Difference between revisions of "EM.Picasso"

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	Figure 2: Edit Port dialog.		Figure 2: Edit Port dialog.
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		+	Click here to learn more about the theory of [[Computing Port Characteristics in Planar MoM]].
	=== Modeling Coupled Ports ===		=== Modeling Coupled Ports ===

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Revision as of 21:17, 3 June 2015

An EM.Picasso Primer

EM.Picasso in a Nutshell

EM.Picasso® is a versatile planar structure simulator for modeling and design of printed antennas, planar microwave circuits, and layered periodic structures. EM.Picasso's simulation engine is based on a 2.5-D full-wave Method of Moments (MoM) formulation that provides the ultimate modeling accuracy and computational speed for open-boundary multilayer structures. It can handle planar structures with arbitrary numbers of metal layouts, slot traces, vertical interconnects and lumped elements interspersed among different substrate layers.

EM.Picasso assumes that your planar structure has a substrate (background structure) of infinite lateral extents. Your substrate can be a dielectric half-space, or a single conductor-backed dielectric layer (as in microstrip components or patch antennas), or simply the unbounded free space, or any arbitrary multilayer stack-up configuration. In the special case of a free space substrate, EM.Picasso behaves similar to EM.Libera's Surface MoM simulator. In all the other cases, it is important to keep in mind the infinite extents of the background substrate structure. For example, you cannot use EM.Picasso to analyze a patch antenna with a finite-sized dielectric substrate, if the substrate edge effects are of concern in your modeling problem. EM.Tempo is recommended for the modeling of finite-sized substrates. Since EM.Picasso's Planar MoM simulation engine incorporates the Green's functions of the background structure into the analysis, only the finite-sized traces like microstrips and slots are discretized by the mesh generator. As a result, the size of EM.Picasso's computational problem is normally much smaller compared to the other techniques and solver. In addition, EM.Picasso generates a hybrid rectangular-triangular mesh of your planar structure with a large number of rectangular cells. This results in very fast computation times that oftentimes make up for the limited applications of EM.Picasso.

An Overview of Planar Method of Moments

The Method of Moments (MoM) is a rigorous, full-wave numerical technique for solving open boundary electromagnetic problems. Using this technique, you can analyze electromagnetic radiation, scattering and wave propagation problems with relatively short computation times and modest computing resources. The method of moments is an integral equation technique; it solves the integral form of Maxwell’s equations as opposed to their differential forms that are used in the finite element or finite difference time domain methods.

In EM.Picasso, the background structure is usually a layered planar structure that consists of one or more laterally infinite material layers always 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 more slots (apertures). Such slots are modeled by magnetic surface currents. 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 a planar MoM simulation, the unknown electric and magnetic currents are discretized as a collection of elementary currents with small finite spatial extents. As a result, the governing integral equations reduce to a system of linear algebraic equations, whose solution determines the amplitudes of all the elementary currents defined over the planar structure's mesh. Once the total currents are known, you can calculate the fields everywhere in the structure.

Click here to learn more about the theory of Planar Method of Moments.

Building a Planar Structure

Understanding the Background Structure

EM.Picasso 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. The background structure in EM.Picasso is called the "Layer Stack-up". 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. Some structures like stripline components require two bounding grounds (PEC half-spaces) both at their top and bottom.

Planar Object Types

EM.Picasso groups objects by their trace type and their hierarchical location in the substrate layer stack-up. All the planar objects belonging to the same trace group are located on the same substrate layer boundary and have the same color. All the embedded objects belonging to the same embedded set lie inside the same substrate layer and have the same color and same material composition.

EM.Picasso provides the following types of objects for building a planar layered structure:

1. PEC Traces: These represent infinitesimally thin metallic planar objects that are deposited or metallized on or between substrate layers. PEC objects are modeled by surface electric currents.
2. Slot Traces: These are used to model slots and apertures in PEC ground planes. Slot objects are always assumed to lie on an infinite horizontal PEC ground plane with zero thickness (which is not explicitly displayed in the project workspace). They are modeled by surface magnetic currents.
3. Conductive Sheet Traces: These represent imperfect metals. They have a finite conductivity and a very small thickness. A surface impedance boundary condition is enforced on the surface of such traces.
4. PEC Via Sets: These are metallic objects such as shorting pins, interconnect vias, plated-through holes, etc. that are grouped together as prismatic object sets. The embedded objects are modeled as vertical volume conduction currents.
5. Embedded Dielectric Sets: These are prismatic dielectric objects inserted inside a substrate layer. You can define a finite permittivity and conductivity for such objects, but their height is always the same as the height of their host layer. The embedded dielectric objects are modeled as vertical volume polarization currents.

Click here to learn more about Planar Traces & Object Types.

Defining the Layer Stack-Up

When you start a new project in EM.Picasso, 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 open EM.Picasso or switched to it from EM.Cube's other modules, 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...

The Stack-up Settings dialog has two tabs: Layer Hierarchy and Embedded Sets. The Layer Hierarchy tab has a table that shows all the background layers in hierarchical order from the top half-space to the bottom half-space. It also lists the material label of each layer, Z-coordinate of the bottom of each layer, its thickness (in project units) and material properties: permittivity (e_r), permeability (µ_r), electric conductivity (s) and magnetic conductivity (s_m). There is also a column that lists the names of embedded object sets inside each substrate layer, if any.

You can add new layers to your project's stack-up or delete its layers, or move layers up or down and thus change the layer hierarchy. To add a new background layer, click the arrow symbol on the Insert...button at the bottom of the dialog and select Substrate Layer from the button's dropdown list. A new dialog opens up where you can enter a label for the new layer and values for its material properties and thickness in project units.

You can delete a layer by selecting its row in the table and clicking the Delete button. To move a layer up and down, click on its row to select and highlight it. Then click either the Move Up or Move Down buttons consecutively to move the selected layer to the desired location in the stack-up. Note that you cannot delete or move the top or bottom half-spaces.

After creating a substrate layer, you can always edit its properties in the Layer Stack-up Settings dialog. Click on any layer's row in the table to select and highlight it and then click the Edit button. The substrate layer dialog opens up, where you can change the layer's label and assigned color. In the material properties section of the dialog, you can change the name of the material and its properties: permittivity (e_r), permeability (µ_r), electric conductivity (s) and magnetic conductivity (s_m). To define electrical losses, you can either assign a value for electric conductivity (s), or alternatively, define a loss tangent for the material. In the latter case, check the box labeled "Specify Loss Tangent" and enter a value for it. In this case, the electric conductivity field becomes greyed out and reflects the corresponding s value at the center frequency of the project. You can also set the thickness of any substrate layer in the project units except for the top and bottom half-spaces.

For better visualization of your planar structure, EM.Picasso 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 button of the Simulate Toolbar, or by selecting Simulate > Computational Domain > Domain Settings... from the Simulate Menu or by right clicking the Virtual Domain item of the Navigation Tree and selecting Domain Settings... from the contextual menu, or using the keyboard shortcut Ctrl+A. 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. If you assign different colors to your substrate layers, you have get a better visualization of multilayer virtual domain box surrounding your project structure.

Defining Traces & Object Sets

When you start a new project in Planar Module, the project workspace looks empty, and there are no finite objects in it. However, a default background structure is always present by default. Objects are defined as part of traces or embedded sets. Once defined, you can see a list of project objects in the Physical Structure section of the navigation tree.

Traces and object sets can be defined either from Layer Stack-up Settings dialog or from the navigation tree.

In the Layer Stack-up Settings dialog, you can add a new trace to the stack-up by clicking the arrow symbol on the Insert button of the dialog. You have to choose from Metal (PEC), Slot (PMC) or Conductive Sheet options. A respective dialog opens up, where you can enter a label and assign a color other than default ones. Once a new trace is defined, it is added, by default, to the top of the stack-up table underneath the top half-space. From here, you can move the trace down to the desired location on the layer hierarchy.

