THE REALIZATION OF MINIMAL TWO-ELEMENT-KIND ONE-PORT NETWORKS by LLOYD JUDSON MASON B.Sc, Mount A l l i s o n University, 1956 B.E., Nova Scotia Technical College, 1958 M.Sc, University of New Brunswick, 1962 A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY i n the Department of E l e c t r i c a l Engineering We accept t h i s thesis as conforming to the required standard Research Supervisor Members of the Committee Head of Department Members of the Department of E l e c t r i c a l Engineering THE UNIVERSITY OF BRITISH COLUMBIA May, 1969 In 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 of the r e q u i r e m e n t s f o r an advanced degree a t the U n i v e r s i t y of B r i t i s h C olumbia, I a g r e e t h a t t h e 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 Study. I f u r t h e r a g r e e 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 u rposes 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 or 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 not 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 A new method of r e a l i z i n g two-element-kind d r i v i n g - J point Impedances i s given and i l l u s t r a t e d by examples. In t h i s method, networks of any desired topology and having a minimum of elements are u t i l i z e d . A transformation to normal coordinates forms the basis of the method and, i n order to determine network element values, evaluation of the associated transformation matrix i s necessary. This matrix i s found by formulating and solving a set of multivariable polynomial equations of second degree. The s o l u t i o n to t h i s set of polynomial equations i s obtained by a numerical perturbation procedure. To i n i t i a t e the procedure, a set of element values i s chosen, and the network of s p e c i f i e d topology i s analysed. The corresponding transfor-mation matrix and driving-point impedance are determined from t h i s a n a l y s i s . The impedance parameters are then perturbed by small amounts i n the d i r e c t i o n of the s p e c i f i e d ones, and the r e s u l t i n g changes i n the transformation matrix are calculated. The process i s continued u n t i l the transformation matrix corresponding to the s p e c i f i e d impedance i s obtained. A detailed d e s c r i p t i o n of the computer program written to carry out the above procedure i s Included. A large number of examples of various complexities, including some canonic structures, have been r e a l i z e d by the method. Examples show the superiority of the numerical method to conventional procedures f o r solving multivariable nonlinear equations. In p a r t i c u l a r , the choice of the i n i t i a l set of element values i s not required to be close to the f i n a l set to achieve convergence to a solution. i i Some r e s t r i c t i o n s on the r e a l i z a b i l i t y of i r r e d u c i b l e complementary tree structures are reported. I t i s shown that the s p e c i f i c a t i o n parameters may have l o c a l extrema at a point where the Jacoblan of the system of polynomial equations vanishes. Examples which support these r e s u l t s are given. i i i TABLE OF CONTENTS Page' LIST OF ILLUSTRATIONS v * ACKNOWLEDGEMENT v i i I. INTRODUCTION 1 1 - 1 . Statement of the Problem.... 1 1 - 2 . The Method 2 II. PREVIOUS WORK ON CANONIC NETWORKS AND KNOWN METHODS OF REALIZATION k 2 - 1 . Necessary Conditions for an RC Network to be Canonic • 4 2 - 2 . Lee's Necessary Condition for an RC Network to be Canonic 8 2 - 3 . Lee Realization Procedures 12 2 - 4. Yarlagadda-Tokad Realization Procedure........ 18 III. MATHEMATICAL BACKGROUND AND RECENT WORK IN MATRIX SYNTHESIS 21 3 - 1 . The Normal Coordinate Transformation 21 3 - 2 . Previous Work. 28 IV. THE REALIZATION PROCEDURE 36 ^ - 1 . Derivation of Equations 36 k-2. An Example ^8 V. A METHOD FOR SOLVING THE SYSTEM OF MODAL EQUATIONS. 51 5 - 1. Standard Techniques 51 5 - 2 . The Method of Perturbation of Specification Parameters 52 5 - 3 * Implementation of the Method on the Digital Computer. 57 5 - ^ . General Description and Flow Chart of Computer Program 6? VI. EXAMPLES DEMONSTRATING THE REALIZATION PROCEDURE... 70 6 - 1 . A Second-Order Network 70 6 - 2 . A Fir s t Cauer Canonic Form 72 6 - 3 . Lee's Unsymmetrical Bridged Tee Network 7^ A Network Not Realizable by Other Methods..... 75 6 - 5 . A Fifth-Order Network. 77 6 - 6 . Discussion 79 i v VII. THE JACOBIAN AND ITS SIGNIFICANCE 80 7-1. The Vanishing Jacobian. 80 7-2. Vanishing Jacobian and Local Extrema of S p e c i f i c a t i o n Parameters....... 82 7 - 3 . Some Examples................. 86 7.3.1. Example I....... 86 7.3.2. Example I I . . 90 7 . 3 . 3 . Example I I I . . . . . . . 91 7.3.4. Example IV 94 7-4, Comparison of Lee's Extremum and S p e c i f i c a t i o n Parameter Extrema................... 96 7 - 5 . Discussion 98 VIII. CONCLUSION 102 BIBLIOGRAPHY 104 APPENDIX I - DESCRIPTION OF COMPUTER PROGRAM 106 Al-T. The Main Program 106 Al-2. The Subroutine ANAL 112 A l - 3 . The Subroutine EVALI 114 APPENDIX II - FORTRAN STATEMENT LIST 115 v LIST OF ILLUSTRATIONS CHAPTER II Page 2-1. RC Tie-set 9 2-2. RC Cut-set 9 2-3. Canonic Networks of Order Three 11 2-4. (a) Lee's Equivalent Network f o r Example Given 14 (b) Example of Lee's Nested L a t t i c e Network 14 2-5. (a) Equivalent Network f o r Lee's Bridged Tee Method 16 (b) Example of Lee's Bridged Tee Method 16 2-6. Canonic R e a l i z a t i o n Cycle Demonstrated by Lee 17 2-7. Equivalent Network f o r Yarlagadda-Tokad Method 18 2-8. Example of Yarlagadda-Tokad Method 20 CHAPTER IV 4-1. Three Possible Relative Locations of Nodes i and j l n the Network Tree 43 4- 2. (a) Network 49 (b) C-graph 49 (c) G-graph 49 CHAPTER V 5- 1. General Flow Chart f o r Computer Program. 68 CHAPTER VI 6- 1. Example of Second-Order Network 70 6-2. Example of F i r s t Cauer Canonic Form 72 6-3. Example of a Third-Order Network 76 6- 4. Example of a Fifth-Order Network 77 CHAPTER VII 7- 1. Irreducible Complementary Tree Structure 86 7-2. Graph of x| versus njl 93 7-3. Second-Order Foster Form 95 v i ACKNOWLEDGEMENTS The f i n a n c i a l assistance given by the National Research Council and the University of B r i t i s h Columbia during the period 1°62 to 1965 i s greatly appreciated. The author would l i k e to thank the various members of the U.B.C. Computing Centre f o r t h e i r very h e l p f u l assistance during the research. Various members of the Department of E l e c t r i c a l Engineering were most co-operative when advice was sought. In p a r t i c u l a r , the time and guidance given by Dr. A.D. Moore i n his capacity as supervisor i s g r a t e f u l l y acknowledged. The typing of the f i n a l draft was done by the author's wife, Thelma, and i s sincerely appreciated. v i i CHAPTER I INTRODUCTION . . * 1-1, Statement of the Problem An important problem in the f i e l d of passive network synthesis i s that of determining r e a l i z a b l l i t y conidtions and finding realization methods for one-port networks with general topological configurations. Realization methods for the canonic structures of Poster 1 and Cauer 2 are well known. Other methods using certain simple cascaded structures have also been developed recently-^*/** ^ A common feature of a l l these methods is that the given function to be realized is developed in some manner so that the element values may be determined by inspection. The chief disadvantage of these procedures i s that the variety of network configurations obtainable i s extremely limited. Hence, there i s a need for much more general realization procedures. These procedures should give the designer direct control of the topology-rand yield networks with the minimum number of elements. The objective of this research has been the development of such realization procedures. For simplicity, the work w i l l be described i n terms of RC (resistance-capacitance) networks, but w i l l apply, with appropriate transformations^, to RL (resistance-inductance) and LC (inductance-capacitance) networks. A concept of paramount importance in network synthesis is that of physical r e a l i z a b l l i t y . A physically realizable RC driving-point Impedance function has the form 2 k m k, Z(s) = k T O + - 2 + 2 3 -—TV. (1.1.1) s 1=1 B + 0 i where s i s the complex frequency var i a b l e ; s = 0, s = - G£» (1=1,2,—m), are poles of Z(s), and the quantities k 1 # (1=0,1,2,—m) are the residues of these poles respectively. The k^, andCT^, are r e a l and non-negative. For purposes of convenience In subsequent passages, any function which has these properties w i l l be referred to as a p-r RC d-p Impedance. Any network which r e a l i z e s Z(s) with the same number of elements as there are independent parameters In the impedance function Is said to be a minimal network. The number of independent parameters needed to completely specify Z(s) i n Eq. (1.1.1) i s 2m+2, which Is the minimum number of elements needed to r e a l i z e Z ( s ) . A canonic RC one-port i s a minimal network able to r e a l i z e any p-r RC d-p Impedance at a spe c i f i e d port or pai r of d r i v i n g terminals. The necessary and s u f f i c i e n t conditions f o r an RC one-port network to be canonic are not known when the network has more than two open-circuit natural frequencies. Seshu and Reed^ and Lee? have given some necessary conditions, but beyond these, very l i t t l e Is known. 1-2. The Method In the sequel, a numerical method f o r the r e a l i z a t i o n of minimal two-element-kind one-port networks with a r b i t r a r y , but s p e c i f i e d , topology i s given and i s i l l u s t r a t e d by examples. The method has been used to r e a l i z e a wide var i e t y of Impedances with many d i f f e r e n t network configurations. In every case tested, the method enabled the soluti o n to be found If such a so l u t i o n 3 existed. Due to the complexity of the problem, no necessary and s u f f i c i e n t conditions f o r a structure to be canonic have been found. That i s , given a network function and a config-uration, i t has not "been possible to say, i n general, whether or not a s o l u t i o n e x i s t s . However, some new re s u l t s on the conditions of r e a l i z a b l l i t y of a p a r t i c u l a r complementary tree Q structure are given. Lee and Murphy0 have shown that the natural frequencies of t h i s network are subject to a constraint. Some examples are given i n the thesis which show that the r e a l i z i n g a b i l i t y of t h i s structure i s much more r e s t r i c t e d than t h i s constraint Indicates. Before describing the approach used here, a b r i e f summary and discussion of the procedures developed by other workers w i l l be given. This summary w i l l serve as a vehicle f o r d efining terms and concepts and f o r producing c e r t a i n examples of canonic r e a l i z a t i o n s with which r e s u l t s obtained by the new numerical procedure can be checked. CHAPTER II PREVIOUS WORK ON CANONIC NETWORKS AND KNOWN METHODS FOR REALIZATION BY CASCADED NETWORKS 2-1. Necessary Conditions f o r an RC Network to be Canonic Seshu and Re ed° have given some information about networks minimal l n the number of reactive elements which Is relevant to canonic RC networks. This information i s given i n Theorem I so that i t may be referred to l a t e r . Theorem I: A one-terminal-pair network without mutual inductance i s minimal In reactive elements i f , and only i f , (a) there are no all-inductance or all-capacitance loops, (b) there are no all-inductance or all-capacitance cut-sets l n the network obtained by shorting the d r i v i n g terminals together. A cut-set Is a set of elements of a network such that the removal of these elements separates the network into two separate parts. This theorem a c t u a l l y applies to RLC networks, but l t also applies to LC, and, consequently, to a l l two-element?** kind networks! Consider the capacitance and resistance networks of an RC network separately. These are referred to c o l l e c t i v e l y as the component networks and i n d i v i d u a l l y as the C-network and R-network. respectively. 4 5 Condition (a) of Theorem I implies that the C-network and R-network contain no loops. Condition (b) means that, when the d r i v i n g terminals are shorted together, the shorted C-network and: shorted R-network are connected networks since the complete network Is connected. A connected network i s one l n which there exists a path between any two nodes of the network. I f , f o r example, the shorted R-network i s not connected, i t would mean that we could f i n d an a l l - c a p a c i t o r cut-set i n the shorted C-network. The same argument applies to the shorted C-network. If the shorted C-network and R-network are connected, then the C-network and R-network e x i s t i n g before the input terminals were shorted must each consist of no more than two parts. Moreover, i f e i t h e r or both of these component networks do consist of two parts, one d r i v i n g terminal i s i n one part and the other d r i v i n g terminal i s located i n the remaining part. This must be so, otherwise the shorted C-network and shorted R-network would not necessarily be connected networks. Hence the o r i g i n a l component networks consist of no more than two separate parts and each part must be a tree since there are no loops. A tree i s defined as a connected network which has no loops. Seshu and Reed^ have also stated the conditions under which separate parts occur. These are paraphrased i n the following theorem. Theorem I I : The driving-point impedance of a passive network without mutual Inductance has (a) a pole at s=0 i f , and only i f , there i s an all-capacitance cut-set such that i t s 6 removal from the network places the d r i v i n g terminals i n d i f f e r e n t connected parts, (b) a pole at s=00 i f , and only i f , there i s an all-inductance cut-set such that i t s removal from the network places the d r i v i n g terminals In d i f f e r e n t connected parts, (c) a zero at s=0 i f , and only i f , there Is an a l l -inductance path between the d r i v i n g terminals, (d) a zero at ssOD i f , and only I f , there Is an all-capacitance path between the d r i v i n g terminals. If we combine the r e s u l t s of Theorem I and Theorem II and apply them to minimal RC one-ports, we obtain the r e s u l t s shown In Table 2.1 f o r the four possible types of p-r RC d-p impedances. Table 2.1 1 Type 10 01 11 00 Behaviour at s=0 pole constant pole constant Behaviour at s=oo • zero constant constant zero No. of connected parts i n R-network 2 1 2 1 No. of connected parts i n C-network j 1 i 2 2 1 The type numbers, designated by binary d i g i t s , are used f o r convenient reference In the sequel. The f i r s t d i g i t Indicates the presence (1) or absence (0) of the term k Q/s lh"Eq. (1.1.1). The second d i g i t indicates the presence (1) or absence (0) of the term k t t In Eq. (1.1.1). The component networks corresponding to type 00 impedances 7 are trees. When the component networks, that i s , the C-network and R-network, "both consist of trees, the composite or complete network i s often c a l l e d a complementary tree structure. The name ar i s e s from the f a c t that, when ei t h e r component network i s removed from the composite network, the remainder or complement i s a tree. I t should be noted that a minimal network appropriate to a type 10 impedance may be generated by taking a complementary tree structure which has a resistance connected d i r e c t l y across the d r i v i n g terminals and removing t h i s resistance. A s i m i l a r procedure i s possible f o r generating a minimal network appropriate to a type 01 impedance, by using a complementary tree structure with a capacitance shunting the input and removing I t . To generate a minimal network appropriate to a type 11 d-p Impedance, a complementary tree structure with both a resistance and capacitance shunting the input terminals i s used. Hence, minimal network structures, corresponding to any of the f i r s t three types of p-r RC d-p Impedances In Table 2.1 may be generated from p a r t i c u l a r forms of complementary tree structures. A p h y s i c a l l y r e a l i z a b l e RC driving-point admittance. the r e c i p r o c a l of a p h y s i c a l l y r e a l i z a b l e RC driving-point Impedance, has the form Y(s) = ho+hoo s + £ i l l . (2.1.1) 1=1 s + 0 i where h 1 # (1=0,1,2,— ,p,oo), andO^, (1=1,2,— ,p), are r e a l and non-negative. 8 The driving-point impedance types 10, 01,and 11 correspond respectively to the cases where (1) the term h Q i s zero, (2) the term h t t s i s zero, and (3) both h Q and h ^ s are zero. Thus, given an RC driving-point impedance which i s not a type 00 impedance, terms of the form h Q or h ^ s , or both, may he added to the admittance to obtain a type 00 impedance. Ph y s i c a l l y t h i s corresponds to adding a resistance of known value or a capacitance of known value, or both, i n shunt with the d r i v i n g terminals. Therefore, even when discussing minimal networks appropriate to impedances which are other than type 00, l t i s s u f f i c i e n t to consider complementary tree structures havingj the proper elements shunting the input. This indicates the generality of complementary tree structures and shows how they may be related to RC minimal networks which have driving-point impedances of any of the four types. 2-2. Lee's Necessary Condition f o r an RC Network to be Canonic 7 9 Lee * has given a necessary condition f o r a two-element- kind network as follows: I f a minimal RC network i s to be canonic, l t i s necessary that i t be reducible to an elementary RC t i e - s e t of the form shown i n Figure 2-1 by successively shorting out structures of t h i s same type. Lee has c a l l e d any complementary tree structure having t h i s property a reducible complementary tree structure. Consequently, i t i s l o g i c a l to denote a complementary tree structure which i s not reducible as Irreducible and t h i s w i l l be done henceforth. 9 Lee has also shown that the natural frequencies of an i r r e d u c i b l e complementary tree structure are subject to the constraint (2.2.1) where and Q m i n are the absolute values of the natural frequencies with the largest and smallest magnitude respectively. Lee's o r i g i n a l proof was given i n terms of RL networks, but the r e s u l t also applies to RC networks. I t Is also possible to make a dual statement, as Lee has pointed out. That Is, f o r an m+1 node complementary tree structure to be canonic, l t i s necessary that i t can be reduced to m+1 i s o l a t e d nodes by successively erasing cut-sets of the form shown i n Figure 2-2. Figure 2-1. RC T i e - s e t . Figure 2-2. RC Cut-set. Lee has shown that these two conditions are equivalent l n the following way: I f one open c i r c u i t s the branches of a complementary tree structure i n opposite order to that i n which one short c i r c u i t s them, the r e s u l t i s the same as i f one short c i r c u i t s t i e - s e t s of the form shown i n Figure 2-1. 