Anhysteretic Curve Research Articles

Anhysteretic strains in ferroelectric ceramics under electromechanical loading

This study investigates anhysteretic strains in PZT ceramics. The anhysteretic curves are associated with a stable balanced state of polarization in the domain structure, excluding dissipative effects related to mechanisms such as domain wall pinning. Anhysteretic measurements are representative of an -ideal- scenario in which the material would undergo no energy loss due to dissipative processes, focusing on the stable and reversible aspects of the domain configuration. The different methodologies employed to measure deformations under electromechanical loading are presented, leading to the introduction of digital image correlation (DIC) as the chosen technique, recognized for its ability to capture detailed information on transverse and longitudinal strain. The article then describes a procedure developed to obtain anhysteretic strain and anhysteretic polarisation for different levels of compressive loadings. The subsequent presentation of the results of the transverse and longitudinal strain analyses provides valuable insights into the reversible and irreversible behavior of the material. They can be used as a basis for the thermodynamical modelling approaches grounded on separating reversible and irreversible contributions or as a validation of existing models describing anhysteretic behavior. The compressive stress affects both the shape of hysteretic and anhysteretic curves. The anhysteretic curve represents a stable equilibrium in the domain structure. Compressive stress reduces strain by affecting the pinning of domain walls. These points justify the interest in studying the effect of compressive stress on the anhysteretic behavior of ferroelectrics.

Designing an Improved Method to Determine the Anhysteretic Curve in Soft Magnetic Materials via the Jiles-Atherton Model

Designing an Improved Method to Determine the Anhysteretic Curve in Soft Magnetic Materials via the Jiles-Atherton Model

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.

Magnetostrictive energy conversion ability of Iron Cobalt Vanadium alloy sheet: Experimental and theoretical evaluation

The increasing demand for autonomous devices has made the concept of energy harvesting a significant industrial and academic point of interest. In this domain, an ideal magnetostrictive material for converting mechanical vibrations into electrical energy in a cost-effective way (i.e. competing with the price of primary batteries) remains to be determined. Iron-Cobalt-Vanadium (Permendur, Co49-Fe49-V2) is a promising candidate: it is a soft ferromagnetic material with high magnetization saturation, high magnetostrictive coefficients, low price, and good availability. In this study, the experimental magnetic characterization and simulation of Permendur sheets under tensile stress were performed, and their energy conversion capabilities were assessed. The conversion ability was predicted using thermodynamic Ericsson cycles from reconstructed anhysteretic curves. A maximum of 10.45 mJ cm−3 energy density was obtained under a tensile stress of 480 MPa and a magnetic excitation of 5.5 kA m−1. Then, an additional estimation was proposed to account for the hysteresis losses. For this, major hysteresis loops at different stress levels were considered, yielding an energy density of 3.52 mJ cm−3. Finally, experimental Ericsson cycles were performed to prove the feasibility of the conversion and corroborate the energy level predictions.

Analysis of energy conversion capability among various magnetostrictive materials for energy harvesting

This work addresses vibrational energy harvesting using magnetostrictive materials. In this field, materials with exceptional magneto-mechanical coupling properties (e.g. Galfenol, Terfenol-D) have attracted significant attention. Only a few magnetostrictive materials have been tested in devices, however, leaving the actual influence of these materials’ properties on the energy harvesting device open to question. This work compares an extensive range of ferromagnetic materials through analysis of their magnetic behavior under static stress. To enable fair comparison of the materials, a model was developed to interpolate their magnetic anhysteretic curves under fixed stress of σ = ±50 MPa. The energy harvesting process was then simulated using a theoretical Ericsson thermodynamic cycle, where the area represents the energy density. This approach estimates the ultimate energy density of the materials using a fair approach, without placing conditions on the applied magnetic field. The correlation between ultimate energy density and the magnetoelastic coefficient show that highly magnetostrictive materials achieve higher ultimate energy densities, as expected. In the low field range, it is however concluded that all materials exhibit energy densities of the same order of magnitude. Secondly, the magnetoelastic coefficient versus excitation field characteristics revealed an optimal bias magnetic field for each material. Finally, for realistic implementation, the paper considers a pre-stress in combination with a bias magnetic field and the small dynamic variations that result from currents induced in surrounding coils. A model was developed and revealed an optimum output energy density that was independent of the geometry and the coil. An energy harvesting figure of merit was then defined to enable a final comparison of the materials, encompassing both material characteristics and realistic applications. Under these working conditions and with all costs considered, some low-magnetostriction materials appeared able to compete with giant magnetostriction materials.

