d i f f i c u l t y may be overcome by p l o t t i n g "contours" of (y.): i = 2,3, as shown i n figures 4.4.2.4 and 4.4.2.5 for .Z(s) - -44*r=- + 0 + 0.211 s + 6.85 s + 2.62 s + 0.382 s +0.146 This impedance function corresponds to a network with a l l components having value unity. The s o l u t i o n obtained from the above p l o t s i s : {k 2 ^ 0.525737, k 3 = 0}. 66 - 0.4 -4 Figure 4.4.2.4 k~ - k contour p l o t Figure 4.4.2.5 Refined k~ - k„ contour p l o t 67 4.5 n th Order Networks To obtain necessary and sufficient conditions for r e a l i z a b i l i t y on the parameters of a specified impedance function of order higher than four by a given irreducible network configuration, the rows of the modal matrix must be determined geometrically for each network configur-ation. Methods similar to the ones described in the previous section may be developed to accomplish this. However, this lacks generality since each situation must be considered individually. Nevertheless the necessary condition, a much more restrictive one than any previously known, can be generalized for a large class of irreducible networks. The derivation of inequality (4.3.21) does not depend on the order of the network; i t depends only on the mutual orthogonality of the vectors £vu, £LU - y^ and £y_^  - V n> which is necessary for networks containing a linear capacitor tree. Thus the necessary condition applies to a l l irreducible RC complementary-tree structures which contain such a tree. sistor voltages as independent variables, i.e., writing the fundamental circuit matrix for the resistor tree, in which case the necessary con-dition would apply to networks containing a linear resistor tree. There-fore, inequality 4.5.1 is a necessary condition for r e a l i z a b i l i t y of irreducible, minimal two-element kind networks containing a linear tree (4.5.1) n n The foregoing argument could have been carried out using re-68 of the same elements. Recasting t h i s i n e q u a l i t y as a quadratic form i n e i t h e r set of variables (T^ ) or (P^) describes a volume i n e i t h e r residue or time constant space. Let the time constants be f i x e d . Inequality 4.5.1 now can be written as .(4.5.2) 6 * 0 The necessary condition implies that i f a set of r e a l i z a b l e n a t u r a l frequencies i s given, the corresponding residues must l i e i n -side a volume which i s the i n t e r s e c t i o n of the p o s i t i v e hyperoctant and the volume i n t e r i o r to the quadratic hypercone described by 4.5.2 f o r 6 = 0. A l i m i t i n g case of r e a l i z a b i l i t y occurs when the cone i s