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[[Image:Splash-static.jpg|right|720px]]<strong><font color="#2603c4" size="4">Electrostatic, Magnetostatic & Thermal Solvers For DC And Low Frequency Simulations</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= EM.Tempo]] [[image:prop-ico.png | link=EM.Terrano]] [[image:planar-ico.png | link=EM.Picasso]] [[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.Ferma_Documentation | EM.Ferma Primer Tutorial Gateway]]''' [[Image:Back_icon.png|30px]] '''[[EM.Cube | Back to EM.Cube Main Page]]'''==Product Overview==

[[EM.Ferma]] is [[EM.Cube]]'s a 3D static solver. It features two distinct electrostatic and magnetostatic simulation engines and a steady-state thermal simulation engine that can be used to solve a variety of static and low-frequency electromagnetic and thermal problems. Both The thermal solver includes both conduction and convection heat transfer mechanisms. All the three simulation engines are based on finite difference solutions of Poisson's equation for electric and magnetic potentialsand temperature.

With EM.Ferma, you can explore the electric fields due to volume charge distributions or fixed-potential perfect conductors, and magnetic fields due to wire or volume current sources and permanent magnets. Your structure may include dielectric or magnetic (permeable) material blocks. Using the thermal simulator, you can solve for the steady-state temperature distribution of structures that include perfect thermal conductors, insulators and volume heat sources. You can also use EM.Ferma's 2D quasi-static mode to compute the characteristic impedance (Z0) and effective permittivity of transmission line structures with complex cross section profiles.

=== Static Modeling [[Image:Info_icon.png|30px]] Click here to learn more about the '''[[Electrostatic & Magnetostatic Field Analysis | Theory of Electrostatic and Magnetostatic Methods ===]]'''.

[[Image:MOREInfo_icon.png|40px30px]] Click here to learn more about the theory of '''[[Electrostatic and Magnetostatic Steady-State_Thermal_Analysis | Theory of Steady-State Heat Transfer Methods]]'''.

== Defining the Physical Structure in EM.Ferma ==<table><tr><td>[[Image:Static7Magnet lines1.png|thumb|270pxleft| EM.Ferma's Navigation Tree.]]The simplest static problems involve a charge source in the free space that produces an electric field, or a current source in the free space that produces a 400px| Vector plot of magnetic field. In such cases, the only applicable boundary conditions are defined at the computational domain boundary. As soon as you introduce a dielectric object next to a charge source or a magnetic (permeable) material next to a current source, you have to deal with a complex boundary value problem. In other words, you need to solve the electric or magnetic Poisson's equation subject to the domain boundary conditions as well as material interface boundary conditions. EM.Ferma used the Finite Difference technique for numerical solution of your static boundary value problem.  === A Note on Material and Source Types distribution in EM.Ferma === In [[EM.Cube]]'s other modules, material types are specified under the "Physical Structure" section of the Navigation Tree, and sources are organized under a separate "Sources" sectioncylindrical permanent magnet. In those modules, the physical structure and its various material types typically represent all the CAD objects you draw in your project. Sources are virtual entities that might be associated with certain physical objects and provide the excitation of your boundary value problem.  In [[EM.Cube]]'s Static Module, materials and sources are all listed under the "Physical Structure" section of the Navigation Tree, and there is no separate "Sources" section. For example, you can define default zero-potential perfect electric conductors (PEC) in your project to model metal objects. You can also define fixed-potential PEC objects with a nonzero voltage, which can effectively act as a voltage source for your boundary value problem. In this case, you will solve the Lapalce equation subject to the specified nonzero potential boundary values. Both types of PEC objects are defined from the same PEC node of the Navigation Tree by assigning different voltage values. Charge and current sources are defined as CAD objects that you must draw in the project workspace.</td>=== Fixed-Potential PEC Objects=== A perfect electric conductor (PEC) is a material with &epsilon;<sub/tr>r</subtable> = 1 and &sigma; = &infin;. Under the static condition, every point on a PEC object has the same electric potential. By default, this is a zero potential, assuming the PEC object is "grounded". You can define a nonzero voltage value for a PEC group. In that case, the PEC object is effective turned into a voltage source. For example, tow parallel PEC plates, one with a zero potential and the other with a nonzero potential represent a simple air-filled capacitor.  To add a new Fixed-Potential PEC group to a project, right-click on "Fixed-Potential PEC Objects" on the Navigation Tree, and select "Insert New PEC..." From the PEC dialog, you can change the default red color and set a value for the "Voltage" in Volts.

{{Note| You can define any solid or surface object === EM.Ferma as a fixed-potential PEC objectthe Static Module of EM.}} Cube ===

In electromagnetic analysis, a general dielectric material is represented by four constitutive material [[parametersImage:Info_icon.png|30px]]: relative permittivity &epsilon;_r, relative permeability &mu;_r, electric conductivity &sigma; and magnetic conductivity &sigma;_mClick here to learn more about '''[[Getting_Started_with_EM. In Cube | EM.Ferma, you can define dielectric materials for electrostatic analysis and magnetic (permeable) materials for magnetostatic analysis from the same section of the Navigation Tree titled "Dielectric/Magnetic Materials". For a dielectric material, you specify the relative permittivity &epsilon;_r and electric conductivity &sigma;. For a magnetic material, you specify the relative permeability &mu;_rCube Modeling Environment]]'''.

