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== Static Modeling Methods ==
<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>
== Static Fields Arising from Steady-State Conduction Currents ==
In an Ohmic conductor, the current density is related to the electric field as follows:
<math> \mathbf{J(r)} = \sigma \mathbf{E(r)} = -\sigma \nabla \Phi(\mathbf{r}) </math>
where σ is the electric conductivity. On the other hand, the continuity equation for a stationary current requires no charge buildup or decay inside a closed region. This means that
<math> \nabla . \mathbf{J(r)} = 0 </math>
These above two equations lead to the Laplace equation inside an Ohmic conductor medium:
<math>\nabla^2 \Phi(\mathbf{r}) = 0</math>
In addition, the boundary condition at a conductor-dielectric interface requires a vanishing normal derivative of the electric potential:
<math> \frac{\partial \Phi}{\partial n} = 0 </math>
At the interface between two contiguous conductors, the normal component of the current density must be continuous.
<math> \hat{\mathbf{n}} . [ \mathbf{J_2(r)} - \mathbf{J_1(r)} ] = 0 </math>
which can be written as:
<math> \sigma_1 \hat{\mathbf{n}} . \mathbf{E_1(r)} = \sigma_2 \hat{\mathbf{n}} . \mathbf{E_2(r)} </math>
== Magnetostatics Analysis==
<math> \frac{\partial^2\psi}{\partial x^2} + \frac{\partial^2\psi}{\partial y^2} + \frac{\partial^2\psi}{\partial z^2} = -f(\mathbf{r}) </math>
When f(<b>r</b>) = 0, one obtains the well-known Laplace equation, which applies to source-free regions.
The second derivative of ψ with respect to the x coordinate can be approximated by the second-order difference:
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In the special case of a uniform grid with Δx = Δy = Δz, it can be shown that in a source-free region:
<math> \psi(i,j,k) = \frac{1}{6} \big[ \psi(i+1,j,k) + \psi(i-1,j,k) + \psi(i,j+1,k) + \psi(i,j-1,k) + \psi(i,j,k+1) + \psi(i,j,k-1) \big] </math>
Two types of domain boundary conditions can be applied:
*Dirichlet boundary condition: ψ = k =const.
*Neumann boundary condition: ∂ψ/∂n = k = const.
In the above, ∂ψ/∂n denotes the normal derivative of the potential at the surface of the domain boundary. [[EM.Ferma]]'s default domain boundary condition for both the electrostatic and magnetostatic solvers is Dirichlet. At the interface between different material media, additional boundary conditions must be applied. These boundary conditions involve electric or magnetic field components. The field components can be expressed as partial derivatives of the potential, i.e. in the form of ∂ψ/∂x, ∂ψ/∂y or ∂ψ/∂z. Using the respective finite difference approximations of these derivatives, one arrives at fairly complicated difference equations involving the constitutive parameters ε, μ and σ, which must be solved simultaneously with the primary potential difference equations.
Note that the electrostatic Poisson and Laplace equations are of the scalar type, while the magnetostatic Poisson and Laplace equations are vectorial. As a result, the size of the numerical problem in the latter case is three times as large as the former case for the same mesh size.
== 2D Quasi-Static Solution of TEM Transmission Line Structures ==
and its characteristic impedance is given by:
<math> Z_0 = \frac{\eta_0}{\sqrt{ \epsilon_{eff}}} = \eta_0 \sqrt{ \frac{C_a}{C} } </math>
where η<sub>0</sub> = 120π Ω is the intrinsic impedance of the free space.