658 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 42, NO. 3, MAY/JUNE 2006
On the Variation With Flux and Frequency of the
Core Loss Coefficients in Electrical Machines
Dan M. Ionel, Senior Member, IEEE, Mircea Popescu, Senior Member, IEEE, Stephen J. Dellinger,
T. J. E. Miller, Fellow, IEEE, Robert J. Heideman, and Malcolm I. McGilp
Abstract—A model of core losses, in which the hysteresis coeffi-
cients are variable with the frequency and induction (flux density)
and the eddy-current and excess loss coefficients are variable only
with the induction, is proposed. A procedure for identifying the
model coefficients from multifrequency Epstein tests is described,
and examples are provided for three typical grades of non-grain-
oriented laminated steel suitable for electric motor manufac-
turing. Over a wide range of frequencies between 20–400 Hz
and inductions from 0.05 to 2 T, the new model yielded much
lower errors for the specific core losses than conventional models.
The applicability of the model for electric machine analysis is also
discussed, and examples from an interior permanent-magnet and
an induction motor are included.
Index Terms—Brushless permanent-magnet (PM) motor, core
loss, electric machine, Epstein test, finite-element analysis (FEA),
induction motor, iron loss, laminated steel.
I. INTRODUCTION
S
INCE its first formulation by Steinmetz more than a hun-
dred years ago [1], the model of power losses in ferro-
magnetic materials has been continuously under study. Jordan
brought a significant contribution by defining the hysteresis and
eddy-current components [2] on which the analysis of electri-
cal machines is still based. Improved models based on these
concepts, e.g., [3] and [4], combined with careful calibration
against experimental data collected from generic motor designs,
have been typically used in industrial practice.
More recently, Bertotti proposed a frequency domain model
including one supplementary term of excess or anomalous
loss [5]. The model, which employs material-dependent con-
stant coefficients, was further extended into the time domain
[6], gained popularity in the electrical machines community,
and was used in various forms in example studies, such as
[7]–[9]. However, the general applicability of the model re-
mained under scrutiny, and a new benchmark study, which
Paper IPCSD-05-112, presented at the 2005 IEEE International Electric
Machines and Drives Conference, San Antonio, TX, May 15–18, and approved
for publication in the IEEE T
RANSACTIONS ON INDUSTRY APPLICATIONS by
the Electric Machines Committee of the IEEE Industry Applications Society.
Manuscript submitted for review July 1, 2005 and released for publication
January 31, 2006.
D. M. Ionel and R. J. Heideman are with the Corporate Technology
Center, A. O. Smith Corporation, Milwaukee, WI 53224-9512 USA (e-mail:
dionel@aosmith.com; rheideman@aosmith.com).
M. Popescu, T. J. E. Miller, and M. I. McGilp are with the SPEED Lab-
oratory, Department of Electrical Engineering, University of Glasgow, Glas-
gow G12 8LT, U.K. (e-mail: mircea@elec.gla.ac.uk; t.miller@elec.gla.ac.uk;
mal@elec.gla.ac.uk).
S. J. Dellinger is with the Electrical Products Company, A. O. Smith Corpo-
ration, Tipp City, OH 45371-1899 USA (e-mail: sdellinger@aosepc.com).
Digital Object Identifier 10.1109/TIA.2006.872941
was conducted by a large number of research groups in Japan,
provided good correlation between a surface permanent-magnet
(PM) brushless motor experimental data and computations
performed with steel models that ignored the anomalous loss
component [10]. In another recent paper, Boglietti et al. [11]
investigated eight different materials at inductions between 0.6
and 1.7 T and frequencies between 10 and 150 Hz, system-
atically identified a zero value for the excess loss coefficient,
and observed that, based on Epstein frame experiments, the
individual contributions of eddy-current and anomalous losses
cannot be separated. In yet another relevant paper, Chen and
Pillay proposed a model with invariable coefficients for the
eddy-current and excess loss and variable hysteresis loss pa-
rameters [12], an approach that combined and extended the
concepts introduced by Hendershot and Miller [3], Bertotti [5],
Slemon and Liu [13], and Miller et al. [14].
This paper brings further original contributions to the subject
by studying three different laminated steels for electric motors
on a wide range of frequencies between 20 and 400 Hz and
inductions from 0.05 to 2 T. A mathematical model fitting
procedure, which results in the coefficients of the core loss
components being variable with frequency and/or induction, is
introduced and proved to yield relatively small errors between
the numerical estimations and the Epstein measurements. The
comparison between the improved model and a conventional
model provides interesting insights into the separation of core
loss components. Also included are two example studies from
a prototype interior permanent-magnet (IPM) machine and an
induction motor.
