FOSTER DISTRIBUTED~LUMPED
NETWORK SYNTHESIS

Thesis for the Degree of Ph. D.
MICHIGAN STATE UNlVERSiTY
G. T. DARYANANI
1958

 

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ABSTRACT

FOSTER DISTRIBUTED-LUMPED
NETWORK SYNTHESIS

by G. T. Daryanani

Distributed network theory is approached from a
basic and unifying standpoint. Sufficient conditions are
deve10ped for the realizability of frequency domain, non-
rational, immittance functions. The networks consist of
distributed and lumped elements and have Foster-type topo-
logies. As a starting point for a comprehensive theory
for distributed network synthesis, they are recommended by
their mathematical tractability.

The approach used is to classify functions by their
singularities. RC and RL networks are the only ones con-
sidered. All singularities lie on the negative real axis
of the complex s-plane. The first class considered consists
of functions that have a discontinuity across a line on the
axis and are holomorphic elsewhere. This class includes
branches of multivalued functions which have branch points
as their singularities. The theory depends on the prop-
erties of an integral with a Cauchy-type kernel evaluated
along the line of discontinuity. The Russian mathematician,
Muskhelishvili, discusses such integrals in his work on

singular integral equations.

G. T. Daryanani

In the second class the functions may have a count-
able (finite or infinite) number of poles. The Mittag-
Leffler theorem gives representations for functions with
infinite numbers of poles which yield Foster-type infinite-
lumped networks.

An Open question, the answer to which is possibly
in the negative, is--can p. r. immittance functions which

are RC, RL realizable have any other singularities?

FOSTER DISTRIBUTED-LUMPED
NETWORK SYNTHESIS

By 6‘

. ,1)-
(\§ \‘

gO‘,5”
GK T. Daryanani

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Electrical Engineering

1968

TO

Carol

ACKNOWLEDGMENTS

The author wishes to thank his major professor,
Dr. J. A. Resh, for his inspiring guidance during the prep-
aration of this thesis.

The author also thanks the other members of his
doctoral committee Drs. Y. Tokad, C. E. Weil, R. O. Barr

and R. C. Dubes for their interest in this work.

iii

TABLE OF CONTENTS

ACKNOWLEDGMENTS C O 9 O O i‘ O O '3 '-

LIST OF FIGURES . . - . . . o . . . . 3 .

LIST OF APPENDICES. . o . . . .

CHAPTER

Io INTRODUCTION . . . . . . . . . . .

1.1 Literature Survey . . . . . . . . .
1.2 Objectives of Thesis. . . . . . . .
1.3 Summary of Chapters . o . . . . .
II. THE CAUCHY INTEGRAL. . . . . . . . . . c .
2.1 Motivation for Cauchy Representation
23 2 Definitions 0 O 0 O C 0 fl 0 0 O C O
2.3 Properties of the Cauchy Integral .

III. FOSTER

IV. FOSTER

DISTRIBUTED NETWORKS. . . .

Non-Negative Density Functions.
Non-Positive Density Functions.
Real Density Functions-—The General

case. 0 0 O I O O B O O O

9

Outline of Synthesis Procedure. .

Positive-Real Transformations

Extensions. . . . . . . . .
LUMPED-INFINITE NETWORKS. .

RC Lumped-Infinite Networks
LC Lumped-Infinite Networks
RL Lumped-Infinite Networks
Examples. . . . . . . . . .

V. CONCLUSIONS 0 O O O O O O C O O C O

APPENDICES. .

BIBLIOGRAPHY.

O 9 O C 7. O O O O O 0 O O O

Q 0 0 O G 0 O C 0 C O O O 0

iv

o

a

O

C

5

O,

Page

iii

vii

l6

17
23

27
31
34
43
47
49
50
53
55
56
59

71

11°

12.

13.

14.
15.

16.

LIST OF FIGURES

Network Satisfying State Description
Equations 2- 3 and 2-4 . . . . . c . . .

The Line of Discontinuity . . . .

Networks for Non-Negative Density Function

(a) Distributed Network
(b) Approximate Finite-Lumped Network. .

Network for Non-Negative Density Function
Using a Transformer . . . . c . . . . . .

Network for Non-Positive Density Function
(qu 3.4) O G O O P O t O O C G 0 O O O 0

Network for Non-Positive Density Function
(qu 3.6) O 0 E. '3 O O O O 9 O O O O O 0 0

Network for Real Density Function--The Gen-

eral Case (Eq. 3.7) . . . . . . . . . . .
Distributed Network for Equation 3.11 .

Roots of %%§l-+ x = 0 . . . . . . . . . c

5)
Network for Example 5 . . . . . , . . . .
Network for Example 6 . c . . . . . . . .

Network for Example 7
(a) in the z Plane Z(z)
(b) in the s Plane Z(s) . . . . o . . . .

Network for an Admittance with a Non-
Negative Density Function . . . . . .

LC Distributed Network for Eq. 3.15 . .
RC Infinite-Lumped Network for Eq. 4.1. .

RL Infinite-Lumped Network for Eq. 4.2. .

Page

10

19

20

25

25

29
37
37
42

42

44

45
45
51

51

Figure Page

17. LC Infinite-Lumped Network for Eq. 4.3 . . . . 54
18. RL Infinite-Lumped Network for Eq. 4.4 . . . . 54
19. RC and RL Realizable Immittance Functions. . . 58
20. Contour for Evaluating Eq. A1.l. . . . . . . . 62

vi

LIST OF APPENDICES

Appendix Page

A. State Description for RC, RL Realizable
Driving Point Impedances. . . . . . . . . . . 60

B. Proof of Real Valuedness of the Density
Function. 0 O O O O O O O O O O O O O O O O O 66

C. A Corollary of Mittag-Leffler's Theorem . . . 69

vii

CHAPTER I
INTRODUCTION

With the development of integrated circuit tech-
nology, distributed networks have taken an important place
in the synthesis of network functions. A problem the net-
work designer faces is to translate a given set of graphi-
cal or tabular data into a mathematical function for which
a network synthesis procedure exists. There is much infor-
mation available on the approximation of rational network
functions. These lead to networks with lumped elements,
which are a sub-class of distributed elements. Distributed
networks can be used to realize functions which are not
rational. Infinite numbers of lumped elements would be
needed to realize non-rational functions. Distributed net-
works are preferred for their ease of fabrication.

The networks considered are linear, time-invariant
and passive. They may consist of distributed elements and

a countable number of lumped elements.

1.1 Literature Survey

 

References 1-3 and 13 deal with the realizability
of uniformly distributed RC line (URC)networks by means of

certain transformations of the frequency variable to a

form suitable for lumped synthesis procedures. Wyndrum[l]
uses positive real transformations to obtain the realiza-
bility conditions for driving point synthesis of 5R5 net-
works with identical RC products. The network sections
have a cascade or a series-parallel structure. O'Shea's[2]
transformations (not p. r.) are more general in that they
yield realizations consisting of an arbitrary interconnec-
tion of 6R5 networks with constant RC products. Rao,

Schaffer and Newcomb[3]

treat the realizability of arbi-
trary n-port connections of URC networks with rationally
related Vrcl products.

[4]

Heizer shows how a class of immittances in ratio-
nal form can be realized using a single tapered distributed
network.

Networks consisting of both distributed lines and
lumped elements are treated in references 5-7. Rao and

Newcomb[6] apply the works of YoulaIS] [10]

and Koga to the
synthesis of arbitrarily interconnected networks consisting
of DRE lines with rationally related /F31 products, lumped
resistors, capacitors, and ideal transformers. Protonotarius

and Wing[7]

suggest a description for a non-uniform RC line
with lumped elements along the line. The networks are
characterized by some analytic properties of the network
functions (ABCD parameters). Conditions are given for a

function to be realizable by a non-uniform RC line with an

RC impedance termination.

