I I
1

I

l

N}
. x
I

l

I‘MUII
1 “M

I

I 1‘
I

'1
I I
:I I

I

MW
I

,‘_,_ 7’—

III"

 

TESTING AN ARTIFICIAL TRANSMISSION
LINE FOR STEADY AND TRANSIENT

BEHAVIOR FOR DIFFERENT TERMINATIONS

Thesis I‘M I‘M 9991‘“ of M. 5.

MICHIGAN STATE COLLEGE

Them-53m I’Mer Rykaia
1949

THESIS

 

This is to m-rtiIg that the

thosis Pntitlml

"Testing an Artificial Transmission Line

for Steady and Transient Behaviour
for Different Terminations'

[prosr-ntmI In]

 

 

Theodore P. Rykala

has hm)" at‘t‘elttt-(l ILHM'II'IIN IuHiIlnn-nt

”I th rcquirmncnts Iur

MOS. (IHJI‘N‘ “ILL-

 

 

 

IIHIP May 19, 1949

0163

TESTING AN ARTIFICIAL TRANSMISSION LINE FOR STEADY AND

TRANSIENT BEHAVIOR FOR DIFFfiRENT TLRMINATIONS

By
Theodore Peter Rykala

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

MASTER OF SCIENCE
Department of Electrical Engineering

1949

THESIS

 

Table of Contents

1. Introduction
2. Solution of Cascade Networks
.3. Matrix Solution of Ladder Networks
4. PrOpagation Function Characteristics
5. Reaponse Function
6. Error Introduced in Lump Lines
7. Transient Solution of a Transmission
Line
8. Transient Solution of an Artificial
Line
9. Response Function Measurements
10. Conclusion
BIBLIOGRAPHY

217920

page

15
15
20
24

28

52

55
46

INTRODUCTION

Transmission phenomena play an important part
in various fields of endeavor. Whether in power,
communications or other applications, the transfer of
energy from one point to another embraces a large
section of Electrical Engineering studies. It is the
purpose of this paper to study the transfer phenomenon
of a lumped parameter artificial line. Various types
of circuits exist for the transmission of signals
ranging from simple transmission lines to complicated
networks of electronic devices. The prime requisite is
to transmit a given signal in a manner whereby it can
be recognized at the receiving end as the same signal

which was placed on the input terminals.

All types of networks can be treated as a
four-point terminal box. The effect of the system
on a given signal can be studied using transient
methods as most signal transmission requires that a
network be in a transient state at all times. Response
of a network to the step function and an impulse
function provides a severe test for the system. These
functions can be represented mathematically quite
easily using Fourier methods and shown to contain the

sum of a number of sinusoidal frequencies. The transient

response can be completely specified in terms of steady

state sinusoidal behavior.

The ratio of the input voltage to the output
voltage is a complex response function of the form/%%?éfif
How this function behaves is the problem.to be met.
First, the ratio of voltages is to be determined; second,
the variation of the response function with frequency
is to be analyzed. Ladder networks are adaptable to
this method of treatment, whereas some other transmission
systems cannot be handled easily mathematically. In
these networks, the transient effects must be handled

eXperimentally.

SOLUTION OF CASCADE NETWORKS

The artificial line is a number of like four-
terminal networks or sections connected in cascade
arrangement. There are forty sections in.the line

to be analyzed. This arrangement is shown below.

a5: 5:

 

 

 

 

o<
§<
AaAAA
fiv‘vvv
N
F

 

 

 

 

 

 

 

 

 

Each section contains inductance (L) and resistance
(R) in series with capacitance (C) in parallel. This

arrangement is called an H section and is represented

 

 

 

 

b e 1 OW o

L E F? L

———W’———Wv~lr W
.. L
/ __c: 1212
I 7

L '9 1:? L

——-’Ym>--—-’VWW— wu~———/WYn—

It is necessary at this point to define certain

characteristics of the section. ‘Y is the admittance

of loop 1. Yiz' is the admittance of loop 2 and is
equal to the admittance of 100p l in the symmetrical
case. Ya. is the mutual admittance between 100ps.
Taking a few sections in the interior of the line and
extending the analytical results will give the over-all

performance of the line.

 

 

 

 

IL-t I‘K I“
9* —<+—- —«&—- ———<>——-
Vt. \ VK Vt * ‘
a -—e~1 ——o—- ———e———-

 

 

 

 

 

 

 

 

 

 

Writing the equations for the current at Junction k -
Ik = " “Vt-I —>/11VK
(I)
1k :- X’ VK + y]: Via-I

Rearranging the identities gives the following equation:
XIII/(«4 "" YMVL +XIVK + )6. ‘44: = 0

 

Vk~t *-

”“7" / )4:-
y.. I“ I 73%“

However, due to the symmetrical conditions of each

>41 = y”

section,

and therefore,

VK + 2%,“, v + Vkuz (1-)

