energies
Article
Analytical Model of a Dual Rotor Radial Flux Wind
Generator Using Ferrite Magnets
Peifeng Xu, Kai Shi *, Yuxin Sun and Huangqiu Zhu
School of Electrical and Information Engineering, Jiangsu University, Zhenjiang 212013, China;
xupeifeng2003@126.com (P.X.); syx4461@ujs.edu.cn (Y.S.); zhuhuangqiu@ujs.edu.cn (H.Z.)
* Correspondence: shikai80614@163.com; Tel.: +86-511-8879-1245
Academic Editor: Frede Blaabjerg
Received: 9 July 2016; Accepted: 18 August 2016; Published: 24 August 2016
Abstract: This paper presents a comprehensive analytical model for dual rotor radial flux wind
generators based on the equivalent magnetic circuit method. This model is developed to predict
the flux densities of the inner and outer air gaps, flux densities of the rotor and stator yokes, back
electromotive force (EMF), electromagnetic torque, cogging torque, and some other characteristics
important for generator design. The 2D finite element method (FEM) is employed to verify the
presented analytical model, fine-tune it, and validate the prediction precision. The results show
that the errors between the proposed analytical model and the FEM results are less than 5% and
even less than 1% for certain parameters, that is, the results obtained from the proposed analytical
model match well the ones obtained from FEM analysis. Meanwhile, the working points at different
temperatures are confirmed to exceed the knee point of the BH curve, which means that irreversible
demagnetization does not occur. Finally, the optimization by FEM with the objective of fully using
the inner space of the generator, decreasing the cogging torque, and reducing the total harmonic
distortion (THD) of back EMF is performed.
Keywords: dual rotor radial flux wind generator; analytical model; equivalent magnetic circuit;
ferrite magnets; finite element method; optimization
1. Introduction
Wind power generation, as the most competitive and clean renewable energy technology,
is attracting more and more attention all over the world. In wind energy conversion systems,
permanent magnet generators (PMGs) have been widely used for their numerous advantages such as
simple structure, high efficiency, reliable operation and no need for an additional power supply due
to the magnet field excitation. Most PMGs contain rare-earth permanent magnet materials, namely,
NdFeB grades, for their high remanence and coercivity values [1–5]. However, the cost of NdFeB
magnets has increased significantly over the last several years and there is an increasing concern about
the stable supply of the raw material for NdFeB [6]. With this consideration, generators with less or
without rare-earth materials are required in wind power applications to reduce the machinery cost [7].
Ferrite magnet material has many merits, such as low weight density, low cost, and stainless
nature, etc. PMGs using ferrite magnets were studied in [8–10]. Due to the relatively low residual flux
density of ferrite magnet material, it is obvious that the performance is dramatically decreased when a
ferrite magnet with 0.4 T of remanence is used instead of a NdFeB magnet [11]. In order to resolve this
problem, the dual rotor permanent magnet (PM) machine using ferrite magnets was proposed [12].
When using NdFeB magnets, the dual rotor permanent magnet (DRPM) machine can significantly
improve the torque density and efficiency owing to the special structure of double rotors and dual
air-gaps. In fact, the improvement of torque density is achieved by doubling the working portion of
the air gap [13,14]. Accordingly, the torque density and efficiency of the DRPM wind generator using
Energies 2016, 9, 672; doi:10.3390/en9090672
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Energies 2016, 9, 672
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ferrite magnets are higher than those of traditional PM generators using ferrite magnets. Qu designed
a dual rotor permanent magnet (DRPM) motor using ferrite magnets in [15]. In [16] he came to
the conclusion that the torque density of a DRPM is almost three times the density of a commercial
induction motor (IM) and 2.2 times the density of a commercial interior PM (IPM) machine with
an efficiency just slightly lower than that of an IPM machine, but higher than that of an IM. Hence,
we may consider transplanting the dual rotor machine using ferrite to wind power generation.
DRPM machines include dual rotor radial flux PM machines and dual rotor axial flux PM
machines [17]. In general, dual rotor radial flux (DRRF) and dual rotor axial flux (DRAF) machines
have similar torque density, torque-to-mass ratio, losses, and efficiency performance. However,
the material cost of DRAF machines is much higher than that of DRRF machines due to more magnets
being required for DRAF machines. Meanwhile, the DRRF machines need and can provide stronger
cooling capability than DRAF machines [18]. As a consequence, this paper proposes a DRRF wind
generator using ferrite magnets.
When performing preliminary design of an electric machine, the equivalent magnetic circuit
(EMC) approach is a popular choice for designers due to its fast and analytical nature [19].
EMC needs empirical coefficients to correct the magnetic saturation and leakage flux conditions.
Those coefficients are easily affected by many factors, including the operation state and magnetic
saturation level, so the precision of the EMC approach is not high enough to use in practice. The finite
element method (FEM), as a numerical method, is a numerically accurate algorithm [20], but is too
time-consuming. Considering the fast performance of EMC and high accuracy of FEM, the analytical
approach combining EMC and FEM should be a good compromise between simplicity and accuracy.
This approach is useful and effective in the preliminary design of an electrical machine.
The main contribution of this paper is the development of a comprehensive analytical model for a
DRRF wind generator based on EMC. The elements of this analytical model are precisely calculated in
cylindrical coordinates to achieve a high accuracy. The proposed model takes into account the iron
saturation, all generator dimensions as well as material properties, and is able to accurately predict
the electromotive force (EMF), torque and cogging torque, as well as numerous other design aspects.
The results of the proposed model are then evaluated and verified by FEM. The errors between the
proposed model and FEM model are less than 5% for flux densities, 3% for EMF, and even as low as 1%
for electromagnetic torque, which are basic parameters used to verify the feasibility of the proposed
model. In addition, the working points at different temperatures are measured to determine whether
the ferrite magnets are demagnetized or not, and the corresponding results indicate that irreversible
demagnetization does not occur.
At the end of this paper, the optimization to achieve specific design objectives, such as making
the best use of the inner space of the DRRF generator, decreasing the cogging torque, and reducing the
total harmonic distribution (THD) of the back EMF, is presented. Consequently, the optimal split ratio
of the inner and outer stator radius, the pole arc ratio of the inner and outer rotors, and the outer slot
opening are determined.
2. Modeling
2.1. Machine Topology
Figure 1 illustrates the configuration and linear topology of the DRRF wind generator. Therein,
a cup stator is sandwiched between two concentric rotors. The radially polarized surface mounted
ferrite magnets are placed on the air-gap sides of the inner and outer rotors. For the sake of a smaller
effective air gap, the stator is slotted on both sides to enhance the power density. The back-to-back
toroidally wound winding is adopted, which brings advantages such as short end winding length, less
copper loss, and simple maintenance. In some sense, the DRRF generator can be seen as one inner