The theory of functions of two complex variables
has been utilized for the synthesis of variable parameter
networks in references 8-10. Anse11[8] applies the two-
variable theory to the synthesis of lossless transmission
lines with commensurate delays (i.e., all the line delays in
the network are whole multiples of some unit delay). The
lumped elements are allowed to have a frequency-dependent

behavior.

1.2 Objectives of the Thesis

 

In this thesis distributed network theory is ap-
proached from a basic and unifying standpoint. The frequency-
domain, non-rational immittance functions considered are
classified. by their singularities. Synthesis procedures are
developed for functions with various types of singularities.

Syntheses are effected by distributed networks with
Foster-type t0pologies. Such tOpologies have not previously
beentinvestigated,yet their study is well-motivated on both
theoretical and applied bases. Physical components with
such models occur in nature, albeit frequently in forms
equivalent to lumped models. For example, a lumped resis-
tor R may be written as R =‘frxdx which has a Foster-
distributed representation. As a starting point for a
comprehensive theory of distributed network synthesis, they

are recommended by their mathematical tractability.

fl 5 I." \|\ Illusi'ls. I zI’F! .3
mildly. wt. , \ I 5... , . ,.

a

RC and RL networks are the only ones considered.
All singularities lie on the negative real axis of the com-
plex s-plane for such network functions. The first class
considered consists of functions that have a discontinuity
across a line (or a union of line segments) on the axis
and are holomorphic elsewhere. This class includes branches
of multivalued functions which have branch points as their
singularities (Ref. 12 p. 59-64). The theory depends on
the prOperties of an integral with a Cauchy-type kernel
evaluated along the line of discontinuity. The Russian
mathematician, Muskhelishvili, discusses this integral in
his monograph on singular integral equations.[ll]
In the second class considered the functions may
have both a line of discontinuity and a countably infinite
number of poles. The Mittag-Leffler theorem (Appendix B)

gives representations for functions with infinite numbers

of poles which yield Foster-type infinite-lumped networks.

1.3 Summary of Chapters

 

Properties of the Cauchy integral and related
definitions are discussed in Chapter 2. In Chapter 3
synthesis procedures are deve10ped for functions with a
line discontinuity. Various network interpretations of
the Cauchy integral are given. Extensions to admittances,
to LC distributed networks, and to positive real transfor-

mations are also covered. Chapter 4 deals with functions

which have a countably infinite number of poles together
with line discontinuities. Sufficient conditions are devel-
Oped for the synthesis of such functions. In the conclusions

some possible extensions and practical aspects of this thesis

are considered.

CHAPTER II

THE CAUCHY INTEGRAL

In this chapter immittance functions with a line

discontinuity are considered.

2.1 Motivation for Cauchy
Representation

 

 

An integral representation with a Cauchy-type ker-
nel forms the basis of the develOpment. Such a represen-
tation is motivated by considering the following state

description of the driving point impedance of a linear,

passive, time-invariant network:[22]
SEGA) = s‘i’(S,t) + i(t) (2.1)
1
v(t) = 2?? .Z(s)W(s,t)ds (2.2)
Er.

where,
i(t) is the input current,

Br. denotes the Bromwich path of integration

v(t) is the voltage developed at the input port,
Z(s) is the driving point impedance,
W(s,t) the state "vector" is the solution to equa-

tion 1 and seC the class of complex numbers (s = o + jw)

For RC and RL networks Z(s) has singularities only
on the negative real axis. The driving point impedance
then has the alternate state description (see Appendix A

for proof).

dW(-O,t) _ _ _ .

aE—-———— — oW( o,t) + i(t) (2.3)

v(t) = 7’f(o) W(-o,t)do (2.4)
0

These equations are the state space description of
a distributed network, one form of which is shown in fig.

2. The voltage at the input port (fig. 2) is

‘T
v(t) = jv(o)do = j;(o)W(-o,t)do
o

The current i(t) is the sum of the currents ir and
1C through any section.

i = i + i
r c

= OV(o)do + 1 d(v(o)do)
ETETEE f(o)do' dt

= W(-o,t) f(o)do.o + f(o)do dW(-o,t)
f(o)do fTo)do dt

 

i(t)

 

 

dW(-o,t)

dt = -oW(—o,t) + i(t)

So the network of figure 1 is seen to have the state de-
scription given by equations 2.3 and 2.4.
The impedance of this network is an integral with

a Cauchy-type kernel

 

 

 

 

i(t)
/\ i(t)
I)
i
v(t) rm = Elam. r 1— v(o)do =
Clo =f(o)d W(-o,t)f(o)do

 

 

 

Figure 1. Network Satisfying State Description Equations
2.3 and 2.4

z... = jSiéi (2.5,

O

The above development indicates that immittance
functions with line discontinuities on the negative real
axis could have a Cauchy-type integral representation which
could lead to a network interpretation of the function.

The conditions under which such functions do have a Cauchy
representation are given in Theorem 2.2. Synthesis pro-
cedures for the integral are discussed in Chapter 3. Some

[11]

definitions required for the development are given in

the next section.

2.2 Definitions

 

Def. 2.1 A line.--The union of a finite number of

 

non-intersecting arcs Lj is called a smooth line L. A
smooth arc possesses a continuous tangent at each point
and is Open or closed. The ends of the arcs Lj will be
denoted by cj (see fig. 2). Infinity may be an end point.

The orientation chosen for the arc is from cj_to c. For RC,

3+l'
RL realisable network functions the arcs will lie on the
negative real axis. The upper half plane is called the

L+ region and the lower half plane the L- region.

Def. 2.2 Cauchy Integral.--Let f(t) be a function
of the point t on a line L, bounded everywhere on L, with
the possible exception of a finite number of points cj

where

Figure 2.

10

The Line of Discontinuity

 

11

K

|f(t)| : t—c O.

c stands for any of the points cj, K and a are positive

constants and a<l. Then
_ 1 f f(t)
F(S) - 2—TT-j- L ——t-S dt (2.6)

where s is any point in the plane not on the line, is called

the Cauchy Integral.

Def. 2.3 Density Function.--In equation 2.6 f(t)/an

 

is called the density function. (This definition differs
from the one in Ref. 11 where f(t) is referred to as the

density function.)

Def. 2.4 Sectionally Holomorphic Function.--A func-

 

tion F(s) is sectionally holomorphic with line of continu-
ity L, if F(s) is holomorphic in the plane not including
L and if F(s) is continuous (in the sense defined below)
on L from the left and right with the possible exception

of the ends near which the following inequality holds

lF(s)| < const.
s-c a

Oia<l (2.7)
F(s) is said to be continuous at t on L from the left if
F(s) tends to a definite limit F+(t) as s approaches t
along any path which remains in L+. A similar definition

holds for F-(t).

Def..2.5 Holder Condition.--A function f(t) is said

to satisfy the H61der (or H) condition on an arc if for

12

any two points t1, t2 on the arc
u
[f(tz) - f(tl)l i Alt2 - tll (2.8)

where A and u are positive constants, A is called the
Holder constant and u the Holder index.

Def. 2.6 Class H.--f(t) belongs to the class H on

 

L, if it satisfies the H (u) condition on each of the
closed arcs Lj of L for some u>0.

Def. 2.7 Class H*.--If f(t) satisfies the H (u)

 

condition on every closed part of L not containing the ends

and if near any end c it is of the form

*
= f (t)
(t-C)
*
where f (t) belongs to the class H, then f(t) is said to

f(t) Oia<l (2.9)

belong to the class H* on L.

