Response Analysis on Nonuniform Transmission Line
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1 SERBIAN JOURNAL OF ELECTRICAL ENGINEERING Vol. No. November Respose Aalysis o Nouiform Trasmissio Lie Zlata Cvetković 1 Slavoljub Aleksić Bojaa Nikolić 3 Abstract: Trasiets o a lossless epoetial trasmissio lie with a pure resistace load are preseted i this paper. The approach is based o the two-port presetatio of the trasmissio lie. Usi Picard-Carso's method the trasmissio lie equatios are solved. The relatioship betwee source voltae ad the load voltae i s-domai is derived. All the results are plotted usi proram packae Mathematica 3.. Keywords: Epoetial trasmissio lie Trasiet aalysis Picard-Carso's method Two-port etwork. 1 Itroductio The trasiet aalysis i the area of power etwork elemets represetable by ouiform trasmissio lies is aii more importace [1-5]. For the trasiet aalysis of ouiform lies the ouiform trasmissio lie ca be treated as a cascadi of ifiitely short semets of the uiform trasmissio lies with differet characteristic parameters [ 3]. The secod techique is based o etedi the cocept of the reflectio ad refractio coefficiets creati lattice diaram [3]. If the focus of iterests is the propaatio of sial or eery o the lie the trasmissio lie ca be studied as a circuit theory model where voltaes ad currets are the variables [1] [] ad [5]. The Laplace trasform is used for obtaii the closed form of sials ad later the s-domai model is trasformed ito the time domai. The aalysis of trasiets i trasmissio lie for differet voltae sources usi circuit theory approach is preseted i this paper. The A B C D parameters of trasmissio lie are derived by Picard-Carso's method. 1 Faculty of Electroic Eieeri Departmet of Theoretical Electrical Eieeri P.O.Bo Niš Serbia & Moteero zlata@elfak.i.ac.yu Faculty of Electroic Eieeri Departmet of Theoretical Electrical Eieeri P.O.Bo Niš Serbia & Moteero as@elfak.i.ac.yu 3 Faculty of Electroic Eieeri Departmet of Theoretical Electrical Eieeri P.O.Bo Niš Serbia & Moteero bojaaik@elfak.i.ac.yu 173
2 Z. Cvetković S. Aleksić B. Nikolić Voltae Respose Network Trasfer Fuctio A ouiform trasmissio lie show i Fi. 1(a) is cosidered. The source voltae is U ad the load voltae is U p Z ad Y p are source impedace ad load admittace respectively. Three cascaded two-port etworks show i Fi. 1(b) ca represet the system show i Fi.1 (a). Equivalet A B C D parameters of the system are []: U CZ DZYp BYp A = I C Yp D ZD B U p D Ip (1) (a) (b) Fi. 1 Nouiform lie ad its correspodi cascaded two-port etworks. The relatioship betwee load ad source voltae i s-domai ca be derived as T U p 1 1 () s = = = u U I = A CZ DZ Y BY () A p ABCD Parameters of Distributed Networks ek Assumi TEM mode of propaatio the behaviour of the trasmissio lie is described i s-domai by Teleraph's equatios: ad ) du d 17 p ( s) I ) = Z (3) d I ) ( s) U ) d where Z ( s) is per-uit leth series impedace ad ( s) leth shut admittace of the distributed etwork. = Y () p Y is the per-uit
3 Respose Aalysis o Nouiform Trasmissio Lie Picard-Carso's method is used to solve trasmissio lie equatios (3) ad (). This method is a powerful method i etti a power series solutio for the distributed etwork because it is easy to calculate poles ad zeros. This method solves differetial equatios by a iterative sequece [1] ad U I ( s) I ( s) d = U Z 1 (5) 1 ( s) U ( s) d = I Y (6) for = 13 K where U ad I are the voltae ad curret at the iput port of trasmissio lie = Fi.. Fi. - Trasmissio lie ad model of elemetary lie leth. Sice the terms iside the iterals are cotiuous ad bouded the sequeces will covere to the true solutios U s = lim U s (7) ad ( ) ( ) Δ ) = lim I ) I. (8) Δ Equatios (5) ad (6) may be writte i the form of U I ) U Z ( s) I ) d = (9) ) I Y ( s) U ) d =. (1) Usi Picard-Carso's method these solutios ca be preseted i the form of two-port parameters [1] 175
