Changes

In general, a structure may support both electric ('''J''') and magnetic ('''M''') currents. The total electric ('''E''') and magnetic ('''H''') fields can be expressed in terms of the electric and magnetic currents in the following way:

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:

where '''L_E''' is the boundary value operator for the electric field and '''L_H''' is the boundary value operator for the magnetic field. For example, '''L_E''' may require that the tangential components of the '''E'''field vanish on perfect conductors:

Or '''L_E''' and '''L_H''' may require that the tangential components of the '''E''' and '''H''' fields be continuous across an aperture in a perfect ground plane:

Given the fact that the dyadic Green’s 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, magnetic currents are always surface current with units of V/m. Electric currents, however, can be surface currents with units of A/m as in the case of metallic traces like microstrip lines, or they can be volume currents with units of A/m² as in the case of perfectly conducting vias. Dielectric inserts are modeled as volume polarization currents that are related to the electric field '''E''' in the following manner:

where k₀ is the free space propagation constant, Y₀=1/Z₀ =1/(120π) is the free space intrinsic admittance, ε_r is the permittivity of the dielectric insert, and ε_b is the permittivity of its background layer. In a 2.5-D formulation, it is assumed that the volume currents have only a vertical component along the Z direction, and their circumferential components are negligible.

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:

where '''f_n^(J)''' and '''f_k^(M)''' are the generalized vector basis functions for the expansion of electric and magnetic currents, respectively, and I_n^(J) and V_k^(M) are the unknown amplitudes of these basis functions, which have to be determined. Substituting these expansions into the integral equations generates a set of discretized integral equations, which can further be converted to a system of linear algebraic equations. This is accomplished by testing the discretized integral equations using the a set of test functions. In the method of moments, the Galerkin technique is typically used, which chooses the expansion basis functions as test functions. This leads to the following linear system:

where

and

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 1: Rooftop or RWG basis functions built over two rectangular, triangular or mixed cells.

The volume polarization currents in 2.5-D MoM have a vertical direction along the Z-axis. These are discretized using prismatic basis functions that have either a rectangular or triangular base with a constant profile along the Z-axis.

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

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 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.

Figure 1: A typical planar layered structure.

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

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 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.

Figure 2: EM.Cube's Materials dialog.

# '''Embedded Dielectric Sets:''' These are prismatic dielectric objects inserted inside a substrate layer. You can define a finite permittivity and conductivity for such objects, but their height is always the same as the height of their host layer. The embedded dielectric objects are modeled as vertical volume polarization currents.

Figure 1: Planar Module's 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.

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

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.

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

where Z_s is the surface impedance of the conductive sheet. If the thickness of the sheet is greater than the skin depth of the metal at the project frequency, then the surface impedance is given by

If the thickness τ of the sheet is less than the skin depth, then the conductive sheet transition boundary condition is used instead, and the surface impedance is given by

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.

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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 objects can be defined either from the Layer Stack-up Settings dialog or directly from the Navigation Tree. In the former case, open the &quot;Embedded Sets&quot; 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.'''

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

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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.

Figure 1: Planar Module's Virtual Domain Settings dialog.

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

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:

where the spatial-domain dyadic Green's functions are a function of the observation and source coordinates, '''r'''and '''r' '''. The MoM matrix elements can indeed be interpreted as interactions between two elementary basis functions '''f_i(r)''' and '''f_j(r')''' on that particular background structure. The spatial-domain dyadic Green's functions can themselves be expressed in terms of the spectral-domain dyadic Green's functions as follows:

where the doubly infinite integration is performed with respect to the spectral variables k_x and k_y. As can be seen from the above expression, the spatial-domain dyadic Green's functions are functions of z, z', as well as (x-x') and (y-y'). The MoM matrix elements can now be transformed into the spectral domain as

where the tilde symbol signifies the Fourier transform of a function defined as

Rectangular cells have simple Fourier transforms. The rooftop basis functions are triangular functions in the direction of current flow and constant in the perpendicular direction. This means that their Fourier transform is a product of a sinc-squared function along one spectral direction and a sinc function along the other. You can see from the figure below that if one deals with a rectangular mesh of identical cells (all equal and parallel), then the interactions among the rooftop basis functions become a functions of the index differences and not the absolute indices:

