# Model of a 3D Magnetic Permeability Tensor Considering Rotation and Saturation States in Materials with Axial Anisotropy

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## Abstract

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## 1. Introduction

**μ**, which is one of the most important properties of magnetic materials. For simple isotropic systems, relative magnetic permeability

**μ**can be approximated by a scalar value. However, such a description is insufficient in the case of the most common technical applications [1], such as electrical steels [2] or amorphous alloys [3]. Since commonly used modern soft magnetic materials [4] are characterised by strongly anisotropic properties [5,6,7], relative magnetic permeability

**μ**has to be described as a 3 × 3 tensor. Another problematic aspect of magnetic modelling is that such materials often work near the saturation state. This case occurs in transductors [8], pressductors [9], and other sensors using ferromagnetic cores [10]. It should be highlighted that when a magnetization process ends in a full saturation state, the directions of the magnetizing field

**H**and the flux density

**B**induced in the material are parallel. However, in the case of anisotropic materials, for smaller values of a magnetizing field

**H**, the angle between the magnetizing field

**H**and the flux density

**B**should be considered in modelling [11].

**μ**, such as commercial COMSOL [12] and ANSYS [13] or open-source ELMER FEM [14]. Apart from the tensor description, one must consider the magnetization process’s nonlinear characteristics related to saturation flux density B

_{s}.

**μ**, which considers uniaxial anisotropy for soft magnetic materials and allows for a saturation state. The model is based on the physical principles describing a magnetizing process [29], in which the saturation state is connected with the rotation of magnetic flux density vector

**B**towards the direction of magnetizing field

**H**. The proposed model can be used for efficient and accurate modelling of three-dimensional inductive components with axial anisotropy. Moreover, since only axial anisotropy occurs in isotropic materials subjected to mechanical stresses [30], the proposed model of a magnetization process can be used for complex magneto-mechanical modelling cases where isotropic magnetic materials are subjected to axial and shear mechanical stresses described by a stress tensor

**σ**.

## 2. Axial Anisotropy of Relative Magnetic Permeability Tensor μ in 3D

_{y}and µ

_{z}determine the permeability with respect to the hard axes Y and Z of magnetization, whereas the permeability with respect to the easy axis X equals µ

_{x}.

**μ**is now given by

**R**can be used [31] as

**R**can be given in various forms. In the most general case, the rotation matrix

**R**can be given in the Euler form as an array of cosine values between the principal XYZ and the rotated X’Y’Z’ coordinate axes [32]:

**R**.

## 3. The Proposed Linear–Rotation–Saturation (LRS) Model of Relative Magnetic Permeability for Materials with Axial Anisotropy

- The linear phase.
- The magnetization rotation phase, when the length of flux density vector
**B**is equal to saturation flux density B_{s}; however, the magnetostatic energy is consumed for the rotation of flux density**B**vector towards the magnetizing field**H**. - The magnetic saturation phase, in which when the length of flux density vector
**B**is equal to saturation flux density B_{s}, it is parallel to the magnetizing field**H**, and saturation is modelled by shrinking the magnetic permeability tensor.

**B**is given as

_{0}is the magnetic constant. In the proposed model, the length of the magnetic flux density vector

**B**grows with the magnetizing field

**H**growth until reaching the value of saturation flux density B

_{s}. However, when the length of flux density

**B**reaches the magnetic saturation value B

_{s}, the material is not yet fully saturated. The second phase of the saturation process is related to the rotation of magnetic flux density

**B**towards the direction of the magnetizing field

**H**.

_{y}and µ

_{z}with respect to the hard axes Y and Z of magnetization are the same [33]. In such a case, simplified B(H) magnetization curves (linear with saturation) are presented in Figure 2.

_{x}and E

_{y}(required for the magnetic saturation in X and Y directions, respectively) can be calculated based on geometric dependencies:

_{s}is the saturation flux density (determined by the chemical composition of the material and being equal for X and Y directions), whereas H

_{sx}and H

_{sy}are the values of the magnetizing field required for the magnetic saturation in X and Y directions, respectively. Finally, the average magnetic anisotropy energy K

_{an}(measured per cubic meter of the material) required for the rotation of magnetization in saturation from the easy axis to the hard axis of the material [34] is given as

**R**is the rotation matrix, and

**H**is given as

_{s}**B**, the value of required energy and the rotation angle need to be calculated. The energy can be described as the difference between the total magnetic energy connected with the magnetizing field

**H**and the magnetic field energy required to reach saturation B

_{s}:

_{an}(ϕ) required for performing this process. For a uniaxial material, their relationship can be described by a formula [11]:

_{an}is the average anisotropy energy density defined in Equation (7) (alternatively the anisotropy coefficient), and ϕ is the angle of magnetization concerning the axis of magnetization. This formula can be simplified to the linear form:

_{rot}of

**B**, one condition needs to be checked. If the energy E

_{rot}passed for rotation is lower than the anisotropic energy E

_{an}(ϕ), so if E

_{rot}< E

_{an}(ϕ), the rotation angle φ

_{rot}equals

_{rot}≥ E

_{an}(ϕ). In full saturation, the rotation angle φ

_{rot}equals the angle ϕ between the easy magnetization axis and the direction of the magnetizing field

**H**.

**B**is parallel to the direction of the magnetizing field

**H**. The length of the magnetic flux density vector

**B**equals the saturation value B

_{s}and cannot increase despite the increase in the magnetizing field

**H**. The visualisation of rotation angles in the saturation process is presented in Figure 3.

**B**can be calculated with the use of a three-dimensional rotation matrix

**R**:

**R**can be obtained in many ways, depending on the complexity of a rotation process. In our case, the rotation by the rotation angle φ