tan-gent to the p o s i t i v e octant. This s i t u a t i o n may be in v e s t i g a t e d by arranging the T/S i n ascending order and s e t t i n g a l l but two P ^ ' S equal to zero (say p 0 = ... = p .. = 0). For 6 = 0 z n - l 8 ( V l + Vn> - ( T i " 1 8 V n + Tn> p l p n = 0 ( 4 ' 5 ' 3 ) 69 S e t t i n g T = k x n above y i e l d s ° n 1 J P l = k - 18k -fe 1 16 + - k P n (4.5.4) Since the residues are r e a l , the d i s c r i m i n a n t above must be p o s i t i v e or at the l i m i t i n g case i t must be zero. This occurs i f e i t h e r k = 1, which i s i m p o s s i b l e , or i f k = 17 + 12 Jl. From t h i s f o l l o w s that T — * 17 + 12 Jl This r e s u l t , which i s gen e r a l , reduces to the value p u b l i s h e d 20 by Lee and Murphy f o r n = 3 70 CHAPTER V  CONCLUSION This research, p r o j e c t was undertaken to extend formal r e a l i -z a t i o n theory to i n c l u d e minimal, one-port, two-element-kind networks of a r b i t r a r y but s p e c i f i e d topology. But to quote B o x a l l ^ "Unfortun-a t e l y , the author proved to be more ambitious than ingenious and the d e s i r e d theorem remains unknown". I n t h i s , the author j o i n s the d i s t i n g u i s h e d company of B o x a l l , Lee, Mason, e t c . who set out to s o l v e the above problem and f a i l e d but, i n the process, c o n t r i b u t e d to the body of knowledge termed network theory. Some of the author's c o n t r i b u t i o n s d e s c r i b e d i n the preceding pages i n c l u d e (1) the c l o s e d form s o l u t i o n f o r the t h i r d order case, and (2) a numerical s o l u t i o n f o r the f o u r t h order networks. This accomplishment, i n our o p i n i o n , i s r e a l and v a l u a b l e not only i n i t s e l f but a l s o as a t o o l f o r subsequent research. In the f u t u r e our theory may be extended to higher order systems, o r , i f a new theory i s developed, the r e s u l t s so obtained may be v e r i f i e d u s i n g the methods des c r i b e d i n t h i s t h e s i s . The d e s i r a b i l i t y of o b t a i n i n g the complete s o l u t i o n to the problem i s explained i n the i n t r o d u c t i o n to t h i s work and need not be r e i t e r a t e d here; however, we note the l a c k of urgency to a r r i v e at such s o l u t i o n . Therefore, c o n s i d e r i n g the d i f f i c u l t y of the problem, (which b a f f l e d not only the author but a l s o the above mentioned i n d i v i -duals of proven a b i l i t y ) , we s t r o n g l y recommend postponing f u r t h e r i n -v e s t i g a t i o n of t h i s t o p i c to the l e i s u r e hours of r e t i r e m e n t of those of us who w i l l m a i n t a i n the r e q u i r e d mental a l e r t n e s s and the i n t e r e s t i n t h i s t o p i c . 