Y’HE LENEAR GRAPH IN SYSEM ANALYSIS Thesis for the Degree of DH. D. MEQHEGAN STE-E UNIVERSITY Wiiliam Alien Blackwell 1958 mum This is to certify that the thesis entitled THE LINEAR GRAPH IN SYSTEM ANALYSIS presented by William Allen Blackwell has been accepted towards fulfillment of the requirements for M degree in AL ‘A . ./-7 . . ’5 . _ _. x Q reA-thfljv‘ . ‘ I. 4-31 Major professor ' Date May 2) 1958 0-169 L I B R A R Y Michigan State University I“ w THE LINEAR GRAPH IN SYSTEM ANALYSIS BY William Allen Blackwell AN ABSTHNCT Submitted to the School for Advanced Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PHILOSOPHY Department of Electrical Engineering 1958 .1 / _/"_n_' . 'l ‘1 I; ‘ _ n / //’ f.“ .. C . ”fit. "/1": / ’. - - Approved ”3+. ,1 aa“¢:1..g . , J-zJJ. c 0 ABSTRACT None of the common systemranalysis techniques are com- pletely satisfactory for the analysis of mixed systems in general. Electromechanical analogies have been used with considerable success, but for complicated systems made up of multiterminal components, a more powerful formulation tech- nique is needed. In recent years, electrical network theory, using the notions of linear graph theory (studied formally under the mathematical designation of topology) has made significant advances in formulation techniques. In electrical network analysis, it has been found that networks, for which equa- tions are virtually impossible to formulate using the conven- tional from-the-diagram node and mesh techniques, can be treated--using a systematic and simple procedure--by distin- guishing between the equations characteristic of the compo- nents and those characteristic of the component connection pattern. The equations, which are characteristic of the con- nection pattern, are called circuit and segregate equations, and are written from a collection of oriented line segments, called a linear graph of the system. The linear graph is useful in the analysis of any system in which one set of measurements sums to zero around closed circuits, and/or one set of measurements at points, areas or -1- regions, sums to zero. With a proper understanding of its role in the analysis of systems, the linear graph can be ob- tained for a particular system by an orderly, logical pro- cedure. The general pattern of formulation of equations, used in electrical network theory, can be extended to the analysis of mixed systems if mathematical forms, different from.those encountered in formal electrical network theory, are admitted for the equations characteristic of the components. When these mathematical forms are used, questions relative to rank of equations arise in a manner not treated in electrical net- work theory . The problem considered is defined in Chapter I. In Chapter II convenient terms are defined, and the background is set, with reSpect to terminology and concepts, for consid- eration of the system.formu1ation problem, In Chapter III an examination is made of the conditions on topological placement of the various component types con— sidered, such that a unique solution to the system equations is possible. A set of general procedures for the systematic reduction of equations to be solved simultaneously is presented in Chap- ter IV. In all cases, the equation-reduction procedures do not involve taking a matrix inverse--depending instead upon explicit forms of equations. A set of procedures for systems containing specific -3- component types is given in Chapters V and VI. The number of equations to be solved simultaneously for each situation is noted. In.many of these procedures, the number of equa- tions to be solved as a simultaneous set--under certain topo- logical arrangements of particular components--is less than would be the case with conventional mesh or node formulation. THE LINEAR GRAPH IN SYSTEH ANALYSIS By dilliam Allen Blackwell A THESIS Submitted to the School for Advanced Graduate Studies of Michigan State University of Agriculture and Applied Science in partial fulfillment of the requirements for the degree of DOCTOR OF PnlLUSUPRY Department of Electrical Engineering 1958 ACKNOWLEDGMENTS The author wishes to express his thanks to Dr. H. E. Koenig for his guidance in preparing this thesis. He is also greatly indebted to Dr. M. B. Read, on whose fundamen- tal work in electrical network analysis, much of this the- sis is based. ii CONTENTS A CKIQ OLJ LEDGMN If S O O O O O O O O O O O O O O O O O 0 LIST OF SYIJIBOLS O O O O O C O I O O O O O O O O O I. II. III. IV. V. VI. VII. INT mDUCTIOII . O O I O O C O O O O O 0 O O O EQUATIONS OF PHYSICAL SYSTEMS . . . . . . . INDEPENDENCE CRITERIA . . . . . . . . . . . GENERAL PROCEDURE FOR FORMULATION . . . . . FOWIULATION FOR SYSTEMS OF PARTICULAR COMPONENTS................. SYSTEM GRAPHS FOR WHICH EACH PART INVOLVES EITHER ACROSS OR THROUGH VARIABLES ONLY . . CONCLUSION . c o o o o o o o o .o o o o o o o LISTOFREFERENCES................. iii ii iv 16 36 1+6 75 81 83 LIST OF SYMBOLS A list of symbols, which are used repeatedly, is given below. Symbol e V P Number of Number of Number of Number of Number of Number of Number of Number of component Number of component Variables Variables Variables Variables Description elements of a graph. vertices of a graph. separate parts of a graph. J terminal equations. K terminal equations. specified x-variable equations. Specified y-variable equations. x-variables not related in the equations. y-variables not related in the equations. related by A terminal equations. related by B terminal equations. related by J terminal equations. related by K terminal equations. x-variable specified function. y-variable specified function. iv 0‘ 3 Description x-variables not related in the component equations. y-variables not related in the component equations. Tree variables, either the unknowns or the complete set, depending on context. Chord variables, either the unknowns or the complete set, depending on context. Tree variables not classified by some other criteria. Chord variables not classified by some other criteria. I. INTRODUCTION Historically, lagrange's eouations have served as the basis for formulating equations for tce analysis of necnani- cal svstens, Electrical network t eory, using he notions of linear graph theorv, developed along corpletely different lines, In electromechanical-system analysis and electronic~ system analysis, signal-flow graphs [1,2] and block diagrams [3,4,5] have been utilized for formulation, Each technique is successful on a certain class of systems, No one technique is completely satisfactory in general for systems which con- tain subsystems of different types, There are certain limitations in the Lagrangian form- ulation, which make it unsatisfactory for formulating equa— tions of complex systems [6], Firestone [7,8] and others recognized some years ago that the form of the equations, descriptive of a mechanical systet, is identical to that used in electrical network theory. It was also recognized that the network-theory technique had some definite advan- tages over the Lagrangian technique for certain types of mechanical systems, Electrical analoEs of mechanical systems have been used in an attempt to exploit these advantages, It is possible to define the techniques of analysis used in electrical network theory in a manner such that they are equally applicable to systems of other types--mechanical, -1- [U thermal, hydraulic, etc, [9,10,11]. Trent [12] has written of the usefulness of linear graph theory, and introduced the notion that the convention— al node and mesh formulation procedures can be extended to include "perfect couplers"--components of the "ideal trans- former" type, I V The objective of this thesis is to extend the work of Trent with respect to the analysis of systems which include "direct couplers," In following this objective, the work of Reed [13,14,15,16] in electrical network theory is used and extended, and the work of Koenig [9,10,11], in establishing foundations of system analysis, is used and extended, When components of the type called by Trent "perfect couplers" and "direct couplers" are included in a system, certain questions as to rank of equations arise, These ques- tions have not occurred in electrical network theory, and hence have not been formally investigated until now, Chap- ter III is devoted to investigating conditions for which a uniqtua solution to the equations of a system is possible, «Sertain procedures in the writing of the equations, for a Syst£nr containing "direct couplers", are very helpful in eStablishing a systematic formulation technique, Extensions 0f Reeci's work in electrical network formulation [13] are re- quired. This is the subject of Chapter IV. 