Changes

[[Image:Splash-planar new.jpg|right|750px720px]]<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:PMOM32Embedded object sets represent short material insertions inside substrate layers.png|thumb|450px|Planar hybrid They can be metal or dielectric. Metallic embedded objects can be used to model vias, plated-through holes, shorting pins and triangular meshes for rectangular patchesinterconnects.]][[Image:PMOM30These are called PEC via sets.png|thumb|450pxEmbedded 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 dialog or directly from the navigation tree. Open the &quot;Embedded Sets&quot; tab of the stack-up dialog. This tab has a table that lists all the embedded object sets along with their material type, the host substrate layer, the host material and their height. To add a new object set, click the arrow symbol on the {{key|Mesh Insert}} button of the dialog and select one of the two rectangular patches at two different planesoptions, '''PEC Via Set''' or '''Embedded Dielectric Set''', from the dropdown list. The lower substrate This opens up a new dialog where first you have to set the host layer has of the new object set. A dropdown list labeled &quot;'''Host Layer'''&quot; gives a higher permittivitylist of all the available finite substrate layers.]][[Image:PMOM31You can also set the properties of the embedded object set, including its label, color and material properties.png|thumb|400px|Keep in mind that you cannot control the height of embedded objects. Moreover, you cannot assign material properties to PEC via sets, while you can set values for the '''Permittivity'''(&epsilon;_r) and '''Electric Conductivity'''(&sigma;) of embedded dielectric sets. Vacuum is the default material choice. To define an embedded set from the navigation tree, right-click on the '''Embedded Object Sets''' item in the '''Physical Structure''' section of the navigation tree and select either '''Insert New PEC Via Set...''' or '''Insert New Embedded Dielectric Set...''' The Planar Mesh Settings respective New Embedded Object Set dialogopens up, where you can set the properties of the new object set.]]=== Understanding As soon as you close this dialog, it takes you to the Planar MoM Mesh ===Layer Stack-up Settings dialog, where you can verify the location of the new object set on the layer hierarchy.

The method of moments (MoM) discretizes all the finite-sized objects of a planar structure (excluding the background structure) into a set of elementary cells<table><tr><td> [[Image:PMOM23. The accuracy of the MoM numerical solution depends greatly on the quality and resolution of the generated meshpng|thumb|550px|EM. As a rule of thumb, a mesh density of about 20Picasso's Layer Stack-30 cells per effective wavelength usually yields satisfactory results. Yet, for structures with lots of fine geometrical details or for highly resonant structures, higher mesh densities may be required. Also, up dialog showing the particular simulation data that you seek in a project also influence your choice of mesh resolutionEmbedded Sets tab. For example, far field characteristics like radiation patterns are less sensitive to the mesh density than field distributions ]] </td></tr></table> === Drawing Planar Objects on a structure with a highly irregular shape and a rugged boundary.Horizontal Work Planes ===

EM.Picasso generates two types of mesh for a planar structure: a pure triangular and a hybrid triangular-rectangular. In both case, EM.Picasso attempts to create a highly regular mesh, As soon as you start drawing geometrical objects in which most the project workspace, the '''Physical Structure''' section of the cells have almost equal areasnavigation tree gets populated. The hybrid mesh names of traces are added under their respective trace type tries to produce as many rectangular cells as possible especially in category, and the case names of objects with rectangular or linear boundariesappear under their respective trace group. In connection or junction areas between adjacent objects or close to highly curved boundariesAt any time, one and only one trace is active in the use project workspace. The name of triangular cells the active trace in the navigation tree is clearly inevitablealways displayed in bold letters. EMAn active trace is where all the new objects you draw belong to.Picasso's By default mesh type , the last defined trace or embedded object set is hybridactive. The uniformity You can immediately start drawing new objects on the active trace. You can also set any trace or regularity of mesh is an important factor in warranting a stable MoM numerical solutionobject set group active at any time by right-clicking on its name on the navigation tree and selecting '''Activate''' from the contextual menu.

The mesh density gives a measure of the number of cells per effective wavelength that are placed in various regions of your planar structure. The higher the mesh density, the more cells are created on the geometrical objects. Keep in mind that only the finite-sized objects of your structure are discretized. The free-space wavelength is defined as <math>\lambda_0 = \tfrac{2\pi f}{c}</math>, where f is the center frequency of your project and c is the speed of light in the free space. The effective wavelength is defined as <math>\lambda_{eff} = \tfrac{\lambda_0}{\sqrt{\varepsilon_{eff}}}</math>, where e_eff is the effective permittivity. By default, [[EMImage:Info_icon.Picassopng|30px]] generates a hybrid mesh with a mesh density of 20 cells per effective wavelength. The effective permittivity is defined differently for different types of traces and embedded object sets. This is Click here to make sure that enough cells are placed learn more about '''[[Building Geometrical Constructions in areas that might feature higher field concentration. 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 the metallic trace. For slot traces, the effective permittivity is defined as the mean (average) of the permittivity of the two substrate layers just above and below the metallic trace. For embedded object sets, the effective permittivity is defined as the largest of the permittivities of all the substrate layers and embedded dielectric setsCubeCAD#Transferring Objects Among Different Groups or Modules | Moving Objects among Different Groups]]'''.

<table><tr><td> [[Image:PMOM44PMOM23B.png|thumb|400px280px|Deleting or curing defective triangular cells: Case 1EM.]][[Image:PMOM42.png|thumb|400px|Deleting or curing defective triangular cells: Case 2Picasso's Navigation Tree populated with planar objects.]]</td>[[Image:PMOM45.png|thumb|300px|Locking the mesh density of an object group from its property dialog.]]</tr>=== Generating, Viewing & Customizing a Planar Mesh ===</table>

You can generate and view a planar mesh by clicking the '''Show Mesh''' [[File:mesh_toolEM.pngPicasso]] button has a special feature that makes construction of planar structures very convenient and straightforward. <u>The horizontal Z-plane of the '''Simulate Toolbar''' active trace or by selecting '''Menu &gt; Simulate &gt; Discretization &gt; Show Mesh''' or using the keyboard shortcut '''Ctrl+M'''. When object set group is always set as the mesh active work plane of the planar structure is displayed in [[EM.Cube]]’s project workspace, its &quot;Mesh View&quot; mode is enabled. In this mode you can perform view operations like rotate view, pan or zoom, but you cannot create </u> That means all new objects or edit existing onesare drawn at the Z-coordinate of the currently active trace. To exit As you change the mesh view modeactive trace group or add a new one, press the keyboard's '''Esc Key''' or click the '''Show Mesh''' [[File:mesh_tool.png]] button once againactive work plane changes accordingly.

Once a mesh is generated, it stays in the memory until the structure is changed or the mesh density or other settings are modified. Every time you view mesh, the one in the memory is displayed. You can force {{Note| In [[EM.Picasso]] to create a new mesh from , you cannot modify the ground up by selecting '''Menu &gt; Simulate &gt; Discretization &gt; Regenerate Mesh''' or by right clicking on the '''Planar Mesh''' item in the '''Discretization''' section Z-coordinate of the navigation tree an object as it is set and selecting '''Regenerate''' from the contextual menu controlled by its host trace.}}

You can change the settings of the planar mesh including the mesh type and density from the Planar Mesh Settings Dialog[[EM. You can also change these settings while in the mesh view mode, and Picasso]] does not allow you can update the changes to view the new meshdraw 3D or solid CAD objects. To open the mesh settings dialog, either click The solid object buttons in the '''Mesh SettingsObject 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. [[File:mesh_settingsEM.pngPicasso]] button extrudes and extends these planar objects across their host layer automatically and displays them as 3D wireframe, prismatic objects. The automatic extrusion of the '''Simulate Toolbar''' or select '''Menu &gt; Simulate &gt; Discretization &gt; Mesh Settingsembedded objects happens after mesh generation and before every planar MoM simulation...''', or You can enforce this extrusion manually by right click on -clicking the '''Planar MeshLayer Stack-up''' item in the '''Discretization''' "Computational Domain" section of the Navigation Tree navigation tree and select selecting '''Mesh Settings...Update Planar Structure''' from the contextual menu, or use the keyboard shortcut '''Ctrl+G'''. You can change the mesh algorithm from the dropdown list labeled '''Mesh Type''', which offers two options: '''Hybrid''' and '''Triangular'''. You can also enter a different value for '''Mesh Density''' in cells per effective wavelength (&lambda;_eff). For each value of mesh density, the dialog also shows the average &quot;Cell Edge Length&quot; in the free space. To get an idea of the size of mesh cells on the traces and embedded object sets, divide this edge length by the square root of the effective permittivity a particular trace or set. Click the '''Apply''' button to make the changes effective.

