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

[[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.

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

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

[[Image:PMOM11.png|thumb|250px|=== EM.Picasso's Navigation Tree.]]In EM.Picasso, as the background structure is usually a layered planar structure that consists Planar Module 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 currentsEM.Cube ===

In a planar MoM simulation, [[EM.Picasso]] is the unknown electric and magnetic currents are discretized as a collection frequency-domain, full-wave '''Planar Module''' of elementary currents with small finite spatial extents'''[[EM. As a resultCube]]''', the governing integral equations reduce to a system of linear algebraic equationscomprehensive, whose solution determines integrated, modular electromagnetic modeling environment. [[EM.Picasso]] shares the amplitudes of 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 the elementary currents defined over the planar structureof [[EM.Cube]]'s mesh. Once the total currents are known, you can calculate the fields everywhere in the structureother computational modules.

[[Image:MOREInfo_icon.png|40px30px]] Click here to learn more about the theory of '''[[Planar Method of MomentsGetting_Started_with_EM.Cube | EM.Cube Modeling Environment]]'''.

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

[[Image:PMOM9EM.png|thumb|270px|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 Add Substrate Layer dialogPlanar MoM simulation engine incorporates the Green's functions of the background structure into the analysis, only the finite-sized traces like microstrips and slots are discretized by the mesh generator.As a result, the size of [[EM.Picasso]]=== Understanding 's computational problem is normally much smaller than that of [[EM.Tempo]]. In addition, [[EM.Picasso]] generates a hybrid rectangular-triangular mesh of your planar structure with a large number of equal-sized rectangular cells. Taking full advantage of all the Background Structure ===symmetry and invariance properties of dyadic Green's functions often results in very fast computation times that easily make up for [[EM.Picasso]]'s limited applications. A particularly efficient application of [[EM.Picasso]] is the modeling of periodic multilayer structures at oblique incidence angles.

EM<table><tr><td> [[Image:ART PATCH Fig12.Picasso is intended for constructing and modeling png|thumb|left|480px|The hybrid planar layered structures. By a planar structure we mean one that contains a background substrate mesh of laterally infinite extents, made up of one or more material layers all stacked up vertically along the Z-axis. Objects of finite size are then interspersed among these substrate layers. The background structure in EM.Picasso is called the &quot;'''Layer Stack-up'''&quot;. The layer stack-up is always terminated from the top and bottom by two infinite half-spaces. The terminating half-spaces might be the free space, or a perfect conductor (PEC ground), or any material medium. Most planar structures used in RF and microwave applications such as microstrip-based components have a PEC ground at their bottom. Some structures like stripline components require two bounding grounds (PEC halfslot-spaces) both at their top and bottomcoupled patch antenna array. ]]</td></tr></table>

=== Planar Object Types =EM.Picasso Features at a Glance ==

EM.Picasso provides the following types <ul> <li> Multilayer stack-up with unlimited number of objects substrate layers and trace planes</li> <li> PEC and conductive sheet traces for building a planar layered modeling ideal and non-ideal metallic layouts</li> <li> PMC traces for modeling slot layouts</li> <li> Vertical metal interconnects and embedded dielectric objects</li> <li> Full periodic structure:capability with inter-connected unit cells</li> <li> Periodicity offset parameters to model triangular, hexagonal or other offset periodic lattice topologies</li></ul>

# '''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.# '''Slot Traces''': These are used to model slots 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 surface magnetic currents.# '''Conductive Sheet Traces:''' These represent imperfect metals. They have a finite conductivity and a very small thickness. A surface impedance boundary condition is enforced on the surface of such traces.# '''PEC Via Sets:''' These are metallic objects such as shorting pins=== Sources, interconnect vias, plated-through holes, etc. that are grouped together as prismatic object sets. The embedded objects are modeled as vertical volume conduction currents.# '''Embedded Dielectric Sets:''' These are prismatic dielectric objects inserted inside a substrate layer. You can define a finite permittivity and conductivity for such objects, but their height is always the same as the height of their host layer. The embedded dielectric objects are modeled as vertical volume polarization currents.Loads &amp; Ports ===

<ul> <li> Gap sources on lines</li> <li> De-embedded sources on lines for S parameter calculations</li> <li> Probe (coaxial feed) sources on vias</li> <li> Gap arrays with amplitude distribution and phase progression</li> <li> Periodic gaps with beam scanning</li> <li> Multi-port and 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 circular polarizations</li> <li> Multi-ray excitation capability (ray data imported from [[Image:MOREEM.png|40pxTerrano]] Click here to learn more about ''' or external files)</li> <li> Huygens sources imported from other [[Planar Traces & Object TypesEM.Cube]]'''.modules</li></ul>

=== Defining the Layer Stack-Up Mesh Generation ===

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 <ul> <li> Optimized hybrid mesh with rectangular 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 triangular cells</li> <li> Regular triangular surface mesh</li> <li> Local meshing of the navigation tree and selecting '''Layer Stack-up Settings...''' from the contextual menu. Or alternatively, you can select the menu item '''Simulate &gt; Computational Domain &gt; Layer Stack-up Settings...'''trace groups</li> <li> Local mesh editing of planar polymesh objects</li> <li> Fast mesh generation of array objects</li></ul>

You can add new layers to your project's stack<ul> <li> 2.5-up or delete its layers, or move layers up or down and thus change the layer hierarchyD mixed potential integral equation (MPIE) formulation of planar layered structures</li> <li> 2. To add a new background layer, click the arrow symbol on the '''Insert...'''button at the bottom 5-D spectral domain integral equation formulation of the dialog periodic layered structures</li> <li> Accurate scattering parameter extraction and select '''Substrate Layer''' from the button'de-embedding using Prony&#39;s dropdown list. method</li> <li> Plane wave excitation with arbitrary angles of incidence</li> <li> A new dialog opens up where you can enter a label for the new layer variety of matrix solvers including LU, BiCG and GMRES</li> <li> Uniform and values for its material fast adaptive frequency sweep</li> <li> Parametric sweep with variable object properties or source parameters</li> <li> Generation of reflection and thickness in project units.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>

After creating a substrate layer, you can always edit its properties in the Layer Stack<ul> <li> Current distribution intensity plots</li> <li> Near field intensity plots (vectorial -up Settings dialog. Click on any layer's row in the table to select amplitude &amp; phase)</li> <li> Far field radiation patterns: 3D pattern visualization and highlight it 2D Cartesian and then click the '''Edit''' button. The substrate layer dialog opens uppolar graphs</li> <li> Far field characteristics such as directivity, where you can change the layer's label beam width, axial ratio, side lobe levels and assigned colornull parameters, etc. In the material properties section </li> <li> Radiation pattern of the dialog, you can change the name an arbitrary array configuration of the material and its 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>). To define electrical losses, you can either assign a value Touchstone-style S parameter text files for electric conductivity (s), or alternatively, define a loss tangent for the materialdirect export to RF. 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 Spice or its Device Editor</li> <li> Huygens surface generation</li> <li> Custom output parameters defined as mathematical expressions 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. standard outputs</li></ul>

For better visualization == Building a Planar Structure in EM.Picasso == [[EM.Picasso]] is intended for construction and modeling of your planar structure, EMlayered structures.Picasso displays By a virtual domain in planar structure we mean one that contains a default orange color to represent part background substrate of the laterally infinite background structure. The size extents, made up of this virtual domain is a quarter wavelength offset from the largest bounding box that encompasses one or more material layers all stacked up vertically along the finite Z-axis. Planar objects of finite size are interspersed among these substrate layers. The background structure in the project workspace. You can change the size of the virtual domain or its display color from the Domain Settings dialog, which you can access either by clicking the '''Computational Domain''' [[File:domain_iconEM.pngPicasso]] button of is called the &quot;'''Simulate ToolbarLayer Stack-up''', or by selecting '''Simulate &gtquot; Computational Domain &gt; Domain Settings...''' The layer stack-up is always terminated from the Simulate Menu or top and bottom by right clicking the '''Virtual Domain''' item of the Navigation Tree and selecting '''Domain Settingstwo infinite half-spaces...''' from The terminating half-spaces might be the contextual menufree space, or using the keyboard shortcut '''Ctrl+A'''a perfect conductor (PEC ground), or any material medium. Keep Most planar structures used in mind that the virtual domain is only for visualization purpose RF and does not affect the MoM simulation. The virtual domain also shows the substrate layers in translucent colors. If you assign different colors to your substrate layers, you microwave applications such as microstrip-based components have get a better visualization of multilayer virtual domain box surrounding your project structurePEC ground at their bottom. Some structures like stripline components are sandwiched between two grounds (PEC half-spaces) from both their top and bottom.

<td> [[Image:PMOM8(1)PMOM11.png|thumb|550pxleft|480px|EM.Picasso's Layer Stack-up Settings dialog with the initial default valuesnavigation tree and trace types.]] </td><td> [[Image:PMOM12.png|thumb|550px|EM.Picasso's Layer Stack-up Settings dialog showing a multilayer substrate configuration.]] </td></tr>

=== Defining Traces &amp; Object Sets the Layer Stack-Up ===

When you start a new project in [[Planar ModuleEM.Picasso]], the project workspace looks empty, and there are no finite objects in it. However, is always a default background structure is always present by default. Objects are defined as part that consists of traces a finite vacuum layer with a thickness of one project unit sandwiched between a vacuum top half-space and a PEC bottom half-space. Every time you open [[EM.Picasso]] or embedded setsswitched 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 definedyou close this dialog, you can see a list of project objects open it again by right-clicking the '''Layer Stack-up''' item in the '''Physical StructureComputational Domain''' section of the navigation treeand selecting '''Layer Stack-up Settings. ..''' from the contextual menu. Or alternatively, you can select the menu item '''Simulate &gt; Computational Domain &gt; Layer Stack-up Settings...'''

Traces and object sets can be defined either from Layer The Stack-up Settings dialog or has two tabs: '''Layer Hierarchy''' and '''Embedded Sets'''. The Layer Hierarchy tab has a table that shows all the background layers in hierarchical order from the navigation treetop half-space to the bottom half-space. It also lists the material composition of each layer, Z-coordinate of the bottom of each layer, its thickness (in project units) and material properties: permittivity (&epsilon;_r), permeability (&mu;_r), electric conductivity (&sigma;) and magnetic conductivity (&sigma;_m). There is also a column that lists the names of embedded object sets inside each substrate layer, if any.

