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MICHIGAN STATE UNWERSETY
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UNIVERSITY
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TRANSFORMATIONS 0F ELECTRIC NETWORKS BY MATRICES:
CASES OF INVARIANT POWER, NON—INVARIANT

POWER, AND INVARIANT INPUT-IMPEDANCE

By
SAKAE mmum

A Thesis
Submitted to the School of Graduate Studies of Michigan
State College of Agriculture and Applied Science
in partial fulfillment of the requirements

for the degree of

MASTER OF SCIENCE

Department of Electrical Engineering

1951

 

Chapter I.

Chapter II.

Chap ter I II 0

Chapter IV 0
Chapter V.
Chapter VI.

Chapter VII.

Contents

Introduction

Derivation of Generalized Transformation
Equation

necessary and Sufficient Conditions for
Invariance of Power

Two Kinds of Transformation Matrices
Invariant Transformations

{on-Invariant Transformations
Transformations of Invariant Input»

Impedance

Page

18
19

Chapter I. Introduction

(1) (2‘2)

Kron published many papers and books about
matrix treatments of electric networks. The core of
his works is congruent transformations, which are ex-
plained in mathematics hooks(4)and the books by Guil—

(5) (C)
lemin . Congruent transformations keep quairatie
forms, such as electric power in the case of electric
networks, invariant, when transformation matrix is non-
singular.

In this thesis the author deals with both invariant
and non-invariant transformations, and defines necessary
and sufficient conditions for each case and the valid
range of applications of each of them. Although almost
all examples are by mesh methods, the same discussions
hold for node methods.

In Chapter VII transformations of invariant input-
impedance are discussed. These are actually reduction
of electric networks to Foster's equivalent forms.

This process by usual methods(5)is very laborious and

tedious. By using the matrix treatments, which are axe

plained in Chapter VII, this process is made orderly
and less tedious.

This papaer is written by the author as the thesis
for Haster's Degree. Professor J. A. Strelzoff taught
the author matrix treatments of electric networks and
made it possible for the author to work on these subjects.

To Professor J. A. Strelzoff the author presents

his sincere thanks.

Chapter II. Derivation of the Generalized Transforma-
tion Equation
The equation of a electric network can be written

in the matrix form of

[9] = [Z] [1] 2.1
Now we change current [i] into new current [it] by the
transformation equation

[1] = [A] [1'] 2.2
Substituting 2.2 into 2.1, we get

[e] = [z] [A][i'] 2.3
Multiplying 2.3 by a matrix [B], we get

[31b] = [s] [z] [A] [1 '1 2 . A
In the process of deriving equation 2.4 we change voltage
[e] into new voltage Esq 'hy the transformation equation'

[e'] = [Elm 2.5
From 2.4 and 2.5 we get i

[6'] = [21sz [1'] . 2-6
Let new impedance matrix for new voltage [e'] and new cur-
rent [1'] be written [2'] , then

[a] = [20 [v] 2.7
From 2.6 and 2.7 we get

[2'] = [B] [Z] [A] 2.8

So we can conclude as following,
When current [i] and voltage fie] are transformed into
new current [1'] and new voltage [e'] by the transfor-
mation equations 2.2 and 2.5, the new impedance matrix
[Z{] for the new current and the new voltage is given
by 2.8.

This is the generalized transformation equation,
which contained congruent transformation [61.3] [C] as

one special case.

Chapter III Necessary and Sufficient Conditions
for Invariance of Power
To reach transformation equations, Kron started
from tho assumotion that power is invariant, th:t is,
powers at each impedance branch of new system are equal
resoectively to those of old system. And he got the follow-
ing results.
When current [i] is transformed into new current
[ii] by the transformation equation
[1] = [CM] 3.1
and voltage [e] is transformed by the equation
[e'] =8 [C1,[e] 3.2
, than impelance [Z] is transformed by the transformation
equation
[2'] : [C1[L][C] 3.3
* From his whole book it Seems that Kron thinks thet
3.1, 3.2 and 3.3 are the necessary and sufficient condi-
tions for invariance of power. But it is not correct as

explained later.

 

When we compare these equations with 2.2, 2.7 and

2.8, we see that

[A] = [0] [B] = [‘31: 3-4

 

......

From what Kron derived, we can see that the equations
3.4 are the necessary conditions for invariance of power.
But it is not yet clear whether these are the sufficient
conditions, too. Let us examine it.
Power for the old system is given by
P: [9}: [1] 3' 5
And power for the new system is niven by
P = [eatfl'] 3.6
From 3.1 we get
[v] _= [c]~l [1] 3.7
From 3.2 we set .
[6']. = ([0].; [9]), = [e]t[C] 3.8
By substituting 3.7 and 3.8 into 3.6 we get
in =[e1.n[c1"[11 =[e1.[11= P 3.9
50 power is invariant for this transformation. This tyne
of transformation is called congruent transformation.
It seems true that 3.4 is the sufficient condition for
invariance of power, too. But this is not sufficient.
In the process above we used[CI1 in the equation 3.7.
So[p] must be non-singular.
’ Then we can conclude that the condition 3.4 and
non—singularity of the transformation matrix EC] are the

necessary and sufficient conditions for invariance of power.

Here there must be some comments about the statement
7)
by Pipe that the transformation matrix [C]( Pipe usesfh] )
need not be non-sinaular. But he insists that only the
shape of the enerqy eouation (not the numerical value itself)
is preserved under the transformation by 3.4. In order that
not only the shaoe of the rncrgy equation but also the

numerical value of the enervy are nreserved, non-singularity

of the transformation matrix [C] hast be ended.

Chapter IV Two Kinds of Transformation Matrices C .

To set up sancaicai equations of electric networks,
Kron started from their nrimitive networks. For Example,

Fig. 4.1 a) is the prinitive network for Fig. 4.1 b) and c).

m

 

 

 

 

 

 

 

 

 

(c)
These are ailinesh.u networks, because they have as

many meshes as number of impedances. In these networks all
emf's in series with each impedances are short-circuited.

So each current in each impedance is not changed by changing
interconnections as a), b) and c). In this case power is
invariant for these trasfoemations. Let's find the trans-
formation matrix [C] between Fig. a) and c) by assuming

reference currents indicated in the figures.

