Microsoft Word - B_30_A_R.doc HUNGARIAN JOURNAL OF INDUSTRIAL CHEMISTRY VESZPRÉM Vol. 38(2). pp. 207-210 (2010) SENSORLESS ROTOR POSITION DETECTION OF PMSM FOR AUTOMOTIVE APPLICATION D. FODOR, H. MEDVE, I. SZALAY, T. KULCSÁR Automotive Electronic Systems Group, Institute of Mechanical Engineering, University of Pannonia Egyetem str. 10. Veszprém, HUNGARY E-mail: fodor@almos.uni-pannon.hu, hunor.medve@gmail.com, ifj_szalay_istvan@yahoo.com, trovo.st@gmail.com For correct operation of the permanent magnet synchronous motors is essential the knowledge of the rotor position, which is usually given by Hall-sensors, encoders or resolvers. The aim of the present work is to set up such a general motor model, which takes into consideration the anisotropies in the magnetic structure of the motor. Keywords: rotor position estimation, PMSM motor model, model validation, sensorless control methods Introduction The electrical drives can be found in every field of the modern life from the cooling fans in computers to the most modern electric tractions. Most of the cases where a variable-speed operation was required, DC motors were used extensively, since the flux and torque can be controlled easily by the field and armature current. However, the main problem of the conventional DC motors are the commutators and the brushes. The commutators have limited capability under high-speed operation. A good alternative to the conventional DC motors is the permanent magnet synchronous motor (PMSM), which are widely used in industrial applications due to their high efficiency, high torque-to-inertia ratio, high reliability, low maintenance and longer lifetime [1]. Position sensors are placed in the motor for accurate control. For position sensing Hall-effect sensors, different encoder types or resolvers are used. The construction of permanent magnet synchronous motors yields the advantages mentioned above but they do not have only bright side. The position sensors have many drawbacks: they increase cost and size of the motor, the operating temperature range is limited and need some external electronic circuits to transmit the correct position. Thus in space restricted applications, in cost effective applications and in hostile environments the necessity of sensorless operation is brought up [1, 3, 6]. In this paper slotless motors will be analyzed, which are high power small size servo motors. These PMSMs cannot be ranked in any conventional group because the rotor is one block magnet. The other special characteristic is the absence of the iron core in stator windings, which results ironless stator windings. This is not a common motor type, so far sensorless control of slotless PMSMs is not discussed in literature. Modelling the PMSMs Electric drives are complex electromechanical systems. Their power supply and their control are usually electrical and their output is usually mechanical. Therefore the modeling usually is done only for the electrical and mechanical phenomena. But if the magnetic structure of the motor is important (anisotropies, saturation, hysteresis and the combination of them is not negligible) then it should be taken into consideration thoroughly. Our sensorless method is based on these magnetic phenomena, so their analysis is very important from both theoretical and experimental side. Geometrical Model Usually more than one coordinate systems are used to describe the motor geometry. One of these systems is fixed to the stator (α – β) and the other to the rotor (d – q). Their origins are the same. The angular position of the (d – q) system in the (α – β) system is defined by the angle θ. The modeled motors have 3 phase windings. The phases and their parameters usually are indexed with a, b and c. By fixing an axis to each phase can be created the 3 phase or natural (a – b – c) coordinate system of the motor. This system is fixed to the stator as the (α – β). It is comfortable to define the (α – β) in a way which grants the equality of the axes α and a. 208 Electrical Model The voltage equations of the phases (the voltage reference point is the start point): (1) The flux in Phase i can be dissolved to flux components that are produced by the windings and the permanent magnet. For Phase a: (2) In the above