-5-

Considering the voltage in section k, the following

equations can be written from.Kirchhoff's Laws:
VIC = ZuIt-t - lurk

C3)
Vic r. in]; — i”l:'*/

or-

zu‘l-I—I - full.- zu ‘Lk + Z» ‘LLH =

dividing by z -

J. _ .— _—_2 .Lk + 4" L’L-H - 0
LI it '
M

The symmetrical prOperties of the section permit the

following equation:

 

........ 2., -- ..
it.-. __ 2 II. + _ o (9)

For the moment let us consider some of the properties
of four-terminal networks. Input and output currents

are obtained by the relationships

I; = X' ET 4’ X2 5-2... (5')
1:; t )4.‘§ +' >1“ EN~

This is called the admittance method. The input and

output voltages can be written by a system, as follows:

E i + ZH,'L3_
f

(L)

I

I
I: = M]: + in. in.
’vu

This system has bmpedance coefficients. Using another
system of coefficients called general circuit coefficients,
the input voltage and current can be written in terms

of output current and voltage.
_ ‘_ I
5-4/2, 6% (7/
I Gil-firs:—

I

The solution of these equations by the method of deter-
minants will give the following relationships that exist

between the various systems:

MI! ‘ = i1!- ~—————.
«Cd ‘ X, lit fl

 

M, 5:, I
"‘ “IQ: :. >//9‘ : -" 7‘ - -—“' (g)

I a 6’
”7M. _ >/ _ it! ~ .42.
w 3" lit .6
Since the line sections are symmetrical -

67:49

It can_now be shown that the coefficients of the I

 

term.and the V term in equations 2 and S are equi-
valent: . 4
a? i
a 22- 2 __._..
it. ._ g m z 5’
\ on
/l7- __ in, '-

m‘.

IEI

or- 32V _ _9‘zam-_ao‘

x.’ in

 

Using this equality, equations 2 and 4 can be written in

the following manner:

- 9»
\4-‘ - QQVK T" Lu ‘ O ( k)
_ .. T“ A : 0
ik—t (Ra ik + it“ L4. a.)

The above equations are of the same nature as the
differential equations of the uniform.line, considering

only the solution to the first approximation.

at .
“fiF‘YF=0

9’1“ ._
2K» ._ \6 .I. : 0

 

The solutions for equations 2a and 4a can therefore
be assumed to be of the exponential form which is the

solution for the uniform line.

—£Y kY
Vt 3 A.6 'I- Ate

G)

In this solution‘»/ plays the role of a prOpegation
function per section in the same manner as the prOpe-

gation function of a line per unit length. Determination

-8-

of ‘X can be accomplished by substitution of this

solution into equations 2a and 4a.

-Uc~:)r (MY - (My din v“
4:5 1‘ 9. ‘a’laL+'4,e late _.

Combining terms - kr if V

-tr t Y - -
(Ate +416 r)é -2«\/;. + (14.6 +A16 e _ o

substitution -

[gt—2&+ CYAV ‘ 0 Q0)

-(b )Y .. Y ‘ -Uu-IIY _
HIE '4.5L(kl)—4’Lo.-Lh+8né +62. _.

(”6326. + e”) ‘1= 0 (H)

As the voltage and current cannot be zero at all times,

it follows that the coefficients must be zero in equations

10 and 11. Y“
6 -,2¢L 4-IE = C7
-Y
éYT-E: ‘= 9‘9»

This can be transformed into hyperbolic cosine-

y -‘b’
:2 I \,.: G +t€ q: Ck. I\IA)

9~

 

The H sections can be placed in a T arrangement by
adding the impedance of the lower series arm to the

upper arm.

-9-

1%. 35/.
ZL

 

 

Cosh \Y can then be evaluated in terms of impedances.

 

Z6- _ let
___I__ __,_ 55,,
@ ti-t

Hence,

 

J:§_
Boundary conditions permit the evaluation of the con-

stants AU A}, B. and B*_. substituting the assumed

solution for V; into equation 5,

Jr by 4" tr -(t+u)Y (Law
As + Ate . 35,,(3f “3,6 ) —Zn_ua,e t-B’é ) Q4)

4)»

The coefficients of the term 6 on each side of the

equation must equal each other if equation 14 is to hold.

/4 = ZWIB\-;ZPL[3\é;Y ‘ 51 (}N\-J&H-ér)

I

-10-

tr“
Likewise the coefficients of the term EL on each side

of equation 14 are equal.

Y‘ r
dwarfs/baa - age” )

Then - A. Y
3', .. 2,, 2,,6 = i‘.
and _ A r i (/5)
e - -
73: -_ z"- 2,, o

This is the characteristic impedance of the line, one
being interpeted as the impedance looking into the
sending end and the other looking into the receiving
end of the line. In terms of the parameters of the

T section -

—— *—

‘ 1h - (/ 6)
3° ALI/13,3: I

 

At the boundary of the network, the following conditions

are'present:

‘ Ti
k3 v°+ ° 3 (I7)
0 \k,‘ Emit

[I

At the input to the network, the voltage and current
are given by substituting k equal to zero into equations

9a and 9b.

