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Transcript
3876
IEEE TRANSACTIONS ON MAGNETICS, VOL. 34, NO. 6, NOVEMBER 1998
Calculation of No-Load Induction Motor Core
Losses Using the Rate-Dependent Preisach Model
Johan J. C. Gyselinck, Luc R. L. Dupré, Lieven Vandevelde, and Jan A. A. Melkebeek, Senior Member, IEEE
Abstract— In this paper the authors present a two-step algorithm for predicting the core losses in an electrical machine. As a
first step, the flux patterns in the cross section of the machine
are calculated by using a time stepped two-dimensional finite
element (FE) model, neglecting hysteresis and eddy currents in
the laminated core. The second step consists in enforcing the
calculated tooth and yoke flux waveforms to a one-dimensional
FE lamination model in which the variation along the thickness
of the induction and of the induced eddy currents is considered.
The hysteretic behavior of the ferromagnetic material is taken
into account by means of a rate-dependent Preisach model. The
outlined procedure is applied to a 3kW squirrel-cage induction
motor with either open or closed rotor slots, the former yielding
elevated flux harmonics. Computation results and measurements
at no-load (phase currents, stator tooth flux, and total iron losses)
are compared.
Index Terms—Electric machines, finite element methods, hysteresis, induction motors, magnetic fields, magnetic losses.
Fig. 1. Lamination model.
I. INTRODUCTION
E
LECTRICAL steels are usually characterized under standard and well defined flux conditions, i.e., alternating
and sinusoidal (50 or 60 Hz) flux using the Epstein frame.
Straightforward extrapolation of Epstein loss data for predicting machine core losses is bound to produce a significant
underestimation of the losses. Indeed, due to the slotting
and—in case of inverter supply—due to the harmonic contents
of the voltage supply, the induction may be considerably
distorted. Furthermore, in some parts of the machine the
induction is rotational rather than alternating [1].
The discrepancy may be overcome by applying global
empirical correction factors, as was the design practice for
many decades, or by using a more physical approach, e.g.,
some kind of loss separation concept [2], [3].
In [4] the rate-independent scalar Preisach hysteresis model
embedded in a one-dimensional (1-D) lamination model is
used for calculating the core losses in a switched reluctance
motor. In this paper a similar method is applied to a squirrelcage induction machine, employing a rate-dependent Preisach
Manuscript received December 31, 1996; revised April 1, 1998. This
work was supported by the Inter-University Attraction Poles for fundamental
research funded by the Belgian State and the Ministry of Economy of the
Flemisch Government in collaboration with OCAS, the Research Centre of
ARBED-SIDMAR-ALZ.
The authors are with Laboratory for Electrical Machines and Power
Electronics, Department of Electrical Power Engineering, University of Gent,
B-9000 Gent, Belgium (e-mail:[email protected]).
Publisher Item Identifier S 0018-9464(98)07331-2.
model and calculating the flux pattern inside the machine core
by means of the finite element (FE) method.
II. PROPOSED METHOD
A. Two-Dimensional FE Machine Model
The heart of the machine model is a two-dimensional (2-D)
FE representation of the cross section. The FE model may
contain two distinct types of electrical conductors: stranded or
filamented conductors on the one hand and massive conductors
which may display skin effect on the other hand. These
conductors are embedded in an electrical network so as to
model voltage supply and the actual interconnection of coil
sides in windings and bars in cages. The end effects due to
end windings and end rings are accounted for by inserting
resistances and inductances in the electrical network.
Anisotropy, hysteresis, and eddy currents in the laminations
are not considered. Magnetic saturation is introduced by means
curve.
of a single valued
The coupled field-circuit system is time stepped by using
efficient solving techniques [5].
B. One-Dimensional FE Lamination Model
The magnetic field in the machine, resulting from the 2
FE time stepped model, is approximated as a uniform and
alternating flux in a number of flux tubes, judiciously chosen
in the motor cross section. These flux waveforms are fed to a
0018–9464/98$10.00  1998 IEEE
GYSELINCK et al.: CALCULATION OF NO-LOAD INDUCTION MOTOR CORE LOSSES
3877
1-D lamination model with thickness 2 , length , and width
(
,
), as shown in Fig. 1. The variation of
, the magnetic field
the magnetic induction
, and the induced current
along the thickness of the lamination (
) due
to an alternating flux along the -direction is governed by the
following differential equations:
(1)
(2)
where is the electrical conductivity.
The Preisach model, well described in literature (e.g., [6]),
and . The
supplies the hysteretic relationship between
represents the relative
Preisach distribution function
density of the elementary dipoles with switching fields and
and accurately describes the static hysteretic behavior of
the material. At higher frequencies rate dependence should be
included in the Preisach model in order to preserve a good
agreement with measurements. The finite rate at which the
dipoles switch can be defined by means of a single, frequency
independent, parameter [7], [8].
