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# Class Note for ECE 6340 with Professor Jackson at UH 3

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Date Created: 02/06/15

ECE 6340 Intermediate EM Waves Fall 2008 Prof David R Jackson Dept of ECE Notes 9 Fields of a Guided Wave Oquot I Theorem Assume E06 y Z E0 6 Guided wave xyzH0 6 72 Then The l subscript denotes transverse to z Fields of 3 Guided Wave cont Proof for Ey Vx ngc E Z 1 8HZ8Hx y jwec 8x 82 1 H or Ey 8 27ij ngc 8x Now solve for Hx VXE jw Fields of 3 Guided Wave cont H 2 1 aEZ Ey x jau ay 52 2 aEZE jaw 0y y Substituting this into the equation for Ey yields the result Multiply by jayjagc k or Fields of 3 Guided Wave cont 8H Z k2 Ey jam 8x The other components may be found similarly Fields of 3 Guided Wave cont Summary of Fields jwe jaEZ 7 deIZ Hy 2 2 2 2 7 k 6x 7 k 6y Wavenumber Property of TEM Wave Assume a TEM wave To avoid having a completely zero field y2 k2 u 7 Z 0 1393 se k2 1a 30 k 2 k2 2 0 Note this holds even for a Z lossy medium Wavenumber Property cont Lossless TL 2 0N LC k a 8 1 1 v 19 The phase velocity is equal to the speed of light in the dielectric Wavenumber Property cont Lossy TL 19 317 meawcoc k ME wi ma jequot Note TEMZ r H assumption JJWLGJWC Milo Jg requires that R 0 0th I EZ 0 r H froringl rewslw JaLGJaCa28 Jlt9 Imaginary part Lsz quot Static Property of TEM Wave The fields of a TEM mode may be written as EONJ E005 6 EtOltx9ye 7Z HOW 2 Emmy 7 J79 17 Theorem Em x y and mU y are 2D static field functions Static Property of TEM Wave cont Proof VXEIO QJQRgt O Ngt 3 a a Ex0 11 2 aEyO aExo 8x 8y Therefore only a Z component of the curl exists We next prove that this must be zero Static Property of TEM Wave cont Nowuse ngvxltgmewgt e 72VxE Ve72gtlt t0 t0 e7ZVgtltE WA z0 7e ZXEzo 2Vx e 2ltVx o Also 2V x g 2 jw jw z 0 Static Property of TEM Wave cont Hence 2VXEZO Therefore Static Property of TEM Wave cont Also V E Z O No charge density in the timeharmonic steady state for a homogeneous medium Therefore V Em 6 72 0 V39Er0e Er0 39Ve7z O V Em 6 72 ErOg 76 72 O Static Property of TEM Wave cont VXEIO xay 9 EIO VDx9 V39Eto 0 up V2q30 16 Static Property of TEM Wave cont V2 x y 0 I constant on A or B E0 0 inside conductors The potential function is unique because of the uniqueness theorem of statics and is the same as the static potential function 17 Static Property of TEM Wave cont The static property tells us why a TEMz wave can exist on a transmission line two parallel conductors Transmission line A nonzero field can exist at DC 18 Static Property of TEM Wave cont The static property also tells us why a TEMz wave cannot exist inside of a waveguide hollow conducting pipe Waveguide No field can exist inside at DC Static Property of TEM Wave cont Similarly V x SO HIO VCD xay TEM Mode Magnetic Field Vx ngc i i i ngc 8x 8y Hx Hy 0 so Ex 7 Hy Eyz o7 we we TEM Magnetic Field cont Also 7 jkz jk Nit 80 g 77 139 mac a mac 1080 1080 so Ex 77Hy E 2 77H y X This can be written as TEM Mode Charge Density TEM Charge Density cont Hence Example Microstrip Line Ignore substrate in order y air to have a TEM mode IOS x 1 our JSZ x pf x 80770 w 25 Example cont Strip in free space with a static charge density no ground plane 172 This was first derived by pm x Maxwell using conformal S wzy x2 mapping Hence In this result I0 is the total current Amps on the strip at z 0 Example Find E 51 Coaxial Cable V2q3 0 0 1361 V0 131 0 27 Example cont Boundary conditions C1 In a C2 V0 cllnbc2 0 so cllna lnb V0 Hence c1 V0 czz cllnbz Volnb In g In E b 7 Therefore CD V0 1np Vomb 2 V0 Imp 119 119 In E b b b b Example cont A8CID Ezoltxay VDxay Ba 0 Example cont This result is valid at any frequency Example cont V VI e ij 39 x IZEaH a 2a IQ jkz e 77a 1n a Example cont Lossless coax assume a no Result Example cont Lossy coax LC 2 ug39 L USS ESS 8quot E 25 I G 2 0C Result TEM Mode Telegrapher s Eqs Telegrapher s Eqs cont x y 4 5 8c 2 a d idy 82 A 82 82 Note v is path A independent in r V X 2 ds the xy plane C S Telegrapher s Eqs cont Use VX O 81 59 g 693 ga 6 ay 62 a 0x 62 5 8 9 WM dy So aZ A at l or Telegrapher s Eqs cont A QCV 2xd xlt dx dygt 3205c idy dy dy m mcvdzbw 4 But L E K i w Li Note L is the magnetostatic DC value 37 Telegwrapher s qus cont If we add R into the equation This is justifiable if the mode is approximately a TEM mode small conductor loss 38 Telegrapher s Eqs cont r Cl I I The contour Cl hugs l I theA conductor l l 39 I I A B Ampere s law 139 dy C C 6 so 4adx ydy c 62 62 Note there is no displacement current through the surface since E2 0 Telegrapher s Eqs cont Now use 6WMg Telegrapher s Eqs cont Hence ai 6 69 dx xd dx d 82 ty Km 6 at a a C dex Qxdylt ydx dy C cl Telegrapher s Eqs cont l dy A Adx5cd I X y I C i QEnc ps 4dxgxdy lt ampqd1 pm p C C Cf ydx gcdy Cf cidl ileak C C pz 2 CV C and G are the static values lleak Gv Telegwrapher s Eqs cont Hence 2 2 CV GV 82 at Telegrapher s Eqs Alternate Derivation Telegrapher s Eqs Alternate Derivation cont

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