Formulations in Transient magnetics

Transient magnetic analysis is the generalized form of computation of electric and magnetic field, induced by direct or time-varying currents (alternating, impulse, etc.), permanent magnets, or external magnetic fields, in linear or nonlinear (ferromagnetic) media, and takes into account eddy current (skin) effect in conductors of electric current.

The formulation is derived from Maxwell's equations for vector magnetic potential A (B = curl A, B - magnetic flux density vector), and scalar electric potential U (E = -grad U, E - electric field vector):
curl(μ-1·curl A) = j + curl Hc
j = σE = -σ·grad U - σ·∂A/∂t,
where μ-1 is an inverse permeability tensor, and σ is electric conductivity. In accordance with the second equation, vector j of the total current in a conductor can be considered as a combination of a source current produced by the external voltage and an eddy current induced by the time-varying magnetic field.

j = j0 + jeddy,
where
j0 = -σ·grad U
jeddy = -σ·∂A/∂t.

The flux density is assumed to lie in the plane of model (xy or zr), while the vector of electric current density j and the vector potential A are orthogonal to it. Only jz and Az in planar or jθ and Aθ in axisymmetric case are not equal to zero. We will denote them simply j and A. Finally, the equation for planar case is

and for axisymmetric case is

,

where components of magnetic permeability tensor μx and μy (μz and μr), components of coercive force vector Hcx and Hcy (Hcz and Hcr) are constants within each block of the model. Source current density j0 is assumed to be constant within each model block in planar case and vary as 1/r in axisymmetric case

Note. Isotropic (μx = μy or μz = μr) but field dependent permeability is assumed in nonlinear case. Magnetization characteristic of material is described by the B-H curve.