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[[Image:Splash-static.jpg|right|800px720px]]<strong><font color="#0d10e52603c4" size="4">Electrostatic and , Magnetostatic & Thermal Solvers For DC And Low Frequency Simulations</font></strong>

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[[Image:STAT7 11.png|thumb|400px| Vector plot of electric field distribution in a microstrip transmission line.]][[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.

[[Image:Tutorial_iconInfo_icon.png|40px30px]] Click here to access learn more about the '''[[EMElectrostatic & Magnetostatic Field Analysis | Theory of Electrostatic and Magnetostatic Methods]]'''.Cube#EM [[Image:Info_icon.Ferma_Tutorial_Lessons png| EM.Ferma Tutorial Gateway30px]] Click here to learn more about the '''[[Steady-State_Thermal_Analysis | Theory of Steady-State Heat Transfer Methods]]'''. <table><tr><td>[[Image:Magnet lines1.png|thumb|left|400px| Vector plot of magnetic field distribution in a cylindrical permanent magnet.]]</td></tr></table>

EM.Ferma is the low-frequency '''Static Module''' of '''[[EM.Cube]]''', a comprehensive, integrated, modular electromagnetic modeling environment. EM.Ferma shares the visual interface, 3D parametric CAD modeler, data visualization tools, and many more utilities and features collectively known as [[Building_Geometrical_Constructions_in_CubeCAD | CubeCAD ]] with all of [[EM.Cube]]'s other computational modules.

[[Image:Info_icon.png|40px30px]] Click here to learn more about '''[[Getting_Started_with_EM.Cube | EM.Cube Modeling Environment]]'''.

[[Image:Info_icon=== Advantages & Limitations of EM.png|40px]] Click here to learn 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 about efficient than the basic functionality 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 [[Building_Geometrical_Constructions_in_CubeCAD | CubeCADEM.Cube]]'''s full-wave modules like [[EM.Tempo]], [[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. <table><tr><td>[[Image:Ferma L8 Fig title.png|thumb|left|400px| Vector plot of electric field distribution in a coplanar waveguide (CPW) transmission line.]]</td></tr></table>

Metal Perfect electric conductor(PEC) solids and surfaces in free space (Electrostatics)</li>

Dielectric objects in free space (Electrostatics)</li>

Magnetic (permeable) objects in free space (Magnetostatics)</li> <li> Perfect thermal conductor (PTC) solids and surfaces (Thermal)</li> <li> Insulator objects (Thermal)</li>

Fixed-potential PEC for maintaining equi-potential metal objects(Electrostatics)</li> <li> Volume charge sources (Electrostatics)</li> <li> Volume current sources (Magnetostatics)</li>

Volume charge Wire current sources(Magnetostatics)</li>

Volume current sourcesPermanent magnets (Magnetostatics)</li>

Wire current sourcesFixed-temperature PTC for maintaining iso-thermal objects (Thermal)</li>

Permanent magnetsVolume heat sources (Thermal)</li>

Electric and magnetic potential intensity plotson planes</li> <li> Temperature and heat flux intensity and vector plots on planes</li> <li> Electric and magnetic energy density, dissipated power density and thermal energy density plots on planes</li>

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 (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 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 (FD) technique to find a numerical solution of your static boundary value problem.

[[EM.Ferma ]] offers six the following types of physical objects:

| 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: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: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: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: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: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:100px;" | line Line and polyline objects

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

Your physical structure in EM.Ferma is typically made up of some kind of source object either in the free space or in the presence of one or more material objects. EM.Ferma's electrostatic and magnetostatic or thermal simulation engines then discretize the entire computational domain including these source and material objects and solve the Laplace or Poisson equations to find the electric or magnetic fields or temperature everywhere in the computational domain.

[[Image:Info_icon.png|40px30px]] Click here to learn more about '''[[Building Geometrical Constructions in CubeCAD#Transferring Objects Among Different Groups or Modules | Moving Objects among Different Groups]]'''.

