Magnetic Equivalent Circuit Model of Interior Permanent

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EVS28
KINTEX, Korea, May 3-6, 2015
Magnetic Equivalent Circuit Model of Interior
Permanent-Magnet Synchronous Machine Considering
Magnetic Saturation
Zaimin Zhong, Shang Jiang1 , Guangyao Zhang
1School
of Automotive Studies, Tongji University, 4800 Cao’an Road, Shanghai, China, [email protected]
Abstract
This paper improves an analytical model for interior permanent-magnet synchronous machine (IPMSM) by
a magnetic equivalent circuit (MEC) approach. The proposed MEC model consists of three major regions:
the stator, the rotor, and the air gap. The conventional reluctance approach is applied to the first two
regions. Since the magnetic field in air gap region is supposed to distribute unevenly, this model employs a
function of rotor electrical angle to describe the flux density distribution. Firstly, the permeability of all
iron is assumed to be infinite, so the reluctance of the iron core is close to zero and can be ignored. Then
the air gap flux density in the direct-axis and quadrature-axis caused by armature currents are calculated in
the d-q reference frame respectively, which are used to acquire the direct and quadrature axes inductances
Ld and Lq. Open-circuit voltage E0 can be obtained by the air gap flux density caused by the magnets. A
finite element model (FEM) is developed to validate the analytical model, and the comparison results
indicate that the accuracy of the analytical model is acceptable only in the condition of low armature
currents. When the armature currents exceed a certain value, magnetic saturation of the iron core must be
taken into account. Therefore, a modification model considering the magnetic saturation is proposed by
adding the reluctance of iron core parts to the previous one. The permeability of iron is nonlinear and
presented as a function by fitting the BH-characteristic curve with a cubic polynomial. Finally, through
numerical calculation, the air gap flux density is derived from the equations of the analytical model. The
result of the modification model is validated with the FEM in the saturated condition.
Keywords: Analytical Model, Magnetic equivalent circuit (MEC), Interior permanent-magnet synchronous machine
(IPMSM), Saturation, Finite element model (FEM)
1
Introduction
The main trend of clean energy automobile is
developing Electric vehicles (EVs). As one of the
key components in EVs, the driving motor
system deserves more attentions. IPMSM is a
type of motor and widely applied in EVs due to
its high power density, reliability, very good
efficiency, and their reduced maintenance costs[1].
In project design phase, it’s important to build an
efficient and accurate model of IPMSM [2][3].
Generally, accurate modelling of electric machines
requires the use of finite-element method[4].
However, finite-element analysis is too time
EVS28 International Electric Vehicle Symposium and Exhibition
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consuming, especially for the parametric studies
at the first stages of the design process. In order
to reduce the pre-design stages duration, MEC
model[6]-[8] is usually applied. Especially when
regardless of saturation, the simulation speed is
much higher due to absence of iterative process.
MEC method divides the magnetic field into a
network of flux tubes with homogenous
distribution of flux density and strength through
each cross section [5]. The accuracy of MEC
model of IPMSM usually meets the design
requirement.
In first section, this paper builds conventional
MEC models of IPMSM with PMs, d- and q-axis
armature currents exciting alone respectively
based on the distribution form of PMs, and
obtains the distribution function of airgap flux
density which takes into counting slotting effects.
Then, the synchronous inductances are computed
based on their definitions and the proposed MEC
model. Furthermore, a MEC network model
considering saturation is presented. At last, both
the conventional and new MEC model are
resolved and validated through comparison with
the simulation results of FEM.
2
2.1
0  m 
Br
Hc
(2)
B
Br
B
H Hc
H
O
Figure 1: Demagnetization curve of PMs
Conventional MEC Model
Assumptions
Considering the complexity of motor structure
and field distribution, the conventional MEC
model is based on the following hypotheses:
(1) The permeability of rotor and stator iron core
is assumed infinite and the saturation is neglected.
(2) Only radial fluxes in airgap are counted.
(3) The MMFs produced by stator windings are
sinusoidally-distributed in circumferential space
around the airgap.
(4) The fields distribute evenly in the surface of
PMs.
2.2
Where, μ m is relative permeability of PM and μ 0 is
the absolute permeability of vacuum.
When B equals to zero, it is deduced as:
MEC Model with PMs Acting
Alone
PMs are made of hard magnetic materials and
usually work along the demagnetization curve.
There are two important quantities associated
with the demagnetization curve that are the
residual flux density B r and the coercive force Hc.
PMs are designed to operate on the linear part of
the demagnetization curve between B r and Hc,
which is illustrated in Figure 1. Such recoil line
is usually approximated by:
B  0 m H  Br
(1)
Figure 2: Rotor configuration of IPMSM with
tangentially magnetized PMs
Based on the rotor configuration and geometric
parameters in Figure 2, the reluctances of PM_1
and PM_2 are calculated by[9]:
Rm1 
L1
2 L2
, Rm 2 
0 mls h1
0 mls h2
(3)
Where denotes the active length of the rotor.
PMs are equivalent to MMF source and the MMF
produced by PM_1 and PM_2 respectively are
calculated by:
Fc1  H c L1 ,
Fc 2  Hc  2L2
(4)
The leakage reluctance of magnetic bridge is
deduced by:
R 
2 L2
0ls (2t2 )
(5)
Since the space MMF produced by PMs is
supposed to be evenly distributed along the airgap,
the airgap can be equivalent to a hollow cylindrical
reluctance element and its resistance is calculated
by:
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R 
1
 0 ls

