Changes

Since its introduction in 2002, EM.Picasso has been successfully used by numerous users around the globe in industry, academia and government. The new EM.Picasso 2013 has been totally reconstructed based on our integrated [[EM.Cube ]] software foundation. This integration has introduced far more powerful CAD utilities, greater geometrical variety, and a vast array of capabilities like parametric sweep, [[optimization]], data visualization and post-processing computations. The new foundation also facilitates import and export of many popular CAD formats and provides a seamless interface with our other simulation tools.

In a planar MoM simulation, the background structure is usually a layered planar structure that consists of one or more laterally infinite material layers. In [[EM.Cube’s Cube]]’s [[Planar Module]], the layered structure is stacked along the Z axis. In other words, the dimensions of the layers are infinite along the X and Y axes. Metallic traces are placed at the boundaries between the substrate or superstrate layers. These are modeled by perfect electric conductor (PEC) traces or conductive sheet traces of finite thickness and finite conductivity. Some layers might be separated by infinite perfectly conducting ground planes. The two sides of a ground plane can be electromagnetically coupled through one or several slots or apertures. Such slots or apertures are modeled by magnetic currents and are realized and represented by perfectly magnetic conductor (PMC) traces. Furthermore, the metallic traces can be interconnected or connected to ground planes using embedded objects. Such objects can be used to model circuit vias, plated-through holes or dielectric inserts. These are modeled as volume polarization currents.

In [[EM.Cube|EM.CUBE]]'s [[Planar Module|Planar module]], magnetic currents are always surface current with units of V/m. Electric currents, however, can be surface currents with units of A/m as in the case of metallic traces like microstrip lines, or they can be volume currents with units of A/m² as in the case of perfectly conducting vias. Dielectric inserts are modeled as volume polarization currents that are related to the electric field '''E''' in the following manner:

The planar integral equations derived earlier can be solved numerically by discretizing the unknown currents using a proper meshing scheme. The original functional equations are reduced to discretized linear algebraic equations over elementary cells. The unknown quantities are found by solving this system of linear equations, and many other [[parameters ]] can be computed thereafter. This method of numerical solution of integral equations is known as the Method of Moments (MoM). In this method, the unknown electric and magnetic currents are represented by expansions of basis functions as follows:

[[EM.Cube’s Cube]]’s [[Planar Module]] is intended for constructing and modeling planar layered structures. By a planar structure we mean one that contains a background substrate of laterally infinite extents, made up of one or more material layers all stacked up vertically along the Z axis. Objects of finite size are then interspersed among these substrate layers. This is somehow different than [[EM.Cube]]'s other computational modules, which are geared for handling arbitrary 3D structures.

In [[Planar Module]], the background structure, called "'''Layer Stack-up'''", may involve one or more material layers of infinite extents along the X and Y axes but of finite thickness along the Z axis. When you start a new project, the background structure has a single vacuum layer. The layer stack-up is always terminated from the top and bottom by two infinite half-spaces. The terminating half-spaces might be the free space, or a perfect conductor (PEC ground), or any material medium. Most planar structures used in RF and microwave applications such as microstrip-based components have a PEC ground at their bottom. [[EM.Cube]]'s default stack-up has a vacuum top half-space and a PEC bottom half-space. Some structures like stripline components require two bounding PEC grounds at both top and bottom.

The finite-sized objects of a planar structure may include metal traces, slots and apertures, vertical vias and interconnects, or dielectric inserts including air voids inside the substrate layers. Metal traces are modeled as electric surface currents. These are planar [[Surface Objects|surface objects]], always parallel to the XY plane, that are defined on metal (PEC) traces and placed at the boundary (interface) plane between two substrate layers. Slots and apertures are modeled as magnetic surface currents on the surface of an infinite PEC plane and provide electromagnetic coupling between its top and bottom sides. These, too, are constructed using planar [[Surface Objects|surface objects]], always parallel to the XY plane, that are defined on slot (PMC) traces and placed at the boundary (interface) plane between two substrate layers. [[EM.Cube]]'s [[Planar Module]] also allows prismatic objects that can be modeled by electric volume currents. These include vertical vias and dielectric inserts, and are called embedded object sets. [[Planar Module|Planar module]] does not allow construction of 3D CAD objects. Instead, you draw the cross section of prismatic objects as planar [[Surface Objects|surface objects]] parallel to the XY plane. [[EM.Cube ]] then automatically extrudes these cross sections and constructs and displays 3D prisms over them. The prisms extend all the way across the thickness of the host substrate layer.

When you start a new project in [[EM.Cube’s Cube]]’s [[Planar Module]], there is always a default background structure that consists of a finite vacuum layer sandwiched between a vacuum top half-space and a PEC bottom half-space. Every time you enter the [[Planar Module|Planar module]], the '''Stack-up Settings Dialog''' opens up. This is where you define the entire background structure. Once you close this dialog, you can open it again by right clicking the '''Layer Stack-up''' item in the '''Computational Domain''' section of the Navigation Tree and selecting '''Layer Stack-up Settings...''' from the contextual menu. Or alternatively, you can select the menu item '''Simulate > Computational Domain > Layer Stack-up Settings...'''

You can also use [[EM.Cube]]'s Material List to define the material properties of a substrate layer. In the Substrate Layer Dialog, click the '''Material''' button to open the '''Material List'''. In the Material List Dialog, pick any material or type the first letter of a material to highlight it. Then click the '''OK''' button or simply hit the '''Enter''' key of your keyboard to close the list and return to the substrate layer dialog.

Figure 2: [[EM.Cube]]'s Materials dialog.

[[EM.Cube’s Cube]]’s [[Planar Module]] groups objects by their material and electromagnetic properties. Each object group shares the same color and same position in the layer stack-up. All the planar objects belonging to the same trace are located on the same substrate layer boundary. All the prismatic objects belonging to the same embedded set lie inside the same substrate layer and have the same material composition. Theoretically speaking, all the objects belonging to a group are governed by the same boundary conditions. [[EM.Cube’s Cube]]’s [[Planar Module]] currently provides the following types of objects for building a planar layered structure:

[[EM.Cube]]'s [[Planar Module]] has a special feature that makes construction of planar structures quite easy and straightforward. '''The active work plane of the project workspace is always set at the plane of the active trace.''' In [[EM.Cube]]'s other modules, all objects are drawn in the XY plane (z = 0) by default. In [[Planar Module]], all new objects are drawn on a horizontal plane that is located at the Z-coordinate of the currently active trace. As you change the active trace or add a new trace, you will also change the active work plane.

Slots and apertures are cut-out and removed metal in an infinite perfectly conducting (PEC) ground plane. When a slot is excited, tangential electric fields are formed on the aperture, which can be modeled as finite magnetic surface currents confined to the area of the slot. Therefore, instead of modeling the electric surface currents on the PEC ground around the slot, one can alternatively model the finite-extent magnetic surface currents on PMC traces. In [[EM.Cube]]'s [[Planar Module]], you define slot objects under PMC traces. A PMC trace at a certain Z-plane implies the presence of an infinite PEC plane at that Z-coordinate. Therefore, you do not need to define an additional PEC plane at that location on the layer stack-up. The slot (PMC) objects provide the electromagnetic coupling between the two sides of this infinite ground plane. By the same token, you cannot place a PEC trace and a PMC trace at the same Z-level, as the latter's ground will short the former. However, you can define two or more PMC traces at the same Z-plane. In this case, all the slot objects lie on the same infinite PEC ground plane. <br />

Embedded object sets represent short material insertions inside substrate layers. They can be metal or dielectric. Metallic embedded objects can be used to model vias, plated-through holes, shorting pins and interconnects. These are called PEC via sets. Embedded dielectric objects can be used to model air voids, thin films and material inserts in metamaterial structures. Embedded magnetic object are not currently supported by [[EM.Cube’s Cube]]’s [[Planar Module]].

To add a new object set, click the arrow symbol on the '''Insert''' button of the dialog and select one of the two options, '''PEC Via Set''' or '''Embedded Dielectric Set''', from the dropdown list. This opens up a new dialog where first you have to set the host layer of the new object set. A dropdown list labeled &quot;'''Host Layer'''&quot; gives a list of all the available finite substrate layers. You can also set the properties of the embedded object set, including its label, color and material properties. Keep in mind that you cannot control the height of embedded objects. Moreover, you cannot assign material properties to PEC via sets, while you can set values for the '''Permittivity'''(&epsilon;_r) and '''Electric Conductivity'''(&sigma;) of embedded dielectric sets. Vacuum is the default material choice. You may use [[EM.Cube]]'s Material List for this purpose, which can be opened up by clicking the '''Material''' button. Once embedded object sets are added to the Embedded Sets table, you can edit their properties at any time by selecting their row and clicking the '''Edit''' button.

After a new embedded object set has been defined and added to the Navigation Tree, it becomes the active trace. You are now ready to create geometrical objects in the new active trace. Remember that [[Planar Module]] does not allow you to draw 3D objects. The solid object buttons in the '''Object Toolbar''' are disabled to prevent you from doing so. Instead, you draw planar [[Surface Objects|surface objects]] as the cross section of embedded sets. [[EM.Cube ]] extends these planar objects across their host layer automatically and displays them as wire-frame, 3D extruded objects. Extrusion of embedded object sets happen after meshing and before every simulation. You can enforce this extrusion manually by right clicking the '''Layer Stack-up''' item in the '''Computational Domain''' section of the Navigation Tree and selecting '''Update Planar Structure...''' from the contextual menu.

# The current release of [[EM.Cube ]] allows any number of PEC via sets collocated in the same substrate layer. However, you can define only one embedded dielectric object set per substrate layer, and no vias sets collocated in the same layer. Note that the single set can host an arbitrary number of embedded dielectric objects of the same material properties.

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:domain_icon.png]] button of the '''Simulate Toolbar''', or by selecting '''Simulate &gt; Computational Domain &gt; 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.

