Topological Analysis and Synthesis of Communication Networks 9780231898553

Citation preview

Topological Analysis and Synthesis of Communication Networks

Topological Analysis and Synthesis

NEW YORK AND LONDON

1962

of Communication Networks WAN HEE KIM Associate Professor o f Electrical Engineering, Columbia University

ROBERT TIEN-WEN CHIEN Member o f Research Staff, I B M Adjunct Assistant Professor o f Electrical Engineering, Columbia University

Columbia University Press

To Chung Sook and Sophie

Copyright © 1962 Columbia University Press Library of Congress Catalog Card N u m b e r :

62-14636

Manufactured in the United States of America

Preface

implies, intrinsically, a collection of linesegments which interconnect points with a specified topological configuration. If each line-segment of a network represents an electrical element such as a resistor, an inductor, a capacitor, and a voltage or a current source, then the network is an "electrical network." On the other hand, if each line-segment of a network characterizes a contact of an electrical switch, then it is called a "switching network" and each line-segment is associated with two states of the contact, whether it is open or closed. One more important category of networks which has had a significant engineering application in recent years is that of the so-called "communication nets," often called "flow networks." Each line-segment of a communication net is associated with a fixed or random varying capacity of information flow or material flow. If the reliability of a communication net is studied and tested, each line-segment of the net may be associated with a probability of reliability (or of failure). For an electrical network, we consider the form of the voltage or current response in time or in frequency through the network. However, the sequence of closing contacts decides the characteristics of a switching network. The major objective of a communication net is the design of a network which will achieve maximum flow or minimum cost. Thus, each category of network described has its own techniques of analysis and synthesis. However, they are all still networks; e.g., they are basically the so-called "connected linear graphs." This book presents a uniform approach to various engineering problems, such as analysis and synthesis of electrical networks, sampled-data feedback control systems, switching networks, and communication nets. Chapter 1 of Part I studies the fundamentals of linear graph theory, always with an eye to their applications. The review of the theory of linear graphs is, however, minimized by including only what is essential for comprehension of the materials presented in the book. The book is designed to study the solutions of engineering problems rather than the theory of linear graphs itself. Topological analysis of RLC networks is the first subject discussed, since it is the THE WORD " N E T W O R K "

vi

Preface

most direct application of the theory. We then extend the analysis techniques to multipoles, i.e., networks containing multiterminal linear devices, such as vacuum tubes and transistors, transformers, gyrators, and circulators. Part II contains rigorous proof and derivation of the various properties of the so-called "signal-flow graphs" and of their applications to the analysis of feedback control systems and single-rate as well as multirate sampled-data systems. Part III discusses the analysis and synthesis of n-port networks. Then, in Part IV, the realization of loop and cut-set matrices is investigated, and a simple algorithm for the realization is proposed as an extension of some of the analytic techniques developed for the synthesis of «-port resistive networks. The realization of loop and cut-set matrices is, however, presented in terms of the synthesis of contact-switching networks. Part V deals with basic contributions already made in the area of communication nets. A complete theory is presented for the analysis and synthesis of one-terminal-pair flow networks. Also discussed are topics such as n-terminal-pair simultaneous-flow problems and applications of linear programming. Parts II and V are more or less self-contained, except that an understanding of a very few fundamental notions of linear graph theory is necessary for their comprehension. Part IV may not be comprehensible without the background of most of the materials presented in Chapter 1 of Part I and also of some developments in Part III. This book is written for use in a course on advanced network theory or as a reference book for those who desire to familiarize themselves with this subject or to pursue further research in this area. For the most part, this book is based on the senior author's class notes for a one-year graduate course which has been given at the Department of Electrical Engineering, Columbia University, since 1957. These notes were based on materials developed by doctoral students and staff members in the Department under research grants from the National Science Foundation. Part V is based mainly on contributions made by Dr. D. T. Tang, Dr. W. Mayeda, Dr. R. E. Gomory, and Dr. T. C. Hu, of the IBM Research Center, Yorktown Heights, New York. The authors acknowledge the sponsorship of the National Science Foundation and International Business Machines Corporation which has made this research possible. The senior author wishes to express his thanks to former students who have made his manuscript possible, including Dr. Omar Wing,

Preface

vii

Dr. C. V. Freiman, and Dr. J. A. Bernstein, particularly to Dr. R. T. Ash, Dr. I. T. Frisch, and also to a number of students in the Columbia University courses EE 305-306 (EEE 8201x and E 8202y) who have made numerous constructive suggestions for the improvement of the presentation of the materials. A number of constructive criticisms by Professor Unger, particularly for Part IV, were greatly helpful. We are grateful to Dr. C. C. Halkias for his careful reading of the manuscript and the proofs. The authors are indebted to the Circuit Theory Group at the University of Illinois, and especially to Professor M. E. Van Valkenburg and Professor S. Seshu (both authors were members of the group at one time), for their numerous contributions to much of the material included in this book. We owe a special debt to many authors who have made significant contributions in the field which we have incorporated into the materials presented here. Finally, we express our special thanks to Mrs. Susan McCauley for typing the manuscript. WAN H. KIM

June, 1962

ROBERT T. CHIEN

Contents

Preface

v

P A R T I. T O P O L O G I C A L ANALYSIS OF ELECTRICAL N E T W O R K S

1

Introduction

3

Chapter 1. Fundamentals of the Theory of Linear Graphs

4

Chapter 2. Analysis of the Ordinary Two-Port Networks

42

Chapter 3. Linear Active Multipoles

63

References

89

PART

II.

FLOW-GRAPH

TECHNIQUES

FOR

THE

SOLUTION

OF

LINEAR A N D SAMPLED-DATA SYSTEMS

93

Introduction

95

Chapter 1. Topological Properties of Signal-Flow Graphs

96

Chapter 2. Flow-Graph Analysis of Sampled-Data Systems Chapter 3. Flow-Graph Analysis of Multirate Sampled-Data Systems

126

References

131

P A R T III.

ANALYSIS A N D SYNTHESIS O F LINEAR A L P O R T

113

NET-

WORKS

133

Introduction

135

Chapter I. Formulation of TV-Port Networks

137

Chapter 2. Analytic Properties of Y- and Z-Matrices

157

Chapter 3. Realization of a ^-Matrix with

179

1) Nodes

Chapter 4. Generalized Realization of a Y- and Z-Matrix

198

References

213

Contents

X

PART IV. SYNTHESIS OF SINGLE-CONTACT SWITCHING NETWORKS AND REALIZATION OF LOOP AND CUT-SET MATRICES

215

Introduction

217

Chapter 1. Fundamentals of the Theory of Contact Networks

218

Chapter 2. Realization of Loop and Cut-Set Matrices

228

References

248

PART V. ANALYSIS AND SYNTHESIS OF COMMUNICATION NETS

249

Introduction

251

Chapter 1. Oriented Communication Nets

253

Chapter 2. Nonoriented Communication Nets

269

Chapter 3. Further Discussion on Communication Nets

295

References

306

Index

308

PART

I

Topological Analysis o f Electrical Networks

Introduction

SYSTEMATIZED development of the modern theory of electrical networks which is the basis of the study of almost all subjects in electrical engineering may be attributed to a number of reasons. One of the most important of these is the abundant supply of available analytic tools. In particular, the formulation of conventional network theory, which is based on ordinary two-terminal devices such as resistors, inductors, and capacitors, has been greatly advanced by the application of linear graphs. Moreover, this particular method is suitable for digital computers. Thus, the topological study of electrical networks has become indispensable to modern engineering. We shall therefore begin with the study of the fundamentals of linear graph theory. We will then formulate the various functions of ordinary networks in topological terms, and then extend the analysis techniques applied to an ordinary network so that they can also be used for the topological study of networks containing mutually coupled devices, as well as linear active devices.

THE H I G H L Y

CHAPTER

1

Fundamentals of the Theory of Linear Graphs

is BELIEVED that the work of Euler 1 in 1736 on the well-known "Königsberg bridge problem" (see Reference 2 for discussion of the problem) was the first formal study of linear graphs. In 1847 Kirchhoff 3 introduced the topological study of electrical networks and laid the groundwork for present network theory. Since 1847 numerous contributions have been made concerning the analytic properties of linear graphs as well as their broad applications to engineering problems. We shall, however, review only those basic concepts and properties of linear graphs which are necessary for subsequent discussions in this book, since the purpose of this book is to show the applications of graph theory to engineering problems, particularly in electrical engineering; the book is not designed for the study of graph theory itself. We will therefore study some of the fundamental concepts of linear graphs, always keeping their applications in mind. For a more extensive and rigorous study of the theory of linear graphs, other books 3 - 8 and papers 9 are recommended. The symbols and terminology used in this chapter follow, for the most part, that of References 5-7, 16, 20a, 32, and 33e. Definition 1. An "edge" (or arc, element, 1 -cell), is a line segment with two end points. An end point of an edge is called a " n o d e " (or vertex, 0-cell). If two nodes of an edge are identified with each other, the edge will be called a "self-looped edge." A node and an edge are "incident" with each other if the node is an end point of the edge. Definition 2. A "linear graph" (or graph, linear complex, 1complex) is a collection of edges with no self-looped edges. An "oriented graph" (or directed graph) is a graph in which the direction of each edge is specified. Otherwise, the graph is "nonoriented" (or nondirected). In the definitions which follow, we shall not specify

IT

The Theory of Linear Graphs

5

whether a graph under consideration is oriented or nonoriented unless necessary. According to Definition 2, a linear graph, or simply a graph, should not contain any isolated points. However, on many occasions, we will consider a single node of a graph to be a subgraph of the graph. If a graphical representation of a network contains self-looped edge(s) it will be referred to as a "signal-flow graph" (see Part II) so that we will be able to distinguish it from a linear graph. Definition 3. The "degree" of a node is the number of edges incident at that node. Definition 4. A graph in which every node is of even degree is called an "Euler graph." If every node of a graph is of the same degree, it is referred to as a "regular graph," and a regular graph in which every node is of degree A: is a "regular graph of degree k . " Definition 5. A " p a t h " between nodes i and j in a graph is an ordered sequence of edges in the graph in which every node is of degree two but nodes i and j are of degree one. If a direction of a path is specified, then the path is called a "directed path." Definition 6. If in a graph there exists a path between every pair of nodes in the graph it is "connected." It is therefore clear that a connected graph cannot contain any isolated edge or point. If there exists exactly one edge between any two nodes of a graph, the graph is a "completely connected" graph or, simply, a "complete graph," since there exists a direct path from one node to any other nodes in the graph. Definition 7. A connected graph is "separable" if it contains at least one subgraph which has only one node in common with its complement. This common node is called a "cut node" or "cut vertex." Definition 8. A " l o o p " (or circuit) is a connected regular graph or subgraph of degree two. If a loop of a connected graph contains all nodes of the graph, it has a special name: "Hamilton loop" (or Hamilton circuit). A collection of loops is called a "union of loops" and if no two loops in the union have an edge (or a node) in common it is a "union of edge-disjoint loops" (or union of node-disjoint loops). A set of edges which form a single loop is referred to as "prime loopedges" or "prime loop-set" while the term "loop-edges" or "loop set" is used for a set of edges which form a union of edge-disjoint loops. 15 Definition 9. A "tree" is a connected graph or subgraph containing all nodes of the graph but no loops. A " b r a n c h " (link) is an edge

6

Topological Analysis of Electrical

Networks

of a tree, and a "chord" (tie) is an edge of the complement of a tree. The complement of a tree of a connected graph is sometimes called a "co-tree." A fc-tree of a connected graph of n nodes is a collection of A: disjoint subgraphs of the graph containing n nodes but containing no loop. A A:-chord, or "Ar-co-tree" is the complement of a fc-tree of a connected graph. The concept of a /r-tree of a graph is illustrated in the following example.

(b)

•d

b*

(d)

•d

b*

c («) Fig. 1.1. Illustration

of a

k-tree

(a) Graph G; (b) some of the trees of G; (c) some of the 2-trees of G; (d) some of the 3-trees of G; (e) a 4-tree of G.

The Theory of Linear

Graphs

7

Example 1. Let us consider a graph G as shown in Fig. 1.1a, where the number of nodes is 4 and the number of edges 6. Some trees, 2-trees, 3-trees, and a 4-tree of G, are given respectively in Figs. 1.1b, 1.1c, l . l d , and l . l e . Note here that an isolated node is considered to be a subgraph of G. Definition 10. A "maximally connected s u b g r a p h " Gm of a graph G is a subgraph of G or the graph itself such that the addition of an edge in the complement of Gm to Gm makes the resultant subgraph no longer connected. Therefore, if G is a connected graph, the maximally connected subgraph of G is the graph itself. If G is a collection of disjoint subgraphs and each subgraph is connected, then each subgraph is a maximally connected subgraph of G. This is, therefore, a more elegant name for a so-called "connected piece." Property 1. Let G be a connected graph of b edges and n nodes. Then: A fc-tree of G has (n — k) edges, where n ^ k; A subgraph of G, which contains n nodes and (n—k) edges but n o loop, is a fc-tree of G. Proof: We shall prove the first property by the induction method. For n — 2, a tree of the graph contains one edge, and a 2-tree of the graph is a collection of two isolated points containing no edges. It is thus clear that the property is true for n = 2. Let the property be true for n = r, i.e. a k-tree of a graph of r nodes contains (r — k) edges and r > k. Since there should exist at least one node of degree one in a Ar-tree of r nodes, because it contains no loop, we add an edge at the node of degree one of the A:-tree such that no loop is formed. Then, the resulting graph contains ( r + 1 ) nodes and (r — k + l) edges but no loop. That is, a A:-tree of ( r + 1 ) nodes has (r-k + 1) edges. This completes the proof of the first property. For the proof of the second property, let us denote a subgraph of G which satisfies the conditions of the second property by Gi. Let us assume that Gi consists of m maximally connected subgraphs ?2 gm and that the number of nodes in each subgraph is i i , «2, . . . , nm, respectively. Since Gi has no loop, each subgraph is its own tree. Thus, m

(number of nodes in Gi) = 2 iand from the first property it is clear that

— "» I

0-1)

8

Topological Analysis of Electrical m

(number of edges in Gi) = 2 ( n < - l ) i -l

=

n—m.

Networks (1.2)

Thus, if m = 1, G\ is a connected subgraph of G and contains n nodes but no loop; G\ is then a tree of G and has (n — 1) edges. If m = 2, Gi consists of two disjoint subgraphs of G and contains n nodes but no loop; G\ is therefore a 2-tree of G and has (n — 2) edges. If m = k, G\ is a collection of k disjoint subgraphs of G and contains n nodes but no loop. Then Gi is a fc-tree of G and has (n—k) edges. Thus, the proof of the second property is completed. Definition 11. A "basic loop" (or fundamental loop,/-circuit or c-circuit) is a loop formed by each chord and some or all branches of a tree of a connected graph. It is therefore clear that a connected graph with n nodes and b edges contains (b — n + 1) basic loops. Definition 12. The "incidence matrix" (or incidence cut-set, or vertex cut-set matrix), denoted by A = [a2, incident at node a, because the product of the ,/th row of Q and the ath column of A must produce a zero. Edge ,¿2 must be incident at another node, say node Then, there should •exist an edge, say edge ¿>3, also incident at node /?. If we continue •this process of reasoning until we have exhausted all nodes of the graph, then it is clear that the set of the edges corresponding to the nonzero entries in the /th row of Q must correspond to a loop or • edge-disjoint union of loops in order to satisfy our assumption. This proves the second property. Since Property 5 is of primary importance in the study of network a

Fig. 1.4. A graph for the illustration of Property 5

14

Topological

Analysis of Electrical

Networks

topology, it needs further investigation. Let us consider a graph given in Fig. 1.4, with f o u r loops and their orientations chosen arbitrarily. Then, the incidence matrix A and the loop matrix of the four chosen loops B are f o u n d t o be

a b

1

2

3

4

5

6

7

8

1

0

0 -1

0

1

0

0

1

1

0

0

0

0

0

0

0 -1

0

0 -1

0

1

-1

c

0 -1

d

0

0 -1

1

1

e -

0

0

0

0 -1

-1

0

0

0

0 -

I II

-1

0 0 -1

0

-1

1

0 -1

1

0

1

0

0

0

0

1 -1

0

0

1 _

III

0

0

0

1

IV .