'Phe application of the general principles outlined in Chapter‘ IV is carried out in Chapters V and V1 for particular II. EQUATILNS OF PHYSICAL sysrers 2,1 Introduction The oriented linear graph, as used in system analysis, could be said to serve as a rack Upon which is placed infor- mation relative to two-point observations made in the system, In electrical network theory, as expounded by Reed [13], two variables, v(t) and i(t), are associated with each element of the graph for the electrical system, These variables, as treated in [14,15], are postulated to satisfy the circuit and segregate equations of the graph, and are assumed to corre- late with the instantaneous voltage and current measurements, reapectively, associated with a pair of designated observa- tion points, In order to use the linear graph for a nore general System analysis, it is desirable to associate with each ele- ment of'the graph of the system a pair of variables, not car- Ihdlug the connotation of voltage and current, Lkgfinition 2,1,1: Across variable: The element vari- able, Munich is associated with an across measurement, is called the across variable. Ifius across variable of a graph eleaent is also desig- nated 8&3 the x-variable, and is postulated to satisfy the circuit enquations of the linear graph, -3- -4- Definition 2,1,2: Series variable: The element vari- able, whieh is associated with a series measurement, is called the series variable, The series variable of a graph element is also called the y-variable, and is postulated to satisfy the segregate eouations of the linear graph, A discussion of across and series measureaents for various physical systems, and tech— niques for relating them to the variables f the graph, is included in [9,11,12], The definitions and theorems of linear graph theory, which are used in this thesis without Specific reference, are taken from Reed [13,16], Reed and Reed [15], and Reed and Seshu [14], 2,2 Components Essential to the application of linear granh theory to the formulation of system eouations is the notion of compo- nents, In electrical network theory, the "building block" is the two—terminal component, Devices with two terminals accessible for measurement (resistors, coils, etc,) are rep- resented in the network graph by one element, However, the need for a graphical representation for larger subnetworks has resulted in such "enuivalent circuits" as Thevenin's and Norton's two-terminal equivalent circuits, and the "tee" and "pi" three-tersinal equivalent circuits, There is no cues- tiOn as to the usefulness of these equivalents, Based on ex- perience, it can be reasonably assumed that any electrical subsystem, which is connected to the remainder of the system at two points, can be represented in a graph of the system by one element, Further, it seems apparent from developments in [10], that any subsystem, which has three terminals in couaon with its complement in the system, can be represented in the system graph by two elements, which are connected, and which do not form a circuit, This notion can be extended to a subsystem with n ter- minals connected to the remaining systei, A rigorous mathe— aatical hovel pnent to Specify the necessary and sufficient conditions for a su table subgragh for an n-terninal coupo- nent is, as yet, lacking, The following postulate, based on experience, was suggested by Koenig [10], It has proved use- ful, and in a wide variety of problems, a contradiction has not been found, rostulate: A subsysten of a physical system, which is connected at n points to its couplement in the system, and which involves p different zinds of measurements (electrical, hydraulic, rotational mechanical, etc.) can be represented in the system graph by some graph G of p parts, such that the Subgraph of G in each part is connected, and contains no cir- cuits, The graph 3 thus contains for an n-teraihal 00mTUHGNt with p different kinds of neasurementS, (n ' P) elements and n vertices, The te'sinology "can be represented in the sys— tem graph by", used in the postulate, is intended to imply that an analysis of a system using a linear g aph, which is made up of suhirapns of the type described for the subsystems, yields a solution which correlate with physical measurements made on the system, Definition 2,2,1: Component subgraph: A graph for a component, Which is sufficient to represent the component in a graph for tu: system, is defined to be a component subgraph, .L The graph G, described in the postulate, is thus a special form of component subgraph, In a physical system the components of the system are connected in some manner, The points of connection are log— ically the observztion points, or terminals, of the compo- nents, Suppose a physical system to be composed of an arbi- o, trary set of subsystems, lor each of which, the couponent subgrayh is known, Definition 2,2,2: System graph: A collection of com- oonent subgraphs, such that the vertites common to two or more subgraphs correspond to coincidence of observation points of correSponding subsystems in the physical system, is called a system graph, The equations for a system fall into two classes: (1) those equations peculiar to the components, and (2) those equations which result from the way the components are con— -'. '1-1‘: i 4%»:- . nected, The equations of the first classification are desig- nated as component eduations The component equations relate \ ; re- (‘ the variables of the component subgraph e ements, Thes lations are assumed independent of the particular system in whicn the component is connected, 2,} Component Lquations A convenient type of component is one such that the variables associated with the coaponent subgraph are related only auong themselves, This a convenient cosponent in an electrical system might be a three-winding transformer, an operational amplifier, or sole connected set of two-terminal if, components, This restricted concept of a component is so useful that it is the only type discussed further, The following defini— tions were suggested by the formulation procedures outlined“ by Reed [13], definition 2,5,1: Component equations: Those mathe- u.atical relations between, and only between, the variables of a cosponent subgraph, are called component eouations, Component equations can be conveniently divided into two types, Definition 2,3,2: Specified-variable equations: The component equations, which equate element variables to Speci- fied functions, are called specified~varieble equationS. Definition 2,3,5: terminal equations: All those con- ponent equations, wnich are not specified-variable enuations, are called terminal equations, The only components known to have utility in system analysis, and hence the only ones considered here, are those for which there are exactly as many component eouations as there are elements in the component swtgrayh, In general, physical systems may be composed of a set of two-terminal components, a set of nultiterminal components, or a combination of the two, As in electrical network theory, the terminal equations, which can be associated with the j'th ‘- two-terminal contonent, are limited to exyressing X3 in 4-. ,- udrULS of y,, or vice versa. If the couponent equation takes the a form of a specified—variable equation, either involving x or then th. other element variable cannot be related in the component equations, Formulation of equations for systems containing two-terminal components is very thoroughly devel- Oped, In contrast to the two—terminal comtonent, there is no practical limit on the number of forms that might be encoun- tered in the descrittion of multitecsinal components, how- ever, there is a particular set of forms, which is of partic— ular iatortance in electrical, mechanical and electromechan- ical Systems, Forms of coiyonent enuations Which are enscon- tered freouently, and which are considered here are shown 2.3.1. his, .Wuj. Typ; Number Descrittion 0 J? A n I‘ = ”‘7/ Q I) I r) \J - a d - J B n, V”. = “”‘5 7 53 ’/’X8 n " t J n ‘/(L/_ . z: i, '/‘./ ii d3! ? 332 .7 - \‘l _ I J n “10 k2 / x / kl v~ . I UK n /X = / d u n /‘/= ’ / 3' I' (1' c, " c , . — ‘ .