 ln(
rg
)
r
2.3
(6)
2p
Where g denotes the thickness of airgap, r
external radius of the rotor, p number of
magnetic pole pairs.
Figure 3 shows the MEC model when PMs
acting alone.
MEC Produced by D-axis Armature
Current
When a symmetrical set of three sinusoidal
currents is inserted to the three-phase stator
winding, the armature currents are expressed as:

ia  2 I s cos( wr t   )

2

ib  2 I s cos( wr t     )
3

4

ic  2 I s cos( wr t    3  )
(11)
Where denotes the effective value of currents,
angular velocity of the rotor, the torque angle.
The currents in dq frame are deduced based on
MMF conservation and power equivalent principle:
I d  3I s  cos  , I q  3I s  sin 
(12)
Considering the previous assumption about MMF
waveform, the MMFs produced by d- and q-axis
armature currents are calculated respectively by:
Figure 3: MEC model when d-axis current exciting
alone
Considering the previous hypotheses, Ampere’s
theorem and flux conservation, the equation set
of MEC model is written in:
 Fc1  Rm1m1  Fc 2  Rm 2m 2  Fm
      
 m1 m 2  

 R  Fm
  2 R  Fm
(7)
Nkw1
I d cos( p ) (13)
p
Nk
Fq ( )  Fqm sin( p )  0.9 s w1 I q sin( p ) (14)
p
Fd ( )  Fdm cos( p )  0.9
Where, Ns denotes the number of series winding
turns of each phase in abc reference frame,
the
winding factor.
When PMs are considered as reluctance elements
without remanence and the armature current in qaxis is neglected, the MEC model of IPMSM is
illustrated in Figure 4.
The airgap flux is given in:
 
( Rm1Fc 2  Rm 2 Fc1 ) R
(8)
Rm1Rm 2 ( R  2R )  ( Rm1  Rm 2 )2 R
The flux density distributed along airgap with
respect to the mechanical angle of the rotor θ is
deduced by:
B f ( ) 


g
(r  )ls
2p
2
;  [
 
, ]
2p 2p
(9)
Due to the symmetry of fields in airgap between
adjacent poles, the flux density with respect to
from
to
is written in:
B f ( )  


g
 (r  )ls
2p
2
(10)
Figure 4: MEC model when d-axis current exciting
alone
Considering the previous hypotheses, Ampere’s
theorem and flux conservation, the equation set of
MEC model is written in:
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u ( )  H d ( )  2 g

