A Coenergy-Based Model
of Anisotropic Magnetic Properties
K. Chwastek, J. Szczygłowski1, W. Wilczyński2
1Czestochowa University of Technology, Czestochowa, Poland
2Institute of Electrical Engineering, Warsaw, Poland
ABSTRACT
Anisotropy affects magnetic properties of steels used in electrical engineering. Designers of magnetic circuits require proper modelling tools to take this phenomenon into account. The concept of coenergy offers an approach, which is thermodynamically coherent and simple to be implemented numerically. The model advanced recently by Biró et al.
is applied to describe anisotropic dependencies at arbitrary angles for grain-oriented steel, grade 097-30N5. The modelled dependencies are in an acceptable agreement with experimental data for engineering purposes. The description makes it possible to predict the magnetization curves at arbitrary angles using a limited number
of measurements. It allows to cut down time and efforts related to sample processing for measurements.
Introduction
The concept of coenergy [1,2] has been recently the subject of renewed interest in the scientific community. Many useful descriptions of complex mechatronic devices, e.g. those based on bond graph theory [3], can be easily developed using this formalism [4,5].
The great potential behind the coenergy-based models (their coherence with the principles of thermodynamics) has been noticed by B. Cornut [6]. A practical approach to describe the anisotropic magnetic properties in soft magnetic materials has been advanced by T. Péra et al. [7]. Recently, the description has been further extended by O. Biró et al. [8]. Their proposal is the subject of study in the present paper.
Even the so-called isotropic steels used in electrical engineering can reveal a significant anisotropy [6,9-11]. One of the indicators in common use is the ratio
where the symbols denote loss density measured either in the rolling direction or in the transverse one, respectively, which is determined in quasi-static conditions at T. For example, the value of for the isotropic electrical steel used for alternators, grade M 330-35A (basic properties: maximum loss density at 1.5 T, 50 Hz – 3.30 W/kg, thickness 0.35 mm, minimum flux density for A/m is 1.49 T, whereas for A/m is 1.6 T) may reach 17 %, as declared by the producer.
Yet, the true values obtained in practical measurements are smaller, about 11 %, for this grade [11].
Other macroscopic anisotropy measures in common use are defined as the differences and , where is arbitrary angle (usually either or ), denotes magnetic induction measured at A/m, whereas denotes magnetic field strength measured at T. The graphs like the one depicted in Fig. 1 are also useful for the designers of magnetic circuits.
Fig. 1. Induction anisotropy for electrical steel, grade 097-30N5.
Source: own measurements
The effect of anisotropy is yet more pronounced for anisotropic steels used in transformers. It is directly related to grain orientation within magnetic material, cf. Fig. 2.
Fig. 2. Magnetic domains in grain-oriented electrical steel as seen in domain viewer. The arrow shows rolling direction of the steel. Black curves outline the grain structure. Source: Wikimedia, Creative Commons license, authors: Stanislaw Zurek,
Harshad V. Patel, Wolfson Centre for Magnetics, Cardiff
The knowledge of directional magnetic properties of electrical steel allows us to tailor and optimize the operation point of magnetic circuits. On the other hand, it is a tedious and laborous task to prepare samples for magnetic measurements, which are cut at different angles from the rolling direction. The deteriorating influence of processing technologies on magnetic properties of electrical steel has been examined in detail by one of the co-authors of the present paper [12, 13].
a) b)
c)d)
Fig. 3. Microstructure of a stator tooth in an electric machine: a) highly deformed edge of the tooth, b) middle part of the tooth, c) at the edge after classical thermal treatment of the whole stator sheet pack, d) at the edge after induction treatment of the tooth-slot zone [12]
Taking the abovegiven into account, it is thus desirable to focus on descriptions, which make it possible to predict the directional dependencies between the vectors: magnetic field strength and magnetic flux density for arbitrary angles, from a limited number of magnetic measurements, carried out for well-specified directions, i.e. for the rolling and the transverse one. The solution is to avail of the coenergy-based approach. On the one hand, it seems much easier to work with scalar coenergy-based relationships than to consider vector dependencies. On the other one, some approaches advanced in literature, e.g. those based on linear interpolation between many data, are incorrect from the point of thermodynamics, as pointed out by B. Cornut [6]. Some first-order derivatives of induction and field are in fact second-order derivatives of a state variable and should be equal, what is neglected by some model developers. The coenergy-based models, like the one described in Ref. [14], are thermodynamically coherent, what makes them useful and important.
