== Simulating Infinite and Finite-Sized Periodic Planar Structures ==
[[Image:PMOM99.png|thumb|300px|EM.Picasso's Periodicity Settings dialog.]]
[[Image:image121.png|thumb|400px|Diagram of an equilateral triangular periodic lattice.]]
[[Image:image122.png|thumb|400px|Modeling a periodic screen using two different types of unit cell.]]
[[Image:pmom_per5_tn.png|thumb|300px|The PEC cross unit cell.]]
[[Image:pmom_per6_tn.png|thumb|300px|Planar mesh of the PEC cross unit cell. Note the cell extensions at the unit cell's boundaries.]]
=== The Infinite Periodic Structures & the Periodic Lattice ===
A periodic structure is made up of identical elements that exhibits a repeated geometric pattern and are arranged in the form of a periodic lattice. The spacing between the elements is denoted by Sx along the X direction and Sy along the Y direction. The number of elements is denoted by Nx along the X direction and Ny along the Y direction (i.e. a total of Nx.Ny elements). If Nx and Ny are finite numbers, you have a finite-sized periodic structure, which is constructed using an "'''Array Object'''". If Nx and Ny are infinite, you have an infinite periodic structure with periods Sx and Sy along the X and Y directions, respectively. An infinite periodic structure in EM.Picasso is represented by a "'''Periodic Unit Cell'''".
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To define a periodic structure, you must open EM.Picasso's Periodicity Settings Dialog by right clicking the '''Periodicity''' item in the '''Computational Domain''' section of the navigation tree and selecting '''Periodicity Settings...''' from the contextual menu or by selecting '''Menu''' '''>''' '''Simulate > 'Computational Domain > Periodicity Settings...''' from the menu bar. In the Periodicity Settings Dialog, check the box labeled '''Periodic Structure'''. This will enable the section titled''"''Lattice Properties". You can define the periods along the X and Y axes using the boxes labeled '''Spacing'''. In a periodic structure, the virtual domain is replaced by a default blue periodic domain that is always centered around the origin of coordinates. Keep in mind that the periodic unit cell must always be centered at the origin of coordinates. The relative position of the structure within this centered unit cell will change the phase of the results.
Besides conventional rectangular lattices, EM.Picasso can also handle complex non-rectangular periodic lattices. For example, many frequency selective surfaces have skewed grids. In order to simulate skewed-grid periodic structures, the definition of the grid has to be generalized. Let us define a periodic structure as a repetition of a basic unit cell at pre-determined locations described by (x<sub>mn</sub>, y<sub>mn</sub>), where m and n are integers ranging from -∞ to +∞. For a general skewed grid, x<sub>mn</sub> and y<sub>mn</sub> can be expressed as:
In many cases, your planar structure's traces or embedded objects are entirely enclosed inside the periodic unit cell and do not touch the boundary of the unit cell. EM.Picasso allows you to define periodic structures whose unit cells are interconnected. The interconnectivity applies only to PEC, PMC and conductive sheet traces, and embedded object sets are excluded. Your objects cannot cross the periodic domain. In other words, the neighboring unit cells cannot overlap one another. However, you can arrange objects with linear edges such that one or more flat edges line up with the domain's bounding box. In such cases, EM.Picasso's planar MoM mesh generator will take into account the continuity of the currents across the adjacent connected unit cells and will create the connection basis functions at the right and top boundaries of the unit cell. It is clear that due to periodicity, the basis functions do not need to be extended at the left or bottom boundaries of the unit cell. As an example, consider a periodic metallic screen as shown in the figure on the right. The unit cell of this structure can be defined as a rectangular aperture in a PEC ground plane (marked as Unit Cell 1). In this case, the rectangle object is defined as a slot trace. Alternatively, you can define a unit cell in the form of a microstrip cross on a metal trace. In the latter case, however, the microstrip cross should extend across the unit cell and connect to the crosses in the neighboring cells in order to provide current continuity.
[[Image:PMOM98.png|thumb|600px|Changing the number of Floquet modes from the Planar MoM Engine Settings dialog.]]
=== Running a Periodic MoM Analysis ===
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In general, a planar structure in [[EM.Cube]]'s [[Planar Module]] is assumed to have open boundaries. This means that the structure has infinite dimensions along the X and Y directions. In other words, the layers of the background structure extend to infinity, while the traces and embedded object sets have finite sizes. Along the Z direction, a planar structure can be open-boundary, or it may be truncated by PEC ground planes from the top or bottom or both. You can define a planar structure to be infinitely periodic along the X and Y directions. In this case, you only need to define the periodic unit cell. [[EM.Cube]] automatically reproduces the unit cell infinitely and simulates it using a spectral domain periodic version of the Green's functions of your project's background structure.
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To define a periodic structure, you must open [[Planar Module]]'s Periodicity Settings Dialog by right clicking the '''Periodicity''' item in the '''Computational Domain''' section of the Navigation Tree and selecting '''Periodicity Settings...''' from the contextual menu or by selecting '''Menu''' '''>''' '''Simulate > 'Computational Domain > Periodicity Settings...''' from the Menu Bar. In the Periodicity Settings Dialog, check the box labeled '''Periodic Structure'''. This will enable the section titled''"''Lattice Properties". You can define the periods along the X and Y axes using the boxes labeled '''Spacing'''. You can also define values for periodic '''Offset''' along the X and Y directions, which will be explained later.
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In a periodic structure, the virtual domain is replaced by a default blue periodic domain that is always centered around the origin of coordinates. Keep in mind that the periodic unit cell must always be centered at the origin of coordinates. The relative position of the structure within this centered unit cell will change the phase of the results.
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[[File:PMOM99.png]]
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Figure 1: [[Planar Module]]'s Periodicity Settings dialog.
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[[File:PMOM98.png]]
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Figure 1: Changing the number of Floquet modes from the Planar MoM Engine Settings dialog.
=== Modeling Periodic Phased Arrays ===