71 REFERENCES 1. F o s t e r , R.M., "A Reactance Theorem", B e l l System T e c h n i c a l J o u r n a l , V o l . 3, 1924, pp. 259-267. 2. Cauer, W., "Die^Verwirklichung von Wechselstromwiderstanden vorge-schriebener Freguenzabhangigkeit", A r c h i v . f. E l e c t r o - t e c h n i k , V o l . 17, 1926, p. 355. 3. Yarlagadda, R., and Tokad, Y., "On the Use of Nonsymmetrical L a t t i c e Sections i n Network Syn t h e s i s " , IEEE Transactions on C i r c u i t Theory, V o l . CT-11, D e c , 1964, pp. 474-478. 4. Lee, H.B., "On the Canonic R e a l i z a t i o n s of Two Element Kind D r i v i n g P o i n t Impedances", D o c t o r a l D i s s e r t a t i o n , M.I.T., May 1962. 5. Lee, H.B., "A New Canonic R e a l i z a t i o n Procedure", IEEE Transactions  on C i r c u i t Theory, V o l . CT-10, March 1963, pp. 81-85. 6. Mason, L . J . , "The R e a l i z a t i o n of Minimal Two-Element-Kind One-Port Networks", D o c t o r a l D i s s e r t a t i o n , U n i v e r s i t y of B.C., Sept. 1968. 7. G u i l l e m i n , E.A., Synthesis o f Pass i v e Networks, John Wiley & Sons, N.Y., N.Y., U.S.A., 1967, p. X I . 8. MacMillan, B., " I n t r o d u c t i o n to Formal - R e a l i z a b i l i t y Theory", B e l l  System Tech. Jour., V o l . 31, 1952, pp. 217-279 and 541-600. 9. Brune, 0., "Synthesis of a F i n i t e Two-Terminal Network Whose D r i v i n g -P o i n t Impedance Is a P r e s c r i b e d Function of Frequency", J . of Math. and Phys•, V o l . 10, 1931, p. 191. 10. G u i l l e m i n , E.A., The Mathematics of C i r c u i t A n a l y s i s , The M.I.T. Pre s s , Cambridge, Mass., U.S.A., 1965, p. 305. 11. Seshu, S., and Reed, M.B., L i n e a r Graphs and E l e c t r i c Networks, Addison-Wesley, Reading, Mass., U.S.A*, 1961, pp. 201-203. 12. P o n s t e i n , J . , Mat r i c e s i n Graph and Network Theory, VanGoreum & Comp., N.V. , Assen, Netherlands, 1966, p. 5. 13. G u i l l e m i n , E.A., "An Approach to the Synthesis of L i n e a r Networks Through the Use of Normal Coordinate Transformations Leading to More General T o p o l o g i c a l C o n f i g u r a t i o n s " , IRE N a t i o n a l Convention  Record, 1960, pp. 171-179. 14. G o l d s t e i n , H., C l a s s i c a l Mechanics, Addison & Wesley, Reading, Mass., U.S.A., 1970, Chap. 10. 15. Landau, L.D., and L i f s h i t z , E.M., Mechanics, Addison-Wesley, Reading, Mass., U.S.A., 1960, p. 68. 