The N-wire exponential transmission line
Transcription
The N-wire exponential transmission line
Scholars' Mine Masters Theses Student Research & Creative Works 1968 The N-wire exponential transmission line Vernon Ray Stanley Follow this and additional works at: http://scholarsmine.mst.edu/masters_theses Part of the Electrical and Computer Engineering Commons Department: Recommended Citation Stanley, Vernon Ray, "The N-wire exponential transmission line" (1968). Masters Theses. Paper 5193. This Thesis - Open Access is brought to you for free and open access by Scholars' Mine. It has been accepted for inclusion in Masters Theses by an authorized administrator of Scholars' Mine. This work is protected by U. S. Copyright Law. Unauthorized use including reproduction for redistribution requires the permission of the copyright holder. For more information, please contact [email protected]. THE N -:'ITRJ~ '~:XPONEN'I'L"lL BY 'l6·!.NON H.. STANLliT.I \C1'f~ A THl~'3IS subr:J5.ttcd to the :fc.cul"i:,y o:f in pr:r·tl::>.l :fu.1.fillqcmt of the Degree of 1968 t'( . .L . ~----·~ . ~ . -·-·-----c~c}~~sor) requircT~J.ent::; :for the ii ABSTHACT This paper deals 1-r.i..th the n-"rire exponentially tapered transmission line. A ch..:'lracteristic im)ecbnce is defined, and the results of terminating in this im"~edance are inver;tigated. A very long line and the condition of no reflect:!.0ns are considered, and the conclusions are com:9ared. Finally, three specific 3-•·Tire line problems are 1rorked 'tvith a ch~racteristic impedance termin'1tion and a unit s·tep input~ \ iii ACKNOi·11EDGEHZNT The author 1·Tishes to express his sincere tlPllrcciation to his major . adVisor, Dr. EdHard C. Bertnolli, for his valuable guidance during the i·Jri ting of this thesis. iv TABLE OF CONTENTS Page ABSTR.IiCT · . ii AC:KNO'h'LEOOENEI:--TT iii LIST OF FIGURES v PROLOGUE TO THESIS 1 THE N-i·HRE EXFONENTIAL 3 TR.li.NSHISSION LINE INTRODUCTION 3 II. DEFINING CHARACTERISTIC IHPEDANCE 3 III. INPUT IHPgD.ANCE 01? VJYRY LONG LINE 5 LINE TERT:JINATED IN CHA..RACTERISTIC HLPEDANCE 6 CONCEPTS OF 'l'R..tt!.rELING 1.JAVES 7 EXANPLE 3-1·1IRE LINE CALCULATION 8 I. IV. V. VI. VITA EXA!·PLE 2 12 EXAJ:.!PLE 3 15 VII. CONCLUSIONS 36 VIII. REFE.RE.tiCES 37 38 v. ___ __ _;._ LIST OF FIGUR.ES 1 2n + 2 Terrri.na1 Ta!)ered Line of Length 2 Equivalent Circuit of Length 3 x of n-1-rlre 2n + 2 21 Terminal Exponential Line of Length d •••••••• d \·Jith Z 0 Termination ••.••.•••••••••• ~r~ple 1. • . . • . . . . . • . . • . . . . • . . . . . . . . • • • Six Terminal LC Ljne With C3 7 Unit Step. Response of a 3'-'t·:rire LC Line. 'Exar:lpl e 1. 1. a = 1......................................... 25 a-::. 1 ....... .......... r.................... 26 a :::. 1 ,. . • . . • • . . • . .. . • • . . • . • • • • • • • • • • • .. • • • • • 27 n =- 0 . . . . . . • . . . . . . • . • ..... ~ . . . • . . • . • • • • • . • • 28 Unit Sten Re >ponse Q' a. 3-1·::irc LC Line. E..~~pl e 1. a = 1. . . . ................................ .} ~ 29 Unit Step Res:~;onse cif a 3-~rire LC Line. ExPJn.ple 1. 