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[[Image:Splash-planar new.jpg|right|800px720px]]<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==

[[Image:PMOM14EM.png|thumb|400px|A typical planar layered structurePicasso]]EM.Picasso^® is a versatile planar structure simulator for modeling and design of printed antennas, planar microwave circuits, and layered periodic structures. [[EM.Picasso]]'s simulation engine is based on a 2.5-D full-wave Method of Moments (MoM) formulation that provides the ultimate modeling accuracy and computational speed for open-boundary multilayer structures. It can handle planar structures with arbitrary numbers of metal layouts, slot traces, vertical interconnects and lumped elements interspersed among different substrate layers.

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.

{{Note|EM.Picasso is the frequency-domain, full-wave '''[[Planar Module]]''' of '''[[EMImage:Info_icon.Cubepng|30px]]''', a comprehensive, integrated, modular electromagnetic modeling environment. EM.Picasso shares Click here to learn more about the visual interface, 3D parametric CAD modeler, data visualization tools, and many more utilities and features collectively known as '''[[CubeCADBasic Principles of The Method of Moments | Theory of Planar Method of Moments]]''' with all of [[EM.Cube]]'s other computational modules.}}

<table><tr><td> [[Image:Info_iconART PATCH Fig title.png|40px]] Click here to learn more about '''[[Getting_Started_with_EM.CUBE thumb| EMleft|480px|3D radiation pattern of a slot-coupled patch antenna array with a corporate feed network.Cube Modeling Environment]]'''.</td></tr></table>

[[Image:Info_icon=== EM.png|40px]] Click here to learn more about Picasso as the basic functionality Planar Module of '''[[CubeCAD]]'''EM.Cube ===

=== An Overview of [[EM.Picasso]] is the frequency-domain, full-wave '''Planar Method Module''' of '''[[EM.Cube]]''', a comprehensive, integrated, modular electromagnetic modeling environment. [[EM.Picasso]] shares the visual interface, 3D parametric CAD modeler, data visualization tools, and many more utilities and features collectively known as [[Building_Geometrical_Constructions_in_CubeCAD | CubeCAD]] with all of Moments ===[[EM.Cube]]'s other computational modules.

The Method of Moments (MoM) is a rigorous, full-wave numerical technique for solving open boundary electromagnetic problems[[Image:Info_icon. Using this technique, you can analyze electromagnetic radiation, scattering and wave propagation problems with relatively short computation times and modest computing resources. The method of moments is an integral equation technique; it solves the integral form of Maxwell’s equations as opposed png|30px]] Click here to their differential forms that are used in the finite element or finite difference time domain methodslearn more about '''[[Getting_Started_with_EM.Cube | EM.Cube Modeling Environment]]'''.

[[Image:PMOM11.png|thumb|250px|=== Advantages & Limitations of EM.Picasso's Navigation Tree.]]In EM.Picasso, the background structure is usually a layered planar structure that consists of one or more laterally infinite material layers always stacked along the Z-axis. In other words, the dimensions of the layers are infinite along the X and Y axes. Metallic traces are placed at the boundaries between the substrate or superstrate layers. These are modeled by perfect electric conductor (PEC) traces or conductive sheet traces of finite thickness and finite conductivity. Some layers might be separated by infinite perfectly conducting ground planes. The two sides of a ground plane can be electromagnetically coupled through one or more slots (apertures). Such slots are modeled by magnetic surface currents. Furthermore, the metallic traces can be interconnected or connected to ground planes using embedded objects. Such objects can be used to model circuit vias, plated-through holes or dielectric inserts. These are modeled as volume polarization currents.Planar MoM Simulator ===

In [[EM.Picasso]] 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 [[EM.Picasso]] to analyze a patch antenna with a finite-sized dielectric substrate. If the substrate edge effects are of concern in your modeling problem, you must use [[EM.Tempo]] instead. On the other hand, since [[EM.Picasso]]'s Planar MoM simulationengine incorporates the Green's functions of the background structure into the analysis, only the unknown electric finite-sized traces like microstrips and magnetic currents slots are discretized as a collection of elementary currents with small finite spatial extentsby the mesh generator. As a result, the governing integral equations reduce to a system size of linear algebraic equations[[EM.Picasso]]'s computational problem is normally much smaller than that of [[EM.Tempo]]. In addition, whose solution determines the amplitudes [[EM.Picasso]] generates a hybrid rectangular-triangular mesh of all the elementary currents defined over the your planar structurewith a large number of equal-sized rectangular cells. Taking full advantage of all the symmetry and invariance properties of dyadic Green's meshfunctions often results in very fast computation times that easily make up for [[EM. Once Picasso]]'s limited applications. A particularly efficient application of [[EM.Picasso]] is the total currents are known, you can calculate the fields everywhere in the structuremodeling of periodic multilayer structures at oblique incidence angles.

<table><tr><td> [[Image:Info_iconART PATCH Fig12.png|40px]] Click here to learn more about thumb|left|480px|The hybrid planar mesh of the theory of '''[[Planar Method of Momentsslot-coupled patch antenna array.]]'''.</td></tr></table>

== Building EM.Picasso Features at a Planar Structure Glance ==

[[Image:PMOM9.png|thumb|270px|EM.Picasso's Add Substrate Layer dialog.]]=== Understanding the Background Structure Definition ===

EM.Picasso is intended for constructing and modeling planar layered structures. By a planar structure we mean one that contains a background substrate of laterally infinite extents, made <ul> <li> Multilayer stack-up with unlimited number of one or more material layers all stacked up vertically along the Z-axis. Objects of finite size are then interspersed among these substrate layers. The background structure in EM.Picasso is called the &quot;'''Layer Stackand trace planes</li> <li> PEC and conductive sheet traces for modeling ideal and non-up'''&quot;. The layer stack-up is always terminated from the top ideal metallic layouts</li> <li> PMC traces for modeling slot layouts</li> <li> Vertical metal interconnects and bottom by two infinite halfembedded dielectric objects</li> <li> Full periodic structure capability with inter-spaces. The terminating half-spaces might be the free spaceconnected unit cells</li> <li> Periodicity offset parameters to model triangular, hexagonal or a perfect conductor (PEC ground), or any material medium. Most planar structures used in RF and microwave applications such as microstrip-based components have a PEC ground at their bottom. Some structures like stripline components require two bounding grounds (PEC half-spaces) both at their top and bottom. other offset periodic lattice topologies</li></ul>

=== Planar Object Types Sources, Loads &amp; Ports ===

EM.Picasso groups objects by their trace type and their hierarchical location in the substrate layer stack<ul> <li> Gap sources on lines</li> <li> De-up. All the planar objects belonging to the same trace group are located embedded sources on the same substrate layer boundary lines for S parameter calculations</li> <li> Probe (coaxial feed) sources on vias</li> <li> Gap arrays with amplitude distribution and have the same color. All the embedded objects belonging to the same embedded set lie inside the same substrate layer phase progression</li> <li> Periodic gaps with beam scanning</li> <li> Multi-port and have the same color coupled port definitions</li> <li> RLC lumped elements on strips with series-parallel combinations</li> <li> Short dipole sources</li> <li> Import previously generated wire mesh solution as collection of short dipoles</li> <li> Plane wave excitation with linear and same material compositioncircular 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>

# '''PEC Traces''': These represent infinitesimally thin metallic planar objects that are deposited or metallized on or between substrate layers. PEC objects are modeled by surface electric currents.<ul># '''Slot Traces''': These are used to model slots <li> Optimized hybrid mesh with rectangular and apertures in PEC ground planes. Slot objects are always assumed to lie on an infinite horizontal PEC ground plane with zero thickness (which is not explicitly displayed in the project workspace). They are modeled by triangular cells</li> <li> Regular triangular surface magnetic currents.mesh</li># '''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 <li> Local meshing of such traces.trace groups</li># '''PEC Via Sets:''' These are metallic <li> Local mesh editing of planar polymesh objects such as shorting pins, interconnect vias, plated-through holes, etc. that are grouped together as prismatic object sets. The embedded objects are modeled as vertical volume conduction currents.</li># '''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 <li> Fast mesh generation of their host layer. The embedded dielectric array objects are modeled as vertical volume polarization currents.</li></ul>

[[Image:Info_icon.png|40px]] Click here to learn more about '''[[=== Planar Traces & Object Types]]'''.MoM Simulation ===

=== Defining the Layer Stack<ul> <li> 2.5-Up ===D 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>

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

The Stack<ul> <li> Current distribution intensity plots</li> <li> Near field intensity plots (vectorial -up Settings dialog has two tabsamplitude &amp; phase)</li> <li> Far field radiation patterns: '''Layer Hierarchy''' 3D pattern visualization and '''Embedded Sets'''2D Cartesian and polar graphs</li> <li> Far field characteristics such as directivity, beam width, axial ratio, side lobe levels and null parameters, etc. The Layer Hierarchy tab has a table that shows all the background layers in hierarchical order from the top half-space to the bottom half-space. It also lists the material label </li> <li> Radiation pattern of each layer, Z-coordinate an arbitrary array configuration of the bottom of each layer, its thickness (in project units) and material properties: permittivity (eplanar structure or periodic unit cell<sub/li>r <li> Reflection and Transmission Coefficients of Periodic Structures</subli>), permeability (µ <subli>r Monostatic and bi-static RCS&nbsp;</subli>) <li> Port characteristics: S/Y/Z parameters, electric conductivity (s) VSWR and magnetic conductivity (sSmith chart<sub/li>m </subli>) Touchstone-style S parameter text files for direct export to RF. There is also a column that lists the names Spice or its Device Editor</li> <li> Huygens surface generation</li> <li> Custom output parameters defined as mathematical expressions of embedded object sets inside each substrate layer, if any.standard outputs</li></ul>

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

You can delete a layer by selecting its row in the table [[EM.Picasso]] is intended for construction and clicking the '''Delete''' buttonmodeling of planar layered structures. To move By a layer up and downplanar structure we mean one that contains a background substrate of laterally infinite extents, click on its row to select and highlight itmade up of one or more material layers all stacked up vertically along the Z-axis. Then click either Planar objects of finite size are interspersed among these substrate layers. The background structure in [[EM.Picasso]] is called the &quot;'''Move UpLayer Stack-up''' or '''Move Down''' buttons consecutively to move the selected &quot;. The layer to the desired location in the stack-up. Note that you cannot delete or move 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.