To add a new dielectric or magnetic material group to a project, right-click on "Dielectric/Magnetic Materials" on the Navigation Tree, and select "Insert New Dielectric..." From the Dielectric Dialog, you can change the default green color of a material group or set the values === Advantages & Limitations of the material [[parameters]]EM. Ferma's Static Simulator ===

{{Note| You can define EM.Ferma computes the electric and magnetic fields independent of each other based on electrostatic and magnetostatic approximations, respectively. As a result, any solid object as "electromagnetic" coupling effects or wave retardation effects are ignored in the simulation process. In exchange, static or quasi-static solutions are computationally much more efficient than the full-wave solutions of Maxwell's equations. Therefore, for low-frequency electromagnetic modeling problems or for simulation of sub-wavelength devices, EM.Ferma offers a dielectric faster alternative to [[EM.Cube]]'s full-wave modules like [[EM.Tempo]], [[EM.Picasso]] or magnetic material object[[EM.Libera]]. EM.Ferma currently provides a fixed-cell brick volume mesh generator. To model highly irregular geometries or curved objects, you may have to use very small cell sizes, which may lead to a large computational problem.}}

<td> [[Image:Static1Ferma L8 Fig title.png|thumb|270pxleft| EM.Ferma's PEC dialog400px| Vector plot of electric field distribution in a coplanar waveguide (CPW) transmission line.]] </td><td> [[Image:Static2.png|thumb|270px| EM.Ferma's Dielectric/Magnetic Material dialog.]] </td>

==EM.Ferma Features at a Glance = Volume Charge Sources ===

To add a new volume current source group to a project, right<ul> <li> Fixed-click on "potential PEC for maintaining equi-potential metal objects (Electrostatics)</li> <li> Volume Currents" on the Navigation Tree, and select "Insert New Current Source..." From the charge sources (Electrostatics)</li> <li> Volume Current Source Dialog, you can change the default brown color of the source group or set the values of the Current Density magnitude and unit direction vector components. The default direction vector is zcurrent sources (Magnetostatics)</li> <li> Wire current sources (Magnetostatics)</li> <li> Permanent magnets (Magnetostatics)</li> <li> Fixed-directed. temperature PTC for maintaining iso-thermal objects (Thermal)</li> <li> Volume heat sources (Thermal)</li></ul>

=== Wire Current Sources Mesh generation ===

EM.Ferma allows you to define idealized wire current sources. You can use this source type to model filament currents or coils. Wire currents are defined using only line and polyline objects. You also need to define a current value I in Amperes and a wire radius r in the project units. The line or polyline object is then approximated as a volume current with a current density of J = I/(<ul> <li> Fixed-size brick&pinbsp;rcells<sup/li>2</supul>) flowing along the line or polyline side's direction. All the wire current sources belonging to the same group have the same color, same current value and same wire radius. The direction of the current can be reversed in wire current sources.

{{Note| If you draw [[Curve Objects|curve objects]] under a wire current group, they will be permanently converted to polyline objects before running <ul> <li> Finite difference solution of Laplace and Poisson equations for the simulation engine.}} electric scalar potential with Dirichlet and Neumann domain boundary conditions&nbsp;</li> <li> Finite difference solution of Laplace and Poisson equations for the magnetic vector potential with Dirichlet domain boundary conditions&nbsp;</li> <li> Calculation of electric scalar potential and electric field</li> <li> Calculation of magnetic vector potential and magnetic field</li> <li> Calculation of electric flux over user defined flux boxes and capacitance</li> <li> Calculation of magnetic flux over user defined flux surfaces and inductance</li> <li> Calculation of electric and magnetic energies, Ohmic power loss and resistance</li> <li> Parametric sweep with variable object properties or source parameters</li></ul>

<math> \mathbf{B(r)} = {\mu} (\mathbf{H(r)} + \mathbf{M(r)} ) </math>== Data Generation &amp; Visualization ===

where <b>M(r)</b> is == Building the magnetization vector. In SI units system, the magnetic field <b>H</b> and magnetization <b>M</b> both have the same units of A/mPhysical Structure in EM. Ferma ==

It can be shown that for magnetostatic analysis, the effect === Variety of the permanent magnetization can be modeled as an equivalent volume current source:Physical Objects in EM.Ferma ===

<math> \mathbf{J_{eq}The simplest static problems involve a charge source in the free space that produces an electric field, or a current source in the free space that produces a magnetic field. In such cases, the only applicable boundary conditions are defined at the boundary of the computational domain. As soon as you introduce a dielectric object next to a charge source or a magnetic (rpermeable)} = \nabla \times \mathbf{Mmaterial next to a current source, you have to deal with a complex boundary value problem. In other words, you need to solve the electric or magnetic Poisson equation subject to the domain boundary conditions as well as material interface boundary conditions. The simplest thermal problem involves one or more thermal plates held at fixed temperatures. Once you introduce material blocks, you have to enforce conductive and convective boundary conditions at the interface between different materials and air. EM.Ferma uses the Finite Difference (rFD)} technique to find a numerical solution of your static boundary value problem. </math>

