Changes

[[Image:Splash-planar new.jpg|right|720px]]<strong><font color="#015865" size="4">Fast Full-Wave Simulator For Modeling Multilayer Planar Structures</font></strong><table><tr><td>[[image:Cube-icon.png | link=Getting_Started_with_EM.Cube]] [[image:cad-ico.png | link=Building_Geometrical_Constructions_in_CubeCAD]] [[image:fdtd-ico.png | link= An EM.Tempo]] [[image:prop-ico.png | link=EM.Terrano]] [[image:static-ico.png | link=EM.Ferma]] [[image:metal-ico.png | link=EM.Libera]] [[image:po-ico.png | link=EM.Illumina]]</td><tr></table>[[Image:Tutorial_icon.png|30px]] '''[[EM.Cube#EM.Picasso_Documentation | EM.Picasso Primer Tutorial Gateway]]''' [[Image:Back_icon.png|30px]] '''[[EM.Cube | Back to EM.Cube Main Page]]'''==Product Overview==

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

Since its introduction in 2002, [[EM.Picasso assumes that your planar structure ]] has a substrate (background structure) of infinite lateral extents. Your substrate can be a dielectric half-space, or a single conductor-backed dielectric layer (as been successfully used by numerous users around the globe in microstrip components or patch antennas)industry, or simply the unbounded free space, or any arbitrary multilayer stack-up configurationacademia and government. In the special case of It has also undergone several evolutionary cycles including a free space substrate, total reconstruction based on our integrated [[EM.Picasso behaves similar Cube]] software foundation to expand its CAD and geometrical construction capabilities. [[EM.LiberaPicasso]]'s Surface MoM simulator. In all the other cases, it is important to keep in mind the infinite extents of the background substrate structure. For example, you cannot use EM.Picasso to analyze a patch antenna integration with a finite-sized dielectric substrate, if the substrate edge effects are of concern in your modeling problem. [[EM.TempoCube]] is recommended for the modeling facilitates import and export of finite-sized substrates. Since EM.Picasso's Planar MoM simulation engine incorporates the Green's functions many popular CAD formats (including DXF export of the background structure into the analysis, only the finite-sized layered traces like microstrips ) and slots are discretized by the mesh generator. As provides a result, the size of seamless interface with [[EM.PicassoCube]]'s computational problem is normally much smaller compared to the other techniques and solver. In addition, EM.Picasso generates a hybrid rectangular-triangular mesh of your planar structure with a large number of rectangular cells. This results in very fast computation times that oftentimes make up for the limited applications of EM.Picassosimulation tools.

=== An Overview [[Image:Info_icon.png|30px]] Click here to learn more about the '''[[Basic Principles of The Method of Moments | Theory of Planar Method of Moments ===]]'''.

The Method <table><tr><td> [[Image:ART PATCH Fig title.png|thumb|left|480px|3D radiation pattern of Moments (MoM) is a rigorous, fullslot-wave numerical technique for solving open boundary electromagnetic problems. Using this technique, you can analyze electromagnetic radiation, scattering and wave propagation problems coupled patch antenna array with relatively short computation times and modest computing resources. The method of moments is an integral equation technique; it solves the integral form of Maxwell’s equations as opposed to their differential forms that are used in the finite element or finite difference time domain methodsa corporate feed network.]]</td></tr></table>

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

The currents in a planar MoM simulation are discretized as a collection of elementary currents with small finite spatial extents[[EM. These elementary currents are called basis functions and obviously have a vectorial nature. The total currents (solution of Picasso]] is the problem) are summations frequency-domain, full-wave '''Planar Module''' of these elementary currents'''[[EM. The basis functions are well defined and easy to calculate; howeverCube]]''', their amplitudes are initially unknown in a MoM problemcomprehensive, integrated, modular electromagnetic modeling environment. Through [[EM.Picasso]] shares the planar MoM solutionvisual interface, you find these unknown amplitudes3D parametric CAD modeler, data visualization tools, and many more utilities and features collectively known as [[Building_Geometrical_Constructions_in_CubeCAD | CubeCAD]] with all of [[EM. Once the total currents are known, you can calculate the fields everywhere in the structureCube]]'s other computational modules.

[[Image:Info_icon.png|30px]] Click here to learn more about the theory of '''[[Planar Method of MomentsGetting_Started_with_EM.Cube | EM.Cube Modeling Environment]]'''.

== Building a = Advantages & Limitations of EM.Picasso's Planar Structure MoM Simulator ===

[[Image:PMOM11.png|thumb|250px|EM.Picasso's Navigation Tree]] assumes that your planar structure has a substrate (background structure) of infinite lateral extents.In addition, the planar 2.5-D assumption restricts the 3D objects of your physical structure to embedded prismatic objects that can only support vertical currents. These assumptions limit the variety and scope of the applications of [[EM.Picasso]]. For example, you cannot use [[Image:PMOM14EM.png|thumb|400px|A typical planar layered structurePicasso]]to analyze a patch antenna with a finite-sized dielectric substrate. If the substrate edge effects are of concern in your modeling problem, you must use [[EM.CubeTempo]]’s instead. On the other hand, since [[EM.Picasso]]'s Planar ModuleMoM simulation engine incorporates the Green's functions of the background structure into the analysis, only the finite-sized traces like microstrips and slots are discretized by the mesh generator. As a result, the size of [[EM.Picasso]] 's computational problem is intended for constructing and modeling planar layered structuresnormally much smaller than that of [[EM. By Tempo]]. In addition, [[EM.Picasso]] generates a hybrid rectangular-triangular mesh of your planar structure we mean one that contains with a background substrate large number of laterally infinite extents, made up equal-sized rectangular cells. Taking full advantage of one or more material layers all stacked up vertically along the Z axis. Objects symmetry and invariance properties of finite size are then interspersed among these substrate layers. This is somehow different than dyadic Green's functions often results in very fast computation times that easily make up for [[EM.CubePicasso]]'s other computational modules, which are geared for handling arbitrary 3D limited applications. A particularly efficient application of [[EM.Picasso]] is the modeling of periodic multilayer structuresat oblique incidence angles.

In <table><tr><td> [[Planar Module]], the background structure, called &quot;'''Layer Stack-up'''&quot;, may involve one or more material layers of infinite extents along the X and Y axes but of finite thickness along the Z axis. When you start a new project, the background structure has a single vacuum layerImage:ART PATCH Fig12. png|thumb|left|480px|The layer stack-up is always terminated from hybrid planar mesh of the top and bottom by two infinite halfslot-spacescoupled patch antenna array. The terminating half-spaces might be the free space, or a perfect conductor (PEC ground), or any material medium. Most planar structures used in RF and microwave applications such as microstrip-based components have a PEC ground at their bottom. [[EM.Cube]]'s default stack-up has a vacuum top half-space and a PEC bottom half-space. Some structures like stripline components require two bounding PEC grounds at both top and bottom.</td></tr></table>

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

=== Defining Layer Stack<ul> <li> Multilayer stack-Up ===up with unlimited number of substrate layers and trace planes</li> <li> PEC and conductive sheet traces for modeling ideal and non-ideal metallic layouts</li> <li> PMC traces for modeling slot layouts</li> <li> Vertical metal interconnects and embedded dielectric objects</li> <li> Full periodic structure capability with inter-connected unit cells</li> <li> Periodicity offset parameters to model triangular, hexagonal or other offset periodic lattice topologies</li></ul>

When you start a new project in [[EM.Cube]]’s [[Planar Module]]=== Sources, 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 &gt; Computational Domain Loads &gtamp; Layer Stack-up Settings...'''Ports ===

The Stack-up Settings dialog has two tabs: '''Layer Hierarchy''' and '''Embedded Sets'''. The Layer Hierarchy tab has a table that shows all the background layers in hierarchical order from the top half-space to the bottom half<ul> <li> Gap sources on lines</li> <li> De-space. It also lists the material label of each layer, Z-coordinate of the bottom of each layer, its thickness embedded sources on lines for S parameter calculations</li> <li> Probe (in project unitscoaxial feed) and material properties: permittivity (esources on vias<sub/li>r <li> Gap arrays with amplitude distribution and phase progression</subli>), permeability (µ <subli>r Periodic gaps with beam scanning</subli>), electric conductivity (s) <li> Multi-port and magnetic conductivity (scoupled port definitions<sub/li>m <li> RLC lumped elements on strips with series-parallel combinations</subli>). There is also a column that lists the names <li> Short dipole sources</li> <li> Import previously generated wire mesh solution as collection of embedded object sets inside each substrate layer, if anyshort dipoles</li> <li> Plane wave excitation with linear and circular polarizations</li> <li> Multi-ray excitation capability (ray data imported from [[EM.Terrano]] or external files)</li> <li> Huygens sources imported from other [[EM.Cube]] modules</li></ul>

You can delete a layer by selecting its row in the table <ul> <li> Optimized hybrid mesh with rectangular 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.triangular cells</li> <li> Regular triangular surface mesh</li> <li> Local meshing of trace groups</li> <li> Local mesh editing of planar polymesh objects</li> <li> Fast mesh generation of array objects</li></ul>

Figure 1: [[Planar Module]]'s Layer Stack<ul> <li> 2.5-up Settings dialogD mixed potential integral equation (MPIE) formulation of planar layered structures</li> <li> 2.5-D spectral domain integral equation formulation of periodic layered structures</li> <li> Accurate scattering parameter extraction and de-embedding using Prony&#39;s method</li> <li> Plane wave excitation with arbitrary angles of incidence</li> <li> A variety of matrix solvers including LU, BiCG and GMRES</li> <li> Uniform and fast adaptive frequency sweep</li> <li> Parametric sweep with variable object properties or source parameters</li> <li> Generation of reflection and transmission coefficient macromodels</li> <li> Multi-variable and multi-goal optimization of structure</li> <li> Remote simulation capability</li> <li> Both Windows and Linux versions of Planar MoM simulation engine available</li></ul>

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

You can also set the thickness of the substrate layer in the project units<ul> <li> Current distribution intensity plots</li> <li> Near field intensity plots (vectorial - amplitude &amp; phase)</li> <li> Far field radiation patterns: 3D pattern visualization and 2D Cartesian and polar graphs</li> <li> Far field characteristics such as directivity, beam width, axial ratio, side lobe levels and null parameters, etc. Note that you cannot change the thickness </li> <li> Radiation pattern of an arbitrary array configuration of the top planar structure or periodic unit cell</li> <li> Reflection and bottom halfTransmission Coefficients of Periodic Structures</li> <li> Monostatic and bi-static RCS&nbsp;</li> <li> Port characteristics: S/Y/Z parameters, VSWR and Smith chart</li> <li> Touchstone-spaces. You can only change their material propertiesstyle S parameter text files for direct export to RF.Spice or its Device Editor</li> <li> Huygens surface generation</li> <li> Custom output parameters defined as mathematical expressions of standard outputs</li></ul>

[[File:PMOM9== Building a Planar Structure in EM.png]]Picasso ==

Figure 1: [[EM.Picasso]] is intended for construction and modeling of 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. Planar Moduleobjects of finite size are interspersed among these substrate layers. The background structure in [[EM.Picasso]]is called the &quot;'''s Substrate Layer dialogStack-up'''&quot;. The layer stack-up is always terminated from the top and bottom by two infinite half-spaces. The terminating half-spaces might be the free space, or a perfect conductor (PEC ground), or any material medium. Most planar structures used in RF and microwave applications such as microstrip-based components have a PEC ground at their bottom. Some structures like stripline components are sandwiched between two grounds (PEC half-spaces) from both their top and bottom.

You can also use <table><tr><td> [[Image:PMOM11.png|thumb|left|480px|EM.Cube]]Picasso'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 navigation tree and return to the substrate layer dialogtrace types.]]</td></tr></table>

Figure 2: When you start a new project in [[EM.Picasso]], there is always a default background structure that consists of a finite vacuum layer with a thickness of one project unit sandwiched between a vacuum top half-space and a PEC bottom half-space. Every time you open [[EM.Picasso]] or switched to it from [[EM.Cube]]'s Materials other modules, the '''Stack-up Settings Dialog''' opens up. This is where you define the entire background structure. Once you close this dialog, you can open it again by right-clicking the '''Layer Stack-up''' item in the '''Computational Domain''' section of the navigation tree and selecting '''Layer Stack-up Settings...''' from the contextual menu. Or alternatively, you can select the menu item '''Simulate &gt; Computational Domain &gt; Layer Stack-up Settings...'''

<table><tr><td> [[Image:PMOM8(1).png|thumb|550px|EM.Cube]]’s [[Planar ModulePicasso's Layer Stack-up Settings dialog with the initial default values.]] groups objects by their material </td></tr></table> You can add new layers to your project's stack-up or delete its layers, or move layers up or down and electromagnetic propertiesthus change the layer hierarchy. Each object group shares To add a new background layer, click the same color arrow symbol on the {{key|Insert…}} button at the bottom of the dialog and same position select '''Substrate Layer''' from the button's dropdown list. A new dialog opens up where you can enter a label for the new layer and values for its material properties and thickness in project units. You can delete a layer by selecting its row in the table and clicking the '''Delete''' button. To move a layer stack-upand down, click on its row to select and highlight it. All Then click either the planar objects belonging '''Move Up''' or '''Move Down''' buttons consecutively to move the same trace are located on the same substrate selected layer boundary. All the prismatic objects belonging to the same embedded set lie inside desired location in the same stack-up. Note that you cannot delete or move the top or bottom half-spaces. After creating a substrate layer and have , you can always edit its properties in the same material compositionLayer Stack-up Settings dialog. Theoretically speaking, all Click on any layer's row in the objects belonging table to a group are governed by select and highlight it and then click the same boundary conditions{{key|Edit}} button. [[EM.Cube]]’s [[Planar Module]] currently provides The substrate layer dialog opens up, where you can change the following types of objects for building a planar layered structure:layer's label and assigned color as well as its constitutive parameters.

# '''Perfect Electric Conductor (PEC) Traces[[Image:''' These represent infinitesimally thin metallic objects that are deposited or metallized on or between substrate layersInfo_icon. PEC objects are modeled by surface electric currents that satisfy the PEC boundary condition.# '''Perfect Magnetic Conductor (PMC) Traces:''' These are used to model slots and apertures in infinite PEC ground planes. PMC objects are always assumed to lie on an infinite horizontal PEC ground plane with zero thickness. They are modeled by surface magnetic currents, enforcing the continuity png|30px]] Click here for a general discussion of tangential fields across the slots or apertures.# '''Conductive Sheet Traces:''' These represent imperfect metals. They have a finite conductivity and a very small thickness. A surface impedance boundary condition is enforced on the surface of such traces.[[Preparing_Physical_Structures_for_Electromagnetic_Simulation# '''PEC Via Sets:''' These are metallic objects such as shorting pins, interconnect vias, plated-through holes, etcAssigning_Material_Properties_to_the_Physical_Structure | Materials in EM. that are grouped together as prismatic object sets. The embedded objects are modeled as vertical volume conduction currents.# Cube]]'''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.

When you start a new project in For better visualization of your planar structure, [[Planar ModuleEM.Picasso]], the project workspace looks empty, and there are no finite objects displays a virtual domain in it. However, a default orange color to represent part of the infinite background structure . The size of this virtual domain is always assumed to exist by defaulta quarter wavelength offset from the largest bounding box that encompasses all the finite objects in the project workspace. Objects are defined as part You can change the size of traces the virtual domain or embedded sets. Once definedits display color from the Domain Settings dialog, which you can see a list of project objects in access either by clicking the '''Physical StructureComputational Domain''' section [[File:domain_icon.png]] button of the Navigation Tree'''Simulate Toolbar''', or using the keyboard shortcut {{key|Ctrl+A}}. Traces Keep in mind that the virtual domain is only for visualization purposes and object sets can be defined either from Layer Stack-up Settings dialog or from its size does not affect the Navigation TreeMoM simulation. The virtual domain also shows the substrate layers in translucent colors. If you assign different colors to your substrate layers, you have get a better visualization of multilayer virtual domain box surrounding your project structure.

In the ''<table><tr><td> [[Image:PMOM12.png|thumb|550px|EM.Picasso's Layer Stack-up Settings''' dialog, you can add showing a new trace to the stack-up by clicking the arrow symbol on the multilayer substrate configuration.]] </td></tr></table> <table><tr><td> [[Image:PMOM9.png|thumb|280px|EM.Picasso'''Insert''' button of the s Add Substrate Layer dialog. You have to choose from '''Metal (PEC)''', '''Slot (PMC)''' or '''Conductive Sheet''' options]] </td><td> [[Image:PMOM9A. png|thumb|440px|A respective dialog opens upmicrostrip-fed, where you can enter slot-coupled patch antenna on a label and assign double-layer substrate with a color other than default ones. Once a new trace is defined, it is added, by default, to PEC ground plane in the top of middle hosting the stack-up coupling slot.]] </td></tr></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 ModuleEM.Picasso]]'s Stackgroups objects by their trace type and their hierarchical location in the substrate layer stack-up Settings dialog. A trace is a group of finite-sized planar objects that have the same material properties, same color and same Z-coordinate. All the planar objects belonging to the same metal or slot trace group are located on the same horizontal boundary plane in the layer stack-up. All the embedded objects belonging to the same embedded set lie inside the same substrate layer and have same material composition.

