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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 === <ul> <li> Optimized hybrid mesh with rectangular and triangular cells</li> <li> Regular triangular surface mesh</li> <li> Local meshing of trace groups</li> <li> Local mesh editing of planar polymesh objects</li> <li> Fast mesh generation of array objects</li></ul>

The Stack-up Settings dialog has two tabs: '''Layer Hierarchy''' and '''Embedded Sets'''<ul> <li> 2. The Layer Hierarchy tab has a table that shows all the background layers in hierarchical order from the top half5-space to the bottom half-spaceD mixed potential integral equation (MPIE) formulation of planar layered structures</li> <li> 2. It also lists the material label 5-D spectral domain integral equation formulation of each layer, Zperiodic layered structures</li> <li> Accurate scattering parameter extraction and de-coordinate embedding using Prony&#39;s method</li> <li> Plane wave excitation with arbitrary angles of the bottom incidence</li> <li> A variety of each layermatrix solvers including LU, its thickness (in project units) BiCG and material properties: permittivity (eGMRES<sub/li>r <li> Uniform and fast adaptive frequency sweep</subli>), permeability (µ <subli>r Parametric sweep with variable object properties or source parameters</subli>), electric conductivity (s) <li> Generation of reflection and magnetic conductivity (stransmission coefficient macromodels<sub/li>m <li> Multi-variable and multi-goal optimization of structure</subli>). There is also a column that lists the names <li> Remote simulation capability</li> <li> Both Windows and Linux versions of embedded object sets inside each substrate layer, if any.Planar MoM simulation engine available</li></ul>

You can delete a layer by selecting its row in the table <ul> <li> Current distribution intensity plots</li> <li> Near field intensity plots (vectorial - amplitude &amp; phase)</li> <li> Far field radiation patterns: 3D pattern visualization and clicking the '''Delete''' button. To move a layer up 2D Cartesian and downpolar graphs</li> <li> Far field characteristics such as directivity, click on its row to select beam width, axial ratio, side lobe levels and highlight itnull parameters, etc. Then click either </li> <li> Radiation pattern of an arbitrary array configuration of the '''Move Up''' planar structure or '''Move Down''' buttons consecutively periodic unit cell</li> <li> Reflection and Transmission Coefficients of Periodic Structures</li> <li> Monostatic and bi-static RCS&nbsp;</li> <li> Port characteristics: S/Y/Z parameters, VSWR and Smith chart</li> <li> Touchstone-style S parameter text files for direct export to move the selected layer to the desired location in the stack-upRF. Note that you cannot delete Spice or move the top or bottom half-spaces.its Device Editor</li> <li> Huygens surface generation</li> <li> Custom output parameters defined as mathematical expressions of standard outputs</li></ul>

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

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

<td> [[FileImage:PMOM36PMOM23B.png|250px]] [[File:PMOM38.pngthumb|250px]] [[File:PMOM37.png280px|250px]] </td></tr><tr><td> Two overlapping planar objects and their triangular and hybrid planar meshesEM. </td></tr><tr><td> [[File:PMOM33.png|250px]] [[File:PMOM35.png|250px]] [[File:PMOM34.png|250px]] </td></tr><tr><td> Edge-connected rectangular Picasso's Navigation Tree populated with 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 of short and long vertical PEC vias connecting two horizontal metallic strips. </td>

=== Refining [[EM.Picasso]] has a special feature that makes construction of planar structures very convenient and straightforward. <u>The horizontal Z-plane of the Planar Mesh Locally ===active trace or object set group is always set as the active work plane of the project workspace.</u> That means all new objects are drawn at the Z-coordinate of the currently active trace. As you change the active trace group or add a new one, the active work plane changes accordingly.

It is very important to apply the right mesh density to capture all the geometrical details of your planar structure{{Note| In [[EM. This is especially true for &quot;field discontinuity&quot; regions such as junction areas between objects of different side dimensionsPicasso]], where larger current concentrations are usually observed at sharp corners, or at you cannot modify the connection areas between metallic traces and PEC vias, Z-coordinate of an object as well as the areas around gap sources and lumped elements, as these create voltage or current discontinuities. For large planar structures, using a higher mesh density may not always be a practical option since it will quickly lead to a very large MoM matrix is set and thus growing the size of the numerical problem. EM.Picasso provides several ways of controlling the mesh of a planar structure locally controlled by its host trace. }}

The Planar Mesh Settings dialog gives a few options for customizing your planar mesh around geometrical and field discontinuities[[EM. You can check the check box labeled &quot;'''Refine Mesh at Junctions'''&quot;, which increases the mesh resolution at the connection area between rectangular Picasso]] does not allow you to draw 3D or solid CAD objects. Or you can check The solid object buttons in the check box labeled &quot;'''Refine Mesh at Gap LocationsObject Toolbar'''&quot;, which may prove particularly useful when gap sources or lumped elements are placed on a short transmission line connected disabled to prevent you from both endsdoing so. Or you can check the check box labeled &quot;'''Refine Mesh at Vias'''&quot;In order to create vias and embedded object, which increases the mesh resolution on the you simply have to draw their cross section of embedded object sets and at the connection regions of the metallic geometry using planar surface CAD objects connected to them. [[EM.Picasso]] typically doubles the extrudes and extends these planar objects across their host layer automatically and displays them as 3D wireframe, prismatic objects. The automatic extrusion of embedded objects happens after mesh resolution locally at generation and before every planar MoM simulation. You can enforce this extrusion manually by right-clicking the discontinuity areas when '''Layer Stack-up''' item in the respective boxes are checked"Computational Domain" section of the navigation tree and selecting '''Update Planar Structure''' from the contextual menu.

You should always visually inspect {{Note| In [[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 some rare cases, extremely small triangular cells may be generated, whose area is a small fraction of the average mesh cell. These cases typically happen at the junctions and other discontinuity regions or at the boundary of highly irregular geometries with extremely fine details. In such cases, increasing or decreasing the mesh density by one or few cells per effective wavelength often resolves that problem and eliminates those defective cells. Nonetheless, EM.Picasso's you can only draw horizontal planar mesh generator offers an option to identify the defective triangular cells and either delete them or cure them. By curing we mean removing a narrow triangular cell and merging its two closely spaced nodes to fill the crack left behind. EM.Picasso by default deletes or cures all the triangular cells that have angles less than 10º. Sometimes removing defective cells may inadvertently cause worse problems in the mesh. You may choose to disable this feature and uncheck the box labeled &quot;'''Remove Defective Triangular Cells'''&quot; in the Planar Mesh Settings dialog. You can also change the value of the minimum allowable cell anglesurface CAD objects.}}

Another way of local mesh control is to lock the mesh density of certain traces or object sets<table><tr><td> [[Image:PMOM23A. The mesh density that you specify in the Planar Mesh Settings dialog is png|thumb|620px|A planar structure with a global parameter and applies to all the traces and embedded object sets in your project. Howevertwo-layer conductor-backed substrate, you can lock the mesh of individual two PEC, PMC and conductive sheet traces or embedded objects sets. In that case, patches located at the locked mesh density takes precedence over the global density. Note that locking mesh tops of object groups, in principle, is different than refining the mesh at discontinuities. In the latter caselower and upper substrate layers, four PEC vias located inside the mesh of connection areas is affected. However, objects belonging to different traces cannot be connected to one another. Therefore, locking mesh can be useful primarily for isolated object groups that may require a higher (or lower) mesh resolution. You can lock substrate layer between the local mesh density by accessing the property dialog of a specific trace or lower patch and bottom ground and an embedded object set and checking dielectric film located inside the box labeled '''Lock Mesh'''. This will enable the '''Mesh Density''' box, where you can accept top substrate layer sandwiched between the default global value or set any desired new valuetwo patches.]] </td></tr></table>

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

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

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

For antennas and Your planar circuits, where you typically define one or more ports, you usually use lumped sources. A lumped structure must be excited by some sort of signal source is indeed a gap discontinuity that is placed induces electric surface currents on the path of an electric or metal parts, magnetic current flowsurface currents on slot traces, where a voltage and conduction or current source is connected to inject a signal. Gap sources are placed across metal or slot traces. Probe sources are placed across polarization volume currents on vertical PEC vias. A de-and embedded objects. The excitation source is a special type of gap source that is placed near you choose depends on the open end of an elongated metal or slot trace to create a standing wave pattern, from which the scattering [[parameters]] can be calculated accurately. To calculate the scattering characteristics of a planar structure, e.g. its radar cross section (RCS), observables you excite it with a plane wave sourceseek in your project. Short dipole sources are used to explore propagation of points sources along a layered structure. Huygens sources are virtual equivalent sources that capture the radiated electric and magnetic fields from another structure possibly in another [[EM.CubePicasso]] computational module and bring them as a new provides the following source to excite your types for exciting planar structure.structures:

{| class="wikitable"|-! scope="col"| Icon! scope="col"| Source Type! scope="col"| Applications! scope="col"| Restrictions|-| style="width:30px;" | [[ImageFile:MOREgap_src_icon.png]]|40px[[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Strip Gap Circuit Source |Strip Gap Circuit Source]] Click here to learn more about '| style="width:300px;" | General-purpose point voltage source (or filament current source on slot traces)| style="width:300px;" | Associated with a PEC rectangle strip|-| style="width:30px;" | [[File:probe_src_icon.png]]| [[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Probe Gap Circuit Source |Probe Gap Circuit Source]]| style="width:300px;" | General-purpose voltage source for modeling coaxial feeds| style="width:300px;" | Associated with an embedded PEC via set|-| style="width:30px;" | [[File:waveport_src_icon.png]]| [[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Scattering Wave Port |Scattering Wave Port Source]]| style="width:300px;" | Used for S-parameter computations| style="width:300px;" | Associated with an open-ended PEC rectangle strip, extends long from the open end|-| style="width:30px;" | [[Planar MoM File:hertz_src_icon.png]]| [[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Hertzian Short Dipole Source |Hertzian Short Dipole Source]]| style="width:300px;" | Almost omni-directional physical radiator| style="width:300px;" | None, stand-alone source|-| style="width:30px;" | [[File:plane_wave_icon.png]]| [[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Plane Wave |Plane Wave Source]]| style="width:300px;" | Used for modeling scattering & computation of reflection/transmission characteristics of periodic surfaces| style="width:300px;" | None, stand-alone source|-| style="width:30px;" | [[File:huyg_src_icon.png]]| [[Glossary of EM.Cube'''s Materials, Sources, Devices & Other Physical Object Types#Huygens Source |Huygens Source]]| style="width:300px;" | Used for modeling equivalent sources imported from other [[EM.Cube]] modules | style="width:300px;" | Imported from a Huygens surface data file|}

Click on each category to learn more details about it in the [[Image:PMOM52.png|thumb|400px|Glossary of EM.PicassoCube's Port Definition dialog.Materials, Sources, Devices & Other Physical Object Types]][[Image:PMOM53.png|thumb|300px|The Edit Port dialog.]][[Image:PMOM51(2).png|thumb|600px|Coupling gap sources in the Port Definition dialog by associating more than one source with a single port.]]=== Defining Ports ===

Ports are used in a planar structure to order For antennas and index the planar circuits, where you typically define one or more ports, you usually use lumped sources for calculation of circuit . [[parametersEM.Picasso]] such as scattering provides three types of lumped sources: gap source, probe source and de-embedded source. A lumped source is indeed a gap discontinuity that is placed on the path of an electric or magnetic current flow, where a voltage or current source is connected to inject a signal. Gap sources are placed across metal or slot traces. A rectangle strip object on a PEC or conductive sheet trace acts like a strip transmission line that carries electric currents along its length (Slocal X direction), . The characteristic impedance of the line is a function of its width (Zlocal Y direction) . A gap source placed on a narrow metal strip creates a uniform electric field across the gap and admittance pumps electric current into the line. A rectangle strip object on a slot trace acts like a slot transmission line on an infinite PEC ground plane that carries a magnetic current along its length (local X direction). The characteristic impedance of the slot line is a function of its width (local Ydirection) [[parameters]]. In [[EMA gap source placed on a narrow slot represents an ideal current source.Cube]]'s A slot gap acts like an ideal current filament, which creates electric fields across the slot, equivalent to a magnetic current flowing into the slot line. Probe sources are placed across vertical PEC vias. A de-embedded source is a special type of gap source that is placed near the open end of an elongated metal or slot trace to create a standing wave pattern, from which the scattering [[Planar Moduleparameters]], you can use the following types of sources to define ports:be calculated accurately.

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_Finite-Sized_Source_Arrays | Using Source Arrays for Modeling Antenna Arrays]]''' 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.

'''A short dipole provides another way of exciting a planar structure in [[EM.Picasso]]. A short dipole source acts like an infinitesimally small ideal current source. You can define any number of ports equal also use an incident plane wave to or less than excite your planar structure in [[EM.Picasso]]. In particular, you need a plane wave source to compute the total number radar cross section of sources in your projecta planar structure.''' The Port List direction of the dialog shows a list of all the ports in ascending order, with their associated sources and the port's characteristic impedance, which incidence is 50S defined by default. You can delete any port by selecting it from the Port List θ and clicking the '''Delete''' button φ angles of the dialog. Keep unit propagation vector 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 intendthe spherical coordinate system. You can change the characteristic impedance The default values of a port by selecting it from the Port List incidence angles are θ = 180° and clicking φ = 0° corresponding to a normally incident plane wave propagating along the '''Edit''' button of the dialog-Z direction with a +X-polarized E-vector. This opens up Huygens sources are virtual equivalent sources that capture the Edit Port dialog, where you can enter a new value radiated electric and magnetic fields from another structure that was previously analyzed in the box labeled '''Impedance'''another [[EM.Cube]] computational module.

<table><tr><td> [[Image:MOREPMOM64A.png|40px]] Click here to learn more about the theory of '''[[Computing Port Characteristics in Planar MoM]]'''. === Modeling Coupled Ports === Sources can be coupled to each other to model coupled strip lines (CPS) on metal traces or coplanar waveguides (CPW) on slot traces. Similarly, probe sources may be coupled to each other. Coupling two or more sources does not change the way they excite a thumb|550px|A multilayer planar structure. It is intended only for the purpose of S parameter calculation. The feed lines or vias which host the coupled sources are usually parallel and aligned with one another and they are all grouped together as containing a single transmission CPW line represented by with a single port. This single &quot;coupled&quot; port then interacts with other coupled or uncoupled ports. You couple two or more sources using the '''Port Definition Dialog'''. To do so, you need to change the default port assignments. First, delete all the ports that are to be coupled from the Port List of the dialog. Then, define a new port by clicking the '''Add''' button of the dialog. This opens up the Add Port dialog, which consists of two tables: '''Available''' sources on the left and '''Associated''' sources on the right. A right arrow ('''--&gt;''') button and a left arrow ('''&lt;--''') button let you move the sources freely between these two tables. You will see in the &quot;Available&quot; table a list of all the sources that you deleted earlier. You may even see more available sources. Select all the sources that you want to couple and move them to the &quot;Associated&quot; table lumped element on the right. You can make multiple selections using the keyboard's '''Shift''' and '''Ctrl''' keys. Closing the Add Port dialog returns you to the Port Definition dialog, where you will now see the names of all the coupled sources next to the name of the newly added portan overpassing metal strip.]] </td></tr>{{Note|It is your responsibility to set up coupled ports and coupled [[Transmission Lines]] properly. For example, to excite the desirable odd mode of a coplanar waveguide (CPW), you need to create two rectangular slots parallel to and aligned with each other and place two gap sources on them with the same offsets and opposite polarities. To excite the even mode of the CPW, you use the same polarity for the two collocated gap sources. Whether you define a coupled port for the CPW or not, the right definition of sources will excite the proper mode. The couple ports are needed only for correct calculation of the port characteristics.}}</table>

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

[[Image:Info_icon.png|40px]] Click here to learn more about the theory of '''[[Computing_Port_Characteristics_in_Planar_MoMPreparing_Physical_Structures_for_Electromagnetic_Simulation#Modeling_Lumped_Elements_in_Planar_MoMModeling_Lumped_Elements_in_the_MoM_Solvers | Modeling Defining Lumped Elements in Planar MoM]]'''.

EM.Picasso allows you to define passive circuit elements[[Image: '''Resistors'''(R), C'''apacitors'''(C), I'''nductors'''(L), and series and parallel combinations of them Info_icon. To define png|40px]] Click here for a lumped RLC circuit in your planar structure, follow these steps: * Open the Lumped Element Dialog by right clicking on the '''Lumped Elements''' item in the '''Sources''' section general discussion of the Navigation Tree and selecting '''Insert New Source[[Preparing_Physical_Structures_for_Electromagnetic_Simulation#A_Review_of_Linear_.26_Nonlinear_Passive_..'''* In the '''Gap Topology''' section of the dialog, select one of the two options: '''Gap on Line''' and '''Gap on Via'''.* In the '''Lumped Circuit Type''' section of the dialog, select one of the two options: '''26_Active_Devices | Linear Passive RLC''' and '''Active with Gap Source'''.* Depending on your choice of gap topology, in the '''Lumped Circuit Location''' section of the dialog, you will find either a list of all the '''Rectangle Strip Objects''' or a list of all the '''PEC Via Objects''' available in the project workspace. Select the desired rectangle strip or embedded PEC via object.* In the box labeled '''Offset''', enter the distance of the lumped element from the start point of the rectangle strip line or from the bottom of the via object, whichever the case. The value of '''Offset''' by default is initially set to the center of the line or via.* In the '''Load Properties''' section, the series and shunt resistance values Rs and Rp are specified in Ohms, the series and shunt inductance values Ls and Lp are specified in nH (nanohenry), and the series and shunt capacitance values Cs and Cp are specified in pF (picofarad). Only the checked elements are taken into account in the total impedance calculation. By default, only the series resistor is checked with a value of 50S, and all other circuit elements are initially greyed out.<br /> 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 '''Lumped Circuit Type''' section of the dialog, the right section of the dialog entitled '''Source Properties''' becomes enabled, where you can you can specify the '''Source Amplitude''' in Volts (or in Amperes in the case of PMC traces) and the '''Phase''' in degrees. Also, the box labeled '''Direction''' becomes relevant in this case which contains a gap source. Otherwise, a passive RLC circuit does not have polarity. If the project workspace contains an array of rectangle strip objects or PEC via objects, the array object will also be listed as an eligible object for lumped element placement. A lumped element will then be placed on each element of the array. All the lumped elements will have identical direction, offset, resistance, inductance and capacitance values. If you define an active lumped element, you can prescribe certain amplitude and/or phase distribution to the gap sources just like in the case of gap and probe sources. The available amplitude distributions include '''Uniform''', '''Binomial'''''', Chebyshev''' and '''Data FileDevices]]'''.