‘_) ’ ‘ RI no, not relattc in cosgcnent eouations, ( ,(q) N n_ not related in CDMLOHGHt enuations, ( fl! ) v 037 ‘ ‘ 0'11 Figure 2,3,1 (0 Confonent-eouation Fytes and Variable oesignstion It should oe noted that, since the system equations _ t . . / 'o- .1 . W/ , are fornulateo in the s-oonain, , “7V, 9, ano %\ Cun- L” tain elenents which are functions of 8, however, no funC-, ( 0 (D tional notation is u d, since t-donain eduations do not at— pear exce,t where s,ecifically Lentioned, In this thesis there are no classifications of terminal eouetions in which siccified functions appear, For exanple, L measurenents on a congenent may indicate tertinal eouations O H: C+ 23’ 'j: ,olloving types: “0 ~= “/‘WX + ”U- " (2.3.2) ( In the formulation y:ocedures cescri bed later (2.3.1) is nanuled in exactly the same manner as a set of A terminal equations, and (2,5,2) fits the sane Forhulation pattern as the B terminal equations. The additional types of terminaI equation classification are not shown because of the more cumtereome forns recuired in the Formal ation procedures ex- 9.111an later, An alternate way of handling 2 components bgzagh with erainal equations of the type of (2.3.1) is by an ecuivalent Frath. In this eouivalent gragh, every e lament with an Xd associated with it, is raglaced by two elements in series-- an an A-e-eaent and a DK-elenent. An eouiv al ent graph for coagonent subgragh, with terminal eouations 3 (2.3.2) is a grant i1 Wnich each element of the cenyonent subgraph, which has a yd associated with it, is renlaced by two eletents in tarallel--a B-elenent and a D ~element. Dhere is no reason to sugpose that satisfactory cor- relation c=n be achieved with ex,1icit terninal equations for all multiterminal componentS, In fact, it is easy to snow that non-ex11icit t-donain Hi fe rcntia l te3ninal equations result for multiteriinal components composed of simple coa- ‘tinations of two-terminal components, In present practice, the ecuivalent of extlicit terminal ecuations is co; only assuxed For the consonants used. The Formulation proceoure, which is taken up later, however, agplies to non-extlicit ter- iinal equations as well as to eXplicit ones, If an (:1 €3.13 called an A-eleivnt. If va went are in terminal 110d resistor has as a con classified as 9n A—e.(uent be a EB-el Laugent, A two-terw ole is a oK—elesent, an N ( Au fnfiofher‘e*:n the graph contains more than one part—~ns may well be the case in system enalisis--tnese enuations are formulated for each tart by ccoosin; a tree for each part If the sys— (4' U} tea grn;h contains ; oer , there are exactly (e - v + y) in— oegencent circuit equations and (v - l) inuegencent segregate equations for the :rcph. fhus, there is a total of e inieter- \. Cent eoustions from the 2.1913490, 1‘*<‘:ga1‘dle;ss of the nu.L‘oer of k. parts. Phe following definition is convenient for use with a svsten grain of more than one nert: Definition 2 A 1: Forest: A fOPBSt F, 0f 9 3r79h G Of L 1 t warts is a subdraph of G such that the elements of F in o 3 each se,siate tart satisfy in tnet part the definitivn of a tree of a connected Lragh, From the definition, it is clear that For a connects} ‘regh the torus "forest" and "tree" are synonomous. For a ufaih Of 9 RaftS, the forest consists of a collection of p aerate Part, Since no enligtity -~-. . s a.” . . :1 -,.A It ,- . Will result, the tari chlrd set is uSed for ~ctr.» tLE oom- J N «I The trogerties of the fundamental seCrecate and cir- cuit equations are so important that, unless otherwise stated, .is narticul r i the symbols )3 and 6;' are tsLen to indicate t r‘ catrix form in this thesis. A designation for the particular sevre ate and circuit equa- L- ‘4‘ (D "V #1 (z‘ (A) H forest, for which the funds; tions are written, is often important. which a particular set of fundamental circuit and/or segre- J ga:e ecu“ U cf- H O :3 U) l U) 2 '3 F“ L §' ('f \I‘ :O :5 to H‘ ('1 J3 r- t It 1' )- L4" [5’ U‘ Q "j{ C H (.1) Cf f.» 8 *4) C) "5 (D (D C" O 5.) The question of ,eriissihle topological locatio: of D- - ‘v‘ ' 3" \ * r: p .“' " " ‘ ’ O A' F‘ closents has teen cuneicereu in [1;] for the case 0L two-tel- J‘J “ v 5‘ A. a \ s. . n ,, ‘~ 0 o)‘ I" “'“‘ minal COutUncfltS, here an element is a U—elcient ii, .nu only A. if, it is also an N—tlcment. The Lenerol theory in this ae- t a (D "x }-.l veloghant EFOCGEQB more snoothlv with a slightly a Viewgoint. The x-variables associated with D -81emUJtS in a sys- .\ ' I ten grath appear only in tho Specified-variable equctions an CH in the circuit equations, If there exists a unioue solution to the equations of the system, the rang of the circuit equa- tions and the Dy—equations must be (e - v + p + n ). Theorem 2,4,1: The circuit ecuations ans the Dx ecua- tions for a system graph have rank (8 - v + p + nx) if, and -14- only if, the DX-elenents form e subset of some forest of the iroof: The circuit equations and the D equations can be written as in which there are (e - v + p + nx) rows in the coefficient matrix. If and only if, @511 contains a set of (e - v + 9) columns, which corre°“0nd the complement of some forest, +Len J Ur P there exists (e - v + p) indetencent columns in. @311, and a nonvsnishing deterninant of order (e - v + p + nv) in the coefficient matrix [14], Theoren 2,4,2: The segregate equations and the Dy equa- tions for a system Cress have rank (v - p + ny) if, and only if, the Dv—elenents form a subcraph of the contlenent of sons forest of the system graph. Proof: Follows tne same settern es the proof for . Theorem 2.4.1. The following theorem is taken from [17]. Theoren 2,4,fi: Let G be a connected graph. Let 51 and S2 oe disjoint subse+s of elements of G such that there exists a tree T1 with the elements of 51 as chords, and there exists a tree T2 with the elenents of S9 as branches, Then there exists a tree T, with the elenents of S as chords and the 1 elements of 59 as branches, In writinb the segregate and circuit equations it is desirable, in general, to obtain a form which exolicit for a set of unknowns in terms of other unknowns and Specified functions, This necessitates using for d3: a formulation forest which includes the DX-elements, and for J2 a formu- lation forest whose complement includes the UV-elements, In the cases considered, no element has both the x-ver- iable and the y-variable Specified, Therefore, Theorem 2.4.3 indicates that, if there is a satisfactory formulation forest for (K and also one for .23, then there exists a formulation forest which is satisfactory for the writing of both sets, 2.5 System Equations It is convenient to elininate the Specified-variable equations from the set of siiultsneous equations for the sys- tem by substituting them into the segregate and circuit equa- tions, Definition 2,5,1: System equations: Let the circuit, segregate, and comoonent equations for a physical system be known. Let all Specified-variable equations be substituted into the circuit and segregate equations. The equations re- sulting are called the system equations, III. INDEPENDENCE CHITERIA 3.1 introduction The system equations in the s-domain are all linear and algebraic. They can be classified in two groups, however, as to type of coefficients. The equations from the graph always have constant coefficients, while the component termi- nal equations may have coefficients which are polynomials in s. The questions of rank and independence may be explored for the circuit and segregate equations, using the standard definitions and theorems of the theory of linear algebraic equations with constant coefficients. However, some defini- tions are now presented in order to make clear what is meant by the terms, rank and independence, in the developments to follow, when applied to linear algebraic equations, which have coefficients which are polynomials in s. Definition_3.l.l: S-matrix: A matrix with elements which are polynomials or ratios of polynomials in s is called an S-matrix. Suppose that a system of Laplace-transformed ordinary Ilinear differential equations, with constant coefficients, is written CQFSX83:I£fiS) -16- -17- where the order of (1(s) is n x n, and iX(s) and £?(s) are . 0, column matrices. It is clear that a solution for /K(t) does not exist if the determinant of (2(a) vanishes for all 3. Therefore the following definition is made. Definition 3.l.2 Rank of an S-matrix: Let (2(5) be an S-matrix. The rank of (She) is the order of the highest- ordered determinant in (jks) which does not vanish identically for all 3. Definition §,1.}; Independent equations: A set of m equations in n unknowns, m.é n, is said to be independent if the coefficient matrix has rank m. Theorem 3.1,;: The rank of the system equations (in the sense of Def. 3.1.2) must be equal to (26 - n x - ny), the num- ber of unknowns, if a unique solution is to exist for some value of s. 3.2 Iopglggical rattprn of N—element_ The material in this section is an extension of the work of Reed and Reed [15]. Their development treats the case of two-terminal components for which a graph element is an N- clement if and only if it is also a D-element. The multi- terminal component requires an extension to handle the situ- ation where N-elements need not be associated with D-elements. Theorem.2;§‘;3 Let Cl be a square n-order matrix which can be partitioned as follows: -18- _. an an: a '" 0 0.. where (111 is of order m.x.p, p 5 m.