equation Фij represents the flux component in Phase i produced by Phase j, and ФMi represents the flux component in Phase i produced by the permanent magnet. The voltage equation of Phase a: (3) The flux derivatives can be dissolved into products of inductances and current derivatives: = (4) The change of the phase flux component induces the following voltage component, which can be considered proportional to the actual rotor speed: (5) Voltage equations for the phases: (6) (7) (8) The flux equation, which describes the relation between the phase fluxes and phase currents: (9) (10) In the development of mathematical background of the sensorless method it is an important task to determine the [L] inductance matrix and its dependencies on the other physical quantities, especially the rotor position. In the magnetic modeling one of the main goals will be to write the inductance matrix into a form which uses the magnetic properties of the motor. The inductance matrix is also important because it is the mathematical object which describes the relationship between the electric and magnetic parts of the model. Mechanical Model The torque equation of the motor: (11) In the above equation Θr is the inertia of the rotor, Mm is the motor torque produced by the phase windings, and the MT is the load torque. The kSωm product is the torque produced by the friction forces. The Mm motor torque can be computed from the equality of the incoming electrical energy and the outgoing mechanical energy (Pelectr = Pmech): (12) (13) Solving the above equation system yields: (14) Magnetic Model The most important physical phenomena were the followings during the mathematical modeling of the motor’s magnetic structure: Saturation: the rotor permanent magnets reluctance is flux dependent. Hysteresis: permanent magnets’ B–H curve is a hysteresis loop. Anisotropies: the rotor permanent magnets reluctance and probably the saturation and the hysteresis are anisotropic quantities. An other phenomenon, that should be considered is the eddy current on the surface of the permanent magnet. For geometrical reason the motor model should be divided into two parts: � stator and air gap model � rotor model. Stator contains the air core windings, the air gap between the windings and the rotor, and the external iron cylinder. The air gap and the air cores can be discussed together as a simple reluctance. The external iron cylinder, the air cores and the air gap can be modeled as constant reluctances. The stator windings are magnetomotive forces in the magnetic circuit. The rotor is a permanent magnet, which is a cylindriform object. The permanent magnet can be modeled as a magnetic circuit that contains reluctances and magnetomotive forces. First, is practical to model separately the reluctances and the magnetomotive forces to create a reluctance model and a magnetomotive model, and then connect them. Supposing the rotor is perfectly cylindriform, the magnetic field of the rotor permanent magnet is symmetrical to the plane defined by the axis and motor shaft. It is assumed that inside the magnet there is no leakage flux [2]. 209 Reluctance Model Hypothesis: The reluctance between two circumferential points opposite each other depends on the point pair and the flux flowing from one to the other. With other words, using the symbols of Fig. 1, the reluctance RAB depends on the angle φ1 and RCD depends on φ3 and both reluctances depend on the actual Ф flux flowing through the permanent-magnet. Figure 1: Reluctance between two surface points of a permanent magnet cylinder So any φ angle and Ф flux defines a R(φ, Ф) “passing through” reluctance value, which can be dissolved into two reluctance components: (15) Where RM(φ, Ф) is the rotor magnet’s “radial” reluctance function, which defines the “radial” reluctance between the center point and the surface point in the direction of the φ angle, when the flux flowing through is Ф. The reluctance between non-opposite surface point can be dissolved to two “radial” reluctances. For example, the reluctance between the points A and C, when Ф flux flows through from point A to point C, is computed in the following way using the RM(φ, Ф) function: (16) The above method has some error, which is greater when