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 Maxwellâs equations as opposed to their differential forms that are used in the finite element or finite difference time domain methods.
In a planar MoM simulation, the background structure is usually a layered planar structure that consists of one or more laterally infinite material layers. In EM.Cubeâs [[Planar Module]], the layered structure is stacked along the Z axis. In other words, the dimensions of the layers are infinite along the X and Y axes. Metallic traces are placed at the boundaries between the substrate or superstrate layers. These are modeled by perfect electric conductor (PEC) traces or conductive sheet traces of finite thickness and finite conductivity. Some layers might be separated by infinite perfectly conducting ground planes. The two sides of a ground plane can be electromagnetically coupled through one or several slots or apertures. Such slots or apertures are modeled by magnetic currents and are realized and represented by perfectly magnetic conductor (PMC) traces. Furthermore, the metallic traces can be interconnected or connected to ground planes using embedded objects. Such objects can be used to model circuit vias, plated-through holes or dielectric inserts. These are modeled as volume polarization currents.
The currents in a planar MoM simulation are discretized as a collection of elementary currents with small finite spatial extents. These elementary currents are called basis functions and obviously have a vectorial nature. The total currents (solution of the problem) are summations of these elementary currents. The basis functions are well defined and easy to calculate; however, their amplitudes are initially unknown in a MoM problem. Through the planar MoM solution, you find these unknown amplitudes. Once the total currents are known, you can calculate the fields everywhere in the structure.
Given the fact that the dyadic Greenâs functions and the incident or impressed fields are all known, one can solve the above system of integral equations to find the unknown currents '''J''' and '''M'''.
In EM.CUBE's [[Planar Module|Planar module]], magnetic currents are always surface current with units of V/m. Electric currents, however, can be surface currents with units of A/m as in the case of metallic traces like microstrip lines, or they can be volume currents with units of A/m<sup>2</sup> as in the case of perfectly conducting vias. Dielectric inserts are modeled as volume polarization currents that are related to the electric field '''E''' in the following manner:
[[File:PMOM5.png]]
= Anatomy Of A Planar Structure =
EM.Cubeâs [[Planar Module ]] is intended for constructing and modeling planar layered structures. By a planar structure we mean one that contains a background substrate of laterally infinite extents, made up of one or more material layers all stacked up vertically along the Z axis. Objects of finite size are then interspersed among these substrate layers. This is somehow different than EM.Cube's other computational modules, which are geared for handling arbitrary 3D structures.
In [[Planar Module]], the background structure, called "'''Layer Stack-up'''", may involve one or more material layers of infinite extents along the X and Y axes but of finite thickness along the Z axis. When you start a new project, the background structure has a single vacuum layer. The layer stack-up is always terminated from the top and bottom by two infinite half-spaces. The terminating half-spaces might be the free space, or a perfect conductor (PEC ground), or any material medium. Most planar structures used in RF and microwave applications such as microstrip-based components have a PEC ground at their bottom. EM.Cube's default stack-up has a vacuum top half-space and a PEC bottom half-space. Some structures like stripline components require two bounding PEC grounds at both top and bottom.
The finite-sized objects of a planar structure may include metal traces, slots and apertures, vertical vias and interconnects, or dielectric inserts including air voids inside the substrate layers. Metal traces are modeled as electric surface currents. These are planar surface objects, always parallel to the XY plane, that are defined on metal (PEC) traces and placed at the boundary (interface) plane between two substrate layers. Slots and apertures are modeled as magnetic surface currents on the surface of an infinite PEC plane and provide electromagnetic coupling between its top and bottom sides. These, too, are constructed using planar surface objects, always parallel to the XY plane, that are defined on slot (PMC) traces and placed at the boundary (interface) plane between two substrate layers. EM.Cube's [[Planar Module ]] also allows prismatic objects that can be modeled by electric volume currents. These include vertical vias and dielectric inserts, and are called embedded object sets. [[Planar Module|Planar module ]] does not allow construction of 3D CAD objects. Instead, you draw the cross section of prismatic objects as planar surface objects 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.
[[File:PMOM14.png]]
== Defining Layer Stack-Up ==
When you start a new project in EM.Cubeâs [[Planar Module]], there is always a default background structure that consists of a finite vacuum layer sandwiched between a vacuum top half-space and a PEC bottom half-space. Every time you enter the [[Planar Module|Planar module]], the '''Stack-up Settings Dialog''' opens up. This is where you define the entire background structure. Once you close this dialog, you can open it again by right clicking the '''Layer Stack-up''' item in the '''Computational Domain''' section of the Navigation Tree and selecting '''Layer Stack-up Settings...''' from the contextual menu. Or alternatively, you can select the menu item '''Simulate > Computational Domain > Layer Stack-up Settings...'''
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 (ε<sub>r</sub>), permeability (μ<sub>r</sub>), electric conductivity (Ï) and magnetic conductivity (Ï<sub>m</sub>). There is also a column that lists the names of embedded object sets inside each substrate layer, if any.
[[File:PMOM8(1).png]]
Figure 1: [[Planar Module]]'s Layer Stack-up Settings dialog.
[[File:PMOM9.png]]
Figure 1: [[Planar Module]]'s Substrate Layer dialog.