=== A {{Note on the Junction Mesh === The integrity of the planar mesh and its continuity in the junction areas where adjacent objects are connected directly affects the simulation results. The most important rule of object connections in | In [[EM.Picasso is that ]], you can only objects belonging to the same trace can be connected to one another. If two objects belong to the same trace (residing on the same Z-plane) and have a common overlap area, EM.Picasso first merges the two objects using the &quot;Boolean Union&quot; operation and converts them into a single object for the purpose of meshing. EM.Picasso's hybrid draw horizontal planar mesh generator has some additional rules: * If two connected rectangular surface CAD objects have the same side dimensions along the common linear edge with perfect alignment, a rectangular junction mesh is produced.* If two connected rectangular objects have different side dimensions along the common linear edge or have edge offset, a set of triangular cells is generated along the edge of the object with the large side.* Rectangular objects that contain gap source or lumped elements, always have a rectangular mesh around the gap area. If an embedded object like an interconnect via is located under or above a metallic trace or connected from both top and bottom, it is critical to create mesh continuity between the embedded object and its connected metallic traces. In other words, the generated mesh must ensure current continuity between the vertical volume currents and horizontal surface currents. EM.Picasso’s planar mesh generator automatically handles situations of this kind and generates all the required connection meshes. }}

<td> [[FileImage:PMOM36PMOM23A.png|250px]] [[File:PMOM38.pngthumb|250px]] [[File:PMOM37.png620px|250px]] </td></tr><tr><td> Two overlapping A planar objects and their triangular and hybrid planar meshes. </td></tr><tr><td> [[File:PMOM33.png|250px]] [[File:PMOM35.png|250px]] [[File:PMOM34.png|250px]] </td></tr><tr><td> Edgestructure with a two-connected rectangular planar objects and their triangular and hybrid planar meshes. </td></tr><tr><td> [[File:PMOM39.png|375px]] [[File:PMOM40.png|375px]] </td></tr><tr><td> Meshes layer conductor-backed substrate, two PEC patches located at the tops of short the lower and long vertical upper substrate layers, four PEC vias connecting located inside the lower substrate layer between the lower patch and bottom ground and an embedded dielectric film located inside the top substrate layer sandwiched between the two horizontal metallic stripspatches. ]] </td>

=== Refining the Planar Mesh Locally EM.Picasso's Special Rules ===

It is very important to apply # PEC ground planes at the right mesh density to capture all the geometrical details top or bottom of your a planar structure. This is especially true for &quot;field discontinuity&quot; regions such are regarded and modeled as junction areas between objects of different side dimensionsPEC top or bottom half-spaces, where larger current concentrations are usually observed respectively.# A PEC ground plane placed in the middle of a substrate stack-up requires at sharp cornersleast one slot object to provide electromagnetic coupling between its top and bottom sides. In this case, or a slot trace is rather introduced at the connection areas between metallic traces and PEC viasgiven Z-plane, as well as which also implies the areas around gap sources presence of an infinite PEC ground.# Metallic and lumped elementsslot traces cannot coexist on the same Z-plane. However, as these create voltage or current discontinuitiesyou can stack up multiple PEC and conductive sheet traces at the same Z-coordinate. For large planar structuresSimilarly, using a higher mesh density may not always 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 practical option since it will quickly lead to suspended metallic trace inside a very large MoM matrix and thus growing dielectric layer (as in the size case of the numerical problemcenter conductor of a stripline), you must split the dielectric layer into two thinner substrate layers and place your PEC trace at the interface between them. # [[EM.Picasso provides several ways ]]'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 controlling the mesh of a planar structure locallymaterial 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 Planar Mesh Settings dialog gives a few options for customizing your planar mesh around geometrical and field discontinuities. You can check the check box labeled &quot;'''Refine Mesh at Junctions'''&quot;, which increases the mesh resolution at the connection area between rectangular objects. Or you can check the check box labeled &quot;'''Refine Mesh at Gap Locations'''&quot;, which may prove particularly useful when gap sources or lumped elements are placed on a short transmission line connected from both ends. Or you can check the check box labeled &quot;'''Refine Mesh at Vias'''&quot;, which increases the mesh resolution on the cross section of embedded object sets and 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.'s Excitation Sources ==

You should always visually inspect EM.Picasso's default generated mesh to see if the current mesh settings have produced an acceptable mesh. You may often need to change the mesh density or other [[parameters]] and regenerate the mesh. Sometimes EM.Picasso's default mesh may contain very narrow triangular cells due to very small angles between two edges. In Your planar structure must be excited by some rare casessort of signal source that induces electric surface currents on metal parts, extremely small triangular cells may be generatedmagnetic surface currents on slot traces, whose area is a small fraction of the average mesh cell. These cases typically happen at the junctions and other discontinuity regions conduction or at the boundary of highly irregular geometries with extremely fine details. In such cases, increasing or decreasing the mesh density by one or few cells per effective wavelength often resolves that problem polarization volume currents on vertical vias and eliminates those defective cellsembedded objects. Nonetheless, EM.Picasso's planar mesh generator offers an option to identify The excitation source you choose depends on 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 behindobservables you seek in your project. [[EM.Picasso by default deletes or cures all ]] provides the triangular cells that have angles less than 10º. Sometimes removing defective cells may inadvertently cause worse problems in the mesh. You may choose to disable this feature and uncheck the box labeled &quot;'''Remove Defective Triangular Cells'''&quot; in the Planar Mesh Settings dialog. You can also change the value of the minimum allowable cell angle.following source types for exciting planar structures:

Another way {| class="wikitable"|-! scope="col"| Icon! scope="col"| Source Type! scope="col"| Applications! scope="col"| Restrictions|-| style="width:30px;" | [[File:gap_src_icon.png]]| [[Glossary of local mesh control is to lock the mesh density of certain EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Strip Gap Circuit Source |Strip Gap Circuit Source]]| style="width:300px;" | General-purpose point voltage source (or filament current source on slot traces or object sets. The mesh density that you specify in the Planar Mesh Settings dialog is )| style="width:300px;" | Associated with a global parameter and applies to all the traces and embedded object sets in your projectPEC rectangle strip|-| style="width:30px;" | [[File:probe_src_icon. However, you can lock the mesh png]]| [[Glossary of individual PECEM.Cube's Materials, PMC and conductive sheet traces or 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 objects setsPEC via set|-| style="width:30px;" | [[File:waveport_src_icon. In that casepng]]| [[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 locked mesh density takes precedence over the global densityopen end|-| style="width:30px;" | [[File:hertz_src_icon. Note that locking mesh png]]| [[Glossary of object groupsEM.Cube's Materials, in principleSources, is different than refining the mesh at discontinuitiesDevices & 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. In the latter case, the mesh png]]| [[Glossary of connection areas is affectedEM. HoweverCube's Materials, objects belonging to different traces cannot be connected to one another. ThereforeSources, locking mesh can be useful primarily Devices & Other Physical Object Types#Plane Wave |Plane Wave Source]]| style="width:300px;" | Used for isolated object groups that may require a higher (or lower) mesh resolutionmodeling scattering & computation of reflection/transmission characteristics of periodic surfaces| style="width:300px;" | None, stand-alone source|-| style="width:30px;" | [[File:huyg_src_icon. You can lock the local mesh density by accessing the property dialog png]]| [[Glossary of a specific trace or embedded object set and checking the box labeled '''Lock Mesh'''EM. This will enable the Cube'''Mesh Density''' boxs Materials, where you can accept the default global value or set any desired new valueSources, 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|}

== Excitation Click on each category to learn more details about it in the [[Glossary of EM.Cube's Materials, Sources ==, Devices & Other Physical Object Types]].

Your For antennas and planar structure must be excited by some sort 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 signal 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 induces 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 parts strip creates a uniform electric field across the gap and magnetic currents pumps electric current into the line. A rectangle strip object on a slot tracestrace acts like a slot transmission line on an infinite PEC ground plane that carries a magnetic current along its length (local X direction). The excitation characteristic impedance of the slot line is a function of its width (local Y direction). A gap source you choose depends 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 observables you seek in your projectslot, equivalent to a magnetic current flowing into the slot line. EMProbe sources are placed across vertical PEC vias.Picasso provides the following A de-embedded source types for exciting planar structures: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.

* [[Planar_MoM_Source_Types#Gap_Sources{{Note|Gap Sources]]* You can realize a coplanar waveguide (CPW) in [[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 SourcesEM.Picasso]]using two parallel slot lines with two aligned, collocated gap sources.}}

For antennas and planar circuits, where you typically define one or more ports, you usually use lumped sources. A lumped source is indeed a gap discontinuity that is placed on the path of an electric or magnetic current flow, where a voltage or current source is connected to inject a signal. Gap sources are placed across metal or slot traces. Probe sources are placed across vertical PEC vias. A de-embedded source is a special type of gap source that is placed near the open end of an elongated metal or slot trace to create a standing wave pattern, from which the scattering [[parametersImage:Info_icon.png|40px]] can be calculated accurately. To calculate the scattering characteristics of a planar structure, e.g. its radar cross section (RCS), you excite it with a plane wave source. Short dipole sources are used Click here 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 learn more about '''[[EM.CubePreparing_Physical_Structures_for_Electromagnetic_Simulation#Modeling_Finite-Sized_Source_Arrays | Using Source Arrays for Modeling Antenna Arrays]] computational module and bring them as a new source to excite your planar structure'''.

A short dipole provides another way of exciting a planar structure in [[Image:Info_iconEM.png|40pxPicasso]] Click here . A short dipole source acts like an infinitesimally small ideal current source. You can also use an incident plane wave to learn more about '''excite your planar structure in [[Planar MoM Source TypesEM.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:Info_iconPMOM64A.png|40px]] Click here to learn more about '''[[Common_Excitation_Source_Types_in_EMthumb|550px|A multilayer planar structure containing a CPW line with a single coupled port and a lumped element on an overpassing metal strip.Cube#Defining_Finite-Sized_Source_Arrays | Using Source Arrays for Modeling Antenna Arrays]]'''.</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 the theory of '''[[Computing_Port_Characteristics_in_Planar_MoMPreparing_Physical_Structures_for_Electromagnetic_Simulation#Modeling_Lumped_Elements_in_Planar_MoMModeling_Lumped_Elements_in_the_MoM_Solvers | Modeling Defining Lumped Elements in Planar MoM]]'''.

