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

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

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

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

[[Image:Info_icon.png|30px]] Click here to learn more about the theory of '''[[Electrostatic and Magnetostatic Steady-State_Thermal_Analysis | Theory of Steady-State Heat Transfer Methods]]'''.

=== EM.Ferma solves as the Poisson equation for the electric scalar potential subject to specified boundary conditions:Static Module of EM.Cube ===

<math>\Delta\Phi(\mathbf{r}) = \nabla^2 \Phi(\mathbf{r}) = EM.Ferma is the low-\frac{\rho(\mathbf{r})}{\epsilon}</math>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.

where &Phi;(<b>r</b>) is the electric scalar potential, &rho;(<b>r</b>) is the volume charge density, and &epsilon; = == Advantages &epsilon;_r &epsilon;₀ is the permittivity Limitations of the mediumEM. Ferma's Static Simulator ===

The <table><tr><td>[[Image:Ferma L8 Fig title.png|thumb|left|400px| Vector plot of electric field boundary conditions at the interface between two material media are:distribution in a coplanar waveguide (CPW) transmission line.]]</td></tr></table>

<math> \hat{\mathbf{n}} == EM. [ \mathbf{D_2(r)} - \mathbf{D_1(r)} ] Ferma Features at a Glance == \rho_s (\mathbf{r}) </math>

<mathul> \hat{\mathbf{n}} \times [ \mathbf{E_2 <li> Perfect electric conductor(rPEC)} - \mathbf{E_1solids and surfaces (rElectrostatics)} ] = 0 </mathli> <li> Dielectric objects (Electrostatics)</li> <li> Magnetic (permeable) objects (Magnetostatics)</li> <li> Perfect thermal conductor (PTC) solids and surfaces (Thermal)</li> <li> Insulator objects (Thermal)</li></ul>

where <mathul> \hat{\mathbf{n}} <li> Fixed-potential PEC for maintaining equi-potential metal objects (Electrostatics)</mathli> is the unit normal vector at the interface pointing from medium 1 towards medium 2, <bli>D Volume charge sources (rElectrostatics)</bli> = &epsilon; <bli>E Volume current sources (rMagnetostatics)</bli> is the electric flux density, <bli>E Wire current sources (rMagnetostatics)</bli> is the electric field vector, and &rho; <subli>s Permanent magnets (Magnetostatics)</subli> <li> Fixed-temperature PTC for maintaining iso-thermal objects (Thermal)</li> <li> Volume heat sources (Thermal)</li></ul> is the surface charge density at the interface.

In a source<ul> <li> Fixed-free region, size brick&rhonbsp;(cells<b/li>r</bul>) = 0, and Poisson's equation reduces to the familiar Laplace equation:

<math>\Delta\Phi(\mathbf{r}) = \nabla^2 \Phi(\mathbf{r}) = 0</math>= 3D Electrostatic & Magnetostatic Simulation ===

Keep in mind that in the absence of an electric charge source, you need to specify a non=== 2D Quasi-zero potential somewhere in your structure, for example, on a perfect electric conductor (PEC). Otherwise, you will get a trivial zero solution of the Laplace equation. Static Simulation ===

<mathul> \mathbf{E(r)} = - \nabla \Phi(\mathbf{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>

== Building the Physical Structure in EM.Ferma solves the Poisson equation for the magnetic vector potential subject to specified boundary conditions:==

<math>\Delta \mathbf{A} (\mathbf{r}) = \nabla^2 \mathbf{A}(\mathbf{r}) = - \mu \mathbf{J}(\mathbf{r}) </math>= Variety of Physical Objects in EM.Ferma ===

where <b>A(r)</b> is the magnetic vector potential, <b>J(r)</b> is the volume current density, and &mu; = &mu;_r &mu;₀ is the permeability of the medium[[EM. The magnetic Poisson equation is vectorial in nature and involves a system of three scalar differential equations corresponding to Ferma]] offers the three components following types of <b>A(r)</b>. physical objects:

The magnetic field boundary conditions at Click on each category to learn more details about it in the interface between two material media are:[[Glossary of EM.Cube's Materials, Sources, Devices & Other Physical Object Types]].

