Multiconductor Transmission Lines and Cables Solver, an Efficient Simulation Tool for PLC Networks Development

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1 Multondutor ranssson nes and Cables Soler, an Effent Sulaton ool for PC Networks Deelopent heophans Callaoudas* and Fawz ssa* * Researh and Deelopent Departent, Eletrté de Frane, F-94 Claart, Frane Eal : heophans.callaoudas@edf.fr, Phone : , Fax : Eal : Fawz.ssa@edf.fr, Phone : , Fax : Abstrat n ths paper, we present an effent tool for the sulaton of lnes or ables n a PC network. he presented sulator s based on the transsson lnes theory and supports an open lbrary of dfferent types of two- or ultondutor lnes and ables used n low oltage () or edu oltage (M) networks. hs sulator helps PC syste desgner to estate at any pont n the lne (or the able) and for any gen frequeny rual paraeters lke sgnal attenuaton and rosstalk effet for any wre and any ode of the lne (or the able). Keywords Power lne ounaton, transsson lne theory, ultondutor, rosstalk, attenuaton, pedane sath, nueral results.. NRODUCON Before tryng to ontrol sgnal propagaton on a oplete PC network, t s neessary to know and to understand sgnal propagaton on straght lne or able segents onsdered as the bas eleents of the whole network. hs s ery useful f we want to predt the behaor of the lne for dfferent frequenes and/or at dfferent ponts. he sulaton tool we propose s a Multondutor ranssson nes Soler for PC sgnals fro the [ MHz - 3 MHz] band on ultondutor or M lnes. A frendly graphal user nterfae allows the settng of eletral boundary ondtons and physal paraeters of the lne to sulate. he user an also hoose the type of sulaton results.e. sulaton outputs ay be urrents or oltages n the phase or odal spaes. After a theoretal fraework dealng wth the resoluton of the ultondutor transsson lnes equatons (see seton ), we present the algorth of our sulator step by step (see seton ). n a last seton, we propose soe nueral sulatons. We hose two representate ases to llustrate our purpose : a twsted sngle phase oerhead lne and a 3-phase underground able.. HEORECA BASS We wll not present here an extended theoretal bass but we wll ntrodue only the an theoretal ponts pleented n our sulator. he dfferental equatons to sole for a sple transsson lne are gen by the well known relatons fro equatons () and () (see e.g. []). () () where and are for the lne pedane and adttane, respetely, x denotes the propagaton dreton. he general for of the solutons are gen aordng to equatons (3) and (4). γ x γ x where ( x) e e γ x γ x ( x) e e ( ) C (3) (4) C / Y s for the haraterst pedane, γ denotes the propagaton onstant, and depend on the boundary ondtons. Hene, for a two-ondutor lne (phase ondutor and neutral ondutor taken as oltage referene), the proble s lassal and the soluton tral. Neertheless, for an N- ondutor lne (wth N>) the proble s uh ore oplated. he for of the equatons s the sae but beause of the N utual pedanes and adttanes between the dfferent ondutors, and beoe NxN atres whereas and beoe Nx etors. ndeed, we hae the generalzed for of the transsson lnes equatons gen by equatons (5) and (6). Y (5) (6)

