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Optimal design and implementation of a permanent magnet
linear vernier machine for direct-drive wave energy extraction
Li, W; Chau, KT; Lee, CHT
IECON 2012 - 38th Annual Conference on IEEE Industrial
Electronics Society, Montreal, QC, 25-28 October 2012, p. 6200 6205
2012
http://hdl.handle.net/10722/225405
Annual Conference on IEEE Industrial Electronics Society.
Copyright © IEEE.
Optimal Design and Implementation of a Permanent
Magnet Linear Vernier Machine for Direct-drive
Wave Energy Extraction
Wenlong Li, K. T. Chau, and Christopher H.T. Lee
Department of Electrical and Electronic Engineering, The University of Hong Kong, Hong Kong
wlli@eee.hku.hk, ktchau@eee.hku.hk, htlee@eee.hku.hk
Abstract- This paper presents a permanent magnet linear vernier
(PMLV) machine which is dedicated for low-speed direct-drive
applications.
Firstly,
the
machine
operation
principle
is
discussed, and the preliminary design approach is formulated.
Then, by applying the analytical calculation method for solving
the
magnetostatic
field
problem,
the
machine
structure,
especially the stator toothed-pole structure which acts on the
field
modulation
function
is
optimized.
Finally,
a
PMLV
machine is prototyped and implemented for direct-drive wave
power generation. Both analytical calculation and experimental
interpolating each other. By integration of the magnetic gear
with the conventional high-speed machine, the integrated
machine can enable low-speed operation but with high power
density. Thus, the magnetic-geared machine becomes a hot
research topic in recent several years [20]-[21]. By extending
its operation principle, the permanent magnet vernier machine
is developed which uses a toothed-pole structure for magnetic
field modulation and exhibits a low-speed and high-torque
nature [22]-[24].
verification are given to verify its performances.
1.
Air-gaps
�f'r"=======:r�===i Field modulation
rings
INTRODUCTION
Linear machines are a class of electromagnetic devices
which operates at a linear oscillatory or progressive motion.
As their rotational counterparts, they can also be used for
mechanical energy and electrical energy inter-conversion. Tn
recent years, development and application of linear machines
are in an accelerated pace in various fields, such as industrial
automation, robotics, power generation, and transportation
[1]-[5].
Low-speed drives, such as wind power generation and
electric vehicle motor drives, attract more and more attention
in recent years [6]-[12]. For the conventional machine
topologies, at the same power rating, low-speed direct drives
usually mean a large physical volume and relatively low
power density compared to the high-speed drives. Tn order to
solve these problems, mechanical gearboxes for speed
reduction and torque transmission are widely applied, thus the
power density of the whole driving system is improved
accordingly. However, the mechanical transmission units
inevitably incur the system complexity, operation cost and
further deteriorate its control performance and reliability. For
solving the problems raised by mechanical gears but still
attaining their merits, the coaxial magnetic gears were
proposed and applied for direct-drive applications [13]-[18].
The coaxial magnetic gear consists of two rotors and one
stationary ring. One rotor operates at a low speed and other at
a high speed. Thus, the high-speed drives can be worked as
the low-speed drives via the coaxial magnetic gear. Fig. 1
shows its linear tubular morphology of the coaxial magnetic
gear [19]. The two movers have surface-mounted PMs, and
the stationary parts consist of iron rings and airspaces
978-1-4673-2421-2/12/$31.00 ©2012
IEEE
Low-speed mover
-speed mover
Fig.
l.Linear tubular magnetic gear.
Air-gap
Fig.
mover
2.Linear tubular vernier machine.
The purpose of this paper is to present a permanent magnet
linear vernier (PMLV) machine. By studying its operation
principle, the design procedure will be introduced. Then, by
6200
analytically modeling the field modulation effect of the
toothed-pole structure, the dimensions of the toothed-pole
structure are optimized and the performance is improved
accordingly. Finally, a PMLV machine will be designed and
prototyped for direct-drive wave energy conversion. By using
analytical and experimental evaluations, the proposed
machine performances will be verified.