0
g

  ls  0 H d ( )(r  )d
2
2p

 B ( )   H ( )
0 d
 d
3.1
(15)
30ls (r  g / 2) Fdm
(16)
g
8 g  0ls (r  )( Rm1 / / Rm 2 / / R )
2
The flux density distributed along airgap with
respect to is deduced by:
Bd ( )  0
Fd ( )  ( Rm1 / / Rm 2 / / R )
(17)
2g
And the flux density with respect to
to
is written in:
Bd ( )   Bd (
2.4

p
 )
from
(18)
MEC
Produced
Armature Current
by
Q-axis
When the field is excited only by q-axis armature
current, the flux doesn’t flow through the PMs.
Therefore, the MEC model is depicted in Figure
5 and the flux density is deduced as:
Bq ( )  0
Fq ( )
2g
;   [

The slots increase the resistance of airgap and
is employed to correct the airgap thickness[11].
1
01 
The airgap flux is given in:
 
Correction of airgap thickness due
to slotting effect
3
]
2p 2p
(19)
1  5 / b01
t1
Kc 
t1  01b01
(20)
(21)
Where, denotes the pitch of the stator tooth,
the inner diameter of the stator,
the number of
stator teeth,
the width of each stator slot.
holds:
t1 
 Di1
Z1
(22)
Based on Carter’s coefficient
thickness is corrected by:
g '  Kc  g
3.2
, the airgap
(23)
Correction of the Flux Density
Distribution along the Airgap
Since stator slot openings change the airgap
reluctance in front of the slots, it is noted that the
presence of stator slots have a large influence on
the air-gap magnetic field distribution and
therefore on the electromagnetic torque. Figure 6
shows the geometric model of the airgap thickness
variation[12].
Figure 6: Airgap variation in slot region
Figure 5: MEC model when q-axis current exciting
alone
3
Modification of MEC Model
Considering Slotting Effect
The slotting influences the magnetic field in two
ways. First, it reduces the total flux per pole, an
effect which is usually accounted by introducing
the Carter’s coefficient [10] into the calculation.
Second, it affects the distribution of the flux
density in the air-gap of the electric machines.
The thickness of slotted airgap region is changed
as:
g ( )  g 
 rw
2
(24)
Based on the distribution of the stator slot
openings, the circumferential variation airgap
thickness
is written in:
EVS28 International Electric Vehicle Symposium and Exhibition
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 
 

  [(k  1)t  t 0 ,(k  1)t  t 0 ]
 g
2
2
g ( )  






 g  r   [(k  1)  t 0 ,(k  1)  t  0 ]
t
t
 2 w
2
2
(25)
Where
and
denote the radian
corresponding to tooth pitch and slot space
respectively, and they are calculated by:
t 
 t  b0
rg
, 0 
b0
rg
(26)
The radius
is the airgap thickness increase
caused by slot openings, and it holds:
rw  [  (k  0.5)t ](r  g )
(27)
The range of k is determined by the slot number.
Then the distribution coefficient of airgap field
due to slotting effect is calculated by:
 ( )  g / g ( )
(28)
The modified distributed flux densities along the
airgap are obtained by multiplying the previous
function of them by distribution coefficient
.
4
Back EMF
Calculation
and
Torque
In order to obtain the synchronization parameters
of the motor, the fast Fourier transform (FFT) is
introduced to formulate the airgap flux density
considering slotting effect. Then we get the
fundamental component of the flux density
distributed along airgap.
 B f 1 ( )  B f 1m  cos( p )

 Bd 1 ( )  Bd 1m  cos( p )
 B ( )  B  sin( p )
q1m
 q1
(29)
Where B f 1m , Bd 1m , Bq1m denote the amplitude
of fundamental wave caused by PMs, d- and qaxis armature currents respectively.
4.1
Back EMF
Airgap flux linkage produced by PMs can be
calculated by the corresponding flux density
function:
E0 
wr f
3
Where
4.2

