ELECTROMAGNETIC FIELD COUPLING TO ARBITRARY WIRE CONFIGURATIONS BURIED IN A LOSSY GROUND: A REVIEW OF ANTENNA MODEL AND TRANSMISSION LINE APPROACH

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1 D. Poljak t al., It. J. Comp. Mth. ad Exp. Ma., Vol., No. (3) 4 63 ELECTROMAGNETIC FIELD COUPLING TO ARBITRARY WIRE CONFIGURATIONS BURIED IN A LOSSY GROUND: A REVIEW OF ANTENNA MODEL AND TRANSMISSION LINE APPROACH DRAGAN POLJAK, KHALIL EL-KHAMLICHI DRISSI & BACHIR NEKHOUL 3 Dpartmt of Elctroic, Uivrity of Split, Split, Croatia. Pacal Ititut, Blai Pacal Uivrity, Clrmot-Frrad, Frac. 3 LAMEL Laboratory, Uivrity of Jijl, Algria. ABSTRACT Th papr dal with two diffrt approach for th aalyi of lctromagtic fild couplig to arbitrary wir cofiguratio burid i a loy mdium: th wir ata thory ad th tramiio li mthod. Th wir ata formulatio dal with th corrpodig t of Pockligto itgrodiffrtial quatio, whil th tramiio li modl i bad o th tlgraphr quatio. Th t of Pockligto quatio i olvd via th Galrki Bubov chm of th idirct boudary lmt mthod, whil th tlgraphr quatio ar tratd uig th chai matrix mthod ad fiit diffrc tchiqu, rpctivly. A umbr of illutrativ computatioal xampl prtaiig to burid multipl li ad groudig ytm i giv i th papr. Kyword: Burid wir, ata modl, tramiio li approximatio, t of pockligto itgrodiffrtial quatio, umrical olutio mthod. INTRODUCTION Th aalyi of lctromagtic fild couplig to thi wir cofiguratio burid i a loy mdium i importat i may lctromagtic compatibility (EMC) applicatio,.g. commuicatio ad powr cabl, gophyical ivtigatio, groudig ytm, tc. Thi problm ca b aalyzd i ithr frqucy or tim domai by uig th tramiio li (TL) modl, or thi wir ata thory (AT) (full-wav modl) [, ] with th lattr big coidrd a a mor rigorou o. Th TL approach i quit plauibl approximatio for log traight coductor with lctrically mall cro ctio but it i ot valid for fiit lgth wir, wir of arbitrary hap ad high frqucy xcitatio. Coqutly, AT ha to b ud. O th othr had, a pricipal drawback of th wir AT applid to burid coductor i rathr high computatioal cot. Uig hacd TL modl it i poibl to ovrcom om limitatio of th modl rtrictio. Thu, a rigorou rlatiohip btw frqucy domai TL quatio ad itgral rlatiohip ariig from th wir AT for th igl wir blow groud ha b rportd i [3]. Th compario btw frqucy domai wir ata modl ad TL modl prtaiig to a igl burid coductor ha b prtd i [4] ad, quit rctly, i [5]. Th aalyi ha b xtdd to multipl burid wir, a wll [6]. Th formulatio ud i [6] ari from th wir AT ad i bad o th t of th Pockligto itgro-diffrtial quatio for half-pac problm. Th TL modl dicud i [4 6] i rlatd to th tlgraphr quatio. Th t of th Pockligto quatio i umrically hadld via th frqucy domai Galrki Bubov chm of th idirct boudary lmt mthod (GB-IBEM) []. Th tlgraphr quatio, ariig from TL modl, ar tratd uig th chai matrix mthod [4 6]. 3 WIT Pr, ISSN: (papr format), ISSN: (oli), DOI:.495/CMEM-V-N-4-63

2 D. Poljak t al., It. J. Comp. Mth. ad Exp. Ma., Vol., No. (3) 43 Furthrmor, a trait aalyi of complx groudig ytm uig both AT ad TL approximatio ha b prtd i [7 9] for a igl horizotal groudig lctrod ad complx groudig ytm, rpctivly. Thi papr rviw th aalyi mthod of lctromagtic couplig to burid wir cofiguratio i th frqucy domai [4 ]. Th full wav approach ad approximat TL approach ar dicud throughout th papr ad a trad-off btw th tchiqu ha b trd out. Modlig of multipl burid coductor ad groudig ytm ar prtd i parat ctio. A umbr of illutrativ computatioal xampl prtaiig to burid coductor ad groudig ytm i giv throughout th papr. BURIED LINES Thi ctio compar th wir AT approach to th TL approach i th modlig of lctromagtic couplig to burid coductor i th frqucy domai. Th AT approach i bad o th t of Pockligto itgro-diffrtial quatio for arbitrary wir. Th prc of a loy half-pac i tak ito accout by ma of approximat rflctio cofficit (RC) approach []. Th rultig itgro-diffrtial quatio ar umrically olvd via a frqucy domai vrio of th GB-IBEM. Th TL modl i th frqucy domai i bad o th corrpodig tlgraphr quatio which ar tratd uig th chai matrix mthod. Th gomtry of itrt i rlatd to multipl horizotal coductor burid i a loy groud, a how i Fig.. Th prt aalyi dal with th frqucy rpo of burid coductor cofiguratio uig th AT ad TL approach, rpctivly.. Ata thory approach: t of coupld Pockligto quatio for arbitrary wir cofiguratio Th t of Pockligto quatio for multipl burid wir of arbitrary hap ca b radily drivd a a xtio of th Pockligto itgro-diffrtial quatio for a igl burid z r E rh ε, m y L ε, m, σ x h h h3 Figur : Th gomtry of horizotal burid li.

