Changes

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.

The Green’s functions are the solutions of boundary value problems when they are excited by an elementary source. This is usually assumed to be an infinitesimally small vectorial point source. In order for Green’s functions to be computationally useful, they must have analytical closed forms like a mathematical expression, or one should be able to compute them using a recursive process. It turns out that only very few boundary value problems have closed-form Green’s functions. Planar layered structures with laterally infinite extents are one of those few cases, which can be represented by recursive dyadic Green's functions.

where ''''''G_EJ''', '''G_EM''', '''G_HJ''', '''GH_M'''''' are the dyadic Green’s functions for the electric and magnetic currents due to electric and magnetic current source, respectively, and '''Eⁱ''' and '''Hⁱ''' are the incident or impressed electric and magnetic fields, respectively. In these equations, '''r''' is the position vector of the observation point and '''r'''' is the position vector of the source point. V is the volume that contains all the sources and the volume integration is performed with respect to the primed coordinates. The incident or impressed fields provide the excitation of the structure. They may come from an incident plane wave or a gap source on a microstrip line, a short dipole, etc. The complexity of the Green’s functions depends on what is considered as the background structure. If you remove all the unknown currents from the structure, you are left with the background structure.

To derive a system of integral equations, we enforce the boundary conditions on the integral definitions of the '''E''' and '''H''' fields as follows:

[[File:PMOM5.png]]

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

Similar expressions can be derived for the T^(EM), U^(HJ) and Y^(HM)elements of the MoM matrix.

The right choice of the basis functions to represent the elementary currents is very important. It will determine the accuracy and computational efficiency of the resulting numerical solution. Rooftop basis functions are one of the most popular types of basis functions used in a variety of MoM formulations. The surface currents (whether electric or magnetic) are discretized using 2D rooftop basis functions shown in the figure below:

Figure 2: Prismatic basis functions built over single triangular and rectangular cells.

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

Figure 1: A typical planar layered structure.

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

You can 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.

Figure 1: [[Planar Module]]'s Layer Stack-up Settings dialog.

You can also set the thickness of the substrate layer in the project units. Note that you cannot change the thickness of the top and bottom half-spaces. You can only change their material properties.

Figure 2: EM.Cube's Materials dialog.

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

Figure 1: [[Planar Module]]'s Navigation Tree.

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.

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.

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.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.

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.

The conductive sheet traces are modeled using the surface impedance boundary condition:

[[File:PMOM18.png]]

[[File:PMOM19(2).png]]

Figure 1: The [[Planar Module]]'s PEC and Conductive Sheet Trace dialogs.

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 />

Figure 1: The [[Planar Module]]'s PMC Trace dialog.

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

Figure 1: [[Planar Module]]'s Layer Stack-up dialog showing the Embedded Sets tab.

<table>

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.

# Terminating PEC ground planes at the top or bottom of a planar structure are defined as PEC top or bottom half-spaces, respectively.

# 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.

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 &gt;''' from the contextual menu. This opens another sub-menu containing '''Planar''' and a list of all the other EM.Cube modules that have already defined object groups. Select '''Planar''' or any other available module, and yet another sub-menu opens up with a list of all the available traces and embedded object sets already defined in your project. Select the desired group, and all the selected objects will move to that group. When selecting multiple objects from the Navigation Tree, make sure that you hold the keyboard's '''Shift Key''' or '''Ctrl Key''' down while selecting a group's name from the contextual menu.

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.

Figure 1: Planar hybrid and triangular meshes for rectangular patches.

Rectangular cells offer a major advantage over triangular cells for numerical MoM simulation of planar structures. This is due to the fact that the dyadic Green's functions of planar layered background structures are space-invariant on the transverse plane. Recall that the elements of the moment matrix are given by the following equation:

Figure 1: Pairs of rooftop basis functions that have identical MoM interactions.

The planar MoM mesh generation process involves three steps:

Once a mesh is generated, it stays in the memory until the structure is changed or the mesh density or other settings are modified. Every time you view mesh, the one in the memory is displayed. You can force EM.Cube to create a new mesh from the ground up by selecting '''Menu &gt; Simulate &gt; Discretization &gt; Regenerate Mesh''' or by right clicking on the '''Planar Mesh''' item in the '''Discretization''' section of the Navigation Tree and selecting '''Regenerate''' from the contextual menu.

