1Spring 2011Notes 34ECE 6345ECE 6345Prof. David R. JacksonECE Dept.2OverviewOverviewIn this set of notes we overview the segmentation methodfor obtaining the input impedance of an arbitrary shaped patch.This is an extension of the cavity model.3Segmentation MethodSegmentation Method0efferckkε=Note: the effective complex wavenumber may be determined iteratively – once the field is known inside the patch, the stored energy and the radiated power may be computed, and from this the Q may be determined. (arbitrary shaped patch)PMCThe perimeter has been extend to account for fringing.4The patch is divided into a number of rectangular-shaped patches. (2)(3)(1)Segmentation Method (cont.)Segmentation Method (cont.)5Along the edges where the rectangular patches connect, we introduce ports. The ports connect the patches together.Segmentation Method (cont.)Segmentation Method (cont.)6For each rectangular patch, we calculate the Z matrix for the ports that connect to it.Segmentation Method (cont.)Segmentation Method (cont.)7To illustrate, consider one patch:[]11 1221 22ZZZZZ⎡⎤=⎢⎥⎣⎦port 1 port 2We can calculate Zmnby using the cavity model.Segmentation Method (cont.)Segmentation Method (cont.)8This figure shows a general two-port system for a rectangular patch. Segmentation Method (cont.)Segmentation Method (cont.)xyPMCport 1port 2LeWe11(, )eexy22(, )eexyport 1:port 2:211110IVZI==(the input impedance at port 1)122220IVZI==(the input impedance at port 2)9Segmentation Method (cont.)Segmentation Method (cont.)xyPMCport 1port 2LeWe222110IVZI==()21 2 2,eezZhExy=−Assume 1210II==10[]()20cos coszm xmememyEBkxLWπ∞=⎛⎞=−⎜⎟⎝⎠∑LeWexy11(, )eexyPMC(1)(2)22(, )eexySegmentation Method (cont.)Segmentation Method (cont.)Use the mode-matching method:()221 20cos coseemxmememyZhB kxLWπ∞=⎛⎞⎡⎤=− −⎜⎟⎣⎦⎝⎠∑so thatNote: Bmis a function of the location of port 111Once the Z matrices of all the patches have been computed, an overall Z matrix of the entire system ZPis calculated, by connecting the individual Z matrices and using circuit theory.Segmentation Method (cont.)Segmentation Method (cont.)[]2Z[]3Z[]1Z[]22×[]22×[]55×Note: The feed is considered one of the ports on patch 1.If there is a single feed, then ZPis a 1×1 matrix (a single number Z11P).12The input impedance is then given bySegmentation Method (cont.)Segmentation Method (cont.)