2 2
g
wr kw1 N s ls B f 1m (r  ) (31)
3p
2
denotes the angular speed of rotor.
Electromagnetic Torque
Similarly, by choosing appropriate integrating
range, the airgap flux linkage produced by the
armature current could be obtained:
d 
2 2
g
 kw1 Nsls Bd 1m (r  )
3 p
2
(32)
2 2
g
q 
 kw1 Nsls Bq1m (r  )
3 p
2
Then the inductance of d- and q-axis is:
g
2 2kw1 Nsls Bd 1m (r  )
2
Ld 

Id
3 pI s
g
q 2 2kw1 Nsls Bq1m (r  2 )
Lq 

Iq
3 pI s
d
(33)
(34)
With the inductance parameters and flux linkage of
PMs, the electromagnetic machine torque can be
presented by the following formula. The first item
in the bracket is the excitation torque produced by
the interaction between the armature currents and
PMs while the second item shows the reluctance
torque caused by the different reluctance between
d- and q-axis.
Te  p[ f I q  ( Ld  Lq ) I d I q ]
5
Simulation
Results
Conventional MEC Model
(35)
of
In the previous sections, a conventional analytical
model of IPMSM in the form of MEC has been
developed. It is necessary to validate this model by
comparing it to professional software results based
on the FEM. The materials in the FEM have linear
behavior, so that the analytical and FE results are
compared under the same working conditions
without saturation.

2
g
f 
kw1 N s ls  2 p B f 1m cos( p )(r  )d

3
2
2p

2 2
g
 kw1 N s ls B f 1m (r  )
3 p
2
5.1
Main Parameters
The motor presented in this paper is an IPMSM
with tangential magnetized, and the main structure
and material parameters are listed in Table 1.
Figure 7 shows its FEM.
(30)
And then the back EMF is:
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Figure 9 and 10 shows the flux density distribution
with respect to the rotor angle in the motor
obtained by MEC model and FEM. They show a
good general agreement between the two results.
Nevertheless, there are still two major differences:
(1) MEC model considers the airgap EMF
distributes sinusoidally while it should be ladder
distribution as for a PMSM with double layer
distributed winding in FEM.
(2) All flux flow through PMs are supposed to be
perpendicular to their surfaces, while there are
fluxes oblique crossing the PMs in FEM, which
results in the difference of flux density around the
angle of 40°.
3
MEC model
FEM
2
Figure 7: FEM of IPMSM
1
Pole pair number
Stator slot number
Stator external diameter
Stator inner diameter
Rotor external diameter
Slot space
Tooth width
Inductances per slot
Size of PM_1
Size of PM_2
Br
5.2
Value
2
36
200mm
118.4mm
117mm
2.2mm
8.13mm
24
5mm*16.5mm
3.5mm*26.2mm
1.23T
1.09978
-2
-3
-50
100
150
MEC model
FEM
2
0
-2
-4
-50
0
50
100
Electrical angle α[°]
150
Figure 10: Airgap flux density produced by q-axis
current
1.5
MEC model
FEM
The inductance matrix Labc in abc frame is
obtained from the FEM simulation results, and the
inductance matrix Ldq in dq frame is calculated by:
0.5
B[T]
50
Electrical angle α[°]
4
The spatial distribution of airgap flux density
produced by PMs is illustrated in Figure 8. It is
noted that the variations between analytical and
FEM simulation results are limited, less than 5%.
Ldq  C T LabcC
0
(36)
Where,
-0.5
-1
-1.5
-50
0
Figure 9: Airgap flux density produced by d-axis current
Comparison of Simulation Results
1
0
-1
B[T]
Quantity
B[T]
Table 1: Main parameters of IPMSM
Labc
0
50
Electrical angle α[°]
100
150
 Laa
  Lba
 Lca
Lab
Lbb
Lcb
Lac 
Lbc 
Lcc 
(37)
Figure 8: Airgap flux density produced by PMs
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 sin( ) 
 cos( )

2
2 
2 cos(   )  sin(   ) 
C
3
3  (38)

