Signal Injection in Sensorless PMSM Drives Equipped With Inverter Output Filter

Abstract

This paper proposes a hybrid observer for sensorless control of permanent-magnet synchronous motor drives equipped with an inverter output LC filter. An adaptive full-order observer is augmented with a high-frequency signal injection method at low speeds. The only measured quantities are the inverter phase currents and the dc-link voltage. The effects of the LC filter on the signal injection are investigated, and it is shown that the filter is not an obstacle to using signal injection methods. The proposed method allows sensorless operation in a wide speed range down to zero speed. Experimental results are given to confirm the effectiveness of the proposed method.

Full text

Page 1

Piippo, A., Salomäki, J., and Luomi, J. (2007). Signal injection in sensorless PMSM
drives equipped with inverter output filter. In Proceedings of the Fourth Power
Conversion Conference (PCC 2007), Nagoya, Japan, pp. 1105-1110.
© 2007 IEEE
Reprinted with permission.
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Signal Injection in Sensorless PMSM Drives
Equipped With Inverter Output Filter
Antti Piippo, Janne Salom¨ aki, and Jorma Luomi
Power Electronics Laboratory
Helsinki University of Technology
P.O. Box 3000, FI-02015 TKK, Finland
Abstract—The paper proposes a hybrid observer for
sensorless control of permanent magnet synchronous motor
drives equipped with an inverter output LC filter. An adap-
tive full-order observer is augmented with a high-frequency
signal injection method at low speeds. The only measured
quantities are the inverter phase currents and the dc-link
voltage. It is shown that the LC filter is not an obstacle
to using signal injection methods. The proposed method
allows sensorless operation in a wide speed range down to
zero speed. Experimental results are given to confirm the
effectiveness of the proposed method.
Index Terms—Inverter output filter, permanent magnet
motors, sensorless control, signal injection.
I. INTRODUCTION
Problems may be encountered in ac motor drives due
to the non-sinusoidal voltage produced by a pulse-width
modulated (PWM) inverter. The high rate of change of
the voltage (i.e. high du/dt) may cause excessive voltage
stresses in the stator winding insulations. It may also cause
leakage currents through the parasitic capacitances of
the stator winding and produce bearing currents. Voltage
harmonics cause acoustic noise and power losses; the
losses caused by eddy currents are a special concern in
high-speed solid-rotor motors.
A common approach to overcome these problems is to
use an inverter output filter [1]–[4]. An LC filter, having
the resonance frequency below the switching frequency,
is a typical choice for the filter topology if a nearly
sinusoidal output voltage is required. However, this kind
of heavy filtering makes the motor control more difficult.
A conventional scalar control method is usually employed,
but when better control performance is needed, a vector
control method must be used. For vector control, the filter
dynamics should be taken into account.
Various methods have been proposed for the vector
control of variable-speed drives equipped with an LC filter.
Methods based on feedforward action and sliding mode
control have been used for compensating the effects of
the filter in a sensorless permanent magnet synchronous
motor (PMSM) drive in [2]. Direct measurement of the
stator voltages and currents has been used for obtaining
feedback signals for the current control and the speed
and position estimation in a sensorless PMSM drive [3].
A feedforward current controller has been used in a
This work was financially supported by ABB Oy and Walter Ahlstr¨ om
foundation.
speed-sensored synchronous reluctance motor drive with
an LC filter [4]. For induction motor drives equipped with
an LC filter, vector control methods based on cascaded
controllers have been proposed in [5], [6].
Due to the LC filter, the voltages and currents at the
motor terminals differ from those of the inverter output.
Frequency converters are equipped with measurements of
output currents, and output voltages are also known. For
vector control, however, the corresponding quantities at
the motor terminals are needed. In [2]–[6], the motor
voltages or currents are measured by additional sensors,
