Anhysteretic Magnetization Research Articles

An enhanced Energy-Based hysteresis model for electrical steel sheets considering mechanical stress

Mechanical stress significantly influences the hysteresis characteristics of silicon steel, which are crucial for the design of electrical equipment cores. This study develops an improved Energy-based hysteresis model that integrates the effects of mechanical stress by modifying the anhysteretic magnetization model to include total average anisotropy energy density and introducing a linear relationship between coercive force and mechanical stress. The model’s performance is demonstrated using 30QG120 oriented silicon steel sheets, where the hysteresis loops under varying stress conditions are successfully simulated and validated against experimental data. The results confirm the model’s accuracy and practical utility in predicting material behavior under mechanical stress, offering a valuable tool for optimizing electrical steel sheet design.

The SF and remanence evaluation of magnetic shields based on SST for low-frequency and degaussing situation

As the dominant shielding functional material, the permalloy sheet primarily determines the static and low-frequency shielding performances of magnetically shielded room (MSR). However, the lack of measurement of shielding sheets for practical use would lead to a non-negligible evaluation error in MSR performance. Therefore, an estimation technique of shielding factor (SF) and the remanence of MSR is proposed in this paper while considering the nonlinear magnetic characteristics of the permalloy sheet tested by a single sheet tester for low-frequency field and degaussing situation (L-D-SST). First, a high-accuracy measurement system, comprising L-D-SST (for exciting magnetic field and sensing the corresponding B and H) and the control system (for applying excitation to L-D-SST and amplifying, filtering, and collecting B and H signals), is established. Weak magnetic fields at low frequencies and decaying alternating demagnetizing field excitations are separately applied to the L-D-SST for basic magnetization curves (BM curves) and an anhysteretic magnetization curve (AM curve) tests. Furthermore, the BM and AM curves are respectively integrated with the FEM algorithm for accurate and reliable estimations of the SF and remanence of MSR in an operational state. In addition, the tested magnetic properties are applied for the optimization of MSR shielding quality.

Linearising anhysteretic magnetisation curves: A novel algorithm for finding simulation parameters and magnetic moments

This paper proposes a new method for determining the simulation parameters of the Jiles–Atherton Model used to simulate the first magnetisation curve and hysteresis loop in ferromagnetic materials. The Jiles–Atherton Model is an important tool in engineering applications due to its relatively simple differential formulation. However, determining the simulation parameters for the anhysteretic curve is challenging. Several methods have been proposed, primarily based on mathematical aspects of the anhysteretic and first magnetisation curves and hysteresis loops. This paper focuses on finding the magnetic moments of the material, which are used to define the simulation parameters for its anhysteretic curve. The proposed method involves using the susceptibility of the material and a linear approximation of a paramagnet to find the magnetic moments. The simulation parameters can then be found based on the magnetic moments. The method is validated theoretically and experimentally and offers a more physical approach to finding simulation parameters for the anhysteretic curve and a simplified way of determining the magnetic moments of the material.

Numerical analysis of the frequency-dependent Jiles-Atherton hysteresis model using the example of Terfenol-D material

<p>The Jiles-Atherton model has been widely used in describing the hysteretic property of a magnetic material or device. However, the calculation errors are not so easily discovered. With a complex expression, the frequency-dependent Jiles-Atherton model should be solved numerically with appropriate settings. This paper proposes an effective solving method for this model and describes some necessary analysis built on the numerical results. In the numerical method proposed in this manuscript, the anhysteretic magnetization was calculated by the secant method, and the trapezoidal rule was utilized to form the implicit function, which can be calculated by the fixed-point iteration. Compared to the other common methods, the proposed one has a friendly expression and fast computation speed. The Terfenol-D material was taken as an example for the numerical analysis. The feasible region was determined and the commonly used approximation that neglects the term of the magnetic field when calculating the magnetic induction intensity was tested. At last, the required number of sampling points per period was reached to guarantee high precision from analyzing its influence on the computation precision. The proposed numerical method is helpful for high-precision solutions of the frequency-dependent Jiles-Atherton model. The results from the numerical analysis can also help users avoid some incorrect calculations when employing this hysteresis model.</p>