It may be noted that a function satisfies the H
condition for all 0 < v i u, if it does so for u. Also,
if fl(t) and f2(t) satisfy the H(u) and H (v) conditions

respectively then fl(t) + f2(t). f1(t) . f2(t) and I_%ET
l

(fl(t) # 0) satisfy the H condition. (The latter two hold

on compact domains.)

2.3 PrOperties of the Cauchy
Integral

 

The next two theorems give the conditions under
which an immittance function has a Cauchy Integral repre-

sentation.

13

Theorem 2.1.--If a function F(s) has the following

 

properties:

(1) F(s) + 0 as s + w

(2) F(s) is sectionally holomorphic with line of
discontinuity L

(3) F(s) satisfies the boundary condition

F-(t) - F+(t) = 2njf(-t)

where f(-t) is a function of the point t on L;
then F(s) is uniquely determined in the entire complex
plane, except where seL.
Proof: Suppose there is another function G(s) that satis-
fies these three properties with the same line of discon-
tinuity and density function. Then G-(t) - G+(t) = 2njf(-t).

Consider W(s) = F(s) - G(s) . F(s) and G(s) are

sectionally holomorphic so W(s) also is. .F.(S.) and G (s) have

the same density function so W+(t) - W7(t)

W'(t>.

F+(t) - F-(t) - G+(t) - G-(t) = 0 or W+(t)
Define

W(s) for s t L

W(s) = W+(t) for t e L

It follows easily from Morera's theorem (Ref. 12 p. 188)
that W(s) is holomorphic in the neighborhood of any point

t on L. Hence W(s) is holomorphic in the entire complex

[12]

plane. By Liouville's theorem W(s) must be a constant.

But W(s) + 0 as s + w. Thus W(s) : O in the entire plane.

Or, W(s) 0 for s t L which implies that F(s) E G(s) for

s t L.

14

Theorem 2.2.--If a function F(s) satisfies the

 

three conditions of theorem 2.1 and in addition if
*
(4) f(-t) belongs to the class H on L then the
Cauchy Integral, with density function -f(-t), is the

unique representation for F(s)

i.e. F(s) = jig-Si dt (2.10)
L

It suffices to show that equation 2.8 with f(-t)eH*
satisfies the three conditions of theorem 2.1. Uniqueness
follows from theorem 2.1. Condition 4 states that f(-t)
is Holder continuous on L and that it satisfies equation
2.9 at the end points.

Consider a function F(s) which satisfies the Cauchy
Integral conditions. By Theorem 2 it has the Cauchy Inte-
gral representation. Consider a network which realizes
this integral. The immittance function of this network
satisfies condition 1, 2, 3 and 4. By Theorem 1, this

immittance function is identical to F(s).

Def. 2.8 Cauchy Integral Conditions.--The following

 

four conditions will henceforth be referred to as the
Cauchy Integral conditions for F(s):

(1) F(s) + 0 as s + w

(2) F(s) is sectionally holomorphic with line of

discontinuity L

15

(3) F(s) satisfies the boundary condition
F-(t) - F+(t) = 2njf(-t) for t on L

(4) f(t) belongs to the class H*

The line L is in general the union of disjoint line
segments in [0,WJ. For convenience of notation L may be
taken to be the whole interval [0,w] if f(-t) is assumed
to be zero where Z(s) does not have a discontinuity. It
should be remembered however that 2(5) and f(-t) must still
satisfy the conditions of equations 2.7 and 2.9 at the end

points of the original line. Then,

0
f -f(-t)dt
t-S

—CX)

_ ”f(x)
_ ([er (2.11)

Z(s)

Equation 2.9 is the form used in the synthesis pro-
cedures. f(-t) may be calculated using the boundary con-
dition Z-(t) - Z+(t) = 2njf(-t). An alternate method which
is often more convenient is to use the Cauchy Inversion

Integral[ll’l4]

f(-t) = IE?— ds (2.12)
L

CHAPTER III

FOSTER-DISTRIBUTED NETWORKS

In Chapter II some properties of the Cauchy

Integral were discussed. The conditions for a function
with a line discontinuity to have a Cauchy Integral rep-
resentation were given. The purpose of this chapter is
to find sufficient conditions for the Cauchy integral to
have a network interpretation. The realizations use dis-
tributed networks having a Foster-type configuration with
resistances, inductances, and elastances.

The discontinuity of an impedance function can be
described by the density function f(-x) defined by the
condition

Z-(x) — Z+(x) = 2njf(-x) for x on L (3.1)
It can be shown that for a linear, passive, time-invariant,
real network the density function is real-valued (see
Appendix B). This suggests that the discontinuity of Z(s)
can be classified according to the sign of the density
function.

Sufficient conditions for the realization of Z(s)
when f(-x) is non-negative are developed in section 1.
Various network interpretations are suggested. The non—

negative and the real density function cases are discussed

l6

17

in sections 2 and 3 respectively. An example illustrates
each case.

The steps for the synthesis procedure in the gen-
eral case are summarized in section 4. In section 5 it
is conjectured that the Cauchy Integral is a basic de-
scription for all RC networks. The facts supporting this
conjecture are discussed.

Extensions to admittance functions and to LC net-

works are given in section 6.

3.1 Non-Negative Density Functions

 

Consider an impedance function Z(s) whose only
singularity is a line of discontinuity on the negative
real axis. If Z(s) satisfies the Cauchy Integral condi-
tions (cf. Def. 2.8) then Z(s) may be represented as in

equation 2.11

Z(s) =J @393 f(x) _>_o (3.2)
o

The density function f(-x) is given to be non-negative
for x e[-w,0]. This is the same as saying f(x) is non-
negative for x e[0,w]. Three RC network interpretations
for Z(s) are given here.

Figure 3(a) is a distributed RC Foster-type real-

ization of equation 3.2. The distributed elements are a

resistance r = £l¥19§

x and an elastance %. = f(x)dx.

X

Both are non-negative elements.

18

The Riemann-sum approximation of this integral is

N
f(x.)
2 1 Ax. (3.3)

. 1
n=1 s+xl

 

This approximation has the finite-lumped network represen-

 

tation of figure 3(b). The lumped elements are rx =
i
f(xi) 1
and 5—. = f(xi)Axi. The approximation becomes exact
x.
1

as Ax + 0.

A transformer with a real, varying turns ratio
could be used to realize the integral of equation 3.2.
Figure 4 illustrates this. The turns ratio corresponding
to each section is f(x)dx which varies with x.
The v-i characteristics of this transformer network are
given by

(X)

v(t) = f vx(t) f(x)dx
O

ix(t) = -i(t) f(x)dx
Here v(t) and i(t) are the primary voltage and current;
vx(t), ix(t) are the secondary voltage and current. (The
Riemann sum approximation of eq. 3.3 may be realized by
taking the turns ratio to be f(xi)Axi.) The secondary
sections consist of a lumped capacitance of 1 farad in
parallel with a lumped resistance of % ohms. Such a
transformer is a new element and is introduced in this

work. The network sections on the secondary side do not

0

Figure 3.

l9

 

 

 

= f(x)dx

 

 

V

 

 

 

 

—£— = f(x)Ax.
c l
x

3(b)

Networks for Non-Negative Density Function

(a) Distributed Network
(b) Approximate Finite-Lumped Network

 

20

 

 

 

 

Z(s) n =f(x)dx
x T
l

 

Figure 4. Network for Non-Negative Density Function Using
a Transformer

21

depend on the density function. They only depend on the
interval over which there is a discontinuity. The density
function of Z(s) determines the turns ratio. This network
representation is conceptually simpler and could be of
analytic value in extensions of this work.

Henceforth the form of fig. 3(a) will be used to
represent network functions.

Note that if Z(s) has a finite limit k as s +w,
the limit is represented as a lumped resistor k in series

with the distributed network.

