Plane wave reflection 2014mk

Nguyễn Công Phương Engineering Electromagnetics Plane Wave Reflection & Dispersion Contents I Introduction II Vector Analysis III Coulomb’s Law & Electric Field Intensity IV Electric Flux Density, Gauss’ Law & Divergence V Energy & Potential VI Current & Conductors VII Dielectrics & Capacitance VIII Poisson’s & Laplace’s Equations IX The Steady Magnetic Field X Magnetic Forces & Inductance XI Time – Varying Fields & Maxwell’s Equations XII Transmission Lines XIII The Uniform Plane Wave XIV Plane Wave Reflection & Dispersion XV Guided Waves & Radiation Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn Plane Wave Reflection & Dispersion Reflection of Uniform Plane Waves at Normal Incidence Standing Wave Ratio Wave Reflection from Multiple Interfaces Plane Wave Propagation in General Directions Plane Wave Reflection at Oblique Incidence Angles Wave Propagation in Dispersive Media Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn Reflection of Uniform Plane Waves at Normal Incidence (1) Ex+1 ( z , t ) = E x+10 e−α1z cos(ωt − β1 z ) + Exs H +ys1 = E x+10e− jk1z = η1 E1+ , H1+ Incident wave + + − jk z Exs = E e x 20 + − jk 2z + H ys = Ex 20e + Boundary c.: E xs + Boundary c.: H xs z =0 z =0 + = E xs + = H xs z =0 z =0 → → µ2 , ε2′ , ε 2′′ µ1 , ε1′, ε2′′ E x+10e− jk1z η2 Region Region x Ex+10 Ex+10 η1 E1− , H1− E+2 , H +2 Transmitted wave Reflected wave = = E x+20 Ex+20 η2 z=0 z → η1 = η2 (unreasonable) − − jk1 z Exs = E e x10 H −ys1 =− η1 E x−10 e jk1z Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn Reflection of Uniform Plane Waves at Normal Incidence (2) Exs1 = E xs ( z = 0) → + − Exs + E xs1 = + E xs ( z = 0) H ys1 = H ys ( z = 0) → + − H ys + H ys1 = → Ex+10 + E x−10 = H +ys → Ex+10 + E x−10 = Ex+20 ( z = 0) → Ex+10 η1 − Ex−10 η1 = Ex+20 η2 E1+ , H1+ Incident wave →Γ= Ex+10 η2 − η1 = η2 + η1 Ex+10 + E x−10 = E x+20 E+2 , H +2 Transmitted wave Reflected wave − + η2 −η1 → Ex10 = E x10 η2 +η1 Ex−10 µ2 , ε2′ , ε 2′′ µ1 , ε1′, ε2′′ E1− , H1− η2 + η2 − Ex10 − E x10 η1 η1 Region Region x z=0 →τ = Ex+20 Ex+10 = z 2η2 =1+ Γ η1 + η2 Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn Reflection of Uniform Plane Waves at Normal Incidence (3) Γ= Ex−10 Ex+10 η −η = η + η1 τ= Ex+20 Ex+10 2η2 = = 1+ Γ η1 + η2 Region is dielectric, region is conductor: η2 = jωµ σ2 + jωε 2' = →τ = → E x+20 =0 Γ = −1 → Ex+10 = −E x−10 − j β1 z j β1 z + − + + Exs1 = Exs + E = E e − E e xs1 x10 x10 Region Region x µ2 , ε2′ , ε 2′′ µ1 , ε1′, ε2′′ E1+ , H1+ Incident wave E1− , H1− E+2 , H +2 Transmitted wave Reflected wave z=0 z Dielectric: jk1 = + j β1 → Exs1 = ( e− j β1z − e j β1z ) Ex+10 = − j 2sin( β1 z ) E x+10 → Ex1 ( z , t ) = 2E x+10 sin( β1z ) sin(ωt ) Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn Reflection of Uniform Plane Waves at Normal Incidence (4) Γ= Ex−10 Ex+10 η −η = η + η1 τ= Ex+20 Ex+10 2η2 = = 1+ Γ η1 + η2 Region is dielectric, region is conductor: Ex1 ( z , t ) = Ex1 = → β1 z = mπ (m = 0, ± 1, ± 2, ) → 2π λ1 z = mπ → z = m λ1 µ2 , ε2′ , ε 2′′ µ1 , ε1′, ε2′′ E1+ , H1+ Incident wave 2E x+10 sin(β1 z )sin(ωt ) Region Region x E1− , H1− E+2 , H +2 Transmitted wave Reflected wave z=0 z x Conductor z = − λ1 z = −λ1 z = − λ1 z=0 Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn z Reflection of Uniform Plane Waves at Normal Incidence (5) Γ= Ex−10 Ex+10 η −η = η + η1 τ= Ex+20 Ex+10 2η2 = = 1+ Γ η1 + η2 H ys1 = H +ys1 = H −ys1 → H ys1 = + Exs η1 =− E1+ , H1+ η1 Incident wave E1− , H1− E+2 , H +2 Transmitted wave Reflected wave z=0 − E xs E x+10 µ2 , ε2′ , ε 2′′ µ1 , ε1′, ε2′′ Region is dielectric, region is conductor: + − H ys + H ys1 Region Region x z η1 (e− jβ1z + e jβ1z ) → H y1( z , t ) = E x+10 η1 cos( β1z ) cos(ωt ) Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn Reflection of Uniform Plane Waves at Normal Incidence (6) Γ= Ex−10 Ex+10 η −η = η + η1 τ= Ex+20 Ex+10 2η2 = = 1+ Γ η1 + η2 Region is dielectric, region is dielectric: µ2 , ε2′ , ε 2′′ µ1 , ε1′, ε2′′ E1+ , H1+ Incident wave η1 & η2 are positive real values, α1 = α2 = Region Region x E1− , H1− E+2 , H +2 Transmitted wave Reflected wave z=0 Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn z Reflection of Uniform Plane Waves at Normal Incidence (7) Ex Given η1 = 100Ω, η2 = 300Ω, Ex+10 = 100 V/ m Find the incident, reflected, and transmitted waves Region Region x µ1 , ε1′, ε2′′ µ2 , ε2′ , ε 2′′ E1+ , H1+ Incident wave E1− , H1− E+2 , H +2 Transmitted wave Reflected wave z=0 Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn z 10 Plane Wave Reflection at Oblique Incidence Angles (3) k1+ + H10 η1 η2 + E10 − E10 θ1 θ1′ Ez+10 Ez−10 k 1− − H10 θ1θ1′ + − jk 1+ r = E10 e E s−1 − − jk 1− r = E10 e = E20e − jk r + = E10 cosθ1e − jk1 ( x cosθ1 + z sinθ1 ) − ' − jk1 ( x cos θ1' − z sin θ1' ) = E10 cos θ1e = E20 cos θ e− jk ( x cos θ2 + z sin θ2 ) E s2 z θ2 θ2 H20 E s+1 E20 Ez 20 k2 x p – polarization, TM + Ezs − Ezs + − j k1+ r = Ez10e − − jk 1− r = E z10 e Ezs = Ez 20e − jk r + − E zs + E zs1 = Ezs (at x = 0) + − → E10 cos θ1e − jk1 z sin θ1 + E10 cos θ1′e − jk1 z sin θ1′ = E20 cosθ e − jk2 z sin θ2 θ1′ = θ1 ′ → k1z sin θ1 = k1 z sin θ1 = k2 z sin θ →  k1 sin θ1 = k2 sin θ → n1 sin θ1 = n2 sin θ Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 31 Plane Wave Reflection at Oblique Incidence Angles (4) + k1+ E10 + H10 η1 η2 − E10 k 1− θ1 θ1′ Ez+10 Ez−10 − H10 θ1θ1′ z θ2 θ2 H20 θ1′ = θ1 k1 sin θ1 = k2 sin θ2 + − E10 cos θ1e − jk1 z sin θ1 + E10 cos θ1′e− jk1 z sin θ1′ = = E20 cos θ2e − jk2 z sin θ + − → E10 cosθ1 + E10 cosθ1 = E20 cos θ2 E20 + − H10 + H10 = H20 (at x = 0) Ez 20 k2 x p – polarization, TM → + E10 cos θ1 η1 p − − E10 cos θ1 η1 p = E20 