By default, the last defined trace or embedded object set is active. You can activate any trace or embedded object set at any time for drawing new objects. You can move one or more selected objects from any trace or embedded object set to another group of the same type or of different type. First select an object in the project workspace or in the Navigation Tree. Then, right click on the highlighted selection and select '''Move To &gt;''' from the contextual menu. This opens another sub-menu containing '''Planar''' and a list of all the other [[EM.Cube ]] modules that have already defined object groups. Select '''Planar''' or any other available module, and yet another sub-menu opens up with a list of all the available traces and embedded object sets already defined in your project. Select the desired group, and all the selected objects will move to that group. When selecting multiple objects from the Navigation Tree, make sure that you hold the keyboard's '''Shift Key''' or '''Ctrl Key''' down while selecting a group's name from the contextual menu.

It is well known that any planar geometry with any degree of complexity can be reasonably discretized using a surface triangular mesh. [[EM.Cube]]'s [[Planar Module]] provides a versatile triangular mesh generator for this purpose. This generates a regular mesh, in which most of the triangular cells have almost equal areas. The uniformity or regularity of mesh is an important factor in warranting a stable numerical solution. A highly incongruous mesh may even produce completely erroneous results. [[EM.Cube]]'s [[Planar Module]] also offers another mesh generator that creates a &quot;Hubrid&quot; planar mesh combining triangular and rectangular cells. Although triangular cells are more versatile than rectangular cells in adapting to arbitrary geometries, many practical planar structures contain a large number of rectangular parts like patch antennas, microstrip lines and components, etc.

[[EM.Cube’s 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:mesh_tool.png]] button of the '''Simulate Toolbar''' or by selecting '''Menu &gt; Simulate &gt; Discretization &gt; Show Mesh''' or using the keyboard shortcut '''Ctrl+M'''. When the mesh of the planar structure is displayed in [[EM.Cube’s Cube]]’s project workspace, its &quot;Mesh View&quot; 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: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 &gt; Simulate &gt; Discretization &gt; 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.

[[EM.Cube]]'s [[Planar Module]], by default, generates a hybrid mesh of your planar structure with a mesh density of 20 cells per effective wavelength. It is important to understand the concept of mesh density (either hybrid or triangular) as used by [[Planar Module]]. It gives a measure of the number of cells per effective wavelength that are placed in various regions of your planar structure. The higher the mesh density, the more cells are created on the geometrical objects. Keep in mind that only the finite-sized objects of your structure are discretized. No mesh is generated for the substrate layers of your background structure. The free-space wavelength is defined as <math>\lambda_0 = \tfrac{2\pi f}{c}</math>, where f is the center frequency of your project and c is the speed of light in the free space. The effective wavelength is defined as <math>\lambda_{eff} = \tfrac{\lambda_0}{\sqrt{\varepsilon_{eff}}}</math>, where e_eff is the effective permittivity.

The effective permittivity is defined differently for different types of traces and embedded object sets. For metal and conductive sheet traces, the effective permittivity is defined as the larger of the permittivity of the two substrate layers just above and below the metallic trace. For slot traces, the effective permittivity is defined as the mean (average) of the permittivity of the two substrate layers just above and below the metallic trace. These definitions of effective permittivity are consistent with the effective propagation constant of [[Transmission Lines|transmission lines ]] realized on such trace types. For embedded object sets, the effective permittivity is defined as the largest of the permittivities of all the substrate layers and embedded dielectric sets. In all cases, for the purpose of calculating the effective wavelength, only the real part of the permittivities are considered. The reason for using an effective wavelength so defined for determination of mesh resolution is to make sure that enough cells are placed in areas that might feature higher field concentration.

Using the generated mesh of a planar structure, [[EM.Cube ]] creates a set of vectorial basis functions that are passed to the input file of the Planar MoM simulation engine. This engine requires edge-based basis functions. The common edges between adjacent cells are used to define edge-based rooftop or RWG basis functions. These elementary basis functions indeed provide the current flow and warrant the continuity among the mesh cells. Therefore, when two objects overlap or share a common edge, the connection between them must be translated into &quot;bridge&quot; basis functions, which carry the information about current flow to the simulation engine.

'''The most important rule of object connections in [[EM.Cube]]'s [[Planar Module]] is that only objects belonging to the same trace can be connected to one another.''' For example, if two objects reside on the same Z-plane and geometrically have a common edge which you can clearly see in the project workspace, but organizationally they belong to two different metal traces, then the bridge basis functions will not be generated between them, and the simulation engine will see them disconnected. If two objects belong to the same trace and have a common overlap area, [[EM.Cube ]] first merges the two objects using the &quot;Boolean Union&quot; operation and converts them into a single object for the purpose of meshing. The mesh of &quot;unioned&quot; areas is usually made up of triangular cells. If two objects reside on the same Z-plane and geometrically overlap with each other but organizationally belong to two different trace groups, incongruous, overlapped cells will be generated that will either blow up the linear system or produce completely wrong simulation results.

When two planar objects belonging to the same trace are connected via a common edge, it is critical to generate a consistent mesh at the connection area and properly transition and merge the meshes of the individual objects. [[EM.Cube]]'s triangular planar mesh generator simply &quot;unions&quot; the two objects and generates a connected mesh. [[EM.Cube]]'s hybrid planar mesh generator, however, behave differently when it comes to the connection between rectangular objects. The rule in this case is the following:

[[EM.Cube]]'s [[Planar Module]] models embedded objects as vertical volume currents. The vectorial basis functions in this case are Z-directed prisms as opposed to rooftop basis functions. If an embedded object is located under or above a metallic trace or connected from both top and bottom, it is critical to create mesh continuity between the embedded object and its connected metallic traces. In other words, the generated mesh must ensure current continuity between the vertical volume currents and horizontal surface currents. [[EM.Cube’s Cube]]’s planar mesh generator automatically handles situations of this kind and generates all the required connection meshes.

Keep in mind that [[EM.Cube’s Cube]]’s Planar MoM engine uses a 2.5-D approximation, whereby only vertical volume currents are assumed inside embedded objects. When the height of an embedded object is small (as should typically be under the 2.5-D assumption), one prismatic cell is placed across the object along the Z-axis. Long PEC vias with a very small radius do also satisfy the 2.5-D assumption. In this case, the long via objects are discretized further along the Z direction and generate multiple stacked cells. Several prismatic cells along the Z-axis may increase the simulation time drastically. This is due to the fact that the host layer is effectively subdivided into a number of sub-layers and the stacked cells are treated as stacked vias embedded inside these sub-layers. As a result, the simulation engine needs to compute all the dyadic Green’s functions accounting for the interactions between all such sub-layers.

The Planar Mesh Settings dialog gives a few more options for customizing your planar mesh around geometrical and field discontinuities. You can check the check box labeled &quot;'''Refine Mesh at Junctions'''&quot;, which increases the mesh resolution at the connection area between rectangular objects. Or you can check the check box labeled &quot;'''Refine Mesh at Gap Locations'''&quot;, which may prove particularly useful when gap sources or lumped elements are placed on a short transmission line connected from both ends. Or you can check the check box labeled &quot;'''Refine Mesh at Vias'''&quot;, which increases the mesh resolution on the cross section of embedded object sets and by extension at the connection regions of the metallic objects connected to them. [[EM.Cube ]] typically doubles the mesh resolution locally at the discontinuity areas when the respective boxes are checked.

You should always visually inspect [[EM.Cube]]'s default generated mesh to see if the current mesh settings have produced an acceptable mesh. You may often need to change the mesh density or other [[parameters ]] and regenerate the mesh. The Planar Mesh Settings dialog gives a few more options for customizing your planar mesh.

As mentioned earlier, highly incongruous meshes should always be avoided. Sometimes [[EM.Cube]]'s default mesh may contain very narrow triangular cells due to very small angles between two edges. In some rare cases, extremely small triangular cells may be generated, whose area is a small fraction of the average mesh cell. These cases typically happen at the junctions and other discontinuity regions or at the boundary of highly irregular geometries with extremely fine details. In such cases, increasing or decreasing the mesh density by one or few cells per effective wavelength often resolves that problem and eliminates those defective cells. Nonetheless, [[EM.Cube]]'s planar mesh generator offers an option to identify the defective triangular cells and either delete them or cure them. By curing we mean removing a narrow triangular cell and merging its two closely spaced nodes to fill the crack left behind.

[[EM.Cube ]] by default deletes or cures all the triangular cells that have angles less than 10º. Sometimes removing defective cells may inadvertently cause worse problems in the mesh. You may choose to disable this feature and uncheck the box labeled &quot;'''Remove Defective Triangular Cells'''&quot; in the Planar Mesh Settings dialog. You can also change the value of the minimum allowable cell angle.

[[EM.Cube]]'s [[Planar Module]] provides different ways of controlling the mesh of a planar structure locally. Earlier you saw how to increase the mesh resolution at the discontinuity regions without affecting the mesh of uniform or regular areas of a planar structure. Another way of local mesh control is to lock the mesh density of certain traces or object sets. The mesh density that you specify in the Planar Mesh Settings dialog is a global parameter and applies to all the traces and embedded object sets in your project. However, you can lock the mesh of individual PEC, PMC and conductive sheet traces or embedded objects sets. In that case, the locked mesh density takes precedence over the global density. Note that locking mesh of object groups, in principle, is different than refining the mesh at discontinuities. In the latter case, the mesh of connection areas is affected. However, objects belonging to different traces cannot be connected to one another. Therefore, locking mesh can be useful primarily for isolated object groups that may require a higher (or lower) mesh resolution.

[[EM.Cube ]] allows you to manually and individually mesh geometrical objects using the concept of polymesh. The Polymesh tool converts a planar surface object to a set of interconnected triangular cells, which is basically identical to its triangular surface mesh. Simply select an object and click the '''Polymesh Tool''' [[File:polymesh_tool_tn.png]] button of '''Tools Toolbar''', or select '''Menu &gt; Tools &gt; Polymesh''', or use the keyboard shortcut '''P'''. You can also right click on a selected object and select '''Polymesh''' from the contextual menu. From the Polymesh Dialog, you can control the mesh resolution through the '''Edge Length''' parameter, which is expressed in project units. Note that unlike the planar mesh generator which uses a frequency-dependent mesh density to drive the mesh resolution, the ploymesh's edge length is fixed and purely geometrical and does not change with the project frequency. '''[[EM.Cube]]'s mesh generator considers a polymesh object as a &quot;final&quot; mesh and reproduces it &quot;As Is&quot; during the meshing process.'''