In the ''<table><tr><td> [[Image:PMOM8(1).png|thumb|550px|EM.Picasso's Layer Stack-up Settings''' dialog, you with the initial default values.]] </td></tr></table> You can add a new trace layers to the your project's stack-up by clicking 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 {{key|Insert…}} button at the bottom of the dialog and select '''InsertSubstrate Layer''' button of from the button's dropdown list. A new dialogopens up where you can enter a label for the new layer and values for its material properties and thickness in project units. You have to choose from can delete a layer by selecting its row in the table and clicking the '''Metal (PEC)Delete'''button. To move a layer up and down, click on its row to select and highlight it. Then click either the '''Slot (PMC)Move Up''' or '''Conductive SheetMove Down''' options. A respective dialog opens up, where you can enter a label and assign a color other than default ones. Once a new trace is defined, it is added, by default, buttons consecutively to move the top of selected layer to the desired location in the stack-up table underneath . Note that you cannot delete or move the top or bottom half-spacespaces. From hereAfter creating a substrate layer, you can move always edit its properties in the trace down Layer Stack-up Settings dialog. Click on any layer's row in the table to select and highlight it and then click the desired location on {{key|Edit}} button. The substrate layer dialog opens up, where you can change the layer hierarchy's label and assigned color as well as its constitutive parameters.

Every time you define a new trace, it is also added under the respective category in the Navigation Tree[[Image:Info_icon. Alternatively, you can define a new trace from the Navigation Tree by right clicking on one of the trace type names and selecting png|30px]] Click here to learn more about '''Insert New PEC Trace[[Preparing_Physical_Structures_for_Electromagnetic_Simulation#Using_EM.Cube.27s_Materials_List | Using EM.Cube's Materials Database]]''or '''Insert New PMC Trace...'''or '''Insert New Conductive Sheet Trace...'''A respective dialog opens up for setting the trace properties. Once you close this dialog, it takes you directly to the Layer Stack-up Settings dialog so that you can set the right position of the trace on the stack-up.

As soon as you start drawing geometrical objects in the project workspace, the Physical Structure section of the Navigation Tree gets populated<table><tr><td> [[Image:PMOM12. The names of traces are added under their respective trace type category, and the names of objects appear under their respective trace grouppng|thumb|550px|EM. At any time, one and only one trace is active in the project workspacePicasso's Layer Stack-up Settings dialog showing a multilayer substrate configuration. An active trace is where all the new objects you draw belong to]] </td></tr></table> <table><tr><td> [[Image:PMOM9. When you define a new trace, it is set as active and you can immediately start drawing new objects on that tracepng|thumb|280px|EM. You can also set any trace active at any time by right clicking its name on the Navigation Tree and selecting Picasso'''Activate''' from the contextual menus Add Substrate Layer dialog.]] </td><td> [[Image:PMOM9A. The name of png|thumb|440px|A microstrip-fed, slot-coupled patch antenna on a double-layer substrate with a PEC ground plane in the active trace is always displayed in bold letter in middle hosting the Navigation Treecoupling slot.]] </td></tr></table>

EM.Picasso has a special feature that makes construction of planar structures quite easy and straightforward. '''The active work plane of the project workspace is always set at the plane of the active trace.''' In [[EM.Cube]]'s other modules, all objects are drawn in the XY plane (z = 0) by default. In [[== Planar Module]], all new objects are drawn on a horizontal plane that is located at the Z-coordinate of the currently active trace. As you change the active trace or add a new trace, you will also change the active work plane.Object & Trace Types ===

You can manage your project's layer hierarchy from the Layer Stack-up Settings dialog[[EM. You can add, delete Picasso]] groups objects by their trace type and move around substrate layers, metallic and slot traces and embedded object sets. Metallic and slot traces can move among their hierarchical location in the interface planes between neighboring substrate layers. Embedded object sets including PEC vias and finite dielectric objects can move from substrate layer into anotherstack-up. When you delete A trace is a trace from the Layer Stackgroup of finite-up Settings dialog, all of its sized planar objects are deleted from that have the project workspacesame material properties, too. You can also delete metallic same color and slot traces or embedded object sets from the Navigation Treesame Z-coordinate. To do so, right click on All the name of planar objects belonging to the same metal or slot trace or object set group are located on the same horizontal boundary plane in the Navigation Tree and select '''Delete''' from the contextual menulayer stack-up. You can also delete all All the traces or object sets of embedded objects belonging to the same type from embedded set lie inside the contextual menu of the respective type category in the Navigation Treesame substrate layer and have same material composition.

By default, the last defined trace or embedded object set is active. You can activate any trace or embedded object set at any time for drawing new objects. You can move one or more selected objects from any trace or embedded object set to another group of the same type or of different type. First select an object in the project workspace or in the Navigation Tree. Then, right click on the highlighted selection and select '''Move To &gt;''' from the contextual menu. This opens another sub-menu containing '''Planar''' and a list of all the other [[EM.CubePicasso]] modules that have already defined object groups. Select '''Planar''' or any other available module, and yet another sub-menu opens up with a list provides the following types of all the available traces and embedded object sets already defined in your project. Select the desired group, and all the selected objects will move to that group. When selecting multiple objects from the Navigation Tree, make sure that you hold the keyboard's '''Shift Key''' or '''Ctrl Key''' down while selecting for building a group's name from the contextual menu.planar layered structure:

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

# Terminating PEC ground planes at the top or bottom 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 object Click on each category 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 of an infinite PEC ground although learn more details about 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 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 [[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types]] allows any number of PEC via sets collocated in the same substrate layer. However, you can define only one embedded dielectric object set per substrate layer, and no vias sets collocated in the same layer. Note that the single set can host an arbitrary number of embedded dielectric objects of the same material properties.

You can define two types of metallic traces in [[Planar Module|Planar moduleEM.Picasso]] does not allow construction of 3D CAD : '''PEC Traces''' and '''Conductive Sheet Traces'''. PEC traces represent infinitesimally thin (zero thickness) planar metal objectsthat are deposited or metallized on or between substrate layers. Instead, you draw the cross section of prismatic PEC objects as planar [[Surface Objects|are modeled by surface objects]] parallel to electric currents. Conductive sheet traces, on the XY planeother hand, represent imperfect metals. [[EM.Cube]] then automatically extrudes these cross sections They have a finite conductivity and constructs and displays 3D prisms over thema very small thickness expressed in project units. The prisms extend all A surface impedance boundary condition is enforced on the way across the thickness surface of the host substrate layerconductive sheet objects.

== Discretizing '''Slot Traces''' are used to model cut-out slots and apertures in PEC ground planes. Planar Structures ==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 and its presence is implied. They are modeled by surface magnetic currents. When a slot is excited, tangential electric fields are formed on the aperture, which can be modeled as finite magnetic surface currents confined to the area of the slot. In other words, instead of modeling the electric surface currents on an infinite PEC ground around the slot, one can alternatively model the finite-extent magnetic surface currents on a perfect magnetic conductor (PMC) trace. Slot (PMC) objects provide the electromagnetic coupling between the two sides of an infinite PEC ground plane.

Besides planar metal and slot traces, [[Image:PMOM32.png|thumb|450px|Planar hybrid and triangular meshes for rectangular patchesEM.Picasso]][[Image:PMOM30allows you to insert prismatic embedded objects inside the substrate layers.png|thumb|450px|Mesh The height of two rectangular patches at two different planessuch embedded objects is always the same as the height of their host substrate layer. Two types of embedded object sets are available: '''PEC Via Sets''' and '''Embedded Dielectric Sets'''. PEC via sets are metallic objects such as shorting pins, interconnect vias, plated-through holes, etc. all located and grouped together inside the same substrate layer. The lower embedded via objects are modeled as vertical volume conduction currents. Embedded dielectric sets are prismatic dielectric objects inserted inside a substrate layer has . You can define a higher finite permittivityand conductivity for such objects.]][[Image:PMOM31.png|thumb|400px|The Planar Mesh Settings dialogembedded dielectric objects are modeled as vertical volume polarization currents.]]=== Understanding the Planar MoM Mesh ===

{{Note|The method height of moments (MoM) discretizes all an embedded object is always identical to the finite-sized objects thickness of a planar structure (excluding the background structure) into a set of elementary cells. The accuracy of the MoM numerical solution depends greatly on the quality and resolution of the generated mesh. As a rule of thumb, a mesh density of about 20-30 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, the particular simulation data that you seek in a project also influence your choice of mesh resolution. For example, far field characteristics like radiation patterns are less sensitive to the mesh density than field distributions on a structure with a highly irregular shape and a rugged boundaryits host substrate layer.}}

The mesh density gives When you start a measure of the number of cells per effective wavelength that are placed new project in various regions of your planar structure[[EM. The higher Picasso]], the mesh densityproject workspace looks empty, the more cells and there are created on the geometrical objects. Keep in mind that only the no finite-sized objects of your structure are discretizedin it. The free-space wavelength However, a default background structure is always present. Finite objects are defined as <math>\lambda_0 = \tfrac{2\pi f}{c}</math>part of traces or embedded sets. Once defined, where f is the center frequency you can see a list of your project and c is objects in the speed '''Physical Structure''' section of light in the free spacenavigation tree. The effective wavelength is Traces and object sets can be defined as <math>\lambda_either from Layer Stack-up Settings dialog or from the navigation tree. In the '''Layer Stack-up Settings''' dialog, you can add a new trace to the stack-up by clicking the arrow symbol on the {eff} = \tfrac{\lambda_0key|Insert}{\sqrt{\varepsilon_{eff}}}</math>, where e_eff is button of the effective permittivitydialog. By defaultYou have to choose from '''Metal (PEC)''', [[EM'''Slot (PMC)''' or '''Conductive Sheet''' options.Picasso]] generates A respective dialog opens up, where you can enter a hybrid mesh with label and assign a mesh density of 20 cells per effective wavelengthcolor. The effective permittivity Once a new trace is defined differently for different types of traces and embedded object sets. This , it is to make sure that enough cells are placed in areas that might feature higher field concentration. For PEC and conductive sheet tracesadded, by default, to the effective permittivity is defined as the larger top of the permittivity of stack-up table underneath the two substrate layers just above and below top half-space. From here, you can move the metallic tracedown to the desired location on the layer hierarchy. For slot tracesEvery time you define a new trace, the effective permittivity it is defined as also added under the mean (average) of respective category in the permittivity navigation tree. Alternatively, you can define a new trace from the navigation tree by right-clicking on one of the two substrate layers just above trace type names and below selecting '''Insert New PEC Trace...'''or '''Insert New PMC Trace...'''or '''Insert New Conductive Sheet Trace...''' A respective dialog opens up for setting the metallic traceproperties. For embedded object setsOnce you close this dialog, it takes you directly to the effective permittivity is defined as Layer Stack-up Settings dialog so that you can set the largest right position of the permittivities of all trace on the substrate layers and embedded dielectric setsstack-up.

