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

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

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

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

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

== Defining the Physical Structure in EM.Ferma ==<table><tr>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 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 ===<td>In [[EMImage:Magnet lines1.Cube]]'s other modules, material types are specified under the "Physical Structure" section png|thumb|left|400px| Vector plot of the Navigation Tree, and sources are organized under magnetic field distribution in a separate "Sources" sectioncylindrical permanent magnet. In those modules, the physical structure and its various material types typically represent all the CAD objects you draw in your project. Sources are virtual entities that might be associated with certain physical objects and provide the excitation of your boundary value problem.  In [[EM.Cube]]'s Static Module, materials and sources are all listed under the "Physical Structure" section of the Navigation Tree, and there is no separate "Sources" section. For example, you can define default zero-potential perfect electric conductors (PEC) in your project to model metal objects. You can also define fixed-potential PEC objects with a nonzero voltage, which can effectively act as a voltage source for your boundary value problem. In this case, you will solve the Lapalce equation subject to the specified nonzero potential boundary values. Both types of PEC objects are defined from the same PEC node of the Navigation Tree by assigning different voltage values. Charge and current sources are defined as CAD objects that you must draw in the project workspace.</td>=== Fixed-Potential PEC Objects=== A perfect electric conductor (PEC) is a material with &epsilon;<sub/tr>r</subtable> = 1 and &sigma; = &infin;. Under the static condition, every point on a PEC object has the same electric potential. By default, this is a zero potential, assuming the PEC object is "grounded". You can define a nonzero voltage value for a PEC group. In that case, the PEC object is effective turned into a voltage source. For example, tow parallel PEC plates, one with a zero potential and the other with a nonzero potential represent a simple air-filled capacitor.  To add a new Fixed-Potential PEC group to a project, right-click on "Fixed-Potential PEC Objects" on the Navigation Tree, and select "Insert New PEC..." From the PEC dialog, you can change the default red color and set a value for the "Voltage" in Volts.

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

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

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

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

<td> [[Image:Static1Ferma L8 Fig title.png|thumb|300pxleft| The PEC dialog400px| Vector plot of electric field distribution in a coplanar waveguide (CPW) transmission line.]] </td><td> [[Image:Static2.png|thumb|300px| The Dielectric/Magnetic Material dialog.]] </td>

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

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

=== Wire Current Sources Mesh generation ===

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

{{Note| If you draw [[[Curve Objects]]] 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>

=== Permanent Magnets2D Quasi-Static Simulation ===

A permanent magnet is typically a ferromagnetic material with a fixed inherent magnetization vector. As a result, it can be used <ul> <li> 2D Finite difference solution of cross section of transmission line structures</li> <li> 3D domain solution as well as 2D solution of a source in an magnetostatic problem. When longitudinally infinite version of the structure defined on a permeable &nbsp;2D plane&nbsp;&nbsp;</li> <li> Calculation of electric potential and electric field distribution</li> <li> Parametric sweep of transmission line&#39;s geometric and material has a permanent magnetization, the following relationship holds: parameters</li> <li> Optimization of transmission line&#39;s parameters for impedance design</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>

where <bul>M(r) </bli> is the magnetization vector. In SI units system, the Electric and magnetic field intensity and vector plots on planes<b/li>H <li> Electric and magnetic potential intensity plots on planes</bli> <li> Temperature and magnetization heat flux intensity and vector plots on planes<b/li>M <li> Electric and magnetic energy density, dissipated power density and thermal energy density plots on planes</bli> <li> both have the same units Animation of Afield and potential plots after parametric sweeps</mli> <li> Graphs of characteristic impedance and effective permittivity of transmission line structures vs. sweep variables</li> <li> Custom output parameters defined as mathematical expressions of standard outputs</li></ul>

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

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

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[[EM. In this case, Ferma]] offers the permanent magnetic can be effectively modeled by an equivalent surface current density on the surface following types of the permanent magnetic objectphysical objects:

<math> \mathbf{J_{Click on each category to learn more details about it in the [[Glossary of EM.Cube'sMaterials,eq}(r)} = \mathbf{M(r)} \times \hat{\mathbf{n}} </math>Sources, Devices & Other Physical Object Types]].

where <math> \hat{\mathbf{n}} </math> Your physical structure in EM.Ferma is typically made up of some kind of source object either in the unit outward normal vector at free space or in the surface presence of the permanent magnet objectone or more material objects. Note that EM.Ferma's electrostatic and magnetostatic or thermal simulation engines then discretize the volume of the permanent magnet still acts as a permeable 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 magnetostatic analysiscomputational 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 add create a new permanent magnet source object group to a project, right-click on its category name in the "Permanent MagnetsPhysical Structure" on section of the Navigation Tree, navigation tree and select one of the "Insert New Permanent Magnet SourceGroup..." From items from the Permanent Magnet Source Dialogcontextual menu. However, if you can change the default purple color of the source group or set the values of the relative permeabilitystart a new EM.Ferma project from scratch, Magnetization magnitude and unit direction vector components. The start drawing a new object without having previously defined any object groups, a new default direction vector is z"Fixed-directedPotential PEC" object group with a zero voltage is created and added to the navigation tree to hold your new geometric object.

