Derivation¶

This page shows how the equation of the Time-Domain Magnetic Model follows from Maxwell's equations.

The model starts from the quasi-static form of Maxwell's equations. This form ignores the displacement current:

\[ \begin{align} \nabla \times \mathbf{H} &= \mathbf{J}, \\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t}, \\ \nabla \cdot \mathbf{B} &= 0, \end{align} \]

where:

\(\mathbf{H}\) is the magnetic field strength (A/m).
\(\mathbf{J}\) is the electric current density (A/m²).
\(\mathbf{E}\) is the electric field (V/m).
\(\mathbf{B}\) is the magnetic flux density (T).
\(t\) is the time (s).

The divergence of a curl is always zero. Thus, the magnetic vector potential \(\mathbf{A}\) (Wb/m) gives the magnetic flux density:

\[ \begin{align} \mathbf{B} = \nabla \times \mathbf{A}. \end{align} \]

For a linear material, \(\mathbf{B} = \mu \mathbf{H}\), where \(\mu\) is the magnetic permeability (H/m). In a conductor, Ohm's law gives \(\mathbf{J} = \sigma \mathbf{E}\), where \(\sigma\) is the electric conductivity (S/m). Put Faraday's law and Ohm's law into Ampère's law. This gives the eddy-current equation:

\[ \begin{align} \nabla \times \left(\frac{1}{\mu} \nabla \times \mathbf{A}\right) + \sigma \frac{\partial \mathbf{A}}{\partial t} = \mathbf{J}, \end{align} \]

where \(\mathbf{J}\) on the right side is the source current density (A/m²), for example the current in a coil.