Frequency-Domain Modeling and Simulation of Coupled Lossy Multiconductor Transmission Lines

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1 IJSI Inenaonal Jounal of opue Scence Issues, ol. 10, Issue 5, No 1, Sepebe 2013 ISSN (Pn): ISSN (Onlne): Fequency-Doan Modelng and Sulaon of oupled ossy Mulconduco Tanssson nes Youssef Mejdoub 1, Hcha Roujaa 1,2, Mohaed Sah 1, and Abdellah Ghaaz 1 1 aboaoy of Eleccal Syses and Telecouncaons, Depaen of Physcs, Faculy of Scences and Technology, ad Ayyad unvesy, P.O. Box 549/40000, Maakesh, Moocco 2 Depaen of Maeal Scences, Poly-dscplnay Faculy, ad Ayyad unvesy, P.O. Box 4162/46000, Saf, Moocco Absac Ths pape pesens a sudy of odelng and sulaon of he lossless and lossy ulconduco anssson lnes n fequency doan. Ths sudy s based on a chaacescs ehod, whch pes odelng he lne as a quadpole whose advanage s no o pesuppose appled chage condons n s exee. Ths pes o be noduced easly n he ccu sulaos such as Spce, Esacap and Sabe. The esuls pesened hee cove wo ypes of ulconduco anssson lnes, he "Rbbon able" and "2We Xalck. Keywods: haacescs ehod, Fequency doan, Mulconduco anssson lne, osses. 1. Inoducon The pobles elaed o he fequency ounng and he effecs of neconnecons n equpens and ndusal applcaons ae vaous (dsoon, aenuaon, cossalk, ec...). Moeove he losses can play a vey poan ole n he degadaon and aenuaon of sgnals ovng hough he lne. onsdeng he faewok of he neequpen elecoagnec copably (EM), skn and poxy effecs becoe oe coplcaed n hgh fequency, whch necessaes, heefoe, a physcal odel adaped o he anssson lnes. The cascade odel [1][2] allows odelng he anssson lne n he fo of a RG ccu. I can also be appled n case of MT lne, whch becoes que coplcaed when usng oe han wo conducos. And esuls no only n oscllaons n e doan (phenoenon of Gbbs), bu needs an poan calculang e, whch akes neffcen. The fne dffeences ehod n e doan (FDTD) [2][3][4] s an analyc ehod. I consss of dvdng he e and he space whee he soluon s seached fo n a newok of pon spaced egulaly o consue a esh. The ajo dffculy hen, s o have a dsbued MT lne odel vald n boh e and fequency doans, wh and whou losses. The odel pesened n hs pape fo odelng and sulaon of he lossless and lossy ulconduco anssson lnes n fequency doan, s vald also n he e doan [5]. Ths pape wll expose a behavo sudy of he lossless and he lossy ulconduco anssson lne MT- n fequency doan, based on he odellng of he MT lnes nuecal ehod wh he help of Bann odel, ha can analyze he MT lne by epesenng n he fo of a quadpole, hs s he chaacescs ehod [2][6][7][8], Ths ehod would pe he analyss of an MT lne whou and wh losses and pesen he advanage o avod pesupposng condons of appled chages o s exees. Ths pes o be noduced easly n ccu sulaos as Spce, Esacap and Sabe. Dvese exaples of applcaons ae pesened o valdae hs odel and o show he neess and o undesand he behavo of a lne MT n he fequency doan. 2. MT nes Modelng 2.1 ossless nes Bann [6] was he fs o popose a nuecal odel of a anssson lne whch peed gvng a sple schea equvalen o he deal lne. Ths schea ncludes wo dpoles. In he npu dpole (z=0,) he enson s deened fo he efleced enson n exees of he lne (z=l) a he peceden e -T. The sae nepeaon s appled o he oupu dpole (T s he delay of he lne). Followng [2][6], he equvalen schea of he lossless lne (R=G=0) s depced n he fg.1 : opygh (c) 2013 Inenaonal Jounal of opue Scence Issues. All Rghs Reseved.