the difference of the two angle (in the above example the difference of φ1 and φ3) is smaller. If the difference is 120° (between the center lines of the phase windings) the error of the method is acceptable. The rotor can be modeled as reluctances connected in star using the RM(φ, Ф) function, see Fig. 2. Figure 2: Reluctance circuit of the permanent magnet rotor. (In the αβ coordinate system) The phase reluctances and their pins are fixed in the standing coordinate system. The value of the phase reluctances can change because the rotor can rotate. Describing the phase reluctances with the RM(φ, Ф) function yields: (17) Complete Magnetic Circuit The motor’s complete magnetic circuit with the inductances and phase windings is shown on Fig. 3: s s sMa θ Mb θ Mc θ Mc θ,Φ Mb θ,Φ Ma θ,Φ Ea t Eb t Ec t g g g L1 θ,Φ L3 θ,ΦL2 θ,Φ Φa Φb Φc Figure 3: The magnetic circuit of the motor The merged phase reluctances: (18) Using the above reluctance components the inductance matrix can be determined. Flux Equations The Фaa flux component, which is produced by the FEa magnetomotive force in Phase a: (19) The Фab flux component, which is produced by the FEb magnetomotive force of Phase b in Phase a: (20) Based on the above equations, flux equations can be written for the other two phases and can be written together in a vectorial form that contains the inductance matrix: (21) 210 Mathematical Description of Standstill Position detection at standstill is critical in sensorless methods, thus it has high priority. During the measurement only one phase winding was excited through the star point of the motor and the difference of the induced voltages in the two other phases was measured. The two induced voltages were not equal, and the amplitude of the difference was depending on the rotor position. The results can be seen on Fig. 4. The above voltage measurements prove the presence of anisotropies. Based on the amplitude of the difference in different rotor positions characteristic curves can be determined which are periodic with 180°, thus cannot give the exact rotor position. Using the previously presented formalism, the following relation can be written for the exciting voltage, current and induced voltages: (22) Similarly to (22) the voltage equations can be written for the other phases. Altogether six equations can be obtained, from these with current measurement three reluctance value can be defined in one specified rotor position. From these the RM reluctance function can be defined. To do this further high precision measurements are necessary. Figure 4: U1: exciting voltage, U2 and U3: voltages induced in Phase 2 and 3, ΔU = U3 - U2, the difference of the induced voltages. The U1 exciting voltage measured on the left-hand y axis. Conclusions According to the phase geometry the phase characteristic curves are shifted with 120° to each other, however they are not really uniform. All three curves have a period of 180°, which means that the angular position only by voltage measurement cannot be done. In that way only the line defined by the rotor magnet poles can be found, but the two poles cannot be distinguished from each other. The line mentioned above can be determined with a relatively small error (5°–10°). The developed mathematical model and formalism makes possible a general description of permanent magnet synchronous motors. The formalism fits in the notation used by the literature. The main goal, to prove the existence of the anisotropies and their measurability has been reached. ACKNOWLEDGEMENTS The work was partly supported by TAMOP-4.2.1/B- 09/1/KONV-2010-0003: Mobility and Environment: Researches in the fields of motor vehicle industry, energetics and environment in the Middle- and West- Transdanubian Regions of Hungary. The Project is supported by the European Union and co-financed by the European Regional Development Fund. REFERENCES 1. P. VAS: Sensorless Vector and Direct Torque Control, Oxford University Press, Inc., New York, 1998. 2. EC-32-118891_08_EN_164.pdf 3. T. KIM, H.-W. LEE, M. EHSANI: Position Sensorless Brushless DC Motor/Generator Drives: Review and Future Trends. Electric Power Applications, IET, 1(4), 2007, 557–564. 