030
1
Vector Control of an Induction Motor Fed by a
PWM Inverter with Output LC Filter
Janne Salomäki and Jorma Luomi
Abstract — This paper introduces a control method for an induction motor that is supplied by a PWM voltage source inverter
through an LC filter. A full-order observer is used to estimate the
system states, and no additional voltage or current measurements
are needed for the vector control of the motor. Simulation and
experimental results are presented confirming the functionality of
the proposed control method.
Index Terms — Vector control, induction motor, LC filter, observer.
I. INTRODUCTION
T
HE voltage generated by a PWM frequency converter consists of sharp-edged voltage pulses. Sudden alteration of
the voltage causes unwanted effects such as bearing currents
and high voltage stresses in motor insulations. The oscillation
at the switching frequency causes additional losses and acoustic noise. These phenomena can be eliminated by adding an
LC filter to the output of the PWM inverter. In addition, the
EMI shielding of the motor cable may be avoided if the output
voltage of the inverter is nearly sinusoidal.
The control of an induction motor becomes more difficult if
an LC filter is used. Usually, a very simple scalar control
method (volts-per-hertz control) is chosen. Although better
control performance is needed in many cases, only few publications deal with the vector control of an induction motor fed
via an LC filter. A deadbeat controller has been used to control
the inductor current and the capacitor voltage [1], the highpass filtered stator voltage has been used to correct the voltage
reference [2], and a multi-loop feedback controller has been
proposed [3]. In these methods, extra current or voltage measurements are needed in addition to the phase current and dc
voltage measurements usual in a frequency converter. A challenge for the motor drive control design is to keep the number
of measurements low in order to obtain cost savings and reliability improvements.
In this paper, a method is presented for the vector control of
an induction motor fed by an inverter with an output LC filter.
A cascade control method is used to control the inverter current, the stator voltage, the stator current and the rotor speed.
The system states are estimated by a full-order observer.
II. THEORY
The principle of the control system is shown in Fig. 1. The
inverter output voltage u A is filtered by an LC filter, and the
induction motor (IM) is fed by the filtered voltage u s . The
inverter current i A , the electrical angular speed ωm of the
rotor, and the dc link voltage u dc are the only measured
quantities, whereas the stator voltage u s and current i s of the
motor are estimated by an observer (the estimated quantities
being marked by ‘^’). The system is controlled by nested control loops in the rotor flux reference frame.
A. Filter and Motor Models
In a reference frame rotating at angular speed ω s , the equations for the LC filter are
di A
dt
du s
dt
iA +
Lf
= − jω s u s +
1
(u − u s )
Lf A
(1)
1
(i − i )
Cf A s
(2)
where L f is the inductance and RLf the series resistance of
the inductor, and C f is the capacitance of the filter.
The motor control is based on the inverse-Γ model of the
induction motor in the rotor flux reference frame. The stator
and rotor voltage equations in this reference frame are
u s = Rs i s +
0 = RR i R +
dψ
dψ
R
dt
s
dt
+ jω sψ
(3)
s
+ j (ω s − ω m )ψ
Lf
u dc
(4)
R
is
IM
ωm
iA
Cf
us
u A,ref
Observer
−
PI
The authors are with Helsinki University of Technology, Power
Electronics Laboratory, P.O. Box 3000, FIN-02015 HUT, Finland (e-mail:
janne.salomaki@hut.fi).
R Lf
= − jω s i A −
+
i A,ref
Fig. 1. Principle of the control system.
−
P
û s
+
PI
u s ,ref
−
iˆ s
+
−
PI
i s ,ref
+
ωm,ref
030
2
ψˆ R
ψ R,ref
ωm,ref
+
iˆ s
i A,ref
isd ,ref
+
u s ,ref +
+
+
jL′s
iˆ s
+
+
+
ω̂s
Motor Control
+
+
u A,ref
+
+
+
isq ,ref
ωm
û s
jC f
jL f
û s
iA
LC Filter Control
Fig. 2. Complex signal flow diagram of the cascade control.
respectively, where ψ and ψ are the stator and
s
R
linkages, respectively, Rs and RR are the stator
resistances, respectively, i R is the rotor current, and
electrical angular speed of the rotor. The stator and
linkages are
rotor flux
and rotor
ω m is the
rotor flux
ψ s = ( Ls′ + LM ) i s + LM i R
(5)
ψ R = LM (i s + i R )
(6)
respectively, where Ls′ denotes the stator transient inductance
and LM is the magnetizing inductance. Based on (1)-(6), the
state-space representation of the system can be written as
shown in (7) and (8) at the bottom of this page. The state vector is x = [i A u s i s ψ R ]T , and the two time constants are
defined as τ σ′ = Ls′ /( Rs + RR ) and τ r = LM / RR .