If the magnetization vector is uniform {| class="wikitable"|-! scope="col"| Icon! scope="col"| Physical Object Type! scope="col"| Applications! scope="col"| Geometric Object Types Allowed! scope="col"| Notes & Restrictions|-| style="width:30px;" | [[File:pec_group_icon.png]]| style="width:200px;" | [[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Fixed-Potential PEC |Fixed-Potential Perfect Electric Conductor (PEC)]]| style="width:300px;" | Modeling perfect metals with a fixed voltage| style="width:100px;" | Solid and constant inside surface objects| style="width:250px;" | Can be considered an electric source if the volume, then its curl fixed voltage is zero everywhere inside the volume except on its boundary surfacenonzero |-| style="width:30px;" | [[File:diel_group_icon. In this casepng]]| style="width:200px;" | [[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Dielectric Material |Dielectric/Magnetic Material]]| style="width:300px;" | Modeling any homogeneous or inhomogeneous material| style="width:100px;" | Solid objects| style="width:250px;" | non-source material|-| style="width:30px;" | [[File:aniso_group_icon.png]]| style="width:200px;" | [[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Volume Charge |Volume Charge]]| style="width:300px;" | Modeling volume charge sources with a fixed charge density or an expression in the permanent magnetic can be effectively modeled by global coordinates (x,y,z) | style="width:100px;" | Solid objects| style="width:250px;" | Acts as an equivalent surface electric source|-| style="width:30px;" | [[File:voxel_group_icon.png]]| style="width:200px;" | [[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Volume Current |Volume Current]]| style="width:300px;" | Modeling volume current sources with a fixed volume current density on vector or expressions in the surface global coordinates (x,y,z) | style="width:100px;" | Solid objects| style="width:250px;" | Acts as a magnetic source|-| style="width:30px;" | [[File:pmc_group_icon.png]]| style="width:200px;" | [[Glossary of the EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Permanent Magnet |Permanent Magnet]]| style="width:300px;" | Modeling permanent magnet sources with a fixed magnetization vector or expressions in the global coordinates (x,y,z) | style="width:100px;" | Solid objects| style="width:250px;" | Acts as a magnetic source|-| style="width:30px;" | [[File:thin_group_icon.png]]| style="width:200px;" | [[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Wire Current |Wire Current]]| style="width:300px;" | Modeling wire current sources| style="width:100px;" | Line and polyline objects| style="width:250px;" | Acts as a magnetic source|-| style="width:30px;" | [[File:pec_group_icon.png]]| style="width:200px;" | [[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Fixed-Temperature PTC |Fixed-Temperature Perfect Thermal Conductor (PTC)]]| style="width:300px;" | Modeling isothermal surfaces with a fixed temperature| style="width:100px;" | Solid and surface objects| style="width:250px;" | Can be considered a thermal source if the fixed temperature is different than the ambient temperature (shares the same navigation tree node as PEC object)|-| style="width:30px;" | [[File:diel_group_icon.png]]| style="width:200px;" | [[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Dielectric Material |Insulator Material]]| style="width:300px;" | Modeling any homogeneous or inhomogeneous material| style="width:100px;" | Solid objects| style="width:250px;" | non-source material (shares the same navigation tree node as dielectric material)|-| style="width:30px;" | [[File:aniso_group_icon.png]]| style="width:200px;" | [[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types#Volume Heat Source |Volume Heat Source]]| style="width:300px;" | Modeling volume heat sources with a fixed heat density or an expression in the global coordinates (x,y,z) | style="width:100px;" | Solid objects| style="width:250px;" | Acts as a thermal source (shares the same navigation tree node as volume charge)|-| style="width:30px;" | [[File:Virt_group_icon.png]]| style="width:200px;" | [[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:100px;" | All types of objects| style="width: 250px;" | None|}

<math> \mathbf{J_{s,eq}(r)} = \mathbf{M(r)} \times \hat{\mathbf{n}} </math>== Grouping Objects by Material or Source Type ===

where <math> \hat{\mathbf{n}} </math> is All the unit outward normal vector at geometric objects in the surface of the permanent magnet project workspace are organized together into objectgroups which share the same properties including color and electric or magnetic parameters. Note It is recommended that you first create object groups, and then draw new objects under the volume active group. To create a new object group, right-click on its category name in the "Physical Structure" section of the permanent magnet still acts as navigation tree and select one of the "Insert New Group..." items from the contextual menu. However, if you start a permeable material in new EM.Ferma project from scratch, and start drawing a new object without having previously defined any object groups, a new default "Fixed-Potential PEC" object group with a zero voltage is created and added to the magnetostatic analysisnavigation tree to hold your new geometric object.

To add a new permanent magnet source group It is important to note that there is a project, rightone-click on "Permanent Magnets" on the Navigation Tree, and select "Insert New Permanent Magnet Source..." From the Permanent Magnet Source Dialog, you can change the default purple color of the source group or set the values of the relative permeability, Magnetization magnitude and unit direction vector components. The default direction vector is zto-directed. one correspondence between electrostatic and thermal simulation entities:

{| class="wikitable"|-! scope= Computational Domain and Discretization"col"| Electrostatic Item! scope="col"| Corresponding Thermal Item|-| style="width:200px;" | Electric Scalar Potential| style="width:200px;" | Temperature|-| style="width:200px;" | Electric Field| style="width:200px;" | Heat Flux Density|-| style="width:200px;" | Perfect Electric Conductor| style="width:200px;" | Perfect Thermal Conductor|-| style="width:200px;" | Dielectric Material| style="width:200px;" | Insulator Material|-| style="width:200px;" | Volume Charge| style="width:200px;" | Volume Heat Source|}

[[Image:fermbc{{Note|Electrostatic and thermal solvers share the same material categories on the navigation tree.png|thumb|250px|Boundary Condition Dialog]]This means that PEC objects are treated as PTC objects, dielectric objects are treated as insulator objects and volume charges are treated as volume heat sources when the thermal solver is enabled.}}

In EM[[Image:Info_icon.Ferma, the Poisson or Laplace equations are solved subject png|30px]] Click here to boundary conditions using the Finite Difference technique. As a result, you need to specify a finite computational domain and then specify the domain boundary conditions. EM.Fermalearn more about 's computational domain defines where the domain boundary condition will be specified. A default domain box is always placed ''[[Building Geometrical Constructions in the project workspace as soon as you draw your first object. The domain can be seen as a blue cubic wireframe that surrounds all of the CAD objects in the project workspaceCubeCAD#Transferring Objects Among Different Groups or Modules | Moving Objects among Different Groups]]'''.