Example l.--Consider the multivalued function

 

Z(s) = i; (0<d<1). Z(s) has branch points at the origin
and infinity. The principal branch of Z(s) determined by
a branch cut along the negative real axis is positive real.
It is this branch that will be realized. This branch is
holomorphic in the whole plane excluding the negative real

axis which is the line of discontinuity.

Check on Cauchy Integral conditions:

(1) i— + 0 as s + m
a

5
(ii) 1 . . .

—— has the negative real aXis as the line of

S -jw/c

discontinuity. The limiting value Z+(x) is —£E e
-x

+jn/d

and Z-(x) is —£E e .

'X

22
At the origin, which is an end point
|Z(s)| = ll-OTl 0<a<1.
s

The end point condition of eq. 2.7 is also satis-
fied at c = w.

Hence Z(s) is sectionally holomorphic.

 

 

+jn/d -jn/d
- + _e _e
(111) Z (x) -Z (x) ——_a_ —oT_
-x -x
= —l- 2j sin 1
a a
-x
s f(-)— l ' 7‘ -mo
0, x - a Sln E xe[ , ]
-x n
f(x) - l sinl xe[0 00]
- d a '
nx

The density function could also be obtained from
the Cauchy Inversion Integral (eq. 2.12).

(iv) That the Holder condition is satisfied by
f(x) on every closed part of L not containing the ends is

shown as follows.
0'I : Ix I“

lea - x - x

2 1 2

which says that xa satisfies the H condition.
This implies that —% and hence —% . (% sin 1) satisfies
x x a

the condition of equation 2.9.
The Cauchy Integral conditions are satisfied. So, the p.r.

branch of £3 has the representation of equation 2. The

s
realization is as in fig. 3 with f(x) = la sin 1 '

O.
1TX

 

xe [0,W]

23

3.2 Non-Positive Density Functions

If an impedance function Z(s) satisfies all the
Cauchy Integral conditions and if its density function is
non-positive, it is not p.r. and hence does not have a
network realization. This is seen by considering the real
part of such a function for 0:9.

_ f(x)dx
Re Z(S) —Rej(: W—

2 f(x) (o+x)dx
(c+x)2 + w2

which is negative for 0:0.

3.2(a) Case 1. A function Z(s) with a non-positive den-
sity function could be realized if Eéil (but not Z(s))
satisfied the Cauchy conditions. The function Zl(s) =

Egig—may be written as

f (x)
Zl(s) =53) = 1 1* dx

 

S S+X

where fl(x) is the density function of Zl(s). Note that

 

fl(x) = f(x) is non-negative.
-x
w s fl(x)
Z(S) = J —S—+-;— dX. (3.4)
This integral form of Z(s) has an RL distributed network
fl(x)dx
realization as shown in figure 5. The inductance x

and the resistance fl(x) dx are non-negative. The Riemann
sum approximate networks with and without the transformer
are obtained as before.

The functions considered in this case need not be

regular at infinity.

24

Example 2. Consider the p.r. branch of sa (O<a<l). The

0.

 

 

density function of Z(s) = s is
l a jnc a-jna
_ = . ( - _
f( x) 2“] ,x e x e )
xa
= ;—-sin “a x e [-w,0]
so f(x) = 1%— sin Nd XE [0,W]
_ Z(s) _ 1 _ .
Zl(s) — s - -§ (8 - l-c). It was shown in example 1
s
that —% satisfied the Cauchy Integral conditions.
s
l . fl . .
f (x) = ——- Sln — lS non-negative.
l wa B

Z(s) thus has the network realization of fig. 5 with

_ l . n
fl”) ’ _.8 51" E . xe [0.001.
TTX

3.2(b) Case 2. -Next, sufficient conditions are obtained
for a function which is regular at infinity. Suppose the
density function f(x) is non-positive and 2(5) has the
limit Z(w) (a constant) as s + w. Let Z(s) = Zl(s) +

Z (w) and suppose

(l) Zl(s) satisfies the Cauchy Integral conditions.

 

fl(x)
(2) Z(m) = if -x dx + Rl where 05Rl<co and

fl(X)

X

 

is integrable.

Then Z(s) may be written as

00 f (x) f (x)
_ . m = 1 1
2(5) - Zl(s) + Z( ) J; dx+JE -x dx+R

 

 

s+x l

(3.5)

 

 

 

 

 

Figure 5.

Figure 6.

Network for Non—Positive Density Function
(Eq. 3.4)

 

 

 

 

Network for Non-Positive Density Function
(Eq. 3.6)

 

26

since Zl(s) has a Cauchy integral representation. The
density function of Zl(s) is the same as that of Z(s) -

so fl (x) is non-positive.
f (x)

 

Since is integrable the two integrals in

eq. 3.5 may be combined

°° (x) f (x)
1 1
bf :27..— + dx + R

 

Z(s) -x 1

 

s+x dx + Rl (3.6)

II
CDL—jS

fl(x)

“X

 

is non-negative. Equation 3.6 is easily seen to
have the RL distributed network realization of figure 6.
In the above Zl(s) satisfies the Cauchy conditions
and has a non-positive density function. So Zl(s) is not
p.r. The addition of a large enough positive constant

(Z(m)) makes the sum (Z(s)) p.r.

2(s+1)

Example 3. An example of such a function is log 5+

Z(s) =log2EI—3 -1ogZ+log :1

f:
=2

27

— 5+ . oo = o =
Here Zl(s) — log (5+ , Z( ) log 2 , R1 0 and
fl(x) = -l.
The realization is as in fig. 6 with fl(x) = -l ;

xe [1,2]. A more general form of example 3 is discussed
in detail later (Example 4).

3.3 Real Density_Functions--
the General Case

In the most general case the density function is
real (see Appendix B). The realization in this case con-
sists of a distributed RC in series with a distributed RL
network. Sufficient conditions for this case are now de-
veloped.

Theorem 3.1.--Let Z(s) have the limit Z(w) (a con-

 

stant) as s + w. Let Z(s) = Zl(s) + Z(w). Let fl(x) =
fl+(x) + fl-(x) where fl+(x) and fl-(x) are the non-
negative and non-positive components of f(x), respectively.

m f '(x)
Let Z(s) =Uf —l§———-dx + R1. Z(s) has a distributed net-
0

work realization if,
(1) Z (s) satisfies the Cauchy Integral conditions
1

+ —
(2) f1 (x) fl (x)

—————— are integrable
x x

(3)0:R1<°°

Proof: Zl(s) has a Cauchy integral representation.

28

 

00 + —
= f fl (1“) + fl (X) dX
O s+x s+x

This integral may be split into two integrals since
f1
x

fl+(x)

X

and are integrable

f (x) f '(x)
21(5) =f —]-‘———— dx +j L dx.

s+x s+x
O

 

2(5) 21(5) + Z(°°)

f” fl+(x) 7.517,.) 751-0.)
—-——— dX + ———-— dX + ——-—— dX+R
0 S+X 0 S+X 0 ‘X

Again by condition 2 the last two integrals may be combined.

co + co —
f (x) s.f (x)/-x
Z(S) ='£—]S-—rX—-— dX ‘1']0‘ :+X dX '1’ R1 (3.7)

1

 

The first integral is realized by an RC Foster-type dis-
tributed network; the second by an RL network and the third
by a lumped resistor. This is depicted in figure 7.

The Riemann approximation of equation 3.7, which
leads to a lumped network, is

N + N - -
f (x.)Ax. f (x.)Ax. f (x.)Ax.\
Z(s) =2; l l J'-+ :5; 1 .1 3 + l 31* :3 + R.
i=1 j=

i s+xj -xj I l

 

 

 

(3.8)

The residues at the poles --xi and -xj are fl+(xi)Axi and

f1_(xj)ij, respectively. In the distributed case of
equation 3.7 fl+(x)dx and fl-(x)dx are seen to play a role

similar to residues in the lumped case. They could be

 

 

 

 

(__/U§\___ ,
f (x)d

"X

 

 

 

 

 

 

 

 

Figure 7.

