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
• by means of a value of the saturation magnetic polarization J_s and a value of the 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 curve. A coefficient adjusts the shape of the curve knee for a better approximation of the experimental curve. With this model, the saturation magnetic polarization J_s and the relative permeability μ_r decrease in an exponential way when the temperature increases (J_s0 and μ_r are multiplied by a temperature coefficient COEF(T)).

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 (at the origin) for T=0°C
• J_s0 is the saturation magnetic polarization for T = 0°C [Tesla]

• a is the knee adjustment coefficient ( 0 < a and a < > 1 )

  The smaller the coefficient, the sharper the knee is.

• COEF(T) is the previously described temperature coefficient

The shape of B(H) curve is presented in the figure below.

Example

An example of B(H) curve dependent on T is presented in the figure below.

It is an analytic curve of saturation + knee adjustment * exponential function of T with: T_curie = 720°, C = 500°C