240
IEEE
TRANSACTIONS
ON
INDUSTRY
APPLICATIONS,
VOL.
31,
NO. 2,
MARCWAPRIL
1995
Transducerless Position and Velocitv Estimation
in Induction and Salient
AC
Machines
Patrick
L.
Jansen,
Member,
ZEEE,
and
Robert
D.
Lorenz,
Senior Member,
ZEEE
Abstract-
This paper presents a viable transducerless rotor
position and velocity estimation scheme for
PWM
inverter driven
induction, synchronous, and reluctance machines with the ca-
pability of providing robust and accurate dynamic estimation
independent of operating point, including zero and very high
speeds, light and heavy loading. The injection of a balanced
three-phase high frequency signal
(500
to
2
kHz) generated by
the inverter, followed by appropriate signal demodulation and
processing combined with a closed-loop observer, enable the
tracking of rotor magnetic saliencies from the machine terminals.
Although rotor magnetic saliency is inherent within reluctance
machines, and most synchronous machines, saliency in the in-
duction machine is introduced via a modulation of the rotor slot
leakage with minimal detrimental effects on the machine per-
formance. ExDerimental verification for the induction machine is
NOMENCLATURE
Denotes estimated and commanded quantities,
respectively.
Superscripts denoting rotor and stator reference
frames.
Net viscous damping.
Band, low pass filter.
Net stator current and voltage vectors; i.e.,
Signal component stator current and voltage
vectors.
Fundamental component stator current and
voltage vectors.
Net rotor
+
load inertia.
Linear controller gains of observer.
Rotor, stator leakage inductance.
Derivative operator.
Steady state impedance seen from the stator
terminals.
Stator flux linkage vector.
Stator transient inductance.
Rotor time constant.
Rotor position.
Injected signal frequency.
Rotor velocity.
Eqds
=
[&,ids].
Paper IPCSD 94-82, approved by
the
Industrial Drives Committee
of
the
IEEE Industry Applications Society
for
presentation at the 1994 Industry
Applications Society Annual Meeting, Denver, CO, October 2-7. This
work
received financial support by the National Science Foundation (NSF) and the
Wisconsin Electric Machines and Power Electronics Consortium (WEMPEC)
of
the University
of
Wisconsin-Madison. Manuscript released
for
publication
October 27, 1994.
P. L. Jansen is with MK Rail Corporation, Boise,
ID
83706 USA.
R. D. Lorenz is with the Department
of
Electrical and Computer Engineer-
IEEE Log Number 9408 18
1.
ing, University
of
Wisconsin, Madison, WI 53706-1572 USA.
I.
INTRODUCTION
HE
elimination of position and velocity
T
ac drives has long been an attractive
transducers in
prospect. The
shaft transducers and the associated signal wiring are a sig-
nificant source of failure and cost, and add to the overall
volume and mass that must be allocated to the motor at
the work site. Numerous approaches have been proposed to
estimate the rotor position and/or velocity from the machine
terminal properties. The most success-although limited-has
been with synchronous and reluctance machines since they
are considerably less complex than the induction machine
and have inherent spatially dependent properties that can be
tracked easily. Position and velocity estimation in induction
machines, which are by far the most common machine type
and thus offer the greatest potential, are complicated due to the
machines’ symmetric smooth rotor, induced rotor currents, and
slip. Previous approaches to motion estimation in induction
machines have had serious limitations in one or more of the
following ways [1]-[13]:
zero and low speed operation is not possible,
position estimation is not possible,
velocity estimation is parameter sensitive,
velocity estimation is valid at steady state only,
estimation is load dependent (e.g., operation at either light
excessive computation/hardware requirements.
Based upon the above shortcomings, two basic requirements
to achieving accurate, parameter insensitive, dynamic position
and velocity estimation independent
of
operating point are:
1) trackable spatial modulation fixed relative to the rotor
2)
appropriate signal injection present throughout the entire
This paper presents a new, viable method of transducerless
position and velocity estimation in induction machines (and
salient ac machines) incorporating these two requirements,
thereby overcoming the limitations of the previous approaches.