Every time you define a new trace, it is also added under the respective category in the Navigation Tree[[EM. Alternatively, you can define a new trace from Picasso]] provides the Navigation Tree by right clicking on one following types 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 objects 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.building a planar layered structure:

{| class="wikitable"|-! scope="col"| Icon! scope= Drawing Planar Objects "col"| Material Type! scope="col"| Applications! scope="col"| Geometric Object Types Allowed|-| style="width:30px;" | [[File:pec_group_icon.png]]| style="width:250px;" | [[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Perfect Electric Conductor (PEC) |Perfect Electric Conductor (PEC) Trace]]| style="width:300px;" | Modeling perfect metal traces on the interface between two substrate layers| style="width:150px;" | Only surface objects|-| style="width:30px;" | [[File:voxel_group_icon.png]]| style="width:250px;" | [[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Conductive Sheet Trace |Conductive Sheet Trace]]| style="width:300px;" | Modeling lossy metal traces with finite conductivity and finite metallization thickness| style="width:150px;" | Only surface objects|-| style="width:30px;" | [[File:pmc_group_icon.png]]| style="width:250px;" | [[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Slot Trace |Slot Trace]]| style="width:300px;" | Modeling cut-out slot traces and apertures on an infinite PEC ground plane | style="width:150px;" | Only surface objects|-| style="width:30px;" | [[File:pec_group_icon.png]]| style="width:250px;" | [[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Embedded PEC Via Set |Embedded PEC Via Set]]| style="width:300px;" | Modeling small and short vertical vias and plated-through holes inside substrate layers| style="width:150px;" | Only surface objects|-| style="width:30px;" | [[File:diel_group_icon.png]]| style="width:250px;" | [[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Embedded Dielectric Object Set |Embedded Dielectric Object Set]]| style="width:300px;" | Modeling small and short dielectric material inserts inside substrate layers| style="width:150px;" | Only surface objects|-| style="width:30px;" | [[File:Virt_group_icon.png]]| style="width:250px;" | [[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Virtual_Object_Group | Virtual Object]]| style="width:300px;" | Used for representing non-physical items | style="width:150px;" | All types of objects|}

As soon as you start drawing geometrical objects Click on each category to learn more details about it in the project workspace, the Physical Structure section [[Glossary of the Navigation Tree gets populatedEM. The names of traces are added under their respective trace type categoryCube's Materials, and the names of objects appear under their respective trace group. At any timeSources, 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 TreeDevices & Other Physical Object Types]].

You can define two types of metallic traces in [[File:PMOM13EM.pngPicasso]]: '''PEC Traces''' and '''Conductive Sheet Traces'''. PEC traces represent infinitesimally thin (zero thickness) planar metal objects that are deposited or metallized on or between substrate layers. PEC objects are modeled by surface electric currents. Conductive sheet traces, on the other hand, represent imperfect metals. They have a finite conductivity and a very small thickness expressed in project units. A surface impedance boundary condition is enforced on the surface of conductive sheet objects.

[[EM.Cube]]'s [[Planar Module]] has a special feature that makes construction of planar structures quite easy and straightforward. ''Slot Traces'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 used to model cut-out slots and apertures in the XY plane (z = 0) by defaultPEC ground planes. In [[Planar Module]], all new slot objects are drawn always assumed to lie on a an infinite horizontal PEC ground plane that with zero thickness, which is located at not explicitly displayed in the Z-coordinate project workspace and its presence is implied. They are modeled by surface magnetic currents. 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 currently active traceslot. As you change In other words, instead of modeling the active trace or add electric surface currents on an infinite PEC ground around the slot, one can alternatively model the finite-extent magnetic surface currents on a new perfect magnetic conductor (PMC) trace, you will also change . Slot (PMC) objects provide the active work electromagnetic coupling between the two sides of an infinite PEC ground plane.

Click here to learn more about Besides planar metal and slot traces, [[Planar Traces & Object TypesEM.Picasso]]allows you to insert prismatic embedded objects inside the substrate layers. The height of such embedded objects is always the same as the height of their host substrate layer. Two types of embedded object sets are available: '''PEC Via Sets''' and '''Embedded Dielectric Sets'''. PEC via sets are metallic objects such as shorting pins, interconnect vias, plated-through holes, etc. all located and grouped together inside the same substrate layer. The embedded via objects are modeled as vertical volume conduction currents. Embedded dielectric sets are prismatic dielectric objects inserted inside a substrate layer. You can define a finite permittivity and conductivity for such objects. The embedded dielectric objects are modeled as vertical volume polarization currents.

=== Managing ObjectsWhen you start a new project in [[EM.Picasso]], the project workspace looks empty, and there are no finite objects in it. However, a default background structure is always present. Finite 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 &amp; Sets ===and object sets can be defined either from Layer Stack-up Settings dialog or from the navigation tree. In the '''Layer Stack-up Settings''' dialog, you can add a new trace to the stack-up by clicking the arrow symbol on the {{key|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. Once a new trace is defined, it is added, by default, to the top of the stack-up table underneath the top half-space. From here, you can move the trace down to the desired location on the layer hierarchy. Every time you define a new trace, it is also added under the respective category in the navigation tree. Alternatively, you can define a new trace from the navigation tree by right-clicking on one of the trace type names and selecting '''Insert New PEC Trace...'''or '''Insert New PMC Trace...'''or '''Insert New Conductive Sheet Trace...''' A respective dialog opens up for setting the trace properties. Once you close this dialog, it takes you directly to the Layer Stack-up Settings dialog so that you can set the right position of the trace on the stack-up.

You Embedded object sets represent short material insertions inside substrate layers. They can manage your project's layer hierarchy from the Layer Stack-up Settings dialogbe metal or dielectric. You Metallic embedded objects can addbe used to model vias, delete and move around substrate layersplated-through holes, metallic shorting pins and slot traces and embedded object interconnects. These are called PEC via sets. Metallic and slot traces Embedded dielectric objects can move among the interface planes between neighboring substrate layersbe used to model air voids, thin films and material inserts in metamaterial structures. Embedded object sets including PEC vias and finite dielectric objects can move from substrate layer into another. When you delete a trace be defined either from the Layer Stack-up Settings dialog, all of its objects are deleted or directly from the project workspace, toonavigation tree. You can also delete metallic and slot traces or Open the &quot;Embedded Sets&quot; tab of the stack-up dialog. This tab has a table that lists all the embedded object sets from along with their material type, the Navigation Treehost substrate layer, the host material and their height. To do soadd a new object set, right click the arrow symbol on the name {{key|Insert}} button of the trace or object set in the Navigation Tree dialog and select one of the two options, '''DeletePEC Via Set''' or '''Embedded Dielectric Set''', from the contextual menudropdown list. This opens up a new dialog where first you have to set the host layer of the new object set. A dropdown list labeled &quot;'''Host Layer'''&quot; gives a list of all the available finite substrate layers. You can also delete all set the traces or properties of the embedded object set, including its label, color and material properties. Keep in mind that you cannot control the height of embedded objects. Moreover, you cannot assign material properties to PEC via sets , while you can set values for the '''Permittivity'''(&epsilon;_r) and '''Electric Conductivity'''(&sigma;) of embedded dielectric sets. Vacuum is the same type default material choice. To define an embedded set from the contextual menu navigation tree, right-click on the '''Embedded Object Sets''' item in the '''Physical Structure''' section of the navigation tree and select either '''Insert New PEC Via Set...''' or '''Insert New Embedded Dielectric Set...''' The respective type category in New Embedded Object Set dialog opens up, where you can set the Navigation Treeproperties of the new object set. As soon as you close this dialog, it takes you to the Layer Stack-up Settings dialog, where you can verify the location of the new object set on the layer hierarchy.

For better visualization of your planar structure, <table><tr><td> [[Image:PMOM23.png|thumb|550px|EM.Cube]] displays a virtual domain in a default orange color to represent part of the infinite background structure. The size of this virtual domain is a quarter wavelength offset from the largest bounding box that encompasses all the finite objects in the project workspace. You can change the size of the virtual domain or its display color from the Domain Settings Picasso's Layer Stack-up dialog, which you can access either by clicking showing the '''Computational Domain''' [[File:domain_iconEmbedded Sets tab.png]] button of the '''Simulate Toolbar''', or by selecting '''Simulate &gt; Computational Domain &gt; Domain Settings...''' from the Simulate Menu or by right clicking the '''Virtual Domain''' item of the Navigation Tree and selecting '''Domain Settings...''' from the contextual menu, or using the keyboard shortcut '''Ctrl+A'''. But keep in mind that the virtual domain is only for visualization purpose and does not affect the MoM simulation. The virtual domain also shows the substrate layers in translucent colors. As you change the colors assigned to the substrate layers, you will see a multilayer virtual domain box surrounding your project structure. </td></tr></table> === Drawing Planar Objects on Horizontal Work Planes ===

[[File:pmom_phys5As 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. The name of the active trace in the navigation tree is always displayed in bold letters. An active trace is where all the new objects you draw belong to. By default, the last defined trace or embedded object set is active. You can immediately start drawing new objects on the active trace. You can also set any trace or object set group active at any time by right-clicking on its name on the navigation tree and selecting '''Activate''' from the contextual menu.png]]

Figure 1[[Image: Info_icon.png|30px]] Click here to learn more about '''[[Planar ModuleBuilding Geometrical Constructions in CubeCAD#Transferring Objects Among Different Groups or Modules | Moving Objects among Different Groups]]'s Virtual Domain Settings dialog''.

By default, the last defined trace or embedded object set is active. You can activate any trace or embedded object set at any time for drawing new objects. You can move one or more selected objects from any trace or embedded object set to another group of the same type or of different type. First select an object in the project workspace or in the Navigation Tree. Then, right click on the highlighted selection and select '''Move To &gt;''' from the contextual menu. This opens another sub-menu containing '''Planar''' and a list of all the other <table><tr><td> [[Image:PMOM23B.png|thumb|280px|EM.Cube]] modules that have already defined object groups. Select Picasso'''Planar''' or any other available module, and yet another sub-menu opens up s Navigation Tree populated 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 planar 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.]] </td></tr></table>

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[[EM. The planar integral equations are then solved approximately on these elementary cells. As this method Picasso]] does not require a discretization of the entire computational domain, it is often computationally much more efficient than differential-based techniques like FEM allow you to draw 3D or FDTD, which mesh the whole domainsolid CAD objects. The accuracy of solid object buttons in 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 '''Object Toolbar''' aredisabled to prevent you from doing so. Low mesh resolutions compromise the accuracy of the numerical solution. On the other handIn order to create vias and embedded object, very high mesh densities may lead you simply have to numerical instability of the method of momentsdraw their cross section geometry using planar surface CAD objects. As a rule of thumb[[EM.Picasso]] extrudes and extends these planar objects across their host layer automatically and displays them as 3D wireframe, a mesh density of about 20-30 cells per effective wavelength usually yields acceptable resultsprismatic objects. Yet, for structures with lots The automatic extrusion of fine geometrical details or for highly resonant structures, higher embedded objects happens after mesh densities may be requiredgeneration and before every planar MoM simulation. Also, You can enforce this extrusion manually by right-clicking the particular simulation data that you seek '''Layer Stack-up''' item in a project will also influence your choice the "Computational Domain" section 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 navigation tree and a rugged boundaryselecting '''Update Planar Structure''' from the contextual menu.

It is well known that any planar geometry with any degree of complexity can be reasonably discretized using a surface triangular mesh. {{Note| In [[EM.CubePicasso]]'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; you can only draw horizontal 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, etcsurface CAD objects.}}

<table><tr><td> [[FileImage:PMOM32PMOM23A.png|800pxthumb|620px|A planar structure with a two-layer conductor-backed substrate, two PEC patches located at the tops of the lower and upper substrate layers, four PEC vias located inside the lower substrate layer between the lower patch and bottom ground and an embedded dielectric film located inside the top substrate layer sandwiched between the two patches.]]</td></tr></table>

Figure 1: Planar hybrid and triangular meshes for rectangular patches=== EM.Picasso's Special Rules ===

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

Rectangular cells offer a major advantage over triangular cells for numerical MoM simulation of planar structures== EM. This is due to the fact that the dyadic GreenPicasso'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:Excitation Sources ==

:<math> Z_{ij}^{(\mu \nu)} = \iiint_{V_i} d\nu f_i^{(\mu)}(r) \cdot \iiint_{V_j}d\nu ' \overline{\overline{G}}_{\mu \nu}(r|r') \cdot f_j^{(v)}(r') </math><!--Your planar structure must be excited by some sort of signal source that induces electric surface currents on metal parts, magnetic surface currents on slot traces, and conduction or polarization volume currents on vertical vias and embedded objects. The excitation source you choose depends on the observables you seek in your project. [[File:PMOM24(1)EM.pngPicasso]]-->provides the following source types for exciting planar structures:

where the spatial{| class="wikitable"|-domain dyadic Green! scope="col"| Icon! scope="col"| Source Type! scope="col"| Applications! scope="col"| Restrictions|-| style="width:30px;" | [[File:gap_src_icon.png]]| [[Glossary of EM.Cube's functions are Materials, Sources, Devices & Other Physical Object Types#Strip Gap Circuit Source |Strip Gap Circuit Source]]| style="width:300px;" | General-purpose point voltage source (or filament current source on slot traces)| style="width:300px;" | Associated with a function PEC rectangle strip|-| style="width:30px;" | [[File:probe_src_icon.png]]| [[Glossary of the observation and source coordinates, '''r'''and '''r' ''EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Probe Gap Circuit Source |Probe Gap Circuit Source]]| style="width:300px;" | General-purpose voltage source for modeling coaxial feeds| style="width:300px;" | Associated with an embedded PEC via set|-| style="width:30px;" | [[File:waveport_src_icon. The MoM matrix elements can indeed be interpreted as interactions between two elementary basis functions png]]| [[Glossary of EM.Cube'''f_i(r)''' and '''f_j(r')''' on that particular background structures Materials, Sources, Devices & Other Physical Object Types#Scattering Wave Port |Scattering Wave Port Source]]| style="width:300px;" | Used for S-parameter computations| style="width:300px;" | Associated with an open-ended PEC rectangle strip, extends long from the open end|-| style="width:30px;" | [[File:hertz_src_icon. The spatialpng]]| [[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Hertzian Short Dipole Source |Hertzian Short Dipole Source]]| style="width:300px;" | Almost omni-domain dyadic Greendirectional physical radiator| style="width:300px;" | None, stand-alone source|-| style="width:30px;" | [[File:plane_wave_icon.png]]| [[Glossary of EM.Cube's functions can themselves be expressed in terms Materials, Sources, Devices & Other Physical Object Types#Plane Wave |Plane Wave Source]]| style="width:300px;" | Used for modeling scattering & computation of the spectralreflection/transmission characteristics of periodic surfaces| style="width:300px;" | None, stand-domain dyadic Greenalone source|-| style="width:30px;" | [[File:huyg_src_icon.png]]| [[Glossary of EM.Cube's functions as followsMaterials, Sources, Devices & Other Physical Object Types#Huygens Source |Huygens Source]]| style="width:300px;" | Used for modeling equivalent sources imported from other [[EM.Cube]] modules | style="width:300px;" | Imported from a Huygens surface data file|}

:<math> \overline{\overline{G}}_{\mu \nu}(r|r') = \frac{1}{(2\pi)^2} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} \tilde{\overline{\overline{G}}}_{\mu \nu} (k_p, z|z') e^{-jClick on each category to learn more details about it in the [[k_x(x-xGlossary of EM.Cube')+k_y(y-y')]} \s Materials, dk_x \Sources, dk_y , \quad {k_p}^2 = {k_x}^2 + {k_y}^2 </math><!--[[File:PMOM26.pngDevices & Other Physical Object Types]]-->.

For antennas and planar circuits, where the doubly infinite integration is performed with respect to the spectral you typically define one or more ports, you usually use lumped sources. [[variablesEM.Picasso]] k_x provides three types of lumped sources: gap source, probe source and k_yde-embedded source. As can be seen from A lumped source is indeed a gap discontinuity that is placed on the above expressionpath of an electric or magnetic current flow, the spatial-domain dyadic Green's functions where a voltage or current source is connected to inject a signal. Gap sources are functions placed across metal or slot traces. A rectangle strip object on a PEC or conductive sheet trace acts like a strip transmission line that carries electric currents along its length (local X direction). The characteristic impedance of z, z', as well as the line is a function of its width (x-x'local Y direction) . A gap source placed on a narrow metal strip creates a uniform electric field across the gap and pumps electric current into the line. A rectangle strip object on a slot trace acts like a slot transmission line on an infinite PEC ground plane that carries a magnetic current along its length (y-y'local X direction). The MoM matrix elements can now be transformed characteristic impedance of the slot line is a function of its width (local Y direction). A gap source placed on a narrow slot represents an ideal current source. A slot gap acts like an ideal current filament, which creates electric fields across the slot, equivalent to a magnetic current flowing into the spectral domain asslot line. 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.

:<math> Z_{ij}^{(\mu \nu)} = \dfrac{1}{(2\pi)^2} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} \tilde{f}_i^{(\mu)} (k_x, k_y) \cdot \tilde{\overline{\overline{G}}}_{\mu \nu} (k_{\rho}, zNote|z') \cdot \tilde{f}_j^{You can realize a coplanar waveguide (\nuCPW)} (k_x, k_y) \, dk_x \, dk_y </math><!--in [[File:PMOM27EM.pngPicasso]]-->using two parallel slot lines with two aligned, collocated gap sources.}}

:<math> \tilde{f}(k_xA short dipole provides another way of exciting a planar structure in [[EM.Picasso]]. A short dipole source acts like an infinitesimally small ideal current source. You can also use an incident plane wave to excite your planar structure in [[EM.Picasso]]. In particular, k_y) you need a plane wave source to compute the radar cross section of a planar structure. The direction of incidence is defined by the θ and φ angles of the unit propagation vector in the spherical coordinate system. The default values of the incidence angles are θ = 180° and φ = \dfrac{1}{(2\pi)^2} \int\limits_{0° corresponding to a normally incident plane wave propagating along the -\infty}^{\infty} \int\limits_{-\infty}^{\infty} f(x,y) e^{j(k_x x Z direction with a + k_y y)} \, dx \, dy </math><!X-polarized E-vector. Huygens sources are virtual equivalent sources that capture the radiated electric and magnetic fields from another structure that was previously analyzed in another [[File:PMOM28(1)EM.pngCube]]-->computational module.

