Figure 2: An example of the 3D mono-static radar cross section plot of a patch antenna.
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== Simulating Infinite and Finite-Sized Periodic Planar Structures ==
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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.
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[[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 Lattice ===
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An infinite periodic structure in EM.Picasso is represented by a "'''Periodic Unit Cell'''". 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.
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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:
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:<math> \begin{align} & x_{mn} = m\Delta x + n \Delta x' \\ & y_{mn} = m\Delta y + n \Delta y' \end{align} </math>
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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 on the right. From this figure, it is evident 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>.
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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.
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=== Running a Periodic MoM Analysis ===
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Once you designate your planar structure to be treated as "periodic" in EM.Picasso's Periodicity Settings dialog, the Planar MoM simulation engine will use a spectral domain solver to analyze it. In this case, the dyadic Green's functions of periodic planar structure take the form of doubly infinite summations rather than integrals.
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[[Image:MORE.png|40px]] Click here to learn more about the theory of '''[[Planar_Method_of_Moments#Periodic_Planar_MoM_Simulation| Periodic Green's functions]]'''.
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You run a periodic MoM analysis just like an a period MoM simulation from EM.Picasso's Run Dialog. Here, too, you can run a single-frequency analysis or a uniform or adaptive frequency sweep, or a parametric sweep, etc. Similar to the aperiodic structures, you can define several observables for your project. If you open the Planar MoM Engine Settings dialog, you will see a section titled "Infinite Periodic Simulation". In this section, you can set the number of Floquet modes that will be computed in the periodic Green's function summations. By default, the numbers of Floquet modes along the X and Y directions are both equal to 25, meaning that a total of 2500 Floquet terms will be computed for each periodic MoM simulation.
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<table>
<tr>
<td>
[[Image:PMOM98.png|thumb|600px|Changing the number of Floquet modes from the Planar MoM Engine Settings dialog.]]
</td>
</tr>
</table>
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=== Modeling Periodic Phased Arrays ===
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Earlier, it was argued that you can calculate the radiation pattern of a finite antenna array by modeling a single isolated element and multiplying its "Element Pattern" by the "Array Factor". This method gives acceptable results only when the inter-element coupling effects are negligible, as it does not take into account such effects. Planar antennas printed on dielectric substrates usually exhibit inter-element coupling effects due to the propagation of the substrate surface wave modes. If your finite-sized array is very large and you cannot afford a straightforward full-wave MoM simulation of it, you can alternatively model it as an infinite array represented by a periodic unit cell. In this case, you calculate the radiation pattern of the unit cell structure and use it as the "Element Pattern" in conjunction with the "Array Factor". The periodic Green's functions, in this case, capture the inter-element coupling effects. What is missing from this picture is the finite edge effects and/or corner effects, if any.
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When a periodic structure is excited using a gap or probe source, it acts like an infinite periodic phased array. All the periodic replicas of the unit cell structure are excited. You can even impose a phase progression across the infinite array to steer its beam. You can do this from the property dialog of the gap or probe source. At the bottom of the '''Gap Source Dialog''' or '''Probe Source Dialog''', there is a section titled '''Periodic Beam Scan Angles'''. You can enter desired values for '''Theta''' and '''Phi''' beam scan angles in degrees. The corresponding phase progressions are calculated and applied to the periodic Green's functions:
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:<math>\Psi_x = -\frac{2\pi S_x}{\lambda_0} \sin\theta \cos\phi</math>
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:<math>\Psi_y = -\frac{2\pi S_y}{\lambda_0} \sin\theta \sin\phi</math>
<!--[[File:PMOM101.png]]-->
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Note that you have to define a finite-sized array factor in the Radiation Pattern dialog. You do this in the '''Impose Array Factor''' section of this dialog. In the case of a periodic structure, when you define a new far field item in the Navigation Tree, the values of '''Element Spacing''' along the X and Y directions are automatically set equal to the value of '''Periodic Lattice Spacing''' along those directions. You have to set the '''Number of Elements''' along the X and Y directions, which are both equal to one initially, representing a single radiator. If you forget to define an array factor, the radiation pattern of the unit cell structure will be displayed, which does not show beam scanning.
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<table>
<tr>
<td> [[Image:PMOM100.png|thumb|300px|Setting the periodic scan angles in EM.Picasso's Gap Source dialog.]] </td>
<td> [[Image:pmom_per9_tn.png|thumb|400px|The 3D radiation pattern of a beam-steered periodic printed dipole array.]] </td>
</tr>
</table>
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=== Exciting Periodic Structures Using Plane Waves ===
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When a periodic structure is excited using a plane wave source, it acts as a periodic surface that reflects or transmits the incident wave. You can model frequency selective surfaces, electromagnetic band-gap structures and metamaterials in this way. [[EM.Cube]] calculates the reflection and transmission coefficients of periodic surfaces or planar structures. If you run a single plane wave simulation, the reflection and transmission coefficients are reported in the Output Window at the end of the simulation. Note that these periodic characteristics depend on the polarization of the incident plane wave. You set the polarization (TMz or TEz) in the '''Plane Wave Dialog''' when defining your excitation source. In this dialog you also set the values of the incident '''Theta''' and '''Phi''' angles.
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At the end of the planar MoM simulation of a periodic structure with plane wave excitation, the reflection and transmission coefficients of the structure are calculated and saved into two complex data files called "reflection.CPX" and "transmission.CPX". These coefficients behave like the S<sub>11</sub> and S<sub>21</sub> [[parameters]] of a two-port network. You can think of the upper half-space as Port 1 and the lower half-space as Port 2 of this network. As a result, you can run an adaptive sweep of periodic structures with a plane wave source just like projects with gap or probe sources. The reflection and transmission (R/T) coefficients can be plotted in EM.Grid on 2D graphs similar to the S [[parameters]]. You can plot them from the Navigation Tree. To do so, right click on the '''Periodic Characteristics''' item in the '''Observables''' section of the Navigation Tree and select '''Plot Reflection Coefficients''' or '''Plot Transmission Coefficients'''. The complex data files are also listed in [[EM.Cube]]'s '''Data Manager''', where you can view or plot them.
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{{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.}}
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[[Image:PMOM102.png|thumb|400px|A periodic planar layered structure with slot traces excited by a normally incident plane wave source.]]
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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.
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[[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° = 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 Sweep'''" option of the '''Frequency Settings Dialog''' when your planar structure has a periodic domain together with a plane wave source.
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=== Modeling Finite-Sized Periodic Arrays ===
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[[Image:MORE.png|40px]] Click here to learn about '''[[Modeling Finite-Sized Periodic Arrays Using NCCBF Technique]]'''.
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