ON THE SYNTHESIS OF N-TERMINAL
R-NETWQRK AND POLYGON TO
STAR TRANSFOEMAf‘EQN

“105:5 For Hm Degree 0‘ M. S.
MICHIGAN STéTE UNE‘JERSETY

Chang - Loi Wang
19 61

 

LIBRARY

Michigan State
University

 

 

 

Qifiliwn‘ 7b . ,

ON THE SYNTHESIS OF N-TERMINAL R-NEIWORK AND
POLYGON ' TO STAR MNSFOEEATION

BY

Chung-Lei Wang

ANABS‘EAC’I‘

Suhnitted to the College or Engineering
Michigan. State University of Agriculture and
Applied Science in partial fulfillment of

the requirements for the degree of

MSW OF SCIENCE

Department of Electrich Engineering
1961

. ‘/,.. .4”? . '
(-3.4 a“ fly: - "J 1/ .
Approveq: -- i ”v“- ‘ '. ....-

ABSTRACT

In this thesis synthesis of n-terminal R-networks is considered.
It is assumed that the terminal representation.of the n-terminal
R-network is given by both the terminal graph and the terminal
equations. The object of this thesis is to calculate the element
values of the R-network directly from.the given specifications.
The method represented in this thesis does not require the tree
'transformation which, generally, is necessary to use in other existing
methods.

As a second topic, in this thesis transformation of a given
polygon R-network into a star R-network is discussed and a set of
necessary and sufficient conditions for the existence of such

transformation is established.

ON THE SYNTHESIS OF N-TERMINALiR-NETWORK.AND

POLYGON T0 STAR TRANSFORMATION

BY

Chung-Lei Hang

.A THESIS

Submitted to the College of Engineering
Michigan State University of Agriculture and
Applied Science in partial fulfillment of

the requirements for the degree of

MASTER OF SCIENCE

Department of Electrical Engineering
1961

ACKNOWLEDGMENT

The author is indebted to his mador professor, Dr. H..E. Koenig,
and to m. Yilmaz Tokad, his thesis -advisor, for their helpful advice
and guidance throughout the preparation of this thesis. The research
leading to this thesis.was supported by the National Science Foundation

under project 6-9735.

CONTENTS

Introduction
.A New Direct.Method for Synthesis of N-Terminal R-Netvork
2.1 Formulation of Thrmdnal Equations for a Direct Solution

2.2 The Existence of K'1

and Related PrOperties
Pblygon to Star Transformation
3.1 Necessary and Sufficient Conditions

3.2 Application

Page

23
23
31

1. INTRODUCTION

Synthesis of n-terminal R-networks are discussed in the literature
by various authorsl-L. If a complete representation of n-terminal
R-network is given in terms of a "terminal graph " and the corresponding
set of "terminal equationss ", the synthesis problem can easily
be solve by transforming the given terminal graph into a Lagrangian
tree. The element values are then detected directly from the
corresponding terminal equations. (See e.g., reference 3).

If the terminal graph is not given it must be determined as
a first step in the synthesis procedure. Cederbauml presents a purely
algebraic method for this determination while Guillemina and
Boiorce and Civallerih have developed "searching" procedures. Brown
and Tokad3 also consider this problem in certain special cases.

If the Coefficient Matrix in the terminal equations contain some
off diagonal entries which are equal to zero, the above methods
do not give a unique solution. To apply these procedures it is
necessary to replace the zero entries by +1 or -1.

In the first part of this thesis the synthesis of n-terminal
R-networks is considered, assuming the terminal graph is known. A
direct method for calculating the element values from the given

terminal equations is presented. The given terminal equations are

lag-y

It is shown thatacan be written easily from the given terminal graph.

put in a new form

Since a—l always exists, the column matrix, H , which contains the
element values of the R-network, can be found by direct solution of a
set of simultaneous linear algebraic equations.

1

This new rearrangement of terminal equations offers the following
advantages over other procedures:

(a) Eliminates the tree transformation process;

(b) In a large R-networks where it is desire to use computering

facilities the above form is ideally suited.

In the second part of the thesis the existance of the polygon to
star transformation is considered along with application to synthesis.
Necessary and sufficient conditions for the existance of such a

transformation are stated in terms of two typaaof terminal representation.

2. A NEW METHOD FOR THE SYNTHESIS OF N-TERMINAL R-NEI'WORKS

2 . l Formulation

Consider the terminal representations of an R-network given in

Figure l .

b 2C4. d

 

 

 

 

 

 

F110;) [GllGlZGBGlh' ‘vl(t)l
| 3 12(t) _ . 622623021+ v2(t)
~11+(t)J ,, o e e Gil-1". Lvll-(t)j
Terminal Graph ‘ Terminal Equations
Figure l

The coefficient matrix 913 the product of three matrices of
the form
19 wedged T (1)
whered is the submatrix of the corresponding cut-set matrix and
ye is a diagonal matrix containingelement conductance values.