<table><tr><td> [[Image:qsource2STAT MAN1.png|thumb|400pxleft|The blue wireframe around the CAD objects defines the extents of the computational domain480px|EM. The specified boundary conditions are applied on the domain wallsFerma's navigation tree. ]]</td></tr></table>

To modify the domain settings, click the Domain button of the Simulate Toolbar or right-click === A Note on "3D Static Domain" entry in the Navigation Tree and select "Domain Settings..." from the contextual menu. In the Domain Settings Dialog, the computational domain can be defined in two different ways: Default and Custom. The default type places an enclosing box with a specified offset from the largest bounding box of your project's CAD objects. The default offset value is 20 project units, but you can change this value arbitrarily. The custom type defines a fixed domain box by specifying the coordinates of its two opposite corners labeled Min Material and Max Source Types in the world coordinate systemEM. Ferma ===

In EM.Ferma allows you to specify the electric potential boundary conditions on the domain box. Two options , materials and sources are available. The Dirichlet boundary condition is the default option all lumped together and is specified as a fixed potential value on listed under the surface "Physical Structure" section of the domain wallsnavigation tree. By defaultIn other words, this value there is 0 Voltsno separate "Sources" section. The Neumann boundary condition specifies the normal derivative of the For example, you can define default zero-potential perfect electric scalar conductors (PEC) in your project to model metal objects. You can also define fixed-potential on PEC objects with a nonzero voltage, which can effectively act as a voltage source for your boundary value problem. In this case, you will solve the surface of the domain walls. This is equivalent Lapalce equation subject to the normal electric field component on the domain walls and its value is specified in V/m. The magnetostatic simulation engine always assumes Dirichlet domain nonzero potential boundary conditions and sets the values . Both types of PEC objects are defined from the magnetic vector potential to zero on all same PEC node of the domain wallsnavigation tree by assigning different voltage values. To modify the boundary conditions, right-click on "Boundary Conditions" in the Navigation TreeCharge and current sources are also defined as geometric objects, and select "Boundary Conditions..." from the contextual menu you have to open draw them in the Boundary Conditions Dialogproject workspace just like other material objects.

===The Static Mesh=EM.Ferma's Computational Domain ==

===The Finite Difference technique discretizes the computational domain using a 3D rectangular grid. EM.Ferma generates a fixed-cell mesh. This means that the extents of the mesh cells along the principal axes are fixed: &Delta;x, &Delta;y, &Delta;z. By default, the mesh cell size is set to one unit project along all the three directions (with &Delta;x Domain Box== &Delta;y = &Delta;z). To modify the cell size, click the Mesh Settings button of the Simulate Toolbar or right-click on "Static Mesh" in the Navigation Tree, and select "Mesh Settings..." from the contextual menu to open the Mesh Settings Dialog. {{Note|To obtain accurate results, it is highly recommended to use a square mesh as much as possible.}}

== Running Static Simulations in In EM.Ferma ==, the Poisson or Laplace equations are solved subject to boundary conditions using the Finite Difference technique. As a result, you need to specify a finite computational domain and then specify the domain boundary conditions. EM.Ferma's computational domain defines where the domain boundary condition will be specified. A default domain box is always placed in the project workspace as soon as you draw your first object. The domain can be seen as a blue cubic wireframe that surrounds all of the CAD objects in the project workspace.

<table><tr> <td> [[Image:Qsource5.png|thumb|left|480px|EM.Ferma has two independent but functionally similar static simulation engines: Electrostatic and Magnetostatic. The electrostatic engine solves the electric form of Poisson's equation for electric scalar potential subject to electric field boundary conditions, in the presence of electric sources (volume charges and fixed-potential PEC blocks) and dielectric material media. The magnetostatic engine solves the magnetic form of Poisson's equation for magnetic vector potential subject to magnetic field boundary conditions, in the presence of magnetic sources (wire and volume currents and permanent magnetic blocks) and magnetic material mediaDomain Settings dialog.]] </td></tr></table>

To run a static simulation, first *EM.Ferma allows you have to open specify the Run Dialogelectric potential boundary conditions on the domain box. This Two options are available. The Dirichlet boundary condition is done by clicking the "Run" button of default option and is specified as a fixed potential value on the Simulate Toolbar, or by selecting the "Run" item surface of the Simulate Menu, or simply using the keyboard shortcut "Ctrl+R"domain walls. The only available simulation engine By default, this value is "Static"0 Volts. Clicking The Neumann boundary condition specifies the Run button normal derivative of this dialog starts a static analysis. A separate window pops up which reports the progress electric scalar potential on the surface of the current simulationdomain walls. This is equivalent to a constant normal electric field component on the domain walls and its value is specified in V/m.

