ABSTRACT

TOPQLOGICAL SYNTHESIS OF N-PORT RESISTIVE NETWORKS
FRQM SHORT CIRCUIT CONDUCTANCE MATRICES
THAT ARE REALIZABLE WITH Two-TREE PORT-STRUCTURES
by

C . G. Jambotkar

Procedures are available in the literature for synthesizing
resistive networks from short circuit conductance matrices which
are realizable with connected (one -tree) port-structures. Little
is known, however, regarding synthesis of resistive networks from
short circuit conductance matrices which are realizable only with
separated (k-tree) port-structures.

In this thesis, a procedure is established for the synthesis
of resistive networks from short circuit conductance matrices which
are realizable with two-tree port—structures. The formulation
presented in the thesis enables the problem to be reduced, in fact,
to the well-known synthesis of resistive networks having linear
port-structures, For a complete resistive network with (n+2) nodes,
the number of the constituent two-terminal resistors is (n+l)(n+2)/2.
in the devised procedure, the conductance values of (n+1) of these
constituent resistors - which are incident at a particular node - have
been considered as parameters. It is established that the indicated
parameters are subject to certain bounds, which, in fact, facilitate

the desired realization of matrices. The problem then is to obtain

C . G. JAIV’IBOT KAR

a SII'it.1i;2iI‘-II' 3 I '. -.: ‘*-~I—, of these parameters within the established
bounds. "I’Izis 3~ I3'1I‘2h the help of a digital computer. Once the
values of tin, I I meters are decided, the complete realization of
the short circuit conductance matrix follows immediately. In
general, the m"; m values of the parameters is not unique, so that
many quIiI'alI-rt I I171 zations are obtainable by means of the devised
procedure.

(3e: mm Hiul'l circuit conductance matrices which belong to

Class are also considered in the thesis. They are

a certain sf» ,L'i
realizable watt II : peeial version of the above procedure, which has
one llllifl‘il‘i, 1 .i'I: ,z‘I- Its-rc- of providing two distinct ”minimal" realizations.
Furtherirmru, tin-I"- III:cessity of machine computations is avoided in
this case.

It is II} in this thesis that every paramount matrix of
order lilll‘ttt‘ II to the above special class. Thus, a new, straight-
forward IrrI ::I-I?:.:,ru- is established for the realization of any third—order
paramo-I‘Int‘ mm" Ir-I'hirh is considered as either a short circuit

condurtanm; :IIIItmz or an Open circuit resistance matrix.

'Ifii";.«”l'y, I ideas are included in this thesis on a possible
LIIII-IrIrileI tat- it’ . ; 1 III-am of realization of Short circuit conductance
matrices V'i‘i’,‘ I In: realizable with k~tree port-structures

TOPOI..~’1)(’.}II".._5-"Il. S‘IUNTI-IESIS OF N-PORT RESISTIVE NETWORKS
I’ R.(”j.‘l_1".\'l ISSUE-RI CIRCUIT CONDUCTANCE MATRICES

THAT ARE I I ~“ 11.91.2175?)le WITH TWO-TREE PORT—STRUCTUR ES

by

C
’ /
J 1" .I"

C'. 1G;"“’Jambotkar

\ i

A THESIS

Submitted to
l\xli(:thigan State University
'17. :if.I_:ft‘7al fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
I‘IIgpartment of Electrical Engineering

1967

ACKNOWLEDGMENTS

The authIm wishes to express, first of all, his deep gratitude
to Dr. Y. Tokad for providing excellent direction to the research
reported in this thesis. The author is also grateful to Dr. H. G.
Hedges for his every interest in the author's doctoral studies in
the capacity of his academic adviser, guidance committee chair-
man, and the acting chairman of the department. Sincere thanks
are due to Prof. T. W. Culpepper, Dr. J. Kateley, and Dr. N. L.
Hills for their willing participation in the author's guidance
committee. Finally, it is acknowledged that this research was
financially supported by the Division of Engineering Research at

Michigan State Unive r sity.

ii

TABLE OF C ONTENTS

Chapter Page
1 INTRODUCTION ............. . . . . 1
10 lo MOtivation. o o o s o o o o o o o o o o o o l
1. 2. Some Basic Concepts and Definitions . . 6
2 MATRICES OF THE N-TH ORDER . . . . . . . 12
2.1.Introduction................ 12
2.2. Some Analytical Considerations . . . . . 13
2.3. A Theorem in Algebra . . . . . . . . . . 29
2. 4. Conductance-Parameter Procedure

for Realization of nth-Order Matrices

with Two-Tree Port-Structures . . . . . 31
The Conductance Parameters . . . . . . 35
The Machine Computations of the

Conductance Parameters . . . . . . . . . 49
Realization of a Special Class of

nth-Order Matrices. . . . . . . . . . . . 50

NN
o
O‘U"
o

N
‘1
O

2.8. Matrices of the Fourth Order . . . . . . 58

2. 9. Realization of Matrices with
k-Tree Port-Structures . . . . . . . . . 79
3 MATRICES OF THE THIRD ORDER . . . . . . 88

3.1.Introduction................ 88
3. 2. Some Properties of Third-Order
Paramount Matrices . , . . . . . . . . . 89
3. 3. Realization of Third-Order
Paramount Matrices . . . . ..... . . 102
3o 40 Example 0 o o o o o o o o o o o o o o o o o 108
4 CONCLUSION . .......... . . . . . . . 112
APPENDIXIoososooooooo'ooooo113
APPENDIX 2 0 O O O O O O O O O C O O O O O O O 116
APPENDIX 3 O O I O O O O C O O O O O O O I O O 133

REFERENCES. . ..... . . . . . ...... 134

iii

lN'l‘lODIICTION

1.1. Motivation

 

Synthesis of a transfornierless n-port resistive network
from its short circuit (s. c.) conductance matrix is an important
topic in network theory. In his classical book, Cauer [CA 1]
presented a complete solution to the relatively less important
.problem of the synthesis of a network which includes ideal
transformers. However, the main problem of transforn'ierless
synthesis remained almost totally unsolved until recently.

The progress ma’de in recent years in this area reveals
that consideration of a network from the tOpOIOgical point of
view offers much insight into the problem. This thesis, also,
is based upon topOIOgical considerations while dealing with one
important part of the whole problem, viz. , the synthesis of
transforrnerless n-port resistive networks from S. c. conductance
matrices that are realizable with two-tree port—structures though
not realizable with connected (one -tree) port-structures.

The general problem of transforrnerless synthesis is a
basic theoretical problem-I. It can be looked upon as the inverse
of the problem of analysis of resistive networks that was solved

long ago by Kirchhoff and l\/laxwell. Knowledge in the area of
synthesis of resistive networks has important applications in fields

such as contact, communication], and probabilistic networks or

sequential machines inasmuch as the weights assigned to edges

v

of the pertinent graphs in these fields are. normally rmnwnegative
real numbers. It is known [ PA 1] that the. solution to the problem
of synthesis of resistive networks does not lend it'selt to direct
extension to the synthesis of the broader class of networks, viz. ,
the RLC networks, as characterized in the s ~domain. However,
it is believed [ DE 1], [LE 1], [RA 1] that if the RLC networks
are characterized by their state equations, then an application of
the topological synthesis of resistive networks may very well.
provide such an extension. Moreover, even it the techniques of
synthesis of resistive networks cannot be extended directly to the
synthesis of RLC networks characterized in the s~domain, those
techniques do have value in regard to the latter problem. For
example, an RLC network displays the properties of a resistive
network for positive, real values of the complex frequency, 5,

so that one of the Obvious necessary conditions for the synthesis
of an RLC network is the R-realizabili'ty of its 5. c. admittance
matrix for all positive, real values of s .

The general problem may be defined precisely as the
“problem of synthesis of a resistive network, it existent, from
an nth—order, real, symmetric matrix considered as a s. <. .
conductance Inatrix. ” The solution to this problem must consist
of two parts: (1) a formulation of the necessary and sufficient
conditions for a matrix to be the s. c. conductance matrix
corresponding to one or more re51st1ve networks, and (2) a
statement of a procedure for syntheSizing at least one of those

networks without using any transformers. The latter includes:

(i) stating the network configuration(s), (ii) specifying the locations
and orientations of ports, and (iii) stating the'values of network
elements in terms of entries of the s. c. conductance matrix.

It is fitting at this stage to review briefly the recent
progress in the area of synthesis of resistive networks. It has
been established [ CE 1] , [CE 2] that a matrix must be paramount
in order that it may be a s. c. conductance matrix, but that
paramountcy is not always sufficient for a matrix to have the
said electrical significance when the matrix order exceeds three.
A procedure is known [ SL 1] for realization of a very special class
of matrices, viz. , the dominant matrices. Satisfactory necessary
and sufficient conditions are known on matrices if they are to
correspond to networks having two special types of port-structures,
viz. , those that form linear (path) trees and starlike (Lagrangian)
trees. In the case of a linear tree, the necessary and sufficient
condition is that a matrix be uniformly tapered [ BI 1] , [GU l] ;
in the case of a starlike tree, the necessary and sufficient
condition is that a matrix be dominant with non-positive off~diagonal
entries [ BO 1] . The articles dealing with these two special port--
structures also state the corresponding realization procedures,
which are straight—forward. It is possible that a matrix is the
s. c. conductance matrix of. a resistive network having a port-
structure that does form a connected graph - the graph must be a
tree in such a case - but then the graph is neither a linear tree
nor a starlike tree. Necessary and sufficient conditions for a

matrix to belong to this broader class have been stated by several

research workers [ GU l] , [BI Z] , [CE 3], [HA1], [BO 2] ,
though, as commented by one of the contributors himself [ PA 1],
none of these sets of necessary and sufficient conditions is fully
satisfactory. Their deficiency lies in the fact that each one of
them requires execution of the complete process of building up
the network; as such, they are all Operational in character. The
possibility of establishing prOper conditions which can be tested
without resorting to a building-up process appears, at present,
rather remote to many research workers, and the few articles
that have been published in the area of R-network synthesis during
the past four years have been directed exclusively toward the
solution of the next and the much more challenging problem of
matrices that may be s. c. conductance matrices realizable with
separated port-structures alone. At the time the research
reported in this thesis was started, the problem remained far
from being solved.

Some elementary ideas were offered by Guillemin [ GU l]
for the first time in 1960 regarding matrices that mlght be
realizable with separated port-structures. A year later, he
elaborated on these ideas [ GU 2] without any claim of having
introduced a practical procedure. A good illustration of this
formidable "augmentation" procedure incorporating trial-and~
error was supplied by Brown and Tokad in one of their articles
[BR 1] , which also presented some further fundamental concepts
in the area. The same idea of augmentation has recently been

pursued by Swaminathan [ SW 1] , who finally formulates some

necessary conditions - named the "supremacy" conditions - that
are applicable in the case of piecewise linear, separated port—
sturctures. Five more articles [BI 3], [El 4], [CE 4], [CE 5], [c1 2]
and four short notes [ B1 5] , [ BI 6] , [NA 1] , [NA 2] are available
in the literature, which considerably aid our understanding of
various important aspects of resistive networks having more than
(n+1) nodes. None of these articles and notes directly cover,
however, the problem of establishing some practical techniques
for synthesizing resistive networks from s. c. conductance
matrices that are realizable with separated port-structures

alone. The more recent one of the two articles by Lupo and
Halkias [HA 2] , [LU l] is of value in the sense just mentioned.

It is directed toward presenting a new method that may apply to

a class of matrices which are realizable with known two-tree
port-structures, the class being defined by the applicability of

the method itself.

The above survey of all the existent literature reveals that
the problem of synthesis of n—port resistive networks is, in fact,
only partially solved. At the same time, as indicated earlier, the
solution to the problem is of much significance, especially in the
context of transformerless synthesis of RLC networks characterized
by their state equations. It was through the realization of the overall
significance of the problem that the author was motivated to carry

out further research in the area of R-network synthesis.

1. 2. Some Basic Concepts and Definitions

 

1. Throughout the thesis, a resistive network will be
represented by a linear graph such that the vertices correspond
to the nodes of the network, the edges correspond to the resistors,
and the edge-weights, to the conductance values.

2. A ”port" is defined as a pair of nodes of a network
accessible for excitations and measurements. A port will be
indicated by an oriented edge in heavy line, the orientation
indicating the polarity of the excitation source. The linear graph
constituted by these edges contains no circuits and is termed as
the "port-structure" (or the ”terminal graph") of the pertinent
nehvork.

3. Let Q denote the s. c. conductance matrix for an
n-port resistive network. Further, let Q' denote the s. c.
conductance matrix for the same network after its original port-
numbers 1, 2, . . . , i, . . . , j, . . . , n are replaced, respectively,
by k,'l, ..., m, ..., c, ..., f (k,l,rn,c,f: n). Then matrices

Q and Q‘ are related by:

(k,1,000,m,000,c,000,f) -\(k,1ooo,nl,coo,c.ooo,f)
I- . 2 9
(1.2.1)

(k,l,...,m,...,c,...,f)

1 denotes an nth—order matrix

where E
derived from the unit matrix by rearranging its columns such that
the entries in positions (k, l), (l, 2), . . . , (m, i), . . . , (c, j), . . . , and

(f, n) are unit entries and the entries in remaining positions are

zero entries.

4. Let Q denote the s. c. conductance matrix for an n-port
resistive network. Further, let (1' denote the s. c. conductance
matrix for the same network after the orientations of some of its
ports i,j, . . ., k (i,j, k E n) are reversed. Then matrices Q and
.Q' are related by:

.,k) k)

Q':U(i’j"' QU(i’jn°°-9
S

3 (1.2.2)

where USU’ j’ ° ° ° ’ M denotes the matrix which results from
reversing the signs of the entries in positions (i, i), (j, j), . . . , and
(k, k) of the nth-order unit matrix. The pre- and post—multiplication
of matrix Q by Usfi’ j’ ° ' ° ’ k) is referred to as the "cross-sign
change operation" on matrix Q.

5. Let Y i [ y..] be a real, symmetric matrix and let

1) n
a matrix, T = [ tij] n’ be defined by:

l fori j

t..= <-l fori:j+1

 

0 otherwise

Then matrix Y is said to be in the uniformly tapered form if each

— 0, each entry, (t)y.(.t) ,

> ' : _
yij _ O and further, w1th yo,j yi, n+1 1.]

in the upper triangular portionf of matrix TYT, viz. ,

(eye)

— _ - > . > .
lJ Yij yi,j+1 Yi-1,j l yi..1,j+1— 0 for J__ 1

(i,j=1,Z,...,n). (1'Z°3)

 

1h.Throughout the thesis, the ”upper triangular portion" of a matrix
will be considered to include all diagonal entries.

A uniformly tapered matrix, Y r [y , can always be

ij] n
realized as a s. c. conductance matrix in the manner shown in
Figure 2. 2.1, where the conductance value of a resistor across

the positive terminal of a port, i, and the negative terminal of

aport, j (i,j = 1,2,...,n), is given by (flyg) in(1.2.3)[GU1].

6. Let
(n) (k)_
Qfl Qi‘z (n)
0*:
szr Q>222 (k)

 

be the s. c. conductance matrix of an (n+k)-port resistive network.
If k of its ports are no longer of interest for any reason, then
the s. c. conductance matrix, Q , corresponding to the first n ports

is given for the same network by:

-l T
- '< _ >:< :'< :f:
Q—Q’r {212sz Q12 (1.2.4)

11

provided Q32 is non-singular [ BR 1] , [KR l] .

7. Let Q“) and Q(2) be the s.c. conductance matrices
corresponding to two different connected (tree) port-structures of
an n-port resistive network. Then matrices Q“) and Q(Z) are

related by:
0(1) = cT 0(2) c

where c is a unimodular matrix [ BR 1] .

(1) and

Consider, now, the s.c. conductance matrices, Q
2 . .
Q( ) , corresponding to two different separated (k-tree) port-

structures of an n-port network such that both port-structures

have the same number of parts and, further, the i—th parts
(i = l, 2, . . . , k) of both port-structures contain the same set of
nodes. Then a congruent relationship holds between matrices

0(1) and Q(2) even in this case as established in the following.

 

Figurel.2.l Figure1.2.2

For the sake of definiteness, suppose that the two port-
structures have three separated parts each as shown in Figure
1.2.1. Let Q?) (Q§2)) denote the s. c. conductance matrix
for the network when measurements are made at ports belonging
to part I alone of the first (second) k-tree port-structure. Since
Q(Il) and Qiz) correspond to two different connected port-

structures of the same network, they must be related by:

(1)_ T (2)
Q1 ‘ C1 Q1 C1

where CI is a unimodular matrix.

(1) (2) (1) (2)
II ' Q11 ' QIII ’ Q111 ’ C11 and C111

in an analogous manner, the following relations can be written:

After defining Q
(1) _" T (7-)
OH ‘ C11 Q11 C11

(1) _ T (2)
QIII ‘ C111 QIII C111

10

Suppose, now, ports (n+1) and (n+2) are added, as shown
in Figure l. 2. 2, to each of the original two k-tree port-structures
so that two connected (one-tree) port- structures are generated.

Note that in both cases ports (n+1), as also (n+2), are connected

between identical pairs of nodes. Let QUV‘ and QR)» denote

the s. c. conductance matrices corresponding to these new connected

J;

"\

port-structures. Then matrices Q(1):‘< and Q( ) must be related

 

 

 

 

 

 

 

 

 

 

by:
r ‘1 — r- "1
CT 0 0(2) 0(2)“? c o
(1),}, 11 1
Q :
(2)" (2%“
_0 U2“ L012 sz__ _0 U2_
where
"" “'1
— —1 (2), (2):]:
CI 0 O Q11 Q12
(74*
C = 0 C O and Q =
II ., WT 2
Q(")' Q( )"\
O O CIII 12 22..
L .4 "
Therefore,
F" .v 1"]
T (Z)n< T (2):,c
C Q11 C C le
<2)* (21*
Q12 C Q22

 

 

Now if ports (n+1) and (n+2) are considered to be no longer of

(1

. . 2
Interest, then S-C. conductance matrices Q‘ ) and Q“ )

corresponding to the two original k—tree port-structures will

ll

be respectively given by:

.-1 T
(1)- T (74* T <2)*=< (2w (2)».
Q ”C Qll C'CQiz sz Q12 C
—1
_ T (2):? (2):}: (2);}; (2)::<
—C Q11 ' Q12 022 Q12 C
and
-1

<2) - (2>* (2)=k <2>==< <2>*
Q — Qll ' Q12 Q22 012

Comparison of the above two expressions establishes the
congruent relationship between matrices Q“) and QB) .

8. A realization of an nth-order s. c. conductance
matrix is termed a "minimal realization" if it contains, at

most, n(n+l )/2 resistors.

Chapter 2

MATRICES OF THE N-TH ORDER

2 . 1. Introduction

 

The present chapter deals, mainly, with the synthesis of
resistive networks from nth-order s. c. conductance matrices
that are realizable with specified two-tree port—structures. In
the beginning, some pertinent analytical aspects of resistive
networks are investigated. The systematic procedure for
realizing the indicated class of s. c. conductance matrices is
then established on the basis of those analytical aspects. Some
machine computations form an integral part of the realization
procedure. Theoretical considerations which facilitate these
computations are presented in one of the sections, followed by
essential details of the method of computations itself. An
interesting version of the above procedure is discussed next.

It has the significant feature of yielding minimal realizations
in the case of a special class of s. c. conductance matrices.
Furthermore, it can be applied easily to solve the pertinent
synthesis problem analytically without the necessity of any
machine computations. Matrices of the fourth order are
considered subsequently in order to illustrate all the foregoing
theory, which covers, in fact, the complete solution to the
problem of realization of matrices with specified two-tree
port-structures. In addition to the above solution, some ideas
are included in this chapter on a possible approach to the problem
of realization of matrices when the port-structure consists of

k trees (3 E k_<_ n).

