CONTROL METHODS FOR SMOOTH OPERATION OF PERMANENT MAGNET SYNCHRONOUS AC MOTORS

Transcription

1 CONTROL METHODS FOR SMOOTH OPERATION OF PERMANENT MAGNET SYNCHRONOUS AC MOTORS Greg Heins A thesis for the degree of Doctor of Philosophy at the School of Engineering and Logistics Charles Darwin University, Australia Submitted on March 31, 2008.

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3 Declaration I hereby declare that the work herein, now submitted as a thesis for the degree of Doctor of Philosophy of the Charles Darwin University, is the result of my own investigations, and all references to ideas and work of other researchers have been specifically acknowledged. I hereby certify that the work embodied in this thesis has not already been accepted in substance for any degree, and is not being currently submitted in candidature for any other degree.

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5 Abstract Many applications require a motor capable of providing a smooth torque output. The use of control methods to ensure smooth operation is attractive because it minimises restrictions on motor design and manufacture. Programmed reference current waveform (PRCW) methods are a commonly proposed control method as they have the ability to work at a range of speeds and load torques, however their experimental implementation to date has been inconclusive. This research compares three published PRCW methods with a standard sinusoidal current waveform. For the test motor considered, all PRCW methods tested are able to reduce the RMS pulsating torque to approximately 3 4% down from the 8 9% created when using a sinusoidal current. While this is a substantial reduction, the results for all methods were too similar to suggest the clear superiority of any method. One method however, the time domain method, produced slightly better results. Further reduction in pulsating torque requires a greater accuracy in motor and controller parameter estimation. To achieve this further reduction, this thesis develops a pulsating torque decoupling (PTD) method where the parameters are determined from the pulsating torque itself. The use of this technique allows calibration of the critical parameters and produces a further reduction of the pulsating torque to approximately 1%. As with the uncalibrated results, the difference between the different PRCW methods is minimal, with the time domain method producing only slightly less pulsating torque. These results suggest PRCW methods can be effective in producing a smooth torque output if the relevant parameters are estimated to a suitable accuracy. The use of the PTD method presented in this thesis can achieve this required accuracy. v

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7 List of publications As of the March 31, 2008, the following papers with contributions from the results presented in this thesis have been published, or are under review: 1. Greg Heins, Friso de Boer, Jeroen Wouters, and Roel Bruns Experimental comparison of reference current waveform techniques for pulsating torque minimization in PMAC motors. In International Electric Machines and Drives Conference (IEMDC 07), volume 1, pages , Antalya, Turkey, Greg Heins, Friso De Boer, and Sina Vafi. Characterisation of the mechanical motor parameters for a PMSM using induced torque harmonics. In Australian Universities Power Electronics Conference, Perth, Greg Heins and Friso De Boer. Modeling of a synchronous permanent magnet motor to determine reference current waveforms. Power Electronics Conference, Brisbane, In Australian Universities vii

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9 Acknowledgments This thesis would not have happened without the guidance and patience of Friso de Boer and Nic Hannekum. It would certainly not be finished yet without the assistance of Charles Darwin University staff members: David Van Munster, Mark Thiele and Jim Mitroy and Eindhoven University of Technology students: Roel Bruns, Jeroen Wouters, Pieter Poels and Erik Grassens. I am also grateful for the support of L institut Francais de Mecanique Avancee students: David Ahounou, Pascal Magnan, Gabriel Caroux, Ben Errard, Nick Ferriere and Florian Barnet, Ecole Superieure d Ingenieurs student: Vincent Lafont and Charles Darwin University students: Jasveer Saini, James Canning and Charles Gammon. Towards the end, somewhere between listening to Up all night by The Waifs and the soundtrack to Mission Impossible, things inevitably went somewhat pear shaped. I was very grateful for the love and support from Kelly Mashford, my folks: Terry and Diana, and my sister Karen. ix

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11 . To Gaz. I have always been a fan of enthusiasm. You have big mobs of it. xi

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13 Contents Declaration iii Abstract v List of publications vii Acknowledgments ix Table of Contents xvi List of Figures xix List of Tables xxi List of Symbols xxviii 1 Introduction Background Research goal Research approach Chapter overview Review: Programmed reference current waveform methods PMAC mathematical model Fundamental control problem PRCW methods Goals and constraints xiii

14 CONTENTS 2.5 Frequency domain method (FDM) Time domain method (TDM) Park-like method (PLM) Experimental implementation Implementation challenges Summary of PRCW methods Review: determination of motor parameters Back EMF Cogging torque Current Rotor Position Sensitivity analysis Summary of methods for determination of motor parameters Theoretical comparison of reviewed methods Methodology Parameter and constraint variation Performance with different back EMFs (without cogging torque) Comparison of cogging torque compensation Upper frequency limit constraint Summary of theoretical comparison of PRCW methods Pulsating torque decoupling approach to motor parameter determination Motor model overview Decoupling of pulsating torque: determination of current imbalance and cogging torque Modifications if only dynamic torque measurement is available System gain determination Sensitivity analysis Summary of PTD approach to motor parameter determination xiv

15 CONTENTS 6 Experimental setup Motor description Review of past implementation problems Mechanical design Current control design Hardware selection and design Data acquisition Experimental setup summary Results Validity of assumptions Current controller Eddy current brake testing Parameter determination Parameter determination - PTD method Comparison of PRCW methods Sensitivity analysis Summary of results Discussion Experimental setup Parameter estimation Comparison of PRCW methods Conclusion Further work References 121 A Hardware design 131 A.1 Hardware overview A.2 Mechanical drawings A.3 Electrical Design xv

16 CONTENTS B Additional Calculations 149 B.1 Design and sizing of eddy current brake B.2 Modal analysis of experimental setup C Additional results 155 C.1 Theoretical sensitivity analysis C.2 Pulsating torque for PRCW methods C.3 Pulsating torque comparison within methods xvi

17 List of Figures 2.1 Fundamental PMAC torque control scheme PRCW control scheme FBD for motor FBD for rotor FBD for stator Current and torque output for sinusoidal back EMF Current and torque output for trapezoidal back EMF Performance of different methods if one phase has a magnitude variation Performance of different methods if one phase has a phase variation Ability of each method to compensate 1Nm of cogging torque at different harmonics Comparison of currents with and without a star connection constraint Ability of each method to compensate 1Nm of cogging torque at different harmonics Block diagram of parameters to be determined Block diagram of parameters to be determined (simplified) Block diagram of parameters to be determined (simplified) α, β, τ cog variation leading to 1% RMS τ meas Stylised motor assembly cross section - full detail see A CDU Experimental Setup (Magnet portion of the eddy current brake has been removed for clarity) xvii

