# Macroscopic hysteresis modelling

The phenomenon of hysteresis has been observed for a long time in many different areas of science and engineering. Examples of hysteresis in material systems include mechanical hysteresis, magnetic hysteresis, ferroelectric hysteresis and many others. In some applications hysteresis can be employed for a useful purpose. Such is the case in systems relying on permanent magnets and in magnetic recording. In other cases such as positioning systems, electrical machines, hysteresis phenomena are often unwanted and consequently must be avoided as much as possible.

In general two types of modelling techniques are used to describe hysteresis processes: physical modelling and phenomenological modelling. In physical modelling, the basic processes involved are simulated in order to be able to describe the basic magnetizing modes. In phenomenological models, the gross behaviour of the material is described mathematically by generating curves, following predefined rules, for the material properties. The latter models are often computationally more efficient than the former, but they do not give any insight into the physical principles involved.

For the design of electromagnetic devices, accurate evaluations of the magnetic field patterns in the device are often essential for a realistic prediction of the performance characteristics of the device. Consequently, in many cases, the magnetic field computations should account for the precise features of the magnetic materials used, including the hysteresis effects. At this point, the macroscopic properties, described by phenomenological models are coupled with the Maxwell’s equations so as to obtain accurate solutions for electromagnetic field problems.

Hysteresis modelling becomes even more important when one aims at the evaluation of properties directly related to hysteresis processes, e.g. the calculation of iron losses in electromagnetic devices. We recall that the hysteresis loss is related to the fact that the relationship between the magnetic induction vector **B** and the magnetic field vector **H** in the material depends on the history of the magnetic field. For these cases a phenomenological hysteresis model describing the **B**-**H**-relation is sufficient as long as the desired accuracy is obtained.

For other applications, such as magnetic hysteretic non-destructive evaluation (NDE), there is a preference for a physical hysteresis model, or a model the parameters of which can be directly related to microstructural features of the material. It is clear that a hysteresis model that has to be used for interpretation of NDE results should have at least a few parameters that are directly related to the metallurgical properties of the material. Changes in the material microstructure lead to a modification of the mechanical properties, which should be identified by the changes of the magnetic hysteresis properties, or the limited number of material parameters defined in the hysteresis model.

Finally, one must distinguish between scalar hysteresis models and vectorial hysteresis models. Indeed, depending on the application, the flux pattern may be unidirectional, i.e. the direction of the magnetic field or the magnetic induction is fixed but the amplitude is changing continuously. For other applications, e.g. in rotating electrical machines, a non-negligible part of the magnetic fields has a rotational character, i.e. the amplitude as well as the direction of the magnetic field and the magnetic induction vector change continuously.

The research at EELAB concerning hysteresis modelling deals with most aspects described above.

## Applications

One of the most widely used models for magnetic hysteresis is the Preisach model. Here, the magnetization curves are constructed from elementary magnetic dipoles, Preisach dipoles, each of them characterized by a hysteretic non-linearity, i.e. a rectangular loop, defined by an ‘up’ switching field a to the +1 state and a ‘down’ switching field b to the –1 state, a>b, see Fig. 1. The density of the Preisach dipoles is represented by the distribution function P(a,b).

The Preisach algorithm allows to describe the quasi-static behaviour of ferromagnetic materials, see Fig. 2. An advanced hysteresis model, such as the Preisach model, reproduces all main features of magnetization, including the ‘return-point-memory’ and the ‘wiping-out’ property.

The research at EELAB aims at the development of new material models and characterization techniques in order to obtain a better perception of the macroscopic behavior of e.g. laminated SiFe-alloys.

The development of new characterization techniques is based on the numerical evaluation of the magnetic behavior of the material starting from Maxwell’s equations in combination with complex constitutive laws, described by e.g. Preisach models. Fig. 3 illustrates the comparison between the measured dynamic magnetization loop and the computed magnetization loop of a steel sheet subjected to a distorted flux pattern, including third and fifth harmonics.

## Relevant publications

Philips D., Dupré L., Melkebeek J., Magneto-dynamic field computation using a rate-dependent hysteresis model, IEEE Transactions on Magnetics 30 (6): 4377-4379 Part 1, NOV 1994

Philips D., Dupré L.R., Melkebeek J., Comparison of Jiles and Preisach hysteresis models in magnetodynamics, IEEE Transactions on Magnetics 31 (6): 3551-3553 Part 2, NOV 1995

Dupré L., Bottauscio O., Chiampi M., Repeeto M, Melkebeek J., Modeling of electromagnetic phenomena in soft magnetic materials under unidirectional time periodic flux excitations, IEEE Transactions on Magnetics 35 (5): 4171-4184 Part 3, SEP 1999

Dupré L., Bertotti G., Basso V., Fiorillo F., Melkebeek J. Generalisation of the dynamic Preisach model toward grain oriented Fe-Si alloys, Physica B 275 (1-3): 202-206 JAN 2000

Makaveev D., Dupré L., De Wulf M., Melkebeek J., Modeling of quasistatic magnetic hysteresis with feed-forward neural networks, Journal of Applied Physics 89 (11): 6737-6739 Part 2, JUN 1 2001

## Contact

For more information: luc.dupre@ugent.be