é n. (:1-1 exists only if 62:1 has rank p. Proof: Use the first m.rows to expand by Laplace's ex- pansion [18] . If the p columns of C211 are not linearly independent the determinant of CZ vanishes. eor 2 2: Given a matrix M of constant coefficients of order p x q, p.g q, which has rank p. Let any set of r independent columns, r < p, be designated by 81' There exists a set of (p - r) columns 82 such that the union of S1 and 82 has rank p. Proof: Follows from Theorem h.22 of [18] . T 60 em 2 : Given a &3 matrix for a connected graph, any p columns, p é (e - v + l), are independent if, and only if, the columns correspond to a subset of some chord set. Proof: a) Sufficient: Follows from.Theorem.lU of [lb]. b) Necessary: By Theorem lu of [1h] , a set of (6 - v + 1) columns are independent only if they correspond to a chord set. Assume that at least one of some set of p inde- pendent columns 31 does not correspond to a chord. By Theorem -19- 3.2.2 there exists in 13 at least one set of (e - v + 1) independent columns S of which S1 is a subset. But this is impossible, since by hypothesis at least one column in S does not correspond to a chord. Corollary 2,2,2: Given azfj matrix for a graph of p parts, any q columns, q 4 (e - v + p), are independent if, and only if, the columns correspond to elements which form.a subgraph of the complement of some forest. Theorem 3,2,Q: Given annfig matrix for a connected graph, any q columns, q é (v - l), are independent if, and only if, the columns correspond to a subset of some tree. Proof: It can be shown [16] that if, and only if, any (v - 1) columns of an 23 matrix correspond to.some tree, the columns are independent. This proves the sufficiency aspect. The necessary proof follows the same pattern as for Theorem 3.2.3. 0 rolls .2. : Given an 95 matrix for a graph of p Parts, any q columns, q S (v - p), are independent if, and only if, the columns correspond to elements which form a sub- Eraph of some forest. Definition 2.2,12 Trivial segregate element: One ele- ment,‘wh.ichforms a segregate set, is called a trivial segre- gate element. IQQQEgEL§;§45; A unique solution for the system.equa- P-..‘ .\\ .v, . ,P my 1,. -20- tions exists only if the Nx-elements can be made a subset of some chord set, and the Ny-elements a subset of some forest. Proof: The system equations can be written 1 F , ‘ * . " F £1: £2 0 an (DJ/Syd CL, 6L. 0 2*; + 0 :0 L911: 3/ C: C) see a: 0 92., Cami ) , . By Theorem.3.2.1 d3ll must have rank nox and 3h must have rank :10 if the inverse is to exist. The columns of 1311 Y must, therefore, correspond to a subset of some chord set by Corollary 3.2.3. The columns of 3umust correspond to a subset of some forest by Corollary 3.2.h. The following lemma is obtainable from developments in [15]. Lemm 2 1: If the fundamental segregate and circuit matrices, in the form of (2.u.1) and (2.u.2), are written for I the same tree of a one-part graph then 611 = - J12. Theorem.§,2,6: For a connected graph containing no trivial segregate elements, if a set of elements S forms a segregate set, the fundamental circuit equations for any tree of the graph can be written where C>< s Xsl 0X32 the adross variables associated with S, :Km contains the chord variables except for <§<32’ and £321 is a column submatrix containing 1 or -1 in each position. Proof: The fundamental segregate matrix for the same formulation tree.is /0 094’. 0 page. By Lemma 3.2.1 dg'll = - SXyIZ’ Therefore a: 0 ex: 2/ 0 1.3,; 3.1;: 0 u and the theorem follows. Theoreml3.2.7: If in a system.graph containing no trivial segregate elements, any segregate set contains only Nx-elements, the C>(n of that set of elements is indeterminate. Proof: By Theorem.3.2.6 the system equations can.be written -22... 4.0.0 0 a 0' x Ym Ow where A n = an <7<1122 and 78 contains all across variables not associated with the segregate of Nx-elements. Let the number of elements in the segregate set be p. With. ix; known, there must exist a non- vanishing determinant of order p in the last p columns if a unique solution is to exist for TXTn. This is impossible since there are only (p-l) rows in Lf323 7L(]. If one Nx-element forms a segregate set the x-variable associated with that element does not appear in the system equations, and the y-variable is zero. The solution for the remainder of the system.graph is unchanged if such an element is omitted from the graph. Theorem 3:2.8: If a set of elements S form a circuit in a connected graph, the fundamental segregate equations for any tree of the graph can be written a o as}; a, ova, 0 y. -23- where fléjs = ysl ye /2 contains the tree variables not in 62/ C2, con- Zb dSZ’ /m tains the chord variables except for ysl, and 1% is a column submatrix with l or -1 entries. Proof: Since the elements of S form a circuit, the com- plement of every tree contains at least one element of S. Furthermore a tree T can be chosen such that only one element of S is in its complement. The fundamental circuit matrix for T can be written as (1.?“ (if): v 0 f}. 0 0 / I By Lemma 3.2.1 i311 = - 42312. Therefore J2! can be written 71 0 do: 0 a a; 0 Theorem.3.242: If in a system graph, any circuit con- tains only N&-elements, the 621; of that set of elements is indeterminate. Proof: By Theorem 3.2.5 the system equations can be writ- ten -2u- a at a: 0 We _0 0 0 0.. 0.2 g... ._. .J where i9 = y n nl ma éu1d. C21a contains all through variables not associated with the circuit of Ny-elements. Let the number of elements in the Ny-element circuit be p. If all variables except (an are known, there must exist a non-vanishing determinant of order p in the first p columns if a unique solution is to ex- is: for gain. But there are only (p - 1) rows in [ a e112]. It should be noted, of course, that even though there is a circuit of Ny-elements or a segregate of Nx-elements in a System graph, Theorems 3.2.7 and 3.2.9 imply nothing about the existence or non-existence of a unique solution for the other variables in the system equations. It has now been established that, if a unique solution exists for a set of system equations, the upper bound on no- relation elements is given by n 5 (e - v + p) and noy é ox (V "‘ p). If (nox + noy) is equal to e, the terminal equations may- be solved independently of the remaining set. For this case the terminal equations consist of (e - nX - 1») equations in (e - nx - ny) unknowns. If the terminal equations are -25- homogeneous, only a trivial unique solution is possible. Thus for a non-trivial solution for the system equations of a system with homogeneous terminal equations (nOX + noy) < e. 3.3 m of the System Equations The rank of the circuit and segregate equations is ex- actly e if the D-elements have an acceptable topological arrangement. The rank of the system equations must be (Ze - nx - my) if a unique solution exists. Thus the rank of the terminal equations must be (e - nx - r57). The maximum rank of the circuit equations and the K terminal equations together fixes an upper limit on the num- ber of K-elements permissible in a system graph. If the K terminal equations and the cir- l‘heorem 3 ,3 .l cuit equations form an independent set, then at most there can be (v - p - nx) lie-elements in the system graph. Proof: The rank cannot be greater than the number of variables related. Therefore, (9 - v + p + 111(2) 4 (e - nx), from which nk2 s (v - p - nx). Likewise, the maximum rank of the segregate equations and the J terminal equations limits the number of J-elements permissible in a system graph. W: If the J terminal equations and the seg- Pegate equations form an independent set, then there can be at most (6 - v + p - 1’5) Jl-elements in the system graph. -26.. Proof: Parallel to that of Theorem 3.3.1. (v - p + njl) S (e - ny), from which njl g (e - v + p - my). It is, of course, desirable to be able to state neces- sary and sufficient conditions for which the segregate and circuit equations, and the terminal equations of type J and K, form an independent set. Necessary conditions are extremely difficult, if not impossible, to show in general. However, sufficiency criteria, of a nature general enough to be widely useful, can be established readily. Thegrem 2.2.3: If in a system graph, all K—elements and Dx-elements can be included in some forest, and all J-elements and D y-elements can be included in the complement of some forest, then the set of circuit equations, segregate equa- tions and terminal equations of type J and K have rank (6 + “3'1 + “k2)° Proof: a. Suppose that all K-elements and Dx—elements are in— °1uded in the formulation forest of the system graph. The circuit equations and the K terminal equations can then be writ ten as follows: v 16.. «6. a; ”x. 0 :1! -"K 0 X; “X... _°X.i Xe contains the chord x-variables, and 7b contains the -27- tree x-variables not associated with K-eloments. This