II. E
PSTEIN FRAME MEASUREMENTS
One of the materials considered in this paper is a widely
available generic M43 fully processed electric steel. The other
two materials are varieties of semiprocessed cold-rolled electric
steel, which after annealing have the main characteristics listed
in Table I and will be denoted as SPA and SPB. All three are
non-grain-oriented steel alloys and are suitable for the high-
volume production of rotating electrical machines.
Samples of the materials were tested in an Epstein frame,
which was built according to ASTM standard [15]. The exci-
tation and measurement system was provided by a Brockhaus
Messtechnik MPG100D 3 Hz to 1 kHz ac/dc hysteresisgraph
equipped with an amplifier rated at peak values of 40 A
and 110 V [16]. The repeatability of the hysteresisgraph is
certified by the instrument manufacturer at 0.1% for magnetic
field measurements and 0.2% for power loss measurements.
0093-9994/$20.00 © 2006 IEEE
IONEL et al.: VARIATION WITH FLUX AND FREQUENCY OF CORE LOSS COEFFICIENTS IN ELECTRICAL MACHINES 659
TABLE I
M
AIN CHARACTERISTICS OF SAMPLE MATERIALS
Fig. 1. Core losses measured in an Epstein frame on a 1sample of SPA
(semiprocessed electric steel of type A).
Fig. 2. Core losses measured in an Epstein frame on a sample of SPB
(semiprocessed electric steel of type B).
Magnetic permeability and core loss measurements (Figs. 1–3)
were performed over a wide range of frequencies in induction
increments of 0.05 T, according to an experimental procedure
suggested in [17]. (The terminology of core loss, rather than
iron loss, and induction, rather than flux density, follows the
relevant ASTM standards [15].)
III. N
EW MODEL FOR SPECIFIC CORE LOSSES
Under sinusoidal alternating excitation, which is typical for
form-factor-controlled Epstein frame measurements, the spe-
cific core losses w
Fe
in watts per pound (or watts per kilogram)
can be expressed by
w
Fe
= k
h
fB
α
+ k
e
f
2
B
2
+ k
a
f
1.5
B
1.5
(1)
Fig. 3. Core losses measured in an Epstein frame on a sample of M43 fully
processed electric steel.
where the first right-hand term stands for the hysteresis loss
component and the second for the eddy-current loss component.
The last term corresponds to the excess or anomalous loss
component, which is influenced by intricate phenomena, such
as microstructural interactions, magnetic anisotropy, nonho-
mogenous locally induced eddy currents. Despite the compli-
cated physical background and based on a statistical study,
Bertotti has proposed the simple expression for the excess
losses, similar to that of the eddy-current losses, but with an
exponent value of 1.5 [5]. In a conventional model, the values
of the coefficients k
h
, α, k
e
, and k
a
are assumed to be constants,
which are invariable with frequency f and induction B.
As the first step of the procedure developed in order to
identify the values of the coefficients, (1) is divided by the
frequency resulting in
w
Fe
f
= a + b
f + c
f
2
(2)
where
a = k
h
B
α
b = k
a
B
1.5
c = k
e
B
2
. (3)
For any induction B at which measurements were taken,
the coefficients of the aforementioned polynomial in
√
f can
be calculated by quadratic fitting based on a minimum of
three points (Fig. 4). During trials, it was observed that a
sample of five points, represented by measurements at the same
induction and different frequencies, is beneficial in improving
the overall stability of the numerical procedure. In this paper,
measurements at one low frequency of 25 Hz (or 20 Hz),
three intermediate frequencies of 60, 120, and 300 Hz, and
one high frequency of 400 Hz were used where available
(Figs. 1–3), and, typically, the values of the fitting residual for
(2) were very close to unity, i.e., r
2
≈ 1, indicating a very good
approximation.
From (2) and (3), the eddy-current coefficient k
e
and the
excess loss coefficient k
a
are readily identifiable. These co-
efficients are independent of frequency, but, unlike those for
the conventional model, they exhibit a significant variation
660 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 42, NO. 3, MAY/JUNE 2006
Fig. 4. Ratio of core loss and frequency w
Fe
/f, as a function of
f
according to (2), for SPA steel.
