Preisach model of hysteresis

Originally, the Preisach model of hysteresis generalized magnetic hysteresis as relationship between 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". 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 formula_1 or formula_2. A sample of iron, for example, may have evenly distributed magnetic domains, resulting in a net magnetic moment of zero.
Mathematically similar model seems 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.
Nonideal relay.
The relay hysteron is the fundamental building block of the Preisach model. It is described as a two-valued operator denoted by formula_3. Its I/O map takes the form of a loop, as shown:
Above, a relay of magnitude 1. formula_4 defines the "switch-off" threshold, and formula_5 defines the "switch-on" threshold.
Graphically, if formula_6 is less than formula_4, the output formula_8 is "low" or "off." As we increase formula_6, the output remains low until formula_6 reaches formula_5—at which point the output switches "on." Further increasing formula_6 has no change. Decreasing formula_6, formula_8 does not go low until formula_6 reaches formula_4 again. It is apparent that the relay operator formula_3 takes the path of a loop, and its next state depends on its past state.
Mathematically, the output of formula_3 is expressed as:
formula_19
Where formula_20 if the last time formula_6 was outside of the boundaries formula_22, it was in the region of formula_23; and formula_24 if the last time formula_6 was outside of the boundaries formula_22, it was in the region of formula_27.
This definition of the hysteron shows that the current value formula_8 of the complete hysteresis loop depends upon the history of the input variable formula_6.
Discrete Preisach model.
The Preisach model consists of many relay hysterons connected in parallel, given weights, and summed. This is best visualized by a block diagram:
Each of these relays has different formula_4 and formula_5 thresholds and is scaled by formula_32. Each relay can be plotted on a so-called Preisach plane with its formula_33 values. Depending on their distribution on the Preisach plane, the relay hysterons can represent hysteresis with good accuracy. As well, with increasing formula_34, the true hysteresis curve is approximated better.
In the limit as formula_34 approaches infinity, we obtain the continuous Preisach model.
The formula_36 plane.
One of the easiest ways to look at the Preisach model is using a geometric interpretation.
Consider a plane of coordinates formula_36. On this plane, each point formula_38 is mapped to a specific relay hysteron formula_39.
We consider only the half-plane formula_40 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 formula_41.
We now present the Preisach density function, denoted formula_42. This function describes the amount of relay hysterons of each distinct values of formula_38. As a default we say that outside the right triangle formula_44.
A modified formulation of the classical Preisach model has been presented, allowing analytical expression of 
the Everett function. 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. 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.
Another approach uses different relay hysteron, closed surfaces defined on the 3D input space. In general spherical hysteron is used for vector hystersis in 3D, and circular hysteron is used for vector hystersis in 2D.