2D analytical model for estimation of eddy
current loss in the magnets of IPM machines
considering the reaction field of the induced
eddy currents
Milind Paradkar, Student Member, IEEE and Joachim Böcker, Senior Member, IEEE
Abstract-- In this paper a closed-form 2D analytical model
developed for the estimation of eddy current loss in interior
permanent magnet machines is presented. To check the
generality of the developed model, two different types of
commonly employed interior permanent magnet machines are
considered. 2D finite-element analysis was used to benchmark
the results from the analytical model. The proposed model can
be directly used to evaluate the eddy current magnet loss for
magnet segmentation in the circumferential direction. The
influence of air-gap length, frequency of carrier harmonic,
reaction field and non-linearity of iron due to saturation on the
magnet loss have been studied in detail and the results are
presented. The effect of axial segmentation has been considered
by incorporating an end-effect factor in the analytical model.
The results from analytical and 2D finite-element analyses are
found to be in concurrence.
Index Terms--permanent magnets, eddy current loss, magnet
segmentation, analytical model, finite element method, interior
permanent magnet synchronous motor
I. INTRODUCTION
I
NTERIOR permanent magnet (IPM) motors are widely
used in industrial applications where the ratio of nominal
speed to maximum speed is in excess of 2 as in applications
like hybrid and pure electric cars [1]. These motors are fed
from an inverter which results in high frequency time
harmonic currents. These harmonics cause eddy current loss
in the magnets resulting in temperature rise and may result in
irreversible demagnetization of the magnets. Hence the
calculation of the eddy current loss and its mitigation through
segmentation can ensure the safety of magnets of the IPM
Milind Paradkar is a research co-worker at the Department of Power
Electronics and Electrical Drives, University of Paderborn, Germany
(email: paradkar@lea.upb.de).
Joachim Böcker is a professor and head of the Department of Power
Electronics and Electrical Drives, University of Paderborn, Germany
(e-mail: boecker@lea.upb.de).
978-1-4799-7940-0/15/$31.00 ©2015 IEEE
machines at different loads and speeds. Many previous
publications have dealt with the calculations of eddy current
loss in the magnets of PM machines, but mostly for surfacemounted PM machines [2]‐[8]. Magnetic field variations in
the magnet are of the form travelling waves in the surfacemounted PM machines while being pulsating waves for IPM
machines. Therefore the analytical calculations of eddy
current loss in the surface mounted PM machines cannot be
directly applied to IPM machines. The eddy current loss
calculation for the IPM machine was investigated in [9]-[13].
Analytical and numerical finite element analyses were
compared in [9]-[12] but did not show good concurrence.
Consideration of air-gap effects on reaction fields of induced
eddy currents is absent, which is necessary for accurate
analysis of eddy currents in the magnets.
In this paper, analytical and numerical finite element analyses
of eddy current loss in the magnet of IPM machines are
presented. To validate the model, two different types of IPM
motors were considered. A 2D analytical model was
developed for eddy current loss in the magnet with the
consideration of the reaction field of induced eddy currents.
Results from the 2D time-stepping finite element method are
used to benchmark the results obtained from the analytical
model. Analytical models help in quick estimation of eddy
current loss and in determination of the appropriate magnet
segmentation required for loss reduction.
II. SPECIFICATIONS OF IPM MOTORS
Fig. 1 shows the two types of IPM motors considered for
analysis [14]. The motors have 8 poles and an internal and
external stator diameter of 160 mm and 269 mm respectively.
The air-gap length and axial length were 0.73 mm and 84 mm
respectively. The magnet in motor 1 and motor 2 has a
thickness of 6.5 mm while the width was 37.8 mm for
motor 1and 18.9 mm for motor 2.
The Cartesian co-ordinate system was chosen for the
1096
analysis as the magnets in an IPM machine are flat shaped.
The following assumptions were made for analytical
calculation of eddy currents in the magnets.
1) The permeability of iron is considered infinite
( r ∞
2) Only the normal component of the flux density is
0
considered x
z
3) Only the axial component of current density, z is
considered x
0
y
4) The effect of stator slotting is neglected.
Fig. 2 Simplified rectangular geometry for the analytical model
3
Fig. 1. IPM Motor 1 and Motor 2 considered for the analysis
The simplified 2D model for the estimation of loss is shown
in Fig. 2. The simplification is justified if the air-gap length is
very much smaller than the air-gap radius. The air-gap length
and magnet thickness is denoted by
and respectively.
For generalization, taking into account the magnet
segmentation in circumferential direction, the magnet width is
denoted by . Applying Ampere’s law for the loop shown in
the Fig. 2 results in (1).
′
If we consider the source field
is changing antiperiodically in axis direction resulting in (4). The variation
of is shown in Fig. 3 [15] [16].