3 44 D. Poljak t al., It. J. Comp. Mth. ad Exp. Ma., Vol., No. (3) wir forcig th cotiuity coditio for th tagtial fild compot alog th thi wir urfac []. Th wir placd i a ifiit loy mdium i firt coidrd, ad th th formulatio i xtdd to a corrpodig half pac problm. Aumig th prfctly coductig wir, th total fild compod from th xcitatio fild E xc ad cattrd fild E ct vaih: E + E = xc ct ( ) o th wir urfac () whr i th uit vctor tagt at th obrvatio poit. Startig from Maxwll quatio ad th Lortz gaug, th cattrd fild ca b xprd i trm of th vctor pottial A : ct E = jwa+ ( A) jwm fc () Th magtic vctor pottial i dfid by th particular itgral: A m () I () g (, ) = d p 4 C (3) whr I( ) i th iducd currt alog th li, i th uit vctor tagt at th ourc poit ad g (, ) i th corrpodig Gr fuctio of th form: jkr g(, ) = (4) R whil R i th ditac from th ourc to th obrvatio poit, rpctivly: ( ) ( ) ( ) R = x x + y y + z z + a (5) whr a dot th wir radiu. Combiig q () (5) lad to th Pockligto itgro-diffrtial quatio for th ukow currt ditributio alog th igl wir of a arbitrary hap iulatd i a uboudd loy mdium [9, ]: xc E ( ) = I( ) k + g(, ) d j4pw ff C (6) Itgral quatio for a ifiit loy mdium (6) ca b xtdd to a ca of a thi wir locatd ar th itrfac btw two mdia by modifyig th krl to accout for th lctric fild rflctig from th itrfac. Th xcitatio fild compot th ca b writt a th um of th icidt fild ic E ad th fild rflctd from th itrfac rf E, i.. xc ic rf E = E + E (7)

4 D. Poljak t al., It. J. Comp. Mth. ad Exp. Ma., Vol., No. (3) 45 whil th rflctd fild rf E i giv by E rf ( ) = j4pw ff k k I ( ) * k + gi (, * ) d + k + k C + I( ) * G (, ) d C (8) Th Gr fuctio g i (,*) ariig from th imag thory i: g i (, * ) jkr = (9) R with R big th ditac from th imag poit to th obrvatio poit, rpctivly, * i th uit vctor tagtial at th ourc poit of th imag wir, whil k ad k ar propagatio cotat of air ad loy groud, rpctivly: k k = wm () = wm () ff Th complx prmittivity of th loy groud ε ff i giv by ff r j = () w whr r ad ar rlativ prmittivity ad coductivity of th groud, rpctivly, ad w i th opratig frqucy. Th krl G (, ) G, = G H G H G H G V G V r r + f f + + r r + (3) ( ) ( ) ( ) ( ) ( ) x z z z z z i a corrctio trm cotaiig th Sommrfld itgral ad ivolv th followig compot for horizotal ad vrtical dipol [, ]: G V r = k V r z V Gz = + k k V z R R (4) (5) H R R Gr = co f k V + k U r (6) H R R G = i k V + k U r r (7) H V Gz = j4pwfccof G r (8)

5 46 D. Poljak t al., It. J. Comp. Mth. ad Exp. Ma., Vol., No. (3) Th Sommrfld itgral trm ar: whr R U g z+ z D( ) J( ) d R V g z+ z D( l) J( ) d = l lr l l (9) = lr l l () D D ( l) k ( l) = g + g g + ( k k ) = ( ) kg + kg g k + k () () ad g = l k; g = l k (3) Fially, combiig q (), (6), (7) ad (8) yild th Pockligto itgro-diffrtial quatio for th ukow currt ditributio alog th igl wir ata of arbitrary hap burid i a loy groud I( ) k + g(, ) d + C xc k k E ( ) = I ( ) * k gi (, * ) d j4pw ff k + k C + I( ) * G (, ) d C (4) To driv th corrpodig t of coupld Pockligto itgro-diffrtial quatio for N W wir of arbitrary hap th ifluc of ach ata ha to b ummarizd, i.. it follow: I( ) m k + g( m, ) d + C NW xc k k Em ( m ) = + I ( ) m * k + gim ( m, *) d + j4pw ff = k + k C + I( ) m * G( m, ) d C m =,,..., N W (5)

