This paper presents a magnetic saliency-based position estimation approach for a permanent magnet (PM) motor fed by a voltage-source pulse width modulation (PWM) inverter. The proposed real-time estimation algorithm detects motor current harmonics and calculates the inductance matrix, including rotor position information. Position estimation can be performed every period of PWM or carrier cycle. An experimental system using an interior permanent magnet (IPM) synchronous motor has been constructed. Experimental results verify that position estimation within 10° in electrical angle is obtained at standstill and at speeds as low as 1 r/min by the proposed approach</p

Akagi, Hirofumi

Ogasawara, Satoshi

Okayama University Scientific Achievement Repository

EngineeringElectrical Engineering fieldsOkayama University Year 1997An approach to real-time positionestimation at zero and low speed for aPM motor based on saliencySatoshi Ogasawara Hirofumi AkagiOkayama University Okayama UniversityThis paper is posted at eScholarship@OUDIR : Okayama University Digital InformationRepository.http://escholarship.lib.okayama-u.ac.jp/electrical engineering/34IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 34, NO. 1, JANUARY/FEBRUARY 1998 163An Approach to Real-Time Position Estimationat Zero and Low Speed for a PMMotor Based on SaliencySatoshi Ogasawara, Senior Member, IEEE, and Hirofumi Akagi, Fellow, IEEEAbstract—This paper presents a magnetic saliency-based posi-tion estimation approach for a permanent magnet (PM) motor fedby a voltage-source pulsewidth modulation (PWM) inverter. Theproposed real-time estimation algorithm detects motor currentharmonics and calculates the inductance matrix, including rotorposition information. Position estimation can be performed everyperiod of PWM or carrier cycle. An experimental system using aninterior permanent magnet (IPM) synchronous motor has beenconstructed. Experimental results verify that position estimationwithin 10 in electrical angle is obtained at standstill and at speedsas low as 1 r/min by the proposed approach.Index Terms—Interior permanent magnet synchronous motor,magnetic saliency, sensorless drive.I. INTRODUCTIONIN RECENT years, significant attention has been paidto removing rotor position sensors from ac motor drivesystems with a position-closed loop [1]–[8]. The rotor positionestimation methods of a permanent magnet (PM) synchronousmotor can be classified into the following two groups:1) based on the back EMF produced in the stator windings[1], [2];2) based on magnetic saliency [3]–[6].Conventional position estimation based on detection of backEMF is suitable for middle- and high-speed applications.However, at low speed and standstill, the EMF is too smallto achieve accurate position estimation. On the other hand,magnetic saliency-based position estimation can be potentiallyemployed at any speed, including zero speed.Lorenz et al., have proposed a sophisticated approach toestimation of flux, position, and speed at zero and low speedin ac motors [4], [5]. The approach is based on tracking themagnetic saliency via inverter-generated high-frequency signalinjection with demodulation incorporating heterodyning and aclosed-loop observer.This paper presents a magnetic saliency-based positionestimation approach for a PM motor fed by a voltage-sourcepulsewidth modulation (PWM) inverter. The proposed real-Paper IPCSD 97–59, presented at the 1996 Industry Applications SocietyAnnual Meeting, San Diego, CA, October 6–10, and approved for publicationin the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Industrial DrivesCommittee of the IEEE Industry Applications Society. Manuscript releasedfor publication August 12, 1997.The authors are with the Department of Electrical Engineering, OkayamaUniversity, Okayama, 700 Japan (e-mail: ogasawara@elec.okayama-u.ac.jp;akagi@elec.okayama-u.ac.jp).Publisher Item Identifier S 0093-9994(98)00880-9.Fig. 1. Model of salient PM motor.time estimation algorithm detects motor current harmonics andcalculates the inductance matrix, including rotor position in-formation. Position estimation can be performed every periodof PWM, without any special signal injection.An experimental system consisting of an interior permanentmagnet (IPM) synchronous motor and a voltage-source PWMinverter has been constructed and tested to verify the effec-tiveness and versatility of the approach to position estimationat zero and low speed. The IPM motor has the magneticsaliency of the -axis inductance being larger than the -axis inductance. Experimental results show that the proposedapproach has the capability of estimating the rotor position atstandstill and at speeds as low as 1 r/min.II. HARMONIC MODEL OF PM MOTORFig. 