Slide 1OverviewTotal Radiated PowerSlide 4Slide 5Slide 6Slide 7Space-Wave and Surface-Wave PowersRadiated Powers and EfficiencyRadiated Powers and Efficiency (cont.)Path for Calculating the Total Radiated Power1Spring 2011Notes 22ECE 6345ECE 6345Prof. David R. JacksonECE Dept.2OverviewOverviewIn this set of notes we use the spectral-domain method to calculate the surface-wave radiation efficiency ersw (radiation efficiency due only to surface-wave loss) of a rectangular microstrip antennasp sp sp swrtot sp sw totspradrad totsw dissr rP P P PeP P P PPPP Pe e� �+� �= =� �� �� �+� �� �� �� �=� �� �� �� �=1swrsw spswreQ Qe� �=� �� �-� �3Total Radiated PowerTotal Radiated PowerswspradPPP ( )( )( ) ( )***2*21Re21Re21 1Re ( , ) ,2 (2 )1 1Re , ,2 (2 )x ysradSx sxSj k x k ysx x x y x ySsx x y x x y x yP E J dSE J dSJ x y E k k e d k dk dx dyJ k k E k k dk dkpp+�+�- +- �- �+�+�- �- �� �= - �� ��� �= - �� ��� �= - �� ��� �� �= -� �� ����� ����%% %4( ) ( )*21 1Re , ,2 (2 )rad sx x y x x y x yP J k k E k k dk dkp+�+�- �- �� �� �= -� �� ����% %The transform of the current is assumed to be a real function of kx and ky.( ) ( )21 1Re , ,2 (2 )rad sx x y x x y x yP J k k E k k dk dkp+�+�- �- �� �= -� ����% %We then haveTotal Radiated Power (cont.)Total Radiated Power (cont.)Note: The transform with the conjugate is not analytic, but the transform without the conjugate is.5DefineNext, use/ 220 01Re2rad x sx t tP E J k dk dpfp+�=-��% %sxxxxJGE~~~/ 2220 01Re2rad xx sx t tP G J k dk dpfp+�=-��%%( )221,2p t xx sx tF k G J k%%fp=-In polar coordinates we haveso thatTotal Radiated Power (cont.)Total Radiated Power (cont.)6Then( )/ 20 0Re ,rad p t tP F k dk dpf+�=��)(1)(1~22tTEtytTMtxxxkDkkkDkkG( )( )0 1 10110 1( ) cotcotTM TM TMt zzz zD k Y jY k hj k hk kwewe= -� �� �= -� �� �� �� �From previous calculation,( )( )0 1 10110 1( ) cotcotTE TE TEt zzzzD k Y jY k hkkj k hwm wm= -� �� �= -� �� �� �� �Total Radiated Power (cont.)Total Radiated Power (cont.)7(same for DTE)00 1001::real, real, compleximaginary, real imaginary, imaginarytTMz zTMz ztk k k Dk k Dkk k= = == =<> = or0Re 0,x txkG k= >%We have the following properties:Hence we have the following property:( )1/ 22 20 0z tk k k= -( )1/ 22 21 1z tk k k= -Total Radiated Power (cont.)Total Radiated Power (cont.)8Space-Wave and Surface-Wave PowersSpace-Wave and Surface-Wave PowersThe TM0 pole gives a real-valued residue contribution.RetkImtk0k1kResjp-SP powerSW power9Radiated Powers and EfficiencyRadiated Powers and Efficiency( )0/ 20 0R e ,ksp p t tP F k dk dpf=��( )/ 20Re Res ,sw p tP j F k dpp f= -�spswrsp swPeP P=+10Radiated Powers and Efficiency (cont.)Radiated Powers and Efficiency (cont.)Alternatively,spswrradPeP=( )/ 20Re ,rad p t tCP F k dk dpf=��The total power comes from integrating along the rectangular path shown on the next slide.( )0/ 20 0Re ,ksp p t tP F k dk dpf=��11Path for Calculating the Total Radiated PowerPath for Calculating the Total Radiated PowerRetkImtk0k1kLCh( )/ 20Re ,rad p t tCP F k dk
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