2 IJSI Inenaonal Jounal of opue Scence Issues, ol. 10, Issue 5, No 1, Sepebe 2013 ISSN (Pn): ISSN (Onlne): Fg.1 : Quadpole epesenaon of he deal lne. Indeed, s easy o show ha eleccal paaees n exees ae lnked by eans of he followng elaons: (0, ) R I(0, ) ( l, T ) R I( l, T ) ( a) ( l, ) R I( l, ) (0, T ) R I(0, T ) ( b) (1) The equaons (1.a) and (1.b) do no depend only on seconday paaees (he chaacesc pedance and he delay of he lne). Hence, he equvalen schea of he fgue 1 s esablshed by pung: ( l, ) ( l, T ) R I( l, T ) ( a) (0, ) (0, T ) R I(0, T ) ( b) ( ) and ( ) ae hen calculaed usng paaees obsevable a a known e T. The feaues ehod can pehaps be spead o he ulconduco lnes case [2][7][9][10]. Ths eques, on he ohe hand, uncoupled popagaon odes on each lne. Ths sepaaon of odes s pleened usng he odal ehod. Keepng n nd elecoagnec couplngs beween he lnes, he pedances ax ae no dagonal. The nees of he odal ehod s hen o uncouple he equaons o be able o dagonalse he ax. To odel he anssson lnes, s necessay o solve he elegaphss equaons. The e doan equaons epesenaon s: RI z I G z I 0 0 (3) (2) [] and [I] epesen, especvely, he ensons and cuens ax. [R], [], [] and [G] epesen, especvely, he essances, nducances, conducance s and capaces ax. These nclude plcly of all nfoaons concenng he ansvese secon, whch pes o chaaceze a uconduco sucue. The coeffcens of hese dffeen ax ae obaned ehe by engneeng paccal echnques [11], o by nuecal ehods [9][10][12]. The odellng of a lossless and lossy MT lnes n fequency doan by he Bann ehod wll be exposed especvely n paagaphs A and B. As we deal wh he lossless lnes case (G=R =0), he MT lnes equaons n he e doan (3) becoe: z z I z z,, I 0 0 (4) The odal ehod noduces fcous paaees and I, by eans of he followng lnea ansfoaons: ( ) T. ( ) I( ) T. I( ) The odal axt and T I (5), ae o be deened o ensue he uncouplng of popagaon odes of he MT lne [2][7][9][10]. If we apply he ansfoaon o he syse of equaons (4), we hen oban:,, z 1 z T 2 Tv 2 z I 1 I T 2 T 2 z 0 0 (6) We can choose boh ax T and T n such way ha he syse (6) could be uncoupled. In ou case of a lossless lne placed n a hoogeneous and soopc edu, we can always fnd he ansfoaon ax whch dagonalse sulaneously he ax and. The wo ansfoaon ax T and T ae bound by he followng elaon [2]: T 1 T (7) The syse of equaon of a lossless lne spells n he odal base : z I z Wh NxN I and 1 T. T. T.. T 2 (8) 1 ae dagonal ax of denson I I (9) The syse of equaons (8) epesens an uncoupled MT lne, whch has a chaacesc pedance R and a opygh (c) 2013 Inenaonal Jounal of opue Scence Issues. All Rghs Reseved.