[[Image:PMOM44Embedded object sets represent short material insertions inside substrate layers.png|thumb|400px|Deleting They can be metal or curing defective triangular cells: Case 1dielectric.]][[Image:PMOM42Metallic embedded objects can be used to model vias, plated-through holes, shorting pins and interconnects.png|thumb|400px|Deleting These are called PEC via sets. Embedded dielectric objects can be used to model air voids, thin films and material inserts in metamaterial structures. Embedded objects can be defined either from the Layer Stack-up Settings dialog or curing defective triangular cells: Case 2directly from the navigation tree.]][[Image:PMOM45Open the &quot;Embedded Sets&quot; tab of the stack-up dialog.pngThis 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|thumb|300px|Locking Insert}} button of the mesh density dialog and select one of an object group the two options, '''PEC Via Set''' or '''Embedded Dielectric Set''', from its property the dropdown list. This opens up a new dialogwhere first you have to set the host layer of the new object set.]]=== Generating, Viewing A dropdown list labeled & Customizing quot;'''Host Layer'''&quot; gives a Planar Mesh ===list of all the available finite substrate layers. You can also set the properties of the embedded object set, including its label, color and material properties. Keep in mind that you cannot control the height of embedded objects. Moreover, you cannot assign material properties to PEC via sets, while you can set values for the '''Permittivity'''(&epsilon;_r) and '''Electric Conductivity'''(&sigma;) of embedded dielectric sets. Vacuum is the 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 respective New Embedded Object Set dialog opens up, where you can set the properties of the new object set. As soon as you close this dialog, it takes you to the Layer Stack-up Settings dialog, where you can verify the location of the new object set on the layer hierarchy.

You can generate and view a planar mesh by clicking the '''Show Mesh''' <table><tr><td> [[FileImage:mesh_toolPMOM23.png]] button of the '''Simulate Toolbar''' or by selecting '''Menu &gt; Simulate &gt; Discretization &gt; Show Mesh''' or using the keyboard shortcut '''Ctrl+M'''. When the mesh of the planar structure is displayed in [[|thumb|550px|EM.Cube]]’s project workspace, its &quot;Mesh View&quot; mode is enabled. In this mode you can perform view operations like rotate view, pan or zoom, but you cannot create new objects or edit existing ones. To exit the mesh view mode, press the keyboardPicasso's '''Esc Key''' or click Layer Stack-up dialog showing the '''Show Mesh''' [[File:mesh_toolEmbedded Sets tab.png]] button once again. </td></tr></table> === Drawing Planar Objects on Horizontal Work Planes ===

Once a mesh is generated, it stays As soon as you start drawing geometrical objects in the memory until project workspace, the structure is changed or '''Physical Structure''' section of the mesh density or other settings navigation tree gets populated. The names of traces are modifiedadded under their respective trace type category, and the names of objects appear under their respective trace group. Every At any time you view mesh, the one and only one trace is active in the memory project workspace. The name of the active trace in the navigation tree is always displayedin bold letters. An active trace is where all the new objects you draw belong to. By default, the last defined trace or embedded object set is active. You can force [[EM.Picasso]] to create a immediately start drawing new mesh from objects on the ground up by selecting '''Menu &gt; Simulate &gt; Discretization &gt; Regenerate Mesh''' active trace. You can also set any trace or object set group active at any time by right -clicking on the '''Planar Mesh''' item in the '''Discretization''' section of its name on the navigation tree and selecting '''RegenerateActivate''' from the contextual menu.

You can change the settings of the planar mesh including the mesh type and density from the Planar Mesh Settings Dialog. You can also change these settings while in the mesh view mode, and you can update the changes to view the new mesh. To open the mesh settings dialog, either click the '''Mesh Settings''' [[FileImage:mesh_settingsInfo_icon.png|30px]] button of the Click here to learn more about '''Simulate Toolbar''' or select '''Menu &gt; Simulate &gt; Discretization &gt; Mesh Settings...''', or by right click on the '''Planar Mesh''' item [[Building Geometrical Constructions in the '''Discretization''' section of the Navigation Tree and select '''Mesh Settings...''' from the contextual menu, CubeCAD#Transferring Objects Among Different Groups or use the keyboard shortcut Modules | Moving Objects among Different Groups]]'''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.

The integrity [[EM.Picasso]] has a special feature that makes construction of the planar mesh structures very convenient and its continuity in the junction areas where adjacent objects are connected directly affects the simulation resultsstraightforward. <u>The most important rule horizontal Z-plane of the active trace or object connections in EM.Picasso set group is that only objects belonging to always set as the same trace can be connected to one anotheractive work plane of the project workspace. If two </u> That means all new objects belong to are drawn at the same trace (residing on the same Z-plane) and have a common overlap area, EMcoordinate of the currently active trace.Picasso first merges As you change the two objects using the &quot;Boolean Union&quot; operation and converts them into active trace group or add a single object for new one, the purpose of meshingactive work plane changes accordingly. EM.Picasso's hybrid planar mesh generator has some additional rules:

* If two connected rectangular objects have the same side dimensions along the common linear edge with perfect alignment, a rectangular junction mesh is produced{{Note| In [[EM.* If two connected rectangular objects have different side dimensions along the common linear edge or have edge offsetPicasso]], a set of triangular cells is generated along you cannot modify the edge Z-coordinate of the an object with the large side.* Rectangular objects that contain gap source or lumped elements, always have a rectangular mesh around the gap areaas it is set and controlled by its host trace.}}

If an embedded object like an interconnect via is located under [[EM.Picasso]] does not allow you to draw 3D or above a metallic trace or connected solid CAD objects. The solid object buttons in the '''Object Toolbar''' are disabled to prevent you from both top and bottom, it is critical doing so. In order to create mesh continuity between the vias and embedded object and its connected metallic traces. In other words, the generated mesh must ensure current continuity between the vertical volume currents and horizontal you simply have to draw their cross section geometry using planar surface currentsCAD objects. [[EM.Picasso’s Picasso]] extrudes and extends these planar mesh generator objects across their host layer automatically handles situations and displays them as 3D wireframe, prismatic objects. The automatic extrusion of embedded objects happens after mesh generation and before every planar MoM simulation. You can enforce this kind extrusion manually by right-clicking the '''Layer Stack-up''' item in the "Computational Domain" section of the navigation tree and generates all selecting '''Update Planar Structure''' from the required connection meshescontextual menu. {{Note| In [[EM.Picasso]], you can only draw horizontal planar surface CAD objects. }}

<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:MOREEM.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.

[[Image:PMOM52.png|thumb|400px|EM.Picasso's Port Definition dialog.]]<table>[[Image:PMOM53.png|thumb|300px|The Edit Port dialog.]]<tr><td> [[Image:PMOM51(2)PMOM64A.png|thumb|600px550px|Coupling gap sources in the Port Definition dialog by associating more than one source A multilayer planar structure containing a CPW line with a single coupled portand a lumped element on an overpassing metal strip.]]</td></tr>=== Defining Ports ===</table>

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

* Gap Sources* Probe Sources* Active Lumped Elements* De-Embedded Sourceselements 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.

Ports are defined in the '''Observables''' section of the Navigation Tree[[Image:Info_icon. Right click on the png|40px]] Click here to learn more about '''Port Definition[[Preparing_Physical_Structures_for_Electromagnetic_Simulation#Modeling_Lumped_Elements_in_the_MoM_Solvers | Defining Lumped Elements]]''' item of the Navigation Tree and select '''Insert New Port Definition...''' from the contextual menu. The Port Definition Dialog opens up, showing the default port assignments. If you have N sources in your planar structure, then N default ports are defined, with one port assigned to each source according to their order on the Navigation Tree. Note that your project can have mixed gap and probes sources as well as active lumped element sources.

'''You can define any number of ports equal to or less than the total number of sources in your project[[Image:Info_icon.''' The Port List of the dialog shows png|40px]] Click here for a list general discussion of all the ports in ascending order, with their associated sources and the port's characteristic impedance, which is 50S by default. You can delete any port by selecting it from the Port List and clicking the '''Delete''' button of the dialog[[Preparing_Physical_Structures_for_Electromagnetic_Simulation#A_Review_of_Linear_. 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 intend26_Nonlinear_Passive_. 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 '''Impedance26_Active_Devices | Linear Passive Devices]]'''.

[[Image:MORE.png{{Note|40px]] Click here to learn more about The impedance of the theory lumped circuit is calculated at the operating frequency of '''[[Computing Port Characteristics in the project using the specified R, L and C values. As you change the frequency, the value of the impedance that is passed to the Planar MoM]]'''engine will change.}}

=== Modeling Coupled Ports Calculating Scattering Parameters Using Prony's Method ===

Sources can be coupled to each other to model coupled strip lines The calculation of the scattering (CPSS) on metal traces or coplanar waveguides (CPW) on slot tracesparameters is usually an important objective of modeling planar structures especially for planar circuits like filters, couplers, etc. SimilarlyAs you saw earlier, probe you can use lumped sources may be coupled like gaps and probes and even active lumped elements to each other. Coupling two or more sources does not change calculate the way they excite a circuit characteristics of planar structure. It is intended only for the purpose of S parameter calculationstructures. The feed admittance / impedance calculations based on the gap voltages and currents are accurate at RF and lower microwave frequencies or when the port transmission lines are narrow. In such cases, the electric or vias which host magnetic current distributions across the coupled sources width of the port line are usually parallel smooth, and aligned with one another and they are all grouped together as quite uniform current or voltage profiles can easily be realized. At higher frequencies, however, a single transmission line represented by a single more robust method is needed for calculating the port. This single &quot;coupled&quot; port then interacts with other coupled or uncoupled portsparameters.

