Jiles–Atherton model


The Jiles–Atherton model of magnetic hysteresis was introduced in 1984 by David Jiles and D. L. Atherton. This is one of the most popular models of magnetic hysteresis. Its main advantage is the fact that this model enables connection with physical parameters of the magnetic material. Jiles–Atherton model enables calculation of minor and major hysteresis loops.
The original Jiles–Atherton model is suitable only for isotropic materials. However, an extension of this model presented by Ramesh et al. and corrected by Szewczyk enables the modeling of anisotropic magnetic materials.

Principles

of the magnetic material sample in Jiles–Atherton model is calculated in the following steps for each value of the magnetizing field :
Original Jiles–Atherton model considers following parameters:
ParameterUnitsDescription
Quantifies interdomain coupling in the magnetic material
A/mQuantifies domain walls density in the magnetic material
A/mSaturation magnetization of material
A/mQuantifies average energy required to break pinning site in the magnetic material
Magnetization reversibility

Extension considering uniaxial anisotropy introduced by Ramesh et al. and corrected by Szewczyk requires additional parameters:
ParameterUnitsDescription
J/mAverage anisotropy energy density
radAngle between direction of magnetizing field and direction of anisotropy easy axis
Participation of anisotropic phase in the magnetic material

Modelling the magnetic hysteresis loops

Effective magnetic field

Effective magnetic field influencing on magnetic moments within the material may be calculated from following equation:
This effective magnetic field is analogous to the Weiss mean field acting on magnetic moments within a magnetic domain.

Anhysteretic magnetization

Anhysteretic magnetization can be observed experimentally, when magnetic material is demagnetized under the influence of constant magnetic field. However, measurements of anhysteretic magnetization are very sophisticated due to the fact, that the fluxmeter has to keep accuracy of integration during the demagnetization process. As a result, experimental verification of the model of anhysteretic magnetization is possible only for materials with negligible hysteresis loop.

Anhysteretic magnetization of typical magnetic material can be calculated as a weighted sum of isotropic and anisotropic anhysteretic magnetization:

Isotropic

Isotropic anhysteretic magnetization is determined on the base of Boltzmann distribution. In the case of isotropic magnetic materials, Boltzmann distribution can be reduced to Langevin function connecting isotropic anhysteretic magnetization with effective magnetic field :

Anisotropic

Anisotropic anhysteretic magnetization is also determined on the base of Boltzmann distribution. However, in such a case, there is no antiderivative for Boltzmann distribution function. For this reason, integration has to be made numerically. In the original publication, anisotropic anhysteretic magnetization is given as:
where
It should be highlighted, that typing mistake happened in the original Ramesh et al. publication. As a result, for isotropic material, presented form of anisotropic anhysteretic magnetization is not coherent with isotropic anhysteretic magnetization given by Langevin equation. Physical analysis leads to conclusion, that equation for anisotropic anhysteretic magnetization has to be corrected to the following form:
In the corrected form, model for anisotropic anhysteretic magnetization was confirmed experimentally for anisotropic amorphous alloys.

Magnetization as a function of magnetizing field

In Jiles–Atherton model, M dependence is given in form of following ordinary differential equation:
where depends on direction of changes of magnetizing field

Flux density as a function of magnetizing field

in the material is given as:
where is magnetic constant.

Vectorized Jiles–Atherton model

Vectorized Jiles–Atherton model is constructed as the superposition of three scalar models one for each principal axe. This model is especially suitable for finite element method computations.

Numerical implementation

The Jiles-Atherton model is implemented in JAmodel, a MATLAB/OCTAVE toolbox. It uses the Runge-Kutta algorithm for solving ordinary differential equations. JAmodel is open-source is under MIT license.
The two most important computational problems connected with the Jiles–Atherton model were identified:
For numerical integration of the anisotropic anhysteretic magnetization the Gauss–Kronrod quadrature formula has to be used. In GNU Octave this quadrature is implemented as quadgk function.
For solving ordinary differential equation for dependence, the Runge–Kutta methods are recommended. It was observed, that the best performing was 4-th order fixed step method.

Further development

Since its introduction in 1984, Jiles–Atherton model was intensively developed. As a result, this model may be applied for the modeling of:
Moreover, different corrections were implemented, especially:
Jiles–Atherton model may be applied for modeling:
It is also widely used for electronic circuit simulation, especially for models of inductive components, such as transformers or chokes.