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><tr><td> [[Image:PMOM11. The substrate layer dialog opens up, where you can change the layerpng|thumb|left|480px|EM.Picasso's label navigation tree and assigned colortrace types. In the material properties section of the dialog, you can change the name of the material and its properties: permittivity (e<sub>r]]</subtd>), permeability (µ<sub>r</subtr>), electric conductivity (s) and magnetic conductivity (s<sub>m</subtable>). 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 any substrate layer in the project units except for the top and bottom half-spaces.

[[Image:Info_icon.png|40px]] Click here to learn more about '''[[=== Defining Materials in EM.Cube]]'''.the Layer Stack-Up ===

For better visualization of your planar structure, When you start a new project in [[EM.Picasso displays a virtual domain in ]], there is always a default orange color to represent part of the infinite background structure. The size that consists of this virtual domain is a quarter wavelength offset from the largest bounding box that encompasses all the finite objects in the vacuum layer with a thickness of one project workspaceunit sandwiched between a vacuum top half-space and a PEC bottom half-space. You can change the size of the virtual domain Every time you open [[EM.Picasso]] or its display color switched to it from [[EM.Cube]]'s other modules, the Domain '''Stack-up Settings Dialog''' opens up. This is where you define the entire background structure. Once you close this dialog, which you can access either open it again by right-clicking the '''Layer Stack-up''' item in the '''Computational Domain''' [[File:domain_icon.png]] button section of the navigation tree and selecting '''Simulate ToolbarLayer Stack-up Settings...'''from the contextual menu. Or alternatively, or by selecting you can select the menu item '''Simulate &gt; Computational Domain &gt; Domain Layer Stack-up Settings...''' from the Simulate Menu or by right clicking the  The Stack-up Settings dialog has two tabs: '''Virtual DomainLayer Hierarchy''' item of the Navigation Tree and selecting '''Domain Settings...''' from the contextual menu, or using the keyboard shortcut '''Ctrl+AEmbedded Sets'''. Keep in mind The Layer Hierarchy tab has a table that shows all the virtual domain is only for visualization purpose and does not affect background layers in hierarchical order from the MoM simulationtop half-space to the bottom half-space. The virtual domain It also shows lists the substrate layers material composition of each layer, Z-coordinate of the bottom of each layer, its thickness (in translucent colors. If you assign different colors to your substrate layersproject units) and material properties: permittivity (&epsilon;_r), you have get permeability (&mu;_r), electric conductivity (&sigma;) and magnetic conductivity (&sigma;_m). There is also a better visualization column that lists the names of multilayer virtual domain box surrounding your project structureembedded object sets inside each substrate layer, if any.

When you start a new project in [[Planar ModuleImage:Info_icon.png|30px]], the project workspace looks empty, and there are no finite objects in itClick here to learn more about '''[[Preparing_Physical_Structures_for_Electromagnetic_Simulation#Using_EM. However, a default background structure is always present by defaultCube. Objects are defined as part of traces or embedded sets27s_Materials_List | Using EM. Once defined, you can see a list of project objects in the Cube'''Physical Structures Materials Database]]''' section of the navigation tree.

Traces and object sets For better visualization of your planar structure, [[EM.Picasso]] displays a virtual domain in a default orange color to represent part of the infinite background structure. The size of this virtual domain is a quarter wavelength offset from the largest bounding box that encompasses all the finite objects in the project workspace. You can be defined either change the size of the virtual domain or its display color from Layer Stack-up the Domain Settings dialog , which you can access either by clicking the '''Computational Domain''' [[File:domain_icon.png]] button of the '''Simulate Toolbar''', or from using the keyboard shortcut {{key|Ctrl+A}}. Keep in mind that the virtual domain is only for visualization purposes and its size does not affect the MoM simulation. The virtual domain also shows the navigation treesubstrate 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.>

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 the Navigation Tree Picasso]] groups objects by right clicking on one of the their trace type names and selecting '''Insert New PEC Tracetheir hierarchical location in the substrate layer stack-up...'''or '''Insert New PMC Trace...'''or '''Insert New Conductive Sheet Trace...'''A respective dialog opens up for setting trace is a group of finite-sized planar objects that have the trace same material properties. Once you close this dialog, it takes you directly to the Layer Stacksame color and same Z-up Settings dialog so that you can set coordinate. All the right position of 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.

As soon as you start drawing geometrical objects in the project workspace{| class="wikitable"|-! scope="col"| Icon! scope="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, the Sources, Devices & Other Physical Structure section of Object Types#Perfect Electric Conductor (PEC) |Perfect Electric Conductor (PEC) Trace]]| style="width:300px;" | Modeling perfect metal traces on the Navigation Tree gets populatedinterface between two substrate layers| style="width:150px;" | Only surface objects|-| style="width:30px;" | [[File:voxel_group_icon. The names png]]| style="width:250px;" | [[Glossary of traces are added under their respective trace type categoryEM.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 the names of finite metallization thickness| style="width:150px;" | Only surface objects appear under their respective trace group|-| style="width:30px;" | [[File:pmc_group_icon. At any timepng]]| style="width:250px;" | [[Glossary of EM.Cube's Materials, one Sources, Devices & Other Physical Object Types#Slot Trace |Slot Trace]]| style="width:300px;" | Modeling cut-out slot traces and only one trace is active in the project workspace. An active trace is where all the new apertures on an infinite PEC ground plane | style="width:150px;" | Only surface objects you draw belong to|-| style="width:30px;" | [[File:pec_group_icon. When you define a new tracepng]]| style="width:250px;" | [[Glossary of EM.Cube's Materials, it is set as active Sources, Devices & Other Physical Object Types#Embedded PEC Via Set |Embedded PEC Via Set]]| style="width:300px;" | Modeling small and you can immediately start drawing new short vertical vias and plated-through holes inside substrate layers| style="width:150px;" | Only surface objects on that trace|-| style="width:30px;" | [[File:diel_group_icon. You can also set any trace active at any time by right clicking its name on the Navigation Tree and selecting 'png]]| style="width:250px;" | [[Glossary of EM.Cube''Activate''' from the contextual menus 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. The name png]]| style="width:250px;" | [[Glossary of the active trace is always displayed in bold letter in the Navigation TreeEM.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|}

EM.Picasso has a special feature that makes construction of planar structures quite easy and straightforward. '''The active work plane of Click on each category to learn more details about it in the project workspace is always set at the plane of the active trace.''' In [[Glossary of EM.Cube]]'s other modulesMaterials, all objects are drawn in the XY plane (z = 0) by default. In [[Planar ModuleSources, Devices & Other Physical Object Types]], all new objects are drawn on a horizontal plane that is located at the Z-coordinate of the currently active trace. As you change the active trace or add a new trace, you will also change the active work plane.

You can manage your project's layer hierarchy from the Layer Stack-up Settings dialog. You can add, delete and move around substrate layers, define two types of metallic and slot traces in [[EM.Picasso]]: '''PEC Traces''' and embedded object sets'''Conductive Sheet Traces'''. Metallic and slot PEC traces can move among the interface planes represent infinitesimally thin (zero thickness) planar metal objects that are deposited or metallized on or between neighboring substrate layers. Embedded object sets including PEC vias and finite dielectric objects can move from substrate layer into anotherare modeled by surface electric currents. When you delete a trace from the Layer Stack-up Settings dialogConductive sheet traces, all of its objects are deleted from on the project workspaceother hand, toorepresent imperfect metals. You can also delete metallic They have a finite conductivity and slot traces or embedded object sets from the Navigation Treea very small thickness expressed in project units. To do so, right click A surface impedance boundary condition is enforced on the name surface of the trace or object set in the Navigation Tree and select '''Delete''' from the contextual menu. You can also delete all the traces or object sets of the same type from the contextual menu of the respective type category in the Navigation Treeconductive sheet objects.

By default, the last defined trace or embedded object set is active'''Slot Traces''' are used to model cut-out slots and apertures in PEC ground planes. You can activate any trace or embedded object set at any time for drawing new Planar slot objects. You can move one or more selected objects from any trace or embedded object set are always assumed to another group of the same type or of different type. First select lie on an object infinite horizontal PEC ground plane with zero thickness, which is not explicitly displayed in the project workspace or in the Navigation Treeand its presence is implied. ThenThey are modeled by surface magnetic currents. When a slot is excited, right click tangential electric fields are formed on the highlighted selection and select '''Move To &gt;''' from aperture, which can be modeled as finite magnetic surface currents confined to the contextual menu. This opens another sub-menu containing '''Planar''' and a list area of all the other [[EMslot.Cube]] modules that have already defined object groups. Select '''Planar''' or any In other available modulewords, and yet another sub-menu opens up with a list instead of all modeling the available traces and embedded object sets already defined in your project. Select electric surface currents on an infinite PEC ground around the desired groupslot, and all one can alternatively model the selected objects will move to that groupfinite-extent magnetic surface currents on a perfect magnetic conductor (PMC) trace. When selecting multiple Slot (PMC) objects from provide the Navigation Tree, make sure that you hold electromagnetic coupling between the keyboard's '''Shift Key''' or '''Ctrl Key''' down while selecting a group's name from the contextual menutwo sides of an infinite PEC ground plane.

