Speed-Controller Design of Induction Motor and Permanent Magnet
Synchronous Motor for High Performance Drives
Sakorn Po-ngam
Power Electronics & Motor Drives Laboratory (PEMD)
Department of Electrical Engineering, Faculty of Engineering
King Mongkut's University of Technology Thonburi
91 Prachauthit Road, Bangmod, Thungkhru District, Bangkok 10140 Thailand.
e-mail:sakornpo@hotmail.com
ABSTRACT
This paper presents a speed-controller design of an
Induction Motor (IM) and a Permanent Magnet
Synchronous Motor (PMSM) for high performance
drive. The PI controller is used for regulating the real
rotor speed to attack the command speed. The same
controller design scheme is proposed based on the
symmetrical optimum (SO) method when it’s
controlled by the decoupling control. Simulation
results are given to verify the validity of the proposed
method.
Keyword: Speed controller design, high performance
drives, induction motor, permanent magnet
synchronous motor
described in Sect.2. Then, the decontrol control of IM
and PMSM are described in Sect.3. Section 4 gives
the proposed technique and simulation result. Finally,
the paper is concluded in Sect.5.
2. DYNAMIC MODEL OF IM AND PMSM
Firstly, we will briefly show the model of IM and
PMSM expressed both on the stator reference frame
and the rotor flux reference frame. These models are
used for the decoupling control in the next section.
2.1 Model of IM on the stator reference frame
The dynamic model of IM viewed from the stator
reference frame is given by equation (1)
1. INTRODUCTION
In recent years, the torque control of the induction
motor (IM) and the permanent magnet synchronous
motor (PMSM) servo drive systems are being used
more and more in various high performance
applications such as industrial robots and numerical
controlled machine tools, etc. This control has been
developed from the conventional vector control, and
thus usually works by controlling the stator current
with a current-controlled inverter [1]-[3]. The vector
control has several disadvantages [4]:
a) current loop of high bandwidth is need, which
causes complexity in the implementation
hard/software, and requires a dedicated high speed
DSP
b) current transducer muse be used, therefore cost
of the drive must be increased.
To overcome such problems, the decoupling control
was proposed in [4]-[7]. In the high performance IM
and PMSM drive, the closed-loop control may be
employed for speed or position controls, therefore
speed or position controllers are designed for system
stability. In this paper, we proposed design guidelines
for speed controller of IM and PMSM drives based
on the symmetrical optimum method. This proposed
technique can be used for the separately excited DC
motor and the Interior Permanent Magnet
Synchronous Motor (IPMSM) too.
The remainder of the paper is organized as follows.
First, the dynamic model of IM and PMSM are
d ⎡ is ⎤ ⎡ A11
⎢ ⎥=⎢
dt ⎣⎢ λr ⎦⎥ ⎣ A21
A12 ⎤ ⎡ is ⎤ ⎡ B1 ⎤
us
⎢ ⎥+
A22 ⎥⎦ ⎣⎢λr ⎦⎥ ⎢⎣ 0 ⎥⎦
(1)
where in
A11 = −
1
σ Ls
( Rs +
Rr M 2
Lr
2
) ∗ I , B1 =
1
σ Ls
A12 =
Rr
pωm
∗I −
*J
ε
Lr ε
,σ = 1 −
A21 =
MRr
∗I
Lr
,ε =
∗I
M2
Ls Lr
σL s Lr
M
⎡1 0⎤
⎡0 − 1⎤
,I = ⎢
⎥ , J = ⎢1 0 ⎥
0
1
⎣
⎦
⎣
⎦
A22 = −ε A12
us : stator voltage space vector is : stator current
space vector
λr :
resistance
Rr : rotor resistance Ls : stator –self
inductance
Lr : rotor –self
rotor flux space vector Rs : stator
inductance
ω m : rotor
speed M : mutual inductance p : numbers of pole
pairs.