The Preisach function is obtained by differentiating twice
. The latter has a simple
the Everett function
physical meaning and is directly (i.e., without fitting) identified
by means of quasi-static measurements on a stacked ring core
is obtained by fitting measured and
[8]. The parameter
calculated dynamic hysteresis loops.
Equation (1) is solved numerically using a FE spatial
discretization and a finite difference time discretization.
III. APPLICATION
The iron losses calculation method presented in this paper
has been applied to a three-phase 3 kW 4-pole induction motor.
The stator has a single layer winding with three slots per pole
and per phase. Two rotors are considered, the 32 unskewed
slots of which are either open or closed.
The motor, delta-connected and with either of the two
0.0004).
rotors mounted, has been tested at no-load (slip
Measurements have been carried out at a reduced sinusoidal
voltage (185 V, 50 Hz instead of the rated 220 V, 50 Hz),
because of the practical limitations of the present measuring
set up for identifying the Everett function.
Enforcing anti-periodicity conditions for both the magnetic
field in the FE mesh (3034 nodes, 5306 triangular elements)
and the electrical network voltages and currents, only one pole
has been modeled. A flux plot at no-load is displayed in Fig. 2.
The electrical network that represents one pole of the squirrel
cage is depicted in Fig. 3.
A. Measurements on the Motor and Calculation
Results with the 2-D FE Model
The stator phase currents and the line voltages have been
measured by means of a data-acquisition system. A single turn
Fig. 2. Flux plot at no load.
Fig. 3. Electrical network for the squirrel cage.
search coil has been fit around a stator tooth. Integration of
the search coil voltage yields the tooth flux.
The waveforms of measured and calculated stator phase
currents and stator tooth flux are shown in Fig. 4 (for sake
of brevity and clarity, the different waveforms have been
phase shifted with respect to each other, as is also the case
in Fig. 7).
The “measured” iron losses are obtained by measuring or
calculating the different components of the power balance. The
results are listed in Table I. Multiplication of the three phase
currents and voltages readily produces the electrical power
input. The resistance of the stator windings was measured
immediately after the tests, allowing an accurate calculation
of the stator joule losses. The joule losses in the rotor cage are
predicted using the 2-D FE model. The mechanical friction
losses have been estimated from the rotor inertia and the
instantaneous deceleration when interrupting the power supply.
B. Identification of the Electrical Steel
Some laminations (VH 800-65D) of the test motor have
been punched and assembled to form a stacked ring core.
Equilines of the measured Everett function are displayed in
Fig. 5.
A good agreement between calculated and measured iron
losses, in the lamination model and in the ring core, respectively, has been observed in a frequency range up to 1 kHz
and for arbitrary waveforms [9].
3878
IEEE TRANSACTIONS ON MAGNETICS, VOL. 34, NO. 6, NOVEMBER 1998
Fig. 5. Measured Everett function
E (H1 ; H2 )
.
(a)
Fig. 6. Flux tubes in stator tooth and yoke.
C. Loss Density Calculation
(b)
Fig. 4. Measured and calculated waveforms of stator phase current and stator
tooth flux.
TABLE I
POWER BALANCE YIELDING THE MEASURED IRON LOSSES
1) Stator: Fig. 6 shows the division of a stator tooth into
six flux tubes, the flux passing through sections 1–7. The
varying cross section and the slot leakage flux are thus
accounted for. A yoke segment is split into four parallel flux
tubes (sections 8–11), considering the lower induction levels
toward the outer stator boundary.
Distinction should further be made between stator teeth in
between slots of the same phase belt (SPB, totaling 24 teeth)
and teeth in between slots of different phase belts (DPB,
12 teeth), the latter displaying a higher but less distorted
induction. As for the yoke segments, such a distinction was
found not to be required.
Some calculated induction waveforms (under the assumption of uniform flux) and hysteresis loops are depicted in
GYSELINCK et al.: CALCULATION OF NO-LOAD INDUCTION MOTOR CORE LOSSES
3879
TABLE IV
TOTAL IRON LOSSES
Fig. 7. Calculated induction waveforms.
TABLE II
STATOR TOOTH IRON LOSS
3) Total Iron Losses: Table IV lists the total iron losses and
the contribution of stator teeth, stator yoke segments, and—if
present—rotor slot bridges. About one third of the stator iron
volume is situated in the teeth (according to Fig. 6). In case
of open rotor slots, the stator teeth produce 40% of the total
iron losses due to the considerable harmonic distortion of the
induction. In case of closed rotor slots, the average loss density
in tooth and yoke do not differ so much.
