Changes
The right choice of the basis functions to represent the elementary currents is very important. It will determine the accuracy and computational efficiency of the resulting numerical solution. Rooftop basis functions are one of the most popular types of basis functions used in a variety of MoM formulations. The surface currents (whether electric or magnetic) are discretized using 2D rooftop basis functions shown in the figure below:
[[File:manuals/emagware/emcube/modules/planar/a-25-d-method-of-moments-primer/meshing-and-discretization-of-planar-structures/image055_tn.png]]
Figure 1: Rooftop or RWG basis functions built over two rectangular, triangular or mixed cells.
The volume polarization currents in 2.5-D MoM have a vertical direction along the Z-axis. These are discretized using prismatic basis functions that have either a rectangular or triangular base with a constant profile along the Z-axis.
[[File:manuals/emagware/emcube/modules/planar/a-25-d-method-of-moments-primer/meshing-and-discretization-of-planar-structures/image065_tn.png]][[File:manuals/emagware/emcube/modules/planar/a-25-d-method-of-moments-primer/meshing-and-discretization-of-planar-structures/image066_tn.png]]
Figure 2: Prismatic basis functions built over single triangular and rectangular cells.
You can manage your project's layer hierarchy from the Layer Stack-up Settings dialog. You can add, delete and move around substrate layers, metallic and slot traces and embedded object sets. Metallic and slot traces can move among the interface planes between neighboring substrate layers. Embedded object sets including PEC vias and finite dielectric objects can move from substrate layer into another. When you delete a trace from the Layer Stack-up Settings dialog, all of its objects are deleted from the project workspace, too. You can also delete metallic and slot traces or embedded object sets from the Navigation Tree. To do so, right click on the name of the trace or object set in the Navigation Tree and select '''Delete''' from the contextual menu. You can also delete all the traces or object sets of the same type from the contextual menu of the respective type category in the Navigation Tree.
For better visualization of your planar structure, EM.Cube displays a virtual domain in a default orange color to represent part of the infinite background structure. The size of this virtual domain is a quarter wavelength offset from the largest bounding box that encompasses all the finite objects in the project workspace. You can change the size of the virtual domain or its display color from the Domain Settings dialog, which you can access either by clicking the '''Computational Domain''' [[File:manuals/emagware/emcube/modules/planar/anatomy-of-a-planar-structure/defining-a-background-structure/domain_icon.png]] button of the '''Simulate Toolbar''', or by selecting '''Simulate > Computational Domain > Domain Settings...''' from the Simulate Menu or by right clicking the '''Virtual Domain''' item of the Navigation Tree and selecting '''Domain Settings...''' from the contextual menu, or using the keyboard shortcut '''Ctrl+A'''. But keep in mind that the virtual domain is only for visualization purpose and does not affect the MoM simulation. The virtual domain also shows the substrate layers in translucent colors. As you change the colors assigned to the substrate layers, you will see a multilayer virtual domain box surrounding your project structure.
[[File:manuals/emagware/emcube/modules/planar/anatomy-of-a-planar-structure/defining-a-background-structure/pmom_phys5.png]]
Figure 1: [[Planar Module]]'s Virtual Domain Settings dialog.
EM.Cube’s [[Planar Module]] offers two mesh generation algorithms for discretizing planar structures: Hybrid and Triangular. The hybrid mesh consists of both rectangular and triangular cells. The hybrid mesh generator creates a kind of “object-centric” mesh that depends on the geometry of each object. It tries to discretize rectangular objects with rectangular cells as much as possible. In certain connection areas, a few triangular cells might be inserted to provide the mesh transition for current continuity. All the non-rectangular objects (circular, polygonal, etc.) are discretized using triangular cells. The triangular mesh generator, on the other hand, discretizes the planar objects with all triangular cells regardless of their shape. The only exceptions are feed lines that contain gap sources or lumped elements, which are always meshed with rectangular cells.
