Issue
Eur. Phys. J. Appl. Phys.
Volume 90, Number 1, April 2020
International Symposium on Electromagnetic Fields in Mechatronics, Electrical and Electronic Engineering (ISEF 2019)
Article Number 10901
Number of page(s) 7
Section Physics of Energy Transfer, Conversion and Storage
DOI https://doi.org/10.1051/epjap/2020200018
Published online 20 May 2020

© EDP Sciences, 2020

1 Introduction

Electrical steel has been the subject of numerous improvements over the past decades, and nowadays steel grades with very low loss characteristics are available, i.e. high-permeability grain-oriented electrical steel (HiB GOES) [1]. This steel grade possesses great magnetic properties in the rolling direction (RD), and is specifically manufactured for low hysteresis loss. Therefore, the core of modern distribution and power transformer is almost exclusively constructed out of HiB GOES.

Generally, a transformer is excited at a frequency of 50 or 60 Hz, with a sinusoidal peak magnetic flux density of 1.4–1.7 T. It is imperative to predict the performance and efficiency, to ensure correct transformer operation and to estimate running costs [2]. This prediction depends on many aspects, e.g. core geometry, applied steel grade, operating conditions, etc. [3]. It is not uncommon for the performance of a new transformer design to be extrapolated based on the performance of previous designs. However, increased insight in the core loss and hysteresis behavior allow accurate predictions without the need of such extrapolations [4,5].

The loss prediction of ferromagnetic materials is a field that has been extensively investigated over the years. Approaches that average the contribution of many individual magnetic particles are common, as the connection between microscopic particles and the consequential macroscopic hysteresis behavior is complex due to the influence of many factors [6]. Additionally, the amount of data to model the orientation and spin of every atom in a lamination of electrical steel is numerically challenging, although microscopic hysteresis behavior can be approximated by volumes of the same magnetization [7]. Therefore, phenomenological models are widely applied, such as the well known Steinmetz equation and Bertotti statistical loss separation theory, which are still in use today [8,9]. The Steinmetz equation is a description of the bulk material loss, and provides no insight in the involved loss mechanisms. The Bertotti loss separation theory describes the loss in separate components. Hence, providing insight in the different loss mechanisms.

To describe magnetic hysteresis in ferromagnetic materials, a well known and widely applied model of phenomenological basis is the Preisach model (PM) of hysteresis [10]. The classical scalar form of the PM was studied in depth by Bertotti and Mayergoyz [1113], who focused on reconstruction of static hysteresis loops, and laid the foundation for dynamic hysteresis prediction, e.g. the eddy current and excess field. Here, rate-dependent elementary Preisach dipoles capture the dynamic behavior by assigning an inertia to every Preisach dipole. Additionally, dynamic hysteresis in the loss separation theory is governed by a phenomenological material characterization model that accounts for the dissipative behavior of the magnetic microstructure of electrical steel [6,14].

Dupré applied this form of the dynamic PM, combined with a lamination model, to predict hysteresis in non-oriented (NO) and grain-oriented (GO) materials for magnetic flux densities up to 1.5 T and frequencies up to 1 kHz [1517]. At the same time, Zirka worked on a congruency based scalar hysteresis model, combined with a rate-dependent magnetic viscosity, that governs the dynamic behavior as a magnetic inertia assigned to the excess field component [18,19]. This model was applied to predict hysteresis in NO and GO materials for magnetic flux densities up to 1.8 T and frequencies up to 1 kHz. These approaches to rate-dependent hysteresis modeling were applied to predict sinusoidal and distorted hysteresis loops, where Dupré focused on distortion by higher harmonics, and Zirka by PWM excitation. For both cases the dynamic PM behavior was determined such that the modeled loss and measured loss were in accordance. However, the parameters were optimized for a single operation point or a limited operating range.

Another Preisach based dynamical hysteresis model is the Loss Surface model [2022]. This model takes on a slightly different approach to predict dynamic hysteresis, that shares some similarity with the viscosity-based approach of Zirka. Namely, by mapping the effect of the dynamic changing magnetic flux density over time in a map, or surface, governed by experimentally determined parameters, similar to how the Everett map captures the hysteresis characteristic. The model was used to predict sinusoidal and distorted hysteresis loops in NO materials for magnetic flux densities up to 1.7 T and frequencies up to 1.5 kHz. The construction of the surface requires an extensive set of measurement data. However, the model predicts the dynamic hysteresis behavior, and corresponding loss remarkably well for the whole surface.

