1
Modeling of Iron Losses in Permanent Magnet
Motors with Field-Weakening Capability
Juliette Soulard, Stephan Meier, and Yung-kang Chin
Abstract—Finite element analysis is usually used to calculate iron
losses in permanent magnet motors under field-weakening
operation. Analytical models for the iron losses taking into
account the rate of change of the flux densities and stator
leakages are presented and compared to FEM results for surfacemounted permanent magnet motors.
Index Terms— Field weakening, iron losses, permanent magnet
machines
I.
INTRODUCTION
Permanent magnet motors tend to replace induction motors
in many applications as they present higher efficiency and
power factor as well as higher specific output power. However
these interesting performances will only be reached if they are
part of the design considerations. In PM motors, copper losses,
core iron losses, stray losses and mechanical losses are the
main components that should be considered. Rotor losses are
usually quite low but should also be considered in high speed
applications.
This article deals with the modeling of the core iron losses
in surface-mounted permanent magnet motors (SMPM) under
field-weakening operation. Nowadays, an easy way to
investigate iron losses is to use finite-element analysis (FEM).
The FEM program we used for this study has a build-in iron
loss calculation module that can be run once the time-transient
simulation results are obtained. However, this can be timeconsuming so analytical models are investigated.
Many articles were already published on the subject and
some of them are shortly presented in section 2. Based on
these previous studies, several models were studied. The first
model was based on the analytical calculations of the no-load
iron losses and an iron losses resistance in the dq equivalent
circuits. The second model takes into account the effects of
load currents in a less global approach. The variations of the
flux densities in the stator teeth and yoke are obtained from the
total air gap flux density (magnets and currents). In a second
step, the armature leakage flux are taken into account.
Comparisons with 2D-FEM results on two four-pole SMPMs
are used to evaluate the performances of the different
analytical models.
Manuscript received March 16, 2002. This work was supported by the
Competence Center in Electric Power Engineering with the Permanent
Magnet Drives program.
The authors are with the Electrical Machines and Power Electronics
division at the Royal Institute of Technology, Stockholm, Sweden
(e-mail: juliette@ ekc.kth.se).
II. DIFFERENT MODELS FOR IRON LOSSES
Manufacturers of iron laminations characterize their
products for sinusoidal magnetic field. The conventional
model separates the iron losses in two terms being the
hysteresis losses, and the eddy current loss. The iron loss
power density is given by:
piron = p hyst + p eddy = k hyst ⋅ Bˆ β ⋅ f + k eddy ⋅ Bˆ 2 ⋅ f 2 (1)
with f the frequency, B̂ the amplitude of the sinusoidal flux
density and β, khyst, keddy, respectively the Steinmetz, hysteresis,
and eddy current constants depending on the core material.
However, the flux densities in iron laminations of electrical
machines are not purely sinusoidal. In [1], it was shown that
considering each harmonic of the flux density was not a proper
approach. Instead, the rate of change of the flux density should
be used in the eddy current loss term:
2
peddy
1
 dB 
= ∫ 2 ⋅ k eddy ⋅   ⋅ dt
T (T )
 dt 
(2)
This method showed good results for the calculations of no
load iron losses assuming the flux densities in the teeth and
yoke are constant or vary linearly with time when the magnets
rotate.
In [2], the same expression of the iron losses was used but
the flux densities were considered to have two orthogonal
components both in the yoke and the teeth (leakage flux).
An improved analytical model for the no-load iron losses is
presented in [3]. It takes into account the minor hysteresis loop
by a correction factor and introduces a third term, the so-called
anomalous or excess losses:
p exc =
k exc
1
dB
⋅
3/ 4
∫
2
T (T ) 2π
dt
( )
3/2
⋅ dt
(3)
Several studies with finite-element analysis concluded that the
flux densities in the different parts of the stator core change
markedly with the operating conditions [4] [5]. In [5], an
analytical model was developed to take into account the load
current effects but the study was only qualitative.
From the different references, it came out that analytical
models of the iron losses may work well if they take into
account the rate of change of the flux densities, the excess
losses as well as the minor hysteresis loops. Evaluating
analytically the iron losses at load conditions will then require
the same properties. Furthermore, the rate of change of the
flux densities in the different parts of the stator should take
2
into account the contribution of the currents. Three successive
models are described in the following parts.
does not exactly follow the same paths as the one from the
magnets. The stator has to be cut in different areas where the
rate of change of the total flux density can be well described.
III. SIMPLE LOAD IRON LOSSES MODEL BASED ON THE NOLOAD IRON LOSSES: MODEL 1
To take into account the effects of the load currents in the first
model, the iron losses are represented by a resistor in parallel
to the induced voltage E and the armature reaction for the qaxis and parallel to the armature reaction only in the d-axis (cf.
fig. 1) [1].