An improved stress-dependent magnetostriction model of silicon steel based on simplified multi-scale and Jiles–Atherton theory

Due to the magnetic domain movement and energy change under magnetic field and stress applied, the macroscopic magnetic properties and magnetostriction properties of non-oriented (NO) silicon steel are influenced. To present this effect of the magnetic domains’ motion mechanism, a stress-dependent magnetostriction must be modeled. This paper proposes an improved stress-dependent magnetostriction model (I-SDM model) for NO silicon steel based on a simplified multi-scale model. Firstly, the stress-dependent expression of the anhysteretic magnetization, obtained by assuming a six-domain structure and considering the volume fraction of magnetic domain energy, is introduced into the inverse Jiles–Atherton (JA) model. In this way, the stress-dependent hysteresis model is constructed. Then, coupling with the improved quadratic domain rotation model that considers pinning effects, the I-SDM model for NO silicon steel is built. Finally, with the parameters obtained from experimental results, the proposed model under varying stresses is simulated and compared with other models. It shows that the proposed model has the highest simulated accuracy for both λ-H loop and λ-B loop with stress applied.

Determining the Anhysteretic Magnetization Curve for a Ferromagnetic Material According to the Limiting Loop Parameters of Its Magnetic Hysteresis

Determining the Anhysteretic Magnetization Curve for a Ferromagnetic Material According to the Limiting Loop Parameters of Its Magnetic Hysteresis

Demagnetization Parameters Evaluation of Magnetic Shields Based on Anhysteretic Magnetization Curve.

To achieve the nearly zero-field environment, demagnetization is an indispensable step for magnetic shields composed of high-permeability material, which adjusts the magnetization of the material to establish magnetic equilibrium with the environmental field and improve the shielding performance. The ideal demagnetization can make the high-permeability material on the anhysteretic magnetization curve to have a higher permeability than on the initial magnetization curve. However, inappropriate parameters of degaussing field cause the magnetization state to deviate from the anhysteretic magnetization curve. Therefore, this article proposes a new assessment criterion to analyze and evaluate the parameters of degaussing field based on the difference between the final magnetization state after demagnetization and theoretical anhysteretic state of the shielding material. By this way, the magnetization states after demagnetizations with different initial amplitude, frequency, period number and envelope attenuation function are calculated based on the dynamic Jiles-Atherton (J-A) model, and their magnetization curves under these demagnetization conditions are also measured and compared, respectively. The lower frequency, appropriate amplitude, sufficient period number and logarithmic envelope attenuation function can make the magnetization state after demagnetization closer to the ideal value, which is also consistent with the static magnetic-shielding performance of a booth-type magnetically shielded room (MSR) under different demagnetization condition.

Magneto-mechanical coupling effect of ferromagnetic materials: Test characteristics and theoretical model

The magneto-mechanical coupling effect of ferromagnetic materials is the result of macroscopic influence of mechanical properties on magnetic properties under the combined action of stress field and applied magnetic field, which essentially can be attributed to the change of material properties caused by the transformation of microscopic magnetic domain structures. The effect of applied magnetic field is mainly manifested as the change of magnetic moment cluster (direction and size) inside microscopic magnetic domain; The stress field mainly causes the movement and rotation of the domain walls inside the ferromagnetic material, and further causes the change of magnetic moment cluster. Taking Q235 low carbon steel as an example, the effects of tensile stress σt on some important magnetic parameters of ferromagnetic materials under different excitation intensities (i.e., H0), such as the maximum value of applied magnetic field Hmax, intrinsic coercive field Hcj, hysteresis loss power per unit volume ph and anhysteretic magnetization Man, were obtained by setting macroscopic magneto-mechanical coupling test and fully considering the influence of demagnetizing field Hd (The standard ring specimen was used for demagnetization correction of the non-standard strip specimen). Meanwhile, the basic magnetization characteristics of ferromagnetic materials were discussed by introducing the basic magnetization curve F(H0, Ban) and considering the effect of σt as the equivalent stress field Hσ based on the theory of microscopic magnetic domain and the principle of thermodynamic. In the range of 635.4 A/m ≤ H0 ≤ 928.6 A/m and 0 MPa ≤ σt ≤ 350 MPa, the theoretical calculation models Man(H0, σt) and Hσ(H0, σt) in the magneto-mechanical coupling effect were obtained, which are in good agreement with the experimental results.