=== Planar ModuleBesides planar metal and slot traces, [[EM.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: 's Rules &amp; Limitations ===''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.

# Terminating PEC ground planes at the top or bottom {{Note|The height of a planar structure are defined 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 an embedded object is always identical to provide electromagnetic coupling between its top and bottom sides. In this case, a PMC trace is rather introduced at the given Z-plane, which implies the presence thickness of an infinite PEC ground although it is not explicitly indicated in the Navigation Tree.# Metallic and slot traces cannot coexist on the same Z-plane. However, you can stack up multiple PEC and conductive sheet traces at the same Z-coordinate. Similarly, multiple PMC traces can be placed at the same Z-coordinate.# Metallic and slot traces are strictly defined at the interface planes between substrate layers. To define a suspended metallic trace in a its host substrate layer (as in the case of the center conductor of a stripline), you must split the dielectric layer into two thinner layers and place your PEC trace at the interface between them.# The current version of the Planar MoM simulation engine is based on a 2.5-D MoM formulation. Only vertical volume currents and no circumferential components are allowed on embedded objects. The 2.5-D assumption holds very well in two cases: (a) when embedded objects are very thin with a very small cross section (with lateral dimensions less than 2-5% of the material wavelength) or (b) when embedded objects are very short and sandwiched between two closely spaced PEC traces or grounds from the top and bottom.# The current release of [[EM.Cube]] allows any number of PEC via sets collocated in the same substrate layer. However, you can define only one embedded dielectric object set per substrate layer, and no vias sets collocated in the same layer. Note that the single set can host an arbitrary number of embedded dielectric objects of the same material properties.}}

[[Image:PMOM31Embedded object sets represent short material insertions inside substrate layers.png|thumb|400px|The Planar Mesh They can be metal or dielectric. Metallic embedded objects can be used to model vias, plated-through holes, shorting pins and interconnects. These are called PEC via sets. Embedded dielectric objects can be used to model air voids, thin films and material inserts in metamaterial structures. Embedded objects can be defined either from the Layer Stack-up Settings dialogor directly from the navigation tree.]]The method Open the &quot;Embedded Sets&quot; tab of moments (MoM) discretizes all the finitestack-sized objects of up dialog. This tab has a planar structure (excluding table that lists all the background structure) into embedded object sets along with their material type, the host substrate layer, the host material and their height. To add a new object set of elementary cells. The accuracy of , click the MoM numerical solution depends greatly arrow symbol on the quality {{key|Insert}} button of the dialog and resolution select one of the generated meshtwo options, '''PEC Via Set''' or '''Embedded Dielectric Set''', from the dropdown list. As This opens up a rule new dialog where first you have to set the host layer of thumb, the new object set. A dropdown list labeled &quot;'''Host Layer'''&quot; gives a mesh density list of about 20-30 cells per effective wavelength usually yields satisfactory resultsall the available finite substrate layers. YetYou can also set the properties of the embedded object set, for structures with lots including its label, color and material properties. Keep in mind that you cannot control the height of fine geometrical details or for highly resonant structuresembedded objects. Moreover, higher mesh densities may be requiredyou 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. AlsoVacuum is the default material choice. To define an embedded set from the navigation tree, right-click on the particular simulation data that you seek '''Embedded Object Sets''' item in a project also influence your choice the '''Physical Structure''' section of mesh resolutionthe navigation tree and select either '''Insert New PEC Via Set. For example..''' or '''Insert New Embedded Dielectric Set...''' The respective New Embedded Object Set dialog opens up, far field characteristics like radiation patterns are less sensitive where you can set the properties of the new object set. As soon as you close this dialog, it takes you to the mesh density than field distributions Layer Stack-up Settings dialog, where you can verify the location of the new object set on a structure with a highly irregular shape and a rugged boundarythe layer hierarchy.

EM.Picasso generates two types of mesh for a planar structure<table><tr><td> [[Image: a pure triangular and a hybrid triangular-rectangular. In both case, EM.Picasso attempts to create a highly regular mesh, in which most of the cells have almost equal areas. The hybrid mesh type tries to produce as many rectangular cells as possible especially in the case of objects with rectangular or linear boundaries. In connection or junction areas between adjacent objects or close to highly curved boundaries, the use of triangular cells is clearly inevitablePMOM23. png|thumb|550px|EM.Picasso's default mesh type is hybrid. The uniformity or regularity of mesh is an important factor in warranting a stable MoM numerical solutionLayer Stack-up dialog showing the Embedded Sets tab. ]] </td></tr></table> === Drawing Planar Objects on Horizontal Work Planes ===

The mesh density gives a measure of the number of cells per effective wavelength that are placed As soon as you start drawing geometrical objects in various regions of your planar structure. The higher the mesh densityproject workspace, the more cells are created on '''Physical Structure''' section of the geometrical objectsnavigation tree gets populated. Keep in mind that only the finite-sized objects The names of your structure traces are discretized. The free-space wavelength is defined as <math>\lambda_0 = \tfrac{2\pi f}{c}</math>added under their respective trace type category, where f is and the center frequency names of your project objects appear under their respective trace group. At any time, one and c only one trace is active in the speed project workspace. The name of light the active trace in the free spacenavigation tree is always displayed in bold letters. The effective wavelength An active trace is defined as <math>\lambda_{eff} = \tfrac{\lambda_0}{\sqrt{\varepsilon_{eff}}}</math>, where e_eff is all the effective permittivitynew objects you draw belong to. By default, [[EM.Picasso]] generates a hybrid mesh with a mesh density of 20 cells per effective wavelength. The effective permittivity is the last defined differently for different types of traces and trace or embedded object sets. This set is to make sure that enough cells are placed in areas that might feature higher field concentrationactive. For PEC and conductive sheet traces, the effective permittivity is defined as the larger of the permittivity of the two substrate layers just above and below You can immediately start drawing new objects on the metallic active 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 You can also set any trace. For embedded or object sets, set group active at any time by right-clicking on its name on the effective permittivity is defined as navigation tree and selecting '''Activate''' from the largest of the permittivities of all the substrate layers and embedded dielectric setscontextual menu.

[[Image:Info_icon.png|40px30px]] Click here to learn more about '''[[Mesh_Generation_Schemes_in_EM.CubeBuilding Geometrical Constructions in CubeCAD#Working_with_Mesh_Generator Transferring Objects Among Different Groups or Modules | Working with Mesh Generator Moving Objects among Different Groups]]'''. <table><tr><td> [[Image:PMOM23B.png|thumb|280px|EM.Picasso's Navigation Tree populated with planar objects.]] </td></tr></table>

[[Image:Info_iconEM.png|40pxPicasso]] Click here to learn more about EMhas a special feature that makes construction of planar structures very convenient and straightforward.Picasso's '''[[Mesh_Generation_Schemes_in_EM<u>The horizontal Z-plane of the active trace or object set group is always set as the active work plane of the project workspace.Cube#The_Triangular_Surface_Mesh_Generator | Triangular Surface Mesh Generator]]'''</u> That means all new objects are drawn at the Z-coordinate of the currently active trace. As you change the active trace group or add a new one, the active work plane changes accordingly.

{{Note| In [[Image:Info_icon.png|40px]] Click here to learn more about EM.Picasso's '''[[Mesh_Generation_Schemes_in_EM.Cube#The_Hybrid_Planar_Mesh_Generator | Hybrid Planar Mesh Generator]]''', you cannot modify the Z-coordinate of an object as it is set and controlled by its host trace.}}

[[Image:Info_iconEM.png|40pxPicasso]] Click here does not allow you to learn more about draw 3D or solid CAD objects. The solid object buttons in the '''Object Toolbar''' are disabled to prevent you from doing so. In order to create vias and embedded object, you simply have to draw their cross section geometry using planar surface CAD objects. [[Mesh_Generation_Schemes_in_EMEM.Cube#General_Rules_of_Planar_Hybrid_Mesh_Generator| General Rules of Planar Hybrid Mesh GeneratorPicasso]]extrudes and extends these planar objects across their host layer automatically and displays them as 3D wireframe, prismatic objects. The automatic extrusion of embedded objects happens after mesh generation and before every planar MoM simulation. You can enforce this extrusion manually by right-clicking the '''Layer Stack-up''' item in the "Computational Domain" section of the navigation tree and selecting '''Update Planar Structure'''from the contextual menu.