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.

Figure 1: Mesh of two rectangular patches at two different planes. The lower substrate layer has a higher permittivity.

[[File:PMOM31.png]]

The Planar Mesh Settings dialog.

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

Figure 2: Edge-connected rectangular planar objects and their triangular and hybrid planar meshes.

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

Figure 1: Mesh of a vertical PEC via connecting two horizontal metallic strips. The shorter via has one prismatic cell along the Z direction, while the longer via is discretized into several stacked cells.

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 &quot;field discontinuity&quot; 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. Sometimes a slightly non-uniform mesh still produces stable numerical results. In other words, you may choose to increase the mesh resolution around the discontinuity regions only.

Figure 1: Refining the planar mesh at the via and surrounding area.

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

Figure 2: Setting the minimum allowable angle for non-defective triangular cells.

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.

Figure 1: Locking the mesh density of an object group from its property dialog.

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

Keep in mind that since a polymesh object it considered a final mesh, its mesh cannot be connected to other objects. In other words, bridge basis functions are not generated if even some of the polymesh edges may coincide with other objects' edges. A polymesh object is treated by the mesh generator as an isolated mesh. However, EM.Cube allows you to connect polymesh objects manually. To do so, bring two or more polymesh objects close to each other so that they have one or more common edges. No face overlaps are allowed in this case. Select the polymesh objects and click the '''Merge Tool'''[[File:manuals/emagware/cubecad/creating-more-complex-objects/merging-open-curves/merge_tool_tn.png]] button of '''Tools Toolbar''' to merge the polymesh objects into a single polymesh object. The new merged polymesh object will provide all the necessary bridge basis functions among the original, separate polymesh objects.

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

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

A gap is an infinitesimally narrow discontinuity that is placed on the path of current flow on a feed line. In planar structures, feed lines are typically in the form of a microstrip, stripline, slotline or coplanar waveguide (CPW). You use rectangle strip objects to construct such feed lines. A gap source can be placed on any rectangle strip object on a PEC, PMC or conductive sheet trace. Depending on the type of the trace on which a gap source is placed, it will have a different physical interpretation.

Figure 1: The [[Planar Module]]'s Gap Source dialog.

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.

Figure 1: The [[Planar Module]]'s Probe Source dialog.

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.

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

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:

Figure 1: The Port Definition dialog.

[[File:PMOM53.png]]

Figure 2: Edit Port dialog.

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 &quot;coupled&quot; port then interacts with other coupled or uncoupled ports.

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

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

Figure 1: Definition of different input impedances at the gap location.

[[File:PMOM60(1).png]]

Figure 4: Input impedance of a probe source on a PEC via connected to a ground plane.

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.

where ['''Z₀'''] and ['''Y₀'''] are diagonal matrices whose diagonal elements are the port characteristic impedances and admittances, respectively.

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:

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

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

* 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.

[[File:PMOM64.png]]

'''Note that 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.

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.

[[File:PMOM73.png]]

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.

[[File:PMOM72.png]]

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.

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.

Figure 2: Analyzing a coplanar waveguide using the line calculator.

A short dipole is the simplest type of radiator, which consists of a short current element of length &amp;DELTA;l, aligned along a unit vector û and carrying a current of I Amperes. The product I&amp;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:

Figure 1: [[Planar Module]]'s Short Dipole Source dialog.

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 &quot;receive&quot; 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]]

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

* RCPz

To define a plane wave source, follow these steps:

Figure 1: [[Planar Module]]'s Plane Wave dialog.

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.

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.

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.

* Frequency Sweep

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

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

Figure 1: [[Planar Module]]'s Simulation Run dialog.

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

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

[[File:PMOM79.png]]

Figure 1: The Planar MoM Engine Settings dialog.

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:

Figure 1: Setting the check box for &quot;Use Optimzied Solvers for Intel CPU&quot; in the Preferences dialog.

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:

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

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:

[[File:PMOM93(1).png]]

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&quot;normals&quot; 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.

