IMPROVED TRAJECTORY CONTROL FOR AN INTERIOR
PERMANENT MAGNET SYNCHRONOUS MOTOR DRIVE WITH
EXTENDED OPERATING LIMIT
M. E. Haque, L. Zhong and M. F. Rahman
School of Electrical Engineering and Telecommunication
The University of New South Wales
Sydney NSW 2052
Australia
E-mail: meh@student.unsw.edu.au
Abstract
This paper presents analysis of trajectory control for an interior permanent magnet (IPM)
synchronous motor drive under PWM current control. An improved trajectory control with
extended speed range has been introduced. The scheme allows the motor to be driven with
maximum-torque-per-ampere characteristic (MTPA) below the base speed and it maintains the
maximum voltage limit of the motor under wide flux-weakening and the motor current limit under
all conditions of operation accurately. Following the analysis of the trajectory control, the results
of extensive simulation and experiment are given.
1. INTRODUCTION
The IPM synchronous motor possesses special
features for adjustable-speed drives which distinguish
it from other classes of ac machines, especially surface
permanent magnet synchronous motor. Because of its
special rotor configuration, pole saliency and
relatively large armature inductance, the IPM
synchronous motor is also more suitable for an
extended speed range operation by the flux-weakening
control. The maximum steady state torque of the IPM
synchronous depends on the continuous armature
current rating. The available output voltage of the
inverter limits the maximum speed attainable at this
torque. In traction and spindle drives, constant power
operation and wide speed range are desirable. With dc
motor drives, these are achieved by the appropriate
reduction of the field current as the speed increases. In
IPM synchronous motor, direct control of the magnet
flux is not available. The air-gap flux, however, can be
weakened by the demagnetizing current in the direct
axis [1-5]. In this paper, the armature current control
method expanding the operating limits is examined
through analysis, modeling and experiment.
2. CONTROL OF IPM MOTOR DRIVE
In a PWM current controlled drives, the currents are
regulated by PI (proportional + integral) controllers.
Traditionally, the current control executes in the
stationary reference frame due to its simplicity. When
the current controllers are viewed in the rotor
reference frame, however, it is seen that there exist
cross-coupling effects between d-axis and q-axis
variables and back-emf disturbance. It is possible to
compensate the back-emf disturbance by using backemf feed-forward compensation. Other advantages of
the latter scheme have been reported in the literature,
for instance, the cross-coupling effect can be easily
compensated. Furthermore, field-weakening control
becomes easier since the d-axis current is directly
controlled. The current control in the rotor flux
reference frame as a means of torque and flux control
has been adopted in this paper. The block diagram of a
PWM current controlled IPM motor drive is shown in
Fig. 1.
3. CONTROL TRAJECTORIES
The voltage and torque equations of an IPMSM may
be expressed in the d-q reference frame as follows:
 vd   r + pLd
v  =  ω L
d
 q 
−ω Lq  id   0 
+
r + pLq   iq  ωλ f 
3
T = P [λ f iq + ( Ld − Lq ) id iq ]
2
(1)
(2)
From equation (1), the steady-state phasor diagram of
an IPMSM shown in Fig. 2 is obtained.
In Fig. 2, β and γ are the leading angles of the stator
current and voltage vectors from the q-axis
respectively and δ is the torque angle. It is clear that
the dq-axis components of the stator current are
id = − I s sin β
iq = I s cos β
(3)
(4)
where Is is the amplitude of the stator current vector.
Substituting (3) and (4) into torque equation (2),
yields the expression for the torque in terms of the
amplitude of the stator current as follows:
* *
ω*
PI-control
with limit
ω
Current control
&
dq-decoupling
*
iq
*
ω Calculation of id
*
id
vdc vqc
Voltage
compensation
PWM
inverter
dq-1
iq id
abc
Te
ib
θ
Tr
IPMSM
θ
Encoder
Figure 1. PWM current controlled IPM motor drive.
ωλf
is
iq
γ
β
id
Phase angle, β
Figure 3. Effects of the current phase angle β.