[[Image:Info_icon.png{{Note|40px]] Click here to learn more about '''In [[Mesh_Generation_Schemes_in_EMEM.Cube#Refining_the_Planar_Mesh_Locally| Refining the Planar Mesh LocallyPicasso]]''', you can only draw horizontal planar surface CAD objects.}}

<td> [[Image:PMOM48FPMOM23A.png|thumb|340px620px|Geometry of A planar structure with a multilayer slottwo-coupled patch array.]] </td><td> [[Image:PMOM48G.png|thumb|380px|Hybrid planar mesh layer conductor-backed substrate, two PEC patches located at the tops of the slot-coupled lower and upper substrate layers, four PEC vias located inside the lower substrate layer between the lower patch arrayand bottom ground and an embedded dielectric film located inside the top substrate layer sandwiched between the two patches.]] </td>

== Excitation Sources = EM.Picasso's Special Rules ===

Your # PEC ground planes at the top or bottom of a planar structure must be excited by some sort are regarded and modeled as PEC top or bottom half-spaces, respectively.# A PEC ground plane placed in the middle of a signal source that induces electric currents on metal parts substrate stack-up requires at least one slot object to provide electromagnetic coupling between its top and magnetic currents on bottom sides. In this case, a slot tracestrace is rather introduced at the given Z-plane, which also implies the presence of an infinite PEC ground. The excitation source you choose depends # Metallic and slot traces cannot coexist on the observables same Z-plane. However, you seek 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 projectPEC trace at the interface between them. # [[EM.Picasso provides the following source types for exciting planar structures]]'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 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.

* [[Planar_MoM_Source_Types#Gap_Sources|Gap == EM.Picasso's Excitation Sources]]* [[Planar_MoM_Source_Types#Probe_Sources|Probe Sources]]* [[Planar_MoM_Source_Types#De-Embedded_Sources|De-embedded Sources]]* [[Planar_MoM_Source_Types#Short_Dipole_Sources|Short Dipole Sources]]* [[Planar_MoM_Source_Types#Plane_Wave_Sources|Plane Wave Sources]]* [[#Huygens Sources|Huygens Sources]]==

For antennas and Your planar circuits, where you typically define one or more ports, you usually use lumped sources. A lumped structure must be excited by some sort of signal source is indeed a gap discontinuity that is placed induces electric surface currents on the path of an electric or metal parts, magnetic current flowsurface currents on slot traces, where a voltage and conduction or current source is connected to inject a signal. Gap sources are placed across metal or slot traces. Probe sources are placed across polarization volume currents on vertical PEC vias. A de-and embedded objects. The excitation source is a special type of gap source that is placed near you choose depends on 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), observables you excite it with a plane wave sourceseek in your project. 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.CubePicasso]] computational module and bring them as a new provides the following source to excite your types for exciting planar structure.structures:

{| class="wikitable"|-! scope="col"| Icon! scope="col"| Source Type! scope="col"| Applications! scope="col"| Restrictions|-| style="width:30px;" | [[ImageFile:Info_icongap_src_icon.png]]|40px[[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Strip Gap Circuit Source |Strip Gap Circuit Source]] Click here to learn more about '| style="width:300px;" | General-purpose point voltage source (or filament current source on slot traces)| style="width:300px;" | Associated with a PEC rectangle strip|-| style="width:30px;" | [[File:probe_src_icon.png]]| [[Glossary of 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.png]]| [[Glossary of EM.Cube's 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;" | [[Planar MoM File:hertz_src_icon.png]]| [[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-directional physical radiator| style="width:300px;" | None, stand-alone source|-| style="width:30px;" | [[File:plane_wave_icon.png]]| [[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Plane Wave |Plane Wave Source]]| style="width:300px;" | Used for modeling scattering & computation of reflection/transmission characteristics of periodic surfaces| style="width:300px;" | None, stand-alone source|-| style="width:30px;" | [[File:huyg_src_icon.png]]| [[Glossary of EM.Cube'''s Materials, 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|}

Click on each category to learn more details about it in the [[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types]]. For antennas and planar circuits, where you typically define one or more ports, you usually use lumped sources. [[EM.Picasso]] provides three types of lumped sources: gap source, probe source and de-embedded source. 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. 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 the line is a function of its width (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 (local X direction). The 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 slot 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.  {{Note| You can realize a coplanar waveguide (CPW) in [[EM.Picasso]] using two parallel slot lines with two aligned, collocated gap sources.}} [[Image:Info_icon.png|40px]] Click here to learn more about '''[[Common_Excitation_Source_Types_in_EM.CubePreparing_Physical_Structures_for_Electromagnetic_Simulation#Defining_FiniteModeling_Finite-Sized_Source_Arrays | Using Source Arrays for Modeling Antenna Arrays]]'''. A 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, 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 = 0° corresponding to a normally incident plane wave propagating along the -Z direction with a +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 [[EM.Cube]] computational module.  <table><tr><td> [[Image:PMOM64A.png|thumb|550px|A multilayer planar structure containing a CPW line with a single coupled port and a lumped element on an overpassing metal strip.]] </td></tr></table>

Lumped elements are components, devices, or circuits whose overall dimensions are very small compared to the wavelength. As a result, they are considered to be dimensionless compared to the dimensions of a mesh cell. In fact, a lumped element is equivalent to an infinitesimally narrow gap that is placed in the path of current flow, across which the device's governing equations are enforced. Using Kirkhoff's laws, these device equations normally establish a relationship between the currents and voltages across the device or circuit. Crossing the bridge to Maxwell's domain, the device equations must now be cast into a from o boundary conditions that relate the electric and magnetic currents and fields. [[EM.Picasso ]] allows you to define passive circuit elements: '''Resistors''' (R), '''Capacitors''' (C), '''Inductors''' (L), and series and parallel combinations of them.

[[Image:Info_icon.png|40px]] Click here to learn more about '''[[Modeling_Lumped_Elements,_Circuits_%26_Devices_in_EM.CubePreparing_Physical_Structures_for_Electromagnetic_Simulation#Defining_Lumped_Elements_in_EM.Picasso_.26_EM.Libera Modeling_Lumped_Elements_in_the_MoM_Solvers | Defining Lumped Elements]]'''.

[[Image:Info_icon.png|40px]] Click here for a general discussion of '''[[Modeling_Lumped_Elements,_Circuits_%26_Devices_in_EMPreparing_Physical_Structures_for_Electromagnetic_Simulation#A_Review_of_Linear_.Cube26_Nonlinear_Passive_.26_Active_Devices | Linear Passive & Nonlinear Active Devices]]'''.

[[Image:PMOM52.png|thumb|400px|EM.Picasso=== Calculating Scattering Parameters Using Prony's Port Definition dialog.]][[Image:PMOM53.png|thumb|300px|The Edit Port dialog.]][[Image:PMOM51(2).png|thumb|600px|Coupling gap sources in the Port Definition dialog by associating more than one source with a single port.]]Method ===

Ports are used in One can calculate the scattering parameters of a planar structure directly by analyzing the current distribution patterns on the port transmission lines. The discontinuity at the end of a port line typically gives rise to order and index a standing wave pattern that can clearly be discerned in the sources for calculation line's current distribution. From the location of circuit [[parameters]] such as scattering (S), impedance (Z) the current minima and admittance (Y) [[parameters]]maxima and their relative levels, one can determine the reflection coefficient at the discontinuity, i. In [[EMe.Cube]]'s [[Planar Module]]the S₁₁ parameter. A more robust technique is Prony’s method, you which is used for exponential approximation of functions. A complex function f(x) can use be expanded as a sum of complex exponentials in the following types of sources to define portsform:

* Gap Sources:<math> f(x) \approx \sum_{n=1}^N c_i e^{-j\gamma_i x} </math>* Probe Sources* Active Lumped Elements* De<!--[[File:PMOM73.png]]--Embedded Sources>

Ports where c_i are defined complex coefficients and &gamma;_i are, in the '''Observables''' section of the Navigation Treegeneral, complex exponents. Right click on From the '''Port Definition''' item physics of the Navigation Tree and select '''Insert New Port Definition...''' from the contextual menu. The Port Definition Dialog opens uptransmission lines, showing the default port assignmentswe know that lossless lines may support one or more propagating modes with pure real propagation constants (real &gamma;_i exponents). If you have N sources in your planar structure, then N default ports are definedMoreover, line discontinuities generate evanescent modes with one port assigned to each source according to their order on pure imaginary propagation constants (imaginary &gamma;_i exponents) that decay along the Navigation Tree. Note that your project can have mixed gap and probes sources line as well as active lumped element sourcesyou move away from the location of such discontinuities.

'''You can define any number of ports equal In practical planar structures for which you want to or less than calculate the total number of sources in your projectscattering parameters, each port line normally supports one, and only one, dominant propagating mode.''' The Port List of the dialog shows Multi-mode transmission lines are seldom used for practical RF and microwave applications. Nonetheless, each port line carries a list superposition of all incident and reflected dominant-mode propagating signals. An incident signal, by convention, is one that propagates along the ports in ascending orderline towards the discontinuity, with their associated sources and where the phase reference plane is usually established. A reflected signal is one that propagates away from the portplane. Prony's characteristic impedance, which is 50S by default. You method can delete any port by selecting it be used to extract the incident and reflected propagating and evanescent exponential waves from the Port List standing wave data. From a knowledge of the amplitudes (expansion coefficients) of the incident and clicking reflected dominant propagating modes at all ports, the '''Delete''' button scattering matrix of the dialog. Keep in mind that after deleting a multi-port, you will have a source in your project without any port assignment and make sure that structure is what you intendthen calculated. You can change In Prony's method, the characteristic impedance quality of a port by selecting it from the Port List and clicking S parameter extraction results depends on the '''Edit''' button quality of the dialogcurrent samples and whether the port lines exhibit a dominant single-mode behavior. This opens up the Edit Port dialog, where you Clean current samples can enter be drawn in a new value in region far from sources or discontinuities, typically a quarter wavelength away from the box labeled '''Impedance'''two ends of a feed line.