10

In imoedance a“_ TL: - fg- id-14'
" z... 1" = - 1"
n z... 1‘: 1"+ 1’2. 1‘“ 4.1
2“ 1*: . i"

I "w- 1*- 1"

From these equations we get the trnasformation matrix C

a’ b’ c’ d' f'
[c]: a ”.1 o -1 -1 0‘}

'b O O -l 0 0.
4.2

 

 

rLo o 0.10

The matrix 4.2 in non-singular.
Equatifin 4.1 and matrix 4.2 give the correspondences
between old currents and new currents. And their corres-
*

pondences are not only geometrical or positional but also

numerical.

 

* Geometrical or positional correspondence means
that corresponding currents flows in the same impedance of
the network. Nemerical correspondence means that values

of these corresponding currents are equal namerically.

11

In 80m: cases correspondence is only fcomctricel, tut not

i- J.

numerical. Luch case: 'rn crylt nri Eczer.

All tran:f:rratien matrice: should be qeemctriCLlly
corresOOuecnt. int {may nec‘ not b; numerically corresnon-
dent/o

Let's examine enother simnle example.

 

 

1 z... 1 2..., 2..., LA 25... z... wad

 

 

(a) (h)

Fig. 4.2 a) anl b) are the Lane network, rhich has two inde—
pendent mcrhes. In a) T‘ {n1 1b are assumed as indenendent,
while in b) fiend id are assumed inieoendent. Only difference
in the difference of reference currents. Cor905fiondences of
these currents are,

If} -? o; ‘ f‘ a - g
In impedance a,“ i a i

-r‘- r’

H

In impedance Z», 1"

12

So the transformation matrix is

/ I

c
[Cl = a l o
4.3
b -1 -1
This is a nonésingular matrix. And the correspondence by
[1] a [c] [1’] 4.4
is not only geometrical but also numerical.

We have Just examined two cases, where transformation
matrices are non-singular. It may seem too early to conclude
that all non-singular transformation matrices are correspon-
dent geometrically and numerically. But this is true. We get
non-singular transformation matrices, only when transformations
are not accompanied by any change of interconnections of impe-
dances. For instance. the transformation of Fig. 3.2 does
not have any change of interconnections of impedances. It has
only change of reference currents. So all currents at all
impedamce branches are invariant at this transformation.
Accordingly the transformation matrix for this case is corres-
pondent geometrically and numerically.

Transformation matrices for tansformations between
different independent reference variables of the same circuit
are generally non-singular and corresponding geometrically

and numerically.

13

It seems likely that the transformation of Fig. 4.1
is accompanied by changes of interconnection of impedances.
But all impedances remain short-circuited even after the
transformation, and the mumber of meshes remains the same.
All currents of imnedance branches remain the Same. So that
the transformation matrixIC]of 4.2 is non-singular and
correspondent geometrically and numerically.

Here we can conclude that all transformation matrices
for transformations, where the number of meshes remains the
same, are non-singular and correspondent geometrically and
numerically, so long as all reference variables are indepen-
dent.

Let's examine the transformation of Fig. 4.3.

" l

T )<
a- 2“, z“, a: z“, 2-“ 2“,, 'z.cc
ea, eb ec. e4’ e5 an.

(L)

as
) F3}. #;3

 

 

(a) is trasformed into (b) by opening one impedance branch 25,.

Reference currents are assumed as indicated in the figure.

14

Then the correspondence between old and new currents are

1" ---)~ 1”

. 4.5
1" -—-> 1/
So the transformation matrix is
1/
[p] = a 1
4.6
c -1

[C115 singular.

It is easily understood that power is not invariant
in this transformation. 50 it seems likely that cengruent
trasnformation [0142] [C] can not be applied in this case, because
Kron proved congruent transformations by assuming that power

is invariant in transformations. Let's try to apply it to this

For Fig. 4.3 (3)

ea_- eh: (Z...+ ‘2“)1‘4» z“ r-

4.7
6&- 8": z“ 1‘ + (255+ Z»)?
This is written in matrix fern as below. ’
[e142] [1]
when. [a]: at — eh [Z] = Z“)- be 2“
8e. - e 255 255+ ZG
4.8
[11 = 1‘
it

I

According to the congruent transformation, new matrices for

15

the new network of Fig. 4.3 (b) are given as following.

[0M2] =[1 -1] z“ + 2“ z“ = [2“, - 2%]

2..., Z» + Z“
[2'] =[CJt [z] [c] = [2... - a} 1 = [2“, + 2“]
l . 4.9
a]: [alt [e] = [l 4] e.. - lie. - e4

80 for the new nctuork the equation of network(e1 =CZ][ij is

e - e = (:4M + z“)! 4.10

a. Ce

This is valid as easily seen from Fig. 4.3 (b). SOECIt[Z] [C]
method can be applied even for the transformations, where
power is not invariant. And this foot is already proved
[by equation 2.8, because we did not assume invariance of
'power to derive equation 2.8. But some limitations are
necessary for applying 2.8 generally,as explained later.
Here we would like to call attention to the fact
that equation 4.5 does not hold numerically, but it does
hold only geometrically. So "-—9" is used instead of "a ".there.
Transformation matrix 4.6 indicates only geometrical corres-
pondence between old and new currents. So it is not correct
to use[i] =[C] [i'] to evaluate one currents group from

values of the other currents group, which are already solved.

16

It is ini*res%inr that consrucnt transformaflisn
can he cwpliofi even to casrs whore transformatihn Nitrix
is corresnonjont only *eametrically but not numerically.
Transforrt?ions, which chance number of imncj"nce
breach ;, are acccm“znfed by transfernation wztricvs, which

are sinvular. In Rhese cases, pose is not invariant and

'1

transfornation matrices are ccrresncnient only rennetrical y,

‘L

but no‘ numrrically. As showed shove, congruent *rsnsfernr‘i‘n
can to t filifif *o this 0:50. Put numlar n? imna‘1rcn
branches 53 ul? Fw chznrmV in'c lezs number. (thermise tkis

-

riefigmi C'IXI ? la” a fixlinfi.

l. ?r{WSfC?HLt10n netrix for transformatirns, which
are not t-.CCv.:m 1.9-; tied by chat". 63 ff” number at" imnaziar‘ce

‘ , ' ‘ I g . ~‘
U “113;: .95, {ST‘C “On-5;” fl... LLI‘.

;. non-sin'riar transformat'cn matrices are corros-

O
J

nnnnt fecnntrically ann numerically.
*rnation matrices fer transformations, which
are accem~anie by channe of number of imnejnnce

branCPQs ( into Smaller nutter), are sinrular.

5.

17

Singular transformation matrices are correspon-
dent only geometrically but not numericallv.

In this case newer is not invariant.