1h 2l~ Unit Step Res~~onse of a 3-vrire LC Line. E::v.Jr~Ple 1 . 13 a :::. 1. • . . . • . . • . • . • . • . . • . . . • .. . . • . • • . . . . • • • 23 Unit Step Response o.f a 3-H:i.re LC Line. ~""l,T,1ple 1. 12 0 ••.•••••••••••••••••• Unit Step Response of a 3-·uire J...C Line. Exa.tnple l. ll ~ 22 Unit Step Response ~ a 3-~rirc LC Line. ~aJrrple 10 22 Six Ter:minal Line ShoH·ing Capacitance and 6 9 20 20 Inductance of B d n •••••••••••• Distributed Line..................................... Line of I.-ength 5 b. 11 n =0 .. •.. ,. . .. . . • . • . • • • . • . • • . • . . . . . • • • • .. . • . . 30 Unit Step Response of a 3-wire LC Lire • Exarrrr.>J.. e 2 • a ::::. 1. ,. . • • • • • • • • • • • . • • • • • • • • • • • .. • • • • .. • • • • • • 31 vi. Pnge Figures 15 Unit ~)tep Res~)onse of a 3-'l·r.i.re LC Line. J!;x3.i"'1plc 2. a = 1. ..... ·• ................................... . 32 a ,. ._ J................ " •••••••.••••••• " • • . .. .. • • • • • • • .. • • 33 17 Unit Step Ttcsl_:lOnse of a 3->·Jire LC Lire • F'x.,-unple 2 • 18 a = 0 .• • • • • • • • • a • • • • • • • • • • • .. • • a • e • • • • • 111 • • • • • • • • • • • • • 34 Unit Stop Response of a 3->·rire LC Line. S;C;s.rrrple 2. a 0 •••••• 'I· • • .. • • • • • • • • .. • • • • • • • • .. • • • • • .. • • • • 35 J.. THESIS PROLOGUE T'nis thesis is concerned vli th tne analysis of the N-1rr.ire tial tr;msmission line. The <mal~:rsis e:~:ponen- that folloviS is considered import;:mt because of the increased al')'l_pha.sis on mul tila'rer, micrvo-electronic, and thin-film circuits ~-~hich make use of uniform ::rod non- uniform distributed parameter structures. Figure 1 shm;rs a pictorial representation of a t~ered N-I·Jire ~ line. The exponential tr,ms,.,:lssion line differs from its classical counternart by the fact that the per unit ir:r:Jedc:.nces and ad.mi ttances are not const.:nt but var-<J as e 2 a.x The taper const;lllt 11 and e:.. 2 a.:x: , an is an arbitrary real constrmt. can be 2ccom:plished by physic~l· respectively. This variation :construction. T'ne distributed -parmneter aP''roach to 2Ilalysis is to subdivide the net-vrork length;-dse j_ntc eler..ent,"'J_ sections of length 2 sh01·Js such a section located at position x. f:l x. Figure The functions zi (x) and Yi (x) shm.m in fig1.rre 2 specify se~ics line .~-:1:ped.ances per meter and shunt admitta.._'l1.ces 9er meter, res!_Jectively. A stl"ai~htforl.·rard application of Kirchhoff's voltage and current laws follovrod by a ljJnit as 6 x approaches zero yields the follo1·r.ing transformed Telegraphist 1 s Equations: iJ V(x,s) -o'X ~(x)f(x,s) iJ I(x,s) "'!..(x)y_(x, s) ax- •·rhcre· the above vectors and matrices are defined as follo"l'rs: (1) (2) 2. V(x,s) -. !_(x,s) ~(x,s) !_(x,s) a col(Vi(x,s)) (3) = col(Ii(s,s)) (4) [ Zij(X) ] (5) = ·= [ ..v·l.J·(;-) ·- ] i , j : ~,2, •••.•• ,N (6) The above Telegraphists Equations (1) and (2) can be vl!'itten in a more compact manner as a ax Y._(x,:) 0 !(x) Y,.(x,s) (7} - I(x,s) !(x) !