energies
Article
Analytical Calculation of D- and Q-axis Inductance
for Interior Permanent Magnet Motors Based on
Winding Function Theory
Peixin Liang 1,2 , Yulong Pei 2 , Feng Chai 1,2, * and Kui Zhao 2
1
2
*
State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150001, China;
liangpx_hit@163.com
Department of Electrical Engineering, Harbin Institute of Technology, Harbin 150001, China;
Peiyulong1@163.com (Y.P.); enjoyingzk@163.com (K.Z.)
Correspondence: chaifeng@163.com; Tel.: +86-451-8640-3480
Academic Editor: K. T. Chau
Received: 30 April 2016; Accepted: 21 July 2016; Published: 25 July 2016
Abstract: Interior permanent magnet (IPM) motors are widely used in electric vehicles (EVs),
benefiting from the excellent advantages of a more rational use of energy. For further improvement
of energy utilization, this paper presents an analytical method of d- and q-axis inductance calculation
for IPM motors with V-shaped rotor in no-load condition. A lumped parameter magnetic circuit
model (LPMCM) is adopted to investigate the saturation and nonlinearity of the bridge. Taking into
account the influence of magnetic field distribution on inductance, the winding function theory (WFT)
is employed to accurately calculate the armature reaction airgap magnetic field and d- and q-axis
inductances. The validity of the analytical technique is verified by the finite element method (FEM).
Keywords: interior permanent magnet motor; d- and q-axis inductances; lumped parameter magnetic
circuit model; winding function theory; armature reaction magnetic field; saturation; nonlinearity
1. Introduction
In recent decades, permanent magnet (PM) motors cover a wide range of industrial applications,
especially for electric vehicles (EVs), attributing to the compact structure, high efficiency, high
power/torque density and excellent dynamic performances [1–6]. As a typical topology of PM motors,
the interior permanent magnet (IPM) motors with a V-shape provide a more superior performance of
the flux-weakening capability and wider constant-power speed range [7–10], achieving considerable
attention. Along with these excellent advantages, IPM motors offer a great alternative opportunity to
contribute for a more rational use of energy, which makes EVs have the potential to provide a perfect
solution to energy security and environmental impacts in the future [11,12].
The d- and q-axis inductances are key parameters for the performance design including
flux-weakening capability, power factor and control strategy [13–15]. For the first stage of the IPM
motor design, the calculation of d- and q-axis inductances in no-load condition is crucial.
For IPM motors, on one hand, the magnetic bridges protect the permanent magnets from flying
away from the rotor. On the other hand, they provide a path for the permanent-magnet leakage
flux and cause saturation. The influence of magnetic bridge saturation is the biggest obstacle for the
calculation of d- and q-axis inductances. In addition, the accurate paths of d- and q-axis fluxes are
another crucial point.
Various researches on inductance with finite element method (FEM) have been successfully
investigated in the past [16], e.g., a 3D FEM model is used to calculate the inductances considering the
end effect and magnetic flux leakage [17]. In [18], taking into account saturation and nonlinearity, the
Energies 2016, 9, 580; doi:10.3390/en9080580
www.mdpi.com/journal/energies
Energies 2016, 9, 580
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cross-coupling inductance characteristics are investigated. FEM can effectively analyze the inductances
accounting
for the saturation, nonlinearity, complex geometry and cross-coupling influence. However,
Energies 2016, 9, 580 2 of 11 FEM is too theory-lacking and time-consuming to provide a physical insight to the motor design,
the cross‐coupling inductance characteristics are atinvestigated. FEM effectively analyze the especially
for the influence
of parameter
variation
the first stages
of can the design
process.
inductances accounting for the saturation, nonlinearity, complex geometry and cross‐coupling Comparatively speaking, analytical methods are often preferred. Lumped parameter magnetic
influence. However, FEM is too theory‐lacking and time‐consuming to provide a physical insight to the circuit model (LPMCM) is a common method. In [19,20], the d- and q-axis inductances are calculated
motor design, especially for the influence of parameter variation at the first stages of the design process. with LPMCMs, which are established based on the d- and q-axis flux paths matching the practical
Comparatively speaking, analytical methods are often preferred. Lumped parameter magnetic operating
point of the control issue. The influence of saturation and nonlinearity is analyzed. In [21],
circuit model (LPMCM) is a common method. In [19,20], the d‐ and q‐axis inductances are calculated the motor
is divided into several mechanical phase regions according to the winding distribution. On
with LPMCMs, which are established based on the d‐ and q‐axis flux paths matching the practical the basis
of
the mechanical phase region and equivalent magnetic circuit, the d- and q-axis inductances
operating point of the control issue. The influence of saturation and nonlinearity is analyzed. In [21], are predicted.
Although LPMCM has advantages in saturation and nonlinearity, the drawback of
the motor is divided into several mechanical phase regions according to the winding distribution. On the method
that
the
calculationphase accuracy
is greatly
affected magnetic by the way
flux paths
are
divided
the basis isof the mechanical region and equivalent circuit, the d‐ and q‐axis and
inductances are are
predicted. Although LPMCM advantages in saturation and nonlinearity, the of
analytical
models
built. On
the other
hand,has though
the LPMCM
expresses
the relationship
drawback of the method is that the calculation accuracy is greatly affected by the way flux paths are permeance, it is complicated to classify the permeance according to their contributions. Especially for
divided and distributed
analytical models are built. On the other ishand, though the LPMCM the to
the motor
