Jiles–Atherton Model

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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.

magnetic hysteresis modeling model

1. Principles

Magnetization $M$ of the magnetic material sample in Jiles–Atherton model is calculated in the following steps ^[1] for each value of the magnetizing field $H$ :

effective magnetic field $H_{e}$ is calculated considering interdomain coupling $α$ and magnetization $M$ ,
anhysteretic magnetization $M_{an}$ is calculated for effective magnetic field $H_{e}$ ,
magnetization $M$ of the sample is calculated by solving ordinary differential equation taking into account sign of derivative of magnetizing field $H$ (which is the source of hysteresis).

2. Parameters

Original Jiles–Atherton model considers following parameters:^[1]

Parameter	Units	Description
$α$		Quantifies interdomain coupling in the magnetic material
$a$	A/m	Quantifies domain walls density in the magnetic material
$M_{s}$	A/m	Saturation magnetization of material
$k$	A/m	Quantifies average energy required to break pinning site in the magnetic material
$c$		Magnetization reversibility

Extension considering uniaxial anisotropy introduced by Ramesh et al.^[2] and corrected by Szewczyk ^[3] requires additional parameters:

Parameter	Units	Description
$K_{an}$	J/m³	Average anisotropy energy density
$ψ$	rad	Angle between direction of magnetizing field $H$ and direction of anisotropy easy axis
$t$		Participation of anisotropic phase in the magnetic material

3. Modelling the Magnetic Hysteresis Loops

3.1. Effective Magnetic Field

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

H_{e} = H + α M

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

3.2. 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.^[3]
Anhysteretic magnetization of typical magnetic material can be calculated as a weighted sum of isotropic and anisotropic anhysteretic magnetization:^[4]

M_{an} = (1 - t) M_{an}^{iso} + t M_{an}^{aniso}

Isotropic

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

M_{an}^{iso} = M_{s} (\coth (\frac{H_{e}}{a}) - \frac{a}{H_{e}})

Anisotropic

Anisotropic anhysteretic magnetization $M_{an}^{aniso}$ is also determined on the base of Boltzmann distribution.^[2] However, in such a case, there is no antiderivative for the Boltzmann distribution function.^[3] For this reason, integration has to be made numerically. In the original publication, anisotropic anhysteretic magnetization $M_{an}^{aniso}$ is given as:^[2]

M_{an}^{aniso} = M_{s} \frac{\int_{0}^{π} e^{E (1) + E (2)} \sin (θ) \cos (θ) d θ}{\int_{0}^{π} e^{E (1) + E (2)} \sin (θ) d θ}

where

E (1) = \frac{H_{e}}{a} \cos θ - \frac{K_{an}}{M_{s} μ_{0} a} \sin^{2} (ψ - θ)

E (2) = \frac{H_{e}}{a} \cos θ - \frac{K_{an}}{M_{s} μ_{0} a} \sin^{2} (ψ + θ)

It should be highlighted, that a typing mistake occurred in the original Ramesh et al. publication.^[3] As a result, for an isotropic material (where $K_{an} = 0)$ ), the presented form of anisotropic anhysteretic magnetization $M_{an}^{aniso}$ is not consistent with the isotropic anhysteretic magnetization $M_{an}^{iso}$ given by the Langevin equation. Physical analysis leads to the conclusion that the equation for anisotropic anhysteretic magnetization $M_{an}^{aniso}$ has to be corrected to the following form:^[3]

M_{an}^{aniso} = M_{s} \frac{\int_{0}^{π} e^{0.5 (E (1) + E (2))} \sin (θ) \cos (θ) d θ}{\int_{0}^{π} e^{0.5 (E (1) + E (2))} \sin (θ) d θ}

In the corrected form, the model for anisotropic anhysteretic magnetization $M_{an}^{aniso}$ was confirmed experimentally for anisotropic amorphous alloys.^[3]

3.3. Magnetization as a Function of Magnetizing Field

In Jiles–Atherton model, M(H) dependence is given in form of following ordinary differential equation:^[5]

\frac{d M}{d H} = \frac{1}{1 + c} \frac{M_{an} - M}{δ k - α (M_{an} - M)} + \frac{c}{1 + c} \frac{d M_{an}}{d H}

where $δ$ depends on direction of changes of magnetizing field $H$ ( $δ = 1$ for increasing field, $δ = - 1$ for decreasing field)

3.4. Flux Density as a Function of Magnetizing Field

Flux density $B$ in the material is given as:^[1]

B (H) = μ_{0} M (H)

where $μ_{0}$ is magnetic constant.