0

0

0

0 -1

1

N o w let us consider the second, third, and fifth columns of A. We multiply ( + 1 ) by all the elements of the first column, and (—1) by the elements of both the second and fifth columns, and add the columns for each row. The resulting column then consists of all zeros. T h a t is, the three columns are linearly dependent, because the edges corresponding to the columns form a loop. It is thus clear that a submatrix of the incidence matrix of a graph is singular if the set of edges corresponding to the columns of the submatrix contains a loop or loops. Bearing this in mind, let us study the following property. Property 6. A square submatrix of order {n — 1) of the incidence matrix A of a connected graph of n nodes is nonsingular if and only if the edges corresponding to the columns of the submatrix constitute a tree of the graph. P r o o f : We k n o w that there always exists a nonsingular submatrix of order (n— 1) since the rank of A is (n— 1) due to Property 2. We also know that a submatrix of order ( n - 1 ) of A is nonsingular if the columns of the submatrix correspond to a set of branches of a tree, because the columns are linearly independent (due to Property 5). For the proof of the sufficient part, we should show that the edges corresponding to the columns of any nonsingular submatrix of order

The Theory of Linear Graphs

15

(«—1) of A constitute a tree. Let us consider a square submatrix of A of order («—1) which is nonsingular, i.e., all the columns of the submatrix are linearly independent. Then, due to Property 5, the set of the edges corresponding to the columns of the submatrix contain no loop. Since the set of (n — 1) edges forms a subgraph of a connected graph but contains no loop, the edges therefore constitute a tree (because of Property 1). The proof of the property can also be obtained without using Property 5 at all. That is, first show that the sum of any k rows of the incidence matrix of a connected graph of b edges and n nodes, k < n, produces a row containing at least one nonzero element. Using this fact, we rearrange the rows and columns and apply the elementary transformation so that one can always reduce the incidence matrix to a triangular matrix with the last row of all zeros (see Reference 7). Property 6 is essential to the topological analysis as well as synthesis of the various types of networks. We shall study an interesting application of Property 6. a

2

3

Fig. 1.5. A connected graph Let us assume that we are given a connected graph in Fig. 1.5. Then A, the incidence matrix of the graph of Fig. 1.5, is found to be a i r

A = 2 3 L

b

e

1 - 1

-1 0

1

d 1

0 1

0

1

0—1

(1.11)

—1_

However, we would not need all three rows of A of Eq. (1.11) to completely characterize the graph. Since each column of the incidence

Topological Analysis of Electrical

16

Networks

matrix contains exactly one — 1 and one 1 we can construct in general the whole matrix of n rows by knowing only (« — 1) rows. This was actually implied by Property 2. This submatrix is denoted by Ai. Thus, for the graph given in Fig. 1.5, the incidence matrix which we will consider would be a

b

1 -1 -1

1

c

d

1

0

0

1

(1.12)

which is obtained from A of Eq. (1.11) by deleting the last row. Now, applying the second condition listed in Property 2 and Property 6, we know that the determinant of every submatrix of Ai is either 1 or — 1 if the columns of each submatrix correspond to a set of edges which form a tree. This property of the incidence matrix of a connected graph actually enables us to compute the number of all possible trees of the graph. Let us evaluate the determinant of the product of Ai and Ai' by using the Binet-Cauchy theorem.! That is, |AiAi'| = 2 product of every corresponding pair of major determinants of Ai and Ai', (1-13) where the summation is over all products of the corresponding majors. Referring to the incidence matrix of the graph of Fig. 1.5 we shall evaluate the determinant of (AiAi') using the Binet-Cauchy theorem: b

c

1 -1

1

a |AiAi'| =

1

1

0

d "

1 -1" a -1

lJ .

1 b

1

0 c

0

1. d

t Binet-Cauchy theorem: Let matrices C and D be of order ( p , q ) and (q,p), respectively, and q > p. Then det(CD) = sum of the products of the corresponding pairs of major determinants of C and D, where a major is a determinant of the highest order formed in a matrix. Or, in terms of the corresponding compound matrices, we have (CD) = CD, where k indicates the order of a compound and the elements of a Ath order compound of a matrix are subdeterminants of order k of the matrix arranged in lexicographical order. See Reference 13.

The Theory of Linear a -1

b +

-1 1

a

c

1

1

1 -1 a

-1

0

1

b

d

b

1 -1 1

1

11-1

1

a

1b

+

c 1 -1

1b

0

0c

1

17

Graphs

+

-1 1

= 0 + 1 + 1 + 1 + 1 + 1 = 5.

a 0c

0 -1

1b

1

1d

0

+

+

1 -1

d 0 1

-1 a

10

-1 d

c

d

1

Olli

0c

0

1||0

1d (1.14)

That is, the total number of trees of the graph of Fig. (1.5) is five. The product of the first corresponding pair of major determinants consists of two subdeterminants; one of which is formed from Ai taking the columns corresponding to edges a and b and the other from Ai' with the rows corresponding to edges a and b. It is, therefore, clear that these determinants both have the same value. Since edges a and b form a loop the majors should bqth vanish, according to Property 5. F o r the other product terms in Eq. (1.14) each set of edges corresponding to the columns o f a major o f Ai (or the rows of a major of Ai') forms a tree. Therefore, both majors o f each pair in the last five products will have the same values o f 1 or — I. Thus each product of the corresponding majors in the evaluation of (AiAi 1 ) will take the value of 0 if the edges corresponding to the columns (or rows) of the major contain a loop(s), or unconnected and of 1 if the set of edges constitute a tree. Thus, the determinant of (AiAi f ) is equal to the number of total trees in a connected graph. Hence, the following property. 11 Property 7. The determinant of (AiAi') is equal to the number of total trees of a connected graph. Ai is the incidence matrix of the graph with one row deleted. Let us denote the incidence matrix o f a connected and oriented graph G by A and a square submatrix of the incidence matrix corresponding to a tree of the graph or called a "tree m a t r i x " ! by A*. The matrix At i t is obtained by deleting row / and column j from A*, and |A{|u is the minor of the determinant of At of (i',7')-position. Property 8. The minor of the determinant of a tree matrix At has the following property (Frisch 2 0 a ) : |Ai|+l)th edge will be

B =

Pll .--Plb

1

Pl2 • • P2b

1

(1.32)

. Pml • • . Pmb That is, B will have the (¿ + l)th column of all ones. Since there exists no path-isolated subgraph in G, and the ring sum of two distinct paths corresponding to the rows of B yields a loop or an edge-disjoint union of loops and the path matrix P1 M'2 K>
] H|

H>4

M'a

4

1

4

3

5

5

2

1

2

!

Let us now introduce the orientation of edges in 2-semi-isomorphic graphs Gi and of ( n + I) nodes and b edges, and let us choose a tree-pair ti and /2 from G\ and G%. If we denote the incidence matrices 9

(a)

(b)

Fig. 1.15. Two-semi-isomorphic

2

*4

(a)

graphs: (a) G\ and (b) GÌ

3 (b)

Fig. 1.16. A tree-pair of G\ and Gz of Fig. 1.15 A tree of (a) C1 and (b) Gi.

34

Topological Analysis of Electrical

Networks

of t\ and ¡2 by Atl and A*,, respectively, then the sign of the common tree-product e of the tree-pair is given by e = |A ( ,A (l '|.

(1.33)

This is obvious because of Property 6. Property 18. The sign of the common tree-product of a tree-pair of 2-semi-isomorphic graphs is determined only by the active edgepairs in the tree-pair. Proof: Consider a tree-pair h and tz of (/i + l) nodes, containing k active edge-pairs and (n—k) ordinary edge-pairs. Then, arrange the columns of the incidence matrices of t\ and ¡2, At ; and At,, such that the first (n — k) columns correspond to the ordinary edge-pairs and the last k columns to the active edge-pairs. Thus, we get ordinary edges

active edges

aii . . . ai,n-k

am

Ai, =

= [Pu Qnn • • • fln.n-At

an,n-k+l • • • &nn

ordinary edges bn • • • b\,n-lc

P12] (1.34a)

active edges I

bin-k+l • • • bm

A,, =

= [Q11 Qw], (1.34b) - bnn • • • bn,n-k

I

¿n,n-ifc+l • • - bnn

where öl.n-t + l • • • a in

Oil . . . ai.n-t

(1.34c)

P12 = an 1 • • • an,-kn

an,n-k+l • • • ann

bn .. .b\,n-k

bi,n-k+l • • • bin

m =

(1.34d)

112 _ bn 1 • • • bn,n-k

- bn,n k + 1 • • • bnn

and since P n and Q n represent the incident relationship of the ordinary edge-pairs P11 = Q n , i.e., an = ba for /' = 1, . . . , n and j = 1, . . . , n-k. Now, assume that the edge 1 corresponding to the first column of A ( i and At, is incident at nodes r and .s, i.e., aT 1, a s i, bri, and b, 1 are

The Theory

of Linear

Graphs

35

nonzero and aTi = bTi = —a,i= —bti, and expand the determinants of At, and At, about the first column. If node r is the principal node of edge 1 of fi, node s is the principal node of edge 1 of /2 and s # r+ 1, then, interchange Jth and ( r + l ) t h rows in the matrices At, and At,, respectively, before expanding the determinants of the matrices At, and At,- If s = r+ 1, then no interchanges of rows are necessary. T h u s we get, using Properties 3 and 8 I AtJ = ( - l J ^ + ' f l r l l A / t J r i I At,I = ( - I ) r + 1 + 1 + « M A / < J S 1 ,

(1.35)

where u is the number of interchanges of rows (equal to 1 or 0), \Mt\ is the minor of the determinant |A

(2.14)

where again the columns of Bn include all chords of a tree and those of B12 correspond to branches of the tree, i.e., Bn is a nonsingular submatrix of order (b — n +1). Here V c and V& represent the chordvoltages and branch-voltages, respectively. From Eq. (2.14), V c = -Bn-^BisVft.

(2.15)

Again using Properties 9 and 13 to Eq. (2.15), we have Vc = -B/..V» = (Ai2 -1 Aii)'Vô v

=

rAu'(Aii-i) L

Ui

,V». J

(2.16a) (2.16b)

t As long as Eqs. (2.12) are valid and the elements of both matrices are real rational numbers, then Eqs. (2.13) follow.

46

Topological Analysis of Electrical

Networks

Using Property 15 we have f r o m Eq. (2.16b) V, = C,«V6.

(2.17)

It is therefore clear f r o m Eqs. (2.7) and (2.16) that a loop matrix of (b—w + 1) independent loops identifies all edge-currents of a n ordinary network if chord-currents of a tree are given, and so d o the incidence and basic cut-set matrices identify all edge-voltages if

(o)

i

a

(b) Fig. 2.1. (a) An ordinary electrical network and (b) the corresponding graph

The Ordinary Two-Port Networks

47

As an illustration, let us consider an ordinary network shown in Fig. 2.1a and the corresponding graph with each edge-weight in terms of edge-current or edge-voltage in Fig. 2.1b. In order to formulate KCL equations of the network from its graphical representation, one may open the driving-point terminals; node a and node d, the reference node of the network. Thus we have 1 0

0

1

0-

- 1 1 1 0

Ail« =

0

0

0 - 1 - 1

" Il

i

h

0

»3

1

0

ii

-

(2.18a)

-

. /5 where a

Ji

12

'3

u

»5

1

0

0

1

0

1

1

0

0

0 -1 -1

1

Ai = b

-1

c

0

(2.18b)

For setting up KVL equations, the driving-point terminals are identified such that the first loop is formed by edges 1 and 2. We therefore have for the orientation of each loop as indicated in the figure, " BVe =

1 -1

.

1

0

0

0 "

0 -1

1

0

0

1 .

0 -1

1

" Vl "

~ e '

V2

0

=

V3

(2.19a)

0

V4 - V5 -

and loop

I •

B = loop II loop III „

Vl

V2

V3

V4

V5

1

1

0

0

0 "

0 -1

1

0

0

1 .

-1

0 -1

1

(2.19b)

Now, let us consider the so-called "node-datum voltage" or "node-pair voltage" for each node with respect to the reference node

48

Topological Analysis of Electrical

Networks

of the network. We shall denote the voltage which appears at node a and is measured with respect to node d by Vmt. Then, we shall study the following expression, 1 -1

0"

0

0

1

Ai'V„ = 0

1 -1

1

0 -1

.0

0

-

Vad

Vad-Vbd

Vbd

Vbd

=

Vbd

~Vai

(2.20)

Vcd Vad—Vcd

1_

- Vcd

The right-hand expression of Eq. (2.20) then will be observed as nothing but the edge-voltages of the network. That is, Vad ~ Vbd = Vbd

Vi

= V2

Vbd-Vcd

=

V3

Vad-Vcd

=

V4

Vcd =

(2.21)

V5 .

It is therefore clear that in general (2.22)

Ai'Vn = Ve.

This expression is called the "node transformation" because the node-datum voltage for each node is found in terms of edge-currents. One may deduce a similar relationship between loop-currents and edge-currents. That is, if we consider the expression '

B'l, =

1 -1 1

0"

0 -1

0 -1

1

0

1

0

_0

0

1 _

l'l - 'II

" »I

»I - »III

»II 1-

»III

=

-1

»III — »II

(2.23)

»II - »III

then we can identify ii —in = n 'I —'HI = h ' i n —»II = »3-

»II = »4. 'Ill = »5

(2.24)

The Ordinary Two-Port

Networks

49

Thus, we find that one may write in general B'lj = Ae,

(2.25)

which is called the "loop transformation." This formula together with the node transformation describes the structure of a network, i.e., the topology of a network. One more set of expressions which characterize the behavior of an electrical network is the so-called "Ohm's law." That is, Zeh{s) = V«(j)

(2.26a)

Y«Ve(i) = Us),

(2.26a')

where Ze and Ye are "edge impedance matrix" and "edge admittance matrix" such that yi

0

0 >'2

Z2

Y« =

Ze =

0

Zb

0

J

yb (2.26b)

and zic and are impedance and admittance of edge k, respectively. Here I€(s) and V«(i) are column matrices such that h(s)

Vi(s)

Ms)

Vz(s)

Us) =

Ye(s) =

L Us)

J

(2.26c)

L Vb(s) J

in which Ik(s) and Fjt(s) are the Laplace transformation of edgecurrent i'jfc(r) and edge-voltage vt(t), respectively. We have thus derived the following set of formulas: Kirchhoff's law

AiWj) = O

(2.27a)

B/V^j) = O

(2.27b)

50

Topological Analysis of Electrical

Networks

Ohm's law Y,V e (j) =

(2.27c)

ZJU.S) = Ve(s)

(2.27d)

V,(J) = AI'V„(5)

(2.27e)

topological relationships h(s)

= B/'I,(5).

(2.27f)

In Eqs. (2.27), if we substitute (c) into (a), then, using (e), we get (AIY,AI«)V,(J) =

O.t

(2.28a)

From (b), (d), and (f), we obtain • (B/Z«B/0II(S) = O.

(2.28b)

In general, Eqs. (2.28) are written for an ordinary network containing independent ideal current and voltage sources as: (AIY,AI«)V»(J) =

J(s)

(2.29a)

(B/Z E B/)II(J) = E(j),

where

J(J)

(2.29b)

Jl

Ei

H

Ei E (, = )

=

(2.29c)

Jn in which Jk and Em are Laplace transforms of/¿(O and em{t), and j>c(t) is the sum of independent current sources connected at node k and em(t) the sum of independent voltage sources contained in loop m. One must note here that if a network under consideration contains both current and voltage sources, then before forming the incidence matrix of the network all the voltage sources should be converted into the corresponding current sources and then all the current sources are open-circuited. For the formulation of a loop matrix all the current sources should be converted into voltage sources, and then all the voltage sources in the network are short-circuited. t The formation of the node admittance matrix in terms of cut-set matrices will be considered in Part III.

The Ordinary Two-Port

Networks

51

Then, the right-hand expression of Eq. (2.28a) or Eq. (2.29a) is the so-called "node admittance matrix" and its determinant is the "node admittance determinant" or simply "node determinant," which is very familiar to us in the analysis of electrical networks. Similarly, the right-hand of Eq. (2.28b) or Eq. (2.29b) is the "loop impedance matrix" and its determinant is the "loop impedance determinant" or simply "loop determinant" of a network, f We therefore have, if we denote the node and loop determinants of a network by A„ and A/, respectively, from Eqs. (2.29) and (2.29) as: An = |AiY,Ai«|

(2.30a)

A, = |B/Z,B/«|.