Figure 3: Near-zone magnetic field map above a microstrip-fed patch antenna.

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:

[[File:PMOM112.png]]

[[File:PMOM113.png]]

[[File:PMOM115.png]]

[[File:PMOM114.png]]

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

[[File:PMOM89.png]]<br />

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

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

[[File:PMOM123.png]]

* 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.

[[File:PMOM124.png]]

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

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.

Figure 1: [[Planar Module]]'s Frequency Settings dialog.

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

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

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

[[File:PMOM121.png]]

[[File:PMOM122.png]]

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

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.

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

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

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

[[File:PMOM138.png]]

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

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

</table>

If you are interested in calculating certain quantities at the end of a simulation, which you do not find among EM.Cube's standard output data, you can define your own custom output. EM.Cube allows you to define new custom output as any mathematical expression that involves the available standard output parameters, numbers, variables and all of EM.Cube's mathematical functions. For a list of legitimate mathematical functions, click the '''Functions [[File:manuals/emagware/cubecad/computing-with-cad-objects/mathematical-functions/functions_icon.png]]'''button of the '''Simulate Toolbar''' or select '''Simulate &gt; Functions...'''from the menu bar, or use the keyboard shortcut '''Ctrl+I''' to open the Function Dialog. Here you can see a list of all the available EM.Cube functions with their syntax and a brief description. To define a custom output, click the '''Custom Output [[File:manuals/emagware/emcube/modules/planar/running-simulations/defining-custom-output-parameters/custom_icon.png]]'''button of the '''Simulate Toolbar''' or select '''Simulate &gt; Custom Output...'''from the menu bar, or use the keyboard shortcut '''Ctrl+K''' to open the Custom Output Dialog. This dialog has a list of all of your custom output parameters. Initially, the list empty. You can define a new custom output by clicking the '''Add''' button of the dialog to open up the '''Add Custom Output Dialog'''. In this dialog, first you have to choose a new label for your new parameter and then define a mathematical expression for it. At the bottom of the dialog you can see a list of all the available standard output parameters, whose number and variety depends on your project's source type as well as the defined project observables. When you close the Add Custom Output dialog, it returns you to the Custom Output dialog, where the parameter list now reflects your newly defined custom output. You can edit an existing parameter by selecting its row in the table and clicking the '''Edit''' button, or you can delete any parameter from the list using the '''Delete''' button.

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

At the end of a planar MoM simulation, a variety of 2D and 3D output data are generated. Some of these can be visualized or graphed directly from the Navigation Tree, while the others can only be accessed from the Data Manager. All of EM.Cube's simulation data are always written into ASCII data files that you can open and inspect or edit. Lists of these 2D and 3D data files appear under Data Manager's various tabs. The generated data also include all of [[Planar Module]]'s legitimate standard outputs that the simulation engine can compute given the specified source and observable types as well as all of your own previously defined custom output parameters. Note that in this release of EM.Cube, all the custom outputs are real-type data. Each custom output is written into a separate real data file with the same name as the parameter's given label and a &quot;'''.DAT'''&quot; file extension. To open data manager, click the '''Data Manager''' [[File:manuals/emagware/emcube/modules/planar/running-simulations/defining-custom-output-parameters/data_manager_icon.png]] button of the '''Simulate Toolbar''' or select '''Simulate &gt; Data Manager''' from the menu bar, or right click on the '''Data Manager''' item of the Navigation Tree and select '''Open Data Manager'''... from the contextual menu. You can also use the keyboard shortcut '''Ctrl+D''' at any time. Select any data file by clicking and highlighting its row in the table and then click the '''Plot''' button to plot its graph in '''EM.Grid'''. You can also view the contents of a data file by selecting its row in th file list and clicking the '''View''' button of the dialog or by simply double-clicking the highlighted row. This opens up a new window containing a convenient spreadsheet that gives a tabular view of the contents of the selected data file. There are a large number of data operations and manipulations that you can perform on the data content including matrix, calculus and statistical calculations as well as computing and plotting new datasets using the &quot;Compute&quot; feature of the spreadsheet. You can make multiple file selection using the keyboard's '''Ctrl''' and '''Shift''' keys.