<math> \hat{\mathbf{n}} . [ \mathbf{B_2(r)} - \mathbf{B_1(r)} ] = 0 </math>== Grouping Objects by Material or Source Type ===

<math> \hat{\mathbf{n}} \times [ \mathbf{H_2(r)} 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- \mathbf{H_1(r)} ] = \mathbf{J_s(r)} </math>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.

where <math> \hat{\mathbf{n}} </math> is the unit normal vector at the interface pointing from medium 1 towards medium 2,| class="wikitable"<b>B(r)</b> |-! scope= &mu"col"| Electrostatic Item! scope="col"| Corresponding Thermal Item|-| style="width:200px;<b>H(r)</b> is the magnetic flux density, <b>H(r)</b> is the magnetic field vector, and <b>J_s</b> is the surface current density at the interface. " | Electric Scalar Potential| style="width:200px;" | Temperature|-| style="width:200px;" | Electric Field| style="width:200px;" | Heat Flux Density|-| style="width:200px;" | Perfect Electric Conductor| style="width:200px;" | Perfect Thermal Conductor|-| style="width:200px;" | Dielectric Material| style="width:200px;" | Insulator Material|-| style="width:200px;" | Volume Charge| style="width:200px;" | Volume Heat Source|}

Once {{Note|Electrostatic and thermal solvers share the magnetic vector potential is computedsame material categories on the navigation tree. This means that PEC objects are treated as PTC objects, dielectric objects are treated as insulator objects and volume charges are treated as volume heat sources when the magnetic field can easily be computed via the equation below: thermal solver is enabled.}}

== Defining the Physical Structure [[Image:Info_icon.png|30px]] Click here to learn more about '''[[Building Geometrical Constructions in EMCubeCAD#Transferring Objects Among Different Groups or Modules | Moving Objects among Different Groups]]'''.Ferma ==

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<table><tr><td> [[Image:STAT MAN1. In such cases, the only applicable boundary conditions are defined at the computational domain boundarypng|thumb|left|480px|EM. 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 PoissonFerma'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 problemnavigation tree. ]] </td></tr></table>

In [[EM.Cube]]'s other modules, material types are specified categorized under the "Physical Structure" section of the Navigation Treenavigation 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 geometric objects you draw in your projectworkspace typically represent material bodies. Sources are virtual entities that might All of [[EM.Cube]] modules except for EM.Ferma require at least one excitation source to be associated with certain physical objects and provide selected from the excitation "Sources" section of your boundary value problemthe navigation tree before you can run a simulation.

In [[EM.Cube]]'s Static ModuleFerma, materials and sources are all lumped together and listed under the "Physical Structure" section of the Navigation Treenavigation tree. In other words, and there is no separate "Sources" section. For example, you can define default zero-potential perfect electric conductors (PEC) in your project to model metal objects. You can also define fixed-potential PEC objects with a nonzero voltage, which can effectively act as a voltage source for your boundary value problem. In this case, you will solve the Lapalce equation subject to the specified nonzero potential boundary values. Both types of PEC objects are defined from the same PEC node of the Navigation Tree navigation tree by assigning different voltage values. Charge and current sources are also defined as CAD geometric objects that , and you must have to draw them in the project workspacejust like other material objects.

