UNVESTÀ EG STU OOGN PTMENTO NGEGNE EETTC Val orgno n - 4 OOGN (T N NYTC SOUTON FO THE CUENT STUTON N TWO -STNS UTHEFO CES COUPE WTH ESSTVE JONT M. Fabbr brac Ulzng gorcal ror of auo/uual nducon coffcn ar aong rand of a -rand urford cabl, oluon for gnral lnar ca of wo cabl conncd roug a r jon gn. PPOTO NTENO EU SETTEME
nd. nroducon..... Cabl Modl.... 5. Jon Modl.... frnc..... nroducon Condr, a own n fgur, wo ual -rand cabl of lng conncd roug a r jon. Moror, au a flowng currn ually drbud on r nu rand and on r ouu rand. (/ (/ (/ NPUT CE ( ( ( ( ( JONT ( ( ( ( ( OUTPUT CE (/ (/ (/ T longudnal ranc (r of rand an rnal olag ( ald o cabl ar aud o b zro. T currn and olag on rnal bwn nu cabl and jon and bwn jon and ouu cabl ar labld w and, rcly.. Cabl Modl Condr, a own n fgur, nu -rand cabl w flowng currn ually drbud on nu rand and forally nown on ouu rand. (/ (/ (/ ( ( ( T longudnal ranc (r and rnal olag ( ar aud o b zro. Conunly, currn flowng n -rand cabl ar dcrbd by followng y: E/UNO/ Sbr d
(P [ G][ M] (, (, (, (, (, (, (, ( ( w,, >, < < w ( and (, and wr [M] and [G] ar followng conan-coffcn crculan yrc ar: [ M ] [ G ] g g g g g g g g g fnng oronoral cral ba b, w,,, a a for [M] and [G]: b ar [M] and [G] can b wrn a b / / b T T T T T T [ M] λ b b λbb λ b b [ G] b b bb b b w gnalu gn by λ λ λ g. - T doan Currn coo and a follow No a (coably condon Probl (P ad a: (P (, (, b (, b (, b ( ( b ( b ( b [ ] ( T ( b ( ( ( ( [ G][ M] (, (, (, (, ( (, ( b w >, < < E/UNO/ Sbr d
fnng g (P u u u u u u (, (, [ G][ M] (, (, [ G][ M] (, (, (, u ( ( b b ( ( w >, < < wr r dno dffrnaon w rc o argun. coong u a follow u (, µ (, b µ (, b (, b Probl (P u ly ddd n followng r robl: (P u µ µ (, µ (,, µ (, µ (, w >, < < (P u λ µ µ µ (, (, λ ( (,, µ (, µ (, w >, < < (P u µ λ µ µ (, (, λ ( (,, µ (, µ (, Probl (P u a l oluon (, µ ± ± λ ± µ ± w >, < < µ, wl robl (P u± can b wrn coacly a: µ ± (, (, λ ( (,, µ (, µ (, ± wc a bn alrady rad [] and g: ± ± µ ± ± wr Γ a wo ualn rrnaon []: ± ± ξ, (, dξ dτ ( τ Γ(, ξ τ w >, < < or Γ (, ξ; n nπ λ nπ nπξ n n E/UNO/ Sbr d
Γ λ (, ξ; 4 π n λ ξ n λ ξ n (T nd a bn drod fro gnalu nc λ λ and. Tu nc ( µ (, T (, b ( (, ( ( ( (, wc l (, (, b ( (, ( T ± ± ± ± ± µ (, ( ( ± ± ξ (, ± ( dξ dτ ( τ Γ(, ξ, τ w >, < < and currn n nl cabl ar (for >, < < : (, ( [ ( b ( b ] dξ dτ ( τ b ( τ ξ [ ] Γ(, ξ τ b b, No a currn n nl cabl ar colly nown f funcon ± ar aalabl. For wa concrn currn n oul cabl, nc forally yrc o nl on, oluon for, and for ± alo, can b oband cangng w and condrng nad of, a follow (* : ± ± (, ξ (, ± ( dξ dτ ( τ Γ(, ξ, τ w >, < < ( and currn n oul cabl ar (for >, < < : (, ( [ ( b ( b ] dξ dτ ( τ b ( τ ξ [ ] Γ(, ξ τ b b, gan, no a currn n oul cabl ar colly nown f funcon ± ar aalabl.. - T doan Volag (* T corrondnc ly donrad, nc dfnng ', on bco ', on bco ', and dra rul o b: ladng uaon naran and boundary condon corrcly ac. and. Tu E/UNO/ Sbr 4 d
