Hysteretic Isotropic Material characterized by a Preisach-type model in Flux

Introduction

This chapter discusses the creation of a material with a B(H) property of types:

Isotropic hysteretic, Preisach model described by 4 parameters of a typical cycle and
Isotropic hysteretic, Preisach model identified by N triplets

in a Flux project.

Materials with these kinds of B(H) property implement a Preisach-type model that allows the consideration of ferromagnetic hysteresis phenomena still during the resolution of the project. Consequently, they allow a direct or a priori evaluation of the iron losses in the bulk of the ferromagnetic material, in contrast with so-called a posteriori approaches like the Loss Surface and Bertotti models.

The following topics are covered in the next sections:

An overview of Preisach-type models and of the specific implementation in Flux.
How to create a material with a Preisach-type B(H) property in a Flux project.
Availability of the Preisach-type model in Flux applications and limitations.
A user example using a Preisach material in Flux.
Bibliographical references on Preisach models for further reading.

The Preisach-type hysteresis model in Flux

The Preisach model is a mathematical tool used to represent physical systems characterized by a hysteretic behavior. In such systems, the output quantity is not only a function of the input: its state at a given time also depends on the history of previous inputs and outputs. The non-linear relation B(H) between the magnetic flux density B in a ferromagnetic sample subjected to a time-varying magnetic field intensity H(t) is an example of hysteretic system whose behavior may be represented by a Preisach-type model.

Preisach-type models may be interpreted in terms of elementary entities called hysterons. In the context of ferromagnetic hysteresis, each hysteron may be regarded as a “magnetic relay” γ_α,β(H) that can exhibit only two magnetization states, depending on its input magnetic field intensity H (Figure 1(a)). For an input H > α, the hysteron switches to a magnetic state γ_α,β(H) = +1. On the other hand, for H < β, γ_α,β(H) = -1. Additionally, each hysteron may be associated to a point (α,β) on a Cartesian half-plane given by α≥β (Figure 1(b)).

Figure 1. Representation of a single hysteron (a) and of the Preisach plane (α,β) for a given magnetization state of a ferromagnetic sample (b).

Consequently, the hysteretic behavior B(H) of a given magnetic sample results from the superposition of the effects of a continuous distribution of hysterons expressed by the following double integral:

B(H) = ∫∫_α≥β [ P(α,β) γ_α,β(H) ] dα dβ.

In the previous expression, the function P(α,β) (known as the Preisach density function) is a property of the ferromagnetic sample and must be identified from experimental magnetic measurements.

Flux implements an optimized, vector Preisach-type model of hysteresis that greatly simplifies the creation of ferromagnetic materials in a project. Indeed, only magnetic measurements performed along a typical hysteresis loop are required for the model identification of a given sample, and two model identification tools are available to help the user in this preliminary step.

Furthermore, the Preisach-type model available in Flux relies on an equivalent Everett function to avoid the direct evaluation of the costly double integral above multiple times at each time step, leading to reduced computation times.

For a comprehensive discussion on the vector Preisach-type model implemented in Flux, please check the references available in the section Further reading at the end of this chapter.

How to create a material with a Preisach-type hysteresis model in Flux

The Preisach-type model available in Flux may be assigned to a material either during its creation or during its modification.

More specifically, the user must set the B(H) property of the material to the appropriate subtype representing its hysteretic behavior and provide the parameters of the Preisach model.

The procedure is provided below:

First, create or edit an existing material in a Flux project. Two methods are available to perform these actions:
1. Using the Physics menu, by selecting the option Material and then New or Edit;
2. Interacting with the Flux Data Tree on the left of the main project view, i.e., in its Physics section, by double-clicking on Material to create a new one or by double-clicking on an existing material in the list to edit it.
Depending on the case, Flux will display either the New Material or the Edit Material window.
In any of the two previous windows, enable the Magnetic Property option that is available in the B(H) tab.
Then, choose one of the two following methods to describe the Preisach-type model from the drop-down menu. The two available options are:
- Isotropic hysteretic, Preisach model described by 4 parameters of a typical cycle and
- Isotropic hysteretic, Preisach model identified by N triplets.

In the case of the Isotropic hysteretic, Preisach model described by 4 parameters of a typical cycle approach, the user must then provide the following parameters of a typical hysteresis cycle (a major cycle in most of the situations) of the material:

its saturation magnetization, measured in teslas;
its remanent flux density, measured in teslas;
its coercive field strength, measured in A/m;
the squareness factor of the typical / major cycle, which is related to its overall shape and varies between 0 and 10.

