Analytic saturation curve * exponential function of T

Presentation

This model defines a B(H) behavior law:

• nonlinear (taking the saturation into consideration)
• for an isotropic material
• through a saturation magnetic polarization J_s and a magnetic susceptibility χ (where χ = μ_r - 1) which decrease in an exponential way when the temperature increases

Mathematical model

This model is a combination of a straight line and a nonlinear equation (based on arctangent function), where the saturation magnetic polarization J_s and the magnetic susceptibility χ decrease in an exponential way when the temperature increases. The temperature coefficient coeff(T) appears only as a multiplying factor of the arctangent function, while not in its argument because it simplifies being applied to both χ and J_s.

The corresponding mathematical formula is written:

where:

• μ₀ is the permeability of vacuum; μ₀ = 4 π 10^-7 H/m
• μ_r0 is the relative permeability of the material (at the origin) for T = 0 °C
• J_s0 is the saturation magnetic polarization for T = 0 °C (T)
• coeff(T) is the temperature coefficient function which describes how the saturation magnetic polarization J_s and the magnetic susceptibility χ decrease when the temperature increases

The shape of B(H) curve for a given temperature is presented in the figure below.

Example

An example of this B(H) analytic saturation curve * exponential function of T is presented in the figure below, where the Curie temperature T_C of the material and the temperature constant C are respectively T_C = 727 °C and C = 100 °C.