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aaahzSensorless Speed and Position Control of Induction MotorsTutorial
Joachim Holtz, Fellow, IEEE42097 Wuppertal – Germany
Abstract — Controlled induction motor drives without me-chanical sensors for speed or motor shaft position have theattraction of low cost and high reliability. The identificationof rotor speed is generally based on measured terminal volt-ages and currents. Dynamic models are used to estimate themagnitude and spatial orientation of the fundamental mag-netic flux waves in the stator or in the rotor. Open loop esti-mators and closed loop observers differ with respect to accu-racy, robustness, and sensitivity against model parametervariations. Dynamic performance and steady-state speed ac-curacy in the low speed range is achieved by exploiting para-sitic effects of the machine. The overview in this paper usessignal flow graphs of complex space vector quantities to pro-vide an insightful description of the systems used in sensor-less control of induction motors.
1. Introduction
Sensorless induction motor drives have reached the statusof a maturing technology in a broad range of applicationsranging from low-cost to high-performance systems [1].Eliminating the speed sensor on the motor shaft representsa cost advantage, which combines favorably with increasedreliability due to the absence of this mechanical componentand its sensor cable. Sensorless drives are well establishedin those industrial applications in which persistent opera-tion at lower speed is not considered essential. Ongoing re-search has focused on providing sustained operation at highdynamic performance in the very low speed range includ-ing zero speed and zero stator frequency. This paper re-views the merits and limits of sensorless control based on asurvey of the available literature.
2. Machine Dynamics
2.1 Basic equations
The graphic representation of dynamic systems by sig-nal flow diagrams is a well-established tool. Its extension tocomplex state variables [2] that represent the sinusoidaldistribution of the magnetic and electrical quantities aroundthe circular air gap of a rotating machine provides extendedinsight into the complexity of this class of dynamic systems[3]. The approach conveys information on the dynamic be-havior of ac machines in an easy-to-understand graphic no-tation.
A machine model that uses by complex space vectorsrelies on the sinusoidal distribution in space of the mag-netic energy. Space harmonics are not considered andhence such models are called fundamental models. The re-lated machine equations are
us s s
sk sj= + +r
dd
iy
yt w (1a)
0 = + + −( )r
ddr r
rk rji
yyt w w , (1b)
where ωk is the angular velocity of the reference frame, andω is the angular velocity of the rotor. The flux linkage equa-tions are
ys s s h r= +l li i (2a)
yr h s r r= +l li i . (2b)
The electromagnetic torque is proportional to the z-com-ponent of the vector product of two state variable space vec-tors, e. g. kr r sy × i z. This expression forms the link to thedynamics of the mechanical system,
t w
tm s s z L=dd Ty × −i , (3)
where τm is the normalized mechanical time constant, andTL is the load torque. Note that time is normalized asτ = ωsRt, where ωsR is the rated stator frequency.
2.2 The complex signal flow diagram
The machine equations depend on the respective spacevectors that are chosen as state variables. Choosing the sta-tor current vector and the rotor flux vector as state variablesconverts the machine equations to
t t w t t wtσ σ σ σ
' 'dd
kr r
ii i uss k s
rr
r r s= – j j+ + −( ) +11y (4a)
t t w w tr
rr k r r h s= – j
dd ly
y y+ −( ) + i (4b)
which follows from (1) and (2). The coefficients in (4) areτσ' = σ ls/rσ, rσ = rs + kr
2 rr, and kr = lh/ lr, where τ r = lr/rris the rotor time constant, and σ = 1 – kskr is the total leak-age coefficient.
The graphic interpretation of (4) is the signal flow dia-gram Fig. 1. This graph exhibits two fundamental structu-res, one on the left-hand side which represents the statorwinding, and one on the right-hand side representing the ro-
rotor
yr
wk
w
us is
stator
yrb
trj
r1s
yr
is
eTkr
lh
t 'sj
kr rst r
LT
t r'ts
tm
Fig. 1. Complex signal flow diagram of the induction motor;state variables: stator current, rotor flux
27th Annual Conference of the IEEE Industrial Electronics Society, IECON, Denver/Co, Nov. 29 - Dec. 2, 2001
aaahztor winding. Each winding is characterized by a first-orderdelay element and a normalized time constant. The timeconstant reappears as an imaginary factor in the internalfeedback path of the respective winding structure, describ-ing the cross-coupling between orthogonal space vectorcomponents of the output to the input. The ω-multiplicatordetermines the angular velocity at which the respectivewinding rotates against the ωk- reference frame [2].
2.3 Rotor field orientation
A fast current control system is usually employed toforce the stator mmf distribution to any desired location andintensity in space, independent of the machine dynamics.The dynamic order then reduces, the system being nowcharacterized by a single complex equation which is ob-tained from (1b) and (2). Referring to synchronous coordi-nates, ω k = ωs, we have
t t w tr
rr r r r h s= j
dd ly y y+ − + i , (5)
where ω r is the angular frequency of the induced rotor volt-ages. Fig. 2 shows that the stator current acts as the forcingfunction. It is commanded by, and almost equal to, the com-plex reference signal is* of the current control loop.
For dynamicallydecoupled control,all space vectors arereferred to in the ro-tating field orienteddq-coordinate sys-tem. Rotor field ori-entation defines thereal axis be alignedwith the rotor fluxvector. The imagi-nary rotor flux com-
ponent yrq is thenzero by definition,and all signals rep-resented by dottedlines in Fig. 2 as-sume zero values.
The balance at the summing point in Fig. 2 defines the con-dition for rotor field orientation
lh q r r rd=i w t y , (6)
which can be satisfied by choosing the velocity of rotationof the reference frame, and thereby ω r, appropriately. Thesignal flow diagram of the motor then assumes the familiar
structure Fig. 3, permitting decoupled control of the ma-chine torque.
Control by rotor field orientation requires on-line esti-mation of the rotor flux vector since the rotor state variablescannot be directly measured in a squirrel cage machine. In aspeed sensorless system, also the speed signal must be ob-tained by estimation techniques.
3. Speed Estimation Based on the FundamentalMachine Model
3.1 Model reference adaptive system
The model reference approach (MRAS) makes use ofthe redundancy of two machine models of different struc-tures that estimate the same state variable on the basis of dif-ferent sets of input variables. The stator model in the upperportion of Fig. 4 serves as a reference model. Its equation isderived in the stationary reference frame, ωk = 0, from (1a)and (2)
yr s s s ss( )= − −∫ u i ir d
d dls t t. (7)
Its output is the estimated rotor flux vector yr.
The rotor model equation is obtained from (1b) and (2)with reference to the stationary reference frame:
t t wtr
rr r r h s= jd
d ly y y+ + i . (8)
The model estimates the rotor flux vector from the mea-sured stator current and from a tuning signal w. The tuningsignal is obtained through a PI controller from an error si-gnal e, which represents the angular difference between thetwo estimated flux vectors. As the error signal e gets mini-mized, the tuning signal approaches the estimated speed w.The rotor model as the adjustable model then produces thesame rotor flux vector as the reference model.
Accuracy and drift problems that are inherent to theopen integration in the reference model at low speed are al-leviated by using a delay element instead of an integrator inthe stator model Fig. 4. This makes the integration ineffec-tive in the frequency range below 1/τ1, and necessitates theaddition of an equivalent bandwidth limiter in the adjustable
w
= j0+
trj
lh
is
wswr
kreT
LT
yr
isyr yrd
t r
tm
flux command
torque command
machine
kr
eT
LT
id
iq
yr
w
lh t r
tm
e
rotor model
stator modelus
rs
1kr
ss lSy
r
Ry r
trj
lhis
w
y r
t rt 1
t1 t1
Fig. 4. Model reference adaptive system for speed estimation.The superscripts of the rotor flux vectors refer to the respectivesource model
Fig. 2. Induction motor signal flowgraph: forced stator currents. The dot-ted lines represent zero signals at ro-tor field orientation.
