Table Of ContentA m u m OF THE INPUT CURRENT TO TIUiNS-MISSION LINES
UNDER AM APPLIED E *M «F ♦ * EU(t) SIN o>t
by
Mohamed Amin Salem
A d is s e rta tio n subm itted In p a r tia l fu lfillm e n t of th e
requirem ents fo r the degree of Dostor of P hilosophy$
in the Department of E le c tr ic a l Engineering
in the Graduate College of the S tate
U niversity of Iowa
June 1951
ProQuest Number: 10583837
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ACKNOWLEDGMENTS
The author wishes to express his g ratitu d e and
ap p rec iatio n to Dr* L* A* Ware* P rofessor of E le c tric a l
Engineering* fo r suggesting the subject and fo r making many
v alu ab le suggestions and c ritic ism s as the work progressed
and h is ap p reciatio n to P rofessor E» M« Lonsdale fo r his
advice and encouragement a t sev eral points*
11
TABLE OF CONTENTS
Page
In tro d u ctio n . a * * , * . * * * * . * * * . * * ® * ® # X
C hapter I f G eneral E xplanation of th e Methods Used • • 3
C hapter I I Smooth L ossless Lin© * a * « ® » » » « » » 7
9
C hapter I I I , An R~C Smooth Lin© 13
C hapter I¥ The Lumped E-C Line * * » « « » * * » • * 17
9
C hapter General D iscussion and Conclusion « < » * » « 21
Appendix I The S olution fo r the L~C Smooth Lin© • • » 25
9
Appendix II* The S olution fo r th e E~C Smooth Line * • 31
Appendix X II Th© Solution fo r tbs R»C Lumped Lin© • • 37
9
F ootnotes • • * « * • • * « • * • • • • • • • • • » * 45
B ibliography a . * * * * * * * ® * * . * . * * ® ® 46
i l l
TABLE OF FIGOTEB
Figure Page
1. L-c Smooth Line H
2. B-C Smooth Line . . . . . . . . . . . . . . . . . 16
3. IMS lamped Mne 20
4. the X Section . 36
5* the 1 Sections . . . . . . . . . . . . . . . . . 38
l v
1
INTRODUCTION
This work deals w ith the determ ination of tfo©
in p u t tra n sie n t and steady cu rren t responses due to the
a p p lic a tio n of a pur© sine wav© voltage on e le c tric tra n s
m ission lin e s • The so lu tio n 1© based on the use of Laplace
tra n sfo rm a tio n s• Three cases are considered*
The f i r s t case deals w ith a smooth lo ss le ss lin e
of d istrib u te d constants of inductive and cap acitiv e re a c t
ances* The in te rn a l impedance of th e supply voltage is con
sid e re d to be equal to th e c h a ra c te ris tic impedance of th©
lin e . The input cu rren t is solved by th© inv ersio n theorem
fo r the Laplace transform ation*
The second case is a smooth transm ission lin e w ith
d is trib u te d c o n sta n ts re sista n c e s and cap acities* The
9
so lu tio n of th e Input cu rren t is developed by the in v ersio n
theorem fo r th© Laplace transform ation*
Th© th ird case deals w ith a lin e composed of
lumped re sista n c e s and ca p ac itie s (R-C c irc u it)* The
cu rren t response term s are determ ined by th© Laplace tra n s -
f ormat io n .
