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Application of the Finite Element Method in Design and Analysis of
Permanent-Magnet Motors
ARASH KIYOUMARSI1, PAYMAN MOALLEM1, MOHAMMADREZA HASSANZADEH2 and
MEHDI MOALLEM3
Department of Electronic Engineering, Faculty of Engineering, University of Isfahan, Isfahan
2
Faculty of Electrical Engineering, Abhar Islamic Azad University, Abhar, Ghazwin
3
Department of Electrical and Computer Engineering, Isfahan University of Technology, Isfahan
IRAN
1
Abstract- In this research, the results of approximate analytical methods and Finite Element Method (FEM),
those are used for prediction of airgap flux density distribution, are compared. In this comparison, permanentmagnet direct-current (PMDC) motors and brushless permanent magnet motors are considered. In addition, a
coupled magnetic field, electrical circuit, and mechanical system program by which the FEM analysis is
accomplished, is briefly discussed. Then, time stepping finite element method is used for the magnetic field
analysis. At last, an example of shape design optimization, i.e., optimal shape design of an interior permanentmagnet (IPM) synchronous motor, is considered.
Key-Words: The Finite Element Method, Brushless Permanent Magnet Motors, DC Motors.
1 Introduction
Prior to the development of reliable high-power
solid-state switching devices, the DC motor was the
dominant electric machine for all variable-speed
motor drive applications. The DC motor turns out to
be the most economical choice in the automotive
industries for cranking motors, wind shield wiper
motors, blower motors and power window motors
[1]. In this paper, first, the magnetic flux density of
both a six-pole, 29-slot PMDC motor and a seriesexited four-pole, 21-slot field-winding DC motor are
obtained based on an iterative analytical method.
Then, the magnetic field is modeled based on a twodimensional field analysis method considering the
effect of rotor slots [1-3]. The results of calculations
are compared with finite element method results and
predicted output characteristics of both motors are
also compared with those obtained by real output
measurements [4-7].
Brushless permanent magnet motors can be
divided into the PM synchronous AC motor (PMSM)
and PM brushless DC motor (PMBDCM). The
former has sinusoidal airgap flux and back EMF, thus
has to be supplied with sinusoidal current to produce
constant torque. The PMBDCM has the trapezoidal
back EMF, so the rectangular current waveform in its
armature winding is required to obtain the low ripple
torque. Generally, the magnets with parallel
magnetization are used in the PMSM while the
magnets with radial magnetization are suitable for the
BDCM [8-9].
Permanent-magnet synchronous motors (PMSM)
have higher torque to weight ratio as compared to
other AC motors. There are different rotor topologies
that divide into two basic types, i.e., Exterior
Permanent-Magnet motors (EPM) and Interior
Permanent-Magnet motors (IPM). Surface–mounted
permanent-magnet synchronous motor (SPMSM)
and inset permanent-magnet
motor, belong to
former and buried or interior–type permanentmagnet synchronous motor (IPMSM) and flux
concentration or spoke–type permanent-magnet
synchronous motor, belong to the latter. Because of
the mechanical and electromagnetic properties of
each type, different topologies have different
advantages and disadvantages used in high-speed
applications [10, 11]. They have different control
strategies and there is usually torques, speed, angular
position and current-control loops in the control
system.
Interior permanent-magnet synchronous motor,
has many advantages over other permanent-magnet
synchronous motors. It has usually larger quadrature
axis magnetizing reactance than direct axis
magnetizing reactance. These unequal inductances in
different axes, enable the motor to have both the
properties of SPMSM and synchronous reluctance
motor (SynchRel) [12]. The total resultant
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instantaneous torque of a brushless permanentmagnet motor has two components, a constant or
useful average torque and a pulsating torque which
causes torque ripple. There are three sources of
torque pulsations. The first is field harmonic torque
due to non-ideal distribution of flux density in the
airgap, i.e., non-sinusoidal in the PMSM or nontrapezoidal in the PM brushless DC motor. The
second is due to the cogging torque or detent torque
caused by the slotted structure of the armature and
the rotor permanent-magnet flux. The third is
reluctance torque, produced due to unequal
permeances of the d- and q-axis. This torque is
produced by the self-inductance variations of phase
windings when the magnetic circuits of direct- and
quadrature-axis are unbalanced [13].
