ARTICLE IN PRESS

Journal of Magnetism and Magnetic Materials 304 (2006) e586–e588 www.elsevier.com/locate/jmmm

Dependence of the power losses of a non-oriented 3% Si-steel on frequency and gauge A. Broddefalk, M. Lindenmo Cogent Power Ltd., Surahammars Bruk, PO Box 201, SE-735 23 Surahammar, Sweden Available online 20 March 2006

Abstract The power losses of a non-oriented 3% Si-steel rolled to gauges between 0.05 and 2 mm and heat-treated thereafter have been measured under sinusoidal polarizations at frequencies between 15 Hz and 10 kHz. The losses were analysed using a loss separation model based on statistical theory. For the thick samples the skin effect caused the model to fail above a certain frequency, while for the very thin samples the model seems to describe the losses well at all frequencies studied. r 2006 Elsevier B.V. All rights reserved. PACS: 75.50.Bb; 75.60.Ej Keywords: Electrical steel; Non-oriented steel; Core loss; Skin effect

The understanding of the dependence of power losses in electrical steels on polarization and frequency is of great importance [1]. A tool for estimating losses at different frequencies is the statistical loss theory [2]. For producers of the steels it gives insight how to reduce the loss. For users of the steels, e.g. electrical machine designers, it helps to estimate the loss at various working conditions. High speed machines and applications where higher harmonics can be expected are two examples of when the core loss above the usual 50 or 60 Hz are of great importance. The power losses of electrical steel laminations can be described using the loss separation [2,3]. The total loss is then expressed as the sum of three terms, the hysteresis loss (Ph ), the classical eddy current loss (Pcl ) and the excess eddy current loss (Pexc ) P ¼ Ph þ Pcl þ Pexc .

(1)

The hysteresis loss per cycle, W h ¼ Ph =f , is independent of frequency. It is mostly related to the microstructure of the steel. The rolling and heat-treatment cycle of the steel will influence the grain size, size distribution of the precipitates and the texture, which all affect the hysteresis loss. The Corresponding author.

E-mail address: [email protected] (A. Broddefalk). URL: http://www.sura.se/. 0304-8853/$ - see front matter r 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.jmmm.2006.02.183

classical eddy current loss is a dynamic loss, which would arise if the steel were perfectly homogeneous. This loss can be derived from Maxwell’s equations. For the slab geometry, sinusoidal induction and full flux penetration, the classical eddy current loss can be written as Pcl ¼

p2 sd 2 ðBp f Þ2 p2 sd 2 ðJ p f Þ2
, 6r 6r

(2)

where s is the electrical conductivity, d is the gauge, Bp is the peak induction, J p is the peak polarization, f is the frequency and r is the mass density. The excess eddy current loss is the dynamic loss related to the magnetic domains. Using statistical theories this loss can be approximated as [3] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8:76 sGSV 0 ðJ p f Þ3=2 . (3) Pexc ¼ r Sinusoidal polarization, full flux penetration and high magnetization rate are assumed [2,3]. G ¼ 0.1356 is a dimensionless coefficient, S is the cross-section area and V 0 is a characteristic field describing the influence of the microstructure on the domain wall process. The excess loss is reduced when the grain size is reduced [4]. The parameter V 0 has a complex dependence on J p [3]. The validity of the model can be studied more easily if the equations are

ARTICLE IN PRESS A. Broddefalk, M. Lindenmo / Journal of Magnetism and Magnetic Materials 304 (2006) e586–e588

250

Table 1 The thickness of the samples in this study and the estimated upper frequency limit for the use of the equations f lim (Hz)

0.0488 0.0970 0.124 0.175 0.195 0.349 0.496 0.636 0.978 2.06

Outside experimental range Outside experimental range 4625 1620 825 225 121 95 47 Outside experimental range

200

P (W/kg)

Gauge (mm)

e587

150 100 50 0

0

0.5

1 Jp (T)

1.5

2

Fig. 2. The total loss for the 0.349 mm sample. f ¼ 1 kHz. : Experimental data. : Calculated data.

The values of f lim are estimates from measurements at J p ¼ 0:5 T.

0.012

0.08

0.011

0.07 10

0.01 0.06 0.009 0.05

0.008

0.04

0.007 0.006 5

10

15

20

0.03 25

f1/2 (Hz1/2) pffiffiffi Fig. 1. ðW  W cl Þ versus f for d ¼ 0:349 mm at J p ¼ 0:5 T () and 1.5 T (). The lines are fits to Eq. (4) for f of lim . The vertical dashed line indicates f lim .