You couple two or more sources using One can calculate the '''Port Definition Dialog'''. To do so, you need to change scattering parameters of a planar structure directly by analyzing the current distribution patterns on the default port assignmentstransmission lines. First, delete all The discontinuity at the ports that are to be coupled from the Port List end of the dialog. Then, define a new port by clicking line typically gives rise to a standing wave pattern that can clearly be discerned in the line'''Add''' button of the dialogs current distribution. This opens up From the Add Port dialog, which consists location of two tables: '''Available''' sources on the left current minima and '''Associated''' sources on the right. A right arrow ('''--&gt;''') button maxima and a left arrow ('''&lt;--''') button let you move their relative levels, one can determine the sources freely between these two tables. You will see in reflection coefficient at the &quot;Available&quot; table a list of all the sources that you deleted earlierdiscontinuity, i. You may even see more available sourcese. Select all the sources that you want to couple and move them to the &quot;Associated&quot; table on the rightS₁₁ parameter. You can make multiple selections using the keyboard's '''Shift''' and '''Ctrl''' keys. Closing the Add Port dialog returns you to the Port Definition dialogA more robust technique is Prony’s method, where you will now see the names which is used for exponential approximation of all the coupled sources next to the name functions. A complex function f(x) can be expanded as a sum of complex exponentials in the newly added port.following form:

:<math> f(x) \approx \sum_{n=1}^N c_i e^{Note|It is your responsibility to set up coupled ports and coupled -j\gamma_i x} </math><!--[[Transmission LinesFile:PMOM73.png]] 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.}}-->

[[Image:PMOM64where c_i are complex coefficients and &gamma;_i are, in general, complex exponents.png|thumb|600px|EM.Picasso's Lumped Element dialog.]][[Image:PMOM49.png|thumb|600px|Defining gap sources on an array From the physics of rectangle strip objects transmission lines, we know that lossless lines may support one or more propagating modes with a Chebyshev amplitude distributionpure real propagation constants (real &gamma;_i exponents).]][[Image:PMOM50.png|thumb|500px|Defining gap source array weights using a data file.]]=== Modeling Lumped Elements in EMMoreover, 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.Picasso ===

Lumped elements are componentsIn practical planar structures for which you want to calculate the scattering parameters, deviceseach port line normally supports one, or circuits whose overall dimensions and only one, dominant propagating mode. Multi-mode transmission lines are very small compared to the wavelengthseldom used for practical RF and microwave applications. As a resultNonetheless, they are considered to be dimensionless compared to the dimensions each port line carries a superposition of a mesh cellincident and reflected dominant-mode propagating signals. In factAn incident signal, by convention, a lumped element is equivalent to an infinitesimally narrow gap one that is placed in propagates along the path of current flowline towards the discontinuity, across which where the device's governing equations are enforcedphase reference plane is usually established. A reflected signal is one that propagates away from the port plane. Using KirkhoffProny's laws, these device equations normally establish a relationship between method can be used to extract the currents incident and voltages across reflected propagating and evanescent exponential waves from the device or circuitstanding wave data. Crossing From a knowledge of the bridge to Maxwellamplitudes (expansion coefficients) of the incident and reflected dominant propagating modes at all ports, the scattering matrix of the multi-port structure is then calculated. In Prony's domainmethod, the device equations must now 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 cast into drawn in a region far from sources or discontinuities, typically a quarter wavelength away from o boundary conditions that relate the electric and magnetic currents and fieldstwo ends of a feed line.

Click here to learn more about the theory of '''<table><tr><td> [[Computing_Port_Characteristics_in_Planar_MoM#Modeling_Lumped_Elements_in_Planar_MoMImage:PMOM71.png| Modeling Lumped Elements in Planar MoMthumb|600px|Minimum and maximum current locations of the standing wave pattern on a microstrip line feeding a patch antenna.]]'''.</td></tr></table>

* Open the Lumped Element Dialog by right clicking on the '''Lumped Elements''' item Ports are used in the '''Sources''' section of the Navigation Tree a planar structure to order and selecting '''Insert New Source...'''* In index the '''Gap Topology''' section sources for calculation of the dialogcircuit parameters such as scattering (S), select one of the two options: '''Gap on Line''' impedance (Z) and '''Gap on Via'''admittance (Y) parameters.* In the '''Lumped Circuit Type''' section of the dialog, select one of the two options: '''Passive RLC''' and '''Active with Gap Source'''[[EM.* Depending on your choice of gap topology, in the '''Lumped Circuit Location''' section of the dialogPicasso]], you will find either a list of all the '''Rectangle Strip Objects''' can use one or a list more of all the '''PEC Via Objects''' available in the project workspace. Select the desired rectangle strip or embedded PEC via object.* In the box labeled '''Offset''', enter the distance following types of the lumped element from the start point of the rectangle strip line or from the bottom of the via object, whichever the case. The value of '''Offset''' by default is initially set sources to the center of the line or via.* In the '''Load Properties''' section, the series and shunt resistance values Rs and Rp are specified in Ohms, the series and shunt inductance values Ls and Lp are specified in nH (nanohenry), and the series and shunt capacitance values Cs and Cp are specified in pF (picofarad). Only the checked elements are taken into account in the total impedance calculation. By default, only the series resistor is checked with a value of 50S, and all other circuit elements are initially greyed out.<br />define ports:

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 '''Active with * Gap Source''' option of the '''Sources* Probe Sources* Active 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.Elements* De-Embedded Sources

If Ports are defined in the project workspace contains an array '''Observables''' section of rectangle strip objects the navigation tree. You can define any number of ports equal to or PEC via objects, less than the array object will also be listed as an eligible object for lumped element placementtotal number of sources in your project. A lumped element will If you have N sources in your planar structure, then be placed on N default ports are defined, with one port assigned to each element of source according to their order on the arraynavigation tree. All the lumped elements will Note that your project can have identical direction, offset, resistance, inductance mixed gap and capacitance values. If you define an probes sources as well as active lumped element, you can prescribe certain amplitude sources on PEC and/slot traces or phase distribution vias. You can also couple ports together to the gap sources just like in the case of gap and probe sources. The available amplitude distributions include '''Uniform''', '''Binomial'''''', Chebyshev''' and '''Data File'''define coupled transmission lines such as coupled strips (CPS) or coplanar waveguides (CPW).

{{Note[[Image:Info_icon.png|The impedance of 40px]] Click here to learn more about the lumped circuit is calculated at the operating frequency of the project using the specified R, L and C values'''[[Glossary_of_EM. As you change the frequency, the value of the impedance that is passed to the Planar MoM engine will changeCube%27s_Simulation_Observables_%26_Graph_Types#Port_Definition_Observable | Port Definition Observable]]'''.}}

If the project workspace contains an array of rectangle strip objects, the array object will also be listed as an eligible object for gap source placement. A gap source will then be placed on each element of the array. All the gap sources will have identical direction and offset. Similarly, if the project workspace contains an array of PEC via objects, the embedded array object will also be listed as an eligible object for probe source placement. A probe source will then be placed on each via object of the array. All the probe sources will have identical direction and offset== EM.Picasso's Simulation Data & Observables ==

However, you can prescribe certain amplitude Depending on the source type and/or phase distribution over the array types of gap or probe sources. By defaultobservables defined in a project, all a number of output data are generated at the gap or probe sources have identical amplitudes end of 1V (or 1A for the slot case) a planar MoM simulation. Some of these data are 2D by nature and zero phasesome are 3D. The available amplitude distributions to choose from include '''Uniform''', '''Binomial''' and '''Chebyshev''' and '''Date File'''output simulation data generated by [[EM. In the Chebyshev case, you need to set a value for minimum side lobe level ('''SLL''') in dB. You Picasso]] can also define '''Phase Progression''' in degrees along all three principal axes. You can view be categorized into 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.following groups:

In the data file option, the complex amplitude are directly read Click on each category to learn more details about it 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 Glossary of all the array element indices with default 1+j0 amplitudes for all of them. You can replace the default complex values with new one and save the Notepad data file, which brings you back to the Gap Source dialog. To import the array weights, click the '''Open Data File''' button, which opens the standard [[Windows]] Open dialog. You can then select the right data file from the one of your folders. It is important to note that the data file must have the correct format to be read by [[EM.Cube's Simulation Observables & Graph Types]]. For this reason, it is recommended that you first create a new data file with the right format using Notepad as described earlier and then save it for later use.

== Running Planar If your planar structure is excited by gap sources or probe sources or de-embedded sources, and one or more ports have been defined, the planar MoM Simulations ==engine calculates the scattering, impedance and admittance (S/Z/Y) parameters of the designated ports. The scattering parameters are defined based on the port impedances specified in the project's Port Definition dialog. If more than one port has been defined in the project, the S/Z/Y matrices of the multiport network are calculated.