=== Defining Source Arrays Calculating Scattering Parameters Using Prony's Method ===

If The calculation of the project workspace contains scattering (S) parameters is usually an array important objective of rectangle strip objectsmodeling planar structures especially for planar circuits like filters, couplers, etc. As you saw earlier, you can use lumped sources like gaps and probes and even active lumped elements to calculate the array object will also be listed as an eligible object for gap source placementcircuit characteristics of planar structures. A gap source will then be placed The admittance / impedance calculations based on each element of the array. All the gap sources will have identical direction voltages and offsetcurrents are accurate at RF and lower microwave frequencies or when the port transmission lines are narrow. SimilarlyIn such cases, if the project workspace contains an array electric or magnetic current distributions across the width of PEC via objectsthe port line are usually smooth, the embedded array object will also and quite uniform current or voltage profiles can easily be listed as an eligible object realized. At higher frequencies, however, a more robust method is needed for probe source placement. A probe source will then be placed on each via object of calculating the array. All the probe sources will have identical direction and offsetport parameters.

However, you One can prescribe certain amplitude and/or phase distribution over calculate the array scattering parameters of gap or probe sources. By default, all a planar structure directly by analyzing the gap or probe sources have identical amplitudes of 1V (or 1A for current distribution patterns on the slot case) and zero phaseport transmission lines. The available amplitude distributions discontinuity at the end of a port line typically gives rise to choose from include '''Uniform''', '''Binomial''a standing wave pattern that can clearly be discerned in the line' s current distribution. From the location of the current minima and '''Chebyshev''' maxima and '''Date File'''their relative levels, one can determine the reflection coefficient at the discontinuity, i.e. In the Chebyshev caseS₁₁ parameter. A more robust technique is Prony’s method, you need to set a value which is used for minimum side lobe level exponential approximation of functions. A complex function f('''SLL'''x) in dB. You can also define '''Phase Progression''' in degrees along all three principal axes. You can view the amplitude and phase be expanded as a sum of individual sources by right clicking on the top '''Sources''' item complex exponentials in the Navigation Tree and selecting '''Show Source Label''' from the contextual menu.following form:

In the data file option, the where c_i are complex amplitude coefficients and &gamma;_i are directly read , in from a data file using a real - imaginary format. When this option is selectedgeneral, you can either improvise the complex array weights or import them from an existing fileexponents. In From the former case click the '''New Data File''' button. This opens up the [[Windows]] Notepad with default formatted data file physics of transmission lines, we know that has a list of all the array element indices lossless lines may support one or more propagating modes with default 1+j0 amplitudes for all of thempure real propagation constants (real &gamma;_i exponents). You can replace the default complex values Moreover, line discontinuities generate evanescent modes with new one and save pure imaginary propagation constants (imaginary &gamma;_i exponents) that decay along the Notepad data file, which brings line as you back to the Gap Source dialog. To import the array weights, click the '''Open Data File''' button, which opens the standard [[Windows]] Open dialog. You can then select the right data file move away from the one location of your folders. It is important to note that the data file must have the correct format to be read by [[EM.Cube]]. For this reason, it is recommended that you first create a new data file with the right format using Notepad as described earlier and then save it for later usesuch discontinuities.

The first step of planning a planar MoM simulation is defining your planar structure<table><tr><td> [[Image:PMOM71. This consists png|thumb|600px|Minimum and maximum current locations of the background structure plus all the finite-sized metal and slot trace objects and possibly embedded metal or dielectric objects that are interspersed among the substrate layersstanding wave pattern on a microstrip line feeding a patch antenna. The background stack-up is defined in the Layer Stack-up dialog, which automatically opens up as soon as you enter the [[Planar Module]]. The metal and slot traces and embedded object sets are listed in the Navigation Tree, which also shows all the geometrical (CAD) objects you draw in the project workspace under each object group at different Z-planes.</td></tr></table>

* Current Distribution* Field Sensors* Far Fields Ports are used in a planar structure to order and index the sources for calculation of circuit parameters such as scattering (Radiation Patterns or Radar Cross SectionS)* Huygens Surfaces* Port Characteristics* Periodic Characteristics, impedance (Z) and admittance (Y) parameters. In [[EM.Picasso]], you can use one or more of the following types of sources to define ports:

If you run a simulation without having defined any observables, no data will be generated at the end of the simulation. Some observables require a certain type of excitation source. For example, port characteristics will be calculated only if the project contains a port definition, which in turn requires the existence of at least one gap or probe or de* Gap Sources* Probe Sources* Active Lumped Elements* De-embedded source. The periodic characteristics (reflection and transmission coefficients) are calculated only if the structure has a periodic domain and excited by a plane wave source.Embedded Sources

=== Planar ModulePorts are defined in the 's Simulation Modes ===''Observables''' section of the navigation tree. You can define any number of ports equal to or less than the total number of sources in your project. If you have N sources in your planar structure, then N default ports are defined, with one port assigned to each source according to their order on the navigation tree. Note that your project can have mixed gap and probes sources as well as active lumped element sources on PEC and slot traces or vias. You can also couple ports together to define coupled transmission lines such as coupled strips (CPS) or coplanar waveguides (CPW).

The simplest simulation type in [[EMImage:Info_icon.Cubepng|40px]] is an analysis. In this mode, Click here to learn more about the planar structure in your project workspace is meshed at the center frequency of the project. '''[[EMGlossary_of_EM.Cube%27s_Simulation_Observables_%26_Graph_Types#Port_Definition_Observable | Port Definition Observable]] generates an input file at this single frequency, and the Planar MoM simulation engine is run once. Upon completion of the planar MoM simulation, a number of data files are generated depending on the observables you have defined in your project. An analysis is a single-run simulation'''.

[[EMImage:Info_icon.Cubepng|40px]] offers a number of multi-run simulation modes. In such cases, the Planar MoM simulation engine is run multiple times. At each engine run, certain [[parameters]] are varied and a collection of simulation data are generated. At the end of a multi-run simulation, you can graph the simulation results in EM.Grid or you can animate the 3D simulation data from the Navigation Tree. For example, in a frequency sweep, the frequency of the project is varied over its specified bandwidth. Port characteristics are usually plotted vs. frequency, representing your planar structureClick here to learn more about '''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. [[EMPreparing_Physical_Structures_for_Electromagnetic_Simulation#Modeling_Coupled_Sources_.Cube26_Ports | Modeling Coupled Ports]]'s [[Planar Module]] currently provides the following types of multi-run simulation modes:''.

Depending on the source type and the types of observables defined in a project, a number of output data are generated at the end of a planar MoM simulation. Some of these data are 2D by nature and some are 3D. The output simulation data generated by [[File:PMOM80EM.pngPicasso]]can be categorized into the following groups:

Figure 1{| class="wikitable"|-! scope="col"| Icon! scope="col"| Simulation Data Type! scope="col"| Observable Type! scope="col"| Applications! scope="col"| Restrictions|-| style="width:30px;" | [[File:currdistr_icon.png]]| style="width:150px;" | Current Distribution Maps| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Current Distribution |Current Distribution]]| style="width:300px;" | Computing electric surface current distribution on metal traces and magnetic surface current distribution on slot traces | style="width:250px;" | None|-| style="width:30px;" | [[File:fieldsensor_icon.png]]| style="width:150px;" | Near-Field Distribution Maps| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Near-Field Sensor |Near-Field Sensor]] | style="width: Selecting 300px;" | Computing electric and magnetic field components on a simulation mode specified plane in the frequency domain| style="width:250px;" | None|-| style="width:30px;" | [[Planar ModuleFile:farfield_icon.png]]| style="width:150px;" | Far-Field Radiation Characteristics| style="width:150px;" | [[Glossary of EM.Cube's Simulation Run dialogObservables & Graph Types#Far-Field Radiation Pattern |Far-Field Radiation Pattern]]| style="width:300px;" | Computing the radiation pattern and additional radiation characteristics such as directivity, axial ratio, side lobe levels, etc.| style="width:250px;" | None|-| style="width:30px;" | [[File:rcs_icon.png]]| style="width:150px;" | Far-Field Scattering Characteristics| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Radar Cross Section (RCS) |Radar Cross Section (RCS)]] | style="width:300px;" | Computing the bistatic and monostatic RCS of a target| style="width:250px;" | Requires a plane wave source|-| style="width:30px;" | [[File:port_icon.png]]| style="width:150px;" | Port Characteristics| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Port Definition |Port Definition]] | style="width:300px;" | Computing the S/Y/Z parameters and voltage standing wave ratio (VSWR)| style="width:250px;" | Requires one of these source types: lumped, distributed, microstrip, CPW, coaxial or waveguide port|-| style="width:30px;" | [[File:period_icon.png]]| style="width:150px;" | Periodic Characteristics| style="width:150px;" | No observable required | style="width:300px;" | Computing the reflection and transmission coefficients of a periodic surface| style="width:250px;" | Requires a plane wave source and periodic boundary conditions |-| style="width:30px;" | [[File:huyg_surf_icon.png]]| style="width:150px;" | Equivalent electric and magnetic surface current data| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Huygens Surface |Huygens Surface]]| style="width:300px;" | Collecting tangential field data on a box to be used later as a Huygens source in other [[EM.Cube]] modules| style="width:250px;" | None|}

To run a If your planar MoM analysis of your project structure, open the Run Simulation Dialog is excited by clicking the '''Run''' [[File:run_icon.png]] button on the '''Simulate Toolbar''' gap sources or select '''Menu''' '''&gt;''' '''Simulate &gt;''' '''Run''' probe sources or use the keyboard shortcut '''Ctrl+R'''. The '''Analysis''' option of the '''Simulation Mode''' dropdown list is selected by default. Once you click the '''Run''' buttonde-embedded sources, the simulation starts. A new windowand one or more ports have been defined, called the '''Output Window'''planar MoM engine calculates the scattering, opens up that reports the different stages of simulation impedance and the percentage admittance (S/Z/Y) parameters of the tasks completed at any timedesignated ports. After the simulation is successfully completed, a message pops up and reports the end of simulation. In certain cases like calculating The scattering [[parameters]] of a circuit or reflection / transmission characteristics of a periodic surface, some results are also reported defined based on the port impedances specified in the Output Windowproject's Port Definition dialog. At If more than one port has been defined in the end of a simulationproject, you need to click the '''Close''' button S/Z/Y matrices of the Output Window to return to the project workspacemultiport network are calculated.

