Wave Equation (2 Week)

Transcription

1 Rfrnc Wav quaion ( Wk 6.5 Tim-armonic filds 7. Ovrviw 7. Plan Wavs in Losslss Mdia 7.3 Plan Wavs in Loss Mdia 7.5 Flow of lcromagnic Powr and h Poning Vcor 7.6 Normal Incidnc of Plan Wavs a Plan Boundaris Inro.

2 PLAN LCTROMAGNTIC WAVS Wirlss applicaions ar possibl bcaus lcromagnic filds can propaga in fr spac wihou an guiding srucurs. Plan wavs ar good approimaions of lcromagnic wavs in nginring problms afr h propaga a shor disanc from h sourc. Inro.

3 Inro.3 Tim-armonic lcromagnics In pas chapr, fild quaniis ar prssd as funcions of im and posiion. In ral applicaions, im signals can b prssd as suprimposiion of sinusoidal wavforms. So i is convnin o us h phasor noaion o prss filds in h frqunc domain, jus as in a.c. circuis. { } j ω,, ( R,,, ( { } j j j ω ω ω,, ( R,, ( R,,, (

4 Inro.4 Tim-armonic lcromagnics Diffrnial form of Mawll s quaions Tim-harmonic Mawll s quaions 0 + B D D J B v ρ 0 + B D D J B v j j ρ ω ω

5 Wav quaions in Sourc-Fr Mdia In a sourc-fr mdia (i..charg dnsi ρ v 0, Mawll s quaion bcoms: jωµ ( σ + jωε 0 0 W will driv a diffrnial quaion involving or alon. Firs ak h curl of boh sids of h s quaion: jωµ jωµ ( σ + jωε Inro.5

6 Sinc 0 in sourc-fr mdia, w obain whr γ γ 0 j ωµ ( σ + jωε Th abov quaion is calld lmhol quaion, and γ is calld h propagaion consan. In rcangular coordinas, lmhol quaion can b dcomposd ino hr scalar quaions: i,, γ 0 i Similarl for h magnic fild, i γ 0 Inro.6

7 Inro.7 lmol quaion and Transmission Lin quaions lmol quaion Transmission Lin quaions In rcangular co-ordinas 0 γ 0 γ I I V V γ γ 0 i i γ i,,

8 Plan wavs in losslss mdia In a losslss mdium, h conduciv currn is ro so σ is qual o ro; hr is no nrg loss and µ, ε ar boh ral numbrs. Thrfor in losslss mdia γ jω µε Plan wav is a paricular soluion of h Mawll s quaion, whr boh h lcric fild and magnic fild ar prpndicular o h propagaion dircion of h wav. L h wav propaga along h -dircion, and h lcric fild is along h -dircion. Inro.8

9 - and -fild vcors for a plan wav propagaing in -dircion Inro.9

10 (a Subsiuing his ino olmhol quaion, and sinc 0, w obain d γ 0 d This scond ordr linar diffrnial quaion has wo indpndn soluions, so h soluion of is ( + o γ +γ + o Th s rm is a wav ravlling in h + dircion, and h nd rm is a wav ravlling in h - dircion. Inro.0

11 γ is calld h propagaion consan. In gnral, γ is compl γ α + jβ whr α, β ar h anuaion and phas consans. Bu in losslss mdia, α 0 β ω µε (µ, ε ar ral So h lcric fild in losslss mdia is givn b: ( + jβ + jβ + a o o Inro.

12 L u p b h phas vloci wih which ihr h forward or backward wav is ravlling, and λ is h wavlngh. Sinc β ω µε, hrfor u p ω β µε λ π β In fr spac (vacuum, ε ε o, µ µ o and h phas vloci is h sam as h vloci of ligh in vacuum (3 0 8 m/s. Inro.

13 Similarl for h magnic fild, can b found b solving jωµ undr h condiions 0 W obain jωµ [ + jβ + jβ ] o + o a [ + jβ jβ ] a β + o o ωµ Inro.3

14 W noic ha h magnic fild is in h -dircion, i.. prpndicular o h lcric fild, and h raio of magniuds of lcric and magnic filds is: + o + o η ωµ β o o η η is calld h inrinsic (or wav impdanc of h mdium. In fr spac η has a valu approimal qual o 377Ω. µ ε Inro.4

15 Th abov plan wav is calld TM (ransvrs lcromagnic wav, bcaus boh h lcric and magnic filds onl is in h ransvrs dircions prpndicular o h propagaion dircion. Ohr wavs (.g. in wavguids can b also: (a T (ransvrs lcric wav - lcric fild iss onl in ransvrs dircion, i.. ro lcric fild in h propagaion dircion. (b TM (ransvrs magnic wav - magnic fild iss onl in ransvrs dircion, i.. ro magnic fild in h propagaion dircion. Inro.5