Every time you define a new trace, it is also added under the respective category in the Navigation Tree. Alternatively, you can define a new trace from the Navigation Tree by right clicking on one of the trace type names and selecting Insert New PEC Trace...or Insert New PMC Trace...or Insert New Conductive Sheet Trace...A respective dialog opens up for setting the trace properties. Once you close this dialog, it takes you directly to the Layer Stack-up Settings dialog so that you can set the right position of the trace on the stack-up.

Drawing & Managing Planar Objects

As soon as you start drawing geometrical objects in the project workspace, the Physical Structure section of the Navigation Tree gets populated. The names of traces are added under their respective trace type category, and the names of objects appear under their respective trace group. At any time, one and only one trace is active in the project workspace. An active trace is where all the new objects you draw belong to. When you define a new trace, it is set as active and you can immediately start drawing new objects on that trace. You can also set any trace active at any time by right clicking its name on the Navigation Tree and selecting Activate from the contextual menu. The name of the active trace is always displayed in bold letter in the Navigation Tree.

EM.Picasso 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.

You can manage your project's layer hierarchy from the Layer Stack-up Settings dialog. You can add, delete and move around substrate layers, metallic and slot traces and embedded object sets. Metallic and slot traces can move among the interface planes between neighboring substrate layers. Embedded object sets including PEC vias and finite dielectric objects can move from substrate layer into another. When you delete a trace from the Layer Stack-up Settings dialog, all of its objects are deleted from the project workspace, too. You can also delete metallic and slot traces or embedded object sets from the Navigation Tree. To do so, right click on the name of the trace or object set in the Navigation Tree and select Delete from the contextual menu. You can also delete all the traces or object sets of the same type from the contextual menu of the respective type category in the Navigation Tree.

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 > 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.

Planar Module's Rules & Limitations

1. Terminating PEC ground planes at the top or bottom of a planar structure are defined as PEC top or bottom half-spaces, respectively.
2. A PEC ground plane placed in the middle of a substrate stack-up requires at least one slot object to provide electromagnetic coupling between its top and bottom sides. In this case, a PMC trace is rather introduced at the given Z-plane, which implies the presence of an infinite PEC ground although it is not explicitly indicated in the Navigation Tree.
3. Metallic and slot traces cannot coexist on the same Z-plane. However, you can stack up multiple PEC and conductive sheet traces at the same Z-coordinate. Similarly, multiple PMC traces can be placed at the same Z-coordinate.
4. Metallic and slot traces are strictly defined at the interface planes between substrate layers. To define a suspended metallic trace in a substrate layer (as in the case of the center conductor of a stripline), you must split the dielectric layer into two thinner layers and place your PEC trace at the interface between them.
5. The current version of the Planar MoM simulation engine is based on a 2.5-D MoM formulation. Only vertical volume currents and no circumferential components are allowed on embedded objects. The 2.5-D assumption holds very well in two cases: (a) when embedded objects are very thin with a very small cross section (with lateral dimensions less than 2-5% of the material wavelength) or (b) when embedded objects are very short and sandwiched between two closely spaced PEC traces or grounds from the top and bottom.
6. 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.

Planar module does not allow construction of 3D CAD objects. Instead, you draw the cross section of prismatic objects as planar 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.

Discretizing Planar Structures

Understanding the Planar MoM Mesh

The method of moments (MoM) discretizes all the finite-sized objects of a planar structure (excluding the background structure) into a set of elementary cells. The accuracy of the MoM numerical solution depends greatly on the quality and resolution of the generated mesh. As a rule of thumb, a mesh density of about 20-30 cells per effective wavelength usually yields satisfactory results. Yet, for structures with lots of fine geometrical details or for highly resonant structures, higher mesh densities may be required. Also, the particular simulation data that you seek in a project also influence your choice of mesh resolution. For example, far field characteristics like radiation patterns are less sensitive to the mesh density than field distributions on a structure with a highly irregular shape and a rugged boundary.

EM.Picasso generates two types of mesh for a planar structure: a pure triangular and a hybrid triangular-rectangular. In both case, EM.Picasso attempts to create a highly regular mesh, in which most of the cells have almost equal areas. The hybrid mesh type tries to produce as many rectangular cells as possible especially in the case of objects with rectangular or linear boundaries. In connection or junction areas between adjacent objects or close to highly curved boundaries, the use of triangular cells is clearly inevitable. EM.Picasso's default mesh type is hybrid. The uniformity or regularity of mesh is an important factor in warranting a stable MoM numerical solution.

The mesh density 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. 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. By default, EM.Picasso generates a hybrid mesh with a mesh density of 20 cells per effective wavelength. The effective permittivity is defined differently for different types of traces and embedded object sets. This is to make sure that enough cells are placed in areas that might feature higher field concentration. For PEC 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. For embedded object sets, the effective permittivity is defined as the largest of the permittivities of all the substrate layers and embedded dielectric sets.

A Note on the Junction Mesh

The integrity of the planar mesh and its continuity in the junction areas where adjacent objects are connected directly affects the simulation results. The most important rule of object connections in EM.Picasso is that only objects belonging to the same trace can be connected to one another. If two objects belong to the same trace (residing on the same Z-plane) and have a common overlap area, EM.Picasso first merges the two objects using the "Boolean Union" operation and converts them into a single object for the purpose of meshing. EM.Picasso's hybrid planar mesh generator has some additional rules:

• If two connected rectangular objects have the same side dimensions along the common linear edge with perfect alignment, a rectangular junction mesh is produced.
• If two connected rectangular objects have different side dimensions along the common linear edge or have edge offset, a set of triangular cells is generated along the edge of the object with the large side.
• Rectangular objects that contain gap source or lumped elements, always have a rectangular mesh around the gap area.

If an embedded object like an interconnect via 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.Picasso’s planar mesh generator automatically handles situations of this kind and generates all the required connection meshes.

Two overlapping planar objects and their triangular and hybrid planar meshes.

Edge-connected rectangular planar objects and their triangular and hybrid planar meshes.

Meshes of short and long vertical PEC vias connecting two horizontal metallic strips.

Generating, Viewing & Customizing a Planar Mesh

You can generate and view a planar mesh by clicking the Show Mesh button of the Simulate Toolbar or by selecting Menu > Simulate > Discretization > Show Mesh or using the keyboard shortcut Ctrl+M. When the mesh of the planar structure is displayed in EM.Cube’s project workspace, its "Mesh View" mode is enabled. In this mode you can perform view operations like rotate view, pan or zoom, but you cannot create new objects or edit existing ones. To exit the mesh view mode, press the keyboard's Esc Key or click the Show Mesh 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.Picasso to create a new mesh from the ground up by selecting Menu > Simulate > Discretization > Regenerate Mesh or by right clicking on the Planar Mesh item in the Discretization section of the navigation tree and selecting Regenerate from the contextual menu.

You can change the settings of the planar mesh including the mesh type and density from the Planar Mesh Settings Dialog. You can also change these settings while in the mesh view mode, and you can update the changes to view the new mesh. To open the mesh settings dialog, either click the Mesh Settings button of the Simulate Toolbar or select Menu > Simulate > Discretization > Mesh Settings..., or by right click on the Planar Mesh item in the Discretization section of the Navigation Tree and select Mesh Settings... from the contextual menu, or use the keyboard shortcut Ctrl+G. You can change the mesh algorithm from the dropdown list labeled Mesh Type, which offers two options: Hybrid and Triangular. You can also enter a different value for Mesh Density in cells per effective wavelength (λ_eff). For each value of mesh density, the dialog also shows the average "Cell Edge Length" in the free space. To get an idea of the size of mesh cells on the traces and embedded object sets, divide this edge length by the square root of the effective permittivity a particular trace or set. Click the Apply button to make the changes effective.

Refining the Planar Mesh Locally

It is very important to apply the right mesh density to capture all the geometrical details of your planar structure. This is especially true for "field discontinuity" regions such as junction areas between objects of different side dimensions, where larger current concentrations are usually observed at sharp corners, or at the connection areas between metallic traces and PEC vias, as well as the areas around gap sources and lumped elements, as these create voltage or current discontinuities. For large planar structures, using a higher mesh density may not always be a practical option since it will quickly lead to a very large MoM matrix and thus growing the size of the numerical problem. EM.Picasso provides several ways of controlling the mesh of a planar structure locally.