-11-

V0: A\+ AL
To 1 B‘+ B?—

The voltage and current at the end of the line are
obtained by setting k equal to n which is 40 for the

particular structure under consideration.

-nv HY
V; 2 /\‘6- +—Abé

_‘ nmr evfl’
‘LK : 3‘ 6 + 8,,
Equation 17 can be written -
- I?
I:% 1 A\+ Kg ‘I— i‘a’cBfl-BL) C )

- r 'vm I—KF ‘KY
0 =A‘EK+A,G - Ame Hie )

Substituting for the constants B' and B1‘ -

/Z:+‘Ze 112-730 _
K 4. )A‘— (.1£D)AL‘; k3

 

Z ‘ £0 “MY qu+ *0 HY-
.. L / 6 + ———-. : .0
< a > I’ K 7: )Ave

0 0

Using determinental methods for the solution of the

above equations, the values of A, and A1_ are found
to be: Y
‘M
/4 __ (Zkflo) 6
I - ~——-——-——-
10 A 3

W

4% z (ii-Me E
it. a 3

 

-12-

where [5 is the value of the determinant of the
above system of equations. The voltage and current
equations for the kth section can be found in terms
of the reflection coefficients. Defining the reflec-
tion coefficient (Ya ) at the sending end -

r '1 -i

 

 

-. is ° (Jo)
S’ 4-},
and the reflection coefficient at the receiving end -
t1 -.-. TV?“
11' 1"}0

The voltage and current at any section is given by

the e uations -
q (K'IC\Y __ (w-IC‘Y)

VL?‘ I53“; 6 +— ilOY‘LE

my.)

(23+k.)(e”- ’37, e

(9')

 

 

.) -‘w-I
IL: Sagag’lém I”)
Lay-mac '- i“; 6"”)

where k is an integer between zero and n. The output

voltage and current is obtained by setting k equal to

n83] -

 

 

-15...

MATRIX SOLUTION OF LADDER NETWORKS [a 3]

Matrix algebra may be used to a great advantage
in the solution of symmetrical networks thus making the
problem easier. Artificial lines are very adopt to
this method as they consist of a group of symmetrical
sections in cascade. The matrix form of the entire
ladder network is -

W

Q

- vfl I a 435‘} vs

)I

 

 

 

 

_J .

‘Lo I [. CL {9' try“
Raising a matrix to the 11% power is laborious
if it had to be accomplished.by direct matrix multi-
plication when n is a large integer. This difficulty
can be overcome using theorems of matrix algebra. In

the case of symmetrical sections the network can be

represented -

A

Pg @I tea/hex {OMB

II

 

(1491 132*“ MY
.— h— o ‘J

 

 

 

 

 

-14-

and -

Q A9, WAIW Chas/LVN

a L so

The general voltage and current in the line is obtained

 

 

 

by finding the inverse of the square matrix on the right
of equation 22. In matrix form the output of a n section

network is -

 

I-VK TH” f T - Vo
LiN L A9 i0

Performing the inversion gives -

I~C34J~THJ '- ibAT“iTXW~’ V5

H

 

 

 

 

 

\ ‘ t k, To
4‘ ‘— SAM/Lick. M\ L

V” L

 

Fe m ” FM” :20qu

 

 

U

-15...

PROPAGATION FUNCTION CHARACTERISTICS

The prOpegation constant and characteristic
impedance play an important role in transmission phe-
nomena and their variations with frequency determine
the fidelity of signal or voltage transmission. Types
of terminations also have effects on the fidelity of
transmission, but these conditions can usually be changed
in such.a manner to eliminate their contribution to

distortion.

T’ is complex in nature having a real and an

imaginary part.
Y = a4 + Jfl

The real part 0\ is the attenuation function and gives
the amount of energy dissipated while passing through
a unit length of line. The imaginary part fl is the
phase function per unit length of line. In the case of
an artificial line the unit length is one section which
approximates a given length of actual line. This
particular artificial line approximates a pair of 104
mil; wires spaced twelve inches apart, having a length
of 6.52 miles. This prOpegation function is of the

general form -

 

Y=Vz/v

-15-

 

01‘-

aH-Jfl = Vt/fwaH/w-JWC) (7‘5)
Squaring both sides of the equation -
when J M76 = ($5 -/.c.w"')+./ ~(RC MG)

Equating the reals and imaginaries -
"‘fl’ =- A’s—Law“

Qdfl ~_ wot/"C +1.6)

The expression for attenuation a4 is-

/.z.
.4— [i- (ERG- Lee“) + Wren/3w ”)(GH: cud]

The expression for phase shift fl is- (C! 4’)

fl L41 (Lee are + M“ W“ ,C “’91”!