EM[[Image:Info_icon.Picasso allows you to define passive circuit elements: png|40px]] Click here for a general discussion of '''Resistors[[Preparing_Physical_Structures_for_Electromagnetic_Simulation#A_Review_of_Linear_.26_Nonlinear_Passive_.26_Active_Devices | Linear Passive Devices]]'''(R), C'''apacitors'''(C), I'''nductors'''(L), and series and parallel combinations of them . To define a lumped RLC circuit in your planar structure, follow these steps:

* Open the Lumped Element Dialog by right clicking on the '''Lumped Elements''' item in the '''Sources''' section {{Note|The impedance of the Navigation Tree and selecting '''Insert New Source...'''* In lumped circuit is calculated at the '''Gap Topology''' section operating frequency of the dialog, select one of project using the two options: '''Gap on Line''' and '''Gap on Via'''.* In the '''Lumped Circuit Type''' section of the dialogspecified R, select one of the two options: '''Passive RLC''' L and '''Active with Gap Source'''C values.* Depending on your choice of gap topology, in the '''Lumped Circuit Location''' section of the dialog, As you will find either a list of all change the '''Rectangle Strip Objects''' or a list of all the '''PEC Via Objects''' available in the project workspace. Select the desired rectangle strip or embedded PEC via object.* In the box labeled '''Offset'''frequency, enter the distance value of the lumped element from the start point of the rectangle strip line or from the bottom of the via object, whichever the case. The value of '''Offset''' by default impedance that is initially set passed to the center of the line or via.* In the '''Load Properties''' section, the series and shunt resistance values Rs and Rp are specified in Ohms, the series and shunt inductance values Ls and Lp are specified in nH (nanohenry), and the series and shunt capacitance values Cs and Cp are specified in pF (picofarad). Only the checked elements are taken into account in the total impedance calculation. By default, only the series resistor is checked with a value of 50S, and all other circuit elements are initially greyed outPlanar MoM engine will change.<br />}}

EM.Picasso allows you to define a voltage source in series with a series-parallel RLC combination and place them across the gap. This is called an active lumped element. If you choose the === Calculating Scattering Parameters Using Prony'''Active with Gap Source''' option of the '''Lumped Circuit Type''' section of the dialog, the right section of the dialog entitled '''Source Properties''' becomes enabled, where you can you can specify the '''Source Amplitude''' in Volts (or in Amperes in the case of PMC traces) and the '''Phase''' in degrees. Also, the box labeled '''Direction''' becomes relevant in this case which contains a gap source. Otherwise, a passive RLC circuit does not have polarity.s Method ===

If the project workspace contains an array The calculation of rectangle strip objects or PEC via objects, the array object will also be listed as scattering (S) parameters is usually an eligible object for lumped element placement. A lumped element will then be placed on each element important objective of the array. All the lumped elements will have identical directionmodeling planar structures especially for planar circuits like filters, offsetcouplers, resistance, inductance and capacitance valuesetc. If As you define an active lumped elementsaw earlier, you can prescribe certain amplitude use lumped sources like gaps and/or phase distribution probes and even active lumped elements to calculate the gap sources just like in the case circuit characteristics of planar structures. The admittance / impedance calculations based on the gap voltages and probe sourcescurrents are accurate at RF and lower microwave frequencies or when the port transmission lines are narrow. The available amplitude In such cases, the electric or magnetic current distributions include '''Uniform'''across the width of the port line are usually smooth, '''Binomial'''''', Chebyshev''' and '''Data File'''quite uniform current or voltage profiles can easily be realized. At higher frequencies, however, a more robust method is needed for calculating the port parameters.

{{Note|The impedance One can calculate the scattering parameters of a planar structure directly by analyzing the lumped circuit is calculated current distribution patterns on the port transmission lines. The discontinuity at the operating frequency end of a port line typically gives rise to a standing wave pattern that can clearly be discerned in the project using the specified R, L and C valuesline's current distribution. As you change From the frequencylocation of the current minima and maxima and their relative levels, one can determine the value of reflection coefficient at the impedance that is passed to discontinuity, i.e. the Planar MoM engine will changeS₁₁ parameter.}}A more robust technique is Prony’s method, which is used for exponential approximation of functions. A complex function f(x) can be expanded as a sum of complex exponentials in the following form:

:<math> f(x) \approx \sum_{n=1}^N c_i e^{-j\gamma_i x} </math><!--[[ImageFile:Info_iconPMOM73.png|40px]] Click here to learn --> where c_i are complex coefficients and &gamma;_i are, in general, complex exponents. From the physics of transmission lines, we know that lossless lines may support one or more about propagating modes with pure real propagation constants (real &gamma;_i exponents). Moreover, line discontinuities generate evanescent modes with pure imaginary propagation constants (imaginary &gamma;_i exponents) that decay along the line as you move away from the location of such discontinuities. In practical planar structures for which you want to calculate the scattering parameters, each port line normally supports one, and only one, dominant propagating mode. Multi-mode transmission lines are seldom used for practical RF and microwave applications. Nonetheless, each port line carries a superposition of incident and reflected dominant-mode propagating signals. An incident signal, by convention, is one that propagates along the line towards the discontinuity, where the phase reference plane is usually established. A reflected signal is one that propagates away from the port plane. Prony's method can be used to extract the incident and reflected propagating and evanescent exponential waves from the standing wave data. From a knowledge of the amplitudes (expansion coefficients) of the incident and reflected dominant propagating modes at all ports, the scattering matrix of the multi-port structure is then calculated. In Prony''[[Modeling Lumped Elementss method, Circuits & Devices the quality of the S parameter extraction results depends on the quality of the current samples and whether the port lines exhibit a dominant single-mode behavior. Clean current samples can be drawn in EMa region far from sources or discontinuities, typically a quarter wavelength away from the two ends of a feed line. <table><tr><td> [[Image:PMOM71.png|thumb|600px|Minimum and maximum current locations of the standing wave pattern on a microstrip line feeding a patch antenna.Cube]]'''.</td></tr></table>

[[Image:PMOM52.png|thumb|400px|EM.Picasso'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.]]=== Defining Independent & Coupled Ports ===

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

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

'''You can define any number of ports equal [[Image:Info_icon.png|40px]] Click here to or less than learn more about the total number of sources in your project.''' The Port List of the dialog shows a list of all the ports in ascending order, with their associated sources and the port's characteristic impedance, which is 50S by default[[Glossary_of_EM. You can delete any port by selecting it from the Cube%27s_Simulation_Observables_%26_Graph_Types#Port_Definition_Observable | Port List and clicking the '''Delete''' button of the dialog. Keep in mind that after deleting a port, you will have a source in your project without any port assignment and make sure that is what you intend. You can change the characteristic impedance of a port by selecting it from the Port List and clicking the '''Edit''' button of the dialog. This opens up the Edit Port dialog, where you can enter a new value in the box labeled '''ImpedanceDefinition Observable]]'''.

[[Image:Info_icon.png|40px]] Click here to learn more about the theory of '''[[Computing Port Characteristics in Planar MoMPreparing_Physical_Structures_for_Electromagnetic_Simulation#Modeling_Coupled_Sources_.26_Ports | Modeling Coupled Ports]]'''.

=== Modeling Coupled Ports =EM.Picasso's Simulation Data & Observables ==

Sources can be coupled to each other to model coupled strip lines (CPS) on metal traces or coplanar waveguides (CPW) Depending on slot traces. Similarlythe source type and the types of observables defined in a project, probe sources may be coupled to each other. Coupling two or more sources does not change a number of output data are generated at the way they excite end of a planar structureMoM simulation. It is intended only for the purpose Some of S parameter calculation. The feed lines or vias which host the coupled sources these data are usually parallel and aligned with one another 2D by nature and they some are all grouped together as a single transmission line represented 3D. The output simulation data generated by a single port. This single &quot;coupled&quot; port then interacts with other coupled or uncoupled ports[[EM.Picasso]] can be categorized into the following groups:

You couple two or more sources using the '''Port Definition Dialog'''{| 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. 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 png]]| style="width:150px;" | Current Distribution Maps| style="width:150px;" | [[Glossary of the dialogEM. Then, define a new port by clicking the Cube'''Add''' button of the dialog. This opens up the Add Port dialog, which consists of two tabless Simulation Observables & Graph Types#Current Distribution |Current Distribution]]| style="width: '''Available''' sources 300px;" | Computing electric surface current distribution on the left metal traces and '''Associated''' sources magnetic surface current distribution on the rightslot traces | style="width:250px;" | None|-| style="width:30px;" | [[File:fieldsensor_icon. A right arrow (''png]]| style="width:150px;" | Near-Field Distribution Maps| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Near-Field Sensor |Near-&gtField Sensor]] | style="width:300px;''') button " | Computing electric and magnetic field components on a left arrow ('''&ltspecified plane in the frequency domain| style="width:250px;" | None|-| style="width:30px;" | [[File:farfield_icon.png]]| style="width:150px;" | Far-Field Radiation Characteristics| style="width:150px;" | [[Glossary of EM.Cube''') button let you move s Simulation Observables & Graph Types#Far-Field Radiation Pattern |Far-Field Radiation Pattern]]| style="width:300px;" | Computing the sources freely between these two tablesradiation pattern and additional radiation characteristics such as directivity, axial ratio, side lobe levels, etc. You will see in the &quot| style="width:250px;Available&quot" | None|-| style="width:30px; table a list " | [[File:rcs_icon.png]]| style="width:150px;" | Far-Field Scattering Characteristics| style="width:150px;" | [[Glossary of all the sources that you deleted earlierEM. You may even see more available sources. Select all Cube's Simulation Observables & Graph Types#Radar Cross Section (RCS) |Radar Cross Section (RCS)]] | style="width:300px;" | Computing the sources that you want to couple bistatic and move them to the &quotmonostatic RCS of a target| style="width:250px;Associated&quot" | Requires a plane wave source|-| style="width:30px; table on the right" | [[File:port_icon. You can make multiple selections using the keyboardpng]]| style="width:150px;" | Port Characteristics| style="width:150px;" | [[Glossary of EM.Cube's '''Shift''' and '''Ctrl''' keys. Closing the Add Simulation Observables & Graph Types#Port dialog returns you to the Definition |Port Definition dialog, where you will now see ]] | style="width:300px;" | Computing the names S/Y/Z parameters and voltage standing wave ratio (VSWR)| style="width:250px;" | Requires one of all these source types: lumped, distributed, microstrip, CPW, coaxial or waveguide port|-| style="width:30px;" | [[File:period_icon.png]]| style="width:150px;" | Periodic Characteristics| style="width:150px;" | No observable required | style="width:300px;" | Computing the coupled sources next to the name reflection and transmission coefficients of the newly added porta periodic surface| style="width:250px;" | Requires a plane wave source and periodic boundary conditions |-| style="width:30px;" | [[File:huyg_surf_icon.png]]| style="width:150px;" | Equivalent electric and magnetic surface current data| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Huygens Surface |Huygens Surface]]| style="width:300px;" | Collecting tangential field data on a box to be used later as a Huygens source in other [[EM.Cube]] modules| style="width:250px;" | None|}