<table><tr><td> [[Image:Info_iconPMOM71.png|40px]] Click here to learn more about thumb|600px|Minimum and maximum current locations of the theory of '''[[Computing Port Characteristics in Planar MoMstanding wave pattern on a microstrip line feeding a patch antenna.]]'''.</td></tr></table>

=== Modeling Defining Independent & Coupled Ports ===

Sources can be coupled Ports are used in a planar structure to each other to model coupled strip lines order and index the sources for calculation of circuit parameters such as scattering (CPSS) on metal traces or coplanar waveguides , impedance (CPWZ) on slot tracesand admittance (Y) parameters. SimilarlyIn [[EM.Picasso]], probe sources may be coupled to each other. Coupling two you can use one or more sources does not change of the way they excite a planar structure. It is intended only for the purpose following types of S parameter calculation. The feed lines or vias which host the coupled sources are usually parallel and aligned with one another and they are all grouped together as a single transmission line represented by a single port. This single &quot;coupled&quot; port then interacts with other coupled or uncoupled to define ports.:

You couple two or more sources using the '''Port Definition Dialog'''. To do so, you need to change the default port assignments. First, delete all the ports that are to be coupled from the Port List of the dialog. Then, define a new port by clicking the '''Add''' button of the dialog. This opens up the Add Port dialog, which consists of two tables: '''Available''' sources on the left and '''Associated''' sources on the right. A right arrow ('''* Gap Sources* Probe Sources* Active Lumped Elements* De--&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 of all the sources that you deleted earlier. You may even see more available sources. Select all the sources that you want to couple and move them to the &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.Embedded Sources

{{Note|It is your responsibility to set up coupled Ports are defined in the '''Observables''' section of the navigation tree. You can define any number of ports and coupled [[Transmission Lines]] properly. For example, equal to excite or less than the desirable odd mode total number of a coplanar waveguide (CPW)sources in your project. If you have N sources in your planar structure, you need then N default ports are defined, with one port assigned to create two rectangular slots parallel each source according to and aligned with each other and place two gap sources their order on them with the same offsets and opposite polaritiesnavigation tree. To excite the even mode of the CPW, you use the same polarity for the two collocated Note that your project can have mixed gap and probes sourcesas well as active lumped element sources on PEC and slot traces or vias. Whether you You can also couple ports together to define a coupled port for the transmission lines such as coupled strips (CPS) or coplanar waveguides (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).}}

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 ModuleImage:Info_icon.png|40px]]Click here to learn more about '''[[Preparing_Physical_Structures_for_Electromagnetic_Simulation#Modeling_Coupled_Sources_. 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-planes26_Ports | Modeling Coupled Ports]]'''.

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]]Picasso's [[Planar Module]] offers the following observables:Simulation Data & Observables ==

If you run {| class="wikitable"|-! scope="col"| Icon! scope="col"| Simulation Data Type! scope="col"| Observable Type! scope="col"| Applications! scope="col"| Restrictions|-| style="width: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 simulation without having defined any observables, no data will be generated at specified plane in the end 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 simulationradiation pattern and additional radiation characteristics such as directivity, axial ratio, side lobe levels, etc. Some observables require a certain type | style="width:250px;" | None|-| style="width:30px;" | [[File:rcs_icon.png]]| style="width:150px;" | Far-Field Scattering Characteristics| style="width:150px;" | [[Glossary of excitation sourceEM. For example, port characteristics will be calculated only if Cube's Simulation Observables & Graph Types#Radar Cross Section (RCS) |Radar Cross Section (RCS)]] | style="width:300px;" | Computing the project contains bistatic and monostatic RCS of a port definition, which in turn requires the existence target| style="width:250px;" | Requires a plane wave source|-| style="width:30px;" | [[File:port_icon.png]]| style="width:150px;" | Port Characteristics| style="width:150px;" | [[Glossary of at least 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 gap of these source types: lumped, distributed, microstrip, CPW, coaxial or probe or dewaveguide port|-embedded source| style="width:30px;" | [[File:period_icon. The periodic characteristics (png]]| style="width:150px;" | Periodic Characteristics| style="width:150px;" | No observable required | style="width:300px;" | Computing the reflection and transmission coefficients) are calculated only if the structure has of a periodic domain and excited by surface| style="width:250px;" | Requires a plane wave sourceand 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|}

=== Planar ModuleClick on each category to learn more details about it in the [[Glossary of EM.Cube's Simulation Modes ===Observables & Graph Types]].

The simplest simulation type in [[EM.Cube]] is an analysis. In this mode, the If your planar structure in your project workspace is meshed at the center frequency of the project. [[EM.Cube]] generates an input file at this single frequencyexcited by gap sources or probe sources or de-embedded sources, and one or more ports have been defined, the Planar planar MoM simulation engine is run once. Upon completion of calculates the planar MoM simulationscattering, a number impedance and admittance (S/Z/Y) parameters of data files the designated ports. The scattering parameters are generated depending defined based on the observables you have port impedances specified in the project's Port Definition dialog. If more than one port has been defined in your the project. An analysis is a single-run simulation, the S/Z/Y matrices of the multiport network are calculated.

[[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 Electric and a collection of simulation data magnetic currents are generated. At the end fundamental output data of a multi-run planar MoM simulation, you can graph the simulation results in EM.Grid or you can animate After the 3D simulation data from numerical solution of the Navigation Tree. For example, in a frequency sweepMoM linear system, they are found using the frequency of the project is varied over its specified bandwidth. Port characteristics are usually plotted vs. frequency, representing your planar structuresolution vector '''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]I]'s [[Planar Module]] currently provides '' and the following types definitions of multi-run simulation modesthe electric and magnetic vectorial basis functions:

* Frequency Sweep* Parametric Sweep* :<math> \mathbf{[[Optimization]X]* HDMR Sweep}_{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:PMOM80Note 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. 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]]

Figure 1: Selecting {{Note|You have to define a simulation mode in [[Planar Module]]'s Simulation Run dialogseparate current distribution observable for each individual trace or embedded object set.}}

To run a planar MoM analysis of your project structure, open the Run Simulation Dialog by clicking the '''Run''' [[File:run_iconEM.pngPicasso]] button on allows you to visualize the '''Simulate Toolbar''' or select '''Menu''' '''&gt;''' '''Simulate &gt;''' '''Run''' or use the keyboard shortcut '''Ctrl+R'''near fields at a specific field sensor plane. The '''Analysis''' option of the '''Simulation Mode''' dropdown list is selected by defaultNote that unlike [[EM. Once you click the Cube]]'''Run''' buttons other computational modules, the simulation startsnear field calculations in [[EM. A new window, called the '''Output Window''', opens up that reports the different stages Picasso]] usually takes a significant amount of simulation and the percentage of the tasks completed at any time. After the simulation This is successfully completed, a message pops up and reports due to the fact that at the end of a planar MoM simulation. In certain cases like calculating scattering , the fields are not available anywhere (as opposed to [[parametersEM.Tempo]] ), and their computation requires integration of a circuit or reflection / transmission characteristics complex dyadic Green's functions of a periodic surface, some results are also reported in the Output Window. At the end of a simulation, you need multilayer background structure as opposed to click the free space Green'''Close''' button of the Output Window to return to the project workspaces functions.

{{Note|Keep in mind that since [[File:PMOM78EM.pngPicasso]]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 (at least one or few wavelengths) away from the scatterers to produce acceptable results.}}

Figure 1: <table><tr><td> [[Planar ModuleImage:PMOM116.png|thumb|left|600px|Near-zone electric field map above a microstrip-fed patch antenna.]]'s Simulation Run dialog</td></tr><tr><td> [[Image:PMOM117.png|thumb|left|600px|Near-zone magnetic field map above a microstrip-fed patch antenna.]] </td></tr></table>

=== Stages Of A Planar Even though [[EM.Picasso]]'s MoM Analysis ===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. 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.

[[EMImage:Info_icon.Cubepng|30px]]'s Planar MoM simulation engine uses a particular formulation of the method of moments called mixed potential integral equation (MPIE). Due Click here to high-order singularities, learn more about 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 theory of using these slowly converging dyadic Green's function, the MPIE formulation uses vector and scalar potentials. These include vector electric potential ''[[Defining_Project_Observables_%26_Visualizing_Output_Data#Using_Array_Factor_to_Model_Antenna_Arrays | Using Array Factors to Model Antenna Arrays ]]'A(r)''', scalar electric potential K^&Phi;'''(r)''', vector magnetic potential '''F(r)''' and scalar magnetic potential K^&Psi;'''(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<table><tr><td> [[Image: matrix fill and linear system inversionPMOM119. In the first stage, the moment matrix and excitation vector are calculated. In the second stage, the MoM system png|thumb|left|600px|3D polar radiation pattern plot of linear equations is inverted using one of the several available matrix solvers to find the unknown coefficients of all the basis functionsa microstrip-fed patch antenna. 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;.</td></tr></table>

A planar MoM simulation involves a number of numerical <table><tr><td> [[parameters]] that take preset default values unless you change themImage:PMOM125. You can access these [[parameters]] and change their values by clicking png|thumb|left|600px|An example of the '''Settings''' button next to the '''Select Engine''' dropdown list in the [[Planar Module]]'s Simulation Run dialog3D monostatic radar cross section plot of a patch antenna. 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.</td></tr></table>

The == Discretizing a 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 Structure in the box labeled '''Max Dimensions for Quasi-Static Analysis'''EM.Picasso ==

