The Ampère’s Law node adds Ampère’s law for the magnetic field and provides an interface for defining the constitutive relation and its associated properties as well as electric properties.

The Material type setting decides how materials behave and how material properties are interpreted when the mesh is deformed. Select Solid for materials whose properties change as functions of material strain, material orientation, and other variables evaluated in a material reference configuration (material frame). Select Nonsolid for materials whose properties are defined only as functions of the current local state at each point in the spatial frame, and for which no unique material reference configuration can be defined. Select From material to pick up the corresponding setting from the domain material on each domain.

Specify the constitutive relation that describes the macroscopic properties of the medium (relating the magnetic flux density B and the magnetic field H) and the applicable material properties, such as the relative permeability.

The equation for the selected constitutive relation displays under the list. For all options, the default uses values From material, or select User defined to enter a different value or expression.

Select a Magnetization model — Relative permeability (the default), B-H curve, Magnetic losses, Remanent flux density, Magnetization, Effective B-H curve, Hysteresis Jiles–Atherton model, Nonlinear permanent magnet, Analytic magnetization curve, or External material.

Select Relative permeability μr (dimensionless) to use the constitutive relation B = μ0μrH. For User defined select Isotropic, Diagonal, Symmetric, or Full and enter values or expressions in the field or matrix. If Effective medium is selected, an Effective Medium subnode is available from the context menu (right-click the parent node) as well as from the Physics toolbar, Attributes menu, which can specify the relative permeability of the mixture.

Select B-H curve |H| (SI unit: A/m) to use a curve that relates magnetic flux density B and the magnetic field H as |H| = f(|B|).

The Magnetic field norm and Magnetic coenergy density settings can take the values From material or User defined.

When User defined is selected, specify a user-defined expression for the magnetic field norm. The direction of the magnetic field is taken to be the same as the direction of the magnetic flux density at each point.

This option introduces a complex relative permeability and it is intended for time-harmonic (frequency domain) studies. Therefore, it is not available for The Magnetic Fields, Currents Only Interface.

Select Magnetic losses μ′ and μ″ (dimensionless) to describe the relative permeability as a complex-valued quantity: μr = μ′ − iμ″, where μ′ and μ″ are the real and imaginary parts, respectively. Note that the time-harmonic Sign Convention requires a lossy material to have a positive material parameter μ″ (see Modeling Losses in the Frequency Domain).

Select Remanent flux density Br (SI unit: T) to use the constitutive relation B = μ0μrecH + Br, where μrec and Br are the recoil permeability and the remanent flux density respectively (the flux density when no magnetic field is present). The recoil permeability μrec is very similar to the relative permeability, and is valid as long as the magnet is subjected to normal operating conditions (it is only valid within the linear region close to the vertical axis H = 0). Br is given by taking the remanent flux density norm (typically, provided by the material) and multiplying it with a normalized direction field specified in the physics: Br = ||Br|| e/||e||.

Select Magnetization M (SI unit: A/m) to use the constitutive relation B = μ0H + μ0M. Enter x and y components. For 3D components, enter x, y, and z components.

Select Effective B-H curve |H|eff (SI unit: A/m) to use an effective curve that provides the local linearized relation between the magnetic flux density B and the magnetic field H in time-harmonic problems.

Select Nonlinear permanent magnet to use a nonlinear BH-relation that is isotropic around a point in H space that is shifted by the coercive field Hc. This constitutive relation is intended for easy modeling of self-demagnetization of soft permanent magnets. It is recommended to keep all input settings at the default From material and use the example material, Nonlinear Permanent Magnet in the AC/DC material library either as is or as a template for defining customized materials. The latter is done by changing the interpolation functions defined in the material. The example material is a generic and approximate representation of AlNiCo 5.

The Direction of magnetization is the only input that normally should be entered in the physics.

Select the Hysteresis Jiles–Atherton model to use in the constitutive relation B = μ0H + μ0M with the magnetization M (SI unit: A/m) computed from the solution of the five parameters Jiles–Atherton model. Specify the five parameters Ms, a, k, c, and α either from the material (default) or as user defined. The example material Jiles–Atherton Hysteretic Material is available in the AC/DC material library. The parameters may be tensor quantities resulting in the modeling of an anisotropic hysteretic material as shown in the application library entry:

In stationary study, the model defaults to use Transient initialization defined by the user. Switch it to Parametric hysteresis when performing Parametric Sweep on time instants. The entry Initial Magnetization is present to set the initial values of Jiles–Atherton variables.

The Discretization section is used to choose the discretization order of Jiles–Atherton variables.

Select this option use the constitutive relation B = μ0(H + M(H) + Mr), where Mr is the remanent magnetization (SI unit: A/m), and the magnetization vector is M(H) is calculated from the magnetic field using a nonlinear relation with possible saturation and hysteresis. This option is available only when the Material type is set to Solid.

All domains have magnetization of the same magnitude |M| = Ms, but the magnetization can have different orientations. The applied magnetic field changes the domain orientation, and the resulting net magnetization is found from the following nonlinear implicit relation:

where the matrix α characterizes the interdomain coupling.

The magnetization shape is characterized by the function L with the following properties:

and can be characterized by the initial magnetic susceptibility matrix χ0.

where χ0 is the magnetic susceptibility in the initial linear region.

Other possible choices of the L function are a hyperbolic tangent, which is sometimes referred to as the Ising model

where cr is the reversibility parameter, and kp is the pining loss parameter. The above can be solved using either a time-dependent analysis or a stationary parametric sweep.

Select External material to use a curve that relates magnetic flux density B and the magnetic field H as |H| = f(|B|) according to an externally coded function.

Specify the External material to use (from the Materials node under Global Definitions). This setting allows using material models or constitutive relations defined in an external library. See Working with External Materials for more information.

This section is described for the Current Conservation feature.

The options Effective medium and Archie’s law require additional subnodes. If Effective medium is selected, an Effective Medium subnode is available from the context menu (right-click the parent node) as well as from the Physics toolbar, Attributes menu. If Archie’s law is selected, add an Archie’s Law subnode in the same way. These subnodes contain additional settings to specify how the material properties are computed. Effective medium models a mixture of materials whose properties are computed by averaging the properties of the components. Archie’s law models a conductive liquid in a nonconductive matrix.

The default Relative permittivity εr (dimensionless) for the media is used From material and defined on the shell domain. For User defined, select Isotropic, Diagonal, Symmetric, or Full based on the characteristics of the permittivity and then enter values or expressions in the field or matrix. If Effective medium is selected, an Effective Medium subnode is available from the context menu (right-click the parent node) as well as from the Physics toolbar, Attributes menu, which can specify the relative permittivity of the mixture.