In IPM synchronous motor, the effective airgap
length on the d-axis is large so the variation of the daxis magnetizing inductance, Lmd, due to magnetic
saturation, is minimal. For the q-axis, there is an
inverse condition, i.e., the effective airgap length on
the q-axis is small and therefore the saturation effects
are significant [14-16].
139
Then, the magnetic field is modeled based on a twodimensional field analysis method considering the
effect of rotor slots [1-3]. The results of calculations
are compared with finite element method results and
predicted output characteristics of both motors are
also compared with those obtained by real output
measurements.
2.1 Analytical Method
Figs. 1-a and 1-b show the frame assembly and
armature of two ideal motors. Fig. 1-c shows
armature and field current densities applied to the
FWDC motor in this study. Having determined the
structure of the magnetic circuit (Fig. 2) and electric
circuit of both motors, the node permeance matrix
and the node magnetic flux source vector can be
found and interactive calculation is used to solve the
equation. The permeability of segment i, for the
(k+1)th iteration, considering magnetic saturation, is
then corrected and determined by the following
expression [5]:
µ ik = µ ik −1 + λki {µ ik − µ ik −1 }
(1)
2 Characteristics of a PMDC and a
FWDC Starter Motor
The influence of magnetic saturation on
electromagnetic field distribution in both permanentmagnet direct-current (PMDC) and field-winding
(wound-field) direct-current (FWDC) motors with
the same output mechanical power, have been
studied. An approximate analytical method and FEM
are used for prediction of airgap flux density
distribution. No-load and rotor-lucked conditions,
according to experimental measurements, and the
FEM and analytical method studies of the motor,
have been studied. A sensitivity analysis has also
been done on the major design parameters that affect
motor performance. At last, these two DC motors are
compared, in spite of their differences, on the basis of
measured output characteristics.
Boules developed a two-dimensional field
analysis technique by which the magnet and armature
fields of a surface–mounted brushless synchronous
machine can be predicted [1,2]. In a comprehensive
proposed model, Zhu et al. presented an analytical
solution for predicting the resultant instantaneous
magnetic field in the radially-magnetized BDCM and
PMDC under any load conditions and commutation
strategies [3]. In this research, first, the magnetic flux
density of both a six-pole, 29-slot PMDC motor and
a series-exited four-pole, 21-slot FWDC motor are
obtained based on an iterative analytical method.
Fig. 1-a
Fig. 1-b
Fig. 1-c
Fig. 1. Frame of ideal DC motor and applied
current densities to the prototype motor
Fig. 3 shows results of this method for FWDC
motor magnetization curve. Results of design
sensitivity analysis are also evident on this figure for
changing the number of turn of armature windings
[1-3].
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140
The no-load flux distribution is also shown in Fig.
7 in details with magnifications. The direction and
magnitude of the flux density distribution vector is
also included in the Fig. 7.
Fig. 2. Magnetic equivalent circuit of the motors
(b)
(a)
Fig. 7. (a) Flux lines, and (b) colored vector
plot of vector field: no-load conditions; FWDC
motor
Fig. 3. Flux per pole of series-exited DC motor
vs. line current
2.2 Finite Element Method
Figs. 8 and 9 show the flux distribution and radial
and tangential components of flux density curve at
no-load without and with the magnetic holder, for the
PMDC motor. Finally, Fig. 10 shows the flux
distribution at full-load with the magnetic holder
[6,7].
Figs. 4 and 5 show the flux distribution and radial
and tangential components of flux density curves at
no-load and rated load, respectively.
(a)
(b)
Fig. 8. (a), Frame of the prototype PMDC motor
and rotor windings, (b), Equipotential lines for
magnetic vector potential: no-load conditions;
PMDC motor
Fig. 4. Flux lines: no-load conditions; FWDC
motor
Fig. 5. Flux lines: rated-load conditions; FWDC
motor
Fig. 6 shows the equi-potential lines for magnetic
vector potential and their color map by considering
the effect of screw and bolt on the stator frame, for
the FWDC motor.