The computational cost of time stepping the 2-D machine
model and the 1-D lamination model depends on the space
and time discretization and on the number of periods (of 20
ms) to reach steady state. We use a 2-D first order triangular
mesh with 3000 nodes and a 1-D second order mesh with 21
nodes, and 6 400 and 3 1000 time steps, respectively. The
measured Preisach distribution function is stored in about 6000
plane. The 2-D calculation and each 1-D
points in the half
calculation take about 60 and 20 CPU minutes, respectively,
on a Alpha Work station 200/166.
D. Discussion
TABLE III
STATOR YOKE IRON LOSSES
Figs. 7 and 8, respectively. The calculated iron loss densities
are presented in Tables II and III.
2) Rotor: At no-load the iron losses in the rotor can be
expected to be negligible. Indeed, the calculated loss density
in the bulk of rotor teeth and rotor yoke is less than 0.02 W/kg.
The ensuing total iron losses, less than 0.1 W, are negligible
compared to the stator iron losses.
However, in case of closed rotor slots, the slots bridges
may contribute significantly to the iron losses. The magnetic
induction in the bridges over the closed slots has a very large
900 Hz component and may be quite rotational. Application of
the lamination model yields local loss densities ranging from
80 up to 500 W/kg for the eight bridges, yielding additional
losses estimated at 5 W.
The difference between measured and calculated iron losses
(Tables I and IV) is 15% and 8% for open and closed rotor
slots, respectively. This can be attributed to the different
assumptions and simplifications implied by “measuring” the
iron losses as described above and by using the presented
motor, lamination, and material model, some of which are
briefly discussed hereafter.
1) A Posteriori Inclusion of Hysteresis and Eddy Currents:
The 2-D field calculations have been done with a single valued
curve. Hysteresis and eddy currents have been accounted
for a posteriori using the 1-D model. In [10] this two step
approach is compared to the 2-D hysteretic case. The BH loops
obtained with these two approaches differ to some extent, as
can be seen in Fig. 9. However, the loss density is observed
to differ only slightly.
2) Alternating versus Rotational Flux: An alternating flux
has been assumed throughout the iron core. This is very
questionable in the rotor slot bridges and in the vicinity of
the stator tooth-yoke interface as can be seen in Fig. 10. This
figure shows the B-loci in five points in the stator, the position
of which is indicated in Fig. 6. Apparently, the induction in
points 3 and 4 is significantly rotational, inconsistent with the
assumption of alternating flux.
Rotational flux causes an increase of the classical eddy current losses. As for the hysteresis losses, the ratio of rotational
to alternating losses depends on the peak induction and may
become smaller than one for peak inductions higher than 1.5 T
[2]. In [1] a 9% increase of induction motor iron losses due
to rotational flux is reported.
3880
IEEE TRANSACTIONS ON MAGNETICS, VOL. 34, NO. 6, NOVEMBER 1998
(a)
(b)
(c)
Fig. 8. Calculated hysteresis loops. (Hs : magnetic field at the surface of the lamination, Ba : magnetic induction averaged out over the thickness
of the lamination).
3) Machining and Assembling Laminations: The Preisach
function has been obtained using a stacked ring core.
Machining and assembling the laminations in motor core and
ring core, respectively, may have considerable effect on the
magnetic properties of the steel [9]. The imperfect insulation
of the laminations has been disregarded as well.
IV. CONCLUSION
Fig. 9. Two-dimensional BH loops with direct and a posteriori hysteresis,
respectively.
A two-step algorithm for predicting the iron losses in a
squirrel cage induction motor has been proposed. The flux
patterns in the cross section of the motor, calculated by means
of the 2-D FE method, are approximated as an alternating and
uniform flux in a number of flux tubes. The flux waveforms
are fed to a 1-D lamination model incorporating the powerful
rate-dependent Preisach hysteresis model.
GYSELINCK et al.: CALCULATION OF NO-LOAD INDUCTION MOTOR CORE LOSSES
3881
[7] G. Bertotti, “Dynamic generalization of the scalar Preisach model of
hysteresis,” IEEE Trans. Magn., vol. 28, pp. 2599–2601, 1992.
[8] L. Dupré, “Electromagnetic characterization of nonoriented electrical
steel” (in Dutch), Ph.D. dissertation, Faculty of Applied Sciences,
University of Gent, Belgium, 1995.
[9] D. Philips, L. Dupré, and J. Melkebeek, “Magneto-dynamic computation
using a rate-dependent Preisach model,” IEEE Trans. Magn., vol. 30,
pp. 4377–4379, Nov. 1994.
[10] R. Van Keer, L. Dupré, and J. Melkebeek, “Computational methods
for the evaluation of the electromagnetic losses in electric machines,”
Archives Computational Methods in Engineering, to be published.
