Electromagnetic Waves I

Transcript

1 Electromgnetic Wves I Jnury, 03. Derivtion of wve eqution of string. Derivtion of EM wve Eqution in time domin 3. Derivtion of the EM wve Eqution in phsor domin 4. The complex propgtion constnt 5. The six component equtions from the vector Lplcin 6. Uniform Plne Wve 7. Prmeters of EM wve in different medi : lossy, lossless, free spce, perfect conductor, good conductor, imperfect conductor Derivtion of Wve Eqution of string For one Dimensionl Wve Y = yx, t The net upwrd force is T x + x, t T x, t = T sinθ x+ x T sinθ x = T sinθ x+ x sinθ x

2 For smll vibrtion, x 0 = θ 0 sinθ tnθ θ Also, Therefore, yx, t tn θ = lim x 0 x yx + x, t yx, t = lim x 0 x sinθ y x = y x Hence, the net upwrd force is y T y x x x+ x x By Newton s Second Lw, net externl force = mss ccelertion = length liner mss density ccelertion y T y y = F = s µ x x t + ϵ x+ x x As the vibrtion is smll, s x y T y x x x+ x x y = µ x t + ϵ Re-rrnge the terms T µ y y x x x+ x x x = y t + ϵ Tke the limit LHS : lim x 0 y y x x x+ x x x = x y x RHS : lim x 0 ϵ = 0 Thus c y x = yx, t T c = t µ For the 3D cse U = ux, y, z, t

3 The Generlized Wve Eqution is u t = c u x + u y + u z In short u t = c u Derivtion of the EM wve Eqution in time domin Consider the time domin Mxwell Eqution in source free region in the medi chrcteriszed by µ, ε, Recll of vector identity E = µ H t H = ε E t E = 0 H = 0 V = V V Tke the curl of Frdy s Lw, nd consider the left hnd side Plug in the Guss s Lw E = E E Then consider the right hnd side Plugin the Ampere Lw E = E = E µ H = µ H t t E = εµ E t t = E εµ t Equte the left hnd side nd right hnd side Therefore In the sme wy, E = E = εµ E t E εµ E t = 0 H εµ H t = 0 Compre the EM wve eqution to the generl wve eqution u t = c u 3

4 The propgtion velocity of the wve is thus In free spce c = µε c = µ0 ε ms 3 Derivtion of the EM Wve Eqution in phsor domin Consider the Phsor form Mxwell Eqution in the medi chrcterized by µ, ε, Recll of vector identity Ẽ = jωµ H Ẽ = 0 H = Ẽ + jωεẽ H = 0 V = V V Tke the curl of Frdy s Lw, nd consider the left hnd side Ẽ = Ẽ Ẽ Plug in the Guss s Lw Ẽ = Ẽ Then consider the right hnd side jωµ Ẽ = H = jωµ H Plugin the Ampere Lw Ẽ Ẽ = jωµ + jωεẽ = jωµ + jωε Ẽ Equte the left hnd side nd right hnd side Therefore Which is wve eqution. Ẽ = Ẽ = jωµ + jωε Ẽ Ẽ jωµ + jωε Ẽ = 0 Apply the sme method to get the wve eqution of H First, tke the curl of the Ampere Lw, nd consider the left hnd side H = H H Apply Guss Lw of zero divergence Then consider the right hnd side H = H 4

5 Plug in the Frdy s Lw H = + jωε Ẽ Equte the left hnd side nd right hnd side H = jωµ + jωε H Therefore H = H = jωµ + jωε H i.e. the wve equtions re H jωµ + jωε H = 0 Let Ẽ jωµ + jωε Ẽ = 0 H jωµ + jωε H = 0 jωµ + jωε = γ γ : the complex propgtion constnt The wve equtions cn be written in more compct form, the phsor f orm vector Helmholtz eqution Ẽ γ Ẽ = 0 H γ H = 0 4 The complex propgtion constnt 4. The terms Where γ = jωµ + jωε = α + jβ γ complex propgtion constnt α ttenution constnt β progttion constnt γ is the complex propgtion constnt. Unit m α is rte of decy s the wve propgte in lossy medium. For lossless medium, α = 0. Unit Np/m β is the propgtion constnt. This is very importnt constnt, phse velocity, wvelength, frequency, cn be derived using β. Unit rd/m 4. Expressing jωµ + jωε s stndrd complex number α + jβ Using the following eqution, α, β cn be evluted γ = r + r + jb = + j Sgn b where r = + b 5