A different type of hysteresis model that is also commonly applied, and of more physical basis, is the Jiles-Atherton model [23]. This model is governed only by a couple parameters that shape the modeled hysteresis loops, and does not use a measured map or surface. Hence, its accuracy is generally lower, especially when a large range of magnetic flux densities is considered, as the model parameters are tuned to this range instead of a single operating point [24]. This leads to errors in the modeledshape of the resulting hysteresis loops, which is why this model is not further researched in this work.

For transformer design it is desirable to predict the loss and dynamic behavior for the complete operating range. Therefore, this work predicts the magnetic loss by application of the loss separation theory with material characterization, for flux densities up to and including 1.9 T, and frequencies up to and including 300 Hz. Note that consideration of a large frequency range aids proper material characterization. Three steel grades with different gauges are considered. Namely: M090-23P, M100-27P, and M105-30P [25]. These steel grades are non domain refined which allows practical loss measurements using an Epstein frame. Additionally, the dynamic hysteresis behavior forsinusoidal concentric hysteresis loops is predicted as governed by the same material characterization. This has not yet been applied for HiB GOES over the whole transformer operating range. The results of this work focus on M090-23P, as this grade is more favorable for use in power transformer cores due to its low loss characteristic.

The procedure described in this work could readily be implemented, either as a post-processing step in case of the statistical loss prediction, or as part of the solving routine for the hysteresis model, in a finite element package. Plenty of measurements should be performed beforehand, over the whole range of magnetic flux densities and frequencies, to ensure proper material characterization. Moreover, hysteresis loop modeling and its corresponding loss prediction has already been done before in case of the Preisach, Loss Surface, and Jiles-Atherton models [24,26].

This paper has the following structure. Section 2 treats the theory of the applied modeling approach. It details about the Bertotti loss separation theory, and material characterization, which is combined with Preisach hysteresis modeling. Section 3 gives the modeling procedure specifically for HiB GOES, and verifies the predicted dynamic hysteresis loops based on the material characterization. This paper closes with the conclusion of Section 4.

2 Modeling theory

This section describes both the statistical loss separation theory and the dynamical hysteresis model used throughout this work.

2.1 Statistical loss separation

The statistical loss separation theory of Bertotti describes the total magnetic loss, PT, by a decomposition into three separate components, e.g. eddy current, hysteresis and excess loss. The decomposition, for sinusoidal flux density waveforms with a peak value and excitation frequency f, is given by (1)

where Pcl is the eddy current or classical loss, Ph the hysteresis loss, and Pex the excess loss. Furthermore, a thin electrical steel sheet is assumed.

The classical loss component describes the dissipative nature of eddy currents, which are induced in a ferromagnetic material in the (de)magnetization process, and counteract the driving source field. Their presence relies heavily on the degree they permeate the material, i.e. skin depth. The skin depth is assumed to be greater than the thickness of the material, and the loss is calculated as follows (2)

where σ is the material conductivity, d the material thickness, and ρ the materialmass density.

The hysteresis loss component describes the additional energy required to change the magnetization direction of a ferromagnetic material, which occurs continuously in the (de)magnetization process. It is attributed to the switching of magnetic domains [6],and the loss is calculated as follows (3)

where is an induction dependent parameter, obtained from quasi-static measurements, that describes the energy enclosed by the hysteresis loops.

The excess loss is a description of the loss residue that is obtained when the modeled eddy current and hysteresis loss are subtracted from the measured total loss, and is attributed to microscopic eddy currents induced as a result of domain wall movement. Here, a set of active magnetic correlation regions, so-called magnetic objects (MO), accounts for the material magnetization. For low magnetization, only a few MO are present that are inhomogeneously dispersed throughout the material. For increasing magnetization, the increased excess field favors the already active regions. Therefore, a MO describes the increased probability that when a Barkhausen jump occurs, this jump occurs in the neighborhood of an already active MO. The dependence between the increasing excess field, Hex, and number of active magnetic objects n is approximated by the linear relation (4)

The parameters n0 and V0 are used to characterize the material, and describe the dissipative behavior of the magnetic microstructure, due to e.g. grain size, crystallographic texture, residual stress, and domain wall movement [6]. Here, n0 represents the number of MO for f →0 Hz, and V0 represents the ability of the excess field to increase the number of active correlation regions with increasing magnetization. To accurately model the loss of the complex domain structure of HiB GOES the parameter n0 is critical [17]. The tilde is used to denote the time averaged parameters which are obtained as follows (5) (6)

where S is the cross section of the lamination, and G is a dimensionless coefficient related to eddy current damping, given by (7)

The excess loss component is calculated as follows (8)

where the parameters n0 and V0 are determined in the frequency domain for fixed values of .

