Lecture 3.

Coupled Finite Element Method / Time-Dependent Magnetic Field
Instructor Dániel Marcsa (lecturer) Lectures: Monday, 14:50 - 16:25 (D201), 16:30 - 17:15 (D105) Office hours: by request	Teaching Assistants: - Office hours: -.

Coupled Finite Element Method

Rigid Body Motion

In electromechanical systems (electric machines, actuators, etc.), the structure is subject to a rigid motion by means of the force and torque. Thus, the change on the geometry, in turn, may strongly influence the magnetic field. Further, due to the motion and time-varying magnetic field, eddy currents are generated in the conducting parts (where [math]\sigma\neq 0[/math]). The so-called induced current (eddy current) in an electrically conductive body

[math]\vec{J}_{M} = \sigma\vec{v}\times\vec{B}[/math],

where [math]\vec{v}[/math] is the velocity of the body.

Electric Circuit Coupling

In most of the cases, the system is voltage fed and the current in the coil is an unknown. To solve this problem, both the field equations and the electric circuit equations must be solved simultaneously. The circuit equation for a coil can be written as

[math]u(t) = R i(t) + N\frac{\text{d}\Phi(t)}{\text{d}t}[/math],

where [math]u(t)[/math] is the voltage on the coil, [math]R[/math] and [math]N[/math] are the coil resistance and the number of turns, [math]\Phi(t)[/math] is the magnetic flux [Wb/m] generated in the solution domain and linked by the coil.

Finally, the matrix equation of strongly coupled problem is

[math]\begin{bmatrix} \mathbf{S}+\frac{\mathbf{N}}{\Delta t} & -\mathbf{P} \\ \frac{\mathbf{Q}}{\Delta t} & \mathbf{R} \end{bmatrix}\begin{bmatrix} \mathbf{A}(t)\\ \mathbf{I}(t) \end{bmatrix} = \begin{bmatrix} \frac{\mathbf{N}}{\Delta t} & \mathbf{0} \\ \frac{\mathbf{Q}}{\Delta t} & \mathbf{0} \end{bmatrix}\begin{bmatrix} \mathbf{A}(t-\Delta t)\\ \mathbf{I}(t-\Delta t) \end{bmatrix}+\begin{bmatrix} \mathbf{0}\\ \mathbf{U}(t) \end{bmatrix}[/math]

where [math]\mathbf{A}[/math] is the vector of unknown magnetic vector potential, [math]\mathbf{I}[/math] is the vector of unknown coil currents and [math]\mathbf{U}[/math] is the vector of voltage sources. [math]\mathbf{S}[/math] is the matrix related to [math]\mu[/math] permeability, [math]\mathbf{N}[/math] is the matrix related to [math]\sigma[/math] conductivity. [math]\mathbf{P}[/math] is the matrix related to the current in coil and [math]\mathbf{Q}[/math] is the matrix related to the coil flux linkage. [math]\mathbf{R}[/math] is the diagonal matrix of coils DC resistance and [math]\mathbf{U}[/math] is the vector of voltage sources.

Time-Dependent Magnetic Field

The most important case for electromagnetic equipment (sensors, actuators, motors, etc.) is the quasistatic case often referred to as the eddy current or magnetodynamic case. For quasistatic electromagnetic field, we can neglect the displacement current density term [math]\partial \vec{D}/\partial t[/math], which gives Maxwell's equations to the following form

[math]\nabla\times\vec{H}=\vec{J}[/math]	Ampere's law,
[math]\nabla\times\vec{E}=-\frac{\partial \vec{B}}{\partial t}[/math]	Faraday's law,
[math]\nabla\cdot\vec{B}(\vec{r},t)=0[/math]	Gauss's law (magnetic).

Magnetic Vector Potential and Electric Scalar Potential

According to [math]\nabla\cdot\vec{B}(\vec{r},t)=0[/math] the magnetic flux density is conservative and therefore can be described by the curl of a vector

[math] \vec{B} = \nabla\times\vec{A}[/math],

where [math]\vec{A}[/math] is the magnetic vector potential [Wb/m]. Substituting this expression into Faraday's law results in

[math]\nabla\times\vec{E}=-\frac{\partial}{\partial t} \left(\nabla\times\vec{A}\right)=-\nabla\times\left(\frac{\partial\vec{A}}{\partial t}\right) \to \nabla\times\left(\vec{E}+\frac{\partial\vec{A}}{\partial t}\right)=\vec{0}[/math],

because rotation (i.e. derivative by space) and derivation by time can be replaced. The curl-less vector field [math]\vec{E}+\partial\vec{A}/\partial t[/math] can be derived from the so-called electric scalar potential [math]V[/math] ([math]\nabla\times\nabla\varphi\equiv0[/math], for any scalar function [math]\varphi=\varphi(\vec{r})[/math], or [math]\varphi=\varphi(\vec{r},t)[/math]),

[math]\vec{E}+\frac{\partial\vec{A}}{\partial t}=-\nabla V[/math],

and the [math]\vec{E}[/math] electric field intensity vector can be described by two potentials as

[math]\vec{E}=-\frac{\partial\vec{A}}{\partial t}-\nabla V[/math].

Substituting expression of [math]\vec{B}[/math] and [math]\vec{E}[/math] into Faraday's law leads to the partial differential equation

[math]\nabla\times\left(\frac{1}{\mu}\nabla\times\vec{A}\right)=\vec{J}_{S}-\sigma\frac{\partial\vec{A}}{\partial t}-\sigma\nabla V+\sigma\vec{v}\times\nabla\times\vec{A}[/math].

If the velocity is a priori known, the additional term remains linear but will lead to a so-called convective term. Therefore, numerical computation will need some upwind technique for stability reasons.