Electric and magnetic currents are the fundamental output data of a planar MoM simulation. After the numerical solution of the MoM linear system, they are found using the solution vector '''[[FileI]''' and the definitions of the electric and magnetic vectorial basis functions:PMOM78.png]]

Figure 1: <math> \mathbf{[[Planar ModuleX]]'s Simulation Run dialog.}_{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>

=== Stages Of A Planar 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 Analysis ===simulation. At the top of the Current Distribution dialog and in the section titled '''Active Trace / Set''', you can select a trace or embedded object set where you want to observe the current distribution.

[[EM.Cube]]'s Planar MoM simulation engine uses {{Note|You have to define a particular formulation of the method of moments called mixed potential integral equation (MPIE). Due to high-order singularities, the dyadic Green's functions separate current distribution observable 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^&Phi;'''(r)''', vector magnetic potential '''F(r)''' and scalar magnetic potential K^&Psi;'''(r)'''. These potentials have singularities of lower orders. As a result, they coverage relatively faster. The speed of their convergence is further increased drastically using special singularity extraction techniqueseach individual trace or embedded object set.}}

A planar MoM simulation consists of two major stages<table><tr><td> [[Image: matrix fill and linear system inversion. In the first stage, the moment matrix and excitation vector are calculated. In the second stage, the MoM system of linear equations is inverted using one of the several available matrix solvers to find the unknown coefficients of all the basis functionsPMOM85new. png|thumb|left|600px|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 observablesmap of a patch antenna. Having determined all the electric and magnetic currents in your planar structure, [[EM.Cube]] can then calculate the near fields on prescribed planes. These are introduced as field sensor observables. The near-zone electric and magnetic fields are calculated using a spectral domain formulation of the dyadic Green's functions. Finally the far fields of the planar structure are calculated in the spherical coordinate system. These calculations are performed using the asymptotic form of the dyadic Green's functions using the &quot;stationary phase method&quot;.</td></tr></table>

A planar MoM simulation involves a number of numerical {{Note|Keep in mind that since [[parameters]] that take preset default values unless you change themEM. You can access these [[parametersPicasso]] and change their values by clicking uses a planar MoM solver, the '''Settings''' button next to calculated field value at the '''Select Engine''' dropdown list in the [[Planar Module]]'s Simulation Run dialogsource point is infinite. In most casesAs a result, you do not need to open this dialog and you can leave all the default numerical parameter values intact. However, it is useful field sensors must be placed at adequate distances (at least one or few wavelengths) away from the scatterers to familiarize yourself with these [[parameters]], as they may affect the accuracy of your numerical produce acceptable 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:PMOM116. The &quot;'''Matrix Fill'''&quot; section of the dialog deals with the operations involving the dyadic Green's functions. You can set png|thumb|left|600px|Near-zone electric field map above a value for the '''Convergence Rate for Integration''', which is 1Emicrostrip-5 by defaultfed patch antenna. 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:PMOM117. You can modify the definition of &quot;Thin Substrate&quot; by entering png|thumb|left|600px|Near-zone magnetic field map above 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 quasimicrostrip-static analysis can be carried out. You can set this threshold in wavelengths in the box labeled '''Max Dimensions for Quasi-Static Analysis'''fed patch antenna.]] </td></tr></table>

In the Even though [[EM.Picasso]]'s MoM engine does not need a radiation box, you still have to define a &quot;Spectral Domain IntegrationFar Field&quot; section of the dialog, you can set a value to '''Max Spectral Radius in k0''', which has a default value of 30observable for radiation pattern calculation. This means that the infinite spectral-domain integrals in the spectral variable k_&rho; are pre-calculated is because far field calculations take time and tabulated up you have to instruct [[EM.Cube]] to perform these calculations. Once a limit of 30k₀, where k₀ planar MoM simulation is finished, three far field items are added under the free space propagation constantFar Field item in the Navigation Tree. These integrals may converge much faster based on are the specified Convergence Rate for Integration described earlier. However, far field component in certain cases involving highly oscillatory integrands&theta; direction, much larger integration limits like 100k₀ might be needed to warrant adequate convergence. For spectral-domain integration along the real k_{far field component in &rhophi;} axis, direction and the interval [0, Nk₀] is subdivided into a large number of sub-intervals, within each an 8-point Gauss-Legendre quadrature is applied&quot;Total&quot; far field. The next parameter, 2D radiation pattern graphs can be plotted from the '''No. Radial Integration Divisions per k₀Data Manager''', determines how small these intervals should be. By default, 2 divisions are used for the interval [0, k₀]. In other words, the length of each integration sub-interval is k₀/2. You can increase the resolution of integration by increasing this value above 2. Finally, instead A total of eight 2D radiation pattern graphs are available: 4 polar and 4 Cartesian integration in graphs for the spectral domainXY, a polar integration is performed. You can set the '''No. of Angular Integration Points'''YZ, which has a default value of 100ZX and user defined plane cuts.

[[FileImage:PMOM79Info_icon.png|30px]]Click here to learn more about the theory of '''[[Defining_Project_Observables_%26_Visualizing_Output_Data#Using_Array_Factor_to_Model_Antenna_Arrays | Using Array Factors to Model Antenna Arrays ]]'''.

Figure 1<table><tr><td> [[Image: The Planar MoM Engine Settings dialogPMOM119.png|thumb|left|600px|3D polar radiation pattern plot of a microstrip-fed patch antenna.]] </td></tr></table>

After the MoM impedance matrix '''<table><tr><td> [Z]''' (not to be confused with [Image:PMOM125.png|thumb|left|600px|An example of the impedance [[parameters3D monostatic radar cross section plot of a patch antenna.]]) and excitation vector '''[V]''' have been computed through the matrix fill process, the planar MoM simulation engine is ready to solve the system of linear equations:</td></tr></table>

:<math> \mathbf{[Z]}_{N\times N} \cdot \mathbf{[I]}_{N\times 1} = \mathbf{[V]}_{N\times 1} </math><!--[[File:PMOM81= Discretizing a Planar Structure in EM.png]]-->Picasso ==

where '''[I]''' is the solution vector, which contains the unknown amplitudes The method of moments (MoM) discretizes all the basis functions that represent the unknown electric and magnetic currents finite-sized objects of finite extents in your a planar structure. In (excluding the above equation, N is the dimension background structure) into a set of elementary cells. Both the linear system quality and equal to the total number resolution of basis functions in the planar generated mesh. [[EM.Cube]]'s linear solvers compute greatly affect the solution vector'''[I]''' accuracy of the above system. You can instruct [[EM.Cube]] to write the MoM matrix and excitation and numerical solution vectors into output data files for your examination. To do so, check The mesh density gives a measure of the box labeled &quot;'''Output MoM Matrix and Vectors'''&quot; number of cells per effective wavelength that are placed in the Matrix Fill section various regions of the Planar MoM Engine Settings dialogyour planar structure. These The higher the mesh density, the more cells are written into three files called momcreated on the finite-sized geometrical objects.dat1As a rule of thumb, exca mesh density of about 20-30 cells per effective wavelength usually yields satisfactory results.dat1 and solnBut for structures with lots of fine geometrical details or for highly resonant structures, higher mesh densities may be required.dat1The particular output data that you seek in a simulation also influence your choice of mesh resolution. For example, respectivelyfar field characteristics like radiation patterns are less sensitive to the mesh density than field distributions on structures with a highly irregular shapes and boundaries.

There are a large number of numerical methods for solving systems of linear equations. These methods are generally divided into two groups: direct solvers and iterative solvers. Iterative solvers are usually based on matrix-vector multiplications. Direct solvers typically work faster for matrices of smal to medium size (N&lt;3,000). <table><tr><td> [[EMImage:PMOM31.Cube]]'s [[png|thumb|400px|The Planar ModuleMesh Settings dialog.]] offers five linear solvers:</td></tr></table>

# LU Decomposition Method# Biconjugate Gradient Method (BiCG)# Preconditioned Stabilized Biconjugate Gradient Method (BCGEM.Picasso provides two types of mesh for a planar structure: a pure triangular surface mesh and a hybrid triangular-STAB)# Generalized Minimal Residual Method (GMRES)# Transpose-Free Quasi-Minimum Residual Method (TFQMR)rectangular surface mesh. In both case, EM.Picasso attempts to create a highly regular mesh, in which most of the cells have almost equal areas. For planar structures with regular, mostly rectangular shapes, the hybrid mesh generator usually leads to faster computation times.