(a)
(b)
Fig. 6. (a) Flux lines and (b) colored contour
plot, effect of stator frame screw: no-load
conditions; FWDC motor
Fig. 9. (a) Equipotential lines for magnetic vector
potential: no-load conditions, PMDC motor,
magnetic holder considered.
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solutions and also winding distribution in the stator
slots.
Table 1. Comparison of FEM and analytical method
for the FWDC motor
FEM
Analytical
Analysis
Method
No-load
ω m [RPM]
5828.84
5263
Te [Nm]
0.033
0.030
96.57e-5
100.2e-5
0.7
0.7025
1.5
1.4
ω m [RPM]
1985.5
1966.1
Te [Nm]
0.3368
0.33065
105.84e-5
103.88e-5
1.064
1.0421
1.622
1.5
ω m [RPM]
1747.9
1768
Te [Nm]
0.533
0.529396
111.648e-5
110.88e-5
1.13
1.1722
1.727
1.6212
ω m [RPM]
Te [Nm]
0
0
1.2644
1.28
3 Brushless Permanent Magnet Motors
Φ P [Wb]
132.432e-5
35.0e-5
In a comprehensive proposed model, Zhu et al.
presented an analytical solution for predicting the
resultant instantaneous magnetic field in the radiallymagnetized BDCM and PMDC under any load
condition and commutation strategy [3].
Recently, Zhu et al.[17] extended Rasmussen’s
model[9] to predict magnetic field due to the
armature
reaction both in the three phase
overlapping and non-overlapping stator windings.
Finally, open–circuit field distribution and load
condition field distribution can be expressed as
relative permeance functions including the field
Bmid − aigap
1.596
1.733
2.08
2.12
Fig. 10. (a) Equipotential lines for magnetic vector
potential: full-load conditions, PMDC motor,
magnetic holder considered.
Φ P [Wb]
Bmid − aigap
Table 1 includes the results of comparison of the
FEM analysis of the FWDC motor and the twodimensional field distribution analytical method. In
this analysis, the average value of the rotor angular
speed, rotor output shaft torque, flux per pole, midairgap flux density distribution waveform and yoke
flux density distribution waveform are considered.
Magnetic holders are devised to prevent the
demagnetization of the PMs during the influence of a
strong armature reaction field on the stator field. The
FEM results have completely validated the operation
of the holders during different load conditions.
These two prototype direct-current motors, i.e., a
PMDC and a FWDC are compared from the point of
view of output characteristics. Results of an
approximate analytical method, a previouslydeveloped two-dimensional field analysis technique
and finite element method are compared for
prediction of airgap flux density distribution and
possible replacement of the FWDC motor with
PMDC motor is briefly studied. The results of a few
experimental measurements are also involved in this
study and the results of all methods are also
compared.
[T]
BYoke [T]
I a = 100 A
Φ P [Wb]
Bmid − aigap
[T]
BYoke [T]
I a = 150 A
Φ P [Wb]
Bmid − aigap
[T]
BYoke [T]
I a = 300 A
BYoke [T]
[T]
Cross section of the 5 HP, 1500 rpm, 4-pole
surface-mounted brushless permanent magnet motor
[10], is shown in Figure 11. Figure 12 shows
comparison of flux lines in radial magnetization
configuration. Figure 13 shows comparison of
analytical and numerical results. The results shows
that the flux density curves of that new improved
model follows the FE curves more closely, especially
at the corners of the stator slots. As a result, the
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analytical model is a powerful tool for torque
pulsation calculations, using magnetic flux density
distribution in the airgap. Also average torque can be
obtained using useful flux per pole and unit
length of stator. The results show that the flux
density curves of improved model follow the FE
curves more closely, especially at the corners of the
stator slots.
Fig. 11. Cross section of the machine
142
the Magnetic Equivalent Circuit(MEC) model of
machine, Fig. 14, which takes into account the non–
linear characteristics of the iron in the machine and
the FEM method is also accomplished and carried
out.