The Effective Field in the T(x) Hysteresis Model

Hysteresis loops constitute the source of important information for the designers of magnetic circuits in power transformers. The paper focused on the possibility to interpret the phenomenological T(x) model in terms of effective field vs. magnetization dependence. The interdependence of anhysteretic curve and hysteresis loops was emphasized. The concept of the anhysteretic plane introduced at the end of the last century by Sablik and Langman was subject to a tangible interpretation within the hyperbolic model framework. A novel geometric interpretation of the “effective field” related to the concept of affine transformation was introduced. It was shown in the paper that minor hysteresis loops of grain-oriented electrical steel may be described with the proposed formalism.

A Physically Based Model for Soft Magnets’ Anhysteretic Curve

A Physically Based Model for Soft Magnets’ Anhysteretic Curve

Modeling of Magnetic Properties of Rare-Earth Hard Magnets

Magnetic properties of hard magnets are currently attracting a great deal of attention. In the paper, the modified Harrison model was used to describe the saturating hysteresis loops of three praseodymium–dysprosium ribbons that differed in their chemical composition and processing conditions. Microstructural studies (TEM and diffraction patterns) were performed for the ribbons under consideration. The Harrison model incorporates a number of physically tangible concepts such as the anhysteretic curve, bifurcations, and bi-stability. The modification of the original approach consisted of adding an additional degree of freedom in the modeling by freeing the restraints present in the original version, in which both coercivity and remanence are functions of temperature only.

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.

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.

Determination of external and internal demagnetizing factors for strip-like amorphous ribbon samples

The extension of results concerning the calculation of the external (geometric) demagnetizing factor obtained for rod like samples to strip-like ribbon samples is investigated. To match the experimental determined demagnetizing factors of strips with various length two fitting parameters have been introduced: (i) the effective width of the strip and (2) the internal demagnetizing factor. It turned out that the equivalent rod diameter calculation from the cross-section is valid up to a limited width/thickness ratio. The experimental demagnetizing factor determined from the steepness of the anhysteretic curve was greatly reduced by a uniaxial stress; where from a lower bound for the internal demagnetizing factor could be obtained. We have found that this internal demagnetizing factor is about three times larger than the geometric one for an as cast amorphous sample with Finemet composition.

Experimental Verification of Isotropic and Anisotropic Anhysteretic Magnetization Models.

The anhysteretic magnetization curve is the key element of modeling magnetic hysteresis loops. Despite the fact that it is intensively exploited, known models of anhysteretic curve have not been verified experimentally. This paper presents the validation of four anhysteretic curve models considering four different materials, including isotropic, such as Mn-Zn soft ferrite, as well as anisotropic amorphous and nanocrystalline alloys. The presented results indicate that only the model that considers anisotropic energy is valid for a wide set of modern magnetic materials. The most suitable of the verified models is the anisotropic extension function-based model, which considers uniaxial anisotropy.

Temperature-dependent anhysteretic behavior of co-doped PZT

The anhysteretic behavior of a soft Pb(Zr,Ti)O3 was measured from 25 °C to 175 °C. The experimental determination of the anhysteretic polarization curve, combined with classical P-E and S-E loop measurements, allows for an experimental separation of the reversible and dissipative contributions to the ferroelectric behavior. This approach offers insight into the different mechanisms originating at the microscopic scale and the contribution to the macroscopic ferroelectric properties. It was found that the reversible anhysteretic susceptibility χa of the unpoled material increases by 30% from room temperature to 150 °C. On the other hand, the effect on the total susceptibility for a null polarization χc increases only by 17% over the same temperature range. Since the difference between χa and χc reflects the dissipative contribution to the macroscopic ferroelectric behavior, this reveals that dissipation reduces the improvement of susceptibility under increasing temperature. This work illustrates the benefits of separating experimentally the reversible and dissipative contributions to describe the ferroelectric behavior, which can serve as a basis for advanced modeling approaches.

The anhysteretic polarisation of ferroelectrics

Measurement and calculation of anhysteretic curves is a well-established method in the field of magnetic materials and is applied to ferroelectric materials here. The anhysteretic curve is linked to a stable equilibrium state in the domain structure, and ignores dissipative effects related to mechanisms such as domain wall pinning. In this study, an experimental method for characterising the anhysteretic behaviour of ferroelectrics is presented, which is subsequently used to determine the anhysteretic polarisation response of polycrystalline barium titanate and a doped lead zirconate titanate composition at room temperature. Various external parameters, such as electric field, stress, and temperature, can significantly affect ferroelectric behaviour. Ferroelectric hysteresis curves can assess the importance of such effects but cannot distinguish their contribution on the different intrinsic and extrinsic mechanisms involved in ferroelectric behaviour. In this work, the influence of compressive stress on the anhysteretic polarisation is measured and discussed. The comparison of the polarization loop to the anhysteretic curve under compressive stress elucidates the effects on the stable equilibrium domain configuration and dynamic effects associated to dissipation.

