United States

Additive manufacturing (AM) is becoming an alternative to traditional manufacturing methods [1]. In the most extended AM method (fused deposition modeling, FDM), a polymeric filament is extruded above its melting point for its deposition into thin layers. There is a high interest in the use of composite filaments, formed by a polymeric matrix with functional fillers, for the AM of functional parts [2]. The quality of the FDM printed parts is highly dependent on the uniformity of the filament diameter. In the case of composite filaments, a homogeneous distribution of the fillers is also crucial to have a reproducible and uniform functionality. Homogeneous soft magnetic PLA composite filaments for FDM have been recently reported, in which a customized feedstock of polymeric capsules filled with the functional fillers is employed [3]. Fig. 1 shows that composite filaments exhibit a soft magnetic behavior. Additionally, they all have a demagnetizing factor of ~ 1/3 (spheres, according to the shape of the gas atomized powders) regardless of the concentrations of fillers (up to 52 wt. %, corresponding to 12 vol. %). The addition of the fillers causes a remarkable shift in the thermal degradation temperature of the polymer. Increasing filler content produces an abrupt drop in the complex modulus of the polymer above its glass transition, which shifts to lower temperatures according to the peak of tan(δ) (Fig. 2). A decrease in the viscosity in liquid state is also observed, which causes a reduction of the extrudability and printability of the filaments for filler contents above 30 wt. %. The reduction of the viscosity of the composites requires the lowering of the temperatures for their extrusion and printing with respect to the typical ones for PLA.&lt;br/&gt;&lt;br/&gt;Work supported by AEI/FEDER-UE (grant PID2019-105720RB-I00), US/JUNTA/FEDER-UE (grant US-1260179), Consejería de Economía, Conocimiento, Empresas y Universidad de la Junta de Andalucía (grant P18-RT-746), and Army Research Laboratory under Cooperative Agreement Number W911NF-19-2-0212.

![](https://assets.underline.io/uploads/markdown_image/1/image/87db25ab60c9820e49c23306bd80571d.jpg)

Fig. 1. Magnetization vs. magnetic field for the raw maraging steel powder and composite filaments.

![](https://assets.underline.io/uploads/markdown_image/1/image/245b4256f10c5e8a2ffe7ab44cd407d4.jpg)

Fig. 2. (a) Complex modulus and (b) tan(δ) of raw PLA and composite filaments.

2022 Joint MMM-INTERMAG

Manufacturing and Characterization of Soft Magnetic Composite Filaments for Additive Manufacturing

This meeting is a great opportunity to explore sessions and workshops designed with your professional development in mind. Present your research and exchange ideas on the latest in teaching and pedagogy. Participate in a wide range of events, activities, plenaries, and so much more, connecting you with peers and colleagues who share your commitment to excellence in physics education.

AAPT WM 22

Additive manufacturing (AM) is becoming an alternative to traditional manufacturing methods [1]. In the most extended AM method (fused deposition modeling, FDM), a polymeric filament is extruded above its melting point for its deposition into thin layers. There is a high interest in the use of composite filaments, formed by a polymeric matrix with functional fillers, for the AM of functional parts [2]. The quality of the FDM printed parts is highly dependent on the uniformity of the filament diameter. In the case of composite filaments, a homogeneous distribution of the fillers is also crucial to have a reproducible and uniform functionality. Homogeneous soft magnetic PLA composite filaments for FDM have been recently reported, in which a customized feedstock of polymeric capsules filled with the functional fillers is employed [3]. Fig. 1 shows that composite filaments exhibit a soft magnetic behavior. Additionally, they all have a demagnetizing factor of ~ 1/3 (spheres, according to the shape of the gas atomized powders) regardless of the concentrations of fillers (up to 52 wt. %, corresponding to 12 vol. %). The addition of the fillers causes a remarkable shift in the thermal degradation temperature of the polymer. Increasing filler content produces an abrupt drop in the complex modulus of the polymer above its glass transition, which shifts to lower temperatures according to the peak of tan(δ) (Fig. 2). A decrease in the viscosity in liquid state is also observed, which causes a reduction of the extrudability and printability of the filaments for filler contents above 30 wt. %. The reduction of the viscosity of the composites requires the lowering of the temperatures for their extrusion and printing with respect to the typical ones for PLA. Work supported by AEI/FEDER-UE (grant PID2019-105720RB-I00), US/JUNTA/FEDER-UE (grant US-1260179), Consejería de Economía, Conocimiento, Empresas y Universidad de la Junta de Andalucía (grant P18-RT-746), and Army Research Laboratory under Cooperative Agreement Number W911NF-19-2-0212.

