IEEE TRANSACTIONS ON MAGNETICS, VOL. 39, NO. 3, MAY 2003
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Magnetic Field Analysis of Permanent Magnet Motor
With Magnetoanisotropic Materials Nd–Fe–B
M. Enokizono, Member, IEEE, S. Takahashi, and T. Kiyohara
Abstract—In this paper, we propose the method to analyze
the magnetization distribution in magnetoanisotropic materials
by using the finite-element method considering the improved
variable magnetization and Stoner–Wohlfarth model. By using
this method, furthermore, the effect of the eddy currents induced
in permanent magnets was analyzed. From the analyzed result,
it is clarified how the magnetization distribution affects the
performance of the surface permanent magnet-type motors.
Index Terms—Anisotropic material, demagnetization factor
tensor, magnetic field analysis, magnetization, Nd–Fe–B, permanent magnet motor, Stoner–Wohlfarth model.
Fig. 1.
Definition of vector relation and notations.
I. INTRODUCTION
I
N GENERAL, magnetic properties of the permanent magnets must be explained with the vector relation between magvector in
netic field strength vector and the magnetization
materials. The magnetic field analysis on products made from
the hard magnetic material with uniaxial anisotropy usually requires a large number of data from two-dimensional magnetizavector and vector are not always partion curves, because
allel to each other in the material. As such analysis is tedious, a
smaller number of material data is more desirable. Furthermore,
though it is necessary to obtain the inside distribution of the
vector in permanent magnets when we carry out the analysis of
electrical machinery, it cannot be given in fact. It is impossible
vector. The distrito measure the inside distribution of the
bution must be treated as an unknown value [1]. However, up to
now, it has been treated as a given value. In the case of the strong
hard magnetic materials such as Nd–Fe–B alloy having a strong
anisotropy, it is impossible to directly measure the inside distribution of vector in the arbitrary direction. In order to analyze
such problems, the demagnetization factor tensor method was
developed [3]. It is significant that the analytical method for the
magnetizing process of the anisotropic hard magnetic material
is established. By using this method, we can treat the magnetivector quantity, in other words, the absolute value of
zation
magnetization and the directional angle.
This paper gives the new magnetic field analysis method,
which requires only two kinds of data from the magnetization
curves measured along the easy direction and hard direction of
the hard magnetic material.
II. FORMULATION
A. Calculation of Initial Magnetization Process
vector induces
When an external magnetic field strength
vector, the volumetric free energy can be expressed by
the
the following:
(1)
charIn the calculation of the free energy, the vector
acteristic curve is required. This curve involves the effect of the
material shape. Therefore, the curve has to be obtained at every
element and consider the demagnetization factor. It can be expressed by the following:
(2)
is the demagnetizing field vector and
is the
where,
effective field vector; subscripts “ ” and “ ” are the magnetic
field components of the magnetizing easy direction and hard
direction, respectively.
curve is required and is exTo calculate (2), the
and
pressed by two kinds of characteristic curves,
curve. The principal axes of the demagnetization
factor are named the -axis and the -axis. A general spheroidal
is shown in
magnetic body in the uniform effective field
Fig. 1. The self-energy is written as follows:
Manuscript received June 18, 2002.
The authors are with the Department of Electrical and Electronic Engineering, the Faculty of Engineering, Oita University, Oita 870-1192, Japan
(e-mail: enoki@cc.oita-u.ac.jp).
Digital Object Identifier 10.1109/TMAG.2003.810422
0018-9464/03$17.00 © 2003 IEEE
(3)
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IEEE TRANSACTIONS ON MAGNETICS, VOL. 39, NO. 3, MAY 2003
Fig. 3.
Fig. 2. Rectangular element.
where and
are the magnetic susceptibility of the easy and
and
are the
hard direction, respectively. The
demagnetizing factor in the -axis and -axis. The , , and
are expressed as shown in Fig. 1.
The -axis makes an angle from the easy axis. The total
,
energy is minimized by the following conditions:
. Then the following can be obtained:
Residual magnetization process.
where
and
are given by [1], is the length to -axis,
the length of the element “ ” for a polygon element, and is the
number of apex on rectangular elements, as shown in Fig. 2. The
and
can be expressed by means of
relationship between
as follows:
the coefficient of demagnetization factor tensor
when
when
(10)
was used instead of the arbitrary vector
This tensor
curve, which could not be measured.
(4)
(5)
Utilizing two kinds of magnetization curves, the
and
curves, it can solve these two simultaneous equations. These curves can be obtained by the measurement of
two directions, “ ” and “ ”. Therefore, the initial magnetizing
process can be analyzed in the above procedure. Equation (5) is
the Stoner–Wohlfarth equation (2) with two kinds of anisotropic
fields, too: the shape magnetic anisotropic field and the intrinsic
anisotropic field.