Rectangular cells have simple Fourier transforms<table><tr><td> [[Image:PMOM64A. 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 png|thumb|550px|A multilayer planar structure containing a product of CPW line with a sinc-squared function along one spectral direction single coupled port and a sinc function along the otherlumped element on an overpassing metal strip. 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:]] </td></tr></table>

:<math> Z_{(i,k)|(j,l)} = Z \Big\langle f_{i,k}(x,y)| f_{j,l}(x', y') \Big\rangle = Z_{(i-j)|(k-l)} </math><!--[[File:PMOM29= Modeling Lumped Elements in EM.png]]-->Picasso ===

In the above equationLumped elements are components, the vectorial rooftop basis functions have explicitdevices, double indices: i and k along or circuits whose overall dimensions are very small compared to the local X and Y directionswavelength. As a result, respectively, for they are considered to be dimensionless compared to the test (observation) basis function, and j and l along the local X and Y directions, respectively, for the expansion (source) basis functiondimensions of a mesh cell. Thus, uniform rectangular cellsIn fact, i.e. structured rectangular cells of identical size aligned a lumped element is equivalent to an infinitesimally narrow gap that is placed in the same directionpath of current flow, can speed up across which the planar MoM simulation significantly due to device's governing equations are enforced. Using Kirkhoff's laws, these symmetry device equations normally establish a relationship between the currents and voltages across the invariance propertiesdevice or circuit. For exampleCrossing the bridge to Maxwell's domain, all the self-interactions are identical regardless of the location of device equations must now be cast into a rooftop basis function. This reduces the matrix fill process for a total of N rooftop basis functions from an N2 process o boundary conditions that relate the electric and magnetic currents and fields. [[EM.Picasso]] allows you to one define passive circuit elements: '''Resistors''' (R), '''Capacitors''' (C), '''Inductors''' (L), and series and parallel combinations of order Nthem.

[[FileImage:PMOM25Info_icon.png|40px]]Click here to learn more about '''[[Preparing_Physical_Structures_for_Electromagnetic_Simulation#Modeling_Lumped_Elements_in_the_MoM_Solvers | Defining Lumped Elements]]'''.

Figure 1[[Image: Pairs Info_icon.png|40px]] Click here for a general discussion of rooftop basis functions that have identical MoM interactions'''[[Preparing_Physical_Structures_for_Electromagnetic_Simulation#A_Review_of_Linear_.26_Nonlinear_Passive_.26_Active_Devices | Linear Passive Devices]]'''.

=== Generating A {{Note|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 Mesh ===MoM engine will change.}}

# Setting The calculation of the mesh properties# Creating 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 viewing probes and even active lumped elements to calculate the mesh# Verifying circuit characteristics of planar structures. The admittance / impedance calculations based on the mesh 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 integritycalculating the port parameters.

[[EM.Cube]]’s [[Planar Module]] offers two mesh generation algorithms for discretizing One can calculate the scattering parameters of a planar structures: Hybrid and Triangularstructure directly by analyzing the current distribution patterns on the port transmission lines. The hybrid mesh consists discontinuity at the end 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 port line typically gives rise to discretize rectangular objects with rectangular cells as much as possible. In certain connection areas, a few triangular cells might standing wave pattern that can clearly be inserted to provide discerned in the mesh transition for line's current continuitydistribution. All From the non-rectangular objects (circularlocation of the current minima and maxima and their relative levels, polygonalone can determine the reflection coefficient at the discontinuity, etci.) are discretized using triangular cellse. The triangular mesh generator, on the other hand, discretizes the planar objects with all triangular cells regardless of their shapeS₁₁ parameter. The only exceptions are feed lines that contain gap sources or lumped elementsA more robust technique is Prony’s method, which are always meshed with rectangular cellsis used for exponential approximation of functions.A complex function f(x) can be expanded as a sum of complex exponentials in the following form:

You can generate and view a planar mesh by clicking the '''Show Mesh''' [[File:mesh_tool.png]] button of the '''Simulate Toolbar''' or by selecting '''Menu &gt; Simulate &gt; Discretization &gt; Show Mesh''' or using the keyboard shortcut '''Ctrl+M'''. When the mesh of the planar structure is displayed in [[EM.Cube]]’s project workspace, its &quot;Mesh View&quot; mode is enabled. In this mode you can perform view operations like rotate view, pan or zoom, but you cannot create new objects or edit existing ones. To exit the mesh view mode, press the keyboard's '''Esc Key''' or click the '''Show Mesh''' [[File:mesh_tool.png]] button once again. Once a mesh is generated, it stays in the memory until the structure is changed or the mesh density or other settings are modified. Every time you view mesh, the one in the memory is displayed. You can force [[EM.Cube]] to create a new mesh from the ground up by selecting '''Menu &gt; Simulate &gt; Discretization &gt; Regenerate Mesh''' or by right clicking on the '''Planar Mesh''' item in the '''Discretization''' section of the Navigation Tree and selecting '''Regenerate''' from the contextual menu. === Planar Mesh Density === [[EM.Cube]]'s [[Planar Module]], by default, generates a hybrid mesh of your planar structure with a mesh density of 20 cells per effective wavelength. It is important to understand the concept of mesh density (either hybrid or triangular) as used by [[Planar Module]]. It gives a measure of the number of cells per effective wavelength that are placed in various regions of your planar structure. The higher the mesh density, the more cells are created on the geometrical objects. Keep in mind that only the finite-sized objects of your structure are discretized. No mesh is generated for the substrate layers of your background structure. The free-space wavelength is defined as <math>f(x) \lambda_0 = approx \tfracsum_{2\pi f}{c}</math>, where f is the center frequency of your project and c is the speed of light in the free space. The effective wavelength is defined as <math>\lambda_{eff} n= \tfrac{\lambda_01}^N c_i e^{-j\sqrt{\varepsilon_{eff}}gamma_i x}</math>, where e_eff is the effective permittivity. The effective permittivity is defined differently for different types of traces and embedded object sets. For metal and conductive sheet traces, the effective permittivity is defined as the larger of the permittivity of the two substrate layers just above and below the metallic trace. For slot traces, the effective permittivity is defined as the mean (average) of the permittivity of the two substrate layers just above and below the metallic trace. These definitions of effective permittivity are consistent with the effective propagation constant of [[Transmission Lines|transmission lines]] realized on such trace types. For embedded object sets, the effective permittivity is defined as the largest of the permittivities of all the substrate layers and embedded dielectric sets. In all cases, for the purpose of calculating the effective wavelength, only the real part of the permittivities are considered. The reason for using an effective wavelength so defined for determination of mesh resolution is to make sure that enough cells are placed in areas that might feature higher field concentration. 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. [[File:PMOM30.png|800px]] Mesh of two rectangular patches at two different planes. The lower substrate layer has a higher permittivity. === Customizing A Planar Mesh === You can change the settings of the planar mesh including the mesh type and density from the planar Mesh Settings Dialog. You can also change these settings while in the mesh view mode, and you can update the changes to view the new mesh. To open the mesh settings dialog, either click the '''Mesh Settings''' [[File:mesh_settings.png]] button of the '''Simulate Toolbar''' or select '''Menu &gt; Simulate &gt; Discretization &gt; Mesh Settings...''', or by right click on the '''Planar Mesh''' item in the '''Discretization''' section of the Navigation Tree and select '''Mesh Settings...''' from the contextual menu, or use the keyboard shortcut '''Ctrl+G'''. You can change the mesh algorithm from the dropdown list labeled '''Mesh Type''', which offers two options: '''Hybrid''' and '''Triangular'''. You can also enter a different value for '''Mesh Density''' in cells per effective wavelength (&lambda;_eff). For each value of mesh density, the dialog also shows the average &quot;Cell Edge Length&quot; in the free space. To get an idea of the size of mesh cells on the traces and embedded object sets, divide this edge length by the square root of the effective permittivity a particular trace or set. Click the '''Apply''' button to make the changes effective. [[File:PMOM31.png]] The Planar Mesh Settings dialog. === Mesh Of Connected Trace Objects === Using the generated mesh of a planar structure, [[EM.Cube]] creates a set of vectorial basis functions that are passed to the input file of the Planar MoM simulation engine. This engine requires edge!-based basis functions. The common edges between adjacent cells are used to define edge-based rooftop or RWG basis functions. These elementary basis functions indeed provide the current flow and warrant the continuity among the mesh cells. Therefore, when two objects overlap or share a common edge, the connection between them must be translated into &quot;bridge&quot; basis functions, which carry the information about current flow to the simulation engine. '''The most important rule of object connections in [[EM.Cube]]'s [[Planar Module]] is that only objects belonging to the same trace can be connected to one another.''' For example, if two objects reside on the same Z-plane and geometrically have a common edge which you can clearly see in the project workspace, but organizationally they belong to two different metal traces, then the bridge basis functions will not be generated between them, and the simulation engine will see them disconnected. If two objects belong to the same trace and have a common overlap area, [[EM.Cube]] first merges the two objects using the &quot;Boolean Union&quot; operation and converts them into a single object for the purpose of meshing. The mesh of &quot;unioned&quot; areas is usually made up of triangular cells. If two objects reside on the same Z-plane and geometrically overlap with each other but organizationally belong to two different trace groups, incongruous, overlapped cells will be generated that will either blow up the linear system or produce completely wrong simulation results. [[File:PMOM36PMOM73.png|250px]] [[File:PMOM38.png|250px]] [[File:PMOM37.png|250px]] Figure 1: Two overlapping planar objects and their triangular and hybrid planar meshes. When two planar objects belonging to the same trace are connected via a common edge, it is critical to generate a consistent mesh at the connection area and properly transition and merge the meshes of the individual objects. [[EM.Cube]]'s triangular planar mesh generator simply &quot;unions&quot; the two objects and generates a connected mesh. [[EM.Cube]]'s hybrid planar mesh generator, however, behave differently when it comes to the connection between rectangular objects. The rule in this case is the following: * If the two connected rectangular objects have the same side dimensions along the common linear edge with perfect alignment, a rectangular bridge mesh is produced.* If the two connected rectangular objects have different side dimensions along the common linear edge or have edge offset, a set of triangular cells is generated along the edge of the object with the large side.* Rectangular objects that contain gap source or lumped elements, always have a rectangular mesh around the gap area. [[File:PMOM33.png|250px]] [[File:PMOM35.png|250px]] [[File:PMOM34.png|250px]] Figure 2: Edge-connected rectangular planar objects and their triangular and hybrid planar meshes. === Mesh of Embedded Objects === [[EM.Cube]]'s [[Planar Module]] models embedded objects as vertical volume currents. The vectorial basis functions in this case are Z-directed prisms as opposed to rooftop basis functions. If an embedded object is located under or above a metallic trace or connected from both top and bottom, it is critical to create mesh continuity between the embedded object and its connected metallic traces. In other words, the generated mesh must ensure current continuity between the vertical volume currents and horizontal surface currents. [[EM.Cube]]’s planar mesh generator automatically handles situations of this kind and generates all the required connection meshes. Keep in mind that [[EM.Cube]]’s Planar MoM engine uses a 2.5-D approximation, whereby only vertical volume currents are assumed inside embedded objects. When the height of an embedded object is small (as should typically be under the 2.5-D assumption), one prismatic cell is placed across the object along the Z-axis. Long PEC vias with a very small radius do also satisfy the 2.5-D assumption. In this case, the long via objects are discretized further along the Z direction and generate multiple stacked cells. Several prismatic cells along the Z-axis may increase the simulation time drastically. This is due to the fact that the host layer is effectively subdivided into a number of sub-layers and the stacked cells are treated as stacked vias embedded inside these sub-layers. As a result, the simulation engine needs to compute all the dyadic Green’s functions accounting for the interactions between all such sub-layers. [[File:PMOM39.png|400px]] [[File:PMOM40.png|400px]] 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. === Refining Mesh At Discontinuities === 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. 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. [[File:PMOM41.png|800px]] Refining the planar mesh at the via and surrounding area. === Checking Mesh Integrity === You should always visually inspect [[EM.Cube]]'s default generated mesh to see if the current mesh settings have produced an acceptable mesh. You may often need to change the mesh density or other [[parameters]] and regenerate the mesh. The Planar Mesh Settings dialog gives a few more options for customizing your planar mesh. As mentioned earlier, highly incongruous meshes should always be avoided. Sometimes [[EM.Cube]]'s default mesh may contain very narrow triangular cells due to very small angles between two edges. In some rare cases, extremely small triangular cells may be generated, whose area is a small fraction of the average mesh cell. These cases typically happen at the junctions and other discontinuity regions or at the boundary of highly irregular geometries with extremely fine details. In such cases, increasing or decreasing the mesh density by one or few cells per effective wavelength often resolves that problem and eliminates those defective cells. Nonetheless, [[EM.Cube]]'s planar mesh generator offers an option to identify the defective triangular cells and either delete them or cure them. By curing we mean removing a narrow triangular cell and merging its two closely spaced nodes to fill the crack left behind. [[File:PMOM44.png|400px]] [[File:PMOM42.png|400px]] Deleting or curing defective triangular cells. [[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. [[File:PMOM43(1).png]] Setting the minimum allowable angle for non-defective triangular cells. === Locking Mesh Of Object Groups === [[EM.Cube]]'s [[Planar Module]] provides different ways of controlling the mesh of a planar structure locally. Earlier you saw how to increase the mesh resolution at the discontinuity regions without affecting the mesh of uniform or regular areas of a planar structure. Another way of local mesh control is to lock the mesh density of certain traces or object sets. The mesh density that you specify in the Planar Mesh Settings dialog is a global parameter and applies to all the traces and embedded object sets in your project. However, you can lock the mesh of individual PEC, PMC and conductive sheet traces or embedded objects sets. In that case, the locked mesh density takes precedence over the global density. Note that locking mesh of object groups, in principle, is different than refining the mesh at discontinuities. In the latter case, the mesh of connection areas is affected. However, objects belonging to different traces cannot be connected to one another. Therefore, locking mesh can be useful primarily for isolated object groups that may require a higher (or lower) mesh resolution. You can lock the local mesh density by accessing the property dialog of a specific trace or embedded object set and checking the box labeled '''Lock Mesh'''. This will enable the '''Mesh Density''' box, where you can accept the default global value or set any desired new value. [[File:PMOM45.png]] Figure 1: Locking the mesh density of an object group from its property dialog. === Local Mesh Control Using Polymesh Objects === [[EM.Cube]] allows you to manually and individually mesh geometrical objects using the concept of polymesh. The Polymesh tool converts a planar surface object to a set of interconnected triangular cells, which is basically identical to its triangular surface mesh. Simply select an object and click the '''Polymesh Tool''' [[File:polymesh_tool_tn.png]] button of '''Tools Toolbar''', or select '''Menu &gt; Tools &gt; Polymesh''', or use the keyboard shortcut '''P'''. You can also right click on a selected object and select '''Polymesh''' from the contextual menu. From the Polymesh Dialog, you can control the mesh resolution through the '''Edge Length''' parameter, which is expressed in project units. Note that unlike the planar mesh generator which uses a frequency-dependent mesh density to drive the mesh resolution, the ploymesh's edge length is fixed and purely geometrical and does not change with the project frequency. '''[[EM.Cube]]'s mesh generator considers a polymesh object as a &quot;final&quot; mesh and reproduces it &quot;As Is&quot; during the meshing process.''' 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. [[File:PMOM46(1).png]] Figure 2: Discretizing a planar surface object using [[EM.Cube]]'s Polymesh tool. Keep in mind that since a polymesh object it considered a final mesh, its mesh cannot be connected to other objects. In other words, bridge basis functions are not generated if even some of the polymesh edges may coincide with other objects' edges. A polymesh object is treated by the mesh generator as an isolated mesh. However, [[EM.Cube]] allows you to connect polymesh objects manually. To do so, bring two or more polymesh objects close to each other so that they have one or more common edges. No face overlaps are allowed in this case. Select the polymesh objects and click the '''Merge Tool'''[[File:merge_tool_tn.png]] button of '''Tools Toolbar''' to merge the polymesh objects into a single polymesh object. The new merged polymesh object will provide all the necessary bridge basis functions among the original, separate polymesh objects. == Excitation Sources == 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. You can excite a planar structure in a number of different ways. The excitation source you choose depends on the observables you seek in your project. [[Planar Module]] provides the following source for exciting planar structures: * [[#Gap Sources|Gap Sources]]* [[#Probe Sources|Probe Sources]]* [[#De-embedded Sources|De-embedded Sources]]* [[#Plane Wave Sources|Plane Wave Sources]]* [[#Short Dipole Sources|Short Dipole Sources]]* [[#Huygens Sources|Huygens Sources]] For antennas and planar circuits, where you typically define one or more ports, you usually use lumped sources. A lumped source is indeed a gap discontinuity that is placed on the path of an electric or magnetic current flow, where a voltage or current source is connected to inject a signal. Gap sources are placed across metal or slot traces. Probe sources are placed across vertical PEC vias. A de-embedded source is a special type of gap source that is placed near the open end of an elongated metal or slot trace to create a standing wave pattern, from which the scattering [[parameters]] can be calculated accurately. To calculate the scattering characteristics of a planar structure, e.g. its radar cross section (RCS), you excite it with a plane wave source. Short dipole sources are used to explore propagation of points sources along a layered structure. Huygens sources are virtual equivalent sources that capture the radiated electric and magnetic fields from another structure possibly in another [[EM.Cube]] computational module and bring them as a new source to excite your planar structure. Click here to learn more about [[Planar MoM Source Types]]. === Defining Source Arrays === If the project workspace contains an array of rectangle strip objects, the array object will also be listed as an eligible object for gap source placement. A gap source will then be placed on each element of the array. All the gap sources will have identical direction and offset. Similarly, if the project workspace contains an array of PEC via objects, the embedded array object will also be listed as an eligible object for probe source placement. A probe source will then be placed on each via object of the array. All the probe sources will have identical direction and offset. However, you can prescribe certain amplitude and/or phase distribution over the array of gap or probe sources. By default, all the gap or probe sources have identical amplitudes of 1V (or 1A for the slot case) and zero phase. The available amplitude distributions to choose from include '''Uniform''', '''Binomial''' and '''Chebyshev''' and '''Date File'''. In the Chebyshev case, you need to set a value for minimum side lobe level ('''SLL''') in dB. You can also define '''Phase Progression''' in degrees along all three principal axes. You can view the amplitude and phase of individual sources by right clicking on the top '''Sources''' item in the Navigation Tree and selecting '''Show Source Label''' from the contextual menu. [[File:PMOM49.png|800px]] Figure 1: Defining gap sources on an array of rectangle strip objects with a Chebyshev amplitude distribution.>

In the data file option, the where c_i are complex amplitude coefficients and &gamma;_i are directly read , in from a data file using a real - imaginary format. When this option is selectedgeneral, you can either improvise the complex array weights or import them from an existing fileexponents. In From the former case click the '''New Data File''' button. This opens up the [[Windows]] Notepad with default formatted data file physics of transmission lines, we know that has a list of all the array element indices lossless lines may support one or more propagating modes with default 1+j0 amplitudes for all of thempure real propagation constants (real &gamma;_i exponents). You can replace the default complex values Moreover, line discontinuities generate evanescent modes with new one and save pure imaginary propagation constants (imaginary &gamma;_i exponents) that decay along the Notepad data file, which brings line as 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 move away from the one location 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 usesuch discontinuities.