*EM.Ferma currently offers three different simulation modes: Analysis, Parametric Sweep and [[Optimization]]provides two options for thermal boundary conditions on the domain box. An "Analysis" The Dirichlet boundary condition is the default option and is specified as a single-shot finite difference solution fixed temperature value on the surface of your static structurethe domain walls. By default, this value is 0&deg;C. The structure Neumann boundary condition specifies the normal derivative of the temperature on the surface of the domain walls. This is first discretized using equivalent to a fixed-cell mesh and constant heat flux passing through the Poisson equation domain walls and its value is solved numerically everywhere specified in your computational domainW/m². The field and potential values at each mesh node are computed A zero heat flux means a perfectly insulated domain box and is known as the specified observables are written into data filesadiabatic boundary condition.

In a To modify the boundary conditions, right-click on "Parametric SweepBoundary Conditions"in the navigation tree, one ore more [[variables]] are varied at and select "Boundary Conditions..." from the specified steps(s)contextual menu to open the Boundary Conditions Dialog. This means that When you must first define one or more [[variables]] switch from the electrostatic-magnetostatic solver to the thermal solver in your projectsEM. [[Variables]] can be associated with CAD object properties like dimensionsFerma's Run Simulation dialog, coordinatesit automatically checks the box labeled '''Treat as a Thermal Structure''' in the Boundary Conditions dialog. Conversely, rotation anglesif you check this box in the Boundary Conditions dialog, etcthe solver type is set to the thermal solver in the Simulation Run dialog. or with material properties or source properties. For each single variable sample or each combination In the "Global Thermal Properties" section of variable samplesthe Boundary Conditions dialog, first all you can set the associated CAD object propertiesvalues of the ambient temperature in &deg;C, material properties or source properties are updated thermal conductivity of the environment in W/(m.K) and the project workspaceconvective coefficient in W/(m². Then is a finite difference solution K). You can also disable the enforcement of your updated static structure is computed and parametric sweep proceeds to the next variable sample or combinationconvective boundary condition on the surface of solid insulator objects.

The <table><tr> <td> [[optimization]] mode requires definition of one or more objectives based on the standard output quantitiesImage:fermbc. At the present time, the [[optimization]] mode is only available for the 2D Quasi-Static Mode of the png|thumb|left|480px|EM.Ferma, which will be discussed separately later's Boundary Conditions dialog. ]] </td></tr></table>

===Static EM.Ferma's Simulation Engine Settings=Data & Observables ==

EM.Ferma currently uses a single iterative linear system solver based on At the stabilized Bi-Conjugate Gradient (BiCG) method to solve end of an electrostatic simulation, the matrix equations which result from electric field vector and electric scalar potential values are computed at all the discretization mesh grid points of Poisson's equationthe entire computational domain. You can specify some numerical [[parameters]] related to At the Bi-CG solver. To do thatend of an magnetostatic simulation, you need to open the Simulation Engine Settings Dialog by clicking magnetic field vector and magnetic vector potential values are computed at all the "Settings" button located next to the "Select Engine" drop-down listgrid nodes. From this dialog you can set At the maximum number end of BiCG iterations, which has a default value of 10,000. You can also set a value for "Convergence Error". The default value for electrostatic analysis is 0.001. For magnetostatic analysisthermal simulation, the specified value temperature and heat flux vector are computed at all the mesh grid points of convergence error is reduced by a factor 1000 automatically. Therefore, the default convergence error in this case is 1e-6entire computational domain.

{{Note|The value of convergence error affect Besides the accuracy electric and magnetic fields and temperature, EM.Ferma can compute a number of your simulation resultsfield integral quantities such as voltage, current, flux, energy, etc. For most practical scenariosThe field components, the default potential values and field integrals are adequate. You written into output data files and can reduce be visualized on the convergence error for better accuracy at screen or graphed in Data Manager only if you define a field sensor or a field integral observable. In the expense absence of longer computation timeany observable defined in the navigation tree, the static simulation will be carried out and completed, but no output simulation data will be generated.}}

At the end {| class="wikitable"|-! scope="col"| Icon! scope="col"| Simulation Data Type! scope="col"| Observable Type! scope="col"| Applications|-| style="width:30px;" | [[File:fieldsensor_icon.png]]| style="width:150px;" | Near-Field Distribution Maps| style="width:150px;" | [[Glossary of an electrostatic simulation, the EM.Cube's Simulation Observables & Graph Types#Near-Field_Sensor_Observable |Near-Field Sensor]] | style="width:450px;" | Computing electric field and magnetic field components, electric scalar potential values are computed at all the mesh grid points and magnitude of magnetic vector potential on a planar cross section of the entire computational domain|-| style="width:30px;" | [[File:fieldsensor_icon. At the end png]]| style="width:150px;" | Electric and Magnetic Energy and Dissipated Power Density Maps | style="width:150px;" | [[Glossary of an magnetostatic simulation, the magnetic field EM.Cube's Simulation Observables & Graph Types#Near-Field_Sensor_Observable |Near-Field Sensor]] | style="width:450px;" | Computing electric and magnetic vector potential values are computed at all energy densities and dissipated power density on a planar cross section of the grid nodescomputational domain |-| style="width:30px;" | [[File:fieldsensor_icon. The field png]]| style="width:150px;" | Temperature and potential values are written into output data files Heat Flux Distribution Maps| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Near-Field_Sensor_Observable |Near-Field Sensor]] | style="width:450px;" | Computing temperature and can be visualized heat flux components on the screen only if you define a field sensor observable. In the absence planar cross section of a defined observable, the static simulation will be carried out and completed, but to action will take place. computational domain |-| style="width:30px;" | [[File:fieldsensor_icon.png]]| style="width:150px;" | Thermal Energy Density Maps | style= Defining "width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Near-Field_Sensor_Observable |Near-Field Sensors Sensor]] | style="width:450px;" | Computing thermal energy density on a planar cross section of the computational domain |-| style="width:30px;" | [[File:field_integ_icon.png]]| style="width:150px;" | Field Integral Quantities| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Static_Field_Integral_Observable | Static Field Integral]] | style="width:450px;" | Computing line, surface and volume integrals of the electric and magnetic fields and heat flux |}