You can also use EM.Cube's Material List to define the material properties of a substrate layer. In the Substrate Layer Dialog, click the '''Material''' button to open the '''Material List'''. In the Material List Dialog, pick any material or type the first letter of a material to highlight it. Then click the '''OK''' button or simply hit the '''Enter''' key of your keyboard to close the list and return to the substrate layer dialog.
== Planar Object Types ==
EM.Cubeâs [[Planar Module ]] groups objects by their material and electromagnetic properties. Each object group shares the same color and same position in the layer stack-up. All the planar objects belonging to the same trace are located on the same substrate layer boundary. All the prismatic objects belonging to the same embedded set lie inside the same substrate layer and have the same material composition. Theoretically speaking, all the objects belonging to a group are governed by the same boundary conditions. EM.Cubeâs [[Planar Module ]] currently provides the following types of objects for building a planar layered structure:
# '''Perfect Electric Conductor (PEC) Traces:''' These represent infinitesimally thin metallic objects that are deposited or metallized on or between substrate layers. PEC objects are modeled by surface electric currents that satisfy the PEC boundary condition.
[[File:PMOM11.png]]
Figure 1: [[Planar Module]]'s Navigation Tree.
== 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 assumed to exist 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.
[[File:PMOM12.png]]
Figure 1: [[Planar Module]]'s Stack-up Settings dialog.
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.
[[File:PMOM13.png]]
EM.Cube's [[Planar Module ]] has a special feature that makes construction of planar structures quite easy and straightforward. '''The active work plane of the project workspace is always set at the plane of the active trace.''' In EM.Cube's other modules, all objects are drawn in the XY plane (z = 0) by default. In [[Planar Module]], all new objects are drawn on a horizontal plane that is located at the Z-coordinate of the currently active trace. As you change the active trace or add a new trace, you will also change the active work plane.
== Modeling Metallic Traces ==
A trace is a group of finite-sized planar objects that have the same conductive properties and same Z-coordinate. In other words, they are located on the same horizontal plane, or at the same vertical level on the layer stack-up. You can define two types of metallic traces in the [[Planar Module]]:
# '''PEC Traces:''' These represent perfect conductor objects that have zero thickness and no editable material properties.
[[File:PMOM19(2).png]]
When you start a new project in [[Planar Module ]] with no traces defined, if you simply draw a new object, a default PEC trace is created and added to the Navigation Tree to hold that object. Alternatively, you can define your own new traces from the Layer Stack-up Settings dialog or directly from the Navigation Tree.
NOTE: Two or more PEC and conductive sheet traces can coexist at the same Z-coordinate. In this case, the Layer Stack-up Settings dialog shows these trace rows stacked up on top of each other between their common top and bottom substrate layers.
</table>
Figure 1: The [[Planar Module]]'s PEC and Conductive Sheet Trace dialogs.
== Modeling Slot Traces ==
Slots and apertures are cut-out and removed metal in an infinite perfectly conducting (PEC) ground plane. When a slot is excited, tangential electric fields are formed on the aperture, which can be modeled as finite magnetic surface currents confined to the area of the slot. Therefore, instead of modeling the electric surface currents on the PEC ground around the slot, one can alternatively model the finite-extent magnetic surface currents on PMC traces. In EM.Cube's [[Planar Module]], you define slot objects under PMC traces. A PMC trace at a certain Z-plane implies the presence of an infinite PEC plane at that Z-coordinate. Therefore, you do not need to define an additional PEC plane at that location on the layer stack-up. The slot (PMC) objects provide the electromagnetic coupling between the two sides of this infinite ground plane. By the same token, you cannot place a PEC trace and a PMC trace at the same Z-level, as the latter's ground will short the former. However, you can define two or more PMC traces at the same Z-plane. In this case, all the slot objects lie on the same infinite PEC ground plane. <br />
[[File:PMOM20.png]]
Figure 1: The [[Planar Module]]'s PMC Trace dialog.
== Defining Embedded Object Sets ==
Embedded object sets represent short material insertions inside substrate layers. They can be metal or dielectric. Metallic embedded objects can be used to model vias, plated-through holes, shorting pins and interconnects. These are called PEC via sets. Embedded dielectric objects can be used to model air voids, thin films and material inserts in metamaterial structures. Embedded magnetic object are not currently supported by EM.Cubeâs [[Planar Module]].
Embedded objects can be defined either from the Layer Stack-up Settings dialog or directly from the Navigation Tree. In the former case, open the "Embedded Sets" tab of the stack-up dialog. This tab has a table that lists all the embedded object sets along with their material type, the host substrate layer, the host material and their height. '''Note that the height of an embedded object is always identical to the thickness of its host substrate layer.'''
[[File:PMOM23.png]]
Figure 1: [[Planar Module]]'s Layer Stack-up dialog showing the Embedded Sets tab.
To add a new object set, click the arrow symbol on the '''Insert''' button of the dialog and select one of the two options, '''PEC Via Set''' or '''Embedded Dielectric Set''', from the dropdown list. This opens up a new dialog where first you have to set the host layer of the new object set. A dropdown list labeled "'''Host Layer'''" gives a list of all the available finite substrate layers. You can also set the properties of the embedded object set, including its label, color and material properties. Keep in mind that you cannot control the height of embedded objects. Moreover, you cannot assign material properties to PEC via sets, while you can set values for the '''Permittivity'''(ε<sub>r</sub>) and '''Electric Conductivity'''(Ï) of embedded dielectric sets. Vacuum is the default material choice. You may use EM.Cube's Material List for this purpose, which can be opened up by clicking the '''Material'''button. Once embedded object sets are added to the Embedded Sets table, you can edit their properties at any time by selecting their row and clicking the '''Edit''' button.