Network for Real Density Function—-The General
Case (Eq. 3.7)

 

30

called "differential residues" defined on the line of dis-

continuity.
E 1 4 n
xamp e ° H s+a.
i=1 1
Z(s) = log K I)
H s+bi
i=1
ai < ai+l , bi < bi+l ai and bi are non-negative.

Z(s) may be written as

n s+a'i s+a"i bi

Z(S) = Z 109 W + log grb—n— + log —n— + logKl (3.9)
i=1 i i a.
1

It is assumed that log Kl > 0.

ai = ai and bi = bi 1f ai : b

a. a. and b. . . .
1 1 1 1 1 1

II
D.
I'“
I'h
D.)
A
0..

The density function is evaluated for a typical term in

equation 3.9

 

z(s) = log (§$§)
2+(.) = ii? <l°9§I§$I§)
m» =3; megs: 53:5)
{(0) _ 2.,(0) = $3; (log [(o+a)(0+b)- jw1[(o+a)2(o+b)-1w1\

[(c+b) 2+012] [(o+a) 2+w 2]

[(o+a)(o+b)-jw][(o+a)(o+b)-jufl\
(c+a)2 (o+b)2 /

_ lim
— w+0 @og

 

31

0 when oe[0,l)
z-(o) - z+(o) = -2ni when oe[l,2]
0 when oe(2,w]
From this it is seen that the density function for z(s) is
-1 when oe[a,b] and is zero elsewhere.
Returning to equation 3.9, the density function
for Z(s) is seen to be

F
+1 xe[ai , bi ]

f(x) = <-l xe[ai , bi ]

 

L0 elsewhere.

It can be shown that the Cauchy Integral conditions are

satisfied. Z(s) may thus be bwritten as u

b1
2(5):]. —dx+ —dx+ ldx +lo K
I S+X u a n X g l
i
1’ b1
_ 1 s/x
Z(S) -:: afm dX + aufm dX + log Kl
a. a.
1 1

This is of the form of equation 3.7 and is realized

using an RL, RC distributed network in series with the

resistor log K (Figure 7.) fl+(x) is +1 for xe [ai ,

l.

' - o
bi ] and fl (x) 13 -l for xe [ai , bi ].

§.4 Outline of Synthesis
Procedure

 

In this section it is assumed that the only singu—

larities of the impedance function Z(s) are

32

line discontinuities. If Z(s) does have poles they are
removed as a rational function and the latter is realized
by conventional lumped synthesis techniques (or possibly,
as in Chapter 4).

In the preceding sections of this chapter synthe-
sis procedures were developed for a driving point impedance

Z(s) = 21(5) + 22(s) (3.10)

where 22(5) had the form of equation 3.4 (section 2a) and
its density function was non-positive. Z2(s) was not re-
quired to be regular at infinity. Zl(s) in the general
case had a real density function and was expressible as in
equation 3.7. Zl(s) was required to be regular at infinity
(section 3). Impedance functions with non-negative den-
sity functions (section 1) and non-positive density func-
tions (section 2b) are special cases of the function Zl(s).
In the following, the sequence of steps required for the
synthesis of Zl(s) and Z2(s) are outlined.

Given a function Z(s) which is not regular at in-
finity the first step is to decompose it in the form of
equation 3.10. If Z(s) + ksa (O<a<l) as s + m, then 22(3)
is identified as ksa and is subtracted from Z(s) to give
21(5). But in general this is a difficult problem and it
has not been deve10ped here. This decomposition is obvi-
ously not needed if Z(s) is regular at infinity.

The steps for realizing Z1(s) are:

(A)

(B)

(C)

(D)

33

Find Zl(W). This must be a non-negative finite
quantity.

z1(s) = zl'(s) + 21(w)

I
Z (5) tends to zero as s + w.

1
Find L (the line of discontinuity). If the multi-
valued form of function Zl'(s) has branch points
as its singularities, these must lie on the neg-
ative real axis. The p.r. branch of Zl'(s) is
determined. This has a line of discontinuity
joining the branch points.
Find the density function fl'(x). The density
function may be evaluated from the boundary con—
dition:

1 '- '+
5?; (Zl (x) - Zl (x)) for x on L.

Zl -(x) and Z1 +(x) will always be defined if L

I
is a branch cut. Alternately, fl (x) may be cal-

culated from the Cauchy Inversion integral
I

I Z (3)
fl (-x) = f —1——-— dx

L X-S

I *
Check fl (x) for the H condition. This involves
checking f(x) for Holder continuity on closed arcs

of L not containing the ends, and checking the

f1'*(t)

I
end po1nt condition fl (t) = ~t-c a (0<a<l)

34

I
(E) Check the end point condition for Z1 (5).

lzl'(s)|<const.

Wu (010(1).

Conditions A to E constitute the Cauchy Integral

conditions.

I .-
(F) Find fl +(x) and fl (x). This may involve finding
I I
the roots of the function fl (x). If f1 (x) is
of constant sign this step is not needed.

I I-
(G) Check the integrability of f1 +(x) and fl (x)

X X

I
on L. This step is omitted if f (x) is of con-

1

stant sign.

oo '-

 

f1 (X)
(H) Check that Zl(w) 31f ——:§——— dx
'-
0 fl (x)dx
Evaluate R1 = Zl(w) -J: ‘-x

Next consider the conditions on 22(5). The func-
Z (s)

 

tion must satisfy conditions A to E as above. Also

the density function must be non-negative on L.
3.5 Positive-Real Transformations

Consider the function

Z(s) = F1(F2(s))

where F2(s) is a lumped RC realizable rational function
and where Fl(z) has a Cauchy Integral representation. The
line of discontinuity L for Fl(z) lies on the negative real
axis and the density function is non-negative. Fl(z) is

then realizable using an RC distributed network. There is

35

no loss in generality in assuming L = [-a, -b].

Z(s) may be written as
b

f (x)
_ 1
2(5) ”i F275; .1. X dX (3.11)

 

From this equation it is easily seen that Z(s) has the RC
distributed realization of figure 8.

It will now be shown that any RC network with such
a Foster configuration (fig. 8) can also be represented in

the form of the basic Cauchy Integral

This fact suggests the conjecture that "the Cauchy Integral

is a basic description of all RC networks."

Theorem 3.2.--If Fl(z) has a Cauchy Integral repre-

 

sentation with a non-negative integrable density function
and line of discontinuity L = [-a, —b], and if F2(s) is
an RC realizable rational admittance function, then Z(s) =

F1(F2(s) has the Cauchy Integral representation.

Z(s) =Jr E——— dx

S+X

within an additive constant. The line of discontinuity
is on the negative real axis of the s-plane and g(x) is

non-negative.

36

The following Lemma is required.