Figure 2: EM.Cube's Animation Controls dialog.

Parametric sweep is EM.Cube's most versatile sweep type. During a parametric sweep, the values of one or more sweep variables are varied over their specified ranges, and the planar MoM simulation is run for each combination of variable samples. If you define two or more sweep variables, the process will then involve nested sweep loops that follow the order of definition of the sweep variables. The topmost sweep variable in the list will form the outermost nested loop, and the sweep variable at the bottom of the list will form the innermost nested loop. Note that you can alternatively run either a frequency sweep or an angular sweep as parametric sweep, whereby the project frequency or the angles of incidence of a plane wave source are designated as sweep variables. Unlike optimization which will be discussed later, parametric sweeps are simple and straightforward and do not required careful advance planning.

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

Optimization is a process in which the values of one or more variables are varied in a systematic way until one or more design objectives are met. The design objectives are typically defined based on the output simulation data and are mathematically translated into an error (objective) function that is to be minimized. Running a successful optimization requires careful advance planning. First you have to make sure that your optimization problem does have a valid solution within the range of your optimization variables. In other words, the design objectives must be achievable for at least one combination of the optimization variable values within the specified ranges. Otherwise, the optimization process will not converge or will exhaust the maximum allowed number of iteration cycles and exit unsuccessfully.

A design objective is a logical expression that consists of two mathematical expressions separated by one of the logical operators: ==, &lt;, &lt;=, &gt; or &gt;=. These are called the left-hand-side (LHS) and right-hand-side (RHS) mathematical expressions and both must have computable numerical values. They may contain any combination of numbers, constants, variables, standard or custom output parameters as well as EM.Cube's legitimate functions. Objectives that involve the logical operator &quot;'''=='''&quot; are regarded a &quot;'''Goals'''&quot;. The RHS expression of a goal is usually chosen to be a number, which is often known as the &quot;'''Target Value'''&quot;. In the logical expression of a goal, one can bring the two RHS and LHS expressions to one side establish an equality of the form &quot;(LHS - RHS) == 0&quot;. Numerically speaking, this is equivalent to minimizing the quantity | LHS - RHS |. During an optimization process, all the project goals are evaluated numerically and they are used collectively to build an error (objective) function whose value is tried to be minimized. Objectives that involve &quot;non-Equal&quot; logical operators are regarded a &quot;'''Constraints'''&quot;. Unlike goals which lead to minimizable numerical values, constraints are rather conditions that should be met while the error function is being minimized.

[[File:PMOM151.png]]

= Periodic Planar Structures &amp; Antenna Arrays =

A periodic structure is one that exhibits a repeated geometric pattern. It is made up of identical elements that are arranged in the form of a periodic lattice. The spacing between the elements is denoted by Sx along the X direction and Sy along the Y direction. The number of elements is denoted by Nx along the X direction and Ny along the Y direction (i.e. a total of Nx.Ny elements). If Nx and Ny are finite numbers, you have a finite-sized periodic structure, which is constructed using an &quot;'''Array Object'''&quot; in EM.Cube. If Nx and Ny are infinite, you have an infinite periodic structure with periods Sx and Sy along the X and Y directions, respectively. An infinite periodic structure in EM.Cube is represented by a &quot;'''Periodic Unit Cell'''&quot;. Periodic structures have many applications including phased array antennas, frequency selective surfaces (FSS), electromagnetic bandgap structures (EBG), metamaterial structures, etc. EM.Cube allows you to model both finite and infinite periodic structures.<br /> <br /> Real practical periodic structures obviously have finite extents. You can easily and quickly construct finite-sized arrays of arbitrary complexity using EM.Cube's &quot;Array Tool&quot;. However, for large values of Nx and Ny, the size of the computational problem may rapidly get out of hand and become impractical. For very large periodic arrays, you can alternatively analyze a unit cell subject to the periodic boundary conditions and calculate the current distribtutions and far fields of the periodic unit cell. For their radiation patterns, you can multiply the &quot;Element Pattern&quot; by an &quot;Array Factor&quot; that captures the finite extents of the structure. In many cases, an approximation of this type works quite well. But in some other cases, the edge effects and particularly the field behavior at the corners of the finite-sized array cannot be modeled accurately. Periodic surfaces like FSS, EBG and metamaterials are also modeled as infinite periodic structures, for which one can define reflection and transmission coefficients. For this purpose, the periodic structure is excited using a plane wave source. Reflection and transmission coefficients are typically functions of the angles of incidence.

