1228 IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO.

5, MAY 2010

Optimization of Magnetic Pole Shifting to Reduce Cogging Torque

in Solid-Rotor Permanent-Magnet Synchronous Motors
Daohan Wang, Xiuhe Wang, Yubo Yang, and Ran Zhang
School of Electrical Engineering, Shandong University, Jinan 250061, China

In this paper, the method of magnetic poles shifting was combined with optimization method to reduce cogging torque in solid-rotor
permanent-magnet synchronous motors. Although the finite-element method (FEM) can calculate the cogging torque accurately, to find
the peak value of cogging torque, the cogging torque for different relative positions between permanent magnets and slots must be cal-
culated; thus, the optimization will take a long time. To reduce optimization time, a novel analytical method was proposed to determine
the initial value and feasible range of the shifting angles. Then the optimization method and FEM were used to minimize the cogging
torque. Two prototype motors were analyzed and optimized, respectively. It was proved that the cogging torque can be greatly reduced
by the proposed method.
Index Terms—Cogging torque, FEM, global optimization algorithm, solid-rotor permanent-magnet synchronous motors.

I. INTRODUCTION

P ERMANENT MOTORS (PMs) offer significant advan-

tages in comparison with conventional motors. Due to the
advantages of line-starting and simple structure, solid-rotor per-
manent-magnet synchronous motors have found more and more
applications in industry. Similar to other kinds of permanent-
magnet motors, cogging torque is an inherent problem and re-
sults in mechanical resonance, vibration, and noise. For sur-
face-mounted permanent-magnet motors, many methods have
been proposed to reduce cogging torque, such as a fractional
number of slots per pole [1], slot skewing [2], magnet skewing Fig. 1. Surface-mounted permanent-magnet motors: 1-stator yoke, 2-PMs,
[3], [4], “goodness” of slot number and pole number combina- 3-rotor, 4-air gap, 5-shaft. (a) Uniformly distributed PMs. (b) Magnetic pole
tion [4], slot or tooth pairing [5], [10], magnet segmentation [6], shifting.
pole-arc coefficient adjusting and its optimization [7], [9]–[11],
adjusting the shape of the slots or the tooth width [8], [13], [14],
auxiliary slots or teeth [12], [14], magnet displacing and shifting netic poles shifted. According to [15]–[17], the shifting angle
[3], [15]–[17], etc. However, there is no study for solid rotor for each magnetic pole from the full symmetric position is the
permanent-magnet motors with the interior permanent magnet multiple of a appropriate smallest deviation angle which de-
(IPM) by now. For the IPM machines, the magnetic saturation pends on the number of pole pairs, the number of slot and the
makes it difficult to analytically predict the cogging torque ac- greatest common divisor between them, the details to obtained
curately. Many methods used in cogging torque reduction for the shifting angles are discussed in Section II.
surface-mounted permanent magnet appear to be effective for In fact, the method proposed by [15]–[17] only aims at de-
solid rotor permanent-magnet motors. In this paper, the mag- creasing the magnitude of these harmonics with magnetic poles
netic pole shifting method will be studied. symmetry, in which the new harmonics caused by magnetic pole
The magnetic pole shifting method is very effective in re- shifting were not taken into account. As known to all, cogging
ducing cogging torque. Through shifting the magnetic poles, torque is caused by some harmonics in the air gap field. How-
the magnetic field distribution is set to be asymmetric to ob- ever, when the magnetic pole shifting method is adopted, new
tain a compensation effect and the cogging torque can be de- orders of cogging torque harmonics are inevitably introduced.
creased. For arranging the magnetic poles to be asymmetric, Thus, in some cases the determined angles were not the best
the most strategic issue is how to determine the shifting an- for reducing cogging torque yet. Furthermore, the method pro-
gles of magnetic poles. References [15]–[17] had performed posed by [15]–[17] only focuses on surface-mounted PM mo-
the study of this method in surface-mounted permanent-magnet tors, in which interval space between two adjacent permanent
motors. Fig. 1(a) shows the motor with uniformly distributed magnets are usually bigger than that in PM motors with inner
permanent magnets, while Fig. 1(b) shows the motor with mag- buried PMs. Therefore, the method to determine the shifting an-
gles according to [15]–[17] can be restricted on PM motors with
inner buried PMs because the shifting angles for magnetic poles
Manuscript received January 09, 2010; revised February 11, 2010; accepted are increasing in multiplied times, and hence may make two ad-
February 12, 2010. First published March 08, 2010; current version published jacent magnetic poles overlap.
April 21, 2010. Corresponding author: X. Wang (e-mail: wangxh@sdu.edu.cn).
Color versions of one or more of the figures in this paper are available online
In this paper, to optimize the magnetic pole shifting angles
at http://ieeexplore.ieee.org. in solid-rotor permanent-magnet synchronous motors, the fol-
Digital Object Identifier 10.1109/TMAG.2010.2044044 lowing studies will be carried out.

0018-9464/$26.00 © 2010 IEEE

WANG et al.: MAGNETIC POLE SHIFTING TO REDUCE COGGING TORQUE IN SOLID-ROTOR PMSMs 1229

1) The rotor structure of a solid-rotor permanent-magnet syn-

chronous motor suitable for magnetic pole shifting was
proposed.
2) In Section II, an analytical method to determine the mag-
netic pole shifting angles for solid-rotor permanent-magnet
synchronous motors, which aimed at eliminating all objec-
tive existent harmonic components, was proposed based
on the energy method; the validity and effectiveness was
verified by FEM. The results showed that the shifting an-
gles obtained by this method not only reduced the cogging
torque greatly but also were smaller than that obtained in
[15]–[17]. However, this method ignored flux leakage and
saturation; thus, the angles were not the best and the cog-
ging torque can not be calculated accurately.
3) In Section III, to obtain the best shifting angles, optimiza-
tion with global optimization method and FEM should be
performed. In the optimization process, to find the peak
Fig. 2. Cross section of solid rotor.
value of cogging torque, the cogging torque corresponding
to many different relative positions between magnetic
poles and armature must be calculated by FEM, thus too
much time was needed. To achieve compromise between With the magnetic poles distribute uniformly, each magnetic
the optimization time and accuracy of determining shifting pole has the same relative position with respect to the stator
angles, the above analytical method was combined with slots. The torque caused by each magnetic pole is in phase with
global optimization method and FEM to perform the the others, and thus the cogging torque is great. If the magnetic
optimization. The above analytical method was used to poles are suitably shifted, the harmonic components of cogging
determine the initial solution and narrow the feasible torque caused by all the magnetic poles are made to be out of
region of shifting angles, FEM was used to calculate the phase, then the harmonic components of cogging torque can be
peak value of cogging torque. In the optimization, a global eliminated or greatly reduced, thus the cogging torque can be
optimization method: zooming method was used. Two greatly reduced.
prototype motors were analyzed and optimized, respec- According to [15]–[17], the shifting angle can be obtained
tively. It was proved the cogging torque can be greatly through the expression as , where
reduced by the proposed optimization. is the shifting angle of PM, is the number of pole pairs,
is the slot number, , is
II. DETERMINING THE SHIFTING ANGLES BY ANALYTICAL the greatest common divisor between and . The angular
METHOD IN SOLID ROTOR PERMANENT-MAGNET choice only aims at decreasing the fundamental harmonic with
SYNCHRONOUS MOTORS magnetic poles symmetry, whereas other harmonic components
still exist. In order to reduce the cogging torque to a great extent,
A. Cogging Torque in Solid Rotor Permanent-Magnet not only the origin harmonics with magnetic poles symmetry but
Synchronous Motors also the introduced harmonics after shifting the magnetic poles
should be eliminated.
The stator of solid rotor permanent-magnet synchronous
In solid rotor permanent-magnet synchronous motors, the
motors is the same as that of three-phase induction motors.
magnetic pole shifting method can be realized as shown
The cross section of the rotor is shown in Fig. 2. The rotor
in Fig. 3. The shapes and dimensions of the PM slots for
core is made from a cast iron, not a laminated one. The slots
all magnetic poles are the same, while that of nonmagnetic
for inserting permanent magnets and nonmagnetic wedges are
wedge changes according to the requirements of magnetic pole
achieved by mill machining.
shifting.
When three-phase voltages are applied to three-phase wind-
ings, a rotating magnetic field is produced, which will induce
eddy current in rotor core and produce starting torque. B. Analytical Analysis of Cogging Torque With Magnetic Pole
Since there existing permanent magnet and stator slots, cog- Shifting
ging torque is caused by the interaction between the magnetic The cogging torque can be defined as the negative derivative
poles and the stator teeth. The periodically varying air-gap re- of the magnetic co-energy with respect to the rotation angle
luctance would cause periodically distributed cogging torque. when there is no load condition, i.e.,
The cogging torque can be expressed in the following form:

(1) (2)

where is the least common multiple of the number of stator Since the permeability of iron is high, the co-energy stored in
slots and the number of poles, the relative position between iron can be ignored in comparison with that in air gap and PMs.
magnetic poles and armature. Since the permanent magnets are inner buried in rotor core, the
1230 IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 5, MAY 2010

Fig. 4. The B () with uniformly distributed magnetic poles.

Fig. 3. Magnetic pole shifting of solid rotor.

variation of the co-energy in PMs can be neglected in compar-

ison with that in the air gap, the cogging torque can be expressed
as follows:
Fig. 5. The B ( ) with magnetic pole shifting.

(3)
Substituting (4), (5), and (6) into (2), the expression of cogging
where and are the co-energy stored in air gap and torque can be analytically formulated as
PM, respectively [7], [18].
The magnetic co-energy stored in air-gap can be expressed as
(7)
[7], [18]

(4) where is the axial length of armature, is the outer radius

of rotor, and is the inner radius of armature.
It can be seen that cogging torque changes periodically with
where the distribution of air gap flux density in an equiv- the relative position between the magnetic poles and armature.
alent slotless machine, the relative position between the mag- During a tooth pitch, the number of periods is the multiple of
netic poles and armature, the deviated angle along circum- which depends on the slot number and pole number
ference of air gap, the volume of the air gap region, the
length of PM in magnetization direction, the permeability of (8)
air, and the distribution of effective air-gap length [18].
If and can be expressed in the where is the greatest common divisor between
Fourier series, the expression of cogging torque can be obtained. and .
According to [18], the Fourier expansion of It can also be seen from (7) that only has effects
can be expressed as on cogging torque. If the value of can be reduced by
magnetic pole shifting, the cogging torque can be reduced.
However, when the magnetic poles are shifted, the dis-
(5) tribution of changes, and the expression of cogging
torque changes. In other words, magnetic pole shifting method
introduces new harmonic components in . Not only the
where is the slot number, and is the Fourier transform harmonic, but also some new harmonics has effects
coefficients of . Detailed expressions of on cogging torque. To investigate the influence of magnetic
and are derived and shown in [18]. poles shifting on cogging torque, the analytical expression of
When the magnetic poles distribute uniformly, the distribu- cogging torque considering the magnetic pole shifting will be
tion of is shown in Fig. 4, where , the given. When the magnetic poles are shifted, the distribution of
width of PM which provides the magnetic flux per pole, the is as shown in Fig. 5.
pole pitch, and the number of pole pairs. The Fourier expansion of with magnetic pole asym-
The Fourier expansion of can be expressed as follows: metry should be carried out at the interval as follows:

(6) (9)
WANG et al.: MAGNETIC POLE SHIFTING TO REDUCE COGGING TORQUE IN SOLID-ROTOR PMSMs 1231

Substituting (4), (5), and (6) into (2), the analytical expression
of cogging torque with magnetic pole shifting can be obtained

(10)

and can be expressed as

(11)

(12)
Fig. 6. Comparion of B +B (6-pole 30-slot).

where is the shifting angle for the magnetic pole.