2.2 Dynamic hysteresis

The classical PM is a phenomenological model that describes rate-independent hysteretic behavior [10]. When applied to magnetic hysteresis, certain aspects are not accounted for. Therefore, to increase the applicability, a generalization step is performed by model extension [13]. Inherently, the classical PM is a static hysteresis model without time dependency, as such, the rate that the input is provided to the model has no effect on the output. However, model extensions allow the inclusion of dynamical hysteresis effects, such as losses due to bulk eddy currents and domain wall movement. This rate-dependency is included by the addition of two dynamical components. These model the dynamical eddy current field and dynamical excess field. Similar to (1), the effective magnetic field strength H(t) is decomposed into three separate field components when assuming a thin sheet and uniform flux density [19] (9)

where t is time, Hh(t) is the static hysteresis component corresponding to the magnetic field of the static PM. Hcl (t) is the simplified eddy current field given by (10)

where B(t) is magnetic flux density, and Hex(t) is the excessfield given by [6] (11)

where the parameters n0 and V0 are functions of .

In the classical PM, the output is only dependent on the model history and is not dependent on the rate of change of the input. Hysteresis is described by two switching variables, α and β, and a set of infinitely many hysteresis operators , the input signal H(t), and a weight function called the Preisach distribution μ(α, β). The mathematical model description is (12)

where B(t) is the output signal. In this work the PM is adopted for magnetic hysteresis in HiB GOES.

The Preisach distribution is either of analytic or numerical basis [27]. In the analytic case it is often described by well known distribution functions, although these are not capable to describe the hysteresis behavior for the full magnetization range, and are therefore often applied for a small range or single operation point. In the numerical case it is often described by a so-called Everett map, which ensures high correspondence between the modeled and measured hysteresis loop, and is applicable to the whole operation range. This is achieved by substitution of the double integral of the Preisach distribution in (12) by the cumulative quantity, ξ, derived as follows [28,29] (13)

where H1 and H2 are the start and end point respectively, and E represents the arranged measured data.

In Preisach modeling the dynamical field components of the PM are governed by a change in magnetic flux density overtime. However, the obtained magnetic flux density is always a consequence of a given magnetic field strength. Moreover, inherent to hysteresis, the effective magnetic field strength that corresponds to the desired change is different for each point in the model history. Therefore, the required input magnetic field strength, Hh, for a given flux density value is determined by inversion of (12) by a binary search algorithm [30].

By analysis of one period, T, of the model B(t) and H(t) components, the total power loss, P(t), is calculatedas follows (14)

3 Modeling procedure and results

This section treats the modeling procedure in a step by step fashion for each steel grade. The results for the grade with the lowest loss, M090-23P, are presented in detail. First the statistical loss theory of (1) is discussed, and the HiB GOES material is characterized. This characterization is coupled with the dynamic hysteresis model to predict the excess field. The parameters for every grade are shown in Table 1. Furthermore, for every grade ρ is 7650 kg m−3 and σ is (0.48e-6)−1 S m−1.

The measurements of this work are obtained by an Epstein frame which is calibrated to comply with the IEC-standard [31]. Two measurement types are applied: AC excitation, where the magnetic flux density waveform is regulated to match a sinusoid at a specific frequency; and quasi-static excitation, where the rate of change of the magnetic flux density over time is limited, i.e. low frequency, such that all dynamical eddy current effects are assumed negligible.

Table 1

Model parameters per steel grade.