18 LIST OF FIGURES 6.3 Full bridge drive topology Star connected drive topology Simulink TM model of current controller Screen shot of Labview TM virtual instrument External drive for measurement of back EMF Raw back EMF over the angular velocity range Mean normalised back EMF Percentage error of normalised back EMF over the angular velocity range Proportionality of torque to current Bode plot of Plant Simulated step response of system Current error Harmonic content of back EMF Error from truncation of back EMF Dynamically measured cogging torque Measured cogging torque harmonics Dynamically measured cogging torque error System transfer function estimation System transfer function estimation (no multiple of 8 harmonics) Decoupling the pulsating torque (time domain) - ( rest is the remaining pulsating torque for which the source is unknown) Decoupling the pulsating torque (frequency domain) - ( rest is the remaining pulsating torque for which the source is unknown) Determined cogging torque for different set-points Cogging torque error for different set-points Comparison of PRCW methods (* refers to the use of the star connection constraint) Uncalibrated sin wave method TDM - without star connection constraint Comparison of PRCW current waveforms (* refers to the use of the star connection constraint) xviii

19 LIST OF FIGURES 7.24 RMS current increase from PRCW methods (* refers to the use of the star connection constraint) Sensitivity of pulsating torque to a α error Sensitivity of pulsating torque to a β error Sensitivity of pulsating torque to a τ cog error Sensitivity of pulsating torque to a ɛ error Pulsating torque due to current controller error A.1 CDU motor test assembly A.2 Overview of electrical components A.3 Overview of current inverter A.4 Closeup of gate drive and MOSFET module A.5 Current sensor board with LEM LTS25NP A.6 Labview interface board A.7 DSP interface board B.1 Assembled eddy current brake B.2 Expected torque from eddy-current brake B.3 Results of modal analysis C.1 Sensitivity of pulsating torque to parameter variation C.2 FDM - without star constraint C.3 FDM - with star constraint C.4 TDM - without star constraint C.5 TDM - with star constraint C.6 SIN C.7 FDM - without star constraint C.8 FDM - with star constraint C.9 TDM - with star constraint C.10 FDM (frequency domain) C.11 TDM (frequency domain) xix

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21 List of Tables 2.1 Back EMF and current harmonic combinations responsible for torque harmonics (including star constraint) Performance of PRCW methods with a sinusoidal back EMF Performance of PRCW methods with a trapezoidal back EMF Performance of PRCW methods with a modified trapezoidal back EMF Induced scaling and offset errors and the found compensating values Average total RMS current usage increase over operating range xxi

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23 List of Symbols ɛ = overall system gain independent of α, β and τ cog, K DSP i K o = K p = = DSP integral gain, system gain, controller gain, K DSP p λ ν o = = DSP controller gain, = Lagrange multiplier, 63% rise time, o p = current inverter offset for phase p, o p = current inverter supply offset for phase p, ω = τ o = θ = Θ θ e = T i = t s = mechanical angular velocity (rad/sec), delay time, mechanical rotor position (rad), = total number of encoder states, electrical rotor position (rad), integral time constant, sample time, u p = torque transducer gain for phase p, xxiii

24 LIST OF SYMBOLS u p = estimated torque transducer gain for phase p, w p = current inverter gain for phase p, w p = estimated current inverter gain for phase p, m = n = P harmonic number of torque series, harmonic number of back EMF and current series, = number of pole pairs, α p = current scaling error in phase p, α = 3 1 vector of current scaling error, G AA (ω) G BB (ω) = input auto spectrum, = output auto spectrum, β p = current offset error in phase p, β C γ 2 (ω) = 3 1 vector of current offset error, = Lagrange cost constraint, = coherence, C 1,θ = Lagrange cost constraint at angle θ, = matrix multiplication operator, G AB (ω) = cross spectrum, C 2,θ = Lagrange cost constraint at angle θ, E θ,p = back EMF for phase p at angle θ (V s/rad), E θ,p = back EMF for phase p at encoder point θ (V s/rad), = element-wise division operator, = element-wise multiplication operator, xxiv

25 LIST OF SYMBOLS f A(ω) = Lagrange minimization function at, = frequency input function, f θ = Lagrange minimisation function at angle θ, B(ω) H(ω) H 1 (ω) = frequency output function, = frequency response function, = alternate frequency response function, I cog = Θ 3 matrix of the current required to compensate for torque (A), Î n = current Fourier series coefficients, I τref = Θ 3 matrix of the current required to achieve the reference torque (A), I = Θ 3 matrix of the current (A) (NOT the identity matrix), I ref = Θ 3 matrix of the reference current (A), i ref θ,p i θe,a = reference current for phase p at angle θ (A), = current in phase a at electrical angle θ e (A), i θ,a, i θ,b, i θ,c i θ,d, i θ,q, i θ,0 = currents in three phase reference frame at at angle θ (A), = currents in rotating reference frame at at angle θ (A), i θ,p = current in phase p at angle θ (A), Î i p ˆK n K ˆK = vector of current coefficients, = Θ 1 vector of the current in phase p (A), = normalised back EMF Fourier series coefficients, = Θ 3 matrix of the back EMF (V s/rad), = matrix of the back EMF coefficients, k θe,a = normalised back EMF for phase a at electrical angle θ e (V s/rad), xxv

26 LIST OF SYMBOLS k θ,a, k θ,b, k θ,c = normalised back EMFs in three phase reference frame at at angle θ (V s/rad), k θ,d, k θ,q, k θ,0 = normalised back EMFs in rotating reference frame at at angle θ (V s/rad), k θ,p = normalised back EMF for phase p at angle θ (V s/rad), k p = Θ 1 vector of the normalised back EMF of phase p (V s/rad), λ 1,θ = Lagrange multiplier at angle θ, λ 2,θ = Lagrange multiplier at angle θ, λ L M N = Θ 1 vector of Lagrange multipliers, = limit chosen for the truncation of the back EMF series, = limit chosen for the torque harmonics to be canceled, = limit chosen for the truncation of the current series, 1 Θ 1 = Θ 1 vector of ones, N pole pair = number of pole pairs, N slot = number of stator slots, o o = 3 1 vector of the current inverter offset, = 3 1 vector of the estimated current inverter offset, 1 = Θ 3 matrix of ones, + = pseudo-inverse operator, Q = Lagrange minimization objective, Q θ = Lagrange minimization objective at angle θ, τ average = average torque output, τ cog = Θ 1 vector of the cogging torque estimate (Nm), xxvi