6 Proof. Using Polr Form For z = + jb z = z z = r exp j tn b γ = z = j r exp b tn = r cos tn b + j r sin b tn b The sign of b is importnt, since tn give out ngle, nd sin ±θ = ± sin θ : r cos tn b + j r sin tn b b > 0 z = r cos tn b j r sin tn b b < 0 Using Sgn nottion on b to combine the eqution b z = r cos tn + jsgnb r sin Turn sin into cos b tn = r cos b tn + jsgnb r cos b tn To get rid of : recll the Double Angle Formul : cos A = cos A cos A = = tn cos b + r + jsgnb cos tn b + r = tn cos b + r + jsgnb tn cos b r For tn θ = b, cos θ = sec θ = + tn θ = + b = r = r r + + jsgnb r r + r r = + jsgnb γ = r + r + jb = + j Sgn b cos A + Thus γ = jωµ + jωε = ω µε + jωµ 6

7 = ω µε b = ωµ R + sgnb = + r = + b = ωµ + ω ε = ω µε + ωε r + = ω µε + ω µε = ω µε + ωε ωε r = ω µε + + ω µε = ω µε + + ωε ωε r + α = Re γ = β = Im γ = sgn b r i.e. γ = jωµ + jωϵ = α + jβ α = ω µε + ωε β = ω µε + + ωε 5 The Lplcin, the six component equtions Reconsider the wve eqution / phsor form vector Helmholtz equtions Ẽ γ Ẽ = 0 H γ H = 0 the equtions include the terms of vector lplcin Ẽ, H Review of Lplcin opertor in rectngulr coordinte = = x ˆx + y ŷ + z ẑ Lplcin on sclr field Ψx, y, z Ψ = Lplcin on vector field Vx, y, z = V x x Therefore, for + V x y x + y + z V = + V x ˆx + z x + y + z V y + V y x y x ˆx + y ŷ + z ẑ Ψ = Ψ x + Ψ y + Ψ z V xˆx + V y ŷ + V z ẑ + V y V z ŷ + z x = x + y + z Sclr + V z y + V z ẑ z Rerrnge Ẽ γ Ẽ = 0 H γ H = 0 7

8 Ẽ = γ Ẽ Expnd the E nd H into component form H = γ H Ẽ xˆx + Ẽ x ŷ + Ẽ x ẑ = γ Ẽ x + γ Ẽ y + γ Ẽ z Hxˆx + Hx ŷ + Hx ẑ = γ Hx + γ Hy + γ Hz Expnd the Lplcin Ẽ x x + Ẽ x y + Ẽ x = γ Hx Ẽ z x + Hx + Hx x y z = γ Hx Ẽ y x + Ẽ y y + Ẽ y = γ Hy Ẽ z y x + Hy y + Hy z = γ Hy Ẽ z x + Ẽ z y + Ẽ z = γ Hz Ẽ z z x + Hz y + Hz z = γ Hz These six equtions contins ll the informtion of ll the components of the propgting EM wve in phsor form in rectngulr coordinte 6 Uniform Plne Wve Not ll EM wve will contins ll the six component. For plne wve : E, H coplnr, nd E H propgtion difection For uniform plne wve : E, H is sme on in the plne to propgtion direction, it only vries in the direction of propgtion Consider uniform plne wve propgting to +z E hs only x component : Ẽ y = Ẽz = 0 H hs only y component : Hx = H z = 0 So the six component equtions re reduced into Ẽ x x, y, z x + Ẽ x x, y, z y + Ẽ x x, y, z z = γ Ẽ x x, y, z Hy x, y, z x + Hy x, y, z y + Hy x, y, z z = γ Hy x, y, z 8

9 Since it is uniform wve, the vrition of mgnitude only occur long the propgtion direction z, so x = y = 0 The equtions re further simplified s Ẽ x z = γ Ẽ z x z Finlly, the PDE cn be reduced into ODE Hy z z = γ Hy z d Ẽ x z dz With the generl solution s = γ Ẽ x z d Hy z dz = γ Hy z Ẽ x z = E e γz + E e γz = E e αz e jβz + E e αz e jβz H x z = H e γz + H e γz = H e αz e jβz + H e αz e jβz Remrk. E, E, H, H re ll complex number, E = E e jθ E Chnge the eqution from phsor bck to time domin, consider the component } E x z, t = Re {Ẽx ze jωt = Re { E e αz e jβz + E e αz e jβz e jωt} = Re { E e jθ E e αz e jβz+ωt + E e jθ E e αz e jωt βz} = Re { E e αz e jωt+βz+θ E + E e αz e jωt βz+θ E } In sme logic E x z, t = E e αz cos ωt + βz + θ E + E e αz cos ωt βz + θ E H x z, t = H e αz cos ωt + βz + θ H + H e αz cos ωt βz + θ H Thus, the wve equtions for uniform plne wve re Ez, t = E x z, tẑ = E e αz cos ωt + βz + θ E + E e αz cos ωt βz + θ E ẑ Hz, t = H x z, tẑ = H e αz cos ωt + βz + θ H + H e αz cos ωt βz + θ H ẑ Consider the E-field, it consists of terms E x z, t = E e αz cos ωt + βz + θ E + E e αz cos ωt βz + θ E Consider the phse st term mgnitude E e αz st term phse ωt + βz + θ E nd term mgnitude E e αz nd terms phse ωt βz + θ E ωt ± βz + θ = constnt When t increse time move, inorder to hve constnt phse, the z should increse / decrese Thus ωt + βz + θ E = constnt time increse z should decrese = move bckwrd ωt βz + θ E = constnt time increse z should increse = move forwrd 9