16 ampl: A uniform plan wav wih a propagas in a losslss mdium (ε r 4, µ r, σ0 in h -dircion. If has a frqunc of 00M and has a maimum valu of 0-4 (V/m a 0 and /8 (m, a wri h insananous prssion for and, b drmin h locaions whr is a posiiv maimum whn 0-8 s. Soluion: γ jω µ µ ε ε sinc vloci of ligh c o r o r jω c µ o ε o µ r ε r π 0 γ j π 3 j Inro.6

17 Inro cos( φ π π + a (a Sing 0-4 a 0 and /8, 6 π φ k π ε ε µ µ η η 60 o r o r a a (b A 0-8, for o b maimum, n n m m ( ± ± π π π 0,,,... n Th -fild:

18 7.4 Plan Wavs in Loss Mdia Th lcric fild in h -dircion is givn b: ( + o γ +γ + o whr γ is in gnral compl for loss mdia γ jωµ ( σ + jωε γ α + jβ So -fild can b wrin as: + o α jβ + α + o jβ a + a Inro.8

19 Subsiu ino Mawll s quaions, h magnic fild is: η [ + γ + γ ] o o a whr h inrinsic impdanc η is compl. η jωµ γ jωµ σ + jωε Th loss can b du o (i conducion loss whr σ is non-ro, (ii dilcric loss whr ε is compl, '' ε ε ' jε Inro.9

20 ampl: Th -fild of a plan wav propagaing in - dircion in sa war is a 00 cos(0 7 π V/m a 0. Th paramrs of sa war ar ε r 7, µ r, σ4 S/m. (a Drmin h anuaion consan, phas consan, inrinsic impdanc,and phas vloci. (b Find h disanc whr h ampliud of is % of is valu a 0. Soluion: (a γ j ωµ ( σ + jωε α + jβ In our cas σ/ωε 00>> so ha α, β ar approimal α β πfµρ 8.89 Inro.0

21 η jωµ σ + jωε jωµ σ η ( + Phas vloci: 6 7 πfµ π (5 0 (4π 0 j ( + j π σ 4 u 7 ω 0 π m/s β 8.89 jπ / 4 (b Disanc whr wav ampliud dcrass o %: α 0.0 ln 00 α Inro.

22 Inro. 7.5 Polariaion: Polariaion dscribs how h -fild vcor varis wih im as h wav propagas. L, b h componns of -fild in h and dircions. Linar polariaion: Th -fild vcor is alwas in h sam dircion., ar ihr in phas or ou of phas. a a a a cos( cos( ( cos( cos( ( m m m m β ω β ω β ω β ω +

23 Circular polariaion: Th -fild vcor roas around h ais of propagaion and has consan ampliud., hav qual magniuds and phas angl diffrnc of π/. ( m cos( ω β a + m cos( ω β ± π / a llipic polariaion: Th -fild vcor roas around h ais of propagaion bu has im-varing ampliud., can hav an valus of magniud and phas angl diffrnc. ( m cos( ω β a + m cos( ω β + δ a Inro.3

24 7.6 Powr ransmission - Poning s Thorm Powr is ransmid b an lcromagnic wav in h dircion of propagaion. I can b drivd from Mawll s quaion in h im domain ha: S ds V Jdv + V ε + µ dv On h righ hand sid, h s rm rprsns h insananous ohmic powr loss, h nd and 3rd rm rprsn h ra of incras of nrgis sord in h lcric and magnic filds rspcivl. So h lf hand sid rprsns h n insananous powr supplid o h nclosd surfac, or Inro.4

25 Wou. ds S P( ( ( Th insananous Poning vcor P (in wa pr sq.m rprsns h dircion and dnsi of powr flow a a poin. In im-varing filds, i is mor imporan o find h avrag powr. W dfin h avrag Poning vcor for priodic signals as P avg T T Pd ( P avg ( d T 0 T 0 Using compl phasor noaion of,, P avg R( * Inro.5

26 For a uniform plan wav, & fild vcors ar givn b: o α jβ a η [ ] α jβ a o η η j θ η Subsiuing in qu. of P avg givs P η o α avg cos θηa Inro.6

27 7.7 Rflcion of Plan Wavs a Normal Incidnc Mdium (σ,ε,µ Mdium (σ,ε,µ i i ak (incidn wav a k r (ransmid wav a k r (rflcd wav a k propagaion dircion dircion - dircion Inro.7

28 Inro.8 Incidn wav: io io i io i a a a γ γ γ η Rflcd wav: ro ro r ro r a a a ( γ γ γ η Transmid wav: o o o a a a γ γ γ η

29 Boundar condiions a h inrfac (0: Tangnial componns of and ar coninuous in h absnc of currn sourcs a h inrfac (rfr o uorials, so ha Rflcion cofficin η Transmission cofficin io io + + ( Γ io ro ro τ ro io ro o io o η η o η o η + η η η + η Inro.9

30 No ha Γ and τ ma b compl, and + Γ τ 0 Γ Similar prssions ma b drivd for h magnic fild. In mdium, a sanding wav is formd du o h suprimposiion of h incidn and rflcd wavs. Sanding wav raio can b dfind as in ransmission lins. Inro.30

31 ND Inro.3