27 LIST OF SYMBOLS τ cog θ e τ cog θ τ cog θ τ cog = cogging torque at electrical angle θ e (Nm), = cogging torque at angle θ (Nm), = estimate of cogging torque at angle θ (Nm), = Θ 1 vector of the cogging torque (Nm), T em = Θ 3 matrix of the electromagnetic torque (Nm), τ em θ e τ em θ e,a τ em θ e,p τ em θ τ em = electromagnetic torque at electrical angle θ e (Nm), = electromagnetic torque produced by phase a at electrical angle θ e (Nm), = electromagnetic torque produced by phase p at electrical angle θ e (Nm), = electromagnetic torque at angle θ (N m), = Θ 1 vector of the electro-magnetic torque (Nm), τ em,p = Θ 1 vector of the electromagnetic torque produced by phase p (Nm), ˆT m = total electromagnetic torque Fourier series coefficients, ˆT m,a = electromagnetic torque Fourier series coefficients for phase a, = 3 1 vector of ones, τ m θ = motor torque at angle θ (Nm), τ m = Θ 1 vector of the motor torque (Nm), τ meas θ = measured torque at angle θ (Nm), τ pulsating = Θ 1 vector of the pulsating torque estimate (Nm), τ pulsating = Θ 1 vector of the pulsating torque (Nm), τ meas = Θ 1 vector of the measured torque (Nm), τ ref θ e τ ref θ = reference torque at electrical angle θ e (Nm), = reference torque at angle θ (Nm), xxvii

28 LIST OF SYMBOLS τ ref = Θ 1 vector of the constant reference torque (Nm), ˆT u u w w = vector of torque coefficients, = 3 1 vector of the torque transducer gain, = 3 1 vector of the estimated torque transducer gain, = 3 1 vector of the current inverter gain, = 3 1 vector of the estimated current inverter gain, X = Θ 6 matrix of torque and back EMF estimate, X y 0 z = Θ 6 matrix of torque and back EMF, = 9 1 vector of scaling and offset errors, = Θ 1 vector of zeros, = Θ 1 vector of residuals, xxviii

29 Chapter 1 Introduction Smooth electric motor torque output is desirable in many applications. Traditionally, for permanent magnet synchronous AC motors this smooth torque has been achieved using carefully designed motors that are manufactured to close tolerances. The potential exists for a well designed control system to avoid these design and manufacture restrictions. One of the most popular control schemes proposed to achieve this goal is the use of programmed reference current waveforms (PRCW). These waveforms are determined from prior knowledge of the motor parameters. To date however, the experimental implementation of these methods has been inconclusive, mainly due to challenges in acquiring the prior knowledge of the motor parameters to a significant accuracy. The goal of this research is to determine if, for motors requiring a smooth torque output, PRCW methods can be serious contenders to replace the current technology of careful motor design and manufacture. To focus the discussion, smooth operation is defined and an overview of various motor types is provided. Justification for using control methods to achieve smooth operation is given and this research is focused on PRCW control methods. The tasks to be completed are listed along with an overview of the content of the following chapters. 1

30 CHAPTER 1. INTRODUCTION 1.1 Background Smooth operation Motors with a smooth output torque are essential for applications that require precise tracking. These applications include arc welding, laser cutting, numerically controlled (NC) machining and antenna tracking [1], [2]. The existence of pulsating torque can have a negative impact on processes. One such example is the impact of pulsating torque on the surface finish when using rotary machine tools [3]. Pulsating torque also has the potential to excite resonances in the mechanical drive-train of a system along with the production of acoustic noise [4], [5] Synchronous alternating current machines Electric machines are classified as either alternating current (AC) or direct current (DC) machines. AC machines can be further classified as asynchronous (induction), or synchronous. DC machines are simple to control, however they require regular maintenance due to the sliding contact of their brushes. AC asynchronous machines (induction machines) are the most widely used motors due to their low cost, however they are less efficient and unable to be controlled as accurately as other types of machines. AC synchronous machines offer a level of control comparable with DC machines while avoiding the maintenance issues associated with brushes. Control however, is more difficult because of their requirement for a variable frequency drive. One category of AC synchronous machines is often referred to as Brushless DC machines, because of the similar topology. Three-phase permanent magnet AC synchronous (PMAC) machines AC synchronous machines can be further differentiated by the way that the rotating (rotor) magnetic field is generated. This can either be by electromagnets or by permanent magnets. Machines that use electromagnets can create a higher power to weight 2

31 1.1. BACKGROUND ratio, however they are less efficient due to the losses associated with the current required in the field windings. The use of electromagnets also requires brushes and slip rings to transfer the field current to the rotor. Synchronous PMAC machines are also classified by the number of independent windings, or phases. To achieve smooth torque production with evenly spaced phase windings, at least 3 phases are required. For the remainder of this thesis, the abbreviation PMAC will refer to three-phase synchronous permanent magnet AC machines. Rationale of studying PMAC machines PMAC machines avoid the maintenance issues associated with DC motors and are superior to other AC motors in terms of controllability and efficiency. With the introduction of high performance rare-earth magnets, PMAC machines have been capable of improved dynamic performance and higher efficiency [6]. Due to the need for sophisticated controllers and in some cases, the cost of rare-earth magnets, PMAC machines have been a more expensive option. In the last few decades however, the price of power electronic components and rare earth magnets has decreased [7]. For smaller motor sizes (up 10-15kW) PMAC machines are increasingly the machine of choice for servo drives and vehicle applications [6] Pulsating torque PMAC machines generate torque by the interaction between the magnetic flux from the permanent magnets in the rotor and the magnetic flux from the electromagnets in the stator (stationary part of the motor). The term pulsating torque refers to any periodic variation in the torque output of a motor. It is created in two ways: cogging torque and torque ripple. Ambiguity exists for the description of pulsating torque mechanisms. This thesis will use the terms defined by Jahns and Soong in their 1996 paper: Pulsating torque minimisation techniques for permanent magnet AC motor drives - a review [3]. 3