If the order ofg is n x n [(n+1) - terminal R-network], then
R-network containes a maximum ofim = .‘igl'il elements, i.e., A is of
order n x m and ye has 111 rows and columns.

For a complete representation the terminal graph (TG) and terminal
equations (TE) are both given and thed matrix can be written easily
from the TG. For simplicity let us consider the example in Figure 1.
Since the R-network has only the vertices which are the terminal
vertices it is complete network as shown in Figure 2. If the elements

of the corresponding R-network are labeled as indicated in Figure 2,

the relevant submatrix A of the cut-set matrix reads

,3 .

1
2
3
h

11 22
'1 o
o 1
o o
L0 o

 

 

33

0
0
l
O

 

at 12
o l
o -l
o o
l ,0

-1'

23

’ O l'-’ H 0

21+

1

° (2)

 

The number of equations relating the entries of j andé/e that

can be obtained is equal to m. If the entries of y e are considered as

unknown and the entries of j are known then there are exactly m

unkown in m equations. let these equations be arranged in the form

CZ

311

822

 

 

SIB-1,111

ll

Gin-1,11

 

J
where 813 's are the element conductance values and

 

(3)

G 'siare the entries

1.1

in j anins a square matrix to be determined from’J . In order to

obtain Orromj , let us write the matrix in the form

.3

 

n1

i 8ll “12

n2'8n3

813----

in ”I

 

“J

(1+)

and

82
ii, = -

L gm

where the subscripts are consistent witl the previous conventions,

 

 

1.3., 31 = 811, 82: 822, 0.00, gm: %‘l’m. Equation (1) call now
be written as

 

 

 

 

 

 

 

 

Fall 512 ' " " Sin] 2‘1 _i E’ll " " ' Snl
g, . 32 812 She
s s - - - s ' g is

_ n1 n2 nm__ L as: _lm mi
rallgl 51282 " ' ' Elmer; Sll ‘ ' ' snl
= 812 8112

: , (5)

8nlgl Sn2g2 - - - smgm J 8Int 8mn

 

We can consider the rows of matrix A as vectors (row matrix) i.e.,

(11: [811, s12, ..., 81m]. Therefore ,8 matrix has the form

‘F-

... mi“ Jul

,1:

(6)

 

 

_%J
let us define a new set of vectors (row matrices) by use of 51's
as follows,

kiJ=aian=a xa1=k

where'-

13

kid

=[

is given by

(7)

With this new notation equation (5) can be written as

kllg

5/

 

with g =

 

 

128 O O 0
keeg O O O

kneg O O O

E”

a

k

nn J

2n8

ing

8

 

(8)

(9)

Therefore, kijg is the sealer (or dot) product of two vectors, i.e.

k

136

m.
2 s s.
k=l ik 3k

gi

(10)

a.

Equation (8) now can be rewritten by considering only the diagonal

and, say, upper off-diagonal entries. The lower off-diagonal entries

‘will give the same equations.

of 9 yields .
FGll

G22

 

‘b

 

 

 

H

 

l ... “$1 tffl

gh‘

WI 00.

 

n-l,nj

Hence, one arrangement of the entries

(11)

or form (11)

C2“ . __.

 

 

where

53:,

n-l n
. ’ J

 

 

This result represents an algorithm.for writingéCZfromv<f.

Example: As an example consider Figure

given by equation (2).

For this example we have

El = (l, o, o,
5% = (o, l, o,
6% = (o, o, 1,
an:

From equation (7) we have

WI
I

ll

WI

22

W
U)
4:-
l

Hence, the rows onmatrix are obtained (see equation 13).

 

 

FGn
G22
n—l,nj
i011 x a1
5% x 5%
5$_l x 65

 

 

0, l, l; l, 0, 0; C )
O) “l: ‘1) "l: l: 1) O )
0) 02"1: 0) l: O: l )

(O: O: O: l: 0) O)‘l: 0: 11"1 )

~(1,0,0,0,l,l,l,0,0,0)

(0: l: 0: 0; l; l: l; l; l: 0 )

" (O: O: 0: O) O) O) O: O: 0.9-1 )

2, thej matrix of which is

(12)

(13)

Properties of the Qmatrix

Let the amatrix be partitioned in the form

[in ‘

W

22

II

I

: H

II

WI
[CS

 

 

 

 

 

 

 

(11*)
2i 3‘12
_ kn-l,nJ
and consider the properties of
- q a. q
ikll i h‘12. {
k22 : kl3
0’ = and a1: separately. (15)
b knnj kn-l’
L
Every row inc? 1 is of the form
' _ — « _ 2 2 2
kii z “i x i = (511’ Sie’ ’ Sim)
=(l811|’ '812l, e e e Isiml) (l6)

i.e., every entry of ii i is equal to the corresponding squared entry

of 51. Therefore we have:

The entries of 01 are either 0 or +1.