_(x,s) 0 The solutions to the Telegraphist's Equations are used as a basis for the analysis that :tollo1·1S Hi th ~~(x~ and 2 -. proportional to e2ax and e- axrespectively. l.!<xi being set 3. THE 'N-:•n:RE EXFONENTIAL THAHSlliSSION LINE ______ ,. I. -Introduction This thesis is concerned i·rith the analysis of the N-•·r.ire exponentr<msmission line. The transfer matrix, Hhich relates nct'I-JOrk input quantities to output quC~ntities, of the general N--vr.ire. ·~::;ponential line, Has ~;ublished recently by Dertnolli and Vandivort(1,2). This transfer matrix is used as the basis for this ?Jlalysis. ~e purpose of this thesis is to define a characteristic i.mpeclan.ce and to find solutions for three specif.'ic transient analysis problems. 3-~nre line The effect of terminating the N-~-rlre line in its characteristic impedance is investigated. II. Dcf:tning Ch:1racteristic Impedallcc The solution of the Telegraphist 1 s Ec1uations v-.'ith ~(x) z e2ax and !(x) = .! e- 2ax = , the :m...'l.trices Z and Y being constant,· is found in the following forr11: col(V(x,s),I(x,s)) ~ - 1rrhere ~ - T(x,s)•col(V(O,s),I(O,s)) - - (8) XJ.o) - col(V21 (s),V22(s), ••••• ,v2n(s)) (9) !(o; - col(I2l(s),I22(s), ••••• ,r2n(s)) (10) are the variables at the receiving end of the line as shown in Figure. 1. The tenn !_(x, s), called the t,r<U'lsfer matrix, relates neh·rork i..'1put quantities to output quantities, and is eiven by the .follouing equation: (1,2) -1 cosh(rx) -a. rsinh(rx) 0 T(x,s) r sinl:.(f.'X} -T ~· - (ll) E e-a."'<: 0 . 't·.rhere E is the u.11.it matrix, and z f a. ::::o r T represents the trro1spose ! ~! of _, !r ) rsinhux I -r.) coshlfx t a Jrr" .r sinhJ:x -T + a 2f_, ..r. It ca..tl be seen from the above equations that vrhen the ta:pering constant tra 11 equals zr::ro, the line reduces to the const<lllt pareJneter case considered by Rice(3). If the above matrix equations are re1r.1ritten in term.s of ex:xmential functions the .follm-ring equations result: (12) and {~ - a ..('-.,.)~0-ly2 + · (E + a_['-T)!-2} -1/2e-axerTx {C~ - af'-T)_?.o-~ - (!£ - ar-T)!L} !(x)= l/2e-a-Xe..r .,-x The impedance matrix (13) ~e2ax is arbitr<'U'ily defined to be. the line's c~qra.cteristic L~)cdance and is oaual to (14) 5. III. Inl!ut ImPedance of' ~ Long ~ Now consider a line of length c as shoHn in Figure 3. A s:i.Inple change of variables will refer position to the sending end. Let x =d - y, where y is the dist~ce from the sending end. Rearranging the above equations gives -1 !. where ! -1 (y,s) ~]. (y) (y ,s) (15) is the inverse matrix of !,(y,s) As the length of the line d gets very large, ! 2 and! 2 will • approach zero. end by setting y I Kaking this approximation and considering the sending = o, results in the following equations: 0 0 Carrying out the indicated multiplication gives (16) 6• . .... --~ and since d is very large, the ·follo~rlng approximation results (18) Tnis equation indicates that the input impedance is . z . :m = y-1( r ! a!)e2ad = ~e2ad (19) Therefore, the input impedance of a very long tapered l:ine · is equal to the characteristic impedance, and the characteristic :impedance could be defined to be the input impedance of a very long line. IV. JJine · Terminated in Characteristic Impedance Figure 4 shm1s a line of length d, terminated at x = 0 axJ x.::.o characteristic impedance, Z 0 e 2 - Clearly the relationship ! 