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 "Array Tool". 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 "Element Pattern" by an "Array Factor" 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.
Â
=== 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<sub>mn</sub>, y<sub>mn</sub>), where m and n are integers ranging from -8 to 8. For a general skewed grid, x<sub>mn</sub> and y<sub>mn</sub> can be described by:
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>.
Â
=== Modeling Finite Antenna Arrays ===
Â
The straightforward approach to the modeling of finite-sized antenna arrays is to use the full-wave method of moments (MoM). This requires building an array of radiating elements using [[EM.Cube]]'s '''Array Tool''' and feeding the individual array elements using some type of excitation. For example, if the antenna elements are excited using a gap source or a probe source, you can assign a certain array weight distribution among the elements as well as phase progression among the elements along the X and Y directions. [[EM.Cube]] currently offers uniform, binomial, Chebyshev and (arbitrary) data file-based weight distribution types. The full-wave MoM approach is very accurate and takes into account all the inter-element coupling effects. At the end of a planar MoM simulation of the array structure, you can plot the radiation patterns and other far field characteristics of the antenna array just like any other planar structure.
Â
The radiation pattern of antenna arrays usually has a main beam and several side lobes. Some [[parameters]] of interest in such structures include the '''Half Power Beam Width (HPBW)''', '''Maximum Side Lobe Level (SLL)''' and '''First Null [[Parameters]]''' such as first null level and first null beam width. To have [[EM.Cube]] calculate all such [[parameters]], you must check the relevant boxes in the "Additional Radiation Characteristics" section of the '''Radiation Pattern Dialog'''. These quantities are saved into ASCII data files of similar names with '''.DAT''' file extensions. In particular, you can plot such data files at the end of a sweep simulation.
Â
[[File:PMOM91.png]]
Â
Figure 1: [[Planar Module]]'s Radiation Pattern dialog.
Â
Another approach to modeling a finite-sized antenna array is to analyze one of its elements and use the "Array Factor" concept to calculate its radiation patterns. This method ignores any inter-element coupling effects. In other words, you can regard the structure in the project workspace as a single isolated radiating element. To define an array factor, open the '''Radiation Pattern Dialog''' of the project. In the section titled "'''Impose Array Factor'''", you will see a default value of 1 for the '''Number of Elements''' along the X and Y directions. This implies a single radiator, representing the structure in the project workspace. There are also default zero values for the '''Element Spacing''' along the X and Y directions. You should change both the number of elements and element spacing in the X and Y directions to define a finite array lattice. For example, you can define a linear array by setting the number of elements to 1 in one direction and entering a larger value for the number of elements along the other direction. Keep in mind that when using an array factor for far field calculation, you cannot assign non-uniform amplitude or phase distributions to the array elements. For that purpose, you have to define an array object with a source array.
=== Interconnectivity Among Unit Cells ===
{{Note|In the absence of any finite traces or embedded objects in the project workspace, [[EM.Cube]] computes the reflection and transmission coefficients of the layered background structure of your project.}}
[[FileImage:PMOM102.png|thumb|400px|A periodic planar layered structure with slot traces excited by a normally incident plane wave source.]]
Figure: A periodic You learned earlier how to use [[EM.Cube]]'s powerful, adaptive frequency sweep utility to study the frequency response of a planar layered structure . Adaptive frequency sweep uses rational function interpolation to generate smooth curves of the scattering [[parameters]] with slot traces excited by a normally incident plane relatively small number of full-wave simulation runs in a progressive manner. Therefore, you need a port definition in your planar structure to be able to run an adaptive frequency sweep. This is clear in the case of an infinite periodic phased array, where your periodic unit cell structure must be excited using either a gap sourceor a probe source. You run an adaptive frequency sweep of an infinite periodic phased array in exactly the same way to do for regular, aperiodic, planar structures.
[[EM.Cube]]'s Planar Modules also allows you to run an adaptive frequency sweep of periodic surfaces excited by a plane wave source. In this case, the planar MoM engine calculates the reflection and transmission coefficients of the periodic surface. Note that you can conceptually consider a periodic surface as a two-port network, where Port 1 is the top half-space and Port 2 is the bottom half-space. In that case, the reflection coefficient R is equivalent to S<sub>11</sub> parameter, while the transmission coefficient T is equivalent to S<sub>21</sub> parameter. This is, of course, the case when the periodic surface is illuminated by the plane wave source from the top half-space, corresponding to 90°< θ =180°. You can also illuminate the periodic surface by the plane wave source from the bottom half-space, corresponding to 0° == Modeling Periodic Structures Using θ < 90°. In this case, the reflection coefficient R and transmission coefficient T are equivalent to S<sub>22</sub> and S<sub>12</sub> [[parameters]], respectively. Having these interpretations in mind, [[EM.Cube]] enables the "'''Adaptive Frequency Sweeps ===Sweep'''" option of the '''Frequency Settings Dialog''' when your planar structure has a periodic domain together with a plane wave source.