Lemma 3.2. If %%2% is a lumped RC realizable admittance

function and if the roots of
3%EL + x = O (x non-negative)
P5)

are -ci(x), then
dc.(x)
‘ai—H’

Proof (of Lemma 3.2):

%%g% is an RC realizable admittance. Its poles

and zeros alternate on the negative real axis. The lowest
critical frequency is a zero and the largest critical fre-
quency is a pole.

g(S)
p(S)

on the negative real axis. They are also the roots of

+ x (x Z 0) is RC realizable. Its roots lie

3121.: - x (s = o+jw)

The plot of (0) versus 0 is shown in figure 9.
PC)

It is evident from figure 9 that as x increases ci(x) in-

crease in magnitude

dci(x)

-E:T'>°°

i.e.,

Theorem 3.2 is now proved.
Proof (of Theorem 3.2):

_ (S) (s) . .
Let F2(s) - p(s where ETET is an RC realizable

 

 

 

 

L:

I‘

(5L4)

 

 

 

fl(x)dx
zxm = 73737

 

 

 

Figure 8. Distributed Network for Equation 3.11

 

 

 

Figure 9. Roots of SLEL + x = O

p(S)

 

—-—— ‘_P—'

 

"X

"X

38

rational impedance function. Since Fl(z) has a Cauchy

Integral representation

j? fl(X)
Z(S) = F1(F2(S)) = a —T§T—¥dx (3.12)
NS)
. l . . .
ConSider (x>0). This function is also RC real-
%%§% + x _

izable. It may be written as

n

1 :5; ri(x)
= 1‘0 (x) + -——....___
%%2} + x 1= 5+0‘i(X)

where -d.(x) are the roots of Eli) + x = 0.
1 p(8)

The residues ri are all non-negative. The poles -ci(x)

are non-positive.

 

Then, n
Z(s) =f Y“ rim + r°(x) f (x) dx
a {£1 s+ai x) l
i= a s+ci(x) a ° 1 '

(3.13)

The decomposition is valid since fl(x) is integrable.

dai(x)
Let ai(x) = yi ; dyi = ——d§—- dx
ddi(x)
Let -a;——— = Ci(X) ; dyi = Ci(X) dX

By the above lemma ci(x) is positive, hence
dy.
dx = l
ci(x)

39

Substituting in equation 3.13,

 

 

n 0‘1 (b) b
Z(s) = :2; I ( ri(x) f1(x) . in + J )r°(x)fl(x)dx
i= a;(a) S + Yi CizxS a

ri(x) fl(X)

Let,
Ci(x)

 

Since ri(x), fl(x), ci(x) are all non-negative

Then, ai j(b)

91 (X) b
2(5) + x dx + “f r°(x)fl(x)dx
[(a) S a

91(x)
Z(s) = j. s + x dx + constant.
LI

 

Here gi(x) is non-negative and each section of the line

discontinuity [-ci(a), -ci(b)] is on the negative real axis.

In the above theorem suppose Fl(z) and F2(z) both
have Cauchy Integral representations and are RC distributed
realizable. Then the conjecture stated above would require
that F1(F2(s)), which is RC realizable, also have the basic
Cauchy Integral representation. This may very well be true

as is indicated by the following.

fl (x)9
Z(s) = f2 (y) dx (3.14)
aC[___ .x
s + yd

 

40

Using a Riemann-sum approximation for F2(s),

. fl (X)
2(5) 3:. [IN ‘ dx

.j :i'f? 2(yi ) “Y9 + X

F2(s) is now in a form which is lumped RC realizable.

 

Hence by the above theorem,

91 (X)
2 ).——+de

As N tends to infinity ci approaches di' thus

EL 9i(xi)(di - C-)

m .
Z(s) = g; S + x 1
i=1 i

It seems reasonable that in the limit this approximation
would have a Cauchy integral representation with line dis-
continuity [-c, -d].

ki(X)

s + x dx

 

2(3) 2

080.:

Another fact supporting the conjecture that the
Cauchy Integral is a basic description of all RC networks
is that a function with a simple pole at p (with residue r)
also has such a representation. The density function is

r°6(x-p), where G(x-p) is a delta function at p.

i... = f ____E_I'OO(X- ) dx
0

s-p s + x

41

Example 5

 

Consider Z(s) =J/(s+:)):i;7
Here Fl(z) is distributed realizable and F2(s) is lumped

realizable, let 2 = (s+ifjgig) be the p.r. transformation.

 

Its reciprocal is lumped RC realizable.

 

_ 1 dx
Z(s) — 1% n/x (z+x)
The network is shown in figure 10.

Example 6

5?.

+a

“11> = 109 (m am

Here Fl(s) and F2(s) are distributed realizable.

let 2 = /5

 

b b
-f 1 «Jr—1 a
Z(s) - a z + x x _ s + x X

a

The network is shown in figure ll. The impedance

Ax.

1 . . . .
z. =/-—— is realized as in example 1. x varies from a
1 s

to b.

Example 7

2(5) = {/5 + 1)(/§ + 3)
/§ (J? + 2)

Here Fl(s) is lumped realizable and F2(Z) is dis-
tributed realizable.
let /3 = 2

then, z(s) = (z ;%;(: :)3)

 

 

 

 

 

r1.
1
1 1
r2i c cli
21
3 *i _ 3
E" ‘ E f(ximxi
l.
i
-—l- = 2 f(x.)Ax.
c i 1
2i
i
r = f(xi)Axi
l. 4
1
r2. = f(xi)Axi
1
Figure 10. Network for Example 5
r :1 Ai'
x. x.
l i

 

 

 

 

 

 

 

 

Figure 11. Network for Example 6

 

 

 

43

The network is given in figure 12.

3.6 Extensions

 

3.6(a) Admittances. The discussion so far has been re-

 

stricted to impedance functions, which yield type I Foster
networks. The sufficient conditions for realizing admit-
tance functions are entirely analogous to the ones devel-
oped in sections 1-3. The Cauchy Integral is realized by
type II Foster networks. For instance, in the case of a

non-negative density function the admittance

G3

Y(s) =_[ f(x) dx

s+x
O

is realized by the network of figure 13.

 

It was mentioned in the last section that the
decomposition of an impedance function which is not regular
at infinity (equation 3.10) can be difficult. A possible
way of avoiding such a decomposition would be to realize
the reciprocal of the impedance function as an admittance
function. The admittance function will be regular at in-

finity and the decomposition is not needed.

3.5(b) LC Distributed Networks. One way of obtaining an

 

LC distributed realization is by considering the trans-

formation

_ 1
21(5) - 7; . Z(fg)

44

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1/4
W—
.f I!
IV
3_ I!
22 1F
l.
22
(a)
1/4
.__.__._/\ /\ ,.____
i .3.
~ 2.5 ———«/V
1
2.5
(b)

 

Figure 12.

Network for Example 7

(a) in the z Plane, Z(z)
(b) in the s Plane, Z(s)

 

 

 

 

45

1— = f(x) dx
x
Y(s)
_l f(x) dx
r x
x

 

Figure 13. Network for an Admittance with a Non-Negative
Density Function

fl(x) dx
1 =._____—___
x x

4...?

 

 

 

 

1+—

l-
E— — fl(x) dx
x

 

 

Figure 14. LC Distributed Network for Eq. 3.15

46

If 21(5) satisfies the Cauchy Integral conditions and

fl(x) is non-negative, then

f fl(x) dx
21(5) = ....___...___

 

O s + x
m s fl(x) dx

Z(s) = .f 2 (3.15)
O s + x

This representation of Z(s) has singularities along con-
jugate lines on the imaginary axis. The LC distributed
realization for Z(s) is shown in figure 14. An example
of such a function is Z(s) = VQEII ( /g§:l has branch

points at +j and =j). Here Zl(s) = 31+l7s and fl(x) is

‘7—7‘1—1 x , xe [1,oo1.

CHAPTER IV

FOSTER LUMPED-INFINITE NETWORKS

In Chapter II the functions considered were.
characterized by line discontinuities. The next class
considered consists of functions which have a countable
number of simple poles on the negative real axis. In gen-
eral, the driving point impedance Z(s) may be the sum of
two impedances 21(5) and 22(5), where Zl(s) is realizable
as a distributed network (as in the last chapter) and Z2(s)
has a countable number of poles.