Of the above list, LU is a direct solver, while the rest are iterative solvers[[Image:Info_icon. BiCG is a relatively fast iterative solver, but it works only for symmetric matrices. You cannot use BiCG for periodic structures or planar structures that contain both metal and slot traces at different planes, as their MoM matrices are not symmetric. The three solvers BCG-STAB, GMRES and TtFQMR work well for both symmetric and asymmetric matrices and they also belong png|30px]] Click here to a class of solvers called learn more about '''Krylov Sub-space Methods[[Preparing_Physical_Structures_for_Electromagnetic_Simulation#Working_with_EM.Cube.27s_Mesh_Generators | Working with Mesh Generator]]'''. In particular, the GMRES method always provides guaranteed unconditional convergence.

[[EMImage:Info_icon.Cubepng|30px]]Click here to learn more about 's [[Planar Module]], by default, provides a &quot;'''Automatic'''&quot; solver option that picks the best method based on the settings and size of the numerical problem. For linear systems with a size less than N = 3,000, the LU solver is used. For larger systems, BiCG is used when dealing with symmetric matrices, and GMRES is used for asymmetric matrices. If the size of the linear system exceeds N = 15,000, the sparse version of the iterative solvers is used, utilizing a row-indexed sparse storage scheme. You can override the automatic solver option and manually set you own solver type. This is done using the '''Solver Type''' dropdown list in the &quot;'''Linear System Solver'''&quot; section of the Planar MoM Engine Settings dialog. There are also a number of other [[parameters]] related to the solversPreparing_Physical_Structures_for_Electromagnetic_Simulation#The_Triangular_Surface_Mesh_Generator | EM. The default value of Picasso'''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 [[parameterss Triangular Surface Mesh Generator]] that are related to the Automatic Solver option. These are &quot;''' User Iterative Solver When System Size &gt;'''&quot; with a default value of 3,000 and &quot;''' Use SParse Storage When System Size &gt;''' &quot; with a default value of 15,000. In other words, you control the automatic solver when to switch between direct and iterative solvers and when to switch to the sparse version of iterative solvers.

If your computer has an Intel CPU, then <table><tr><td> [[EMImage:PMOM48F.Cube]] offers special versions png|thumb|left|420px|Geometry of all the above linear solvers that have been optimized for Intel CPU platforms. These optimal solvers usually work 2a multilayer slot-3 time faster than their generic counterpartscoupled patch array. When you install ]] </td></tr><tr><td> [[EMImage:PMOM48G.Cube]], png|thumb|left|420px|Hybrid planar mesh of the option to use Intelslot-optimized solvers is already enabledcoupled patch array. However, you can disable this option (e.g. if your computer has a non-Intel CPU). To do that, open the [[EM.Cube]]'s Preferences Dialog from '''Menu &gt; Edit &gt; Preferences''' or using the keyboard shortcut '''Ctrl+H'''. Select the Advanced tab of the dialog and uncheck the box labeled &quot;''' Use Optimized Solvers for Intel CPU'''&quot;.</td></tr></table>

<table><tr><td> [[FileImage:PMOM82PMOM48H.png|thumb|left|420px|Details of the hybrid planar mesh of the slot-coupled patch array around discontinuities.]]</td></tr></table>

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

In a frequency sweep, the operating frequency of a planar structure is varied during each sweep run. [[EM.Cube]]Picasso's [[Planar Module]] offers two types hybrid planar mesh generator tries to produce as many rectangular cells as possible especially in the case of frequency sweep: Uniform and Adaptiveobjects with rectangular or linear boundaries. In a uniform frequency sweepconnection or junction areas between adjacent objects or close to highly curved boundaries, the frequency range and the number of frequency samples triangular cells are specified. The samples are equally spaced over used to fill the frequency range. At the end of each individual frequency run, the output data are collected "irregular" regions in a conformal and stored. At the end of the frequency sweep, the 3D data can be visualized and/or animated, and the 2D data can be graphed in EM.Gridconsistent manner.

To run The mesh density gives a uniform frequency sweep, open the '''Simulation Run Dialog''', and select the '''Frequency Sweep''' option from the dropdown list labeled '''Simulation Mode'''. When you choose the frequency sweep option, the '''Settings''' button next to the simulation mode dropdown list becomes enabled. Clicking this button opens the '''Frequency Settings''' dialog. The '''Frequency Range'''is initially set equal to your project's center frequency minus and plus half bandwidth. But you can change the values measure of '''Start Frequency'''and '''End Frequency''' as well as the '''Number number of Samples'''. The dialog offers two options for '''Frequency Sweep Type''': '''Uniform''' or '''Adaptive'''. Select the former type. It is very important to note cells per effective wavelength that are placed in a MoM simulation, changing the frequency results in a change various regions of the mesh of the your planar structure, too. This The effective wavelength is because the mesh density defined as <math>\lambda_{eff} = \tfrac{\lambda_0}{\sqrt{\varepsilon_{eff}}}</math>, where e_eff is defined in terms of the number of cells per effective wavelengthpermittivity. By default, during a frequency sweep, [[EM.CubePicasso]] fixes the generates a hybrid mesh with a mesh density at the highest frequency, iof 20 cells per effective wavelength.e., at the &quot;End Frequency&quot;The effective permittivity is defined differently for different types of traces and embedded object sets. This usually results in a smoother frequency response. You have the option is to fix the mesh at the center frequency of the project or let [[EM.Cube]] &quot;remesh&quot; the planar structure at each frequency sample during a frequency sweep. You can make one of these three choices using the radio button sure that enough cells are placed in the '''Mesh Settings''' section of the dialog. Closing the Frequency Settings dialog returns you to the Simulation Run dialog, where you can start the planar MoM frequency sweep simulation by clicking the '''Run''' buttonareas that might feature higher field concentration.

Frequency sweeps are often performed to study * For PEC and conductive sheet traces, the frequency response effective permittivity is defined as the larger of a planar structure. In particular, the variation permittivity of scattering [[parameters]] like S₁₁ (return loss) the two substrate layers just above and S₂₁ (insertion loss) with frequency are of utmost interestbelow the metallic trace. When analyzing resonant structures like patch antennas or planar filters over large frequency ranges* For slot traces, you may have to sweep a large number of frequency samples to capture their behavior with adequate details. The resonant peaks or notches are often missed due to the lack of enough resolution. [[EM.Cube]]'s [[Planar Module]] offers a powerful adaptive frequency sweep option for this purpose. It effective permittivity is based on defined as the fact that mean (average) of the frequency response permittivity 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 two substrate layers just above 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 below the scattering [[parameters]]metallic trace. Then* For embedded object sets, it increases the number of samples gradually by inserting intermediate frequency samples in a progressive manner. At each iteration cycle, all effective permittivity is defined as the possible rational functions largest of higher orders are tried out. The process continues until adding new intermediate frequency samples does not improve the resolution permittivities of all the &quot;S_ij&quot; curves over the given frequency range. In that case, the curves are considered as having convergedsubstrate layers and embedded dielectric sets.

You must have defined one or more ports for your planar structure run an adaptive frequency sweep<table><tr><td> [[Image:PMOM32. Open the Frequency Settings dialog from the Simulation Run dialog and select the '''Adaptive''' option png|thumb|360px|A comparison of '''Frequency Sweep Type'''. You have to set values for '''Minimum Number of Samples''' triangular and '''Maximum Number planar hybrid meshes 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 0rectangular patch.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><td> [[EMImage:PMOM30.Cube]] reaches the specified maximum number png|thumb|360px|Mesh of iterations and the convergence criterion two rectangular patches at two different substrate planes. The lower substrate layer has not yet been met, the program will ask you whether to continue the process or exit it and stopa higher permittivity.]] </td></tr></table>

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

== Working with Planar The integrity of the planar mesh and its continuity in the junction areas directly affects the quality and accuracy of the simulation results. EM.Picasso's hybrid planar mesh generator has some rules that are catered to 2.5-D MoM Simulation Data ==simulations:

[[Image:PMOM130.png|thumb|400px|Changing * If two connected rectangular objects have the graph type by editing same side dimensions along their common linear edge with perfect alignment, a data file's propertiesrectangular junction mesh is produced.]][[Image:PMOM131.png|thumb|300px|EM* 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.Picasso's Smart Fit dialog.]][[Image:PMOM133(2).png|thumb|300px|The S₁₁ parameter plot of * Rectangle strip objects that host a two-port structure in magnitude-phase formatgap source or a lumped element always have a rectangular mesh around the gap area.]][[Image:PMOM132(2).png|thumb|300px|The smoothed version of * If two objects reside on the same Z-plane, belong to the same trace group and have a common overlap area, they are first merged into a single object for the S₁₁ parameter plot purpose of meshing using the two&quot;Boolean Union&quot; operation. * Embedded objects have prismatic meshes along the Z-port structure using [[EMaxis.Cube]]'s Smart Fit.]]=== Planar Module's Output Simulation Data ===* 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.