You can generate and view a planar mesh by clicking the '''Show Mesh''' [[File:manuals/emagware/emcube/modules/planar/mesh-generation/the-planar-mom-mesh/mesh_tool.png]] button of the '''Simulate Toolbar''' or by selecting '''Menu > Simulate > Discretization > Show Mesh''' or using the keyboard shortcut '''Ctrl+M'''. When the mesh of the planar structure is displayed in EM.Cube’s project workspace, its "Mesh View" mode is enabled. In this mode you can perform view operations like rotate view, pan or zoom, but you cannot create new objects or edit existing ones. To exit the mesh view mode, press the keyboard's '''Esc Key''' or click the '''Show Mesh''' [[File:manuals/emagware/emcube/modules/planar/mesh-generation/the-planar-mom-mesh/mesh_tool.png]] button once again.
Once a mesh is generated, it stays in the memory until the structure is changed or the mesh density or other settings are modified. Every time you view mesh, the one in the memory is displayed. You can force EM.Cube to create a new mesh from the ground up by selecting '''Menu > Simulate > Discretization > Regenerate Mesh''' or by right clicking on the '''Planar Mesh''' item in the '''Discretization''' section of the Navigation Tree and selecting '''Regenerate''' from the contextual menu.
=== Customizing A Planar Mesh ===
You can change the settings of the planar mesh including the mesh type and density from the planar Mesh Settings Dialog. You can also change these settings while in the mesh view mode, and you can update the changes to view the new mesh. To open the mesh settings dialog, either click the '''Mesh Settings''' [[File:manuals/emagware/emcube/modules/planar/mesh-generation/changing-mesh-type-resolution/mesh_settings.png]] button of the '''Simulate Toolbar''' or select '''Menu > Simulate > Discretization > Mesh Settings...''', or by right click on the '''Planar Mesh''' item in the '''Discretization''' section of the Navigation Tree and select '''Mesh Settings...''' from the contextual menu, or use the keyboard shortcut '''Ctrl+G'''. You can change the mesh algorithm from the dropdown list labeled '''Mesh Type''', which offers two options: '''Hybrid''' and '''Triangular'''. You can also enter a different value for '''Mesh Density''' in cells per effective wavelength (?<sub>eff</sub>). For each value of mesh density, the dialog also shows the average "Cell Edge Length" in the free space. To get an idea of the size of mesh cells on the traces and embedded object sets, divide this edge length by the square root of the effective permittivity a particular trace or set. Click the '''Apply''' button to make the changes effective.
[[File:PMOM31.png]]
Figure 2: Discretizing a planar surface object using EM.Cube's Polymesh tool.
Keep in mind that since a polymesh object it considered a final mesh, its mesh cannot be connected to other objects. In other words, bridge basis functions are not generated if even some of the polymesh edges may coincide with other objects' edges. A polymesh object is treated by the mesh generator as an isolated mesh. However, EM.Cube allows you to connect polymesh objects manually. To do so, bring two or more polymesh objects close to each other so that they have one or more common edges. No face overlaps are allowed in this case. Select the polymesh objects and click the '''Merge Tool'''[[File:manuals/emagware/cubecad/creating-more-complex-objects/merging-open-curves/merge_tool_tn.png]] button of '''Tools Toolbar''' to merge the polymesh objects into a single polymesh object. The new merged polymesh object will provide all the necessary bridge basis functions among the original, separate polymesh objects.
== Excitation Sources ==
Lumped elements are components, devices, or circuits whose overall dimensions are very small compared to the wavelength. As a result, they are considered to be dimensionless compared to the dimensions of a mesh cell. In fact, a lumped element is equivalent to an infinitesimally narrow gap that is placed in the path of current flow, across which the device's governing equations are enforced. Using Kirkhoff's laws, these device equations normally establish a relationship between the currents and voltages across the device or circuit. Crossing the bridge to Maxwell's domain, the device equations must now be cast into a from o boundary conditions that relate the electric and magnetic currents and fields. EM.Cube's [[Planar Module]] allows you to define passive circuit elements: '''Resistors'''(R), C'''apacitors'''(C), I'''nductors'''(L), and series and parallel combinations of them as shown in the figure below:
[[File:manuals/emagware/emcube/modules/planar/excitation-sources/using-lumped-circuits/image106.png]]
Figure 1: A series-parallel RLC combination that can be modeled as a lumped circuit in [[Planar Module]].