4. Vectorized Jiles–Atherton Model

Vectorized Jiles–Atherton model is constructed as the superposition of three scalar models one for each principal axis.^[6] This model is especially suitable for finite element method computations.

5. 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.^[7]

The two most important computational problems connected with the Jiles–Atherton model were identified:^[7]

numerical integration of the anisotropic anhysteretic magnetization $M_{an}^{aniso}$
solving the ordinary differential equation for $M (H)$ dependence.

For numerical integration of the anisotropic anhysteretic magnetization $M_{an}^{aniso}$ 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 $M (H)$ dependence, the Runge–Kutta methods are recommended. It was observed, that the best performing was 4-th order fixed step method.^[7]

6. 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:

frequency dependence of magnetic hysteresis loop in conductive materials ^[8]^[9]
influence of stresses on magnetic hysteresis loops ^[10]^[11]^[12]
magnetostriction of soft magnetic materials ^[10]^[13]

Moreover, different corrections were implemented, especially:

to avoid unphysical states when reversible permeability is negative ^[14]
to consider changes of average energy required to break pinning site ^[15]

7. Applications

Jiles–Atherton model may be applied for modeling:

rotating electric machines ^[16]
power transformers ^[17]
magnetostrictive actuators ^[18]
magnetoelastic sensors ^[19]^[20]
magnetic field sensors (e. g. fluxgates) ^[21]^[22]

It is also widely used for electronic circuit simulation, especially for models of inductive components, such as transformers or chokes.^[23]