6 D. Poljak t al., It. J. Comp. Mth. ad Exp. Ma., Vol., No. (3) 47 A approximat implifid form of Gr fuctio cotaiig rflctio cofficit, dducd from th rigorou approach ivolvig th Sommrfld itgral, for ovrhad wir ha b rportd i []. Similarly, for th ca of multipl burid wir th itgral quatio t (5) bcom [6]: L * k m gm( m, ) + m N w xc * Em ( m ) = RTM k m gim ( m, * ) j4 pw + + ff m * (6) = * + ( RTE RTM ) m pm k pm gim ( m, *) I ( ) d pm * m =,,..., NW whr R TM ad R TE ar th rflctio cofficit for th ca of travr magtic ad travr lctric polarizatio, rpctivly, giv by [6]: R TM = q co i q + q co i q (7) R TE = co i q q + q co i q (8) xc whr I ( ) i th ukow currt ditributio alog th -th wir, Em () th xcitatio fuctio o th m-th wir, g,m ( m, ) th fr pac Gr fuctio, whil g i,m ( m, *) ff ari from th imag thory ad =. Th pricipal advatag of th RC approach vru rigorou Sommrfld approach i implicity of th formulatio ad apprciably l computatioal cot. Grally, RC approach produc rult roughly withi % of tho obtaid via rigorou Sommrfld itgral approach [].. Ata thory approach: umrical olutio Th t of itgral quatio (6) i hadld via a fficit GB-IBEM. Th c of th mthod ha b prtd i dtail i []. Som pcial fatur rlatd to ioparamtric lmt implmtatio hav b dicud rctly i []. Th ukow currt I () z alog th -th wir gmt i xprd i trm of liarly idpdt bai fuctio f i, with ukow complx cofficit I i :

7 48 D. Poljak t al., It. J. Comp. Mth. ad Exp. Ma., Vol., No. (3) ad th u of ioparamtric lmt yild: T ( ) i i( ) { } { } i= I = I f = f I (9) T ( z) i i() z { } { } i= I = I f = f I (3) whr i th umbr of local od pr lmt. A liar approximatio ovr a wir gmt i ud i thi work ad th corrpodig hap fuctio ar giv by: z + z f = f = (3) a thi choic wa provd to b optimal for umrical tratmt of variou wir tructur []. Applyig th wightd ridual approach ad implmtig th Galki Bubov procdur th t of Pockligto quatio i traformd ito a ytm of algbraic quatio. Prformig om mathmatical maipulatio, th followig matrix quatio i obtaid: Nw N = i= df jm ( m ) dfi( ) gm ( m, ) d dm d m m d Δl Δl + k m f jm( m) fi() gm( m, ) d dm Δlm Δl df jm ( m ) dfi( ) (, * ) gim m d dm + k k d * lm l m d Δ Δ + + k + k + k m * fjm( m) fi( ) gim ( m, * ) d dm Δlm Δl + m fjm( m) fi() Gm( m, ) d dm Δlm Δl tr = j4 pw E ( ) f ( ) d ; m =,,..., N ; j =,,..., N ff m m jm m m W Δlm m {} I i (3) whr N m i th umbr of lmt o th m-th ata ad N i th umbr of lmt o th -th ata. Equatio (3) ca alo b, for covic, writt i th matrix form: Nw N [ Z] { I} = { V }, m =,,..., ; j =,,..., i m N ji j w Nm (33) = i= whr [Z ji ] i th mutual impdac matrix for th j-th obrvatio gmt o th m-th wir ad i-th ourc gmt o th -th ata.

