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 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 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. 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 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 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. 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 ,eta }} . 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 eta } 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 eta } —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 ,eta }} 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 ,eta }} is expressed as: y ( x ) = { 1  if  x ≥ β 0  if  x ≤ α k  if  α < x < β {displaystyle y(x)={egin{cases}1&{mbox{ if }}xgeq eta \0&{mbox{ if }}xleq alpha \k&{mbox{ if }}alpha <x<eta 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<eta } , it was in the region of x ≤ α {displaystyle xleq 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<eta } , it was in the region of x ≥ β {displaystyle xgeq eta } . 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} .