To establish additional properties let '51 I be defined as the vector

(row matrix),

A

lair = [| Bill , ' 512' , ,[ sim'] (l7)

and the matrix [A I as
H51!“i
l 3h!
[3! -- 3
MW _
then 01 can be written as the form
fin " H 51H
0 = 12 | a2|

I

 

 

w

j Ll énH

Therefore we have the important result:

385'

 

 

 

 

The 01 matrix is obtained from J by simply changing
the signs of all its negative entries.

let us consider now the other submatrix ofO i.e., r i:

ll
W

0 2 _ln (2°)

“‘1 000

 

 

_ n-l‘JUi
It can be seen from equations (2) and (1+) that if i f J and l S i,

JED, then 81),: 0 for} B 1,2,000, i-l’ 1 + l, 000 110
This implies that in equation (7) the first n entries are equal to zero

under the above conditions (i.e., 1 ;£ .3 and l f i, .15 n).

Therefore we have a. second important result:

The O 2 matrix has the fol lowing form:

(n) nSn-l)
_2
02: [O K Jnn-l

-2

where K15 a square matrix of order

ngn-l)

“2

With the above results they matrix in equation (13) can be written

as follows:

_ ., _ = _____ (21)
l
I V.
0, K O X
, -l , -1
In Section 2.2 it is snown that K exists and thereforeO always

exists and can be calculated as follows:

 

“-7—:‘— (m
0,)< _l

i.e., to invertO it is necessary to know only the inverse of )f

 

Existence of 0-1 permits us to write equation (12) or (3) as

 

f 811 - i Gll i

822 “I Gr22

g ==52 g (m
{gm-1:31} L Gn-i-1,nj

 

 

 

Therefore from the given terminal presentation in the form of the
terminal graph and coefficient matrix% Ocan be written directly

and the element values of the R-network can be found directly by

10

equation (23).

A.

2.2 The Existence of )(ul and Related Properties

Consider thea matrix (21) in Section 2.1. It has already been
indicated in Section 24that existence orQ'l is implied by the
existence ofX'l.

The proof that x-1 exists is based on the induction method.
Let us consider an R~network whose terminal graph consisting of two
elements as it is shown in Figure 1 (For a simplest TG, which is a
single line-segment, the order ofxmatrix is zero, hence this case

is omitted).

 

The submatrix of the cut-set matrix is
1 l 0

A - 2

II
b— g.

2 O 1

From equation (21 ) in Section 2.1

 

 

’1 0| 1 ' 1b

' 3

Q- 0 1| 1 null l
““‘”"T"‘ ...,.--

t0 0 i "ll. __° I 2_

 

 

Therefore
det 7(2- --1 a -l;‘0
i.e.)<2 is a non-singular matrix.
For the terminal graph consisting of three elements we have
two possible configurations;
ll

a) If the terminal graph is in the form of a Lagrangian tree

then from Figure 2 we have

 

 

 

ll 22
l f l o
A = 2 O l
3 B O 0
Therefore
F l o o l 1
o 1 o i l
a O O l I O
= O O O I. l
O O O l O
_ o o o : 0
which gives 1 0 O
deity?) = o l o = -l ,é o

O O -l

and x3-1 exists .

33

12

 

13

b) If the terminal graph is in the form of a path then from

Figure 3 we have

11

 

22

33

12

 

23

 

and

 

 

[ l o o l 1 1 o ‘
o 1 o : 1 1 1
2-3-112_i_1
(:2 = o o o : l. 1 o
o o o I o 1 o
_ o o o | o 1 l ,
Therefore
r 1 1 o
det.>(g = o 1 o = 1 e o
o 1 1

 

 

i.e., K34". exists.

Note here that labeling of the elements of the terminal graph
does not effect the existence of )‘(3-1. This can be seen as follows:
Suppose we have interchanged the labeling of any two elements of the
terminal graph, say 2 and 3. This changes will effect to the matrix
in such a way that only the second and third row of )J are interchanged.
'This interchange will be reflected only as an interchanging in the
same rows of X3 (in this case first and second rows of X3 are
interchanged). Therefore det X3 is not alter except in sign.