(2.30b)

Now, applying our familiar Binet-Cauchy theorem in the evaluation of each determinant of Eqs. (2.30) it follows that A„ = sum of products of all pairs of corresponding major determinants of (AiY«) and Ai e (2.31) Aj = sum of products of all pairs of corresponding major determinants of (B/Z e ) and B / . (2.31b) As one may already have recognized in the expression of Eq. (2.31a), if the network under consideration consists entirely of one-ohm resistors then Y e becomes a unit matrix and then each determinant is equal to the "total number of trees" (see Property 7). However, this time we have the effect of diagonal matrix to consider. Because of diagonal matrix Ye the evaluation of the node determinant will give us the sum of products of admittance of branches of a tree for all possible trees of a network. So, if we call the product of the branch admittances of a tree a "tree admittance product" or simply a "tree-product," then An = sum of tree admittance products for all possible trees of an ordinary network

(2.32)

Similarly one can deduce from Eq. (2.31b), using Property 10, that t Since the incidence matrix is a u n i m o d u l a r matrix, E q . (2.28a) o r E q . (2.29a) is called a " u n i m o d u l a r c o n g r u e n c e . " By the s a m e t o k e n , if E q . (2.28b) or E q . (2.29b) is written in terms of a basic loop matrix, i.e. (B/Z«B/'), then it is called a " u n i m o d u l a r c o n g r u e n c e " because a basic l o o p matrix is a u n i m o d u l a r matrix. This will be thoroughly discussed in Part III. i This f o r m u l a was originally proposed by Maxwell 2 2 and is t h e r e f o r e called the " M a x w e l l f o r m u l a . "

52

Topological Analysis of Electrical Networks Aj = sum of chord impedance products for all possible trees of an ordinary network TV. (2.33)

Although we have thus derived two topological formulas for the analysis of a network the first is preferred to the second. There are a number of reasons to support this practice. We shall, however, see one reason which should be self-evident. That is, it is necessary to find all possible trees of a network to use either formula, but the second formula requires more work to identify each set of chords of a tree even after each tree has been found. There has actually been a considerable amount of work 23 27 done on digital computer analysis of an ordinary network based on the formula of Eq. (2.32). We now extend the formula of Eq. (2.32) in order to complete the topological analysis method of a network. A cofactor of the nodedeterminant of an ordinary network A„, A„, r can be deduced to be f A„(J = |Ai_,Y e Ai 'I = sum of tree-products for all possible trees which are common to both subnetworks Niir) and N{,r) ; (2.34) Ai_ ( is the submatrix of Ai with z'th row deleted and this corresponds to the subnetwork of NitT) of network N with node i and reference node r identified. The reference node is the node which was deleted in forming Ai. Let us define 330 : T = sum of all tree (complete tree) admittance products in a network. A tree admittance product is the product of admittance of all branches of a tree. The tree admittance product of a node is defined to be unity. •fiabc...) — s u m Q f a n t r e e admittance products in the network derived from the original one with a number of nodes, say nodes a, b, and c, . . . , made coincident. Assume a ^ b ^ c ^ . . .; otherwise T (abc. • •) = o. For example, T«"»»- • •> = 0. 7-(q'dV...)(g-i»'e'...) _ s u m o f a l l t r e e a d m i t t a n c e products in the network derived from the original one with a number of groups of nodes, say groups of nodes (abc . . .) and (a'b'c' . . .) made coincident, respectively. Assume a ^ b # c ± . . . , and a' # b' ^ c' ^ . . .; otherwise the tree-product vanishes. For example, 7*(° " • • • ) = 0. j(abe...) n (o'6 c ' . . . ) = s u m tree admittance products 0f an common to both T^abc- • > and r'3>'4 + >1>3>5 + )2W4 + J 2 J 3 J 5 j~(ab) j s the s u m Fig. 2b, and

0

f aii

tree

;

(2.35) admittance products of the subnetwork of

T(ab) = )'2)'i + y2}'b + >'3>'4 + >'3>'5 + }'4)'5 \

(2.36)

r ( c d ) is the sum of all tree admittance products of the subnetwork of Fig. 2c, and Tcd)

= vi>'2 + yi j'3+>'U'4 +;'2>'3 + J 2 J 4 ;

(2.37)

54

Topological Analysis of Electrical

Turned) ¡ s t h e s u m r«">> and TM, and

0f

Networks

tree admittance products common to both Taunton = y 2 y 4 -

(2.38)

(ab){ed)

T is the sum of all tree admittance products of the subnetwork of Fig. 2d, and jta&HCtf) = y 2 + y 3 + y 4 (2.39) Finally,

T(abcd) =

i by definition.

(2.40)

Again, let A n = the node determinant of an ordinary network; A„ (j = the cofactor of the node determinant for the ( i m p o s i t i o n ; A„(i kl = the algebraic complement of the second-order minor of the node determinant. (2.41) Then, from Eqs. (2.34) and (2.32), A„ = T An(> = 7™

for i = j

A„w =

for i # j,

(2.42)

where the rth node is the reference node of the network. It can also be verified that = T('ik)

for / = j and k = I

A n ., „ = 7™*>n"-«> r

r k)

for / = j and k * I

A„(J t ( = F 'A1 • • yhh yM •• — }'hn

-yn • •

(>'ir+;'u)..

+

-yhi •

)'hh )'hr • • —y'hn

-yn • • ~yih )'}T . -yn 1 • ••

—

• •

y'nh y'nr • •

-yjn ynn (2.46)

= A„(, + A V ,

where A' n is the node-determinant of the same network with the ith node as its reference node. Therefore, from Eqs. (2.42) and (2.43) one has A„(i = A'„ ri =

THr)mP.

(2.47)

The rest of the identities can be proved by a process similar to that used for the proof of identity 1. 33c Some of the identities can be derived in terms of two-tree admittance products. 8

56

Topological Analysis of Electrical

Networks

The various network functions of an ordinary network are expressible in terms of the topological identities just derived. For a twoport network with four terminals as shown in Fig. 2.3, the openlz

I V,o

2

N

v 3

0 Fig. 2.3. A four-terminal

network

circuit impedance parameters of the network are easily verified as rni

^

l

U21

Z22J

j

A

T

A

».,

A n L A» lt —An,j

"I

A„ it + A „ 3 ; i - 2 A „ i J

7110)

4[

»..-A»„

7110)0(20)

7110)0(20) _ 7^(10)0(30)

_ j(io)n(30>

y(23)

(2.48)

The short-circuit admittance parameters are also directly derived f r o m Eq. (2.48) and are given by >11

y 12"

.)>21 yzz.

l 77101(23)

7123) 7ii0)n(30) _

7110)0(30) _

7110)1-1(20)

Fig. 2.4. A three-ten.anal

711010(20)

7110)

(2.49)

network

For the three-terminal network of Fig. 2.4, the formulas r f Eqs. (2.48) and (2.49) are reduced to "211

212"

1

.221

Z22.

T

7110) 711010(20)

71101012' 7^(20)

The Ordinary

Two-Port

Networks

->ii

vi2~| _

i

J'21

V22J ~ T i m 2

57

r

_ 7-'1>'3>'4 +J'2J3>'5. (2.52) •

1, »

1, -V

V

+

V,o

\

Fig. 2.5. A four-terminal

Fig.

(c) 2.6. Subnetworks

N it

network

used for Example

necessary to evaluate network of Fig. 2.5 (a) /VI'0'; (b) yV; (c) /V'2 + >'3 + >'s) + >'3(>'2 + J 5 ) (yz+yf,)(yl+y2)+yly2

(2.53b)

(>-3 + >'4)(>'l+>' 2 )+>'l>'2

(2.53c)

(j4+>.5)(>1+>2)

=

(2.53d)

y^+yz+y^+yw

=

f(10)n(30)

=

(2.53a)

y1y2 + y1y3 +

(2.53e)

y2y3+y2yi.

(2.53f)

W e have, therefore, zn = (>'i + yi)()'2++>5) Zl2

=

Z21

+ y3(y-i+>'s)/ T

= j(i0)n(20)_7Xi0)n(30)/

( y ^ - y ^ / T

(2.54)

z 2 2 = r' 23 » = (>'4+>' 5 )(3'i+>' 2 )/r. Let us consider the c o f a c t o r A B i i of the n o d e d e t e r m i n a n t A„ of the n e t w o r k . Then, AB]i = r m = (>'1 + > ' 4 ) ( > ' 2 + > ' 3 + y s ) + y s ( y 2 + j 5 ) • (2.55) If we define a "2-tree a d m i t t a n c e p r o d u c t " (or simply a 2-tree p r o d u c t ) as the p r o d u c t of the admittances of the branches which constitute a 2-tree, with o u r convention that the tree-product of a single n o d e is unity, then each tree-product t e r m in Eq. (2.55) c o r r e s p o n d s to a 2-tree of the n e t w o r k as shown in Fig. 2.7. Since all the tree-products which a p p e a r in Eq. (2.55) c o r r e s p o n d t o all the 2-tree p r o d u c t s of network N, one can express the c o f a c t o r A B i i in terms of 2-tree p r o d u c t s of N instead of in t e r m s of treep r o d u c t s of the subnetworks. Each of the 2-trees s h o w n in Fig. 2.7 has two separated pieces such that one part c o n t a i n s n o d e 1 and the other n o d e 0; the reference node. This may be clearly u n d e r s t o o d f o r the following reasons. The p r o d u c t terms in Eq. (2.55) are the trees of A" 1 0 ' which is the s u b n e t w o r k of N with nodes 1 and 0 identified. T h e r e f o r e , the n u m b e r of nodes of the s u b n e t w o r k is one less t h a n that of the original one. T h e edges connected between nodes 1 and 0 would n o t a p p e a r in the subnetwork. If we choose a 2-tree of the original n e t w o r k such t h a t one part of the 2-tree contains n o d e 1 and the other n o d e 0, then the edge(s) connected between nodes 1 and 0 will never be c o u n t e d . In general, a tree of a network N (or a connected g r a p h ) of ( « + 1) nodes contains n branches. But a 2-tree of the same n e t w o r k has

The Ordinary Two-Port

Networks

59

(n— 1) branches while the number of branches of a tree of subnetwork N{il) of N is also (n— 1). Thus, denoting the sum of 2-tree products of a network such that one part of each of the 2-trees contains node i and the other node j by Ttj, we have the following identity: r"" = 0 T«» = TUi

for i = j

(2.56)

for / / j.

Extending the concept of a 2-tree product defined in Eq. (2.56) one obtains jumki)

=

T(kij

for / # j # k.

(2.57)

The substitution of Eqs. (2.56) and (2.57) into Eq. (2.44) yields the following identities in terms of 2-tree products of a network: -

'

I

' V W ^ • 2

Fig. 2.7. Two-tree products which appear in Eq. (2.55) (a) .»•!>•»; (b) .»xvj; (c) ,vi.vs; (d) a n ; (e)

(f) w s ; '2aia2gm+^ij3gpaia2+7i>'2gpaia2+>'i>'2>'3aia2. (3.19) Extending the formula of Eq. (3.18) into a cofactor of (p, ^ - p o s i t i o n , An , of the node-determinant, A n , of a network, one obtains A„ p? = ]>«'< x a common tree-product w< of current graph G/ i and the corresponding voltage graph Gv { Q r ) , (3.20) where G / ( p r ) is the subgraph of the current graph G j of the network derived from G with node p and the reference node r made coincident, and Gv {QT) is the subgraph of the corresponding voltage graph Gv with node p and the reference node made coincident. The sign for a common tree «'3 = 2a 3 .

(3.21)

In order to evaluate the cofactor of the node-determinant A„ u by the topological method proposed, we must first obtain the subgraphs of current and voltage graphs of the circulator, G / ( l n > and

Linear Active

Multipoles

81

Qv(in) xhey are shown in Fig. 3.20. Inspecting the current and voltage subgraphs of Fig. 3.20, there exist two complete trees «i and «2 as shown in Fig. 3.21. Since U2 consists of the ordinary edges, the sign of tree products is positive. For u\ active edge-pairs and 1x3 (shown in Fig. 3.22) will have signs of — 1 and + 1 , respectively. If we interchange the

Cl.nl

(b)

Fig. 3.20. Subgraphs of the graphs of Fig. 3.14: (a) G/, (b)

(b)

(0)

Fig. 3.21. Tree-pairs of the subgraphs of Fig. 3.20 T r e e - p a i r (a) « 1 ; (b) 1/2.

,.o 2

i k «s

(0)

(b)

Fig. 3.22. Active edge-pairs in tree-pair «1 of Fig. 3.21 E d g e - p a i r (a) a 2 ; (b) a 3 .

Gva"

82

Topological

Analysis

of Electrical

Networks

principal nodes of the edges in the pair «3 only once, they will be the same as the principal nodes of the edges in the pair x 2 . T h e r e f o r e , the sign associated with the c o m m o n tree-product of Fig. 3.22a, is f o u n d to be €', = ( - l ) i ( - l ) ( l ) = 1.

(3.22a)

Hence, and

An,, = aij2+V2>'3 = 2a 2

(3.22b)

Z n = A„ u /A„ = 1/a.

(3.22c)

TOPOLOGICAL FORMULAS AND TRANSFORMATIONS

OF

NETWORK

FUNCTIONS

Let us denote the g r a p h corresponding to the m a t h e m a t i c a l equivalent circuit of network N by G, and the c u r r e n t a n d voltage graphs of the equivalent circuit by G / a n d G v , respectively. N o w , let us define 3 3 e - { : G(ii) GUi> T e T] [ \ T v £

Ty(mn>

T/UJ) e

A the s u b g r a p h of G with n o d e s ; and j m a d e coincident; A the subgraph of G with all the nodes except nodes / and j m a d e coincident; A sum of all tree-products in G ; A sum of all tree-products which are c o m m o n to b o t h G1 and G y with the p r o p e r sign f o r each tree-product defined by E q . (3.18); A sum of all tree-products which are c o m m o n to b o t h G ¡ ( i i ) and G v ( m n ) with the p r o p e r sign for each t r e e - p r o d u c t ;

1~i f") 7 Y < m n > A sum of all tree-products which are c o m m o n to b o t h G / ' ^ and G y < m n > with p r o p e r sign for each tree-product. Then, f r o m Eq. (3.17) and (3.20), A„ = T j H 7V. A„ i; = 77«" Pi where the rth node is the reference n o d e of N.

(3.23a) (3.23b)

Linear Active

Multìpoles

83

If N consists entirely of the ordinary elements, then Eqs. (3.23) may be reduced to A„ = T kn v q = rfJ»rtn(«rt

(3.24)

as we found in the previous chapter. Example 14. Given: the tube circuit shown in Fig. 3.23a. the transfer function V%V\. 3

Find

2

(a)

(b)

Fig. 3.23. (a) Tube circuit and (b) its mathematical

equivalent circuit G

We first find the mathematical equivalent circuit of the network as shown in Fig. 3.24 in terms of current and voltage elements, where gPi and gp, are the plate conductance and g m , and gmj are the transconductances of the first and second tubes, respectively. If we denote the node-determinant of the network by A n , then the ratio of V2 and V\ may be expressed as: V2I Vi = A,,,/A„„ = [7V 1 » n

7V 2 5 ) ] I [7/«» f l TV 15 '].

(3.25)

In order to evaluate Eq. (3.25) it is necessary to derive the subgraphs of the current and voltage graphs G/ ( 1 5 ) , G k | = (-l)«[7i f | Tv'" n > ],

(3.28)

(3.29)

where A pi r v < i n > ) . . . - ( r ; < i » > PI 7V) e

e

_(7'/ P|7'K)-_

(Tl p| r K )

It should be clear that if a network under consideration contains only the ordinary elements, Eq. (3.30) is reduced to

•3

~)'3

Vl+>'2 +V3

(3.32a)

K'

P'

g'p+y'i+y's

-g'p-y'i-g'r,

-y'3+g'm'

K'

-g'p-y't

g'p+y'i+y'2+g'm

-y'2-g'm

• (3.32b)

— v'a —y' 2 y'z+y'z By applying the formula of Eq. (3.30), i

XZ + . . . + aznXn = }'2

flnlXl + a-nzxz +... +

xn

(1.1)

= in-

Any independent set of n linear equations in n unknowns can be manipulated into the above form. In matrix form Eq. (1.1) becomes u / X = Y,

(1.2)

Topological

97

Properties

where 1 021

=

_ ani

an .

•

a\n ' (1.3a)

1 . • a 2n an 2. . 1 .

and " y1 (1.3b)

>•2

X2

X =

Y =

- Xn

_

. yn

-

Definition 1. The "signal-flow g r a p h " associated with Eq. (1.1) is an oriented graph with a node corresponding to each unknown x< (/ = 1 , 2 , . . . , « ) and a node corresponding to each nonzero constant y}. We shall refer to the node corresponding to xt as " n o d e / " and the node (if any) corresponding to y j as "node y j . " If i j and aij / 0, there is a directed edge with weight ( — aij) incident on nodes i and j. The direction of the edge is toward node /. If y j ^ 0, there is a directed edge with weight 1 incident on nodes j and yj, with the direction of the edge toward node j. For example, consider the following set of equations: xi-fxz-exz

= y\

- a x i + x z — dxi = 0 -bx2+x3-gxi

- >'3

(1.4)

—CX3+X4 = 0. The signal-flow graph corresponding to Eq. (1.4) is shown in Fig. 1.1. Definition 2. A "proper edge" of a signal-flow graph is an edge not incident on any of the nodes y j for j = 1,2, . . . ,n and a " p r o p e r n o d e " is a node corresponding to each unknown x< for i — 1,2 For the remainder of this chapter, the term "edge" will mean " p r o p e r edge" and the term " n o d e " will mean "proper n o d e " unless otherwise specified. In other words, we shall restrict our attention to the graph obtained f r o m the original signal-flow graph by removing nodes yj { j = 1,2, . . ., n) and the edges incident on these nodes.

98

Flow-Graph

Techniques

Fig. 1.1. Signal-flow

and Linear and Sampled-Data

graph corresponding

Systems

to Eq. (1.4)

Consider a signal-flow graph with b edges, denoted by b\, bz, . . . , bb and n nodes, denoted by 1,2, . . . , n. Definition 3. The "exit matrix," denoted by Ji of the signal-flow graph, is a matrix whose columns correspond to the edges of the graph and whose n rows correspond to the nodes of the graph. The element mi) of M is defined by: m i j = 1, m i j = 0,

if edge bj is incident at node /, with the direction of bj away f r o m node /'; otherwise.

Definition 4. The "entrance matrix," denoted by 9>, of the signalflow graph is a matrix whose b columns correspond to the edges of the graph and whose n rows correspond to the nodes of the graph. The element d a of is determined by: d(j = 1, d t j .= 0,

if edge bj is incident at node /, with the direction of bj toward node /; otherwise.

For the graph of Fig. 1.1, we find that a

b

c

d

e

/

g

1

1

0

0

0

0

0

0-

2

0

1

0

0

0

1 0

3

0

0

1

0

1

0

0

4 .0

0

0

1

0

0

1 _

Topological

99

Properties a

b

c

d

e

lrO

0

0

0

1

@ = 2 3

1 0 0 0

4L0

g 10-1

1 0 0 0

1 0 0

f

0 1 0

0

0

1

0

0

0.