Fig. 14. Magnetic Equivalent Circuit model
(Non-Linear elements are in black)
Fig. 12. Equipotential lines: flux distribution (opencircuit condition) at time t=0 (Radial Magnetization)
The fundamental component of flux density
distribution is obtained using MEC analysis
(B1:MEC), the two-dimensional Cartesian-based
coordinates
method(B3:2DR)[8],
the
twodimensional polar-based coordinates method with
stator slot effects (B4:2DS) [9], and FEM analysis
(B5:FEM), are compared in Table 2. The figures
show the peak of fundamental component of flux
density distribution considering both the parallel and
radial magnetization. The good agreement of the
analytical method with FE results has proved validity
of this method for fast calculation of back-EMF and
torque ripples.
Table 2. Comparison of different field analysis
methods
Parallel
B1(MEC)
-------B3(2DP)
0.8262
B4(2DS)
0.8138
B5(FEM)
0.7818
The flux densities are in Tesla.
Fig. 13. Flux
magnetization
density
distribution:
radial
A comparison between lumped–parameter and
distributed-parameter flux calculations , i.e., using
Radial
0.8200
0.8385
0.8260
0.7953
4
Interior
Permanent
Synchronous Motor
Magnet
Kim et al. [18,19], in a Recent comprehensive
proposed method, have presented a shape design
optimization method to reduce cogging torque of an
IPM synchronous motor using the continuum
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sensitivity analysis combined with FEM. They used
the finite element nodes on the rotor outer surface as
control points and used the B-spline curves to relate
design variables vector and control points vector to
each other. There is only a slight difference between
initial shape and final shape of rotor outer surface.
They also presented an optimal shape design method
for reducing the higher back-EMF harmonics
generated in IPM synchronous motor; however, they
did not evaluate the optimal design at different load
conditions.
The proposed optimal design method in this
part is industrially applicable to the rotor and motor
drive operation is considered too. The optimal shape
is evaluated at different load conditions which shows
good improvement in all loading conditions.
Electrical circuits’ equations of different stator
windings and rotor mechanical motion equations are
coupled to magnetic field equations, to obtain a
comprehensive model for the IPM motor drive.
The optimal shape design is obtained based on
addition of three circled-type holes that are drilled in
the rotor iron. Motor-drive operation is also
discussed.
4.1 Steady-State Operating Curves of IPM
Synchronous Motors
The main torque control strategies for the speeds
lower than base speed operation are zero d-axis
current, maximum torque per unit current (MTPC),
maximum efficiency, unity power factor and constant
mutual flux linkage. The main control strategies for
the speeds higher than base speed operation are
constant back-EMF and six-step voltage. The MTPC
control strategy provides maximum torque for a
given current. This minimizes copper losses for a
given torque; however, it does not optimize the total
losses. Maximum torque per current curve is shown
by curve 6 in Fig.15. The operating point, A, shown
in this figure is for below base speed. The path
ABCD, shows the above speed operation. Shape
design optimization of the motor-drive has been done
in these five operating points, considering field
weakening on path ABCD [15]. These are shown in
Fig. 15 by curves 8 to 10. According to motor-drive
limits given by these curves, two basic operating
conditions can be identified: infinite-speed operation
and finite-speed operation. Below and above the
rated speed, there are three control modes, current
limited, current-voltage and voltage limited regions.
The motor-drive here is finite maximum speed drive
because the infinite speed operating point lies outside
the current-limited circle.