In the above equation, the vectorial rooftop basis functions have explicit, double indices: i and k along the local X and Y directions, respectively, for the test (observation) basis function, and j and l along the local X and Y directions, respectively, for the expansion (source) basis function. Thus, uniform rectangular cells, i.e. structured rectangular cells of identical size aligned in the same direction, can speed up the planar MoM simulation significantly due to these symmetry and the invariance properties. For example, all the self-interactions are identical regardless of the location of a rooftop basis function. This reduces the matrix fill process for a total of N rooftop basis functions from an N2 process to one of order N.

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

EM.Cube’s Planar Module offers two mesh generation algorithms for discretizing planar structures: Hybrid and Triangular. The hybrid mesh consists of both rectangular and triangular cells. The hybrid mesh generator creates a kind of “object-centric” mesh that depends on the geometry of each object. It tries to discretize rectangular objects with rectangular cells as much as possible. In certain connection areas, a few triangular cells might be inserted to provide the mesh transition for current continuity. All the non-rectangular objects (circular, polygonal, etc.) are discretized using triangular cells. The triangular mesh generator, on the other hand, discretizes the planar objects with all triangular cells regardless of their shape. The only exceptions are feed lines that contain gap sources or lumped elements, which are always meshed with rectangular cells.

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.

Due to the different definitions of effective wavelength in different parts of your planar structure, you will see different mesh resolutions. For example, if you structure has several substrate layers with different permittivities, the mesh of metal traces on layers with a higher permittivity value will feature more cells than the mesh of metal traces on layers with a lower permittivity value even though the mesh density value is the same for the whole structure.

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

= Customizing A Planar Mesh =

The Planar Mesh Settings dialog.

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

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

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

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

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.

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

You have access to every single node of a polymesh object and you can change its coordinates arbitrarily. You do this by opening the property dialog of a polymesh object and selecting a certain node index in the box labeled '''Active Node'''. You can also select a node by hovering the mouse over the node to highlight it and then click to select it. A red ball appears on the current active node. You can delete the nodes arbitrarily using the '''Delete''' button of the dialog, which results in lowering the mesh resolution at the location of the deleted node. Or you can insert new nodes in the faces of a polymesh object. To insert a node, first you have to select a face. Change the '''Mode''' option by selecting the '''Face''' radio button and then select the right '''Active Face''' index. A red triangular border appears around the selected face. You can also simply click on the surface of a face and select it using the mouse. With the desired face selected, click the '''Insert''' button of the dialog to create a new node at the centroid of the selected face. You can adjust the coordinates of the newly inserted node from the three X, Y and Z '''Coordinate''' boxes. Note that immediately after the insertion of a new node, the label of these coordinate boxes changes to &quot;'''New Node'''&quot; and they show the relative local X, Y and Z offsets with respect to the original node position. Once you close the Polymesh Dialog, the new node is added to the existing node list and can be edited later like the other polymesh nodes. By inserting a new node, you increase the mesh resolution locally and selectively.

Figure 2: Discretizing a planar surface object using EM.Cube's Polymesh tool.

= Excitation Sources =

* You can also change the default label as well as the default color of the gap source using the '''Color''' button of the dialog and selecting the desired color from the color palette.

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

Unlike gap sources, whose offset parameter determines their exact location on their host line, the offset parameter of a probe source is not relevant except for long host vias. In the case of a short via that is discretized using a single prismatic element across its host substrate layer, the probe gap is always placed at the middle of its height. Longer vias may have a mesh that consists of two or more stacked prismatic elements. In this case, the probe source's offset determines which prismatic element will host the probe gap discontinuity at its middle.

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

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

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

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

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

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

Figure 1: The Port Definition dialog.

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

Figure 2: Edit Port dialog.