cos θ η2 p where η1 p = η1 cos θ1, η p = η2 cos θ Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 32 Plane Wave Reflection at Oblique Incidence Angles (5) + k1+ E10 + H10 η1 η2 − E10 k 1− θ1 θ1′ Ez+10 Ez−10 − H10 θ1θ1′ z θ2 θ2 H20 E20 Ez 20 k2 x p – polarization, TM + − E10 cos θ1 + E10 cos θ1 = E20 cosθ + E10 cosθ1 η1p − − E10 cos θ1 η1 p = E20 cos θ2 η2 p −  η2 p − η1 p E10 Γ p = + = E10 η2 p + η1 p  → τ = E20 = 2η p  p E+ η +η 2p 1p 10  Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 33 Plane Wave Reflection at Oblique Incidence Angles (6) Γs = τs = E y−10 E y+10 E y 20 E y+10 η2 s − η1s = η 2s + η1s 2η 2s = η2 s +η1s η1 η1s = cos θ1 η2 η2 s = cos θ + k1+ H10 + E10 η1 η2 − H10 k1− θ1 θ1′ H z+10 Ez−10 − E10 θ1θ1′ z θ2 θ2 E20 H20 H z 20 k2 x s – polarization, TE Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 34 Plane Wave Reflection at Oblique Incidence Angles (7) Ex A uniform plane wave is incident from air onto glass at an angle of 30o from the normal Find the fraction of the incident power that is reflected and transmitted for (a) p – polarization, & (b) s – polarization Given glass refractive index n2 = 1.45 sin 30o n1 sin θ1 = n2 sinθ → θ = asin = 20.2o 1.45 η1 p = η1 cos30o = 377 × 0.866 = 326 Ω η1 = η2 = µ1 µr1µ0 µ0 = = ε1 ε r1ε ε0 µ2 µr µ0 µ0 = = ε2 ε r 2ε ε r 2ε → η1 = εr η η2 → = n2 η2 n2 = ε r η 377 → η2 = = = 260 Ω n2 1.45 → η2 p = η2 cos θ2 = 260 cos20.2o = 244 Ω Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 35 Plane Wave Reflection at Oblique Incidence Angles (8) Ex A uniform plane wave is incident from air onto glass at an angle of 30o from the normal Find the fraction of the incident power that is reflected and transmitted for (a) p – polarization, & (b) s – polarization Given glass refractive index n2 = 1.45 η1 p = 326 Ω, η2 p = 244 Ω η2 p − η1 p 244 − 326 Γp = = = − 0.144 η2 p + η1 p 244 + 326 Preflected Pincident = Γ p = ( −0.144)2 = 0.021 Ptransmitted = − Γ p = − ( −0.144)2 = 0.979 Pincident Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 36 Plane Wave Reflection at Oblique Incidence Angles (9) Ex A uniform plane wave is incident from air onto glass at an angle of 30o from the normal Find the fraction of the incident power that is reflected and transmitted for (a) p – polarization, & (b) s – polarization Given glass refractive index n2 = 1.45 η1 377 = = 435 Ω o cos θ1 cos30 η 260 = 277 Ω η2 s = = o cos θ2 cos20.2 η −η 277 − 435 Γ s = s 1s = = −0.222 η2 s + η1s 277 + 435 Preflected = Γs = (−0.222)2 = 0.049 Pincident Ptransmitted = − Γs = − ( −0.222) = 0.951 Pincident η1s = + k1+ E10 θ1 + H10 η1 η2 θ1′ − E10 k1− Ez+10 Ez−10 − H10 θ1θ1′ z θ2 θ2 H20 E20 Ez 20 k2 x p – polarization, TM Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 37 Plane Wave Reflection at Oblique Incidence Angles (10) Total reflection: Γ = ΓΓˆ = cos θ = − sin