[]2Z[]3Z[]1Z11PinZZ=port 113This is a variation of the segmentation method, in which we have a rectangular patch shape in which a set of smaller rectangular patches have been removed.Desegmentation MethodDesegmentation Method14We next show how to combine two Z matrices into a single Zoverall matrix.Segmentation Method (cont.)Segmentation Method (cont.)cThe ports p and q are the external ports, while c and d represent the internal (connected) ports.Zα⎡⎤⎣⎦Zβ⎡⎤⎣⎦pqd[]Zpq15Segmentation Method (cont.)Segmentation Method (cont.)[]ppp p pc cVZIZIαα⎡⎤ ⎡⎤⎡⎤ ⎡⎤=+⎣⎦ ⎣⎦⎣⎦ ⎣⎦cZα⎡⎤⎣⎦Zβ⎡⎤⎣⎦pqd[]Zpq[]qqqqqddVZIZIββ⎡⎤ ⎡⎤⎡⎤ ⎡⎤=+⎣⎦ ⎣⎦⎣⎦ ⎣⎦[][]cccccppVZIZIαα⎡⎤ ⎡⎤⎡⎤=+⎣⎦⎣⎦ ⎣⎦[][]dddddqqVZIZIββ⎡⎤ ⎡⎤⎡⎤=+⎣⎦⎣⎦ ⎣⎦(1)(2)(3)(4)16Segmentation Method (cont.)Segmentation Method (cont.)cZα⎡⎤⎣⎦Zβ⎡⎤⎣⎦pqd[]ZpqAlso, from Kirchhoff’s laws we have[][]cdII=−[][]cdVV=(5)(6)17Segmentation Method (cont.)Segmentation Method (cont.)Substituting (3) and (4) into (6) and using (5) yields[][]cc c cp p dd c dq qZI Z I Z I Z Iαα β β⎡⎤ ⎡⎤ ⎡⎤ ⎡⎤⎡⎤⎡⎤+=−+⎣⎦⎣⎦⎣⎦ ⎣⎦ ⎣⎦ ⎣⎦Hence[]cc dd c cp p dq qZZ I Z I Z Iαβ α β⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤+=− +⎣⎦⎣⎦⎣⎦⎣⎦⎣⎦so that[]()1cccdd cppdqqIZZ Z I Z Iαβ α β−⎡⎤⎡⎤⎡⎤⎡⎤⎡⎤=+ − +⎣⎦⎣⎦⎣⎦⎣⎦⎣⎦(7)18Segmentation Method (cont.)Segmentation Method (cont.)Next, insert (7) into (1):()()1p pp p pc cc dd cp p dq qVZIZZZ ZIZIαααβαβ−⎡⎤ ⎡⎤⎡ ⎤ ⎡⎤ ⎡⎤⎡⎤ ⎡⎤ ⎡⎤ ⎡⎤=+ +−+⎣⎦ ⎣⎦ ⎣⎦ ⎣⎦⎣⎦ ⎣⎦⎣ ⎦ ⎣⎦ ⎣⎦Similarly, from (2) we have()1q qq q qd cc dd dq q c p pVZIZZZ ZIZIββαββα−⎡⎤ ⎡⎤⎡ ⎤ ⎡⎤ ⎡⎤⎡⎤ ⎡⎤ ⎡⎤ ⎡⎤=++−+⎣⎦ ⎣⎦ ⎣⎦ ⎣⎦⎣⎦ ⎣⎦⎣ ⎦ ⎣⎦ ⎣⎦(8)(9)From the definition of the Z matrix of the overall system, we can also writeppp p pq qVZIZI⎡⎤ ⎡⎤⎡⎤⎡⎤⎡⎤=+⎣⎦⎣⎦⎣⎦⎣⎦ ⎣⎦qqppqqqVZIZI⎡⎤ ⎡⎤⎡⎤⎡⎤⎡⎤=+⎣⎦⎣⎦⎣⎦⎣⎦ ⎣⎦(10)(11)Comparing (8) and (9) with (10) and (11), we have the following results:19Segmentation Method (cont.)Segmentation Method (cont.)()1pp pp pc cc dd cpZZZZZZαααβα−⎡⎤⎡⎤⎡ ⎤⎡⎤⎡⎤=− +⎣⎦⎣⎦⎣⎦⎣ ⎦⎣⎦()1pq pc cc dd dqZZZZZααβ β−⎡⎤⎡ ⎤⎡⎤⎡⎤=+⎣⎦⎣⎦⎣ ⎦⎣⎦()1qp qd cc dd cpZZZZZβαβ α−⎡⎤⎡ ⎤⎡⎤⎡⎤=+⎣⎦⎣⎦⎣ ⎦⎣⎦()1qq qq qd cc dd dqZZZZZZββαββ−⎡⎤⎡⎤⎡ ⎤⎡⎤⎡⎤=− +⎣⎦⎣⎦⎣⎦⎣ ⎦⎣⎦20Segmentation Method (cont.)Segmentation Method (cont.)To find the input impedance, assume that the feed port is located on patch α, and that all other patches have been reduced (one pair at a time) to a single connecting patch β. Zβ⎡⎤⎣⎦Zα⎡⎤⎣⎦feedcd21Segmentation Method (cont.)Segmentation Method (cont.)()111in in c cc dd cZZ Z ZZ Zαα αβ α−⎡⎤⎡ ⎤ ⎡ ⎤=− +⎣⎦⎣ ⎦ ⎣ ⎦first of the previous set of equations then specializes to In this equation Zinαis the input impedance of the patch that the feed is connected to (patch α) when this patch is not connected to the other