A CLA$3 <35 TEME DOMAEH MODELS
FQR THEE. WWRKAL SQLUTSON G¥
TRANSMESSION MNE, PROBLEMS

Them ‘0» Hm Degree of DB. D.
MICHIGAN STATE UNEVERSITY
Thomas Lynn Drake
£964

THES!S

LIBRARY

Michigan Stan:
University

 

This is to certify that the
thesis entitled

A CLASS OF TIME DOMAIN MODELS
FOR THE NUMERICAL SOLUTION OF
TRANSMISSION LINE PROBLEMS

presented by

Thomas Lynn Drake

has been accepted towards fulfillment
of the requirements for

M degree in JL

airway

Mariel/professor
Date Au u t 1 0

0-169

ABSTRACT

A CLASS OF TIME DOMAIN MODELS
FOR THE NUMERICAL SOLUTION OF
TRANSMISSION LINE PROBLEMS

by Thomas Lynn Drake

The requirements of modern technology necessitate
extremely large and complex systems which contain trans-
mission lines as components. These systems no longer are
composed of components which are described by linear
equations. Therefore, time domain techniques must be used
to analyze these systems. The equations which describe
these large and complex systems are most easily solved by
numerical techniques with large scale digital computers.

The specific subject of this thesis is the derivation
of three general classes of time domain models, using differ-
ence equations, which give the numerical solution to the
lossless transmission line of finite length. In addition to
giving the numerical solution along the line, any model re-
lates the voltage and current variables, corresponding to the
transmission line linear graph representation, in a manner
such that each graph element which represents each port of
the transmission line can be formulated in the system graph
as either a branch or a chord. The system for which the
transmission line is a component may be nonlinear as well as

linear.

Thomas Lynn Drake

One of the main results of this thesis other than
deriving the three classes of time domain models is the ap-
proach which is used in performing these derivations. In-
stead of directly approaching the general transmission line
problem, three transmission line problems, each having
certain identifying characteristics, are first treated by
standard numerical methods. The superposition principle,
even though the mathematical description of the boundaries
may be nonlinear, is then applied to combine the results for
these specific transmission line problems to obtain these
three classes of time domain models. This general approach
by-passes a number of the difficulties which are normally en—

countered by convential techniques.

A CLASS OF TIME DOMAIN MODELS
FOR THE NUMERICAL SOLUTION OF

TRANSMISSION LINE PROBLEMS

BY

Thomas Lynn Drake

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

1964

ACKNOWLEDGMENT

The author wishes to express his indebtedness to
committee chairman, Dr. R. J. Reid for many valuable sug-
gestions and patient encouragement; and to department chair—

‘man. Dr. L. W. Von Tersch for his unfailing support while

this thesis was being written.

ii

Chapter

II.

III.

IV.

TABLE OF CONTENTS

INTRODUCTION . . . . . . . . . . . . . . .

1.0 Introduction

1.1 Superposition Principle

1.2 Thesis Outline

INFINITE TRANSMISSION LINE . . . . . . . .

2.0 Introduction

2.1 Basic Numerical Solution

2.2 Periodic Initial Conditions

2.3 Aperiodic Initial Conditions

2.4 Numerical Differentiation

2.5 Time Domain Models

2.6 Methods of Numerical Solution for
the Time Domain Models

2.7 Special Properties of Difference
Equation Solution

2.8 Conclusion

TRANSMISSION LINE OF TYPE 3 . . . . . . .

3.0 Introduction

3.1 Transformation Matrix C

3.2 Transformation of Time Domain Models
Belonging to Class 1

3.3 Transformation of Class 2 Time Domain
Models

3.4 Transformation of Difference Methods

3.5 Conclusion

LAUNCHING NUMBERS . . . . . . . . . . . .

hbbbbbb J>J>n§
mmqmmpu Nt—‘O

Introduction

Approximate Fourier Integral

Approximate Fourier Integral and
Series Solution

Concept of Consistent Initial Values

Derivation of g(j,pAt)

Interpretation of z (m,-vpt')

Fourier Integral Comparison

Fourier Series Comparison

Problem of Type 2

Conclusion

iii

Page

14

48

61

Chapter

V.

VI.

VII.

VIII.

REMAINING CASES . . . . . . . . . .

U'IU‘IUlU'!
WNl-‘O

Introduction

Chapter III Validity
Chapter IV Validity
Conclusion

INTERPOLATION FORMULAS . . . . . . .

6.0

Introduction

6.1 At Least One Boundary Condition

6.2
6.3

Specified Zero
Both Boundary Conditions Not
Identically Zero
Conclusion

GENERAL TRANSMISSION LINE PROBLEM . .

7.0

Introduction

Classes A and B

Class C

Discussion of Models

Both Graph Elements Are Not Chords
Example Problem

Conclusion

CONCLUSION . . . . . . . . . . . . . .

8.0
8.1
8.2

Conclusion
Limitations
Additional Problems

LIST OF REFERENCES . . . . . . . . .

iv

Page

113

123

130

160

164

LIST OF FIGURES

Figure Page

1. Oriented linear graph for example problem . . 148

2. Graph of solutions for k=4 and transformation
constants specified by Eqs. 7.5.8 . . . . . 154

3. Graph of solutions for k=8 and transformation
constants specified by Eqs. 7.5.8 . . . . . 155

4. Graph of solutions for k=4 and transformation
constants specified by Eqs. 7.5.9 . . . . . 156

5. Graph of solutions for k=8 and transformation
constants specified by Eqs. 7.5.9 . . . . . 157

I. INTRODUCTION

1.0 Introduction

A lossless parallel transmission line of length L is

described mathematically by the telegrapher's equations

_%vgx,t) =lbet
x t
_bet=cvat
- x t

‘where 1 and c are the inductance and capacitance parameters
for the line. For a given set of initial conditions and an
appropriate boundary condition for each end of the line,
there are a number of techniques, both analytical and numeri-
cal, discussed in the literature for finding the solution to
these equations.

Most analytical techniques for finding the solution
to these equations are applicable only to a restricted class
of problems, generally those for which the mathematical de-
scriptions of the boundaries are linear. Numerical methods
on the other hand, are by no means so restricted. In fact,
most numerical methods of solution allow the mathematical de—
scriptions of the boundaries to be nonlinear as well as
linear. Hence, the subject of the thesis is concerned with

numerical methods of solution for these.equations.

l

 

Any numerical technique consists of a finite set of
equations, obtained by some method, for which the numerical
solution to these equations approximates V(xj,tk) and

I(§p,tk) respectively on the two sets of points in the x,t-
plane, (Xj'tk) and (xp,tk), where the two sets of points,
(xj) and (RP), are finite. The two sets of points, (xj,tk)

and (ip'tk) are respectively called the voltage and current

net points. In addition, the two sets of points, (xj) and

(§P)' are respectively called the voltage and current node

points. Normally, the sets of points, (tk), (xj), and (RP),

are chosen such that the points belonging to any given set are
equally spaced. Hence, if,Ax and At are the increments of the
variables x and t, the sets of voltage and current net points

are given by xj=x0+ij, tk=kA¢, where j=0,1,2,...,J and
k=0,1,2,..., and §p=§0+pr, tk=kDm, where p=0,l,2,...,P and

k=0,1,2,...,. Generally, x0 and E0 are related either as

or x nxoiAx/Z.

x0:320 o
The numerical techniques which are discussed in the
literature can be placed in one of three general categories.
The first such category contains numerical techniques which
employ a Fourier series. This technique simply uses numerical
techniques to find the Fourier coefficients of the series in-
stead of the normal analytical techniques. This method of

solution generally can only be used in cases where the mathe-

matical descriptions of the boundaries are linear.

The second category contains the methods which ap-
Proximate the transmission line by a passive lumped parameter
electrical equivalent network. The resulting network equa-
tions can then be solved by some numerical technique. This
approximate equivalent network approach is probably the most
'widely used by the engineering profession. The biggest
shortcomings of this approach are that all physically real-
izable passive networks either require a large number of ele—
ments to be a good approximation, an exceedingly small At to
solve the network equations numerically, or both.

The last category contains finite-difference methods
for the approximate numerical solution for problems of this
kind. This is the general approach which has been studied
extensively by applied mathematicians. The partial deri-
vatives are first approximated by difference equations at
the net points and the resulting difference equations are
then solved. Instead of attempting to solve the telegrapher's
equations, the literature is primarily concerned with solving
the wave equation by finite-difference methods. Summaries
of these methods are given by Richtmyer (l), Kunz (2), and
Fox (3). These methods are derived on the basis that each
end of the line is terminated in a voltage (current) source
which has a zero source impedance (admittance). If the line
is terminated differently, most of the properties which are
derived for these methods are no longer valid. Therefore,

their biggest shortcoming is that most practical problems

encountered in electrical engineering do not have boundary
conditions for which these methods were intended.

The specific subject of the thesis is the derivation
of three general classes of numerical time domain models,
using difference equations, which give the approximate nu-
merical solution to the lossless transmission line of finite
length. The equations, corresponding to any given time do—
main model, are generally not realizable as a passive lumped
parameter electrical network. In addition, these three
classes of models are applicable for the practical problems
encountered in electrical engineering. The mathematical
descriptions of the transmission line boundaries are allowed
to be nonlinear.

The mathematical development of the thesis treats a
normalized set of telegrapher's equations which are obtained

by introducing a change of variable. By letting

V(x,t) = yl/c E(x,t),

the normalized telegrapher's equations can be written as

._ b:(x,t) = VIE bigx‘tz
__O:(x,t) = VIE'OEflX,t).

The resulting set of equations has the properties that the

characteristic impedance has been transformed to unity and

the phase velocity vp remains unchanged. The quantity Vlc
is easily recognized to be 1/vp.

Each numerical time domain model is derived on the
basis that the transmission line, having finite length, is to
be a two port component in a given system. For any given
model, the set of equations which define this model are such
that the transmission line terminals can be represented as a
two part linear graph with one element in each part. In ad-
dition to giving the approximate solution at the node points,
these equations relate the voltage and current variables,
corresponding to the transmission line linear graph represen—
tation, in a manner such that each graph element can be
formulated in the system graph as either a branch or a chord.

Unless otherwise stated, the mathematical development
assumes that both linear graph elements are chords. Chapter
5 shows that the remaining three cases, one by symmetry, imr
mediately follow from this case.

For this case, the increment Ax is chosen as Ax=L/k,
where k is a positive integer. The set of voltage node

points, (xj), are defined as szij, where j=0,1,2,...,k.
In addition, the set of current node points, (Rb), are de-
fined as E§=pr+Ax/2, where p=0,l,2,...,k-l. These sets of

node points are defined differently for the cases where
either one or both graph elements are not in the cotree of

the system graph.

The node points in a given set are equally spaced.
Hence, standard numerical methods which use equally spaced
data points are applicable for the derivation of the time
domain models. On the other hand, these two sets of node
points are purposely defined so that the intersection of the
two sets is the null set. The basic reason for defining the
node points in this manner is that the derivations and the
construction of any time domain model is greatly simplified.
This will become evident during the derivation of these

methods.

1.1 Ԥuperposition Principle.

All three classes of time domain models are derived
on the basis of certain mathematical properties of the theo-
retical solution to the normalized telegrapher's equations.
One important property on which the derivations are based is
the superposition principle. Hence, certain specific appli-
cations of this principle must be discussed first in order
to indicate the develOpment of the three classes of time do—
main models.

For the purposes of this discussion, let us assume
that the transmission line is a component in a given system.
In addition, let us assume that the entire system can be
represented as a two part linear graph containing two ele—
ments in each part for which the linear graph which repre-

sents the transmission line is the cotree of the system graph.

As far as the solution of normalized telegrapher's
equations is concerned, the initial and boundary conditions

are given as

E(x,0) = hl(x) 0<x<L
Ibgm =hflx) OEXSL
E(0,t) = Esl(I(0,t),t) Ost<oo
E(L,t) = E52(I(L,t),t) Ost<oo

where hl(X) and h2(x) are specified functions of x. The
functions Esl(I(0,t),t) and ESZ(I(L,t),t) are determined in

some manner for tfi: 0 by the mathematical description of the
system components which correspond to the branches of the
tree.

Because of the linearity of the telegrapher's equa—

tions the superposition principle can be applied at t=tO as

long as the boundary conditions for the telegrapher's equa-
tions are properly defined. This property is used to sub—
divide the transmission line problem at t=tO into three prob-
lems, each having certain identifying characteristics.

Let us consider the three transmission line problems
which are described by the following sets of initial and

boundary conditions.

0<x<L 1.1.0

I
O

El(x,t0)

l
O

Il(x,t0) O<fo

l
O

El(L,t) t fit-=00

O
El(0,t) = Esl(I(O,t),t) toft<00
11(01t0) = El(0lt0)
E2(X,t0) = O O<X<L l.
E2(L,t) = E82(I(L,t),t) toft<oo
I2(X,t0) = O OfX<L
E2(O,t) = 0 toft<oo
12(L,t0) = E2(L,t0)
Eflxfib)=fm&tw 0<x<L Ll.
I3(X,t0) = I(x,to) O<X<L
E3(O,t) = E3(L,t) = O toft<oo

I3(O,t0) = I(O,t0) - E(0,t0)

I3(L,t0) = I(L,t0) + E(L,t0)

It is quite clear that the solution for E(x,t) and I(x,t)

for t_>_tO is given as
E(x,t) = El(x,t) + E2(x,t) + E3(x,t)
I(x,t) = Il(x,t) + 12(x,t) + I3(x,t)

where all current orientations are from x=0 to x=L.

Upon examining these three transmission line problems,
one finds that there are two transmission line problems which
have zero initial conditions and one identically zero boundary
condition. The remaining problem has the boundary conditions
specified as zero but has initial conditions which may be
nonzero. For the purposes of simplifying the discussion
throughout the thesis, the transmission line problems which
have the identifying characteristics the same as the problems
which are described by Eqs. 1.1.0, 1.1.1, or 1.1.2 are
classified respectively as problems of type 1, type 2, or
type 3.

At t=tOAt, the same subdivision process can be ap-

plied to the solution of each of the two transmission line
problems which are described by Eqs. 1.1.0 and 1.1.1. These

two solutions at t=tO+At define two problems of type 3, a

problem of type 1, and a problem of type 2. This subdivision
process in effect states that the solution for E(x,t) and

I(x,t) for tit +At can be considered the sum of the solu-

0
tions of a problem of type 1, a problem of type 2, and three
problems of type 3.

Let us consider the two problems of type 3 which

were defined at t=tO+At. Both of these problems are deter—

mined respectively by some process which converts the
boundary conditions E(0,t) and E(L,t) for the interval

1: 5t4=t0+At into initial conditions for these two problems

0

of type 3 which were defined at t=tO+At.

10

At t=t0+At, the three problems of type 3 can be com-

bined by superposition to define one problem of type 3. As

a result, the solution for E(x,t) and I(x,t) for t33t0+At

can also be considered as the sum of the solutions of a
problem of type 1, type 2, and type 3.

The specific approach of the thesis is then first to
derive two general classes of time domain models which are
applicable for obtaining the approximate numerical solution
for E3(x,t) and I3(x,t) at the defined node points. Next, a
method for converting the boundary conditions E(0,t) and

E(L,t) for the interval tojft<=tO+At into initial conditions
in order to define two problems of type 3 at t=tO+At is de-

rived. It is quite clear that a time domain model for the
problem of type 3 and the method for converting the boundary
conditions into initial conditions can be combined by super-
position to obtain a time domain model which is valid for
the original problem. This combination process introduces

three classes of time domain models.

1.2 The§is Outline.

According to the discussion as given in section 1.1,
it is necessary to derive a time domain model which is appli—
cable for the problem of type 3. In order to obtain these
models, the fact is used that a problem of type 3 has the
same solution as an infinite line with certain periodic

initial conditions for the interval 0:5x:EL. Therefore,

11

instead of directly approaching the problem of type 3, the
infinite transmission line is treated extensively in
Chapter 2.

In Chapter 2, two classes of time domain models are
derived for the infinite line. One such class of models is
derived on the basis of a basic numerical solution which is
defined in this chapter. The other such class of models is
derived by obtaining a set of ordinary differential equa-
tions in t by approximating the partial derivatives with re-
spect to x at certain defined node points by difference
methods.

Chapter 3 shows that both classes of time domain
models which were derived for the infinite line can be trans-
formed into the finite line problem of type 3. As a result,
all the properties which are valid for the infinite line,
having periodic initial conditions, are also valid for the
problem of type 3. One important result of this chapter is
the derivation of a matrix which is called the transformation
matrix which transforms all the properties of the infinite
line to this finite line problem of type 3.

In Chapter 4, a class of methods is derived which
transforms the boundary condition, E(0,t), for a problem of
type 1 for the interval toiit-=t0+bm into an initial condi-
tion at t=t0+At. Each method derives a set of launching
numbers, one for each node, such that the initial condition

at each node point at t=t0+At is the product of E(0,nAt) and

12

a launching number. Because of the similarity existing be—
tween a problem of type 1 and type 2, this class of methods
for obtaining these launching numbers is also valid for the
problem of type 2. The specific approach of this chapter is
the use of the superposition principle in order to show that
for a given problem of type 1, its solution at t=pAt is the
sum of the solutions of p problems of type 3 which are
staggered in time for 0'=x:EL. This class of methods is
then derived on the basis that the sum of these p solutions
for the p problems of type 3 must approximate the solution
to the problem of type 1 at t=pAt in a certain sense.

Since the results of Chapters 3 and 4 are valid only
for the case where both graph elements which represent the
transmission line are formulated as chords, Chapter 5 shows
that these results are also valid for the other three cases
which arise in the formulation. In order to obtain this re-
sult, the node points must be redefined such that the node
point at either x=0 or x=L is a voltage (current) node point
when the linear graph element which represents this port is
formulated as a chord (branch).

Since there is no current node point at either x=0
or x=L, there is no information present in these equations
about the approximate solution to I(x,t) at these points.

In order to provide this information, Chapter 6 derives some
properties which are valuable for determining an interpo-

lation formula which is applicable for determining the

13

solution for the currents at these points. In addition,
these special properties are also applicable in certain situ-
ations for determining the solution of either E(x,t) or
I(x,t) at other values of x than the defined node points.

The principle objective of Chapter 7 is to apply the
superposition principle to combine the results of Chapters
2, 3, 4, 5, and 6 to obtain three classes of time domain
models which are applicable to the general transmission line
problems which occur in electrical engineering. Since there
are two classes of numerical time domain models defined for
the problem of type 3, the three classes of models are created '
by the method for which the results of Chapter 3 are inte-
grated with the results of Chapter 4. In addition, a trans-
mission line problem is worked to illustrate the concepts

which are presented in this chapter.

II. INFINITE TRANSMISSION LINE

2.0 Introduction.

An infinite lossless parallel transmission line is
described mathematically by the normalized telegrapher's
equations. In order to distinguish the infinite line case
from the finite transmission line case, the variables,
V(x,t) and S(x,t), are used in place of E(x,t) and I(x,t) to
represent the solution to these partial differential
equations.

The specific subject of this chapter is the defi—
nition of two general classes of time domain models, which
use difference methods, for the purpose of obtaining an ap-
proximate numerical solution to the infinite line. One such
class of models is derived on the basis of a basic numerical
solution while the other class of models is obtained by ap-
proximating the partial derivatives with respect to x at
certain node points and obtaining a set of ordinary linear
differential equations. In addition, certain properties of
the approximate solution such as periodicity will be
investigated.

Once the initial condition functions, V(x,0) and
S(x,0), are specified for -a><rx‘<oo, the analytical solution

for V(x,t) and S(x,t) can be immediately written for all x

14

15

 

 

 

 

and t as
V(x—v t,0) + S(x—v 12.0)
V(x+vpt,0) - S(x+vpt,0)
2
V(x—th,0) + S(x—vpt,0)
S(x,t) = , 2 +
S(x+vpt,0) — V(x+vpt,0) .
2

Since the analytical solution for infinite line problem can
be immediately written from the specified initial conditions,
it may now seem of limited value to pursue the definition of
the two general classes of time domain models. The only
reason for treating the infinite line time domain models is
that all of these models and their properties are also valid
for the finite line problem of type 3. The main advantage of
treating the infinite line over the finite line is that there
are no boundaries which must be taken into consideration.
Hence, standard numerical methods which use more accurate
central differences can be used for the purpose of deriving
these models.

The voltage and current node points are defined re—

spectively as xj=ij, j=0hilhi2h13,..., and xp=pr-Ax/2,

p=0ailh12hi3,...,. In order to simplify the notation, the
functions f(2n,t) and:f(2nrl,t) are respectively the approxi-
mate solution for V(nAx,t) and S(nAx—Ax/2,t) which are ob—

tained by some time domain model.

16

Each initial condition is allowed to be periodic in
x with period 2L or aperiodic in x. If an initial condition
is aperiodic in x, where either V(x,0) or S(x,0) is given by
h(x), for eoo<=x~=oo, h(x) is required to satisfy the Fourier
integral theorem and to be differentiable almost everywhere.
When V(x,0) or S(x,0) is periodic, this initial condition is

required to be differentiable almost everywhere.

2.1 Basic Numerical Solution.

The theoretical solution for f(n,t) for all t and

n=0hilh12hi3,..., in terms of the specified initial condi—

 

 

tions is
V(nAx/z-v t,0) + (-1)n V(nAx/2+v t,0) 2.1.0
f(n,t) = v .1 J: 2- a--- ..
S(nAx/Z—vpt,0) + (-1)n+1 S(nAx/2+vpt,0)

2

Let us assume that the specified initial condition
V(x,0) is known only at the voltage node points. In addition.
let us assume that S(x,0) is only known at the current node
points. The values of V(x,0) and S(x,0) at values of x
other than at their respective node points can be obtained
by means of interpolation. One method of interpolation is

to define two infinite series, Pv(x) and PS(x), which pass

through f(2n,0) and f(2n-l,0) respectively for
naqilhi2h13,...,. It is quite evident from the definition

of f(n,0) that Pv(x) and Ps(x) are the infinite series

l7

approximations to V(x,0) and S(x,0). Hence, an approximate
solution for f(n,t) can be obtained by replacing V(x,0) and

S(x,0) in Eq. 2.1.0 by Pv(x) and Ps(x). This approximate

solution for f(n,t) is then given by

n
Pv(nAx/2-vpt) f (-1) PV(nAx/2+vpt)

f(n,t) = 2 + 2.1.1

 

PS(nAx/2-vpt) f,('l)n+l P§(nAx/2+vpt)
2

 

It is quite clear that there are a number of infinite
series which pass through the prescribed points but are not

equal at the remaining values of x. For example, Pv(x) is

given be the two infinite series

P (x) = f(2n 0).§in2 7T(XAAx-n)
v ' 7T2(X/Ax-n)2
nz—oo

°° ' 7T; éA 2
' Sln X X‘n
PV(X) z: f(2n,0) ”(X x-n)

n=- 00

where both series equal f(2n,0) at x=nAx. In order to

correct this ambiguity, the exact form of Pv(x) and Ps(x)

must be defined. Once these two infinite series are uniquely
defined, Eq. 2.1.1 will be defined as the basic numerical

solution for f(n,t).