rotor generator nested inside one outer rotor generator.
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B1
αp2θp
C1
Energies 2016, 9, 672
PMs
B1
hpm2
g2
Rir2
A2
Ror1
Ris
C1
θp
PMs
A1
Ros
C2
Ror1
Ris
θp
Ros
A
A1
hpm2
g1
g2
B
C
C
C
A
αbo2
bo1
α
B
A
B
hs2 r
dys
Stator
CoreCore
Outer
Rotor
hs1
C
C
C
A
Inner
RotorCore
Core
Stator
θ
g1
B2
αbo2
αAp2θp B
θ dyr2
hpm1 A
dyr1
Rir2
A2
r 3 of 18
Outer Rotor Core
dyr2
dys
hs2
hs1
αp1θp
αbo1
(a)
hpm1
(b)
dyr1
Inner Rotor Core
Figure 1. Configuration and linear topology of the proposed dual rotor radial flux (DRRF) wind
C2 Configuration and linear topology of the
Figure 1.
proposed dual rotor radial flux (DRRF) wind
generator: (a) Configuration; (b) Linear topology. αp1θp
B2
generator: (a) Configuration;
(b) Linear topology.
(a)
(b)
The variables defined in Figure 1 are:
Figure
1. Configuration
and linear
topology of the proposed dual rotor radial flux (DRRF) wind
The variables
Figure
1 are:
Ror1defined
outerin
radius
of the
inner rotor
generator: (a) Configuration; (b) Linear topology.
Rir2
inner radius of the outer rotor
Ror1
outer
radius
of the
rotor
Ris
inner radius
of inner
the stator
defined
in the
Figure
1 are:
Rir2 The variables
inner
radius
of
rotor
Ros
outer radius
ofouter
the stator
θp outer
poleradius
pitch
angle
Ris
inner
radius
of the
stator
Ror1
of the
inner rotor
m
number
of the
phase
Rir2outer
inner
radius
of the
outer rotor
Ros
radius
of
stator
p
machine
pole
pairs
Rispole inner
θp
pitchradius
angle of the stator
innerradius
or outer
slot stator
number
RosNs outer
of the
m
number
of arc
phase
αp1,2
pole
ratio of the inner/outer rotor
θp
pole pitch angle
p
machine
pole pairs
L
effective
stack length
m
number of phase
h
pm1,2
thickness
of the
inner/outer magnets
Ns
or outerpole
slot
number
p innermachine
pairs
α
bo1,2
the
inner/outer
slot
opening angle
αp1,2
arc ratio
of the
rotor
Nspole inner
or outer
slotinner/outer
number
hs1,2
height of the inner/outer slots
L
effective
stack
length
αp1,2
pole
arc ratio of the inner/outer rotor
g1,2
length of the inner/outer air gap
effective
stack
length
hpm1,2 L thickness
of the
magnets
dyr1,2 thickness
ofinner/outer
the
inner/outer
rotor yoke
hpm1,2
thickness of the
magnets
αbo1,2
the inner/outer
slotinner/outer
opening angle
αbo1,2
theofinner/outer
slot opening
2.2. Analytical
Model
hs1,2
height
the inner/outer
slots angle
hs1,2
height of the inner/outer slots
g1,2 The linear
length
of the inner/outer
airthe
gap
translational
topology and
corresponding EMC of the DRRF wind generator are
g1,2
length of the inner/outer air gap
presented
and
shown
in
Figure
2a,b,
respectively,
where the leakage flux, iron saturation as well as
dyr1,2
thickness of the inner/outer rotor yoke
thickness
of theinto
inner/outer
machinedyr1,2
dimensions
are taken
account. rotor yoke
2.2. Analytical
Model
2.2. Analytical
Model
Outer rotor
Rr 2
Φm2 2 10
Φr2
The linear
translational
topology
the
EMC
DRRF
generator
2R
2Rwind
The linear
translational
topologyand
and
the corresponding
corresponding
EMC
ofof
thethe
DRRF
wind
generator
are are
Outer
4
R
R
PM
presented
andand
shown
in in
Figure
where2the
theleakage
leakage
flux,
iron
saturation
as
well
presented
shown
Figure2a,b,
2a,b,respectively,
respectively,
flux,
iron
saturation
as
well
as as
Fwhere
F
R
Φ
8
machine
dimensions
taken
into
account.
machine
dimensions
areare
taken
into
account.
2R
2R
Φmr2
pm2
pm2
mr 2
+
g2
Outer rotor
Outer
Inner
PM
PM
pm2
7
pm1
FFpm2
+
Rmr 2
2
1
Rmr1
8
2Rpm1
2Rg2
Inner rotor
(a)
pm2
9
g2
RR
syr 2
Φs
Φg2 2
Φr2
Φg1 2
2Rg1
5
4
Rmm1
R3mm2
Φmm1
Rmr 2
ΦRmm2
mr1
2Rpm2
F-pm1F
+
pm2
2Rpm1
Φmr1
Φg2 2 Φm1 2
Φg1 2 11
2Rg1
Stator
+
mm2
6
Φg2 2
Φg1 2 11
Φm2 2 10
2Rg1
Φmr2
2R
+
Stator
mr 2
mm2
+
pm2
6
Rr1
 r1
2Rg2
Φg2 2
Φs
(b)Rsy
Φg1 2
2Rg1
5
7
Figure 2. The linear translational topology and the corresponding
equivalent
magnetic
circuit (EMC)
Φmm1
Rmm1
Fpm1 -+
Fpm1
Inner translational
of the DRRF wind generator: (a) The linear
topology;
(b)
The
corresponding
EMC.
1
3
R
R
+
PM
Inner rotor
mr1
2Rpm1
Φm1 2
(a)
mr1
2Rpm1
Φmr1
9
Rr1
 r1
(b)
Figure 2. The linear translational topology and the corresponding equivalent magnetic circuit (EMC)
Figure 2. The linear translational topology and the corresponding equivalent magnetic circuit (EMC)
of the DRRF wind generator: (a) The linear translational topology; (b) The corresponding EMC.
of the DRRF wind generator: (a) The linear translational topology; (b) The corresponding EMC.
Energies 2016, 9, 672
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In this EMC, Φm1 and Φm2 are the fluxes leaving the inner and outer magnets respectively, Φmr1
and Φmr2 are the inner and outer leakage fluxes between the magnet and rotor yoke, Φr1 and Φr2
are the fluxes going through the inner and outer rotor yokes, Φmm1 and Φmm2 are the leakage fluxes
between the adjacent inner and outer magnets, Φg1 and Φg2 are the fluxes passing through the inner
and outer air gaps, and Φs is the flux passing through the stator yoke. The equivalent magneto motive
forces (MMFs) of the inner and outer magnets are respectively given by:
Fpm1 = hpm1 | Hc |; Fpm2 = hpm2 | Hc |
(1)
where Hc is the coercivity of the ferrite magnets.
The reluctance of the inner magnet can be calculated from [21]:
Rpm1 =
1
µ0 µr
Z R +h
or1
pm1
Ror1
hpm1
dr
1
=
ln 1 +
rαp1 θp L
µ0 µr αp1 θp L
Ror1
(2)
Similar to the inner magnet, the reluctance of the outer magnet is:
Rpm2 =
hpm2
1
ln 1 +
µ0 µr αp2 θp L
Rir2 − hpm2
(3)
where µr is the relatively recoil permeability of the ferrite magnets.
The half circle shaped model is one of the most precise techniques for modeling flux flowing in
the inner and outer air gaps as depicted in Figure 3a,b. At each side of the inner and outer air gaps, the
length of the fringing fluxes Li (r) and Lo (r) can be calculated by:
Li (r ) =
Lo (r ) =
q
2
( ge1 /2)2 − r − Ror1 − hpm1 − ge1 /2
q
(4)
2
( ge2 /2)2 − Rir2 − hpm2 − ge2 /2 − r (5)
ge1 = Kc1 g1 ; ge2 = Kc2 g2
(6)
where ge1 and ge2 are the effective lengths of the inner and outer air gaps taking into account the stator
slotting, Kc1 and Kc2 are the inner and outer Carter’s coefficients respectively [21]:
τs1
Kc1 =
τs1 −
2αbo1 Ris
π
Ris
tan−1 αbo1
−
2g
1
g1
αbo1 Ris ln
1+
αbo1 Ris
2g1
2 τs2
Kc2 =
τs2 −
2αbo2 Ros
π
Ros
tan−1 αbo2
2g2
τs1 =
−
g2
αbo2 Ros ln
2πRis
2πRos
; τs2 =
Ns
Ns
where τs1 and τs2 are the inner and outer slot pitches.
1+
αbo2 Ros
2g2
2 (7)
(8)
(9)
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αp2θp
Stator
e1/2
Li(r)
pm1+g
r-(Ror1+h
ge1/2
Ris
dr
r
)
hpm2 Rir2
Li(r)
ge1
Lo(r)
ge2/2
hpm1
Rir2-hpm2
Outer P
-ge2/2-r
dr
M
Lo(r)
ge2
r
Inner P
M
Stator
Ror1 αp1θp
(a)
(b)
dx
g
e1
β1
L1
x
β1
dx
M
Inner P
Inner P
M
(1-αp1)θp
n
otor Iro
Inner R
(c)
(d)
(1-αp2)θp
Outer P
x
n
otor Iro
Outer R
M
β2
Outer P
x
β2
dx
ge2
dx
M
x
L2
(e)
(f)
Rr21
Rr22
α p2θ
p
Rr22
(1-αp2)θp
/2
Rs
(1-αp1)θp
Rr11
αp1θ
/2
p
Rr12
Rr12
(g)
Figure 3. The flux path for calculating: (a) The effective inner air gap reluctance; (b) The effective
Figure 3. The flux path for calculating: (a) The effective inner air gap reluctance; (b) The effective outer
outer air gap reluctance; (c) The inner magnet to magnet leakage reluctance; (d) The inner magnet to
air gap reluctance; (c) The inner magnet to magnet leakage reluctance; (d) The inner magnet to inner
inner rotor iron leakage reluctance; (e) The outer magnet to magnet leakage permeance; (f) The
rotor iron leakage reluctance; (e) The outer magnet to magnet leakage permeance; (f) The outer magnet
outer magnet to iron leakage reluctance; (g) The rotor yokes reluctances.
to iron leakage reluctance; (g) The rotor yokes reluctances.
The effective reluctances of the inner and outer air gaps can be calculated by the trapezoidal
The as:
effective reluctances of the inner and outer air gaps can be calculated by the trapezoidal
integral
integral as:
Z R
 hpm1
ge1
+ ge1
1or1 +Rhor1pm1
ddr
r
1
Rg1 = Rg1 
(10)
(10)

