HEJ, HU ISSN 1418-7108 Manuscript no.: ANM-981030-A

Maxwell equations

Let us first recall the principal equations. We get from the Maxwell equations in the magnetostatic case the relations

 

$${ \bf rot }{\vec{\rm {H}}} = {\vec{\rm {J}}}$$ (1)

$${\vec{\rm {H}}} = \nu(\vert{\vec{\rm {B}}}\vert) {\vec{\rm {B}}}$$ (2)

$$\div {\vec{\rm {B}}} = 0$$ (3)

with $\nu(\vert{\vec{\rm {B}}}\vert) = 1 / \mu_0 \mu_r(\vert{\vec{\rm {B}}}\vert)$ in the nonlinear model, cf. [8]. Here, the magnetic field strength is denoted by ${\vec{\rm {H}}}$, the magnetic induction by ${\vec{\rm {B}}}$, and ${\vec{\rm {J}}}$ is the current density. Since (3) holds, we can introduce the magnetic vector potential ${\vec{\rm {A}}}$ by

$${ \bf rot }{\vec{\rm {A}}} = {\vec{\rm {B}}}.$$ (4)

In magnetostatics, the Coulomb gauge

$$\div {\vec{\rm {A}}} = 0$$ (5)

is standard. Thus, we end up with the equations

  

$${ \bf rot }\left( \nu \left( \vert{ \bf rot }{\vec{\rm {A}}}\vert \right) { \bf rot }{\vec{\rm {A}}} \right) = {\vec{\rm {J}}}$$ (6)

$$\div {\vec{\rm {A}}} = 0$$ (7)

for the unknown vector potential ${\vec{\rm {A}}}$. If permanent magnets are involved, ${\vec{\rm {J}}}$ must be replaced by ${\vec{\rm {J}}} + { \bf rot }{\vec{\rm {H}}}_0$. According to material properties, the coefficient $\nu$ depends on the position $x$. With the unknown ${\bf u} = {\vec{\rm {A}}}$, equations (6) and (7) can be rewritten as

  

$${ \bf rot }\left( \nu \left( x, \vert{ \bf rot }{\bf u}\vert \right) { \bf rot }{\bf u} \right) = {\bf f}$$ (8)

$$\div {\bf u} = 0.$$ (9)

We can formulate a linear problem by the equations (10) and (9),

$${ \bf rot }\left( \nu \left( x \right) { \bf rot }{\bf u} \right) = {\bf f},$$ (10)

which is a good approximation for the nonlinear problem if we can neglect saturation effects in ferromagnetic materials.

HEJ, HU ISSN 1418-7108 Manuscript no.: ANM-981030-A