Dynamic magnetization models for soft ferromagnetic materials with coarse and fine domain structures

https://doi.org/10.1016/j.jmmm.2015.06.082Get rights and content

Highlights

  • Critical analysis of a ferromagnetic-material loss-separation principle.

  • This is to warn materials-science engineers about the inaccuracies resulting from this principle.

  • A transient model having a single dynamic component is proposed.

Abstract

We consider dynamic models, both numerical and analytical, that reproduce the magnetization field H(B) and the energy loss in ferromagnetic sheet materials with different domain structures. Conventional non-oriented (NO) and grain-oriented (GO) electrical steels are chosen as typical representatives of fine-domain and coarse-domain materials. The commonly-accepted loss separation procedures in these materials are critically analyzed. The use of a well-known simplified (“classical”) expression for the eddy-current loss is identified as the primary source of mistaken evaluations of excess loss in NO steel, in which the loss components can only be evaluated using the Maxwell (penetration) equation. The situation is quite different in GO steel, in which the loss separation is uncertain, but the total dynamic loss is several times higher than that explained by any version (numerical or analytical) of the classical approach. To illustrate the uncertainty of the loss separation in GO steel, we show that the magnetization field, and thus the total loss, in this material can be represented with equal accuracy using either the existing three-component approach or our proposed two-component technique, which makes no distinction between classical eddy-current and excess fields and losses.

Introduction

The reliable estimation of energy loss in ferromagnetic sheet materials remains an important and much-discussed problem, both theoretically and practically. It is crucial for designers of electrical machines and transformers, as well as for materials-science engineers engaged in the optimization of morphological properties of electrical steels such as texture and grain size. Although these discussions are largely due to an incomplete understanding of the underlying physics, many are the result of making no distinction between ferromagnetic materials with dissimilar domain structures, i.e., dynamic correlation sizes. Typical examples of such significantly different materials include the conventional non-oriented (NO) and grain-oriented (GO) electrical steels, as considered in this paper.

Because of its small magnetic anisotropy, NO steel, also called dynamo steel, is commonly used in electrical motors and generators, which are applications where the magnetic flux changes direction in the plane of the sheet. Due to its spatially random domain structure and small grain size, NO steel approaches a homogeneous isotropic material, in which the magnetization dynamics can be described, to a good first approximation, by classical Maxwell equations [1], [2], [3].

The situation is quite different in anisotropic GO steel, which is the major core material for large transformers, reactors, and other devices in which the flux is predominantly unidirectional. Soon after its invention by Goss [4] in 1934 and its industrialization by ARMCO in the 1940s, it was recognized that the total loss in this material is anomalously higher than could be explained using the classical approach, even when using an accurate hysteresis model to link the magnetic field H and induction B, when solving the appropriate Maxwell's equations. Although the problem of the anomalous (or excess) loss has been the focus of intense research for several decades [5], [6], its resolution remains almost unaltered in the sense that no reliable deterministic loss theory has emerged [7], [8]. For this reason, a statistical loss model developed in the 1980s [7], [9] is widely used as an engineering alternative. The distinguishing feature of this phenomenological (semi-empirical) approach is the loss separation principle whereby, irrespective of material domain structure, the total loss W (in J/m3 per cycle) is decomposed into hysteresis loss Whyst, classical loss Wclas, and excess loss WexcW=Whyst+Wclas+Wexc.

Although misconceptions in the loss separation approach have been reported in the literature [8], [10], [11], and there is no room for Wclas in numerous physical models of coarse-domain materials (fairly complete reviews can be found in Refs. [5], [12], [13], [14] this three-term loss representation has found a wide range of applications due to its simplicity and functionality. Besides, the loss separation is definitely possible in fine-domain materials (NO steels) if, of course, an appropriate numerical tool [2], [3], [15] is employed. Unfortunately, this is seldom the case, and frequently no distinction is made between GO and NO steels in the engineering literature [16], [17], [18]. This has led to an admixture of separate notions relating to different materials. Here we attempt to analyze both these different materials in a single paper.