2.2 Model of IM on the rotor flux reference frame
As our aim is to control flux and torque of the motor,
it is, therefore, more convenient to transform the
above IM’s model on to the rotor flux reference
frame, and we obtain the following model:
Stator dynamic:
Rs .isd + σ Ls
R M2
d
isd − ωmRσ Ls isq − r 2 ( imR − isd ) = usd
dt
Lr
(2)
2
d
M
isq + ωmRσ Ls isd + ωmR
imR = usq (3)
dt
Lr
Rotor flux dynamic:
R
d
imR = r isd − imR
(4)
dt
Lr
Rs .isq + σ Ls
(
)
R ⎡ isq ⎤
dρ
= ωmR = p.ωm + r ⎢ ⎥
dt
Lr ⎣ imR ⎦
Motor torque:
M2
TM = p.
imR .isq
Lr
where
[ ]d , [ ]q
(5)
(6)
denote the d and q components on
the rotor flux reference frame, and ρ , ωmR , imR are the
rotor flux angle, the rotor flux frequency and the rotor
flux magnetizing current, respectively.
2.3 Model of PMSM on the stator reference frame
The linear model of PMSM viewed from the stator
reference frame is given by equation (7) [8]
⎡ R
d ⎡ i ⎤ ⎢− I
=
L
⎢ ⎥
dt ⎢⎣ λ ⎥⎦ ⎢
⎣⎢ 0
ω⎤
⎡ i ⎤ ⎡1/ L ⎤
u
L⎥ ⎢ ⎥+⎢
⎥ ⎢⎣ λ ⎥⎦ ⎣ 0 ⎥⎦
J ω ⎦⎥
−J
(7)
The conventional decoupling control for IM has been
previously introduced in [5]-[6]. However, this
conventional decoupling control ignores the stator
dynamic, and thus is imperfect. To achieve a
complete decoupling stator dynamics, the decoupling
control of IM as shown in equations (11)- (12) and
the decoupled stator dynamic shown in equations
(13)- (14) [4].
Decoupling Control:
(
)
M2
usd* = Rs .isd∗ − ωmRσ Ls iˆsq + Rr 2 iˆsd − iˆmR (11)
Lr
M2 ˆ
usq* = Rs .isq∗ + ωmRσ Ls iˆsd + ωmR
imR
Ls
(12)
Decoupled stator dynamic:
(
)
(13)
(
)
(14)
d ˆ
R
isd = s isd∗ − iˆsd
σ Ls
dt
R
d ˆ
isq = s isq∗ − iˆsq
dt
σ Ls
where ‘ ^ ’ , ‘ * ’ denotes the estimated value and
the command valued, respectively. The decoupling
control can be calculated from the rotor quantities
( ωmR , iˆmR ) estimated from equations (4) and (5)
together with the estimated current ( iˆsd , iˆsq ) calculated
from the decoupled stator dynamic (13) and (14). The
decoupling control block diagram of IM as shown in
Fig.1.
AC 3 φ
where λ = Ψ e J ρ , Ψ : permanent magnet field flux,
R : stator resistance L : stator inductance. ρ and
ω are the rotor angle and rotor speed (in electrical
degree), respectively.
Flux
Command
isd*
ω
*
m
+
-
Speed
Controller
(PI)
isq*
Decoupling Control IM
Drives
Eq (4)-(5) & (11)-(14)
u
*
Voltage-Source Inverter
VSI
Space Vector
PWM
ωm
Load
IM
2.4 Model of PMSM on the rotor/rotor flux reference
Fig.1: Block diagram of decoupling control IM drives
frame
The linear model of PMSM viewed from the rotor
reference frame is derived by [9] as shown in
equations (8) - (10).
