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Static or quasi-static approximations of [[Image:Back_icon.png|30px]] '''[[A_Review_of_Maxwell%27s_Equations_%26_Computational_Electromagnetics_(CEM) | Back to 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.Equations Page]]'''

At very low frequencies, as &omega;&rarr;0 and <math>q = -k&rarr;0, the [[Basic_Electromagnetic_Theory#Electric_and_Magnetic_Potentials | Helmholtz equation for the electric scalar potential]] reduces to the Poisson equation subject to specified boundary conditions:\nabla T(\mathbf{r})</math>

where q is the heat flux density with units of W/m<mathsup>\Delta\Phi(\mathbf{r}) = \nabla^2 \Phi</sup>, T(\mathbf{<b>r}) = -\frac{\rho(\mathbf{r})}{\epsilon}</mathb>) is the temperature expressed in &deg;C or &deg;K, &nabla; is the gradient operator and k is the thermal conductivity with units of W/(m.K). It can be shown that the distribution of temperature is governed by the heat diffusion equation subject to the appropriate boundary conditions:

where &Phi;(<bmath>\nabla^2 T(\mathbf{r</b>}) is the electric scalar potential expressed in Volts [v], &rho;- \frac{1} {\alpha}\frac{\partial T}{\partial t} = \left(<b>r</b>\frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right) is the volume charge density expressed in C/m³, and &epsilon; - \frac{1} {\alpha}\frac{\partial T}{\partial t} = &epsilon;<sub>- \frac{w(\mathbf{r})}{k} </submath> &epsilon;₀ is the permittivity of the medium having the units of F/m.

The electric field boundary conditions at where &alpha; = k/(&rho;_Vc_p) is the interface between two material media are:thermal diffusivity with units of m²/s, &rho;_V is the volume mass density having units of kg/m³, c_p is the specific heat capacity of the medium having units of J/(kg.K), and w(<b>r</b>) is the volume heat source density with units of W/m³.

<math> \hat{\mathbf{n}} . [ \mathbf{D_2(r)} In the steady- \mathbf{D_1(r)} ] = \rho_s (\mathbf{r}) </math>state regime, the time derivative vanishes and the diffusion equation reduces to the Poisson equation:

<math> \hat{nabla^2 T(\mathbf{n}r} ) = - \times [ \mathbffrac{E_2w(r)} - \mathbf{E_1(r})} ] = 0 {k} </math>

where <math> \hat{\mathbf{n}} </math> is the unit normal vector at the interface pointing from medium 1 towards medium 2In a source-free region,<b>D(r)</b> = &epsilon;<b>E(r)</b> is the electric flux density expressed in C/m², <b>E(r)</b> is the electric field vector expressed in V/m, and &rho;_s is the surface charge density at the interface having Poisson equation reduces further to the units of expressed in C/m². Laplace equation:

In a source-free region, &rho;(<bmath>\nabla^2 T(\mathbf{r</b>}) = 0, and Poisson's equation reduces to the familiar Laplace equation: </math>

<math>\Delta\PhiThe steady-state heat diffusion equations are elliptic partial differential 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 (\mathbf{r}FD) = \nabla^2 \Phi(\mathbf{r}) = 0</math>method.

Once The simplest thermal boundary condition at the electric scalar potential is computed, walls of the electric field can easily be computed via computational domain is the equation belowDirichlet boundary condition:

<math> \mathbf{E(r)} = - \nabla \PhiT(\mathbf{r})= T_0 </math>

== The Solution Neumann boundary condition constrains the rate of Poisson's Equation ==heat flow through the domain boundary walls:

The general form of Poisson's equation for a scalar quantity &Psi;(<bmath>-k \frac{\partial T}{\partial n} = -k \mathbf{\hat{n}} . \nabla T(\mathbf{r}) = q_s </bmath>) can be expressed as:

where q<mathsub>\Delta\Psi(\mathbf{r}) = \nabla^2 \Psi(\mathbf{r}) = -f(\mathbf{r}) s</mathsub>is the heat flux passing through the boundary walls.

where f is a general arbitrary function of the position vector If q<bsub>rs</bsub>. In = 0, the case of Neumann boundary condition is also known as the adiabatic boundary condition, which represents a unit point source, the above equation reduces toperfectly insulated surface:

<math>\Deltafrac{\Psi(\mathbfpartial T}{r}) = \nabla^2 \Psi(\mathbf{rpartial n}) = -\delta(\mathbf{r}) 0 </math>

where &delta;(<b>r</b>) is At the three-dimensional Dirac delta function. The solution to interface between the above equation is called the Green's function surface of the Poisson equation for a certain boundary value problem solid object and is denoted by G(<b>r|r^&prime;</b>). Thenair, the solution to the general Poisson equation can convective boundary condition must be expressed in the terms of the Green's function as followsenforced:

<math> -k \Psifrac{\mathbfpartial T}{(r)\partial n} = -h \int\int\int_V Gleft[ T(\mathbf{r|r^{\prime}}) f(\mathbf{r^- T_{\primeinfty}}) dv^{\prime} right] </math>

== Free-Space Electric Field where T_&infin; is the ambient temperature, and Scalar Potential ==h is the coefficient of convective heat transfer having units of W/(m².K).