4
·
,
1
·
, , , ,…
· sin
·
2
· cos
·
·
4
where, Bs,m is the maximum value of the Bs along the width of
the magnet.
1
is the sum of source field from the stator
where ′
and reaction field from the magnet eddy currents (all in
is the effective air-gap length
direction) and
(
. If the magnitude of
can be assumed to be
constant along the direction, then (1) simplifies to (2).
2
Applying Faraday’s law leads us to (3).
Fig. 3. Source field variation in a magnet segment
The considered
confines the induced eddy current within
the magnet segment. Combining (2) and (3), the resulting
differential equation for sinusoidal variation of
as in (4),
results as
5
1097
Equation (5) is solved using Fourier series method resulting
in solution of as
·
,
, , , ,…
·
·
·
·
2
· sin
· cos
·
·
6
The induced eddy currents hence result as
4
·
·
,
·
· sin
·
·
, , , ,…
· sin
·
·
7
·
350
Analytical
300
250
200
150
Inductance
-limited
Resistance-limited
0
1
2
3
4
5
6
7
8
Circumferential magnet segment number, m
Fig. 4. Loss comparison from FEM and Analytical for Motor 1
| |
·
FEM
50
450
·
·
, , , ,…
·
·
·
·
·
,
FEM
400
8
Analytical
350
Eddy current magnet loss in W
4
is estimated as 15 mT.
400
100
·
2
If the length of the magnet in the axial direction axis) is
represented by , the eddy current loss in each segmented
magnet is then computed as
·
2·
,
450
·
Eddy current loss in W
4
6.25 A. The value of
The eddy current loss in magnet segment of an IPM machine
as given by (8) is proportional to the width, thickness and
magnet conductivity. It also varies as square of the flux
density and frequency of the source field. The second term in
, represents the effect of
the denominator of (8),
reaction field on the eddy current loss. If this is neglected it
can lead to wrong results as will be seen in the next section.
300
250
200
150
100
50
0
1
III. RESULTS AND DISCUSSION
The validity of the analytical model is tested by comparing
the loss results obtained by time-stepping 2D finite element
method. On a fundamental time harmonic current with an
amplitude of 250 A at a frequency of 80 Hz at 1200 min-1, a
carrier harmonic with an amplitude of 6.25 A (2.5 % of
fundamental) at 10 kHz on the rotor side is analyzed for loss
in the magnets of the two IPM motors 1 and 2. The stator and
rotor lamination was considered linear with a relative
permeability of 5000. The magnet length
was taken as
84 mm, same as the lamination length. The conductivity of
the magnet was considered 625000 S/m for the analysis. The
flux density , is evaluated by a magneto-static FE analysis
with d-flux excitation corresponding to current amplitude of
2
3
4
Circumferential magnet segment number, m
Fig. 5. Loss comparison from FEM and Analytical for Motor 2
Fig. 4 and 5 show the comparison of the total eddy current
loss in the magnets for various segmentation number,
in
the circumferential direction, as obtained from analytical and
time stepping 2-D FE analyses for motor 1 and motor 2.
There is a good concurrence with a maximum difference of
8 %. It is observed from Fig. 4 that with increase in magnet
segmentation in the circumferential direction from 1 to 2 the
total eddy current loss actually increases nearly by a factor of
2. This region is the inductance-limited region where the
induced eddy currents in the magnets are limited by the
magnet inductance [6]. Increasing the magnet segmentation,
1098
in this region leads to increase in eddy current loss. Beyond a
certain limit, the eddy current loss starts decreasing with the
increase in segmentation number. This region is the
resistance-limited region where the induced eddy currents in
the magnets are limited by the resistance. For motor 1, the
resistance-limited region is for
2. It is obvious from the
figure that it is better to remain the magnets unsegmented
than segmenting if the chosen
lies in range 2-5. The
reduction in magnet loss due to segmentation can only be
realized for
6. For the case of motor 2, increasing
segmentation reduces the loss as it already in resistancelimited region for
1.
Eddy current magnet loss in W
350
300
250
200
150
FEM
100
Analytical, Air-gap
length considered
Analytical, Air-gap
length neglected
50
0
1
2
3
4
5
6
7
Circumferential magnet segment number, m
Fig. 6. Influence of air-gap length - Motor 1
400
350
Eddy current magnet loss in W
B. Influence of carrier frequency
The region corresponding to inductance or resistance
limitation is determined by the frequency of the carrier
harmonic with all the other parameters remaining constant.
