ECE 6345ECE 6345Spring 2011Prof. David R. JacksonECE Dept.Notes 151OverviewOverviewIn this set of notes we calculate a CAD formula for the directivityyand gain of the rectangular patch antenna.2DirectivityDirectivity()()2Definition of directivity:()()()22,, 4 ,,,4rrspspSr rSrDPPθφπ θφθφ==⎛⎞⎜⎟⎝⎠For typical substrate thicknesses we usually have24rπ⎜⎟⎝⎠For typical substrate thicknesses, we usually have)0,0(maxDD=()()24,0,000rrS rDπwhere3()()0,0rspDP=Directivity (cont.)Directivity (cont.)dipspspPpP=The space-wave radiated power of the patch is (from Notes 12) ppThe radiated power density from the patch in the far field is () ()2,0,0 ,0,0 (0,0)hexrrSr S r A=()()(1,0)22cos2,, sinc22xsx x y yLkWAJkkWLkLπθφπ⎡⎤⎛⎞⎢⎥⎜⎟⎛⎞⎡⎤⎝⎠⎢⎥==⎜⎟⎢⎥⎢⎥⎝⎠⎣⎦⎛⎞ ⎛ ⎞where2222xLkπ⎢⎥⎝⎠⎣⎦⎛⎞ ⎛ ⎞−⎢⎥⎜⎟ ⎜ ⎟⎝⎠ ⎝ ⎠⎣⎦2so that4() () () ()(1,0) (1,0)20, 0 0, 0 ,sx sxpatchSAJJxydSIlWLπ== ==∫Directivity (cont.)Directivity (cont.)The radiated power density is then() ()2,0,0 ,0,0 (0,0)hexrrSr S r A=Note: the superscript “hex” denotes a unit-amplitude horizontal electric dipole in ()2,0,0 ( )(00)hexrpatchdipSr IlSr==the x direction. The superscript “dip” denotes the dipole that has the same dipole moment as (,0,0)rSr=the patch.Hence,24(,0,0)(0 0)diprrS rDπ=5(0,0)dipspDpP=Directivity (cont.)Directivity (cont.)Hence)0,0(1)0,0(dipDpD =pWe now calculate the directivity of the dipole:24(,0,0)(0,0)dipdiprdiprS rDPπ=spP()221dip dip dipSr E Eθφ⎡⎤=+⎢⎥⎣⎦where0(, , ) ( ) cos ( )Er IlE Gθθφ φ θ=()0,,2rSr E Eθφθφη=+⎢⎥⎣⎦whereand6()00(, , ) ( ) cos ( )(, , ) ( ) sin ( )Er IlE GEr IlE Fθφθφ φ θθφφθ=−adDirectivity (cont.)Directivity (cont.)Hence()22 2 2221() i ()dipSIlEGFθφ φ θ φ θ⎡⎤()22 2 22200,, cos()sin()2diprSrIl E G Fθφ φ θ φ θη⎡⎤=+⎣⎦()2cos()Gθθθθ=⎛⎞where()()101()sec1cot()rNjkhNθθθμ⎛⎞−⎜⎟⎝⎠()()012cos( )1cot()()rFjkhNNθεθθθ=⎛⎞−⎜⎟⎝⎠71()Nθ⎝⎠Directivity (cont.)Directivity (cont.)At broadside we have()22()2201 0 1 01 1sinkhN kh n khn khθθ=−==11()secrNnθθε11rrrrμμμ==cos()rrrεθε