Keywords: ferromagnetic hysteresis, stochastic model.

In this paper a two-dimensional vectorial model of magnetic hysteresis is described. Nonlinear material models have great practical importance since they appear in many real-world engineering computational problems. The main concept behind the recent model lies in the stochastic representation of the memory of the material. With the aid of this kind of memory handling the magnetic state of the material can be described easily and the calculation of the magnetization vector becomes very simple even in the case of rotating magnetic field excitations.

Many vector hysteresis models can be found in the literature with their own advantages and disadvantages. These models are mainly grouped around two different approaches. One of them is based on building the vector model from several scalar models aligned to different directions of the space [1,2], the other approach is based on the nucleation theory and energetic considerations like the Stoner-Wolfarth model.

The model presented in this paper follows a rather different approach, a phenomenological one. The investigation of the vector hysteresis phenomenon is carried out on an 'intuitive' basis. The goal is the deduction of the main properties of the vectorial hysteresis behavior from a set of quite simple rules. The main concept behind this approach is the way of handling the magnetic history of the material, rotation of individual magnetic domains and rotation of the applied magnetic field vector H.

For this model the material is considered as a collection of freely rotating magnetic domains with dipole-moments. Each domain has its own orientation and elementary magnetic moment m. For the sake of simplicity it is assumed that this magnetic moment has the same value for each domain. The memory structure of the material is treated as a probability density function (or discrete probability distribution) for the interval , since the magnetization vector M can be represented as the vectorial sum of the statistically distributed domains in the space. This function holds for the probabilities of the alignment of the domains to certain space directions. Theoretically this function can be continuous, but in the case of its application in i.e. a field calculation problem, it is computationally more convenient to treat it as a discrete distribution on .

This memory handling method enables the possible states of the material to be well described, and shows that the model has the so called infinite memory which according to the research is more closely related to physical background of hysteresis phenomena than local memory structure [2]. Thus it has an advantage of this kind of description.

The memory function represents the M vector in space so it is straightforward that the components of the M are

The presented vector-model is based on a scalar hysteresis in one sense, namely the magnetization process is governed by a scalar hysteresis curve. The scalar hysteresis model applied in this case is a simple model based on statistical considerations. A simple magnetic domain can be handled as a delayed relay operator [3] (elementary hysteron) with two states corresponding to the magnetized and the demagnetized state and the value of the switching field (up/down) is statistically distributed among the domains of the material and the expected value of the switching field is the same for all domains [4].

The presented two-dimensional vector hysteresis model has been developed in a fully intuitive way and on a phenomenological basis to provide a reasonable description of the vector hysteresis phenomena in two dimensions. Furthermore the model can be extended to three dimensions with not much difficulty, and various hysteresis characteristics can be modeled with the help of it. The model shows good agreement with known experimental results of vector hysteresis phenomenon and can be easily implemented into field calculation problems.

1

I. D, Mayergoyz, Mathematical Models for Hysteresis, Springer-Verlag, New York, 1991.

2

A. Ivanyi, Hysteresis Models in Elecromagnetic Computation, Akadémiai Kiadó, Budapest, 1997.

3

A. Visintin, Differential Models of Hysteresis, Springer-Verlag, 1994.

4

Z. Sari, A. Ivanyi, Statictical Approach of Hysteresis, Phisica B 372 45-48, 2006. doi:10.1016/j.physb.2005.10.015

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