with
winding,
the
amount
of teeth
too much,
which
makes itexpresses more difficult
relationship of permeance, it is complicated to classify the permeance according to their contributions. classify
the flux paths and calculate the inductance based on permeance. Furthermore, the accurate
Especially for the motor with distributed winding, the amount of teeth is too much, which makes it flux used for the inductance calculation is the summation of the main and leakage magnetic field.
more difficult to classify the flux paths and calculate the inductance based on permeance. However, the LPMCM calculates the main permeance, neglecting the influence of leakage permeance,
Furthermore, the accurate flux used for the inductance calculation is the summation of the main and which
is what causes calculation error.
leakage magnetic field. However, the LPMCM calculates the main permeance, neglecting the This
paper presents an analytical method for d- and q-axis inductance calculation in no-load
influence of leakage permeance, which is what causes calculation error. conditionThis based
on LPMCM
andanalytical windingmethod functionfor theory
(WFT).
is adapted
to the
calculation
paper presents an d‐ and q‐axis LPMCM
inductance calculation in no‐load of magnetic
and
WFT
is employed
divide(WFT). accurate
paths of
and q-axis
fluxes
condition bridge
based saturation,
on LPMCM and winding function totheory LPMCM is dadapted to the and calculation of magnetic bridge saturation, and WFT is employed to divide accurate paths of d‐ and q‐
consider the influence of magnetic field distribution. The d- and q-axis flux linkages produced by
axis fluxes and consider the influence of magnetic field distribution. The d‐ and q‐axis flux linkages a very
small current are calculated, based on which d- and q-axis inductances can be obtained. The
FEMproduced by a very small current are calculated, based on which d‐ and q‐axis inductances can be results verify the validity of the proposed method.
obtained. The FEM results verify the validity of the proposed method. 2. Analytical Method
2. Analytical Method A 48-slot/8-pole IPM motor with a V-shaped rotor is instanced to verify the proposed method.
Figure 1 A 48‐slot/8‐pole IPM motor with a V‐shaped rotor is instanced to verify the proposed method. shows the model with magnetic bridges, where α is the magnet pole-arc to pole-pitch ratio, β
Figure 1 shows the model with magnetic bridges, where α is the magnet pole‐arc to pole‐pitch ratio, is the barrier arc to pole-pitch ratio, Rs is the stator bore radius, Rr is the rotor outer radius, tb is the
β is the barrier arc to pole‐pitch ratio, Rs is the stator bore radius, Rr is the rotor outer radius, tb is the bridge thickness, wbar is the barrier width, lbar1 and lbar2 are the barrier length, wm is the PM width, lm
bridge thickness, wbar is the barrier width, lbar1 and lbar2 are the barrier length, wm is the PM width, lm is is the
PM length in magnetization direction, wr is the rib width,
lr is the rib length. For IPM motors,
the PM length in magnetization direction, wr is the rib width, l
r is the rib length. For IPM motors, the the saturation
mainly focuses on the bridge. Here, we assume that the permeability of stator and rotor
saturation mainly focuses on the bridge. Here, we assume that the permeability of stator and rotor (except
the
bridge)
core is infinite.
(except the bridge) core is infinite. Figure 1. Model of a 48‐slot/8‐pole Interior permanent magnet (IPM) motor. Figure 1. Model of a 48-slot/8-pole Interior permanent magnet (IPM) motor.
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3 of 11 Energies 2016, 9, 580 3 of 11 3 of 11 2.1. 2.1.
Saturation
Calculation
Saturation
Calculation
Energies 2016, 9, 580 2.1. Saturation Calculation The leakage fluxes pass the bridges causing different saturation characteristics depending on the
The leakage fluxes pass the bridges causing different saturation characteristics depending on the
2.1. Saturation Calculation The leakage fluxes pass the bridges causing different saturation characteristics depending on the rotor
shape.
For For
the sake
of simplicity,
the rotor
is divided
into into
two two
pieces:
saturation
regions
withwith
a a
2.1. Saturation Calculation rotor
the sake
of simplicity,
the rotor
is divided
pieces:
saturation
regions
eristics
racteristics
ion characteristics
depending
depending
on
depending
the
on
theshape.
on
the
rotor shape. For the sake of simplicity, the rotor is divided into two pieces: saturation regions with a constant
permeability
andand
non-saturation
region
with
infinite
permeability
shown
in Figure
2. The
The leakage fluxes pass the bridges causing different saturation characteristics depending on the permeability
non-saturation
region
with
infinite
permeability
shown
in Figure
2. The
wo
es:
aturation
saturation
pieces:regions
saturation
regions
withconstant
regions
with
a The leakage fluxes pass the bridges causing different saturation characteristics depending on the a with
a
constant permeability and non‐saturation region the
with infinite permeability shown in Figure 2. The saturation
regions
of
bridges
are
widened,
where
equivalent
bridge
width
can
be
expressed
as as
rotor shape. For the sake of simplicity, the rotor is divided into two pieces: saturation regions with a rotor shape. For the sake of simplicity, the rotor is divided into two pieces: saturation regions with a saturation
of bridges are widened, where the equivalent bridge width can be expressed
ermeability
lity
shown
shown
in Figure
in
shown
Figure
2.inThe
2.
Figure
The 2. regions
The
saturation regions of bridges are widened, where the equivalent bridge width can be expressed as constant permeability and non‐saturation region with infinite permeability shown in Figure 2. The constant permeability and non‐saturation region with infinite permeability shown in Figure 2. The idth
ridge
can can
bewidth
expressed
be expressed
can beasexpressed
as
as
=w
wbw “w
wwbbar
`bar2t+b 2tb
(1) (1)
saturation regions of bridges are widened, where the equivalent bridge width can be expressed as saturation regions of bridges are widened, where the equivalent bridge width can be expressed as (1) b
bar  2tb (1) (1)