Fig. 3. Signal flow at rotor field orientation
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rotor model. Below the cut-off frequency 1/τ1 ≈ 1 Hz, speedestimation becomes necessarily inaccurate. Even so, rever-sal of speed through zero during a fast transient process ispossible. However, if the drive is operated at zero stator fre-quency for more than a few seconds, the estimated flux goesastray and speed control is lost.
The speed control system is shown in Fig. 5. The esti-mated speed w is supplied by the model reference adaptivesystem Fig. 4. The speed controller generates a rotor fre-quency signal wr, which controls the stator current magni-tude
i ls
rd
sr2
r2=
y w t1+ ˆ , (9)
and its phase angle
δ = s r rw t w td∫ + ( )−tan ˆ1 . (10)
Equation (9) is derived from the condition (6) for rotor fieldorientation, while (10) is obtained from the steady-state so-lution isd = yrd/lh of (8) in field coordinates.
It is a particular feature of this approach that, providedthe same value of τr is used in the rotor model (8) and in thecontrol functions (9) and (10) of the speed control systemFig. 5, accurate orientation of the injected current vector isachieved even if value of τr differs from the actual rotortime constant of the machine. If the tuning controller in Fig.4 maintains nearly zero error, the rotor modelexactly replicates that dynamic relationship be-tween the stator current vector and the rotor fluxvector which exists in the actual motor [4].However, the accuracy of speed estimation, re-flected in the feedback signal ω to the speedcontroller, does depend on the error in τ r. Pa-rameter mismatch of the reference model is an-other source of inaccuracy.
Good dynamic performance of the system isreported by Schauder above 2 Hz stator fre-quency [4].
3.2 Feedforward control of stator voltages
In the approach of Okuyama et al. [5], thestator voltage reference in field coordinates,us*(F), is basically generated as a feedforwardsignal. The components of this signal are de-
rived from (4a) under the assumption of steady-state condi-tions, d/dτ ≈ 0, from which yrd = lh isd follows, and usingthe approximation ω ≈ ωs:
u r i l isd sd s s sd= –σ w s (11a)
u r i l isq sq s s sd= +s w (11b)
The d-axis current isd is replaced by its reference valueisd*. The resulting feedforward signals are represented bythe framed equations in Fig. 6. The signals depend on ma-chine parameters, which creates the need for error compen-sation by a superimposed control loop. An isd controller en-sures primarily the error correction of usd*, thus governingthe machine flux. The signal isq* that represents the torquereference is obtained as the output of the speed controller.The estimated speed ω is composed of the stator frequencyωs and the estimated the rotor frequency ω r; the latter isproportional to the torque building current isq. Since thetorque increases when the velocity of the revolving field isincreased, ωs and hence the field angle δ can be derivedfrom the isq controller.
Although the system so far described is equipped withcontrollers for both stator current components, isd and isq,the internal cross-coupling between the input variables andthe output variables of the machine flux is not eliminatedunder dynamic conditions. This means that the desireddecoupled machine structure of Fig. 3 is not established.The reason is that the q-axis current is only indirectly con-trolled through acceleration or deceleration of the rotatingreference frame.
To demonstrate this, the dynamic behavior of the ma-chine is modelled by studying the signal flow graph Fig. 1for small deviations from the state of correct field orienta-tion. The reduced flow graph Fig. 7 shows that the d-axisrotor flux is then constant, denoted as yrd 0, while a nonzeroq-axis rotor flux indicates a deviation. It is now assumedthat the mechanical speed ω changes. A decrease of ω, forinstance, increases ωr and hence produces negative dyrq/dτ.Simultaneously, the component – kr ω yrd 0 of the q-axisback-emf, which acts on the stator winding through the ma-chine coefficient kr/rσ τr in Fig. 7, is increased. Then ir rises,delayed by τsr', and restores dyrq/dτ to zero after some time.Before that, yrq has assumed a nonzero value, and field ori-
speed contr.
mains
d
field statorcoordinates
3~M
*w
wrˆ
wsw
*is*is
is
y r
us
~~PWM
CRejd
Fig. 5. Speed and current controller for MRAS estimator;CR PWM: current regulated pulsewidth modulator
mainsfield statorcoordinates
3~M
speed controller
i controllersd
i controllersq
d
d
yk2= lh tr rd 0
yk1 =1 kr rs rd 0
B
A
rs *isd −ws sls isq
rs ws lsisq + *isd
*isd
isd
kq
*usd
*usq
wrˆ w
*w *isq
k2
isq
k1
*us
is
ws
us
'ts
ejd
e-jd
~~PWM
Fig. 6. Feedforward control of stator voltages, rotor flux orientation.The feedforward signals are synthesized in the shaded frames.
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entation is lost.A similar effect occurs on a change of ωs* which instan-
taneously affects dyrq/dτ, while such disturbance is can-celled only after a delay of τsr' by the feedforward adjust-ment of usq through ωs.
Both undesired perturbations are eliminated by the addi-tion of a signal proportional to – disq/dτ to the stator fre-quency input of the machine controller. This compensationis marked A in Fig. 6 and Fig. 7.
The mechanism of maintaining field orientation needsfurther explanation. In the dynamic structure Fig. 1, the sig-nal – jωτryr, which essentially contributes to the back-emfvector, influences upon the stator current derivative. A mis-alignment between the reference frame and the rotor fluxvector produces a nonzero yrq value, giving rise to a back-emf component that changes isd. Since the feedforward con-trol of usd* is determined by (11a) on the assumption of ex-isting field alignment, this deviation will invoke a correctingsignal from the isd controller. The signal is made to influ-ence, through a gain constant kq, upon the quadrature volt-age usq* (channel B in Fig. 6 and Fig. 7) and hence on isq aswell, causing the isq controller to accelerate or deceleratethe reference frame to reestablish accurate field ali-gnment.
Torque rise time of this scheme is reported around15 ms; speed accuracy is within ± 1% above 3% ratedspeed and ± 12 rpm at 45 rpm [5].
5.5 Rotor field orientation with improved stator
model
A sensorless rotor field orientation scheme basedon the stator model is described by Ohtani [6]. Theupper portion of Fig. 8 shows the classical structurein which the controllers for speed and rotor flux gen-erate the current reference vector iqs* in field coordi-nates. This signal is transformed to stator coordinatesand processed by a set of fast current controllers. Apossible misalignment of the reference frame is de-tected as the difference of the measured q-axis currentfrom its reference value iq*. This error signal feeds aPI controller, the output of which is the estimated me-chanical speed. It is added to an estimated value ω r
of the rotor frequency, obtained with refer-ence to the condition for rotor field orienta-tion (29), but computed from the referencevalues iq* and yr*. The reason is that themeasured value iq is contaminated by in-verter harmonics, while the estimated rotorflux linkage vector yr is erroneous at lowspeed. The integration of ωs provides thefield angle δ.
The stator model is used to estimate therotor flux vector yr. The drift problems of anopen integration at low frequency areavoided by a band-limited integration bymeans of a first-order delay. This entails a se-vere loss of gain in yr at low stator fre-quency, while the estimated field angle lagsconsiderably behind the actual position of therotor field.