Let the operator érfor k=0,1,2,..., be defined as

18

6k£(n,t) .-. (-l)i(]:) f(n—k+2i,t) 2.1.2

1::

where (E) is a binomial coefficient. Then the operator

6k.” has the property

6j+kf<nrt> = éjmkflmtn = (Sk((5jf(n,t)). 2.1.3

If AI corresponds to the standard central difference operator
(2) for fixed t, this Operator, AF, is related to the

operator 6k by

Akf(n.t) = (-1)k (5kf(n.t).

The most accurate interpolation polynomials for a
given interval are those which lead to interpolation at or
near the middle of the interval. These types of formulas
are called central-difference formulas. One such formula is
Stirling's interpolation formula (2) which represents an
interpolation polynomial that is based on tabulated values

symmetrically placed with respect to x0. Hence if x0=nAx
and §=(x-x0)/Ax, one method for finding Pv(x) is by using an

infinite degree Sterling's polynomial interpolation formula.

This formula for PV(§) can be written as

19
pv(3’c) = f(2n,0) - x#6f(2n,0) + 2.1.4

k-l
7T (x2 -j 2)

i=0? (2k): 62kf(2“'0) '

 

kTT: (x2 -j 2)
x 1(2k-1)!#6

2k-1f n,0)

=2

where

2k-l 2k—l
#62k—1fl2mo) = (5 f(2n+l,0) :- 6 fun-1,0) ,

It is quite clear that Pv(§)=f(2n+2i,0) for x=i.

Another type of central difference formula is a
Bessel's interpolation formula (2). This interpolation
formula is also based on tabulated values symmetrically placed

with respect to xl=xO+Ax/2. By letting v=(x—xl)/Ax,
Pv(v) can be written in terms of a Bessel's interpolation

formula as

Pv(v) = #6Of(2n+l,0) - v6f(2n+1,0) + 2.1.5

k 2 2
7T [v ~(2j-l) /4]
El (2k): I” liézkflznfl'ol ‘

20
k

7T [v2—<2j-1)2/4

(2k+1),—~ m— 62k+1f(2n+1,0)

where

2k 2k
uézkf(2n+l.0) = d f(2n+2.oré 6 jf(2n,0) ,

For v=i—l/2, Pv(v) = f(2n+2i,0).
Consider the even function with respect to t which

is given by

Py(nijvpt) + Pv(nAx+vpt)
2

 

By using Eq. 2.1.4, this function is given by

Pv(nAx-v2t) + Pv(nAx+vpt)

*2“ = f(2n,0) +

k-l ~ 2 ‘2
7T [(vpt/Ax) -j ]
.lio 2k
(2k), 6 f(2n,0)
k:
In the same manner, the odd function with respect to t, given

by

Pv(nAx+Ax(2-vpt) —‘Pv(nAx+Ax/2fvpt)
-. 2 _1

 

can be written terms of Eq. 2.1.5 as

21

Py(nAx+Ax/2~vpt) - Pv(nAx+Ax/2+vpt) v t

 

 

2 = *E; f(2n+l,0) +
co k 2 . 2
V." V t ,7T (th/Ax) -(2J—1) /4
P 41:1 ‘ , 2k+l ,
L Ax i (2k+1): (S f(2n+1,0)
k=l

If the same interpolation formulas are used to find

Ps(x), then an approximate solution for f(n,t) is given by‘

2
f(n,t) = f(n,0) + v6f(n,0) + 525-.- (52f(n,0) 4- 2.1.6

 

2 2 2
V(V 3:141 63f(n'0) + l—(fi—i-fl 64f(n,0) +...+...

where v=vptAAx. It will later become evident that both

Stirling's and Bessel's formula give the same infinite series

for Pv(x). Since Eq. 2.1.6 is obtained directly from
Eq. 2.1.1 by replacing Pv(x) and Ps(x) by either their

Stirling's or Bessel's formula representation, Eq. 2.1.6

must be the basic numerical solution for f(n,t). When
Bessel's and Sterling's formulas are used to obtain inter-
polation polynomials of finite degree, Bessel's and Stirling's
interpolating polynomials respectively pass through an even
and an odd number of points. Hence, these polynomials are
generally different in the finite degree case. For the in-
finite degree case, these infinite series as given by

Eqs. 2.1.4 and 2.1.5 give the same function. In order to

show this result, it is necessary to write an alternate form

for Eqs. 2.1.4 and 2.1.5.

22

Consider the polynomial p§n(x) of degree 2n which is

 

 

given by
g H “2]
. 1-——-———-
p?n(x) a ~1)3—;n! n! x p=l P%AX2 .
J . (n+3): (n—j): ij-x

This polynomial has the property that

P§n(X) = 0 x=pr, p=n,-n+l,...,j—l
2n . .

Pj (X) = 0 . x=pr, p=j+1,j+2,...,n

p§n(x) = l x=ij

By letting i=x-3Ax. an interpolating polynomial which
passes through f(23+2i,0) for i=0hilhi2,...hin, can be

written as
p2n(§) = f(2fi,0) pg“(§) +

n

[f(2'fi+2j,0) pinfit) + f(23-2j,0) p3?(§)] .

i=1

The polynomial p2n(§) is identical to all interpolation poly-

nomials of degree 2n which pass through the prescribed points.

Hence, p2“(§) is identical to a Stirling's approximation
which is centered about x=HAx. In the limit as n—+-oo,
p2n(§) is identical to the Stirling's approximation for

Pv(§) as given by Eq. 2.1.4.

Let us consider the limit as n-a-oo of

23

Zn-

f(2n+2j, 0) pj

(X )

for fixed j. According to Knopp (4), this limit is defined

and is given by

2n — . sinll §-' x x
lim f(2n+2j, O) pj (x) = f(2n+23,0) -————J-:JA—%fi%—— -
n~e-oo 71(x JAx x

Hence, the limit as n—e-aa of p2n(x) must be given as

. 2n - - Sin 773? x
11m p (x) = f(2n.0)'—-—?f14Le-+
n-era3 sz
oo _ A

— . sin fllg— ij)z x
Z [f(2n+23,0) 7T(x-ij)/Ax+

_ . si' 7T 32+ 'AX X -
f(2n~23,0) (§+ij)AAx]

An interpolation polynomial which passes through
f(2fi+2i,0) for i=-n,-n+1,...,-l,0,+l,...,n,n+1, can also be

written as

2n+2

 

 

 

(n+1) p (X)
p2n+1 — 0
(X) = f(2n,0) n+1+x x +
n
[ (n+j+1> pj“+2(i)
f(2n+2j'0)ummnfin+1+x/Ax +
j=1
_ , (n-j+l) p23+2(x)]
“Zn-2L0) mm. +
_ (2n+2) p2n+2(§)
f(2n+2n+2,0) n+1+§AAx -
2n+l -

This polynomial, p (x), of degree Znfil is identical to a

Bessel's approximation which is centered about x=hAxtAx/2.

24

The limit as n-—>oo is

lim p2n+1(§<) = f(2'fi,0) J—éfi—Smfl: : +

n—s-oo

.- . sin 32-' Ax
i [f(2n+23.0) W +

i=1
. - . sin 3':+jA Ax
f(zn'ZJ'O) (2+ij) Ax] ‘

It can be concluded that Eqs. 2.1.4 and 2.1.5 are equivalent

definitions of Pv(x).

Using this form of the interpolation formulas, Pv(x)

and Ps(x) respectively can be written as shown in Eqs. 2.1.7

sin x x
PV(X) = f(0,0)'———%¥§é%; + 2.1.7
. sin x-'Ax x . sin x+' x
2 [“23'0) (x-ij) Ax + “’23'0’ Wm;- x) x]

i=1

sin x—Ax 2 Ax

. sin x- Ax—Ax 2 X
[f(23+1:0) (x—ij—Ax/Z) AX +
j=1
. gin mx+jAXTé_léA_-AX 2 Ax
f(-2]+l.0) 7TKX+jAX' x 2) X]

where —oo <X<OO.

 

An alternate form of the basic numerical solution for
f(n,t) which is identical to Eq. 2.1.6 can now be obtained by

Substituting Eqs. 2.1.7 into Eq. 2.1.1.

Sections 2.2 and 2.3 will use the expression for Pv(x)

and Ps(x) as given by Eqs. 2.1.7 to show that these

25

approximating functions are defined for all x. Therefore,

the basic numerical solution must also be defined for all x.

2.2 Periodic Initial Conditions.
In section 2.1, Pv(x) and Ps(x) are given by Eqs.

2.1.7. If an initial condition function is periodic in x
with period 2L, it will be shown that the approximating
function is defined for all x and that the approximating
function is a Fourier series on 2k sample points.

If the initial condition V(x,0) is periodic in x
with period.21u Ax is defined to be Ax=L/k, where k is a
ypositive integer. For this choice of Ax, the functions

f(2j,0) are given as

f(2j+2nk,0) = PV(ij,0)

for j=0,1,2,...,2k-1, and n=0,il,:2,...,. Hence, Pv(x) can

be written in terms of f(2j,0) for j=0,1,2,...,2k-l, as

 

 

2k-l
. _s_iun ”(x/Ax-j) I 1
Pvm = z “23"” 277k lx/zkAx-j/zk +

j=o

 

 

l 1
[x/2kAin/2k-i + x/2kAx-j/2k+i]

'MS

1

.1.

.According to Knopp (4), the infinite series portion of the

expression is given by

26

 

 

00
_____ll;___.+ .rl + .1 .11
x/2kAx-j/2k x/2kAx-j/2k-i x/2kAx—j/2k+i =
i=1

cot TT(x/2kAx-j/2k) .

 

Therefore,
2k—l
PV(X) = Z f(szO) Sjéffi ”(X/AX’J.) COt ”(x/ZW’j/Zk)
i=0

In order to simplify this expression for Pv(x), let

2 = (x/ZkAx) - (j/Zk)

_ ginLZkTTz) cos 7T2 .
y(z) - sinTTz

Upon‘substituting,

2k—l
_ f(2j,0) y(z) ,
,PV(X) I :E: 2k

j=0

 

But y(z) can also be written as

2 .
y(z) = COS ”gins”; ”ZR-'1 2 + cos 7T2 cos 7T(2k-l)z .

This expression for y(z) can be simplified by apply-

 

ing Lagrange's identity (5). Hence,

k-l
Y(z) = cos2 7T2 [1 + 2 Zcos 2m7Tz] +

m=l

cos 7T2 cos 7T(2k-1)z

27

By means of trigonometric identities,

k—l
y(z) = l + 2 E: [cos ZTTmz + cos 27Tkz]
m=1
k—l
= l + 2 Z [cos m7Tx/L cos m7Tij/L] +
m=1
k-l
2 Z[sin mTTx/L sin m’ijAx/L] + cos k7Tx/L cos TTj .
m: .

pv(x) can now be written as

k-l k-l
A0 .
Pv(x) = -2- + Am cos mTTx/L + Bm Sln m 7Tx/L +
m=1 =1
-2-- cos k7Tx/L
where
Zk—l
Am =3]: Z f(2j,0) cos m7Tij/L
i=0
2k—l
Bm =_]_._ E: f(2j,0) sin m7Tij/L .
I k

i=0

This is a Fourier series on 2k sample points (6).

28

If the initial condition S(x,0) is periodic in x
with period 2L and Ax=L/k, the same argument can be applied

to show that the approximating function Ps(x) can also be

written as a Fourier series on 2k sample points.

Since the initial conditions are differentiable
almost everywhere, in the limit as Ax-—+-0, the approximating
functions will approach the actual initial conditions al-

most everywhere.

2.3 Aperiodic Initial Conditions.

The purpose of this section is to show that the
approximating functions for the aperiodic case are defined
and that each approximating function approximates a Fourier
integral of the initial condition.

In section 2.0, an initial condition was required to
satisfy the hypothesis of the Fourier integral theorem (5)
for the aperiodic case. Hence at every point x(—a3<:x«=oo),
where V(x,0) has a right and left-hand derivative, V(x,0) is
represented by its Fourier integral as follows:

vg+o,0)g+rv(x—0_Loi _
2 _

29

Consider the function V(x,0) which is given by

 

OO
Wm) 71% V(x’.0) Si“ xT-rff’XMA—x dx' -
—00

It is clear that V(x,0) and V(x,0) have the same frequency
spectrum for 05 O(<7T/Ax. For O(>777Ax, the function V(x,0)
contains no spectral components. In addition, if 7TKAX is
sufficiently large, V(x,0) is a good approximation to
V(x,0).

If the expression for V(x,0) is numerically inte-
grated using the trapezoidal rule (6), an approximate ex-

pression for V(x,0) is given by

 

oo
- é . gin 'TKijAx)/Ax ,
V(x,0) - V(ij,0) mx-ij)7LT:E
j='°°

At x=ij for j=0,_-_I-_l,_4_-2,j-_3,..., \7(x,0) is identically equal
to V(x,0).

Since V(x',0) satisfies the hypothesis of the
Fourier integral theorem, the approximate expression for
V(x,0) must be an absolutelyconvergent series. Upon re-
arranging this series, it can be seen that the approximate

expression for V(x,0) and the expression for Pv(x) which

30

is given by Eq. 2.1.7 are identical. Hence, Pv(x) is an

approximate Fourier integral of V(x,0). In the limit as

Ax——v0, Pv(x) becomes the exact Fourier integral of V(x,0).

The same procedure can be applied to S(x,0) to show

the validity of the approximating function Ps(x).

2.4 Numerical Differentiation.

The standard approach for finding a set of difference
equations which give the approximate numerical solution to
an equation containing partial derivatives is to approximate
the partial derivatives with some numerical differentiation
scheme. A technique similar to this will be used to solve
the telegrapher's equations numerically. Hence, the purpose
of this section is to define a method of numerical
differentiation.

Consider the function g(x,t) which is given by

V(x+ij+A§/24t) — V(-x+jgx+Ax/2,t)
g(xlt) = *2 '

At t=t g(x,to) is an odd function with respect to x. In

OI
addition, at x=0, the partial derivative with respect to x
of g(x,t) is identical to the partial derivative with re-

spect to x of V(x+ij+/2,t). Therefore, the specific ap-

proach is to derive a numerical differentiation formula which

31

approximates the partial derivative of g(x,t) with respect

to x at x=0.

Once the functional values of g(x,to) at
)fiiAx/ZhiBAx/2hiSAx/2,...hi(2n+l) Ax/2, are given, g(x,to)

can be approximated with a Bessel's interpolation polynomial
of degree 2n+l which passes through the prescribed points.

Hence, g(x,t0)=p2n+1(x), where p2n+1(x) is interpolating

polynomial. This polynomial in terms of the known values of

V(x+mAx+Ax/2,t0) can be written as

p2n+l(X) = - fi6f(2m+l.to) "'

 

n k 2 2
7T [(x/Ax) -(2j—1) M] 2,,“
3.31 0 f(2m+1,to) ..
Ax (2k+l)!
=1

It is evident that the interpolating polynomial is an odd

function of x.

The derivative of p2n+l(x) evaluated at x=0 is de-

fined as a 2n+l degree approximation for the partial deriva-

tive of V(ij+Ax/2,t0) with respect to x. This approximation

is given by

n+1 n1 2
.71 (21-3)
) = - ”=1 v . . 2m“'1f(2j+1,t )
Ax (-2)’“’1 (am-1): 6 °

 

p2n+1(O

m__. 2.4.0

613 ( jAX+Ax/2 . to)
= bxw _ fl

 

32

In order to get an idea of the error which results by
using this numerical differentiation formula, a remainder
term will be derived. For the derivation, it will be assumed
that the (2n+3)th partial derivative of g(x,to) with respect
to x exists in the interval of interpolation,
-(2n+l)Ax/2§xf(2n+l)Ax/2. The function g(x,to) is given

exactly for all x in this interval by
g(x,to) = p2n+l(x) + h(X)

where h(x) is unknown function of x. Since p2n+1(x) is
equal to g(x,to) at x=0,iAx/2hi3Ax/2,...,i(2n+1)Ax/2, h(x)

must have real roots at these points. Therefore, h(x) must
be a function which is identically zero or an odd function
which has at least 2n+3 real roots in the interval of
interpolation.

The partial derivative of g(x,to) with respect to x

in this interval is given exactly by

bg(X.t0) '
FOX — p2n+l

 

(x) + h'(X)

where pén+l(x) is a polynomial of degree 2n. By Rolle's
theorem (7), h'(x) has at least 2n+2 real roots (21) such

that ‘zi «<(2n+1)Ax/2. Therefore, the partial derivative

 

of g(x,to) with respect to x can now be written as

33

 

 

2n+2
'(x—z.)
09(xito)__ . 1:: 11 2.4.0
bx " p2n+1(X) + k(x) "t (26-1-2):

Consider the function w(y) which is defined by

 

2n+2
(y-2-)
()— 69(y't°)- ' ()-k(')''"'-'''""""""""'i7=T1 l -
WY "' By p2n+l Y x (2n+2)!

By virtue of Eq. 2.4.0, w(y) has at least 2n+3 real roots

x.zl,...,22n+2. Upon choosing x in the interval of interpo-

lation such that these roots are distinct, Rolle's theorem
can be applied to show that w'(y) has at least 2n+2 distinct

real roots (2i) such that [Bil < (2n+l)Ax/2. Therefore by

successive applications of Rolle's theorem, k(x) can be
determined. Hence,

(2n+3)
b E(éi'to)
by(2n+3) fl w

 

k(x) =

where -(2n+l)Ax/2 < g < (2n+1)Ax/2.
The partial derivative of g(x,to) can now be written
as

2n+2

bg(x,t0) ézn+3’g(§.to) 77 ”'21)

l __ .31 ,
ax "' = P2n+1(X) + bxl2n+3) (2n¥2$:

Since our main interest is the partial derivative of g(x,to)

 

 

with respect to x at x=0, this partial derivative is given by

34

 

 

 

2n+2
(2n+3) 7T ,(_zi)
bg<OItO) I b 9(§:to) i=1
bxi’ z p2n+2(0) + bxi2n+§) _fi_p (2n+2):
But
2n+2
7T 2i 2
1:1 2) :3 1Ax)2:x~:2 LLLDLGYHLZn-i-l) < 2M
n+ 1

 

 

(AXE ,
(2)2n+2 (2) <4) <6)°°'(2n+2) " (2) “*3

It can now be concluded that

 

 

“YBKFV - p2n+l(0) " Ox(2n+3) 22n+3

whenever the (2n+3)th partial derivative of g(x,to) exists.

The same argument can be used to obtain a numerical

differehtiation formula for

ingsfiézlfl
X

at x=0 and t=t0.

2.5 Time Domain Models.

The preliminary material which was derived in the
previous sections can now be applied to obtain two general
classes of time domain models which represent the infinite

line. A time domain model is a set of equations for

35

which the numerical solution is f(n,jAt), where n=0hilhi2,
i3,..., and j+0,l,2,3,..., and for which the solution
f(n,jAt) approximates the solution of the infinite line at
the defined net points for certain choices of,Ax and At.

One class of models is derived on the basis of ob-
taining a set of ordinary differential equations in t by re—
placing the partials with respect to x with difference
equations. The second class of models is derived on the
basis of the previously derived basic numerical solution.

.Qefinition: Class l-Numerical Time Domain Models:

A time domain model belongs to class 1 if and only if the
time domain model can be obtained by the following steps.
1. Write the appropriate telegrapher's equations at the
defined voltage and current node points.
2. Approximate both the partials for j=0,il,i2,...,

0V(ij+Ax/;,_g_)_ bstij, t)

ex and Ox

, by means of a 2n+l

degree numerical differentiation formula as given by
Eq. 2.4.0. For any given model, n is an integer
equal to or greater than zero which is fixed for all
j.

3. Replace the partial derivatives with respect to x
with their respective difference equation approxi-
mations.

4. The resulting set of equations is a set of ordinary

differential equations in time.

36

It is quite clear that the initial conditions on the
line at t=0 provide the necessary initial conditions to solve

these equations.

The definition of the class l—numerical time domain
model gives a set of rules for the construction of a class 1
model. These rules in effect say that any model which be—

longs to class 1 can be written as

m+l In

i7:(2i-3)2
w- 52m‘1f(j,t) 2.5.0
1(2m- 1)' y

for all j and t where 2n+l is the degree of the numerical

 

differentiation formula. If these resulting differential

equations were solved by some analytical means, the error of

the solution would be of the order of (Ax)2n+2.

In order to illustrate class 1, let us look at some

particular examples of models from this class.

Example 1: n=0

. V
SQJJJEl.-._E . ._
dt - Ax 5f(3't) J-Ozilli2._t3,...,

Example 2: n=1

0 V 3 a
(j,t) .
df =~B 0mm JS—U-L—lf t j=o._~:1,s2,

df AX 24

These examples respectively are obtained by employing a lst

and 3rd order numerical differentiation formula.