{| borderclass="0wikitable"|-! scope="col"| Electrostatic Item! scope="col"| Corresponding Thermal Item|-| style="width:200px;" | Electric Scalar Potential| style="width:200px;" | Temperature|-| style="width:200px;" | Electric Field| style="width:200px;" | Heat Flux Density|-| style="width:200px;" | Perfect Electric Conductor| style="width:200px;" | Perfect Thermal Conductor

| valignstyle="topwidth:200px;"|Dielectric Material[[File:vsource.png|thumb|left|250px|EM.Ferma's Voltage Source Dialog]]| valignstyle="bottom"|[[Filewidth:qsource.png|thumb|left|250px|EM.Ferma's Charge Source Dialog]]| valign=200px;"top"|[[File:isource.png|thumb|left|250px|EM.Ferma's Current Source Dialog]]Insulator Material

== Computational Domain Once a new object group node has been created in the navigation tree, it becomes and Discretization==remains the "Active" object group, which is always listed in bold letters. When you draw a new geometric object such as a box or a sphere, its name is added under the currently active object group. There is only one object group that is active at any time. Any group can be made active by right-clicking on its name in the navigation tree and selecting the '''Activate''' item of the contextual menu.

[[Image:fermbcInfo_icon.png|thumb30px]] Click here to learn more about '''[[Building Geometrical Constructions in CubeCAD#Transferring Objects Among Different Groups or Modules |250px|Boundary Condition DialogMoving Objects among Different Groups]]'''.

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 === A Note on Material and then specify the domain boundary conditions. Source Types in 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. ===

In [[Image:qsource2EM.png|thumb|400px|The blue wireframe around Cube]]'s other modules, material types are categorized under the CAD objects defines the extents "Physical Structure" section of the computational domain. The specified boundary conditions navigation tree, and sources are applied on organized under a separate "Sources" section. In those modules, all the domain wallsgeometric 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.

To modify the domain settingsIn EM.Ferma, click materials and sources are all lumped together and listed under the Domain button of the Simulate Toolbar or right-click on "3D Static DomainPhysical Structure" entry in section of the Navigation Tree and select "Domain Settings..navigation tree.In other words, there is no separate " from the contextual menuSources" section. In the Domain Settings DialogFor example, the computational domain you can be defined in two different ways: Default and Custom. The define default type places an enclosing box with a specified offset from the largest bounding box of zero-potential perfect electric conductors (PEC) in your project's CAD to model metal objects. The default offset value is 20 project unitsYou can also define fixed-potential PEC objects with a nonzero voltage, but you which can change this effectively act as a voltage source for your boundary value arbitrarilyproblem. The custom type defines a fixed domain box by specifying In this case, you will solve the coordinates Lapalce equation subject to the specified nonzero potential boundary values. Both types of its two opposite corners labeled Min PEC objects are defined from the same PEC node of the navigation tree by assigning different voltage values. Charge and Max current sources are also defined as geometric objects, and you have to draw them in the world coordinate systemproject workspace just like other material objects.

===EM.Ferma's Computational 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.Domain Box===

===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 Static Mesh===domain can be seen as a blue cubic wireframe that surrounds all of the CAD objects in the project workspace.

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 sizedomain settings, click the Mesh Settings Domain button of the Simulate Toolbar or right-click on "3D Static MeshDomain" entry in the Navigation Tree, and select "Mesh Domain Settings..." from the contextual menu to open . In the Mesh Domain Settings Dialog, the computational domain can be defined in two different ways: Default and Custom. {{Note|To obtain accurate resultsThe 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, it is highly recommended to use but you can change this value arbitrarily. The custom type defines a square mesh as much as possiblefixed domain box by specifying the coordinates of its two opposite corners labeled Min and Max in the world coordinate system.}}

== Running Static Simulations in <table><tr> <td> [[Image:Qsource5.png|thumb|left|480px|EM.Ferma =='s Domain Settings dialog.]] </td></tr></table>

