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# Class Note for ECE 6341 with Professor Jackson at UH

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This 31 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 37 views.

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Date Created: 02/06/15

Fields in a Source Free Region Example radiation Z I from an aperture Assume the A 2 2 following choice A Z14ZC9 ya Z V AZ k AZ 2 O of vector A potentials E F x y Z VZFZ 25 O Example cont Introduce Fourier transform 212ltkxakyaz I I Amyze quotx ky dxdy CXD CXD 1 2702 I I flagkyzgte quotx quoty dkxdky CXD CXD AZ x y Z 1 M 62 VA sz k2 k2 k2 Z WWII y 622 gt 21 kx ky z e quotx quoty dkxdky Z CXD CXD 0 Example cont 6 822 Hence 2 2 2 k kx ky AZ 0 1 Define k2 k2 kx2 kyz2 Then 8212 822 1sz 0 Example cont Solution 212 kx ky Z 212 kx ky 0 e szz Similarly 133kka z 12kxky0e szz Example cont Hence 1 21 k k 0e quotx kyyquotzzdk dk Z x y x y Azx9y9Z Fzxyz 27102 f 132kxIcy0e j kx quotyyquotzzdkxdky This is a representation of the potentials as a spectrum of plane waves Example cont Derivative property 65142 1 2 J J jkx1212kx 9 ky 9 e jkxxkyyk22dkxdky x 72 OO 00 Z 2712 i i J kx112kwkyazejkxxkyydkxdky Also 6A2 1 a 2 2 Jewx k ukxdky X 7 6A2 6x Similarly for they and Z derivatives Hence jkx1azkxakyoz Example cont Apply BC s at Z 0 E 1 6212 18FZ x jau8 8x82 8 8y 2 E Z 1 8 AZ 1 85 y jaw8 8382 8 6x Hence 1 1 Ex ka szAz 10418 8 1 1 Ey JkyX szMz jwy8 8 jkyFz jk z Example cont Hence we have zero subscript denotes z 0 Exo kx ky kaZ jAZOUCX ky J ijZOUCx ky jams 8 k2 N k EyOkxkyijEJA20kxkJ ij20kxkJ We have two equations in two unknowns Zzoj ZO Hence we can solve for 1420911720 10 Example cont In the space domain 1 27 J JZzokx kye ijZe fkxXkyJdkxdky OO 00 OO AZ 96 y 2 2 and similarly for FZxy Z We can then find the fields from the TEZ TMZ equations 11 mpte Radiation uritde A 7Z39x Em x y 1E0 005 7 Find the complex power radiated by the aperture Example cont ExO 220 0 jays 8 kk 39 Ey0 y szzO jkijzO jays 8 Hence From the second equation 12 13 Example cont k Hence kz EyO 420 X 6sz zj k Icy 133 420 kz jamgky Hence Example cont Eyoac ky Complex Power Example cont Example cont Parseval s theorem Hence Pc l 1 2 JJEyo odkxdk 227r OOOO y k 1 L K zzgl zgl Z31 57109 Z x 377 09f K 77 0x I Ty 3 mfzyxy2f 21F I 2f T g JO gimp Z X 377 mf I f n 02 zx 20x V 3131 1 1 I 1 I E eoueH z x 2 f 77 24 31f lm I 14 21f T xg 29x9 grimf 9 n z z sz I VQ I OXH puu MON X H 391u00 eldwexa Example cont 08 1 kij2 or HxOZ Ey0k k2k2ky2 k2 1 1 oooo PC 2 j IEyonodkxdc 227r OOOO y Hence 1 1 2 we 1 kzla 2 2 x z 11 JV 5211 k2 Note 2 2 k k k is real 18 Examp 8 cont k y Polar coordinates kl kx k cos p kpW ky kp sin New notation Eyo kx9 ky gt EyO kpj Example cont k k2 kj kj real 52 kg k2 k22 k2 kj k ky2 k AISO 00 00 27m Hodkxdky Hokp dkp 61 Hence mm 0 0 I a 8 2 jkp sin2 cos2 dkpdg MM 1 1 2 27roo 375 J N Example cont gem FCCj kp sinz 0032 i 1 k2 k2 492 Eye lgj 2 F lgj dicde where k2 imaginary kp gt k Fkp imaginary kp gt k Example cont k y visiblespace circle Imaginary power Equivalent Circuit I y Z 9 I E inc 4N ref VV I E ZS Impedance Surface yo 23 I EltEx 3 dS S Equivalent Circuit cont e C n e H y 1W d m 1m w Jx W 2390 z a N y H F 2s 0 0 Ey 1 2 a2 a2 a2 V I fo o w b2 bu2 b2 b2 s s PC TEN Model I Z Jr 20 W I ZL Z TEN modeling equations x Ey x y 2 VZ COS 7 x Hx x y 2 12 COS a 25 TEN Model cont VZ Eyx y z Z 0 Hz Hxltxyz TE 0 ZO k 100 Hence TEN Model 20 ZOTE 770 TEN Model cont TEN Model cont 41 1 n H ab2 27 00 where foej F kpj g fjkp sin2gjz cos2 Eyowp f kp vkpdi YL GL JBL visiblespace region invisiblespace region TEN Model cont WIEWW if J if 1 mg 2 g 2 2 k ijpsm k cos 4 2 fkpdkpdi Eyoacp f Ami dkpdi Explicitly accounting for the 2 term the result can be written as G Wm k m a e 0 9 9 L 6117227 ooyp 1p k2k p Bili22 2Fltk a 1 died L ab 2 27 0k 1 p p k2 k2 where F1 kpj kz foej mgkp sin2g ZJZ cos2 MM ECE 6341 Spring 2009 Prof David R Jackson ECE Dept Notes 2

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