12

13

2. 2. Some Analytical Considerations

In order to investigate certain analytical aspects of resistive

] The

networks, consider a uniformly tapered matrix, Y = [ y n'

ij
graphs of its (n+1)-node realization as a s. c. conductance matrix is
shown in Figure 2.2.1. The edges indicated by heavy lines in this
figure represent the n ports, and these edges constitute a linear
tree. With reference to this tree, let cii (i = 1, 2, . . . , n) denote
the sum of conductance values of the edges which belong to the
cut-set defined by branch i, and let cij (i,j : 1, 2, . . . , n; i )4 j)

denote the sum of conductance values of the edges which are common

to the cut-sets defined by branches i and j. Then, as is well-known,

yij ‘2 cij (1,j:l,Z,...,n) (2.2.1)

Consider another resistive network derived from the one in
Figure 2. 2.1 by adding, as shown in Figure 2. 2. 2, (n+1) resistors
which have their respective non-negative conductance values
= gk (k=1,2,...,n+l).

Let Q = [EU] n denote the s. c. conductance matrix for this
derived network corresponding to the port-structure indicated in
Figure 2. Z. 2 itself. The following discussion will be directed,
then, toward establishing a relation between matrices Y and 6.

For the derived network, consider a port-structure which
is obtained by augmentation of the original port-structure with a

port, h*, as shown in Figure 2. 2. 3. Let Q* = [q denote

2:2]
1] n+1
the uniformly tapered s. c. conductance matrix corresponding to this

augmented port-structure. Then, the entries of matrix Q* can

be written in terms of the entries of matrix Q as follows:

’K

7V".

0
’K
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where

O<p1<p2— "(911-1 p>ph—ph+1— 3pm:0 ”'2'”
1 6. C1lJ =—1j+p1pJ (i,3-1,2, ,h-l)

qf‘j :aiJ-l pile (i:1,2,...,h-1;j=h+1,h+2,...,n+1)

q:- ‘Ei_1,j_1 pi_1pj_1 (i,j:h+1,h+2,...,n+1)

Cifij =ppj (j=1,2,...,h-1)

qfij =plpj_1 (j =h+l,h+2,...,n+1)

.i 2

qhh =p

Consider a network which is obtained by shorting port h* in

Figure 2. 2. 3. The s. c. conductance matrix, Q = [23.] for

ij n’
this new network can be obtained by deleting the h-th row and the
h-th column of matrix Q* .

Thus, from (2. 2. 2a),

5 = 6+PPT

where (2.2.4)

T
P ‘[P1 P2 ph-l phph+l pn]

This network is shown in Figure 2. 2. 4. The edges corresponding
to the n ports are shown, as before, in heavy lines, and they
constitute a linear tree. Let :11 (i : 1, 2, . . . , n) denote the sum
of conductance values of the edges which belong to the cut-set
defined by branch i, and let Eij (i,j : l, 2, . . . , n; i315 j) denote

the sum of conductance values of the edges which are common to

the cut-sets defined by branches i and j. Then,

l9

qij = cij (1,j=1,2,...,n) (2.2.5)

By comparing Figures 2. 2.1 and 2, 2. 4 at this stage, one

can write the relation:

where

0.: 1,2,...,11—1;j i,i+1,...,li—1)

ll

ij k=Z gn-h+k

aij = O (i:1,2,...,h-l;j:h,h+l,...,n)
n
aij = 1:] gk-h+l (j = h, h+1, ...,n;1=h,h+1,...,j)

After substituting yij for c..

and at. for 3.. in view of(2.2.l)
1.1 1.1 1.1

and (2. 2. 5), one can write the above relation in a matrix form as

follows:
E3 = ir+ A (2.2.6)

where

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Thus, from (2. 2. 4) and (2. 2. 6), we obtain the following
result:

Y=Q-A+PPT (2.2.8)

In order to establish expressions for the entries of matrix

PPT in terms of gk (k=1,2,...,n+l) and qij, considera

matrix) T = [tij] n+1; defined by:
F
l for i=j
t..= j-l fori=j+l
1)
0 otherwise

 

K.

Pre-pand post-multiplying the uniformly tapered matrix Q’l<
in (2. 2. 2) by T, we obtain matrix TQ*T, whose entries yield
(cf. section 1. 2. 5) the following expressions for the conductance

values gk (k=1,2,...,n+l) in Figure 2.2.3:

From entry (1, h),

gn_h+2 = - qlh + p1 (p-ph) (2.2.9)

From entry (c, h),

gn-h+c+l : - qch + qc-l, h +(pC-PC_1)(P~ph)

(c:2,3,...,h~1) (2.2.10)

From entry (h, h),
gn+1 = qh_1,h +(p-ph_1) (p-ph) (2. 2.11)
From entry (h+1, k),

gk-h : qh, k-l "' ql’lk +(pk-pk—1)(p-ph)

(k=h+1,h+2,...,n) (2.2.12)

22

From entry (h+l , n+1),

5’th = 3hr, - pn (p-ph) (2.2.13)
(Note: (2. 2.10) is to be delected if h = 2).
Let the relation in (2. 2.13) be rewritten as:

p7n (p-ph) = 21hr, - gn_h+1 (2. 2.14)

As inspection of Figure 2. 2. 2 will indicate that (Ehn - gn_h+1);é 0
except in a highly degenerate case, so that, in general, pnsé O
and (p-ph)# 0.

Assuming, then, pntfé 0,1. (2.2.14) can be rewritten as:

ahn ' gn.h+1
_ e 2.2.15
9 ph pn ( )

 

We proceed to prove the following relation using mathematical

 

induction:
“ >31 p“ 2 2 6
Pk‘ (qhk " izkgi-hH) — _ ( - -1 )
qhn gn-h+1
(k : h+1,h+2, ...,n)
For that, we shall require the trivial identity:
_ ' P1.l
pn 7' (qhn ' gn-h+1) — (2°2°17)

 

qhn - gn -h+1

Assuming (2.2.16), and substituting (2. 2.15) - (2. 2.16) into (2. 2.12),

 

vaen in the highly degenerate case, where Elm - gn-h+l = O,

the same final results which we propose to establish can, in
fact, be arrived at by starting with another suitable equation
out of (Z. Z. 9) - (2. 2.12), rather than (2. 2.13), and by
modifying the whole treatment appropriately.

23

we have,for k = h+1,h+2,...,n ,

n P

g ==§ -§T + [E- - E g l n -p
k-h h, k-1 hk hk i=k i-h+l — _ k-l
qhn gn-h+l

 

 

 

qhn - gn-h+1 (2 2 l8)
pn
_- _—- +—- g _ qhn"gn—h+1
‘ qh,k-1 qhk qhk ‘izk gi.h+1 pk-l pn
01'
p

n
‘— n
pk-l"(qh,kel ‘ fEk_1gi-h+1)—- (Z'Z°19)

qhn - gn-h+l

 

(k: h+l,h+2,...,n)

In view of (Z. 2.17) and (2. 2.19), the hypothesis of (2. 2.16) is, in
fact, proved.

As a result of the relation just proved, (2. 2.19) holds for
k = h+l, h+2, . . . , n. In particular, for k : h+l , (2. 2.19) yields:

— n p“ 2 2 2
“fini' ifhgiarnl - _ (° ° 0)
qhn gn.h+1

 

ph

We are justified, therefore, in extending the lowest possible value

of k from (h+1) to h in(2.2.16) so as to cover(2.2.20). Thus,

_ n Pn
qhn gn-h+l

 

(kzh,h+1,...,n)

We prove, next, the following relation using mathematical

induction onc e again:

k P

_ — n
Pk ‘ (qkh + if, gn-h+i+1) - (2. 2.22)

qhn - gn-h+l
(k=1.2..-..h-2)

 

24

Substituting (2. 2.15) into (2. 2. 9), we obtain:

qhn _ gn-h+l

gn-h+2 : - qlh + pl pn

 

01'

P
_ — n
p1 — (q1h + gn-h+2) _ (2. Z. 23)

qhn - gn-h+1

 

From(2.2.10), we have, for c =k+l, (c=2,3,...,h-1),

gn-h+k+2 Z ‘ q((+1, h + qkh + (pk,r1 - pka - ph) (2.2.24)

Further, assuming (2. 2. 22) and substituting (2. 2.15) and (2. 2. 22)

into (2. 2. 24), we obtain:

MW

gn-h+k+2 : ‘ C11<+1,h + qkh + (pk+1 ' [ qkh +1 1 gn—h+i+1]

I3n Clhn - gn--h+l
) P

 

 

qhn - gn-h-H n

qhn - gn-h+1 _ - k

z ‘ q((+1, h + qkh l p((+1 p qkh ‘1‘?) gn.h+i+1

n

 

or
k+1 P

pk+1: (qk+l, h + Elgn-h+i+l) a g
hn - n-h+l

n

 

(2. 2.25)

In view of (2. 2. 23) and (2. 2. 25), the hypothesis of (2. 2.22) is, in
fact, proved.
As a result of the relation just proved, (2. 2.25) holds for

k =1, 2, . . ., h-2 . , In particular, for k = h-2 , (2. 2. 25) yields;

25

 

(_ hil ) pm (2 2 26)
= C1 + . g . _ . .
9 1-
1011-1 h-l h -1 n.h+i+1 q g
hn - n-h+l

We are justified, therefore, in extending the highest possible value

of k from (h-Z) to (h-l) in (2.2.22) so as to cover (2.2.26).

 

 

 

 

Thus,
_ k Pn
pk —(c1kh + £1 gn_h+i+1) E - g (2.2.27)
hn n-h+1
(k =1,2,...,h-1)
Substituting, next, (2. 2. 20) into (2. 2.15), we obtain:
n p 21 - g
— hn n-h+1
= - 2 . n + 2. 2.28
P (qhh izhgl-hH) E _ g pn ( )
hn n-h+l
n (21- g )2 P
— h ’ -h+1
= (qhh " .2 gi-h+1 + n 2n ) — n (2°2'29)
”h q - g
n hn n—h+l

In view of (2.2.15), (2.2.26), and (2.2.29), (2.2.11) can be written

 

 

 

 

 

as:
' F - . )2
_ "' +(— _ £31 + qhn g’n—h+1
gn+1 " qh..1,h .qhh izh gi..h+1 p2
‘ n
_ — hil pn , qhn ‘ gn—h+1
qh—1,h ' 1-1 gn.h+i+1 - p
— qhn - gn—h+1 n
01‘
— 2
(qhn gn-h+l) _ “g1 _—
2 ’ 1:1 g1 qhh
pn
ahn ‘ gn-h+1 n+1 — 1/2
._ :1: ..
. p ( .231 g, qhh) (2.2.30)

26
Note, in passing, that (Shh - gn-hH) > O and pr1 > 0 implies:
> 0 (2. 2. 31)

Substituting the positive root in (2. 2. 30) into (2. 2. 29), (2. 2. 27),

and (2. 2. 21), we have, respectively:

_ 71“: n21 _ _1/2 2 Z 3
pk ' (qkh + i=1 gn-h+i+1) ( i=1 g1 ' qhh) ( - -3 )
(k : 1: 2: ah'l)
_ n n51 _ -1/Z
Pk ' (qhk ‘ 1:21. gi-h+1) (i=1 gi ‘ qhh) (2° 2°34)

(k2h,h+l,...,n)

The negative root in (2.2.30) is neglected since p and pk
(k =1, 2, . . . , n) must be non-negative (cf. (2. 2. 3)).

By introducing the above expressions for pk (k : 1, 2, . . . , n)
into (2. 2. 8), the following significant relation between matrices

Y and Q is finally established:

_ n+1
Y = Q-A+(Z gk

—- -— +
lo'hoi1 (2.2.35)
kzl ' °

where matrix A is as defined in (2. 2. 7), and

 

TThe notations M h and Mh denote, respectively, the

h-th column and h-th row of a matrix, M .

27

F-_ "l
q1h + gn—h+2

q2h+ gn.h+2 + gn—h+3

qh-l, h + gn-h+2 + gn-h+3 + H ' + gn

 

 

[6' ]T=Zi"h = -_ ------------------ (2.2.36)
qhh g1 g2 ' gn-h+1
qh+1, h ' g2 g3 ' gn.h+1
qnh - gn-h+l
_ .1

Note that the above relation (2.2.35) between matrices Y and 5

can be reproduced directly by referring to Figure 2. 2. 2.

An Important Special Case

 

When the conductance values (k = 1, 2, . . . , n) of the

gk

corresponding resistors are constrained to zero, the network in
Figure 2. 2. 2 reduces to the one in Figure 2. 2. 5. For convenience

of notation, we shall let gO E g Now, an examination of the

n+1 °

foregoing analytical considerations will reveal that the desired
relation between matrices Y and Q for this special case can,

in fact, be obtained from (2. 2. 35) by setting gk 2 0 (k :1, 2, . . . , n)
therein. That is, for this special case, we have:

'6

Y = §+(1/§)6h h

where 3? = g0 - Ehh is positive (cf. (2. 2.31)), and [Qh ]T = 6h

(2.2.37)

denotes the h-th column of matrix Q .

Z9

2. 3. A Theorem in Algebra

 

In this section, we shall establish a theorem in algebra,
which will find its application in the subsequent discussion.

Theorem: For a given real, symmetric matrix,

Y = [ yij] n’ and a set of real numbers, ak (k = 0,1, 2, . . . , n),
there exists a unique real, symmetric matrix, Q = [ qij] n’
which satisfies the relation:
_ '1 l I
Y ‘ Q+(ao-qhh) Q.th. (2'3'1)

where qhhaé a0 and

[Q' ]T

: ' Z <
.h Qh. [q1h+al q2h+a2 qnh+an]’h n

Proof: Let there exist another real, symmetric matrix,

Q = [21.1.1] n’ satisfying the relation:

(2. 3. 2)

where qhhaé a0 and

=-Q—' =[E +a +a q

1h 1 q2h 2

[Q'

<
.h +a],h n

nh n —

Equating, then, the entries in positions (h, h) of the

right-hand sides of (2.3.1) and (2.3. 2)) we have:

 

 

-1 2 — — -1 — 2
qhh + (a6' qhh) (qhh+ah) ‘ qhh + (a0 'qhh) (qhh+ah)
2 2 — —2 —2 — 2
aoqhh ' qhh + qbh + Z ahqhh + ah : a'oqhh ‘ qhh :qhh+ 2 ah qhh+ ah
a0 " qhh a - "

o qhh

— 2 — 2
-- (ao-qhh)([ao+2 ah] qhh+ah) ‘ (ao‘qhh)([ 210+ 2 21h] qhh + ah)

3O

01‘

_ 2 _
' (a0 + 2 ah) Clhh qhh ' ah qhh

2
ao(aO + 2 ah) qhh + aoah

_ 2- '-
‘ ao(ao + Z a ah ‘ (ao + zah) qbh Clhh ' ah qhh

h) qhh + ao
' a (a + 2 a )(q - El— ) + 3,2 (q - a ) : O
- o o o h hh hh h hh hh

. 2
1.e., [ao(a0+2ah) +a 0 (2-3-3)

Since, in general, a0(aO + 2 ah) + affix-‘- O, the equation (2.2. 3)

implies that:

(2. 3. 4)

qhh qhh

n

<
Equating, now, the entries in position (i, h), i; h’ of the

right-hand sides of equations (2. 3.1) and (2. 3. 2), we have:
q +(a -q >‘1<q +a)(q +a)=E +(a E )‘NE +a)<’qI +a)
ih ohh ihihhh ih ohh ihihhh

Noting that qhh = E and multiplying both sides of the above

hh

equation by (aO - qhh) , we further obtain:

a

a0 qih ‘ qhh qih + qhh qih + ah qih + ai qhh + ai h

aoqih ‘ qhh qih + C1hh qih + ah qih + ai qhh + ai ah

(a0 + ah) (qih - 31h) = o (2. 3. 5)

Since, in general, a0 + ahsé o, the equation (2. 3. 5) implies that:

— . n
qih — qih’ 1 h (2.3.6)

‘KIA

Thus, from (2. 3. 4) and (2. 3. 6), we can write the identity:

thQh. E Qith. (2.3.7)

31

The matrix identity
Q E '6

follows when the right-hand sides of (2. 3.1) and (2. 3. 2) are compared/
keeping in view the equalities established in (2. 3. 4) and (2. 3. 7).

Having proved thus the uniqueness of matrix Q which satisfies
the relation stated in (2. 3.1), we are now ready to establish the
procedure for synthesizing resistive networks from certain
s. c. conductance matrices.

2. 4. Conductance-Parameter Procedure for Realization of
n-th Order Matrices with Two—Tree Port-Structures

 

 

Let Q“) = [ qu] n denote a paramount matrix to be realized
as the s. c. conductance matrix with a specified two-tree port-
structure. If both the trees are not linear, then, as explained in
section 1. 2. 7, an appropriate congruent transformation can be
applied to matrix Q”) . This appropriate congruent transformation

)

can yield a matrix, Q = CTQ(l C/ which would correspond to the

bilinear port-structure shown in Figure 2. 4.1. Having obtained

matrix Q, we can proceed to realize the same; for the realizations

(l)

of matrices Q and Q are identical except for their port-

structures.
1 c h-l h k n
o—)——o-—->—-0——>—-o 0—9—0- ->——o—>——O
A B E F G H L M

The port-numbering is in the natural order.

Figure 2. 4.1

32

Consider the matrix equation

n+1 _1 ' '
Y = Q -A+(k2:31gk -qhh) Q.th. (2.4.1)
where matrix A is as defined in (2. 2. 7), and
qlh + gn.h+2 7

qZh + gn.h+2 + gn-h+3

qh-1,h+gn-h+2+gn-h+3+ +gn
:Q. = __________________ (2.4.2)

 

 

Suppose a set of non-negative parameters gk (k = 1, 2, . . . , n+1)
is found, which, when substituted in (2. 4.1), gives a uniformly
tapered form to matrix Y . This uniformly tapered matrix Y can
be realized readily as shown in Figure 2. 2.1. Suppose, from this
realization of matrix Y, another network is derived, as shown in
Figure 2. 2. 2, by adding (n+1) resistors which have their respective
conductance values equal to the parameters gk (k = 1, 2, . . . , n+1) .
Then, if Q denotes the s. c. conductance matrix for the derived

network, these matrices Y and Q must be related by (2.2.35).

33

Further, by virtue of the theorem proved in section 2. 3, equations
(2. 2. 35) and (2. 4.1) together must imply the identity of matrices
Q and Q. This leads to the significant conclusion that the above
derived network must, in fact, have been a realization of matrix
Q itself.

The above considerations at once indicate the different steps
of the “conductance-parameter" procedure being devised for
synthesizing resistive networks from s. c. conductance matrices
which are realizable with specified two-tree port—structures. The
procedure is almost evident already; it is presented below explicity
for sake of completeness.

Let Q”) = [ quD] n denote a paramount matrix to be realized
as the s. c. conductance matrix with a specified two-tree port-
structure. If both the trees are not linear, apply an appropriate

(1)

congruent transformation to matrix Q and obtain a matrix,
Q = CTQ(1)C) which corresponds to the bilinear port-structure
shown in Figure 2. 4.1.

Find a set of non-negative parameters gk (k = l, 2, . . . , n+1)
which, when substituted in (2. 4.1), gives a uniformly tapered form
to matrix Y defined thereinjr

Realize the uniformly tapered matrix Y with (n+1) nodes
as shown in Figure 2. 2.1.

Consider (n+1) resistors which have their respective

conductance values equal to the parameters gk (k = l, 2, . . . , n+1),

 

+A method for finding such a set of parameters by machine is
indicated later on.

