Space Vector Modulated Direct Torque Control of PM
Synchronous Motor with Initial Rotor Estimation
Ying Yan, Jianguo Zhu, and Youguang Guo
Faculty of Engineering, University of Technology, Sydney, Australia
Email: yingyan@eng.uts.edu.au
ABSTRACT
The direct torque controlled (DTC) permanent magnet
synchronous motor (PMSM) has become attractive
because of its simplicity and fast torque response.
However, mechanical sensors have caused problems to
the system. It is possible to overcome the disadvantage
if the initial rotor position is known. For the initial rotor
position detection, the structural and magnetic
saturation saliencies are frequently used. Since the
conventional PMSM model does not incorporate the
saturation saliency, this paper presents a nonlinear
model of PMSMs which incorporates both the structural
and saturation saliencies, which will enable the
numerical simulation of new rotor position detection
algorithms.
In this model, the self and mutual
incremental inductances of the phase windings are
expressed as functions of the stator currents and rotor
position. The incremental inductances of a surface
mounted PMSM are measured.
In addition to
mechanical sensors, there are other problems of DTC,
such as large torque and flux ripples. In order to
overcome these disadvantages, space vector modulated
(SVM) DTC has been applied. Based on the proposed
motor model, the dynamic performance of a SVM DTC
drive system is simulated. To verify the effectiveness of
the proposed model, the simulation results are compared
with those obtained by the conventional PMSM model in
the Simulink power system toolbox library. Finally, an
initial rotor position estimation scheme is investigated.
Experiments based on the scheme are carried out and
the effectiveness of the scheme is verified.
1. INTRODUCTION
The direct torque controlled (DTC) permanent magnet
synchronous motor (PMSM) drive system has become
competitive compared with other types of drive system,
because of its simplicity and sensorless control
algorithm. However, the practical application of the
DTC system is handicapped by the difficulty of starting
under full load if the initial rotor position is unknown.
Another disadvantage of DTC is the large torque and
flux ripples in the steady state. In order to overcome
these disadvantages, many efforts have been made on the
initial rotor position detection and torque and flux ripple
minimization. On one hand, the structural and magnetic
saturation saliencies, which exist even in the surface
mounted PMSM, have been used for initial rotor position
estimation. On the other hand, space vector modulation
(SVM) has been applied to achieve a fixed switching
frequency and low torque and flux ripples [1-2].
Since the conventional PMSM model does not
incorporate the saturation saliency, when developing a
new method for the initial rotor position detection, it is
not possible to numerically simulate the method. This
paper presents a nonlinear model of PMSMs which
incorporates both the structural and saturation saliencies.
In this model, the self and mutual incremental
inductances of the three phase windings are expressed in
Fourier series as functions of the stator currents and rotor
position.
The experimental measurement of the
incremental inductances is outlined. Based on the
proposed model, the SVM DTC scheme is simulated.
To verify the effectiveness of the new model, the
simulation results are compared with those obtained by
the conventional PMSM model in the Simulink power
system toolbox library. In order to start the DTC drive
with the full load, an initial rotor position estimation
scheme to inject high voltage pulse is analysed. An
experiment based on the scheme is carried out, and the
effectiveness of the scheme is verified.
2. MODELLING
SALIENCIES
OF
PMSM
CONSIDERING
In a PMSM, the field produced by the rotor magnets is
the dominant component which determines the operating
point of the magnetic core. Although the characteristic
of the magnetic core is nonlinear, the magnetic circuit
can be considered as piecewisely linearized around the
operating point P at a given rotor position. Thus, the
total flux linkage of phase a, ψa, can be separated into
two components [3]:
ψ a = ψ as ( ims ,θ ) + ψ af (θ )
(1)
where θ is the rotor position, and ψas and ψaf are the flux
linkage components produced by the magnetization
component of the stator current ims, and rotor magnets,
respectively. The flux linkage ψas can be further
separated into components excited by individual stator
phase currents, ψaa, ψab, and ψac.
ψ a = ψ aa + ψ ab + ψ ac + ψ af
(2)
Under the assumption of linearization, the flux linkage
components can be considered as proportional to the
corresponding currents: ψaa=Laa(ia, θ)ia, ψab=Lab(ib,
θ)ib, and ψac=Lac(ic, θ)ic.
The proportionality coefficients Laa, Lab, and Lac, which
are determined by the gradient of the magnetization
curve at the operating point, are defined as the self and
mutual incremental inductances of the phase a winding.