=== Running A Planar MoM Analysis ===
To run a planar MoM analysis of your project structure, open the Run Simulation Dialog by clicking the '''Run''' [[File:manuals/emagware/emcube/modules/planar/running-planar-mom-simulations/running-a-planar-mom-analysis/run_icon.png]] button on the '''Simulate Toolbar''' or select '''Menu''' '''>''' '''Simulate >''' '''Run''' or use the keyboard shortcut '''Ctrl+R'''. The '''Analysis''' option of the '''Simulation Mode''' dropdown list is selected by default. Once you click the '''Run''' button, the simulation starts. A new window, called the '''Output Window''', opens up that reports the different stages of simulation and the percentage of the tasks completed at any time. After the simulation is successfully completed, a message pops up and reports the end of simulation. In certain cases like calculating scattering parameters of a circuit or reflection / transmission characteristics of a periodic surface, some results are also reported in the Output Window. At the end of a simulation, you need to click the '''Close''' button of the Output Window to return to the project workspace.
[[File:PMOM78.png]]
Figure 1: Selecting port characteristics data to plot from the Navigation Tree.
You can also see a list of all the port characteristics data files in EM.Cube's Data Manager. To open data manager, click the '''Data Manager''' [[File:manuals/emagware/emcube/modules/planar/running-simulations/defining-custom-output-parameters/data_manager_icon.png]] button of the '''Simulate Toolbar''' or select '''Simulate > Data Manager''' from the menu bar, or right click on the '''Data Manager''' item of the Navigation Tree and select '''Open Data Manager'''... from the contextual menu. You can also use the keyboard shortcut '''Ctrl+D''' at any time. Select any data file by clicking and highlighting its row in the table and then click the '''Plot''' button to plot the graph. By default, the S parameters are plotted as double magnitude-phase graphs, while the Y and Z parameters are plotted as double real-imaginary part graphs. The VSWR data are plotted on a Cartesian graph. You can change the format of complex data plots. In general complex data can be plotted in three forms:
# Magnitude and Phase
All the radiation- and scattering-related standard outputs are available only if you have defined a radiation pattern far field observable or an RCS far field observable, respectively. The standard output parameters DGU and ARU are the directive gain and axial ratio calculated at the certain user defined direction with spherical observation angles (?, f). These angles are specified in degrees as '''User Defined Azimuth & Elevation''' in the "Output Settings" section of the '''Radiation Pattern Dialog'''. The standard output parameters HPBWU, SLLU, FNBU and FNLU are determined at a user defined f-plane cut. This azimuth angle is specified in degrees as '''Non-Principal Phi Plane''' in the "Output Settings" section of the '''Radiation Pattern Dialog''', and its default value is 45°. The standard output parameters BRCS and MRCS are the total back-scatter RCS and the maximum total RCS of your planar structure when it is excited by an incident plane wave source at the specified ?<sub>s</sub> and f<sub>s</sub> source angles. FRCS, on the other hand, is the total forward-scatter RCS measured at the predetermined ?<sub>o</sub> and f<sub>o</sub> observation angles. These angles are specified in degrees as '''User Defined Azimuth & Elevation''' in the "Output Settings" section of the '''Radar Cross Section Dialog'''. The default values of the user defined azimuth and elevation are both zero corresponding to the zenith.
If you are interested in calculating certain quantities at the end of a simulation, which you do not find among EM.Cube's standard output data, you can define your own custom output. EM.Cube allows you to define new custom output as any mathematical expression that involves the available standard output parameters, numbers, variables and all of EM.Cube's mathematical functions. For a list of legitimate mathematical functions, click the '''Functions [[File:manuals/emagware/cubecad/computing-with-cad-objects/mathematical-functions/functions_icon.png]]'''button of the '''Simulate Toolbar''' or select '''Simulate > Functions...'''from the menu bar, or use the keyboard shortcut '''Ctrl+I''' to open the Function Dialog. Here you can see a list of all the available EM.Cube functions with their syntax and a brief description. To define a custom output, click the '''Custom Output [[File:manuals/emagware/emcube/modules/planar/running-simulations/defining-custom-output-parameters/custom_icon.png]]'''button of the '''Simulate Toolbar''' or select '''Simulate > Custom Output...'''from the menu bar, or use the keyboard shortcut '''Ctrl+K''' to open the Custom Output Dialog. This dialog has a list of all of your custom output parameters. Initially, the list empty. You can define a new custom output by clicking the '''Add''' button of the dialog to open up the '''Add Custom Output Dialog'''. In this dialog, first you have to choose a new label for your new parameter and then define a mathematical expression for it. At the bottom of the dialog you can see a list of all the available standard output parameters, whose number and variety depends on your project's source type as well as the defined project observables. When you close the Add Custom Output dialog, it returns you to the Custom Output dialog, where the parameter list now reflects your newly defined custom output. You can edit an existing parameter by selecting its row in the table and clicking the '''Edit''' button, or you can delete any parameter from the list using the '''Delete''' button.