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: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.

In a typical electromagnetic simulation in [[EM.Cube]]'s [[Planar Module]], you define a planar structure that consists of a layered background structure with a number of finite-sized metal and slot traces and possibly embedded metal or dielectric objects interspersed among the substrate layers. The planar structure is then excited by some sort of a signal source that induces electric currents on metal parts and magnetic currents on slot traces. The method of moments (MoM) solver computes these unknown electric and magnetic currents by discretizing the finite-sized objects. The induced currents, in turn, produce their own electric and magnetic fields which coexist (are superposed) with the impressed electric and magnetic fields of the signal source. From a knowledge of the near fields, [[EM.Cube ]] calculates the port characteristics of the planar structure, if any ports have been defined. From a knowledge of the far fields, [[EM.Cube ]] calculates the radiation or scattering characteristics of the planar structure.

For antennas and planar circuits, where you typically define one or more ports, you usually use lumped sources. A lumped source is indeed a gap discontinuity that is placed on the path of an electric or magnetic current flow, where a voltage or current source is connected to inject a signal. Gap sources are placed across metal or slot traces. Probe sources are placed across vertical PEC vias. A de-embedded source is a special type of gap source that is placed near the open end of an elongated metal or slot trace to create a standing wave pattern, from which the scattering [[parameters ]] can be calculated accurately. To calculate the scattering characteristics of a planar structure, e.g. its radar cross section (RCS), you excite it with a plane wave source. Short dipole sources are used to explore propagation of points sources along a layered structure. Huygens sources are virtual equivalent sources that capture the radiated electric and magnetic fields from another structure possibly in another [[EM.Cube ]] computational module and bring them as a new source to excite your planar structure.

Another way of exciting a planar structure is by placing a gap on the path of a vertical current on a PEC via. This represents a filament source, which is used to model coaxial probe excitation. A probe source can be placed only on a PEC via object. Most planar [[Transmission Lines|transmission lines ]] are fed using SMA connectors. The outer conductor of the coaxial line is connected to the ground and its inner conductor is extended across the substrate layer and connected to a metallic line. [[EM.Cube]]'s [[Planar Module]] models a coaxial probe as an infinitesimal gap discontinuity placed across a thin via, representing an ideal voltage source in series with a lumped impedance. When the impedance is zero, the gap acts like an ideal lumped source and creates a uniform electric field across the via. The source pumps vertical electric current into the probe. If the voltage source is shorted (having a zero amplitude), then the gap acts like a shunt lumped element across the via.

In the data file option, the complex amplitude are directly read in from a data file using a real - imaginary format. When this option is selected, you can either improvise the complex array weights or import them from an existing file. In the former case click the '''New Data File''' button. This opens up the [[Windows ]] Notepad with default formatted data file that has a list of all the array element indices with default 1+j0 amplitudes for all of them. You can replace the default complex values with new one and save the Notepad data file, which brings you back to the Gap Source dialog. To import the array weights, click the '''Open Data File''' button, which opens the standard [[Windows ]] Open dialog. You can then select the right data file from the one of your folders. It is important to note that the data file must have the correct format to be read by [[EM.Cube]]. For this reason, it is recommended that you first create a new data file with the right format using Notepad as described earlier and then save it for later use.

Ports are used in a planar structure to order and index the sources for calculation of circuit [[parameters ]] such as scattering (S), impedance (Z) and admittance (Y) [[parameters]]. In [[EM.Cube]]'s [[Planar Module]], you can use the following types of sources to define ports:

{{Note|It is your responsibility to set up coupled ports and coupled [[Transmission Lines|transmission lines ]] properly. For example, to excite the desirable odd mode of a coplanar waveguide (CPW), you need to create two rectangular slots parallel to and aligned with each other and place two gap sources on them with the same offsets and opposite polarities. To excite the even mode of the CPW, you use the same polarity for the two collocated gap sources. Whether you define a coupled port for the CPW or not, the right definition of sources will excite the proper mode. The couple ports are needed only for correct calculation of the port characteristics.}}

If your planar structure has two or more sources, but you have not defined any ports, all the lumped sources excite the structure locally and contribute to the excitation vector needed for the MoM solution of the problem. However, when you assign N ports to the sources, then you have a multiport structure that is characterized by an N×N admittance matrix (instead of a single Y_in parameter), or an N×N impedance matrix, or an N×N scattering matrix. To calculate these matrices, [[EM.Cube ]] uses a binary excitation scheme in conjunction with the principle of linear superposition. In this binary scheme, the structure is analyzed N times. Each time one of the N port-assigned sources is excited, and all the other port-assigned sources are turned off.

In the case of gap sources on metal traces and probe sources on PEC vias, turning a source off means shorting a series voltage source. The electric currents passing through these sources are then found at each port location, and the admittance [[parameters ]] are found as follows:

In the case of gap sources on slot traces, turning a source off means opening a shunt filament current source. The magnetic currents passing through the source locations, and thus the voltages across them, are then found at all ports, and the impedance [[parameters ]] are found as follows:

The N solution vectors that are generated through the N binary excitation analyses are finally superposed to produce the actual solution to the problem. However, in this process, [[EM.Cube ]] also calculates all the port characteristics. Keep in mind that the impedance (Z) and admittance (Y) matrices are inverse of each other. From the impedance matrix, the scattering matrix is calculated using the following relation:

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:

[[EM.Cube]]'s [[Planar Module]] allows you to define a voltage source in series with a series-parallel RLC combination and place them across the gap. This is called an active lumped element. If you choose the '''Active with Gap Source''' option of the '''Lumped Circuit Type''' section of the dialog, the right section of the dialog entitled '''Source Properties''' becomes enabled, where you can you can specify the '''Source Amplitude''' in Volts (or in Amperes in the case of PMC traces) and the '''Phase''' in degrees. Also, the box labeled '''Direction''' becomes relevant in this case which contains a gap source. Otherwise, a passive RLC circuit does not have polarity.

The calculation of the scattering (S) [[parameters ]] is usually an important objective of modeling planar structures especially for planar circuits like filters, couplers, etc. As you saw earlier, you can use lumped sources like gaps and probes and even active lumped elements to calculate the circuit characteristics of planar structures. The admittance / impedance calculations based on the gap voltages and currents are accurate at RF and lower microwave frequencies or when the port [[Transmission Lines|transmission lines ]] are narrow. In such cases, the electric or magnetic current distributions across the width of the port line are usually smooth, and quite uniform current or voltage profiles can easily be realized. At higher frequencies, however, a more robust method is needed for calculating the port [[parameters]].

One can calculate the scattering [[parameters ]] of a planar structure directly by analyzing the current distribution patterns on the port [[Transmission Lines|transmission lines]]. The discontinuity at the end of a port line typically gives rise to a standing wave pattern that can clearly be discerned in the line's current distribution. From the location of the current minima and maxima and their relative levels, one can determine the reflection coefficient at the discontinuity, i.e. the S₁₁ parameter. A more robust technique is Prony’s method, which is used for exponential approximation of functions. A complex function f(x) can be expanded as a sum of complex exponentials in the following form:

where c_i are complex coefficients and &gamma;_i are, in general, complex exponents. From the physics of [[Transmission Lines|transmission lines]], we know that lossless lines may support one or more propagating modes with pure real propagation constants (real &gamma;_i exponents). Moreover, line discontinuities generate evanescent modes with pure imaginary propagation constants (imaginary &gamma;_i exponents) that decay along the line as you move away from the location of such discontinuities.

In practical planar structures for which you want to calculate the scattering [[parameters]], each port line normally supports one, and only one, dominant propagating mode. Multi-mode [[Transmission Lines|transmission lines ]] are seldom used for practical RF and microwave applications. Nonetheless, each port line carries a superposition of incident and reflected dominant-mode propagating signals. An incident signal, by convention, is one that propagates along the line towards the discontinuity, where the phase reference plane is usually established. A reflected signal is one that propagates away from the port plane. Prony's method can be used to extract the incident and reflected propagating and evanescent exponential waves from the standing wave data. From a knowledge of the amplitudes (expansion coefficients) of the incident and reflected dominant propagating modes at all ports, the scattering matrix of the multi-port structure is then calculated. In Prony's method, the quality of the S parameter extraction results depends on the quality of the current samples and whether the port lines exhibit a dominant single-mode behavior. Clean current samples can be drawn in a region far from sources or discontinuities, typically a quarter wavelength away from the two ends of a feed line.

[[EM.Cube]]'s [[Planar Module]] provides de-embedded sources for the exclusive purpose of accurate S parameter calculation based on Prony's method. A de-embedded source is indeed a gap source that is placed close to an open end of a feed line. The other end of the line is typically connected to a planar structure of interest. Like gap sources, de-embedded sources can be placed only on rectangle strip objects. '''During mesh generation, [[EM.Cube ]] automatically extends the length of a port line that hosts a de-embedded source to about two effective wavelengths.''' This is done to provide enough length for formation of a clean standing wave current pattern. The effective wavelength of a transmission line for length extension purposes is calculated in a similar manner as for the planar mesh resolution. It is defined as <math>\lambda_{eff} = \tfrac{\lambda_0}{\sqrt{\varepsilon_{eff}}}</math>, where &epsilon;_eff is the effective permittivity. For metal and conductive sheet traces, the effective permittivity is defined as the larger of the permittivities of the two substrate layers just above and below the metallic trace. For slot traces, the effective permittivity is defined as the mean (average) of the permittivities of the two substrate layers just above and below the metallic trace. The host port line must always be open from one end to allow for its length extension. You have to make sure that there are no objects standing on the way of the extended port line to avoid any unwanted overlaps.