*EM.Ferma allows 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 the a constant normal electric field component on the domain walls and its value is specified in V/m.  *The magnetostatic simulation engine always assumes Dirichlet domain boundary conditions and sets the values of the magnetic vector potential to zero on all the domain walls.  *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 on the surface of the domain walls. By default, this value is 0&deg;C. The Neumann boundary condition specifies the normal derivative of the temperature on the surface of the 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. To modify the boundary conditions, right-click on "Boundary Conditions" in the Navigation Treenavigation 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 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 vector and electric scalar potential 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 values are computed at all the grid nodes. At the end of a thermal simulation, the temperature and heat flux vector are computed at all the mesh grid points of the entire computational domain.  Besides the electric and magnetic fieldsand 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 written into output data files and can be visualized on the screen or graphed in EM.Grid Data Manager only if you define a field sensor or a field integral observable. In the absence of any observable defined in the navigation tree, the static simulation will be carried out and completed, but no output simulation data will be generated.

| style="width:30px;" | [[File:fieldsensor_icon.png]]| style="width:150px;" | Near-Field Distribution Maps| style="width:150px;" | '''[[Glossary of EM.Cube's Simulation Observables& Graph Types#Near-Field Sensor Field_Sensor_Observable |Near-Field Distribution MapsSensor]] | style="width:450px;" | Computing electric and magnetic field 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 Field_Sensor_Observable |Near-Field Sensor]]''' | style="width:450px;" | Computing electric and magnetic field components, electri scalar potential energy densities and magnitude dissipated power density on a planar cross section of magnetic vector potential the computational domain |-| style="width:30px;" | [[File:fieldsensor_icon.png]]| style="width:150px;" | Temperature and 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 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:150px30px;" | '''[[Glossary of EMFile:field_integ_icon.Cube's Simulation Observables#Static Field Integral 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 |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]].

| 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

[[EM.Ferma ]] currently offers three different simulation modes as follows:

| style="width:120px;" | [[#Running an Electrostatic or Magnetostatic Analysis| Analysis]]

| style="width:120px;" | [[Parametric Modeling & Sweep Simulations in EMParametric_Modeling_%26_Simulation_Modes_in_EM.Cube#Running Parametric Sweep Simulations in EMRunning_Parametric_Sweep_Simulations_in_EM.Cube | Parametric Sweep]]

| style="width:120px;" | [[Planning & Performing Optimization in EMParametric_Modeling_%26_Simulation_Modes_in_EM.Cube#Performing_Optimization_in_EM.Cube | Optimization]]

An "Analysis" is the simplest simulation mode of 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 computational domain. The field and potential values at each mesh node are computed=== Running an Electrostatic, and the specified observables are written into data files.Magnetostatic or Thermal Analysis ===

EM.Ferma currently uses To run a single iterative linear system solver based on the stabilized Bi-Conjugate Gradient (BiCG) method to solve the matrix equations which result from the discretization of Poisson's equation. You can specify some numerical parameters related to the Bi-CG solver. To do thatstatic simulation, first you need have to open the Simulation Engine Settings Run Dialog . This is done by clicking the "SettingsRun" button located next to of the Simulate Toolbar, or by selecting the "Select EngineRun" drop-down list. From this dialog you can set item of the maximum number of BiCG iterationsSimulate Menu, which has a default value of 10,000. You can also set a value for or simply using the keyboard shortcut "Convergence ErrorCtrl+R". The default value There are two available options for electrostatic analysis is 0the simulation engine: '''Electrostatic-Magnetostatic Solver''' and '''Steady-State Thermal Solver'''.001. For magnetostatic analysis, Clicking the specified value Run button of convergence error is reduced by this dialog starts a factor 1000 automaticallystatic analysis. Therefore, A separate window pops up which reports the default convergence error in this case is 1e-6.  {{Note|The value progress of convergence error affect the accuracy of your current simulation results. For most practical scenarios, the default values are adequate. You can reduce the convergence error for better accuracy at the expense of longer computation time.}}

<td> [[Image:Qsource7Ferma L1 Fig11.png|thumb|left|480px600px|EM.Ferma's Static Engine Settings Simulation Run dialog.]]</td>

== In EM.Ferma you don't have to choose between the electrostatic or magnetostatic simulation engines. The 2D Electrostatic Simulation Mode==program looks at the types of 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. 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 time. By default, the electrostatic solver is enabled.

