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The distribution of temperature can be modeled by the heat diffusion equation[[Image:Back_icon.png|30px]] '''[[A_Review_of_Maxwell%27s_Equations_%26_Computational_Electromagnetics_(CEM) | Back to Maxwell's Equations Page]]'''

<math>\Delta T(\mathbf{r}) = \nabla^2 T(\mathbf{r}) = -\frac{\rho(\mathbf{r})}{\epsilon}</math>Heat Diffusion Equation ==

where &Phi;(<b>r</b>) is the electric scalar potential expressed in Volts [v], &rho;(<b>r</b>) is the volume charge density expressed in C/m³, and &epsilon; = &epsilon;_r &epsilon;₀ is the permittivity The Fourier law of heat conduction relates the medium having heat transfer rate to the units of F/m. temperature variation:

Under the static assumptions, Maxwell's equations reduce to elliptic <math> \nabla^2 T(\mathbf{r}) - \frac{1} {\alpha}\frac{\partial T}{\partial t} = \left( \frac{\partial^2 T}{\partial x^2} + \frac{\partial^2 T}{\partial y^2} + \frac{\partial^2 T}{\partial z^2} \right) - \frac{1} {\alpha}\frac{\partial T}{\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 t} = - \frac{w(FD\mathbf{r}) method.}{k} </math>

where &alpha; == Electrostatics Analysis==k/(&rho;_Vc_p) is the 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.

At very low frequenciesIn the steady-state regime, as &omega;&rarr;0 the time derivative vanishes and k&rarr;0, the [[Basic_Electromagnetic_Theory#Electric_and_Magnetic_Potentials | Helmholtz diffusion equation for the electric scalar potential]] reduces to the Poisson equation subject to specified boundary conditions:

<math>\Delta\Phi(\mathbf{r}) = \nabla^2 \PhiT(\mathbf{r}) = -\frac{\rhow(\mathbf{r})}{\epsilonk}</math>

where &Phi;(<b>r</b>) is the electric scalar potential expressed in Volts [v]In a source-free region, &rho;(<b>r</b>) is the volume charge density expressed in C/m³, and &epsilon; = &epsilon;_r &epsilon;₀ is the permittivity of the medium having Poisson equation reduces further to the units of F/m. Laplace equation:

<math> \hat{\mathbf{n}} The steady-state heat diffusion equations are elliptic partial differential equations. [ \mathbf{D_2These 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 (rFD)} - \mathbf{D_1(r)} ] = \rho_s (\mathbf{r}) </math>method.

<math> \hat{\mathbf{n}} \times [ \mathbf{E_2(r)} - \mathbf{E_1(r)} ] = 0 </math>= Thermal Boundary Conditions ==

where <math> \hat{\mathbf{n}} </math> is the unit normal vector The simplest thermal boundary condition at the interface pointing from medium 1 towards medium 2,<b>D(r)</b> = &epsilon;<b>E(r)</b> is walls of the electric flux density expressed in C/m², <b>E(r)</b> computational domain is the electric field vector expressed in V/m, and &rho;_s is the surface charge density at the interface having the units of expressed in C/m². Dirichlet boundary condition:

In a source-free region, &rho;(<bmath>T(\mathbf{r}) = T_0 </bmath>) = 0, and Poisson's equation reduces to the familiar Laplace equation:

Keep in mind that in the absence of an electric charge source, you need to specify a non<math>-zero potential somewhere in your structure, for example, on a perfect electric conductor k \frac{\partial T}{\partial n} = -k \mathbf{\hat{n}} . \nabla T(PEC\mathbf{r}). Otherwise, you will get a trivial zero solution of the Laplace equation. = q_s </math>

Once the electric scalar potential where q_s is computed, the electric field can easily be computed via heat flux passing through the equation below: boundary walls.

If q<mathsub> \mathbf{E(r)} = - \nabla \Phi(\mathbf{r})s</mathsub>= 0, the Neumann boundary condition is also known as the adiabatic boundary condition, which represents a perfectly insulated surface:

<math>\frac{\partial T}{\partial n} == The Solution of Poisson's Equation ==0 </math>

The general form At the interface between the surface of Poisson's equation for a scalar quantity &Psi;(<b>r</b>) can solid object and air, the convective boundary condition must be expressed asenforced:

<math>-k \Deltafrac{\Psi(partial T}{\mathbf{rpartial n}) = -h \nabla^2 \Psileft[ T(\mathbf{r}) = -f(T_{\mathbf{rinfty}) \right] </math>

where f is a general arbitrary function of the position vector T<bsub>r&infin;</bsub>. In is the case of a unit point sourceambient temperature, and h is the above equation reduces to:coefficient of convective heat transfer having units of W/(m².K).

where &delta;(<bmath>\left[ -k \frac{\partial T}{\partial n} + h T(\mathbf{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\right]_\Omega = f(<b>\mathbf{r|r<sup>&prime;}) </supmath></b>). Then, the solution to the general Poisson equation can be expressed in the terms of the Green's function as follows:

where &Omega; is the boundary surface and f(<mathb> \Psi\mathbf{(r)} = \int\int\int_V G(\mathbf{r|r^{\prime}}) f(\mathbf{r^{\prime}}) dv^{\prime} </mathb>) can be an arbitrary function in general.

== Free-Space Electric Field The Analogy between Thermal and Scalar Potential Electrostatic Equations ==

There are very few boundary value problems for which you can find closedLet us now compare the steady-form analytical Green's functions. One of them is the electric scalar potential due state thermal Poisson equation to a charge distribution in the free space. The potential Green's function for a free-space medium is given byelectrostatic Poisson equation:

<math> G_{\Phi}nabla^2 T(\mathbf{r|r^{\prime}}) = -\frac{1w(\mathbf{r})}{4k} \pi | quad \leftrightarrow \quad \nabla^2 \Phi(\mathbf{r }) = - r^\frac{\primerho(\mathbf{r})} | {\epsilon} </math>

If One can see a one-to-one correspondence between the electrostatic and thermal quantities: Temperature T(<b>r</b>) is analogous to the electric scalar potential &Phi;(<b>r</b>), the volume charge heat source density w(<b>r</b>) is denoted by analogous to the volume charge density &rho;(<b>r</b>), then and the electric scalar potential thermal conductivity k is given by:analogous to the permittivity &epsilon;.

Similarly, one can establish an analogy between the heat flux <mathb>q</b> \Phi\mathbf{(<b>r</b>)} = \frac{1}{4\pi\epsilon} \int\int\int_V \frac{\rhoand the static electric field <b>E</b>(\mathbf{<b>r^{\prime}})}{ | \mathbf{r - r^{\prime}} | } dv^{\prime} </mathb>):

The electric field due to table below summarizes the charge distribution is then given byanalogous thermal and electrical quantities:

<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 =class="wikitable" 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 &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)} ! scope= 0 </math>"col"| Thermal Item These above two equations lead to the Laplace equation inside an Ohmic conductor medium: <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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