1 shows the simplified model of a PM motor withmagnetic saliency. The equations describing the PM motorare given by(1)where(2)0093–9994/98$10.00  1998 IEEE164 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 34, NO. 1, JANUARY/FEBRUARY 1998Fig. 2. Voltage vectors of inverter output.The equations can be simplified by applying the space vectortheory:(3)Here, the inductance matrix is represented by(4)and it contains the rotor position information of the PM motor.When the PM motor is driven by a voltage-source PWM in-verter, the motor current contains a small amount of harmoniccurrent. The voltage and current vectors can be separated intothe fundamental and harmonic components as follows:(5)(6)Because the resistive voltage drop can be disregarded, theharmonic equivalent circuit can be approximated by the in-ductance:(7)The above equation indicates that the inductance matrix can becalculated from the harmonic components of the motor voltageand current vectors, so that the rotor position can be estimated.III. HARMONIC VOLTAGE AND CURRENT VECTORSA three-phase voltage-source inverter can output eight dis-crete voltage vectors, including two zero vectors, as shown inFig. 2. The PWM inverter has the capability of producing anoutput voltage vector by selecting the discrete voltage vectorsselectively in a modulation period. The average voltage vectoris(8)where means the time ratio of to the modulation period:(9)Fig. 3. Variation of motor current.Therefore, the harmonic voltage vector is equal to a dif-ference between the inverter output voltage vector and theaverage output voltage vector:(10)On the other hand, applying the approximation of (7) impliesthe motor current changes linearly, as shown in Fig. 3. Sincethe motor current includes both the harmonic and fundamentalcomponents, the two components should be separated fromeach other. The current variation for modulation period isshown by(11)where are the current variations for the intervalsof respectively. Assuming that the fundamentalcomponent changes linearly in a modulation period, the har-monic component of the current variation can be separated bythe following equation:(12)As a result, we can extract only harmonic components of thevoltage and current vectors.IV. ESTIMATION OF INDUCTANCEMATRIX AND ROTOR POSITIONFrom (7), a relationship between the harmonic componentsof the voltage vector and the current variation vector is givenby(13)Lumping the above equations together over a modulationperiod gives the following equation:(14)The transposed equation is......(15)OGASAWARA AND AKAGI: REAL-TIME POSITION ESTIMATION FOR A PM MOTOR 165Therefore, the inductance matrix can be calculated as follows:......(16)Here, the “ ” in the above equation is the left pseudoinverseoperator [9], and it performs the following calculation:(17)Therefore, the rotor position can be calculated from theinductance matrix:(18)The inverse matrix of must be existent, otherwise wecannot calculate the left pseudoinverse matrix in (16). Thematrix of is calculated as follows:......(19)If all or components in are zero, the determinantof becomes zero. Similarly, if all of them are lin-early dependent, the inverse matrix is not existent becauseTherefore, PWM control should be modifiedso that all of them are not linearly dependent vectors.Fig. 4(a) shows a timing chart in a conventional sinusoidalPWM scheme. Comparison of the voltage referencesand with a triangular carrier determines the output voltagevector. In this case, and are output vectorsfor and respectively. The following equationexpresses the average voltage vector during a modulationperiod:(20)where(21)For example, when the average voltage vector exists on thecentroid of a triangle formed by and in Fig.4(b), the time ratios are(22)In this case, and are linearly independentof each other, except that is linearly dependent on As(a)(b)Fig. 4. Conventional sinusoidal PWM scheme. (a) Comparing voltage ref-erences with triangle. (b) Voltage vectors.a result, we can calculate the inductance matrix and estimatethe rotor position.However, it is not necessary to output when the averagevoltage vector exists on the axis, i.e., In thiscase, and are linearly dependent on each other,because moves in the opposite direction of or Dueto it is impossible to calculate the inductancematrixIn addition, is close to unity, and and arevery small, when the average voltage vector exists nearthe origin in the – coordinates. In this area, the positionestimation cannot be performed with enough accuracy, becauseall components in are zero or a small value. The factmentioned above shows that the conventional sinusoidal PWMscheme is not applicable to the proposed position estimation.V. MODIFICATION OF PWM CONTROLIn order to perform precise position estimation, the authorspropose a novel PWM scheme using redundant voltage vec-tors. The average voltage vector during a modulation periodcan be expressed by(23)(24)Here, which takes a value of 1 or 0 during the modulationperiod, indicates whether is selected as an output vector166 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 34, NO. 1, JANUARY/FEBRUARY 1998(a) (b)(c)Fig. 5. PWM scheme using redundant voltage vectors. The average voltagevector exists on the  axis. (a) PWM pattern. (b) Harmonic voltage vectors.(c) Trajectory of harmonic current vector.or not, respectively. Lumping the equations together gives thefollowing equation:...(25)In the above equation, has to be decided so that thePWM inverter outputs the average voltage vector during themodulation period. No general solution to (25) exists becausethe number of unknown variables is more than the numberof equations. However, introduction of a right pseudoinversematrix [9] allows a solution to as follows:...