The flux linkages of phases b and c can be obtained
similarly. The voltage equations of the three-phase
stator windings can be written as:
L ( i,θ ) = a0 ( i ) + ∑ ( am ( i ) cos mθ + bm ( i ) sin mθ ) (5)
⎡ia ⎤ ⎡ eaf + eaθ ⎤
⎥
⎢
⎥
⎢i ⎥ + ⎢ e + e ⎥
d
0 ib + [ LT ]
bθ ⎥
dt ⎢ b ⎥ ⎢ bf
⎥⎢ ⎥
⎢⎣ ic ⎥⎦ ⎢⎣ ecf + ecθ ⎥⎦
Rc ⎥⎦ ⎢⎣ ic ⎥⎦
For a group of measured phase inductances, L(ik,θj), the
coefficients of the corresponding Fourier series can be
obtained by nonlinear curve fitting, where k and j refer to
the various experimental currents and rotor positions,
respectively. The nonlinear model of the self and mutual
inductances can be readily incorporated into the PMSM
model presented in section 2.
⎡ va ⎤ ⎡ Ra
⎢v ⎥ = ⎢ 0
⎢ b⎥ ⎢
⎢⎣ vc ⎥⎦ ⎢⎣ 0
0
Rb
0
0 ⎤ ⎡ia ⎤
(3)
where
⎡
∂Laa
⎢ Laa + ia
∂ia
⎢
⎢
∂L
[ LT ] = ⎢ Lab + ia ab
∂ia
⎢
⎢
∂L
⎢ Lac + ia ac
∂ia
⎣
Lab + ib
∂Lab
∂ib
Lbb + ib
∂Lbb
∂ib
Lbc + ib
∂Lbc
∂ib
∂Lac ⎤
⎥
∂ic ⎥
∂L ⎥
Lbc + ic bc ⎥
∂ic ⎥
∂L ⎥
Lcc + ic cc ⎥
∂ic ⎦
Lac + ic
where ωr=dθ/dt is the mechanical angular speed of the
rotor, va, vb, and vc are the terminal voltages, Ra, Rb, and
Rc the winding resistances, ia, ib, and ic the currents of
phases a, b, and c, eaf=ωrdψaf/dθ, ebf=ωrdψbf/dθ, and
ecf=ωrdψcf/dθ the electromotive forces (emfs) induced by
the rotor magnets, and eaθ=ωr(ia∂Laa/∂θ+ib∂Lab/∂θ
+ic∂Lac/∂θ), ebθ=ωr(ia∂Lab/∂θ +ib∂Lbb/∂θ+ic∂Lbc/∂θ), and
ecθ= ωr(ia∂Lac/∂θ+ ib∂Lbc/∂θ+ ic∂Lcc/∂θ) the three phase
emfs induced by the variation of flux linkage due to the
saliencies, respectively.
The electromagnetic torque of the PMSM can be
obtained by taking the partial derivative of the system
co-energy, Wf′, with respect to the rotor position angle θ,
i.e. T=∂Wf′/∂θ, and it can be derived as:
T=
∂ψ af
∂ψ bf
∂ψ cf
∂Laa i ∂L i ∂L i
+
+
∂θ
∂θ
∂θ
∂θ 2 ∂θ 2 ∂θ 2
2∂Lab
2∂Lac
2∂Lbc
(4)
+
iaib +
iaic +
ibic
∂θ
∂θ
∂θ
ia +
ib +
ic +
2
a
2
bb b
2
cc c
The electromagnetic torque has two components: one
produced by the stator currents and rotor magnets, and
the other by the saliencies.
3.
NONLINEAR INCREMENTAL INDUCTANCE
3.1.
ANALYTICAL EXPRESSION
INDUCTANCE
OF
NONLINEAR
As discussed above, the self and mutual inductances of
the stator windings with structural and magnetic
saturation saliencies are nonlinear functions of the stator
currents and rotor position, which require a large amount
of data to describe numerically. The incorporation of the
analytical expressions of these inductances could
simplify the implementation of the numerical simulation
model and reduce significantly the amount of
parameters.
The relationship between the phase inductance and the
rotor position can be expressed by a Fourier series [4].
The different sets of coefficients of the Fourier series, a0,
am, and bm (m=1, 2, …, n), can be obtained with different
currents, and expressed as functions of corresponding
currents.
n
m =1
3.2.