Though this paper focuses on induction machines, the pro-
posed approach is applicable to all ac machines capable of
exhibiting a form of rotor magnetic saliency.
or heavy loading only), and/or
position, e.g., rotor magnetic saliency, and
operating range, including zero speed.
11.
HIGH
FREQUENCY
ROTOR
SALIENCY
IN
INDUCTION
MACHINES
The steady state per phase equivalent circuit of a symmet-
rical induction machine can be drawn as shown in Fig. l(a).
0093-9994/9.5$04.00
0
199.5 IEEE
JANSEN AND LOREN2 TRANSDUCERLESS POSITION AND VELOCITY ESTIMATION IN INDUCTION AND SALIENT AC MACHINES
I
I
(a)
(b)
Fig.
1.
(a) Steady state per phase induction machine equivalent circuit.
(b)
Approximate equivalent circuit
for
stator current modeling under high
frequency signal injection.
Fig.
2.
Illustration of instantaneous currents and
flux
paths for
high-frequency excitation over one pole pitch of a four-pole induction
machine.
The impedance seen from the stator terminals is:
where
For high-excitation frequencies, i.e.,
w;
>>
w,,
the stator
transient reactance dominates the stator impedance such that
z,
x
jWiOL,.
(3)
The high-frequency stator currents are then governed predom-
inately by the stator and rotor leakage reactances as indicated
by the approximate circuit in Fig. l(b). Since the stator and
ro-
tor leakage inductances are generally of comparable magnitude
in conventional machine designs, a spatial modulation in the
rotor leakage inductance should appear as a readily detectable
form of magnetic saliency for high-frequency signals. The
high-frequency model in Fig. l(b) implies that nearly all high-
frequency air gap flux is confined to the rotor surface as rotor
leakage flux as illustrated in Fig.
2
[15].
A
spatial modulation
wide
slot
openings
i/
(low lizakage)
narrow slot openings
(high leakage)
I
I
~
24
1
Fig.
3.
Rotor of a four-pole squirrel-cage induction motor with spatially
variant rotor leakage inductance created via modulation in
the
width of the
slot openings.
of the rotor leakage inductance can therefore be achieved via a
periodic variation in the rotor slot opening widths as shown in
Fig.
3.
The wide slot openings create high-reluctance leakage
flux paths, and hence a low inductance, whereas the narrow
openings create low-reluctance paths and a high inductance.
Completely closed rotor slot bridges are undesirable due to
saturation effects which will be discussed in Section
V.
Since stator windings are designed to accentuate electro-
magnetic phenomena spanning one pole pitch, the spatial
modulation should also have a period
of
one pole pitch.
This highlights a fundamental problem that has faced many
prior approaches that have attempted to track the effects of
individual rotor slots, as the stator windings are designed to
attenuate, not accentuate, the effects of slotting. With the rotor
saliency symmetric about each pole, the absolute rotor position
is unique over each pole pitch; i.e.,
90
mechanical degrees for
a four-pole machine.
A
modulation in the rotor leakage should have little
detrimental effect on torque or main flux because the rotor
impedance is dominated by the rotor resistance, which remains
uniform, and not the rotor leakage at the low slip frequencies
characteristic of the fundamental excitation. Furthermore, the
desired rotor leakage modulation can be designed to exist only
at high slip frequencies as proposed in Section
V.
111.
TRACKING
ROTOR
SALIENCIES
This section outlines an approach to tracking saliencies
in the rotor leakage inductance via high frequency signal
injection.
242
IEEE
TRANSACTIONS
ON
INDUSTRY
APPLICATIONS, VOL.
31,
NO.
2,
MARCWAPRIL
1995
A.
High Frequency Induction Machine Dynamic Model
reference frame is
The stator voltage equation in matrix form and the stationary
where
p
is the derivative operator. At high frequencies,
in which case, based
on
the above steady state analysis, the
stator flux linkage is predominately leakage flux and
For a symmetrical stator, the stator leakage inductance matrix
in the stationary frame is simply
(7)
For a rotor containing a spatial modulation in the rotor leakage
inductance as depicted in Fig.
3,
the rotor leakage inductance
matrix can be expressed in the rotor reference frame by
position invariant elements as
where
Llqr
#
Lldr.
Transforming to the stationary refer-
ence frame, cross coupling in addition to position dependent
inductances are introduced; i.e.,
L1r
+
AL1,
COS
207-
-ALi,
sin
20,
-AL1,
sin
20,
L1,
-
AL1,
cos
20,
L,",
=
where
8,
is the rotor position in electrical radians and
For an applied balanced polyphase high-frequency
signal
consisting of
where
wi
>>
w,.,
the resulting stator currents can be seen to
consist of both position invariant and variant components;
I;o
sin
wit
+
141
sin
(20,
-
wit)
fio
cos
wat
+
121
cos
(28,
-
Wit)
where
and
Fig.
4.
dyning to track rotor magnetic saliencies (LPF
=
low pass filter).
A
closed-loop position and velocity observer incorporating hetero-
B.
Signal Demodulation
A demodulation technique involving heterodyning can be
utilized to extract the position information from the sta-
tor currents. Multiplication of
the
currents
ZtSi
and
Z&i
by
cos
(28,
-
wit)
and sin
(28,
-
wit),
respectively, followed by
differencing; i.e.,
(14)
(where
8,
is the estimated rotor position in electrical radians)
results in two terms:
E
=
Zlsi
cos
(20,
-
wit)
-
i&
sin
(28,
-
wit)
E
=
I;o
sin
[2(wit
-
e,)]
+
Iil
sin
[2(0,
-
Or)].
(15)
The first term in (15) is at frequency
2(wi
-
Ij,)
(where
8,
=
SIj,
dt)
and contains no useful position information.
The second term, however, contains the desirable- position
information and approaches both dc and zero as
0,
-+
8,.
With the selection of signal frequencies such that
w;
>>
G,,
the first term can be easily suppressed via low pass filtering.
The remaining heterodyned and filtered signal is then in the
form of a linear position error; i.e.,
Ej
M
1i1
sin
[2(0,
-
e,)]
=2241(8,
-
8,)
-+
0
as
S,
+
8,
(16)
which can be used as a corrective error input to a Luenberger
style position and velocity observer as shown in Fig. 4 [14].
By driving
Ej
4
0,
the linear controller in Fig. 4 consisting
of gains
K1,
Kz,
and
K3
forces convergence pf the estimated
rotor position on the actual position, i.e.,
8,
-+
8,.
Both
estimated rotor position and velocity are obtained from the
observer. Because the velocity estimate,
Ij,,
taken after the
summation retains the true derivative relationship to
Or,
it is
more accurate than the estimate
Iji,
although noise and high-
frequency components that remain in
Ej
are directly imparted
upon it via the
K1
gain.
Though not required, an estimate of the electromagnetic
torque,
T,,
developed by the induction machine is used
as
feedfoward to drive an estimated mechanical system model
to improve the observer estimation dynamic accuracy. In a
field oriented system, the commanded torque can also be used.
The torque feedfoward reduces phase lag associated with the
feedback controller
[
141.
The closed-loop tracking scheme is similar to that com-
monly used in resolver-to-digital converters (with the excep-
tion of torque feedfoward). It thus shares many of the same
attributes. The accuracy of the position and velocity estimates
JANSEN AND LORENZ: TRANSDUCERLESS POSITION AND VELOCITY ESTIMATION IN INDUCTION AND SALIENT AC MACHINES
__
243
are independent of the actual leakage inductance magnitudes.
The coefficient
1;1
containing the inductance modulation am-
plitude acts merely as a gain in the closed-loop observer;
it
does not alter the value to which the observer converges. The
closed-loop architecture has low pass filtering attributes that
make the estimates highly impervious to noise. Like resolver-
to-digital converters, the observer can be implemented within
a single ASIC.
During operation, the stator current will also contain com-
ponents at the fundamental excitation frequency and at inverter
harmonics. It can be shown that the heterodyning process
shifts these components by
&(2&
-
U;)
in the frequency
domain. Thus with sufficient spectral spread these components
are also suppressed by the LPF, though some suppression is
recommended prior
to
heterodyning. The additional dynamics
of the LPF must, of course, be considered during the design
of the observer’s controller.