=== Fixed-Potential PEC Objects=EM.Ferma's Computational Domain ==

A perfect electric conductor (PEC) is a material with &epsilon;_r = 1 and &sigma; = &infin;. Under the static condition, every point on a PEC object has the same electric potential. By default, this is a zero potential, assuming the PEC object is "grounded". You can define a nonzero voltage value for a PEC group. In that case, the PEC object is effective turned into a voltage source. For example, tow parallel PEC plates, one with a zero potential and the other with a nonzero potential represent a simple air-filled capacitor. =The Domain Box===

To add a new Fixed-Potential PEC group In EM.Ferma, the Poisson or Laplace equations are solved subject to boundary conditions using the Finite Difference technique. As a project, right-click on "Fixed-Potential PEC Objects" on the Navigation Treeresult, you need to specify a finite computational domain and select "Insert New PECthen specify the domain boundary conditions.EM.Ferma's computational domain defines where the domain boundary condition will be specified." From A default domain box is always placed in the PEC dialog, project workspace as soon as you draw your first object. The domain can change the default red color and set be seen as a value for blue cubic wireframe that surrounds all of the "Voltage" CAD objects in Voltsthe project workspace.

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

=== Dielectric<table><tr> <td> [[Image:Qsource5.png|thumb|left|480px|EM.Ferma's Domain Settings dialog.]] </Magnetic Materials === td></tr></table>

To add a new dielectric or magnetic material group *EM.Ferma allows you to a project, right-click on "Dielectric/Magnetic Materials" specify the electric potential boundary conditions on the Navigation Tree, and select "Insert New Dielectricdomain box.Two options are available.." From The Dirichlet boundary condition is the Dielectric Dialog, you can change default option and is specified as a fixed potential value on the surface of the domain walls. By default green color , this value is 0 Volts. The Neumann boundary condition specifies the normal derivative of a material group or set the values electric scalar potential on the surface of the material [[parameters]]domain walls. This is equivalent to a constant normal electric field component on the domain walls and its value is specified in V/m.

{{Note| You can define any solid object as a dielectric or *The magnetostatic simulation engine always assumes Dirichlet domain boundary conditions and sets the values of the magnetic material objectvector potential to zero on all the domain walls.}}

You can define volume charge sources with a specified charge density in C/m³ confined to certain region of your project. You use [[EM.Cube]]'s [[Solid Objects|solid objects]] to define volume charge sources. All the charge sources belonging to the same group have the same color and same charge density value. The charge density can be positive or negative. To add a new charge source group to a projectmodify the boundary conditions, right-click on "Volume ChargesBoundary Conditions" on in the Navigation Treenavigation tree, and select "Insert New Charge SourceBoundary Conditions..." From from the Charge Source 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 can change check this box in the default purple color 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 source group or Boundary Conditions dialog, you can set the values of the Charge Densityambient 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.

You can define volume current sources with a specified current density in A/m² confined to certain region of your project. Note that current density is a vectorial quantity and has a magnitude and unit direction vector. You use [[== EM.Cube]]Ferma's [[Solid Objects|solid objects]] to define volume current sources. All the volume current sources belonging to the same group have the same color and same current density magnitude and unit vector.Simulation Data & Observables ==

To add a new volume current source group to a project, right-click on "Volume Currents" on At the Navigation Treeend of an electrostatic simulation, the electric field vector and select "Insert New Current Source..electric scalar potential values are computed at all the mesh grid points of the entire computational domain." From At the Volume Current Source Dialogend of an magnetostatic simulation, you can change the default brown color of magnetic field vector and magnetic vector potential values are computed at all the source group or set grid nodes. At the values end of a thermal simulation, the Current Density magnitude temperature and unit direction heat flux vector components. The default direction vector is z-directedare computed at all the mesh grid points of the entire computational domain.

