Relevance of microstructure and texture to the accuracy and interpretation of 1 and 2 directional characterisation and testing of grain-oriented electrical steels

Abstract

This paper is intended to stimulate discussion of some effects of not properly accounting for material microstructure in some established methods of measurement and prediction of losses in electrical steel laminations. Aspects of methods which have been used for many years are briefly discussed and some constraints or pitfalls are raised which users should be aware of. The basic cause of losses is summarised in order to set the scene for discussion of the analysis of losses into classical eddy current, hysteresis and anomalous loss. The possible presence and consequences of transverse flux in grain-oriented (GO) steels is raised followed by an explanation of some effects of the strong anisotropy of GO steel on flux density, magnetic field and loss measurement in stacks and single strips of GO steel. The presentation concludes with some questions on how circular rotational magnetisation can be made and precautions needed when assessing the effect of flux harmonics on the losses.

Keywords

Electrical steel losses grain-oriented silicon iron anisotropy

1. Introduction

Many methods of magnetic measurement and analysis or prediction of the performance of electrical machine cores are directly or indirectly related to the texture and size of grains in the steels. Furthermore, the fundamental reorganisation of domains during magnetisation can also have a significant effect. These parameters are rarely included in material or core performance prediction models.

The effect of not accounting for these factors is not well quantified. It is always assumed that their effects are small and this is often the case. However when magnetised at high flux density or frequency or when flux harmonics are present, their influence might be greater than expected.

The purpose of this contribution is to show and explain why caution is necessary in this area particularly when dealing with grain-oriented (GO) steel. The influences of its large grains and strong magnetocrystalline anisotropy which strongly control its unique domain structure is described in the context of how it might affect the interpretation of some types of measurements or analysis. Obviously the fundamental principles of magnetism apply on a magnetic domain scale but there is no reason we can simply extrapolate what we believe happens at this level to practical measurements and analysis of material properties of engineering relevance.

Specific topics to be discussed include (a) the presence of a component of loss in GO steel due to transverse flux which is no detected in the Epstein frame measurement or when using the Single Sheet Tester (SST) (b) whether the instantaneous magnetisation can really be made to be perfectly constant in magnitude and rotate at constant velocity needed for rotational loss measurement (c) the influence of texture on the magnitude and direction of the localised flux density $B(t)$ and $H(t)$ (d) the relevance of the stacking method in Epstein testing of strips cut at angles to the rolling direction (RD) (e) the need to take domain magnetisation more fully into account to improve the range of application and reliability of loss analysis in terms of eddy current and hysteresis components and (f) cautions necessary when assessing losses under severe harmonic flux conditions.

Some of the above phenomena affect the accuracy of routine magnetic measurements and can contribute to the building factors of transformer cores. Others affect the way in which we understand and make use of magnetisation processes in magnetic measurements. The practical relevance and importance of most of these factors are probably generally low. However, this paper is intended to increase awareness that established methods of analyses are not always based on solid foundations and sometimes not fully relevant to today’s increasing demands for accurate magnetic characterisation over wide ranges of magnetisation conditions. Too often users of magnetic test systems are not aware of the limitations or assumptions which have to be made when interpreting their results. It is hope that some of the content of this paper will stimulate the consideration of more knowledge-based characterisation of GO steel, in particular, taking account of the influence of its unique microstructure.

2. Cause, analysis and prediction of losses

The fundamental, well established, relationships between components of surface magnetic field $H(t)$ and flux density $B(t)$ are made use of in several methods of measuring or predicting the magnetic losses of GO steel. These are based on the fundamental physics of magnetism and magnetic circuit analysis including Maxwell’s equations and Ampère’s law. Examples where these principles are used include processing of signals from dB(t)/dt and $H(t)$ sensors to obtain $B$ - $H$ curves or losses, analysing losses in terms of hysteresis and eddy current components and treating losses under distorted magnetisation conditions as the sum of a series of harmonic components under both one or two dimensional magnetisation conditions.

Figure 1.

Variation of hysteresis, classical eddy current and excess loss per cycle with frequency in a typical GO steel.

