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APPLICATEON OF FOSTER'S REACTANCE
THEOREM TO THE DESKSN QF ELECTRIC
WAVE HL'E'ERS

Thesis for tho Dograo of M. S.
MICHEGAN STATE COLLEGE
Fuad Labib Abboud

'ft‘rE/E
37.1%?

IHFSIS

This is to certify that the

thesis entitled

"Application of Foster's ?eactance Theorem to the
Desigv of Electric Wave Filters"

presented by
Fuefi Lébib Abboud

has been accepted towards fulfillment
of the requirements for

_l.'~_i..§._ degree in J21...—

s ./ A1313. a»;

J Major professor] /

 

Date 5115319)“ LL. 3-954

 

APPLICATION OF FOSTER'S REACTANCE THEOREM
TO THE DESIGN OF ELECTRIC WAVE FILTERS

By
Fald Labib Abboud

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
1954

9’27—54'

TABLE OF CONTENTS

IntrOduCtj-on O O C O O O O O O O O O O O O O O O O O 1

CHAPTER I : Foster's Reactance Theorem

1.1 A Statement of the Theorem . . . . . . . . . . . . 2
1.2 Discussion . . . . . . . . . . . . . . . . . . . 2
1.3 Example . . . 4
1.4 An Abbreviated Method for Determining the Analytic

Form of the Reactance Function for a given Non-

Dissipative Two-Terminal Network . . . . . 12
1.5 Forms of the Possible Driving-Point Reactance.

Function According to Foster . . . . . . . . . . . 19

CHAPTER II : Electric Filters

2.1, Fundamental Behavior of Filters . . . . . . . . . 22
20 2 CODStant-K Filters 0 o e e - e o o o o e o e o 26
2.3 Determination of Attenuation and Transmission

Regions 0 O O O O O I O O O O O O O O O O O O O O 27
2.4 Design Procedure . . . . . . . . . . . . . . . . . 32
2 o 5 Exmples o e o e o o o o o e e e o e o e e o o o o 34
2 o 6 The Ill-Derived T Section 0 o o o o o e e o e o o o 44
2.7 Termination with.m-Derived Half-Sections . . . . . 51
2 O 8 Example 0 O O O O I 0 O O O O O O O O O O O O O 0 54
2 O9 conc1u81on O O O O O O O O O O O O O O O O O O 0 O 59

Bibliography 62

338677

PREFACE

This thesis is intended to discuss and illustrate the
extent of application of Foster's reactance theorem in the
design of electric wave filters, and to investigate an abb-
reviated method for determining the analytic form.of the
reactance function for a given non-dissipative two-terminal
network which is discussed in volumes I and II of the book
" Communication Networks " by Mr. Ernst A. Guillemin.

The writer illustrates in this thesis that while this
abbreviated method applies well to many networks, it fails
to apply in its present form to many others. Therefore,
the writer has set a rigid procedure for the application of

the above method to any two-terminal network.

The writer wishes to extend his thanks and express his
appreciation to Dr. Joseph A. Strelzoff for his instructions
and suggestions during this year of advanced study which
made the development of this thesis possible.

INTRODUCTION

Filters in general are designed as non-dissipative
structures. The presence of resistance in the actual phy-
sical filters only modifies the behavior predicted on a
non-dissipative basis a little, hence the theory and design
of such four terminal networks is based upon the behavior
of structures with negligible or no ohmic resistance and
therefore, this discussion will take up only purely reactive

networks.

CHAPTER I
FOSTER'S REACTANCE THEOREM
1.1 A Statement of the Theorem

The most general driving-point impedance ( two-termi—
nal impedance ) Z (HO, obtainable by means of a finite res-
istanceless network is a pure reactance which is an odd ra-
tional function of the frequency and which is completely
determined, except for a constant factor B, by assigning
the resonant and anti-resonant frequencies, subject to the
condition that they alternate and include both zero and
infinity. Any such impedance may be physically constructed
either by combining, in parallel, resonant circuits having
impedances of the form [ in / ( iCp i‘:l, or by combining,

in series, anti-resonant circuits having impedances of the
-t
form [10p / ( in )] .

1. 2 Discussion

For a network driven from the first mesh, the general

form of the driving-point impedance function is

(w‘- wl' )(u‘w: ) . . - - (“"“:"" ) (1)

 

Z W 2 H x
H ( ) J“ “‘(u‘- “I )(N‘ ‘Ut)ooeo(w“ “filth-1)

which can also be written as
2W) : _.._....... (2)

in which D (u) is the determinant of the network and B “(00)
is its minor after eliminating the first row and the first

column.

The zeros of the driving—point impedance which locate
the resonant frequencies are determined by the roots of
D (so : O, and the poles which locate the anti-resonant

frequencies are determined by the roots of 8,,(u0 : 0.

Fbr any possible combination of inductors and capaci-
tors to be realized physically in the form of a driving-
point impedance, it is necessary that the slope of the imp-

edance function versus frequency be everywhere positive i.e.,

d Z,|(u)
Jd‘”

>0 for _.o<w<ao (3)

which requires that the poles and zeros alternate as expre-

ssed by

o=w.<u|<u1“"<wzn_1(Uzn_'< M (4)

l .3 Example

The following example will illustrate the reactance

theorem.

Let us find the impedance function Z "(02) of Fig. 1-1

and determine the equivalent network according to Foster's

 

 

   

theorem.
L1. '0’ ‘1
or
m
C
th(‘“’)"’ L: .41: i Lc‘
I. g T IOM;

loath 1; )

Lzfi '“1 £3

 

 

 

 

[2”;

The mesh equations for Fig. 1-1 are

 

 

 

 

E: JuL|I‘- JuL.I1 / O
1.
o : -3 L I x J (L {L ) / ,____ I - I
“I l u | 3 J” 003 1 3&5. 3
{bizc,l
1 ' 1 '
o: o - I2 / JuL./ I3
cht Jw(C'C7' )
(C./C. )
L J

 

The driving-point impedance for Fig. 1-1 can now be

 

 

 

 

 

 

 

written
JuL. -JuL. °
-ij JuLal 1 - 1
0 - 1
D0» JuC. {JuL /
2“ (w):B 2 Cb
"M JuLg,‘ 1 - 1
“Ca. J”CI
- 1 JuLt/ l
JHC' JNCL
where L“, L‘/ L
C4 0:0; and
CL C
0: C1.
Expanding the two determinents we have
P
u“(LzL‘-LL )-JEL:AI_._9 -34.; / 1 '32;
Z"(N) :JuL, C CB ‘ Cacb Ct
u
u(LL ) nag-i gal {__1___ r -l
1. H " cc:b c.‘

