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[[Image:Splash-static.jpg|right|750px720px]]<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 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.

{{Note|EM.Ferma is the low-frequency '''Static Module''' of '''[[EMImage:Info_icon.Cubepng|30px]]''', a comprehensive, integrated, modular electromagnetic modeling environment. EM.Ferma shares Click here to learn more about the visual interface, 3D parametric CAD modeler, data visualization tools, and many more utilities and features collectively known as '''[[CubeCADElectrostatic & Magnetostatic Field Analysis | Theory of Electrostatic and Magnetostatic Methods]]''' with all of [[EM.Cube]]'s other computational modules.}}

[[Image:Info_icon.png|40px30px]] Click here to learn more about the '''[[Getting_Started_with_EM.CUBE Steady-State_Thermal_Analysis | EM.Cube Modeling EnvironmentTheory of Steady-State Heat Transfer Methods]]'''.

[[Image:Info_icon.png|40px]] Click here to learn more about the basic functionality of '''[[CubeCAD]]'''.<table><tr>=== Static Modeling Methods ===<td>[[Image:Info_iconMagnet lines1.png|40px]] Click here to learn more about the theory of '''[[Electrostatic and Magnetostatic Methods]]'''. == Defining the Physical Structure in EM.Ferma == [[Image:Static7.pngthumb|thumbleft|270px400px| EM.Ferma's Navigation Tree.]]The simplest static problems involve a charge source in the free space that produces an electric Vector plot of magnetic field, or a current source distribution in the free space that produces a magnetic fieldcylindrical permanent magnet. 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 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" section. 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. </td>In EM.Ferma, materials and sources are all listed under the "Physical Structure" section of the navigation tree. In other words, 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, and you have to draw them in the project workspace just like other material objects. === 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, meaning the PEC object is "grounded". You can define a nonzero voltage value for a PEC group. In that case, all the PEC objects belonging to that group are effectively turned into voltage sources. For example, two 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 EM[[Image:Info_icon.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"png|30px]] Click here to learn more about '''[[Getting_Started_with_EM. For a dielectric material, you specify the relative permittivity &epsilon;_r and electric conductivity &sigma;Cube | EM. For a magnetic material, you specify the relative permeability &mu;_rCube Modeling Environment]]'''.

{{Note| Excluding surface and curve CAD objects, you can define any solid CAD object as a dielectric or magnetic material object=== Advantages & Limitations of EM.}} Ferma's Static Simulator ===

EM.Ferma computes the electric and magnetic fields independent of each other based on electrostatic and magnetostatic approximations, respectively. As a result, any "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 faster alternative to [[Image:Info_iconEM.png|40pxCube]] Click here to learn more about '''s full-wave modules like [[Defining Materials in EM.CubeTempo]]''', [[EM.Picasso]] or [[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 CAD 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>

<mathul> \mathbf{B(r)} = {\mu} (\mathbf{H(r)} + \mathbf{M(r)} ) <li> Finite difference solution of Laplace and Poisson equations for the temperature with Dirichlet and Neumann domain boundary conditions&nbsp;</li> <li> Calculation of temperature and heat flux density</li> <li> Calculation of thermal energy density on field sensor planes</li> <li> Calculation of thermal flux over user defined flux boxes</li> <li> Calculation of thermal energy</li></mathul>

It can be shown that for magnetostatic analysis<ul> <li> Electric and magnetic field intensity and vector plots on planes</li> <li> Electric and magnetic potential intensity plots on planes</li> <li> Temperature and heat flux intensity and vector plots on planes</li> <li> Electric and magnetic energy density, the effect dissipated power density and thermal energy density plots on planes</li> <li> Animation of the permanent magnetization can be modeled as an equivalent volume current source:field and potential plots after parametric sweeps</li> <li> Graphs of characteristic impedance and effective permittivity of transmission line structures vs. sweep variables<math/li> \mathbf{J_{eq}(r)} = \nabla \times \mathbf{M(r)} <li> Custom output parameters defined as mathematical expressions of standard outputs</li></mathul>

If == Building the magnetization vector is uniform and constant inside the volume, then its curl is zero everywhere inside the volume except on its boundary surfacePhysical Structure in EM. In this case, the permanent magnetic can be effectively modeled by an equivalent surface current density on the surface of the permanent magnetic object: Ferma ==

