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Anomalous rotational eddy current loss in electrical steel sheets
Journal of Magnetism and Magnetic North-Holland Publishing Company
ANOMALOUS
Materials
38 (1983) 301-304
ROTATIONAL
301
EDDY CURRENT
LOSS IN ELECTRICAL
STEEL SHEETS
W. BRIX and K.A. HEMPEL Institut fiir Werkstoffe
der Elektrotechnik,
Technical University Aachen, Fed. Rep. Germany
The rotational eddy current loss of different qualities of electrical steel sheets has been determined with a dynamical torque magnetometer. The measured values of the losses are higher than the losses calculated with a classical formula. The difference grows with increasing ratio of grain size to sheet thickness. In alternating fields this phenomenon has been explained theoretically by the Pry and Bean model.
1. Introduction
From a classical point of view the magnetic losses in electrical steel sheets due to alternating fields can be separated into hysteresis losses and eddy current losses. Because of their different frequency dependence the separation can be performed by measuring the losses at different frequencies. In a plot of energy loss vs. frequency one may expect a straight line, given by the formula P/f=h+wf.
(1)
P if
h fFig. 1. Energy loss vs. frequency in alternating magnetic fields (measured values; - - - - classically calculated values).
0304-8853/83/0000-0000/$03.00
0 1983 North-Holland
This is drawn as a dashed line in fig. 1. In this formula h corresponds to the hysteresis energy and wf to the eddy current energy loss which can be calculated using the sheet thickness and the electrical conductivity of the sample material as well as the frequency and the amplitude of the sinusoidal magnetic induction. Experimental results, however, lead to a convex plot, which is given as a solid line in fig. 1. This curve shows a higher slope than the calculated one [ 11.The difference between the two curves is called “anomalous eddy current loss”. This loss can be explained by considering the domain structure of the material. Because of the complexity of this structure quantitative calculations have been done only for special cases [2,3]. The result is, that the ratio of the losses calculated with the domain model to the classically calculated losses increases with increasing ratio of domain wall spacing to sheet thickness. In magnetic fields rotating in the plane of the sheet one finds losses which are different from those measured in alternating magnetic fields [4] discussed before. Just like the losses in alternating fields these so-called rotational power losses may be separated into rotational hysteresis loss and rotational eddy current loss. In this paper we give a report on measurements of anomalous eddy current losses in rotating magnetic fields. It will be shown that the results of the Pry and Bean [2] calculation for alternating fields can be used for rotating fields, too.
302
Table 1 Important
W. Bnx. K.A. Hempel / Anomalous
rotational
data of the samples
Sample material
230-50 a 100-35 A HF20” ORSI H h
Mass
Sheet
Average
density
thickness
diameter
(p/kgmm3)
(d/mm)
(a/pm)
7700 7600 7600 7650
0.5 1 0.34 0.17 0.28
69 196 219 21393
’ Nonoriented material, produced by the Stahlwerke b Grain oriented material, produced by the Thyssen
Bochum AG.
2. Experiments For the measurements of rotational power loss at various frequencies the dynamical torque magnetometer described in a previous paper [5] has been used. With this apparatus measurements have been performed in a frequency range between 25 and 100 Hz. Furthermore, measurements at frequencies of about 0.01 Hz have been made on the non-oriented sheets. Sample discs of 60 mm diameter were punched from the sheet and stress relief annealed subsequently. Samples of different grain sizes were chosen from three qualities of nonoriented material and one of grain oriented sheet. This selection was done because the domain wall spacing increases with increasing grain size. The average grain diameter was determined using the technique given in ref. [6]. Important data of the samples are given in table 1.
grain
s l?