 

 

These expressions for attenuation and phase
shift are very cumbersome but it is possible to deter-
mine the effect of frequency variation. However, they
will not give the entire story over a wide range of
frequencies as the parameters R, G, L, and C do not

remain constant.

Experimental data show. that resistance variation
with frequency in Open wires runs from 250% to 400%
over a frequency range of 500 to 50,0000ps. The
exact percentage depends upon the physical character-
istics of the wire. Inductance variation decreases
4% in a frequency range of 0 to 50,000 eps. Capacitance

which theoretically should not vary a measurable amount

-17-

increases about 4% in a frequency range of 0 to 50,000 ops.
Leakage parameter (G) varies the greatest amount; in

a frequency range of 1000 to 50,000 cps, this variation

is 5000%, almost a linear function of frequency. With

good insulation, however the leakage is still negligible.[}€3

These same characteristics exist in the artificial
line as is shown in the table below. The parameter
values are computed from impedance measurements obtained

by bridge methods.

 

 

frequency, C796 cps 1290 ops 2000
R 56.8 J... 56.9 .9— 56.7m—
L 22.7 wk 22.6wk 23.0 Mk
G 1.5,...» remake 1.4.14.4:
c .042»; .042”; .0435“;

 

It would seem that prediction of line behavior
is a h0peless prOposition with all of these variables
infielved. However, an engineering solution can be
obtained by certain approximations, and the effect
of various parameters can be determined by deviations
from an ideal case. Most engineering formulation falls
in this category. The ideal conditions for a trans-

mission system are constant attenuation for all

-18-

frequencies and linear variation With frequency for

the phase function.
9‘ = I T6 (.4 7)

fl = w VIE:—

Equation 26 can be put into the following form:
e< z. //F—6— [4/4)
/9 i: “J/QTEF ;/&(*9

where /./'V) and [A (‘3‘) are deviations from the ideal case.

/,(4) 1- fi(/ “)1"?- //+ 1(M"+»")¥V: 1‘7) J;

(2. .0)

 

 

i_ y
/, (v) = 7%‘(/- 531+ //+i(»hm¢"+ if“) L

The terms in the above equations are defined as-

46/ PC: + /L6
2— L6 69C.

 

 

 

 

 

407 :
_,_ 4°C .61: ) (9.7)
”w = 2.. 12‘— ‘“ .76.
“J
3‘ " 6’6

LC.
The leakage parameter cannot be neglected at this

point, even though it is small. It is quite evident
that a relationship exists between the two error

functions l/Zfiv) and i(;(“) . This is shown by letting
- .1 ...l
j? " z ( 4‘)
11 =' IEV‘V—I *‘,¥

then-

-19-
Equation 29 can now be written- I
I __——-—— ‘9
[35%) =: TC$'(VTM‘¥-/""&7)‘L

, (30)
A“) = 7510/‘7‘47‘i‘fl L

 

or -
[zigfly . /[>(«) -=. uvu. (St)
The value of m for the artificial line is-

, 51.7 (.0 542.110“) ’{gz.71(6’)(l.i‘m8"
_.- y:— +- a
“WI 3 )— (22.7x163)("°“\°b) A

5.7 (.0414 to")

 

 

 

 

M 2. 4! 2’
Conditions which give a distortionless line
are -i? = ‘%% . This makes the value of m.be one

and the value of n be zero. Also -

/’(_q) = [Cf-V) :1 /

The equations for distortionless propagation function
are then obtained by substituting these conditions

into equation 28. It is apparent that distortion can
be expected on the artificial line.

-20-

RESPONSE FUNCTION

The response function is a variable of frequency
which is the outgrowth of the ratio of output voltage
to the input voltage. Let us now consider one section
of the artificial line which is terminated in the
characteristic impedance. The response function of one

section is -
_Vhfl 2 a... 5W”... We“ (3.)
tht)

For the distortionless case -

/UflwQ-= €¢-/fi%§

and

The total effect of the line can then be determined
by raising the amplitude function to the n th.power

and adding the phase characteristics of the n sections.

V,,(t) 1 6-“ 5’6 é-wafl—E, (33)
Va“)

In any system.having phase shift, the output signal
may be small for a time, after which it increases
rapidly giving rise to a time delay. This delay depends

on the slope of the phase function versus frequency

-21-

curve. {A z 2%)

90d

For the distortionless line, the time delay is constant.

't 7: v» l-C.
" . (34)
£4 =- 40 /(2a7x16’)(.o4u10"‘ ‘= / 23 no ,c—u...

 

The Fourier Transforms are strong tools in chang-
ing functions from frequency to time variables and
reversing the procedure. This enables the manipulation

of such equations as -

V,(t) - Mu) Wt) (35‘)

which are often used in circuits. These transforms are-

no
“it
V(t) 2 JMI-tr/ éJ v(w)4w
-o(>

.e w (3")
vac). f 6"“ 7Eva) ed-

“«:13

Applying these transforms to a pulse function
will enable the prediction of a particular network.
Pulse response of networks is proving to be very sat-
isfactory in determining the transient behavior of the
network. Consider the pulse of width 17 applied at
time equal to zero. Since the Fourier Transforms take
in negative values to time, the function of positive
time is reflected to the left of the origin. This
does not have any physical significance but is necessary

to be able to use this mathematical tool.

