Sensorless control of induction motor drives

Sensorless Control of Induction Motor DrivesJOACHIM HOLTZ ,FELLOW,IEEE Invited PaperControlled induction motor drives without mechanical speed sensors at the motor shaft have the attractions of low cost and high reliability.To replace the sensor,the information on the rotor speed is extracted from measured stator voltages and currents at the motor terminals.Vector-controlled drives require estimating the magni-tude and spatial orientation of the fundamental magnetic flux waves in the stator or in the rotor.Open-loop estimators or closed-loop observers are used for this purpose.They differ with respect to ac-curacy,robustness,and sensitivity against model parameter varia-tions.Dynamic performance and steady-state speed accuracy in the low-speed range can be achieved by exploiting parasitic effects of the machine.The overview in this paper uses signal flow graphs of complex space vector quantities to provide an insightful description of the systems used in sensorless control of induction motors.Keywords—Adaptive tuning,complex state variables,identifi-cation,induction motor,modeling,observers,sensorless control,vector control.N OMENCLATUREAll variables are normalized unless statedotherwise.Unity vectorrotators.Stator phaseaxes.Denominator.Function of complex spaceharmonics.Field positionvector.Observer tensor.Direct axis currentsignal.Nonnormalized rms phasecurrent.Mutual inductance.Manuscript received September 3,2001;revised March 11,2002.The author is with the Electrical Machines and Drives Group,Wuppertal University,42119Wuppertal,Germany (j.holtz@).Publisher Item Identifier 10.1109/JPROC.2002.800726.Rotor inductance.Number of rotorbars.Instantaneous reactivepower.Effective transientresistance.Sector indicatorvector.Loadtorque.DC linkvoltage.Rotor-inducedvoltage.Rotor slot harmonicsvoltage.Zero sequencevoltage.Leakage-dependent zero sequencevoltage.Nonnormalized rms phasevoltage.Switching statevectors.Circumferential positionangle.Errorangle.Stator currentangle.Rotor positionangle.Total leakagefactor.Normalizedtime.Mechanical timeconstant.Rotor time constant.0018-9219/02$17.00©2002IEEEPROCEEDINGS OF THE IEEE,VOL.90,NO.8,AUGUST 20021359SubscriptsComponents in stator coordinates.Phases,winding axes.Average value.Synchronous coordinates.Maximum value.Minimum value.Positive sequence.Per phase value.Rated value.-coordinates.Originates from stator(rotor)model.Average value.Peak amplitude.Marks transient time constants.(a)(b)Fig.2.Stator winding with only phase a energized.(a)Symbolic representation.(b)Generated current density distribution.magnetic flux density,the flux linkages,and the current den-sities(magnetomotive force,MMF)are sinusoidal.Linear magnetics are assumed while iron losses,slotting effects,and deep bar and end effects are neglected.To describe the space vector concept,a three-phase stator winding is considered,as shown in Fig.2(a)in a symbolic representation.The winding axis ofphasein statorphaseby90.As the phase currents vary with time,the generated current density profile displaces in pro-portion,forming a rotating current density wave.The superposition of the current density profiles of the in-dividual phases can be represented by the spatial addition of the contributing phase currents.For this purpose,the phase currents need to be transformed into space vectors by im-parting them the spatial orientation of the pertaining phase axes.The resultingequationis the real axis of the ref-erence frame.It is normally omitted in the notation of(1)to characterize the real axis by the unityvector.As a complex quantity,the spacevector represents the si-nusoidal current density distribution generated by the phase current.Such distribution is represented in Fig.2(b).In the second term of(1),is a unity vector that indicates the direction of the winding axis ofphaseis the space vector that represents the sinusoidal current density distribution generated by the phase current .Likewisedoes represent the current density distri-bution generated by,with.Being a complex quantity,the stator current space vector,as illustrated in Fig.3.Its amplitude is proportional to.The scaling factor2/3in(1)reflects the fact that the total current density distribution is obtained as the superposition of the current density distributions of three phase windings while the contribution of only two phase windings,spaced 90Fig.4.Flux density distribution resulting from the stator currentsin