Figure 1 shows the traditional way of separating the loss per cycle of magnetisation in electrical steel into three components. The DC hysteresis loss per cycle is assumed constant. When added to the calculated classical loss per cycle, the sum is always less than the total measured value so a third component, initially called the anomalous loss but now more often called the excess loss, is added to account for this difference. It was assumed that this loss component is present due to domain walls movement at power frequencies through the material or to general reconfiguration of domain structures both causing additional eddy current and hysteresis losses. If the flux density varies sinusoidally with time, the total loss is often expressed in terms of the hysteresis, classical eddy current and excess components in forms such as

$\displaystyle W_{tot}=w_{hys}+\frac{d^{2}\pi^{2}B^{2}_{m}}{6\rho}f+k_{exc}B_{m% }^{1.5}f^{0.5}$ (1)

where $d$ is the sheet thickness, $B_{m}$ is the peak flux density, $\rho$ is the resistivity, $f$ is the magnetising frequency and $k_{exc}$ is a constant, sometimes quantified to help make the sum of the three components equal to the measured loss of a material.

The approach is useful but there are pitfalls which users should be aware of. Firstly, it is well known that the number of domain walls in motion under dynamic magnetisation changes with $f$ . Hysteresis is caused when micro eddy currents are formed as each moving domain wall is impeded by material defects (such as inclusions, dislocations etc.) as it moves through the material due to some external influence such as an external magnetic field or mechanical stress. Hence there is no justification for assuming that hysteresis loss produced in this way at power frequency should be equal in magnitude to the static hysteresis loss, $W_{hys}$ . The fact that the true hysteresis probably increases with increasing frequency was pointed out many years ago in a widely respected text book [1] but it seems to haves been mainly ignored or forgotten. The number of domain walls and the distance they move during a magnetising cycle vary with magnetising frequency hence there is no physical justification for including $W_{hys}$ in Eq. (1) [2]. It is sometimes argued that the “extra” hysteresis loss at power frequency is included in the excess loss term and called dynamic hysteresis so including $W_{hys}$ is actually valid. However, this might keep the equation simple and by giving $k_{exc}$ an appropriate value the calculated loss can agree with experimental measurements but the fundamental argument remains that it is not compatible with known domain magnetisation processes.

The second point to note in the use of the separation method is that the classical eddy current loss is derived and calculated based on assumptions which are gross approximations. It was found [3] that good agreement occurred provided relative permeability $\mu_{r}$ is less than 5000 and the magnetising frequency is less than 100 Hz. Quantitative accuracy is poor for thin or high permeability materials, high flux density or when flux harmonics are present. This particularly applies to GO steel but it should be noted that even in non-oriented (NO) electrical steel the true classical loss $W_{clas}$ can vary by $-$ 30% to $+$ 80% of the loss calculated from Eq. (1) [4]. Such results add to the argument that the separation between excess and classical losses in GO is somewhat artificial and not based on any physical foundation [5].

There are many approaches to predicting B-H loop shapes and losses. These include well known Preisach, Jiles-Atherton, vector and dynamic hysteresis and statistical models. Most require DC characteristics, thickness, resistivity, plus at least one measured ac loss value in order to model B-H characteristics and hence losses. However they all ignore micro-magnetic behaviour which causes poor reliability especially for thin material or magnetisation extremes. In many methods, it is basically necessary to solve the following well known diffusion equation under dynamic conditions for unidirectional magnetisation.

$\displaystyle\sigma\frac{\partial B}{\partial t}=\frac{\partial^{2}H}{\partial x% ^{2}}$ (2)

where $\sigma$ is the material conductivity, $B$ and $H$ are values of flux density and field at depth $x$ in a lamination This is again based on Maxwell’s equations describing the relationship between $B$ and $H$ in a homogenous magnetic material containing no grains or microstructure of any sort and certainly no magnetic domains. It should also be noted that the equation was originally derived and solved based on experimental observations and deductions made before the concept of magnetic domains was even proposed and it invariable gives lower values of loss than measured values. Equations based on Eq. (2) are valuable but indiscriminate use for large grain size, highly anisotropic material, such as GO steel, might result in unquantified errors.

Figure 2.