(1)
w  w  2t b
wbb  wbar
bar  2tb (1) (1) Figure 2. Simplified model of rotor. Figure
2. Simplified
model
of rotor.
Figure
2. Simplified
model
of rotor.
Figure 2. Simplified model of rotor. Figure 2. Simplified model of rotor. or.
As shown in Figure 3, the fluxes, produced by magnets, pass the magnet, magnetic bridges, and AsConsidering shown
in Figure
3, fluxes,
the
fluxes,
produced
by magnets,
pass
the
magnet,
magnetic
bridges,
As shown
in Figure
3, the
produced
magnets,
the magnet,
magnetic
bridges,
air‐gap. the flux paths around the by
magnets, an pass
LPMCM is employed to analyze the andand
As shown in Figure 3, the fluxes, produced by magnets, pass the magnet, magnetic bridges, and As shown in Figure 3, the fluxes, produced by magnets, pass the magnet, magnetic bridges, and air-gap.
Considering
paths
around
magnets,
an LPMCM
is employed
to analyze
agnet,
st,the
magnetic
magnet,
magnetic
bridges,
magnetic
bridges,
and
bridges,
and
and
air-gap.
Considering
the the
fluxflux
paths
around
the the
magnets,
an LPMCM
is employed
to analyze
the the
saturation and nonlinearity of bridges. Figure 4 depicts the LPMCM of one‐half of two magnets with air‐gap. Considering the paths around the magnets, an LPMCM is employed to analyze the air‐gap. Considering the flux flux paths around the magnets, an LPMCM is employed to analyze the saturation
and
nonlinearity
of
bridges.
Figure
4
depicts
the
LPMCM
of
one-half
of
two
magnets
with
PMCM
mployed
is employed
is to
employed
analyze
to analyze
the
to
analyze
the
the
saturation
and
nonlinearity
of
bridges.
Figure
4
depicts
the
LPMCM
of
one-half
of
two
magnets
with
opposite poles and the associated bridges behind the magnet halves, where R
m is the reluctance of saturation and nonlinearity of bridges. Figure 4 depicts the LPMCM of one‐half of two magnets with saturation and nonlinearity of bridges. Figure 4 depicts the LPMCM of one‐half of two magnets with opposite
poles
and
the
associated
bridges
behind
the
magnet
halves,
where
R
m
is
the
reluctance
of
half
CM
ne-half
of
of two
one-half
of two
magnets
magnets
of opposite
two
with
magnets
with
with
poles
and
the
associated
bridges
behind
the
magnet
halves,
where
R
is
the
reluctance
of
magnet, R
g is the reluctance of one‐half of the per pole air gap, Rb is the reluctance of bridge, Φ
r, Φ0, m
opposite poles and the associated bridges behind the magnet halves, where R
opposite poles and the associated bridges behind the magnet halves, where Rmm is the reluctance of is the reluctance of magnet,
R
gthe
isofthe
reluctance
one-half
perpole
poleair
airgap,
gap,
the reluctance
reluctance of
of bridge,
Φr, Φ0,
where
ealves,
Rm isRwhere
Rmreluctance
m
the
is the
reluctance
ismagnet,
the
of
of
Rg bis
reluctance
of of
one-half
of of
thethe
per
RR
Φ
greluctance
and Φ
are the remanent magnet flux, magnet leakage flux, air‐gap flux and bridge leakage flux, b bisisthe
magnet, R
g is the reluctance of one‐half of the per pole air gap, R
b is the reluctance of bridge, Φ
r, Φ0, r
magnet, R
g is the reluctance of one‐half of the per pole air gap, Rb is the reluctance of bridge, Φr, Φ0, g
and
Φ
b
are
the
remanent
magnet
flux,
magnet
leakage
flux,
air-gap
flux
and
bridge
leakage
flux,
Rreluctance
ctance
b is theof
reluctance
bridge,
of bridge,
Φof
r0,, respectively. Φ
Φ
bridge,
0
r
,
Φ
0
,
r
,
Φ
0
,
are
the
remanent
magnet
flux,
magnet
leakage
flux,
air-gap
flux
and
bridge
leakage
Φ
b
Φggg and Φ
and Φbb are the remanent magnet flux, magnet leakage flux, air‐gap flux and bridge leakage flux, are the remanent magnet flux, magnet leakage flux, air‐gap flux and bridge leakage flux, respectively.
and
lux
ir-gap
and
bridge
flux
bridge
leakage
andleakage
bridge
flux,
leakage
flux,
flux,
flux,respectively. respectively.
respectively. Figure 3. Flux line distribution in no‐load condition. Figure 3. Flux line distribution in no‐load condition. Figure 3. Flux line distribution in no‐load condition. Figure
3. Flux line distribution in no-load condition.
Figure 3. Flux line distribution in no-load condition.
Φg
Rg
Rg
Φ
g
Φg
R
R
Rgg
Rgg
Φb Φg
R
Rb Rg
Rb
g
Φ
Φbb
R
R
Rbb Φ
Φ0
Rbb
0
Φb
Rb
R
b
Φ
Φ
Φr
RmΦ00
Φr
Φ00 Rm
Φ
0
Φ
R
R
Φ
Φ
Φrr
Rmm
R0m
Φr
n.
condition.
r
.
m
Φr
RΦ
mr
Rm
r
Rm
Φr
Figure 4. Magnetic circuit model. Figure 4. Magnetic circuit model. Figure 4. Magnetic circuit model. Figure
4. Magnetic circuit model.
Figure 4. Magnetic circuit model.
Φr
Energies 2016, 9, 580
4 of 11
Energies 2016, 9, 580 4 of 11 In the LPMCM, the permeability of bridge is a nonlinear uncertain parameter, which is the basic
of the solution for the all fluxes. The iterative method is effective for this problem. Figure 5 shows the
In the LPMCM, the permeability of bridge is a nonlinear uncertain parameter, which is the basic iterative process:
of the solution for the all fluxes. The iterative method is effective for this problem. Figure 5 shows the iterative process: 1.
Firstly, the initial flux density of bridge is supposed as B .
b
2.
1.
2.
3.
3.
4.
4.
Secondly, calculate all the reluctances and fluxes according
Firstly, the initial flux density of bridge is supposed as B
b. to the material B–H profile and the
law of magnetic flux continuity.
Secondly, calculate all the reluctances and fluxes according to the material B–H profile and the law of magnetic flux continuity. Thirdly, compare the obtained flux density of bridge Bb1 with Bb .
Thirdly, compare the obtained flux density of bridge B
with B
b. Finally, adjust Bb and repeat this process until the errorb1is
satisfied.
Finally, adjust Bb and repeat this process until the error is satisfied. Initial flux density Bb
Based on material B-H profile, calculate all reluctances
Calculate all fluxes
Modify Bb
Calculate flux density Bb1
N
Check error
Y
Obtain the permeability
Figure 5. Flow chart of the iterative process. Figure 5. Flow chart of the iterative process.
All the parameters can be expressed as: All the parameters can be expressed as:
lm
Rm 
lm
Rm “ 0  r wm L
µ0 µr w m L
RRg “
g
 RRr
RRss ´
r
 RpRs s`RRrr q 2π
2
µ
0 0
2
4p LL
(2) (2)
2
4p
wb
Rb “
w
Rb  µ pBb qb ¨ tb L   Bb   tb L
φr “ Br w
mL
φb “
r RBmr w
RmgL φr
Rm Rb ` Rb R g ` R g Rm
Rm Rg
b 
r Rm R
Rb Rφgb Rg Rm
Bbb1 “
tb L
(3)
(3) (4)
(4) (5)
(5) (6)
(6) (7)

where L is effective length of the rotor and stator,
Bb1  Brb is the remanence of magnet, p is the number
(7) tb L µ0 and µ (Bb ) are the permeability of air and
of pole pairs, µr is the relative permeability of magnet,
bridge, respectively.
where L is effective length of the rotor and stator, Br is the remanence of magnet, p is the number of pole Inductance
pairs, μr is the relative permeability of magnet, μ0 and μ (Bb) are the permeability of air and 2.2.
Calculation
bridge, respectively. When the current is very small, the influence of armature-reaction magnetic field on saturation
can
be
neglected, and the d- and q-axis inductances in no-load condition are obtained by:
2.2. Inductance Calculation #
φ
When the current is very small, the influence of armature‐reaction magnetic field on saturation L d “ I d ` L0
d
(8)
can be neglected, and the d‐ and q‐axis inductances in no‐load condition are obtained by: φ
Lq “ Iqq ` L0
d

 Ld  I  L0
d



L  q  L
0
 q I q
(8) Energies 2016, 9, 580
5 of 11
where φd and φq are the main flux linkages in d‐ and q‐axis, Id and Iq are the currents in d‐ and q‐axis, respectively. L0 is the leakage inductance. where
φd and φq are the main flux linkages in d- and q-axis, Id and Iq are the currents in d- and q-axis,
Taking into account the influence of magnetic field distribution, the WFT is used to accurately respectively.
L0 is the leakage inductance.
calculate the d‐ and q‐axis inductances. For the clear expression of magnetic motive force (MMF) and intopotential, account the
of magnetic
field
WFT isare used
to accurately
rotor Taking
magnetic the influence
initial condition is that all distribution,
the angular the
positions expressed with calculate
the
dand
q-axis
inductances.
For
the
clear
expression
of
magnetic
motive
force (MMF)
electrical degree. and rotor
magnetic potential, the initial condition is that all the angular positions are expressed with
Based on the winding function, the armature‐reaction MMF of each phase can be expanded into electrical degree.
a Fourier series as [22,23]: Based on the winding function, the armature-reaction MMF of each phase can be expanded into a
2 Nk wv I
Fourier series as [22,23]: 
cos  s  cos t   
 Fa  

$ 
ř 2Nkvp
wv I
’
Fa “
q cos pωt ` ϕq
’
vpπ cos pνθ
’
2  
2 
 s


’
ν 2 Nk wv I cos
&  Fb ř
` `   s 2π ˘˘  cos` t    2π ˘ (9) 
2Nk wv I
33 
vp cos ν θs ´ 33  cos
Fb “  vpπ
ωt
`
ϕ
´


(9)
ν
’
´ ´
¯¯
´
¯
’

ř 2Nk
’
I I
2 Nk
4   cos ωt ` ϕ ´
44π
 
4π

wvwv
’
cos
% FcF“
cos ν θs s´ 3   cos
3
c   vpπ
 t   
ν
3
3
vp







where N is the amount of winding turns in series of stator phase, kwv
is the winding factor, I is the
wv is the winding factor, I is the where N is the amount of winding turns in series of stator phase, k
amplitude
of
phase
current,
ϕ
is
the
current
phase
measured
from
the
a-axis,
ω is the rated electrical
amplitude of phase current, ϕ is the current phase measured from the a‐axis, ω is the rated electrical angular velocity, and θ s s is the angular position in the stator reference frame measured from the axis is the angular position in the stator reference frame measured from the axis of
angular velocity, and θ
phase
a
shown
in
Figure
6.
of phase a shown in Figure 6. (a) (b)
Figure 6. Model with different rotor position: (a) d‐axis overlaps with a‐axis; (b) q‐axis overlaps Figure 6. Model with different rotor position: (a) d-axis overlaps with a-axis; (b) q-axis overlaps
with a‐axis. with a-axis.
The synthetic armature‐reaction MMF is obtained by: The synthetic armature-reaction MMF is obtained by:
Fs ( s , t )  Fa  Fb  Fc
Fs pθ
, tq “ F ` F ` F
řs F cos( va s  kbv (t c  ))
v
“ Fcospvθs ` k v pωt ` ϕqq
where where
2.2.1. D-Axis Magnetic Potential
(10) (10)
v
$
NkNk
wv
F“
’
’
F 3 3vpπ Iwv I
’
& 
k v “ ´1,vpfor v “ 6pm ´ 1q ` 1

6( m
’
k v k“v 1,1,
f orforv v“6m
´11)  1 ’
’
% 
for v  6m  1
m k“v 
1,1,2, 3...

 m  1, 2, 3...
(11)
(11) When ϕ = 0, t = 0 and the d-axis overlaps with the a-axis, the flux passes the d-axis path shown in
Figure 7.
As the permeability of the non-saturation region in rotor is much larger than that of the magnet
and barrier, the magnet and barrier can be regarded as equal-potential area. The magnetic potential
Energies 2016, 9, 580 Energies 2016, 9, 580 6 of 11 6 of 11 2.2.1. D‐Axis Magnetic Potential 2.2.1. D‐Axis Magnetic Potential When ϕ = 0, t = 0 and the d‐axis overlaps with the a‐axis, the flux passes the d‐axis path shown When ϕ = 0, t = 0 and the d‐axis overlaps with the a‐axis, the flux passes the d‐axis path shown in Figure 7. Energies 2016, 9, 580
6 of 11
in Figure 7. As the permeability of the non‐saturation region in rotor is much larger than that of the magnet As the permeability of the non‐saturation region in rotor is much larger than that of the magnet and barrier, the magnet and barrier can be regarded as equal‐potential area. The magnetic potential and barrier, the magnet and barrier can be regarded as equal‐potential area. The magnetic potential distribution
of the rotor in the d-axis is shown in Figure 8. In one-half of the electrical period, the rotor
distribution of the rotor in the d‐axis is shown in Figure 8. In one‐half of the electrical period, the rotor distribution of the rotor in the d‐axis is shown in Figure 8. In one‐half of the electrical period, the rotor magnetic
potential
can be expressed as
magnetic potential can be expressed as magnetic potential can be expressed as ¯
´
˘
βπ 
˘
 ` 
 
 ` 
απ 
απ
sin jj
sin
j
sin
j
sin
j
´sin
ÿ
ÿ




sin
j
2
2
4
4







4 sin  j 22  cos pjθs q ` U 4 sin  j 2   sin  j 2 cos pjθs q
(12) (12)
FFd pθ
s q “ Ud1

 cos  j s  d ( s )  U d 1 4
π  jj 2  cos  j s   U dd22 4π   2  jj
2
(12) 