An improvement is brought about by thefollowing considerations. The transfer function of an inte-grator is
˜ ˜ ˜yr ir ir= = +
+1 1 1
111s sssu u
ττ (12)
where yr and uir are the Laplace transforms of the respec-tive space vectors, and uir is the rotor induced voltage inthe stator windings. The term in the right is expanded by afraction of unity value. This expression is then decomposedas
˜ ˜ ˜ ˜ ˜y y yr ir ir r1 r2= + + + ⋅ = +τ
τ τ1
1 111
11
s s su u . (13)
One can see from (12) that the factor uir /s on the rightequals the stator flux vector yr , which variable is now sub-stituted by its reference value, yr*:
˜ ˜ ˜ *y yr ir r= + + + ⋅τ
τ τ1
1 111
1s su . (14)
This expression is the equivalent of the pure integral of uir ,on condition that yr = yr
* . A transformation to the time
control system machine
w
toB
A
rs ws lsisq + *isd
*usd
kq
k1
*isd
isd
ws*w s
*usq
r1s lh
kr rstr
isq
yr
is
eT
LT
kr
wswr
tr
yrd 0
yrqt r'ts
'ts
tm
yk1 =1 kr rs rd 0
Fig. 7. Compensation channels (thick lines at A and B) for the speed sensorlesscontrol system Fig. 6
flux controller
speed contr.
iq
field statorcoordinates
ryestimator
i - controllerq
N
PWMCR
ac mains
usd
*w
*yr
wrˆ
ws
w
*yr
*yrtr
*id
*iq
(S)*is
is
y r us
3~M
ejd ~~
*iq
d
d
is
lm
1
Fig. 8. Speed sensorless control based on direct isq estimation and ro-tor field orientation. CRPWM: Current regulated pulsewidth modula-tor; N: Numerator
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domain yields two differential equations
τ τ τ τ τ1
r1r1 1 s s s s s
sdd r r
dd
y y+ = − −
u i
i' , (15)
where uir is expressed by the measured values of the termi-nal voltages and currents using the system equations (1)and (2), and
τ τ1
r2r2 r
Sddy y y+ = *( ) . (16)
It is specifically marked here by a superscript that yr*(S) is
referred to in stator coordinates and hence is an ac variable,the same as the other variables.
The signal flow graph Fig. 9 shows that the rotor fluxvector is synthesized by the two components yr1 and yr2,according to (15) and (16). The high gain factor t1 in theupper channel lets yr1 dominate the estimated rotor fluxvector yr at higher frequencies. As the stator frequency re-duces, the amplitude of us reduces and yr gets increasinglydetermined by the signal yr2 from the lower channel. Sinceyr
* is the input variable of this channel, the estimated valueof yr is then replaced by its reference value yr
* in asmooth transition. Finally, we have yr ≈ yr
* at low frequen-cies which deactivates the rotor flux controller in effect.However, the field angle δ as the argument of the rotor fluxvector is still under control through the speed controller andthe iq-controller, although the accuracy of δ reduces. Fieldorientation is finally lost at very low stator frequency. Onlythe frequency of the stator currents is controlled. The cur-rents are then forced into the machine without reference tothe rotor field. This provides robustness and certain stabil-ity, although not dynamic performance. In fact, the q-axiscurrent iq is directly derived in Fig. 9 as the current compo-nent in quadrature with what is considered the estimated ro-tor flux vector
i zq
r s
r=
׈
ˆ
y i
y, (17)
independently of whether this vector is correctly estimated.Equation (17) is visualized in the lower left portion of thesignal flow diagram Fig. 9.
As the speed increases again, rotor flux estimation be-comes more accurate and closed loop rotor flux control isresumed. The correct value of the field angle is readjusted
as the q-axis current, through (17), now relates to the cor-rect rotor flux vector. The iq-controller then adjusts the es-timated speed, and in consequence also the field angle for arealignment of the reference frame with the rotor field.
At 18 rpm, speed accuracy is reported to be within ± 3rpm. Torque accuracy at 18 rpm is about ± 0.03 pu. at 0.1pu. reference torque, improving significantly as the torqueincreases. Minimum parameter sensitivity exists at τ1 = τr[6].
3.3 Stator field orientation
An alternative approach is the orientation of the synchro-nous reference frame to the stator flux vector. Employingfast stator current control makes the stator current vector theforcing function, and a complex first-order system results inwhich the stator flux vector is a state variable. The machineequation is obtained from (1) and (2),
t t w(t t t tr
ss r s r s s r s
ss sj ( )
dd l l
dd l
y y y+ = − − + +' 'ii
i , (18)
which defines the signal flow graph Fig. 10. This structureis less straightforward than its equivalent at rotor flux orien-tation, Fig. 2. The condition for stator flux orientation, ysq =0, can be read from the balance at the upper summing pointin Fig. 10
l
did i l is r
sqsq r r sd s sd=t t w t y s' +
−( ), (19)
Fig. 9. Rotor flux estimator for the drive control system in Fig. 8;N: Numerator
r y
x2+ y2
Niq
rs
t1
ts'
is
us
iru
yrˆ
t1 ejδ*
ryy *(S)r
δ
yr1
yr2
τ1
12
ysflux command
torque command machine
tr'
ls
tr'
eT
LT
wwrˆ
ws
wr
isq
isd
t r
t r'
t r'
Fig. 11. Machine control at stator flux orientation using an exter-nal dynamic decoupler
Fig. 10. Induction motor signal flow graph, forced stator curents;state variables: stator current, stator flux. The dotted lines repre-sent zero signals at stator field orientation.
weT
t yrrj σ = sls isy jjrk tr
trj t'j r
lsis
LT
ws wr
ys
ys
is
trtr'
tm
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taking into account that the dotted lines represent signals ofzero magnitude at stator flux orientation. The dynamicstructure is then simplified as shown in the shaded area ofFig. 11.
The torque command has now an undesired influence onthe stator flux. Xu et al [7] propose a decoupling arrange-ment that is shown in the left of Fig. 11. It eliminates thecross-coupling between the q-axis current and the statorflux. The decoupling signal depends on the rotor frequency,which is estimated based on (19):
w t
ty sr
sr
rsq
sq
sd s sd=
ldid i
l i
' τ +− . (20)
Fig. 11 shows that the internal influence of isq is canceled bythe external decoupling signal.
The angular mechanical velocity can be expressed as
ˆ tanw d
t tyys
sq
sd=
dd
dd=
−1 , (21)
from which
w
y tss
ss
z=
12 ⋅ ×y yd
d (22)
is obtained. The stator flux vector ys is generated by openintegration as shown in the signal flow graph Fig. 12. Driftand accuracy problems are minimized by employing a fastsignalprocessor, self-calibrating A/D converters of highsampling rates, and automated parameter initialization [7].Smooth operation is reported at 30 rpm with rated loadtorque.
3.4 Adaptive observer
The accuracy of the open loop estimation models descri-bed in the previous chapter reduces as the mechanical speedreduces. The limit of acceptable performance depends onhow precisely the model parameters can be matched to thecorresponding parameters in the actual machine. It is par-ticularly at lower speeds that parameter deviations have sig-nificant influence on the steady-state and dynamic perfor-mance of the drive system.
The robustness against parameter mismatch and signalnoise can be improved by employing closed loop observersfor the estimation of the state variables.
A full order observer can be constructed from the ma-chine equations (4). The stationary coordinate system ischosen, ωk = 0, from which the machine model in the twoupper frames in Fig. 13 results.