![](https://assets.underline.io/uploads/markdown_image/1/image/87db25ab60c9820e49c23306bd80571d.jpg)

Fig. 1. Magnetization vs. magnetic field for the raw maraging steel powder and composite filaments.

![](https://assets.underline.io/uploads/markdown_image/1/image/245b4256f10c5e8a2ffe7ab44cd407d4.jpg)

Fig. 2. (a) Complex modulus and (b) tan(δ) of raw PLA and composite filaments.

technical paper

The 15th Joint MMM-INTERMAG Conference (2022 Joint) is sponsored jointly by AIP Publishing and the IEEE Magnetics Society. Members of the international scientific and engineering communities interested in recent developments in fundamental and applied magnetism are invited to attend and contribute to the technical sessions. The technical program will include invited and contributed papers in oral and poster sessions, invited symposia, a plenary session, and an evening session, with about 1500 presentations overall. This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

The 2022 Joint Conference will offer both in-person and prerecorded on-demand content.

For tickets to the event, please register at the following link: https://magnetism.org/register

Please register for the Joint MMM-Intermag conference to join the event.

This Conference provides an outstanding opportunity for worldwide participants to meet their colleagues and collaborators and discuss developments in all areas of magnetism research.

Amorphous ferromagnetic glass-coated microwires present various properties suitable for technological applications, such as great magnetic softness, ultra-fast magnetization switching and domain wall propagation, inductive heating capabilities or Giant Magnetoimpedance, GMI, effect [1, 2]. Thus, they have been widely studied in the past few years [1]. As-prepared Fe and Co based microwires differ in some of these properties. Fe microwires are observed to present magnetic bistability and rather low GMI values, while Co ones possess nearly non-hysteretic magnetization curves and higher GMI [1], making the latter a more attractive candidate for GMI related technologies. However, Co is a critical raw material, making Fe-based microwires preferable for large scale applications. Thus, search of appropriate postprocessing for magnetic softness and GMI improvement is essentially relevant. Among the prospective methods allowing these goals are stress-annealing and Joule heating. As the latter is more effective for tuning of GMI in Co-rich microwires [1], systematic studies of Joule heating on magnetic properties and GMI of Fe-rich ones are scarce. Accordingly, influence of Joule heating on magnetic properties and GMI effect of Fe75B9Si12C4 glass-coated microwires was studied. The GMI ratio, ΔZ/Z, is defined as: ΔZ/Z = [Z(H) - Z(Hmax)] / Z(Hmax) A decrease in coercivity of Joule heated samples is observed, while hysteresis loop maintain rectangular shape (Fig 1). However, a remarkable increase in ΔZ/Z is observed (Fig 2). Observed GMI ratio improvement has been discussed considering magnetic anisotropy by induced magnetic field during Joule heating and internal stress relaxation. Remarkable GMI effect improvement observed in Joule heated Fe-rich microwires, along the fact that Fe is a more common, cheap metal than Co, makes them suitable for magnetic sensor applications.

![](https://assets.underline.io/uploads/markdown_image/1/image/0c6d500e18157a87372c9dde525ab3a3.jpg)

Fig. 1. Hysteresis loops of Fe75B9Si12C4 microwires Joule heated at 20mA for different times. Inset: coercivity dependence on annealing time.

![](https://assets.underline.io/uploads/markdown_image/1/image/4f8abb050b87d288134fc3e6171b6a1e.jpg)

Fig. 2. GMI ratio of as-prepared (a) and Joule heated for 3min (b) Fe75B9Si12C4 microwires measured at different AC frequencies.

Effect of Joule Heating on GMI and Magnetic Properties of Fe-rich Glass-Coated Microwires