jj
jj
cos  j s   U d 2 
cos  j s  Fd ( s )  U d 1 
 j
j
 j
j
where U
d1 and U
are the magnetic potentials shown in Figure 7. where
Ud1
and Ud2
are
the
magnetic
potentials
shown
in
Figure
7.
d2
where Ud1 and Ud2 are the magnetic potentials shown in Figure 7. Figure 7. Flux line distribution in the d‐axis. Figure
7. Flux line distribution in the d-axis.
Figure 7. Flux line distribution in the d‐axis. a-axis
a-axis
Ud1
Ud1
Ud2
Ud2
a pole pitch
a pole pitch
απ
απ
βπ
βπ
θs
θs
Figure 8. Magnetic potential distribution in the d‐axis. Figure 8. Magnetic potential distribution in the d‐axis. Figure
8. Magnetic potential distribution in the d-axis.
According to the continuity of magnetic flux, the flux through the air‐gap is equal to that passing According to the continuity of magnetic flux, the flux through the air‐gap is equal to that passing the rotor. The path of the d‐axis flux in the rotor consists of the magnet, barrier and bridge. Based on According to the continuity of magnetic flux, the flux through the air-gap is equal to that passing
the rotor. The path of the d‐axis flux in the rotor consists of the magnet, barrier and bridge. Based on the path, the fluxes in the rotor can be calculated by the
rotor. The path of the d-axis flux in the rotor consists of the magnet, barrier and bridge. Based on
the path, the fluxes in the rotor can be calculated by the path,
the fluxes in the rotor
by

 2  w L 2  l  l  L 2   B  t L 
0 can
Rs be
 Rcalculated
r
2
( Fs ( s ,0)  Fd ( s ))
Ld s  U d 1  0 r m  2 0 lbar1  lbar 2 L  2 0  Bb tb L 
 2 0 ´
$ ş
 b b  b  ¯

R
R

p
R
R
2
lmr wm L  0  barw1bar bar 2   0 w

0
s
r


2απ
2 ( F ( ,0)  F ( ))
s

2µ µp B qt L
2µ0 plbar1 `lbar2 q L
µr 0
2µ0 µr wm L
Rs `Ld
Rr  s  U d 1 
’
(13) &22 απ spFss pθs , 0qd´ sFd pθ
` wb 0 w b  b
p s qq
Rs ppRRr´
 R r q2 R 2 Ldθs “ Ud1 Blm  t Llm ` wbar wbar
´
s
b

2
0
b
b
0
s
r
(13)  ş2βπ( Fs ( s ,0)  Fd ( s ))
(13)
 s  U d 2  µ BµpBt qLt L
 22 pF pθ , 0q ´ F pθpqq Rs 0 Rµ0r  Rs R2s R`rRLd
’
r
0 0
wbb  bb b
% 2απ
Ldθ
“
U
(
F
(

,0)
F
(

))
Ld

U


s
s
s
s
2
w
d
sd
s
d 2 d2
pp R ´ R q
 s s


2
2
s
p  Rs  R
r
r
2
wb
b
Substituting (10) and (12) into (13), the magnetic potentials can be obtained as: Substituting
(10) and (12) into (13), the magnetic potentials can be obtained as:
Substituting (10) and (12) into (13), the magnetic potentials can be obtained as: $
 v

vαπ
2 FP2FP
ř
 vp
q
1 sin
1 sin
2 
’

2
’ Ud1 “ 2 FP
` P3 ` P4 `P5 q
U d 1   v vpP1 2sin
&
´ ´ 2¯P
¯
vβπ  
v v  P2  P3  P
’
Ud1  
FP1 sin 42 ´5 sinp vαπ
q
ř
2
P4  P5 
’
% Ud2 v“v  P2 P3 vvp
 0.5P5 `P6 q  v   v

sin
sin
FP

1
  v  
 v  

FP1  sin  2   sin  2  
U



d
2
where
2 P  P  2  
 v  0.5
v
6
µ0
R5s ` R
U d 2  
r
v  0.5P5  P6  L
P1 “
v

p pRs ´ Rr q
2
where where µ0
R s ` Rr
P2 “
Lαπ
p pRs ´ Rr q
2
P3 “
2µ0 µr wm L
lm
(14)
(14) (14) (15)
(16)
(17)
0
s
r
P2 
L 0 R  Rs 
2 Rr L P2  p  Rs 
r
p  Rs  Rr  2
2  w L
P3  2 0 r wm L P3  0 lmr m lm
2 0  lbar1  lbar 2  L
P4  2   l  l  L P4  0 barw1bar bar 2 2µ0 plbar1
` lbar2 q L
wbar
P4 “
2  w bar
B t L
P5  2 0   Bb  tb L 0 w b
P  2µ0 µ pBb b q tbb L P5 5“
wb
w
0
Rs  bRr      
P6 
L  ´αq
R
Rss 
00
2`RRr rLLpβ
2  π p  Rµ
s  Rr 
P6P“
6 
pppR
22
 Rss´ Rrr q 22
Energies 2016, 9, 580
(16) (16) (17) (17) 7 of 11
(18) (18) (18)
(19) (19) (19)
(20) (20)
(20) 2.2.2. Q-Axis Magnetic Potential
2.2.2. Q‐Axis Magnetic Potential 2.2.2. Q‐Axis Magnetic Potential When φ = 0, t = 0 and the q-axis overlaps with the a-axis, the flux passes the q-axis path shown in
When ϕ = 0, t = 0 and the q‐axis overlaps with the a‐axis, the flux passes the q‐axis path shown When ϕ = 0, t = 0 and the q‐axis overlaps with the a‐axis, the flux passes the q‐axis path shown Figure
9. Similarly, the magnetic potential distribution of rotor in the q-axis is shown in Figure 10, and
in Figure 9. Similarly, the magnetic potential distribution of rotor in the q‐axis is shown in Figure 10, in Figure 9. Similarly, the magnetic potential distribution of rotor in the q‐axis is shown in Figure 10, can be expressed in a Fourier series as
and can be expressed in a Fourier series as and can be expressed in a Fourier series as ´
¯
´
¯
π 
βqπ 
 j1p1´αq
 jp11´
sin
´
sin
sin