143
Fig. 15. Operating limits for prototype IPM
synchronous motor: family of maximum torque-perampere curves, constant torque curves, rated stator
current curve and voltage-limit ellipses
4.2 Modeling of Motor-Drive System
4.2.1 Magnetic Field Model
Rotor mesh and stator mesh are coupled together
at a slip interface to allow for rotation which is a
cylindrical slip surface in 3D and a circular slip path
in two-dimension in the middle of the air gap. So,
there is a weak boundary condition enforced on this
interface. Using Lagrange multiplier and Kth rotor
variable, Ark as magnetic vector potential at node k on
rotor, rotor can be coupled to the corresponding
stator variables Asi, as:
4
Ark = ∑ Asi N si (k )
(1)
i =1
The local virtual work method is used for torque
calculation in FEM method. In the rotor PM, the
magnetic field equation can be expressed as:
∇ ×ν∇ × A = ν∇ × M
(2)
In the stator conductors, the magnetic field
equation is given by:
∇ ×ν∇ × A = − J s + σ
∂A
∂t
(3)
And in the rotor and stator iron regions is
expressed as:
∇ × ν∇ × A = σ
∂A
∂t
(4)
4.2.2 The Electrical Circuit Model [20-22]
The voltage equations of the phase stator
windings are given by:
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[Vab ] = [
[Vbc ] = [
published by WSEAS Press
d
(λ a − λb )] + [ Ra ]I a − [ Rb ]I b
dt
(5)
d
(λb − λ c )] + [ Rb ]I b − [ Rc ]I c
dt
(6)
4.2.3 Mechanical System Model
Two equations of the rotation of rotor are given as
follows,
J eq (
dω r
) + Bω r = Te − TL
dt
(7)
dδ
= ωe − ωr
dt
(8)
144
results (contour plots of the resultant absolute value
of flux density distribution) of the time stepping
FEM for this IPM synchronous motor.
4.3 Shape Optimization Method
To reduce the pulsation torque, which consists of
cogging torque and torque ripples, it is necessary to
redistribute the flux in rotor. For flux pattern
optimization, it needs to change the iron and air
combination in a practical approach. In this research,
small holes have been drilled in the flux path at rotor
surface (Fig. 18). The place and radius of these holes
are found to minimize the torque pulsations [22-40].
The above-mentioned equations are coupled in the
two-dimensional circuit-field-torque coupled time
stepping finite element method. This model is used to
optimize the shape of the rotor of motor as will be
discussed in the following section. This analysis is
done by the algorithm shown in Fig.16.
Start
Calculation of static magnetic
field and initial mechanical angle
of the rotor position
Coupling of magnetic field equations
and electric circuit equations
Fig. 18. The model used to optimize the IPM
synchronous motor
Calculation of electromagnetic field
The vector of design variables is indicated by:
Calculation of flux linkages, three phase
Currents and instantaneous torque
t = t + ∆t
Solving the two motion equations, i.e.,
rotation of rotor equations(speed and
torque angle equations)
The rotation of rotor: Moving mesh,
modification of element coordinate
systems on each permanent magnet
X = [ ρ1 ρ 2 ρ 3 r1 r2 r3 θ1 θ 2 θ 3 ]T
(9)
Χ = [ X 1 X 2 X 3 ... X n ]T .
(10)
Where parameters ρ i , ri , θ i are shown in Fig.18.
The design variables are subject to constraint with
upper and lower limits, that is:
X i ≤ Xi ≤ X i
i = 1,2,3,..., n, n = 9
Write related new constraint equations according
to new node positions corresponding to mid
airgap length
(11)
g i ( Χ) ≤ g i i = 1,2,..., m1
No
t > tmax
hi ≤ hi ( Χ) i = 1,2,..., m2
Yes
wi ≤ wi ( Χ) ≤ wi i = 1,2,..., m3
Stop
Fig. 16. Transient Finite Element Method
At last, convergence of the procedure is checked
by the velocity, magnetic vector potential and backEMF errors. Fig. 17 (at the end of chapter) shows the
(12)
A first order optimization method is used and this
constrained problem is translated into an
unconstrained one using penalty functions. Each
iteration is composed of sub-iterations that include
search direction and gradient (i.e., derivatives)
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computations. The unconstrained objective function
is formulated as follows:
⎛ f ⎞ ⎛ n
⎞
Q ( Χ, q ) = ⎜⎜ ⎟⎟ + ⎜ ∑ ΡX ( X i ) ⎟ +
⎠
⎝ f 0 ⎠ ⎝ i =1
(13)
m1
m2
m3
⎛
⎞
q * ⎜ ∑ Ρg ( g i ) + ∑ Ρh (hi ) + ∑ Ρw ( wi ) ⎟
i =1
i =1
⎝ i =1
⎠
And for each optimization iteration ( j ) , a search
direction vector, d , is calculated. The next iteration
( j + 1) is obtained from the following equation
[18,19]:
Χ ( j +1) = Χ ( j ) + s j d ( j )
(14)
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Fig. 19. Diagram for applying desired currents of the
IPM synchronous motor [12]
S j is the line search step size and d ( j ) is given
by:
d ( j ) = −∇Q ( Χ ( j ) , qk ) + rj −1d ( j −1)
Where,
rj −1 =
[∇Q(Χ
( j)
(15)
]
T
, qk ) − ∇Q( Χ ( j −1) , qk ) ∇Q( Χ ( j ) , qk )
∇Q( Χ ( j −1) , qk )
2
.
(16)
In this paper, the objective function is defined as:
Ψ=
Tmax − Tmin
Tavg
(17)
Fig. 20. Cross section of the IPM synchronous motor
based on different phase mmf axes
4.4 Motor-Drive Control Strategy
The d-q components of the reference currents that
were calculated according to the method described in
section 4-1, are used for estimation of phase currents.
So the reference currents, i *A (t ) , i B* (t ) and iC* (t )
will be applied to the FEM model, according to Figs.
19,20. For example, i *A (t ) can be estimated as:
i *A (t ) =
(
2 *
− id cos(θ ) + iq* sin(θ )
3
)
(18)
Fig.21 shows equi-potential lines for the magnetic
vector potential solutions at a time obtained by time
stepped FEM. Fig. 22 presents different wave forms
of motor torque, corresponding to different rotor
shapes, for the operational point A. Fig. 23 shows the
spectrum of curves shown in Fig. 22. In this Fig.,
curve 1 is the motor torque and the rotor has no
holes, curve 2 presents the output torque of motor
when three equal-area circles are created on the rotor
and curve 3 shows the rotor torque when three
optimized circles are drilled into the rotor.
Fig. 21. Equipotential lines: flux distribution (full
load condition) at time t=0.556 S, after creating
optimal circular holes
It can be seen that there is a valuable
improvement both in performance index, torque
pulsations and saliency ratio. This new shape of the
rotor with optimized holes, has the advantage of
increasing the maximum speed for the field
weakening region from almost 1.83 p.u. to 1.96 p.u.
above the base speed. Optimal shape design of rotor
has large effect on reduction of torque pulsations of
IPM synchronous motor.
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Fig. 22. Comparison of electromagnetic torque
calculated by FEM for different rotor structures in
point A, curve 1: no holes on the rotor, curve 2:
same holes on the rotor and curve 3: optimized holes
on the rotor
146
In this part, the shape design optimization is
carried out by drilling internal circular holes of
optimal radius in the flux path at rotor surface. The
torque curves of the optimized motor show lower
pulsating torque and higher average torque. Another
advantage is that the field weakening region has been
extended for optimized motor. Although the shape
optimization is done at nominal operation point, the
performance evaluation of optimized motor at other
operation conditions shows improvement too. The
new shapes are easily applicable in the factory by
drilling the holes of different radius at predetermined
positions.
5
Interior
Permanent-Magnet
Synchronous-Induction Motor
In this part, a synchronous-induction motor has
been considered and a rotor cage for self starting of
the IPM synchronous motor is included in the rotor.
Fig. 24 shows this new topology. Time stepping
FEM is used to simulate this new machine. Fig. 25
shows the equipotential lines for this motor. Figs. 26
and 27 show the results obtained for the motor at
startup from standing using a three-phase 50Hz
voltage source under 15 N.m. load torque obtained
by time stepped finite element method and d-q model
respectively. In these waveforms it is evident that the
results of two dimensional filed modeling, i.e., time
stepping FEM has contained the rotor and stator slots
and rotor saliency affected by permanent magnet
shapes. Using a cage on the rotor of a permanent
magnet motor has this advantage that the motor can
be started directly as an induction motor and it needs
not to have a frequency control process for starting
[23-24].
Fig. 23. Spectrum of three curves shown in Fig.22,
curve 1: no holes on the rotor, curve 2: same holes on
the rotor and curve 3: optimized holes on the rotor
Fig. 24. Interior permanent-magnet synchronousinduction motor
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147
At last, a new topology, i.e., Interior PermanentMagnet Synchronous-Induction Motor has also been
completely studied. Among these different rotor
topologies for these PM machines, it can be seen that
the last one is the best from the point of view of
efficiency and pulsating torque.
Acknowledgment
Fig. 25.
Equipotential lines for magnetic vector
potential: full-load conditions, interior permanentmagnet synchronous induction motor
The authors would like to really appreciate decent
considerations of all people who engaged on different
parts of this work. Unless they had accompanied and
had helped us, the work could not have been finished
and finalized. At last, we would say that the work
presents the results of almost 8 years of
interpretations and main conclusions of different
research and educational projects in the field of FEM
analysis of PM machines.
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[3] Z. Q. Zhu, D. Howe, "Instantaneous magnetic
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[4] J. J. Cathey, Electric machines, analysis and
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[5] M. Cheng, et al., "Nonlinear varying-network
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[6] US Steel Manual, Non-Oriented Electrical
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[7] ANSYS 5.6 Theory and APDL Reference
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[8] W. Cai, D. Fulton and K. Reichert, "Design of
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[1]
Fig. 26. Electromagnetic torque obtained by time
stepping finite element method
Fig. 27.
Electromagnetic torque obtained by
equivalent d- and q-axes model of this synchrousinduction machine.
6 Conclusion
In this research, different DC and AC permanentmagnet motors have been analyzed via both
analytical and numerical methods. The time stepping
Finite Element Method (FEM) has been used as a
numerical method. The PM motors that are
considered in this research include PMDC motor,
brushless AC and DC permanent magnet motors and
interior type permanent magnet synchronous motor.
10
FINITE ELEMENTS
published by WSEAS Press
[10] T. L. Skvarenina, The Power Electronics
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design", IEEE Transactions on Magnetics, Vol.
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A. Kiyoumarsi, M. Hassanzadeh and M.
Moallem, "A new analytical method on field
calculation for interior permanent-magnet
synchronous motors", International Journal of
Scientia Iranica, Vol. 13, No. 4, pp. 364-372,
October 2006.
B. Mirzaeian-Dehkordi, A. Kiyoumarsi, P.
Moallem and M. Moallem, "Optimal design of
switching-circuit parameters for switched
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algorithms", Electromotion, Vol. 13, No. 3,
pp.213-220, July - September 2006.
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permanent-magnet
synchronous
motors", Electromotion, Number 1, pp.3-8,
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[34]
[35]
[36]
[37]
[38]
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published by WSEAS Press
permanent-magnet motors", presented in the
12th Symposium of Power Electronics–Ee
2003, Appeared
in CD Conference
Proceedings, Paper No. T3-1.3, Novasid,
Serbia & Montenegro, November 5-7, 2003.
A. Kiyoumarsi, M. Hassanzadeh and M.
Moallem, "Eccentric magnetic field analysis of
electric machines", presented in ACEMP’2004,
International Aegean Conference on Electrical
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26-28, Istanbul, Turkey, May 2004.
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Hassanzadeh, "Calculation of magnetizing
inductances in interior permanent-magnet
synchronous motors", presented in OPTIM‘04
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Electronic Equipment, pp. 145-148, Poiana
Brasov, Romania, May 20-23, 2004.
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analytical methods on field calculation for
interior
permanent-magnet
synchronous
motors", presented in ICEM 2004, Appeared in
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International
Conference
on
Electrical
Machines, Poland, September 2004.
M. Hassanzadeh, A. Kiyoumarsi and M.
Moallem, "Accurate methods for calculation of
magnetizing inductances in interior permanentmagnet synchronous motors," presented in
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Conference, San Antonio, TX, May 15-18,
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Presented in Electromotion 2005,
6th
International Symposium on Advanced
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in CD Conference Proceedings, Paper No.
OS1/5, , Lausanne, Switzerland, September 2729, 2005.