Johan J. C. Gyselinck graduated in electrical and mechanical engineering
from the University of Gent, Belgium, in 1991.
He joined the Laboratory for Electrical Machines and Power Electronics,
Belgium, in 1993 as a Research Assistant. His research interests concern the
finite element analysis of electrical machines.
Fig. 10.
B-loci in stator iron (SPB points defined in Fig. 6).
Measurements and calculations have been done using two
rotors with either open or closed rotor slots. The former
displays an elevated tooth flux harmonic distortion.
A good agreement between calculated stator currents and
tooth fluxes has been observed in both cases. The iron losses
calculation method presented in this paper has produced losses
that are 15% and 8%, respectively, less than the measured
losses.
Luc R. L. Dupré graduated in electrical and mechanical engineering in 1989
and received the Ph.D. degree in 1995, both from the University of Gent,
Belgium.
He joined the Laboratory for Electrical Machines and Power Electronics,
Belgium, in 1989 as a Research Assistant. His research interests mainly
concern the finite element analysis of electrical machines, advanced modelling,
and characterization of magnetic materials (electrical steels).
Lieven Vandevelde graduated in electrical and mechanical engineering in
1992 and received the Ph.D. degree in 1997, both from the University of
Gent, Belgium.
He joined the Laboratory for Electrical Machines and Power Electronics,
Belgium, in 1992 as a Research Assistant. His research interests mainly concern noise and vibrations of electrical machines, magnetic force computations,
and finite element analysis of electrical machines.
Dr. Vandevelde is a member of the Koninklijke Vlaamse Ingenieursvereniging (K.VIV).
ACKNOWLEDGMENT
The authors gratefully acknowledge Brook Hansen, Huddersfield, U.K., for supplying the motor.
REFERENCES
[1] R. Findlay, N. Stranges, and D. Mackay, “Losses due to rotational flux
in three phase induction motors,” in Proc. 2nd Int. Workshop Electric
Magn. Fields 1994, Leuven, Belgium, pp. 25–28.
[2] G. Bertotti, A. Boglietti, M. Chiampi, D. Chiarabaglio, F. Fiorillo, and
M. Lazzari, “An improved estimation of iron losses in rotating electrical
machines,” IEEE Trans. Magn., vol. 27, pp. 5007–5009, Nov. 1991.
[3] M. Jamil, P. Baldassari, and N. Demerdash, “No-load induction motor
core losses using a combined finite element state space model,” IEEE
Trans. Magn., vol. 28, pp. 2820–2822, Sept. 1992.
[4] D. Philips and L. Dupré, “A method for calculating switched reluctance
motor core losses,” in Proc. Int. Aegean Conf. Elect. Mach. Power
Electron. 1992, Kusadasi, Turkey, May 27–29, 1992, pp. 124–129.
[5] J. Gyselinck and J. Melkebeek, “Numerical methods for time stepping
coupled field-circuit systems,” in Proc. ELECTRIMACS 1996, SaintNazaire, France, Sept. 17–19, 1996, vol. 1, pp. 227–234.
[6] D. Philips, L. Dupré, J. Cnops, and J. Melkebeek, “The application of the
Preisach model in magnetodynamics: Theoretical and practical aspects,”
J. Magnetism Magn. Mater., vol. 133, pp. 540–543, 1994.
Jan A. A. Melkebeek (M’82–SM’84) graduated in electrical and mechanical
engineering in 1975, received the Doctor in Applied Sciences degree in 1980,
and the “Doctor Habilitus” in electrical and electronical power technology in
1986, all from the University of Gent, Belgium.
He was a Visiting Professor at the Université Nationale de Rwanda
in Butare, Rwanda, Africa, in 1981 and a Visiting Assistant Professor
at the University of Wisconsin, Madison, in 1982. Since 1987, he has
been a Professor in Electrical Engineering (Electrical Machines and Power
Electronics) at the Engineering Faculty of the University of Gent, as well as
the Director of the Laboratory for Electrical Machines and Power Electronics
(the former Laboratory for Industrial Electricity). Since 1993, he has also been
the Head of the Department of Electrical Power Engineering. His teaching
activities and research interests include electrical machines, power electronics,
variable frequency drives, and control systems theory applied to electrical
drives.
Dr. Melkebeek is a member of the Koninklijke Vlaamse Ingenieursvereniging (K.VIV), a member of the board of the “Technologisch instituut” of the
K.VIV, a member of the Koninklijke Belgische Vereniging van Elektrotechnici
(KBVE-SRBE), a member of the Belgian Federation for Automatic Control
(BIRA-IBRA), and a fellow of the IEE. He also serves as a member of
the IEEE-IAS Electric Machines Committee and of the IEEE-PES Machine
Theory Subcommittee.