</table>
Figure 2: The [[Planar Module]]'s PEC Via Set and Embedded Dielectric Set dialogs.
To define an embedded set from the Navigation Tree, right click on the '''Embedded Object Sets''' item in the '''Physical Structure''' section of the Navigation Tree and select either '''Insert New PEC Via Set...''' or '''Insert New Embedded Dielectric Set...''' The respective New Embedded Object Set dialog opens up, where you set the properties of the new object set. As soon as you close this dialog, it takes you to the Layer Stack-up Settings dialog, where you can examine the location of the new object set on the layer hierarchy.
After a new embedded object set has been defined and added to the Navigation Tree, it becomes the active trace. You are now ready to create geometrical objects in the new active trace. Remember that [[Planar Module ]] does not allow you to draw 3D objects. The solid object buttons in the '''Object Toolbar''' are disabled to prevent you from doing so. '''Instead, you draw planar surface objects as the cross section of embedded sets. EM.Cube extends these planar objects across their host layer automatically and displays them as wire-frame, 3D extruded objects.''' Extrusion of embedded object sets happen after meshing and before every simulation. You can enforce this extrusion manually by right clicking the '''Layer Stack-up''' item in the '''Computational Domain''' section of the Navigation Tree and selecting '''Update Planar Structure...''' from the contextual menu.
== Planar Module's Rules & Limitations ==
[[File:manuals/emagware/emcube/modules/planar/anatomy-of-a-planar-structure/defining-a-background-structure/pmom_phys5.png]]
Figure 1: [[Planar Module]]'s Virtual Domain Settings dialog.
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.
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 planar integral equations are then solved approximately on these elementary cells. As this method does not require a discretization of the entire computational domain, it is often computationally much more efficient than differential-based techniques like FEM or FDTD, which mesh the whole domain. The accuracy of the MoM numerical solution depends greatly on the quality of the generated mesh. The mesh density gives a measure of how electrically small these elementary cells are. Low mesh resolutions compromise the accuracy of the numerical solution. On the other hand, very high mesh densities may lead to numerical instability of the method of moments. As a rule of thumb, a mesh density of about 20-30 cells per effective wavelength usually yields acceptable 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 will 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.
It is well known that any planar geometry with any degree of complexity can be reasonably discretized using a surface triangular mesh. EM.Cube's [[Planar Module ]] provides a versatile triangular mesh generator for this purpose. This generates a regular mesh, in which most of the triangular cells have almost equal areas. The uniformity or regularity of mesh is an important factor in warranting a stable numerical solution. A highly incongruous mesh may even produce completely erroneous results. EM.Cube's [[Planar Module ]] also offers another mesh generator that creates a "Hubrid" planar mesh combining triangular and rectangular cells. Although triangular cells are more versatile than rectangular cells in adapting to arbitrary geometries, many practical planar structures contain a large number of rectangular parts like patch antennas, microstrip lines and components, etc.
[[File:PMOM32.png]]
# Verifying the mesh for integrity
EM.Cubeâs [[Planar Module ]] offers two mesh generation algorithms for discretizing planar structures: Hybrid and Triangular. The hybrid mesh consists of both rectangular and triangular cells. The hybrid mesh generator creates a kind of âobject-centricâ mesh that depends on the geometry of each object. It tries to discretize rectangular objects with rectangular cells as much as possible. In certain connection areas, a few triangular cells might be inserted to provide the mesh transition for current continuity. All the non-rectangular objects (circular, polygonal, etc.) are discretized using triangular cells. The triangular mesh generator, on the other hand, discretizes the planar objects with all triangular cells regardless of their shape. The only exceptions are feed lines that contain gap sources or lumped elements, which are always meshed with rectangular cells.
You can generate and view a planar mesh by clicking the '''Show Mesh''' [[File:manuals/emagware/emcube/modules/planar/mesh-generation/the-planar-mom-mesh/mesh_tool.png]] button of the '''Simulate Toolbar''' or by selecting '''Menu > Simulate > Discretization > Show Mesh''' or using the keyboard shortcut '''Ctrl+M'''. When the mesh of the planar structure is displayed in EM.Cubeâs project workspace, its "Mesh View" mode is enabled. In this mode you can perform view operations like rotate view, pan or zoom, but you cannot create new objects or edit existing ones. To exit the mesh view mode, press the keyboard's '''Esc Key''' or click the '''Show Mesh''' [[File:manuals/emagware/emcube/modules/planar/mesh-generation/the-planar-mom-mesh/mesh_tool.png]] button once again.
== Planar Mesh Density ==
EM.Cube's [[Planar Module]], by default, generates a hybrid mesh of your planar structure with a mesh density of 20 cells per effective wavelength. It is important to understand the concept of mesh density (either hybrid or triangular) as used by [[Planar Module]]. It gives a measure of the number of cells per effective wavelength that are placed in various regions of your planar structure. The higher the mesh density, the more cells are created on the geometrical objects. Keep in mind that only the finite-sized objects of your structure are discretized. No mesh is generated for the substrate layers of your background structure. The free-space wavelength is defined as λ<sub>0</sub> = 2Ïf/c, 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 λ<sub>eff</sub> = λ<sub>0</sub>/âε<sub>eff</sub>, where ε<sub>eff</sub> is the effective permittivity.
