A 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 All rights reserved INFORMATION TO ALL USERS The quality of this reproduction is dependent upon the quality of the copy submitted. In the unlikely event that the author did not send a complete manuscript and there are missing pages, these will be noted. Also, if material had to be removed, a note will indicate the deletion. uest ProQuest 10583837 Published by ProQuest LLC (2017). Copyright of the Dissertation is held by the Author. All rights reserved. This work is protected against unauthorized copying under Title 17, United States Code Microform Edition © ProQuest LLC. 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Box 1346 Ann Arbor, Ml 48106 - 1346 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