37

Definition: Class 2-Numerical Time Domain Models:
A numerical time domain model belongs to class 2 if and only
if the model can be obtained by the following steps.
1. Given the basic numerical solution for f(j,t) which
is given by Eq. 2.1.6 and is written in terms of
f(j.0).
2. With this expression for f(j,t), write the solutions

for f(j,At), f(j,0), f(j,-At),..., and f(j,-j2At).
3. Derive a relationship for finding f(j,At) in terms
of f(j,0), f(j,-At),..., and f(j,-j2At).
4. Truncate this relationship. If (5kf(j,0) p(v),
where v=vat/Ax and p(v) is a polynomial in v, would

be the next nonzero term of this expression if the

series were extended, the remainder term can be ob-

k 0(k)V(-§ ,0)

 

tained by replacing (5kf(j,0) by (Ax)

ax“)
(k)
when k+j is even or by (Ax)k._3L__§i4£_LQL

Ox(k)

when k+j is odd where ij-kAx/2 <g§ <=ij+kAx/2.
5. Use the same truncated expression to find f(j,At)
for all j.

For j2=0, a model belonging to class 2 is just an ex—

pression for f(j,At) obtained by truncating the basic numeri-

cal solution for f(j,At) after n terms. For j2¢0. the model

is still a truncated version of the basic numerical solution

38

for f(j,At) except that certain terms of the best numerical
solution have been replaced by known values of f(j,—At),...,

and f(j,-j2At).

Some examples of models from class 2 are:

Example 3:
v At
f(j,At) = fume) +—§;{— (Sumac) j=0._4:_1,_t2,....
Example 4:
v At
f(j,At) = f(j,0) +—§;— 5f(j,0) +...+...

vat
f(jI-At) = f(j,0) — Ax (5f(j.0) +...+...

 

By subtracting these two expressions which eliminates all

even terms,

 

 

2v At
f(j,At) = f(j,-At)+' A: 5f(j,0) +...+...
Truncating,
f(j,At) = fun-At) +2323: 6f(j.0)
Example 5:
fl. _ . VpAt . M2 2 .
3,At) - f(],0) + Ax (5f(3,0) +_2(Ax)26 f(3,0) +...

39

f(j.-At) = mac) “£3;- 5f(j.0) +4??? (5 mm) +...
2 x

By adding the two expressions which eliminates all odd terms,

f(j,At) is now given by

f(j,At) = 2 f(j.0) - f(j.-At) +

2
(v At)
-—jL———-(52f(j,0) +...+...

2(Ax)2

Truncating,

, 2
f(j,At) '= 2 f(j,0) - f(j.-At) +ߤ 62mm .
2(Ax)
This example from class 2 is the model which is given in most
numerical analysis books (1,2,3) as an approximation to the
‘wave equation.

For a model belonging to either class 1 or class 2,
the defining expression for f(j,t) for all j is independent
of j for fixed.Ax and At. If the nodes were divided into
mutually disjoint sets, different models from a given class
could be used to represent each set. From a mathematical
point of view, this is a perfectly valid way of obtaining a
time domain model for the line. In addition, two much larger
classes of models could be obtained. The only problem of de-
fining models in this manner is that the resulting models

would be far more complex in form and not nearly as practical

40

for computer solution. There is no apparent advantage in
doing this. Hence, this thesis only considers models be-
longing to the presently defined classes even though the re—
sults of the thesis are generally valid for the more compli—

cated models.
2.6 Methods of Numerical Splution for the Time Domain Models.

Now that two general classes of time domain models
are defined, this section will discuss certain numerical
techniques for finding their solutions..

First of all, let us consider the models belonging
to class 1. It is quite clear that any numerical technique
for solving linear ordinary differential equations can be
used to solve the differential equations which are defined
'by any model belonging to this class. Because of the

linearity of the equations and the special properties of the

operators (5p, given by Eq. 2.1.3, these equations can be
converted to difference equations by treating each equation
individually once the method of solution is stated.

Let us now consider one differential equation belong—
ing to a given model for this class. This equation can be

written as

. V
df(],t) P 3 2m+1 .

H(Ax,At) f(j,t)

41

where d2i+l is a constant for i=0,l,2,...,m, and H(Ax,At) is

an operator which is a function of Ax and At. If H(Ax,At)
is considered a constant, then any numerical technique of
solution that uses the values of f(j,nAt), f(j,(n-1)At),...,

and f(j,(n-j2)At) to find f(j,(n+l)At) can be written as

j2 mi
f(j.(n+l)At) = Z Z bi p(At) Hp(Ax.At) f(j.(n-i)At)
l=0 p=0

where bi,p(At) for l=0,l,2,3,...,]2, and p=0,l,2,...,mi, is

a function of At. By treating Hp(Ax,At) as a polynomial in

(5, f(j,(n+l)At) is given by

J

2
f(j,(n+1)At) = ai(Ax,At,6) f(j,(n—i)At) 2.6.0
l=0

where ai(Ax,At,éS) for i=0,l,2,...,j2, is an operator which

is a function of Ax,At, and(5. Equation 2.6.0 is simply the
resulting difference equation which is obtained from this
differential equation once the method of numerical solution
is specified.

If the same numerical technique is used to solve
each differential equation belonging to this model, then the
difference equation, Eq. 2.6.0, is valid for all j. In fact,
there is no apparent reason for using different methods on

different equations belonging to this model.

 

42

A first order model belonging to class 1 is given by

df(j,t) __:p_ .

for j=0hilhi2,...,. Some examples that use the Runge-Kutta
methods (2) to solve this model are shown to illustrate the

construction process for obtaining Eq. 2.6.0.

Example 1: Runge Kutta first order:

(5

H(Ax,At) =

51:5

For this technique of solution,

f(j,(n+l)At) [1 + At H(Ax,At)] f(j,nAt)

v At
f(j,nAt) + 7%;- 6f(j,nAt).

Example 2: Runge Kutta second order:

v At
_ .2..—
H(AX,At) - AX 6

 

2
f(j,(n+l)At) [1 + At H(Ax,At) + At Eggx,At)] f(j,nAt)

2
v At (vat) 2 .
l+-§}T—+—ZZ?—— (S f(j,nAt)

The definition of a class 2 model gives a difference
equation of the form
j2
f(j,At) = ai(Ax,At,(S) f(j,-iAt) -. 2.6.1

i=0

 

43

where ai(Ax,At,(5) is an operator which is a function of

Ax,At, and(5. Since f(j,At) for j=0hil“:2,..., determines a
new initial value problem at t=At, this formula may be used
to find f(j,2At). Hence, Eq. 2.6.1 can be used recursively
to find f(j,nAt) for n=0,l,2,...,.

For certain models from class 1 and certain numerical
methods of solution for these models, there are models from

class 2 which give the same sets of difference equations.

 

But there are also certain cases where this is not true. For

 

example, consider the model from class 1 given by a third
degree approximation to the partials.

- V
film-155 [Of(j,t) - 63f(j,t)/24]

A Runge Kutta first order method for solving these resulting

differential equations gives the following difference equations.

f(j,(n+l)At) =
v At
f(j,n t) + 7%}— [O£(j,nAt) - (53f(j,nAt)/24]

Since this difference equation gives f(j,(n+l)At) in terms
of f(j,nAt), any model belonging to class 2 which would give
the same difference equation must be obtained by truncating
the basic numerical solution, Eq. 2.1.6. It is quite clear
that Eq. 2.1.6 can not be truncated to give the same dif-

ference equation.

44

A number of these methods derive a difference equation
expression for f(n.(P+l)At) in terms of f(j,pAt),..., and

f(j,(p-j2)At) for n=0uilhi2,..., and j=0hilL12,...,. Such
difference methods are called difference methods with (j2+2)
levels (1) in t. For p=0 and jzfio, where are the values of
f(j,-At),..., and f(j,-j2At) obtained in the case where only

f(j,0) is specified? This question can only be answered by
saying that there is insufficient information present to find
f(j,At).

Let us consider a difference method which gives
f(j,(p+1)At) in terms of f(j,pAt) for all j. If f(j,0) are
specified for all j, then the values of f(j,t) for t=At,...,
(p+l)At, can be calculated for all j by this difference
method. Since the values of f(j,(p+1)At) are known for
m=0,l,2,...,p+l, a different difference method can be used
to find f(j,mAt) for m=p+2,p+3,...,. This new difference

method can be multi-level (j2#0) difference method in t.

2.7 §pecial Properties of Difference Equation Solution.

When the specified initial conditions have special
properties, the solution to any model belonging to class 1
or class 2 must also have special properties. This section
will derive some of these properties which will be used in
later chapters.

First, let the initial condition functions, V(x,0)
and S(x,0), be respectively an odd and an even function with

respect to x=mAx. For p=0,l,2,..., the relationships

45

(Spf(2j+2m,0) = 6pf(—2j+2m,0)

(Spf(2j+l+2m,0) =—6pf(-2j-l+2m,0)‘

must be true for all j. Hence for j2=0, the solution any
difference equation for f(j,At) which is given by either
Eq. 2.6.0 or 2.6.1 must have the property

6pf(2j+2m,At) = —6pf(—2j+2m,At)

6pf(2j+l+2m,At) = 5pf(-2j-1+2m,At)

for all j. By means of induction, it can be seen that

Eqs. 2.7.0 must be true for all j and n=0,l,2,.

00,0

6pf(2j+2m,nAt) = (Spf(-2j+2m,nAt) 2.7.0

(5Pf(2j+1+2m,nAt) =-6pf(—2j-l+2m,nAt)

For the multilevel difference equations, given by
Eqs. 2.6.0 and 2.6.1, where f(j,nAt), f(j,(n-l)At),...., and
f(j,(n-j2)At) satisfy Eqs. 2.7.0 for all j, the solution for

f(j,(n+l)At) must satisfy Eqs. 2.7.0.

This same argument can be applied to the case where
V(x,0) and S(x,0) are respectively an even and an odd
function with respect to x=mAx+Ax/2. The result of this

argument is given by Eqs. 2.7.1.

(5pf(2j+2m+l , nAt)

- (Spf(—2j+2m+l,nAt) 2.7.1

5Pf(2j+2m+2 , nAt)

6pf(-2j+2m.nAt)

46

Now let us consider the case where V(x,0) and S(x,0)
are periodic in x with period L. In addition, letAx be
chosen such that Ax=L/k, where k is a positive integer. Be—

cause of this periodicity, the functions f(j,0) are given by

f(2j+2nk,0) = V(ij,0)

f(2j+l+2nk,0) = S(ij+Ax/2,0)

where n=0,ilhi2,..., and j=0,1,2,...,k—1. These equations

also imply that the following is also valid

épf(j+2nk,0) = (Spf(j,0)

for j=0,1,2,...,2k-l, p=0,l,2,..., and n=0hilhi2,...,. Hence
for j2=0, the solution to either Eq. 2.6.0 or 2.6.1 must have

the property that

(Spf(j+2nk,At) -_- (Spf(j,At).

By induction, this argument can be used to show that Eqs.

2.7.2 are valid for j=0,1,2,...,2k—l, p=0,l,2,..., and all n.

(Spf(j+2nk,mAt) = (Spf(j.mAt) ' 2.7.2

Equation 2.7.2 is also valid when multilevel differ—

ence equations are used to find f(j,p t).

2.8 Conclusion.

Two classes of infinite line time domain models have

been derived. Any model which belongs to class 1 is obtained

47

by approximating the partial derivatives with respect to x

at certain node points and obtaining a set of ordinary linear
differential equations. On the other hand, any model which
belongs to class 2 was derived on the basis of a basic numeri-
cal solution which was defined in section 2.1.

In addition to deriving these two classes of time
domain models, sections 2.2 and 2.3 showed that the basic
numerical solution is defined and can be written as either a
Fourier series on 2k sample points, an approximate Fourier
integral, or a combination of both. Whereas, section 2.7
showed that whenever the specified initial conditions had
special properties, the solution to any model by the methods
of numerical solution which are given in section 2.6 must al-
so have certain special properties.

The findings of this chapter are extensively used in

the remaining chapters of this thesis.

III. TRANSMISSION LINE OF TYPE 3

 

A problem of type 3 is a lossless parallel trans-
mission line of length L for which a boundary condition at
each end of the line is defined to be zero for t20 but has
specified initial conditions at t=0. The line is defined
to start at x=0 and to end at x=L. This chapter will
specifically treat the case where the boundary conditions
E(0,t) and E(L,t) are specified to be identically zero for
tZO .

If E(x,0) and I(x,0) are the specified initial con-
ditions for this problem of type 3, an infinite line for

which the initial conditions are

V(x+2nL,0) = E(x,0) Ofxfl
V(—x+2nL,0) = -E(x,0) 0:x:L
S(x+2nL,O) = I(x,0) ijfL
S(—x+2nL,0) = I(x,0) OijL

where n=0,_-I;l,_-t2,:3,..., has the same solution for Offo as
the problem of type 3 for all t. Hence, an approximate
numerical solution can be obtained for this problem of type 3
by solving the infinite line problem for certain specified

initial conditions.

48

49

Consider the functions V(x,0) and S(x,0). Both of
these functions are periodic in x with period 2L. If (V
f(j,nAt) is the approximate numerical solution to this in-
finite line by any difference method defined in Chapter 2,

then according to Eqs. 2.7.2, when Ax=L/k, the relationship
f(i+4pk,nAt) = f(i,nAt)

where i=0,l,2,...,4k-l, and p=0,il,.i2,..., is true.
Since V(x,0) and S(x,0) are also an odd and an even
function of x respectively with respect to x=0, Eqs. 2.7.0

state that this relationship further reduces to

f(2i+4pk,jAt) = f(Zi,jAt) 3.0.0
f(-2i+4pk,jAt)= —f(2i,jAt)
f(2i+l+4pk,jAt) = f(21+l,jAt)

f(-2i—l+4pk,jAt) = f(2i+l,jAt)

where p=0,j-_l,_-I;,..., and i=0,l,2,...,k—l. This relation—
ship also states that f(2pk,jAt) for all p is identically
zero independent of the difference method.

Because of these special infinite line properties,
any model or difference method derived in Chapter 2 can be
reduced to a set of 2k equations for which the numerical
solution is f(p,jAt), where p=0,l,2,...,2krl. Since f(0,t)
is identically zero, the actual equations which need be

solved can be further reduced to (2k-l) equations.

50

If f(i,jAt) or f(i,t) is replaced by g(i.jAt) or
g(i,t) for i=0,l,2...,2k, these resulting (2k+l) equations
are defined as a time domain model for the given problem of
type 3. The functions g(2i,t) for i=0,l,...,k, and g(21-l,t)
for i=1,2,...,k, are defined as the approximate numerical
solutions for E(iAx,t) and I(iAwax/2,t) respectively.
Since the functions, g(0,t) and g(2k,t), must be identically
zero, only the remaining (2k-1) equations need by solved.
Hence, this chapter will investigate methods for deriving

these (2k-l) equations and their methods of solution.

3.1 Egansformation Matrix C.

The time domain models and difference methods de-
rived in Chapter 2 are all in terms of the operator (5m

for m=0,l,2,..... Because of the special infinite line

properties stated in sections 3.0 and 2.7, (5mf(j,t) for
jaoxih:2,..., can be written in terms of f(p,t) for

p=0,l,2,...,2k. This section will derive a transformation

matrix C such that if c? j is a typical entry of Cm, then

I

2k—1

6mf(p,t) = chJ f(j,t)

j=1

for p=l,2,...,2k-l.
If f(i,t) is substituted for f(i,jAt) in Eqs. 3.0.0,

then the relationship

51

(Smf(2i+4pk,t) = 6m£(2i,t) . 3.1.0
5mf(-2i+4pk,t) = -6mf(2i,t)

(Smf(21+1+4pk,t) = (5mf(2i+l,t)

5’“f(-2i-1+4pk,t) = 6mf(21+l,t)

where i=0,l,2,...,k, and p=0,_tl,_-I;2,..., is valid. Since

5m£(2-pk,t) e -5’“£(0,t)
6mf(-2k+4pk,t) = ~6mf(2k,t)

for p=O,_-i_-_l,_-f_2, 6mf(2pk,t) must be identically zero for
all t.

Let us consider 6m+1f(i,t) for i=l,2,...,2k-l.

According to Eqs. 2.1.3, 6m+lf(i,t) can be written as
5m+lf(i,t) a 6mf(i-1,t) - (Smf(i+1,t).

But
6mf(0,t) .-. 6mf(2k,t)=0.

Hence 6m+lf(i,t) is given as

5m+lf(1,t) 6mf(2,t)

§m+lf(i,t) (Smfli-Lt) - 6mf(i+l,t) i=2,3, . . .,2k-2

6m+1f(2k~l,t) = 6m£(2k-2,t).

52

This gives a direct linear relationship between (5m+lf(i,t)

and 6mf(i,t) for i=1,2,...,2k-1, which is independent of m.

This linear relationship can be expressed by means of a

(2k—l)x(2kv1) matrix C.

entry ci,j

c -1

i,i+l ‘
, . a
ci+l,1

c. . = O
1,]

This resulting matrix is

By definition,

6m+1f(l,t)

6m+lf(2,t)

 

Lu—

5m+1f(2k-2, t)

5m+1f (2k—l , t)

of C is

-1

 

—

It is quite clear that a typical

1:112, coeoZk-l
i=1,2,...,2k-l

i#j+1. j#i+l.

3.1.1

called the transformation matrix.

6mf(l,t)

6mf(2,t)

 

L.—

5mf(2k-2,t)

5’“£(2k-1,t)

.q

 

3.1.2

Hence (5mf(i,t) for i=l,2,...,2k-l, can be written in terms

of 60f(i,t)=f(i,t) as

53

 

 

 

~6mf(l,t) I —f(l,t) 7 3.1.3
6mf(2,t) f(2,t)

' = cm
5m£(2k—2,t) f(2k-2,t) ;
b6‘“f(2k-l,t)J Lf(2k-l,t)-J .

 

It is quite clear that this is a very convenient means of ex—
pressing (Smf(i, t).

The matrix C has at least one nonzero entry and at
most two nonzero entries in each row or column. This
property indicates that special methods can be used to calcu-

late Cm+1. If C? is the ith row of Cm, then it is quite clear

that
cW+l = cw - c? i=2,3,...,2k—2 3.1.4
1 1—1 1+1
63131 = Cgk—Z.

In effect, this says that if this recursion method is em-

ployed to calculate Cm+l from Cm and C, then 2(4k2~6k+2)'ad-
ditions and no multiplications would be required. If the
product was taken in the ordinary sense, then (2k-l)(4k2-4k+l)

2

additions and (4k -4k+l) multiplications are required. If the

time to perform a multiplication is equal to or greater than

54

the time required to perform an addition, the matrix product
in the ordinary sense takes at least k times longer to exe-

cute than the recursion formula.

3.2 Transformation of Time Domain Models Belonging to_Class 1.

Given a time domain model, belonging to class 1, for
the infinite line. There are two approaches for obtaining a
set of (2k—l) equations which give the approximate numerical
solution to the problem of type 3. The first approach is to
directly transform the infinite line time domain model, given
by Eqs. 2.5.0, to the finite line problem of type 3. This
process would give (2k-l) ordinary linear differential equa—
tions for which the solution approximates E(x,t) and I(x,t)
at the defined nodes. A numerical method of solution then
can be defined to solve these resulting differential equa—
tions.

The second approach is to define a method of numeri—
cal solution for the infinite line model. The resulting set
of difference equations, given by Eqs. 2.6.0, can then be
transformed to the finite line problem. As far as the end
result is concerned, both approaches are identical. This
section treats the first approach.

A time domain model for the infinite line belonging
to class 1 which is obtained by using a (2n+l)th degree
difference equation approximation for the partial deriva-'

tives with respect to x can be written as

55

. 2 _
df(j,t)_ [i=1 (21'3) ] (52m lf(j,t)
dt Ax (- 2)m l(Zm- 1)!

where j=0hil.:2, .,. Because of the special infinite line

properties given by Eqs. 3.1.0, the solution to this infinite
set of differential equations can be obtained by solving the

finite set of (2k—l) differential equations which are

pLTrl(2i-3)2] '
df‘llt )_ _:vp i 62m.1f(jlt)

 

Ax (— 2)m—l (2m-1)!

where j=l,2,...,2k—l.
If F(t) is a (2k-l)xl matrix

1N1,t)

F(t) =

 

 

f(2k-1,th

with a typical entry of f(i,t), then these (2k-l) differential

equations can be written in matrix form as

n+1 m 2
Vp[ 7T (Zi— 3) ]
gi(t) = i=1 C2m'1F(t).

 

Ax (.2)m'1(2m-1):
m=1

56

Let the (2k-l)xl matrix G(t) be defined as

A

r
g(l,t) 3.2.1

9(2.t)

G(t) =

g(Zk—th)

 

 

g(2k-l,t) .

By replacing F(t) by G(t), a set of (2k-l) differential equa-
tions are obtained for which g(i,t), where i=l,2,...,2k—l, is
the solution. Since g(i,t) is the approximate solution for
E(x,t) and I(x,t) at the defined node points, this resulting
set of equations is a time domain model for the problem of
type 3 obtained by transforming the infinite line time domain
models belonging to class 1.

Since this time domain model is obtained directly
from the infinite line time domain model, the mathematical
properties are identical to those which were derived for the
infinite line. Probably the most important advantage of trans-
forming the infinite line model or difference method is that
all the approximations for the partial derivatives are ob—
tained by means of central difference formulas of degree
(2n+1). It is not obvious by approaching the problem of
type 3 directly that central difference approximations can

be used at all nodes.

57

Any numerical technique can be used to find the numeri-
cal solution to these equations. One convenient procedure
for converting these differential equations to difference

equations is as follows. If the (2k-l)x(2k—l) matrix An is

 

given by
+ m
n 1v [Wm-39]
A __ P i=1 2m-1 3.2.2
n - m-l C
Ax (-2) (2m-l):

m=1

then this model can be expressed in matrix notation as

dG(t) _ An G(t). 3.2.3
dt "

If G(t) and An are treated respectively as a single variable

and a constant, then any numerical technique of solution that
uses the values of G(jAt), G((j-l)At),..., and G((j-jzAt), to
find G((n+l)At) can be written as

m .
l

j2
G((j+l)At) = Z Z bi p(At) hfi G((j—flAt) 3.2.4
i=0 p20
where bi j(At) for i=0,l,2,...,j2, and p=0,l,2,...,mi, is a
function of the increment At. By treating G(t) and An as

matrices, Eqs. 3.2.4. give us the desired difference

equations.