Just like other Click on each category to learn more details about it in the [[Glossary of EM.Cube|EM.CUBE's Simulation Observables & Graph Types]] Modules, EM.Ferma has a Field Sensor observable, which plots 3D visualizations of electric and magnetic field components on a specified plane. However, unlike the other modules, EM.Ferma field sensors have two additional plots for electric scalar potential and magnitude of the magnetic vector potential. These are called the "EPot" and "HPot" nodes on the Navigation Tree. To define a Field Sensor, right-click on "Field Sensors" in the Navigation Tree and select "Insert New Observable..." from the contextual menu. The Field Sensor dialog allows the user to select the direction of the sensor (X, Y, Z), visualization type, and whether E-field output or H-field output will be shown during a sweep analysis.

The E-fields and H-fields are computed at each mesh node within the specified 2D Field Sensor plane. In other words, the resolution of the Field Sensor is controlled by the mesh resolution.<table><tr> === Defining Field Integrals === It is often needed to compute integrals of the electric or magnetic fields to define other related quantities<td> [[Image:Ferma L1 Fig15. The following table shows some of widely used png|thumb|left|640px|Electric field integrals in electrostatics and magnetostatics. In EM.Ferma, you can define a path integral along a line segment that is parallel to one distribution of the three principal axes, or a loop integral spherical charge on a rectangle that is parallel to one of the principal planeshorizontal field sensor plane. You can also define flux planes or flux boxes]] </td></tr> <tr> <td> [[Image:Ferma L1 Fig16. All this is done from the same Field Integral Dialog. To define png|thumb|left|640px|Electric scalar potential distribution of a Field Integral, right-click spherical charge on "Field Integrals" in the Navigation Tree and select "Insert New Observablea horizontal field sensor plane..." from the contextual menu. The Integral Type drop-down list gives nine options as listed in the ]] </td></tr></table below:>

| <math> W_E = \frac{1}{2} \int \int \int_V \mathbf{D(r)} . \mathbf{E(r)} dv = \frac{1}{2} \int \int \int_V \epsilon \vert \mathbf{E(r)} \vert ^2 dv </math>

| <math> W_H = \frac{1}{2} \int\int\int_V \mathbf{B(r)} . \mathbf{H(r)} dv = \frac{1}{2} \int\int\int_V \mu \vert \mathbf{H(r)} \vert ^2 dv </math>

| <math> P_{ohmic} = \int\int\int_V \mathbf{J(r)} . \mathbf{E(r)} dv = \int\int\int_V \sigma \vert \mathbf{E(r)} \vert ^2 dv </math>

| <math> C = Q/V = \Phi_E/V = \int\int_{S_o} \epsilon \mathbf{E(r)} . \mathbf{ds} / \int_C \mathbf{E(r)} . \mathbf{dl} </math>| capacitance.DAT|-! scope="row"| Capacitance (Alternative)| <math> C = 2W_E/V^2 = 2 \int \int \int_V \epsilon \vert \mathbf{E(r)} \vert ^2 dv / \left( \int_C \mathbf{E(r)} . \mathbf{dl} \right)^2</math>

| inductancemutual_inductance.DAT|-! scope="row"| Resistance| <math> R = V/I_{cond} = - \int_C \mathbf{E(r)} . \mathbf{dl} / \int\int_S \sigma \mathbf{E(r)} . \mathbf{ds} </math>| resistance.DAT|-! scope="row"| Resistance (Alternative 1)| <math> R = V^2/P_{ohmic} = \left( \int_C \mathbf{E(r)} . \mathbf{dl} \right)^2 / \int\int\int_V \sigma \vert \mathbf{E(r)} \vert ^2 dv </math>| resistance.DAT|-! scope="row"| Resistance (Alternative 2)| <math> R = P_{ohmic}/I_{cond}^2 = \int\int\int_V \sigma \vert \mathbf{E(r)} \vert ^2 dv / \left( \int\int_S \sigma \mathbf{E(r)} . \mathbf{ds} \right)^2</math>| resistance.DAT|-! scope="row"| Thermal Flux| <math> \Phi_T = \int\int_{S_o} \mathbf{q(r)} . \mathbf{ds} </math>| flux_T.DAT|-! scope="row"| Thermal Energy| <math> W_T = Q = \int \int \int_V \rho_V c_p \left( T\mathbf{(r)} - T_{env} \right) dv </math>| energy_T.DAT

In == Discretizing the above table, C represents an open curve (path), C_o represents a closed curve (loop), S represents an open surface like a plane, S_o represents a closed surface like a box, and V represents a volumePhysical Structure in EM. In the case of mutual inductance, S' represents an open surface or plane passing through the second (coupled) inductor, and &Phi;'_H represents the magnetic flux linkage due to the magnetic field of the first inductor passing through the second inductor. Ferma ==

===The domain of the field integral is set using the "Integration Box Coordinates" section of the Field Integral dialog. Box domains are specified by the coordinates of two opposite corners. Voltage Path requires a line; therefore, two of the coordinates of the two corners must be identical. Otherwise, an error message will pop up. For example, (0, 0, 0) for Corner 1 and (10, 0, 0) for Corner 2 define a Z-directed line segment. Current Loop requires a rectangle; therefore, one of the coordinates of the two corners must be identical. For example, (0, 0, 0) for Corner 1 and (10, 10, 0) for Corner 2 define a rectangle in the XY plane.  After the completion of a static simulation, the result of the field integrals are written into ".DAT" data files. These files can be accessed using [[EM.Cube]]'s Data Manager. Static Mesh===