EXPERIMENTAL MEASUREMENT
INDUCTANCES
OF
PHASE
In its normal operating state, the total flux linkage of a
PMSM is composed of the flux produced by the rotor
PMs and stator currents, ψt= ψr+ψs, and hence the phase
inductances are related to both of them. In order to
estimate the saturation effect at various stator fluxes, DC
offsets, idc, are used to emulate the effect of the threephase currents. A method for measuring the phase
inductances is designed to express the relationship
between the inductance variation and the saturation of
magnetic core.
The experiment is carried out on a 6-pole surface
mounted PMSM, and the rotor of the motor is locked by
a dividing head. Various DC offsets are injected to one
of the stator windings. Meanwhile, a small AC current
is applied to one phase while the other two are opencircuited. With this method, the self and mutual
inductances of phases a, b and c at different rotor
positions and stator currents can then be obtained.
It can be concluded that the inductances are the functions
of amplitudes and angles of both stator and rotor fluxes,
L(θs, θr,│Φs│,│Φr│), where θs is the position of the
stator rotating flux, θr the position of rotor flux, and
│Φs│ and │Φr│ are the amplitudes of the stator and
rotor fluxes, respectively. Since the amplitude of the
rotor flux produced by the permanent magnets can be
considered as a constant, the inductances can be
simplified as functions of θs, θr, and │Φs│, and rewritten
as L(θs, θr,│Φs│).
In order to simplify the inductance expressions, the angle
between the stator flux, Φs, and the stator winding axis
of phase a, denoted by α, and the angle between the
stator and rotor fluxes, denoted by δ, as shown in Figure
1, are used in the inductance model instead of θs and θr.
The phase a inductance, Lsa(α, δ, ims), at any stator and
rotor fluxes (α=0º~360º, δ=0º~360º, and ims=0~Irated)
can be estimated by linear interpolation of the
inductance expression measured by injecting various DC
currents to phase a at α=0º, α=120º, and α=240º with
δ=0º~360º, and ims=0~Irated. Due to the symmetry of the
three phase windings, the inductances of phases b and c
at any rotor position and stator current can be estimated
based on the phase a inductance expression by shifting
the rotor position by ±120º, respectively. A similar
procedure can be applied to estimate the mutual
inductances. Therefore, the structural and saturation
saliencies of the motor magnetic filed could be mapped
out in terms of the three phase self and mutual
inductances. Figures 2 and 3 illustrate Lsa and Lmab, at
different rotor positions with a DC offset of 5.5A applied
to different α.
B
B
Z
electromagnetic torque, ψs and Tem, are obtained as
follows:
ψ α ( tn+1 ) = ( uα ( tn ) − Rsiα ( tn ) ) Ts + ψ α ( tn )
Z
Phase a axis
Phase a axis
Φr
Φs
N
α
δ
A
A
X
δ
Φr
N
ψ s ( tn ) = ψ α ( tn ) + ψ β ( tn )
(8)
(a)
2
⎛ ψ β ( tn ) ⎞
⎟⎟
⎝ ψ α ( tn ) ⎠
(9)
ϕ s = arctan ⎜⎜
Y
C
(b)
B
(7)
2
X
Phase c axis
Y
C
ψ β ( tn+1 ) = ( u β ( tn ) − Rs iβ ( tn ) ) Ts + ψ β ( tn )
Φs
Phase b axis
Phase c axis
Phase b axis
S
αb
S
(6)
Tem ( t n ) =
Z
3
P ψ
2 (
i − ψ β iα )
(10)
α β
Phase a axis
where Rs is the stator winding resistance, Ts the sampling
time, ψα and ψβ are the actual stator flux, and ψs and φs
the actual stator flux amplitude and position.
S
A
δ
Phase b axis
X
Φs
αc
N
Phase c axis
Φr
ψsr
Y
C
(c)
Figure 1: (a) Rotor and stator fluxes linking phase
a with a DC offset in phase a; (b) Rotor and stator
fluxes linking phase b with a DC offset in phase a;
(c) Rotor and stator fluxes linking phase c with a
DC offset in phase a.
ωref
PI Controller
Tref
ωe
Terror
Tem
Vr
Predictive
Controller φvr
ψs φs
ψs and Tem
Calculator
udc
Voltage
Modulator
is
udc, Va, Vb, Vc
ia, ib, ic
θ
d/dt
PWM
Inverter
PMSM Model
with Saliencies
0.0104
Figure 4: DTC of PMSM based on space vector
modulation.