If 22(5) has a finite number of poles, classical
finite-lumped network synthesis techniques may be used to
realize it. This chapter considers the realization of
functions which have countably infinite numbers of poles.
Given a function which has a sequence of poles pn with
corresponding residues rn , the problem is to find a
representation for this function which leads to a network

interpretation. A simple choice might have been the sum

w r
of the principal parts i.e.. F(z) ==:; 25:25— . But this
n
n=l
function does not always converge. (Consider for instance
Eéfi . This function does not converge anywhere in

47

48

the 2 plane.) The answer to this problem is found in a
corollary of Mittag-Leffler's theorem (see Appendix C).
This theorem gives representations which lead to Foster-
type networks consisting of infinite numbers of lumped
elements.

The countably infinite lumped realization of some
irrational functions have been treated in references 17-

k[l7]

l9. Halija uses a Newton and Halley rational form

of /E which leads to an RC infinite lattice structure.

[18]

Stiegletz approximates /§ as a rational function whose

poles and zeros alternate on the negative real axis yielding

[19] shows that a con-

a Foster-type RC network. Dutta-Roy
tinued fraction expansion of ea, -l:d:l, gives an infinite
RC ladder structure which is the approximant to an RC dis-
tributed line. The function sa occurs in the design of

constant-argument immittances which have many applications.

. The image admittance of a constant K filter with a cut-off

frequency mo is l + E: . This can be realized using an
‘/ 2
w

0

[15] so and J1 + $2 belong to the class

infinite structure.
of functions that were treated in Chapter 3.

In section 1 sufficient conditions are developed
for an RC infinite-network realization. Sections 2 and

3 consider the LC and RL cases respectively. Two examples

are given in section 4.

49

4.1 RC Infinite-Lumped Networks

Theorem 4.1.-—A driving point impedance function

 

Z(s) having a sequence of distinct, simple poles {pn}
(1im pn = 0c) and a corresponding sequence of residues {rn}

n+oo
can be realized by a network consiting of a countably

infinite number of lumped resistors and capacitors if,
1. all the poles lie on the negative real axis

2. the residues are all real and non-negative

r
I n

H91 pn

Proof: Condition (3) is used to show that Z(s) has a form

similar to the one for k=0 in Mittag-Leffler's theorem.

For Isl : R, E— + 0 uniformly as n + w. So for Isl : R

pn
and n > N, where N is some integer.
r r . r

< 2'3!
_ -pn

 

 

 

r
Hence, Z(s) = Z S_n (4.1)
n= pn

converges uniformly and absolutely on compact sets not
w r

containing any of the poles. :2 11

has sim 1e oles
s-pn P P

 

n=l

50

at pn, n = 1,2,...w, with the corresponding residues rn.
The network of figure 15 is a Foster lumped-

infinite RC realization for Z(s) represented as in eq. 4.1.
Since Iffl! 1 0 as n + w, the impedance of sections

r n
s-p decreases uniformly to zero as n + m. So
n

 

Zn(S) =

a finite network approximatumnnmy beattained by omitting
the sections Zn(s) beyond some N.

A similar set of conditions holds for an admittance

 

function Y(s). The RL network representation for an admit-
tance m
rin
Y(s) = s— (4.2)
n=l pn 1

is shown in figure 16.
4.2 LC Infinite Lumped Networks

Theorem 4.2.--A driving point impedance function

 

Z(s) having two sequences of distinct, simple poles

{+jpn} and {-jpn} (1im pn = w), the corresponding sequences

n-«no

of residues for both sequences being rn . can be realized
by a network consisting of a countably infinite number of
lumped inductors and capacitors if,

1. all the poles are on the imaginary axis i.e., p

n
is real
2. the residues are all real and non-negative
3o Iw'ii < m
n=1 p

51

 

 

 

 

 

 

 

-rn
r:-----
n
I!
I)
2(5) 1
—=r
c n
Figure 15.

RC Infinite-Lumped Network for Eq. 4.1

 

 

 

 

 

 

Efiryure 16. RL Infinite-Lumped Network for Eq. 4.2

52

Proof: Z(s) will be shown to have a form similar to that
of k=1 in the Mittag-Leffler theorem

r r r S
n + n _ n.

s-jpn jpn pnCS-ipn)

 

 

For lsl < R, 3%- + 0 uniformly as n + m. So, for lsl ; R

and n > N, where N is some integer,

 

 

 

r r .2r.R
Isn '+‘nl= n lil lag—l
3p Jpn 2(1_ ) p
jpn n
+ n Fl < 2R l——-I < w
- ls- Jpn Jpn - n21 p 21
n

Hence, Z1 (5) - =2 (S-jpn+ +% will converge absolutely

and uniformly on compact sets not containing any of the

 

poles. m
rn
Similarly Z2 n:;1(sfjpn - 35_lWill also con-
verge.
SO NS) = (s) + 22?.)

 

7"“: (4-3)

converges.

53

This representation of Z(s) (eq. 4.3) has the required poles

at : jpn, n =l,2,...x: and the corresponding residues rn.
Figure 17 is an LC lumped infinite realization for

2(5). The %: capacitor is present if Z(s) has a pole at

the origin with residue r3.

Admittances are treated as in the last section.
4.3 RL Infinite Lumped Netwcrgs

Theorem 4.3.--A driving point impedance Z(s) having

 

a sequence of distinct, Simple poles pn (1im pn = 00) and
n+w
a corresponding sequence of residues rn , can be realized
by a network consisting of a countably infinite number of
lumped resistors and inductors if,

1. all the poles lie on the negative real axis

2. the residues are all real and non-positive

m r

3, T l—E—Zul < no
A p
n=l n

Proof: Proceeding as in Theorem 4.2 it is seen that Z(s) =

 

00
r r
n n
- §_ * —- converges.
n=l pn pn

w rn +lrn/pnl(s-pgi

 

 

5'?
n=l n
:: rn/pn s
n=l S - pn
= l
_ _ 2 - (4.4)
n—l pn/r + pn /r . l/s

l/ro R _

.____ll_ _-________ __ __ _ .___

 

 

 

2(8) +

Figure 17. LC Infinite-Lumped Network for Eq. 4.3

 

 

 

 

 

 

 

rn
1:7
pn
m
R—___ _.__
-r
n
r:_.__
pn

 

Figure 18. RL Infinite Lumped Network for Eq. 4.4

 

55

This representation of Z(s) has poles at pn, n=l,2,...oo
and the corresponding reSidues rn

Figure 18 is an infinite RL realization for Z(s).

4.4 Examples

 

The following two examples yield LC infinite real-
izations.
Example 8: 2(8) = coth s

cot s has simple poles at s = :jkn with residue 1

at each pole.

:ikw!’2 < cc
k=1

cot s has a pole at the origin so by Mittag-
Leffler’s theorem +m

- l - __
cot s - g- + Z (“s-k" + kw)
kz—on

 

Hence coth s -j cot {-js)

._ l (_.__J’__.._ + __l'._)
' g + s+jkn s-jkN'

2s

2 2
v

_ l.
S c2+k

Y‘
L
k=1
Therrealization is of the form of figure 17 with

r°=1;r=1;pk=kw

CHAPTER V
CONCLUSIONS

Foster-type network representations were found for
a class of driving point immittances. The distributed
networks of Chapter III have a finite lumped network re-
presentation, attained by considering the Riemann-sum ap-
proximation of the Cauchy Integral. The lumped-infinite
networks of Chapter IV could also be approximated by using
a finite number of sections, since the impedance of the
sections converged to zero as n + w. .fi

Fabrication procedures for such networks have not
been proposed in this work. It would be desirable to find
a transformation leading to an equivalent Cauer RC infinite
network, since this could be implemented by a single non-
uniform RC distributed line. In the case of lumped net-
works whenever a Foster form exists a Cauer form does also
exist. Both are canonical forms. This indicates that such
a transformation should exist for lumped networks. The
Riemann-sum approximation of the Cauchy Integral leads to
a lumped network and the transformation postulated could
be applied to it. In order to know how closely the trans-

formed Cauer network represented the original integral,

it would be necessary to estimate the approximations

56

 

 

57

incurred in forming the Riemann sum and in applying the
transformation. The former approximation could involve
forming a sequence of Riemann-sums which converged to the
integral. The convergence of the corresponding sequence
of Cauer networks would need investigation.