3.1 Statistical loss prediction

Following the order of Section 2.1, first the eddy current component of (2) is determined for every peak magnetic flux density and frequency. Note that the approach to average the eddy current contribution, instead of modeling the field in the lamination, does have a small influence on the accuracy of the total predicted loss, but becomes more critical for higher frequencies, i.e. small skin depths. The second loss component is the hysteresis loss. The enclosed loop energy, that describes the static energy at low frequency, is determined for a set of sinusoidal concentric hysteresis loops for different peak magnetic flux densities at a frequency 1 Hz, see Figure 1a. The total hysteresis loss is determined by multiplication of the enclosed loop energy by the desired frequency according to (3). To describe the dissipative behavior of the magnetic microstructure of M090-23P under (de)magnetization, the material characterization of (4) is applied. First the loss residue is calculated and the evolution of the time averaged excess field of (5), and the time averaged amount of MO of (6) are obtained, see Figure 1b. Linear regression is applied to the different data points in frequency for each to obtain the relation of (4), where the static amount of active magnetic objects, n0, is equal to the y-intercept, and the evolution of the excess field, V0, is equal to the slope of the fitted line. The evolution of n0 and V0 vs. peak magnetic flux density is shown in Figure 2a and 2b respectively. Note that it is critical that the hysteresis loss is correctly determined, and sufficient data points in frequency are applied for the linear regression to be accurate. This relies on properly performed quasi-static measurements with an accurately controlled measurement speed. Additionally, it is important to use a sufficient amount of laminations for the total loss measurements, such that a stack height compliant with the IEC-standard is reached. The evolution of both parameters is approximated by a polynomial of degree 5 to ensure accurate excess loss and field prediction [14], see Table 2. Moreover, it is worthy to note that for HiB GOES n0 is non-zero at f = 0 Hz due to its oriented steel microstructure. The final loss component is the excess loss of (8). This component is calculated for any operation point, as governed by the material characterization, and completes the loss separation, see Figure 3a.

The model results are in good agreement with the measurement data for the whole studied range of magnetic flux densities and frequencies. The error is within a maximum of 7.18 %, and on average −0.34 %, see Figure 3b. It is concluded that magnetic loss is underestimated especially for high peak magnetic flux densities. This is likely due to the error introduced by the function fit for n0 and V0. The same trend is observed for the othersteel grades, see for the maximum, and for the average error of the applied statistical loss prediction in Table 3.

thumbnail Fig. 1

(a) The static hysteresis energy of M090-23P as determined by quasi-static measurements vs. peak magnetic flux density, (b) material characterization parameters n0 and V0 determined by linear regression, shown for selected peak magnetic flux densities.

thumbnail Fig. 2

(a) The evolution of n0 vs. peak magnetic flux density, (b) the evolution of V0 vs. peak magnetic flux density.

Table 2

Material characterization polynomials per steel grade.

thumbnail Fig. 3

(a) Measured and modeled magnetic loss components for M090-23P for a frequency of 50 Hz vs. peak magnetic flux density, (b) error characteristic between the measured and modeled total magnetic loss, shown for selected frequencies.

Table 3

Model error of the magnetic loss for the complete operating range per steel grade.

3.2 Dynamic hysteresis prediction

Following the order of Section 2.2, first the dynamic eddy current field is determined for a sinusoidal loop for every peak magnetic flux density and frequency by (10). The hysteresis field is calculated next by inversion of the static PM of (12), as governed by the measured Everett map of (13), see Figure 4a. The modeled loops reproduce the measured loops exactly. An additional advantage of the Everett map is that it accurately models the “wasp-waisted” tendency of the loops [19]. The dynamic excess field of (11) is governed by the parameters determined from the statistical loss separation theory. See Figure 4b, for the AC measured and dynamic modeled hysteresis loops for a frequency of50 Hz. It should be remarked that the Everett map is only a valid description for sinusoidal concentric hysteresis loops, for distorted loops an alternative description such as a congruency-based hysteresis model is required [32].

The modeled hysteresis loops are more accurate for lower peak magnetic flux densities, and deviate with increasing magnetization as the excess field is underestimated. For the most outer hysteresis loop a large deviation from the modeled loop is observed. This error is partly due to the measurement setup regulator, which was not capable to keep the applied magnetic field properly controlled, which results in an overshoot of the magnetic field. However, the inherent shape of the excess field, as governed by (11), also contributes to the deviation. Overall, the modeled result closely matches the measured loss, but due to the regulator overshoot the measured loss at high peak magnetic flux densities is considerably larger than what was predicted, see Figure 5a. This trend is also observed for the whole studied range of magnetic flux densities and frequencies, see Figure 5b. The overall error is within a maximum of 7.86 %, and on average about 0.26 %. The same trend is observed for the other two steel grades, see Figure 6. In Table 3 the maximum error , and average error of the applied dynamic hysteresis prediction are given.

thumbnail Fig. 4

(a) Static modeled and measured sinusoidal concentric hysteresis loops of M090-23P for several peak magnetic flux densities, (b) dynamic modeled and measured sinusoidal hysteresis loops of M090-23P for several peak magnetic flux densities at a frequency of 50 Hz.

thumbnail Fig. 5

(a) Modeled and measured dynamic magnetic loss of M090-23P at a frequency of 50 Hz vs. peak magnetic flux density, (b) error characteristic between the measured and modeled dynamic magnetic loss of M090-23P, shown for selected frequencies.

thumbnail Fig. 6

(a) Error characteristic between the measured and modeled dynamic magnetic loss of M100-27P vs. peak magnetic flux density, shown for selected frequencies, (b) error characteristic between the measured and modeled dynamic magnetic loss of M105-30P vs. peak magnetic flux density, shown for selected frequencies.