 

Substituting the values from Fig. 1-1 and simplifying

gives

Z..(“) = J

N

—

u.“ “.1“ 25 “1441s L2 x1601
u“. w(1..l25 x10 ) / (1. 67x10“)

L-

1
36

 

 

d

 

 

The roots of the numerator are

 

z 7 / 1 2 Ie
0* = (4.25 x10 ) év (4.25 x10 ) - 4(2 x10)

2

 

6 '1
3 5.4)!10 and 3.71X10

The roots of the denominator are

 

7 /
(4.125x10 ) 5V (4.125 x13 )1- 4(1.67x10“)
2

E

 

 

6 7
4.51.):10 and 3.67): 10

and the final form of the driving-point impedance for

 

(- 7
zuo») = 10,31 Luz-5.1;: 10 Lou‘- 3:711:10 1]
0 («£4.54 at 10‘)(w‘-3.67x 10")

This shows that the resonant frequencies as character-

ized by the zeros of the system occur at

fl: 0 ; f3: 369 cps ; f3: 969 cps

while the anti-resonant frequencies as characterized by the

poles of the system occur at

£3339Cps; £3961cps; f=°°
1 l. 5

The reactance function having these properties is ill-
ustrated in Fig. 1-2
2(0)) ‘1

I
l
l
I
a

A

W

“L:

E
c

U... “a

 

-— --d-.———---

I
l
l
I
f

Now let us determine the equivalent network that will
have the same driving-point impedance.

The driving-point impedance for this example was found
to have the following form

Z"(u)=JwH (3-3,) («J—J.) ~ (5)

(wk wt) (3-..?)

 

=J~H % 513313 [mt/ub-(uflubl #1 u H“): «03- ~31)

(u‘- J.) («f-wt)

7- 5
=3... 1 1 % “ (41.24 - 42.5)10 A20 - 1g.€:7)1o”
35 (w‘- 4.54x10‘)(u‘- 3.67 1101)

By partial fraction expansion this becomes

Z"(W):ju 1 1 I‘mA‘ )‘______A________1
35 l u‘- 1.54mi w‘- 3.67x10"

Equating the last two equations and solving for Al and
A2 gives

A1 = -8.6x105 ; A2 = -3.9 X105

and finally

 

 

5 5

I 2
N -4.54 no6 w -3.67 no"

The last equation shows that Foster's network is com-
posed of three branches in series, the first is an inductor

whose reactance is

1
Jué'd

the second is a parallel branch of L and C whose reactance is

-3“, 1 3.9 x105

 

7.

and the third again is an L C parallel branch whose reacta-

nce is

5
“dbl—J; . 3.6310

60 u‘ -/..54 x 106

 

The form of Foster's equivalent network as determined

by the partial fraction expansion is shewn in Fig. 1-3

 

 

 

 

 

L1 L4,
L0
C1 Cl.
t zll(u) ‘
Fig. 1-3

The individual elements are determined from the follow-

ing formulas which can be easily derived

 

LO=H
=—3"’Y11 k=024
Ck [mg “:03“ ”’
11;: .1.__
In: (:3

The following conclusions may be drawn from the above
example, for proofs, the reader is referred to the refer-
ences listed under bibliography :

a. From inspection of Fig. 1-1, it is seen to be con-
structed of six elements while the equivalent net-
work, Fig. 1-3, involved five elements. Similar
examples will show that the networks obtained by
Foster's method always contain the least number of
elements by means of which any driving-point imped-

ance can be realized.

Co

10

It is seen from equation (5) that the degree of the
determinental equation is five, which is equal to
the number of elements in the equivalent network
of Fig. 1-3.

The least number of elements by means of which any
two-terminal, linear, reactive network may be rea-
lized, and therefore the degree of the determinen-
tal equation of this equivalent network cannot
exceed 2n where n is the number of meshes in the
network.

The least number of elements by means of which a
given driving—point impedance may be realized equals

one more than the sum of its internal zeros and poles.

In the light of what paragraph (d) above suggests, it

is interesting to point out the following :

1.

In chapter two, article eleven, page 66 of his book
"Networks Lines and Fields", Prentice-Hall, Inc.,
1949, Mr. John D. Ryder states :

"In general, a network may have a total of resonant
and anti-resonant points not exceeding its number
of meshes plus one".

In section three, paragraph 21, page 201 of the
Radio Engineers' Handbook, first edition, ninth
impression, McGraw-Hill Book Company, Inc., 1943,

Mr. Frederick Emmons Terman makes the following

statement :

11

"The sum of the number of poles and number of zeros
is one less than the number of independent meshes
of the network".
We notice immediately that the above two statements are
not in line with each other, and in addition, neither of them
agrees with condition (d) above. That the statements (1)

and (2) above are incorrect may be seen from the following

 

 

illustration.
20») 2(a)
(a) (b)
7.0») o

I
.l
I

.,\
i

————d-—---

 

(C)
Fig. 1-4

Without resorting to mathematical analysis it can be
seen that the network (a) above can be realized by network

(b) according to the reactance theorem and that both have

12

a reactance function as illustrated by (c), Fig. 1-4. While
the network (a) is made of four meshes, we find from (c) that
the reactance function has a total of eight zeros and poles
or six internal zeros and poles, neither of which corresponds
to either of the statements in (l) or (2) while condition

(d) holds as stated.

1.4 An Abbreviated Process fppgDetermining the Analytic Form
of the Reactance Function for a Given Non-dissipative Twp;

terminalryetwork;

The determination of the equivalent network by Foster's
method in the above example was relatively simple. Had the
network involved five or more meshes the above method would
have been rather lengthy and laborious.

Recalling that the least number of elements by means of
which any given driving-point impedence function may be real-
ized is equal to the degree of the determinental equation of
the equivalent network, it seems that if we can find a simple
method by means of which we can determine the degree of the
determinental equation of any driving-point impedance, then
we can determine Foster's equivalent network without much
effort.

In section four, chapter five of his book "Communication
Networks", volume two, Mr. Ernst A. Guillemin gives such an

illustration in which Mr. Guillemin refers to a method outlined

13

in section seven, chapter five of volume one for the deter-
mination of the degree of the determinental equation of any
network. Mr. Guillemin then explains that the roots of the
determinental equation are equal to the number of the internal
zeros and poles plus one, and this number, he explains in sec-
tion five, chapter five, volume two, is equal to the least
number of elements by means of which any given driving-point
reactance function may be realized. Finally, in section
four, chapter five, volume two, Mr. Guillemin states that
this method of setting up the reactance function is applic-
able to any case.