-22-

l Wt)

 

I
I!
I
I
I
I

 

 

 

 

a ———-- t
~t, I i

Applying the transform to the pulse of unit magnitude-

why-.1: detat- naught. <37)

 

This transform of the input voltage multiplied by the
response function will give the transform of the output

voltage.

- - I '
\AIw): SKIRTSG ““4 9 thi (3‘?)

LL)

Although in the distortionless line, all frequencies
are transmitted, filter theory shows that a cut off

frequency exists for the artificial line.

9 2 ¢
we: Ir: 2 i. a = ‘Mr’iX/o 44-04 (a?)
Q I’(22.7u5’ )(.0 +1! [5" 5‘9

 

Beyond this frequency, the amplitude response is zero.

The transformation integral then becomes:

~ u)
.. Gum C- 3414.000", J‘Uzt" éd)
%&)- e
21' _u, tag
a

5240

 

 

 

é-~flfia§ u%.l_" Clay
vh(19 -; ‘7’ .//r “J(ghuJTJg)/Q«Jutr+5fuaduttdg)‘gudg36mn
'“Q.

-23...

The second term of the integrand is an odd function
and the first term is an even function. When the
integration is performed, the odd functions drOp out
so they can be immediately eliminated. The first term

can be rewritten -

3: If“ Rhea, I - a... th-ia «AI

 

 

th - .19 DJ
en {My grasps) 5:57 highest)
KG): 7— o T Lew 7,— 0 — 0,—— A»

~KQ

v1“) :§—-— [2 We (12- 'Lcwfi) - S. we“ “; "(,8

Cut off frequency plays an important role in

(i I)

fidelity of transmission. As the width of the pulse

is decreased, the fidelity of transmission increases.

Reflections caused by terminations other than
the characteristic impedance will give rise to greater
echos. The response function is more complex in nature
and great difficulty is encountered in solving the
equation. In terms of the reflection coefficients,

the response function at the k section is -

 

V g E(u~k)x+ night-H8
K - __°_ 4
73 ‘ 3360 EM— 3’2 ‘5‘“, (+1)

3. HYW) 6J6

-24-

ERROR INTRODUCED IN LUMP LINES

 

The question arises as to the extent of the
error introduced by lumping line parameters and over
what range of frequencies the artificial line will
approximate an actual line. Let us first consider
the prOpagation function. In the actual line -
Y£ = r I/(ff’+JwL)(G1—ch) (43)

where /L is the length of the line. The prOpagation

 

function of the artificial line is given by the product
of the number of sections in cascade and the propagation

per section.

F
~I

t ‘ I
III = 52w M —— CM)
4:,
The series and shunt impedances of the symmetrical

H section take the form-

SZ = é (IPA/wk)

 

. 44‘
*1“ — 4f: (:6;IKIUJCI <’ )
z - ’k/
t
The value of 'A can be determined by substitution
4V9;
.._..__. _ 4 ‘61
J I 1 -_-_ ——_—
....-.__..’- .1 I J/é(fflwb)1(6+JwC-) 1k; (6(6)

4‘1.

-25-

hence- ‘ . "xi
XTKJ" 2%)La.49“4~'2f;; <9L7)
Using the series expansion for
. " g 4 _
m = mitts-é) +£(I:)--- - 3
3’2 2 n CHI-X1 (if)

The error in the propagation function is-

I ~. --’— (é—é : 7:- éffif’uwbxsflm) 6"")

r e

The characteristic impedance of the actual line is-

. :m w 7* -
17" ‘ €+JwE <30)

 

 

For a symmetrical section the characteristic impedance
is-

 

220’- )“, V7+ 7:1— é")

Substituting equation 46 for

 

', f 1! 7’; ‘-

23, *3. V/f (3:) (4”)
Using the binomial expansion for the radical-
'_ _I_ 71 I- . ,6 4~_'_ __

Jo ‘ ADI—213:.) "EVE; J (55)

z;‘- ¥OCI+ 3:]

The error in the characteristic impedance is-

I. =- flit-E 1 RETA’HWYWCI (54)

-25-

The error in the characteristic impedance is
of Opposite sign and is three times the error in the
proPagation function. For a given frequency the error
is prOportional to the length squared and inversely
prOportional to the number of sections squared. As
the number of sections is increased for a given length
the error approaches zero. The limiting case where

n is infinite is the smooth line.

Considering a given length and number of sections,
the error increases with frequency. At high frequencies,
the resistance and conductance are negligible, hence
the error increases as the square of the frequency.

