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# Class Note for ECE 6340 with Professor Jackson at UH 2

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This 28 page Class Notes was uploaded by an elite notetaker on Friday February 6, 2015. The Class Notes belongs to a course at University of Houston taught by a professor in Fall. Since its upload, it has received 14 views.

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Date Created: 02/06/15

ECE 6340 Intermediate EM Waves Fall 2005 Prof David R Jackson ECE Dept Notes 17 General Plane Waves Assume Eltxyzgtgowltxyzgt jkxxkyykzz where WOC y Z 6 Helmholtz Eq V2Ek2 E 9 so V2wk2w0 which gives kxz kyz kzz 0 separation or equation General Plane Waves cont Denote Then gyxyz 6 and separation equation General Plane Waves cont we can also write 0 Iquot wltxyzgte j 39te The vector gives the direction of most rapid phase change The gvector gives the direction of most rapid attenuation VgtltE jw Vx zjwa Now look at Maxwell s equations General Plane Waves cont A a A a A a V y g 8x 8y 82 iojknjx jicjk jamp Hence jkgtlt jau 1 Similarly V X E 82 Note 880 in general jkgtlt jw8 2 General Plane Waves cont Note 880 in general General Plane Waves cont Equations 1 and 2 imply that Power Flow XE IN 1 2 1 VgtltE Jaw Power Flow cont 1 1 gtxlt so SEWE0XEXE0 aw lwlzgox x Use AX gtltQ A Q Q439 so onEgtltEo50 EJE 50EEJ S lwr Eo 39E0E E0 Power Flow cont lwm oEiwEO EgtE5 Assume E0 real vector or a real vector times a constant gtXlt EOE Zk E k 0 Hence Power Flow cont so lt Z gt2 w 02 Reg 20 Denote EZIB jg Power Flow cont Then Note The conclusion is also valid if we assume that lc real vector times a complex constant Direction Angles First assume lc real vector The direction angles k k sin 6 COS 6 1 are defined by x ky k51n651n k2 ZkCOSQ Direction Angles cont 0056EL k tan y k X Even when kx ky k2 become complex these equations define the direction angles which may be complex Homogeneous Plane Wave In this case kkisin cos isin 9in cos k39 jkquotisin cos isin sin cos 1 Hence where 5zgsin cosg ysin65in gcos Example Z Plane wave x x y S An infinite current sheet at z O launches a plane wave 39k k is 46 1U yy kxky ereal kx2 Icy2 k 2 k02 Example cont Part a kx 2051 ky 2051 1 k2 k02 025 kg 025 k022 iizko Choose kz 0707 0 Then 5k0g052050707 PE Power flow y Example cont 1 k2 W 4k02 9k02gt2 zijQJ Choose 2 Z jko M Then kkoilt2gt2lt3gt lt NE koilt2gt2lt3gt2lt0gt gko ltd gt Example Z Power flow in lane 30 P y x k2 7r COSHZ Z jVIZ so 6 l956 rad co 2 Note the inverse y cosine should be tan 3 chosen so that the x sin is correct to give SO 56310 the correct kx Wave is evanescent in the Z direction Example cont propagating evanescent Free space acts as a lowpass filter Radiation fxrom Waveg ui39ide PEC Z I y gt WG Ey 7239x Ey xy 0 cos a Radiation from Waveguide cont Fourier transform pair 00 00 kxxky Eyltkxkyz HEyltxyzgteJ ydxdy OO 00 1 00 00 jkxxkyygt Eyxyz27r2 I IEykxkyze dkxdky OO 00 Radiation from Waveguide cont 0000 39kxxky Eyxyz m2 j JEykxkyze J y dkxdky 2 2 V Ey k Ey O 62E 82E 62E a 5 ayzy a 2yk2Ey0 x Z H ence 2E 2 2 2 kx Ey ky Ey azzy k Ey O Radiation from Waveguide cont 2E 2 ykZEy O 2 62 39kZZ Eykxkyz Eykxky0e 1 Hence 2722 y 00 OO 3 7g kx2ky2ltk02 Hi gt w kx2ky2gtk02 Eyxyz 1 E kxIcy0e quotx quotyyquotzzdkxdky Theorems Theorem 1 EEU2 Theorem 2 If PW is homogeneous IEIZ 772 l lz ltIossygt IEI2 772 Iossless Theorem 3 If medium is lossless a0 Example E ie kxchyJchz 2wx y Z kx 2k0 Note It can be seen that Icy 0 k2 M k0 Find and compare its magnitude with that of E Example cont Vx jw jkgtltE ja L aw z 20 j gtlt010wxayaz W E Ex Example cont i ilt2gt ie mu 770 l l i22 M 212 w 770 J7 W 770 IE 770 Note The field magnitudes are not related by no

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