θ2 n1 sin θ1 = n2 sin θ 2  n1  → cos θ = −   sin2 θ1  n2  → η2 p = j η2 p η p = η cos θ2 n If sin θ1 > n1 η1 p = η1 cos θ1 → η1 p > η1 > η1 p − j η2 p η2 p −η1 p j η2 p −η1 p Z →Γp = = =− =− → Γ pΓˆ p = η2 p +η1 p j η2 p +η1 p Zˆ η1 p + j η2 p → If sin θ1 ≥ n2 then total reflection n1 → θ1 ≥ θc = asin Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn n2 n1 38 Plane Wave Reflection at Oblique Incidence Angles (11) Ex Compute n1 so that total reflection occurs at the back surface n1 45o n2 = Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 39 Plane Wave Reflection at Oblique Incidence Angles (12) Total transmission: Γ = Γs = η −η Γ s = s 1s η2 s +η1s → η2 s = η1s η η2 η1 η1s = → = cos θ1 cos θ cos θ1 η2 η 2s = n1 sin θ1 = n2 sin θ cos θ2   n 2  → η2 1 −   sin θ1    n2     − = η1 1 − sin θ1    − 2 Γ p = → η2 n2  n1  2 −   sin θ1 = η1 1− sin θ1 → sin θ1 = sin θ B = n12 + n22  n2  Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 40 Plane Wave Reflection at Oblique Incidence Angles (13) Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 41 Plane Wave Reflection & Dispersion Reflection of Uniform Plane Waves at Normal Incidence Standing Wave Ratio Wave Reflection from Multiple Interfaces Plane Wave Propagation in General Directions Plane Wave Reflection at Oblique Incidence Angles Wave Propagation in Dispersive Media Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 42 Wave Propagation in Dispersive Media (1) Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 43 Wave Propagation in Dispersive Media (2) β (ω ) = k = ω µ 0ε (ω ) = n (ω ) ( ω ω c ωb Ec, net ( z , t ) = E0 e − j β a ze − jωa t + e − j βb ze − jωb t ) ω0 ωa ∆ω = ω0 − ω a = ωb − ω0 β ∆β = β − β a = βb − β ( → Ec , net ( z , t ) = E0 e− jβ ze jω0t e j ∆β z e − j∆ ωt + e− j∆β ze j∆ ωt βa β0 βb ) = E0e − j β0 z e jω0t cos(∆ω t − ∆β z ) → Enet ( z , t ) = Re[Ec, net ] = 2E0 cos(∆ωt − ∆β t) cos(ω0t − β 0t ) Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 44 Wave Propagation in Dispersive Media (3) Enet ( z, t ) = 2E0 cos( ∆ωt − ∆β t ) cos(ω0t − β0t ) ω vp, sm = vg(ω0) ωb v p, carrier = v p, envelope ω0 β0 ∆ω = ∆β ω0 vp, sb ωa β βa β0 βb ∆ω d ω lim = = v g (ω0 ) ∆ω→0 ∆β dβ ω Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 45 ... Uniform Plane Wave XIV Plane Wave Reflection & Dispersion XV Guided Waves & Radiation Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn Plane Wave Reflection & Dispersion Reflection. .. ( Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn ) 11 Plane Wave Reflection & Dispersion Reflection of Uniform Plane Waves at Normal Incidence Standing Wave Ratio Wave Reflection. .. Γ Plane Wave Reflection & Dispersion - sites.google.com/site/ncpdhbkhn 16 Plane Wave Reflection & Dispersion Reflection of Uniform Plane Waves at Normal Incidence Standing Wave Ratio Wave Reflection