set of equations has rank (e - v + p + 111(2) because of the tri~ angular submatrix in the leading position, with unity elements on the main diagonal. b. Suppose that all J-elements and Dy-elements are in- cluded in the complement of the formulation forest of the system graph. The segregate equations and J terminal equa- tions can be written: " '1 I 71:92; Qjaz 3J4» ef'f- :0 0 ; u 4 0 ’2; L “J These equations have rank (v - p + nJl) by the same reasoning as that used in (a). The hypotheses for Theorem 3.3.3 are relatively simple. For a. given system graph the independence criteria outlined there are easy to apply, and thus may be quite useful. How- ever, the sufficient conditions for independence can be made 1988 restrictive, by sacrificing some of the simplicity of the hYpOtheses. Theorem 2,3,4: If for some system graph, 1. The K-elements form a subset of some forest T1, 2. The Dx-elements and KZ-elements form a subset of some forest T2, 3 . There are no circuits of K1’ K2, and Dx-elements -28.. (all three, and only all three, types), then there exists a forest for which the fundamental circuits associated with the Ill-elements include no KZ-elements. Proof: Take the forest T2 which includes as a subset all elements of type Dx and K2. If T2 includes the Kl-ele- ments, Theorem 3.3.3 holds. If not, each fundamental circuit a :1. involving a Kl-element either: (1) does not involve a K2-ele- ment, or (2) involves at least one KZ-element. If (1) there “H's" ONA'L - is no problem. If (2), there must be at least one other ele- ment involved since Tl includes all K-elements. If a hz—ele- ment is involved, either: (a) only Dx-elements in addition to the K-elements are also involved, or (b) some other type elenient, say an A-element is included. If (a), then the hYpothesis is violated. If (b), a forest T3 can be chosen 30 as to include the Til-element as a branch with the A-element in its complement. The form of the proofs of Theorems 3.3.5 and 3.3.7 was Suggested by Koenig [10]. From this form the hypothesis of Theorems 3.3.1; and 3.3.6 were devised. Theorem 2.3.3: If the hypothesis of Theorem 3.3.14 is satiszfi‘ied then the fundamental circuit equations and the K- terrainal equations are independent for all VK. Proof: By Theorem3.3.l+ there exists a forest for which the fundamental circuits for the K1 chord elements involve no h2-e:‘l-ements. Therefore the fundamental circuit equations for t his tree and. the K terminal equations can be written r .. :- ‘“ " ‘ o x. u 1 ‘K, 0 ‘79 g” (5.. 0 :71 0 7H,, :0 __ \b'u ([322. 623: O 71‘) ~352— where OXKZ : [kn 0K2] -7Kn" “X“. Elementary Operations will reduce the matrix to one with an identity submatrix in the trailing position. Theorem 2,3,6: If for some graph G, l. The J-elements form a subset of the complement of Some forest Tl’ 2. The J1 and Dy-elements form a subset of the comple- men-t; of some forest T2, 3. There are no segregates of J1, J2, and Ny-elements (all three, and only all three, types), then there exists a Ibreast for which the fundamental segregates for the J2 branch 61€‘>I‘r1ents include no Jl-clements. Proof: Take the forest T2 which has in its complement all elements of type J1 and Dy' If the complement includes all the JZ-elements, Theorem 3.3.3 holds. If not, each funda- mental segregate involving a JZ-element either: (1) (1093 not in"Olve a Jl-element, or (2) involves at least one Jl-element. If (1) there is no problem. If (2) there must be at least JV'IIJJIII»: .I Eli. i i i. i -30- one other element since the complement of T1 includes all J—elements. If a Jl-element is involved either: (a) only Dy-elements in addition to J-elements are involved or (b) some other type element, say an A-element, is included. If (a ) the hypothesis is violated, if (b) a forest T3 can be chosen so as to include the A-element as a branch and the J2- element in its complement. Theorem 3,3,2: If the hypothesis of Theorem 3.3.6 is satisfied, then the fundamental segregate equations and the IF?“ fii i- i ulfiflfikw J terminal equations are independent for all g . Proof: m Theorem 3.3.6 there exists a forest for which the fundamental segregates for the J.2 branch elements include no J -elements. Therefore the fundamental circuit equations 1 for this forest and the J terminal equations can be written T (0") '1 -9” 1 0 —7I I C’ ‘62 0 / b l U f. 7/! O :22): 9J2, 52/55 Vyjz’ L0 a; o 5%.. at, , If the order of the variables is rearranged so as to bring the third column to the leading position, then elemen- tar-y Operations will reduce the leading submatrix to an rim ! .1 n' N -31- identity matrix. A situation of considerable importance occurs when the system graph is in separate parts and when J-elements and K- elements are distributed such that: (l) the Kl-elements are in one set of parts--the KZ-elements in the other, and (2) the Jl-elements are in one set of parts--the J2-elements in the other. Theorems 3.3.5 and 3.3.7 apply, of course, to graphs of 86Paratie parts. However, a more general set of conditions can be given, for the particular distribution of J-elements and K-elements just stated, such that the segregate and cir- cuit equations and the J and K terminal equations are inde- Pe nd ent . The em . 8: If for some graph G, l. The graph is in two separate parts, such that Kl- elements are in part 1 and K2-elements in part 2: 2. The Did-elements form a subset of some tree of part 1, 3. The lie-elements and DxZ-elements form a subset of some tree of part 2, than the fundamental circuit equations and the K terminal equa- tions are independent for all 6K. Proof: équa'b ions can be written as: A K" J ..‘.‘Jm The fundamental circuit equations and K terminal -32- F 0 -7<, -0 ‘a E-K. 0 0 FOX. 8,, .8 0 0 : 7/4 0 0 7,... a. a. o 0 5 0 7,1 0 ”X... - 0 o €53. 08.; 0 0 ”U J “>91 Where 7K2: [kl Y2] Vim" WW The Iii-elements are in the chord set of part 1, and the Ki- elements in the tree together with the DIE-elements. Thus the only restriction on the topological arrangement of the K1- elements is that they be contained in part 1. Thggrem 3,3,9: If for some graph G, l. The graph is in two separate parts, such that J1- eleInents are in part 1 and JZ-elements in part 2, 2. The Dya-elements form a subset of some chord set of part 2, 3. The Jl-elements and Dyl-elements form a subset of some chord set of part 1, than the fundamental segregate equations and the J terminal equations are independent for all g. Proof: The fundamental segregate equations and J tenni- nal equations can be written as: “‘1 ”a 0 0 : are, of, 0 0 ~ (“847 0 ”a 0 I 0 0 pic .3; 9.. I :0 0 0 a ll 0 0 J33 S2334 _%f ,0 0-?KH 0 0 $2) 70/. Bid/WI: where 10/1,] : [g %] yjz. 02/1 2' The J'Z-elements form a subset of some tree of part 2, and the Jé—elements together with the Dye-elements form a subset of the chord set. If the third row is multiplied by 7 and added to the fourth row the result is a triangular submatrix, With 1 on the main diagonal, in the leading position. Thus the only requirement on the topological arrangement of the J2'--elements is that they be restricted to part 2. As a result of Theorems 3.3.8 and 3.3.9, a theorem can be stated for the graph of two parts in which all K-elements 3P6 also J-elements, and vice-versa. W: .If for some graph, containing no K- eleI'Iients nor J-elements which are not JK-elements l. The graph is in two separate parts such that the J'Kl- eleInents are in part 1, and the JKZ-elements in part 2: 2. The JKl-elements and the Dyl-elements form a subset or some chord set of part 1: ~3h- 3. The Dxl-elements form a subset of some tree of part 2, h. The JKZ-elements and the DXZ-elements form a subset of some tree of part 2, 5. The Dya-elements form a subset of some chord set of part 2, then.the fundamental circuit and segregate equations, together with the JK terminal equations, form an independent set for all g and 7(. Proof: Follows from.Theorems 3.3.8 and 3.3.9 since all conditions in the hypotheses of both theorems are satisfied. Theorems 3.3.8 - 3.3.10 can all be extended to the case of a system graph of more than two parts by replacing the term "tree" by "forest" and the term "part" by "set of parts". The proofs of independence have been based on the fact that fundamental £5 and g9) matrices are used, and that these matrices are written for the particular tree used to locate tepologically the J-elements and the K-elements. The next step is to show that, if these equations are independent, then independence is assured for any full-rank set of circuit and segregate equations. For this development let ‘— fill BIZ 6!} F 7m _ (5o -7< ‘21 C) fine -_ (g _7(. “XL-0 It I.‘.I)l lu.....|(\ I I..I CHI —- r ‘1’ ‘* and 32%| 9&2 flu 62/ .518! 