3.9;.

”1
MI

E)

58

Some examples of difference equations derived from a
first order model belonging to class 1 are:

Example 1: Runge Kutta first order:

v At
G((j+l)At) = [u + -§;‘— 0] G(jAt)

Example 2: Runge Kutta second order:

vAt (vp At)2
G((j+1)At) = U + —-P—- c +
2 Ax

2 02. G(jAt)

J

3.3 Transformation o ass 2 T'me Domain Models.

 

It was shown in section 2.1 by Eqs. 2.1.6 that the

basic numerical solution for the infinite line is

2
f(j,t) = f(j,o) + v6f(j,o) .,. v2 £413.21 .,.

2!
3 . 4 .
mat/4, fistula . v2(v2-1) 4.541491 .

where j=0hilh12,..., and v=th/Ax. Because of the special

infinite line relations given by Eqs. 3.1.0, it is only
necessary to consider the basic numerical solution for f(i,t)
for i=1,2,...,2k—1.

If the transformation matrix C and the matrix F(t)
are used, it is quite evident that this basic numerical solu—

tion can be written in matrix notation as

59

2

F(t) = F(O) + v c 9(0) + v2 S—g-f-Ql + 3.3.0
2 c3}? 0
V(v -l/4) 3. +

If F(t) is replaced by G(t), the result is an alternative form

of the basic numerical solution at the defined nodes for the

problem of type 3.

The definition of a class 2 time domain model which
is given in section 2.5 can be applied to this form of the
basic numerical solution. The resulting equations are identi-

cally the transformed class 2 infinite line time domain model.

3.4 Transformation of Difference Methods.

Instead of transforming the time domain models as
shown in sections 3.2 and 3.3, the difference methods which
are given in section 2.6 by Eqs. 2.6.0 and 2.6.1 can be
transformed to the finite line case.

A difference method. given by either Eqs. 2.6.0 or
2.6.1, can be written as

32
f(j,(n+l)At) = zai(Ax.At.6) f(j,(n-imt)

i=0
‘where ai(Ax,Am,(5) is an operator. This operator can be ex—

pressed as

m .
l

_ ' P
ai(Ax,At,CS) - Z di’p(AX.At)6
=0

60

where 3i p(Ax,At) is a constant for fixed.Ax and At.

I

If the same procedure is used as was given in sec-
tions 3.2 and 3.3, this transformed difference can obviously

be written as

G(jAt) = di’p(Ax.At) cp G((j-i>At>.

3.5 Conclusion.

The time domain models which are presented in this
section are obtained directly from the infinite line time
domain models by means of a finite-infinite line transfor-
mation. Therefore, the mathematical properties of these
models are identical to those derived for the infinite line
models. The most important advantage of obtaining the models
in this manner is that all the approximations for the partial
derivatives with respect to x are obtained by more accurate

central difference formulas.

IV. LAUNCHING NUMBERS

4.0 Introduction.

A problem of type 1 is a transmission line problem
which has zero initial conditions and one identically zero
boundary condition. One such problem of type 1 has the

initial and boundary conditions specified as

Ebgm =Ihmm =0 0<fo . 4JLO
E(L,t) = O Oft<GD

I(0,0) = Esl(0)

EULU =Esflt) Oft<oo

where Esl(t) is a specified function of t.

In order to indicate the development of this chapter,
let us consider the problem of type 1 for which the boundary

and initial conditions are specified as follows:

En,l(x’0) = In,l(x’0) = O 0<fo
Emluufl =0 Ojt<a>
In,l(0’0) = Esl(nAt)
En,l(0’t) = O t<0
= Esl(t+nAt) o:t<At
=0 tiAt

61

62

Since En l(0,t) is identically zero for tfjAt, it is possible

to define a problem of type 3 at t=At for which the initial

and boundary conditions are

En(x,0) = En,l(x’At) 0:5fo 4.0.1
In(x,0) = In,l(X’At) Offo
En(0,t) = En(L,t) = O ()ft:<CD

such that

En,l(x,t+At) En(x,t) 4.0.2

In,l(x’t+At) = In(X,t)

fm:0:t<a>am10:x:L.

Since the lossless parallel transmission line is a
linear problem, the superposition principle may be applied
at any time. Hence, it is quite evident that the relation-
ship

P

E(x,t) = Z En I(x,t-nAt) 4.0.3
=0

P
I(x,t) = Z In l(x,t-nAt)
n=0

is valid for pAtft" (p+1)At. If Eqs. 4.0.2 are substituted
into Eqs. 4.0.3, then the solution for E(x,t) and I(x,t) can

be written as

63

p—l

E(x,t) = E: En(X,t-(n+l)At) + Ep’l(x,t—pAt)
n=0
p-l

I(x,t) :20 In(x,t-(n+l)At) + Ip’l(x,t-pAt)

where pAtft< (p+l)At. At t=pAt, the solutions for E(x,pAt)

and I(x,pAt) are given as

p-l

E(x,pAt) = Z En(x,(p-n-1)At) 0<x:L 4.0.4
n=0
p—l

I(x,pAt) = Z In(x,(p-n—1)At) 0<fo
n=O

E(0,pAt) = Esl(pAt)

p—l
I(O,pAt) = Z) In(0,(p-n-1)At) + Esl(pAt).
n:

Equations 4.0.4 simply state that the solutions for E(x,pAt)
and I(x,pAt), where 0<x__<_L, can be obtained by summing the
solutions for p problems of type 3.

The general method of numerical solution can now be
stated for the problem of type 1 which is specified by the
initial and boundary conditions given by Eqs. 4.0.0. The

nonzero boundary condition, Esl(t), at t=nAt is used to de-

fine a problem of type 3 at t=(n+l)At. The numerical method
of solution for a problem of type 3 was covered in detail in

Chapter 3. According to Eqs. 4.0.4, the numerical solution

64

at the interior node points at t=pAt is the superposition of

the numerical solutions of p problems of type 3. If gn(j,0).

j=0,1,2,...,2k, where AxeL/k, are the initial conditions to
the problem of type 3 which was defined at t=(n+l)At, this

chapter specifically derives a class of transformations for

which
gn(3.0) = bj I(Ax,At) E81(nAt) j=0,l,...,2k 4.0.5
such that
p—l
g(j.PA¢) = :E: gn(j.(p-n-1)At) j=l,2,...,2k 4.0.6
n=0

9(0.pAt) = Esl(pAt)

are the approximate numerical solutions to E(x,pAt) and
I(x,pAt) at the defined net points. Any transformation from

this class of transformations defines the bj 1(Ax,At), where

j=0,1,2,...,2k, which are employed in Eqs. 4.0.5. These

quantities, bj l(Ax,At), are defined as launching numbers.

The specific approach for deriving this class of
transformations is to use the basic numerical solution, given

by Eqs. 3.3.0, to find gn(j,(p-n—1)At), and then compare the

functions g(j,pflm), given by Eqs. 4.0.6, for all p, to an
approximate Fourier integral, an approximate Foureir series,

and an analytical solution for this problem of type 1.

65

4.1 'gpproximate Fourier Integral.

Part of this chapter is based on an approximate
Fourier integral. Hence, this section is devoted to the
derivation of the approximate Fourier integral.

The Fourier integral transform pairs are

00
g(CX) = [f(X') COS(O((X'-x)) dx'

"'00

0(
f(x) = lim(l/7T)[g(0(') dO('
0

0<-oo

where g(CX) and f(x) at present bear no relationship to the

functions g(n,t) or f(n,t).

Let us approximate g(CX) by 5(cx) where 5(cx) is

given as
6(a) = g(ou o:o<: TT/AX
5(00 = 0 CX>7T/Ax

It is quite clear that this is a good approximation ifAAX is
chosen sufficiently small. If g(CX) is used in place of

g(CX), the transform pairs are

00
6(0() =[f(X') COS(O((X'-X)) dX' 0:0(fTT/Ax

00

66

5(():) = 0 CX:> 7TXAX

7T/Ax
7f(X) = (1/7T)[ §(0() dO<
0

where f(x) is an approximation to f(x) . If §( (X) is elimin-

ated from these equations. f(x) can be written as

oo 7T/Ax
f(X) = (l/TT) / f(X') dx' [coed O('(X'-X)) dO<'

'—oo 0

oo
= [f(x') gin TRx'-x)/ x dx' .

 

H(X'-X)/ x .

-00

If this integral is numerically integrated by means of the

trapezoidal rule, the function f(x) is then approximated by

 

w .
f(x) = £3; E f(nAx') El“ $23k32§3§ 4.1.0
-oo

where Ax' may or may not equal Ax. Equation 4.1.0 is defined
as the approximate Fourier integral of f(x).

If f(x) is given by Eqs. 4.1.0, it is quite clear
that f(x) approximates f(x) for Ax and Ax' (generally
Ax'SAx) sufficiently small. Since

s'n x'-x A

(n x'-x /Ax

contains no spectral components for O(>7T/Ax, f(x) can have

67

no spectral components for CX>47XAX. In addition, when
Ax'=Ax, this is the same Fourier integral approximation as

was given in section 2.3.

4.2 ,Approximate Fourier Integral and Series Solution.

The main purpose of this section is to derive an ap-
proximate Fourier integral and series solution for E(x,pAt)
and I(x,pAt). In fact, special forms of an approximate
Fourier integral and series solution for E(x,pAt) and
I(x,pAt) are derived which will be used in later sections
of this chapter. The reasons for developing these special
forms will become evident in later sections.

First of all, let us consider the problem of type 1
for which the nonzero boundary condition is given by the

function E:l(t) where

p <
Esl(t) Esl(t) o_t<pAt 4.2.0

=0. tipm:

If the analytical solution for this problem is given by

Ep(x,t) and Ip(x,t), it is quite clear that

Ep(x,t) = E(x,t) 0ft<pAt, osfo
Ip(x,t) = I(x,t) Oft<pAt, 0:5fo
Ep(x,pAt) = E(x,pAt) 0<fo
Ip(x,pAt) = I(x,pAt) 0<fo

68

is valid. In addition for tE:pAt, this problem corresponds
to a problem of type 3 where Ep(0,t) and Ep(L,t) are identi—

cally zero.
For tE:pAm, the analytical solution for Ep(x,t) and

Ip(x,t) can be written as

00

p _ p _ _ _ p _

E (x,t) — ZESl(t x/vp 2nL/vp) isslum/vp 2nL/vp)
n=-0‘0 H's-'00

00

p _ p _ _ p _

I (X,t) _i Esl(t x/vp 2nL/vp) +2 Esl(t+x/vp 2nL/vp).
n=-OO n=--—oo

Both the functions Ep(x,t) and Ip(x,t) are periodic in both
x, {-00 < X<oo) and t, (—00 < t< 00) even though the solutions

are valid for Offo and pAtft<oo. This forces Ep(x,t)

and Ip(x,t) to satisfy the finite-infinite line transfor-
mation as discussed in Chapter 3.

If we let t'=t-pA¢ and define the function Mp(-vpt')

as
p_ .___p .,
M ( vpt ) Esl(t +pAt), 4.2.1

then the solution for Ep(x,t') and Ip(x,t') for t'3:0 can be

written as

69

OO
Ep(x,t') = ZMp(—vpt'+x+2nL) - i Mp(-vpt'-X+2nL) 4.2.2

n=-OO =-OO

{XL
Ip(x,t') = MP(-vpt'+x+2nL) +,Z_,MP(-th'—X+2nL).
n=-OO nz—oo
By letting
oo
Vp(x) = :E:Mp(x+2nL), 4.2.3
n=—co

Ep(x,t') and Ip(x,t') can now be expressed as

Ep(x,t') = Vp(-vpt'+x) - Vp(-vpt'-x) 4.2.4
Ip(x,t') = Vp(-v t'+x) + Vp(-V t'—X)
P P
where Vp(x) is periodic in x with period 2L.

Since Vp(x) is periodic in x with period 2L, this

function can be expanded in a Fourier series as

Vp(x) = 38/2 + [SE cos(nTTx/L) + 4.2.5

n:

SE sin(n7Tx/L)]

where

7O

2L
331: (l/L)[Vp(x) cos(n7Tx/L) dx
0
2L
bi = (l/L) /PVp(x) sin(n7Tx/L) dx.
0

If x=vpt and the trapezoidal rule is used to numerically

integrate the integrals to find the coefficients, it can

easily be shown that

t
3" = zip-$— Mp(jvat) cos(n7ijpAt/L) 4.2.6

J=l

v At
-p = . . _
bn “BI—L i: Mp(jvat) Sin(n7ijpAt/L),
j=l
The function Np(Ax',—vpt') is defined as the approxi-

mate Fourier integral of MP(—vpt'). According to Eqs. 4.1.0,

the approximate Fourier integral of MP(-vpt') is given as

Np(Ax',-vpt') = 4.2.7

v At sin TT(nv At+v t')/Ax'
_£L_. P P- P
M (nvat) TT(nv§At+vpt')/Ax'

 

71

where,Ax'k'=L. The increment Ax' may or may not be equal to
the increment Ax used to solve the transmission line problems
of type 3. Hence an approximate Fourier integral solution

can be obtained by first rearranging the absolute convergent

series which are given in Eqs. 4.2.2 and then be replacing

Mp(-vpt') by its approximate Fourier integral

E§(x,t') = Np(Ax',-vpt'+x) - Np(Ax',-vpt'—x) + 4.2.8
00

ii: [NP(AX',-vpt'+x+2nL) + Np(Ax',-vpt'+x-2nL{] -
n=1
00

:E: [NPQAX',—vpt'-x+2nL) + Np(Ax',—vpt'—x—2mLJ
n=1
I§(x,t') = NP(Ax',-vpt'+x) + NPLAx',-vpt'-x) +
co

[Hp(Ax',-vpt'+x+2nL) + Np(Ax',-vpt'+x-2nLJ +

:3
18 ,'.'.L

[Np(Ax' ,-vpt'-—x+2nL) + Np(Ax' ,-vpt'—x-2nL)]

:3
ll
l-J

where E§(x,t') and-IE(x,t') are respectively the approximate
Fourier integral solutions for Ep(x,t') and IP(x,t') when
t'EZO.

By letting the function V§(x) be defined as

00
vim = Np(Ax',x) + Z [Np(Ax',x+2nL) + Np(Ax',x—2nL)]

n=1 4.2.9

72

the approximate solution for E§(x,t') and I§(x,t') can be

written as

p. p_ . _p_ ._
Ef(x,t ) Va( vpt +x) Va( v t x) 4.2.10

P

P I_p_ I p- '.
If(x,t ) — Va( vpt +x) + Va( vpt x).

This function is a periodic function in x with period 2L.

Therefore, it can be represented as a Fourier series. If
Np(Ax',x+2nL) is replaced by its defining equation, Eq.

4.2.7, then V§(x) can be written as

P
sin 7T( nvPAt-x)/Ax '

VP(X) * VpAt ‘_fi MP(nv At) 1"— +
a " Ax' p TT(nvat-x)/Ax'

n=1

 

w \
'7 sin 7T(nvat-x—2iL)/Ax' sin ”(anAt*X+ZiL)/Ax'
L 77(nvat—x-2iL)/Ax' + 7TKnvat-x+2iLLZ§§T

 

 

 

 

i=1

But according to section 2.2, Vg(x) can now be written as

k'—l
V§(x) = ag/Z + :2: [afi cos(nTTx/L) + 4.2.11

m=1

bi sin(n7Tx/L)] + [a£. cos(k'7Tx/L) + bp. sin(k'7Tx/L)] /2..

73

 

 

where
vat
ai = L EZ: Mp(nvat) cos(m7Thvat/L). 4.2.12
m=1
P
v At
b: = pL E Mp(nvat) sin(m7Tnvat/L).
m=1

If the Fourier coefficients which are given by Eqs.
4.2.12 are compared to the coefficients as given by Eqs.

4.2.6, it can be concluded that these are the same co-

efficients. Therefore, by truncating Vp(x) after k' terms

and obtaining the coefficients by numerical integration, the
result is the function V§(x).

Let us examine briefly the solutions for E§(x,t')
and I§(x,t') at t'=0. If Eqs. 4.2.11 is substituted into

Eqs. 4.2.10, the solutions at t'=0 are

k'-l
E§(x,0) = Z bi sin(n 7Tx/L) + bp, sin(k'7Tx/L) 4.2.13
n=1
k'—1
Irf)(x,0) = a8 + 2 Z a5: cos(nTTx/L) + ai, cos(k'7Tx/L)
n=1

At the defined node points, the k'th term is zero and need

not be considered. If E§1(t) is discontinuous at tapAt, it

74

is quite likely that Vp(x) is discontinuous at x=2nL, where
n=0“11“12,...,. As a result, these approximations as given
by Eqs. 4.2.13 can tend to oscillate about the actual solu—
tions. These oscillations decreases slowly as we move away
from the discontinuity. This prOperty can not be overlooked
and is taken into consideration in this and a later section.

One method which helps to correct this situation is

the introduction of Lanczos' Cffactors (6). If E§(x,t') and

Ig(x,t') for t'=0 are truncated after the n'th term, the

introduction of these factors in effect replace these trun-

cated approximations by smoothed function which are given as

x+L/2n'
"'p _ I I I
Ef(x,0) — .n /L Ef'n.(x ,0) dx
x-L/Zn'
x+L/2n'
‘p _ I P I I
If(x,0) — n /L If,n'(x ,0) dx
x—L/2n'
where

nl
Eg’n.(x',0) = 2 bi sin(n7Tx/L)

n=1

nl

P l P P
If’n.(x ,0) a0 + 2 :E:an cos(n7Tx/L)

n=1

75

and n'<:k'. Once this integration or averaging process has

been performed, E§(x,0) and I§(x,0) are given by

nl
E§(x,0) = 2 Unb‘: sin(n7Tx/L) 4.2.14
n=1
nl
’P _ P P
If(x,0) — a0 + 2:E: caan cos(n7Tx/L)
n=1
where

sin 7Tn 2n' .
CL ='—-—%7TE§§HT% 4.2.15

4.3 Concept of Consistent Initial Values.

First of all, let us consider the transmission line
problem of type 3 for which the initial conditions at the

defined nodes, gn(j,0), where j=0,1,2,...,2k, are specified.
If the solution for gn(j,(p-n-1)At) is obtained by some

method described in Chapter 3 from these specified initial
values, this chapter assumes that the increments Ax and At

are chosen such that gn(j,(p—n—1)At) approximates the basic

numerical solution as given in section 3.3 for all p such
that pf:n+l. Hence, the basic numerical solution is employed

to express gn(j,(p-n-1)At) in this chapter.

76

If Ax and At are not chosen sufficiently small, re-

gardless of how the launching numbers, bj l(Ax,At) are chosen,

I

it is not likely that g(j,pAt) is an approximate solution to
E(x,pAt) and I(x,pAt) at the node points. In order to avoid
this situation when comparing the functions g(j,t) to either
the approximate Fourier solutions or the analytical solutions
for E(x,t) and I(x,t) which are given in section 4.2, the
assumption is always made that these increments have been
prOperly chosen.

The set of initial values gn(j,0) for n=0,l,2,...,p—l,

and j=0,1,2,...,2k, is a set of consistent initial values if
and only if the following conditions are satisfied:

1. Let gn(j,t) be obtained by using the basic numerical

solution as defined in section 3.3.
2. There exists a 3; such that if Ax-zAx, there corres~

ponds a At which depends on Ax such that if At<=At,

then g(j,pAt) approximates Ep(x,t') and Ip(x,t') for
all p at the defined node points when t'=0.
It is clear that the transformation defined by the

launching numbers bj l(Ax,At) defines a valid transformation
only if the set of initial values gn(j,0) are a set of con-

sistent initial values.

4.4 Derivation of g(j.PAt2.
Before the function g(j,pAt) can be compared to the

function Ep(x,t) and Ip(x,t) as given by Eqs. 4.2.2 or the

77

Fourier approximations to these functions at t'=0, g(j,pAt)
must be placed in the pr0per form so that a comparison can be
made. This section develops such a form.

Given the initial values gn(j,0) for j=0,1,2,...,2k,

and n=0,l,2,.... whether these values are consistent initial

values or not. Since the functions gn(j,t), j=0,l,...,2k,

are the approximate solutions to a problem of type 3 at
certain node pointS‘WhiCh are obtained from the initial

values gn(j,0), j=0,1,2,...,2k, the functions gn(0,t) and
gn(2k,t) are required to be identically zero for all t. If
we let fn(j,t)=gn(j,t) for j=0,1,2,...,2k, and use the re—

lationships given by Eqs. 3.1.0, the problem of type 3 has
been transformed to the infinite line situation which was

described in Chapter 2. The functions fn(j,t) are used in
place of f(j,t) for identification purposes.

Consider the function zp(j,t) which is defined as

—1
zp(j,t) = fn(j,t—(n+l)At) 4.4.0

n=0
where j=ohi1,:2,...,. If we have a set of consistent initial
values, the functions zp(j+4mk,t) for j=0,1,2,...,2k, and
m=0h11h12,..., would give us the desired approximate numeri-

cal solution at t=pAt to either EP(ij,t) or IP(ij+Ax/2,t),

depending on whether j is odd or even.

