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. 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 h {\displaystyle h} or − h {\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.
Nonideal relay The relay hysteron is the fundamental building block of the Preisach model. It is described as a two-valued operator denoted by R α , β {\displaystyle R_{\alpha ,\beta }} . Its I/O map takes the form of a loop, as shown:
Above, a relay of magnitude 1, α {\displaystyle \alpha } defines the "switch-off" threshold, and β {\displaystyle \beta } defines the "switch-on" threshold. Graphically, if x {\displaystyle x} is less than α {\displaystyle \alpha } , the output y {\displaystyle y} is "low" or "off." As we increase x {\displaystyle x} , the output remains low until x {\displaystyle x} reaches β {\displaystyle \beta } —at which point the output switches "on." Further increasing x {\displaystyle x} has no change. Decreasing x {\displaystyle x} , y {\displaystyle y} does not go low until x {\displaystyle x} reaches α {\displaystyle \alpha } again. It is apparent that the relay operator R α , β {\displaystyle R_{\alpha ,\beta }} takes the path of a loop, and its next state depends on its past state. Mathematically, the output of R α , β {\displaystyle R_{\alpha ,\beta }} is expressed as:
y ( x ) = { 1 if x ≥ β 0 if x ≤ α k if α < x < β {\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 k = 0 {\displaystyle k=0} if the last time x {\displaystyle x} was outside of the boundaries α < x < β {\displaystyle \alpha <x<\beta } , it was in the region of x ≤ α {\displaystyle x\leq \alpha } ; and k = 1 {\displaystyle k=1} if the last time x {\displaystyle x} was outside of the boundaries α < x < β {\displaystyle \alpha <x<\beta } , it was in the region of x ≥ β {\displaystyle x\geq \beta } . This definition of the hysteron shows that the current value y {\displaystyle y} of the complete hysteresis loop depends upon the history of the input variable x {\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 α {\displaystyle \alpha } and β {\displaystyle \beta } thresholds and is scaled by μ {\displaystyle \mu } . With increasing N {\displaystyle N} , the true hysteresis curve is approximated better.
In the limit as N {\displaystyle N} approaches infinity, we obtain the continuous Preisach model.
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