(26)where(27)As a result of the calculation, one can decide a suitable PWMpattern under any combination of during the modulationperiod.Fig. 5(a) shows a PWM pattern produced when the averagevoltage vector exists on the axis. Note that isselected during an interval of time, different from beingselected during the corresponding interval in the conventional(a) (b)(c)Fig. 6. PWM scheme using redundant voltage vectors. The average voltagevector exists on the origin of the – coordinate. (a) PWM pattern. (b)Harmonic voltage vectors. (c) Trajectory of harmonic current vectors.sinusoidal PWM scheme. As shown in Fig. 5(b), andproduce -axis components in the harmonic current vectors,while and produce -axis components. Fig. 5(c) showsthe trajectory of the harmonic current vector. The trajectorystarts from the origin toward the left at the beginning of thefirst modulation period. Since the harmonic current vectorchanges toward the direction near that of the correspondingharmonic voltage vector, the trajectory draws a rectangle andreturns to the origin at the end of the first period. None ofthe vectors and are linearly dependent inthe first period. The rotor position, therefore, can be estimatedby the proposed algorithm, based on the harmonic componentof the current variation, independent of the location of theaverage voltage vector. Note that the sum of the harmoniccurrent vectors in the first modulation period shown in Fig. 5is not equal to a zero vector. In order to compensate for thedeviation, the sequence of selecting the voltage vectors shouldbe modified as shown in the second modulation period.Fig. 6 shows a PWM scheme where the average voltagevector exists on the origin of the - coordinate. This situationcorresponds to the case of position estimation at standstill.Each voltage vector is output for 1/6 of the modulationperiod. Note that the inverter selects neither nor Thesequence is scheduled so that the trajectory of the harmoniccurrent vectors starts from the origin at the beginning of themodulation period and returns to the origin at the end. Sincenone of the harmonic current vectors are linearly dependent,the position estimation can be performed even at a standstill.OGASAWARA AND AKAGI: REAL-TIME POSITION ESTIMATION FOR A PM MOTOR 167Fig. 7. Configuration of experimental system.TABLE IIPM MOTOR PARAMETERSVI. EXPERIMENTAL RESULTFig. 7 shows the configuration of the experimental system,consisting of a V40 CPU and a digital signal processor (DSP)(ADSP-2101). The V40 CPU supervises the whole operationof the system, and the DSP calculates the PWM pattern, theinductance matrix, and the estimated rotor position.The counter board is capable of synthesizing the PWMsignal of the inverter, following the PWM pattern calculated bythe DSP. The PWM signal in a modulation period consists ofsix voltage vectors in any order. The detected motor currentsare transformed from three-phase to – components. Thecounter board also generates the sample/hold signals by whichthe transformed motor currents are held just before everyswitching, so that the switching noises do not affect thesignals being held. Each current variation in (11) is appliedto an A/D converter through a differential amplifier. Thedetection circuit consisting of the sample/hold amplifiers andthe differential amplifiers makes a significant contribution toprecisely detecting each current variation.Table I shows the parameters of the PM motor used in theexperimental system. Since the tested motor is an IPM motor,it has the saliency of the -axis inductance being larger than-axis inductance. The DSP can perform all the calculationsfor the inductance matrix and rotor position, as well as PWMcontrol within a modulation period of 333 s. In the followingexperimental results, the PWM pattern shown in Fig. 6 is usedto achieve the rotor position estimation at standstill and in thelow-speed range.Fig. 8. Experimental result at standstill.Fig. 9. Experimental result at 1 r/min. (a) Estimated positiontan 1(L12 + L21=L11   L22): (b) L12 + L21: (c) L11   L22:Fig. 8 shows the experimental result of the position estima-tion proposed in this paper at standstill. A circle “ ” and thecorresponding vertical bar mean the average value and existingrange of estimated position, respectively, obtained from 20trials at a rotor position. The experimental result shows thatthe real-time position estimation is achieved at standstill withsuch satisfactory accuracy as to add a position-closed loop toa PM motor drive system. Note that the vertical axis is not ,but It would be possible to distinguish the polarity of therotor magnetic pole by means of another technique based onmagnetic saturation [3].Fig. 9 shows another experimental result at 1 r/min. Theconventional position estimation based on back EMF wouldnot be applicable at such a low speed. In this experiment,the inverter is controlled to continue to select the zero averagevoltage vector, and the rotor of the IPM motor is rotated by thedc generator, which is directly connected to the shaft. Theseexperimental