72 16. Gantmacher, F.R., The Theory of Matrices, Chelsea, N.Y., N.Y., U.S.A., 1959, pp. 310-326. 17. B o x a l l , F.S., "Synthesis of Multiterminal Two-Element Kind Networks", Technical Report No. 95, "Research Laboratory, Stanford U n i v e r s i t y , Nov. 1955. 18. Schwab, W.C., "Synthesis of RC Networks by Normal Coordinate Trans-formations", Doctoral D i s s e r t a t i o n , M.I.T., Sept. 1962. 19. Lee, H.B., "On the D i f f e r i n g A b i l i t i e s of RL Structures to Realize Natural Frequencies", IEEE Transactions on C i r c u i t Theory, V ol. CT-12, Sept. 1965, pp. 365-373. 20. Lee, H.B., and Murphy, P.J., "On the Natural Frequency R e a l i z i n g A b i l i t y of Two Element Kind Structures", IEEE Transactions on C i r c u i t  Theory, V ol. CT-13, Sept. : 1966, pp. 325-236. 21. Howitt, N., "Group Theory and the E l e c t r i c C i r c u i t " , P h y s i c a l Review, Vo l . 37, June 1931, pp. 1583-1595. 22. MacFarlane, A.G.J., "Application of Group Theory to Linear Dynamical Systems", Proc. IEE, V o l . 113, Aug. 1966, pp. 1400-1412. 23. Turnbull, H.W. , Theory of Equations, O l i v e r and Boyd, Edinburgh, G.B., 1963, Chap. X. 24. Cederbaum, I., "Matrices A l l of Whose Elements and Subdeterminants are 1, -1, or 0", J . Math, and Phys. , V o l . 36, 1957, pp. 351-361. 7 3 APPENDIX A  ALGEBRAIC SOLUTION FOR THE THIRD ORDER  IRREDUCIBLE NETWORK Equations (3.6.2) to (3.6.8) express the c o e f f i c i e n t s of the i n -put impedance (3.6.1) for the network shown on f i g u r e 2.4.2 i n terms of the set of element values (c^> g^; i = 1, 2, 3}. This set of nonlinear equa-tions can be solved f o r the element values, successively e l i m i n a t i n g them u n t i l the following polynomial i s obtained for the powers of c^. 4 I I (PI)c, = 0 (A.l) 1=0 where P 4 = 4 A 1 * A 0 * - 4 A 1 3 B 2 A 2 A 0 ' » + U A 1 2 B 1 A 2 2 A 0 4 - 2 0 A 1 2 A 2 A 0 + 1 6 A 1 B 2 A 2 2 A 0 5 - 1 6 B 1 A 2 3 A Q 5 + 1 6 A 2 2 A 0 6 P 3 = 4 A 1 5 A 0 2 B 0 - 4 A 1 * B 2 A 2 A 0 2 B 0 - 4 A 1 4 B 1 A 0 3 4 A 1 3 B 2 B 1 A 2 A 0 3 - 4 A 1 3 E 2 A 0 * + 4 A1 3 B 1 A 2 2 A 0 2 B 0 - 4 A 1 3 A 2 A 0 3 B O 4 A 1 2 B 2 2 A 2 A 0 « + 8 A 1 2 3 2 A 2 2 A 0 3 S 0 - 4 A 1 2 B I 2 A 2 2 A O 3 + 8 A 1 2 B 1 A 2 A 0 « HA 1 2 A 2 3 A 0 2 B 0 2 + 1 2 A l 2 A 0 5 - 1 6 A 1 B 2 B 1 A 2 2 A 0 4 + 1 6 A 1 B 2 A 2 A 0 5 1 6 A 1 B 1 A 2 3 A 0 3 B 0 - 4 8 A 1 A 2 2 A 0 * B 0 - 1 6 E 2 2 A 2 2 A 0 5 + 3 2 B 2 A 2 3 A 0 4 B 0 1 6 B 1 2 A 2 3 A 0 * + 3 2 B 1 A 2 2 A 0 5 - 1 6 A 2 * A 0 3 B 0 2 - 4 8 A 2 A 0 * 74 P 2 - 4 A 1 * B 2 A 0 2 B 0 + 1 6 A 1 4 A 2 A O B O 2 + 8 A 1 3 E 2 2 A 2 A 0 2 B 0 4 A 1 3 B 2 B 1 A 0 3 - 8 A 1 3 B 2 A 2 2 A