To show whether or not this method is applicable to any
case, the example given in section seven, chapter five of
volume one is repeated here for illustration.

Consider the network shewn in Fig. 1—5

 

 

 

 

AF

 

 

 

 

14

We begin by following the contour of mesh number one. If
it contains an inductance and a capacitance besides resistance
we weight it two. If it contains only inductance or only
capacitance besides resistance, we weight it one. If it con—
tains only resistance, we weight it zero. If the contour
contains several inductances or capacitances besides resistance
the weight is also two except that in this case only one
inductance and one capacitance need be checked off in order
to give the mesh that weight.

In the same way we continue with the other meshes, bearing
in mind that a coil or condenser that has already been check-
ed off for a previous mesh does not count. Also, the prese-
nce of resistance is not essential unless the mesh contains
only capacitance. Having thus weighted all meshes, the total
number of modes ( the degree of the determinental equation )
equals the sum of all the weights.

The total weights for Fig. 1-5 add up to nine as shewn

below :

Mesh f Weight

0&1!wa

7
Total weights

\OIN H l-‘ O H N N

15

It would seem for an instant that we can apply this
method indiscriminately to any two terminal network in order
to obtain Foster's equivalent network. The following two
examples will show that this is not the case.

Let us analyze Fig. 1—6

 

 

LI 2 L2
2040) l 3 L3
l C:
I 2 f .3
Fig. 1-6

It is readily seen that the total weights for the above
network add up to four.

The mesh equations for the above network are

E = Jlell - JuLlIz o

O
H

- I L L 1 - L 1
J~L11#[Jw<1%2)/ Wk [3w #3752113

0
II

o - [J “2,13%62] 12 ,t [10.)(L2/L3) ,4 $2] I,

The driving-point impedance can be written as follows

 

 

ijl -j le O

4le JN(L1%L2) / __1___ «my __1___.)

30002 JinCz

o -(JwL2/ 1... ) wag/L3); 1
_ wC2 JWCZ
2,.GA)I-

04 L [L )/ l - wL / 1

~(JNL2/ __1___) swag/L3H __1__

 

 

Expanding the two determinents and taking into account

all the initial conditions assigned to the network we have

- " 2L L

hence the determinental equation is of the fourth degree,
which corresponds also to the total weights of the network,

and therefore may deduce that the least number of elements by
means of which the network in Fig. 1-6 may be realized is eqyal
to four. If we attempt a partial fraction expansion of 211(0))
however, we obtain the following

 

Z (w) : JNL l I _f A
11 1 E “a,” 117143 3
C21-

where L 3 L1L2 f L1L3 / L2L3

17

The above form of the driving-point reactance function
shows that Foster's equivalent network is composed of two
branches in series, the first is an inductor, the second is an
inductor and a capacitor in parallel making a total of only
three elements and hence there is a discrepency.

As another example consider Fig. 1-7

 

 

 

In the above network the total weights add up to seven,
but upon expanding the driving-point impedance function by
partial fractions we will find that the equivalent network
according to Foster is composed of only six elements.

Many other examples will show that while it is true that
the number of modes (the degree of the determinental equation)
of the equivalent network obtained by Foster's method is equal
to the number of elements in that equivalent network, it is not
always correct to assume that the least number of elements by
means of which any two terminal network.may be realized is equal

to the degree of the determinental equation of the original

18

network. Yet the method outlined earlier for determining
the degree of the determinental equation could be applied to
find the least number of elements by means of which any given
two-terminal network may be realized according to Foster, if
we follow this line of reasoning.

For a network having at least one independent inductor
and one independent capacitor in each of its meshes, the total
number of weights in that network is equal to the degree of
the determinental equation and hence to the least number of
elements in the equivalent network, and this number is equal
to Zn, where n is the number of independent meshes. This
condition is rather too narrow. In order to apply the method
outlined earlier to any network, we should follow these steps:

1. Disregard any mesh that contains only inductors or
only capacitors around its contour except the mesh
from which the network is driven.

2. Number the remaining meshes each of which (except
the first) should have at least one capacitor and
one inductor (not necessarily independent) around
its contour.

3. Count the total number of weights of the numbered
meshes. If this number is even, then the elements
are made of equal number of inductors and capacitors.
If this number is odd, then the number of inductors
should equal to one more or one less than the num-

ber of capacitors. Any discrepency is discarded.-

19

1,5 Forms of the Possible Driving-Point Reactance Function
According t9 Foster

The four different forms that the driving-point impedance
can assume are given below :
a. If the network offers zero impedance at both zero and
infinite frequencies, the total number of weights is
an even number composed of equal number of inductors

and capacitors as shewn in Fig. 1-8

2(uq
0

I

I

I

l

l
T

21(u9 “H

E
c

 

 

 

———-q,——--
l

(a) (b)
Fig. 1-8

The mathematical expression.for this function is

1. 1
z (w) = -JwH 0" "“1) (6)
(u‘-w:)(u" ant.)

b. Should the network offer infinite impedance both at
zero and infinite frequencies, the total number of
weights is again even with the number of inductors
equal to the number of capacitors. This is shewn

in Fig. 1-9

20

42(90

T ..

(a) (b)
Fig. 1-9

UN“

::’(oi)

/;w
“4’

 

“--“ P-“_--

V‘ __.....
p
8

GI

The mathematical expression for this condition is

expressed by

Z (W): .1003 (“z-“T)(W"“:)(¢~_’l:w:) (7)
(«1‘3 w:)(w‘_u:)

 

c. If a network offers zero impedance at zero frequency
and infinite impedance at infinite frequency, we
obtain the following

ZN)

fi

B--

Z (N) w,‘

 

 

 

(a) (b)
Fig. 1—10

21

This condition is expressed by

z (.0): sum (of-w: )(wz-wi) (a)
(.9- at) (what)

 

d. When the network offers infinite impedance at zero
frequency and zero impedance at infinite frequency,

the fourth possibility obtained is shewn below

 

 

 

 

2(w)
- I
-' I j. /I
I I
I l
1 l
2‘“) l i“: “3 We ,w
| I
I I
l I
(a) (b)
Fig. 1-11
which is expressed by
1 I 1, I-
Z (w) = -JNH (w ‘2‘ )(W -N3) (9)

 

2.
w"(w‘-wt>(w‘- we)

Before closing this discussion, it should be noted that
the reactance functions discussed above may be realized phy-
sically in other fundamental forms than those given, but these
other forms will not be presented here since this is outside

the scope of this treatise.

