Equation System
To derive the equation system in this physics interface, start with Faraday’s law:
The electric field is computed from the current density, using an appropriate constitutive relation:
where Je is an externally generated current density, v is the relative velocity of the domain in the physics interface’s or node’s frame, and Em(J) is a (possibly nonlinear) relation sometimes called E-J characteristic, which describes the electric field in the local (material) frame as a function of the current density. The total current density J is finally obtained from Ampère’s Law:
Combining these equations give the first equation in the system applied by the Magnetic Field Formulation interface:
This equation must be modified in different studies and according to the chosen constitutive relation. In stationary studies, the equation becomes:
while in frequency domain is:
The term involving the velocity only applies in the 2D and 2D axisymmetric formulations.
The second equation in the system is the magnetic Gauss’ law:
This equation is added with two purposes:
•
Make the problem numerically stable when using curl elements. The feature adding this equation can only be applied when the magnetic field is discretized using curl elements, that is, in 3D components, and 2D or 2D axisymmetric components when the in-plane components are solved for.
•
In magnetostatics, the magnetic flux density B does not appear in the first equation. To allow modeling of magnetic materials and permanent magnets, it is necessary to include B (with its constitutive relation) through the Magnetic Gauss’ Law node.
E-J Characteristic for the Electric Field Computation
The Em(J) characteristic introduced in the previous section can be specified in different ways in the Faraday’s Law node. The most common case is a linear relation between Em and J. In magnetostatics and in time-dependent studies (in the quasistatic approximation) this relation is specified by the resistivity or the electrical conductivity:
σ and ρ can be tensor quantities, in which case the inverse is intended as the inverse of a tensor. In frequency-domain, the displacement current is included, giving the following relations:
If the E-J characteristic is nonlinear, such in the case of a superconductor, the function can be expressed explicitly or as a material property.
Superconducting Wire: Application Library path ACDC_Module/Devices,_Superconducting/superconducting_wire