epstein frame measurement based determination of original non-degraded and fully degraded magnetic...
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Epstein Frame Measurement Based Determination
of Original Non-Degraded and Fully Degraded
Magnetic Characteristics of Material Submitted to
Laser CuttingMadeleine Bali1, Student Member, IEEE , Herbert De Gersem2, Annette Muetze1, Senior Member, IEEE
1 Graz University of Technology, Electric Drives and Machines Institute
Inffeldgasse 18/I, A-8010 Graz, Austria2 Technische Universität Darmstadt, Institut f ̈ur Theorie Elektromagnetischer Felder
Schlossgartenstraße 8, D-64289 Darmstadt, Germany
[email protected], [email protected], [email protected]
Abstract—The degrading effect of laser cutting on steel sheetmaterial, and thus on the material’s magnetic characteristics, ismuch less understood than that of mechanical cutting. Further-
more, the estimated degrading influence on magnetic propertiesis still difficult to determine and not sufficiently known. Inthis paper, the magnetic characteristics of the degraded andnon-degraded zones are computed from data obtained by twoEpstein frame measurements using sample strips of differentwidths. Subsequently, these new characteristics are inserted intoa finite-element model, which accounts for arbitrary geometries.The simulation results for the influence of laser cutting onthe magnetic characteristics of the stator lamination stacks areverified by measurements.
Index Terms—Electrical steel sheets, ferromagnetic material,
finite element method, laser cutting, magnetostatics, manufactur-
ing, material degradation.
I. INTRODUCTION
To date, the electromagnetic characteristics of electrical
steel sheets used in electrical machines are implemented in
finite element simulations by magnetization and loss curves
obtained from Epstein frame measurements [1] and/or pro-
vided by manufacturers. The literature generally agrees that
these data sets differ from those of the finished machine (e.g.
[2]), as the magnetic deterioration due to the manufacturing
steps is not taken into account. Commonly, these differences
are considered, for example, by correction or building factors
[3]–[5], which generally do not include the variabilities be-
tween different materials, machine designs (stator and rotor
cross sectional areas), and manufacturing techniques.
This paper focuses on modeling the degrading effect of laser
cutting, in particular solid state laser cutting, in contrast to
the majority of works that have focused on the degradation
effect due to mechanical cutting [6]–[9]. It reviews a method
to estimate the losses for different geometries and applies this
method to samples of solid state laser-cut stator lamination
stacks, thereby further investigating possible differences bet-
ween the modeling of mechanical and laser cutting.
Fig. 1. Arrangement of steel strips in the Epstein frame, based on [1].
I I . PERFORMED M EASUREMENTS
The BH - and specific loss curves of two samples with thesame length, y , and thickness, wz , but with different widths(w1= 30mm and w2 = 7.5mm) are determined by Epsteinframe measurements (Fig. 2). The Epstein frame measure-
ments have been based on the standards IEC 60404-2 [1] and
IEC 60404-10 [10], and the setup has been presented in detail
in [6].
In the case of the ‘wide’ samples (width of w1) 16 electricalsteel sheet strips, in the case of the ‘small’ samples (width
of w2) 64 strips are placed into the frame (Fig. 2). Note thatthe transversely-cut and longitudinally-cut samples are placed
in two opposite coils (Fig. 1) [1].
All samples of one material are taken from the same mother
coil (MC) since material deviations from MC to MC may be
larger than the influence of the manufacturing process [11].
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Fig. 2. Correlation between measurements and identification procedure.
III. REVIEW: IDENTIFICATION OF N ON-D EGRADED ANDFULLY D EGRADED BH -C HARACTERISTICS
A. Assumptions and simplifications
The following assumptions and simplifications have been
made:
• The laser cutting process induces equal degradation pro-
files on both sides of the sample.
• The degradation zone is homogeneous and has a specified
depth, d (Fig. 2).• The deteriorated zones on the short sides of the Epstein
samples is neglected as d y (Fig. 2).
The fractions of degraded material in samples 1 and 2 are
denoted as γ 1 (2d / w1) and γ 2 (8d / w2) respectively.Only very few results have been published on the degra-
dation depth of laser cutting, for example in [12]. This is,
among other reasons, also due to a lack of understanding of
the detailed deterioration of the material which is caused by
the laser cutting technique. In contrast to mechanically-cut
samples, laser cutting induces thermal stresses due to the fast
heating followed by a rapid cooling [13], [14]. Thus, not every
measurement method used to identify the degradation depth
Fig. 3. Identifying the non-degraded and degraded BH -curves with the BH -curves of small and wide samples [6].
known from mechanically-cut samples may be applicable
to laser-cut samples. For example, the determination of the
changed grain sizes is not suitable, as according to [15] laser
cutting does not change the grain size of the material. Further-
more, the laser technique and the laser settings influence the
degree, and thus, the depth of deterioration.