22

CHAPTER II
ELECTRIC FILTERS
2.1 Fundamental_Behavior of Filters

Electric wave filters are four terminal networks which
discriminate between currents of different frequencies, tran-
smitting those currents which lie within a certain range of
frequencies and attenuating all others.

For our purposes here let us study the fundamental be-

havior of filters by presenting some examples.

Example 1

 

Filter Zr

 

 

 

Considering the terminating impedances Z8 and Zr of
Fig. 2-1 to be resistive, let us analyze the behavior of the
filter when it takes the form of a shunt capacitor as shewn

in Fig. 2-2(a).

 

 

 

____41nnu___.
c JI ' ‘—
<a> (b)

23

If Z5: 0 and E is not changed by the current through the
generator, it can be easily seen that the capacitor will have
no effect on the voltage across Zr' If 25 fl 0 and E is non-
sinusoidal, we find that the same fraction of each harmonic
of generated voltage appears across the load resistor Zr if
the capacitor is not in the circuit. When the capacitor is
in the circuit, we see that the voltage across Z1. is the same
as it was with the capacitor removed only for zero frequency
while for all other frequencies the voltage across Z1. is red-
uced, and as the harmonic order increases, it is increasingly
effective in suppressing the reaction on the load of the gen-
erated voltage. Hence, this capacitor has less effect on
lower freqyencies than on higher frequency harmonics and ap-
pears to pass the lower frequency effects of the generator
more effectively.

Now let us replace the shunt capacitor by the series
inductor appearing in Fig. 2-2(b). Here we find that at
zero frequency the inductor offers no impedance and therefore
does not affect the load action of a d-c component of the
generated voltage, whereas at infinite frequency the series
inductor offers infinite impedance and acts as an open circuit
seperating the generator from the load as did the shunt
circuit capacitor.

Thus, at zero and infinite frequencies the series ind-

uctor and shunt capacitor behave the same, both appear to pass

24

the lower effects of the generator mere effectively and there-

fore both are referred to as low-pass filters.

Example 2

 

IL
r

C L

 

 

(a) (b)
Fig. 2-3

~

By the same reasoning applied in example 1, we can see
that a series capacitor and a shunt inductor as in Fig. 2-3
will have opposite effects to the shunt capacitor and series
inductor.

At zero frequency the series capacitor offers infinite
impedance, thus preventing any d-c component of generated
voltage from appearing across the load while at infinite fre-
quency it has no effect.

The shunt inductor acts as a short circuit at zero freq-
uency, preventing any effect on the load while at infinite
frequency it acts as an open circuit and has no effect on the

load. Therefore both are referred to as high-pass filters.

Example 2

The four terminal network in Fig. 2-4(a) acts as a short

circuit at the frequency of resonance and therefore when placed

25

between the generator and the load in Fig. 2-1 will connect
the generator directly to the load at that frequency while

at zero and infinite frequencies the filter offers infinite
impedance and the generator will be isolated from the load,

or the circuit will be open.

gar—4+-
L c

 

 

 

(a) (b)
Fig. 2-3

At the frequency of resonance we also find that the net-
work in Fig. 2-4(b) offers infinite impedance, thus connecting
the generator directly to the load while at zero and infinite
frequencies the inductor and the capacitor respectively short
circuit the generator and therefore both networks are called

band-pass filters.

Example 4

43—

C. L

 

(5
.4

 

(a) (b)
Fig. 2-5

26

Finally consider the network in Fig. 2—5(a). Applying
this four terminal network to Fig. 2-1, we will see that the
generator will be disconnected from the load at the resonant
frequency, and at both zero and infinite frequencies the gen-
erator will be connected directly to the load.

If the network in Fig. 2-5(b) was used instead, the gen-
erator will be shorted at the resonant frequency but will be
across the load at zero and infinite frequencies and hence

both networks of Fig. 2-5 are known as band-elemination filters..

2 2 Constant-K Filters

 

Most filters are designed as symmetrical T or symmetrical
1T networks. Since T and'lT networks can be made equivalent,
it is immaterial which is used in this discussion.

Consider the following symmetrical T network

Zt/Z 2V1

T2.

This network is evidently a low-pass filter because it
is constructed on a low-pass arrangement of inductors and

capacitors.

27

From the discussion in the beginning of this chapter we
recall that at zero frequency, the impedance of the series
inductor was zero while the impedance of the shunt capacitor
was infinite. Since in addition we find that at infinite
frequency the series inductor offered infinite impedance but
the shunt capacitor offered zero impedance, in other words,
due to the fact that Z1 had zeros when Z2 had poles and vice
versa, it follows from Foster's reactance theorem that the
series inductor and the shunt capacitor are potentially re-
ciprocal, and by proper choice of L and C, they can be made
reciprocal with respect to any constant, say k2 (where k is

a real positive constant), hence the name "constant-k filters".
2a3Determipation of thefigransmission and Attenuation Regions

Fig. 2-7 consists of a symmetrical T network connected

between a generator and a load whose impedance Zr = Z0

 

 

Fig 0 2'7

The loop equations for the above network are

28

E = g g; l 22 3 Il - 22 12 (1)

C
I

- - 22 11 / g g; l 22 ; 2O 3 12 (2)

Upon solving the above equations for Il a nd 12, and

defining £;_= e' we obtain

 

I2
Zl/Z/Z 1
1;=§'__f °-e (3)
from which we get
§i=22<e‘-1>-zo (4)

Substituting the value of Z0 from equation (4) into the

equation for the characteristic impedance of the T network

 

/ 2
2: /zz%?1
° V 12 4 (5)
gives
22 (c."..1)2.z1 3:0 (6)

Equation (6) can be simplified to

1‘ 1' Y
e - 2e / 1 = E; e (7)

z2
a~nd after arranging terms we get

e.‘ 2f ed;- 1 )1 ?_J.__ (8')

222

29

or cosh"= 1 ,1 i (10)
222
where '5 -_-_ a( + a. [.3 (10)

and therefore
cosh¥= coshwcosp/ J sinhasinp (11)

Y is the propagation constant,o<the attenuation constant and
B the phase constant.

