Eddy current analysis

Only time-harmonic eddy current problems have been tested within this category. The domain in the two-dimensional problems is a square lying in the first quadrant of the plane; in the three-dimensional problems the domain is a cuboid lying in the first octant in space. For the differential equation $\nabla \times \left({\mu }^{-1}\nabla \times \mathbf{A}\right)+i\omega {\sigma }^E\mathbf{A}=\mathbf{J}$, the solution sought is $\mathbf{A}={\mathbf{A}}_R+i{\mathbf{A}}_I$, where ${\mathbf{A}}_R={\mathbf{A}}_I=-y{\mathbf{e}}_1+x{\mathbf{e}}_2$. For this solution the first term in the differential equation vanishes. Therefore, a nonuniform body load (CJNU) of $\mathbf{J}=i\omega {\sigma }^E\mathbf{A}$ is applied everywhere in the domain. Nonzero boundary conditions (distributed surface magnetic vector potential) on the outer boundary and symmetry boundary conditions on the symmetry planes are also specified.

Both time-harmonic and transient eddy current problems have been tested within this category. The domain for some of the two-dimensional problems is a quarter of a circle lying in the first quadrant of the plane; in the three-dimensional problems the domain is a quarter of a cylinder lying in the first octant in space, with the axis of the cylinder aligned along the global $Z$-direction. Surface current loads $\mathbf{K}=\left(2/\mu \right){\mathbf{e}}_{\theta }$ are specified on the outer boundary as a Neumann-type boundary condition. Symmetry boundary conditions are specified on the symmetry planes. The analytical solution in this case is ${\mathbf{A}}_R={\mathbf{A}}_I=r{\mathbf{e}}_{\theta }$ for the time-harmonic problem, which is the same as that of the CCBL problems. The solution (real only) for the transient problem is identical.

The problems testing the transient eddy current procedure have similar domains (except for the two-dimensional problems with input file names beginning with ccsc_2d_, which consist of stand-alone electromagnetic elements subjected to boundary conditions/loading). In some of the problems the magnetic properties are defined to be different in different regions of the model; in particular, linear properties are used in one region while nonlinear properties are used in another region. The surface current loading results in a constant magnetic field within the domain, but the magnetic flux density varies based on the material behavior. Both isotropic and orthotropic magnetic behavior have been tested.

A few problems also test motional effects on the solution. In all cases a uniform translational velocity is applied. The magnetic field remains the same as the problem without motion, but the electric fields are modified due to the motional effects.

Magnetic permeability of $\mu =1.25663706144\times {10}^{-6}$ H/m or N/A² for free space is used throughout for all the time-harmonic problems and in the regions with linear magnetic behavior for the transient problems. For regions with nonlinear magnetic behavior, the response is defined in terms of a B–H curve describing the strength of the magnetic flux density as a function of the strength of the magnetic field. Table 1 provides the B–H curve used in these tests.

A small electrical conductivity (compared to that of a metal) of ${\sigma }^E$ = 1.0 or 0.58 S/m is used.

$\omega =2\pi f$ rad/s, $f=$ 50 or 60 Hz.

Transient problems are loaded with a surface current magnitude that varies with time. Some of the problems use a sinusoidal time variation.