There are very few The convective boundary value problems for which you can find closed-form analytical Green's functions. One of them condition 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 byspecial case of Robin boundary condition:

<math> G_\left[ -k \frac{\Phipartial T}{\partial n} + h T(\mathbf{r|r^{\prime}}) = -\frac{1}{4right]_\pi | Omega = f(\mathbf{r - r^{\prime}} | } ) </math>

If the volume charge density is denoted by where &rhoOmega;is the boundary surface and f(<b>r</b>), then the electric scalar potential is given by:can be an arbitrary function in general.

<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>= The Analogy between Thermal and Electrostatic Equations ==

where V is Let us now compare the volume containing steady-state thermal Poisson equation to the charge distribution. electrostatic Poisson equation:

One can see a one-to-one correspondence between the electrostatic and thermal quantities: Temperature T(<mathb> \mathbf{E(r</b>)} = - \nabla \is analogous to the electric scalar potential &Phi;(\mathbf{<b>r}</b>) = \frac{1}{4\pi\epsilon} \int\int\int_V \frac{\mathbf{, the volume heat source density w(<b>r - r^{\prime}} }{ | \mathbf{r - r^{\prime}} |^3 } \</b>) is analogous to the volume charge density &rho;(\mathbf{<b>r^{\prime}}) dv^{\prime} </mathb>), and the thermal conductivity k is analogous to the permittivity &epsilon;.

where &sigma; 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)} | class= 0 </math>"wikitable" |-These above two equations lead to the Laplace equation inside an Ohmic conductor medium:! scope="col"| Thermal Item <math>\nabla^2 \Phi(\mathbf{r}) ! scope= 0</math>"col"| Corresponding Electrical Item 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} | style= 0 </math>"width:200px;" | Temperature 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)} ] | style= 0 </math> which can be written as"width: <math> \sigma_1 \hat{\mathbf{n}} . \mathbf{E_1(r)} = \sigma_2 \hat{\mathbf{n}} . \mathbf{E_2(r)} </math> == Magnetostatics Analysis== At very low frequencies, as &omega200px;&rarr;0 and k&rarr;0, the [[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:Electric Scalar Potential <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; | style= &mu"width:200px;_r &mu;₀ 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>. " | Heat Flux Density The magnetic field boundary conditions at the interface between two material media are| style="width:200px;" | Electric Field <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)} ] | style= \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"width:200px;<b>H(r)</b> is the magnetic flux density, <b>H(r)</b> is the magnetic field vector, and <b>J_s</b> is the surface current density at the interface. " | Perfect Thermal Conductor Once the magnetic vector potential is computed, the magnetic field can easily be computed via the equation below:  <math> \mathbf{H(r)} | style= \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"width: 200px;" | Perfect Electric Conductor|-<math> \mathbf{B(r)} | style= {\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"width:200px;" | Insulator Material <math> \mathbf{J_{eq}(r)} | style= \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"width:  <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 &mu200px; in the magnetostatic analysis." | Dielectric Material|-| style== 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"width:200px;" | Volume Heat Source <math> \mathbf{\overline{\overline{G}}_A}(\mathbf{r|r^{\prime}}) style= -\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"width: <math> \mathbf{A(r)} = \frac{\mu}{4\pi} \int\int\int_V \frac{ \mathbf{J(r^{\prime})}}{ 200px;" | \mathbf{r - r^{\prime}} | } dv^{\prime} </math>Volume Charge 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>

The general form of Poisson's equation for any potential &psi; temperature can be expressed as:

<math> \frac{\partial^2\psiT}{\partial x^2} + \frac{\partial^2\psiT}{\partial y^2} + \frac{\partial^2\psiT}{\partial z^2} = -f(\mathbf{r}) </math>

The second derivative of &psi; T with respect to the x coordinate can be approximated by the second-order difference:

<math> \frac{\partial^2\psiT(\mathbf{r})}{\partial x^2} \approx \frac{\psiT(x+\Delta x,y,z)-2\psi2T(x,y,z)+\psiT(x-\Delta x,y,z)}{(\Delta x)^2} </math>

The potential twmperature at the point (x,y,z) can be expressed in terms of the potentials temperatures at its six neighboring grid points along the principal axes. This creates a 7-point computational molecule shown in the figure below:

[[Image:FD 7Point7PointA.png|thumb|left|480px| The 7-point computational molecule used by the finite difference solver.]]

<math> \psiT(i,j,k) = \frac{1}{6} \big[ \psiT(i+1,j,k) + \psiT(i-1,j,k) + \psiT(i,j+1,k) + \psiT(i,j-1,k) + \psiT(i,j,k+1) + \psiT(i,j,k-1) \big] </math> Two types of domain boundary conditions can be applied: *Dirichlet boundary condition: &psi; = k =const.*Neumann boundary condition: &part;&psi;/&part;n = k = const. In the above, &part;&psi;/&part;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 &part;&psi;/&part;x, &part;&psi;/&part;y or &part;&psi;/&part;z. Using the respective finite difference approximations of these derivatives, one arrives at fairly complicated difference equations involving the constitutive parameters &epsilon;, &mu; and &sigma;, 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 Z₀ and effective permittivity &epsilon;_eff. 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_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: <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 &eta;₀ = 120&pi; &Omega; 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:

where c is the speed of light in the free space*Dirichlet boundary condition: T = T₀ =const.*Neumann boundary condition: &part;T/&part;n = -q_s0/k = const.*Adiabatic boundary condition: &part;T/&part;n = 0.*Convective boundary condition: &part;T/&part;n = h(T-T_&infin;)/k.

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