2 = in its. =Z-o • Z 0 !, 2 holds and the follo'Wi.ng equ.,."ltion may be developed from (12) (20) Therefore, a cho"tracteristic impedance termination has elj.minated the e- Cx term• In a si.rnilar manner from {13) r 'T.. e x~-a.X]: -2 (21) and from equations(20) and (21) T V(x) - e rx~ 0 e 2axe-!:x!,(x) (22) 7. Equation (22) can be shov-m to reduce to the form . V(x) = Z e 2 axr(x) -o This equation indiqates that the voltagc. an<;!. current are e 2a:x. related through Z -o Conce'\")ts V. It' ~ (23.) a.eyvmere along the line. Traveling Waves the or5.ginal. solution to the exponential line problem equation (8) is re·written in terms of exponentials the follo1·ring equations Hill result: =l/2erx e-ax·y_(x) ·+ 1I 2e-rx - {c~- a.r-')y 2+r-• ! ! 2} r.( ~! 2 J l { + a r-•)! cr ) ! ! - r-• ! ! 2 1 } (24) } + (E! + a\1· ) - l'e I } (f T~ )Y J.. 2- (~- aff':T)- I.z T-f [ 01 T 2 (25) Ii' the exponen-tial terms in equations {24) and (25) are considered to be traveling HaVeS, on the line and e- elimina~e ~·Jhen Y. 2 r X the e rx term "tti.ll travel from left to right term Hill travel from right to left. reflections, set the coefficient cf e-C x equal to zero • .this is done, the f'ol.lovr.i..ng relationship results: = (f +a!)-l Z f V,.,= Z I_ 2 =Y-l (f! a!)!_ 2 To 8. It is, therefore, se:n that in order to el~~ate reflections, the line must be terni.nated in its characteristic L-:1pccb.nce, and a stipuJ.ation of zero reflections cmLld be used to define the characteristic impedD.:nce. This agrees with the previous section 1-Jhich e-rx concluded there nns no ter:rn in' the current or voltage equations 1rfucn the tcrm.in..--:ttion uas the characteristic :i.rapeda.nce. It is therefore concluded that the concepts of tr2.veling i·raves, a very long line, and characteristic termination a..:,'e all three compatible in defining the character5 st,ic VI. in1ped..:mce: of a tapered line. Exa..rn.ple .3-i'Iire JJine Calculation The most general case is considered first, and the circlut to be analyzed is that o:f Figure capacitances and farads/raeter induct~ces L 1= L 2 = s. In this case, the distributed are chosen to be C 1 L .3 =1 = c 2 == :c .3 = 1 henr;:.rs/meter. ~ere.f9re, z = r s [2 . -ll and. -1 2J (27) 2. is fO'lL11d using the Va1ues above as : 2. .f=ZY+ a 2 E = (2~) Since one input is an open circuit this forces I 12 eqtml zero. The voltage V 11 is chosen to be a vu~it to step or 1/s in the Laplace tr<:.ns:form dona:in. Since the netuork is terninated in its chnracteristic i..-,pedance the voltage ru1d current are related ~treugn Z e 2ax I(x) -o x::: d at the sending end this gives Y,.(x) :: Since (30) !