Preisach model of hysteresis

In electromagnetism, the Preisach model of hysteresis is a model of magnetic hysteresis. Originally, it generalized hysteresis as the relationship between the magnetic field and magnetization of a magnetic material as the parallel connection of independent relay hysterons. It was first suggested in 1935 by Ferenc (Franz) Preisach in the German academic journal Zeitschrift für Physik.^[1] In the field of ferromagnetism, the Preisach model is sometimes thought to describe a ferromagnetic material as a network of small independently acting domains, each magnetized to a value of either ${\displaystyle h}$ or ${\displaystyle -h}$. A sample of iron, for example, may have evenly distributed magnetic domains, resulting in a net magnetic moment of zero.

Mathematically similar models seem to have been independently developed in other fields of science and engineering. One notable example is the model of capillary hysteresis in porous materials developed by Everett and co-workers. Since then, following the work of people like M. Krasnoselkii, A. Pokrovskii, A. Visintin, and I.D. Mayergoyz, the model has become widely accepted as a general mathematical tool for the description of hysteresis phenomena of different kinds.^[2][3]

Nonideal relay

The relay hysteron is the fundamental building block of the Preisach model. It is described as a two-valued operator denoted by ${\displaystyle R_{\alpha ,\beta }}$. Its I/O map takes the form of a loop, as shown:

Above, a relay of magnitude 1, ${\displaystyle \alpha }$ defines the "switch-off" threshold, and ${\displaystyle \beta }$ defines the "switch-on" threshold.

Graphically, if ${\displaystyle x}$ is less than ${\displaystyle \alpha }$, the output ${\displaystyle y}$ is "low" or "off." As we increase ${\displaystyle x}$, the output remains low until ${\displaystyle x}$ reaches ${\displaystyle \beta }$—at which point the output switches "on." Further increasing ${\displaystyle x}$ has no change. Decreasing ${\displaystyle x}$, ${\displaystyle y}$ does not go low until ${\displaystyle x}$ reaches ${\displaystyle \alpha }$ again. It is apparent that the relay operator ${\displaystyle R_{\alpha ,\beta }}$ takes the path of a loop, and its next state depends on its past state.

Mathematically, the output of ${\displaystyle R_{\alpha ,\beta }}$ is expressed as:

${\displaystyle y(x)={\begin{cases}1&{\mbox{ if }}x\geq \beta \\0&{\mbox{ if }}x\leq \alpha \\k&{\mbox{ if }}\alpha <x<\beta \end{cases}}}$

Where ${\displaystyle k=0}$ if the last time ${\displaystyle x}$ was outside of the boundaries ${\displaystyle \alpha <x<\beta }$, it was in the region of ${\displaystyle x\leq \alpha }$; and ${\displaystyle k=1}$ if the last time ${\displaystyle x}$ was outside of the boundaries ${\displaystyle \alpha <x<\beta }$, it was in the region of ${\displaystyle x\geq \beta }$.

This definition of the hysteron shows that the current value ${\displaystyle y}$ of the complete hysteresis loop depends upon the history of the input variable ${\displaystyle x}$.

Discrete Preisach model

The Preisach model consists of many relay hysterons connected in parallel, given weights, and summed. This can be visualized by a block diagram:

Each of these relays has different ${\displaystyle \alpha }$ and ${\displaystyle \beta }$ thresholds and is scaled by ${\displaystyle \mu }$. With increasing ${\displaystyle N}$, the true hysteresis curve is approximated better.

An example of hysteresis modeled with different numbers, N, of hysterons.

In the limit as ${\displaystyle N}$ approaches infinity, we obtain the continuous Preisach model.^[4][5]

${\displaystyle y(t)=\iint _{\alpha \geq \beta }\mu (\alpha ,\beta )R_{\alpha \beta }x(t)d\alpha d\beta }$

Preisach plane

One of the easiest ways to look at the Preisach model is using a geometric interpretation. Consider a plane of coordinates ${\displaystyle (\alpha ,\beta )}$. On this plane, each point ${\displaystyle (\alpha _{i},\beta _{i})}$ is mapped to a specific relay hysteron ${\displaystyle R_{\alpha _{i},\beta _{i}}}$. Each relay can be plotted on this so-called Preisach plane with its ${\displaystyle (\alpha ,\beta )}$ values. Depending on their distribution on the Preisach plane, the relay hysterons can represent hysteresis with good accuracy.

We consider only the half-plane ${\displaystyle \alpha <\beta }$ as any other case does not have a physical equivalent in nature.

Next, we take a specific point on the half plane and build a right triangle by drawing two lines parallel to the axes, both from the point to the line ${\displaystyle \alpha =\beta }$.

We now present the Preisach density function, denoted ${\displaystyle \mu (\alpha ,\beta )}$. This function describes the amount of relay hysterons of each distinct values of ${\displaystyle (\alpha _{i},\beta _{i})}$. As a default we say that outside the right triangle ${\displaystyle \mu (\alpha ,\beta )=0}$.

A modified formulation of the classical Preisach model has been presented, allowing analytical expression of the Everett function.^[6] This makes the model considerably faster and especially adequate for inclusion in electromagnetic field computation or electric circuit analysis codes.