j
j
2
2
4 ÿsin
 1     
 1     
2
2
(21)
Fq pθs q “ Uq4
pjθs q
(21)  j sin  j
cos
cos  j s  Fq ( s )  U q 4π sin  j
2
2
(21) j




j
j
cos  j s  Fq ( s )  U q  
 j
j
where
Uq is the magnetic potential shown in Figure 9. is the magnetic potential shown in Figure 9.
where U
where Uq is the magnetic potential shown in Figure 9. Figure 9. Flux line distribution in the q‐axis. Figure 9. Flux line distribution in the q-axis.
Figure 9. Flux line distribution in the q‐axis. a-axis
a-axis
Uq
Uq
βπ
βπ
a pole pitch
a pole pitch
απ
απ
θs
θs
Figure 10. Magnetic potential distribution in the q‐axis. Figure 10. Magnetic potential distribution in the q‐axis. Figure 10. Magnetic potential distribution in the q-axis.
As shown in Figure 9, the flux is divided into two parts in bridge region. The two parts operate in
parallel, and the length of each path is 0.5 wb . The flux in the q-axis can be calculated by
ż
p1´αqπ
2
p1´βqπ
2
`
µ0
R s ` Rr
Fs pθs , 0q ´ Fq pθs q
Ldθs “ Uq
p pRs ´ Rr q
2
˘
ˆ
µ0 µ pBb q tb L µ0 µ pBb q tb L
`
0.5wb
0.5wb
˙
(22)
Energies 2016, 9, 580
8 of 11
From Equation (17), we can get
Uq “
´ ´
¯
´
¯¯
vp1´αqπ
v p 1´ β q π
´ sin
ÿ FP1 sin
2
2
v p2P5 ` P6 q
v
(23)
2.2.3. D-and Q-Axis Inductances
The armature reaction airgap magnetic fields in d-axis can be obtained by
µ
Bd “ pFs pθs , 0q ´ Fd pθs qq pR ´0R q
s
r
¯
´
˜
¸
βπ
απ
sin j 2 ´sinp j απ
ř
2 q
µ
4 ř sinp j 2 q
4ř
“
Fcos pvθs q ´ Ud1 π
cospjθs q ´ Ud2 π
cospjθs q pR ´0R q
j
j
s
r
v
j
(24)
j
The d-axis inductance can be calculated by
$
& Ld “ Lmd ` L0
ş2π ř
%
0
Lmd “
ν
Rs `Rr
2Nk wv
vpπ cospνθs q Bd
2
Ldθr
(25)
I
where Lmd is the main inductance in the d-axis. L0 consists of end-winding leakage inductance, slot
leakage inductance and tooth-tip leakage inductance. The detailed computation processes of the
leakage inductance are expressed in [24]. The end-winding leakage inductance can be calculated by
Lew “ µ0 N 2 q p2lw λh ` wew λw q
4m
Q
(26)
where q is the number of stator slots per pole and phase, lw is the average length of the end winding,
m is the number of phases, Q is the number of slots, wew is the coil span, λh is the empirical axial
permeance factor and λw is the empirical span permeance factor of the end winding.
The slot leakage inductance is
4m
L u “ µ0 N 2 L
λu
(27)
Q
where λw is the slot leakage permeance factor, which is relative to slot dimensions [13,24].
The tooth-tip leakage inductance is
Ltt “ µ0 N 2 L
4m
k tt λtt
Q
(28)
where λtt is the tooth-tip leakage permeance factor, and ktt is a factor that takes into account the
presence of different phases in a slot [24].
The armature reaction airgap magnetic fields in the q-axis can be obtained by
`
˘ µ
Bq “ Fs pθs , 0q ´ Fq pθs q pR ´0R q
s ´ r
¯
´
¯
¸
˜
p1´βqπ
´sin j 2
sin j p1´αqπ
ř
ř
2
µ
4
“
Fcos pvθs q ´ Uq π
cos pjθs q pR ´0R q
j
s
r
v
(29)
j
The q-axis inductance can be calculated by
$
& Lq “ Lmq ` L0
ş2π ř
%
Lmq “
0
ν
where Lmq is the main inductance in the q-axis.
2Nk wv
Rs `Rr
vpπ cospνθs q Bq
2
I
Ldθr
(30)
Energies 2016, 9, 580
9 of 11
3. Results and Analysis
The armature reaction airgap magnetic fields and inductances in the d- and q-axis are calculated
by the proposed method, and the validity is verified by FEM. The FEM results are obtained by the
commercial software ANSYS (version 16.0, ANSYS, Pittsburgh, PA, USA). The main parameters of the
motors are listed in Table 1. Based on these parameters, the saturation of bridge is firstly investigated.
Table 2 depicts the results of the analytical method and FEM. The comparison shows that the proposed
method is very effective for bridge saturation.
Table 1. Main parameters of the 8-pole/48-slot motor.
Symbol
Parameter
Results and Unit
p
Q
I
N
Br
µr
α
β
Rs
Rr
tb
wbar
lbar1
lbar2
wm
lm
wr
lr
Number of pole pairs
Number of slots
Amplitude of phase current
Amount of winding turns in series of stator phase
Remanence of magnet
Relative permeability of magnet
Magnet pole-arc to pole-pitch ratio
Barrier arc to pole-pitch ratio
Stator bore radius
Rotor outer radius
Bridge thickness
Barrier width
Barrier 1 length
Barrier 2 length
PM width
PM length in magnetization direction
Rib width
Rib length
4
48
1A
32
1.25
1.024
0.742
0.886
91 mm
90 mm
1 mm
5 mm
5 mm
5.9 mm
20 mm
6 mm
8 mm
8.8 mm
Table 2. Permeability of bridge.
Method
Value
Unit
10´5
Proposed method
Finite element method (FEM)
H/m
H/m
4.40 ˆ
4.52 ˆ 10´5
Taking into account the influence of the armature-reaction magnetic field on saturation, a very
small current is employed to calculate no-load d- and q-axis inductances. Freeze permeability is