Loss difference (%)

W-Wcl (J/kg)

15

5 0 -5 -10 -15

0

0.5

1 Jp (T)

1.5

2

Fig. 3. ðPmeasured  Pcalculated Þ=Pmeasured versus J p at selected f 4f lim . : 0.496 mm sample measured at 600 Hz. : 0.349 mm sample measured at 1 kHz. : 0.195 mm sample measured at 2.5 kHz.

written as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffi 8:76 sGSV 0 J 3=2 P  Pcl p ¼ W  W cl ¼ W h þ f. (4) f r pffiffiffi A plot of ðW  W cl Þ versus f is expected to be linear if the model holds. The gauges of the samples used in this study are listed in Table 1. The steel contained 3.0 wt% Si and 0.4 wt% Al. The cold-rolled coils were heat-treated to give the same nominal microstructure. The samples were measured in an Epstein frame using a PMS 3000 digital magnetic measurement system [5] at frequencies between 15 Hz and 10 kHz. The measurements were done in steps of polarizations of 0.1 T, using sinusoidal waveforms, up to the limits of the measurement system. pffiffiffi ðW  W cl Þ versus f is plotted in Fig. 1 for J p ¼ 0:5 and 1.5 T for the 0.349 mm sample. Above 225 Hz, the experimental data points do not lay on the calculated straight line. This frequency is denoted as f lim . When deriving the above equations full flux penetration was assumed, but at high frequencies the skin effect is no longer

negligible. The local induction varies across the thickness, which makes the loss separation invalid [6,7]. The dynamic losses will not have the same dependence of J p on f and the hysteresis loss per cycle will not even be frequency independent. It is noted that the curves for 0.5 and 1.5 T deviates from the line differently. For further examination, values along the fitted line above f lim are compared with the measured values. When plotting the measured loss (Pmeasured ) and the calculated loss (Pcalculated ) versus the polarization at a frequency above f lim , see Fig. 2, it is noted that the calculated values overestimate the losses at low polarizations and underestimate them at high polarizations. Similar observations have been made before [8]. Fig. 3 shows the difference between the experimental loss data and the extrapolated loss data, for some of the samples, at frequencies above f lim , depending on the polarization. It is noted that the cross-over from overestimation to underestimation of the loss is just below 1.2 T for the considered samples. f lim increases with decreasing gauge. In Table 1 the estimated value for f lim at J p ¼ 0:5 T

ARTICLE IN PRESS flux penetration was assumed. When this requirement is not fulfilled, the equations give relatively poor estimates of the total loss. Above f lim , extrapolations overestimate the loss for low J p and underestimate them for high J p . Attempts have been made to correct the expressions when the skin effect is apparent [9], but the new expressions include many parameters which do not appear to have a physical meaning. This paper gives estimates on how far the loss separation expressions can be extrapolated for a given lamination thickness. References

is given for the different samples. There is no clear dependence of J p on f lim , as Fig. 1 indicates. It is noted that for d ¼ 0:978 mm, the skin effect is already apparent at 50 Hz. Theoretical calculations have given the same result [6]. For the 2.06 mm sample, pffiffiffif lim could not be estimated since ðW  W cl Þ versus f does not show a linear regime in the studied frequency range. For the thin samples (dp0:10 mm), ðW  W cl Þ appears to be linear with pffiffiffi f even up to 10 kHz, at least for polarizations up to 0.3 T, see Fig. 4. Though statistical loss theory is a good tool for estimating the power losses in electrical steel laminations, one has to be aware of its limitations. To derive the equations, several approximations were used [2], but full

[1] P. Beckley, Electrical Steels for Rotating Machines, The Institution of Electrical Engineers, London, 2002. [2] G. Bertotti, Hysteresis in Magnetism, Academic Press, San Diego, 1998. [3] E. Barbisio, F. Fiorillo, C. Ragusa, IEEE Trans. Mag. 40 (2004) 1810. [4] G. Bertotti, G. Di Schino, A. Ferro Milone, F. Fiorillo, J. Phys. 46 (1985) 385; G. Bertotti, J. Phys. 46 (1985) 389; G. Bertotti, J. Magn. Magn. Mater. 54–57 (1986) 1556. [5] Cogent Power Ltd., PO Box 30, Newport, South Wales NP19 0XT, United Kingdom. [6] C. Appino, G. Bertotti, O. Bottauscio, F. Fiorillo, P. Tiberto, D. Binesti, J.P. Ducreux, M. Chiampi, M. Repetto, J. Appl. Phys. 79 (1996) 4575. [7] A. Boglietti, M. Chiampi, M. Repetto, O. Bottauscio, D. Chiarabaglio, IEEE Trans. Magn. 34 (1998) 1240. [8] V. Basso, G. Bertotti, O. Bottauscio, F. Fiorillo, M. Pasquale, M. Chiampi, M. Repetto, J. Appl. Phys. 81 (1997) 5606. [9] N. Derebasi, I. Kucuk, A.J. Moses, Sensors Actuators A 106 (2003) 101.