Progress in Electromagnetic Research Symposium 2004, Pisa, Italy, March 28 - 31
Comparison of Methods to Calculate Iron Losses
Caused by Harmonic Fields in Rotating Machines
K. Yamazaki and S. Watari
Dept. of Electric Engineering, Chiba Institute of Technology,
2-17-1, Tsudanuma, Narashino, Chiba 275-0016, Japan
e-mail: yamazaki@pf.it-chiba.ac.jp
Abstract
In this paper, we examine the accuracy of several calculation methods for iron loss of rotating
machines using the finite element method. Two calculation methods are compared with the measurements.
One is based on the Fourier transformation of the flux density waveforms at each finite element. The
other is the direct calculation from the waveforms without Fourier transformation. It is clarified that
the errors of the calculation methods vary due to the kinds of machines and that the method based on
the Fourier transformation overestimate the hysteresis loss in case of highly saturated machines.
I. Introduction
Owing to the progress of the computers and the numerical methods, it has become possible to
calculate the iron loss considering the harmonic and rotational electromagnetic fields in the rotating
machines. Various calculation methods have been proposed and applied in recent years.
There are two major methods for the harmonic losses calculation. One is based on the Fourier
transformation of the waveforms of the flux density at each finite element. The other is the direct
calculation from the waveforms without Fourier transformation (1)-(4). The first method is very useful
to understand the each harmonic loss component. However, this method has one problem. In the strict
scene, the hysteresis loss cannot be decomposed into harmonic components because of its nonlinear
characteristics. Consequently, the total calculated iron loss may include error.
In this paper, we compare the measured and the calculated iron losses of several rotating
machines for the purpose of clarifying the accuracy of the calculation methods.
II. Calculation Method
A) Outline of the calculation
Figure 1 shows the outline of the common procedure for the iron loss calculation of rotating
machines. First, the electromagnetic field of the machine is calculated by the nonlinear time-stepping
finite element method considering the magnetic saturation of the core and the movement of the rotor.
The governing equation is as follows:
∇ × (ν∇ × A) = Jc − σ
∂A
+ v 0∇ ×
∂t
(1)
where A is the magnetic vector potential, Jc is the current of the coil, M is the magnetization of the
magnet, ν is the reluctivity and σ is the conductivity.
At this step, the eddy current and the hysteresis phenomena in the core are neglected because it
will require vast of computer resources to take the lamination structure of the core into account
directly by the finite element analysis. The iron loss of the core, which are the eddy current and the
hysteresis losses in the laminated electrical steel plates, are calculated in the second step with the time
variation of flux density at each finite element. In this case, the effects of the reaction field caused by
the iron loss in the laminated core are neglected in the total procedure. However it can be considered
that they are small compared with the total field.
B) Methods to calculate iron loss
The core loss per weight induced by the single frequency alternating magnetic field can be expressed
as follows:
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Progress in Electromagnetic Research Symposium 2004, Pisa, Italy, March 28 - 31
w i = w ie + w ih = K e f 2 B max 2 + K h fB max
γ
(2)
where wie, wih are the eddy current and the hysteresis losses per weight, Ke, Kh and γ are the
experimental constants obtained by the Epstein frame, f is the frequency of the alternating magnetic field
and Bmax is the maximum flux density during the time interval.
However, as we will discuss in Chapter III, the magnetic field in rotating machines are not single
frequency alternating field even when the sinusoidal power supply is utilized. As the stator and the rotor
structures produce much time harmonics due to the rotation, the harmonic eddy current losses and the
minor hysteresis losses are generated in the core. To consider these effects, the method based on the
Fourier transformation of the flux density waveform at each finite element are often applied:
We = ∫
∑ K D(nf ) {B
e
iron
2
2
mr , n
+ Bmθ , n }dv (3)
Wh = ∫
2
h
iron
n
∑ K D(nf ){B
mr , n
γ
+ Bmθ , n }dv
γ
(4)
n
where We and Wh are the total eddy current and hysteresis losses, D is the density of the core, n is the
order of the harmonics, Br,n and Bθ,n are the radial and the peripheral component of the nth harmonic flux
density.