===Two Simulation EnginesDomain Boundary Conditions===

*EM.Ferma has two independent but functionally similar static simulation engines: Electrostatic and Magnetostatic. The electrostatic engine solves allows you to specify the electric form of Poisson's equation for electric scalar potential subject to electric field boundary conditions, in on the presence of electric sources (volume charges domain box. Two options are available. The Dirichlet boundary condition is the default option and is specified as a fixed-potential PEC blocks) and dielectric material mediavalue on the surface of the domain walls. By default, this value is 0 Volts. The magnetostatic engine solves Neumann boundary condition specifies the magnetic form normal derivative of Poisson's equation for magnetic vector the electric scalar potential subject on the surface of the domain walls. This is equivalent to magnetic a constant normal electric field boundary conditions, in component on the presence of magnetic sources (wire domain walls and volume currents and permanent magnetic blocks) and magnetic material mediaits value is specified in V/m.

In EM.Ferma you don't have to select any specific *The magnetostatic simulation engine. The program looks at always assumes Dirichlet domain boundary conditions and sets the values of 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 vector potential to zero electric fieldson all the domain walls.

To run a static simulation, first you have to open *EM.Ferma provides two options for thermal boundary conditions on the Run Dialogdomain box. This The Dirichlet boundary condition is done by clicking the "Run" button default option and is specified as a fixed temperature value on the surface of the Simulate Toolbardomain walls. By default, or by selecting this value is 0&deg;C. The Neumann boundary condition specifies the "Run" item normal derivative of the Simulate Menu, or simply using temperature on the keyboard shortcut "Ctrl+R"surface of the domain walls. The only available simulation engine This is "Static". Clicking the Run button of this dialog starts equivalent to a static analysisconstant heat flux passing through the domain walls and its value is specified in W/m². A separate window pops up which reports the progress of zero heat flux means a perfectly insulated domain box and is known as the current simulationadiabatic boundary condition.

=== 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 Modes ===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.

<table><tr> <td> [[Image:fermbc.png|thumb|left|480px|EM.Ferma currently offers three different simulation modes: Analysis, Parametric Sweep and [[Optimization's Boundary Conditions dialog.]]. 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 and potential values at each mesh node are computed and the specified observables are written into data files.</td></tr></table>

In a "Parametric Sweep", one ore more [[variables]] are varied at the specified steps(== EM.Ferma'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.Simulation Data & Observables ==

The [[optimization]] mode requires definition At the end of one or more objectives based on an electrostatic simulation, the standard output quantitieselectric field vector and electric scalar potential values are computed at all the mesh grid points of the entire computational domain. At the present timeend of an magnetostatic simulation, the [[optimization]] mode is only available for magnetic field vector and magnetic vector potential values are computed at all the 2D Quasi-Static Mode of the EMgrid nodes.FermaAt the end of a thermal simulation, which will be discussed separately laterthe temperature and heat flux vector are computed at all the mesh grid points of the entire computational domain.

EM.Ferma currently uses a single iterative linear system solver based on offers the stabilized Bi-Conjugate Gradient (BiCG) method to solve the matrix equations which result from the discretization following types 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. output simulation data:

{{Note|The value class="wikitable"|-! scope="col"| Icon! scope="col"| Simulation Data Type! scope="col"| Observable Type! scope="col"| Applications|-| style="width:30px;" | [[File:fieldsensor_icon.png]]| style="width:150px;" | Near-Field Distribution Maps| style="width:150px;" | [[Glossary of EM.Cube's Simulation Observables & Graph Types#Near-Field_Sensor_Observable |Near-Field Sensor]] | style="width:450px;" | Computing electric and magnetic field components, electric scalar potential and magnitude of magnetic vector potential on a planar cross section of convergence error affect the accuracy 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 your simulation resultsEM. For most practical scenarios, 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 default values are adequatecomputational domain |-| style="width:30px;" | [[File:fieldsensor_icon. You can reduce 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 convergence error for better accuracy at 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 expense computational domain |-| style="width:30px;" | [[File:field_integ_icon.png]]| style="width:150px;" | Field Integral Quantities| style="width:150px;" | [[Glossary of longer computation timeEM.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 |}}

== Working with Static Click on each category to learn more details about it in the [[Glossary of EM.Cube's Simulation Data ==Observables & Graph Types]].