The effective permittivity is defined differently for different types of traces and embedded object sets. For metal and conductive sheet traces, the effective permittivity is defined as the larger of the permittivity of the two substrate layers just above and below the metallic trace. For slot traces, the effective permittivity is defined as the mean (average) of the permittivity of the two substrate layers just above and below the metallic trace. These definitions of effective permittivity are consistent with the effective propagation constant of transmission lines realized on such trace types. For embedded object sets, the effective permittivity is defined as the largest of the permittivities of all the substrate layers and embedded dielectric sets. In all cases, for the purpose of calculating the effective wavelength, only the real part of the permittivities are considered. The reason for using an effective wavelength so defined for determination of mesh resolution is to make sure that enough cells are placed in areas that might feature higher field concentration.
Using the generated mesh of a planar structure, EM.Cube creates a set of vectorial basis functions that are passed to the input file of the Planar MoM simulation engine. This engine requires edge-based basis functions. The common edges between adjacent cells are used to define edge-based rooftop or RWG basis functions. These elementary basis functions indeed provide the current flow and warrant the continuity among the mesh cells. Therefore, when two objects overlap or share a common edge, the connection between them must be translated into "bridge" basis functions, which carry the information about current flow to the simulation engine.
'''The most important rule of object connections in EM.Cube's [[Planar Module ]] is that only objects belonging to the same trace can be connected to one another.''' For example, if two objects reside on the same Z-plane and geometrically have a common edge which you can clearly see in the project workspace, but organizationally they belong to two different metal traces, then the bridge basis functions will not be generated between them, and the simulation engine will see them disconnected. If two objects belong to the same trace and have a common overlap area, EM.Cube first merges the two objects using the "Boolean Union" operation and converts them into a single object for the purpose of meshing. The mesh of "unioned" areas is usually made up of triangular cells. If two objects reside on the same Z-plane and geometrically overlap with each other but organizationally belong to two different trace groups, incongruous, overlapped cells will be generated that will either blow up the linear system or produce completely wrong simulation results.
<table>
== Mesh of Embedded Objects ==
EM.Cube's [[Planar Module ]] models embedded objects as vertical volume currents. The vectorial basis functions in this case are Z-directed prisms as opposed to rooftop basis functions. If an embedded object is located under or above a metallic trace or connected from both top and bottom, it is critical to create mesh continuity between the embedded object and its connected metallic traces. In other words, the generated mesh must ensure current continuity between the vertical volume currents and horizontal surface currents. EM.Cubeâs planar mesh generator automatically handles situations of this kind and generates all the required connection meshes.
Keep in mind that EM.Cubeâs Planar MoM engine uses a 2.5-D approximation, whereby only vertical volume currents are assumed inside embedded objects. When the height of an embedded object is small (as should typically be under the 2.5-D assumption), one prismatic cell is placed across the object along the Z-axis. Long PEC vias with a very small radius do also satisfy the 2.5-D assumption. In this case, the long via objects are discretized further along the Z direction and generate multiple stacked cells. Several prismatic cells along the Z-axis may increase the simulation time drastically. This is due to the fact that the host layer is effectively subdivided into a number of sub-layers and the stacked cells are treated as stacked vias embedded inside these sub-layers. As a result, the simulation engine needs to compute all the dyadic Greenâs functions accounting for the interactions between all such sub-layers.
== Locking Mesh Of Object Groups ==
EM.Cube's [[Planar Module ]] provides different ways of controlling the mesh of a planar structure locally. Earlier you saw how to increase the mesh resolution at the discontinuity regions without affecting the mesh of uniform or regular areas of a planar structure. Another way of local mesh control is to lock the mesh density of certain traces or object sets. The mesh density that you specify in the Planar Mesh Settings dialog is a global parameter and applies to all the traces and embedded object sets in your project. However, you can lock the mesh of individual PEC, PMC and conductive sheet traces or embedded objects sets. In that case, the locked mesh density takes precedence over the global density. Note that locking mesh of object groups, in principle, is different than refining the mesh at discontinuities. In the latter case, the mesh of connection areas is affected. However, objects belonging to different traces cannot be connected to one another. Therefore, locking mesh can be useful primarily for isolated object groups that may require a higher (or lower) mesh resolution.
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 =
== Exciting In a typical electromagnetic simulation in EM.Cube's [[Planar Structures ==Module]], you define a planar structure that consists of a layered background structure with a number of finite-sized metal and slot traces and possibly embedded metal or dielectric objects interspersed among the substrate layers. The planar structure is then excited by some sort of a signal source that induces electric currents on metal parts and magnetic currents on slot traces. The method of moments (MoM) solver computes these unknown electric and magnetic currents by discretizing the finite-sized objects. The induced currents, in turn, produce their own electric and magnetic fields which coexist (are superposed) with the impressed electric and magnetic fields of the signal source. From a knowledge of the near fields, EM.Cube calculates the port characteristics of the planar structure, if any ports have been defined. From a knowledge of the far fields, EM.Cube calculates the radiation or scattering characteristics of the planar structure.