On the other hand, the core loss can be calculated directly from the flux density waveform without
the Fourier transformation (1)-(4). The eddy current loss in the steel plate can be expressed by the timevariation of the magnetic field as follows
wie =
h 2σcore 1
12 D T
∫
T
0
dB 2
dt
dt
(5)
where h is the thickness, σcore is the conductivity of the steel plate and κ is the modified coefficient of
the eddy current loss due to the effect of grains in the steel plate. In this case, the total eddy current loss
can be calculated with the result of the finite element analysis and the coefficient Ke as.
Wie =
1 N  Brk + 1 − Brk  2  Bθk + 1 − Bθk  2
K eD
∑  ∆t  +  ∆t  dv
2π 2 ∫core N k =1 
(6)
where N is the number of time step per one time period, ∆t is the time interval. This expression
includes the effect of the time-harmonics. The hysteresis loss including harmonics can be also
calculated by counting the local minimum and maximum peaks of the waveforms as
Np θ

KhD NE ∆Vi  Npr
ij 2

(
)
(Bmθij )2 
Wih =
B
mr
+
∑
∑
∑

T i =1 2  j =1
j =1

i
i
(7)
where ∆Vi is the volume of i-th finite element, Bmrij, Bmθij are the amplitudes of each hysteresis loop,
Npri, Npθi are the number of peaks in the time-variation of Br and Bθ. Fig.2 shows the outline of the
method to determine the amplitudes of each hysteresis loop involving major and minor ones.
Finite Element Method
Iterative calculation for magnetic saturation of core
Re-mesh at each time step due to movement of rotor
Br,Bθ
1
2Bm1
(2Bmax)
Time series data of flux density
5 2Bm5
2Bm6 6
Iron Loss Calculation
2Bm3 3
Eddy current loss of core considering harmonics Wie
2Bm4
(2Bmax)
Hysteresis loss of core considering minor loops Wih
2
2 2Bm
Iron loss considering harmonics Wi = Wie
Wih
4
Figure 1. Outline of calculation
Figure 2. Determination of amplitudes of major and minor
hysteresis loops
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Progress in Electromagnetic Research Symposium 2004, Pisa, Italy, March 28 - 31
III. Comparisons between Measured and Calculated Results
The two calculation methods are applied to two kinds of machines, which are the surface
permanent magnet motor and the induction motor. The measured and the calculated iron losses are
compared to clarify the advantages and disadvantages of the calculation methods.
Figure 3, 4 show the field distribution of the permanent magnet and induction motors. The motors are
manufactured and measured by IEE-Japan. Although the outline of the permanent magnet motor is round
shape and the induction motor is rectangular, the shape of the stator slot is identical. The electrical steel
plates applied to the stators and the rotors cores are also identical. The number of the steel is 50A1300
defined by Japanese industrial standard. The iron loss coefficients determined by the Epstein frame test are
Ke=3.098X10-4 and Kh=6.013X10-2. On the other hand, the rotor structures of the motors are much different.
The permanent magnet motor has no rotor slots, whereas the induction motor has 34 rotor slots with the
aluminum cage. As a result, the flux density waveform of the induction motor includes high order timeharmonics caused by the movement of the rotor, especially at the stator teeth top. On the contrary, the
waveforms of the permanent magnet motor are nearly rectangular wave due to the magnetic saturation and
they include no slot harmonic field. In this case, 3, 5, 7th harmonics are mainly included in the waveform.
Figure 5, 6 show the experimental and the calculated iron losses of the motors. The experimental and
calculated results agree well. In the experiment, the iron loss is obtained from the no-load loss. As a result,
several components except for the eddy current and the hysteresis loss of the laminated core is included in
the experimental iron loss, for instance, the harmonic eddy current losses of the magnet in the case of the
permanent magnet motor and the harmonic eddy current losses of the aluminum cage in the case of the
induction motor. These losses cannot be separated from the core losses by the measurement. Thus, these
components are calculated directly by the finite element method and added to the calculated results. As the
air gap of the permanent magnet is relatively large, the time-variation of the flux density at the rotor is
nearly DC filed and the rotor iron loss is negligible. On the other hand, in the case of the induction motor,
the rotor iron loss caused by the stator slot harmonics cannot be neglected because of the small air-gap. The
underestimation of the calculated results in the case of the induction motor is mainly caused by the neglect
of the inter-bar current, which pass through the rotor cage to the rotor core.
