Preisach model of hysteresis

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The Preisach model of hysteresis generalizes hysteresis loops as the parallel connection of independent relay hysterons. It was first suggested in 1938 by P. Preisach in the German academic journal, "Zeitschrift für Physik". Since then, it has become a widely accepted model of hysteresis.

The Preisach model is especially accurate in the field of ferromagnetism, as the ferromagnetic material can be described as a collection of small domains, each magnetized to a value of either h or − h. A sample of iron, for example, may have randomly distributed magnetic domains, resulting in a net magnetic field of zero.

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[edit] Nonideal Relay

The relay hysteron is the fundamental building block of the Preisach model. It is described as a two-valued operator denoted by Rα,β. Its I/O map takes the form of a loop, as shown:

Image:Preisach_Relay.svg

Above, a relay of magnitude 1. α defines the "switch-off" threshold, and β defines the "switch-on" threshold.

Graphically, if x is less than α, the output y is "low" or "off." As we increase x, the output remains low until x reaches β--at which point the output switches "on." Further increasing x has no change. Decreasing x, y does go low until x reaches α again. It is apparent that the relay operator Rα,β takes the path of a loop, and its next state depends on its past state.

Mathematically, the output of Rα,β is expressed as:

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

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

[edit] 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:

Image:Preisach_Model.PNG

Each of these relays has different α and β thresholds and is scaled by μ. Depending on their distribution on the Preisach plane, the relay hysterons can represent hysteresis with good accuracy. As well, with increasing N, the true hysteresis curve is approximated better.

Image:Discrete_Preisach_Model.PNG

In the limit as N approaches infinity, we obtain the continuous Preisach model.

[edit] The αβ plane

One of the easiest ways to look at the preisach model is using a geometric interpretation. Consider a plane of coordiantes αβ. On this plane, each point (αi,βi) is mapped to a specific relay hysteron R_{\alpha_{i},\beta_{i}}.

We consider only the half-plane α < β 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 α = β.

We now present the Preisach Density Function, denoted μ(α,β). This function describes the amount of relay hysterons of each distinct values of (αi,βi). As a default we say that outside the right triangle μ(α,β) = 0.

[edit] External links