(1.6)

are always assumed to be ordered The rows and columns of Ji and in the same way. Notice that { J i — & ) is the incidence matrix of linear graph theory, which we defined in Part I. Definition 5. The "edge-weight matrix," denoted by W, of a signal-flow graph with b edges is a diagonal matrix of order b whose rows and columns correspond to the edges of the graph. The element Wij of W is defined by: w(f = 0, if i ^ j; W(( = the weight of edge bi. The ordering of the columns of W is the same as the ordering of the columns of Ji and Ql. For the graph of Fig. 1.1. the edge-weight matrix of the graph is found to be 'a

0

0

0

0

0-

0 b

0 0

0

0

0

0

c

0

0

0

0

0

0

W = 0 0 0 i / 0 0 0 0

0

0

0

e

0

0

0

0

0

-0

0

0

0

0

0

(1.7) 0

/ 0 0

s-

The three matrices Ji, @>, and W completely characterize a signalflow graph. The basis for this statement is the following theorem. Theorem 1. Suppose that a signal-flow graph with matrices Ji, S>, and W describes the set of equations u&X = Y, where c*/ is the coefficient matrix of the form Eq. (1.3a). Then, u-®wjir< =

¿a,

where JC is the transpose of Ji and U is a unit matrix whose order is equal to that of Proof: Let q t j be the element in row i and column j of the matrix ( U - S » W ^ ( ) - Then, if i * j, it is clear that

100

Flow-Graph Techniques and Linear and Sampled-Data qtj = -

Systems

b b y 2 ditwtrmjr. t - l r-1

(1.8)

Since W is a diagonal matrix, the expression for q r e d u c e s to qu = -

b 2 dit w ** m )tk^1

(1-9)

It follows from the definition of the three matrices that dikWkktnjk = Wkk = the weight of edge bk, if bk leaves node j and enters node /'; (1.10a) dtkWkkfftjk = 0, otherwise. (1.10b) By construction of the graph, there cannot be more than one edge between any pair of nodes. Hence, if an edge bk leaves node j and enters node i, then —qu is the weight of bk. If there is no such edge, qij = 0. Therefore, qi, bi, . . . , bk).

Topological

Properties

101

Proof: We consider the matrix Ji. The proof f o r 3 is identical. First suppose that each node of the set (1, 2, . . . , k) has exactly one exit-edge from the set (bj, ¿2, • • - , bt). Then there is exactly one nonzero element si in row 1 of S. There is exactly one nonzero element sz in row 2 of S ; sz c a n n o t be in the same column as j j , f o r Ji (and therefore S) can have n o m o r e t h a n one nonzero element per column. Continuing in this way. there is exactly one nonzero element st in row k of S, and st is not in the same column as any of the elements Ji, S2, . . . , Hence the determinant of S is ( ± 5i, sz st) = ± 1 , a n d S is nonsingular. N o w suppose S is nonsingular. If node / (1 ^ / < k), has no exit-edges from the set (¿>1, bz, . . . , bt), then row / of S consists entirely of zeros, contradicting the hypothesis of nonsingularity. If node / has two exit-edges bT and bn (1 ^ r, n < k), then columns r and n of S are identical, again contradicting the nonsingularity of S. Hence, the lemma. The proof of L e m m a 1 shows that all nonsingular submatrices of Ji and 3 have determinant ± 1, i.e., they are u n i m o d u l a r matrices. Theorem 2. C o r r e s p o n d i n g square submatrices S i and S2 of Ji and 3, respectively (submatrices defined by the same rows and columns) are both nonsingular if and only if the edges corresponding to the columns of S i and S2 f o r m a feedback loop or a set of nontouching feedback loops. Proof: For convenience, assume that the edges corresponding to the columns of S i a n d S2 are ordered ¿1, ¿2, . . . , bt and that the nodes corresponding to the rows of S i and S2 are ordered 1, 2, . . . , k. By L e m m a 1, S i and S2 are both nonsingular if and only if each n o d e of the set (1,2, . . . , Ac) has exactly one exit-edge and one entrance-edge f r o m the set (¿»1, ¿2, . . . , bt), However, a subgraph in which each n o d e has exactly one exit-edge a n d exactly one entrance-edge is a feedback loop if the subgraph is connected, or a set of n o n t o u c h i n g feedback loops if the subgraph is not connected. Hence, the theorem. In the graph of Fig. 1.1, consider the feedback loop f o r m e d by edges b, c, and d. F r o m Eqs. (1.5) and (1.6), the corresponding nonsingular submatrices of Ji and 3 are b e d

b e d 21" 1

0

0"

0

1

0

4L0

0

1.

Si = 3

2ro 0 S2 = 3

1

0

4L 0

1

0 1.

(1.12)

102

Flow-Graph

MASON'S

Techniques

GENERAL

and Linear

GAIN

and Sampled-Data

Systems

FORMULA

The solution to Eq. (1.1) may be obtained directly from the associated signal-flow graph by use of Mason's general gain formula. In many situations, including the analysis of linear feedback control systems, the use of Mason's formula is far more efficient than the standard algebraic methods of solution. In a later section we shall give a proof of Mason's formula based on linear graph theory. We first establish two preliminary lemmas. Lemma 2. Let Si and S2 be corresponding nonsingular submatrices of j f i and S>, respectively. Let L\, Lz, . . . , Lr be the (nontouching) feedback loops formed by the edges corresponding to the columns of Si and S2. Let N be the number of loops in the set (L\, Lz, . . . , Lr) which have an even number of edges. Then Si and S2 have the same determinant if and only if N is even. Proof: Consider a feedback loop L\ for i = 1,2, . . . , r. Suppose that the edges of Li are ordered bn, bt2, . . . , bin, and that the nodes of Lt are ordered so that the edge bij leaves node ij and enters node ')+1 U — 1» 2, . . . , m). Let S „ be the submatrix of Si formed by columns bn, bit, • • • , bin, and rows i 1, iz, . . . , /»,• Let Sz< be the corresponding submatrix of S2. Then we have bt 1

il

r

H

bi 1

biz...bin4

1

0...0

0

1...0

Si( =

biz.-.bin.

/'l - 0

0. . 1

iz

1

0 . .0

'3

0

1. .0 (1.13)

S2Î =

iVL

0

0...1

In, L

0

0...0

S k and S2< are, of course, nonsingular. Then S21 may be made to coincide with S n by («0 is f o r m e d f r o m Q by deleting rows k + 1, . . . , n and J f formed from M

To

. . . , n f r o m R and

is

0

in the same w a y .

Since the minor | n . . .

r**| is the determinant o f the product o f the

t w o matrices ( ® 0 W ) and J l 0 K the B i n e t - C a u c h y theorem applies. T h e matrix ( ® 0 W ) is f o r m e d f r o m 3>0 by multiplying each column o f

by

the weight o f the edge corresponding to that column. Hence, a square submatrix o f

is nonsingular if and only if the c o r r e s p o n d i n g

submatrix o f 2>0 is nonsingular.

By T h e o r e m 2, a typical term in the

is ± W1H2 . . . H-*, where H < is the weight o f

expansion o f

edge bi, and edges b\, . . . , b* f o r m a feedback l o o p or a set o f nontouching feedback l o o p s L\, . . . , Lr incident on nodes 1 , 2 , . . .

,k.

Since all possible combinations o f nodes appear in Eq. (1.22), w e have established the f o r m u l a , Eq. (1.19) except f o r considerations o f sign. T o complete the p r o o f , consider T a b l e 2. T h e table shows that the Table 2. Sign associated

with a set of nontouching

Sign of wi . . . wie in I he expansion of |ni • . . Ortl (by Lemma 3) 1 2 3 4

even even odd odd

Sign of . . . rti| in Eq. (1.22)

feedback

loops

Sign of vvi . . . m> in the expansion of A

even odd even odd

sign associated with a set o f nontouching feedback loops is positive if and only if the number o f loops in the set is even, exactly as stated in the f o r m u l a Eq. (1.19). Thus, the p r o o f is complete. Definition

8.

A " d i r e c t p a t h " or " f o r w a r d p a t h " f r o m n o d e i t o

node j o f a signal-flow graph is a connected, ordered sequence o f edges whose initial node (first node relative to the o r d e r i n g ) is i and

Topological

Properties

107

whose final node (last node relative to the ordering) is j. Node / has one exit-edge but no entrance-edges; node j has one entrance-edge but no exit-edges, and all other nodes have exactly one exit-edge and exactly one entrance-edge. The " f o r w a r d path p r o d u c t " or the "weight of a forward p a t h " is the product of the weights of the edges constituting a forward path. Theorem 4. (Due to Mason.) Suppose a signal-flow graph G characterizes a set of equations oi/X = Y, where a / is the coefficient matrix of the form of Eq. (1.3a). Let s / i j be the cofactor of the element at] of | o / | , Then if i ^ j, stfi] is given by

J*u = 2

k

W(Pk )bk ,

(1.24)

where W(Pk) is the weight of the forward path P t f r o m n o d e / to node j, and A* is the formula, Eq. (1.19), evaluated for the g r a p h obtained f r o m G by deleting all the nodes of f t and all edges incident at these nodes. The summation is over all forward paths from n o d e i to node j. The cofactor j ^ « is given again by Eq. (1.19) evaluated for the graph obtained from G by deleting node / and all edges incident at node /. As an illustration, for the graph of Fig. 1.1, s/u

= abc

s/n

= 1 —bed—eg

J&12 = a(l —eg) = s/32 =

(1.25)

\-af ea+cd.

Proof of Theorem 4: Without loss of generality let us consider the cofactor sfn \ for n # 1. Then s/„i may be written as follows: a 12

«13- • • flln

J / . 1 = ( - 1 ) 71 + 1

(1.26)

Q fln-l.n

108

Flow-Graph Techniques and Linear and Sampled-Data

where l Q = a22

3

n-l

a 23

• • .02,n-l

1

• . • Û3,n-1

Systems

(1.27)

On-1,2

an-1,3

... 1

Let us now expand s / n i by the Cauchy expansion of a bordered determinant. We obtain stn\

= ( - l)»+i[(- l)»ai»0 + C-1)»-

1

n-l ^ a{„affQij], U=2

(1.28)

where Qa is the cofactor of the element in row i and column j of Q for i ,j = 2, 3, 1. N o w let tjt = — a t j . Then t n is the weight of the edge b)i leaving node j and entering node i of G. With the substitution tji = —at], Eq. (1.28) becomes n-l (1.29) J&nl = tnlQ+ 2 'ni'nQil2 The subgraph G' corresponding to Q is the original graph G with nodes 1 and n removed, as well as all edges incident at these nodes. In particular, G' does not contain edges bni or bji (i,j= 1,2, . . . , n). (See Fig. 1.2.) I|l= -Oil —fr— (b)

(o)

Fig. 1.2. Edges which are not contained in G' (a) Edge ¿ n i ; (b) edge bji. The terms tniQ and /„ / kT\ 2 g(nT)f>(t — nT) = 2 f[nT+—h(t-nT). n——ao n——oo \ pi

(3.2)

Thus 2 _ / ( . r

+ T

) . ( . - . r - - ) .

p.»

The output of the system of Fig. 3.1b is then given by /*(>3 which f o r m a tree of the parent network, we have K V L equations, V4 Bf\e

=

J'5

>'1

V2

>'3

rl

0 -1

-1

- n

.0

1 - 1 - 1

oj

r

V* 1 Ks Fi

= 0.

(129)

Vz V3 .

+ In other words, there should be no node in a f o r m u l a t e d n-port with only current sourses connected to it.

Formulation of N-Port

Networks

151

If we form the basic cut-sets with respect to the same tree, then they are given by the basic cut-set matrix C/ J4

ys

ji

>'2

J3

~ 1

l

l

0

0

1

l

0

1

0

. 1

0

0

0

1.

C/ =

(1.30)

Thus, K C L equations for the basic cut-sets are found to be Cfh

where

h

=

=

(1.31a)

CfJe,

14

0

is

0 and

il

J, =

(1.31b)

M

h

jb 2

¡3

L jb3 J

The right-hand expression of which each row represents the connected across the branches expression by a column matrix

Eq. (1.31a) is a column matrix in algebraic sum of the current sources of the tree. We therefore replace the such that

J = C,J„

(1.32)

where an element of J , jt, is the current source connected across the &th node-pair. Referring to the same network, one also can readily verify the relationship between the edge-voltages and branch-voltages, as shown previously that (1.33a)

Ve = C / V „

and Vi

Vi

v5 Ve =

Vi

v2

and

V6 =

V2

(1.33b)

V3

V3 Using Eqs. (1.31), (1.32), and (1.33) with Ohm's law, we obtain J = Cfh = CfYeVe = C/Y«C/'Vft

(1.34a)

Analysis and Synthesis of Linear N-Port

152

Networks

and 0

>'4 Ye =

0

)'i

0

0

0

0

0

0

0

0

0

0

n

0

0

0

>'2

0

0

0

0

0

V3

(1.34b)

It is thus clear that the short-circuit admittance matrix of the network, Y, is given by: Y = C/Y«C/' 0 1 1 1 0 1 1 0

yi+>'4 4-^5

1

0

0

0 - -1

0

0

1

0

1 0

0

1

1-

0 "

0

>'5 0

0

0

0

yi 0

1 .

0

0

0

X2 0

0

1

0

.0

0

0

0 V3 - _0

0

1 „

y*+ys

>'4+>'5

yz +>'4 +>'5

>'4

>'4

1 0

>'4 yi }'3

(1.35)

+>'4

Thus far, we have considered the derivation of multiports based on the short-circuit admittances. Let us consider an ordinary network with b elements and (n + 1) nodes. Then, with respect to trees tk and tr of the network, let C/* and C/ r be the basic cut-set matrices, respectively. We then have, using Eq. (2.17c) of Part I, Ve = C/t'Vft*

(1.36a)

V< = Q/Vftr,

(1.36b)

where the elements of the network are arranged in the same order for both systems. V « and Vt,r are column matrices of the branchvoltages, i.e., port-voltages of the «-ports formed based on trees tk and tr, respectively. If we again denote the submatrices of C/* ( and C / r ( corresponding to each set of the branches of the chosen trees tT and tk, respectively, by a and p, then the following relationships, which are similar to those derived for the transformation of port-currents, can be directly obtained

Formulation ofN-Port

Networks VU

153 = aVbr

(1.37a)

Vfrr = ß V u '

(1.37b)

a = ß-i

(1.37c)

= a'C/r

(1.37d)

C/r = ß'C/t

(1.37e)

Cft

and o and p are nonsingular unimodular matrices, therefore, d e t a = d e t p = ± 1.

(1.38)

Thus we have shown that the transformation matrix between two complete sets of independent port-voltages is a nonsingular unimodular matrix with its determinant equal to ± 1. For the short-circuit admittance matrices Y* and Y r of both «-ports we have Y* = C / t Y e C / t '

(1.39a)

Yr = C/rYiC/r'.

(1.39b)

Substituting Eqs. (1.37d) and (1.37e) into Eq. (1.39) we obtain Y* = a'Yra

(1.40a)

Yr = p'Yi-p.

(1.40b)

This set of equations is dual to Eq. (1.20) which relates two sets of independent chord-currents, i.e., port-currents, and their corresponding open-circuit impedance matrices. Thus far we have discussed the derivation of an n-port from a network of (n + 1) nodes on the basis of basic cut-sets with respect to a tree of a network. The short-circuit admittance matrix Y of the «-port is given by Y = C/YeC/1,

(1.41)

where Y e is the edge admittance matrix. It may not always be necessary, however, to identify each portvoltage with a branch-voltage of a tree of the parent network in order to form an n-port. In other words, as long as the port-structure is chosen so that the port-voltages are independent of each other, then each port-voltage is not necessarily identical with a branchvoltage of a tree of a network into which the n-port is described. F o r instance, let us consider the network given in Fig. 1.9a, where the orientation of each edge indicates the direction of an edge-current, and let us describe a four-port into the network as indicated in

154

Analysis and Synthesis

of Linear N-Port

Networks

Fig. 1.9b. Then, since there exists no edge connected between nodes 1 and 2 on the parent network, the port-structure of the four-port is not based on an actual tree of the original network, even though

(c) Fig. 1.9.

(d)

Illustration

of the derivation of a multiport fictitious tree of a network

based on a

(a) An o r d i n a r y n e t w o r k of five n o d e s ; (b) a f o u r - p o r t is described into t h e network of (a); (c) the fictitious tree on which the f o u r - p o r t of (b) is der.ved; (d) the network of (a) is modified with a fictitious edge.