32 CHAPTER 1. INTRODUCTION Cogging torque In a PMAC machine, the electromagnets that make up the stator usually have steel cores. Copper windings fill the space between consecutive steel cores. Cogging torque is the variation in torque due to the permanent magnets on the rotor having a much greater attraction to the steel cores than to the copper windings. Even without any current flowing in the motor, torque needs to be applied to cog from where the magnets align to one set of cores to where the magnets align to the next set of cores. Torque ripple The interaction between the magnetic flux from the rotor and stator can be characterised by the shape of the voltage induced in each phase winding by motor motion (back EMF). Torque is created in PMAC machines through the interaction of the back EMF waveform in each phase and the current waveform in that phase. For a three phase machine, if the back EMF from the rotor is purely sinusoidal and the current in each of the three phases is also purely sinusoidal then the torque will be constant. Torque ripple is the term applied to any periodic variation in torque created when either of these waveforms deviate from sinusoids Control methods for smooth operation Torque ripple is minimised by optimising the interaction between the phase currents and the phase back EMF waveforms. This optimisation can be achieved by either mechanical means (altering the back EMF shape) or by electrical means (altering the current waveform) [3]. There are two major disadvantages of motor modification methods: design tradeoffs and the accurate manufacturing process required. Design trade-offs Sometimes the modification of motor design to reduce cogging torque can have a negative impact on torque ripple [8](p1293). Efforts to reduce either cogging torque or torque ripple can also have a negative effect on average torque [9], [10]. 4

33 1.1. BACKGROUND Manufacturing tolerances Jahns and Soong [3] noted that: techniques which require a high accuracy of assembly, magnetisation, magnet placement or dimensions may prove to be impractical for lowcost, high volume production. If pulsating torque is to be minimised by mechanical means, high accuracy manufacture is required, limiting the practicality for low-cost, high volume production. Control method benefits In contrast to the motor design approach, the control based approach allows motors to be designed for maximum average torque and minimum manufacturing cost. Ripples can be removed that were created by a design that maximises average torque or by manufacturing inaccuracies. A control based approach can use feedback information from a torque transducer, however the associated cost and complexity is usually prohibitive. This research will focus on reducing pulsating torque using only current and position feedback information Programmed reference current waveform (PRCW) methods Jahns and Soong [3], characterised control methods for minimising pulsating torque into five categories: 1. commutation torque minimisation; 2. speed loop disturbance rejection; 3. high speed current regulator saturation; 4. estimators and observers; and 5. programmed current waveform control. Of these categories, commutation torque minimisation is only relevant to motors with a trapezoidal back EMF driven by a square wave current, high speed current regulator saturation is only relevant for high speed operation and speed loop disturbance rejection is only relevant for low speed operation. 5

34 CHAPTER 1. INTRODUCTION Most estimator and observer methods require a very high resolution position signal or they will not work well at low speeds [11],[12]. They can also have difficulty coping with load fluctuations unless adaptive control is used [13]. Some research has been done on this problem by using adaptive control [9], [14], [15]. Control systems benefit most from adaptive control when parameters are time varying. This research only considers the implementation of programmed current waveform methods under a set of operating conditions where the parameters are time invariant. Under different operating conditions, parameters may vary and adaptive control may be beneficial. This is a subject for further work (see section 9.1). Of the methods suggested, only PRCW control has the potential to work at all speeds and work independently of the applied load. As such, research focuses on these methods Previous implementation of PRCW methods Despite the potential of PRCW methods in theory, Jahns and Soong [3] commented that: experimental verification of the proposed harmonic injection techniques is generally weak. Since 1996, further work has been done on PRCW methods however no experimental verification has been published that is much more convincing. In 2004, Bianchi and Cervaro [10] suggested that The suppression of the torque ripple of SPM machines is a problem that is not completely solved. The major challenges to the experimental implementation of these methods are: 1. the determination of motor parameters to a suitable accuracy; 2. accurate torque measurement; 3. the provision of a smooth load to the motor; and 4. the presence of mechanical resonances in the experimental setup. 1.2 Research goal In view of the limited experimental implementation and verification of PRCW methods, this research will use a motor with an inherently high pulsating torque and a well 6

35 1.3. RESEARCH APPROACH designed experimental setup to compare the different PRCW methods. In doing so, it will be determined if any of the methods can achieve a smooth torque comparable to that of a motor specifically designed for that goal. 1.3 Research approach Completion of the research goal involved the following tasks: 1. A literature review of: published PRCW methods for minimising pulsating torque; and existing methods for determination of motor parameters. 2. A theoretical comparison of the published methods to determine the conditions under which each is the preferred method; 3. A theoretical evaluation of alternative motor parameter determination methods; 4. Design of an experimental setup that overcomes the challenges described in section 1.1.6; and 5. Experimental comparison of: motor parameter determination techniques; different PRCW methods; and motor parameter sensitivities. 1.4 Chapter overview To explain the tasks outlined above, chapter 2 discusses published PRCW methods and chapter 3 describes the parameter determination which is critical to the success of these methods. Chapter 4 theoretically compares the published PRCW methods based on various back EMF and cogging torque motor parameters. Chapter 5 proposes a pulsating torque decoupling (PTD) approach for accurate motor parameter determination and theoretically analyses that method. 7

36 CHAPTER 1. INTRODUCTION Chapter 6 discusses the challenges and solutions involved in creating the experimental setup and chapter 7 reports the results of the experiments. Results are discussed in chapter 8 and conclusions are made in chapter 9. The drawings and calculations for the detailed design of the test rig and computer controlled power supply are presented in the appendices along with additional results. 8

37 Chapter 2 Review: Programmed reference current waveform methods The introduction presented PMAC motors, the problem of pulsating torque and focused this research on programmed reference current waveform (PRCW) methods. This chapter reviews past research into these control methods to provide a background for the theoretical comparison discussion in chapter 4, and for the experimental setup design described in chapter 6. A mathematical model for a PMAC motor is presented and the general control problem is considered. Three published PRCW methods are discussed. To allow later comparison, each method has been explained in a common nomenclature which may differ slightly from the original published nomenclature. The last section of the chapter reviews the experimental implementation of PRCW methods and discusses the challenges previous researchers have highlighted from their experimental work. 2.1 PMAC mathematical model Control of PMAC motors relies on a mathematical model. This model is derived from a relationship between mechanical and electrical power and is based on several assumptions. 9