B. Cascade Control
Figure 2 illustrates the proposed cascade control of the system in the rotor flux reference frame. In the LC filter control,
the innermost control loop governs the inverter current i A by
means of a PI controller, and the stator voltage u s is governed
by a P-type controller in the next control loop. In both control
loops, decoupling terms are used to compensate the cross-couplings caused by the rotating reference frame.
The motor control is based on vector control and forms two
outermost loops of the cascade control. The stator current is
controlled by a PI-type controller with cross-coupling compensation, and the rotor speed is governed by a PI-controller. In
addition, a PI-type rotor flux controller is used.
C. Observability
The observability of the system can be investigated using
the observability matrix
M o = [C C A C A 2
C A 3 ]T
(9)
The system is observable if the rank of the observability matrix
is equal to the number of states [4]. The observability was
checked for various angular speeds of the rotor and the reference frame, indicating that the system is observable.
⎡ Rlf
⎤
1
− jω s −
0
0
⎢−
⎥
Lf
⎢ Lf
⎥
⎡ 1 ⎤
⎢
⎥
1
1
⎢L ⎥
− jω s
−
0
⎢
⎥
⎢ f ⎥
Cf
Cf
⎥ x + ⎢ 0 ⎥u A
x = ⎢
⎢
⎞ ⎥
1
1 ⎛ 1
1
⎢ 0 ⎥
⎜⎜ − jω m ⎟⎟ ⎥
−
− jω s
0
⎢
⎢ ⎥
′
′
′
τ
τ
L
L
s
s⎝ r
σ
⎠ ⎥
⎢
⎢⎣ 0 ⎥⎦
1
⎢
⎥
0
0
−
− j(ω s − ω m )⎥
RR
B
⎢
τ
⎣
r ⎦
A
i A = [1 0 0 0]x
C
(7)
(8)
030
3
15
where estimated states are marked by the symbol ‘^’ and
K = [k 1 k 2 k 3 k 4 ]T is the observer gain vector.
The digital implementation of the full-order observer based
on the conventional forward Euler discretization causes instability at higher speeds. Other alternatives, such as the backward Euler method or the bilinear transformation, hold the
stability but are more complicated to implement. A simple
symmetric Euler method has been found to be an effective and
reliable discretization method in electromechanical simulations
[5] and full-order flux observers [6]. The corresponding discretization of the full-order observer (10) is given in the Appendix.
The observer gain vector K can be selected in many ways.
The selection can be based on the pole placement method, or a
simple constant gain can be used, as will be presented in the
following.
1) Pole placement: The dynamics of the estimation error
~
x = x − xˆ are given by
~
x = ( A − KC)~
x
i A (s)
=
k1
k
≈ 1 ,
s + R Lf L f + jω s + k1 s + k1
5
0
−5
−10
−15
−15 −10 −5 0 5 10 15
Real Axis (p.u)
0.1
0
−0.1
−0.2
−0.2
−0.1
0
0.1
Real Axis (p.u)
(a)
(12)
0.2
(b)
Fig. 3. Observer poles obtained by pole placement as rotor speed changes
from –1 to 1 p.u. and slip frequency changes from –0.05 to 0.05 p.u.: (a) pole
plot and (b) its magnification in the neighborhood of the origin.
15
0.2
10
5
0
−5
−10
−15
−15 −10 −5 0 5 10 15
Real Axis (p.u)
0.1
0
−0.1
−0.2
−0.2
−0.1
0
0.1
Real Axis (p.u)
(a)
(11)
In the pole placement method, the poles of the estimation error
dynamics are placed to desired locations.
Because the system model is time variant, the pole placement should be carried out at every calculation step. However,
this would increase the computing time of the processor dramatically. A practical solution is to use gain scheduling: the
observer gain vector is calculated in advance as a function of
the angular speed of the rotor, and interpolation between
tabulated values is used during the operation. The poles also
depend on the angular slip frequency ωr = ω s − ω m as the
estimated rotor flux reference frame is used. If the gains obtained for no-load operation are used, the slip frequency affects only the imaginary parts of the observer poles, as illustrated in Fig. 3 for the example values used in the next sections. Therefore, the assumption of ωr = 0 in the pole placement method does not cause stability problems.
