Uniform plane wave 2014 mk

Nguyễn Công Phương Engineering Electromagnetics The Uniform Plane Wave Contents I II III IV V VI VII VIII IX X XI XII XIII XIV XV Introduction Vector Analysis Coulomb’s Law & Electric Field Intensity Electric Flux Density, Gauss’ Law & Divergence Energy & Potential Current & Conductors Dielectrics & Capacitance Poisson’s & Laplace’s Equations The Steady Magnetic Field Magnetic Forces & Inductance Time – Varying Fields & Maxwell’s Equations The Uniform Plane Wave Transmission Lines Plane Wave Reflection & Dispersion Guided Waves & Radiation Uniform plane wave - sites.google.com/site/ncpdhbkhn The Uniform Plane Wave Wave Propagation in Free Space Wave Propagation in Dielectrics The Poynting Vector Skin Effect Wave Polarization Uniform plane wave - sites.google.com/site/ncpdhbkhn Wave Propagation in Free Space (1) ∂E ∇ × H = ε0 ∂t ∂H ∇ × E = − µ0 ∂t ∇ E = ∇.H = Uniform plane wave - sites.google.com/site/ncpdhbkhn Wave Propagation in Free Space (2) E = Exa x Ex = E ( x, y , z ) cos(ω t + ϕ ) e jωt = cos ω t + j sin ωt → Ex = Re  E ( x, y , z )e j (ω t +ϕ )  = Re  E ( x , y , z )e jϕ e jω t  Exs = E ( x , y , z )e jϕ E s = Exs a x Ex = Re  E xs e jωt    Uniform plane wave - sites.google.com/site/ncpdhbkhn Wave Propagation in Free Space (3) Ex Find the time – varying function of the vector field: E s = 100 30oa x + 20 − 50oa y + 40 210oa z V/ m If f = MHz E s = 100e ( j 30o → Es (t ) = 100e = 100e a x + 20 e j 30o − j 50 o a x + 20e j (2π 106 t +30 o ) a y + 40e − j 50o j 210o a y + 40e a x + 20e a z V/ m j 210o ) az e j 2π 106 t j (2π 106 t −50o ) a y + 40e j (2π 10 t + 210 o ) az → E(t ) = 100 cos(2π 106 t + 30o )a x + + 20 cos(2π 106 t − 50o )a y + 40 cos(2π 106 t + 210o )a z Uniform plane wave - sites.google.com/site/ncpdhbkhn Wave Propagation in Free Space (3) Ex = E ( x, y, z ) cos(ωt + ϕ ) ∂E x ∂ → = [ E ( x, y, z ) cos(ωt + ϕ ) ] = −ω E ( x, y, z )sin(ωt + ϕ ) ∂t ∂t Re  jω Exs e jω t  = Re jω  E ( x, y, z )e jω t  e jϕ { } = Re  jω E ( x, y, z )e j (ωt +ϕ )  = Re {ω E ( x, y, z) j [ cos(ωt + ϕ ) + j sin(ωt + ϕ )]} = Re {ω E ( x, y, z ) [ j cos(ωt + ϕ ) − sin(ωt + ϕ )]} = −ω E ( x, y, z ) sin(ωt + ϕ ) ∂E x → = Re  jω E xs e jωt    ∂t Uniform plane wave - sites.google.com/site/ncpdhbkhn Wave Propagation in Free Space (4) Ex = E ( x, y, z ) cos(ωt + ϕ ) ∂E x = Re  jω E xs e jωt  ↔ jω E xs   ∂t ∂E ∇ × H = ε0 ∂t ∂H ∇ × E = − µ0 ∂t ∇ E = ∇.H = ∇ × H s = jωε 0E s ∇× Es = − jωµ0 H s ∇ E s = ∇.H s = Uniform plane wave - sites.google.com/site/ncpdhbkhn Wave Propagation in Free Space (5) ∇ × E s = jà0 H s ì ì E s = ì ( jà H s ) = jà0 ì H s ì H s = jωε E s → ∇ ×∇ × E s = ω µ0ε 0E s ∇ × ∇× E s = ∇ (∇ E s ) − ∇ 2Es ∇ E s = → ∇ (∇.Es ) = → ∇ 2E s = −k 02E s k0 = ω µ0ε (wavenumber) ∇ Exs = −k02 Exs ∂ Exs ∂ Exs ∂ Exs 2 → + + = − k E d E xs xs 2 2 → = − k ∂x ∂y ∂z Exs dz