* In the box labeled '''Offset''', enter the distance of the phase reference plane from the end of the feed line object. The value of '''Offset''' by default is initially set to zero, meaning that the S [[parameters ]] are calculated at the plane passing through the end of the feed line. Type in a new offset value or use the spin buttons to move the source arrow along the line away from its end. As you change the offset value, you can see the source arrow move along its host object.

[[EM.Cube]]'s [[Planar Module]] provides a simple calculator for analyzing planar [[Transmission Lines|transmission lines]]. It is based on the frequency domain finite difference (FDFD) technique. You can find the characteristic impedance, effective permittivity and guide wavelength of a TEM or quasi-TEM transmission line defined based on your project's background structure. Therefore, any arbitrary stack-up configuration with any number of substrate layers can be considered.

[[EM.Cube]]'s [[Planar Module]] provides the following polarization options:

The next step is to decide on the excitation scheme. If your planar structure has one or more ports and you seek to calculate its port characteristics, then you have to choose one of the lumped source types or a de-embedded source. If you are interested in the scattering characteristics of your planar structure, then you must define a plane wave source. Before you can run a planar MoM simulation, you also need to decide on the project's observables. These are the simulation data that you expect [[EM.Cube ]] to generate as the outcome of the numerical simulation. [[EM.Cube]]'s [[Planar Module]] offers the following observables:

The simplest simulation type in [[EM.Cube ]] is an analysis. In this mode, the planar structure in your project workspace is meshed at the center frequency of the project. [[EM.Cube ]] generates an input file at this single frequency, and the Planar MoM simulation engine is run once. Upon completion of the planar MoM simulation, a number of data files are generated depending on the observables you have defined in your project. An analysis is a single-run simulation.

[[EM.Cube ]] offers a number of multi-run simulation modes. In such cases, the Planar MoM simulation engine is run multiple times. At each engine run, certain [[parameters ]] are varied and a collection of simulation data are generated. At the end of a multi-run simulation, you can graph the simulation results in EM.Grid or you can animate the 3D simulation data from the Navigation Tree. For example, in a frequency sweep, the frequency of the project is varied over its specified bandwidth. Port characteristics are usually plotted vs. frequency, representing your planar structure's frequency response. In an angular sweep, the &theta; or &phi; angle of incidence of a plane wave source is varied over their respective ranges. [[EM.Cube]]'s [[Planar Module]] currently provides the following types of multi-run simulation modes:

To run a planar MoM analysis of your project structure, open the Run Simulation Dialog by clicking the '''Run''' [[File:run_icon.png]] button on the '''Simulate Toolbar''' or select '''Menu''' '''&gt;''' '''Simulate &gt;''' '''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.

[[EM.Cube]]'s Planar MoM simulation engine uses a particular formulation of the method of moments called mixed potential integral equation (MPIE). Due to high-order singularities, the dyadic Green's functions for electric fields generated by electric currents as well as the dyadic Green's functions for magnetic fields generated by magnetic currents have very slow convergence behaviors. Instead of using these slowly converging dyadic Green's function, the MPIE formulation uses vector and scalar potentials. These include vector electric potential '''A(r)''', scalar electric potential K^&Phi;'''(r)''', vector magnetic potential '''F(r)''' and scalar magnetic potential K^&Psi;'''(r)'''. These potentials have singularities of lower orders. As a result, they coverage relatively faster. The speed of their convergence is further increased drastically using special singularity extraction techniques.

A planar MoM simulation consists of two major stages: matrix fill and linear system inversion. In the first stage, the moment matrix and excitation vector are calculated. In the second stage, the MoM system of linear equations is inverted using one of the several available matrix solvers to find the unknown coefficients of all the basis functions. The unknown electric and magnetic currents are linear superpositions of all these elementary solutions. These can be visualized in [[EM.Cube ]] using the current distribution observables. Having determined all the electric and magnetic currents in your planar structure, [[EM.Cube ]] can then calculate the near fields on prescribed planes. These are introduced as field sensor observables. The near-zone electric and magnetic fields are calculated using a spectral domain formulation of the dyadic Green's functions. Finally the far fields of the planar structure are calculated in the spherical coordinate system. These calculations are performed using the asymptotic form of the dyadic Green's functions using the &quot;stationary phase method&quot;.

A planar MoM simulation involves a number of numerical [[parameters ]] that take preset default values unless you change them. You can access these [[parameters ]] and change their values by clicking the '''Settings''' button next to the '''Select Engine''' dropdown list in the [[Planar Module]]'s Simulation Run dialog. In most cases, you do not need to open this dialog and you can leave all the default numerical parameter values intact. However, it is useful to familiarize yourself with these [[parameters]], as they may affect the accuracy of your numerical results.

The Planar MoM Engine Settings Dialog is organized in a number of sections. Here we describe some of the numerical [[parameters]]. The &quot;'''Matrix Fill'''&quot; section of the dialog deals with the operations involving the dyadic Green's functions. You can set a value for the '''Convergence Rate for Integration''', which is 1E-5 by default. This is used for the convergence test of all the infinite integrals in the calculation of the Hankel transform of spectral-domain dyadic Green's functions. When the substrate is lossy, the surface wave poles are captured in the complex integration plane using contour deformation. You can change the maximum number of iterations involved in this deformed contour integration, whose default value is 20. When the substrate is very thin with respect to the wavelength, the dyadic Green's functions exhibit numerical instability. Additional singularity extraction measures are taken to avoid numerical instability but at the expense of increased computation time. By default, a thin substrate layer is defined to a have a thickness less than 0.01&lambda;_eff, where &lambda;_eff is the effective wavelength. You can modify the definition of &quot;Thin Substrate&quot; by entering a value for '''Thin Substrate Threshold''' different than the default 0.01. The parameter '''Max Coupling Range''' determines the distance threshold in wavelength between the observation and source points after which the Green's interactions are neglected. This distance by default is set to 1,000 wavelengths. For electrically small structures, the phase variation across the structure may be negligible. In such cases, a fast quasi-static analysis can be carried out. You can set this threshold in wavelengths in the box labeled '''Max Dimensions for Quasi-Static Analysis'''.

After the MoM impedance matrix '''[Z]''' (not to be confused with the impedance [[parameters]]) and excitation vector '''[V]''' have been computed through the matrix fill process, the planar MoM simulation engine is ready to solve the system of linear equations:

where '''[I]''' is the solution vector, which contains the unknown amplitudes of all the basis functions that represent the unknown electric and magnetic currents of finite extents in your planar structure. In the above equation, N is the dimension of the linear system and equal to the total number of basis functions in the planar mesh. [[EM.Cube]]'s linear solvers compute the solution vector'''[I]''' of the above system. You can instruct [[EM.Cube ]] to write the MoM matrix and excitation and solution vectors into output data files for your examination. To do so, check the box labeled &quot;'''Output MoM Matrix and Vectors'''&quot; in the Matrix Fill section of the Planar MoM Engine Settings dialog. These are written into three files called mom.dat1, exc.dat1 and soln.dat1, respectively.

There are a large number of numerical methods for solving systems of linear equations. These methods are generally divided into two groups: direct solvers and iterative solvers. Iterative solvers are usually based on matrix-vector multiplications. Direct solvers typically work faster for matrices of smal to medium size (N&lt;3,000). [[EM.Cube]]'s [[Planar Module]] offers five linear solvers:

[[EM.Cube]]'s [[Planar Module]], by default, provides a &quot;'''Automatic'''&quot; solver option that picks the best method based on the settings and size of the numerical problem. For linear systems with a size less than N = 3,000, the LU solver is used. For larger systems, BiCG is used when dealing with symmetric matrices, and GMRES is used for asymmetric matrices. If the size of the linear system exceeds N = 15,000, the sparse version of the iterative solvers is used, utilizing a row-indexed sparse storage scheme. You can override the automatic solver option and manually set you own solver type. This is done using the '''Solver Type''' dropdown list in the &quot;'''Linear System Solver'''&quot; section of the Planar MoM Engine Settings dialog. There are also a number of other [[parameters ]] related to the solvers. The default value of '''Tolerance of Iterative Solver''' is 1E-3, which can be increased for more ill-conditioned systems. The maximum number of iterations is usually expressed as a multiple of the systems size. The default value of '''Max No. of Solver Iterations / System Size''' is 3. For extremely large systems, sparse versions of iterative solvers are used. In this case, the elements of the matrix are thresholded with respect to the larges element. The default value of '''Threshold for Sparse Solver''' is 1E-6, meaning that all the matrix elements whose magnitude is less than 1E-6 times the large matrix elements are set equal to zero. There are two more [[parameters ]] that are related to the Automatic Solver option. These are &quot;''' User Iterative Solver When System Size &gt;'''&quot; with a default value of 3,000 and &quot;''' Use SParse Storage When System Size &gt;''' &quot; with a default value of 15,000. In other words, you control the automatic solver when to switch between direct and iterative solvers and when to switch to the sparse version of iterative solvers.

If your computer has an Intel CPU, then [[EM.Cube ]] offers special versions of all the above linear solvers that have been optimized for Intel CPU platforms. These optimal solvers usually work 2-3 time faster than their generic counterparts. When you install [[EM.Cube]], the option to use Intel-optimized solvers is already enabled. However, you can disable this option (e.g. if your computer has a non-Intel CPU). To do that, open the [[EM.Cube]]'s Preferences Dialog from '''Menu &gt; Edit &gt; Preferences''' or using the keyboard shortcut '''Ctrl+H'''. Select the Advanced tab of the dialog and uncheck the box labeled &quot;''' Use Optimized Solvers for Intel CPU'''&quot;.

You can click on any current plot to visualize it in the project workspace. A legend box at the upper right corner of the screen shows the color map scale as well as the minimum, maximum, mean and standard deviation of the current data and its units. To exit the 3D plot view and return to [[EM.Cube]]'s normal view, hit the keyboard's '''Esc Key'''.