0.0102
Lsa (H)
0.01
0.0098
0.0096
0 degree
120 Degree
240 Degree
100 Degree
200 Degree
0.0094
0.0092
0
8
16
24
32
40
48
56
64
72
80
88
96
104
112
120
Rotor Position (Mech. Deg.)
Figure 2: Measured and computed Lsa at different α.
0.004
0.0035
In the SVM-DTC scheme without switching table and
hysteresis controllers, a predictive controller, including a
proportional integral (PI) controller and a voltage vector
calculation block, is used to produce the reference stator
voltage vector. The PI controller generates the load
angle increment required to minimize the instantaneous
error between reference and actual torque. Afterwards,
the load angle increment and reference stator flux
amplitude, ∆δ and ψsr, are input to the voltage vector
calculation block and used to calculate the amplitude and
position of the reference stator voltage vector, vr and φvr,
as the following [1]:
0.003
Lmab (H)
(11)
vr = vsr2 α + vsr2 β
0.0025
⎛ vsr β ⎞
⎟
⎝ vsrα ⎠
0.002
0.0015
120Degree
0.001
(12)
ϕvr = arctan ⎜
0Degree
50Degree
The α- and β-axis components are obtained by:
100Degree
0.0005
0
0
8
16
24
32
40
48
56
64
72
80
88
96
104
112
120
Rotor Position (Mech. Deg.)
Figure 3: Measured and calculated Lmab at different α.
4.
NUMERICAL SIMULATION OF DTC DRIVE
4.1.
SVM DTC ALGORITHM
Figure 4 shows a schematic block diagram of the SVM
DTC, which is based on the two-phase stationary α-β
reference frame. In the simulation system, the stator
currents, ia, ib, and ic are calculated by the proposed
model. The currents are then transformed by the Clark
transformation into components (iα, iβ), and the uα and uβ
are obtained by the combination of udc and the switching
state of the PWM inverter, Va, Vb and Vc in the previous
sampling period. The stator flux linkage and the
vsrα =
vsr β =
ψ sr cos (ϕ s + ∆δ ) −ψ s cos ϕ s
Ts
ψ sr sin (ϕ s + ∆δ ) − ψ s sin ϕ s
Ts
+ Rsiα
(13)
+ Rsiβ
(14)
Under the operating conditions without flux weakening,
the flux amplitude produced by the rotor PMs is referred
to the stator flux reference amplitude, ψsr.
In the scheme, the selection of reference stator voltage
vectors is not based on the actual position of the flux
linkage. It aims to eliminate the error between the actual
and reference flux linkages at the end of each sampling
time. The reference stator voltage vector is obtained by
the available PWM voltage vectors. For example, if the
reference stator voltage vector is between the vectors of
V2 (110) and V3 (010), V2, V3 and zero voltage vectors,
V0 (000) are selected. The method to calculate the time
durations (T1, T2 and T0) corresponding to the voltage
vectors V2, V3 and V0 has been presented in [5].
6
Tem (Nm)
5
SIMULATION OF SVM DTC DRIVE USING
PROPOSED MODEL
4.2.
Based on the proposed model and the parameters
identified by experiments, the dynamic simulation of the
PMSM SVM DTC drive system is conducted. Figures
5-9 show the simulated starting-up performance of the
system corresponding to a speed command of 1000 rpm
and a load torque of 1.0 Nm with the known initial rotor
position, while the parameters of the speed loop PID
controller are carefully tuned for the optimal
performance. The rotor speed, electromagnetic torque,
stator flux trajectory, and stator flux are shown in
Figures 5-8, respectively, and the stator currents are
shown in Figure 9.
4
3
2
1
0
0.0
0.1
0.2
0.3
0.4
0.5
Time (s)
Figure 6: Simulated electromagnetic torque with
SVM DTC using the proposed model.
0.20
0.15
0.10
0.05
0.00
-0.05
-0.10
-0.15
SIMULATION OF SVM DTC DRIVE USING
PRESET SIMULINK MODEL
4.4.
SIMULATION OF CONVENTIONAL DTC DRIVE
USING PROPOSED MODEL
In order to illustrate the effectiveness of the proposed
model and SVM-DTC, the dynamic simulation of
conventional DTC was carried out using the proposed
PMSM model under the same conditions and references.
Figure 14 shows the electromagnetic torque, and Figure
15 the stator flux. It can be seen that the ripple of the
stator flux has been diminished in the SVM-DTC
scheme compared with the conventional DTC based on
the proposed model. More torque and stator flux ripples
appear by the proposed model due to the incorporation
of the inductance variation.