requiring hardware modifications in the motor drive. If
the motor quantities are estimated instead, as in [7]–[9]
for induction motor drives, the additional measurements
are avoided and a filter can be added to an existing
drive. Recently, a similar method has been proposed for
sensorless PMSM drives [10].
The above mentioned sensorless techniques are based
on fundamental-excitation methods, which do not allow
sustained operation at low speeds. As a solution to this
problem, high-frequency (HF) signal injection methods
have been developed for sensorless PMSM drives without
a filter [11]–[13]. However, such methods have not been
previously applied to ac motor drives equipped with an
inverter output LC filter.
In this paper, a hybrid observer is proposed where a
speed-adaptive full-order observer [10] is augmented with
a pulsating high-frequency (HF) signal injection technique
[14] at low speeds. Signal injection methods are shown to
be feasible even with the filter if the frequency of the
injected voltage signal is suitably chosen. The proposed
system is validated by means of simulations and laboratory
experiments.
II. FILTER AND MOTOR MODELS
Fig. 1 shows a PMSM drive system equipped with an
LC filter. The inverter output voltage uAis filtered by the
LC filter, resulting in a nearly sinusoidal stator voltage us.
The inverter output current iAand the dc-link voltage udc
are the only measured quantities.
The PMSM and the LC filter are modeled in the d-q
reference frame fixed to the rotor. The d axis is oriented
along the permanent magnet flux, whose angle in the stator
reference frame is θm in electrical radians. The stator
voltage equation is
us= Rsis+˙ψs+ ωmJψs
(1)
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Lf
Cf
is
s
us
A
is
A
us
s
udc
ωm,ref
Control
PMSM
Inverter
Diode bridge
Fig. 1.
Superscript s denotes the stator reference frame.
PMSM drive system equipped with three-phase LC filter.
where us = [usd
[isd
isq]Tthe stator current, ψs= [ψsd
stator flux, Rsthe stator resistance, ωm=˙θmthe electrical
angular speed of the rotor, and
usq]Tis the stator voltage, is =
ψsq]Tthe
J =
?
0
1
−1
0
?
The stator flux is
ψs= Lsis+ ψpm
(2)
where ψpm= [ψpm 0]Tis the permanent magnet flux
and
?
0
Ls=
Ld
0
Lq
?
is the inductance matrix, Ldand Lqbeing the direct- and
quadrature-axis inductances, respectively. The electromag-
netic torque is given by
Te=3p
2ψT
sJTis
(3)
where p is the number of pole pairs.
The LC filter equations are
uA− us= RLfiA+ Lf˙iA+ ωmLfJiA
iA− is= Cf˙ us+ ωmCfJus
where iA =
?iAd
?uAd
tance and RLf the series resistance of the filter inductor,
and Cf is the filter capacitance.
(4)
(5)
iAq
?Tis the inverter current, uA =
?Tthe inverter output voltage, Lf the induc-
uAq
III. CONTROL SYSTEM
Fig. 2 shows a simplified block diagram of the control
system (the estimated quantities being marked by the
symbol ˆ). The cascade control and speed-adaptive full-
order observer are implemented in the estimated rotor ref-
erence frame. The estimated rotor positionˆθmis obtained
by integrating ˆ ωm. The inverter current, the stator voltage,
and the stator current are controlled by PI controllers, and
cross-couplings due to the rotating reference frame are
compensated [5]. A maximum torque per current method
is used for calculating the stator current reference [15].
The rotor speed is governed by a PI controller with active
damping.
IV. OBSERVER STRUCTURE
In the following, the speed-adaptive full-order observer
proposed in [10] is augmented with an HF signal injection
technique to stabilize the observer at low speeds. The two
methods are combined in a fashion similar to [16]. The
observer gain is modified for better compatibility with the
HF signal injection method.
A. High-Frequency Signal Injection
The HF signal injection method is based on [14]. A
carrier excitation signal
uc= ˆ uccos(ωct)
?
1
0
?
, (6)
+?+?
Stator
current
control
PMSM
uc
Error
Signal
ε
Speed
control
Inverter
current
control
Stator
voltage
control
Adaptive
full-order
observer
us,ref
is,ref
uA,ref
iA,ref
ˆis
ˆ us
iA
ˆθm
ωm,ref
ˆ ωm
eJθ
e−Jθ
Fig. 2. Simplified block diagram of the control system. Double lines indicate vector quantities whereas single lines indicate scalar quantities. Vector
quantities on the left-hand side of coordinate transformations are in the estimated rotor reference frame and on the right-hand side in the stator