In the present work, we combine magnetometry and magnetic small-angle neutron scattering to study the influence of the microstructure on the macroscopic magnetic properties of a nanocrystalline Ni bulk sample, which was prepared by straining via high-pressure torsion. As seen by magnetometry, the mechanical deformation leads to a significant increase of the coercivity compared to nondeformed polycrystalline Ni. The neutron data reveal a significant spin-misalignment scattering caused by the high density of crystal defects inside the sample, which were created by the severe plastic deformation during the sample preparation. The corresponding magnetic correlation length, which characterizes the spatial magnetization fluctuations in real space, indicates an average defect size of 11 nm, which is smaller than the average crystallite size of 60 nm. In the remanent state, the strain fields around the defects cause spin disorder in the surrounding ferromagnetic bulk, with a penetration depth of around 22 nm. The range and amplitude of the disorder is systematically suppressed by an increasing external magnetic field. Under certain assumptions, the analysis of the penetration depth allows one to determine the magnitude of the effective magnetic anisotropy. For HPT Ni, we find an increased anisotropy (factor of 4), presumably due to magnetoelastic interactions, relative to the single-crystalline ground state. Our findings are supported and illustrated by micromagnetic simulations, which, for the particular case of nonmagnetic defects (holes) embedded in a ferromagnetic Ni phase, further highlight the role of localized spin perturbations for the magnetic microstructure of defect-rich magnets such as high-pressure torsion materials. This work was supported by the financial support from the National Research Fund of Luxembourg (Pride MASSENA Grant No. 10935404 and CORE SANS4NCC Grant), from KAKENHI (Grant No. 19K05102) and from the Japan Science and Technology Agency (Grant No. JPMJSK1511).

Revealing defect-induced spin disorder in nanocrystalline

Magnetic wires are one of the most convenient materials allowing to study domain wall, DW, propagation. From the viewpoint of technological applications, fast, reliable, and controllable DW propagation is essential [1]. Therefore, great efforts have been made to control the single DW propagation [1]. Amorphous perfectly cylindrical microwires with positive magnetostriction coefficient can present a fast (DW velocity, v, of the order of 1 km/s) single DW propagation [2]. In viscous regime below the Walker breakdown field a uniform v is usually assumed if magnetic fielf, H is constant. We observed that in Fe75B9Si12C4 magnetic microwire stress-annealing at variable annealing temperature, T, rather different hysteresis loops for different sections of such microwire are observed (Fig. 1a). In as-prepared microwires DW propagation with the same velocity is observed in different section of the microwire (see Fig. 1b). However, in a media with graded magnetic anisotropy v is essentially non-uniform (Fig. 1c): faster DW propagation is observed in the region with moderate stress-annealing induced magnetic anisotropy (see Fig. 1c). Such character of induced anisotropy is evidenced by the dependence of the inner axially magnetized core radius, Rc. on sample length, L (see Fig. 1d). Evaluated Rc(L) dependence and its correlation with T(L) reflects graded magnetic anisotropy obtained by stress-annealing at variable Tann. Higher DW velocity in the region with induced magnetic anisotropy is explained by the transverse character of stress-annealing induced magnetic anisotropy which affects the travelling domain wall in a similar way as application of transversal bias magnetic field. Obtained graded magnetic anisotropy allows engineering the DW dynamics of Fe-rich microwires and observation of essentially non-uniform singe DW propagation in media with graded magnetic anisotropy.

![](https://assets.underline.io/uploads/markdown_image/1/image/6cc90572f9480697e48013153a1184c0.jpg)

Graded magnetic anisotropy appearing as a modification of hysteresis loop character over the microwire length, L, obtained by stress-annealing at variable T (a), v(H) dependencies in as-prepared microwire (b) and in the region with graded magnetic anisotropy (c) and Rc (L) dependence in microwire with graded anisotropy.

Domain wall propagation in Fe-rich magnetic microwires with graded magnetic anisotropy

The D03 Fe3Sn phase shows higher magnetization and smaller magnetocrystalline anisotropy compared to Fe3Si based on the comprehensive first principles calculations [1]. A new computation-based alloy design method, motivated by the work of Villars [2], was used to design a new alloy system (Fe1-x-yCoxSny)87Nb3B9Cu1 (4&lt;x&lt;27, 5&lt;y&lt;15) to achieve D03 Fe3Sn phase with the help of the melt-spinning process. Differential scanning calorimetry was used to determine the primary crystallization temperatures of the Fe79-xCoxSn8Nb3B9Cu1 alloys (x=4, 9, 14, 19 and 24), which have peak temperatures from 405°C to 420°C at a heating rate of 10°C /min. The amorphous ribbons were encapsulated in fused quartz ampoules and annealed in tube furnace at 450°C, 500°C, or 550 °C for 3600 seconds. X-Ray Diffraction (XRD) has been used to analyze structural properties of as-spun and heat-treated ribbons of all alloy compositions. Partial crystalline exists during the melt-spinning process for these series alloys except the Fe60Co20Sn7Nb3B9Cu1 as-spun ribbons, which has a single broad scattering peak of XRD diffractograms indicating an amorphous phase. XRD diffractogram of the heat-treated ribbons are comprised of a B2 phase, which has a primitive cubic structure. The crystallite sizes for each heat-treated sample were estimated by Scherrer broadening to have values between 6 and 11 nm. The magnetic properties of as-spun and heat-treated ribbons were studied by vibrating sample magnetometry. Saturation magnetization and coercivity were determined from room temperature magnetic hysteresis loops. The highest magnetization value 171.4 Am2/kg was achieved by Fe60Co22Sn5Nb3B9Cu1 alloy annealed at 500°C while the coercivity value is 67.3A/m.