2) Simple constant gain: The observer becomes relatively
simple when the real-valued gain K = [k1 0 0 0]T is selected. By assuming that the estimated inverter current does
not interact with the estimated stator voltage, the transfer
function from the inverter current to its estimate can be written
as
iˆ A ( s)
10
Imaginary Axis (p.u)
(10)
0.2
Imaginary Axis (p.u)
xˆ = A xˆ + Bu A + K (i A − iˆ A )
Imaginary Axis (p.u)
The most essential part of the control is a full-order observer, which is implemented in the estimated rotor flux reference frame, i.e., in a reference frame where ψˆ = ψˆ R + j 0 .
R
The observer is defined as
Imaginary Axis (p.u)
D. Full-Order Observer
0.2
(b)
Fig. 4. Observer poles obtained by constant gain as rotor speed changes from
–1 to 1 p.u. and slip frequency changes from –0.05 to 0.05 p.u.: (a) pole plot
and (b) its magnification in the neighborhood of the origin.
where the last approximation holds if k1 >> R Lf L f + jω s .
Thus the gain k1 is approximately the bandwidth of the inverter current estimator, and it can be selected so that the estimator is at least twice as fast as the controller of the inverter
current. Figure 4 illustrates the observer poles as the speed and
the slip of the motor are varied. All poles stay in the left halfplane, indicating that the observer is stable.
III. SIMULATION RESULTS
The behavior of the system was investigated by means of
computer simulations with Matlab/Simulink software. The data
of a 2.2-kW four-pole induction motor (400 V, 50 Hz), given
in Table I, were used for the simulations. The sampling frequency of 5 kHz was equal to the switching frequency. The LC
filter was designed to have a cutoff frequency of 566 Hz in order to meet a rule of thumb that the cutoff frequency should be
about one decade below the switching frequency and one decade above the nominal fundamental frequency [7]. The fundamental-frequency voltage drop in the filter inductance was
chosen to be less than 5 % of the nominal voltage in the nominal operating point [8]. The filter parameters are also given in
Table I. The bandwidths of the controllers were 500 Hz for the
inverter current, 250 Hz for the stator voltage, 150 Hz for the
4
Motor Parameters
Stator resistance Rs
Rotor resistance R R
Stator transient inductance Ls′
Magnetizing inductance LM
Total moment of inertia J
Rated speed nN
Rated current I N
Rated torque TN
3.67 Ω
1.65 Ω
0.0209 H
0.264 H
0.0155 kgm 2
1430 r/min
5.0 A
14.6 Nm
1
0.5
0
0
2
4
6
8
10
0
2
4
6
8
10
0
2
4
6
8
10
2
T/TN
TABLE I
PARAMETERS OF THE MOTOR AND THE LC FILTER
ω (p.u.)
m
030
1
0
stator current, 15 Hz for the rotor speed, and 3 Hz for the rotor
flux. The poles of the observer were set to (–7.1 ± 11.8) p.u.,
−2.7 p.u. and –0.02 p.u., the base value of the angular frequency being 2π ⋅ 50 rad/s . The reference value of the inverter
voltage u A, ref was used in the observer and control instead of
uA .
Figure 5 shows an example of simulated sequences, consisting of accelerations, nominal load torque steps at various
speeds, and a deceleration ramp to standstill. The results show
that the operation of the system is successful. The rotor speed
is in accordance with its reference, and the torque behaves as
expected. The rotor flux linkage is reduced by 15 % at the
highest speed because of field weakening. The voltage and
current waveforms are illustrated in detail in Fig. 6. The stator
voltage and current are nearly sinusoidal.
IV. EXPERIMENTAL RESULTS
The experimental setup is illustrated in Fig. 7. The 2.2-kW
four-pole induction motor was fed by a frequency converter
controlled by a dSPACE DS1103 PPC/DSP board. The parameters of the experimental setup correspond to those given
in Table I. Three 3.3-µF filter capacitors were used in delta
connection, giving the per-phase capacitance value of 9.9 µF.
The measured rotor speed was used as a feedback signal for
the control. The shaft torque was measured using a HBM
T10F torque flange for monitoring purposes. A permanent
magnet servo motor was used to provide load torque.
Figure 8 presents experimental results corresponding to the
simulations shown in Fig. 5. Constant motor parameters were
used in the observer and control. The measured performance
corresponds to the simulation results rather well. The total
moment of inertia of the setup was 2.2 times the inertia of the
induction motor, which explains a large part of the difference
between the measured shaft torque and the electromagnetic
torque reference during the accelerations. However, the electromagnetic torque reference is far too high in no-load operation at the highest speed, at which the main flux saturation has
decreased because of the field weakening. Therefore, the
measured saturation characteristic of the magnetizing inductance was added to the observer and control. The correspond-
0.5
0
t (s)
Fig. 5. Simulation results showing a sequence with speed and load changes.