reference frame. The speed control includes the calculation of the stator current reference according to the maximum torque per current method.
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TABLE I
SYSTEM DATA
Motor Data
Nominal voltage UN
Nominal current IN
Nominal frequency fN
Nominal torque TN
Stator resistance Rs
Direct-axis inductance Ld
Quadrature-axis inductance Lq
Permanent magnet flux ψpm
Total moment of inertia
370 V
4.3 A
75 Hz
14.0 Nm
3.59 Ω
0.036 H
0.051 H
0.545 Vs
0.015 kgm2
LC Filter Parameters
Inductance Lf
Capacitance Cf
Series resistance RL f
5.1 mH
6.8 µF
0.1 Ω
Control System Parameters
Inverter current control bandwidth
Stator voltage control bandwidth
Stator current control bandwidth
Speed control bandwidth
Speed adaptation bandwidth αfo
Bandwidth αiat zero speed
2π·600 rad/s
2π·400 rad/s
2π·200 rad/s
2π·5 rad/s
2π·100 rad/s
2π·5 rad/s
having amplitude ˆ ucand angular frequency ωc, is super-
imposed on the inverter voltage reference in the estimated
rotor reference frame. An HF current response is detected
on the q axis of the estimated rotor reference frame,
amplitude modulated by the rotor position estimation error.
The current signal is band-pass filtered, demodulated,
and low-pass filtered to obtain an error signal, which is
approximately
ε ≈ Kεsin(2˜θm)
where Kεis the signal injection gain and˜θm= θm−ˆθm
is the estimation error of the rotor position. Without the
inverter output LC filter, the signal injection gain would
be
Kε=ˆ uc
ωc
(7)
Lq− Ld
4LqLd
(8)
The error signal ε is used as a correction in the adaptive
full-order observer as will be described in the following
subsection.
To illustrate the effect of the filter, frequency responses
of the system were calculated numerically. Parameters
given in Table I were used for this example. The inverter
d-axis current response to the inverter d-axis voltage is
shown in Fig. 3(a) for the rotor position estimation error
˜θm = 10◦. The amplitude response has a notch at the
parallel resonance frequency of the filter capacitor and the
stator inductance, and a peak at the LC filter resonance
frequency. Above the frequency f ≈ 500 Hz, the LC
filter amplifies the response as compared to the response
obtained for the PMSM without the filter.
Fig. 3(b) shows the inverter q-axis current response to
the inverter d-axis voltage for the rotor position estimation
error˜θm= 10◦. This response is important for the pul-
sating HF signal injection method. Compared to Fig. 3(a),
the parallel resonance of the filter capacitance and the d-
axis inductance cannot be seen in the response, but the LC
filter resonance peak is visible. At higher frequencies, the
10−4
10−2
100
100
101
102
103
104
−90
0
90
|iAd/uAd|
? {iAd/uAd} (◦)
f (Hz)
(a)
10−6
10−4
10−2
100
100
101
102
103
104
−360
−180
0
|iAq/uAd|
? {iAq/uAd} (◦)
f (Hz)
(b)
Fig. 3.
response to inverter d-axis voltage; (b) inverter q-axis current response
to inverter d-axis voltage. Dashed lines show the response of the PMSM
only, solid lines the response of LC filter and PMSM.
Frequency responses of the system: (a) inverter d-axis current
amplitude response decays rapidly. In HF signal injection,
it is reasonable to select the frequency of the carrier
excitation signal below the resonance frequency. If the
excitation frequency is selected close to the resonance
frequency, the HF current amplitude, and hence also the
signal injection gain Kε, are increased by the LC filter. At
the frequency of 500 Hz, the LC filter increases the signal
injection gain Kεby a factor of 1.65 in this example.
B. Speed-Adaptive Full-Order Observer
The adaptive full-order observer is based on the dy-
namic models of the PMSM and the LC filter. The inverter
current serves as the feedback signal for the observer, and
the electrical angular speed of the rotor is estimated using
an adaptation mechanism. The stator flux is selected as
the state variable representing the electrical dynamics of
the motor by inserting the stator current solved from (2) to
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(1). When the dynamic equations of the filter are included,
the observer is defined by
˙ˆ x =ˆA ˆ x +ˆB
?
uA
ˆψpm
?T
+ K(iA−ˆiA)
(9)
where ˆ x =?ˆiA
the permanent magnet flux estimateˆψpmare considered as
inputs to the system. The system matrices and the observer
gain matrix in (9) are