Development of an (Fe, Sn)-based Nanocrystalline Soft Magnetic Alloy

Ultra-Rapid Annealing (URA) has been demonstrated as a viable means of producing nanocrystalline soft magnetic materials with a saturation magnetic polarization comparable to that of Fe-Si steels (1.85 to 2.02 T) while also maintaining low coercivity (2.5 to 9.3 A/m)[1–3]. However, the high heating rates (&gt; 104 K/s) and short annealing times (&lt; 3 s) utilized by URA compared to conventional annealing techniques, along with the effect of latent heating during primary crystallization [4], has created new material production challenges. Specifically, the requirement for batch processing by URA places a limit on the commercial viability of this technique. In this work, a Continuous Ultra-Rapid Annealing (CURA) process has been demonstrated whereby a ribbon undergoes URA in a reel-to-reel continuous process. Annealing time (0.1 to 10 s) and tension (5-40 MPa) utilized by CURA are carefully controlled to allow for the processing of brittle nanocrystalline alloys. The microstructural and magnetic properties of nanocrystalline (Fe0.8Co0.2)86B14 produced by CURA have been characterized. Figure 1 shows DC hysteresis loops acquired from nanocrystalline (Fe0.8Co0.2)86B14 produced by the batch URA [1,2] and the newly developed CURA processes. The coercivity was reduced from 14.9 A/m to 8.8 A/m and the core loss at 1.5 T and 50 Hz was reduced from 0.62 W/kg to 0.50 W/kg by applying the CURA process. The CURA process has also been utilized to produce a nanocrystalline (Fe0.8Co0.2)86B14 toroidal core weighing 0.16 kg. This core was machined to produce a stator core for a commercially available 220 W electric motor which demonstrated a 10-25% relative improvement in efficiency over an unmodified motor with a conventional Fe-Si steel stator core [5].

![](https://assets.underline.io/uploads/markdown_image/1/image/c02f56623f189e8fb02e1aa0efeed0ac.jpg)

DC hysteresis loops acquired from nanocrystalline (Fe0.8Co0.2)86B14 produced by conventional Ultra-Rapid Annealing (URA) and the newly developed Continuous Ultra-Rapid Annealing (CURA) process.

![](https://assets.underline.io/uploads/markdown_image/1/image/72718e73d3acbc83299d81db6d6ad39f.jpg)

A photo of the nanocrystalline (Fe0.8Co0.2)86B14 electric motor stator core produced by CURA after winding of the excitation coils and mounting to a commercially available 220 W motor frame.

Nanocrystalline soft magnetic materials produced by Continuous Ultra-Rapid Annealing (CURA)

Perpendicular magnetic anisotropy (PMA) is a property of magnetic films which allows for high-density data storage in spintronic applications. In rare earth iron garnets, PMA typically originates from magnetoelastic anisotropy either as a result of film strain due to lattice mismatch with a single crystal garnet substrate or from thermal mismatch upon annealing1. However, PMA derived from magnetoelastic anisotropy requires specific combinations of strain state and magnetostriction coefficients, so there is interest in finding new ways to control PMA in thin films. In this study we demonstrate that PMA can also be obtained from a dominant growth-induced anisotropy (GIA) using europium thulium iron garnet (EuTmIG) as a model material. By varying the ratio of Eu to Tm, the lattice parameter of the garnet can be changed, resulting in films which are under compression or tension on a gadolinium gallium garnet (GGG) substrate, as shown in Fig. 1 with high resolution x-ray diffraction (HRXRD) analysis. Vibrating sample magnetometry (VSM), shown in Fig. 2, indicates that PMA is present in all films regardless of strain state. We propose that PMA in the zero-strain film is attributed to GIA due to the preferential ordering of different rare earth ions on inequivalent lattice sites within the garnet2. Additionally, we quantify GIA and spin transport in the films with spin hall magnetoresistance measurements on Pt/EuTmIG hall crosses. This work shows that precise engineering of the structure of these high quality films creates new methods for control of garnet thin film properties for spintronic applications.