Since 21 and 22 are pure reactances, their ratio is real
and therefore equation (11) can be real only if the reactive

term is equal to zero
sinhet sinp 3 0 (12)

and this relation is true if either sinh-t or sin p is equal
to zero.
For sinhcx= 0

0‘: O and cosha<= 1
this condition gives

cosp = l / Z1 (13)

and therefore

-1<€1x£1__)<1
222
01‘

-l 4 cosh < l (14)

30

Since in this region the attenuation is zero, it must be a
transmission region.

For sinfl = 0

=1“ and cos/3: -1
When cos = / 1 f3: 0
and cosh-a: E 1 / E2. ; 7 1 (15)
222
For cosfs= --1 3:11’
and El/EL_3<-1 (16)
222

The regions expressed by equations (15) and (16) are attenua-

tion regions. Therefore transmission regions exist for
-1< cosh1< l

and outside this region attenuation exist.
The frequencies at which the network changes from a trans-
mission to an attenuation region and vice versa are called cut-

off frequencies. These frequencies occur such that

cosh¥= If 1
Therefore 1 ,1 _Z_I_L___ = -l
222
gives EL = -1 (as one cutoff condition) (17)
422
and l / 2.1 = ,l 1

2Z2

31
gives Zl = 0 (as the second cutoff condition)
422 (18)
Hence the transmission region is defined by

-1 5 Z1 5 o (19)
422
The transmission region can be expressed in a different
form which will be useful in the following analysis.
From the relation

Zsinh2X=coshX-l

N

substitution in equation (9) yields

 

. /
512ml: / z1 (20)
Z 422
2
Let / 1 = If 3x (21)
' V 422 k

where Xk may be complex.

For the ratio / Zl to be real, the expansion

432
sinh 1 3 cosh‘lsinfl- J sinhflcos .6—
2 Z 2 2 2

requires that the imaginary part vanish. This may be so if
either sinh % or cos—g- is equal to zero.
For 511mg:

at: O and coshf‘i'J

32

hence sin£= Xk (22)
2
For cosfi-‘e 0
{3:317 and sinPi=Ifl
a..
and coshnz- IXkl (23)

The transmission range can now be expressed

-1 s xks 1 (24)
and the attenuation range by

lxkl :1 (25)

In this last derivation, negative at was discarded because

such values are impossible in a passive structure.

a .4 Design Procedure

We have already defined the constant-K filter to be a
filter whose component impedances are reciprocal to a real
positive constant. Let us write this statement in the form

of an equation
Z Z ' R2 (26)
l 2 “

If the characteristic impedance of the T network of Fig.2-l
is resistive and if the network is terminated with this char-
acteristic impedance Z0, it can be easily shewn that the input
impedance will be equal to 20, and since the T network is

purely reactive we can secure maximum power transfer. The

33

nominal value of the characteristic impedance is R, defined
by equation (26).

From equations (21) and (26) we obtain the following
relation

xk - E;__ (27)
21R

and for reactive networks, it is seen that Xk is a real func-
tion of frequency similar to Z1.

The usual practice in the design of constant-K filters
is to choose the value of the characteristic impedance R
equal to the resistance the filter is to work into and out
of. Next we need to design Z1 only, for when this is det-
ermined, 22 can be evaluated by reciprocation. 21 is det-
ermined from equations (27) and (24). Equation (27) deter-

mines the cutoff frequencies which correspond to
21 = f 233 (28)

The point Xk = 0 corresponds to 21 = O, i.e., to a zero for
the reactance Z1, therefore 21 must have as many zeros as the
filter is to have transmission regions. Since a reactance
function has either a zero or a pole at the origin and at
infinity, that is, since Zl cannot have a finite non-zero
value at either of the extremities of the frequency range, it
follows that a pass-band starting at the origin or terminat-
ing at infinity requires that Z1 have a zero at the origin or
at infinity respectively.

34

These regions are called external, to distinguish them
from internal regions for which both boundaries lie at finite
non-zero frequencies. Thus Z1 must have one zero for each
internal pass—band and a zero at the origin or at infinity for
an external pass-band according to whether the latter is loc-
ated at the origin or at infinity respectively. This deter-
mines the structure of 21.

Any reactance function is determined by specifying the
locations of its internal poles and zeros plus one additional
information which.may be the value of the reactance at one
other frequency. That is, the number of determining factors
is equal to the number of internal zeros, plus the number of
internal poles, plus one. This number equals the least num-
ber of elements by means of which the reactance function may
be realized according to Foster's reactance theorem, and from
the above discussion we see that this number coincides with
the number of cutoff frequencies or boundaries between trans-

mission and attenuatuon regions.

2.5 Examples on the Design of Constant-K Filters
a. Low-pass Filters

In article 2.2 we saw that a symmetrical T network con-
structed on a low-pass arrangement of inductors and capacitors

may be used as a low-pass filter, Fig. 2-8(a).

35

25/2 Zh/Z

 

 

 

(‘22
.§ \ . Z.
\l
hPOuS‘ \ Aft".
Z ’4 ‘~ 3* f
Z2- 2
Z
(a) (b)
Fig. 2-8
The equation expressing the transmission region
-1 < 21 g 1 (18)

.. 422
is demonstrated by Fig. 2-8(b).

The cutoff frequency is the point of intersection of Z1
and -422 since at that point Z1 8 -422. Hence a transmiss-
ion range starts at the frequency at which Z1 = O and ends
at the frequency at which Z1 = -422, or the transmission
range extends from f = 0 to f = fc and the frequencies above
fc lie in an attenuation range.

It is seen however from Fig. 2-8(b) that the network (a)
does not discriminate sharply between frequencies above and
below to, the response changes gradually. In the ideal case
it is desired that the transition between the transmission
and attenuation regions be abrupt. In practice, this ideal

condition can be approached, for a sharper frequency discri-

36

mination can be obtained by cascading two or more identical
sections and connecting them between the generator and the
load. This form of network is called a ladder-type filter.
Still another approach to the ideal case may be had by making
‘use of the m—derived type of filters the analysis of which
will be given in article 2-6.

In Fig. 2-8(a) we have

21 22 = K2 (29)

or 21 22 = juoL. 1 = L = R2 (30)

JuJC C
‘where R = [Q'was defined as the nominal value of the charac-
C
teristic impedance.

The characteristic impedance of a T network is

20 .: V/ 2122 (1 x g?)