024 __ 0 , . $3 Thegrem.3.2.ll: For some graph of p parts let 43; and 6 2 be circuit matrices with rank (6 - v + p). f. and 032 a a have the same rank. Proof: Follows from the fact that 681 and .82 are re- lated by a non-singular transformation. Thegrem 2,3,12: For a graph of p parts let 5J1 and ,Q8 2 be segregate matrices with rank (v - p). 93. and 9!: have the same rank. .3 .13 Proof: Parallel to that of Theorem 3.3.ll. Thus if, and only if, independence exists for some set of full-rank circuit and segregate equations, the equations are independent for any set of full-rank circuit and segre- gate equations . IV. GEEEBAL PROCEDURE FOR FORMULATION h.l lgtggductign If the graph of a physical system is known, together with the necessary number of component equations, all infor- mation required for simultaneous solution is available. How- ever, for complex systems, the number of simultaneous equa- tions is very large. Even when computers, either digital or analog, are to be used, a great saving in time can usually be gained by a reduction in the equations to be solved simul- taneously. With regard to formulation of the system.equations, the system graph serves just one purpose-~that of providing a systematic method of writing a set of independent equations in a convenient explicit form. When this has been accomplished, the graph has served its purpose and the analysis procedure is based only on the form of the system equations. In general, the graph of a system.may contain elements with all types of component equations. The formulation pro- cedure is, of course, influenced by the type of elements pre- sent. HOwever, a general procedure can be stated, utilizing the explicit form of at least some of the system equations, to effect a reduction in the number of equations to be solved simultaneously. There is no assurance that this procedure yields the smallest set of simultaneous equations, which it -36— -37- is convenient to obtain, nor that the procedure is the simplest possible. It is, however, relatively simple to accomplish, and it handles systems containing the types of components dis- cussed. In order to facilitate the reduction process, certain procedures in the writing of the circuit and segregate matrices are helpful. These procedures are examined next. Lina Circuit Equations for System Grgph Containing Nx-Elements The following is an extension of a formulation technique developed by Reed [13] for two-terminal components in which an element is an Nx-element if, and only if, it is also a Dy- element. If there are Nx-elements in the system graph, then the only system equations involving these variables are the cir- cuit equations. It has been shown that a complete solution to the system equations can only be obtained if the Nx-elements Discussion is limited to the If a formulation ”‘6 a subset of some chord set. case for which a complete solution exists. 1breast is chosen, such that the Nx-elements are a subset of the Chord set, and if the equations and variables are properly sequenced, then the circuit equations can be written r—' -— £11 @2212 O OXJ "‘ 0 "(QZI 0/322 7/! 77 (Ll-'2.l) ”X.U a: specified across functions; .— 0 Where the variables are: 0X1" across variables related by some type of terminal equa— -38- tions; cxn, no-relation across variables. Since 0X n is related only by the second row of Eq. 14-.2.l, and the number of equations is equal the number of variables in 0X11, this set of equations can be "set aside)) in that they need not be solved simultaneously with the other equations of the system. Thus, for simultaneous solution, the circuit equations are reduced to (ti/yd + $2 VT :0 (#0202) The unknown variables in Eq. 14.2.2 number (6 - nx - nox). LL .3 The Se re 'ate at ns for S stem Gr h Con i Ell—Elements The following is an extension of a formulation technique developed by Reed [13] for two-terminal components in which an element is an Ely-element if, and only if: it 13 also a 131:” element . If the system graph contains Ny-elements, then the series Variables @n are related only in the segregate equations. If a complete solution to the system equations exists, Ny-ele- merits must form a subgraph of some forest. If a formulation r0Pee-1: is chosen such that the Ny-elements are included as a subset of the forest elements, and if the variables and equa- tions are properly sequenced, then the segregate equations can be written all, -39- .— 0 ex ex. “5/ i0 0 sin ,9st g/ 9—] (h.3.l) where the variables are: 1/” no-relation series variables; QT, series variables related by some type of terminal equa- tions; 7%, , specified through functions. The only equations, in the entire set of system equations, that involve [911, are those represented by the first row of Eq. 14.3.1. This, these equations can also be set aside. There- fore the contribution of the segregate equations to the set that must be solved simultaneously is: 91:2 ”7 + 23 (/ (9 (Li-“3.2) In this set the rmmber of unknown variables is (e - ny - noy)' {LIL The Substitution Procedure In reducing a set of n simultaneous equations in n un- knowns to a smaller simultaneous set, the equivalent of the following must be done: Some set of (n - m) equations are used to obtain an explicit relationship for some (n - m) un- known variables, in terms of the remaining m variables. Then the (n - m) variables are eliminated in the remaining m equa- tions - If the n equations are independent, this can always be done for any subset of unknown variables. Therefore, whether the reduction process is a useful one or not, depends in general upon the degree of difficulty with which the ex- -uo- plicit relations can be obtained. If matrix inverses must be taken before the substitution process can be carried out, it may well be that inverting the original set of n equations would be less tedious than reducing the set before inverting. In the system formulation, as considered in this thesis, the terminal equations are explicit in form. Therefore, by substitution of the terminal equations into the circuit and segregate equations, the number of simultaneous equations can always be reduced to (e - nox - nOy)‘ The circuit and segregate equations are not, in general, explicit forms. However, the fact that the fundamental cir- cuit and segregate equations are explicit, make them very use- ful in any procedure involving reduction of simultaneous sys- tem.equations. As noted earlier, because of this explicit relationship, the ikfn and ‘2/h variables can be solved for, and since they appear in none of the remaining equations, the simultaneous set is reduced by (nox - noy)‘ The system equa- tions can be shown in the form: 9.. 7% 0 0745 2 :0: :0 (his) 1/ 7... _ "J where 7m and in are the chord x-variables, and 09b and Jain the forest y-variables. These equations are not homo- -h1- geneous, as the matrix form might seem to imply. The Dx- element x~variables are assumed to be contained in.(nox + nOy)’ this procedure would be preferable to that of substituting the terminal equations into the circuit and segregate equa- tions. ms 0 ord F rmul A third reduction technique utilizes the explicit form of the second row of Eq. h.h.l, and any terminal equations explicit in the x-variables. It is evident that the forest y-variables ‘ij, can be expressed in terms of the chord y- variables iéfly. Let the second row be substituted into the terminal equations. Let m.terminal equations be explicit in X- If these m.terminal equations are then substituted into the fourth row, the set of simultaneous equations remaining -ug- are (e - n.x - n.y - m) + (e - v + p - nox) = (26 ' V + P ' nx - ny - nox - m) in number. If m.= (e - n.x - my), a common case for electrical cir- cuits, this reduces to (e - v + p - nox) simultaneous equa- tions, conventionally called "mesh equations” in electrical network theory. For this case the terminal equations are of type A and the problem of formulation has been treated by Reed [13], h.6 Tree Fgrmulatign A fourth reduction technique utilizes the explicit form of the fourth row of Eq. u.u.i, and any terminal equations explicit in the y-variables. Using the fourth row, the chord x-variables, C§1-"‘t:.ing a diagonal, or otherwise simple matrix form. the>~ . £363 on see, the terminal equations are assumed to be eitn tV‘ .ieee (Dr A or B, depending on convenience. In the Special cases that follow, the chord and tree The for: I“‘J:Léltion procedUies are pie esented for the host part. -46- -47- rdeocedures are valid for systes gra,hs of here than one port, of? course, However, the enuation counts, unless otherwise Sixecified, are based on a graph of one part. b I Q . I. U . J r '? entldliy, 118 is to allow easy corpslison of the number of simultane- t '\' Outs equations for the particular fornugetion procedure, to be Hmuie with the nuuber of conventional mesh and node equations, ‘ ‘Nhixzn would result for the same graph and appropriate termi— .nal, equations, If the term "tree" is rr;luced by "forest”, etuzrl rrocedure a p;ies to a system graph of separate parts. Each s,ecisl formulation procedure is presented in out— cable, if the sage forsulation H- ;1rra fern, as much as pract trace is used for both circuit and segregate matrices, one is Ubtffiixnabie from the other; immerefore in most of the cases written is the one into which the final L- \J Grainined, the only one u ‘1‘ " . . ' -- ‘¢*Jeftitution is made, i r) , _ - 3-“- gyros of Components Considered In order to facilitate the present tion of formulation i‘i4536klures for systems containinf Specific types of cos 0- rip) 11 t C! a , _ K ‘ f‘ x . - . ' -—\ ‘5 ‘~° it is convenient to oefine some couponent typis, Definition 5,2,1: HUN-component: A Ken-component is D-x‘“ ~ 3‘. . . loiLUlelg L01”... one -15 ‘ , . +, , -L(Jr wnicn one component equations have the 0\/ I ’ r1 9/ Ate ‘ 9%th], (“Jul = 54/61 flfk2 not related in the component equations, -48.. An electronic exclifier, with outcut voltate taken 'ro- i. LNDPtional to input voltage, and input current ne¢lected for ""3 \.