In The method of moments (MoM) discretizes all the &quot;Spectral Domain Integration&quot; section finite-sized objects of a planar structure (excluding the dialog, you can background structure) into a set a value to '''Max Spectral Radius in k0''', which has a default value of 30elementary cells. This means that Both the infinite spectral-domain integrals in the spectral variable k_&rho; are pre-calculated quality and tabulated up to a limit resolution of the generated mesh greatly affect the accuracy of 30k₀, where k₀ is the free space propagation constantMoM numerical solution. These integrals may converge much faster based on The mesh density gives a measure of the specified Convergence Rate for Integration described earlier. However, number of cells per effective wavelength that are placed in certain cases involving highly oscillatory integrands, much larger integration limits like 100k₀ might be needed to warrant adequate convergencevarious regions of your planar structure. For spectral-domain integration along The higher the real k_&rho; axismesh density, the interval [0more cells are created on the finite-sized geometrical objects. As a rule of thumb, Nk₀] is subdivided into a large number mesh density of subabout 20-intervals, within each an 8-point Gauss-Legendre quadrature is applied. The next parameter, '''No. Radial Integration Divisions 30 cells per k₀''', determines how small these intervals should beeffective wavelength usually yields satisfactory results. By default, 2 divisions are used But for the interval [0structures with lots of fine geometrical details or for highly resonant structures, k₀]higher mesh densities may be required. In other words, the length The particular output data that you seek in a simulation also influence your choice of each integration sub-interval is k₀/2. You can increase the mesh resolution of integration by increasing this value above 2. FinallyFor example, instead of 2D Cartesian integration in far field characteristics like radiation patterns are less sensitive to the spectral domain, mesh density than field distributions on structures with a polar integration is performed. You can set the '''No. of Angular Integration Points''', which has a default value of 100highly irregular shapes and boundaries.

<table><tr><td> [[FileImage:PMOM79PMOM31.png|thumb|400px|The Planar Mesh Settings dialog.]]</td></tr></table>

Figure 1EM.Picasso provides two types of mesh for a planar structure: The Planar MoM Engine Settings dialoga pure triangular surface mesh and a hybrid triangular-rectangular surface mesh. In both case, EM.Picasso attempts to create a highly regular mesh, in which most of the cells have almost equal areas. For planar structures with regular, mostly rectangular shapes, the hybrid mesh generator usually leads to faster computation times.

=== Planar Module[[Image:Info_icon.png|30px]] Click here to learn more about 's Linear System Solvers ===''[[Preparing_Physical_Structures_for_Electromagnetic_Simulation#Working_with_EM.Cube.27s_Mesh_Generators | Working with Mesh Generator]]'''.

After the MoM impedance matrix '''[Z[Image:Info_icon.png|30px]] Click here to learn more about ''' (not to be confused with the impedance [[parametersPreparing_Physical_Structures_for_Electromagnetic_Simulation#The_Triangular_Surface_Mesh_Generator | EM.Picasso's Triangular Surface Mesh Generator]]) and excitation vector '''[V]''' have been computed through the matrix fill process, the planar MoM simulation engine is ready to solve the system of linear equations:.

:<mathtable><tr><td> \mathbf{[Z]}_{N\times N} \cdot \mathbf{[IImage:PMOM48F.png|thumb|left|420px|Geometry of a multilayer slot-coupled patch array.]}_{N\times 1} = \mathbf{[V]}_{N\times 1} </mathtd><!--/tr><tr><td> [[FileImage:PMOM81PMOM48G.png|thumb|left|420px|Hybrid planar mesh of the slot-coupled patch array.]]--</td></tr></table>

where '''<table><tr><td> [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[Image:PMOM48H. In the above equation, N is the dimension png|thumb|left|420px|Details of the linear system and equal to the total number of basis functions in the hybrid planar mesh. [[EM.Cube]]'s linear solvers compute the solution vector'''[I]''' of the above systemslot-coupled patch array around discontinuities. 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.</td></tr></table>

There are a large number of numerical methods for solving systems of linear equations. These methods are generally divided into two groups: direct solvers and iterative solvers. Iterative solvers are usually based on matrix-vector multiplications. Direct solvers typically work faster for matrices of smal to medium size (N&lt;3,000). [[EM.Cube]]'s [[=== The Hybrid Planar Module]] offers five linear solvers:Mesh Generator ===

Of the above list, LU is The mesh density gives a direct solver, while measure of the rest number of cells per effective wavelength that are iterative solversplaced in various regions of your planar structure. BiCG The effective wavelength is a relatively fast iterative solverdefined as <math>\lambda_{eff} = \tfrac{\lambda_0}{\sqrt{\varepsilon_{eff}}}</math>, but it works only for symmetric matriceswhere e_eff is the effective permittivity. You cannot use BiCG for periodic structures or planar structures that contain both metal and slot traces at different planesBy default, as their MoM matrices are not symmetric[[EM.Picasso]] generates a hybrid mesh with a mesh density of 20 cells per effective wavelength. The three solvers BCG-STAB, GMRES and TtFQMR work well effective permittivity is defined differently for both symmetric and asymmetric matrices and they also belong to a class different types of solvers called '''Krylov Sub-space Methods'''traces and embedded object sets. In particular, the GMRES method always provides guaranteed unconditional convergenceThis is to make sure that enough cells are placed in areas that might feature higher field concentration.

[[EM.Cube]]'s [[Planar Module]], by default, provides a &quot;'''Automatic'''&quot; solver option that picks the best method based on the settings * For PEC and size of the numerical problem. For linear systems with a size less than N = 3,000conductive sheet traces, the LU solver effective permittivity is used. For defined as the 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 permittivity of the iterative solvers is used, utilizing a row-indexed sparse storage scheme. You can override the automatic solver option two substrate layers just above 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 below the Planar MoM Engine Settings dialogmetallic trace. There are also a number of other [[parameters]] related to * For slot traces, 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 effective permittivity is usually expressed defined as a multiple of the systems size. The default value mean (average) of '''Max No. of Solver Iterations / System Size''' is 3. For extremely large systems, sparse versions of iterative solvers are used. In this case, the elements permittivity of the matrix are thresholded with respect to two substrate layers just above and below the larges elementmetallic trace. The default value of '''Threshold for Sparse Solver''' is 1E-6* For embedded object sets, meaning that all the matrix elements whose magnitude effective permittivity is less than 1E-6 times defined as the large matrix elements are set equal to zero. There are two more [[parameters]] that are related to largest of the Automatic Solver option. These are &quot;''' User Iterative Solver When System Size &gt;'''&quot; with a default value permittivities 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 all the automatic solver when to switch between direct substrate layers and iterative solvers and when to switch to the sparse version of iterative solversembedded dielectric sets.

If your computer has an Intel CPU, then <table><tr><td> [[EMImage:PMOM32.png|thumb|360px|A comparison of triangular and planar hybrid meshes of a rectangular patch.Cube]] offers special versions of all the above linear solvers that have been optimized for Intel CPU platforms. These optimal solvers usually work 2-3 time faster than their generic counterparts. When you install </td><td> [[EMImage:PMOM30.Cube]], the option to use Intel-optimized solvers is already enabledpng|thumb|360px|Mesh of two rectangular patches at two different substrate planes. However, you can disable this option (e.g. if your computer The lower substrate layer has a non-Intel CPU)higher permittivity. To do that, open the [[EM.Cube]]'s Preferences Dialog from '''Menu &gt; Edit &gt; Preferences''' or using the keyboard shortcut '''Ctrl+H'''. Select the Advanced tab of the dialog and uncheck the box labeled &quot;''' Use Optimized Solvers for Intel CPU'''&quot;.</td></tr></table>

[[Image:PMOM127.png|thumb|400px|Settings adaptive frequency sweep parameters The integrity of the planar mesh and its continuity in the junction areas directly affects the quality and accuracy of the simulation results. EM.Picasso's Frequency Settings Dialoghybrid planar mesh generator has some rules that are catered to 2.]]=== Running Uniform and Adaptive Frequency Sweeps ===5-D MoM simulations:

In a frequency sweep* If two connected rectangular objects have the same side dimensions along their common linear edge with perfect alignment, the operating frequency of a planar structure rectangular junction mesh is varied during each sweep runproduced. [[EM.Cube]]'s [[Planar Module]] offers * If two types of frequency sweep: Uniform and Adaptive. In a uniform frequency sweepconnected rectangular objects have different side dimensions along their common linear edge or have edge offset, a set of triangular cells is generated along the frequency range and edge of the number of frequency samples are specifiedobject with the larger side. The samples are equally spaced over * Rectangle strip objects that host a gap source or a lumped element always have a rectangular mesh around the frequency rangegap area. At * If two objects reside on the end of each individual frequency runsame Z-plane, belong to the output data are collected same trace group and stored. At have a common overlap area, they are first merged into a single object for the end purpose of meshing using the frequency sweep, &quot;Boolean Union&quot; operation. * Embedded objects have prismatic meshes along the 3D data can be visualized and/Z-axis.* If an embedded object is located underneath or animatedabove a metallic trace object or connected from both top and bottom, it is meshed first and the 2D data can be graphed in EM.Gridits mesh is then reflected on all of its attached horizontal trace objects.

