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. Magnetization formula_1 of the magnetic  material sample in Jiles–Atherton model is calculated in the following steps for each value of the magnetizing field formula_2: Parameters. Original Jiles–Atherton model considers following parameters: Extension considering uniaxial anisotropy introduced by Ramesh et al. and corrected by Szewczyk requires additional parameters: Modelling the magnetic hysteresis loops. Effective magnetic field. Effective magnetic field formula_10 influencing on magnetic moments within the material may be calculated from the 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 formula_13 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 formula_10: Anisotropic. Anisotropic anhysteretic magnetization formula_16 is also determined on the base of Boltzmann distribution. However, in such a case, there is no antiderivative for the Boltzmann distribution function. For this reason, integration has to be made numerically. In the original publication, anisotropic anhysteretic magnetization formula_16 is given as: where It should be highlighted, that a typing mistake occurred in the original Ramesh et al. publication. As a result, for an isotropic material (where formula_21), the presented form of anisotropic anhysteretic magnetization formula_16 is not consistent with the isotropic anhysteretic magnetization  formula_13 given by the Langevin equation. Physical analysis leads to the conclusion that the equation for anisotropic anhysteretic magnetization formula_16 has to be corrected to the following form: In the corrected form, the model for anisotropic anhysteretic magnetization formula_16 was confirmed experimentally for anisotropic amorphous alloys. Magnetization as a function of magnetizing field. In Jiles–Atherton model, M(H) dependence is given in form of following ordinary differential equation: where formula_28 depends on direction of changes of magnetizing field formula_29 (formula_30 for increasing field, formula_31 for decreasing field) Flux density as a function of magnetizing field. Flux density formula_32 in the  material is given as: where formula_34 is magnetic constant. Vectorized Jiles–Atherton model. Vectorized Jiles–Atherton model is constructed as the superposition of three scalar models one for each principal axis. 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 formula_16 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 formula_36 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: Applications. 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.