<math> \mathbf{J_{s,eq}(r)} = \mathbf{M(r)} \times \hat{\mathbf{n}} </math>== Variety of Physical Objects in EM.Ferma ===

where <math> \hat{\mathbf{n}} </math> is 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 unit outward normal vector only applicable boundary conditions are defined at the surface boundary of the permanent magnet computational domain. As soon as you introduce a dielectric objectnext to a charge source or a magnetic (permeable) material next to a current source, you have to deal with a complex boundary value problem. Note that In other words, you need to solve the volume of electric or magnetic Poisson equation subject to the permanent magnet still acts domain boundary conditions as well as a permeable material in 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 magnetostatic analysisinterface between different materials and air. EM.Ferma uses the Finite Difference (FD) technique to find a numerical solution of your static boundary value problem.

To add a new permanent magnet source group to a project, right-click on "Permanent Magnets" on the Navigation Tree, and select "Insert New Permanent Magnet Source[[EM..." From Ferma]] offers the Permanent Magnet Source Dialog, you can change the default purple color following types of the source group or set the values of the relative permeability, Magnetization magnitude and unit direction vector components. The default direction vector is z-directed.physical objects:

{| class="wikitable"|-! scope= Computational Domain "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 Discretizationsurface objects| style="width:250px;" | Can be considered an electric source if the fixed voltage is nonzero |-| 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 |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 global coordinates (x,y,z) | style="width:100px;" | Solid objects| style="width:250px;" | Acts as an 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 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:pmc_group_icon.png]]| style="width:200px;" | [[Glossary of 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|}

In EM.Ferma, the Poisson === Grouping Objects by Material 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. Source Type ===

To modify the domain settings, click the Domain button Your physical structure in EM.Ferma is typically made up of some kind of source object either in the Simulate Toolbar free space or right-click on "3D Static Domain" entry in the Navigation Tree and select "Domain Settingspresence of one or more material objects.EM.." from the contextual menu. In the Domain Settings Dialog, Ferma's electrostatic and magnetostatic or thermal simulation engines then discretize the entire computational domain can be defined in two different ways: Default including these source and Custom. The default type places an enclosing box with a specified offset from the largest bounding box of your project's CAD material 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 and solve the coordinates of its two opposite corners labeled Min and Max Laplace or Poisson equations to find the electric or magnetic fields or temperature everywhere in the world coordinate systemcomputational domain.

EM.Ferma allows you It is important to specify the electric potential boundary conditions on the domain box. Two options are available. The Dirichlet boundary condition note that there is the default option and is specified as a fixed potential value on the surface of the domain walls. By default, this value is 0 Volts. The Neumann boundary condition specifies the normal derivative of the electric scalar potential on the surface of the domain walls. This is equivalent one-to the normal electric field component on the domain walls -one correspondence between electrostatic and its value is specified in V/m. The magnetostatic thermal simulation engine always assumes Dirichlet domain boundary conditions and sets the values of the magnetic vector potential to zero on all the domain walls. To modify the boundary conditions, right-click on "Boundary Conditions" in the Navigation Tree, and select "Boundary Conditions..." from the contextual menu to open the Boundary Conditions Dialog.entities:

<table>{| class="wikitable"<tr> |-<td> [[Image! scope="col"| Electrostatic Item! scope="col"| Corresponding Thermal Item|-| style="width:Qsource5.png200px;" |thumbElectric Scalar Potential|350pxstyle="width:200px;" |EM.Ferma's Domain Settings dialog.]] </td>Temperature<td> [[Image|-| style="width:fermbc.png200px;" |thumbElectric Field|350pxstyle="width:200px;" |EM.Ferma's Boundary Conditions dialog.]] </td>Heat Flux Density</tr>|-</table>| 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:Qsource4.png|thumb|350px{{Note|EMElectrostatic and thermal solvers share the same material categories on the navigation tree.Ferma's Mesh Settings dialogThis 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.]]===The Static Mesh===}}