8 D. Poljak t al., It. J. Comp. Mth. ad Exp. Ma., Vol., No. (3) 49 Th u of ioparamtric lmt rult i th followig xprio for mutual impdac matrix: d d Z D D g d T m m ( m, ) z [ ] = { } { } ji j i d d + k f f g d + T m m { } { } m( m, ) z j i T d d m { } { } D D gim ( m, * ) + j i k k + k + k T d d + m + k m * { f} { f } gim( m, * ) j i d d + f f G d T m m m( m, ) z { } { } j i (34) Not that matric {f} ad {f } cotai th hap fuctio whil {D} ad {D } cotai thir drivativ. Th voltag vctor i giv by: d V j E f d m N j N (35) { } m ic m = 4 pw ( ) ( ) j ff m m jm m dx xm, =,,..., W ; =,,..., m ad ca b valuatd i th clo form []..3 Tramiio li approximatio Voltag ad currt alog th multipl burid coductor how i Fig. iducd by a xtral fild xcitatio ca b dtrmid from th fild-to-tramiio li matrix quatio i th frqucy domai [5]: d V ˆ( x ) Z ˆ. I ˆ( x ) V ˆ F ( x ) dx + = (36) d I ˆ( x ) Y ˆ. V ˆ( x ) I ˆ F ( x ) dx + = (37) Th procdur for th amt of logitudial impdac matrix [ Z ] ad th travral admittac matrix [ Y ˆ] ar dicud i dtail i []. Th olutio of th frqucy domai TL quatio i bad o th chai matrix dicud i [3]. Th pr uit lgth paramtr R, L, C ad G of burid coductor ar valuatd uig th modal quatio availabl i [3] ad ar frqucy dpdt. Such a approach i mor accurat tha u of wll-kow Polaczck formula []. Th tadard TL ad modifid TL (MTL) [5] approach ar both ud, rpctivly. It i worth mphaizig that th TL thory ca hadl th problm of lctromagtic couplig dirctly i tim but without coidrig th impact of frqucy o th paramtr

9 5 D. Poljak t al., It. J. Comp. Mth. ad Exp. Ma., Vol., No. (3) pr uit lgth (i th xprio of Z impdac). Th dirct tim domai olutio of TL quatio rquir th umrical calculatio of a covolutio itgral which i a rathr tdiou tak a th corrctio trm iclud th fiit coductivity of th oil withi th impdac Z. Morovr, thi trm i ot availabl i th clod form ad th ivr Fourir traform (IFT) algorithm ha to b ud..4 Computatioal xampl Numrical rult prtd i thi ctio ar rlatd to variou cofiguratio of thr burid coductor. Th lctric fild of th tramittd pla wav xcitig multipl wir cofiguratio at crtai burial dpth z i [6]: E E tr jk z = Γ TM (38) whr E i th fild amplitud ad th poit of rfrc i locatd at th itrfac of two mdia z =. Not that E = V/m i all xampl to follow. Alo, all umrical rult obtaid via GB-IBEM (abbrviatd a BEM i figur to follow), tadard TL ad MTL, rpctivly ar compard to th rult calculatd via NEC (umrical lctromagtic cod) [4] with Sommrfld approach ad RC approximatio, rpctivly. Figur how th firt cofiguratio of itrt. Th radiu of all coductor i.5 mm, th ditac btw ighborig coductor i 6 mm, ad th burial dpth i m. Th umrical rult for th currt iducd at th ctr of th middl wir (cofiguratio No ) ar how i Fig. 3. Th lgth of coductor i 5 m ad th groud paramtr ar: =. S/m ad r =. Figur 4 how th currt iducd at th ctr of th middl wir (cofiguratio No ) for th coductor lgth of 5 m with highr groud coductivity ( =. S/m), whil th prmittivity i th am ( r = ). Th cod cofiguratio of itrt i how i Fig. 5. Th radiu of all coductor i.5 mm, th ditac btw ighborig coductor ar d = 36 mm, d = 8 mm, whil th burial dpth ar h = m, h =.97 m. Th umrical rult for th currt iducd at th ctr of th middl wir (cofiguratio No ) ar how i Fig. 6 for th coductor lgth of 5 m with th coductivity =. S/m ad prmittivity r =. Th umrical rult for th currt iducd at th ctr of th middl wir (cofiguratio No ) ar how i Fig. 7 for th coductor lgth 5 m. Th groud paramtr ar =. S/m ad r =. D h d d Figur : Cofiguratio No : D =.5 mm, d = 6 mm, h = m.

10 D. Poljak t al., It. J. Comp. Mth. ad Exp. Ma., Vol., No. (3) 5 5 I [ma] 5 5,,5,9,3,7,,5,9 f [MHz] Figur 3: Th frqucy rpo at th ctr of th middl wir (cofiguratio No, L = 5 m, =. S/m). I [ma] ,,5,9,3,7,,5,9 f [MHz] BEM MTL TL NEC-Som NEC-RC Figur 4: Th frqucy rpo at th ctr of th middl wir (cofiguratio No, L = 5 m, =. S/m). D h BEM MTL TL NEC-Som NEC-RC h d d d Figur 5: Cofiguratio No : D =.5 mm, d = 36 mm, d = 8 mm, h = m, h =.97 m.