You learned earlier how to use [[EM.Cube]]'s powerful, adaptive frequency sweep utility to study the frequency response of a planar structure. Adaptive frequency sweep uses rational function interpolation to generate smooth curves of the scattering [[parameters]] with a relatively small number of full-wave simulation runs in a progressive manner. Therefore, you need a port definition in your planar structure to be able to run an adaptive frequency sweep. This is clear in the case of an infinite periodic phased array, where your periodic unit cell structure must be excited using either a gap source or a probe source. You run an adaptive frequency sweep of an infinite periodic phased array in exactly the same way to do for regular, aperiodic, planar structures.=== Modeling Finite Antenna Arrays ===
The straightforward approach to the modeling of finite-sized antenna arrays is to use the full-wave method of moments (MoM). This requires building an array of radiating elements using [[EM.Cube]]'s Planar Modules also allows you to run an adaptive frequency sweep '''Array Tool''' and feeding the individual array elements using some type of periodic surfaces excitation. For example, if the antenna elements are excited by using a plane wave gap source or a probe source. In this case, the planar MoM engine calculates the reflection and transmission coefficients of the periodic surface. Note that you can conceptually consider assign a periodic surface certain array weight distribution among the elements as a two-port network, where Port 1 is well as phase progression among the top half-space elements along the X and Port 2 Y directions. [[EM.Cube]] currently offers uniform, binomial, Chebyshev and (arbitrary) data file-based weight distribution types. The full-wave MoM approach is very accurate and takes into account all the bottom halfinter-spaceelement coupling effects. In that case, At the reflection coefficient R is equivalent to S<sub>11</sub> parameterend of a planar MoM simulation of the array structure, while you can plot the transmission coefficient T is equivalent to S<sub>21</sub> parameterradiation patterns and other far field characteristics of the antenna array just like any other planar structure. This is,  The radiation pattern of courseantenna arrays usually has a main beam and several side lobes. Some [[parameters]] of interest in such structures include the '''Half Power Beam Width (HPBW)''', '''Maximum Side Lobe Level (SLL)''' and '''First Null [[Parameters]]''' such as first null level and first null beam width. To have [[EM.Cube]] calculate all such [[parameters]], you must check the case when relevant boxes in the periodic surface is illuminated by the plane wave source from the top half-space, corresponding to 90°<quot; Additional Radiation Characteristics&thetaquot; = 180°section of the '''Radiation Pattern Dialog'''. You These quantities are saved into ASCII data files of similar names with '''.DAT''' file extensions. In particular, you can also illuminate plot such data files at the periodic surface by the plane wave source from the bottom halfend of a sweep simulation. Another approach to modeling a finite-space, corresponding sized antenna array is to 0° = analyze one of its elements and use the &thetaquot; Array Factor<quot; 90°concept to calculate its radiation patterns. This method ignores any inter-element coupling effects. In this caseother words, you can regard the reflection coefficient R and transmission coefficient T are equivalent to S<sub>22</sub> and S<sub>12</sub> [[parameters]], respectively. Having these interpretations structure in mindthe project workspace as a single isolated radiating element. To define an array factor, [[EMopen the '''Radiation Pattern Dialog''' of the project.Cube]] enables In the section titled "'''Adaptive Frequency SweepImpose Array Factor'''" option , you will see a default value of 1 for the '''Frequency Settings DialogNumber of Elements''' when your planar along the X and Y directions. This implies a single radiator, representing the structure has in the project workspace. There are also default zero values for the '''Element Spacing''' along the X and Y directions. You should change both the number of elements and element spacing in the X and Y directions to define a periodic domain together finite array lattice. For example, you can define a linear array by setting the number of elements to 1 in one direction and entering a larger value for the number of elements along the other direction. Keep in mind that when using an array factor for far field calculation, you cannot assign non-uniform amplitude or phase distributions to the array elements. For that purpose, you have to define an array object with a plane wave sourcearray.
Click here to learn about [[Modeling Finite-Sized Periodic Arrays Using NCCBF Technique]].