# Modeling the Hysteresis Loop of Ultra-High Permeability Amorphous Alloy for Space Applications

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## Abstract

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^{®}MM-5Co. The measurement stand is capable of accurately measuring minor and major hysteresis loops for such a material together with exemplary measurement results. The main source of the measurement error is highlighted, which includes the Earth’s field influence. The results of hysteresis loop modeling with the original Jiles-Atherton model and with two of its modifications are given. In all cases, the parameters of the Jiles-Atherton model were identified in two-step identification on the basis of a differential evolution optimization algorithm. The results indicate that both the original and modified Jiles-Atherton models are suitable for modeling the ultra-soft amorphous alloy. However, the hysteresis model’s parameters vary significantly.

## 1. Introduction

^{®}MM-5Co amorphous alloy was chosen for the core of the device [4]. Reference [5] describes a low-noise (<1 pT) sensor based on similar amorphous material. Trying to apply this amorphous material in three-axis sensors was met with problems of excitation voltage unbalance. This could happen if the pieces of tape for each core are taken from different bobbins (even annealed in the same batch) or even from different parts of the same bobbin. For instance, from the inner, middle, and upper part of the bobbin. The following investigation is a follow-up of this project.

## 2. Materials and Methods

#### 2.1. Ultrahigh Permeability Sample

_{67}Fe

_{3}Cr

_{3}B

_{12}Si

_{15}. It is a novel magnetic material developed as the modified version of the Co-based amorphous alloy MELTA

^{®}MM-5Co [11] with a low saturation flux density B

_{s}, nearly zero magnetostriction λ, a Curie temperature of T

_{C}= 460 K, and remarkably soft magnetic properties. Magnetic cores made of Co

_{67}Fe

_{3}Cr

_{3}B

_{12}Si

_{15}ribbons exhibit very high relative permeability. Its initial relative permeability µ

_{i}exceeds 180,000 and 135,000 for 1 and 10 kHz frequencies, respectively. Moreover, maximal relative permeability µ

_{max}exceeds 150,000 and 250,000 for these frequencies. It also exhibits low core losses of 0.2 and 4.42 W/kg for saturation at 1 and 10 kHz frequencies, respectively.

_{67}Fe

_{3}Cr

_{3}B

_{12}Si

_{15}amorphous alloy can be improved by inducing magnetic anisotropy and by annealing in the magnetic field. After such processing, the hysteresis loop changes its shape and becomes squarer or flat depending on the direction and magnitude of the induced magnetic anisotropy.

_{2}atmosphere. At the cooling stage (approximately at 250 °C), a transverse DC (Direct Current) field of 40 kA/m was applied until the core reached 100 °C. Then it was removed from the oven. The field was generated by a solenoid with a DC current. The material was in 25–30 mm bobbins, but only the small parts of the tape from the bobbins were rewrapped to 32 mm diameter supports (Figure 1).

_{67}Fe

_{3}Cr

_{3}B

_{12}Si

_{15}cores are mainly amorphous. To investigate this, Differential Scanning Calorimetry (DSC 404 F1 Pegasus

^{®}, NETZSCH Group, Selb, Germany) tests were conducted and the results are presented in Figure 2. Each thermogram of differential scanning calorimetry has only a maximum, which is a sign of eutectic crystallization typical for this kind of alloy. A glass transition temperature T

_{g}and the temperature of the onset of crystallization T

_{onset}are shown by the arrows in Figure 2.

_{onset}is high enough to confirm the amorphous structure of the samples used in the following investigation. Moreover, the literature review hints at a slight increase of the crystallization temperature for amorphous ribbons annealed under the influence of a magnetic field [12,13,14,15,16]. The rate of the crystallization, however, was higher.

#### 2.2. Measurement Method

## 3. Results

_{m}= 3 A/m) were taken under increasing external homogeneous magnetic field. Figure 4 presents the influence of this field on maximal induction B

_{m}for fields perpendicular to the sample axis (designated X and Y). It was found when taking measurements that omitting the Earth’s field (~50 μT) cancellation induces errors as high as 10%. Fields parallel to the sample axis (Z axis) had a negligible effect in the typical Earth field range.

#### 3.1. Jiles-Atherton Hysteresis Model and Its Modifications

_{s}is the saturation magnetization of magnetic materials, K

_{an}is the average uniaxial anisotropy energy density, a quantifies the domain wall density, φ

_{1}= (ψ − θ), and φ

_{2}= (ψ + θ) where ψ is the angle between the magnetization direction and the easy axis of magnetized material. According to the Bloch model, an effective magnetizing field is defined by H

_{e}= H + αM where α Bloch’s is inter-domain coupling and M is the total magnetization of the sample. For the isotropic materials where K

_{an}= 0, Equation (1) reduces to the Langevin function below.

_{irr}, irreversible magnetization, c, reversibility of the magnetization process, k, quantifies the average energy required to the break pining site, δ, determines if magnetizing the field increases or decreases while δ

_{M}is necessary for the avoidance of unphysical stages of the Jiles-Atherton model for minor loops where incremental susceptibility becomes negative [19].