Lm.ω.Id
Rcu
Uq
Iq
In this section, a second model that takes the armature reaction
into account is presented. The approach is to describe the
change of rate of the flux densities in the stator teeth and yoke.
Using FEM simulations, the flux density variations in the teeth
and yoke of the stator are compared under field-weakening
conditions to the obtained waveforms. The losses are then
compared in the different parts for motor A (cf. fig. 2 and
appendix) at a given speed.
E
Riron
Lm.ω.Iq
Rcu
Ud
IV. TEETH AND YOKE IRON LOSS MODEL: MODEL 2
Id
Riron
Fig. 1 Equivalent dq-diagram with iron loss resistor Riron
The iron loss resistance Riron consists of two parallel
resistances, one for the eddy current and one the hysteresis
losses. The eddy current and hysteresis resistances are
calculated from the no-load iron losses analytical model. The
no-load iron losses were obtained from a set of improved
analytical models based on general FEM-calculations that
were derived in a paper from Mi, Slemon and Bonert [2].
The eddy current and hysteresis resistance are then calculated
at no-load conditions (no current) as:
Reddy (ω b ) =
3⋅ E2
p eddy (teeth) ⋅ Vt + p eddy ( yoke) ⋅ V y
(4)
Rhyst (ω b ) =
3⋅ E
p hyst (teeth) ⋅ Vt + p hyst ( yoke) ⋅ V y
(5)
Fig. 2: One pole of the type of SMPMs that are investigated.
1) Losses in the stator teeth
Our model is based on the fact that the flux in the stator teeth
is the superposition of the flux produced by the magnets and
the stator currents and can be obtained from the air gap flux.
Figure 3 shows the air gap flux density created by the magnets.
The magnet flux is assumed to be trapezoidal. The height of
the magnet flux atop of the magnet is Bm and the edge is given
by the pole anJOH DQGWKHVWDWRUVORWSLWFK s.
Bm (T)
2
model 1
FEM
As the induced voltage E is proportional to the electrical
frequency f and the eddy current losses are proportional to the
square of the electrical frequency (Peddy~f2), the eddy current
resistance Reddy doesn’t depend on the frequency. In contrast,
the hysteresis losses are only proportional to the electrical
frequency (Physt~f). The hysteresis resistance Rhyst therefore
depends on the frequency and must be calculated for fieldweakening applications according to:
Rhyst = Rhyst (ω b ) ⋅
f
fb
(6)
As it can be seen in figure 5 (model 1), this simple model does
not give satisfactory results for the investigated motor. A
global model is not accurate enough because the armature flux
2α
τs
θ (rad)
Fig. 3 Air gap flux density created by the magnets
The air gap flux density created by the three-phase stator
currents is assumed to be sinusoidal and the peak value is
noted
B̂δ ,arm . This sinusoidal waveform is shifted according
3
to the d- and q-components of the stator current. All the air
JDS IOX[ ZLWKRXWOHDNDJHZLWKLQD VWDWRUVORWSLWFK s ideally
passes through the corresponding stator tooth. The flux density
in the teeth Bst FDQEHREWDLQHGDVIROORZV
1
Bst (θ ) =
⋅
bts
θ +τ s
∫θ (B (θ )+ Bˆ δ
m
, arm
)
⋅ cos(θ + ψ ) ⋅ dθ
(7)
where the magnet flux and the flux by the stator current are
integrated over one slot pitch and ψ is the angle between the
current and the no-load voltage.
A comparison with the simulated flux density by FEM
shows that the analytical flux density is slightly lower (cf. fig.
4). This is due to the fact that the leakage flux is neglected.
One minor hysteresis loop is pointed out. These minor
hysteresis loops contribute to the hysteresis losses. Since their
detection and inclusion in the calculation is quite complex,
their contribution to the losses is neglected.
Bst (T)
2) Stator yoke losses
The stator yoke losses are more difficult to describe than the
teeth losses. The model is based on the partition of the flux
density in the radial r-DQGWDQJHQWLDO -components.
It was noticed that the radial component of the flux density
Bsy_r in the middle of the stator yoke has the same waveform as
the teeth flux density. The gradient of the flux density in the
stator yoke with the radius can be approximated as cubic
resulting in the following relationship:
Bsy _ r (θ ) =
Bst (θ )
(9)
7
For the calculation of the radial component iron losses, it is
assumed, that the r-component does only appear in the area
that is in the prolongation of the stator teeth. Figure 5 shows
the flux distribution in the middle of the stator yoke in the
extension of a stator tooth from the FEM simulation. It can be
noticed that the radial flux density is over-estimated by the
analytical model.