3 IJSI Inenaonal Jounal of opue Scence Issues, ol. 10, Issue 5, No 1, Sepebe 2013 ISSN (Pn): ISSN (Onlne): delayt : R, T (10) 2.2 ossy nes The losses can play a vey poan ole n he degadaon and aenuaon of sgnals anspoed by lne. The losses ae caused boh by a non-null conducvy and of he loss of he polazaon edu o pefec conducos. The losses pesened by pefec conducos ae habually oe sgnfcan han hose owng o he anssson lne. Fo hs eason, we ofen suppose ha he suoundng edu s lossless (G = 0) n he MT lne equaons. The essance due o pefec conducos s epesened n he essance ax by a lengh uny [R]. The syse of equaons (3) becoes: RI z I z l I 0 0 (11) The equaons (11) expessed n he fequency doan by aplace opeao p gves us: [ R][ I] p[ ][ I] [ Z][ I] z I p[ ][ ] [ Y][ ] z (12) In geneal, a lne can be odeled by s seal pedance Z R jw, and paallel adance Y jw, (The deleccs losses ae supposed o be neglgble G=0). Fo he wokng fequences well supeo o a chaacesc fequency of he lne equal R / 2 (low losses hypohess) [2][7], and usng a fs developen ode, we oban: Z R jw RR R w 2 jw whee R (13) The chaacesc pedance n ha case s equvalen o chaacesc pedance R ouned seally wh a capacy 2 pf, when he fequency nceases Z R. R becoes equal o R. Wh he sae appoxaon, he consan of popagaon becoes: R j jw 2R To ake no accoun he aenuaon of he wave he e e l s noduced, s enough, hen, o odfy Bann s geneaos of he quadpole epesenaon of he deal lne (cf Fg 2). They becoe [2][7] : ( ) ( ) ( ). e ( ). e l l Fg.2 : Quadpole epesenaon of he low loss lne. (4) Fo a lossy ulconduco lne, we do he sae as n he case of a lossless MT lne. Theefoe, n he case of coupled low loss lnes, he chaacesc pedance and he delay ae gven by: Z RR 2 jw R, R (15) Thas supposes ha he essance s losses efeed o he R ax whose non-dagonal es wee null. Also, s enough o bng he followng odfcaons o he geneaos of he deal lnes, gven by elaons (2.a) (2.b), whch becoe: l T l T Z I l T a (, ) (, ) (, ) ( ) (0, ) T (0, T ) Z I (0, T ) ( b) (16) l Wh T T e s he aenuaed odal ax. The Elecc paaees o he exees of he MT lne ae lnked by he followng elaons: l Z I l a (, ) (, ) (0, ) ( ) Z I l b (0, ) (0, ) (, ) ( ) 3. Sulaon & Resuls (17) To vefy he valdy of hs odel n fequency doan, we sulae wo ypes of ulconduco anssson lnes, he "Rbbon able" and "2We Xalck n ESAAP opygh (c) 2013 Inenaonal Jounal of opue Scence Issues. All Rghs Reseved.

4 IJSI Inenaonal Jounal of opue Scence Issues, ol. 10, Issue 5, No 1, Sepebe 2013 ISSN (Pn): ISSN (Onlne): ccu sulao [13]. In hs paagaph, we ake a copason beween sulaon esuls n he leaue fo exaple 1[2] and exaple 2 [10] and ou sulaon esuls usng he ESAAP ccu sulao n fequency doan n he lossless and lossy. Ths odel s vald also n he e doan [5]. 3.1 Exaple 1: Rbbon able We consde a anssson lne of hee conducos wh 2 of lengh and lgneous paaees R, and, fed by a ensonal geneao, he n and he ou chages ae equal o 50 Ohs, as ndcaed on he fgue / nh R / Angle (Dege) ossless ossy -100 Fequency (Hz) b ) Phase. Fg.4 : The nea-end cossalk volage sulaed wh Esacap pf / a ) Magnude. a ) densons. b) Elecc epesenaon. Fg.3 : Rbbon ables anssson lne o 3 conducos. Magneud () Fequency (Hz) a ) Magnude. ossless osses b) Phase. Fg.5 : aldaon esul of he nea-end cossalk volage [2]. Sulaon esuls: The expeenal esuls fo Ref [2] ae copaed o he pedcons of ou ehod, ove fequency ange of 1 khz o 100 MHz. As fo fgue (4.a), fo F<100 khz he losses n he lne conducos ae pedonan. Fo F>100KH he nducve effec becoes pedonan ove essve effec. In he Fgue 4, he aplude and he phase of nea-end cossalk, he pedced volage nduced a he souce sde of conduco n.1, ae dsplayed. The coespondng values fo Ref. [2] ae dsplayed fo copason on fgue 5. As exaple, he low-fequency value n Fgue (4.a) s ~ - 48dB, coespondng o ~ 4 n Fgue (5.a). opygh (c) 2013 Inenaonal Jounal of opue Scence Issues. All Rghs Reseved.