The first subplot shows the rotor speed (solid) and its reference (dashed). The
second subplot shows the electromagnetic torque (solid) and its reference
(dashed) normalized by the rated torque TN . The third subplot shows the
rotor flux linkage (solid) and its estimate (dashed).
500
Voltage (V)
8.0 mH
9.9 µF
0.1 Ω
0
-500
4.6
4.61
4.62
4.63
4.64
4.65
4.66
4.67
4.68
4.61
4.62
4.63
4.64
Time (s)
4.65
4.66
4.67
4.68
10
Current (A)
LC Filter Parameters
Inductance L f
Capacitance C f
Series Resistance RLf
ψ (p.u.)
R
1
5
0
-5
-10
4.6
Fig. 6. Voltage and current waveforms from the simulation shown in Fig. 5.
The first subplot shows the inverter output voltage (phase-to-phase) and the
stator voltage (phase-to-phase). The second subplot shows the inverter current
and the stator current.
Freq.
converter
LC filter
PM
servo
IM
Freq.
converter
Speed
Torque
dSPACE
Fig. 7. Experimental setup. Control of induction motor (IM) is investigated,
and permanent magnet (PM) servo motor is used as loading machine. Measured shaft torque is used only for monitoring.
ing experimental results are shown in Fig. 9, and the measured
voltage and current waveforms are illustrated in detail in Fig.
10. The results are in good agreement with the simulation results when the main flux saturation is taken into account.
5
500
1
0.5
0
0
2
4
6
8
10
Voltage (V)
ω (p.u.)
m
030
0
-500
1
0
2
4
6
8
10
ψ (p.u.)
R
1
0.5
0
2
4
6
8
10
Fig. 8. Experimental results showing a sequence with speed and load changes
as observer gain is obtained by pole placement and constant motor parameters
are used in observer and control. The first subplot shows the rotor speed
(solid) and its reference (dashed). The second subplot shows the measured
shaft torque (solid) and the electromagnetic torque reference (dashed). The
third subplot shows the rotor flux reference (solid) and the estimated rotor
flux (dashed).
0.03
0.04
0.05
0.06
0.07
0.08
0
0.01
0.02
0.03
0.04
Time (s)
0.05
0.06
0.07
0.08
0
-5
Fig. 10. Experimental result showing voltage and current waveforms as the
rotation speed is 25 Hz and the load torque is 14.6 Nm. The first subplot
shows the inverter output voltage (phase-to-phase) and the stator voltage
(phase-to-phase). The second subplot shows the inverter current and the stator
current.
1
0.5
0
T/TN
0.5
0
2
4
6
8
10
2
4
6
8
10
0
2
4
6
8
10
0
2
4
6
8
10
1
0
0
0
2
4
6
8
10
ψ (p.u.)
R
1
1
1
0.5
0
t (s)
0.5
0
0
2
2
T/TN
0.02
1
0
ψ (p.u.)
R
0.01
5
-10
t (s)
ω (p.u.)
m
Current (A)
0
0
0
10
ω (p.u.)
m
T/TN
2
0
2
4
6
8
10
t (s)
Fig. 9. Experimental results showing a sequence with speed and load changes
as observer gain is obtained by pole placement and main flux saturation is
taken into account. The explanations of the curves are as in Fig. 8.
Experiments were also carried out using an observer with a
constant gain k1 = 2π ⋅ 1000 s −1 . The main flux saturation was
taken into account in the observer and control. The results are
shown in Fig. 11. The performance is nearly equal to that of
the more complicated observer.
Fig. 11. Experimental results showing a sequence with speed and load
changes as the observer gain is constant. The explanations of the curves are as
in Fig. 8.
experimental results show that the proposed control method
operates correctly. The observer gain can be selected by means
of pole placement, or a constant gain can be used.
REFERENCES
[1]
[2]
V. CONCLUSIONS
When the inverter output voltage is filtered by an LC-filter,
the vector control of an induction motor can be based on
nested control loops. The system states can be estimated by a
full-order observer, requiring only the measurements of inverter current, dc voltage, and rotor speed. Simulation and
[3]
[4]
[5]
M. Kojima, K. Hirabayashi, Y. Kawabata, E.C. Ejiogu, and T.
Kawabata, “Novel vector control system using deadbeat controlled
PWM inverter with output LC filter,” in Conf. Rec. IEEE/IAS Annu.
Meeting, vol. 3, Pittsburgh, PA, Oct. 2002, pp. 2102-2109.
A. Nabae, H. Nakano, and Y. Okamura, “A novel control strategy of the
inverter with sinusoidal voltage and current outputs,” in Proc. IEEE
PESC’94, vol. 1, Taipei, Taiwan, June 1994, pp. 154-159.
R. Seliga and W. Koczara, “Multiloop feedback control strategy in sinewave voltage inverter for an adjustable speed cage induction motor
drive system”, in Proc. EPE 2001, Graz, Austria, Aug. 2001, CD-ROM.
G.F. Franklin, J. D. Powell, and A. Emami-Naeini, Feedback Control of
Dynamic Systems, 4th ed., Prentice Hall, Upper Saddle River, NJ, 2002.
J. Niiranen, “Fast and accurate symmetric Euler algorithm for electromechanical simulations,” in Proc. Electrimacs’99, Lisboa, Portugal,
Sept. 1999, pp. 71-78.
030
[6]
[7]
[8]
6
M. Hinkkanen and J. Luomi, “Parameter sensitivity of full-order flux
observers for induction motors,” IEEE Trans. Ind. Applicat. vol. 39,
July/Aug. 2003, pp. 1127-1135
J. Steinke, C. Stulz, and P. Pohjalainen, “Use of a LC filter to achieve a
motor friendly performance of the PWM voltage source inverter,” in
Proc. IEEE IEMDC’97, Milwaukee, WI, May 1997, pp. TA2/4.1TA2/4.3.
C. Xiyou, Y. Bin, and G. Yu, “The engineering design and the optimization of inverter output RLC filter in AC motor drive system,” in Proc.
IEEE IECON'02, vol. 1, Sevilla, Spain, Nov. 2002, pp. 175-180.
APPENDIX
DIGITAL IMPLEMENTATION
The full-order observer (10) is discretized using the symmetric Euler method [5]. In contrast to the forward Euler
method, the new state values are used when available. The
discretized observer is given below in (13), where the sample
time is denoted by Ts .
⎡ R Lf
⎤
1
1
~
~
iˆAd (n + 1) = iˆAd ( n) + Ts ⎢−
iˆAd (n) + ωˆ s (n)iˆAq (n) −
uˆ sd ( n) +
u Ad (n) + k1d iAd (n) − k1q iAq (n) ⎥
Lf
Lf
⎢⎣ L f
⎥⎦
⎡ R Lf
⎤
1
1
~
~
iˆAq (n + 1) = iˆAq (n) + T s ⎢ −
iˆAq (n) − ωˆ s (n)iˆAd ( n + 1) −
uˆ sq (n) +
u Aq ( n) + k1q iAd (n) + k1d iAq ( n) ⎥
Lf
Lf
⎢⎣ L f
⎥⎦
(13a)
(13b)
⎡ 1
⎤
1 ˆ
~
~
uˆ sd ( n + 1) = uˆ sd (n) + Ts ⎢
iˆAd (n + 1) + ωˆ s (n)uˆ sq ( n) −
i sd (n ) + k 2 d i Ad ( n) − k 2 q i Aq ( n)⎥
Cf
⎢⎣ C f
⎥⎦
(13c)
⎡ 1
⎤
1 ˆ
~
~
uˆ sq (n + 1) = uˆ sq (n) + Ts ⎢
iˆAq (n + 1) − ωˆ s ( n)uˆ sd (n + 1) −
i sq (n) + k 2 q i Ad ( n) + k 2 d i Aq (n )⎥
Cf
⎢⎣ C f
⎥⎦
(13d)
⎡ 1
⎤
1 ˆ
1
~
~
iˆsd (n + 1) = iˆsd (n) + Ts ⎢ uˆ sd ( n + 1) −
i sd (n) + ωˆ s (n)iˆsq ( n) +
ψˆ R ( n) + k 3d iAd (n) − k 3q iAq ( n)⎥
τ σ′
L s′ τ r
⎣ L s′
⎦
(13e)
⎡ 1
⎤
ω ( n)
1 ˆ
~
~
iˆsq (n + 1) = iˆsq (n) + Ts ⎢ uˆ sq (n + 1) −
i sq (n) − ωˆ s ( n)iˆsd (n + 1) − m ψˆ R ( n) + k 3q i Ad (n) + k 3d i Aq (n)⎥
′
′
′
L
τ
L
s
σ
⎣ s
⎦
(13f)
⎡
1
⎣
τr
ψˆ R ( n + 1) = ψˆ R (n) + T s ⎢ R R iˆsd (n + 1) −
ωˆ s (n + 1) =
~
~
⎤
ψˆ R (n) + k 4 d iAd ( n) − k 4 q iAq (n)⎥
~
~
iˆsq (n + 1) R R + k 4 q i Ad ( n) + k 4 d i Aq (n)
ψˆ R ( n + 1)
⎦
+ ω m (n)
(13g)
(13h)