~\ ‘ 3 1‘ ~ .‘I 7 ‘ ‘ . .,H .- . 1r'-y.7 '7‘ U‘ :J4I,use o analvsis, is an clan,le of F nun-comyonent. Definition 5,2,2: nix—contonent: A KER-contonent is ODx? for wkich the co ,1nent eouations hive the follOWinC ferns: / '6‘: ' .7 f 0‘. . [I :X 'D = 7<\ .:K \ 9 . ‘ -' 1 «I :: // /< ., k f) ting not relates n the cougonent eouations, H” . An examgle of a KEN-colgonent is an opeivtionel ampli— f‘ . — J. Lier' of an ana;oa coipu_er, Lefinition §,2,7: JDN—contonent: A JpN—congonent is fink; fWDI'WniCn the conconent eouations have the followin¢ fOFbfi= \ _- a" a\ I 0\ {/1 / l 57 \' ' '0’ a = C ' x/v’ 3 I) , lK \ f) = K 2’) ‘ 1:7 ‘1‘. (C— V ijl not related in the comgonent equations. v __§finition 5.2i4: JAl-com,onentz A JAR-congonent is o o . . . . .q a ne ~C>I“ wnicn the cottonent enuations have tue followinp forks: a 0x 3 ‘231 " 332' 32 " 9' J32 / ile not related in the component equations, I:f~. for a common-base transistor, the collector current arr: the emitter voltage are assimeu protortional to emitter CLerent, t"e transistor can be classified as a JAN-component, .5“ Definition 3,2,5: JK-COngonent: A JK-con;onent is one TV)? which the cosgonent enuations have the following ferns: 6X“ ._. ’Kaxks’ 47(6/k2 ‘ 7 (viz: 55.3 Systems of A and hub Someonents Chord formulation is used: In the formulation tree: All Ny-olesents, wnich in— cludees the hp~elements. In the chord set: All Dv-elenents and Nx-eiements, in- , -. .J ClUCLini’. ttie Kg -ele:nents. J- CirCint;equationS: r— I ‘ '1 F'- ‘5 as. gt» 0 0 . 1.”, «82,15 ’U 0 .-__~___T___.-...-__- + ______ by p32 0 “U A, \A “6:0 (5.3.1) 33 big3 £3 bxtvlic it form of the seé‘regste equations from the circuit equat'1_()rjs. ”2,4, a: tin #332 ’34“ (5.5.2) -744 c,» cut/U- _, o , - . In (5.),2) 1%” contains the unsgecifiou tree Y’V5P1531e3~ LU 'L‘E‘I‘iinal equations after substitution from seJ‘egzite eoustions: m \\ \_/ i 0- I ,Q’ _.I [- ‘ b1 0 b1 0 ..'> V Circuit eouations after substitution of terminal eouations: as: W; as; ea] ’23 + m tr 03»: t=o 1322 “u. di,"/J+‘U J3 e L Libs (5.3.5) Unknowns: 67g. and 7k“ LL: " Number of equations: (6 - V t 1 " 110,). '1 - ‘ ‘ - v > oijentS: -n is the number of DyNY—elenents in the byttem OX j r;,-,",. ~ \ V profit; for t;‘,1S CE-be. : 7‘ - ‘- ‘7 "4 Saffstems of b and KDm Comgonents "I" 1'- .: .c \ . +I‘ee iormulatlon is used. In the formulation tree: All Dh-eleulents, and all Pos- 2' 81"“1 ‘.t , ‘ “9 Jx-elencnts, If all K-elements cannot :38 gut into the Sen": 4" , r , , Lb i. orumulation tree with the xix-elements, then a. formula- tion a _ - t1 E‘e should be chosen such that the maximum nunoer of P-a J; Ii terminal equations relate only tree variables, In the chord set: All Nx-elements, In this procedure all K-elements are assumed to be in the formulation tree. Segregate ecuations: JM—l flan (Jim J .2 i Jaw (5.4.1) 2,... J, _ J L’14PJ 492‘; 'Expluiwzit form of the circuit equations from the segregate aqua, ti one : e? I ..' t ' F d ' 73: [44 J24 J34 9244] (5.4.2) .2333 :3 CZKb contains the unknown tree x—variables. Where Lerm111£1]_ equations after substitution from segresate equa- tions 3 ya: GKGXM (5'14’3) ”gsmfiwg J: J; Jllxl (5.1.... Terminal ecuations after elimination of oxke: ”lg. :WLJ; (stn J1] 7t (5.4.5) a 74m ,4. Ms; Wu.» J] (3) JZ£+ . iXKi 52%; (5.4.6) UllknO‘an; 7k]. and X12). Number of eouations: (v - 1 - noy) "' (V ' l ' nk2 ' nx) 5-5 ivstems of A and KBN Components Chord formulation is used, In the formulation tree: All Nv-elements, which in- elude S t he lira-elements , In the chord set: All Dy-elements and NX-eleznents. . .r I .Dr. J ' 'pl‘ .Uw C ircuit enuati ons: #3 ’ , _ __“. __:,_--___ + -1395- ”Xdzo (5.5.1) 0 its; U523 0524 7K2 [:26 Explicit form of the segregate ecuations from (5.5.1): 0ng 6'; 622. fig“, (5. 5.2) 5! Q4“ : 33 £32; (7: m ”2,1. 03,; (1:; _. U A l— _ Terminal ecuations after substitution from (5.5.2): 7.. =3aU3fi 68.1.] {gt (5.5.5) ”76/". X, = X‘XK, (5.5.4) [(83 fig] 76!.) :WX. (5.5.5) .24,” Circuit equations after substitution of eXplicit terminal eQuati One . -54- [fix+njty¢gi[a;nfl‘b +agxg=0 (5.5.6) Final enuations: . ,v I . .‘ / a ‘%7 7+ t” in QJJ~ 0 :0 f (K4.ka VI l-j f‘ J" /:(\ J" J, 6 .. 23 ’33 ~52: 20 1-44 (ti-124-1(A 11224 £24m asyd (5.5.7) Number of ecuations: (e - v + l - my + ny1)° If W51 3 (2;? is known, or easily obtained, the K terminal enuations can be written 7... 2 3k (25.5) For this form, the terminal ecuations after substitu- tion from the segregate enuations are: d 7&2 31K 0 613’ 62'3. (yd (5.5.9) OXKn : 3) 0 at; CB; 52am -ixal _ C) ‘3“_i Circuit equations after substitution of (5.5.9): [it—2. e .25.] 5.). 0 6... a: in. C/ Q} + szfxd : O 6') “J I V 3' 0 £14. (1824 ’2’") 1/) P _ 0 a0 4 (5.5.10) F— Unknowns: '72? .. . Number of ecuations: (e - v + l - n0?) = (e - V t l - Dy). 5.6 Systems of B and KBN components Tree formulation is used, In the formulation tree: All Ny-elements, and all E- clements possible. In any case the maximum number of K tarni- nal eouations possible should relate tree variables only. In the chord set: All Nn-elements: Segregate equations: Fa, Fa. I I t1 0 C) £341 :5; 23. (5.6.1) .. .0 _ __?/l_ _O__ __ :zg‘. [(sz + fié: ‘24" :2 O a V‘ g. a... Explicit form of the circuit equations from (5.6.1): ><= [22,; l; 93,; J4] x 39 N 432% (5.6.2) C where ij contains all tree x-variables not in. ’le' >8 3‘xe ___ ‘K u my Terminal equations after elimination of 7(ygz Us L W; [ 237; HQKW‘JL) 24g \4 ‘ZJK, = ”W. 0X,“ A U1 0 0\ (5.6.4) .6) Segregate eauations after substitution of (5.6.5) and (5.6.6): . . . F - [U r :23?» a?” (JZ:K+;J3; 22$?“ bxa 133‘ ., O )Mva + J‘ 9V1; [ ) :1 y“ + 2345 24d -0 WW v.- (5.6.7) Unknowns: ixkl and :Kb, Number of equations: (v - l - noy) = (v - l - nX — nk2). The form of (5.6.7) is not agpealing, but no good pur- pose is served by rewritin; to ¢et all variables tOgether, since this ; eatly Complicates the general form, The case of kal = 0&1 :22" As might be exuected, there is no apgreciable change in the is not considered here, ‘1 (Ii. form of the Pinsl ccuations. If <%fl‘7(kl_18 replaced by (figkl’ and 3% ternate enuations are obtained, is replaced by 637lcajk1 in (5.6.7) the al- J. ’0 1 f 5.7 §gstems of KEN Congonents A special case, which is important enough to merit some attention, is the type which includes the analog-computer SYStem. If each Operational amplifier is represented by a Lagrangian-tree subgraph, With the common vertex correSpond- ing to ground, the graph of the system is separable at the ground vertex, Lech nonseparable part consists of two or more elements in parallel. In each sernrable part there is one, and only one, element of the following types: (1) DVNV- 45 J ~58- el 8&8? it, or (2) an ele ment regresenting the output of an Operational amplifier. Thus, the Specified-voltage elements and Operational-amplifier output elements in general make up a Lagrangian tree of the system graph. f In the formulation tree: All K2-elements, since ng2 is not related in the COu onent equations, and all Dx-elements, In the chord set: All Ll-elehents, The complete set of system equations: 0 (‘74,, i F0 j _-0._ 74x2 *9" + 0 : O O 63.. U _J _ "J fluvial (5.7.1) The first three rows may be set aside. The fourth row may be substituted into the fifth row, or vice versa, There- fore, the final equations are: (71+ M6,? )+°X,<, 68.3%: 0 (5.7.2) 01‘ (chi/”Q‘— «’21)sz -+ “K KEGQXd: 0 (5.7.3) Unknowns: (éKkl or C§ nk2) Comments: The segregate equations, and the 5 terminal equa- tions, do not affect +he x-variable solution for this type of system. 