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EM.Ferma

3,010 bytes removed, 03:27, 3 June 2015
With EM.Ferma, you can explore the electric fields due to volume charge distributions or fixed-potential perfect conductors, and magnetic fields due to wire or volume current sources and permanent magnets. Your structure may include dielectric or magnetic (permeable) material blocks. You can also use EM.Ferma's 2D quasi-static mode to compute the characteristic impedance (Z0) and effective permittivity of transmission line structures with complex cross section profiles.
=== Static Modeling Methods===
Click here to learn more about the theory of [[Electrostatic and Magnetostatic Methods]].
 
=== Electrostatics Analysis===
 
EM.Ferma solves the Poisson equation for the electric scalar potential subject to specified boundary conditions:
 
<math>\Delta\Phi(\mathbf{r}) = \nabla^2 \Phi(\mathbf{r}) = -\frac{\rho(\mathbf{r})}{\epsilon}</math>
 
 
where &Phi;(<b>r</b>) is the electric scalar potential, &rho;(<b>r</b>) is the volume charge density, and &epsilon; = &epsilon;<sub>r</sub> &epsilon;<sub>0</sub> is the permittivity of the medium.
 
 
The electric field boundary conditions at the interface between two material media are:
 
<math> \hat{\mathbf{n}} . [ \mathbf{D_2(r)} - \mathbf{D_1(r)} ] = \rho_s (\mathbf{r}) </math>
 
 
<math> \hat{\mathbf{n}} \times [ \mathbf{E_2(r)} - \mathbf{E_1(r)} ] = 0 </math>
 
 
where <math> \hat{\mathbf{n}} </math> is the unit normal vector at the interface pointing from medium 1 towards medium 2,
<b>D(r)</b> = &epsilon;<b>E(r)</b> is the electric flux density, <b>E(r)</b> is the electric field vector, and &rho;<sub>s</sub> is the surface charge density at the interface.
 
 
In a source-free region, &rho;(<b>r</b>) = 0, and Poisson's equation reduces to the familiar Laplace equation:
 
<math>\Delta\Phi(\mathbf{r}) = \nabla^2 \Phi(\mathbf{r}) = 0</math>
 
 
Keep in mind that in the absence of an electric charge source, you need to specify a non-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.
 
 
Once the electric scalar potential is computed, the electric field can easily be computed via the equation below:
 
<math> \mathbf{E(r)} = - \nabla \Phi(\mathbf{r})</math>
 
 
=== Magnetostatics Analysis===
 
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>
 
 
where <b>A(r)</b> is the magnetic vector potential, <b>J(r)</b> is the volume current density, and &mu; = &mu;<sub>r</sub> &mu;<sub>0</sub> is the permeability of the medium. The magnetic Poisson equation is vectorial in nature and involves a system of three scalar differential equations corresponding to the three components of <b>A(r)</b>.
 
 
The magnetic field boundary conditions at the interface between two material media are:
 
<math> \hat{\mathbf{n}} . [ \mathbf{B_2(r)} - \mathbf{B_1(r)} ] = 0 </math>
 
 
<math> \hat{\mathbf{n}} \times [ \mathbf{H_2(r)} - \mathbf{H_1(r)} ] = \mathbf{J_s(r)} </math>
 
 
where <math> \hat{\mathbf{n}} </math> is the unit normal vector at the interface pointing from medium 1 towards medium 2,
<b>B(r)</b> = &mu;<b>H(r)</b> is the magnetic flux density, <b>H(r)</b> is the magnetic field vector, and <b>J<sub>s</sub></b> is the surface current density at the interface.
 
 
Once the magnetic vector potential is computed, the magnetic field can easily be computed via the equation below:
 
<math> \mathbf{H(r)} = \frac{1}{\mu} \nabla \times \mathbf{A} (\mathbf{r})</math>
== Defining the Physical Structure in EM.Ferma ==
Kazem Sabet
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