In a typical electromagnetic simulation in EM.Cube's Planar Module, you define You can excite a planar structure that consists of a layered background structure with in a number of finite-sized metal and slot traces and possibly embedded metal or dielectric objects interspersed among the substrate layersdifferent ways. The planar structure is then excited by some sort of a signal excitation source that induces electric currents you choose depends on metal parts and magnetic currents on slot traces. The method of moments (MoM) solver computes these unknown electric and magnetic currents by discretizing the finite-sized objectsobservables you seek in your project. The induced currents, in turn, produce their own electric and magnetic fields which coexist (are superposed) with [[Planar Module]] provides the impressed electric and magnetic fields of the signal following source. From a knowledge of the near fields, EM.Cube calculates the port characteristics of the for exciting planar structure, if any ports have been defined. From a knowledge of the far fields, EM.Cube calculates the radiation or scattering characteristics of the planar structure.structures:
You can excite a planar structure in a number of different ways. The excitation source you choose depends on the observables you seek in your project. Planar Module provides the following source for exciting planar structures:Â * Lumped Sources with three varieties: '''[[#Gap Sources''']], '''[[#De-embedded Sources''' ]] and '''[[#Probe Sources''']]* '''[[#Plane Wave Sources''']]* '''[[#Short Dipole 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.
[[File:PMOM47.png]]
Figure 1: The [[Planar Module]]'s Gap Source dialog.
== Probe Sources ==
Another way of exciting a planar structure is by placing a gap on the path of a vertical current on a PEC via. This represents a filament source, which is used to model coaxial probe excitation. A probe source can be placed only on a PEC via object. Most planar transmission lines are fed using SMA connectors. The outer conductor of the coaxial line is connected to the ground and its inner conductor is extended across the substrate layer and connected to a metallic line. EM.Cube's [[Planar Module ]] models a coaxial probe as an infinitesimal gap discontinuity placed across a thin via, representing an ideal voltage source in series with a lumped impedance. When the impedance is zero, the gap acts like an ideal lumped source and creates a uniform electric field across the via. The source pumps vertical electric current into the probe. If the voltage source is shorted (having a zero amplitude), then the gap acts like a shunt lumped element across the via.
To define a probe source, follow these steps:
[[File:PMOM48.png]]
Figure 1: The [[Planar Module]]'s Probe Source dialog.
== Defining Source Arrays ==
== 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
== 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), C'''apacitors'''(C), I'''nductors'''(L), and series and parallel combinations of them as shown in the figure below:
[[File:manuals/emagware/emcube/modules/planar/excitation-sources/using-lumped-circuits/image106.png]]
Figure 1: A series-parallel RLC combination that can be modeled as a lumped circuit in [[Planar Module]].
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:
[[File:PMOM64.png]]
Figure 1: The [[planar Module]]'s Lumped Element dialog.
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'''.
== De-Embedded Sources ==
EM.Cube's [[Planar Module ]] provides de-embedded sources for the exclusive purpose of accurate S parameter calculation based on Prony's method. A de-embedded source is indeed a gap source that is placed close to an open end of a feed line. The other end of the line is typically connected to a planar structure of interest. Like gap sources, de-embedded sources can be placed only on rectangle strip objects. '''During mesh generation, EM.Cube automatically extends the length of a port line that hosts a de-embedded source to about two effective wavelengths.''' This is done to provide enough length for formation of a clean standing wave current pattern. The effective wavelength of a transmission line for length extension purposes is calculated in a similar manner as for the planar mesh resolution. It is defined as λ<sub>eff</sub> = λ<sub>0</sub>/âε<sub>eff</sub>, where ε<sub>eff</sub> is the effective permittivity. For metal and conductive sheet traces, the effective permittivity is defined as the larger of the permittivities of the two substrate layers just above and below the metallic trace. For slot traces, the effective permittivity is defined as the mean (average) of the permittivities of the two substrate layers just above and below the metallic trace. The host port line must always be open from one end to allow for its length extension. You have to make sure that there are no objects standing on the way of the extended port line to avoid any unwanted overlaps.
[[File:PMOM72.png]]
[[File:PMOM74.png]]
Figure 2: The [[Planar Module]]'s De-embedded Source dialog.
In a planar project with de-embedded sources, if you do not define any ports, the feed lines will simply be extended, and the exciting gap sources will be placed at the open ends of these extended lines. Note that if you define a de-embedded source along with a port definition in your project, then all the other port-assigned sources of your project must be of the same de-embedded type. You can define de-embedded sources for coplanar waveguides (CPW) on slot traces. To do so, you need to place two collocated, de-embedded sources with identical offsets (same phase reference plane), same source amplitudes but 180° phase difference. Note that for CPW structures, setting the number of Prony modes to 2 can get you more accurate results. In this case, the two extracted Prony modes will include the incident and reflected, odd and even, propagating modes of the CPW.
== Using the Line Calculator ==
EM.Cube's [[Planar Module ]] provides a simple calculator for analyzing planar transmission lines. It is based on the frequency domain finite difference (FDFD) technique. You can find the characteristic impedance, effective permittivity and guide wavelength of a TEM or quasi-TEM transmission line defined based on your project's background structure. Therefore, any arbitrary stack-up configuration with any number of substrate layers can be considered.