Adding an error compensator to the model establishesthe observer. The error between the model current and themachine current is ei = is – i s. It is used to generate correc-tive inputs to the dynamic subsystems of the stator and therotor, which yields the observer equations from (4):
t t w t t w t
w
s s s
s
'
ˆ
dd
kr
r
ii i
u G
ss k s
rr
r r
s r s
= –j ' j
( )
+ + −( )
+ −
1
1
y
e(23a)
t t w w t wr
rr k r r h s r s= – j ( )d
d ly y y+ −( ) + −i G ˆ e (23b)
Kubota et al. [8] select the complex gain factorsG(ω) such that the two complex eigenvalues of theobserver λλλλλ1,2 obs = k . λλλλλ1,2 mach, where λλλλλ1,2 mach arethe machine eigenvalues, and k > 1 is a real constant.The value of k scales the observer by pole placement tobe dynamically faster than the machine. Given thenonlinearity of the system, the resulting complex gainsGs(ω) and Gr(ω) in Fig. 13 depend on the estimatedangular mechanical speed ω [8].
The signal ω is also required to adapt the rotorstructure of the observer to the mechanical speed of themachine. The signal is obtained through a PI controller,primarily from the current error ei. More specifically,the term |yr × εεεεε i|z represents the torque error. It is com-puted to include also the sign of the deviation betweenthe estimated speed and the actual speed. Minimumspeed is reported as 0.034 p.u. or 50 rpm.
Fig. 12. Stator flux, speed and rotor frequency estimator for thecontrol of the system in Fig. 14; N: Numerator
isq
N
isd
ls
y s
is
w rˆ
d
N
us
flux estimator
rs
ys2ˆ ys
w ws
wr -estimator
e-jd
t r'
xx2 + y2
ss l
tr
y sddt
Fig. 13. Full order nonlinear observer
statorrotor
errorcompensator
speed adaptation
trj
kr rst r
us is'tσ t rt rlm
ei
is
y r
ei
w
w
Gs )(w Gr )(w
w
∆Te
r1s
1
2
ˆ
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0
0.1
0
0.1
isα,
w
isβ
iq
0 t 1 s
offset
isq
w
1 st
isq
w
0
0
0.5
0 1 st
w 100rpm
1
0.5
0
0
0
isq 1
,ys
isq
w
ysbysa
Fig. 15. Effect of data acquisition errors;(a) gain unbalance in the current acquisition channels, (b) dc offsetin one of the current signals
Fig. 16. Speed reversal from – 60 rpm to + 60 rpm; the estimatedstator flux signal is limited to its nominal value
0
40
80
120
160
0 2 3 5 V
IC
ADiode
IGBT
25° 125°C
UCEuth
rdiff
4
Fig. 17. Forward characteristics of the power devices
4. Models for Operation at Very Low Speed
The important information on the field angle and the me-chanical speed is conveyed by the induced voltage of thestator winding, independent of the respective method that isused for sensorless control. The induced voltage ui =us – rsis is not directly accessible by measurement. It mustbe estimated, either directly from the difference of the twovoltage space vector terms us and rsis, or indirectly when anobserver is employed.
In the upper speed range above a few Hz stator fre-quency, the resistive voltage rsis is small as compared withthe stator voltage us of the machine, and the estimation ofui can be done with good accuracy. Even the temperaturedependent variations of the stator resistance are negligible athigher speed. The performance is exemplified by the oscil-logram Fig. 14, showing a speed reversal between ±4500rpm that includes field weakening. If operated at frequen-cies above the critical low speed range, a sensorless ac driveperforms as good as a vector controlled drive with a shaftsensor; even passing through zero speed in a quick transi-tion is not a problem.
As the stator frequency reduces at lower speed, the statorvoltage reduces almost in direct proportion, while the resis-tive voltage rsis maintains its order of magnitude. It be-comes the significant term at low speed. It is particularly the
stator resistance rs that determines the estimation accuracyof the stator flux vector. A correct initial value of the statorresistance is easily identified by conducting a dc test duringinitialization [9]. Considerable variations of the resistancetake place when the machine temperature changes at vary-ing load. These need to be tracked to maintain the systemstable at low speed.
4.1 Data acquisition errors
Also data acquisition errors become significant at lowspeed as the signal level of the induced voltage reduces.Current transducers convert the machine currents to voltagesignals which are subsequently digitized by analog-to-digi-tal (A/D) converters. Parasitic dc offset components super-impose to the analog signals appear and as ac components offundamental frequency after their transformation to syn-chronous coordinates. They act as disturbances on the cur-rent control system, thus generating a torque ripple, Fig.15(a).
Unbalanced gains of the current acquisition channelsmap a circular current trajectory into an elliptic shape. Themagnitude of the current vector then varies at twice the fun-damental frequency, producing undesired torque oscilla-tions as shown in Fig. 15(b).
Deficiencies like current signal offset and gain unbal-
rpm4000
0 10 st2 4 6
isq
1
1
ys
w
0
0
0
ys
isq
w
Fig. 14. Stator flux oriented control without speed sensor; speedreversal from – 4500 rpm to + 4500 rpm with field weakening
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0 Re
jImsi
0 Re
jImsi
0 Re
jImsi
ai
c– i
Udb– i
ai
b– iUd
ai
b– i
0u
1u
2u
c– i c– i
switching state vector u 1 2 0switching state vector u switching state vector u
Fig. 18. Effect at pulsewidth modulation of the forward voltages of the power semiconductors
ia
ib
jIm
is
sec(is)
ic
sector 1
Fig. 19. The six possible locations of the sector indicator sec(is);the dotted lines indicate the transitions at which the signs of therespective phase currents change
ance have not been very detrimental so far. A lower speedlimit for persistent operation is anyway imposed by drift anderror problems of the flux estimation schemes. Data acquisi-tion errors may require more attention as new solutions ofthe flux integration problem gradually evolve, Section 4.4.
The basic limitation is owed to unavoidable dc offsetcomponents in the stator voltage acquisition channels.These accumulate as drift when being integrated in a fluxestimator. Limiting the flux signal to its nominal magnitudeleads to waveform distortions, Fig. 16. The field transforma-tion angle as the argument of the flux vector gets modulatedat four times the fundamental frequency, which introduces aripple component in the torque producing current iq. The re-sulting speed oscillations may eventually render the systeminstable as the effect is more and more pronounced as thestator frequency reduces.
4.2 PWM inverter model
Also the voltage distortions introduced by the nonlinearbehavior of the PWM inverter become significant at lowspeed. They are caused by the forward voltage of the powerdevices. The respective characteristics are shown in Fig. 17.They can be modeled by an average threshold voltage uth,
and an average differential re-sistance rd as marked by thedotted line in Fig. 17. A moreaccurate model is used in [10].The differential resistance ap-pears in series with the machinewinding; its value is thereforeadded to the stator resistance ofthe machine model. Againstthis, the influence of the thresh-old voltage is nonlinear whichrequires a specific invertermodel.