10 Before proceeding, i t i s necessary to define terms. By the order of a complementary tree structure, we s h a l l mean the number of elements i n each component network. This number i s always one les s than the number of nodes i n the network since the number of branches i n a tree i s always one less than the number of nodes. The order i s also equal to the number of open-circuit natural frequencies of the network, which i s the same as the degree of the denominator of the driving-point impedance. The degree of the denominator w i l l be referred to as the order of the driving-point impedance. Hence the network order and the driving-point impedance order are the same. Although Lee has conjectured that r e d u c i b i l i t y i s a necessary and s u f f i c i e n t condition f o r a network to be canonic, i t has yet to be proven that t h i s i s the case f o r networks of order greater than three. Lee's condition i s necessary and s u f f i c i e n t f o r complementary tree structures of order l e s s than or equal to three, however. Because a l l complementary tree structures of order two are reducible and since a l l structures of t h i s order are canonic, i t can be said the condition i s necessary and s u f f i c i e n t f o r networks of order two. Lee has shown that the condition i s s u f f i c i e n t f o r complementary tree structures of order three. The argument proceeds as follows: f i r s t , there i s only one i r r e d u c i b l e complementary tree structure of order three. This structure i s given In Figure 7-1. A l l other complementary tree structures of order three are reducible. Of those structures of order three which are reducible, some are 11 Foster and Cauer networks or combinations of these and some are not. The remaining reducible complementary tree structures, which are not Foster and Cauer networks or combinations thereof, are s i x i n number and are shown i n Figure 2-3. O 6-(b) -II-© A <J \ \ ^ I o i \\ (d) AN-( ) Figure 2-3. Canonic Networks of Order Three. 1 2 Note that networks (b) and (d) may be obtained from (a) and (c) respectively by Interchanging resistances and capacitances. Lee 7 was able to show that these structures are able to r e a l i z e a l l physically r e a l i z a b l e type 00 driving-point impedances of order three. Hence, a l l s i x structures are canonic and, since a l l Poster and Cauer networks or combinations of these are canonic, we see that a l l reducible complementary tree structures of order three are^canonic. This proves the s u f f i c i e n c y of r e d u c i b i l i t y f o r networks of order three. Beyond the above conditions nothing i s known about the necessary and s u f f i c i e n t conditions f o r a network of order greater than three to be canonic. I t i s worthwhile noting that the inequality i n ( 2 . 2 . 1 ) Is a l e a s t lower bound i n the sense that, given a p a r t i c u l a r i r r e d u c i b l e complementary tree structure, the actual lower bound f o r C max/^min m a y b e m u c n greater than four. For example, Lee and Murphy^ have shown that, f o r the network of Figure 7 - 1 , the lower bound i s ? " M A X ^ 3 3 - 9 7 . ( 2 . 2 . 2 ) O m l n In the next two sections, some recently published canonic r e a l i z a t i o n procedures^* t J are described and i l l u s t r a t e d by examples. These methods are In addition to the well-known Poster and Cauer canonic r e a l i z a t i o n procedures. 2-3 Lee R e a l i z a t i o n Procedures 3 7 Lee^ , f has given a canonic network r e a l i z a t i o n procedure In which the r e s u l t i n g network i s a sequence of l a t t i c e s . This method w i l l be demonstrated using the impedance 13 M S ) s+1.00 s+20.0 + s+80.0* ( 2 . 3.1) F i r s t , the impedance function i s expanded to determine the element values of the equivalent network of Figure 2 - 4 ( a ) . The shunt capacitance and shunt resistance r e s u l t when the constant at s=0 and the pole at s=COof the admittance function are r e a l i z e d . This leaves a remainder impedance Z^(s). The geries resistance and capacitance of Figure 2 -4(a) r e s u l t from removing the appropriate terms from Z-^(s) leaving a remainder impedance Z£. The turns r a t i o of the i d e a l transformer i s determined from the formula p = _ 2 _ ( 2 . 3 . 2 ) where G and C are shown i n Figure 2 - 4(a). The quantities and -f><2 a r e z e r o s °f zj?2* s n o w n * n F i S 1 1 1 , 6 2 - 4 ( a ) . The impedance z 2 2 a t terminals 2-21 of Figure 2-Ma) i s now calculated from The r e c i p r o c a l of z 2 2 1 8 then expanded i n p a r t i a l f r a c t i o n form and the element values of the l a t t i c e of Figure 2 -4(b) are found using t h i s expansion. Next, r e f e r r i n g to Figure 2-Ma), we see that Z = /° 2Z». r / r For the impedance of (2 . 3.1) Z r i s r e a l i z e d by a p a r a l l e l RC network connected "between the terminals 2 and 2' of Figure 2 - 4(b). I t should be noted that t h i s network has the same form as that of Figure 2 - 3 ( f ) . 14 G=0.462-LT f—sAj 0.230f z -o-0. 489V< I O-=0.0121f (a) 22 r '22 2 -C-Plgure 2-4. (a) Lee's Equivalent Network f o r Example Given, (b) Example of Lee's Nested L a t t i c e Network. For higher order Impedances, the sequence of steps In ' going from Z(s) to Z r(s) i s repeated and leads to a set of "nested l a t t i c e s " , with each l a t t i c e within the preceding one. At each step only four elements are generated i n order to reduce the number of c o e f f i c i e n t s by: four; so the r e s u l t i n g network Is minimal. Moreover, Lee has shown that a l l p-r RC d-p impedances of type 00 can be r e a l i z e d ; so the network i s canonic. The method lacks complete generality i n the sense that, i f the given Impedance i s not of type 00, i t must f i r s t be reduced by removing 15 appropriate terms. Since t h i s can always be done, the method Is seen to be applicable to a l l . f o u r types of impedances. Lee has shown that a dual procedure which applies to type 11 Impedances i s also possible. In another method, Lee 5 makes use of a cascaded network of unsymmetrical bridged tee sections. The method w i l l be i l l u s t r a t e d by r e a l i z i n g the impedance of Eq. (2.3.1). The element values of the equivalent network of Figure 2-5(a) are f i r s t determined. The quantities C 0 and G Q are the f i r s t two elements In a f i r s t Cauer canonic form. Removing these, a remainder impedance Z^ i s obtained. The quantity ° C Q = -G//OC, where G,yO, and C are shown i n Figure 2-5(a), i s found by solving the polynomial equation zl<°<o> + W l ^ o * + | § ^ o z l ^ o ) + F ^ o Z l ^ o ) - 0 (2.3.4) «o where 1 d Z ^ s ) Z n(<*„) = — 1 — l ^ o ' ds s = °< 0-This equation r e s u l t s when the two-port of Figure 2-5(a) with terminal pairs 1-11 and 2-21 i s made equivalent to the bridged tee with terminal pairs 1-1* and 2-2* of Figure 2-5(b). Knowing a l l the element values, the remainder impedance can be calculated. We f i n d that Z r = s + 20.1 which i s r e a l i z e d as the p a r a l l e l RC combination shown connected across the terminal p a i r 2-2* i n Figure 2-5(b). 16 The element values of the bridged tee of Figure 2-5(b) are determined by f i n d i n g the driving-point impedance, z 2 2 , a n d expanding i t Into a second Cauer canonic form. The procedure i s repeated i f the order of the remainder Impedance i s greater than two. The r e s u l t i n g network i s a cascade of bridged tee sections. At each step four elements are generated i n order to reduce the number of c o e f f i c i e n t s by four. The network Is therefore minimal and, as Lee has shown, a l l p-r RC d-p Impedances of type 00 can be r e a l i z e d using t h i s network configuration. Consequently, the network i s canonic. / o—p—»*v\/ X '22 '2 ( 4 t o-/ o A g ^ O . ^ t r g2=o.502ir c ^ o . 247f , c 0=2.84f c2=0.0127f~""^ 3 *1 :22 (b) 2' > s .=57.2 t r Figure 2 - 5 . (a) Equivalent network f o r Lee's Bridged Tee Method, (b) Example of Lee's Bridged Tee Method. ^ 17 The method has the disadvantage that, i n f i n d i n g the turns r a t i o of the Ideal transformer In Figure 2-5(a), It i s necessary to solve a polynomial equation. This becomes very laborious when high order Impedances are being r e a l i z e d , as It i s necessary to solve t h i s equation each time the cycle i s executed. A variant of the procedure i s obtained by i n t e r -changing resistances and capacitances. The r e s u l t i n g network i s also canonic. The topology of the network of Figure 2-5(b) i s the same as that of the network of Figure 2-3(c), while i t s variant, that i s , the network obtained by interchanging resistances and capacitances, Is the same, t o p o l o g i c a l l y , as the network of Figure 2-3(d). Lee^ has indicated that type 11 impedances can be synthesized by using procedures which are dual to the above method and i t s variant. L e e 7 has also demonstrated t h e o r e t i c a l l y that any type GO p-r RC d-p impedance can be r e a l i z e d using the cycle of Figure 2-6. He was not able to give a method f o r determining element values, however. In the next section, a method i s described which does t h i s . Figure 2-6. Canonic Realization Cycle Demonstrated by Lee. 18 2-4. Yarlagadda-Tokad Realization Procedure Yarlagadda and Tokad* 0 have published a procedure which i s s i m i l a r to Lee's f i r s t method In that i t makes use of the unsymmetrieal l a t t i c e s of Figure 2-6. However, i n contrast to Lee's method, the procedure can be applied d i r e c t l y to any of the four types of p-r RC d-p impedances. We s h a l l i l l u s t r a t e the procedure by r e a l i z i n g the driving-point impedance of Eq. (2.3.1)» used i n previous examples. The f i r s t step i n t h i s procedure i s to remove two Brune sections from Z(s) to leave a remainder impedance Z r as shown i n Figure 2-7. Then i t i s shown that the two-port with terminal pairs 1-1» and 2-2' i s equivalent to the two-port with terminal pairs 1-1' and 2-2' of Figure 2-6 i f [Z(OC 0) + 2*0Z,(o<:0Jl [ z(-c* 0) - 2 0 ^ ' ( - o ^ ) ) = Z(* 0)Z(-°* 0) (2.4.1) where z'{o(0) i s the f i r s t d e rivative of Z(s) with respect to s and s replaced by Q , To obtain a r e a l i z a b l e network, Figure 2-7. Equivalent Network f o r Yarlagadda-Tokad Method. 19 Eq. (2.4.1) must have at least one r e a l negative root, °^ 0» In the general case, there may not be such a solu t i o n and so the method would f a i l . But, since Lee has shown that the network of Figure 2-6 i s canonic f o r type 00 impedances, we see that Eq. (2.4.1) must always have a negative r e a l root f o r type 00 p-r R C d-p impedances and hence, the Yarlagadda-Tokad method i s a canonic procedure when applied to type 00 Impedances. Upon f i n d i n g the negative r e a l root of Eq. (2.4,1), we then f i n d G - [ , C - [ , and oC , shown i n Figure 2-7 using the formulae Cn = _ 1 , (2.4.2) otgZ (od G) t Next Z(s), as shown i n Figure 2-7, i s found, i i _ Then G2. C2, and/? are calculated using Eq. (2.4.2), but with G { , C{, QC , and Z(s) replaced by G 2 . C 2 . /3 , and Z(s) respectively. F i n a l l y , Z p i s found. To determine the elements of the l a t t i c e of Figure 2-6, z l l t the d-p impedance at terminal p a i r 1-1 * of Figure 2-7 with Z r disconnected, i s calculated. This Impedance i s then expanded i n a second Foster canonic form to obtain the element values of the l a t t i c e of Figure 2-6. The r e s u l t of performing these steps on (2.3.1) i s shown i n Figure 2-8, 20 Figure 2-8. Example of Yarlagadda-Tokad Method. We note i n passing that t h i s network has the same topology as Figure 2-3(e). Hence, the only reducible complementary tree structures of order three f o r which there are no r e a l i z a t i o n procedures are those of Figure 2-3(a) and (b). I t i s evident that the Yarlagadda-Tokad method also suffers from the disadvantage of being very involved from a computational point of view. This would be even more obvious f o r higher order Impedances where the cycle would have to be repeated a number of times. In the next chapter the method of normal coordinate transformations i s discussed and the attempts by other workers to use t h i s method are summarized. CHAPTER III MATHEMATICAL BACKGROUND AND RECENT WORK IN MATRIX SYNTHESIS 3 - 1 . The formal Coordinate Transformation The lack of s i m i l a r i t y of the r e a l i z a t i o n techniques of the previous chapter, and the l i m i t e d range of c i r c u i t configurations to which they apply, indicate the importance of a concept i n terms of which a l l such r e a l i z a t i o n s can be described. This concept, which serves as the basis f o r our formulation of the problem, i s that of transformations to normal coordinates. Synthesis methods using normal coordinate transformations have been investigated by o t h e r s ^ " ^ . These w i l l be described b r i e f l y a f t e r developing the basic rel a t i o n s h i p s involved. The normal coordinate method i s applicable to systems containing two kinds of elements and i s nearly always associated with a l o s s l e s s system undergoing small o s c i l l a t i o n s - ^ . Recently successful attempts have been made to modify the procedure so as to handle systems with d i s s i p a t i o n ^ . The normal coordinate method i s an analysis procedure. Since our objective i s synthesis, we must attempt to make the method work i n reverse Instead of i n the usual manner. Although t r a d i t i o n a l l y i t applies to the analysis of l o s s l e s s systems, i t also applies to any two-element-kind network (RC or RL). Due to the great i n t e r e s t i n RC networks at present, we s h a l l phrase our d e s c r i p t i o n i n terms of RC networks. Choosing any set of Independent voltage variables, the equilibrium equations f o r an RC network with m + 1 nodes can be written as 21 22 Y(s)V «* I (3.1 .D where Y(s) = sC + G. (3.1.2) The matrices C and G are respectively the capacitance and conductance parameter matrices. The quantities V and I are m-dlmenslonal voltage and current column vectors. To e f f e c t a normal coordinate transformation, a nonsingulaf matrix M must be found such that M^ YM = sDi + D2 (3.1.3) where and D 2 are diagonal matrices. Thus we must have M such that fTCM « Di (3.1.4) and M GM = D2. (3.1.5) That Is, we,have the problem of simultaneous diagonallzatlon of two matrices. For the s i t u a t i o n where one matrix i s positive d e f i n i t e , the procedure i s well knownl^. But the parameter matrices are, i n general, pos i t i v e seml-deflnlte. MacMillan 1? has shown that the lmmlttance matrix Is posit i v e r e a l . Using t h i s f a c t , B o x a l l 1 1 has shown that a transformation matrix M can be found to dlagonalize the two posit i v e seml-deflnlte matrices 12 C and G simultaneously. Through d i f f e r e n t arguments, Schwab has also shown that M can always be found. Since C and G are pos i t i v e seml-deflnlte, and D 2 may have the form »1 (3.1.6) O i o 23 and D 2 O | O o! D O I (3.1.7) U r ( J i s the unit matrix, i t s order being equal to the rank, r c , of C. D c i s a diagonal matrix. I t s order being equal to the rank r g of G. Let Y d ( s ) • sDx + D 2 . (3.1.8) Boxall has shown that + D 2 i s pos i t i v e d e f i n i t e ; hence i t cannot have zero elements on the diagonal. The matrix Y d ( s ) has the form Y d ( s ) = s u n i 1 O i o 0 i 1 p ! O o i D ° , (3.1.9) where U ' and U„ are unit matrices of order n_ = m-r_ and n g n g g n c » m-r c respectively. Y d p ( s ) i s diagonal and has the form Y d (s) = P (s + <*i) (. + cr2)^ (s + c r y . (3.1.10) where p = r c + r g - nu 24 Taking the inverse of Y(s) i n Eq. ( 3 . 1 . 3 ) . we have Z = Y" 1 = MZ d(s)M t ( 3 .1 .11 ) where Zd(s) = Y d ( s ) ; . ( 3 .1 .12 ) The matrix Z d(s) i s also diagonal and i s written as Z d(s) = \-s-I-\ i I I ° I Z d p ( s )| O i u n. where z d <s> P 1 _ s + o"i s + cr 2 s + OV (3 .1 .13 ) (3.1.14) Using Eq. ( 3 . 1 . 1 1 ) , Eq. ( 3 . 1 . 1 3 ) . and Eq. (3 .1.14), i t can be shown that a t y p i c a l element z^j of Z has the following form: P n ar m m is _ _ m 1 J k=l s k=nT+l ( s+CT n g . k + 2 ) k^+1 ^ . 1 . 1 5 ) 25 The usual expression f o r the p a r t i a l f r a c t i o n expansion of the open-circuit impedance of an RC network has the form (3.1.16) Z U " K » + T + s + <T k * Comparing Eq. (3.1.15) and Eq. (3.1.16), we have the re l a t i o n s n g K 0 = ID mik mjk» k=l m Koo ° mik mJk» k=p+l (3.1.17) (3.1.18) and K k ~ mik mjk« (3.1.19) This expression holds only i f a l l CT^ i n are d i f f e r e n t . I f some are equal, w i l l equal the sum of products ^ 1 ^ ^ ^ pertaining to these equal roots. In one-port synthesis, Is given i n the form K. P K 11 ^ + ^ 2 ^ where the Oj. are a l l d i s t i n c t . (3-1.20) To r e a l i z e t h i s impedance with a network of minimum complexity, the simplest form f o r Z d(s) i s 1 s Zfl(s) = (s + 0V) . (3.1.21) 26 A l t e r n a t i v e l y Zd(s) could have been assumed such that some or a l l diagonal entries are repeated. But t h i s requires that the order of Z d(s) be larger than that of Eq. (3.1.21) which, i n turn, w i l l lead to a network with more nodes than the network corresponding to Za(s) of Eq. (3.1.21). to Eq. (3.1.19) i n c l u s i v e , we see that the f i r s t row Of M can be determined to within a fa c t o r of +1 If Z (s) Is of the form shown i n Eq. (3.1.21). We can assume the fa c t o r +1 without loss of generality. To show t h i s , consider that we have determined M In some manner and that the f i r s t row of M has a l l p o s i t i v e qu a n t i t i e s . Then we may change to any other pattern of +1 or -1 i n the f i r s t row of M by postmultiplying by a diagonal matrix U' where U 1 has as diagonal elements the combination of +1 or -1 corresponding to the sign pattern to which we wish to change. Thus, i f we wish to change the element of the J t n column of M to -mjj. a -1 would be placed i n row j of U». The new M-^ i s Further, upon comparing Eq. (3.1.20) with Eq. (3.1.17) Mx = MU'. (3.1.22) Now using Eq. (3.1.11) we have Z = MU'ZdU'M*. (3.1.23) But Zd = U'ZdU', (3.1.24) so that we obtain Eq. (3.1.11) again. 2? Now Di and D 2 w i l l have the forms and D l -D 2 = "1 a, (3.1.25) (3.1.26) Consequently C and G both have rank m-l. I f the term i s missing i n z ^ , the zero i n the l a s t diagonal p o s i t i o n of D 1 i s missing and so C has rank m. I f the term K Q/s i s missing, the zero i n the f i r s t diagonal p o s i t i o n of D 2 i s missing so that the rank of G i s m. I f both terms K a n d K D/s are missing, the zero i n both and D 2 i s missing so that both C and G are non-singular. If we have. z n ( s ) expressed as a r a t i o of two polynomials, then the general expression i s 28 In terms of the above r a t i o n a l function, the four types of impedances have been l i s t e d i n Table 3-1 together with the rank of the corresponding parameter matrices and the behaviour at s=0 and s= OO. Note that both C and G are nonsingular f o r Type 00 impedances. As discussed previously, the f i r s t row of M can be determined. But i n order to determine C and G, we must determine the remaining m-l rows of M i n such a way that C and G are not only r e a l i z a b l e , but r e a l i z a b l e with the minimum number of 11 13 14 elements. Previous work by Boxall , Guillemin , Duda , and Schwab-^ has been concerned with determining the matrix M and the conditions on M such that a physically r e a l i z a b l e network r e s u l t s . In the next section, we s h a l l describe some of t h i s work. 3-2. Previous Work Boxall 1" 1" attempted to use the normal coordinate method to synthesize RC common-ground n-ports, p a r t i c u l a r l y f o r the case n > l . In t h i s case, more than one row of M can be determined. I f the given impedances have non-compact sets of poles, another problem, not present i n the canonical one-port case, i s introduced. By inspecting Eq. ( 3 « 1 « 1 5 ) » I t c a n be seen that non-compact sets of poles r e s u l t only i f there are repeated roots i n Zd . Thus, i n order to determine the elements of M from the residues, they must f i r s t be factored into compact sets. We s h a l l not discuss i t further because l t does not d i r e c t l y concern us i n the case of one-port synthesis. Table 3.1 29 Classification of RC Driving-Point Impedance Functions Type 10 Type 01 Type 11 Type 00 n _ k=0 K k=0 bQ=0 m=n+l to* 5 m=n bo=0 m=n ac#> V ° m=n+l Behaviour at s=0 i n f i n i t e constant Infinite constant Behaviour at s=CO zero constant constant zero Rank of C m m-1 m-1 m Rank of G m-1 m m-1 m The main d i f f i c u l t y experienced by a l l workers using the normal coordinate method is that of determining the remain-ing rows of M after a l l information from the given impedance functions has been used. No one has solved this problem satisfactorily. The method proposed by Boxall and independently by Schwab is as follows: the remaining rows are assumed arb i t r a r i l y , but such that M i s nonsingular. A nodal admittance matrix is found which i s , i n general, not realizable. By means of elementary row and column operations, a new modal matrix i s found which is realizable. 