To access the Line Calculator, first you have to select a metal (PEC) trace or a slot (PMC) trace in the Navigation Tree. Right click on the name of a trace and select '''Line Calc...''' from the contextual menu to open the Line Calc Dialog. You can analyze a metal strip line on any PEC trace or a coplanar waveguide (coupled slot lines) on any PMC trace. The 2D line structure to be analyzed by the FDFD method consists of the background structure of your project with a metal strip or CPW located at the Z-plane of your selected trace. Depending on whether your open the Line Calc dialog from a metal trace or a slot trace, a picture of a microstrip line or a CPW line appear at the top of the dialog, respectively. In the former case, you have to specify '''Strip Width''' in the project units. In the latter case, you have to specify '''Slot Width''', too. Keep in mind that the strip width is equal to the spacing between the two slot lines minus the width of individual slot lines. Clicking the Analyze button of the dialog evokes the FDTD simulator, and calculated results are reflected in the boxes labeled '''Zo''', '''Effective Permittivity''' and '''Guide Wavelength'''.
Figure 2: Analyzing a coplanar waveguide using the line calculator.
== Exciting Planar Structures With Short Dipole Sources ==
A short dipole is the simplest type of radiator, which consists of a short current element of length &DELTA;l, aligned along a unit vector û and carrying a current of I Amperes. The product I&DELTA;l is often called the dipole moment and gives a measure of the radiator's strength. A short dipole in the free space generates an azimuth-symmetric, almost omni-directional, far field. However, the radiated fields of a short dipole above a layered planar background structure are greatly altered by the presence of the substrate layers. Note that the electric and magnetic field radiated by a short dipole in the presence of a layered background structure are indeed nothing but the dyadic Green's functions of that structure:
[[File:PMOM110.png]]
Figure 1: [[Planar Module]]'s Short Dipole Source dialog.
== Exciting Planar Structures With Plane Waves Wave Sources ==
You can excite a planar structure with an incident plane wave to explore its scattering characteristics such as radar cross section (RCS). Exciting an antenna structure with an incident plane wave is equivalent to operating it in the "receive" mode. Plane wave excitation in the [[Planar Module ]] is particularly useful for calculation of reflection and transmission coefficients of periodic surfaces. Note that the incident plane wave in your project bounces off the layered background structure and part of it also penetrates the substrate layers. The total incident field that is used to calculate the excitation vector of the MoM linear system is a superposition of the incident, reflected and transmitted plane waves at various regions of your planar structure:
[[File:PMOM111.png]]
where η<sub>0</sub> = 120Ï is the characteristic impedance of the free space, '''k<sub>1</sub>''' and '''k<sub>2</sub>''' are the unit propagation vectors of the incident plane wave and the wave reflected off the topmost substrate layer, respectively, and '''ê<sub>1</sub>''' and '''ê<sub>2</sub>''' are the polarization vectors corresponding to the electric field of those waves. R is the reflection coefficient at the interface between the top half-space and the topmost substrate layer and has different values for the TM and TE polarizations.
EM.Cube's [[Planar Module ]] provides the following polarization options:
# TMz
[[File:PMOM77.png]]
Figure 1: [[Planar Module]]'s Plane Wave dialog.
= 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
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 Ï 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
[[File:PMOM80.png]]
Figure 1: Selecting a simulation mode in [[Planar Module]]'s Simulation Run dialog.
== Running A Planar MoM Analysis ==
[[File:PMOM78.png]]
Figure 1: [[Planar Module]]'s Simulation Run dialog.
== Stages Of A Planar MoM Analysis ==
== 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<sub>eff</sub>, where λ<sub>eff</sub> 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'''.
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:
# LU Decomposition Method
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'''".
[[File:PMOM84.png]]
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. '''Note that 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:
[[File:PMOM90.png]]
Figure 1: [[Planar Module]]'s Field Sensor dialog.
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:
[[File:PMOM88.png]]
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).
[[File:PMOM116.png]]
[[File:PMOM118.png]]
Figure 1: [[Planar Module]]'s Radiation Pattern dialog.
[[File:PMOM119.png]]
== 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 Range'''is initially set equal to your project's center frequency minus and plus half bandwidth. But you can change the values of '''Start Frequency'''and '''End Frequency''' as well as the '''Number of Samples'''. The dialog offers two options for '''Frequency Sweep Type''': '''Uniform''' or '''Adaptive'''. Select the former type. It is very important to note that in a MoM simulation, changing the frequency results in a change of the mesh of the structure, too. This is because the mesh density is defined in terms of the number of cells per effective wavelength. By default, during a frequency sweep, EM.Cube fixes the mesh density at the highest frequency, i.e., at the "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.
[[File:PMOM126.png]]
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 S<sub>11</sub> (return loss) and S<sub>21</sub> (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<sub>ij</sub>" curves over the given frequency range. In that case, the curves are considered as having converged.
'''Note that to run an adaptive frequency sweep, you must have defined one or more ports for your planar structure.''' 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.'''
[[File:PMOM127.png]]
Figure 1: Settings adaptive frequency sweep parameters in [[Planar Module]]'s Frequency Settings Dialog.
== Examining Port Characteristics ==
== 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. '''Note that interpolant order more than 15 will suffer from numerical instabilities even if you have a very large number of data samples.'''
[[File:PMOM131.png]]
Figure 1: [[Planar Module]]'s Smart Fit dialog.
[[File:PMOM133(2).png]]
== 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)
== Standard vs. Custom Output ==
At the end of a planar MoM simulation, a number of computed quantities are designated as "Standard Output" parameters and can be used for various post-processing data operations. For example, you can define design objectives based on them, which you need for optimization. The table below gives a list of all the currently available standard output parameters in EM.Cube's [[Planar Module]]:
<table>
== Viewing & Visualizing Various Output Data Types ==
At the end of a planar MoM simulation, a variety of 2D and 3D output data are generated. Some of these can be visualized or graphed directly from the Navigation Tree, while the others can only be accessed from the Data Manager. All of EM.Cube's simulation data are always written into ASCII data files that you can open and inspect or edit. Lists of these 2D and 3D data files appear under Data Manager's various tabs. The generated data also include all of [[Planar Module]]'s legitimate standard outputs that the simulation engine can compute given the specified source and observable types as well as all of your own previously defined custom output parameters. Note that in this release of EM.Cube, all the custom outputs are real-type data. Each custom output is written into a separate real data file with the same name as the parameter's given label and a "'''.DAT'''" file extension. To open data manager, click the '''Data Manager''' [[File:manuals/emagware/emcube/modules/planar/running-simulations/defining-custom-output-parameters/data_manager_icon.png]] button of the '''Simulate Toolbar''' or select '''Simulate > Data Manager''' from the menu bar, or right click on the '''Data Manager''' item of the Navigation Tree and select '''Open Data Manager'''... from the contextual menu. You can also use the keyboard shortcut '''Ctrl+D''' at any time. Select any data file by clicking and highlighting its row in the table and then click the '''Plot''' button to plot its graph in '''EM.Grid'''. You can also view the contents of a data file by selecting its row in th file list and clicking the '''View''' button of the dialog or by simply double-clicking the highlighted row. This opens up a new window containing a convenient spreadsheet that gives a tabular view of the contents of the selected data file. There are a large number of data operations and manipulations that you can perform on the data content including matrix, calculus and statistical calculations as well as computing and plotting new datasets using the "Compute" feature of the spreadsheet. You can make multiple file selection using the keyboard's '''Ctrl''' and '''Shift''' keys.
[[File:PMOM144.png]]
[[File:PMOM91.png]]
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.
[[File:PMOM99.png]]
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<sub>mn</sub>, y<sub>mn</sub>), where m and n are integers ranging from -â to â. For a general skewed grid, x<sub>mn</sub> and y<sub>mn</sub> can be described by:
x<sub>mn</sub> = m&DELTA;x + n&DELTA;x'<br /> y<sub>mn</sub> = n&DELTA;y + m&DELTA;y'
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 &DELTA;y = L and &DELTA;y' = 0. 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 &DELTA;x = L and &DELTA;x' = L/2. To sum up, an equilateral triangular grid can be described by &DELTA;x = L, &DELTA;x' = L/2, &DELTA;y = L and &DELTA;y' = 0. 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, &DELTA;x' = &DELTA;x/2 and &DELTA;y'=0. A Hexagonal lattice (with alternating rows) is a special case of triangular lattice in which &DELTA;y = â3&DELTA;x/2.
== 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.
[[File:PMOM100.png]]
Figure 1: Setting the periodic scan angles in [[Planar Module]]'s Gap Source dialog.
[[File:manuals/emagware/emcube/modules/planar/periodic-planar-structures/modeling-periodic-phased-arrays/pmom_per9_tn.png]]
== 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 Ï 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.
[[File:PMOM103.png]]
Figure 1: [[Planar Module]]'s Angle Settings dialog.
== Modeling Periodic Structures Using Adaptive Frequency Sweeps ==
== Running a NCCBF Simulation ==
In the current release of EM.Cube's [[Planar Module]], the NCCBF MoM solver works with any number of distinct, finite-sized arrays if they are excited with one of the following three source types:
# Gap Sources
Note that you can have several coexisting finite arrays with different element spacings (or different periodicities). You can also have regular (aperiodic) objects coexisting with your collection of finite arrays. In that case, the NCCBF process will create entire-domain basis functions for the elements of the finite arrays, while the regular method of moments will apply to the aperiodic portions of your planar structure. This flexibility makes NCCBF a very versatile and powerful technique.
There are a few rules that must be followed and observed when planning a NCCBF simulation. '''Each finite-sized array must be constructed using an EM.Cube "Array Object". Additionally, each array object must stand alone in a dedicated trace or embedded object set of its own.''' In other words, if an array object belongs to a trace or embedded object set that contains other objects, it will be excluded from the NCCBF process and will get a regular MoM treatment. Keep in mind that [[Planar Module ]] allows you to define different traces located at the same Z-plane, although the objects belonging to these separate traces cannot be connected to one another according to the planar meshing rules. Similarly, you can define two or more PEC via sets hosted by the same substrate layer. Therefore, if your planar structure contains finite arrays and aperiodic objects, you have to group them into separate traces or embedded object sets.