Fig. 18 illustrates the in-verter topology over a switch-ing sequence of one half cycle.The three phase currents ia, iband ic, flow either through anactive device, or a recovery di-
ode, depending on the switching state of the inverter. Thedirections of the phase currents, however, do not change in alarger time interval of one sixth of a fundamental cycle.Also the effect of the threshold voltages does not change asthe switching states change in the process of pulsewidthmodulation. The inverter always introduces voltage compo-nents of identical magnitude uth to all three phases, while itis the directions of the respective phase currents that deter-mine their signs. Writing the device voltages as a voltagespace vector defines the threshold voltage vector
uth th a th b2
th c= sign( ) + a sign( ) + a sign( )u i u i u i , (24)
where a = exp(j2p/3). To separate the influence of the statorcurrents, (24) is expressed as
u sec ith th s= 2 ( )u ⋅ , (25)
where
sec i( ) =12 sign( ) + a sign( ) + a sign( )s a b
2ci i i( ) (26)
is the sector indicator [11], a complex nonlinear function ofis(t) of unity magnitude. The sector indicator marks the re-spective ± 30°-sector in which is is located. Fig. 19 showsthe six discrete locations that the sector indicator sec(is) canassume in the complex plane.
The reference signal u* of the pulsewidth modulatorcontrols the stator voltages of the machine. It follows a cir-cular trajectory in the steady-state. Owing to the thresholdvoltages of the power devices, the average value uav of thestator voltage vector us, taken over a switching cycle, de-scribes trajectories that result distorted and discontinuous.Fig. 20(a) shows that the fundamental amplitude of uav isless than its reference value u* at motoring, and larger atregeneration, Fig. 20(b). The voltage trajectories exhibitstrong sixth harmonic components in addition. Since thethreshold voltage does not vary with stator frequency as thestator voltage does, the distortions are more pronouncedwhen the stator frequency, and hence the stator voltage, arelow. They may even exceed the commanded voltage inmagnitude, which then makes correct flux estimation andstable operation of the drive impossible. Fig. 21 demon-strates how the voltage distortion caused by the inverter in-
aaahz
Fig. 21. Current waveform distortions and speed oscillationscaused by the threshold voltage of the inverter devices; sensorlesscontrol at 2 Hz stator frequency, bipolar power transistors used inthe inverter
w
0 0.6t
0.2 0.4 1 s0.8
0
0.5
0
0.1
0
0.1
isα, isβ
iq
iq
w
isα
isβ
troduces oscillations in the current and the speed signals.Using the definitions (25) and (26), an estimated value
us of the stator voltage vector is obtained from the PWMreference voltage vector u*
ˆ *u u u is th d s= − − r , (27)
where the two substracted vectors on the right represent theinverter voltage vector. The inverter voltage vector reflectsthe respective influence of the threshold voltages throughuth, and of the resistive voltage drops of the power devicesthrough rd is. A signal flow graph of the inverter model (27)is shown in the left hand side of Fig.22.
Note that uth is the threshold volt-age of the power devices, while uth isthe resulting threshold voltage vector.We have therefore from (25) the un-usual relationship |uth| = 2 uth. Thereason is that, unlike in a balancedthree-phase system, the three phasecomponents in (24) have the samemagnitude, which is unity.
4.3 Identification of the invertermodel parameters
The threshold voltage uth can beidentified during self-commissioningfrom the distortions of the reference
voltage vector u* [10, 11]. In this process, the componentsua* and ub* of the reference voltage vector are acquiredwhile the current controllers inject sinusoidal currents ofvery low frequency into the stator windings. In such condi-tion, the machine impedance is dominated by the stator re-sistance. The stator voltages are then proportional to the sta-tor currents.
Deviations from a sinewave of the reference voltagesthat control the pulsewidth modulator are therefore causedby the inverter. They are detected by substracting the funda-mental components from the reference voltages, which thenyields square wave like, stepped waveforms as shown inFig. 23. The fundamental components are extracted fromsets of synchronous samples of ua* and ub* by Fast FourierTransform.
The differential resistance of the power devices, rd in(27), establishes a linear relation between the load currentand its influence on the inverter voltage. Functionally, itadds to the resistance rs of the stator windings and henceinfluences also upon the transient stator time constant of theinduction motor, and on the design parameters of the currentcontrollers. The value (rs + rd) is estimated by an on-linetuning process described in Section 4.5.
4.4 Stator flux estimation
The inverter model (27) is used to compensate the non-
0
jIm
*u
*u
is
Re
thuuav
uav
0Re
jIm
*u
*u
uav
is
thu
uav
Fig. 20. The effect of inverter nonlinearity. The trajectories uavrepresent the average stator voltage(switching harmonics excluded)
a) at motoring
b) at generating
is
ws
ysˆ
uoffˆ
k1
sy
*sy
1
ysˆ
ejd--u*
d
rs
rd
inverter model stator flux estimator
sec(is)
d
ejd
u i
tan –1
ˆ rs from (67)tf
th2u
u s
Fig. 22. Signal flow graph of the inverter model and the stator flux estimator
aaahz
Fig. 24. Vector diagram illustrating the estimation of thestator resistance; S marks stationary reference frame (a,b )and C marks the current reference frame (x,y)
0
S
is x
jIm
γ
us
ui
rs is
ys
d
C
jy
Re
γ − dusy
linear distortions introduced by the power devices. Themodel estimates the stator voltage vector us that prevails atthe machine terminals, using the reference voltage vector u*of the pulsewidth modulator as the input variable. The in-verter model thus enables a more accurate estimation of thestator flux linkage vector. This signal flow graph is shownin the left hand side of Fig. 22.
The right hand side of Fig. 22 shows that the stator fluxvector is obtained by pure integration [11], thus avoiding theestimation error and bandwidth limitation associated withusing a low pass filter. The method necessarily incorporates
the identification of a time-varying vector uoff that repre-sents the offset voltages.
The defining equation of the stator flux estimator is
ˆ ˆ ˆ ˆys s s s off= − +( )∫ u i ur dτ , (28)
where us is the estimated stator voltage vector (27), and
ˆ ˆ* ˆuoff 1 s s
j= −( )k ey y δ (29)
is the estimated offset voltage vector, while δ is the esti-mated stator field angle. The offset voltage vector uoff in(28) is determined such that the estimated stator flux vector
ys rotates close to a circular trajectory of radius ys*, whichfollows from (28) and (29). The integrator drift is thuseliminated, while the essential information on the fieldangle δ = arg(ys) is maintained.
The stator field angle δ is computed as
ˆ tanδ = ( )−1 y ys sb a , (30)
which is symbolized by the arctan function block in Fig. 22.The magnitude of the stator flux linkage vector is then ob-tained by
ˆ ˆ ˆys s
j= −y e δ . (31)
This value is used in (29) to determine the vector of the ac-tual offset voltage. The stator frequency signal is computedby
ω δτs = d
d
ˆ, (32)
from which the angular mechanical velocity w is calculated
ˆ ˆω ω ω= −s r . (33)
4.5 Stator resistance estimation
An important measure to improve the low-speed per-formance is the accurate on-line adaptation of the statorresistance, which is the most relevant parameter in sensor-less control. Kubota et al [12] use the observer structureFig. 13 to determine the component of the error vector eiin the direction of the stator current vector, which is pro-portional to the deviation of the model parameter rs fromthe actual stator resistance. The identifying eqation istherefore
ˆ ˆr ds i s= − ( )∫1
1τ τe io (34)
The identification delay of this method is reported as 1.4 s.A faster algorithm relies on the orthogonal relationship
in steady-state between the stator flux vector and the in-duced voltage [11]. The inner product of these two vectorsis zero:
ˆ ( ) ˆ ˆ ( ) ˆ ˆy ys i s s s sq q ro ou u i= −( ) = 0 . (35)
The stator flux vector in this equation must not depend onthe stator resistance rs to facilitate the estimation of rs. Anexpression ys (q) is therefore derived from the instanta-neous reactive power q = u is s z× , which notation describesthe z-component of the vector product of the stator voltageand current vector.
