Theoeal mehods fo alulang eleomagne felds fom lghnng dshage ajeev Thoapplll oyal Insue of Tehnology KTH Sweden ajeev.thoapplll@ee.kh.se
Oulne Despon of he poblem Thee dffeen mehods fo feld alulaons - Dpole and monopole mehods fo feld alulaons - Non-unqueness of feld omponens - Felds n ems of appaen hage hd mehod Speal ase of eun soke wh speed of lgh Aknowledgemen: V.A akov and M. Uman Unvesy of Floda Ganesvlle ll
Despon of he poblem ghnng g eun soke speed -x 8 ms Cuen se me < -6 s Dsbued soue fas hangng g n boh spae and me Mehods of fndng exa expessons fo emoe ele and magne felds v τ- τ < v. d d E B θ 3
El f ld f d l Ele felds fom a dpole Sa Z j β Z e j + m Z E - j - E ω β β θ π 3 os y P θ e - j + m jz - E - j - E ω β θ β β θ π β 3 sn 4 y φ e j + m j H - j - E ω β φ β θ π β sn 4 x j β π 4 d adaon Induon o Ae he feld omponens unque? nemedae
Ele felds fom lne soues ne soue seveal dpoles onneed end o end Is possble o defne sa nduon and adaon omponens unquely? How do we defne sa feld? - dsane 3? - The pa of ele feld gven by he gaden of he sala poenal? How do we defne adaon feld? - dsane? - Assoaed wh ae of hange of uen aeleang hages? - ae of hange of veo poenal?
Two mehods fo fndng ele felds Monopole mehod Cuen Densy Cuen densy Chage Densy fom onnuy equaon Veo Sala Poenal Veo Poenal fom oen Poenal ondon Sala a Poenal Ele Feld Dpole mehod Expl use of oen ondon Ele Feld Expl use of onnuy equaon Jefmenko
ubnsen e al. 989 showed ha fo a exendng sep pulse boh gve denal ele felds numeally. Safaenl and Mna [99] showed ha fo exendng sep pulse boh expessons ae analyally equvalen. Thoapplll and akov showed analyally ha fo any dsbuon of uen and hages on a lghnng hannel he dpole and monopole mehods gve he same ele felds
Dpole and monopole mehods fo feld alulaons - Dpole mehod Expl use of oen τ τ ondon o fnd sala A θ τ poenal fom veo 4 poenal φ oen ondon A + d ˆ φ θ Adτ A θ E φ A B A 8
Monopole mehod fo feld o opo e e od o e d alulaons - The monopole mehod mehod The onnuy equaon explly used o fnd ˆ 4 τ τ τ θ d A p y hage densy fom uen * ρ. ons ρ + * 4 4 d Q ρ φ φ A E φ Q 9
Expessons fo sala poenal Dpole mehod 4 3 d d + + τ τ θ φ Monopole mehod * 4 4 + d Q ρ φ o opo e e od 4 4 d Q τ τ Boh ae analyally equvalen
Sample alulaon usng he wo mehods p g lghnng eun soke feld a gound D l h d v Dpole mehod v 3 3sn ˆ V d d E τ τ α + 3sn ˆ d b α sn ˆ d α 3 Sa Monopole mehod 3 * ˆ V d E ρ Induon adaon 3 * ˆ V d d ρ ρ adaon ˆ d
Non-unqueness of feld omponens Numeal example - 5 m 3 EV_C & EV_CE EQ_ C 5 m Ele Fe eld Vm EQ_CE EI_CE E_CE E_C - EI_C 4 6 8 Tme μs
Non-unqueness of feld omponens Numeal example - m 6 EV_C & EV_CE km Ele Feld V Vm 4 EI_C EQ_C EI_CE EQ_CE E_C E_CE 4 6 8 Tme moseond
Non-unqueness of feld omponens Numeal example - 3 m 3. km.5 EV_C & EV_CE le Feld Vm E..5..5 E_C & E_CE EI_C & EI_CE. 4 6 8 Tme μs EQ_C & EQ_CE
Ele feld a gound plane Dpole mehod Boh he gaden of he sala poenal and he me devave of he veo poenal onbue o he adaon feld em. Monopole mehod adaon em s ompleely gven by he me devave of he veo poenal. Tme devave of he veo poenal onbue o he nduon feld em. Eleosa and nduon ems ae gven ompleely by he gaden of he sala poenal No one-o-one oespondene beween feld omponens. Howeve oal feld s he same
