1 Spring 2011 Notes 29 ECE 6345 Prof. David R. Jackson ECE Dept.2 Overview In this set of notes we use the spectral-domain method to find the input impedance of a rectangular patch antenna. This method uses the exact spectral-domain Green’s function, so all radiation physics, including surface-wave excitation, is automatically included (no need for an effective permittivity). It does not account for the probe inductance (the way it is formulated here), so the CAD formula for probe inductance is added on at the end.3 Spectral Domain Method xzL00( , )xyhsxJizJr01 [ ]IASet 0xE Syx ,  0izxsxxJEJE Syx ,S is the patch surface This is the “Electric Field Integral Equation (EFIE)” The probe is viewed as an impressed current.4 Spectral Domain Method (cont.) Let     ,,, cossx x xxJ x y A B x yxB x yL 0ix x x x zA E B E JPick a “testing” function T(x,y):    ( , ) 0( , ) ( , ) 0ix x x x zSix x x x zSST x y A E B E J dSA T x y E B dS T x y E J dSThe EFIE is then5 Spectral Domain Method (cont.) Galerkin’s Method: xT x,y B x,y   ,,,,iix x zzxSxxxx x xSB x y E J dSJBABBB x y E B dS   (The testing function is the same as the basis function.)   ( , ) , 0ix x x x x x zSSA B x y E B dS B x y E J dSThe solution for the unknown amplitude coefficient Ax is then Hence6 Spectral Domain Method (cont.) The input impedance is calculated as  iz z z z sxE E J E JThe total field comes from the patch and the probe: 20*202212ininizzVizzVPZIE J dVIE J dV(The probe current is real and equal to 1.0 [A].)7 Spectral Domain Method (cont.) Hence  i i iin z z z z z sxVVZ J E J dV J E J dV  Then we have  iin probe z z sxVZ Z J E J dV,i i i iprobe z z z z zVZ J E J dV J J   Define or  ,iiin probe x z z x probe x x zVZ Z A J E B dV Z A B J   8 Spectral Domain Method (cont.) We have from reciprocity that Hence ,,iin probe x x ziprobe x z xZ Z A B JZ A J B,,,izxiin probe z xxxJBZ Z J BBB  or 2,,izxin probexxJBZZBB9 Spectral Domain Method (cont.) Denote ,xx x xZ B B,ixz z xZ J B2xzin probexxZZZZThen we have Note: The subscript notation on Zij follows the usual MoM convention. 2zxin probexxZZZZUsing reciprocity again, Note: Zzx is easier to calculate than Zxz.10 Spectral Domain Method (cont.) Note: The probe impedance may be approximately calculated by using a CAD formula: probe pZ jX0001ln ln ln/p r r rhXa                 0.57722()Euler's constantThis result comes from a probe inside of an infinite parallel-plate waveguide. (Calculating Zprobe exactly from the spectral-domain method would be more difficult.)11 Spectral Domain Method (cont.) Hence, integrating over the patch surface, we have  ,xx x x x x xSZ B B E B B dS      212xyj k x k yx x xx x x yE B G B e dk dk       21, , ,2xx xx x y x x y x x y x yZ G k k B k k B k k dk dk     The next goal is to calculate the reactions Zxx and Zxz in closed form. From previous SDI theory, we have For the patch-patch reaction we have x xx xE G Bso12 Spectral Domain Method (cont.)     221,,2xx xx x y x x y x yZ G k k B k k dk dk Since the Fourier transform of the basis function (cosine function) is an even function of kx and ky, we can write    22001,,xx xx x y x x y x yZ G k k B k k dk dkor13 Spectral Domain Method (cont.) Converting to polar coordinates, we have    /22201,,xx xx t x t t tCZ G k B k k dk d  RetkImtk0k1kLCRhNote: The path must extend to infinity.14 Spectral Domain Method (cont.)  22cos2, sinc2222xx x y yxLkWB k k LW kLk                  From previous calculations, we have15 Spectral Domain Method (cont.)   000,,izx z x zhZ E B x y z J dzzxG~To calculate use 11yxzHHEj x yso that  11z x y y xE jk H jk Hj  For the patch-probe reaction we have     000021,,2xyj k x k yz zx x x yE x y z G z B e dk dk  where z zx sxE G Jso16 Spectral Domain Method (cont.)           cos sincos sincos sinsin cos cos sinx u vTE TMTE TMi sv i suTE TMi sx i sxH H HIII J I JI J I J              Using spectral-domain theory, we have17 Spectral Domain Method (cont.)       sin cossin cossin cossin sin cos cosy u vTE TMTE TMi sv i suTE TMi sx i sxH H HIII J I JI J I J        18 Spectral Domain Method (cont.)  2222sincoscossin cossinsin coscos cos sincosTEi sxx y y x tTMi sxTEi sxtTMi sxTMt i sxTMt sx iIJjk H jk H jkIJIJjkIJjk I Jjk J I     Hence we have cossinxtytkkkkNote:19 Spectral Domain Method (cont.) Hence we have we then identify that     11cosTMzx t iG z jk I zj 11cosTMz t sx iE jk J Ijz zx sxE G JUsing20 Spectral Domain Method (cont.) Hence 1( ) ( )cos ( )TM TMi i zI z I h k z h  From TL theory, we have the property that   111cos ( )cos ( )TMzx t i zG z jk I h k z hj  (The short circuit at z = -h causes the current to have a zero derivative there.)21 Spectral Domain Method (cont.) For the field due to the patch basis function we then have            00000000221211,,21211cos ( ) cos ( )2xyxyxyj k x k yz z x yj k x k yzx x x yj k x k yTMt i z x x yE x y z E z