1Spring 2011Notes 30ECE 6345Prof. David R. JacksonECE Dept.2OverviewIn this set of notes we use the spectral-domain method to find the mutual impedance between two rectangular patch antennas.3GeometryxyxyLLWWxzLhrL1I2I4Mutual Impedance FormulationAssume two arbitrary antennas, to be general.+-1I1V+-2I2V1 11 1 12 22 21 1 22 2V Z I Z IV Z I Z IThe two-port system is described by a Z matrix. 1iJ2iJ5Mutual Impedance Formulation (cont.)The self impedance Z11is calculated.+-1I1V+-2V1 11 1 12 22 21 1 22 2V Z I Z IV Z I Z I11 inZZ(The presence of open-circuited antenna 2 does not significantly affect the input impedance of antenna 1.) 211110IVZI20I 6+-1I1V222110IVZI1 11 1 12 22 21 1 22 2V Z I Z IV Z I Z IMutual Impedance Formulation (cont.)The mutual impedance Z21is calculated.+-2V20I 71IE1= electric field produced by the feed current I1, in the presence of antenna 1 and antenna 2.+-2C2V221CV E drThe open-circuit voltage V2is obtained by integrating the electric field from current I1over the path C2. Mutual Impedance Formulation (cont.)81I+-2C2V22112212211,CiViiV E drE J dVIJJI  2I2iJThe open-circuit voltage V2is put in the form of a reaction. Mutual Impedance Formulation (cont.)1iJ91I2 1 221221,1,iiant iV J JIJJI2I2iJThe equivalence principle is used to replace antenna 1 with its surface current. 1I1antJThe field produced by current I1exciting antenna 1 is the same as that produced by the current on antenna 1.(The antenna current is that excited on antenna 1 when it is in the presence of open-circuited antenna 2.)Mutual Impedance Formulation (cont.)102 1 222122121,1,1,ant ii antant antV J JIJJIJJI2I2iJReciprocity is invoked, and then the equivalence principle. 1I1antJ2I2antJThe field produced by current I2exciting antenna 2 is the same as that produced by the current on antenna 2.Mutual Impedance Formulation (cont.)(The antenna current is that excited on antenna 2 when it is in the absence of antenna 2.)112 2 121221,1,ant antant antV J JIJJIReciprocity is invoked one more time. +-1I1V1antJ2I2antJMutual Impedance Formulation (cont.)122 1 221,ant antV J JI+-1I1V1antJ2I2antJ21 1 2121,ant antZ J JIIThe mutual impedance is thenMutual Impedance Formulation (cont.)Note:21 12ZZ1312 1 2121,ant antZ J JIISummary1I1antJ2I2antJJ1ant= current on antenna 1, when excited by current I1in the presence of open-circuited antenna 2.J2ant= current on antenna 2, when excited by current I2in the absence antenna 1.Mutual Impedance Formulation (cont.)14Mutual Impedance Between PatchesxyxyLLWW(1) (2)12121,sx sxZ J JIIAssume 121 [A]II15Mutual Impedance Between Patches (cont.)(1) (2) (1) (2) (1) (2) 2 (1) (2)12, , ,sx sx x x x x x x xZ J J A A B B A B B     1,2 (1) (2),xx x xZ B BDenoteRecall thatzxxxxZAZ21,212zxxxxxZZZZHence2 1,212 x xxZ A ZThen we haveNote: Formulas for Zzxand Zxxwere given in Notes 29.16Mutual Impedance Between Patches (cont.)Calculation of reaction Zxx1,2between patch basis functions:Hence, integrating over the surface of patch 2, we have   11212xyj k x k yx x xx x x yE B G B e dk dk x xx xE G B      121,221, , ,2xx xx x y x x y x x y x yZ G k k B k k B k k dk dk    17Mutual Impedance Between Patches (cont.)      121,221, , ,2xx xx x y x x y x x y x yZ G k k B k k B k k dk dk         21,,xyj k x k yx x y x x yB k k B k k e  From the Fourier “shifting” theorem, we haveHence we have     1,2 (1) (2) 221, , ,2xyj k x k yxx x x xx x y x x y x yZ B B G k k B k k e dk dk       Note: The “1” superscript is dropped henceforth.18Since the integrand is an even function of kxand ky, we can writeConverting to polar coordinates, we have     21,2 (1) (2) 22001, , ,2xyj k x k yxx x x xx x y x x y t tZ B B G k k B k k e k dk d      Mutual Impedance Between Patches (cont.)     /21,2 (1) (2) 22001, , , cos cosxx x x xx x y x x y x y t tZ B B G k k B k k k x k y k dk d                       0 0 0 000000000000002cos 2cos2cos4cos cosy y y yxxxxyyyyj k y j k y j k y j k yj k x j k x j k x j k xj k y j k yxxj k y j k yxxye e e e e e e ek x e k x ek x e ek x k y     Note:Quadrant 1 Quadrant 2 Quadrant 3 Quadrant 419Final form of mutual reactionMutual Impedance Between Patches (cont.)     /21,2 (1) (2) 2201, , , cos cosxx x x xx x y x x y x y t tCZ B B G k k B k k k x k y k dk d     RetkImtk0k1kLCRh20Summary     /21,2 2201, , cos cosxx xx x y x x y x y t tCZ G k k B k k k x k y k dk d   21,212zxxxxxZZZZ   /22201,,xx xx x y x x y t tCZ G k k B k k k dk d    /22121000( ) cos sincsin cosTMzx t i x zCx y tjhZ k I h B k hk x k y dk d 21ResultsD. M. Pozar, "Input Impedance and mutual coupling of rectangular microstrip antennas,“ IEEE Trans. Antennas Propagat., Vol. AP-30. pp. 1191-1196, Nov.