3.1 Maximum Torque Per Ampere (MTPA)
Trajectory
q-axis
vs
ri s
T
ia
dq
•
Speed
observer
Torque, N m
* *
vd vq
*
iq
To obtain fast transient response and high torque, the
current phase angle β must be controlled to develop
the maximum torque. The relationship between the
amplitude of the stator current and the phase angle β
for the maximum torque can be derived by setting the
derivative of (5) with respect to β to zero.
λs
δ
λf
d-axis
3
3
d
T = − P λ f I s sin β + P( Lq − Ld ) I s2 cos2 β = 0 (6)
2
2
dβ
Figure 2. Phasor diagram of an IPM motor.
3
3
T = P λ f I s cos β + P( Lq − Ld ) I s2 sin 2 β
2
4
(5)
where
id, iq – d-and q-axes armature current,
vd,vq – d- and q-axis armature voltage,
Ld, Lq – d- and q- axes axis inductance,
λf – magnet flux,
r – armature resistance,
Is – amplitude of the armature current
P – number of pole pairs,
p – differential operator d/dt,
T – Torque and
ω - rotor speed.
The first term of (5) is the excitation torque Te and the
second term is the reluctance torque Tr. Figure 3
shows T, Te and Tr as functions of the current phase
angle β for the motor in Table 1 when the amplitude
of the stator current is kept at the rated value. As is
seen from the fig. 3, Te is maximum at β = 0 and Tr is
maximum at β = 45o. Therefore, the total torque, T, is
maximum within the range of 0o < β < 45o. For the
IPM motor, Lq-Ld may be large, so the reluctance
torque is no longer negligible. From the above
analysis, to operate an IPM synchronous motor at high
torque and efficiency, id should be determined by (4)
with β corresponding to the maximum torque for a
given stator current.
By substituting equation (3) and (4) into equation (6),
one obtains
3
3
− P λ f I s sin β + P( Lq − Ld ) I s2 (cos2 β − sin2 β ) = 0
2
2
(7)
From equation (7), one can obtain
id =
λf
2( Lq − Ld )
−
λ 2f
4( Lq − Ld ) 2
+ iq2
(8)
Equation (8) implies that the maximum torque-perampere (MTPA) is obtained if id is determined by
equation (8) for any iq. . It should be noted here that
the torque is not directly proportional to iq. This is
why the torque control via current control is called
indirect torque control.
3.2 Current and Voltage Limit Trajectories
When an IPM synchronous motor is fed from an
inverter, the maximum stator current and voltage are
limited by the inverter/motor current and dc-link
voltage ratings respectively. These constraints can be
expressed as
I s = id2 + iq2 ≤ I sm
(9)
Vs = vd2 + vq2 ≤ Vsm
(10)
where Ism and Vsm are the available maximum current
and voltage of the inverter/motor.
Substituting equation (1) at steady state into equation
(10), one can obtain
Vs = (rid − ω Lq iq )2 + ( riq + ω Ld id + ωλ f )2 ≤ Vsm
(11)
Equation (11) can be simplified as equation (12) if the
stator resistance is neglected.
( Lqiq )2 + ( Ld id + λ f ) 2 ≤ (
Vsm 2
)
ω
(12)
id can be calculated from equation (12) as (13) if
resistance is neglected.