Fig. 8 and 9 show the variation of the loss for different carrier
frequencies for motor 1 and motor 2 respectively. With
increasing harmonic frequency the loss curve shifts to the
right. It can be seen from Fig. 10 that for a 20 kHz harmonic
for motor 2, there is an inductance-limited region along with
the resistance-limited which is not seen at 5 kHz and 10 kHz.
Fig. 10 and 11 show the variation of the flux density and
the induced current density in the magnet along the width of
the magnet for a carrier amplitude of 6.25 A at different
carrier frequencies for motor 1. The effect of the reaction
field can be clearly noticed. At low frequencies where the
reaction field effect is negligible, the flux density of the
source field is uniformly distributed and the induced current
density varies linearly
1
, both along the width of
the magnet. However at higher frequencies the reaction field
is much stronger thus causing the flux density and induced
current density variation along the width of the magnet to be
no longer uniform and linear along the magnet width. With
increasing frequency both the fields are concentrated at the
edges of the magnet.
400
8
FEM
Analytical, Air-gap
length considered
Analytical, Air-gap
length neglected
300
250
200
150
100
50
0
1
2
3
Circumferential magnet segment number, m
Fig. 7. Influence of air-gap length - Motor 2
10 kHz Analytical
800
5 kHz Analytical
700
20 kHz FEM
600
10 kHz FEM
5 kHz FEM
500
400
300
200
100
0
1
2
3
4
5
6
7
8
Circumferential magnet segment number, m
Fig. 8. Variation of loss with carrier frequency-Motor 1
1099
4
20 kHz Analytical
900
Eddy current magnet loss in W
A. Influence of air-gap length
In earlier works on eddy current loss estimation in IPM
motors the effect of air-gap length was not considered [10]
[12]. This leads to inaccurate results especially if the air-gap
length is not too small when compared to the magnet
thickness. The influence of air-gap length is studied by
replacing the effective air-gap length
by
in (8). The effect on loss with and without the consideration
of the air-gap length is seen in Fig. 6 and Fig. 7 for motor 1
and motor 2 respectively. Neglecting the air-gap length leads
to under-estimation of eddy current loss as the reaction effect
is smaller.
450
20 kHz Analytical
900
10 kHz Analytical
Eddy current loss in W
800
C. Influence of the reaction field
If the reaction field is neglected (3) results as in (9)
5 kHz Analytical
20 kHz FEM
700
·
10 kHz FEM
600
5 kHz FEM
500
9
defined as in (4), resulting in solution of
With
400
4
300
·
,
·
· sin
·
, , , ,…
as
·
2
200
100
· sin
·
·
10
0
The power loss in each magnet segment results as in (12)
1
2
3
4
Circumferential magnet segment number, m
Fig. 9. Variation of loss with carrier frequency-Motor 2
·
2·
16
·
| |
14
4
12
1 Hz
Flux density in mT
10
5 kHz
6
10 kHz
4
20 kHz
2
0
-18.9
-12.6
-6.3
-2
0
6.3
12.6
18.9
Magnet width in mm
Fig. 10. Flux density variation along magnet width- Motor 1
Induced current density in kA/mm2
3000
2000
1000
0
-18.9
-12.6
, , , ,…
1kHz
8
-6.3
0
-1000
6.3
12.6
18.9
1 Hz
1 kHz
-2000
-3000
Magnet width in mm
5 kHz
10 kHz
20 kHz
Fig. 11. Induced current density variation along magnet width- Motor 1
·
·
·
·
·
·
,
11
Fig. 12 and 13 show the effect on loss with and without
the consideration of the reaction field on the evaluated total
eddy current loss. It is observed that neglecting the effect of
reaction field results in over-estimation of calculated loss,
especially in the inductance-limited region. In the resistancelimited region, however, the difference is not significant.
D. Influence of iron saturation
The analytical model assumed the relative iron
permeability as infinite and hence it was possible to obtain a
closed form expression for the eddy current loss. 2D FE
analysis has been used to calculate the loss taking the nonlinearity into consideration. For stator and rotor, lamination
material M330-35A was used [17]. Fig 14 and 15 show the
variation of loss with the consideration of non-linear effects
of iron lamination for motor 1 and motor 2 respectively. It has
to be noted that at the operating condition of 250A
fundamental peak, the current loading corresponds to
1500 A/cm RMS. So at the chosen operating point, the motor
is in a highly saturated condition. For motor 1, the eddy
current loss with saturation increases when compared to the
linear case in the range of 2-27 % depending on the
segmentation number . For the motor 2, however, the loss
decreases when compared to the linear case and is in the
range of 3-10 %. Hence a generalization of the saturation
effect on the loss cannot be made. It has to be noted that the
pattern of variation of the loss with segmentation number
in the circumferential direction remains unchanged.