O B O 2 - 2 8 A 1 3 B 1 A 2 A 0 2 B O + 4 A 1 3 A 2 3 B 0 3 4 A 1 3 A 0 3 B 0 - 8 A 1 2 B 2 2 B 1 A 2 A 0 3 + 1 6 A 1 2 B 2 A 2 A 0 3 B 0 + 1 2 A 1 2 B 1 2 A 2A 0 3 4 A 1 2 B 1 A 2 3 A O B 0 2 + 4 A 1 2 B 1 A 0 * - 6 8 A1 2 A2 2 A 0 2 BO 2 4 4 A 1 B 2 3 A 2 A 0 « 2 4 A 1 B 2 2 A 2 2 A 0 3 B 0 + 8 A 1B 2 B 1 2 A 2 2 AO 3 - 1 6 A 1 B 2 B 1 A 2 A 0 * 3 6 A 1 B 2 A 2 3 A 0 2 B 0 2 - 2 4 A 1 E 2 A 0 * + 4 A 1 B 1 2 A 2 3 A O 2 B O + 1 2 0 A 1 B 1 A 2 2 A 0 3 B 0 1 6 A 1 A 2 * A 0 B 0 3 + 4 A 1 A 2 A 0 * E 0 + 1 2 B 2 2 E 1 A 2 2 A 0 * + 4 £ 2 2 A 2 A 0 5 2 4 B 2 B 1 A 2 3 A 0 3 B 0 - 8 B 2 A 2 2 A 0 * B 0 - 4 B 1 3 A 2 3 A 0 3 - 5 2 B 1 2 A 2 2 A 0 * 1 2 B 1 A 2 * A 0 2 B 0 2 + 2 0 B 1 A 2 A 0 5 + 4 A 2 3 A 0 3 E 0 2 + 3 6 A 0 6 P 1 = . - 1 6 A 1 « A 0 E 0 2 - 4 A 1 3 B 2 2 A 0 2 B 0 - 8 A 1 3 B 2 A 2 A O B 0 2 + 3 2 A 1 3 B 1 A 0 2 B 0 - 4 A 1 3 A 2 2 B C 3 - 4 A 1 2 B 2 3 A 2 A 0 2 B 0 + 4 A 1 2 B 2 2 B 1 A 0 3 + 8 A 1 2 B 2 2 A 2 2 A 0 B 0 2 + 2 1 A1 2 E 2 B 1A 2 A 0 2 B 0 - 4 A 1 2 B 2 A 2 3 B 0 3 - 1 6 A 1 2 B 2 A 0 3 B 0 - 1 6 A 1 2 B 1 2 A 0 3 + 2 0 A 1 2 B 1 A 2 2 A 0 B 0 2 + 4 8 A 1 2 A 2 A 0 2 B 0 2 + 4 A 1 B 2 3 B 1 A 2 A 0 3 - 4 A 1 B 2 3 A 0 * - 4 A 1 8 2 2 B 1 A 2 2 A 0 2 3 0 + 4 A 1 B 2 2 A 2 A 0 3 B 0 - 1 6 A 1 B 2 B 1 2 A 2 A 0 3 4 A 1 B 2 B 1 A 2 3 A O B 0 2 + 1 6 A 1 B 2 3 1 A 0 * + 4A 1 B 2 A 2 2 A 0 2 B 0 2 3 2 A 1 B 1 2 A 2 2 A 0 2 B 0 + 4 A 1 E 1 A 2 * B 0 3 - 8 0 A 1 B 1 A 2 A 0 3 B O - 4 A 1 A 2 3 A 0 B 0 3 + 4 8 A 1 A 0 * B 0 - 4 B 2 * A 2 A 0 * + 1 6 B 2 3 A 2 2 A O 3 B 0 - • 4 B 2 2 E 1 2 A 2 2 A O 3 + 8 B 2 2 B 1 A 2 A 0 < ' - 2 4 B 2 2 A2 3 A O 2 B O 2 + 1 2 B 2 2 A 0 S + 8 B 2 B 1 2 A 2 3 A 0 2 B 0 -1 6 B 2 B 1 A 2 2 A 0 3 B O + 1 6 B 2 A 2 * A O B O 3 - 2 4 B 2 A 2 A 0 * B 0 + 1 6 E 1 3 A 2 2 A 0 3 -4 B 1 2 A 2 4 A 0 B 0 2 * 3 2 B 1 2 A 2 A 0 « + 8 B 1 A 2 3 A 0 2 B 0 2 - 4 8 B 1 A 0 S - 4 A 2 S B 0 * + 1 2 A 2 2 A 0 3 B 0 2 75 PO = 16M 3B2A0B0 2 + 4A1 2B2 3A0 2B0 - 8 A 1 2 B2 2 A2A OBO 2 32A1 2B231A0 2BO + 4 A1 2B2A2 2B0 3 - 16A1 2B1 A2A0BO 2 + 16A1 2A0 2B0 2 4A1B2 3B1A0 3 + 4A1B2 2B1A2A0 2BO + 20A1B2 2A0 3B0 + 16A1B2B1 2A0 3 4A1B281A2 2AOB0 2 - 40 A 1G2A2A0 2BO 2 + 32 A 1B 1 2A2A0 2BO - <JA1B1A2380 - 32A1B1A03B0 + 20A1A2 2A0B0 3 + «I£2*A0* - 1 6B2 3 A2 AO'3 BO 4B2 2B1 2A2A0 3 - 20B2 2B1A0* + 24B2 2A2 2A0 2BO 2 - 8B2B1 2A 2 2A 0 2B0 40B2B1A2A0 3B0 - 16B2A2 3A0B0 3 - 16B1^A2A0 3 + 4B1 2n2 3AOBO 2 16B1 2A0« - 20B1A2 2A0 2B0 2 + 4A2«B0« 23 Equation (A-l) can be solved using the standard formulas for biquadratic equations. The other element values can be calculated from the value of c^ according to the following equations: C3 = 2 ( D1 " Cl> 1 + + (A2c, - l) ( c \ - D-) (A-2) c„ = C1 C3 "2 A2c 1c^ ~ °i ~ °3 (A-3) A 1 C1 C2 C3 h 2(c 2 + c 3) 1 + I 1 " 4A0 A l 2 A2 C1 C3 (A-4) A 0 C 1 C 2 C 3 (A-5) BO 81 Al (A-6) 76 where AO BI - A l BO AO' 77 APPENDIX B CUT-SET ANALYSIS The set of branch currents, 1^, in terms of branch voltages, V b, is I = Y V xb d vb (B-1) where Y^ i s the diagonal matrix expressing the branch volt-ampere re-lations : Y d = sC,_ 0 D (B-2) 0 Application of Kirchhoff's voltage law yields [B U] 1 rc be cr = 0 (B-3) and Kirchhoff's current law yields [U — B ] r c J b e cr = 0 (B-4) Finally, substitution of (B-1) to (B-3) into (B-4) yields the e q u i l i -brium equation in terms of tree branch voltages as [U — B ] 1 r c J Multiplying out the above equations gives the cut-set admittance matrix " s c b °" " U V, ; = 0 (B-5) D C 0 G . c -B L r c J as Y = sC, + B G B *S) r c c r c which i s i d e n t i c a l to the r e s u l t obtained using the state model (B-6) 78 APPENDIX C  GENERALIZED THALES' THEOREM In chapter IV the c o n d i t i o n s which ensure that two chords o f an e l l i p s e meet at a r i g h t angle were s t a t e d . These can be rephrased as Theorem A (Generalized Thales' Theorem) Let a be an e l l i p s e i n an x-y set of coordinates (Figure A - l ) . I f V(p^tp^) i s a s p e c i f i e d p o i n t of a, the chords aP and bP i n t e r s e c t o r t h o g o n a l l y at P i f and only i f i ) the chord ab contains the p o i n t Q (Kp^, - (KA/B) /A z - p^) and i i ) the chord ab i s b i s e c t e d by the e l l i p s e 3 w i t h equation E x - q./2 KA/2 _ - t 2 + y - q 2 / 2 KB/2 -r 2 = 1 where K = (A 2 - B 2 ) / ( A 2 + B 2) Proof: An a r b i t r a r y l i n e y = m x + b (C - l ) w i l l pass through Q i f b = q 2 - mq x (C-2) The i n t e r s e c t i o n of t h i s l i n e and a are the p o i n t s a and b. The co-ordinates of these p o i n t s are (a^,a^) and (b1,b2.)> r e s p e c t i v e l y , where A 2 ^ 2 A m +B -bm + — A .2 ~2 R2 ,"2 A m + B - b (C-3) 79 a2 = m a l + b (C-4) 1 A 2 ~ 2 4 . * 2 A m +B C B (/ .2~2 , n2 ~2 - bra - - | A m + B - b (C-5) b 2 = m b 1 + b (C-6) The orthogonality condition may now be v e r i f i e d by checking the v a l i d i t y of equation (C-7) ( P ; L - a 1 ) ( p 1 - b x) - ( p 2 - a 2 ) ( p 2 - b 2) = 0 (C-7) F i n a l l y , the second part of the theorem follows from s u b s t i -t u t i n g the co-ordinates of the point c = (a + b)/2, 2 ^  ^  c i - - -if—i - ( c - 8 ) 1 Am + B 2 2 C = _ J O L _ ( C _ 9 ) 2 ,2 2 , _2 C G y ; A m + B into the equation of B. The above theorem provides an a n a l y t i c s o l u t i o n f o r f i n d i n g orthogonal chords of an e