http://dx.doi.org/10.11142/jicems.2015.4.1.70
70
Journal of International Conference on Electrical Machines and Systems Vol.4, No.1, pp.70~75, 2015
A Linear Consequent Pole Stator Permanent
Magnet Vernier Machine
Chunhua Zou*, Yi Du*, and Feng Xiao*
Abstract – The linear permanent magnet (PM) vernier machine is very suitable for directdrive applications. This paper proposes a linear consequent pole stator PM vernier
(LCPSPMV) machine, which can offer high thrust force due to the use of magnetic gear
effect. The key of the proposed machine is the using of consequent pole. Thus the volume of
the PM can be reduced, and the initial cost can be reduced consequently. Based on finite
element analysis, the static characteristics of the proposed machine including the PM flux
linkage, zero load EMF, detent force and thrust force are accurately analyzed. Compared
with other linear stator PM vernier (LSPMV) machines, the proposed machine can provide
higher zero load EMF, lower detent force and enhanced power density at the same conditions
of rated speed, winding turns, electric loading.
Keywords: Consequent pole, Stator permanent magnet, Vernier machine, High power
density
1. Introduction
Permanent magnet (PM) synchronous machines (PMSMs)
have been developed rapidly during the past 30years since
the development of rare-earth PM materials, especially
modern neodymium-iron-boron (NdFeB) magnets. The PM
machine is distinguished from other machine types for its
high power density, high efficiency, and low rotor inertia
characteristics [1]. As a result, PMSMs are used universally
in demanding applications such as hybrid-electric automobile
vehicles and electrically-powered servo machine tools[2, 3].
Direct-drive application is one of the broad application
areas in which PM machines have been achieving great
success during recent years [4, 5]. It is widely recognized
that linear PMSMs are attractive candidates for linear
direct-drive applications due to the elimination of
gearboxes and other mechanical transmission components
between the machine and the load [6]. Thus, compared with
the rotational machine systems, the linear PMSM drive
systems have a simple structure and a quick speed
response [7]. However, conventional linear direct-drive
machines usually require a large number of poles and
windings to produce high thrust force at low speed.
Linear vernier machine can be preferred to simplify the
machine structure and reduce the slot number due to the
merits of high thrust force and low speed. In [8], a linear
* School of Electrical and Information Engineering, Jiangsu University,
China. (duyie@ujs.edu.cn)
Received 31 March 2014; Accepted 15 November 2014
stator PM vernier (LSPMV) machine with concentrated
windings has been proposed, which can offer high thrust
force at low speed while reducing the number of slots and
thus copper loss [9]. The high thrust force feature of the
LSPMV machine is achieved based on the magnetic gearing
effect, in which a small movement of the secondary causes
a huge PM flux change, and at the same time, the high
speed travelling armature field in stator can be converted to
the low speed travelling field in secondary [10]. However,
it suffers from the disadvantage of high cost for its massive
use of PMs.
This paper introduces a linear stator PM vernier machine
with consequent poles, so that the total amount of PM
material is expected to be reduced without sacrificing the
high thrust force [11]. Besides, the consequent pole
structure reduces fringing flux and thus increasing the flux
linkage [12]. In order to confirm the advantages, a
quantitatively comparison between the linear consequent
pole stator PM vernier (LCPSPMV) machine and the
LSPMV machine is performed in the conditions of the same
rated speed, winding turns and electric loading. The
advantages mentioned above are explained in the following
sections by finite element analysis (FEA).
2. Proposed Machine and Operation Principle
Fig. 1 (a) shows the topology of the proposed machine,
and the existing LSPMV machine is plotted in Fig. 1 (b) for
comparison. It can be seen that the proposed machine is
A Linear Consequent Pole Stator Permanent Magnet Vernier Machine
composed of a flat stator and a flat secondary. The stator
consists of an iron core with salient poles wound with 3phase armature windings and a number of PMs which are
inset on the surface of the stator teeth. The proposed
machine adopts three PMs and two ferromagnetic poles on
the surface of each stator tooth. Compared with the existing
LSPMV machine, the consumption of PM material can be
reduced and PMs can be easily mounted. By adopting the
consequent pole structure, the proposed LCPSPMV machine
also belongs to a doubly salient PM machine [13-14]. The
secondary adopts a simple iron core with salient poles so
that it is very robust to transmit high thrust force.
Fig. 1. Comparison of topologies
⎛ 2πppmeff x ⎞
F ( x) = F1 cos ⎜
⎟
l
⎝
⎠
(1)
Where x is the secondary position, ppmeff is the effective
number of PM pole pairs, F1 is the amplitude of the
fundamental component of MMF and l is the active length
of the machine.
Permeance coefficient changes with the moving of the
secondary due to its toothed structure. Hence, both the
moving speed and corresponding secondary position can
influence the permeance coefficient. Only considering the
fundamental component, the permeance coefficient can be
written as
2πnr ( x − vt )
(2)
l
Where P0 is the direct current component in permeance
coefficient, P1 is the amplitude of the fundamental
component of the permeance coefficient, nr is the active
number of secondary tooth and v is the secondary speed.
P ( x, t ) = P0 + P1 cos
Then the air gap magnetic flux density can be governed
by
2πppmeff x
B ( x, t ) = F ( x) P ( x, t ) = F1 P0 cos
+
l
2π( ppmeff + nr ) x − 2πnr vt
1
F1 P1 cos
+
2
l
2π(nr − ppmeff ) x − 2πnr vt
1
(3)
F1 P1 cos
2
l
In (3), the first term represents a static magnetic field
that cannot induce voltage in the stator windings. The
second term and the third term represent two magnetic field
components produced by PMs and modulation of secondary
teeth, with a short wavelength and a long wavelength,
respectively. Since the secondary and the magnetic field
harmonics have the same speed in electrical degree, the
traveling speed of long wavelength magnetic field is
considerablely higher than that of the short wavelength
magnetic field. Thus, the induced electromotive force (EMF)
produced by the long wavelength magnetic field is expected
to be high. Hence, the third term is selected to be effective
harmonic to match with the pole pairs of stator windings
and the pole-pair number of the armature winding can be
expressed as
(4)
pwi =| nr − ppmeff |
Gr =
In order to simplify the analysis of the two machines, the
magnetic resistance saturation of the steel are considered to
be negligible in this paper. Only considering the fundamental
component, the effective magnetic motive force (MMF)
produced by PMs can be expressed as
71
nr
pwi
vflux = Gr v
(5)
(6)
Where pwi is the number of stator winding pole pairs, Gr is
the gear ratio, and vflux is the speed of the travelling
magnetic field.
In the proposed machine, the number of secondary tooth
nr=17, the effective number of PMs ppmeff=18. According to
(4), the pole pairs produced by stator windings should be
pwi=1.
3. Optimization Design
The magnets pole ratio can significantly affect the
performances of the proposed machine, such as no-load
EMF. Hence, it is very important to optimize the ratio
between the PM width and the stator tooth shoe width.
The optimization variables includes width fr and ratio kr
of the PMs, ferromagnetic pole width fa and ratio ka and
stator tooth shoe width ws, which are defined as
f
kr = r
(7)
ws
ka =
fa
ws
(8)
72
Chunhua Zou, Yi Du, and Feng Xiao
Since there are three PMs and two ferromagnetic poles in
one fixed width stator tooth, so
3kr + 2ka = 1
(9)
The PM width should be larger enough to produce a high
MMF. However, the ferromagnetic material is easy to be
saturated, which can highly affect the value of thrust force.
So the variation of the no-load EMF and the thrust force
with respect to kr are analyzed individually.