R

h
pm1
rαp1
θp +2
(r )
µ0 L μR0or1L+hor1
rα
+ L2L
p1 p
i i (r )
pm1
Rir2 −hpm2
1
dr
Rg2 =
1 Rir2  hpm2
dr
µ
L
rα
θ
Rg20  Rir2 −hpm2 − ge2 p2 p + 2Lo (r )
μ 0 L Rir2  hpm2  ge2 rα p2θp +2 Lo (r )
Z
(11)
(11)
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The arc shaped model of the leakage permeance between two adjacent inner magnets is shown in
Figure 3c. And the leakage permeance of the inner magnets will be:
Λmm1 =
1
Rmm1
= µ0 L
Z g
e1
0
dx
2β1 x + ( Ror1 + hpm1 + x )(1 − αp1 )θp
β1 = π − arccos
ge1 /2
Ror1 + hpm1
(12)
(13)
So Λmm1 can be further expressed as:
Λmm1
"
#
2β1 ge1 + (1 − αp1 )θp
µ0 L
=
ln 1 +
2β1 + (1 − αp1 )θp
( Ror1 + hpm1 )(1 − αp1 )θp
(14)
According to the model shown in Figure 3d, the leakage permeance between the inner magnet
and inner rotor iron can be calculated from:
Λmr1 =
1
Rmr1
= µ0 L
Z L
1
0
dx
2β1 x + hpm1
L1 (αp1 ) = min ge1 , ( Ror1 + hpm1 )(1 − αp1 )θp /2
(15)
(16)
and then Λmr1 can be further expressed as:
Λmr1
"
#
2β1 L1
µ0 L
=
ln 1 +
2β1
hpm1
(17)
For two adjacent outer magnets, according to Figure 3e:
ge2 /2
Rir2 − hpm2
(18)
dx
2β2 x +( Rir2 −hpm2 − x )(1−αp2 )θp
h
i
g [2β −(1−αp2 )θp ]
µ0 L
ln 1 + ( Re2 −h2 )(1−α
2β2 −(1−αp2 )θp
)
θ
pm2
p2 p
ir2
(19)
β2 = arccos
so the leakage permeance of the outer magnets is:
Λmm2
=
=
1
Rmm2
= µ0 L
R ge2
0
Similarly, according to Figure 3f the leakage permeance between the outer magnet and outer rotor
iron can be calculated by:
Z L2
1
dx
= µ0 L
(20)
Λmr2 =
Rmr2
2β2 x + hpm2
0
L2 (αp2 ) = min ge2 , ( Rir2 − hpm2 )(1 − αp2 )θp /2
(21)
and Λmr2 can be further expressed as:
Λmr2 =
µ0 L
2β L2
ln 1 + 2
2β2
hpm2
(22)
As shown in Figure 3g, the total reluctances of the inner and outer rotors are:
Rr1 = Rr11 + 2Rr12 ; Rr2 = Rr21 + 2Rr22
(23)
Energies 2016, 9, 672
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Reluctances adjacent to the inter-polar regions are:
Rr11 =
( Ror1 − dyr1 /2)(1 − αp1 )θp
µ0 µr1 A11
(24)
Rr21 =
( Rir2 + dyr2 /2)(1 − αp2 )θp
µ0 µr1 A21
(25)
A11 = dyr1 L; A21 = dyr2 L
(26)
where A11 and A21 are the areas passing through the corresponding yoke flux flow. Making use of the
average area of the surfaces, reluctances adjacent to the magnets are:
Rr12 =
0.5αp1 θp ( Ror1 − dyr1 /2)
µ0 µr2 ( A11 + A12 )/2
(27)
Rr22 =
0.5αp2 θp ( Rir2 + dyr2 /2)
µ0 µr2 ( A21 + A22 )/2
(28)
A12 = 0.5αp1 θp Ror1 L; A22 = 0.5αp2 θp Rir2 L
(29)
where µr1 , µr2 are the relatively permeabilities of the corresponding iron parts. They change with the
flux densities inside the iron parts.
The reluctance of the stator yoke is:
Rsy =
Ns ( Ris + kRos )/(k + 1)
h
; k = s1
2p
µ0 µr1 dys L
hs2
(30)
Applying the Kirchhoff’s circuit law (KCL) and Kirchhoff’s voltage law (KVL) to the loops and
nodes in Figure 2, some relationships can be obtained:
Rpm1 Φm1 + Rmr1 Φmr1 = Fpm1 ; Rpm2 Φm2 + Rmr2 Φmr2 = Fpm2
(31)
Rmm1 Φmm1 − 2Rmr1 Φmr1 + Rr1 Φr1 = 0; Rmm2 Φmm2 − 2Rmr2 Φmr2 + Rr2 Φr2 = 0
(32)
2Rg1 Φg1 − Rmm1 Φmm1 + Rs Φs = 0; 2Rg2 Φg2 − Rmm2 Φmm2 + Rs Φs = 0
(33)
0.5Φg1 + Φmm1 + Φmr1 = 0.5Φm1 ; 0.5Φg2 + Φmm2 + Φmr2 = 0.5Φm2
(34)
Φmr1 + Φr1 = 0.5Φm1 ; Φmr2 + Φr2 = 0.5Φm2
(35)
0.5Φg1 + 0.5Φg2 = Φs
(36)
By calculating these equations, all fluxes flowing into the circuit elements can be obtained. As a
result, the flux densities corresponding to reluctances Rr11 , Rr12 , Rr21 , Rr22 and Rsy can be calculated by:
Br11 =
Φr1
Φr1
; Br12 =
A11
( A11 + A12 )/2
(37)
Br21 =
Φr2
Φr2
; Br22 =
A21
( A21 + A22 )/2
(38)
Bs =
Φs
dys L
(39)
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18
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According to (37) and (38), a conclusion can be drawn that the flux densities adjacent to the
According
to (37)
(38),
a conclusion
can be
drawn
that the
adjacent
to the
inter-polar regions
are and
higher
than
those adjacent
to the
magnets,
andflux
thedensities
flux densities
within
the
inter-polar
regions
are
higher
than
those
adjacent
to
the
magnets,
and
the
flux
densities
within
the
inner and outer air gaps are respectively:
inner and outer air gaps are respectively:
Φg1
Φg2
; Bg2 
Bg1 
(40)
Φ
Φ
g1
g2
Bg1 = rα p1θp +2 Li (r)  L; Bg2 = rα p2θp +2 Lo (r ) L
(40)
rαp1 θp + 2Li (r ) L
rαp2 θp + 2Lo (r ) L
2.3.
2.3. Back
Back Electromotive
Electromotive Force
Force (EMF)
(EMF) Derivation
Derivation
Before
Beforethe
theEMF
EMFcan
canbe
befound,
found,ititisisnecessary
necessarytotoobtain
obtainthe
thefundamental
fundamentalcomponents
componentsB1g1
B1g1and
andBB1g2
1g2
of
the
flux
densities
in
the
inner
and
outer
air
gaps.
When
the
leakage
flux
is
neglected,
all
the
air
gap
of the flux densities in the inner and outer air gaps. When the leakage flux is neglected, all the air gap
flux
flux flows
flows into
into the
the air
air gap
gap through
throughthe
themagnet
magnetsurface.
surface. In
In this
this situation,
situation, the
the air
air gap
gap flux
flux density
density is
is an
an
approximate
square
wave
as
shown
in
Figure
4
in
green,
and
its
fundamental
component
obtained
by
approximate square wave as shown in Figure 4 in green, and its fundamental component obtained by
Fast
Figure 4b
4b in
in red,
red, where
whereααppisisthe
thepole
polearc
arcratio
ratioofofthe
therotor.
rotor.
Fast Fourier
Fourier Transform (FFT) is shown in Figure
0.3
0.3
0.2
0.2
αp θ p
0.1
B1g(T)
Bg(T)
0.1
0
0
-0.1
-0.1
αpθp
-0.2
-0.2
-0.3
-0.3
0
60
120
180
240
Electrical angle(deg)
300
360
0
60
120
180
240
Electrical angle(deg)
(a)
300
360
(b)
Figure 4. The air gap flux density waveform under a pair of poles of conventional surface mounted
Figure 4. The air gap flux density waveform under a pair of poles of conventional surface mounted
pole structure and its fundamental component: (a) The air gap flux density; (b) The fundamental
pole structure and its fundamental component: (a) The air gap flux density; (b) The fundamental
component of the air gap flux density.
component of the air gap flux density.
Based on FFT, the fundamental components of the inner and outer air gap flux densities, B1g1
Based on FFT, the fundamental components of the inner and outer air gap flux densities, B1g1 and
and B1g2 can be calculated by:
B1g2 can be calculated by:
2 2
 1 α p1θp Bg1 ; B1g2  2 √2 α p2 sin 1 α p2θp  B1g1  √ α p1 sin
Bg2
(41)
2 π
1
1
2
2 2
Blg1 =
αp1 sin  2αp1 θp  Bg1 ; Blg2 = π αp2 sin 2 αp2 θp Bg2
(41)
π
2
π
2
For an alternating current (AC) machine, it is well known that the induced root mean square
Forvoltage
an alternating
current
(AC) machine,
it is well
known
(RMS)
per phase
of a concentrated
winding
is given
by: that the induced root mean square
(RMS) voltage per phase of a concentrated winding is given by:
Ef  4.44 fNph Φf
(42)
Ef = 4.44 f Nph Φf
(42)
where the subscript “f” denotes the fundamental, f is the frequency, Nph is the total number of turns
in series
phase, “f”
anddenotes
Φf is thethe
fundamental
flux
where
theper
subscript
fundamental,
f isper
thepole.
frequency, Nph is the total number of turns in
For
the
sake
of
dual
air
gap
structure,
the
phase
EMF
series per phase, and Φf is the fundamental flux per pole. of the DRRF wind generator is the sum of
the EMF
induced
theair
flux
linkage
in both
inner EMF
and outer
gaps.
Given
Nph/2 turns
both
For the
sake ofby
dual
gap
structure,
the phase
of theair
DRRF
wind
generator
is thefor
sum
of
inner
and
outer
slots,
the
induced
RMS
phase
voltage
is:
the EMF induced by the flux linkage in both inner and outer air gaps. Given N /2 turns for both
ph
inner and outer slots, the induced RMS phase voltage is:
Nph
Ef =Ef1 +Ef2  4.44 f
(Φf1 +Φf2 )
N2ph
Ef = Ef1 + Ef2 = 4.44 f
(Φf1 +Φf2 )
2
2
2
Φ f1 =√ 2 B1g1τ p1 L ; Φ f2 = √ 2 B1g2 τ p2 L
2 π
2π
Φf1 =
2Blg1 τp1 L; Φf2 =
2Blg2 τp2 L
π
π
where τp1, τp2 are the inner and outer pole pitches, respectively:
(43)
(43)
(44)
(44)
Energies 2016, 9, 672
9 of 18
where τ p1 , τ p2 are the inner and outer pole pitches, respectively:
τp1 =
2πRis
πRis
πRos
=
; τp2 =
2p
p
p
(45)
When the fundamental distribution factor, the pitch factor, the skew factor for the distributed
fractional-pitch, and skewed magnets are applied, the voltage is modified to:
√
Ef = 4.44 2Kw f Nph ( B1g1 Ris + B1g2 Ros ) L p
(46)
where Kw is the winding factor for the fundamental.
2.4. Voltage Equation and Torque Derivation
When neglecting the slot opening height, the average coil length per turn is given by:
Lc = 2L + 2dys + 2hs1 + 2hs2 + g1 + g2
The phase resistance Rs is:
$Lc Nph
CAw
(48)
mCNph
Aslot Kcu
; Nc =
Nc
Ns
(49)
Rs =
Aw =
(47)
where $ is the winding material resistivity, C is the number of parallel circuits of the winding, Aw is
the available bare wire area, Kcu is stator bare copper filling factor, and Nc is the number of conductors
in each slot. For the DRRF generator, due to the adoption of back-to-back toroidally wound winding,
the area of the inner slot should equal to that of the outer one Aslot in order to take full advantage of
the inner space.
The voltage of the DRRF wind generator in synchronously rotating dq reference frame can be
expressed by:
dψsq
dψsd
ud = Rs id +
− ωψsq ; uq = Rs iq +
+ ωψsd
(50)
dt
dt
where ud and uq are the d and q axis stator voltages, id and iq are the d and q axis stator currents, ψsd
and ψsq are the d and q axis flux linkages of the stator, and ω is the angular electrical speed.
The flux linkage equations can be expressed as:
ψsd = Ld id + ψm ; ψsq = Lq iq
(51)
where Ld and Lq are the d and q axis inductances, and ψm is the permanent magnet flux linkage.
The electromagnetic torque can be expressed as:
Tem =
3
3
p(ψsd iq − ψsq id ) = p(ψm iq − ( Ld − Lq )id iq )
2
2
(52)
From (52), it can be seen that the electromagnetic torque consists of two parts: the first part is the
torque produced by the magnets, and the second part is the reluctance torque generated due to the
saliency effect.
2.5. Cogging Torque Derivation
Cogging torque is caused by the interaction of the rotor magnetic flux and angular variations in
stator magnetic reluctance. In detail, cogging torque is produced by the magnetic energy of the field
Energies 2016, 9, 672
10 of 18
due
to 2016,
the magnets
with the mechanical angular position of the rotor. For the DRRF generator,
the
Energies
9, 672
10 of
18
cogging torques of the inner and outer rotors are:
dg1
Rg1
dRg2
1 22 dR
1 1 2dR
g2
; T
Tcog1
=−
Tcog1
  Φ2g1
Φg1
Tcog2=−
 Φ
Φg2
; cog2
g2
22
dθ
22
dθ
dθ
dθ
(53)
(53)
Equation
torques
areare
caused
by by
the the
stator
slotted
on both
sides,sides,
and
Equation (53)
(53)implies
impliesthat
thatthe
thecogging
cogging
torques
caused
stator
slotted
on both
that
torques
always
exist.
It should
be be
noted
that
thethe
airair
gap
reluctances
and the
thatcogging
the cogging
torques
always
exist.
It should
noted
that
gap
reluctancesRg1
Rg1and
andRRg2g2
here
are
not
the
average
value
from
Equations
(10)
and
(11).
The
inner
air
gap
reluctance
accounting
here are not the average value from Equations (10) and (11). The inner air gap reluctance accounting
for
for the
the angular
angular position
position of
of the
the rotor
rotor relative
relative to
to the
the stator
stator teeth
teeth will
will be
be shown
shown in
in Figure
Figure 5.
5. When
When the
the
center
line
of
the
inner
ferrite
magnet
aligns
with
the
center
line
of
the
inner
tooth,
θ
=
0.
The
air
gap
center line of the inner ferrite magnet aligns with the center line of the inner tooth, θ = 0. The air gap
corresponding
corresponding to
to one
one half
half magnet
magnet pole
pole can
can be
bedivided
dividedinto
intothree
threeparts,
parts,PP11,, PP22,, and
and P
P3..
r
Stator Core
Tooth
Slot
Tooth
hs1
2π
 αbo1
Ns
αbo1
dx
g1
P3 P2
x
hpm1
P1
αp1θp
θ+θp/2
Ror1
θ
θ=0
Figure 5. The inner air gap reluctance for cogging torque calculation.
Figure 5. The inner air gap reluctance for cogging torque calculation.
The air gap reluctance of P1 part can be calculated by:
The air gap reluctance of P1 part can be calculated by:
Ror1  hpm1  g1
1 R  h  g1
dr
1
Z R +h or1 +pm1
R