_{rot}causes the movement of the magnetic flux density vector

**B**towards the magnetic field strength vector

**H**. As both vectors emanate from the origin of the XYZ system, both share a common plane, which can be defined by a surface normal originating from the same origin. To simplify the case, the common plane for both vectors can be defined by a unit normal

**u**(obtained by normalising surface normal), which equals the vector product of the normalised magnetic flux density vector

_{n}**u**and the normalised magnetizing field

_{B}**u**, so that

_{H}**B**and

**H**vectors lie in the X’Y’ plane and the unit normal

**u**is the rotation axis Z’. In such a way, the rotation can be performed for vectors located anywhere in the body. Thus, the rotation corresponding to the rotation angle φ

_{n}_{rot}about the rotation axis

**u**= [u

_{n}_{nx}, u

_{ny}, u

_{nz}] can be performed by using the rotation matrix defined as follows:

**R**can be used to perform the rotation in Formula (14). The graphical summary of the proposed process needed in the linear–rotation–saturation (LRS) model is presented in Figure 4. The whole magnetization process conducted for a specific range of values for magnetic field strength

**H**and permeability tensor components μ

_{x}, μ

_{y}and μ

_{z}is also presented in the Supplementary Materials as a video file.

**H**and

**B**are not parallel to each other, rotation is performed. In this stage (Figure 4b), the length of

**H**is reduced to the value H

_{s}, as presented in Formula (9). The magnetic permeability tensor must be modified using the rotation matrix in Formula (17). In this phase, the permeability tensor is described as

**H**and

**B**are aligned with the hard magnetization direction, the magnetic permeability tensor is given as follows:

## 4. Modelling of Magnetization Curves

**H**. The presented magnetic flux density vector

**B**is a projection on the direction of the magnetizing field

**H**. Such a case occurs in a model where the magnetizing coil is coaxial with the sensing coil placed around the magnetic core. An example of such a model is a fluxgate sensor in Foerster configuration [37]. The magnetization curves for different angles between the magnetizing field

**H**and the easy axis of magnetization are presented in Figure 5.

**H**is parallel to either the minor or the major axis of the relative magnetic permeability tensor, rotation does not appear. The area between the curves for easy and hard magnetization axes represents the rotation state area, where the rotation state is represented by the plot curvature between the not-saturated and the saturated state.

- Induced during the thermomagnetic annealing (Experiment in Ref. [38]).

## 5. Practical Implementation

**B**(

**H**) dependence should be considered, depending on the working point on the magnetization curve. On the other hand, the

**B**(

**H**) relation is always monotonous due to physical reasons. As a result, the successive approximation method can be applied to the assessment of

**ν**(

**B**) dependence.

**B**(

**H**) curve to have a discontinuous derivative at the transition point. Since FEM solutions are based on the methods of solving differential equations, solving the problem of discontinuous derivatives is a mathematical problem. It is solved in different ways, with both classical and numerical approaches. In our numerical case, several solutions have been provided regarding this issue [42,43]. The discontinuity problem in FEM modelling is addressed primarily in the cases of mechanical cracks [44]. The analysis should be provided individually for each modelling case in terms of convergence for discontinuities. One example is the Discontinuous Galerkin formulation [45]. FEM modelling tools enable using different techniques for solving differential equations, which should be adjusted to a specific modelling case.

## 6. Conclusions

**R**-based description can be easily implemented into finite element software, such as open-source ELMER FEM or other commercially available alternatives. In the case of such implementation, the relative permeability tensor can be modelled, which creates new possibilities for modelling the 3D magnetostatic and magnetodynamic systems.

## Supplementary Materials

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The visualisation of relative magnetic permeability tensor

**μ**for axial anisotropy in three dimensions.

**Figure 2.**Simplified B(H) magnetization curves of axially anisotropic magnetic material, magnetized in the direction of the easy axis (red) and hard axis (blue) of magnetization.

**Figure 3.**The angles between magnetizing field

**H**, easy-magnetization axis X’, and flux density

**B**in the rotation plane X’Y’ (magnetizing field H – red arrow, flux density B before rotation – blue arrow, flux density B after rotation – magenta arrow).

**Figure 4.**Three phases of the magnetization process for material with uniaxial anisotropy: (

**a**) linear magnetization region, (

**b**) material saturated—flux density vector

**B**not parallel to magnetizing field

**H**, (

**c**) fully saturated material. (magnetizing field H – red arrow, flux density B before rotation – blue arrow, flux density B after rotation – magenta arrow).

**Figure 5.**The magnetizing curves for different angles ϕ between magnetizing direction and easy-magnetization axis.

**Figure 6.**The magnetizing curves for different angles ϕ between magnetizing direction and easy-magnetization axis.

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**MDPI and ACS Style**

Kopala, D.; Ostaszewska-Liżewska, A.; Råback, P.; Szewczyk, R.
Model of a 3D Magnetic Permeability Tensor Considering Rotation and Saturation States in Materials with Axial Anisotropy. *Materials* **2023**, *16*, 3477.
https://doi.org/10.3390/ma16093477

**AMA Style**

Kopala D, Ostaszewska-Liżewska A, Råback P, Szewczyk R.
Model of a 3D Magnetic Permeability Tensor Considering Rotation and Saturation States in Materials with Axial Anisotropy. *Materials*. 2023; 16(9):3477.
https://doi.org/10.3390/ma16093477

**Chicago/Turabian Style**

Kopala, Dominika, Anna Ostaszewska-Liżewska, Peter Råback, and Roman Szewczyk.
2023. "Model of a 3D Magnetic Permeability Tensor Considering Rotation and Saturation States in Materials with Axial Anisotropy" *Materials* 16, no. 9: 3477.
https://doi.org/10.3390/ma16093477