![](https://assets.underline.io/uploads/markdown_image/1/image/4ca3d887128a3f5963a08aed3388a341.jpg)

Fig. 1. HRXRD of the (444) symmetric reflection of ~25 nm EuIG, TmIG and Eu0.7Tm2.3Fe5O12 films on (111) GGG substrates show tensile, compressive, and near zero strain states, respectivley, as shown by the relative position of the film and substrate peaks.


![](https://assets.underline.io/uploads/markdown_image/1/image/92fab549d605d7cf75d3aa45236e89fd.jpg)

Fig 2. PMA is evident in out-of-plane VSM of all films, regardless of the magnitude of magnetoelastic anisotropy.

Growth-Induced Perpendicular Magnetic Anisotropy in Thin Film Rare Earth Iron Garnets

Colloidal magnetic nanoparticles (MNPs) of metal oxides are increasingly important for applications such as biomedical imaging, sensing, drug delivery, and hyperthermia. These exploit the very soft magnetism found in nanoscale ferrimagnets, often ferrites, which yield a strong collective response to applied alternating (AC) magnetic fields while remaining dispersed in media. Their synthesis relies on reproducible, scalable, colloidal solution chemistry routes. Further, the size, shape, structure, and composition are tunable, serving as levers to design and manipulate the magnetic behavior of these functional MNPs. Here, we report colloidal MNPs specifically designed for remote 3D thermometry; a technique that employs MNPs to measure and image temperature remotely with high spatial resolution throughout a three-dimensional volume. This metrology is based on the temperature-dependent nature of the particles’ magnetization, their dispersibility throughout a liquid or solid of interest, and the ability of applied AC magnetic fields to penetrate through many materials. A series of shape- and size-controlled colloidal MNPs based on ferrites were synthesized via the thermal decomposition of organometallic precursors. Further, we exploited compositional doping of transition metal elements and exchange coupling between core-shell magnetic heterostructures to generate improved temperature-dependent magnetization (200-400 K at &lt;0.001 T applied AC field). Extensive structural characterization was performed to better understand the underlying factors influencing the magnetic response. Results collected from X-ray photoelectron and Raman spectroscopies, and high-resolution electron microscopies revealed correlations between the nanoscale atomic structure of these particles with their magnetic performance. Finally, we used a home-built AC magnetic particle spectrometer to determine the magnetization-dependence of these nanothermometers as a function of temperature, frequency, and field amplitude. We show that designed complex metal oxide MNPs display tunable thermosensitivity and AC field signal response, which represents a potential pathway to the implementation of a practical 3D thermal imaging platform based on magnetometry.

Designing Magnetic Metal Oxide Colloidal Nanoparticles for Spatially Resolved Thermometry

Yttrium Iron Garnet is the benchmark material for ferromagnetic resonance; it is up to today the only magnetic material that is being integrated in on-shell electronic devices for its radiofrequency properties that combine high resonance quality factor (104) and frequency tunability. It is the goal of YIG magnonics to make use of this potential for a large variety of applications ranging from beyond CMOS computation to radiofrequency front-end and back-end analog signal processing. YIG devices have however a sever drawback that is that they should include thermal stabilization. This is a necessity since YIG has a relatively low Curie temperature (~550K) which results in a high sensitivity of its magnetization to temperature (~0.4 mT /Kelvin). In the present study we show a novel approach to solve this long standing issue by engineering the magnetic anisotropy in Bi substituted YIG. We evidence for those films that vanishing effective magnetization yields high thermal stability over very wide temperature range: from 260K to 400K, where the frequency thermal drift is 50 times smaller than that of YIG while keeping the extremely low damping and insulating character of YIG [1] For instance, we show that this was instrumental to the excitation of coherent spin-waves using SOT described in Ref. 2. FMR frequency can be written for a thin film in the in-plane magnetization configuration as: f= γ√(Ha+(Ms-Hu)). Where γ is the gyromagnetic factor, Ha is the applied magnetic field, Ms is the saturation magnetization and Hu is the out-of-plane magnetic anisotropy field. The effective magnetization is defined as Meff=Ms-Hu . It arises from this equation that the only temperature dependent terms are Ms(T) and Hu(T). Our observation show that here those term can be engeniered to have identical temperature dependences yielding a vanishing effective magnetization. This observation is unexpected as in most magnetic material the magnetic anisotropy has a much steeper temperature dependence than the magnetization. We anticipate that fabricating a material whose FMR frequency is temperature independent open new opportunities for the field of spintronics.