The rotor equation in terms of is and ys is obtained from(1) and (2)
t t t tsr
ss r sr s
sr rs
srs= j j' '
dd r ri
i i u+ − + −
+ω ω1 1 1y (36)
were tsr' = sls/rsr and rsr = (rs + krrr ). Equation (36) is nowexternally multiplied by the vector is, from which
u i
ii i i is s s
ss s s s
rs sj = j× − × − × −
×σ ωσ ωl
dd lt t
1 y (37)
is obtained. This operation eliminates the stator and the ro-tor resistances from (36) where these parameters are con-tained in tsr'. Taking the z-component of all terms in (37)and assuming field orientation, ysd = ys and ysd = 0, we
5 st
0 4321
0.003
0
– 0.003
0.003
0
– 0.003
uα*–u1α
uβ*–u1β
uth43
uth43
Fig. 23. The distortion voltage generated by the inverter; compo-nents in stationary coordinates
aaahz
0 31 2 5 st
4
0.05
0iq
0.4
0rs
rs
iq
Fig. 27. Identification of the stator resistance, demonstrated by a25% step increase of the resistance value
have
ˆ ( )ys
q d d q s s s qd
dq
dq
r
qu i u i l i l i
did i
did
ii=
−( ) − + −
+
ωσ σ τ τ
ω τ
2
(38)
The stator flux value thus defined does not depend on thestator resistance.
To reduce the on-line computation time for the estima-tion of rs, (35) is transformed to a reference frame thataligns with the current vector. The current reference frame(xy-frame) rotates in synchronism and is displaced with re-spect to stationary coordinates by the phase angle g(t) ofthe stator current, as shown in Fig. 24. We have is(C) =is(S).exp(–jg) and consequently isx = is and isy = 0. Of thesuperscripts, (S) refers to stator coordinates and (C) refers tocurrent coordinates.
The estimated value of the stator resistance is obtainedas the solution of (34) in current coordinates
ˆ
ˆˆ
ˆˆ
ˆ ˆ ( ) sin ˆr
u u
iu q
is
sxsy
sxsy
sx
sx s s
s=
−=
− −( )yy yω γ δ
(39
using the geometric relationships
ˆ
ˆ tan ˆyy
sy
sx= −( )γ δ (40)
and
u usy i= −( )ˆ cos ˆγ δ , (41)
which can be taken from the vector diagram Fig 24. Wehave furthermore in a steady-state
ˆ ( )u qi s= ω y . (42)
The estimated stator resistance value rs from (39) isthen used as an input signal to the stator flux estimator Fig.22. It adjusts its parameter through a low pass filter. The fil-ter time constant Tf = wsRtf is about 100 ms.
4.6 Low speed performance achieved by improved models
The oscillogram Fig. 25 demonstrates the effect of theaforementioned methods on the dynamic performance atvery low speed, exemplified by a speed reversal from –10rpm to +10 rpm (fs = ws /2p = ± 0.33 Hz, ws = ± 0.007). Therecorded components ysa and ysa of the estimated statorflux linkage vector exhibit sinusoidal waveforms withoutoffset, drift or distortion, and smooth crawling speed isachieved. Fig. 26 shows the response to load step changes ofrated magnitude while the speed is maintained constant at 5rpm. This corresponds to operating at a stator frequency of0.16 Hz (ws = 0.003) during the no-load intervals. Finally,the performance of the stator resistance identificationscheme is demonstrated in Fig. 27. The stator resistance isincreased by 25% in a step change fashion. The disturbancecauses a sudden deviation from the correct field angle,which temporarily produces an error in iq. The correct valueof rs is identified after a short delay, and iq readjusts to itsoriginal magnitude.
4.7 Low speed estimation by field weakening
At very low stator frequency, the induced voltage issmall and its influence on the measured terminal quantitiesis difficult to detect. Depenbrock [13] proposes not reduc-
t0 20 s4 8 12 16
2p
0.02
0
0
1
0
w
ysαysβ
d
w
ysα
d
ysβ
– 0.02
Fig. 25. Speed reversal at 10 rpm, fundamental frequency f1 =ws/2p = ±0.33 Hz which corresponds to ws = ± 0.007
0 20 st
4 8 12 16
1
0
–1
0
2π
ia, ib
1
0
d
iq
Fig. 26. Constant speed operation at 5 rpm (f1 = ws/2p = ±0.16 Hz,ws = ± 0.003), with load step changes of rated magnitude applied.
aaahz
0 15t
5 10 25 s20
0
0.1
0.2
wr
w
0.1
0.2
0
0
0.5
1
0
0.05
ys
Tel
w
wr
Tel
ys
Fig. 29. Quasi steady-state transition through zero speed at lowload; stator and rotor frequency are kept off the region |ws|, |wr| <ws min by field weakening and by forcing a residual torque
ing the stator frequency below a certain minimum level ws
min, a level that still permits identifying the mechanicalspeed. At values below that level, the speed is controlledthrough the magnetic excitation of the machine. The methodmakes use the fact that the slip, or rotor frequency, increasesat field weakening. This is demonstrated by inserting (20)into (33) and considering steady-state, d/dt = 0, from which
ω ω ω ω τ σ= − = − −( )s min r s min
s q
r sd s d
l i
l iy (43)
is obtained. The equation is used to demonstrate how con-trolled operation at lower speed w < ws min is achieved whileoperating the machine at constant stator frequencyws = ws min. For this purpose, field weakening is introducedby reducing isd. This makes ysd reduce after a time delaythat depends on tr' and tr, Fig. 10. The rotor frequency termon the right in (43) then increases as the denominator de-creases, and also the numerator increases as the product ysdisq is constant at a given load, provided that field orientationexists.
The following oscillograms illustrate the method. Fig.28 shows controlled operation at locked rotor while thetorque is continuously varied from positive to negative val-
ues. Since w = 0, ws = wr follows. The stator frequency re-duces as Tel reduces until ws min is reached and field weak-ening begins. As the machine torque becomes negative, thestator frequency is abruptly changed from ws min to –ws minwhich makes the rotor frequency also change its sign. Thetorque magnitude subsequently increases until the machineexcitation has reached its nominal value. Thereafter, thetorque is again controlled through the stator frequency.
When operating at very low speed at light or zero load,the level of field weakening must be very small. Establish-ing the required slip to maintain the stator frequency highenough for speed estimation may then become difficult. Fig.29 shows that a small torque component, although not com-manded, is intentionally introduced to increase the slip.This, and also the time delay required for changing the ma-chine flux is tolerable for certain applications, e. g. in rail-way traction drives [10].
5. Limitations
Considering the special case of operation at very lowstator frequency, ωs → 0, the following equation can be di-rectly read from the signal flow diagram Fig. 1 :
˜ ˜yr
h
r k rs=
s jl
t w w t+ + ( )−1i , (44)
where yr and is are the Laplace transforms of the respec-tive space vectors.