38 CHAPTER 2. REVIEW: PROGRAMMED REFERENCE CURRENT WAVEFORM METHODS Assumptions and Constraints 1. Back EMF is proportional to angular velocity: Maxwell s equation states that the back EMF across a phase is proportional to the speed of the machine [16]. 2. Torque produced is proportional to phase currents: The Lorentz force equation states that the instantaneous torque produced by a particular phase winding is proportional to the phase currents [16]. 3. DC bus voltage: For the purposes of this comparison, speed limits imposed by a finite voltage source are not considered General Model Equation For a three phase synchronous motor, electrical power is given by the product of the current and voltage. Mechanical power is given by the product of the torque and the speed. Assuming an efficiency of 100%, the relationship between the mechanical and electrical power is given by [17](eq 8.1): where: τ em θ ω = p=a,b,c i θ,p E θ,p (2.1) θ = mechanical rotor position (rad) τ em θ = electromagnetic torque at angle θ (Nm) ω = mechanical angular velocity (rad/sec) i θ,p = current in phase p at angle θ (A) E θ,p = back EMF for phase p at encoder point θ (V s/rad). This equation is simplified by assuming the back EMF is proportional to angular velocity, allowing E n to be defined by a normalised waveform scaled by the mechanical rotor speed. ( E θ,p ω = k θ,p ). Equation 2.1 becomes: where: τ em θ = p=a,b,c 10 i θ,p k θ,p (2.2)

39 2.2. FUNDAMENTAL CONTROL PROBLEM k θ,p = normalised back EMF for phase p at angle θ (V s/rad). If cogging torque is considered then the total torque is: τ m θ = τθ em = p=a,b,c + τ cog θ (2.3) i θ,p k θ,p + τ cog θ where: τ m θ τ cog θ = motor torque at angle θ (Nm) = cogging torque at angle θ (Nm) It should be noted that this equation is based on the mechanical position and mechanical angular velocity. Some of the methods below are based on the electrical position and angular velocity. In a complete mechanical revolution of the motor, there will be an electrical revolution for each pair of poles. Therefore: θ e = P θ (2.4) where: θ e = electrical rotor position (rad) θ = mechanical rotor position (rad) P = number of pole pairs 2.2 Fundamental control problem The most basic form of the control problem considered has three inputs to the controller, torque reference, current feedback and position feedback. The output of the controller is the EMF applied to the ends of each of the three phases. This control scheme is shown in figure

40 CHAPTER 2. REVIEW: PROGRAMMED REFERENCE CURRENT WAVEFORM METHODS Figure 2.1: Fundamental PMAC torque control scheme 2.3 PRCW methods PRCW methods split the fundamental control problem into two parts. A current reference generator uses predetermined information about the back EMF and cogging torque to determine the current for a particular position. The current controller ensures that the actual current follows the current commanded by the current reference block. This scheme is shown in figure 2.2. Figure 2.2: PRCW control scheme Previous researchers have taken a number of different approaches, of which the most popular are: 1. Frequency domain method (FDM) where the normalised back EMF (k) is defined as a Fourier series which leads to a definition of the current reference (i) as a 12

41 2.4. GOALS AND CONSTRAINTS Fourier series [18], [16], [19], [20], [21] and [10]. 2. Time domain method (TDM) where k is defined as an array of values for each encoder value θ. This array is then used to determine i as an array [22]. 3. Park-like method (PLM) where a modification of field oriented control is used to determine i in a rotating reference frame [23], [24], [11],[25], [26] and [27]. 2.4 Goals and constraints The primary goal of each of these methods is to minimise pulsating torque. This may include the requirement to suppress cogging torque. For the FDM and the TDM there is an additional constraint to minimise RMS current. Depending on the topology of the motor there may be an additional star connection constraint to ensure that the sum of all currents at any time is zero. 2.5 Frequency domain method (FDM) This analysis follows that presented by Hung and Ding [16]. It is similar to that presented by Le-Huy [18], Hanselman [19] and Bianchi and Cervaro [10]. Chapman, Sudhoff and Whitcomb [20], [21] describe this method as the minimum current no ripple method. They present a further addition to this method which they describe as the minimum current minimum ripple method [20], [21]. This addition allows the designer to place more emphasis on either the ripple minimisation constraint or the current minimisation constraint. This modification however, adds considerable complexity to the analysis and studies have shown [20], [28] that there is negligible benefit to be gained by allowing more torque ripple in an attempt to further minimise the current Assumptions This method requires several further assumptions to those outlined in section 2.1.1: 1. All phases have the same back EMF and current waveform shapes and are 120 o out of phase. 13

42 CHAPTER 2. REVIEW: PROGRAMMED REFERENCE CURRENT WAVEFORM METHODS 2. The back EMF waveform is identical between pole pairs Parameter descriptions Back EMF For this method, the back EMF is converted into an exponential Fourier series (trigonometric Fourier series in [20]). It should be noted that this analysis is done in electrical degrees rather than mechanical degrees. where: k θe,a = n= ˆK n e jnθ e (2.5) k θe,a = normalised back EMF for phase a at electrical angle θ e (V s/rad) θ e = electrical rotor position (rad) ˆK n = normalised back EMF Fourier series coefficients n = harmonic number of back EMF and current series Current It is assumed that the current and electromagnetic torque can also be represented as a Fourier series. where: i θe,a = Î n e jnθ e (2.6) n= i θe,a = current in phase a at electrical angle θ e (A) Î n = current Fourier series coefficients k θe,b and i θ e,b can be found by replacing θ e with (θ e 2π 3 ). k θ e,c and i θ e,c found by replacing θ e with (θ e + 2π 3 ). can be Electromagnetic torque in each phase It is also assumed that the electromagnetic torque produced by each phase can be represented by a Fourier series. 14