Note: An identification tool is available to help the user obtain the four parameters above from measurement data representing the typical / major hysteresis cycle of a given material. For further information on this identification tool, the user is referred to the following documentation topic: material identification.

On the other hand, in the case of the Isotropic hysteretic, Preisach model identified by N triplets approach, the user must provide instead a table containing N triplets (a_i, b_i, c_i) that are measured respectively in teslas, A/m and A/m.

The triplets (a_i, b_i, c_i) correspond to the parameters of the following function:

B d (H) = μ_{0} H + \sum_{i = 1}^{N} a_{i} atan (\frac{H + c_{i}}{b_{i}})

which represents the descending branch of the major hysteresis loop of the material. Their values may be obtained from measured data with the help of a least-squares fitting procedure.

Note: A second identification tool is also available to help the user perform the data fitting procedure required to obtain parameters (a_i, b_i, c_i) from a set of (B,H) measurements. For additional information on this identification tool, the user is referred to the following documentation topic: material identification.

Availability and limitations

The Preisach-type model for the hysteretic relationship (B, H) of a magnetic material is only available for Transient Magnetic applications in Flux 2D and 3D.

Moreover, a material containing the Preisach-type model may only be assigned to magnetic non-conducting regions and to solid conductor regions. In the case of solid conductor regions, the material will also require a J(E) property, and under such conditions the Preisach model will also account for dynamical hysteretical effects, leading to increased losses at higher frequencies.

The following constraints should also be observed in projects containing regions characterized by a material using the Preisach-type model implemented in Flux:

Transient initialization by file is not yet supported.
Transient initialization by static computation is not yet supported.
Assigning a compressible mechanical set to such regions is not yet supported.

Example of application

Let us consider the well-known TEAM 32 benchmark problem shown schematically in Figure 2 below. The device is composed by a double-fed, three-limbed core made from a ferromagnetic material whose hysteretic behavior we would like to investigate with the help of Flux 3D.

Figure 2. Schematic 2D representation of the double-fed magnetic circuit of the TEAM 32 problem considered in this Flux 3D example.

The voltage sources feeding the circuit impose sinusoidal voltages at a frequency of 10 Hz and with equal amplitudes of 14.5 volts to the excitation coils, but with a 90^o phase shift between each other. This leads to the establishment of a rotating magnetic flux density at certain positions of the device (e.g., at position C1 shown in Figure 2). The numerical computation of this time-varying magnetic flux density is sometimes used as a benchmark problem for Preisach models. For a complete description on this specific example (including geometric dimensions and additional circuit and material data), please refer to the TEAM problem repository maintained by the International Compumag Society.

In Flux 3D, the magnetic core may be represented by a Magnetic non-conducting volume region. To account for its hysteretic behavior in a Transient Magnetic application, a material with a B(H) property of type Isotropic hysteretic, Preisach model identified by N triplets must be assigned to this region. With the help of the appropriate Preisach identification tool provided in Flux, it may be shown that the material used in the TEAM 32 device corresponds to the following triplets (a_i, b_i, c_i) for a data fitting with N = 2:

Table 1. Triplets for the Preisach material of the TEAM 32 problem for a centered, symmetric (B,H) cycle with a saturation magnetic flux density of 1.3 T at 10 Hz.
i	a_i (T)	b_i (A/m)	c_i (A/m)
1	0.5043	11.08	59.37
2	0.4162	130.19	114.39

The two excitation coils may be conveniently represented by non-meshed coil magnetic sources. These are fed by voltage sources connected to their corresponding coil conductor components in a coupled electric circuit. Figure 3 shows the resulting geometry in Flux.

Figure 3. The meshed device of the TEAM 32 problem inside an infinite box of a Flux 3D project.

After solving the project, the rotating character of the magnetic flux density may be verified with the help of sensors positioned at the spot C1 (shown in Figure 2). The hysteretic behavior of fields B and H may also be verified with the help of a sensor placed at the centermost position of the device, that is, the origin of the coordinate system shown in Figure 2. These results are available in Figure 4.

Figure 4. The rotating magnetic flux density during a cycle at position C1 (a) and the hysteresis loop at the centermost position of the magnetic circuit (b). Part (b) also shows the (B,H) measurements used for the identification of the input triplets (a_i,b_i,c_i).

A sensor may also be employed to compute the total magnetic power absorbed by the magnetic core of the device at a given time step. In this case, the user must create a sensor of type Predefined, choose Magnetic Power and perform the computation using the Magnetic non-conducting volume region representing the core as the computation domain. The results are shown in Figure 5 below.

Figure 5. Magnetic power absorbed by the volume region representing the magnetic circuit of the TEAM 32 device evaluated with a Magnetic power sensor in Flux 3D.