Experimental studies on the performance of magnetic shields under different magnetization conditions

In recent decades, magnetic shields have provided basic experimental environments for the measurements of extremely weak magnetic fields represented by the biological magnetic signal. Excellent shielding performances, including the low residual field and high shielding factor (SF), are necessary to ensure the quality of these weak magnetic signals and avoid the interference of external magnetic fields. The magnetic shielding performance of the same device can be affected by different degaussing and test conditions, which remains to be systematically studied. In this paper, experiments with variable magnetization conditions, including different degaussing orders, test fields and environmental fields, are established in a nearly zero-field space to simulate the different situations during measurement. The residual field and SF of the cubic shielding device are tested in these cases. Meanwhile, these shielding performances are analyzed from the perspective of the magnetization state and calculated based on the magnetic properties which are tested and fitted by the Jiles–Atherton model. The results show the influence of these different conditions on the shielding performances of the cubic device, consistent with the numerical calculation. Under the same environmental field, the different degaussing order and test field lead to completely different residual field and shielding performance, respectively. The influence of the Earth’s magnetic field on the SF can be ignored due to its tiny equivalent bias field determined by the anhysteretic magnetization curve.

Magnetostriction property modeling of silicon steel considering stress-induced and magnetocrystalline anisotropy

This paper proposed an improved magnetostriction model for correlation of anisotropy in non-oriented (NO) silicon steel based on the free energy, which considers stress-induced and magnetocrystalline anisotropy. Firstly, the free energy model, which includes stress-induced anisotropy energy, the energy of magnetic field, and the anisotropic energy of magnetic crystals, is incorporated into the anhysteretic magnetization parameter Man. Then, to obtain the magnetic field and proposed model parameters related to stress-induced and magnetocrystalline anisotropy, the magnetostrictive strain loops at different magnetization directions of NO silicon steel are measured. Finally, based on the parameters obtained from experimental data of the proposed model, magnetostrictive strain loops under varying magnetization directions are simulated. This improved magnetostriction model can be applied to the calculation of the vector magnetostriction of the motor core.

Numerical Solving Method for Jiles-Atherton Model and Influence Analysis of the Initial Magnetic Field on Hysteresis

The Jiles-Atherton model was widely used in the description of the system with hysteresis, and the solution for the model was important for real-time and high-precision control. The secant method was used for solving anhysteretic magnetization and its initial values were optimized for faster convergence. Then, the Fourth Order Runge-Kutta method was employed to solve magnetization and the required computation cycles were supplied for stable results. Based on the solving method, the effect of the nonzero initial magnetic field on the magnetization was discussed, including the commonly used linear model of the square of magnetization under the medium initial value. From computations, the proposed secant iteration method, with supplied optimal initial values, greatly reduced the iterative steps compared to the fixed-point iteration. Combined with the Fourth Order Runge-Kutta method under more than three cycles of calculations, stable hysteresis results with controllable precisions were acquired. Adjusting the initial magnetic field changed the result of the magnetization, which was helpless to promote the amplitude or improve the symmetry of magnetization. Furthermore, the linear model of the square of magnetization was unacceptable for huge computational errors. The proposed numerical solving method can supply fast and high-precision solutions for the Jiles-Atherton model and provide a basis for the application scope of typical linear assumption.

An Easily Used Phenomenological Magnetization Model and Its Empirical Expressions Based on Jiles-Atherton Parameters.