Frequency sweeps are often performed to study the frequency response of a planar structure<table><tr><td> [[File:PMOM36. In particular, the variation of scattering png|250px]] [[parametersFile:PMOM38.png|250px]] [[File:PMOM37.png|250px]] like S<sub/td>11</subtr> (return loss) and S<subtr>21</subtd> (insertion loss) with frequency are of utmost interest. When analyzing resonant structures like patch antennas or Two overlapping planar filters over large frequency ranges, you may have to sweep objects and a large number comparison of frequency samples to capture their behavior with adequate details. The resonant peaks or notches are often missed due to the lack of enough resolutiontriangular and hybrid planar meshes. </td></tr><tr><td> [[EMFile:PMOM33.Cubepng|250px]]'s [[Planar Module]] offers a powerful adaptive frequency sweep option for this purposeFile:PMOM35. It is based on the fact that the frequency response of a physical, causal, multiport network can be represented mathematically using a rational function approximation. In other words, the S [[parameterspng|250px]] of a circuit exhibit a finite number of poles and zeros over a given frequency range. [[EMFile:PMOM34.Cubepng|250px]] first starts with very few frequency samples </td></tr><tr><td> Edge-connected rectangular planar objects and tries to fit rational functions of low orders to the scattering [[parameters]]. Then, it increases the number of samples gradually by inserting intermediate frequency samples in a progressive mannercomparison their triangular and hybrid planar meshes. At each iteration cycle, all the possible rational functions of higher orders are tried out. The process continues until adding new intermediate frequency samples does not improve the resolution of the &quot;S<sub/td>ij</subtr></table>&quot; curves over the given frequency range. In that case, the curves are considered as having converged.

You must have defined one or more ports for your planar structure run an adaptive frequency sweep. Open the Frequency Settings dialog from the Simulation Run dialog and select the '''Adaptive''' option of '''Frequency Sweep Type'''. You have to set values for '''Minimum Number of Samples''' and '''Maximum Number of Samples'''. Their default values are 3 and 9, respectively. You also set a value for the '''Convergence Criterion''', which has a default value of 0.1. At each iteration cycle, all the S <table><tr><td> [[parametersFile:PMOM39.png|375px]] are calculated at the newly inserted frequency samples, and their average deviation from the curves of the last cycle is measured as an error. When this error falls below the specified convergence criterion, the iteration is ended. If [[EMFile:PMOM40.Cubepng|375px]] reaches the specified maximum number </td></tr><tr><td> Meshes of iterations and the convergence criterion has not yet been met, the program will ask you whether to continue the process or exit it short and stoplong vertical PEC vias connecting two horizontal metallic strips.</td></tr></table>

{{Note|For large frequency ranges, you may have to increase both === Refining the minimum and maximum number of samples. Moreover, remeshing the planar structure at each frequency may prove more practical than fixing the mesh at the highest frequency.}}Planar Mesh Locally ===

== Working with EMIt 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 connected objects, where larger current concentrations are usually observed at sharp corners, or at the junction areas between metallic traces and PEC vias, as well as the areas around gap sources and lumped elements, which create voltage or current discontinuities.Picasso Simulation Data ==

[[Image:PMOM130The Planar Mesh Settings dialog gives a few options for customizing your planar mesh around geometrical and field discontinuities.png|thumb|400px|Changing The check box labeled &quot;'''Refine Mesh at Junctions'''&quot; increases the graph type by editing 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 data fileshort transmission line connected from both ends. The check box labeled &quot;'s properties''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 Output Simulation Data ===default generated mesh to see if the current mesh settings have produced an acceptable mesh.

Depending on the source type and the types of observables defined in a projectSometimes EM.Picasso's default mesh may contain very narrow triangular cells due to very small angles between two edges. In some rare cases, extremely small triangular cells may be generated, whose area is a number small fraction of output data are generated the average mesh cell. These cases typically happen at the end junctions and other discontinuity regions or at the boundary of a planar MoM simulationhighly irregular geometries with extremely fine details. Some of these data are 2D In such cases, increasing or decreasing the mesh density by nature one or few cells per effective wavelength often resolves that problem and some are 3Deliminates those defective cells. The output simulation data generated by 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 Mesh Settings dialog. You can be categorized into also change the following groups:value of the minimum allowable cell angle.

* '''Port Characteristics''': S{{Note| Narrow, Z and Y [[Parameters]] and Voltage Standing Wave Ratio (VSWR)* spiky triangular cells in a planar mesh are generally not desirable. You should get rid of the either by changing the mesh density or using the hybrid planar mesh generator'''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), etcs additional mesh refinement options.* '''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}}

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 == Running Planar MoM engine calculates the scattering, impedance and admittance (S/Z/Y) [[parameters]] of the designated ports. The scattering [[parameters]] are defined based on the port impedances specified in the project's Port Definition dialog. If more than one port has been defined Simulations in the project, the S/Z/Y matrices of the multiport network are calculatedEM. Picasso ==

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

If you run an analysis, the port characteristics have single complex values, which you can view using [[EM.CubePicasso]]'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 itemsoffers five Planar MoM simulation modes: '''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]].

In particular, it may be useful to plot the S_ii {| class="wikitable"|-! scope="col"| Simulation Mode! scope="col"| Usage! scope="col"| Number of Engine Runs! scope="col"| Frequency ! scope="col"| Restrictions|-| style="width:120px;" | [[parameters#Running a Single-Frequency Planar MoM Analysis | Single-Frequency Analysis]] on a Smith chart| style="width:270px;" | Simulates the planar structure "As Is"| style="width:80px;" | Single run| style="width:250px;" | Runs at the center frequency fc| style="width:80px;" | None|-| style="width:120px;" | [[Parametric_Modeling_%26_Simulation_Modes_in_EM. To change Cube#Running_Frequency_Sweep_Simulations_in_EM.Cube | Frequency Sweep]]| style="width:270px;" | Varies the format operating frequency of the planar MoM solver | style="width:80px;" | Multiple runs | style="width:250px;" | Runs at a data plot, select it specified set of frequency samples or adds more frequency samples in the Data Manager and click its '''Edit''' buttonan adaptive way| style="width:80px;" | None|-| style="width:120px;" | [[Parametric_Modeling_%26_Simulation_Modes_in_EM. In Cube#Running_Parametric_Sweep_Simulations_in_EM.Cube | Parametric Sweep]]| style="width:270px;" | Varies the Edit File Dialog, choose one value(s) of one or more project variables| style="width:80px;" | Multiple runs| style="width:250px;" | Runs at the options provided in 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 dropdown list labeled '''Graph Type'''value(s) of one or more project variables to achieve a design goal | style="width:80px;" | Multiple runs | style="width:250px;" | Runs at the center frequency fc| style="width:80px;" | None|-| style="width:120px;" | [[Parametric_Modeling_%26_Simulation_Modes_in_EM.Cube#Generating_Surrogate_Models | HDMR Sweep]]| style="width:270px;" | Varies the value(s) of one or more project variables to generate a compact model| style="width:80px;" | Multiple runs | style="width:250px;" | Runs at the center frequency fc| style="width:80px;" | None|}

You can set the simulation mode from [[Image:Info_iconEM.png|40pxPicasso]] Click here to learn more about '''[[Data_Visualization_and_Processing#Graphing_Port_Characteristics | Graphing Port Characteristics]]'''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 [[Image:Info_iconEM.png|40pxPicasso]] Click here to learn more about '''[[Data_Visualization_and_Processing#Rational_Interpolation_of_Port_Characteristics | Rational Interpolation of Scattering Parameters]]'''.simulation and involves the following steps:

Electric and magnetic currents are the fundamental output data of To run a planar MoM simulation. After the numerical solution analysis of the MoM linear systemyour project structure, they are found using open the Run Simulation Dialog by clicking the solution vector '''Run'''[I[File:run_icon.png]] button on the ''' Simulate Toolbar''' or select '''Menu > Simulate > Run''' or use the keyboard shortcut {{key|Ctrl+R}}. The '''Single-Frequency Analysis''' option of the '''Simulation Mode''' dropdown list is selected by default. Once you click the {{key|Run}} button, the simulation starts. A new window called the "Output Window" opens up that reports the different stages of simulation and the definitions percentage of the electric tasks completed at any time. After the simulation is successfully completed, a message pops up and magnetic vectorial basis functions: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.

:<mathtable> \mathbf{[X]}_{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} </mathtr><!--td> [[FileImage:PMOM83Picasso L1 Fig18.png|thumb|left|480px|EM.Picasso's Simulation Run dialog.]]--</td></tr></table>

{{Note|A planar MoM simulation involves a number of numerical parameters that take preset default values unless you change them. You have can access these parameters and change their values by clicking the '''Settings''' button next to define a separate current distribution observable for each individual trace or embedded object setthe '''Select Engine''' drop-down list in [[EM.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'''. In the &quot;Spectral Domain Integration&quot; section of the dialog, you can set a value to '''Max Spectral Radius in k0''', which has a default value of 30. This means that the infinite spectral-domain integrals in the spectral variable k_&rho; are pre-calculated and tabulated up to a limit of 30k₀, where k₀ is the free space propagation constant. These integrals may converge much faster based on the specified Convergence Rate for Integration described earlier. However, in certain cases involving highly oscillatory integrands, much larger integration limits like 100k₀ might be needed to warrant adequate convergence. For spectral-domain integration along the real k_&rho; axis, the interval [0, Nk₀] is subdivided into a large number of sub-intervals, within each an 8-point Gauss-Legendre quadrature is applied. The next parameter, '''No. Radial Integration Divisions per k₀''', determines how small these intervals should be. By default, 2 divisions are used for the interval [Image:Info_icon0, k₀]. In other words, the length of each integration sub-interval is k₀/2. You can increase the resolution of integration by increasing this value above 2. Finally, instead of 2D Cartesian integration in the spectral domain, a polar integration is performed. You can set the '''No. of Angular Integration Points''', which has a default value of 100. [[EM.png|40pxPicasso]] Click here to learn more about 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 [[Data_Visualization_and_Processing#Visualizing_3D_Current_Distribution_Maps | Visualizing 3D Current Distribution MapsEM.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.

