# Electromagnetic analysis¶

This section discuss the electromagnetic theories that are applied in the WELSIM application.

## Electromagnetic field fundamentals¶

The electromagnetic fields are governed by the well-known Maxwell's equations $$\eqref{eq:ch4_theory_maxwell1}$$-$$\eqref{eq:ch4_theory_maxwell4}$$12.

\begin{align} \label{eq:ch4_theory_maxwell1} \nabla\times\mathbf{H}=\mathbf{J}+\dfrac{\partial\mathbf{D}}{\partial t}=\mathbf{J}_{S}+\mathbf{J}_{e}+\mathbf{J}_{V}+\dfrac{\partial\mathbf{D}}{\partial t} \end{align}
\begin{align} \label{eq:ch4_theory_maxwell2} \nabla\times\mathbf{E}=-\dfrac{\partial\mathbf{B}}{\partial t} \end{align}
\begin{align} \label{eq:ch4_theory_maxwell3} \nabla\cdot\mathbf{B}=0 \end{align}
\begin{align} \label{eq:ch4_theory_maxwell4} \nabla\cdot\mathbf{D}=\rho \end{align}

where $$\mathbf{H}$$ is the magnetic field intensity vector, $$\mathbf{J}$$ is total current density vector, $$\mathbf{J}_{s}$$ is the applied source current density vector, $$\mathbf{J}_{e}$$ is the induced eddy current density vector, and $$\mathbf{J}_{VS}$$ is the velocity current density vector, $$\mathbf{D}$$ is the electric flux density vector (this term is also called electric displacement), $$\mathbf{E}$$ is the electric field intensity vector, $$\mathbf{B}$$ is the magnetic flux density vector, and $$\rho$$ is the electric charge density.

The above field governing equations contian the constitutive relations:

$\mathbf{D}=\epsilon\mathbf{E}+\mathbf{P}$

and

$\mathbf{B}=\mu\mathbf{H}$

where $$\mathbf{P}$$ is the polarization density, and $$\mathbf{M}$$ is t he magnetization. In many materials the polarization density can be approximated as a scalar multiple of the electric field. $$\mu$$ is the magnetic permeability matrix. For example, if the magnetic permeability is a function of temperature,

$\mu=\mu_{0}\left[\begin{array}{ccc} \mu_{rx} & 0 & 0\\ 0 & \mu_{ry} & 0\\ 0 & 0 & \mu_{rz} \end{array}\right]$

For the permanent magnets, the constitutive relation of magnetic field becomes

$\mathbf{B}=\mu\mathbf{H}+\mu_{0}\mathbf{M}_{0}$

where $$\mathbf{M}_{0}$$ is the remanet intrinsic magnetization vector.

Similarly, the consitutive relations for the related electric fields are:

$\mathbf{J}=\sigma[\mathbf{E}+\mathbf{v}\times\mathbf{B}]$
$\sigma=\left[\begin{array}{ccc} \sigma_{xx} & 0 & 0\\ 0 & \sigma_{yy} & 0\\ 0 & 0 & \sigma_{zz} \end{array}\right]$
$\epsilon=\left[\begin{array}{ccc} \epsilon_{xx} & 0 & 0\\ 0 & \epsilon_{yy} & 0\\ 0 & 0 & \epsilon_{zz} \end{array}\right]$

where $$\sigma$$ is the electrical conductivity matrix, $$\epsilon$$ is the permittivity matrix, and $$v$$ is the velocity vector.

## Electrostatics¶

The WELSIM application introduces electric scalar potential to solve the electrostatic problems. When the time-derivetive of magnetic flux density is neglected from the full Maxwell's equations. The governing equations are reduced to

\begin{align} \label{eq:ch4_theory_govern_eqn_electrostatic} \nabla\times\mathbf{H}=\mathbf{J}+\dfrac{\partial\mathbf{D}}{\partial t} \end{align}
$\nabla\times\mathbf{E}=\mathbf{0}$
$\nabla\cdot\mathbf{B}=0$
$\nabla\cdot\mathbf{D}=\rho$

Since the electric field $$\mathbf{E}$$ is irrotational and can be expressed as the function of electric scalar potential

$\mathbf{E}=-\nabla \varphi$

where $$\varphi$$ is the electric scalar potential and has units of Volts in the SI system. Inserting this definition into the Gauss's Law gives:

$-\nabla \cdot \epsilon\nabla\varphi = \rho - \nabla \cdot \mathbf{P}$

which is Poisson's equation for the electric potential , where we have assumed a linear constitutive relation between $$\mathbf{D}$$ and $$\mathbf{E}$$ of the form $$\mathbf{D}=\epsilon\mathbf{E}+\mathbf{P}$$.

### Boundary Conditions¶

For an electric material interface, the continuious conditions for $$\mathbf{E}$$, $$\mathbf{D}$$, and $$\mathbf{J}$$ are

$E_{t1}-E_{t2}=0$
$J_{1n}+\dfrac{\partial D_{1n}}{\partial t}=J_{2n}+\dfrac{\partial D_{2n}}{\partial t}$
$D_{1n}-D_{2n}=\rho_{s}$

where $$E_{t}$$ is the tangential components of $$\mathbf{E}$$, $$J_{n}$$ is the normal components of $$\mathbf{J}$$, $$D_{n}$$ is the normal components of $$\mathbf{D}$$, and $$\rho_{s}$$ is the surface charge density.