Formulation ofN-Port

155

Networks

the ports form a "tree-like structure." This port-structure is, however, based on the tree-like structure as shown in Fig. 1.9c, which contains a fictitious edge (indicated by a dotted line segment) corresponding to a current generator. This structure will be referred to as a "modified" or "fictitious tree" of the parent network. Then, the basic cut-set matrix based on the modified tree, the "modified basic cut-set matrix," denoted by C/ m , of the network of Fig. 1.9a, or the basic cut-set matrix of the modified network of Fig. 1.9d, is found to be a

b

e

/

g

1 0

0

o-

d D

1 0

0

1

1

1 0

1 0

1 0

1

1 0

0

1 0

0

0

-1

1

C/ m =

c

.0

1 0

0

0

0

(1.42)

1.

and the short-circuit admittance matrix of the four-port, Y, is then given by (1.43)

Y — C/mYemC/n

where Y e m is the edge-admittance of the modified network. That is, Y e m includes a zero diagonal element corresponding to the fictitious edge which is a current generator with zero admittance. If the zero diagonal element in the modified edge-admittance matrix Y em should be eliminated in Eq. (1.43), the column corresponding to edge D must also be removed from C/ m - We therefore have C/m with the fifth column deleted,

C/m_, =

a

b

c

d

e

/

g

1

1 0

0

0

0

0

1

1

1

1

1 0

0

1 0

1

1 0

.0

0

1 0

0

1 0 0

(1.44a)

1

and Eq. (1.43) is reduced to Y = C/m_jYeC/m_,®,

(1.44b)

where Y« is the edge-admittance of the network of Fig. 1.9a. We now investigate the matrix C/m more carefully. The first, third, and fourth rows of the matrix correspond to cut-sets of the original network. The second row, however, corresponds to a union

156

Analysis and Synthesis of Linear N-Port

Networks

of two cut-sets; edges a, b, and edges c, d, e. It is thus clear that the formula stated in Eq. (1.41) is not always applicable to an n-port described into a network of ( « + 1 ) nodes, unless the port-structure of the n-port is based on a tree of the parent network. We therefore need to introduce an extended concept of a cut-set and cut-set matrix which includes the union of cut-sets. The following definitions and concepts were due to Reed. 7 Definition 1. A "seg" is a set of edges in a connected graph G, such that: The removal of the edges in the set separates G into two disjoint groups of nodes, Ni and JV2; N o edge in the set has both of its nodes in either Ni or N2; Ni or N2 is not necessarily a connected subgraph of G. Definition 2. A "seg matrix", C„ = [r J ( / ] is defined by a set of edges and segs of a graph such that: c, = 1 , if edge j is included in seg i and the orientations of both edge j and seg /' coincide; c,fj = — 1, if edge j is included in seg / but the orientation of edge j and seg i are opposite; c 'U = -gih and

n g' for i = 1, 2 . . ., n

(2.3a) (2.3b)

(2.4a)

g(l = ~yu, for ' * j. (2.4b) Equations (2.3) and (2.4) show that matrix Y corresponding to a star-tree has the property that the main diagonal elements are not less than the sum of the absolute values of all other elements on the same row; that is, yu > 2 j-i

(2.5)

Analytic Properties of Y- and Z-Matrices

159

A ^-matrix satisfying condition (2.5) is called a "dominant matrix," and we have the following theorem. Theorem I. The short-circuit admittance matrix of a resistive /¡-port, every port having a common node as the reference node, is a dominant matrix with nonpositive off-diagonal elements. Proof: If the network contains only (n + 1) nodes, then the shortcircuit admittance matrix of the network is actually the node admittance matrix and the theorem is therefore true. If the network contains more than (n + l) nodes, then star-mesh transformations in the unused nodes will transform the network into an equivalent network with only (n + 1) nodes. Thus, the theorem is proved. The short-circuit admittance matrix of an RLC network with (n + 1) nodes and with a star-tree port-structure also gives rise to a dominant matrix when s takes a positive, real value. That is, in general, we have every element y t ] of the short-circuit admittance matrix Y of the form yti = Ci)S +gi) + (1 / Lijs),

(2.6a)

and Y itself is given by

Y = jC + G + ( l / j ) r ;

(2.6b)

C is the capacitance matrix, G the conductance matrix, and r = [1 ¡L\. All three matrices are dominant. If we let s take a positive real value, say k, then an RLC network is equivalent to a resistive network, and Y(£) is therefore dominant because of Theorem 1. Dominant matrices play an important role in the theory of «-port networks because the dominant property is a sufficient condition for the r e a d a b i l i t y of a X-matrix. The study of a dominant matrix and its application to the synthesis of multiport networks are presented in detail in later chapters.

LINEAR-TREE

PORT-STRUCTURE

In the previous section we examined the star-tree port-structure and the dominant short-circuit admittance matrix associated with it. In this section we will deal with another important port-structure, a so-called "linear-tree" port-structure. Most of the material in this section is based on the work of Guillemin, 5 b and also of Biorci and Civalleri. 8 » Let us again consider a completely connected resistive network with six nodes as shown in Fig. 2.2. The network shown in Fig. 2.2

160

Analysis and Synthesis of Linear N-Port

Networks

is identical to the one of Fig. 2.1, except that the port-structure, that is, the port-structure presently under consideration, is a "lineartree." First, we note that if the ports are numbered and oriented as shown in Fig. 2.2, all elements of the basic cut-set matrix correspond-

'+

^v-f-

Fig. 2.2. A linear-tree port-structure

^

of a completely connected

+ network

ing to the linear-tree of the port-structure are positive. The basic cut-set matrix C/ with respect to the tree is then gli gl2 gl3 gl4 gl5 g22 g23 g24 g25 g33 g34 g3S g44 g4S g55 1

1

1

1

1

2

0

1 1

1

C/= 3

0

0

4

0

0 0

5

0

0

1 1 0

1 0 0 0 0 0 0 0 0 0 0 1

1

1

1

1

0

0

0

0

0

1 1 1 1 1 1 0

1

0

0

1 1 0

0

0

0

0

1 0

0

0

0

0

(2.7)

1 1 1 1 0 0

1 0

1

1 .

If we form the congruence Y = C / G w h e r e G c is a diagonal matrix of edge-conductances, then for the elements of Y we have yn = gl5+gl4+gl3+gl2+gll >'12 = gl5+gl4+gl3+gl2 >'13 = gl5+gl4+gl3 >14 = gl5+gl4

Analytic Properties of Y- and

Z-Matrices

161

>'15 = £15

(2.8)

>'22 = gl5 +gl4 + g25 +£24 +g23 +£l3 +g22 +gl2 >23 = gl5+gl4+g25+g24+g23+£l8 >24 - £15 +#14 +g25 +g24 >25 = #15 +g25, and similarly for other rows. Here we observe the following properties of Y: The elements of Y are non-negative. The following relationships are valid: yn ^ yi2 >

^ >m

^22 ^ >23 ^ . . . ^ )'2n

>n-l,n-l > >n-l,n

(2.9a)

and >nn ^ >n-l,n >

^ >ln

>n-l,n-l ^ >n-2,n-l ^ . . . > >l,n-l

>22

>12-

(2.9b)

A matrix with properties of Eq. (2.9) is called a "tapered matrix," and we may now state the results in the form of a theorem. Theorem 2. The short-circuit admittance matrix Y of a resistive nport network with a linear-tree port-structure numbered and oriented in one direction (as shown in Fig. 2.2) is a tapered matrix. 5b • 8 »• b, That is, y{j > 0,

for a// i

(2.10a)

yu > yt.uu

for j > i

(2.10b)

yn

for 1 < / < j.

(2.10c)

vt-ij,

Proof: If an «-port network under consideration contains n +1 nodes, then the theorem follows directly from the previous discussion. If the network contains more than n + 1 nodes, then star-mesh trans-

162

Analysis and Synthesis

of Linear N-Port

Networks

formation applied to the unused nodes will result in an equivalent network with n + 1 nodes. This completes the proof. If any of the linear-tree ports have an opposite orientation than that shown in Fig. 2.2, then the corresponding row and column of Y is multiplied by —1. Also, if the sequence of the p o r t s is different f r o m the sequence shown in Fig. 2.2, then the corresponding rows and columns of Y are transposed. F o r example, if we interchange the positions of ports z and j, then the z'th row and z'th column are transposed with the y'th row and column. In order to investigate further the properties of a tapered matrix appropriate to a linear-tree, let us transform matrix Y corresponding to a linear-tree to another matrix corresponding to a star-tree. In Fig. 2.3, we show the new port-structure as well as the old one. T h e

Fig. 2.3. Transformation

of a star-tree port-structure port-structure

into a linear-tree

subscripts " L " and " S " stand for "linear-tree-port" and "star-treep o r t , " respectively. Then, we have YVl = JL,

(2.11)

where Y is the tapered matrix, and Vj, and J c are column matrices of the linear-tree port-voltages and port-currents, respectively. For the same parent network, we may choose a star-tree upon which to base a five-port as shown in Fig. 2.3. Let us denote the short-circuit admittance matrix of the resultant five-port by Y. Then, as shown in the previous section, Y is a dominant matrix and

YVs = Js,

(2.12)

Analytic Properties of Y- and

Z-Matrices

163

where V s and J s are column matrices of the star-tree port-voltages and port-currents, respectively. Since the two systems of the five-ports were derived from the same network, the port-voltages and port-currents of the two systems should be related by (2.13)

V l = T Vi

and T is a nonsingular transformation matrix. Then, as was shown in the linear transformation of port-voltages of the previous chapter, the short-circuit admittance matrix of the five-port of the star-tree structure is found to be Y = T'YT,

(2.14)

where T was found in the previous chapter to be a nonsingular unimodular matrix. As an illustration, let us consider two systems of five-ports which are related by 0

0

y si

1

0

0

V S2

0

1

0

K.S-3

Vla

0

- 1

0

Vsi

. Vf.5 .

0

0

1

vSi

"

Vli VL2 VL3

=

-1

(2.15)

Using Eq. (2.14) one obtains the short-circuit admittance coefficients of the five-port with star-tree port-structure in terms of those of the network with linear-tree port-structure. They are shown in Eq. (2.16):

vO '15 g22 = >'22 +>'13 ~ >12 ~ >23 g23 = }'23 + yi4—yi3—y24, and so on for the remaining conductances. In general, we can write with stipulation that for ( / - 1) = 0, yi-ij — y t ~ = 0 and for (y'-i-l) greater than the order of the y-matrix then >> V i - i , i + vu+1,

for all i ^ j.

(2.19)

P r o o f : If the network under consideration contains n + 1 nodes, then the theorem follows from Eq. (2.18), because all conductances, i.e., gn for all / and j, are non-negative. If the network contains more than n + 1 nodes, then star-mesh transformations applied to the unused nodes will result in an equivalent network with n + 1 nodes. Thus, the theorem is proved. It is obvious that Theorems 2 and 3 are valid for RLC n-ports for positive real values of s. Example 3. Let us assume that the network of Fig. 2.2 is formed

166

Analysis and Synthesis of Linear N-Port

Networks

by unit conductances. Then, the short-circuit admittance matrix of the network, Y, is reduced to

Y =

r 5

4

3

2

1

4

8

6

4

2

3

6

9

6

3

2

4

6

8

4

L 1 2

3

4

5

(2.20)

This matrix is uniformly tapered. However, if the sequence of the ports is changed to (2, 1, 3, 4, 5) from (1, 2, 3, 4, 5), then, by interchanging the first and second rows and columns in Y, we obtain

Y21345

=

8

4

6

4

2

4

5

3

2

1

6

3

9

6

3

4

2

6

8

4

2

1 3

4

5

(2.21)

This new matrix is not tapered because the branches of the tree are not arranged in an ordered sequence. PROPERTIES FORMATIONS

OF

UNIMODULAR

CONGRUENT

TRANS-

In the previous sections we derived some of the properties of the short-circuit admittance and open-circuit impedance matrices of RLC networks without ideal transformers. For our investigation of Z- and /-matrices it is essential at this point to consider some of the properties of unimodular congruent transformations and also the properties of paramount matrices which result from these transformations. The material presented in this section is based on the work of Cederbaum. 6 We have shown that the Y-matrix of an RLC n-port with (/i + l) nodes is of the form Y = C f Y e CfK

(2.22)

where C/ is the basic cut-set matrix corresponding to a tree portstructure, and Y e is a diagonal matrix of edge-admittances. Similarly,

Analytic Properties of Y- and

Z-Matrices

167

for the Z-matrix of an RLC network with n independent loops we have Z = BfZeBf',

(2.23)

where B/ is the basic loop matrix with respect to the tree portstructure, and Ze is a diagonal matrix of edge-impedances. Equations (2.22) and (2.23) are called " u n i m o d u l a r congruences" because the matrices C/ and B/ are unimodular or £-matrices. If we consider only resistive networks, then the diagonal elements of Y r and Ze are always positive and real numbers. In order to extend our discussions to RLC networks in general, we shall assume that the complex variable s takes only positive real values. We can now examine properties of the congruence K = P D P ' , where P is a unimodular matrix and D is a diagonal matrix with positive real elements. Theorem 4. Each principal minor of the matrix K = P D P ( , where D is a diagonal matrix with positive elements and P is a u n i m o d u l a r matrix, is not less than the absolute value of any other minor built f r o m the same rows (or columns). 6 b P r o o f : Let us denote the elements of a unimodular matrix P, of order n by b, by p n such that p a = ± 1 or 0 for / = 1 , 2 , . . . , « and j = 1,2, . . ., b. Also denote the diagonal elements of D by du dz db- Then elements of K are found to be ktt = pn2di +pi22d2 + ... +pib2db

(2.24a)

k i j = p n p n d i +pizpizdz+ . . . +p(bpjbdb-

(2.24b)

F r o m Eqs. (2.24) we see that \k(j\

= \pnp)idi + ... +ptbpjbdb\ 2

2

< \pn di+pi2 d2+

\pnpn\kt}\,

for i J

= 1,2...,«.

(2.26)

N o w we use the Binet-Cauchy theorem which states that the c o m p o u n d of a product of matrices is equal to the product of the c o m p o u n d s . We therefore have for the /ith compound K = p»»Dp«A)_ (A)

(2.27)

The elements of D are obviously positive. The elements of P ( A ) are submatrices of a unimodular matrix, hence they are equal to ± 1 or 0. Using these facts, we can deduce that the main diagonal

168

Analysis and Synthesis of Linear N-Port

Networks

elements of K(A) are not smaller than the absolute value of any other element on the same row and column. An element on the main diagonal of K(*> is a principal minor, of order h, of K, and an element on the same row (column) of K(A> is a minor of order h built from the same rows (columns) of K which were included in K 0.

(3.4)

Then, referring to Eq. (3.2) and the discussions used in the proof of

182

Analysis and Synthesis of Linear N-Port

Networks

Theorem 9, there should exist a chord belonging to all three segs /', j, and k which is impossible since i and j are star-connected. Thus, the theorem is proved. Corollary 2. If Y is an admittance matrix of an /l-port resistive network with n +1 (n ^ 4 ) nodes, then there must exist at least two pairs of different indices (/,_/') and (k, m) such that either yuyptyp) ykmy'qkyqm

> 0,

for all p = 1, 2, . . ., n

0,

for all q = 1, 2, . . ., n

(3.5a)

(ports i,j series-connected; ports k, m series-connected) or J U W P ; < 0,

for all/? /

y'kmyqkyqm < 0,

for all q /

ij,

(i.5b)

k,m

(ports i,j star-connected; ports k, m star-connected) or ynypO'pl > 0 , ykmyqkyqm 0,

for all p = 1, 2, . . ., n, for all q ^ k,m

(3.5c)

(ports i,j series-connected; ports k, m star-connected). 1 3 a Since the proof of the corollary is evident from Lemma 1 and Theorems 9 and 10, it is left to the reader. The application of the corollary requires knowledge only of the signs of the elements of a ^-matrix. In both pairs of columns (i,j) and (k, m) in Y the elements in each row are required to be associated with the same or always opposite signs. The main diagonal elements and their counterparts in the other column may be exempted from this rule since p # i j and q # k, m. On the basis of the previous theorems we now give a systematic procedure for obtaining a tree-port-structure (if one exists) appropriate to the short-circuit admittance matrix Y of a resistive «-port with n + 1 nodes. Let us assume that matrix Y contains no zero elements. Step 1: By applying Theorems 9 and 10 or Corollary 2, first find the pairs of ports (or branches) which are series- or star-connected. There must be at least two such pairs; if not, the procedure can be terminated here. Step 2: Start at any one pair of series-connected or star-connected branches of the tree and short-circuit one of the branches which is equivalent to crossing out the corresponding row and column of matrix Y. Apply Corollary 2 and determine if the remaining branch is series-connected or star-connected to any other branch. Repeat the process until arriving at another pair of series- or star-connected

Realization of a Y-Matrix

183

with (n+ 1) Nodes

branches going through all branches. If the tree has more than two such pairs, then one may arrive eventually at a branch which is neither series- nor star-connected to any of the remaining branches. Step 3: Start at another series- or star-connected pair and repeat Step 2. By repeated applications of Steps 2 and 3, one should eventually go through every tree-branch. If it is impossible to continue this process until one goes through every tree-branch, then the sign pattern is nonrealizable and the procedure may be discontinued. If the sign pattern is realizable, then proceed with Step 4. Step 4: Arbitrarily assign the direction of one port, say port /; then the direction of any other port k is obtained from the sign of the element y The following examples will illustrate the previous procedure. Example 6. Determine the port-structure appropriate to the following sign pattern of a short-circuit admittance matrix: —

+

+

+

•

+

+

-

+

--

+

--

+

+

+

+

+

+

+

-

+

+

-

•

Sign pattern o f Y =

+

+ +

•

_

-

+

.