For wa concrn olag n nl cabl, ang no accoun a and dcoong a follow followng rlaon ar oband: Moror, nc ψ ψ w >, < < [ G] (, (, (, ψ (, b ψ(, b ψ (, b ± ± ψ ± (, (, w >, < < (, b (, [ (, (, ] T T (, b (, (, (, (, (, (, (, and ( (,, ( (, olag a nd of nl cabl can b ad a: ( ψ (, ( ψ (, ψ (, Now, nc olag ( and ( a nd of nl cabl ar gn n r of ψ ± (,, a ar rlad o aal dra of ± (, wc ar dndn fro currn a nd of nl cabl (rd n r of ± (, aarn a obl o oban a olag- currn caracrc of nl cabl, a can b n a an ac rolar coonn. For wa concrn olag n oul cabl, nc forally yrc o nl on, oluon for ( and (, can b oband cangng w and condrng nad of, a follow: ( ψ (, ( ψ (, ψ (, (wr ψ ± a o b aluad rfrrng o aal dra of ± aluad for oul cabl To oban olag-currn caracrc for nl cabl, dfn: ξ (, ξ Γ( ξ d,, Tu, ubuon g: w >, < < E/UNO/ Sbr 5 d
± ψ ± ± (, (, ( dξ dτ ( τ Γ(, ξ, τ ± ± ( dτ ( τ (, τ τ [ ] ( dτ ( τ (, τ ( τ ± ± and, nc ± (, g: ± ± ± ξ τ >, < < ± (, τ ( τ ( τ d, w >, < < ψ ± ± Moror, n a way olag-currn caracrc for oul cabl, ar oband: ± (, τ ( τ ( τ d, w >, < < ψ ± ± ( gn du o -dffrnaon: foono a. 4 Fnally, olag a nd of nl cabl can b ad a:, ( dτ ( τ ( τ [ ] ( τ ( dτ ( τ ( τ, (wr nd on a bn drod for bry and olag a bgnnng of oul cabl rul o b:, ( dτ ( τ ( τ [ ] ( τ, ( dτ ( τ ( τ. - alac doan Currn wa own, currn flowng n nl -rand cabl ar dcrbd by followng robl (P: (P [ G][ M] (, (, (, (, ( b (, ( E/UNO/ Sbr d w >, < <
coo and a follow w coably condon (, (, b (, b (, b ( ( b ( b ( b [ ] ( T ( b ( ( ( ( Probl (P ly ddd n followng r robl: (P (, (P λ, (P,, λ (,, (, (, ( (, ( ( (,, (, ( (, (, (,, (,, (, ( Probl (P a l oluon (, (P ± ± ± λ ± ± ± (, (, ( w >, < < w >, < < w >, < <, wl robl (P ± can b wrn coacly a: (,, (,, (, ( lyng alac ranfor o robl (P ± g: (P ± ± ± λ ± ± ± ± ± (, (, (,, (, ( ± ± ± n wc can b aly old o g: (, ( Condr now oluon for ± (, gn bfor: ± ± n ± ( ± λ ± ( λ ± ± ± w < < w >, < < ξ (, ± ( dξ dτ ( τ Γ(, ξ, τ w >, < < Nong a ngraon of conoluon nd can b aly alac-ranford: E/UNO/ Sbr 7 d