(a) Typical field variation on the surface of a sheet of GO steel in a demagnetised state. (b) Magnetisation conditions in a single grain with magnetisation $M$ directed along the [001] direction showing the resultant surface field aligned in a different direction to the external field due to the demagnetising field. $\theta$ is the angle between the RD and the [001] direction of the grain.

3. Transverse flux in GO steels

It is well known that the localised magnetic field on the surface of GO steel varies in magnitude and direction particularly from grain to grain such as shown in Fig. 2a [6]. It used to be thought that there might be direct correlation between the surface field variation and the orientation of individual grains but now it appears more likely that it depends also on the demagnetisation factors of individual grains as shown in Fig. 2b. From Eq. (1) the following commonly used expression can be obtained to calculate the total loss under two dimensional (2-D) magnetisation

$\displaystyle P=\frac{1}{\rho T}\int^{T}_{0}\left(H_{x}\frac{dB_{x}}{dt}+H_{y}% \frac{dB_{y}}{dt}\right)dt$ (3)

where $T$ is the period of magnetisation and $x$ and $y$ refer to orthogonal directions of components of field and flux density. It is clear from Fig. 2 that there can be components of $H$ and dB/dt in individual grains which would not be detected by sensors used for measuring unidirectional magnetisation along the external field direction. The effect in a well oriented GO steel is shown in Fig. 3 [7]. The magnitude of the loss due to the transverse component of magnetisation is not very significant but at the same time around 5% of the true total loss is not included in the measured value.

Figure 3.

Variation of the loss with flux density along the RD due to transverse components of flux density in a strip of commercial GO steel magnetised along its RD.

The effect of the transverse component is more apparent in a laboratory produced, very poorly oriented steel as shown in Fig. 4 [8]. When the overall flux density along the RD is 1.10 T, the longitudinal component shown in Fig. 4b varies between 0.60 T and 1.30 T and the transverse component shown in Fig. 4c lies within a range between 0.05 T and 0.30 T producing a significant component of loss.

Figure 4.

Longitudinal and transverse flux density components in a laboratory produced strip of poorly oriented GO steel. (a) Grain and static domain structure. (b) Distribution of longitudinal flux density. (c) Transverse flux density.

It is also found that when the same strip of poorly oriented steel is magnetised in a DC field that domains in different grains are activated at different field levels. For example domains in grain 8 begin to contribute to the overall magnetisation at around 0.7 T whereas domains in grain 1 do not begin to contribute until 1.2 T [8].

It may be argued that the loss due to the transverse components of field and flux density is included in the excess loss component in Eq. (1) but clearly it is not quantified or its physical nature is not accounted for. The diffusion equation applies within individual grains but its true effect and implementation is not understood. It can be further argued that equations such as Eq. (3) commonly used to solve 2-D magnetisation problems are somewhat different from the diffusion equation hence subject to greater errors. If this is the case our understanding of the basic magnetisation and loss mechanisms in GO steel is still superficial and perhaps wrong.

An implication of these transverse components in loss measurements is that the use of field sensors and loss computations based on the diffusion equation should be reviewed. Even if there effects are negligible under normal circumstances, their existence should be recognised. No steels have far more complex domain structures than GO steel so it is possible that transverse components can occur in a similar way within the individual grains. The extent to which this might affect measurements on both types of material using dB/dt and $H$ sensors might be clarified by accurate, direct comparisons of losses measured using sensors and using an absolute thermal method

Figure 5.

(a) Two ways of assembling strips cut at an angle to the RD in the Epstein frame. (b) Variation of loss measured in strips cut at various angles to the RD using different stacking methods at 0.8 T, 1.2 T and 1.6 T.

4. Influence of anisotropy of GO steel on B, H and losses in stacks and strips

There is often a call for measurement of losses in GO steel under unidirectional magnetisation at a fixed angle to the RD. It is not widely appreciated that the method in which strips are stacked in an Epstein frame to carry out such measurements affects the result. Figure 5 shows the result of testing strips cut at angles to the RD in an Epstein frame [9]. In practice the effect is found to be greatest in high permeability steels. The loss using the (a) type stacking gives results closest to values measured on single strips whereas the (c) type stacking gives results closest to completely random stacking. The reason for the difference in measured loss using the different stacking methods is probably partly due to the fact that demagnetising fields referred to earlier can affect the magnitude and direction of localised $B$ and $H$ differently in the two stacking methods and partly due to different transverse flux paths in the parallel and cross cut stacking assemblies [10].