30 The method has the disadvantage that l t Is not always obvious what elementary operations must be performed to obtain a r e a l i z a b l e admittance matrix. This i s e s p e c i a l l y true f o r higher order cases. No rules can be given f o r carrying out the procedure f o r i t Is s t r i c t l y a t r i a l - a n d - e r r o r method. Boxall and Schwab have also discussed a procedure based on the theory of equivalent networks. The given Impedance matrix i s r e a l i z e d by an equivalent network using some conventional n-port synthesis procedure. The admittance matrix Y 2 f o r t h i s equivalent p h y s i c a l l y r e a l i z a b l e network i s related to the admittance matrix Y of the desired network by Y 2 = A ] C (3.2.1) where C i s a r e a l nonsingular matrix. The transformation of Eq. (3.2.1) may be effected i n practice by elementary row and column operations as suggested by Boxall, or, as Schwab has suggested, by solving i n e q u a l i t i e s on a computer. Ac t u a l l y , Howitt 1^ f i r s t suggested t h i s method i n 1932. More recently Yarlagadda 1^ has also attempted to synthesize LC networks by transforming from one equivalent network to another. His approach d i f f e r s from those of previous workers i n that he compares the networks through t h e i r state models. Unfortunately, he also obtains systems of nonlinear equations which must somehow be solved. He does show that the equations may sometimes be reduced to one polynomial equation;, but he does not indicate whether t h i s i s always possible or not. 31 A reasonably simple procedure f o r constructing the parameter matrices pertinent to a one-port RC network has been given by Schneider 1 7, His method i s based on wri t i n g the modal matrix as the product of a number of simpler matrices; the elements of these matrices can then be determined so as to lead to r e a l i z a b l e parameter matrices. Generally, however, .the method Is not minimal In the number of elements. lie Duda • has shown how the Poster and Cauer forms may be obtained f o r Type 00 impedances using the normal coordinate approach. Consider the f i r s t Foster canonic network. I f the system of node-to-datum voltage i s assumed i n which the datum node i s one of the d r i v i n g terminals, then M has the following form: j K i JT2 fK~m M= J K ^ JTTM X J K 3 fKm o\ 1 (3.2.2) Km where Kj i s the residue f o r the i t n pole. For a type 00 RC impedance, the capacitances i n a f i r s t Cauer network form a s t a r l i k e tree. A s t a r l i k e tree i s one i n which a l l branches of the tree have one node i n common. The common node i s conveniently taken as the datum node i n a node-to-datum system of voltages. I f the capacitance voltages are selected as the node-to-datum system of voltages, Duda has shown that M = DQ (3.2.3) 32 where D and Q are diagonal and orthogonal matrices respectively. The corresponding admittance matrix has the form Y = sCe+G ( 3 .2 .4 ) where C e i s diagonal and G i s po s i t i v e d e f i n i t e . The simultaneous diagonallzation may be carried out i n the following manner. Let c l c 2 V 'm ( 3 .2 .5 ) where c^, — c m are the capacitance values. Also l e t D = *1 d2 4n ( 3 . 2 .6 ) where dj - 1 / \Tci f o r i = 1 , 2 , m. ( 3 . 2 .7 ) A congruent transformation using the matrix D i s then performed on Y. Therefore L^YD = sU+H ( 3 . 2 .8 ) where H = D*GD „ ( 3 .2 .9 ) and U i s the unit matrix. An orthogonal matrix Q i s next found such that Q HQ = D2 (3 .2 .10 ) 33 where D 2 i s diagonal. Since H i s p o s i t i v e d e f i n i t e , D 2 w i l l he of the form: (3.2.11) where - O ^ * ~^2»"~~t " ^m a r e ^ n e natural frequencies of the network. I f Eq. (3.2.8) i s transformed congruently using Q, we obtain Q^D^YDQ = sU+D2. (3.2.12) Comparing t h i s equation with Eq. (3.1.3). we see that M has the form shown by Eq. (3*2.3)• To synthesize a Cauer I network, Duda gives the following method. The vector which has as i t s components the m elements of the i t n row of Q i s denoted by t ^ . A s i m i l a r d e f i n i t i o n holds f o r the 1 t h row of M. We have _ _ 2 Mi-Mi = dxTJi'TJ! (3.2.13) by v i r t u e of Eq. (3-2.3). Since the rows of Q are orthogonal, Eq. (3.2.13) determines d]_. Then q^ may be determined since (3.2.14) Next define P i " * 1 2 a 2 » - *lmaJ* (3.2.15) The remaining rows of Q are determined from the following formulae: 34 T 2 - P ^ W ^ l q 2 = 7 2/|7 2) T 3 = V^ i 'VV^ 'V^ i 3 = f 3/1^3! . ( 3 . 2 . 1 6 ) Tm = Pm-l" ( qm-l , pm-2 )^m-2- (^m-l. iPm-l)qm-l q m = f m / | f m | m m • m* From t h i s set of equations, i t can be shown that H has the form: TTn n h-i 0 \ ~~ 11 12 N Q h 1 2 h 2 2 n 2 3 \ \ ^ H = \ h 2 3 h 3 3 h 34 % \ H 3 4 hl±L ( 3 . 2 . 1 7 ) where a l l non-zero, off-diagonal elements are negative. I f Eq. (3.2.9) i s considered, we see that G has the same form as H. Therefore, G w i l l have a form appropriate to a ladder network analyzed on a node-to-datum system of voltages. The elements of D are determined by f i r s t f i n d i n g W = H - 1K = QD^VK (3.2.18) where W and K are column vectors with t y p i c a l elements w^ and k^ respectively. The elements d^ of Eq. (3.2.6) are calculated from w^ by d< = w i ( 3 . 2 . 1 9 ) 35 The elements of K are selected a r b i t r a r i l y . D ifferent choices r e s u l t i n d i f f e r e n t networks, not a l l of which are canonic, but a l l are; physically r e a l i z a b l e . The Cauer I network r e s u l t s i f k t = 0 f o r 1=1,2, m-l (3.2.20) km " 1 ' Duda also states that i f one of the natural frequencies i s zero, then Eq. (3.2.18) cannot be used since D 2 i s singular. The constants d^ are calculated by taking them proportional to the cofactors of the elements of the l a s t column of H. The proportionality constant i s chosen so that dj assumes i t s correct value as determined by Eq. (3.2.13). The success of t h i s method depends primarily on being able to determine an appropriate orthogonal matrix Q. This seems to be possible only f o r ladder networks where the capacitances form a s t a r l i k e tree. Attempts were made to extend the procedure to networks with other than l a d d e r - l i k e topologies. For the general s i t u a t i o n , however, no set of equations equivalent to the set Eq. (3.2.16) could be found. Thus, i t was necessary to take a d i f f e r e n t approach. This procedure w i l l be explained i n the next chapter. CHAPTER IV THE REALIZATION PROCEDURE 4-1. Derivation of Equations In conventional r e a l i z a t i o n procedures the network topology Is Inherently s p e c i f i e d by the method being used. That Is, each method Is coupled with a certain s t r u c t u r a l form. This Is seen to be true of the various methods f o r synthesizing two-element-kind d-p Impedances described In Chapter I I . For example, when r e a l i z i n g a given Z(s) using the f i r s t Cauer canonic form, we know that the element values of t h i s network can be obtained by inspection i f we expand Z(s) i n continued-f r a c t i o n form beginning with the highest powers of the complex frequency, s. In contrast, workers who have attempted to use coordinate transformations have not t r i e d to specify a desired topology; instead, they have been w i l l i n g to accept any physically r e a l i z a b l e network. In f a c t , by placing no r e s t r i c t i o n s on the topology, workers hoped to generate a host of r e a l i z a t i o n s , each one having a d i f f e r e n t structure. We have applied the method of normal coordinates to formulate the r e a l i z a b l l i t y conditions f o r minimal networks of s p e c i f i e d structure. The formulation i s systematic and i s applicable to any minimal r e a l i z a t i o n including the canonic forms discussed i n Chapter I I . I t always results i n a set of algebraic equations of second degree. As w i l l be shown l a t e r , those equations are amenable to solution by numerical methods, If a solut i o n e x i s t s . 3 6 37 The following discussion applies to the r e a l i z a t i o n of type 00 impedances only. However, as discussed i n Chapter I I , the networks appropriate to the remaining three types of impedances can "be obtained by considering reducible complementary tree structures with the appropriate elements shunting the input. Consequently, i f an impedance other than type 00 i s given, terms of the form hco and h©s are added to i t s r e c i p r o c a l to obtain a higher order Impedance which Is type 00 . S i m i l a r l y , to the desired network graph, we add the appropriate elements i n shunt with the input. By the res u l t s of Theorem I and Theorem II of Chapter I I , we now must have a complementary tree structure. However, we s t i l l must check to see i f i t i s reducible i f we wish to meet the necessary conditions f o r the network to be canonic. For a type 00 impedance, Eq. ( 3 .1 .4 ) and Eq. (3 .1 .5 ) become and where M CM = U M GM = D2 (4 .1 .1 ) (4 .1 .2 ) D 2 ° 2 \ ^m (4 .1 .3 ) Next we write A eCeA c and (4 .1 .4 ) (4.1.5) . 38 where A c and A g are the transposed nonsingular reduced incidence matrices f o r the C-graph and G-graph respectively, and C e and G e are of the form: c l and c e " 'm Ge -S l 82 N (4.1.6) (4.1.7) The diagonal elements c^, g^ are the element values f o r the capacitances and conductances respectively. Substituting Eq. (4.1.4) and Eq. (4.1.5) into Eq. (4.1.1) and Eq. (4.1.2), we have: H tA*C e A eM--U (4.1.8) and M*AgGeAgM = D 2 (4.1.9) Let S^ and R be the inverses of C L and G^ 6 6 6 6 respectively. Then, from Eq. (4.1.8) and Eq. (4.1.9). we have: S e = AcMM^c and R e = AgMTM^Ag (4.1.10) (4.1.11) 39 where T = D i 1 = t 2 'm ( 4 . 1 . 1 2 ) and t i = f o r 1 = 1,2,-—m. ( 4 . 1 . 1 3 ) Eq. ( 4 . 1 . 1 0 ) and Eq. ( 4 . 1 . 1 1 ) lead to a system of m(m-l) algebraic equations of second degree i n the m(m-l) unknown elements of M. I f the topology i s s p e c i f i e d , the matrices A c and A g are known. The equations a r i s e i n the following manner. The matrices and R e are diagonal. Hence t h e i r off-diagonal elements are zero. Consider the ( i , o ) p o s i t i o n of S, This element, when expressed i n terms of the elements of A c and M, i s equal to the inner product of the i row of ACM with the row th th of ACM. The i row of ACM i s Z_y ^ UrM^ where M k i s the k row k=l of M and a l k i s the ( i , k ) t h entry i n A c. The equation i s then t t h m m a i A • 72 ajpMp = 0 . k=l p=l When written i n terms of the elements of M t h i s equation becomes m m m ZD S &ik mkn 53 ajpNpn = 0' n=l k=l p=l (4.1.14) We obtain §m(m-l) equations from S e. The remaining §m(m-l) equations are obtained from R e of Eq. ( 4 . 1 . 1 1 ) . These equations are of the form 40 m m m ^ tn 2 3 kintal 13 bjpmpn = 0 (4.1.15) n=l k=l p=l where b i k i s the ( i , k ) t h entry i n A g. A c t u a l l y Eq. (4.1.14) and Eq. (4.1.15) reduce to a much simpler form. I t i s possible to write the equations down i n t h e i r simplest form without r e f e r r i n g to Eq. (4.1.10) and Eq. (4.1.11). This i s done by r e f e r r i n g e i t h e r to the network i t s e l f or to i t s parameter matrices S and R which are the inverses of C and G respectively. Thus S = A ^ S e U c 1 ) * = MM^ (4.1.16) and R = A g 1 R e ( A g 1 ) t = MTM* (4.1.1?) which are equivalent to Eq. (4.1.10) and Eq. (4.1.11) respectively. Since both S e and R e are diagonal, and since elements of A c and A g are e i t h e r 0, 1, or -1, we see that the elements of S and R may be expressed i n terms of the elastances s* = — and the resistances 1 C i TA = respectively. 1 S i For reference purposes, i t i s customary to assign a reference d i r e c t i o n to each capacitance branch and each resistance branch. One node i n the network i s designated as the reference node and i s numbered 0. Since each component network i s a tree, there i s only one path from each node i n the component network to node 0. We adopt the convention that the reference d i r e c t i o n f o r each branch Is taken such that i t points toward node 0 i n i t s respective component network. Hence, i f we pick any p a r t i c u l a r node i n the C-graph and: follow the path to node 0, we f i n d that we are always proceeding i n the p o s i t i v e sense f o r each capacitance In that path. 41 Now, from elementary network theory, i t i s easy to show that A c represents the matrix of transformation from the capacitance voltage variables to those of the node-to-datum system adopted f o r the complete network. Thus, suppose a set of node-to-datum voltages has been adopted. The r e s u l t i n g system of equilibrium equations written i n matrix form i s YV = I (4.1.18) where Y = sC+G. The matrix C i s not, i n general, diagonal. I t can be made to be diagonal, however, If the node-to-datum system of voltages i s transformed to those of the capacitances of the network. Thus suppose we had adopted those capacitance voltages as vari a b l e s . Then the equilibrium equations are * C V C = Ic where Y c - s C e + G C Let V c = A CV (4.1.19) then A&oAcV = Aclc = I. Hence * - A c Y c A c and C = A*C eA c which agrees with Eq. (4.1.4). This shows that A c i s the s p e c i f i e d matrix of transformation. A s i m i l a r argument can be 42 made regarding A g , namely, that i t i s the matrix of transformation from the node-to-datum system of voltages to r e s i s t i v e branch voltages. -1 -1 Prom t h i s discussion, we conclude that A c and A g r e l a t e the node-to-datum system of voltages to those of the capacitances and resistances respectively. For convenience i n the following discussion, we s h a l l denote the path which exists between node i and node zero In each component network by path 1. The ( i , J ) element of S has the form s ^ = Z > b l q b j q S q where b^q i s the (i,q) entry In A ^ and s q the diagonal element i n the q t n p o s i t i o n of S e. Now b l q i s +1 only i f capacitance c q i s In the path between node i and node 0. Hence S ^ equals the sum of the elastances which are common to both path i and path j . Si m i l a r l y s i i (a diagonal element of S) equals the sum of the elastances i n path 1. Now three separate situations may a r i s e with respect to the r e l a t i v e locations of nodes i , j , and 0. These are I l l u s t r a t e d i n Figure 4-1. We s h a l l denote these three cases by I, V, and Y f o r obvious reasons. 43 6 o 6 O Figure 4-1. Three Possible Relative Locations of Nodes i and" j l n the Network Tree. Case I i s the s i t u a t i o n where a l l capacitances In path j are contained i n path i . I f S J J were the ( i , j ) t h element of S, we should have But r e c a l l that S = MM^ . Hence, the above r e l a t i o n s h i p becomes m (4.1.20) 22 m j k ( m J k - m i k ) = 0 k=l (4.1.21) when expressed i n terms of m^j, the elements of M. I t i s possible to give a physical i n t e r p r e t a t i o n of Eq. (4.1.20). I f a unit impulse of current i s applied between node j and node 0, the voltage between node i and node 0 i s s±y But t h i s i s the same voltage as appears at node j which i s SJJ since there are no loops. Eq. (4.1.20) i s a representation of t h i s f a c t . 44 Case V corresponds to the case where there are no elements common to the paths i and j. Hence S 1 J = 0 or, In terms of ^ y we have m J2 Nik* 1;)]* = ° ' (4.1.22) Case Y corresponds to the s i t u a t i o n where only some of the elastances i n each path are common to each loop. This gives r i s e to the rel a t i o n s h i p s i j - s h j = 0. (4.1.23) P h y s i c a l l y t h i s r e l a t i o n s h i p means that when a unit Impulse of current Is applied to node j , the voltage at node i i s s^y but t h i s equals s^j since there are no loops. Expressing t h i s r e l a t i o n s h i p i n terms of the n^y we have m Z ) m;jk( mik~ mhk) = °« (4.1.24) k=l Equations (4.1.21) and (4.1.22) are act u a l l y s p e c i a l cases of Eq. (4.1.24). I f j=h i n Eq. (4.1.24), then Eq. (4.1.21) i s obtained. I f we adopt the convention that 111^=0 when h=0, we see that Eq. (4.1.24) reduces to Eq. (4.1.22). A s i m i l a r d e r i v a t i o n of equations pertaining to the G-graph i s possible, r e s u l t i n g i n the.following three equations corresponding to the I, V, and Y types discussed above: m £ 2 tkmikUjk-mik) = 0, (4.1.25) k=l m ZD t k m i k m j k = °» (4.1.26) 45 m TD t k ( m l k - m h j ) n i j k = 0. (4.1.27) Here, Equations (4.1.25) and (4.1.26) are sp e c i a l cases of Eq. (4.1.27). Thus, the two general equations f o r the C-graph and G-graph are £ > K k - m h k > * 0 k - °. (4.1.28) m "ZD t k ( ra l k-mhk) mik = °» (4.1.29) k=l respectively. For each subgraph, the number of equations equals the number of combinations of m objects ( l n t h i s case ports) taken two at a time,: i . e . ^im(m-l). Thus there Is a t o t a l of m(m-l) equations i n a l l , and t h i s i s the number of unknowns i n the matrix M, since the f i r s t row can be determined from a knowledge of the given driving-point impedance function. We s h a l l r e f e r to the system of m(m-l) equations of the form given i n Eq. (4.1.28) and Eq. (4.1.29) as the system of modal equations i n the sequel. A further question might be asked regarding the suf f i c i e n c y of the system of modal equations. Thus, i f we f i n d a r e a l s o l u t i o n to t h i s system, does i t mean that we w i l l necessarily obtain r e a l p o s i t i v e element values? From Eq. (4.1.10) and Eq. (4.1.11) we see that the diagonal elements of S e and R e can never be negative since each Is equal to the squared magnitude of a r e a l vector. Moreover, i f the determinant of M i s zero, then the determinants of S e and R e w i l l also be zero i n d i c a t i n g 46 that one or more of the diagonal elements of S e and R e Is zero which Is c l e a r l y unacceptable. Conversely, i f none of the diagonal elements of S e and R e are zero, the determinant of M cannot be zero. Hence we must also require that M be nonsingular i f we are to obtain n o n - t r i v i a l solutions and indeed t h i s i s a condition which was sp e c i f i e d at the beginning of Chapter I I I . That i s , i n simultaneous diagonallzation of two matrices, the dlagonalizing matrix i s nonsingular. Hence, t h i s condition, together with the condition that the system of modal equations be s a t i s f i e d , are the necessary and s u f f i c i e n t conditions f o r a p h y s i c a l l y r e a l i z a b l e minimal network to e x i s t . Another i n t e r e s t i n g i n t e r p r e t a t i o n of the equations derived i n t h i s section can also be given. We s h a l l describe case I, but the ideas can e a s i l y be extended to cases V and Y. Consider a common terminal two-port with the terminals denoted by i , J , and 0 where node 0 Is the common terminal. Let the open-circuit impedance at port j be n k* • U~A and the open-circuit t r a n s f e r impedance with respect to ports 1 and J be q n k. Now the tran s f e r elastance n Sa * = lim sz, . = T ) k r , 1 0 s-»o© 1 J q=l 1 J and the driving-point elastance 47 s-»oo q=l Then,If a unit Impulse of current i s applied at port j , we have q=l and 2L m - CJqt v 3(t) = £ ) kjje Hence and Vi(0) = ^ k ^ = s i a n q i f s i j = S J J » -we see that v 1(0) = Vj(0). Hence, for case I, the i n i t i a l value of the voltage generated at port i due to a unit impulse being applied at port j equals the i n i t i a l value of the voltage at port i . By a similar procedure, i t may be shown t h a t | i f r U = rJ"J» the f i n a l value of the voltage generated at port J due to a unit step being applied at port j equals the f i n a l value of the voltage at port i . Eq. (4.1.28) and Eq. (4.1.29) may also be expressed in terms of the residues. Let K i j k = mik mjk» (4.1.30) 48 where K j j k i s the residue of the pole at s — C ^ i n the transfer function Z j j , Then Eq. (4.1.28) and Eq. (4.1.29) take the form g <Kljk - Khjk> " 0. (4.1.3D and 2? t k ( K 1 J k - K h j k ) = 0. (4.1.32) These equations are l i n e a r . But the residues must also s a t i s f y the compactness r e l a t i o n 2 ^ K i i k K j j k - K i J k = 0 (4.1.33) as can be e a s i l y shown by substituting Eq. (4.1.30) into Eq. (4.1.33). Because t h i s system i s of much higher order than the system of modal equations, the l a t t e r i s preferred.. 