 == Modeling Transmission Lines Using EM.Ferma== [[Image:qstatic.png|thumb|300px|Setting up The Finite Difference technique discretizes the computational domain using a Transmission Line simulation3D rectangular grid.]] ===2D Electrostatic Simulation Mode=== EM.Ferma's electrostatic simulation engine features generates a 2D solution mode where fixed-cell mesh. This means that the model is treated as a longitudinally infinite structure in extents of the direction normal to specified "2D Solution Plane". More than one 2D solution plane may be defined. In that casemesh cells along the principal axes are fixed: &Delta;x, &Delta;y, multiple 2D solutions are obtained&Delta;z. A 2D solution plane By default, the mesh cell size is defined based on a "Field Sensor" definition that already exists in set to one unit project along all the projectthree directions (with &Delta;x = &Delta;y = &Delta;z). To explore EM.Ferma's 2D modemodify the cell size, click the Mesh Settings button of the Simulate Toolbar or right-click on "2D Solution PlanesStatic Mesh" in the Navigation Tree , and select "2D Domain Mesh Settings..." from the contextual menu. In the 2D Static Domain dialog, enable the checkbox labeled "Treat Structure as Longitudinally Infinite across Each 2D Plane Specified Below". The user is then able to Add or Edit 2D Solution Plane definitions to open the solution listMesh Settings Dialog. In the Add/Edit 2D Solution Plane dialog, you can choose a name other than the default name and select one of the available field sensor definitions in your project.  At the end of a 2D electrostatic analysis, you can view the electric field and potential results on the respective field sensor planes. It is assumed that your structure is invariant along the direction normal to the 2D solution plane. Therefore, your computed field and potential profiles must be valid at all the planes perpendicular to the specified longitudinal direction.  === 2D Quasi-Static Solution of Transmission Lines === At lower microwave frequencies (f < 10GHz), it is usually possible to perform a 2D electrostatic analysis of a transmission line structure and compute its characteristics impedance Z₀ and effective permittivity &epsilon;_eff. This "quasi-static approach" involves two steps: <ol><li>First, you have remove all the dielectric materials from your structure and replace them with free space (or air). Obtain a 2D electrostatic solution of your "air-filled" transmission line structure and compute its capacitance per unit length C_a.</li><li>Next, obtain a 2D electrostatic solution of your actual transmission line structure with all of its dielectric parts and compute its true capacitance per unit length C.</li></ol> Then effective permittivity of the transmission line structure is then calculated from the equation:

and its characteristic impedance is given by<table><tr> <td> [[Image:Qsource4.png|thumb|350px|EM.Ferma's Mesh Settings dialog.]] </td></tr></table>

<table><mathtr> Z_0 = \eta_0 \sqrt{ \frac{C_a}{C} } <td> [[Image:Qsource2.png|thumb|360px|Geometry of a spherical charge source and the enclosing domain box.]] </td><td> [[Image:Qsource3.png|thumb|360px|Fixed-cel mesh of the spherical charge object.]] </td></tr></mathtable>

where &eta;₀ = 120&pi; &Omega; is the intrinsic impedance of the free space== EM. Ferma's Simulation Modes ===

The guide wavelength {| class="wikitable"|-! scope="col"| Simulation Mode! scope="col"| Usage! scope="col"| Number of your transmission line at Engine Runs! scope="col"| Frequency ! scope="col"| Restrictions|-| style="width:120px;" | [[#Running an Electrostatic or Magnetostatic Analysis | Analysis]]| style="width:270px;" | Simulates the physical structure "As Is"| style="width:100px;" | Single run| style="width:200px;" | N/A| style="width:150px;" | 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:100px;" | Multiple runs| style="width:200px;" | N/A| style="width:150px;" | 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 given frequency f is then calculated fromdesign goal | style="width:100px;" | Multiple runs | style="width:200px;" | N/A| style="width:150px;" | None|}

<math> \lambda_g = \frac{\lambda_0}{\sqrt{\epsilon_{eff}}} = \frac{c}{f\sqrt{\epsilon_{eff}}} </math>= Running an Electrostatic, Magnetostatic or Thermal Analysis ===

and its propagation constant To run a static simulation, first you have to open the Run Dialog. This is given done byclicking the "Run" button of the Simulate Toolbar, or by selecting the "Run" item of the Simulate Menu, or simply using the keyboard shortcut "Ctrl+R". There are two available options for the simulation engine:'''Electrostatic-Magnetostatic Solver''' and '''Steady-State Thermal Solver'''. Clicking the Run button of this dialog starts a static analysis. A separate window pops up which reports the progress of the current simulation.

<mathtable> \beta = k_0\sqrt{\epsilon_{eff}} = \frac{2\pi f}{c}\sqrt{\epsilon_{eff}} <tr> <td> [[Image:Ferma L1 Fig11.png|thumb|left|600px|EM.Ferma's Simulation Run dialog.]] </td></tr></mathtable>

where c An "Analysis" is the speed simplest simulation mode of light EM.Ferma. It is a single-shot finite difference solution of your static problem. The physical structure of your project workspace is first discretized using a fixed-cell mesh and the Poisson equation is solved numerically everywhere in the free spacecomputational domain. The field and potential values at each mesh node are computed, and the specified observables are written into data files. The other available simulation modes, parametric sweep and optimization, involve multiple runs of the static solvers.