{{Note|It is your responsibility Click on each category to set up coupled ports and coupled learn more details about it in the [[Transmission LinesGlossary of EM.Cube's Simulation Observables & Graph Types]] properly. For example, to excite the desirable odd mode of a coplanar waveguide (CPW), you need to create two rectangular slots parallel to and aligned with each other and place two gap sources on them with the same offsets and opposite polarities. To excite the even mode of the CPW, you use the same polarity for the two collocated gap sources. Whether you define a coupled port for the CPW or not, the right definition of sources will excite the proper mode. The couple ports are needed only for correct calculation of the port characteristics.}}

If Electric and magnetic currents are the project workspace contains an array fundamental output data of rectangle strip objects, a planar MoM simulation. After the array object will also be listed as an eligible object for gap source placement. A gap source will then be placed on each element numerical solution of the array. All MoM linear system, they are found using the gap sources will have identical direction solution vector '''[I]''' and offset. Similarly, if the project workspace contains an array definitions of PEC via objects, the embedded array object will also be listed as an eligible object for probe source placement. A probe source will then be placed on each via object of the array. All the probe sources will have identical direction electric and offset.magnetic vectorial basis functions:

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

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

The first step of planning a planar MoM simulation is defining your planar structure[[EM. This consists of Picasso]] allows you to visualize the background structure plus all the finite-sized metal and slot trace objects and possibly embedded metal or dielectric objects near fields at a specific field sensor plane. Note that are interspersed among the substrate layersunlike [[EM. The background stack-up is defined in the Layer Stack-up dialogCube]]'s other computational modules, which automatically opens up as soon as you enter the near field calculations in [[Planar ModuleEM.Picasso]]usually takes a significant amount of time. The metal and slot traces and embedded object sets are listed in This is due to the Navigation Treefact that at the end of a planar MoM simulation, which also shows all the geometrical fields are not available anywhere (CADas opposed to [[EM.Tempo]]) objects you draw in , and their computation requires integration of complex dyadic Green's functions of a multilayer background structure as opposed to the project workspace under each object group at different Z-planesfree space Green's functions.

The next step is to decide on the excitation scheme{{Note|Keep in mind that since [[EM. If your Picasso]] uses a planar structure has one or more ports and you seek to calculate its port characteristicsMoM solver, then you have to choose one of the lumped source types or a de-embedded source. If you are interested in calculated field value at the scattering characteristics of your planar structure, then you must define a plane wave sourcepoint is infinite. Before you can run As a planar MoM simulationresult, you also need to decide on the project's observables. These are field sensors must be placed at adequate distances (at least one or few wavelengths) away from the simulation data that you expect [[EM.Cube]] scatterers to generate as the outcome of the numerical simulationproduce acceptable results. [[EM.Cube]]'s [[Planar Module]] offers the following observables:}}

If Even though [[EM.Picasso]]'s MoM engine does not need a radiation box, you run 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 without having defined any observablesis finished, no data will be generated at three far field items are added under the end of Far Field item in the simulationNavigation Tree. Some observables require a certain type of excitation source. For example, port characteristics will be calculated only if These are the project contains a port definitionfar field component in &theta; direction, which the far field component in turn requires &phi; direction and the existence of at least one gap or probe or de-embedded source&quot;Total&quot; far field. The periodic characteristics (reflection and transmission coefficients) 2D radiation pattern graphs can be plotted from the '''Data Manager'''. A total of eight 2D radiation pattern graphs are calculated only if available: 4 polar and 4 Cartesian graphs for the structure has a periodic domain XY, YZ, ZX and excited by a user defined plane wave sourcecuts.

=== Planar Module[[Image:Info_icon.png|30px]] Click here to learn more about the theory of 's Simulation Modes ===''[[Defining_Project_Observables_%26_Visualizing_Output_Data#Using_Array_Factor_to_Model_Antenna_Arrays | Using Array Factors to Model Antenna Arrays ]]'''.

The simplest simulation type in <table><tr><td> [[EMImage:PMOM119.Cube]] is an analysis. In this mode, the planar structure in your project workspace is meshed at the center frequency png|thumb|left|600px|3D polar radiation pattern plot of the projecta microstrip-fed patch antenna. [[EM.Cube]] generates an input file at this single frequency, and the Planar MoM simulation engine is run once. Upon completion of the planar MoM simulation, a number of data files are generated depending on the observables you have defined in your project. An analysis is a single-run simulation.</td></tr></table>

[[EM.Cube]] offers When a number of multi-run simulation modes. In such casesplanar structure is excited by a plane wave source, the Planar MoM simulation engine is run multiple timescalculated far field data indeed represent the scattered fields of that planar structure. At each engine run, certain [[parametersEM.Picasso]] are varied and a collection can also calculate the radar cross section (RCS) of simulation data are generateda planar target. At Note that in this case the end of RCS is defined for a multifinite-run simulation, you can graph the simulation results sized target in EM.Grid or you can animate the 3D simulation data from the Navigation Tree. For example, in a frequency sweep, the frequency presence of the project is varied over its specified bandwidth. Port characteristics are usually plotted vs. frequency, representing your planar an infinite background structure's frequency response. In an angular sweep, the The scattered &theta; or and &phi; angle components of incidence the far-zone electric field are indeed what you see in the 3D far field visualization of a plane wave source is varied over their respective rangesradiation (scattering) patterns. Instead of radiation or scattering patterns, you can instruct [[EM.CubePicasso]]'s [[Planar Module]] currently provides to plot 3D visualizations of &sigma;_&theta;, &sigma;_&phi; and the following types of multi-run simulation modes:total RCS.

* Frequency Sweep<table>* Parametric Sweep<tr>* <td> [[OptimizationImage:PMOM125.png|thumb|left|600px|An example of the 3D monostatic radar cross section plot of a patch antenna.]]</td>* HDMR Sweep</tr></table>

[[File:PMOM80== Discretizing a Planar Structure in EM.png]]Picasso ==

Figure 1: Selecting The method of moments (MoM) discretizes all the finite-sized objects of a simulation mode planar structure (excluding the background structure) into a set of elementary cells. Both the quality and resolution of the generated mesh greatly affect the accuracy of the MoM numerical solution. The mesh density gives a measure of the number of cells per effective wavelength that are placed in [[Planar Module]]'s Simulation Run dialogvarious regions of your planar structure. The higher the mesh density, the more cells are created on the finite-sized geometrical objects. As a rule of thumb, a mesh density of about 20-30 cells per effective wavelength usually yields satisfactory results. But for structures with lots of fine geometrical details or for highly resonant structures, higher mesh densities may be required. The particular output data that you seek in a simulation also influence your choice of mesh resolution. For example, far field characteristics like radiation patterns are less sensitive to the mesh density than field distributions on structures with a highly irregular shapes and boundaries.

=== Running A <table><tr><td> [[Image:PMOM31.png|thumb|400px|The Planar MoM Analysis ===Mesh Settings dialog.]] </td></tr></table>

To run EM.Picasso provides two types of mesh for a planar MoM analysis of your project structure, open the Run Simulation Dialog by clicking the '''Run''' [[File:run_icon.png]] button on the '''Simulate Toolbar''' or select '''Menu''' '''&gt;''' '''Simulate &gt;''' '''Run''' or use the keyboard shortcut '''Ctrl+R'''. The '''Analysis''' option of the '''Simulation Mode''' dropdown list is selected by default. Once you click the '''Run''' button, the simulation starts. A new window, called the '''Output Window''', opens up that reports the different stages of simulation a pure triangular surface mesh and the percentage of the tasks completed at any timea hybrid triangular-rectangular surface mesh. After the simulation is successfully completedIn both case, a message pops up and reports the end of simulationEM. In certain cases like calculating scattering [[parameters]] of Picasso attempts to create a circuit or reflection / transmission characteristics of a periodic surfacehighly regular mesh, some results are also reported in which most of the Output Windowcells have almost equal areas. At the end of a simulationFor planar structures with regular, mostly rectangular shapes, you need to click the '''Close''' button of the Output Window hybrid mesh generator usually leads to return to the project workspacefaster computation times.