3
2
2 
cos(   )  sin(   ) 
3
3 

The results of synchronous inductances obtained
from MEC are compared with the results from
FEM in Table 2.
6.1.1
Reluctance Calculation of Stator Iron
Core
The stator iron core is composed of yoke, tooth
and tooth shoe from view of flux flow paths, as is
shown in Figure 12.
The reluctances of three components of the stator
are calculated based on their shapes, geometric
parameters and flux directions.
Table 2: Comparison of synchronous motor
parameters
Quantity
Back
EMF(V)
Ld(mH)
Lq(mH)
MEC
model
184.88
FEM
variation
183.86
0.55%
103.7
288.5
111.5
272.2
-7%
4.9%
Figure 12: Geometric figure of the stator
 flux tube of stator yoke reluctance:
The curves of electromagnetic torque versus
rotor position from MEC model and FEM are
illustrated in Figure 11. Due to the results in
Table 2, the offset of the two torques mainly
derives from the error of Ldq.
MEC model
FEM
Te[N.m]
-100
0
100
200
Torque angle β[°]
300
400
Figure 11: Electromagnetic torque versus rotor
position characteristic
6.1
(39)
 flux tube of tooth reluctance:
hd
u0uFels ld
(40)
 f ln(lepld )   lep 
Rdb  
//
 u u l (l  l )   u u l b  (41)
 0 Fe s ep d   0 Fe s 
0
-50
6
rc 2
)
rc1
 Flux tube of tooth shoe relctance:
50
-150
u0uFels (ln
Rdh 
150
100
d
Rc 
MEC Model
Saturation
for
Magnetic
6.1.2 Nonlinear Relative Permeability
The relative permeability for each point of a
ferromagnetic material is not constant but depends
on the actual magnetic field strength H
respectively the actual magnetic flux density B of
this point. The communication curve H(B) of
ferromagnetic material DW310_35_2DSF0.950 is
shown in Figure 13.
Nonlinear Reluctance Elements
When working in saturated region, the
reluctances of elements with nonlinear
permeability characteristics change with respect
to the flux density. Saturation usually happens in
stator iron core, so this paper take stator yoke,
teeth and teeth shoes as nonlinear reluctance
elements.
Figure 13: Fitting curve of relationship of B and H
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In order to obtain the analytical MEC model, the
relationship H(B) in saturated region is
approximated with a cubic polynomial function
by fitting.
(42)
Because the relative permeability is get by
dividing flux density by magnetic field strength,
it is deduced that:
Since the MMF produced by armature currents is
supposed to be sinusoidal wave along the airgap,
the values of the MMF elements are calculated as:
(44)
Where
is the amplitude of the fundamental
harmonic.
Then, the system state equation of the MEC
network model is obtained based on Kirchhoff
laws:
(45)
AX  b
(43)
6.2
Reluctance Network Model with
D-axis Currents
The constant vector is:
b
The proposed MEC network model of the
IPMSM in Figure 7 is depicted in Figure 14.
0.5fd1
Rc
Rc
Rc
Rc
Rc
Rc
Rc
0.5fd2
0.5fd3
0.5fd4
0.5fd5
0.5fd6
0.5fd7
0.5fd8
0.5fd9
0.5fd10
Rd
Rd
Rd
Rd
Rd
2Rd
Rg
Rg
Rg
Rg
Rg
2Rg
Rd
Rd
2Rg
Rg
Rg
Rg
Linear reluctance element
Nonlinear reluctance element
Figure 14: Reluctance network model each pole
  Rdg
1
 Rc  1 Rm
2
2
1
1
R  R
A   2 c 2 m
1
1
 Rc  Rm
2
2