Figure 6.

Local deviation of magnetic field and flux density from the axis of a strip of GO steel cut at 45 ${}^{\circ}$ to the RD at low (left) and higher (right) magnetisation. ( $H_{1}$ and $H_{2}$ are orthogonal components of field along and at 90 ${}^{\circ}$ to the RD and $H_{r}$ is the resultant. $B_{1}$ , $B_{2}$ and $B_{r}$ are corresponding values of flux density).

When single strips of GO steel are cut and magnetised at angles to the RD, magnetisation occurs based on the process depicted in Fig. 2b. The strip is cut at angle $\theta$ to the RD so if it is assumed that the material is well oriented then the main component of magnetisation (apart from in a very high field) is along the [001] directions of the grains producing the demagnetising field as shown. The measured flux density is effectively along the strip axis. Figure 6 shows some practical results from measurements on 300 mm long by 100 mm wide strip of GO steel cut at 45 ${}^{\circ}$ to the RD [11].

Clearly when magnetising single strips of GO steel cut at angles to the RD, the direction and magnitude of the flux density and field varies in a complex manner according the nominal flux density so loss measurement needs careful interpretation. Furthermore the results of such tests are likely to vary with the geometrical aspect ratio of the strips. Conflicting findings on this aspect of the measurements have been reported so further clarification and quantification of the mechanism is needed. It is fortunate that the effect is expected to become far less significant at high flux density but the author is not aware of any published experimental or other verification that this is the case.

5. Production of circular rotational flux density in laboratory experiments

Circular (or pure) rotational flux density in a sheet of electrical steel refers simply to the situation when the flux is of constant magnitude rotates at a constant angular velocity in the plane of a sheet. To obtain this condition in the laboratory, often a sheet of electrical steel is magnetised using orthogonal magnetising coils in which the magnitude and phase of the currents are controlled such that the voltage outputs. dB(t)/dt , of orthogonal search coils vary sinusoidally with time, have the same peak value and are 90 ${}^{\circ}$ out of phase. In simple mathematical terms, this is equivalent to a vector of constant magnitude rotating at constant velocity.

Figure 7.

Domain structure on the surface of a sheet of GO steel subjected to a slow (DC) rotational magnetic field when the flux density is at (a) 0 ${}^{\circ}$ (b) 25 ${}^{\circ}$ (c) 28 ${}^{\circ}$ (d) 31 ${}^{\circ}$ (e) 34 ${}^{\circ}$ and (f) 37 ${}^{\circ}$ to the RD.

Figure 8.

Simple vector representation of clockwise rotational magnetisation in a perfectly oriented crystal of GO at instances when flux density is along (a) [001] (OA) (b) [110] (OC) and (c) at angle $\theta$ to [001] showing contributions Ba and Bc to the total flux density OC.

Such circular magnetisation is usually set up as a convenient reference for comparing rotational losses of materials under known magnetising conditions. However it is questionable whether such conditions are actually achievable in practice in GO steel even when the voltage outputs of search coils satisfy the condition given in the previous paragraph. because of its complex domain structure and large magnetocrystalline anisotropy. Figure 7 [12] gives an indication of the complex changes in the domain structure observed on the surface of a grain in sheet of GO steel when subjected to a slowing rotating (DC) field. At the instant when the field is applied along the RD, typical slab domains are present directed along the [001] axis of the grain as expected. (The curved line in the photograph is part of a search coil which should be ignored). When the flux density has rotated to 25 ${}^{\circ}$ to the RD, small closure domains appear and at 31 ${}^{\circ}$ to the RD the surface is mainly covered by these domains. From this we can deduce that the majority of the magnetisation is caused by growth of domains oriented along the [001] crystal axis closest to the RD while the rotating field forces the resultant flux density to rotate from 0 ${}^{\circ}$ to about 25 ${}^{\circ}$ to the RD. When the flux density is directed at much larger angles to the [001] direction, internal transverse domains are present oriented along [010] and [100] directions. The appearance of the surface closure domains indicates this process occurs which is very similar to the process occurring under compressive stress in GO steel.