[[FileImage:PMOM78Info_icon.png|30px]]Click here to learn more about '''[[Preparing_Physical_Structures_for_Electromagnetic_Simulation#Working_with_EM.Cube.27s_Mesh_Generators | Working with Mesh Generator]]'''.

Figure 1: [[Planar ModuleImage:Info_icon.png|30px]]Click here to learn more about '''[[Preparing_Physical_Structures_for_Electromagnetic_Simulation#The_Triangular_Surface_Mesh_Generator | EM.Picasso's Simulation Run dialogTriangular Surface Mesh Generator]]'''.

<table><tr><td> [[EMImage:PMOM48H.Cube]]'s Planar MoM simulation engine uses a particular formulation png|thumb|left|420px|Details of the method hybrid planar mesh of moments called mixed potential integral equation (MPIE). Due to highthe slot-order singularities, the dyadic Green's functions for electric fields generated by electric currents as well as the dyadic Green's functions for magnetic fields generated by magnetic currents have very slow convergence behaviorscoupled patch array around discontinuities. Instead of using these slowly converging dyadic Green's function, the MPIE formulation uses vector and scalar potentials. These include vector electric potential '''A(r)''', scalar electric potential K]] <sup/td>&Phi;</suptr>'''(r)''', vector magnetic potential '''F(r)''' and scalar magnetic potential K<sup>&Psi;</suptable>'''(r)'''. These potentials have singularities of lower orders. As a result, they coverage relatively faster. The speed of their convergence is further increased drastically using special singularity extraction techniques.

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

A planar MoM simulation involves The mesh density gives a measure of the number of numerical [[parameters]] cells per effective wavelength that take preset default values unless you change themare placed in various regions of your planar structure. You can access these [[parameters]] and change their values by clicking the '''Settings''' button next to the '''Select Engine''' dropdown list in The effective wavelength is defined as <math>\lambda_{eff} = \tfrac{\lambda_0}{\sqrt{\varepsilon_{eff}}}</math>, where e_eff is the effective permittivity. By default, [[Planar ModuleEM.Picasso]]'s Simulation Run dialoggenerates a hybrid mesh with a mesh density of 20 cells per effective wavelength. In most cases, you do not need to open this dialog The effective permittivity is defined differently for different types of traces and you can leave all the default numerical parameter values intactembedded object sets. However, it This is useful to familiarize yourself with these [[parameters]], as they may affect the accuracy of your numerical resultsmake sure that enough cells are placed in areas that might feature higher field concentration.

The Planar MoM Engine Settings Dialog * For PEC and conductive sheet traces, the effective permittivity is organized in a number of sections. Here we describe some defined as the larger of the numerical [[parameters]]. The &quot;'''Matrix Fill'''&quot; section permittivity of the dialog deals with two substrate layers just above and below the operations involving the dyadic Green's functionsmetallic trace. You can set a value for the '''Convergence Rate for Integration'''* For slot traces, which the effective permittivity is 1E-5 by default. This is used for defined as the convergence test mean (average) of all the infinite integrals in the calculation permittivity of the Hankel transform of spectral-domain dyadic Green's functions. When the two substrate is lossy, layers just above and below the surface wave poles are captured in the complex integration plane using contour deformationmetallic trace. You can change the maximum number of iterations involved in this deformed contour integration* For embedded object sets, whose default value is 20. When the substrate effective permittivity is very thin with respect to defined as the wavelength, the dyadic Green's functions exhibit numerical instability. Additional singularity extraction measures are taken to avoid numerical instability but at the expense largest 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 permittivities of &quot;Thin Substrate&quot; by entering a value for '''Thin Substrate Threshold''' different than all the default 0.01. The parameter '''Max Coupling Range''' determines the distance threshold in wavelength between the observation substrate layers 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'''embedded dielectric sets.

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<subtable>&rho;</subtr> are pre-calculated and tabulated up to a limit of 30k<subtd>0</sub>, where k₀ is the free space propagation constant[[Image:PMOM32. These integrals may converge much faster based on the specified Convergence Rate for Integration described earlierpng|thumb|360px|A comparison of triangular and planar hybrid meshes of a rectangular patch. However, in certain cases involving highly oscillatory integrands, much larger integration limits like 100k_{0]] </subtd> might be needed to warrant adequate convergence. For spectral-domain integration along the real k<subtd>&rho;} axis, the interval [0, Nk₀] is subdivided into a large number [Image:PMOM30.png|thumb|360px|Mesh of sub-intervals, within each an 8-point Gauss-Legendre quadrature is appliedtwo rectangular patches at two different substrate planes. The next parameter, '''Nolower substrate layer has a higher permittivity. Radial Integration Divisions per k<sub>0]] </subtd>''', determines how small these intervals should be. By default, 2 divisions are used for the interval [0, k<sub>0</subtr>]. In other words, the length of each integration sub-interval is k<sub>0</subtable>/2. You can increase the resolution of integration by increasing this value above 2. Finally, instead of 2D Cartesian integration in the spectral domain, a polar integration is performed. You can set the '''No. of Angular Integration Points''', which has a default value of 100.

Figure 1: The Planar MoM Engine Settings dialogintegrity of the planar mesh and its continuity in the junction areas directly affects the quality and accuracy of the simulation results.EM.Picasso's hybrid planar mesh generator has some rules that are catered to 2.5-D MoM simulations:

:<mathtable><tr><td> \mathbf{[Z[File:PMOM36.png|250px]]}_{N\times N} \cdot \mathbf{[I[File:PMOM38.png|250px]]}_{N\times 1} = \mathbf{[V[File:PMOM37.png|250px]]}_{N\times 1} </mathtd><!--/tr><tr><td> Two overlapping planar objects and a comparison of their triangular and hybrid planar meshes. </td></tr><tr><td> [[File:PMOM81PMOM33.png|250px]][[File:PMOM35.png|250px]] [[File:PMOM34.png|250px]] </td></tr><tr><td> Edge--connected rectangular planar objects and a comparison their triangular and hybrid planar meshes. </td></tr></table>

where '''[I]''' is the solution vector, which contains the unknown amplitudes of all the basis functions that represent the unknown electric and magnetic currents of finite extents in your planar structure. In the above equation, N is the dimension of the linear system and equal to the total number of basis functions in the planar mesh. <table><tr><td> [[EMFile:PMOM39.Cubepng|375px]]'s linear solvers compute the solution vector'''[I]''' of the above system. You can instruct [[EMFile:PMOM40.Cubepng|375px]] 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 </td></tr><tr><td> Meshes of the Planar MoM Engine Settings dialog. These are written into three files called mom.dat1, exc.dat1 short and soln.dat1, respectivelylong vertical PEC vias connecting two horizontal metallic strips.</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 [[=== Refining the Planar Module]] offers five linear solvers:Mesh Locally ===

Of the above list, LU is The Planar Mesh Settings dialog gives a direct solver, while the rest are iterative solvers. BiCG is a relatively fast iterative solver, but it works only few options for symmetric matrices. You cannot use BiCG for periodic structures or customizing your planar structures that contain both metal mesh around geometrical and slot traces field discontinuities. The check box labeled &quot;'''Refine Mesh at different planes, as their MoM matrices are not symmetricJunctions'''&quot; increases the mesh resolution at the connection area between rectangular objects. The three solvers BCG-STAB, GMRES and TtFQMR work well for both symmetric and asymmetric matrices and they also belong to check box labeled &quot;'''Refine Mesh at Gap Locations'''&quot; might be particularly useful when gap sources or lumped elements are placed on a class of solvers called short transmission line connected from both ends. The check box labeled &quot;'''Krylov Sub-space MethodsRefine 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. In particular, EM.Picasso typically doubles the GMRES method mesh resolution locally at the discontinuity areas when the respective boxes are checked. You should always provides guaranteed unconditional convergencevisually inspect EM.Picasso's default generated mesh to see if the current mesh settings have produced an acceptable mesh.

[[Sometimes EM.Cube]]Picasso's [[Planar Module]], by defaultmesh may contain very narrow triangular cells due to very small angles between two edges. In some rare cases, provides extremely small triangular cells may be generated, whose area is a &quot;'''Automatic'''&quot; solver option that picks small fraction of the best method based on average mesh cell. These cases typically happen at the settings junctions and size other discontinuity regions or at the boundary of the numerical problemhighly irregular geometries with extremely fine details. For linear systems with a size less than N = 3,000In such cases, increasing or decreasing the LU solver is used. For larger systems, BiCG is used when dealing with symmetric matrices, mesh density by one or few cells per effective wavelength often resolves that problem and GMRES is used for asymmetric matriceseliminates those defective cells. If the size of the linear system exceeds N = 15Nonetheless,000, the sparse version of the iterative solvers is used, utilizing a row-indexed sparse storage schemeEM. You can override Picasso's planar mesh generator offers an option to identify the automatic solver option defective triangular cells and manually set you own solver typeeither delete them or cure them. This is done using the '''Solver Type''' dropdown list in the &quot;'''Linear System Solver'''&quot; section of the Planar MoM Engine Settings dialog. There are also By curing we mean removing a number of other [[parameters]] related narrow triangular cell and merging its two closely spaced nodes to fill the solverscrack left behind. The default value of '''Tolerance of Iterative Solver''' is 1E-3, which can be increased for more ill-conditioned systemsEM. The maximum number of iterations is usually expressed as a multiple of the systems size. The Picasso by default value of '''Max No. of Solver Iterations / System Size''' is 3. For extremely large systems, sparse versions of iterative solvers are used. In this case, deletes or cures all the elements of the matrix are thresholded with respect to the larges element. The default value of '''Threshold for Sparse Solver''' is 1E-6, meaning triangular cells that all the matrix elements whose magnitude is have angles less than 1E-6 times 10º. Sometimes removing defective cells may inadvertently cause worse problems in the large matrix elements are set equal to zeromesh. There are two more [[parameters]] that are related You may choose to disable this feature and uncheck the Automatic Solver option. These are box labeled &quot;''' User Iterative Solver When System Size &gt;Remove Defective Triangular Cells'''&quot; with a default in the Planar Mesh Settings dialog. You can also change the value of 3,000 and &quot;''' Use SParse Storage When System Size &gt;''' &quot; with a default value of 15,000. In other words, you control the automatic solver when to switch between direct and iterative solvers and when to switch to the sparse version of iterative solversminimum allowable cell angle.