**Step 1**: calculate anhysteretic parameters: M

_{s}, a, α, and K

_{an}.

**Step 2**: introduce hysteresis and find hysteresis parameters k and c.

#### 3.2. Modeling Results

^{2}, which exceeds 0.992. This indicates that the P. Cheng et al. version of the Jiles-Atherton model is the most suitable for modeling the magnetic characteristics of the investigated ultra-high permeability alloy for space applications. Modeled hysteresis loops compared with the measurement results are presented in Figure 7. Modeling accuracy is better than the extended uncertainty of the measurement. Therefore, it is treated as negligible.

## 4. Conclusions

^{®}MM-5Co, were given. The obtained measurement data were used for the Jiles-Atherton model parameter identification in its basic and modified forms. The presented results confirm that the Jiles-Atherton model is suitable for modeling the magnetic characteristics of Co

_{67}Fe

_{3}Cr

_{3}B

_{12}Si

_{15}ultra-high permeability magnetic material. Moreover, for modeling the Co

_{67}Fe

_{3}Cr

_{3}B

_{12}Si

_{15}amorphous alloy for space applications, the most adequate results may be achieved by utilizing the J-A model modification proposed by P. Cheng et al. High accuracy of the model given by Equation (7) is confirmed by the determination coefficient R

^{2}, which exceeds 0.992. The reason may be due to the fact that the P. Cheng et al. version is believed to better reflect the physical processes behind ferromagnetic hysteresis.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Typical sample used in the investigation, a ring-shaped core with a small cross-section, with 400 sensing windings, ensuring high measurement signal. Magnetization was provided with a straight magnetizing rod, which ensured a homogenous magnetizing field in the sample.

**Figure 2.**Differential scanning calorimetry investigation of the Co

_{67}Fe

_{3}Cr

_{3}B

_{12}Si

_{15}amorphous alloy.

**Figure 3.**Schematic diagram of the developed Ferrograph system [17].

**Figure 4.**Dependence of measured maximal induction B

_{m}value on the external homogeneous field. Values normalized with a B

_{max}for an external field equal 0 ± 0.01 μT in all three axes. Each of the axes were investigated separately. The obtained values for fields in the plane of the ring sample (X and Y) are similar while the influence of the transverse field (Z axis, along the axis of the ring core) is negligible.

**Figure 5.**The measurement results for the ring-shaped core made of the amorphous Co

_{67}Fe

_{3}Cr

_{3}B

_{12}Si

_{15}alloy was annealed for 1 h at 440 °C in the presence of magnetic field H, which is equal to 40 kA/m applied in the direction of the amorphous alloy ribbon. The maximal points of the minor hysteresis loops follow the normal magnetization curve (blue line—hysteresis loop for magnetizing field H

_{m}= 0.5 A/m; green line—H

_{m}= 1.5 A/m; red line—H

_{m}= 3 A/m).

**Figure 6.**An anhysteretic magnetization curve (black line) is determined in Step 1 on the base of major hysteresis loop measurement results (red line).

**Figure 7.**Modeling results (black lines) according to Equation (7). The measurement results are presented as red lines.

Parameter | Unit | Jiles-Atherton | Venkataraman et al. | Cheng et al. |
---|---|---|---|---|

M_{s} | A/m | 219,600 | 216,600 | 237,200 |

a | A/m | 0.269 | 0.268 | 1.669 |

α | - | 2.824 × 10^{−8} | 1.151 × 10^{−8} | 6.268 × 10^{−6} |

K_{an} | J/m^{3} | 2.383 | 32.08 | 1230 |

ψ | rad | 0 | 0 | 0 |

k | A/m | 0.318 | 0.322 | 0.263 |

c | - | 0.0001 | 0.000 | 0.459 |

R^{2} | - | 0.984 | 0.984 | 0.992 |

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**MDPI and ACS Style**

Nowicki, M.; Szewczyk, R.; Charubin, T.; Marusenkov, A.; Nosenko, A.; Kyrylchuk, V.
Modeling the Hysteresis Loop of Ultra-High Permeability Amorphous Alloy for Space Applications. *Materials* **2018**, *11*, 2079.
https://doi.org/10.3390/ma11112079

**AMA Style**

Nowicki M, Szewczyk R, Charubin T, Marusenkov A, Nosenko A, Kyrylchuk V.
Modeling the Hysteresis Loop of Ultra-High Permeability Amorphous Alloy for Space Applications. *Materials*. 2018; 11(11):2079.
https://doi.org/10.3390/ma11112079

**Chicago/Turabian Style**

Nowicki, Michał, Roman Szewczyk, Tomasz Charubin, Andriy Marusenkov, Anton Nosenko, and Vasyl Kyrylchuk.
2018. "Modeling the Hysteresis Loop of Ultra-High Permeability Amorphous Alloy for Space Applications" *Materials* 11, no. 11: 2079.
https://doi.org/10.3390/ma11112079