The Planar Mesh Settings dialog gives a few options for customizing your planar mesh around geometrical and field discontinuities. You can check the check box labeled "Refine Mesh at Junctions", which increases the mesh resolution at the connection area between rectangular objects. Or you can check the check box labeled "Refine Mesh at Gap Locations", 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 "Refine Mesh at Vias", which increases the mesh resolution on the cross section of embedded object sets and at the connection regions of the metallic objects connected to them. EM.Picasso typically doubles the mesh resolution locally at the discontinuity areas when the respective boxes are checked.

You should always visually inspect EM.Picasso'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. Sometimes EM.Picasso'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.Picasso'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.Picasso 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 "Remove Defective Triangular Cells" in the Planar Mesh Settings dialog. You can also change the value of the minimum allowable cell angle.

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. You can lock the local mesh density by accessing the property dialog of a specific trace or embedded object set and checking the box labeled Lock Mesh. This will enable the Mesh Density box, where you can accept the default global value or set any desired new value.

Excitation Sources

Your planar structure must be excited by some sort of a signal source that induces electric currents on metal parts and magnetic currents on slot traces. The excitation source you choose depends on the observables you seek in your project. EM.Picasso provides the following source types for exciting planar structures:

• Gap Sources
• Probe Sources
• De-embedded Sources
• Short Dipole Sources
• Plane Wave Sources
• Huygens Sources

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.

Click here to learn more about Planar MoM Source Types.

Defining Source Arrays

If the project workspace contains an array of rectangle strip objects, the array object will also be listed as an eligible object for gap source placement. A gap source will then be placed on each element of the array. All the gap sources will have identical direction and offset. Similarly, if the project workspace contains an array of PEC via objects, the embedded array object will also be listed as an eligible object for probe source placement. A probe source will then be placed on each via object of the array. All the probe sources will have identical direction and offset.

However, you can prescribe certain amplitude and/or phase distribution over the array of gap or probe sources. By default, all the gap or probe sources have identical amplitudes of 1V (or 1A for the slot case) and zero phase. The available amplitude distributions to choose from include Uniform, Binomial and Chebyshev and Date File. In the Chebyshev case, you need to set a value for minimum side lobe level (SLL) in dB. You can also define Phase Progression in degrees along all three principal axes. You can view the amplitude and phase of individual sources by right clicking on the top Sources item in the Navigation Tree and selecting Show Source Label from the contextual menu.

Figure 1: Defining gap sources on an array of rectangle strip objects with a Chebyshev amplitude distribution.

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.

Figure 2: Defining gap source array weights using a data file.

Defining Ports

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:

• Gap Sources
• Probe Sources
• Active Lumped Elements
• De-Embedded Sources

Ports are defined in the Observables section of the Navigation Tree. Right click on the Port Definition item of the Navigation Tree and select Insert New Port Definition... from the contextual menu. The Port Definition Dialog opens up, showing the default port assignments. If you have N sources in your planar structure, then N default ports are defined, with one port assigned to each source according to their order on the Navigation Tree. Note that your project can have mixed gap and probes sources as well as active lumped element sources.

Figure 1: The Port Definition dialog.

You can define any number of ports equal to or less than the total number of sources in your project. The Port List of the dialog shows a list of all the ports in ascending order, with their associated sources and the port's characteristic impedance, which is 50S by default. You can delete any port by selecting it from the Port List and clicking the Delete button of the dialog. Keep in mind that after deleting a port, you will have a source in your project without any port assignment and make sure that is what you intend. You can change the characteristic impedance of a port by selecting it from the Port List and clicking the Edit button of the dialog. This opens up the Edit Port dialog, where you can enter a new value in the box labeled Impedance.

Figure 2: Edit Port dialog.

Click here to learn more about the theory of Computing Port Characteristics in Planar MoM.

Modeling Coupled Ports

Sources can be coupled to each other to model coupled strip lines (CPS) on metal traces or coplanar waveguides (CPW) on slot traces. Similarly, probe sources may be coupled to each other. Coupling two or more sources does not change the way they excite a planar structure. It is intended only for the purpose of S parameter calculation. The feed lines or vias which host the coupled sources are usually parallel and aligned with one another and they are all grouped together as a single transmission line represented by a single port. This single "coupled" port then interacts with other coupled or uncoupled ports.

You couple two or more sources using the Port Definition Dialog. To do so, you need to change the default port assignments. First, delete all the ports that are to be coupled from the Port List of the dialog. Then, define a new port by clicking the Add button of the dialog. This opens up the Add Port dialog, which consists of two tables: Available sources on the left and Associated sources on the right. A right arrow (-->) button and a left arrow (<--) button let you move the sources freely between these two tables. You will see in the "Available" table a list of all the sources that you deleted earlier. You may even see more available sources. Select all the sources that you want to couple and move them to the "Associated" table on the right. You can make multiple selections using the keyboard's Shift and Ctrl keys. Closing the Add Port dialog returns you to the Port Definition dialog, where you will now see the names of all the coupled sources next to the name of the newly added port.

It is your responsibility to set up coupled ports and coupled 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.

Figure 1: Coupling gap sources in the Port Definition dialog by associating more than one source with a single port.

Calculating Port Characteristics At Gap Discontinuities

A gap source on a metal trace and a probe source on a PEC via behave like a series voltage source with a prescribed strength (of 1V and zero phase by default) that creates a localized discontinuity on the path of electric current flow. At the end of a planar MoM simulation, the electric current passing through the voltage source is computed and integrated to find the total input current. From this one can calculate the input admittance as

[math] Y_{in} = \frac{I_{in}}{V_s} = \frac {\int_W \hat{y} \cdot \mathbf{J_s} \, dy} {V_s} [/math]

for gap sources on metal traces, where the line integration is performed across the width of the metal strip, and

[math] Y_{in} = \frac{I_{in}}{V_s} = \frac {\int_S \hat{z} \cdot \mathbf{J_p} \, ds} {V_s} [/math]

for probe sources on PEC vias, where the surface integration is performed over the cross section of the via. On the other hand, a gap source on a slot trace behaves like a shunt current source with a prescribed strength (of 1A and zero phase by default) that creates a localized discontinuity on the path of magnetic current flow. At the end of a planar MoM simulation, the magnetic current passing through the current source is computed and integrated to find the total input voltage across the current filament. From this one can calculate the input impedance as

[math] Z_{in} = \frac{V_{in}}{I_s} = \frac{\int_W \hat{y} \cdot \mathbf{M_s} \,dy} {V_s} = \frac{\int_W E_y \, dy}{V_s} [/math]

Note that the input admittance or impedance defined at a gap source port is referenced to the two terminals of the voltage source connected across the gap as shown in the figure below. This is different than the input admittance or impedance that one may normally define for a microstrip port, which is referenced to the substrate's ground.

Definition of different input impedances at the gap location.

To resolve this problem, you can place a gap source on a metal strip line by a distance of a quarter guide wavelength (λ_g/4) away from its open end. Note that (λ_g = 2p/ß), where ß is the propagation constant of the metallic transmission line. As show in the figure below, the impedance looking into an open quarter-wave line segment is zero, which effectively shorts the gap source to the planar structure's ground. The gap admittance or impedance in this case is identical to the input admittance or impedance of the planar structure.

Placing a gap source a quarter guide wavelength away from the open end of a feed line to effectively short it to the ground at the gap location.

The same principle applies to the gap sources on slot traces. The figure below shows how to place two gap sources with opposite polarities a quarter guide wavelength away from their shorted ends to calculate the correct input impedance of the CPW line looking to the left of the gap sources. Note that in this case, you deal with shunt filament current sources across the two slot lines and that the slot line carry magnetic currents. The end of the slot lines look open to the magnetic currents, but in reality they short the electric field. The quarter-wave CPW line acts as an open circuit to the current sources.