But still [0M2] [0] method can be used for singu—
lar transformation matrices.

But we can not evaluate [i] from [1'] , using

[1] = [c] [1'] ..

18

Chapter V. Invariant Transformations.
is exnlained in Chester III, the necessary and
sufficient conditiots I‘or invariance o- Dower are

[A] = [c] [B] = [c],: 5.1
And [9] must be non-singular.

In this case formulas of transformation between
old circuit ecuation

[91“[2131 ‘5 . 2
and new circuit equation

[e'] = [2'] [1'] 5.3
are given as following.

[1] = [c] [1'] 5.1.

W [cit] 5.5

[Z'l= [CL [21E] 5.6
Because [C] is non-sin ".Ult't,

[1'1= [CM] 5.7
he can evaluate [i] from [iq, using 5.A, beceuce 5.4 is
corresoondent numerically.

(1)
Host of examoles in Kron's book are invariant
transfo nations, althoufin some non-invariant trans forma-
tions are mixed, without explanations. So it is not neces-

sary to present here more such examples.

19

Chapter VI Ron~invariant Transformations

When at least one of the nece.sary conditions for
invariant transformations is not satisfied, the transfor—
mation is non—invariant. Power is not kept invariant at
this transformation. to the sufficient condition for non-
invariant transformation is ;

l. [A] = [c1 an; [B] a [ch are not satisfiei.
or 2. Transformation matrix [C] is sinwular.

Accordin? to which necessary condition for invariance
of power is not satisf'ed, there are several kinds of non-

invarinnt transformations.
(1) Cases where [A] 2 [C] and [B] = [Clt are not satisfied.

As derived in Chapter II, when old and new variables
are related hy
[i] = [A] [1'] 6.1
[eq 3 [B][bj] 6.2
, the new impedance matrix [Zflis given by
[2’] = [B] [z] [A] 6.3
So lens as transformation equations (.1 and (.2 are numeri—

cally correspongmnt, [A] an?[3] can be arhitrary. Numerical

2O

corespondences of 0.1 and 6.2 mean that matricesfiflland [S]
are non-singular, as explained in Chapter IV.

Let's show several examples of this case.

 

 

 

 

T Z“ T Z“, Z“. t 2A4. 25" T 2“
Lo- . Lb ie’ 25'
ea. eh ea, eh. Cg, cc
(1) ( 5)
Fag. A. 1

The network of Fir. 6.1 (a) gives the followina matrix

equation.
[e] = [Z] [1]
, where
E6] = at - eD
L. - .J
[1] : 1'~
6.5
is
[Z] = Z... + 2.. z“
2... 255+ 2“

Now if independent variables [1] are changed into [1d,
which are indicated in fig. 6.1 (b), then the transformation

ma trix [C] is

L.)

21

[c]

O
6.6

N
0‘ m
I.
...:
I
H

Here we choose

 

 

[a] = [c] and [B] = [U] 6.7
, where [U] is a unit matrix.
Then
[2'] = [51(2) [A1 = [Z] [c]
= ”2&4- z“ 2.. "'1 0
Am fi*‘+ an .1 -l
= " z» '2...
6.8
L‘Z.5 “(255+ zcc.)
[e'] = [B183] = [e] 6.9
So finally we get as [8']: [Z'Hi']
_ I
ea- - 8‘ 3 La. - Z“ 1“
, 6.10
eb ......c -2... -(z..,+ z“) 1‘
Fro. 6.10 we get
4 e.(z..+ 2..) - e.z.. - ex...
1 z
zaazbb + zbb zen. + Z“. zap
6.11

1 4‘2“; - ebzmi» e¢(ZM+ Z»)

r =

 

z+zbz+zz
5b 5a

cl. 0.0.

22

Now let's use the usual C Z G method. Then

to]. [Z] [c] = 1 -1 2 -Z..

‘4—

0 -1 Jab-(2,32%)

= The 2.. 2..

 

255 255+ Zc

[C]t[e] = . 1 -1 at - e,

= €.' eb
-e,+ s.

{.0 finally we get as [an = [z'] [1']

[e '1

 

1
ea, - ab = Zu‘i' Z.” Z” 1‘
6.12
-eb+ ec Z“, be+ L 1d
From 6.12 we get
' eav( zbb+ zen.) - glazes". 80..be
i‘ a
2.12.“? szufi’ sz‘b
6.13

1Q, .90.sz- ebzmm" oc.( za~+ 26b)
3

 

ZuLZH;+ ZHDZR_+ Zu’fiub
Final results of both 6.11 and 6.13 coincide with each
other as exnected. To reach the final result 6.11, we
used [A]:[C] and [B] = [U] . To reach the final result

6.13, we used(}] = [C] and [B] = [Clt . So in the latter

23

we must handle more operations, such as [CL [2] [C] and

[Gide] . In the former method we can omit these operations,

so it is much easier to reach the final answer. This is I ”
so simple case that difference of amount of work involved
in both methods is not so great. Let's show another more

complicated case.

 

4:3 .
\‘Jbb (a) 133. 6.2

 

 

In fig. 6.2 there are given circuits of Wheatstone's
bridge. At first independent currents are riven as indicated

in fig. (a), and then circuit equations are given as holow.

r “t. ” - 91
a .. z + 2m + z“ -2“ - -2“, F1
3* a 6.14
0 -2“ 2.2+ 2“ + Q. -2” 1"
_o_ L..th .2” 2.. + Z... + 2,5 _1"4

 

 

 

 

 

 

To get the balance condition of Wheatstone's bridge, we nust
get current in impedance Z” . So let's assume 1P , ifiland 1‘,

as new independent currents, as indicated in fig. (b).

24

Then we get tTe transformatijn matrix [C]
f0 3' a!
[C] = g r-0 1 01

a O O 1 6.15

 

 

bL—l o l

.1

Assuming [A3 = [C] and [B] = [U] , we get

 

 

I ’- . ‘1
[z] =[z] [c] = z“, zfi+ z“ + 23‘ -(Z.. + z“)
L-(ZH, + z“ + a) .2“, Zn. + Z“
80 we get new circuit equation as below.
'e,'= '2‘“L Z”+Z¢+Zu 42;,4-52111‘“
o z” -21 2.1+ z“, 13’ 6.17
1.0.. L‘(Z* + Z“ + Z# ) '2“ 2"" + 2‘4. 111

 

 

 

 

 

 

From 6.17 we get
I e (a; Zn.” Zuvz,‘)
f = ’ 6.18
D

 

D = z 2W2“ + 2.12.... 2,3 2.12... 233+ zwzm zw+

zuz“ 2,, + z“ z“ z” + zwzuz”; 2,“,2fl 2” +

zzzu+2k2hzfi+zzz+z22+

ob as up u. 5’. W Mr.”-
2‘2 2 + z‘gzuz + 2. 2,7,2”; zuzfi z”

”19 33 “'

From 6.18 we get the well-known balance condition

Z Z Z Z 6.19

av ii- 55 (c,

Now let's try the usualUCL [Z][C] method.