* '''Port Characteristics'''<table><tr><td> [[File: S, Z and Y PMOM36.png|250px]] [[ParametersFile:PMOM38.png|250px]] and Voltage Standing Wave Ratio (VSWR)* '''Radiation Characteristics'''[[File: Radiation Patterns, Directivity, Total Radiated Power, Axial Ratio, Main Beam Theta and Phi, Radiation Efficiency, Half Power Beam Width (HPBW), Maximum Side Lobe Level (SLL), First Null Level (FNL), Front-to-Back Ratio (FBR), etcPMOM37.png|250px]] </td>* '''Scattering Characteristics''': Bi-static and Mono-static Radar Cross Section (RCS)</tr>* '''Periodic Characteristics''': Reflection and Transmission Coefficients<tr>* '''Current Distributions''': Electric <td> Two overlapping planar objects and magnetic current amplitude and phase on all metal and slot traces a comparison of their triangular and embedded objectshybrid planar meshes. </td>* '''Near-Field Distributions'''</tr><tr><td> [[File: Electric and magnetic field amplitude and phase on specified planes PMOM33.png|250px]] [[File:PMOM35.png|250px]] [[File:PMOM34.png|250px]] </td></tr><tr><td> Edge-connected rectangular planar objects and a comparison their central axestriangular and hybrid planar meshes. </td></tr></table>

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

At It is very important to apply the end right mesh density to capture all the geometrical details of a your planar MoM simulation, the values of S/Z/Y [[parameters]] and VSWR data are calculated and reported in the output message windowstructure. The S, Z and Y [[parameters]] are written into output ASCII data files of complex type with a This is especially true for &quot;'''.CPX'''field discontinuity&quot; extension. Every file begins with a header consisting of a few comment lines that start with the &quot;#&quot; symbol. The complex values regions such as junction areas between connected objects, where larger current concentrations are arranged into two columns for usually observed at sharp corners, or at the real junction areas between metallic traces and imaginary parts. In the case of multiport structuresPEC vias, every single element of as well as the S/Z/Y matrices is written into a separate complex data file. For exampleareas around gap sources and lumped elements, you will have data files like S11.CPX, S21.CPX, ..., Z11.CPX, Z21.CPX, etc. The VSWR data are saved to an ASCII data file of real type with a &quot;'''.DAT'''&quot; extension called, VSWR.DATwhich create voltage or current discontinuities.

If you run an analysis, the port characteristics have single complex values, which you can view using [[EMThe Planar Mesh Settings dialog gives a few options for customizing your planar mesh around geometrical and field discontinuities.Cube]]The check box labeled &quot;'s data manager. However, there are no curves to graph. You can plot ''Refine Mesh at Junctions'''&quot; increases the S/Z/Y [[parameters]] and VSWR data when you have data sets, which are generated mesh resolution at the end of any type of sweep including a frequency sweepconnection area between rectangular objects. In that case, the The check box labeled &quot;.CPX'''Refine Mesh at Gap Locations'''&quot; files have multiple rows corresponding to each value of the sweep parameter (e.g. frequency). [[EM.Cube]]'s 2D graph data might be particularly useful when gap sources or lumped elements are plotted in EM.Grid, placed on a versatile graphing utility. You can plot the port characteristics directly short transmission line connected from the Navigation Treeboth ends. Right click on the The check box labeled &quot;'''Port DefinitionRefine Mesh at Vias''' item in &quot; increases the '''Observables''' mesh resolution on the cross section of embedded object sets and at the Navigation Tree and select one connection regions of the items: '''Plot S [[Parameters]]''', '''Plot Y [[Parameters]]''', '''Plot Z [[Parameters]]''', or '''Plot VSWR'''metallic objects connected to them. In EM.Picasso typically doubles the first three cases, another sub-menu gives a list of individual port [[parameters]]mesh resolution locally at the discontinuity areas when the respective boxes are checked. You should always visually inspect EM.Picasso's default generated mesh to see if the current mesh settings have produced an acceptable mesh.

Sometimes EM.Picasso's default mesh may contain very narrow triangular cells due to very small angles between two edges. In particularsome rare cases, it extremely small triangular cells may be useful to plot the S_ii [[parameters]] on generated, whose area is a Smith chartsmall fraction of the average mesh cell. To change These cases typically happen at the format junctions and other discontinuity regions or at the boundary of a data plothighly irregular geometries with extremely fine details. In such cases, select it in increasing or decreasing the Data Manager mesh density by one or few cells per effective wavelength often resolves that problem and click its ''eliminates those defective cells. Nonetheless, EM.Picasso'Edit''' buttons planar mesh generator offers an option to identify the defective triangular cells and either delete them or cure them. In By curing we mean removing a narrow triangular cell and merging its two closely spaced nodes to fill the Edit File Dialog, choose one of crack left behind. EM.Picasso by default deletes or cures all the options provided triangular cells that have angles less than 10º. Sometimes removing defective cells may inadvertently cause worse problems in the dropdown list mesh. You may choose to disable this feature and uncheck the box labeled &quot;'''Graph TypeRemove Defective Triangular Cells'''&quot; in the Planar Mesh Settings dialog. You can also change the value of the minimum allowable cell angle.

The adaptive frequency sweep described earlier is an iterative process, whereby the Planar MoM simulation engine is run at a certain number of frequency samples at each iteration cycle. The frequency samples are progressively built up, and rational fits for these data are found at each iteration cycle. A decision is then made whether to continue more iterations. At the end of the whole process, a total number of scattering parameter data samples have been generated, and new smooth data corresponding to the best rational fits are written into new data files for graphing. <table><tr><td> [[EMImage:PMOM44.png|thumb|left|480px|Deleting or curing defective triangular cells: Case 1.Cube]]'s </td></tr><tr><td> [[planar ModuleImage:PMOM42.png|thumb|left|480px|Deleting or curing defective triangular cells: Case 2.]] also allows you to generate a rational fit for all or any existing scattering parameter data as a post-processing operation without a need to run additional simulation engine runs.</td></tr></table>

You can interpolate all the scattering [[parameters]] together or select individual [[parameters]]. You do this post-processing operation from the Navigation Tree. Right click on the '''Port Definition''' item == Running Planar MoM Simulations in the '''Observables''' section of the Navigation Tree and select Smart Fit. At the top of the Smart Fit Dialog, there is a dropdown list labeled '''Interpolate''', which gives a list of all the available S parameter data for rational interpolation. The default option is &quot;All Available [[Parameters]]&quot;. Then you see a box labeled '''Number of Available Samples''', whose value is read from the data content of the selected complex .CPX data file. Based on the number of available data samples, the dialog reports the '''Maximum Interpolant Order'''. You can choose any integer number for '''Interpolant Order''', from 1 to the maximum allowedEM. Picasso ==

{{Note|Interpolant order more than 15 will suffer from numerical instabilities even if you have a very large number of data samples=== EM.}}Picasso's Simulation Modes ===

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

{| class="wikitable"|-! scope="col"| Simulation Mode! scope= Visualizing Current Distributions "col"| Usage! scope="col"| Number of Engine Runs! scope="col"| Frequency ! scope="col"| Restrictions|-| style="width:120px;" | [[#Running a Single-Frequency Planar MoM Analysis | Single-Frequency Analysis]]| style="width:270px;" | Simulates the planar structure "As Is"| style="width:80px;" | Single run| style="width:250px;" | Runs at the center frequency fc| style="width:80px;" | None|-| style="width:120px;" | [[Parametric_Modeling_%26_Simulation_Modes_in_EM.Cube#Running_Frequency_Sweep_Simulations_in_EM.Cube | Frequency Sweep]]| style="width:270px;" | Varies the operating frequency of the planar MoM solver | style="width:80px;" | Multiple runs | style="width:250px;" | Runs at a specified set of frequency samples or adds more frequency samples in an adaptive way| style="width:80px;" | None|-| style="width:120px;" | [[Parametric_Modeling_%26_Simulation_Modes_in_EM.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;" | [[Parametric_Modeling_%26_Simulation_Modes_in_EM.Cube#Performing_Optimization_in_EM.Cube | Optimization]]| style="width:270px;" | Optimizes the value(s) of one or more project variables to achieve a design goal | style="width:80px;" | Multiple runs | style="width:250px;" | Runs at the center frequency fc| style="width:80px;" | None|-| style="width:120px;" | [[Parametric_Modeling_%26_Simulation_Modes_in_EM.Cube#Generating_Surrogate_Models | HDMR Sweep]]| style="width:270px;" | Varies the value(s) of one or more project variables to generate a compact model| style="width:80px;" | Multiple runs | style="width:250px;" | Runs at the center frequency fc| style="width:80px;" | None|}

Electric and magnetic currents are You can set the fundamental output data of simulation mode from [[EM.Picasso]]'s "Simulation Run Dialog". A single-frequency analysis is a planar MoM single-run simulation. After All the numerical solution other simulation modes in the above list are considered multi-run simulations. If you run a simulation without having defined any observables, no data will be generated at the end of the MoM linear systemsimulation. In multi-run simulation modes, they certain parameters are found using the solution vector '''[I]''' varied and a collection of simulation data files are generated. At the definitions end of a sweep simulation, you can graph the electric and magnetic vectorial basis functions:simulation results in EM.Grid or you can animate the 3D simulation data from the navigation tree.