Let us discuss the detailed difference between the results of the calculation methods. In the case of the
permanent magnet motor, the result calculated with the Fourier transformation is larger than the direct
calculation from the flux density waveform whereas the results are nearly same in the case of the induction
motor. This error is caused by the misestimated hysteresis loss caused by the harmonic fields. Figure 7, 8
show the calculated hysteresis losses by the two methods. In the case of the permanent magnet motor, the
fundamental component obtained by the FFT is already larger than the total hysteresis loss obtained by the
direct calculation because the waveform is nearly rectangular wave. Furthermore, the 3, 5, 7th harmonic
losses are added. However, they do not produce the hysteresis loss because they do not cause the
minor loops. As a result, the calculation method using the Fourier transformation overestimates the
total hysteresis loss in the case of the permanent magnet motor.
IV. Conclusions
The two calculation methods for the harmonic iron loss calculation of the rotating machines are
applied to two kinds of machines. It is clarified that the errors of the calculation methods vary due to
the kinds of machines. Although the method based on the Fourier transformation is useful to understand
each harmonic loss component, it overestimates the hysteresis loss in case of highly saturated machines. It
can be said that the method based on the direct calculation from the flux density waveform can be applied
to various types of rotating machines.
REFERENCES
1.
2.
3.
4.
K.Atallah, Z. Q. Zhu and D. Howe. ”An Improved Method for Predicting Iron Losses in Brushless Permanent Magnet
Motor ”, IEEE Trans. Magn., vol.28, no.5, pp.2997-2999, 1992.
1. S. L. Ho, W. N. Fu and H. C. Wong: “Estimation of Stray Losses of Skewed Rotor Induction Motors Using Coupled
2-D and 3-D Time Stepping Finite Element Methods”, IEEE Trans. Magn, vol.34, no.5, pp.3102-3105, 1998.
Sadowski, J. P. A. Bastos, M. V. Ferreira Da Luz, P. Kuo-Peng and M. Lajoie-Mazenc, “The Rain-Flow Method for
Evaluation of Iron Losses in Electrical Machines”, IEEE Trans. Magn, vol.36, no.4, pp.1923-1927, 2000.
K. Yamazaki, “Torque and Efficiency Calculation of an Interior Permanent Magnet Motor Considering Harmonic Iron
Losses of both Stator and Rotor”, IEEE Trans. Magn, vol.39, no.3, pp.1460-1463, 2003.
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Progress in Electromagnetic Research Symposium 2004, Pisa, Italy, March 28 - 31
1
1
0
-1
0
45
90
135
180
Rotor Position (deg)
-2
-3
0
-1
0
45
-2
-3
Stator Teeth Top
Br
1
Br
2
Br
T)
2
90
135
180
Flux Density(T)
3
Flux Density
Br
2
Flux Density
Flux Density
3
0
0
5
10
15
20
Tim e(m s)
1
0
0
5
10
-1
15
20
Time(ms)
Rotor Positoin (deg)
-1
Stator Teeth Center
-2
Stator Teeth Top
Stator Teeth Center
Figure 4. Field of induction motor.
Figure 3. Field of permanent magnet motor.
30
30
Magnet eddy
current loss
Magnet
Eddy
20
Rotor Cage
eddy
current
loss
Magnet
Eddy
Iron Loss W)
Iron Loss W)
20
Stator
hysteresis loss
10
Rotor
hysteresis loss
Stator
hysteresis loss
10
Rotor eddy
current loss
Stator eddy
current loss
0
Exp.
1
(Average)
Cal.
2
(Direct)
Stator eddy
current loss
0
Cal.
3
(FFT)
Cal.
2
(Direct)
Cal.
3
(FFT)
Figure 6. Measured and calculated loss of IM.
18
9
16
8
14
7
Hysteresis loss (W)
Hysteresis loss (W)
Figure 5. Measured and calculated loss of PM.
Exp.
1
(Average)
12
10
8
6
6
5
4
3
4
2
2
1
0
0
All1
2
31
4
53
6
75
8
9
1
All
10
12
16 11 18
2
Time Harmonics
OrderOrder
of time of
harmonics
Direct Cal.
31
4
5
3
6
75
8
9
10 11 12
16
18
OrderOrder
of timeofharmonics
Time Harmonics
Using FFT
Direct Cal.
Figure 7. Calculated hysteresis loss of PM.
Using FFT
Figure 8. Calculated hysteresis loss of IM.
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