classPermanentMagnets

Information

Concept of permanent magnet modeling

High performance permanent magnets (e.g. NdFeB) operate on their largest hysteresis loop. This characteristic is nearly linear in the second quadrant between remnant flux density B_r and coercive field strength H_c. As an example, typical values are B_r = 1.3 T and H_c = 1000 kA/m. This proves that the slope of the hysteresis loop is nearly fully saturated with a relative permeability μ_r = B_r / H_c / μ₀ = 1.

At normal operational temperatures, the bend below that demagnetization could occur is left from coercive field strength at negative flux density. The permanent magnet's energy density is the maximum of the product B x H along this characteristic. It can be calculated as B_r x H_c / 4 = 325 kJ/m³ for the above mentioned example.

Using geometry, flux density is proportional to magnetic flux and magnetic field strength is proportional to magnetomotive force. The point of operation can be found at the intersection (1) between the magnet's hysteresis loop and the characteristic of the rest of the machine's magnetic circuit in d-axis. The magnet can be represented as a linear source with open-circuit magnetomotive force mmf₀ resulting from H_c, short-circuit flux ψ_sc resulting from B_r and an inner magnetic resistance R_i = ψ_sc / mmf₀ equivalent to that of air. If this inner magnetic resistance R_i is added to the nonlinear characteristic of the rest of the machine's magnetic circuit R_m, the permanent magnet can be characterized by its open-circuit mmf₀ (2).

Fig. 1: Magnetic characteristic and

In Modelica.Electrical.Machines this open-circuit mmf₀ is modeled as a constant equivalent excitation current source, in Modelica.Magnetic.FundamentalWave as a constant magnetic potential difference. In both models, the permanent magnet's inner magnetic resistance R_i is assumed to be comprised in the d-axis inductance L_d.

The remnant flux density B_r and coercive field strength H_c are known to be dependent on temperature, with a relative temperature coefficient of approximately α_B = - 0.1 %/K for B_r and α_H = - 0.5 %/K for H_c. This results in a temperature dependency of the inner magnetic resistance R_i = ψ_sc / mmf₀ and therefore a temperature dependency of d-axis inductance L_d.

Assuming the same temperature coefficient for B_r and H_c, the inner magnetic resistance R_i and furthermore the d-axis inductance L_d would be invariant with respect to temperature. However, this temperature effects are not taken into account in the Modelica Standard Library.

Further reading

[VAC] Vacuumschmelze, Rare Earth Permanent Magnets.