The Finite Difference technique discretizes the computational domain using Once a 3D rectangular grid. EM.Ferma generates a fixed-cell mesh. This means that new object group node has been created in the extents of navigation tree, it becomes and remains the mesh cells along the principal axes are fixed: &Delta;x"Active" object group, &Delta;y, &Delta;zwhich is always listed in bold letters. By defaultWhen you draw a new geometric object such as a box or a sphere, its name is added under the mesh cell size currently active object group. There is set to only one unit project along all the three directions (with &Delta;x = &Delta;y = &Delta;z)object group that is active at any time. To modify the cell size, click the Mesh Settings button of the Simulate Toolbar or Any group can be made active by right-click clicking on "Static Mesh" its name in the Navigation Tree, navigation tree and select "Mesh Settings..." from selecting the '''Activate''' item of the contextual menu to open the Mesh Settings Dialog. {{Note[[Image:Info_icon.png|To obtain accurate results, it is highly recommended 30px]] Click here to use a square mesh as much as possiblelearn more about '''[[Building Geometrical Constructions in CubeCAD#Transferring Objects Among Different Groups or Modules | Moving Objects among Different Groups]]'''.}}

<tr> <td> [[Image:Qsource2STAT MAN1.png|thumb|350pxleft|Geometry of a spherical charge source and the enclosing domain box480px|EM.]] </td><td> [[Image:Qsource3.png|thumb|350px|Fixed-cel mesh of the spherical charge objectFerma's navigation tree.]] </td>

== Running Static Simulations = A Note on Material and Source Types in EM.Ferma ===

In [[Image:Qsource6.png|thumb|400px|EM.FermaCube]]'s Simulation Run dialogother modules, material types are categorized under the "Physical Structure" section of the navigation tree, and sources are organized under a separate "Sources" section.In those modules, all the geometric objects you draw in your project workspace typically represent material bodies. All of [[EM.Cube]] ===Two Simulation Engines===modules except for EM.Ferma require at least one excitation source to be selected from the "Sources" section of the navigation tree before you can run a simulation.

In EM.Ferma has two independent but functionally similar static simulation engines: Electrostatic , materials and Magnetostatic. The electrostatic engine solves sources are all lumped together and listed under the electric form "Physical Structure" section of Poisson's equation for electric scalar the navigation tree. In other words, there is no separate "Sources" section. For example, you can define default zero-potential subject to perfect electric field boundary conditions, in the presence of electric sources conductors (volume charges and PEC) in your project to model metal objects. You can also define fixed-potential PEC blocks) and dielectric material mediaobjects with a nonzero voltage, which can effectively act as a voltage source for your boundary value problem. The magnetostatic engine solves In this case, you will solve the magnetic form of Poisson's Lapalce equation for magnetic vector potential subject to magnetic field the specified nonzero potential boundary conditions, in values. Both types of PEC objects are defined from the presence same PEC node of magnetic the navigation tree by assigning different voltage values. Charge and current sources (wire are also defined as geometric objects, and volume currents and permanent magnetic blocks) and magnetic you have to draw them in the project workspace just like other material mediaobjects.

In == EM.Ferma you don't have to select any specific simulation engine. The program looks at the types sources and material objects present in your project workspace and then it determines whether an electrostatic analysis or a magnetostatic analysis or possibly both should be performed. When there are only electric sources present, you will get nonzero electric fields and zero magnetic fields. When there are only magnetic sources present, you will get nonzero magnetic fields and zero electric fields.s Computational Domain ==

To run a static simulation, first you have to open the Run Dialog. This is done by clicking 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". ===The only available simulation engine is "Static". Clicking the Run button of this dialog starts a static analysis. A separate window pops up which reports the progress of the current simulation. Domain Box===

EMTo modify the domain settings, click the Domain button of the Simulate Toolbar or right-click on "3D Static Domain" entry in the Navigation Tree and select "Domain Settings.Ferma currently offers three .." from the contextual menu. In the Domain Settings Dialog, the computational domain can be defined in two different simulation modesways: Analysis, Parametric Sweep Default and [[Optimization]]Custom. An "Analysis" is The default type places an enclosing box with a single-shot finite difference solution specified offset from the largest bounding box of your static structureproject's CAD objects. The structure default offset value is first discretized using 20 project units, but you can change this value arbitrarily. The custom type defines a fixed-cell mesh and the Poisson equation is solved numerically everywhere in your computational domain. The field and potential values at each mesh node are computed box by specifying the coordinates of its two opposite corners labeled Min and Max in the specified observables are written into data filesworld coordinate system.