Fig.3.contributing phase currents,and can be readily reconstructed as the projectionsof,as illustrated in Fig.4.It is convenient to choose theflux linkage wave as a system variable instead of the fluxdensity wave as the former contains added information onthe winding geometry and the number of turns.By defini-tion,a flux linkage distribution has the same spatial orien-tation as the pertaining flux density distribution.The statorflux linkage distribution in Fig.4is therefore represented bythe spacevector90ofpole pairs to the two-pole equivalent machine that is shownin the illustrations.It has been found convenient to normalizetimeas is the rated stator frequency ofthe machine.A rotating coordinate system is chosen to establish thevoltage equations of the induction motor.This coordinatesystem rotates at an angular statorvelocity-coordinatesystemisis the mo-tion-induced voltage that results from the varying displace-ment of the winding conductors with respect to the referenceframe.In the rotor,this displacementis is theangular mechanical velocity of the rotor,and hence the rotorvoltage equationis.Therefore,two flux linkageequations(6)are needed to establish completeness.In(6)and(7),is the angularmechanical velocity of therotor,component of the vector product of two state variables,forinstance,as(9)whenand(10b)The coefficients in(10)are the transient stator timeconstantand the rotor timeconstant,whereis an equivalent resis-tance,and is the coupling factor of the rotor.The selected coordinate system rotates at the electricalangular statorvelocity.The same time constant reap-pears asfactor.The resultingsignal,being determined by the leakage induc-tances and the winding resistances both in the stator and therotor.The dynamics of the rotor flux are governed by thelarger rotor timeconstant if the rotor is excited by thestator currentvector(11)in which thecomponent predominatesover un-less the speed is very low.A typical value of the normalizedrotor time constantisis close to unity in the base speed range.The electromagnetic torque as the input signal to the me-chanical subsystem is expressed by the selected state vari-ables and derived from(6),(7),and(9)as(12)D.Speed Estimation at Very Low Stator FrequencyThe dynamic model of the induction motor is used to in-vestigate the special case of operation at very low stator fre-quency,Fig.6.Induction motor at zero stator frequency;signal flowgraph in stationary coordinates.purpose.The angular velocity of this reference frame is zero and,hence,depends predominantly on the load torque.Partic-ularly,if the machine is fed by avoltageand.Its Laplace transformis obtained with reference to(13)as(14)As.Hence,we have from(14)(15)The right-hand side of(15)is independentofof the rotor does not exert an influence on the stator quan-tities.Particularly,they do not reflect on the stator current asthe important measurable quantity for speed identification.It is concluded,therefore,that the mechanical speed of therotor is not observableatorand,and,respectively.Its eigenbe-havior is characterized by oscillatory components of varyingfrequencies which make the system difficult to control.To illustrate the problem,a large-signal response is dis-played in Fig.7(a),showing the torque–speed characteristicat direct-on-line starting of a nonenergized rgedeviations from the corresponding steady-state characteristiccan be observed.During the dynamic acceleration process,the torque initially oscillates between its steady-state break-down value and the nominal generating torque.The initial oscillations are predominantly generated fromthe electromagnetic interaction between the two windingsystems in the upper portion of Fig.5,while the subsequentlimit cycle around the final steady-state pointat(a)(b)Fig.7.Dynamic behavior of the uncontrolled induction motor.(a)Large-signal response:direct on-line starting compared with the steady-state characteristic.(b)Small-signal response:speed oscillations following a step change of the statorfrequency.Fig.8.Constant V/Hz control.III.C ONSTANT V/H Z C ONTROL A.Low Cost and Robust DrivesOne way of dealing with the complex and nonlinear dy-namics of induction machines in adjustable speed drives is avoiding excitation at their eigenfrequencies.To this aim,a gradient limiter reduces the bandwidth of the stator fre-quency command signal as shown in Fig.8.The band-limited stator frequency signal then generates the stator voltage ref-erencemagnitudewhile its integral determines the phaseangleconst.