The feedback signal that acts from the rotor to the statorin Fig. 1 is composed of two components. The signal can beexpressed as yrb = yr (1– jω τr), and its Laplace transform isobtained with reference to (5):
˜ ˜yrb h
r
r k rs=
js j
l1
1−
+ + ( )−wt
t w w t i . (45)
As ωs approaches zero, the feeding voltage vector usapproaches zero frequency when observed in the stationaryreference frame, ωk = 0. As a consequence, all steady-statesignals tend to assume zero frequency, and the Laplace vari-able s → 0. Hence we have from (45)
lim ˜ ˜
s rbh r
rs= r→
⋅0
y l kσ t i (46)
0 60t
20 40 100 s80
ws
w
0.2
0
0
1
ys Tel
0
– 0.2
– 0.4
0.4
0.2
0.1
0.3
0.1
0.2
0
0.3
ys
wr
w
Tel
Fig. 28. Locked rotor test to demonstrate torque control by field weakening; stator androtor frequency are kept off the region |ws|, |wr| < ws min for speed identification
aaahzsince the aforementioned two components of yrb cancel.The right-hand side of (46) is independent of ω , indicatingthat, at zero stator frequency, variations of the mechanicalangular velocity ω the rotor exert no influence on the stator.Particularly, they do not reflect on the stator current as theimportant measurable quantity for speed identification.Hence the mechanical speed of the rotor is not observable atωs = 0.
The situation is different when operating near zero statorfrequency. The aforementioned steady-state signals are nowlow frequency ac signals which get modified in phase angleand magnitude when passing through the τ r-delay elementon the right-hand side of Fig. 1. Hence, the cancelation ofthe two components of yrb is not perfect. Particularly athigher speed is a voltage of substantial magnitude inducedfrom the rotor in the stator. Its influence on measurablequantities at the machine terminals can be detected: the rotorstate variables are then observable.
The stator frequency must have a minimum value abovezero to ensure that the voltage induced in the stator windingsis sufficiently high, such reducing the influence of parame-ter mismatch and noise to an acceptable level. The inabilityto operate below this level constitutes a basic limitation forsensorless control of induction machines. This limitationholds if the machine model used for speed estimation re-flects only the effects of flux linkages with the fundamentalfields. Speed estimation at very low stator frequency is pos-sible, however, if other phenomena like rotor saliency andslotting effects are exploited. Such methods bear a promisefor speed identification at very low speed including sus-tained operation at zero stator frequency.
6. Estimation Based on Parasitic Effects
6.1 Speed estimation based on space harmonics
Secondary effects of the machine magnetics offer a po-tentiality for speed estimation. Zinger et al. [14] exploit ro-tor slot harmonics in the airgap field, which modulate thestator flux linkage with a frequency proportional to the rotorspeed. On condition that the number of rotor slots is not amultiple of three, which is true for most machines, the de-sired slot harmonic signals (uh in Fig. 30) can be separatedfrom the much larger fundamental emf by taking the sum of
the three phase voltages in a wye connected winding. Thiseliminates all nontriplen voltage components, including thefundamental, while the slot harmonic voltages add up. Fig.30(a) shows that saturation induced triplen harmonics can-not be eliminated by this technique.
An adaptive bandpass filter can be employed to suppressall components other than the slot harmonics. The centerfrequency of this filter is made to track the slot harmonicfrequency wsl using a phase locked loop (PLL). This fre-quency is a multiple of the mechanical rotor speed. Al-though of high accuracy, the slot harmonics speed signal isinappropriate for direct feedback in a high-performancespeed control system [15]. This is owed to its low-frequencyripple content originated by the low number of rotor slots. Adistorted feedback signal would deteriorate the dynamicperformance of the drive system.
0 50 mst25
– 2
2 V
0
uhr, u1
uh
u1
– 2
2 V
0
uf
uf
a)
b)
Fig. 30. Measured zero sequence voltages; (a) rotor slot harmonicvoltage uh and fundamental phase voltage u1 (at reduced magni-tude for comparison, fundamental frequency f1 = 25 Hz); (b) signaluf after bandpass filtering;
N
w r -estimatorlsσ
flux estimator
rs
e-jd
t1
1
us
wrˆ
ws
w
isq
isd
y s
is
(F)is
w rsl
wsl
wrˆ D
wrˆ
(S)y s(F)y s
d
d
rr
e-jd
Fig. 31. Machine model (upper portion), speed estimator androtor resistance adaption based on rotor slot harmonics
Fig. 32. Effect of parameter adaptation at different values ofspeed; left: speed error without parameter adaptation, right: withadaptation
0 2 s t
Dw
1
wsR
adaptation on
w∗ = 0.5
w∗ = 1
w∗ = 2
0
1
2
3%
0
1
2%
0
1
2%
aaahzIn a different approach, Jiang [15] uses the accuratespeed signal for parameter adaptation so as to improve therotor frequency estimation. The method also simplifies therotor speed estimator. As the speed loop reacts much slowerthan the current control, the derivative disq/dτ in (20) can beneglected. Equation (20) is then rewritten as
ˆ ˆw y sr r
sq
sd s sd= −r
il i
, (47)
which demonstrates that, at given load and magnetization,the estimated rotor frequency ωr is proportional to the rotorresistance. The rotor resistance estimator is shown in thelower portion of Fig. 31. The adaptation of the rotor resis-tance r r is based on two different rotor frequency signals. Afirst rotor frequency signal is obtained as wr sl = ws – wsl,where wr sl is the rotor slot harmonics signal. This signal isaccurate and serves as a reference. The second signal is theestimated rotor frequency generated by the machine modelon the basis of (47). The difference between the two signalsserves as error indicator. Fig. 31 shows that only the magni-tudes of the two signals are taken. This avoids that the signof the error signal Dwr inverts in the generator mode.
The rotor frequency error Dwr is then lowpass filtered toattenuate its ripple content and to eliminate the dynamic er-ror in wsl during acceleration. The filtered signal feeds a PI-controller, the output of which represents the model rotor re-sistance r r for rotor frequency estimation as per (47).
Fig. 32 demonstrates the effectiveness of the rotor resis-tance adaptation at rated load torque and different speed set-tings. The speed error that exists without rotor resistance ad-aptation is almost eliminated. The residual error Dw /wsR asreferred to rated speed is less than 0.002 p.u. The overshootof the w* = 2 curve is a secondary effect which is due to theabsence of a torque gain adjustment at field weakening.
6.2 Position estimation through leakage inductancemeasurement
A position sensing concept based on parasitic effectswas proposed by Jiang [16]. His approach relies on thevariation of the total leakage inductance as a function of therotor position. The conductors of a squirrel cage rotor con-sist of a finite number of discrete bars. The rotor bars arelocated in rotor slots that are uniformly distributed aroundthe rotor circumference. The mutual inductance between atthis coarse structure and the stator winding is not constant.As the angular position of the rotor is displaced for one rotorslot pitch, the mutual inductance varies between a minimumand a maximum value. This variation is reflected in corre-sponding changes of the total leakage inductance. The effectpermits counting the number of rotor slots that pass by asthe rotor rotates, just by taking frequent measurementsamples of the leakage inductance.
In most machines, the number of rotor bars is not a mul-tiple of three. Therefore, in a three phase stator winding, therespective maximum values of the leakage inductances ofthe three phases are displaced and equally distributed withinone rotor slot pitch. While the leakage inductance of onephase may assume its maximum value in a given rotor posi-tion, the leakage inductances of the other two phases areclose to their minimum values. This introduces an unbal-
ance in a wye connected stator winding that is utilized tomeasure the absolute angular rotor position within one rotorslot pitch.