In a "Parametric Sweep", one ore more <table><tr> <td> [[variables]] are varied at the specified steps(Image:Qsource5.png|thumb|left|480px|EM.Ferma's)Domain Settings dialog. This means that you must first define one or more [[variables]] in your projects. [[Variables]] can be associated with CAD object properties like dimensions, coordinates, rotation angles, etc. or with material properties or source properties. For each single variable sample or each combination of variable samples, first all the associated CAD object properties, material properties or source properties are updated in the project workspace. Then is a finite difference solution of your updated static structure is computed and parametric sweep proceeds to the next variable sample or combination.</td></tr></table>

[[Image:Qsource7.png|thumb|400px|*EM.Ferma's Static Engine Settings dialogallows you to specify the electric potential boundary conditions on the domain box. Two options are available. The Dirichlet boundary condition is the default option and is specified as a fixed potential value on the surface of the domain walls. By default, this value is 0 Volts. The Neumann boundary condition specifies the normal derivative of the electric scalar potential on the surface of the domain walls. This is equivalent to a constant normal electric field component on the domain walls and its value is specified in V/m.]] ===Static Simulation Engine Settings===

EM.Ferma currently uses a single iterative linear system solver based on *The magnetostatic simulation engine always assumes Dirichlet domain boundary conditions and sets the stabilized Bi-Conjugate Gradient (BiCG) method to solve the matrix equations which result from the discretization values of Poisson's equation. You can specify some numerical [[parameters]] related to the Bi-CG solver. To do that, you need magnetic vector potential 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, zero on all the default convergence error in this case is 1e-6domain walls.

{{Note|*EM.Ferma provides two options for thermal boundary conditions on the domain box. The Dirichlet boundary condition is the default option and is specified as a fixed temperature value of convergence error affect on the accuracy surface of your simulation results. For most practical scenarios, the domain walls. By default values are adequate, this value is 0&deg;C. You can reduce The Neumann boundary condition specifies the convergence error for better accuracy at normal derivative of the expense temperature on the surface of longer computation timethe domain walls. This is equivalent to a constant heat flux passing through the domain walls and its value is specified in W/m². A zero heat flux means a perfectly insulated domain box and is known as the adiabatic boundary condition.}}

== Working with Static To modify the boundary conditions, right-click on "Boundary Conditions" in the navigation tree, and select "Boundary Conditions..." from the contextual menu to open the Boundary Conditions Dialog. When you switch from the electrostatic-magnetostatic solver to the thermal solver in EM.Ferma's Run Simulation Data ==dialog, it automatically checks the box labeled '''Treat as a Thermal Structure''' in the Boundary Conditions dialog. Conversely, if you check this box in the Boundary Conditions dialog, the solver type is set to the thermal solver in the Simulation Run dialog. In the "Global Thermal Properties" section of the Boundary Conditions dialog, you can set the values of the ambient temperature in &deg;C, thermal conductivity of the environment in W/(m.K) and the convective coefficient in W/(m².K). You can also disable the enforcement of the convective boundary condition on the surface of solid insulator objects.

At the end of an electrostatic simulation, the electric field and electric scalar potential values are computed at all the mesh grid points of the entire computational domain<table><tr> <td> [[Image:fermbc. At the end of an magnetostatic simulation, the magnetic field and magnetic vector potential values are computed at all the grid nodespng|thumb|left|480px|EM. The field and potential values are written into output data files and can be visualized on the screen only if you define a field sensor observableFerma's Boundary Conditions dialog. In the absence of a defined observable, the static simulation will be carried out and completed, but to action will take place. ]] </td></tr></table>

=== Defining Field Sensors =EM.Ferma's Simulation Data & Observables ==

[[Image:Qsource8.png|thumb|350px|EM.Ferma's Field Sensor dialog.]] Just like other [[EM.Cube|EM.CUBE]] Modules, EM.Ferma has a Field Sensor observable, which plots 3D visualizations At the end of electric and magnetic field components on a specified plane. Howeveran electrostatic simulation, unlike the other modules, EM.Ferma electric field sensors have two additional plots for vector and electric scalar potential and magnitude values are computed at all the mesh grid points of the entire computational domain. At the end of an magnetostatic simulation, the magnetic field vector and magnetic vector potential. These values are called computed at all the "EPot" and "HPot" grid nodes on . At the Navigation Tree. To define end of a Field Sensorthermal simulation, right-click on "Field Sensors" in the Navigation Tree temperature and select "Insert New Observable..." from heat flux vector are computed at all the contextual menu. The Field Sensor dialog allows the user to select the direction mesh grid points of the sensor (X, Y, Z), visualization type, and whether E-field output or H-field output will be shown during a sweep analysisentire computational domain.