<td> [[Image:PMOM84PMOM79.png|thumb|300pxleft|720px|EM.Picasso's Current Distribution Planar MoM Engine Settings dialog.]] </td><td> [[Image:PMOM85(1).png|thumb|420px|The current distribution map of a patch antenna.]] </td>

=== Visualizing the Near Fields =Modeling Periodic Planar Structures in EM.Picasso == [[EM.Picasso]] allows you to simulate doubly periodic planar structures with periodicities along the X and Y directions. Once you designate your planar structure as periodic, [[EM.Picasso]]'s Planar MoM simulation engine uses a spectral domain solver to analyze it. In this case, the dyadic Green's functions of periodic planar structure take the form of doubly infinite summations rather than integrals.  [[Image:Info_icon.png|30px]] Click here to learn more about the theory of '''[[Basic_Principles_of_The_Method_of_Moments#Periodic_Planar_MoM_Simulation | Periodic Green's functions]]'''.

[[File:PMOM90.png|thumb|320px{{Note|[[Planar Module]]'s Field Sensor dialog.]] EM.Picasso allows you to visualize the near fields at a specific field sensor plane. Note that unlike [[EM.Cube]]'s other computational modules, near field calculations in EM.Picasso usually takes a significant amount of 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 [[EM.Tempo]]), can handle both regular and their computation requires integration of complex dyadic Green's functions of a multilayer background structure as opposed to the free space Green's functionsskewed periodic lattices.}}

[[Image:Info_icon.png|40px]] Click here to learn more about '''[[Data_Visualization_and_Processing#The_Field_Sensor_Observable | === Defining a Field Sensor Observable]]'''Periodic Structure in EM.Picasso ===

An infinite periodic structure in [[Image:Info_iconEM.png|40pxPicasso]] Click here to learn more about is represented by a &quot;'''Periodic Unit Cell'''&quot;. To define a periodic structure, you must open [[Data_Visualization_and_Processing#Visualizing_3D_Near-Field_Maps | Visualizing 3D Near Field MapsEM.Picasso]]'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'''. 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.

<td> [[Image:PMOM116PMOM99.png|thumb|360px300px|Near-zone electric field map above a microstrip-fed patch antennaEM.]] </td><td> [[Image:PMOM117.png|thumb|360px|Near-zone magnetic field map above a microstrip-fed patch antennaPicasso's Periodicity Settings dialog.]] </td>

=== Computing Radiation Pattern 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 Planar Structures ===the unit cell. [[EM.Picasso]] 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 periodic domain. In other words, the neighboring unit cells cannot overlap one another. However, you can arrange objects with linear edges such that one or more flat edges line up with the domain's bounding box. In such cases, [[EM.Picasso]]'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 a periodic metallic screen as shown in the 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 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.

Even though EM.Pplanar 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 <table><tr><td> [[EMImage:image122.Cube]] to perform these calculations. Once png|thumb|400px|Modeling 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 periodic screen using two different types of eight 2D radiation pattern graphs are available: 4 polar and 4 Cartesian graphs for the XY, YZ, ZX and user defined plane cutsunit cell.]] </td></tr></table>

<table><tr><td> [[Image:Info_iconpmom_per5_tn.png|40pxthumb|300px|The PEC cross unit cell.]] Click here to learn more about the theory of '''</td><td> [[Computing_the_Far_Fields_%26_Radiation_CharacteristicsImage:pmom_per6_tn.png| Far Field Computations]]''thumb|300px|Planar mesh of the PEC cross unit cell. Note the cell extensions at the unit cell's boundaries.]] </td></tr></table>

[[Image:Info_icon.png|40px]] Click here to learn more about the theory of '''[[Data_Visualization_and_Processing#Using_Array_Factors_to_Model_Antenna_Arrays | Using Array Factors to Model Antenna Arrays ]]'''=== Exciting Periodic Structures as Radiators in EM.Picasso ===

[[Image:Info_iconWhen a periodic planar structure is excited using a gap or probe source, it acts like an infinite periodic phased array.png|40px]] Click here All the periodic replicas of the unit cell structure are excited. You can even impose a phase progression across the infinite array to learn more about steer its beam. You can do this from the property dialog of the gap or probe source. At the bottom of the '''[[Data_Visualization_and_Processing#Visualizing_3D_Radiation_Patterns | Visualizing 3D Planar Gap Circuit Source Dialog''' or '''Gap Source Dialog''', there is a button titled '''Periodic Scan...'''. You can enter desired values for '''Theta''' and '''Phi''' beam scan angles in degrees. To visualize the radiation patterns of a beam-steered antenna array, you have to define a finite-sized array factor in the Radiation Patterns]]Pattern dialog. You do this in the '''Impose Array Factor''' section of this dialog. The values of '''Element Spacing''' along the X and Y directions must be set equal to the value of '''Periodic Lattice Spacing'''along those directions.

<table><tr><td> [[Image:Info_iconPeriod5.png|40pxthumb|350px|Setting periodic scan angles in EM.Picasso's Gap Source dialog.]] Click here to learn more about '''</td><td> [[Data_Visualization_and_Processing#2D_Radiation_and_RCS_Graphs Image:Period5_ang.png| Plotting 2D Radiation Graphsthumb|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>

<td> [[FileImage:PMOM118Period7.png|thumb|300px360px|EM.Picasso's Radiation Pattern dialogpattern of an 8×8 finite-sized periodic printed dipole array with 0&deg; phi and theta scan angles.]] </td><td> [[Image:PMOM119Period8.png|thumb|420px360px|3D polar radiation Radiation pattern plot of a microstripbeam-steered 8×8 finite-fed patch antennasized periodic printed dipole array with 45&deg; phi and theta scan angles.]] </td>

=== Radar Cross Section of Planar Exciting Periodic Structures Using Plane Waves in EM.Picasso ===

When a periodic planar structure is excited by using a plane wave source, it acts as a periodic surface that reflects or transmits the calculated far field data indeed represent the scattered fields of that planar structureincident wave. [[EM.Picasso]] can also calculate calculates the radar cross section (RCS) reflection and transmission coefficients of a periodic planar targetstructures. Note that in this case the RCS is defined for If you run a finitesingle-sized target frequency plane wave simulation, the reflection and transmission coefficients are reported in the presence Output Window at the end of an infinite background structurethe simulation. The scattered &theta; and &phi; components Note that these periodic characteristics depend on the polarization of the far-zone electric field are indeed what you see in incident plane wave. You set the 3D far field visualization of radiation polarization (scatteringTMz or TEz) patternsin the '''Plane Wave Dialog''' when defining your excitation source. Instead of radiation or scattering patterns, In this dialog you can instruct [[EMalso set the values of the incident '''Theta''' and '''Phi''' angles.Picasso]] to plot 3D visualizations 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 &sigmaquot;_{reflection.CPX&thetaquot;}, and &sigmaquot;_{transmission.CPX&phiquot;} and the total RCS.

{{Note|In the absence of any finite traces or embedded objects in the project workspace, [[Image:Info_iconEM.png|40px]] Click here to learn more about '''[[Data_Visualization_and_Processing#Visualizing_3D_RCS | Visualizing 3D RCSPicasso]]'''computes the reflection and transmission coefficients of the layered background structure of your project.}}

<table><tr><td>[[Image:Info_iconPMOM102.png|40pxthumb|580px|A periodic planar layered structure with slot traces excited by a normally incident plane wave source.]] Click here to learn more about '''</td></tr></table> === Running a Periodic MoM Analysis === You run a periodic MoM analysis just like an aperiodic MoM simulation from [[Data_Visualization_and_Processing#2D_Radiation_and_RCS_Graphs | Plotting 2D RCS GraphsEM.Picasso]]'s 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.

<td> [[File:PMOM124.png|thumb|300px|EM.Picasso's Radar Cross Section dialog]] </td><td> [[Image:PMOM125PMOM98.png|thumb|420px600px|An example Changing the number of Floquet modes from the 3D mono-static radar cross section plot of a patch antennaPlanar MoM Engine Settings dialog.]] </td>

You learned earlier how to use [[EM.Cube]]'s powerful, adaptive frequency sweep utility to study the frequency response of a planar structure. Adaptive frequency sweep uses rational function interpolation to generate smooth curves of the scattering parameters with a relatively small number of full-wave simulation runs in a progressive manner. Therefore, you need a port definition in your planar structure to be able to run an adaptive frequency sweep. This is clear in the case of an infinite periodic phased array, where your periodic unit cell structure must be excited using either a gap source or a probe source. You run an adaptive frequency sweep of an infinite periodic phased array in exactly the same way to do for regular, aperiodic, planar structures. [[EM.Cube]]'s Planar Modules also allows you to run an adaptive frequency sweep of periodic surfaces excited by a plane wave source. In this case, the planar MoM engine calculates the reflection and transmission coefficients of the periodic surface. Note that you can conceptually consider a periodic surface as a two-port network, where Port 1 is the top half-space and Port 2 is the bottom half-space. In that case, the reflection coefficient R is equivalent to S<psub>11</sub> 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°&nbsplt;&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 S<sub>22</psub>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. <!--=== Modeling Finite-Sized Periodic Arrays === [[Image:Info_icon.png|40px]] Click here to learn about '''[[Modeling Finite-Sized Periodic Arrays Using NCCBF Technique]]'''.--> <br /> <hr> [[Image:Top_icon.png|48px30px]] '''[[EM.Picasso#An_EM.Picasso_Primer Product_Overview | Back to the Top of the Page]]''' [[Image:Tutorial_icon.png|30px]] '''[[EM.Cube#EM.Picasso_Documentation | EM.Picasso Tutorial Gateway]]'''

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