References

Jiles, D. C.; Atherton, D.L. (1984). "Theory of ferromagnetic hysteresis". Journal of Applied Physics 55 (6): 2115. doi:10.1063/1.333582. Bibcode: 1984JAP....55.2115J. https://dx.doi.org/10.1063%2F1.333582
Ramesh, A.; Jiles, D. C.; Roderick, J. M. (1996). "A model of anisotropic anhysteretic magnetization". IEEE Transactions on Magnetics 32 (5): 4234. doi:10.1109/20.539344. Bibcode: 1996ITM....32.4234R. https://dx.doi.org/10.1109%2F20.539344
Szewczyk, R. (2014). "Validation of the anhysteretic magnetization model for soft magnetic materials with perpendicular anisotropy". Materials 7 (7): 5109–5116. doi:10.3390/ma7075109. PMID 28788121. Bibcode: 2014Mate....7.5109S. http://www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&;artid=5455830
Jiles, D.C.; Ramesh, A.; Shi, Y.; Fang, X. (1997). "Application of the anisotropic extension of the theory of hysteresis to the magnetization curves of crystalline and textured magnetic materials". IEEE Transactions on Magnetics 33 (5): 3961. doi:10.1109/20.619629. Bibcode: 1997ITM....33.3961J. https://zenodo.org/record/1232138. ;
Jiles, D. C.; Atherton, D.L. (1986). "A model of ferromagnetic hysteresis". Journal of Magnetism and Magnetic Materials 61 (1–2): 48. doi:10.1016/0304-8853(86)90066-1. Bibcode: 1986JMMM...61...48J. https://dx.doi.org/10.1016%2F0304-8853%2886%2990066-1
Szymanski, Grzegorz; Waszak, Michal (2004). "Vectorized Jiles–Atherton hysteresis model". Physica B 343 (1–4): 26–29. doi:10.1016/j.physb.2003.08.048. Bibcode: 2004PhyB..343...26S. https://dx.doi.org/10.1016%2Fj.physb.2003.08.048
Szewczyk, R. (2014). Computational problems connected with Jiles–Atherton model of magnetic hysteresis. 267. 275–283. doi:10.1007/978-3-319-05353-0_27. ISBN 978-3-319-05352-3. https://dx.doi.org/10.1007%2F978-3-319-05353-0_27
Jiles, D.C. (1994). "Modelling the effects of eddy current losses on frequency dependent hysteresis in electrically conducting media". IEEE Transactions on Magnetics 30 (6): 4326–4328. doi:10.1109/20.334076. Bibcode: 1994ITM....30.4326J. https://zenodo.org/record/1232132. ;
Szewczyk, R.; Frydrych, P. (2010). "Extension of the Jiles–Atherton model for modelling the frequency dependence of magnetic characteristics of amorphous alloy cores for inductive components of electronic devices". Acta Physica Polonica A 118 (5): 782. doi:10.12693/aphyspola.118.782. Bibcode: 2010AcPPA.118..782S. https://dx.doi.org/10.12693%2Faphyspola.118.782
Sablik, M.J.; Jiles, D.C. (1993). "Coupled magnetoelastic theory of magnetic and magnetostrictive hysteresis". IEEE Transactions on Magnetics 29 (4): 2113. doi:10.1109/20.221036. Bibcode: 1993ITM....29.2113S. https://zenodo.org/record/1232130. ;
Szewczyk, R.; Bienkowski, A. (2003). "Magnetoelastic Villari effect in high-permeability Mn-Zn ferrites and modeling of this effect". Journal of Magnetism and Magnetic Materials 254: 284–286. doi:10.1016/S0304-8853(02)00784-9. Bibcode: 2003JMMM..254..284S. https://dx.doi.org/10.1016%2FS0304-8853%2802%2900784-9
Jackiewicz, D.; Szewczyk, R.; Salach, J.; Bieńkowski, A. (2014). "Application of extended Jiles–Atherton model for modelling the influence of stresses on magnetic characteristics of the construction steel". Acta Physica Polonica A 126 (1): 392. doi:10.12693/aphyspola.126.392. Bibcode: 2014AcPPA.126..392J. https://dx.doi.org/10.12693%2Faphyspola.126.392
Szewczyk, R. (2006). "Modelling of the magnetic and magnetostrictive properties of high permeability Mn-Zn ferrites". Pramana 67 (6): 1165–1171. doi:10.1007/s12043-006-0031-z. Bibcode: 2006Prama..67.1165S. https://dx.doi.org/10.1007%2Fs12043-006-0031-z
Deane, J.H.B. (1994). "Modeling the dynamics of nonlinear inductor circuits". IEEE Transactions on Magnetics 30 (5): 2795–2801. doi:10.1109/20.312521. Bibcode: 1994ITM....30.2795D. https://dx.doi.org/10.1109%2F20.312521
Szewczyk, R. (2007). "Extension of the model of the magnetic characteristics of anisotropic metallic glasses". Journal of Physics D: Applied Physics 40 (14): 4109–4113. doi:10.1088/0022-3727/40/14/002. Bibcode: 2007JPhD...40.4109S. https://dx.doi.org/10.1088%2F0022-3727%2F40%2F14%2F002
Du, Ruoyang; Robertson, Paul (2015). "Dynamic Jiles–Atherton Model for Determining the Magnetic Power Loss at High Frequency in Permanent Magnet Machines". IEEE Transactions on Magnetics 51 (6): 7301210. doi:10.1109/TMAG.2014.2382594. Bibcode: 2015ITM....5182594D. https://www.repository.cam.ac.uk/handle/1810/246907. ;
Huang, Sy-Ruen et al. (2012). "Distinguishing internal winding faults from inrush currents in power transformers using Jiles–Atherton model parameters based on correlation voefficient". IEEE Transactions on Magnetics 27 (2): 548. doi:10.1109/TPWRD.2011.2181543. https://dx.doi.org/10.1109%2FTPWRD.2011.2181543
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