4-2. An Example We s h a l l demonstrate the method of formulating the system of modal equations using the network of Figure 4-2(a). The elastance and resistance matrices f o r t h i s network are S and s l + s 2 s 2 s 2 s 2 s 2 s 2 s 2 s 2 s 2+s^ • l + * 3 r 3 49 Figure 4-2. (a) Network, (b) the C-graph, (c) the G-graph. We obtain the following set of equations from an inspection of these parameter matrices, or from the physical Interpretation i n terms of impulse and step response given previously. £2 (m2k - mik)m2k = 0, k=l 5p (m2k~ m l k ) m 3 k = °« (m3k - *2k)m2k = °* k=l (a) (b) (c) (d) (4.2.1) 53 ^k^m3k ~ m l k ^ m 3 k = ®» 50 and 3 P ^ l k ^ k = °- <f> k=l The m l k, k=l ,2.3. and the t k , k=l ,2,3, are determined from the given driving-point impedance. I t i s obvious that one so l u t i o n of the system (4.2.1) i s m 2 k = 0, m3k = ®' f > o r k = 1» 2»^* B u t t n i s leads to a singular modal matrix M and thus l t i s not an acceptable solution. It i s worthwhile noting that the equations are always of second degree or less regardless of the network complexity. The method of equating l i t e r a l c o e f f i c i e n t s on the other hand, leads to equations i n which the degree Increases as the network complexity increases. I t i s also true that, because the equations can be conveniently expressed by the matrix equations (4.1.10) and (4.1.11), i t i s easier to program t h i s system of equations;on the d i g i t a l computer than the equivalent set obtained by equating l i t e r a l c o e f f i c i e n t s . In t h i s chapter i t has been shown that, by specifying the network topology, a system of nonlinear equations involving the unknown elements of the modal matrix can be formulated. It has been shown that these equations may be written down d i r e c t l y i n t h e i r simplest form by an inspection of the given network. A physical i n t e r p r e t a t i o n of these equations has also been given. In the next chapter, a numerical procedure f o r solving the system of modal equations w i l l be given. CHAPTER V A METHOD FOR SOLVING THE SYSTEM OF MODAL EQUATIONS 5-1. Standard Techniques Commonly used procedures f o r solving nonlinear systems of equations are the Newton-Raphson Method and the Method of 21 Steepest Descent . However, our Investigations have shown that these standard techniques are not well suited to the system of modal equations derived In Chapter IV. Several examples of t h i r d order networks were tested using the Newton-Raphson Method. Except f o r a few i s o l a t e d cases, the method was found to "be unsatisfactory because of the d i f f i c u l t y of choosing an i n i t i a l approximation s u f f i c i e n t l y close to the sol u t i o n point. This problem can be rea d i l y appreciated when i t i s r e a l i z e d that t h i r d order networks lead to a system of s i x equations i n s i x varia b l e s . S u f f i c i e n t conditions f o r the convergence of the Newton-Raphson process are well known'' . To meet these conditions, an i n i t i a l approximation extremely close to the so l u t i o n point i s needed. The problem of picking a good s t a r t i n g point was complicated further by the fac t that the method tended to converge to t r i v i a l solutions. A t r i v i a l s o l u t i o n refers to a sol u t i o n such that the determinant of M i s zero. The Method of Steepest Descent was also investigated. In most examples the rate of convergence to a soluti o n was slow. 22 A v a r i a t i o n of t h i s method by Fletcher and Powell was found to improve the rate of convergence. However, the problem of picking a s t a r t i n g point so as to avoid t r i v i a l solutions was s t i l l present. 51 52 I t was evident that, I f t r i v i a l solutions were to be avoided, a sp e c i a l technique would be necessary. This technique i s explained i n the next section. 5-2. The Method of Perturbation of S p e c i f i c a t i o n Parameters Rec a l l that the two general equations f o r the C-graph and G-graph are m 2D ( m i k ~ m h k ^ m j k = ®» (5*2.1) k=l and m lp tkWk ~ mhk^mjk = °» (5.2.2) respectively. These equations r e s u l t from topological considerations only. Suppose the system of modal equations, each l i k e e i t h e r Eq. (5.2.1) or Eq. (5.2.2), i s written as follows: F l ( u i , u 2 u n,x) = 0, F 2 ( u l » u 2 V x ) = °« i (5.2.3) i i i F n ( u l f u 2 u n » x > = °* The u^ represent the n=m(m-l) unknown elements of M appearing i n these equations and x Is a parameter to be explained l a t e r . Suppose that these equations are s a t i s f i e d f o r the values x=x°, u-j^^,U2=u°>.——-u n=u°. The functions Fj. are continuous and possess f i r s t p a r t i a l derivatives which are continuous In the neighbourhood of the soluti o n point. F i n a l l y , suppose that the Jacoblan 53 3 F i c>un i 3 un does not vanish at t h i s s o l u t i o n point. Under these conditions 21 the Implicit Function Theorem t e l l s us that there exists one and only one system of continuous functions u x = ^ ( x ) i i i »n = $ n ( x ) which s a t i s f y the system (5*2.3) and which reduce to u°, u^ u^ fo r x = x Q. The power series expansion of the functions uj_(x) has the form: 0 4. / ^ l \ (x-x Q) + I (&0 (x-x 0) + — v . (5.2.4) To f i n d the p a r t i a l derivatives \ d ^ 7 0 VTx2/ 0 etc, each of the equations i n (5*2.3) i s d i f f e r e n t i a t e d with respect to x. The following system of equations Is obtained: 4^ +Z?^ 4^ -0,(1-1,2, n):. (5.2.5) O x j ^ l O^j ox Rearrange Eq. (5.2.5) as follows: 54 is n 0 « 1 2>pl " 3 ^ 3 F n c ^ n j x i 1 1 i i » . 5 * J (5.2.6) This system becomes l i n e a r In the p a r t i a l derivatives when the remaining p a r t i a l derivatives ^ F i A 3 F1 *• and * are evaluated '9uj Ox at the soluti o n point. Since, by hypothesis, the Jacobian i s not zero, we can solve f o r the f i r s t - o r d e r p a r t i a l derivatives The second order p a r t i a l derivatives are obtained by d i f f e r e n t i a t i n g each member of (5*2.5). We obtain dx2 £ i T u 7 5)x2 j=lO*a*j ,2r j=l k=l d u j d u k \ O x / \0 x y (5-2.7) Rearranging Eq. (5.2.7). we have ~ d F l * P 1 1 r a S d x * a u x j d u n I i S Fn & x 2 » 1 1 a u x 3 u n L ^ x * 'n (5.2.8) 55 c i = - d x 2 3*1 * X * U J * x o=l k=l c)uj du k Vex y This system i s also l i n e a r i n the | v when evaluated at the solu t i o n point. Hence the second-order p a r t i a l derivatives may be determined since 1J| does not vanish at the solution point. The process may be repeated to solve f o r as many of the higher-order p a r t i a l derivatives as desired. The above method i s applied to our problem i n the following manner. We know the topology of the network. Next we assume a set of values f o r the network elements. The network i s then analysed to f i n d the appropriate modal matrix MQ and the diagonal matrix of inverse natural frequencies T Q . The f i r s t row of M 0 and the diagonal elements of T 0 contain elements which do not,: i n general, correspond to those of the given d r i v i n g -point impedance. Suppose the desired elements of the f i r s t row f f f of M are designated as x^, x 2 ,— - X j ^ and the values of the f i r s t o o row of M 0 are designated x^ x m. S i m i l a r l y , the diagonal f f f elements of the desired matrix T are denoted by x m + 1 . x m + 2 x 2 m o o o and the elements of T are xm+-^, x M + 2 - — x 2 m * W e s n a l l denote the elements of the f i r s t row of M and the diagonal elements of T as s p e c i f i c a t i o n parameters. Now we express these s p e c i f i c a t i o n parameters i n terms of x, a so-called i n t e r p o l a t i o n parameter. We, i n e f f e c t , eliminate a l l the m-^ and the t j , 1=1,2, m by the following formulae. Let 56 = (l-x) x ^ + xx* (5.2.9) t 4 - ( l - x ) x m + 1 + x x m + i . As x varies from 0 to 1 the s p e c i f i c a t i o n parameters are seen to vary from those corresponding to the assumed network element values to those of the desired network. I f we v i s u a l i z e a 2m-dimenslonal space i n which the coordinate axes are the s p e c i f i c a t i o n parameters, we see that, as x varies from 0 to 1, each value of x corresponds to a point on the straight l i n e which connects the point (x°, x 2—" xZm) *° t h e P ° i n t (*i» x2""" x2m^ i n t h i s space. Hence, i f we denote the remaining elements of M by U j » the equations take the form shown i n Eq. (5«2.3)» The value of o o o x Q= 0 i n t h i s case and the quantities u^, u ^ , — u n are obtained from the l a s t m-l rows of M . This set of values i s indeed a solu t i o n of the set of equations (5*2.3) since we have assumed a set of element values and analysed the network to obtain the o u i • The procedure i s now straightforward. The various p a r t i a l derivatives can be evaluated, and the values of the u^ can be calculated f o r x=l using Eq. (5.2.4). Thus we have found the values of the or, i n other words, the unknown elements of M corresponding to the desired driving-point impedance. The elements of the f i r s t row of M are now equal to the desired values as are the diagonal elements of T. Using Eq. (4.1.10) and Eq. (4.1.11), we can then f i n d the desired element values S„ and R^. 57 In a p r a c t i c a l case, we may f i n d that x=l l i e s outside the region of convergence of one or more of the series expressions given i n Eq. (5 .2.4) or, i t may turn out that, since we s h a l l only use a f i n i t e number of terms i n each s e r i e s , the VL^ are Inaccurate i f we evaluate each at x=l. In t h i s s i t u a t i o n , we may perform the i n t e r p o l a t i o n i n s p e c i f i c a t i o n parameter space i n a series of steps. Thus we might l e t x=0.05 and f i n d the Uj^ f o r t h i s value of x. We should then have a set of xi^ corresponding to a set of s p e c i f i c a t i o n parameters which are 5% of the distance along the l i n e j o i n i n g the i n i t i a l point to the f i n a l , or desired, point. This new point could then be treated as a new I n i t i a l point since the U£ and the corresponding set of parameters would constitute a s o l u t i o n to the equations. This process Is then repeated u n t i l the s p e c i f i c a t i o n parameters are perturbed to thoseocorresponding to the desired ones. The only r e s t r i c t i o n placed on the procedure i s that the Jacobian | j | must not vanish at any starting-point along the l i n e . More w i l l be said about t h i s i n Chapter VII. For the moment, i t i s s u f f i c i e n t to say that t h i s r e s t r i c t i o n does not pose a problem i n most cases. 5 - 3 . Implementation of the Method on the D i g i t a l Computer. Although the procedure described i n the previous section i s conceptually straightforward, the programming of the technique i n the form i n which i t i s described i s not. The greatest disadvantage of the technique i s that the expressions f o r the p a r t i a l derivatives of various orders of each of the functions F^ must be programmed. Each time a network with a d i f f e r e n t topology i s used these p a r t i a l 58 derivatives change. I t appears d i f f i c u l t to write subroutines to f i n d p a r t i a l derivatives of general expressions. Since a method with more f l e x i b i l i t y was desired, a new seheme was developed to allow the method to be adapted more e a s i l y to the computer. We begin with the two matrix equations Sg = AcMMtA* (5.3.1) and R e <= AgHTM^Ag. (5-3.2) We expand M into the series M = MQ + fiMi + 6 2M 2 + (5.3.3) where M^, ( k « l , 2 , — ) , i s designated as the k -order correction matrix. Next t h i s series i s substituted Into Eq. (5.3.1) and Eq. (5.3.2). We obtain S e = ACMQM*A*+ €[A cM 0(A cM 1) t+A cM 1(A cM 0 ) t J +62 [A cMi (ACM^) t+A cM Q (A CM 2 ) t+A(JM2 (A CM 0) *] + £ 3 [ A c M o (A CM 3) t+A cM 3(A cM 0) AcMi(A cM 2) ^ +ACM2(ACM^) + (5-3.4) We also allow T to be represented i n the form T = T 0 + 6T X. (5.3.5) 59 Then Eq. (5.3.2) "becomes R e - W o M o A g + t LA gM 0T G(A gM 1) t+A gM 1T 0(A gM 0) t] + e [ A g « 0 T 1 ( A g l ! 0 ) t J + € 2 C W o ( V l ) t + V o T o ( A g M 2 ) t + A g M 2 T o ( V o ) t + £ 3 L W o ( A g M 3 ) t + A s M 3 T o ^ V o ) t + AgMiT c (AgM 2) t+A gM 2T 0(A^) t+A gM 2T 1(AgM Q)*+ AgM 0T 1(AgM 2) t+AgM 1T 1(A gM 1 ) t J + (5.3-6) We now require that the c o e f f i c i e n t matrix of each power of £ be diagonal l n both Eq. (5.3.4) and Eq. (5*3.6). This requirement leads to sets of l i n e a r equations which are equivalent to the l i n e a r systems obtained In Section (5-2) as w i l l be shown. The technique i s s i m i l a r to that described l n Section (5-2) i n that a set of element values i s picked and the corresponding modal matrix MQ and the diagonal matrix of inverse natural frequencies T Q are determined. Hence the two matrices A C M 0 M Q A * i n Eq. (5*3*4) and AgM 0T 0MoA g l n Eq. (5*3.6) are already diagonal. We determine the f i r s t row of i n Eq. (5.3.3) by assuming that the f i r s t row of a l l other higher order matrices M 2 f M 3 , etc. i s zero, and from a knowledge of the desired values f o r the elements of the f i r s t row of M . S i m i l a r l y , i n Eq. (5*3*5)» knowing the-desired inverse natural 60 frequencies, T i can be determined. We now determine the remaining entries i n Mi by requiring the c o e f f i c i e n t matrices of £ to be diagonal i n each equation. Let W = A M . ( A M ) * . C 1 C O The c o e f f i c i e n t matrix of € i n Eq. ( 5 .3 .4 ) i s W+W*. I f W j j i s the ( i , j ) element of W, we require that wjj+ W j i = 0 (ijfj) i f t h i s c o e f f i c i e n t matrix i s to be diagonal. I f we l e t H = A CM 0 we obtain Z} (aikhjp+ajichipjekp = 0 (5 -3 .7 ) k,p where h±y and e ^ are the ( i , j ) elements of A c, H, and M3, respectively. L e t t i n g F = AgM0 we may show, i n a s i m i l a r manner, that the c o e f f i c i e n t matrix of € i n Eq. ( 5 . 3 .6 ) i s diagonal i f , f o r a l l i and j , ( l ^ j ) , we have Z) V b i k f d P + b ; ) k f i P ) e k p + 2?Atpfipf jp - 0 ( 5 -3 .8 ) k;p p where the ( i , J ) t n element of A g and F are b j j and f ^ j respectively. The terms t p and Atp are the diagonal elements of the p t n row of T Q and T^ respectively. 61 I t Is evident that we obtain the same number of equations as before, namely, n=m(m-l) and that there are n unknowns, e k p , since e k p Is known when k=l. These equations are l i n e a r In e k p and so may be arranged i n the form J E X = B x (5-3.9) where J i s a r e a l nonsymmetric matrix of order n by n and each element I s given by = a i k h J q + a j k h i q or = VWjq^JkW' The r i g h t hand side, B l t of Eq. (5*3.9) i s a column vector where each element has either the form ^ = - £ ( a i i h j p + a n h i P ) e i P or \=- ID s^n'jp + bn fip ) eip- Z>ViP fjp-The vector E^ consists of the unknown elements e k p . Hence, provided J i s nonsingular, the vector E^ may be determined and, consequently, a l l elements of can be found. The next step i s to f i n d M 2 so that the c o e f f i c i e n t matrix of £ 2 i n both Eq. (5.3.4) and Eq. (5.3.6) i s diagonal. The f i r s t row of M 2 i s set equal to zero as previously discussed. Hence we have n=m(m-l) unknowns as before. In a manner s i m i l a r to that described above, we may obtain a set 62 of n l i n e a r equations i n the n unknown elements of M 2 using Eq. (5.3.4) and Eq. (5.3.6). The system ean be written i n the form J E 2 = B 2 (5.3.10) where J i s the same matrix as appears i n Eq. (5*3.9); E^ i s the vector of unknown elements from M 2 and B 2 i s a vector involving, among others, the matrix M^. The matrix M 3 i s determined i n a s i m i l a r fashion. The appropriate l i n e a r system Is of the form JE3 = B3. (5.3.11) The process may be continued to Include as many terms In Eq. (5*3.3) as accuracy demands. At each step, the right-hand column vector involves a l l previously calculated correction matrices, and hence, the procedure i s recursive. The c o e f f i c i e n t matrix J i s always the same and, from a computational point of view, t h i s i s an advantage. Only one matrix inversion i s needed, namely, that of J . Once J 1 i s found, the correction vectors E^, 1=1,2, — , are found using the r e l a t i o n E t = J~XB±. (5-3.12) In p r actice, only a f i n i t e number of terms can be included In the series of Eq. (5«3.3). Suppose the f i n a l term to be Included i s To ensure that M i s s u f f i c i e n t l y accurate, we require the elements of M n to be small i n magnitude compared to those of M. The elements of M n are proportional to t Vi the n power of e l p and Atp, (p = l , 2 , — m ) . By making e l p 63 and A tp small, the elements of M can be made small. The computation i s then carried out using a f r a c t i o n of the t o t a l increments e, and At such that the elements of M are small l p P n i n magnitude. A modal matrix M i s found which i s closer to that of the modal matrix of the desired network. By repeating the process several times, the modal matrix i s gradually changed to the one desired. I t i s not evident that the numerical procedure described i n t h i s section i s the equivalent of the procedure discussed i n Section 5-2. To show t h i s equivalence, consider the two types of equations written i n the form F i j = D a i p a j q m p r m q r (5-3.13) P.Q»r and F i j = 12 t r b l p b j q m p r m q r ' ( 5 - 3 . 1 4 ) p.q.r F i r s t we note that i f we consider the series expansion of Eq. (5.3-3). and i f € = x-x Q then Eq. (5-3-3) when considered element by element, has the same form as the series i n Eq. (5.2.4). Also, from Eq. (5-3-3). we notice that ^ = i _ y M r! d€r C = 0 Thus each element of M„ Is of the form r 1_ cfokp , k>l. €= 0 Because u ± = m k p f o r k>l, Eq. ( 5 - 2 . 