10 Consider the phse E x z, t = E e αz cos ωt + βz + θ E + E }{{} e αz cos ωt βz + θ E }{{} Bckwrd wve F orwrd wve ωt ± βz + θ = constnt The phse velocity is hence z = ± β constnt ωt θ Thus, the mgnitude of phse velocity is v p = dz dt ẑ = ±ω β ẑ And thus, the wvelength is v p = ω β λ = v p f = ω /β f = πf/β f = π β Finlly, the intrisic impednce is defined s E since E is relted to V nd H is relted to I H Consider fowrd wve of the phsor form eqution Ẽ = Ẽx + Ẽy + Ẽz = E 0 e γz ˆx + 0ŷ + 0ẑ where E 0 is rel, not complex no phse Apply the Frdy s Lw Ẽ = jωµ H H = jωµ Ẽ = jωµ ˆx ŷ ẑ x y z E 0 e γz 0 0 Since Ẽ hs only x component, so Ẽy = Ẽz = 0 Since Ẽ is only z-dependent, Ẽz, so x = y = 0 H = jωµ ˆx ŷ ẑ 0 0 z E 0 e γz 0 0 = jωµ 0 z E 0 e γz 0 ŷ = E 0e γz jωµ z = γ jωµ E 0e γz ŷ = H 0 e γz ŷ where H 0 = γ jωµ E 0 H 0 = γ jωµ E 0 mens the mgnitude of H-field nd E-field inphse 0 ŷ

11 The intrinsic impednce is defined s η = E 0 H 0 = jωµ γ = jωµ jωµ + jωε = Remrk. In free spce, = 0 no conductivity jωµ 0 µ0 η 0 = = 0π Ω jωε 0 ε 0 Express the η in polr form Becuse of the η 0, it is better to extrct η = µ ε out jωµ + jωε = µ ε jω + jω ε jωµ + jωε For rtionliztion of the denomintor, extrct the ω term out in the denomintor µ jω µ η = ε + jω = jω = = ε ε jω + + j j j For the term j In polr form j = j Therefore j η = η e jθη = = exp j tn exp j tn Smll summry of properties of uniform plne EM wve

12 E H E H = propgtion direction, nd E H propgtion direction E, H mgnitude is sme for the plne, it only chnge long propgtion direction For lossless medi, E nd H inphse : when E is mx, H is lso mx, when E is min, H is lso min the digrm bove For lossy medi, E nd H re not inphse : phse ngle differ by θ η The solution of wve equtions consist of both forwrd wve nd bckwrd wve Ẽ = η H ˆk, H = η k Ẽ, where ˆk is wve vector tht in direction of propgtion 7 The prmeters of EM wves 7. Generl prmeters in lossy medium > 0, µ = µ r µ 0, ε = ε r ε 0 γ = jωµ + jωε α = ω µε + ωε β = ω µε + + ωε v p = ω β = µε + + ωε λ = π β = 4π µε + + ωε jωµ η = + jωε = 4 + exp j tn Ẽz = E 0 e γz ˆx Ez, t = E 0 e αz cos ωt βz ˆx Hz = ηe 0 e γz ŷ Hz, t = η E 0 e αz cos ωt βz + θ η ŷ 7. For lossless medi = 0, µ = µ r µ 0, ε = ε r ε 0 γ = ω µϵ = jω µε pure imginry number α = 0 β = ω µε v p = ω β = µε

