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Loss Separation in Nonoriented Electrical Steels Sergey E. Zirka1 , Yuriy I. Moroz1 , Philip Marketos2 , and Anthony J. Moses2 Department of Physics and Technology, Dnepropetrovsk National University, Dnepropetrovsk 49050, Ukraine Wolfson Centre for Magnetics, School of Engineering, Cardiff University, Cardiff CF24 3AA, U.K. The accuracy of conventional loss separation technique as applied to nonoriented electrical steel is evaluated. It is shown that the main reason of inaccuracy of the loss separation carried out analytically is the use of simplified equation for the classical eddy-current loss originated from the theory developed for a linear magnetic media. The error of the order of 10% in classical loss at 1.5 T results in errors of 25 and 50% in excess loss at 50 and 100 Hz, respectively. Index Terms—Classical loss, loss separation, saturation wave model, skin effect.
I. INTRODUCTION
I
T IS important for metallurgists working on magnetic properties of electrical steels and engineers involved in the loss , into components prediction to subdivide the total loss, called hysteresis loss, , classical eddy-current loss, , . The problem considered in the paper and excess loss, arises from the fact that there are no methods for measuring these components individually at a given frequency , and their values are usually estimated by approximate analytical expressions [1], [2]. For example, in the case of sinusoidal magnetic , the value of (in per induction of peak value cycle) is expressed as
(1) where is the strip thickness and is the electrical resistivity. is identified with the static loss , The contribution of or measured under found by extrapolating the total loss to almost static conditions (at millihertz frequencies in modern is found as systems). Finally, the value of (2) Here,
is the excess loss found by the subtraction, and designates hysteresis loss at (so-called static loss). It is usually neglected that (1) originates from the theory [3] developed for a ferromagnetic medium with a linear dependence and is valid in the range of magnetization frequencies where skin-effect is negligible. Neither of these conditions is met in reality, so the use of (1) can lead to doubtful quantitative estimates of the classical loss and, especially excess loss, found by the subtraction technique. In this study attention is confined to nonoriented (NO) electrical steel. It focuses mainly on the classical loss due to its domplus ) inant contribution to the whole dynamic loss (
that can lead to large relative errors in even for moderate . The choice of NO steel is also made because its errors in magnetic domains are much smaller than the sheet thickness. This allows the processes in NO laminations to be described by the magnetodynamic model (MDM) [4] which is a solver of the classical Maxwell equation
(3) combined with a dynamic hysteresis model. In this paper, the MDM [4] is applied to NO steel ( m, mm) for which its error in the total loss prediction is within 2% of values measured in the frequency range of 50 to 600 Hz by means of a versatile Epstein frame [5]. This fact along with the ability of the MDM to reproduce the shapes of dynamic hysteresis loops under sinusoidal and nonsinusoidal flux densities makes the model a reliable source of ref, , erence data for total loss and its three components, (superscript designates that these values are found and numerically). Using this model we corroborate and explain the can differ from fact ([6], [7]) that eddy-current loss calculated through (1). The ability of the MDM [4] to evaluate and individually (in the Preisach type models [6], [7] they are inseparable) enables us to comment on the errors in excess loss calculated through (2). We do not discuss factors influencing the excess loss. Our aim is to increase the awareness of developers of excess loss theories and loss prediction methods to use (1) and (2) with caution. II. TWO ANALYTICAL EXPRESSIONS FOR CLASSICAL LOSS The key role in discussing numerical results in Section III is to compare two analytical expressions for evaluating classical eddy-current loss. The first originated in the nineteenth century [3] and is a full form of the “truncated” equation (1):
(4) Here the skin-effect function
Manuscript received June 15, 2009; revised September 03, 2009; accepted September 14, 2009. Current version published January 20, 2010. Corresponding author: S. E. Zirka (e-mail:
[email protected]). Digital Object Identifier 10.1109/TMAG.2009.2032858 0018-9464/$26.00 © 2010 IEEE
(5)
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where , , is the relaand tive magnetic permeability. It is important that at (in this limit one arrives at formula (1)). It is often stated (e.g. in [1]) that (1) is equally suitable for both linear and nonlinear magnetic materials. As an argument for this statement, the obvious fact is given that (1) is indepen[1]. The insuffident of the material magnetization law ciency of such an argument is seen from (4), (5) where a constant permeability is used. Besides, the inapplicability of (1) and (4) to nonlinear magnetic material will become clear in the course of the analysis of the second expression for eddy-current loss:
(6) This equation arises from the saturation wave model [8], [9] and determines the energy loss at the same sinusoidal induction as in (1), but in a magnetic material characterized by a steplike [1]. It should magnetization curve with maximum value be noted that in contrast to the low-frequency formula (1), expression (6) is valid for the whole frequency range. There is no (in the sense skin effect in the “steplike medium” at of different maximum flux densities at the surface and in the a strong middle of the sheet). On the contrary, at skin effect is observed even at very low frequencies. It is worth remarking that the analysis of (6) in [1] has been , and the regime of restricted to the case where medium and low flux densities is outside the scope of [1]. By comparing (1) and (6), we obtain the ratio
Fig. 1. Static loops of NO steel.