Vector Preisach model

The vector Preisach model is constructed as the linear superposition of scalar models.^[7] For considering the uniaxial anisotropy of the material, Everett functions are expanded by Fourier coefficients. In this case, the measured and simulated curves are in a very good agreement.^[8] Another approach uses different relay hysteron, closed surfaces defined on the 3D input space. In general spherical hysteron is used for vector hysteresis in 3D,^[9] and circular hysteron is used for vector hysteresis in 2D.^[10]

Applications

The Preisach model has been applied to model hysteresis in a wide variety of fields, including to study irreversible changes in soil hydraulic conductivity as a result of saline and sodic conditions,^[11] the modeling of soil water retention^{[12][13][14][15]} and the effect of stress and strains on soil and rock structures.^[16]

See also

• Jiles–Atherton model
• Stoner–Wohlfarth model

References

1. Preisach, F (1935). "Über die magnetische Nachwirkung". Zeitschrift für Physik. 94 (5–6): 277–302. Bibcode:1935ZPhy...94..277P. doi:10.1007/bf01349418. S2CID 122409841.
2. Smith, Ralph C. (2005). Smart material systems : model development. Philadelphia, Pa.: SIAM, Society for Industrial and Applied Mathematics. p. 189. ISBN 978-0-89871-583-5.
3. Visintin, Augusto (1994). Differential models of hysteresis. Berlin, Heidelberg: Springer Berlin Heidelberg. ISBN 978-3-662-11557-2.
4. Mayergoyz, I.D.; Friedman, G. (1988). "Generalized Preisach model of hysteresis". IEEE Transactions on Magnetics. 24 (1). Institute of Electrical and Electronics Engineers (IEEE): 212–217. Bibcode:1988ITM....24..212M. doi:10.1109/20.43892. ISSN 0018-9464.
5. Mayergoyz, I. D. (1991). "The Classical Preisach Model of Hysteresis". Mathematical Models of Hysteresis. New York, NY: Springer New York. pp. 1–63. doi:10.1007/978-1-4612-3028-1_1. ISBN 978-1-4612-7767-5. S2CID 118969949.
6. Szabó, Zsolt (February 2006). "Preisach functions leading to closed form permeability". Physica B: Condensed Matter. 372 (1–2): 61–67. Bibcode:2006PhyB..372...61S. doi:10.1016/j.physb.2005.10.020.
7. Mayergoyz, I.D. (2003). Mathematical models of hysteresis and their applications (1st ed.). Amsterdam: Elsevier. ISBN 978-0-12-480873-7.
8. Kuczmann, Miklos; Stoleriu, Laurentiu. "Anisotropic vector Preisach model" (pdf). Journal of Advanced Research in Physics. 1 (1): 011009. Retrieved 3 August 2016.
9. Cardelli, Ermanno; Della Torre, Edward; Faba, Antonio (2010). "A General Vector Hysteresis Operator: Extension to the 3-D Case". IEEE Transactions on Magnetics. 46 (12): 3990–4000. Bibcode:2010ITM....46.3990C. doi:10.1109/tmag.2010.2072933. S2CID 31552464.
10. Cardelli, Ermanno (2011). "A general hysteresis operator for the modeling of vector fields". IEEE Transactions on Magnetics. 47 (8): 2056–2067. Bibcode:2011ITM....47.2056C. doi:10.1109/tmag.2011.2126589. S2CID 25965526.
11. Kramer, Isaac; Bayer, Yuval; Adeyemo, Taiwo; Mau, Yair (2021-04-14). "Hysteresis in soil hydraulic conductivity as driven by salinity and sodicity – a modeling framework". Hydrology and Earth System Sciences. 25 (4): 1993–2008. Bibcode:2021HESS...25.1993K. doi:10.5194/hess-25-1993-2021. ISSN 1027-5606.
12. Flynn, D; Rasskazov, O (2005-01-01). "On the integration of an ODE involving the derivative of a Preisach nonlinearity". Journal of Physics: Conference Series. 22 (1): 43–55. Bibcode:2005JPhCS..22...43F. doi:10.1088/1742-6596/22/1/003. ISSN 1742-6588.
13. Flynn, Denis; Mcnamara, Hugh; O'kane, Philip; PokrovskÜ, Alexei (2006-01-01), Bertotti, Giorgio; Mayergoyz, Isaak D. (eds.), "Chapter 7 - Application of the Preisach Model to Soil-Moisture Hysteresis", The Science of Hysteresis, Oxford: Academic Press, pp. 689–744, doi:10.1016/b978-012480874-4/50025-7, ISBN 978-0-12-480874-4, retrieved 2022-02-07
14. O’Kane, J. P.; Flynn, D. (2007-01-17). "Thresholds, switches and hysteresis in hydrology from the pedon to the catchment scale: a non-linear systems theory". Hydrology and Earth System Sciences. 11 (1): 443–459. Bibcode:2007HESS...11..443O. doi:10.5194/hess-11-443-2007. ISSN 1027-5606.
15. McNamara, H. (January 2014). "An estimate of energy dissipation due to soil-moisture hysteresis". Water Resources Research. 50 (1): 725–735. Bibcode:2014WRR....50..725M. doi:10.1002/2012wr012634. ISSN 0043-1397. S2CID 129547567.
16. Guyer, Robert A. (2006-01-01), Bertotti, Giorgio; Mayergoyz, Isaak D. (eds.), "Chapter 6 - Hysteretic Elastic Systems: Geophysical Materials", The Science of Hysteresis, Oxford: Academic Press, pp. 555–688, doi:10.1016/b978-012480874-4/50024-5, ISBN 978-0-12-480874-4, retrieved 2022-02-07

External links

• University College, Cork Hysteresis Tutorial
• Budapest University of Technology and Economics, Hungary Matlab implementation of the Preisach model developed by Zs. Szabó.
• Python implementation of Preisach Model.
• Matlab implementation of Preisach Model.