13 η = λ = π β = µ ε π ω µε rel, no ngle Ẽz = E 0 e βz ˆx Ez, t = E 0 cos ωt βz ˆx Hz = 7.3 In free spce = 0, µ 0, ε 0 µ µ ε E 0e βz ŷ Hz, t = ε E 0 cos ωt βz ŷ γ = jω µ 0 ε 0 α = 0 β = ω µ 0 ε 0 v p = ω β = µ0 ε 0 = c λ = π β = π ω = c µ 0 ε 0 f µ0 η 0 = 0π ε 0 Ẽz = E 0 e βz ˆx Ez, t = E 0 cos ωt βz ˆx Hz = 0πE 0 e βz ŷ Hz, t = 0πE 0 cos ωt βz ŷ 7.4 Perfect conductor =, µ = µ r µ 0, ε = ε r ε 0 γ = α = β = v p = ω β = = 0 η = λ = π β = π = 0 jωµ + jωε = jωµ = 0 Ẽz = E 0 e z ˆx = 0 Ez, t = E 0 e z cos ωt βz ˆx = 0 Hz = 0E 0 e z ŷ = 0 Hz, t = 0E 0 e z cos ωt βz + θ η ŷ = 0 Remrk. It mens no field propgting inside perfect conductor 3

14 7.5 Good conductor lrge : α = β = ω µε + ω µε ωε ωε, µ = µ rµ 0, ε = ε r ε 0 γ = jωµ + jωε µε µε = ω ωε ωε ω ωε µ ωµ = ω ω = = πfµ v p = ω β ω 4πf = πfµ µ λ = π β π 4π = πfµ fµ η = 4 + exp j tn 4 exp j π = + j = µω + j πfµ πfµ = + j = π ej 4 Ẽz = E 0 e γz ˆx µε µε Ez, t E 0 exp ω z cos ωt ω z ˆx ωε ωε Hz πfµ π ej 4 E0 e γz ŷ Hz, t 7.6 Imperfect dielectric smll : Tylor Approximtion will be used frequenctly here Remrk. ω µε + ωε πfµ E 0e αz cos ωt βz + π ŷ 4 ωε, µ = µ rµ 0, ε = ε r ε 0 γ = jωµ + jωε µε ω = 0 is not good,so use Tylor Series t zero fx = f0 + f 0x + f 0! x + f 3 0 x ! Tylor Expnd + x for smll x t x = 0. Since x is smll, so terms lrger thn x 3 cn be ignore ctully x cn lso be ignored, but it is not deleted for better pproximtion + x + x 8 x 4

15 Thus Tylor Approximte ωε ωε 8 ωε α = ω µε + ω ωε = 4 µε + 4 ωε 8 ωε µ = ε 4 ωε µ ε 4 ωε ωε gin this time up to power only α µ ε 8 ωε β = ω µε µε + + ω ωε = ω µε + 4 ωε 8 ωε 4 ωε 6 ωε β = ω µε + 4 ω µε + 4 ωε } 6 {{ ωε } 4 ignore Tylor ω ωε µε + 8 ωε v p = ω β µε + = µε + 8 ωε 8 ωε Tylor prroximtion on the term + 8 ωε + x + x x=0 + + x x x=0 }{{} 0 x + + x x=0 x! i.e. So + x x + 8 ωε 8 ωε Thus v p = µε 8 ωε 5

16 Finlly λ = π β π µε + π µε 8 ωε 8 ωε η = + 4 exp j tn Remrk. exp j tn exp j tn 0 = e j0 = is not good The Tylor pproximtion of η is very long, for simplicity, ignoe ll terms tht power 3 Consider Tylor pproximtion on tn x Let y = tn x tn y = x sec y dy dx = dy dx = sec y = + tn y = + x And Thus d dx tn x = d dx + x = x It is zero when x = 0 so no x term + x tn x tn x }{{ x=0 } x x=0x + tn x x = tn x + x x=0 } {{! } x=0 Actully it is fmous pproximtion tht, for smll ngle θ, sin θ tn θ θ For the exp term exp j tn exp j = exp j Since e x + x + x! Then for exp j tn + j + 4 +, consider + x 4! x = + j 8 + x 4 + x x + x 4 x=0 + + x 3 x=0 x + 7 3! }{{} 0 6 x=0 x

17 Finlly Where re η = + x x = + + } 4 {{ } ignore + η = η 4 exp j µ tn ε + j 8 µ + j ε 8 µ + ε 8 η = tn 8 Ẽz = E 0 e γz ˆx Ez, t E 0 exp µ z cos ωt ω ε 8 ωε µε + z ˆx 8 ωε Hz ηe 0 e γz ŷ Hz, t η E 0 exp µ z cos ωt ω ε 8 ωε µε + z + η ŷ 8 ωε 7