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A periodic structure is one made up of identical elements that exhibits a repeated geometric pattern. It is made up of identical elements that 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 &quot;'''Array Object'''&quot; 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 &quot;'''Periodic Unit Cell'''&quot;.  Periodic structures have many applications including phased array antennas, frequency selective surfaces (FSS), electromagnetic bandgap structures (EBG), metamaterial structures, etc. [[EM.Cube]] allows you to model both finite and infinite periodic structures.<br /> <br /> Real practical periodic structures obviously have finite extents. You can easily and quickly construct finite-sized arrays of arbitrary complexity using [[EM.Cube]]'s &quot;Array Tool&quot;. However, for large values of Nx and Ny, the size of the computational problem may rapidly get out of hand and become impractical. For very large periodic arrays, you can alternatively analyze a unit cell subject to the periodic boundary conditions and calculate the current distribtutions and far fields of the periodic unit cell. For their radiation patterns, you can multiply the &quot;Element Pattern&quot; by an &quot;Array Factor&quot; that captures the finite extents of the structure. In many cases, an approximation of this type works quite well. But in some other cases, the edge effects and particularly the field behavior at the corners of the finite-sized array cannot be modeled accurately. Periodic surfaces like FSS, EBG and metamaterials are also modeled as infinite periodic structures, for which one can define reflection and transmission coefficients. For this purpose, the periodic structure is excited using a plane wave source. Reflection and transmission coefficients are typically functions of the angles of incidence. === Regular vs. Generalized Periodic Lattices === Besides conventional rectangular lattices, [[EM.Cube]]'s [[Planar Module]] 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. A periodic structure is a repetition of a basic structure (unit cell) at pre-determined locations. Let these locations be described by (x_mn, y_mn), where m and n are integers ranging from -8 to 8. For a general skewed grid, x_mn and y_mn can be described by: :<math> \begin{align} & x_{mn} = m\Delta x + n \Delta x' \\ & y_{mn} = m\Delta y + n \Delta y' \end{align} </math> where <math>\Delta x</math> is the primary offset in the X direction (X Spacing) controlled by index m and <math>\Delta x'</math> is the secondary offset in the X direction (X Offset) controlled by index n. The meanings of <math>\Delta y</math> (Y Spacing) and <math>\Delta y'</math> (Y Offset) are similar with the roles of indices m and n interchanged. To illustrate how to use this definition, consider an example of an equilateral triangular grid with side length L as shown in the figure below. 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>.