Placing two oppositely polarized gap sources a quarter guide wavelength away from the short end of a CPW line to effectively create an open circuit beyond the gap location.

The case of a probe source placed on a PEC via that is connected to a ground plane is more straightforward. In this case, the probe source's gap discontinuity is placed at the middle plane of the PEC via. If the via is short, it is meshed using a single prismatic element, which is connected to the ground from one side and to the metal strip line from the other. Therefore, the probe admittance or impedance is equal to that of the structure at a reference plane that passed through the host via.

Input impedance of a probe source on a PEC via connected to a ground plane.

Exciting Multiport Structures Using Linear Superposition

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:

[math] I_m = \sum_{n=1}^N Y_{mn} V_n, \quad \quad Y_{mn} = \frac{I_m}{V_n} \bigg|_{V_k=0, k \ne n}[/math]

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:

[math] V_m = \sum_{n=1}^N Z_{mn} I_n, \quad \quad Z_{mn} = \frac{V_m}{I_n} \bigg|_{I_k=0, k \ne n}[/math]

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:

[math] \mathbf{[S] = [Y_0] \cdot ([Z]-[Z_0]) \cdot ([Z]+[Z_0])^{-1} \cdot [Z_0]} [/math]

where [math]\mathbf{[Z_0]}[/math] and [math]\mathbf{[Y_0]}[/math] are diagonal matrices whose diagonal elements are the port characteristic impedances and admittances, respectively.

Modeling Lumped Elements In Planar MoM

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), Capacitors(C), Inductors(L), and series and parallel combinations of them as shown in the figure below:

Figure 1: A series-parallel RLC combination that can be modeled as a lumped circuit in Planar Module.

Lumped elements are conceptualized in a similar way as gap or probe sources. They are indeed considered as infinitesimally narrow gaps placed in the path of current flow, across which Ohm's law is enforced. If a lumped element is placed on a PEC or conductive sheet trace, it is treated as a series connection. The boundary condition at the location of the lumped element is:

[math] V_{gap} = Z_L I_{in} \quad\quad \int_{\delta} \hat{x}\cdot \mathbf{E_{gap}} \, dx = Z_L \int_W \hat{y} \cdot \mathbf{J_s} \, dy [/math]

where Z_L is the total impedance across the two terminals of the series element. If the lumped element is placed on a slot trace, it is treated as a shunt connection that creates a current discontinuity. In this case, the magnetic current across the gap is continuous, and the boundary condition at the location of the lumped element is:

[math] I_{gap} = Y_L V_{in} \quad\quad \int_{\delta} J_Y^{fila} \, dx = Y_L \int_W E_y \, dy [/math]

[math] \int_{\delta} \hat{x}\cdot\hat{n} \times (\mathbf{H_{gap}^+ - H_{gap}^-}) \, dx = Y_L \int_W \hat{y}\cdot\mathbf{M_s} \, dy [/math]

where Y_L is the total admittance across the two terminals of the shunt element. If a lumped element is placed on a PEC via that is connected to a metal strip from one side and to a PEC ground plane from the other end, it is indeed as a series connection across a gap discontinuity at the middle plane of the via. If the via is short, it is meshed using a single prismatic element. In that case, the lumped element in effect shunts the metal strip to the ground. The boundary condition at the location of the lumped element across the PEC via is:

[math] V_{gap} = Z_L I_{in} \quad\quad \int_{\delta} \hat{z}\cdot \mathbf{E_{gap}} \, dz = Z_L \int_S \hat{z} \cdot \mathbf{J_p} \, ds [/math]

Figure 1: Using a shunt lumped element on a PEC via to terminate a metallic strip line.

Defining Lumped Circuits

To define a lumped RLC circuit in your planar structure, follow these steps:

• Open the Lumped Element Dialog by right clicking on the Lumped Elements item in the Sources section of the Navigation Tree and selecting Insert New Source...
• In the Gap Topology section of the dialog, select one of the two options: Gap on Line and Gap on Via.
• In the Lumped Circuit Type section of the dialog, select one of the two options: Passive RLC and Active with Gap Source.
• Depending on your choice of gap topology, in the Lumped Circuit Location section of the dialog, you will find either a list of all the Rectangle Strip Objects or a list of all the PEC Via Objects available in the project workspace. Select the desired rectangle strip or embedded PEC via object.
• In the box labeled Offset, enter the distance of the lumped element from the start point of the rectangle strip line or from the bottom of the via object, whichever the case. The value of Offset by default is initially set to the center of the line or via.
• In the Load Properties section, the series and shunt resistance values Rs and Rp are specified in Ohms, the series and shunt inductance values Ls and Lp are specified in nH (nanohenry), and the series and shunt capacitance values Cs and Cp are specified in pF (picofarad). Only the checked elements are taken into account in the total impedance calculation. By default, only the series resistor is checked with a value of 50S, and all other circuit elements are initially greyed out.
  

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.

If the project workspace contains an array of rectangle strip objects or PEC via objects, the array object will also be listed as an eligible object for lumped element placement. A lumped element will then be placed on each element of the array. All the lumped elements will have identical direction, offset, resistance, inductance and capacitance values. If you define an active lumped element, you can prescribe certain amplitude and/or phase distribution to the gap sources just like in the case of gap and probe sources. The available amplitude distributions include Uniform', Binomial', Chebyshev and Data File.

The impedance of the lumped circuit is calculated at the operating frequency of the project using the specified R, L and C values. As you change the frequency, the value of the impedance that is passed to the Planar MoM engine will change.

Calculating Scattering Parameters Using Prony's Method

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 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. 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:

[math] f(x) \approx \sum_{n=1}^N c_i e^{-j\gamma_i x} [/math]

where c_i are complex coefficients and γ_i are, in general, complex exponents. From the physics of transmission lines, we know that lossless lines may support one or more propagating modes with pure real propagation constants (real γ_i exponents). Moreover, line discontinuities generate evanescent modes with pure imaginary propagation constants (imaginary γ_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 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.

Figure 1: Minimum and maximum current locations of the standing wave pattern on a microstrip line feeding a patch antenna.

Running Planar MoM Simulations

The first step of planning a planar MoM simulation is defining your planar structure. This consists of the background structure plus all the finite-sized metal and slot trace objects and possibly embedded metal or dielectric objects that are interspersed among the substrate layers. The background stack-up is defined in the Layer Stack-up dialog, which automatically opens up as soon as you enter the Planar Module. The metal and slot traces and embedded object sets are listed in the Navigation Tree, which also shows all the geometrical (CAD) objects you draw in the project workspace under each object group at different Z-planes.

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:

• Current Distribution
• Field Sensors
• Far Fields (Radiation Patterns or Radar Cross Section)
• Huygens Surfaces
• Port Characteristics
• Periodic Characteristics

If you run a simulation without having defined any observables, no data will be generated at the end of the simulation. Some observables require a certain type of excitation source. For example, port characteristics will be calculated only if the project contains a port definition, which in turn requires the existence of at least one gap or probe or de-embedded source. The periodic characteristics (reflection and transmission coefficients) are calculated only if the structure has a periodic domain and excited by a plane wave source.

Planar Module's Simulation Modes

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 θ or φ 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:

• Frequency Sweep
• Parametric Sweep
• Angular Sweep
• R/T Macromodel
• Huygens Sweep
• Optimization
• HDMR

Figure 1: Selecting a simulation mode in Planar Module's Simulation Run dialog.

Running A Planar MoM Analysis

To run a planar MoM analysis of your project structure, open the Run Simulation Dialog by clicking the Run button on the Simulate Toolbar or select Menu > Simulate > Run or use the keyboard shortcut Ctrl+R. The Analysis option of the Simulation Mode dropdown list is selected by default. Once you click the Run button, the simulation starts. A new window, called the Output Window, opens up that reports the different stages of simulation and the percentage of the tasks completed at any time. After the simulation is successfully completed, a message pops up and reports the end of simulation. In certain cases like calculating scattering parameters of a circuit or reflection / transmission characteristics of a periodic surface, some results are also reported in the Output Window. At the end of a simulation, you need to click the Close button of the Output Window to return to the project workspace.