4 Conclusion

Magnetic loss prediction and hysteresis modeling in the RD of three grades of HiB GOES for magnetic flux densities up to and including 1.9 T and frequencies up to and including 300 Hz has been researched in this work.

It has been concluded that the statistical loss separation theory is capable to accurately describe the magnetic loss of three grades of HiB GOES. In the eddy current loss prediction some error was introduced by the assumption that the skin depth was greater thanthe lamination thickness. Additionally, some error was introduced by the assumption that there are no dynamic eddy current effects when the quasi-static measurements are performed at 1 Hz. However, as the excess loss parameters are determined based on the loss residue, these mismatches are readily compensated, when present. For M090-23P this resulted in a maximum error of the predicted magnetic loss of 7.18 % and an average error of −0.34 %. To improve the eddy current loss prediction, it is recommended to model the flux density distribution in the steel lamination instead of using an average. To improve the measured static hysteresis energy, it is recommended to control the rate of change of the flux density over time, such that the dynamic eddy current effects are minimized and the true static hysteresis energy is obtained.

It has been found that static hysteresis was exactly reproduced by the classical PM that applied a measured Everett map. Additionally, dynamic hysteresis was predicted by two dynamic field components, of which the excess field was governed by the parameters obtained by the statistical loss separation theory. The coupled method is capable to accurately describe the magnetic loss of three grades of HiB GOES for sinusoidal concentric hysteresis loops with a maximum of error of 7.86 % and an average error of 0.26 %. It is inherent to the coupled method that the total magnetic loss was closely predicted. At the same time, the prediction of the modeled excess field remains a large source of error, as well as the regulator overshoot.

Author contribution statement

This work was written by B.D., who performed the measurents and modeling and generated the results. The analysis was carried out in cooperation with T.O., who made contributions and improvements to the work. The work was reviewed by T.O. and E.L. All authors have read and agreed to the published version of the manuscript, they declare no conflict of interest.

Acknowledgements

This paper is part of a project that has received funding from the European Union’s Horizon 2020 research and innovation programme under grant agreement No 769953.

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Cite this article as: Bram Daniels, Timo Overboom, Elena Lomonova, Coupled statistical and dynamic loss prediction of high-permeability grain-oriented electrical steel, Eur. Phys. J. Appl. Phys. 90, 10901 (2020)

All Tables

Table 1

Model parameters per steel grade.

Table 2

Material characterization polynomials per steel grade.

Table 3

Model error of the magnetic loss for the complete operating range per steel grade.

All Figures

thumbnail Fig. 1

(a) The static hysteresis energy of M090-23P as determined by quasi-static measurements vs. peak magnetic flux density, (b) material characterization parameters n0 and V0 determined by linear regression, shown for selected peak magnetic flux densities.

In the text
thumbnail Fig. 2

(a) The evolution of n0 vs. peak magnetic flux density, (b) the evolution of V0 vs. peak magnetic flux density.

In the text
thumbnail Fig. 3

(a) Measured and modeled magnetic loss components for M090-23P for a frequency of 50 Hz vs. peak magnetic flux density, (b) error characteristic between the measured and modeled total magnetic loss, shown for selected frequencies.

In the text
thumbnail Fig. 4

(a) Static modeled and measured sinusoidal concentric hysteresis loops of M090-23P for several peak magnetic flux densities, (b) dynamic modeled and measured sinusoidal hysteresis loops of M090-23P for several peak magnetic flux densities at a frequency of 50 Hz.

In the text
thumbnail Fig. 5

(a) Modeled and measured dynamic magnetic loss of M090-23P at a frequency of 50 Hz vs. peak magnetic flux density, (b) error characteristic between the measured and modeled dynamic magnetic loss of M090-23P, shown for selected frequencies.

In the text
thumbnail Fig. 6

(a) Error characteristic between the measured and modeled dynamic magnetic loss of M100-27P vs. peak magnetic flux density, shown for selected frequencies, (b) error characteristic between the measured and modeled dynamic magnetic loss of M105-30P vs. peak magnetic flux density, shown for selected frequencies.

In the text

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