1Spring 2011Notes 12ECE 6345ECE 6345Prof. David R. JacksonECE Dept.2OverviewOverviewIn this set of notes we derive a general expression for the “p factor”that is used to determine the space-wave power radiated by the rectangular patch.In the next set of notes we will evaluate the integrals that appear and actually develop a final closed-form CAD expression for the p factor.3Definition of the p factor:dipspspPPp ≡= power radiated by the actual rectangular patch= power radiated by a dipole that has the equivalent dipole momentdipspPspPThe The ppFactorFactor4The The ppFactor (cont.)Factor (cont.)We then havepPPdipspsp=From Notes 11, we have()2222000 126dipsp rPWLkhk cημππ⎛⎞⎛⎞=⎜⎟⎜⎟⎝⎠⎝⎠1241112/51cnn=− +1,0cossxxJLπ⎛⎞=⎜⎟⎝⎠assumingLWxy52/2200(, , ) sinsp rPSr r ddππθφθθφ=∫∫()()()10hex ,ppsxxyEr,θ,Er,θ,Jk,kφφ=Calculation of the space-wave radiated power:where22012r θSEEη⎡⎤=+⎢⎥⎣⎦φhexpEand= far field of unit-amplitude horizontal electric dipole in the xdirection.The The ppFactor (cont.)Factor (cont.)and6ThenDenote()()yxsxkkJA ,~,0,1=φθ() ()()hexppEr,θ,Er,θ,Aθ,φφφ=and()()2hexrrSr,θ,SAθ,φφ=The The ppFactor (cont.)Factor (cont.)7We also haveHenceNote that2/22200(,) sinhexsp rPSA r ddππθφθθφ=∫∫()2/22200() ,, sindip hexsp patch rPIl Srr ddππθφθθφ=∫∫()()()/2 /21,0/2 /21,0(, )2(0,0) 0,0LWsxpatchLWsxIl J x y dy dxJAWLπ++−−====∫∫The The ppFactor (cont.)Factor (cont.)8Hence()()2/222002/22200,, (,) sin,, (0,0) sinhexrhexrSr A r ddpSr A r ddππππθφθφ θθφθφθθφ=∫∫∫∫The The ppFactor (cont.)Factor (cont.)9Note: depends on but not the substrate parameters. This may be written as2/222 222002/222 22200()sin () cos (,)sin()sin () cos (0,0)sinFGAddpFGAddππππθφθ φθφ θθφθφθ φ θθφ⎡⎤+⎣⎦=⎡⎤+⎣⎦∫∫∫∫),(φθA,WLThe The ppFactor (cont.)Factor (cont.)10The patch array factor is()22cos2,sinc2222xyxLkWAWLkLkπθφπ⎡⎤⎛⎞⎢⎥⎜⎟⎛⎞⎡⎤⎝⎠⎢⎥=⎜⎟⎢⎥⎢⎥⎝⎠⎣⎦⎛⎞ ⎛ ⎞−⎢⎥⎜⎟ ⎜ ⎟⎝⎠ ⎝ ⎠⎣⎦00sin cossin sinxykkkkθφθφ==,0WL→As() ()2,AWLθφπ→()(),0,0AAθφ→so1p →HenceThe The ppFactor (cont.)Factor
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