ECE 107: Electromagnetism

Transcription

1 ECE 07: Elcomagnsm S 8: Plan wavs Insuco: Pof. Valy Lomakn Dpamn of Elccal and Compu Engnng Unvsy of Calfona, San Dgo, CA 92093

2 Wav quaon Souc-f losslss Maxwll s quaons Apply cul = jωμ ε = = jωε μ = 2 2 = jωμ H + ω με = ( E ) 0 2 jωε Hlmholz quaon (HE) k = k = k = ω με wavnumb In Casan coodnas ( + ) = k E xyz,, 2

3 Plan wavs n f spac () Radaon fom a unfom sufac cun x= y = 0 = xˆ, = yˆ Maxwll s qs. duc o TL quaons! d dz d x = y = dz Hlmholz quaons duc o Soluons jωμ jωε y x + = 0, + k y = 0 dz + jkz E = = x d H x y 2 k 2 x 2 d E dz x + jkz x x0, y = μ ε, E = J 2 x0 s0 y J J s ˆ = x s z= 0 J s0 y x z 3

4 Plan wavs n f spac (2) Plan wavs soluons a smla o TL soluons! Fquncy doman xpssons Tm doman xpssons Paams Phas vlocy ( + jkz jkz ) x0 x0 ( z) = xˆ + + ( ) ˆ x0 jkz z = y x0 vp = με = ω k ( μ0ε0 = 2 0 ) Wavlngh λ = 2π k = vp f Chaacsc (nnsc) mpdanc jkz ( + + ) x0 ω φ + x0 ω φ E(, z) = xˆ E cos( kz+ ) + E cos( + kz+ ) (, ) ˆ x + z cos( ω kz φ ) x H = y + cos( ω+ kz+ φ ) 8 = μ ε ( = μ ε =20 π Ω)

5 Plan wavs n f spac (3) Equvalnc bwn plan wavs and TL wavs Rplac V, I H, Z 0 Th sulng wavs wll hav h sam bhavo E I = x Z0 ẑ y = V x V 5

6 Plan wavs n f spac (4) Rlaon bwn & and popagang dcon ˆ ˆ ˆ ˆ L k = xsnθ cosϕ+ ysnθsnϕ+ zcosθ b h popagaon dcon of a plan wav wh θ & ϕ bng angls Gnal soluons Fld-dcon laons = k ˆ = kˆ Plan wavs a TEM wavs E kˆ, H kˆ, E H fom a gh-handd pl EHk,, ˆ jk k ˆ jk k ˆ 0, 0 = = o = k ˆ = kˆ Plan wavs sasfy Maxwll s and Hlmholz quaons 6

7 Plan wavs n f spac (5) Exampls ˆ + jkz k = zˆ, x = x y = x ˆ + jkz k = zˆ, = = y y x y Any combnaon of h wo abov Paalll polazaon (TM fld) kˆ = xˆsnθ + zˆcos θ, = yˆ = x θ z θ + jk( xsnθ + z cos θ ) y + ˆ ˆ jk( xsnθ + z cos θ ) y ( cos sn ) Ppndcula polazaon (TE fld) kˆ = xˆsnθ + zˆcos θ, = yˆ = ( xˆcosθ + zˆsn θ) + jk( xsnθ + z cos θ ) y + y jk( xsnθ + z cos θ ) = xˆ E x + kˆ = zˆ Hˆ = yˆ H y + = yˆ E y + = xˆ H x + ˆx E θ = yˆ H + = yˆ E y + y ˆx θ H ˆk ẑ ˆk ẑ 7

8 Plan wavs n f spac (6) Plan wavs n lossy mda σ ε εc = ε j = ε + jε ω wav quaon 2 2 γ = 0 Consd kˆ = zˆ E = xˆe, x Skn dph Dlccs vs. conducos ε ε good dlcc (lag δ s ) ε ε δ s good conduco (small ) 8

9 Plan wavs n f spac (7) Pow flow () Poynng vco S s h pow dnsy (pow p un aa) I shows h sngh AND dcon of pow flow Pow hough a sufac Avag pow dnsy Ths s analogous o TLs! 9

10 Plan wavs n f spac (8) Pow flow (2) Exampl: Plan wavs n a losslss mdum 0

11 Plan wavs n f spac (9) Why a plan wavs (PWs) mpoan? PWs a smpl soluons of Maxwll s quaons allowng lanng many mpoan wav pops Flds adad by annnas a local plan wavs Plan wavs allow canoncal soluon of sval mpoan poblms,.g. flcon fom an nfac Any fld/cun can b psnd (xpandd) as an ngal/summaon of a s of plan wavs Plan wav psnaons allow fo solvng many mpoan poblms