C. Calculation of Residual Magnetization
decreases, both
and
As the applied magnetizing field
decrease. Finally, when the
becomes zero, the
and
change the
and , respectively, as shown in Fig. 3. The
residual angle is the angle due to the residual magnetization
from the easy direction. It occurs when the direction of
is not parallel to the easy direction. The relationship between
and can be expressed experimentally as follows [4]:
(11)
is the square ratio
in the easy direcwhere,
tion. The residual magnetization is given from the Stoner–Wohlfarth equation (2) by the following:
B. Calculation of Demagnetization Factor Tensor
As shown in Fig. 2, when the rectangular element is magnetized uniformly, the magnetic charge induced in an element
“ ” is expressed by the following:
(12)
(6)
The rectangular element is more useful than the triangular eleand
ment [1]. The demagnetizing field component
which the magnetic charge
makes for the point
are shown as follows:
(7)
(8)
(9)
(13)
is the anisotropic magnetic field strength to the
where
can be obtained by using
magnetization . The value of
the Stoner–Wohlfarth equation. In this way, the magnetizing
process can be analyzed in all the elements. As a result, the final
inside distribution of the residual magnetization in the permanent magnets can be analyzed, and then the magnetic field induced by the permanent magnets can be calculated.
By introducing Kirchhof’s law, the exciting circuit equation
for calculating the distribution of initial magnetization is given
as follows [5], [6]:
(14)
ENOKIZONO et al.: MAGNETIC FIELD ANALYSIS OF PERMANENT MAGNET MOTOR
1375
(a)
(b)
(c)
Fig. 6. Top part of teeth. (a) Model 1. (b) Model 2. (c) Model 3.
Fig. 4. Analytical model of magnetizer.
Fig. 7. Initial magnetization curve of Nd–Fe–B material.
Fig. 5. Analytical model of surface-type permanent magnet motor.
where is the interlinkage flux to the winding and
initial electric charge of the capacitor. The current
.
expressed using the charge as
is the
can be
D. Calculation of Demagnetization Process
with the from the easy axis, when the external
For the
applies to direction of angle
, the
is shown as
field
follows:
Fig. 8.
Equivalent circuit of magnetizer.
Fig. 9.
Flux distribution of magnetizer.
(15)
(16)
In the demagnetization curve of the easy axis, is a gradient of
the demagnetization characteristic curve in the second quadrant.
E. Fundamental Equation of Permanent Magnet Motor
The fundamental equation of the surface permanent typed
motor is given by
(17)
is the magnetic vector potential.
and
are the
where
reluctivity and the conductivity, respectively. Fig. 4 shows a
full model of a magnetizer. The magnetizer is used to magnetize the four poles hard magnetic materials, Nd–Fe–B alloy.
The pole pieces are made of steel, and its conductivity is
S/m. Fig. 5 shows the model of the permanent
magnet motor. Fig. 6(a), (b), and (c) shows magnetizer models,
which have the different shape of the top part of teeth. The
initial magnetizing curve of two directions, as shown in Fig. 7,
are used in this analysis. Fig. 8 shows the equivalent circuit
, the total
of the magnetizer and the capacitance is 2400
is 0.031 . The charging voltage is 3187 V.
resistance
III. ANALYTICAL RESULTS
Fig. 9 shows the flux distribution in the magnetizer at magnetizing state (Model 1). It looks like a successful magnetizing
state from the flux distribution. However, as shown in Fig. 10,
the value of the eddy current density induced in permanent
magnet was different from each other at each time steps. The
time change of the eddy current was drawn at positions A, B,
and C as shown in Fig. 11. The eddy current increases in the
boundary neighborhood of the magnetic pole. Fig. 12 shows
the inside distributions of the magnetization vector after the
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Fig. 10.
IEEE TRANSACTIONS ON MAGNETICS, VOL. 39, NO. 3, MAY 2003
Fig. 14.
Value of eddy current density with time.
Characteristic curve of the cogging torque.
motor was analyzed. Fig. 14 shows that the characteristic curve
of the cogging torque was obtained for each model. The least
pulsation of cogging torque was obtained when the magnetizer
Model 2 was used.
IV. CONCLUSION
Fig. 11.
Calculated points.
(a)
(b)
(c)
Fig. 12. Distribution of residual magnetization vector. (a) Model 1. (b) Model
2. (c) Model 3.
In this paper, finite-element method introduced the improved
VSWM method considering eddy current for anisotropic permanent magnet was presented. We have carried out the magnetic
field analysis of the permanent magnet motor considering the
distribution of the residual magnetization vector. As a result, it
has been shown that it is important to consider the magnetization process of permanent magnets. The presented method is
very useful in designing permanent magnet motors. However,
this method does not considers the effect of the dc biased based
on the permanent magnet. It will also be the next problem.
From the applied results, in order to develop the high performance permanent motor, it was found that controlling the distribution of the magnetization by using the magnetizer was very
important.
REFERENCES
(a)
(b)
(c)
Fig. 13. Flux distribution of permanent magnet motor. (a) Model 1. (b) Model
2. (c) Model 3.
demagnetizing process in the region of the permanent magnet.
As shown in this figure, the pattern of magnetizing distribution
is different due to the construction of the magnetizer. Fig. 13
shows the magnetic flux distribution of the permanent motor
which was incorporated the above analyzed permanent magnet,
to investigate the influence of the magnetization vector. By
using these results, the characteristic of the permanent magnet
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