Electromagnetics¶

Magnetostatic formulation¶

The magnetostatic solve assumes a single non-zero vector potential component \(A_z(x, y)\). The governing PDE is

\[ \nabla \cdot (\nu \nabla A_z) = -J_z, \]

with \(\nu = 1/\mu\). For uniform materials the expression reduces to the familiar Poisson form

\[ \nabla^2 A_z = -\mu J_z. \]

Recovered field components follow from central differences of \(A_z\):

\[ B_x \approx \frac{A_{i,j+1} - A_{i,j-1}}{2\Delta y}, \qquad B_y \approx -\frac{A_{i+1,j} - A_{i-1,j}}{2\Delta x}. \]

The magnetic field intensity combines the solved field with magnetisation contributions:

\[ \mathbf{H} = \nu \mathbf{B} - \frac{1}{\mu_r} \mathbf{M}. \]

Permanent magnets appear as an equivalent bound current density

\[ J_{m,z} = \partial_x M_y - \partial_y M_x, \]

which is added to \(J_z\) during rasterisation.