(3.6)

+

+ J

Then, applying Step 1, we see that for only ports 5 and 6 we have yseyakyek > o. Thus ports 5 and 6 are series-connected. We need not proceed any further because it is clear from Corollary 2 that no tree-port-structure exists corresponding to the sign pattern of Eq. (3.6). Example 7. Determine the port-structure appropriate to the sign pattern of a matrix Y of order six, given in Eq. (3.7):

+

Sign pattern of Y =

+ + + + +

+ + -

+

+

+

+

-

+

+

+

+

+ + (3.7)

184

Analysis and Synthesis

of Linear N-Port

Networks

In order to make the procedure of finding the port-structure more systematic, it may be necessary to build a table of signs for ytjytkyjk as shown in Table 1. In Table I, the signs of the product yuytkyik are given for i,j, k = 1, 2, 3, 4, 5, 6. In the positions corresponding to k = i or k = j an asterisk is placed because we are not interested in those signs. By Step 1, we find that ports (2, 6) and (4, 5) are series-connected and that ports ( 1 , 3 ) are star-connected. Now, short-circuit port 1. This is equivalent to deleting the first column of Table 1. Then Table 1. Signs of the triple product \

(yayiky/k)

k

j 1,2 1,3 1,4 1,5 1,6 2,3 2,4 2,5 2,6 3,4 3,5 3,6 4,5 4,6 5,6

l

->

3

4

5

6

\ *

*

_

_

_

+

*

—

*

-

-

-

*

-

-

*

+

-

*

-

-

+

*

+

-

-

*

*

* -

*

-

*

+

-

-

*

*

-

—

*

+ + + +

*

•

-

+ —

-

*

-

+ -

-

*

-

+ +

*

Star

•

*

+ + +

+ +

-

+

*

—

-

-

*

*

*

+

*

+

*

+

*

*

*

Series

Series

port 3 is neither series- nor star-connected to any other port. As the last step: (a) Short-circuit ports 1 and 6; i.e., consider Table 1 with columns 1 and 6 deleted and do not take into account

Fig. 3.1. Port-structure

corresponding

to the sign pattern of Eq. (3.7)

Realization

of a Y-Matrix

with (n+ 1) Nodes

185

the rows which contain 1 or 6. Then ports 2 and 3 are star-connected; (b) Short-circuit port 2; then ports 3, 4, and 5 are series-connected. Hence, the tree may be constructed as shown in Fig. 3.1. It must be noted that the sequence of ports (6, 2) and (4, 5) cannot be determined solely on the basis of the sign pattern. The submatrix corresponding to ports 6, 2, 4 and 5 must be realized with a lineartree using known methods. Thus far, we have developed the procedure of deriving the portstructure. Once the port-structure is known we use the results in the previous chapter to transform the tree into a star-tree. If the resulting matrix is dominant the realization can be accomplished by inspection. Let us illustrate the final realization of a given matrix. Example 8. Realize the following matrix with a resistive network of seven nodes: " 12 -5

5

1

2

2

-2

-2

6

12 - 5

-2

-2

13 - 5

1 -5

5

5 -2

-5

13

6 -2

2 -2

-2

6

7 -2

6 -2

-2

_ 2

-2

7

The sign pattern of the matrix is the same as the sign pattern of Example 7; thus the port-structure is also the same. The order of ports 4 and 5 can be obtained by considering the submatrix

12 Yi45

=

5

2

5 13

6

2

7

6

(3.9)

This matrix is uniformly tapered and the sequence of ports 1, 4, and 5 is as shown in Fig. 3.2. The same procedure can be followed for ports 1, 2, and 6. Now we arbitrarily select a reference node and obtain the star-tree port-structure with respect to this node, which is shown by dotted lines in Fig. 3.2. We proceed to write the transformation matrix between the two sets of port-voltages:

186

Analysis and Synthesis

^

yI v r

Fig. 3.2.

Transformation

- Vi v2 Vz v4 v5 - Ve -

A

+

\

\

Y

\

0V

of Linear N-Pori

n

v

Networks

=

3

\

of the port-structure tree port-structure

of Fig. 3.1 into a star-

-

Vi

0

0

0

-1

0

0

-1

o

1

0

Vz'

0

0

-1

1

0

Vz

0

0

0

1

-1

Vi

0

0

0

0

1

-1

1

0

0

0

-

(3.10a)

Ks' -

V6' _

where K< is the port-voltage of port i of the tree while Vt' is the port-voltage of port k of the star-tree. In matrix form we have V = TV'

(3.10b)

and T is the coefficient matrix of Eq. (3.10 a). W e therefore transform matrix Y to Y corresponding to the star-tree by T : 7 - 1 - 2 1 Y = T'YT =

8 -3

2 -3

12

0 - 1 - 1 0 2

0

0

- 2

-1

0

0

-1 -3

2

4 -1

0

0 - 3 - 1 0 - 2

0

8

(3.11)

1 7

N o w we can see that matrix Y is dominant with negative offdiagonal elements and can therefore be realized by inspection. (See

Realization

of a Y - M a t r i x with (n + 1) Nodes

187

Fig. 3.3.) The ports were determined in Fig. 3.2. This completes the realization of the given network. In the previous example we considered a matrix with no zero elements; from our discussion of a star-tree and a linear-tree portstructure two special classes of matrices can be realized almost by inspection. The first class is the class of dominant matrices with

Fig. 3.3. Final realization of the Y-matrix of Eq. (5.5) negative off-diagonal elements. This is the ordinary node admittance matrix and its realization is obvious. The second class is the class of matrices with all positive elements. If such a matrix can be made uniformly tapered then the conductance of each edge of a network to be realized can be obtained using the equation gii = yts+yt-t.ui

->'i-ij-.v'U = ±[g+sC

+ (\lsL)],

(3.20)

where g, C, and L are non-negative real numbers. Equation (3.20) follows from Eq. (3.19) and the sign-rule theorem. The realization of Y can now be based on the following observation: For real and positive values of the complex variable s an RLC n-port behaves as a resistive network. We arbitrarily assign, then, a real positive value to the variable s resulting in a matrix Y, and proceed as in the previous section. Of course, we must find the value of the various resistors, inductors, and capacitors required for the realization. This can be done in two different ways depending on the nature of the matrix Y. Case 1. Y contains no zero elements. If matrix Y contains no zero elements, then Approach 1 illustrated in the previous section can be used to obtain the port-structure. The transformation to a star-tree port-structure can now be applied to ¥ and the resulting matrix Y* must be realizable as the node admittance matrix. This is illustrated in the following example. Example 12. Realize the following y-matrix with an RLC network with exactly seven nodes:

194

Analysis and Synthesis of Linear N-Port 1 +6j + 35

+

2

3 s+s

2

3Î+ s

s

75 +

-2s-

2t +

-2s

3 — 2s— s

2s

-2s

-2s

s

S

2

—2s

4Î + s

-2s

-2s

1 + 6s + s

-2S

4j+-

3

-2s-

2 5

— 2s

3 6 - 2 j - - 1 + 5i+ s s

2

2s

Network

65

-2s

6s

Is

-2s

-2s

-2s

1 +4J + 5.

(3.21) Let s = 1, then matrix Y becomes

r 12 5 Y =

1 13 - 5

1 -5

-2

-2

6

12 - 5

-2

-2

5 -2

-5

13

6

-2

2 -2

-2

6

7

-2

6 -2

-2

-2

2

(3.22)

7.

The matrix of Eq. (3.22) is the same as the matrix which was considered in Example 8. The port-structure of the network corresponding to the matrix is therefore the same as the port-structure obtained in that example (see Fig. 3.2). The transformation of the port-structure into a star-tree port-structure applied to the matrix results in the following matrix:

Realization

of a Y-Matrix

1 + 4 j H— s — 1

Y* =

-2.Ï

0

with (n+ 1) Nodes

-1

1 + 3s + s 3

-2s 1

s

s

1+55+s

3

— 2s

0

0 3

- 1

s

1 3+ s

.v 0

-2s

3

1

0

195

s

-I

-2s

-2s

- I

1+.Ï+S

-5 Is

0

T h e matrix Y* is the ordinary node a d m i t t a n c e matrix and can be realized by inspection with the port-structure shown in Fig. 3.3. The final realization is shown in Fig. 3.11.

C a s e 2 : Y c o n t a i n s some zero elements. If Y contains zero elements, t h e n A p p r o a c h 2 of the previous section should be used t o o b t a i n a seg m a t r i x C«. T h u s , we have Y = C.DCV and

Y =

C.DCV,

(3.24)

196

Analysis and Synthesis of Linear N-Port

Networks

where D is obtained f r o m D by replacing the complex variable s with a positive real number. In other words, first derive Y from Y by the substitution of a positive real number in s, and then apply Cederbaum's decomposition technique to get C». This procedure is illustrated in the following example. Example 13. Realize the following /-matrix with an RLC network with four nodes:

Y =

2+

2s

25

4 2 s+s

3

(3.25)

s

3

3 25 + s

5

Let 5 = 1 ; then Y becomes

Y =

'4

2

0"

2

6

3

.0

3

5.

(3.26)

However, this matrix was considered in Example 10. Thus we have 2 + 2s2s 2s

4 25+5

0

(iiOOOO

"110"

10 10 0

0(f:0 0 0

0 11

1 1 0 10

0 0 f/30 0

1 0 0

0 1 0 0 1 J 0 0 0 (li 0

0 10

0 3 S

2j+

3

-

.001.

5

(3-2 1 ) From Eq. (3.37) we solve for the unknowns clu d-i. 0 for j = jw. Next we write Y(j) as a sum of matrices and that each matrix corresponds to a particular pole. The general expansion is of the f o r m :

-kn{x)ki2{œ).. (a0)

(CC)

A:21 A-22

km™ ~ ••

-A:nl(°0)A:»2,œ,.. • knn(00) ~/cll • • 1

k2i{0)k2z{0) •. • *2„

s -A-„2(0,A:„2(0). •

206

Analysis and Synthesis of Linear N-Port -Arn'f/tiz«1»

+

a

2s

*i„

W

k2l k22 ...

+ IOi

10

kin

(4.11)

2

-k

n

^k

n z

» \ . . knnU)

-

w h e r e each matrix in the r i g h t - h a n d expression is called a " r e s i d u e m a t r i x . " E a c h d o m i n a n t residue matrix in E q . (4.11) is realized separately in a m a n n e r similar t o the realization of a d o m i n a n t real m a t r i x . F o r the matrix with the f a c t o r l/s, each edge is an i n d u c t a n c e ; for the matrix with the f a c t o r s, each edge is a capacitance; a n d , f o r the matrices with the f a c t o r 2 j / ( j 2 + tu 0, a n d then a p p l y i n g C e d e r b a u m ' s a l g o r i t h m . T h a t is, Z(A:) = B , Z e ( £ ) B s l .

(4.17)

Let us illustrate t h e p r o c e d u r e by considering t h e f o l l o w i n g example. E x a m p l e 17. Realize t h e F o s t e r matrix as an o p e n - c i r c u i t imp e d a n c e matrix,

(4.18)

Z =

W e then d e c o m p o s e t h e m a t r i x into the f o r m

Z =

1

1

0

1

1

1

0

0

1

1

0

1

0

0

1

1

1

0

1

1

0

0

1

0

0

1

LO

0

0 0 BjZfBj

1

(4.19)

Let us a u g m e n t the matrix of Eq. (4.19) by inserting t h e missing c o l u m n s until the m a t r i x is c o n v e r t e d into a modified basic l o o p matrix B / m a s : 7 4 5 6 8 9 3 2

B•fm —

1

1

0

0

1

0

0

0"

1

1

1

1

0

1

0

0

1

0

1

0

0

0

1

0

0

0

1

1

0

0

0

1.

(4.20)

Realization

of a Y - and Z-

Matrix

209

A n d t h e c o r r e s p o n d i n g modified basic cut-set matrix is 1 2 3 4 5 6 7 8 9 1 0 0 0 0 -1 -1 - 1 - 1

C/m =

0

1

0

0

0 -1

-1 - 1

0

0

0

1

0

0 -1

-1

0

0

0

0

1

0

0

-1 -1

0

0

0

0

1

0-

1

0

(4-21)

-1

0 -1

T h e short-circuit a d m i t t a n c e matrix Y = C / C / (here we a s s u m e all elements having a resistance of I o h m , elements 6, 7, 8, 9 a r e of c o u r s e fictitious) is 3 3 2

Y — CfmCfrn

—

4

2

2

2

3

1

2

1

4

1

1

2

(4.22)

A p e r m u t a t i o n of t h e r o w s a n d c o l u m n s 1, 2, 3, 4, 5, into 3, 2, 1, 4, 5 results in t h e m a t r i x 2 1 r -3 2

Y32145 =

2

4

3

2

1

2

3

5

3

2

1

2

3

4

2

_1

1

2

2

3^

s

(4.23)

+

•AA/V Fig. 4.7.

Realization

of the Y-matrix of Eq. (4.24) where all are one ohm

resistors

Analysis and Synthesis of Linear N-Port Networks

210

This matrix is u n i f o r m l y t a p e r e d a n d the realization is shown in Fig. 4.7. P o r t s 1, 2, 3, a n d 4 of the i m p e d a n c e matrix correspond t o the impedanceless c h o r d s 6, 7, 8, a n d 9. If t h e five p o r t s of Fig. 4.7 are open-circuited a n d p o r t s 1, 2, 3, a n d 4 are inserted in series with t h e impedanceless c h o r d s 6, 7, 8, a n d 9 t h e n the f o u r - p o r t n e t w o r k realizing the F o s t e r matrix is o b t a i n e d as shown in Fig. 4.8.

A / W

Fig. 4.8. Final realization of the Z-matrix of Eq. {4.18) N o t e here t h a t the matrix c o n s i d e r e d in this example was also examined in E x a m p l e 16. It was s h o w n t h e r e t h a t this matrix is not realizable as a short-circuit a d m i t t a n c e matrix with any n u m b e r of nodes. W e t h u s c o n c l u d e t h a t the conditions on a matrix for realization as a short-circuit a d m i t t a n c e matrix or as an open-circuit i m p e d a n c e matrix are n o t the same in each case.

Fig. 4.9.

Realization

of a 2 by 2 paramount

matrix

T h e p r o b l e m of realizing a Z - m a t r i x by a n e t w o r k with more than n i n d e p e n d e n t loops is similar t o the p r o b l e m of realizing a Ymatrix with m o r e t h a n n + 1 nodes. T h e process of augmenting a Z - m a t r i x can be used, a n d the p r o c e d u r e is similar to the one discussed in the previous section.

Realization

of a Y- and Z - M a t r i x

211

In the previous section, the dominance condition was given as sufficient for the realization of a short-circuit admittance matrix; in the case of an open-circuit impedance matrix no procedure exists for realizing a dominant matrix. Tellegen has shown that all 2 x 2 and 3 x 3 p a r a m o u n t matrices are realizable as open-circuit impedance matrices. The realization is shown in Figs. 4.9 and 4.10.

mount

matrix

The element values of Fig. 4.10 can be obtained by solving for the resistances in terms of the matrix elements, and the voltage polarities and numbering of the ports can be selected so that each one of the resistances is non-negative. T h u s we shall let Rn > R22 with no loss in generality, then Rc = Rn — R22 Ra — R22 — R21

(4.24)

Rb = R22 + R2I' Finally, before concluding this section, we give a theorem which is very useful in the investigation of the realization of a special class of Z-matrices. Theorem ¡4. A Z-matrix of order n, whose elements for some value of A:, which satisfying the condition |z*y| = Zjj, j = 1, 2, . . ., n, is realizable only if it can be realized as the impedance matrix corresponding to a system of basic loops of some resistive n e t w o r k . 8 ' Since the proof of the theorem is given in Reference 6f, the following example is given as an illustration of the theorem.

212

Analysis and Synthesis

of Linear N-Port

Networks

Example 18. Let us consider the matrix Z =

3

1

1 5

2

3

"

4

5

2

4

6

6

-3

5

6

57/7 -

(4.25)

This is a p a r a m o u n t matrix satisfying Theorem 14. Since z*4 = z* for k = 1, 2, 3, 4. If we try to decompose Z into the unimodular congruence using C e d e r b a u m ' s algorithm, we find after five steps that the procedure diverges. The failure of the decomposition proced u r e shows that this matrix satisfying Theorems 12 and 14 can be neither an admittance nor an impedance matrix of a resistive 4-port. Thus far we have investigated the various known realization methods for a short-circuit admittance matrix as well as an opencircuit impedance matrix. T h e synthesis of a multiport is actually accomplished by the realization of a basic cut-set matrix or a basic loop matrix, i.e., unimodular matrices, in terms of a connected graph. There exists, however, an interesting and important contribution by So i a to the realization of a loop impedance matrix of a network. We shall briefly review the work of So, since it is directly related to the subject of our discussion in this part. Let us consider a real matrix M = [mij] of order n by n. Definition 5: M is "definite d o m i n a n t " if n

W« > ^ \mij\, J=i

i^j-

Definition 6: A real matrix M of order n(n > 4) is "definite s l a n t - d o m i n a n t " if the rows of M may be arranged in such an order that (a) the first four rows are definite d o m i n a n t ; (b) any row 4 < / < n is definite ( n - / + 3) dominant. That is, ma is greater than the sum of the absolute values of any (n — i + 3) of the (n— 1) elements of the /-¿th row. Definition 7: A realization of M is coherent if there exist no two loops l t and Ij such that /< and / j agree in orientation in one resistor Ra but oppose in orientations in another resistor Rf,. So has obtained a synthesis procedure which, when applied to a definite slant-dominant matrix, either leads to a coherent realization or furnishes proof that no such realization exists. If M is not definite

References

213

s l a n t - d o m i n a n t , application of the p r o c e d u r e may or may n o t lead to a n e t w o r k , even if M is coherently realizable. In order to use So's results for the realization of a loop matrix B of order n xb a n d r a n k n with elements ± 1 a n d O's, where B is not necessarily u n i m o d u l a r , we must limit ourselves to the sub-class of B matrices which c o u l d be coherently realizable. T h a t is, if B = [¿JJ], then bikblk=+\,Q

(4.26a)

bikb}k=-1,0,

(4.26b)

or

for all i,j, k a n d i / j. If B satisfies E q . (4.26), then we may f o r m M = BB'.