PRELIMINARY DESIGN
II.
The concept of vernier machine lies in its toothed-pole
stator configuration which resembles to the field modulation
rings in the magnetic gear, as shown in Fig. 2. Due to the
distinct difference in the permeability of ferromagnetic
material and air space, the toothed-pole structure can enable
different pole-pair numbers of magnetomotive forces (MMFs)
of stator and mover to develop a steady force. In order to
realize this function, the pole-pair numbers of MMFs of PMs
and armature winding and the stator tooth number should
satisfy the following relationship:
(1)
Z = I Z ]±p l
2
where ZI is the number of teeth of the stator, Z2 and P are the
pole-pair numbers of PMs of the mover and armature winding
respectively.
When AC current with an angular frequency w is fed into
the armature winding, the resultant magnetic field of armature
excitation rotates at a speed of wlP, and the mover speed is
w1Z2. By borrowing the concept of gear ratio in magnetic
gears, the gear ratio of this machine is Z2IP.
A.
Mover Design
The mover configuration with PMs can be classified into
three categories depending on the location of PMs, namely
surface-mounted PM (SPM), surface-inset PM (STPM), and
interior PM (TPM). Tn this design, only the SPM topology is
considered and mathematically modeled.
Under the current frequency f in the armature winding, the
mover speed v can be expressed as:
Z
v=v..1 =2frm _2
p
(2)
where VI is the synchronous speed, Tm is the pole-pitch of PMs
on the mover. For a given operation speed and current
frequency, the PM pole-pitch can be easily determined.
By using the Ampere's circuital law and assuming that the
magnetic drop occurs only in the air-gap, the fundamental
component of air-gap flux density Bagl can be expressed as:
Bagl =
4
lr
B lh m
sin(wmlr)
+
hm
2Tm
fLlgo
(3)
Tt =
Stator Design
According to (1), the stator tooth pitch Tt can be written in
terms of the PM pole-pitch:
(4)
The developed thrust force of the linear machine is
determined by the electric loading which is limited by the
cooling method. Most of the heat is caused by the copper loss
expressed as:
, = 2mNI2p
P(u
(5)
l�f
Ac
where m is the phase number, N is the turns of coil of each
phase, I is the phase current, p is the copper resistivity, lef is
the conductor effective length, and Ac is the cross-sectional
area of the conductor.
The heat produced by armature winding dissipates to the
air space via the stator yoke. The relationship of the total
copper loss with the coil temperature rise T and the heat
transfer coefficient h can be expressed as [25]:
(6)
Pc" = 2PThTl�f
According to (5) and (6), the machine electric loading can
be determined, depending on the cooling method and its
temperature limit.
C
Thrust Estimation
Based on the air-gap flux density and the electric loading in
the windings, the pull-out thrust force can be estimated from
the magnetic field interaction:
.
4m 1m NI
(7)
BgJ hm SIll 2 (� Z )dz
Fem 1
_
lr
Tm
9tJ
where RI is the amplitude of the fundamental component of
magnetic reluctance of the toothed-pole structure.
Therefore, given the basic design requirements, the initial
specifications of the linear machine can be determined
according to the above equations.
OPTIMAL SIZING
Ill.
As shown in Fig. 3, the toothed-pole structure of the
vernier machine takes the field modulation function which
inherently governs the machine operation. Therefore, it is
necessary to study the underlaid relationship between its
structure dimensions and the field modulation effect. Tn order
to take a physical insight of contributions of the toothed-pole
structure for the field modulation, an analytical calculation is
carried out. By taking the toothed-pole structure as an
anisotropic material, the calculation can be easily handled. Tn
the calculation, the tubular linear machine configuration is
considered. The permeabilities of the toothed-pole structure
along z and r direction are described as:
where BI is the PM remanence, hm is the PM thickness, !-lr is
the PM relative recoil permeability, go is the air-gap length,
Will is the PM length, and Till is the PM pole-pitch.