The E-fields Besides the electric and H-magnetic fields and temperature, EM.Ferma can compute a number of field integral quantities such as voltage, current, flux, energy, etc. The field components, potential values and field integrals are computed at each mesh node within written into output data files and can be visualized on the specified 2D Field Sensor planescreen or graphed in Data Manager only if you define a field sensor or a field integral observable. In other words, the resolution absence of any observable defined in the Field Sensor is controlled by navigation tree, the mesh resolutionstatic simulation will be carried out and completed, but no output simulation data will be generated.

EM.Ferma offers the following types of output simulation data:  {| class="wikitable"|-! scope="col"| Icon! scope="col"| Simulation Data Type! scope="col"| Observable Type! scope="col"| Applications|-| style="width:30px;" | [[ImageFile:Info_iconfieldsensor_icon.png]]|40pxstyle="width:150px;" | Near-Field Distribution Maps| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Near-Field_Sensor_Observable |Near-Field Sensor]] Click here to learn more about defining | style="width:450px;" | Computing electric and magnetic field sensor observables for ''components, electric scalar potential and magnitude of magnetic vector potential on a planar cross section of the computational domain |-| style="width:30px;" | [[File:fieldsensor_icon.png]]| style="width:150px;" | Electric and Magnetic Energy and Dissipated Power Density Maps | style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Near-Field_Sensor_Observable |Near-Field Sensor]] | style="width:450px;" | Computing electric and magnetic energy densities and dissipated power density on a planar cross section of the computational domain |-| style="width:30px;" | [[Data_Visualization_and_ProcessingFile:fieldsensor_icon.png]]| style="width:150px;" | Temperature and Heat Flux Distribution Maps| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Visualizing_3D_NearNear-Field_Maps Field_Sensor_Observable | Visualizing 3D Near-Field Sensor]] | style="width:450px;" | Computing temperature and heat flux components on a planar cross section of the computational domain |-| style="width:30px;" | [[File:fieldsensor_icon.png]]| style="width:150px;" | Thermal Energy Density Maps| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Near-Field_Sensor_Observable |Near-Field 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 |} Click on each category to learn more details about it in the [[Glossary of EM.Cube''s Simulation Observables & Graph Types]].

<td> [[Image:Qsource9Ferma L1 Fig15.png|thumb|450pxleft|640px|Electric field distribution of a spherical charge on a horizontal field sensor plane.]] </td></tr> <tr><td> [[Image:Qsource10Ferma L1 Fig16.png|thumb|450pxleft|640px|Electric scalar potential distribution of a spherical charge on a horizontal field sensor plane.]] </td>

=== Defining Field Integrals === It is often needed to compute integrals of the electric or magnetic fields to define other related quantities. The following table shows some below list the different types of widely used field integrals in electrostatics and magnetostatics. In EM.Ferma, you can define a path integral along a line segment that is parallel to one of the three principal axes, or a loop integral on a rectangle that is parallel to one of the principal planes. You can also define flux planes or flux boxes. All this is done from the same Field Integral Dialog. To define a Field Integral, right-click on "Field Integrals" in the Navigation Tree and select "Insert New Observable..." from the contextual menu. The Integral Type drop-down list gives nine options as listed in the table belowtheir definitions: 

| <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. Static Mesh===

After The Finite Difference technique discretizes the completion of computational domain using a static simulation, 3D rectangular grid. EM.Ferma generates a fixed-cell mesh. This means that the result extents of the field integrals mesh cells along the principal axes are written into "fixed: &Delta;x, &Delta;y, &Delta;z.DATBy default, the mesh cell size is set to one unit project along all the three directions (with &Delta;x = &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 " data filesMesh Settings. These files can be accessed using [[EM.Cube]]'s Data Manager." 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.}}

== Modeling Transmission Lines Using EM[[Image:Info_icon.png|30px]] Click here to learn more about '''[[Preparing_Physical_Structures_for_Electromagnetic_Simulation#Working_with_EM.Cube.27s_Mesh_Generators | Working with Mesh Generator]]'''.Ferma==

[[Image:qstaticInfo_icon.png|thumb30px]] Click here to learn more about the properties of '''[[Glossary_of_EM.Cube%27s_Simulation-Related_Operations#Fixed-Cell_Brick_Mesh |300px|Setting up a Transmission Line simulationEM.Ferma's Fixed-Cell Brick Mesh Generator]]'''.