=== Using An "Analysis" is the simplest simulation mode of EM.Ferma to Simulate 2D Transmission Lines ===. 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 computational 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.

You can use [[EM.Ferma]] to perform a quasi-static analysis of multi-conductor transmission line structures, which usually provides good results at lower microwave frequencies (f < 10GHz). [[EM.Ferma]] computes the characteristics impedance Z₀ and effective permittivity &epsilon;_eff of your TEM or quasi-TEM transmission line. The "quasi-static approach" involves two steps as described above, which [[EM.Ferma]]'s 2D Quasi-===Static mode automatically performs to calculate &epsilon;_eff and Z₀. Simulation Engine Settings===

To perform 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 simulationGauss-Seidel solver. The default solver type is BiCG. You can specify some numerical parameters related to the BiCG solver. To do that, first draw your structure in you need to open the project workspace just like 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 typical 3D structuredefault value of 10,000. Define You can also set a value for "Field SensorConvergence Error" observable in . The default value for electrostatic analysis is 0.001. For magnetostatic analysis, the navigation tree so as to capture the cross section specified value of your structure as your desired transmission line profileconvergence error is reduced by a factor 1000 automatically. Therefore, the default convergence error in this case is 10^-6.

Next, define a "2D Solution Plane" in {{Note|The value of convergence error affect the navigation tree based on accuracy of your existing field sensor. When defining the 2D plane, check the box labeled "Perform 2D Quasi-Static Simulation". If an analysis is run with this option checked, the characteristic impedance Z₀ and effective permittivity &epsilon;_eff will be computed for the corresponding 2D Solution Plane. The simulation results are written to two output data files named "solution_plane_Z0.DAT" and "solution_plane_EpsEff.DAT", respectivelyFor most practical scenarios, where "solution_plane" is the default name of your 2D planevalues are adequate.  Many 2D quasi-static solutions You can be obtained in reduce the same analysis, convergence error for example, when your design contains many types of transmission lines. At the end of a quasi-static analysis, the electric field components and scalar potential better accuracy at the selected 2D planes will still be computed and can be visualized. In the case expense 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.Gridlonger computation time.}}

[[Image:Qsource16Qsource7.png|thumb|left|480px|A field sensor and 2D solution plane defined for a microstrip lineEM.Ferma's Static Engine Settings dialog.]]

EM.Ferma's electrostatic simulation engine features a == The 2D solution mode where your physical model is treated as a longitudinally infinite structure in the direction normal to specified "2D Solution Plane". More than one 2D solution plane may be defined. In that case, multiple 2D solutions are obtained. A 2D solution plane is defined based on a "Field Sensor" definition that already exists in your project.Quasi-Static Simulation Mode==

EM.Ferma's electrostatic simulation engine features a 2D solution mode where your physical model is treated as a longitudinally infinite structure in the direction normal to specified "2D Solution Plane". A 2D solution plane is defined based on a "Field Sensor" definition that already exists in your project. To explore EM.Ferma's 2D mode, right-click on '''2D Solution Planes''' in the "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 "Treat Structure as Longitudinally Infinite across Each 2D Plane Specified Below". This would enable you Reduce the 3D Domain to add new a 2D Solution Plane definitions to the list or edit the existing ones". In The first field sensor observable in the Add/Edit 2D Solution Plane dialog, you can choose a name other than navigation tree is used for the default name and select one definition of the available field sensor definitions in your project2D solution plane.  At the end of a 2D electrostatic analysis, you can view the electric field and potential results on the respective field sensor planesplane. 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. A 2D structure of this type can be considered to represent a transmission line of infinite length. EM.Ferma also performs a quasi-static analysis of the transmission line structure, and usually provides good results at lower microwave frequencies (f < 10GHz). It computes the characteristics impedance Z₀ and effective permittivity &epsilon;_eff of the multi-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.

<td> [[Image:Qsource14.png|thumb|left|640px450px|The 2D static domain dialog.]] </td>

<td> [[Image:Qsource15Qsource16.png|thumb|left|550px480px|The "Add/Edit A field sensor and 2D Solution Plane" dialogsolution plane defined for a microstrip line.]] </td>

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