In the presented modeling approach, a non-degraded area in
the middle of the sheet and a homogeneously degraded zone
with a deterioration depth of 2.1mm from the cut edge isassumed. Thus, a better comparison with the results presented
in [6], in which mechanically-cut samples are investigated,
is possible, when the same degradation depth is used. Note
that the proposed technique also allows to set other degrada-
tion depths, and the computed material characteristics result
accordingly.
B. Basic identification procedure
The identification procedure has been discussed at length
in [6] and is reviewed here briefly for comprehensiveness.
The basic idea of this procedure is presented in (Fig. 3).
The measurement series (H 1, B1) and (H 2, B2) are obtainedindependently from the ‘wide’ and ‘small’ samples in the
Epstein frame (see also Fig. 2). Of course, it is ensured that
these samples result from the same material and MC.
Subsequently, data points (H, Bnd) and (H, Bdg) are calcu-
lated from 1− γ 1 γ 11− γ 2 γ 2
BndBdg
=
B1B2
. (1)
The subscripts 1 and 2 label the measured data of the twosamples with different widths (see also Section II). Eq. (1)
presents the parallel connection of the flux paths through the
non-degraded (nd) and degraded (dg) zones. Note, the values
B1, B2, Bnd and Bdg all correspond to the same magnetic fieldstrength H .
A few generalizations are implemented to improve the
applicability in practice:
1) Interpolation and sampling,
2) Rayleigh region,3) Full-saturation region.
A more detailed explanation is reviewed from [6] in Ap-
pendix A.
The obtained curves (H, Bnd) and (H, Bdg) can easily beintegrated into a finite element calculation of, for example, a
machine consisting of a material with a fully degraded zone
at the cut edge and a non-degraded zone in the middle of the
sample.
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IV. RESULTS
In the following, the identification procedure is applied to
three different materials
• M270-35A,
• M400-50A,
• M800-65A,
commonly used in electric machine design, at three different
frequencies• 50Hz• 250Hz• 500Hz.
Thus, an extensive verification of this method is possible.
The permeability of sample 1 is substantially better than that
of sample 2, because w1 > w2. Thus, the relative degradedvolume (V degraded/V total sample volume) in sample 2 is larger thanin sample 1 (Fig. 2).
As already shown in [6], piecewise cubic splines in com-
bination with a repairing procedure of the Rayleigh and full
saturation regions represent the measured curves accurately
(see Fig. 4). This, along with the basic identification procedure,
provides the results presented in Fig. 5. The sampling pointsfor the magnetic field strength are taken from the measure-
ments of the first sample (see Appendix A).
0 2000 4000 6000 8000 10000 12000 140000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
22
Magnetic field strength (A/m)
M a g n e t i c f l u x d e n s i t y ( T )
BH1 (measured)
BH1 (calculated)
BH2 (measured)
BH2 (calculated)
Fig. 4. Measured and identified BH -curves, M270-35A at 50 Hz.
0 2000 4000 6000 8000 10000 12000 140000
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
1.8
2
Magnetic field strength (A/m)
M a g n e t i c f l u x d e n s i t y ( T )
first sample
second sample
nondegraded
degraded
Fig. 5. Identified BH -curves for the degraded and non-degraded materialzone, M270-35A at 50Hz.
As expected, the BH -curves are stacked such that for allvalues H of the magnetic field strength holds, Bnd(H ) >B1(H ) > B2(H ) > Bdg(H ), in all regions of the charac-teristic (Fig. 5).
Note that the model may produce unusual (unphysical)
behavior in the knee point area of the identified non-degraded
BH -curve by producing an extrema (see Fig. 5). This mayoccur when the difference of the magnetic induction or the
relative permeability of both measured samples is very largeat a respective magnetic field strength H : In contrast tomechanically-cut samples (see [16]), the relative permeabil-
ity of the small laser-cut samples is more degraded at the
small magnetic field strength area (see Fig. 6). Addition-
ally, the relative permeability of laser-cut samples at small
magnetic field strengths is, with decreasing sample width or
increasing deteriorated zone, degraded to a plateau (see e.g.
Fig. 6 and [16]). Hence, the difference of relative permeabil-
ities of samples 1 and 2 is significantly larger for laser-cutsamples. Therefore, such artificial extrema may result if the
method is applied to laser-cut samples of significantly different
widths.