Scaling of Anhysteretic Curves for LaFeCoSi Alloy near the Transition Point

Scaling of Anhysteretic Curves for LaFeCoSi Alloy near the Transition Point

An Improved Method for Acquisition of the Parameters of Jiles-Atherton Hysteresis Scalar Model Using Integral Calculus

An accurate modeling of materials is essential to obtain reliable results in fields calculation. The Jiles-Atherton approach is widely used for modeling the magnetic hysteresis and depends on its set of five parameters to properly represent material. In this article is proposed an original methodology for obtaining this set of parameters avoiding the derivatives rough calculation and using the calculation of integrals. From the model equations, a new methodology with two nonlinear algebraic systems of five equations in five unknowns is obtained. The initial magnetization curve, the anhysteretic curve and filtering data are not necessary. The proposed methodology also does not restrict the search space of parameters. The parameters assume values in the interval (0,∞). Calculated data were compared with experimental data to validate the methodology. The simulations showed that the proposed method can obtain an accurate set of parameters from a single experimental hysteresis loop and with low computational effort.

Integration of the hysteresis in models of asymmetric three‐phase transformer: finite‐element and dynamic electromagnetic models

This paper deals with the models elaboration for the asymmetric three-phase transformers. The first model uses the principle of the hysteresis integration in two-dimensional finite-element method (FEM) with magnetic field calculations. This method is based on inverse Jiles–Atherton hysteresis model. The second one is a dynamic electromagnetic model (DEM) based on Tellinen hysteresis model. The DEM and FEM models used in ferromagnetic cores have been modified by replacing the anhysteretic curve by the hysteresis loop. A simulation in time domain is carried out and the simulated results are compared with those obtained experimentally to validate the proposed approach. An error calculation is performed to show the accuracy of results obtained by DEM and FEM models.

Analytical model for magnetic anisotropy of non-oriented steel sheets

Purpose – Recent investigations on magnetic properties of non-oriented (NO) steel sheets enhance the comprehension of the magnetic anisotropy behaviour of widely employed electrical sheets. The concept of energy/coenergy density can be employed to model these magnetic properties. However, it usually presents an implicit form which requires an iterative process. The purpose of this paper is to develop an analytical model to consider these magnetic properties with an explicit formulation in order to ease the computations. Design/methodology/approach – From rotational measurements, the anhysteretic curves are interpolated in order to extract the magnetic energy density for different directions and amplitudes of the magnetic flux density. Furthermore, the analytical representation of this energy is suggested based on statistical distribution which aims to minimize the intrinsic energy of the material. The model is finally validated by comparing measured and computed values of the magnetic field strength. Find...

Robust solution procedure for the discrete energy-averaged model on the calculation of 3D hysteretic magnetization and magnetostriction of iron–gallium alloys

A reformulation of the Discrete Energy-Averaged model for the calculation of 3D hysteretic magnetization and magnetostriction of iron-gallium (Galfenol) alloys is presented in this paper. An analytical solution procedure based on an eigenvalue decomposition is developed. This procedure avoids the singularities present in the existing approximate solution by offering multiple local minimum energy directions for each easy crystallographic direction. This improved robustness is crucial for use in finite element codes. Analytical simplifications of the 3D model to 2D and 1D applications are also presented. In particular, the 1D model requires calculation for only one easy direction, while all six easy directions must be considered for general applications. Compared to the approximate solution procedure, it is shown that the resulting robustness comes at no expense for 1D applications, but requires almost twice the computational effort for 3D applications. To find model parameters, we employ the average of the hysteretic data, rather than anhysteretic curves, which would require additional measurements. An efficient optimization routine is developed that retains the dimensionality of the prior art. The routine decouples the parameters into exclusive sets, some of which are found directly through a fast preprocessing step to improve accuracy and computational efficiency. The effectiveness of the model is verified by comparison with existing measurement data.

Anhysteretic Functions for the Jiles–Atherton Model

The Jiles–Atherton (JA) model of ferromagnetic hysteresis usually bases on the Langevin function as its anhysteretic part. This leads to a problem, since for some known materials, the anhysteretic curve may not be modeled in view of fact, that the coupling parameter $\alpha $ , as determined from their major hysteresis loop, is too large, i.e., greater than the value permissible for the Langevin function. Therefore, a new function is required in order to omit this mathematical dilemma. Here, we present a set of simple functions, with their knees depending on one parameter only. Also, a more complicated function, with knee location depending on two parameters is analyzed, however, the Brillouin function again does not solve the difficulty with $\alpha $ . Therefore, within the frame of the JA model, a new function is proposed, making possible to have the initial differential susceptibility arbitrarily small. In addition, the strengthening of the effective field is considered and the permissible values of the second coupling parameter $\beta $ , in respect to the Brillouin and Langevin functions, are presented. Finally, our new function is successfully used to model the measured anhysteretic curve.