Th© so lu tio n s of the th ree above cases are d e te r
mined com pletely and the steps are also explained in th©
appendices® Moreover a l l theorems or fundam ental re la tio n s
are e ith e r explained or referre d to in the references*
2
HISTORICAL BACKGROUND
Very few a r tic le s have been published in the fie ld
of th© p resen t in v e s tig a tio n « J* R» Carson and 0* J «
$ob@l% ' d iscuss , In th e ir a r tic le , th e th e o re tic a l behavior
of th e wav© f i l t e r s In th© tra n sie n t s ta te on th© a p p lica tio n
of pure sin e wave ©*m*f* Th© method used in th e so lu tio n
employed th© H eaviside o p eratio n al calcu lus • They did not
co n sid er re sista n c e s in th e ir f i l t e r elem ents and used only
pure Inductive and cap acitiv e re a c ta n c e s• However, in th is
th e s is , r e s is tiv e elem ents w ill be treated® G enerally, most
o f th o se who w rote about netw orks, wav© f i l t e r s , smooth
lin e s and a r t i f i c i a l lin e s do not tr y , s p e c ific a lly , th©
p rese n t problem and re fe rre d to i t as d if f ic u lt to solve
even w ith approxim ations* Th© so lu tio n s of th© cases given
here ar© system atic and c le a r and give accurate responses
w ith no approxim ations*
3
C hapter I
GEHERiiL EXPLANATION OF THE METHODS USED
la solving the d if f e r e n tia l equations of e le c tr ic a l
c ir c u its * I t is p o ssib le to express the q u a n titie s involved
la term s of a secondary v ariab le* In term s of th is secondary
v a ria b le th e problem can be solved alg eb raically * Then by
9
transform ing back to the o rig in a l Independent v a ria b le ^ th e
so lu tio n to th e o rig in a l d if fe re n tia l equation is obtained*
The transform ation which makes these operations p o ssib le is
c a lle d th e Laplace transformation*
The Laplace tran sform ations transform f ( t ) a
9
fu n ctio n of j: in to F ($}f a fu n ctio n of some new v aria b le
according to th e equation
co
/ t i t )©-S tdt a F(s>* (1)
0
For example I f
9
f ( t ) * e -a t (a)
th en
oo
rstdt - /e“(a+s^dt * = _A_ (3)
-L -Ta+s) J a+s
0 0
T herefore F ( s ) =? «JL* Thus the Laplace tran sfo rm ation
a*s
allow s @”a^to be expressed as -JL-* The q u a n titie s e“a^ and
aHrS
—i - are th e re fo re called Laplace transform s* S p e c ific a lly
a Hf*s
4
is c a lle d the Laplace transform of a'ma*' $ w hile e~a^ is
04-S
c a lle d th© inverse transform of «*i-*
a*s
Equation <1) is ca lle d th e d ire c t Laplace tra n s
form ation* The p h y sical sig n ifican ce of £ in equation (1)
is determ ined by th© problem a t hand, which w ill he tim e in
th© cases under consideration* On the other hand, the
p h y sic al significanc© of js is ju s t the sig n ifican ce which we
fin d fo r i t because of i t s re la tio n to t, in equation ( )*
1
As we s h a ll use i t , £ is re s tric te d to re a l values
9
but j? w ill be alloi'ved to talc© on complex values® However,
th© r e a l p art of s, w ill be required to be larg e enough to
make the in te g ra l in equation (1) abso lu tely convergent® In
accordance w ith the fo rego ing , i t w ill be convenient
fre q u en tly to re fe r to t as the re a l v ariab le and s, as the
complex v aria b le in th® Laplace transform ation*
A fundam ental form ula which is used in the
so lu tio n of one of th© cases is expressed by th© follow ing
theorem s» If
0
and f (t ) - 0 when t ^ 0
<PO
th e n
provided th a t b is a p o sitiv e constant*
Th© d ire c t Laplace transform ation and the preceding
theorem are used in solving th© case of a smooth lin e of
in d u ctiv e and cap ac itiv e reactances*
The inv ersio n theorem fo r the Laplace tra n s -
form ation is an in te g ra l form ula toy which f ( t ) may to©
obtained from F(s)® Th© general statem ent of the inversion
theorem is as follow s! If F(s) is defined toy th© Laplace
t ran sf orraat ion
(1)
0
th en (4)
y~3
Equation (4) can be expressed also as
f ( t ) =s F (s)e s*^ds (5)
2tr3
Equations (1) and (4) and th e ir equivalent equations (1) and
(5) nr© c a lle d inversion theorems* Therefor© f ( t ) is the
9
sum of th© residii.es of the fun ction F (s)e s^ w ith resp ect to
a l l i t s sin g u lar points*
Equation (5) i t s e l f is an e x p lic it form ula fo r
c a lc u la tin g the inverse of the Laplac© transform ation^ and
i t is th e re fo re ca lle d the inverse transform ation* I t can
be used fo r fin d in g f ( t ) from F(s) when the given form of
F (s) cannot toe tre a te d w ith the aid. of av ailab le ta b le s of