ξ ξ ± ± ± ± (, ( ( dξ Γ(, ξ, ( dξ Γ(, ξ, y coaron can b dducd a n( λ (, ξ, n( λ ξ dξ Γ wr nd a bn drod for bry. To oban alac-ranfor of caracrc funcon, dffrna la uaon w rc o : Tang no accoun dfnon and nc { } / wr (, { (, }, rul: co( λ (, ξ, λ n( λ ξ ξ Γ d ξ (, ξ Γ( ξ d,, ( λ ( λ. Tu, fnally (, co λ n (, co( λ co( λ (, λ (, λ n( λ n( λ.4 - alac doan Volag wa own bfor, olag a nd of nl cabl and a bgnnng of oul cabl can b ad a -conoluon. Trfor, y can b aly ranford n alac doan: ( ( (, [ ] (, ( ( ( ( ( (, [ ] (, ( ( (.5 Pror of funcon wa own bfor, currn-olag caracrc of nl and oul cabl ar uarzd n funcon. n con o ror of funcon ar gn. Sarng fro dfnon: Γ (, ξ; n nπ λ nπ nπξ n n ( < ξ < and < < and ang no accoun a ξ (, ξ Γ( ξ d,, E/UNO/ Sbr 8 d
ξ nπξ dξ n ( nπ nπξ nπξ nπξ n co co nπ ( nπ g λ ( λ (, co co nπ co co( nπ n nπ nπ n nπ nπ, < < n ordr o oban a dffrn (and w far conrgnc rrnaon of funcon condr a co ( co( [ co( co( ] and a llc a funcon ϑ ad followng rrnaon (wr uaon r ar non-ocllang[]: Trfor ϑ ( u, n n co nu π n ( u nπ a g (, n ϑ π λ π nπ λ n, con π π λ λ n n ϑ π λ π n nπ λ, π λ con π λ n ( λ λ λ n n,, < < π n n arcular, (, bco: ( λ, (, n nπ Corrondngly, n alac doan (, n (, nπ λ (, n λ π n λ ( n a followng r dffrn rrnaon []: λ λ n ( λ co ( λ, n λ ( E/UNO/ Sbr 9 d
. Jon Modl. T doan Modl Condr, a own n fgur, r jon w flowng currn nown forally on nu and on ouu rand. ( ( ( ( ( ( ( ( ( ( T KC on cnral nod g: and, dfnng ( ( ( ( ( ( ( ( ( ( ( ( ( [ Ω] KC can b coacly wrn a: Now no a ( ( ( [ Ω] ( [ Ω ] b b [ ] Ω b b b [ ] Ω b b b coong, and a follow and nong a (coably condon followng uaon old: ( ( b ( b ( b ( ( b ( b ( b ( ( b ( b ( b ( ( ( E/UNO/ Sbr d
E/UNO/ Sbr d ( ( ( ( ( ( ( ( ( ( T y can b aly old for ±, ladng o: ( ( [ ] ( ( [ ] T KT on lf and rg d of jon g: ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( and, dfnng ( ( [ ] [ ] [ ] { },, dag, KT can b coacly wrn a: ( [ ] ( [ ] ( ( [ ] ( [ ] ( Moror, no a [ ] [ ] T T b b ( ( ( ( ( ( T la rory ld u o wr: ( b b ( a ru for Tu, dcoong and a follow ( ( ( ( β β β b b b ( ( ( ( β β β b b b can b aly ron n β β β β β β Fnally, dfnng [ ] T, b b ρ, w,,,, and,,
KT can b wrn a: β ρ, ρ, ρ, ρ, ρ, ρ, β ρ, ρ, ρ, ρ, ρ, ρ, β ρ, ρ, ρ, ρ, ρ, ρ, β ρ, ρ, ρ, ρ, ρ, ρ, ρ, ρ, ρ, No a, nc ± wa rd n r of ± and ±, and la uaon can b daly old for, rulng y of four uaon, conanng ±, ± and β ±, β ±, can b old rodng currn-olag,.. -β, caracrc of nu and ouu cabl. For wa concrn coffcn, a l bu cubro calculaon lad o (w, : ρ, (, ρ ρ, ρ, ( ρ, ρ, Furror, can b aly donrad, nc [ ] yrc, an ρ, ρ,. Trfor: ρ, ( ρ ρ,, ( ρ, ρ, ( ρ, ( 4 To rduc z of olng y la uaon old for : ρ, ρ, ρ ρ Conunly, dfnng (for, ±,, ρ, ρ,, ρ, ρ, and ubung, followng rducd y (conanng only ±, ± and,,, found:,, ρ, ρ, ρ, [ ( ( ] [( ( ],, ρ, ρ, ρ, [ ( ( ] ( (,, ρ, ρ, ρ, [ ( ( ] [( ( ],, ρ, ρ, ρ, ( ( For wa concrn coffcn, a l bu cubro calculaon lad o: (, ( ( 4,,, lr for can b oband nroducng ar-ualn ranc: [ ] [ ] [( ( ] E/UNO/ Sbr d
