WAVEGUIDES AND SYSTEMS Parallel-plate waveguide: TE case Plane wave interference satisfies boundary conditions x̂ kz θi ŷ θi ko E kz = kosin θi λo x ẑ ko ( E k o = ω μo εo λx = 2π/kx = λo/cosθi Null lines λo kx = kocos θi x ẑ vp = vo/sinθi vg = vosin θi λz = λo/sinθi ) E = yˆ Eo e jk x x − jk z z − Eo e− jk x x − jk zz = yˆ 2jEo sink x x ⋅ e− jk z z ⇒ k x = m π a ∂E y ⎞ ∇ ×E 1 ⎛ ∂E y ˆ ˆ H=− =− −x ⎜z ⎟ jωμ jωμ ⎝ ∂x ∂z ⎠ = x=a ⇒ m E 2Eo (xˆ ⋅ k z sink x x − zˆ ⋅ jk x cosk x x)e− jk zz jωμ x=0 λx =a 2 m=3 L16-1 TE10 WAVEGUIDE MODE Add Sidewalls to TE1 Parallel Plate Waveguide ⇒ TE10: E Surface charge σs x̂ a - σs = n̂ ⋅ εE + + - Power flow b ŷ ẑ H Surface current Js Js = nˆ × H Antennas can be coupled through holes or slots; slots at angles can radiate x̂ a b ŷ H Sidewall slots that don’t block current L16-2 MODE PATTERNS - TE TE modes: ( ) π x ⋅ e − jk zz E = y2jE sin m ˆ o a () 2 − k 2x () ( ) ω c = 2 − mπ a H(t, x,z) ω3,0 = 3πc/a E(t, x,z) ŷ S ⇒ a=λx/2 ω2,0 = 2πc/a Slope = vg ω1,0 = πc/a Slope = vp Slope = c 0 ⇒ kz v phase = ω kz TE2 mode: S 2 = 0 for 'cutoff frequency' ωmn ω TE1 mode: x̂ kz = ω c a=λx ẑ v group = dω → 0 at ωco = m πc dk z a ω < ωco → k z = jα Evanescent L16-3 RECTANGULAR WAVEGUIDE MODES a x̂ ẑ ẑ x̂ ŷ b TE11 b TE10 a a E H ŷ x̂ E b a b H ŷ x̂ H E ŷ E H ẑ ẑ ŷ H x̂ E kz = a b () ( ) ( ) ω c 2 − mπ a ω a TM11 No TEoo,TMoo, TMmo,TMon x̂ ẑ ŷ b E TE20 H TM11TE11 TE01 ω20 ω11 ω01 TE10 ω10 ẑ 2 − nπ b 2 TE(M)mn : a < b Dispersion Relations slope=vg=dω/dkz k z = ω με slope=vp =ω/kz kz L16-4 RECTANGULAR WAVEGUIDE DESIGN Modes and Dimensions: TEm0 modes: a = mλx/2 b E Em0 = yˆ ⋅ Eo 2jsink x x e − jk zz kz = (k o2 − k x 2 ) = a ωo2 2 − ( mπ )2 = 0 if ωo = mcπ a a c TEmn modes: kx = mπ/a, ky = nπ/b ⇒ k z = k − k − k = 2 o Cutoff Frequencies: Want b ≤ a/2 so that f01 ≥ f20 = c/a (f01 = c/2b) fm0 = mc 2a m 4 3 ; f0n = 2 x 2 y x ( ) ( ) ωo2 mπ − a c2 2 − nπ b 2 nc 2b Evanescence Single propagating mode 2 (if a ≥ 2b) [TE10] 1 (Two modes would interfere, 0 producing nulls in frequency) a = λ10/2 ⇐ c/2a = f10 Propagation TE20 TM20 TE10 f 2c/2a = f20 ⇒ a = λ10 c/2b = f01 L16-5 CAVITY RESONATOR DESIGN Derivation of resonant frequencies fmnp z b x y a d≥a≥b TEmn waveguide modes: λy ωo2 λx mπ 2 2 2 a=m , b=n ⇒ k z = ko − k x − k y = − 2 2 a c2 ( ) ( ) 2 − nπ b 2 Shortest side Longest side TEmnp resonator modes: a = mλx/2, b = nλy/2, d = pλz/2 pπ ⇒ k z = k o2 − k 2x − k 2y = 2π = ⇒ λz d ⇒ ωmnp = c ( maπ ) + ( nbπ ) 2 2 2 2 ωmnp c 2 ⎛ pπ ⎞ + ⎜ ⎟ ⇒ fmnp ⎝ d ⎠ ( ) ( ) ⎛ pc ⎞ = ( mc ) + ( nc ) + ⎜ ⎟ 2a 2b ⎝ 2d ⎠ − mπ a 2 − nπ b 2 2 2 ⎛ pπ ⎞ −⎜ ⎟ = 0 ⎝ d ⎠ 2 2 L16-6 CAVITY RESONATOR PERTURBATION Work = force • distance = Δw Fe [N/m2] <⎯fe > = ρsE/2 (attractive) = εE2/2; <Fe> = εοE2/4 = <We>[J/m3] ⎯E Fm [N/m2] =⎯Js ×⎯Hμo/2(repulsive) Cross-section = μH2/2; <Fm> = μoH2/4 = <Wm>[J/m3] At resonance <we> = <wm> = wT/2 wT = n hf, so ΔwT = nhΔf Δf [Hz] = ΔwT/nh = f (ΔwT)/wT Δf [Hz] = f •(Δwm – Δwe)/wT Thus pushing in the wall where Wm dominates does work and raises fmnp; where We dominates fmnp drops <⎯fm > ⎯H <⎯fe > <We>, <Wm> L16-7 WAVEGUIDE SYSTEMS Physics of Matching: Choose waveguide that propagates only one mode at f. Structural discontinuities cause reflections; tuning cancels them. Given some reflecting obstacle: Cancel reactance with [= f(freq.)] near obstacle Cancel reflection with offset [magnitude and phase (Δ)] Δ Example: Matches coupler Waveguide coupler Horn transition Matches transition Horn aperture L16-8 WAVEGUIDES AND SYSTEMS Parallel-plate waveguide: TE case Plane wave interference satisfies boundary conditions kz θi θi ( kz = kosin θi λo ko x ẑ ko x̂ E E k o = ω μo εo λx = 2π/kx = λo/cosθi Null lines λo kx = kocos θi x ẑ vp = vo/sinθi vg = vosin θi λz = λo/sinθi ) E = yˆ Eo e jk x x − jk z z − Eo e− jk x x − jk zz = yˆ 2jEo sink x x ⋅ e− jk z z ⇒ k x = m π a ∂E y ⎞ ∇ ×E 1 ⎛ ∂E y ˆ ˆ H=− =− −x ⎜z ⎟ jωμ jωμ ⎝ ∂x ∂z ⎠ = x=a ⇒ m E 2Eo (xˆ ⋅ k z sink x x − zˆ ⋅ jk x cosk x x)e− jk zz jωμ x=0 λx =a 2 m=3 L16-9 MIT OpenCourseWare http://ocw.mit.edu 6.013 Electromagnetics and Applications Spring 2009 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms.