Cogging Torque Measurement, Moment of Inertia Determination and Sensitivity Analysis of an Axial Flux Permanent Magnet AC motor P.W. Poels DCT 2007.147 Traineeship report Traineeship performed at the Charles Darwin University, Darwin, Australia Coach(es): dr. ir. F. de Boer G. Heins Supervisor: dr.ir. M. Steinbuch Technische Universiteit Eindhoven Department Mechanical Engineering Control Systems Technology Group Eindhoven, June, 2008 ii Abstract Many technical applications require a smooth torque. An axial flux permanent magnet (PM) AC motor is used to achieve this with control methods. Required are motor param- eters, such as the moment of inertia. This parameter is determined by calculation with help of the CAD drawings. To verify the result, an experimental setup is designed. The resulting difference of 6.6% between the calculated and experimental determinded value of the moment of inertia is explained with the help of a sensitivity analysis. One of the properties of the type of motor used to achieve smooth torque is the presence of cogging torque. To compensate for cogging torque, this parameter needs to be measured. To be able to do this, a measurement method is designed. To explain the resulting RMSerror a sensitivity analysis of the calibration method is made. This is done theoretically and verified experimentally. The remaining RMSerror of 0.8 − 1.5% is caused by the current sensor and control errors. iii iv Samenvatting Voor vele technische toepassingen is een constant koppel vereist. Door enkele regel me- thodes toe te passen op een axiale flux permanente magneet (PM) AC motor wordt dit bereikt. Hiervoor is het nodig om de motor parameters te weten, zoals het massatraaghei- dsmoment. Met behulp van de CAD tekeningen wordt deze parameter berekend. Met een experiment wordt de berekende waarde geverifieerd. Het verschil van 6.6% tussen deze twee waarden wordt verklaard aan de hand van een foutenanalyse. Een van de eigenschappen van het type motor dat gebruikt wordt is cogging. Door deze te meten kan hiervoor gecompenseerd worden. Voor deze meting is een opstelling bedacht. Om de resulterende RMSfout te verklaren wordt een foutenanalyse van de kalibratie methode gemaakt. Allereerst gebeurt dit theoretisch. Hierna zijn de antwoorden experi- menteel geverifieerd. De overgebleven RMSfout van 0.8 − 1.5% wordt veroorzaakt door de stroomsensor en regelfouten. v vi Contents Abstract iii Samenvatting v Table of contents vii Nomenclature ix 1 Introduction 1 1.1 Motor Setup .................................. 2 1.2 Report overview ................................ 2 2 Determination of the Moment of Inertia 5 2.1 Determination of the moment from the CAD drawings ........... 5 2.2 Experimental determination of the Moment of Inertia ........... 7 2.2.1 Method of determining the Moment of Inertia experimentally ... 7 2.2.2 Experimental Setup .......................... 8 2.2.3 Results of the experiment ....................... 10 2.3 Comparison and sensitivity analysis of the results .............. 13 2.3.1 Sensitivity analysis of the experiment ................ 13 2.3.2 Improvements of the experiment ................... 18 3 Measurement of the Cogging Torque 19 3.1 Defining problem ............................... 19 3.2 Measurement method ............................. 20 3.3 Results ..................................... 21 vii Contents 4 Sensitivity Analysis of the Calibration 25 4.1 Calibration Method ............................... 25 4.1.1 Determination of ~y and ~z over operating range ........... 28 ∗ ~∗ ∗ 4.1.2 Determination of ~α , β and ~τcog .................. 28 4.1.3 Compensation for ~α, β~ and ~τcog ................... 28 4.2 Theoretical Sensitivity Analysis ........................ 29 4.3 Measurements for verification of the calibration method .......... 30 4.4 Comparison of the Theoretical and practical sensitivity ........... 34 5 Conclusion 39 Bibliography 39 viii Nomenclature Symbols from chapter "Determination of the Moment of Inertia" δ = Logarithmic increment θ = Rotation of the electric motor (rad) ζ = Damping ratio F = Force (Nm) J = Moment of inertia (kg · m2) K = Spring stiffness (Nm) L = Distance to the rotation point (m) M = Mass attached to a sping (kg) m = Mass of a spring (kg) T = Driving torque of the electric motor (Nm) t = (Period) time (s) x = Elongation of the spring (m) x1 = Maximum of the first oscillation x5 = Maximum of the fifth oscillation Symbols from chapter "Measurement of the Cogging Torque" 2p = Number of motor poles τcog = Cogging torque (Nm) ix Nomenclature HCF = Highest Common Factor Np = Number of periods of the cogging torque in a slot pitch rotation Q = Number of stator slots Symbols from chapter "Sensitivity Analysis of the Calibration" τ¯m = Φ × 1 vector of the mean torque δ = torque sensor scaling error = current inverter scaling error τrated = Rated torque of the electric motor (Nm) N = Number of measurements points RMSerror = Root Mean Square