If your computer has an Intel CPU{{Note| Narrow, then [[EMspiky triangular cells in a planar mesh are generally not desirable.Cube]] offers special versions You should get rid of all the above linear solvers that have been optimized for Intel CPU platforms. These optimal solvers usually work 2-3 time faster than their generic counterparts. When you install [[EM.Cube]], either by changing the option to use Intel-optimized solvers is already enabled. However, you can disable this option (e.g. if your computer has a non-Intel CPU). To do that, open the [[EM.Cube]]'s Preferences Dialog from '''Menu &gt; Edit &gt; Preferences''' mesh density or using the keyboard shortcut hybrid planar mesh generator'''Ctrl+H'''. Select the Advanced tab of the dialog and uncheck the box labeled &quot;''' Use Optimized Solvers for Intel CPU'''&quot;s additional mesh refinement options.}}

<table><tr><td> [[FileImage:PMOM82PMOM44.png|thumb|left|480px|Deleting or curing defective triangular cells: Case 1.]]</td></tr><tr><td> [[Image:PMOM42.png|thumb|left|480px|Deleting or curing defective triangular cells: Case 2.]]</td></tr></table>

[[Image:PMOM127.png|thumb|400px|Settings adaptive frequency sweep parameters == Running Planar MoM Simulations in EM.Picasso's Frequency Settings Dialog.]]=== Running Uniform and Adaptive Frequency Sweeps ===

In a frequency sweep, the operating frequency of a planar structure is varied during each sweep run. [[=== EM.Cube]]Picasso's [[Planar Module]] offers two types of frequency sweep: Uniform and Adaptive. In a uniform frequency sweep, the frequency range and the number of frequency samples are specified. The samples are equally spaced over the frequency range. At the end of each individual frequency run, the output data are collected and stored. At the end of the frequency sweep, the 3D data can be visualized and/or animated, and the 2D data can be graphed in EM.Grid.Simulation Modes ===

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

Frequency sweeps are often performed to study the frequency response {| class="wikitable"|-! scope="col"| Simulation Mode! scope="col"| Usage! scope="col"| Number of Engine Runs! scope="col"| Frequency ! scope="col"| Restrictions|-| style="width:120px;" | [[#Running a Single-Frequency Planar MoM Analysis | Single-Frequency Analysis]]| style="width:270px;" | Simulates the planar structure. In particular, "As Is"| style="width:80px;" | Single run| style="width:250px;" | Runs at the variation of scattering center frequency fc| style="width:80px;" | None|-| style="width:120px;" | [[parametersParametric_Modeling_%26_Simulation_Modes_in_EM.Cube#Running_Frequency_Sweep_Simulations_in_EM.Cube | Frequency Sweep]] like S₁₁ (return loss) and S₂₁ (insertion loss) with | style="width:270px;" | Varies the operating frequency are of utmost interest. When analyzing resonant structures like patch antennas or the planar filters over large frequency ranges, you may have to sweep MoM solver | style="width:80px;" | Multiple runs | style="width:250px;" | Runs at a large number specified set of frequency samples to capture their behavior with adequate details. The resonant peaks or notches are often missed due to the lack of enough resolution. adds more frequency samples in an adaptive way| style="width:80px;" | None|-| style="width:120px;" | [[EMParametric_Modeling_%26_Simulation_Modes_in_EM.Cube#Running_Parametric_Sweep_Simulations_in_EM.Cube| Parametric Sweep]]'s [[Planar Module]] offers a powerful adaptive frequency sweep option for this purpose. It is based on | style="width:270px;" | Varies the fact that the frequency response value(s) of a physical, causal, multiport network can be represented mathematically using a rational function approximation. In other words, one or more project variables| style="width:80px;" | Multiple runs| style="width:250px;" | Runs at the S [[parameters]] of a circuit exhibit a finite number of poles and zeros over a given center frequency range. fc| style="width:80px;" | None|-| style="width:120px;" | [[EMParametric_Modeling_%26_Simulation_Modes_in_EM.Cube#Performing_Optimization_in_EM.Cube | Optimization]] first starts with very few frequency samples and tries to fit rational functions | style="width:270px;" | Optimizes the value(s) of low orders one or more project variables to achieve a design goal | style="width:80px;" | Multiple runs | style="width:250px;" | Runs at the scattering center frequency fc| style="width:80px;" | None|-| style="width:120px;" | [[parametersParametric_Modeling_%26_Simulation_Modes_in_EM.Cube#Generating_Surrogate_Models | HDMR Sweep]]. Then, it increases | style="width:270px;" | Varies the number value(s) of samples gradually by inserting intermediate frequency samples in one or more project variables to generate a progressive manner. At each iteration cycle, all the possible rational functions of higher orders are tried out. The process continues until adding new intermediate frequency samples does not improve the resolution of the &quotcompact model| style="width:80px;S_ij&quot" | Multiple runs | style="width:250px; curves over " | Runs at the given center frequency range. In that case, the curves are considered as having converged.fc| style="width:80px;" | None|}

You must have defined one or more ports for your planar structure run an adaptive frequency sweep. Open can set the Frequency Settings dialog simulation mode from the [[EM.Picasso]]'s "Simulation Run dialog and select the '''Adaptive''' option of '''Frequency Sweep Type'''Dialog". You have to set values for '''Minimum Number of Samples''' and '''Maximum Number of Samples'''A single-frequency analysis is a single-run simulation. Their default values All the other simulation modes in the above list are 3 and 9, respectivelyconsidered multi-run simulations. You also set If you run a value for simulation without having defined any observables, no data will be generated at the '''Convergence Criterion''', which has a default value end of 0the simulation.1. At each iteration cycleIn multi-run simulation modes, all the S [[certain parameters]] are calculated at the newly inserted frequency samples, varied and their average deviation from the curves a collection of the last cycle is measured as an errorsimulation data files are generated. When this error falls below At the specified convergence criterionend of a sweep simulation, you can graph the iteration is ended. If [[simulation results in EM.Cube]] reaches Grid or you can animate the specified maximum number of iterations and 3D simulation data from the convergence criterion has not yet been met, the program will ask you whether to continue the process or exit it and stopnavigation tree.

{{Note|For large === Running a Single-Frequency Planar MoM Analysis === A single-frequency ranges, you may have to increase both analysis is the minimum and maximum number simplest type of samples[[EM. Moreover, remeshing Picasso]] simulation and involves the planar structure at each frequency may prove more practical than fixing the mesh at the highest frequency.}}following steps:

== Working with EM* Set the units of your project and the frequency of operation. Note that the default project unit is '''millimeter'''. * Define you background structure and its layer properties and trace types. * Construct your planar structure using [[Building_Geometrical_Constructions_in_CubeCAD | CubeCAD]]'s drawing tools to create all the finite-sized metal and slot trace objects and possibly embedded metal or dielectric objects that are interspersed among the substrate layers.* Define an excitation source and observables for your project.* Examine the planar mesh, verify its integrity and change the mesh density if necessary.* Run the Planar MoM simulation engine.* Visualize the output simulation data.Picasso Simulation Data ==

To run a planar MoM analysis of your project structure, open the Run Simulation Dialog by clicking the '''Run''' [[ImageFile:PMOM130run_icon.png]] button on the '''Simulate Toolbar''' or select '''Menu > Simulate > Run''' or use the keyboard shortcut {{key|thumb|400px|Changing Ctrl+R}}. The '''Single-Frequency Analysis''' option of the graph type by editing a data file's properties''Simulation Mode''' dropdown list is selected by default.]]=== EMOnce you click the {{key|Run}} button, the simulation starts.Picasso's A new window called the "Output Simulation Data ===Window" opens up that reports the different stages of simulation and the percentage of the tasks completed at any time. After the simulation is successfully completed, a message pops up and reports the end of simulation. In certain cases like calculating scattering parameters of a circuit or reflection / transmission characteristics of a periodic surface, some results are also reported in the output window.

