B(H) law: Linear hard magnetic material

Presentation

This model (Linear hard magnetic described by HcB and Br) defines a linear B(H) dependence.

Linearity is considered in the direction of magnetization, as well as in the transversal directions, but the remanent flux density B_r is only applied in the direction of magnetization. In fact, this magnet model is noted as unidirectional because the mathematical model and the direction of magnetization are dissociated. The mathematical model defines the demagnetization curve, while the direction of magnetization is defined by the orientation in the region.

Mathematical model

The mathematical formula used in Flux for the demagnetization curve is:

where:

• B_r is the remanent flux density (T)
• H_cB is the normal coercivity (A/m)

In transversal directions one can write:

where μ_r⊥ is the transversal relative permeability which is automatically computed through B_r and H_cB values:

Direction of magnetization

The direction of permanent magnet magnetization is “dissociated” from the model. In the software, it is defined separately.

The various possibilities available to the user are shown in the table below. (The basic plane is a plane XOY.)

Unidirectional	Radial positive	Radial negative	Orthoradial positive	Orthoradial negative

CR is the radiality center.

Mathematical model with thermal dependence

This linear hard magnetic model can also account - in the direction of magnetization - for thermal dependence through the reversible temperature coefficient α which applies to both Br and HcB, so that the B(H) slope remains unchanged, i.e. the B(H) characteristics for different temperatures are parallel. As a consequence, there is no thermal dependence in the transversal directions.

The thermal dependence of Br (the same law applies for HcB) can be expressed as:

where:

• T_ref is the reference temperature (degrees Celsius °C or Kelvin K)
• Br_ref is the remanent flux density Br of the magnet at the temperature T_ref (T)
• α is a reversible temperature coefficient for the remanent flux density Br and for the normal coercivity HcB (degrees^-1)

Here is the Br(T) curve in the figure below:

Dialog Box

Without Thermal dependence	With Thermal dependence