B.
2Z
2 Tm
Z]
fL,p
_=
=
fLfeT,
fL,p - I = flo
+
+
b, (flo - fLfe )
'
bt
(fLte - flo)
Tt
(8)
(9)
where !-lo is the air space permeability, !-lIe is the stator iron
core permeability, Tt is the tooth pitch and bt is the tooth
6201
In the region TIT:
2
2
a rp111(r,z) fllpJ arp111(r,z)
a rp111(r,z) - (14)
0
+
+ flip
flip r
2
2
oOlr
az
ar
r
where Mr is the PM residual magnetization vector in the r
direction which can be expressed as:
width.
???????????
Mover
, ???????????
z
Air-gap
_
CD
pp
Mr(r,z) = ( _0_1 +r) I Mn sin(mnz)
r
n�l
(15)
where
1
Po= --2ro +hm
Fig. 3. Toothed-pole structure.
r
/
hI
g0
x
o at e -poIe
)<
\, }!r;t<;,t�r; ;�g�oxnx
Y' x y x h xl x
Air-gap region
PM
Fig.
X
region
1
� =----,--2-­
ro +rohm
(2n -1)
OJn =---lr
Tm
4B1
M =
flolr(2n - 1)
jll
r3
11
r2
The unique solutions of the above partial differential
equations are governed by the boundary conditions. There are
four boundaries formed by three regions and its conditions
are given by:
(17)
B/z(ro,z) = 0
rl
ro
z
4.Analytical model in cylindrical coordinates.
By solving Maxwell's equations in each region, the
magnetic field can be analytically obtained [26]-[28]. In order
to ease the computation, some assumptions have been taken
into consideration: the permeability of the back iron of the
mover and the yoke of the stator is considered to be infinite;
the relative recoil permeability of PMs is considered to be
unity; the saturation effect of the ferromagnetic material is
ignored.
As shown in Fig. 4, the three regions are the region I (PM),
region IT (air-gap) and region TIT (toothed-pole structure). The
flux density B and field intensity H in the above three regions
can be expressed as:
In the region I:
(10)
BJ = flo(HJ + Mre,)
In the other two regions where no PMs involved:
BII ,III = flII ,III H11,111
(16)
(11)
where Moes is the PM residual magnetization vector. It should
be noted that the permeability in region TIT is not the same in
the r and z directions.
For the optimization of toothed-pole structure dimensions,
the magnetic field excited by PMs is evaluated. Thus, there is
no current involved, and the magnetic scalar potential is
applied for solving the Maxwell's equations. The governing
Maxwell's equations in the aforementioned three regions are
expressed as:
In the region I:
(12)
In the region IT:
(13)
BIr(rpz) = BIIr(rl'z)
HIz (rl'z)= HIIz(rpz)
(18)
HIIz(r2,z) = H111z(r2,z)
BJ//z(r3,z) 0
(21)
BJ/r(r2,z) = BJ//r(r2,z)
=
(19)
(20)
(22)
Using separation of variables, the general solution of the
Laplace's equations can be easily obtained. By ignoring the
right term of (12), the Poisson's equation can be reduced to
the Laplace's equation. Thus, the general solution can be
simplified as:
lj// (r,z) =
+X·
I [a�Io(kl1r) +h};Ko(knr)]x sin(kl1z)
11=1
(23)
where 100 and KoO are the modified Bessel functions of the
first kind and the second kind of order zero respectively. The
special solution of this Poisson's equation is:
� - 2POMn sm(mnz)
P
(24)
lj/ (r,z ) = L...,
2
r
11=1
OJn
•
Therefore, the solution of (12) is the summation of (23) and
(24).