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

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

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

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

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

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

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

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

Figure 2: Placing a gap source a quarter guide wavelength away from the open end of a feed line to effectively short it to the ground at the gap location.

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

Figure 3: Placing two oppositely polarized gap sources a quarter guide wavelength away from the short end of a CPW line to effectively create an open circuit beyond the gap location.

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

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

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

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

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

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: 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:

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

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

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

* In the '''Load Properties''' section, the series and shunt resistance values Rs and Rp are specified in Ohms, the series and shunt inductance values Ls and Lp are specified in nH (nanohenry), and the series and shunt capacitance values Cs and Cp are specified in pF (picofarad). Only the checked elements are taken into account in the total impedance calculation. By default, only the series resistor is checked with a value of 50Σ, and all other circuit elements are initially greyed out.<br />

Figure 1: The planar Module's Lumped Element dialog.

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

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

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

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

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_eff = λ₀/√ε_eff, where ε_eff is the effective permittivity. For metal and conductive sheet traces, the effective permittivity is defined as the larger of the permittivities of the two substrate layers just above and below the metallic trace. For slot traces, the effective permittivity is defined as the mean (average) of the permittivities of the two substrate layers just above and below the metallic trace. The host port line must always be open from one end to allow for its length extension. You have to make sure that there are no objects standing on the way of the extended port line to avoid any unwanted overlaps.

Figure 1: The mesh of a patch antenna excited with a de-embedded source. Note the feed line extension in the mesh view.

* In the '''Prony Mode Extraction''' section, you can specify the '''Number of Prony Modes''', which refers to the number of positive-negative exponential pairs that are extracted from the standing wave current data. The default value is 1 and represents the dominant quasi-TEM incident/reflected signal pair.

Figure 2: The Planar Module's De-embedded Source dialog.

When your background structure involves a slot (PMC) trace, then there is an infinite PEC ground plane at the plane of the slot trace. In that case, when you analyze a strip line on a metal trace, you must keep in mind that your stack-up configuration will be truncated by the slot's ground plane just for purpose of Z_o calculation. A typical case of this type is a slot-coupled patch antenna fed by a microstrip line underneath the slot. From the point of view of the Line Calculator, the microstrip line lies on a substrate layer that is backed by the slot's ground plane and it does not see the substrate layer lying above the slot plane.

Figure 1: Analyzing a metal strip line using the line calculator.

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:

To define a short dipole source, follow these steps:

* In the section titled '''Source Properties''', you can change the values of the dipole's '''Amplitude'''(in A), '''Phase'''(in degrees) and '''Length''' in the project's length units. A new dipole, by default, is Z-directed. You can change its orientation by entering the components of its unit vector in the three boxes labeled '''Direction Unit Vector'''.

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:

where η₀ = 120π is the characteristic impedance of the free space, '''k₁''' and '''k₂''' are the unit propagation vectors of the incident plane wave and the wave reflected off the topmost substrate layer, respectively, and '''ê₁''' and '''ê₂''' 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.

* The direction of the Plane Wave is determined by the incident '''Theta''' and '''Phi''' angles expressed in the spherical coordinate system in degrees. You have to choose the '''Polarization''' of the plane wave from the four options: '''TM_z''', '''TE_z''', '''LCP_z'''and '''RCP_z'''. The components of the unit propagation vector are shown based on your choice of the angles of incidence. The components of the normalized E- and H-field vectors are also displayed based on your choice of polarization.

Figure 1: Planar Module's Plane Wave dialog.

* HDMR

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

== Running A Planar MoM Analysis ==

Figure 1: Planar Module's Simulation Run dialog.

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

Figure 1: The Planar MoM Engine Settings dialog.