When an induction motor is fed from a pulsewidthmodulated power converter, the frequent changes betweendifferent switching states produce high di/dt values in thethree phase currents. A zero sequence system cannot exist inthe stator currents since the start point of the stator windingis not connected. However, given the unbalance of the leak-age inductances in the three phases, the induced voltagesacross the leakage inductances do exhibit a zero sequencecomponent. The resulting zero sequence voltage can be eas-ily calculated from the three measured phase voltages:
u u u uσ = + +a b c . (48)
Measuring the rotor displacement within one slot pitchthrough its effect on the zero sequence voltage of the statorwinding has the benefit of producing a high signal to noise
0 2 st1
pa 0
2V
–2V
0
0pc
pb
Fig. 33. Phase components pa, pb and pc of the position signalmeasured at 0.1 Hz stator frequency
0 0.05
0pα
4 V
0.1 s
0pα
4 V
.2 1 s0 .8.6.4
0t
5
0
pα
4 V
10 s2 rpm
200 rpm
20 rpm
Fig. 34. Position signal pa measured at different values of the rotorspeed
aaahz
level, since all fundamental voltage components get can-celled in (48). Sampling rates of around 1 kHz ensure a highdynamic bandwidth of the acquired signal.
The switching state vector that is active in a particularsampling instant determines the direction of change in thecomplex plane of the harmonic current vector. Hence thecurrent must change in that very phase winding that has itswinding axis aligned with the active switching state vector.For example, if one of the switching state vectors ul(+ – –),or u4(– + +), is active, the harmonic current changes inphase a. The measured zero sequence voltage then relates tothe leakage inductance of phase a. In a similar way, switch-ing state vectors u3(– + –), or u6(+ – +), produce currentchanges in phase b, while the vectors u5(– – +), or ul(+ + –),affect the harmonic current in phase c. The taken samples ofthe zero sequence voltage then refer to phase b, or to phasec, respectively. Processing of the sampled zero sequencevoltages as described in [17] yields three phase-related posi-tion signals pa, pb and pc as shown in Fig. 33. The oscillo-gram was recorded from a 56-slot machine at steady-stateoperation and no-load. The stator frequency is 0.1 Hz whichcorresponds to 3 rpm or 0.2% of rated speed.
Note that the amplitudes of the position signals do notdepend on speed. This makes the approach particularly suit-
able for the low speed range and for operation at standstill.Fig. 34 shows the position signal pa at speed values varyingin a range of 1:100. The amplitudes of the acquired signalsare always the same.
The three phase signals pa, pb and pc can be convertedinto two othogonal components pα and pβ that uniquely rep-resent the absolute rotor position within an angular displace-ment of a rotor slot pitch. The signals permit determiningthe incremental rotor position angle with the help of an up/down counter. A synthesized position signal then consists ofthe quasi-continuous position angle that gives a high resolu-tion within a rotor slot pitch, and the discrete, signed incre-ments that indicate the number of rotor bars that have passedby. The position signals pα and pβ exhibit a relatively highfrequency as compared with the mechanical frequency ofthe motor shaft. These signals are displayed at crawlingspeed in Fig. 35. The rotor position is represented by theshaft angle dmech
The high angular resolution and the smoothness of thesignal permit high-accuracy position measurement. Thiscombines with the high dynamic bandwidth of the positionsignal, enabling unrestricted sensorless control in the lowspeed range as demonstrated in the following oscillograms.
A 7.5-kW induction motor of a machine tool spindle
0 20 st10
pα0
2V
–2V
0
2V
–2V
0
180º
– 180º
pβ
dmech
Fig. 35. Components pα and pβ of the position signal and the rotorposition angleδmech, recorded over a full motor revolution at0.2 % rated speed, or 0.1 Hz
t
dmech0
p/2
0 0.2 0.4 0.6 0.8 1 s
pα
pβ
0
4V
0
4V
100 ms
Fig. 36. Sensorless position control; traces from top: motor shaftangle δmech, rotor position signal pα, rotor position signal pβ.
0 20 st10
0
10 rpm
–10
0
180º
– 180º
w
d
Fig. 37. High-accuracy speed control at crawling speed; reversalfrom – 0.2 % to + 0.2 % of rated speed (± 0.1 Hz stator frequency)
0 5 st2.5
–π
πδf
0wmech
100rpm
0
–1iq–2
Fig. 38. Persistent operation at zero stator frequency with 120%rated torque applied, initiated and terminated by transient com-mands; traces from top: mechanical speed ωmech, normalizedtorque-building current iq, field angle df.
aaahzdrive was used for the experiments. Closed-loop sensorlessposition control was implemented using underlying controlloops for speed and stator current. Fig. 36 shows the charac-teristic signals for a positioning cycle executing periodicvariations of the motor shaft angle of ±90°.
Sensorless speed control at crawling speed is achievedby deriving a precise speed signal from the available posi-tion information. The oscillogram Fig. 37 shows the dy-namic performance during speed reversal around ±10 rpm.At this speed, the motor needs as much as 20 seconds to per-form one revolution of the motor shaft.
Sustained operation at zero stator frequency and full dy-namic performance at 120% torque overload is demon-strated in Fig. 38. In this experiment, the machine is initiallyoperated at no-load. The speed reference is set to 50 rpm, avalue that corresponds to the slip speed at 120% torqueoverload. The drive system is then subjected to a 120% stepchange of the load torque. Such disturbance causes a tran-sient increase in speed which is subsequently compensatedby a steeply rising negative torque current iq. The motor re-gains its original speed after the transient. The drive systemmaintains the following steady-state by adjusting the statorfrequency to zero. Hence the angular position of the statorfield does not change, which is indicated by a the constantvalue ϑ f of the field angle in the lower curve: the drive sys-tem operates at zero stator frequency. Its ability to operate athigh dynamic performance is demonstrated after some sec-onds by the a sudden removal of the load torque. The result-ing drop in speed causes a fast response of the torque currentiq, and the original no-load operation is reestablished.
7. Summary
Controlled induction motor drives without mechanicalsensors extract the information on speed or motor shaft po-sition from measured stator voltages and currents at the mo-tor terminals. The majority of speed identification methodsrely on the fundamental field model of the machine. Theyprovide very good results when the switched stator voltagewaveform is measured at high bandwidth, or when a nonlin-ear model is used to reconstruct the stator voltage from theinternal voltage reference signal of the control system.
The accuracy of field angle and speed estimation de-pends on the parameter settings in the model. The accuracyat which the model parameters match the machine proper-ties dominates the estimation error at low speed. Stabilityand steady-state accuracy of speed sensorless operation tendto degrade in the low speed range. The dynamic perfor-mance depends on the accuracy of field angle estimation,which is also parameter dependent.
Robust estimation techniques and parameter identifica-tion by self-commissioning or by on-line tuning offer thepotential of reducing the estimation errors. The limit of tol-erable performance can be then extended to very low speed;stable operation at zero stator frequency, though, is not pos-sible with the fundamental model of the machine, whichthen becomes unobservable.
Exploiting parasitic effects can furnish additional infor-mation on the state of the machine. Rotor slot harmonicvoltages are used for parameter tuning to achieve high speedaccuracy in the steady-state. Precise and high-bandwidth
position information is acquired utilizing the dependence ofthe total leakage inductance on the rotor position angle. Themethod opens a possibility to control the motor speed andposition even in the low speed range and at zero stator fre-quency.
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10. Th. Frenzke, F. Hoffman, H. G. Langer, „Speed SensorlessControl of Traction Drives – Experiences on Vehicles”, 8thEuropean Conference on Power Electronics and ApplicationsEPE, Lausanne, (1999), CD ROM.
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12. H. Kubota, K. Matsuse, „Speed Sensorless Field OrientedControl of Induction Motor with Rotor Resistance Adapta-tion”, IEEE Transactions on Industry Applications, Vol. 30,No. 5, 1994, pp. 1219- 1224.
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