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 offers the project units. The line or polyline object is then approximated as a volume current with a current density following types of J = I/(&pi;r²) 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. output simulation data:

To add a new wire current source group to a project, right-click on {| class="Wire Currentswikitable" on the Navigation Tree, and select "Insert New Current Source..." From the Wire Current Source Dialog, you can change the default brown color of the source group or set the values of the Current and Wire Radius. There is also a check box for "Reverse Current Direction". Note that this will reverse the direction of all the wire currents belonging to the same group. When you draw a line or polyline object under a wire current group in the Navigation Tree, you will notice that direction arrows are placed on the drawn CAD object. You can draw any curve object in the project workspace and convert it to a polyline using [[EM.Cube]]'s Polygonize Tool.  {{Note| If you draw [[[Curve Objects]]] under a wire current group, they will be permanently converted to polyline objects before running the simulation engine.}} -! scope=== Permanent Magnets==="col"| Icon A permanent magnet is typically a ferromagnetic material with a fixed inherent magnetization vector. As a result, it can be used as a source in an magnetostatic problem. When a permeable material has a permanent magnetization, the following relationship holds:   <math> \mathbf{B(r)} ! scope= {\mu} (\mathbf{H(r)} + \mathbf{M(r)} ) </math>"col"| Simulation Data Type  where <b>M(r)</b> is the magnetization vector. In SI units system, the magnetic field <b>H</b> and magnetization <b>M</b> both have the same units of A/m.  It can be shown that for magnetostatic analysis, the effect of the permanent magnetization can be modeled as an equivalent volume current source:  <math> \mathbf{J_{eq}(r)} ! scope= \nabla \times \mathbf{M(r)} </math>  If the magnetization vector is uniform and constant inside the volume, then its curl is zero everywhere inside the volume except on its boundary surface. In this case, the permanent magnetic can be effectively modeled by an equivalent surface current density on the surface of the permanent magnetic object:   <math> \mathbf{J_{s,eq}(r)} = \mathbf{M(r)} \times \hat{\mathbf{n}} </math>  where <math> \hat{\mathbf{n}} </math> is the unit outward normal vector at the surface of the permanent magnet object. Note that the volume of the permanent magnet still acts as a permeable material in the magnetostatic analysis.  To add a new permanent magnet source group to a project, right-click on "Permanent Magnetscol" on the Navigation Tree, and select "Insert New Permanent Magnet Source..." From the Permanent Magnet Source Dialog, you can change the default purple color of the source group or set the values of the relative permeability, Magnetization magnitude and unit direction vector components. The default direction vector is z-directed. | Observable Type  {| border! scope="0col"| Applications

| valignstyle="topwidth:30px;"|[[File:vsourcefieldsensor_icon.png|thumb|left|250px|EM.Ferma's Voltage Source Dialog]]| valignstyle="bottomwidth:150px;"|Near-Field Distribution Maps| style="width:150px;" | [[File:qsource.png|thumb|left|250px|Glossary of EM.FermaCube's Charge Source DialogSimulation Observables & Graph Types#Near-Field_Sensor_Observable |Near-Field Sensor]]| valignstyle="topwidth:450px;"|[[File:isource.png|thumb|left|250px|EM.Ferma's Current Source Dialog]]Computing electric and magnetic field components, electric scalar potential and magnitude of magnetic vector potential on a planar cross section of the computational domain