2 After deraton of both these equatons and puttng the results utually of eah one to the other, we obtan equatons (7) and (8). ( ) ( ) (7) (8) Consderng the aboe syste, eah dfferental equaton ontans oltage (or urrent) ters ssued fro dfferent ondutors. he only way to sole the s to dagonalze and Y atres. hs an be done by usng lnear transforaton atres, let us all the and, so that equaton (9) s heked. ( ) ( ) (9) and Y atres hae the sae egenalues. s a dagonal atrx defned by equaton () and ontanng all the propagaton egenodes on ts dagonal. to sulate, aong dfferent types proposed wthn a physal support lbrary (underground able, M oerhead lne, oerhead lne wth bare or twsted wres, et). Step : For a gen frequeny (to be defned by the user) and, f oerhead lnes, for a gen ground ondutty (to hoose aong preset alues or to defne by the user), alulate prary paraeters of the lne. Mrror ages odfed theory (see [,3]) s used when ground s taken as oltage referene. For underground ables odellng, we nuerally soled aplae s equaton. Step 3 : Defne the length of the lne and the nuber Np of spae saplng ponts. Hene, we wll hae a oputng step of /(Np-). Step 4 : Defne the boundary ondtons.e. oltages n the nput of the lne (on eah one of the three phases) and ternal loads. Step 5 : Calulate egenetors for urrents and oltages after odal analyss and resoluton of the transsson lne equatons n the odal spae. he ntegraton onstants are deterned usng the boundary ondtons. Step 6 : Calulate urrent and oltage for eah phase usng the results of step 5. hs s done by applyng lnear transforaton operators thus allowng a base hange fro odal spae to phase spae (see opanon paper [3] and equatons (9)-()). γ r () γ N he soluton has the general for of equatons () and () n phase spae. x ( e e ) x ( e e ) ( x) () ( x) () t s nterestng to note that equatons () and () put n lght the obnaton of two knds of wae, soe progresse and regresse ones. he ehans of the odal analyss s the followng (see detals n appendx) : equatons (5) and (6) are rewrtten n odal spae usng and. Hene, odal urrents and oltages are alulated. Fnally, phase urrents and oltages are obtaned thanks to nerse lnear transfors. Moreoer, urrents and oltages are lnear obnatons of the exted odes dependng on lne haratersts and boundary ondtons.. HE AGORHM he algorth s pleented usng the sx opleentary followng steps. Step : Defne the physal paraeters of the lne to sulate.e. hoose the type of the lne Fg.. Graphal user nterfae of the sulator.. NUMERCA SMUAONS A. wsted -phase oerhead lne et us onsder the followng nput data: ne type : a twsted one-phase able of x 6 ² ooper ondutors, Ground ondutty :.5 S/ Frequeny : 5 MHz njeton : n dfferental ode oad : 5 Ohs between the two ondutors (phase-neutral) Neutral-ground onneton : through a Ohs resste load. hs lne wll produe two propagaton egenodes (dfferental and oon odes). We obtan after sulaton the followng results for odal oltages :

3 Dff Modal oltages [dbµ] ogether Modal oltages [dbµ] Co x [] Fg.. Modal oltages. Note the leel dfferene between the dfferental ode really exted and the oon ode ndued. he sulator operates the lnear transforaton to obtan oltages n the phase spae. Neutral Phase Phase oltages [dbµ] x [] Fg. 3. Phase and neutral oltages n the lne. Note the standng wae phenoenon due to the ternal pedane sath. B. Underground 3-phase able Cable type : HN33S33 (underground) Frequeny : 5 MHz ne length : njeton : between neutral and phase oads : 5 Ohs between neutral and phase, Ohs between neutral and phase, 7 Ohs between neutral and phase 3. hs able leads to three propagaton egenodes naed together, syetr and ross desrbed n the opanon paper [3]. After odal analyss the sulator ges the followng results for odal oltages : Syetr Cross x [] Fg. 4. Modal oltages. he oltages aross the able for eah one of the 3 phases wth referene to the neutral are gen below. Phase Phase Phase3 4 Phase oltages [dbµ] x [] Fg. 5. Phase oltages wth referene to the neutral ondutor. Note oltages on phase and phase3 ondutors. hey are obtaned by rosstalk. hs s why they are lower than phase oltage (only phase s really fed). he leel dfferene between phase and phase3 s due to asyetrally dstrbuted loads.. CONCUSONS We presented an effent tool for PC lnes or ables sulaton. hs sulator s partularly useful beause t helps PC syste desgner to estate at any pont n the lne and for any gen frequeny rual paraeters lke sgnal attenuaton and rosstalk effet on eah one of the wres and for eah propagaton egenode.. APPENDX n ths appendx, we analytally present how to pass fro the phase spae to the ode spae, sole the lne propagaton proble and fnally pass n the phase spae. et us reall equatons (7) and (8). ( ) (7)

