Figure 1: Planar hybrid and triangular meshes for rectangular patches.
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=== The Rectangular Mesh Advantage ===
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Rectangular cells offer a major advantage over triangular cells for numerical MoM simulation of planar structures. This is due to the fact that the dyadic Green's functions of planar layered background structures are space-invariant on the transverse plane. Recall that the elements of the moment matrix are given by the following equation:
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:<math> Z_{ij}^{(\mu \nu)} = \iiint_{V_i} d\nu f_i^{(\mu)}(r) \cdot \iiint_{V_j}d\nu ' \overline{\overline{G}}_{\mu \nu}(r|r') \cdot f_j^{(v)}(r') </math>
<!--[[File:PMOM24(1).png]]-->
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where the spatial-domain dyadic Green's functions are a function of the observation and source coordinates, '''r'''and '''r' '''. The MoM matrix elements can indeed be interpreted as interactions between two elementary basis functions '''f<sub>i</sub>(r)''' and '''f<sub>j</sub>(r')''' on that particular background structure. The spatial-domain dyadic Green's functions can themselves be expressed in terms of the spectral-domain dyadic Green's functions as follows:
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:<math> \overline{\overline{G}}_{\mu \nu}(r|r') = \frac{1}{(2\pi)^2} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} \tilde{\overline{\overline{G}}}_{\mu \nu} (k_p, z|z') e^{-j[k_x(x-x')+k_y(y-y')]} \, dk_x \, dk_y , \quad {k_p}^2 = {k_x}^2 + {k_y}^2 </math>
<!--[[File:PMOM26.png]]-->
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where the doubly infinite integration is performed with respect to the spectral [[variables]] k<sub>x</sub> and k<sub>y</sub>. As can be seen from the above expression, the spatial-domain dyadic Green's functions are functions of z, z', as well as (x-x') and (y-y'). The MoM matrix elements can now be transformed into the spectral domain as
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:<math> Z_{ij}^{(\mu \nu)} = \dfrac{1}{(2\pi)^2} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} \tilde{f}_i^{(\mu)} (k_x, k_y) \cdot \tilde{\overline{\overline{G}}}_{\mu \nu} (k_{\rho}, z|z') \cdot \tilde{f}_j^{(\nu)} (k_x, k_y) \, dk_x \, dk_y </math>
<!--[[File:PMOM27.png]]-->
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where the tilde symbol signifies the Fourier transform of a function defined as
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:<math> \tilde{f}(k_x, k_y) = \dfrac{1}{(2\pi)^2} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} f(x,y) e^{j(k_x x + k_y y)} \, dx \, dy </math>
<!--[[File:PMOM28(1).png]]-->
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Rectangular cells have simple Fourier transforms. The rooftop basis functions are triangular functions in the direction of current flow and constant in the perpendicular direction. This means that their Fourier transform is a product of a sinc-squared function along one spectral direction and a sinc function along the other. You can see from the figure below that if one deals with a rectangular mesh of identical cells (all equal and parallel), then the interactions among the rooftop basis functions become a functions of the index differences and not the absolute indices:
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:<math> Z_{(i,k)|(j,l)} = Z \Big\langle f_{i,k}(x,y)| f_{j,l}(x', y') \Big\rangle = Z_{(i-j)|(k-l)} </math>
<!--[[File:PMOM29.png]]-->
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In the above equation, the vectorial rooftop basis functions have explicit, double indices: i and k along the local X and Y directions, respectively, for the test (observation) basis function, and j and l along the local X and Y directions, respectively, for the expansion (source) basis function. Thus, uniform rectangular cells, i.e. structured rectangular cells of identical size aligned in the same direction, can speed up the planar MoM simulation significantly due to these symmetry and the invariance properties. For example, all the self-interactions are identical regardless of the location of a rooftop basis function. This reduces the matrix fill process for a total of N rooftop basis functions from an N2 process to one of order N.
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[[File:PMOM25.png]]
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Figure 1: Pairs of rooftop basis functions that have identical MoM interactions.
=== Generating A Planar Mesh ===
You couple two or more sources using the '''Port Definition Dialog'''. To do so, you need to change the default port assignments. First, delete all the ports that are to be coupled from the Port List of the dialog. Then, define a new port by clicking the '''Add''' button of the dialog. This opens up the Add Port dialog, which consists of two tables: '''Available''' sources on the left and '''Associated''' sources on the right. A right arrow ('''-->''') button and a left arrow ('''<--''') button let you move the sources freely between these two tables. You will see in the "Available" table a list of all the sources that you deleted earlier. You may even see more available sources. Select all the sources that you want to couple and move them to the "Associated" table on the right. You can make multiple selections using the keyboard's '''Shift''' and '''Ctrl''' keys. Closing the Add Port dialog returns you to the Port Definition dialog, where you will now see the names of all the coupled sources next to the name of the newly added port.
{{Note|It is your responsibility to set up coupled ports and coupled [[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|transmission lines]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]] properly. For example, to excite the desirable odd mode of a coplanar waveguide (CPW), you need to create two rectangular slots parallel to and aligned with each other and place two gap sources on them with the same offsets and opposite polarities. To excite the even mode of the CPW, you use the same polarity for the two collocated gap sources. Whether you define a coupled port for the CPW or not, the right definition of sources will excite the proper mode. The couple ports are needed only for correct calculation of the port characteristics.}}
[[File:PMOM51(2).png|800px]]