[[Image:fermbc.png|thumb|250px|Boundary Condition Dialog]] ===The Domain Box=== In EM.Ferma, table below list 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.  [[Image:qsource2.png|thumb|400px|The blue wireframe around the CAD objects defines the extents of the computational domain. The specified boundary conditions are applied on the domain walls. ]] To 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..." from the contextual menu. In the Domain Settings Dialog, the computational domain can be defined in two different ways: Default and Custom. The default type places an enclosing box with a specified offset from the largest bounding box of your project's CAD objects. The default offset value is 20 project units, but you can change this value arbitrarily. The custom type defines a fixed domain box by specifying the coordinates of its two opposite corners labeled Min and Max in the world coordinate system.  ===Domain Boundary Conditions=== 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. ===The Static Mesh=== The Finite Difference technique discretizes the computational domain using a 3D rectangular grid. EM.Ferma generates a fixed-cell mesh. This means that the extents of the mesh cells along the principal axes are fixed: &Delta;x, &Delta;y, &Delta;z. By default, the mesh cell size is set to one unit project along all the three directions (with &Delta;x = &Delta;y = &Delta;z). To modify the cell size, click the Mesh Settings button of the Simulate Toolbar or right-click on "Static Mesh" in the Navigation Tree, and select "Mesh Settings..." from the contextual menu to open the Mesh Settings Dialog. {{Note|To obtain accurate results, it is highly recommended to use a square mesh as much as possible.}} == Running Static Simulations in EM.Ferma == ===Two Simulation Engines=== 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. 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.  === Simulation Modes === EM.Ferma currently offers three different simulation modes: Analysis, Parametric Sweep and [[Optimization]]. An "Analysis" is a single-shot finite difference solution of your static structure. The structure is first discretized using a fixed-cell mesh and the Poisson equation is solved numerically everywhere in your computational domain. The field integrals and potential values at each mesh node are computed and the specified observables are written into data files. In a "Parametric Sweep", one ore more [[variables]] are varied at the specified steps(s). 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. The [[optimization]] mode requires definition of one or more objectives based on the standard output quantities. At the present time, the [[optimization]] mode is only available for the 2D Quasi-Static Mode of the EM.Ferma, which will be discussed separately later.  ===Static Simulation Engine Settings=== 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.  {{Note|The value of convergence error affect the accuracy of your 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.}}  === Observables in EM.Ferma === 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. At the end of an magnetostatic simulation, the magnetic field and magnetic vector potential values are computed at all the grid nodes. 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 observable. In the absence of a defined observable, the static simulation will be carried out and completed, but to action will take place.  === Defining Field Sensors === Just like other [[EM.Cube|EM.CUBE]] Modules, EM.Ferma has a Field Sensor observable, which plots 3D visualizations of electric and magnetic field components on a specified plane. However, unlike the other modules, EM.Ferma field sensors have two additional plots for electric scalar potential and magnitude of the magnetic vector potential. These are called the "EPot" and "HPot" nodes on the Navigation Tree. To define a Field Sensor, right-click on "Field Sensors" in the Navigation Tree and select "Insert New Observable..." from the contextual menu. The Field Sensor dialog allows the user to select the direction of the sensor (X, Y, Z), visualization type, and whether E-field output or H-field output will be shown during a sweep analysis.  The E-fields and H-fields are computed at each mesh node within the specified 2D Field Sensor plane. In other words, the resolution of the Field Sensor is controlled by the mesh resolution. === 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 of widely used field integralstheir definitions:

| <math> V = - \int_C \mathbf{E(r)} . \mathbf{dl} </math>| voltage.DAT

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

| <math> W_H = \frac{1}{2} \int\int\int_V \mu \vert \mathbf{H(r)} \vert ^2 dv </math>| energy_H.DAT|-! scope="row"| Ohmic Power Loss| <math> P_{ohmic} = \int\int\int_V \sigma \vert \mathbf{E(r)} \vert ^2 dv </math>| ohmic.DAT

| <math> C = Q\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>| capacitance.DAT|-! scope="row"| Self-Inductance

|}  In 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 volumeinductance.  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 five options: {| class="wikitable"DAT

! scope="colrow"| Field Integral TypeSelf-Inductance (Alternative)! scope="col"| Output Data File<math> L = 2W_M/I^2 = 2 \int \int \int_V \mu \vert \mathbf{H(sr)} \vert ^2 dv / \left( \oint_{C_o} \mathbf{H(r)} . \mathbf{dl} \right)^2</math>| inductance.DAT

! scope="row"| Voltage PathMutual Inductance| voltage<math> M = \Phi_H^{\prime}/I = \int\int_{S^{\prime}} \mu \mathbf{H(r)} . \mathbf{ds} / \oint_{C_o} \mathbf{H(r)} . \mathbf{dl} </math>| mutual_inductance.DAT