4 ) and Eq. (5-3-3) are equivalent. Next i t w i l l be shown that, d i f f e r e n t i a t i n g Eq- (5.3.13) and Eq. ( 5 - 3 - 1 4 ) with respect to € i n accordance 6k with Eq. (5.2.5), we obtain Eq. (5.3.7) and Eq. (5»3.8) respectively. F i r s t consider Eq. (5.3.13). We have = Eaika J qmqp+5a i raj km rp. But and Hence C a0q mqp ~ h J P D ^j^ r p = hip' 0. | ^ - a i k h J P + a 0 k h i p -F^j can be written as But F i J s 22 ( S aiiajq'airnqr + £ aip^n npr mlr) r q p E aip ajq mpr mqr-pfl.qf-l £ 1 1 = r -2!5* dx r d m l r c) and o f m l r = (l-x)m l r+xm l r where m ° r i s the i n i t i a l value of m l r and m ^ l s .the desired or f i n a l value. Therefore - D ( E a i l a j q m q r + a i q a j l m q r ) ( m l r - m l r ) • We defined e-^r previously as f o e l r = m l r - m l r 65 so that "Y^ " S < a n V + a J l h l r ) e : i r -Now substitute into the equation k ^ l We obtain D ( a i l h j r + a j l h i r ) e l r + Z3 ( a i k h j p + a ; ) k h i p ) e k p ~ 0 r k,P k#L where ekp = s " ^ » and t h i s may be written i n the form O x E ( a i k h J p + a j k h i p > e k p = °. (5-3.16) k,p which i s the same as Eq. (5*3.7). Next we consider Eq. (5.3.14). The term c)Fl.1 w i l l be found f i r s t . We have t r = (l-x)t r+xt£ o f where t r i s the i n i t i a l value of t r and t r i s the desired or f i n a l value of t r . By using a Fi.i _ n a F u v i r + r aFi.i av ^x Y c ) m l r d x r T f r dx we f i n d that d F U = E M b i l f j r + b n f i r ) e l r + TJ b i p b l q m p r m q r ^ t r " t r ^ P.q.r = £ t r ( b l x f j r ^ J l ^ i r ^ l r r + S f i r f J r ^ y 66 where A t = t f - t°. r r r The second term i n (5.3*15) can be found i n a manner very s i m i l a r to that used i n deriving Eq. (5.3.I6). Thus, the *\ F expression f o r *i1 i s the same as found f o r Eq. (5*3.13)» but with a t j and h 1 ;j replaced by b j j and respectively, and the whole term multiplied by t p . Thus Eq. (5*3.15) gives us which i s the same as Eq. (5*3.8). We could go on to the higher order equations, Eq. (5.2.8) : f o r example, and show that the method of t h i s section f o r f i n d i n g the higher order correction terms i s equivalent to the previous method. However, the procedure i s e s s e n t i a l l y the same as given above; f o r the sake of brevity, t h i s w i l l not be carr i e d out here. I t becomes evident as higher order correction terms are added that the matrix equations (5*3.4) and (5*3*6) are much more compact and more e a s i l y written down than the corresponding d i f f e r e n t i a t e d equations given In Section 5-2. Hence, there i s a notational advantage as well as a computational one i n handling the systems of equations i n t h i s manner. d m k p 2D t q(b l kf j q+b J kf l q)e k q + 2 } f l q f j q A t q (5.3.17) 67 5 - 4 . General Description and Flow Chart of Computer Program A flow chart showing the basic steps i n the computer program i s given i n Figure 5-1• A general d e s c r i p t i o n of the program i s given here. However, a more detailed d e s c r i p t i o n appears i n Appendix I and a Fortran program l i s t i n g i s supplied In Appendix I I . The f i r s t step i s to read the input data as shown by block 1. Using the values of the assumed network elements, the network i s analysed to determine the natural frequencies and modal matrix as shown i n block 2. Block 3 consists of computing and i n v e r t i n g the Jacobian matrix. Block 4 evaluates the correction matrixes M]^ , E2* M 5 using an assumed step s i z e . In block 5 the step s i z e i s adjusted so that the r a t i o of the element with the largest absolute value i n to the element with the largest absolute value i n M equals a s p e c i f i e d value. This ensures that the magnitudes of the elements of are small compared to those of M. Block 6 tests the step size to determine i f i t i s too small. I f so, control i s transferred to block 9; otherwise i t continues with block 7. This i s a safeguard to prevent an excessive number of cycles. For, i f the step size becomes too small, t h i s indicates that a large number of cycles i s required. This r a r e l y happens and so the t r a n s f e r from block 6 to block 9 and consequently, from block 10 to block 1 3 t i s seldom executed. But, i f i t i s carried out, the f i n a l network element values as printed out i n block 13 w i l l not be the correct ones,as the required number of steps w i l l not have been completed. 68 w u 53 o M 8 w S o o READ INPUT DATA I ANALYZE INITIAL 2 NETWORK co M w w o Q B o _ J L COMPUTE, INVERT 3 JACOBIAN MATRIX M PQ I COMPUTE Mi,M2,—-M5 -4 USING ASSUMED STEP SIZE UPDATE MODAL 8 MATRIX AND NATURAL FREQ. I NO YES I ADJUST STEP SIZE IS STEP SIZE TOO SMALL? I YES NO HAS FINAL NETWORK BEEN REACHED? YES I COMPUTE ELEMENT 9 VALUES, ANALYZE IS STEP SIZE TOO SMALL? {DETERMINED IN BLOCK 6) 10 YES i NO COMPUTE SPEC. ERROR 11 * IS SPEC. ERROR TOO LARGE? 12 ^ NO .... PRINT OUTPUT DATA 13 EXIT Figure 5-1. General Flow Chart for Computer Program. 69 Block 7 determines i f the s p e c i f i c a t i o n parameters equal the desired ones. I f the desired s p e c i f i c a t i o n parameters have not been reached, control i s transferred to block 8. Block 8 updates the modal matrix M and the natural frequencies also. Control i s then transferred to block 3 and the sequence i s repeated. I f the desired s p e c i f i c a t i o n parameters have been reached, control i s transferred to block 9» The network element values are calculated using the f i n a l modal matrix. Using these element values, the network i s then analysed to determine the s p e c i f i c a t i o n parameters. Because M always has some associated error, the s p e c i f i c a t i o n parameters so obtained w i l l not equal the desired ones. A measure of t h i s error i s obtained by c a l c u l a t i n g the square root of the sum of the squared differences between the desired s p e c i f i c a t i o n parameters and those a c t u a l l y found. This i s designated as the s p e c i f i c a t i o n parameter error i n the sequel. Block 11 calculates the s p e c i f i c a t i o n parameter error. Block 12 then tests t h i s error and, i f i t i s larger than a sp e c i f i e d amount, control i s transferred back to block 3 where another complete cycle, designated a correction cycle, i s executed. The network element values determined i n block 9 now play the r o l e of the assumed network element values which were o r i g i n a l l y read i n from data cards. I f the s p e c i f i c a t i o n parameters are s u f f i c i e n t l y accurate, control i s transferred to block 13 where the output data i s printed. CHAPTER VI EXAMPLES DEMONSTRATING THE REALIZATION PROCEDURE In t h i s chapter the numerical procedure given previously i s i l l u s t r a t e d by several examples. For each of these examples the maximum s p e c i f i c a t i o n error (defined i n Section 5-4) was 0 . 0 0 1 . This quantity was found to be s u f f i c i e n t l y small that a l l s p e c i f i c a t i o n parameters i n the examples to follow were accurate to f i v e s i g n i f i c a n t f i g u r e s . For convenience, however, a l l r e s u l t s have been rounded to three s i g n i f i c a n t f i g u r e s . 6 - 1 . A second-Order Network We s h a l l demonstrate the procedure using the second-order network of Figure 6 - 1 . This second Cauer canonic form i s chosen to show the simple form of the system of modal equations. I t also i l l u s t r a t e s the power of the method to avoid t r i v i a l solutions. Figure 6 - 1 . Example of Second-Order Network. The modal matrix and the diagonal matrix of inverse natural frequencies are written as x l x2 X3 M = , T = - 1 U 2 respectively. 70 The system of modal equations f o r the network Is: F i = uiCu^Xx) + u 2 ( u 2 - x 2 ) = 0 , F 2 = X-J^X^U-J^ +x 2x i } iU2 = 0 , I t Is evident that the solution to t h i s system of equations can be found without the a i d of a numerical procedure. Yet i t i s i n s t r u c t i v e to apply the proposed numerical technique to t h i s system. We wish to r e a l i z e the driving-point impedance = 1.00 16.0 . z l l " s+1.00 s+10.0 The parameters are x 1 « 1 .00 , x 2 = 4 . 0 0 , Xj = 1 .00, x^ = 0 .100. The assumed set of network element values was c i = 0.0954 farads, c 2 = O.398 farads. Si = 0.102 mhos, g 2 « 1.86 mhos. These element values correspond to the driving-point impedance . . . s 9.00 + 4.00 . 1 1 s+1.00 s+5.00 The s o l u t i o n was found to be U ! = -1.24, u 2 « 3 .10 . This sol u t i o n was reached i n four steps. The element values corresponding to t h i s s o l u t i o n are c^ = 0.172 farads, c 2 = O.O895 farads, g 1 = 0.385 mhos, g 2 = 0.400 mhos. It Is evident that u-^ = 0 , u 2 = 0 , i s also a sol u t i o n to Fj_ t and F 2 above. But t h i s i s a t r i v i a l s olution since i t gives a modal matrix which has a zero determinant. 72 I f one were using conventional numerical methods, such as the Newton-Raphson or Steepest Descent Methods, to solve the equations F]=0, F2=0 p a r t i c u l a r care would have to be taken i n s e l e c t i n g an i n i t i a l point. Otherwise, the so l u t i o n obtained might be the t r i v i a l one. This problem does not exist when using the proposed procedure. This i s because we s t a r t with a n o n - t r i v i a l solution to a set of equations which have parameters d i f f e r e n t from those required. We gradually perturb these parameters to those desired and the corresponding s o l u t i o n i s gradually changed to the desired one. 6-2. A F i r s t Cauer Canonic Form Next we s h a l l demonstrate the method using the f i r s t Cauer canonic form of Figure 6-2. This example was chosen to show that, even i n cases f o r which the assumed set of element values i s v a s t l y d i f f e r e n t from the f i n a l set, the solution i s obtained within a reasonable number of i t e r a t i o n s . Figure 6-2. Example of F i r s t Cauer Canonic Form. We require t h i s network to r e a l i z e the d r i v i n g -point impedance Z 1 1 s 1.00 + 10.0 + 5.QQ 5 2 1 1 8+1.00 s+8.00 s+8.10 73 In t h i s example and several others given i n t h i s chapter, the element values have been normalized so that the smallest natural frequency and the corresponding residue are unity. The modal matrix i s M = x l x2 x 3 u l u 2 n3 (6.2.1) U ^ u^ u^ and the diagonal matrix of Inverse natural frequencies Is T = x 6 (6.2.2) The system of modal equations i s F 2 c V i =0 F 5 - I? x j u ^ u j - x ^ = 0 = X D ^ U J U J =0 F6 = X J U J U J - X J ) =0 where j = i+3.• The following set of element values was assumed: c-j^ = 0.0473 farads, c 2 = 0.727 farads, c^ = 0.727 farads. g^ = 2.00 mhos, g 2 = 2.00 mhos, Sj m 2.00 mhos. 74 These element values correspond to the driving-point Impedance „ - 1.00 . 0.470 + 19.7 1 1 s+1.00 s+7.04 s+45.2 ' The element values r e a l i z i n g the sp e c i f i e d impedance were found to be c i = 0.0625 farads, C2 = 1.242 farads, 03 = 39.2 farads, g l = 0.475 mhos, g2 = 1.32 mhos, g3 = 314. mhos. The solu t i o n corresponding to t h i s set of element values Is u i 0 . 8 6 8 , u 2 - -0.169. U3 = -0.149 u^ = 0.00414, u^ = -0.0931. u 6 = 0.130. The solu t i o n was reached i n fo r t y - f o u r steps, and was executed i n t h i r t y - s i x seconds. This example demonstrates that impedances which have poles close together can be r e a l i z e d using the proposed procedure. 6-3. Lee's Unsymmetrical Bridged Tee Network We also applied the technique to the problem of synthesizing Lee's bridged tee network using the impedance of Eq. (2.3.1), namely 1.00 . 0.640 . 81.0 T 11 T H s+1.00 s+20.0 s+80.0 * This example i s chosen to demonstrate the r e l a t i v e ease with which a solut i o n to the unsymmetrical bridged tee network can be achieved compared to the method given by Lee. The network and element values f o r t h i s impedance are given i n Figure 2-5 ( D ) . These element values were found to be the same as those obtained i n Chapter II when rounded to the three s i g n i f i c a n t f i g u r e s . 75 The system of modal equations f o r t h i s network i s as follows: P i - V u ^ ) =0, Fk = X j U j U j - X i ) = G 3 3 F 2 - g «j xl =0 . F 5 - I D z i x j ( u r U l ) = 0 3 3 F 3 = p 1 u i u J = G ' F 6 = g x J u j ( u i - x i ) = °» where j> = i+3. The u A and x A are defined as i n Eq. (6.2.1) and Eq. (6.2.2). To test our synthesis procedure, we selected the following set of element values: c i = 0.028? farads, c 2 = 0.287 farads, C3 = 1.44 farads g l = 0.0703 mhos, g 2 = 0.352 mhos, g^ = 7.03 mhos. These values correspond to the driving-point Impedance Z T 1 = 1.00 + ?5»? + 1.80 . 2 1 1 s+1.00 s+2.72 s+5.38 The s o l u t i o n was found to be ui = -1.00, u 2 = 0.779* U3 = 8.77, VL^ = -0.000234, u^ = 0.591, ug = -0.0525. The solution was obtained i n fourteen steps. 6-4. A Network Not Realizable by Other Methods We s h a l l apply the method to r e a l i z e the impedance, - = 1.00 + 0.667 . 1.00 . z l l e+10.0 s+30.0 s+50.0 76 using the network of Figure 6-3. This network Figure 6-3. Example of a Third-Order Network. i s the only t h i r d order network f o r which no r e a l i z a t i o n procedure has been found to date. This example shows that the method can be applied to networks which cannot be synthesized by other methods. The system of modal equations f o r t h i s network i s 3 * i - D 1=1 u i x i = °» 3 22 x j u i ^ u i - 3 C i ) - °» 3 F 2 - ID 1=1 u l u J = °» F 5 = 3 £ ? x j u ^ u j - x ^ = 0, 3 F 3 = ZD f i = i U ^ U j - X ^ = 0, F 6 3 - ZD x^ujCui -X i ) = 0, where J = i+3. The x^ and u^ are defined as i n Eq. (6.2.1) and Eq. (6.2.2). We selected the following set of element values f o r the s t a r t i n g point: c^ = 0.500 farads, c 2 - 20.0 farads c^ = 1.00 farads, g^ = 50*0 mhos, gg = 10.0 mhos, g^ = 100. mhos. 77 These values correspond to a driving-point impedance of 1 1 s+0.445 s+53.8 s+208. F o r t t h i s example, the element values were not normalized to make the smallest natural frequency and i t s residue equal to unity. The following s o l u t i o n was reached i n thirty-two steps: u x = 0.647. u 2 = -0 .322, u^ = -O .383, u^ = 0.0387, = -0.216, u 6 - 0.247. The element values obtained from t h i s solution are c^ = 0.391 farads, c 2 = 1.49 farads, c^ = 9.13 farads, g-L = 10.6 mhos, g 2 = 22.0 mhos, g^ = 34l. mhos. As a matter of i n t e r e s t , the time f o r solution using the IBM 360/50 was approximately 40 seconds. This excludes the time required f o r compiling. 6-5. A Fifth-Order Network An example of a f i f t h - o r d e r network i s now given. The network topology i s shown i n Figure 6-4. SI Figure 6-4. Example of a Fifth-Order Network. 78 The s p e c i f i c driving-point Impedance was z,, = 1-00 i 3-00 + s .oo + i o *0 + 1^0 . s+1.00 s+5.00 s+20.0 s+40.0 s+100. The following set of element values was used f o r the s t a r t i n g point * 0.00126 farads, c 2 = 0.00126 farads, Cj = 0.126 farads, c^ = 0.126 farads, c^= 0.126 farads g 1 = 0.430 mhos, g 2 = 0.430 mhos, = 0.430 mhos, g^ - 0.430 mhos, g^ = 0.430 mhos. These values give the driving-point impedance Z i i SA LOO + 0.985 . 1.18 . 403. . 1180. J- L s+1.00 s+1.71 s+5.70 s+163. s+2930.* The equations f o r t h i s network are twenty i n number and w i l l not be l i s t e d here f o r the sake of brevity. The s o l u t i o n was found i n 23 steps and the element values corresponding to t h i s solution are c^ = 0.0328 farads, c 2 • 0.282 farads, 03 = 0.179 farads, 04 = O.258 farads, 05 = 0.0495 farads, g^ = 1.32 mhos, g 2 = 9.64 mhos, - 0.670 mhos, g^ = 0.648 mhos, g^ = 1.54 mhos. The time required f o r solut i o n was I.67 minutes. This i s a reasonably short period of time when i t i s re a l i z e d that, at each of the twenty-three steps, a 20 x 20 matrix was inverted. 79 This example demonstrates that, even f o r the more complex networks, the amount of computing time i s not excessive. This i s true even when using i n i t i a l network element values which are quite d i f f e r e n t from the f i n a l values as i s the case i n t h i s example. 6-6. Discussion Extensive t e s t i n g of the method has made one fact c l e a r . While i t i s desirable to s t a r t with network element values close to the f i n a l ones, t h i s i s not at a l l necessary. This i s evident from the examples given i n t h i s chapter. The only disadvantage Is that more computing time may be required. Examples of driving-point Impedances with residues and natural frequencies spread over a wide range were tested. For example, one driving-point impedance i n which the r a t i o of largest residue to smallest residue was 5 x 10^ was r e a l i z e d without d i f f i c u l t y . Another impedance i n which the r a t i o of the largest natural frequency to the smallest natural frequency k was 5 x 10 was r e a l i z e d without any trouble. The method has been tested on approximately three hundred examples. Networks of various topologies up to the f i f t h order were tested. Except f o r about f i v e examples, the method yielded the correct r e s u l t i n a l l eases. Upon further in v e s t i g a t i o n , i t was found that each of these f a i l u r e s occurred at a point where the Jacobian was zero. Some of these examples and a discussion of the s i g n i f i c a n c e of the vanishing of the Jacobian are given i n the next chapter. CHAPTER VII THE JACOBIAN AND ITS SIGNIFICANCE 7-1. The Vanishing Jacobian It w i l l be r e c a l l e d that the Theorem on Implicit Functions requires that the Jacobian be non-zero at the solut i o n point. So f a r we have not discussed the consequences of a vanishing Jacobian If i t does occur. As shown i n Section (5-3). we obtain a sequence of sets of l i n e a r equations which we l i s t below i n the form JE^ = J E 2 = B2» (7.1.D i JEp = Bp, where p i s the number of terms included i n the series expansion of each u^. I f | j | =0, we must consider whether or not we can f i n d a n o n - t r i v i a l s o l u t i o n to each set of equations i n t h i s sequence. ok Lanczos discusses the compatibility of l i n e a r equations and shows that, f o r a system Jx = b (7.1.2) to have a solu t i o n when | j \ = 0, i t must s a t i s f y the relationship y*b = 0, (7.1.3) where y i s a s o l u t i o n of the system J*y = 0. (7.1. k) In words, the right side of a given set of l i n e a r equations has to be orthogonal to any solu t i o n of the adjoint homogeneous system. I f J i s of rank n-r, there are r independent n o n - t r i v i a l 80 81 solutions to the adjoint system (7.1.4) and each must satisfy Eq. (7.1.3). We now consider the case where the rank of J is n-1. Hence r=l. Hence there i s only one non-trivial solution to Eq. (7.1.4). But y must satisfy p relationships, namely, y^Bi = 0, y ^ o « 0, j (7.1.5) 7 % = 0. since there are p systems of equations. Now i f a l l terms i n the series are taken into account, p i s in f i n i t e and so we see that the only way for y to meet a l l these conditions would be for a l l B 4 = 0, 1 = 1,2, n. But B^ = 0 only i f the specification increments elk» &tk» & = 1,2,—m), are a l l zero. To show this, the f i r s t member of (7*1.1) becomes JE1 = 0. (7.1.6) Since the rank of J i s n-1, we see that there i s one non-trivial solution for E^. But i f E^ = 0, we see from Eq. (5.3.4) and Eq. (5*3.6) that B 2 = 0. Hence, the only acceptable solution to Eq. (7.1.7) Is E]_ = 0. By similar arguments we can deduce that a l l Ei =0, ( i = 1,2, p). This means that i f , by some coincidence, we pick an i n i t i a l set of element values which lead to a zero Jacobian, we cannot move away from this position by perturbing specification parameters since there i s no solution other than the t r i v i a l one. 82 Hence the procedure would f a i l . The simple remedy to t h i s s i t u a t i o n i s to choose a new set of element values and st a r t again. I t should be noted that f i n d i n g a set of values which leads to a vanishing Jacobian i s not easy and t h i s s i t u a t i o n i s ra r e l y experienced. The procedure w i l l also f a i l , i f at any point along the path i n specification-parameter space, the Jacobian vanishes. Again, t h i s problem may be overcome by picking a new i n i t i a l network and repeating the procedure. More w i l l be said about t h i s l a t e r . The above arguments seem to Indicate that, since the l i n e a r equations become incompatible f o r a vanishing Jacobian and f o r non-zero s p e c i f i c a t i o n increments, no solu t i o n exists f o r the new set of s p e c i f i c a t i o n parameters. Hence a boundary seems to exi s t i n s p e c i f i c a t i o n parameter-space which cannot be crossed and s t i l l maintain a solu t i o n which s a t i s f i e s the o r i g i n a l functions. In the next section, we demonstrate that the s p e c i f i c a t i o n parameters may have l o c a l extrema at a point where the Jacobian vanishes. 