In this paper, a simple magnetization model convenient for engineering applications is presented based on the expressions of the first-order LTI system model. Considering the trade-off between the nonlinearity of anhysteretic magnetization and the hysteresis width, the proposed model employs two different equations with different magnetic field amplitudes. Furthermore, the proposed model utilizes the first-order LTI system model with a low magnetic field amplitude and a simple nonlinear function, based on the amplitude-frequency function, with a high magnetic field amplitude. Two important characteristic parameters for engineering applications, namely, amplitude and the equivalent phase lag, were exacted and analyzed to validate the computation precision of the proposed model. Then, the model was verified through comparisons to the validated Jiles-Atherton model. For easy use, similar to a physics-based model instead of a fitting method, empirical expressions for the model parameters were given, and applicable ranges of these equations were determined using the parameters of the Jiles-Atherton model. Finally, an example of the magnetization model applied to an on/off type device was computed to further verify the effectiveness of the proposed model with quite a simple expression.

Cyclostratigraphic calibration of the ca. 1.56 Ga carbon isotope excursion and oxygenation event recorded in the Gaoyuzhuang Formation, north China

Recent studies of the upper Gaoyuzhuang Formation (Fm) (ca. 1600–1540 Ma) in north China have suggested that Earth underwent an oxygenation event associated with a negative δ 13 C carb excursion and this event may have led to the expansion of eukaryotic life. However, the duration of this negative δ 13 C carb excursion, which is crucial to understand its origin, remains uncertain. Here, we report a floating astronomical timescale based on Milankovitch cycles extracted from the carbonate-dominated strata of the Member 3 of the Gaoyuzhuang Fm in north China, in which the oxygenation event is recorded. High-resolution magnetic susceptibility, anhysteretic remnant magnetization and saturation isothermal remnant magnetization series were obtained from the stratigraphic sections in the Jixian and Yanqing regions. Rock magnetic experiments suggest that the magnetic parameters are dominated by the detrital materials whose flux was likely controlled by orbitally forced climatic changes. Our results for the first time reveal well-constrained eccentricity, obliquity, and precession cycles from the Gaoyuzhuang Fm and demonstrate that the ca. 1560 Ma negative δ 13 C carb excursion and associated transient oxygenation event indicated by I/(Ca + Mg) had durations of ~1.7 Myr and ~6.2 Myr, respectively. The ca. 1560 Ma δ 13 C carb excursion is comparable with the short-term Neoproterozoic excursions in respect of duration, rather than with the long-term ones such as the ca. 574–567 Ma Shuram excursion. The similarity may indicate a common mechanism in origin of the short-term excursions in the Mesoproterozoic and Neoproterozoic time, which deserve testing by more investigations. • Cyclostratigraphic analyses revealed Milankovitch cycles from the ca. 1.56 Ga Gaoyuzhuang Formation • Durations of the δ 13 C carb excursion and the oxygenation event in the ca. 1.56 Ga are 1.7 Myr and 6.2 Myr, respectively • The duration of this δ 13 C carb excursion are comparable to the short-term ones in the Neoproterozoic

On the anhysteretic magnetization of soft magnetic materials

Electrical steels are still the materials of choice for large-scale transformers and most electric motors. Yet, they may present a nonhomogeneous magnetic nature which prevents describing accurately their anhysteretic magnetization with the Langevin-Weiss model. Although interpolation and extrapolation methods may be used to model any anhysteretic curve, a simple and physically-based model would be of great value for fundamental and applied research. Inspired in the law of partial volumes for gas mixtures, we proposed a law of partial magnetizations for magnetic mixtures. In a two-component system, the model leads to the double Langevin-Weiss function. We also introduced a graphical method and a fitting approach to analyze and model anhysteretic magnetization curves. A semi-log magnetization derivative plot is central to this end. We validated our strategy through well-motivated examples using published data on soft magnets. The single Langevin-Weiss function provided an accurate description of the magnetization of isotropic and anisotropic magnetically homogeneous materials: a soft ferrite and a nanocrystalline alloy, respectively. For modelling a magnetization transverse to the material’s preferred direction, the key is to allow a negative molecular field constant. The double Langevin-Weiss function was suitable for less homogeneous materials, such as a grain-oriented electrical steel magnetized along the rolling direction and a non-oriented electrical steel. Moreover, a highly-grain-oriented electrical steel magnetized transverse to the rolling direction, which exhibits a constricted hysteresis loop, could be modeled in excellent agreement with data. The key for the latter, has been to allow an antiparallel arrangement of the mean magnetization of both components.