Figure 1: Planar Module's Simulation Run dialog.

Stages Of A Planar MoM Analysis

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^Φ(r), vector magnetic potential F(r) and scalar magnetic potential K^Ψ(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 "stationary phase method".

Setting Numerical Parameters

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 "Matrix Fill" 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 0.01λ_eff, where λ_eff is the effective wavelength. You can modify the definition of "Thin Substrate" 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.

In the "Spectral Domain Integration" section of the dialog, you can set a value to Max Spectral Radius in k0, which has a default value of 30. This means that the infinite spectral-domain integrals in the spectral variable k_ρ are pre-calculated and tabulated up to a limit of 30k₀, where k₀ is the free space propagation constant. These integrals may converge much faster based on the specified Convergence Rate for Integration described earlier. However, in certain cases involving highly oscillatory integrands, much larger integration limits like 100k₀ might be needed to warrant adequate convergence. For spectral-domain integration along the real k_ρ axis, the interval [0, Nk₀] is subdivided into a large number of sub-intervals, within each an 8-point Gauss-Legendre quadrature is applied. The next parameter, No. Radial Integration Divisions per k₀, determines how small these intervals should be. By default, 2 divisions are used for the interval [0, k₀]. In other words, the length of each integration sub-interval is k₀/2. You can increase the resolution of integration by increasing this value above 2. Finally, instead of 2D Cartesian integration in the spectral domain, a polar integration is performed. You can set the No. of Angular Integration Points, which has a default value of 100.

Figure 1: The Planar MoM Engine Settings dialog.

Planar Module's Linear System Solvers

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:

[math] \mathbf{[Z]}_{N\times N} \cdot \mathbf{[I]}_{N\times 1} = \mathbf{[V]}_{N\times 1} [/math]

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 "Output MoM Matrix and Vectors" 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<3,000). EM.Cube's Planar Module offers five linear solvers:

1. LU Decomposition Method
2. Biconjugate Gradient Method (BiCG)
3. Preconditioned Stabilized Biconjugate Gradient Method (BCG-STAB)
4. Generalized Minimal Residual Method (GMRES)
5. Transpose-Free Quasi-Minimum Residual Method (TFQMR)

Of the above list, LU is a direct solver, while the rest are iterative solvers. BiCG is a relatively fast iterative solver, but it works only for symmetric matrices. You cannot use BiCG for periodic structures or planar structures that contain both metal and slot traces at different planes, as their MoM matrices are not symmetric. The three solvers BCG-STAB, GMRES and TtFQMR work well for both symmetric and asymmetric matrices and they also belong to a class of solvers called Krylov Sub-space Methods. In particular, the GMRES method always provides guaranteed unconditional convergence.

EM.Cube's Planar Module, by default, provides a "Automatic" 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 "Linear System Solver" 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 " User Iterative Solver When System Size >" with a default value of 3,000 and " Use SParse Storage When System Size > " 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 > Edit > Preferences or using the keyboard shortcut Ctrl+H. Select the Advanced tab of the dialog and uncheck the box labeled " Use Optimized Solvers for Intel CPU".

Running a Frequency Sweep

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 Rangeis initially set equal to your project's center frequency minus and plus half bandwidth. But you can change the values of Start Frequencyand 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 "End Frequency". 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 "remesh" 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.

Figure 1: Planar Module's Frequency Settings dialog.

Adaptive Frequency Sweep

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 "S_ij" 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.

For large frequency ranges, you may have to increase both the minimum and maximum number of samples. Moreover, remeshing the planar structure at each frequency may prove more practical than fixing the mesh at the highest frequency.

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

Examining Port Characteristics

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:

[math]\mathbf{ [S] = [Y_0] \cdot ([Z]-[Z_0]) \cdot ([Z]+[Z_0])^{-1} \cdot [Z_0] }[/math]

[math]\mathbf{ [Y] = [Z]^{-1} } [/math]

[math]\mathbf{ [Z] = [\sqrt{Z_0}] \cdot ([U]+[S]) \cdot ([U]-[S])^{-1} \cdot [\sqrt{Z_0}] }[/math]

where [math]\mathbf{[U]}[/math] is the identity matrix of order N, [math]\mathbf{[Z_0]}[/math] and [math]\mathbf{[Y_0]}[/math] are diagonal matrices whose diagonal elements are the port characteristic impedances and admittances, respectively, and [math]\mathbf{[\sqrt{Z_0}]}[/math] is a diagonal matrix whose diagonal elements are the square roots of port characteristic impedances. The voltage standing wave ratio (VSWR) of the structure at the first port is also computed:

[math]\text{VSWR} = \frac{|V_{max}|}{|V_{min}|} = \frac{1+|S_{11}|}{1-|S_{11}|}[/math]

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 ".CPX" extension. Every file begins with a header consisting of a few comment lines that start with the "#" 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 ".DAT" 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 ".CPX" 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.

Figure 1: Selecting port characteristics data to plot from the Navigation Tree.

You can also see a list of all the port characteristics data files in EM.Cube's Data Manager. To open data manager, click the Data Manager button of the Simulate Toolbar or select Simulate > Data Manager from the menu bar, or right click on the Data Manager item of the Navigation Tree and select Open Data Manager... from the contextual menu. You can also use the keyboard shortcut Ctrl+D at any time. Select any data file by clicking and highlighting its row in the table and then click the Plot button to plot the graph. By default, the S parameters are plotted as double magnitude-phase graphs, while the Y and Z parameters are plotted as double real-imaginary part graphs. The VSWR data are plotted on a Cartesian graph. You can change the format of complex data plots. In general complex data can be plotted in three forms:

1. Magnitude and Phase
2. Real and Imaginary Parts
3. Smith Chart

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.

Figure 3: Changing the graph type by editing a data file's properties.

Figure 4: The S₁₁ parameter plotted on a Smith Chart graph in EM.Grid.

Rational Interpolation Of Scattering Parameters

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 "All Available Parameters". 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.

Interpolant order more than 15 will suffer from numerical instabilities even if you have a very large number of data samples.

You can use the Update button of the dialog to generate the interpolated data for a given order. The new data are written to a complex data file with the same name as the selected S parameter and a "_RationalFit" suffix. While this dialog is still open, you can plot the new data either directly from the Navigation Tree or from the Data Manager. If you are not satisfied with the results, you can return to the Smart Fit dialog and try a higher or lower interpolant order and compare the new data.

Figure 1: Planar Module's Smart Fit dialog.

Figure 2: The S₁₁ parameter plot of a two-port structure in magnitude-phase format.

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

Working with Planar MoM Simulation Data

Planar Module's Output Simulation Data

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)
• Radiation Characteristics: Radiation Patterns, Directivity, Total Radiated Power, Axial Ratio, Main Beam Theta and Phi, Radiation Efficiency, Half Power Beam Width (HPBW), Maximum Side Lobe Level (SLL), First Null Level (FNL), Front-to-Back Ratio (FBR), etc.
• Scattering Characteristics: Bi-static and Mono-static Radar Cross Section (RCS)
• Periodic Characteristics: Reflection and Transmission Coefficients
• Current Distributions: Electric and magnetic current amplitude and phase on all metal and slot traces and embedded objects
• Near-Field Distributions: Electric and magnetic field amplitude and phase on specified planes and their central axes

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 ".CPX" extension. Real-valued quantities are written into real data files with a ".DAT" extension. Polar 2D radiation pattern data and some other radiation characteristics are written into angular data files with a ".ANG" 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 (θ, φ). The 3D radiation patterns are written into a file with a ".RAD" extension. This file contains the complex values of the θ- and φ-components of the far-zone electric field (E_θ and E_φ) as well as the total far field magnitude as functions of the spherical observation angles θ and φ. The 3D RCS patterns are written into a file with a ".RCS" extension. This file contains the real values of the θ- and φ-polarized RCS values as well as the total RCS as functions of the spherical observation angles θ and φ. The current distributions are written into data files with a ".CUR" 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 ".SEN" 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.