34

and add these resistors to the above realization of matrix Y in the
manner shown in Figure 2. 2. 2. The new network thus obtained is,

(1)

in fact, the desired realization of matrix Q , the accompanying
port-structure to be considered being, of course, the original
two-tree port-structure.

It can be noticed that the network shown in Figure 2. 2. 2
forms a full polygon of (n+2) nodes. The network is thus of the
most general form for that many nodes. Further, whenever a
set (or sets) of non-negative parameters gk (k = l, 2, . . . , n+1)

exists so as to provide a uniformly tapered form to matrix Y in

(2. 4.1), it can be computed by machine as indicated later. In

(1)

, is, indeed, realizable

view of these facts, if a certain matrix) Q
with a specified two-tree port-structure, it can always be realized
by the procedure stated above. In other words, given the two-tree
port-structure, computation of a set of non-negative conductance-
parameters gk (k = 1, 2, . . . , n+1) so as to provide a uniformly
tapered form to matrix Y in (2. 4.1) is the necessary and sufficient
condition for realizability of any given nth-order matrix, Q“) .
Before prooeding to the considerations of the conductance
parameters, let us reiterate here one well-known result, viz. ,
for an nth-order matrix, Q, which is realizable with a bilinear

Q(1,2,...,h-1)

port-structure shown in Figure 2. 4.1, the submatricesl‘

(h,h+1,...,n)

and 0 must both be uniformly tapered. Now, given

 

1- In this thesis, the principal submatrix formed by rows and columns
i, j, . . . , k (i, j, k _<_ n) of an‘nth-order symmetric matrix, Q, will
be denoted by the symbol Q(1’J’ ' ' ' ’ k

35

a matrix, Q”) , this prOperty of the related matrix Q can be
checked easily; as such, we note that it is desirable to do so first
of all by way of checking a fundamental necessary condition when-
ever a matrix, Q“) , is posed for realization with a specified

two-tree port -structure.

2. 5. The Conductance Parameters

 

In this section we shall establish upper and lower bounds on
the conductance values which have been regarded as parameters in
the previous section. These bounds are valuable in the machine
computations of the parametric conductance values themselves.

Let a matrix, T = [tij] n, be defined by:

 

r .
l for i = j
tij: fi-lfori=j+l
0 otherwise
L

By definition, matrix Y in (2. 4.1) would be in uniformly tapered
form if, and only if, the n(n+l)/2 entries in the upper triangular
portion of matrix TYT are non-negative. Let these entries be

denoted by (t) yigt) . From (2. 4.1),

n+1 1

TYT : TQT -TAT+(k2:?lgk-qhh) TQ T (2.5.1)

I I
. h h.
Let (”qg) and (Haigt) denote, respectively, the entries in
positions (i, j) of the upper triangular portions of matrices TQT

and TAT, and let (t)qih and q'(jt) denote, respectively, the

entries in positions (i, l) and (l, j) of column and row matrices

TQ'h and Qh T. Then, through matrix-multiplications,

36

where
qi n” qO ‘2 O (i,j = 0,1,2,. .,n+1)
Also,
(t)(t) _ ._ ,. _
aij ‘L‘n-h+i+l (i - l,2,...,h-l, j — h-l)
(t), (t) . , . _ . - _
01‘} - ‘LJ -})+1 (1 — h, J - h, h+1, o o o , 1'1)
(wail?) 0 otherwise

(forj: 1: 19,121,29-00an)

or, in matrix form,

(h-l)
F— ! | ""
\ ,gn-h+2,
\ O | | 0
\~ lgn-h+3l
‘\\ I - I
\ I ’ l
\ I - I t
TuiT - ~\ lg I (2.5.3)
\ I “‘1 I
\' K
\gn \ \ ________
\
] \
(‘) \ g1 g2 ° gn—h+1
\ —————————
\.
\
\ O
\
\
\
\
\
c \ _l

 

 

 

1.

The symbol X is used to imply that the entries in the lower
triangular portion of the matrix are of no interest in the
discussion and hence have been omitted.

37

 

 

Further,
(t) , . . . - _
qih ”in ' qi—l, h (1 ‘1'2"'°’n)
and
(t) .
I -_ (. I _ I :
th ” 'Jlij qh,j+1 (J 1' 2’ ’n)
where, by deiinition, qo,h = qh, n+1 = O ,
With reference to (2. 4. 2),
(t) . - . _ \
qih “ qih ’ qi..1., h + gn-h+i+1 (1 ‘1’2'°°"h‘1)
(qO h = O by definition)
(t) . _ n r (2. 5.4)
qhh ’ 11in ‘ C1h..1,h ‘ (53:1 gk
(t) . _
q1h — q1h - q1_1,h+ g1.“h (1 — h+1,h+2,...,n)
J
and
(t) l
I -- __ _ ° : -
th ‘ (in) qh,j+1 gn.h+j+2 (J 1'2" 'h 2)
.(1') ~ + £1 (2 5 5)
qh,h.1 ”uh-1 ' C1hh ks) gk ? ° °
(1') .
I - , _ _ .__
th “ C‘hj qh, j+1 g)--h+1 (J h' “1' ' ' "1“)
(qh n+1 : O by definition) J
Cmnpa ring (2. 5. 4) and (2. 5. 5), we may observe that:
(14.“ (thy (j = 1, 2,...,n-l) (2.5.6)

it] " j+1, 11

Obviously, entries in the upper triangular portion of matrix

TQ' th T can he obtained from (2. 5. 4) and (2. 5. 5) by considering
(t) C .(t)

I
the products qih 111.].

f()r jii; 1,j=1,2,..o,n.

38
Now, from (2. 5.1), matrix Y would be uniformly tapered
if, and only if,

(t) (t) (tin) “1 -1(t) . .(t)
ij ij + (£1 gk ‘ qhh) qih th ->— 0 (2'5'7)

(iii: i.j= 1,2,...,n)

For sake of illustration, let us consider the particular case i : j = 1

and 11> 2. Then,

(t) (t) (t) (1:) 1‘“ -1 (t) .. .(t)
q11 " a11 H131 gk ‘qhh) q1h th ->- 0 (2'5°8)

where, through (1". 5. 2) — (2. 5. 5),

(t) (t) _
q11 ’ qll (112
(was) __ O
11 (2.5.9)
(t) . - ,
(11h q1h + En-h+2
(”(13) : — ..
1hi qhi qh2 gn-h+3

Substituting (2. 5. 9) into (2. 5. 8), we have:

n + 1
X1

-1
qi1 q12 l (1,4 51.- qhh) (qlh+gn-h+2)(th th gn-h+3) —

(t) (t)

This inequality is given by consideration of entry y11 . We

(11) (t)
yij

0 (2.5.10)

observe that each of the n(n+l)/2 entries in the upper
triangular portion of matrix TYT offers a similar inequality.
Thus, from what has been stated in section 2. 4, if a matrix, Q,
is realizable with a specified bilinear port-structure, the problem

of realization reduces, now, to solving these n(n+l)/2 nonlinear

simultaneous inequalities involving the (n+1) non-negative parameters

39

gk (k:1, 2, . . . , n+1). A computer method is indicated in section 2. 6

for solving these inequalities. Note, incidentally, that more than

one set of non-negative parameters will exist, in general, which

gk
will satisfy the indicated simultaneous inequalities.
We proceed, next, to establish upper and lower bounds on

the parameters gk .

Recall relation (2. 2. 3), viz. ,

OEPIEPZEH' Eph_1_<_p_>_ph_>_ph+1_>_°" _>_ :Pn: 0'

We shall have occasion to refer to this relation a few times in what
follows.

From (2.2.9),
gn_h+2 : ' qlh + p1 (p "' ph)

Since pl 3 0 and (p - ph) : 0, therefore,

gn.h+2 3 'q1h (2°5'11)
From (2.2.10),
gn—h+c+1 : ” ach + ac—l, h + (pc " pc-l) (P ‘ ph)
(c : 2, 3, , h-l)
Since (pC - pC_1): 0 and (p - ph) i 0, therefore,
gn-h+c+l _>_ -Ech+Ec-l,h (2.5.12)

From (2. 2.11),

gn+l = qh_1, h + (p - ph_1) (p - ph)

40

Since (p - ph_1) _>_ 0 and (p - ph) : 0, therefore,

> _
gn+1 _ qh-1,h (Z. 5.13)

From (2.2.12),
gk_h : qh’ k-l " qhk + (pk ' pk-l) (P‘Ph)
(k = h+l,h+2,...,n)

Since (pk—1 ' Pk) Z 0 and (P - ph) _>_ 0, therefore,
gk-h E 511,1.-1 551* (k = h+l,h+2,...,n) (2.5.14)
From (2.2.13),

gn-h+1 : Ehn ' Pn (P - Ph)

Since pn_>_ O and (p - ph) _>_ 0, therefore,

< ._
gn-h+l _ qhn (2. 5.15)
Also, from (2.2.31),
n+1 _
>
kél gk qhh (2.5.16)

Recall having seen in section 2. 4 that in'the context of
realization of a matrix, Q, by the conductance-parameter
procedure, matrices Q and Q are, in fact, identical. Hence
we may rewrite the above bounds in terms of entries qij of matrix
Q, rather than in terms of entries an. of matrix Q. Thus, from

(2.5.11) - (2.5.16), we have, respectively:

41

 

2min 2 "11h (45-17)
gn_h+c+1 3 - qch + qc_1,h (c = 2,3, ,h-l) (2.5.18)
gn+1 3 (1114,}, (2.5.19)
< " 2'.’
gk-h _ qh,k-1 qhk (k h+1,h+2,..,n) (2.5.20)
gn_h+1 _<_ qhn (2.5.21)
nil-l 2
>
ks, gk qhh (2.5.2 )

Note: (2. 5.18) is to be deleted if h = 2 .

We shall proceed to establish further significant bounds on
the conductance parameters gk (cf. (2. 5. 27) - (2. 5. 30) below).
For that, we note that consideration of (2. 5. 4) along with the above
inequalities (2. 5.17) - (2. 5. 22) leads to the following information

concerning column matrix TQ' h :

[From (3.5.17) - (3.5.18)] (t)qih—>- 0 (i = 1.2. .114)

[From (3. 5.20)] (thih 5 o (i = h+l, n+2, . ..,n) (2.5. 23)
t , >
()qhh < 0

Again, considering (2. 5. 5) along with (2. 5.17) - (2. 5. 2.2),

we get the following information concerning row matrix Q' hT:

[From (3.5.18)] qilgt) 5 0 (j = 1, 2,...,h-2)
[From (3.5.20) and (3.5.21)] qith) _>_ 0 (j = h, h+1,...,n) (2.5.24)
((15) _<_
qh,h-1 > O

42

Recall, in passing, the result in (2. 5.6), viz. ,

(1.;-

Now, by virtue of (2. 5. 23) - (2. 5. 24), the sign-pattern

matrix for TQ‘ Q’
. ll 11.

_s
G)

(11) Gr)

 

 

(h)

 

1.

and a non-positive entry.

(1)

I
qj+1,h

'1‘ is established as:

_,_ (h-l)
\ :69:
\
\@ I®l
\ I l
\l .I
\l .I
\121
\,.L
\EF/ \—
\@@
"\
X

 

‘—

(j:1,2,..

(2.5.25)

pertinent entry may be positive, negative, or zero.

.,n-1)

 

The symbols 6) and 9 denote, respectively, a non-negative
The symbol (9 implies that the

43

In the above sign-pattern matrix of TQ' th T, the entries
in positions (i, h-l) (i = l, 2, . . . , h-l) are non-positive and the
entries in positions (h, j) (j = h, h+l, . . . , n) are non-negative when

n
2 gk 2 O . On the other hand, the entries in

qhh - qh--l,h - kzl
positions (i, h-l) (i = l, 2, . . . , h-l) are non-negative and the

entries in positions (h, j) (j = h, h+1, . . . , n) are non-positive

n
- _ < ' -
when qhh qh-l, h 13.31 gk __ 0 . Note also that the Sign pattern
. +1 -l
t _ I I -
ma r1x for (:1 gk qhh) TQ.th. T 18 the same as that for

n+1
TQith.T since, according to (2.5.22), 1:21 gk > qhh'

Consider the entries in positions (i, h-l) (i = 1, 2, , . . , h-l)
and (h, j) (j = h, h+l, . . . , n) of matrix TYT given by (2. 5.1).
We recall that the non-negative conductance parameters gk
(k = l, 2, . . . , n+1) are to be chosen such that the indicated entries —
along with the rest of the entries in the upper triangular portion of
matrix TYT - are non-negative. Then, with reference to (2. 5. 2),
(2. 5. 3), (2. 5. 7), and (2. 5. 26), we can infer the following:

(1) Whenever, in the given matrix, Q,

+ q1.1,h

qi,h-l ’ qi,h-l ‘ (11,11 ' qi-l,h-l
is non-positive for even one i amongst i = l, 2, . . . , h-l,
conductance parameters gk (k = 1, 2, . . . , n) must be
subject to the lower bound:

I1
23 g

- .. <
qhh qh-l,h k=1 k -— O

i.e.,

 

(2. 5.27)

44

further, conductance parameters gk (k = l, 2, . . - . n-h+l)

must be subject to the upper bounds:

- " >
qh,j qh,j+1 qh-l,j +qh-1,j+1 _ gj_h+1 (2.5.28)

 

(j =h,h+1,...,n)

Bounds in (2. 5. 28) are, of course, additional to those stated in

(2. 5. 20) - (2. 5. 21) for the same parameters. These bounds are
easily derived by noting that (2. 5. 27) implies, as stated earlier,
that the entries in positions (h, j) (j = h, h+l, . . . , n) of matrix
(:é: gk - qhh)—1TQ:th. T are non-positive. Bounds in (2. 5. 28)
follow when (2. 5. 7) is considered along with (2. 5. 2) and (2. 5. 3)
for i = h; j = h,h+1,...,n.

(2) Whenever, in the given matrix, Q,

(t) (t) _
th ‘ qh,j ’ qh,j+1 ' qh-l,j + qh-l,j+l

is non-positive for even one j amongst j = h, h+1, . . . , n,
conductance parameters gk (k = 1, 2, . . . , 11) must be

subject to the upper bound:

n
- - >
qhh qh-1,h 1:1 gk — 0
1.e.,
n
>3 (2.5.29)

g <_q -q ;
kzl k hh h-1,h

further, parameters gk (k = n-h+2, n-h+3, . . . , 11) must be

subject to the upper bounds:

.. .. >
qi,h-l qi,h qi-l,h-1+qi-1,h— gn-h+i+1 (2.5.30)

 

(i=1,2,...,h—1)

45

The inequality in (2. 5. 29) obviously suggests the bounds:

>

qhh 'qh-1,h _ 8k (k=1,2,....n) (2.5.31)

Thus, for gk (k = n-h+2, n-h+3, . . . , n-l) , we have the bounds

in (2. 5. 30) as additional to those in (2. 5. 31). For the derivation

of (2. 5. 30), we only need to note one of the implications of (2. 5. 29)
stated earlier, viz. , the entries in positions (i, h-l) (i = l, 2, . . . , h-l)
of matrix (ilgll gi - qhh)-l TQith. T are non-positive. Bounds in

(2. 5. 30) follow when (2. 5. 7) is considered along with (2. 5. 2) and
(2.5.3) for i : l,2,...,h-l; j = h-l .

In addition to the bounds established thus far, some further
conditional upper bounds stated in (2. 5. 37) below can be established
for conductance parameters gk (k = n-h+2, n-h+3, . . . , n). But
let us first enunciate one significant necessary condition, which
follows directly from the above discussion, for the realizability of
a given matrix, Q , accompanied by a specified bilinear port-

structure:

If, for a given matrix, Q ,

~.. .. <
€11,114 qi,h qi-l,h-1+qi-1,h O

for even one i amongst i : l, 2, . . . , h-l, then the matrix

Q is realizable with the bilinear port-structure only if

.. .. >
qh,j qh,j+l qh-1,j + qh-1,j+l — 0

for each j amongst j = h, h+l, . . . , n. On the other hand)

if

 

.. .. <
qh,j qh,j+l qh-l,j+qh-l,j+l

46

for even one j amongst j = h, h+l, . . . , n, then the matrix

Q is realizable with the bilinear port-structure only if

- .. >
i,h-1 (11,11 C114,114 +qi-l,h—O

Cl

 

for each i amongst i: 1, 2, . . . , h-l .
To establish the bounds stated in (2. 5. 37) below, let us

consider entries (t)y(t) (i=1, 2, .. .,h-1) as given by (2.5.7).

i, h-l
Recall that it is our aim to select conductance-parameters gk

(k = 1, Z, . . . , n+1) such that these entries in the upper triangular

portion of matrix TYT are non-negative. Thus, we must have:

I(

(t) (t) _(t)a(t) +65“ -1 It)q,hqht)11
1 1 , -

.. >
(11,114 ,h-l k:1gk qhh) — 0

Therefore, from (2. 5. 2) - (2. 5. 5),

(11,114 ' (in. ' (11-1, h-l + qi-l,h ‘ gn-h+i+l

 

 

 

 

n
+ (CI-(h - (ii-1, h + gn,_h,m-L+1)(qh,h_1 - qhh +k§1 gk) > o
n51 _
k=l gk - qhh
where, we recall, qOh : (10,114 : 0 by deflnltion. Or,
n+1
- + ‘ (qih - qi-l, h + gn-h+i+1)(k§1gk ‘ qhh)
qi,h-1 qi-1,),-1 n+1
1:1 gk ' qhh
n
+ (qih - (ii-1:11 i gn-h+i+1)(qh,h-l " qhh + 151 gk) > o
nxfl _.
2.4 g .- g
k2] k hh
OI‘
- + (qih - qi-Lh + gn-h+i+1)(qh,h-1 ' gn+l) > 0
C1i.h-1 (ii—1,114 n+1 2 _

E 5 -q
k=l k hh

47
Hence,

(qi, h ' qi-1,h l gn-h+i+l)(gn+l ' qh, 11-1)
q. - q. >
1,h-1 1-1,h-l — n+1

,3, gk " qhh

 

(2.5.32)

Consider entry (flyifij) (j: h, h+1, . . . ,n) as given by (2. 5.7).