Changing the labeling of the R graph elements does not alter the
det X3 except in sign, since it. corresponds to a change between
the columns of the matrix i.e., changes between the columns. of )13
matrix. It is obvious that the above preperty applies also for a
terminal graph having more than three elements. '

Proceeding with the induction proof, assume that )‘fn-l exists

-1

for a terminal graph having 11 elements. We shall show that K n +1

13

also exists.

Consider an R-network whose system.graph is represented by Gh and

the terminal graph by Tn, as in Figure h(a).

 

Figure A

Suppose a new vertex a.n+2 is added to the graph and all the
vertices a1(i = l, 2, ... n+1) are connected to this vertex by a set
of new elements (1, n+1), (2, n+1),...., (n+1, n+1), alsoa new element
(n+1) is added to the terminal graph Th. By doing this we here obtained
again a complete R-graph for which the terminal graph (Th+l) contains

(n+1) elements. Let the,<£matrix for Oh and Th be written as

 

 

l .
xi = : U. 11

n

then for the new graph Gn+1 and Th+1the matrix will have the form

' (n) (1) ( .213211_.) New elements (n)
1 f“ - * y -
. 11 I O 1 /<)_ l
g I ' l l l 12
: .. l . I 1.
. n ---:———% ------ n ————— Q)
n+1 , [<8 , .
. O I 1-. ' 21 I A 22-]

Since the last cut-set corresponding to the element (n+1) contains
only the element (1, n+1), ..., (n+1, n+1), the last row of”a_has the

properties such that/2221 = O and/8.22 = i l 1 1 ... 1]. Since the

1%

element orientations of the new elements in Gn+l are chosen as in
Figure Mb) the entries of )3 22 are all +1. If we change some of
the orientations of these elements the corresponding entries ofA 22
becomes -1. It will be seen later that changes in the orientations
of these elements are immaterial as far as the inverse of ”n+1 is
concerned.

Now from equation (15) in Section 2.1 we can construct the
Kn+l matrix. It is easyto see that

Xn {112
l

X22
withX22 =A12 since/322 =[ 1 1 1 1].

' -1
If we can show that )5 12 is nonsingular thenx 22 exists. This

Xn+1 = I-..

0

implies that Xn+l is nonsingular, since by elementary transformation
matrix equivalent tOXn+l can be obtained in the form
>< ‘ 0
n
0 K 22

which has an inverse.
Now the problem is reduced to show that A 12 is nonsingular. From

equation (1) it can be seen that ,4 12 is a square matrix of order n.

Let us consider the R-network in Figure Ma). For this network we

can disregard Gn and consider only Tn since-2'3n and a set of

correSponding terminal equations constitute terminal representation

of Gn' Therefore, we have a system graph containing TnU (n+1) =

Tn+1

and the elements (1, n+1), --- (n+l,n+l) as in Figure ’+(b). If Tn+l

is chosen as a tree in this graph, then the cut-set matrix is of form

15

(l,n+l)...(n+l,n+l)

 

- I V ' l“
l t I ' x I0
2 I I A I0
C = . 2’[m-ll C12 = 2’414-1' '2' '3 (3)
3 I _ __.—19. -
n+lL I . Fl 1 000 1'1
-.Ji L. I I (L

 

 

 

On the other hand, the chord-set in Figure 1+(h) also constitute
a tree of the graph, i.e. , all through and across variables of Tn+1
can be expressed in terms of the corresponding variables for the 1
elements (l,n+1), ..., (n+1,n+1), and vice versa. In other words

considering matrix (3) we have
U T
Q = 0 (4)

C
)- a

[u 6-2]

 

 

If Tn +1 is taken as chord set then we have
' T
IQ c
[[21 012] § = 0 (5)
T _

L 1

 

 

Equations (1+) and {5) give

3T=-C12Qc
QC {312%

respectively which imply that

CmeDm=Qm€12=ZL

Hence

. --1
C12 =0 i: or $12 =67 :1: , therefore [Q42] exists

and the proof that is a nonsingular matrix is established.