adopted to calculate the armature-reaction magnetic field. Figure 11 shows the waveforms of airgap
flux density in the d- and q-axes. It is observed that the analytical results match well with that of
FEM. Based on Equations (25) and (30), the d- and q-axis inductances in the no-load condition can be
obtained shown in Table 3. The analytical data and simulation data tally well, which indicates that the
proposed method is quite accurate for d- and q-axis inductance calculations.
Table 3. The inductances of the two methods.
Item
Proposed Method
FEM
Unit
D-axis inductance
Q-axis inductance
0.524
1.267
0.513
1.271
mH
mH
0.004
0.003
0.002
0.001
0.000
-0.001
-0.002
-0.003
-0.004
FEM
Analytical
0
60
120
180
240
Electrical Angle (Deg)
Flux Density (T)
Flux Density (T)
small current is employed to calculate no‐load d‐ and q‐axis inductances. Freeze permeability is adopted to calculate the armature‐reaction magnetic field. Figure 11 shows the waveforms of airgap flux density in the d‐ and q‐axes. It is observed that the analytical results match well with that of FEM. Based on Equations (25) and (30), the d‐ and q‐axis inductances in the no‐load condition can be obtained shown in Table 3. The analytical data and simulation data tally well, which indicates that Energies 2016, 9, 580
10 of 11
the proposed method is quite accurate for d‐ and q‐axis inductance calculations. 300
360
0.010
0.008
0.006
0.004
0.002
0.000
-0.002
-0.004
-0.006
-0.008
-0.010
FEM
Analytical
0
60
120
180
240
Electrical Angle (Deg)
(a) 300
360
(b)
Figure 11. Comparison of airgap flux density: (a) in the d‐axis; (b) in the q‐axis. Figure 11. Comparison of airgap flux density: (a) in the d-axis; (b) in the q-axis.
4. Conclusions
Table 3. The inductances of the two methods. Item Proposed Method
FEM
Unit In this paper, a novel analytical method of d- and q-axis inductance calculation in a no-load
D‐axis inductance 0.524 0.513 mH condition is proposed and applied to IPM motors with V-shaped rotors. Taking bridge saturation into
Q‐axis inductance 1.267 1.271 mH consideration, the rotor is divided into two pieces: saturation regions with a constant permeability and
non-saturation regions with infinite permeability. The LPMCM and iterative method effectively solve
4. Conclusions the saturation and nonlinearity of the bridge. The rotor magnetic potentials in the d- and q-axis are
In this paper, a novel of the
d‐ and inductance calculation a no‐load investigated
according
to theanalytical flux paths.method Based on
WFT,q‐axis the influence
of magnetic
fieldin distribution
condition is proposed and applied to IPM motors with V‐shaped rotors. Taking bridge saturation on inductance is considered. The analytical results of bridge permeability, armature-reaction magnetic
into consideration, the rotor is divided into with
two the
pieces: saturation with confirms
a constant field and
d- and q-axis
inductances
match well
ones obtained
by regions FEM, which
the
permeability non‐saturation validity of theand proposed
model. regions with infinite permeability. The LPMCM and iterative method effectively solve the saturation and nonlinearity of the bridge. The rotor magnetic potentials Acknowledgments: This study was carried out as a part of the industrial application technology in electric
in the d‐ and q‐axis are investigated according to the flux paths. Based on the WFT, the influence of vehicles and supported by the National Natural Science Foundation of China (51477032) and Self-Planned Task
magnetic field distribution on inductance is considered. The analytical results of bridge permeability, (NO. SKLRS201608B) of the State Key Laboratory of Robotics and System (HIT).
armature‐reaction magnetic field and d‐ and q‐axis inductances match well with the ones obtained by Author Contributions: This paper is the results of the hard work of all authors. Peixin Liang, Yulong Pei and
FEM, which confirms the validity of the proposed model. Feng Chai conceived and designed the proposed method. Kui Zhao built the FEM models and analyze the data.
Peixin Liang wrote the paper. All authors gave advices for the manuscripts.
Acknowledgments: study was carried out as a of the industrial application technology in electric Conflicts of Interest:This The authors
declare
no conflict
ofpart interest.
vehicles and supported by the National Natural Science Foundation of China (51477032) and Self‐Planned Task (NO. SKLRS201608B) of the State Key Laboratory of Robotics and System (HIT). References
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