5 IJSI Inenaonal Jounal of opue Scence Issues, ol. 10, Issue 5, No 1, Sepebe 2013 ISSN (Pn): ISSN (Onlne): Physcal nepeaon: Fo (4.a) a low fequency (R- D >w), he essve couplng s pedonan, whle a needae fequency, he classc nducve X-alk becoes pedonan (slope +20dB/dec). A hgh fequency, he volage s led by he lne s chaacesc pedance. The shap esonance a F~65MHz s he λ/2 esonance of 2 cable, shfed down by nceased eleccal lengh due o exa capacance of delecc gans. The λ/2 esonance s exced because of low R enaons, copaed o coon ode pedance Zc~ 130Ω. 3.2 Exaple 2: 2We Xalk Angle (Dege) ossless ossy Fequency (Hz) b) Phase Fg.7 : The nea-end cossalk volage sulaed wh Esacap 2000 We consde a 2we-xalk anssson lne wh lengh and of delecc consan ε =1, fed by a ensonal geneao, he n and he ou chages ae equal o 50 Ohs, as ndcaed on he fgue 6. a ) Magnude a) Geoec confguaon b ) Elecc odel of he lne Fg.6 : confguaon of he lne 2we-xalk. Fgue 6 pesens he geoec confguaon of he lne whose paaees ae: / Magneud nh R / pf / Fequency (Hz) a ) Magnude ossless ossy b) Phase Fg.8 : aldaon esul fo he nea-end cossalk volage [2]. Sulaon esuls: The pedcons of ou ehod ae copaed o expeenal esuls (Ref.[2]), fo fequency ange F=10 khz -1GHz. A F<200KH he conduco losses becoe pedonan. In he followng fgues, he aplude and phase of neaend cossalk volage on conduco n. 1 ae shown, fo copason beween EMAP pedcon esuls and efeence canoncal esuls (Ref. [2]). Fo exaple, he low-fequency value n Fg.(7.a) s dB, equvalen o 3.3 n he fgue (8.a). Physcal nepeaon: The sae consdeaons of pevous exaple n he Secon 3.2 ae applcable. The fequency of he shap esonance s now shfed o F~400MH copaed o pevous one F~ 65MH wh b) opygh (c) 2013 Inenaonal Jounal of opue Scence Issues. All Rghs Reseved.

6 IJSI Inenaonal Jounal of opue Scence Issues, ol. 10, Issue 5, No 1, Sepebe 2013 ISSN (Pn): ISSN (Onlne): scalng alos sla o lengh ao (2 agans 0.25 ). The nceased capacance due o PB subsae, copaed o we delecc gans, explans he dffeence beween he fequency and lengh aos. 4. oncluson In hs wok, we show ha he nees of he chaacescs ehod odelng he MT lnes n fequency doan. Ths ehod can be noduced easly n ccu sulaos as Spce, Esacap and Sabe, bu s only vald n low loss MT lne n he e and he fequency doans. The fuue wok wll be devoed o he developen odel of boh eal losses usng a odel based on he Pade appoxaon, and he effec of dsubng EM wave dsubed on a MT lne. Refeences [1].W. HO, "Theoy and copue aded analyss of lossless anssson lnes", IBM. J. Res. Dev, ol. 16, No. 3, 1973, pp [2]. R. Paul, Analyss of Mulconduco Tanssson nes, Wley sees n Mcowave and Opcal Engneeng, Ka hang, Sees Edo, [3] A. ASSADI-HAGHI, "onbuon au développeen de éhodes d opsaon sucuelle pou la concepon asssée pa odnaeu de coposans e de ccus hypeféquences", Ph.D. ouncaons Opques e Mcoondes, Unvesé de oges Ecole Docoale Scences e Technques Faculé des scences e Technques XIM- Dépaeen Mnaco, [4] K. Afoo A. Abdpou, A. Tavakol and M. Movahhed, "Te-doan analyss of lossy acve anssson lnes usng FDTD ehod", In. J.oun. (AEU), ol. 63, No. 3, 2009, pp [5] Y. Mejdoub, H. Roujaa and A. Ghaa "Tansen analyss of lossy ulconduco anssson