:<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)} Running a Single-Frequency Planar MoM Analysis == \sum_{k=1}^K V_k^{(M)} \mathbf{f_k^{(M)} (r)} \end{cases} </math><!- A single-frequency analysis is the simplest type of [[File:PMOM83EM.pngPicasso]]-->simulation and involves the following steps:

* Set the units of your project and the frequency of operation. Note that currents are complex vector quantitiesthe default project unit is '''millimeter'''. Each electric or magnetic current has three X, Y * Define you background structure and Z components, its layer properties and each complex component has a magnitude and phasetrace types. You can visualize * Construct your planar structure using [[Building_Geometrical_Constructions_in_CubeCAD | CubeCAD]]'s drawing tools to create all the surface electric currents on finite-sized metal (PEC) and conductive sheet traces, surface magnetic currents on slot (PMC) traces and vertical volume currents on the PEV vias trace objects and possibly embedded metal or dielectric objects. 3D color-coded intensity plots of electric and magnetic current distributions that are visualized in interspersed among the substrate layers.* Define an excitation source and observables for your project workspace.* Examine the planar mesh, superimposed on verify its integrity and change the surface of physical objectsmesh density if necessary.* Run the Planar MoM simulation engine.* Visualize the output simulation data.

In order to view the current distributions, you must first define them as observables before running the To run a planar MoM simulation. To do thatanalysis of your project structure, right click on open the Run Simulation Dialog by clicking the '''Current DistributionsRun''' item in [[File:run_icon.png]] button on the '''ObservablesSimulate Toolbar''' section of the Navigation Tree and or select '''Insert New Observable...Menu > Simulate > Run'''or use the keyboard shortcut {{key|Ctrl+R}}. The Current Distribution Dialog opens up. At the top of the dialog and in the section titled '''Active Trace / SetSingle-Frequency Analysis''', you can select a trace or embedded object set where you want to observe option of the current distribution. You can also select the current map type from two options: '''Confetti''' and '''ConeSimulation Mode'''dropdown list is selected by default. The former produces an intensity plot for current amplitude Once you click the {{key|Run}} button, the simulation starts. A new window called the "Output Window" opens up that reports the different stages of simulation and phasethe percentage of the tasks completed at any time. After the simulation is successfully completed, while a message pops up and reports the latter generates end of simulation. In certain cases like calculating scattering parameters of a 3D vector plotcircuit or reflection / transmission characteristics of a periodic surface, some results are also reported in the output window.

<table><tr><td> [[FileImage:PMOM84Picasso L1 Fig18.png|thumb|left|480px|EM.Picasso's Simulation Run dialog.]]</td></tr></table>

Once A planar MoM simulation involves a number of numerical parameters that take preset default values unless you close change them. You can access these parameters and change their values by clicking the current distribution dialog, the label of the selected trace or object set is added under '''Settings''' button next to the '''Current DistributionsSelect Engine''' node of drop-down list in [[EM.Picasso]]'s Simulation Run dialog. In most cases, you do not need to open this dialog and you can leave all the Navigation Treedefault numerical parameter values intact. However, it is useful to familiarize yourself with these parameters, as they may affect the accuracy of your numerical results.

{{Note|The Planar MoM Engine Settings Dialog is organized in a number of sections. Here we describe some of the numerical parameters. The &quot;'''Matrix Fill'''&quot; section of the dialog deals with the operations involving the dyadic Green's functions. You have can set a value for the '''Convergence Rate for Integration''', which is 1E-5 by default. This is used for the convergence test of all the infinite integrals in the calculation of the Hankel transform of spectral-domain dyadic Green's functions. When the substrate is lossy, the surface wave poles are captured in the complex integration plane using contour deformation. You can change the maximum number of iterations involved in this deformed contour integration, whose default value is 20. When the substrate is very thin with respect to define 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 separate current distribution observable thin substrate layer is defined to a have a thickness less than 0.01&lambda;_eff, where &lambda;_eff is the effective wavelength. You can modify the definition of &quot;Thin Substrate&quot; by entering a value for each individual trace or embedded object '''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 setto 1,000 wavelengths. For electrically small structures, the phase variation across the structure may be negligible. In such cases, a fast quasi-static analysis can be carried out. You can set this threshold in wavelengths in the box labeled '''Max Dimensions for Quasi-Static Analysis'''.}}

At In the end &quot;Spectral Domain Integration&quot; section of the dialog, you can set a planar MoM simulationvalue to '''Max Spectral Radius in k0''', which has a default value of 30. This means that the current distribution nodes infinite spectral-domain integrals in the Navigation Tree become populated by the magnitude spectral variable k_&rho; are pre-calculated and phase plots tabulated up to a limit of 30k₀, where k₀ is the three vectorial components of free space propagation constant. These integrals may converge much faster based on the electric (specified Convergence Rate for Integration described earlier. However, in certain cases involving highly oscillatory integrands, much larger integration limits like 100k₀ might be needed to warrant adequate convergence. For spectral-domain integration along the real k_&rho; axis, the interval [0, Nk₀] is subdivided into a large number of sub-intervals, within each an 8-point Gauss-Legendre quadrature is applied. The next parameter, '''JNo. Radial Integration Divisions per k₀''') and magnetic (, determines how small these intervals should be. By default, 2 divisions are used for the interval [0, k₀]. In other words, the length of each integration sub-interval is k₀/2. You can increase the resolution of integration by increasing this value above 2. Finally, instead of 2D Cartesian integration in the spectral domain, a polar integration is performed. You can set the '''MNo. of Angular Integration Points''') currents as well as the total electric and magnetic currents, which has a default value of 100.

[[Image:MOREEM.png|40pxPicasso]] Click here to learn more about provides a large selection of linear system solvers including both direct and iterative methods. [[EM.Picasso]], by default, provides a &quot;'''Automatic'''&quot; solver option that picks the best method based on the settings and size of the numerical problem. For linear systems with a size less than N = 3,000, the LU solver is used. For larger systems, BiCG is used when dealing with symmetric matrices, and GMRES is used for asymmetric matrices. You can instruct [[Data_Visualization_and_Processing#Visualizing_3D_Current_Distribution_Maps | Visualizing 3D Current Distribution MapsEM.Cube]]to write the MoM matrix and excitation and solution vectors into output data files for your examination. To do so, check the box labeled &quot;'''Output MoM Matrix and Vectors'''&quot; in the Matrix Fill section of the Planar MoM Engine Settings dialog. These are written into three files called mom.dat1, exc.dat1 and soln.dat1, respectively.

<table><tr><td> [[FileImage:PMOM85(1)PMOM79.png|800pxthumb|left|720px|EM.Picasso's Planar MoM Engine Settings dialog.]]</td></tr></table>

Figure 2: The current distribution map of a patch antenna== Modeling Periodic Planar Structures in EM.Picasso ==

[[File:PMOM86(2)EM.png|800pxPicasso]]allows you to simulate doubly periodic planar structures with periodicities along the X and Y directions. Once you designate your planar structure as periodic, [[EM.Picasso]]'s Planar MoM simulation engine uses a spectral domain solver to analyze it. In this case, the dyadic Green's functions of periodic planar structure take the form of doubly infinite summations rather than integrals.

Figure 3[[Image: Vectorial (cone) visualization of Info_icon.png|30px]] Click here to learn more about the current distribution on a patch antennatheory of '''[[Basic_Principles_of_The_Method_of_Moments#Periodic_Planar_MoM_Simulation | Periodic Green's functions]]'''.

Once you close the Field Sensor dialog, its name is added under the '''Field Sensors''' node of the Navigation Tree. At the end of === Defining a planar MoM simulation, the field sensor nodes Periodic Structure in the Navigation Tree become populated by the magnitude and phase plots of the three vectorial components of the electric ('''E''') and magnetic ('''H''') field as well as the total electric and magnetic fields definedEM.Picasso ===

Note that unlike An infinite periodic structure in [[EM.CubePicasso]]is represented by a &quot;'s other computational modules''Periodic Unit Cell'''&quot;. To define a periodic structure, near field calculations in the you must open [[Planar ModuleEM.Picasso]] usually takes substantial time's Periodicity Settings Dialog by right clicking the '''Periodicity''' item in the '''Computational Domain''' section of the navigation tree and selecting '''Periodicity Settings...''' from the contextual menu or by selecting '''Menu''' '''&gt;''' '''Simulate &gt; 'Computational Domain &gt; Periodicity Settings...''' from the menu bar. In the Periodicity Settings Dialog, check the box labeled '''Periodic Structure'''. This will enable the section titled''&quot;''Lattice Properties&quot;. You can define the periods along the X and Y axes using the boxes labeled '''Spacing'''. In a periodic structure, the virtual domain is due to replaced by a default blue periodic domain that is always centered around the fact origin of coordinates. Keep in mind that the periodic unit cell must always be centered at the end origin of coordinates. The relative position of a planar MoM simulation, the fields are not available anywhere (as opposed to structure within this centered unit cell will change the [[FDTD Module]]), and their computation requires integration phase of complex dyadic Green's functions (as opposed to [[MoM3D Module]]'s free space Green's functions)the results.

