Preisach model of hysteresis
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
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] The relay hysteron is the fundamental building block of the Preisach model.
It is described as a two-valued operator denoted by
Its I/O map takes the form of a loop, as shown:
This definition of the hysteron shows that the current value
of the complete hysteresis loop depends upon the history of the input variable
The Preisach model consists of many relay hysterons connected in parallel, given weights, and summed.
, the true hysteresis curve is approximated better.
approaches infinity, we obtain the continuous Preisach model.
[4][5] One of the easiest ways to look at the Preisach model is using a geometric interpretation.
is mapped to a specific relay hysteron
Each relay can be plotted on this so-called Preisach plane with its
Depending on their distribution on the Preisach plane, the relay hysterons can represent hysteresis with good accuracy.
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
We now present the Preisach density function, denoted
This function describes the amount of relay hysterons of each distinct values of
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.
[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] 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.