[2'] = [01 [z] [:1

_

 

 

 

 

 

 

. o o -11 ’ z“ 2,, + zm + z“ 42.. + z-)
1 o 0 2+, -2“, zm+ zm
Lo 1 lj L-(Zu. + Z“ + Z” ) -Za 25.3" Zn 1.
=fi'2‘b+-zu_+ g” Z...L -(ZH,+ 2&1) 1r
Zn: 2.3, + zcc + 2-,, --(zcc + z“) 6.20
-(Zu.*' z“) 4ch + Z“) Zai’ Zn. * 3.. + Zak

[9'] = [(3]t [a] = —O O --l" To; 3 P0 '1
1 o o o a, 6.21
0 1 1 0 0

h— -L _ —1 L- -L

Now we can set up [6'] = [2'] [1']

 

 

 

 

 

 

 

 

 

 

 

 

0-1: 2H: + 241+ 2,1 Z“ —-(be+ z“) W ifl
8.} Z“ 233-+ 22¢ + Z“ '(zcc + zit) 1" 6.22
L0 J L-(bef z“) -(zu + z“) z“. z» 1 Z... + Mir/l

From 6.22 we get the same final results as (.18, which proves
validity «[2] [c] mehtod. [z] [c] method does not involve
operations 6.20 and 6.21, so it is much easier to get final
results.

Sometimes circuit equations are set up, but some

26

terlinals of emf‘s are not accessible, so these emf's can
not be measured. In such a case we must change enf's, which
appear in circuit equations, into other emf's, which are

accessible. Let's Present one example of such a case.

 

 

Pea: 6. .3

For the network of fig. 6.3 circuit equations are given

as below.
=3 at - e 3 Zn I -Z 1"
1’ " 5" b 6.23
3“, a o,L + a, Zu+ Zcc 2.. i

To use 6.23, no load voltages between terminals 1-8 and
A-A' must be measured. But it is possible to measure no
load voltages between A-A' and B—B'. Then voltages must
be transformed into measurable no load voltages between
14' and 843'.

No load voltage between A—A' 9M = a“. + co

6.21.

" B-B' ow: ob + ea

27

The relations between old and new emf's are

e~v = e + e = 0 1 e

a. a a.

-e,= O 1 cm3

shy e + e

b C a.

So transformation matrix [B] is

[B] a O l

6.26
-l 1
Then if assumed [A] = [U],
[2'] = mm = o 1 zu’ —z.,.,
-1 1 Z-“+ Zcc Zcc
= ZML+ %. Zn
6.27
-zct 256 + Zea.

So we get new circuit equation form 6.25 and 6.27.

A.

3.. + e: ZM + Z“ Z“ i
6.28
e. + e -Z.. Z». + 2.. 1"

This equation contains only measurable emf's, so we can
evaluate currents by solving 6.28.
This is the case where [A] is a unit matrix and
[B] is trnasforaation matrix between old and new emf's.
Sometimes we find that a given circuit equation is
not adequate, because some of the emf's used in the equation

is not accessible and cannot be measured, or because we want

28

to use other independent currents than those used in the
given circuit equation. In such a case we must transform
both emf's and currents used into other independent emf‘s
and currents, which are suitable to our purposes. Then
old and new variables lust be related in some way. These
relations can be expressed by

[11 = [11 [M 6.29

[e']= [B3 [6] A 6.30
Generally [A] and [B] are not related by the relation [B]: [LJt
as is the case with congruent transfbrnations. Let's show

one such example-

 

 

 

For the network shown in fig. 6.4 (a) a circuit equa-

tion is set up as below.

29

 

 

 

 

 

 

”a” - e: = ”z“, 4.. o o 1 ’1“
ea - ed. 0 0 Z“ -2“ 1b
' 6.31
ea, z”) Z.” + z” 2“ + Z” Z” Z” i
d.
_3a. A _za% 23%- 223 24d- + 234‘ L1

But four currents, which are indicated in fig 6.4 (b), are
wanted. And four no load voltages, which are used in 6.31,
cannot be measured, but four no-load voltages between 1-2,
4-5, 2-3, and 5-6 can be measured. These four currents,
which are wanted, and four measurable no-lcad voltage are
assumed as new reference variables. Then relations between
old and new currents are,

i°' I i"

1" = 1‘"

1° a .1“ 4'”

1‘ a -1“ -16
So the transformation matrix [A] for currents is,

[A] a F 1 o o 0']
0 l O 0

6.32
-l O -l O

 

 

o -l 0 -lJ

_

Measurable four no—load voltages are,

 

= 6a,...
.. eb -
2-. at
3 ed

Fem - e:]= -- 0
eb - e.i -l
ec_ 0
fat I. L0

 

 

-1

O

.l

O

l

l

O

0

.fi

-1

 

 

l
l

6-33
- e
‘L 6.3!.

 

So the transformation matrix [B] for voltages is,

[a] =

 

1" o .1
-1 o
0 1
L_o o

l

1

0

0

.17
-l
1

l

 

—

6.35

Impedance matrix for new variables is given by [B][Z][KL

that is,

[z] [A] =

 

 

 

o 7 _1
-2... o
2,, -1
zu+ 231 L0

30

005]
100
0.10
.104

 

[B] [Z] [A] =

fl

 

2.1 -25., 0
~an 24¢ -l.._
z“; z” z” -2”
Lo .z‘.‘L -2”
r 2.1 + z“ + 2;}
ZH
-Z..
__0

 

31

 

So the new circuit equation, which is suitable to our purposes,

is given from 6.34 and 6.36 as below.

 

3

 

 

" Z“; 2,“ + Z” 2,,
7m

.2cc o
o -2“

zbb+zd+gf o

Zcc O
z...

-(zcc + 23% ) -283,

.2 .(de+ 233.

a?
6. 37

We can check easily validity of 6.37 by inspecting fie. 6.4

and 6.37.

All exanoles in this article are the case where the

condition [£043] and [B]=[C]t are not satisfied.