Y nd u, Y Y ( Y ( Y Y Y,, Y Y, Y. Sy oluon n alac doan Coulng jon uaon, wrn n alac doan a follow,, ρ ρ ρ,,, [ ( ( ] [( ( ],, ρ ρ ρ,,, [ ( ( ] [( ( ],, ρ ρ ρ,,, [ ( ( ] [( ( ],, ρ, ρ, ρ, [ ( ( ] [( ( ] w nu and ouu cabl caracrc lad o (afr rordrng,,,,,,,,,, ρ ρ ρ,,,,,,,,, ρ,, ρ, ρ,,,,,,,, ρ,,, ρ ρ ρ,,,,,,,,,,, ρ ρ ow ow ow ow ; rd n ordr o ylfy ar, l u rfor followng oraon: ( ( ; nd ( ( ( ow ( ow4 ; 4 ( ow4 ( ow. T lad o: E/UNO/ Sbr d
ρ ρ,, ρ ρ,,,,,,,,,,,,,, ρ, ρ, ρ ρ,,,,, E/UNO/ Sbr 4 d,,,,,,,,,,,,,,,,,,,,,,,, ρ ρ ρ ρ,,,,,,,,,, ρ, ρ, ρ ρ,,, Forunaly, nold coffcn can b ly rd n r of rnc, a follow,, 4Y Y Y,,,, Y, Y (w, 4 ρ, ρ, ρ,,,,, ρ, ρ, ρ, ρ ρ ρ ρ,,,, ρ ρ ρ ρ,,,, ( ρ, ( ρ ρ ( ρ, ρ, (,,, ρ T ron can b furr lfd dfnng followng an ranc Y Y
E/UNO/ Sbr 5 d Y Y Y Y Fnally, ullyng all row by /, followng y oband: ( ( ( ( ( ( ( ( Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y ( ( ( ( ( ( ( ( ( ( ( ( Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Now dfnng (,, oluon of y can b aly found followng Crar rul: ( ( ( ( ( (,,, w, and ± wr funcon ar dfnd a follow (all ar drnan:
E/UNO/ Sbr d ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( ( Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y
( ( ( ( ( ( ( ( ( Y ( ( ( ( Y ( ( ( ( Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y ( ( Y Y ( Y Y ( Y Y ( Y Y ( ( ( ( Y ( ( ( ( Y ( ( ( ( Y Y Y Y Y Y Y Y Y Y aarn fro dfnon a ( a olynoal of four dgr n, wl ( ar olynoal of rd dgr n. Moror, condrng uaon ( wc clarly nold n nron of alac ranforaon of oluon found, can b n a can b wd a cular uaon for a yrc ral ar. Trfor, gnalu ar l ral. To lfy followng calculaon aud a gnalu (dnod by,,, 4 ar dnc,.. d ( for,,, 4 d Moror, Grgorn or [4] rod bound on,.. for ac gnalu on of followng rlaon afd (w, : ( ( ( ( Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y Y E/UNO/ Sbr 7 d
for gong on w bac-ranforaon of oluon found, l u udy uaon co( λ (, λ n λ T cang of arabl λ lad o uaon co ( wr can b condrd a a ral o unnown, nc λ own n fgur (wr aud a <, oluon of uaon, dnod by ξ ( for,,,, ar boundd a π < ξ ( < ( π, for < and,,, f o a lar uaon occur; an dffrnc daaranc of roo ξ ( f >. T or oluon, dnod a bfor by ξ ( for,,, ar boundd a π < ξ ( < π, for > and,, n any way, rul a: lξ ( π. No a cal ca, a lad o an lc oluon for ξ (, anngl, nc a null gnalu of ar l ngulary and u a robl ll-od. 5 4 8 4-5 E/UNO/ Sbr 8 d