Figure 2: Viewing the contents of a mesh-type 3D data file in Data Manager.

Visualizing Current Distributions

Electric and magnetic currents are the fundamental output data of a planar MoM simulation. After the numerical solution of the MoM linear system, they are found using the solution vector [I] and the definitions of the electric and magnetic vectorial basis functions:

[math] \mathbf{[I]}_{N\times 1} = \begin{bmatrix} I^{(J)} \\ \\ V^{(M)} \end{bmatrix} \quad \Rightarrow \quad \begin{cases} \mathbf{J(r)} = \sum_{n=1}^N I_n^{(J)} \mathbf{f_n^{(J)} (r)} \\ \\ \mathbf{M(r)} = \sum_{k=1}^K V_k^{(M)} \mathbf{f_k^{(M)} (r)} \end{cases} [/math]

Note that currents are complex vector quantities. Each electric or magnetic current has three X, Y and Z components, and each complex component has a magnitude and phase. You can visualize the surface electric currents on metal (PEC) and conductive sheet traces, surface magnetic currents on slot (PMC) traces and vertical volume currents on the PEV vias and embedded dielectric objects. 3D color-coded intensity plots of electric and magnetic current distributions are visualized in the project workspace, superimposed on the surface of physical objects.

In order to view the current distributions, you must first define them as observables before running the planar MoM simulation. To do that, right click on the Current Distributions item in the Observables section of the Navigation Tree and select Insert New Observable.... The Current Distribution Dialog opens up. At the top of the dialog and in the section titled Active Trace / Set, you can select a trace or embedded object set where you want to observe the current distribution. You can also select the current map type from two options: Confetti and Cone. The former produces an intensity plot for current amplitude and phase, while the latter generates a 3D vector plot.

Figure 1: The Planar Module's Current Distribution dialog.

Once you close the current distribution dialog, the label of the selected trace or object set is added under the Current Distributions node of the Navigation Tree.

You have to define a separate current distribution observable for each individual trace or embedded object set.

At the end of a planar MoM simulation, the current distribution nodes in the Navigation Tree become populated by the magnitude and phase plots of the three vectorial components of the electric (J) and magnetic (M) currents as well as the total electric and magnetic currents defined in the following manner:

[math] | \mathbf{J_{tot}} | = \sqrt{|J_x|^2 + |J_y|^2 + |J_z|^2}[/math]

[math] | \mathbf{M_{tot}} | = \sqrt{|M_x|^2 + |M_y|^2 + |M_z|^2}[/math]

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.

Figure 2: The current distribution map of a patch antenna.

Figure 3: Vectorial (cone) visualization of the current distribution on a patch antenna.

Computing The Near Fields

Once all the current distributions are known in a planar structure, the electric and magnetic fields can be calculated everywhere in that structure using the dyadic Greens's functions of the background structure:

[math] \begin{align} \mathbf{E(r) = E_{inc}(r)} + & \sum_{n=1}^N I_n^{(J)} \iiint_V \overline{\overline{G}}_{EJ}(r|r') \cdot f_n^{(J)}(r') \, d\nu' + \\ & \sum_{k=1}^K V_n^{(M)} \iiint_V \overline{\overline{G}}_{EM}(r|r') \cdot f_k^{(M)}(r') \, d\nu' \end{align} [/math]

[math] \begin{align} \mathbf{H(r) = H_{inc}(r)} + & \sum_{n=1}^N I_n^{(J)} \iiint_V \overline{\overline{G}}_{HJ}(r|r') \cdot f_n^{(J)}(r') \, d\nu' + \\ & \sum_{k=1}^K V_n^{(M)} \iiint_V \overline{\overline{G}} {HM}(r|r') \cdot f_k^{(M)}(r') \, d\nu' \end{align} [/math]

The above equations can be cast into the spectral domain as follows:

[math] \begin{align} \mathbf{E(r) = E_{inc}(r)} + \frac{1}{(2\pi)^2} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} \bigg[ & \sum_{n=1}^N I_n^{(J)} \tilde{\overline{\overline{G}}}_{EJ}(k_{\rho}, z|z') \cdot \tilde{f}_n^{(J)}(k_x, k_y) + \\ & \sum_{k=1}^K V_n^{(M)} \tilde{\overline{\overline{G}}}_{EM}(k_{\rho}, z|z') \cdot \tilde{f}_k^{(M)}(k_x, k_y) \bigg] \, dk_x \, dk_y \end{align} [/math]

[math] \begin{align} \mathbf{H(r) = H_{inc}(r)} + \frac{1}{(2\pi)^2} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} \bigg[ & \sum_{n=1}^N I_n^{(J)} \tilde{\overline{\overline{G}}}_{HJ}(k_{\rho}, z|z') \cdot \tilde{f}_n^{(J)}(k_x, k_y) + \\ & \sum_{k=1}^K V_n^{(M)} \tilde{\overline{\overline{G}}}_{HM}(k_{\rho}, z|z') \cdot \tilde{f}_k^{(M)}(k_x, k_y) \bigg] \, dk_x \, dk_y \end{align} [/math]

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_ρ 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, 2π]. You also saw some of the numerical parameters related to this spectral-domain integration scheme.

When the observation plane is placed very close to the radiating J and M currents, the Green's functions exhibit singularities, which translate to very slow convergence or divergence of the integrals. You need to be careful to place field sensors at adequate distances from these radiating sources.

Visualizing The Near Fields

In order to view the near field distributions, you must first define field sensor observables before running the planar MoM simulation. To do that, right click on the Field Sensors item in the Observables section of the Navigation Tree and select Insert New Observable.... The Field Sensor Dialog opens up. At the top of the dialog and in the section titled Sensor Plane Location, first you need to set the plane of near field calculation. In the dropdown box labeled Direction, you have three options X, Y, and Z, representing the"normals" to the XY, YZ and ZX planes, respectively. The default direction is Z, i.e. XY plane parallel to the substrate layers. In the three boxes labeled Coordinates, you set the coordinates of the center of the plane. Then, you specify the Size of the plane in project units, and finally set the Number of Samples along the two sides of the sensor plane. The larger the number of samples, the smoother the near field map will appear.

In the section titled Output Settings, you can also select the field map type from two options: Confetti and Cone. The former produces an intensity plot for field amplitude and phase, while the latter generates a 3D vector plot. In the confetti case, you have an option to check the box labeled Data Interpolation, which creates a smooth and blended (digitally filtered) map. In the cone case, you can set the size of the vector cones that represent the field direction. At the end of a sweep simulation, multiple field map are produced and added to the Navigation Tree. You can animate these maps. However, during the sweep only one field type is stored, either the E-field or H-field. You can choose the field type for multiple plots using the radio buttons in the section titled Field Display - Multiple Plots. The default choice is the E-field.

Once you close the Field Sensor dialog, its name is added under the Field Sensors node of the Navigation Tree. At the end of a planar MoM simulation, the field sensor nodes in the Navigation Tree become populated by the magnitude and phase plots of the three vectorial components of the electric (E) and magnetic (H) field as well as the total electric and magnetic fields defined in the following manner:

[math] |\mathbf{E_{tot}}| = \sqrt{|E_x|^2 + |E_y|^2 + |E_z|^2} [/math]

[math] |\mathbf{H_{tot}}| = \sqrt{|H_x|^2 + |H_y|^2 + |H_z|^2} [/math]

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).

Near-zone electric field map above a microstrip-fed patch antenna.

Near-zone magnetic field map above a microstrip-fed patch antenna.

