Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 101Spring 2011Notes 12ECE 6345ECE 6345Prof. David R. JacksonECE Dept.2OverviewOverviewIn this set of notes we derive a general expression for the “p factor” that is used to determine the space-wave power radiated by the rectangular patch.In the next set of notes we will evaluate the integrals that appear and actually develop a final closed-form CAD expression for the p factor.3Definition of the p factor:dipspspPPp = power radiated by the actual rectangular patch= power radiated by a dipole that has the equivalent dipole momentdipspPspPThe The pp Factor Factor4The The pp Factor (cont.) Factor (cont.)We then havepPPdipspspFrom Notes 11, we have( )222 200 0 126dipsp rP WL k h k chmp p� �� �=� �� �� �� �12 41 11 2 / 51cn n= - +1,0cossxxJLp� �=� �� �assumingLWxy52 / 220 0( , , ) sinsp rP S r r d dp pq f q q f=��( ) ( )( )1 0hex ,p p sx x yE r,θ, E r,θ, J k ,kf%=Calculation of the space-wave radiated power:where22012rθS E Eη� �= +� �� �fhexpEand= far field of unit-amplitude horizontal electric dipole in the x direction.The The pp Factor (cont.) Factor (cont.)and6ThenDenote yxsxkkJA ,~,0,1( ) ( ) ( )hexp pE r,θ, E r,θ, A θ,f f=and( ) ( )2hexr rS r,θ, S A θ,f =The The pp Factor (cont.) Factor (cont.)7We also haveHenceNote that2 /2220 0( , ) sinhexsp rP S A r d dp pq f q q f=��( )2 /2220 0( ) , , sindip hexsp patch rP Il S r r d dp pq f q q f=��( )( ) ( )/2 /21,0/2 /21,0( , )2(0,0) 0, 0L WsxpatchL WsxIl J x y dy dxJ A WLp+ +- -== = =� �%The The pp Factor (cont.) Factor (cont.)8Hence( )( )2 /2220 02 /2220 0, , ( , ) sin, , (0,0) sinhexrhexrS r A r d dpS r A r d dp pp pq f q f q q fq f q q f=����The The pp Factor (cont.) Factor (cont.)9Note: depends on but not the substrate parameters. This may be written as2 / 22 2 22 20 02 / 22 2 22 20 0( ) sin ( ) cos ( , ) sin( ) sin ( ) cos (0,0) sinF G A d dpF G A d dp pp pq f q f q f q q fq f q f q q f� �+� �=� �+� �����),(A,W LThe The pp Factor (cont.) Factor (cont.)10The patch array factor is( )2 2cos2, sinc2 22 2xyxLkWA WL kLkpq fp� �� �� �� �� � � �� �� �=� �� �� �� � � �� � � �-� �� � � �� � � �� �00sin cossin sinxyk kk kq fq f==, 0W L �As( ) ( )2,A WLq fp�( ) ( ), 0,0A Aq f �so1p �HenceThe The pp Factor (cont.) Factor
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