A nonlinear magneto-elastoplastic coupling model based on Jiles–Atherton theory of ferromagnetic materials

As seen in the Jiles–Atherton (J–A) model and its modified form, the linear relationship between the magnetization coefficient and the stress deviates significantly from the experimental results. It is necessary to introduce many parameters that are difficult to obtain or unknown to describe the effect of elastoplastic deformation on magnetization or hysteresis, such as shape coefficient, pinning coefficient, and molecular field coefficient. In this paper, a new nonlinear magneto-elastoplastic model for ferromagnetic materials is established based on the magneto-mechanical coupling effect, and both the sixth-order term of magnetization and the nonlinear equation of the magnetization coefficient are introduced into the magnetostriction equation. In the models established in this paper, the elastoplastic deformation equivalent magnetic field is introduced into the effective magnetic field, and the Frohlich–Kennelly equation is used to describe the anhysteretic magnetization. After comparing the prediction results of different models with the available experimental results, it is observed that the proposed model in this paper exhibits superior prediction ability for magnetostrictive strain, magnetization, and hysteresis phenomena under different stresses. This paper has also analyzed the mechanism of the effect of elasto-plastic loading and residual stress on the hysteresis in different models as well as the differences between them. The determination coefficient of the proposed model in this paper is closer to 1, which is better than the existing models, indicating that it has a better fitting effect and is of great significance to the development of quantitative non-destructive testing technology.

Anhysteretic Magnetization Effect on the Centered and Non-Centered Minor Hysteresis Loops in Jiles-Atherton Model

In this paper, an accurate evaluation of minor hysteresis loops using the modified Jiles-Atherton model is presented. This model is based on the anhysteretic magnetization, which is given in most cases by the Langevin equation. The anhysteretic magnetization is characterized by three parameters, the mean field parameter α, the shape parameter of anhysteretic magnetization curve a and the saturation magnetization Ms. The parameters influencing the minor hysteresis loops are a and α. These parameters are expressed usually in the form of simple power laws and they connect the minor loop parameters to the major ones. These expressions are applicable in both centered and non-centered minor loops cases. In the centered minor loops, the parameter k is introduced in order to adjust the width of the minor loops regarding the level of the magnetic excitation. The coefficients of the proposed expressions (γ, β and σ) are obtained by optimization procedure. The proposed approach is validated using measured minor loops in both cases. A close agreement is obtained between modeled and measured ones.

Deciphering syn- and post-emplacement processes in shallow mafic dykes using magnetic anisotropy

Dykes form key pathways for the transport and emplacement of magma within the crust. We have identified syn- and post-emplacement processes recorded across a ~2 m thick basaltic dyke on the Isle of Skye, Scotland. We measured the rock magnetic properties, anisotropy of magnetic susceptibility (AMS) and anisotropy of anhysteretic magnetisation (AARM) across two dyke-thickness profiles spaced 13 m apart along the dyke strike (sites G5 and G6). At 20–25 cm intervals, our samples are very closely spaced compared to standard sampling protocols. Our results show that the dyke's magnetic fabrics originate from two mineral groups: titanomagnetite, which is abundant in the central dyke region at site G5, and iron sulphides (pyrite and pyrrhotite) that dominates the margin regions at both sites. The titanomagnetite occurs in unaltered dyke rock; its magnetic fabrics are primary, having formed during magma solidification, and record lateral magma flow. The pyrrhotite occurs in jointed and hydrothermally altered dyke rock; its petrological and magnetic fabrics are secondary, having originated from a sulphide-rich fluid which infiltrated cooling joints oriented perpendicular to the dyke margins and locally modified the primary magnetic fabrics. At site G6, pyrrhotite also occurs in the dyke centre, suggesting that locally the post-emplacement sulphide-rich fluid permeated into this region; this site is located close to a branch in the dyke and has increased joint frequency, which could explain the enhanced alteration. The presence of syn- (primary) and post-emplacement (secondary) fabrics was only identified due to our high frequency sampling regime and use of both AMS and AARM techniques. We highlight that future magnetic anisotropy studies of dykes may benefit from high sample frequency combined with sampling along-strike and across-thickness. Using both AMS and AARM techniques to detect more variations in magnetic fabrics can reveal more complete syn- and post-emplacement dyke histories.