 

 

 

 

/_— 2
or Z = / L l — 93LC (31)
° V c E 4 3
Making use of the relation Z1: -4Z2 we obtain
rc : 1 ’ (32)
1f «£6
_ /
'therefore Zo - / ;_ E 1- (f f; (33)
v c .9
(Dr z = R / 1- x2 (34)
0 V

37

where x = f (35)
f0
Equation (34) shows that for frequencies below cutoff,
the characteristic impedance is a pure resistance varying
from R: L for low frequencies to zero for f : fc, and for
C
frequencies above cutoff the characteristic impedance is a

pure imaginary. The variations of the characteristic imped-
ance with frequency are illustrated in Fig. 2—9.

 

 

 

Applying equations (22) and (23) to the network treated

here, we obtain the following conditions

0(: 0
fl = 2 sin" _1; )
fc
and «x l
= 2 cosh" f
(fc) ( rceruo) (37)

(G : 1T

38

The above two conditions are demonstrated in Fig. 2-10
‘* 6

rr _______ B /

 

 

 

 

 

 

 

I
r3 i ‘<
I
I
1 °‘ 2.9
Fig. 2-10

b. High-Pass Filters

""1 *““ +
272 2'/z

252
(a) (b)

'Fig. 2-11

Fig. 2-11(a) is a high-pass filter because it is const-
ructed on a high-pass arrangement of inductors and capacitors.
This condition is illustrated by Fig. 2-11(b).

The characteristic impedance for this network is

ZO = B 1 - (38)

1
x2

39

and the variations of this equation with frequency are illus-
trated in Fig. 2-12 below
2.

R .___._ _____

Immatnary

 

The variations of the attenuation and phase character-

istics are

°‘ = 2 cosh'l £9
(1‘ ) (o<r<rc) (39)

(r0 < r < w) (40)

(3 " -2 sin"1(_t§_)

These conditions are illustrated in Fig. 2-13
«fl

 

 

40

c. Band-Pass Filters

 

 

 

 

 

(a) I (b)
Fig. 2-14

In the above network

 

7.
21 = J (‘0 £1.01 - 1.; (41)
U01
and
z2 = J“L2 (42)
1-... L2 02
therefore
1.
2122 = 32 : - Lg (€d L101 - 1g (43)
C1 1- (,3 L2 C2

If the anti-resonant frequency of the shunt arm is made

to correspond to the resonant frequency of the series arm, then

an? L1 c1 = u} L2 02 (44)
01‘
L1 cl : L2 02 (45)
and hence
2122 2 £2 = 21 = R2 (46)

C1 2

41

At the cutoff frequencies
Z1 3 “422
multiplying both sides of the above equation by Zl gives

212 : -42122 = -4R2
or
21 : Z 233
so that
21 at lower cutoff frequency : ~21 at upper cutoff frequency
The variations of the characteristic impedance with fre—
quency are shewn in Fig. 2-15 and given by equation (47)

 

 

 

 

Zo ‘-' Wan:- ‘>(u*-w.‘> <47)
OJ (Na-NI)
I Real I
I I 1
+R"":" "l"" Imag
fCo l
2' '9“
l I
- l I
I l
I
Mm,
Fig. 2-15

The attenuation and phase characteristics are

1, 1.
0‘ I 2 cosh‘l w“°"L

__—

w<w.-w.) (rc2 <r <00) <48)
(3 = w

42

 

 

1 2. 1
0t : 2 cosh” “’0 "' 03
(5 = --'W
and
6K : 0
w" u"
- n -1 __l ’ 4L-
3 .. L sin ‘0 (Down-NI.) (fcl<f<f02) (50)
The above variations are illustrated in Fig. 2-16

I I :>/
«I: '
4“.» I P
I
I

 

d. Band-Elimination Filters

This filter is obtained by interchanging the series and
shunt arms of Fig. 2-14(a), as illustrated in Fig. 2-17

 

 

N

 

(a) (b)
Fig. 2-17

43

At the cutoff frequencies

Z1 3 “422
and
Z2:15.15
2

Equation (51) and Fig. 2-18 give the variations of the

characteristic impedance with frequency

 

 

 

 

2o = Rv<u--s;‘><w--LI:> (51>
(ME-N")
rumma-
+ I l '
REA‘ : : :
R ‘ I : : Real
I
z I F I
£3 I l;¢1
l I l
- l I :
I I
V : :‘filrms'
Fig. 2-18

The attenuation and phase characteristics are given below
aton rzro (52)

(3 : -1T
(fo<f<fcg) (53)

A

_ 2 cosh"1 “(”VNL)

‘
us}*-00

51

 

 

 

 

 

 

 

 

44
Oi: O
(3— o sin-1 Nauru) (fc2<f<°°) (54)
- k IMJ=¢;:
at : 0
- 2 Sin-l 03(U1-UJI) } (O <f<fCl) (55)
p - Unz'O-T‘
(3 = 77
x .. } (£01411 £0) (56)
i
..L....__
I
I at
If“ f
M
5 E I
Fig. 2-19

2.6 The m-Derived T Section

We recall from the previous article that the constant-K
filters do not discriminate sharply between frequencies above
and below cutoff, and that the characteristic impedance is
not constant over the pass-band so that a satisfying impedance
match is not possible.

Where impedance matching is not important, the attenuation

45

near cutoff may be built up by cascading several constant-K
sections and in this case, if every section had the same
characteristic impedance, the sections remain matched at all
frequencies. The propagation constant will assume the value
nflx, where n is the number of sections in cascade. The char-
actersstic bmpedance however still does not improve.

A.more economical way than cascading several constant-K
sections to improve the attenuation near cutoff is by use of
the so-called m-derived filter which will now be discussed.

Referring to the constant—K low-pass filter shewn in
Fig. 2—8(a), let us derive from this one an improved filter
that will have the same allocation of attenuation and trans-
mission regions, and that will have the same characteristic
impedance.

Let us indicate the series and shunt arms of the derived
filter by 21 and 25 respectively.

Since the derived filter in question is a low-pass filter,
the series arm must remain to be an inductor, either a frac-
tion of, equal to, or a multiple of Z1 of the constant-K

filter and hence let us assume
I

m to be determined. In order that the characteristic imped-
ances of the derived and prototype sections be equal, the

following must be true

46

(mzl)2 / mzlzé = El? / 2122 (58)
4
4
from which
2; : .22. ,z 1-1112 21 (59)
m 1.111

Equation (59) shows that the shunt arm of the derived
network is made of an inductor and a capacitor in series hav-
ing impedances equal to 3:33 21 and 52 respectively.