12
Ebcample 1: Six-Vertex Terminal Graph

16

 

 

 

 

 

 

 

Figure 5

For Figure 5 the matrix is

53 51+

52

12 31 32 #1 he #3 51

5

 

L0
fram'which

 

l7

Example 2: Six-Vertex- Path Terminal Graph

 

 

 

 

I

 

1 2 3 u 5 12 13 23 1h 2% 3h 15
1 o o o o 1 1 o 1 o o ' 1
o 1 o o o 1 1 1 1 1 o 1
[,8 = o o 1 o o o 1 1 1 1 1 1
o o o 1 o o o o 1 1 1 1
(_o o o o 1 o o o o o o 1
and
1 1 o 1 o o 1 o o 01
__T___|
o . 1 o 1 o o 1 o o o
I l
o . 1 1 l 1 1 o 1 1 o o
*“u “ ‘ ‘1
o o o I 1 o o | 1 o o o
I
)(;.=r o o o | 1 1 o l 1 1 o o
o o o I 1 1 1 I 1 1 1 o
+-————l——————
o o o o o o ' 1 o o o
o o o o o o ' 1 1 o o
o o o o o o l 1 1 1 o
o o o o o o I 1 1 1 1J

 

 

General form of K-matrix and its inverse.

One can choose the labeling of the elements in the graph GH
18

25

O

HHH

35

#5

l-‘l-‘OOO

 

(-corresponds to R-network) such that the7< matrix assumes a simple form.
We already know that 7(n+l-can be written as in equation ( 3) by labeling

the new elements in Gn+ as discussed earlier i.e.,

l

_
—

Zrn a€n+l

= (6)
Xn—tl O 7<(n+1 )

 

 

where 7((n+l) is used to replace 7&2 in equation (3). If we do the

same thing for the vertices in Gn we would have e.g.,

[BI/ml 55h

“'1

7(n=

 

 

__ ° 71?“?

Continuing this labeling process, Kn then can be put in the form

RI (1) 51(12 £13 ‘" 33h
0' X (2) aI023 "" 9621;
7< _ ' 0 31(3)

-P

I
I
l l
I l

O XII-1,1)

L. O O O x0204.

where the order of’}( (i) is exactly equal to 1. Also it has been

 

 

established that 7(a) has an inverse

The inverse of (Xn will be in the form

'r -1 t
7((1) 70 ----- 7Dln
-1
7. -1 . W a a.
o o o o 7((n)':

 

 

Where<7C;J can be calculated in terms of‘3K(i) and but these

ij’

expressions for are not convenient to calculate the inverse of

13

~><g, since the complexity of these expressions increase as the indices

1 and 3 increase. For'example the expressions for some of thggaij in

-1
terms of;><(i) andéZZij are.

0.2 owl JR )(2'1
on 13 = Wit-€13 Xgl + X342 Xe'lefewgl
@ n- ‘X {12¢ n X151 + Xl'loire Xz'lofinX 1'1
+ Xl-lb{13 X3'lcsZ3uX u-l

- X1-16812X2-12Z23 X52? 311% h-1
03 23 " X2-10{ 23 X3.1
9 211 X {lieux {l + Xa-lsfla Xflgfln K 1:1

The Labeling of Graph Elements

 

In order to put:)<n in the form indicated in equation 7 the labeling
of the elements of the terminal graph as well as the R-graph elements
must be chosen carefully.

a) Labeling of the TErminal graph elements

Guilleminlo has described a method by which a tree can be assumed
to "grow". This procedure can be used effectively here. To illustrate,

let us consider the tree in Figure 7(a).

20

 

 

 

Figure 7

In Guillemin's terminology Figure 7(c) is called a "one-year-old treeV,
Figure 7(b) is called a "two—year-old tree". Therefore the tree in
Figure 7(a) is a "three-year-old tree". »The labeling of the latter
is obtained by first labeling Figures 7(b) and (c).
b) Labeling of the R-graph elements. .
To label the R-graph elements the following procedure is used.
1. Consider the end element of the terminal graph with the labeling
of highest number.' (This number in Figure 7(a) is 9).
2. Since the element considered in (l) is an end element, consider the

vertex of this element which is incident to this element only ( in

Figure 9(a) this vertex is a lo).

3. Consider the elements between the vertices alO and a1, a2, a3,..... a9

let us temporarily indicate these elements by the symbol (alo,ai)

P

cl

where i = l, 2, ...., 9.

h. Consider vertices a1(i # 10) which is incident to only one element
(e.g., a7, a8, ah, a9) and the correSponding elements'(a10, a7),
(a10,a8), ..... (alO’ a9). Label these elements as (g3). Where

5 is the labeling of the end elements at which the selected vertices
are incident. For example (alo, a7) is labeled as(g7) while

element (alO’ an) is labeled as (g3). ‘

5. Remove all the labeled elements and the corresponding end elements.
Repeat the same thing for the new end elements.

6. After labeling all the elements which are incident to the vertex
considered in (2), remove all these elements and the element considered
in (l). _

7. ~Repeat the same thing for the second highest labeled element

and continue to this process to establish a labeling for all the

elements of the R-graph.