Calculation of the near-zone fields (fields at the vicinity of the unknown currents) is done at the post-processing stage and in a Cartesian coordinate systems. These calculations involve doubly infinite spectral-domain integrals, which are computed numerically. As was mentioned earlier, [[EM.Cube]]'s planar MoM engine rather uses a polar integration scheme, where the radial spectral variable k_&rho; is integrated over the interval [0, Mk₀], M being a large enough number to represent infinity, and the angular spectral variable t is integrated over the interval [0, 2&pi;]. You also saw some of the numerical [[parameters ]] related to this spectral-domain integration scheme.

Note that unlike [[EM.Cube]]'s other computational modules, near field calculations in the [[Planar Module]] usually takes substantial time. This is due to the fact that at the end of a planar MoM simulation, the fields are not available anywhere (as opposed to the [[FDTD Module]]), and their computation requires integration of complex dyadic Green's functions (as opposed to [[MoM3D Module]]'s free space Green's functions).

where &eta;₀ = 120&pi; is the characteristic impedance of the free space. As can be seen from the above equations, the far fields have the form of a TEM wave propagating in the radial direction away from the origin of coordinates. This means that the far-field magnetic field is always perpendicular to the electric field and the propagation vector, which in this case happens to be the radial unit vector in the spherical coordinate system. In other words, one only needs to know the far-zone electric field and can easily calculate the far-zone magnetic field from it. In [[EM.Cube]]'s mixed potential integral equation formulation, the far-zone electric field can be expressed in terms of the asymptotic form of the vector electric and magnetic potentials '''A''' and '''F''':

Even though the planar MoM engine does not need a radiation box, you still have to define a &quot;Far Field&quot; observable for radiation pattern calculation. This is because far field calculations take time and you have to instruct [[EM.Cube ]] to perform these calculations. To define a far field, right click the '''Far Fields''' item in the '''Observables''' section of the Navigation Tree and select '''Insert New Radiation Pattern...'''. The Radiation Pattern Dialog opens up. You may accept the default settings, or you can change the value of '''Angle Increment''', which is expressed in degrees. You can also choose to '''Normalize 2D Patterns'''. In that case, the maximum value of a 2D paten graph will have a value of 1; otherwise, the actual far field values in V/m will be used on the graph.

The 2D radiation pattern graphs can be plotted from [[EM.Cube]]'s '''Data Manager'''. A total of eight 2D radiation pattern graphs are available: 4 polar and 4 Cartesian graphs for the XY, YZ, ZX and user defined plane cuts.

When a planar structure is excited by a plane wave source, the calculated far field data indeed represent the scattered fields of that planar structure. [[EM.Cube ]] can also calculate the radar cross section (RCS) of a planar target:

'''Note that in this case the RCS is defined for a finite-sized target in the presence of an infinite background structure.''' The scattered &theta; and &phi; components of the far-zone electric field are indeed what you see in the 3D far field visualization of radiation (scattering) patterns. Instead of radiation or scattering patterns, you can instruct [[EM.Cube ]] to plot 3D visualizations of &sigma;_&theta;, &sigma;_&phi; and the total RCS. To do so, you must define an RCS observable instead of a radiation pattern. Follow these steps:

In a frequency sweep, the operating frequency of a planar structure is varied during each sweep run. [[EM.Cube]]'s [[Planar Module]] offers two types of frequency sweep: Uniform and Adaptive. In a uniform frequency sweep, the frequency range and the number of frequency samples are specified. The samples are equally spaced over the frequency range. At the end of each individual frequency run, the output data are collected and stored. At the end of the frequency sweep, the 3D data can be visualized and/or animated, and the 2D data can be graphed in EM.Grid.

To run a uniform frequency sweep, open the '''Simulation Run Dialog''', and select the '''Frequency Sweep''' option from the dropdown list labeled '''Simulation Mode'''. When you choose the frequency sweep option, the '''Settings''' button next to the simulation mode dropdown list becomes enabled. Clicking this button opens the '''Frequency Settings''' dialog. The '''Frequency Range'''is initially set equal to your project's center frequency minus and plus half bandwidth. But you can change the values of '''Start Frequency'''and '''End Frequency''' as well as the '''Number of Samples'''. The dialog offers two options for '''Frequency Sweep Type''': '''Uniform''' or '''Adaptive'''. Select the former type. It is very important to note that in a MoM simulation, changing the frequency results in a change of the mesh of the structure, too. This is because the mesh density is defined in terms of the number of cells per effective wavelength. By default, during a frequency sweep, [[EM.Cube ]] fixes the mesh density at the highest frequency, i.e., at the &quot;End Frequency&quot;. This usually results in a smoother frequency response. You have the option to fix the mesh at the center frequency of the project or let [[EM.Cube ]] &quot;remesh&quot; the planar structure at each frequency sample during a frequency sweep. You can make one of these three choices using the radio button in the '''Mesh Settings''' section of the dialog. Closing the Frequency Settings dialog returns you to the Simulation Run dialog, where you can start the planar MoM frequency sweep simulation by clicking the '''Run''' button.

Frequency sweeps are often performed to study the frequency response of a planar structure. In particular, the variation of scattering [[parameters ]] like S₁₁ (return loss) and S₂₁ (insertion loss) with frequency are of utmost interest. When analyzing resonant structures like patch antennas or planar filters over large frequency ranges, you may have to sweep a large number of frequency samples to capture their behavior with adequate details. The resonant peaks or notches are often missed due to the lack of enough resolution. [[EM.Cube]]'s [[Planar Module]] offers a powerful adaptive frequency sweep option for this purpose. It is based on the fact that the frequency response of a physical, causal, multiport network can be represented mathematically using a rational function approximation. In other words, the S [[parameters ]] of a circuit exhibit a finite number of poles and zeros over a given frequency range. [[EM.Cube ]] first starts with very few frequency samples and tries to fit rational functions of low orders to the scattering [[parameters]]. Then, it increases the number of samples gradually by inserting intermediate frequency samples in a progressive manner. At each iteration cycle, all the possible rational functions of higher orders are tried out. The process continues until adding new intermediate frequency samples does not improve the resolution of the &quot;S_ij&quot; curves over the given frequency range. In that case, the curves are considered as having converged.

You must have defined one or more ports for your planar structure run an adaptive frequency sweep. Open the Frequency Settings dialog from the Simulation Run dialog and select the '''Adaptive''' option of '''Frequency Sweep Type'''. You have to set values for '''Minimum Number of Samples''' and '''Maximum Number of Samples'''. Their default values are 3 and 9, respectively. You also set a value for the '''Convergence Criterion''', which has a default value of 0.1. At each iteration cycle, all the S [[parameters ]] are calculated at the newly inserted frequency samples, and their average deviation from the curves of the last cycle is measured as an error. When this error falls below the specified convergence criterion, the iteration is ended. If [[EM.Cube ]] reaches the specified maximum number of iterations and the convergence criterion has not yet been met, the program will ask you whether to continue the process or exit it and stop.

Figure 1: Settings adaptive frequency sweep [[parameters ]] in [[Planar Module]]'s Frequency Settings Dialog.

If your planar structure is excited by gap sources or probe sources or de-embedded sources, and one or more ports have been defined, the planar MoM engine calculates the scattering, impedance and admittance (S/Z/Y) [[parameters ]] of the designated ports. The scattering [[parameters ]] are defined based on the port impedances specified in the project's Port Definition dialog. If more than one port has been defined in the project, the S/Z/Y matrices of the multiport network are calculated. Note that the S/Z/Y matrices of an N-port structure are related to each other through the following equations:

At the end of a planar MoM simulation, the values of S/Z/Y [[parameters ]] and VSWR data are calculated and reported in the output message window. The S, Z and Y [[parameters ]] are written into output ASCII data files of complex type with a &quot;'''.CPX'''&quot; extension. Every file begins with a header consisting of a few comment lines that start with the &quot;#&quot; symbol. The complex values are arranged into two columns for the real and imaginary parts. In the case of multiport structures, every single element of the S/Z/Y matrices is written into a separate complex data file. For example, you will have data files like S11.CPX, S21.CPX, ..., Z11.CPX, Z21.CPX, etc. The VSWR data are saved to an ASCII data file of real type with a &quot;'''.DAT'''&quot; extension called, VSWR.DAT.

If you run an analysis, the port characteristics have single complex values, which you can view using [[EM.Cube]]'s data manager. However, there are no curves to graph. You can plot the S/Z/Y [[parameters ]] and VSWR data when you have data sets, which are generated at the end of any type of sweep including a frequency sweep. In that case, the &quot;.CPX&quot; files have multiple rows corresponding to each value of the sweep parameter (e.g. frequency). [[EM.Cube]]'s 2D graph data are plotted in EM.Grid, a versatile graphing utility. You can plot the port characteristics directly from the Navigation Tree. Right click on the '''Port Definition''' item in the '''Observables''' section of the Navigation Tree and select one of the items: '''Plot S [[Parameters]]''', '''Plot Y [[Parameters]]''', '''Plot Z [[Parameters]]''', or '''Plot VSWR'''. In the first three cases, another sub-menu gives a list of individual port [[parameters]].

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:data_manager_icon.png]] button of the '''Simulate Toolbar''' or select '''Simulate &gt; 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:

Figure 2: [[EM.Cube]]'s Data Manager showing a list of the port characteristics data files.

In particular, it may be useful to plot the S_ii [[parameters ]] on a Smith chart. To change the format of a data plot, select it in the Data Manager and click its '''Edit''' button. In the Edit File Dialog, choose one of the options provided in the dropdown list labeled '''Graph Type'''.

The adaptive frequency sweep described earlier is an iterative process, whereby the Planar MoM simulation engine is run at a certain number of frequency samples at each iteration cycle. The frequency samples are progressively built up, and rational fits for these data are found at each iteration cycle. A decision is then made whether to continue more iterations. At the end of the whole process, a total number of scattering parameter data samples have been generated, and new smooth data corresponding to the best rational fits are written into new data files for graphing. [[EM.Cube]]'s [[planar Module]] also allows you to generate a rational fit for all or any existing scattering parameter data as a post-processing operation without a need to run additional simulation engine runs.