id = −
λf
Ld
+
1
Ld
2
Vsm
ω2
− ( Lq iq ) 2 = −
λf
Ld
+
1
Ld
id 1 (13)
where
id 1 =
2
Vsm
− ( Lqiq ) 2
ω2
(14)
3.3 Voltage Limited Maximum Output Trajectory
The armature current vector i3(id3,,iq3) producing
maximum output power under the voltage limit
condition is derived as follow [6]:
id = −
λf
Ld
− ∆id
(15)
2
 Vsm 

 − ( Ld ∆id )
 ω 
iq =
ρ Ld
(16)
V 
− ρλ f + ( ρλ f ) 2 + 8( ρ − 1) 2  sm 
 ω 
∆id =
4( ρ − 1) Ld
where, ρ =
2
(17)
Lq
Ld
The current vector trajectory of the voltage limited
maximum output is shown in Fig. 4. The rotor speed
iq, A
Maximum torque per
ampere trajectory
Voltage limited
maximum output id= -λf/Ld
trajectory
G
B
D
A2
ω = ωp
A3
C
E
4. CONTROL MODE SELECTION
The MTPA and current limit trajectories are
independent of the rotor speed and are only
determined by the motor parameters and inverter
current rating. However, the voltage limit trajectory
varies with the change in rotor speed. It is seen that
the voltage limit trajectory is an ellipse, which
contracts when the rotor speed increases. When the
rotor speed increases infinitely, the voltage limit
ellipse becomes a point on the d axis. If this point is
inside the current limit circle, theoretically, the motor
has an infinite maximum speed, otherwise, has a finite
maximum speed. The intersection of MTPA, current
limit trajectories is the operation point A1 at which the
motor has the rated current and voltage and will
produce the rated torque at the base speed. The
control modes, i.e., maximum torque-per-ampere and
flux weakening controls, are selected according to the
analysis of fig. 4. The q-axis commanded current i q* is
determined by the outer control loop and the d-axis
commanded current i *d is decided by equation (8) in
maximum torque-per-ampere control mode, or by (12)
in flux weakening control mode. Whether MTPA
mode or the field weakening mode should be selected
is determined by the rotor speed and the load.
According to the rotor speed, the motor operation is
divided into three sections, that is, below the base
speed ωb, above the crossover speed ωc and between
the base and crossover speeds. The crossover speed ωc
is the speed at which the back-emf voltage of the
unloaded motor equals to the maximum voltage. As a
result, the control mode is selected as follows.
4.1 Operation below the base speed
Voltage limit
trajectory
ω = 1500 rpm = ωbase
ω = 2200 rpm
ω = 2400 rpm= ωc
ω = 2800 rpm= ω1
ω > 2800 rpm
A1
ωp is the minimum speed for the voltage-limited
maximum-output operation. Below this speed, the
voltage-limited maximum-output operating point can
not be reached, because the voltage-limited maximumoutput trajectory intersects the voltage limit trajectory
outside the current limit circle. If (λ f Ld ) > I sm , the
voltage-limited maximum output trajectory is outside
the current-limit trajectory. Therefore, voltage-limited
maximum-output trajectory needs not to be
considered.
O
id, A
Figure 4. Control trajectories in id-iq plane.
It is seen in fig.4 that the voltage limit ellipse
corresponding to the operation below the base speed is
larger than the one for base speed. Therefore, if the
stator current vector is controlled according to the
maximum torque-per-ampere trajectory and satisfying
the current limit, which is the trajectory A1O, it must
satisfy the voltage limit since the current vector is
inside the voltage limit ellipse. Therefore, maximum
torque-per-ampere mode is selected for constant
torque operation.
The d- and q-axis currents, idA1 and iqA1, with which
the maximum torque is produced, are determined by
(8) and (9) when Is = Ism.
idA1 =
λf
2( Lq − Ld )
−
λ 2f
4( L − L)2
2
2
+ I sm
− idA
1
(18)
Solving (12), one gets
idA1 =
λf
4( Lq − Ld )
−
λ 2f
16( Lq − Ld )2
+
2
I sm
2
2
2
iqA1 = I sm
- idA1
(19)
(20)
This is the operating point at which the motor
produces the maximum torque and the limit value of
the outer control loop for constant torque operation is
then determined by (20).
4.2. Operation above the crossover speed
To run the motor above the crossover speed, the flux
λf, which is the flux linkage along the rotor d axis, has
to be reduced. Although the flux is already reduced
with the MTPA control, the voltage limitation is no
longer satisfied when the rotor speed is above the
crossover speed. The stator current vector is therefore
controlled according to the voltage limit trajectory
instead of MTPA trajectory. Thus, id and iq are
determined according to the voltage and current limit
equation (12) and (9). The limit value, idv and iqv, of
the outer control loop for such a field weakening
operation is therefore determined by these two
equations with Is = Ism, which is inversely proportional
to the rotor speed.
idv = −
λ f Ld
a
+
1
λ 2f L2d − a b
a
(21)
where
a = L2d − L2q
2 2
b = I sm
Lq + λ 2f −
(22)
2
Vsm
ω2
(23)
current vector moves from A3 to E (fig. 4) along the
voltage-limited maximum output trajectory.
