calculating electromagnetic

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1 Theoeal mehods fo alulang eleomagne felds fom lghnng dshage ajeev Thoapplll oyal Insue of Tehnology KTH Sweden

2 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

3 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

4 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

5 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?

6 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

7 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

8 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

9 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

10 Expessons fo sala poenal Dpole mehod 4 3 d d + + τ τ θ φ Monopole mehod * d Q ρ φ o opo e e od 4 4 d Q τ τ Boh ae analyally equvalen

11 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

12 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 Tme μs

13 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 Tme moseond

14 Non-unqueness of feld omponens Numeal example - 3 m 3. km.5 EV_C & EV_CE le Feld Vm E E_C & E_CE EI_C & EI_CE Tme μs EQ_C & EQ_CE

15 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

16 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

17 elaon beween eaded uen and eaded hage? * ρ?. ons O O ρ Thoapplll akov Uman 997 7

18 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

19 elaon beween appaen hage densy and eaded uen ρ d - τ d τ d v+ 9

20 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

21 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.

22 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

23 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

24 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.

25 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 [] 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] ubnsen M. and M. A. Uman On he adaon feld un-on em assoaed wh avellng uen dsonnues n lghnng J. Geophys. es [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 [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 [6] Thoapplll. and V. A. akov On dffeen appoahes o alulang lghnng ele felds J. Geophys. es [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 [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