Evidence for Temperature Independent Ferromagnetic Resonance Frequency in Bismuth Substituted Iron Garnets

In alloy based soft magnetic materials various factors such as alloy composition, phase, synthesis process and annealing conditions affect their magnetic properties. Co, Fe and Ni based alloy thin films exhibit high magnetic moment and high permeability which mark their importance in memory devices and MEMS. The magnetization reversal mechanism in thin films plays a significant role for such technological applications. Here, binary FeCo and ternary (FeCo)(100-x) Nix alloy thin films were deposited on Indium Tin Oxide coated glass substrate by electrochemical deposition. The composition of as deposited alloy was confirmed by EDAX. The surface morphology analysed by SEM, confirmed uniformly dispersed flakes all over the substrate in FeCo films. The increasing content of Ni in the alloy results in the formation of irregular shaped granular structure. Bulk magnetic properties were characterized by VSM hysteresis. As the Ni content increases the coercivity as well as remanence decreases (fig.1) and this is probably due to decrease in anisotropy with added Ni content. The ternary alloy tends to exhibit permalloy like properties but with higher magnetization. The decrease in remanence for higher Ni content marks the change in easy axis orientation from film-plane direction in FeCo to an added minor fraction of out-plane orientation in (FeCo)75Ni25. The plot of coercivity versus angle (θ) between easy axis of magnetization (film plane) and applied field along with the appropriate fitting model to the magnetization reversal is shown in fig.2. It is clear that for lower Ni content, a good fit to the experimental data for angular variation of coercivity is obtained using Kondorsky model, indicating domain reversal mechanism arising due to domain wall depinning from a local defect. The origin of these pinning centres could be the presence of an oxide layer on top of the film surface or at the ITO-alloy film interface. The magnetization reversal mechanism in FeCoNi alloy is predominantly influenced by the Ni concentration. A deeper understanding of this behaviour would be provided by micromagnetic simulation.

![](https://assets.underline.io/uploads/markdown_image/1/image/e2a88a2d12347264f1d1b30df6fbf6b1.jpg)

Change in Hc and Mr/Ms with increasing concentration of Ni


![](https://assets.underline.io/uploads/markdown_image/1/image/4ed8f861136ff255ce3d617ed218ed56.jpg)

Angular variation in Hc fitted with modified Kondorsky Model

Magnetization Reversal Behavior in Electrodeposited FeCoNi Thin Films

Developing a model of magnetic materials, such as electrical steel, that allows to account for both their nonlinear and anisotropic behavior remains a challenge. The approach presented in this paper is based on the use of a Gaussian function whose cumulative distribution function’s shape is very similar to that of a first magnetization curve. This approach has been introduced in the 90s for the study of isothermal remanent magnetization acquired by rock samples containing magnetic minerals [1,2,3]. The main drawback of this approach lies in the fact that a given magnetization curve is derived from one Gaussian function only, which is itself defined by three fixed parameters only [1], leading to non-negligible differences when compared to experimental data. Moreover, this approach has never been adapted to account for the anisotropic behavior of electrical steel. In this paper, we propose to enhance this method by using a function which results from the sum of four Gaussian functions, as illustrated in Fig. 1. To account for the anisotropy, all the parameters used to define the Gaussian functions depend on the angle between the magnetic field H and the rolling direction of the steel, denoted θ. Fig. 2 shows the comparison between first magnetization curves for three different magnetization directions (θ) obtained from experimental data (M530-50A electrical steel), from the original model based on the use of one Gaussian function only and from the proposed enhanced model. The work presented in this paper shows multiple originalities and improvements. Firstly, it shows that using more than one Gaussian function allows for a better approximation of first magnetization curves (Fig. 2). Secondly, it shows that considering θ-dependent parameters to define the Gaussian function allows the method to account for magnetic anisotropy (Fig. 2). Thirdly, it highlights that those parameters follow a polynomial trendline.

![](https://assets.underline.io/uploads/markdown_image/1/image/a840a400f9aa59752141c240e934913c.jpg)

Fig. 1 First magnetization curve derived from a Gaussian function defined as the sum of four subfunctions.

![](https://assets.underline.io/uploads/markdown_image/1/image/553c7c268068ad44177edd6923859cff.jpg)

Fig. 2 Comparison between curves obtained from experimental data (markers), from the original model (dashed) and from the enhanced proposed model (solid).

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