Electrical conductivity ( o/IO6 Sm- ‘)
0.14 0.58 I .2Y 76.4
2.50 1.82 1.59 2.22
AG
ing fields (see fig. 1) is weaker in rotating fields. Similar results were found by other authors [7]. Therefore, the extrapolated values (_/-+ 0) represent the rotational hysteresis energy. The difference between these values and the measured energy at a given frequency corresponds to the related rotational eddy current energy. In figs. 3 to 6 the rotational power loss at 50 Hz vs. induction is given for the four samples (a). Curves (b) denote the rotational hysteresis loss
3. Results and discussion As an example fig. 2 shows the measured rotational loss energy vs. frequency at different inductions for the material 230-50. The circles correspond to the dynamical measurements and the triangles to the quasi-static measurements, respectively. Using linear regression for the dynamical measurements, the straight lines in the figure were constructed. The extrapolation of the straight lines to zero frequency is in good accordance with the statically measured values. From this we may conclude that the curvature of the frequency dependence of the energy loss which appears in alternat-
0
25
50
100 Hz
75 fo-
Fig. 2. Rotational loss energy vs. frequency tions for the material 230-50 (0 : dynamical static measurements).
at different inducmeasurements; A:
W. Brix, K.A. Hempel / Anomalous rotational
,
303
w
kg
a
pr 5 4
l-
0
0.3
0.6
0.9
1.2
1.5
1
0
0.3
0.6
0.9
1,2
B-
1,5
T
B-
Fig. 3. (a) Rotational power loss at 50 Hz vs. induction for sample 230-50; (b) rotational hysteresis loss; (c) measured rotational eddy current loss; (d) rotational eddy current loss calculated with (2).
Fig. 5. (a) Rotational power loss at 50 Hz vs. induction for sample HF20; (b) rotational hysteresis loss; (c) measured rotational eddy current loss; (d) rotational eddy current loss calculated with (2).
I I’ I
It
1
.
-
I I(
PC i
0
0.3
0.6
0.9
12
1.5
T
BFig. 4. (a) Rotational power loss at 50 Hz vs. induction for sample 100-35; (b) rotational hysteresis loss; (c) measured rotational eddy current loss; (d) rotational eddy current loss calculated with (2).
0
0.3
0.6
0.9
1.2
1.5
T
BFig. 6. (a) Rotational power loss at 50 Hz vs. induction for sample ORSI H; (b) rotational hysteresis loss; (c) measured rotational eddy current loss (d) rotational eddy current loss calculated with (2).
W. Brix. K.A.
304
Hempel/ Anomalous rotational
and curves (c) the rotational eddy current loss, respectively. Let p be the density of the sample material, B the modulus and o the angular velocity of the rotating magnetic induction, u the electrical conductivity and d the thickness of the sheet, then the classical rotational eddy current loss can be calculated using the formula [8] p,,
= $B2u20d2.
(2)
These values are given in curves (d). The difference between curves (c) and (d) may be called the anomalous rotational eddy current loss. Obviously the ratio of the anomalous rotational eddy current loss to the rotational eddy current loss calculated with (2) increases with increasing ratio of average grain diameter to sheet thickness. This behaviour corresponds clearly to that predicted by the Pry and Bean calculation. Thus for rotating as well as for alternating mag-
netic field grains avoid anomalous
should be kept small in order to eddy current losses.
Acknowledgement The authors are indebted Bochum AG and the Thyssen the samples.
to the Stahlwerke AG for supplying
References [II F. Brailsford and R. Fogg, Proc. IEE 113 (1966) 1562. PI R.H. Pry and C.P. Bean, J. Appl. Phys. 29 (1958) 532. [31 H.J. Williams, W. Shockley and C. Kittel, Phys. Rev. 80 (1950) 6. [41 F. Brailsford, J. IEE 83 (1938) 566. 151 W. Brix, J. Magn. Magn. Mat. 26 (1982) 193. [61 R.L. Fullman, Trans. AIME 197 (1953) 447. [71 A. Cecchetti et al.. IEEE Trans. Magn. MAG-14 (1978) 356. PI T. Yamaguchi and K. Narita, El. Eng. Japan 96 (1976) 15.