ln
g
or1
pm1
1
R
h
+g
1

1
1
or1 +
μ 0 L Ror1  hpm1 h   2π Ndr
Ror1pm1
 hpm1 1
 α bo1 i =
h   2π Ns   α bo1 i ln
R1 =
s
2π/N
α
2π/N
α
(
s )− bo1
s )− bo1
µ0 L Ror1 +hpm1
− θ rθ r µ0μL0 L (
− θ θ Ror1 + hpm1
2 2
2 2




(54)
(54)
The
The air
air gap
gap reluctance
reluctance of
of PP22 part can be calculated by:
by:
R  g+hhs1 + g1 +hs1
1 R1or1 R hpm1
dr dr
h
i
= µ0 L R or1+1 h pm1
(2π/N )−α
θ+ 12 α
−   αs2 bo1
p1 θpN
μ 0 L Ror1  hpm1or1 pm1
 r
2π

1
s
bo1
1θ  α p1θp i Ror1 +hpm1 + g1 +
 rhs1
h
=
2 s )−αbo1 ln
R2or1 +hpm1
p − (2π/N

µ0 L θ+ 21 αp1 θ
2
Ror1  hpm1  g1  hs1
1
 of P part can be calculated by: ln
The air gap permeance
Ror1  hpm1
3 1
 2π Ns   α bo1 
μ 0 L θ  α p1θp 

Z θ + 1 θp 
2µ Ldx
2 L

4hs1 + 4g1 + 2πθ + πθp
2
2µ
0
0
Λ3 =
=
ln
1
1
π
4h + 4g1 + 2πθ + παp1 θp
θ+ 2 αp1 θp hs1 + g1 + 2 πx
The air gap permeance
of P3 part can be calculated by: s1
R2R2
1one half inner magnet pole is:4 h  4 g  2πθ  πθ
The air gap reluctance of
θ θp
μ 0 Ldx
2μ L
s1
1
p
Λ3   12
 0 ln
θ α p1θp
1
π
hs1 hpm1
4 g+1g1 2πθ  πα p1θp
R4
or1 +
1x
hs1  gh1(2π/N
 π
i ln
Rg1 = R1 + R2 +2 Λ13 =
Ror1 +hpm1
2 s2)−αbo1 −θ
µ0 L
Ror1 +hpm1 + g1 +hs1
1
i ln
+ h 1 of one
The air gap reluctance
halfs )−
inner
pole
is:
αbo1 magnet
(2π/N
Ror1
+hpm1
µ0 L θ+ 2 αp1 θp −
1
Rg1  R1  R2 

Λ3
1
2
ln
Ror1  hpm1  g1
+
π
(55)
(55)
(56)
(56)
(57)
4h +4g1 +2πθ+πθp
2µ0 Lln 4h s1
s1 +4g1 +2πθ+παp1 θp
Ror1  hpm1
  2π Ns   α bo1