Stator dynamic:
⎡ud ⎤
⎡id ⎤
d ⎡id ⎤ ⎡ −ω Liq ⎤ ⎡ 0 ⎤
⎢u ⎥ = R ⎢i ⎥ + L ⎢i ⎥ + ⎢
⎥+
dt ⎣ q ⎦ ⎣ ω Lid ⎦ ⎣⎢ωΨ ⎦⎥
⎣ q⎦
⎣ q⎦
3. DECOUPLING CONTROL FOR IM AND
PMSM
3.1 Decoupling control for IM
⎡ud∗ ⎤
⎡id∗ ⎤ ⎡ −ω Liˆq ⎤ ⎡ 0 ⎤
=
R
⎢ ∗⎥
⎢ ∗⎥ + ⎢
⎥+⎢
⎥
⎢⎣ uq ⎥⎦
⎢⎣iq ⎥⎦ ⎣ ω Liˆd ⎦ ⎣ωΨ ⎦
(15)
Decoupled stator dynamic:
(9)
Motor torque:
TM = p.Ψiq
The decoupling control of PMSM is proposed in [9]
as shown in equation (15) and the decoupled stator
dynamic shown in equation (16).
Decoupling Control:
(8)
Rotor dynamic:
d ⎡ ρ ⎤ ⎡ω ⎤
=
dt ⎢⎣ Ψ ⎥⎦ ⎢⎣ 0 ⎥⎦
3.2 Decoupling control for PMSM
(10)
∗
d ⎡iˆd ⎤
R ⎡iˆd ⎤ R ⎡id ⎤
=
−
+
⎢ ⎥
⎢ ⎥
⎢ ⎥
dt ⎢⎣iˆq ⎥⎦
L ⎢⎣iˆq ⎥⎦ L ⎢⎣iq∗ ⎥⎦
(16)
The zero- id control method is suited for the
maximum torque/amp ratio of the PMSM, therefore,
id* = 0 ⇒ iˆd 0 . From the result, the decoupled
stator dynamic in equation (16) can be expressed in
equations (17) and (18).
iˆd = 0
iˆq =
1
iq*
Ls / R + 1
[10]. As such,
(18)
control of IM drive and 2.76, 70 for the PMSM drive,
respectively. The phase margin of these systems is
about 44.8 ° as shown in Fig. 5-6.
Where ‘s’ is the differentiator operation. The
decoupling control block diagram of PMSM as
shown in Fig.2.
ωm*
Speed
Controller
(PI)
+
-
u
Decoupling Control
PMSM Drives
Eq (9),(15) & (17)-(18)
*
q
i
*
Bode Diagram
Gm = Inf dB (at Inf rad/sec) , Pm = 44.8 deg (at 77.9 rad/sec)
100
50
Magnitude (dB)
AC 3 φ
Flux
Command
id* = 0
k p , ki are about 1.3, 42 for the speed
(17)
Voltage-Source Inverter
VSI
Space Vector
PWM
0
-50
-100
-120
1
p
ρ
ω
Load
PMSM
Phase (deg)
ωm
d
dt
Fig.2: Block diagram of decoupling control PMSM
drives
-150
-180
0
10
1
10
2
10
3
4
10
10
Frequency (rad/sec)
Fig.5: Bode diagram of open-loop speed control IM
drive
Bode Diagram
4. SPEED CONTROLLER DESIGN FOR IM
AND PMSM
Gm = Inf dB (at Inf rad/sec) , Pm = 44.8 deg (at 130 rad/sec)
ωm* (s) +
-
k
kp + i
s
Isq* (s)
1
(σLs / Rs )s +1
Iˆsq (s)
k
TL (s)
Tm (s) -
+
1 ωm (s)
•
Js
Speed controller
100
Magnitude (dB)
50
0
-50
-100
-120
Phase (deg)
The main objective of this paper is to present the
design guidelines of the speed controller for IM and
PMSM drives when it’s controlled by decoupling
control method. The speed control loop of IM and
PMSM as shown in Fig.3 and Fig.4, respectively.