[[File:PMOM50In 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.png|800px]]

Figure 2<table><tr><td> [[Image: Defining gap source array weights using PMOM71.png|thumb|600px|Minimum and maximum current locations of the standing wave pattern on a microstrip line feeding a data filepatch antenna.]] </td></tr></table>

=== Defining Independent & Coupled Ports ===

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

Ports are defined in the '''Observables''' section of the Navigation Treenavigation tree. Right click on the '''Port Definition''' item You can define any number of ports equal to or less than the Navigation Tree and select '''Insert New Port Definition...''' from the contextual menu. The Port Definition Dialog opens up, showing the default port assignmentstotal number of sources in your project. 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 Treenavigation tree. Note that your project can have mixed gap and probes sources as well as active lumped element sourceson PEC and slot traces or vias. You can also couple ports together to define coupled transmission lines such as coupled strips (CPS) or coplanar waveguides (CPW).

[[FileImage:PMOM52Info_icon.png|40px]]Click here to learn more about the '''[[Glossary_of_EM.Cube%27s_Simulation_Observables_%26_Graph_Types#Port_Definition_Observable | Port Definition Observable]]'''.

Figure 1[[Image: The Port Definition dialogInfo_icon.png|40px]] Click here to learn more about '''[[Preparing_Physical_Structures_for_Electromagnetic_Simulation#Modeling_Coupled_Sources_.26_Ports | Modeling Coupled Ports]]'''.

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

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 [[File:PMOM53EM.pngPicasso]]can be categorized into the following groups:

Figure 2{| class="wikitable"|-! scope="col"| Icon! scope="col"| Simulation Data Type! scope="col"| Observable Type! scope="col"| Applications! scope="col"| Restrictions|-| style="width: Edit 30px;" | [[File:currdistr_icon.png]]| style="width:150px;" | Current Distribution Maps| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Current Distribution |Current Distribution]]| style="width:300px;" | Computing electric surface current distribution on metal traces and magnetic surface current distribution on slot traces | style="width:250px;" | None|-| style="width:30px;" | [[File:fieldsensor_icon.png]]| style="width:150px;" | Near-Field Distribution Maps| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Near-Field Sensor |Near-Field Sensor]] | style="width:300px;" | Computing electric and magnetic field components on a specified plane in the frequency domain| style="width:250px;" | None|-| style="width:30px;" | [[File:farfield_icon.png]]| style="width:150px;" | Far-Field Radiation Characteristics| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Far-Field Radiation Pattern |Far-Field Radiation Pattern]]| style="width:300px;" | Computing the radiation pattern and additional radiation characteristics such as directivity, axial ratio, side lobe levels, etc. | style="width:250px;" | None|-| style="width:30px;" | [[File:rcs_icon.png]]| style="width:150px;" | Far-Field Scattering Characteristics| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Radar Cross Section (RCS) |Radar Cross Section (RCS)]] | style="width:300px;" | Computing the bistatic and monostatic RCS of a target| style="width:250px;" | Requires a plane wave source|-| style="width:30px;" | [[File:port_icon.png]]| style="width:150px;" | Port dialogCharacteristics| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Port Definition |Port Definition]] | style="width:300px;" | Computing the S/Y/Z parameters and voltage standing wave ratio (VSWR)| style="width:250px;" | Requires one of these source types: lumped, distributed, microstrip, CPW, coaxial or waveguide port|-| style="width:30px;" | [[File:period_icon.png]]| style="width:150px;" | Periodic Characteristics| style="width:150px;" | No observable required | style="width:300px;" | Computing the reflection and transmission coefficients of a periodic surface| style="width:250px;" | Requires a plane wave source and periodic boundary conditions |-| style="width:30px;" | [[File:huyg_surf_icon.png]]| style="width:150px;" | Equivalent electric and magnetic surface current data| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Huygens Surface |Huygens Surface]]| style="width:300px;" | Collecting tangential field data on a box to be used later as a Huygens source in other [[EM.Cube]] modules| style="width:250px;" | None|}

Sources can be coupled to each other to model coupled strip lines (CPS) on metal traces If your planar structure is excited by gap sources or coplanar waveguides (CPW) on slot traces. Similarly, probe sources may be coupled to each other. Coupling two or more de-embedded sources does not change , and one or more ports have been defined, the way they excite a planar structure. It is intended only for MoM engine calculates the purpose of scattering, impedance and admittance (S parameter calculation/Z/Y) parameters of the designated ports. The feed lines or vias which host the coupled sources scattering parameters are usually parallel and aligned with one another and they are all grouped together as a single transmission line represented by a single defined based on the portimpedances specified in the project's Port Definition dialog. This single &quot;coupled&quot; If more than one port then interacts with other coupled or uncoupled portshas been defined in the project, the S/Z/Y matrices of the multiport network are calculated.

You couple two or more sources using the '''Port Definition Dialog'''. To do so, you need to change the default port assignments. First, delete all the ports that Electric and magnetic currents are to be coupled from the Port List fundamental output data of the dialoga planar MoM simulation. Then, define a new port by clicking After the '''Add''' button numerical solution of the dialog. This opens up the Add Port dialogMoM linear system, which consists of two tables: '''Available''' sources on they are found using the left and solution vector '''Associated[I]''' sources on the right. A right arrow ('''--&gt;''') button and a left arrow ('''&lt;--''') button let you move the sources freely between these two tables. You will see in the &quot;Available&quot; table a list definitions of all the sources that you deleted earlier. You may even see more available sources. Select all the sources that you want to couple electric and move them to the &quot;Associated&quot; table on the right. You can make multiple selections using the keyboard's '''Shift''' and '''Ctrl''' keys. Closing the Add Port dialog returns you to the Port Definition dialog, where you will now see the names of all the coupled sources next to the name of the newly added port.magnetic vectorial basis functions:

:<math> \mathbf{{Note|It is your responsibility to set up coupled ports and coupled [[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|[[Transmission Lines|transmission linesX]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]]] properly. For example, to excite the desirable odd mode of a coplanar waveguide }_{N\times 1} = \begin{bmatrix} I^{(CPWJ), 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.}\\ \\ V^{(M)} \end{bmatrix} \quad \Rightarrow \quad \begin{cases} \mathbf{J(r)} = \sum_{n=1}^N I_n^{(J)} \mathbf{f_n^{(J)} (r)} \\ \\ \mathbf{M(r)} = \sum_{k=1}^K V_k^{(M)} \mathbf{f_k^{(M)} (r)} \end{cases}</math>

[[File:PMOM51Note 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 (2PEC)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. At the top of the Current Distribution 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.png|800px]]

Figure 1: Coupling gap sources in the Port Definition dialog by associating more than one source with {{Note|You have to define a single portseparate current distribution observable for each individual trace or embedded object set.}}

A gap source on a metal trace and a probe source on [[EM.Picasso]] allows you to visualize the near fields at a PEC via behave like a series voltage source with a prescribed strength (of 1V and zero phase by default) specific field sensor plane. Note that creates unlike [[EM.Cube]]'s other computational modules, near field calculations in [[EM.Picasso]] usually takes a localized discontinuity on the path significant amount of electric current flowtime. At This is due to the fact that at the end of a planar MoM simulation, the electric current passing through the voltage source is computed fields are not available anywhere (as opposed to [[EM.Tempo]]), and integrated their computation requires integration of complex dyadic Green's functions of a multilayer background structure as opposed to find the total input currentfree space Green's functions. From this one can calculate the input admittance as

:<math> Y_{in} = \frac{I_{Note|Keep in}}{V_s} = \frac {\int_W \hat{y} \cdot \mathbf{J_s} \, dy} {V_s} </math><!--mind that since [[File:PMOM54EM.Picasso]] uses a planar MoM solver, the calculated field value at the source point is infinite. As a result, the field sensors must be placed at adequate distances (1at least one or few wavelengths)away from the scatterers to produce acceptable results.png]]-->}}

:<math> Y_{in} = \frac{I_{in}}{V_s} = \frac {\int_S \hat{z} \cdot \mathbf{J_p} \Even though [[EM.Picasso]]'s MoM engine does not need a radiation box, ds} {V_s} </math><!--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 [[File:PMOM55EM.pngCube]]-->to perform these calculations. 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 &theta; direction, the far field component in &phi; direction and the &quot;Total&quot; far field. The 2D radiation pattern graphs can be plotted from the '''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.

for probe sources on PEC vias, where the surface integration is performed over the cross section of the via[[Image:Info_icon. On png|30px]] Click here to learn more about the other hand, a gap source on a slot trace behaves like a shunt current source with a prescribed strength (theory 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 '''[[Defining_Project_Observables_%26_Visualizing_Output_Data#Using_Array_Factor_to_Model_Antenna_Arrays | Using Array Factors to find the total input voltage across the current filamentModel Antenna Arrays ]]'''. From this one can calculate the input impedance as

:<mathtable> Z_{in} = \frac{V_{in}}{I_s} = \frac{\int_W \hat{y} \cdot \mathbf{M_s} \,dy} {V_s} = \frac{\int_W E_y \, dy}{V_s} </mathtr><!--td> [[FileImage:PMOM56PMOM119.png|thumb|left|600px|3D polar radiation pattern plot of a microstrip-fed patch antenna.]]--</td></tr></table>

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.Picasso]] can also calculate the radar cross section (RCS) of a planar target. Note that in this case the input admittance or impedance RCS is defined at for a gap source port is referenced to finite-sized target in the two terminals presence of an infinite background structure. The scattered &theta; and &phi; components of the voltage source connected across the gap as shown far-zone electric field are indeed what you see in the figure below3D far field visualization of radiation (scattering) patterns. This is different than the input admittance Instead of radiation or impedance that one may normally define for a microstrip portscattering patterns, which is referenced you can instruct [[EM.Picasso]] to plot 3D visualizations of &sigma;_&theta;, &sigma;_&phi; and the substrate's groundtotal RCS.

<table><tr><td> [[FileImage:PMOM59(1)PMOM125.png|800pxthumb|left|600px|An example of the 3D monostatic radar cross section plot of a patch antenna.]]</td></tr></table>

Definition of different input impedances at the gap location== Discretizing a Planar Structure in EM.Picasso ==

To resolve this problem, you can place a gap source on a metal strip line by a distance The method of a quarter guide wavelength moments (&lambda;_g/4MoM) away from its open end. Note that discretizes all the finite-sized objects of a planar structure (&lambda;_g = 2p/ßexcluding the background structure), where ß is into a set of elementary cells. Both the propagation constant quality and resolution of the metallic transmission linegenerated mesh greatly affect the accuracy of the MoM numerical solution. As show The mesh density gives a measure of the number of cells per effective wavelength that are placed in various regions of your planar structure. The higher the figure belowmesh density, the impedance looking into an open quartermore cells are created on the finite-wave line segment is zerosized geometrical objects. As a rule of thumb, which effectively shorts the gap source to the planar structure's grounda mesh density of about 20-30 cells per effective wavelength usually yields satisfactory results. The gap admittance But for structures with lots of fine geometrical details or impedance for highly resonant structures, higher mesh densities may be required. The particular output data that you seek in this case is identical a simulation also influence your choice of mesh resolution. For example, far field characteristics like radiation patterns are less sensitive to the input admittance or impedance of the planar structuremesh density than field distributions on structures with a highly irregular shapes and boundaries.

<table><tr><td> [[FileImage:PMOM60(1)PMOM31.png|800pxthumb|400px|The Planar Mesh Settings dialog.]]</td></tr></table>

Placing EM.Picasso provides two types of mesh for a gap source planar structure: a quarter guide wavelength away from the open end of pure triangular surface mesh and a feed line to effectively short it hybrid triangular-rectangular surface mesh. In both case, EM.Picasso attempts to create a highly regular mesh, in which most of the ground at cells have almost equal areas. For planar structures with regular, mostly rectangular shapes, the gap locationhybrid mesh generator usually leads to faster computation times.

The same principle applies to the gap sources on slot traces[[Image:Info_icon. The figure below shows how png|30px]] Click here 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 sourceslearn more about '''[[Preparing_Physical_Structures_for_Electromagnetic_Simulation#Working_with_EM. Note that in this case, you deal Cube.27s_Mesh_Generators | Working 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 sourcesMesh Generator]]'''.

[[FileImage:PMOM61(1)Info_icon.png|800px30px]]Click here to learn more about '''[[Preparing_Physical_Structures_for_Electromagnetic_Simulation#The_Triangular_Surface_Mesh_Generator | EM.Picasso's Triangular Surface Mesh Generator]]'''.

Placing two oppositely polarized gap sources a quarter guide wavelength away from the short end <table><tr><td> [[Image:PMOM48F.png|thumb|left|420px|Geometry of a CPW line to effectively create an open circuit beyond multilayer slot-coupled patch array.]] </td></tr><tr><td> [[Image:PMOM48G.png|thumb|left|420px|Hybrid planar mesh of the gap locationslot-coupled patch array.]] </td></tr></table>

The case of a probe source placed on a PEC via that is connected to a ground plane is more straightforward<table><tr><td> [[Image:PMOM48H. In this case, the probe source's gap discontinuity is placed at the middle plane png|thumb|left|420px|Details 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 hybrid planar mesh of the structure at a reference plane that passed through the host viaslot-coupled patch array around discontinuities.]] </td></tr></table>

Input impedance EM.Picasso's hybrid planar mesh generator tries to produce as many rectangular cells as possible especially in the case of a probe source on a PEC via connected objects with rectangular or linear boundaries. In connection or junction areas between adjacent objects or close to highly curved boundaries, triangular cells are used to fill the "irregular" regions in a ground planeconformal and consistent manner.

The mesh density gives a measure of the number of cells per effective wavelength that are placed in various regions of your planar structure. The effective wavelength is defined as <math>\lambda_{eff} === Exciting Multiport Structures Using Linear Superposition ===\tfrac{\lambda_0}{\sqrt{\varepsilon_{eff}}}</math>, where e_eff is the effective permittivity. By default, [[EM.Picasso]] generates a hybrid mesh with a mesh density of 20 cells per effective wavelength. The effective permittivity is defined differently for different types of traces and embedded object sets. This is to make sure that enough cells are placed in areas that might feature higher field concentration.

If your planar structure has two or more sources* For PEC and conductive sheet traces, but you have not the effective permittivity is defined any ports, all as the lumped sources excite the structure locally and contribute to the excitation vector needed for larger of the MoM solution permittivity of the problemtwo substrate layers just above and below the metallic trace. However* For slot traces, when you assign N ports to the sources, then you have a multiport structure that effective permittivity is characterized by an N×N admittance matrix defined as the mean (instead of a single Y_in parameteraverage), 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 of the principle permittivity of linear superpositionthe two substrate layers just above and below the metallic trace. In this binary scheme* For embedded object sets, the structure effective permittivity is analyzed N times. Each time one defined as the largest of the N port-assigned sources is excited, and permittivities of all the other port-assigned sources are turned offsubstrate layers and embedded dielectric sets.

In the case <table><tr><td> [[Image:PMOM32.png|thumb|360px|A comparison of gap sources on metal traces triangular and probe sources on PEC vias, turning a source off means shorting planar hybrid meshes of a series voltage sourcerectangular patch. The electric currents passing through these sources are then found at each port location, and the admittance ]] </td><td> [[parametersImage:PMOM30.png|thumb|360px|Mesh of two rectangular patches at two different substrate planes. The lower substrate layer has a higher permittivity.]] are found as follows:</td></tr></table>

:<math> I_m = \sum_{n=1}^N Y_{mn} V_n, \quad \quad Y_{mn} = \frac{I_m}{V_n} \bigg|_{V_kGeneral Rules of Planar Hybrid Mesh Generator ===0, k \ne n}</math><!--[[File:PMOM57.png]]-->

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 integrity of the source locations, planar mesh and thus its continuity in the voltages across them, are then found at all ports, junction areas directly affects the quality and accuracy of the impedance [[parameters]] simulation results. EM.Picasso's hybrid planar mesh generator has some rules that are found as followscatered to 2.5-D MoM simulations:

:<math> V_m = \sum_{n=1}^N Z_{mn} I_n* If two connected rectangular objects have the same side dimensions along their common linear edge with perfect alignment, \quad \quad Z_{mn} = \frac{V_m}{I_n} \bigg|_{I_k=0a rectangular junction mesh is produced.* If two connected rectangular objects have different side dimensions along their common linear edge or have edge offset, k \ne n}</math>a set of triangular cells is generated along the edge of the object with the larger side.<!* Rectangle strip objects that host a gap source or a lumped element always have a rectangular mesh around the gap area.* If two objects reside on the same Z--[[File:PMOM58plane, belong to the same trace group and have a common overlap area, they are first merged into a single object for the purpose of meshing using the &quot;Boolean Union&quot; operation.png]]* Embedded objects have prismatic meshes along the Z-->axis.* If an embedded object is located underneath or above a metallic trace object or connected from both top and bottom, it is meshed first and its mesh is then reflected on all of its attached horizontal trace objects.