4. J. PERSSON, M. MARKOVIC, Y. PERRIARD: A new standstill position detection technique for non-salient PMSM's using the magnetic anisotropy method (MAM); Industry App. Conf., 2005. Fourtieth IAS Annual Meeting. Conference Record of the 2005, Vol 1, 2-6 Oct. 2005, 238–244. 5. J. SHANLIN, Z. JIBIN, Z. HONGLIANG, S. JING: A novel method of detecting for rotor position of a sensorless brushless DC motor, ICEMS. International Conference on Electrical Machines and Systems, 2007. 8-11 Oct. 2007, 797–800. 6. M. LINKE, R. KENNEL, J. HOLTZ: Sensorless position control of permanent magnet synchronous machines without limitation at zero speed, IECON 02, Industrial Electronics Society, IEEE 2002 28th Annual Conference of the, 5-8 Nov. 2002, Vol.1, 674–679. << /ASCII85EncodePages false /AllowTransparency false /AutoPositionEPSFiles true /AutoRotatePages /None /Binding /Left /CalGrayProfile (Dot Gain 20%) /CalRGBProfile (sRGB IEC61966-2.1) /CalCMYKProfile (U.S. Web Coated \050SWOP\051 v2) /sRGBProfile (sRGB IEC61966-2.1) /CannotEmbedFontPolicy /Error /CompatibilityLevel 1.4 /CompressObjects /Tags /CompressPages true /ConvertImagesToIndexed true /PassThroughJPEGImages true /CreateJobTicket false /DefaultRenderingIntent /Default /DetectBlends true /DetectCurves 0.0000 /ColorConversionStrategy /CMYK /DoThumbnails false /EmbedAllFonts true /EmbedOpenType false /ParseICCProfilesInComments true /EmbedJobOptions true /DSCReportingLevel 0 /EmitDSCWarnings false /EndPage -1 /ImageMemory 1048576 /LockDistillerParams false /MaxSubsetPct 100 /Optimize true /OPM 1 /ParseDSCComments true /ParseDSCCommentsForDocInfo true /PreserveCopyPage true /PreserveDICMYKValues true /PreserveEPSInfo true /PreserveFlatness true /PreserveHalftoneInfo false /PreserveOPIComments true /PreserveOverprintSettings true /StartPage 1 /SubsetFonts true /TransferFunctionInfo /Apply /UCRandBGInfo /Preserve /UsePrologue false /ColorSettingsFile () /AlwaysEmbed [ true ] /NeverEmbed [ true ] /AntiAliasColorImages false /CropColorImages true /ColorImageMinResolution 300 /ColorImageMinResolutionPolicy /OK /DownsampleColorImages true /ColorImageDownsampleType /Bicubic /ColorImageResolution 300 /ColorImageDepth -1 /ColorImageMinDownsampleDepth 1 /ColorImageDownsampleThreshold 1.50000 /EncodeColorImages true /ColorImageFilter /DCTEncode /AutoFilterColorImages true /ColorImageAutoFilterStrategy /JPEG /ColorACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /ColorImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000ColorACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000ColorImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasGrayImages false /CropGrayImages true /GrayImageMinResolution 300 /GrayImageMinResolutionPolicy /OK /DownsampleGrayImages true /GrayImageDownsampleType /Bicubic /GrayImageResolution 300 /GrayImageDepth -1 /GrayImageMinDownsampleDepth 2 /GrayImageDownsampleThreshold 1.50000 /EncodeGrayImages true /GrayImageFilter /DCTEncode /AutoFilterGrayImages true /GrayImageAutoFilterStrategy /JPEG /GrayACSImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /GrayImageDict << /QFactor 0.15 /HSamples [1 1 1 1] /VSamples [1 1 1 1] >> /JPEG2000GrayACSImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /JPEG2000GrayImageDict << /TileWidth 256 /TileHeight 256 /Quality 30 >> /AntiAliasMonoImages false /CropMonoImages true /MonoImageMinResolution 1200 /MonoImageMinResolutionPolicy /OK /DownsampleMonoImages true /MonoImageDownsampleType /Bicubic /MonoImageResolution 1200 /MonoImageDepth -1 /MonoImageDownsampleThreshold 1.50000 /EncodeMonoImages true /MonoImageFilter /CCITTFaxEncode /MonoImageDict << /K -1 >> /AllowPSXObjects false /CheckCompliance [ /None ] /PDFX1aCheck false /PDFX3Check false /PDFXCompliantPDFOnly false /PDFXNoTrimBoxError true /PDFXTrimBoxToMediaBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXSetBleedBoxToMediaBox true /PDFXBleedBoxToTrimBoxOffset [ 0.00000 0.00000 0.00000 0.00000 ] /PDFXOutputIntentProfile () /PDFXOutputConditionIdentifier () /PDFXOutputCondition () /PDFXRegistryName () /PDFXTrapped /False /CreateJDFFile false /Description << /ARA /BGR /CHS /CHT /CZE /DAN /DEU /ESP /ETI /FRA /GRE /HEB /HRV (Za stvaranje Adobe PDF dokumenata najpogodnijih za visokokvalitetni ispis prije tiskanja koristite ove postavke. 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