EM<table><tr> <td> [[Image:Qsource2.Ferma's electrostatic simulation engine features png|thumb|360px|Geometry of a 2D solution mode where spherical charge source and the model is treated as a longitudinally infinite structure in the direction normal to specified "2D Solution Plane"enclosing domain box. More than one 2D solution plane may be defined]] </td><td> [[Image:Qsource3. In that case, multiple 2D solutions are obtained. A 2D solution plane is defined based on a "Field Sensor" definition that already exists in png|thumb|360px|Fixed-cel mesh of the projectspherical charge object.]] </td></tr></table>

To explore == Running Static Simulations in EM.Ferma's 2D mode, right-click on "2D Solution Planes" in the Navigation Tree and select "2D Domain 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 the solution list. 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=== EM. 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. Ferma's Simulation Modes ===

At lower microwave frequencies (f < 10GHz), it is usually possible to perform a 2D electrostatic analysis {| class="wikitable"|-! scope="col"| Simulation Mode! scope="col"| Usage! scope="col"| Number of a transmission line 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 and compute its characteristics impedance Z₀ and effective permittivity &epsilon"As Is"| style="width:100px;<sub>eff<" | Single run| style="width:200px;" | N/sub>A| style="width:150px;" | None|-| style="width:120px;" | [[Parametric_Modeling_%26_Simulation_Modes_in_EM.Cube#Running_Parametric_Sweep_Simulations_in_EM. This Cube | Parametric Sweep]]| style="quasiwidth: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|-static approach| style=" involves two stepswidth: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:100px;" | Multiple runs | style="width:200px;" | N/A| style="width:150px;" | None|}

<ol><li>First=== Running an Electrostatic, you have remove all the dielectric materials from your structure and replace them with free space (Magnetostatic 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>Thermal Analysis ===

Then effective permittivity [[EM.Ferma]] has three independent but functionally similar static simulation engines: Electrostatic, Magnetostatic and Thermal. The electrostatic engine solves the electric form of Poisson's equation for electric scalar potential subject to electric field boundary conditions, in the transmission line structure is then calculated from 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: <math> \epsilon_{eff} = \frac{C}{C_a} </math>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 media. The thermal engine solves the thermal form of Poisson's equation for steady-state temperature subject to thermal boundary conditions, in the presence of heat sources (volume sources and fixed-temperature PTC blocks) and insulator material media.

and its characteristic impedance 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: <math> Z_0 = \eta_0 \sqrt{ \frac{C_a}{C} } </math>'''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.

where &eta;<subtable>0</subtr> = 120&pi; &Omega; is the intrinsic impedance of the free space<td> [[Image:Ferma L1 Fig11. png|thumb|left|600px|EM.Ferma's Simulation Run dialog.]] </td></tr></table>

In EM.Ferma you don't have to choose between the electrostatic or magnetostatic simulation engines. The guide wavelength program looks at the types of sources and material objects present in your transmission line project workspace and then it determines whether an electrostatic analysis or a magnetostatic analysis or possibly both should be performed. When there are only electric sources present, you will get nonzero electric fields and zero magnetic fields. When there are only magnetic sources present, you will get nonzero magnetic fields and zero electric fields. On the other hand, since the electrostatic and thermal solvers share the same navigation resources, you can run only one of the two engines at a given frequency f time. By default, the electrostatic solver is then calculated from:enabled.

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 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<mathsup> \beta = k_0\sqrt{\epsilon_{eff}} = \frac{2\pi f}{c}\sqrt{\epsilon_{eff}} -6</mathsup>.

where c is {{Note|The value of convergence error affect the speed accuracy of light in your simulation results. For most practical scenarios, the free spacedefault values are adequate. You can reduce the convergence error for better accuracy at the expense of longer computation time. }}

<table><tr> <td> [[Image:Qsource7.png|thumb|left|480px|EM.Ferma's 2D Quasi-Static mode automatically performs the two-step process described above and calculates &epsilon;Engine Settings dialog.]]<sub/td>eff</subtr> and Z<sub>0</subtable>. So you don't need to modify your structure in the first step.