:<mathtable><tr><td> \mathbf{[S] = [Y_0File:PMOM36.png|250px] \cdot ([Z]-[Z_0]) \cdot ([ZFile:PMOM38.png|250px]+[Z_0])^{-1} \cdot [Z_0[File:PMOM37.png|250px]]} </mathtd><!--/tr><tr><td> Two overlapping planar objects and a comparison of their triangular and hybrid planar meshes. </td></tr><tr><td> [[File:PMOM63PMOM33.png|250px]][[File:PMOM35.png|250px]] [[File:PMOM34.png|250px]] </td></tr><tr><td> Edge--connected rectangular planar objects and a comparison their triangular and hybrid planar meshes. </td></tr></table>

where <mathtable>\mathbf{[Z_0]}</mathtr> and <mathtd>\mathbf{[Y_0[File:PMOM39.png|375px]] [[File:PMOM40.png|375px]]}</mathtd> are diagonal matrices whose diagonal elements are the port characteristic impedances </tr><tr><td> Meshes of short and admittances, respectivelylong vertical PEC vias connecting two horizontal metallic strips.</td></tr></table>

=== Modeling Lumped Elements In Refining the Planar MoM Mesh Locally ===

Lumped elements are components, devices, or circuits whose overall dimensions are It is very small compared important to apply the wavelength. As a result, they are considered to be dimensionless compared right mesh density to capture all the dimensions geometrical details of a mesh cellyour planar structure. In factThis is especially true for &quot;field discontinuity&quot; regions such as junction areas between connected objects, a lumped element is equivalent to an infinitesimally narrow gap that is placed in the path of where larger current flowconcentrations are usually observed at sharp corners, across which or at the device's governing equations are enforced. Using Kirkhoff's laws, these device equations normally establish a relationship junction areas between the currents metallic traces and voltages across the device or circuit. Crossing the bridge to Maxwell's domainPEC vias, as well as the device equations must now be cast into a from o boundary conditions that relate the electric areas around gap sources and magnetic currents and fields. [[EM.Cube]]'s [[Planar Module]] allows you to define passive circuit lumped elements: '''Resistors'''(R), C'''apacitors'''(C), I'''nductors'''(L), and series and parallel combinations of them as shown in the figure below:which create voltage or current discontinuities.

[[File:image106The Planar Mesh Settings dialog gives a few options for customizing your planar mesh around geometrical and field discontinuities. The check box labeled &quot;'''Refine Mesh at Junctions'''&quot; increases the mesh resolution at the connection area between rectangular objects. The check box labeled &quot;'''Refine Mesh at Gap Locations'''&quot; might be particularly useful when gap sources or lumped elements are placed on a short transmission line connected from both ends. The check box labeled &quot;'''Refine Mesh at Vias'''&quot; increases the mesh resolution on the cross section of embedded object sets and at the connection regions of the metallic objects connected to them. EM.Picasso typically doubles the mesh resolution locally at the discontinuity areas when the respective boxes are checked. You should always visually inspect EM.Picasso's default generated mesh to see if the current mesh settings have produced an acceptable mesh.png]]

Figure 1: A series-parallel RLC combination that can Sometimes EM.Picasso's default mesh may contain very narrow triangular cells due to very small angles between two edges. In some rare cases, extremely small triangular cells may be modeled as generated, whose area is a lumped circuit small fraction of the average mesh cell. These cases typically happen at the junctions and other discontinuity regions or at the boundary of highly irregular geometries with extremely fine details. In such cases, increasing or decreasing the mesh density by one or few cells per effective wavelength often resolves that problem and eliminates those defective cells. Nonetheless, EM.Picasso's planar mesh generator offers an option to identify the defective triangular cells and either delete them or cure them. By curing we mean removing a narrow triangular cell and merging its two closely spaced nodes to fill the crack left behind. EM.Picasso by default deletes or cures all the triangular cells that have angles less than 10º. Sometimes removing defective cells may inadvertently cause worse problems in [[the mesh. You may choose to disable this feature and uncheck the box labeled &quot;'''Remove Defective Triangular Cells'''&quot; in the Planar Module]]Mesh Settings dialog. You can also change the value of the minimum allowable cell angle.

Lumped elements are conceptualized {{Note| Narrow, spiky triangular cells in a similar way as gap or probe sourcesplanar mesh are generally not desirable. They are indeed considered as infinitesimally narrow gaps placed in You should get rid of the path of current flow, across which Ohmeither by changing the mesh density or using the hybrid planar mesh generator's law is enforcedadditional mesh refinement options. 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:}}

<table><tr><td> [[Image:PMOM44.png|thumb|left|480px|Deleting or curing defective triangular cells: Case 1.]]<math/td> V_{gap} = Z_L I_{in} \quad\quad \int_{\delta} \hat{x}\cdot \mathbf{E_{gap}} \, dx = Z_L \int_W \hat{y} \cdot \mathbf{J_s} \, dy </mathtr><!--tr><td> [[FileImage:PMOM67PMOM42.png|thumb|left|480px|Deleting or curing defective triangular cells: Case 2.]]--</td></tr></table>

where Z_L is the total impedance across the two terminals of the series element== Running Planar MoM Simulations in EM. 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:Picasso ==

:<math> I_{gap} = Y_L V_{in} \quad\quad \int_{\delta} J_Y^{fila} \, dx = Y_L \int_W E_y \, dy </math>= EM.Picasso's Simulation Modes ===

:<math> \int_{\delta} \hat{x}\cdot\hat{n} \times (\mathbf{H_{gap}^+ - H_{gap}^-}) \, dx = Y_L \int_W \hat{y}\cdot\mathbf{M_s} \, dy </math><!--[[File:PMOM70(1)EM.pngPicasso]]-->offers five Planar MoM simulation modes:

where Y_L is the total admittance across the two terminals {| class="wikitable"|-! scope="col"| Simulation Mode! scope="col"| Usage! scope="col"| Number of the shunt element. If a lumped element is placed on a PEC via that is connected to Engine Runs! scope="col"| Frequency ! scope="col"| Restrictions|-| style="width:120px;" | [[#Running a metal strip from one side and to a PEC ground plane from Single-Frequency Planar MoM Analysis | Single-Frequency Analysis]]| style="width:270px;" | Simulates the other end, it is indeed as a series connection across a gap discontinuity planar structure "As Is"| style="width:80px;" | Single run| style="width:250px;" | Runs at the middle plane of the viacenter frequency fc| style="width:80px;" | None|-| style="width:120px;" | [[Parametric_Modeling_%26_Simulation_Modes_in_EM. If Cube#Running_Frequency_Sweep_Simulations_in_EM.Cube | Frequency Sweep]]| style="width:270px;" | Varies the via is short, it is meshed using operating frequency of the planar MoM solver | style="width:80px;" | Multiple runs | style="width:250px;" | Runs at a single prismatic elementspecified set of frequency samples or adds more frequency samples in an adaptive way| style="width:80px;" | None|-| style="width:120px;" | [[Parametric_Modeling_%26_Simulation_Modes_in_EM. In that case, Cube#Running_Parametric_Sweep_Simulations_in_EM.Cube | Parametric Sweep]]| style="width:270px;" | Varies the lumped element in effect shunts value(s) of one or more project variables| style="width:80px;" | Multiple runs| style="width:250px;" | Runs at the metal strip center frequency fc| style="width:80px;" | None|-| style="width:120px;" | [[Parametric_Modeling_%26_Simulation_Modes_in_EM.Cube#Performing_Optimization_in_EM.Cube | Optimization]]| style="width:270px;" | Optimizes the value(s) of one or more project variables to achieve a design goal | style="width:80px;" | Multiple runs | style="width:250px;" | Runs at the groundcenter frequency fc| style="width:80px;" | None|-| style="width:120px;" | [[Parametric_Modeling_%26_Simulation_Modes_in_EM. The boundary condition at Cube#Generating_Surrogate_Models | HDMR Sweep]]| style="width:270px;" | Varies the location value(s) of one or more project variables to generate a compact model| style="width:80px;" | Multiple runs | style="width:250px;" | Runs at the lumped element across the PEC via iscenter frequency fc| style="width:80px;" | None|}

:<math> V_{gap} = Z_L I_{in} \quad\quad \int_{\delta} \hat{z}\cdot \mathbf{E_{gap}} \, dz = Z_L \int_S \hat{z} \cdot \mathbf{J_p} \, ds </math><!--You can set the simulation mode from [[File:PMOM68EM.pngPicasso]]'s "Simulation Run Dialog". A single-frequency analysis is a single->run simulation. All the other simulation modes in the above list are considered multi-run simulations. If you run a simulation without having defined any observables, no data will be generated at the end of the simulation. In multi-run simulation modes, certain parameters are varied and a collection of simulation data files are generated. At the end of a sweep simulation, you can graph the simulation results in EM.Grid or you can animate the 3D simulation data from the navigation tree.

=== Running a Single-Frequency Planar MoM Analysis === A single-frequency analysis is the simplest type of [[File:PMOM69EM.pngPicasso]]simulation and involves the following steps:

Figure 1: Using a shunt lumped element on a PEC via * Set the units of your project and the frequency of operation. Note that the default project unit is '''millimeter'''. * Define you background structure and its layer properties and trace types. * Construct your planar structure using [[Building_Geometrical_Constructions_in_CubeCAD | CubeCAD]]'s drawing tools to terminate a metallic strip linecreate all the finite-sized metal and slot trace objects and possibly embedded metal or dielectric objects that are interspersed among the substrate layers.* Define an excitation source and observables for your project.* Examine the planar mesh, verify its integrity and change the mesh density if necessary.* Run the Planar MoM simulation engine.* Visualize the output simulation data.

[[File:PMOM64.png|thumb|400px|Lumped Element dialog]]To define a lumped RLC circuit in your planar structure, follow these steps: * Open the Lumped Element Dialog by right clicking on the '''Lumped Elements''' item in the '''Sources''' section of the Navigation Tree and selecting '''Insert New Source...'''* In the '''Gap Topology''' section of the dialog, select one of the two options: '''Gap on Line''' and '''Gap on Via'''.* In the '''Lumped Circuit Type''' section of the dialog, select one of the two options: '''Passive RLC''' and '''Active with Gap Source'''.* Depending on your choice of gap topology, in the '''Lumped Circuit Location''' section of the dialog, you will find either a list of all the '''Rectangle Strip Objects''' or a list of all the '''PEC Via Objects''' available in the project workspace. Select the desired rectangle strip or embedded PEC via object.* In the box labeled '''Offset''', enter the distance of the lumped element from the start point of the rectangle strip line or from the bottom of the via object, whichever the case. The value of '''Offset''' by default is initially set to the center of the line or via.* In the '''Load Properties''' section, the series and shunt resistance values Rs and Rp are specified in Ohms, the series and shunt inductance values Ls and Lp are specified in nH (nanohenry), and the series and shunt capacitance values Cs and Cp are specified in pF (picofarad). Only the checked elements are taken into account in the total impedance calculation. By default, only the series resistor is checked with a value of 50S, and all other circuit elements are initially greyed out.<br /table> [[EM.Cube]]'s [[Planar Module]] allows you to define a voltage source in series with a series-parallel RLC combination and place them across the gap. This is called an active lumped element. If you choose the '''Active with Gap Source''' option of the '''Lumped Circuit Type''' section of the dialog, the right section of the dialog entitled '''Source Properties''' becomes enabled, where you can you can specify the '''Source Amplitude''' in Volts (or in Amperes in the case of PMC traces) and the '''Phase''' in degrees. Also, the box labeled '''Direction''' becomes relevant in this case which contains a gap source. Otherwise, a passive RLC circuit does not have polarity. If the project workspace contains an array of rectangle strip objects or PEC via objects, the array object will also be listed as an eligible object for lumped element placement. A lumped element will then be placed on each element of the array. All the lumped elements will have identical direction, offset, resistance, inductance and capacitance values. If you define an active lumped element, you can prescribe certain amplitude and/or phase distribution to the gap sources just like in the case of gap and probe sources. The available amplitude distributions include '''Uniform''', '''Binomial'''''', Chebyshev''' and '''Data File'''. {{Note|The impedance of the lumped circuit is calculated at the operating frequency of the project using the specified R, L and C values. As you change the frequency, the value of the impedance that is passed to the Planar MoM engine will change.}} === Calculating Scattering Parameters Using Prony's Method === The calculation of the scattering (S) [[parameters]] is usually an important objective of modeling planar structures especially for planar circuits like filters, couplers, etc. As you saw earlier, you can use lumped sources like gaps and probes and even active lumped elements to calculate the circuit characteristics of planar structures. The admittance / impedance calculations based on the gap voltages and currents are accurate at RF and lower microwave frequencies or when the port [[Transmission Lines|transmission lines]] are narrow. In such cases, the electric or magnetic current distributions across the width of the port line are usually smooth, and quite uniform current or voltage profiles can easily be realized. At higher frequencies, however, a more robust method is needed for calculating the port [[parameters]]. One can calculate the scattering [[parameters]] of a planar structure directly by analyzing the current distribution patterns on the port [[Transmission Lines|transmission lines]]. The discontinuity at the end of a port line typically gives rise to a standing wave pattern that can clearly be discerned in the line's current distribution. From the location of the current minima and maxima and their relative levels, one can determine the reflection coefficient at the discontinuity, i.e. the S<subtr>11</sub> 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: :<mathtd> f(x) \approx \sum_{n=1}^N c_i e^{-j\gamma_i x} </math><!--[[FileImage:PMOM73Picasso L1 Fig18.png]]--> where c_i are complex coefficients and &gamma;_i are, in general, complex exponents. From the physics of [[Transmission Lines|transmission lines]], we know that lossless lines may support one or more propagating modes with pure real propagation constants (real &gamma;_i exponents). Moreover, line discontinuities generate evanescent modes with pure imaginary propagation constants (imaginary &gamma;_i exponents) that decay along the line as you move away from the location of such discontinuities. In practical planar structures for which you want to calculate the scattering [[parameters]], each port line normally supports one, and only one, dominant propagating mode. Multi-mode [[Transmission Linesthumb|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. [[File:PMOM71.pngleft|480px|800px]] Figure 1: Minimum and maximum current locations of the standing wave pattern on a microstrip line feeding a patch antenna. == Running Planar MoM Simulations == The first step of planning a planar MoM simulation is defining your planar structure. This consists of the background structure plus all the finite-sized metal and slot trace objects and possibly embedded metal or dielectric objects that are interspersed among the substrate layers. The background stack-up is defined in the Layer Stack-up dialog, which automatically opens up as soon as you enter the [[Planar Module]]. The metal and slot traces and embedded object sets are listed in the Navigation Tree, which also shows all the geometrical (CAD) objects you draw in the project workspace under each object group at different Z-planes. The next step is to decide on the excitation scheme. If your planar structure has one or more ports and you seek to calculate its port characteristics, then you have to choose one of the lumped source types or a de-embedded source. If you are interested in the scattering characteristics of your planar structure, then you must define a plane wave source. Before you can run a planar MoM simulation, you also need to decide on the project's observables. These are the simulation data that you expect [[EM.Cube]] to generate as the outcome of the numerical simulation. [[EM.Cube]]'s [[Planar Module]] offers the following observables: * Current Distribution* Field Sensors* Far Fields (Radiation Patterns or Radar Cross Section)* Huygens Surfaces* Port Characteristics* Periodic Characteristics If you run a simulation without having defined any observables, no data will be generated at the end of the simulation. Some observables require a certain type of excitation source. For example, port characteristics will be calculated only if the project contains a port definition, which in turn requires the existence of at least one gap or probe or de-embedded source. The periodic characteristics (reflection and transmission coefficients) are calculated only if the structure has a periodic domain and excited by a plane wave source. === Planar Module's Simulation Modes === The simplest simulation type in [[EM.Cube]] is an analysis. In this mode, the planar structure in your project workspace is meshed at the center frequency of the project. [[EM.Cube]] generates an input file at this single frequency, and the Planar MoM simulation engine is run once. Upon completion of the planar MoM simulation, a number of data files are generated depending on the observables you have defined in your project. An analysis is a single-run simulation. [[EM.Cube]] offers a number of multi-run simulation modes. In such cases, the Planar MoM simulation engine is run multiple times. At each engine run, certain [[parameters]] are varied and a collection of simulation data are generated. At the end of a multi-run simulation, you can graph the simulation results in EM.Grid or you can animate the 3D simulation data from the Navigation Tree. For example, in a frequency sweep, the frequency of the project is varied over its specified bandwidth. Port characteristics are usually plotted vs. frequency, representing your planar structure's frequency response. In an angular sweep, the &theta; or &phi; angle of incidence of a plane wave source is varied over their respective ranges. [[EM.Cube]]'s [[Planar Module]] currently provides the following types of multi-run simulation modes: * Frequency Sweep* Parametric Sweep* Angular Sweep* R/T Macromodel* Huygens Sweep* [[Optimization]]* HDMR [[File:PMOM80.png]] Figure 1: Selecting a simulation mode in [[Planar Module]]Picasso's Simulation Run dialog. === Running A Planar MoM Analysis === To run a planar MoM analysis of your project structure, open the Run Simulation Dialog by clicking the '''Run''' [[File:run_icon.png]] button on the '''Simulate Toolbar''' or select '''Menu''' '''&gt;''' '''Simulate &gt;''' '''Run''' or use the keyboard shortcut '''Ctrl+R'''. The '''Analysis''' option of the '''Simulation Mode''' dropdown list is selected by default. Once you click the '''Run''' button, the simulation starts. A new window, called the '''Output Window''', opens up that reports the different stages of simulation and the percentage of the tasks completed at any time. After the simulation is successfully completed, a message pops up and reports the end of simulation. In certain cases like calculating scattering [[parameters]] of a circuit or reflection </ transmission characteristics of a periodic surface, some results are also reported in the Output Window. At the end of a simulation, you need to click the '''Close''' button of the Output Window to return to the project workspace.td> [[File:PMOM78.png]] Figure 1: [[Planar Module]]'s Simulation Run dialog. === Stages Of A Planar MoM Analysis === [[EM.Cube]]'s Planar MoM simulation engine uses a particular formulation of the method of moments called mixed potential integral equation (MPIE). Due to high-order singularities, the dyadic Green's functions for electric fields generated by electric currents as well as the dyadic Green's functions for magnetic fields generated by magnetic currents have very slow convergence behaviors. Instead of using these slowly converging dyadic Green's function, the MPIE formulation uses vector and scalar potentials. These include vector electric potential '''A(r)''', scalar electric potential K<sup>&Phi;</suptr>'''(r)''', vector magnetic potential '''F(r)''' and scalar magnetic potential K<sup>&Psi;</suptable>'''(r)'''. These potentials have singularities of lower orders. As a result, they coverage relatively faster. The speed of their convergence is further increased drastically using special singularity extraction techniques. A planar MoM simulation consists of two major stages: matrix fill and linear system inversion. In the first stage, the moment matrix and excitation vector are calculated. In the second stage, the MoM system of linear equations is inverted using one of the several available matrix solvers to find the unknown coefficients of all the basis functions. The unknown electric and magnetic currents are linear superpositions of all these elementary solutions. These can be visualized in [[EM.Cube]] using the current distribution observables. Having determined all the electric and magnetic currents in your planar structure, [[EM.Cube]] can then calculate the near fields on prescribed planes. These are introduced as field sensor observables. The near-zone electric and magnetic fields are calculated using a spectral domain formulation of the dyadic Green's functions. Finally the far fields of the planar structure are calculated in the spherical coordinate system. These calculations are performed using the asymptotic form of the dyadic Green's functions using the &quot;stationary phase method&quot;.