The first step Electric and magnetic currents are the fundamental output data of planning a planar MoM simulation is defining your planar structure. This consists After the numerical solution of the background structure plus all the finite-sized metal and slot trace objects and possibly embedded metal or dielectric objects that MoM linear system, they are interspersed among found using the substrate layers. The background stack-up is defined in the Layer Stack-up dialog, which automatically opens up as soon as you enter the [solution vector '''[Planar ModuleI]]. The metal ''' and slot traces and embedded object sets are listed in the Navigation Tree, which also shows all definitions of the geometrical (CAD) objects you draw in the project workspace under each object group at different Z-planes.electric and magnetic vectorial basis functions:

The next step is to decide on the excitation scheme. If your planar structure has one or more ports and you seek to calculate its port characteristics, then you have to choose one of the lumped source types or a de-embedded source. If you are interested in the scattering characteristics of your planar structure, then you must define a plane wave source. Before you can run a planar MoM simulation, you also need to decide on the project's observables. These are the simulation data that you expect :<math> \mathbf{[[EM.CubeX]] to generate as the outcome of the numerical simulation. [[EM.Cube]]'s [[Planar Module]] offers the following observables:}_{N\times 1} = \begin{bmatrix} I^{(J)} \\ \\ V^{(M)} \end{bmatrix} \quad \Rightarrow \quad \begin{cases} \mathbf{J(r)} = \sum_{n=1}^N I_n^{(J)} \mathbf{f_n^{(J)} (r)} \\ \\ \mathbf{M(r)} = \sum_{k=1}^K V_k^{(M)} \mathbf{f_k^{(M)} (r)} \end{cases} </math>

* Note that currents are complex vector quantities. Each electric or magnetic current has three X, Y and Z components, and each complex component has a magnitude and phase. You can visualize the surface electric currents on metal (PEC) and conductive sheet traces, surface magnetic currents on slot (PMC) traces and vertical volume currents on the PEV vias and embedded dielectric objects. 3D color-coded intensity plots of electric and magnetic current distributions are visualized in the project workspace, superimposed on the surface of physical objects. In order to view the current distributions, you must first define them as observables before running the planar MoM simulation. At the top of the Current Distribution* Field Sensors* Far Fields (Radiation Patterns dialog and in the section titled '''Active Trace / Set''', you can select a trace or Radar Cross Section)* Huygens Surfaces* Port Characteristics* Periodic Characteristicsembedded object set where you want to observe the current distribution.

If you run {{Note|You have to define a simulation without having defined any observables, no data will be generated at the end of the simulation. Some observables require a certain type of excitation source. For example, port characteristics will be calculated only if the project contains a port definition, which in turn requires the existence of at least one gap separate current distribution observable for each individual trace or probe or de-embedded source. The periodic characteristics (reflection and transmission coefficients) are calculated only if the structure has a periodic domain and excited by a plane wave sourceobject set.}}

The simplest simulation type in [[EM.CubePicasso]] is an analysis. In this mode, allows you to visualize the planar structure in your project workspace is meshed near fields at the center frequency of the projecta specific field sensor plane. Note that unlike [[EM.Cube]] generates an input file at this single frequency's other computational modules, and the Planar MoM simulation engine is run oncenear field calculations in [[EM. Upon completion Picasso]] usually takes a significant amount of time. This is due to the fact that at the end of a planar MoM simulation, a number of data files are generated depending on the observables you have defined in your projectfields are not available anywhere (as opposed to [[EM. An analysis is Tempo]]), and their computation requires integration of complex dyadic Green's functions of a single-run simulationmultilayer background structure as opposed to the free space Green's functions.

{{Note|Keep in mind that since [[EM.CubePicasso]] offers uses a number of multi-run simulation modes. In such casesplanar MoM solver, the Planar MoM simulation engine calculated field value at the source point is run multiple timesinfinite. At each engine run, certain [[parameters]] are varied and a collection of simulation data are generated. At the end of As a multi-run simulationresult, you can graph the simulation results in EM.Grid field sensors must be placed at adequate distances (at least one or you can animate the 3D simulation data few wavelengths) away from the Navigation Treescatterers to produce acceptable results. For example, in a frequency sweep, the frequency of the project is varied over its specified bandwidth. Port characteristics are usually plotted vs. frequency, representing your planar structure's frequency response. In an angular sweep, the &theta; or &phi; angle of incidence of a plane wave source is varied over their respective ranges. [[EM.Cube]]'s [[Planar Module]] currently provides the following types of multi-run simulation modes:}}

* Frequency Sweep<table>* Parametric Sweep<tr>* <td> [[OptimizationImage:PMOM116.png|thumb|left|600px|Near-zone electric field map above a microstrip-fed patch antenna.]]</td>* HDMR Sweep</tr><tr><td> [[Image:PMOM117.png|thumb|left|600px|Near-zone magnetic field map above a microstrip-fed patch antenna.]] </td></tr></table>

Even though [[File:PMOM80EM.pngPicasso]]'s MoM engine does not need a radiation box, you still have to define a &quot;Far Field&quot; observable for radiation pattern calculation. This is because far field calculations take time and you have to instruct [[EM.Cube]] to perform these calculations. Once a planar MoM simulation is finished, three far field items are added under the Far Field item in the Navigation Tree. These are the far field component in &theta; direction, the far field component in &phi; direction and the &quot;Total&quot; far field. The 2D radiation pattern graphs can be plotted from the '''Data Manager'''. A total of eight 2D radiation pattern graphs are available: 4 polar and 4 Cartesian graphs for the XY, YZ, ZX and user defined plane cuts.

Figure 1[[Image: Selecting a simulation mode in Info_icon.png|30px]] Click here to learn more about the theory of '''[[Planar ModuleDefining_Project_Observables_%26_Visualizing_Output_Data#Using_Array_Factor_to_Model_Antenna_Arrays | Using Array Factors to Model Antenna Arrays ]]'s Simulation Run dialog''.

To run When a planar MoM analysis of your project structureis excited by a plane wave source, open the Run Simulation Dialog by clicking calculated far field data indeed represent the '''Run''' scattered fields of that planar structure. [[File:run_iconEM.pngPicasso]] button on can also calculate the '''Simulate Toolbar''' or select '''Menu''' '''radar cross section (RCS) of a planar target. Note that in this case the RCS is defined for a finite-sized target in the presence of an infinite background structure. The scattered &gttheta;''' '''Simulate and &gtphi;''' '''Run''' or use the keyboard shortcut '''Ctrl+R'''. The '''Analysis''' option components of the '''Simulation Mode''' dropdown list is selected by default. Once far-zone electric field are indeed what you click see in the '''Run''' button, the simulation starts. A new window, called the '''Output Window''', opens up that reports the different stages 3D far field visualization of simulation and the percentage of the tasks completed at any timeradiation (scattering) patterns. After the simulation is successfully completed, a message pops up and reports the end Instead of simulation. In certain cases like calculating radiation or scattering patterns, you can instruct [[parametersEM.Picasso]] to plot 3D visualizations of a circuit or reflection &sigma;<sub>&theta;</ transmission characteristics of a periodic surfacesub>, some results are also reported in the Output Window. At the end of a simulation, you need to click the '''Close''' button of the Output Window to return to &sigma;_&phi; and the project workspacetotal RCS.

<table><tr><td> [[FileImage:PMOM78PMOM125.png|thumb|left|600px|An example of the 3D monostatic radar cross section plot of a patch antenna.]]</td></tr></table>

Figure 1: [[== Discretizing a Planar Module]]'s Simulation Run dialogStructure in EM.Picasso ==

=== Stages Of A Planar The method of moments (MoM Analysis ===) discretizes all the finite-sized objects of a 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 various 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.

<table><tr><td> [[EMImage:PMOM31.png|thumb|400px|The Planar Mesh Settings dialog.Cube]]'s Planar MoM simulation engine uses a particular formulation of the method of moments called mixed potential integral equation (MPIE). Due to high-order singularities, the dyadic Green's functions for electric fields generated by electric currents as well as the dyadic Green's functions for magnetic fields generated by magnetic currents have very slow convergence behaviors. Instead of using these slowly converging dyadic Green's function, the MPIE formulation uses vector and scalar potentials. These include vector electric potential '''A(r)''', scalar electric potential K<sup/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 EM.Picasso provides two types of mesh for a planar MoM simulation consists of two major stagesstructure: matrix fill a pure triangular surface mesh and linear system inversiona hybrid triangular-rectangular surface mesh. In the first stageboth case, the moment matrix and excitation vector are calculatedEM. In the second stagePicasso attempts to create a highly regular mesh, the MoM system of linear equations is inverted using one in which most of the several available matrix solvers to find the unknown coefficients of all the basis functionscells have almost equal areas. 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 For planar structurestructures with regular, mostly rectangular shapes, [[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 mesh generator usually leads to faster computation times.

A planar MoM simulation involves a number of numerical [[parameters]] that take preset default values unless you change themImage:Info_icon. You can access these [[parameterspng|30px]] and change their values by clicking the '''Settings''' button next Click here to the learn more about '''Select Engine''' dropdown list in the [[Planar Module]]Preparing_Physical_Structures_for_Electromagnetic_Simulation#The_Triangular_Surface_Mesh_Generator | EM.Picasso's Simulation Run dialog. In most cases, you do not need to open this dialog and you can leave all the default numerical parameter values intact. However, it is useful to familiarize yourself with these [[parametersTriangular Surface Mesh Generator]], as they may affect the accuracy of your numerical results'''.

The Planar MoM Engine Settings Dialog is organized in a number of sections. Here we describe some of the numerical <table><tr><td> [[parameters]]Image:PMOM48F. The &quot;'''Matrix Fill'''&quot; section png|thumb|left|420px|Geometry 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 1Emultilayer slot-5 by defaultcoupled patch array. This is used for the convergence test of all the infinite integrals in the calculation of the Hankel transform of spectral-domain dyadic Green's functions. When the substrate is lossy, the surface wave poles are captured in the complex integration plane using contour deformation. You can change the maximum number of iterations involved in this deformed contour integration, whose default value is 20. When the substrate is very thin with respect to the wavelength, the dyadic Green's functions exhibit numerical instability. Additional singularity extraction measures are taken to avoid numerical instability but at the expense of increased computation time. By default, a thin substrate layer is defined to a have a thickness less than 0.01&lambda;]] <sub/td>eff</subtr>, where &lambda;<subtr>eff</subtd> is the effective wavelength[[Image:PMOM48G. You can modify the definition png|thumb|left|420px|Hybrid planar mesh of &quot;Thin Substrate&quot; by entering a value for '''Thin Substrate Threshold''' different than the default 0.01. The parameter '''Max Coupling Range''' determines the distance threshold in wavelength between the observation and source points after which the Green's interactions are neglected. This distance by default is set to 1,000 wavelengths. For electrically small structures, the phase variation across the structure may be negligible. In such cases, a fast quasislot-static analysis can be carried out. You can set this threshold in wavelengths in the box labeled '''Max Dimensions for Quasi-Static Analysis'''coupled patch array.]] </td></tr></table>

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:PMOM48H. These integrals may converge much faster based on png|thumb|left|420px|Details of the specified Convergence Rate for Integration described earlier. However, in certain cases involving highly oscillatory integrands, much larger integration limits like 100k₀ might be needed to warrant adequate convergence. For spectral-domain integration along hybrid planar mesh of the real k_&rho; axis, the interval [0, Nk₀] is subdivided into a large number of subslot-intervals, within each an 8-point Gauss-Legendre quadrature is appliedcoupled patch array around discontinuities. The next parameter, '''No. 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 dialogEM.Picasso's hybrid planar mesh generator tries to produce as many rectangular cells as possible especially in the case of objects with rectangular or linear boundaries. In connection or junction areas between adjacent objects or close to highly curved boundaries, triangular cells are used to fill the "irregular" regions in a conformal and consistent manner.