78

The basic numerical solution for fn(m,t) can be ob-

tained by substituting Eqs. 2.1.7 into Eq. 2.1.1. If this

basic numerical solution is used to represent fn(m,t) and
upon introducing the change of variable t'=t-pAt and i=p—n—1,

the function zp(m,t') can be written as follows:

 

p—l
' ' At+v t')/AX
p ' _ Y“ 811‘! F(ivp P
z (m't ) “ '1‘?) fp-i-1(m’0) TI’(ivat+vpt')/Ax +
1:

fp_ i-l (m+l , 0) sin 7T( ivPAt+Ax/2+vpt' )/Ax

 

 

 

 

 

2 7T(ivat+Ax/2+vpt')/A§ -
fP- i_l(m+l, 0) sin ”(ivat-Ax/2+vpt' )/A)t +
2 7T( ivat—Ax/2+vpt' )/Ax
OO
fp_i_l(m+2j,0) sin 7T(ivat-ij+vpt' )/Ax +
2 7T(ivat+ij+vpt' VIE
j=1
fp_i_l(m-2j ,0_) _s_i_n F(iVPAt+ij+th' )/Ax +
2 7TKivat+ij+vpt')/Ax
00
fp—i—1(m+2jflgl sin flRivat+ij+vpt')/Ax +
2 , 7T(ivat+ij+vpt' )7Ax
j=1
fp_i_l(m-2j,0) sin flRivat-ij+th')/sz +

 

2 7T(ivat-ij+vpt' )/Ax

OO
5&1_l(m+2j+1, 0) sin TRiVPAt+ij+Ax/2+vpt' )/Ax

2 7T(ivat+ij+Ax/2+vpt' )/Ax
n=1

1 (m— 2j+1 , 0) sin m ivat-ij+Ax/2+:Pt ' )/Ax
2 TT( ivat- ij+Ax/2+vpt' )/Ax

f? i_

79

 

00
f . (m+2j+1, 0) sin 7T(ivp At-ij—Ax/2+v t' )/Ax
P-i— 1
2 771v: At--ij-Ax/2-I-vp t')7Ax+
j=1

fpuiulm"2j+1 , 0) sin TR ivPAt+ij-Ax/2+vpt' )/Ax
2 m ivat+ij-Ax/2+vpt' )/Ax

This expression for zp(m,t') as given by Eqs. 4.4.1

is quite complicated. The expression can be simplified
greatly if two new functions, Jp(j,x) and Hp(j,x), and their

properties are introducted. Let Jp(j,x) be defined as

Jp(j,x) = 0 xf-vat/Z
Jp(j,x) = 0 vat(n-l/2) EX<vat(n+l/2)
where j=0hilui2,...,. It is quite clear that
P - .
J (3.x) =f___ (3.0).
x=nvat p n l

The function Hp(j,x) is the approximate Fourier integral, de-

fined by Eqs. 4.1.0, of Jp(j,x). Therefore,

p-l
Vp At sin 7T(nvp At-x)/Ax

p .
H (j,X)= T f...p_n 1” 0) 7'[’(nvp pAtv-x)/Ax

n=0

4.4.2

 

Because of the relationships derived in Chapter 3, given by

Eqs. 3.1.0, the following relationships for Hp(j,x) are

valid for j=0hilhi2,

80

Hp(j+4ik,x) = Hp(j,x) i=0,_-1-_l,_-1-_2,..., 4.4.3
Hp(2j+l,x) = -Hp(—2j-l,x)

HP(2j.x) = Hp(-2j.x)

Let us now employ Eqs. 4.4.2 and 4.4.3 to simplify

Eqs. 4.4.1. It is quite clear that the expression for zp(m,t')

as given by Eqs. 4.4.4 is valid.

2k—1

p _ . _
zp(m't.) = 2VAxt Z H (n, vpt +(m n)Ax/2) + 4.4.4
m=1 I
(—1)“+’“ Hp(n,—vpt'+(n—m)Ax/2) +
(--1)n+1 Hp(n,-vpt'+(m+n)Ax/2) +

m+1

(—1) Hp(n,-vpt'-(n+m)Ax/2) +

00
Z [Hp(n,—vpt'+(m-n)Ax/2+2iL) +

i=1

Hp (n,—vpt'+ (m-n)Ax/2-21L)] +

(_l)n+m

'M8

H
II
l-‘

[Hp(n,-vpt'+(n-m)Ax/2+2iL) +
Hp(n,-vpt'+(n—m)Ax/2-2iL)] +
00

“+12 [Hp(n, -vp t' +(m+n)Ax/2+2iL) +

Hp(n,—vpt '+ (m+n)Ax/2-2iL)] +

81

00
(-1)"‘+1 2 [Hp(n,—vpt'-(n+m)Ax/2+2iL) +
l=l
Hp(n,-vpt'-(n+m)Ax/2-2iL)]

This expression places zp(m,t') directly in terms of Hp(n,x)

for n=l,2,...,2k-l.

For simplification purposes, let us introduce two

new functions z§(m,-vpt'+2iL) for j=l,2, and i=0hiln12,...,

These functions are given as

2k—1
zp(m,-vp t+2iL) ='§4§E~':E: Hp(n,-vp t' +(m-nJAx/2+21L)
44.5
n=1
P I - _
22(m,—vpt +21L) — 4.4.6

Zk-l
'EiiA— 2E: (' _1)n+1 HP(n, vp t' +(m+n)Ax/2+2iL)

Therefore, Eqs. 4.4.7 is obtained by substituting these
functions into Eqs. 4.4.4.

P I = p _ I P _ I
z (m,t ) zl(m, vpt ) + 22(m,.vpt ) + 4.4.7

q

l p . p .
(-l)m+ [z (-m.-v t ) + z (-m.-v t ) +
1 P 2 P J

1

00
2:: [21 p(m,-vp t' +2iL) + zp(m,-vpt'— —ZiL) +

 

p.
3.:

[zp(m,-vp t' +2iL) + zp(m,—vpt' -21L{] +

l"-
|-‘

82

oo
(‘1)m+1 Z [zp-m,~vpt'+ZiL) + z§(—m,-vpt'-21L)] +

I-'

00
_ 111+]. P _ _ I - P _ _ '—
( 1) 2E: [zz( m, vpt +2iL) + 22( m, vpt ZLLJ

Before this relationship, given by Eqs. 4.4.7, can be
investigated, the initial values, gn(j,0) or fn(j,0), must be
defined in some manner so that a comparison can be made.

Let gn(j,0) be defined by Eqs. 4.0.5. That is,
gn(j.0) = bjll(Ax,At) E81(nAt).
If this expression is substituted into Eqs. 4.4.2, the
function Hp(j,x) is now given as
Hp(j,X) =

P'1
v At sin 11(nv At-x)/Ax
P .
bj,1(Ax'At) Ax E E31“P'“'lmt) mépAti-x’fifix

n=0

 

 

If we let x=-vpt'+vat and compare the resulting expression

to Eqs. 4.2.7, it is evident that Eq. 4.4.8 is valid.

Hp(j.—vpt'-v t) = bj Ax,At) NP(Ax,-vpt') 4.4.8

p .1‘

If Eqs. 4.4.8 is substituted into Eqs. 4.4.5 and

4.4.6, then z§(m,~vpt'+ZiL) and zg(m,-vpt'+2iL) are now

given as
p_. =
zl(m, vpt +ZiL) 4.4.9
wl
x b 1(Ax,At) Np(Ax,—v t'+(m—n)Ax/2+2iL+v At)
2v t n, P p

P -1

83

P _ I - =
22(m, vpt +21L) , 4.4.10
2k—l

2%:78? E (-1)n+lbn(Ax, At) Np(Ax,-vpt'+(m+n)Ax/2+2iL+vat) .

n=1
Therefore, z§(m,-vpt'+2iL) and z§(m,-vpt'+2iL) can be placed
directly in terms of the approximate Fourier integral of
Mp(—vpt') which is given by Eq. 4.2.7.
The function zp(m,t') can also be expressed in terms
of the quantity V§(x) which was defined by Eq. 4.2.9. In
order to show this result, let us define the functions

y§(m,-vpt') and y§(m,—vpt') as

P _ I = P _ I
yi(m, vpt ) zi(m, vpt ) + 4.4.11
00
P _ I - p _ I_
2E:[zi(m, vpt +21L) + zi(m, vpt ZiLfl.
i=1

By means of Eqs. 4.2.9, 4.4.9, and 4.4.10, it can easily be

shown that y§(m,-vpt') and y§(m,-vpt') can be written as

y§(m,—vpt') = 4.4.12

2k—l

Egg? an’flAxAt) v§(-vpt'+(m-n)Ax/2+vat)

n=1

84

P I
," t = o o
2k—l
Ax b (Ax At) Vp(-v t'+(m+n)Ax/2+v At)
2vp t n,l ’ a p p '
[1:
Therefore,
P I _ _ I _ I
z (m,t ) _ y§(m, vpt ) + y§(m, vpt ) + 4.4.14

(-1)m+l [y§(-m.-vpt') + y§(-m,-vpt')] .

The desired expressions for zp(m,t') have now been
derived. In section 4.5, these expressions will be ins

terpreted. In sections 4.6 and 4.7, these expressions for

zp(m,t') will be compared to both the analytical solution
and the approximate Fourier solutions to obtain the launching

numbers.

 

4.5 Interpretation of zp(m,-vpt').

wv ww—vvvr‘rfv—wfi

In section 4.2, an analytical solution for Ep(x,t')
and Ip(x,t'), given by Eqs. 4.2.2, was derived. In section

4.4, a number of special forms for zp(m,t') were derived.
The objective of this section is to compare these two solu—
tions for the purposes of interpreting the results of section

4.4.

85

First of all, the function wp(m,t') is introduced
for notation purposes. Let this function be defined as
follows:

wp(2m,t') = Ep(mAx,t') m=0,1,2,...,k 4.5.0

wp(2m+1,t') = Ip(mAx+Ax/2,t') m=0,1,2,...,2k-1

Therefore,
00
wp(m,t') ==§E:MP(—vpt'+mAx/2+2nL) + 4.5.1
nz—OO

oo
(~-1)m+1 2E:MP(-vpt'-mAx/2+2nL).

n=— 00

If this problem was a semi-infinite transmission
line which was defined to start at x=0 and to extend to in-

finity in the positive x direction, the solution for

wp(m,t') would then be given as

wp(m,t') = Mp(-vpt'+mAx/2)

where m=0,l,2,...,. Therefore, for the finite line, the
remaining terms which appear in Eq. 4.5.1 for n positive re—
sults from reflections which occur at x=0 and x=L. The terms
which appear in Eq. 4.5.1 for negative n are always zero for

OijL and were introduced to obtain periodicity in x and t' .

86
According to Eq. 4.4.7, zp(m,t') can be written as
p I __._. P _ I p _ I
z (m,t ) zl(m, vpt ) + 22(m, vpt ) +

q

A
l-‘

m+1 p _ _ . p _ _
) [zl( m, vpt ) + zz( m, vpt')] +

[V18

P _ I _ _
[21(m, vpt +2nL) + 25m, vpt' 2nL)] +

:3
II
P

[\48

[z§(m,-v t'+2nL) + z§(m,—v t'—2nLJ -+

n=1
r
(—l)m+l Lz§(-m,-vp t' +2nL) + zp(- -m, ,-vpt'—2nL) +
n=1
(—1)m+l i [zp(-m -v t'+2nL) + zp(-m -v t'-2nL)] J
2 I p 2 I p .

:3
I!
H

'I
In order to better interpret the functions z§(m,~vpt'+2nL)
and z§(m,-vpt'+2nL), let us look at zp(m,t') for the semi-
infinite transmission line case. For this case, zp(m,t') is

given as

P I _ P _ I P _ I
z (m,t ) — [21(m, vpt ) + 22(m, vpt )] + 4.5.2

_m+lp__. p__.
( 1) [z1( m, vpt ) + z2( m, vpt )] .
If z§(m,—vpt') and zg(m,~vpt') are respectively defined by
Eqs. 4.4.9 and 4.4.10, then for m=0,1,2,..., and t'ito, it

can be shown that z§(—m,-vpt') is approximately zero. For

the finite line case, it can then be concluded that the re-
maining terms in the expression for n positive result from

the reflections at x=0 and x=L. For n negative, the terms

87

have no effect on the solution for 05fo and are added to
obtain periodicity in x.

Let us analyze the result which is given by Eq. 4.5.2.
First of all, z§(m,—vpt') and z§(m,-vpt') for m=0,l,2,3,...,

are incident waves traveling on this semi—infinite trans-

mission line. The remaining terms are reflected waves.

Therefore, z§(-m,—vpt') for m=0,l,2,..., can be interpreted

as a false reflected wave which is created by the definition
of the p semi-infinite line problems. By looking at Eq.

4.4.10, it can be seen that the incident wave which is given
by z§7m,-vpt') results from the reflection of z§(-m,—vpt') at
x=0. The function z§(-m,-vpt') was introduced in the process

of transforming the semi-infinite line to the infinite line
situation.
The main objective of this chapter is to find a set

of launching numbers, bj l(Ax,At), j=l,2,...,2k-l, such that

zp(m,0) s wP(m,0) 4.5.3

for all p and m=0,l,2,...,2k. Since both these functions
represent the solutions to problems of type 3, this relation-
ship is satisfied at m=0 and m=2k independent of the launch-
ing numbers.

If Eq. 4.5.3 is satisfied, then the two semi—infinite
line situations which have been discussed must also be satis-

fied. Therefore, it can be concluded that the relationship

88

Mp(mAx/2) é z§(m,0) + z§(m,0) + 4.5.4

(—l)m+l[z§(-m,0) + z§(-m,0)]

for m=l,2,...,2k-l, and p=0,l,2,..., must be satisfied.
Conversely, if this relationship is satisfied, then Eq.
4.5.3 must also be satisfied for the finite line case.

In a later section, this relationship, Eq. 4.5.4,

will be used to obtain the launching numbers.

4.6 Fourier Integral Comparison.

The object of this section is to find a set of

launching numbers such that Eq. 4.5.4 is satisfied. Since
Mp(x) is generally discontinuous at x=0, then it is quite

clear that Np(Ax,x) is the approximate Fourier integral of a

discontinuous function. Therefore, in the vicinity of x=0,

NP(Ax,x) could be a very poor approximation to Mp(x) and

probably the assumption that
Np(Ax.x) _._ M1900

is invalid in the vicinity of this discontinuity. It will
be shown in this section that if this comparison is correctly
made that it is possible to avoid this problem.

Normally, the transmission line is found to be a two
port component in a larger system. For this system, only

the terminal variables, E(0,t), I(0,t), E(L,t), and I(L,t),

89

are generally of interest. For example, when E(0,t) is
specified in some manner, the approximate numerical solution
for I(0,t) is generally required. At present there is no
information regarding this variable. In Chapter 6, it will
be shown that this variable is obtained by interpolation from
the values of current which are known at the defined current
node points located near x=0. Therefore, it is necessary
that in deriving these launching numbers that I(x,t) must be
determined very accurately at the defined current node
points which are used for interpolation purposes. In adv
dition, the launching numbers must be chosen such that the
terminal variables at x=L can be determined quite accurately.
As long as these terminal characteristics are determined
accurately, the accuracy is not as important at the interior
voltage nodes.

Before we can proceed with the comparison which is
indicated in Eq. 4.5.4, it is necessary to consider the

function y(x). This function is defined as
y(x) = Np(Ax,x+vam) + Np(Ax,—x+vat). 4.6.0

If Eq. 4.2.7 is substituted into this equation, y(x) can be

expressed now as

9O

 

p—l

v At sin.TT(nv At-x)/Ax

_ _E__ P P

Y(x) - Ax EM ((n+l)vat) ”(nvat—x)7Ax +
n=0

 

p—l
v At sin Tf(nv At+x)AAx
_E__ P p
Ax M ((n+l)vat) TT (nvat+X)/Ax '
n=0

By defining fip(x) as

Rpm) = Mp(vat+x) 0:X<oo 4.6.1

Mp(vat—x) —oo<_x f 0

and EP( x,x) as the approximate Fourier transform of fip(x),

y(x) can now be expressed as

V

At .
P Sln gixéfix . 4.6.2
Ax MP(vat) x Ax

The function Ep(x) does not have a discontinuity at x=0.

 

y(X) = 31p(Ax,x) +

Before we proceed to derive the launching numbers,

let us introduce the quantities b. which are defined as

3.1

- .._Ax_
bj,l — v At bj,1(Ax’At)° 4.6.3
P
These quantities are called transformation constants. There-

fore, instead of deriving the launching numbers bj l(Ax,At),

the quantities bj 1 will be derived. It will become evident

I

that the discussion is greatly simplified as a result of the

91

introduction of these transformation constants.

Let us now assume that

for j=0,1,2,...,k. By substituting Eqs. 4.4.9 and 4.4.10 in-
to Eq. 4.5.4, then for m=l,2,3,...,k, it can be shown that

this relationship reduces to

k

b
Mp((2m-1)Ax/2) =Z—3-‘3-3—l-‘i Y((m-n)Ax) + 4.6.4

n=1

k ~13
E 49-344 y( (m+n-1)Ax) .

n=1
In effect, Eq. 4.5.4 is being studied at the defined current

nodes.
If the increments Ax and At are chosen sufficiently

small, then the approximation

fip<Ax.x) :2 Rpm

is a valid approximation. In addition, if
vat/AX <¢= 1

then the approximation

y(X) é Rpm

92

is also valid. Therefore, for m=l,2,3,...,k, the relationship

as given by Eq. 4.6.4 reduces to

k

b ..
Mp((2m—1)Ax/2) s Eligibl- EP((m-nmx) + 4.6.5

n=1

k
B
Z—Z-fl-EJ-J‘ fip((m+n-1)Ax) .
n=1

For discussion purposes, let us further assume that

b2j-1,1 =
for j=:j' where j' is a positive integer such that
lfj'fk.

For this assumption, let us first look at the relation which

is given by Eq. 4.6.5 for mf:j'. Since the arguments of
fip(x) are always positive, the function fip(x) can now be re-

placed by Mp(x+vat). By introducing the change of variable

x' = (2m-l)Ax/2 + vam,

this relation now reduces to

93

i32n—l l
Mp(x'-vat) é '———§—L- Mp(x'—(2n—1)Ax/2) + 4.6.6
n=1

3 _
: b
anl,1 MP(X|+(2n—1)AX/2)o

n=1
This relationship states that the transformation constants
are certain constants of an interpolation formula which

gives Mp(x'—vat) in terms of MP(x'+(2n-l)Ax/2) and

MP(x'—(2n-l)Ax/2) for n=l,2,...,j'.

’ I
Let P3 (2) be an interpolation polynomial of degree

(2j'-l) which passes through the following points

Pj'(z)' = Mp(x'+z)
z=(2n-l)Ax/2 z=(2n-l)Ax/2

Pj'(Z) = Mp(X'+Z)
z=—(2n—l)Ax/2 z=—(2n—1)Ax/2

 

where n=l,2,3,...,j'. This polynomial can be written by
means of a Bessels' central difference interpolation formula

as

.. 2_
1:3 (z) =uy0 + VOYO + Q zi/fluézyo + . -+-

where

v = -zAAx

94

2p+l
<52p+ly0 = :E:—ln (2§+l) Mp(X"(2P+l)AX/2+nAx)
n=0
2
LU52PY0 = 'ln (2i) Mp(x'+Ax/2—pr+nAx)/2 +
n=0

Ef:—ln (2E) Mp(x'-Ax/2-pr+nAx)/2 .

n=0

This polynomial is an interpolating polynomial of degree

(2j'-l) which has the pr0perty that

Mp(x'+z) é pj'(z).

In addition, the coefficients are in terms of MP(x'+nAx/2)
for neilni2hi5,...ui(2j'-l).

Since it has already been assumed that vat<u=Ax,

the assumption can be made that

MP(x'—vat) é MP(x').

Therefore, the transformation constants can be obtained by

‘l
replacing Mp(x'—va ) with P3 (0) in Eq. 4.6.6 and then
equating coefficients. Therefore, these constants can be
found by means of the expression
jl
j' b2n- l l p
P (0) ————-L-IM (x'- (2n—l)Ax/2) 4.6

:ji}—;£-;Li MP(X'+(2n-1)AX/2)
n=1

.7

95

where
j' n
. 2
' 77 (21—3) 2n—1
i=1 6 yO . 4.6.8

pj'm) = .
(-4>“"1(2(n—1>):

Some examples which use the results of this derivation

are as follows.

W: 3" = 1
Bl 1[Mp(x'+Ax/2) + Mp(x'—Ax/2)] /2~=Pl(0)

where

Pl(0) = [Mpix'+AX/2) + MP(x'-Ax/2)]/2

Therefore,

U‘l
1|
1.:

1,1
and

b1,l(AX’At) = vat/Ax.

Example 2: j'=2
bl l [MP(X'+AX/2) + Mp(x'-Ax/2fl,/2 +
B

3 1 [Mp(x'+3Ax/2) + MP(x'-3Ax/2)]/2 = 92(0)

where

 

96
2 _ 5 IMP(X'+Ax12) + MPLX'sAX/Zfl
P (0) _ 8 _

MP(x'+3Ax/2) + MP(x'—3Ax/2)

 

8

Therefore,

31,1 = (5/4)

53,1 = -(1/4)
and

5v At
bl,l(Ax’At) =-ZjE;—
.. Knit.

b3’l(AXIAt) "’ " 4AX

If the transformation constants are determined by
Eqs. 4.6.7 and 4.6.8, then Eq. 4.6.5 is satisfied for
m=j',j'+l, j'+2,...,k. Let us now look at this relationship,
Eq. 4.6.5, for m=l,2,...,j'—l, when the transformation con-
stants have been determined by Eqs. 4.6.7 and 4.6.8. It can

be shown that the relationship now becomes

m

b
Mp((2m-1)Ax/2) 5-- Elia—LL;- -MP( (m-n)Ax+vat) + 4.6.9

n=1

3
B

:E:-—;E§l*i MP((m+n—l)Ax+vat) +

n$1_ '
bms11
'-—-§-‘-.Mp((n-m)Ax+vat) .

n=m+l

97

Since the transformation constants have already been
determined, this relationship can no longer be interpreted as
an interpolation formula as was the case of Eq. 4.6.6. But
interpreting this formula in a different manner, it can be
shown that Eqs. 4.6.7 and 4.6.8 are still valid for these
constants.