At the end of an electrostatic simulation, the electric <table><tr> <td> [[Image:Ferma L1 Fig15.png|thumb|left|640px|Electric field and electric scalar potential values are computed at all the mesh grid points distribution 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 a spherical charge on the screen only if you define a horizontal field sensor observableplane. In the absence of a defined observable, the static simulation will be carried out and completed, but to action will take place. ]] </td></tr> === Defining Field Sensors ===<tr>  Just like other <td> [[EMImage:Ferma L1 Fig16.Cubepng|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 thumb|left|640px|Electric scalar potential and magnitude distribution of the magnetic vector potential. These are called the "EPot" and "HPot" nodes a spherical charge 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 horizontal field 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.]] </td></tr>=== 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 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 below:>

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

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

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

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

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

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

===The domain of the field integral is set using the "Integration Box Coordinates" section of the Field Integral dialog. Box domains are specified by the coordinates of two opposite corners. Voltage Path requires a line; therefore, two of the coordinates of the two corners must be identical. Otherwise, an error message will pop up. For example, (0, 0, 0) for Corner 1 and (10, 0, 0) for Corner 2 define a Z-directed line segment. Current Loop requires a rectangle; therefore, one of the coordinates of the two corners must be identical. For example, (0, 0, 0) for Corner 1 and (10, 10, 0) for Corner 2 define a rectangle in the XY plane. Static Mesh===

After The Finite Difference technique discretizes the completion of computational domain using a static simulation, the result of the field integrals are written into "3D rectangular grid.DAT" data files. These files can be accessed using [[EM.Cube]]'s Data Manager.   == Modeling Transmission Lines Using EM.Ferma== [[Image:qstatic.png|thumb|300px|Setting up generates a Transmission Line simulationfixed-cell mesh.]] ===2D Electrostatic Simulation Mode=== EMThis means that the extents of the mesh cells along the principal axes are fixed: &Delta;x, &Delta;y, &Delta;z.Ferma's electrostatic simulation engine features a 2D solution mode where By default, the model mesh cell size is treated as a longitudinally infinite structure in the direction normal set 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 unit project along all the projectthree directions (with &Delta;x = &Delta;y = &Delta;z). To explore EM.Ferma's 2D modemodify the cell size, click the Mesh Settings button of the Simulate Toolbar or right-click on "2D Solution PlanesStatic Mesh" in the Navigation Tree , and select "2D Domain Mesh Settings..." from the contextual menu. In the 2D Static Domain dialog, enable the checkbox labeled "Treat Structure as Longitudinally Infinite across Each 2D Plane Specified Below". The user is then able to Add or Edit 2D Solution Plane definitions to open the solution listMesh Settings Dialog. In the Add/Edit 2D Solution Plane dialog, you can choose a name other than the default name and select one of the available field sensor definitions in your project.  At the end of a 2D electrostatic analysis, you can view the electric field and potential results on the respective field sensor planes. It is assumed that your structure is invariant along the direction normal to the 2D solution plane. Therefore, your computed field and potential profiles must be valid at all the planes perpendicular to the specified longitudinal direction.  === 2D Quasi-Static Solution of Transmission Lines === At lower microwave frequencies (f < 10GHz), it is usually possible to perform a 2D electrostatic analysis of a transmission line structure and compute its characteristics impedance Z₀ and effective permittivity &epsilon;_eff. This "quasi-static approach" involves two steps: <ol><li>First, you have remove all the dielectric materials from your structure and replace them with free space (or air). Obtain a 2D electrostatic solution of your "air-filled" transmission line structure and compute its capacitance per unit length C_a.</li><li>Next, obtain a 2D electrostatic solution of your actual transmission line structure with all of its dielectric parts and compute its true capacitance per unit length C.</li></ol> Then effective permittivity of the transmission line structure is then calculated from the equation:

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

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

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

The guide wavelength {| class="wikitable"|-! scope="col"| Simulation Mode! scope="col"| Usage! scope="col"| Number of your transmission line at Engine Runs! scope="col"| Frequency ! scope="col"| Restrictions|-| style="width:120px;" | [[#Running an Electrostatic or Magnetostatic Analysis | Analysis]]| style="width:270px;" | Simulates the physical structure "As Is"| style="width:100px;" | Single run| style="width:200px;" | N/A| style="width:150px;" | None|-| style="width:120px;" | [[Parametric_Modeling_%26_Simulation_Modes_in_EM.Cube#Running_Parametric_Sweep_Simulations_in_EM.Cube | Parametric Sweep]]| style="width:270px;" | Varies the value(s) of one or more project variables| style="width:100px;" | Multiple runs| style="width:200px;" | N/A| style="width:150px;" | None|-| style="width:120px;" | [[Parametric_Modeling_%26_Simulation_Modes_in_EM.Cube#Performing_Optimization_in_EM.Cube | Optimization]]| style="width:270px;" | Optimizes the value(s) of one or more project variables to achieve a given frequency f is then calculated fromdesign goal | style="width:100px;" | Multiple runs | style="width:200px;" | N/A| style="width:150px;" | None|}

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

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

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

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

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

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

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

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

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

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

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

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

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

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