Suppose E does not vary with x or y xs Uniform plane wave - sites.google.com/site/ncpdhbkhn Wave Propagation in Free Space (6) d E xs = −k02 E xs dz → Exs = E x e− jk0 z → Ex ( z, t ) = Ex cos(ωt − k0 z ) E ′x ( z, t ) = E x′ cos(ωt + k z ) k0 = ω µ0ε µ0ε = 2.998 × 108 ≈ × 108 m/s → k0 = ω c  E x ( z, t ) = E x cos[ω (t − z / c )] →  E ′x ( z, t ) = E ′x cos[ω (t + z / c )] Uniform plane wave - sites.google.com/site/ncpdhbkhn 10 The Poynting Vector (4) Ex = Ex 0e−α z cos(ωt − β z) η = η θη → S z = Ex H y = Ex20 η → Hy = Ex η e−α z cos(ωt − β z − θη ) e−2α z cos(ωt − β z ) cos(ωt − β z − θη ) Ex20 −2α z = e [cos(2ωt − β z − 2θη ) + cos θη ] 2η → S z, av 1 E x20 −2α z ˆ  W/m = e cosθη = Re E s × H s η E s = Ex e − jβ z a x ˆ = Ex e jβ z a = Ex e jθη e j β z a H s y y ηˆ η Uniform plane wave - sites.google.com/site/ncpdhbkhn 26 The Uniform Plane Wave Wave Propagation in Free Space Wave Propagation in Dielectrics The Poynting Vector Skin Effect Wave Polarization Uniform plane wave - sites.google.com/site/ncpdhbkhn 27 Skin Effect (1) σ σ jk = jω µε ′ − j ≈ jω µε ′ − j = j − jωµσ ωε ′ ωε ′ − j = − 90o 1 − 90 = − 45 = −j 2 o o  1  → jk = j  −j  ωµσ = ( j1 + 1) π f µσ = α + jβ 2  → α = β = π f µσ → Ex = Ex0 e−α z cos(ω t − β z ) = Ex 0e − z π f µσ cos(ωt − z π f µσ ) Uniform plane wave - sites.google.com/site/ncpdhbkhn 28 Skin Effect (2) E x = E x 0e − z Ex z =0 π f µσ cos(ω t − z π f µσ ) Dielectrics = Ex cos ω t J x = σ E x = σ E x 0e − z δ= δ Cu = π f µσ π f µσ = z α = Conductor cos(ωt − z π f µσ ) β 0.066 f δ Cu ; 50 Hz = 9.3 mm δ Cu; 10,000 MHz = 6.61 × 10−4 mm Uniform plane wave - sites.google.com/site/ncpdhbkhn 29 Skin Effect (3) α=β= β= δ = π f µσ 2π λ → λ = 2πδ ω vp = β → v p = ωδ Uniform plane wave - sites.google.com/site/ncpdhbkhn 30 Skin Effect (4) Ex Consider an MHz wave propagating in seawater, σ = S/m, ε’r = 81 σ = = 8.9 × 10 ≫1 − 12 ωε ′ (2π × 10 )(81)(8.85 × 10 ) 1 δ= = = 0.25 m − π f µσ (π × 10 )(4π × 10 )(4) λ = 2πδ = 1.6 m v p = ωδ = (2π × 106 )(0.25) = 1.6 × 106 m/s Uniform plane wave - sites.google.com/site/ncpdhbkhn 31 Skin Effect (5) η= µ µ jωµ = = ε ε ′ − jε ′′ σ + jωε ′ →η = σ ≫ ωε ' j = 45 o →η = − z π f µσ Ex = Ex e 45o σδ = σδ + j δ= jωµ σ π f µσ σδ cos(ω t − z π f µσ ) = E x0 e − z /δ cos(ω t − z / δ ) Ex =η Hy → Hy = σδ Ex z π  e − z / δ cos  ωt − −  δ 4  Uniform plane wave - sites.google.com/site/ncpdhbkhn 32 Skin Effect (6) Ex = Ex 0e Hy = Sav − z /δ σδ Ex e cos(ωt − z / δ ) − z /δ z π  cos  ω t − −  δ 4  Conductor L Dielectrics ˆ  = Re  E s × H s S Jx0 z |Jxs| σδ E x20 −2 z / δ π  → Sav = e cos   2 4 = σδ E x20 e −2 z / δ SL, av = ∫ S z , av dS = ∫ x b L1 ∫ 0 σδ Ex20e −2 z / δ b y δ dxdy = σδ bLE x20 z =0 Uniform plane wave - sites.google.com/site/ncpdhbkhn 33 Skin Effect (7) SL, av = σδ bLE x20 J x = σ Ex → SL , av = δ bLJ 2x 4σ I=∫ ∞ b ∫0 x Conductor L Dielectrics