lnes odel based by he chaacescs ehod", Inenaonal Jounal of Engneeng and Technology IJET, ol. 3, No. 1, 2011, pp [6] F. H. Bann, "Tansen Analyss of ossless Tanssson nes", Poc. IEEE, ol. 55, No. 11, 1967, pp [7] H. ROUIJAA, "Modélsaon des gnes de Tanssson Mulconduceus pa a éhode des Appoxanes de Pade : Appoche ccu", Ph.D. Géne Elecque, Unvesé de Do d Econoe e des Scences d Ax- Maselle (Ax-Maselle III), Ma [8] A. Dounavs, A. Pohwala and A. A Beyg, "Passve acoodels of lossy ulconduceu anssson lnes based on he ehod of chaacescs", IEEE Tans. Mcowave Theoy, ol.32, No. 1, 2009, pp [9] M. Kane, Ph. Selne and Ph Auol, "Déenaon des paaèes lnéques des câbles ulflaes", n 6èe olloque Inenaonal EM-92, 1992,pp [10] M. KANE, "Modèles analyques ognaux pou la déenaon des paaèes lnéques des lgnes e câbles ulflaes pacouus pa des sgnaux lage bande", Ph.D. Géne Elecque, école Docoale de yon des Scences pou ngéneu : Eleconque ; Elecoechnque ; Auoaque, [11] S. BAZZOI, "aacésaon e Sulaon de la Suscepblé des cus Inégés face aux Rsques dinducons engendées pa des Mco-ondes de Foe Pussance", Ph.D Eleconque, Unvesé des Scences e Technologes de lle (lle-1), ocobe [12]. R. PAU, "Inoducon o Elecoagnec copably", wley sees n Mcowave and Opcal Engneeng, Ka hang, Sees Edo [13]. Inzol, H. Roujaa, G. Akoun and J. Robe "Epap2000- Esacap, un oul de sulaon négan le couplage chapccu", n 11èe olloque nenaonal & Exposon su la EM, Genoble, Youssef MEJDOUB was bon n Moocco, n He s eceved he DESA (6 yeas sudy afe he baccalaueae, equvalen o Mase) n Telecouncaons & Newok fo ad Ayyad Unvesy, Maakesh Moocco, n He s a PhD suden n Eleccal Syses and Telecouncaons aboaoy SET a ad Ayyad Unvesy Maakech - Moocco. Hs eseach nees ncludes he elecoagnec copably, ulconduco anssson lnes and elecouncaons. Mohaed SAIH obaned hs dploa n Eleccal Engneeng fo ad Ayyad Unvesy Moocco n He s a eseache ebe of he Eleccal Syses and Telecouncaons aboaoy SET and pepang hs docoae hess. Hs eseach nees ncludes he elecoagnec copably and ulconduco anssson lnes. Hcha ROUIJAA s a Pofesso of physcs, aached o ad Ayyad Unvesy, Maakesh Moocco. He s obaned hs PhD Thess on \" Modelng of Mulconduco Tanssson nes usng Pade appoxan ehod: cu odel \", n 2004, fo Ax- Maselle Unvesy - Fance. He s assocae ebe of Eleccal Syses and Telecouncaons aboaoy SET a he ad Ayyad Unvesy. Hs cuen eseach neess concen elecoagnec copably and ulconduco anssson lnes. Abdellah GHAMMAZ eceved he Doco of Eleconc degee fo he Naonal polyechnc Insu (ENSEEIHT) of Toulouse, Fance, n In 1994 he wen back o ad Ayyad Unvesy of Maakech Mooco. Snce 2003, he has been a Pofesso a he Faculy of Scences and echnology, Maakech, Mooco. He s a ebe of Eeccal Syses and Telecouncaons aboaoy SET a he ad Ayyad Unvesy. Hs eseach neess n he feld of elecoagnec copably, ulconduco anssson lnes, elecouncaons and anennas. opygh (c) 2013 Inenaonal Jounal of opue Scence Issues. All Rghs Reseved.

1 Constant Real Rate C 1

1 Constant Real Rate C 1 Consan Real Rae. Real Rae of Inees Suppose you ae equally happy wh uns of he consumpon good oday o 5 uns of he consumpon good n peod s me. C 5 Tha means you ll be pepaed o gve up uns oday n eun fo 5 uns