ε11()()rrrrNnθμ==2(0) (0)1t()rGFjkhε==Hence811cot( )rrjkhμ−Wth hDirectivity (cont.)Directivity (cont.)We then have()()22 22201,0,0(0)cos sin2rSr IlE Gφφ=+()()00220,, ()21424rIlφφηωμ⎛⎞=⎜⎟⎛⎞⎝⎠()201241cotrrrkhηπεμ⎜⎟⎛⎞⎝⎠+⎜⎟⎝⎠We can re-write this using:000kωμη=⎛⎞()()22200212,0,041rSr Ilkrkhηπε⎛⎞⎜⎟⎛⎞⎜⎟=⎜⎟⎜⎟⎛⎞⎝⎠⎜⎟⎜⎟9()211cotrrkhεμ⎛⎞⎝⎠+⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠Directivity (cont.)Directivity (cont.)From previous calculations in Notes 11:p()()22220dipPIlkhkη⎛⎞⎜⎟()()22000 16dipsrPIlkhkcημπ⎛⎞≈⎜⎟⎝⎠10Directivity (cont.)Directivity (cont.)To summarize so far, we have()2()24,0,0(0,0)rdipdipsprS rDPπ=⎛⎞with()()22200212,0,041cotrrSr Ilkrkhηπε⎛⎞⎜⎟⎛⎞⎜⎟=⎜⎟⎜⎟⎛⎞⎝⎠+⎜⎟⎜⎟⎜⎟()22220diη⎛⎞()11cotrkhμ+⎜⎟⎜⎟⎜⎟⎝⎠⎝⎠11()2222000 16dipsp rPIlkhk cημπ⎛⎞=⎜⎟⎝⎠Directivity (cont.)Directivity (cont.)The result is⎡⎤22201111 1(0,0) 31cot()diprrDkh ckhεμ⎡⎤⎢⎥⎛⎞⎛⎞⎢⎥=⎜⎟⎜⎟⎢⎥⎝⎠⎝⎠+⎢⎥⎣⎦Thi bittrμ⎢⎥⎣⎦This may be re-written as 22⎡⎤⎢⎥⎛⎞⎛⎞⎛⎞2211221011tan( )11(0,0) 3tan ( )diprrrkh kDkh k ckhεμμ⎢⎥⎛⎞⎛⎞⎛⎞⎢⎥=⎜⎟⎜⎟⎜⎟⎢⎥⎝⎠⎝⎠⎝⎠+⎢⎥⎣⎦12⎣⎦Directivity (cont.)Directivity (cont.)or2⎡⎤⎢⎥⎛⎞⎛⎞212111tan( )1(0,0) 3tan ( )diprrrrkhDkh ckhεεμμ⎢⎥⎛⎞⎛⎞⎢⎥=⎜⎟⎜⎟⎢⎥⎝⎠⎝⎠+⎢⎥⎣⎦rμ⎣⎦or()2111(0,0) 3tancdipDkhμ⎡⎤⎢⎥⎛⎞⎢⎥=⎜⎟⎢⎥⎝⎠()2111tan()rrckhμε⎜⎟⎢⎥⎝⎠+⎢⎥⎣⎦13where()()tantancxxx≡Directivity (cont.)Directivity (cont.)Since the substrate is assumed to be thin, we can further approximate this as13(0,0)dipDc≈1c12/51241112/51cnn=−+whereNote: for,11>>n(0,0) 3dipD≈14Directivity (cont.)Directivity (cont.)For the patch we have⎡⎤⎢⎥⎛⎞()2121111113(0,0) 3tanc1tan()rDkhpcpckhμε⎢⎥⎛⎞⎢⎥≈≈⎜⎟⎢⎥⎝⎠+⎢⎥⎣⎦rε⎢⎥⎣⎦whereand()()()2204211032apkWkW=+⎛⎞++⎜⎟0 16605a=1241112/51cnn=− +with()()()24 0220256015aakWckL++⎜⎟⎝⎠⎛⎞+⎜⎟⎝⎠2420.166050.007610.0914153aac=−==−15()()2222 0 05170ac kW kL⎝⎠⎛⎞+⎜⎟⎝⎠2GainGainThe gain of the patch is related to the directivity as ()0, 0 (0, 0)rGDe=whererQeQ=spQ111 1 1111 1 1d c sp swQQ Q Q Q=++ +and16CAD formulas for all of the Q factors were presented in Notes