Since the solutons to the governing equation are non-unique, we must impose a Dirichlet boundary condition at least at one node in the domain to get the physical solution. The Dirichlet condition could be a fixed piecewise voltage value on certain nodes. In addition, the normal derivative boundary condition $$\hat{n}\cdot\mathbf{D}$$ such as surface charge density can be imposed on the boundary.

### Matrix Forms¶

The electric scalar potential algorithm is applied in the WELSIM application for solving electrostatic problems. The governing equations are reduced to the following:

$-\nabla\cdot\left(\epsilon\nabla V\right)=\rho$

The matrix equation for an electrostatic analysis is derived from Equation $$\eqref{eq:ch4_theory_govern_eqn_electrostatic}$$:

$\left[K^{VS}\right]\left\{ V_{e}\right\} =\left\{ L_{e}\right\}$

where

$\left[K^{VS}\right]=\intop_{V}\left(\nabla\left\{ N\right\} ^{T}\right)^{T}\epsilon\left(\nabla\left\{ N\right\} ^{T}\right)dV$
$\left\{ L_{e}\right\} =\left\{ L_{e}^{n}\right\} +\left\{ L_{e}^{c}\right\} +\left\{ L_{e}^{SC}\right\}$
$\left\{ L_{e}^{c}\right\} =\int_{V}\rho\left\{ N\right\} ^{T}dV$
$\left\{ L_{e}^{sc}\right\} =\int_{V}\rho_{s}\left\{ N\right\} ^{T}dV$

### Vector magnetic potential¶

The WELSIM application applies the vector magnetic potential method for the magentostatic analysis. Considering the neglected electric displacement currents, the full Maxwell's equations can be reduced to

$\nabla\times\mathbf{H}=\mathbf{J}$
$\nabla\times\mathbf{E}=-\dfrac{\partial\mathbf{B}}{\partial t}$
$\nabla\cdot\mathbf{B}=0$

A numerical solution can be achieved by introducing potentials to the governing equations. The proposed magnetic vector potential $$\mathbf{A}$$ and electric scalar potential $$V$$ have the following characteristics:

$\mathbf{B}=\nabla\times\mathbf{A}$
$\mathbf{E}=-\dfrac{\partial\mathbf{A}}{\partial t}-\nabla V$

In addition, the Coulomb gauge condition is introduced to ensure the uniqueness of the vector potential, as shown in the following equations.

$\nabla\times\mathbf{v}\nabla\times\mathbf{A}-\nabla v_{e}\nabla\cdot\mathbf{A}+\sigma\left\{ \dfrac{\partial\mathbf{A}}{\partial t}\right\} +\sigma\nabla V-\mathbf{v}\times\sigma\nabla\times\mathbf{A}=\mathbf{0}$
$\nabla\cdot\left(\sigma\left\{ \dfrac{\partial\mathbf{A}}{\partial t}\right\} -\sigma\nabla V+\mathbf{v}\times\sigma\nabla\times\mathbf{A}\right)=\mathbf{0}$
$\nabla\times\mathbf{v}\nabla\times\mathbf{A}-\nabla v_{e}\nabla\cdot\mathbf{A}=\mathbf{J}_s+\nabla\times\dfrac{1}{\mathbf{v}_{0}}\mathbf{v}\mathbf{M}_{0}$

where matrix invarient $$v_{e}$$ is $$v_{e}=\frac{1}{3}\mathrm{tr}(v)=\frac{1}{3}(v_{11}+v_{22}+v_{33})$$.

### Edge-element magnetic vector potential¶

Due to the limitation of node-based vector magnetic potential algorithm2, WELSIM application uses the edge-based finite element for the magnetic vector potential algorithm.

The governing equation for the edge finite element method is given below.

$\nabla\times\mathbf{v}\nabla\times\mathbf{A}+\sigma\left\{ \dfrac{\partial\mathbf{A}}{\partial t}+\nabla V\right\} +\epsilon\left(\left\{ \dfrac{\partial^{2}\mathbf{A}}{\partial t^{2}}\right\} +\nabla\left\{ \dfrac{\partial V}{\partial t}\right\} \right)=\mathbf{0}$
$\nabla\cdot\left(\sigma\left(\left\{ \dfrac{\partial\mathbf{A}}{\partial t}\right\} +\nabla V\right)+\epsilon\left(\left\{ \dfrac{\partial^{2}\mathbf{A}}{\partial t^{2}}\right\} +\nabla\left\{ \dfrac{\partial V}{\partial t}\right\} \right)\right)=\mathbf{0}$
$\nabla\times\mathbf{v}\nabla\times\mathbf{A}=\mathbf{J}_{s}+\nabla\times\dfrac{1}{\mathbf{v}_{0}}\mathbf{v}\mathbf{M}_{0}$

The uniqueness of these equations is ensured by the tree gauging procedure, which sets the edge-flux degrees of freedom related to the spanning tree of the finite element mesh to zero.

1. John D. Jackson, Classical Electrodynamics, 3rd edition, Wiley.

2. Jian-Ming Jin, The Finite Element Method in Electromagnetics, 2nd edition, Wiley-IEEE Press.