(orand )thus obtained constitute the referencevectorof the stator voltage,which in turn controls a pulsewidth modulator (PWM)to generate the switching sequence of the inverter.Overload protection is achieved by simply inhibiting the firing signals of the semiconductor devices if the machine currents exceed a permitted maximum value.Sincediffers from the referencespeedwhen the machine is loaded.The difference is the slip frequency,equal to the electricalfrequencyFig.9.Drive control system for moderate dynamic requirements.The machine dynamics are represented here in terms of thestatevariables.The system equations are derivedin the stationary reference frame,letting(17b)where is a transient rotor time constantand,computed in stationary coordinatesas(19)from the measured orthogonal stator current componentsand in stationary coordinates,whereand,a control input variable.The active statorcurrent is proportional to the torque.Accordingly,itsreferencevalue-controller,and on the ac-tive stator current,which is proportional the rotor fre-quency.The nominalvalue,thus.The estimated speed isthen,with itsreference signal limited to prevent overloading the inverterand to avoid pull-out of the induction machine if the loadtorque is excessive.Fig.9shows that anexternalas estimated values.1366PROCEEDINGS OF THE IEEE,VOL.90,NO.8,AUGUST2002Fig.10.Rotor model in stator coordinates.Suitable models for field angle estimation are the model of the stator winding (see Fig.11)and the model of the rotor winding shown in Fig.10.Each model has its merits and drawbacks.A.The Rotor ModelThe rotor model is derived from the differential equation of the rotor winding.It can be either implemented in stator co-ordinates or in field coordinates.The rotor model in stator co-ordinates is obtained from (10b)in a straightforward manner byletting(21)Fig.10shows the signal flow graph.The measured valuesof the stator current vectorare the input signals to the model.The output signal is the rotorflux linkagevector,marked by thesuperscript .Themagnitude(22)where thesubscriptsmark the respective compo-nents in stator coordinates.The resultismarks the angular orientation of therotor flux vector.It is always referred to in stator coordinates.The functions (23)are modeled at the output of the signal flow graph Fig.10.In a practical implementation,these func-tions can be condensed into two numeric tables that are read from the microcontroller program.The accuracy of the rotor model depends on the correct set-ting of the model parameters in (21).It is particularly rotortimeconstantthat determines the accuracy of the esti-mated field angle,the most critical variable in a vector-con-trolled drive.The other model parameter is the mutual induc-tance .It acts as a gain factor as seen in Fig.10and does not affect the field angle.It does have an influence on the magnitude of the flux linkage vector,which is less critical.B.The Stator ModelThe stator model is used to estimate the stator flux linkage vector or the rotor flux linkage vector,without requiring a speed signal.It is therefore a preferred machine modelfor(a)(b)Fig.11.Stator model in stationary coordinates;the idealintegrator is substituted by a low-pass filter.(a)Signal flow graph.(b)Bode diagram.sensorless speed control applications.The stator model is de-rived by integrating the stator voltage equation (4)in statorcoordinates,.One of the two model equations (24)or (25)can be used to estimate the respective flux linkage vector,from which the pertaining field angle and the magnitude of the flux linkage are obtained.The signal flow diagram Fig.11(a)illustrates rotor flux estimation according to (25).The stator model (24)or (25)is difficult to apply in prac-tice since an error in the acquiredsignalsandin Fig.11(a).The resulting runaway of the output signal is a fundamental problem of an open integration.A negative,low-gain feedback is therefore added which stabilizes the integrator and prevents its output from increasing without bounds.The feedback signal converts the integrator into aHOLTZ:SENSORLESS CONTROL OF INDUCTION MOTOR DRIVES1367first order delay having a low cornerfrequency ,and thestator models (24)and (25)become(26)and(27)respectively.The Bode diagram [Fig.11(b)]shows that the first-order delay,or low-pass filter,behaves as an integrator for frequen-cies much higher than the corner frequency.It is obvious that the model becomes inaccurate when the frequency reduces to values