0 T
Z...
“23%
-(Zdu- Zeal
z” 2.. o
Zhb+ Zm+ Z” O 244.
o -(zu_ + z ) J”
‘2“. - z” -(ZM+ ..
6.36

 

]

 

 

 

32

Here it should be emphasized that both [A] and [B]
are non-singular and give numerical correspondences between
old and new variables. Sometimes [A] or [B] is a unit
matrix, which is apparently non-singular. Cases, where [A]

or [B] is singular, are treated in the fellowing article.

(2) Cases where transformation matrix [C] is singular.

In Chapter IV we explained that transformations,
which changes number of impedance branches, are generally
accompanied by singular transformation matrices. And
singular matrices relate old and new variables only geo-
metrically but not numerically. In Chapter III we expalained
that power is not invariant even when [CLfZ] [O] transform-

ationis used, if'[C]is singular. Let's show examples.

._, Ly

 

IF—e
F
e

9a.. ec ea-

 

 

l
)<
'23).» 1 Zoe. L2”. 2.5:, ,2 Zoe
ed
(a) (b) i

33

The circuit equation for figure 6.5 (a), which is the same

as figure 4.3, is

e... =3 Z + Z 25', Le"
a bb (1.28
Lo

at 2H, be+ zcc
In figure 6.5 (b) branch of §~is broken, and one new cur-
rent if is enough to describe the circuit behaviors. Then
the correspondence between old and new currents is
lk-d’ it
6.39
1 4 ~31."
This is a geometrical corresuondence but not numerical.
Be it is numerically wrong when we write [i]: [CJIiflas
below.
I
i 1 i1
6.40
if -1 ‘
So it seems wrong to use the C Z 0 method for this
case, because we use numerically wrong equation [1] = [CJ[1q .

But it was already shown in Chapter IV that the [CLIZIIE]

method is valid even for this case. Let's try here once more.

2] [C] = z“; z z 1 = 2...
L "" b” ‘ 6.41

34

[2'] = [chm m = [1 .1] _ z.“ = [hi 2“] 6.1.2
L.zct.
[e'] = [OLE-#1 = [l -1] 9J3 [9..." cc] 6.43
92‘

 

So we get as the new circuit equation,

e - 9., = (Zn-0’ Z“) 11’ 6.11;.

O.

This is evidently correct for the circuit of figure 6.5
(b)-
Next let's try the case where [A] = [C] and [B] is

a unit matrix. From 6.41 we get the result at once. That is,

eUL 3 thvgfij
6.45
ea ~Lm

Expanded we zet,
e 8 QM‘11’
I 6.46
e = -@m 1‘
This is evidently incorrect for the circuit of figure 6.5
(b).
Next let's try the case where [B] = [CLrand [A] is

a unit matrix.

[Olin] = [1 '1] Zao-+ zen, 2'56 3 [2a - Zea]
“ 6.47
z z + 2

H. H, cc

35

From 6.43 and 6.47 we get,

_ L
(e... - g- [2.0. - or] W
That is,
at - eb = an 1L - ZULic 6.49
This result is not wrong, but it is not adequate to solve.
the circuit of figure 6.5 (b), because it contained two
currents for the one-mesh circuit.

Let's examine the case where a 3-meshes circuit is

transformed into a 2~mczice'circuit as shown in figure 6.6.

 

 

I

 

 

X
L am. is ca is da 85 “. 2%, 3; z“; 2“
at a; Eh. 2‘
. f m
(a) Fig, 6. l,

The circuit equation for figure 6.6 (a) is

“a: 3 [2“) z» 7‘5». 2% _ Vivi

ea 21* 255+ a 2H, 1" 6.53

_o _L 1 2“, 2,, 256+ 2,, 3*

 

 

 

 

 

 

36

In figure 6.6 (b) branch of Z is broten and new currents
_ be
are assumed as shown in the figure. Then geoxctrical

corresponcences between old and new currents are given by

’fi='1 0'1 1"

I

1° 0 1 1‘ 6.51

1“ E1 .1

b -

 

 

 

 

Equation (.51 woes not correspond numerically. From 6.51
we get a singular transformation matrix EC] .

[c]='1 07

 

 

 

 

 

o 1 6.52
_-1 —1_
Then
2] [C]_== '2”; z“, z“, z.,., l" 1 o“
z... 2.5+ z“ 7..., o 1
_z... z,,,, . 255+ 24$ _.1 -1_
2 F2“, 0 l
o z“ 6.53
-Z.u .zu_

 

 

[z']=[01t[z] [c] = z“; 2...; 2M

2M zc‘ + 2“

37

 

 

[e'] = cue] = 1 o -1 ”9;: 1.,
- 6.55
O l -1 8L ea
—O-‘
80 we get as [e'] = [Z'] [1']
ea. z“_+ g, z“ 1“-
I I 6056
8c 244. ch + Z“ it

This result is evidently correct for the circuit of figure
6.6 (b).

Next let us examine the case where CK] = [C] and
[B] is a unit matrix. From 6.53 we at once as the new
circuit eduatiou,
"eja ' z“, 0T1“
e. 0 1a 1" 6.5
L9.. L‘ZEL “zed

This result is eviiently incorrect for the circuit of

 

 

 

 

figure 6.6 (b).
Let us examine the case where [B] = [Ck and [A] is

a unit matrix.

38

[0M2]: 1 o .1 _zu+z» 2.. 2.. ‘1
o 1 .1 2., zm+zbb z“,

 

Us 2., z“, + 24

= z“ o .2413
6.58
O Zu_ —Z&L

_

 

From 6.55 and 6.58 we get as the new circuit equation,

 

 

eL = Zap O - Did]
a. o z“ .2“ r’ 6.59
1" ‘
That is,

a z 1" z 10”
e” ”’ ,- ”L 6.60
qL=3 Z“ i‘ - Zagr“
This result is not wrong, but is not adequate to solve
the circuit of figure 6.6 (b), because it contains more
numbers of reference currents than numbers of independent
meshes.
From two examples above we lay conclude for the
case of singular transformation matrix DC] as following.
(i) The method of '[e'] 8 [01b] , [i] = [C] [1']

and [z']=[c1.[Z] [c] . is valid.

39

(ii) The method of '[e']=[e'l, [i]=[C] [i'} and
EZf] = [Z][C]' is wrong.

(iii) The method of '[e'] 2 [C1,[e] , [1'] =[i] and
[2f]: [CL[Z] ' is not wrong, but is not adequate
for solving the problems, because it contains

more currents than numbers of independent meshes.