200 400 600 800 1000 1200
1000
2000
3000
4000
5000
6000
7000
Magnetic field strength in A/m
R e l a t i v e p e r m e a b i l i t y
M27035A_30_250Hz_mc
M27035A_75_250Hz_mc
M27035A_30_250Hz_lc
M27035A_75_250Hz_lc
Fig. 6. The difference in relative permeability due to the different used samplesizes is larger for laser-cut (lc) than for mechanically-cut (mc) samples at smallmagnetic field strengths.
V. REVIEW: IDENTIFICATION OF L OS S C URVES FOR
NON-D EGRADED AND F ULLY D EGRADED M ATERIAL
A. Loss model
The loss model used in this identification procedure is
described by
ploss = |c1|B + c2B2 , (2)
with ploss the loss density and c1 and c2 the parameters which
need to be identified by regression. The expression is quitesimple but approximates the loss density very accurately.
B. Basic identification procedure
The identification of the loss curves for the non-degraded
and degraded material requires several steps and depends on
the previously obtained BH -curves, Bnd(H ), and Bdg(H ).With this procedure, including both the loss model and the
measurement data, the parameters for the loss model for
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the degraded and the non-degraded material zones can be
determined.
Fig. 7. Identifying the non-degraded and degraded loss curves from the losscurves of small and wide samples [6].
The identification procedure is reviewed from [6]:
1) Choose a set of sampling points for the magnetic field
strength H sample.2) Evaluate the (H, B∗nd)- and (H, B
∗
dg) curves.
3) Calculate the corresponding averaged magnetic flux den-sities, B∗1 and B∗
2 , in both samples, using eq. (1).
4) Evaluate the measured loss curves for B∗1 and B∗
2 , leading
to p1 and p2.5) Evaluate the loss model for a set of model parameters
and for the points B∗nd and B∗
dg.
6) Calculate the corresponding averaged loss densities p∗1and p∗2 in both samples, using:
1− γ 1 γ 11− γ 2 γ 2
pnd pdg
=
p∗1 p∗2
. (3)
7) Compare p∗1 and p∗
2 with p1 and p2.
Steps 1 to 4 are carried out in advance. Steps 5 to 6 are
implemented in a procedure which is given as an input toan optimization routine minimizing the error in step 7.
VI. RESULTS
The loss identification procedure is applied to the measure-
ment results for ‘wide’ and ‘small’ samples of all investigated
material samples. The modeled curves approximate the mea-
sured ones sufficiently well (as illustrated in Fig. 8).
The resulting loss densities for the same magnetic flux
density are ordered as pnd < p1 < p2 < pdg (Fig. 9). Thisapplies to all investigated samples.
VII. APPLYING THE I DENTIFIED C URVES TO A
LASER-C UT S TATOR G EOMETRY
A. Stator measurement setup
The laminations of the investigated stator stack are derived
from the same MC as the investigated Epstein samples and
have been cut by the same laser cutting method. This has been
done to avoid possible additional deviations due to different
production batches.
0 0.5 1 1.50
5
10
15
20
25
30
35
40
45
50
Magnetic flux density (T)
L o s s d e n s i t y ( W / k g )
sample 1: measurement
sample 1: approximation
sample 2: measurement
sample 2: approximation
Fig. 8. Measured and identified loss curves for laser-cut samples, M400-50Aat 250Hz supply frequency.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80
10
20
30
40
50
60
70
80
Magnetic flux density (T)
L o s s d e n s i t y ( W / k g )
sample 1
sample 2
nondegraded
degraded
Fig. 9. Measured and computed loss curves for laser-cut samples, M400-50Aat 250Hz supply frequency.
As for the mechanically-cut stator samples in [6], a pri-
mary and secondary coil are wound around the stator yoke
(yoke height = 12mm). Hence, the magnetic flux is onlyconsidered in the yoke; and an alternating field, similar to
the Epstein samples, can be assumed. Similar to the Epstein
frame measurements (see Section II), the form factor of the
secondary voltage is controlled to be in the range 1.111±1%.Additionally, the same power amplifier and power analyzer as
for the Epstein frame measurements are used (see Section II).
The lamination stacks are demagnetized before any readings
are taken.
B. Simulation
The obtained BH -curves for the fully degraded and non-degraded zones of the Epstein samples are applied to the
stator geometry implemented in a finite element simulation(FEMM [17] coupled with Matlab [18]), see Fig. 10. The same
degradation depth, d, is set at the cut edges as for the Epsteinsamples (see Section III).