You can interpolate all the scattering [[parameters ]] together or select individual [[parameters]]. You do this post-processing operation from the Navigation Tree. Right click on the '''Port Definition''' item in the '''Observables''' section of the Navigation Tree and select Smart Fit. At the top of the Smart Fit Dialog, there is a dropdown list labeled '''Interpolate''', which gives a list of all the available S parameter data for rational interpolation. The default option is &quot;All Available [[Parameters]]&quot;. Then you see a box labeled '''Number of Available Samples''', whose value is read from the data content of the selected complex .CPX data file. Based on the number of available data samples, the dialog reports the '''Maximum Interpolant Order'''. You can choose any integer number for '''Interpolant Order''', from 1 to the maximum allowed.

Figure 3: The smoothed version of the S₁₁ parameter plot of the two-port structure using [[EM.Cube]]'s Smart Fit.

Depending on the source type and the types of observables defined in a project, a number of output data are generated at the end of a planar MoM simulation. Some of these data are 2D by nature and some are 3D. The output simulation data generated by [[EM.Cube]]'s [[Planar Module]] can be categorized into the following groups:

* '''Port Characteristics''': S, Z and Y [[Parameters ]] and Voltage Standing Wave Ratio (VSWR)

At the end of an analysis, the 2D quantities usually have a single value that is written into an ASCII data file. Complex-valued quantities are written into complex data files with a &quot;'''.CPX'''&quot; extension. Real-valued quantities are written into real data files with a &quot;'''.DAT'''&quot; extension. Polar 2D radiation pattern data and some other radiation characteristics are written into angular data files with a &quot;'''.ANG'''&quot; extension. In this latter file type, polar data are stored as functions of an angle expressed in degrees. At the end of a sweep simulation of one of the many types available (frequency, angular, parametric, etc.), the ASCII output data files are populated with rows that correspond to the samples of the sweep variable(s). If a sweep simulation involves N sweep [[variables]], then the first N columns of the output data files show the samples of those sweep [[variables]]. All the 2D data files are listed in the '''2D Data Files''' tab of [[EM.Cube]]'s '''Data Manager'''. You can view the contents of these data files by selecting their row in the data manager and clicking the '''View''' button of the dialog.

3D output data, on the other hand, are defined as functions of the space coordinates and are usually of vectorial nature. Cartesian-type and mesh-type data such as current distributions and near-field field distributions are expressed as functions of the Cartesian (X, Y, Z) coordinates. Spherical-type data like far-field radiation patterns and RCS are expressed as functions of the spherical angles (&theta;, &phi;). The 3D radiation patterns are written into a file with a &quot;'''.RAD'''&quot; extension. This file contains the complex values of the &theta;- and &phi;-components of the far-zone electric field (E_&theta; and E_&phi;) as well as the total far field magnitude as functions of the spherical observation angles &theta; and &phi;. The 3D RCS patterns are written into a file with a &quot;'''.RCS'''&quot; extension. This file contains the real values of the &theta;- and &phi;-polarized RCS values as well as the total RCS as functions of the spherical observation angles &theta; and &phi;. The current distributions are written into data files with a &quot;'''.CUR'''&quot; extension. They contain the real and imaginary parts of the X, Y and Z components of electric ('''J''') and magnetic ('''M''') current on each cells together with the definition of all the node coordinates and node indices of the cells. The near-field distributions are written into data files with a &quot;'''.SEN'''&quot; extension. They contain the amplitude and phase of the X, Y and Z components of electric ('''E''') and magnetic ('''H''') fields as functions of the coordinates of sampling points. All the 3D data files are listed in the '''3D Data Files''' tab of [[EM.Cube]]'s '''Data Manager'''. You can view the contents of these data files by selecting their row in the data manager and clicking the '''View''' button of the dialog.

Figure 1: The 3D Data Files tab of [[EM.Cube]]'s Data Manager.

At the end of a planar MoM simulation, a number of computed quantities are designated as &quot;Standard Output&quot; [[parameters ]] and can be used for various post-processing data operations. For example, you can define design objectives based on them, which you need for [[optimization]]. The table below gives a list of all the currently available standard output [[parameters ]] in [[EM.Cube]]'s [[Planar Module]]:

In the table above, SijM, etc. means the scattering parameter observed at port i due to a source excited at port j. Similar definitions apply to all the S, Z and Y [[parameters]]. If your planar structure has N ports, there will be a total of N² scattering [[parameters]], a total of N² impedance [[parameters]], and a total of N² admittance [[parameters]]. Additionally, there are four standard output [[parameters ]] associated with each of the individual S/Z/Y [[parameters]]: magnitude, phase (in radians), real part and imaginary part. The same is true for the reflection and transmission coefficients of a periodic planar structure excited by a plane wave source. Each coefficient has four associated standard output [[parameters]]. These [[parameters]], of course, are available only if your planar structure has a periodic domain and is also excited by a plane wave source incident at the specified &theta; and &phi; angles.

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 (&theta;, &phi;). These angles are specified in degrees as '''User Defined Azimuth &amp; Elevation''' in the &quot;Output Settings&quot; 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 &quot;Output Settings&quot; 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 &theta;_s and &phi;_s source angles. FRCS, on the other hand, is the total forward-scatter RCS measured at the predetermined &theta;_o and &phi;_o observation angles. These angles are specified in degrees as '''User Defined Azimuth &amp; Elevation''' in the &quot;Output Settings&quot; 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:functions_icon.png]]'''button of the '''Simulate Toolbar''' or select '''Simulate &gt; 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:custom_icon.png]]'''button of the '''Simulate Toolbar''' or select '''Simulate &gt; 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.

Figure 1: [[EM.Cube]]'s Custom Output dialog.

Figure 2: Defining a new custom output using the available standard output [[parameters]].

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 &quot;'''.DAT'''&quot; file extension. To open data manager, click the '''Data Manager''' [[File:data_manager_icon.png]] button of the '''Simulate Toolbar''' or select '''Simulate &gt; 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 &quot;Compute&quot; feature of the spreadsheet. You can make multiple file selection using the keyboard's '''Ctrl''' and '''Shift''' keys.

Figure 1: [[EM.Cube]]'s Data Manager dialog showing an angular file selected and highlighted for further action.

[[EM.Cube]]'s 3D output simulation data usually have a vectorial nature and are defined as functions of the Cartesian or spherical space coordinates. At the end of a planar MoM simulation, you can view 3D visualizations of the vectorial output data such as current distributions, near-field field distributions, far-field radiation patterns and RCS in [[EM.Cube]]'s project workspace by clicking on the corresponding observable entries in the Navigation Tree. When you run a sweep simulation of some sort, multiple 3D plots appear on the Navigation Tree representing all the sweep variable samples. You can animate these 3D visualization plots very conveniently from the Navigation Tree. To do so, right click on an observable's name in the Navigation Tree and select the '''[[Animation]]''' item from the contextual menu. Make sure that you right click on the observable's parent node, not on one of its child components corresponding to the sweep variable samples. The 3D plot in the project workspace starts to animate and continues forever until to stop it. A new window called &quot;''' [[Animation ]] Controls Dialog'''&quot; opens up at the lower right corner of the [[EM.Cube ]] desktop. This dialog allows you to control the [[animation ]] speed using a box labeled '''Rate''', whose value multiplied by 100 milliseconds indeed gives the frame duration. You can speed up the [[animation ]] or slow it down from the default rate of one frame per 300ms. The box labeled '''Sample''' show the current frame's plot label at any time. You can pause the [[animation]], rewind it to the first frame, fast-forward it to the last frame or manually step it through back and forth using the movement buttons marked with the symbols |&lt;, &lt;&lt;, ||, &gt;&gt;,&gt;|.

Figure: [[EM.Cube]]'s [[Animation ]] Controls dialog.

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 &quot;expression&quot; 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 &quot;formal&quot; option, first you open the [[Variables]] Dialog by clicking the '''[[Variables]]''' [[File:variable_icon_tn.png]] button of the '''Simulate Toolbar''' or selecting '''Menu &gt; Simulate &gt; [[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 &quot;Add Variable Dialog&quot;. 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]].

Figure 1: [[EM.Cube]]'s [[Variables]] dialog.

Once you finish the definition of a new variable, its name and syntax (Definition) are added to the &quot;Variable List&quot;. You can also see the '''Current Value''' of every variable at any time in the Variable as dialog. Note that at this stage, you have simply defined one or more [[variables]], but you have not yet associated them with actual objects or project properties. Wherever you see a numeric value for a parameter in a dialog, e.g. the length of a rect strip object in its property dialog, you can replace the numeric value with a variable name or a mathematical expression using the names of the currently available [[variables]]. From this moment on, that parameter or quantity becomes tied up with the associated variable. This means that every time you change the value of that variable, the value of the associated object parameter or project property will change accordingly. You can change the value of a variable directly from the [[Variables]] Dialog using the '''Edit''' button or indirectly during a parametric sweep. In the former case, you have to click the '''Update''' button of the [[Variables]] dialog to make the changes effective. Alternatively, you can define new [[variables]] directly from the property dialogs of CAD objects, trace and object set dialogs, stack-up dialog or many other [[EM.Cube ]] dialogs. If you replace the numeric value of a parameter with a text-string name that has not already been defined as a variable, then a new variable by that name is created and added to the &quot;Variable List&quot;. The numeric value of the associated parameter at the time of replacement is taken as the &quot;current Value&quot; of the newly created variable. In this way, you can easily and quickly define [[variables]] associated with the design [[parameters ]] that you intend to sweep in your project.

In a parametric sweep, you can vary the values of one or more &quot;Independent&quot; project [[variables]]. In other words, you designate one or more independent [[variables]] as sweep [[variables]] and specify how they should vary (be sampled) during the sweep simulation process. In [[EM.Cube ]] you can define three types of sweep [[variables]]: '''Uniform''', '''Discrete''' and '''Random'''. Each sweep [[variables]] can be defined as one of these three types, and you can mix sweep [[variables]] of different types in a multivariable parametric sweep. A uniform sweep variable is defined by a &quot;Start&quot; and &quot;End&quot; value and is incremented by a predetermined &quot;Step&quot; value during a sweep. A discrete sweep variable is defined by a discrete set of values and takes on these values by the order of their list during a sweep. A random sweep variable, on the other hand, takes on random values during a sweep according to a specified probability distribution. [[EM.Cube ]] currently offers two random distribution types. '''Uniform Distribution''' is defined by &quot;Minimum&quot; and &quot;Maximum&quot; values, while '''Normal (Gaussian) Distribution''' is defined by a &quot;Mean&quot; and &quot;Standard Deviation&quot;. You need to specify the &quot;Number of Samples&quot; for both random variable types.