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

After * For PEC and conductive sheet traces, the MoM impedance matrix '''[Z]''' (not to be confused with effective permittivity is defined as the impedance [[parameters]]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 excitation vector '''[V]''' have been computed through below the matrix fill processmetallic trace. * For embedded object sets, the planar MoM simulation engine effective permittivity is ready to solve defined as the system largest of linear equations:the permittivities of all the substrate layers and embedded dielectric sets.

:<mathtable><tr><td> \mathbf{[Z]}_{N\times N} \cdot \mathbf{[IImage:PMOM32.png|thumb|360px|A comparison of triangular and planar hybrid meshes of a rectangular patch.]}_{N\times 1} = \mathbf{[V]}_{N\times 1} </mathtd><!--td> [[FileImage:PMOM81PMOM30.png|thumb|360px|Mesh of two rectangular patches at two different substrate planes. The lower substrate layer has a higher permittivity.]]--</td></tr></table>

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

There are a large number The integrity of numerical methods for solving systems of linear equations. These methods are generally divided into two groups: direct solvers the planar mesh and iterative solvers. Iterative solvers are usually based on matrix-vector multiplications. Direct solvers typically work faster for matrices its continuity in the junction areas directly affects the quality and accuracy of smal to medium size (N&lt;3,000)the simulation results. [[EM.Cube]]Picasso's [[Planar Module]] offers five linear solvershybrid planar mesh generator has some rules that are catered to 2.5-D MoM simulations:

# LU Decomposition Method* If two connected rectangular objects have the same side dimensions along their common linear edge with perfect alignment, a rectangular junction mesh is produced.# Biconjugate Gradient Method (BiCG)* If two connected rectangular objects have different side dimensions along their common linear edge or have edge offset, a set of triangular cells is generated along the edge of the object with the larger side.# Preconditioned Stabilized Biconjugate Gradient Method (BCG* Rectangle strip objects that host a gap source or a lumped element always have a rectangular mesh around the gap area.* If two objects reside on the same Z-STAB)plane, belong to the same trace group and have a common overlap area, they are first merged into a single object for the purpose of meshing using the &quot;Boolean Union&quot; operation. # Generalized Minimal Residual Method (GMRES)# Transpose* Embedded objects have prismatic meshes along the Z-Free Quasi-Minimum Residual Method (TFQMR)axis.* If an embedded object is located underneath or above a metallic trace object or connected from both top and bottom, it is meshed first and its mesh is then reflected on all of its attached horizontal trace objects.

<table><tr><td> [[EMFile:PMOM36.Cubepng|250px]]'s [[Planar ModuleFile:PMOM38.png|250px]], by default, provides a &quot;'''Automatic'''&quot; solver option that picks the best method based on the settings [[File:PMOM37.png|250px]] </td></tr><tr><td> Two overlapping planar objects and size of the numerical problem. For linear systems with a size less than N = 3,000, the LU solver is used. For larger systems, BiCG is used when dealing with symmetric matrices, and GMRES is used for asymmetric matrices. If the size comparison of the linear system exceeds N = 15,000, the sparse version of the iterative solvers is used, utilizing a row-indexed sparse storage scheme. You can override the automatic solver option their triangular and manually set you own solver typehybrid planar meshes. This is done using the '''Solver Type''' dropdown list in the &quot;'''Linear System Solver'''&quot; section of the Planar MoM Engine Settings dialog</td></tr><tr><td> [[File:PMOM33. There are also a number of other png|250px]] [[parametersFile:PMOM35.png|250px]] related to the solvers. The default value of '''Tolerance of Iterative Solver''' is 1E-3, which can be increased for more ill-conditioned systems. The maximum number of iterations is usually expressed as a multiple of the systems size. The default value of '''Max No. of Solver Iterations / System Size''' is 3. For extremely large systems, sparse versions of iterative solvers are used. In this case, the elements of the matrix are thresholded with respect to the larges element. The default value of '''Threshold for Sparse Solver''' is 1E-6, meaning that all the matrix elements whose magnitude is less than 1E-6 times the large matrix elements are set equal to zero. There are two more [[parametersFile:PMOM34.png|250px]] that are related to the Automatic Solver option. These are &quot;''' User Iterative Solver When System Size &gt;'''&quot; with a default value of 3,000 </td></tr><tr><td> Edge-connected rectangular planar objects 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 comparison their triangular and iterative solvers and when to switch to the sparse version of iterative solvershybrid planar meshes.</td></tr></table>

If your computer has an Intel CPU, then <table><tr><td> [[EMFile:PMOM39.Cubepng|375px]] offers special versions of all the above linear solvers that have been optimized for Intel CPU platforms. These optimal solvers usually work 2-3 time faster than their generic counterparts. When you install [[EMFile:PMOM40.Cubepng|375px]], the option to use Intel-optimized solvers is already enabled. However, you can disable this option (e.g. if your computer has a non-Intel CPU). To do that, open the [[EM.Cube]]'s Preferences Dialog from '''Menu &gt; Edit &gt; Preferences''' or using the keyboard shortcut '''Ctrl+H'''. Select the Advanced tab </td></tr><tr><td> Meshes of the dialog short and uncheck the box labeled &quot;''' Use Optimized Solvers for Intel CPU'''&quot;long vertical PEC vias connecting two horizontal metallic strips.</td></tr></table>

[[Image:PMOM127It is very important to apply the right mesh density to capture all the geometrical details of your planar structure.png|thumb|400px|Settings adaptive frequency sweep parameters in EM.Picasso's Frequency Settings Dialog.]]=== Running Uniform This is especially true for &quot;field discontinuity&quot; regions such as junction areas between connected objects, where larger current concentrations are usually observed at sharp corners, or at the junction areas between metallic traces and Adaptive Frequency Sweeps ===PEC vias, as well as the areas around gap sources and lumped elements, which create voltage or current discontinuities.

In a frequency sweep, the operating frequency of The Planar Mesh Settings dialog gives a few options for customizing your planar structure is varied during each sweep run. [[EM.Cube]]'s [[Planar Module]] offers two types of frequency sweep: Uniform mesh around geometrical and Adaptivefield discontinuities. In a uniform frequency sweep, The check box labeled &quot;'''Refine Mesh at Junctions'''&quot; increases the frequency range and mesh resolution at the number of frequency samples are specifiedconnection area between rectangular objects. The samples check box labeled &quot;'''Refine Mesh at Gap Locations'''&quot; might be particularly useful when gap sources or lumped elements are equally spaced over the frequency rangeplaced on a short transmission line connected from both ends. At The check box labeled &quot;'''Refine Mesh at Vias'''&quot; increases the end of each individual frequency run, mesh resolution on the output data are collected cross section of embedded object sets and stored. At at the end connection regions of the frequency sweep, metallic objects connected to them. EM.Picasso typically doubles the 3D data can be visualized and/or animated, and mesh resolution locally at the 2D data can be graphed in discontinuity areas when the respective boxes are checked. You should always visually inspect EM.GridPicasso's default generated mesh to see if the current mesh settings have produced an acceptable mesh.

To run a uniform frequency sweep, open the '''Simulation Run Dialog''', and select the '''Frequency Sweep''' option from the dropdown list labeled '''Simulation Mode'''Sometimes EM. When you choose the frequency sweep option, the Picasso'''Settings''' button next s default mesh may contain very narrow triangular cells due to the simulation mode dropdown list becomes enabledvery small angles between two edges. Clicking this button opens In some rare cases, extremely small triangular cells may be generated, whose area is a small fraction of the '''Frequency Settings''' dialogaverage mesh cell. The '''Frequency Range'''is initially set equal to your project's center frequency minus and plus half bandwidth. But you can change These cases typically happen at the values of '''Start Frequency'''junctions and '''End Frequency''' as well as other discontinuity regions or at the '''Number boundary of Samples'''highly irregular geometries with extremely fine details. The dialog offers two options for '''Frequency Sweep Type''': '''Uniform''' In such cases, increasing 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 decreasing the mesh density is defined in terms of the number of by one or few cells per effective wavelengthoften resolves that problem and eliminates those defective cells. By defaultNonetheless, during a frequency sweep, [[EM.Cube]] fixes the Picasso's planar mesh density at generator offers an option to identify the highest frequency, idefective triangular cells and either delete them or cure them.eBy curing we mean removing a narrow triangular cell and merging its two closely spaced nodes to fill the crack left behind., at EM.Picasso by default deletes or cures all the &quot;End Frequency&quot;triangular cells that have angles less than 10º. This usually results Sometimes removing defective cells may inadvertently cause worse problems in a smoother frequency responsethe mesh. You have the option may choose to fix disable this feature and uncheck the mesh at the center frequency of the project or let [[EM.Cube]] box labeled &quot;remesh'''Remove Defective Triangular Cells'''&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 '''Planar Mesh Settings''' section of the dialog. Closing the Frequency Settings dialog returns you to the Simulation Run dialog, where you You can start also change the planar MoM frequency sweep simulation by clicking value of the '''Run''' buttonminimum allowable cell angle.