It seems rather stranse to reach the right results
of (i) and. (iii), even when we use the numerically wrong
relation [1] = [C] [i'] . Let us consider the reason. for it.

In the method (iii) we use [o']=[C]t[e], but do not use
[i] = [C] [1'] . And it is easily understood from the two
examples above that [e'1=[C]¢[e] is numerically correct,
while [i]: [C] [i'l is numerically wrong. In figure 6.5

(b) the new mesh emf' e' is

e' a eL- eL 6.61
, while
[Clerc] =8 [1 ~12] e = [9... — ea] 6.62
a .

From 6.61 and 6.62 we see that [e'] = [Ck [e] is numerically
correct. finother thing, which should be emphasized, is the

fact that all emf's, which appear in matrix equations, are

AO

mesh—emf's. Hesh-emf's mean that they are emf's which
act around closed meshes. Then we choose new emf's for
transformed circuits, we must choose mssh—emf's. The
transformation [e']=[C]t[e] gives automatically mesh-emf's,
as new emf‘s [e’] , because [B] = [Ch is a transformation

*
matrix between old and new mesh—currents . And even when

 

‘* Here it should be stated that branch—currents can

be considered as mesh-currents, so long as these branch-

(8)

currents are independent.

 

[11 = [C] [i'] is correspondent only geometrically, Le') =[Clt [e]
is correspondent geometrically and numerically. This is‘
the reason why method (iii) gives correct results, while
method (ii) gives wrong results.

When we applied method (ii) to figure 6.5, we got
6.45, which is wrong. In 6.45 emf's are not mesh-emf's,
but branch-emf's or open emf's. So it is impossible for
them to give correct circuit equations, which are actually

the Second Kirchhoff's law.

Pre-multiplication of [231:0] by [Clt is equivalent
to transforming of open emf's into mesh-emfs, and it sets
up the Second Kirchhoff's Law correctly. So method (1)
gives correct results.

It is not necessary that [A] = [C] and [B];[Clt
are satisfied, to reach correct results. For instance,
in method (iii) [Blsfclt and [A] is a unit matrix. fie can
use any transfbrmation matrices [A] and [B] , so long as
[i] = [A][iT] gives geometrical correspondences between
old and new independent currents and [eTl=[B][e] gives
geometrical and numerical correspondences between new and
old independent mesh-emf's. Let us apply the case where
[B] 75 ”[A]{ to figure 6.6. We assume that new currents are
the same as shown in figure 6.6 (b), but new emf's are
those acting meshes of 2...," Z“ and Z“: d, . Then

[9'] =[B] [e] is

ev-ecz 1 -1 0 To:
em - O
l O -_ eg 6.63
l—no-J

 

 

[Zlfl] = [Z][C] is given by 6.53. Then

[z']=[31[21m= 1 -1 o z”, 07'

1 o .1 o zcc

 

3 zeal. .ZCL

6 0 6'4
ZW+ 24¢ de

From 6.63 and 6.64 we get as the new circuit equation,

I

3., " e. 3 Zea. .2“; 1L
, 6. 65
ea, Zola—‘- zdd. Zr“. 1‘.

It is easily seen from figure 6.6 (b) that 6.65 is correct.
Here it shall be mentioned that we must be carerl in setting
up [e'] = [B][e] as shown in 6.63. That is, when there is

no emf in some branches, some simbol (here we use 0) must

be substituted to get the right [3),

'Ve

Now we can conclude about transformations by singu—
lar matrices es felling.
(i) The method of n [e'] = [c149] ,[i] = [01(1']
and [2']: [C]J2][Dl “ can be used correctly.
(ii) The method of ' [cf] = [Blfe] , [i] = [A][ii]
and [2C] 2 [B][Z][A] " can be used correctly, so

lonfi as all emf's and currents assumed are indecen-

 

43

*
dent mesh—emf‘s and infependont mesh-currents.

(iii) The method of " [1] = [C][1€l and [2f] = [Z][C]”
is not correct.

(1v) In all cases powers :r: not invariant.

 

* Breech-currents can he considerei as mosh—cur—

(8)

rents, so long as these branch—currents are indenenflent.

 

Chapter VII Transformations of Invariant Input-

Impedance.

So far "Invariance of Power" has meant that not
only input-power at terminals but aISo all powers consumed
or stored in all imoedances are invariant. But at some
practically very important aoplications of circuit trans-
formations only input-nower at terminals and consequently
input-impedance are kept invariant. Transformations between
equivalent circuits are the case. So far transformation
matrices contain only integers as their elements, mostlyizl
or 0. But transformation matrices of invariant input—impe-
dance can have non-integer elements as shown later.

Here it is not attempted to discuss whole asnects
of transformations of invariant input-impedance, but one
interesting example is to be explained; that is a new method
of reduction of networks to their Foster's Forms by means of
matrices.

As Guillemin explained very nicely in his book, any
dissipationless network can be reduced to four equivalent

canonic forms, which have least number of elements. Two of

 

45

them are Foster's forms and other two are Cauer's forms.
Reductions to Cauer's forms are easier than to Foster's
forms, because amdular determinant (or its minor) must
be solved, to reduce to Foster's forms, but it is not
necessary for Cauer's forms.

Here let us try other approach to Foster's forms.

69%: fl
Ts” T

¢ -“n

Fia.‘1 1

 

 

 

Figre 7.1 shows one of Foster's forms. Independent
currents are aSSumed as shown in the figure. Then the

circuit equation is,

"' ' '1
«fl: L.A+S——J o .o W 1"
e’ o LzA-‘I'LSXz' - -. - o 11’
7.1
I 5.. u,
Led L o o - - IAN-71E

 

 

 

 

 

 

.4. ..L...
, where k‘dt’ A-j alt.

46

This equation has a diagonal or normal forms. So the

inductance matrix [L] is,

 

 

[L]="L, o o-H-o
0 L1 0.00.0
7.2
O O L300000
L-—O O OOOOOL’L
The susceptance matrix [S] is,
[51:75. 0 o-°--o1
o s, o----o
7.3

o o 83 O O O O o

 

 

o o 0°°°°s

I.— "z...
7.2 and 7.3 have diagonal or normal forms.