Jx0 z J x dydz |Jxs| J x = J x 0e− z / δ cos (ωt − z / δ ) → J xs = J x 0e− z / δ e− jz / δ b = J x e−(1+ j) z / δ → Is = ∫ ∞ b →I = ∫0 J x 0e J x0 bδ −(1+ j ) z / δ dydz = J x 0be π  cos  ωt −  4  −(1+ j ) z / δ y δ ∞ J x 0bδ −δ = 1+ j 1+ j Uniform plane wave - sites.google.com/site/ncpdhbkhn 34 Skin Effect (8) I= J x 0bδ π  cos ωt −    4 I J π  = x cos  ωt −  bδ 4  → SL = ( J ′)2 bLδ → J′= x Conductor L Dielectrics σ J x20 π  = bLδ cos  ωt −  2σ 4  → SL, av = J x 0bLδ 4σ Jx0 z |Jxs| b y δ (if the current is distributed uniformly throughout < z < δ) SL, av = J x 0bLδ 4σ (if the total current is distributed throuthout < z < ∞) Uniform plane wave - sites.google.com/site/ncpdhbkhn 35 Skin Effect (9) L L R= = σ S σ 2π aδ RCu, 1MHz, a =1mm, l =1km 103 = = 41.5 Ω −3 −3 (5.8 × 10 )(2π )(10 )(0.066 × 10 ) Uniform plane wave - sites.google.com/site/ncpdhbkhn 36 The Uniform Plane Wave Wave Propagation in Free Space Wave Propagation in Dielectrics The Poynting Vector Skin Effect Wave Polarization Uniform plane wave - sites.google.com/site/ncpdhbkhn 37 Wave Polarization (1) • In the previous sections, E & H are supposed to lie in fix directions • However, the directions of E & H within the plane perpendicular to az may change as functions of time and position • λ, vp, S, … • The instantaneous orientation of field vectors • Wave polarization: its electric field vector orientation as a function of time, at a fixed point in space • H can be found from E Uniform plane wave - sites.google.com/site/ncpdhbkhn 38 Wave Polarization (2) E s = ( Ex 0a x + E y 0a y )e−α z e− jβ z y Ey0 H s = ( H x 0a x + H y 0a y )e−α z e− j β z E H Sz , av Hy0  E y0 E x0  −α z − jβ z = − ax + ay e e η  η  Hx0 ˆ ] = Re[ Es × H s = Re  Ex Hˆ y (a x × a y ) + E y Hˆ x (a y × a x )  e −2α z  Ex Eˆ x E y Eˆ y  −2α z = Re  + e az  ηˆ ηˆ   1 = Re    E x0 + E y  e−2α z a z W/m2 ηˆ    Uniform plane wave - sites.google.com/site/ncpdhbkhn Ex0 x 39 Wave Polarization (3) E s = ( E x 0a x + E y 0a y )e− j β z → E( z, t ) = Ex cos(ωt − β z )a x + E y cos(ωt − β z + ϕ )a y → E( z, 0) = E x cos( β z )a x + E y cos( β z − ϕ )a y E(z, 0) Ex0 Ey0 a ϕ β Observer location b z Wave travel Uniform plane wave - sites.google.com/site/ncpdhbkhn 40 ... Equations The Uniform Plane Wave Transmission Lines Plane Wave Reflection & Dispersion Guided Waves & Radiation Uniform plane wave - sites.google.com/site/ncpdhbkhn The Uniform Plane Wave Wave Propagation... z / c )] Uniform plane wave - sites.google.com/site/ncpdhbkhn 11 The Uniform Plane Wave Wave Propagation in Free Space Wave Propagation in Dielectrics The Poynting Vector Skin Effect Wave Polarization... 2ωε ′  Uniform plane wave - sites.google.com/site/ncpdhbkhn 21 The Uniform Plane Wave Wave Propagation in Free Space Wave Propagation in Dielectrics The Poynting Vector Skin Effect Wave Polarization