Model description
Manufacturers of electrical steel usually provide the dependencies between the amplitudes just for two orthogonal directions: when the samples are subjected to magnetization along the rolling direction and for the transverse orientation. For these directions the vectors and are collinear. The measurements in these two directions provide the basis for formulation of the model, which is capable to predict for intermediate directions.
Whereas the best magnetization conditions are obtained for the rolling direction (angle , cf. Fig. 4, the opposite is not true for the transverse direction. Actually, the worst magnetization conditions are obtained for the angle
Fig. 4. The considered geometry. Index I =1, 2 denotes the rolling
and the transverse direction, respectively.
In order to describe the contours traced by flux density vector for any fixed value of , Biró et al. [8] have proposed to use the relationship
,(1)
where are unit vectors, whereas and denote nonlinear scalar dependencies in the rolling (i = 1) and the transverse direction (i = 2) , respectively, whereas n is a model parameter. Biró et al. [8] suggest assuming
The relationship (1) has to be supplemented with another one, which determines the angle . In the aforementioned Reference it is assumed, that static permeability in the rolling direction is solely the function of flux density in that direction, what allows us to write
(2)
or, in equivalent form .
The international standard IEC 60404 prescribes the conditions for magnetic measurements, e.g. the shape of flux density waveform should be controlled to be sine. This is achieved by the use of additional feedbacks in measurement setups.
The choice of flux density as the input variable in the developed model is a logical consequence of the aforementioned regulations. The sought dependencies are for given angle. The design of measurement setup implies
Modelling
Measurements were carried out for samples of anistropic steel, grade 097-30-N5 using the Epstein device. The basic properties of the examined steel are given in Table I.
Table 1 Basic magnetic properties of the examined steel
Grade / Thickness / Maximum loss density@ 1,7 T, 50 Hz / Minimum polarization J
@ 800 A/m
097-30-N5 / 0,30 mm / 1,50 / 1,75
The samples were cut into stripes 3050,5 mm 300,2 mm at different angles from the rolling direction. The measurements were carried out in accordance with the international standard IEC 404 – 2 part 2 1996 – 03: Methods of measurement of the magnetic properties of electrical steel sheet and strip by means of an Epstein frame.
The coenergy-based model was implemented in a popular spreadsheet software. In order to interpolate the measurement data for two orthogonal angles
the freeware add-on [15] was found useful. The exponent n was kept fixed to 1.4.
The results of modelling the dependencies for chosen values of angle are depicted in Figure 5. The values of angle varied in the range for the depicted cases and were lower for lower values of flux density B and angle
Fig. 5. The results of modelling for grain-oriented steel 097-30-N5.
On the basis of the presented results it can be stated, that the coenergy-based model is able to reproduce the directional dependencies in anisotropic steel reasonably well. The discrepancies between the model and experimental data, in particular for could be the result of arbitrary choice of the exponent n = 1.4.
Conclusions
A simple description of anisotropic magnetic properties of electrical steel based on coenergy concept has been presented. The model makes it possible to predict the dependencies for arbitrary values of angle on the basis of a limited number of measurements, carried out in two orthogonal directions. It can be easily implemented in a popular spreadsheet software. The model reproduces correctly the typical behaviour of anisotropic steel – the worst magnetic properties are obtained for closeto
The proposed description is easier in implementation than the existing models of rotational losses. It also allows us to avoid troublesome measurements for intermediate directions of samples.
References
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