Analytic saturation curve + knee adjustment * exponential function of T

Presentation

This model defines a B(H) behavior law:

• nonlinear (taking the saturation and the knee adjustment into consideration)
• for an isotropic material
• through a saturation magnetic polarization J_s and a relative permeability μ_r which decrease in an exponential way when the temperature increases

Mathematical model

This model consists, like the previous one, of a combination of a straight line and a nonlinear equation (based on square root function), where the saturation magnetic polarization J_s and the relative permeability μ_r decrease in an exponential way when the temperature increases. The temperature coefficient coeff(T) appears as multiplying factor of the square root function, as well as in its argument through the quantity H_a.

The corresponding mathematical formula is written:

with:

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)
• a is the knee adjustment coefficient (a > 0 and a ≠ 1); the smaller the coefficient, the sharper the knee is
• coeff(T) is the temperature coefficient function which describes how the saturation magnetic polarization J_s and the relative permeability μ_r decrease when the temperature increases

The shape of B(H) curve for a given temperature and several knee adjustment coefficients is presented in the figure below.

Example

An example of this B(H) analytic saturation curve + knee adjustment * 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 = 500 °C.