around the corner frequency.The gain is then reduced and,more importantly,the90,unlessthe switching frequency of the PWM inverter is lower than about 1kHz.The other two time constants of the machine (Fig.5),the rotor timeconstant and the mechanicaltimeFig.12.Induction motor signal flow graph at forced stator currents.The dotted lines represent zero signals at rotor field orientation.constant ,are much larger in comparison.The current con-trol therefore rejects all disturbances that the dynamic eigen-behavior of the machine might produce,thus eliminating the influence of the stator dynamics.The dynamic order reduces in consequence,the system only being characterized by the complex rotor equation (10b)and the scalar equation (8)of the mechanical subsystem.Equations (10b)and (8)form a second-order system.Referring to synchronouscoordinates,of the current control loop.To achieve dynamically decoupled control of the now de-cisive systemvariables,a particular synchronous coordinate system is defined,having its real axis aligned with the rotor flux vector [8].This reference frame is the rotor fieldoriented-componentcomponent of therotor flux vector must be forced to zero.Hence,the(29)which is put into effect byadjusting-axis currentand,hence,is independently controllable.Also,the rotorflux is independently controlled bytheFig.13.Signal flow graph of the induction motor at rotor field orientation.order.The control concept also eliminates the nonlinearities of the system and inhibits its inherent tendency to oscillate during transients,illustrated in Fig.7.B.Model Reference Adaptive System Based on the Rotor FluxThe model reference approach (MRAS)makes use of the redundancy of two machine models of different structures that estimate the same state variable on the basis of different sets of input variables [9].Both models are referred to in the stationary reference frame.The stator model (26)in the upper portion of Fig.14serves as a reference model.Its output is theestimated rotor fluxvectorindicatesthatoriginates from the stator model.The rotor model is derived from (10b),where(30)This model estimates the rotor flux from the measured stator current and from a tuning signal in Fig.14.The tuning signal is obtained through a proportional-integral (PI)controller from a scalar errorsignalbetween the two estimated flux vectors.As the errorsignaland necessitatesthe addition of an equivalent bandwidth limiter in the input of the adjustable rotor model.Below the cutofffrequencyof (21)in field coordinates,whereFig.16.Model reference adaptive system for speed estimation;reference variable:rotor-induced voltage.known.Evenif).Thus,an errorin(33)which is a quantity that provides information on the rotor flux vector from the terminal voltage and current,without the need to perform an ing (33)as the reference model leaves (21)as=r =k =lrl lo r r re re l re r l re l r r l r re r r ke r l r r r r r ro r r r l lo r l le r r r r l r ro r r rer l l re re r re r r l le r r r r rr r r l le r l r r l le r r r r re r l r l r l r l l r r l r r l re ro re r r l lrro r l rr lpensation channels (thick lines at A and B )for the sensorless speed control system in Fig.17;k r .misalignment of the field-oriented reference frame.It is now assumed that the mechanicalspeedincreases-axiscomponentof the rotor-induced voltage is increased,which is the back-EMF that acts on the stator.The consequence is that rises,delayed by the transient statortimeconstant ,whichrestoresto its original zero value after the delay.Before this readjustment takes place,though,which instanta-neouslyaffects,while this disturbance is compen-sated only after a delayof by the feedforward adjustmentofthrough to the stator fre-quency input of the machine controller.This compensation channel ismarked,which essentially contributes to back-EMFvector,influences the stator current derivative.A misalign-ment between the reference frame and the rotor flux vector produces anonzerois determined by (35a)on the assumption of existing field alignment,such a deviation will invoke a correctingsignal from thecontroller.This signal is used to influ-ence,through a gainconstant(channel as well,causing the controller to accelerate or decelerate the refer-ence frame to reestablish accurate field alignment.Torque rise time of this scheme is reported around 15ms;speed accuracy iswithin 12rpm at 45rpm [11].E.Rotor Field Orientation With Improved Stator Model A sensorless rotor field orientation scheme based