A schematic vector representation of the rotational process is given in Fig. 8. If the field is rotating slowly in a clockwise direction, initially the flux density OA is along the [001] direction of the crystal and the driving field $H_{a}$ is along the same direction. The flux density at this instant is due to domain growth along that direction. When the field has rotated to drag the flux to lie at $\theta$ to the [001] direction then the flux density is made of two components, Oa and Oc, due to domain growth in the [001] and [010] directions respectively. It should be noted that no domains are oriented at $\theta$ to the [001] direction despite this being the direction of the resultant flux density. It should also be noted that resultant B and driving H are not co-linear and the angle between them varies throughout one cycle of rotation.

Hence it can be seen from these domain structures that when the measured resultant flux density is at an angle $\theta$ to the RD of the strip the actual flux within the steel is still mainly along the RD and transverse direction. The flux does not physically smoothly rotate in the way that the vector representation implies. Unpublished work by the author showed that the e.m.f.s in search coils placed at 30 ${}^{\circ}$ angles do not all have sinusoidal outputs when the field is controlled to force the e.m.f.s in an orthogonal pair of the coils are forced to satisfy the conditions for circular magnetisation. This suggests that magnetisation is not perfectly circular due to the complex domain magnetisation process. It is not known whether this produces any significant deviation from pure rotational flux.

It is useful to briefly consider a frequently quoted basic expression for the rotational hysteresis loss in a sheet given as

$\displaystyle P_{r}=\int^{2\pi}_{0}B_{r}H_{r}\sin\alpha d\theta$ (4)

where $\alpha$ is the instantaneous angle between $B_{r}$ and $H_{r}$ . Equation (3) is itself derived from Eq. (4) which in turn is derived from the torque exerted on a sample subjected to a rotating field where all domains contribution to the magnetisation are oriented parallel to that of $B_{r}$ . This clearly does not occur in GO material so the absolute validity of such expressions is not certain.

6. Loss measurement and prediction under highly distorted flux density

Electrical steels are being increasingly used under severe flux harmonic distortion conditions. Waveforms containing up to the 50 ${}^{\rm th}$ harmonic are found in industrial applications of GO steel today. Such distortion can increase losses by orders of magnitude so it is important to be able to accurately measure or predict the effect. Expressions such as Eq. (5) have been used with reasonable success to predict or measure losses

$\displaystyle P_{\textit{total}}=0.5\Sigma\omega_{n}B_{n}H_{n}\sin\theta_{n}$ (5)

where where $B_{n}$ and $H_{n}$ are the peak values of the $n^{th}$ harmonic components of flux density and magnetic field, $\theta_{n}$ is the phase angle between them and $\omega_{n}$ is the angular frequency of the $n^{th}$ harmonic. This is reasonably accurate for low distortion but the summation of this sort becomes inaccurate as harmonic levels increase because of the severe non-linearity of the B-H relationship in GO steel.

Today the more common way of analysis is simply to sum the instantaneous values of $H$ and dB/dt over a magnetising cycle. However it is uncertain whether high frequency flux harmonics are effectively equivalent to an oscillation of particular parts of domain walls whose overall distribution is not greatly affected or manifests itself as a form of skin effect which could have a very complex effect on the domain walls. Whatever the case may be, it seems possible that under high distortion even processing from instantaneous values of $H$ and dB/dt breaks down.

7. Summing up

This paper covers the presentation made at the 1 and 2 DM Workshop with the intention of provoking and stimulating discussion of the influence of microstructure in GO steel on some established measurement and performance prediction methods based on ideal magnetic theory. In practice microstructure and domain dynamics do have a large influence on losses particularly under rotational conditions and their effects are neglected or not properly quantified in many traditional approaches which cannot deal with modern material and the wide ranges of magnetisation conditions they experience in practice today.

The effect of ignoring them has not been quantified, it might be trivial in some cases, but unexpectedly large in others. However at present we do not have any better approach so we still need to attempt solutions based on Maxwell’s equations adapted to suit GO steel where possible, but it is important to be more aware of the possible errors this brings.

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