4 ( ) (8) Y (8) he goal of the odal analyss s to obtan a dagonal for of and Y usng lnear transforatons. Hene t wll be possble to get the expressons of odal oltages and urrents at any pont of the lne. et us all and the atres of the lnear transforaton so that ( ) and ( ). he two atres and Y hae the sae egenalues sne and are syetral.e. ( ) and Y ( ). he propagaton onstants are the square root of eah dagonal ter of. So atrx ontans on ts dagonal all the propagaton egenodes (see equaton ()). γ () γ N Equatons (7) and (8) are soled n the odal spae. Usng and lnear transforaton atres, oltage and urrent etors n the odal spae are gen by equatons (3) and (4). (3) (4) Equatons (5) and (6) rewrtten n the odal spae yelds equatons (5) and (6). Y he dagonal atres Y Y (5) (6) and are alled pedane atrx and adttane atrx respetely. t s easy to show that ther produt ges the atrx aordng to equaton (7). Y Y (7) We an fnally dere the atrx of the haraterst pedane of the odes as gen by equaton (8). Moreoer, equaton (9) shows the exstng relatons between the odal haraterst pedane and adttane atres. Y Y (9) Fro (5)-(7), we obtan dfferental equatons () and (). hese are the equalent of equatons (7) and (8) n the odal spae. N () () As atrx s dagonal, dfferental equatons () and () an be easly soled and hae general soluton fors gen by equatons () and (3). x ( x) e e () x ( x) e e (3) Fro (), (3), (5) and (6), we an dere equaton (4). x x ( e e ) Y ( e e ) (4) Besdes, the ntroduton of the dagonal atrx of odal haraterst adttane atrx Y leads to equaton (5). x ( e e ) x e e Y (5) An dentfaton of progresse and regresse ters of equatons (5) yelds equatons (6) and (7). Y (6) Y (7) So we obtan for the relaton gen by equaton (8) where only and our. x ( e e ) ( x) Y (8)

5 At x (nput of the lne), the oltage etor (known boundary ondtons n the phase spae) s related to the equaton () through the lnear transforaton atrx as shown aordng to equaton (9). ( ) ( x ) (9) On the other hand, at the end of the lne (for x ) the oltage etor s lnked to the urrent one through the onneted loads atrx aordng to equaton (3). ( x ) ( x ) (3) By replang n ths equaton and wth ther expressons gen by () and (8) we an dere a relaton between and : ( Y ) ( Y ) e (3) We an defne the atrx S defned by equaton (3). ( Y ) ( Y ) S (3) We an rewrte equaton (3) as : Se (33) Fnally, fro (9) and (33) we obtan after operatons the oplete expresson of : ( Se ) Se (34) and we dere the followng relaton for : (35) Equatons (34) and (35) are portant sne they allow the alulaton of oltages and urrents at any pont of the propagaton axs n the odal spae (see equatons () and (8)) and n the phase spae (see equatons (3) and (4)). t s nterestng to note that S atrx s dagonal and eah dagonal ter s represents a reflexon oeffent defned n the odal spae. Atually, for a sngle ode syste S beoes a salar and an be wrtten aordng to equaton (36). ( Y ) ( Y ) s (36) By rearkng that Y /,we dere the well known followng relaton for s oeffent : s hs s oherent wth our atrx for generalzaton. he algorth fnally wrtes :. For a gen lne or able known. and Y are Y, we obtan. By dagonalzaton of (seondary paraeters) usng the lnear transforaton atres and. 3. We alulate S atrx usng equaton (3). 4. he boundary ondtons defned n phase spae (ternal loads and nput oltages ) are used for the alulaton of and usng equatons (34) and (35). 5. We alulate the solutons and of the dfferental equatons for oltages and urrents n odal spae usng equatons () and (8). 6. We ultply the by the lnear transforaton atres and to obtan oltages and urrents and n the phase spae : (37) (38). REFERENCES [] P. Cobes, ranssson en espae lbre et sur les lgnes, Edtons Dunod, 983. [] C. Gary, Approhe oplète de la propagaton ultflare en haute fréquene par utlsaton des atres oplexes, EDF Bulletn de la DER, no. ¾, pp. 5-, 986. [3] F. ssa, An effent ool for Modal Analyss of Multondutor ranssson nes for PC Networks Deelopent, Pro. nternatonal Syposu on PC and ts Applatons, Athens (Greee),.