It can be seen from (9) that, when the magnetic poles dis- addition, it is not certain that (15) has solutions. If (15) has no
tribute uniformly, , is 0, while is real solutions, the problem is changed to the following form as

(16)
(13)
It can be seen that is not zero only when is the mul-
tiple of . When the slot number per pole is integer , As regards the effects of reducing cogging torque as well as the
the cogging torque can be reduced effectively by the method complexity of calculation, the value of is set to be 4 which
proposed in [15]–[17], because no harmonic components are in- indicate that the lower 4 orders of harmonic components are al-
troduced after the magnetic poles are shifted. However, if the lowed to be eliminated. Due to the complexity for solving (15)
slot number per pole is not integer , new orders of and (16), in this paper, the universal global optimization soft-
harmonic components are introduced because both and ware 1STOPT is adopted to solve (15) and (16).
are not zero after the magnetic poles are shifted to be For a 6-pole 30-slot prototype motor , there exist
asymmetric, thus it is possible that the cogging torque cannot which satisfy (15). Hundreds of solutions can be obtained.
be reduced effectively. Under this circumstance, in order to re- The smallest solutions are adopted because the shifting angles
duce cogging torque, all objective existent harmonic compo- are constraint by the slot width of nonmagnetic slot wedge in
nents should be considered to be reduced. rotor. The comparison of with that of uni-
formly distributed magnetic poles is shown in Fig. 6. The unit
C. Determination of Shifting Angles by Analytical Method of Y-axis in Fig. 6 and Fig. 7 is per unit, with the magnitude of
th order harmonic divided by the magnitude of fundamental.
To reduce the cogging torque, the magnitude of all objective
From Fig. 6, it can be seen that for , all the orders of cog-
existent harmonic components should be minimized theoreti-
ging torque harmonics exist before shifting the magnetic poles.
cally. According to (9), the magnetic pole shifting angles have
After the magnetic poles are shifted by the proposed method,
direct effects on the magnitude of each harmonic of cogging
all the lower orders ( 4) of cogging torque harmonics are elim-
torque. The magnitude of the harmonic can be formulated
inated absolutely since there exist solutions for (15).
as
For a 6-pole 27-slot prototype motor , there exists
(14) no which can satisfy (15), but which satisfy (16) can be
obtained. The comparison of with that of uni-
As we know, the higher the order of cogging torque harmonics, formly distributed magnetic poles is shown in Fig. 7. It can be
the smaller its magnitude. The magnetic pole shifting angles seen from Fig. 7, according to (10), only the cogging torque
which can eliminate the lower harmonics can reduce the harmonics which order is multiple of 2 exist when the mag-
cogging torque to a great extent. So we can make the lower order netic poles distribute uniformly. Shifting magnetic poles engen-
to be zero by suitable selection of the shifting angles as dered new orders of cogging torque harmonics which were in-
follows: existent when the magnetic poles distribute uniformly, but they
are greatly reduced by adopting the shifting angles from solving
(16).
(15)
From the studies above, for whichever prototype motor, in-
teger slots per pole or fractional slots per pole, the overall mag-
where is an integer that means the lower orders of harmonics nitude of cogging torque harmonics both can be greatly reduced
which is equal or less than are considered to be eliminated. To by the analytical method proposed. The detailed solutions of
obtain the values of , (15) should be solved. Equation (15) is for 6-pole 30-slot and 6-pole 27-slot prototype motor obtained
a very complicated nonlinear equation and difficult to solve. In by the proposed method are shown in Table II.
1232 IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 5, MAY 2010

Fig. 7. Comparion of B +B (6-pole 27-slot). Fig. 8. Distribution of magnetic field calculated by FEM for 22 kw, 6-pole,
30-slot solid-rotor PM prototype motor.