The Opposite sign of the errors indicatesthat some
compensation exists; however, the highest essential
frequency to be passed will determine the number of

sections to be used, allowing a certain limit of error.

The error introduced at the highest frequency
passed can be easily calculated. At 10,000 cps the
resistance and leakage terms are small enough to be

neglected. The error in the characteristic impedance

is -

_ - 3mg
iii-(L53) (37:16:97 ((12 4 I/O 310421,?)
It]

(.29

[I

-27-

The characteristic impedance is approximately 690 ohms
for a wide range of frequencies in the audio range.

Hence the percentage error for the artificial line is-

fl /0'D -=. 3'7/70’

490

-28-
TRANSIENT SOLUTION OF A TRANSMISSION LINE

Methods of transient solution vary depending on
the type of input voltage function. In general the
Laplacian and Fourier transforms are superior to other
methods of attack. However in the case of the unit
step function, the results of Laplacian transforms
lead to the Heaviside expansion theorem.in the eval-
uation of the inverse of the Laplace. Since it is
desired to compare the transient solution of the
artificial to the actual transmission line, the Opera-

tional methods of Heaviside will suffice.

The solution of the wave equation for the trans-

mission line is - ['03

W 4%
V T. 14,6 +4L6

-L_(: KIA. 51%;!
VAL-‘- £0 Ale 1.

where the constants are evaluated by boundary conditions
of the line terminations. These equations can also
be written in terms of the parameters of the line.

Using the following notations-
I

V‘rzz‘
azi

2L

 

-29-

The general transmission flquations then become-

~ - use
'vwat’e

~T¥
1%: AV” —.I//:::—: (415} - A"- 6 (56“)

where-

 

t= I: WW)“

Considering the boundary of the line terminated in

a short circuit:

V¢iva 0'1- 1L.-

V-o a} 1L~—/€

'

Hence the equations of voltage and current at point

1% from the transmitter when the unit step voltage

is applied becomeMY(2_ 4L)

V1: V3, MEI.

V E at MYI’Q'S") j
1“: 7:}: [9494; MY!-

Solving these equations with Heavisides expansion formula

 

which is - Pf.
74: IL (0) 21L) e
The determinental solution of -

.39fi)‘~0 3 ‘vaJL‘YIZ

13" Y3 :J 47’

\/

-50-

 

: J‘A'fi‘.‘ “L. ‘— 1' i
X T w-I/(lo+/0)—0‘

solving for p -

 

(I’Wo) - r ‘- Agit” >4,
fl -/ If-
V'H’flt

Next taking the derivative of .29.)

9a )_ 90“
7% ._ MYQ '57;

= 0,24%
(F7) ~7-

For \):(9 , the first term of the expansion is -

ll

 

- I? - 14)
(o) M a b (2 i) = W
1(0) : MAW/e MeQZ

Substituting these equations in the expansion formula

the develOped solution for the voltage at point x is-

 

 

 

Maw—«I Z J 01%» urO- THE/”fa “2"“!
LI iifl. 6011/...) 5;: Crate“) Ii;—

which can be simplified to-

M J(,(?-#) 9V VTTF 732‘?

 

- ¥ 4mm‘( It; {-4. [den/‘1)
I'VE Miffi-‘j—Té/Z‘ {fl fl fl I!

f9 9%")

-31-

This is the complete snlution for the voltage.
The first term gives the steady state condition and
the summation term gives the transient state. The
summation extends for all integral values of A. from

zero to infinity.

The current eXpression can be obtained in the

same manner and is-
I ._ \/ wz(cv«9_ ”67*: apex—6‘
3mm T

zfévopkjflpt'+ 6%atéhyfiit__ (3;:E:£§:§;-
fl: (WW) fl“

Applying the condition of negligible conductance

<63)

 

or leakage, the above equations can be reduced to-

\} “BR! 0. _E)___g_____ Vé-“fzgwegia .ee/e‘efi e743:
fie (“rt")

 

 

-52-

TRANSIENT SOLUTION OF AN ARTIFICIAL LINE

The general equations applicable to any artificial'

line are - [:3] V: 9 (“ways
L z ‘_

F’ _iw,p&J«Vfi4LJ~w&Y <:6~f)

 

l
W‘ 1 + _.L
. ‘Y I 351p

However, the input and output are the most considered
formulas. For the artificial line of negligible leak-

age, the series and shunt impedances take the form-

21 2 #10+ R‘
‘L
Z1,= CF)
and- (06)

MY ~— I+-':.C'|°<’“F+R5

The determinental solution is-

m») -=. AM YMW" 0

hence
54~Aak‘* 5’
or- ,..
‘ )‘~ J’tn- »¢'=’/1I3 --
- _7;:
then-

solving for {3

-35-

 

lo - -45. i Vane ”Hod (pane—i)
rec

 

 

1 .. 0.. t J yayw’géfl- -a'v
x.