In order to show that these transformation constants
also give the desired results at x=(2m—l)Ax/2 where
m=1,2,...,j', it is necessary to briefly discuss the subject
of linear filters (6). Such linear filters are used pri-
marily in data smoothing applications. For discussion pur—
poses, let us suppose that we have a time-varying function
h(t) and decide to smooth this function. By introducing a,

linear filter, the smoothed function h(t) is given as

pl

h(t) = :E:dj‘h(t+ij)

j=-p'

where

pl
:E:dj = 1. 4.6.10

It can be seen that this function, h(t), is a weighted
average of the functional values h(t+jAX) for j=0hilhi2,...nip2
IIt can be seen that the transformation constants,

determined by Eqs. 4.6.7 and 4.6.8, are independent of the

functional values of Mp(x). If we look at these equations

 

98

for the case when Mp(x) is a constant, it can be concluded
that these transformation constants must also have the

property that
B . =' 1. 4.6.11

Therefore, the requirement which is given by Eq. 4.6.10 is
satisfied. It can be concluded that for m=1,2,...,j'—1, the
approximation, given by Eq. 4.6.4, is satisfied.
So far in this discussion it has been shown that if
b2j,l = 0 j=l,2,...,k—l

b2j"l,l = O 3:]

(VpAt)/AX <<l

and if the remaining transformation constants are determined
by eqs. 4.6.7 and 4.6.8, then Eq. 4.5.4 is satisfied when—
ever m=l,3,5,...,2k-l. In other words, Eq. 4.5.4 is satis—‘
fied at the current nodes. Let us keep the transformation
constants defined in the same way and let us investigate
Eq. 4.5.4 at the voltage node points.

The functions z§(m,—vpt') and z§(m,-vpt') are both
linear combinations of discontinuous functions. But at the

current nodes, these discontinuous functions can be combined

to produce continuous functions. At the voltage node points,

this is no longer the case. Even though this creates some

99

minor problems, the relationship, given by Eq. 4.5.4, can
Still be investigated.

For the time being, let us assume that the assumption
Mp(x) é Np(Ax,X)

is valid for all x. For this assumption, it can be shown

that

"0
O

z§(-m,0) m=2,3,...,2k-2

ll-
0

z§(-m,0) m=2j',2j'+2,...,2k-2
Therefore, by introducing the change of variable

i = mAx+vat,

Eq. 4.5.4 reduces to

j '
- B2n-l l —
Mp(X-VpAt) = '—-*§—L— Mp(X-(2n—1)Ax/2) +
n=1

3
i b
'-g£%l‘l Mp(x+(2n-1)Ax/2).

n=1

for m=2',2j'+2,...,2k-2. If we compare this result with Eq.
4.6.6, it can be seen that these equations are the same equa-
tions except that the variable x' has been replaced by i.
Since the transformation constants satisfy Eq. 4.6.6, they
must also satisfy this result. For m=2j',2j'+2,...,2k—2, the

discontinuity presents no problem since we are not evaluating

X

i q

H

‘0

I

.1

IL.‘

be CO:

0.41“. I
. | VIC
9 NJ . Q a a,»
I 2 . l
I u at” Fri

.ic.

:r

7er
4‘.

til

i
\
U

wiicl

$3621

+...

O
\

AT 1

lOO

Np(Ax,X) in the vicinity of the discontinuity. It can now
be concluded that these transformation constants satisfy Eq.
4.5.4 at the voltage nodes which are located at x=nAx, where
n=j',j'+l,...,k—l. In fact if j'=l, this equation is satis-
fied at all the voltage node points.

If we investigate Eq. 4.5.4 at the voltage nodes
which are located at x=nAx, where n=1,2,...j'—l, it can be

seen that z§(—2n,0) is no longer zero. In fact the sign

which appears in front of z§(-2n,0) is now a negative sign

 

where in the current case, the sign was positive. It can
then be concluded that Eq. 4.5.4 is not satisfied at these

nodes. For the assumption that

upon =2- Np(Ax.X),

let us investigate this error. At x=nAx, zg(—2n,0) can be

written as

k
b .
z§(—2n,0) = ) -—31§lei Mp((j-n)Ax-AX/2+vat).
j=n+l

If g(2n,t') and Mp(nAx) are respectively the actual solution
and the desired solution at x=nAx, where n=l,2,...,k-l, then

it can be shown that

g(2n,0) e Mp(nAX) — 2 z§(-2n,0).

101

By means of linear filter theory, it can be shown that

k
4 P _ '-
g(2n,0) — M (nAx) l :E:b2j-l,l 4.5.12

j=m+l

where Mp(nAx) is the desired solution. This says the actual

 

:
solution is the true solution multiplied by a constant. ;
So far in this section, the discussion has been based
on the assumption that '
_ I}
b2j,l = 0 j=l,2,...,k—l

For discussion purposes, let us assume that

b2j-l,l = O 3:] +1,j +2,...,k
'— _ '_l I __
b2j,l — 0 j—n +1,n +2,...,k l
Mp(x) é Np(Ax,x)

and that the remaining constants have not been determined.
Let us look at Eq. 4.5.4 for the case where m is greater

than both j' and n'. For this choice of m,

ll
0

zll’(—m,0)

ll
0

z§(-m,0)

Upon making the change of variable

102

X' = mAm/Z + vat,
Eq. 4.5.4 now reduces to

jl
B ‘
Mp(x'-vat) 4- £111.31; Mp(x'—(2n-1)Ax/2) +

n=1

B2n—1 l p

-——3-4— M (x'+(2n-l)Ax/2) +
n=1
n' _

b2n l

'——§L— Mp(x'-nAX) -
n=1
nl

i52n 1

'——§‘- Mp(x'+nAX).
n=1

The method for obtaining these constants will be to consider
separately the terms which contain the constants BZj-l 1'

Let us first look at the terms which contain the con-

stants sz 1' If we look at these terms from the viewpoint

Of linear filter theory, the values of these terms should be
approximately zero independent of the value of the constants.
An alternate way of looking at these terms is to define a

Polynomial Pn.(z) of degree 2n' which passes through the

Points

Pn.(z)’ = nAx Mp(x'+nAX)
z=nAx

 

103

where n=0hilh12,...h:n'. Let us look at the function Pn.(z)/z.

It is quite clear that this function passes through the

following points.

P I(z)/z = Mp(x'+nAx) n=1,2,...,n'
n z=nAx

Pn,(z)/z = Mp(x'-nAx) n=1,2,...,n'
=-— nAX

 

Since the limit of this function as z—vo is defined, these
constants can be obtained by equating the limit as 29.0 of
this interpolation polynomial to the terms which contain the

constants 52' This function has the property

JIl'

lim P .(z)/z = lim P ,(z) - P .(—z) /22
24.0 n z—bo [ n n ]

where
[Pn.(z)-Pn}(-z)]/22 = Mp(x'+z) - MP(x'—z).

Therefore, the limit as z—>0 must be approximately zero.
Hence, the same conclusion is obtained as was the conclusion
obtained by looking at these terms from the filter theory
point of View.

Since the sum of the terms which contain the sz 1's

I

is approximately zero, the terms which contain the constants

BZj-l 1 must be defined in the same manner as was the case

when the constants, sz 1’ were zero.

104

If we look at Eq. 4.5.4 for m less than both j' and

n‘, we have the same terms as we had in the case where sz l

but with some addition terms added since the constants sz l

are no longer zero. Let the function h((2j—l)Ax/2), 2j—ls m,
be the sum of these addition terms at the current nodes.

Hence,

nl
B
h((2j-1)Ax/2) £2 __2_;_1_,_l_ Mp((2j-1+2n)Ax/2) +

n=1
.1 _

b2n l p
L ”‘2“ M ((2j-1—2nmx/2) -

n=1
'-1

b

'-2%*l MP((2n—2j+l)Ax/2).

n=1

This result states that Eq. 4.5.4 can only be satisfied at
these current nodes when the sz 1's are identically zero.

Therefore, these constants must be zero in order to correctly

determine I(0,t).
4.7 Fourier Series Comparison.

In section 4.2, an approximate Fourier series solu-
tion for Ep(x,t') and Ip(x,t'), given by Eqs. 4.2.13, was de-

rived. In section 4.4, some special forms for zp(m,t') were
derived. The object of this section is to compare the ap-

proximate Fourier series solution as given by Eq. 4.2.13 to

105

zE)(m,t') which is given by Eq. 4.4.15. Once this comparison

is; performed, a set of launching numbers can be obtained.

First, let us consider the functions E§(x,t') and

IE;(x,t') which are given as

E§(x,t') = y§(2x/Ax,-vpt') + y§(2xflAx,-vpt') - 4.7.0
y§(-2xflAx,-vpt') — y§(-2xAAx,—vpt')
I§(x,t') = y§(2xAAx,—vpt') + y§(2xAAx,-vpt') +

p I p '
Yl(-2XAAXI-vpt ) + y2(-2x/Ax,-vpt )

vv11ere y§(2xAAx,—vpt') and y§(2x/Ax,-vpt') are respectively

given by Eqs. 4.4.12 and 4.4.13. It is quite clear that

E§(mAx/2,t') zp(m,t') m=0,2,4,6,...,2k

zp(m,t') m=l,3,5,...,2k—l

I§(mAx/2,t')

ls true.

By substituting Eqs. 4.2.12, 4.4.12 and 4.4.13 into
Ig‘lis. 4.7.0 and letting t'=0, E§(x,t') and I§(x,t') can now

13GB 'written as

p
I3(X.

valeere

B =
n
A =
n

106

sin(n7Tvat/L) sin(nTTx/L) + 4.7.1

n cos(n7Tvat/L) sin(nTTx/L) +
n cos(n7Tvat/L) sin(nfo/L) -
sin(nTTvat/L) sin(nTTx/Li]+
(Ak/Z) bi cos(kTTvat/L) sin(kYTx/L) —

p . .
(AR/2) ak Sin(k7Tvat/L) 51n(k7Tx/L)

k-l
+2

n=1

0) (A0 ag)/2 [An a: cos(nTTvat/L) cos(dn&/L)+

cos(n7Tx/L)

l

sin(n7Tvat/L)

cos(nTTx/L)

+

n cos (n flvat/L)

+

p
(Ak/Z) ak

p

w
1...:

2E: 2j 1 sin(n

L.)
|-‘

b
.k
b2j-1,1

Z

l

sin(nTTvat/L)

cos(k7Tvat/L)

sin(kTTvat/L)

jTTAx/L)

cos(nTTx/LJ

cos(k7Tx/L)

cos(k7Tx/L)

cos(anR2j-1)Ax/2L).

+

.7.

107

Since the kth term is zero at the defined node points, these
terms can be disregarded in the comparison.

The transformation constants would be best chosen if

Ep(x,0)

P
E3(x,0)

Ip(x,0) I§(X,0)

at the defined node points. This requirement is satisfied
only when the following relationship is satisfied for

n=0,l,2,...,k-l.

00
*P 'P ’ _
bn + :E: [-b2kj—n + b2kj+n] _ 4.7.3
B [afi sin(dnvat/L) + b5 cos(dnvat/L)] +

n

00
-p p p _
an + Z [a2kj—n + a2kj+n] ‘
1

J:

A [bi cos(n77§/pAt/L) — arp1 sin(rflTvat/L)]

An [as cos(n77vat/L) + bi sin(n7Tvat/L)] _

B [bi cos(nn%pAt/L) - afi sin(mnvat/L)]

n

For a given Ax and At, there are no constants An and Bn such

that this relationship is satisfied for all p. Hence, this
relationship as it stands does not provide a means for obtain—

ing the transformation constants.

 

108

An alternate approach is to use the approximate

Fourier series expressions for Ep(x,t') and Ip(x,t') which
are given by Eqs. 4.2.13. If we truncate these expressions

after the (k—l)th term, then it is possible to obtain an ap-

proximate Fourier series expression for Ep(x,t') and Ip(x,t')

’ 57

which contains the same number of terms as does E§(x,t') and
I§(x,t'). Since Ep(x,t') and Ip(x,t') for t'=0 are generally

discontinuous functions at x=0, these truncated series may

 

tend to oscillate about the actual solution. In order to ?'
correct this situation, let us introduce the Lanczos'(j
factors which are discussed in section 4.2.
By performing a term by term comparison of the re—
spective series, it is possible to obtain the following re—

lationship between the coefficients.

ll
ED

Unbfi n a}; sin(n‘lTvat/L) + bi sin(anvat/L)] +

4.7.4

An [bi cos(n7Tvat/L) - ai sin(n7Tvat/L)]
O’nag = An [a2 cos(n7Tvat/L) +b§1 sin(n7Tvat/L)] +
Bn [bi cos(n7Tvat/L) - afi sin(n7Tvat/L)]

Upon performing the indicated mathematics, the

quantities Bn and An are given as

109

B = o’n sin(n'ITvat/L) n=1,2,...,k—l 4.7.5

A = (in cos(n7Tvat/L) n=0,l,2,...,k—l

By substituting the defining equations for Bn and An’ Eqs.

4.7.2, into these relationships, it is possible to solve for

the transformation constants bj 1' Once these transformation

constants have been determined, the launching numbers,
bj l(Ax,At), can be determined by Eq. 4.6.3.
Let us investigate Eqs. 4.7.5. If the assumption

that

v t L <‘= l
pA/

is made, then the assumptions that

"0
O

sin(n7Tvat/L) n=1,2,...,k-l

ll
H

cos(n7Tvat/L) n=0,l,2,...,k—l

are valid, Therefore,

B é O n=1,2,...,k-l
n

An = 6n. n=0,l,2,...,k—l

According to Eqs. 4.7.2, the quantity Bn is a linear combin-

I

ation of the transformation constants sz 1' Upon solving

for b we find that

2j,l’

b2j,l = o. 3:1,2,...,k-1

110

This is the same result as was obtained in section 4.6.
In order to find the remaining transformation con—
stants, it is necessary to take the inverse of a (k)x(k)

matrix for which the quantity An corresponds to a row of

this matrix. By investigating this matrix, it can be seen
that these constants depend onwa or the size of the matrix.
For this reason, the transformation constants or launching

numbers as determined by section 4.6 are more useful.

4.8 Problem of Type 2.

A problem of type 2 is a transmission line problem
which has zero initial conditions and one identically zero
boundary condition at x=O. One such problem of type 2 has

the initial and boundary conditions specified as

E2(x,0) 12(x,0) = O O_<_X<L 4.8.0

0 O§t<oo

E2(O,t)
-12(L,O) = 382(0)

E2(L’t) = E52(t) Oft-coo

where E52(t) is a specified function of t. The current

orientations for this problem are from x=0 to x=L. Because
of the similarity existing between this problem and a problem
of type 1, it is possible to use methods which have been de—
rived to obtain a set of launching numbers which are valid
for transforming the boundary condition for the interval

nAtft<(n+l)At to an initial condition.

111
By letting

El(L-X,t) = E2(X,t)

Il(L-Xlt) “I2(X,t)

this problem of type 2 has been transformed to a problem of

type 1 for which the solutions are El(x,t) and Il(x,t).

Therefore, if bj 2(Ax,At), j=l,2,...,2k-l, are the launching

numbers for the problem of type 2 which is described by

Eqs. 4.8.0, these launching numbers are given by

_ _ j
bj’2(Ax,At) _ ( 1) b2k_j’l(Ax,At) 4.8.1

where j=l,2,...,2k—l.

4.9 Conclusion.

Two classes of methods are derived which transform
the boundary condition, E(0,t), for a problem of type 1 for

the interval tOfEt¢=tO+At into an initial condition at
t=tO+At. Each method derives a set of launching numbers, one

for each node, such that the initial condition at each node

point at t=tO+At is the product of E(O,nAt) and a launching

number. This class of methods is also valid for the problem
of type 2.

Of the two classes of methods which are given in
sections 4.6 and 4.7, the general method as given by section

4.6 is the preferred method. There are two basic reasons

 

112

for preferring this general method. First of all, any
launching number can be placed in terms of a transformation
constant by Eq. 4.6.3. For any given method from section
4.6, these transformation constants are independent of Ax
where as this is not the case in section 4.7. Secondly, the
sum of the solutions of the problems of type 3 is compared
to the analytical solution whereas in section 4.7, this com-

parison is made to an approximate Fourier series solution.

 

V. REMAINING CASES

5.0 Introduction.

In Chapters 3 and 4, the derivations are performed
with the assumption that both boundary conditions are speci-
fied voltages. The main object of this chapter is to demon—
strate that if the functions g(j,t) and f(j,t) are defined
in a slightly different manner, the derivations are equally
valid for cases for which the boundary conditions are either
both specified currents or a specified voltage and a speci-
fied current.

Let us consider the case where both boundary condi-

tions are specified currents. In this situation, Isl(t) and
Isz(t) are now specified boundary conditions instead of
Esl(t) and E52(t)' Let Ax=L/k, where k is a positive integer

equal to or greater than 1. In addition, let the functions
g(2n,t) for n=0,l,2,...,k, and g(2n+l,t) for n=0,l,2,...k—l,
be respectively the approximate numerical solutions for
I(nAx,t) and E(nAx+Ax/2,t). For the function g(j,t) defined
in this manner, this case is simply the dual of the case al—
ready presented. That is, the derivations for this case are
identical to the results derived in Chapters 3 and 4 except
that all voltages have been replaced by currents and all

113

114

currents have been replaced by voltages. Therefore, the re-
sults are valid for the case where both boundary conditions
are specified currents.

The remaining portion of this chapter specifically

treats the case where one boundary condition is a Specified

31

voltage and the other is a specified current.

5.1 Chapter 3 Validity.

This section will specifically show that for the

case where the boundary conditions E(0,t) and I(L,t) are

 

specified to be identically zero, the results derived in
Chapter 3 are still valid if the function g(j,t) is defined
in a slightly different manner. This case corresponds to a
lossless parallel transmission line for which a short is
located at x=0 and an open is located at x=L.

An infinite line for which the initial conditions are

V(x+4nL,0) = E(x,0) 5.1.0
V(-x+4nL,O) = ~E(x,0)

V(—x+2nL,0) = E(x,0)

V(x+2nL,O) =~E(x,0)

S(x+4nL,O) = I(x,0)

S(-x+4nL,O) = I(x,0)

S(—x+2nL,0) =-{(x,0)

S(x+2nL,0) =-I(x,0)

where 013fo and n=0,j—_l,i2,:3,..., has the same solution

for 0:5x;fL as this problem. Hence, an approximate numerical

115

solution can be obtained for this problem by solving the in—
finite line problem for the initial conditions given by
Eqs. 5.1.0.

Both the functions V(x,0) and S(x,0) have special

properties as was the case in Chapter 3. First of all, both

functions are periodic in x with period 4L. Secondly,
V(x,0) and S(x,0) are respectively an odd and an even function
of x with respect to x=0. In addition to these two properties,

V(x,0) and S(x,0) are respectively an even and an odd

 

function of x with respect to x=L. .;
Let the increment Ax be given as Ax=2L/k', where k'

is a positive odd integer. Because of these special proper-

ties, the functions f(n,t) have some very special relation-

ships. By means of Eqs. 2.7.0 and 2.7.1, one of these

special relationships can be written as

(Smf(-21,t)=-(Smf(2i,t) 5.1.1
(Smf(2i+1,t)=5mf(~21-1,t)

(Smf(2i+k' ,t)=-(§mf(-Zi+k' ,t)

(Smf(2i+1+k' ,t)=6mf(—2i-l+k' ,t)

where i=0,il,:2,..., and m=0,l,2,...,. Eqs. 5.1.1 in effect

state that Eqs. 5.1.2 are also valid.

6mf(0,t)=0 m=0,l,2,..., 5.1.2

(5mf(k',t)=0

116

Let us now define the functions g(2n,t), n=0,l,2,...,(k'-l)/2,
and g(2n+l,t), n=0,l,2,...,(k'-l)/2, respectively as the

approximate numerical solutions for E(nAx,t) and I(nAx+Ax/2,tL

Since

6m9(i,t) = (Smf(i.t) i=0,l,2,...,k',

it is quite clear that if the quantity 2k is replaced by k'.

the proof for Cm and its properties as given in Chapter 3

are also valid for this case. Hence, the result is restated

as follows:

P'—

 

 

~ ‘1
9(1.t) (5mg(l.t) 5.1.3
g(z't) (5mg(2.t)
g(k'-l,t) 5mg(k'-1,t)

 

 

In addition, the remaining properties except possibly those

given in section 3.0 are also valid.

5.2 Chapter 4 Validity.

This section specifically treats the case where the

boundary and initial conditions are given as

I(X,0) = 0 0<Xf_L 5.2.0

E(x,0)

O Oft<00

I(L,t)

'7

 

117

E(0,t)

Esl(t) 0__t<oo

1(0,0) = Esl(0) .

This case corresponds to a lossless parallel transmission
line with a voltage driver at x=O and an open circuit at x=L.
The object of this section is to show that the results of F
Chapter 4 are valid for this case.

Let us first consider the lossless parallel trans—
mission line of length 2L for which the initial and boundary

conditions are given as

 

E*(x,0) = 1*(2L,0) = o 0<x<2L 5.2.1
1*(0,0) = 1*(2L,0) = E (0)
$1
* *
E (O,t) = E (2L,t) = Esl(t). Oft<oo

It can be easily verified that the solutions for the trans-
mission line problem which is described by Eqs. 5.2.0 are
identical to the solutions for this problem for Offo.

Since this problem, defined by the boundary and initial
conditions as given by Eqs. 5.2.1, is a linear problem, the
superposition principle can be applied. Hence, it is only
necessary to solve the problem of type 1, having a length of
2L, for which the boundary and initial conditions are given

by Eqs. 5.2.2.

E:(x,0) = I:(x,0) = o 0<x:2L 5.2.2

EI(2L.t) = 0 Oft<oo

118

*
Il(0,0)

Esl(0)

*
El(o.t> = Eslm 0_t<oo

It is quite clear that the solutions for E*(x,t) and 1*(x,t)

are given as

E*(x,t)

E:(x,t) + E:(2L-x,t) 5.2.3

1*(x,t)

* *
Il(x,t) - 11(2L-x,t).

The problem of type 1 for which the boundary and
initial conditions are given by Eqs. 5.2.2 satisfies all
the requirements of Chapter 4. If Ax=2L/k', where k' is a

positive integer, the launching numbers bj 1(Ax,At),

j=l,2,...,2k'-l, are valid for this problem.