ANOMALOUS
Materials
38 (1983) 301-304
ROTATIONAL
301
EDDY CURRENT
LOSS IN ELECTRICAL
STEEL SHEETS
W. BRIX and K.A. HEMPEL Institut fiir Werkstoffe
der Elektrotechnik,
Technical University Aachen, Fed. Rep. Germany
The rotational eddy current loss of different qualities of electrical steel sheets has been determined with a dynamical torque magnetometer. The measured values of the losses are higher than the losses calculated with a classical formula. The difference grows with increasing ratio of grain size to sheet thickness. In alternating fields this phenomenon has been explained theoretically by the Pry and Bean model.
1. Introduction
From a classical point of view the magnetic losses in electrical steel sheets due to alternating fields can be separated into hysteresis losses and eddy current losses. Because of their different frequency dependence the separation can be performed by measuring the losses at different frequencies. In a plot of energy loss vs. frequency one may expect a straight line, given by the formula P/f=h+wf.
(1)
P if
h fFig. 1. Energy loss vs. frequency in alternating magnetic fields (measured values; - - - - classically calculated values).
0304-8853/83/0000-0000/$03.00
0 1983 North-Holland
This is drawn as a dashed line in fig. 1. In this formula h corresponds to the hysteresis energy and wf to the eddy current energy loss which can be calculated using the sheet thickness and the electrical conductivity of the sample material as well as the frequency and the amplitude of the sinusoidal magnetic induction. Experimental results, however, lead to a convex plot, which is given as a solid line in fig. 1. This curve shows a higher slope than the calculated one [ 11.The difference between the two curves is called “anomalous eddy current loss”. This loss can be explained by considering the domain structure of the material. Because of the complexity of this structure quantitative calculations have been done only for special cases [2,3]. The result is, that the ratio of the losses calculated with the domain model to the classically calculated losses increases with increasing ratio of domain wall spacing to sheet thickness. In magnetic fields rotating in the plane of the sheet one finds losses which are different from those measured in alternating magnetic fields [4] discussed before. Just like the losses in alternating fields these so-called rotational power losses may be separated into rotational hysteresis loss and rotational eddy current loss. In this paper we give a report on measurements of anomalous eddy current losses in rotating magnetic fields. It will be shown that the results of the Pry and Bean [2] calculation for alternating fields can be used for rotating fields, too.
302
Table 1 Important
W. Bnx. K.A. Hempel / Anomalous
rotational
data of the samples
Sample material
230-50 a 100-35 A HF20” ORSI H h
Mass
Sheet
Average
density
thickness
diameter
(p/kgmm3)
(d/mm)
(a/pm)
7700 7600 7600 7650
0.5 1 0.34 0.17 0.28
69 196 219 21393
’ Nonoriented material, produced by the Stahlwerke b Grain oriented material, produced by the Thyssen
Bochum AG.
2. Experiments For the measurements of rotational power loss at various frequencies the dynamical torque magnetometer described in a previous paper [5] has been used. With this apparatus measurements have been performed in a frequency range between 25 and 100 Hz. Furthermore, measurements at frequencies of about 0.01 Hz have been made on the non-oriented sheets. Sample discs of 60 mm diameter were punched from the sheet and stress relief annealed subsequently. Samples of different grain sizes were chosen from three qualities of nonoriented material and one of grain oriented sheet. This selection was done because the domain wall spacing increases with increasing grain size. The average grain diameter was determined using the technique given in ref. [6]. Important data of the samples are given in table 1.
grain
s l?