12 Plan wavs a boundas () Movaon Poblms nvolvng boundas a m n ou vy day lf Boundas can lad o nfnc poblms and w nd o know h ffcs Boundas can b usd o gud EM flds along hm,.g. fbs Th a many applcaons of wav phnomna occung on boundas Poblm of plan wav scang fom a plana bounday can b solvd analycally 2

13 Plan wavs a boundas (2) Nomal ncdnc () W hav alady sablshd a clos smlay bwn plan wav and TL wavs Ths smlay can b xndd o h poblm of plan wav scang fom an nfac 3

14 Plan wavs a boundas (3) Nomal ncdnc (2) 4

15 Plan wavs a boundas (4) Nomal ncdnc (3) Fom TL quvalnc h flcon coffcn E0 2 Γ= = E Fom h bounday condon E0 + E0 = E0 and fom h TL analogy, h ansmsson coffcn s gvn by E Γ= τ τ = = E0 2 + Fo non-magnc mda μ = μ = μ 2 0 ε ε 2 ε =, = Γ=, τ = ε ε2 ε + ε2 ε + ε2 5

16 Plan wavs a boundas (5) Nomal ncdnc (4) All sul fo sandng wavs n TLs apply h as wll! Sandng wav ao Fo machd mda = 2 Γ= 0, S = Fo a PEC wall 2 = 0 Γ=,S = Elcc fld maxma Elcc fld mnma 6

17 Plan wavs a boundas (6) Nomal ncdnc (5): TL Analogy 7

18 Plan wavs a boundas (7) Nomal ncdnc (6): Pow flow N avag pow flow n h fs mdum N avag pow flow n h scond mdum Pow laons S = S av av2 8

19 Plan wavs a boundas (8) Oblqu ncdnc () Wha s dffn? Th a fld componns nomal and angnal o h nfac Rflcon and ansmsson dpnd on h sa of polazaon of h ncdn fld A gnal polazaon s wn as a sum of paalll (TM) and ppndcula (TE) polazaons Rflcng pops fo h paalll and ppndcula polazaons a sudd spaaly Angl of ansmsson s dffn fom angl of ncdnc 9

20 Plan wavs a boundas (9) Oblqu ncdnc (2): Snll s law Consd a plan wav a an ncdnc angl θ = jk( xsnθ+ z cos θ) 0 = jkx snθ jkz cosθ 0 E E E Th flcd and ansmd flds a jk xsn jk z cos θ E = E 0 E = E 0 Th bounday condons a z=0 θ θ jk2xsn 2 cos ; jk z ( jk xsnθ jk ) xsnθ zˆ E + E = zˆ E zˆ E + E = zˆ E ( ) 2 To sasfy h bounday condons, h phas along h bounday has o b machd! jk xsnθ jk xsnθ = = 2 jk xsnθ θ jk xsnθ 20

21 Plan wavs a boundas (0) Oblqu ncdnc (3): Snll s law (con d) Fom h phas machng condon k snθ = k snθ = k snθ 2 Dfn ndcs of facon Snll s law (gnal fom) n snθ = n snθ = n snθ 2 Snll s law of flcon θ = θ Snll s law of facon snθ n u snθ = n = u p2 2 p n = k k, n = k k

22 Plan wavs a boundas () Oblqu ncdnc (4): Snll s law (con d) Tansmsson fom and no a dns mdum Dns mda hav lag n Tansmsson no a dns mdum n > n θ < θ 2 Incdnc fom a dns mdum n2 < n θ > θ Toal nnal flcon (o b connud ) π n2 n θ 2 = snθc = snθ = 2 n θ = π 2 n θ c = sn ( n2 n) - ccal angl θ > θc - oal nnal flcon (no fld can popaga n h scond mdum; fuh dscusson wll follow sholy) 22

23 Plan wavs a boundas (2) Oblqu ncdnc (5): Ppndcula polazaon Incdn fld = yˆ 0 jk ( xsnθ + z cos θ ) = ( xˆcosθ + zˆsn θ) Rflcd fld 0 jk ( xsnθ + z cos θ ) = yˆ 0 jk ( xsnθ + z cos θ ) = ( xˆcosθ + zˆsn θ) Tansmd fld = yˆ jk ( xsnθ + z cos θ ) = ( xˆcosθ + zˆsn θ) 0 2 jk ( xsnθ z cos θ ) jk ( xsnθ + z cos θ ) 2 23