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LIST OF SYMBOLS
Bocb
Flux density due to magnet at stator inside
surface
Open-circuit flux density; Boules’ method
Boco
Open-circuit flux density; Only assuming λ0
Bocn
Bsum
b0
P
Qs
Open-circuit flux density; New method
Radial component of flux density; obtained
by FEM analysis
Absolute resultant open-circuit flux
Stator slot-opening
Number of poles
Stator slot number
rg
Mid-airgap radius
wS
Equivalent stator slot-opening
wT
Stator tooth width
α
Angular displacement between the stator
mmf and rotor mmf
Magnet arc angle
Bmagnet
Br
αm
αp
Pole-shoe arc angle
α ′p
Pole arc angle
α sa
λ
Angular displacement between the stator slot
axis and the axis of the coils of phase A
Relative permeance function
Λ ref
Reference permeance
~
12
~
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150
APPENDIX I
APPENDIX II
PM and field-winding DC motor parameters
Brushless PM motor parameters
Stator outer radius (rso)
Stator yoke depth (hy)
Stator tooth depth (ht)
Stator slot opening (bo)
Stator Lamination material
Rotor Lamination material
Magnet arc angle ( 2α m )
97.1 mm
17.4 mm
17.2 mm
5.1 mm
M36,26 Gage
M19,26 Gage
Magnet material
Remanence (Bres)
Relative recoil permeability (µr)
Nd-Fe-B, N33
1.1 Tesla
1.05
PMDC Motor Data
Stator
Dsi = 59.0mm , Dso = 79.0mm ,
hm = 7mm , α m = 35.19°mech ,
Br = 0.45T , H c = 300 KA m , P=6
Rotor
Lr = 60mm , Dri = 38.6mm ,
Dro = 58.5mm , ω slot = 2.7mm ,
Dshaft = 16.6mm , N a = 2.turns
o
60 mechanical
APPENDIX III
IPM synchronous motor parameters
Field-Winding DC Motor Data:
Stator
h p = 8.6mm , Dso = 89.5mm , Dsi = 78.2mm
α p = 68°mech , α ′p = 37°mech , N f = 8 .5,
P=4, Acond . = 4.5 * 1. mm 2
Stator outer radius (rso)
Stator yoke depth (hy)
Stator tooth depth (ht)
Stator slot opening (bo)
Magnet arc angle ( 2α m )
Stator lamination material
Rotor lamination material
Permanent-magnet material
Nd-Fe-B, Remanence (Bres)
97.1 mm
17.4 mm
17.2 mm
5.1 mm
2×37
mechanical degrees
M36, 26 Gage
M19, 26 Gage
N33
1.1 T
Rotor
Lr = 48mm , Dri = 60.5mm , Dro = 39.0mm
Dshaft = 17.45mm , ω slot = 2.7mm ,
ω brush = 5 mm , lbrush = 10. mm ,
N a = 2turns , Dcond . = 2.1mm
Stator lamination material
Rotor lamination material
M36,26 Gage
M19,26 Gage
APPEND IV
Back EMF calculations by FEM
The instantaneous flux Φ trough a coil of the
winding is given by integrating the flux density over
the coil area, which is bounded by the contour C.
Φ(t ) = ∫ B(r ,θ , t ) ⋅ ds = ∫ (∇ × A) ⋅ ds = ∫ A ⋅ dl (App.IV-1)
S
S
C
Furthermore the instantaneous flux linkage of the
winding is found as:
λ wind (t ) =
P 1
∗ ∗ N s ∗ Φ (t )
2 c
(App.IV-2)
And the instantaneous value of the induced noload voltage is given by:
e(t ) =
13
d
λ wind (t )
dt
(App.IV-3)
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t=1.0 ms
t=2.0 ms
t=3.0 ms
t=4.0 ms
t=5.0 ms
t=6.0 ms
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t=7.0 ms
t=8.0 ms
t=9.0 ms
t=10.0 ms
t=11.0 ms
t=12.0 ms
152
Fig. 17. Magnetic vector potential produced by means of time stepped finite element method
15