[[File:PMOM141.png]]
=== Viewing & Visualizing Various Output Data Types ===
At the end of a planar MoM simulation, a variety of 2D and 3D output data are generated. Some of these can be visualized or graphed directly from the Navigation Tree, while the others can only be accessed from the Data Manager. All of EM.Cube's simulation data are always written into ASCII data files that you can open and inspect or edit. Lists of these 2D and 3D data files appear under Data Manager's various tabs. The generated data also include all of [[Planar Module]]'s legitimate standard outputs that the simulation engine can compute given the specified source and observable types as well as all of your own previously defined custom output parameters. Note that in this release of EM.Cube, all the custom outputs are real-type data. Each custom output is written into a separate real data file with the same name as the parameter's given label and a "'''.DAT'''" file extension. To open data manager, click the '''Data Manager''' [[File:manuals/emagware/emcube/modules/planar/running-simulations/defining-custom-output-parameters/data_manager_icon.png]] button of the '''Simulate Toolbar''' or select '''Simulate > Data Manager''' from the menu bar, or right click on the '''Data Manager''' item of the Navigation Tree and select '''Open Data Manager'''... from the contextual menu. You can also use the keyboard shortcut '''Ctrl+D''' at any time. Select any data file by clicking and highlighting its row in the table and then click the '''Plot''' button to plot its graph in '''EM.Grid'''. You can also view the contents of a data file by selecting its row in th file list and clicking the '''View''' button of the dialog or by simply double-clicking the highlighted row. This opens up a new window containing a convenient spreadsheet that gives a tabular view of the contents of the selected data file. There are a large number of data operations and manipulations that you can perform on the data content including matrix, calculus and statistical calculations as well as computing and plotting new datasets using the "Compute" feature of the spreadsheet. You can make multiple file selection using the keyboard's '''Ctrl''' and '''Shift''' keys.
[[File:PMOM144.png]]
Parametric sweep is EM.Cube's most versatile sweep type. During a parametric sweep, the values of one or more sweep variables are varied over their specified ranges, and the planar MoM simulation is run for each combination of variable samples. If you define two or more sweep variables, the process will then involve nested sweep loops that follow the order of definition of the sweep variables. The topmost sweep variable in the list will form the outermost nested loop, and the sweep variable at the bottom of the list will form the innermost nested loop. Note that you can alternatively run either a frequency sweep or an angular sweep as parametric sweep, whereby the project frequency or the angles of incidence of a plane wave source are designated as sweep variables. Unlike optimization which will be discussed later, parametric sweeps are simple and straightforward and do not required careful advance planning.
Before you can run a parametric sweep, first you have to define one or more variables in your EM.Cube project. A variable is a mathematical entity that has a numeric value. This numeric value can be changed at your discretion at any time. You can define a variable either directly as a number or as a mathematical expression that may involve other previously defined variables. Even in the latter case, an "expression" variable has a numeric value at any time. You can designate almost any numeric quantity or parameter in EM.Cube as a variable. Or alternatively, you can associate a variable with almost anything in EM.Cube. This includes all the geometrical properties of CAD objects like coordinates, rotation angles, dimensions, etc. as well as material properties of object groups and background structure, source parameters, project frequency, mesh density, and unit cell periods in the case of a periodic structure. You can define a variable either in a formal manner using EM.Cube's Variable Dialog or directly from the project workspace or from the Navigation Tree. In the former "formal" option, first you open the Variables Dialog by clicking the '''Variables''' [[File:manuals/emagware/cubecad/computing-with-cad-objects/defining-variables/variable_icon_tn.png]] button of the '''Simulate Toolbar''' or selecting '''Menu > Simulate > Variables...'''or using the keyboard shortcut '''Ctrl+B'''. By default, the variable list is initially empty. To add a new variable, click the '''Add''' button to open the "Add Variable Dialog". Choose a '''Name''' for your new variable. In the box labeled '''Definition''', define your new variable either as an independent variable with a numeric value or as a dependent variable using a mathematical expression that involves previously defined variables.