7-2. Vanishing Jacobian and Local Extrema of S p e c i f i c a t i o n Parameters We s h a l l explain the theory as i t applies i n two dimensions, and then extend i t to the higher dimensional case. Consider a curve F(x,y) = 0 (7.2.1) i n two dimensions. Let - H - 0 (7.2.2) at a point jP.0 with coordinates x Q, y Q on t h i s curve. We s h a l l assume that ^Z^> 0 without loss of generality. I f x i s considered as the independent v a r i a b l e , we see that, upon d i f f e r e n t i a t i n g 83 Eq. (7.2.1) with respect to x, the derivative cannot be dx found. I f we d i f f e r e n t i a t e Eq. (7.2.1) with respect to y, however, we obtain and, consequently, ^ Z + ^ l | 2 = 0. (7.2.3) 0 y a x dy This gives, by v i r t u e of Eq. (7.2.2) dx dy i s = 0 at P 0. That Is, i f x i s considered as a function of y, x has a l o c a l extremum at P Q. The nature of the extremum i s determined j 2 by the second derivative . D i f f e r e n t i a t i n g Eq. (7.2.4) and, dy 2 using Eq. (7.2.2), we f i n d that ( 7 - 2 - 5 ) Since 0 we see that ^ F 70 gives a l o c a l maximum while ' 0 X yZ -.2 2 d < 0 gives a l o c a l minimum. I f ^ ff = 0, and assuming "dy 2 d y 2 ^ F / 0, we have an i n f l e c t i o n point which i s neither a dy3 r minimum nor a maximum. Hence, we conclude that a vanishing Jacobian -^F-oy indicates the presence of an extremum when x i s considered as a function y. The nature of t h i s extremum i s determined by the second p a r t i a l d e r i v a t i v e . 84 Next consider the system of equations (5.2.3). This set may be considered to represent a curve i n a space of n+1 dimensions. Let the Jacobian [ j | , defined i n Section 5-2, / o o o o x vanish at a point P Q which has coordinates (u^Ug.--— u n.x ). This prevents us from solving f o r the p a r t i a l derivatives >)U, ^u.. *NU * x ^x 7>x ° Suppose that we consider u^ to be the independent variable as opposed to x. Then we have, upon d i f f e r e n t i a t i n g each equation i n (5.2.3) with respect to u^. ^2 duJ *ul dx ^ul i l l 3Hl + i l l JL2 =o. (1=1,2,—n). (7.2.6) Suppose that the corresponding Jacobian Uil = i l l OX 5)U 2 I I I 2>Fn i ! i i i i 3F n does not vanish at P, bx £ u n This i s the same as the Jacobian (7.2.7) of Section 5-2 except the f i r s t column has been replaced by b p i the p a r t i a l derivatives fcx equations as follows z>x i l s ifjk I I i£n (1=1,2,—n). Rearrange the o F , i i c>ul (7.2.8) Solving f o r using Cramer's r u l e , we f i n d that * x = 0. c ) u l (7 .2 .9 ) at P Q. This shows that x i s a l o c a l extremum when considered as a function of u^. As i n the two-dimensional case, the nature of t h i s extremum i s determined by the second p a r t i a l > 2 V d e r i v a t i v e ° * To f i n d t h i s we d i f f e r e n t i a t e each of the equations i n (7 .2 .6 ) with respect to u i . A f t e r rearranging and making use of Eq. (7 .2 .9 ) we obtain r,323c" ^x 1 * u n i ^u2 a l 1 1 3 P n i i • o n I 1 = + a n "ox (7.2.10) where J=l k=l ^ U J ^ U k 3 U 1 ^ u l We use the convention that = 1 i f j=l i n evaluating t h i s o u l quadratic form. A disadvantage of t h i s technique i s that l o c a l and global extrema cannot always be distinguished. It i s necessary to have more knowledge about the function x such as i t s convexity or concavity to determine i f the extrema are global. Convexity or concavity i s e a s i l y established i n most two dimensional cases, but l t i s often very d i f f i c u l t to e s t a b l i s h i n higher dimensional cases. We s h a l l see, i n the examples that follow, that t h i s prevents us from d e f i n i t e l y establishing 86 whether or not the local extrema are also global extrema. The parameter x, as explained i n Chapter V, i s the interpolation parameter. In general i t i s some linear combination of the specification parameters. In particular i t may be equal to one specification parameter or at least be proportional to i t . This would be so, for example, when the i n i t i a l network has a l l i t s specification parameters except one, say, X j , equal to those of the desired network. The procedure of moving from the i n i t i a l to the f i n a l point i n specification- parameter space consists of moving along the coordinate axis corresponding to X j . In the next section we shall apply the technique to various examples. 7-3. Some Examples 7.3.1. Example I We shall now examine the network of Figure 7-1 which is the only irreducible complementary tree structure of third-order. Figure 7-1. Irreducible Complementary Tree Structure. the form The system of modal equations f o r t h i s network takes F i = & V V x i ) s °-3 F 0 = £2 u ^ U j - X j ^ ) * 0, '3 = *g uj(ul"% ) - 0 . = ZD u i x l x 1 = ° » 1=1 J ( 7 .3 .1 ) ED XiXjUy.^) - 0, where J=l+3. The u^ and xj are defined as In Eq. (6 . 2 . 1 ) and Eq. ( 6 . 2 . 2 ) . The Jacobian matrix f o r t h i s set of equations Is given by 2 u l ~ x l 2 u 2 ~ x 2 2U3-X3 0 0 0 0 0 0 2U2(,-Xij_ 2u^-x 2 2U5-X3 J = u 4 u 5 u 6 u 1-x 1 u 2 - x 2 U3-X3 X 2 X 5 x 3 x 6 0 0 0 . ( 7 . 3 .2 ) *iFk u 5 x 5 U 6 X 6 u l x 4 U 2 X 5 U 3 X 6 A so l u t i o n to the set (7.3.1) i s : u° = -0.137, u g = 1.90, u° = 2 . 0 4 , u g = 1.07, u° = 1.58, u g = 0.509, x° = 1.00, x g = 0.760, x°j = 3.17, x g = 1.00, x°5 = 0.0450, x g = 0.0112. This solu t i o n corresponds to a Jacobian of 0.381x10"^ which i s small compared to the elements .of J . The variable u^ i s taken to be the independent vari a b l e . The parameter x i s considered to be any one of the six s p e c i f i c a t i o n parameters. That i s , while one i s being considered, the remaining f i v e are fix e d at the values given above. F i r s t i t was v e r i f i e d that the f i r s t p a r t i a l d e r i vatives 0 1» (1=1,2,—-6) were zero, thus establishing that the X j given above are extreme values when considered as functions of U]_. Next the second p a r t i a l derivatives c> 2x T i were calculated to determine the type of extremum. c)uj A c t u a l l y i t i s only necessary to determine the sign of each of these second p a r t i a l d e r i v a t i v e s . The resu l t s are shown i n Table 7-1. Table 7-1 I 1 2 3 4 5 6 3 2 x x C>U 2 - + + + - -Xj i s max min min min max max 89 This shows that x^, X 5 , x£ are l o c a l maxima while x 2» x^, Xfy are l o c a l minima. Since x^. x^ t xg are the reciprocals of the natural frequencies, we see that the frequencies corresponding to x^ and x^ are actually minima. We must keep i n mind that when we are considering any one of the X j , the remaining ones are f i x e d . They are, i n t h i s sense, conditioned maxima and minima. I f we change any one of the remaining f i v e parameters e i t h e r the Jacobian i s no longer zero or x has a d i f f e r e n t maximum value. I t i s believed the above r e s u l t s are global extrema on the basis of the following. The network of Figure 7-1 may be synthesized by f i r s t r e a l i z i n g the shunt conductance g^ In the normal manner and then using the Yarlagadda-Tokad method to r e a l i z e an unsymmetrical l a t t i c e terminated i n the capacitance C g . Referring to the Yarlagadda-Tokad method of Chapter I I , we r e c a l l that, f o r a s o l u t i o n to e x i s t , Eq. (2 .4 .1 ) must have a r e a l negative root. We t r i e d to use t h i s method to synthesize an impedance i n which a l l of the parameters, except Xg, were the same as given i n the above example. We used x^ = 0 .0125. Eq. (2 .4 .1 ) was found to have no r e a l negative root. Hence, no r e a l i z a t i o n e x i s t s . This i s i n keeping with what we predicted previously since xg = 0 . 0 1 1 2 was a maximum. Similar tests were carried out on the remaining parameters. In each case no r e a l i z a t i o n was found. On the basis of these findings, i t i s believed that the extrema found above are global extrema. 90 7.3.2. Example II o, As discussed i n Section 2-2, Lee and Murphy have shown that x k / x 6 ^33.97 f o r the network of Figure 7-1. This minimum i s an absolute one i n the sense that i t i s the lowest value the r a t i o can have f o r any set of x^ and u^. Lee and Murphy have also given the element values f o r which t h i s r a t i o i s attained. These values, when normalized so that smallest natural frequency and the corresponding residue are unity, are as follows: c 1 = c 2 = c^ = 0.0858, (7.3.3) g-L = g 2 = = 0.5000 mhos. The s o l u t i o n and the s p e c i f i c a t i o n parameters f o r t h i s set of values are: u° = -0.707, u| = 2.42, U3 = 4.12, ujj = 1.71. u0; = 2.42, ug = 1.71, x° = 1.00, x?> = 0, x^ = 5.83, xg = 1.00, x° = 0.172, x° = 0.0294. 5 0 The Jacobian f o r t h i s set of values was found to be IJI = -0.540x10 ~* which again may be assumed to be zero. Note that the residue corresponding to the natural frequency 1/x^ i s zero. This means only two natural frequencies are observable at the d r i v i n g terminals and i s c l e a r l y a special case. The method outlined i n the previous section was used to determine the types of extrema. Unfortunately, the method led to inconclusive r e s u l t s f o r a l l x except x and x.. 91 This i s because the Jacobian | J l | of Eq. (7.2.7) was zero when xl» x 2» x 3» a n < * x 5 w e r e being tested. This peculiar phenomenon Is thought to be due to x 2 being zero. The Yarlagadda-Tokad method was used to determine what was happening i n these cases. When each X j , (1=1,3,5) was varied about x£ while the remaining X j were held constant, i t was found that no solution exists i n each case. The method was successful when x^ and xg were tested. I t was found that xjj, was a minimum and x D a maximum. This i s as expected since xg = 0.0294, which i s the r e c i p r o c a l of 33.97, and, as Lee and Murphy have shown, i s the maximum value f o r xg under any conditions. This must include the s p e c i a l situations being considered here. 7.3.3. Example III The l a s t two examples discussed above were concerned with a complementary tree structure which i s i r r e d u c i b l e . Examples of reducible complementary tree structures which are known to be canonic and having a vanishing Jacobian f o r c e r t a i n sets of element values have been observed. Upon further i n v e s t i g a t i o n l t was found that these points correspond to a sudden disappearance of the s o l u t i o n from the r e a l domain. As suggested previously, as we proceed from the i n i t i a l to f i n a l network, we are, i n e f f e c t , following a solution. I t i s believed that the solution becomes complex at t h i s point. This Is what happens i n two dimensions, f o r example, when we have a straight l i n e and a c i r c l e . If the l i n e intersects the c i r c l e , we have two r e a l solutions. But i f the coordinate 92 of the centre of the c i r c l e are changed, so that the l i n e becomes tangent to the c i r c l e , the corresponding Jacobian w i l l vanish at the s o l u t i o n point. I f the coordinates of the centre of the c i r c l e are changed further, the l i n e w i l l no longer touch the c i r c l e and both solutions become complex. Hence the vanishing Jacobian marks the t r a n s i t i o n from the r e a l to the complex domain. I t i s believed, on the basis of conducted t e s t s , that the system of modal equations displays a s i m i l a r phenomenon when the Jacobian Ul vanishes. We now give an example. The network of Section 6-4 was found to have a vanishing Jacobian f o r the following set of element values: c-j^ - 0.641 farads, c 2 = 0.829 farads, = 1.48 farads, g^ = 2.39 mhos, g 2 = 2.23 mhos, g^ = 3.62 mhos. This set gives the following driving-point impedance: z 1 T = If 00 + 0-0983 + 1.14 (7.34) 1 1 s+1.00 s+2.08 s+11.8 ' Kf.W) Using the method of Section 7-2, i t was found that x 2, which i s the square root of the residue of the pole at s = -2.08, was a maximum. Since It i s known that t h i s network Is canonic, i t was concluded that t h i s value of x 2 was a l o c a l maximum. It Is informative, however, to determine the nature 2 of the curve of x 2 versus u^, f o r example. We used the proposed synthesis procedure to generate points In the neighbourhood of 2 x 2 = 0.0983. The r e s u l t s are shown plotted In Figure 7-2. The solu t i o n consists of two branches. The upper branch ends at 2 x 2 = 0.0983 as expected. The lower branch, which overlaps the 2 upper branch, ends at x 2 = 0.068. The Jacobian also vanishes at t h i s point. We see that there i s always at least one solution 1 1 1 i 1 i : j 1 1 1 i l r n f { 1 — i — i 1 i I M -I 1 1 1 1 : O 1 1 _ »_. i 1 \ IT i 1 i |! - I 1 i 1 I | i PC 1 1 i 1 i or IX - • 1 1 1 *r } '— i 1 1 i i 1. l. 1 i/ /"V-7 p, 1 j I 1 1 f Tt 1 1 1 / I I t 1 /l . 1 . f« s. 11 | CO 1 1 j i O • i W i i I i 1 i 1 1 / 1 1 i 1 1 1 i 1 i 1 I* i i 1 1 L I 1 1 i - rh ! 1 1 1 q 1 wa H 'rf. -: — 1 U Ul oJ 1 i 1 1 o -* 1 8 1 i 1 1 1 1 n i 1 1 ! 1 ! n K l 1 -O-I o i > l O i I Q ! O i o I 1 IQl 1 1 l O i < 2 • '1 4-1. - I C O J n i cn 1 — o o O P l O 0 i !<D 1 1 o - a 1 s 1 1 r i 1 •I ! 1 1 i 1 i 1 1 i 1 1 i i 1 1 i i 1 1 ! i i 93 ra (0 U > -U CM CM O ft T H E UN1VERS 1TY OF TORONTO PRESS 2 and, i n the region 0.068 <x 2<0.0983. there are two solutions. 2 Hence If X2 = 0.08 there are two sets of element values. These are c 1 « 0.586 farads, c^ = 0.778 farads, c^ = I . 9 6 farads, g 1 = 2.63 mhos, Sg = 1 . 9 0 mhos, g^ = 4.38 mhos, corresponding to the solut i o n from the upper branch, and c 1 « 0.764 farads, - 1.17 farads, c^ = 1.10 farads, g^ ^ = 1.66 mhos, g 2 - 3.18 mhos, g^ = 4.55 mhos, corresponding to the solut i o n from the lower "branch. Both sets of element values give the driving-point Impedance - 1.00 t 0.0800 . 1.14 ( 7 , M 2 1 1 s+1.00 s+2.08 + s+11.8* W.J.5) I f an impedance i s to be synthesized f o r which there are multiple solutions, the solution that i s obtained i s , of course, governed by the i n i t i a l network. Thus, i f one uses an i n i t i a l network which i s closer to the solutions.on the upper branch of Figure 7-2, the solution on the upper branch w i l l be found. 7.3.4. Example IV It i s worth pointing out that the Jacobian |J| may vanish f o r situations which are somewhat t r i v i a l . For example, the appropriate equations f o r the second-order Foster form of Figure 7-3 are F l = U 1 ( U 1 - X 1 ) + U 2 ( U 2 " X 2 ) = ° » (7.3.6) F 2 = u 1 ( u i - x 1 ) x 3 + U 2 ( U 2 - * 2) X4 = °» where u^ and U2 are the unknown elements of the modal matrix and the x^, (i=l,2,3» k), are the parameters obtained from the driving-point impedance Z(s) = S+I/X3 s+l/xk Figure 7-3. Second-Order Foster Form. The Jacobian f o r the system (7.3 .6') i s |J| = 2 u l " x l 2u 2-x 2 which i s (2ui-xi)x3 (2u2-X2)xu | j | = (2u 1-x 1) (2u 2-x 2) (x k-x 3) (7.3.7) We see that i f X 3 = x^ the Jacobian vanishes. In t h i s s i t u a t i o n both natural frequencies are the same and also both F -L and F 2 become the same. But c l e a r l y the network i s not minimal i n the number of elements since, i f X 3 = x^, the driving-point Z(s) can be r e a l i z e d by just two elements. Such situations as t h i s may be remedied by requiring that x 3 ? x ^ . This does not l i m i t the procedure i n any way. 96 7-4 . Comparison of Lee's Extremum and S p e c i f i c a t i o n Parameter Extrema To show more c l e a r l y that the extrema which we have discussed i n the above examples are conditioned extremum, whereas Lee's extremum i s an absolute one, we s h a l l examine b r i e f l y a second method f o r determining the absolute extrema. We r e f e r to the system of equations given i n ( 7 . 3 . 1 ) f page 87. Without loss of generality, we can f i x x i and x4 at unity. This can always be done by amplitude and frequency s c a l i n g . Hence the s i x equations (7 .3 .1 ) r e a l l y involve four parameters, namely, Xg, x^, x^, and Xg. It i s also convenient to use and Fg to eliminate two of the u^, say and ujj,, from a l l equations. This i s possible since F^ and F5 are l i n e a r i n the u^. We are l e f t with four equations F-, F„, F_, and F_, which involves four u, and four x,. J- J 5 1 1 We can now think of these four equations as defining the x± i n terms of the Uj_ i m p l i c i t l y . Again^using the theory discussed i n Section 5 -2 , we can solve f o r the p a r t i a l derivatives ^ . This i s done by use of the equations - k f i + 5^ J ! i Jgk = 0 ( 1=1 .2 ,3 ,4 ) , (7 % 4 % 1) k=l S*k ( 3=1 ,2 ,3 ,4 ) . Now, at the s o l u t i o n point, provided 3 F i Jx"= a * ! 1 1 1 1 ( 7 . 4 . 2 ) 97 does not vanish, we can solve f o r the various p a r t i a l derivatives ^ 1 . There w i l l be sixteen i n a l l . b u 3 In the previous examples we considered what happens when a l l x^ except Xj are fixe d and we were able to show. In general, that X j i s a l o c a l extremum i n such cases. Here we are considering what happens to x^ when the remaining x^ are eliminated from the p a r t i c u l a r function, say F-^ . This elimination i s done by use of the remaining functions i n the set. In other words, we are fi n d i n g the series expansion of o JL / >>xT\ o *I = X i + £ P * - M ( u r u j + (7.4.3) o by the method described i n Section 5-2 except that the roles of the X J L and u^ have been interchanged. Therefore, when we fi n d that: ° 1 (3=1,2,— n), are zero at a p a r t i c u l a r point, c)u3 t h i s i s true no matter what the values of the remaining x^ turn out to be. Several sets of element values which correspond to a zero value f o r the Jacobian of (7.3.2) were tested using t h i s technique. I t was found that the set of values given i n Example II leads to zero values f o r the set of p a r t i a l derivatives (3=2,3.5.6). Since xjj=l, we see that xg i s actually the 3 CT inverse of the r a t i o ~ m a x » Also x^=0.0294=1/33.97. This CTmln extremum i s the same as that of Lee and Murphy. No other set of element values which corresponds to an extremum of any of the s p e c i f i c a t i o n parameters X g . x ^ , x^, and xg was found. This demonstrates the difference between the two methods. 7-5» Discussion For the network of Figure 7-1 the equation \J\ - 0, when written as a function of u^ and x^ , i s very complex as can be seen from a study of ( 7 « 3 » 2 ) . In general t h i s equation describes a manifold i n a space of twelve dimensions. Example I consists of an inv e s t i g a t i o n of only one point on t h i s manifold. Example II dealt with another point. Several other points f o r which |J) = 0 were also investigated. A l l re s u l t s showed a common feature. That i s , every point investigated which s a t i s f i e d the equation IJI = 0 corresponded to extrema of the s p e c i f i c a t i o n parameters. To es t a b l i s h that t h i s i s true f o r every point on the | j | = 0 manifold, using the method given i n Section 7-2, requires that every point on t h i s manifold be investigated. This i s c l e a r l y Impossible. Unsuccessful attempts were made to develop a mathematical proof showing that extrema of the s p e c i f i c a t i o n parameters occur at | j | = 0 In the general case. However, i t i s f e l t that, i n view of the r e s u l t s discussed above, there are some grounds f o r making the following conjecture: the boundary of the r e a l i z a b i l l t y f o r the complementary tree structure of Figure 7-1 coincides with the vanishing of the Jacobian of the system:-( 7 . 3 . D . 