WAVEGUIDES AND SYSTEMSTE10 WAVEGUIDE MODEMODE PATTERNS - TERECTANGULAR WAVEGUIDE MODESRECTANGULAR WAVEGUIDE DESIGNWAVEGUIDE SYSTEMSWAVEGUIDES AND SYSTEMSL16-1WAVEGUIDES AND SYSTEMSPlane wave interference satisfies boundary conditionskz=kosin θivg=vosin θikx=kocos θivp=vo/sinθi=ωμεoookλoλo()− − −−π=−=⋅⇒=jk x jk xjk z jk z jk zzz zoxxxxooEe e 2jˆˆyE E y E e sink x k maλ=xma2x=0x=a−∂∂⎛⎞∇×=− =− −⎜⎟ωμ ωμ ∂ ∂⎝⎠=⋅ −⋅ωμyyjk zozzx x xEEE1ˆˆHzxjjxz2Eˆˆ (xksinkx zjkcoskx)ejENull linesˆzm = 3Parallel-plate waveguide: TE case⇒EEλz= λo/sinθiλx= 2π/kx= λo/cosθixxˆzθiˆxkokoθikzyˆL16-2TE10WAVEGUIDE MODEAdd Sidewalls to TE1Parallel Plate Waveguide ⇒ TE10:Surface charge σssˆnEσ= ⋅εˆxˆzˆyEH--abPower flow+Surface currentsˆJnH=×sJHˆxˆySidewall slots thatdon’t block currentab+Antennas can be coupled through holes or slots; slots at angles can radiateL16-3MODE PATTERNS - TE()−π=⋅jkzzoEy2jEsinmxeˆaTE modes:()()()ωωπ=−=−ω2222zxmnmkkcca = 0 for 'cutoff frequency' phasezvkω=group cozdcv0 at mdk aωπ=→ ω=co zkjω<ω → = αEvanescentkzωω1,0= πc/aω2,0= 2πc/aω3,0= 3πc/a0Slope = cSlope = vpSlope = vgTE1 mode:ˆxE(t,x,z)H(t,x,z)ˆya=λx/2⇒STE2mode:ˆza=λx⇒SL16-4RECTANGULAR WAVEGUIDE MODESTE11ω10ω20ω11ω01zk=ωμεTE10TE01TE20ωslope=vp=ω/kzslope=vg=dω/dkzkzDispersion Relations()()()222zmnkcabωππ=−−mnTE(M):a b<TM11ˆxTE11ˆzˆxˆyEHHabˆyˆzbaEˆyTM11ˆzˆxˆzˆxˆyEEHHbbaaNo TEoo,TMoo,TMmo,TMonTE10ˆxˆyEEHHabˆxˆzaˆyˆzbL16-5Modes and Dimensions:Cutoff Frequencies:TEm0modes: a = mλx/2()−=⋅ωππ=−=− =ωjk zzm0ox222 2ozox o2ˆEyE2jsinkx emmck k k ( ) 0 if = aacm43210TE20TM20fTE10PropagationEvanescencea = λ10/2 ⇐ c/2a = f102c/2a = f20 ⇒ a= λ10c/2b = f01Want b a/2 so that f01≥ f20= c/a (f01= c/2b)Single propagating mode(if a ≥ 2b) [TE10](Two modes would interfere, producing nulls in frequency)m0 0nmc ncf ; f2a 2b==≤RECTANGULAR WAVEGUIDE DESIGNaxbETEmnmodes: kx= mπ/a, ky = nπ/b ⇒()()ωππ=−−= − −222222ozoxy2mnkkkkabcL16-6TEmnwaveguide modes:()()λωλππ==⇒=−−=−−222y222oxzoxy2mnam , bn k k k k22 abcCAVITY RESONATOR DESIGNabd ≥ a ≥ bxyzTEmnpresonator modes: a = mλx/2, b = nλy/2, d = pλz/2 ()()()() ()()ωππ⎛⎞πππ⇒= −−== ⇒ − − − =⎜⎟λ⎝⎠π⎛⎞ ⎛⎞ππ⇒ω = + + ⇒ = + +⎜⎟ ⎜⎟⎝⎠ ⎝⎠2222mnp222zoxy2z2222 22mnp mnppp2mn kkkk 0dabdcppcmn mcnc c fabd 2a2b2dDerivation of resonant frequencies fmnpLongest sideShortest sideL16-7CAVITY RESONATOR PERTURBATIONWork = force • distance = ΔwFe[N/m2] = ρsE/2 (attractive)= εE2/2; <Fe> = εοE2/4 = <We>[J/m3]Fm[N/m2] =⎯Js×⎯Hμo/2(repulsive)= μH2/2; <Fm> = μoH2/4 = <Wm>[J/m3]At resonance <we> = <wm> = wT/2wT= n hf, so ΔwT=nhΔfΔf[Hz] =ΔwT/nh = f (ΔwT)/wTΔf [Hz] = f •(Δwm– Δwe)/wTThus pushing in the wall where Wmdominates does work and raisesfmnp; where Wedominates fmnpdrops⎯E⎯H<⎯fe><We>, <Wm><⎯fe><⎯fm>Cross-sectionL16-8Example:Matches couplerWaveguide couplerHorn transitionHorn apertureMatches transitionPhysics of Matching:Choose waveguide that propagates only one mode at f.Structural discontinuities cause reflections; tuning cancels them.WAVEGUIDE SYSTEMSGiven some reflecting obstacle:Cancel reactance with[= f(freq.)] near obstacleCancel reflection with offset[magnitude and phase (Δ)]ΔL16-9WAVEGUIDES AND SYSTEMSPlane wave interference satisfies boundary conditionskz= kosin θivg= vosin θikx= kocos θivp= vo/sinθi=ωμεoookλz= λo/sinθiNull linesλoλx= 2π/kx= λo/cosθiˆzλoθiˆxkokoθiˆz()− − −−π=−=⋅⇒=jk x jk xjk z jk z jk zzz zoxxxxooEe e 2jˆˆyE E y E e sink x k maλ=xma2x=0x=a−∂∂⎛⎞∇×=− =− −⎜⎟ωμ ωμ ∂ ∂⎝⎠=⋅ −⋅ωμyyjk zozzx x xEEE1ˆˆHzxjjxz2Eˆˆ (xksinkx zjkcoskx)ejEkzm = 3Parallel-plate waveguide: TE case⇒xxEEMIT OpenCourseWare http://ocw.mit.edu 6.013 Electromagnetics and Applications Spring 2009 For information about citing these materials or our Terms of Use, visit:
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