</tr>
</table>
== 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 Φ(<b>r</b>) is the electric scalar potential expressed in Volts [v], ρ(<b>r</b>) is the volume charge density expressed in C/m<sup>3</sup>, and ε = ε<sub>r</sub> ε<sub>0</sub> is the permittivity of the medium having the units of F/m.
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> = ε<b>E(r)</b> is the electric flux density expressed in C/m<sup>2</sup>, <b>E(r)</b> is the electric field vector expressed in V/m, and ρ<sub>s</sub> is the surface charge density at the interface having the units of expressed in C/m<sup>2</sup>.
In a source-free region, ρ(<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>
== The Solution of Poisson's Equation ==
The general form of Poisson's equation for a scalar quantity Ψ(<b>r</b>) can be expressed as:
<math>\Delta\Psi(\mathbf{r}) = \nabla^2 \Psi(\mathbf{r}) = -f(\mathbf{r}) </math>
where f is a general arbitrary function of the position vector <b>r</b>. In the case of a unit point source, the above equation reduces to:
<math>\Delta\Psi(\mathbf{r}) = \nabla^2 \Psi(\mathbf{r}) = -\delta(\mathbf{r}) </math>
where δ(<b>r</b>) is the three-dimensional Dirac delta function. The solution to the above equation is called the Green's function of the Poisson equation for a certain boundary value problem and is denoted by G(<b>r|r<sup>′</sup></b>). Then, the solution to the general Poisson equation can be expressed in the terms of the Green's function as follows:
<math> \Psi\mathbf{(r)} = \int\int\int_V G(\mathbf{r|r^{\prime}}) f(\mathbf{r^{\prime}}) dv^{\prime} </math>
== Free-Space Electric Field and Scalar Potential ==
There are very few boundary value problems for which you can find closed-form analytical Green's functions. One of them is the electric scalar potential due to a charge distribution in the free space. The potential Green's function for a free-space medium is given by:
<math> G_{\Phi}(\mathbf{r|r^{\prime}}) = -\frac{1}{4\pi | \mathbf{r - r^{\prime}} | } </math>
If the volume charge density is denoted by ρ(<b>r</b>), then the electric scalar potential is given by:
<math> \Phi\mathbf{(r)} = \frac{1}{4\pi\epsilon} \int\int\int_V \frac{\rho(\mathbf{r^{\prime}})}{ | \mathbf{r - r^{\prime}} | } dv^{\prime} </math>
where V is the volume containing the charge distribution.
The electric field due to the charge distribution is then given by:
<math> \mathbf{E(r)} = - \nabla \Phi(\mathbf{r}) = \frac{1}{4\pi\epsilon} \int\int\int_V \frac{\mathbf{r - r^{\prime}} }{ | \mathbf{r - r^{\prime}} |^3 } \rho(\mathbf{r^{\prime}}) dv^{\prime} </math>
== 2D Quasi-Static Solution of TEM Line Structures ==
At lower microwave frequencies (f < 10GHz), multi-conductor transmission line structures usually support either a dominant transverse electromagnetic (TEM) propagating mode or a dominant quasi-TEM propagating mode. These modes are almost non-dispersive, and their behavior can be regarded as frequency-independent. As a result, it is usually possible to perform a 2D electrostatic analysis of a transmission line structure and compute its characteristics impedance Z<sub>0</sub> and effective permittivity ε<sub>eff</sub>. The "quasi-static approach" to modeling of a TEM transmission line 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<sub>a</sub>.</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:
<math> \epsilon_{eff} = \frac{C}{C_a} </math>
and its characteristic impedance is given by:
<math> Z_0 = \eta_0 \sqrt{ \frac{C_a}{C} } </math>
where η<sub>0</sub> = 120π Ω is the intrinsic impedance of the free space.
The guide wavelength of your transmission line at a given frequency f is then calculated from:
<math> \lambda_g = \frac{\lambda_0}{\sqrt{\epsilon_{eff}}} = \frac{c}{f\sqrt{\epsilon_{eff}}} </math>
and its propagation constant is given by:
<math> \beta = k_0\sqrt{\epsilon_{eff}} = \frac{2\pi f}{c}\sqrt{\epsilon_{eff}} </math>
where c is the speed of light in the free space.
== 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 μ = μ<sub>r</sub> μ<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> = μ<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>
== Free-Space Magnetic Field and Vector Potential ==
The magnetic vector potential stratifies the vectorial form of Poisson's equation. Therefore, the free-space Green's function for the magnetic vector potential has the same functional dependence as that of the scalar electric potential but in a dyadic form, which is given by:
<math> \mathbf{\overline{\overline{G}}_A}(\mathbf{r|r^{\prime}}) = -\frac{1}{4\pi | \mathbf{r - r^{\prime}} | } \mathbf{\overline{\overline{I}}} </math>
where is <math> \mathbf{\overline{\overline{I}}} </math> is the unit dyad.
If the volume current density is denoted by <b>J(r)</b>, then the magnetic vector potential is given by:
<math> \mathbf{A(r)} = \frac{\mu}{4\pi} \int\int\int_V \frac{ \mathbf{J(r^{\prime})}}{ | \mathbf{r - r^{\prime}} | } dv^{\prime} </math>
where V is the volume containing the current distribution.
The magnetic field due to the current distribution is then given by:
<math> \mathbf{H(r)} = \frac{1}{\mu} \nabla \times \mathbf{A} (\mathbf{r}) = \frac{1}{4\pi} \int\int\int_V \mathbf{J(r^{\prime})} \times \frac{ \mathbf{r - r^{\prime}} }{ | \mathbf{r - r^{\prime}} |^3 } dv^{\prime} </math>
== Differential Form of Maxwell's Equations & the Yee Cell ==