C.
Signal Generation and Frequency Selection
The selection of the injected signal frequency
is
primarily
dictated by four concerns: conductor and lamination skin
effects, spectral spread relative to fundamental excitation,
spectral spread relative to inverter harmonics, and the signal
generation hardware.
The frequency must be high enough for the skin effect in
the rotor bars to force a major portion
of
the high-frequency
flux crossing the air gap to be confined near the rotor surface
(as rotor leakage flux) where the modulation occurs, but low
enough such that lamination skin effects do not significantly
reduce the ratio of rotor to stator leakage. Depending upon
the precise rotor barklot shape, rotor conductor skin effects
can be significant well below 100 Hz, while lamination skin
effects can be expected to become significant beyond 400-2k
Hz depending upon lamination thickness. At 10-20+ kHz, the
lamination skin effects are generally extensive.
The injected signal can be generated via either dedicated
circuitry or via the same inverter that is producing the funda-
mental excitation. The inverter is the preferred generator based
upon cost and reliability. Switching frequencies of 10-20’
kHz are becoming increasingly common for small- to medium-
size drives. Since the inverter switching harmonics are load
dependent and not a balanced polyphase set, they are generally
not suitable for the desired signal generation. However, a
balanced, polyphase 500-2k Hz signal with low harmonic
distortion can be synthesized in addition to the fundamental
excitation with 5-20+ kHz PWM switching. Sufficient spectral
spread relative to the fundamental excitation and inverter
switching frequencies
is
thus also achieved. Fig.
5
illustrates
such a scheme.
The LPF in Fig.
5
suppresses the high-frequency signal
component from the measured currents to minimize interaction
with the current regulator. Alternatively, the current command
can be augmented to include an estimate of the signal current,
thereby achieving decoupling via command feedfoward.
The proposed demodulation scheme assumes the injected
signal voltage is a balanced polyphase set of fixed ampli-
tude. Inverter deadtime and bus voltage variations will cause
d’
S*
I
ACMachine
Vqdsi
-
with
Rotor
Saliency
/-
I
LPF
hi&
Fig.
5.
A
high-frequency signal injection scheme utilizing a
PWM
voltage source inverter
to
simultaneously synthesize
both
fundamental
and
high-frequency components.
undesirable modulation of the signal voltages resulting in
estimation errors. The minimization and/or compensation of
such effects must be addressed in the system implementation.
Large-drive systems with low-switching frequencies
(300-2k Hz) are not capable of synthesizing signal voltages
of frequencies much beyond
60-400
Hz. With these systems,
either external signal generation circuitry is required or
the signal injection scheme must be limited to zero and
low-speed operation. In the latter case, a back-EMF based
closed-loop estimation scheme such as that in [13] could be
incorporated to obtain field orientation and velocity estimation
over a wide-speed range. With sufficient current regulator
bandwidth, the signal injection and demodulation scheme can
also be implemented with voltages and currents transposed;
i.e., a balanced polyphase signal current is applied, followed
by heterodyning of the resulting signal voltages.
Iv.
MACHINE
DESIGN
FOR
TRACKING
ROTOR
SALIENCY
A.
Saturation Ezects
Localized saturation of the rotor slot bridge or opening will
induce a spatial modulation in the rotor leakage inductance
similar to that intentionally introduced in Fig. 3. The resulting
saliency, however, will align with the position of the rotor
current vector, and not the rotor position. Likewise, saturation
of the main flux path, particularly within the stator, will induce
a saliency aligned with the flux vector. Both situations can be
shown to add an additional term to the error signal of (16)
in the form:
~f
M
lil
sin
[2(8,
-
e,)]
+
I;2
sin
[2(8,
-
e,)]
(17)
where
1~
and
19,
correspond to the magnitude and position of
the saturation-induced saliency.