To define sweep [[variables]] and run a parametric sweep, open the '''Simulation Run Dialog''', and select the '''Parametric Sweep''' option from the dropdown list labeled '''Simulation Mode'''. When you choose the parametric sweep option, the '''Settings''' button next to the simulation mode dropdown list becomes enabled. Clicking this button opens the '''Parametric Sweep Settings''' dialog. The '''Sweep [[Variables]] List''' is initially empty. On the left side of the dialog you see the &quot;'''Independent [[Variables]] Table'''&quot;, which lists all the available independent [[variables]] of your project. Select an independent variable from the table and use the right arrow ('''--&gt;''') button of the dialog to move it to the Sweep [[Variables]] List. Before moving the variable to the new location, the &quot;Define Sweep Variable Dialog&quot; opens up, where you have to define the attributes of your new sweep variable. In this dialog, you need to choose the type of the sweep variable using the three radio buttons labeled '''Uniform''', '''Discrete''' and '''Random'''. Depending on your choice, the proper section of the dialog becomes enabled, where you can define the range of your sweep variable and other relevant [[parameters]].

The Parametric Sweep Settings dialog also features another useful button labeled '''Dry Run''', which runs a &quot;fake&quot; sweep [[animation]]. During a dry run, the sweep [[variables]] are varied sample by sample and all of their associated [[parameters ]] in the project workspace are updated at each run. However, the simulation engine is not called during a dry run, and no numerical computations take place at all. Also, the Variable Dry Run dialog appears on the screen which shows the changing values of all the [[variables]] at all times. This dialog works in a similar way as the [[Animation ]] Controls Dialog described earlier. You can change the speed of the updates or control them manually using the motion buttons. The dry run process continues forever until to stop it by clicking the close (X) button of the Dry Run dialog of simply hitting the keyboard's '''Esc Key'''. At ach update of a dry run, you can see how the CAD objects in your planar structure change. This is very useful to inspect the integrity of your structure and your defined [[variables]] before an actual simulation run. If you run a dry run while [[EM.Cube ]] is in the mesh view mode, then the planar mesh of your structure is updated for each combination of the sweep variable samples during the dry run process. Once you are satisfied with the choice and definition of your sweep [[variables]], close the Parametric Sweep Settings dialog to return to the Simulation Run dialog, where you can start the planar MoM parametric sweep simulation by clicking the '''Run''' button.

Figure 3: [[EM.Cube]]'s Parametric Sweep Settings dialog.

Figure 5: [[EM.Cube]]'s Variable Dry Run dialog.

An [[optimization]] process in [[EM.Cube ]] involves several steps as follows:

# Define project (design) objectives using a combination of standard and custom output [[parameters]].

# Choose the [[optimization]] algorithm type and set its relevant [[parameters]].

# Examine the optimal values of the participating [[variables]] and the updated values of their associated [[parameters]].

To define an objective, open the '''Objectives Dialog''' either by clicking the '''Objectives''' [[File:objective_icon.png]] button of the '''Simulate Toolbar''', or by selecting '''Menu &gt; Simulate &gt; 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 '''&quot;=== (Equal To)&quot;'''. 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.

Figure 1: [[EM.Cube]]'s Objectives dialog.

Figure 2: Defining a new objective using a list of available output [[parameters]].

To define [[optimization]] [[variables]] and perform an [[optimization]], open the '''Simulation Run Dialog''', and select the '''[[Optimization]]''' option from the dropdown list labeled '''Simulation Mode'''. When you choose the [[optimization]] option, the '''Settings''' button next to the simulation mode dropdown list becomes enabled. Clicking this button opens the '''[[Optimization]] Settings''' dialog. This is a large dialog with several distinct sections. In the section titled &quot;'''[[Optimization]] Algorithm'''&quot;, you can choose one of [[EM.Cube]]'s three currently available optimizers: '''Powell's Method''', '''Basic Genetic Algorithm''' and '''Fast Pareto Genetic Algorithm'''. For all three optimizers you have to set the '''Maximum Number of Iterations''', which has a default value of 5. For the two genetic algorithms, basic GA and Pareto, you also need to set the '''Population Size''' as a multiple of 5. The default population size is 50.

In the [[Variables]] section of the dialog, you designate the [[optimization]] [[variables]]. This is very similar to how you assign sweep [[variables]] in a parametric sweep as discussed earlier. The '''[[Optimization]] [[Variables]] List''' is initially empty. On the left side of the dialog you see the &quot;'''Independent [[Variables]] Table'''&quot;, which lists all the available independent [[variables]] of your project. Select an independent variable from the table and use the right arrow ('''--&gt;''') button of the dialog to move it to the [[Optimization]] Variable List. Before moving the variable to the new location, the &quot;Define [[Optimization]] Variable Dialog&quot; opens up, where you have to set the '''Minimum'''and '''Maximum'''values of your new [[optimization]] variable. Once you finish the definition of an [[optimization]] variable, its name and Min/Max values are added to the [[Optimization]] [[Variables]] List. Note that you can change your mind and remove an [[optimization]] variable from the list. To do so, select its name or row from the list and use the left arrow (&lt;--) button to move it back to the Independent [[Variables]] Table. You can also change the Min/Max values of an [[optimization]] variable after it has been defined. Select the variable and click the '''Edit''' button of the dialog to change those value. If you check the box labeled &quot;''' Update [[Variables]] with Optimal Values'''&quot; (as it is always checked by default), [[EM.Cube ]] will automatically replace the definitions (and current values) of all the participating [[optimization]] [[variables]] in the '''[[Variables]] Dialog''' with their computed optimal values and will update all the associated [[parameters ]] in the project workspace. This, of course, will happen only if the [[optimization]] process successfully converges. Sometimes, the [[optimization]] process may get trapped in a local minimum. You may encounter this problem primarily when using the Powell method. You will notice that the values of the [[optimization]] [[variables]] soon get &quot;saturated&quot; and remain constant afterwards. [[EM.Cube ]] lets you exit such local traps if you check the box labeled &quot;'''Exit Local Min Trap'''&quot;. There is a box underneath, labeled '''Tolerance''', that becomes enabled and has a default value of 0.01. This means that exit a local minimum trap when the value of the [[optimization]] variable stays within 1% error after successive [[optimization]] runs. In that case, the [[optimization]] process ends forcibly before having achieved convergence.

Finally, in the section titled &quot;'''Error Function'''&quot; of the [[Optimization]] Dialog you build the mathematical form of your objective function. Keep in mind that [[EM.Cube]]'s [[optimization]] is a numerical process. Therefore, you can rarely minimize your error function to zero literally. You need to set a &quot;'''Maximum Error'''&quot; value for the objective function, which terminates the process as &quot;converged&quot; when it is reached. The default value of Maximum Error is 0.01. In two tables labeled '''Goals''' and '''Constraints''', you see a list of all the project objectives that have been split between the two tables according to their types. Many [[optimization]] problems involve a single, straightforward goal. Others may involve multiple goals subject to multiple constraints. In those cases, you have assign weights to your goals and constraints. If all of your goals and constraints have the same level of importance, then you assign equal weights to them. You can do this easily by clicking the '''Distribute''' button of the dialog. Or you may assign individual weights manually. However, you have to make sure that all the weight adds up to unity. The objective function of the [[optimization]] problem is constructed from the goals and constraints using the specified weights. This can be done in two different ways: as '''Linearly Weighted Goals''' or as '''Mean Square Weighted Goals.''' You set these options from the dropdown list labeled '''Weight Type'''. Once you are satisfied with the choice and definition of your [[optimization]] [[variables]], [[optimization]] algorithms and goals weights, close the [[Optimization]] dialog to return to the Simulation Run dialog, where you can start the planar MoM [[optimization]] process by clicking the '''Run''' button.

Figure 3: [[EM.Cube]]'s [[Optimization]] dialog.

A periodic structure is one that exhibits a repeated geometric pattern. It is made up of identical elements that 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.

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.

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 &quot;Additional Radiation Characteristics&quot; 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.

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.

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:

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>.

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. In [[EM.Cube]]'s [[Planar Module]], you can define periodic structures whose unit cells are interconnected. Interconnectivity applies only to PEC, PMC and conductive sheet traces, and embedded object sets are excluded. Note that in a periodic planar structure, your objects cannot cross the periodic domain. However, you can arrange objects with linear edges such as one or more flat edges line up with the domain's bounding box. In such cases, [[EM.Cube]]'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.

The above doubly infinite periodic Green's functions are said to be expressed in terms of &quot;Floquet Modes&quot;. The exact formulation involves an infinite set of these periodic Floquet modes. During the MoM matrix fill process for a periodic structure, a finite number of Floquet modes are calculated. By default, [[EM.Cube]]'s planar MoM engine considers M_x = M_y = 25. This implies a total of 51 modes along the X direction and a total of 51 modes along the Y direction, or a grand total of 51² = 2,601 Floquet modes. You can increase the number of Floquet modes for your project from the Planar MoM Engine Settings Dialog. In the section titled &quot;Periodic Simulation&quot;, you can change the values of '''Number of Floquet Modes''' in the two boxes designated X and Y.

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.

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 &quot;reflection.CPX&quot; and &quot;transmission.CPX&quot;. These coefficients behave like the S₁₁ and S₂₁ [[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.