In contrast to the Epstein samples, the stator lamination
stacks have been pressed (around 4MPa) and glued. Thisinfluence of pressing and gluing is added to the computed
values, so that the computed and measured loss data are
comparable. The increase of specific losses due to the pressing
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Fig. 10. Application of computed magnetic characteristics to arbitrarygeometries.
and gluing is in the range of experimentally determined values
that have been presented in the literature which consider the
pressing of steel sheets into the thickness direction and is based
on [19]. A more detailed explanation of this has been given
in [6].
C. Results
Selected computed and measured results of the investigated
stator samples are presented in Figs. 11-13. In all cases, the
measured and computed specific losses of the investigatedstator lamination stacks are in good agreement, for different
materials as well as different frequencies. Thus, the proposed
identification procedure for laser-cut laminations can also
serve as a good tool in the design process of electrical
machines to obtain more accurate data of changed magnetic
characteristics of electric steel sheets due to cutting.
VIII. SUMMARY
A method which considers the influence of cutting that
had been proposed for mechanically-cut samples [6] has also
been applied to laser-cut samples. This method is verified
by measurements on several stator lamination stacks withdifferent materials and at different frequencies. The computed
specific losses for the stator lamination stacks correspond
with the measured ones. This method has been validated for
frequencies up to 500Hz.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60
10
20
30
40
50
60
70
80
90
Magnetic flux density (T)
P o w e r l o s s d e n s i t y ( W / k g )
measured
computed excl. pressing
computed incl. pressing
Fig. 11. Measured and computed loss curves of the investigated statorlamination stack at 50Hz, material M270-35A.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80
10
20
30
40
50
60
Magnetic flux density (T)
P o w e r l o s s d e n s
i t y ( W / k g )
measured
computed excl. pressing
computed incl. pressing
Fig. 12. Measured and computed loss curves of the investigated statorlamination stack at 250Hz, material M400-50A.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.80
1
2
3
4
5
6
7
8
Magnetic flux density (T)
P o w e r l o s s d e n s i t y ( W / k g )
measured
computed excl. pressing
computed incl. pressing
Fig. 13. Measured and computed loss curves of the investigated statorlamination stack at 50Hz, material M800-65A.
ACKNOWLEDGEMENT
The authors would like to thank Dr. M. Braun of Dr.-Ing.
Ernst Braun GmbH and Mr. A. Peter of Kienle & Spiess
GmbH for making the measurement equipment and material
samples available.
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APPENDIX A
Generalizations implemented in the basic identification pro-
cedure, reviewed from [6]:
1) Interpolation and sampling. For the identification proce-
dure there is the need to obtain matching values for H 1
and H 2. This is achieved by interpolating the samplingpoints. As shown in [6], cubic splines provide the best re-
sults with regard to accuracy respecting all magnetization
curve characteristics. This is in contrast to other well-
known fitting functions as e.g., the Brauer curve [20],
the Bertotti curve [21] or the Langevin curve [22]. The
sampling points for the magnetic field strength H aretaken from the measurements of the first sample.
2) Rayleigh region. The measurements for small fields may
lack accuracy. As a consequence, the Rayleigh effect is
only marginally represented in the measurement curves.
Under these circumstances, eq. (1) may lead to invalid
results, e.g., Bdg may become negative. Two remedies
were already shown in [6].Note, even when the (H 1, B1)-curves and (H 2, B2)-curves are “cleaned” to correctly represent the Rayleigh
region, this quality is not carried over by the identification
procedure represented by eq. (1) to the (H, Bnd)- and(H, Bdg)-curves. There, the clean-up procedure shouldbe repeated. In fact, it makes more sense to align the
Rayleigh region for the (H, Bnd)- and (H, Bdg)-curves,because these are assumed to correspond to individual
materials, whereas the (H, B1)- and (H, B2)-curves areparticularly averaged material properties.
3) Full-saturation region. It is necessary to clearly define the
extrapolation of the BH -characteristics for large fields.
• Extrapolating from the last measurement point witha differential permeability, µ0, is based on physicalunderstanding, but may introduce a discontinuity in
the differential permeability which may hamper the
convergence of the Newton method applied later on.
As long as the (H 1, B1)-curve lies above the (H 2, B2)-curve, this procedure guarantees consistent results for
(H, Bnd) and (H, Bdg).• Another technique extends the BH -curves according
to the slope determined by the two last measurement
points. This approach may be inappropriate when the
small variations of the magnetic flux density are not
sufficiently resolved by the measurements. Moreover,
it should be verified that the final slope for (H 1, B1)is larger or equal to that of (H 2, B2).
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