VJ 171_
' I I '
I I '

'I

 

Fka.‘7.2,

Figure 7.2 shows the other form of Foster's equivalent

circuits. Independent junction voltages are assumed.as

47

shown in the figure. Then the circuit equation is,

 

 

rI": rogue-r} o - ~ - - o T W,"
I 0 qu+§é . . - 0 V; 7.4
_I: _ o o ' ° ' QA-i'ffi VJJ

 

 

 

 

This circuit equation has diagonal or normal admittance

matrix. Then the capacitance matrix [C] is,

 

 

[c]=’c, o o-°-ol
0 C 0 ° ° ' O
" 7.5
0 O C;"° ° 0
o o 0... on
The reciprocal inductance matrix fl_] is,
[HAVE 0 0- - 'ol
0 r2 0. o .0
7.6

O 0 ,30000

_0 o 0' - Tn,

 

 

7.5 and 7.6 have diagonal or normal ferms, too.

A8

?hen a dissinetionless network is aiven, we Can
easily write down its matrices of [L] and [S] by means
of mesh method, and its matrices of [C] and U") by means
of node method. And if we can diagonelize these matrices
in such a way as transformed circuit equations have the
forms of 7.1 or 7.4, then we can write down its Foster's
equivalent circuit as shown above.

Let us assume that a given two-terminal network
has following [L] and [S] .
[L] 2 FL” er' . . . L’J'

Luv in ' ° ° Lzu

 

 

7.7

LL“. I'M; o o 0 LI":

[8) = ’5” sn- - - - s":
82' S21. 0 o 0 82m 7.8

 

 

H1 tum—j

Then the next step is to find such a transformation
matrix [C] as
[CLIL] [C] gives a diagonal form.

flfltEflID] gives a diagonal form.

49

Now we can see that the problem is simultaneous reduc-
tion of two matrices [L] and [S] to their diesenal forms.
Guillemin treats very nicely simultaneous diaconali7a-
tion of two matrices in his book.(10) But we can not
apply to this case directly what is exnlained in his ‘ ck.
Let us explain necessary procedures.
Using [L] and [S] of 7.7 and 7.8,

\XU-l + [s] I = o 7.9
has generally n roots of7\.
7.9 can be exoanded as below.
AL” + S" AL,1+ 8,1. 0 0 0 KLmi- 8,,L = O
KL1,+ St, AL,,_+ S2,: ' ° ' ALm-O- Sm: 7.10

5
O O I O O O O O O C O O O O O O

KLM + SM KL)..." 81:; . . . Aland" Sun

 

 

And 7.10 is the determinant of n set of homogeneous
equations below.
(RLH + §)x'+ (Mu-Ir 8,1)11‘0- . . + (ALmi- SJ)“: 0

(KI.fl + g')x,+ (AL,,+ 5,933+ . . + 0ng 5.,ng 0 7 11

(M... + ax.“ N.» sax» - . +01...» sex: 0

50

7.11 has solutions when 7.? and 7.10 are satisfied.

So 7.11 has n sets of solutions for n roots ofik., which
satisfy 7.9 and 7.10. But these n sets of solutions

for 7.11 are not unique, but only proportionalities
among each components 1,, x1, 33.... are determined.

If cofactors of the determinant 7.10 for the sth root
of K's are denoted by gig , then one set of solution is

given by s

Keg

£5: ,_
4 fl.) + (Huh - - -+ (no?

(for k=l,2, n)

7.12

 

, where the index i is arbitrary but must, of course,
be the same for all sets of.l's.
Then

55:4; = a!“ (for k = l, 2, n) 7.13
satisfy 7.11, where 95 is a arbitrary constant. 7.11

can be expressed by using matrix as below.

(MI-l +[S] )-[x]= 0 7.14
, where [x] =‘i,”h
x1
J;E.

 

 

51

Substituting 7.13 into 7.14, we get
Amfips g]: —[.81[(p5125)] 7.15
There are n equations 0? this form for nlis. And these
can be combined into one matrix ecuation by defining
a new matrix, which is called a modal matrix. The modal
matrix for this case is,
(«$1, 2,; - - mi

El 211. . . . [In
7.16

 

 

1;, 2h} . . . ILnJ

L

,where the first column elements are given by 7.12 for
s = 1. Then n equations of the form of 7.15 can be
combined into

[L1 [men/d = - [81(26le 7.17
, where [A] is the diagonal matrix with n latent roots
of}\ls diagonal elements, and [p] is the diagonal matrix
with arbitrary constnat g, g,....ntas diagonal elements.
7.17 is pre—multiplied on both sides by transpose of
[£1th . then

[DELHI-l [£1 [pH/U: _[flt[z1t[5][£][p] 7,13
In 7.18 [flt[x]t[L][£][p] and [p],[£]t[S][:f/] [p] are symmetri-

cal, because [L].and [S] are symmetrical. But the left-

52

hand side of 7.18 is not symmetrical due to post-mul-
tiplication by DA], which is a diagonal matrix, if
[fikECLIL][Kfl[fi] is not a diagonal matrix. So only one
possibility is that both [ply]. [1.105103] and

[pltmltlbj [IHp] must be diagonal matrix(li) in order for

7.18 to be correct. That is

EPMJCML] [1:] [p] = [In] 7.19
[pltfiltEE-Hflm = [Dz] 7,20

, where [DJ and [D2] are diagonal matrices.
From 7.19 and 7.20 we know that

[c] = [:6] [p] 7.21

is the matrix which diagonalizes both [L1 and [S] simul-
aneously. But [C] is not unique, because it contains
the diagonal matrix [Q] , which has arbitrary diagonal
elements.

So the next problem is to choose from 7.21 the
proper matrix [I] , which keeps the input—impedance
invariable. This proper diagonalizing matrix be desig-
nated by [F]. Then the relation between old current [1]
and new current [1'] is given by

[11 = [F1 [1'] 7.22

53

It is easily seen from figure 7.1 that the following
relation must exist between old and new currents for
invariance of inout-imoedcnce. That is

i =i"+1”+- . - +1"! 7.23
,where i is the input-current of the original network,
which is counted as the first mesh—current.
From 7.22 and 7.23 we can see that [F] must have the
form as below.

[F]: "1 1.... 1'7
z” sh: . . . 13‘

7.24

 

 

f“ f“. . . . 'fimJ

_

For [C] given by 7.21 to be identical to [F] given by

7.24, [p] must be the diagonal matrix as below.
[p1="'1/z.,o-----o ‘

O 1/4212. o o o o O

ooooooooo 7.25

 

 

[F] =[x] [P]

=

r12., 2,; ' - 2,:
[1, 21‘. o 0 an

 

 

P

_

 

LE,” (12,. o o ’[nu-J

‘&%Zr'éyzz° ° 66%62

EH/III [flu '

1 1. . . . 1 T

MIL-d

 

 

h

r-llg

.‘ II

o 1/,g,z...o

0.00-01/fimJ

From 7.12 we get the following relation.