Bsy_r (T)
minor
loop
FEM
model 2
FEM
θ(rad)
θ(rad)
model 2
Fig. 4 FEM and analytical flux density variation in the stator teeth (model 2)
The stator teeth iron loss density in transient magnetic
DSSOLFDWLRQVRYHURQHFRPSOHWHSHULRG LVREWDLQHGIURP
dp teeth =
+
τ
1
2
⋅ ∫ piron (θ ) ⋅ dθ = k hyst ⋅ max(Bst (θ ) ) ⋅ f ⋅ k j
τ 0
2
3/ 2
2
τ
d
1 
 dB (θ )  
 dB (θ ) 
⋅ ∫ σ iron ⋅ lam ⋅  st
  ⋅ k j ⋅ dθ
 + k exc ⋅  st
12  dθ 
τ 0 
 dθ  
(8)
TABLE 1: ITEMIZATION OF THE STATOR TEETH LOSSES
Stator teeth losses
Hysteresis loss
Eddy current loss
Excess loss
Total losses
FEM
38.4 W
145.1 W
30.6 W
214.0 W
Model 2
25.4 W (-33.9%)
124.1 W (-14.4%)
25.6 W (-16.2%)
175.1 W (-18.2%)
Table 1 shows an itemization of the stator teeth losses. The
analytically calculated losses are higher than the FEM
simulated ones. The reason is that the analytical flux density is
lower and the strong distortions in the teeth shoes (due to the
leakage flux from the current) are not included in the analytical
model.
Fig. 5 Radial component of the stator flux distribution
The calculation of the tangential component of the stator flux
distribution is based on the assumption, that the total air gap
flux is guided through the stator teeth to the stator yoke and
that it splits there and closes on both sides over the adjacent
poles. ThH -component of the flux density in the middle of the
stator yoke is then obtained by:
B sy _ θ (θ ) =
D −δ
⋅
4 ⋅ hsy
2π / p +θ
∫ (B (α ) + Bˆ δ
m
θ
, arm
)
⋅ cos(α + ψ ) ⋅ dα
(10)
where D is the air gap diameter, δ the air gap and hsy the stator
yoke thickness.
Figure 6 shows the FEM and analytically calculated tangential
component of the flux density in the middle of the stator yoke.
It can be noticed again, that the flux density is analytically
over estimated.
For the studied case, the analytical model overestimates the
total yoke iron losses by 75% compared with FEM. This is due
to the fact that the flux levels (of both radial and tangential
component) are overestimated. The reason is that the leakage
flux through the tooth shoe and the stator slot are not
considered. Thus the flux density in the yoke from the current
alone is much too low. For field-weakening operation, the flux
from the stator current opposes the magnet flux. When this
4
opposing flux is calculated too small, the resulting flux is too
high and therefore the losses are overestimated.
Bsy_θ (T)
Furthermore, a mean value Bleak is added to the flux density in
the teeth to take into account the leakage flux Φ shoe and
Φ slot . Bleak is obtained by integrating the air gap flux density
created by the current shifted by 90° with the corrected value
of
Bˆ δ ,arm . Equation 7 becomes:
Bst (θ ) =
FEM
1
⋅
bts
θ +τ s
∫
θ
π 

 Bm (θ ) + Bˆδ , arm ⋅ k gap ⋅ cos(θ + γ ) + Bˆ leak ⋅ cos(θ + γ − )  ⋅ dθ
2 

(11)
θ(rad)
with Bˆ leak = k teeth ⋅ Bˆ δ , arm ⋅ k gap /q s and
model 2
k teeth =
Φ t 2 + Φ t1
.
Φ g1
Figure 8 shows the obtained flux density in the middle of a
tooth at 178 Hz.
Fig. 6 Tangential component of the stator flux distribution
Bst (T)
V. TEETH AND YOKE IRON LOSS WITH LEAKAGE CORRECTION
FACTORS: MODEL 3
model 3
To include the leakage flux created by the currents through the
shoes and the stator slots, correction factors are introduced in
the calculation of the flux densities in the air gap, stator yoke
and teeth. The losses in the shoe of each slot are also added to
the teeth and yoke losses. The correction factors are obtained
from an equivalent reluctance model of half a pole magnetized
by the current in one slot (cf. fig. 7).
Φy
Φt1
Rt2
Rt1
Φshoe
Φt2
Φg1
Rg2
Φshoe
multiplying
defined by:
Φ gref
Φ shoe
b
⋅ ts .
Φ t 2 + Φ t1 hsw
The radial component of the flux density density in the yoke
Bsy_r is not corrected because it is derived from Bst that is
already corrected. For Bsy_θ, equation 10 is corrected by
corrected by multiplying the former Bˆ δ , arm by kgap, which is
, with
k shoe = 2 ⋅
(12)
Rshoe
Rg1
The flux created by the current on the chosen half slot goes
through the first tooth and a part of that flux leaks through the
slot and the shoe. The other part of the flux goes through the
other teeth under the half pole and does not leak (Rt2 and Rg2).