99 In view of the previous discussion i n Section 7-2 on the vanishing Jacobian* i t i s pertinent to point out that the method w i l l also handle driving-point functions which lead to zero Jacobians. It i s able to do t h i s because, at the f i n a l step, the Jacobian matrix Involved i s that of a network which has impedance parameters which d i f f e r from the desired ones by the step Increment i n each case. Hence, while we must f i n d the U J L corresponding to the network with the vanishing Jacobian, the Jacobian involved i n f i n d i n g these u i t i s that which occurs at the beginning of the l a s t step. Several cases which led to a vanishing Jacobian using the network of Figure 7-1 were tested. The networks were r e a l i z e d without d i f f i c u l t y i n each case. Several examples were also tested where i t was necessary to jump from one sol u t i o n branch to another sol u t i o n branch. This would be necessary, f o r example, If we selected a s t a r t i n g network which l i e s on the upper branch of Figure 7-2, and we wished to r e a l i z e an impedance which has the same 2 parameters as the impedance of Eq. (7.3*5) except x 2 ?0.0983. As explained In Appendix I, the program automatically adjusts the step s i z e so that the r a t i o of the element with the largest absolute value In the f i f t h - o r d e r correction matrix to the element with the largest absolute value i n the modal matrix equals a s p e c i f i e d value, Ek, I f Ek i s small, 0.0001 or l e s s , a small step size r e s u l t s . I f , at the same time, a small value of iJl i s encountered, the step size may become so small that progress towards the s o l u t i o n point ceases. 100* The problem may be solved by s e l e c t i n g a larger value f o r E4, say 0.01. This allows a larger step size and has, i n a l l cases tested, been found s u f f i c i e n t to allow the Jump to take place. Allowing a l a r g e r step size increases the error somewhat. But t h i s e r r o r , i s , i n e f f e c t , erased when the computer executes a correction cycle as shown i n Figure 5-1* There i s no way to ensure that t h i s technique w i l l be successful i n every case. I f i t f a i l s , l t i s probably advisable to select a new i n i t i a l set of element values and begin again rather than t r y to pick a new value f o r E4. I f one t r i e s to use the program to r e a l i z e an impedance not r e a l i z a b l e by a given topology, an o s c i l l a t i o n about the |j|=0 point occurs. This was found to be the case when t e s t i n g the network of Figure 7-1, f o r example. This points up an a d d i t i o n a l use f o r the proposed procedure, namely, that of investigating further the boundaries of r e a l i z a b i l i t y f o r two-element-kind networks. For example, one may f i x a l l s p e c i f i c a t i o n parameters except one, say X j , and then use the computer program to f i n d solutions f o r various values of X j . I f , f o r some value of X j , a solu t i o n cannot be found, t h i s indicates that eit h e r the boundary of r e a l i z a b i l i t y or the end of a p a r t i c u l a r solution branch has been reached. Further Investigation would then determine which i s the case. This was done during the i n v e s t i g a t i o n of the network of Figure 7-1 and served as a check on the r e s u l t s obtained by the Yarlagadda-Tokad method. The method may be the only way to carry out s i m i l a r research on higher order networks i n which the topology i s such that previously developed methods cannot be used. CHAPTER VIII CONCLUSION A numerical procedure f o r the r e a l i z a t i o n of two-element-kind driving-point impedances using minimal networks with a r b i t r a r y , but s p e c i f i e d , topology has been given. This method i s based upon the theory of normal coordinate transformations and i t i s shown that a set of second order nonlinear equations may be obtained when the topology i s s p e c i f i e d . This system of equations i s solved by a perturbation method and requires an estimate of the network element values. The procedure has been programmed on the d i g i t a l computer. Numerous examples of various network configurations and driving-point impedance parameters have been used to test the procedure. A disadvantage of the method i s that no guarantee can be given that the numerical procedure w i l l converge to a solution. A second disadvantage i s that the number of equations equals n(n-l) where n i s the order of the network. Thus f o r networks of order greater than s i x or seven, i t i s doubtful i f the method i s p r a c t i c a l . The procedure i s more general i n the sense that, whereas standard synthesis procedures are limited to s p e c i f i c topologies, t h i s one i s not. We can r e a l i z e a l l of the topologies which previously developed methods use, and many others as w e l l . The chief advantages of the method are that the designer has d i r e c t control over:the topology and the method i s amenable to computer programming. 102 A conjecture regarding the r e a l i z a b i l i t y boundary of a c e r t a i n complementary tree structure Is given. On the basis of several examples i t i s f e l t that the boundary of r e a l i z a b i l i t y of t h i s network coincides with the vanishing of the Jacobian of the system of nonlinear equations. No rigorous mathematical proof i s given, however. Future work should be concerned with determining i f the above conjecture i s v a l i d and whether or not i t applies to higher order networks. A method to te s t f o r the existence of n o n t r i v i a l solutions to the system of nonlinear equations would also be, of great value. The p r i n c i p a l contributions of t h i s work are a new formulation of the problem and the development of a numerical method of general a p p l i c a b i l i t y f o r r e a l i z a t i o n of minimal two-element-kind driving-point impedances. 104 BIBLIOGRAPHY 1. F o s t e r , R.M., "A R e a c t a n c e T h e o r e m " , B e l l S y s t e m T e c h n i c a l J o u r n a l . V o l . 3. 1924, 259-267. 2. C a u e r , W., " D i e V e r w i r k l i c h u n g v o n W e c h s e l s t r o m w i d e r s t a n d e n v o r g e s c h r i e b e n e r F r e q u e n z a b h a n g l g k e i t " , A r c h l y , f . E l e c t r o t e c h n i k . V o l . 17, 1926, 355. 3 . 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L e e , H.B., "On t h e C a n o n i c R e a l i z a t i o n s o f Two E l e m e n t K i n d D r i v i n g P o i n t I m p e d a n c e s " , 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. 8. L e e , H.B. a n d M u r p h y , P . J . , "On t h e N a t u r a l F r e q u e n c y R e a l i z i n g A b i l i t y o f Two E l e m e n t K i n d S t r u c t u r e s " , I E E E T r a n s a c t i o n s o n C i r c u i t T h e o r y . V o l . CT - 1 3 , S e p t , , 1966, 325-326. 9. L e e , H.B., "On t h e D i f f e r i n g A b i l i t i e s o f RL S t r u c t u r e s t o R e a l i z e N a t u r a l F r e q u e n c i e s " , I E E E T r a n s a c t i o n s o n C i r c u i t T h e o r y . V o l . CT - 1 2 , S e p t . , 1965, 365-373. 10. Y a r l a g a d d a , R. a n d T o k a d , Y., "On t h e Use o f N o n s y m m e t r i c L a t t i c e S e c t i o n s i n N e t w o r k S y n t h e s i s " , I E E E T r a n s - a c t i o n s o n C i r c u i t T h e o r y . V o l . CT - 1 0 , D e c . 1964, 474-2478. 11 . B o x a l l , F . S., " S y n t h e s i s o f M u l t i t e r m i n a l T w o - E l e m e n t -K i n d N e t w o r k s " , T e c h n i c a l R e p o r t N o . 95, E l e c t r o n i c s R e s e a r c h L a b o r a t o r y , S t a n f o r d U n i v e r s i t y , Nov., 1955. 12. Schwab, W.C., " S y n t h e s i s o f RC N e t w o r k s by N o r m a l C o o r d i n a t e T r a n s f o r m a t i o n s " , D o c t o r a l D i s s e r t a t i o n , M.I.T., S e p t . 1962. -105 13. G u i l l e m i n , E.A., "An A p p r o a c h t o t h e S y n t h e s i s o f L i n e a r N e t w o r k s t h r o u g h t h e Use o f N o r m a l C o - o r d i n a t e T r a n s -f o r m a t i o n s L e a d i n g t o More G e n e r a l T o p o l o g i c a l C o n f i g u r a t i o n s " , I R E N a t i o n a l C o n v e n t i o n R e c o r d . I960, 171-179. 14. Duda, R., " M a t r i x S y n t h e s i s o f T w o - E l e m e n t - K i n d D r i v i n g -P o i n t I m p e d a n c e s " , M.I.T. Q u a r t e r l y P r o g r e s s R e p o r t . No. 56, J a n . 15, I960, 245-254. 15. H o w i t t , N., " E q u i v a l e n t E l e c t r i c a l N e t w o r k s " , P r o c e e d i n g s o f t h e I R E . V o l . 20, J u n e , 1932, 1042-1051. 16. Y a r l a g a d d a , R., " S t a t e - m o d e l A p p r o a c h t o t h e S y n t h e s i s o f LC N e t w o r k s a n d t h e C a n o n i c LC N e t w o r k T r a n s f o r m a t i o n s " D o c t o r a l D i s s e r t a t i o n , M i c h i g a n S t a t e U n i v e r s i t y , 1964. 17. S c h n e i d e r , A . J . , "RC D r i v i n g - P o i n t Impedance R e a l i z a t i o n b y L i n e a r T r a n s f o r m a t i o n s " , I E E E T r a n s a c t i o n s o n C i r c u i t T h e o r y . V o l . CT-13, S e p t , , 1966, 265-271. 18. G o l d s t e i n , H., C l a s s i c a l M e c h a n i c s . A d d i s o n - W e s l e y , R e a d i n g , M a s s . , U.S.A., 1950, C h a p t e r 10. 19. M a c M l l l a n , B., " I n t r o d u c t i o n t o F o r m a l R e a l i z a b i l i t y T h e o r y " , B e l l S y s t e m T e c h n i c a l J o u r n a l . V o l . 31, 1952, 217-279. 541-600. 20. H a z o n y , D., E l e m e n t s o f N e t w o r k S y n t h e s i s . R e i n h o l d P u b l i s h i n g C o r p o r a t i o n , New Y o r k , 1963, 140. 21. S a a t y , T . L., a n d Bram, J . , N o n l i n e a r M a t h e m a t i c s . M c G r a w - H i l l B o o k Company, New Y o r k , 1964, C h a p t e r s I a n d I I . 22. F l e t c h e r , R., a n d P o w e l l , M.J.D., "A R a p i d l y C o n v e r g e n t D e s c e n t M e t h o d f o r M i n i m i z a t i o n " , C o m p u t e r J o u r n a l . V o l . 6, J u n e , 1963, I63-I68. 23. S o k o l n i k o f f , I . S . , A d v a n c e d C a l c u l u s . M c G r a w - H i l l B o ok Company, New Y o r k , 1939, C h a p t e r X I I . , 24. L a n c z o s , C , A p p l i e d A n a l y s i s . P r e n t i c e - H a l l , E n g l e w o o d C l i f f s , N . J . , U.S.A., 1964, 149-156. 106 APPENDIX I DESCRIPTION OF COMPUTER PROGRAM Al-1 The Main Program A Fortran statement l i s t i n g of the main program plus the subroutines i s given i n Appendix I I . At the extreme l e f t -hand edge of these l i s t s of statements i s a column of numbers c a l l e d ISN used to i d e n t i f y the various statements i n the program. We s h a l l use these numbers as a convenient reference when describing the function of the various statements. The program w i l l handle up to f i f t h - o r d e r networks as can be seen "by the dimension statement ISN 0001. Of course higher-order networks can be handled by increasing the dimension of the various matrices i n t h i s dimension statement. The f i r s t quantities to be read are IP(I), (1=1,8), (ISN 0004) according to format 812, (ISN 0005). These numbers are used to control print-out of various quantities as w i l l be explained i n the sequel. ISN 0006 reads the following quantities according to format 212, 5F10.0 (ISN 0007): M - the network order NORMAL - used i n the subroutine ANAL and, i f p o s i t i v e , causes the network elements to be scaled so that the lowest natural frequency and the corresponding residue are equal to unity. SX - the assumed number of steps to be taken i n moving from the assumed network element values to the desired ones. This value i s not c r i t i c a l and i s used to prevent io7 numbers from becoming too large, which may happen i f the assumed network i s f a r from the desired one. A t y p i c a l value Is 100. E l - the maximum r e l a t i v e error i n the modal matrix. This i s explained more f u l l y below. E2 - the same as E l , except that t h i s value i s used on the correction cycle. E3 - allowable s p e c i f i c a t i o n error. I f the square root of the sum of the squared differences between the desired s p e c i f i c a t i o n parameters and those actually found at the end of the program i s less than E3, the sol u t i o n i s considered s u f f i c i e n t l y accurate. Otherwise the machine executes a correction cycle i n which i t uses the network element values found on the f i r s t cyle as I n i t i a l network element values. E4 - minimum f r a c t i o n a l step s i z e . I t w i l l be r e c a l l e d that f o r each step taken on the l i n e connecting the i n i t i a l and desired networks, a modal matrix corresponding to a phy s i c a l l y r e a l i z a b l e network i s found. The lo c a t i o n on t h i s l i n e a f t e r each step is.given by an in t e r p o l a t i o n parameter; i t i s denoted as XC i n the program. XC has a:value of zero at the point correspond-ing to the i n i t i a l point; i t i s unity at the desired or f i n a l point. The change i n XC from the previous step Is calculated a f t e r each step Is completed. I f t h i s change i s les s than E4, the procedure i s interrupted and control i s transferred as i s explained below. ISN 0011 and ISN 0012 pr i n t out these various quantities. 108 ISN 0014 to ISN 0018 read i n the transposed incidence matrices A and B f o r the capacitance and resistance trees repectlvely according to format 5F10.0 (ISN 0016). The statements ISN 0019 through ISN 0025 read, respectively, the assumed capacitance values, CE(I), the assumed conductance values, GE(I), the required residues, R(I), and the required natural frequencies S(I) according to format 5P15.? (ISN 0003). ISN 0027, and 0028 calculate some quantities used l a t e r . ISN 0029 through ISN 0042 have to do with p r i n t i n g out the data which has been read i n . ISN 0043 through ISN 0047 do the following: the square roots of the residues are found and stored i n R(I) since these square roots are equal to the elements i n the f i r s t row of the modal matrix. The reciprocals of the desired natural frequencies are found and placed i n S( I ) . ISN 0048 through ISN 0059 f i n d the inverse of A which i s stored i n AI. This is.'done using MINV, a subroutine from the IBM S c i e n t i f i c Subroutine Package. ISN 0060 c a l l s the subroutine ANAL which Is described i n Section Al-2. This program analyzes the network using the assumed network element values. The arguments l n t h i s subroutine are: A l , the inverse of the transposed incidence matrix of the capacitance tree; B, the transposed incidence matrix of the resistance tree; CE, the capacitance values; GE, the conductance values; D(I), the natural frequencies of the assumed network; P, the modal matrix of the assumed network; and NORMAL, the quantity which determines whether or not the network element 109 values are to "be scaled as explained above. ISN 0061 through ISN 0075 have to do with the print-out of various quantities calculated l n the subroutine. The statements ISN 0076 through ISN 0082 compute the parameter increments P1(1,I) and S1(I); P l ( l . I ) i s the difference between desired value of the f i r s t row of the modal matrix and the assumed value divided by SX. S1(I) i s the difference between the desired inverse natural frequency and the I n i t i a l value. We note that the matrices PI, P2, etc. correspond to the matrices Ml» M2* e t c » o f Eq-» (5»3«3). Hence the f i r s t row of the matrices P 2» etc. are set to zero. ISN 0083 and 0084 places the reciprocals of the i n i t i a l eigenvalues i n the p o s i t i o n S ( I ) . Statements ISN 0085 to 0087 do the following; the in t e r p o l a t i o n parameter, XC, i s set to zero; the number of cycles, N2, i s set to zero; IL, the function of which w i l l be explained below, i s set to zero. ISN 0088 through 0121 evaluate the Jacobian matrix, PD, and also f i n d the vector Y which corresponds to the right-hand side of Eq. (5.3.9). I f IP(7) i s po s i t i v e , the matrix PD i s printed out i n ISN 0125. The inverse of PD i s found i n ISN 0126 to ISN 0134 by c a l l i n g MINV from the IBM S c i e n t i f i c Subroutine Package. If IP(6) i s po s i t i v e , the vector Y i s printed i n ISN 0136. In ISN 0137 the subroutine EVALI i s ca l l e d and the unknown elements of PI are found. 110 I f IP(1) i s p o s i t i v e . PI i s printed out i n ISN 0140. Statements ISN 0 l 4 l through 0 1 5 9 are f o r evaluating the right-hand side of Eq. ( 5 . 3 . 1 0 ) which are now stored i n Y. P2, the second order correction matrix, i s evaluated by c a l l i n g EVALI i n ISN 0 1 6 0 . P2 i s printed out i f IP(2) i s po s i t i v e i n ISN O I 6 3 . Statements ISN 0 1 6 4 through ISN 0 2 3 2 , which calculate P3» P4, and P5» are e s s e n t i a l l y a r e p e t i t i o n of the statements used i n f i n d i n g PI and P2. ISN 0 2 3 3 to ISN 0243 f i n d the element i n P5 with the largest absolute value and the element i n P with the largest absolute value. These quantities are stored i n X 5 M and XM respectively. ISN 0244 then computes the r a t i o X;5M/XM and stores l t i n XM. The f i f t h root of El/XM Is found i n ISN 0245 and stored i n X 5 . The quantity X5/SX i s the same as C l n Eq. ( 5 . 3 . 3 ) . In ISN 0246 the old value of XC i s stored i n XP. ISN 0247 finds the new value of XC. We note that X 5 was calculated so that X5M/XM equals E l . This value of X 5 i s then used i n ISN 0247. However, t h i s value may be so large that XC becomes greater than unity which means we have passed by our desired network. ISN 0248 tests f o r t h i s , and i f XC i s greater than or equal to unity, control Is transferred to ISN 0249 where X 5 i s recalculated so that XC becomes unity. XC i s then set to unity i n ISN 0 2 5 0 . I f XC i s less than unity, control i s trans-ferred to ISN 0 2 5 1 . ISN 0 2 5 1 through ISN 0 2 5 4 now finds the new values f o r P and S. Note that quantity X5 i s not divided by SX i n these statements since t h i s has, i n e f f e c t , already":been done i n I l l c a l c u l a t i n g P1(I) and S1(I) previously. If IP(8) i s p o s i t i v e , the new P i s printed out l n ISN 0257. In ISN 0258, N2 i s incremented by one. ISN 0259 p r i n t s XC, the in t e r p o l a t i o n parameter, N2, the number of cycles, and DET, the Jacobian. In ISN 0261 the change l n XC from the previous cycles " i s compared with Ek, I f the change i s greater than E4, control i s transferred back to ISN 0266 where XC i s compared with unity. If XC Is s t i l l l e s s than unity, control i s transferred back to ISN 0088 where another step Is executed. I f XC equals unity, control Is transferred to ISN 0267• I f the change i s less than E4, control i s transferred to ISN 0262. The message TOO MANY CYCLES REQUIRED i s then printed out i n ISN 0263. IL i s set to minus one i n ISN 0264. Control i s now transferred to ISN 0267. Statements ISN 0267 through ISN 0283 calculate the element values and the network i s again analyzed using subroutine ANAL. If IL i s non-negative, which means that the change i n XC i s greater than E4, control i s transferred to ISN 0285. ISN 0285 to ISN 0288 calculate the square root of the sum of the squares of the differences between the f i n a l s p e c i f i c a t i o n parameters and the desired ones. This i s XI. Next, i n ISN 0289. XI i s compared with E3. I f XI i s l e s s than, or equal to, E3, control i s trans-ferred to ISN 0294. I f XI i s greater than E3, a correction cycle must be executed. On the correction cycle E l Is replaced by E2 i n ISN 0290. E2 i s usually smaller than E l to ensure that more accuracy i s achieved on the correction cycle. Next the desired 112 values of the natural frequencies, which were stored i n SD(I) previously, are now read into S(I) i n ISN 0292. Control i s then transferred hack to ISN 0076 where a correction cycle Is executed. I f IL i s negative, the change i n XC i s less than E4, and so control i s transferred d i r e c t l y to ISN 0294. From ISN 0294 to the end of the main program, the f i n a l network element values, modal matrix, natural frequencies, and residues are printed out. We note that i f ISN 0294 has been reached because IL was negative, the f i n a l network element values, etc. w i l l not be the correct ones as t h i s constitutes a f a i l u r e of the numerical procedure. Al - 2 The Subroutine ANAL. This subroutine finds the eigenvalues and the modal matrix of the network i n the following manner. The admittance matrix Y has the form Y = sC + G, (Al.2.1) where C = A^eA, (Al.2.2) and G = B^eB, (Al.2.3) are the parameter matrices. A and B are the transposed incidence matrices of the capacitance tree and resistance tree respectively. The matrices Cg and G e are diagonal and contain the element values as diagonal elements. Statements ISN 0004 through ISN 0010 f i n d the matrix G i n Eq. (Al . 