{{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.}}

The reflection and transmission characteristics of a period surface as functions of the incidence angle are often of great interest. For that purpose, you can run an angular sweep of your periodic structure, where you normally fix the &phi; angle and sweep the &theta; angle from 180 to 90 degrees for one-sided surfaces and from 180 to 0 degrees for two-sided surface. To run an angular sweep, open the [[Planar Module]]'s '''Simulation Run Dialog''' and select the '''Angular Sweep''' option from its '''Simulation Mode''' dropdown list. This enables the '''Settings''' button, which opens up the '''Angle Settings Dialog'''. First, you must choose either Theta or Phi as the '''Sweep Angle'''. Then you can set the '''Start''' and '''End''' values of the selected incidence angle as well as the '''Number of Samples'''. At the end of an angular sweep simulation, you can plot the reflection and transmission coefficients 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 reflection and transmission coefficients of the structure are saved into two complex data files called &quot;reflection.CPX&quot; and &quot;transmission.CPX&quot;. These data files are also listed in [[EM.Cube]]'s '''Data Manager''', where you can view or plot them.

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.

[[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 S₁₁ parameter, while the transmission coefficient T is equivalent to S₂₁ 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°&lt; &theta; = 180°. You can also illuminate the periodic surface by the plane wave source from the bottom half-space, corresponding to 0° = &theta; &lt; 90°. In this case, the reflection coefficient R and transmission coefficient T are equivalent to S₂₂ and S₁₂ [[parameters]], respectively. Having these interpretations in mind, [[EM.Cube ]] enables the &quot;'''Adaptive Frequency Sweep'''&quot; option of the '''Frequency Settings Dialog''' when your planar structure has a periodic domain together with a plane wave source.

The array factor approach works well when the inter-element coupling is negligible or when the array contains a very large number of elements. However, finite edge and corner effects cannot be modeled accurately using a simple array factor. A full-wave approach is needed where all the elements are discretized properly, and their interactions are incorporated into the final solution. [[EM.Cube]]'s Planar MoM simulation engine lets you analyze finite-sized antenna arrays in a rigorous, full-wave manner. As you saw earlier, you can even introduce a source array with arbitrary (amplitude and phase) weights and realize a complex, non-uniformly excited, finite-sized antenna array. Note that if the array contains a total of N_F = N_x . N_y elements, and each radiating element involves a total of N_B vectorial basis functions, the numerical solution of the problem will produce a system of N = N_B. N_Flinear equations. As an example, consider a rectangular patch antenna element that involves 240 X-directed and 240 Y-directed rooftop basis functions, i.e. N_B= 480. Now consider a not-so-large, 8 × 8 array of these patch radiators, i.e. N_F= 64. The resulting linear system will have an enormous size of N = 30,720. Keep in mind that, unlike the sparse matrices of the Finite Element Method (FEM), MoM linear systems are dense by nature and typically ill-conditioned. The inversion of dense matrices of such sizes or larger takes a significant amount of computation time even if you use fast iterative solvers.

In the current release of [[EM.Cube]]'s [[Planar Module]], the NCCBF MoM solver works with any number of distinct, finite-sized arrays if they are excited with one of the following three source types:

There are a few rules that must be followed and observed when planning a NCCBF simulation. '''Each finite-sized array must be constructed using an [[EM.Cube ]] &quot;Array Object&quot;. Additionally, each array object must stand alone in a dedicated trace or embedded object set of its own.''' In other words, if an array object belongs to a trace or embedded object set that contains other objects, it will be excluded from the NCCBF process and will get a regular MoM treatment. Keep in mind that [[Planar Module]] allows you to define different traces located at the same Z-plane, although the objects belonging to these separate traces cannot be connected to one another according to the planar meshing rules. Similarly, you can define two or more PEC via sets hosted by the same substrate layer. Therefore, if your planar structure contains finite arrays and aperiodic objects, you have to group them into separate traces or embedded object sets.

To run an NCCBF simulation, open the '''Simulation Run Dialog''', and then open the'''Planar MoM Engine Settings Dialog'''. In the &quot;'''Finite Array Simulation'''&quot; section of the latter dialog, check the box labeled &quot;'''NCCBF Matrix Compression'''&quot;. This box is unchecked by default. Checking it enables the NCCBF Settings button. Click this button to open the NCCBF Settings Dialog. The dialog features a &quot;List of Unit Cells Used for NCCBF Matrix Compression&quot;. This list initially empty. To add unit cells to it, click the '''Add''' button of the dialog to open the &quot;'''Add Unit Cell Dialog'''&quot;. This dialog has two tables: Available Unit Cells on the left side and Associated Unit Cells on the right side. The left table shows a list of all the available, legitimate array objects in your project workspace. Remember that for an array object to be eligible for NCCBF compression, it has to stand alone on a dedicated trace or embedded object set, whichever applies. Select an array object from the left table and use the right arrow button (--&gt;) to move it to the right table to associate it with the new NCCBF unit cell. You can associate more than one array object with the same NCCBF unit cell. In this case, the parent elements of all the associated array objects collectively constitute the NCCBF unit cell. The NCCBF unit cell is the planar structure that is analyzed separately, first, as a stand-alone isolated element, and next, as a periodic unit cell, to generate the NCCBF entire-domain basis function solutions. It is therefore very important that the array objects be positioned carefully with respect to the origin of coordinated and relative to one another to form the correct NCCBF unit cell. Once you move one or more array object names to the &quot;Associated&quot; table on the right, you can move them back to the &quot;Available&quot; table on the left using the left arrow (&lt;--) button. You can also instruct [[EM.Cube ]] to use only the isolated element solution by unchecking the box labeled &quot;'''Include Periodic Solution of Unit Cell'''&quot;. Once you are satisfied with the definition of your NCCBF unit cell, close the dialog to return to the NCCBF Settings dialog. Here you see the name of the newly added NCCBF unit cell in the list along with the Number of Solutions and the names of all the associated array objects for each NCCBF unit cell. You can modify each row using the '''Edit''' button or remove it from the list using the '''Delete''' button. Close the NCCBF Settings dialog to return to the Planar MoM Engine Settings dialog, and close the latter to return to the Simulation Run dialog, where you can now start the NCCBF simulation by clicking the '''Run''' button.

[[EM.Cube]]'s [[Planar Module]] treats array objects in a special way. That is why you need to use array objects with certain rules for NCCBF simulations. In general, if the mesh of your planar structure involves a total of N vectorial basis functions, the MoM matrix will contain a total of N² elements. Instead of computing the entire N² basis interactions, the Planar MoM simulation engine takes advantage of the inherent symmetry properties of the dyadic Green's functions and camputes the diagonal elements of the matrix and all the elements below the diagonal. This amounts to N.(N+1)/2 basis interactions. In many cases, the MoM matrix is symmetric, and the elements above the diagonal are simply mirror-image of the below-diagonal elements. In planar structures that involve both metal and slot traces, there will be sign reversals for some interactions.

In many cases, especially in the areas that contain sizable numbers of rectangular mesh cells, the basis functions are naturally grouped into distinct sets that are called domains. As you saw earlier in the discussion of planar mesh generation, uniform domains with identical rectangular cells bring significant savings during the matrix fill process. Using the concept of domains renders the MoM matrix as a block matrix, whose blocks represent the interactions among the domains. The diagonal blocks therefore correspond to self-domain interactions. By a similar argument, if your planar structure is made up of N_D domains, then a total of N_D . (N_D +1)/2 domain interactions (or matrix blocks) are computed. An [[EM.Cube ]] array object consists of N_F identical geometrical elements. If the array object belongs to a trace that has other objects in it, then by the planar mesh generator's rules, the elements of the array object are merged with the other objects on the same trace using the &quot;Union&quot; Boolean operation. If some array elements possibly have connections with other objects, such connections are taken care of in the meshing process. '''However, if an array object stands alone in a dedicated trace, then only the parent (first) element is meshed, and it mesh is copied and cloned for all the other elements of the array.''' This produces a total of N_F identical domains of vectorial basis functions. A direct consequence of this is identification of only N_F unique domain-pair interactions or matrix blocks. In the absence of these symmetries, a total of at least N_F . (N_F +1)/2 domain interactions (or matrix blocks) must be computed. To better illustrate such matrix fill savings, let us consider the previous, not-so-large, 8 × 8 array of patch radiators, i.e. N_F= 64. It was previously assumed that each rectangular patch antenna element involves 240 X-directed and 240 Y-directed rooftop basis functions, i.e. N_B= 480. The numerical solution of this structure produces a linear system of total size N = N_B. N_F= 30,720. The total number of complex-valued elements of this matrix is 9.44E+08. This is the total number of highly sophisticated multi-dimensional integrals that you need to compute during a brute-force matrix fill process. For the sake of generality of the argument, here we ignore the huge additional savings that rectangular cells offer, and we assume that each unique domain-pair interaction involves N_B² = 230,400 elements, except for the self-domain interaction which requires N_B . (N_B +1)/2 = 115,440 integral computations. This amounts to a total of (N_F -1) . N_B² + N_B . (N_B +1)/2 = 1.46E+07 integral computations, which is roughly N_F(64) times fewer and faster than a brute-force matrix fill process.

To remedy the limitation that stand-alone array objects on dedicated traces cannot be connected to anything and therefore would severely limit the geometrical complexity of individual elements, [[EM.Cube ]] allows you to build arrays of composite objects. A &quot;'''Composite Object'''&quot; in [[EM.Cube ]] is a group of objects that are tied together only for the purpose of organization. As a result, the mesh of a composite object is the same as that of its constituent member objects (including any possible connections), had they not been grouped together. In a similar manner to simpler array objects, if a composite array stands alone in a dedicated trace or embedded object set, then only the parent composite object is meshed, and its mesh is copied and cloned for all the other composite elements of the array object. Furthermore, all the unique domain-pair interactions are identified during the matrix fill process, and lead to a major saving in computation time. '''Note that you can assign gap or probe sources to composite arrays in a similar way as you excite simpler array objects.''' If your planar structure involves a composite array, whose composite parent element has constituent members: Object1, Object2, ..., then the Gap Source dialog or Probe Source dialog will include the names of all the eligible constituent members (rectangle strips or PEC via objects) that can host the respective source types. '''Complex composite arrays hosting gap or probe source arrays or illuminated by plane wave sources are great candidates for NCCBF simulation.'''