by the layer-by-layer flux reversal obtained by the MDM at . As seen in Fig. 2, the layerwise flux reversal takes , but also at . Similarly place not only at , there is to the process in the “steplike medium” at nothing like skin-effect in both these cases, the peak flux density being the same at any depth in the sheet. As can be expected from comparison of (1) and (6) , the conventional equation (1) should give an underestimated value with . respect to the eddy-current loss The percentage error of (1) will be evaluated as
(8)
(7) which shows that the classical loss calculated through the steplike - curve at a high induction is 50% higher than that in a linear medium [1]. However, at , (1) and (6) give the same result; and finally, at , classical loss in this highly nonlinear medium can be substantially lower than that in the linear material. These findings can be verified numerically by solving classical equation (3) for both linear and almost steplike static curve. Therefore, the losses calculated through (1), (4) and (6) all should be called classical eddy current loss. III. LOSS COMPONENTS IN NO STEEL Although (6) describes an idealized media, it can be used to explain qualitatively the loss peculiarities inexplicable from the viewpoint of the linear theory [3]. A. Classical Loss at High Flux Densities Let us first compare the static major loop of the NO steel used in this study with a square-loop (steplike) approximation shown in Fig. 1. It is obvious that the square-loop representation seems more accurate than any linear approximation of the major loop. This means that the magnetization process calculated numerically should resemble the layerwise flux reversal characterizing saturation wave model [8]. This is corroborated
The negative ( 9%) value of in Table I at means that the value of is indeed underestimated. This contradicts the linear theory according to which the eddy-current loss calculated from (1) can only be overestimated (if funcin (4) is omitted). tion B. Classical Loss at Low Flux Densities It may appear that the positive errors in Tables I and II (overestimated values of ) are explained by at the absence of the skin-effect function in (1). However, these errors remain also unexplained in the case when we use (4) instead of (1). To show this, we should determine the perfor the static minor loop with the tip coordinates meability and . The permeability needed (this gives to calculate may be determined as the ratio ) or calculated through the differential perwhose average value over the static meability loop is 0.0386 H/m. At , these permeabilities yield equal to 0.999 and 0.981, and at the values of is equal to 0.996 and 0.931. Using these values of decrease the errors 24% and 43% in the tables above only to 22% and 36%, respectively. and This means that there is no merit in using function the linear theory as a whole to calculate classical loss at low, as well as high induction. However, the use of formula (6) where
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TABLE III COMPUTED LOSS COMPONENTS AND ERRORS AT B
= 1:5 T, 60 Hz
The only way to calculate classical loss exactly is to use the MDM [4] or Maxwell solvers [6], [7]. C. Hysteresis and Excess Losses The change of hysteresis loss with frequency is characterized in this paper by the ratio
(9) The error in the excess loss is evaluated as
(10)
Fig. 2. Flux densities at equidistant points from the surface to the middle of the sheet (solid curves). Dashed curve is the sinusoidal average induction B versus time.
