Slide 1OverviewQ of the PatchQ of the Patch (cont.)Calculation of QdCalculation of Qd (cont.)Calculation of QcCalculation of Qc (cont.)Slide 9Slide 10Calculation of QspCalculation of Qsp (cont.)Calculation of QSWCalculation of QSW (cont.)Slide 15Slide 16Slide 17Comparison of QsSlide 19Slide 20Slide 21Slide 22Slide 231Spring 2011Notes 3ECE 6345ECE 6345Prof. David R. JacksonECE Dept.2OverviewOverviewThis set of notes discusses the Q of a patch, and its different components, as well as the radiation efficiency of the patch.  Defining Q and its components: Qd, Qc, Qsp, Qsw Calculating Qd, Qc, Qsp, Qsw Calculating the radiation efficiency3QQ of the Patch of the Patch0SaveDUQPw=01aveDSPQ Uw=ave sp sw c dD D D D DP P P P P= + + +ShrefeedThe patch is allowed to have an arbitrary shape.soNote: Here 0 denotes the real part of the complex resonance frequency.4QQ of the Patch (cont.) of the Patch (cont.)0 0 0 01sp sw c dD D D DS S S SP P P PQ U U U Uw w w w= + + +1 1 1 1 1sp sw c dQ Q Q Q Q= + + +HenceorNote: A smaller Q is a more dominant one!Note: Combining Q terms is like combining resistors in parallel.5Calculation of Calculation of QQdd22212412S S E H EVSU U U U UE dVh E dSee= = + =�=�=��( )20201212dDVSP E dVh E dSw ew e��=��=��We have equal time-average stored energies at resonance.6Calculation of Calculation of QQdd (cont.)(cont.)001212dhQheewew e��= =����tan1dQHenceThereforedQ0k hThe dielectric Q factor is independent of substrate thickness.7Calculation of Calculation of QQcc2 21 12 2P Gc P GD s sS SP R H dS R H dS= +� �( )2212ave P GD s sSavesSP R R H dSR H dS= +=��Hence8Calculation of Calculation of QQcc (cont.)(cont.)2S S E H HU U U U U= = + =2201414HVrSU H dVh H dSmmm==��For the stored energy,where92002124rScavesSh H dSQR H dSmmw� �� �� �=��0000kHenceUseso0 00 01( )2rcavesk hQRmmme=Calculation of Calculation of QQcc (cont.)(cont.)100 0( )2c ravesk hQRhm� �� �=� �� �� �� �HencecQ0k hCalculation of Calculation of QQcc (cont.)(cont.)The conducting Q factor becomes more important as the substrate gets thinner.11Calculation of Calculation of QQspsp1 03 116 /erspeLQpc W hel� �� �� ��� �� �� �� �� �� � rrnnnc1412115/211Rectangular Patch:( )( )( )( ) ( ) ( )2 4220 2 4 02 2 22 0 2 2 0 031 210 5601 15 70e ee e eap k W a a k Wc k L a c k W k L� �= + + +� �� �� � � �+ +� � � �� � � �(The derivation is given later.)12Calculation of Calculation of QQspsp (cont.)(cont.)2420.166050.007610.0914153aac=-==-wherespQ0/h lThe radiation Q factor becomes more important as the substrate gets thicker.13Calculation of Calculation of QQSWSWspswrsp swPeP P�+11sp swswswr sp spP PPe P P+= = +11sw spswrP Pe� �= -� �� �Surface wave radiation efficiency:Hence14Calculation of Calculation of QQSWSW (cont.)(cont.)00SswswSspspUQPUQPww==Now look at the Q’s :1111spswsp swswrswrswrPQQ Peee==-=-Hence151swrsw spswreQ Qe� �=� �� �-� �sw hedr re e�( )11 /hedsphedrhed hedhed hedsp swsw spPeP PP P= =++HenceApproximate patch by dipole:horizontal electric dipole (HED)Calculation of Calculation of QQSWSW (cont.)(cont.)hreThis holds for any shape patch.Note: The space-wave and surface-wave Qs are equal when the surface-wave efficiency is 50%.)16( )( )( )22 20 120333 302 20 11801 160 1hedsp rhedsw rp k h cp k hnp mlp ml�� �� �� �� -� �� �� �� �� �12 41 11 2 / 51cn n= - +rrn1Calculation of Calculation of QQSWSW (cont.)(cont.)whereFor the HED we have (derivation given later):173021113 1 11 ( ) 14sw hedr rre ek hcnpm� =� �� �+ -� �� �� �� �HenceCalculation of Calculation of QQSWSW (cont.)(cont.)This holds for any shape patch (assuming the HED approximation is accurate).18Comparison of Comparison of QsQs0/h lQswQspQcQdQQc dominatesQd dominatesQsp dominates19sp sprtot sp sw c dP PeP P P P P� =+ + +0siiUQPw=Radiation EfficiencyRadiation EfficiencyNote thatHence1iiPQ�2011 1 1 1sprsp sw c dQeQ Q Q Q=+ + +HenceorRadiation Efficiency (cont.)Radiation Efficiency (cont.)orQQespr11sprQQe 21Also, we can writeRadiation Efficiency (cont.)Radiation Efficiency (cont.)sp sp swrsp sw totsprr totP P PeP P PPPP P� �+� �=� �� �� �+� �� �� �� �=� �� �� �� �spswrrdissrrtotPePPeP==Define22Radiation Efficiency (cont.)Radiation Efficiency (cont.)andsw dissr r re e e=swrrspQeQ=dissrrQeQ=ThenwhereswsprQQQ11123Radiation Efficiency (cont.)Radiation Efficiency (cont.)2.210.82.2r =