Steady State AC Magnetic: solved equations (introduction)


The equations used for the solving are:

  • Maxwell's equations (for a magnetic system)
  • the constitutive equations of the matter

The computation conditions for a Steady state AC Magnetic application are the following:

  • the state variables are time dependent: d/dt ≠ 0 (steady state sinusoidal: sinusoidal time dependence of the current sources)
  • the computation concerns only the B, H and E fields (the D field is not computed).

    The equations of the electric fields E and D and of the magnetic fields B, H cannot be decoupled.

Equations and conditions

In the previously defined conditions of computation, the equations are summarized as follows:

E: electric field strength (in V/m)

B: magnetic flux density (in T)

H: magnetic field strength (in A/m)

J: current density (in A/m2)

σ : conductivity (in S)

μ : permeability (in H/m)

The main equation for magnetic materials, can be put in form B(H) or H(B) as presented below.

μr : relative permeability

μ0 : vacuum permeability


νr : reluctivity νr =1/μr

ν0 : vacuum reluctivity ν0 =1/μ0


To solve these equations, two models are used:

  • the vector model, which uses: the magnetic vector potential (written )

  • the scalar model, which uses: magnetic scalar potentials (written ϕtot or ϕred)

Model and 2D or 3D application

The two models (vector and scalar) are proposed:
  • the vector model for the 2D applications
  • the scalar model for the 3D applications