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

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

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

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:

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

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

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

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

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

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

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

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:

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

Calculation of the near-zone fields (fields at the vicinity of the unknown currents) is done at the post-processing stage and in a Cartesian coordinate systems. These calculations involve doubly infinite spectral-domain integrals, which are computed numerically. As was mentioned earlier, EM.Cube's planar MoM engine rather uses a polar integration scheme, where the radial spectral variable k_ρ is integrated over the interval [0, Mk₀], M being a large enough number to represent infinity, and the angular spectral variable τ is integrated over the interval [0, 2π]. You also saw some of the numerical parameters related to this spectral-domain integration scheme. '''Note that when the observation plane is placed very close to the radiating J and M currents, the Green's functions exhibit singularities, which translate to very slow convergence or divergence of the integrals.''' You need to be careful to place field sensors at adequate distances from these radiating sources.

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

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:

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).

Figure 2: Near-zone electric field map above a microstrip-fed patch antenna.

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:

where η₀ = 120π is the characteristic impedance of the free space. As can be seen from the above equations, the far fields have the form of a TEM wave propagating in the radial direction away from the origin of coordinates. This means that the far-field magnetic field is always perpendicular to the electric field and the propagation vector, which in this case happens to be the radial unit vector in the spherical coordinate system. In other words, one only needs to know the far-zone electric field and can easily calculate the far-zone magnetic field from it. In EM.Cube's mixed potential integral equation formulation, the far-zone electric field can be expressed in terms of the asymptotic form of the vector electric and magnetic potentials '''A''' and '''F''':

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

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

== Visualizing The Far Fields ==

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

Figure 1: Planar Module's Radiation Pattern dialog.

Figure 2: 3D polar radiation pattern plot of a microstrip-fed patch antenna.

Figure 2: 3D vectorial (cone) radiation pattern plot of a microstrip-fed patch antenna.

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

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

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

Figure 1: Plana Module's Radar Cross Section dialog.

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

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

Figure 1: Planar Module's Frequency Settings dialog.

'''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.'''

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:

where ['''U'''] is the identity matrix of order N, ['''Z₀'''] and ['''Y₀'''] are diagonal matrices whose diagonal elements are the port characteristic impedances and admittances, respectively, and ['''√Z₀'''] is a diagonal matrix whose diagonal elements are the square roots of port characteristic impedances. The voltage standing wave ratio (VSWR) of the structure at the first port is also computed:

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

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

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

# Magnitude and Phase

# Smith Chart

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

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

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

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

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

Figure 1: Planar Module's Smart Fit dialog.

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

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

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

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

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

All the radiation- and scattering-related standard outputs are available only if you have defined a radiation pattern far field observable or an RCS far field observable, respectively. The standard output parameters DGU and ARU are the directive gain and axial ratio calculated at the certain user defined direction with spherical observation angles (θ, φ). These angles are specified in degrees as '''User Defined Azimuth &amp; Elevation''' in the &quot;Output Settings&quot; section of the '''Radiation Pattern Dialog'''. The standard output parameters HPBWU, SLLU, FNBU and FNLU are determined at a user defined φ-plane cut. This azimuth angle is specified in degrees as '''Non-Principal Phi Plane''' in the &quot;Output Settings&quot; section of the '''Radiation Pattern Dialog''', and its default value is 45°. The standard output parameters BRCS and MRCS are the total back-scatter RCS and the maximum total RCS of your planar structure when it is excited by an incident plane wave source at the specified θ_s and φ_s source angles. FRCS, on the other hand, is the total forward-scatter RCS measured at the predetermined θ_o and φ_o observation angles. These angles are specified in degrees as '''User Defined Azimuth &amp; Elevation''' in the &quot;Output Settings&quot; section of the '''Radar Cross Section Dialog'''. The default values of the user defined azimuth and elevation are both zero corresponding to the zenith.

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

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

== Viewing &amp; Visualizing Various Output Data Types ==

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

Figure 2: Data Manager's spreadsheet showing the contents of an angular data file.