=== Setting up a Transmission Line The 2D Quasi-Static Simulation =Mode==

To perform a transmission line EM.Ferma's electrostatic simulation, first draw engine features a 2D solution mode where your physical model is treated as a longitudinally infinite structure in the project workspace just like a typical 3D structuredirection normal to specified "2D Solution Plane". Define A 2D solution plane is defined based on a "Field Sensor" observable definition that already exists in your project. To explore EM.Ferma's 2D mode, right-click on '''2D Solution Planes''' in the Navigation Tree so as "Computational Domain" section of the navigation tree and select '''2D Domain Settings...''' from the contextual menu. In the 2D Static Domain dialog, check the checkbox labeled "Reduce the 3D Domain to capture a 2D Solution Plane". The first field sensor observable in the cross section navigation tree is used for the definition of your structure as your desired transmission line profilethe 2D solution plane.

Next, define At the end of a "2D Solution Plane" in electrostatic analysis, you can view the Navigation Tree based electric field and potential results on your existing the field sensorplane. When defining It is assumed that your structure is invariant along the direction normal to the 2D solution plane. Therefore, check your computed field and potential profiles must be valid at all the box labeled "Perform planes perpendicular to the specified longitudinal direction. A 2D Quasi-Static Simulation"structure of this type can be considered to represent a transmission line of infinite length. If an EM.Ferma also performs a quasi-static analysis is run with this option checkedof the transmission line structure, and usually provides good results at lower microwave frequencies (f < 10GHz). It computes the characteristic characteristics impedance Z₀ and effective permittivity &epsilon;_eff will be computed for of the corresponding 2D Solution Planemulti-conductor TEM or quasi-TEM transmission line. 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 2D quasi-static solutions can be obtained in the same analysis,for example, when your design contains many types of <table><tr> <td> [[Transmission Lines|transmission lines]]Image:Qsource14. At the end of a quasi-static analysis, the electric field components and scalar potential at the selected png|thumb|left|450px|The 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.Gridstatic domain dialog.]] </td> </tr> === Optimizing a Transmission Line ===</table>

In an [[optimizationImage:Info_icon.png|30px]] simulation, Click here to learn more about the values theory of one or more '''[[variablesElectrostatic_%26_Magnetostatic_Field_Analysis#2D_Quasi-Static_Solution_of_TEM_Transmission_Line_Structures | 2D Quasi-Static Analysis of Transmission Lines]] are varied over their specified ranges, and a design objective is tested at each simulation run. A design objective is typically a logical expression that sets an expression equal to a target value. 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, click the "Objectives" button of the Simulate Toolbar, or select the "Objectives" item of the Simulate Menu, or simply use the keyboard shortcut "Ctrl+J". In the Objectives Dialog, you can add new objective or edit the existing objectives'''.

For a step-by-step demonstration (including transmission line [[optimization]]), take a look at this video on our YouTube channel: [http://www.youtube.com/watch?v=Iiu9rQf1QI4 EM.CUBE Microstrip Optimization]<table><tr> == Simulation Examples / Gallery == {| border="0"|-| valign="top"|<td> [[FileImage:ScreenCapture1Qsource17.png|thumb|left|350px480px|Classic Example: Two oppositely charged spheres.]]| valign="top"|[[File:iarray.png|thumb|left|350px|H-Field from array Electric field distribution of current loopsthe microstrip line on the 2D solution plane.]]</td>|-</tr> |}{| border="0"|-| valign="top"|<tr> <td> [[FileImage:ustripQsource18.png|thumb|left|350px|Potential near microstrip conductor from a quasistatic simulation.]]| valign="top"|[[File:ustrip2.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>|} == Version History ==</table>

* [http[Image://en.wikipedia.org/wiki/Electrostatics Wikipedia: ElectrostaticsTop_icon.png|30px]]* '''[[http://www.youtube.com/watch?v=Iiu9rQf1QI4 YouTube: EM.Ferma Optimization Example.#Product_Overview | Back to the Top of the Page]* [http://www.emagtech.com/content/emferma More about EM.Ferma.]'''

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