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

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

[[File:PMOM79EM.pngPicasso]]provides a large selection of linear system solvers including both direct and iterative methods. [[EM.Picasso]], by default, provides a &quot;'''Automatic'''&quot; solver option that picks the best method based on the settings and size of the numerical problem. For linear systems with a size less than N = 3,000, the LU solver is used. For larger systems, BiCG is used when dealing with symmetric matrices, and GMRES is used for asymmetric matrices. 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.

Figure 1<table><tr><td> [[Image: The PMOM79.png|thumb|left|720px|EM.Picasso's Planar MoM Engine Settings dialog.]] </td></tr></table>

=== Modeling Periodic Planar Module's Linear System Solvers =Structures in EM.Picasso ==

After the MoM impedance matrix '''[Z[EM.Picasso]]''' (not allows you to be confused simulate doubly periodic planar structures with periodicities along the impedance X and Y directions. Once you designate your planar structure as periodic, [[parametersEM.Picasso]]) and excitation vector '''[V]''' have been computed through the matrix fill process, the planar s Planar MoM simulation engine is ready uses a spectral domain solver to solve analyze it. In this case, the dyadic Green's functions of periodic planar structure take the system form of linear equations:doubly infinite summations rather than integrals.

:<math> \mathbf{[Z]}_{N\times N} \cdot \mathbf{[IImage:Info_icon.png|30px]}_{N\times 1} = \mathbf{[V]}_{N\times 1} </math><!--Click here to learn more about the theory of '''[[File:PMOM81.pngBasic_Principles_of_The_Method_of_Moments#Periodic_Planar_MoM_Simulation | Periodic Green's functions]]-->'''.

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. {{Note| [[EM.CubePicasso]]'s linear solvers compute the solution vector'''[I]''' of the above system. You can instruct [[EM.Cube]] to write the MoM matrix handle both regular 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, respectivelyskewed periodic lattices.}}

Of the above list, LU An infinite periodic structure in [[EM.Picasso]] is represented by a direct solver, while the rest are iterative solvers&quot;'''Periodic Unit Cell'''&quot;. BiCG is To define a relatively fast iterative solverperiodic structure, but it works only for symmetric matricesyou must open [[EM. You cannot use BiCG for periodic structures or planar structures that contain both metal Picasso]]'s Periodicity Settings Dialog by right clicking the '''Periodicity''' item in the '''Computational Domain''' section of the navigation tree and slot traces at different planes, as their MoM matrices are not symmetricselecting '''Periodicity Settings. The three solvers BCG-STAB..''' from the contextual menu or by selecting '''Menu''' '''&gt;''' '''Simulate &gt; 'Computational Domain &gt; Periodicity Settings...''' from the menu bar. In the Periodicity Settings Dialog, GMRES and TtFQMR work well for both symmetric and asymmetric matrices check the box labeled '''Periodic Structure'''. This will enable the section titled''&quot;''Lattice Properties&quot;. You can define the periods along the X and they also belong to a class of solvers called Y axes using the boxes labeled '''Krylov Sub-space MethodsSpacing'''. In particulara periodic structure, the GMRES method virtual domain is replaced by a default blue periodic domain that is always provides guaranteed unconditional convergencecentered 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.

<table><tr><td> [[Image:PMOM99.png|thumb|300px|EM.Cube]]Picasso's [[Planar Module]], by default, provides a &quot;'''Automatic'''&quot; solver option that picks the best method based on the settings and size of the numerical problem. For linear systems with a size less than N = 3,000, the LU solver is used. For larger systems, BiCG is used when dealing with symmetric matrices, and GMRES is used for asymmetric matrices. If the size of the linear system exceeds N = 15,000, the sparse version of the iterative solvers is used, utilizing a row-indexed sparse storage scheme. You can override the automatic solver option and manually set you own solver type. This is done using the '''Solver Type''' dropdown list in the &quot;'''Linear System Solver'''&quot; section of the Planar MoM Engine Periodicity Settings dialog. There are also a number of other [[parameters]] related to the solvers. The default value of '''Tolerance of Iterative Solver''' is 1E-3, which can be increased for more ill-conditioned systems. The maximum number of iterations is usually expressed as a multiple of the systems size. The default value of '''Max No. of Solver Iterations </td></ System Size''' is 3. For extremely large systems, sparse versions of iterative solvers are used. In this case, the elements of the matrix are thresholded with respect to the larges element. The default value of '''Threshold for Sparse Solver''' is 1E-6, meaning that all the matrix elements whose magnitude is less than 1E-6 times the large matrix elements are set equal to zero. There are two more [[parameters]] that are related to the Automatic Solver option. These are &quot;''' User Iterative Solver When System Size &gt;'''&quot; with a default value of 3,000 and &quot;''' Use SParse Storage When System Size &gt;''' &quot; with a default value of 15,000. In other words, you control the automatic solver when to switch between direct and iterative solvers and when to switch to the sparse version of iterative solvers.tr></table>

If your computer has an Intel CPUIn many cases, then [[EM.Cube]] offers special versions your planar structure's traces or embedded objects are entirely enclosed inside the periodic unit cell and do not touch the boundary of all the above linear solvers that have been optimized for Intel CPU platformsunit cell. These optimal solvers usually work 2-3 time faster than their generic counterparts. When you install [[EM.CubePicasso]]allows you to define periodic structures whose unit cells are interconnected. The interconnectivity applies only to PEC, PMC and conductive sheet traces, and embedded object sets are excluded. Your objects cannot cross the option to use Intel-optimized solvers is already enabledperiodic domain. In other words, the neighboring unit cells cannot overlap one another. However, you can disable this option (e.g. if your computer has a non-Intel CPU). To do arrange objects with linear edges such that, open one or more flat edges line up with the domain's bounding box. In such cases, [[EM.CubePicasso]]'s Preferences Dialog from '''Menu &gt; Edit &gt; Preferences''' 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 using bottom boundaries of the keyboard shortcut '''Ctrl+H'''unit cell. Select As an example, consider a periodic metallic screen as shown in the Advanced tab figure on the right. 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 dialog 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 uncheck connect to the box labeled &quot;''' Use Optimized Solvers for Intel CPU'''&quot;crosses in the neighboring cells in order to provide current continuity.

<table><tr><td> [[FileImage:PMOM82image122.png|thumb|400px|Modeling a periodic screen using two different types of unit cell.]]</td></tr></table>

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 === Exciting Periodic Structures as Radiators in EM.Grid.Picasso ===

To run When a uniform frequency sweepperiodic planar structure is excited using a gap or probe source, open it acts like an infinite periodic phased array. All the '''Simulation Run Dialog''', and select periodic replicas of the '''Frequency Sweep''' option from the dropdown list labeled '''Simulation Mode'''unit cell structure are excited. When you choose You can even impose a phase progression across the frequency sweep option, the '''Settings''' button next infinite array to the simulation mode dropdown list becomes enabledsteer its beam. Clicking You can do this button opens from the '''Frequency Settings''' property dialogof the gap or probe source. The '''Frequency Range'''is initially set equal to your project's center frequency minus and plus half bandwidth. But you can change At the values bottom of the '''Start FrequencyPlanar Gap Circuit Source Dialog'''and or '''End FrequencyGap Source Dialog''' as well as the , there is a button titled '''Number of SamplesPeriodic Scan...'''. The dialog offers two options You can enter desired values for '''Frequency Sweep TypeTheta''': and '''UniformPhi''' or '''Adaptive'''. Select the former type. It is very important to note that beam scan angles in a MoM simulation, changing the frequency results in a change of the mesh of the structure, toodegrees. This is because To visualize the mesh density is defined in terms radiation patterns of the number of cells per effective wavelength. By default, during a frequency sweepbeam-steered antenna array, [[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 you have the option to fix define a finite-sized array factor in 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 sweepRadiation Pattern dialog. You can make one of these three choices using the radio button do this in the '''Mesh SettingsImpose Array Factor''' section of the this dialog. Closing The values of '''Element Spacing''' along the Frequency Settings dialog returns you X and Y directions must be set equal to the Simulation Run dialog, where you can start the planar MoM frequency sweep simulation by clicking the value of '''RunPeriodic Lattice Spacing''' buttonalong those directions.

<table><tr><td> [[FileImage:PMOM126Period5.png|thumb|350px|Setting periodic scan angles in EM.Picasso's Gap Source dialog.]] </td><td> [[Image:Period5_ang.png|thumb|350px|Setting the beam scan angles in Periodic Scan Angles dialog.]] </td></tr><tr><td> [[Image:Period6.png|thumb|350px|Setting the array factor in EM.Picasso's Radiation Pattern dialog.]]</td></tr></table>

Figure 1: <table><tr><td> [[Planar ModuleImage:Period7.png|thumb|360px|Radiation pattern of an 8×8 finite-sized periodic printed dipole array with 0&deg; phi and theta scan angles.]]'s Frequency Settings dialog</td><td> [[Image:Period8.png|thumb|360px|Radiation pattern of a beam-steered 8×8 finite-sized periodic printed dipole array with 45&deg; phi and theta scan angles.]] </td></tr></table>

=== Adaptive Frequency Sweep Exciting Periodic Structures Using Plane Waves in EM.Picasso ===

Frequency sweeps are often performed to study the frequency response of When a periodic planar structure. In particularis excited using a plane wave source, it acts as a periodic surface that reflects or transmits the variation of scattering incident wave. [[parametersEM.Picasso ]] like S₁₁ (return loss) calculates the reflection and S₂₁ (insertion loss) with frequency are transmission coefficients of utmost interestperiodic planar structures. When analyzing resonant structures like patch antennas or planar filters over large frequency ranges, If you may have to sweep run a large number of single-frequency samples to capture their behavior with adequate details. The resonant peaks or notches plane wave simulation, the reflection and transmission coefficients are often missed due to reported in the lack Output Window at the end of enough resolutionthe simulation. [[EM.Cube]]'s [[Planar Module]] offers a powerful adaptive frequency sweep option for this purpose. It is based Note that these periodic characteristics depend on the fact that the frequency response polarization of a physical, causal, multiport network can be represented mathematically using a rational function approximationthe incident plane wave. You set the polarization (TMz or TEz) in the '''Plane Wave Dialog''' when defining your excitation source. In other words, this dialog you also set the S [[parameters]] values of a circuit exhibit a finite number of poles the incident '''Theta''' and zeros over a given frequency range'''Phi''' angles. [[EM.Cube]] first starts with very few frequency samples and tries to fit rational functions At the end of low orders to the scattering [[parameters]]. Then, it increases the number planar MoM simulation of samples gradually by inserting intermediate frequency samples in a progressive manner. At each iteration cycleperiodic structure with plane wave excitation, all the possible rational functions reflection and transmission coefficients of higher orders the structure are tried outcalculated and saved into two complex data files called &quot;reflection. The process continues until adding new intermediate frequency samples does not improve the resolution of the CPX&quot;S_ijand &quot; curves over the given frequency rangetransmission. In that case, the curves are considered as having convergedCPX&quot;.

You must have defined one or more ports for your planar structure run an adaptive frequency sweep. Open {{Note|In the Frequency Settings dialog from the Simulation Run dialog and select the '''Adaptive''' option absence 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 any finite traces or embedded objects in the '''Convergence Criterion'''project workspace, which has a default value of 0.1. At each iteration cycle, all the S [[parametersEM.Picasso]] are calculated at computes the newly inserted frequency samples, reflection and their average deviation from the curves transmission coefficients 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 layered background structure 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 stopyour project.}}

{{Note|For large frequency ranges, you may have to increase both the minimum and maximum number of samples<table><tr><td>[[Image:PMOM102. Moreover, remeshing the png|thumb|580px|A periodic planar layered structure at each frequency may prove more practical than fixing the mesh at the highest frequencywith slot traces excited by a normally incident plane wave source.}}]]</td></tr></table>

Figure 1: Settings adaptive frequency sweep You run a periodic MoM analysis just like an aperiodic MoM simulation from [[parameters]] in [[Planar ModuleEM.Picasso]]'s Frequency Settings Run Dialog.Here, too, you can run a single-frequency analysis or a uniform or adaptive frequency sweep, or a parametric sweep, etc. Similar to the aperiodic structures, you can define several observables for your project. If you open the Planar MoM Engine Settings dialog, you will see a section titled "Infinite Periodic Simulation". In this section, you can set the number of Floquet modes that will be computed in the periodic Green's function summations. By default, the numbers of Floquet modes along the X and Y directions are both equal to 25, meaning that a total of 2500 Floquet terms will be computed for each periodic MoM simulation.

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) You learned earlier how to use [[parameters]] of the designated portsEM. The scattering [[parametersCube]] 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 projectpowerful, adaptive frequency sweep utility to study the S/Z/Y matrices frequency response of the multiport network are calculateda planar structure. Note that Adaptive frequency sweep uses rational function interpolation to generate smooth curves of the S/Z/Y matrices scattering parameters with a relatively small number of an Nfull-wave simulation runs in a progressive manner. Therefore, you need a port definition in your planar structure are related to each other through be able to run an adaptive frequency sweep. This is clear in the following equations: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.

:<math>\mathbf{ [S] = [Y_0EM.Cube] \cdot ([Z]'s Planar Modules also allows you to run an adaptive frequency sweep of periodic surfaces excited by a plane wave source. In this case, the planar MoM engine calculates the reflection and transmission coefficients of the periodic surface. Note that you can conceptually consider a periodic surface as a two-[Z_0]) \cdot ([Z]+[Z_0])^{-port network, where Port 1} \cdot [Z_0] }is the top half-space and Port 2 is the bottom half-space. In that case, the reflection coefficient R is equivalent to S<sub>11</mathsub>parameter, while the transmission coefficient T is equivalent to S₂₁ parameter. This is, of course, the case when the periodic surface is illuminated by the plane wave source from the top half-space, corresponding to 90°&lt; &theta; = 180°. You can also illuminate the periodic surface by the plane wave source from the bottom half-space, corresponding to 0° = &theta; &lt; 90°. In this case, the reflection coefficient R and transmission coefficient T are equivalent to S₂₂ and S₁₂ parameters, respectively. Having these interpretations in mind, [[EM.Cube]] enables the &quot;'''Adaptive Frequency Sweep'''&quot; option of the '''Frequency Settings Dialog''' when your planar structure has a periodic domain together with a plane wave source.