Since it is desired to have each entry (”3413.) (j : h,h+1, . . . ,n) non-

negative, we can write through (2. 5. 2) - (2. 5. 5):

gm ' (111,341 ‘ qh-l,j + qh.1,j+1 ' gj.h+1
I1
(qhh ‘ C111.1,11 ' 1331 gk)(th ‘ qh, 1+1 ‘ gthfl)
n51 '—
kzl gk " qhh

+

 

where, we recall, 2 O by definition. Therefore,

qh-l,n+l : qh,n+l

 

 

 

n+1
_ (a g. - qhhmhj - <1th 61-h“)
.. q . tq - +
h-1,J h-l,_]+l n+1
>3 g - q
k=l k hh
n
+ (qhh ' ql’l-l,h "131 gk)(th - qh.j+1 - gi-htl) > 0
n+1 —
E g -q
kzl k hh
or
.. q . + q + (gntl 7 qh-1.h)(th_- qhfl.“ - gj'hH) > 0
11-1,, h-1,j+1 n+1 —

Zg-q
kzlk hh

47a

Hence,

(gmL‘ qh-1,h)(th ' thjH ‘ gthH)
1
1 gk ’ qhh

> -
— qh-1,j qh-1,j+1

”flail

(j = h,h+1,...,n) (2.5.33)

. . . . _ >
Suppose, 1n the given matrix, Q , qh_1, J qh_1, j+1 0 for at
least one j amongst j = h, h+l, .. . , n . Let j enote such a j;

' A - > O "’
1. e. , qh-1,j qh—l,j+l O . Then, With reference to (2. 5.19)

(2. 5. 22), we see that the term (th - (ah/5+1 - g3:_h+1) - as well as
each of the other two terms on the left-hand side of (2. 5. 33) - is
strictly positive. Further, from (2. 5.17) - (2. 5.18), the term

.. > - = _ . .
(qih qi-1,h + gn-h+i+l)— O for 1 1,2,..., h l. Multiplylng,

then, both sides of (2. 5. 33) by the non-negative quotient

+ we have:

(qih ' Gil—1,11 gn.h+i+1)/(qh’j‘ “ qh,’j‘+1 " g’j—hH)’

(qih — qi-1,h + gn-h+i+1)(gn+1 - qh'l: h)
n+1
k2, gk ' qhh

. _ . + . (\- /.\ l
_>_ (th (11-1.1. gn-hflHMqh‘l'J qh'L'I“ . (2.5.34)

(”113‘ ' qh,’j‘+1 ' g’j-hH)
(i : 1,2,...,h-1)
We note that the right-hand side of (2. 5. 32) is the same as the

left-hand side of (2. 5. 34), so that

(qih ' (ii-1,11 + gn-h+i+l)(qh-1,A ‘ (lb-1,35%)
(1. h - q. > J
hj h,j+1 j-h+1

48

Therefore,

(qi,h-l ’ qi-1,h-l)(q1fL- qh,’i+1 "ff-M1) > (q _ q + g )
“lb—1,3" qh-1,’j‘+1l — 1h 1-1,h n—h+1+l

(2. 5. 35)

Recall having seen toward the end of section 2. 4 that in matrix Q ,

(1,2,...,h-1) (h,h+1,...,n) must be uniformly

and Q

submatrices Q
tapered. This implies, among other things, that qi h-l - qi_1 h-l Z O

': .. E - A > <"‘<
(1 1,2,...,h1 O) and th qh,j+l—O’h—J—n'

; qo,h-l

(qh n+1 5 0). Next, by our choice, conductance parameters

g3~_h+1 2 0; further, through (2.5.20) - (2.5.21), thf‘ - qh,’j+1

- g3\_h+1 _>_ O; and finally, as hypothesized earlier,

qh_1,’j‘ - qh-l,’j+1 > .0 . Hence, we can write:
(‘11, h-l ’ qi-IJh-lnqh’j‘ ' “lb/5+1)
(qh-l,’j‘ ‘ qh-l,’j‘+1)

 

> (“11,114 ' qi-1,h-1)(qh’j‘ " qh,j‘+1 firm)

(gin-1,3" ' qh-i,’j‘+1)
(2.5.36)

The left-hand side of (2. 5. 35) being equal to the right—hand side of

(2. 5. 36), we have:

((11,114 " qi-1,h-1)(qifi ‘ qh,/i+l) > (q _ q + g )
(qh_1,lj\ - qh-l,/j+l) — 1h 1-l,h n-h+1+l

01')

49

((11,114 " qi-l, 11-1“th ' qh, 1+1) _ q + q > g
(qh-1,j .. qh-l,’j\+l) 1h 1-1,h -— n-h+1+l
i=1,2,...,h-1;

h: j: n, such that

.. >
(cab-1’? qh_1,3.\+1) 0 (2.5.37)

 

Note: If there does not exist even one value of index j between h

and n such that qh-l 3>- qh__1 lj+1 > 0, then, of course, the

bounds in (2. 5. 37) will not hold.

2. 6. The Machine Computations of the Conductance Parameters

It is seen in the previous section that in the realization of

nth-order matrices with two-tree port-structures, a set of

n(n+1 )/2 nonlinear simultaneous inequalities involving the (n+1)

conductance parameters gk (k = l, 2, . . ,n+l) must be solved.

In the absence of an analytical method to solve such a set of

inequalities in general, a numerical method must be used to

obtain one or more solutions. The idea which forms the basis

of the numerical method can be explained as follows:
The n(n+1)/2 simultaneous inequalities are of the form:
(t) t
ij) 3 0

where (t)ylgt) is an algebraic expression involving (n+1)

arameters (k =1, 2, . . ., n+1) (cf. (2. 5.10)). We select a
P 8k

random set of the (n+1) parameters within the bounds established
(tiygg) . If each (t )yi(.t) : 0

earlier and evaluate each expression J

for the selected set of parameters, the set is, indeed, a solution

50

In general, however,

satisfying the simultaneous inequalities.
(t )yijt) such that (t)yi(jC) < O for the

there will be some
Let these particular

randomly selected set of parameters.

(Halli?) Then a systematic search

expressions be denoted by

for a suitable set of parameters gk is started with the aim of
.t) I . A suitable set of the

minimizing the expression 2) “)qu
parameters will have been located when >3 I (t Wig”) attains,

in fact, zero value. The logic diagram of a typical program

capable of the aforesaid systematic search within the bounds is

given in Appendix 3.

2. 7 Realization of a S_pecia1 Class of nth-Order Matrices
So far, we considered different aspects of the conductance-

procedure capable of realizing s. c. conductance

parameter
We shall discuss, now,

matrices with two-tree port-structures.

a special class of nth-order matrices which can be realized

through a special version of the conductance-parameter procedure

One important feature of the special versionf is that it
avoids the necessity of computations by machine. It is also

significant that the procedure readily offers infinitely many

equivalent realizations of matrices belonging to the special class,

two of these realizations being assured to be minimal.

 

f For convenience, we shall often refer to the special version
of the conductance-parameter procedure by the term ”the

special procedure. “

4‘1

lJy‘

I - " ~‘
...,. -... .- ., . . x, _ ,
T., ‘_{_=3I3.Tii,=rll,~. (TH :Dd-Wni 41:1 {Ti-1ft”, l‘l: (...; — g u..)

dli'l’n't‘f :1 p'fi".fi._Yn-')l'.¥?j 1‘1'4'j?“:"

‘1

1‘:

., _ .' . . ’,'-(, ,_ __ _ ..
ziiatriv. “Win a :pt-‘Ci .1: c t

iii);iL-sl”r'-.-n"f"-‘.Ic. 'I‘ l null-

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I... . - -. . ,. ‘31'.. .-... ., --. .... . .- ..
51,101.13er 1‘; J.--fli l“ 1111;; all, '11,):2] ., :L.‘ .1110] Upllafez Lungzhetht

.. . I l . .
7".1115rorma‘t‘«,c)'ri to z‘ri...af:r:». Q‘ ) (1"..1('l olutain a ii'iatrix, Q, J: (3 Q

{.9-

. .' ‘ ' I . . . — 5. . 4. .‘
"-1 and Ffe 11‘ 2+: 2 -.‘~:t1'i.s.rtu1 e stiim'n 1:1

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A 15 E: i" (ii ll L M

.-, , . - V ‘
llitff {JUPIHI‘1.:l'11111“IHw‘ iri 1. 1.))? ""de‘flilkLL order.

’4

Letting Q ‘ (OI ‘* 1‘. :1’3. :(zi'l: 1hr: h..+}~; ("olnuni (row) of

l

I; first a matrix, Y,

L,
...:
....

matrix Q, Iind a f7u.)5‘l*.",‘»"'v EJFIZ‘I-i-‘Y'lv‘i'x (r, x,

cl e l"; ’1‘ v d h

4

v I
o o
assertive a :..::vv,l.rr.»1~:m-.' ta)?" :i.
I
‘ - I A L I I
I\( ’1 :At bl' ‘1 ' 1 r “ I’i I‘ I I I l i) 1‘ (it ‘
.
- . 1., )r'.\ , I_" I r. ’ l
(S 23' It; -) K I l}._'t ' . o o - o
‘ ,. .1 _ , r , -. ,;_ * .5, '., .7 . 1,.
. I iii‘. 3.1 2 " )5 l ,1, "‘ .. a ’ «..UII', 1‘i-I, -ilctlinlf‘l I‘.')£')\\IJ

.,, ‘-,.- , )')r“ .‘ , ‘1 I
3. £1gL’.Y‘(3 .... L... _, a. lgblv"..=£ ‘.‘,‘j i- "It: (:1 \. ‘l‘(.lll~-._l,1.1v’.‘: ‘\(:1.ll‘.C,

' refit: Is a realization of matrix

Q if ’he (”ii-1:73.211. twrr-Lw. -‘ IAN:?-:.z"v1;("*."m':> Is 1‘«misicltkrt-Cl instead

\ )1
ix)

E.v-._dei':tljx, a 3'11&l.'!‘:“~ Q( ) K’lll belong to the special class

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' ’ V“ - a .. 1 'I - .~- 7. ~~ . A "- . ~ " v : r'\ ~ ‘~ ~ I '- ' " fl ‘7' I '73 V ‘-
fnw‘: Ill-”(’5 T“‘:1112:I.i it: \*.I ’. .iH' dirt (If )JT‘JLL‘ClUIC ll, dd’cl “:lL-J 1i,

1
|

(Z):
a p-firj.‘l.';i.\.‘t‘ waitzr- (...: i zass.{"";<:‘(l to the parameter x in (3.. 7.1)
sax-l“. that tine ;-:Tn.=.:,ri‘v. "i: (ivflmnd lilH‘l‘Cll’l assumes a uniforml’f
tap-i red torm.

‘Ne <)I.-s(=.=“<:r= (1:23.11 3‘2? (: port h and the particular
“sis-£1.22 lya v11“?! liS ( 1))er -.: c yalue go are. in series in
Figine 3. 3. '3. ‘\s .1511." i, positions can always be inter-
Vbarigc—‘d w‘tl =:‘-.1:; dim-91113111111 the 1:0 tutork electrically. Thus, in
the‘ 1421.1:zaij‘v-t-“i. of a .Iri.31‘ri~.-., C)(1 ) , by the above procedure, the

port-str1,;:.~‘.'iii e «or C133}w‘."l.!i:?’- to matrix Q can freely be

considered to be: as either :12 Figure 3.7.1 or Figure 2. 7. 2.

l (‘ i ‘: ~ l l i k D
(I}-n—-.-—.j~-"-au-a._-(}.4 .~ 9. c..; .-. tnvf‘rv_..au.v'. gin-n. .. . ’_,m—a--7———-—O (>— - -) . W)
A l’; P: l“ G H L 1V1

. , . .(i .3 I.)
.: I (__J-’ll lilC SUlJI‘lIdit1‘;(‘(_I.S Q ’ ’ ‘ ’ ' 3

3 ed Q" ’ i ' ’ ' ° ° ’ ’1 mt" 5:11;. . Q must be uniformly tapered.

indi'ud, giver. a Justus, Q :' , {his property of the i‘clfItI-d Juanita

Q can he: ,‘riw’. ivy-(l tc1,;,;)."; : 1', “GNU HUlC that it 1:5 desirable.

to do so ti z-s‘: by we. , i a}. :"iv‘1‘; :3. Mindan‘iental nel‘-:.'-ssary cm‘iditim‘:

for reahzability (:1) thr- .r:2.-'t:3 31h ihe above special procedure.
“ii-he. c; :--1:«.Clit1t;n‘1 1'"? 2’: m . r 5,. med implies, among other things,

".1 - , . 1'. .z .
(he 1'01le 1.110 Fr [(11) (17.:

k4

‘ ,x’ .{' (<1; 1' T'
0 .: ql ,r' ...; o o o ; q — o o o ,.... (41; :‘ ’) q

In .- l’li’l,ll"'.. >qkh— ”:q‘ith:0
(2. 7.2)
This relatirm will be exploited later on.
Let a I‘natrix, T -: it: il n’ be defined by
i l for ilj
tij :- 4' --1 Mr is j +1 (2.7.3)

l 0 otherwise

Pre- and post-Inuitiplying the matrix equation (2. 7.1) by T, we

11a vet
1‘11“ TQT +(1/x)TQ th T (2.7.4)
g . (t) (t) _ , . . .
11x.- entries, qij , 11‘: UN) upper triangular portion of matrix
TQT are given by:
(t) _(t) _ , -
‘11., " “1) "11,331 ' (11-1,) + “ii-1,141 (2' 7° 5)
(J > 1, 1:1: 1,Z,-.-,n)
where, by (iel'i'z'Iiti-rm,
.. .. ' ' - ‘)
(11’1”.l -- (1.1,; - 0 (1,j -- 0,1,2,..,,n+l)

. . .. t t .
131.11 (.1161, 13"."1; tri‘"! ‘zi .8, (l/-) ( )qihqhg) 9 1n the upper

(ram-gigflar portioi: m.‘ the: matrix (l/x) TQ 101 T can be obtained
‘ . i 1.

l" I“ -. Im:

 

54

qkh " qk-1,h(: W)

qnh - qn-l,h(: z)

_ _l

 

 

As discussed earlier, a matrix, Q“), belongs to the
special class being considered if, and only if, a positive value
can be assigned to parameter x such that matrix Y in (2. 7.1)
is uniformly tapered, i. e. , each entry in the upper triangular

portion of matrix TYT, viz. 3

(0,0) :quii) +(1/x) (0,, (t)
‘1] ij 1

beconoes non ~11egative.

Now, in View of (2. 7. 2), the sign—pattern matrix for

(l/x) TO 110‘)» T in (2.7.6) can be written for x > O as follows:

55

6969

 

 

' (h-l),
(h) 69 [9 O 9:69 69 69] (2.7.8)
O
O
O
1.e.,
(h-l) (h)
_ ‘ ‘l
\ l
\\ G I G
\ |
\ l
\ I
\ I
"@L \\L ---------- (arm
(h+1) \
\
X \\ Q
\
\
\\
\
\
\
__ \ _

 

 

With reference to (2. 7. 7), the sign-pattern matrix in (2. 7. 9) implies,

in the first place, one necessary condition that the entries in the

upper triangular portions of both submatrices (TQT)(1’ 2' ° ° o a h-I)

(h+l, h+2, . ..,n)

and (TQT) must be non-negative. However, this

56

condition is actually implied by the necessary condition already

(1,2,.-..h)

established, viz., the submatrices Q

Q(h9 h+190003n)

and

of matrix Q must be uniformly tapered. Now,
in view of (2. 7. 9), consideration of each entry (t)yi(jt) , as given
by (2. 7. 7), in the upper triangular portions of submatrices
(TYT)(1' 2’ ° ° ° ’ h-l) and (TYT)(h+1’ h+2’ ' ° ' ’ n) establishes an
individual lower bound on the positive parameter x, while
consideration of each of the remaining entries lays) in the
upper triangular portion of matrix TYT establishes an individual
upper bound on the parameter x . As an example, consider the

entries (fly (t) and (03,191) .. From (2. 7. 5) - (2. 7. 7), and

 

11
(2.7.9),
my)? = (c111 - (412’ +(1/x)q1h(q1h - <12.) 3 o
1.6.,
((111 - (112) 3 (l/x) q1h(q2h - qlh)
or
q ( - )
1h qZh qlh (2.7.10)
(qll " (112)
Again,
(12), (t)
Yih = (qlh ' ql,h+1) +(1/X)qlh(qhh ' qh+1,h)->- 0
(2.7.11)

The entry (03(1):) would, in fact, be automatically non-negative
without imposing any bound whatsoever on the positive parameter

x if the given matrix, Q“) , is such that

57

(t) (t) 2

>
qlh — 0

(qlh 'q1,h+1)

This follows from noting the non-negative character of the term

(t) (t) _ .
(1/x) qlhth _ (1/x)q1h(qhh - qh+l, h) . (cf. 2. 7. 9)). It is
clear that similar statements can be made with regard to

individual bounds imposed on x through consideration of entries

(t)yij) (i :1, 2, . . . , h; j: h, h+l, . . . , n). To continue the
illustration, suppose the given matrix, Q”), is such that:

(t) (t)

qlh Z q1h‘ql,h+1 0'

Accordingly, from (2. 7.11), we have:

(l/X) q111mm: ' qh+1,h) 3 Cl1,h+1 ‘ qlh
01‘

q (q - q )
1h hh h+1,h > x (2.7.12)
q1,h+1 " qlh
For a given matrix, Q“) , let x1 and x2 denote,
respectively, the least upper bound and the greatest lower bound

on parameter x.. Then, in view of the statement immediately

following (2. 7. 6), the inequality
(2.7.13)

is a necessary and sufficient condition for matrix Q“) to belong

to the special class considered in this section. If the inequality
in (2. 7.13) is satisfied, parameter x can evidently be assigned
any value within the bounds x1 and x2 , and each value of x

will offer a different equivalent realization of matrix Q“) . All

 

 

58

these infinitely many equivalent realizations would contain, at most,

1 u n
- n(n+l) +1 resistors. In fact, if x is a551gned a value either

2
equal to x

l
exactly —2— 11(n+l) resistors would be obtained.

1 or x, , a distinct minimal realization containing
‘4

, 2,8. Matrices of the Fourth Order

 

The various considerations stated previously are the most
general in the sense that they hold for any particular order, n, of
a s. c. conductance matrix which is realizable with a specified
two-tree port-structure. Now we shall apply some of these
considerations to matrices of the fourth order mainly with the
intention of illustrating them.

It was indicated earlier that the problem of matrix
realization with a two-tree port-structure is easily reducible to

the one of realization with a bilinear port-structure. Hence we

 

shall be considering only the bilinear port-structures in what

follows. As shown in Figure 2. 8.1, only two distinct bilinear

 

 

 

Figure 2. 8.1

port-structures are possible in the case of four-port networks.
We shall consider either of these port-structures separately so
that the problem of realization of fourth-order matrices with

two-tree port-structures will have been dealt with completely.

 

59

. 1
Case 1. Suppose, for a given fourth-order matrix, Q( ),

the port-structure is specified as indicated in Figure 2. 8. 2.

Figure 2. 8. 2

To obtain a matrix, Q, corresponding to the port-structure shown
in Figure 2- 8. 3. first we pre- and post-multiply matrix 0(1)

by a suitable El matrix, viz. , 131(2’ 3’ 4’ 1) in the present

example, and then pre- and post-multiply the resultant matrix

by a suitable Us matrix, viz. , US(Z) in the present example,

' s

so that Q : US(Z)E . We note

1
that the realizations of matrices Q“) and Q are identical

except for the port-numbering and port-orientations.

Figure 2. 8. 3

Let Y = [yij] 4 denote a fourth-order uniformly tapered

matrix. Its realization as a s. c. conductance matrix with five
notes is shown in Figure 2. 8. 4, where the edges indicated by

heavy lines represent the ports. Let another network be derived,

as shown in Figure 2. 8. 5, from the one in Figure 2. 8. 4 by adding

 

 

60

five resistors which have their respective non-negative conductance-

values equal to gk (k = l - 5).

 

 

 

 

Figure 2. 8. 4

 

 

M

 

\
U'I
4E9

LA)

 

 

 

Figure 2. 8. 5

61

If matrix Q = [25. .] denotes the s. c. conductance matrix

1j 4
for the derived network, then by noting that n = 4 and h = 2 and by
applying the final result of the generalized discussion in section 2. 2,

we can write the following relation between matrices Y and Q.