Depending on the source type and the types of observables defined in a project, a number of output data are generated at the end of a planar MoM simulation<table><tr><td> [[Image:Picasso L1 Fig18. Some of these data are 2D by nature and some are 3D. The output simulation data generated by png|thumb|left|480px|EM.Picasso can be categorized into the following groups:'s Simulation Run dialog.]] </td></tr></table>

* '''Port Characteristics''': S, Z and Y [[=== Setting Numerical Parameters]] and Voltage Standing Wave Ratio (VSWR)* '''Radiation Characteristics''': Radiation Patterns, Directivity, Total Radiated Power, Axial Ratio, Main Beam Theta and Phi, Radiation Efficiency, Half Power Beam Width (HPBW), Maximum Side Lobe Level (SLL), First Null Level (FNL), Front-to-Back Ratio (FBR), etc.* '''Scattering Characteristics''': Bi-static and Mono-static Radar Cross Section (RCS)* '''Periodic Characteristics''': Reflection and Transmission Coefficients* '''Current Distributions''': Electric and magnetic current amplitude and phase on all metal and slot traces and embedded objects* '''Near-Field Distributions''': Electric and magnetic field amplitude and phase on specified planes and their central axes===

If your planar structure The Planar MoM Engine Settings Dialog is excited 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 gap sources or probe sources or dedefault. This is used for the convergence test of all the infinite integrals in the calculation of the Hankel transform of spectral-embedded sourcesdomain dyadic Green's functions. When the substrate is lossy, and one or more ports have been definedthe 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 planar MoM engine calculates substrate is very thin with respect to the scatteringwavelength, impedance and admittance (Sthe 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;<sub>eff</Zsub>, where &lambda;<sub>eff</Y) [[parameters]] sub> 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 designated portsdefault 0.01. The scattering [[parameters]] are defined based on parameter '''Max Coupling Range''' determines the port impedances specified distance threshold in wavelength between the projectobservation and source points after which the Green's Port Definition dialoginteractions are neglected. If more than one port has been defined in the projectThis distance by default is set to 1,000 wavelengths. For electrically small structures, the S/Z/Y matrices of phase variation across the multiport network are calculatedstructure 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'''.

At In the end &quot;Spectral Domain Integration&quot; section of a planar MoM simulation, the values of S/Z/Y [[parameters]] and VSWR data are calculated and reported in the output message window. The Sdialog, Z and Y [[parameters]] are written into output ASCII data files of complex type with you can set a &quot;value to '''.CPXMax Spectral Radius in k0'''&quot; extension. Every file begins with , which has a header consisting default value of a few comment lines 30. This means that start with the infinite spectral-domain integrals in the spectral variable k_{&quotrho;#&quot; symbol. The complex values} are arranged into two columns for the real pre-calculated and imaginary partstabulated up to a limit of 30k₀, where k₀ is the free space propagation constant. In These integrals may converge much faster based on the case of multiport structuresspecified Convergence Rate for Integration described earlier. However, every single element of 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 Sreal k<sub>&rho;</Zsub> axis, the interval [0, Nk<sub>0</Y matrices sub>] is written subdivided into a separate complex data file. For examplelarge number of sub-intervals, you will have data files like S11within each an 8-point Gauss-Legendre quadrature is applied.CPXThe next parameter, S21'''No.CPXRadial Integration Divisions per k₀''', determines how small these intervals should be...By default, Z11.CPX2 divisions are used for the interval [0, Z21k₀].CPXIn other words, etcthe length of each integration sub-interval is k₀/2. The VSWR data are saved to an ASCII data file You can increase the resolution of real type with integration by increasing this value above 2. Finally, instead of 2D Cartesian integration in the spectral domain, a &quot;polar integration is performed. You can set the '''No.DATof Angular Integration Points'''&quot; extension called, VSWR.DATwhich has a default value of 100.

If you run an analysis, the port characteristics have single complex values, which you can view using [[EM.CubePicasso]]'s data managerprovides a large selection of linear system solvers including both direct and iterative methods. However, there are no curves to graph. You can plot the S/Z/Y [[parametersEM.Picasso]] and VSWR data when you have data sets, which are generated at the end of any type of sweep including by default, provides a frequency sweep. In that case, the &quot;.CPX'''Automatic'''&quot; files have multiple rows corresponding to each value solver option that picks the best method based on the settings and size of the sweep parameter (enumerical problem.gFor linear systems with a size less than N = 3,000, the LU solver is used. frequency)For larger systems, BiCG is used when dealing with symmetric matrices, and GMRES is used for asymmetric matrices. You can instruct [[EM.Cube]]'s 2D graph to write the MoM matrix and excitation and solution vectors into output data are plotted in EMfiles for your examination.GridTo do so, a versatile graphing utility. You can plot the port characteristics directly from the Navigation Tree. Right click on check the box labeled &quot;'''Port DefinitionOutput MoM Matrix and Vectors''' item &quot; in the '''Observables''' Matrix Fill section of the Navigation Tree and select one of the items: '''Plot S [[Parameters]]''', '''Plot Y [[Parameters]]''', '''Plot Z [[Parameters]]''', or '''Plot VSWR'''Planar MoM Engine Settings dialog. In the first These are written into three casesfiles called mom.dat1, another sub-menu gives a list of individual port [[parameters]]exc.dat1 and soln.dat1, respectively.

In particular, it may be useful to plot the S<subtable>ii</subtr><td> [[parameters]] on a Smith chartImage:PMOM79. To change the format of a data plot, select it in the Data Manager and click its '''Edit''' buttonpng|thumb|left|720px|EM. In the Edit File Dialog, choose one of the options provided in the dropdown list labeled '''Graph Type''Picasso's Planar MoM Engine Settings dialog.]] </td></tr></table>

[[Image:Info_icon.png|40px]] Click here to learn more about '''[[Data_Visualization_and_Processing#Graphing_Port_Characteristics | Graphing Port Characteristics]]'''== Modeling Periodic Planar Structures in EM.Picasso ==

[[Image:Info_iconEM.png|40pxPicasso]] Click here allows you to learn more about '''simulate doubly periodic planar structures with periodicities along the X and Y directions. Once you designate your planar structure as periodic, [[Data_Visualization_and_Processing#Rational_Interpolation_of_Port_Characteristics | Rational Interpolation of Scattering ParametersEM.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.

Electric and magnetic currents are the fundamental output data of a planar MoM simulation. After the numerical solution of the MoM linear system, they are found using the solution vector '''{{Note| [I[EM.Picasso]''' ] can handle both regular and the definitions of the electric and magnetic vectorial basis functions:skewed periodic lattices.}}

Note that currents are complex vector quantities. Each electric or magnetic current has three X, Y and Z components, and each complex component has === Defining 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 Periodic Structure in the project workspace, superimposed on the surface of physical objectsEM.Picasso ===

In order to view the current distributionsAn infinite periodic structure in [[EM.Picasso]] is represented by a &quot;'''Periodic Unit Cell'''&quot;. To define a periodic structure, you must first define them as observables before running the planar MoM simulationopen [[EM. To do that, Picasso]]'s Periodicity Settings Dialog by right click on clicking the '''Current DistributionsPeriodicity''' item in the '''ObservablesComputational Domain''' section of the Navigation Tree navigation tree and select selecting '''Insert New ObservablePeriodicity Settings...'''from the contextual menu or by selecting '''Menu''' '''&gt;''' '''Simulate &gt; 'Computational Domain &gt; Periodicity Settings. The Current Distribution Dialog opens up. At .''' from the top of menu bar. In the dialog and in Periodicity Settings Dialog, check the section titled box labeled '''Active Trace / SetPeriodic Structure''', you can select a trace or embedded object set where you want to observe the current distribution. You can also select This will enable the current map type from two options: section titled'''Confetti'&quot;'' Lattice Properties&quot;. You can define the periods along the X and Y axes using the boxes labeled '''ConeSpacing'''. The former produces an intensity plot for current amplitude and phaseIn a periodic structure, while the latter generates virtual domain is replaced by a 3D vector plotdefault 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.

Once you close the current distribution <table><tr><td> [[Image:PMOM99.png|thumb|300px|EM.Picasso's Periodicity Settings dialog, the label of the selected trace or object set is added under the '''Current Distributions''' node of the Navigation Tree. ]] </td></tr></table>

{{Note|You have In many cases, your planar structure's traces or embedded objects are entirely enclosed inside the periodic unit cell and do not touch the boundary of the unit cell. [[EM.Picasso]] allows you to define a separate current distribution observable for each individual trace or periodic structures whose unit cells are interconnected. The interconnectivity applies only to PEC, PMC and conductive sheet traces, and embedded object setsets 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.}}

At the end of a planar MoM simulation, the current distribution nodes in the Navigation Tree become populated by the magnitude and phase plots of the three vectorial components of the electric ('''J''') and magnetic ('''M''') currents as well as the total electric and magnetic currents.<table><tr><td> [[Image:Info_iconimage122.png|40px]] Click here to learn more about '''[[Data_Visualization_and_Processing#Visualizing_3D_Current_Distribution_Maps thumb| Visualizing 3D Current Distribution Maps400px|Modeling a periodic screen using two different types of unit cell.]]'''.</td></tr></table>

<td> [[Image:PMOM84pmom_per5_tn.png|thumb|300px|EM.Picasso's Current Distribution dialogThe PEC cross unit cell.]] </td><td> [[Image:PMOM85(1)pmom_per6_tn.png|thumb|450px300px|The current distribution map Planar mesh of a patch antennathe PEC cross unit cell. Note the cell extensions at the unit cell's boundaries.]] </td>

=== Visualizing the Near Fields Exciting Periodic Structures as Radiators in EM.Picasso ===

In order to view the near field distributionsWhen a periodic planar structure is excited using a gap or probe source, you must first define field sensor observables before running it acts like an infinite periodic phased array. All the planar MoM simulationperiodic replicas of the unit cell structure are excited. To You can even impose a phase progression across the infinite array to steer its beam. You can do that, right click on this from the property dialog of the gap or probe source. At the bottom of the '''Field SensorsPlanar Gap Circuit Source Dialog''' item in the or '''ObservablesGap Source Dialog''' section of the Navigation Tree and select , there is a button titled '''Insert New ObservablePeriodic Scan...'''. The Field Sensor Dialog opens up. At the top of the dialog and in the section titled You can enter desired values for '''Sensor Plane LocationTheta''', first you need to set the plane of near field calculation. In the dropdown box labeled and '''DirectionPhi'''beam scan angles in degrees. To visualize the radiation patterns of a beam-steered antenna array, you have three options X, Y, and Z, representing the&quot;normals&quot; to define a finite-sized array factor in the XY, YZ and ZX planes, respectivelyRadiation Pattern dialog. The default direction is Z, i.e. XY plane parallel to You do this in the substrate layers. In the three boxes labeled '''CoordinatesImpose Array Factor''', you set the coordinates section of the center this dialog. The values of the plane. Then, you specify the '''SizeElement Spacing''' of along the plane in project units, X and finally Y directions must be set equal to the value of '''Number of SamplesPeriodic Lattice Spacing''' along the two sides of the sensor plane. The larger the number of samples, the smoother the near field map will appearthose directions.