As/«Exs 3

5

t/ x-

(fl—

0 . . . 0

54

 

7.26

7.27

s
, where Kiais cofactor of the determinant 7.10 for the

8th root of

[F] =
I I 2. z n. n-
K-u/ Kit KQ/ Kc; . ' xiz/ Kt/

nK's. So [F] can be expressed as below.

1 l

 

{1&1 ii, Ew/ ii,

1

it"! i},

1'

 

db

7.28

7.26 or 7.28 gives the transformation matrix [F] , which

reduces [L] and [B] of a given network to their diagonal

forms and keeos the inputcimoedemce invariant.

55

Here it should be explained that the new voltage
matrix obtained after the transformation by [F] has the
form of the left—hand side of 7.1. The given network
is a two-terminal passive network, so its voltage matrix
has the form as below.

[e]t= [e o o - - - 0] 7.2.9
Then the new voltage [e'] is
[e'] -.-. [rue]: F. l

O
u

7.30

 

 

M
So we can see that the transformed circuit equation
has the form of 7.1,whioh is the circuit equation of
one of Foster's equivalent circuits. Now we can say
that we have reduced the given network to one of its
Foster's equivalent circuits. .28 is the better formula
of the transformation matrix than 7.26, because calcu-
lation of 7.12 is not necessary for 7.28.

Let us derive here some very useful relations.

Putting 7.19 and 7.20 into 7.18, we get

[DJ [A] = - [Dz] 7.31

56

Three matrices containtd in 7.71 are diaeonal. So we
get from 7.31,

1.

..J

I -.2 ... .
where c,sano assere diagonal elements 0? [JJ_nnf [Li]

.I sth latent root of ecnation 7.10.

(‘0
...1
LL
u?
I J.

.31 or 7.32: we can derive [73,] or [£32] from each.

It is easily seen from 7.32 that A‘must be neevtive
or zero, because dgsand d; must be both positivgt for
the equivalent network to be physically realizable.
That Atare negative or zero was anticipated from the

(12‘)
fact that [L] and [S] are both positive definite.

We started from equation 7.9, where A.is atta-
ched to [L] . If [L] is singular, some of K’s are
infinity. But.Ns must be finite. In this case A~must
be attached to [5] instead of [L] ; that is

“Ll +A[s]} a o 7.73
At least either of [L] or [S] is non-singular, because
these are matrices for solvine electric networks.
According to whic. of [L] or [S] is non-singular, we
choose 7.9 or 7.93. Otherwise the procedures remain

*
Including zero.

57

the same as shove.

All procedures 0? reducing a given network to
its Foster's equivalent form shown in figure 7.2 are
the same as exuleined so far in this chapter. But in
this case the original matrix equation of a riven net-
work must be written by the node method, using capaci-
tances and reciprocal inductances. And capacitance
matrix [C] and reciprocal inductance matrix [Tflare
diefionelized simultaneously. That is 7.9 and 7.33 is
substituted by

me] + [r1‘ = 0 or Ho] +Mr1|= o 7.31.
, according to which 0? [C] or Uilis non-singular.

n3 tnc Same. 7.24 remains t3?

*1.

The form oi 7.12 rams
same, because in this case 7.22 3 replaced by
,I r I
V ==V,+V2+~--+V,, 7.35
h 4‘

So we can use the same 7.;6 or 7..8 ,or this case, using

n roots of A.for 7.34.

 

 

Example.
Wt LL21 SLr-l
Lb Li Lb=2 sb=2
fiL 55 Ln = 3
o—e -I- LL 3 4

Let us reduce the network of figure 7.3 into the form

cf resonant components in parallel.

[Z] = L&_+ Ls --L‘,

[s] ’s

M

 

|>\[L] + [3| =

 

Solving 7.38 we get

}\= 0,

A2: ..

Cofoctors of 7.38 for

K" =£
2,
K s 1.32

1

KIZ

.2

o“ : _ 3 -2
Lb + LC 0 -2 5
0 LJ L_o o
0']: ' 1 -1 0—1
+ s, '5. -1 3 -2
ss_ _0 -2 2
3 + 1 -2 — 1 o
-2 - 1 5 + 3 -2
o -3: 4 +
1.068, A3: -0341

 

 

 

 

these his are

8.)

3
ii, = -3175 Km: 0.232

= 2

’3 - -Z’O272

is: 0.636

Substituting these values into 7.28, we get

[1’]: 1

1

L1

 

l

-1

1
-0.0637

-0020

 

‘

This is the transformation matrix.

 

 

07'

O

 

7.36

7.38

7.39

7.40

7.41

 

59

 

Then
[u] = [runs] = ’m 0 0 l
0 26.21 0 7.42
_0 0 3.432]
Lni
[5 '1 = [Flt] B‘] = "o o o 7
0 .9.3 0 7.43
_P O 1.149d

 

From 7.42 and 7.43 we can write at

 

once the Foster's

equivalent form es shown in figure

L' =
a: s; z
. ‘ T _T L.

J

 

II

 

Fig.7.“-

7.4.

7.0 s; = 0
26.21 s; = 23.0
3.435 s; = 1.169

Using equations 7.31 or 7.32, 7.43 can be derived from

7.39 and 7.42 as below.

 

 

[s'] a - "9.0 0 0 _i
0 26.21 0
L0 0 3.435]
= "o 0 o "
o 28.0 o
L o 0 1 .161

 

 

This is the same as 7.43.

T—o

 

O

O

 

O
O
-O.341
..JL
7.44

(1)

(3)
(A)
(5)
(6)
(7)

(8)
(9)
(10)
(11)
(12)

(13)

60

References

Kron: Tensor Analysis of Networks

Kron: Short Course in Tensor Analysis

Slepian: A. I. E. E. , Vol. 56, P. 617 (1937 May)

Bocher: Introéuction to Richer filgebra

Guillemin: The Hathematics of Circuit Analysis

Guillvmin: Communication Networks, Vol. II, Chap. VI

Pipe: "Transformation Theory of General Static

Polyphaae Networks", Trans. of A. I. E. B. Vol. 59

Kron: Tensor Analysis of Networks, p. 152

Guilleminz Communication Betworks, Vol. II, Chap, VI

Guillenin: The Hathematica of Circuit Analysis, 9. 156
' ' p. 163
' “ p. 158

' Communication [ctworks, Vol. II, Chap. V