The driving point immittance functions of passive,
linear, time-invariant networks are characterized by their
singularities. Functions with a countable number of poles
and a line discontinuity were treated. An open question,
the answer to which is possibly in the negative, is--can
positive real driving point impedances which are RC, RL
realizable have any other singularities? This question is
illustrated in figure 19 as question 1. The shaded regions
indicate the class of functions for which realizations have
been deve10ped. All p.r. functions with a finite number of
poles and line discontinuities. Necessary and sufficient
conditions would be needed for all such functions to be
realizable.

Extensions of this work to RLC realizable functions
will involve conjugate linesof discontinuity and countable

numbers of conjugate poles in the negative half plane.

58

POSITIVE REAL IMMITTANCE FUNCTIONS
RC AND RL REALIZABLE

 

L' E DIS’ONT-NU'

 

Figure 19. RC and RL Realizable Immittance Functions

 

APPENDICES

APPENDIX A

State Description for RC, RL Realizable
Driving Point Impedances

61

A state description for the driving point impedance
of a linear, time-invariant, passive network is given by
equations 2.1 and 2.2. For RC and RL networks the singu-
larities of Z(s) are on the negative real axis. In the
following it is shown that equations 2.3 and 2.4 are a
state description for RC, RL realizable impedance functions.

The contour drawn in figure 20 excludes the neg-

ative real axis. The path Br is broken up into three parts

B1’ 32 and B3.
5%3- { Z(s)W(s,t)ds = 5%3Q{Z(S)W(s,t)ds+jrz(s)W(s,t)ds+
Br 1 B2
j’Z(s)Y(s,t)ds (Al.1)

 

3:

By Cauchy's Theorem (Ref. 12, p. 163)

fZ(s)‘P(s,t)ds = -fZ(s)‘i’(s,t)ds-[Z(s)‘l’(s,t)ds-[ Z(s)‘1’(s,t)ds

B C C + C +

l l 3 4

(A1.2)

The solution to equation 2.1 is

W(s,t) = fesVVT) i(T)dT

—00

Assuming i(T) = 0 for t < -T and i(T) :.M for all t :

iii-T

62

 

 

 

 

 

 

 

 

 

w
A
I
s-plane
1P1
n +
*4 y 6
_m w. - - O
. wd
\
C4
/\
32
Figure 20. Contour for Evaluating Eq. Al.l

 

A
2
('D
(D
d-
S
(D .
l
(n
'—3
Q.-
'-3

lT(s,t)l

 

= lgles(T+t)- 1)] (A1.3)

Estimating the integral along C1 in equation Al.2 where

s = Re36 or s = c + jw
ds = jRejede and Isl + w

by equation Al.3

| Z(s)W(s,t)dsl §”{l2(s)%(es(T+t)-l)dsl
C C1

1
+
< lZés) es(T t)R del
‘c
l
:M f|2(s)e°(T+t) del
C1

0 varies from 0 to -m. It is required that 2(5) + 0 as
s + w (cf. Theorem 2.1). Thus this integral tends to zero.

Equation Al.2 reduces to

f Z(s)W(s,t)ds = -"{ Z(s)W(s,t)ds - l, Z(s)W(s,t)ds (Al.4)
+ +

Bl C3 C4

Similarly considering the integral along B2

] Z(s)T(s,t)ds = - j. Z(s)W(s,t)ds - j. Z(s)W(s,t)ds. (Al.5)

32 C3 C4

64

Consider the integral along B3
+‘6
I] Z(s)¥(s,t)dsl < lZ(s)W(s,t)dsl (Al.6)
B3 -j5
2(5) and W(s,t) are analytic in the right half plane.
Hence they are bounded along the interval of integration.
As 6 + O the integral along 83 tends to zero.
Another fact that is obtained from the analyticity of 2(5)

and W(s,t) in the right half plane is that

fZ(s)‘P(s,t)ds = - fZ(s)‘P(s;,t)ds (Al.7)
+ _

4 C4
Equations Al.4, Al.5, Al.6 and Al.7 are substituted in

C

equation Al.l to give
—%3- Jl2(s)W(s,t)ds = - J[ Z(s)W(s,t)ds - Z(s)W(S.t)ds
Br C3 3 (Al.8)
On the line C3+, o varies from - w to O. W(s,t)
is analytic in the entire s-plane, so W(s,t) = W(o,t).

Let Z(s) = z+(o) on C3+.

0n the line C3-, 0 varies from -w to 0. As before
W(s,t) = W(o,t) and let Z(s) = Z-(o) here. Then by equation

2.2 and Al.8,

v(t) = - J[§%3-Z+(U)W(a,t)da - Jr
c + c '
3 3

In

Trj Z (O)W(Clt)do

N

65

Define f(-0) = §%§. (Z—(o) - Z+(o))

as in page 14.
Then,

v(t)

II
SK‘SO

f(-o) W (o,t)do

j f(o) T (-o,t)do (Al.9)
0

Equation 2.1 reduces to

9.15%LEL = -o w (-o,t) + i(t) (Al-1)

Equations Al.9 and Al.lO are the reduced state description

of the driving point impedance Z(s).

APPENDIX B

Proof of Real Valuedness of Density Function

67

Consider a one-port linear, time-invariant, passive,

real network. Let the input i(t) Ioep°t be applied at
time tO (po is a point in the complex frequence p plane).

The output can be written as

v(t) = Z(p°)Ioep°t ts [to,w) (A2.l)
Where Z(p) is the driving point impedance function.
The assumption made in writing the solution as in equation
A2.l is that no transients occur as the excitation is ap—
plied at time to and after. In the case of distributed
networks the apprOpriate distribution of initial voltages
and currents need to be established to assure that this
particular solution alone is excited. This can always be
brought about as long as pO is not a singular point.
Consider a real input
i(t) = Ioep°t + Ice-5°t
the output is real since the network is real. That is,
Z(po) Ioep°t + Z(E.) Ice5°t (A2.2)

is real. Hence the expression in equation A2.2 is equal

to its own conjugate

ETFIT IerOt + 575:7 Ioep°t (A2.3)
If it assumed that I°#0 then equation A2.2 and A2.3 are
equal only if

5757 = 2(5) (A2.4)

The above discussion may be found in references 15 and 16.

68

Consider the boundary condition defining the den-

sity function

Z-(x) - Z+(x) 2nj f(-x)

 

 

Z(E) for lwl > o . Z-(x) = Z+(x)

By equation A2.4 2(5)

and Z-(x) - Z+(x) is pure imaginary. Hence f(-x) is real

valued.

APPENDIX C

A Corollary of Mittag-Leffler's Theorem

70

Given a sequence of distince complex numbers zn
having no limit point in the finite plane (arranged so that

lz lzn+l|)' if there exists an integer k g 0 such that

(l) :2 lznl-k = w and

n=l

(2) Z lznl-k'l < o.
n:

then there exists a meromorphic function f(z) having prin-

n |

 

Cipal parts z_zn at z = 2n for n = O,l,2...m.

The function f(z) may be taken to be

00

k-l
_ l E l 1 z z .
f(Z) -E'+ _ (z_z +E- +-——+... 1f k>0
n— n n z z

n n

 

(The % term is present if there is a pole at the origin.)

00

Ifk=0,f(z)=§ r; .
n
n:

The series for f(z) converges absolutely and uni-

 

formly on compact sets not containing any of the poles.

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