Let g; l(j,O), j=0,1,2,...,2k', be the initial con—

ditions to a problem of type 3 which was defined at

t=(n+1)At. Chapter 4 showed that if

* . _ ._ '—
gn,1(3:0) — bj 1(Ax.At) Esl(nAt) 3_1,2,...,2k 1

ll
0

*
gn’1(0,0)

* . _
gn’1(2k IO) — OI

then
p—l
g:(j.pAt) = Zg;’l(j.(p—n-1)At) j=l,2,...,2k'
n=O
91(0,pAt) = Esl(pAt)

 

119

are the approximate numerical solutions to E:(x,pAt) and
II(x,pA¢) at the defined node points. Therefore, if g*(j,t),

j=0,1,2,...,2k', are the approximate numerical solutions for
E*(x,t) and I*(x,t) at the same node points, then the solu-

tion for g*(j,t), j=0,1,2,...,2k', can be written as

9*(j.t) = 91(j.t) + (-l)Jg:(2k'-j,t). 5.2.5

The result, given by Eq. 5.2.5, states that the i

 

numerical solution for E*(x,pAt) and I*(x,pAt) at the in-
terior node points is the sum of 2p problems of type 3. By
applying the superposition principle, the two problems of
type 3 which were defined at t=(n+l)At can be combined to

form one problem of type 3 such that if

* _ _ j
bj’l(Ax,At) — bj l(Ax,At) + ( l) b2k'—j(AX’At) 5.2.6
and
§n(j,0) = b; l(Ax,At) Esl(nAm) j=l,2,...,2k'—l
§n(0.0) = o

gn(2k ,0) = 0

then
p-l

Z §n(j.(p-n-1)At) j=l,2,...,2k'-l

n=0
*
9 (0.13411) = Esl(pAt)

9*(j,pAt)

9*(2k'.p4¢) = Esl(p4¢)

120

gives the same solution for g*(j,t) as does Eq. 5.2.6.
Let us now transform the problem of type 3 for which

the initial conditions are §n(j,0) to the infinite line case.
If fn(m,0), m=0uilhi2,..., are the initial condition for this

infinite line problem, when k' is odd, it can be easily

 

Bis
shown that the solutions fn(m,t) satisfy Eqs. 5.1.1. There- 1%
fore, if
. __ * -_ I_ 1,
gn(j,0) — bj’l(Ax,At) Esl(nAt) j-l,2,...,k 1 .J
gn(0,0) = 0
gn(k',0) = 0.
then
p-l
g(j'pAt) = Z gn(j:(P"n"]—)At) j=llzlooolk'
n=0
9(0.pAt) = 231(pAt)

are the numerical solutions for E(ij/2,pAt),
j=0I214’ooo’(k'-l), and I(jAX/prAt)p j=l,3poo0(k'o In ad-
dition, if the same time domain model is used to obtain

gn(j,(p-n-1)At) as is used to obtain §n(j,(p—n-1)At), then

Eq. 5.2.7 must be true.

g(j.pAt) = g*(j.pAt) j=0,1,2,...,k' 5.2.7

121
For the case where

bj ’1(Ax,At) = 0

for j=k'+l,k'+2....,2k', the launching numbers as given by

Eq. 5.2.6 are the same as those given in Chapter 4.

W
,.L._"' “ '

If the duality principle is applied to this result,

it can be easily seen that this result is equally valid for

j=fi7r1

the case where E(L,t) is specified to be identically zero

 

and I(O,t) is a specified function of t. :J

5.3 Conclusion.

It can be concluded that if the functions g(j,t) are
defined in a slightly different manner, the results of
Chapters 3 and 4 are equally valid for the three cases which
arise when both boundary conditions are not specified
voltages.

For the case when both boundary conditions are speci-
fied voltages (currents), the increment Ax is chosen as
Ax=L/k, where k is a positive integer. The functions
g(2j,t), j=0,1,2,...,k, and g(2j+l,t), j=0,1,2,...,k—1, are
defined respectively as the approximate solutions for
E(ij,t) (I(ij,t)) and I(ij+Ax/2,t) (E(ij+Ax/2,t).

When the boundary conditions at x=0 and x=L are re-
spectively a specified voltage (current) and a specified
current (voltage), the increment Ax is chosen as Ax=2L/k',

where k' is a positive odd integer. The functions g(2j,t),

122

j=0,1,2,...,(k'-l)/2, and g(2j+l.t), j=0,1,2,...,(k'-l)/2,
are defined respectively as the approximate numerical solu—
tions for E(ij,t) (I(ij,t)) and I(ij+Ax/2,t)
(E(ij+Ax/2,t)). In addition, the launching numbers for

this case are defined by Eq. 5.2.6.

;fl;7

 

VI. INTERPOLATION FORMULAS

6.0 Introduction.

In Chapters 3 and 4, there has been no mention of F3
I(O,t) or I(L,t). For a practical problem, the approximate
numerical solution for these two variables are generally of

particular interest. In addition to these two variables, a

 

given problem might require the knowledge of the approximate
numerical solution for E(x,t) or I(x,t) at some value of x
other than at a node point. This chapter specifically de-
rives methods for obtaining these results.

If the knowledge of E(x,t) or I(x,t) are required at

t=to and x=x0. it is assumed that there is no information

present concerning the specified boundary conditions for

t=>tO except in the case where a boundary condition is speci-

fied to be identically zero for all t.

There are two general methods which can be employed
to find E(x,t) and I(x,t) at some value of x other than at a
node point. One such method is the use of interpolation or
extrapolation formulas. The other method is to numerically
solve the appropriate telegrapher equation at the point of
interest. The use of an interpolation or extrapolation
formula is by far more convenient to apply. Hence, this

chapter will only treat this method.

123

124

The most obvious approach is simply to apply any
standard interpolation or extrapolation formula for which
g(j,t) are known quantities. For values of x near the ends
of the line, it is quite clear that this approach does not
result in central difference formulas. Hence, this chapter
derives some methods for which central difference formulas Ea

can by employed for every value of x for certain special

n.’ ' "'

cases .

6.1 At Least One Boundary Condition Specified Zero. E

For the case for which at least one boundary condi-

 

tion is specified as zero, the solutions for problems of this
type have certain Special properties. These special pro-
perties can be employed to obtain central difference interpo—
lation formulas for values of x near the end of the line for
which the boundary condition is specified as zero.

Let us first consider the transmission line of length

L for which the initial and boundary conditions are given as

follows:
E(0,t) = Esl(t) Oft-ccn 6.1.0
E(x,0) = hl(x) O<fo
Ihmm =h2m0 Ofx<L
quU =0 Oft<m

It is quite obvious that the transmission line of length 2L

for which the boundary and initial conditions are given by

125

Eqs. 6.1.1 has the same solution for Ofifo. Because of the

*
E*(2L-x,0) = E (x,0) = E(x,0) 0<x<L 6.1.1
-I*(2L—x,0) = 1*(x,0) = I(x,0) Offo
E*(2L,t) = E*(0,t) = Esl(t) 0ft<co

symmetry of this equivalent problem, the result as given by

Eqs. 6.1.2 must be valid. This result in effect states that

E(x,t) = E*(x,t) E*(2L—x,t) 0:5x:L 6.1.2

I(x,t) = I*(x,t) —I*(2L-x,t) ijfL

 

the solution E(x,t) and I(x,t) can be considered respectively
as an even and an odd function of x with respect to x=L.
New, let us consider the transmission line of length

L for which the initial and boundary conditions are given as

follows:
EHLU =Egflt) Oft<® 6Ju3
E(x,0) = hl(x) 0<x<L
I(x,0) = h2(x) Offo
EUHU =0 Oft<a>

An equivalent problem which gives the same solution for

Offo is given as follows:

-E*(2L—x,0) E*(x,0)

E(x,0) O<X<L 6.1.4

ll

I*(2L-—x,0) 1*(x,0) = I(x,0) 0‘:fo

—E*(2L,t) = E(0,t) = Esl(t) 0<t<ao

126

Because of the symmetry of this problem with respect to x=L,

the result as given by Eqs. 6.1.5. must be valid. For this

E(x,t) = E*(x,t) a E*(2L—x,t) 015fo 6.1.5

I(x,t) = I*(x,t) = I*(2L—x,t) 0:3fo

case, the solutions E(x,t) and I(x,t) can be considered re-
spectively an odd and an even function of x with respect to
x=L.

It can be concluded that when E(x,t) is specified to
be identically zero at a boundary, then E(x,t) and I(x,t) at

t=t0 can be considered respectively an odd and an even

function of x with resPect to this boundary. Likewise, when
I(x,t) is specified to be zero at a boundary, E(x,t) and

I(x,t) at t=t0 can be considered respectively an even and an

odd function of x with respect to the boundary. These
properties are independent of the boundary condition at the

other end of the line.

6.2 Both Boundary Conditions Not Identically Zero.

 

This section derives some preperties about the solu—
tion for the case for which the conditions of section 6.1 are
not satisfied.

Let us consider the transmission line of length L

for which the initial and boundary conditions are given as

E(0,t) = Esl(I(0,t),t) Oft<cn 6.2.0

E(x,0)= hl(x) 0<x<L

 

127

I(x,0) = h2(x) OEXEL

E(L,t) = E82(I(L,t),t) O§t<ool

where the boundary condition on each end of the line is al-
lowed to be a function of the other boundary variable. It is

quite clear that if either Esl(I(0;t),t) or E82(I(L.t),t) are

not identically zero, this problem does not satisfy the con—
ditions as stated in section 6.1. If the specified boundary
conditions are a function of I(O,t) and I(L,t), it is
absolutely necessary that the numerical solution for these
two variables be known. Hence, it is necessary to derive
some properties about I(x,t) which can be used for interpo-
lation purposes.

The transmission line itself is linear. Hence, the

superposition principle may be applied at tatO in order to

divide this problem into two problems for which the sum of
the solutions of the two problems gives the solution to the
original problem. The initial and boundary conditions for

the two resulting problems are given as follows:

E1(O,t) = Esl(I(0,t),t) test-zoo 6.2.1
'El(x,t0) = o 0<x<L

Il(x,t0) - 0 Offo

E1(L.t) = 0 toft<oo

 

128
E2(0,t) = 0 toft<ao 6.2.2
12(X.t0) == I(x,to) OSXEL
E2(X,t0) a E(x,to) O<X<L
E2(L,t) = ESZ(I(L,t),t) toft<oo

The initial condition I2(x,t0) for 0::fo contains all the

__ .—
'I'
I
-1

information about I(x,to). Since the problem for which the

initial and boundary conditions are given by Eqs. 6.2.2

 

satisfies the conditions of section 6.1, the function I(x,to) LJ

can be considered an even function of x with respect to x=0.

A similar argument can be used to show that I(x,to) can be

considered an even function of x with respect to x=L as far
as interpolation is concerned.

Let us now consider the case where I(O,t) and I(L,t)
are specified functions of t. The duality principle can be
applied to show that E(x,t) can be considered for interpo-
lation purposes as an even function of x with respect to
both x=0 and x=L. For the remaining case when E(0,t) and
I(L,t) are both specified in some manner, I(x,t) and E(x,t)
can be considered even functions respectively with respect to

x=0 and x=L.

6.3 Conclusion.

The results presented in this chapter have indicated

that if one variable (either E(x,t) or I(x,t)) is specified

129

at a boundary in some manner, the remaining variable can be
considered an even function of x with respect to this boundary
as far as interpolation is concerned. This specified variable
can be a function of the remaining variable. In addition,

the specified variable has no special property as far as be-
ing an even or an odd function of x with respect to this

boundary except when it is specified to be identically zero.

 

VII. GENERAL TRANSMISSION LINE PROBLEM

7.0 Introduction

The main objective of this chapter is to demonstrate
how the concepts of Chapters 3, 4, 5, and 6 can be combined
to derive three classes of time domain models which can be
applied to a very general transmission line problem. For
demonstration purposes, let us consider the transmission line
problem for which the initial and boundary conditions are

given as follows.

I(x,0) == hl(x) Offo 7.0.0
E(X,0) = h2(x) 0<x<L
E(0,t) = Esl(I(0,t),t) Oft<cn
E(L,t) = E52(I(L't)’t) Gift-=00

This problem allows E(0,t) and E(L,t) to respectively depend
on I(O,t) and I(L,t) in some prescribed manner. This de-
;pendence may be either linear or nonlinear.

Before any attempt is made to apply the results of
the previous chapters, let us discuss certain preliminary
aspects of the numerical solution. First of all, the incre—
ment Ax is chosen as Ax=L/k, where k is a positive integer.

In addition, g(2n,t), n=0,l,2,...,k, and g(2n+l,t),

130

 

f F
"“229

13'1

n=0,l,2,...,k—l, are respectively the approximate numerical
solutions for E(nAx,t) and I(nAx+Ax/2,t).

Since E(0,t) and E(L,t) are dependent on I(O,t) and
I(L,t), it is necessary to employ an interpolation formula
to calculate these variables. The results of Chapter 6 can
be directly applied in this case for the purposes of obtain-
ing such an interpolation formula. Hence, I(O,t) and I(L,t)
can be written as shown in Eqs. 7.0.1. The same formula

need not be used

I(O,t), a. rg'(1,t) 1 7.0.1

1,1 ’ ' i,i ' i,k

I(L,t) d2,l . . d . . . d2,k g(3,t)

g(Zi’llt)

 

 

g(Zk-llt)

for both I(O,t) and I(L,t). It is quite clear that this

interpolation formula is independent of the difference method

employed to find the solution for g(j,t).

The functions g(0,t) and g(2k,t) must be specified as

9(0.t) = Esl(I(0.t).t) 0_<_t<co 7.0.2

g(2k,t) = E52(I'(L,t),t) Oftcoo

independent of the method of solution for g(j,t),

 

132

j=l,2,...,2k—l. At t=0, the specified values of I(O,t) and
I(L,t) should be used to calculate these boundary conditions.
At all other values of t, the interpolation formula as given
by Eqs. 7.0.1 must be used to determine the values of I(O,t)
and I(L,t) in Eqs. 7.0.2.

In order to indicate how the results of the previous
chapters can be applied to this problem to obtain the solu-
tion for g(j,t), the superposition principle will be used.

At t=nAx, this problem may be subdivided into three separate
problems for which the initial and boundary conditions are

given as follows.

En(x,nAt) = E(x,nAt) 0<x<L 7.0.3
In(x,nAt) = I(x,nAt) Offo

En(0,t) = 0 nAtft<co

En(L,t) "' 0 nAt§t<OO

En'l(x,nAt) = 0 0<fo 7.0.4
In’l(x,nAt) = 0 Offo

Emluufl =0 r$tft<®

En’1(0,t) .. Esl(I(0,t),t) nAtft<oo

Emzhnm =0 Ofx<L * 7JL5

In,2(x'0) = 0 OEX<L

 

133

0 nAtEt-coo

En,2(0't)

En’2(L,t) = E52(I(L't)’t) nAtft<CD

The solution to the original problem for tf:nAt is simply
the sum of the solutions to these three problems. That is,

E(x,t) and I(x,t) are given by

E(x,t) = En(x,t) + En,l(x’t) + En’2(x,t) 7.0.6

I(x,t) = In(x,t) + In,1(x’t) + In 2(x,t)

I

for tfinAt. The orientation of the three currents in Eqs.
7.0.6 are assumed to be from x=0 to x=L. The problem,
described by Eqs. 7.0.3, satisfies the conditions of Chapter
3. The remaining two problems, described by Eqs. 7.0.4 and
7.0.6, each satisfy the conditions of Chapter 4 as long as

the boundary conditions, En 1(0,t) and En 2(L,t), for tEZnAt

are properly defined.

Let the functions gn(j,t), gn l(j,t), and gn 2(j,t)

be respectively the numerical solutions to these three

problems for which the initial and boundary conditions are
given by Eqs. 7.0.3, 7.0.4, and 7.0.5. It is quite clear
that the following relationships must be true, independent

of the method of numerical solution.

g(j,t) = gn(j,t) + 9n,1(j't) + 9n 2(j,t) 7.0.7

134

 

gn(j,nAt) = g(j,nAt) j=l,2,...,2k—l 7.0.8

gn(0,t) = gn(2k,t)= 0 nAtft<<D

gn'1(j.nAt) = 0 j=l,2,...,2k-l 7.0.9

9n,1(0.t) = g(0.t) nAtft<oo

gn,l(2k't) = O nAtEt<oo R
1

gn’2(j,nAt) = 0 . j=l,2,...,2k-l 7.0.10 .

9n,2(0.t) = 0 nAtft<oo j

gn’2(2k,t) = g(2k,t) nAtft<oo

The numerical procedure for finding g(j,(n+l)At),

j=0,1,2,...,2k-l, is essentially finding gn(p,(n+l)At),
9n 1(P.(n+l)At). and 9n 2(p,(n+l)At) for p=l,2,...,2k-l, by

the methods indicated in the previous chapters. According to
the results of Chapter 4, the quantities gn l(p,(n+l)At) and

gn 2(p,(n+l)At), p=1,2,...,2k-l, respectively depend only on
the values of gn l(0,nA.t) and gn 2(2k,nAt). Once all of

these values are determined, Eqs. 7.0.7 can be used to find
g(j,(n+l)At) for j=l,2,...,2k—l. By means of the interpo-
lation formula, Eqs. 7.0.1, and Eqs. 7.0.2, g(0,(n+l)At) and
g(j(n+l)At) can be determined. Since the functions
g(j,(n+l)At) corresponds to the approximate numerical solution
for E(x,(n+l)At) and I(x,(n+l)At) at certain prescribed node
points, the whole process of superposition can be repeated

to find g(j.(n+2)At).

135

The next sections Ofthis chapter indicate in detail
how the results of Chapters 3 and 4 can be employed to obtain
three classes time domain models for which g(j,t),

j=o,l,...,2k, is the solutions.

7.1 Classes A and B.

This section indicates how the results of Chapters 3,
4, and.6 can.be employed to obtain two classes of time do-
main models which use difference methods in both the x and t
dimensions. In order to obtain these models, it is first

necessary to find the solution for gn(j,(n+l)At),

gn 1(j,(n+l)At), and gn 2(j,(n+l)At). Once this result is

shoWn, this section demonstrates that the difference methods
for each of these three transmission line problems can be
combined to produce two classes of time domain models which
represent the transmission line.

Since the transmission.line problem, described by Eqs.
7.0.8, is a problem of type 3, the results of Chapter 3 are

valid for obtaining gn(j,(n+1)At) for j=l,2,...,2k-1, from

the specified initial conditions which are given at t=nAt.
There are two possible techniques for obtaining this result.
One technique uses any difference method, given by Eqs.
3.2.4, which is derived for a class 1 time domain model. The
other technique uses any class 2 time domain model. Regard-

less of the technique, the matrix equation, Eqs. 7.1.0, must

 

136

be valid for obtaining gn(j,(n+l)At). The matrix A; is

n— r— d!
gn(1,(n+l)At) T gn(1,nAt) * 7.1.0
gn(2. (n+1)At) gn(2,nAt)
*
= A0
gn(2k*l,(n+l)At) gn(2k—l,nAt)

 

 

 

 

determined by a given two level difference method which uses

either of the two indicated techniques. The entries of A;

are functions of both Ax and At.
The results of Chapter 4 can be directly applied to
the remaining two transmission line problems, described by

Eqs. 7.0.9 and 7.0.10, in order to obtain gn l(j,(n+l)At)
and gn 2(j,(n+l)At) for j=l,2,...,2k-l. The results of

Chapter 4 states that there are two sets of launching numbers,

(bj 1(Ax,At)) and (bj 2(Ax,At)), such that Eqs. 7.1.1 are

true for j=1,...,2k—l.

9n'1(j:(n+l)At) lajll(Ax,At) gn l(O.nAt) 7.1.1

9n’2(j.(n+l)At) bj 2(Ax,At) gn 2(2k,nA.t)

In Chapter 4, there is a general method for determining these
launching numbers. These launching numbers may be chosen by
the same specific method or by a different method. If the
same specific method is employed, then these two sets are

related by

 

137

_ j '-
bj’2(Ax,At) - (—l) b2k-j,l(Ax’At) j-l,2,...,2k-l.

Before gn l(0,(n+l)At) and gn 2(2k,(n+l)At) can be de-

termined, I(O,(n+l)At) and I(L,(n+l)At) must be known. By
means of Eqs. 7.0.7 and 7.0.1, I(O,(n+l)At) and I(L,(n+l)At)

are given as follows.

I(O,(n+l)At) d Ign(l,(n+l)At) 1 7.1.2

1,1

I(L,(n+l)At) d d gn(3,(n+l)At)

2,1 ' ° 2,k

gn(2k—l,(n+l)At)

 

 

 

K— _
._ .T _ _1\
9n,1(l.(n+l)At) gn'2(l,(n+l)At)
9n'1(3.(n+l)At) gn 2(3,(n+1)At)

1 >

 

 

 

 

 

gn l(2k—l,(n+l)At) (?n'2(2k-l'(n+l)Atl

‘J I

Once these variables are known, gn l(O,(n+l)At) and
gn 2(2k,(n+l)At) can be determined by Eqs. 7.0.2.

New that the functions gn(j,(n+l)At), gn l(j,(n+l)At),
and gn’2(j,(n+1)At) are known for j=0,1,2,...,2k, g(j,(n+l)AtL

is given by Eqs. 7.0.7. With this knowledge of g(j,(n+l)At),
three new transmission line problems can be created at
t=(n+l)At). Therefore, the entire process can then be re-

peated for finding g(j,(n+2)At).

.7 l ‘Tf’
9‘ '—:—"= .

 

 

138

The discussion has indicated how the results of

Chapters 3, 4, and 6 are applied to determine g(j,(n+l)At)

from g(j,nAt). It is clear that a number of the indicated

steps can be combined. Hence, the general form of the equa-

tions required to obtain a difference equation solution can

be written as follows.

rg(l,(n+l)At)
g(2,(n+l)At)

bl,1(Ax,At)

b2’1(Ax,At)

 

b2k_l(Ax,At)

I(O,(n+l)At)

I(L,(n+l)At)

g(0,(n+l)At)

g(2k—l,(n+l)AtU

—q __

g(l.nAt)
9(2.nAt)

0 .

 

 

—

g(2k—1,nAt)
J- .4

T

 

In 2(Ax,At) 9(0.nAt)

g2,2

 

b2k’l’2(Ax,At:

d1,1 . . d1,k

d d

2,1 ° ' 2,k

(Ax,At) g(2k,nAt)

~-<3(l.(n.+1)At)

g(3,(n+l)At)

 

= Esl(I(0,(n+l)At),(n+l)At)

g(2k,(n+l)At) = E52(I(L,(n+l)At),(n+1)At)

Lg(2k-l,(n+1)At)

7.1.3

_

7.1.45

 

7.1.5

139

Equations 7.1.3 and 7.1.4 are simply a restatement of Eqs.
7.0.1 and 7.0.2.

