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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. 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 '''[[EM.Cube#EM.Ferma_Tutorial_Lessons Electrostatic & Magnetostatic Field Analysis | EM.Ferma Tutorial GatewayTheory of Electrostatic and Magnetostatic Methods]]'''.

[[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]]'''.

=== Advantages & Limitations of EM.Ferma's Static Simulator === [[Image:Info_iconEM.png|40pxFerma]] Click here to learn 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, [[Building_Geometrical_Constructions_in_CubeCAD | CubeCADEM.Ferma]] offers a faster alternative to [[EM.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.

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. [[EM.Ferma ]] uses the Finite Difference (FD) technique to find a numerical solution of your static boundary value problem.

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

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 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 everywhere in the computational domain.

All the geometric objects in the project workspace are organized together into object groups which share the same properties including color and electric or magnetic parameters. It is recommended that you first create object groups, and then draw new objects under the active group. To create a new object group, right-click on its category name in the "Physical Structure" section of the navigation tree and select one of the "Insert New Group..." items from the contextual menu. However, if you start a 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 navigation tree to hold your new geometric object.

In [[EM.Cube]]'s other 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]] 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]], materials and sources are all lumped together and 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 also defined as geometric objects, and you have to draw them in the project workspace just like other material objects.

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.

[[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 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. 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.

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. Besides the electric and magnetic fields, [[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 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.

[[EM.Ferma ]] offers the following types of output simulation data:

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: Δx, Δy, Δz. By default, the mesh cell size is set to one unit project along all the three directions (with Δx = Δy = Δz). 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.

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

[[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 media.

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.

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, and the specified observables are written into data files.

[[EM.Ferma ]] currently uses 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 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 1e-6.

[[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". 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.

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 to add new 2D Solution Plane definitions to the list or edit the existing ones. 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.

You can also 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). For that purpose, check the box labeled "Perform 2D Quasi-Static Simulation" when defining the 2D solution plane. [[EM.Ferma ]] computes the characteristics impedance Z₀ and effective permittivity &epsilon;_eff of your 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. 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.