Slide 1OverviewTL ModelTL Model (cont.)Planar Waveguide ModelSlide 6Slide 7Slide 8Fringing ExtensionsEffective Loss TangentInput ImpedanceInput Impedance (cont.)Probe CorrectionAlternative (Edge Admittance)Alternative (Edge Admittance) (cont.)Slide 16Slide 17Slide 18Slide 19Slide 20Another Alternative (Edge Admittance Network)1Spring 2011Notes 25ECE 6345ECE 6345Prof. David R. JacksonECE Dept.2OverviewOverviewIn this set of notes we investigate the transmission line (TL) model for the input impedance of a rectangular patch antenna.Assumption: The patch is operating in the usual (1,0) mode: the patch acts as a wide microstrip line of width W.3TL ModelTL ModelLWxy0 0( , )x yxz0ILre4TL Model (cont.)TL Model (cont.)Ignore variation in the y direction:ˆ( )s sxJ x J x�(quasi-TEM mode)0Z0I0 0ex x L= +D0x =2x L L= + D2eL L L= + DFringing extensions are added at both ends.5Planar Waveguide ModelPlanar Waveguide Model0z zsx e y eE h E hVZI J W H W- -= = =0 effehZWh� �=� �� �Planar waveguide model for wide microstrip line:yzeffrePMCheWrer r rje e e� ��= -Note: Fringing is accounted for by the effective width and the effective permittivity.6The characteristic impedance and phase constant are given by0 01efferhZWhe� �=� �� �0effrkb e=From a knowledge of the characteristic impedance and the phase constant, we can then determine the effective width and the effective permittivity.Planar Waveguide Model (cont.)Planar Waveguide Model (cont.)71 112 21 12effr rrhWe ee+ -� � � �= +� � � �� � � �� �+� �� �The effective permittivity and characteristic impedance are then given by1001.393 0.667 ln 1.444effrW WZh hhe-� �� �= + + +� �� �� �� �Pozar, Microwave Engineering, Wiley, 1998, p. 162These are accurate for W / h > 1Planar Waveguide Model (cont.)Planar Waveguide Model (cont.)8Alternative (the approach that we will use): We keep the original permittivity, and extend the edges.Note: This is consistent with the Hammerstad formula:( )02 2rcfL Le=+ D(The Hammerstad formula appears to work best when used with the actual permittivity instead of the effective permittivity in the frequency formula.)Planar Waveguide Model (cont.)Planar Waveguide Model (cont.)yz2eW W W= + DrePMCheW9Fringing ExtensionsFringing Extensionsln 4W hp� �D =� �� �We then have the following results:0.2620.3000.4120.2580.813effreffrWhL hWhee� �+� �� �+D =� �� �-� �� �+� �(Hammerstad formula)1 112 21 12effr rrhWe ee+ -� � � �= +� � � �� � � �� �+� �� �(Wheeler formula)10Effective Loss TangentEffective Loss Tangent( )1effrc r effjle e�= -1 1 1 1 1d c sp swQ Q Q Q Q= + + +( )1taneffefflQd� =We account for all losses (and radiation) by means of an effective loss tangent:1tandlQd= �Recall that11Input ImpedanceInput Impedance0( )effcZcomplex0I0 0ex x x L= = +D0x =ex L=2eL L L= + D0 01effcefferchZWhe� �=� �� �( )1effrc r effjle e�= -12Input Impedance (cont.)Input Impedance (cont.)( ) ( )( )0 0 0 0tan tanTL eff eff e eff eff ein c c c c eY jY k x jY k L x= + -0001effceffceff effc rcYZk k e==whereFrom transmission-line theory we have that1/TL TLin inZ Y=Note: The effective permittivity accounts for all losses and radiation.13Probe CorrectionProbe Correctionwhere0effcZ0x =ex L=pL0ex x=inZTLin in pZ Z jX= +00 01ln ln ln/p r r rhXah m g p mel l� �� � � �= - - -� �� � � �� � � �� �p pX Lw=0.57722( )BgEuler's constant14Alternative (Edge Admittance)Alternative (Edge Admittance)For0effcZ0x =ex L=0ex x=inZedgesG0effcZwe now use1 1disseffd clQ Q= +edgesGpL0 01effcefferchZWhe� �=� �� �( )1eff dissrc r effjle e�= -The edge admittances account for radiation effects (radiation into space and surface waves).15Alternative (Edge Admittance) (cont.)Alternative (Edge Admittance) (cont.)Also, we can express the radiated power in terms of the radiation Q asedgesGmodels radiation (space-wave + surface-wave)22212 ( 0 )2( 0 )edgerad sedges zP G VG E h� �=� �� �=We assume that the two edge voltages are approximately the same in magnitude.0 02s Errad radU UQP Pw w= =02EradrUPQw=1 1 1r sp swQ Q Q� �= +� �� �� �where16Alternative (Edge Admittance) (cont.)Alternative (Edge Admittance) (cont.)202001414eE r zVLe r zU E dVW h E dxe ee e�=� ��=� �� ���02EradrUPQw=2000214eLrad e r zrP W h E dxQwe e� �� ��=� �� �� �� ��The time-average stored electric energy is given byHence we have17Alternative (Edge Admittance) (cont.)Alternative (Edge Admittance) (cont.)( )220 001 1 10 cos2eLrad e r zsp sw exP W h E dxQ Q Lpw e e� �� ��= +� �� �� �� �� ��( )20 01 1 102 2erad e r zsp swLP W h EQ Qw e e� �� ��= +� �� �� �� �� �Therefore we haveThe field inside the patch is approximately described by a cosine function.Evaluating the integral, we have18Alternative (Edge Admittance) (cont.)Alternative (Edge Admittance) (cont.)( )0 01 1 14edgee es rsp swW LGQ Q he w e� �� ��= +� �� �� �� �� �( )20 01 1 102 2erad e r zsp swLP W h EQ Qw e e� �� ��= +� �� �� �� �� �22( 0)edgerad s zP G E h=We then equate these two expressions for the radiated power:The result for the edge conductance is19Alternative (Edge Admittance) (cont.)Alternative (Edge Admittance) (cont.)( )0201 1 14edgee erssp swW LG k hQ Q heh� ��� �� �= +� �� �� �� �� �� �� �0 0 0 0/kw e h=Usingthe final result for the edge conductance is20Alternative (Edge Admittance) (cont.)Alternative (Edge Admittance) (cont.)( )( )( )( )( )( )0 000 010 000 0tantantantanedge eff eff es c ceffin p ceff edge eff ec s cedge eff eff es c c eeffceff edge eff ec s c eG jY k xZ jX YY jG k xG jY k L xYY jG k L x-�� �+�� �= +�+� ��� ���� �+ -�� �+�� �+ -�� ��0 0 001 1eff effc ceff effc rc ehY ZZ Whe� �= =� �� �The input impedance is( )01eff dissrc r effeff effc rcjlk ke ee�= -=Note: The effective permittivity accounts for only material losses (not radiation).21Another Alternative Another Alternative (Edge Admittance Network)(Edge Admittance Network)0x =ex L=pL0edgesG12Y0edgesG now accounts for power radiated by a single edge (without mutual