Figure 7: Simulated stator flux trajectory with SVM
DTC using the proposed model.
0.174
0.172
Stator Flux (Wb)
In order to validate the proposed model, the dynamic
simulation was also carried out using the preset PMSM
model in the Simulink power system toolbox library
under the same conditions and references. Figures 10-13
show the rotor speed, electromagnetic torque, stator flux,
and stator currents, respectively. It can be seen that
there is a difference between the electromagnetic
torques, stator currents and stator flux obtained by the
two models. More torque and flux ripples appear in the
results based on the proposed model, as the effect of
inductance variation or the structural and saturation
saliencies has been taken into account.
Flux Trajectory (Wb)
-0.20
-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20
0.168
0.166
0.164
0.160
0.0
0.1
0.2
0.3
0.4
0.5
Time (s)
Figure 8: Simulated stator flux with SVM DTC
using the proposed model.
8
6
4
2
0
-2
-4
Ia
Ib
-6
-8
0.0
0.1
0.2
0.3
0.4
0.5
Time (s)
Figure 9: Simulated stator currents with SVM DTC
using the proposed model.
1200
1000
1000
800
Speed (rpm)
Rotor Speed (rpm)
0.170
0.162
Current (A)
4.3.
800
600
600
400
400
200
200
0
0.0
0.1
0.2
0.3
0.4
0.5
Time (s)
Figure 5: Simulated rotor speed with SVM DTC
using the proposed model.
0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Time (s)
Figure 10: Simulated rotor speed with SVM DTC
using preset Simulink model.
8
7
5.
AN ALGORITHM FOR DETECTING INITIAL
ROTOR POSITION
5.1.
SCHEME FOR INITIAL ROTOR POSITION
ESTIMATION
6
Tem (Nm)
5
4
3
2
1
0
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Time (s)
Figure 11: Simulated electromagnetic torque with
SVM DTC using preset Simulink model.
0.174
0.172
Stator Flux (Wb)
0.170
0.168
0.166
0.164
0.162
0.160
0.00
0.05
0.10
0.15
0.20
0.25
0.30
Time (s)
Figure 12: Simulated stator flux with SVM DTC
using preset Simulink model.
10
Current (A)
5
0
-5
Ia
Ib
-10
0.00
0.05
0.10
0.15
0.20
0.25
Figure 13: Simulated stator currents with SVM
DTC using preset Simulink model.
10
Tem (Nm)
It has been found that the winding inductance is a
function of rotor position and stator currents. With this
feature, the initial rotor position can be estimated by
injecting voltage pulses with high amplitude to the stator
windings [6-7]. At an unknown initial rotor position,
one of the three phase terminals, say phase a, is applied
with a positive pulse while the terminals of the other two
phases are applied with a negative pulse. After a certain
time interval, the positive pulse is followed by a negative
pulse and the negative pulse by a positive pulse to bring
down the currents in the phase windings to zero. For
each phase, the corresponding phase currents are
measured at each locked rotor position and the motor is
mechanically rotated from one position to the next. The
three phase peak currents at different rotor positions
could be obtained.
0.30
Time (s)
The experimental three phase peak currents can be
modelled by an average value, I0, plus some offset
values, ∆I01 and ∆I02, as a function of rotor position, θ,
as the following:
8
I a = I 0 + I 01 cos (θ ) + I 02 cos ( 2θ )
(15)
6
2π ⎞
2π ⎞
⎛
⎛
I b = I 0 + I 01 cos ⎜θ −
⎟ + I 02 cos ⎜ 2θ +
⎟
3 ⎠
3 ⎠
⎝
⎝
2π ⎞
2π ⎞
⎛
⎛
I c = I 0 + I 01 cos ⎜θ +
⎟ + I 02 cos ⎜ 2θ −
⎟
3 ⎠
3 ⎠
⎝
⎝
(16)
4
2
0
0.0
0.1
0.2
0.3
0.4
0.5
Time (s)
Figure 14: Simulated electromagnetic torque with
conventional DTC using the proposed model.
0.174
0.172
Stator Flux (Wb)
In the DTC scheme, it is very difficult to start the motor
under full load because no current or emf information is
available for determining the rotor position before
starting. Therefore, a sensorless method to determine
the initial rotor position is necessary for full load starting
of PMSM drive systems. Many approaches have been
proposed by various researchers to estimate the initial
rotor position. Most of them are effective for the motor
with large structural saliency, e.g. the interior PMSM.