error of the motor torque ~α∗ = 3 × 1 vector of current scaling error estimate αp = current scaling error in phase p ~α = 3 × 1 vector of current scaling error β~∗ = 3 × 1 vector of current offset error estimate βp = current offset error in phase p β~ = 3 × 1 vector of current offset error • = element-wise multiplication operator ∗ I = Φ × 3 matrix of the current estimate (A) I = Φ × 3 matrix of the current (A)(NOT the identity matrix) iφ,p = current in phase p at encoder point φ (A) K = Φ × 3 matrix of the back EMF (V s/rad) kφ,p = normalised back EMF for phase p at encoder point φ (V s/rad) x Nomenclature ∗ ~τcog = Φ × 1 vector of the cogging torque estimate (Nm) cog τφ = cogging torque at encoder point φ (Nm) ~τcog = Φ × 1 vector of the cogging torque (Nm) ~τem = Φ × 1 vector of the electro-magnetic torque (Nm) ∗ ~τm = Φ × 1 vector of the estimate of motor torque (Nm) ∗ τm,φ = estimate of motor torque at encoder point φ (Nm) ∗ ~τp = Φ × 1 vector of the pulsating torque estimate (Nm) ∗ ~τr = Φ × 1 vector of the estimated mean torque (Nm) X = Φ × 6 matrix of torque and back EMF ~y = 9 × 1 vector of scaling and offset errors ~z = Φ × 1 vector of residuals xi Nomenclature xii Chapter 1 Introduction Many electric motor applications require a constant torque, especially applications that require precise tracking. These processes are for instance laser cutting and numerically controlled machining. Pulsating torque (any kind of variation in the torque output of the motor) can have a negative effect on, for example, the surface finish when using rotary machine tools. Also pulsating torque can excite resonances in the drive-train of the appli- cation. This produces acoustic noise as well. A smooth torque output can be achieved by using a programmed reference current wave- form. This method has the ability to work at different speed and torque set points. This minimizes restrictions on motor design and manufacture. When limiting pulsating torque mechanically, accurate manufacturing is required. This limits the practicality for low-cost, high volume production. Research on this subject is performed at the Charles Darwin University (CDU) in Darwin, Australia. The goal of the CDU electric motor research program is to create an output torque with a maximum RMSerror of 1%. The contribution to this research explained in this report consists of three parts: • For control purposes, the moment of inertia of the motor has to be known. With the help of CAD drawings of the electric motor, the moment of inertia is calculated. To verify this result, the moment of inertia is determined experimentally. • One of the properties of the electric motor used, is the presence of cogging torque. To achieve a smooth output torque, compensation for the cogging torque is added to the control scheme. Therefore the cogging torque is measured. • To improve the result of a programmed reference current waveform, a calibration 1 Chapter 1. Introduction method is designed. The sensitivity of this calibration method is analyzed theoreti- cally and verified practically. In this analysis also the cause for the remaining torque ripple is explained. 1.1 Motor Setup The type of motor used for this research is a Permanent Magnet Synchronous AC motor. The first natural frequency of the motor mounted on a force table is 700 Hz. The stator consists of 48 slots and the rotor has 16 poles. The motor has a rated torque of 6Nm and a rated voltage of 24V . The position of the axle is measured with a 12-bit (4096 states), gray code, absolute en- coder. The torque is measured with piezo electric reaction torque sensors. An eddy current brake is installed to apply different torque set points. The defined set points are 1,2,3,4 and 5 Nm and 0.5, 0.6, 0.7, 0.8, 0.9 and 1.0 Hz. This provides a mesh which consists of 30 measurements. Data acquisition is done with the help of Labview. In figure 1.1 a picture of the electric motor can be seen. In this picture the magnets of the eddy current brake have been removed. 1.2 Report overview The contribution to the research consist of three parts, which are divided into three chap- ters. The determination of the moment of inertia is described in chapter 2. In chapter 3 the cogging torque is measured. The sensitivity analysis of the calibration method is explained in chapter 4. The conclusions and recommendations of the three parts are com- bined in chapter 5. 2 1.2. Report overview Figure 1.1: The electric motor with the different components named. The magnets of the Eddy Current brake have been removed. 3 Chapter 1. Introduction 4 Chapter 2 Determination of the Moment of Inertia The goal of this experiment is to determine the moment of inertia of the electric motor. This motor parameter is necessary in the control scheme. First of all the moment of inertia is calculated based on the CAD drawings. To verify this calculation, a experimental setup is designed to determine the moment of inertia experimentally.