Taking the derivative of 217°)

 

£22321): 14.21,ka MVW_%§;
However,
saxgée Megs»; chttm'
then-
DYJ ): m1“ (LC‘: +£RQ) WWY
6°

g 7ik ( +4. Cra,Afl'
0° P 3

As P approaches zero, the value of MY approaches

one. The series expansion for

CU4ISY': l‘+ %: “' ' " "" "

substitution for y‘ gives -
(25,1...‘6 '1 I 4- {LOUCfV‘I—ec/a):

and- ‘k
. - X

a. 2‘ “L, Y MKY ‘- ‘2'?“

Hana 11- C63

Substituting into the eXpansion formula
t
er

 

 

 

 

 

 

 

_ V

J—‘KIVZnLQMJM‘U _i
at?

_r.‘_\/_.,_ave‘“'t My4wmgi“

"' ‘ “R 191., Gafifrh/filrxueg’; '4‘—

-54-

The frequency of the damped transient oscillation is-

»-v/‘JT' a.
tap I V4VMWZTC' “

 

If woois very large
diam—411: «2 .4117...-

s»"‘ew
then the current takes the form of-

 

 

 

 

- 1: , -.. t.

V 6 a 44w m «- 4—D
Iz—M' rig—f (- ‘e y 1"” _—
7"? 1L Ctr/.77" /v'k/¢Wb .. 4.1“

The summation on /L is from.zero to infinity. This
equation is of the same form as that of the uniform
line. Hence in all transient solutions, the artifi-
cial line approximates the uniform line if the number

of sections is large.

SM

6:»:

-55...

BESPONSE FUNCTION MEASUREMENTS

The response function, a complex quantity
relating the input voltage to the output voltage, is
readily obtained by measuring the amplitude and phase
shift. The amplitude response function of the artificial
line was obtained by using an audio oscillator as a
voltage source and making output voltage measurements

with a vacuum tube voltmeter.

A DuMont type 274 cathode ray oscilloscOpe

was used to determine the phase shift.

 

 

 

 

 

 

r
R *L-_________---

O \ Lin]: 1,,

R r _____________

Chadd {on (Phase Skflf Meant” mails

 

 

If two voltages of the same frequency are placed
on the vertical and horizontal plates of the cathode
ray tube, the resultant pattern on the screen will be
a straight line if the two waves are in phase or 180
degrees out of phase. When a phase shift of 90 or
270 exists an ellipse will appear on the screen having
its major axis in a vertical position. The angle

of the major axis determines the phase shift. [:1]

Line Termination -

Frequency

500
1000
2000
5000
4000

5000

cps
cps
cps

cps

Voltage Measurements

V0
10 v
10
10
10
10
10

-35-

690 ohms resistance

V
1.5 v
1.9
2.1
1.4
1.7

1.2

Vg/V,
.15
.19
.21
.14
.17
.12

-57...

Voltage Measurements

Line Termination - 1400 ohms resistance

Fre quency V0 V1 Vn/V‘XI
500 cps 10 v 1.6 v .16
1000 10 2.0 .20
2000 10 5.0 .30
2500 10 2.2 .22
5000 10 1.4 .14
3500 10 1.8 .18
4000 10 5.2 .52
4500 10 2.4 .24

5000 10 1.5 .15

-38-

Phase Measurement 3

Lint Termination - 690 ohms resistance

Fre quency Phase Shi ft
1000 Cp 3 . 25 1'
2000 .50 1T
5000 . .40 ‘fl'
4000 .75 17'

5000 1.254f

-59-

Phase Measurements

Line Termination 1400 ohms resistance

Frequency Phase Shift
1000 cps .40'fl‘
2000 .8 1r
5000 .4 1“
4000 .8 1r

5000 1.2 1r

i

 

 

 

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-44-

The amplitude response function with the line
terminated in the characteristic impedance exhibits
distortion in that it oscillates about a mean line
at approximately 0.19. This is expected in that the
ratiog does not equal the ratio % which is a
necessity if transmission of a signal is to be perfectly
distortionless. The fact that these curves indicate
the existence of echos is shown by comparing the curves
of the two terminations. With.the termination of twice
the characteristic impedance, the amplitude fUnction
deviates farther from-a straight line as the termination
causes greater reflection which will give rise to more
echos. It would be possible to eliminate this distortion

if a means for varying the inductance in each section

of the line existed.

The phase function did not show as great an
oscillation difference in the two terminations as the
terminations contained no reactive components in their
impedances. However, phase distortion is also present

in the line.

-45-

The output of an impressed pulse function can
now be computed with-the use of equation 41. The
mean amplitude for the frequency range of 1000 to
5000 cps is taken from-the curves to be about 0.19.
The general tendency for the curve is to decrease
slightly but a flat reSpohse assumption will not
introduce toogreat an error. Tables for the 8101-)

function make plotting the response possible.

 

 

 

 

 

 

 

 

 

 

 

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I. II.»

 

 

‘I
o.