The 2D output data include real or complex quantities like various port, radiation, scattering and periodic characteristics. At the end of an analysis, most .CPX and .DAT data files have a single complex or real value, respectively. in other words, there are no curves to plot. Exceptions are Cartesian 2D radiation pattern or RCS data files along the principal and user define phi-cut planes, as well as polar 2D radiation pattern or RCS data files of angular type with a &quot;'''.ANG'''&quot; file extension. These files contain the radiation pattern or RCS data as a function of some relevant angle in the specified plane. 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). You can plot graphs of the 2D output data files that contain more than one row in '''EM.Grid'''. Each data file has a default graph type. Real data are plotted on EM.Grid's Cartesian graphs. Complex data files with a &quot;'''.CPX'''&quot; extension are plotted on double Cartesian graphs of &quot;'''Magnitude-Phase'''&quot; type, showing the magnitude in dB and phase in radians. You can change the complex data's graph type to the &quot;'''Real-Imaginary'''&quot; or &quot;'''Smith Chart'''&quot; by selecting its entry in the Data Manager and clicking the '''Edit'''button to open the &quot;Edit File Properties Dialog&quot;. Angular data files like polar 2D radiation patterns or RCS, by default, are plotted on EM.Grid's &quot;'''Polar'''&quot; graphs. Note that real data can be graphed on bar charts, too, just as angular can alternatively be graphed on polar stem charts.

Figure 1: A 2D radiation pattern polar graph in EM.Grid.

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

Figure 1: Animating 3D radiation patterns as the send of a frequency sweep.

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 1: EM.Cube's Variables dialog.

Figure 2: Defining a new independent variable.

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

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

Figure 4: Defining the type and range of a sweep variable.

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

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.

Figure 1: EM.Cube's Objectives dialog.

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

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

Figure 3: EM.Cube's Optimization dialog.

The radiation pattern of antenna arrays usually has a main beam and several side lobes. Some parameters of interest in such structures include the '''Half Power Beam Width (HPBW)''', '''Maximum Side Lobe Level (SLL)''' and '''First Null Parameters''' such as first null level and first null beam width. To have EM.Cube calculate all such parameters, you must check the relevant boxes in the &quot;Additional Radiation Characteristics&quot; section of the '''Radiation Pattern Dialog'''. These quantities are saved into ASCII data files of similar names with '''.DAT''' file extensions. In particular, you can plot such data files at the end of a sweep simulation.

Figure 1: Planar Module's Radiation Pattern dialog.

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

Figure 1: Planar Module's Periodicity Settings dialog.

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

Figure 1: Diagram of an equilateral triangular periodic lattice.

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

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

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

where

In the above equations, '''J₀₀(r)''' and '''M₀₀(r)''' are the periodic unit cell's electric and magnetic currents that are repeated everywhere in space on a rectangular lattice with periods S_x and S_y along the X and Y directions, respectively. k_x00 and k_y00 are the periodic propagation constants along the X and Y directions, respectively, and they are given by:

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

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

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

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

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

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

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

At the end of the planar MoM simulation of a periodic structure with plane wave excitation, the reflection and transmission coefficients of the structure are calculated and saved into two complex data files called &quot;reflection.CPX&quot; and &quot;transmission.CPX&quot;. These coefficients behave like the S₁₁ and S₂₁ parameters of a two-port network. You can think of the upper half-space as Port 1 and the lower half-space as Port 2 of this network. '''As a result, you can run an adaptive sweep of periodic structures with a plane wave source just like projects with gap or probe sources.''' The reflection and transmission (R/T) coefficients can be plotted in EM.Grid on 2D graphs similar to the S parameters. You can plot them from the Navigation Tree. To do so, right click on the '''Periodic Characteristics''' item in the '''Observables''' section of the Navigation Tree and select '''Plot Reflection Coefficients''' or '''Plot Transmission Coefficients'''. The complex data files are also listed in EM.Cube's '''Data Manager''', where you can view or plot them. '''Note that in the absence of any finite traces or embedded objects in the project workspace, EM.Cube computes the reflection and transmission coefficients of the layered background structure of your project.'''

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

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

Figure 1: Planar Module's Angle Settings dialog.

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

Figure 1: Planar MoM's NCCBF Settings dialog.

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

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

Figure 1: Assigning gap sources to the elements of a composite array object.