43 2.5. FREQUENCY DOMAIN METHOD (FDM) where: τ em θ e,a = m= ˆT m,a e jnθ e (2.7) τ em θ e,a = electromagnetic torque produced by phase a at electrical angle θ e (Nm) ˆT m,a = electromagnetic torque Fourier series coefficients for phase a m = harmonic number of torque series The combination of the current and back EMF harmonics can be modeled as a convolution sum so: ˆT m,a = n= ˆK m n Î n (2.8) Total electromagnetic torque The total electo-magnetic torque produced will be a sum of the three phase torques. τ em θ e = m= ˆT m,a e jmθ e + ˆT m,b e jm(θ e 2π 3 ) + ˆT m,c e jm(θ e+ 2π 3 ) (2.9) From the assumption that the back EMF and current harmonics are the same for all phases: ˆTm,a = ˆT m,b = ˆT m,c, equation 2.9 can be rewritten as: τ em θ e = m= Which can be simplified to: By noting that : τ em θ e = ( ˆT m,a (e jmθ e + e jm(θ e 2π 3 ) + e jm(θ e+ 2π 3 )) (2.10) m= ( ˆT m,a 1 + 2cos( 2πm ) 3 ) e jmθ e (2.11) 1 + e jn( 2π 3 ) + e jn(+ 2π 3 )) = 1 + 2cos( 2πm 3 ), equations 2.11 and 2.8 can be combined to determine an expression for the coeffients of the total torque harmonics: ˆT m = n= ( ˆK m n Î n 1 + 2cos( 2πm ) 3 ) 15 (2.12)

44 CHAPTER 2. REVIEW: PROGRAMMED REFERENCE CURRENT WAVEFORM METHODS Now: ( ) 2πm cos = 3 n = ±3, ±6, ±9... (2.13) 3 ( ) 2πm cos = 0 n ±3, ±6, ±9... (2.14) 3 So: ˆT m = 3 n= ˆK m n Î n n = ±3, ±6, ±9... (2.15) As expected, consideration of equations 2.8 and 2.15 suggests that the total torque is three times the torque produced by each phase Torque ripple and RMS current minimisation The relationship between back EMF, current and generated torque harmonics can be seen clearly in Table 2.1 which is a modification of Table 1 presented by Favre, Cardoletti and Jufer [29]. The 0 harmonics are those responsible for average torque creation. Those marked with an x would create torque harmonics, however a star connection constraint ensures that there are no current harmonics that are multiples of three. It is worth noting not only which harmonics will combine to create torque ripple, but conversely, which harmonics must be present in order to cancel out particular cogging torque harmonics. For example if the back EMF is purely sinusoidal, the fifth and seventh current harmonics are required to cancel out a sixth harmonic in the cogging torque. Limits of series So far the limits of the Fourier series have been considered infinite. In practice, a limit needs to be chosen for each of the series. To ensure that equation 2.15 is not under-constrained, the limit of the current harmonics (N) must be greater or equal to the limit of the back EMF harmonics (L). N can be equal to L as long as L is not a multiple of three. 16

45 2.5. FREQUENCY DOMAIN METHOD (FDM) Table 2.1: Back EMF and current harmonic combinations responsible for torque harmonics (including star constraint) ˆK n x x x x x x x x x x Î n 9 x x x x x x x x x x x x x x x

46 CHAPTER 2. REVIEW: PROGRAMMED REFERENCE CURRENT WAVEFORM METHODS The largest torque harmonic that can be cancelled is equal to the sum of the current and back EMF harmonics (M = N + L). This is particularly important to consider when cogging torque harmonics need to be cancelled. Matrix representation Equation 2.15 can be represented in matrix form by: ˆK Î = ˆT (2.16) where: ˆK = matrix of the back EMF coefficients Î = vector of current coefficients ˆT = vector of torque coefficients As long as the system of equations defined in equation 2.16 is over-constrained, an additional constraint to ensure the current is minimised can be added. By using the minimum norm solution, the optimal current can be defined as: Îoptimal = ˆK T ( ˆK ˆK T ) 1 ˆT (2.17) Cogging torque suppression If cogging torque suppression is required, the first element of ˆT should be the average torque required and the remaining elements should be the negative of the cogging torque Fourier coefficients. In this way the torque ripple should cancel out the cogging torque Star connection constraint In the frequency domain, the addition of a star connection constraint requires that there are no current harmonics that are multiples of three. If this is required, the rows of ˆK that correspond to these harmonics should be removed. 18

47 2.6. TIME DOMAIN METHOD (TDM) 2.6 Time domain method (TDM) Parameter descriptions This method follows the analysis presented by Wu and Chapman [22]. The back EMF is defined as an array of values for each discrete value of θ. This array is then used directly in calculations to define the reference current Torque ripple and RMS current minimisation The optimisation is done using the method of Lagrange multipliers where the objective Q is defined and subjected to a cost constraint C. The optimal solution is found by differentiating the resulting function with respect to i. Objective to be minimised (RMS current): Q θ = p=a,b,c 1 2 (i θ,p) 2 (2.18) Cost constraint to be minimised: deviation of electromagnetic torque from reference torque (τ em θ - τ ref θ ): Overall function: C 1,θ = p=a,b,c k θ,p i θ,p τ ref θ (2.19) so: f θ = p=a,b,c f θ = Q θ + λ 1,θ C 1,θ (2.20) 1 2 (i θ,p) 2 + λ 1,θ k θ,p i θ,p τ ref θ (2.21) p=a,b,c This function will be minimised with respect to i θ,p when f θ i θ,p = 0. Taking this partial derivative of equation 2.21, expressions are found for optimal current: i θ,p = λ 1,θ k θ,p (2.22) 19

48 CHAPTER 2. REVIEW: PROGRAMMED REFERENCE CURRENT WAVEFORM METHODS λ 1,θ, however is still unknown. To find the optimal value of λ 1,θ requires setting f θ = 0, which gives: λ 1,θ τ ref θ = p=a,b,c This implies that the cost function is satisfied. k θ,p i θ,p (2.23) To find an expression for λ 1,θ that is independent of i θ,p, equation 2.22 needs to be substituted into equation 2.23, to give: τ ref θ = λ 1,θ p=a,b,c which when rearranged to solve for λ 1,θ gives: (k θ,p ) 2 (2.24) Now equation 2.25 can be substituted into equations 2.22 to find the optimal currents: λ 1,θ = τ ref θ p=a,b,c (k θ,p) 2 (2.25) i θ,p = τ ref θ ( k θ,p p=a,b,c (k θ,p) 2 ) (2.26) Cogging torque suppression If cogging torque suppression is required, τ ref θ should be defined as the average torque required minus the cogging torque. As in the FDM, this should ensure that the torque ripple will cancel out the cogging torque Star connection constraint If a star connection constraint is to be considered, it can be written in matrix form as: i θ,p = 0 (2.27) p=a,b,c This equation can be formulated as a Lagrange cost constraint that needs to be minimised: 20