Due to the leakages, the flux created by the currents in the air
gap is lower than previously calculated. The previous value is
Φ g 2 + Φ g1
B shoe (θ ) = (B̂δ ,arm (θ ) ⋅ k shoe ) 2 + B m (θ )
with
Fig. 7: Reluctance model for the leakage flux.
k gap =
The iron losses in the shoes are now included in the model. It
is assumed that the flux density in the shoe is a combination of
the flux density in the air gap created by the magnet and the
one created by the current corrected by the factor kshoe:
2
Rslot
Rt1
Φg2
θ(rad)
Fig. 8 Flux density in the stator teeth
nI
Ry
FEM
Φ gref =
nI
R g1 + R g 2
k yoke =
Bˆ δ ,arm by kyoke. kyoke is defined as:
Φy
Φ gref
(13)
5
Iron losses (W)
Bsy_θ (T)
Model 2
model 3
FEM
θ(rad)
Model 3
FEM
Model 1
Fig. 9 Tangential component of the flux density in the stator yoke (model 3)
Figure 9 shows the obtained flux density for the tangential
component in the middle of the yoke with model 3.
Table 2 compares the iron losses at 178 Hz with and without
the leakage model derived in this section. It can be noticed that
the prediction of the iron losses in the different parts of the
machine is more accurate with the use of the model including
the leakage flux (model 3).
Frequency (pu)
Fig. 10 Comparison of different analytical iron loss models for motor B.
TABLE 4: IRON LOSSES FOR MOTOR B WITH FEM AND MODEL 3.
Iron losses (W)
50 Hz
TABLE 2: IRON LOSSES AT 178 HZ WITH AND WITHOUT THE LEAKAGE
135 Hz
CORRECTION FACTORS
Iron loss (W)
Model 2
Model 3
FEM
teeth
175
186
-
shoe
23
-
teeth & shoe
209
214
yoke
69
39
40
total
245
248
254
Table 3 compares the results from model 3 with the simulated
FEM iron losses. The described iron loss model gives a total
value of the stator iron losses that is less than 17 % different
from the FEM results over the all speed range.
TABLE 3: COMPARISON OF THE FEM AND ANALYTICAL IRON LOSSES AT
DIFFERENT FREQUENCIES FOR MOTOR A
Frequency (Hz)
Model 3 (W)
FEM
(W)
50
135
116
114
159
167
178
248
259
242
382
364
306
560
511
VI. COMPARISON OF THE DIFFERENT MODELS FOR MOTOR B
Figure 10 shows a comparison of the different analytical iron
loss models to the FEM results for another SMPM design
noted motor B. A better iron losses model was obtained with
the inclusion of the stator leakage. The model works well for
higher frequencies. At lower frequencies around the base
speed, there is an overestimation of the iron losses, especially
in the stator yoke (see table 4).
From the results, it is concluded that model 3 works well
enough to compare the performances of different motors
during the first stage of a design as long as the motor is not
highly saturated.
219 Hz
304 Hz
FEM
Model 3
FEM
Model 3
FEM
Model 3
FEM
Model 3
teeth
shoe
41
74
145
243
2
18
54
116
teeth &
shoe
43
43
120
92
221
199
370
359
yoke
total
60
84
47
64
52
59
65
61
102
127
167
156
273
257
434
420
VII. CONCLUSIONS
Different analytical models to predict the core iron losses in
SMPM motors under field-weakening conditions were
presented. The model including the leakage flux from the
currents gives the best results compared with FEM
simulations. Such an analytical model makes it easy to take
into account the iron losses in the early stage of the design
when high efficiency should be reached.
The next stage of this study is to compare the results
obtained with the best analytical model with measurements on
different prototypes.
APPENDIX
TABLE 5: PARAMETERS OF MOTOR A AND B
Parameters
Outer stator diameter Dy
Inner stator diameter D
Height of the stator yoke hsy
Height of the stator slot hss
Number of stator slots Qs
Active machine length l
Air gap length
Stator tooth width bts
Slot wedge height hsw
Magnet thickness hm
Base frequency
Magnet remanent flux density
Motor A
150 mm
60 mm
13.5 mm
32.5 mm
24
200 mm
1 mm
3.62 mm
1.4 mm
3.14 mm
50 Hz
1.1 T
Motor B
170 mm
91.8 mm
18.6 mm
20.5 mm
24
160 mm
1 mm
4.8 mm
1.4 mm
2.4 mm
50 Hz
1.1 T
6
ACKNOWLEDGMENT
The authors thank the support team at Cedrat for their help
with the use of Flux2D for the finite-element simulations.
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[1]
[2]
[3]
[4]
[5]
[6]
[7]
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