Computing The Far Fields

Unlike differential-based methods, MoM simulators do not need a radiation box to calculate the far field data. The far-zone fields are calculated directly by integrating the currents on the traces and across the embedded objects using the asymptotic form of the background structure’s dyadic Green's functions:

[math] \mathbf{E^{ff}(r)} = \iiint_V \mathbf{ \overline{\overline{G}}_{EJ,ff}(r|r') \cdot J(r') } \, d\nu ' + \iiint_V \mathbf{ \overline{\overline{G}}_{EM,ff}(r|r') \cdot M(r') } \, d\nu '[/math]

[math] \mathbf{H^{ff}(r)} = \dfrac{1}{\eta_0} \mathbf{ \hat{r} \times E^{ff}(r) }[/math]

where η₀ = 120π 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:

[math]\mathbf{E^{ff}}(x,y,z) = j k_0 \eta_0 \hat{r} \times [\hat{r} \times \mathbf{A}(r \to \infty)] + j k_0 \hat{r} \times \mathbf{F}(r \to \infty)[/math]

The asymptotic form of these vector potentials are calculated using the "Method of Stationary Phase" when k₀r → ∞. In that case, one can use the approximation:

[math] k_0 |\mathbf{r-r'}| \approx k_0 (r - \mathbf{\hat{r} \cdot r'}) [/math]

After applying the stationary phase method, one can extract the spherical wave factor exp(-jk₀r)/r from the far-zone electric field, leaving the rest as functions of the spherical angles θ and φ. In other words, the far field is normalized to r, the distance from the field observation point to the origin. It is customary to express the far fields in spherical components E_θ and E_φ. Note that the outward propagating, TEM-type, far fields do not have radial components, i.e. E_r = 0.

[math] \mathbf{E_{\theta}}(\theta, \phi) = \cos\theta \cos\phi E_x + \cos\theta \sin\phi E_y - \sin\theta E_z [/math]

[math] \mathbf{E_{\phi}}(\theta, \phi) = -\sin\phi E_x + \cos\phi E_y [/math]

Visualizing The Far Fields

Even though the planar MoM engine does not need a radiation box, you still have to define a "Far Field" 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.

Once a planar MoM simulation is finished, three far field items are added under the Far Field item in the Navigation Tree. These are the far field component in θ direction, the far field component in φ direction and the "Total" far field. The 3D plots can be viewed in the project workspace by clicking on each item. The view of the 3D far field plot can be changed with the available view operations such as rotate view, pan, zoom, etc. If the structure blocks the view of the radiation pattern, you can simply hide or freeze the whole structure or parts of it. In a 3D radiation pattern plot, the fields are always normalized to the maximum value of the total far field for visualization purpose:

[math]|\mathbf{E_{ff,tot}}| = \sqrt{ |E_{\theta}|^2 + |E_{\phi}|^2 }[/math]

Figure: 3D polar radiation pattern plot of a microstrip-fed patch antenna.

Figure: 3D vectorial (cone) radiation pattern plot of a microstrip-fed patch antenna.

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.

Radar Cross Section of Planar Structures

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:

[math] \sigma_{\theta} = 4\pi r^2 \dfrac{|E_{\theta}^{scat}|^2}{|E^{inc}|^2}, \quad \sigma_{\phi} = 4\pi r^2 \dfrac{|E_{\phi}^{scat}|^2}{|E^{inc}|^2}, \quad \sigma = \sigma_{\theta} + \sigma_{\phi} = 4\pi r^2 \dfrac{|E_{tot}^{scat}|^2}{|E^{inc}|^2} [/math]

Note that in this case the RCS is defined for a finite-sized target in the presence of an infinite background structure. The scattered θ and φ 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 σ_θ, σ_φ and the total RCS. To do so, you must define an RCS observable instead of a radiation pattern. Follow these steps:

• Right click on the Far Fields item in the Observables section of the Navigation Tree and select Insert New RCS... to open the Radar Cross Section Dialog.
• The resolution of RCS calculation is specified by Angle Increment expressed in degrees. By default, the θ and φ angles are incremented by 5 degrees.
• At the end of a planar MoM simulation, besides calculating the RCS data over the entire (spherical) 3D space, a number of 2D RCS graphs are also generated. These are RCS cuts at certain planes, which include the three principal XY, YZ and ZX planes plus one additional constant f-cut. This fourth plane cut is at φ = 45° by default. You can assign another φ angle in degrees in the box labeled Non-Principal Phi Plane.

At the end of a planar MoM simulation, in the far field section of the Navigation Tree, you will have the θ and φ components of RCS as well as the total radar cross section. You can view a 3D visualization of these quantities by clicking on their entries in the Navigation Tree. The RCS values are expressed in m². The 3D plots are normalized to the maximum RCS value, which is also displayed in the legend box.

Figure 2: An example of the 3D mono-static radar cross section plot of a patch antenna.

Periodic Planar Structures & Antenna Arrays

Finite Arrays vs. Infinite Periodic Structures

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 "Array Object" 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 "Periodic Unit Cell". 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.

Real practical periodic structures obviously have finite extents. You can easily and quickly construct finite-sized arrays of arbitrary complexity using EM.Cube's "Array Tool". However, for large values of Nx and Ny, the size of the computational problem may rapidly get out of hand and become impractical. For very large periodic arrays, you can alternatively analyze a unit cell subject to the periodic boundary conditions and calculate the current distribtutions and far fields of the periodic unit cell. For their radiation patterns, you can multiply the "Element Pattern" by an "Array Factor" that captures the finite extents of the structure. In many cases, an approximation of this type works quite well. But in some other cases, the edge effects and particularly the field behavior at the corners of the finite-sized array cannot be modeled accurately. Periodic surfaces like FSS, EBG and metamaterials are also modeled as infinite periodic structures, for which one can define reflection and transmission coefficients. For this purpose, the periodic structure is excited using a plane wave source. Reflection and transmission coefficients are typically functions of the angles of incidence.

Modeling Finite Antenna Arrays

The straightforward approach to the modeling of finite-sized antenna arrays is to use the full-wave method of moments (MoM). This requires building an array of radiating elements using EM.Cube's 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 "Additional Radiation Characteristics" section of the Radiation Pattern Dialog. These quantities are saved into ASCII data files of similar names with .DAT file extensions. In particular, you can plot such data files at the end of a sweep simulation.

Figure 1: Planar Module's Radiation Pattern dialog.

Another approach to modeling a finite-sized antenna array is to analyze one of its elements and use the "Array Factor" concept to calculate its radiation patterns. This method ignores any inter-element coupling effects. In other words, you can regard the structure in the project workspace as a single isolated radiating element. To define an array factor, open the Radiation Pattern Dialog of the project. In the section titled "Impose Array Factor", you will see a default value of 1 for the Number of Elements along the X and Y directions. This implies a single radiator, representing the structure in the project workspace. There are also default zero values for the Element Spacing along the X and Y directions. You should change both the number of elements and element spacing in the X and Y directions to define a finite array lattice. For example, you can define a linear array by setting the number of elements to 1 in one direction and entering a larger value for the number of elements along the other direction. Keep in mind that when using an array factor for far field calculation, you cannot assign non-uniform amplitude or phase distributions to the array elements. For that purpose, you have to define an array object with a source array.

Defining A Periodic Domain

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.

To define a periodic structure, you must open Planar Module's Periodicity Settings Dialog by right clicking the Periodicity item in the Computational Domain section of the Navigation Tree and selecting Periodicity Settings... from the contextual menu or by selecting Menu > Simulate > 'Computational Domain > Periodicity Settings... from the Menu Bar. In the Periodicity Settings Dialog, check the box labeled Periodic Structure. This will enable the section titled"Lattice Properties". You can define the periods along the X and Y axes using the boxes labeled Spacing. You can also define values for periodic Offset along the X and Y directions, which will be explained later.

In a periodic structure, the virtual domain is replaced by a default blue periodic domain that is always centered around the origin of coordinates. Keep in mind that the periodic unit cell must always be centered at the origin of coordinates. The relative position of the structure within this centered unit cell will change the phase of the results.

Figure 1: Planar Module's Periodicity Settings dialog.