The general solution of the region IT has the same form as
that of the region I:
+co
lj/ (r,z) = I [a:,r10 (kJ) + h:,rKo(kl1r)]x sin(knz)
rr
n=1
(25)
By using the same approach, the general solution of region
TIT is described as:
+oc
lj/JJJ(r, z) = I [a:,rr10 (mJ) +h:,rrKO(ml1r)]x sin(kl1z) (26)
n=l
where
6202
�
_kn
mn- �
V fltp_r
(27)
After calculating the magnetic field in each region, the
machine performance can be evaluated analytically. For
assessing the field modulation of the toothed-pole structure,
the Pth order harmonic component of flux density in the
stator back-iron is investigated. The flux density of r and z
components of region TIT can be expressed as:
oqJ111(r,z)
_
(28)
BIIII' - flip -r
u'"r
oqJ111(r,z)
(29)
BIIIz - _flip - z
'"
renewable energy sources in the future. Its low-frequency and
reciprocating motion nature makes it hard to harvest directly.
Till now, people usually capture this kind of energy by
intermediate devices or facilities and then use conventional
electric generators for electricity production [29]. This
approach has low conversion efficiency and high initial cost.
Therefore, the direct-drive approach attracts more and more
attention [30]. For direct-drive energy harvesting, a generator
of high thrust density is highly expected. The PMLV machine
which possesses the magnetic gear effect is a suitable
candidate for this application [31].
uz
The unknown coefficients can be determined from the
boundary conditions (17)-(22) as given by:
a111 = _Ko(mnrj) bIll
(30)
n
n
10(mnrj)
1
b�II
where
=
XCOI](mnr2)Ko(mnr3)-10(mnr2)Ko(mnrj)
+
[10 (aV2)-Xl](OJnr2)]w10(mnr3)
. . .
. . .
. . .
0.4
0.3
�0.2
p::l
(31)
0.1
XCOKj(mnr2)1 o(mnrj) + 10(mnrJ)Ko (mnr2)
[10(OJnr2)-XI j(OJnr2)]WI0(mnrJ)
S
=
T=
V
=
. .
. . .
. . .
. . .
. . .
. . .
. . .
o
_ �flIPJfllp_Z
C0flo
MJ2PoI](OJn'i) +OJJo(OJnro)]
OJ�10(OJnro)
(a)
...... hlr,=O.4
2PoM JI0(OJnrj)-OJ"Io(OJ"ro)]
0.3
OJ�lo(OJnro)
_._.-
h/,,=0.55
- h/"tt=O.7
/1
//
.
.
" ,
"
"
.
'.
I](OJnr])Ko(OJnro) +10(OJnro)K](OJnr])
10(OJnrj)Ko(OJnro)-10(OJnro)Ko(OJnrj)
cO
'0. 0.2
ill
Kj(OJnrj) + VKo(OJnrj)
I](OJn'i)-Ufo(OJn'i)
S-TV
=
W
Ij(OJnrj)-Vio(OJnrj)
v=
0.1
-------
x=
V/o(OJnr2)+Ko(OJnrJ
VI](OJnrJ-K](OJnr2)
0.2
(32)
where 110 and KIO are the modified Bessel functions of the
first kind and the second kind of order one respectively.
Fig. 5 shows the Pth harmonic component flux density
versus the PM remanence according to the variation of the
tooth width b, and slot depth h, over the tooth pitch i,. It can
be observed that when the tooth width equals to one half of
the tooth pitch and the slot depth equals about 0.55 of the
tooth pitch, the field modulation efficiency of the toothed­
pole structure is optimum.
TV.
TMPLEMENTATlON FOR DIRECT-DRIVE WAVE POWER
GENERATION
Because of its clean, sustainable, and tremendous features,
the oceanic wave energy is accepted as one of the promising
0.4
0.8
0.6
(b)
Fig. 5. Pth harmonic component of flux density versus tooth width and slot
depth. (a) 3-D view. (b) 2-D view.
