ECE 6345ECE 6345Spring 2011Prof. David R. JacksonECE Dept.Notes 61OverviewOverviewIn this set of notes we look at two different models for calculating the radiation pattern of a microstrip antenna: Electric current model Magnetic current modelWllktt diff tbtt tiWe also look at two different substrate assumptions: Infinite substrateTruncated substrate (truncated at the edge of the patch)Truncated substrate (truncated at the edge of the patch).2Review of Equivalence PrincipleReview of Equivalence Principlenew problem:esJesMoriginal problem:(, )EH(, )EH,in inεμ+-rεPEC,out outεμ,out outεμ(0,0)ˆnSS()()ˆ0ˆeseJnH=× −3()ˆ0esMnE=−× −Review of Equivalence PrincipleReview of Equivalence PrincipleA common choice:The electric surfaceesJesMThe electric surface current sitting on the PEC object does not radiate, and can be ignored.(, )EHPECˆesJnH=×PECˆesMnE=−×,out outεμˆS4ˆnSModel of Patch and FeedModel of Patch and FeedzpatchhrεxInfinite substratepatchprobexAS(aperture)coax feed5Model of Patch and FeedModel of Patch and Feedhrε(, )EHxASSSzero fieldsˆeMzE=−×Aperture:Magnetic frill model:sMzE=×Aperture:hεesMMagnetic frill model:6hrεxsElectric Current Model: Electric Current Model: Infinite SubstrateInfinite SubstratehεSinfinite substrate(, )EHhrεxinfinite substrateThe surface Szero fieldsˆˆ0ˆesteMnEnEJnHJ=−× =−× ==× =Note: The direct radiation from the frill is ignored.“hugs” the metal.ssJnHJ=× =frill is ignored.topsJbhrεxprobesJbotsJ7xElectric Current Model: Electric Current Model: Infinite Substrate Infinite Substrate (cont )(cont )Letpatch top botsssJJJ=+(cont.)(cont.)ssshpatchsJprobeJhrεxpsJhtop viewprobesJpatchsJ8Magnetic Current Model: Magnetic Current Model: Infinite Substrate Infinite Substrate hrε(, )EHhStSesMrxzero fieldsbSsSKeep ground plane and substrate in the zero-field region. Remove patch, probe and frill current.ˆ0, ,ˆ0estbeMnE r SSJnH rS=− × = ∈+× ∈0,0,sbestJnH rSJrS=+ ×=∈≈∈(weak fields)90,eshJrS≈∈(approximate PMC)Magnetic Current Model:Magnetic Current Model: Infinite Substrate Infinite Substrate (cont )(cont )Exact model:(cont.)(cont.)esJhrεxesMesJesJesMApproximate model:hrεxesMesMˆesMnE=−×10Note: The magnetic currents radiate inside an infinite substrate above a ground plane.Magnetic Current Model: Magnetic Current Model: Truncated SubstrateTruncated SubstraterεNxbSApproximate model:Now we remove the substrate inside the surface.Approximate model:esMNote: The magnetic currents radiate in free space above a ground plane. x11gpgpElectric Current Model: Electric Current Model: Truncated SubstrateTruncated SubstrateSThe patch and probe are replaced by surface tbfrεxcurrents, as before.Next, we replace the dielectric patchJpatch top botsssJJJ=+with polarization currents.patchsJrε12xElectric Current Model: Electric Current Model: Truncated SubstrateTruncated Substrate(cont )(cont )HjEωε∇× =(cont.)(cont.)()00jEjEωεε ωε=−+()1polJj E()01polrJj Eωεε=−εpatchsJprobeJpolJIn this model we have three separate 0εxpobesJJelectric currents.13Comments on ModelsComments on ModelsInfinite SubstrateInfinite Substrate The electric current model is exact (if we neglect the frill), but it requires knowledge of the exact patch and probe currentsrequires knowledge of the exact patch and probe currents. The magnetic current model is approximate, but fairly simple.  For a rectangular patch, both models are fairly simple if only the (1,0) gp, yp y(,)mode is assumed.  For a circular patch, the magnetic current model is much simpler (it does not involve Bessel functions)does not involve Bessel functions).14Comments on Models (cont.)Comments on Models (cont.)Truncated SubstrateTruncated SubstrateThe electric current model is exact (if we neglect the frill) but itThe electric current model is exact (if we neglect the frill), but it requires knowledge of the exact patch and probe currents, as well as the field inside the patch cavity (to get the polarization currents). It is a complicated model.p The magnetic current model is approximate, but very simple. This is the recommended model.  For the magnetic current model the same formulation applies as for the infinite substrate – the substrate is simply taken to be air.15TheoremTheoremAssumptions: 1)The electric and magnetic current models are based on the1)The electric and magnetic current models are based on the fields of a single cavity mode corresponding to an ideal cavity with PMC walls.2)The probe current is neglected in the electric current model2)The probe current is neglected in the electric current model. Then the electric and magnetic models yield identical resultsThen the electric and magnetic models yield identical results at the resonant frequency of the cavity mode. (This is true for either infinite or truncated substrates).16Theorem (cont.)Theorem (cont.)Electric-current model:esJˆesJzH=−×hrεxMagnetic-current model:hεeMeMˆesMnE=−×hrεxsMsM17s(E, H) = fields of resonant cavity modeTheorem (cont.)Theorem (cont.)Proof:We start with an ideal cavity having PMC walls on the sides. This cavity will support a valid non-zero set of fields at the resonance frequency f0of the mode.ideal cavityPECrεxPMCPEC(, )EH()()00ff EH≠Atx18()()0,0,0ff EH=≠AtProofProofPECSPMC()EHInfinite substratexSPMC(,)EHEquivalence principle:Put (0, 0) outside SThe PEC and PMC walls have been keep (E, H) inside Sremoved in the zero field region. We keep the substrate and ground plane.xS(, )EH()0, 019Proof (cont.)Proof (cont.)ˆesiJnH=׈eMENote the inward pointing normalˆinesiMnE=−×pgieMesJx(, )EHesM()0, 0Note: The electric current on the ground is neglected (it does not radiate).20gg( )Proof (cont.)Proof (cont.)Exterior Fields:0eessEJ EM++⎡⎤ ⎡ ⎤+=⎣⎦⎣⎦ˆe patch JsssiJnHJ J=×= =(note the inward pointing normal)sssi(The equivalent electric current is the same as the electric current in the electric current model.) pointing normal)ˆesiMnE=−×()ˆˆsinEnE=+ ×=−− ×(The equivalent current is the negative of the magnetic current in the magnetic current model )21MsM=−the magnetic current model.)Proof (cont.)Proof (cont.)Hence0JMssEJEM++⎡⎤⎡ ⎤+−=⎣⎦ ⎣ ⎦ss⎣⎦ ⎣ ⎦orJMssEJEM++⎡⎤⎡⎤=⎣⎦ ⎣ ⎦ss⎣⎦ ⎣ ⎦22Theorem for Truncated SubstrateTheorem for Truncated SubstrateProof for truncated model:PECSPMC(, )EHPECxPECReplace the dielectric with polarization