 

 

Figure 8

3. POLYGON TO STAR TRANSFORMATION
3.1 Necessary and Sufficient Conditions

In this section the transformation of a.polygon-connected network
into an equivalent star-connected network is considered. A set of
necessary and sufficient conditions for the existence of such a
transformation is presented in terms of the relationships between
the entries of a matrix defined later. These conditions are also
interpreted in terms of the elements of the given polygon-connected
network. For the sake of simplicity in the proof only R-networks
are considered. ‘HOwever, the result applies more generally. An
alternate statement of necessary and sufficient conditions is also
stated.

Consider a.polygonal R-network having the nodes A1, A2, ... A.n
and the element conductance values G13, (1, J = 1, 2, . . . n; 1 #3),
where GiJ corresponds to the element between the nodes A1 and AJ'

To characterize the prOperties of this R-network, an additional isolated

, is chosen as the reference node. A star-like tree

5

node, A.n+1

(Lagrangian tree) terminal graph , T, having Ai's (i = l, 2, . . ., n)
as its end vertices is selected. This terminal graph is given in
Figure l, and the corresponding terminal equations are given in (l)
and (2) '

A. A

 

 

 

 

 

Iii“) I I311 ‘112 ° ' ° 8Lin 3 “(10:)"

12(t) al2 a.22 . . . a2n v2(t)

. : . . (l)
~in(t)J raln a.2n . . . ahn__ vh(t)J

 

 

 

or
3s) sa’wt) (2)
If all the elements of T are oriented toward (or away from)
An+1’ it is well known6 that all the off diagonal entries in the
coefficient matrix, A, in Figure l, are negative and their magnitudes
are equal to the conductance values of the elements, i.e., -313 = G1J(if J).

The diagonal entries, aii’ are equal to the sum of the conductance

n
values of all the elements incident to node A,, i.e., QB. = 2 G .
More specifically, since a.ii = - Z aiJ, then the coefficient

* 3:1
matrix, A, is strictly dominant. jfl

Consider a star-connected R-network having the terminal nodes
Bl’ B , . . ., Rn and the element conductance values gi(i =1, 2, ...,n).
Where gi corresponds to the element between the nodes B~ and the

i
center node B . An additional isolated node B is chosen as
n+1 n+2
the reference node and a starlike tree terminal graph, Tl’ having

B1 (1 = l, 2, ..., n+1) as its end nodes is selected. It can be seen
easily that coefficient matrix appearing in the terminal equations

for this star-connected R-network is as follows

 

For:matrix.A since dominantcy condition is satisfied with equality

- n
sign, 2 a13 = O, the word "strictly cominant" is used. A.strictly

J=1
dominant matrix actu ly is an "indefinite admittance" matrix7 or an
"equicofactor matrix ."

2h

 

 

Is I _ ,
1 I 81
82 I ' 82
o I 0
I
’ I (3)
° l
. I .
l
3n I 'Ign
__________ I___._
.131 '32 ’ 8n I z _
n
where Z a Z gk. In order to characterize the properties of this
' kal

R-network at the terminal nodes, Bi (1 a 1, 2, ..., n), corresponding

to a terminal graph, T , having the same form as T in Figure 1,

2
the current variable in+l(t) is set equal to zero. Therefore, from
Eq. (3) we obtain the coefficient matrix of the terminal equations

corresponding to T .

 

 

 

 

2
81 g1
82 82
- - l. ° [g . . . g 1 (h)
. Z . ‘ l g2 n
L. .
L 8w M
Since terminal graphs T and Th are identical, the two networks are

equivalent when the matrices in Eqs. (1) and (h) are identical, i.e.,

CZZ:EZ§?7 - (5)

or
s s
a,=-%H-(i#n (@
and g 2
6‘ii = 31 ' 'IT’ (7)

The following relation is easily established from Eq. (6)

11:511.,
”13 '31 1‘1 (i = 1, 2, ..., n; 0% 5.1) (8)

where ai-l is a positive real constant. Since - 8'13 = G13 (1 f J)
Eq. (It) is actually Rosen's theorem9, i.e., Eq. (6) expresses the
element values of the polygon-connected network in terms of the
element values of the star-connected network. (Star to polygon
transformation). -

To establish the inverse relation, i.e. , the polygon to star
transformation, solve Eq. (8) for g i

g:L a (11-1 g:L (i a l, 2, ..., n) (9)
Hence, if the element vaiue gl is obtained the values of all other
gi follow from Eq. (9). ‘

Consider sq. (7) for i = 1 and substitute for z the identity
2 = gl (1 + (1"- + . . . + an-l)° We have then