Once you close the Field Sensor dialog, its name is added under the '''Field Sensors''' node of the Navigation Tree. At the end of a planar MoM simulation, the field sensor nodes in the Navigation Tree become populated by the magnitude and phase plots of the three vectorial components of the electric ('''E''') and magnetic ('''H''') field as well as the total electric and magnetic fields defined.<table><tr>Note that unlike <td> [[Image:Period5.png|thumb|350px|Setting periodic scan angles in EM.Cube]]Picasso's other computational modules, near field calculations in the [[Planar Module]] usually takes substantial timeGap Source dialog. This is due to the fact that at the end of a planar MoM simulation, the fields are not available anywhere (as opposed to the [[FDTD Module]]), and their computation requires integration of complex dyadic Green's functions (as opposed to </td><td> [[MoM3D ModuleImage:Period5_ang.png|thumb|350px|Setting the beam scan angles in Periodic Scan Angles dialog.]]'s free space Green's functions).</td></tr><tr><td> [[Image:Info_iconPeriod6.png|40px]] Click here to learn more about thumb|350px|Setting the array factor in EM.Picasso'''[[Data_Visualization_and_Processing#Visualizing_3D_Near-Field_Maps | Visualizing 3D Near Field Mapss Radiation Pattern dialog.]]'''.</td></tr></table>

<td> [[File:PMOM90.png|thumb|300px|[[Planar Module]]'s Field Sensor dialog.]] </td><td> [[Image:PMOM116Period7.png|thumb|420px360px|NearRadiation pattern of an 8×8 finite-zone electric field map above a microstrip-fed patch antennasized periodic printed dipole array with 0&deg; phi and theta scan angles.]] </td><td> [[Image:PMOM117Period8.png|thumb|420px360px|Near-zone magnetic field map above Radiation pattern of a microstripbeam-steered 8×8 finite-fed patch antennasized periodic printed dipole array with 45&deg; phi and theta scan angles.]] </td>

=== Visualizing the Far-Field Radiation Patterns Exciting Periodic Structures Using Plane Waves in EM.Picasso ===

Even though EM.Pplanar MoM engine does not need When a radiation boxperiodic planar structure is excited using a plane wave source, you still have to define it acts as a &quot;Far Field&quot; observable for radiation pattern calculationperiodic surface that reflects or transmits the incident wave. This is because far field calculations take time and you have to instruct [[EM.CubePicasso ]] to perform these calculationscalculates the reflection and transmission coefficients of periodic planar structures. To define If you run a far fieldsingle-frequency plane wave simulation, right click the '''Far Fields''' item reflection and transmission coefficients are reported in the '''Observables''' section Output Window at the end of the Navigation Tree and select simulation. Note that these periodic characteristics depend on the polarization of the incident plane wave. You set the polarization (TMz or TEz) in the '''Insert New Radiation Pattern...Plane Wave Dialog'''when defining your excitation source. The Radiation Pattern Dialog opens up. You may accept the default settings, or In this dialog you can change also set the value values of the incident '''Angle IncrementTheta''', which is expressed in degrees. You can also choose to and '''Normalize 2D PatternsPhi'''angles. In that case, At the maximum value end of a 2D paten graph will have a value the planar MoM simulation of 1; otherwisea periodic structure with plane wave excitation, the actual far field values in V/m will be used on reflection and transmission coefficients of the graphstructure are calculated and saved into two complex data files called &quot;reflection.CPX&quot; and &quot;transmission.CPX&quot;.

Once a planar MoM simulation is finished, three far field items are added under {{Note|In the Far Field item absence of any finite traces or embedded objects in the Navigation Tree. These are the far field component in &theta; directionproject workspace, the far field component in &phi; direction and the &quot;Total&quot; far field. The 2D radiation pattern graphs can be plotted from [[EM.CubePicasso]]'s '''Data Manager'''. A total computes the reflection and transmission coefficients of eight 2D radiation pattern graphs are available: 4 polar and 4 Cartesian graphs for the XY, YZ, ZX and user defined plane cutslayered background structure of your project.}}

<table><tr><td>[[Image:Info_iconPMOM102.png|40px]] Click here to learn more about the theory of '''[[Computing_the_Far_Fields_%26_Radiation_Characteristicsthumb| Far Field Computations580px|A periodic planar layered structure with slot traces excited by a normally incident plane wave source.]]'''.</td></tr></table>

You run a periodic MoM analysis just like an aperiodic MoM simulation from [[Image:Info_iconEM.png|40pxPicasso]] Click here to learn more about '''[[Data_Visualization_and_Processing#Visualizing_3D_Radiation_Patterns | Visualizing 3D Radiation Patterns]]'''s Run Dialog. [[Image:Info_iconHere, too, you can run a single-frequency analysis or a uniform or adaptive frequency sweep, or a parametric sweep, etc.png|40px]] Click here Similar to learn more about '''[[Data_Visualization_and_Processing#2D_Radiation_and_RCS_Graphs | Plotting 2D Radiation Graphs]]''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:PMOM118.png|thumb|300px|EM.Picasso's Radiation Pattern dialog.]] </td><td> [[Image:PMOM119PMOM98.png|thumb|450px600px|3D polar radiation pattern plot Changing the number of a microstrip-fed patch antennaFloquet modes from the Planar MoM Engine Settings dialog.]] </td>

=== Radar Cross Section You learned earlier how to use [[EM.Cube]]'s powerful, adaptive frequency sweep utility to study the frequency response of Planar Structures ===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.

When a planar structure is [[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 calculated far field data indeed represent planar MoM engine calculates the scattered fields reflection and transmission coefficients of the periodic surface. Note that planar structure. [[EM.Picasso]] you can also calculate the radar cross section (RCS) of conceptually consider a planar targetperiodic surface as a two-port network, where Port 1 is the top half-space and Port 2 is the bottom half-space. Note In that in this case , the RCS reflection coefficient R is defined for a finite-sized target in equivalent to S₁₁ parameter, while the presence of an infinite background structuretransmission coefficient T is equivalent to S₂₁ parameter. The scattered 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°&thetalt; and &phitheta; components of = 180°. You can also illuminate the farperiodic surface by the plane wave source from the bottom half-zone electric field are indeed what you see in the 3D far field visualization of radiation (scattering) patterns. Instead of radiation or scattering patternsspace, you can instruct [[EM.Picasso]] corresponding to plot 3D visualizations of 0° = &sigmatheta;&lt; 90°. In this case, the reflection coefficient R and transmission coefficient T are equivalent to S_&theta;22, &sigma;and S_&phi;12 and the total RCSparameters, respectively. To do soHaving these interpretations in mind, you must define an RCS observable instead [[EM.Cube]] enables the &quot;'''Adaptive Frequency Sweep'''&quot; option of the '''Frequency Settings Dialog''' when your planar structure has a radiation pattern by following these steps:periodic domain together with a plane wave source.

* Right click on the '''Far Fields''' item in the '''Observables''' section of the Navigation Tree and select '''Insert New RCS...''' to open the Radar Cross Section Dialog.* The resolution of RCS calculation is specified by '''Angle Increment''' expressed in degrees. By default, the &theta; and &phi; angles are incremented by 5 degrees.* At the end of a planar MoM simulation, besides calculating the RCS data over the entire (spherical) 3D space, a number of 2D RCS graphs are also generated. These are RCS cuts at certain planes, which include the three principal XY, YZ and ZX planes plus one additional constant f<!-cut. This fourth plane cut is at &phi; -= 45° by default. You can assign another &phi; angle in degrees in the box labeled '''Non== Modeling Finite-Principal Phi Plane'''.Sized Periodic Arrays ===

[[Image:Info_icon.png|40px]] Click here to learn more about '''[[Data_Visualization_and_Processing#Visualizing_3D_RCS | Visualizing 3D RCSModeling Finite-Sized Periodic Arrays Using NCCBF Technique]]'''.-->

<tablehr><tr><td> [[FileImage:PMOM124Top_icon.png|thumb|300px|EM.Picasso's Radar Cross Section dialog30px]] </td><td> '''[[Image:PMOM125EM.pngPicasso#Product_Overview |thumb|450px|An example Back to the Top of the 3D mono-static radar cross section plot of a patch antenna.Page]] </td></tr></table>'''

<p>&nbsp;</p>[[Image:Top_iconTutorial_icon.png|48px30px]] '''[[EM.PicassoCube#An_EMEM.Picasso_Primer Picasso_Documentation | Back to the Top of the PageEM.Picasso Tutorial Gateway]]'''

[[Image:Back_icon.png|40px30px]] '''[[EM.Cube | Back to EM.Cube Main Page]]'''