ECE 6345ECE 6345Spring 2011Prof. David R. JacksonECE Dept.Notes 141OverviewOverviewIn this set of notes we derive a CAD formula for the space-wave Q factor(Q)fth t l th i th i ldi df lf(Qsp) of the rectangular patch, using the previously-derived formula for the space-wave radiated power.2CAD Formula for CAD Formula for QQspsp0sspspUQPω⎛⎞=⎜⎟⎜⎟⎝⎠Recall02pHspUPω⎝⎠⎛⎞=⎜⎟⎜⎟⎝⎠sp⎝⎠dipPpP=We have, from Notes 12, ()2222000 126spsprPpPpWL k h k cημππ⎛⎞⎛⎞=⎜⎟⎜⎟⎝⎠⎝⎠1241112/51cnn=− +6ππ⎝⎠⎝⎠10xπ⎛⎞11nn3Where it is assumed that1,0cossxxJLπ⎛⎞=⎜⎟⎝⎠(The origin is at the center of the patch.)CAD Formula for CAD Formula for QQsp sp (cont.)(cont.)To calculate the stored magnetic energy, useˆcosxHyLπ⎛⎞=⎜⎟⎝⎠2014HryUHdVμμ=∫The stored magnetic energy is then20414VryShHdSμμ=∫∫/220/21cos4SLrLxhW dxLπμμ−⎛⎞=⎜⎟⎝⎠∫/20142LrLhWμμ−⎝⎠⎛⎞=⎜⎟⎝⎠4()018rhWLμμ=CAD Formula for CAD Formula for QQsp sp (cont.)(cont.)Hence⎛⎞0000221128822rrsphWLhWLQPωμμμμωη⎛⎞⎛⎞⎜⎟⎜⎟⎝⎠==⎜⎟⎛⎞⎛⎞⎜⎟⎜⎟()222000 126sprPpWLkhk cημππ⎛⎞⎛⎞⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠Simplify, using 1kLπ=0 rrkLμεπ=1πor5so01rrkLπμε=CAD Formula for CAD Formula for QQsp sp (cont.)(cont.)and0011kπω⎛⎞==⎜⎟⎜⎟This eliminates k0and ω0and leaves only theTh lt i000 00 rrLμεμε με⎜⎟⎜⎟⎝⎠ω0and leaves only the patch dimensions.3111LQπ⎛⎞⎛⎞⎛⎞⎛⎞⎜⎟⎜⎟⎜⎟⎜⎟The result is018sp rQWkhc pε⎛⎞=⎜⎟⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠⎝⎠⎝⎠0118 1()Wkh cp⎛⎞⎛⎞=⎜⎟⎜⎟⎝⎠or601()3sp
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