(d) Substituting the value o.f f Vu and I 12 and evaJ.uating 0 yields the follmrl.ng equa.tion: 1/s v (29) 0 2ads -a+ 12 .J a+2 3s2' 2 -1 -1 2 Iu (31) 0 Simple algebra yields the solution: (32) v 12 = -l/2s Attention is no1-r turned to the receiving end, and the reader (33) 10. is referred to Figure follolr.ing eqtritiO~l.. 5. Again, because of the ~ 0 termination, the can be ~r.ritten: (34) and a straight for..rard substitution yields the folloi·Ting solution: I ~2 =0 (35) I 21 = (36) T = 21 Now that =. 2 [ I .. L22 ! J has been .fotmd, 2 can be found through the relationship: v -2 z = -o 1 -2 =[vn] v (37) 22 This gives the solutions: v 21::: e -ad 2 + 3s 2' e -d)a (38) s v 22 = -V2 21 The solutions in the time domain are (4,5): (39) ll. rlhere J 1 (:x) in a Bessel function of the first Jd_"!d a..Dd order, and U(t) is the stc~) first i'tmctj_on. For the case Hhere a ::. 0 the folloirlng sin.pJ.ific:.::.ticns result: _ t;'ds Ill v 21 I21 v - -e.- - 22 - the other. V2.I'iables rcr;tc:ti.11inG the s:;r'le. In the tine clon<-dn I 1l (t)::: ~3/2 U(t) V 21. (t) - I 21 (t) :-:: D/2 U(t-{3 d) 7 22(t) =. -1/2 U(t U(t - fJ' d) -6 d) 12., Since the time domain solutions could not be found in a closed finite form, a va.J..ue o:r the tapering constant nan must ·1;>e chosen so the ~.ntegration can be carried out nrunericaJJ.y. A value of 1 ·is chosen for "a" and the plot of the responses is sho1-m in f-igures 7 ,8,9 ,and 10. t\Tith a= 0 , the constant parmnetor case, the responses are sho-vm in figm--es 11,12, and 13. There is a l!k'U'ked cU.fference betHe en the constant parruneter and 'k"lpered current response, the tapered current response being unbom1def. It should also be pointed out· that the tapered line acts as a transformer. The tapered line . introduced time delay as did the constant line case, but j.n the t.:=tpercd case the load responses are di:r£erent in ma&.nitude from the source quantities as opposed to the constant r,1agni tude in·'.the constant line case. As·: seen from the ana.lytic solutions, for very large time the source and load variables -vdll be approximately eqt"k"il. E:x:anmle 2 The above problem is repeated Hi th the parameters changed to C 1 =- C 2 1 2 = = farad/meter and 1 1 = 1 3 1, C 3 = 0 0 he:nr;Vs/metcr. = 1, The circuit is agnin terminated in its characteristic impedance. Hith these ne..r parameters the Z nnd Y matrices take on the values z !. 1 8[ 0 ·a] 1 13. ?.. and r is i'Ol.md using the above values as ZY + a 2 . E (h5) AgainJ the ch.2.1'acteristic 'llapeclance is foundJ and it turns out to. be a particuJ.arly simple form. ~0 (h6) and Hhen the analysis is carried out in a proccdm·e parallel to the previous e);:,::;rpleJ the follo~ring solutions result: In (4?) v v 12 22:::. 0 (h8) I 21 I 12 et~9) I 22 -:- (50) 0 _d v 21 :::: e -ad ,\J<l I 2 e ---s + 21 s- The solutions in the tirne_ domc:d.n are: (51) · .LLJ.. = e-2ad [~at (52) )r 21 ( t::: e -2ad L-e at (53) I ll (t) (54) Again a value o:f 11 a.U must be chosen, a : ;: 1, and the integration carried ou:t numerical~y. and 16. The results are sho'tm in .figures The ca~>e for a ::.0 is sh01,m in figures 17 and 18. 