TABLE I COMPUTED LOSS COMPONENTS AND ERRORS AT 50 Hz
TABLE II COMPUTED LOSS COMPONENTS AND ERRORS AT 100 Hz
As seen in Tables I and II, this error is negative when is is 2 to 4 times greater than , relative erpositive. As are substantially higher than . This circumstance is rors a hindrance in developing the excess loss theory. For example, the method of the thickness reduction [2] can lead to inaccurate . One might think that the relatively small revalues of duction of [2] (see Table III) introduces almost the same error calculated by (1). However, this leads to substantially in larger changes in the error . Of course the data in Table III are only illustrative since they were obtained by means of the MDM fitted to the steel specified here (a value of resistivity of given in [2] was only used). Notice that subtrac0.205 tion technique based on (1) was also employed in [2] even at 150 become much and 400 Hz. At these frequencies the errors greater than those in Table III which can lead to dubious conclusions in the development of theories explaining excess loss. When discussing the use of (1) and (2) we can compare the loss prediction techniques [10] and [11] developed by Fiorillo and co-workers. Whereas it was considered in [10] that paramof the statistical loss theory [1] is a constant (indepeneter dent of ), it is stressed in the later paper [11] that the loss requires novel experimental prediction at any novel value of and . This can also be caused by the estimates of both approximate nature of (1) used in both these techniques and by the necessity to compensate the error in the classical loss calculated through (1). IV. EXCESS LOSS NUMERICAL EVALUATION
gives an error of 40% at 50 Hz and 31% at ). Hence, the real value of classical 100 Hz (both at loss always lies between the values determined by (1) and (6).
The purpose of this section is to draw attention to an error in evaluating excess loss which can arise from another version of the subtraction technique. To explain the problem we compare dynamic loops shown in Fig. 3. The inner loops are obtained by solving (3) where and are linked by a static hysteresis
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of excess loss to the total loss. Speaking figuratively, this loss is concentrated mainly in the “boots” of the dynamic loops calculated by the MDH and observed in experiments (see Fig. 3). It is remarkable that the same “boots” are present in dynamic loops of grain-oriented steel [13] where their areas (i.e., excess loss contribution) are much more significant than those in NO steel. V. CONCLUSION
Fig. 3. Dynamic loops calculated by SHM-solver (dashed lines) and DHMsolver (solid lines) at B p = 0.5, 1.0, and 1.5 T.
Using an accurate magnetodynamic model [4], the relative and 0.5, 1.0, and 1.5 T were errors of (1) at found to be 24%, 4%, and 9%, respectively. When using equal the subtracting technique, this leads to the errors in , the errors in to 36%, 7%, and 26%. At increase to 74%, 18%, and 53%. The signs and the values of these errors cannot be explained in the framework of the theory developed for a linear magnetic material and require engaging the saturation wave theory [8] for their explanation. These large errors show that (1), like any other simple equation of this type, is not a reliable basis for evaluating excess loss through (2). The use of numerical models like those in [4], [6], [7] is necessary for the accurate loss separation. This does not mean, however, that (1) cannot be used in loss prediction schemes like that in [11] but it should be viewed as an approximate conditional expression. REFERENCES
model (SHM) [12]. Corresponding SHM-solver (comparatively simple numerical scheme [12, Eq. (8)]) predicts the total loss which is the sum of only two components, namely hysteresis loss and eddy-current loss. The outer loops, whose shape and area almost coincide with measurements, have been oband tained using the MDM which is a solver of (3) where are linked by a dynamic hysteresis model (DHM). The total calculated with the MDM is the sum of hysteresis loss . loss, eddy-current loss, and excess loss, It might appear that excess loss can be evaluated as the difwhich is equivalent to the area beference tween solid and dashed curves in Fig. 3. However, this subtraction always gives underestimated excess loss. For example, for in Fig. 3 the values of at 50, the loops with 100, 200, and 400 Hz are larger than corresponding differences by 17, 30, 48, and 70%, respectively. The reason for this is that introducing dynamic component into the static hysteresis model, i.e., the substitution of SHM by DHM in the solver of (3) is accompanied by a decrease of eddy-current loss calculated by the solver. So the subtraction technique is too crude for evaluating excess loss not only when using (1) and (2) but also in the subtraction method described in this section. In other words, the real excess loss is greater than the difference between the measured total loss and total loss calculated by the SHM-solver. The close proximity of the loops calculated by the SHMsolver and the MDM (also close to measured loops [4]) indicate that (3) solved even with the static hysteresis model is an acceptable physical prototype for the MDM reproducing excess loss. The proximity of these loops illustrates the minor contribution
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