The m-derived low-pass filter an: the reactance curve

demonstrating its performance are shewn below

+ LF_ I .
Pa 5: -—\¢:‘-n”fl' Z.

2!-=—"”"'1:FC:;:::::::z‘ 5’

\
\ I
- \‘h Zr.

 

 

 

 

(a) (b)
Fig. 2-20

The shunt arm in Fig. 2-20(a) is seen to be a series
resonant circuit with resonance above cutoff. At this fre-
quency the shunt arm appears as a short circuit and the atten-
uation becomes infinite and by making the frequency at the
point of infinite attenuation (fee) close to the cutoff fre-

quency (fc), the attenuation near cutoff can be made high.

47

Since 22 in the constant-K filter is opposite in sign
to 21’ the same relation should remain in the two series im-
pedances of the shunt arm of the m-derived filter. This
can be true if l;m3_ in equation (59) is positive, therefore
(l-mz) and m must4ge positive, thus giving the limits on the
value of m

0<m<l (a»

We have seen in Fig. 2-20 that the frequency of reson-
ance occurs above cutoff, this follows from the requirement
that below fc the shunt arm appears capacitive. Therefore

at the resonant frequencu

 

--I lm—Z— le

and for the low-pass filter we get

 

 

1 = 1--1m‘2 21rf,.L (62)
zfir..mc 4m
hence
1" V(1-m2) LC

Since the cutoff frequency for the low-pass filter is

: ;_.._. 6
fc 11 V1.5 (4)

the frequency of infinite attenuation will be

f° (65)

fan

 

V (l_m2)

48

from which

 

/ ~2
"‘3 V11?)

(66)

From equation (66) m may be found for any specified foo .
The variation of attenuation over the stop band for a
low—pass m-derived filter may be found from the relation
cosh$= l / Z1
22, (9)

Following the same procedure applied in the derivation
of the constant-K filter we find that in an attenuation region

cosf3= {51

When cos/3 = )l l, (3: O
and

cosh“: l )1 EL
222

1 _ mm].
2 _1___- uL 1—m2
[umC 4m )1
1 - 012 In2 LC
2 1 - “2 LC 1-m2
[ (———. )]

Applying the relations (61), (66), and (64) to the above

 

 

we get

49

 

 

 

 

 

 

2r2 m2
cosh“: l - fcI 2 __ (6'7)
1 - 3-2 (1- 2)
fc
When cos/3: -1, (3:11
and 2&2- m2
1. 2
cosh“: ; - l (68)
£— (1 m2)
1 " f 2 -.
c
and when 0‘ 3 O, coshat: l
and
- Z
cosp- l )1 __l__
222 21‘?2 m2
:2
: 1 - ° (69)
2
1 - f 2 (1.-m2)
1‘c
The above conditions are demonstrated in Fig. 2-21
for m: 0.6
°< (3
I I
I
I «I ..
1T .. -
l
f I
I
I
f
{c {to

Fig. 2-21

50

Fig. 2-21 shows thatinfinite attenuation is achieved at
fao but that the attenuation above fa falls to low values.
If high attenuation is desired over the whole attenuation
band it is necessary to use the above section in series with
a prototype section to provide high attenuation at frequen-
cies much higher than cutoff. Such a combination of net—
works is known as a composite filter.

A condition that was imposed on the m-derived filter
was that its characteristic impedance be equal to that of
the prototype. The proof is as follows

Zom = VZoczsc

 

 

 

— m2l l-m Z1 / Z2
: \/(m_z_1 ,l _1__-m2 z1 ,1 22) L21 1‘ 2 ”P 431.111..)
41" m 2 mZ____1_/1____-m2 21/22
2. Tm. m
3"
: V/glzz / §%_, : Z01 (70)

Applying the transformation relations developed above
to the other three prototype networks lead to m—derived net-
works. Since the procedure is the same, these networks will
not be derived here and the reader is referred to the refer-
ences listed in the bibliography for immediate reference.

We recall that the characteristic impedance of the con-
stant-K filter was not constant over the pass band, the same

can also be said for the m-derived filter since Zom = 201 .

51

To improve the response of the characteristic impedance,i.e.,
to make it nearly constant over the transmission range, half
sections of the m-derived filters are used. The character—
istic impedance for one end of this network can be made to
have the same value as that of the constant-K filter and the
characteristic impedance of the other end will have nearly

the desired properties.
2.7 Termination with m-Derived Half Sections

Here we will be concerned with the design of an L section
(half section) such that it changes its characteristics with
frequency in such a way that the filter will be approximately
matched to its load at all frequencies over the transmission

range.

E %

2.5-. -2“, 2e: -—’ «22:.

--‘" %
. M

 

 

 

 

Fig 0 2'22

Referring to the inverted L sections of an m-derived

T section in Fig. 2-22 and solving for Zab we get

52

Vzaboc Zabsc

Zab

 

___________ _ 2
l-mz z]. 1‘ ?_Z_2_ / Ill-_Z—l-
2111

E’

14122 ’/ 2?.2.)2 E
g

m 2
= [1 / (l-mz) 31.] __ZIZZ
4Z2 1 / 2.1..
422
: 1 1—2 Z1 2 71
[ /( ”1.2.2. M. < >

and hence Zab is found to be a function of Z0." modified by
a value which varies with m. For the low-pass filter equa-

tion (71) becomes

f2
Zéb = a [1- (1—m2) f2] (72)
c

 

 

 

 

«L ----—-c—-

53

Fig. 2-23 shows that by using the value m=0.6 for the
L section, a nearly constant value of Zab equal R is obtained
for most of the pass band.

The image hmpedance at the terminals c,d in Fig. 2-22

is found as follows

 

zcd

\’ choc chsc

\/(m_§1 2‘ 1:1112 2 / 22.2)?151
2 2m 1 m 2

 

 

filzz (1 )1 [27:23.) 1' ZOT (73)

Similarly we find that Zef 2 201 and Zgh = Zab‘ There-
fore a generator of internal impedance R.may be connected to
terminals a,b and a load of value R to terminals g,h and bet—
ween the c,d and e,f terminals a constant-K and an m-derived
T sections designed for a value R.may be inserted and obtain
a satisfactory match over the largest range of the transmiss-
ion band and also obtain maximum power transfer. The char-
acteristic impedance will be nearly constant and equal to the
value R except near cutoff. These half sections are referred
to as End or Terminating half-sections.