4 Z. Cvetković S. Aleksić B. Nikolić From equatios (11) ad (1) the ) = U ζ I = = 176 U ζ (11) ) = U ψ 1 I 1 ψ = = B C D I. (1) ( ) = ψ B( ) A parameters are: = = = A ζ. (13) ( ) C = ψ 1 ad ( ) = = = 1 D ζ. (1) ζ = 1 ad ψ = 1 (15) are values chose as iitial values for iteratio starti. The other terms i the summatios are foud by evaluati the followi iterals iteratively: Substituti is obtaied: ζ = Z ψ 1 d ad ψ = Y ζ 1 d. (16) A B C D parameters ito () the voltae respose i s-domai ( s) T ( s) U ( s) U u p =. (17) The time domai output sials are the iverse Laplace's trasforms of the s-domai sials. 3 Numerical Results for the Epoetial Trasmissio Lie A lossless epoetial trasmissio lie of kow characteristic impedace k 1 ZC ( s) = ZC e with k = l M where M deotes a taper ratio of epoetial trasmissio lie of leth d which is defied as M = Z C d / Z C is ob- d served i this paper. Z C ad Z C d are the characteristic impedaces at the source ad load eds respectively Z C = l / c. Z ( s) ad Y ( s) are defied as: Z k ( s) = l s ad ( ) k Y s c s e =. (18) e
5 Respose Aalysis o Nouiform Trasmissio Lie l is uit leth iductace ad c is uit leth capacitace. Starti from (15) ad usi equatios (16) a few first terms of series are []: l c ζ s ls ( ) = ( e k cs k 1) ψ ( ) = ( e 1) 1 k ( ( ) ζ ( ) = 1 e k k ψ ( ) ( k) ψ ( k) 177 k 1 ( k) ( 1 ( e k k ) lcs = ( k e ( 1 e k ) k ) 3 3 ( ) 3 l cs ζ = ( k) ( k e ( ( 1 e k ) k ) 3 lc s 3 ( ) = 3 ( k) k ( 6e ( e ) k ) k l c s 1 ζ = 3 k k k k ( 6 e ( e ) k ) e k l c s = 1 ψ 3 ( k) M Whe M = the umerical results for A B C D parameters are: A ( d ) ψ ( ) = ψ ψ ψ L = = d 1 ( τ).3188( sτ). 139 ( s ) s τ B ( d ) ξ ( d ) = ζ ζ ζ = = L 3 [ 1.7sτ.3855 ( sτ). ( sτ) ] 5 Z C 1 Z C ( d ) ψ ( d ) = ψ ψ ψ C = = L 3 [.7138sτ.1197 ( sτ).11887( sτ) ] 5 6
6 Z. Cvetković S. Aleksić B. Nikolić D ( d ) ξ ( ) = ζ ζ ζ L = = d 1 ( τ) ( sτ) ( s ) s τ where τ = d l c is the time costat. For the case whe Z = 5Ω Z = 1Ω ad p 6 u ( t) = h( t) (19) the uit step voltae respose is preseted i Fi. 3. The steady-state value of the load voltae is epectedly equal to the fial value that at the source ed which / 1 Z Y. [ 3]. 1 p is ( ) 6667 Fi. 3 Uit step voltae respose. Whe the source voltae is of epoetial form at ( t) = e h( t) u () time depedece of voltae respose for a = 5τ is show i Fi.. Whe the source is rectaular sial u ( t) = h( t) h( t a) (1) of uit amplitude ad duratio a = 1τ trasiet respose output voltae is preseted i Fi. 5. Whe the source voltae is u ( t) = t[ h( t) h( t a) ] () time depedece of voltae respose for a = 5 τ is show i Fi. 6. I the case of source voltae 178
7 Respose Aalysis o Nouiform Trasmissio Lie ( t) = th( t) ( t a) h( t a) u (3) time depedece of voltae respose for a = 5 τ is show i Fi. 7. Fi. Epoetial trasiet voltae respose. Fi. 5 Voltae respose o the sial (1). Fi. 6 Voltae respose o the sial (). 179
8 Z. Cvetković S. Aleksić B. Nikolić 5 Refereces Fi. 7 Voltae respose o the sial (3). [1] S. M. Ghausi ad J. J. Kelly: Itroductio to Distributed-Parameter Networks Holt Riehart ad Wisto Ic New York. [] C. W. Huse ad C. H. Hechtma: Trasiet Aalysis of Nouiform Hih-Pass Trasmissio Lies IEEE Tras. Microwave Theory Tech. Vol. 38 No pp [3] Z. Ž. Cvetković: Odziv a kostatu pobudu kod vodova bez ubitaka 11 Telekomuikacioi forum TELFOR 3 CD 9.6 Beorad 3. [] Z. Ž. Cvetković: Aaliza homoeih i ehomoeih vodova u s-domeu Proceedis of XLVIII ETRAN Coferece Vol. 1 pp [5] P. Peres I. Boatt ad A. Lopes: Trasmissio Lie Modeli: A Circuit Theory Approach Society for Appl. Mathem Rev. Vol. No. Jue 1998 pp
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