The same argument can be directly applied for differ-
ence methods are greater than two level. The end result can
be obtained by replacing Eqs. 7.1.3 by Eqs. 7.1.6 where the

matrix G(t) is defined by Eqs. 3.2.1. The other sets of

J2
G((n+l)At) = 2A: G((n-i)At) + 7.1.6
k=0
Pbl’l(Ax,At) bl 2(Ax,At) g(0,nAt)
b2’l(Ax,At) b2’2(Ax,At) g(2k,nAt)
b (Ax,At)
*2k-1,l b2k-1,1(Ax’AtU

 

 

equations remain the same.
New that certain results of the previous chapters
have combined, it is possible to define two classes of time

domain models which represent the transmission line.

Definition: Class ArTime Domain Model:
A time domain model belongs to class A if and only if the
model can be obtained by the following steps.
1. The model is described by Eqs. 7.1.3 or 7.1.6, and
7.1.4.
2. The matrices A3, A3,..., and A32 are obtained by a
difference method in both x and t which is derived

from a class 1 time domain model.

 

140

Definition: Class B-Time Domain Models:
A time domain model belongs to class B if and only if the
model can be obtained by the following steps.
1. The model is described by Eqs. 7.1.3 or 7.1.6, and
7.1.4.
2. The matrices A6, A:,..., and Agz correspond to a
time domain model belonging to class 2.
The only difference between the two classes of models

is the technique for which the matrices A*, A1,..., and AI

32

are obtained. For a model belonging to class A, these
matrices have a class 1 time domain model as a basis. Class
B on the other hand have the class 2 time domain models as
its basis. Since these two classes are not mutually dis—

joint, it is possible for a model to belong to both classes.

7.2 Class C.

This section combines the results of Chapters 3, 4,
and£5 to derive a set of (2k—1) linear ordinary differential
equations for the solutions are g(j,t) for j=l,2,...,2k~l.

Let us first consider the problem of type 3 which is

described by Eqs. 7.0.8.' The approximate solution, gn(j,t),

to this problem can be found by solving a set of (2k—1)

ordinary differential equations which are given by Eqs. 3.2.3.

These equations for a (2p+l) degree approximation to the

partial derivations can be written as

 

 

141
dGn(t)
“a?” = Ap Gum
where the matrices Gn(t) and Ap are respectively defined by
Eqs. 3.2.1 and 3.2.2.

If Eqs. 7.1.1 are restated, these equations state

that

gn’l(j.(n+1)At) bj’1(Ax,At) gn’l(0,nAt) j=l,...,2k-l

9n’2(j.(n+1)At) bj’2(Ax,At) gn’2(2k,nAt) j=l,...,2k—l

are true for the remaining two problems. This relationship

uses the fact that gn l(j,nAt) and gn 2(j,nAt) for

j=l,2,...,2k-l are zero. For any given method for finding the

launching numbers, bj l(Ax,At) and bj 2(Ax,At), Chapter 4

showed that these launching numbers can be written as

bj’l(Ax,At) = (VpAt/Ax) bj,1 j=l,2,...,2k-l 7.2.1
b. x,At = t x B. '—1,2,...,2k-l
3,2(A ) (va /A) 3’2 3
where Ej l,and bj 2 are constants independent ofllx or At.

If Eqs. 7.2.1 are substituted into Eqs. 7.1.1, the

following relationships can be obtained.

 

 

gnll(j.(n+l)At) - gnJl(j,nAt) v __
At -"""'—_'= x bj,1 9n,1‘°'nA¢)

9nL2(j.(n+l)At) - gn 2(j,nAt) .XE _
= Ax bj,2 9n,2(2k'nAt)

At

 

142
But

dgn,1(j’nAt) . gn l(j,(n+1)At) - gn l(j,nAt)
dt = At

 

 

dgn12(j,nAt) . gn 2(j,(n+l)At) - gn 2(j,nAt)
dt = At

 

 

Therefore,

 

 

dg (j,nAt) v
n,l . E -

dg (j,nAt) v
n,l ; p -

If this relationship, given in Eq. 7.2.2, is combined with
Eqs. 7.2.0, the solution for g(j,t), j=l,2,...,2k—l, can be

found by solving the following (2k-l) ordinary differential

equations.
V __ _
.Q§1£1 _ ._E ' t
dt ‘ Ap G(t) + Ax b1,1 b1,2 g(0,t) 7.2.3
b2,1 b2,2 g(2k,t)
szk-1,1 b2k-1,f_a__

 

 

In addition to Eqs. 7.2.3, it is necessary to use
Eqs. 7.0.1 and 7.0.2 to properly define the quantities
g(0,t), g(2k,t), I(O,t) and I(L,t). The third class of time

domain models can now be defined.

 

143

.Qefinition: Class C—Time domain Models:
A time domain model belongs to class C if and only if the

model is described by Eqs. 7.0.1 and 7.2.3.

7.3 Discussion of Mggels.

Let us assume that this parallel line of length L is
a single component in a given system. Each port can be
represented by a linear graph of one part which contains

exactly one element. The variables el(t) and il(t) can be

associated with the graph representing the port at x=0. For

the port x=L, the variables e2(t) and 12(t) are associated

with its linear graph representation. These graph variables
are associated with the transmission line variables as shown

in qu. 7.3.0 where the minus sign is required because of

the
el(t) = E(0,t) 7.3.0
il(t) = I(O,t)

assumed current orientations.

Let us assume that a time domain model which belongs
to either class A or class B is employed to relate these
graph variables. If we assume that the increments,Ax and,At

are properly chosen for this transmission line model, it is

 

 

144

quite clear that this model defines the relationship

11((n+1)At) h3(el(nAt),e2(nAt),nAt) 7.3.1

12((n+l)At) h4(el(nAt),e2(nAt),nAt)

where the functions h3(el(t),e2(t),t) and h4(e1(t),e2(t),t)

are given by Eqs. 7.3.1, may or may not be the desired rela—
tionship to represent the transmission line as a two port
component. The answer to this question is determined by the

methods of numerical solution which is to be used to solve the

 

system for which the transmission line is a two port
component.

For discussion purposes, let us assume that Eqs.
7.3.1 are the desired relationships. The most difficult
problem which occurs when using the results of section 7.1
is the problem of choosing the increment At once,Ax has been
chosen. For a given model for the transmission line, the
increment which is required is a function of this model and
a function of the mathematical description of the entire
system. 'For example, if At is chosen such that the trans-

mission 1ine model is stable for the case when e1(t) and

e2(t) are specified functions of t and are independent of

il(t) and 12(t), there is a strong possibility that the equa-

tions which describe the transmission line are unstable for
the case when el(t) and e2(t) depend on il(t) and 12(t) in

some manner .

145

The transmission line time domain model which belongs
to class C relates these graph variables in an entirely dif-
ferent manner. In fact, this class of models gives the rela-

tionship

dil(t) _
EfiT——* = h3(el(t),e2(t),t) 7.3.2 E-

di2(t) _ 531..
dt _ Fl4(eJ-(t)7e2(t)lt)

where the functions h3(el(t),e2(t),t) and h4(e1(t),e2(t),t)

 

are determined by Eqs. 7.2.3 and 7.0.1. The method of . E5;
numerical solution which is used to solve the differential
equations, given by Eqs. 7.2.3, can be chosen so that it is
the same as the method of numerical solution which is em—
polyed to solve the entire system. For example, if the other
components in the system are described by a set of differen—
tial equations, then numerical method for solving these
differential equations can also be employed to solve Eqs.
7.2.3.

One advantage of using a class C model is that some
technique of numerical solution can be used which auto-

matically decides on the increment At,

7.4 Bgth Graph glements arefinot Chords.

All of the discussion so far presented in this
chapter is applicable to transmission line problems for which

the transmission line linear graph representation has been

 

146

formulated as chords. The findings of Chapters 5 and 6 can
be directly applied to the results of sections 7.0, 7.1, 7.2.
and 7.3 to show that by modifying the equations which
describe the models, the models are also applicable to the
other cases which arise in formulation.

Upon applying the findings of Chapters 5 and 6, the

results are stated as follows:

 

Both Graph elements are branches:

 

l. Ax = L/k, where k is a positive integer. . -
2. The functions g(2j,t), j=0,1,2,...,k, and g(2j+l,t), -gJ
j=0,1,2,...,k—l, are respectively defined as the ap—
proximate numerical solutions for I(ij,t) and
E(ij+Ax/2,t) .
3. In Eqs. 7.0.1, replace I(O,t) and I(L,t) respectively
by E(0,t) and Eu;,t).
4. Upon performing these modifications, the three

classes of time domain models are valid for this case.

One qrgph element is a chord and the other is a branch:

1. Ax=2L/k', where k' is a positive odd integer.

2. When the port at x=0 is formulated as a chord
(branch), the functions g(2j,t), j=0,1,2,...,(k'—1)/2,
and g(2j+l,t), j=0,1,2,...,(k'wl)/2, are defined re-
spectively as the approximate numerical solutions for
E(ij,t)(I(ij,t)) and I(ijtAx/2,t)(E(ij+Ax/2,t)).

3. Replace the quantity 2k by k' in all equations in

sections 7.0, 7.1, and 7.2.

 

147

4. In Eqs. 7.0.1, replace I(L,t)(E(0,t)) by

E(L,t)(I(0,t)).

 

5. Use Eqs. 5.2.6 to recalculate the launching numbers.
6. Upon performing these modifications, the three
classes of time domain models are valid for this

case.

7.5 Example Problem.

In order to illustrate the concepts which are pre—

sented in this chapter, let us work a simple problem. This

 

problem consists of a lossless transmission line of length L
which is terminated respectively at x=0 and x=L with a speci—
fied voltage source and a matched load. In addition, it will
be assumed that the initial conditions on the line are speci-
fied to be zero.

For simplification purposes, the parameters of the

transmission line are specified as

Vl/c = 1 ohm

vp = 1 meter/second

L = 1 meter.

This choice of parameters forces the telegrapher's equations
to be in normalized form. In addition, the computations re-
quired to obtain any time domain model is simplified.

The linear graph for this problem can be drawn as
shown in Fig. l where elements 2 and 3 are associated

respectively

148

Fig. l Oriented linear graph for example problem.

with the transmission line ports at x=0 and x=L. Elements 1
and 4 are associated respectively with the specified voltage
source and the matched load. If we let elements 1 and 4 be
the tree for this graph, the f—cut set, f—circuit, and ele-

ment equations are respectively:

 

 

 

 

1 0 1 OF. bil(t)1 7.5.0
o 1 0 1_J i4(t)
= 0
12m
i3(t).
-1 0 1 0 Elm)- 7.5.1
0 -1 o 1 e4(t)
= o
e2(t)
f3”).
el(t) - 2t Oft20.5 7.5.2

 

.119

149

-m;+4 luSstszfi

e2(t) = E(0,t)

12(t) = I(O,t)

e3(t) = E(th)
i3(t) = -I(L,t)
e4(t) = i4(t)

Before any attempt is made to solve these equations, the
time domain model which is used to represent the trans-
mission line must be specified.

Since there are an infinite number of time domain
models which belong to each class of models, the transmission
line portion of the problem will be represented by a model
which belongs to class C because it is described by differ—
ential equations. To further illustrate some of the con-
cepts, this problem will be worked four times using two dif-
ferent models and 2 choices oszx for each model.

For this problem, the incrementzAx must be chosen as
Ax=l/L, where k is a positive integer, independent of the.
method of numerical solution. In addition, the functions
g(2j,t), j=0,1,2,...,k, and g(2j+l,t), j=0,1,2,...,k—l, are
defined respectively as the numerical solution for E(ij,t)

and I(ij+ x/2,t).

150

According to section 7.2, any model which belongs to

class C can be written in general terms as

 

 

 

 

 

e121- —— — — ~
dt _ Ap G(t) + (VP/Ax) bl’l bl’z g(0,t)1
b2,1 b2,2 tg(2k,t)
. . 7.5.3 i1
LbZk-l,l b2k-1,2_ E .
I(O,t) d1 1 . . d1 1 d1 k g(l,t) 7.5.4 "7*
I(L,t) d2 1 . . d2 1 . . d2 k g(2i-1,t)
g(2k—l,t)
L .—

 

 

Once the matrix AP' the interpolation formulas, and the

launching numbers or transformation constants are chosen,

the time domain model has been defined. In order to solve
this resulting time domain model, it is only necessary to

specify Ax, At, the method of numerical solution, and the

boundary conditions.

In order to find the terminal variable I(O,t), let
us define an interpolation formula in terms of the functions
g(l,t) and g(3,t). According to Chapter 6, the function
I(x,t) can be assumed to be an even function of x with re-

spect to x=0. Therefore, if

151

I(x,t) a + a x

such that

g(l,t) = a0 + a2 (Ax/2)2

9(37t) = a + a (3Ax/2)2

0 2
then

I(O,t) 9 g(l,t)/8 — g(3,t)/8. 7.5.5

By using the same interpolation formula to find I(L,t) in

terms of g(2k-l,t) and g(2k-3,t), I(L,t) is then given as
I(L,t) = 9 g(2k—l,t)/9 — g(2k—3,t)/8. , 7.5.6

The interpolation formulas will remain the same when defining
both time domain models.

The matrices Ap, p=0,l,2,3,..., are defined by Eqs.

3.2.2. Therefore, once the value of p is chosen, this portion
of the domain model has been specified. For the two specific
models which will be solved numerically, p is chosen as zero.

Therefore,

A0 = k C 7.5.7

where C is the transformation matrix which is defined by
Eqs. 3.1.1 and k is a scalar.
Finally, two sets of transformation constants must

be specified in order to completely specify these two time

152

domain models. One class of methods for obtaining these

transformation constants is derived in Section 4.6. If the

results are
section 4.6

such set is

III
II

used from the two examples which are given in

and the results of section 4.8 are applied, one

1 7.5.8
o j=2,3,...,2k—l

— -1
0 j=l,2,...,2k-2

and the other set is

U‘I
II

U” 0‘!
II ll

6‘
II

5/4 7.5.9
-1/4
0 j=4,5,...,2k—l
0
= -5/4
l/4
0
0. j=l,2,...,2k—4

 

 

153

Let us now solve Eqs. 7.5.0, 7.5.1, and 7.5.2 for

i2(t) and i4(t). The numerical solution for i2(t) and i4(t)

along with the analytical solutions for these variables are
shown for the interval 05t54 in Figures 2, 3, 4,.and 5
where the information concerning each technique of solution

is given as follows:

Figure 2:

1. At = 1/80 seconds.
2. k = 4.
3. Transformation constants specified by Eqs. 7.5.8.

4. Runge Kutta fourth order method of solution.

Figure 3:
1. At = 1/80 seconds.
2. k = 8.
3. Transformation constants specified by Eqs. 7.5.8.

4. Runge Kutta fourth order method of solution.

Figure 4:

1. At = 1/80 seconds.
2. k = 4.
3. Transformation constants specified by Eqs. 7.5.9.

4. Runge Kutta fourth order method of solution.

Figure 5:
1. At = l/80 seconds.

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158

3. Transformation constants specified by Eqs. 7.5.9.

4. Runge Kutta fourth order method of solution.

Let us now discuss certain aspects of the numerical

solutions for i2(t) and i4(t) which are given by Figures 2,

3,‘L and 5. If Figure 2 is compared to Figure 3 and Figure
4 is compared to Figure 5, the effect of the increment.Ax on
the numerical solutions is clearly shown. Hence, if Ax was
chosen even smaller, the accuracy of the approximation would
be improved.

Since the input current, i2(t), for all of the models
is obtained by interpolation from the calculated values of
current at certain interior node points, the time delay
property of the equations causes the calculated input current
to lag the theoretical solution. This time delay property is
corrected at the expense of overshoot to some extent by using
the transformation constants which are specified by Eqs.
7.5.9. This effect is also reduced by a smaller Ax since
the time delay between the input and the first current node
is reduced.

For tEEZ, the numerical solution for i2(t) is not

zero where theoretically this current is zero. The reason
for this result is that the time domain models do not truly
have a characteristic impedance equal to unity. As a result,
reflections occur at x=0 and x=L. This effect is also ob-
served in the solution for i4(t). The reduction of this ef-

fect by decreasing.Ax is shown by comparing the results for

 

 

159

k=4 to k=8. This phenomena is generally present in all trans-

mission line approximations whether they are those derived in
the thesis or not.

The variable i4(t) corresponds to the output current.
Upon observing i4(t), we see that the model gives us the cor—
rect time delay. In addition, we see that i4(t) begins to

rise before the theoretical solution begins to rise. This
property can generally be expected from all transmission
line approximations.

It can then be concluded that for sufficiently small

Ax these models provide the desired representation.

7.6 Conclusion.

The findings of the previous chapters have been com-
bined in this chapter by superposition to obtain the three
classes of time domain models which are applicable to the
transmission line problems encountered in electrical
engineering. In addition, a numerical example was worked to
show certain aspects of the numerical solution which are

obtained by the use of these models.

 

+1.1

 

VIII. CONCLUSION

8.0 Conclusion

Three classes of time domain models have been de-
rived which are directly applicable to the practical problems
which are encountered in electrical engineering. Any model
which belongs to either class A or class B is defined by a
finite set of difference equations in both the x and t
dimension. On the other hand, a model which belongs to
class C is described by a finite set of ordinary differential
equations which are in normal form.

For the case where the transmission line linear
graph representation is formulated as chords, the general
form of the equation for any model which belongs to class A
or class B are given by Eqs. 7.1.3 or 7.1.6, and 7.1.4. The

general method for determining the matrices Ag, i=0,1,....j2.

which appear in these equations is one of the principle re—
sults of Chapter 3. The result of Chapter 4 is the deri—
vation of a general method for obtaining the launching num-
bers which appear in these equations. The remaining terms
which appear in these equations correspond to the interpo-
lation formulas used to find the numerical solution for
I(O,t) and I(L,t). Chapter 6 derives certain properties

which are useful in determining these interpolation formulas.

160

 

161

The general form of the equations which describe any
model belonging to class C are given by Eqs. 7.0.1 and 7.2.3.
The terms appearing in these equations are found by applying
the results of Chapter 2, 3, 4, and 6 for the case when the
transmission line linear graph representation is formulated
in the cotree.

For the other three cases which result when both of
the linear graph elements which represent the transmission
line are not formulated as chords, the results of Chapter 5
and section 7.4 must be applied to slightly modify the equa-
tions which represent any model even though the general form
of the equations remains essentially the same.

One of the main results of this thesis other than de-
riving these classes of time domain models is the approach
which was used in the derivations. This approach consisted
of the specific use of the superposition principle to sub—
divide the general transmission problem into 3 transmission
line problems which are easily treated individually. One of
these resulting problems was treated by a finite—infinite
line transformation which provided an effective means for
approximating all of the partial derivatives with respect to
x by more accurate central difference formulas. The remaining
problems are treated by a transformation which transforms the
boundary conditions for a given interval to initial condi-
tions. This transformation is defined by a set of launching

numbers.

 

162

8.1 Limigagions.

It must be pointed out that the telegrapher's equa—
tions or the wave equation describe a two conductor trans—
mission line only when the algebraic sum of the currents in
any cross—section is zero. This condition must be satisfied
before any of the models can be expected to be a valid repre-
sentation. This requirement generally places some restric—
tions on the system for which one or more transmission lines

are components.

8.2 Additional Problems.

This thesis derives an infinite number of time domain
models. One subject which warrants further study is the
choice of a best model or models. It would first be neces-
sary to formulate a criteria for making this choice.

One additional problem would be to extend the find—
ings of this thesis to the lossy transmission line. By
making the appropriate change of variable, it is clear that
the distortionless line immediately follows. For the lossy
lines which are not distortionless, some of the findings of
this thesis can be easily extended while others are not.

First of all, the derivations in Chapters 2 and 3
which treat the time domain models, belonging to class 1,
can be readily modified to include lossy lines by the intro-
duction of the appropriate dissapation terms which appear in
the telegraphers equations. On the other hand, modification

of the derivations associated with time domain models which

 

163_

i
belong to class 2 can be quite dpmplex. The reason for this
is that the extention of the basic numerical solution to in-
clude lossy lines results in a very complicated expression.
Since the derivations in Chapter 4 are based on cer-
tain forms of this basic numerical solution, there is no easy

way to modify the launching numbers to include lossy lines.

The transformation constants which are given by

bl’l = 1

3,1 = o 3:2,3,...,2k-1
bmelfl l
153’2 = 0 3:1,2,...,2k-2

are valid for this lossy case.

Finally, the results in Chapters 5, 6, and 7 are
valid whenever the best numerical solution does not enter in
any way into the derivations. It can be concluded in order
to extend a large portion of the thesis, it would be neces-
sary to first investigate the basic numerical solution which
is applicable or the lossy infinite line and to secondly ob—
tain the Fourier approximations, used in Chapter 4, which
are valid for the lossy line. Once these items are deter—
mined, the same approach as was used in the thesis can be

'applied.

 

REFERENCES

Richtmyer, R. D.,.Qifference Methods for Initial-value

Problems, Interscience Publishers, Inc., 1957.

Kunz, K. 8., Numerical Apalysis, McGraw—Hill Book Co.,
1957.

Fox, P., "The Solution of Hyperbolic Partial Differential
Equations by Difference Methods,” Mathematical Methods
for Digital Computers, ed. Anthony Ralston and Herbert

S. Wilf, John Wiley and Sons, 1960, pp. 180-188.

Knopp, K., Theory of Functions, Part 2, trans. Frederick
Bagemihl, Dover Book Publishers, Inc., 1947.

Churchill, R. V., Fourier Series and Boundary Value

Problems, McGraw-Hill Book Co., 1941.

Hamming, R. W., Numerical Methggs for Scientists and
Engineers, McGraw~Hill Book Co., 1962.
Goffman, C., Real Functions, Rinehart and Company, Inc.,

1960.

164

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