That is, from (2.2.7), (2.2.35), and (2.2.36),

 

 

 

 

5
_— ' —I —l
Y_Q-A+(k§1gk-q22) QOZQZ. (2.8.1)
where
f—- —
I
g4 O 0 O
\\
g1+g2 g3 g2 g3 g3
A = (2.8.2)
g2+g3 g3
g3
_ ..J
and
l—__ —l
q12 + g4
q22'gl'gz‘g3 __ l_ — —
5, —. _ q12| q22 ' g1 q32 9142
.2 2. “ I

q42 ‘ g3 (2.8.3)

 

 

We shall proceed, now, to apply the important results of the
discussion in section 2. 5 to the conductance-parameters g1 through

g5 . Thus, from (2. 5.2),

62

(gm-C112) (qu-ql3) (q13'q14) (€114)

\
\

\ \(qzz'q23'q12m13) (q23‘q24‘q13+q14) (q24'q14)

\

 

 

 

 

TQT = \ \
(2.8.4)
From (2.5.3),
I- l '1
g4 . O 0 O
x
\ \— — — -— — —— _—
\\ g1 82 g3
TAT = x- —————— (2.8.5)
\ 0 0
\
x \\ 0
From (2. 5.4) - (2. 5.5),
T
rqiz + g4

TQIZQ'2.T : [-u -v -w q42-g3]

 

 

(132 ‘ qzz + g1 (EV)
_ (2. 8. 6)
(142 ' C132 + 82 (l W)
L. _I
Applying, next, the results in (2. 5.17) - (2. 5. 22) to the present
case, the following bounds are established on gk (k = l - 5):
g4: -q12 (2.8.7)
g5 qu (2. 8. 8)

gl 5, qzz " C123 (2. 8. 9)

 

 

63

83 __ <1,24 (2.8.11)
5
k-Zl gk > C122 (2.8.12)

From (2. 5. 26), the sign-pattern matrix for TQ' 2Q2 T is

obtained a s

Reje e a"
vase
\ _____ (2.8.13)
X \\\ 6
\GJ

 

 

In the above sign-pattern matrix, the entry in position (1, l)

(2, 3), and (2, 4)

is non-positive and the entries in positions (2, 2),

are non-negative when q22 - qu _>_ gl + g‘Z + g3 + g4. On the other

hand, the entry in position (1,1) is non-negative and the entries in

positions (2, 2), (2, 3), and (2, 4) are non-positive when q‘22 - q12

_E g1 + g2 + g3 + g4. Further, by virtue of (2. 8.12), the sign-pattern

-1
. , - ' , A
matrix for (1:15,, q23) TQ’ 2Q2. T IS the same as that for

' 1 I
matr1x TQ' ZQZ. T .
From (2. 5. 29) - (2. 5. 30) and the associated generalized state-
ment, if, in matrix Q, q22 - qZ3 - q12 + ql3 _<_ O and/or q23 - q“24 - (113

+ ql4 E 0 and/or (124 - q14 : 0 , then conductance-parameters g1

through g4 are subject to the upper bounds:

1 < - '
glJrgZ~l-g3+g4_q22 qu’ (2.8.14)

64

< (2.8.15)

g4 — qll ‘q12
A conditional upper bound on g4 is further obtained through
(2. 5. 37) as follows:

A (2.8.16)
(qu ' q1,€+1)

 

>
q12 — g4

A
where j is any index between 2 and 4 such that, in matrix Q,

.. >
qu q1, j+1 O '
As an example, suppose, in matrix Q, q13 - q14 > 0; then

the upper bound on g4 is given by:

 

- - q i 8
q13 q14 12 4

Further, if, say, q12 - q13 > O , then an additional upper bound
is:

(q11)(qZZ-q23)

 

- - q 3 8 -
q12 q13 12 4

. 1
Case 11. Suppose, for a given fourth-order matrix, Q( ),

the port-structure is specified as indicated in Figure 2. 8. 6.

\/W

0

Figure 2. 8. 6

65

To obtain a matrix, Q , corresponding to the port-structure

shown in Figure 2. 8. 7, we first pre- and post-multiply matrix

E(3919 4’ 2)

0(1) by a suitable E matrix, viz. , in the present

1
example, and then pre- and post—multiply the resultant matrix by
a suitable US matrix, viz., US(Z) in the present example, so

1 2 3 4

0—9—°-—9'—° o-—-)—-o—-)—o

Figure 2. 8. 7

Let Y = [ yij] 4 denote a fourth-order uniformly tapered
matrix. Its realization as a s. c. conductance matrix with five
nodes is shown in Figure 2. 8. 4. Let another network be derived,
as shown in Figure 2. 8. 8, from the one in Figure 2.8. 4‘by adding
five resistors which have their respective non-negative conductance
values equal to gk (k = l - 5). If matrix Q = [5194 denotes the
s. c. conductance matrix for the derived network, then by noting
that n : 4 and h = 3 and by applying the final result of the
generalized discussion in section 2. 2, we can write the following

relation between matrices Y and Q.

That is, from (2.2.7), (2.2.35), and (2.2.36),

5 _1 __

Y=Q-A+(E:lgk-q33) (2.8.17)

where

and

 

66

 

 

 

 

 

 

 

' ..
83 I
| O
83 + g4|
\
g1 + g2 g2
g2
_ _ I _
q13 q23 I q33
I
l -
+g3 + g3 + g4. g1

(2. 8.18)

(2. 8.19)

We shall proceed, now, to apply the important results of

the discussion in section 2. 5 to the conductance parameters gl

through g5. Thus from (2.5.3),

 

From (2. 5. 4) - (Z. 5- 5) .

T013031"

,_
q13 +g3

q23'q13+g4 (Eu)

q33

q43

“Q33+g1(:-W)

 

—

 

(o 'g3l o 0'1
\ l I
\ |
Lg4| O O
\ \
TAT = \\\ _____
\g1 g2
X \-———
\ 0
_ \ _l

‘qz3'g1'g2'g3‘g4(iv)

(2.8.20)

['u "V "W C143 " gz]

(2.8.21)

 

Applying, next, the results in (2. 5.17) - (2. 5. 22), the

following bounds are established on gk (k = 1 - 5):

7171301

g3 _>. 'q13

> ..
— q23 +q13

853

OD
p—n
I /\

(IO
N
I A

(2. 8.

(2.8.

(2. 8.

(2. 8.

(2. 8.

(2. 8.

22)
23)
24)
25)
26)

27)

From (2. 5. 26), the sign-pattern matrix for TQ' 3Q§ T is

obtained as:

 

 

l l _l

i; l®l 69 69

\‘e{ e o
\ \ \._ _ _ __; (2.8.28)

\® ®

\ _____

\e

In the above sign-pattern matrix, the entries in positions
(1, 2) and (2, 2) are non-positive and the entries in positions (3, 3)
and (3, 4) are non-negative when q}3 - q23 _>_ g1 + g2 + g3 + g4.
On the other hand, the entries in positions (1, 2) and (2, 2) are non-
negative and the entries in positions (3, 3) and (3, 4) are non-
positive when q33 - qZ3 E g1 + g2 + g3 + g4. Further, by
virtue of (2. 8. 27), the sign-pattern matrix for (15:31 gk - q33)
TQ: 3Q'3. T is the same as that for matrix TQ: 3Q‘15. T .

From (2. 5. 27) - (2. 5. 28) and the associated generalized
statement, if, in matrix Q, q12 - ql3 _<_ 0 and/or
C122 - q23 - q12 + q13 5 0, then conductance-parameters gl
through g4 are subject to the bounds:

g1+ g2 + g3 + g4 ->- q33 ' q23 (2‘8‘29)
< .- .. . .
g1 — q33 q34 q23 +q24 (2 8 30)
and
(2.8.31)

g2 -<— C134 ' q24

Again, from (2. 5.29) - (2. 5. 30) and the associated statement,
. . . _ _ < _ <
1f, 1n matr1x Q , c133 q34 qZ3 + q24 _ 0 and/or q}4 qZ4 _ 0 ,

then parameters g1 through g4 are subject to the bounds:

69

< -
g1+ g2 + g3 + g4 — q33 q23 (2°8°32)
< ..
g3 __ q12 q13 (2.8.33)
< .. ..
g4 — q22 q23 q12+q13 (2'8'34)

It is evident that a matrix Q is, in fact, not realizable with
the bilinear port-structure shown in Figure 2. 8. 7 if (i) q12 - q13 < 0
' ' .. - .. < .. <
s1mu1taneously w1th q33 q34 q23 qZ4 0 and/ or q34 q24 0 ,

and/or if (11) - q12 + ql3 < 0 simultaneously with

C122 ' q23
.. .. .. < .. <
q33 q34 q23 q24 O and/or q34 q24 0‘
Next, application of (2. 5. 37) offers the following bounds on

parameters g3 and g4. Thus, with i: 1,

 

(q )(q 4- q A )
qu q2,j+1

With 1: 2,

(<1 - q )(q 4‘ - q 4‘ )
qu q2,J‘+1

A
where j is either equal to 3 or 4 such that, in matrix Q,

A

A >
J qZ,j+l 0'

q2
As an example, suppose, in matrix Q, q23 - q24 > 0;

then the upper bounds on g3 and g4 are given by:

(C112) (Q33 ' C134)

 

- 2.8.37
(C123 _ C124) C113 83 ( )

and

(q -q )(q -q )
22 12 33 34 (2,8,38)

 

+ >
(q23 -2124) C123 C113 — g4

70

However, if matrix Q is such that qZ3 - q24: 0 as well as

q24 _<_ 0 , then the bounds in (2. 8. 35) - (2. 8. 36) do not hold.
Example 1. Consider the s. c. conductance matrix

F

12 l -6 0T

1) 1 10 l 4

(2.8.39)
-6 l 11 2

 

 

O 4 2 9 J

the port-structure being specified as shown in Figure 2. 8. 9.

3 1
W C

Via;

Z
C?’O

Figure 2. 8. 9

We apply the conductance-parameter procedure for

synthesizing a six-terminal resistive network from this s. c.

conductance matrix. The matrix, Q, which corresponds to the

bilinear port-structure shown in Figure 2. 8.10 is given by:

s 1 1 s

1 Z 3
W

Figure 2. 8.10

71

 

 

Thus,
1- 7
11 6 2 1
6 12 0 -1
Q = (2. 8. 40)
Z O 9 4
1 -1 4 10
__ _I
We observe that both the submatrices QU’ 2) and Q(3’ 4) are

uniformly tapered so that one fundamental necessary condition
for realizability of the original matrix, Q“) , is indeed satisfied.
We also observe that this example is a direct illustration of
Case 11 discussed above, so that the realization of matrix Q
must assume the form shown in Figure 2. 8. 8.

With reference to (2. 8.17), we can write:

5 _1
.. _ I I
TYT — TQT - TAT 1. ( El gk q33) TQO3Q3.T

Applying (Z. 8. 4), (Z. 8. 20), and (2. 8. 21) to the present example,

 

 

 

 

 

wehave:
i— “ F 'l
\5\ 4 1 1 \O g3 0 0
‘\8 0 -2 \\g4 o 0
TYT: \ - \\
‘4 5 \g1 g2
\
x \ 6 X \\ 0
i. \4 _. \ _
i_2 +g3
-2+ (Eu)
+ l g4
g: _) [-u--v- -w- 4-g2]
g -9 9-8 -8 -8 -8 (:v
1<=lk 1 2 3 4 (2.8.41)
-5+g1 (3w)

 

 

 

 

72

(1)

The problem of realization of matrix Q now depends upon
finding a suitable set of parameters gk (k = 1 - 5) such that

each entry in the upper triangular portion of the above matrix,

TYT , is non-negative. The following bounds are obtained on

gk through (2.8.22) - (2.8.38):
[From (2.8.24)]: g4 3 2
[From (2.8.26)]: g1 : 5
[From(2.8.27)]: g2 _<_ 4
[From(2.8.37)]: g3 5 28
[From (2. 8.38)]: g4 : 32

A machine search within these bounds, using the principles

indicated in section 2. 6, yields one suitable set of parameters

gk as follows:
81 =1-005252, gZ = 0.000000, g3 : 0.209919,
g4 = 8.070859, and g5 = 11.802151.

Another suitable set with integral values can be easily obtained

from the above set as:
g1:1,g220,g320,g4:8’andg5:12.

Substituting these values in (2. 8. 41), we get:

 

 

 

 

 

 

—I — r— ‘1
F5 4 1 1 [O O 0 0 2
\\ \

\8 0 -2 \8 0 0 1 6

TYT: \ .. \\ +fi [-6 0 4 4]

\\4 5 \l O O
‘ \
\
_ X \f; _X \0_I :4-
P 2 .21
\4 4 1? 1—3-
\
\
_ \O 2 O
- \
\
\3 5
\
\ Z
X \\ 4‘3"
_ \ .4

 

 

The realization of matrix Q“) is readily obtained as shown in

Figure 2. 8.11 (cf. Figures 2. 8. 8 and 2. 8. 9).

 

 

 

 

 

Figure 2. 8.11

74

We shall consider, now, another matrix and apply to it the
special version of the conductance-parameter procedure discussed
in section 2. 7. It will be seen that the matrix does, in fact, belong
to the special class of matrices realizable with the special
procedure, with the result that its two distinct minimal realizations

can be obtained very easily.

Example 2: Consider the s. c. conductance matrix [SL 1]

 

F 2 33
(2(1) = (2.8.42)

3 5 6 18

L -

 

 

Figure 2. 8.12

(2’ 3’ 4’ 1)) we obtain matrix

Pre- and post-multiplying Q“) by E1

Q :: 1531(2’ 3’ 4' 1)Q(1) E1(Z' 3’ 4’ 1) corresponding to the bilinear

port-structure shown in Figure 2. 8.13. Thus,

Figure 2. 8.13

We note that here n : 4 and h = 2.

9
matrices QU’“) and Q(2’3’4)

 

 

5 1q

6 2
18 3

3 7

(2.8.43)

We also note that the sub-

are both in the uniformly tapered

form so that the fundamental necessary condition for realizability

of Q“)

with the special procedure is satisfied.

Applying the relation in (2. 7. 4) to the present example,

we have:

T Y T

 

= TQT +(1/x) TQ 2Q2 T

 

F—

+
NIH

 

4
11
-9

-4

-I

 

m

[-119 4 2]

It is our aim to assign, if possible, a positive value to the

parameter x

position of Inatrix

above equation may be written as:

so that each entry,

(’6) (t)
Yij !

T YT becomes non-negative.

in the upper triangular

Signwi s e, the

T\+' - + +7 "I — l + + + 7
\l \\ I
\' + + + \' + + +
\ \
TYT: \-""——— + \ —————
\ + + \ - -
\ \
\ \
X \\+ X \\_
\ \
__ 5.. L. \—

 

 

 

 

Evidently, the upper bound on x is imposed only through

(9,412 , while an individual lower bound

(1:) (t) (t) (t)
y11 ’ y33 ‘

consideration of entry
on x is imposed through consideration of entries

(t)y(t) and (t) (t) . Thus,

 

 

 

 

 

34’ V44
Hwy???) -l+4;{9:0, or x_<_ 36
[(tlylltll] . 8-4X11_>_0, or x_>_-l-Z-1-
[(Uygtgl 11-9;{4 :0, or xiii—i)-
[(t)y(3tzi] 1- 9x 2 _>_‘_ 0 , or x: 18
[(9,122] 4-4;230, or i._>_ 2
Therefore, we have the least upper bound on x (3 x1) = 36
and the greatest lower bound on x (5 x2) = 18 .

The compatibility of these two bounds directly implies that matrix

Q( ) can, in fact, be realized by the "special procedure. " Infinitely

(1)

many equivalent realization: of matrix Q can be obtained by
assigning different values to x within the bounds 18 and 36 . We
shall consider here three realizations corresponding to the values

18, 36, and 24; the first two realizations will be the minimal ones

containing exactly ten resistors.

77

For x =18, we have:

 

 

 

 

F H
5 8 4
59 1 49 19
\\ 1 4 Z
\ — —— —
\1\52 29 29
TYT = \ (2.8.44)
X \9\ O
\ 5
\ \36
L. \x-d
For x = 36, we have:
t' 1
7 4 Z
6— 0 4— l—
\\9 9 9
‘ 3 2 ll
\ _ _ __
\r24 19 118
TYT : \ (2.8.45)
‘\10 —1—
\ 2
X \\ 31
\\ 9
L \ _
For x : 24, we have:
1 1 Z l
\66 7 43 13
\
\ 14.1. 1.5. 11.}.
\ 8 6 12
_ \
TYT _ .\ 9}- 1. (2.8.46)
\\2 4
X \ \ 3E
\ 3
._ \ —

 

 

The realizations shown in Figures 2. 8.14 - Z. 8.16 follow
when we recall from sections 2. 2 and 2. 7 that the conductance value,

go, of the pertinent resistor is given by ((122 + x) , i.e. , (15 + x).

78

 

 

 

 

C 33 D

 

 

 

 

 

 

 

 

 

Figure 2. 8.16

79

2. 9. Realization of Matrices with k-Tree Port-Structures

The problem of synthesis of resistive networks from s. c.

conductance matrices which are realizable with two-tree port-

structures was dealt with thoroughly in the preceding sections.

However, certain matrices may be realizable, exclusively or

otherwise, with port-structures forming k-trees (n 3. k _>_ 3).

Some ideas on a possible approach to the problem of realization

of these matrices are presented in the following.

We shall establish first one useful result in matrix

algebra.

Theorem 1: Let real, non-singular matrices Y = [y

and Q = [ qij]n

where all entries of matrix R = [ r

diagonal entrie s

rbb’ rff’

be related by:

.,r

l

11

ijn

(b< f< ...<1< n).

]

ijn

(2.9.1)

are zero excepting certain

Then

the relation in (2. 9.1) has the alternate form:

F—

 

q

 

P

I'

 

" C1
bb bb

..qu

' q1b

 

 

 

' qbf ' qb1 Qb.
1
- q . . - q Q
rff ff fl f.
' _1_ _ Q
" qlf " r11 q11 l.

80

provided the inverse matrix on the right-hand side of the
equation (2. 9. 2.) exists;

Proof: Let d denote the number of non-zero entries of
the diagonal matrix R .

Let the n x (1 matrix P z [ pij] be such that

.—

pbl 21’9"" :pld:

L-

1, the rest of the entries being zero.

Let the n x n diagonal matrix R be such

(S)_ (S)

_[rlj ]
(S)_ -1 (S)_ -1 (S)_ -1

that rbb -— rbb , rff — rff , ..., r11 — r11 , the rest of the

entries being zero. (Matrix R(S) represents thus the semi-

inverse of matrix R.)

Let R : [rij] denote the d x d non-singular diagonal
. T . ~ _ ~ __ ~ _
matr1x P RP. Ev1dently, r11 — rbb’ r22 — rff, ..., rdd — r11.
Further, we have 131-1 : PTR(S)P and PR PT 2 R .
. . T -l
Consuler matrix P QY .
T - T -l
p oyl = P o(o -R)
: PT - PTQR
= 'fi'1 RPT - PTQPRPT
= ("fi'1 - PTQP)R Pr

= (PTR(S) P - PTQP)RPT

= (PT[ R‘s) - o] p) ’13 pT

Therefore,

pT o 2 (PT [ R(S) - o] P) "1i PT Y (2.9.3)

 

t The alternate form in (Z. 9. 2) is valuable in that it facilitates
computations.

81

Now,
- '1
1
-— -q -q -q
rbb bb bf b1
_1. _ -
' qu rff qff qfl
PT[R(S) —Q] p -.- . . -, . (2.9.4)
_1_ -
' qlb ' qlf r11 q11
L. a

 

 

Let this be non-singular (hypothesis). Note, further, that

l -R)-1 implies R = (Q-1 - Y-l), so that

Y=(Q'
QRY = (Y-Q) (2.9.5)

Pre-multiplying both sides of the relation in (Z. 9. 3) by the non-

singular matrix (PT[ R(S) - Q] P)"1 and interchanging the sides, we

have:

RPTY = (PT[R(S) - Q]P)‘1PTQ
Therefore,

QPRPTY = QP(PT[R(S)-Q]P)-1PTQ
1.e.,

QRY -_- QP(PT[R(S)-Q]P)-1PTQ

From (2. 9. 5), then,

Y -Q = OP(PT[R(S) -o] P)‘1PTQ

01'

Y = Q+QP(PT[R(S)-Q] P)‘1PTQ. (2.9.6)

82

Note that
QP = Q.b Q.f Q.1 (2.9.7)
and
r- a
Qb.
Qf.
pTo = . (2.9.8)
Ql.
L —i

 

 

In view of (2. 9. 4), (2. 9. 7), and (2. 9. 8), the relation in (2. 9. 6) is,
in fact, the same as that in (2. 9. 2). This proves the theorem.
An obvious corollary of the above theorem is that if every

diagonal entry of matrix R is non-zero, then the relation

Y = (0-1 - R)”1 has the alternate form:

Y = Q +o(R‘1-Q)'lo (2.9.9)

Consider, now, an n-port network, A , such that its 8. c.