Fig. 5. Variation of the eddy-current loss component coefficient k
e
with
magnetic induction; k
e
is invariable with frequency.
with the induction (Figs. 5 and 6). The following third-order
polynomials were employed for curve fitting of k
e
and k
a
:
k
e
= k
e0
+ k
e1
B + k
e2
B
2
+ k
e3
B
3
(4)
k
a
= k
a0
+ k
a1
B + k
a2
B
2
+ k
a3
B
3
. (5)
For k
e
, the best r
2
was obtained for SPB with a value of
0.98, followed by SPA at 0.87 and M43 at 0.75. For k
a
, r
2
varied from 0.883 for M43 to 0.82 for SPB and down to 0.78
for SPA. The discrete variations of k
e
and k
a
at high induction
are noticeable in Figs. 5 and 6, and these could be attributed,
at least in part, to the fact that less than five fitting points were
available for fitting (2). The use of a lower order polynomial in
(4) and (5) is not recommended, as it leads to a poorer data fit
with a considerably lower r
2
.
One possible explanation for the variation of k
e
and k
a
with
induction—the two coefficients having somehow complemen-
tary trends (see Figs. 5 and 6), i.e., k
e
substantially increasing
and k
a
substantially decreasing with B, respectively, after k
e
has experienced a minimum value in the range of 0.3–0.5 T and
k
a
a local maximum around 0.5–0.7 T—could lay in the 1.5
fixed exponent value of the anomalous loss component and/or
Fig. 6. Variation of the excess (anomalous) loss component k
a
with magnetic
induction; k
a
is invariable with frequency.
in the fact that the separation in-between the eddy-current
and anomalous losses is questionable, this being a hypothesis
already advanced by other authors [11] based on a different
analysis than ours. On the other hand, it should be mentioned
that yet other authors [12], by following a similar frequency
separation procedure as per (2) and (3), were able to identify
constant valued coefficients k
e
and k
a
—a result that we have
not experienced on any of the three steels reported in this paper
or on any other steels that we have studied.
In order to identify the coefficients k
h
and α, which can be
traced back to Steinmetz’s original formula, further assump-
tions have to be made regarding their variation. An improved
model, in which α is a first-order polynomial of flux density,
has already been in use for a number of years in a commercially
available motor design software [4]. Recently, in [12], a second-
order polynomial has been proposed for α, and in our new
formulation, the following third-order polynomial is employed:
α = α
0
+ α
1
B + α
2
B
2
+ α
3
B
3
. (6)
Substituting (6) in (3) and applying a logarithmic operator
leads to an equation
log a = log k
h
+
α
0
+ α
1
B + α
2
B
2
+ α
3
B
3
log B (7)
with five unknowns, namely k
h
and the four polynomial coeffi-
cients of α. The coefficient a represents the ratio of hysteresis
loss and frequency, which is calculated from (2) by substituting
the values of b and c from (3) and making use of the analytical
estimators (4) and (5), which greatly reduce numerical insta-
bilities. The plot of log a against induction at a set frequency
indicates three intervals of different variation types, which,
for the example shown in Fig. 7, can be approximately set
to induction ranges of 0.0–0.7, 0.7–1.4, and 1.4–2 T. For a
given frequency and induction range, (7) is solved by linear
regression using at least five induction values, i.e., log B.The
discrete values of the hysteresis loss coefficient k
h
and the
average values ¯α for the three materials studied are listed in
Tables II–IV.
It is interesting to note that the aspect of the log a curves
plotted in Fig. 7 also provides support to an observation made
IONEL et al.: VARIATION WITH FLUX AND FREQUENCY OF CORE LOSS COEFFICIENTS IN ELECTRICAL MACHINES 661
Fig. 7. Logarithm of the ratio of hysteresis loss and frequency for SPA steel;
curves for different frequencies are overlapping.
TABLE II
H
YSTERESIS LOSS COEFFICIENTS FOR SPA STEEL
TABLE III
H
YSTERESIS LOSS COEFFICIENTS FOR SPB STEEL
by other authors in [10], where a two-step approximation of
k
h
and α was proposed without the disclosure of any other
details. In our model, an estimation with three induction steps
is employed for k
h
and α.
While other numerical models with some type of variable
hysteresis coefficients have already been published, e.g., [3],
[10], [12], and [14], a phenomenological theory to support such
TABLE IV
H
YSTERESIS LOSS COEFFICIENTS FOR M43 STEEL
a mathematical formulation is not yet unanimously accepted.
One possible explanation can lay in the fact that the area of
the quasi-static magnetization loop, which is a measure of
the hysteresis losses, is influenced by the dynamic losses [5],
[6] and that the instability of the magnetic domains at the
microscopic level is a nonlinear and complicated function of
magnetization and frequency.