However, in a surface mounted PMSM, the structural
saliency is very small and the rotor position has to be
deduced from the saturation saliency.
0.170
0.168
0.166
0.164
0.162
0.160
0.0
0.1
0.2
0.3
0.4
0.5
Time (s)
Figure 15: Simulated stator flux with conventional
DTC using the proposed model.
(17)
where Ia, Ib and Ic are the peak currents and θ the rotor
position, and I0, ∆I01 and ∆I02 can be obtained by curve
fitting the experimental results.
The peak current curves of phases a, b and c at different
rotor positions are stored in the memory of the
microprocessor in the form of a look up table. When the
rotor is standstill, the gate signals of 100, 010, and 001
are sent to the PWM inverter to apply high voltage
pulses to the three phase stator winding terminals, and
record the peak phase currents. Possible rotor positions
can be readily found by solving inversely (15)-(17) using
a sectional interpolation method. For phase a, the
possible rotor positions corresponding to the peak
current Iap can be obtained as θa1 θa2, θa3 and so on, and
similarly, for phases b and c, the possible rotor positions
corresponding to Ibp and Icp can be obtained as θb1 θb2,
θb3, θc1 θc2, θc3 and so on. Among all these possible rotor
positions, the real rotor position is given by the common
angle obtained of all the three phases.
EXPERIMENTAL RESULTS
5.2.
Peak current (A)
An experiment with the above scheme is carried out. In
the experiment, the high voltage pulse is applied to the
surface mounted PMSM through a PWM inverter, and
the gate signals of the inverters are controlled by a
dSPACE DS1104 system. The phase currents are
measured by current probes and an oscilloscope. Figure
16 plots the three phase peak currents at different rotor
positions. The estimated electrical rotor angle that is
obtained with the above algorithm is plotted as a
function of the actual mechanical rotor angle, as shown
in Figure 17. The actual rotor angle was measured using
an optical encoder attached to the motor shaft and the
dividing head used to fix the rotor. It can be seen that
this algorithm can produce satisfactory results.
stator phase windings, the effect of magnetic saturation
becomes evident.
The proposed model is implemented for simulation of a
drive system with the SVM DTC scheme. The dynamic
performance of the drive system is simulated and the
results are compared with those obtained by the preset
model in the Simulink power system toolbox library.
More torque and flux ripples appear for the proposed
model due to the inductance variation caused by
saliencies. The steady state performance obtained from
the SVM-DTC system has been improved because of the
smaller stator flux ripple compared with that from
conventional DTC system. The scheme for detecting the
initial rotor position by injecting high voltage pulse has
been discussed. A simple numerical algorithm to
estimate the initial rotor position from the peak stator
current variation has been investigated, and the
effectiveness of this algorithm has been verified by the
experimental results.
6.0
REFERENCES
5.5
[1]
D. Swierczynski and M. P. Kazmierkowski,
“Direct Torque Control of Permanent Magnet
Synchronous Motor (PMSM) Using Space
Vector modulation (DTC-SVM)-Simulation
and Experimental Results,” The 28th IEEE
Annual Conf. of the Industrial Electronics
Society, 5-8 Nov. 2002, pp751-755.
[2]
Lixin Tang and M. F. Rahma, “A New Direct
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[3]
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Permanent
Magnet
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5.0
4.5
A
B
C
4.0
3.5
3.0
0
20
40
60
80
100
120
Rotor position (Degree)
Estimated Rotor Electrical Angle (Degree)
Figure 16: Peak current versus rotor position with
high voltage pulse.
360
340
320
300
280
260
240
220
200
180
160
140
120
100
80
60
40
20
0
Theoretical Rotor Angle
Estimated Rotor Angle
0
10
20
30
40
50
60
70
80
90
Actual Rotor Mechanical Angle (Degree)
100
110
120
Figure 17: Estimated rotor electrical angle versus
actual rotor mechanical angle.
6.
CONCLUSIONS
A nonlinear PMSM model incorporating both the
structural and magnetic saturation saliencies has been
presented in this paper. In the model, the saliencies are
reflected by the variation of the stator winding
inductances with respect to the rotor position and stator
phase currents. The incremental inductances of a surface
mounted PMSM are measured, and expressed with the
Fourier series as the functions of rotor position and stator
currents. The small structural saliency in the surface
mounted PMSM is revealed in the self and mutual
inductance profiles versus rotor position without any DC
offset, but when DC offset currents are applied in the