‘11

l-

 

31 = (10)

QIH

whereo=l+(x_|_+...+an_ Sinceo>l,andall<O,Eq.(10)

1'
implies 81 > O. From Eq. (9) we also have in general g:L > O.
From the above discussion, the following theorem can be stated:
Theorem: For a given n-node R-network to have an I
equivalent n-branch star R-network the necessary and sufficient
conditions are: ' [assuming the terminal representations of this
network is given by the terminal graph as shown in Fig. l and the

terminal equations (1)]

H
II

lpoooo’nsail
° (11)
2,....,n;i;é,j '

J -

C4.
ll

26

where 03-1 is a positive real constant.
Proof: Necessity: Follows immediately from the fact that gi > O
in Eq. (8).
Sufficiency: Given 01-1 >'O, we have from.Eq. (10), gl >’O
and from.Eq, (9), g1 >'O, i.e., there exists a unique star equivalent.
The condition (11) implies the following property of the
polygon connected network:
Let A1 and A.J be any two nodes of the network. The
ratio of the conductance of the elements between each one
of these nodes and any other node Ak (for k = l, 2, ..., n;
k # k, k f J) must be the same.

Take any vertex as reference node as shown in Fig. 2 by the heavy

line.

 

 

Figure 2

Denote the new terminal equation in matrix notation by
I I /
s=5 == (libr' (12)

where Ii'l I v'l

fi

3': 5

 

 

 

 

i
To establish the coefficient matrix Q , apply a tree transformation
to take the original set of voltages to the new set. I
V= 77/"
From.the circuit equations of Fig. 2 we have

_— 'I ‘

1 o o..o'

ooooH [—1

7:

00.00
I—'
O
o

0.000

0

Lo........ 0

 

 

 

 

H

I_
l
I

It is well-known that

J ‘ 7:9 (13)
Hence from .Equations (12) and (13)

3= U ‘=7't.Q2/‘=7'tafy/
or a' s Th 7- (11+)

Let theQ matrix be written in the partitional form

Q = (15)

 

 

 

 

 

 

I' 2 "
g _a. __ ...... -8148-
1 z 2: * 2:
2
- as - Es. ...... - 82311-1
2 82 2 g:
I
where Q = I , : (16)
11 , I 2 _
8n-lgn _ g2gn-l _ _ _ _ _ gn-l _
z 2 8n-1 2 ~

 

28

0-12 =

 

I.

r glgn

 

Z
g2gn
Z

 

gn-lgn

 

2I

2

8
3 (222 = gn - -§%-

 

Thus for the coefficient matrix in Equation (1%) we have

where

 

t
a = 7&7- = .
t l 6:>
A Q12 22 O /
an anA + 12
t t t t
A 11 +412 “tau" (112)“ + A 412* 22
2 . \ 4' \ 8 a n V.
-E‘1_-§i§2%1 .I (-..-s... o
2: 2; 2: 2:-
2 s
- 5.6.2. .. EL _ gegn“1 1 - 82 n o
2: 82 2: z + 2:
s s s a s 2 s s
l n-l 2 n-l ... n-1 _ n-l r.
' 2: ' 2"- gn-l - Z I .lI I Z .01

 

U 0 6411012 UA

 

 

 

 

 

 

 

 

 

 

 

 

. . t t _
similarly A Q11 +Q12 ; [o o 0]

At Q12 + 22 - [0]
Equation (12) now reads
I11 I I I I Ivi ‘
3:2 all I 0 Y2
: = _.__ __'____ .. :
_iJn-l . Vn--l
i'n J O I O I I VI] J

 

 

 

 

 

 

- 2;;

 

and the last equation in the set of no concern, i.e. the nth element

can be deleted from.the terminal representation giving

q. r T
Ii'l v.1

' '
12 V2

5 =Q11 :

I
n-Jj LV n-l I
(2111 13 a submatrix 0fC:?, and is a dominant matrix but not strictly

 

 

 

 

dominant in the sense discussed earlier. An equivalent statement
of the theorem.for a given n-terminal complete polygon using an n—terminal
Langrangian tree can now be given. The necessary and sufficient
condition on the conductance matrix of an n-terminal polygon such
that the given polygon has an equivalent star network are:
(1) .Except for the diagonal element, all the ratios of the
‘ off-diagonal elements in the same row to the corresponding

elements in the first row must be the same, e.g.,

323': 32%.: ... = a2zn-l =
8-13 an ”Ln—1 “1

(2) The ratio of any row sum to the first row sum must give

the same ratio as the condition in (l)

e.g.
n-l
f ”‘21
1-1 _
n-l — al
2 a
J=l 13

If the above conditions are satisfied we can use Equation (10)
to calculate the corresponding star element values.. From Equation (16)

'we can determine 01’ 02 ... oh_2 as before and a%_l is obtained from

30

the ratio of any column sum to the corresponding absolute value of

the element value in the first row.

n-l
2 “J2
e.g. a g: ESL—_—
n'l I“.12
3.2 Applications

Example 1
Does the polygon in Figure 3 have

an equivalent star network?

