Figure 2: An example of the 3D mono-static radar cross section plot of a patch antenna.
== Simulating Infinite and Finite-Sized Periodic Planar Structures & Antenna Arrays ==
[[Image:image121.png|thumb|400px|Diagram of an equilateral triangular periodic lattice.]]
[[Image:image122.png|thumb|600px|Modeling a periodic screen using two different types of unit cell.]][[Image:pmom_per5_tn.png|thumb|400px|The PEC cross unit cell.]][[Image:pmom_per6_tn.png|thumb|400px|Planar mesh of the PEC cross unit cell. Note the cell extensions at the unit cell's boundaries.]]=== Finite Arrays vs. 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'''" in [[EM.Cube]]. 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.Cube]] is represented by a "'''Periodic Unit Cell'''".
From the figure, it is obvious that the y coordinate of each row is fixed and identical, thus <math>\Delta y = L</math> and <math>\Delta y' = 0</math>. While in each row the spacing between adjacent elements is L, there is an offset of L/2 between the consecutive rows. This results in <math>\Delta x = L</math> and <math>\Delta x' = L/2</math>. To sum up, an equilateral triangular grid can be described by <math>\Delta x = L</math>, <math>\Delta x' = L/2</math>, <math>\Delta y = L</math> and <math>\Delta y' = 0</math>. In an [[EM.Cube]] [[Planar Module]] project, the secondary offsets are equal to zero by default, implying a rectangular lattice. You can change the values of the secondary offsets using the boxes labeled '''X Offset''' and '''Y Offset''' in the '''Periodicity Settings Dialog''', respectively. Triangular and Hexagonal lattices are popular special cases of the generalized lattice type. In a triangular lattice with alternating Rows, <math>\Delta x' = \Delta x/2</math> and <math>\Delta y' = 0</math>. A Hexagonal lattice (with alternating rows) is a special case of triangular lattice in which <math>\Delta y = \sqrt{3\Delta x / 2}</math>.
=== Interconnectivity Among Unit Cells ===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.
In many casesAs an example, your planar structure's traces or embedded objects are entirely enclosed inside the consider a periodic unit cell and do not touch metallic screen as shown in the boundary of figure on the right. The unit cellof this structure can be defined as a rectangular aperture in a PEC ground plane (marked as Unit Cell 1). In [[EMthis case, the rectangle object is defined as a slot trace.Cube]]'s [[Planar Module]]Alternatively, you can define periodic structures whose a unit cells are interconnected. Interconnectivity applies only to PEC, PMC and conductive sheet traces, and embedded object sets are excluded. Note that cell in the form of a periodic planar structure, your objects cannot microstrip cross on a metal trace. In the periodic domain. Howeverlatter case, you can arrange objects with linear edges such as one or more flat edges line up with the domain's bounding box. In such caseshowever, [[EM.Cube]]'s planar MoM mesh generator will take into account the continuity of the currents microstrip cross should extend across the adjacent connected unit cells cell and will create connect to the connection basis functions at crosses in the right and top boundaries of the unit cell. It is clear that due neighboring cells in order to periodicity, the basis functions do not need to be extended at the left or bottom boundaries of the unit cellprovide current continuity.
As an example, consider the periodic structure in the figure below that shows a metallic screen or wire grid. 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. [[File:image122.png]] Figure 1: Modeling a periodic screen using two different types of unit cell. [[File:pmom_per3_tn.png|400px]] [[File:pmom_per4_tn.png|400px]] Figure 2: The PMC aperture unit cell and its planar mesh. [[File:pmom_per5_tn.png|400px]] [[File:pmom_per6_tn.png|400px]] Figure 3: The PEC cross unit cell and its planar mesh. Notice the cell extensions at the unit cell's boundaries. === Defining A Running a Periodic Domain MoM Analysis ===
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.