A first theme of the paper is to point out the dangers concealed in the inappropriate application of loss separation to NO steels. These result from an uncritical use of the well-known approximation for the classical eddy-current loss [5], [7], [9]Wclas=σ(dπBp)26f

This is derived by assuming a uniform sinusoidal induction, with peak value Bp and frequency f, in a magnetically linear ferromagnetic sheet of thickness d and conductivity σ.

The widespread use of Eq. (2) in the literature has often led to distorted pictures of real processes, and has resulted in numerous “correction” techniques [19], [20], [21], often using the ideas of skin-effect and skin depth, which lose their original meanings being applied to magnetically nonlinear media [22].

A separate theme of this paper is to demonstrate that, in the case of GO steels, while retaining the framework of the phenomenological approach, dynamic hysteresis loops, and hence total losses, can be reproduced quite accurately without splitting the dynamic loss into “classical” and “excess” components. This result highlights the uncertainty inherent in the idea of separating out the dynamic loss components [14] and serves to partly reconcile its followers and opponents.

In Section 2, we consider the erroneous but widely held view that dynamic losses in electrical steels of any type, such as GO and NO, can be described by means of diffusion-like equations or their circuit equivalents (Cauer networks). This idea has endured for a long time, starting with Refs. [23], [24] and reaching recent works [25], [26], [27]. Typical errors encountered in both using and ignoring the diffusion (penetration) equations are considered in Section 3. In Section 4 we briefly discuss the physical validity of the loss separation principle (1), as applied to GO steels, and propose alternative versions of the field- (and loss-) separation, having somewhat different underlying concepts, but the same abilities as the transient models [28] that are based on Eq. (1).

Section snippets

Classical loss models

In the case of a thin steel strip (with d«l in Fig. 1), the classical approach reduces to the solution of the one-dimensional penetration equation [2], [3]σBt=2Hx2

This links the magnetic field Hz(x, t) to the magnetic induction Bz(x, t), both directed along the z-axis of a strip with conductivity σ, the x-axis being normal to the strip surface. In posing this problem, all eddy currents are considered y-directed, that is, edge effects are neglected. Only two of their innumerable contours are

Typical errors in evaluating excess loss in magnetic materials with fine domain structure

To analyze the errors arising from simplifying assumptions about excess loss, we can rely on the results obtained with finite-difference solver [3] or finite-element solvers [2], [15] of Eq. (3). The successful application of the FDS to different NO steels [3], [31] operating in a wide frequency range under arbitrary excitations (sinusoidal or non-sinusoidal) confirms its physical validity and allows us to consider the FDS as a reliable tool for evaluating both total loss and its individual

Domain loss models

Any uncertainties about the existence of excess loss in electrical steels [20], [27] disappear when the “classical tools” (3) or (4) are applied to GO steel. This can be seen in Figs. 3 and 6 which show that dynamic loops constructed with these equations have substantially smaller areas than measured loop 1. The GO steel represented in Fig. 3 (it is similar to steel M4 and steel 27Z130 [43]) and the high-permeability grain-oriented (HGO) steel in Fig. 6 will be referred to as Steel 1 (d=0.255 

Concluding remarks

In this paper, we have tried to outline the situation with loss separation in soft ferromagnetic materials, which is a subject of continuing debate among physicists and engineers. Due to difficulties in their precise classification, we have chosen conventional non-oriented and grain-oriented steels as very different materials in the sense of their domain structures and their different domain sizes with respect to the thickness of the sheet.

We have emphasized that the penetration (diffusion)

Acknowledgments

R. Harrison thanks the Natural Sciences and Engineering Research Council of Canada for support under Discovery Grant 1340-2005 RGPIN. The work of S. Steentjes supported by the Deutsche Forschungsgemeinschaft (DFG) and carried out in the research project “Improved modeling and characterization of ferromagnetic materials and their losses”.

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