To
avoid position estimation errors created by saturation,
the effects of the additional term must be attenuated via a
compensation or decoupling scheme, or preferably by proper
machine design. Simply selecting the minimum slot opening
widths
to
avoid localized saturation under the highest loading
can greatly reduce estimation errors.
Note that
(17)
suggests the intentional tracking of the flux
vector (rather than rotor position) in a heavily saturated though
244
IEEE
TRANSACTIONS
ON
INDUSTRY
APPLICATIONS, VOL.
31,
NO.
2.
MARCWAPRIL
1995
d
shallow slot openings
1/
1
deepslotopenings
(low leakage)
d
filled slot openings
unfiied slot oDenings
Fig.
6.
Rotor of a four-pole induction motor with a modulation in the rotor
slot opening depths
to
create a spatially variant rotor leakage inductance With
a nearly uniform magnetizing inductance.
Fig.
7.
Rotor
of
a four-pole induction motor with a modulation in the slot
opening fill to create a spatially variant rotor leakage inductance that is seen
only at high slip frequencies.
otherwise symmetric induction machine may be possible,
thereby enabling torque control at zero and low speeds without
position estimation.
B.
Modification
of
Existing Machines
the q-axis. Since the available cross section of the slot opening
walls is considerably larger for leakage flux along the q-axis,
the q-axis leakage inductance seen at high frequencies will be
considerably larger than the d-axis inductance, even though
In machines with normally closed rotor slots, a simple
machining process using a slitting saw can be used to open
the rotor slots to varying widths. All existing induction motor
product lines with closed rotor slots are thus potential candi-
dates for implementation of the proposed estimation scheme.
However, in addition to modulating the leakage inductance,
a variation in the slot opening width may also adversely im-
pact the magnetizing inductance, possibly inducing reluctance
torque pulsations and/or decreasing the average inductance.
C.
Design
of
New Machines
With new machine designs, considerably more flexibility
exists for creating a spatial modulation of the rotor leakage
inductance. The rotor slot opening depths and fills, and the
entire slot shapes, can be varied in addition to the slot
openings. The rotor slot leakage can be designed to account
for a greater portion of the total leakage in the machine at
high frequencies.
Rather than varying the slot opening widths, the depths
of
the rotor slot openings can be varied as illustrated in Fig. 6.
With this design,
the
magnetizing inductance remains rela-
tively unaffected. Furthermore, modulation amplitudes con-
siderably higher than by simply varying slot width can be
achieved. However, the asymmetry in the rotor lamination may
be undesirable from a manufacturing standpoint, especially for
large machines that are punched a single slot at a time.
The leakage inductance modulation can also be achieved by
varying
the
fill
of
the slot opening as shown in Fig.
7.
High-
frequency currents and flux are forced to the air gap surface
along the d-axis, but only to the top of the rotor bars along
the lamination is uniform. The fundamental components will
not
see
a saliency because the rotor currents, which are at low
slip frequencies, are uniformly distributed throughout the rotor
bar. In this case, the corresponding low-frequency leakage
inductance is dictated more by the shape of the lamination,
and not the rotor bar or slot fill.
The variation in slot opening fill is easily achieved in large
machines with rotor cages fabricated of individual copper bars.
In cast cage rotors, spacers can be incorporated within the
casting process to restrain the flow of aluminum. Alternatively,
for lower volume production, the bars can be cut back in a
milling process after casting.
An
increase in inverter harmonic losses is one potential
drawback
of
this method of achieving saliency.
V.
EXPERIMENTAL
VERIFICATION
This section demonstrates the viability of creating detectable
rotor impedance modulations in existing induction machines,
followed by a laboratory implementation of the proposed
demodulation scheme.
A.
Introduction
of
Rotor Saliency
A
spatial modulation in the rotor slot leakage inductance
was introduced in a conventional three-phase, 5hp, four-pole
induction motor with closed rotor slots. Slot openings were cut
using slitting saws on a horizontal milling machine as detailed
in the appendix.
Fig.
8
illustrates the resulting spatial modulation in the mea-
sured locked-rotor terminal impedance under 60-Hz, single-
phase, line-line excitation (herein defined to be the stator