]

conductance matrix, Y = [ yij 11’ corresponding to some particular
port-structure is non-singular. Let ES (5 = 1, 2, . . . , n) denote
the n voltage-generators exciting the network.

Consider another n-port network, N, obtained from the
above network by adding non-negative resistors, rSS , in series
with generators ES (5 = l, 2, . . . , n). Let 6 = [-q—ij] n denote

the s. c. conductance matrix for the derived network N, and let

a diagonal matrix, R , be formed with resistor values rSS as its

83

diagonal entries in the same sequence. If matrix 6 is non-singular,
. —-1 - . - - -

matrices Q and Y 1 are obv1ously the open-Circuit re51stance

matrices for networks N and A respectively. Now, in view of

the above discussion, it is clear that
= Y + R , (2. 9.10)

so that

l -1

Y = ('6' -R) . (2.9.11)

We observe that s.c. input conductance ass (5 = 1, 2, . . - ,n)

is given by s. c. input conductance yss itself if rSS = O . Again,

q

ss is given by the series combination of conductances yss and

rss whenever rSS > 0. As a consequence, the following
relation holds:

-1 _
> : 0
SS 958 (8 1.2,....n, rssaéO). (2.9.12)

Suppose, now, that network A has exactly (n+1) nodes, so
that the n voltage-generators Es (s = 1, 2, . . . ,n) constitute a
tree. On the other hand, if network N is obtained from network
A by adding a positive resistor, rmm’ in series with at least
one generator, Em (1 E m E n) , which is represented by an
internal branch, m, then the n voltage—generators exciting

network N must necessarily constitute a forest. This concept

is illustrated in Figure 2. 9.1, where figures (a) and (b) depict

85

networks A and N respectively.

After establishing a simple result in matrix algebra at this
point, we shall be in a position to consider the problem proper of
realization of s. c. conductance matrices with k-tree port-structures.

Theorem 2: Let an nth-order real, non-singular matrix

Y be related to nth-order, real, non-singular matrices Q and

6 by:
Y = (o"1 -R)'1 (2.9.13)
and
Y = (6'1 -R)'1. (2.9.14)
Then Q E 6. (2.9.15)

Proof: Equating the right—hand sides of equations (2. 9.13)
and (2. 9.14), the identity in (2. 9.15) follows immediately.

Let Q = [ qij] n denote a paramount matrix to be realized
as a s. c. conductance matrix. If matrix Q is not realizable with
either a connected or a two-tree port-structure, then realization
must be tried with a k-tree port-structure (n _>_ k _>_ 3). Excepting
in the very special case where matrix Q is dominant, no technique
is available in the literature for realizing matrix Q as specified
above. In this situation, the trial-and-error technique established
below has some value. Besides the fact that by means of this
technique we may be able to realize certain matrices with k-tree
port—structure, it could perhaps lead to a precise method of
realization of all s. c. conductanceimatrices which are realizable

with k-tree port-structures.

 

 

 

 

 

 

86
Suppose, then, that a matrix, R = [ rij] n’ is found with
all entries zero excepting certain diagonal entries rbb’ rff, . , r11,
(b < f < . . . < l < n) such that a matrix, Y, as given by:
r" T (- ~) ‘1 l" "‘
—i- - q - q - 9 Q
rbb bb bf bl b.
_L Q
‘ qu rff ' qff " qfl f.
Y: Q + Q.bQ. f O 0 Q1. . . O
_1. Q
‘ q1b ' qlf ° r ' q11 l.
11
(2. 9.16)

is realizable, through the known techniques, as a s. c. conductance

matrix with one or two-tree port-structures.

r I‘

bb’ ff’ '°" 11

Evidently, entrie s

r of matrix R must be such that the inverse matrix

on the right-hand side of (2. 9.16) exists; further, as will become

clear below, they are subject to the condition r23

b, f, ..., l).

qSS

>

If matrices Q and Y are non-singular, then,

through Theorem 1, the relation in (2. 9.16) is, in fact, an alternate

form of the relation:

--1

-R)

(2. 9.17)

Let an n-port network, A , be the realization of s. c.

conductance matrix Y with ES (5

voltage -generators .

=1,2,

...,n)

denoting the n

Let another n-port network, N, be obtained

from network A by adding non-negative resistors having the above

values r
s s

, in series with the corresponding generators ES

87

If 6 = [an] denotes the s. c. conductance matrix for network N,
then, as seen earlier, matrices Y and 6 must be related by
(Z. 9.11). Further, as stated in Theorem 2 above, the equations
(2. 9.11) and (2. 9.17) together imply the identity of matrices Q
and 6. Hence, network N must, in fact, be a realization of
matrix Q . It should be noted that the relation in (2. 9.12) and the
relevant discussion offer a useful guideline for the selection of the
(s = b, f, ..., 1) must

diagonal matrix R . Stated explicitly, rSS

be chosen such that r.1 > q
85 ss

Chapter 3

MATRICES OF THE THIRD ORDER

3 .1. Introduction

 

Tellegen has proved [ TE 1] that a matrix of order 5 3
is realizable either as a s. c. conductance matrix or as an o. c.
resistance matrix if, and only if, the matrix is paramount. He
has also given canonical structures of realization in each case.
Recently, Cederbaum has shown [ CE 4] that every paramount
matrix of order three can also be realized with a network
which is t0pologically optimal in accordance with the criteria
specified in his paper. The problem of synthesis of resistive
networks from matrices of the second and the third order can
thus be regarded as, in essence, solved. However, the
problem is reinvestigated in this chapter with an entirely
fresh approach, which besides being interesting in itself, has
the feature of offering two distinct minimal realizations
amongst infinitely many continuously equivalent realizations
for any third-order paramount matrix considered as either
a s. c. conductance matrix or an o. c. resistance matrix.
Further, extremely simple computations are involved in the
application of the new procedure as is illustrated by means
of an example toward the end of the chapter.

We shall begin by establishing certain useful properties

of third-order paramount matrices in the following.

88

89

3. 2. Some Properties of Third -Order Paramount Matrices

 

Lemma: Let 0(1) :[ qgh3 be a real, symmetric matrix
(1)

q13 and qgll) are negative, the rest of the

such that the entries
entries being non-negative. Then by applying a cross—sign change
operation to, and/or by interchanging some rows and the corresponding
columns of matrix Q“) , it is always possible to obtain a matrix,

Q = [ qij] 3 , such that the relations stated in (3. 2.1) and (3. 2. 2)

below hold simultane ously:

l
q11 q23 + q12 [9131 2. (3.2.1)

>912 q231’922 iql3|
(3.2.2)

 

q33 q12+q23 lq13l J

Proof: Consider the following four possible cases separately:

 

Case 1:
(1) (1) (1) 2
q11 q23 +"112 'q13 I -3 (1) (1) (3'2'3)
quzq q23 q22 [913'
q(13) q(112) q(213) |q(113)| i , (3.2.4)
Case 2:
(1) (1)+ q(1) (1)
q l I > (3.2.5)
“1.423.412 q” "‘ (1)(1) q(1), (1),
q12 q23+ q22 q13
qgfqig+qgnqflm < (1&6)
Case 3:
9§W1)qg3 q(lfl)|1?l < 1 (3.2.7)

q(1) q+(1)
>q 12 q23 q22 'q13l
($342+qnwl )l

 

I V

(3.2.8)

 

 

Case 4:

Case 1:
trivially

or of int.

Case 2:

i.e.,

Noting

We Set

90

Case 4:
q(111)q(213) + q(112) ' qil3)l < 1 1 1 1 (3' 2° 9)
> $12423) + quZ’ l q‘13’l
q(313)q(112) + q(.213) I q(113)| < J (3'2'10)

 

Case 1: Letting Q = (2(1) , the relations in (3. 2.1) - (3. 2. 2) follow
trivially without the necessity of any cross-sign change operation

)

or of interchanging any rows and columns of matrix Q(1 .

Case 2: With the notations defined in section 1. 2, let

Q _ U21) E<1,3.2)Q(1)E<1,3.2>U(1)

 

l 1 s
1.e.,
r— -1 — 1
(1) I (”I _ (1)
q11 qu q13 q11 q13 q12
_ (1) (1)
qzz q23 ‘ q33 q23 (3°2°11)
(1)
. q33 qzz
L _. _ ._

 

 

 

Noting that I (113' = qglz) and applying (3. 2.11) to (3. 2. 5) - (3. 2.6),

we get:

>
q11‘123+|q13l q12 — (3°2'12)

> lq13l q23 + C133‘112

 

qzz lq13I +(4.23 q12. (3'2°13)

The relation in (3. 2.13) is the same as that in (3. 2. 2), while
the one in (3. 2.1) follows from comparison of the left-hand sides of

(3. 2.12) and (3. 2.13). Clearly, both the inequalities in (3. 2.1) and

 

 

 

T
while the
hand 5 Rh
i"equallit

ߣ pres.

Cage 4:

91
(3. 2. 2) hold as strict inequalities in the present case.

Case 3: Let Q : US) 131(2912 3) 0(1) Ef2,1, 3) U(S3)

 

 

 

 

1.e.,
f‘ " "' fl
(1) (1) (1)
C111 q12 C113 C122 C112 ' (123
_ (1) (1)
q22 C123. - €111 |q13| (3.2.14)
(1)
q33 Q33
h — — —
Noting that |q13| = c193) and applying (3.2.14) to (3.2.7) - (3. 2.8),
we get:
I l + q 7 (3 2 15)
q2.2 q13 q12 23 - -
> q121‘313l +q11‘123
>
(133(1inr IC113' C123 ‘1 (3.2.16)

 

The relation in (3. 2.15) is the same as that in (3. 2.1),
while the one in (3. 2. 2) follows from the comparison of the left-
hand sides of (3. 2.15) and (3. 2.16). As inCase 2, both the
inequalities in (3. 2.1) and (3. 2. 2) hold as strict inequalities in

the present case.

Case 4: Let

(1(2): U<1>E(1,3,2>Q(1) E(1.3..2)U(1)
s 1 1 s

92

1.e.,
P (2) (Z) (2)-1 7(1) (1)1
C111 q12 q13 q11 lq13l "112
(2) (1(2) _ (1) (1)
q22 C123 " q33 q23 (3'2°17)
(2) (1)
C133 q22

 

 

 

 

Noting that lq(123)l = q“) and applying (3. 2.17) to (3. 2. 9) - (3. 2.10),

12
we get:
qgl) <12? + 1(1)? 1 q)? < (3.2.18)
2 2
"1123)l C1223) + q(33) C112)
C1222) | q(123)l+ c5223) (1:22) < (3.2.19)

Two possible cases will be considered separately:

Case 4(a):

1:13:12) :22 2 <2): q+§i31 ‘2; #2 (3.2.20)
Case 4(b):

q(1.21) (15223)” q(12))q (Z) < ($222ng 23H +413) (1‘12) (3.2.21)

Case 4(a): Letting 0(2) 2 Q, we observe that the relation (3. 2.20)
is the same as that in (3. 2.1), while the one in (3. 2.19) is the same

as that in (3. Z- 2) holding as a strict inequality.
2
Case 4(b): Let Q = US) 1331(2’1’3)Q(2)E1( ,1, 3) U(S3)

Therefore,

93

" ‘ ”' (2) (2) (2)“
q11 q12 q13 q22 q12 ' q23
_ (Z) (Z)
q22 q23 ‘ q11 lq13l (3°2'22)
(Z)
q33 q33
— n—J — _-

 

 

 

 

Noting that l (113’ = q(223) and applying (3. 2. 22) to (3. 2.19) and

(3. 2. 21), we get:

(3.2.23)

 

<
q11 q23 + q13I q12 q23 lq13l + q33 q12

q22 lq13I 1'q23 q12 << q11 q23+ Iq13l q12 (3'2°24)

The inequality in (3. 2. 24) is the same as the strict inequality
in (3. 2.1). Again, the strict inequality in (3. 2. 2) follows from
observing the identity of the left- and the right-hand sides of (3. 2. 23)
and (3. 2. 24) respectively and then comparing the right- and the
left-hand sides of the same two inequalities.

The lemma is proved thus in all the four possible cases.

Before proceeding to the main theorem, we consider one
Q(1)

special case where the original matrix, , is such that

= / =
q11 q12 and/0r q33 q23

in the matrix Q which is obtained as indicated in the lemma. In

this special case, an inspection of (3. 2.1) - (3. 2. 2) will reveal

that q11 = a12 implies:

qZZ : q12 (: Q11) (Since l (113' 9‘4 0) (3.2.25)

94

while q33 = qz3 implies:

C122 = C123 (= q33) (since I C1131 95 0) (3.2.26)

We shall have occasion to refer to this special case later on.

Theorem: Let 0(1) = [ q(l)] 3 be a paramount matrix such

11
that all its entries qS) are non-negative excepting, possibly, the

entries q(ll3) and qgll) . Then by applying a suitable cross-sign
change Operation to, and/ or by interchanging some rows and the

corresponding columns of matrix 0(1) , it is always possible to

obtain a matrix, Q, such that a matrix, Y, defined by:
Y = Q+(1/x)Q 2Q2 (3.2.27)

assumes a uniformly tapered form for some positive value of the

parameter x .

Proof: Let
r"- "1
1 O O
T = -l 1 0 (3.2.28)
0 -1 1
.. ..J

 

 

Then, from (3. 2. 27),

95

TYT = TQT + (1/x)TQ 2Q2 T

P

 

 

 

 

 

 

+q1z(912'922) +q1z(qzz“q32) + q12932
q11'q12 x q12'913 x q13 x
x + +(922'912H‘122‘q32) _ +(922‘912)q32
q22'932'912 q13 x q23 q13 _ x
x x + (q32'922)q32
q33'q32 x

(3.2.29)

Consider the following two possible cases separately:

Case 1: All the entries in matrix (2(1)

Case 2: The entries q(ll3) and qgll)

a re non -negative .

are negative, the rest

0.“).

of the entries being non-negative in matrix

Case 1: Let Q = E1Q(1)E

1 such that q13 = min (qij)

1,]
Then
> > < <
q11—‘112—913—‘3123—‘133
and
< >
C112 __ c122 _ C123. (3.2.30)
sothat:
- > - > — >
q11 qu—O’ q12 913—0' q23 q13-0
(3.2.31)
- > - — >
q33 923~0' q22 q12__>_0,andq22 923n-0

Consider two further possible cases separately:

Case 1(a): qZZ - q32 - q12 + ql3 : O (3.2.32)

96

Case 1(b): - qu + q13 < O (3. 2.33)

q22 " q32

Case 1(a): In this case, we observe incidentally that matrix Q itself
is in the uniformly tapered form.

In view of (3. 2. 31)-(3. 2. 32), the entries in positions (1, 2),
(1, 3), (2, 2), and (2, 3) of the matrix TYT in (3. 2. 29) are non-
negative for every positive value of x. Thus, matrix Y is uniformly

tapered if:

from the entry I q11 - q12 + (1/x) q12(q12 -q22) Z O
in position (1,1)
in (3.2.29)

q (q -q )
or x> 12 22 12 (3.2.34)

q11 "112

 

- - >
q33 q32 +(1/x)(q32 C122) C132 — 0
from the entry

in position (3, 3)

. (q -q )q
m(3.2.29) or X: 22 32 32 (3.2.35)

q33 ' q32

 

It is evident that a positive parameter x can always be chosen

such that the conditions in both (3. 2. 34) and (3. 2. 35) are satisfied.

(x *00 as q11 -q12-'O and/or q33 -q32-’0.)

Case 1(b): In view of (3. 2. 31), the entires in positions (1, 2), (1, 3),
and (2, 3) of the matrix TYT in (3. 2. 29) are always non-negative
for every positive value of x. Thus, matrix Y is in the uniformly
tapered form if a positive value can be assigned to x such that

the entries in positions (1,1), (3, 3), and (2, 2) in (3. 2. 29) are
non-negative. Consideration of the entries in positions (1,1) and

(3, 3) implies the same constraints on x as stated in (3. 2. 34) and

97

and (3. 2. 35), while from the entry in position (2, 2), we have:

- .. - - >
q22 q32 q12 + q13 + (l/szz C112"“«122 932) — 0
or

((122 " (112“qu "' (132)

 

. . i x (3.2.36)
' q22 + q32 + q12 ‘ q13

In order to establish the compatibility of this upper bound on x
with the lower bounds in (3. 2. 34) - (3. 2. 35), we observe that the
paramountcy of the original matrix, 0(1) , implies the paramountcy

of matrix Q so that:

2
.. > ..
911922 q12 — q11923 q12 q13 (3°2°37)

and

2

.. > -

q22 q33 q23 — q33 q12 q32 q13 (3'2'38)
. . 2

Adding the quant1ty(- q22 q12 + q32 q12 - q32 q11 + qlz) to both

sides of (3.2.37), we get:

2
q11‘3122 ' q22912 + q3qu2 ‘ Cl32911?- ' q22912 + C132‘112 + q12 " q12‘113

(922 ‘ q32)(q11 ‘ 912) 3 (' q22 + q32 + q12 ' C113)q12 ”'2' 39)

Note that, in view of (3.2.37), the hypothesis q22 - qZ3 - qu - q13 < 0

implies q11 3'4 qlz; for if q11 = qu’ then from (3.2.37),

q11(‘122 “112) 3 q11(‘123 “913)“

Because of paramountcy, must be positive and hence

q11
_ - >
q22 Cl12 923+q13—0°

98
From (3. 2. 39), then,

q22"‘132 > q12
“922+q32+q12'ql3 '- q11’q12

 

 

Multiplying both sides of this inequality by the non~negative quantity

(q22 - qu)’ we have:

'qzz+q32+q12‘q13 " q11 -q12

 

. . 2
Adding the quantlty (- qlzq33 - q22q32 + q12q32 + q32) to both

sides of (3.2.38), we get:

2
.. .. > - .-
q22‘1133 q12933 C122932 + C1120132 — q22932 + q32 + q12932 q32‘113
i.e.,
(922 ‘ q12)(q33 ‘ q32) ->- (‘ q22 + q32 + q12 " C113) C132
(3.2.41)

. . . _ _ <
Note that, in v1ew of (3.2.38), the hypothe31s q22 q23 q12 + ql3 0
implies q33;£ q32; for if (133 = q32, then from (3.2.38),
q33(922 ' 923) 3 q33 (912 ' 913) '

B ecause of paramountcy, q33 must be positive and hence

C122 "123 ‘912 +q13 ->— 0'
From (3.2.41), then,

 

q22 "3112 > q32
‘922+q32+q12 ‘q13 "' q33 ”(132

Multiplying both sides of this inequality by the non-negative quantity

99
(qzz - 9132).

(922 ‘ q1z)(qzz " 932) > (922 ‘ q32’0132

_ (3.2.42)
‘922+q32+q12'ql3 q33 ‘q32

 

 

The relations in (3. 2. 40) and (3. 2. 42) establish the compatibility
of the upper bound on x in (3. 2. 36) with the lower bounds on x
in (3. 2. 34) - (3. 2. 35). This proves the theorem in the present

case, 1(b).