on the stator model is described by Ohtani [12].The upper por-tion of Fig.19shows the classical structure in whichtheFig.19.Sensorless speed control based on direct iin i r i i i is r r r r i r r re r r A i i i re r r is i r ri r ro r i r r i is i i is ir is i ro rr r r i i i i r i i i ro r i Bo in i r An im r is r i i r i r r i in r risis ro r i in r i i r rig i is r i i i r i isrr N N N DU N DRFig.20.Rotor flux estimator for the structure in Fig.19.N :numerator,D :denominator.This expression is the equivalent of the pure integralof ,on conditionthat.A transformation to the time domain yields two differentialequations(39)whereisreferred to in stator coordinates and,hence,is an ac variable,the same as the other variables.The signal flow graph in Fig.20shows that the rotor fluxvector is synthesized by the twocomponentsand ,ac-cording to (39)and (40).The high gainfactor in the upperchannelletsdominate the estimated rotor fluxvector at higher frequencies.As the stator frequency reduces,the am-plitudeofat low frequencies which deactivatesthe rotor flux controller in effect.However,the fieldanglereduces.Field orientation is finally lost at verylow stator frequency.Only the frequency of the stator currents is controlled.The currents are then forced into the machine without reference to the rotor field.This provides robustness and certain stability,although not dynamic performance.In fact,the-axis current,through (41),now relates to the correctrotor flux vector.The controller then adjusts the estimated speed and,in consequence,the field angle for a realignment of the reference frame with the rotor field.At 18rpm,speed accuracy is reported to bewithin0.03p.u.at 0.1p.u.ref-erence torque,improving significantly as the torque increases.Minimum parameter sensitivity existsat(42b)These equations represent the machine model.They are vi-sualized in the upper portion of Fig.21.The model outputs the estimatedvaluesresent the stator and the rotor in the machine model.Theequations of the full-order observer are then established inaccordance with(42).We have(43b)Kubota et al.[13]select the complex gain factors,where(44)which can be interpreted as a stator current component thatreduces the influence of model parameter errors.The fieldtransformation angle as obtained from the reduced-orderobserver is independent of rotor resistance variations[17].The complex gainFig.23.Reduced-order nonlinear observer.The MRAS block contains the structure shown in Fig.14;k= +(10 )= .VI.S TATOR F IELD O RIENTATIONA.Impressed Stator CurrentsControl with stator field orientation is preferred in combi-nation with the stator model.This model directly estimates the stator flux ing the stator flux vector to define the coordinate system is therefore a straightforward approach.A fast current control system makes the stator current vector a forcing function,and the electromagnetic subsystem of the machine behaves like a complex first-order system, characterized by the dynamics of the rotor winding.To model the system,the stator flux vector is chosen as the state variable.The machine equation in synchronous coordi-nates,(45)where is the transient rotor time constant.Equa-tion(45)defines the signal flow graph shown in Fig.24.This first-order structure is less straightforward than itsequivalent at rotor field orientation(Fig.12),although wellinterpretable:since none of the state variables in(45)hasan association with the rotor winding,such a state variableis reconstructed from the stator variables.The leakageflux.Thus,thesignalis obtained,which,although reduced in magnitudeby.Here,the velocityfactoralso explains why the rotor time constant characterizesthis subsystem,although its state variable is the stator fluxlinkagevector=.Fig.25.Machine control at stator flux orientation using adynamic feedforward decoupler.be an input.Infact,that is added from its output.To establish stator flux orientation,the stator flux linkagevector.Therefore,the-axis signals at the summingpoint-axiscurrent and the stator flux.The decoupling signal dependson the rotorfrequencyFig.26.Estimator for stator flux,field angle,speed,and rotor frequency.The estimator serves to control the system in Fig.27.N :numerator,D :denominator.for stator field orientation (46),andlettingrotate.Inaccuracies of signal acquisitionare further discussed in Section VII.The stator field angle is obtained as the integral of the statorfrequency(50)that appear as triplen harmonics with respect to the funda-mental