49 2.6. TIME DOMAIN METHOD (TDM) C 2,θ = i θ,p (2.28) p=a,b,c Now there are two constraints to minimise in the Lagrange function which becomes: f θ = p=a,b,c 1 2 (i θ,p) 2 + λ 1,θ k θ,p i θ,p τ ref θ + λ 2,θ p=a,b,c p=a,b,c i θ,p (2.29) As before, the function will be minimised with respect to i θ,p when f θ i θ,p = 0. Taking this partial derivative of equation 2.29, expressions are found for optimal current: i θ,p = p=a,b,c To find the optimal value of λ 2,θ requires setting p=a,b,c ( λ 1,θ k θ,p λ 2,θ ) (2.30) f θ λ 2,θ = 0, which gives: i θ,p = 0 (2.31) Again, this implies that the cost function is minimised. Substituting equation 2.30 into equation 2.23 gives: τ ref θ = p=a,b,c τ ref θ = λ 1,θ k θ,p ( λ 1,θ k θ,p λ 2,θ ) (2.32) p=a,b,c k θ,p 2 λ 2,θ p=a,b,c and substituting equation 2.30 into equation 2.31 gives: k θ,p (2.33) ( λ 1,θ k θ,p λ 2,θ ) = 0 (2.34) p=a,b,c λ 1,θ p=a,b,c k θ,p 3λ 2,θ = 0 (2.35) λ 1,θ and λ 2,θ can be found from equations 2.32 and 2.34 to give: λ 1,θ = 3τ ref θ 3 p=a,b,c k θ,p 2 ( p=a,b,c k θ,p) 2 (2.36) 21

50 CHAPTER 2. REVIEW: PROGRAMMED REFERENCE CURRENT WAVEFORM METHODS λ 2,θ = τ ref θ p=a,b,c k θ,p 3 p=a,b,c k θ,p 2 ( p=a,b,c k (2.37) θ,p) 2 Which can be substituted back into equation 2.30 to find an expression for the optimal current: i θ,p = τ ref θ ( 3k θ,p p=a,b,c k θ,p 3 p=a,b,c k θ,p 2 ( p=a,b,c k θ,p) 2 ) (2.38) 2.7 Park-like method (PLM) The PLM is a variation on field oriented control where the current is described with reference to a rotating coordinate system. This discussion will follow the most recent implementation of this technique which was done by Park et al. in 2001 [27]. A similar technique was described by Grenier et al. [23] Lu et al. [24] and Chen and Sekiguchi [25][26] Parameter descriptions In this method, all the parameters are converted from their three phase coordinates (a, b, c) to a rotating reference frame (d, q). k θ,d k θ,q = C k θ,a k θ,b k θ,0 k θ,c (2.39) where: sin(θ) sin(θ C = 2 2π 3 ) sin(θ + 2π 3 ) cos(θ) cos(θ 3 2π 3 ) cos(θ + 2π 3 ) (2.40) Torque ripple minimisation In a rotating reference frame, the general torque equation stated in equation 2.2 can be rewritten as: 22

51 2.8. EXPERIMENTAL IMPLEMENTATION τ em θ = 3 2 (k θ,di θ,d + k θ,q i θ,q + k θ,0 i θ,0 ) (2.41) The torque produced by i d and i 0 should be zero. Substituting this into equation 2.41 and rearranging gives the following expression for the i q current required to provided the desired reference torque: i θ,q = 2 3 τ em θ k θ,q (2.42) If it is desired to know the currents in the three phase reference frame then the current can be converted back by multiplying the d, q reference currents by C 1 : k θ,d k θ,q k θ,0 = C k θ,a k θ,b k θ,c (2.43) Unlike the other two methods, this method does not include a current minimisation constraint Cogging torque suppression Like the other two methods, cogging torque suppression is done by defining a reference electromagnetic torque that is the negative of the cogging torque Star connection constraint The PLM ensures that i 0 = 0 so there will always be a star connection constraint. 2.8 Experimental implementation As suggested in the introduction, experimental implementation of PRCW methods is limited. This is partly due to a lack of a quantitative measure of the remaining pulsating torque. If the torque is actually measured, most authors resort to presenting figures and commenting that the pulsating torque has been significantly reduced. This section will consider the goal for pulsating torque, and discuss the challenges faced by previous PRCW method researchers when implementing their methods experimentally. 23

52 CHAPTER 2. REVIEW: PROGRAMMED REFERENCE CURRENT WAVEFORM METHODS Pulsating torque goal The goal for smooth operation of a PMAC motor is not well defined. Jahns and Soong [3] stated that for motor modification methods such as skewing, about 1% of rated torque is the lowest achievable in practice. Grcar et al. [9] suggested that for high performance drives, a torque pulsation under 1-2% is typically considered as the desired objective. Liu et al. [30] discussed a power steering application with a peak-peak pulsating torque requirement of less than 2-5%. Specifications of commercially available high performance motors also vary considerably. Trust Automation [31] claims a 0.3% ripple torque for their SE700 motors. For their BL series motors, Malivor [32] do not quote a pulsating torque however provide data on reluctance torque (cogging torque) as 3.5-6%. Parker Motion [33] (p 154) quotes 5% ripple peak-peak for their Dynaserv system and ThinGap [34] quotes 0.045% ripple. That value however, is not based on measurements but rather on the harmonic distortion of the back EMF. Additional confusion is created by multiple methods for calculating the percentage pulsating torque. Often the method is not quoted. When the method is quoted, it is usually the ratio of peak-peak torque to rated torque. Peak-peak torque however, is susceptible to large variations with noise. To avoid that issue, for this research, the RMS torque is used as suggested by Gieras and Wing [6] (p245). However, rather than divide the RMS torque by the average torque as they did, division is done by the maximum torque. The justification for this is that at low torque set points, the use of average torque creates distortion. This is because, for the motor used, a large proportion of the pulsating torque is related to cogging which does not scale with average torque. T pulsating = T error RMS (2.44) T max While the RMS gives a lower pulsating torque than the peak-peak value, a rough comparison is to consider the relationship for a pure sine wave where the peak to peak value is 2 2 times larger than the RMS value. Assuming the figures given above for high performance drives are peak-peak values then a RMS value of 1% would fit somewhere in the range of motors surveyed and so is the goal of this reaserch. 24