Case 2: Let Q denote the matrix obtained from 0(1) such
that the inequalities in (3. 2.1) and (3. 2. 2) hold simultaneously
(1)

(cf. Lemma). We observe that, as in matrix Q , the entries in
positions (1, 3) and (3,1) of matrix Q are negative, the rest of the
entries being non-negative. This can be readily ascertained by

inspecting the relations in (3. 2.11), (3. 2.14), (3. 2.17), and (3. 2. 22).

Therefore, we have:

q11->- C1123 q13E q23-E 0133
and (3.2.43)
q125- C1223 933'
so that:
q11 “(112->- 0' q12 “9133 0’ q23 q13— 0

(3.2.44)
q33 "1233 0' q22 ”112—>- 0' (122 “123?. 0

In view of (3. 2. 44), the entries in positions (1, 2) and (2, 3)
Of the matrix TYT in (3. 2. 29) are always non-negative for every

Positive value of x. Thus, matrix Y is in the uniformly tapered

 

100

form if a positive value can be assigned to x such that the entries
in positions (1,1), (2, 2), (3, 3), and (1, 3) of TYT are non-negative.
In fact, the entry in position (2, 2) is non-negative for every positive
value of x whenever C122, - q32 - q12 + q13 _>_ 0. Only if
q22 - q32 - q12 + ql3 < O, the entry in position (2, 2) yields the
bound stated in (3. 2. 36). The entries in positions (1,1) and (3, 3)
respectively imply the bounds stated in (3. 2. 34) and (3. 2. 35)
provided (111% q12 and c133 aé qZ3 . If qll = q12 , then through
(3. 2.25), q22 = q12(= qll); for, by hypothesis, q13 is negative;
1. e. , )ql3l # O . Now, an inspection of (3. 2. 34) will reveal
that the same relation does hold as an equality for all positive
values of x. Again, if q33 = q23 , then through (3.2.26),
qZZ = qZ3 (= q33) . An inspection of (3. 2. 35) will reveal that the
same relation does hold as an equality for all positive values of
x.

Consideration of the entry in position (1, 3) of TYT in (3. 2. 29)

yields the following constraint:

q12 q32
- M1131 + ‘T" Z 0
or
q q
-—12———3’—2— _ (3.2.45)
1q13|

Matrix Y can be made uniformly tapered if, and only if,
a. positive value can be assigned to x such that the upper bounds
On x as given by (3. 2. 36) and (3. 2. 45) are compatible with the

lower bounds given by (3. 2. 34) and (3. 2. 35). Inasmuch as the

 

101

proof for case 1(b) is independent of the sign of q13 , it applies
directly to the present case, proving thus the compatibility of the
bound in (3. 2. 36) with those in (3. 2. 34) and (3. 2. 35).

Consider, now, the bound in (3. 2. 45) in relation with the
bounds in (3. 2.34) and (3. 2.35). If either q11 = q12 or q33 = q23 ,
then, in view of the earlier discussion, the compatibility of the
upper bound in (3. 2. 45) with the lower bounds in (3. 2. 34) and
(3. 2. 35), respectively, is established trivially. In the following
discussmn, we shall assume, therefore, (111% q12 and q33=74 q23 .

Rearranging (3. 2.1) - (3. 2. 2), we have:

q11‘132'0112 9323 q22,913I '912 lq13l (3°2'46)
.. > ..

q33 q12 q12 q32— 922' q13I q32 lq13I (3'2°47)
From (3. 2. 46),

- > ..
(911 C112) €132 __ ((122 C112) |q13l

 

01'
C132 > q22'912
- q -q
lq13l 11 12

Multiplying both sides of this inequality by the non-negative factor

(112’ we get

 

q12 q32 C112 (922 "312) (3 2 48)
)q13l _ q11 "112

From (3. 2. 47),

>

(933 ‘ 932) 912 — (922 ' 932) lq13l

102

 

01‘
q12 > q22 ' q32
lql3| " C133 ‘ q32

Multiplying both sides of this inequality by the non-negative factor

q32, we get:

 

q q (q - q )q
12 32 _ 22 .321 32 (31.49)
1:113) C133 32

The relations in (3. 2. 48) and (3. 2. 49) establish the compatibility
of the upper bound on x in (3. 2. 45) with the lower bounds on x
in (3. 2. 34) - (3. 2. 35). This proves the theorem in the present

case, 2.

3. 3. Realization of Third-Order Paramount Matrices

 

We shall establish first a new proof for the fact that the
property of paramountcy is, indeed, sufficient for realizing a
third-order matrix as either a s. c. conductance matrix or an
o. c. resistance matrix.

Without loss of generality, let the zero entries in a matrix,
if any, be regarded as positive. Then, we observe that any third-
order symmetric matrix must have one of the eight sign-patterns

s hown below:

 

 

Gem
folj
(C1
Crz
Cr.

p0

 

 

 

 

 

 

 

 

 

)— fl r— -- r— T r— '—
+ + + + + - + + + + - +
+ + + + + - + +
L + +_J + +
... L.- ._ .... m ..
(l) (2) (3) (4)
(3. 3.1)
r- f' m )— '- "‘
+ — j — — + + + -1 + - -
+ + + - + - + -
+ + + +

 

 

 

 

 

 

 

 

(5) (6) (7) (8)

Let the sign-patterns numbered (1), (5), (6), and (7) be considered
to belong to a group, I, and the rest of the sign-patterns in (3. 3.1),
to another group, 11. Now, if a matrix, 0(2) , has a sign-pattern
which belongs to Group I (Group II), then it is always possible to
reduce the problem of realization of that matrix 0(2) to the
problem of realization of a matrix, 0(1), whose sign-pattern is
of type (1) (type (2)) and whose every entry has the absolute value
equal to that of the corresponding entry in matrix Q(2) . This
follows from the fact that any sign-pattern belonging to Group I
(Group II) can be converted to that of type (1) (type (2)) by a proper
cross-sign change Operation; and we have already seen that a
cross-sign change operation has the electrical equivalence of
port-reorientations.
It is sufficient, therefore, to consider third-order matrices
which have the sign-patterns of types (1) and (2) only. Let (2(1)
cienotea third-order paramount matrix of either type (1) or (2). In

the previous section it was seen that by applying a suitable cross-

 

104

sign change operation to, and/or by interchanging some rows and

(1)

the corresponding columns of matrix Q , it is always possible

to obtain a matrix, Q , such that a matrix, Y, defined by

Y = Q+(l/x)Q.ZQ2. (3.3.2)

assumes a uniformly tapered form for some positive value of the
parameter x.

Let, now, the uniformly tapered matrix Y be realized
as a s. c. conductance matrix with four nodes (Figure 3. 3.1). Let
a new network be derived from this realization of matrix Y by
adding, in the manner shown in Figure 3. 3. 2, a resistor which

has its conductance-value = gO .

A E

 

 

 

 

 

 

 

 

 

Figure 3.3.1 Figure 3.3.2

If 6 denotes the s. c. conductance matrix for the derived

network, matrices Y and 6 must be related by:

Y = 6+ (1/x)6.262. , (32ng -2322 > 0) (3.3.3)

( Cf. the discussion on pp.2"7 —'.’.“-). Further, through the theorem

Proved in section 2. 3, the relations in (3. 3. 2) and (3. 3. 3) together

 

 

 

must
in F1

C0115

the f

for r

cond

as 51
shou'

abov

105

must imply the identity of matrices Q and 6. Hence the network
in Figure 3. 3. 2 must have been a realization of matrix Q itself
considered as the s. c. conductance matrix, realization of the

(2)

original matrix Q following immediately. This establishes
the fact that the property of paramountcy is, indeed, sufficient
for realizing a third-order matrix considered as the s. c.
conductance matrix.

Note that the networks in Figures 3. 3.1 and 3. 3. 2 are planar;
as such, their duals are existent [WHl] . These dual networks are
shown in Figures 3. 3. 3 and 3. 3. 4 respectively. Now, it is established
above that every third-order paramount matrix can be realized as
a s. c. conductance matrix in the manner shown in Figure 3. 3. 2,
with appropriate re-numbering and/ or re-orienting of some ports.

It is obvious, then, that every third-order paramount matrix can
also be realized as an o. c. resistance matrix in the manner shown
in Figure 3. 3. 4, with appropriate re-numbering and/ or re-orien‘ting

of some ports.

 

 

 

Figure 3.3.3 Figure 3.3.4

 

be re

resis

CC

106

The earlier discussions, including those in the previous
section, indicate the new procedure for realizing any third-order
paramount matrix. The procedure is stated below explicitly in
four steps:

1. Let Q(Z) denote a third-order paramount matrix to
be realized as either a s.c. conductance matrix or an o. c.
resistance matrix.

Apply, if necessary, a suitable cross-sign change operation
to matrix Q(Z) so as to obtain a matrix, Q“) , which has the

sign-pattern either of type (1) or type (2) indicated below:

 

 

 

+ + +3 + + -
+ + + + (3.3.4)
+ +
_. .J _ ...

 

(1) (2)

(cf. 3. 3.1). This can always be done easily. If the sign-pattern

(1) belongs to type (2), proceed to step 2(b) below.

of matrix Q
2(a). Interchange suitable rows and the corresponding

columns of matrix 0(1) so as to obtain a matrix, Q, which has

the smallest entry in its position (1, 3). Proceed to step 3 below.

2(b). Compute the quantities A : (1(111)q(213q(12) I (1:13))

1 1 1 l 1
B= q(33)q(12)+q 213)Iq(131)l.andc=q(12)qg3)+q(2 Z)qI(1:)1I-
(i) If A_>_ (1)
C , let Q = Q and proceed to step 3.

E)

(ii) If A>‘I
‘ ' (3, let Q — U(1)E(1’3’Z)Q(1)
B<j s 1

1 s

and proceed to step 3.

 

 

(iii)

Iiva)

(ivb)

107

(iii) If A< B]
> s l l s
— and proceed to step 3.
(iv) If A < A 2
let Q : U(1)E(1:3:Z)Q(1)E(1,3, )U(1)
s 1 1 s
J and compute:
__ A A A A
D ‘ q11923 + lCl13| C112
A A A A
and E“122I‘113I +q23 q12
(iva) If D > E , let Q = Q and roceed to step 3.
— P

and proceed to step 3.

3. Let Y = Q + (gO - (122)-1 Q. 202. . Select any one value
of gO > q22 such that matrix Y assumes a uniformly tapered form.
(This is always possible.)

4. Realize the uniformly tapered matrix Y as shown in
Figure 3. 3.1 (Figure 3. 3. 3) . From this realization of Y, obtain
the realization of matrix Q as shown in Figure 3. 3. 2 (Figure 3. 3. 4).
Realization of the original matrix, Q(Z), as the s. c. conductance
matrix (0. c. resistance matrix) will follow when cross-sign change
operations and/ or interchanges of the rows and the corresponding
columns carried out in the earlier steps are taken into consideration
for the purpose of assigning proper numberings and orientations

to the ports.

108

3 . 4. Example

Consider the realization of the paramount matrix

 

I-' -!
7 -2 1

0(2) = -2 12 3 (3.4.1)
1 3 5

 

both as a s. c. conductance matrix and an o. c. resistance matrix
by the procedure established in the previous section.

1. We observe that reversing the signs of the first row
and the first column results in a sign-pattern of type (2) defined

in (3.3.4). Let, therefore, Q“) = U(S1)Q(Z)U(Sl) ; i.e.,

 

 

P 7 2 -17
0(1) = 2 12 3 (3.4.2)
-1 3 5
c _.
203)- A = (7)(3)+(2)(1) = 23
= (5)(2) +(3)(1) = 13

c z (2)(3)+(12)(1)= 18

Since A > C > B, let

Q _ U(1) E(1.3.Z)Q(1) E(1.3.Z)U(Sl)

s 1 1
r —
7 1 -2
= 1 5 3 (3.4.3)
-2 3 12
... _1

 

 

109

“1(1 Q (3.4.4)

3. Y=Q+(g 22

0"922)
In order to obtain a parameter gO > q22 such that this
matrix Y assumes a uniformly tapered form, pre- and post-

multiply the above relation by:

 

 

 

 

 

 

 

 

 

 

1 o 0
'r z -1 1 0 (3.4.5)
0 -1 1
_ .4
so that
TYTzzTQT +( -q )JTQ Ci'r
go 22 .2 2.
I 7 7 7
q11‘q12 q12"q13 q13 q12
- _ 1 q q q
‘ X q22 q23'912+ql3 q23'q13 +(gO-q22) q22'q12 12 22 32
'922 "932
X X q33”q23 q32‘q22
7 “ ‘ (3.4.6)
F H) r— -1
6 3 -2 1
= x -1 5 + ( 1_ 5) 4 [}4 2 :{J
g0
x x 9 -2
r- -1 )— m
6 3 -2 -4 2 3
1
= x _1 5 + x 8 12 (3.4.7)
(sci-5)
x x 9 x x --6
_ ..I ._ ..1

 

 

 

 

110

F c—

From the entry : (g _ 5) i

0443.
IV
U1

WIN

in position (1,1)
.. _1

—

I-From the entry

0:)
I
m
V
\olcr
H.
m
GO
0
Iv
U1
w|N

in position (3, 3)
... .4

 

 

r F—
From the entry

3::
I
U1
IA
le
H
(D
00
|/\
0‘

in position (1, 3)

r
Fromth ntr
88 y = (g -5):

balm

in position (2, 2)
_ —I

 

 

It follows, therefore, that matrix Y in (3. 4. 4) will assume a uniformly
tapered form if gO is assigned any value within the lower and upper
bounds of 5%- and 617 respectively. We shall assign the limiting
values to g0 and obtain two distinct minimal realizations of matrix
Q(2) considered both as a s. c. conductance matrix and an o. c.
resistance matrix. Figures 3. 4.1 and 3. 4. 2 depict the realizations
when Q(2) is considered as a s. c. conductance matrix. Figures

3. 4. 3 and 3. 4. 4 depict the realizations when Q(Z) is considered

as an o. c. resistance matrix.

For gO = 5%- , (3.4.7) yields:

7— 1'1
0 6 2-2—
TYT: x 11 23 (3.4.8)
x x 0
L... _I

 

 

 

For

For a

we ha

111

For go=6-:—, (3.4.7) yields:

 

" 1 1 '7
33 4-3— 0
TYT = x 4%- 13 (3.4.9)
x x 5_I

 

For assigning the prOper numberings and orientations to the ports,

we have only to observe that:

(1, 3, 2)Q(1)E(11.3, 2’11

_ (1) (1)
Q Us E1

S

s 1 s

U11) E11, 3, 2) Um
s 1

S

 

 

 

 

 

 

 

 

U V

Figure 3. 4. 4

Figure 3. 4. 3

Chapter 4
CONCLUSION

A complete solution is presented in this thesis to the problem
of synthesis of n-port resistive networks from short circuit conduc-
tance matrices which are realizable with two-tree port-structures.
The fact that the conductance values of (n+1) constituent resistors
themselves are considered as parameters provides excellent
control over the maximum number of resistors which constitute
the network. This is directly illustrated by the attainment of
”minimal" realizations in the case of a certain class of matrices
described in section 2. 7 of the thesis.

It is possible that certain nth-order short circuit conductance
matrices may be realizable only with k-tree port-structures
(3 E k f n) . It would be an interesting problem for further
research to investigate whether the "conductance-parameter"
approach presented in the thesis can be extended to the realization

of these matrices.

112

APPENDIX 1

An interesting "special case" was discussed on pp. 27-28.
If matrices Y = [ y..] and Q = [3. .] as considered there
11 n 1] n
are non-singular, then we can observe that the above special
case can also be regarded as a special case of the considerations
on p. 82. As a consequence, matrices Y and Q must be related
by:

l -1

Y = (6' -R) (A.1.1)

where matrix R = [ rij] n is defined by:

>0fori=j=h hconstant<n.

ii
= 0 otherwise
\

That the relation in (2. 2. 37) is, in fact, an alternate form
of the above relation, (A. l . 1), can be easily proved by applying
Theorem 1 on p. 79 . However, we present below an interesting
direct proof for the indicated fact by exploiting the theory of

functions of matric es:

Proof: The relation in (A. 1.1) can be written as:

Y = (Q-1[U-QRI)-1
: (U-ER)‘16
-_~ (U-B)'16 (A.1.2)

113

 

 

where
,_ _ ._
0 . . . 0 q1h 0 . 0
— 0 . O q‘2h 0 . . . 0
B ._. QR : rhh (A.1.3)
0 o E o . o
nh
_ J
Let @(B) = (U .13)"1 (A.l.4)
so that the relation in (A.l. 2) assumes the form:
Y' = §(13) ES (2x.1. 5)

Note that (n-l) eigenvalues of matrix B are zero, the n-th eigen-

value being = rhh qhh'
Let t rhh 'q'hh .

By definition, the minimal polynomial of matrix B = m()\) = (RU -BI/p()\)

where p(>\) denotes the highest common factor amongst all the

entries of the adjoint of matrix (X U - B) [KO 1] . In the present

case, it can be easily seen that p01) : kn-Z . Therefore,

mo.) = in‘1 (x - t)/>.n'2 : i (i - t)

Hence the Lagrange-Sylvester interpolation polynomial [ KO 1] for

h-t X
t ' §(O)+'t—' Wt)

 

150) on the spectrum of matrix B = h()\) =

_x-t i. -1
_ _t 1+t (1-1)

X x
~tt+1+771277

 

1+(1 -1)‘1).

ll

Therefore,

11(3) = 3(3) = U+(1-t)-1B

so that, from (A. 1. 2) - (A. 1. 5),

Y

6+(1-r

[U+(1-t)'1

5+(1 —1:)'1 B

hh qhh)

BIQ

6

—1
I‘

hh

 

L

Io...

0...

_ _ -1... ...
Q ”LU/X1111 ' qhh) Q.th.

(A.1.6)

Since gO = l/rhh and go - Shh = E, the relation in (A. 1.6) is, in

fact, identical to the one in (2. 2. 37).

Q. E. D.

 

DI

APPENDIX 2

It is seen that the conductance-parameter procedure
established in section 2. 4 has, at its basis, the relation (2. 2. 35').
Several similar relations can be discovered so as to build the
procedure upon them. The derivations of these relations are
fundamentally of the same nature as in section 2. 2. We shall
state these relations directly in the following with the purpose of

making them available for ready reference.

 

 

-1 _.
Ref.: Figures A.2.1 and A.2.2.
n+1 _ _1__' _I
Y z Q‘AHZI gk qh-l h-l) Q h-l h—l
where
— T _ Q1 (h-l)
1 __ t _ __
[Oh-1] ’ Q.h-l ‘ "'
QZ (n-h+l)
c. _J
_ (1) _ (2) .
Q1 — [qil] and Q2 — [qil] W1th
(1) =" .. >1: (1:12 11-1)
q11 91,114 k:1gh-k ’
(2) _ n-i+l
qil = qi,h-1 +k§1 gh+k-1 (1=h,h+l,...,n)

Further, A = [ a. ] with
1] n

(1: 1,2,...,h-1;j =i,i+1,...,h-1)

116

118

h,h+1,...,n)

a” : O (1:1,Z,...,h'1;j

11
_ nij+1

aij k=1 gh+k.1 (J

h,h+1,...,n;i h,h+1,...,j)

Ref.: Figures A.2.3 -A.2.5.

