Slide 1OverviewCAD Formula for QspApproximation of “p”Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 171Spring 2011Notes 17ECE 6345ECE 6345Prof. David R. JacksonECE Dept.2OverviewOverviewIn this set of notes we derive a closed-form CAD formula for the space-wave Q of the circular patch, Qsp assuming that the substrate is thin.We start with the expression for Qsp derived in Notes 16.3CAD Formula for CAD Formula for QQspspThe formula that was derived previously is0/ 1h l �We wish to approximate Qsp assuming that 0012c rspxk hQpeI� �� �� �=( )21121111 0.238695cx xx�� -�Bwhere4Approximation of “Approximation of “pp””( )( )/ 20/ 2000cC dIpIC dppq qq q� =��( )( )( ) ( ) ( ) ( )/ 22 22 2 21 1 0 00sin tanc sin sinc o incI k h N Q J k a P J k a dpq q q q q q q� ��= +� ��From the definition of p, The terms in this expression were found to be:( )( )( ) ( )/ 22 220 0 101sin tanc4Ι k h N P Q dpq q q q q� �= +� ��5From Notes 9 we also haveFor a thin substrate, we use 0 1tanc( ( )) 1k hN q �( ) ( )( )10 121( )sec1 tan ( )TMrQNj k h Nq qq qqe= - G =� �+� �� �( )0 112cos( ) cos (1 ( ))cos1 tan ( )( )TErPj k h NNqq q qm qqq= - G =� �+� �� �( ) 2cos( ) 2PQq qq��For a thin substrate we then haveApproximation of “Approximation of “pp” (cont.)” (cont.)6For the denominator of the p function we have043I =Using the thin-substrate approximations, ( )( )( ) ( )/ 22 220 0 101sin tanc4Ι k h N P Q dpq q q q q� �= +� ��( )( )/2200/2201sin 4 cos 14sin cos 1Ι ddppq q qq q q� �� +� �= +��This yieldsApproximation of “Approximation of “pp” (cont.)” (cont.)7The formula for the p function then becomes( ) ( ) ( ) ( )/ 22 22 21 0 003sin sin sin4incp Q J k a P J k a dpq q q q q q� ��� +� ��( )( )2cos2PQq qq��Usingwe have( ) ( )/ 22 2 21 0 003 sin sin cos sinincp J k a J k a dpq q q q q�� �� +� ��Approximation of “Approximation of “pp” (cont.)” (cont.)8( ) ( )/22 2 21 0 003 sin sin cos sinincp J k a J k a dpq q q q q�� �� +� ��Note that p is only a function of the patch radius now,and not the substrate parameters.Also,1p �as0a �To summarize so far,Approximation of “Approximation of “pp” (cont.)” (cont.)This follows from( )/2204sin cos 13dpq q q+ =�( )2114J x��( )214incJ x �0x �asand9( )( ) ( )2 2/ 21 12003 s in cosJ x J xp J x dx xpq q q� �� � � �� �� - +� � � �� �� � � �� ��Using( ) ( )( )11 0J xJ x J xx�= -we haveThe goal is to approximate this integral for the p factor in closed form.We now approximate the Bessel function terms appearing in this expression. 0sinx k aθ=whereApproximation of “Approximation of “pp” (cont.)” (cont.)10From Abramowitz and Stegun, we have the following approximations:( )2 4 6 8 10 120 0 2 4 6 8 10 12024466881010121.00.249999970.0156249484.340008 106.77456 106.6799 103.951 10J x a a x a x a x a x a x a xaaaaaaa----= + + + + + +==-==- �= �=- �= �error( )0 2 4 6, , ,a a a a110.001 1.8412x x�< � =85 10 0 3x-< � < �+We keeperror forApproximation of “Approximation of “pp” (cont.)” (cont.)11errorWe keeperror for2 4 6 8 10 1210 2 4 6 8 10 1202456789101112( )0.50.0624999830.00260414485.424265 106.75688 105.3788 102.087 10J xb b x b x b x b x b x b xxbbbbbbb----= + + + + + +==-==- �= �=- �= �31.3 10 , 3 3x-< � - � �( )0 2 4 6, , ,b b b b110.001 1.8412x x�< � =Approximation of “Approximation of “pp” (cont.)” (cont.)12DefineWe then haven n nc a b= -( )2 2/ 2 / 23 32 2 3 2 22 20 00 03 sin sin 3 sin sin sinn n n nn nn np c A d b A dp pq q q q q q q= =� � � �= + -� � � �� � � �� �� �where0A k a=Approximation of “Approximation of “pp” (cont.)” (cont.)2sin cosq q13In order to evaluate the integrals that appear, we use23 3 32 2 2 2 2 22 2 20 0 03 32 2 2 22 20 0sin sin sinsinn n n n m mn n mn n mn m n mn mn mb A b A b Ab b Aq q qq= = =+ += =� � � �� �=� � � �� �� � � �� �=� � ���23 3 32 2 2 2 2 22 2 20 0 0sin sinn n n m n mn n mn n mc A c c Aq q+ += = =� �=� �� �� ��Similarly,Approximation of “Approximation of “pp” (cont.)” (cont.)14Hence( )/ 23 32 12( )2 20 00/ 2 / 23 32( ) 2( ) 1 2( ) 32 20 00 03 sin3 sin sinn mn mn mn mn m n m n mn mn mp c c A db b A d dpp pq qq q q q+ ++= =+ + + + += ==� �+ -� �������� �Approximation of “Approximation of “pp” (cont.)” (cont.)15Define/ 20sinnnS dpq q=�( )( )22 12 !12 1 !mmmS mm+= �+013579111315/ 21.02 / 38 /1516 / 35128 / 315256 / 6931024 / 30032048 / 6435SSSSSSSSSp=========General formula:Values:Approximation of “Approximation of “pp” (cont.)” (cont.)16We then have( )3 3 3 32( ) 2( )2 2 2( ) 1 2 2 2( ) 1 2( ) 30 0 0 03 3n m n mn m n m n m n m n mn m n mp c c A S b b A S S+ ++ + + + + += = = == + -�� ��6220kkkp A e==�Rearranging, we havewhere( )( )( ) ( )( )( ) ( ) ( )( ) ( )( )( )( )2 20 0 1 0 1 32 0 2 3 0 2 3 52 24 2 0 4 5 2 0 4 5 76 0 6 2 4 7 0 6 2 4 7 92 28 4 2 6 9 4 2 6 9 1110 4 6 11 4 6 11 132 212 6 13 6 13 153 36 63 2 3 23 2 2 3 2 23 2 3 26 63 3e c S b S Se c c S b b S Se c c c S b b b S Se c c c c S b b b b S Se c c c S b b b S Se c c S b b S Se c S b S S= + -= + -= + + + -= + + + -= + + + -= + -= + -Approximation of “Approximation of “pp” (cont.)” (cont.)17Hence we havewhere( )620 20kkkp k a e==�024364851071210.4000000.07857107.27509 103.81786 101.09839 101.47731 10eeeeeee----==-==- �= …