Electrical conductivity ( o/IO6 Sm- ‘)
0.14 0.58 I .2Y 76.4
2.50 1.82 1.59 2.22
AG
ing fields (see fig. 1) is weaker in rotating fields. Similar results were found by other authors [7]. Therefore, the extrapolated values (_/-+ 0) represent the rotational hysteresis energy. The difference between these values and the measured energy at a given frequency corresponds to the related rotational eddy current energy. In figs. 3 to 6 the rotational power loss at 50 Hz vs. induction is given for the four samples (a). Curves (b) denote the rotational hysteresis loss
3. Results and discussion As an example fig. 2 shows the measured rotational loss energy vs. frequency at different inductions for the material 230-50. The circles correspond to the dynamical measurements and the triangles to the quasi-static measurements, respectively. Using linear regression for the dynamical measurements, the straight lines in the figure were constructed. The extrapolation of the straight lines to zero frequency is in good accordance with the statically measured values. From this we may conclude that the curvature of the frequency dependence of the energy loss which appears in alternat-
0
25
50
100 Hz
75 fo-
Fig. 2. Rotational loss energy vs. frequency tions for the material 230-50 (0 : dynamical static measurements).
at different inducmeasurements; A:
W. Brix, K.A. Hempel / Anomalous rotational
,
303
w
kg
a
pr 5 4
l-
0
0.3
0.6
0.9
1.2
1.5
1
0
0.3
0.6
0.9
1,2
B-
1,5
T
B-
Fig. 3. (a) Rotational power loss at 50 Hz vs. induction for sample 230-50; (b) rotational hysteresis loss; (c) measured rotational eddy current loss; (d) rotational eddy current loss calculated with (2).
Fig. 5. (a) Rotational power loss at 50 Hz vs. induction for sample HF20; (b) rotational hysteresis loss; (c) measured rotational eddy current loss; (d) rotational eddy current loss calculated with (2).
I I’ I
It
1
.
-
I I(
PC i
0
0.3
0.6
0.9
12
1.5
T
BFig. 4. (a) Rotational power loss at 50 Hz vs. induction for sample 100-35; (b) rotational hysteresis loss; (c) measured rotational eddy current loss; (d) rotational eddy current loss calculated with (2).
0
0.3
0.6
0.9
1.2
1.5
T
BFig. 6. (a) Rotational power loss at 50 Hz vs. induction for sample ORSI H; (b) rotational hysteresis loss; (c) measured rotational eddy current loss (d) rotational eddy current loss calculated with (2).
W. Brix. K.A.
304
Hempel/ Anomalous rotational
and curves (c) the rotational eddy current loss, respectively. Let p be the density of the sample material, B the modulus and o the angular velocity of the rotating magnetic induction, u the electrical conductivity and d the thickness of the sheet, then the classical rotational eddy current loss can be calculated using the formula [8] p,,
= $B2u20d2.
(2)
These values are given in curves (d). The difference between curves (c) and (d) may be called the anomalous rotational eddy current loss. Obviously the ratio of the anomalous rotational eddy current loss to the rotational eddy current loss calculated with (2) increases with increasing ratio of average grain diameter to sheet thickness. This behaviour corresponds clearly to that predicted by the Pry and Bean calculation. Thus for rotating as well as for alternating mag-
netic field grains avoid anomalous
should be kept small in order to eddy current losses.
Acknowledgement The authors are indebted Bochum AG and the Thyssen the samples.
to the Stahlwerke AG for supplying
References [II F. Brailsford and R. Fogg, Proc. IEE 113 (1966) 1562. PI R.H. Pry and C.P. Bean, J. Appl. Phys. 29 (1958) 532. [31 H.J. Williams, W. Shockley and C. Kittel, Phys. Rev. 80 (1950) 6. [41 F. Brailsford, J. IEE 83 (1938) 566. 151 W. Brix, J. Magn. Magn. Mat. 26 (1982) 193. [61 R.L. Fullman, Trans. AIME 197 (1953) 447. [71 A. Cecchetti et al.. IEEE Trans. Magn. MAG-14 (1978) 356. PI T. Yamaguchi and K. Narita, El. Eng. Japan 96 (1976) 15.