WaveguidesMicrowave EngineeringEE 172Dr. Ray KwokWaveguides - Dr. Ray KwokWave Equations( ) ( )( ) ( )2222222tEEEEEEtEHttHEtEHtHE∂∂µε=∇−∇=∇−⋅∇∇=×∇×∇∂∂µε−=×∇∂∂µ−=∂∂µ−×∇=×∇×∇∂∂ε=×∇∂∂µ−=×∇rrrrrrrrrrrrrrMaxwell’s source-free equationsSimilar to the H-field equations….222tHH∂∂µε=∇rrwave equationv2= 1/µεWaveguides - Dr. Ray KwokPhasor Equationsωε=×∇=⋅∇ωµ−=×∇=⋅∇⇔∂∂=×∇=⋅∇∂∂−=×∇=⋅∇EjH0HHjE0EtDH0HtBE0Drrrrrrrrrrrr)rkt(jo)rkt(joeH)t,r(HeE)t,r(Errrrrrrrrr⋅−ω⋅−ω==wave equationssource-free Maxwell’s equationsWaveguides - Dr. Ray KwokGuided Waves (along z-axis)yExEHjEjxEHjEjyEHje)y,x(Ee)y,x(Ee)y,x(EzyxzˆyˆxˆEHjExyzxzyyzxzjzzjyzjx∂∂−∂∂=ωµ−β−∂∂−=ωµ−β+∂∂=ωµ−∂∂∂∂∂∂=×∇ωµ−=×∇β−β−β−rrr[][ ])zt(jzyx)zt(jzyxe)y,x(Hzˆ)y,x(Hyˆ)y,x(Hxˆ)t,z,y,x(He)y,x(Ezˆ)y,x(Eyˆ)y,x(Exˆ)t,z,y,x(Eβ−ωβ−ω++=++=rryHxHEjHjxHEjHjyHEje)y,x(He)y,x(He)y,x(HzyxzˆyˆxˆHEjHxyzxzyyzxzjzzjyzjx∂∂−∂∂=ωεβ−∂∂−=ωεβ+∂∂=ωε∂∂∂∂∂∂=×∇ωε=×∇β−β−β−rrrβ = 2π/λgguided wavelengthWaveguides - Dr. Ray KwokTransverse fields∂∂β−∂∂ωµβ−=∂∂β+∂∂ωµβ−−=∂∂β+∂∂ωεβ−−=∂∂β−∂∂ωεβ−=yExHkjExEyHkjEyHxEkjHxHyEkjHzz22yzz22xzz22yzz22xOne can rewrite the transverse fields by combining the 6 equations g22vkλπ=βλπ=ω=µεω=i.e. The wave is completely determined once Ez& Hzare known !!Waveguides - Dr. Ray KwokTEM waves∂∂β−∂∂ωµβ−=∂∂β+∂∂ωµβ−−=∂∂β+∂∂ωεβ−−=∂∂β−∂∂ωεβ−=yExHkjExEyHkjEyHxEkjHxHyEkjHzz22yzz22xzz22yzz22xµε=λ=λ=λβ=1fvkgEz=0 & Hz= 0≠ 0 →independent of freq= distortionlessWaveguides - Dr. Ray KwokTEM field equationsEz=0 & Hz= 00yExEHjEjEjxEHjEjEjyEHjxyzxxzyyyzx=∂∂−∂∂=ωµ−β−=β−∂∂−=ωµ−β=β+∂∂=ωµ−0yHxHEjHjHjxHEjHjHjyHEjxyzxxzyyyzx=∂∂−∂∂=ωεβ−=β−∂∂−=ωεβ=β+∂∂=ωε222xyxkjHjjEjHj=β=µεωωεβ−β=β=ωµ−[][ ]0H0yHxH0E0yExED2xyD2xy=×∇⇒=∂∂−∂∂=×∇⇒=∂∂−∂∂rrsource-free2D staticequationsSolution: E & H = 0 inside conducting shellunless there is another conducting surface inside.i.e. No TEM waves inside hollow waveguide !!!!How can we see light through hollow waveguide???Waveguides - Dr. Ray KwokCommon TEM waveguidesEHkHEkCoaxial CablesMostly TEM wavesExcept at very high frequencies•Other modes become possible•Displacement current comparable to conduction current•Becomes expensive & lossyMicrostrip linesQuasi-TEM wavesIn general, more lossyBut inexpensive & smallerExcept at very