The straightforward approach to the modeling of finite-sized antenna arrays is to use the full-wave method of moments (MoM). This requires building an array of radiating elements using EM.Cube's '''Array Tool''' and feeding the individual array elements using some type of excitation. For example, if the antenna elements are excited using a gap source or a probe source, you can assign a certain array weight distribution among the elements as well as phase progression among the elements along the X and Y directions. EM.Cube currently offers uniform, binomial, Chebyshev and (arbitrary) data file-based weight distribution types. The full-wave MoM approach is very accurate and takes into account all the inter-element coupling effects. At the end of a planar MoM simulation of the array structure, you can plot the radiation patterns and other far field characteristics of the antenna array just like any other planar structure.

Another approach to modeling a finite-sized antenna array is to analyze one of its elements and use the &quot;Array Factor&quot; 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 &quot;'''Impose Array Factor'''&quot;, 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.

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.

Figure 1: [[Planar Module]]'s Periodicity Settings dialog.

x_mn = m&amp;DELTA;x + n&amp;DELTA;x'<br /> y_mn = n&amp;DELTA;y + m&amp;DELTA;y'

Figure 1: Diagram of an equilateral triangular periodic lattice.

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.

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

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

[[File:PMOM96(1).png]]

[[File:PMOM97.png]]

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

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 &quot;Element Pattern&quot; by the &quot;Array Factor&quot;. 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 &quot;Element Pattern&quot; in conjunction with the &quot;Array Factor&quot;. 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.

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

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.

Figure 1: A periodic planar layered structure with slot traces excited by a normally incident plane wave source.

[[File:PMOM103.png]]

Figure 1: [[Planar Module]]'s Angle Settings dialog.

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.

Previously, you saw how the concept of &quot;Array Factor&quot; is used to approximate the far field radiation pattern of a finite-sized array of radiators. The total radiation pattern can be expressed as the product of the array factor and the &quot;Element Pattern&quot;. The array factor captures the topology of the array lattice and depends on the number of elements along the X and Y directions as well as the element spacing along those directions. As for the choice of element pattern, you saw two extreme cases. In the &quot;'''Isolated Element'''&quot; option, you compute the radiation pattern of a single stand-alone radiator and completely ignore any coupling effects from the neighboring elements. This option is readily available in the Radiation Pattern Dialog of the Far Field observable. In the &quot;'''Periodic Element'''&quot; option, you analyze a periodic version of the radiating element with periods equal to the element spacing. The computed radiation pattern of the periodic unit cell in this case captures the coupling effects from an infinite number of elements.

EMAG Technologies Inc. has recently developed a novel technique, called '''Numerically Constructed Characteristic Basis Functions (NCCBF)''', which generates physics-based entire-domain basis functions for the elements of a finite-sized array. These &quot;sophisticated&quot; basis functions are linear combinations of the &quot;'''Isolated Element'''&quot; solutions and &quot;'''Periodic Element'''&quot; solutions. Unlike the array factor method, which is a post-processing calculation of far-field data, the NCCBF method generate a full-wave MoM solution with entire-domain basis functions. Considering the same example of the patch antenna array discussed earlier, the NCCBF method generates a total of N_B= 4 entire-domain basis functions on each patch element: an isolated X-directed solution, a periodic X-directed solution, an isolated Y-directed solution, and a periodic Y-directed solution. The same approach applies equally well to triangular RWG basis functions and is not limited to rectangular cells. As a result, the new MoM linear system has a dimension of N = N_B. N_F = (4)(64) = 256. In other words, the NCCBF method compresses the original MoM matrix of size N = 30,720 to one of significantly reduced size N = 256 (i.e. a compression factor of 120x).

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:

Figure 2: Planar MoM's &quot;Add Unit Cell&quot; dialog.

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