Tb use the criterion

 

of Theorem.1 derive the

Figure 3
conductance matrix using an 5 vertex terminal graph with the 5th

vertex taken as the ccmmon vertex. Then we have

 

 

f 21L --12 4+ -8 \I

-12 21 -3 -6

Q = .. 1+ - 3 9 -2
-. a - 6 -2 16 I

§

It is easy to see that the condition of Theorem 1 is satisfied and

the element values in the equivalent 5 vertex star configuration are

 

31

or g =--——— =—-—-=
l a .3.
1'5 s
g = g = 3 x ho = 30
2 0,11. E-
- 1 _
g3=a2gl—Exho_1o
g4 = at3gl = %»x #0 = 20

and the equivalent star network is given in Figure h.
A:
A2
4-0 3°

 

 

Example 2
Does the polygon-connected

network in Figure 5 have
an equivalent star-connected

network?

  

Using additional vertex
.A6 as reference node then the

conductance matrix.

(9 -1 -1 -I-I- -3 I

II

I
H
I
H
0'.)
I
4:-
I
[D

 

 

-3 -3 -2 -l2 2OI

We see immediately that the ratios between the elements of the first

and the second rows are not the same and we need not go further to

conclude that there is no equivalent star-connected network for this

given polygon-connected network

Example 3 A1 A2 . A3
2 e3
Given Terminal Graph I
and Terminal Equation Ah
as shown in Figure 6. Terminal
Graph

Can this be synthesized

as a star-connected R-network?

 

 

I
O\

[10.5 -1.5‘I v I

 

 

 

1
-6 12 - 2 v2
I_-1.5 -2 n.5I v3J

Terminal Equation

Figure 6

In this condition we simply use the alternate theorem. Since

condition (1) and (2) of this theorem is satisfied there exists a

5.vertex star network the element values of which are given by

Equation (10).

 

 

E'11
g:
1 l__1_
0
now all — 10.5
-2 h
a].='105=-3-
-2 l
Q2=z=§
a _12-6-2_1I_2_
3‘1-61'6'3
.. i 2: 31.1.0.
0 — 1 + 3 + 3 + 3 3
_ 10.5 105
31 1.3- 7 -15
10 h
82=algl"§' 15:20
- -1 =
83-0331-3Xl5 5
sh=a3sl §x15 =10

33

The star network is given in Figure 7.

h
/5

31+

‘5‘”111 4:11..

10.

REFERENCES

I. Cederbaum, "- Applications of Matrix Algebra to Network Theory"
IRE Trans. on Circuit Theory, Vol-CT-6, Special Suppl. PP 127-137,
MAW, 1959

E. A. Guillemin, "On the Analysis and Synthesis of Single-Element
Kind Networks " IRE Trans. on Circuit Theory, Vol-CT-T,

September, 1960.

D. P. Brown and Y. Tokad "On the Synthesis of R Network" IRE
Trans. on Circuit Theory, Vol-CT-B, March, 1961

G. Biorci and P.P. Civalleri, "On the Synthesis of Resistive
N-Port Network" IRE Trans. on Circuit Theory, Vol-CT-B, March 1961.
H. E. Koenig and W. A. Blackwall, Electromechanical Systems

Theo , McGraw-Hill Book Company, Inc., 1961.

P. Slepian and- L. Weinberg, "Applications of Paramount and
Dominant Matrices, "Proc. Natl. Electronics Conference, Vol. 11+,
pp. 611-630, October, 1958.

J. Shekel, "Voltage Reference Node, " Wireless Engineers, Vol. 31,
pp. 6-10; January, 195%.

G. E. Sharpe and B. Spain, "On the Solution of Networks by Means
of the Equicofactor Matrix, " IRE Trans. on Circuit Theory,

Vol. CT-7, pp. 230-239, September, 1960.

A. Rosen, "A New Network Theorem," IEE Journal, Vol. 62, pp. 91-198,
192h.

E. A. Guillemin, "How to Grow your Own Trees from Given Cut-Set
or Tei-set Matrices", IRE Trans. on Circuit Theory, Vol. CT-6,

pp. 110-126; may, 1959.

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