14, 15, Notice· · should again be taken of the unbounded nature of the response currents for .the a~ 1 case, and of the ti·ansformer effect betueen load and so1trce varia1)1es. Novr considering the r.8.r:::e a ::. 0 I l l = 1/s v 12 = v I 2l =- 22 -.:.. 0 = I 12::. I 22 v 22 v 21 = 1 e-sd s 0 e -sd a e -sd s Or equivalently i..'rl the time dornain gives the solutions: 15 •. I 11 -::: U(t) U(t - d) U(t - d) v 21 (t) U(t - d) Exanrole 3 -~-- This eY. .a.mple represents a prG.ctical situation in which the parrunoters to.lce on the values C 1 :::. C 2 meter &"ld L 1 ::::: L 2 =- ~ 1, C 3 :::. 0 farads/ L 3 =. 1 hel:1...rys/meter. 'rhe circuit is sh01·m in figure 6 J 'Hhere it is ngain terminated in its characteristic inpedance. ~L'he characteristic :i.mped<.mce must. again be found but it presents more op a oroblem this diagonal. To find~ eigenva.lues of' r 2. 0 tir~e since some of the matrices are not Sylvester's Theorem (6) Hill be used. a.rc found as f'ollo-vTs : 0 The .J..o. which gives the folloHing solutions:. s 2" + a 2 a 2 (56) In this special case Sylvester t s Theorem may be H:ri tten as cr -;t 2. (57) Tr 'j::.! Jtt: ~ Where f( L ) is c>.:ny function of r r 2 and -\ :repreGents the resrective ~ eieenv<liue of _[' • fu.n.ction of r The cllaracte:ristic inpedance is expressed ·as a ?... in the follovring r:lD...'I'Jller: (58) Applic~tion of Sylvester 1 s Theorem;yields the follo-vr:ing for! o• ( ~ a2 + s2'-t ~a2 -t- 3s2 ' -2a) ( ~a2+ 2s z s 2' -i a 2+ 3s 2 I } 2s - (59) -o (~ a2+ s 2' -Ja2+ 3s 2 ') 2s ( Ja2+ s21 +-4 a2 + 2s 3s2' - 2a) 17. The circuit is again .:malyzed using the same !"'rocedure as. used in the previous e:xa.mple. ·· The sending end is. malyzed first,, <2nd ·the follo1;·r.Lng values are found: 2 8 -2ad In (60) v 12 (61) Again, going back. to the originitl cqrotion and using the relation'!.. 2 ~ 0 I 2 -vre get: (62) Since "t·re r..no't·T the va~ue of the m.."':.trix !(d), vTe need to find ! 2• This is fotm.d r.tost easily by mldl:lg usc d: Sylvester's 'l'heoren . . . . ~~ to take the inverse of the (!IJ.<;-mtity -vrhich is a fi.mction of I ~·Jii.cn this is carried out, the folloHing solution results: ad I 21 = e . (e-d\r a2+ s2 + . ~a2 ~ s 2 '+ ~ a2 + e-d 3s2 ~) + 3s2' .) (63) 18. e-d~ a.2.,_ e-ad I 22 ~~ 2~ s2 1 + s2, - e..d ~ a2~-3;2' ~2:;_---37' (64) -2a The matrix:! 2 in aga:i..n ffnmd using the rr3lation v, -.c_2 ·= [vv J 21 22 (65) - i"Thich eives, I 2 \ -d ( '\1 a + 3s2 - a) e ~a2+3s 21· 2a) and v (~~2 + 3s2 22 _ a) ~-d·!a2 ; 3l -2a ) If He again assume a case. = o, th:is reduces to the constant The ab9ve solutions then reduce to the follo-vr.ing simvlified forms: 2 1 I 11= - - - l+IT V 12 I ')., s (iY - 2 ) 1/s 1 e -sd+ 6 -YJ'd.s par~eter 19. 1 e I 22 11 I')' v 21 l+"fJ' v 22 ru1d 1 s ff e-D~d s- _13' e~ 13' sd e -sd 1+ lJ s the characteristic impedance Z - (1 t YJ) z -o e - 13"Sd e-sd+ l -- -sd - -0 ·is resistive, . (1 - {3) (68) 1/2 (1-t f3 1 ) 20. r'll(s) +o-----=~---~ vll(s) +o -+ I12(s) Q~---'3>------7 ifln\s) v •.;.o----;;;>::;;:...---1 (S) Zh X <-J. ·--------------~ 1 J.