Regular vs. Generalized Periodic Lattices

Besides conventional rectangular lattices, EM.Cube's Planar Module can also handle complex non-rectangular periodic lattices. For example, many frequency selective surfaces have skewed grids. In order to simulate skewed-grid periodic structures, the definition of the grid has to be generalized. A periodic structure is a repetition of a basic structure (unit cell) at pre-determined locations. Let these locations be described by (x_mn, y_mn), where m and n are integers ranging from -8 to 8. For a general skewed grid, x_mn and y_mn can be described by:

[math] \begin{align} & x_{mn} = m\Delta x + n \Delta x' \\ & y_{mn} = m\Delta y + n \Delta y' \end{align} [/math]

where [math]\Delta x[/math] is the primary offset in the X direction (X Spacing) controlled by index m and [math]\Delta x'[/math] is the secondary offset in the X direction (X Offset) controlled by index n. The meanings of [math]\Delta y[/math] (Y Spacing) and [math]\Delta y'[/math] (Y Offset) are similar with the roles of indices m and n interchanged. To illustrate how to use this definition, consider an example of an equilateral triangular grid with side length L as shown in the figure below.

Figure 1: Diagram of an equilateral triangular periodic lattice.

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].

Interconnectivity Among Unit Cells

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.

As an example, consider the periodic structure in the figure below that shows a metallic screen or wire grid. The unit cell of this structure can be defined as a rectangular aperture in a PEC ground plane (marked as Unit Cell 1). In this case, the rectangle object is defined as a slot trace. Alternatively, you can define a unit cell in the form of a microstrip cross on a metal trace. In the latter case, however, the microstrip cross should extend across the unit cell and connect to the crosses in the neighboring cells in order to provide current continuity.

Figure 1: Modeling a periodic screen using two different types of unit cell.

Figure 2: The PMC aperture unit cell and its planar mesh.

Figure 3: The PEC cross unit cell and its planar mesh. Notice the cell extensions at the unit cell's boundaries.

Periodic MoM Simulation

In the case of an infinite periodic planar structure, the field equations can be written in the following form:

[math] \mathbf{E(r) = E^{inc}(r)} + \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} \bigg[ \iiint_V \mathbf{ \overline{\overline{G}}_{EJ}(r|r') \cdot J_{mn}(r') } \, d\nu' + \iiint_V \mathbf{ \overline{\overline{G}}_{EM}(r|r') \cdot M_{mn}(r') } \, d\nu' \bigg] [/math]

[math] \mathbf{H(r) = H^{inc}(r)} + \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} \bigg[ \iiint_V \mathbf{ \overline{\overline{G}}_{HJ}(r|r') \cdot J_{mn}(r') } \, d\nu' + \iiint_V \mathbf{ \overline{\overline{G}}_{HM}(r|r') \cdot M_{mn}(r') } \, d\nu' \bigg] [/math]

where

[math]\mathbf{J_{mn}(r) = J_{mn}}(x,y,z) = \mathbf{J_{00}}(x+m S_x, y+n S_y, z) e^{j(m k_{x00} S_x + n k_{y00} S_y)}[/math]

[math]\mathbf{M_{mn}(r) = M_{mn}}(x,y,z) = \mathbf{M_{00}}(x+m S_x, y+n S_y, z) e^{j(m k_{x00} S_x + n k_{y00} S_y)}[/math]

and

[math] -\infty \lt m, n \lt \infty [/math]

In the above equations, [math]\mathbf{J_{00}(r)}[/math] and [math]\mathbf{M_{00}(r)}[/math] are the periodic unit cell's electric and magnetic currents that are repeated everywhere in space on a rectangular lattice with periods S_x and S_y along the X and Y directions, respectively. [math]k_{x00}[/math] and [math]k_{y00}[/math] are the periodic propagation constants along the X and Y directions, respectively, and they are given by:

[math] k_{x00} = k_0 \sin\theta \cos\phi [/math]

[math] k_{y00} = k_0 \sin\theta \sin\phi [/math]

where θ and φ are the beam scan angles in the case of periodic excitation of lumped sources, or they are the spherical angles of incidence in the case of a plane wave source illuminating the periodic structure. Using the infinite summations, one can define periodic dyadic Green's functions in the spectral domain in the following manner:

[math] \mathbf{ \overline{\overline{G}}_{\mu \nu}^{PER} (r|r') } = \frac{1}{S_x S_y} \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} \mathbf{ \tilde{\overline{\overline{G}}}_{\mu \nu} } (k_x, k_y, z|z') e^{-j[k_{xm}(x-x') + k_{yn}(y-y')]} [/math]

where

[math] k_{xm} = k_{x00} + \frac{2\pi m}{S_x} \quad \text{and} \quad k_{ym} = k_{y00} + \frac{2\pi m}{S_y} [/math]

The above doubly infinite periodic Green's functions are said to be expressed in terms of "Floquet Modes". 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 "Periodic Simulation", you can change the values of Number of Floquet Modes in the two boxes designated X and Y.

Figure 1: Changing the number of Floquet modes from the Planar MoM Engine Settings dialog.

Modeling Periodic Phased Arrays

Earlier, it was argued that you can calculate the radiation pattern of a finite antenna array by modeling a single isolated element and multiplying its "Element Pattern" by the "Array Factor". This method gives acceptable results only when the inter-element coupling effects are negligible, as it does not take into account such effects. Planar antennas printed on dielectric substrates usually exhibit inter-element coupling effects due to the propagation of the substrate surface wave modes. If your finite-sized array is very large and you cannot afford a straightforward full-wave MoM simulation of it, you can alternatively model it as an infinite array represented by a periodic unit cell. In this case, you calculate the radiation pattern of the unit cell structure and use it as the "Element Pattern" in conjunction with the "Array Factor". The periodic Green's functions, in this case, capture the inter-element coupling effects. What is missing from this picture is the finite edge effects and/or corner effects, if any.

When a periodic structure is excited using a gap or probe source, it acts like an infinite periodic phased array. All the periodic replicas of the unit cell structure are excited. You can even impose a phase progression across the infinite array to steer its beam. You can do this from the property dialog of the gap or probe source. At the bottom of the Gap Source Dialog or Probe Source Dialog, there is a section titled Periodic Beam Scan Angles. You can enter desired values for Theta and Phi beam scan angles in degrees. The corresponding phase progressions are calculated and applied to the periodic Green's functions:

[math]\Psi_x = -\frac{2\pi S_x}{\lambda_0} \sin\theta \cos\phi[/math]

[math]\Psi_y = -\frac{2\pi S_y}{\lambda_0} \sin\theta \sin\phi[/math]

Note that you have to define a finite-sized array factor in the Radiation Pattern dialog. You do this in the Impose Array Factor section of this dialog. In the case of a periodic structure, when you define a new far field item in the Navigation Tree, the values of Element Spacing along the X and Y directions are automatically set equal to the value of Periodic Lattice Spacing along those directions. You have to set the Number of Elements along the X and Y directions, which are both equal to one initially, representing a single radiator. If you forget to define an array factor, the radiation pattern of the unit cell structure will be displayed, which does not show beam scanning.

Figure 1: Setting the periodic scan angles in Planar Module's Gap Source dialog.

Figure 2: The 3D radiation pattern of a beam-steered periodic printed dipole array.

Exciting Periodic Structures Using Plane Waves

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 "reflection.CPX" and "transmission.CPX". 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.

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.

Figure: A periodic planar layered structure with slot traces excited by a normally incident plane wave source.

Characterizing Periodic Surfaces Using Angular Sweeps

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 φ angle and sweep the θ 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 "reflection.CPX" and "transmission.CPX". These data files are also listed in EM.Cube's Data Manager, where you can view or plot them.

Figure 1: Planar Module's Angle Settings dialog.

Modeling Periodic Structures Using Adaptive Frequency Sweeps

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°< θ = 180°. You can also illuminate the periodic surface by the plane wave source from the bottom half-space, corresponding to 0° = θ < 90°. In this case, the reflection coefficient R and transmission coefficient T are equivalent to S₂₂ and S₁₂ parameters, respectively. Having these interpretations in mind, EM.Cube enables the "Adaptive Frequency Sweep" option of the Frequency Settings Dialog when your planar structure has a periodic domain together with a plane wave source.

Click here to learn about Modeling Finite-Sized Periodic Arrays Using NCCBF Technique.

 

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