. Soluon n doan ( (, fnng Ψ (, w, and ± ( (, oluon n alac doan wrn a: ( ( Ψ (, w, and ± and forally nrd n doan a follow ( ( τ To rfor nron of funcon ( auon ad, lad o: dτ Ψ ( τ, w, and ± ( ( Ψ no a y afy Jordan a [5]. Tu, w ξ 4 λ Ψ (, w, and ± d u( [ (, ] ξ ( d λ wr (nfn rdu a bn drcly aluad w ad of Hoal rul, nc ngnalu wa aud o b dnc. T funcon u( dfnd a, f < and orw. T nold dffrnal ly aluad a: d d (ang no accoun a Ψ ( ( [ ( ] ( d d, ξ [ λ] ξ ( [ co ] ξ ( λ λ n ξ ξ ( ( d λ λ ξ λ ( ξ λ [ co - ( co ] ξ ( co ξ ( ( n ξ ( n ξ ( λ co ξ co ξ ( and fnally d ( ( ( ( ( n ξ ξ ξ n ξ ( ξ 4 ( ξ ( ( co u λ ( n ξ ( ξ ( n ξ ( ξ ( No a, nc ol caracrz an analycal funcon, of funcon ( Ψ ad alo followng rrnaon: Ψ ( λ ( (, w, and ± ξ ( n ξ ( ( co ξ ( n ξ ( ξ ( ξ ( 4 u w, and ± λ, E/UNO/ Sbr 9 d
To col ran, l u rrn currn n nl and oul cabl (roug r - coonn. Condr fr nl cabl. wa own bfor a (, dτ ( τ (, τ [ ( (, ]( ψ w >, < < and ± (, d ψ (, w >, < < and ± wr ar dno conoluon oraon. Furror, dffrnaon of a rou rul lad o []: ( τ ( dτ Ψ ( τ Ψ ( ( w ± and ubung: ( ( ψ, ( (, ( Ψ w >, < < and ± Subung agan: ( (, d ( (, ( Ψ w >, < < and ±, : (o Trfor, w bgn aluang conoluon of Ψ ( w ( [ Ψ ( (, ]( λ λ λ n 4 4 4 ( ( ( ( ( ( nπ co co u u u ( co ξ ( co ξ ( co ( nπ ξ ( ( n ξ ( ( n ξ ( ξ ( ξ ( n ξ ( ( n ξ ( ξ ( ξ ( λ ( n ξ ( ξ ( n ξ ( ξ ( 4 λ ( ( u ξ n n nπ co co nπ co co co n ( n ξ ( ( n ξ ( ξ ( ξ nπ ( co( nπ ( nπ ( nπ ( co ξ nπ ξ ( ( λ λ λ ( ξ λ ( ξ λ λ ( nπ ( ( nπ ( ξ ξ λ nπ λ nπ λ λ nπ λ No a r n brac wa rouly dlod a rrnaon of Ψ ( and (, Trfor. dτ, rodd α β α β (o α β ατ β No a [ ] ( τ α β E/UNO/ Sbr d
[ Ψ ( (, ]( λ Now ang no accoun a g (,,, [ Ψ ( (, ]( Saal ngraon now g d [ Ψ ( (, ]( Fnally, dfnng lad o F (, 4 n ( ( for u ( co nπ co co ( ξ, λ for λ 4 4 4 ( ( 4 nπ ( ( ( ( ( ( ( ( u ξ, n u ( u ( ξ co ξ ( u ( λ ( n ξ ( ξ ( n ξ ( ξ ( ( nπ Ψ λ ξ nξ coξ nξ ( ( τ co ξ co ξ nπ λ ( co Ψ ( ξ nπ λ [ ξ ( / ] n ξ ( ( co[ ξ ( / ] ( nξ ( ξ ( ( n[ ξ ( / ] ( ( nξ ( ξ ( ( n[ ξ ( / ] ( nξ ( ξ ( co ξ ( co ξ ( ξ, λ and nπ, for,n λ (, for ( ξ ( λ ( ξ λ ( n[ ξ ( / ] ( n ξ ( ξ ( ( ξ λ ( ( ( ξ λ (, dτ F (, τ w >, < < and ± ( ( ( ( For wa concrn oul cabl, nc forally yrc o nl on, oluon for ±, can b oband cangng w and condrng nad of, a follow ( ( ( ( τ (, dτ F (, τ w >, < < and ± frnc [] M. Fabbr, nalycal Soluon for currn drbuon n a urford cabl w N rand, nrnal or E-U (,. []. Couran,. Hlbr, Mod of Maacal Pyc, nrcnc, NY,9, ol.,. ; ol.,. 75. [].S. Gradyn,.M. yz, Tabl of ngral, Sr, and Produc, cadc Pr, ondon, 98,. 45. [4] Y. Saad, ra Mod for ar nar Sy, nrnaonal Toon Publng, ondon, 99,. 9. [5] G. rfn, Maacal Mod for Pyc, cadc Pr, ondon, 985,. 4-48. [] G. oc, Gud o alcaon of alac Tranfor,. Van Norand d., ondon, 9,.8. E/UNO/ Sbr d