μ0 L 
 θ
2


R
 hpm1  g1  hs1
1
π
or1

ln

4
h

4
g1  2πθ  πθp
R

h


2π
N

α

s1
1
or1
pm1
s
bo1
2μ 0 L ln
μ 0 L θ  α p1θp 

4hs1  4 g1  2πθ  πα p1θp
2
2


Similarly, the air gap reluctance of one half outer magnet pole is:
(57)
Energies 2016, 9, 672
11 of 18
Similarly, the air gap reluctance of one half outer magnet pole is:
Rg2
=
+
µ0 L
h (2π/N1)−α
s
2
bo2 −θ
i ln
Rir2 −hpm2
Rir2 −hpm2 − g2
1
µ0 L
h
(2π/Ns )−αbo2
θ+ 12 αp2 θp −
2
i ln
Rir2 −hpm2
Rir2 −hpm2 − g2 −hs2
+
π
(58)
4h +4g2 +2πθ+πθp
2µ0 Lln 4h s2
s2 +4g2 +2πθ+παp2 θp
The other part of the inner and outer air gap reluctances can be obtained by symmetry.
The cogging torques have a close relation to the tooth harmonic and are affected by the tooth number
under a pole pair, and the frequency of cogging torques increases with the slot number. The number of
the pulsations of cogging torque in the number of slots can be expressed by:
Ncog =
2p
HCD( Ns , 2p)
(59)
where the denominator is the highest common divider (HCD) of the slot number and pole number.
The spatial period of cogging torque Tsp can be obtained by the least common multiple (LCM):
Tsp =
360◦
LCM( Ns , 2p)
(60)
2.6. Design Consideration
During the generator designing process, a number of technical and economic requirements
must be taken into account, such as torque, efficiency, material cost, etc. It is often difficult for all
these requirements to be satisfied simultaneously. Due to the low remanence, the ferrite magnets
are liable to be affected by a demagnetizing field, which results in a serious deterioration of machine
performance. Accordingly, it is important to limit the flux density inside the ferrite magnets to avoid
the irreversible demagnetization.
3. Model Evaluation
In this section, characteristics of the DRRF wind generator are obtained and evaluated by the
proposed analytical model and are compared with those extracted from FEM. The FEM analysis is
an auxiliary approach to validate the analytical model, slightly adjust it, and prove its predictions.
The primary design specifications and parameters are given in Table 1.
Table 1. Design specifications of the dual rotor radial flux (DRRF) wind generator.
Specification
Values
Unit
Rated Power
Rated Speed
Rated Voltage
Outer Diameter of Stator
Inner Diameter of Stator
Stack Length
Air Gap Length
1.5
375
180
200
86
200
0.5
kW
rpm
V
mm
mm
mm
mm
Flux line distribution and flux density distribution for both no load and full load conditions are
illustrated in Figure 6. It can be found that the flux densities within the portions corresponding to Rr11
and Rr21 are higher than those corresponding to Rr12 and Rr22 , which verifies the correctness of the
analysis result in Figure 3g. By making a comparison between the distributions at no load and full
load conditions, it can be seen that some distortions in the flux lines and flux density have appeared
for the sake of the armature reaction.
Energies 2016, 9, 672
Energies 2016, 9, 672
12 of 18
12 of 18
1.5
1.4
1.3
1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
(a)
(b)
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1.0
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
(c)
(d)
Figure 6. Flux line distribution and flux density distribution at no load and full load: (a) Flux line at
Figure 6. Flux line distribution and flux density distribution at no load and full load: (a) Flux line at no
no load; (b) Flux density at no load; (c) Flux line at full load; (d) Flux density at full load.
load; (b) Flux density at no load; (c) Flux line at full load; (d) Flux density at full load.
The comparison shown in Table 2 demonstrates that all the designed parameters closely agree
The
comparison
shownvalues,
in Tableand
2 demonstrates
that all them
the designed
closely
agreethat
to
to the FEM
measurement
the errors between
are lessparameters
than 5%. This
implies
the
FEM
measurement
values,
and
the
errors
between
them
are
less
than
5%.
This
implies
that
the
the model derived in Section 2 is acceptable as a predictor of machine performance.
model derived in Section 2 is acceptable as a predictor of machine performance.
Table 2. Comparison of the designed and finite element method (FEM) measured values for the
Table
Comparison
of the designed and finite element method (FEM) measured values for the DRRF
DRRF2.wind
generator.
wind generator.
Parameter
Designed Value FEM Value
Flux density
in
inner
air
gap,
B
g1
(T)
0.24
0.23Value
Parameter
Designed
Value
FEM
Flux density in outer air gap, Bg2 (T)
0.273
0.265
Flux density in inner air gap, Bg1 (T)
0.24
0.23
Flux density in inner rotor core, Br11 (T)
0.4
0.39
Flux density in outer air gap, Bg2 (T)
0.273
0.265
Flux density in inner rotor core, Br12 (T)
1.02
0.97
Flux density in inner rotor core, Br11 (T)
0.4
0.39
Flux density
in outer
Br21 (T)
0.376
0.36
Flux density
in inner
rotor rotor
core, core,
Br12 (T)
1.02
0.97
Flux density
in outer
Br22 (T)
1.28
1.22
Flux density
in outer
rotor rotor
core, core,
Br21 (T)
0.376
0.36
Flux in
density
stator
core,
s (T)
1.52
1.45
Flux density
outer in
rotor
core,
Br22B(T)
1.28
1.22
Flux density in stator core, Bs (T)
1.52
1.45
For the no load case, the magnetic fields in the inner and outer air gaps are all generated by
ferrite
However,
the loadfields
case,inthe
not
only by the
ferrite
Formagnets.
the no load
case, thefor
magnetic
themagnetic
inner andfields
outerare
airproduced
gaps are all
generated
by ferrite
magnets
but
also
the
armature
current.
The
armature
current
tends
to
create
a
magnetic
field
with
magnets. However, for the load case, the magnetic fields are produced not only by the ferrite magnets
its
axis
90°
away
from
the
magnetic
axis,
which
is
called
armature
reaction.
The
effect
of
armature
but also the armature current. The armature current tends to create a magnetic field with its axis 90◦
reaction
is to varyaxis,
the which
amplitude
andarmature
phase ofreaction.
the magnetic
fieldofinarmature
the air gaps.
Figure
away
fromfield
the magnetic
is called
The effect
reaction
field7
shows
how
flux
densities
in
the
inner
and
outer
air
gaps
are
altered
by
the
armature
current.
By
is to vary the amplitude and phase of the magnetic field in the air gaps. Figure 7 shows how flux
comparison,
the
flux
densities
decrease
in
the
one
side
of
magnets
while
increase
in
the
other
side
densities in the inner and outer air gaps are altered by the armature current. By comparison, the flux
when the generator operates at full load. Due to the low remanence of the ferrite magnet material
and the thickness of magnets, the effect of armature reaction is not very severe.
Energies 2016, 9, 672
13 of 18
densities decrease in the one side of magnets while increase in the other side when the generator
operates at full load. Due to the low remanence of the ferrite magnet material and the thickness of
magnets,
the9, effect
of armature reaction is not very severe.
Energies 2016,
672
13 of 18
Energies 2016, 9, 672
0.6
0.6
0.4
no load
full
load
no load
0.6
no load
full
load
no load
0.4
0.2
0.6
0.4
full load
full load
0.4
0.2
0.2
0
Bg2(T)
Bg2(T)
Bg1(T)
Bg1(T)
13 of 18
0
-0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.2
-0.4
-0.4
-0.6
-0.4
-0.6
-0.6 0
0
15
15
30
45
60
Electrical
angle(deg)
30
45
60
Electrical
(a) angle(deg)
75
90
75
90
-0.6
0
15
0
15
30
45
60
Electrical angle(deg)
30
45
60
Electrical
(b) angle(deg)
(a)
75
90
75
90
(b)
Figure 7. The air gap flux densities at no load and full load: (a) The inner air gap flux densities;
Figure 7. The air gap flux densities at no load and full load: (a) The inner air gap flux densities;
Figure
7. Theair
airgap
gapflux
flux
densities at no load and full load: (a) The inner air gap flux densities;
(b)
The outer
densities.
(b) The outer air gap flux densities.
(b) The outer air gap flux densities.
The developed back EMF of the proposed DRRF wind generator at the rated speed of 375 rpm
The
of
generator
at
of
Thedeveloped
developed
back
EMF
ofthe
theproposed
proposed
DRRF
wind
generator
atthe
ratedspeed
speeddiscrepancy
of375
375rpm
rpm
is shown
in Figure 8.back
TheEMF
results
are
still
prettyDRRF
close wind
to
each
other, and
the rated
maximum
is
shown
in
Figure
8.
The
results
are
still
pretty
close
to
each
other,
and
the
maximum
discrepancy
is shownthem
in Figure
The 3%.
results
still in
pretty
close
to each
other, of
and
maximum
between
is less8.than
As are
shown
Figure
8, the
waveform
thethe
EMF
is nearlydiscrepancy
sinusoidal
between
them
less
3%.
As
shown
in
waveform
of
sinusoidal
between
themisisisabout
lessthan
than
3%.
Aspeak
shown
inFigure
Figure8,8,the
thethe
waveform
ofthe
theEMF
EMF
isnearly
nearly
sinusoidal
and
the THD
2.4%.
The
value
obtained
by
analytical
model
isisabout
185
V, which
and
the
THD
is
about
2.4%.
The
peak
value
obtained
by
the
analytical
model
is
about
185
V,
which
is
and
the THD
is about
2.4%.
The value.
peak value obtained by the analytical model is about 185 V,
which
is
slightly
larger
than the
design
slightly
larger
than
the
design
value.
is slightly larger than the design value.
200
FEM
Prop.
FEM Model
200
100
Prop. Model
e / Ve / V
100
0
0
-100
-100
-200
-200
0
0.005
0
0.005
0.01
t/s
0.01
t/s
0.015
0.02
0.015
0.02
Figure 8. Phase electromotive force (EMF) of the proposed DRRF generator at no load.
Figure8.8.Phase
Phaseelectromotive
electromotiveforce
force(EMF)
(EMF)of
ofthe
theproposed
proposedDRRF
DRRFgenerator
generatoratatno
noload.
load.
Figure
The electromagnetic torque and its pulsating torques of the DRRF wind generator at full load
The electromagnetic
and its value
pulsating
torques
of the DRRF
windcalculated
generator at
load
are exhibited
in Figure 9.torque
The average
of the
electromagnetic
torque
byfull
FEM
is
The electromagnetic torque and its pulsating torques of the DRRF wind generator at full load are
are exhibited
in Figure
9. The
average
valuemodel
of theis electromagnetic
calculated
by FEM
is
−39.6
Nm and that
obtained
by the
analytical
−39.7 Nm, whichtorque
are very
close to each
other.
exhibited in Figure 9. The average value of the electromagnetic torque calculated by FEM is −39.6 Nm
−39.6
Nm and
that obtained
analytical
modelinisthe
−39.7
Nm, which are
very close
to them
each other.
As
shown
in Figure
9b, thereby
arethe
pulsating
torques
electromagnetic
torque.
One of
is the
and that obtained by the analytical model is −39.7 Nm, which are very close to each other. As shown
As shown
in Figure
areispulsating
torques
the electromagnetic
torque. One
of them
is the
cogging
torque,
and9b,
thethere
other
the ripple
torqueinproduced
by the interaction
of the
harmonic
in Figure 9b, there are pulsating torques in the electromagnetic torque. One of them is the cogging
cogging torque,
and
other isinthe
ripple
torque
by the
interaction
of thegenerator.
harmonic
components
of the
fluxthe
densities
both
air gaps
and produced
the sinusoidal
current
in the DRRF
torque, and the other is the ripple torque produced by the interaction of the harmonic components of
components
the flux
densities
in both
air gapsmaximum
and the sinusoidal
current
in the DRRF
generator.
The
torque of
ripple
(the
difference
between
value and
minimum
value)
of the
the flux densities in both air gaps and the sinusoidal current in the DRRF generator. The torque ripple