24 Plan wavs a boundas (3) Oblqu ncdnc (6): Ppndcula polazaon Bounday condons zˆ ( + ) = zˆ zˆ ( + ) = zˆ y + y = y = 0 E 0cos E 0cos E 0cos x + x = x + = 2 Solv and oban h coffcns! E Γ = = E cosθ cosθ 2 + cosθ cosθ θ θ θ τ E = = +Γ = E cos θ 2 + cosθ cosθ 24

25 Plan wavs a boundas (4) Oblqu ncdnc (7): Paalll polazaon Incdn fld x z ( ˆcos ˆ = θ sn θ) 0 = yˆ 0 jk ( xsnθ + z cos θ ) Rflcd fld x z Tansmd fld ( ˆcos ˆ = θ + sn θ) 0 = yˆ 0 jk ( xsnθ + z cos θ ) x z 2 ( ˆcos ˆ = θ sn θ) 0 = yˆ 0 jk ( xsnθ + z cos θ ) 2 jk ( xsnθ + z cos θ ) jk ( xsnθ z cos θ ) jk ( xsnθ + z cos θ ) 2 Th sgn s chosn o kp h sam x componn of h lcc fld! 25

26 Plan wavs a boundas (5) Oblqu ncdnc (8): Paalll polazaon Bounday condons zˆ ( + ) = z ˆ zˆ ( + ) = z ˆ x + x = x 0 cosθ + 0 cosθ = 0 cosθ y + y = y = Solv and oban h coffcns! E θ cosθ Γ = = E θ cosθ 0 2cos 0 2 cos + τ E 0 22 cos = = +Γ = E 0 2 cos + θ θ cosθ 26

27 Plan wavs a boundas (6) Oblqu ncdnc (9): TL quvalnc Obsvaons Bounday condons a gvn n ms of angnal componns Tangnal componns dpnd on h ncdnc angl and polazaon Equvalnc fs o scond mdum Dfn h chaacsc mpdanc va h angnal componns,,, E an y 0,2,2 = = = =,,, cos an x 0 cosθ θ,,,,, an x 0 cosθ,,2 = = = =,,,,2 cos, an y 0 θ Rplac Z0 & Z02 n a TL juncon by & 2 fo h ppndcula and & fo h paalll polazaons 2, ˆ, = y E y ˆx, E θ, ˆ, = y y ˆx Dffncs: jk xsn θ phas popagaon dncal fo ALL flds θ H ˆk ẑ ˆk ẑ 27

28 Plan wavs a boundas (7) Oblqu ncdnc (0): TL quvalnc (con d) Unfd xpssons 2 Γ = =, 2,, 2, τ,, 2 +,, 2 +, Pacula cass Ppndcula polazaon cosθ cosθ cosθ,2,2 = Γ = τ cosθ, 2 = 2 Paalll polazaon + + cosθ cosθ cosθ cosθ 2cosθ cos 2 2cos,2,2 cos θ θ =, Γ = = 2cosθ + cosθ 2cosθ + cosθ28 θ τ

29 Plan wavs a boundas (8) Oblqu ncdnc (): Pops Coffcns dpnd on h polazaon (fo a gnal polazaon, h fld s wn as a sum of wo polazaons wh dffn pops) Th coffcn dpnds on h angl of ncdnc Th coffcns do no dpnd on h fquncy fo losslss cas Dpndng on h angl, h coffcns can b al o COMPLEX vn n h losslss cas! 29

30 Plan wavs a boundas (9) Oblqu ncdnc (2): Bws angl Consd h cas of machd mpdanc = Γ = = Γ = Ths s oband und h Bws angl 2 μ : sn ε2 μ2ε = θ B = cosθ cosθ 2 ( μ μ ) fo non-magnc maals μ= μ2 = μ0, θb, dos no xs! : cosθ = cosθ snθb = fo non-magnc maals 0 ( ) 2 ε μ ( ε μ ) ( ε ε2) θb = an ε2 ε 30

31 Plan wavs a boundas (20) Oblqu ncdnc (3): Toal nnal flcon Incdnc fom a dns mdum n > n2 (.g. fom wa o a) Ccal angl θ = θc = ( n2 n) θ = π θ > θc cos θ =± j cos θ 2& 2 a puly acv (loads)!!! jφ j sn 2 Γ =, Γ = - phas shf a h nfac Γ = Γ = - oal nnal flcon!!! φ = φ ( θ), φ = φ ( θ) - angula dpndnc!!! φ φ - dpndnc on polazaon!!! φ Γ φ 3

32 Plan wavs a boundas (2) Oblqu ncdnc (4): Th phnomnon of oal nnal flcon s usd o mak fbs! 32

33 Plan wavs a boundas (22) Oblqu ncdnc (5): Summay 33