-45-

CONCLUSION

Although experimental results of transient
behavior have not been obtained in photographic
form, the sinusoidal behavior of networks present
the best method of determining pulse and step function
responses. In order that fidelity of transmission
can be observed, more equipment is needed. A multi-
trace oscilloscOpe with provisions for micro-second
timing is a necessity for thorough studies of transient
behavior. Time delays and variations of pulse widths
on fidelity can only be observed when both input
Iand output can-be viewed simulttfineously. Equipment
of this nature would be a great asset but it would

also mean a large investment.

Various photographs have been obtained of trans-
mission phenomena with respect to given types of re-
sponse function curves and the results will apply to
the response curves for the M. S. C. Artificial Line.

This artificial line is well suited for the demonstra-

tion of transmission line theory.

l.
2.

5.

6.

9.

10.

11.

12.

BIBLIOGRAPHY

A. Books
Bartlett, Artificial Transmission Line. Wiley.

Bewley, L. V., Traveling Waves gn_Transmission
Systems. New York: Wiley, 1955.

 

 

Carson, J. R., Electric Circuit Theory and the
Operational Calculus. New York: McGraw-Hill,
1926.

 

Guillemin, Ernst A., Communication Networks.
New York: Wiley, 1947, pp. 55-185, 461-507,
Vol. II.

Kennelly, A. E., Electric Lines and Nets. New
York: McGraw-Hill, 1928. 426 pp.

King, Mimmo, and Wing, Transmission Lines, Antennas
and Wave Guides. New York: mcCranHill.

 

 

Schulz, E. H ., L. T. Anderson, Ex eriments in
Electronics and Communication Engineering.
New York: Harper, 1945. pp. 1 -1 3.

Terman, Frederick Emmons, Radio Engineers' Handbook.

New York: Meoraw-H111,1945. pp. 135-273,
959-950.

 

 

 

Ware, Lawrence A., and Henry R. Reed, Communication
Circuits. New Y ork: Wiley, 1944. pp.‘72-89,
90-104, 265-276.

B. Periodical Articles

Carlson, "Transient Oscillations," Transactions 32
AIEE, pp. 567, 1919.

 

Carson, J. R., "Rigorous and Approximate Theories
of Electrical Transmission Along Wires," Bell
System Technical Jou£., pp. 1-52, 1925.

 

Brune, 0., "Reflections of Transmission Line Surges
at a Terminal Impedance," General Electric
Review, 32:258-265, 1929.

 

15.

14.

15.

16.

17.

18.

19.

20.

21.

22.

25.

24.
25.

26.

Green, E. I., "Transmission Characteristics of
Open-Wire Telephone Lines," Bell System
Tgphnical Jour., 1950.

 

 

Ku, Yo Ho, To C. T330, and Y. 01111, "Tran31ent
Analysis of Artificial Lines," Journal
gprlec. Eng., China, 1:507-515, 1950.

 

Rohats, "Impulse Testing Technique," General
.Electric Review, 55:558, 1952.

 

Barnes, J. L., and C. T. Prendergast, " 0n the
Time Admittances of Transmission Networks,"
M.I.T, gourn a1 2: Math. and Physics, 11:
27:72, 1952.

 

 

Alford, A., "A Method of Calculating Transmission
Properties of Electrical Networks," 1,3,E,
Proceedings, 21:1210-1220, 1955.

 

Weber, Ernst, " Sinusoidal Traveling Waves,"
Transactions of A.l,§,§,, p 245, 1956.

 

Woodruff, L.F., "Transmission Line Transients

_——_—

 

p 591, 1958.

Peterson, H. A., "An Electric Circuit Transient
Analyzer," General Electric Review, 42:594,
1959.

Bellaschi, P. L., and A. J. Palermo, "Analysis
of Transient Voltages in Networks," Transactions

Gebelein, J. P., "Circuit Constants for Production
of Impulse Test Waves," General Eleftric Review,
45:572, 1940.

Pipes, Louis A., "The Matrix Theory of Four-Terminal
Networks," Philos. Mag., 50:570-595, 1940.

 

Hoodley, "Transient Analysis," Communications, 1945.

Rohats, N., "Instrument for Analyzing Transients,"
General Electric Review, 45:121, 1945.

King, "Transmission Line Theory and Its Application,"
Jour. 2: Applied Physics, 14:577-600, 1945.

27.

28.

29.

50.

51.

Margoules, S., and P. Fourmarier, "Localization
of Faults on Overhead Lines by Means of Impulse
Waves," Conference Internationle des Grands
Reseaux Electriques, 1948.

 

 

Cunningham, "Construction and Tests of An Artificial

 

50:245.

Weber, E., "Transients in Finite Artificial Line,"

 

Bewley, L. V., "Traveling Waves Initiated by
Switching," Transaction of A .I.§._E_., 58:18.

 

DiToro, M. J., "Phase and Amplitude Distortion
in Linear Networks," Proceedingd g: I.§.§.,
56:24, 1948.

 

   

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