11 5 D. Poljak t al., It. J. Comp. Mth. ad Exp. Ma., Vol., No. (3) 5 I [ma] 5 5,,5,9,3,7,,5,9 f [MHz] BEM MTL TL NEC-Som NEC-RC Figur 6: Th frqucy rpo at th ctr of th middl wir (cofiguratio No, L = 5 m, =. S/m). I [ma] ,,5,9,3,7,,5,9 f [MHz] BEM MTL TL NEC-Som NEC-RC Figur 7: Th frqucy rpo at th ctr of th middl wir (cofiguratio No 3, L = 5 m, =. S/m). Though all wavform ar alik, th magitud vary igificatly. Grally, th umrical rult obtaid via diffrt approach agr bttr for highr valu of groud coductivity ad logr wir. 3 GROUNDING SYSTEMS Thi ctio dal with a amt of th trait bhavior of diffrt groudig grid cofiguratio uig both AT ad modifid TL modl, rpctivly. Th AT approach i bad o th t of homogou frqucy domai Pockligto itgro- diffrtial quatio, with groud air itrfac ffct big tak ito accout through th xact Sommrfld itgral formulatio. Th t of homogou Pockligto quatio i olvd via th GB-IBEM [] faturig liar ioparamtric lmt. Fially, th corrpodig trait rpo i obtaid by ma of Ivr Fat Fourir Traform (IFFT) algorithm. Th MTL modl i bad o th corrpodig tlgraphr quatio. Th partial diffrtial quatio for th trait voltag, ariig from th tim domai TL quatio, i olvd via th fiit diffrc tchiqu dirctly i tim domai.

12 D. Poljak t al., It. J. Comp. Mth. ad Exp. Ma., Vol., No. (3) 53 Th phyical problm of itrt how i Fig. 8 i rlatd to th quar groudig grid rgizd by a lightig chal. Svral groudig grid cofiguratio with dimio varyig from m to 3 3 m, with or without additioal vrtical lctrod ar aalyzd i th papr. All grid coit of wir coductor with radiu a = 5 mm, ad burid at d =.5 m dpth. Figur 9 how variou grid cofiguratio. Two valu of oil coductivity ar coidrd: σ =. S/m ad σ =. S/m. I both ca rlativ prmittivity i ε r = 9. I all ca th currt ijctio poit i locatd at th ctr of th grid. 3. Ata thory approach: t of homogou Pockligto Itgro-diffrtial quatio for groudig ytm Th currt flowig alog th groudig grid ar govrd by th t of coupld Pockligto itgro-diffrtial quatio for wir of arbitrary hap [9]. Th full wav aalyi of groudig ytm xcitd by th currt ourc how o th lft-had id of xprio (5) vaih ad th rultig t of Pockligto quatio implifi rducig to th homogou o. air Ijctio of th lightig trok arth Squar groudig grid Figur 8: Squar groudig grid ubjctd to a lightig trok. Cofig. Cofig. Cofig. 3 m Cofig. 4 5 m m m 3 m Figur 9: Diffrt groudig grid cofiguratio.

13 54 D. Poljak t al., It. J. Comp. Mth. ad Exp. Ma., Vol., No. (3) I( ) m k + g( m, ) d + C k k + I( ) m * k + gim( m, * ) d + = + I( ) m * G( m, ) d C m =,,..., N NW = k + k C W (39) Not that th xcitatio i tak ito accout ito formulatio through th forcig coditio [5]: I = I (4) g whr I g dot currt grator ad I currt at th ijctio poit. Furthrmor, at a juctio coitig of two or mor gmt th cotiuity proprti of th lctric fild mut b atifid [6], which i govrd by applyig th Kirchhoff currt law: ad th cotiuity quatio: Ik = (4) k= I I I = = = (4) at juctuio at juctuio at juctuio Coditio (4) ur th dicotiuiti i charg pr uit lgth to b ruld out i paig from o coductor to aothr acro th juctio. O th othr had, at th coductor fr d, th total currt vaih. 3. Ata thory approach: th valuatio of th iput impdac pctrum Th iput impdac i giv by th ratio: Z i Vg = (43) I g whr V g ad I g ar th valu of th voltag ad th currt at th drivig poit. Oc calculatig th currt ditributio, a fdig poit voltag i obtaid by itgratig th ormal lctric fild compot from ifiity to th lctrod urfac: Vg r = Edl (44) It i worth otig that th dirct calculatio of (44) i vry tim coumig. O th othr had, by carfully chooig a itgratio path, th computatioal cot ca b apprciably

