View source for Preisach model of hysteresis

{{short description|Model of magnetic 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 Preisach|Ferenc (Franz) Preisach]] in the German academic journal {{lang|de|[[Zeitschrift für Physik]]}}.<ref>{{cite journal|first=F |last=Preisach |title=Über die magnetische Nachwirkung|journal=Zeitschrift für Physik|volume=94 |pages=277&ndash;302 |year=1935 |issue=5–6 |doi=10.1007/bf01349418|bibcode=1935ZPhy...94..277P|s2cid=122409841 }}</ref> In the field of [[ferromagnetism]], the Preisach model is sometimes thought to describe a ferromagnetic material as a network of small independently acting [[Magnetic domain|domains]], each magnetized to a value of either <math>h</math> or <math>-h</math>. 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.<ref>{{cite book|last1=Smith|first1=Ralph C.|title=Smart material systems : model development|date=2005|publisher=SIAM, Society for Industrial and Applied Mathematics|location=Philadelphia, Pa.|isbn=978-0-89871-583-5|page=189}}</ref><ref>{{cite book|last1=Visintin|first1=Augusto|title=Differential models of hysteresis|date=1994|publisher=Springer Berlin Heidelberg|location=Berlin, Heidelberg|isbn=978-3-662-11557-2}}</ref>

==Nonideal relay==
The relay hysteron is the fundamental building block of the Preisach model. It is described as a two-valued [[Operator (mathematics)|operator]] denoted by <math>R_{\alpha,\beta}</math>. Its I/O map takes the form of a loop, as shown:

[[Image:Preisach Relay.svg]]

Above, a relay of magnitude 1, <math>\alpha</math> defines the "switch-off" threshold, and <math>\beta</math> defines the "switch-on" threshold.

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

Mathematically, the output of <math>R_{\alpha,\beta}</math> is expressed as:

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

Where <math>k=0</math> if the last time <math>x</math> was outside of the boundaries <math>\alpha<x<\beta</math>, it was in the region of <math>x\leq\alpha</math>; and <math>k=1</math> if the last time <math>x</math> was outside of the boundaries <math>\alpha<x<\beta</math>, it was in the region of <math>x\geq\beta</math>.

This definition of the hysteron shows that the current value <math>y</math> of the complete hysteresis loop depends upon the history of the input variable <math>x</math>.

==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:

[[Image:Preisach Model.PNG]]

Each of these relays has different <math>\alpha</math> and <math>\beta</math> thresholds and is scaled by <math>\mu</math>. With increasing <math>N</math>, the true hysteresis curve is approximated better.

[[Image:Discrete Preisach Model.PNG|frame|center|An example of hysteresis modeled with different numbers, N, of hysterons.]]

In the limit as <math>N</math> approaches infinity, we obtain the continuous Preisach model.<ref>{{cite journal | last1=Mayergoyz | first1=I.D. | last2=Friedman | first2=G. | title=Generalized Preisach model of hysteresis | journal=IEEE Transactions on Magnetics | publisher=Institute of Electrical and Electronics Engineers (IEEE) | volume=24 | issue=1 | year=1988 | issn=0018-9464 | doi=10.1109/20.43892 | pages=212–217| bibcode=1988ITM....24..212M }}</ref><ref>{{cite book | last=Mayergoyz | first=I. D. | title=Mathematical Models of Hysteresis | chapter=The Classical Preisach Model of Hysteresis | publisher=Springer New York | publication-place=New York, NY | year=1991 | isbn=978-1-4612-7767-5 | doi=10.1007/978-1-4612-3028-1_1 | pages=1–63| s2cid=118969949 }}</ref>

:<math>y(t)=\iint_{\alpha \geq \beta}\mu(\alpha,\beta)R_{\alpha\beta}x(t) d \alpha d \beta</math>

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

We consider only the half-plane <math>\alpha<\beta</math> 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 <math>\alpha=\beta</math>.

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

A modified formulation of the classical Preisach model has been presented, allowing analytical expression of 
the  Everett function.<ref>{{cite journal|last1=Szabó|first1=Zsolt|title=Preisach functions leading to closed form permeability|journal=Physica B: Condensed Matter|date=February 2006|volume=372|issue=1–2|pages=61–67|doi=10.1016/j.physb.2005.10.020|bibcode=2006PhyB..372...61S}}</ref> This  makes  the  model  considerably  faster  and  especially  adequate  for inclusion in [[electromagnetic field]] computation or [[Network analysis (electrical circuits)|electric circuit analysis]] codes.