The
torque
ripple
(the
difference
between
maximum
value
and
minimum
value)
of the
electromagnetic torque is 3.69%.
(the difference between maximum value and minimum value) of the electromagnetic torque is 3.69%.
electromagnetic
torque
is outer
3.69%.parts of the DRRF wind generator are designed with the same slot
When the inner
and
When the inner and outer parts of the DRRF wind generator are designed with the same slot
When
theand
inner
and
of the DRRF
wind
generator
are inner
designed
slot
opening
angle
pole
arcouter
ratio,parts
the cogging
torques
produced
by the
and with
outerthe
air same
gaps will
opening angle and pole arc ratio, the cogging torques produced by the inner and outer air gaps will be
opening
angle
and pole
ratio,
cogging
torques
produced
bytheir
the inner
and outer air
gaps will
be
in phase,
as shown
in arc
Figure
10,the
and
the total
cogging
torque is
sum. According
to Equation
in phase, as shown in Figure 10, and the total cogging torque is their sum. According to Equation (60),
be in the
phase,
as shown
Figure
10, and the
total
cogging
torque
is
their
sum.
According
to
Equation
(60),
spatial
periodin of
the cogging
torque
is
2.5°.
It
can
be
seen
that
the
cogging
torque
is
the spatial period of the cogging torque is 2.5◦ . It can be seen that the cogging torque is satisfyingly
(60),
the
spatial
period
of
the
cogging
torque
is
2.5°.
It
can
be
seen
that
the
cogging
torque
is
satisfyingly low, since the wind turbine may never start with high cogging torque.
low, since the wind turbine may never start with high cogging torque.
satisfyingly low, since the wind turbine may never start with high cogging torque.
Energies 2016, 9, 672
14 of 18
Energies 2016, 9, 672
Energies 2016, 9, 672
14 of 18
14 of 18
Energies 2016,
9, 672
5
-25
-15
-25
T /TNm
T / Nm
/ Nm
T /TNm
T / Nm
/ Nm
-38
-39
-39
FEM
FEM
Prop. Model
Prop.
FEM Model
Prop. Model
-15
-5
-15
-35
-25
-35
-45
-35
-45 0
0
-45
0
14 of 18
-38
-38
50
0-5
5
-50
0.01
0.01
0.01
0.02
0.02
t/s
t/s
0.02
(a)
(a)t / s
0.03
0.03
0.04
0.04
0.03
0.04
-39
-40
-40
-40
-41
-41
-41
-42
-42 0.03
0.03
-42
0.03
FEM
FEM
Prop. Model
Prop.
FEM Model
0.034Prop.0.036
Model 0.038
0.034 t / s0.036
0.038
t/s
(b) 0.036
0.034
0.038
(b)t / s
0.032
0.032
0.032
0.04
0.04
0.04
Figure9.9.Electromagnetic
Electromagnetic
and its pulsating
torques
of proposed
the proposed
generator
atload:
full
(a) torqueand
(b)DRRF
Figure
torques
of the
DRRF
generator
at fullat
Figure
9. Electromagnetictorque
torque anditsitspulsating
pulsating
torques
of the
proposed
DRRF
generator
full
load:
(a)
Electromagnetic
torque;
(b)
Pulsating
torques
in
the
electromagnetic
torque.
(a)
Electromagnetic
torque;
(b) Pulsating
in the
electromagnetic
torque.DRRF
load:
(a)9.Electromagnetic
torque;
(b)
Pulsating
torques
in
theof
electromagnetic
torque.generator at full
Figure
Electromagnetic
torque
and
itstorques
pulsating
torques
the proposed
load: (a) Electromagnetic torque; (b) Pulsating torques in0.2
the electromagnetic torque.
0.1
0.1
0.1
0
0
0
-0.1
-0.1 0
0
-0.1
0
FEM
FEM
Prop. Model
Prop.
FEM Model
Prop. Model
0.5
1
1.5
0.5Mechanical
1 angle
1.5/deg
/deg
0.5Mechanical
1 angle1.5
Mechanical angle /deg
2
2
2.5
2.5
2
2.5
Tcog/
Tcog/
NmNm
Tcog/
Nm
Tcog/
Tcog/
NmNm
Tcog/
Nm
0.2
FEM
FEM
Prop. Model
Prop.
FEM Model
0.2
0.1
0.1
Prop. Model
0.1
0
0
0
-0.1
-0.1
-0.1
-0.2
-0.2 0
0
-0.2
0
0.5
1
1.5
0.5 Mechanical
1
1.5 /deg
angle
Mechanical angle /deg
0.5
1(b)
1.5
Mechanical
(b) angle /deg
2
2
2.5
2.5
2
2.5
(a)
(a)
Figure 10. One cycle of
DRRF wind generator:
(a)cogging torque versus magnet rotor position of the(b)
Figure 10. One cycle of cogging torque versus magnet rotor position of the DRRF wind generator:
(a) Inner
torque;
(b) Outer
cogging
torque.
Figure
10.cogging
One cycle
of cogging
torque
versus
magnet rotor position of the DRRF wind generator:
(a)
Inner10.
cogging
torque;
(b) Outer
cogging
torque.
Figure
One cycle
of cogging
torque
versus
magnet rotor position of the DRRF wind generator:
(a) Inner cogging torque; (b) Outer cogging torque.
(a) Inner cogging torque; (b) Outer cogging torque.
Since the ferrite magnet is vulnerable to demagnetization, this risk is estimated by calculating
Since the ferrite magnet is vulnerable to demagnetization, this risk is estimated by calculating
the flux
density
of ferrite
magnets.
To analyze
the irreversible demagnetization
characteristics,
five
Since
the
magnet
is
to
this
by
Since
the ferrite
ferrite
magnet
is vulnerable
vulnerable
to demagnetization,
demagnetization,
this risk
risk is
is estimated
estimated
by calculating
calculating
the flux
density
of ferrite
magnets.
To analyze
the irreversible demagnetization
characteristics,
five
differentdensity
measuring
points
in the To
inner and outer
magnets are
chosen respectively
as shown in
the
of
magnets.
the
demagnetization
characteristics,
the flux
flux density
of ferrite
ferrite
magnets.
To analyze
analyze
the irreversible
irreversible
demagnetization
characteristics,
five
different
measuring
points
in the inner
and outer
magnets are
chosen respectively
as shownfive
in
Figure
11.
The
flux
densities
of
these
points
under
no
load
operation
at
−20
°C
in
radial
direction,
different
measuring
points
in
the
inner
and
outer
magnets
are
chosen
respectively
as
shown
in
Figure
11.
different
points inofthe
inner
andunder
outer no
magnets
are chosen
respectively
as direction,
shown in
Figure
11.measuring
The flux densities
these
points
load operation
at −20
°C in radial
which
are
the magnetization
direction
of the
ferrite
magnets,
are◦ Cshown
in Figure
12. which
It can be
seen
The
flux
densities
of densities
these points
underpoints
no
load
operation
at −operation
20
in radial
direction,
areseen
the
Figure
11.
Themagnetization
flux
of these
under
no load
atin
−20
°C in12.
radial
direction,
which
are
the
direction
of
the
ferrite
magnets,
are shown
Figure
It can
be
that
the
flux
density
of
point
C
1 is closer to the demagnetization limit value than other points. This
magnetization
direction
of the
magnets,
are shown
in Figure
12.
It
be other
seen
flux
which
thedensity
magnetization
direction
of to
thethe
ferrite
magnets,
are limit
shown
in can
Figure
12. Itthat
can the
be This
seen
that
theare
flux
of point
C1ferrite
is closer
demagnetization
value
than
points.
means that
pointC Cis
1 is easier to be demagnetized. So point C1 is selected to confirm the working
density
of
point
closer
to
the
demagnetization
limit
value
than
other
points.
This
means
that
1
that thethat
fluxpoint
density
point Cto1 is
to the demagnetization
valuetothan
otherthe
points.
This
means
C1 of
is easier
becloser
demagnetized.
So point C1 islimit
selected
confirm
working
points atisdifferent
temperatures,
andSothe
working
points are
shown the
in Figure
13. The FEM
results
point
easier
be
demagnetized.
C1 is selected
to C
confirm
working
at working
different
1that
meansCat
pointtoCtemperatures,
1 is easier to be
So point
1 is selected
to confirm
points
different
anddemagnetized.
thepoint
working
points
are
shown
in Figure
13. points
The the
FEM
results
demonstrate that
the
DRRF
wind
generator
is safe at
all temperatures
even at −20
°C.
temperatures,
andthe
the
working
points
Figure
13. The
FEM
demonstrate
the
points at different
temperatures,
and are
the shown
working
are
shown
in results
Figure
13.°C.
The FEMthat
results
demonstrate
that
DRRF
wind
generator
is safeinatpoints
all temperatures
even
at −20
◦
DRRF
wind
generator
is
safe
at
all
temperatures
even
at
−
20
C.
demonstrate that the DRRF wind generator is safe at all temperatures even at −20 °C.
D2
D2
E2
C2
E2
C2
B2
D
B2
E2
A22
AC
2
B2
A2
D1
D1
B1
D
B11
B1
E1
E1
E1
C1
C1
A1 1
AC
1
A1
Figure 11. Different measuring points for analyzing the irreversible demagnetization characteristics.
Figure 11. Different measuring points for analyzing the irreversible demagnetization characteristics.
Figure
11. Different
points for
for analyzing
analyzing the
the irreversible
irreversible demagnetization
demagnetization characteristics.
characteristics.
Figure 11.
Different measuring
measuring points
Energies 2016, 9, 672
Energies 2016, 9, 672
Energies 2016, 9, 672
15 of 18
15 of 18
15 of 18
0
0
0.4
-0.1
0.4
-0.1
Flux density
[T] [T]
Flux density
Flux density
[T] [T]
Flux density
0.5
0.5
0.3
0.3
0.2
0.2
0.1
0.1
0
0
0
0
Point A1
Point B1
Point A1
Point C1
Point B1
Point D1
Point C1
Point E1
Point D1
Demag limit
Point E1
15
30 limit
Demag
Rotor position [mech. deg.]
15
30
Rotor position [mech. deg.]
Point A2
Point B2
Point A2
Point C2
Point B2
Point D2
Point C2
Point E2
Point D2
Demag limit
Point E2
Demag limit
-0.2
-0.2
-0.3
-0.3
-0.4
-0.4
(a)
(a)
45
- 0.5
45
- 0.5
0
30
15
Rotor position [mech. deg.]
30
15
Rotor position [mech. deg.]
0
(b)
(b)
45
45
Figure 12. Flux densities (radial direction) of different locations of the magnets: (a) Points of the
Figure 12. Flux densities (radial direction) of different locations of the magnets: (a) Points of the inner
Figure
12. Flux(b)densities
direction)
inner magnet;
Points of(radial
the outer
magnet.of different locations of the magnets: (a) Points of the
magnet; (b) Points of the outer magnet.
inner magnet; (b) Points of the outer magnet.
BH curve
-20℃
Magnetic
Flux Flux
Density
(T) (T)
Magnetic
Density
BH curve
0.4
0.4
20℃
-20℃
60℃
20℃
100℃
60℃
100℃
Working Point
-20℃
Working Point
20℃
-20℃
60℃
20℃
100℃
60℃
100℃
0.2
0.2
0
-300000
0
-300000
-200000
-200000
-100000
Magnetic Field (A/m)
-100000
Magnetic Field (A/m)
0
100000
0
100000
Figure 13. The working points of measuring point C1.
Figure
Figure 13.
13. The working points of measuring point C
C11.
Ef (V)
Ef (V)
4. Optimization by Finite Element Method (FEM)
4. Optimization
Optimizationby
byFinite
FiniteElement
ElementMethod
Method(FEM)
(FEM)
A set of equations describing the generator performance are derived in Section 2 and their
set
equations
describing
performance
are derived
in Section
2 and
their
A set
ofofproven
equations
thethe
generator
performance
area derived
in DRRF
Section
2 andgenerator
their
validity
validity
are
by describing
FEM
in Section
3.generator
However,
designing
satisfying
wind
still
validity
are by
proven
FEM in
Section
3. However,
designing
a satisfying
DRRFgenerator
wind generator
still
are
proven
FEM by
in some
Section
3.
However,
designing
a satisfying
DRRF wind
still needs
needs
optimization
of
design
parameters.
needsDue
optimization
ofdesign
some
parameters.
optimization
of some
parameters.
to the
adoption
of design
back-to-back
toroidally wound winding, in order to take full advantage
to the
the
adoption
ofback-to-back
back-to-back
toroidally
wound
winding,
order
to
take
full
advantage
to
adoption
of
toroidally
winding,
inin
order
toslot.
take
full
advantage
of
of theDue
inner
space,
the area
of
the inner slot
shouldwound
equal
to
that of the
outer
According
to (46),
of
the
inner
space,
the
area
of
the
inner
slot
should
equal
to
that
of
the
outer
slot.
According
to
(46),
the
inner
space,
the
area
of
the
inner
slot
should
equal
to
that
of
the
outer
slot.
According
to
(46),
we
we define the split ratio as λ = Ris/Ros, and then:
we define
λ is=/R
Risos
/R, osand
, and
then:
define
the the
splitsplit
ratioratio
as λas
=R
then:
Ef = 4.44 2Kw fNph ( B1g1 λ  B1g2 )Ros L p
(61)
√
p
E
=
4.44
2
K
fN
(
B
λ