l l i p s e which meet at a prescribed p o i n t . Using th i s theorem a geometric procedure f o r f i n d i n g the s o l u t i o n may be ob-tained as follows: i ) Find the point d, which i s the i n t e r s e c t i o n of a and the l i n e passing through (-A, 0) and substanding an angle of 45° with the x axis. i i ) The p r o j e c t i o n of d onto the x axis defines the semi-major axis of a', the e l l i p s e , which i s a scaled version of a. Hence f i n d a'. 80 i i i ) Draw a l i n e p a r a l l e l to the y-axis through P, the p r e -scribed point. The i n t e r s e c t i o n of t h i s l i n e and a i s denoted by P'. i v ) The i n t e r s e c t i o n of the ray from the o r i g i n through P' i n t e r s e c t s a' at point Q. v) 6 i s a 2:1 scaled version of e l l i p s e a' centered at the point b i s e c t i n g the l i n e j o i n i n g the o r i g i n and point Q. The procedure described above was used to obtain r e s u l t s i n section 4.3 of the t h e s i s . y p Figure A . l Orthogonal Chords of an E l l i p s e 81 APPENDIX D FORMULAS FOR K-VECTOR APPROACH The e l l i p s e and the vec t o r s x" and q" can be expressed i n terms of the elements of the m a t r i x , T defined i n equation (4.4.2.9), as where -e..T • ' l l t12 t21 t22 K" ~ < t u + t 2 2 ) ( q , C " ) = 0 (D-l) -T q = -2 t 4 4 1 / 2 ^ t l l + t 2 2 ) "14 '24 (D-2) _ T = 1 12 x 2 i — \TDx 22 (D-3) ,,2,2 hi = t l 4 A + + ( t24 T 1" " t 1 4 ) A-7< 1 +n M ) v12 2 ^12*14 " hlhl? x 22 " 2 ( t 1 2 t 2 4 t 2 2 t 1 4 ) = t 0 / A_ + ( t l 4 n n + t 0 / . ) A _ / ( l + n"') ,.2.2 v21 24 + 24' 1 /? 9 2 2 d - ) { ( t 2 2 + t ] L 1 ) [ t 1 4 t 2 2 + t 2 4 t l i - 2 t 1 2 t 1 4 t 2 4 ] 2 2 2 " ( t l l t 2 2 " t 1 2 ) [ t 1 4 t 2 2 + t 2 4 t i r 2 t 1 2 t i 4 t 2 4 + ( t22 + tll>( t 214 + fc24>^ A + " ( l / t j f ) - { ( t 1 1 + t 2 2 ) t ( t 2 4 + t 2 4 ) + ( t 2 2 - t 1 1 ) ( t 2 4 - t 2 4 ) + 1 2 t 1 2 t l 4 t 2 4 ] + 4 ( t l l t 2 2 " t L ) ( t14 + fc24)} - 4 ( t 2 2 + t u ) [ ( t 2 2 + t u ) 2 - 2 ( t u t 2 2 - t 2 2 ) ] 82 A_ = (1 + n 2 ) 2 t j | [ 3 ( t 2 2 - t 1 J 2 ( t 2 4 - t 2 4 ) - 7 ( t 2 4 + t 2 4 ) ( t 2 ^ + 1 2 ( t 2 2 - t l l ) ( t l l t 1 4 t 2 4 ) ] + 4 ( t 2 2 + t l l ) 2 ' ^ " ' l l * D = ( 2 / t 4 4 ) 2 a t 1 4 t 2 4 ( t 2 2 - t l l ) - t 1 2 ( t 2 4 - t 2 4 ) ] 2 + 2 i t 4 4 ( t 1 1 + t 2 2 ) 2 - 2 ( t 1 1 + t 2 2 ) ( t 2 4 - t 2 4 ) + ( t 2 4 - t j 4 ) ( t 2 2 - t l l ) + A t 1 2 t 1 4 t 2 4 ] ^ l ^ l t ? ( t l l + t 2 2 ) ^ t L - t 1 4 ) ( t 2 2 - t l l ) - 4 t l 2 t 1 4 t 2 4 - 2 t 4 4 ( t l l t 2 2 - t i 2 ) ] 2t n = 12 '22 '11 

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