It must be mentioned that the analytical method ignores the

effects of saturation and flux leakage, so the shifting angles where are the pole shifting angles. The first mag-
given is not accurate and not the most suitable. However, they netic pole does not shift, thus . is the fea-
are close to the optimal shifting angles. The optimization by sible region of the th pole shifting angle. It can be determined
FEM can be carried out in the region around the determined as follows:
shifting angles. Analytical method can be used to shorten the
feasible region of optimizations, and thus the optimization time (19)
can be saved.
where is the shifting angle of the th magnetic pole obtained
III. OPTIMIZATION OF COGGING TORQUE by the above analytical method, the width of feasible region
BY ZOOMING ALGORITHM for the shifting angle of the th magnetic pole.
The air-gap flux per pole and shifting angles are used as the
A. The Zooming Algorithm optimization constraints. The optimization can be formulated as
The Zooming algorithm is a global optimization algorithm follows:
for general NLP problems, which is useful for engineering op-
timization applications. The basic idea of the algorithm is to ex-
plore the entire feasible domain in a systematic way for a global (20)
minimum. The algorithm avoids searching near any local min-
imum point and all the points leading to it, thus increasing the
chance of finding a new local minimum in the unexplored re- where , are the flux per pole in the air gap before and after
gion. After searching the whole region, the minimal local min- the optimization, is chosen to be 0.05.
imum found is the global minimum. The local phase uses a The peak value of cogging torque is calculated by FEM. At
zooming strategy, which is defined by adding the following ad- first, the distribution of magnetic field corresponding to different
ditional constraint to the problem: relative positions between the magnetic pole and armature is
calculated. Then the cogging torque corresponding to different
(17) relative positions is calculated by Maxwell tensor method on
the basis of the distribution of magnetic field, and the max-
The reduction parameter is less than 1 for a positive imum cogging torque is the peak value. Since the cogging torque
and greater than 1 for a negative . In the algorithm is set changes periodically with a period of a slot pitch, calculating the
to be a small number 0.05–0.15. The reduction factor should be cogging torque for different positions between stator and rotor
specified and adjusted during the solution process so as to find within a slot pitch is enough.
the global optimum and decrease the CPU time. The details for The distribution of magnetic field calculated by FEM for a
zooming algorithm are shown in [19]. 22 kw, 6-pole, 30-slot solid-rotor PM prototype motor is shown
in Fig. 8.
B. The Optimization of Cogging Torque
C. Results of Optimization
The objective function is the peak value of cogging torque
for solid-rotor permanent-magnet motor, which is calculated by The software for cogging torque optimization of solid-rotor
FEM. The design variables are magnetic pole shifting angles PM motor is developed. In this paper, a 6-pole 30-slot and 6-pole
27-slot solid-rotor PM motor are taken as examples. The main
(18) parameters of the prototype motors are shown in Table I. The
WANG et al.: MAGNETIC POLE SHIFTING TO REDUCE COGGING TORQUE IN SOLID-ROTOR PMSMs 1233

TABLE I
PARAMETERS OF PROTOTYPE MOTOR

TABLE II
OPTIMAL RESULTS

Fig. 9. The comparison of cogging torque (6-pole, 27-slot). Fig. 10. The comparison of cogging torque (6-pole, 30-slot).

differences between them are the slot number. Calculation by

IV. CONCLUSION
FEM indicates that the shifting angles obtained by the analytical
method can greatly reduce cogging torque, thus the optimized In this paper, the method of magnetic pole shifting is adopted
results are quite close to the analytical results. For the prototype to reduce cogging torque in solid-rotor permanent-magnet syn-
motors, was set to be 1 and the feasible region was set to be chronous motors. To obtain the best shifting angles, optimiza-
to make sure the optimal solution is included tions with global optimization method and FEM are performed
in the feasible region. to minimize cogging torque. The following conclusions can be
The optimal results of shifting angles are shown in Table II. drawn.
The comparison of cogging torque is shown in Fig. 9 and 1) An analytical method which aims at eliminating all objec-
Fig. 10, respectively. The Y-axis in Fig. 9 and Fig. 10 is the tive existent harmonic components was proposed to deter-
ratio of cogging torque calculated to the rated output torque. It mine the shifting angles for solid rotor permanent-magnet
can be seen that the cogging torque can be greatly reduced by synchronous motors. Calculation by FEM showed that the
optimization. In addition, as a result of the narrowed feasible shifting angles obtained by this method can reduce cogging
region by analytical method, the optimization time can be saved torque greatly.
60%–75% compared with that without using the proposed 2) The optimization method and FEM were combined to min-
method. imize the cogging torque, in which the analytical method
1234 IEEE TRANSACTIONS ON MAGNETICS, VOL. 46, NO. 5, MAY 2010

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