(4.27)

If in Eq. (4.27) M is definite s l a n t - d o m i n a n t , then So's procedure can be used. Since this p r o c e d u r e is complicated a n d requires the construction of a n u m b e r of tables, interested readers should refer to Reference 16.

REFERENCES 1. Y. Oono, "Synthesis of a finite 2n-terminal network as the extension of Brane's two-terminal network theory," J. Elec. Commun. Eng., Japan, (1948). 2. B. MacMillan, "Introduction to formal readability theory," Bell System Tech. J., 31 (1952), 217-79, 541-600. 3. I. W. Sandberg, "Synthesis of active N-port networks," Bell System Tech. J., 40 (1961), 761-83. 4. H. J. Carlin and D. C. Youla, "Network synthesis with negative resistors," Proc. IRE, 49 (1961), 907-20. 5a. E. A. Guillemin, Introductory Circuit Theory. New York, John Wiley & Sons, 1953. 5b. E. A. Guillemin, "On the analysis and synthesis of single element kind networks," IRE Trans. CT-1 (1960), 303-12. 6a. I. Cederbaum, " O n networks without ideal transformers," IRE Trans. CT-3 (1956), 179-82. 6b. I. Cederbaum, "Matrices all of whose elements and subdeterminants are 1, - 1 , or 0 , " J. Math. Phys., 36 (1958), 351-61. 6c. I. Cederbaum, "Conditions for the impedance and admittance matrices of N-ports without ideal transformers," IEE (London), Monograph 276R, January, 1958.

214

Analysis

and Synthesis

of Linear N-Port

Networks

6d. I. Ccderbaum, "Applications of matrix algebra to network theory," IRE Trans. CT-6, Special Supplement (1959), 127-37. 6e. I. Cederbaum, " O n matrices with some form of dominance of the main diagonal," The Matrix and Tensor Quart., 9 (1959), 29-39. 6f. I. Cederbaum, "Topological considerations in the realization of resistive N-port networks," IRE Trans. CT-9 (1961), 324-29. 7. M. B. Reed, "The seg: a new class of subgraphs," IRE Trans. CT-8 (1961), 17-22. 8a. G. Biorci and P. P. Civalleri, "Alcune considerazioni sulla sintesi dei multipoli resistivi," Atti accad. sci. Torino, 94 (1959-1960). 8b. G. Biorci and P. P. Civalleri, " O n the conductance matrices with all all-positive elements," IRE Trans. CT-8 (1961), 76-77. 8c. G. Biorci and P. P. Civalleri, " O n the synthesis of resistive N-portnetworks," IRE Trans. CT-8 (1961), 22-28. 8d. G. Biorci, "Sign matrices and the realizability of conductance matrices," IEE (London), Monograph No. 424E, December, 1960. 9. B. D. H. Tellegen, "Theorie der Wisselstromen," Deel IV, Theorie der Electrische Netwerken, P. NoordhofT N.V., Gronigen, Djakarta, 1952, 166-68. 10. P. Slepian and L. Weinberg, "Synthesis applications of paramount and dominant matrices," Proc. Natl. Electronics Con/., 14 (1958), 611-30. 11a. P. R. Bryant, Ph.D. thesis, Cambridge University, Cambridge, England, 1959. l i b . P. R. Bryant, "Discussion on conditions for the impedance and admittance matrices of N-ports without ideal transformers," Proc. IEE (London), Part C, 106 (1959), 116. 12. P. P. Civalleri, " A direct procedure for the synthesis of resistive (n + l)-poles," IEE (London), Monograph No. 464E, August, 1961. 13a. C. C. Halkias, I. Cederbaum, and W. H. Kim, "Synthesis of resistive N-ports with (n+ 1) nodes," IRE Trans. CT-9 (1962), 69-73. 13b. C. C. Halkias, "Synthesis of N-port networks," Ph.D. thesis, Columbia University, New York, 1962. 14. D. P. Brown and Y. Tokad, " O n the synthesis of R networks," IRE Trans. CT-% (1961), 31-39. 15. R. M. Foster, see L. Weinberg, "Circuit theory progress report," J. Research Natl. Bur. Standards, Part D, Radio Propagation, 64D (1960). 687-706. 16. Hing Cheong So, "Realization of loop-resistance matrices," Ph.D, thesis, University of Illinois, Urbana, Illinois, 1960; Int. Tech. Rept. No. 17, contract No. DA-11-Q22-ORD-1983, University of Illinois, 1960.

PART

IV

Synthesis of Single-Contact Switching Networks and Realization of Loop and Cut-Set Matrices

Introduction

O U R PREVIOUS STUDY of the synthesis of an ordinary RLC multiport the reader must have noted that the realization of a shortcircuit admittance matrix or open-circuit impedance matrix is accomplished by synthesizing a connected graph based on a cut-set or loop matrix. The network synthesis problems, whether they are RLC networks, contact networks, or flow-nets, are actually reduced to the realization of a graph. In particular, the synthesis of a class of unate switching network (the so-called "single contact network") in minimal form is directly related to the realization of a nonoriented connected graph. We shall therefore discuss the synthesis techniques of a single-contact network in terms of the realizability of a real matrix of integer mod 2 as a loop or cut-set matrix of a connected graph. Various authors have proposed a number of algorithms for the realization of a loop or cut-set matrix, both with and without relation to the applications of the techniques to network synthesis problems. The method presented here is based on development of the synthesis techniques of a resistive multiport network described previously. This method will be briefly compared to some other known techniques. We shall first review some fundamental concepts of the theory of switching networks. Then, we shall develop a simple synthesis method f o r a single-contact switching network in terms of the realizability of a path, loop, or cut-set matrix. It may therefore be necessary f o r the reader to have a clear understanding of some concepts of the theory of linear graphs defined in Part I and the realization techniques of a multiport network discussed in Part III. FROM

CHAPTER

1

Fundamentals of the Theory of Contact Networks

and the most basic types of switching networks are the so-called "contact networks." Each element of a contact network is a contact switch which has two states—open and closed. If a contact is in the open state there exists no conduction of current through the element, while if it is closed there will be a conductive path in either direction through the switch. It is further assumed that the terminals of a contact can be reversed without affecting the operation of a contact network. It should be understood that although each state of the contact is determined by some means, whether electrical, magnetical, or mechanical, the means will be considered external to the network. In other words, the conduction of currents through the various paths of a contact network depends only on the combination of the state of each contact of the network. Such a network is called a "combinational switching network." We assign to a contact a binary variable x taking the value 0 when the contact is open and the value 1 when the contact is closed. With each variable x, we associate a symbol x' to denote the corresponding contact which is open while x is closed, and closed when x is open. This symbol x' is called the "complement" of variable x. We also denote the output of a contact network, or the output of the current detector, t by F which is a binary function of the binary variables associated with the contacts of the network. A function /"is usually called a "switching function." A switching function is normally expressed in Boolean algebra which has three fundamental operations defined by:

THE SIMPLEST

f It is assumed that a current detector is connected between the input and the output of a network in order to measure the conduction of currents through the network.

The Theory of Contact

Networks

219

sum: xt+xj

= 0,

if Xi = X) = 0

x t + X j = 1, product:

(1.1a)

otherwise;

XiXj = 1 ,

i f Xl

=

X]

=

x i x j = 0,

otherwise;

1

(1.1b)

complementation: (*)' = 0,

(1.1c)

if* = 1

(*)' = !, if JC = 0. The contact network diagrams corresponding to the addition and multiplication of Boolean operations are shown in Fig. 1.1. The -Xr •

Xf

-xr (o)

(b)

Fig. 1.1. Parallel connection for Xt + Xj (a) and series connection xtXj (b)

for

parallel connection of x and its complement x' is equivalent to a direct connection, i.e., a shorted wire, since x+x' = 1. The series connection of .r and x' is equivalent to an open-wire since their

X' (a)

(b)

Fig. 1.2. Physical implications for the sum (a) and the product (b) of a binary variable and its complement product is always zero. They are illustrated in Fig. 1.2, where the symbol ~ is used to indicate an "equivalence class." If more than one path exists between the input and output terminals of a contact network, the output of the network may be expressed by the sum of the currents flowing through the various paths. Since each path for current flow should result only when every contact comprising the path is in the closed state, we therefore set the switching function describing the output equal to the sum of products of

220

Realization

of Linear

Graphs

the variables associated with each path. When a switching f u n c t i o n is expressed by the sum of products of variables, it is said to be in "normal form." As an illustration, let us consider a graphical representation o f a

(0)

(b)

(c)

Fig. 1.3. Graphically 2-isomorphic and electrically equivalent networks

contact

contact network shown in Fig. 1.3a, in which each edge represents a contact with a binary variable assigned. Then the switching function F which describes all the conducting paths between terminals a and b of the network is:

F = .V1.V2 +.Vl.V3-V'3J»r6 +*i;*-3.Y5X6 +.V3.Y4.Y6 -f .V4.T5.Y6 +.v'l.\7 + -Y2.T3.V4 = .Y1.V2 +.V1.V3.T5X6 +X'3.Y4.Y6 + .Y4-TE.Y6 +.v'l.v7 -I-.Y2.Y3.Y4,

( 1 .2)

where ,vi.T3.t'3.T6 = 0, since xt.x't = 0 for all /. In Fig. 1.3a, if we operate on edges .v? and .y'! such that the resulting graph becomes 2-isomorphic to the original one as shown in Fig. 1.3b, then the new network will have the same switching function with respect to terminals a and b. The two networks are therefore "electrically equivalent" or simply "equivalent" with each other. In Fig. 1.3b, we again reverse the connection of the subnetwork consisting of edges T'3, .T5, and .V6 and obtain another equivalent network as shown in Fig. 1.3c. Thus the networks shown in Fig. 1.3 are equivalent contact networks and are topologically 2-isomorphic with each other. In order to generate a class of 2-isomorphic graphs with respect to a given graph, the graph must be isolated into two

The Theory of Contact

221

Networks

s u b g r a p h s by cutting it at two nodes. In other words, the graph should contain a subgraph(s) which has only two c o m m o n nodes with its complement of the graph. For this reason, Gould (Reference 15 of Part I) named such a type of subgraph a "two-terminal subg r a p h " and it plays an important role in his realization techniques. W h e n a binary variable x takes the value of one, its complement x' takes zero. The complement of a function of binary variables, F, is defined in the same way, such that if F = 1, then F' = 0 or vice versa. The complement function F' of a switching function F is obtained by the so-called "dual operations": {xtxj)' = x'i+x't

(l-3a)

(xt+x}y

(1.3b)

= xix'i

(*')' = x.

(1.3c)

As an illustration, let us consider the switching function of the network of Fig. 1.3a, i.e., Eq. (1.2). Then the complement of F is readily obtained as: F' = (*' +X2)(Xl +x'7)(*'l +.v'3

+x'6)(*3 + x' 4 + *'g)(x'4X'5 +x'e)

x(*'2+x'3+x'4).

(1-4)

O n e more fundamental concept in the theory of switching network is the so-called "canonical f o r m " of a switching function. The canonical form of a switching function F which is in a normal form can be obtained by the following procedure: (a) multiply each p r o d u c t in F which contains neither xt nor x\ as a factor by the sum ( x i + x ' i ) for every product term and for every variable in F; (b) remove every redundant term. These steps are obviously valid because v." i 2 {/(«'» "l)~Jl"i> }

2

"i)} = - F,

{ / ( " ' . « ; ) - / ( « ; . " < ) } = 0,

Wi is the input;

"i is the output;

otherwise.

(1.3)

3(0)

Fig. 1.2. A possible flow pattern of the net of Fig. 1.1 In what follows the flow of any edge shall be written in parentheses following the capacity of the edge. F o r instance, a possible flow pattern for the graph in Fig. 1.1 is shown in Fig. 1.2.

Oriented Communication THE MAX-FLOW

Nets MIN-CUT

255 THEOREM

A fundamental property of the oriented communication net is contained in a theorem formulated by Ford, Fulkerson, and Dantzig. l a , b ' 2 It is called the " M a x - F l o w Min-Cut Theorem." 1 3 This theorem answers the question: "Suppose a flow of magnitude f i s supplied at node i and taken away at node j. What is the maximum value of /"obtainable through the n e t ? " This question is a basic one and is analogous to the driving-point problem in electrical network theory. Definition 2: Let m, . . ., n* be a set of distinct nodes of a graph G. If for each i (/' = 1,2 k— 1), (m, n n0) > 0. t N o t e here that a " c u t " is not in general a " c u t - s e t " as defined in Part I. A f t e r the removal of a cut the remaining g r a p h may still be connected.

256

Analysis

and Synthesis

of Communication

Nets

First, we shall show that m is in Gi. Suppose n2 is in Gi; then there is a path from «1 to m such that all forward edges (where the edge orientation agrees with the path orientation) have flows lower than the edge capacity, and all reverse edges (where the edge orientation disagrees with the path orientation) have a positive flow. Thus there exists a positive quantity t such that if flows in forward edges are increased by an amount equal to t they are still under the edge capacity. Also, reducing flows in reverse edges by an amount equal to e will not make them negative. However, by doing this we have increased the flow by an amount equal to e > 0. This contradicts our original_assumption that the flow was maximal. Therefore belongs to G1. Let us now focus our attention to the set of edges, S, that lead from Gi to Gi. If 5 is not empty, for each edge e< in 5, M )

= Qet).

(1.4)

However, for any node n< in G1, 2 {f(m, nj)-f{n], )

m)} = F,

^

2 {/(" t"i), or both. But since ttj = C'ti + C"t],

(1-31)

now C'tj + C'a

> t'tl +

t",j

(132)

or hi > t'a + t",!,

(1.33)

which is a contradiction. SYNTHESIS

OF A TERMINAL

MATRIX

In this section we show that for a graph containing three nodes or less, the conditions stated in Theorems 2 and 4 are necessary and sufficient for the r e a d a b i l i t y of a terminal matrix. F o r a graph

Oriented Communication

Nets

263

containing f o u r nodes or more, these conditions are no longer sufficient. A counter example of order four has been found.f The realization techniques of low-order cases have their applications in the realization of higher-order matrices, if they can be partitioned in some special way as will be shown later in this section. If a graph contains two nodes only, then each of the two cuts contains exactly one edge. The edge matrix is thus the same as the terminal matrix and the synthesis is trivial. The terminal matrix of a graph containing three nodes is 3 by 3 a n d the partition must be o n e - t w o as follows:

'l

r ® T =

'21 - '31

©

r© or

! '23

1 1

'32

! ©

'21

-

_ '31

i '12 1 1 -1 '1 1 1 J® 1 1 '32 I ®

(1.34)

We shall consider the partition on the left. The synthesis methods for these two cases are analogous. First we write T as the sum of two c o m p o n e n t matrices: ® T =

t

i 21

«31

o

©

i / 23 ©

r CD

+

t 32

h L h

'i

'i

t'zi, we have x0 = t'n-t'23 }'o

= max{(/'31-/'32),(i , 31-/'21 + ''23)}.

(1-40) (1.41)

Oriented

Communication

Nets

265

If f'31 ^ /'21, we have -v0 = max{(/'2i - t'zz), (r'21 — f'31 + '32)} }'o = i'31-i'32.