:<math>\mathbf{ [Y] !--= [Z]^{== Modeling Finite-1} } </math>Sized Periodic Arrays ===

:<math>\mathbf{ [Z] = [\sqrt{Z_0}] \cdot ([U]+[S]) \cdot ([U]-[S])^{-1} \cdot [\sqrt{Z_0}] }</math><!--[[FileImage:PMOM121Info_icon.png]]--> where <math>\mathbf{[U]}</math> is the identity matrix of order N, <math>\mathbf{[Z_0]}</math> and <math>\mathbf{[Y_0]}</math> are diagonal matrices whose diagonal elements are the port characteristic impedances and admittances, respectively, and <math>\mathbf{[\sqrt{Z_0}]}</math> is a diagonal matrix whose diagonal elements are the square roots of port characteristic impedances. The voltage standing wave ratio (VSWR) of the structure at the first port is also computed: :<math>\text{VSWR} = \frac{|V_{max}|}{|V_{min}|} = \frac{1+|S_{11}|}{1-|S_{11}|}</math><!--[[File:PMOM122.png40px]]--> 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 Click here to an ASCII data file of real type with a &quot;learn about '''.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]]. [[File:PMOM128.png]] Figure 1: Selecting port characteristics data to plot from the Navigation Tree. You can also see a list of all the port characteristics data files in [[EM.Cube]]'s Data Manager. To open data manager, click the '''Data Manager''' [[File:data_manager_icon.png]] button of the '''Simulate Toolbar''' or select '''Simulate &gt; Data Manager''' from the menu bar, or right click on the '''Data Manager''' item of the Navigation Tree and select '''Open Data Manager'''... from the contextual menu. You can also use the keyboard shortcut '''Ctrl+D''' at any time. Select any data file by clicking and highlighting its row in the table and then click the '''Plot''' button to plot the graph. By default, the S [[parameters]] are plotted as double magnitude-phase graphs, while the Y and Z [[parameters]] are plotted as double real-imaginary part graphs. The VSWR data are plotted on a Cartesian graph. You can change the format of complex data plots. In general complex data can be plotted in three forms: # Magnitude and Phase# Real and Imaginary Parts# Smith Chart [[File:PMOM129.png]] 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'''. [[File:PMOM130.png]] Figure 3: Changing the graph type by editing a data file's properties. [[File:PMOM134.png|800px]] Figure 4: The S₁₁ parameter plotted on a Smith Chart graph in EM.Grid. === Rational Interpolation Of Scattering Parameters === The adaptive frequency sweep described earlier is an iterative process, whereby the Planar MoM simulation engine is run at a certain number of frequency samples at each iteration cycle. The frequency samples are progressively built up, and rational fits for these data are found at each iteration cycle. A decision is then made whether to continue more iterations. At the end of the whole process, a total number of scattering parameter data samples have been generated, and new smooth data corresponding to the best rational fits are written into new data files for graphing. [[EM.Cube]]'s [[planar Module]] also allows you to generate a rational fit for all or any existing scattering parameter data as a post-processing operation without a need to run additional simulation engine runs. You can interpolate all the scattering [[parameters]] together or select individual [[parameters]]. You do this post-processing operation from the Navigation Tree. Right click on the '''Port Definition''' item in the '''Observables''' section of the Navigation Tree and select Smart Fit. At the top of the Smart Fit Dialog, there is a dropdown list labeled '''Interpolate''', which gives a list of all the available S parameter data for rational interpolation. The default option is &quot;All Available [[Parameters]]&quot;. Then you see a box labeled '''Number of Available Samples''', whose value is read from the data content of the selected complex .CPX data file. Based on the number of available data samples, the dialog reports the '''Maximum Interpolant Order'''. You can choose any integer number for '''Interpolant Order''', from 1 to the maximum allowed.  {{Note|Interpolant order more than 15 will suffer from numerical instabilities even if you have a very large number of data samples.}} You can use the '''Update''' button of the dialog to generate the interpolated data for a given order. The new data are written to a complex data file with the same name as the selected S parameter and a &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. [[File:PMOM131.png]] Figure 1: [[Planar Module]]'s Smart Fit dialog. [[File:PMOM133(2).png|400px]] Figure 2: The S₁₁ parameter plot of a two-port structure in magnitude-phase format. [[File:PMOM132(2).png|400px]] Figure 3: The smoothed version of the S₁₁ parameter plot of the two-port structure using [[EM.Cube]]'s Smart Fit. == Working with Planar MoM Simulation Data == === Planar Module's Output Simulation Data === Depending on the source type and the types of observables defined in a project, a number of output data are generated at the end of a planar MoM simulation. Some of these data are 2D by nature and some are 3D. The output simulation data generated by [[EM.Cube]]'s [[Planar Module]] can be categorized into the following groups: * '''Port Characteristics''': S, Z and Y [[Parameters]] and Voltage Standing Wave Ratio (VSWR)* '''Radiation Characteristics''': Radiation Patterns, Directivity, Total Radiated Power, Axial Ratio, Main Beam Theta and Phi, Radiation Efficiency, Half Power Beam Width (HPBW), Maximum Side Lobe Level (SLL), First Null Level (FNL), Front-to-Back Ratio (FBR), etc.* '''Scattering Characteristics''': Bi-static and Mono-static Radar Cross Section (RCS)* '''Periodic Characteristics''': Reflection and Transmission Coefficients* '''Current Distributions''': Electric and magnetic current amplitude and phase on all metal and slot traces and embedded objects* '''Near-Field Distributions''': Electric and magnetic field amplitude and phase on specified planes and their central axes At the end of an analysis, the 2D quantities usually have a single value that is written into an ASCII data file. Complex-valued quantities are written into complex data files with a &quot;'''.CPX'''&quot; extension. Real-valued quantities are written into real data files with a &quot;'''.DAT'''&quot; extension. Polar 2D radiation pattern data and some other radiation characteristics are written into angular data files with a &quot;'''.ANG'''&quot; extension. In this latter file type, polar data are stored as functions of an angle expressed in degrees. At the end of a sweep simulation of one of the many types available (frequency, angular, parametric, etc.), the ASCII output data files are populated with rows that correspond to the samples of the sweep variable(s). If a sweep simulation involves N sweep [[variables]], then the first N columns of the output data files show the samples of those sweep [[variables]]. All the 2D data files are listed in the '''2D Data Files''' tab of [[EM.Cube]]'s '''Data Manager'''. You can view the contents of these data files by selecting their row in the data manager and clicking the '''View''' button of the dialog. 3D output data, on the other hand, are defined as functions of the space coordinates and are usually of vectorial nature. Cartesian-type and mesh-type data such as current distributions and near-field field distributions are expressed as functions of the Cartesian (X, Y, Z) coordinates. Spherical-type data like far-field radiation patterns and RCS are expressed as functions of the spherical angles (&theta;, &phi;). The 3D radiation patterns are written into a file with a &quot;'''.RAD'''&quot; extension. This file contains the complex values of the &theta;- and &phi;-components of the far-zone electric field (E_&theta; and E_&phi;) as well as the total far field magnitude as functions of the spherical observation angles &theta; and &phi;. The 3D RCS patterns are written into a file with a &quot;'''.RCS'''&quot; extension. This file contains the real values of the &theta;- and &phi;-polarized RCS values as well as the total RCS as functions of the spherical observation angles &theta; and &phi;. The current distributions are written into data files with a &quot;'''.CUR'''&quot; extension. They contain the real and imaginary parts of the X, Y and Z components of electric ('''J''') and magnetic ('''M''') current on each cells together with the definition of all the node coordinates and node indices of the cells. The near-field distributions are written into data files with a &quot;'''.SEN'''&quot; extension. They contain the amplitude and phase of the X, Y and Z components of electric ('''E''') and magnetic ('''H''') fields as functions of the coordinates of sampling points. All the 3D data files are listed in the '''3D Data Files''' tab of [[EM.Cube]]'s '''Data Manager'''. You can view the contents of these data files by selecting their row in the data manager and clicking the '''View''' button of the dialog. [[File:PMOM138.png]] Figure 1: The 3D Data Files tab of [[EM.Cube]]'s Data Manager. [[File:PMOM139.png|800px]] Figure 2: Viewing the contents of a mesh-type 3D data file in Data Manager. === Visualizing Current Distributions === Electric and magnetic currents are the fundamental output data of a planar MoM simulation. After the numerical solution of the MoM linear system, they are found using the solution vector '''[I]''' and the definitions of the electric and magnetic vectorial basis functions: :<math> \mathbf{[I]}_{N\times 1} = \begin{bmatrix} I^{(J)} \\ \\ V^{(M)} \end{bmatrix} \quad \Rightarrow \quad \begin{cases} \mathbf{J(r)} = \sum_{n=1}^N I_n^{(J)} \mathbf{f_n^{(J)} (r)} \\ \\ \mathbf{M(r)} = \sum_{k=1}^K V_k^{(M)} \mathbf{f_k^{(M)} (r)} \end{cases} </math><!--[[File:PMOM83.png]]--> 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. [[File:PMOM84.png]] Figure 1: The [[Planar Module]]'s Current Distribution dialog. Once you close the current distribution dialog, the label of the selected trace or object set is added under the '''Current Distributions''' node of the Navigation Tree.  {{Note|You have to define a separate current distribution observable for each individual trace or embedded object set.}} At the end of a planar MoM simulation, the current distribution nodes in the Navigation Tree become populated by the magnitude and phase plots of the three vectorial components of the electric ('''J''') and magnetic ('''M''') currents as well as the total electric and magnetic currents defined in the following manner: :<math> | \mathbf{J_{tot}} | = \sqrt{|J_x|^2 + |J_y|^2 + |J_z|^2}</math> :<math> | \mathbf{M_{tot}} | = \sqrt{|M_x|^2 + |M_y|^2 + |M_z|^2}</math><!--[[File:PMOM87.png]]--> 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'''. [[File:PMOM85(1).png|800px]] Figure 2: The current distribution map of a patch antenna. [[File:PMOM86(2).png|800px]] Figure 3: Vectorial (cone) visualization of the current distribution on a patch antenna. === Computing The Near Fields === Once all the current distributions are known in a planar structure, the electric and magnetic fields can be calculated everywhere in that structure using the dyadic Greens's functions of the background structure: :<math> \begin{align} \mathbf{E(r) = E_{inc}(r)} + & \sum_{n=1}^N I_n^{(J)} \iiint_V \overline{\overline{G}}_{EJ}(r|r') \cdot f_n^{(J)}(r') \, d\nu' + \\ & \sum_{k=1}^K V_n^{(M)} \iiint_V \overline{\overline{G}}_{EM}(r|r') \cdot f_k^{(M)}(r') \, d\nu' \end{align} </math> :<math> \begin{align} \mathbf{H(r) = H_{inc}(r)} + & \sum_{n=1}^N I_n^{(J)} \iiint_V \overline{\overline{G}}_{HJ}(r|r') \cdot f_n^{(J)}(r') \, d\nu' + \\ & \sum_{k=1}^K V_n^{(M)} \iiint_V \overline{\overline{G}} {HM}(r|r') \cdot f_k^{(M)}(r') \, d\nu' \end{align} </math><!--[[File:PMOM92(2).png]]--> The above equations can be cast into the spectral domain as follows: :<math> \begin{align} \mathbf{E(r) = E_{inc}(r)} + \frac{1}{(2\pi)^2} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} \bigg[ & \sum_{n=1}^N I_n^{(J)} \tilde{\overline{\overline{G}}}_{EJ}(k_{\rho}, z|z') \cdot \tilde{f}_n^{(J)}(k_x, k_y) + \\ & \sum_{k=1}^K V_n^{(M)} \tilde{\overline{\overline{G}}}_{EM}(k_{\rho}, z|z') \cdot \tilde{f}_k^{(M)}(k_x, k_y) \bigg] \, dk_x \, dk_y \end{align} </math> :<math> \begin{align} \mathbf{H(r) = H_{inc}(r)} + \frac{1}{(2\pi)^2} \int\limits_{-\infty}^{\infty} \int\limits_{-\infty}^{\infty} \bigg[ & \sum_{n=1}^N I_n^{(J)} \tilde{\overline{\overline{G}}}_{HJ}(k_{\rho}, z|z') \cdot \tilde{f}_n^{(J)}(k_x, k_y) + \\ & \sum_{k=1}^K V_n^{(M)} \tilde{\overline{\overline{G}}}_{HM}(k_{\rho}, z|z') \cdot \tilde{f}_k^{(M)}(k_x, k_y) \bigg] \, dk_x \, dk_y \end{align} </math><!--[[File:PMOM93(1).png]]--> Calculation of the near-zone fields (fields at the vicinity of the unknown currents) is done at the post-processing stage and in a Cartesian coordinate systems. These calculations involve doubly infinite spectral-domain integrals, which are computed numerically. As was mentioned earlier, [[EM.Cube]]'s planar MoM engine rather uses a polar integration scheme, where the radial spectral variable k_&rho; is integrated over the interval [0, Mk₀], M being a large enough number to represent infinity, and the angular spectral variable t is integrated over the interval [0, 2&pi;]. You also saw some of the numerical [[parameters]] related to this spectral-domain integration scheme. {{Note|When the observation plane is placed very close to the radiating J and M currents, the Green's functions exhibit singularities, which translate to very slow convergence or divergence of the integrals. You need to be careful to place field sensors at adequate distances from these radiating sources.}} === Visualizing The Near Fields === [[File:PMOM90.png|thumb|300px|[[Planar Module]]'s Field Sensor dialog]]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. In the section titled Output Settings, you can also select the field map type from two options: '''Confetti''' and '''Cone'''. The former produces an intensity plot for field amplitude and phase, while the latter generates a 3D vector plot. In the confetti case, you have an option to check the box labeled '''Data Interpolation''', which creates a smooth and blended (digitally filtered) map. In the cone case, you can set the size of the vector cones that represent the field direction. At the end of a sweep simulation, multiple field map are produced and added to the Navigation Tree. You can animate these maps. However, during the sweep only one field type is stored, either the E-field or H-field. You can choose the field type for multiple plots using the radio buttons in the section titled '''Field Display - Multiple Plots'''. The default choice is the E-field. Once you close the Field Sensor dialog, its name is added under the '''Field Sensors''' node of the Navigation Tree. At the end of a planar MoM simulation, the field sensor nodes in the Navigation Tree become populated by the magnitude and phase plots of the three vectorial components of the electric ('''E''') and magnetic ('''H''') field as well as the total electric and magnetic fields defined in the following manner: :<math> |\mathbf{E_{tot}}| = \sqrt{|E_x|^2 + |E_y|^2 + |E_z|^2} </math>:<math> |\mathbf{H_{tot}}| = \sqrt{|H_x|^2 + |H_y|^2 + |H_z|^2} </math><!--[[File:PMOM88.png]]--> Note that unlike [[EM.Cube]]'s other computational modules, near field calculations in the [[Planar Module]] usually takes substantial time. This is due to the fact that at the end of a planar MoM simulation, the fields are not available anywhere (as opposed to the [[FDTD Module]]), and their computation requires integration of complex dyadic Green's functions (as opposed to [[MoM3D Module]]'s free space Green's functions). [[File:PMOM116.png|800px]] Near-zone electric field map above a microstrip-fed patch antenna. [[File:PMOM117.png|800px]] Near-zone magnetic field map above a microstrip-fed patch antenna. === Computing The Far Fields === Unlike differential-based methods, MoM simulators do not need a radiation box to calculate the far field data. The far-zone fields are calculated directly by integrating the currents on the traces and across the embedded objects using the asymptotic form of the background structure’s dyadic Green's functions: :<math> \mathbf{E^{ff}(r)} = \iiint_V \mathbf{ \overline{\overline{G}}_{EJ,ff}(r|r') \cdot J(r') } \, d\nu ' + \iiint_V \mathbf{ \overline{\overline{G}}_{EM,ff}(r|r') \cdot M(r') } \, d\nu '</math> :<math> \mathbf{H^{ff}(r)} = \dfrac{1}{\eta_0} \mathbf{ \hat{r} \times E^{ff}(r) }</math><!--[[File:PMOM112.png]]--> where &eta;₀ = 120&pi; is the characteristic impedance of the free space. As can be seen from the above equations, the far fields have the form of a TEM wave propagating in the radial direction away from the origin of coordinates. This means that the far-field magnetic field is always perpendicular to the electric field and the propagation vector, which in this case happens to be the radial unit vector in the spherical coordinate system. In other words, one only needs to know the far-zone electric field and can easily calculate the far-zone magnetic field from it. In [[EM.Cube]]'s mixed potential integral equation formulation, the far-zone electric field can be expressed in terms of the asymptotic form of the vector electric and magnetic potentials '''A''' and '''F''': :<math>\mathbf{E^{ff}}(x,y,z) = j k_0 \eta_0 \hat{r} \times [\hat{r} \times \mathbf{A}(r \to \infty)] + j k_0 \hat{r} \times \mathbf{F}(r \to \infty)</math><!--[[File:PMOM113.png]]--> The asymptotic form of these vector potentials are calculated using the &quot;'''Method of Stationary Phase'''&quot; when k₀r &rarr; &infin;. In that case, one can use the approximation: :<math> k_0 |\mathbf{r-r'}| \approx k_0 (r - \mathbf{\hat{r} \cdot r'}) </math><!--[[File:PMOM115.png]]--> 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 &theta; and &phi;. 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_&theta; and E_&phi;. Note that the outward propagating, TEM-type, far fields do not have radial components, i.e. E_r = 0. :<math> \mathbf{E_{\theta}}(\theta, \phi) = \cos\theta \cos\phi E_x + \cos\theta \sin\phi E_y - \sin\theta E_z </math>:<math> \mathbf{E_{\phi}}(\theta, \phi) = -\sin\phi E_x + \cos\phi E_y </math><!--[[File:PMOM114.png]]--> === Visualizing The Far Fields === [[File:PMOM118.png|thumb|300px|[[Planar Module]]'s Radiation Pattern dialog]]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. 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 &theta; direction, the far field component in &phi; 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: :<math>|\mathbf{E_{ff,tot}}| = \sqrt{ |E_{\theta}|^2 + |E_{\phi}|^2 }</math><!--[[File:PMOM89.png]]--> [[File:PMOM119.png|800px]] Figure: 3D polar radiation pattern plot of a microstrip-fed patch antenna. [[File:PMOM120.png|800px]] Figure: 3D vectorial (cone) radiation pattern plot of a microstrip-fed patch antenna. The 2D radiation pattern graphs can be plotted from [[EM.Cube]]'s '''Data Manager'''. A total of eight 2D radiation pattern graphs are available: 4 polar and 4 Cartesian graphs for the XY, YZ, ZX and user defined plane cuts. === Radar Cross Section of Planar Structures === [[File:PMOM124.png|thumb|300px|Planar Module's Radar Cross Section dialog]] When a planar structure is excited by a plane wave source, the calculated far field data indeed represent the scattered fields of that planar structure. [[EM.Cube]] can also calculate the radar cross section (RCS) of a planar target: :<math> \sigma_{\theta} = 4\pi r^2 \dfrac{|E_{\theta}^{scat}|^2}{|E^{inc}|^2}, \quad \sigma_{\phi} = 4\pi r^2 \dfrac{|E_{\phi}^{scat}|^2}{|E^{inc}|^2}, \quad \sigma = \sigma_{\theta} + \sigma_{\phi} = 4\pi r^2 \dfrac{|E_{tot}^{scat}|^2}{|E^{inc}|^2} </math><!--[[File:PMOM123.png]]--> '''Note that in this case the RCS is defined for a finite-sized target in the presence of an infinite background structure.''' The scattered &theta; and &phi; components of the far-zone electric field are indeed what you see in the 3D far field visualization of radiation (scattering) patterns. Instead of radiation or scattering patterns, you can instruct [[EM.Cube]] to plot 3D visualizations of &sigma;_&theta;, &sigma;_&phi; and the total RCS. To do so, you must define an RCS observable instead of a radiation pattern. Follow these steps: * Right click on the '''Far Fields''' item in the '''Observables''' section of the Navigation Tree and select '''Insert New RCS...''' to open the Radar Cross Section Dialog.* The resolution of RCS calculation is specified by '''Angle Increment''' expressed in degrees. By default, the &theta; and &phi; angles are incremented by 5 degrees.* At the end of a planar MoM simulation, besides calculating the RCS data over the entire (spherical) 3D space, a number of 2D RCS graphs are also generated. These are RCS cuts at certain planes, which include the three principal XY, YZ and ZX planes plus one additional constant f-cut. This fourth plane cut is at &phi; = 45° by default. You can assign another &phi; angle in degrees in the box labeled '''Non-Principal Phi Plane'''. At the end of a planar MoM simulation, in the far field section of the Navigation Tree, you will have the &theta; and &phi; 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. [[File:PMOM125.png|800px]] Figure 2: An example of the 3D mono-static radar cross section plot of a patch antenna. == Periodic Planar Structures &amp; Antenna Arrays == === Modeling Finite Arrays vs. Infinite Periodic Structures === A periodic structure is one that exhibits a repeated geometric pattern. It is made up of identical elements that are arranged in the form of a periodic lattice. The spacing between the elements is denoted by Sx along the X direction and Sy along the Y direction. The number of elements is denoted by Nx along the X direction and Ny along the Y direction (i.e. a total of Nx.Ny elements). If Nx and Ny are finite numbers, you have a finite-sized periodic structure, which is constructed using an &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;'''Sized 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. === Modeling Finite Antenna Arrays === The straightforward approach to the modeling of finite-sized antenna arrays is to use the full-wave method of moments (MoM). This requires building an array of radiating elements using [[EM.Cube]]'s '''Array Tool''' and feeding the individual array elements using some type of excitation. For example, if the antenna elements are excited using a gap source or a probe source, you can assign a certain array weight distribution among the elements as well as phase progression among the elements along the X and Y directions. [[EM.Cube]] currently offers uniform, binomial, Chebyshev and (arbitrary) data file-based weight distribution types. The full-wave MoM approach is very accurate and takes into account all the inter-element coupling effects. At the end of a planar MoM simulation of the array structure, you can plot the radiation patterns and other far field characteristics of the antenna array just like any other planar structure. The radiation pattern of antenna arrays usually has a main beam and several side lobes. Some [[parameters]] of interest in such structures include the '''Half Power Beam Width (HPBW)''', '''Maximum Side Lobe Level (SLL)''' and '''First Null [[Parameters]]''' such as first null level and first null beam width. To have [[EM.Cube]] calculate all such [[parameters]], you must check the relevant boxes in the &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. [[File:PMOM91.png]] Figure 1: [[Planar Module]]'s Radiation Pattern dialog. Another approach to modeling a finite-sized antenna array is to analyze one of its elements and use the &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. === Defining A Periodic Domain === In general, a planar structure in [[EM.Cube]]'s [[Planar Module]] is assumed to have open boundaries. This means that the structure has infinite dimensions along the X and Y directions. In other words, the layers of the background structure extend to infinity, while the traces and embedded object sets have finite sizes. Along the Z direction, a planar structure can be open-boundary, or it may be truncated by PEC ground planes from the top or bottom or both. You can define a planar structure to be infinitely periodic along the X and Y directions. In this case, you only need to define the periodic unit cell. [[EM.Cube]] automatically reproduces the unit cell infinitely and simulates it using a spectral domain periodic version of the Green's functions of your project's background structure. To define a periodic structure, you must open [[Planar Module]]'s Periodicity Settings Dialog by right clicking the '''Periodicity''' item in the '''Computational Domain''' section of the Navigation Tree and selecting '''Periodicity Settings...''' from the contextual menu or by selecting '''Menu''' '''&gt;''' '''Simulate &gt; 'Computational Domain &gt; Periodicity Settings...''' from the Menu Bar. In the Periodicity Settings Dialog, check the box labeled '''Periodic Structure'''. This will enable the section titled''&quot;''Lattice Properties&quot;. You can define the periods along the X and Y axes using the boxes labeled '''Spacing'''. You can also define values for periodic '''Offset''' along the X and Y directions, which will be explained later. In a periodic structure, the virtual domain is replaced by a default blue periodic domain that is always centered around the origin of coordinates. Keep in mind that the periodic unit cell must always be centered at the origin of coordinates. The relative position of the structure within this centered unit cell will change the phase of the results. [[File:PMOM99.png]] Figure 1: [[Planar Module]]'s Periodicity Settings dialog. === Regular vs. Generalized Periodic Lattices === Besides conventional rectangular lattices, [[EM.Cube]]'s [[Planar Module]] can also handle complex non-rectangular periodic lattices. For example, many frequency selective surfaces have skewed grids. In order to simulate skewed-grid periodic structures, the definition of the grid has to be generalized. A periodic structure is a repetition of a basic structure (unit cell) at pre-determined locations. Let these locations be described by (x_mn, y_mn), where m and n are integers ranging from -8 to 8. For a general skewed grid, x_mn and y_mn can be described by: :<math> \begin{align} & x_{mn} = m\Delta x + n \Delta x' \\ & y_{mn} = m\Delta y + n \Delta y' \end{align} </math> where <math>\Delta x</math> is the primary offset in the X direction (X Spacing) controlled by index m and <math>\Delta x'</math> is the secondary offset in the X direction (X Offset) controlled by index n. The meanings of <math>\Delta y</math> (Y Spacing) and <math>\Delta y'</math> (Y Offset) are similar with the roles of indices m and n interchanged. To illustrate how to use this definition, consider an example of an equilateral triangular grid with side length L as shown in the figure below. [[File:image121.png]] Figure 1: Diagram of an equilateral triangular periodic lattice. From the figure, it is obvious that the y coordinate of each row is fixed and identical, thus <math>\Delta y = L</math> and <math>\Delta y' = 0</math>. While in each row the spacing between adjacent elements is L, there is an offset of L/2 between the consecutive rows. This results in <math>\Delta x = L</math> and <math>\Delta x' = L/2</math>. To sum up, an equilateral triangular grid can be described by <math>\Delta x = L</math>, <math>\Delta x' = L/2</math>, <math>\Delta y = L</math> and <math>\Delta y' = 0</math>. In an [[EM.Cube]] [[Planar Module]] project, the secondary offsets are equal to zero by default, implying a rectangular lattice. You can change the values of the secondary offsets using the boxes labeled '''X Offset''' and '''Y Offset''' in the '''Periodicity Settings Dialog''', respectively. Triangular and Hexagonal lattices are popular special cases of the generalized lattice type. In a triangular lattice with alternating Rows, <math>\Delta x' = \Delta x/2</math> and <math>\Delta y' = 0</math>. A Hexagonal lattice (with alternating rows) is a special case of triangular lattice in which <math>\Delta y = \sqrt{3\Delta x / 2}</math>. === Interconnectivity Among Unit Cells === In many cases, your planar structure's traces or embedded objects are entirely enclosed inside the periodic unit cell and do not touch the boundary of the unit cell. In [[EM.Cube]]'s [[Planar Module]], you can define periodic structures whose unit cells are interconnected. Interconnectivity applies only to PEC, PMC and conductive sheet traces, and embedded object sets are excluded. Note that in a periodic planar structure, your objects cannot cross the periodic domain. However, you can arrange objects with linear edges such as one or more flat edges line up with the domain's bounding box. In such cases, [[EM.Cube]]'s planar MoM mesh generator will take into account the continuity of the currents across the adjacent connected unit cells and will create the connection basis functions at the right and top boundaries of the unit cell. It is clear that due to periodicity, the basis functions do not need to be extended at the left or bottom boundaries of the unit cell. As an example, consider the periodic structure in the figure below that shows a metallic screen or wire grid. The unit cell of this structure can be defined as a rectangular aperture in a PEC ground plane (marked as Unit Cell 1). In this case, the rectangle object is defined as a slot trace. Alternatively, you can define a unit cell in the form of a microstrip cross on a metal trace. In the latter case, however, the microstrip cross should extend across the unit cell and connect to the crosses in the neighboring cells in order to provide current continuity. [[File:image122.png]] Figure 1: Modeling a periodic screen using two different types of unit cell. [[File:pmom_per3_tn.png|400px]] [[File:pmom_per4_tn.png|400px]] Figure 2: The PMC aperture unit cell and its planar mesh. [[File:pmom_per5_tn.png|400px]] [[File:pmom_per6_tn.png|400px]] Figure 3: The PEC cross unit cell and its planar mesh. Notice the cell extensions at the unit cell's boundaries. === Periodic MoM Simulation === In the case of an infinite periodic planar structure, the field equations can be written in the following form: :<math> \mathbf{E(r) = E^{inc}(r)} + \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} \bigg[ \iiint_V \mathbf{ \overline{\overline{G}}_{EJ}(r|r') \cdot J_{mn}(r') } \, d\nu' + \iiint_V \mathbf{ \overline{\overline{G}}_{EM}(r|r') \cdot M_{mn}(r') } \, d\nu' \bigg] </math>  :<math> \mathbf{H(r) = H^{inc}(r)} + \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} \bigg[ \iiint_V \mathbf{ \overline{\overline{G}}_{HJ}(r|r') \cdot J_{mn}(r') } \, d\nu' + \iiint_V \mathbf{ \overline{\overline{G}}_{HM}(r|r') \cdot M_{mn}(r') } \, d\nu' \bigg] </math><!--[[File:PMOM94.png]]--> where :<math>\mathbf{J_{mn}(r) = J_{mn}}(x,y,z) = \mathbf{J_{00}}(x+m S_x, y+n S_y, z) e^{j(m k_{x00} S_x + n k_{y00} S_y)}</math> :<math>\mathbf{M_{mn}(r) = M_{mn}}(x,y,z) = \mathbf{M_{00}}(x+m S_x, y+n S_y, z) e^{j(m k_{x00} S_x + n k_{y00} S_y)}</math> and :<math> -\infty < m, n < \infty </math><!--[[File:PMOM95(1).png]]--> In the above equations, <math>\mathbf{J_{00}(r)}</math> and <math>\mathbf{M_{00}(r)}</math> are the periodic unit cell's electric and magnetic currents that are repeated everywhere in space on a rectangular lattice with periods S_x and S_y along the X and Y directions, respectively. <math>k_{x00}</math> and <math>k_{y00}</math> are the periodic propagation constants along the X and Y directions, respectively, and they are given by: :<math> k_{x00} = k_0 \sin\theta \cos\phi </math> :<math> k_{y00} = k_0 \sin\theta \sin\phi </math><!--[[File:PMOM96(1).png]]--> where &theta; and &phi; are the beam scan angles in the case of periodic excitation of lumped sources, or they are the spherical angles of incidence in the case of a plane wave source illuminating the periodic structure. Using the infinite summations, one can define periodic dyadic Green's functions in the spectral domain in the following manner: :<math> \mathbf{ \overline{\overline{G}}_{\mu \nu}^{PER} (r|r') } = \frac{1}{S_x S_y} \sum_{m=-\infty}^{\infty} \sum_{n=-\infty}^{\infty} \mathbf{ \tilde{\overline{\overline{G}}}_{\mu \nu} } (k_x, k_y, z|z') e^{-j[k_{xm}(x-x') + k_{yn}(y-y')NCCBF Technique]} </math> where:<math> k_{xm} = k_{x00} + \frac{2\pi m}{S_x} \quad \text{and} \quad k_{ym} = k_{y00} + \frac{2\pi m}{S_y} </math><!--[[File:PMOM97.png]]--> 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. [[File:PMOM98.png]] Figure 1: Changing the number of Floquet modes from the Planar MoM Engine Settings dialog. === Modeling Periodic Phased Arrays === Earlier, it was argued that you can calculate the radiation pattern of a finite antenna array by modeling a single isolated element and multiplying its &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. When a periodic structure is excited using a gap or probe source, it acts like an infinite periodic phased array. All the periodic replicas of the unit cell structure are excited. You can even impose a phase progression across the infinite array to steer its beam. You can do this from the property dialog of the gap or probe source. At the bottom of the '''Gap Source Dialog''' or '''Probe Source Dialog''', there is a section titled '''Periodic Beam Scan Angles'''. You can enter desired values for '''Theta''' and '''Phi''' beam scan angles in degrees. The corresponding phase progressions are calculated and applied to the periodic Green's functions: :<math>\Psi_x = -\frac{2\pi S_x}{\lambda_0} \sin\theta \cos\phi</math> :<math>\Psi_y = -\frac{2\pi S_y}{\lambda_0} \sin\theta \sin\phi</math><!--[[File:PMOM101.png]]--> 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. [[File:PMOM100.png]] Figure 1: Setting the periodic scan angles in [[Planar Module]]'s Gap Source dialog. [[File:pmom_per9_tn.png]] Figure 2: The 3D radiation pattern of a beam-steered periodic printed dipole array. === Exciting Periodic Structures Using Plane Waves === When a periodic structure is excited using a plane wave source, it acts as a periodic surface that reflects or transmits the incident wave. You can model frequency selective surfaces, electromagnetic band-gap structures and metamaterials in this way. [[EM.Cube]] calculates the reflection and transmission coefficients of periodic surfaces or planar structures. If you run a single plane wave simulation, the reflection and transmission coefficients are reported in the Output Window at the end of the simulation. Note that these periodic characteristics depend on the polarization of the incident plane wave. You set the polarization (TMz or TEz) in the '''Plane Wave Dialog''' when defining your excitation source. In this dialog you also set the values of the incident '''Theta''' and '''Phi''' angles. At the end of the planar MoM simulation of a periodic structure with plane wave excitation, the reflection and transmission coefficients of the structure are calculated and saved into two complex data files called &quot;reflection.CPX&quot; and &quot;transmission.CPX&quot;. These coefficients behave like the S₁₁ and S₂₁ [[parameters]] of a two-port network. You can think of the upper half-space as Port 1 and the lower half-space as Port 2 of this network. As a result, you can run an adaptive sweep of periodic structures with a plane wave source just like projects with gap or probe sources. The reflection and transmission (R/T) coefficients can be plotted in EM.Grid on 2D graphs similar to the S [[parameters]]. You can plot them from the Navigation Tree. To do so, right click on the '''Periodic Characteristics''' item in the '''Observables''' section of the Navigation Tree and select '''Plot Reflection Coefficients''' or '''Plot Transmission Coefficients'''. The complex data files are also listed in [[EM.Cube]]'s '''Data Manager''', where you can view or plot them. {{Note|In the absence of any finite traces or embedded objects in the project workspace, [[EM.Cube]] computes the reflection and transmission coefficients of the layered background structure of your project.}} [[File:PMOM102.png]] Figure: A periodic planar layered structure with slot traces excited by a normally incident plane wave source. === Characterizing Periodic Surfaces Using Angular Sweeps === The reflection and transmission characteristics of a period surface as functions of the incidence angle are often of great interest. For that purpose, you can run an angular sweep of your periodic structure, where you normally fix the &phi; angle and sweep the &theta; angle from 180 to 90 degrees for one-sided surfaces and from 180 to 0 degrees for two-sided surface. To run an angular sweep, open the [[Planar Module]]'s '''Simulation Run Dialog''' and select the '''Angular Sweep''' option from its '''Simulation Mode''' dropdown list. This enables the '''Settings''' button, which opens up the '''Angle Settings Dialog'''. First, you must choose either Theta or Phi as the '''Sweep Angle'''. Then you can set the '''Start''' and '''End''' values of the selected incidence angle as well as the '''Number of Samples'''. At the end of an angular sweep simulation, you can plot the reflection and transmission coefficients from the Navigation Tree. To do so, right click on the '''Periodic Characteristics''' item in the '''Observables''' section of the Navigation Tree and select '''Plot Reflection Coefficients''' or '''Plot Transmission Coefficients'''. The reflection and transmission coefficients of the structure are saved into two complex data files called &quot;reflection.CPX&quot; and &quot;transmission.CPX&quot;. These data files are also listed in [[EM.Cube]]'s '''Data Manager''', where you can view or plot them. [[File:PMOM103.png]] Figure 1: [[Planar Module]]'s Angle Settings dialog.