0




0
ˆRsˆL−1
s


respectively, where I is the 2x2 unit matrix. The factors
kidand kiq(i = 1, 2, 3) are scalar gain parameters. The
estimate for the stator current, required for the controller
feedback, is calculated based on (2):
ˆ us
ˆ ψs
?T. The inverter voltage uAand
ˆA =

−ˆRLfˆL−1
ˆC−1
fI
fI
−ˆL−1
0
fI0
−ˆC−1
−ˆRsˆL−1


f
ˆL−1
s
I
s



− (ˆ ωm− ωε)
J
0
0
0
J
0


0
0
J
(10)
ˆB =

ˆL−1
fI
0
0
ˆC−1
f
ˆL−1
s

(11)
K =
k1dI + k1qJ
k2dI + k2qJ
k3dI + k3qJ


(12)
ˆis=ˆL−1
s(ˆ ψs−ˆ ψpm)
(13)
The speed adaptation law is
ˆ ωm= −kp(iAq−ˆiAq) − ki
?
(iAq−ˆiAq)dt
(14)
where kp and ki are nonnegative adaptation gains. Ac-
cording to [16], these gains are selected as
kp=
2αfo
ˆψpm/ˆLq
,ki=
α2
fo
ˆψpm/ˆLq
(15)
where the design parameter αfo corresponds to the ap-
proximate bandwidth of the speed adaptation.
In (10), the speed correction term ωε is based on the
HF signal injection method. It is obtained from the error
signal ε by means of a PI mechanism
ωε= γpε + γi
?
εdt
(16)
where γp and γi are nonnegative gains. The gains are
selected as [17]
γp=
αi
2Kε,γi=
α2
6Kε
i
(17)
where αiis the approximate bandwidth of the PI mecha-
nism. In (17), the influence of the LC filter on the signal
injection gain Kεshould be taken into account.
The effect of the signal injection method is reduced
linearly as the rotor speed increases, reaching zero at the
transition speed ω∆. Both the HF excitation voltage ˆ uc
−1
0
1
f(ˆ ωm), g(ˆ ωm)
ˆ ωm
−ω∆
0
ω∆
Fig. 4.
and proposed function g(ˆ ωm) (solid) as functions of estimated rotor
speed.
Function f(ˆ ωm) (dotted), original function g(ˆ ωm) (dashed),
and the approximate bandwidth αi of the PI mechanism
in (16) are decreased, i.e.
ˆ uc= f(ˆ ωm)ˆ uc0,αi= f(ˆ ωm)αi0
(18)
where ˆ uc0 and αi0 are the zero-speed values of the
HF excitation voltage and the PI mechanism bandwidth,
respectively. The function f(ˆ ωm) is shown in Fig. 4 as a
function of the estimated rotor speed.
In [10], the observer gain was selected as
K =



k1dI
0
k3dI + k3qsign(ˆ ωm)
?
???
g(ˆ ωm)
J



(19)
When the observer is augmented with the signal injection
method, however, zero-speed operation becomes unstable
due to the discontinuity of the signum function. It is better
to use a smoother speed-dependent function
K =



k1dI
0
k3dI + k3q(2/π)atan(ksˆ ωm/ω∆)
?
where ksdefines the steepness of the g(ˆ ωm) function. A
value ks= 5 is used in the following. This modification
is illustrated in Fig. 4.
???
g(ˆ ωm)
J



(20)
V. RESULTS
The proposed method was investigated by means of
simulations and laboratory experiments. The data of the
six-pole interior-magnet PMSM (2.2 kW, 1500 rpm) and
the LC filter, and the cascade control bandwidths are given
in Table I. The base values for the voltage, current, and
angular speed are defined as UB =
√2IN, and ωB = 2πfN, respectively. The electromag-
netic torque is limited to 22 Nm, which is 1.57 times the
nominal torque TN. The high-frequency carrier excitation
signal has a frequency of 500 Hz and an amplitude of
30 V. The transition speed ω∆ = 0.13 p.u. The digital
implementation of the adaptive full-orderobserver is based
on a simple symmetric Euler method [18], [7].
The experimental setup is illustrated in Fig. 5. The
PMSM is fed by a frequency converter that is controlled
by a dSPACE DS1103 PPC/DSP board. Mechanical load is
?2/3UN, IB =
1108