Jiles–Atherton model

Effective magnetic field ${\displaystyle H_{\text{e}}}$ influencing on magnetic moments within the material may be calculated from the following equation:[1]

${\displaystyle H_{\text{e}}=H+\alpha M}$

This effective magnetic field is analogous to the Weiss mean field acting on magnetic moments within a magnetic domain.[1]

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.[4]
Anhysteretic magnetization of typical magnetic material can be calculated as a weighted sum of isotropic and anisotropic anhysteretic magnetization:[5]

${\displaystyle M_{\text{an}}=(1-t)M_{\text{an}}^{\text{iso}}+tM_{\text{an}}^{\text{aniso}}}$

Isotropic anhysteretic magnetization ${\displaystyle M_{\text{an}}^{\text{iso}}}$ 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 ${\displaystyle H_{\text{e}}}$:[1]

${\displaystyle M_{\text{an}}^{\text{iso}}=M_{\text{s}}\left(\coth \left({\frac {H_{\text{e}}}{a}}\right)-{\frac {a}{H_{\text{e}}}}\right)}$

Anisotropic anhysteretic magnetization ${\displaystyle M_{\text{an}}^{\text{aniso}}}$ is also determined on the base of Boltzmann distribution.[3] However, in such a case, there is no antiderivative for the Boltzmann distribution function.[4] For this reason, integration has to be made numerically. In the original publication, anisotropic anhysteretic magnetization ${\displaystyle M_{\text{an}}^{\text{aniso}}}$ is given as:[3]

${\displaystyle M_{\text{an}}^{\text{aniso}}=M_{\text{s}}{\frac {\displaystyle \int _{0}^{\pi }\!e^{E(1)+E(2)}\sin \theta \cos \theta \,d\theta }{\displaystyle \int _{0}^{\pi }\!e^{E(1)+E(2)}\sin \theta \,d\theta }}}$

where ${\displaystyle {\begin{aligned}E(1)&={\frac {H_{\text{e}}}{a}}\cos \theta -{\frac {K_{\text{an}}}{M_{\text{s}}\mu _{0}a}}\sin ^{2}(\psi -\theta )\\[4pt]E(2)&={\frac {H_{\text{e}}}{a}}\cos \theta -{\frac {K_{\text{an}}}{M_{\text{s}}\mu _{0}a}}\sin ^{2}(\psi +\theta )\end{aligned}}}$

It should be highlighted, that a typing mistake occurred in the original Ramesh et al. publication.[4] As a result, for an isotropic material (where ${\displaystyle K_{\text{an}}=0)}$), the presented form of anisotropic anhysteretic magnetization ${\displaystyle M_{\text{an}}^{\text{aniso}}}$ is not consistent with the isotropic anhysteretic magnetization ${\displaystyle M_{\text{an}}^{\text{iso}}}$ given by the Langevin equation. Physical analysis leads to the conclusion that the equation for anisotropic anhysteretic magnetization ${\displaystyle M_{\text{an}}^{\text{aniso}}}$ has to be corrected to the following form:[4]

${\displaystyle M_{\text{an}}^{\text{aniso}}=M_{\text{s}}{\frac {\displaystyle \int _{0}^{\pi }\!e^{\frac {E(1)+E(2)}{2}}\sin \theta \cos \theta \,d\theta }{\displaystyle \int _{0}^{\pi }\!e^{\frac {E(1)+E(2)}{2}}\sin \theta \,d\theta }}}$

In the corrected form, the model for anisotropic anhysteretic magnetization ${\displaystyle M_{\text{an}}^{\text{aniso}}}$ was confirmed experimentally for anisotropic amorphous alloys.[4]

In Jiles–Atherton model, M(H) dependence is given in form of following ordinary differential equation:[6]

${\displaystyle {\frac {dM}{dH}}={\frac {1}{1+c}}{\frac {M_{\text{an}}-M}{\delta k-\alpha (M_{\text{an}}-M)}}+{\frac {c}{1+c}}{\frac {dM_{\text{an}}}{dH}}}$

where ${\displaystyle \delta }$ depends on direction of changes of magnetizing field ${\displaystyle H}$ (${\displaystyle \delta =1}$ for increasing field, ${\displaystyle \delta =-1}$ for decreasing field)

Flux density ${\displaystyle B}$ in the material is given as:[1]

${\displaystyle B(H)=\mu _{0}M(H)}$

where ${\displaystyle \mu _{0}}$ is magnetic constant.