14 D. Poljak t al., It. J. Comp. Mth. ad Exp. Ma., Vol., No. (3) 55 rducd. Thu, i th ca of horizotal arragmt of wir, th bt path i vrtical, i.. ovr th z-axi. Th iput impdac pctrum i multiplid with th currt pctrum ad th frqucy rpo of groudig ytm i obtaid. Fially, th trait rpo i calculatd by ma of th IFT. 3.3 Ata thory approach: umrical olutio Th t of Pockligto itgro-diffrtial quatio (39) i umrically hadld via th GB-IBEM. Th boudary lmt olutio tchiqu ud i thi work i a xtio of th mthod applid to igl wir ca ad prtd lwhr,.g. i []. Udrtakig th procdur alrady prtd i Sctio II.B th t of Pockligto quatio (39) i trafrrd to th ytm of quatio [9]: N N = i= [ ] { } ji i w Z I =, m =,,..., Nw; j =,,..., Nm (45) whr [Z] ji i th mutual impdac matrix for th j-th obrvatio gmt o th m-th ata ad i-th ourc gmt o th -th ata, whr N w i th total umbr of wir, N m th umbr of lmt o th m-th coductor ad N th umbr of gmt o th -th coductor. Implmtatio of ioparamtric lmt yild th followig xprio for th mutual impdac matrix: d d Z D D g d T m m ( m, ) z [ ] = { } { } ji j i d d + k f f g d + T m m { } { } m( m, ) z j i T d d m { } { } D D gim ( m, * ) + j i k k + k + k T d d + m + k m * { f} { f } gim( m, * ) j i d d + f f G d T m m m( m, ) z { } { } j i (46) Matric {f} ad {f } cotai th hap fuctio whil {D} ad {D } cotai thir drivativ. Th xcitatio fuctio i th form of th currt ourc I g i tak ito accout a a forcd coditio at th crtai od i of th groudig ytm [9]: I =I (47) Th wir juctio ar tratd through th Kirchhoff currt law i it itgral ad diffrtial form, rpctivly, rlatd to th cotiuity of iducd currt ad charg at th juctio (4) (4). i g

15 56 D. Poljak t al., It. J. Comp. Mth. ad Exp. Ma., Vol., No. (3) 3.4 Modifid tramiio li mthod approach Withi th framwork of th MTL approach o glct th travr propagatio ffct th groudig ytm i imulatd by ma of a complx twork [9]. Th corrpodig couplig quatio for th calar pottial ad th currt i th tim domai for o-dimioal ca ar rportd by th Agrawal modl [7]: u (, l t) i(, l t) + Rilt (, ) + L = El (, lt) l t ilt (, ) u(, lt) + Gu(, lt) + C = l t (48) (49) whr l = x or y. E l (l, t) i th tagtial compot of th lctric fild xcitatio. Not that combiig th two tlgraphr quatio, (48) ad (49), th iducd currt or voltag ca b limiatd ad th cod ordr partial diffrtial quatio for ithr voltag or currt i obtaid. If th propagatio occur i two-dirctio; x ad y, th corrpodig two-dimioal partial diffrtial quatio for trait voltag i giv by: u u u u E E x + RGu ( RC + LG) LC = + x y t t x y y (5) whr th lctric fild cotitut th xcitatio ourc, ad R, L, C ad G ar pr uit lgth paramtr of th itrcoctd coductor. For th groudig grid, th pr uit lgth paramtr ar calculatd takig ito accout th oil air itrfac ffct. Thr ar variou approach for th amt of th paramtr,.g. uig th formula uggtd by E.D. Sud [8] or by Y. Liu [9]. 3.5 Fiit diffrc olutio of th pottial diffrtial quatio for trait iducd voltag Partial diffrtial quatio (5) i olvd umrically uig th fiit diffrc tchiqu. Th patial dicrtizatio of d ordr diffrtial oprator at crtai poit (i, j) uig th fiit diffrc approximatio i how i Fig.. Nd Nd Δ x u (i,j) u (i,j+) u (i-) u (i,j-) Δ y u (i+,j) Figur : Spatial dicrtizatio of th quar grid.

16 D. Poljak t al., It. J. Comp. Mth. ad Exp. Ma., Vol., No. (3) 57 Th fiit diffrc approximatio of patial ad tmporal drivativ at crtai poit (i, j) ca b writt, a follow: u = x Δx ( u ) ( u ) + ( u ) i+, j i, j i, j (5) u y = Δy ( u ) ( u ) + ( u ) ij, + ij, ij, (5) u t = Δt ( u ) ( u ) ij, ij, (53) u t = Δ t ( ) u ( u ) + ( u ) ij, ij, ij, (54) E x x = Δx ( E x) ( Ex) i+, j i, j (55) E y ( E y) ( Ey) y = Δy ij, + ij, (56) Subtitutig (5) (56) ito (5) yild th followig rlatio: ( RC + LG) LC RG ( Δx) ( Δy) Δt ( Δt) LC ( ) ( ) u ( Ex) ( Ex) Δt x i j i ( Ey) ( Ey) ( ) ( u ) ij, + ( ) + ( ) + ( Δx) ( Δx) ( Δy) u u u i+, j i, j i, j+ + = ( Δy) ( RC+ LG) 4LC ( u ) ( u ) ij,, k Δt ij, ( Δt) + + Δ + Δ y ij, + ij, +,, j (57) which ca alo b xprd i th matrix form: A A k A N U B A k Akk A kn U k = B k A A A U B N Nk NN N N (58)