B
)
R
L
(61)
1g1
1g2 ) Ros
Ef = f4.44 2Kw fwNphph( Blg1
λ + Blg2
(61)
os L p
How λ affects Ef is shown in Figure 14, where it can be found that when λ is 0.43, Ef reaches the
How λ
affects
EEf isis
shown
Figure
it it
can
be
found
that
when
λ isλλ0.43,
Ef reaches
the
maximum
reason
isinin
given
as14,
follows:
on
the
one
hand,
the
smaller
the
How
λvalue.
affectsThe
shown
Figure
14,where
where
can
be
found
that
when
isis,0.43,
Elarger
f
f reaches
maximum
value.
The
reason
is
given
as
follows:
on
the
one
hand,
the
smaller
λ
is,
the
larger
the
whole
area of value.
the inner
outer
slots as
is,follows:
so thereonare
conductors
for producing
induced
the
maximum
The and
reason
is given
themore
one hand,
the smaller
λ is, the larger
the
whole
area
of
the
inner
and
outer
slots
is,
so
there
are
more
conductors
for
producing
induced
voltage.
On
the
other
hand,
the
smaller
λ
is,
the
smaller
R
is
is,
and
consequently
there
is
less
amount
whole area of the inner and outer slots is, so there are more conductors for producing induced voltage.
voltage.
Onfor
the
other
theλsmaller
λ is, theR smaller
Ris is, and consequently there is less amount
of magnet
producing
induced
On
the other
hand,
thehand,
smaller
is,voltage.
the smaller
is is, and consequently there is less amount of magnet
of
magnet
for
producing
induced
voltage.
for producing induced voltage.
140
Torque ripple can generate acoustic noise, vibrations and mechanical stress in the wind turbine,
140
thus, its minimization is highly required for wind power generations. Furthermore, the cogging
120 structures, as an important component of the pulsating torque, must
torque caused by the tooth-slot
120
be minimized to better start wind power generator. There are several solutions to reduce the cogging
torque, such as shifting the
slot opening, shaping the magnets, skewing the stator or the magnets,
100
100
varying slot opening angular width, and varying the pole arc ratio etc. By way of reducing manufacture
cost, pole arc ratio design and
80 shifting the slot opening are adopted to reduce the cogging torque.
0.4
0.5
0.6
0.7
80 0.3
0.3
0.4 The Split
0.5Ratio, λ0.6
0.7
The Split Ratio, λ
Figure 14. The back EMF is a function of the split ratio λ.
Figure 14. The back EMF is a function of the split ratio λ.
Ef = 4.44 2Kw fNph ( B1g1 λ  B1g2 )Ros L p
(61)
How λ affects Ef is shown in Figure 14, where it can be found that when λ is 0.43, Ef reaches the
maximum value. The reason is given as follows: on the one hand, the smaller λ is, the larger the
whole area of the inner and outer slots is, so there are more conductors for producing induced
Energies
2016, 9, On
672 the other hand, the smaller λ is, the smaller Ris is, and consequently there is less amount
16 of 18
voltage.
of magnet for producing induced voltage.
140
120
Torque
Energies
2016, 9, ripple
672
16 of 18
Ef (V)
Energies 2016, 9, 672
can generate
100 acoustic noise, vibrations and mechanical stress in the wind turbine,
16 of 18
thus, its minimization is highly required for wind power generations. Furthermore, the cogging
torque
caused
by the
as anvibrations
importantand
component
of the
pulsating
torque,
must
Torque
ripple
cantooth-slot
generate
acoustic noise,
mechanical
stress
in the wind
turbine,
80 structures,
0.3
0.4
0.5
0.6
0.7
be
minimized
to
better
start
wind
power
generator.
There
are
several
solutions
to
reduce
the
thus, its minimization is highly required for wind power generations. Furthermore, the cogging
The Split Ratio, λ
cogging
torque,
such
as
shifting
the
slot
opening,
shaping
the
magnets,
skewing
the
stator
or
the
torque caused by the tooth-slot structures, as an important component of the pulsating torque, must
magnets,
varying
slot opening
angular
width,
varying
theseveral
pole
arc
ratio etc.
waythe
of
be
minimized
to better
start wind
power
generator.
There are
solutions
to By
reduce
Figure
Theback
backEMF
EMF
is aaand
function
ratio
λ.
Figure
14.14.
The
is
functionofofthe
thesplit
split
ratio
λ.
reducingtorque,
manufacture
polethe
arcslot
ratio
designshaping
and shifting
the slotskewing
openingthe
arestator
adopted
to
cogging
such ascost,
shifting
opening,
the magnets,
or the
reduce thevarying
coggingslot
torque.
magnets,
opening angular width, and varying the pole arc ratio etc. By way of
Since
the energy
stored
in the
air
gap
is more
than
that that
in
the
inner
air gap,
the
total
cogging
Since
the energy
stored
in outer
thearc
outer
airdesign
gap
isand
more
than
in
theopening
inner
airare
gap,
the total
reducing
manufacture
cost,
pole
ratio
shifting
the
slot
adopted
to
cogging
torque
of the
DRRF
is by
dominated
byportion.
the outerTherefore,
portion. Therefore,
outer
torque
of thethe
DRRF
generator
is generator
dominated
the outer
the outerthe
pole
arc pole
ratio αp2
reduce
cogging
torque.
arc optimized
ratio
is energy
firstly
optimized
then αthe
αp1
is secondly
optimized.
Ingap,
fact,
the
pole
Sinceαp2
the
stored
in and
the ratio
outer
air
gap
isratio
more
than
that inInthe
inner
total
is firstly
and then
the inner
is
secondly
optimized.
fact,
theair
pole
arcthe
ratio
affects
p1 inner
arc ratio
affects of
not
only
the
but
also the
THD
of
EMF.
For the
DRRF
generator,
how
cogging
torque
the
DRRF
generator
is dominated
by
the
outer
portion.
Therefore,
the
outer pole
not only
the cogging
torque
butcogging
also thetorque
THD
of EMF.
For
the
DRRF
generator,
how
different
αp1 and
different
αp2p1 isand
αtorque
p2 optimized
affect(peak-to-peak
the and
cogging
(peak-to-peak
value,
THD
ofin
EMF
is 15.
arc
ratio
firstly
then torque
the
inner
ratio αp1 and
is secondly
optimized.
In fact,
the
pole
αp2 affect
theαcogging
value,
pk2pk)
THDpk2pk)
of
EMFand
is shown
Figure
shown
inaffects
Figurenot
15.only
To obtain
a sinusoidal-induced
voltage
with
low For
cogging
torque,
compromise
arc
ratio
the
cogging
torque
but
also
the
THD
of
EMF.
the
DRRF
generator,
how
To obtain a sinusoidal-induced voltage with low cogging torque, compromise selection of parameters
selection of
is considered.
With
this consideration,
the optimal
value
αp2 isof0.67,
different
αp1parameters
and αp2 affect
the cogging
torque
(peak-to-peak value,
pk2pk)
andofTHD
EMFand
is
is considered. With this consideration, the optimal value of αp2 is 0.67, and αp1 is 0.69.
αp1 is 0.69.
shown
in Figure 15. To obtain a sinusoidal-induced voltage with low cogging torque, compromise
selection of parameters is considered. With this consideration, the optimal value of αp2 is 0.67, and
2
0.6
10
3
Tcog
Tcog
αp1 is 0.69.
0.5
1
5
7.5
2.5
5
0
0.50.5
0.55
0.6
0.65
0.7
The Outer Pole Arc Ratio, αp2
(a)
0
0.5
0.55
0.6
0.65
0.7
The Outer Pole Arc Ratio, αp2
0
0.752.5
0
0.75
THD
0.5
0.6
Tcog
THD
0.4
0.5
0.3
0.4
2.5
3
2
2.5
THD of EMF
THD of EMF
1
1.5
Tcog (pk2pk)/
Nm
Tcog (pk2pk)/
Nm
Tcog
THD
7.5
10
THD of EMF
THD of EMF
Tcog (pk2pk)/
Nm
Tcog (pk2pk)/
Nm
THD
1.5
2
21.5
0.2
0.30.5
0.2
0.5
0.55
0.6
0.65
0.7
The Inner Pole Arc Ratio, αp1
(b)
0.55
0.6
0.65
0.7
The Inner Pole Arc Ratio, αp1
1
0.75
1.5
1
0.75
Figure 15. Cogging torque and THD as a function of the pole arc ratio: (a) The influence of outer
Figure 15. Cogging torque
THD as a function of the pole arc ratio: (a) (b)
The influence of outer pole
(a)and
pole arc ratio αp2; (b) The
influence of inner pole arc ratio αp1.
arc ratio αp2 ; (b) The influence of inner pole arc ratio αp1 .
Figure 15. Cogging torque and THD as a function of the pole arc ratio: (a) The influence of outer
The arc
cogging
waveforms related to different pole arc ratios are demonstrated in Figure
pole
ratio αtorque
p2; (b) The influence of inner pole arc ratio αp1.
16. Itcogging
is showntorque
that thewaveforms
maximum value
of cogging
torque
is significantly
reduced
by optimizing
the 16.
The
related
to different
pole
arc ratios are