! scope="row"| Current LoopResistance| current<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"| Flux PlaneResistance (Alternative 1)| flux_H<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"| Flux BoxResistance (Alternative 2)| flux_E<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"| Energy BoxThermal Flux| energy_E<math> \Phi_T = \int\int_{S_o} \mathbf{q(r)} . \mathbf{ds} </math>| flux_T.DAT & energy_H|-! 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

The domain of == Discretizing 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 Physical Structure in the XY planeEM. Ferma ==

{{Note| If you define The Finite Difference technique discretizes the computational domain using a single Flux Box observable and 3D rectangular grid. EM.Ferma generates a single Voltage Path observable for your static projectfixed-cell mesh. This means that the extents of the mesh cells along the principal axes are fixed: &Delta;x, &Delta;y, &Delta;z. By default, the Capacitance mesh cell size is calculated and written 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 "capacitanceStatic Mesh" in the Navigation Tree, and select "Mesh Settings...DAT" data filefrom 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.}}

{{Note[[Image:Info_icon.png| If you define a single Flux Plane observable and a single Current Loop observable for your static project, the Inductance is calculated and written 30px]] Click here to "inductance learn more about '''[[Preparing_Physical_Structures_for_Electromagnetic_Simulation#Working_with_EM.DAT" data fileCube.27s_Mesh_Generators | Working with Mesh Generator]]'''.}}

== Modeling Transmission Lines Using [[Image:Info_icon.png|30px]] Click here to learn more about the properties of '''[[Glossary_of_EM.Cube%27s_Simulation-Related_Operations#Fixed-Cell_Brick_Mesh | EM.Ferma=='s Fixed-Cell Brick Mesh Generator]]'''.

<table><tr> <td> [[Image:qstaticQsource4.png|thumb|300px350px|Setting up a Transmission Line simulationEM.Ferma's Mesh Settings dialog.]]</td></tr></table>

== Running Static Simulations in EM.Ferma's electrostatic simulation engine features a 2D solution mode where the 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 the project.==

To explore === 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. Simulation Modes ===

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]] currently offers three different simulation modes as follows:

{| class="wikitable"|-! scope="col"| Simulation Mode! scope= 2D Quasi"col"| Usage! scope="col"| Number of Engine Runs! scope="col"| Frequency ! scope="col"| Restrictions|-Static Solution | style="width:120px;" | [[#Running an Electrostatic or Magnetostatic Analysis | Analysis]]| style="width:270px;" | Simulates the physical structure "As Is"| style="width:100px;" | Single run| style="width:200px;" | N/A| style="width:150px;" | None|-| style="width:120px;" | [[Parametric_Modeling_%26_Simulation_Modes_in_EM.Cube#Running_Parametric_Sweep_Simulations_in_EM.Cube | Parametric Sweep]]| style="width:270px;" | Varies the value(s) of one or more project variables| style="width:100px;" | Multiple runs| style="width:200px;" | N/A| style="width:150px;" | None|-| style="width:120px;" | [[Parametric_Modeling_%26_Simulation_Modes_in_EM.Cube#Performing_Optimization_in_EM.Cube | Optimization]]| style="width:270px;" | Optimizes the value(s) of Transmission Lines one or more project variables to achieve a design goal | style="width:100px;" | Multiple runs | style="width:200px;" | N/A| style="width:150px;" | None|}

At lower microwave frequencies (f < 10GHz)=== Running an Electrostatic, it is usually possible to perform a 2D electrostatic analysis of a transmission line structure and compute its characteristics impedance Z₀ and effective permittivity &epsilon;_eff. This "quasi-static approach" involves two steps:Magnetostatic or Thermal Analysis ===