Fig. 2 shows the variation of the peak no-load EMF with
respect to the PM ratio. It can be seen that when kr equals to
0.21, and the peak no-load EMF can reach its maximum
value. Fig. 3 shows the performance characteristics of the
thrust force versus kr. It indicates that the thrust force can
achieve maximum value when kr is 0.21, which is
corresponding to the result in Fig. 2. Therefore, the PM
ratio kr is selected as 0.21. According to (9), it can be
calculated that ka equals to 0.185.
thrust force is maximum when the PM thickness is 5mm.
Thus, the PM thickness is designed as 5mm.
The parameters of secondary tooth, such as ctip and croot,
can also influence the performance characteristics of the
proposed machine, where ctip is defined as the ratio of
secondary tooth tip width to the secondary pole pitch, and
croot is defined as the ratio of secondary tooth root to the
secondary pole pitch as shown in Fig. 5. In [8], authors
have exhibited that secondary tooth dimensions can largely
affect the average thrust force of the machine. The value of
the ctip has significant impact on the average thrust force,
while the influence of croot is negligible. Based on these
conclusions, thrust force optimization of the proposed
machine can be simplified to design the value of the croot to
equal to the value of the ctip. Fig. 6 shows the average thrust
force with respect to the value of ctip. It can be observed
that the thrust force reaches maximum value when the ctip
equals to 0.3. Hence, ctip is chosen as 0.3.
Fig. 2. Variation of peak no-load EMF with respect to kr
Fig. 5. Module of secondary
Fig. 3. Variation of thrust force with respect to kr
PM thickness is another factor which has great influence
on the thrust force. The waveform of the thrust force versus
PM thickness is plotted in Fig. 4, which indicates that the
Fig. 4. Variation of thrust force with respect to PM
thickness
Fig. 6. Variation of thrust force with respect to ctip
Table 1 compares the detail design parameters of the
two machines. From Table 1, it can be found that the
proposed machine only requires 75% PM volume of that of
its counterpart, which resulting a cost reduction. This is
actually due to the fact that the proposed machine employs
many ferromagnetic poles. In addition, the equivalent
magnetic reluctance of the proposed machine is much less
than that of the existing one due to the adoption of the
consequent poles, so the proposed machine can
significantly improve the power density.
A Linear Consequent Pole Stator Permanent Magnet Vernier Machine
Table 1. Design parameters of existing LSPMV machine
and proposed LCPSPMV machine.
Existing
Proposed
Item
machine
machine
Number of phases
3
3
Rated speed (m/s)
1
1
Stack length (mm)
100
100
Number of secondary tooth
17
17
Winding turns per phase
142
142
Secondary pole pitch (mm)
21.18
21.18
Active stator length (mm)
360
360
Air gap length (mm)
1
1
Magnet remanence (T)
1.2
1.2
Number of PMs
30
18
Magnet volume (cm3)
120
90
73
harmonics of both machines due to the modulation of the
secondary teeth, in which the main harmonics include the
1st, 6th, 12th, 18th and so on. It is important to emphasize that
the 1st harmonic, which is the effective harmonic, can
significantly affect the performance of the machine. In
Fig. 8 (c), it can be found that the 1st harmonic of the
proposed LCPSPMV machine is 42.9% higher than that of
the existing one. In other words, the performance of the
proposed machine can be greatly enhanced.
4. Finite Element Analysis
According to the aforementioned design procedure, a 3phase 6/17 pole LCPSPMV machine is designed. The
performances and characteristics of the machine are
analyzed by the FEM.
The no-load magnetic field distributions of the two
machines are depicted in Fig. 7. As can be seen, the pole to
pole leakage of the LCPSPMV machine is much smaller
than that of the existing one. So the flux linkage of the
LCPSPMV machine is much higher than that of the
LSPMV machine.
Fig. 8. No-load air gap flux density with secondary position
Fig. 7. No-load magnetic field distribution
The operation fundamental of the LCPSPMV machine is
magnetic field modulation function of the secondary teeth.
So it is very important to analyze the no-load flux density in
the air gap and its harmonics. Fig. 8 (a) shows the no-load
air gap flux density waveform of the LCPSPMV machine
and the no-load air gap flux density waveform of the
LSPMV machine is plotted in Fig. 8 (b). Fig. 8 (c) depicts
the corresponding harmonics analysis results. It can be
observed that there are a number of asynchronous space
Fig. 9 compares the detent force of both machines. It can
be seen that the peak to peak detent force of the LCPSPMV
machine is much smaller than that of the existing one,
which offering smoother thrust force, especially in low
speed operations. Fig. 10 shows the corresponding
harmonics of both machines. It can be found that the main
harmonics of the LSPMV machine include the 6th, 7th and
so on, in which the 6th harmonic occupies the largest
proportion. Compared with the 6th harmonic, all the other
harmonics are negligible. As a consequence, the detent
force of the LSPMV machine presents six periods.
Similarly, the most important harmonics of the LCPSPMV
are 2nd and 6th, while the 2nd harmonic is much larger than
74
Chunhua Zou, Yi Du, and Feng Xiao
the 6th one. Therefore, the detent force of the LCPSPMV
machine presents two periods. But the 6th harmonic is not
small enough to be ignored, so there are six small periods
hide inside the two large periods. All these factors made the
period of the detent force not very conspicuous.
Fig. 9. Detent force waveforms
existing machine under brushless AC operation with the
maximum value of 6A are 710N and 530N, respectively. It
indicates that the proposed machine possesses the better
thrust force performance.
Fig. 12. Thrust force waveforms
5. Conclusion
Fig. 10. Harmonics analysis of detent force
Fig. 11 compares the no-load EMF waveforms under the
conditions of the same armature winding turns and speed. It
can be found that the no-load EMF of the LCPSPMV
machine is 85V, while the no-load EMF of the existing
LSPMV machine is 60V. Which means, the proposed
machine generates higher no-load EMF than the existing
one, which is corresponding to the analysis result in Fig. .
Therefore, the thrust force of the proposed machine is larger
than that of the LSPMV machine when both machines
operate at the same speed and electric loading. As shown in
Fig. 12, the thrust forces of the proposed machine and the
Fig. 11. No-load EMF waveforms
In this paper, a LCPSPMV machine is proposed for
direct-drive applications, and it artfully integrates PMs and
ferromagnetic poles together. This machine offers the
advantages of a robust structure and enhanced performances.
An existing vernier machine is comparatively designed for
evaluation. Compared with its counterpart, the proposed
machine offers higher no-load EMF, lower detent force and
higher thrust force in the same conditions. Based on the
finite element software, the validity of the proposed
machine is verified.
Acknowledgements
This work was supported and funded in part by grants
(Project No. 51307072 and 61174055) from the National
Natural Science Foundation of China, the grant from the
priority academic program development of Jiangsu higher
education institution.
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Chunhua Zou received the B.Sc. degree
in electrical engineering from Jiangsu
University, Zhenjiang, China, in 2013,
where she is currently working toward the
M.Sc degree. Her areas of interest include
design and electromagnetic field analysis
of electric machine.
75
Yi Du received the B.Sc. and M.Sc.
degrees in electrical engineering from
Jiangsu University, Zhenjiang, China, in
2002 and 2007, respectively, and the
Ph.D. degree in electrical engineering
from Southeast University, Nanjing, China,
in 2013. His research interests include electric machine
design, modeling, and control.
Feng Xiao received the B.Sc. and M.Sc.
degrees in electrical engineering from
Jiangsu University, Zhenjiang, China, in
2002 and 2007, respectively. She is
currently working toward the Ph.D.
degree at Jiangsu University, Zhenjiang,
Jiangsu, China. Her areas of interest include design and
electromagnetic field computation of permanent magnet
motor.