statorvoltage .As all triplen harmonics form zero sequence systems,they can be easily separated from the much larger fundamental voltage.The zero sequence voltage is the sum of the three phase voltages in a wye-connected statorwindingare the triplen harmonicsthat originate from the saturation-dependent magnetization of the iron core.These contribute significantly to the zero sequence voltage as exemplified in the upper trace of the os-cillogram in Fig.28.To isolate the signal that represents theHOLTZ:SENSORLESS CONTROL OF INDUCTION MOTOR DRIVES1375Fig.28.Zero sequence component u of the stator voltages,showing rotor slot and saturation harmonics.Fundamental frequency ff fi fr u u fr uu f fi f funf fi f u uu u u u f u fr f u fo u fe u f f us fo u u u fr f fe fr u fi u fr uf fo fr u f fr ufiFig.29.Accurate speed identification based on rotor slot harmonicsvoltages.Fig.30.Effect of parameter adaptation shown at different values of operating speed.Left-hand side:without parameter adaptation;right-hand side:withadaptation.s to eliminate dynamic errors during accel-eration at low speed.The filtered signal feeds a PI controller,the output of which eliminates the parameter errors in a sim-plified rotor frequencyestimator,it also corrects other parameter errors in (47),such as variations of the total leakageinductanceis nevertheless maintained.Fig.30demonstrates how the rotor resistance adaptation scheme operates at different speed settings [20].The oscil-lograms are recorded at nominal load torque.Considerable 1376PROCEEDINGS OF THE IEEE,VOL.90,NO.8,AUGUST 2002Fig.31.Stator flux-oriented control without speed sensor.Speedreversal from04500rpm to+4500rpm with field weakening.speed errors,all referred to the rated speed,can be ob-served without rotor resistance adaptation.When the adap-tation is activated,the speed errors reduce to less than0.002p.u.The overshoot of the4500rpm that includes field weakening.If operated at frequenciesabove the critical low-speed range,a sensorless ac drive per-forms as well as a vector-controlled drive with a shaft sensor;even passing through zero speed in a quick transition is nota problem.As the stator frequency reduces at lower speed,the statorvoltage reduces almost in direct proportion,while the resis-tive voltageerage differential resistance,as marked by the dotted lineHOLTZ:SENSORLESS CONTROL OF INDUCTION MOTOR DRIVES1377Fig.33.Speed reversal from 060rpm to +60rpm;the estimated stator flux signal is limited to its nominalvalue.Fig.34.Forward characteristics of the power devices.in Fig.34.A more accurate model is used in [22].The differ-ential resistance appears in series with the machine winding;its value is therefore added to the stator resistance of the machine model.Against this,the influence of the threshold voltage is nonlinear which requires a specific inverter model.Fig.35illustrates the inverter topology over a switching sequence of one half cycle.The three phase currents andto all three phases,while it is the di-rections of the respective phase currents that determine their signs.Writing the device voltages as a voltage space vector (3)defines the threshold voltagevector(54)where.To separate the influence of the stator currents,(54)is expressedas(55)whereof unity magnitude.The sector indicator marks the re-spectivecanassume in the complex plane.The referencesignalof the stator voltagevectoris less than its referencevalueand of the resistive voltage drop of the power devicesthroughis the threshold voltage of the power devices,whilecan be identified during self-commissioning from the distortions of the reference voltagevectorand of the reference voltage vector are acquired while the current controllers inject sinusoidal currents of very low fre-quency into the stator windings.In such condition,the ma-chine impedance is dominated by the stator resistance.The stator voltages are then proportional to the stator currents.Deviations from a sinewave of the reference voltages that control the pulsewidth modulator are therefore caused by the inverter.They are detected by substracting the fundamental components from the reference voltages,which then yields square wave like,stepped waveforms as shown in Fig.40.The fundamental components are extracted from sets of syn-chronous samplesofand by a fast Fourier transform.The differential resistance of the powerdevices,in (57),establishes a linear relation between the load current and its1378PROCEEDINGS OF THE IEEE,VOL.90,NO.8,AUGUST 2002。