The preceding illustrates the advantages of the m-derived
sections especially when cascaded with a prototype section.

High attenuation at other frequencies in the attenuation band

54

may still be obtained by cascading as many m-derived sections
as required.

A further improvement however can be achieved by deri—
vation of another m-section from the first m-section Just
as the latter was derived from the prototype. These sect-
ions are called "double m-derived" or "mm—derived" sections.
The procedure is similar to that used in deriving the m-der-
ived from the prototype section and therefore will not be

repeated here since this is not the purpose of this discussion.

2,8 Example

The following example will illustrate the advantages of
cascading a prototype section and an m-derived section ter-
minated with half sections, the whole network being a compo-
site low-pass filter.

Design a composite low-pass filter with a 2000 ohms res-
istance termination. A cutoff frequency of 3000 cycles and
very high attenuation at 3840, 5000, andao cycles are required.

First we proceed with the design of the prototype. The
cutoff frequency for a constant-K filter is that at which
Z1 2 ~422 and for a low-pass filter this becomes

“cL=.4_
“EC
2 2
or 17 fc LC : l (74)

but R : (L
c

55

and hence L 3 R20

substituting this value of L in equation (74) gives for the

value of the shunt capacitor

 

 

Therefore for the prototype section

L =2 R = 2000 : 0.213 henry
11' to «x 3000

and C 2 l 2 l = 0.053 mfd
1r fcR 1r x 3000 X 2000

This section is shewn in Fig. 2-24

L/ 2 l-/z

 

Fig. 2-24

56

The m-derived section providing high attenuation at
3840 cycles will have a value of m given by the equation

f- (fa-I2
F (332%

Therefore this section may be used for the terminating half

 

B
u

 

sections, for we saw earlier that terminating half sections
using the value m2 0.6 provide a nearly constant value of
image impedance equal to R over about 85 per cent of the pass

band. The component values for the half sections are then

 

 

 

 

m; : 0.610.213 = 0.0639 henry
2 2
l-mZL : 1-0.26 X 0.213 = 0.114 henry
Zn 1.2
and mg : 0.6X0.053 = 0.0159 mfd
2 2
These end sections are shewn below
mL/z mL/z
————f0"¢‘.
2
I-nI
ZrnL -2331'
"‘C 2.4.9.
I 7- 2 T

 

 

Fig. 2-25

57

To achieve high attenuation at 5000 cycles the m'-der-

ived section requires an m' having the value

‘/1- (3000f-
5000

“1‘0036 = 008

The component values for the m'-derived section are

 

m!

 

m1; = 0.8X0.213 : 0.0852 henry
2 2
1.231"?L : 1- 0.36 x0.213 = 0.0426 henry
4111' 3.2
and m'C = 0.8x0.053 = 0.0424 mfd

The composite filter thus derived is shewn below

21!.- I. L 'I. .
1'? “i."m film"

    
 

If

It

 

 

_I--__----
‘—-|

58

Of course the series arms between any two sections in Fig.
2-26 may be combined to form one physical inductor.
The attenuation of each section and of the whole compo-

site network of Fig.2-26 are shewn below

cx (nepers)
5‘

 

 

 

In the above curves:

dash-dot curve : response of constant—K.mid-section

dashed curve ' response of m-derived mid-section

dotted curve : response of m-derived terminations

solid curve response of composite filter.

59

2.9, Conclusion

In the examples given in a rticle 2-5 the application
of Fester's reactance theorem in the design of the reacta-
nce arms of the constant-K filter networks was illustrated.
In designing the series arm for example, Z1 must have as
many zeros as the filter is to have transmission regions,
or one zero for each internal pass band and a zero at the
origin or at infinity for an external pass band as the case
may be. Thus a series arm of any form may be reduced by
means of Foster's reactance theorem to an equivalent network
containing the least number of elements as discussed in
Chapter I. With 21 determined, 22 is its reciprocal with
respect to R2 .

The procedure in the design of the reactance arms of
the m-derived filter follows the same as that of the constant-
K filter in so far as the application of the reactance theo-
rem is concerned. Hewever, the shunt arm in the m-derived
section is not the reciprocal of the series arm, it is rather
derived from it as discussed earlier.

Therefore, the arms of a constant-K and an m-derived
filter do contain the least number of elements according to
Foster, for they were designed as two-terminal networks.

We should wonder at this point whether or not a compo-
site filter may still be reduced to contain less number of

elements. We have seen earlier that each section in the

60

composite network contains least number of elements, and
that we can use either the constant-K or the m-derived sec-
tion to give us the required pass and stop bands, the cutoff
point and the characteristic impedance. The reason for cas-
cading these sections however was to build up the attenuation
in the stop band at the cutoff and higher frequencies, for
each.m-derived section can be designed to boost the attenua-
tion at some frequency above cutoff, and hence the composite
network in the example of article 2-8 does contain the least
number of elements designed for a cutoff frequency of 3000
cycles and to have high attenuation at 3840, 5000, and cycles,
and we know that that network can be reduced to a single
section, a prototype for example, and still have the same
allocation of attenuation and transmission regions, cutoff
frequency and characteristic impedance, but of course they
will not be as high as desired at frequencies above cutoff
in the stop band as offered by the composite network, neither
will the characteristic impedance be nearly constant over
the transmission range. Therefore, to achieve these impro-
vements we cascade a prototype with as many m-derived sect-
ions as required, each of which contains least number of
elements.

Now to try and apply Foster's and Cauer's theorems in-
order to further reduce the number of elements in either the

constant-K or the m-derived sections is not possible because

61

these theorems are first limited to the treatment of net-
works involving only two kinds of elements namely R,C; R,L;
and L,C networks and second, the question regarding the
equivalence of networks with respect to more than one pair
of terminals is completely outside the scope of the above

theorems.

62

BIBLIOGRAPHY

Foster, R.M. A Reactance Theorem. B.S.T.J., April,l924.

Campbell, G.A. Physical Theory of the Electric wave Filter.
B.S.T.J., November, 1922.

Zobel, 0.J. Theory and Design of Uniform and Composite
Electric Wave Filters. B.S.T.J., January,l923.

Guillemin, E.A. Communication NetworksI Vol.1. John
Wiley and Sons, New York, 1949.

Guillemin, E.A. Communication Networks, Vol.lL John
Wiley and Sons, New York, 1949.

Ryder, J.D. Networks Lines and Fields. Prentice-Hall, Inc.

Reed, M.B. Alternating-Current Circuit Theory. Harper and
Brothers.

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