Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18Slide 191Spring 2011Notes 13ECE 6345ECE 6345Prof. David R. JacksonECE Dept.2OverviewOverviewIn this set of notes we perform the algebra necessary to evaluate the p factor in closed form, and to simplify the final result.3Approximation of “Approximation of “pp””2 / 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� �+� �=� �+� �����From Notes 12 we have4Also assume thatApproximation of “Approximation of “pp””Assume00h /λ �( )( )02 21 0( ) 2 cos2( ) sinrrF j k hjG n k hq m qq qe�� �� -� �� �2 21sinr rn e m q= >>Then( )( )00( ) 2 cos( ) 2rrF j k hG j k hq m qq m��Then we have(see Notes 11)This implies that the patch is fairly small (or that the angles of significant radiation are small).5ThereforeApproximation of “Approximation of “pp” (cont.)” (cont.)Hence( )( )2 222 2 2 2 2 20sin ( ) cos ( ) 4 cos sin cosrF G k hf q f q m q f+ � +( )( )2 /222 2 20 02 /222 2 20 0cos sin cos ( , ) sincos sin cos (0,0) sinA d dpA d dp pp pq f q f q q fq f q q f+�+����Note: The p factor is now only a function of the patch dimensions – not the substrate.6The patch array factor is( )2 2cos2, sinc2 22 2xyxLkWA WL kLkpq fp� �� �� �� �� � � �� �� �=� � � �� �� � � �� � � �-� �� � � �� � � �� �00sin cossin sinxyk kk kq fq f==  WLA20,0Approximation of “Approximation of “pp” (cont.)” (cont.)whereAlso,7Denominator of p expression:( )( )22 / 22 2 20 02/ 220222cos sin cos sin2cos 1 sin2 1132 43D WL d dWL dWLWLp ppq f q q fpp q q qppppp� �= +� �� �� �= +� �� �� � � �= +� � � �� � � �� �� �=� �� �� �� ����Approximation of “Approximation of “pp” (cont.)” (cont.)8Hence( )242 / 22 2 2 22 20 0cos32cos sin cos sinc sin4 2 22 2xyxLkWp k d dLkp ppq f q q fpp� �� �� �� �� � � �� �� �� +� � � �� �� � � �� � � �-� �� � � �� � � �� ���( )2/ 2 / 22 2 2 220 0cos32cos sin cos sinc sin2212xyxLkWp k d dLkp pq f q q fpp� �� �� �� �� �� �� �� +� �� �� �� �� �� �-� �� �� �� �� �� ���UsingApproximation of “Approximation of “pp” (cont.)” (cont.)( ) ( )2 /2 /2 /20 0 0 04d d d dp p p pq f q f=�� ��and factoring out a ( /2)-4, we have9Next, use [Abromowitz & Stegun]2 42 42 42 4sin1cos 1xa x a xxx b x b x� + +� + +20x24240.166050.007610.496700.03705aabb=-==-=Approximation of “Approximation of “pp” (cont.)” (cont.)wherefor2xLx k=2yWx k=Note: These are not Taylor series, but are approximations that are more uniformly accurate over the entire range.10where we have used use a Taylor series for2 42 42 42cos 2 21 121xb x b x x xxp pp� �� � � �� �� + + + +� �� � � �� �� � � �� �� �� �-� �� �1221 x2 42 4 222 2 4cos 4 4 16121xx b x b bxp p pp� � � �� + + + + +� �� �� � � �� �-� �� �The cosine term may thus be approximated as We then have (keeping terms up to x4)Approximation of “Approximation of “pp” (cont.)” (cont.)11Define2 224 4 22 444 16c bπc b bπ π= += + +2740 09141537 884 10 0c .c .-=-= � �2 42 42cos121xc x c xxp� + +� �-� �� �ThenApproximation of “Approximation of “pp” (cont.)” (cont.)The numerical values are12We then have( )/ 2 / 22 2 20 022 40 02 422023cos sin cos1 sin sin sin sin2 21 sin cos sin2pk W k Wa ak Lc d dp pq fpq f q fq f q q f� +� �� � � �� + +� �� � � �� � � �� �� �� �� �� +� �� �� �� �� ���Approximation of “Approximation of “pp” (cont.)” (cont.)13Neglect the following terms:2 24 4 2 2, ,a a a c( )( )/ 2 / 22 2 20 02 420 02 2 42023cos sin cos1 2 sin sin 2 sin sin2 21 2 sin cos sin2pk W k Wa a ak Lc d dp pq fpq f q fq f q q f� +� �� � � �� + + +� �� � � �� � � �� �� �� �� �� +� �� �� �� �� ���We then haveApproximation of “Approximation of “pp” (cont.)” (cont.)14Next, we also neglect the following terms:We then have22224, caca( )( )/ 2 / 22 2 20 02 420 02 2 42 2 20 0 02 2 23cos sin cos1 2 sin sin 2 sin sin2 22 sin cos 4 sin sin sin cos2 2 2sinpk W k Wa a ak L k W k Lc a cd dp pq fpq f q fq f q f q fq q f� +�� � � �� + + +�� � � �� � � ����� � � �� �+ +�� � � �� �� � � �� ������Approximation of “Approximation of “pp” (cont.)” (cont.)15Expanding, we have{( ) ( )/ 2 / 22 2 20 02 24 3 2 2 2 30 02 24 42 6 5 2 2 4 2 50 02 4 2 422 2 2 3023sin cos sin cos sin2 sin sin cos 2 sin cos sin2 22 sin sin cos 2 sin cos sin2 22 cos sin cos sin2pk W k Wa ak W k Wa a a ak Lcp pf q q f qpf q q f qf q q f qf q= +� � � �+ +� � � �� � � �� � � �+ + + +� � � �� � � �� �+� �� ���}24 3022 2 2 24 2 5 2 4 2 50 0 0 02 2 2 22 cos sin24 sin cos sin cos 4 cos sin sin2 2 2 2k Lck W k L k W k La c a c d dq f qf q q f q q f� �+� �� �� �� � � �� �+ +� �� � � �� �� �� � � �� �Approximation of “Approximation of “pp” (cont.)” (cont.)16220sin4dppf =�2403sin16dppf =�4403cos16dppf =�All of the  integrals may now be done in closed form: 220cos4dppf =�22 20sin cos16dppf f =�2605sin32dppf =�22 40sin cos32dppf f =�24 20sin cos32dppf f =�Approximation of “Approximation of “pp” (cont.)” (cont.)1720sin 1dpq q =�2302sin3dpq q =�2201cos sin3dpq q q =�All of the  integrals may also be done in closed form: 2508sin15dpq q =�23 202sin cos15dpq q q =�25 208sin cos105dpq q q =�Approximation of “Approximation of “pp” (cont.)” …