<ol><li>First[[EM.Ferma]] has three independent but functionally similar static simulation engines: Electrostatic, you have remove all the dielectric materials from your structure Magnetostatic and replace them with free space (or air)Thermal. Obtain a 2D The electrostatic solution engine solves the electric form of your "airPoisson's equation for electric scalar potential subject to electric field boundary conditions, in the presence of electric sources (volume charges and fixed-filled" transmission line structure potential PEC blocks) and compute its capacitance per unit length C_adielectric material media.</li><li>NextThe magnetostatic engine solves the magnetic form of Poisson's equation for magnetic vector potential subject to magnetic field boundary conditions, obtain a 2D electrostatic solution in the presence of your actual transmission line structure with all magnetic sources (wire and volume currents and permanent magnetic blocks) and magnetic material media. The thermal engine solves the thermal form of its dielectric parts 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 compute its true capacitance per unit length Cinsulator material media.</li></ol>

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

<mathtable> \epsilon_{eff} = \frac{C}{C_a} <tr> <td> [[Image:Ferma L1 Fig11.png|thumb|left|600px|EM.Ferma's Simulation Run dialog.]] </td></tr></mathtable>

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 its characteristic impedance the Poisson equation is given by: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.

<math> Z_0 = \eta_0 \sqrt{ \frac{C_a}{C} } </math>==Static Simulation Engine Settings===

where &eta;₀ = 120&pi; &Omega; is {{Note|The value of convergence error affect the intrinsic impedance accuracy of 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. }}

== The guide wavelength of your transmission line at a given frequency f is then calculated from:2D Quasi-Static Simulation Mode==

and its propagation constant is given by<table><tr> <td> [[Image:Qsource14.png|thumb|left|450px|The 2D static domain dialog.]] </td></tr> </table>

where c is the speed of light in the free space.  EM.Ferma's 2D Quasi-Static mode automatically performs the two-step process described above and calculates &epsilon;<subtable>eff</subtr> and Z₀. So you don't need to modify your structure in the first step.  === Setting up a Transmission Line Simulation === To perform a transmission line simulation, first draw your structure in the project workspace just like a typical 3D structure. Define a "Field Sensor" observable in the Navigation Tree so as to capture the cross section of your structure as your desired transmission line profile.  Next, define a "2D Solution Plane" in the Navigation Tree based on 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 Z<subtd>0</sub> and effective permittivity &epsilon;_eff will be computed for the corresponding 2D Solution Plane. The results are written to two output data files named "solution_plane_Z0.DAT" and "solution_plane_EpsEff.DAT", respectively, where "solution_plane" is the default name of your 2D plane.  Many 2D quasi-static solutions can be obtained in the same analysis,for example, when your design contains many types of [[Transmission LinesImage:Qsource17.png|transmission lines]]. At the end thumb|left|480px|Electric field distribution of a quasi-static analysis, the electric field components and scalar potential at microstrip line on the selected 2D planes will still be computed and can be visualizedsolution plane. In the case of a parametric sweep, the data files will contain multiple data entries listed against the corresponding variable samples. Such data files can be plotted in EM.Grid. === Optimizing a Transmission Line === In an [[optimization]] simulation, the values of one or more [[variables]] 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;<sub>eff</subtd> and Z<sub>0</subtr>. Two possible objectives are "Z<subtr>0</subtd> == 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] == Simulation Examples / Gallery == {| border="0"|-| valign="top"|[[FileImage:ScreenCapture1Qsource18.png|thumb|left|350px480px|Classic Example: Two oppositely charged spheres.]]| valign="top"|[[File:iarray.png|thumb|left|350px|H-Field from array Electric scalar potential distribution of current loops.]]|-|}{| border="0"|-| valign="top"|[[File:ustrip.png|thumb|left|350px|Potential near the microstrip conductor from a quasistatic simulationline on the 2D solution plane.]]</td>| valign="top"|</tr>[[File:ustrip2.png|thumb|left|350px|Electric field near microstrip conductor from a quasistatic simulation. This Field Sensor's view mode has been set to Vector mode.]]|-|}</table>

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

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