1
1
 2 Rc  2 Rm
6.3
fd 5 
Rdg  Rc
Rc
Rc
1
3
Rdg  Rm  Rc
2
2
1
3
Rm  Rc
2
2
1
3
Rm  Rc
2
2
1
3
Rm  Rc
2
2
1
3
Rm  Rc
2
2
1
5
Rdg  Rm  Rc
2
2
1
5
Rm  Rc
2
2
1
5
Rm  Rc
2
2
1
3
Rm  Rc
2
2
1
5
Rm  Rc
2
2
1
7
Rdg  Rm  Rc
2
2
1
7
Rm  Rc
2
2
Model Solution and Simulation
Results
The algorithm for solution[13] of the MEC
network model is concluded based on the
researches on the nonlinear equations’ solution:
First, the dimensions and material properties of
motor are initialized. Variables and airgap
permeances and flux sources are calculated.
Then, using a constant permeability for iron the
permeance matrix of system is developed and
solved. Flux density values in different parts of
the motor are then computed and permeabilities
of saturated parts are updated using curve of iron
parts and flux densities are recomputed for these
regions.
T
(46)
(47)
The coefficient matrix A is listed in equation (48):
The five state varibles correspond to the radial
fluxes through five stator teeth, and the airgap flux
density along each stator tooth can be obtained
based on its definition. Futhermore, all reluctances
are replaced by their corresponding formulas with
geometric parameters and the relative permeability
of stator iron core is expressed as the fitted
function. Then, a nonlinear equation set to
calculate distributed airgap flux density is obtained.
Rm2
Magnetomotive force source
fd 4
Where, the states denote:
Rm1
Flux tube
fd 3
T
Rc
2Rd
fd 2
X  1 2 3 4 5 
Rc
Rd
1
 fd1
2
Rc