4.3 Operation between the base and crossover
speeds
(3-24)
When the rotor speed is in the range from ωb to ωc,
the control mode is determined by the load. If the
motor is unloaded, it may operate near the crossover
speed with MTPA control. When the motor is fully
loaded, it has to be controlled according to the voltage
limit trajectory right above the base speed. For
instance, if the motor runs at 2200 rpm, the
corresponding voltage limit trajectory is BCO as
shown in fig. 4. Thus, if the motor is heavily loaded
and the current vector is along the trajectory BC, it has
to be controlled according to the voltage limit
trajectory. Otherwise, the motor can be still controlled
under MTPA control along the trajectory CO. The
determination of the control mode is based on the
calculated id from both equation (8) and (13). If the
calculated id from equation (9) is smaller than the one
calculated from (13), the current vector is controlled to
the MTPA trajectory for constant torque operation.
Otherwise, voltage limit trajectory is used to control
the current vector for field weakening operation. The
limit value for the outer control loop is still
determined by equation (24).
5.
MODELING AND EXPERIMENTAL
RESULTS
Modeling and experiment were performed on an IPM
synchronous motor. The specification of the motor
used is shown in Table 1. The rated and crossover
speeds for this motor are 1500 and 2400 rpm,
respectively. The sampling time of the inner control
loop is 150 µs and that of the outer control loop is 750
µs. The drive system was implemented on a
TMS320C31 digital signal processor with a clock
speed of 33 MHz. The rotor position and speed were
obtained from an incremental encoder with 5000
pulses per revolution. A three-phase insulated gate
bipolar transistor (IGBT) intelligent power-module is
used for an inverter, which is supplied at a dc link
voltage of 570 V.
than ω1. The trajectory for id = −λ f / Ld has shown in
Figure 5 shows the modeling results of speed, iq, and
id with respect to a step change in speed reference
from 0 to 1500 rpm. It is seen from these figures that
the current vector is at the intersection of MTPA and
current limit trajectories, i.e., point A1 during
acceleration and moves along the MTPA trajectory
when the speed approaches its reference. Since the
load torque is taken to be zero, the current vector
finally settles down at origin, O.
figure 4. If the rotor speed ω ≥ ω p , id and iq has to be
determined from equation (15) and (16) and the
Figure 6 shows the dynamic responses with respect to
a step change of speed reference from 0 to 2800
2
2
iqv = I sm
- idv
(24)
When the rotor speed is in the range from ωc to ω1, id
is determined from equation (13). However, as the
rotor speed exceeds ω1 for which id1 is –ve and id
becomes a complex number. So id should be
calculated as id = −λ f / Ld for a rotor speed greater
Speed, rpm
Speed, rpm
Time, sec.
Time, sec.
(a) Speed response.
(a) Speed response.
iq, A
iq, A
0.4
Time, sec.
Time,sec
(c) iq response
(b) iq response.
id, A
id, A
Time, sec.
(c) id respnse.
Time, sec.
(c) id respnse.
iq, A
MTPA
iq, A
MTPA
Current limit
Current
lim it
A
O
id, A
(d) Locus of current vector.
id , A
(d) Locus of current vector.
Figure 6. Modeling results for flux-weakening
operation under trajectory control ωr >ωb.
Figure 5. Modeling results for constant torque
operation under trajectory control ωr ≤ ωb.
rpm. It is clear that the smooth transition between
constant torque and field weakening controls occurs
when the rotor speed exceeds the base speed (1500
rpm). It is also clear that the stator current vector
moves along the maximum-torque-per armature
current trajectory when the speed is below the base
speed and shifts to the voltage limit ellipse when the
speed increases above base speed. Figure 7 shows the
torque and power versus speed characteristics of the
motor. It is seen that field weakening occurs once the
speed goes above the base speed (1500 rpm). It is seen
that constant power operation is possible with the
trajectory control discussed in this paper.
T o r q u e, N m
P o w e r, W
2
800
400
0
0
1000
2000
3000
4000
0
1000
2000
3000
Speed, rpm
4000
Speed, rpm
Figure 7. Torque and power versus speed.