high frequencies•Other modes become possibleWaveguides - Dr. Ray KwokCommon TE or TM waveguidesab ≈ a/2RHollow waveguidesWRxxx standardWR284 2.84” x 1.34” 2.60 – 3.95 GHz S-bandWR229 2.29” x 1.15” 3.30 – 4.90 GHz “C-band”WR75 0.75” x 0.375” 10 – 15 GHz “Ku-band”WR42 0.42” x 0.170” 18 – 26.5 GHz K-bandWR28 0.28” x 0.140” 26.5 – 40 GHz Ka-bandWaveguides - Dr. Ray KwokTM waves (Hz=0)[]( )( )222y2x2cz2cz22z2222ztjmnzz2cz22222c22222222)zt(jzyxbnamkkkEkEbnamEyxeybnsinxamsinBEEkEyxEkEkEEkEEEkEe)y,x(Ezˆ)y,x(Eyˆ)y,x(Exˆ)t,z,y,x(Eπ+π=+=−=π−π−=∂∂+∂∂ππ=−=∂∂+∂∂−≡−β=∇−=β−∇−=∇++=β−ω⊥⊥β−ωrrrrrrrrrxyzso that Ez= 0 when x = 0 or x = aand when y = 0 or y = bi.e. tangential E = 0 on the wallsFrom here, all other components of E & H can be determined.m = 1, 2, 3, … n = 1, 2, 3, …Waveguides - Dr. Ray KwokCut-off frequency222c22bnamkkπ+π=≡β−If k ≤ kc, β becomes imaginary and therefore no wave propagation !!!!2mn,c2mn,c222mn,cvf22bnamkπ=λπ=π+π=Or equivalently, if f ≤ fcor λ ≥ λc, no propagation !!!!λ>−λ=λλ−λ=λλπ=λπ−λπβ=−2c2cg2g2c222c2ff11222kkHEkββββλλλλg> λλλλλλλλaTop viewWaveguides - Dr. Ray KwokTM modes - impedanceTMyxz2czz22yz2czz22xZkkkv1kkkHExEkjyHxEkjHxEkjxEyHkjE≡ηβ=εµβ=εµεβ=εβ=ωεβ=ωεβ=∂∂ωε−=∂∂β+∂∂ωεβ−−=∂∂β−=∂∂β+∂∂ωµβ−−=Waveguides - Dr. Ray KwokTE waves (Ez=0)[]( )22222cztjmnzz2cz2222)zt(jzyxbnamkkeybncosxamcosAHHkHyxe)y,x(Hzˆ)y,x(Hyˆ)y,x(Hxˆ)t,z,y,x(Hπ+π=β−=ππ=−=∂∂+∂∂++=β−ωβ−ωrxyzso that Ex= 0 when y = 0 or y = band Ey= 0 when x = 0 or x = ai.e. tangential E = 0 on the wallsm = 0, 1, 2, 3, … n = 0,1, 2, 3, … but not both = 0∂∂ωµβ−=∂∂ωµβ−−=xHkjEyHkjEz22yz22xsame cut-off as TM modesWaveguides - Dr. Ray KwokTE modes - impedanceTEyxz2czz22yz2czz22xZkkkHEyHkjyHxEkjHyHkjxEyHkjE≡ηβ=ωµβ=βωµ=∂∂β−=∂∂β+∂∂ωεβ−−=∂∂ωµ−=∂∂β+∂∂ωµβ−−=Waveguides - Dr. Ray KwokExample – WR752mn,c2mn,c222mn,cvf22bnamkπ=λπ=π+π=What are the first 5 modes propagate in WR75?a = 0.75”, b = 0.375”0.224”0.300”0.416”0.5”m=30.237”0.335”0.530”am=20.247”0.364”0.671”2am=10.25”b2bm=0n=3n=2n=1n=0λc,mn52.8239.3728.3923.62m=349.8035.2122.2715.75m=247.9032.4717.617.874m=147.2431.5015.75m=0n=3n=2n=1n=0fc(GHz)c = 11.811 GHz-in f7.87GHz15.7517.6122.323.628.4TE10No signalTE20TE01TE11TM11TE21TM21TE30more & closerTransmission