1Spring 2011Notes 25ECE 6345Prof. David R. JacksonECE Dept.2OverviewIn 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 ModelLWxy00( , )xyxz0ILr4TL Model (cont.)Ignore variation in the y direction:ˆ()s sxJ x J x(quasi-TEM mode)0Z0I00ex x L  0x 2x L L  2eL L L  Fringing extensions are added at both ends.5Planar Waveguide Model0zzsx e y eE h E hVZI J W H W  0 effehZWPlanar waveguide model for wide microstrip line:yzeffrPMCheWrr r rj   Note: Fringing is accounted for by the effective width and the effective permittivity.6The characteristic impedance and phase constant are given by001efferhZW0effrk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.)7111221 12effrrrhW         The effective permittivity and characteristic impedance are then given by1001.393 0.667ln 1.444effrWWZhh   Pozar, Microwave Engineering, Wiley, 1998, p. 162These are accurate for W / h > 1Planar 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: 022rcfLL(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.)yz2eW W W  rPMCheW9Fringing Extensionsln4WhWe then have the following results:0.2620.3000.4120.2580.813effreffrWhLhWh(Hammerstad formula)111221 12effrrrhW         (Wheeler formula)10Effective Loss Tangent 1effrc r effjl1 1 1 1 1d c sp swQ Q Q Q Q    1taneffefflQWe account for all losses (and radiation) by means of an effective loss tangent:1tandlQRecall that11Input Impedance0()effcZ complex0I00ex x x L   0x exL2eL L L  001effcefferchZW 1effrc r effjl12Input 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 rcYZkkwhereFrom transmission-line theory we have that1/TL TLin inZYNote: The effective permittivity accounts for all losses and radiation.13Probe Correctionwhere0effcZ0x exLpL0exxinZTLin in pZ Z jX0001ln ln ln/p r r rhXa                 ppXL0.57722()Euler's constant14Alternative (Edge Admittance)For0effcZ0x exL0exxinZedgesG0effcZwe now use11disseffdclQQedgesGpL001effcefferchZW 1eff dissrc r effjlThe edge admittances account for radiation effects (radiation into space and surface waves).15Alternative (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 sedgeszP G VG E hWe assume that the two edge voltages are approximately the same in magnitude.002sErrad radUUQPP02EradrUPQ1 1 1r sp swQ Q Qwhere16Alternative (Edge Admittance) (cont.)202001414eE r zVLe r zU E dVW h E dx02EradrUPQ2000214eLrad e r zrP W h E dxQThe time-average stored electric energy is given byHence we have17Alternative (Edge Admittance) (cont.) 220001 1 10 cos2eLrad e r zsp sw exP W h E dxQ Q L   2001 1 1022erad e r zsp swLP W h EQQ  Therefore we haveThe field inside the patch is approximately described by a cosine function.Evaluating the integral, we have18Alternative (Edge Admittance) (cont.) 001 1 14edgeeesrsp swWLGQ Q h   2001 1 1022erad e r zsp swLP W h EQQ  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.) 0201 1 14edgeeerssp swWLG k hQ Q h0 0 0 0/k  Usingthe final result for the edge conductance is20Alternative (Edge Admittance) (cont.)      00000100000tantantantanedge 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 0011eff effcceff effc rc ehYZZWThe input impedance is 01eff dissrc r effeff effc rcjlkkNote: The effective permittivity accounts for only material losses (not radiation).21Another Alternative (Edge Admittance Network)0x exLpL0edgesG12Y0edgesGnow accounts for power radiated by a single edge (without mutual