Let matrix Y = [ yij] n be obtained from Y such that the
former is the s. c. conductance matrix for the same resistive
network which is shown in Figure A. 2. 3, the port-numberings
being altered as indicated in Figure A. 2. 4. Matrix Y is uniformly
tapered and 2 El YEI , where the exact nature of matrix E1 can
be readily decided by comparing the port-structures associated

with matrices Y and Y. Then

_ n+1 __ _1_ __
_ _ _ l I
Y ‘ E1QE1 A+(k§1 gk qll) 0.101.

 

 

where
U21 ] = (2.1 = _,_ _
QZ (n-h+1)
L _
(1)
1 2 .
Q1 = [q:1)] and Q2 = [(1:1)] With
(1) - " - é (1:12 h-l)
C111 ‘ q11.1,1 k:1 gh-k '
(Z) _ n—i+1

+ 2 g (1=h,h+l,...,n)

q : q -
11 n+h-1,1 k3]. h+k-1

120

Further, A = [a..] with
13 n

i
aij = 1:1 gh-k (1=1,2,. ,h-l,J=1,1+l,...,h-1)
aij = 0 (121,2, ,h-l,J=h,h+l,...,n)
n-'+l
aij = kgl gh+k-l (J=h,h+1,...,n;1=h,h+1,...,_])

Ref.: Figures A.2.6 - A.2.8.

Let matrix Y = [ yij] n be obtained from -Y_ such that the
former is the s. c. conductance matrix for the same resistive
network which is shown in Figure A. 2. 6, the port-numberings
being altered as indicated in Figure A. 2. 7. Matrix Y is uniformly
tapered and = E1 Y El , where the exact nature of matrix F.l can

be readily decided by comparing the port-structures associated

with matrices Y and "1?". Then

_ n+1 __ _1_' —I
Y = ElQEl -A+(Z: gk-qnn) Q nQn
where
r- '1
__ T _ Q1 (h-l)
[Q' l = Q' = -..-
n. n
02 (n-h+1)
L— _-

 

 

 

 

 

 

 

 

 

 

 

 

 

122

(2) _ n-i+1

qil : qn+h-i,n 1:1 gn-h—k+2 (1 : 11’ h+1””’n)

Further, A = [ a..] with
13 n

i
: Z) I: - “'33. I ... '
aij kzl gn-h+1+k (1 1,2,...,h1,J 1,1+1, ,h 1)
aij = O (i:1,2,...,h-l;j=h,h+1,...,n)
n-j+l .
aij = kzzl gn-h-k+2 (J=h,h+1,...,n;1=h,h+1,...,J)

Ref.: Figures A. 2. 9 - A. 2.13.

The derivation of the relation stated in this section incorporates
some ides that are somewhat different from those used in section 2. 2.
These ideas will be applicable even in deriving the relations stated
in the following three sections, and have been described in sufficient
details below.

Let Y = [2].] n denote the s. c. conductance matrix for the
network shown in Figure A. 2. 9. Let matrix Y = Us EIYE1 US be
obtained from Y such that the former is the s. c. conductance
matrix for the same network after the port-numberings and port-
orientations are altered as indicated in Figure A.l. 10. Matrix Y
is uniformly tapered and the exact forms of matrices US and E1
can be readily determined by comparing the port-structures
associated with matrices Y and Y.

Let Q = [21.1.1] n denote the s. c. conductance matrix for the
network derived from the one in Figure A. 2. 9 as shown in Figure A. 2.11.

Then in establishing the relation between matrices Y and Q, it is

A
necessary to consider the s. c. conductance matrix Q”!< correspondlng to the

125
derived network whose port-structure is modified as shown in Fig. A. 2.12.
A uniformly tapered matrix, Q* , can be easily obtained from
6* by suitably interchanging some of its rows and the corresponding
columns and by applying cross-sign change operations. Let the
network shown in Figure A. 2.13 be obtained from the one in
Figure A. 1.12 by shorting port h’i‘ . Then, as in section 3. 2,
the s. c. conductance matrix, 5, corresponding to this last
network is required to be considered in the derivation of the
relation, which is stated below:

n+1

_ — _ "1—1 —1
Y ‘ UsElQ I‘31Us 'A + (131 gk " qhh) 0.11 h.
where
P _-
Q1 (h-l)
.... T _
[Q' ] 2 Q, 2 — c.- c—
. .h
Q2 (n—h+1)
L- .—

 

 

1 11 Z
q(l) : -a + PE g (i=12 h-1)
11 h—i,h k:1 n-k+2 '
(2) _ i-ZI:1+1 .
q11 ‘ ‘ q1,11 ’ k:1 gn..h+k (1 ‘ h'h+1"'°’n)

Further, A : [ a. .] with
1] n

i
ij 13:1 gn-k+2 (1:1,2,...,h-1;]=1,1+1,...,h-l)

a

aij = 0 (i:1,2,...,h-1;j:h,h+1,...,n)

j-n+1

aij = kél gn-h+k (J h,h+1,...,n;1:h,h+1,..-,J)

126
-5-

Ref.: Figures A. 2.14 - A. 2.16.

Let matrix Y = [ yij] n be obtained from —Y— such that the
former is the s. c. conductance matrix for the same resistive
network which is shown in Figure A. 2.14, the port-numberings
and port-orientations being altered as indicated in Figure A. 2.15.
Matrix Y is uniformly tapered and = US E1 Y E1 US , where the

exact nature of matrices US and El can be readily determined

by comparing the port-structures associated with matrices Y

and -Y_ . Then

 

 

n+1
_ "' " “1—1 _I
Y ‘ Us I‘31 Q E1 Us ‘A + (If gk ' qh-1,h-l) Q.h-lQh-1
where I— _.
__ T _ Q1 (h-l)
l ._ l ...
[Q -1.] ' 0.11-1 ‘ ——-
QZ (n-h+1)
_ (1) _ (2) ,1
Q1 — [ qil] and Q2 - [qi1 I whth
(1(1):; _ g g (,:12 11-1)
11 1,h-1 k:1“h-1( ’ " '
(Z) — " 1}?“ (° — h h+1 )
q11 “ qn+h-i,h—1 k:1 gn+k 1’ ’ “'“n
Further, A :[aij] with
.15 . . . .
aij =k=1gh_k (1:1,2,...,h-1,3—1,1+1,...,h-l)
= 0 (i:l,2,...,h-—l;j=h,h+1,...,n)

128

j-n+l

ij kzl gn+k (J=h,h+1,...,n;J=h,h+1,...,J)

-6—

Ref.: Figures A. 2.17 - A.2. 18.

Let matrix Y = [ yij] n be obtained from 7 such that the

former is the s. c. conductance matrix for the same resistive
network which is shown in Figure A. 2.17, the port-numberings
and port-orientations being altered as indicated in Figure A. 2.18.

Matrix Y is uniformly tapered and = US E1_Y_EIUS , where the

exact nature of matrices Us and E1 can be readily determined

by comparing the port-structures associated with matrices Y and

 

 

Y. Then
_ n+1 _1_ -_
_ _-- 1 I
Y ‘ UsEIQElus ‘A ”1331 gk “111) 0.101.
where
)— —1
_ T _ Q1 (h-l)
1 _. l ._
[01.] _ Q.l _ 7”
Q2 (n-hrl)
L. .....1
1 (2) -
Q1 = [(111)] and Q.2 = [qil I w1th
(1)___ +21“, (1:12 .. h-l)
qll qh-i,1 k 1 h-k ’ , ,
. :_ Z) (i=h,h+1,...,J)

 

I...

130
Further, A = [a..] with
13 n

ij k21gh'k (1:1,2,...,h—l;_]=1,1+1,...,h-1)

aij = 0 (i:1,2,...,h-l;j=h,h+1,...,n)
J-n+1

o = z .: .0. ..z 0.. .

a1] k:l gn+k (J 11: h+1’ ’n’ 1 h’ h+1’ ’3)

Ref.: Figures A. 2.20 - A. 2.22.

Let matrix Y = [ yij] n be obtained from —Y_ such that the
former is the s. c. conductance matrix for the same network which
is shown in Figure A. 2. 20, the port-numberings and port-orientations
being altered as indicated in Figure A. 2. 21. Matrix Y is uniformly
tapered and : U8 El YE US , where the exact nature of matrices

1

Us and E1 can be readily determined by comparing the port-

structures associated with matrices Y and Y. Then

 

 

n+1
_ — — '17—: 7")
Y—UsElQElUs-A+(k§l gk-qnn) Q.n n
where
Q1 (h-l)
' — — - _—
[Qn.] 7 Q n 7
02 (n-h+l)
L. L
(1) _ (3) -
Q1 — [q11 ] and Q2 — [qil ] w1th
(1) — + >1: —1 2 hl
911‘ "11,11 kZIgn-k+2 (1‘ .. ' ' "I

132

(2) — i-fin-f-l

qil : " qn+h_i,n '13; gn_h+k (1 = h, h+1,...,n)

Further, A : [a. .] with
ljn
% 2 hl ' "1 hi
aiJ k:1gn-k+2 (1:1' ' ’J"1’1+"'°’ ')
aij = O (1:1,2,...,h-l;j:h,h+1,...,n)
j?“ ' hhl '
a1] - kzl n-h+k (J—h,h+l,...,n,1— , + ,...,J)

APPENDD<5

 

 

irntialize
indices

l

 

 

 

function
data

l

1rntialize

 

 

 

 

\Neights

 

 

 

 

renornufliza
weights

 

 

 

 

end

 

 

punch
output

 

 

T

fetrli
and
arrange
extrenni

 

 
    

terrnination

   

is

 

 

required

 

incrernent
extrernurn ,

 

generate
initial
search
p01nt

J.

adjust

 

 

 

 

 

 

increment
secondary
search
index

 

 

 

A

 

 

find
secondary
search
point

 

 

T

 

utilize
2nd order
function

 

 

weights

 

 

 

 

 

generate
search
vector

 

 
   

search vector

yes

(5.— yes

 

 

store
extremuin

 

index

 

 

 

point

 

 

 

approxi-
mation

 

 

 

T

 

perforni
contracting
search

 

 

}nerforrn
expanding
search

 

 

 

      
     

is
terrnination
required

Logic Diagram of the Computer Program
"Generalized Random Extremum Analysis Technique"
(Courtesy of Instrument Division, Lear Siegler, Inc. ,
Grand Rapids, Michigan)

122

 

 

[B11]

[BI 2]

[BI 3]

[B14]

[B1 5]

[BI 6]

[B17]

[BI 8]

[BI 9]

[B01]

[Bo 2]

G.

F.

REFERENCES

. Biorci and P. P. Civalleri, ”Alcune Considerazioni Sulla

Sintesi Dei Multipoli Resistivi, " Atti Accad. Sci. Torino,
vol. 94; 1959-1960.

. Biorci and P. P. Civalleri, "Conditions for the Realizability

of a Conductance Matrix, " IRE Trans. on Circuit Theory,
vol. CT-8, pp. 312-317; September, 1961.

. BiorCi, P. P. Civalleri, and G. Fiorio, ”Degeneracy in

n-Port Networks, " Proc. First Annual Allerton Conference
on Circuit and System Theory, Monticello, Illinois;
November 15-17, 1963.

Biorci and P. P. Civalleri, ”About a Basic Theorem on
Resistive Networks, " 1963 IRE International Convention
Record, pt. 2, pp. 83-90.

Biorci and R. M. Foster, ”Some Considerations on the
n-Port Problem, " IEEE Trans. on Circuit Theory,
vol. CT-IO, pp. 305-307; June, 1963.

. Biorci, ”Existence of Irreducible Single-Element-Kind

n-Ports, ” IEEE Trans. on Circuit Theory, vol. CT-13,
pp. 106-108; March, 1966.

. BiorCi, ”Sign Matrices and Realizability of Conductance

Matrices," Proc. IEEE Mono. No. 424E, vol. 108,
pt. C, pp. 296-299;1961.

Biorci and P. P. Civalleri, "On the Synthesis of Resistive
n-Port Networks, " IRE Trans. on Circuit Theory,
vol. CT-8, pp. 22-28; March, 1961.

Biorci and P. P. Civalleri, "A Contribution to the
Synthesis of Resistive Three-Ports, " Alta Frequenza,
vol. 30, pp. 714-717; October, 1961.

S. Boxall, "Synthesis of Multiterminal Two-Element-
Kind Networks, ” Tech. Report No. 95, Electrical
Research Laboratory, Stanford University;
November 1, 1955.

T. Boesch, "On the Synthesis of Resistor n-Ports, "
Microwave Research Institute, Polytechnic Institute of
Brooklyn, Report No. PIBMRI-1068-62, Memo. No. 71,
August 29, 1962.

134

135

[BR 1] D. P. Brown and Y. Tokad, "On the Synthesis of R Networks, "
IRE Trans. on Circuit Theory, vol. CT-8, pp. 31-39;
March, 1961.

[BR 2] P. R. Bryant, "Discussion on Conditions for the Impedance
and Admittance Matrices of n-Ports Without Ideal
Transformers,” Proc. IEE (London), pt. C, p. 116;
March, 1959.

[CA 1] W. Cauer, "Synthesis of Linear Communication Networks, "
(Book), volumes I and II, 2nd edition, McGraw-Hill
Book Co. , New York; 1958.

[CE 1] I. Cederbaum, ”Conditions for the Impedance and Admittance
Matrices of n-Ports Without Ideal Transformers, "
Proc. IEE, Mono. No. 276R, vol. 105, pt. C,
pp. 245-251; January, 1958.

[CE 2] I. Cederbaum, ”Topological Considerations in the Realization
of Resistive n-Port Networks, " IRE Trans. on Circuit
Theory, vol. CT-8, pp. 324-329; September, 1961.

[CE 3] I. Cederbaum, ”Applications of Matrix Algebra to Network
Theory, " IRE Trans. on Circuit Theory, vol. CT-6,
special suppl. pp. 127-137; May, 1959.

[CE 4] I. Cederbaum, “Paramount Matrices and Realization of
Resistive 3-Port Networks, " Proc. IEE (London),
vol. 110, pp. 1960-1964; November, 1963.

[CE 5] I. Cederbaum, ”On Equivalence of n-Port Networks, "
IEEE Trans. on Circuit Theory, vol. CT—12, pp. 338-
344; September, 1965.

[CE 6] I. Cederbaum, ”On Networks Without Ideal Transformers, "
IRE Trans. on Circuit Theory, vol. CT-3, pp. 179-182;
September, 1956.

[CE 7] I. Cederbaum, "A Generalization of the 'No-Amplification'
Property of Resistive Networks, " IRE Trans. on Circuit
Theory, vol. CT-5, p. 224; September, 1958.

[CE 8] I. Cederbaum, ”Paramount Matrices and Synthesis of
Resistive n-Ports, " IRE Trans. on Circuit Theory,
vol. CT-8, pp. 28-31; March, 1961.

[CE 9] I. Cederbaum, “On Duality and Equivalence, " IRE Trans.
on Circuit Theory, vol. CT-8, pp. 487-488; December,
1961.

[CIH

[GIN

[DEl]

[F01]

[GUI]

[GU2]

[GU3]

[GU4]

[HA2]

[HA3]

[Kin

P.

P.

E.

E.

E.

C.

C.

C.

136

P. Civalleri, ”A Direct Procedure for the Synthesis of
Resistive (NH)-Poles," Proc. IEE, Mono. No. 464E,
vol. 109, pt. C, pp. 76~82; March, 1962.

P. Civalleri, “Linear Degeneracies in n-Port Networks, ”
Proc. Second Annual Allerton Conference on Circuit
and System Theory, Monticello, Illinois; September
28-30, 1964.

. Dervisoglu, "Realization of the A Matrix of Half-Degenerate

RLC Networks, " Ph. D. Thesis, University of Illinois,
1964.

. M. Foster, "An Open Question, " IRE Trans. on Circuit

Theory, vol. CT-8, p. 175; June, 1961.

. A. Guillemin, "On the Analysis and Synthesis of Single-

Element-Kind Networks, " IRE Trans. on Circuit Theory,
vol. CT-7, pp. 303-312; September, 1960.

A. Guillemin, "On the Realization of an nth-Order G
Matrix, " IRE Trans. on Circuit Theory, vol. CT-8,
pp. 318-323; September, 1961.

A. Guillemin, "On the Analysis and Synthesis of Single-
Element-Kind Network, ” Mass. Inst. Tech. , Cambridge,
Quart. Prog.Rept. No. 56, pp. 213-235; January, 1960.

A. Guillemin, ”An Approach to the Synthesis of Linear
Networks Through Use of Normal Coordinate Trans-
formations Leading to More General TOpological
Configurations, " IRE Trans. on Circuit Theory, vol.
CT-7, special suppl., pp. 40-48; August, 1960.

C. Halkias, I. Cederbaum, and W. H. Kim, "Synthesis
of Resistive n-Port Networks with n+1 Nodes, " IRE
Trans. on Circuit Theory, vol. CT-9, pp. 69-73;
March, 1962.

C. Halkias and F. J. Lupo, HA New Approach to the
Realization of Resistive n-Ports, " 1963 Proc. Sixth
Midwest Symposium on Circuit Theory, pp. D1 -D15.

C. Halkias, ”Synthesis of n-Port Networks, ” Ph. D.
Thesis, Columbia University, 1962.

W. H. Kim and R. T. Chien, "Topological Analysis and

Synthesis of Communication Networks, ” (Book),
Columbia University Press, New York, 1962.

[SW1]

[TEl]

[WE 1]

[WH 1']

137

H. E. Koenig, Y. Tokad, and H. K. Kesavan, “Analysis
of Discrete Physical Systems, "' (Book), McGrawuHill
Book Co. , New York, 1967.

G. Kron, "Tensor Analysis of Networks, " (Book), Chap. X,
p. 242, John Wiley and Sons, New York, 1939.

M. D. Levy and D. P. Brown, ”Time Domain Synthesis of
a Class of RLC Networks, " Proc. Third Annual Allerton
Conference on Circuit and System Theory, Monticello,
Illinois; 1965.

F. J. Lupo and C. C. Halkias, "Synthesis of n-Port Networks
on Two-Tree Pc.>rt-Stru(_,~tures, " IEEE Trans. on Circuit
Theory, vol. CT-o12, pp. 571-577; December, 1965.

K. K. Nambiar, "On n—Port Networks: A Matrix Theorem
and an Open Problem, " IEEE Trans. on Circuit Theory,
vol. CT-10, p. 454, September, 1963.

K. K. Nambiar, "On the Realization of Singular R Matrices, "
IEEE Trans. on Circuit Theory, vol. CT-ll, pp. 421-
423, September, 1964.

"The Realization of n-Port Networks Without Transformers -
A Panel Discussion, ” IRE Trans. on Circuit Theory,
vol. CT-9, pp. 202-214; September, 1962.

D. J. Rauch, "On the. Realization of Time Domain Models
of Real Linear Bielement Systems, " Ph. D. Thesis,
Michigan State University, 1963.

P. Slepian and L. Weinberg, "Synthesis Applications of
Paramount and Dominant Matrices, " Proc. National
Electronics C(‘inference, (:Zhicago, Illinois; vol. 14,
pp. 1-20; October 13-15, 1958.

K. R. Swaminathan and I. T. Fris<‘:h, "Net essary Conditions
for the Realizability of n-Port Resistive Networks with
More Than (n+1) Nodes, ” IEEE Trans. on Circuit Theory,
vol. CT-IZ, pp. 520-627; December, 1965.

B. D. H. Tellegen, "Theorie Der Electrische Netwerken, "
(Book), Chap. IV, p. 166, P. NoordhoffN.V.,
Gronigen, Djakarta; 1952.

L. Weinberg, "Network Analysis and Synthesis, " (Book),
Chap. 8, p. 348, Mt-Graw-Hill Book Co., New York;
1962.

H. Whitney, "Planar Graphs, " Fundamental Math., vol. 21,
pp. 73-84;193’>.