<table><tr><td> [[Image:MOREPMOM99.png|40px]] Click here to learn more about thumb|300px|EM.Picasso'''[[Data_Visualization_and_Processing#Visualizing_3D_Near-Field_Maps | Visualizing 3D Near Field Mapss 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. [[File:PMOM116EM.png|800pxPicasso]]allows you to define periodic structures whose unit cells are interconnected. The interconnectivity applies only to PEC, PMC and conductive sheet traces, and embedded object sets are excluded. Your objects cannot cross the periodic domain. In other words, the neighboring unit cells cannot overlap one another. However, you can arrange objects with linear edges such that one or more flat edges line up with the domain's bounding box. In such cases, [[EM.Picasso]]'s planar MoM mesh generator will take into account the continuity of the currents across the adjacent connected unit cells and will create the connection basis functions at the right and top boundaries of the unit cell. It is clear that due to periodicity, the basis functions do not need to be extended at the left or bottom boundaries of the unit cell. As an example, consider a periodic metallic screen as shown in the figure on the right. The unit cell of this structure can be defined as a rectangular aperture in a PEC ground plane (marked as Unit Cell 1). In this case, the rectangle object is defined as a slot trace. Alternatively, you can define a unit cell in the form of a microstrip cross on a metal trace. In the latter case, however, the microstrip cross should extend across the unit cell and connect to the crosses in the neighboring cells in order to provide current continuity.

Near-zone electric field map above <table><tr><td> [[Image:image122.png|thumb|400px|Modeling a microstrip-fed patch antennaperiodic screen using two different types of unit cell.]] </td></tr></table>

<table><tr><td> [[FileImage:PMOM117pmom_per5_tn.png|800pxthumb|300px|The PEC cross unit cell.]] </td><td> [[Image:pmom_per6_tn.png|thumb|300px|Planar mesh of the PEC cross unit cell. Note the cell extensions at the unit cell's boundaries.]]</td></tr></table>

Near-zone magnetic field map above a microstrip-fed patch antenna=== Exciting Periodic Structures as Radiators in EM.Picasso ===

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

<table><tr><td> [[FileImage:PMOM118Period5.png|thumb|300px350px|[[Planar Module]]Setting periodic scan angles in EM.Picasso's Radiation Pattern Gap Source dialog.]]</td>Even though EM.Pplanar MoM engine does not need a radiation box, you still have to define a &quot;Far Field&quot; observable for radiation pattern calculation. This is because far field calculations take time and you have to instruct <td> [[EMImage:Period5_ang.png|thumb|350px|Setting the beam scan angles in Periodic Scan Angles dialog.Cube]] to perform these calculations</td></tr><tr><td> [[Image:Period6. To define a far field, right click png|thumb|350px|Setting the '''Far Fields''' item array factor in the '''Observables''' section of the Navigation Tree and select '''Insert New Radiation Pattern..EM.Picasso'''. The s Radiation Pattern Dialog opens updialog. You may accept the default settings, or you can change the value of '''Angle Increment''', which is expressed in degrees. You can also choose to '''Normalize 2D Patterns'''. In that case, the maximum value of a 2D paten graph will have a value of 1; otherwise, the actual far field values in V]] </m will be used on the graph.td></tr></table>

Once a planar MoM simulation is finished, three far field items are added under the Far Field item in the Navigation Tree<table><tr><td> [[Image:Period7. These are the far field component in png|thumb|360px|Radiation pattern of an 8×8 finite-sized periodic printed dipole array with 0&thetadeg; direction, the far field component in &phi; direction and the theta scan angles.]] </td><td> [[Image:Period8.png|thumb|360px|Radiation pattern of a beam-steered 8×8 finite-sized periodic printed dipole array with 45&quotdeg;Total&quot; far fieldphi and theta scan angles. ]] </td></tr></table>

[[Image:MORE.png|40px]] Click here to learn more about the theory of '''[[Data_Visualization_and_Processing#Far-Field_Observables | Far Field Computations]]'''=== Exciting Periodic Structures Using Plane Waves in EM.Picasso ===

When a periodic planar structure is excited using a plane wave source, it acts as a periodic surface that reflects or transmits the incident wave. [[Image:MOREEM.png|40pxPicasso ]] Click here to learn more about calculates the reflection and transmission coefficients of periodic planar structures. If you run a single-frequency plane wave simulation, the reflection and transmission coefficients are reported in the Output Window at the end of the simulation. Note that these periodic characteristics depend on the polarization of the incident plane wave. You set the polarization (TMz or TEz) in the '''[[Data_Visualization_and_Processing#Visualizing_3D_Radiation_Patterns | Visualizing 3D Radiation Patterns]]Plane Wave Dialog'''when defining your excitation source. In this dialog you also set the values of the incident '''Theta''' and '''Phi''' angles. At the end of the planar MoM simulation of a periodic structure with plane wave excitation, the reflection and transmission coefficients of the structure are calculated and saved into two complex data files called &quot;reflection.CPX&quot; and &quot;transmission.CPX&quot;.

{{Note|In the absence of any finite traces or embedded objects in the project workspace, [[Image:MOREEM.png|40px]] Click here to learn more about '''[[Data_Visualization_and_Processing#2D_Radiation_and_RCS_Graphs | Plotting 2D Radiation GraphsPicasso]]'''computes the reflection and transmission coefficients of the layered background structure of your project.}}

<table><tr><td>[[FileImage:PMOM119PMOM102.png|800pxthumb|580px|A periodic planar layered structure with slot traces excited by a normally incident plane wave source.]]</td></tr></table>

Figure: 3D polar radiation pattern plot of === Running a microstrip-fed patch antenna.Periodic MoM Analysis ===

You run a periodic MoM analysis just like an aperiodic MoM simulation from [[File:PMOM120EM.png|800pxPicasso]]'s Run Dialog. Here, too, you can run a single-frequency analysis or a uniform or adaptive frequency sweep, or a parametric sweep, etc. Similar to the aperiodic structures, you can define several observables for your project. If you open the Planar MoM Engine Settings dialog, you will see a section titled "Infinite Periodic Simulation". In this section, you can set the number of Floquet modes that will be computed in the periodic Green's function summations. By default, the numbers of Floquet modes along the X and Y directions are both equal to 25, meaning that a total of 2500 Floquet terms will be computed for each periodic MoM simulation.

Figure: 3D vectorial (cone) radiation pattern plot of a microstrip-fed patch antenna.<table><tr><td>The 2D radiation pattern graphs can be plotted from [[EMImage:PMOM98.Cube]]'s '''Data Manager'''. A total png|thumb|600px|Changing the number of eight 2D radiation pattern graphs are available: 4 polar and 4 Cartesian graphs for Floquet modes from the XY, YZ, ZX and user defined plane cutsPlanar MoM Engine Settings dialog.]]</td></tr></table>

=== 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.

[[File:PMOM124EM.png|thumb|300px|Planar ModuleCube]]'s Radar Cross Section dialogPlanar Modules also allows you to run an adaptive frequency sweep of periodic surfaces excited by a plane wave source. In this case, the planar MoM engine calculates the reflection and transmission coefficients of the periodic surface. Note that you can conceptually consider a periodic surface as a two-port network, where Port 1 is the top half-space and Port 2 is the bottom half-space. In that case, the reflection coefficient R is equivalent to S₁₁ parameter, while the transmission coefficient T is equivalent to S₂₁ parameter. This is, of course, the case when the periodic surface is illuminated by the plane wave source from the top half-space, corresponding to 90°&lt; &theta; = 180°. You can also illuminate the periodic surface by the plane wave source from the bottom half-space, corresponding to 0° = &theta; &lt; 90°. In this case, the reflection coefficient R and transmission coefficient T are equivalent to S₂₂ and S₁₂ parameters, respectively. Having these interpretations in mind, [[EM.Cube]]enables the &quot;'''Adaptive Frequency Sweep'''&quot; option of the '''Frequency Settings Dialog''' when your planar structure has a periodic domain together with a plane wave source.

When a planar structure is excited by a plane wave source, the calculated far field data indeed represent the scattered fields of that planar structure. [[EM.Picasso]] can also calculate the radar cross section (RCS) of a planar target. Note that in this case the RCS is defined for a finite<!-sized target in the presence of an infinite background structure. The scattered &theta; and &phi; components of the far-zone electric field are indeed what you see in the 3D far field visualization of radiation (scattering) patterns. Instead of radiation or scattering patterns, you can instruct [[EM.Picasso]] to plot 3D visualizations of &sigma;_&theta;, &sigma;_&phi; and the total RCS. To do so, you must define an RCS observable instead of a radiation pattern by following these steps:=== Modeling Finite-Sized Periodic Arrays ===

* Right click on the '''Far Fields''' item in the '''Observables''' section of the Navigation Tree and select '''Insert New RCS[[Image:Info_icon...''' png|40px]] Click here to open the Radar Cross Section Dialog.* The resolution of RCS calculation is specified by learn about '''Angle Increment[[Modeling Finite-Sized Periodic Arrays Using NCCBF Technique]]''' expressed in degrees. By default, the &theta; and &phi; angles are incremented by 5 degrees.* At the end of a planar MoM simulation, besides calculating the RCS data over the entire (spherical) 3D space, a number of 2D RCS graphs are also generated. These are RCS cuts at certain planes, which include the three principal XY, YZ and ZX planes plus one additional constant f-cut. This fourth plane cut is at &phi; = 45° by default. You can assign another &phi; angle in degrees in the box labeled '''Non-Principal Phi Plane'''.>

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