To run an NCCBF simulation, open the '''Simulation Run Dialog''', and then open the'''Planar MoM Engine Settings Dialog'''. In the "'''Finite Array Simulation'''" section of the latter dialog, check the box labeled "'''NCCBF Matrix Compression'''". This box is unchecked by default. Checking it enables the NCCBF Settings button. Click this button to open the NCCBF Settings Dialog. The dialog features a "List of Unit Cells Used for NCCBF Matrix Compression". This list initially empty. To add unit cells to it, click the '''Add''' button of the dialog to open the "'''Add Unit Cell Dialog'''". This dialog has two tables: Available Unit Cells on the left side and Associated Unit Cells on the right side. The left table shows a list of all the available, legitimate array objects in your project workspace. Remember that for an array object to be eligible for NCCBF compression, it has to stand alone on a dedicated trace or embedded object set, whichever applies. Select an array object from the left table and use the right arrow button (-->) to move it to the right table to associate it with the new NCCBF unit cell. You can associate more than one array object with the same NCCBF unit cell. In this case, the parent elements of all the associated array objects collectively constitute the NCCBF unit cell. The NCCBF unit cell is the planar structure that is analyzed separately, first, as a stand-alone isolated element, and next, as a periodic unit cell, to generate the NCCBF entire-domain basis function solutions. It is therefore very important that the array objects be positioned carefully with respect to the origin of coordinated and relative to one another to form the correct NCCBF unit cell. Once you move one or more array object names to the "Associated" table on the right, you can move them back to the "Available" table on the left using the left arrow (<--) button. You can also instruct EM.Cube to use only the isolated element solution by unchecking the box labeled "'''Include Periodic Solution of Unit Cell'''". Once you are satisfied with the definition of your NCCBF unit cell, close the dialog to return to the NCCBF Settings dialog. Here you see the name of the newly added NCCBF unit cell in the list along with the Number of Solutions and the names of all the associated array objects for each NCCBF unit cell. You can modify each row using the '''Edit''' button or remove it from the list using the '''Delete''' button. Close the NCCBF Settings dialog to return to the Planar MoM Engine Settings dialog, and close the latter to return to the Simulation Run dialog, where you can now start the NCCBF simulation by clicking the '''Run''' button.
== Symmetries, Array Objects & Composite Arrays ==
EM.Cube's [[Planar Module ]] treats array objects in a special way. That is why you need to use array objects with certain rules for NCCBF simulations. In general, if the mesh of your planar structure involves a total of N vectorial basis functions, the MoM matrix will contain a total of N<sup>2</sup> elements. Instead of computing the entire N<sup>2</sup> basis interactions, the Planar MoM simulation engine takes advantage of the inherent symmetry properties of the dyadic Green's functions and camputes the diagonal elements of the matrix and all the elements below the diagonal. This amounts to N.(N+1)/2 basis interactions. In many cases, the MoM matrix is symmetric, and the elements above the diagonal are simply mirror-image of the below-diagonal elements. In planar structures that involve both metal and slot traces, there will be sign reversals for some interactions.
In many cases, especially in the areas that contain sizable numbers of rectangular mesh cells, the basis functions are naturally grouped into distinct sets that are called domains. As you saw earlier in the discussion of planar mesh generation, uniform domains with identical rectangular cells bring significant savings during the matrix fill process. Using the concept of domains renders the MoM matrix as a block matrix, whose blocks represent the interactions among the domains. The diagonal blocks therefore correspond to self-domain interactions. By a similar argument, if your planar structure is made up of N<sub>D</sub> domains, then a total of N<sub>D</sub> . (N<sub>D</sub> +1)/2 domain interactions (or matrix blocks) are computed. An EM.Cube array object consists of N<sub>F</sub> identical geometrical elements. If the array object belongs to a trace that has other objects in it, then by the planar mesh generator's rules, the elements of the array object are merged with the other objects on the same trace using the "Union" Boolean operation. If some array elements possibly have connections with other objects, such connections are taken care of in the meshing process. '''However, if an array object stands alone in a dedicated trace, then only the parent (first) element is meshed, and it mesh is copied and cloned for all the other elements of the array.''' This produces a total of N<sub>F</sub> identical domains of vectorial basis functions. A direct consequence of this is identification of only N<sub>F</sub> unique domain-pair interactions or matrix blocks. In the absence of these symmetries, a total of at least N<sub>F</sub> . (N<sub>F</sub> +1)/2 domain interactions (or matrix blocks) must be computed. To better illustrate such matrix fill savings, let us consider the previous, not-so-large, 8 Ã 8 array of patch radiators, i.e. N<sub>F</sub>= 64. It was previously assumed that each rectangular patch antenna element involves 240 X-directed and 240 Y-directed rooftop basis functions, i.e. N<sub>B</sub>= 480. The numerical solution of this structure produces a linear system of total size N = N<sub>B</sub>. N<sub>F</sub>= 30,720. The total number of complex-valued elements of this matrix is 9.44E+08. This is the total number of highly sophisticated multi-dimensional integrals that you need to compute during a brute-force matrix fill process. For the sake of generality of the argument, here we ignore the huge additional savings that rectangular cells offer, and we assume that each unique domain-pair interaction involves N<sub>B</sub><sup>2</sup> = 230,400 elements, except for the self-domain interaction which requires N<sub>B</sub> . (N<sub>B</sub> +1)/2 = 115,440 integral computations. This amounts to a total of (N<sub>F</sub> -1) . N<sub>B</sub><sup>2</sup> + N<sub>B</sub> . (N<sub>B</sub> +1)/2 = 1.46E+07 integral computations, which is roughly N<sub>F</sub>(64) times fewer and faster than a brute-force matrix fill process.