(1.42) (1.43)

N o t e that if r'31 = t'21 < (f'32 + ^23), the solution is not unique. All the points on the straight line-segment where two curves coincide are solutions. F u r t h e r m o r e , x0 and y0 must assume positive values. To show this, we consider the different cases separately. If f'31 > t'21, and x0 = (''21 — ''23) < 0, then f'21 < min(/'3i, r'23), violating Theorem 4. If f'31 > t'21, and ya = max{(f' 3 i-f'32), (i'3i — i'21 + i'23)} < 0, then (f'21 — ''23) > ''31 > t'21 which means r'23 < 0, contradicting the assumption that / > 0. One can show similarly that if f'31 < t'21, both x0 and y0 must be non-negative. If f'31 = i'21 < (f'32-l-/'¡¡a), then the line-segment where two curves coincide will pass t h r o u g h the first q u a d r a n t since f'31 = f'21 > 0 by assumption. The realization of a uniform matrix T u is a cycle oriented in either direction. T h e graph may also contain two cycles oriented in different directions (Fig. 1.7). We note that in three-node cases the

(o)

Fig. 1.7. Realization

(b)

of a uniform

(c)

matrix

minimum cuts of T' always correspond to minimum cuts of T u since the capacity of all cuts is equal. When the same approach is used in higher-order cases, a set of simultaneous equations similar to Eqs. (1.36) and (1.37) can be written f o r the unknown edges connecting two lower-order subgraphs which were already realized. It is found that although the

266

Analysis and Synthesis

of Communication

Nets

successive approximation method converges to the set of solutions, the triangular relationship of Theorem 4, may not guarantee that the solutions be non-negative. However, in general, the technique of decomposition could be used. Theorem 6. When a matrix T is realizable and T = T' + U,

(1.44)

where T is realizable as G' and U is a uniform matrix of value t then T may be realized as

G =

G'+Gu,

(1.45)

where Gu is a realization of U such that all simple cuts in G have capacity tu, and minimum cuts of G' correspond to simple cuts in GuP r o o f : The proof of this theorem follows directly from Theorem 5. The techniques of realizing a low-order ^-matrix can be conveniently applied to a 7-matrix of higher order, in general, if it can be partitioned. This is done by rearranging the nodes in such a fashion that the following conditions are satisfied: 1. Each submatrix corresponding to a subcollection of nodes lying along the diagonal line is square and contains elements with values no smaller than the value of any of the elements in an offdiagonal submatrix. We refer to these as "dominant conditions" of partitioning of a terminal matrix. 2. Each ofT-diagonal submatrix is a constant matrix. We see that the theorems obtained for nodes apply to submatrices. For instance, tu > min(//K, txj), as required by Theorem 4. This generalization is immediate since all the minimum cuts corresponding to the off-diagonal matrices must stay outside the subgraphs corresponding to submatrices by Condition 1. Theorem 7. A matrix T satisfying the dominant conditions is realizable as a terminal matrix if: (a) treating these submatrices along the diagonal line as nodes, the matrix T is realizable; (b) each submatrix along the diagonal line is realizable. Proof: The sufficiency of these two conditions to imply realizability is proved by constructing the synthesis procedure demonstrated in the following example. Example 2: The following matrix is to be realized:

Oriented

Communication '

Nets

267

5

will be a cut-set which leaves Gi and G4 connected, and similarly for G2 and G3. The graph is therefore cut into two parts by the cut-set (Si©S2). A more rigorous a r g u m e n t is also possible. Suppose Si cuts the graph G into two s u b g r a p h s Gi a and Gib, and S2 cuts G into Gz a and G2b. Let Gi denote the s u b g r a p h c o m m o n to Gi a and G>a

Nonoriented Communication Nets

273

G2 denote the subgraph c o m m o n to Gi 0 and Gzb G3 denote the subgraph c o m m o n to Gib and G20 G4 denote the subgraph c o m m o n to Gib and GzbIt is seen that Gi, G2, G3, and G4 are connected subgraphs. Also, if any edge is contained in one, it will not be contained in any of the other three subgraphs. Let us focus our attention on Gz and G3. After the removal of cut-set Si or S2, Gi is not connected with G3. Since we have a completely connected graph, there exists an edge e which connects some node in Gz and some node in G3, provided both Gz and G3 are nonempty. This edge e is in both Si and S2, consequently not in (Si@S2). Thus Gz and G3 are connected after the cut-set (Si@S2) is removed. The same argument may be applied to Gi and G4. It then follows that in all cases there will be at most two separated nonempty subgraphs after the cut-set (Si©S2) is removed from the graph. Definition 5. A function Ftj(Sk) is defined for each pair of nodes i, j and each simple cut-set St such t h a t : Fij(Sk) = — 1, if i and j are not connected with the removal of S t ; Fij(Sic) = 1, if / and j are connected with the removal of St. (2.12) Theorem 12. For a completely connected graph, Fi}{Sk@SL)

= Fi,(Sk)FtASL).

(2.13)

In essence, this theorem says that if both St and Si cut nodes i and j or d o not cut i and j, then the set ( S t © S t ) will not cut i,j. If one and only one of St or Sl cuts i,j, then the set (Sk@Si) will also cut i,j. P r o o f : We shall first show that implies

Fn(St)

= Fa( Si.)

Fu(Sk@SL)

= 1.

(2.14) (2.15)

In a completely connected graph, there exists an edge e connecting / and /. If Fij(St) = Fij(SL.), either e is contained in both S t or Sl or e is contained in neither. In both cases, (St©Sz.) will not contain e. Therefore / and j will be connected with the removal of the cut-set (St©Si.). On the other hand, if

Fij(Sk) = — Fi ¡{Si.)

(2.16)

274

Analysis and Synthesis of Communication

Nets

we cannot also have F(j{St®SL)

= 1.

(2.17)

For, without loss of generality, we may assume FiliSk)

= 1

(2.18)

F0(Sl) = - 1 , if, at the same time, Ftl(S/c ® Sl) = 1.

(2.19)

It follows from the above that Ftj(Sk)

which implies

= F^S^Sl),

(2.20)

Fi](St ® S* © SL) = Fij(SL) = 1.

(2.21)

Since it is originally assumed that f«(Si,)=-l,

(2.22)

Fuis* ® SL) = - 1.

(2.23)

we must conclude that Therefore, in all cases, ^ ( S * © ^ ) = Ftl(Sk)Fii(SL).

(2.24)

Corollary 2. Let SG„G, denote the cut-set which cuts a connected graph G into G\ and Gz. For a complete graph, Sc,.c, © (2.25) C j = SGS,G,, such that: (a) subgraphs Gi, G3, G5 all contain a given reference node; (b) subgraph G5 contains all nodes which are common to subgraphs Gi and G3, or subgraphs G2 and G4, and only these nodes. This corollary follows directly from Theorem 12. Example 3: Figure 2.4 shows a completely connected graph of four nodes. The cut-sets are S i = (a,b,f);

S2 = (a, c, d)\

Ss = (b, c, d,f)\

S3 = (b,c,e);

S» = (a,c,e,fy,

S4 =

S, = (a,b,d,e).

{d,e,f)\ (2.26)

Applying the notation in Theorem 12 F 2 3 ( 5 I © 5 2 ) = F 23 (S 5 ) = - 1

(2.27)

FzziS^F^Sz) = ( ! ) ( - ! ) = - 1 .

(2.28)

Nonoriented

Communication

Also

It does follow that

275

Nets

Si = (a, b, /) = Si,234

(2.29)

$2 = (a, C, d) = Sl34,2-

(2.30)

Ss = Si@S2

= (b,c,d,f)

3

2

4

(2.31)

= Si2,34.

\ / f

Fig. 2.4. A completely connected graph of four nodes The terminal-capacity / - M 5 )

(1^3

(d)

Fig. 2.7. Computation

(c)

(e) of terminal-capacities

(f)

for Example 6

Nonoriented

Communication

Nets

283

and there is indeed a cut-set, namely, the incident cut-set of C3, which has the desired property. Corollary 6. Let S be the minimum cut-set which separates G into G' and G"; then the terminal-capacity tij is not changed when all edges in G" are shorted, provided that i and j are both in G'. Proof: Since shorting never decreases the flow capacity, this follows directly from Theorem 16. With successive use of Corollary 6, the computation of all terminal capacities can be done in a rather simple manner. We shall illustrate this with the following example. Example 6. We are given the nonoriented communication net as shown in Fig. 2.7a, and we find that Si is a minimum cut-set for /15. For computing terminal-capacities 112, /13, and /23, the reduced graph in Fig. 2.7c is used. For computing ¿45, the graph in Fig. 2.7e is used. The terminal capacity for each pair of nodes of the net is given in Fig. 2.7f.

SYNTHESIS OF NONORIENTED COMMUNICATION WITH MINIMUM TOTAL EDGE-CAPACITY

NETS

As in most design problems, the realization of a T-matrix is not unique. This has been demonstrated in the last section. In a situation like this, one naturally inquires whether it is possible to impose some further constraints on the realization. In this section we shall discuss realization of matrices with the additional constraint that the total edge-capacity of the realization must be a minimum. 9 Other constraints are discussed in Chapter 3. First we shall derive a bound for total edge-capacity required for the realization of a given terminal matrix. We observe that for any realization the capacities of the edges, e is the index of partitioning. This results from the fact that when all entries in T are equal, Ip = 1, in which case n edges are needed, and from the fact that an extra edge is added each time a new ring is formed. As an illustration the matrix of Eq. (2.39) is realized as shown in Fig. 2.19. Method of decomposition

matrices8

of

This method is based on the decomposition of terminal matrices as discussed earlier in this chapter. The given terminal matrix is written as a sum of two terminal matrices of which one is a uniform matrix as: T = Ti + T u ,

(2.68)

where tu, the uniform element in 1 u, is equal to the minimum element of tij, t0- Since the zero elements of Ti will indicate where the minimum cut-set will be in the realization of Ti, one may realize T u in such a way that the minimum cut-sets of the realization of T « correspond to the minimum cut-sets of the realization of Ti. The realization of each nonzero part of Ti is accomplished by further decomposition and realization. By successive application of the above process, one simply obtains an expansion of a terminal matrix into a sum of uniform matrices as: T =

(2.69)

where the T„.'s are realized and combined in such a way that all their minimum cut-sets correspond to each other. Example 7. Realize the matrix,

T =

5

5

4

4

5

(2)

6

4

4

5

6

(3)

4

4

4

4

4

©

6

_ 4

4

4

6

(5)

(2.70)

Nonoriented Communication Nets

293

First we write T as T = TUi + T„t + TUl + T Uj ,

(2.71)

where r(D 4 T«, =

T«, =

@

4

4

4

4 "

©

4

4

4

4

4

4

4

©

4

4

4

©

4

. 4

4

4

4

©-

r © 0

0

0

0

o -

©

0

0

0

0

0

0

0

©

0

0

0

©

2

. 0

0

0

2

© -

(2.72)

(2.73)

© (d)

(e)

Fig. 2.20. Minimal realization of the terminal matrix of Example 7

294

Analysis

and Synthesis

1 r(D 1 i 0) 1 i 1 CD T u , == 0 0 0

T«, ==

. 0

0

0

r® 0

0

0

0 0 _0

(D 1 l (D 0 0 0

0

of Communication

0

0 -

0

0

0

0

©

0

0

(D-

0 0

0 -

0

0

©

0

0

Nets

(2.74)

0 (2.75)

CD-

T h e realization f o r each u n i f o r m matrix T U ( is shown in Figs. 2.20a, b, c, a n d d, a n d the over-all realization is o b t a i n e d by a d d i t i o n of t h e individual realizations in p r o p e r m a n n e r as s h o w n in Fig. 2.20e.

CHAPTER

3

Further Discussion on Communication Nets

APPLICATION

OF LINEAR

PROGRAMMING

IN C H A P T E R 2 we discussed the synthesis of nonoriented communication nets under some constraints. Several methods for the synthesis of communication nets with minimal total edge-capacity were discussed in detail. Here a more general case will be considered; that is, the case of synthesis with minimum linear cost. A cost cij is defined as the cost for installing a unit of edge-capacity between the nodes i and j. The total cost of the net is then

Ct

=

i.l

'ZCUEIJ,

for i , j = 1, 2

n\

i Ï j.

(3.1)

Since Ct is seen to be a linear function of the edge-capacities linear programming techniques turn out to be very useful. 7,10 The unknowns of the problem are the [n(n—1)/2] e and j = p +1, . . ., n. Applying this principle to all cut-sets, we have a total of ( 2 n _ 1 —1) inequalities of the form v n 2

2

et

(-1 1-JM-1

*

^

m a x { f i ( P + I ) , F I ( P + 2 ) , . . . , tin • • -, 'P(J>+D • • -, Ipn}-

(3-2)

296

Analysis

and

Synthesis

of

Communication

Nets

In addition to the above, we have en

>

0

i j

=

1, 2 , . . . , n .

(3.3)

The problem can now be stated in the linear programming f o r m as follows. "Minimize the total system cost 2

c

*iev

u

subject to the set of linear constraints of (3.2) and (3.3)." The simplex method or other modified methods can be used to find the o p t i m u m solution. As an example, consider finding a four-node net satisfying a terminal matrix

T =

©

9

7

8

8

7

9

(D

7

7

7

(3-4)

©J

such that the total system cost ei2 + 2ei3

+ 2eu

+ 2 ^ 2 3 + Se24

+

(3.5)

l e ^

is minimum. The constraints on the unknown are: e u + ei3 + e u

Js m a x { i i 2 , i i 3 , ' 1 4 }

£21 + ^23 +P24

=

9

= 8

Ss m a x { / 2 l , ' 2 3 , (24}

^31+^32+^34

^

m a x { ? 3 i , t32,

'34}

=

9

^41+^42+^43

>

m a x { ? 4 i , Uz,

Ì43} =

7

+ e23 + e24

>

m a x { i i 3 , tu,

123,

(3.6)

¿24} =

9

£ 1 2 + £ 1 4 + ^ 3 2 + ^34 ^

m a x { ? ! 2 , '14, '32, '34} =

8

^12+^13 + ^42+^43

m a x { / i 2 , '13, '42, '43}

9

e\3+eu

=

and en > 0,

i, j — 1 , 2 , . . . , « ;

(3.7)

The edge-capacity matrix E which realizes the given T and at the same time minimizes the system cost is

Further Discussion on Communication - 0

4

4

Nets

297

1.5

3.5"

©

4

0

1.5

4

(D

3.5

.3.5

0

3.5

(3.8)

The corresponding system cost is 29. The network is shown in Fig. 3.1.

3.

Fig. 3.1. Realization of minimum cost of the terminal matrix of Eq. (3.4) The number of inequalities is (2 n _ 1 —1) and the number of unknowns is n(n— l)/2. For any realistic n, the simplex method becomes unmanageable even with a digital computer. The large number of zeros in the set of inequalities suggests that a more efficient method for finding the optimal solution is possible. Some attempts in this direction have been made and more efficient algorithms are now available. However, even with the improved methods, this approach is useful only if it is incorporated with other techniques. INDEX

OF A COMMUNICATION

NET

Another interesting problem in the synthesis of nonoriented communication nets is the case where the terminal matrix is not given. Situations like this occur when not enough information is available, as in the case of military warning systems or other types of systems for emergency purposes. Under these circumstances, an average measure of the system's capability may be obtained in terms of the index of the communication net. The index of a communication net, I, is defined as 9 1

= 2 (>J

for

'> J =

1

- 2, ... n.

(3.9)

298

Analysis

and Synthesis

of Communication

Nets

It is seen t h a t the index is a f u n c t i o n of the realization. S u p p o s e o n e is given only the total edge-capacity and not the terminal matrix. T h e n one would be interested in obtaining a net such that the index is maximized, assuming we have a fixed n u m b e r of nodes. Before investigating possible ways of maximizing the index, let us first observe a very special type of c o m m u n i c a t i o n net. Suppose we have a completely connected graph with edges of equal capacity e\ then all terminal-capacities would also be equal to each other. In fact, f o r all terminal-capacities, hi = t = (n-\)e.

(3.10)

In the meantime, the total edge-capacity of the net is equal to Et

= [n(n-\)l2]e

= NT\2

(3.11)

I = (n— \ )ET-

(3.12)

N o w suppose one submatrix of a terminal matrix has the following form: T

rA M r , .

il •]•

(3,3

>

where A is a u n i f o r m matrix of value IA and has u nodes, B is a u n i f o r m matrix of value IB and has v nodes, and Ti is a matrix which is u n i f o r m and of value t\. W i t h o u t loss of generality we may assume h

IA < tR.

(3.14)

T h e m i n i m u m total edge-capacity f o r this submatrix is £> 0 = J(i/M+V/b),

(3.15)

while the index of this submatrix is /o = [u(u-l)l2]tA

+ [v(v-\)l2]tR

+ uvti.

(3.16)

Solving Eq. (3.15) for ¡R and substituting it into Eq. (3.16), o n e obtains /o =

{V-\)ET0

+ (UIAI2)(U+V).

(3.17)

This means that if we increase IA, the index will always increase with the total edge-capacity held constant. T h e highest possible value for /A is reached when it is equal to tB, and this value is 2

EtJ(U+V).

(3.18)

Further Discussion on Communication

Nets

299

Thus if the total edge-capacity is held constant, the maximum value for the index is ( / o W = (v —

\ )ET„

+ U(W+ V)[2£T 0 /("+ V)] =

ET0(U+

v — 1). (3.19)

By smoothing out the terminal-capacities we have succeeded in increasing the index to a maximum while making all terminalcapacities equal in the meantime. Due to the fact that in any terminal matrix the smallest partitions are always uniform in value, the above procedure may be applied repetitively so that any terminal matrix can eventually have the maximum possible index for the given total edge-capacity; in the meantime all the terminal-capacities will have the same value. SIMULTANEOUS CATION NETS

FLOWS

IN NONORIENTED

COMMUNI-

Up to now, it was assumed that at any instant there is in the communication net only one source and one destination. In reality, this is a rather rare case. In general, simultaneous flows occur and their magnitude may vary with time. However, as we shall show in this section, many of the concepts discussed are very useful even in the treatment of simultaneous flows.11 The requirements for simultaneous flow are conveniently represented in the form of a set of matrices:

r(D

n2»>...

rin,tr

R k = 1, 2, . . m, for simultaneous flow, then for every collection of nodes y, C(y) of E satisfies C(y) > C(y).

(3.31)

Proof: Assume the theorem is false. Then C(y) < C(y)

(3.32)

for some y. From Eq. (3.30) it follows that C(y) < C maxt{C(*>(a ©£)}.

(3.39)

Analysis and Synthesis

302

of Communication

Nets

Theorem 20. A tree-network satisfies the given set of simultaneous flow requirement matrices if and only if the (n— 1) simple cut-sets satisfy the inequality of (3.32). Proof: The "only i f " follows directly from Theorem 18. F o r the " i f , " let the (n— 1) simple cut-sets of a given tree satisfy the inequality of (3.31). For each k write E = 2 ÏW*'. (3.40) i'22+^)]. (3.45) Therefore

dYab/dg = yi22l(}'22+g)2

= positive number.

(3.46)

The identification of nodes can only increase the value of This leads to the conclusion that tab > Yab.

Yab. (3.47)

Another analogy may be made between the readability of resistive networks and communication nets. If we remember our study of the realization of a 7-matrix in terms of a resistive multiport network in Part III, we see that the following two conditions are necessary and sufficient for the realization of a symmetric matrix Y = [yu], with positive real elements, possibly after rows and columns are interchanged, as the short-circuit admittance matrix of an n-port resistive network with (n + 1) nodes and a linear-tree port-structure. Namely, Y must be symmetric and be uniformly tapered. That is, yu > >'(,(+1 ^ >'