demonstrated
in Figure
outer
pole
arc
ratio
α
p2
.
The
inner
pole
arc
ratio
α
p1
is
also
very
important
in
this
optimization.
In
this
Thethat
cogging
torque waveforms
related
to different
arc ratios arereduced
demonstrated
in Figure the
It is shown
the maximum
value of
cogging
torque pole
is significantly
by optimizing
example,
whenthat
αp2the
= 0.67
and αp1value
= 0.65,
the
maximum
value
of cogging
torquebyreduces
to 40%
16.
It
is
shown
maximum
of
cogging
torque
is
significantly
reduced
optimizing
the
outer pole arc ratio αp2 . The inner pole arc ratio αp1 is also very important in this optimization.
compared
to
that
with
α
p2
=
0.67
and
α
p1
=
0.69.
outer pole arc ratio αp2. The inner pole arc ratio αp1 is also very important in this optimization. In this
In this example, when αp2 = 0.67 and αp1 = 0.65, the maximum value of cogging torque reduces to 40%
example, when αp2 = 0.67 and αp1 = 0.65, the maximum value of cogging torque reduces to 40%
compared 1.2
to that with αp2 = 0.67 and αp1 = 0.69.
0.2
0.6
1.2
0.1
0.2
Tcog / Nm
Tcog / Nm
Tcog / Nm
Tcog / Nm
compared to that with αp2 = 0.67 and αp1 = 0.69.
0.60
0.10
-0.6
0
-1.2
-0.6
0
-1.2
0
-0.1
0
αp2=0.6
αp2=0.67
11.25
22.5
33.75
αp2=0.6
Mechanical angle / deg
αp2=0.67
45
11.25
22.5
33.75
Mechanical angle / deg
45
(a)
-0.2
-0.1
0
-0.2
0
αp1=0.65
αp1=0.69
11.25
22.5
33.75
αp1=0.65angle / deg
Mechanical
αp1=0.69
45
(b)
11.25
22.5
33.75
45
Mechanical
angle
/ deg of outer
Figure 16. Comparison of cogging torques for the different pole arc ratio:
(a) The
influence
(a) influence of inner pole arc ratio αp1.
pole arc ratio αp2; (b) The
(b)
Figure 16. Comparison of cogging torques for the different pole arc ratio: (a) The influence of outer
Figure
16.the
Comparison
of cogging
fortorque
the different
poleas
arcthe
ratio:
The influence
outer
For
slotted
structure,
thetorques
cogging
decreases
slot(a)
opening
angularofwidth
pole
arc ratio
αp2; (b)
The influence
of inner pole
arc ratio
αp1.
pole
arc ratio
; (b)
Theslot
influence
of angular,
inner pole
arcarc
ratio
αp1 .of the outer slot opening is larger than
decreases.
Forαp2
the
same
opening
the
length
that of
inner
one. structure,
In order tothe
reduce
the cogging
torque, theasouter
can
Forthe
the
slotted
cogging
torque decreases
the slot
slot opening
openingangular
angularαbo2
width
decreases. For the same slot opening angular, the arc length of the outer slot opening is larger than
that of the inner one. In order to reduce the cogging torque, the outer slot opening angular αbo2 can
Energies 2016, 9, 672
17 of 18
For the slotted structure, the cogging torque decreases as the slot opening angular width decreases.
For the same slot opening angular, the arc length of the outer slot opening is larger than that of the
Energies 2016, 9, 672
17 of 18
inner one. In order to reduce the cogging torque, the outer slot opening angular αbo2 can be designed
tobebedesigned
smaller than
αbo1
, as shown
Figure
17a. Figure
17b compares
the cogging
torque the
at different
to be
smaller
than in
αbo1
, as shown
in Figure
17a. Figure
17b compares
cogging
αtorque
.
It
can
be
seen
that
there
is
a
big
decline
in
the
maximum
cogging
torque
by
changing
αbo2
bo2
at different αbo2. It can be seen that there is a big decline in the maximum cogging torque
by
◦
◦
from
2.3
to
1.9
.
This
implies
that
changing
the
opening
angular
is
a
valid
approach
to
reduce
the
changing αbo2 from 2.3° to 1.9°. This implies that changing the opening angular is a valid approach
cogging
torque
of the DRRF
generator.
But it should
be known
thisbe
method
cannot
completely
to reduce
the cogging
torque
of the DRRF
generator.
But it that
should
known
that this
method
eliminate
the cogging
torque. the cogging torque.
cannot completely
eliminate
αb o2
0.8
Tcog / Nm
0.4
0
-0.4
-0.8
αb o1
(a)
αbo2=2.3
αbo2=1.9
0
11.25
22.5
33.75
Mechanical angle / deg
45
(b)
Figure 17. (a) Different slot opening angular αbo2 < αbo1; (b) Comparison of cogging torques for
Figure 17. (a) Different slot opening angular αbo2 < αbo1 ; (b) Comparison of cogging torques for
different slot opening angular αbo2.
different slot opening angular αbo2 .
5. Conclusions
5. Conclusions
In this paper, a comprehensive analytical model for a DRRF wind generator is developed based
In this paper, a comprehensive analytical model for a DRRF wind generator is developed based
on EMC. Accurate calculations were performed, and results with rather high accuracy are obtained.
on EMC. Accurate calculations were performed, and results with rather high accuracy are obtained.
The analytical model can precisely estimate the flux density distribution in the inner and outer air gaps
The analytical model can precisely estimate the flux density distribution in the inner and outer air gaps
as well as rotor and stator cores, back EMF waveform, both average and ripples of electro-magnetic
as well as rotor and stator cores, back EMF waveform, both average and ripples of electro-magnetic
torque, cogging torque, and other parameters. The results provided by the proposed model match
torque, cogging torque, and other parameters. The results provided by the proposed model match well
well those from FEM analysis. That is, the analytical model developed in this paper can be employed
those from FEM analysis. That is, the analytical model developed in this paper can be employed in
in the preliminary design and analysis of the generator throughout the whole design process.
the preliminary design and analysis of the generator throughout the whole design process. Moreover,
Moreover, the working points of the ferrite magnets exceed the knee of the BH curve at different
the working points of the ferrite magnets exceed the knee of the BH curve at different temperatures,
temperatures, which means the irreversible demagnetization does not occur at full load. Finally, the
which means the irreversible demagnetization does not occur at full load. Finally, the optimization is
optimization is carried out by FEM to make the best use of the inner space of the generator, reduce
carried out by FEM to make the best use of the inner space of the generator, reduce the cogging torque,
the cogging torque, and decrease the THD of EMF. Accordingly, the optimal split ratio, pole arc
and decrease the THD of EMF. Accordingly, the optimal split ratio, pole arc ratio of the inner and outer
ratio of the inner and outer rotor, as well as the outer slot opening has been obtained.
rotor, as well as the outer slot opening has been obtained.
Acknowledgments: This work was supported in part by the National Natural Science Foundation of China
Acknowledgments: This work was supported in part by the National Natural Science Foundation of China under
under
grant 51407085,
the Postdoctoral
Science Foundation
China
under
Award
No. 2015M571685,
the grant
grant
51407085,
the Postdoctoral
Science Foundation
of China of
under
Award
No.
2015M571685,
the grant from
the
from
the
Priority
Academic
Program
Development
of
Jiangsu
Higher
Education
Institution
and
Priority Academic Program Development of Jiangsu Higher Education Institution and Jiangsu University Jiangsu
Senior
Talents
Special
Project
under
awardProject
13JDG111.
University
Senior
Talents
Special
under award 13JDG111.
Author
contributed
to this
work
by collaboration.
Peifeng
Xu, Kai
Sun and
AuthorContributions:
Contributions:All
Allauthors
authors
contributed
to this
work
by collaboration.
Peifeng
Xu, Shi,
Kai Yuxin
Shi, Yuxin
Sun
Huangqiu Zhu are the main authors of this manuscript.
and Huangqiu Zhu are the main authors of this manuscript.
Conflicts of Interest: The authors declare no conflict of interest.
Conflicts of Interest: The authors declare no conflict of interest.
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