Based on the measurement of core losses w
Fe
at different
inductions B
k
and frequencies f
i
, the calculation of the eddy
currents, k
e
, excess, k
a
, and hysteresis, k
h
and α, coefficients
is summarized by the following computational procedure:
Start
For each B
k
For each f
i
Compute the ratio w
Fe
(f
i
,B
k
)/f
i
EndFor
Curve fit (2)
Compute k
e
(B
k
) and k
a
(B
k
) with (2) and (3)
EndFor
Polyfit k
e
(B) with (4) and k
a
(B) with (5)
For each B
k
Compute a = k
h
B
α
k
from (2) and (3) using (4)
and (5)
Compute log a; see (6) and (7)
EndFor
Plot log a versus B and identify curve
inflexions
Define intervals of B for k
h
and α
For each B interval with a minimum of
five values of B
k
For each f
i
Solve (7) for k
h
, α
0
, α
1
, α
2
and α
3
For each B
k
Compute α with (6)
EndFor
Compute average ¯α for B interval
EndFor
EndFor
End
662 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 42, NO. 3, MAY/JUNE 2006
Fig. 8. Relative error between the calculated and the Epstein measured core
loss at the frequencies used in the numerical model fitting for SPA steel.
Fig. 9. Relative error between the calculated and the Epstein measured core
loss at frequencies not used in the numerical model fitting for SPA steel.
The new core loss model covers frequencies up to 400 Hz
and a very wide induction range between 0.05 and 2 T, and yet,
the relative error between the estimated and measured specific
core losses is very low, as shown in Fig. 8 for SPA steel. The
results in Fig. 8 were produced using the actual value of α at
each set B, as per (6). The errors for the SPB and M43 steel,
which are not included here for brevity, are even lower.
The model was also used to estimate losses at frequencies
not employed in the curve-fitting procedure, and an example is
provided in Fig. 9. In this case, analytically fitted values, as per
(4) and (5), were used for k
e
and k
a
, and linearly interpolated
values from Tables II–IV were employed for k
h
and average
¯α. The errors are still well within limits considered satisfactory
for most practical engineering applications and considerably
lower than those provided by other known models, which
represents, in our opinion, a remarkable result.
IV. C
OMPARISON WITH CONVENTIONAL MODELS
The comparison of the new model with the conven-
tional model provides some interesting observations and, most
Fig. 10. Relative error between the values estimated by a conventional model
with constant coefficients and Epstein measured core losses for SPA steel. The
y-axis scale limits are ten times larger than in Figs. 8 and 9.
notably, shows that the new model can be regarded as an
extension of the classical theory rather than a contradiction of
it. For example, conventional values for the power coefficient
α from the hysteresis loss formula are typically in the range of
1.6–2.2 T. In Tables II–IV, with the new coefficient values, this
approximately corresponds to low frequencies and midrange
inductions.
According to conventional models, the eddy-current loss,
which is often referred as “classical” loss, can be estimated with
a constant value coefficient calculated as
k
e
=
π
2
σδ
2
6ρ
v
(8)
based on the electrical conductivity σ, the lamination thickness
δ, and the volumetric mass density ρ
V
. For the materials
considered, SPA, SPB, and M43, the classical values of k
e
correspond on the nonlinear curves shown in Fig. 5 to an
induction of approximately 1.3, 1.5, and 1.7 T, respectively.
Analytical estimations or typical values are not available for
k
h
and k
a
.
As a comparative exercise, coefficient values were selected
to be constant, for the hysteresis losses equal to the values
corresponding to 60 Hz and the 0.7–1.4 T range (see Table II)
and for the eddy-current and excess losses equal to the
values at 1.5 T (see Figs. 5 and 6), i.e., the actual val-
ues for the SPA steel are k
h
=0.0061 W/lb/Hz/T
α
, where
α =1.9412, k
e
=1.3334 × 10
−4
W/lb/Hz
2
/T
2
, and k
a
=
2.7221 × 10
−4
W/lb/Hz
1.5
/T
1.5
. In this case, the very large
errors and the numerical oscillations, which fall around the
selected reference point of 1.5 T, exemplified in Fig. 10, are
not a surprise and are in line with previous studies published by
other authors, e.g., [10].
Selecting different but constant values for the four coef-
ficients may change the induction around which the errors
oscillate and even reduce the maximum error but will not be
able to bring this within acceptable limits for a wide range
of frequencies and inductions due to the inherent limitations