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

You must have defined one <table><tr><td> [[Image:PMOM44.png|thumb|left|480px|Deleting or more ports for your planar structure run an adaptive frequency sweep. Open the Frequency Settings dialog from the Simulation Run dialog and select the '''Adaptive''' option of '''Frequency Sweep Type'''. You have to set values for '''Minimum Number of Samples''' and '''Maximum Number of Samples'''. Their default values are 3 and 9, respectively. You also set a value for the '''Convergence Criterion''', which has a default value of 0.curing defective triangular cells: Case 1. At each iteration cycle, all the S [[parameters]] are calculated at the newly inserted frequency samples, and their average deviation from the curves of the last cycle is measured as an error. When this error falls below the specified convergence criterion, the iteration is ended. If </td></tr><tr><td> [[EMImage:PMOM42.png|thumb|left|480px|Deleting or curing defective triangular cells: Case 2.Cube]] reaches the specified maximum number of iterations and the convergence criterion has not yet been met, the program will ask you whether to continue the process or exit it and stop.</td></tr></table>

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

== Working with Planar MoM = EM.Picasso's Simulation Data Modes ===

[[Image:PMOM130.png|thumb|400px|Changing the graph type by editing a data file's propertiesEM.Picasso]]=== offers five Planar Module's Output Simulation Data ===MoM simulation modes:

Depending on the source type and the types {| class="wikitable"|-! scope="col"| Simulation Mode! scope="col"| Usage! scope="col"| Number of observables defined in Engine Runs! scope="col"| Frequency ! scope="col"| Restrictions|-| style="width:120px;" | [[#Running a project, a number of output data are generated Single-Frequency Planar MoM Analysis | Single-Frequency Analysis]]| style="width:270px;" | Simulates the planar structure "As Is"| style="width:80px;" | Single run| style="width:250px;" | Runs at the end center frequency fc| style="width:80px;" | None|-| style="width:120px;" | [[Parametric_Modeling_%26_Simulation_Modes_in_EM.Cube#Running_Frequency_Sweep_Simulations_in_EM.Cube | Frequency Sweep]]| style="width:270px;" | Varies the operating frequency of a the planar MoM simulation. Some solver | style="width:80px;" | Multiple runs | style="width:250px;" | Runs at a specified set of these data are 2D by nature and some are 3Dfrequency samples or adds more frequency samples in an adaptive way| style="width:80px;" | None|-| style="width:120px;" | [[Parametric_Modeling_%26_Simulation_Modes_in_EM. The output simulation data generated by Cube#Running_Parametric_Sweep_Simulations_in_EM.Cube | Parametric Sweep]]| style="width:270px;" | Varies the value(s) of one or more project variables| style="width:80px;" | Multiple runs| style="width:250px;" | Runs at the center frequency fc| style="width:80px;" | None|-| style="width:120px;" | [[EMParametric_Modeling_%26_Simulation_Modes_in_EM.Cube#Performing_Optimization_in_EM.Cube | Optimization]]'| style="width:270px;" | Optimizes the value(s ) of one or more project variables to achieve a design goal | style="width:80px;" | Multiple runs | style="width:250px;" | Runs at the center frequency fc| style="width:80px;" | None|-| style="width:120px;" | [[Planar ModuleParametric_Modeling_%26_Simulation_Modes_in_EM.Cube#Generating_Surrogate_Models | HDMR Sweep]] can be categorized into | style="width:270px;" | Varies the value(s) of one or more project variables to generate a compact model| style="width:80px;" | Multiple runs | style="width:250px;" | Runs at the following groupscenter frequency fc| style="width:80px;" | None|}

* '''Port Characteristics''': S, Z and Y You can set the simulation mode from [[ParametersEM.Picasso]] 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), Fronts "Simulation Run Dialog". A single-tofrequency analysis is a single-Back Ratio (FBR)run simulation. All the other simulation modes in the above list are considered multi-run simulations. If you run a simulation without having defined any observables, etcno data will be generated at the end of the simulation.* '''Scattering Characteristics''': BiIn multi-static run simulation modes, certain parameters are varied 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 axesa collection of simulation data files are generated. At the end of a sweep simulation, you can graph the simulation results in EM.Grid or you can animate the 3D simulation data from the navigation tree.

=== Examining Port Characteristics Running a Single-Frequency Planar MoM Analysis === A single-frequency analysis is the simplest type of [[EM.Picasso]] simulation and involves the following steps:

If * Set the units of your planar structure is excited by gap sources or probe sources or de-embedded sources, project and one or more ports have been defined, the planar MoM engine calculates the scattering, impedance and admittance (S/Z/Y) [[parameters]] frequency of operation. Note that the designated portsdefault project unit is '''millimeter'''. The scattering * Define you background structure and its layer properties and trace types. * Construct your planar structure using [[parametersBuilding_Geometrical_Constructions_in_CubeCAD | CubeCAD]] are defined based on 's drawing tools to create all the port impedances specified in 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's Port Definition dialog. If more than one port has been defined in * Examine the projectplanar mesh, verify its integrity and change the S/Z/Y matrices of mesh density if necessary.* Run the multiport network are calculatedPlanar MoM simulation engine.* Visualize the output simulation data.

At the end of To run a planar MoM simulationanalysis of your project structure, open the values of S/Z/Y Run Simulation Dialog by clicking the '''Run''' [[parametersFile:run_icon.png]] and VSWR data are calculated and reported in button on the output message window'''Simulate Toolbar''' or select '''Menu > Simulate > Run''' or use the keyboard shortcut {{key|Ctrl+R}}. The S, Z and Y [[parameters]] are written into output ASCII data files '''Single-Frequency Analysis''' option of complex type with a &quot;the '''.CPXSimulation Mode'''&quot; extensiondropdown list is selected by default. Every file begins with a header consisting of a few comment lines that start with Once you click the &quot;#&quot; symbol{{key|Run}} button, the simulation starts. The complex values are arranged into two columns for A new window called the real "Output Window" opens up that reports the different stages of simulation and imaginary parts. In the case of multiport structures, every single element percentage of the S/Z/Y matrices tasks completed at any time. After the simulation is written into successfully completed, a separate complex data filemessage pops up and reports the end of simulation. For example, you will have data files In certain cases like S11.CPX, S21.CPX, ..., Z11.CPX, Z21.CPX, etc. The VSWR data are saved to an ASCII data file calculating scattering parameters of real type with a &quot;'''.DAT'''&quot; extension calledcircuit or reflection / transmission characteristics of a periodic surface, VSWR.DATsome results are also reported in the output window.

If you run an analysis, the port characteristics have single complex values, which you can view using <table><tr><td> [[EMImage:Picasso L1 Fig18.Cube]]'s data manager. However, there are no curves to graph. You can plot the S/Z/Y [[parameters]] and VSWR data when you have data sets, which are generated at the end of any type of sweep including a frequency sweep. In that case, the &quot;.CPX&quot; files have multiple rows corresponding to each value of the sweep parameter (e.g. frequency). [[png|thumb|left|480px|EM.Cube]]Picasso's 2D graph data are plotted in EMSimulation Run dialog.Grid, a versatile graphing utility. You can plot the port characteristics directly from the Navigation Tree. Right click on the '''Port Definition''' item in the '''Observables''' section of the Navigation Tree and select one of the items: '''Plot S [[Parameters]]''', '''Plot Y [[Parameters]]''', '''Plot Z [[Parameters]]''', or '''Plot VSWR'''. In the first three cases, another sub-menu gives a list of individual port [[parameters]].</td></tr></table>

The adaptive frequency sweep described earlier is an iterative process, whereby the Planar MoM simulation engine Engine Settings Dialog is run at organized in a certain number of frequency samples at each iteration cyclesections. Here we describe some of the numerical parameters. The frequency samples are progressively built up&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''', and rational fits which is 1E-5 by default. This is used for these data the convergence test of all the infinite integrals in the calculation of the Hankel transform of spectral-domain dyadic Green's functions. When the substrate is lossy, the surface wave poles are found at each iteration cyclecaptured in the complex integration plane using contour deformation. A decision You can change the maximum number of iterations involved in this deformed contour integration, whose default value is then made whether 20. When the substrate is very thin with respect to continue more iterationsthe wavelength, the dyadic Green's functions exhibit numerical instability. At Additional singularity extraction measures are taken to avoid numerical instability but at the end expense of the whole processincreased computation time. By default, a total number of scattering parameter data samples thin substrate layer is defined to a have been generateda thickness less than 0.01&lambda;_eff, and new smooth data corresponding to where &lambda;_eff is the best rational fits are written into new data files effective wavelength. You can modify the definition of &quot;Thin Substrate&quot; by entering a value for graphing'''Thin Substrate Threshold''' different than the default 0. [[EM01.Cube]]The parameter '''Max Coupling Range''' determines the distance threshold in wavelength between the observation and source points after which the Green's [[planar Module]] also allows you interactions are neglected. This distance by default is set to generate 1,000 wavelengths. For electrically small structures, the phase variation across the structure may be negligible. In such cases, a rational fit fast quasi-static analysis can be carried out. You can set this threshold in wavelengths in the box labeled '''Max Dimensions for all or any existing scattering parameter data as a postQuasi-processing operation without a need to run additional simulation engine runsStatic Analysis'''.