1100
700
450
Analytical with reaction field
Analytical without reaction
field
500
400
300
200
100
1
FEM Non-linear
300
Analytical
250
200
150
100
50
0
2
3
4
5
6
7
8
Analytical with reaction field
Analytical without reaction
field
140
120
100
80
60
40
20
0
1
2
3
Circumferential magnet segment number, m
4
Fig. 13. Loss variation with and without reaction field – Motor 2
500
FEM-NonLinear
Analytical
400
350
4
depends on the width and depth of a magnet segment [4].
Using the above method, for one segment in circumferential
direction i.e.
1 and for various magnet segments in axial
direction, , the loss results obtained by analytical model is
presented. The number of magnet segments in axial direction,
, was varied from 1 to 21. For each , the value of
was
computed using (9) and the modified magnet conductivity
was calculated as
FEM-Linear
450
3
E. Influence of axial segmentation
As the model has been developed for 2D, the analytical
model considers only the segmentation in the circumferential
direction. This is also true for 2D FE analysis. In some cases
it may be beneficial to have axially segmented magnets. In
that case it is necessary to do a 3D analysis and a 2D analysis
would not suffice. However, 3D FE analysis is time-intensive
and demands huge computational power especially if the
model size is large. Previous works have proposed the usage
of an end-effect factor in 2D simulations to include the axial
segmentation effect [4]. This factor, which is less than 1, is
then used to modify the conductivity of the magnet to account
for the magnet segmentation in the axial direction.
Effectively, the magnet conductivity is reduced to account for
the axial segmentation effect. The effect factor
3
9
4
180
160
2
Circumferential magnet segment number, m
Fig. 15. Loss variation with non-linear iron – Motor 2
Circumferential magnet segment number, m
Fig. 12. Loss variation with and without reaction field – Motor 1
Eddy current magnet loss in W
350
1
0
Eddy current magnet loss in W
FEM Linear
400
Eddy current magnet loss in W
Eddy current magnet loss in W
600
300
250
200
625000 S/m
150
100
50
0
1
2
3
4
5
6
7
Circumferential magnet segment number, m
Fig. 14. Loss variation with non-linear iron – Motor 1
8
Fig. 16 shows the eddy current magnet loss in motor 1 and
motor 2 for
1 and varying from 1 to 21. Only the 3D
FE analysis of motor 1 is available at this time. It is observed
that in the inductance-limited region, the magnet loss
estimated using the end effect factor is higher than calculated
using 3D FE analysis. In the resistance-limited region the
difference reduces.
1101
V. ACKNOWLEDGMENT
450
Eddy current loss in W
This work is funded by the Federal
Ministry for Economic Affairs and
Energy (BMWi) within the ATEM
program. The authors are responsible
for the contents of this publication.
Motor 1
400
Motor 2
350
3D FEM - Motor 1
300
For circumferential
segmentation, m =1
250
VI. REFERENCES
200
[1]
150
100
[2]
50
[3]
0
1
3
5
7
9 11 13 15 17 19 21
Axial magnet segment number, n
Fig. 16. Loss variation with axial magnet segmentation, n for m =1
[4]
F. Loss with inverter excitation
The proposed analytical model can be extended to calculate
the magnet loss under inverter excitation. For each known
harmonic current the loss is computed and summed to get an
initial estimate of the total eddy current loss.
IV. CONCLUSIONS
[5]
[6]
[7]
This paper proposed a 2D analytical model for quick
estimation of eddy current loss in the magnets of IPM motors
taking into account the reaction field of the induced eddy
currents. The proposed model was analyzed for two different
types of commonly used IPM machines taking into account
the effect of circumferential magnet segmentation. The
analysis showed that an unskilled choice of the segmentation
number can actually lead even increased magnet loss in the
machine resulting in higher magnet temperature and thereby
resulting in reduced performance. It was observed that the
reaction field, if neglected, results in big errors in the
estimation of magnet loss especially in the inductance-limited
region. The effect of air-gap length, which was previously
neglected, has a significant effect on the magnet loss. The
actual loss was higher with the air-gap length considered. The
effect of carrier frequency on the magnet loss was also
analyzed. Choice of switching frequency can be a major
factor in deciding the eddy current loss in the magnets. The
effect of iron saturation on eddy current loss was not
conclusive. Analysis showed that in one case it actually
resulted in reduced loss. The effect of axial segmentation was
considered by using an end-effect factor and results show
good agreement in the resistance-limited region. However,
considering the time taken by a 3D FE analysis, it is still a
good bargain. Having proved the validity of the proposed
analytical model at different switching frequencies, the
analysis can be extended to evaluate the magnet eddy current
loss with inverter excitation.
[8]
[9]
[10]
[11]
[12]
[13]
[14]
[15]
[16]
[17]
1102
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