Infeenes Indvdual feld omponens - sa nduon and adaon - ae no unque Toal ele feld s unque Dffeenes beween feld omponens ae sgnfan a lose dsanes and neglgble a fa dsanes Cauon has o be exesed n nepeng measuemen esuls o n makng appoxmaons n alulaons 6
elaon beween eaded uen and eaded hage? * ρ?. ons O O ρ Thoapplll akov Uman 997 7
elaon beween wo defnons of eaded hage densy os θ ρ ρ * + os θ oal hage densy a eaded me oal hage densy a eaded me as seen by emoe obseve appaen hage densy 8
elaon beween appaen hage densy and eaded uen ρ d - τ d τ d v+ 9
Felds a gound n ems of appaen g pp hage densy 3 * d E ρ ε π * an 3 d - - ρ ε π an 3 d d v - - ρ ε π * d ρ ε π M d d v ρ ε π 3 d d v ρ ε π
Pulse popagaon on a veal anenna above pefe gound exa fomulaon Wha happens f he eun soke speed s speed of lgh and f he uen avels whou any aenuaon and dspeson? I s poved n Thoapplll e al. ha he exa geneal expesson wh effe of pefe gound nluded edues o E θ θˆ θ sn θ B θ ˆ φ θ sn θ θ ˆ θˆ Cooay and Cooay also deve same esuls sang fom he felds of a movng pon hage.
Pulse popagaon on a veal anenna exa fomulaon - onnued Poynng veos adally-deed deed wh ogn a pon hage Wave mpedane 377 Ω Pon soue Pefe ondung plane
Smlay o he soluon of nfne onal anenna Spheal TEM soluon [Shelkunoff 95] Spheal TEM soluon [Thoapplll e al.] Pon soue Pon soue Pefe ondung plane
Infeenes A sem-nfne ondung we of vanshng adus pependula o a ondung plane all onduos beng pefe suppo spheal TEM f he only soue s a pon soue a he boom of he we. The uen eleased fom he pon soue avels unaenuaed wh he speed of lgh. The Poynng veo and enegy flow s n he adal deon fom he soue a he boom of he anenna. The wave mpedane s he fee-spae mpedane 377 Ω a all dsanes fom he anenna.
SOME EFEENCES FO MOE INFOMATION [] Uman M. A. D. K. Man and E. P. Kde The eleomagne adaon fom a fne anenna Am. J. Phys. 43 33-38 975. [] ubnsen M. and M.A. Uman 989. Mehods fo alulang he eleomagne felds fom a known soue dsbuon: Applaon o lghnng IEEE Tans. Eleomagn. Comp. 3 83-89. 89 [3] ubnsen M. and M. A. Uman On he adaon feld un-on em assoaed wh avellng uen dsonnues n lghnng J. Geophys. es. 95 37-373 99. [4] Thoapplll. V. A. akov and M. A. Uman Dsbuon of hage along he lh lghnng hannel: ll elaon o emoe ele and magne felds fld and do eun-soke models J. Geophys. es. 6987-76 997. [5] Thoapplll. Uman M.A. and akov V.A. Teamen of eadaon effes n alulang he adaed eleomagne felds fom he lghnng dshage J. Geophys. es. 3 93-93 93 998. [6] Thoapplll. and V. A. akov On dffeen appoahes o alulang lghnng ele felds J. Geophys. es. 6 49-45. [7] Thoapplll. and V.A. akov On he ompuaon of ele felds fom a lghnng dshage n me doman IEEE EMC Inenaonal Symposum Moneal Canada Aug. 3-7. [8] Thoapplll. Compuaon of eleomagne felds fom lghnng dshage Chape n he book The ghnng Flash ed V. Cooay The Insuon of Eleal Engnees ondon 3. [9] Thoapplll. M.A. Uman N. Theehay Ele and magne felds fom a semnfne anenna above a ondung plane J. Eleosas 6 9-4.