Improvement on parameter identification of modified Jiles-Atherton model for iron loss calculation

The physical behaviour of a magnetic material can be characterized by Jiles-Atherton (J-A) model where some model parameters are generally identified by optimization techniques. For identification of model parameters using optimization techniques, an error criterion based on the error between the measured and calculated magnetic flux density (B) or magnetic field strength (H) is commonly considered where the relative error in the calculation of iron loss is ignored. Consequently, the calculated iron loss from B-H loop sometimes highly differs from its experimental value. In this paper, the error criteria for J-A model’s parameter identification are designed as the combination of the relative iron loss error criterion and the general existing error criterion. Furthermore, a modified J-A model is proposed to improve the agreement between experimental and calculated results especially at the low magnetic induction levels by introducing a scaling factor in the anhysteretic magnetization. The proposed modified J-A model and the effectiveness of the error criteria for its parameter identification are tested by comparing calculated results with the experimental results as well as recently works in the literature.

On the modelling of the anhysteretic magnetization of homogeneous soft magnetic materials

The analysis and optimization of devices with soft magnetic cores, such as inductive sensors and electric machines, rely on a simple and precise modelling of their nonlinear anhysteretic magnetization. The Langevin-Weiss equation of state has the advantages of being physically-based and capable of accurately describing isotropic materials. In fact, it has even been adopted as the backbone of the popular Jiles-Atherton model to describe magnetic hysteresis. However, many studies show the difficulty of retrieving the modelling parameters given an experimental anhysteretic curve, for which stochastic optimization methods are generally used.In this work, we present a simple and deterministic method for finding the parameters of the Langevin-Weiss model: the saturation magnetization, the temperature dependent a parameter, and the molecular field constants. To this end, we optimize a problem with a nonlinear solver where the fitting variables are instrinsically bounded. We validated the approach by analyzing three widely used soft magnets; a grain-oriented silicon steel, a Mn-Zn soft ferrite, and a Fe-based nanocrystalline alloy with induced anisotropy.We demonstrate that, contrary to what has previously been assumed, the Langevin-Weiss function can provide a good description of the anhysteretic magnetization of any homogeneous material, even materials with strong transverse uniaxial anisotropy. The key, which seems to have been overlooked, is to allow the Weiss molecular field to be negative.

One-dimensional magneto-mechanical model for anhysteretic magnetization and magnetostriction in ferromagnetic materials

By suggesting that the relationship between magnetization and magnetostriction meet an even-degree polynomial relationship, an ideal magnetization model to analyze the effects of stress on the magnetostriction and magnetization of ferromagnetic materials is established by Jiles in 1995. Based on this classic model and its modification, researchers have conducted extensive research on the characterization, design, optimization and application of ferromagnetic materials. By proposing a new fitting expression for a quadratic piecewise function with a clearer physical meaning, a generalized magneto-mechanical model based on the effective field theory is established in this paper. By comparing with the existing experimental results and theoretical results of the existing models, it is confirmed that the proposed model in this paper can more accurately describe the nonlinear response of the magnetostriction and magnetization of the material under stress and applied ambient magnetic field. A detailed analysis of the magneto-mechanical effects of three types of ferromagnetic materials, such as ferrous materials, magnetostrictive materials and inverse magnetostrictive materials, demonstrates the widespread effectiveness of the newly established model in this paper.