5.8 Systems of A and JDN Components Chord formulation is used. In the formulation tree: All hv—elcments. In the chord set: All DV—elements, and all J-elesents if po (1') sible. In any case, the maximuu possible number of J terminal equations should relate only chord variables. In this procedure, the J-elements are assumed to be a subset of the complement of the formulation tree. Circuit equations: as. 0 0 ol 7’] ti; at. "J 0 0 ‘>< c8 - ., 1. _ _'__ - _. __ ________ J2 + ___5'_ Kd —— 0 (5030‘) $3?! 0 {U 0 9X3, 535' LC“! 0 O 02/! ‘— 9>3 -52- may». < Segregate equations after substitution of terminal equations: ”lg-2+ J. WLJ.’ XX. 93;] ’ a. x .4) U1 0 \i) fld+i$g _. a. .=o lg ii. 74 (509.5) Unknowns: (2332 and OXb. Numloer of equations: (v - 1 - n 0y) = (v - l - nx). 5.1(3 Systems of A and JK Comnonents Chord formulation is used. In the formulation tree: All Ny-elements. In the chord set: All Dy-elements, and all JK-elements if PCDSSlble. In any case, the maximum possible number of J terflfilrial equations should relate only chord variables. In this procedure, it is assumed that all JK-elements are 1.r1 the chord set. CirCL11.t equations: 333 32>? 7% (5.10.1) EXplicit form of the segregate equations from (5.10.1): — 71a: [(3, £2; 433: (3:) ”U... (5.10.2) O, ”a. (4 -674” Terminal equations after substitution of (5.10.2): ‘Xa: 3165 62:, fl 354] 3% (5.10.3) 74... :9?” (5.10.4) (XKZ: Y7K] (501005) Terminal equations after elimination of {Zak}: ngjictu' (We) a] 74. (5.10.6) fix“:- 0K?“ (5.10.7) -64- Circuit equations after substitution of (5.10.6) and (5.10.7): F W ' , I I I F U — a. file. (age) a] a. 10 a. 6‘" gm -+ U ’XK'+ ‘55:: r>(y1 in the second row. Therefore, another sub- stitution can be made which reduces the number of simultane- ous equations by n A direct substitution procedure in the kl’ general form is very unwieldy. However, the equivalent of substitution for c>" .. Q... 6“ and the aetrix to be inverted is symmetrical, if /{B is sym- metrical. 5.12 A formulation problem of considerable practical impor- tance arises freouently in electromechanical the syste; graph consists of at least two separate subtraphs. systems, §ystems of A, B and JK Components--Separate Graph where The terminal ecuations for the electrical subgraph are most likely to tions for plicit in the y-variables. graphs is procedure Koenig [10]. usually of the "direct" type. be explicit in the x-variables. The The terminal equa- the mechanical subgraph are more likely to be ex- The COUpling between the two sub- formulation shown here is an extension of one prOposed by If the electrical-subgraph variables are denoted by subscript l, noted by subscript 2, ten as: F I I l ‘21 922: (9 C) I C) C) I 0 _Q____0_ i- it}- 4.11: 0 i a 3 j- : 0 0 ,'-‘U 0 :‘K. an: o 0 I 0 ~74 1 "k. $2 3-?4 0 I o 0 E74,, 272: 97225 0 74' 0 0 :74. “22' 0 :‘o ‘. : a 424i a“ t 0 0 1 o o 5 0 0 i 5 01 O ”K. K. .. 62;] .7 . T .4544 £3235 {gm 0 and the mechanical-subgraph variables are de- then the system eauations can be writ- (5.12.1) d“ O -69- where the equations involving no-relation variables are omitted. If the first row and the last row are substituted into the terminal equations, the terminal equations remain explicit, and hence can be substituted into the second and seventh rows. In this case, chord formulation in used in part 1, and tree formulation in part 2. In this procedure there is no particular advantage in partitioning to show the JK—element variables distinct from those remaining. Therefore, suppose the terminal equations to take the following form: OXICL : :95 22’“ + 6K0 728 (5.12.2) V7 3' .> 6‘ , f 2 J" .’ / :’ 7 I + *} ./ '9) In the formulation forest: DX-elements. In the chord set: Dy-elements. Circuit enuations, part 1: fig" ._ .612. __O_-_ 7S —0 ‘E “'3" 7" 7.. (5.12.4) Explicit form of the segregate equations, part 1, from (5.12.4): ' (3’ ‘0 D :2 [ fig (022] (gm. (5.12.5) O -70- Segregate equations, part 2: — —- -2/-_-_1222_. 247a 2% .. 0 (5.12.6) 0 2J2: DJ; 7425 - 24241 Explicit form of the circuit equations, part 2 (7125 = [9332’ 422.] 724 (5.12.7) 7.. Terminal equations after substitution of (5.12.5) in part 1, and of (5.12.7) in part 2: yin. :: 0?“ 7 '54 3 ) . /_., l L) r- J ' , I) ~~~h —> 1 l _ ’ 1" 0 n 0 q 44 ,2, ( L ; f l _. O F (X 0 . (,« 4.. 0 /r,C I 4 1 . ( C 5 ( ' ‘ f I ,, ‘l - I ) l , '.' C 0 C “u u A... 4 0 //c 1 I / M l l C ‘I, i 1 I . ( . 0 0 9 0 ' (‘4 2,. f 1" 1 7.. ' 1.5).- ‘T-‘(" 4 i l j I l .4 . 1/ O O 0 O ’ (Tn /J . (0 ’X I _» i ‘ (1.725726 ) L. ( J ‘J ‘ ‘J ' (5.15.1) Subscripts l and 2 denote A-elewent and E-element vari- ables, reapectively. Coefficient matrix of (5.13.1) after substitution of the first row into rows 3 and 4, and substitution of row 8 into rows 5 and 6: . 1 r“ / U j I -4: a f C O 3 ?_ — ‘ ___ -fi 2 ~_ .— -_____ _ - _ - _. ..__ __ f C’ “J ,’ . JJ'a O O O 0 O 0 (31C - 31423) ~ 1 t 24:4 ”3/{ 0 L) 0 - ,,. 7 \ . " .- 0 0 (it #:1ct5‘1’?) _ 61"“ 0 L) 5” (.2 4" /'~ v j ‘ /__ - ’ \ O J J 0 ’ /I"tc ,. 2, )fi fr“. :2) O O a) 421 -av. ._ , , 0 44. (”a - .44..-.) 0 0 ( 0 , _ .0. u ”0— _ 0 _. ,,_ _‘_f”_ , _ QED/2_ _- “1/ _ _.() ! 0 0 0 0 (824 {-127.22 0 ’21 J (5.13.2) Coefficient matrix after substitution in (5.13.2) of row 2 into row 5, and of row 7 into row 4: - p h l fl F " <’ o J; x 0 O _ 0 0 _ -, "I _ - .i. _.-_ ,- __ -_i,, .i,__ _ -_ ,V W. C ’ I I j “J“ n O 0 [j I O 0 (’1; :t d? a) It (29" (l 0 0 O I E» .m. 0 O |(3c" “ct 92,3) ‘5 ct 1M» - 5:: “(13.2 i O O | _ l ’ j u ; O 0 ; 9823 - 1 34 ' )tEC we. (#2 " 7f“ .522) 0 0 0 l 0 3/ - ”c ”32' (“9h 746’} 0 I? g Q ”l 0 . ,C? 0_. i @Q , raj: hm%f- 0 I é-i _ O 0 0 0 (gel @732 ‘ J “5( (5.13.3) At this point, either of two substitutions may be made conveniently. Which is used would depend in general on the number of simultaneous equations to solved in each case. 1. If row 3 is substituted into rows 4, 5 and 6, the eduation count is (e - v + l - n + nt9)’ where nt2 is the ox rnndber of B-elements in the formulation tree. The variables in the simultaneous equations are the chord y—variables and the B-element x-variables. Therefore, if B-elements can be lnlt into some chord set with the Dy-elements. only (e - v + 1 - nox) equations result. 2. If row 6 is substituted into rows 3, 4 and 5, the equation count is (v - l - n0y + n10), wnere nlc is the num— ber‘cxf A—elements in the tree complement. The variables in ‘the simultaneous equations are the tree x-veriables and the -74- A—eiement y-variaoles. 1f, therefore, the A—elelents can be put into the formulation tree with the DK-elements, only (v - l - noy) equations result. ‘1 ?‘ 1"er ‘ ‘{-T:_ ' ‘ 'JRJ!’ 62L. VI. SYSTEM GRAPES FOR WHICH EACH PART INVOLVES EITHER ACROSS 0R SERIES VARIABLES ONLY 6.l System Grath of Cne Part Consider first a system graph, with which is associated only x—variables. The number of unknowns is (e - nx), there— fore there must be (e — nx) indegendent equations, if a u- nique solution exists. If the Dy-elements form a subset of some tree, the indegenuent circuit eduations nuiher (e - v + l). The additional (v - l - ny) equations must be K terminal equa- n must be exactly (v — l - nx). From this, it tions. Thus k2 follows that nk1 = (e — v + l - n01). This is so restrictive as +0 make it a trivial case. For the one-pert grabh invol— ving only Y-variahles, a similar situation exists. Thus, be- cause of a lack of practical use, the one-uart gragh does not seem to merit further consideration. The followina formulation procedures are written for £1 two-part system éraph. They aggly to a system gravh of Q ,‘ ,_ ' ‘ _ V ' ‘ 1'. .' ‘\ ' i more than two lJarts if the t'c‘ru. "part' is replaced by 'set of parts." 6,22 System Gragh of Two Iarts--Across Variables In the formulation forest: he DX-elements. In the chord set: NX-elenents. -75- hwy:- _._ .— .‘Zv? ‘ 1:. a“! v: 1-?- i‘gh System equauions: I'K. °l<. ”K. 'K. . 0 o 0/ 0 I . , I m. i din “(1 O O l O 0 Arm (5,71,, 1 a I ,, , o 0 (52. c1 . O 0 ‘M + £27?“ = 0 _ _ , _ _ _ I' - .. a __ __ ... .i 63; C632 0 0 I 0 0 Km; 637%5” I _;ffl ;- a O 0 I543 £44: 0 “y J Km L547 A/dz‘I (5.2.1) The last two rots may be set aside. The terminal equa— tions number (v - 2 - nx), since the svstem equations must nuaber (e - n7). whether they are explicit or not, the equa— tions to be solved simultaneously can be reduced to (v - 2 ~ nx) by substituting the circuit equations into thei, yielding I smut. + (kg-MAM. + mm. + Kata ——- 0 (6.2 .2) If the variables of part l can be eyhressed explicitly in ixarms of the variables in part 2, then a different broca- duxwalsay yield fewer equations. Suppose the teruinal equations to have the form ”a o w, a, xf’ 0 4:1 77;, "K- 7,." :0 (6.2.3) -77- iquation (6,2.3)--after the circuit equations of part 2 have been substituted in--to;ether with the circuit equations of part 1: F III 0 (Ike—[KM 823 )1 P‘Xtu—I PKG 627;)Idz—I o “u (712,-lt.a,l ”x... :— t-vamz (6.2.4) 0132:, [M O [‘thw _f|47