== Vector Preisach model ==

The vector Preisach model is constructed as the linear superposition of scalar models.<ref>{{cite book|last1=Mayergoyz|first1=I.D.|title=Mathematical models of hysteresis and their applications|date=2003|publisher=Elsevier|location=Amsterdam|isbn=978-0-12-480873-7|edition=1st}}</ref> For considering the uniaxial [[anisotropy]] of the material, Everett functions are expanded by [[Fourier transform|Fourier]] coefficients. In this case, the measured and simulated curves are in a very good agreement.<ref>{{cite journal |last1=Kuczmann |first1=Miklos |last2=Stoleriu |first2=Laurentiu |title=Anisotropic vector Preisach model |url=http://stoner.phys.uaic.ro/jarp/index.php/jarp/article/download/24/10 |format=pdf |journal=Journal of Advanced Research in Physics |volume=1 |issue=1 |pages=011009 |access-date=3 August 2016}}</ref>
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,<ref>{{cite journal |last1=Cardelli |first1=Ermanno |last2=Della Torre |first2=Edward |last3=Faba |first3=Antonio |date=2010 |title=A General Vector Hysteresis Operator: Extension to the 3-D Case |journal=IEEE Transactions on Magnetics |volume=46 |issue=12 |pages=3990–4000 |doi=10.1109/tmag.2010.2072933|bibcode=2010ITM....46.3990C |s2cid=31552464 }}</ref> and circular hysteron is used for vector hysteresis in 2D.<ref>{{cite journal |last=Cardelli |first=Ermanno |date=2011 |title=A general hysteresis operator for the modeling of vector fields |journal=IEEE Transactions on Magnetics |volume=47 |issue=8 |pages=2056–2067 |doi=10.1109/tmag.2011.2126589|bibcode=2011ITM....47.2056C |s2cid=25965526 }}</ref>

== 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,<ref>{{Cite journal|last1=Kramer|first1=Isaac|last2=Bayer|first2=Yuval|last3=Adeyemo|first3=Taiwo|last4=Mau|first4=Yair|date=2021-04-14|title=Hysteresis in soil hydraulic conductivity as driven by salinity and sodicity – a modeling framework|url=https://hess.copernicus.org/articles/25/1993/2021/|journal=Hydrology and Earth System Sciences|language=English|volume=25|issue=4|pages=1993–2008|doi=10.5194/hess-25-1993-2021|issn=1027-5606|doi-access=free|bibcode=2021HESS...25.1993K }}</ref> the modeling of soil water retention<ref>{{Cite journal|last1=Flynn|first1=D|last2=Rasskazov|first2=O|date=2005-01-01|title=On the integration of an ODE involving the derivative of a Preisach nonlinearity|journal=Journal of Physics: Conference Series|language=en|volume=22|issue=1|pages=43–55|doi=10.1088/1742-6596/22/1/003|issn=1742-6588|doi-access=free|bibcode=2005JPhCS..22...43F}}</ref><ref>{{Citation|last1=Flynn|first1=Denis|title=Chapter 7 - Application of the Preisach Model to Soil-Moisture Hysteresis|date=2006-01-01|url=https://www.sciencedirect.com/science/article/pii/B9780124808744500257|work=The Science of Hysteresis|pages=689–744|editor-last=Bertotti|editor-first=Giorgio|place=Oxford|publisher=Academic Press|language=en|doi=10.1016/b978-012480874-4/50025-7|isbn=978-0-12-480874-4|access-date=2022-02-07|last2=Mcnamara|first2=Hugh|last3=O'kane|first3=Philip|last4=PokrovskÜ|first4=Alexei|editor2-last=Mayergoyz|editor2-first=Isaak D.}}</ref><ref>{{Cite journal|last1=O’Kane|first1=J. P.|last2=Flynn|first2=D.|date=2007-01-17|title=Thresholds, switches and hysteresis in hydrology from the pedon to the catchment scale: a non-linear systems theory|url=https://hess.copernicus.org/articles/11/443/2007/|journal=Hydrology and Earth System Sciences|language=English|volume=11|issue=1|pages=443–459|doi=10.5194/hess-11-443-2007|issn=1027-5606|doi-access=free|bibcode=2007HESS...11..443O }}</ref><ref>{{Cite journal|last=McNamara|first=H.|date=January 2014|title=An estimate of energy dissipation due to soil-moisture hysteresis|url=https://doi.org/10.1002/2012WR012634|journal=Water Resources Research|volume=50|issue=1|pages=725–735|doi=10.1002/2012wr012634|bibcode=2014WRR....50..725M |s2cid=129547567 |issn=0043-1397}}</ref> and the effect of stress and strains on soil and rock structures.<ref>{{Citation|last=Guyer|first=Robert A.|title=Chapter 6 - Hysteretic Elastic Systems: Geophysical Materials|date=2006-01-01|url=https://www.sciencedirect.com/science/article/pii/B9780124808744500245|work=The Science of Hysteresis|pages=555–688|editor-last=Bertotti|editor-first=Giorgio|place=Oxford|publisher=Academic Press|language=en|doi=10.1016/b978-012480874-4/50024-5|isbn=978-0-12-480874-4|access-date=2022-02-07|editor2-last=Mayergoyz|editor2-first=Isaak D.}}</ref>

==See also==
* [[Jiles–Atherton model]]
* [[Stoner–Wohlfarth model]]

==References==
{{Reflist}}

==External links==
* [https://web.archive.org/web/20051102025906/http://euclid.ucc.ie/hysteresis/ University College, Cork] Hysteresis Tutorial
* [https://sourceforge.net/projects/hysteresis/files/ Budapest University of Technology and Economics, Hungary] Matlab implementation of the Preisach model developed by Zs. Szabó.
* [https://github.com/isaackramer/python-preisach/] Python implementation of Preisach Model.
* [https://github.com/jhart12/Preisach_Model_MATLAB] Matlab implementation of Preisach Model.

[[Category:Magnetic hysteresis]]
[[Category:Hysteresis]]