1
3
Rm  Rc 
2
2


1
5
Rm  Rc 
2
2


1
7
Rm  Rc 
2
2

1
9 
Rdg  Rm  Rc 
2
2 
(48)
Last, the convergence of flux density is then
provided through an iterative process.
However, since the high accuracy of fitting for the
permeability of the iron core only holds when the
flux density B doesn’t exceed 2.1T, the error will
be not accepted in working conditions of large
currents. Therefore, this paper introduces the
constrained optimization method to solve the
nonlinear MEC equations.
The MEC equation set in (45) is used to define the
error vector f:
f  [ f1 , f 2 , f3 , f 4 , f5 ]T  AX  b
(49)
The objective function is based on least squares
criterion, and holds:
Minimize
EVS28 International Electric Vehicle Symposium and Exhibition
f12  f 2 2  f32  f 42  f52 (50)
8
Considering the limitation of the flux density B
in the iron core due to the fitting accuracy, the
constraint condition holds:
B(Fe)≤2.1
(51)
The optimization process is implemented using
the optimization toolbox of Matlab, and it takes
13 iterations to satisfy the convergence criteria
within a few seconds. Noted that the MEC
method saves much time compared to FEA.
Figure 15 and Figure 16 illustrate the comparison
of airgap flux densities with 15A and 200A
armature currents from MEC model and FEM
density. At last, the two proposed MEC models are
validated by comparing the solution results with
the simulation results of FEM, which indicates the
feasibility of the conventional MEC model and the
limitation of the proposed network model
considering saturation.
References
[1]
Renyuan Tang. Modern Permanent Motor: Theory
and Design, ISBN 978-7-111-06010-9, Beijing:
China Machine Press, 1997.
[2]
Tiegna H, Amara Y, Barakat G. Overview of
analytical models of permanent magnet electrical
machines for analysis and design purposes,
Mathematics and Computers in Simulation, ISSN
03784754, 90(2013): 162-177.
[3]
Vido L, Gabsi M, Chabot F, et al. Interior
Permanent-Magnet Synchronous Machine Design
by Reluctants Networks Approach for Hybrid
Vehicle Applications, In: 3rd IET International
Conference on Power Electronics, ISBN 0-86341609-8, 2006, 541-545.
[4]
F. Magnussen, P. Thelin, and C. Sadarangani,
Performance evaluation of permanent magnet
synchronous machines with concentrated and
distributed windings including the effect of field
weakening, ISSN 0537-9989, PEMD 2004(2):
679-685.
[5]
J.R. Hendershot, T.J.E. Miller. Design of Brushless
Permanent-Magnet
Motors,
ISBN
9780984068708, Motor Design Books LLC, 2010.
[6]
Hsieh M, Hsu Y. A Generalized Magnetic Circuit
Modeling Approach for Design of Surface
Permanent-Magnet Machines. IEEE Transactions
on Industrial Electronics, ISSN 0278-0046 2012,
59(2): 779-792.
[7]
Sheikh-Ghalavand B, Vaez-Zadeh S, Hassanpour
Isfahani A. An Improved Magnetic Equivalent
Circuit Model for Iron-Core Linear PermanentMagnet Synchronous Motors, IEEE Transactions
on Magnetics, ISSN 0018-9464, 2010, 46(1): 112120.
[8]
Han S, Jahns T M, Soong W L. A Magnetic Circuit
Model for an IPM Synchronous Machine
Incorporating Moving Airgap and Cross-Coupled
Saturation Effects, 2007 International Electric
Machines & Drives Conference, ISBN 1-42440742-7, 2007, 21-26.
[9]
Perho J. Reluctance network for analysing
induction machines, ISBN 951-666-620-5, Finland:
Helsinki University of Technology, 2002.
1.5
MEC network
FEM
1
B[T]
0.5
0
-0.5
-1
-1.5
-1
-0.5
0
0.5
1
Electrical angle α[°]
1.5
2
2.5
Figure 15: Airgap flux density with 15A current
3
MEC network
FEM
2
B[T]
1
0
-1
-2
-3
-1
-0.5
0
0.5
1
Electrical angle α[°]
1.5
2
2.5
Figure 16: Airgap flux density with 200A current
It is noted that the airgap flux densities from the
two simulation methods have a certain deviation.
The reasons can be concluded as following:
 The error exits in fitting function of the
permeability of iron core.
 The linkage fluxes are not taken into
consideration when modelling.
7
Conclusion
This paper presents a simple MEC model and a
MEC network model for IPMSM applied in
unsaturated and saturated regions, respectively.
Both models take into account the slotting effect.
The synchronous inductances and back EMF are
obtained by applying FFT to the airgap flux
[10] Dajaku, G. and D. Gerling, Stator Slotting Effect
on the Magnetic Field Distribution of Salient Pole
Synchronous Permanent-Magnet Machines. IEEE
Transactions on Magnetics, ISSN 0018-9464, 2010,
EVS28 International Electric Vehicle Symposium and Exhibition
9
46(9): 3676-3683.
[11] Zhiguang Hu. Analysis and computation of
motor field, ISBN 978711128798, Beijing, China
Mechanical Press, 1989, 57-79.
[12] Zhu, Z.Q. and D. Howe, Instantaneous magnetic
field distribution in brushless permanent magnet
DC motors. III. Effect of stator slotting.
Magnetics, IEEE Transactions on, ISSN 00189464, 1993. 29(1): 143-151.
[13] Zhang M, Macdonald A, Tseng K, et al.
Magnetic equivalent circuit modeling for interior
permanent magnet synchronous machine under
eccentricity
fault:
Power
Engineering
Conference
(UPEC),
48th
International
Universities, INSPEC Accession Number:
14043562, 2013: 1-6.
Authors
Zaimin Zhong, is professor of
automotive electronics at school of
automotive studies, Tongji University.
He received bachelor’s degree in 1995
and doctor’s degree in 2000 for
automotive engineering from Beijing
Institute of Technology, and has more
than ten years in research on design
and control of electrified mechanical
transmission applied in EV and HEV.
Shang Jiang, is studying for doctor’s
degree of automotive electronics at
school of automotive studies, Tongji
University. He received the bachelor’s
degree of communication and
transportation from Central South
University in 2012. His research
interests include modelling of electric
machines and powertrain control.
Guangyao Zhang, is studying for
master’s degree of automotive
electronics at school of automotive
studies, Tongji University. He
received the bachelor’s degree of
automotive
engineering
from
Tsinghua University in 2013. His
research interest is analysis of
vehicular PMSM.
EVS28 International Electric Vehicle Symposium and Exhibition
10
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