You can interpolate all the scattering [[parameters]] together or select individual [[parameters]]. You do this post-processing operation from In the Navigation Tree. Right click on the '''Port Definition''' item in the '''Observables''' &quot;Spectral Domain Integration&quot; section of the Navigation Tree and select Smart Fit. At the top of the Smart Fit Dialogdialog, there is you can set a dropdown list labeled value to '''InterpolateMax Spectral Radius in k0''', which gives has a list default value of all 30. This means that the available S parameter data infinite spectral-domain integrals in the spectral variable k_&rho; are pre-calculated and tabulated up to a limit of 30k₀, where k₀ is the free space propagation constant. These integrals may converge much faster based on the specified Convergence Rate for rational interpolationIntegration described earlier. The default option is However, in certain cases involving highly oscillatory integrands, much larger integration limits like 100k₀ might be needed to warrant adequate convergence. For spectral-domain integration along the real k_{&quotrho;All Available} axis, the interval [[Parameters]0, Nk₀]&quot;. Then you see is subdivided into a box labeled large number of sub-intervals, within each an 8-point Gauss-Legendre quadrature is applied. The next parameter, '''Number of Available SamplesNo. Radial Integration Divisions per k₀''', whose value is read from determines how small these intervals should be. By default, 2 divisions are used for the data content of interval [0, k₀]. In other words, the selected complex .CPX data filelength of each integration sub-interval is k₀/2. Based on You can increase the number resolution of available data samplesintegration by increasing this value above 2. Finally, instead of 2D Cartesian integration in the dialog reports the '''Maximum Interpolant Order'''spectral domain, a polar integration is performed. You can choose any integer number for set the '''Interpolant OrderNo. of Angular Integration Points''', from 1 to the maximum allowed.  {{Note|Interpolant order more than 15 will suffer from numerical instabilities even if you have which has a very large number default value of data samples100.}}

You can use the [[EM.Picasso]] provides a large selection of linear system solvers including both direct and iterative methods. [[EM.Picasso]], by default, provides a &quot;'''UpdateAutomatic''' button &quot; solver option that picks the best method based on the settings and size of the dialog to generate numerical problem. For linear systems with a size less than N = 3,000, the interpolated data LU solver is used. For larger systems, BiCG is used when dealing with symmetric matrices, and GMRES is used for a given orderasymmetric matrices. The new data are written You can instruct [[EM.Cube]] to a complex data file with write the same name as the selected S parameter MoM matrix and a excitation and solution vectors into output data files for your examination. To do so, check the box labeled &quot;'''_RationalFitOutput MoM Matrix and Vectors'''&quot; suffix. While this dialog is still open, you can plot in the new data either directly from Matrix Fill section of the Navigation Tree or from the Data ManagerPlanar MoM Engine Settings dialog. If you These are not satisfied with the resultswritten into three files called mom.dat1, you can return to the Smart Fit dialog exc.dat1 and try a higher or lower interpolant order and compare the new datasoln.dat1, respectively.

<td> [[Image:PMOM131PMOM79.png|thumb|300pxleft|720px|EM.Picasso's Smart Fit Planar MoM Engine Settings dialog.]] </td><td> [[Image:PMOM133(2).png|thumb|300px|The S₁₁ parameter plot of a two-port structure in magnitude-phase format.]] </td><td> [[Image:PMOM132(2).png|thumb|300px|The smoothed version of the S₁₁ parameter plot of the two-port structure using [[EM.Cube]]'s Smart Fit.]] </td>

=== Visualizing Current Distributions =Modeling Periodic Planar Structures in EM.Picasso ==

Electric and magnetic currents are [[EM.Picasso]] allows you to simulate doubly periodic planar structures with periodicities along the fundamental output data of a X and Y directions. Once you designate your planar structure as periodic, [[EM.Picasso]]'s Planar MoM simulationengine uses a spectral domain solver to analyze it. After the numerical solution of the MoM linear systemIn this case, they are found using the solution vector dyadic Green'''[I]''' and the definitions s functions of periodic planar structure take the electric and magnetic vectorial basis functions:form of doubly infinite summations rather than integrals.

:<math> \mathbf{[I]}_{N\times 1} = \begin{bmatrix} I^{(J)} \\ \\ V^{(M)} \end{bmatrix} \quad \Rightarrow \quad \begin{cases} \mathbf{J(r)} = \sum_{n=1}^N I_n^{(J)} \mathbf{f_n^{(J)} (r)} \\ \\ \mathbf{M(r)} = \sum_{k=1}^K V_k^{(M)} \mathbf{f_k^{(M)} (r)} \end{cases} </math><!--[[FileImage:PMOM83Info_icon.png|30px]]-->Click here to learn more about the theory of '''[[Basic_Principles_of_The_Method_of_Moments#Periodic_Planar_MoM_Simulation | Periodic Green's functions]]'''.

{{Note that currents are complex vector quantities| [[EM. Each electric or magnetic current has three X, Y and Z components, and each complex component has a magnitude and phase. You Picasso]] 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 handle both regular and magnetic current distributions are visualized in the project workspace, superimposed on the surface of physical objectsskewed periodic lattices.}}

Once you close the current distribution dialog, the label of the selected trace or object set is added under the '''Current Distributions''' node of the Navigation Tree=== Defining a Periodic Structure in EM. Picasso ===

{{Note|You have to An infinite periodic structure in [[EM.Picasso]] is represented by a &quot;'''Periodic Unit Cell'''&quot;. To define a separate current distribution observable for each individual trace periodic structure, you must open [[EM.Picasso]]'s Periodicity Settings Dialog by right clicking the '''Periodicity''' item in the '''Computational Domain''' section of the navigation tree and selecting '''Periodicity Settings...''' from the contextual menu or embedded object setby selecting '''Menu''' '''&gt;''' '''Simulate &gt; 'Computational Domain &gt; Periodicity Settings...''' from the menu bar. In the Periodicity Settings Dialog, check the box labeled '''Periodic Structure'''. This will enable the section titled''&quot;''Lattice Properties&quot;. You can define the periods along the X and Y axes using the boxes labeled '''Spacing'''. In a periodic structure, the virtual domain is replaced by a default blue periodic domain that is always centered around the origin of coordinates. Keep in mind that the periodic unit cell must always be centered at the origin of coordinates. The relative position of the structure within this centered unit cell will change the phase of the results.}}

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''<table><tr><td> [[Image:PMOM99.png|thumb|300px|EM.Picasso') currents as well as the total electric and magnetic currentss Periodicity Settings dialog.]] </td></tr></table>

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. [[Image:MOREEM.png|40pxPicasso]] Click here allows you to learn define periodic structures whose unit cells are interconnected. The interconnectivity applies only to PEC, PMC and conductive sheet traces, and embedded object sets are excluded. Your objects cannot cross the periodic domain. In other words, the neighboring unit cells cannot overlap one another. However, you can arrange objects with linear edges such that one or more about ''flat edges line up with the domain's bounding box. In such cases, [[Data_Visualization_and_Processing#Visualizing_3D_Current_Distribution_Maps | Visualizing 3D Current Distribution MapsEM.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.

<td> [[Image:PMOM184image122.png|thumb|300px400px|EM.Picasso's Current Distribution dialog.]] </td><td> [[Image:PMOM85(1).png|thumb|300px|The current distribution map Modeling a periodic screen using two different types of a patch antennaunit cell.]] </td>

=== Visualizing <table><tr><td> [[Image:pmom_per5_tn.png|thumb|300px|The PEC cross unit cell.]] </td><td> [[Image:pmom_per6_tn.png|thumb|300px|Planar mesh of the Near Fields ===PEC cross unit cell. Note the cell extensions at the unit cell's boundaries.]] </td></tr></table>

In order to view the near field distributions, you must first define field sensor observables before running the planar MoM simulation. To do that, right click on the '''Field Sensors''' item === Exciting Periodic Structures as Radiators in the '''Observables''' section of the Navigation Tree and select '''Insert New Observable...'''. The Field Sensor Dialog opens up. At the top of the dialog and in the section titled '''Sensor Plane Location''', first you need to set the plane of near field calculation. In the dropdown box labeled '''Direction''', you have three options X, Y, and Z, representing the&quot;normals&quot; to the XY, YZ and ZX planes, respectively. The default direction is Z, i.e. XY plane parallel to the substrate layers. In the three boxes labeled '''Coordinates''', you set the coordinates of the center of the plane. Then, you specify the '''Size''' of the plane in project units, and finally set the '''Number of Samples''' along the two sides of the sensor plane. The larger the number of samples, the smoother the near field map will appearEM.Picasso ===

Once you close When a periodic planar structure is excited using a gap or probe source, it acts like an infinite periodic phased array. All the Field Sensor periodic replicas of the unit cell structure are excited. You can even impose a phase progression across the infinite array to steer its beam. You can do this from the property dialogof the gap or probe source. At the bottom of the '''Planar Gap Circuit Source Dialog''' or '''Gap Source Dialog''', its name there is added under the a button titled '''Field SensorsPeriodic Scan...''' node of the Navigation Tree. At You can enter desired values for '''Theta''' and '''Phi''' beam scan angles in degrees. To visualize the end radiation patterns of a planar MoM simulationbeam-steered antenna array, you have to define a finite-sized array factor in the field sensor nodes Radiation Pattern dialog. You do this in the Navigation Tree become populated by the magnitude and phase plots '''Impose Array Factor''' section of the three vectorial components this dialog. The values of the electric ('''EElement Spacing''') along the X and magnetic (Y directions must be set equal to the value of '''HPeriodic Lattice Spacing''') field as well as the total electric and magnetic fields definedalong those directions.

Note that unlike <table><tr><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:MOREPeriod6.png|40px]] Click here to learn more about '''[[Data_Visualization_and_Processing#Visualizing_3D_Near-Field_Maps thumb| Visualizing 3D Near Field Maps]]''350px|Setting the array factor in EM.Picasso's Radiation Pattern dialog.]] </td></tr></table>

<td> [[File:PMOM90.png|thumb|300px|[[Planar Module]]'s Field Sensor dialog.]] </td><td> [[Image:PMOM116Period7.png|thumb|300px360px|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|300px360px|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:MOREPMOM102.png|40px]] Click here to learn more about the theory of '''[[Data_Visualization_and_Processing#Far-Field_Observables thumb| 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:MOREEM.png|40pxPicasso]] Click here 's Run Dialog. Here, too, you can run a single-frequency analysis or a uniform or adaptive frequency sweep, or a parametric sweep, etc. Similar to 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|300px600px|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:MOREInfo_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|30px]] '''[[EM.Picasso's Radar Cross Section dialog#Product_Overview | Back to the Top of the Page]] </td>''' <td> [[Image:PMOM125ngTutorial_icon.png|thumb30px]] '''[[EM.Cube#EM.Picasso_Documentation |300px|An example of the 3D mono-static radar cross section plot of a patch antennaEM.Picasso Tutorial Gateway]] </td></tr></table>'''

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