EM.Ferma's 2D Quasi-===Static mode automatically performs the two-step process described above and calculates &epsilon;_eff and Z₀. So you don't need to modify your structure in the first step. Simulation Engine Settings===

=== Setting up EM.Ferma offers two different types of linear system solver for solving the matrix equations that result from discretization of Poisson's equation: an iterative solver based on the stabilized Bi-Conjugate Gradient (BiCG) method and a Transmission Line Gauss-Seidel solver. The default solver type is BiCG. You can specify some numerical parameters related to the BiCG solver. To do that, you need to open the Simulation ===Engine Settings Dialog by clicking the "Settings" button located next to the "Select Engine" drop-down list. From this dialog you can set the maximum number of BiCG iterations, which has a default value of 10,000. You can also set a value for "Convergence Error". The default value for electrostatic analysis is 0.001. For magnetostatic analysis, the specified value of convergence error is reduced by a factor 1000 automatically. Therefore, the default convergence error in this case is 10^-6.

To perform a transmission line {{Note|The value of convergence error affect the accuracy of your simulationresults. For most practical scenarios, first draw your structure in the project workspace just like a typical 3D structuredefault values are adequate. Define a "Field Sensor" observable in You can reduce the Navigation Tree so as to capture convergence error for better accuracy at the cross section expense of your structure as your desired transmission line profilelonger computation time. }}

Next, define a "2D Solution Plane" in the Navigation Tree based on your existing field sensor<table><tr> <td> [[Image:Qsource7.png|thumb|left|480px|EM. When defining the 2D plane, check the box labeled "Perform 2D Quasi-Ferma's Static Simulation"Engine Settings dialog. If an analysis is run with this option checked, the characteristic impedance Z]]<sub/td>0</subtr> and effective permittivity &epsilon;<sub>eff</subtable> will be computed for the corresponding 2D Solution Plane. The results are written to two output data files named "solution_plane_Z0.DAT" and "solution_plane_EpsEff.DAT", respectively, where "solution_plane" is the default name of your 2D plane.

Many == The 2D quasiQuasi-static solutions can be obtained in the same analysis,for example, when your design contains many types of [[Transmission Lines|transmission lines]]. At the end of a quasi-static analysis, the electric field components and scalar potential at the selected 2D planes will still be computed and can be visualized. In the case of a parametric sweep, the data files will contain multiple data entries listed against the corresponding variable samples. Such data files can be plotted in EM.Grid. === Optimizing a Transmission Line =Static Simulation Mode==

In an [[optimization]] EM.Ferma's electrostatic simulation, engine features a 2D solution mode where your physical model is treated as a longitudinally infinite structure in the values of one or more [[variables]] are varied over their direction normal to specified ranges, and a design objective is tested at each simulation run"2D Solution Plane". A design objective 2D solution plane is typically defined based on a logical expression "Field Sensor" definition that sets an expression equal to a target valuealready exists in your project. To explore EM.Ferma currently offers two standard outputs: &epsilon;_eff and Z₀. Two possible objectives are "Z₀ == 50" or "sqrt(&epsilon;_eff) == 1.5". To define an objective's 2D mode, right-click on '''2D Solution Planes''' in the "ObjectivesComputational Domain" button section of the Simulate Toolbar, or navigation tree and select '''2D Domain Settings...''' from the "Objectives" item of contextual menu. In the Simulate Menu2D Static Domain dialog, or simply use check the keyboard shortcut checkbox labeled "Ctrl+JReduce the 3D Domain to a 2D Solution Plane". In The first field sensor observable in the Objectives Dialog, you can add new objective or edit navigation tree is used for the definition of the existing objectives2D solution plane.

For a step-by-step demonstration (including transmission line <table><tr> <td> [[optimizationImage:Qsource14.png|thumb|left|450px|The 2D static domain dialog.]]), take a look at this video on our YouTube channel: [http:</td></www.youtube.comtr> </watch?v=Iiu9rQf1QI4 EM.CUBE Microstrip Optimization]table>

{| border="0"<table>|-<tr> | valign="top"|<td> [[FileImage:ScreenCapture1Qsource16.png|thumb|left|350px480px|Classic Example: Two oppositely charged spheresA field sensor and 2D solution plane defined for a microstrip line.]]| valign="top"|</td>[[File:iarray.png|thumb|left|350px|H-Field from array of current loops.]]</tr>|-</table>|}{| border="0"<table>|-| valign="top"|<tr> <td> [[FileImage:ustripQsource17.png|thumb|left|350px480px|Potential near Electric field distribution of the microstrip conductor from a quasistatic simulationline on the 2D solution plane.]]</td>| valign="top"|</tr> <tr> <td> [[FileImage:ustrip2Qsource18.png|thumb|left|350px480px|Electric field near scalar potential distribution of the microstrip conductor from a quasistatic simulation. This Field Sensor's view mode has been set to Vector modeline on the 2D solution plane.]]</td>|-</tr>|}</table>

* [http[Image://en.wikipedia.org/wiki/Electrostatics Wikipedia: ElectrostaticsTutorial_icon.png|30px]* ] '''[http://www[EM.youtubeCube#EM.com/watch?v=Iiu9rQf1QI4 YouTube: Ferma_Documentation | EM.Ferma Optimization Example.Tutorial Gateway]* [http://www.emagtech.com/content/emferma More about EM.Ferma.]'''

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