[[EMImage:Top_icon.Cubepng|30px]]'s Planar Modules also allows you to run an adaptive frequency sweep of periodic surfaces excited by a plane wave source''[[EM. In this case, the planar MoM engine calculates the reflection and transmission coefficients of the periodic surface. Note that you can conceptually consider a periodic surface as a two-port network, where Port 1 is the top half-space and Port 2 is the bottom half-space. In that case, the reflection coefficient R is equivalent Picasso#Product_Overview | Back to S₁₁ parameter, while the transmission coefficient T is equivalent to S₂₁ parameter. This is, Top of course, the case when the periodic surface is illuminated by the plane wave source from the top half-space, corresponding to 90°&lt; &theta; = 180°. You can also illuminate the periodic surface by the plane wave source from the bottom half-space, corresponding to 0° = &theta; &lt; 90°. In this case, the reflection coefficient R and transmission coefficient T are equivalent to S₂₂ and S₁₂ [[parametersPage]], respectively. Having these interpretations in mind, [[EM.Cube]] enables the &quot;'''Adaptive Frequency Sweep'''&quot; option of the '''Frequency Settings Dialog''' when your planar structure has a periodic domain together with a plane wave source.

Click here to learn about [[Modeling Finite-Sized Periodic Arrays Using NCCBF TechniqueImage:Tutorial_icon.png|30px]]'''[[EM.Cube#EM.Picasso_Documentation | EM.Picasso Tutorial Gateway]]'''

<p>&nbsp;</p>[[Image:Back_icon.png|30px]] '''[[EM.Cube | Back to EM.Cube Main Page]]'''