results show the effectiveness and versatility ofthe position estimation proposed in this paper.VII. CONCLUSIONThis paper has proposed an approach to real-time posi-tion estimation based on magnetic saliency. A novel PWMscheme using redundant voltage vectors has been proposed168 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 34, NO. 1, JANUARY/FEBRUARY 1998to achieve the position estimation. Experimental results havedemonstrated that the proposed approach has the capacity ofestimating the rotor position with an error of less than 10 inelectrical angle at standstill and at extremely slow speeds, aslow as 1 r/min. The proposed approach is applicable to other acmotors with magnetic saliency, such as synchronous reluctancemotors. In addition, it identifies inductance parameters in realtime for motor control, without requiring user input of machineconstants. The authors plan to construct a position-sensorlessIPM motor system with a position-closed loop based on theproposed approach in the next stage.REFERENCES[1] K. Iizuka, H. Uzuhashi, M. Kano, T. Endo, and K. Mori, “Microcom-puter control for sensorless brushless motor,” IEEE Trans. Ind. Applicat.,vol. IA-21, pp. 595–601, May/June 1985.[2] S. Ogasawara and H. Akagi, “An approach to position sensorless drivefor brushless dc motors,” IEEE Trans. Ind. Applicat., vol. 27, pp.928–933, Sept./Oct. 1991.[3] N. Matsui and T. Takeshita, “A novel starting method of sensorlesssalient-pole brushless motor,” in Proc. IEEE-IAS Annu. Meeting, 1994,pp. 386–392.[4] P. L. Jansen and R. D. Lorenz, “Transducerless position and velocityestimation in induction and salient AC machines,” in Proc. IEEE-IASAnnu. Meeting, 1994, pp. 488–495.[5] P. L. Jansen, M. J. Corley, and R. D. Lorenz, “ Flux, position, andvelocity estimation in AC machines at zero and low speed via trackingof high frequency saliency,” in Proc. EPE’95, Seville, Spain, 1995, vol.3, pp. 154–159.[6] S. Kondo, A. Takahashi, and T. Nishida, “Armature current locus basedestimation method of rotor position of permanent magnet synchronousmotor without mechanical sensor,” in Proc. IEEE-IAS Annu. Meeting,1995, pp. 55–60.[7] T. Matsuo and T. A. Lipo, “Rotor position detection scheme forsynchronous reluctance motor based on current measurements,” IEEETrans. Ind. Applicat., vol. 31, pp. 860–868, July/Aug. 1995.[8] K. D. Hurst and T. G. Habetler, “Sensorless speed measurement usingcurrent harmonic spectral estimation in induction machine drives,” IEEETrans. Power Electron., vol. 11, pp. 66–73, Jan. 1996.[9] S. Wolfram, Mathematica—A System for Doing Mathematics by Com-puter, 2nd ed. Reading, MA: Addison-Wesley, 1991.Satoshi Ogasawara (A’87–M’93–SM’97) was bornin Kagawa Prefecture, Japan, in 1958. He receivedthe B.S., M.S., and Dr.Eng. degrees in electrical en-gineering from Nagaoka University of Technology,Niigata, Japan, in 1981, 1983, and 1990, respec-tively.From 1983 to 1992, he was a Research Associateat Nagaoka University of Technology. Since 1992,he has been with the Department of ElectricalEngineering, Okayama University, Okayama, Japan,where he is currently an Associate Professor. Hispresent reseach interests are ac motor drives systems and static powerconverters.Dr. Ogasawara received the 1996 Prize Paper Awards of the IndustrialPower Converter Committee and Industrial Drive of the IEEE IndustryApplications Society in 1996 and 1997. He is a member of the Institute ofElectrical Engineers of Japan.Hirofumi Akagi (M’87–SM’94–F’96) was born inOkayama-city, Japan, in 1951. He received theB.S. degree from Nagoya Institute of Technology,Nagoya, Japan, in 1974 and the M.S. and Ph.D.degrees from Tokyo Institute of Technology, Tokyo,Japan, in 1976 and 1979, respectively, all in elec-trical engineering.In 1979, he joined Nagaoka University of Tech-nology, Niigata, Japan, as an Assistant Professorin the Department of Electrical Engineering, wherehe later became an Associate Professor. In 1987,he was a Visiting Scientist at the Massachusetts Institute of Technology,Cambridge, for ten months. Since 1991, he has been a Full Professor in theDepartment of Electrical Engineering, Okayama University, Okayama, Japan.From March to August 1996, he was a Visiting Professor at the Universityof Wisconsin, Madison, and the Massachusetts Institute of Technology. Hisresearch interests include ac motor drives, high-frequency resonant invertersfor induction heating and corona discharge treatment, and utility applicationsof power electronics, such as active filters, static var compensators, andFACTS devices.Dr. Akagi is the recipient of seven IEEE Industry Applications Society andCommittee Prize Paper Awards, including the IEEE Industry ApplicationsSociety 1991 First Prize Paper Award. He is a Distinguished Lecturer of theIEEE/IAS for 1998–1999.

An approach to real-time position estimation at zero and low speed for a PM motor based on saliency

Okayama University Digital Information Repository

S. Ogasawara

H. Akagi

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An approach to real-time position estimation at zero and low speed for a PM motor based on saliency

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