-150
-180
Fig.3: Block diagram of speed control loop IM drive
10
0
1
10
10
2
3
10
10
4
Frequency (rad/sec)
ωm* (s) +
-
kp +
ki
s
Iq* (s)
1
(L / R)s +1
Iˆq (s)
k′
TL (s)
Tm (s) -
+
1 ωm (s)
•
Js
Speed controller
Fig.6: Bode diagram of open-loop speed control
PMSM drive
In the next section, we will simulated by using the
Matlab/ Simulink and the machine’s parameters of the
simulated system are given in the appendix.
Fig.4: Block diagram of speed control loop PMSM
drive
M2
imR (6) , k ′ = p.Ψ
Lr
(10) and J is the system inertia. The open-loop
transfer functions of these systems are given by
equations (16) - (17).
4.1 Simulation results for speed control of IM and
PMSM drives
In the Fig.3. and Fig.4., k = p
ω m*
100 rpm
1000
100 rpm
1000
⎞⎛ 1 ⎞
⎛ k p s + ki ⎞ ⎛
k
FoL ( s ) = ⎜
⎟⎜ ⎟
⎟⎜
s
⎝
⎠⎝ (σ Ls / Rs ) s + 1 ⎠ ⎝ Js ⎠
⎛ k s + ki ⎞⎛
k
⎞⎛ 1 ⎞
′ (s) = ⎜ p
FoL
⎟⎜
⎟⎜ ⎟
s
⎝
⎠ ⎝ ( L / R ) s + 1 ⎠ ⎝ Js ⎠
0
(16)
ωm
5.78
isq
20 A
isd
20 A
isu
(17)
10 A
0
time: 0.5 s / div
Since the control block diagram in Fig.3 and Fig.4
have two poles at the origin and same control block
diagram, therefore, we can design the speed controller
by using the Symmetrical Optimum (SO) method
Fig.7: Small step speed response of IM drive
1 sec
0.5 sec
1000 rpm-
ω m*
100 rpm
Tm
10 Nm
0-
isq
Symmetrical Optimum (SO) method, the response of
speed and torque both IM and PMSM are very well.
The system stability is confirmed by the maximum
phase margin of the system. Simulation results are
given to verify the validity of the proposed method.
10 A
05.78 A-
isd
isu
1 A
10 A
APPENDIX
Induction motor’s parameter
2HP , 220 / 380 V , 50 Hz , 6.3 / 3.7 A , 1430 rpm , 4 poles
isd = 5.788 A (rated ), isq = 9.25 A (rated ), J = 0.021 kg im2
isq = 9.25 A (rated ) , M = Lr = 93.4 mH
0-
Lr = 93.4 mH , Ls = 104.9 mH , Rr = 0.963 Ω , Rs = 2.15 Ω.
time: 0.5 s / div
Permanent magnet synchronous motor’s parameter
Fig.8: Simulation results at step load (1000 rpm) of
IM drive
0.5 s
ω*m
L = 4.3mH , Ψ = 0.11Wb , J = 0.01547 kg − m 2 .
REFERENCES
100rpm
1000rpm -
2 Hp, 200V , 200 Hz , 3.4 A, 3000 rpm, 8 poles, R = 1.355Ω ,
ωm
100rpm
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1000rpm -
0-
0-
isq*
6A
i
6A
sq
[2] J.W. Finch et al. “Scalar to vector: general principles
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[3] H. Matsugae et al. “DSP-based all digital, vector
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time: 0.5 s / div
[4] S. Sangwongwanich and S. Suwankawin, “A SpeedSensorless IM Drive With Modified Decoupling”,
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Fig.9: Small step speed response of PMSM drive
1s
1000rpm-
ωm
100rpm
*
0-
isq
i
sq
6A
6A
0-
isu
3A
0-
ρ
360
0-
time: 1 s / div
Fig.10: Simulation results at step load (1000 rpm) of
PMSM drive
Fig.7-8 and Fig.9-10 shown the small step speed
response and step load change of IM drive and
PMSM drive, respectively. From the simulation
results, we can see that the actual speed can be closed
to the command speed with fast the torque response.
5. CONCLUSION
In this paper, the speed controller design guidelines
for IM and PMSM drives are proposed. By using the
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