[[File:PMOM146(1).png]]
A design objective is a logical expression that consists of two mathematical expressions separated by one of the logical operators: ==, <, <=, > or >=. These are called the left-hand-side (LHS) and right-hand-side (RHS) mathematical expressions and both must have computable numerical values. They may contain any combination of numbers, constants, variables, standard or custom output parameters as well as EM.Cube's legitimate functions. Objectives that involve the logical operator "'''=='''" are regarded a "'''Goals'''". The RHS expression of a goal is usually chosen to be a number, which is often known as the "'''Target Value'''". In the logical expression of a goal, one can bring the two RHS and LHS expressions to one side establish an equality of the form "(LHS - RHS) == 0". Numerically speaking, this is equivalent to minimizing the quantity | LHS - RHS |. During an optimization process, all the project goals are evaluated numerically and they are used collectively to build an error (objective) function whose value is tried to be minimized. Objectives that involve "non-Equal" logical operators are regarded a "'''Constraints'''". Unlike goals which lead to minimizable numerical values, constraints are rather conditions that should be met while the error function is being minimized.
To define an objective, open the '''Objectives Dialog''' either by clicking the '''Objectives''' [[File:manuals/emagware/emcube/modules/planar/running-simulations/optimization-defining-design-objectives/objective_icon.png]] button of the '''Simulate Toolbar''', or by selecting '''Menu > Simulate > Objectives...''' from the Menu Bar, or using the keyboard shortcut '''Ctrl+J'''. The objectives list is initially empty. To add a new objective, click the '''Add''' button to open up the '''Add Objective Dialog'''. At the bottom of this dialog, you can see a list of all the available EM.Cube output parameters including both standard and custom output parameters. This list may vary depending on the types of sources and observables that you have already defined in your project. You can enter any mathematical expressions in the two boxes labeled '''Expression 1''' and '''Expression 2'''. The Available Output Parameter List simply helps you remember the syntax of these parameters. You should also select one of the available options in the dropdown list labeled '''Logical Operator'''. The default operator is '''"=== (Equal To)"'''. As soon as you finish the definition of an objective, its full logical expression is added to the Objective List. You can always modify the project objectives after they have been created. Select a row in the Objective List and click the '''Edit''' button of the dialog and change the expressions or the logical operator. You can also remove an objective from the list using the '''Delete''' button.
[[File:PMOM151.png]]
where &DELTA;x is the primary offset in the X direction (X Spacing) controlled by index m and &DELTA;x' is the secondary offset in the X direction (X Offset) controlled by index n. The meanings of &DELTA;y (Y Spacing) and &DELTA;y' (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.
[[File:manuals/emagware/emcube/modules/planar/periodic-planar-structures/regular-vs-offset-periodic-lattices/image121.png]]
Figure 1: Diagram of an equilateral triangular periodic lattice.
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:manuals/emagware/emcube/modules/planar/periodic-planar-structures/interconnectivity-among-unit-cells/image122.png]]
Figure 1: Modeling a periodic screen using two different types of unit cell.
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<td align="left">[[File:manuals/emagware/emcube/modules/planar/periodic-planar-structures/interconnectivity-among-unit-cells/pmom_per3_tn.png]]</td><td align="left">[[File:manuals/emagware/emcube/modules/planar/periodic-planar-structures/interconnectivity-among-unit-cells/pmom_per4_tn.png]]</td>
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<td align="left">[[File:manuals/emagware/emcube/modules/planar/periodic-planar-structures/interconnectivity-among-unit-cells/pmom_per5_tn.png]]</td><td align="left">[[File:manuals/emagware/emcube/modules/planar/periodic-planar-structures/interconnectivity-among-unit-cells/pmom_per6_tn.png]]</td>
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Figure 1: Setting the periodic scan angles in [[Planar Module]]'s Gap Source dialog.
[[File:manuals/emagware/emcube/modules/planar/periodic-planar-structures/modeling-periodic-phased-arrays/pmom_per9_tn.png]]
Figure 2: The 3D radiation pattern of a beam-steered periodic printed dipole array.