(x+l'q:,s) Axz,(x) +~·-------~----~~ . ( X+ l).tx;s') ~------~----~---~---~+ V:1 l ~ : ·~ ) t\ ~, S ) .,__ _ _--1----::T----=:J---~ + V'2 <z, s > ~-----~--~----r---~+ • • • • • • • • Ax Zn(x) -~o------------V\1\i\(---------T...1-t....,...... , .... ,.... #""\ ~ .... .t--~-~,_......,..a.,. 1"1:.:.-- ... .I..L. -L' T - ...... _...._,__ X~-1 '"'----.../!'a- 'tr.l-~- + 21. 1 d <.: t- (7\- V11cs) Yz I ( s) -t- + Vu(s) V,'l.(s) f!J- +- r}--·----------------~---:-----o +-· Ven (5) ---·-c- -<$:-----~1. )::. 1='1~- 3 2 n ·r- 0 F ?-· '[EI~MtNAL U:.. N G; IH d •- E.'i-PO!-.fE.l\J\iAL . II 22. .IZI(,/1 r--·-l--:-,-,(~-----------·-__A_I\\E.I::_I~l?~.~---· - o--==-=-------··--------·-------·----.::::-:..:..7 ___t c---~ . t Vzlh) Vii(S) l -------------b----~ ~~-~-----·- 1- -0 ------<.:~ ..'• • 0-----------------------------------FiG. 4· r ------··· . Li N E. . L I d 0 F L 1:: l\1 G, \ ~ · --·(j·--. W I n~ d I M•=.-.·r:.r=<s ~,;.-, ----··--·-·· . -""'"""--r----·1- -.- ..... :)~ 1 . . L_ . _· ..•... _.rv"""'"-·-··j··------:'l---·- ..•.. -----c·------1--..o----· ...... . ---------- -----c----..----- C II ~-r- V;?l '-;. r ··-:>- L lz= o . cz '"T- . Tc3 Vn l.....J f • -"'1:'"\f"\1""\- FtC,. . ,• ... r ·------""0 Ct\Pi\C..ITANCE. 5 f.\ tJ D i 1'-.1 D U C. T 1\ 1~ C f:_ .0 F '!=_ 1-. 1\ M P L E.. i . '-...J• Zo Tor:m~na S>---- :~'igure 6. Six Terminal LC Line ~{ith c3.,=- o. t:i n <.) J ,.J \ fl!. $ f {q .:!. tL 0 uJ (J} 2 0 n. ~ tr Q. U1 II ~ • "\"I 1:1 J a. ·:i <( .,t. JJ} !:: z :::;:, ,.... . <!J lJ.. ~ 2 0 <J \- U} «) -t4 w 2 J l.!.l V) ....., uJ ~ i= 25. J .J LJ Jt - s I f!) < IJ.. 0 w !/) ":2. .....\ II rd • ~ uJ .J CL a. JJl. <( Ct. IJ..I 0 !,}) lt lJ.l 1(/) z:::>- 1- ~ ~ ,...... tlj 0 • "Z 'Z uJ V) v 0 (.) tiJ ..J ul :i: != ~ t.S IL. • ;!1'l J .J tJ ~ <:.... ' rf) . >((. u. 0 w ~ tl rd liJ z 0 1).. tJ) -.-1 uJ •JJ Q! -' a. 0.. <:(; •.tJ 0 ., ~ ~ l- w {I) 1- w z z ::> J . 0" 6 iC ul .s.: f- . 0 "-I. (.) ....l 0 - ,j w oc 3r 11) 1:) ,... \) r.:!: ""'II I!. rd Q (ll 0 "Z Ill <J z 0 - U) Ill 0 (/) uJ ~ 0 a. "i- 1.:1 :Jl a; i= (!.. 1:1 l- <I) <) >Q 2 :::> . ... ~ ..." .... u! J 0.. z4. ~ uJ ~ J- uJ 0 0 • (}1 G: z ...1 28. 2 3 ·seconds t ~XL'X:i p 1 e At<\PS 1. 11 a 11 := Q tJN\T STE.'? 'Rf:.SPCI\lS\:.. 0\= LI~E.~ ·a= o, E:.i.A~<~PI-F_ 1~ I21 ( t} 1 .866 1 2 t seconds 3 4 11. 3- \ 1JiP-E LC . 0 0 uJ to ...l 11 a:: r;l. $ 0 I !Y) _.... (/') D <( :z u... (..) 0 c w tl} u.! U) z 0 !l.. ':/} ~ " .o .... uJ ..J c. w 2 cr <( 0. :>< IU tr.l 1- (/) ,_ -z td :z :::> _l 0 30. r21 ( t) VoLT.S 1 . . 866 1 2 t 3 seconds E:<o.l:Jple 1. "a"= 0 v22 ( t) \/oL-TS Fl~. 13 t)N\T STl:.P RE.f)PONS\:. OF Ll Nt. .. 2 -.5 t seconds Exnmple 1. Q •. 3 - 1 11 au =0 A 3·-WIRE. t •] I I I !,-,. 'ill II) <J !7. 10 J l ! -J I.!J :;I] 'v IJj lui 0:: '$ ...... I ("() .:( I.L () ll rd • <\l t_,j I~ I 0 \.'\ V) z ul 1).. :I} D.. 0 uJ ~ 11. u.l J- V) 1.:: z ::> _j :z <( ·,(, 1.!.1 ., 0 I~ . w -z J- . () . rt) -..1:- cl- u.. . Q N u -l U.l ct ~ I ft) • <! -M IJ.. II 0 rU !IJ <I) <Q (\f U) uJ J r.L Ill ~ n. tr.J ,.,.. 0. :E <! -.,.<: fJ] 1r.J ...... lJ) 1- ::z ::;) .. f}~ ul 2 "'l;:::. ....J 0 "Z tJ u! fl) o...., . b) 0 .r.- 0 <.) .-I ul (t - < I . fY) .<:£ ..