Based on the above design approach, a PMLV machine is
dimensioned and prototyped. For easing the fabrication, the
single-sided flat type is adopted. The toothed-pole structure
dimensions are set, namely the tooth pitch it 12 mm, tooth
width bt 6 mm, and slot depth ht 6 mm. The number of
teeth of the stator ZI is 27, pole-pair number of PMs Z2 is 24
and pole-pair number of armature MMFs P is 3.
Tn order to assess the performance of the proposed
machine, the finite element analysis (FEA) is applied for the
field computation and a set-up is built for testing the
prototype machine. Fig. 6 shows the test-bed of the proposed
machine which consists of a DC reducer motor, the prototype
machine and some mechanical components for rotational-
6203
=
=
=
linear motion inter-conversion. The DC reducer motor serves
as the prime mover. By using the cam and connecting rod, the
rotational motion can be converted into linear motion which
imitates the incident wave force. When the DC reducer motor
operates at a constant speed, the prototype machine operates
in an oscillatory mode via the cam and the linkage.
Fig. 7 shows the magnetic field distributions at no-load
condition. It can be seen that there are multi-pole-pair MMFs
at the mover side, but at the stator side, only several pole­
pairs are found which confirms the field modulation function
of the toothed-pole structure. As indicated by the arrow in
Fig. 7, even with a tiny movement of the mover, the flux
linkage in the armature winding is changed dramatically
which confirms the high force production.
When the prototype machine operates in the oscillatory
mode, the no-load induced voltage waveform is measured as
shown in Fig. 8. It can be observed that the amplitude of the
voltage depends on the mover instantaneous velocity. Also,
the measured no-load induced voltage waveform at 0.5 mls is
compared with the simulated no-load induced voltage
waveform under the same mover velocity. As shown in Fig. 9,
the agreement is very good.
Fig. 10 depicts the static force waveforms when the 3phase winding is fed with the rated current and without
current. It shows that the cogging force as a parasitic force
component is low compared with the rated thrust. Fig. 11
shows the static force characteristic at different currents.
50.00rns
f �
J
0.00V
... . ........ ....
.. ....... .. . --. ....... ...
.
.
.. .... .............. ....
.....: .... : .... .: .... : .....: .... -: .....: .....: .... .: ..5.V./
. di�.....
:
:
:
:
:
: : 50: ms/�iv
Fig.
t!Ial!!! 5.00V
8.Measured no-load induced voltage waveform.
5.eeerns
r
-B
f �
T
e.eev
: ---� ----:-- Simulated waveform:
... . : . . .. : . �. 'Measw-e'd wavefoI'Ili : .. . . : .... : ... .
.
.
..
..
.
..
..
..
..
..
.. . . : . . .. . . . .. . . . .... . . . .... . _ . . . .. . . . .. . . . .... . .� .Y!9!'!.. .
:
:
.: .: .: S mslaiv
.: .: .:
Fig.
t!Ial!!! 5.00V
9.Comparison of no-load induced voltage waveforms.
300 .-------�
200
�
100
b
�
o
�
0
-100
-200
-300+-------,-�--_.--�
5
15
o
10
20
Position (mm)
Fig. 10. Rated static force and cogging force waveforms.
300 .-------�
•
250
�
200
en
150
f-;
100
]
•
•
50
o
Fig.
7.Magnetic field distributions.
2
4
6
Current (A)
Fig. II. Static force characteristic at different currents.
6204
8
10
brushless machine,"
V.
CONCLUSION
This paper has presented the design, optimization and
application of a PMLV machine. Firstly, the basic
specifications of the machine are determined by the
traditional design approach. Then, by using the analytical
calculation method, the relationship of the machine
performance and the toothed-pole structure dimension is
unveiled. According to the analytical result, a PMLV machine
is designed and prototyped. Finally, the prototype machine is
implemented for the direct-drive wave power generation.
Both the numerical analysis and experimental results have
been used to verify the machine performances.
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This work was supported and funded by a research grant
(Project No. HKU 71071 IE) of the Research Grants Council
in Hong Kong Special Administrative Region, China.
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