17 58 D. Poljak t al., It. J. Comp. Mth. ad Exp. Ma., Vol., No. (3) whr [A] dot th cofficit matrix, [u ] i th ukow voltag vctor, [B] i th tir right-had id ad N i th total umbr of od. Th diagoal lmt of matrix [A] ar giv by: A kk ( RC + LG) LC = RG ( Δx) ( Δy) Δt ( Δt) (59) whil th o-diagoal lmt of matrix [A] ar a follow: A kl A kl = if i th adjact od k i x dirctio (6a) Δ ( x) = if i th adjact od k i y dirctio (6b) Δ ( y) Th lmt of vctor [B] ar: A = lwhr. (6c) kl ( RC+ LG) 4LC LC Bk = ( u ) + u + Δt ij, ( Δt) Δt ( ) ( ) ( E x) ( Ex) + ( E y) ( Ey) Δx i+, j i, j Δy ij, + ij, (6) Th olutio of th partial diffrtial quatio (5) rquir th kowldg of th coditio at th grid dg, a idicatd i Fig.. I th ca of a idirct lightig trik, at a crtai poit o th bordr of th grid Fig. th followig quatio i to b ud [7]: h (, ) ( ) (, ) z, u l t = G I l t Δ t + E (l t) (6) whr G i th quivalt coductac of corrpodig od, whil: I(l, t Δt) i th travral currt kow at itat (t Δt), ad E z i th z compot of th lctric fild i oil. Cll i ox dirctio L R z Cll i oy dirctio y x G C a- bordr of th grid. b- rprtatio by π-cll Figur : (a) Coditio at th grid dg; (b) Equivalt lctrical twork of groudig grid.

18 D. Poljak t al., It. J. Comp. Mth. ad Exp. Ma., Vol., No. (3) 59 V (Volt) igma=. AM MTL AM MTL AM 3 MTL 3 AM 4 MTL t () x 6 Figur : Trait fdig-poit voltag for dry oil. Th prc of a two-mdia cofiguratio i tak ito accout calculatig th liar paramtr of th lctrical circuit for th ca of th groudig lctrod [8, 9]. Such a tratmt of o-homogou mdia i idtical to th ca of TL with groud rtur path. At ach calculatio tp, th dtrmiatio of iducd voltag provid th valuatio of th currt iducd alog itrcoctd coductor of groudig grid by umrical itgratio of th tlgraphr quatio (48). Not that th ca of th dirct impact of th lightig trik o th prtd procdur i ud by imply impoig E =. Thrfor, i thi ca th voltag diffrtial quatio (q (5)) i th frqucy domai bcom: u u + RG+ jw ( RC+ LG) LCw u = x y (63) whr u i th pottial i ay poit of th grid. I thi ca, at th ijctio od th valu of th currt i kow (lightig trik grator) providig th amt of th corrpodig voltag. Th iput impdac i dfid a follow: whr k i th ijctio od idx. 3.6 Numrical rult Z i ukt (,) = (64) Ikt (,) I all computatioal xampl th lightig currt i xprd by th doubl xpotial fuctio with paramtr: I =.43 ka, a =.794 6, b = Figur how th trait voltag at th fdig poit calculatd by AT ad TL approach, rpctivly, for all four grid cofiguratio cario ad oil coductivity σ = ms/m, whil Fig. 3 how th rlatd trait impdac of th groudig ytm. Th rult obtaid by diffrt approach for grid typ ad, agr rathr atifactorily, ad rlativly good agrmt ca b alo obrvd for typ 3, whil major diffrc

19 6 D. Poljak t al., It. J. Comp. Mth. ad Exp. Ma., Vol., No. (3) occur for typ 4 grid. Som diffrc appar at vry arly tim itat (from 8 to 7 ) corrpodig to th high frqucy cott of th iput igal pctrum, which caot b accuratly prdictd by th MTL mthod. Coqutly, thr ar om diffrc i rathr arly tim itat, a i viibl from Fig., i.. MTL fail to accuratly prdict th arly tim bhavior of groudig grid. A imilar cocluio ca b draw for th trait impdac, how i Fig. 3, a wll. Gratr th grid iz, th wor agrmt btw th rult i achivd. Th MTL mthod would b xpctd to work bttr if th wir ar logr, but it i ot th ca i thi particular grid cofiguratio. Such a bhavior ca b xplaid i du to th fact that th part of th grid bhav a igl ata ad thr ar may rflctio from dicotiuiti which MTL fail to tak ito accout. Thi ffct i mor vidt for low coductiv oil tha for highr o ad what ca b i Fig. 4 ad 5, prtig th trait voltag ad impdac at ijctio poit calculatd for groud coductivity σ =. S/m. Th agrmt btw th rult obtaid via diffrt approach i foud to b atifactory, particularly for latr tim itat which corrpod to lowr frqucy part of th Zt (Ω) igma=. AM AM MTL MTL AM 3 MTL 3 AM 4 MTL t () x 6 Figur 3: Trait impdac for dry oil. V (Volt) igma=. AM MTL AM MTL AM 3 MTL 3 AM 4 MTL t () x 6 Figur 4: Trait fd-poit voltag for wt oil.