Figures 8 and 9 show the experimental results for
constant torque operation and flux-weakening
operation, respectively. The experimental results
conform to the modeling results quite well.
iq, A
Speed, rpm
2
1600
0.1
id, A
id, A
1.5
0
1200
1
-0.1
1
800
-0.2
0.5
400
-0.3
0
0
0
0.1
0.2
0.3
0.1
0.4
0.2
0.3
-0.4
0.4
0.1
T i m e, s e c.
T i m e, s e c.
(a) speed response.
0.2
0.3
0.4
0
-1
0
1
T i m e, s e c.
(b) iq response.
(c) id response.
iq, A
(d) Locus of current vector.
Figure 8. Experimental results for constant torque operation under trajectory control.
i q, A
Speed, rpm
iq, A
id, A
3000
2
0
1.5
-0.5
1
-1
0.5
1.5
2000
1
0.5
1000
0
0
0
-0.5
0.2
0.4
0.6
Time, sec
(a) speed response.
-1.5
0.2
0.4
Time, sec
(b) iq response.
0
0.2
0.4
0.6
Time, sec.
(c) id response.
-1.5 -1
-0.5
0
0.5
1
1.5
id, A
(d) Locus of current vector.
Figure 9. Experimental results for flux-weakening operation under trajectory control.
REFERENCES
6. CONCLUSION
This paper analyses the control trajectories for an IPM
synchronous motor. The control trajectories, namely
the maximum torque-per-ampere, current and voltage
limit trajectories, have been derived and expressed in
terms of id and iq, and then drawn in the id -iq plane.
Based on the analysis of these trajectories, the
selection of constant torque and flux-weakening
modes has been discussed. It has been established that
the maximum torque-per-ampere trajectory should be
selected for constant torque operation and the voltage
limit trajectory should be selected for flux-weakening
operation. It has been shown by the modeling and
experimental results that flux-weakening is very well
incorporated in this drive. The transition between
constant torque and flux-weakening is very smooth. It
is also shown that constant power operation is possible
with the improved trajectory control discussed in this
literature.
TABLE 1. Parameters of the IPM Motor used.
Number of pole pairs P
Stator resistance
r
Magnet flux linkage ϕf
d-axis inductance
Ld
q-axis inductance
Lq
Phase voltage
V
Phase current
I
Base speed
ωb
Crossover speed
ωc
Rated torque
Tb
2
19 Ω
0.447 Wb
0.3885 H
0.4755 H
240 V
1.6 A
1500 rpm
2400 rpm
2 Nm
[1] B. Sneyers, D.W Novotny, and T. A. Lipo,
“Field-weakening in buried permanent magnet ac
motor drives”, IEEE Trans. Ind. Appl., vol.21,
pp. 398-407, Mar./Apr. 1985.
[2] T. M. Jahns, “Flux-weakening regime operation
of an interior permanent magnet synchronous
motor drivem,” IEEE Trans. on Ind. Appl., vol.
23, no. 4 pp. 398-407, 1987.
[3] T. Sebastian and G.R. Slemon, “Operating limits
of inverter-driven permanent magnet motordrives”, IEEE Trans. Ind. Appl., vol. 23, pp. 327333, Mar./Apr. 1987.
[4] S. Morimoto, M. Sanada, Y. Takeda, “Widespeed operation of interior permanent magnet
synchronous motors with high-performance
current regulator”, IEEE Trans. on Ind. Appl.,
vol.30, no.4, pp. 920-926, 1994.
[5] M. F. Rahman, L. Zhong and K. W. Lim, “A
DSP based instantaneous torque control strategy
for interior permanent magnet synchronous
motor drive with wide speed range and reduced
torque ripples”, IEEE Industry applications
society Annual Meeting, vol. 1, pp. 518-524,
1996.
[6] S. Morimoto, M. Sanada, Y. Takeda, “Expansion
of operating limits for permanent magnet motor
by current vector control considering inverter
capacity”, IEEE Trans. on Ind. Appl., vol., no. 5,
26, pp. 866-871, 1990.