2 .3 ) . We then f i n d a vector SESQ(I) which contains the re c i p r o c a l square roots of the elements of CE(I). This i s done i n ISN 0011 and 0012. 113 The inverse of A, which i s AI, i s now multiplied by SESQ to obtain HI and i s performed i n ISN 0013 through 0015. The matrix G i s then premultiplied by Hl^ and post-m u l t i p l i e d by HI to obtain H. Thus H = Hl^GHl. (Al.2.4) This Is done i n ISN 0016 through 0029. The eigenvalues and eigenvectors of H are found using the subroutine EIGEN available from the IBM S c i e n t i f i c Subroutine Package. This subroutine calculates the eigenvalues and eigenvectors of any r e a l symmetric matrix H of dimension M. The eigenvectors appear i n the matrix Q, and the eigenvalues appear i n D(I). The statements ISN 0042 through 0054 arrange the eigen-values i n increasing order of magnitude, and arrange the correspond-ing eigenvectors i n Q accordingly. The modal matrix P Is now calculated using P = H1Q (Al.2.5) i n ISN 0055 through 0061. The statements ISN 0062 through 0066 arrange P so that the f i r s t element i n each column i s p o s i t i v e . This i s done so as to prevent the necessity of residues passing through zero as the network i s perturbed from the i n i t i a l s t a r t i n g point to the f i n a l one. Statements ISN OO67 through to the end normalize the element values, the natural frequencies, and the modal matrix, so that the smallest natural frequency and the corresponding residue i s unity i f NORMAL i s p o s i t i v e . 114 Al-3 The Subroutine EVALI This subroutine calculates the unknown elements U of each correction matrix PI, P2, etc. This i s done i n ISN 0004 through ISN 000?. The elements U(I) so calculated are then placed i n the l a s t M-l row of PG which corresponds to any one of the correction matrices PI, P2, etc. l n the main program. This i s done i n ISN 0008 through ISN 0011. APPENDIX I I FORTHAN STATEMENT L I S T A F o r t r a n s t a t e m e n t l i s t i n g o f t h e m a i n p r o g r a m a n d s u b r o u t i n e s i s g i v e n o n t h e f o l l o w i n g p a g e s . IV G LEVEL 1, MOD 0 MAIN DATE =' 68122 09 /58 /25 PAGE 000] C ' RC NETWORK SYNTHESIS BY USING PERTURBATION OF SPEC PARAMETERS J . . . . . . DIMENSION A (5-, 5) , B ( 5 , 5 ) , C E ( 5 ) , G E { 5 ) , R ( 5 ) , S ( 5 ) , D { 5 ) ,P ( 5, 5 ) . . „ • 1 ,P1 (5 ,5 ) , P 2 ( 5 , 5 ) , IP{10) , P 3 ( 5 , 5 ) , P 4 ( 5 , 5 ) , P 5 ( 5 , 5 ), S l ( 5 ) » "" 2Y(20) , G ( 5 , 5 ) , H ( 5 , 5 ) , G I ( 5 , 5 ) , G 2 ( 5 , 5 1 , G 3 ( 5 , 5 ) , G 5 ( 5 , 5 ) , H 1 ( 5 , 5 ) , 3 H 2 ( 5 , 5 ) , H 3 ( 5 , 5 ) , H 4 ( 5 , 5 ) , H 5 ( 5 , 5) fPD ( 2 1 , 2 1),Q ( 5 , 5),V ( 5 , 5 ) * : ftGi.J_5.»_5J_} RI (5) . SOC 5} .AI ( 21 ,21 ) .WIJ4QQ) . ; 5W2(400),PV(400) COMMON.M,N,PD 6 FORMATi5F15.7) 120 READ V, ( IP( I ) , 1=1 , 8) 4 "'" FORMAT (312) READ 1,M,NORMAL,SX,El,E2,E3,E4 1 FORMAT(212,5F10.0) 258 FQ R M A TJ_1_H IJ : _ PRINT. 25 8 254 FORMAT( / , 10X,40HMAX RELATIVE ERROR IN MODAL MATRIX #E 1#F 15. 8, / ,' 110X,60HMAX RELATIVE ERROR IN MODAL MATRIX ON CORRECTION CYCLE#E2 2 # F 1 5 . 8 , / , I 0 X , 2 iHSPICIFICATI ON ERR0R#E3#F15.8,/, 3i0X,25HMiN FRACTIONAL STEP SI ZE'#F 15. 8 ,/) " ~ PRINT 2 5 4 , E 1 , E 2 , E 3 , E 4 ' ' PRINT 10,SX 1.0 FORMAT ( / , IPX, 21HASSUMED NO. Or ST£PS#F15.8) _ _ 3 READ 5 , < A( I , J ) , J = l ,M ) 5 FORMAT(5F10.0) 00 7 1=1.M 7 READ 5 , ([J( I , J ) , J = l , M ) READ 6 ,<CE< 1),1=1,M) READ 6, , ( GE { I ) ,1=1,M ) PR INT 9 DO 310 I = 17* 310 PRINT 49, I , C E M ) , T ,GE( I ) READ 6 , ( R ( I ) , 1 = 1 , MJ READ 6 , (S( I ) ,1=1,M) N=M*(M-l) ' M1=M-1. M2=2*M PRINT 11 11 F U P A f l / , 10X .42HTRATJ5PUSED I NT IDEN'CE RAflTlX FQT<~Z~GkirPH IS, / T DO 13 1 = 1',M 13 PRINT 3 9 , I A ( I , J ) , J = l , M ) 39 FORMAT(10X,5F15.8) PR INT " 1 5 " "• " " " ' ~ ' " " ~ ~ ~ ~ " H M O N IKIRAN IV G LEVEL 1, MOO 0 MAIN DATE = 68122 09/58/25 PAGE 000 035 15 FORMAT!/, 10X, 42HTRANSPOSED INCIDENCE MATRIX FOR G GRAPH I S , / ) 036 . .. ; DO 17 1 = 1, M • . . .. i037 17 PRINT 39,(B(I» J ) , J = 1» M) 038 PRINT 19 '039 19 FORMAT(/, 10X,46HREQUI RED RESIDUE AND NATURAL FREQUENCY VECTORS,/) JJAQ Q0__2„.L__I^-UM : : 104 1 21 PRINT 2 3, 1,R( I) , I,S( I ) 042 23 FORMAT(10X,2HR2I2 ,2H<#F15.8,10X,2HSZI 2,2H<#F15.8) 0 43 DO 2 5 1 = 1 ,M '044 R( I)=SQRT(R( I ) ) . . . \ .... .. „.; 0 45 R 1 ( I ) = R ( I ) 046 SO(I ) = S( 1 ) 047 25 S ( I ) = 1 . / S ( I ) DAS U£LJKKI_L_L_!! : 049 DO 3 00 J=1,M 050 300 A I ( I , J ) = A ( I , J ) 051 DO 323 1 = 1,M • 0 5 2 . 1 ... . DO 323 J = 1,M . .• _ . . _ _ _ _ _ _ 053 C A L L LOG ( I , J , IJ,M,M,0) " ' " ~ " 0 54 32 3 P V ( I J ) = A J{ I , J ) 055 C A L L M I N V ( P V,M,DET,W1,W2) 0_5_6 DO 325 I=1,M ; 057 .DO 325 J - 1 , M 053 C A L L L O C ( ] , J , I J , M , M , 0 ) 059 325 AI( I , J ) = PV( I J ) 060 _ _ C A L L A N A U A I , 8, CE,GE,D,P, NORMAL) _ 061 PRINT 2 -062 2 FORMAT!/, 10X , 41HN0RMAL.I ZED I N I T I A L NETWORK ELEMENT VALUES,/) 063 9 'FORMAT(/, 10 X , 30 H I N I TIAL NETWORK ELEMENT VALUES,/) 0_64 DO 5 1 i = l,M : 065 51 PRINT 4 9 , 1 ,CE( I ) , I , G E ( I ) 066 49 FaRMAT(10X,2HCXI2,2H<.#F15.8,10Xt2HG£I2-,2H<#F15.8) 06 7 PRINT 2 9 06 8 29. FORMAT (/ , 10 X , 34HE l.GENVALUES OF ..IN I J.I .A L... NETWORK ARE) _ 069 00 3 1 1=1,M 0 70 31 PRINT 3 3 , 1 , 0 ( 1 ) 071. 33 FORMAT! 1 OX , 2 HDSI 2 , 2H<#F 1 5 . 8) 07.2 PRINT 35 ; : "073 35 FORMAT ( 10X , 35HEIGENVECTQRS OF I N I T I A L NETWORK A R E ) 074 DO 37 1=1,M 075 37 PRINT 39 , ( P ( I , J ) , J = l , M ) 076 _.. 24.6 DO.. 82 ,.I<=1 ,.M _ ; . . „ 077 P 2 ( 1 , I ) = 0 . H 3RTAAN IV G LEVEL 1, MOD 0 MAIN DATE * 68122 09/58/25" PAGE OOOJ D073 0079 3080 3081 !)082 3083 3084" 3065 3086 3087 3088 3089 3090 3091. 3wr 3093 3094 3095 3096 3097 3098 3099 82 P3(1 ,1 )=0 . P 4 ( 1 , I ) = 0 . P 5 ( 1 , I ) = 0 . P 1 ( 1 , I ) = < R U ) - P < 1 , I ) ) / S X sn i ) = i s m - i . / D ( n >/sx DO 200 I - 1 , M n u c r 3101 3102 3103 3104 3105 3106 31(37 )l<JrV 3109 )1 10 n n )112 U 13 n i b )116 »117 )118 J119 200 93 14 s u r = i . / D i i ) XC=0. N2=0 IL=1 DU 14 ,1=1,M DO 14 J=1,M G(I .J)=0. H( I , JJ =0 G( I , J ) = G( I , J ) + A ( I , K ) * P ( K , J ) H ( I , J }=H ( I » J ) + 0 ( 1 , K ) * P ( K , J ) DO 100 I=1,M DO 100 J = l , M 0 ( I , J ) = 0 . VI I ,J l=0 . DO LOO K = b M QTT,"J1 = 0( i ; i ) + G ( I ,K)*G( J . K ) 100 V ( I , J ) = V ( I , J ) + H {ItK)*S(KI*H<J,K) 210 K 1=I K2=2 00 18 1=1,Ml IR=I+1 DO 18 J=IR,M DQ 20.K=1.M T F T K - l ) 22 ,22 ,24 Y(K1)=0. Y(K2)=0. DO 26 L=l ,M Y ( K l ) = Y ( K l ) - ( A ( I , l ) * G { J , L ) + A < J , 1 ) * G ( I , L ) ) * P 1 ( 1 , L > Y(K2 > = Y { K 2 ) - < B { I , 1 ) * H t J , L ) + B < J , l ) * H ( I , L J ) * S { L ) * P 1 < 1 , U 1 H ( I , L ) * S 1 ( L ) * H ( J , L ) GO TO 2 i l 22 26 T4 (30 30 L = l 7 f i ' Ll=(K-2)*M+l P D ( K 1 , L I ) = A ( I , K ) * G ( J , L ) + A < J , K ) * G ( I , L ) 30 P 0 ( K 2 , L 1 ) = ( B ( I , K ) * H ( J , L ) + B ( J , K ) * H ( I , U ) * S U ) 20 CONTINUE co R TKAN IV G LEVEL 1. MOD 0 MAIN DATE = 68122 0 9 / 5 8 / 2 5 PAGE 0 0 0 120 K l = K l + 2 ' 121 18 K2=K2+2 122 160 F0 RM AT(10X, 10F15.8) 12 3 I F { I P ( 7 )) 154,154,156 124 156 DO 158 I = l . N I 2 4 1 5R PRINT 1 6 0 , ( P D ( I f . l ) . J = l ,N» • • 126 154 DO .3 2 7 1= 1 , N 127 DO 327 J=.1,N 128 CALL LOG( I , J , IJ.N,N,0) L29 ... 32 7. . PV( I J ) = P D ( I , J ) . . . . _ _ 130 CALL M I N V ( P V , N t D E T f W l , W 2 ) 131 DO 3 29 1=1,N 132 DO 329 J=1,N 1 3 3 CALL LOG( I , J , I J . N . N . O ) 134 329 P 0 ( I , J ) = P V ( I J ) 135 I F ( J P ( 6 ) J 2 0 2 ,202,204 136 204 PRINT 1 6 0 , ( Y ( I ) , 1 = 1 , N ) 137 .20 2 CALL EV A L K Y , PI.) , . 13 6. I F { I P ( 1 ) ) 164,164,166 139 L66 DLT 162 I=l..M 140 162. PR I NT 1.60 , ( PI ( I , J ) , J=l,M) 1 4 1 164 oo 36 i = i , M 142 DU 36 J - l ,M 143 G 1 ( I , J ) = 0. 144 HI ( I , J ) = 0. • 145 DO 36 K=1,M 146 G1 ( I , J ) = G 1 ( IV J")+ A ( I , K )'* P1 (K, J ) 147 3 6 H 1 ( I , J ) = H 1 { I , J ) + 8 ( I , K ) * P 1 ( K , J ) 143 K l = l L49 K2 = 2 150 DO 3 8 1=1,Ml 151 IR-I+1 152 DO 3 6 J=IR,M 153 Y( K l J - 0 . _ . ' ._ .... 154 Y ( K 2 ) = 0 . 155 DO 4 0 K =I,M 156 Y ( K 1 ) = Y ( K 1 ) - G l i I , K ) * G 1 U , K ) 157 40 Y ( K 2 ) = Y ( K 2 ) - H 1 ( I , K ) * S ( K ) * H 1 ( J . K l - H K I , K ) * S 1 ( K ) * H < J . K ) L - H K J , K ) * S 1 ( K ) * H ( I , K ) 158 K l = K l + 2 1 59 38 K2=K2+2 160 .. . . _CALL. EVAL1..(.Y,.P2)_ 161 I F ( I P ( 2 ) > 4 2 , 4 2 , 1 7 0 H o o o L U o < QL CM "s. 00 m r> o fM (NJ -I ao LU < < o — LU > -J IS > II CM QL O r - W O O NO II II II II I—in UN •> OCCCfMCM OfNI r»r-fNi CM CM + + •CM CM -<XO ll II |l >£—-•» ~ n • • ^ H H * — < r — I O C M C M ^ UN * a: It II C M C c C O -> x * 5^ 00 •* ->-» *-t •> x— *~ *X —a-->— •> t-^ * O — fM •M-—X — f M * V.X — "I * w ^ r - l fM » V ) 0 - s * — C M : * : « -it i—i - > ~ w —^x fM— I •«• •«• ^ ^^-> • » * w MMl — w # 1— «• 4 0^ OX* I I — - E — - 1 • ~4fMl/> • . n • . 0 0 ^ > > - x : f M f M II II II II • + + —»«-». O — «-• «-J fM rHtMin—trM—v:^: i^ bcr vr^ fM ti II . — o — . x - i >>o>->- i i c : : * : a: fM m >j- UN so x o o * H fM ro --j- UN O O O Q O f J Q O n Q O O O Q o m Q0 •tf - o r ^ - c o o © -jf\jirn f - K r » - f " - x o c c c o -rnr\J —> 0 .UN w >-CM -CL — U N - ^ X J E : II • • • • t J~HCHHOO <j II | ! II II lij>-*fv. C t - i i — -o-o • • _ j > — zmir . M i - ' <JLOf * O a r <" !Cc ' w « aooox tv«x -O00 ,„ r-r-<M j-m-or-co^c x x c o x x x x o • w w mm * * * : * <co + + •mm H O I II II II -5 - J *0 • mt-« <-*t-> II Omm—« • O UN n n - J O X + X CM UN M UN II II M C 0 C F ^ r v r o ^ j - L n s o r ^ x c> c?* o c*-c* o o o o o c; o n n i n o n r i c - i l ' T - i o , ' CM ^-5 —x <M# w — I ^ ^ w *</> ** w i C o>-< # — -1 — ^ o-> I »»CN 5<:x — ^ : * * o x It •r-l CM . . || w w II II II II o •"'*'"** r-ifM»0—'fM o C^Or -n fMrn o o o o CM fM CM CM crr> o o UJ < CL fM ao o < < a o 5 : > < _ _ a _ X on i * ! CM - j • j ao - >~< ! v O — i m i I — I — « *K -)r~>. — X ro-H-i I — « • _ — 1/) r - H # , 0 0 — ' V 1- 1 fM — X X 1 I — —»_i — ---3 -3 — — rM CM X X». — _ _ — — _ 4 •£{• o . _ II ~5 0 0 CO —• T— r—r » « a o »- v >-CM 5 : 0 . i —1 " + 1 __CM » • + — 1—<.—1 ~ _ ^ CM—I II X X — 1 I I * r - l CM 3 H " C M > N T - '~T H I LL O CM OLC S__J — • ! — * 1 -j CM< ac - to. o o c o o CM CM CM C\ f—'.r-l O C I—" C II II II II II II —>~> ~ : . O D O I O C * ^ c_> r—4 o 1 r*^ -f\JC\ir\ j rMCMCM!\!(\ — II - t - i X ^ ~4 r—l II II • • — - } Q O - I II II a + C D — — CM v C —• v C H r\j 11 11 CMC-oca—«--o CM I r» r>_» rr> r"N r"> . 1 1.-«-. C> O •—* *-* — 1 — - ) » — « CV CNJ C\i r\i! cv t\j c\f r\i ryj t\j<\j O O O o ! o o a _ _ r _ r _ n r ^ n r i n r ~ . r i n — x x— . .* ^ -00 •oO—« —>-* _C —#. x —• r-l CO — — X — X I * _ -— r-~3 _ - ) — ^—m — C M X —x# « t - M - — X - ~ _ I _ — 0 - 5 1 — _— —«* » . •» _ X — . - * — — —3 . - .or ir - l — _ X X -4- — I I — O 00 — — _ # «• _ _ • — • ~ - - ~ > _ _-}-3 — ^ _ - ^ - r M —"-»>—irn X - - - I I * — 1 < « • ^ — CJ5~5X— I — I _ _ — O CNJ t/! r-H 00 - I _c: •>'.- _ i w> * k>-i_>-_^-~_—' _ 11 • 11 • + ----1 — C M — •— — itC • „ r O _ C M — f M _ ! — O — I ^ X r - r _ >- I >-. I X I _ r H r—IfMCO o r-CM •tn' C u rfM Q.r-1-J — II J t P l r -- C . II CM CM IfMfM CM CM CM < CM> + LU CM __J II —J c — — r - O — — — O t / ) r _ ! v t a . C M — a D < II LOi_, X C i u . a r - O 0 0 ; I— CO" ^ < — ' l| ac _ i O . X : CJsOfM CO r-Uj<Qi o C M C M co rr. fTi m r o d CM CM CM CM CM O J (Nl fM -*-JCM 1 — tr,-» -3a.r_.00 — — x : -4-0000 1! CD CC _ O0 —|<<XX II II — II 000_ L L : 5_ C X X r - X O O c o r o r o r o CM CM 0 J rM CM OUTRAN IV G LfcVEL 1, MOO 0 MAIN DATE « 68122 0 9 / 5 8 / 2 5 PAGE 000 0241 0242 0243 0244 0245 0_2A6 0247" 0248 0240 0250 9251 0252 0253 Q2_54_ 0255 32 56 3257 3258 3259 3260 32 61 >263 3264 3265 3266 3267 3268 1269 303 IF(X5M-XG) 304,104,104 304 X5M=XG 104 CONTINUE XM=X5M/XM X5=(E1/XM)**0.2 XP=XC_ 312 314 313 309 XC=XC+X5/SX 1F(XC-XP-E4) 312,312,313 PRINT 314 FORMATt/,10X,24HTOO MANY C Y C L E S R E Q U I R E D , / ) Il=-1 GO TO 91 IFIXC-1.) 308,309,309 X5=(i,-XP)*SX 308 306 331 T5T XC= 1 00 306 1 = 1,M S( I )=S1(I)*X5 + S(I) DO 306 J=l , lY P(I» J ) - P I I » J)+P1(I , J ) * X 5 + P 2 { I , J ) * X 5 * * 2 + P 3 ( i , J ) * X 5 * * 3 + P 4 ( I , J ) * X 5 * * 14+P5(I,J)*X5**5 IF(IP<8}} 330 ,330,331 DO 332 I=1,M PRINT 39,(P< l , J ) , J = 1 , M J 330 N2=N2+1 P R I N T 3 0 7 , X C , N 2 t O E T 307 F 0 R M A T ( / , 1 0 X , 3 H X C # F 1 5 . 8 , 1 0 X , 3 H N 2 # I 5 , 1 0 X , 4 H D E T # F 1 5 . 8 , / ) I F(XC - 1 . ) 9 3 , 9 1 , 9 1 91 DO 90 1=1,M DO 90 J=1,M G( I, J ) = 0 . : ; T2TTJ-3271 3272 3273 )274 )275 3276 J277 TZTW 32 79 J2 80 5281 3282 FT( I, J J =0 . DO 90 K=1,M G ( I ,J)=G( I , J ) + A ( I , K ) * P < K , J ) 90 H I I , J ) = H ( I , J » + B ( I i K ) * P ( K , J I DO 92 1=1,M CE ( I)=0. Gt ( I)=0. 00 92 J=1,M C E ( I )=CErTT+G( It J ) * * 2 92 G E ( I ) = G E < I ) + H I I , J ) * * Z * S ( J ) DO 109 1=1,M C E { I ) = L . / C E ( I ) 109 G E { I)=1 . / G E ( I ) OR TRAM IV G LEVEL 1, MUD 0 MAIN DATE = 68122 09/58/25 PAGE 00< 0283 CALL ANAL ( A I , B ,CE ,-GE *D , P , NORMAL 1 02 64. IF ( I D 185,32 1,321 : 0285 321 X1=0. 02 86 DO 240 .1 = 1, M 0287 240 X1=X1 + (P ( I , I ) - R l { I ) )'**2 + (D( I ) - S 0 ( I ) 1**2 _2__8_ X1=SQRT(X1) 0289 IF ( X 1 - E 3 ) 185, 185,244 0290 244 E1 = E2 0291 250 00 243 1=1,M 0292 __248 ; . S( I) = l./.S0( I ) .„ 02 93 GO TO 246 0294 . 18 5 PRINT 187 0295 187 FORMAT!/,5X,15HMODAL MATRIX I S , / ) 0_2_96 D0__1 89 I =_1_,_M 0297 L89 PRINT 39, (P ( I , J) ,J=l,M) 02 9 8 PRINT 110 0299 ' 110 FORM AT{/, 10X,28HF INAL NETWORK ELEMENT VALUES,/) 0300 . _ 00 111 1 = 1,M . _ „_ •_ _ 0301 111 PRINT 4 9 , I ,CE! 1 ) , I , G E ( I ) 03 02 DO 113 1=1,M 0 3 0 3 R( I ) = P ( 1 , I ) * * 2 0_3 04 113 S ( I ) = D ( I ) 03 05 PRINT 115 0306 115 FORMAT!/,10X,38HFINAL RESIDUES AND NATURAL FREQUENCIES,/) 0307 DO 117 1=1,M 0308 _ 11 7 _ PRINT 2 3 , I , R ( I ) , I , S ( I ) . 03 09 315 GO TO 12 0 "" '. 0310 END D R l k A N IV G L E V E L It MOD 0 ANAL DATE * 68122 0 9 / 5 8 / 2 5 PAGE 000; 0001 0002 0003 CLCLOA. SUBROUTINE ANAL(A I , B , C E , G E , D , P , N O R M A L ) DIMENSION A I ( 2 1 , 2 1 ) , 8 ( 5 , 5 ) f C E ( 5 ) , G E ( 5 ) , 0 ( 5 ) , P ( 5 , 5 ) , G { 5 , 5 J , 1 S E S O ( 5 ) , H 1 ( 5 , 5 ) , H 2 ( 5 , 5 ) , H ( 1 0 , 1 0 ) , Q ( 1 0 , 1 0 ) , T Y ( 1 0 ) , A E I 1 5 ) , 2 B F ( 2 5 ) COMMON M , N , P D DO 9 1 = 1,M • ; : 0005 0006 0007 1)008 0009 0010 3011 2DAZ. 11 12-DO 9 J=1 ,M G ( I f J ) = 0 . DO I I 1=1,M DO 11 J = l , M DO 11 .K=1,M G ( I , J ) = G ( I , J ) + B ( K , I ) * G E ( K ) * B ( K , J ) DO 13 1=1,M S F S Q ( I ) = 1 . / S Q R T I C E M ) 1 3013 3014 3015 3016 3017 3018 3019 1025 302 1 30 22 302 3 3024 3025 3026 3027 1H2£L DO 15 1 = 1 ,M 00 15 J = l , M 15 H l ( I , J ) = A I ( I » J ) * S E S Q ( J ) 00 17 1=1,M DO 17 J=1 ,M 17 H 2 ( I f J ) = 0 . DO 19 I=1,M DO 19 J=JUM DO 19 K = l , M 19 H2( I , J ) = H2( I , J ) + G U , K ) * H 1 ( K , J ) DO 21 1=1,M DO 21 J = l , M 21 H( I , J ) = C. DO 23 1=1,M DO 23 J = l f M DO 2 3 K=1.M 23 H ( I , J)=H( I, J ) + H . l ( K , I ) * H 2 ( K , J ) DO 30 • 1 = 1,M DO 30 J - 1 , I C A L L LOC ( I, J , I J , M , M , 1) . .. .... 30 A F ( I J ) = H ( I , J ) C A L L E I G E N ( A E , B E , M , 0 ) DO 40 1=1,M XALI. L Q C U f l f U f H t M f l l 1029 3030 3031 3032 3033 30 34 3035 lQ3i»_ 3037 3038 3039 3040 3041 40 0 ( I ) = A E ( I J ) DO 50 I=1,M DO 50 J=1 ,M C A L L L O C ( I , J , I J , M , M , 0 ) 50 Q ( I , J ) = 8 E ( I J ) URAN IV G LEVEL 1 , MUD 0 DATE = 6 8 1 2 2 097~58/25 PAGE 0 0 0 2 1.6 7 >68 169 '70 7 1 n r '73 '74 75 76 •77 78 JL7_ 6 9 65 2 5 27 3 ^ 7 1 4 3 45 4 7 41 M2 = 00 11 = DO I F ( T E M M-6 5 I + 6 5 D ( P = D ! I D { J D O TY( Q ( K 0 I K C O N D O D O P ( I D O 00 D O P ( I D O f F{ TJfl P ( J C O N IF( no G E ( C E ( X 1= / — ) = 6 9 K ) , I , J T I 2.5_ 2 5 , J 27 27 2 7 , J L P ( 1 1=1 ,M2 1 J = J )-D J J . D( J T E M K = = Q( ) = Q ) = T N O E 1,M ( I ) ) 6 7 , 6 5 , 6 5 , I ) K , J <K) X2= D O D ( I D O P { I RET END 7 , I T I N O 4 5 I ) I ) D ( T ' T 4 7 ) = 4 7 t J UR J = )=0 1 = J = K = ) = P 1 = 1 J U i J = l ) = -M U E R M A r — = P ( = P( 1 ) T7T 1 = D( I J = ) =P N 1 , M ,M ,M ,M I » J M ) 3 •|sr ( J , ) 4 tM' , 1 ) , 1 ) )+Hl( I , K ) * 0 ( . K , J ) I ) 1 , 4 1 , 4 3 ... ... . **2* G E U 5 / D ( 1 ) **2*CE{I) ,M / X I ,M . . . I , J ) / X 2 GK TKAN IV G LfcVEL 1, MUD 0 EVALl DATE = 68122 09/58725 PAGE 00< 0001 SUBROUTINE EVALl (Y-,PG) 0 0 0 2 DIMENSION PD ( 21 »2 1 ) » U ( 20 J ».Y.(20 ) » PG { 5 , 5 ) 0 0 0 3 ' COMMON M,N,PD 00 04 DO 3 2 ' 1=1 ,N 0 0 0 5 . U(I ) = 0 . 00_0_6 _n__3 2__J=J__N ._ 0 0 0 7 32 U"( I ) =U ( I }+P0( I , J ) *Y( J ) 0 0 0 8 DO 34 1=1,M 00 0'.' DO 3 4 J=2tM 00 10 K=JJ-2)*.M+1 0 0 1 1 34 PG ( J » I ) = U ( K J " ' . " 0 0 1 2 RETURN 0013 END
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Realization of minimal two-element-kind one-port networks Mason, Lloyd Judson 1969
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Title | Realization of minimal two-element-kind one-port networks |
Creator |
Mason, Lloyd Judson |
Publisher | University of British Columbia |
Date Issued | 1969 |
Description | A new method of realizing two-element-kind driving-point Impedances is given and illustrated by examples. In this method, networks of any desired topology and having a minimum of elements are utilized. A transformation to normal coordinates forms the basis of the method and, in order to determine network element values, evaluation of the associated transformation matrix is necessary. This matrix is found by formulating and solving a set of multivariable polynomial equations of second degree. The solution to this set of polynomial equations is obtained by a numerical perturbation procedure. To initiate the procedure, a set of element values is chosen, and the network of specified topology is analysed. The corresponding transformation matrix and driving-point impedance are determined from this analysis. The impedance parameters are then perturbed by small amounts in the direction of the specified ones, and the resulting changes in the transformation matrix are calculated. The process is continued until the transformation matrix corresponding to the specified impedance is obtained. A detailed description of the computer program written to carry out the above procedure is Included. A large number of examples of various complexities, including some canonic structures, have been realized by the method. Examples show the superiority of the numerical method to conventional procedures for solving multivariable nonlinear equations. In particular, the choice of the initial set of element values is not required to be close to the final set to achieve convergence to a solution. Some restrictions on the realizability of irreducible complementary tree structures are reported. It is shown that the specification parameters may have local extrema at a point where the Jacobian of the system of polynomial equations vanishes. Examples which support these results are given. |
Subject |
Electric networks -- Computer programs |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2011-06-21 |
Provider | Vancouver : University of British Columbia Library |
Rights | For non-commercial purposes only, such as research, private study and education. Additional conditions apply, see Terms of Use https://open.library.ubc.ca/terms_of_use. |
DOI | 10.14288/1.0104057 |
URI | http://hdl.handle.net/2429/35606 |
Degree |
Doctor of Philosophy - PhD |
Program |
Electrical and Computer Engineering |
Affiliation |
Applied Science, Faculty of Electrical and Computer Engineering, Department of |
Degree Grantor | University of British Columbia |
Campus |
UBCV |
Scholarly Level | Graduate |
AggregatedSourceRepository | DSpace |
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