20 D. Poljak t al., It. J. Comp. Mth. ad Exp. Ma., Vol., No. (3) 6 Zt (Ω) igma=. AM MTL AM MTL AM 3 MTL 3 AM 4 MTL t () x 6 Figur 5: Trait impdac for wt oil. Zf (Ω) cofig cofig cofig 3 cofig 4 igma= f (Hz) Figur 6: Th impdac pctrum for σ =. S/m. pctrum. For vry arly tim itat th rult ar alik, although th trait voltag pak, for all ca of grid cofiguratio ar omwhat highr. Comparig th rult for both valu of groud coductiviti it ca b oticd that th pak valu of th voltag i advacd for th ca of highr coductivity. Alo, th valu of th pak voltag ar prtty much alik rgardl of th grid iz. It i wll-kow that for vry arly tim itat th highr frqucy part of th impdac pctrum i importat. A th grid dity (mh) rmai th am for all grid cofiguratio that part of th frqucy pctrum i uchagd rgardl of th grid iz, a how i Fig. 6. Dpdig o th coductivity that part of pctrum tart at diffrt frqucy valu. I th ca of σ =. S/m th frqucy i about 3 MHz (Fig. 6), whil for σ =. S/m it i abov 3 MHz (Fig. 7). Thu, o coclud that i high coductivity viromt ffctiv lgth of th groudig wir bcom vry hort at highr frquci. Thrfor, vry arly tim bhavior i idtical for all cofiguratio ic th vry high frqucy pctrum part i prtty much th am. Alo, it hould b otd that th aalyi of rult prtd i Fig. 6 ad 7, rpctivly i rlatd to th rult obtaid by applyig th ata approach oly, a th MTL rult ar obtaid dirctly i tim.

21 6 D. Poljak t al., It. J. Comp. Mth. ad Exp. Ma., Vol., No. (3) Zf (Ω) igma=. cofig cofig cofig 3 cofig f (Hz) x 7 Figur 7: Th impdac pctrum for σ =. S/m. 4 CONCLUDING REMARKS Th prt papr rviw th modl ad mthod for th aalyi of th lctromagtic fild couplig to burid wir tructur. Thu, th multipl burid li ad groudig ytm hav b aalyzd by ma of both th wir ata modl ad (TL) mthod. Th ata modl i bad o th t of coupld Pockligto quatio for wir of arbitrary hap, whil th TL modl dal with th corrpodig t of tlgraphr quatio. Th tadard TL approximatio ad modifid TL approach ar both ud. I th ca of groudig ytm th t of Pockligto quatio i homogou. Th t of Pockligto itgro-diffrtial quatio for burid li ha b olvd via th GB-IBEM, whil th TL quatio ar tratd uig th chai matrix mthod. Th obtaid umrical rult hav b compard to th rult obtaid via th TL ad MTL approach, a wll a th umrical lctromagtic cod (NEC). Th umrical rult for burid multipl li obtaid via diffrt approach agr mor atifactory for highr valu of groud coductivity ad logr wir. Th t of homogou Pockligto quatio for groudig ytm i olvd via th GB- IBEM, a wll. Th corrpodig MTL quatio ar olvd uig th fiit diffrc mthod. Obrvig th obtaid umrical rult for groudig ytm, o coclud that MTL mthod fail to prdict accurat rult for th vry arly tim itat of th trait impdac, particularly for lowr valu of oil coductiviti. At latr tim itat, a good agrmt btw th mthod ca b oticd, although diffrc ar highr a th grid iz icra for th ca of low coductivity oil. For th highr coductiviti cario, a agrmt btw rult obtaid via diffrt mthod i rathr atifactory. REFERENCES [] Tch, F., Iaoz, M. & Carlo, F., EMC Aalyi Mthod ad Computatioal Modl, Joh Wily ad So: Nw York, 997. [] Poljak, D., Advacd Modllig i Computatioal Elctromagtic Compatibility, Joh Wily ad So: Nw York, 7. doi: [3] Poljak, D. t al., Gralizd form of tlgraphr quatio for th lctromagtic fild couplig to burid wir of fiit lgth. IEEE Tra. EMC, 5(), pp , 9. [4] Poljak, D., Doric, V., Sic, S., El Khamlichi Drii, K. & Krroum, K., Elctromagtic fild couplig to burid wir: compario of frqucy domai wir ata ad tramiio li modl, ICECom 7. Cofrc Procdig, 9th Itratioal Cofrc o Applid Elctromagtic ad Commuicatio, pp. 3 6, 7.

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