-I 11' lL 0 w ~ t.IJ {\! 0 u.l :z Q... <fl w ~ ,.... U') ...I 0 "Z 0.. z 0 c) <:! ·>< a. IU u.l ~ l!.l '-J !- u.l (J') 1"'Z ::> l:: ,: t.J 2 .J . () ~ .s . c.5 l.l. '.n ..:. Ill (\). H a. 'f: <( ....9 \ ~· ' l_ n .)LJ.e ,. o ~~ I • I /'·'. '1 r---------------------------------------------------- 2 1 3 seconds t Example 2. "~"::::. 0 UNIT ST£~ RE'SPOI'tSE. O!=" A 3-'f./IRE fiC1. l7 LtNE.. .. E~AMPLE. 1 3 2 1 t seconds 2.. 3 ::=.0., LC VII. Conclusions The soJ_utior.s found in this thesis are :for the e:-Qonentiruly tapered tral"l.smission line and can be used .for a. constant parameter line by setting the taper constant "a11 equal to zero. Although the ex.a.mples Harked here 'iere .for LC lines, the general equations developed in the first part of fue thesis a})l')ly equally l·rell to eh~onential axrr line. It is ap-parent that anc"llysis of the problem is much simplified . - '1-Jhcn the J.ine is terminated in the characteristic irape<'l.ance, since this €limina:tes the reflec"Wd uaves of both current and vol ta.ge • It should be noted from the e::camples, that the current responses increased without bonnd in the tapered cases, but 1rere bounded for the constant pnrameter cases. A"l.other .. i.rn.portant dif.ference beti·reen the tapered line and the constcmt para."neter line is that the ta:x:1red line acts as a transformer. The responses at t.~e in time, but the ma.gni tude is also different. load are not only delayed ':Jhen the tc;.pering constant nan equals zero, this rednccs to the constant parameter line case and it is seen that the transforr:~er between the source <.>nd load variables. effect no longer holds VIII. Hcferences 1. C.A. Vondivort and B.C. J3ertnolli, 11 De'termining the Trr~ns:fer !!:·; trix of Tapered ~·:ul ti ~·Tire Trnns:-nl ssion Lines", 10 th Midwest S:mposium qn Circuit Theory, Purdue University, 1'-~ay 1967 2. E. 0. Bertnolli, "Anr:tlynis of the t:r-;nre l'~.x:ponentinl Line", IE3E Proceedings vo1.55,No.?,pp.l225,July,l967 .3. Rice, s.o., "Steady State Solutions of TransmisGion Line Equations", The Bell. 8,;"8te~ Technicn..1 Journal, VOL. 20, No. 2,pp. 131-17", April 1941 4. P.A. ~··cc.)1lum, B.F. Brow·n, '"Loplace Trnnsfor::1 Tables a:-:d The;)rems, Holt, H.ienhart, ·.nnston,l965 5. G.E. Roberts,H. ·Kaufman, Table of Lenlnce Trannforms, :?. Saunders Co., 1966 - 1;{. 6. P. i·L Do russo, R ...J. :~oy, St2. te Vs.ri :1 bles for .El~f:ineers, John Wiley ond sons, Inc.,l96~ 7. J. J. K8rak~, sh, Tr~.nsmission Hncr.tillan Co., 1950 8. Fra: er, Dunc1'=ln, 9.nd Coll~~r, ].le•nentnry ~1atrices, Ca:u•.~ridge: Cambridge University Fress ,. 195'7 L~nos ~ Filter ~:ct;-rorks, 132976 . _./V• Vita The author nas born on January 19, 19h3 in Oak Hill, H. Va. He received his primary and ~econdary education at Fayetteville, ·H. Va., end then enroJJ.ed in ':lest Vj_rginia Ll"lstitute of Technology in 1961 at Hontgome~J, 1·1,. Va~ He received his fuchelor of Science in Electrical Engii1eering in Juno 1966. has been a f'u1.1 tirae student at the University of r·1issom"'i at Rolla since Se1")tember 1966. He