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== Static Modeling Methods ==
Static or quasi-static approximations of Maxwell's equations can be reliably applied in two different scenarios: at low frequencies from DC to a few Megahertz, or when the total electrical size of your physical structure is a fraction of the wavelength, and wave retardation effects are negligible. In the latter case, your physical structure is effectively considered as a lumped device. Under those conditions, the electric and magnetic fields decouple from each other. Electric fields can be computed from charge sources or their equivalents and magnetic fields can be computed from current sources or their equivalents.
Under the static assumptions, Maxwell's equations reduce to elliptic partial differential equations known as the Poisson and Laplace equations. These equations can be solved analytically only for a few canonical geometries with very simple boundary conditions. For most practical and realistic problems, you need to utilize a numerical technique and seek a computer solution. The Poisson and Laplace equations can be solved numerically using the finite difference (FD) method.
== Electrostatics Analysis==
At very low frequencies, as ω→0 and k→0, the [[EM.Ferma]] solves the Poisson Basic_Electromagnetic_Theory#Electric_and_Magnetic_Potentials | Helmholtz equation for the electric scalar potential ]] reduces to the Poisson equation subject to specified boundary conditions:
<math>\Delta\Phi(\mathbf{r}) = \nabla^2 \Phi(\mathbf{r}) = -\frac{\rho(\mathbf{r})}{\epsilon}</math>
<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 Transmission Line Structures Fields Arising from Steady-State Conduction Currents ==
At lower microwave frequencies (f < 10GHz), multi-In an Ohmic 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 the current density is usually possible related 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 stepsthe electric field as follows:
<olmath><li>First, you have remove all the dielectric materials from your structure and replace them with free space \mathbf{J(or airr). Obtain a 2D electrostatic solution of your "air} = \sigma \mathbf{E(r)} = -filled" transmission line structure and compute its capacitance per unit length C<sub>a\sigma \nabla \Phi(\mathbf{r}) </submath> where σ is the electric conductivity.</li><li>NextOn the other hand, obtain the continuity equation for a 2D electrostatic solution of your actual transmission line structure with all of its dielectric parts and compute its true capacitance per unit length Cstationary current requires no charge buildup or decay inside a closed region.</li></ol>This means that
Then effective permittivity of the transmission line structure is then calculated from the equation:<math> \nabla . \mathbf{J(r)} = 0 </math>
These above two equations lead to the Laplace equation inside an Ohmic conductor medium: <math> \epsilon_nabla^2 \Phi(\mathbf{effr} ) = \frac{C}{C_a} 0</math>
and its characteristic impedance is given byIn addition, the boundary condition at a conductor-dielectric interface requires a vanishing normal derivative of the electric potential: <math> Z_0 = \eta_0 \sqrt{ \frac{C_a}{C} } </math>
where η<submath>\frac{\partial \Phi}{\partial n} = 0</submath> = 120π Ω is the intrinsic impedance of the free space.
The guide wavelength At the interface between two contiguous conductors, the normal component of your transmission line at a given frequency f is then calculated from:the current density must be continuous. <math> \hat{\mathbf{n}} . [ \mathbf{J_2(r)} - \mathbf{J_1(r)} ] = 0 </math>
<math> \lambda_g = \frac{\lambda_0}{\sqrt{\epsilon_{eff}}} = \frac{c}{f\sqrt{\epsilon_{eff}}} </math>which can be written as:
and its propagation constant is given by: <math> \beta = k_0sigma_1 \sqrthat{\epsilon_mathbf{effn}} = . \fracmathbf{2\pi f}{cE_1(r)}= \sigma_2 \sqrthat{\epsilon_mathbf{effn}} . \mathbf{E_2(r)} </math> where c is the speed of light in the free space.
== Magnetostatics Analysis==
At very low frequencies, as ω→0 and k→0, the [[EM.Ferma]] solves the Poisson Basic_Electromagnetic_Theory#Electric_and_Magnetic_Potentials | Helmholtz equation for the magnetic vector potential ]] reduces to the vector Poisson equation subject to specified boundary conditions:
<math>\Delta \mathbf{A} (\mathbf{r}) = \nabla^2 \mathbf{A}(\mathbf{r}) = - \mu \mathbf{J}(\mathbf{r}) </math>
<math> \mathbf{H(r)} = \frac{1}{\mu} \nabla \times \mathbf{A} (\mathbf{r}) </math>
The relationship between the magnetic flux density and magnetic field vectors is rather different inside permeable materials that have a permanent intrinsic magnetization. Examples of such materials are ferromagnetic material that are used as permanent magnets. When a permeable material has a permanent magnetization, the following relationship holds:
<math> \mathbf{B(r)} = {\mu} \big[ \mathbf{H(r)} + \mathbf{M(r)} \big] </math>
where <b>M(r)</b> is the magnetization vector. In the 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)} = \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 that case, the permanent magnetization 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 with a relative permeability of μ in the magnetostatic analysis.
== Free-Space Magnetic Field and Vector Potential ==
<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>
== The Finite Difference Technique ==
The general form of Poisson's equation for any potential ψ can be expressed as:
<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:
<math> \frac{\partial^2\psi(\mathbf{r})}{\partial x^2} \approx \frac{\psi(x+\Delta x,y,z)-2\psi(x,y,z)+\psi(x-\Delta x,y,z)}{(\Delta x)^2} </math>
Similar expressions can be written for the second derivative with respect to the y and z coordinates.
In the finite difference method, the computational domain is discretized using a 3D rectangular grid as shown on the figure below. The grid spacing along the three principal coordinate axes is denoted by Δx, Δy and Δz, respectively. In this grid, the coordinates of any point (x,y,z) in the space can be expressed as x = iΔx, y = jΔy and z = kΔz. Therefore, every point can simply be represented by an index triplet (i,j,k).
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[[Image:FD grid.png|thumb|left|480px| The 3D rectangular grid used to mesh the computational domain.]]
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The potential at the point (x,y,z) can be expressed in terms of the potentials at its six neighboring grid points along the principal axes. This creates a 7-point computational molecule shown in the figure below:
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[[Image:FD 7Point.png|thumb|left|480px| The 7-point computational molecule used by the finite difference solver.]]
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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 ==
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 = \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.
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.
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