53 2.9. IMPLEMENTATION CHALLENGES 2.9 Implementation challenges The most significant challenge facing the implementation of PCRW methods is the determination of motor parameters to a suitable accuracy [3]. This is discussed in detail in chapter 3. The other challenges, as outlined in the introduction are: 1. accurate torque measurement; 2. the provision of a smooth load to the motor, and 3. the presence of mechanical resonances in the experimental setup Torque measurement In the absence of a suitable torque sensor, many PRCW method experimental setups rely on recreating the torque from the measured currents and back EMF waveforms. Unfortunately, this neglects any cogging torque or pulsating torque created from parameter variations in the windings. These factors have the potential to have a significant impact on the effectiveness of programmed current waveform methods. When the torque is measured, one potential issue is achieving resolution. Li et al. [35] highlighted a problem with traditional methods of testing for torque ripple. A torque sensor with a large enough range to deal with the maximum load will not necessarily have the resolution to measure the torque ripple. Another issue with torque measurement is adequate bandwidth. All the sensors used for PRCW research have been in-line torque sensors, either strain gauge or surface acoustic wave technology. The surface acoustic wave technology sensor used by Wu and Chapman [22] had a bandwidth up to 1 khz. Aghili, Buehler and Hollerbach [36] used a Himmelstein strain gauge sensor which can have a bandwidth somewhere between Hz depending on the signal type [37] and the signal conditioner used [38]. With bandwidths in this range, the speed of the motor is limited if higher harmonics are to be measured. Beccue et al. [39] suggest the use of a piezoelectric polymer for use in PMAC motors in preference to the the other methods discussed. Their justification however, was cost on the assumption that this type of sensor would be installed in all motor installations. As PRCW methods seek to avoid a torque sensor in all installations and 25

54 CHAPTER 2. REVIEW: PROGRAMMED REFERENCE CURRENT WAVEFORM METHODS only use a torque sensor for initial calibration and validation, cost is a less significant issue Load application To apply a load, previous PRCW method researchers have either used: DC motors, hysteresis brakes, eddy current brakes, hydraulic motors or friction brakes. DC motors (generator) DC motors provide a load by operating as a generator and turning kinetic energy into electrical energy. The electrical energy then needs to be either fed back into the power supply for the test motor or dissipated in a resistor bank. The torque applied can be readily controlled by adjusting the current flowing out of the generator. Some issues can arise however, when this method is used in a setup for measuring pulsating torque. Qian, Panda and Xu [40] described how the pulsating torque from the load DC motor interfered with the measurement of the pulsating torque from the test motor. This can be minimised with careful choice of load motor. However, because of the commutations in a DC motor, it will always be an issue to some extent. Hysteresis Hysteresis is a property of magnetic materials where the flux density is a function of previous field intensity across the material [17]. In a hysteresis brake, a disk made of magnetic material rotates between a series of magnetic poles. Poles previously induced in the disk interact with the stationary magnetic poles and torque is transmitted until the disk begins to slip. Once slipping, the poles in the disk move as the material is magnetised and demagnetised [41], [42] and [43]. Energy is dissipated due to the hysteresis of the material. Load torque is adjusted by varying the strength of the applied magnetic field. Hysteresis motors have the benefit that the torque applied is independent of speed so torque can be applied down to zero speed. This means however, that if speed is to be regulated then a control system is required to do so. Wu and Chapman [22] found that the controller from the commercially available hysteresis brake was not able to 26

55 2.9. IMPLEMENTATION CHALLENGES keep a constant low speed. Another problem associated with hysteresis brakes is the potential for residual cogging torque to be created if the brake is incorrectly shut down [44](p53). As hysteresis brakes are unable to drive the motor, another drive source is required to determine back EMF. Any pulsating torque from the drive motor is not an issue however, because the back EMF is normalised with velocity. Eddy current In an eddy current brake a highly conducting disk (such as copper or aluminium) rotates in a magnetic field. The magnetic field induces eddy currents in the disk. These eddy currents in turn create a magnetic field that opposes the original magnetic field. As the current produced is a function of the velocity the torque produced is a function of motor speed [43], [45]. This proportionality of torque to speed, limits the applicability of eddy current brakes as they cannot be used for zero speed testing ( locked rotor ). Another issue is that due to their inherent damping they have a large mechanical time constant so are unsuitable for high frequency dynamic testing. They do however have the benefit that the torque applied is completely smooth. Because eddy current brakes are unable to create motion, like hysteresis brakes, another drive motor is required when measuring the back EMF. Hydraulic motors Hydraulic motors can be used to apply a load in a similar way to a DC motor [36]. Instead of controlling the torque by the amount of current, the pressure is regulated. As with DC motors, if the motor is not chosen carefully it can create pulsating torque. Due to the peripheral equipment required, hydraulic motors are usually more complex and expensive than other options. Prony friction brake Another device used for research into PRCW methods is a Prony brake test device which was used in [46]. This is a rudimentary device made of a drum immersed in 27

56 CHAPTER 2. REVIEW: PROGRAMMED REFERENCE CURRENT WAVEFORM METHODS cooling water with a belt running over it. Accurate control of this device is difficult and the authors noted that there was a significant once per revolution component due to the brake device Mechanical resonance problems In their 2005 paper [22], Wu and Chapman suggest that mechanical resonances in the experimental setup obscured the fluctuating torque that they were attempting to measure. Unless measures are taken to avoid this problem, other experimental setups would suffer from the same problem. The use of flexible couplings to connect in-line torque transducers is a particular concern as their flexibility creates additional system dynamics that are difficult to model Summary of PRCW methods Three PRCW methods have been presented: the time domain method (TDM) the frequency domain method (FDM) and the park-like method (PLM), each claiming to minimise pulsating torque. The FDM and the TDM also attempt to minimise RMS current. The performance of each of these methods is compared for various back EMF and cogging torque waveforms in chapter 4. Methods of describing pulsating torque were discussed and levels of torque in industrial drives considered. This allowed a goal for smooth operation to be defined. The challenges facing successful experimental implementation were presented. The largest challenge facing all of these methods is the accuracy of the determination of the motor parameters. Published methods for determination of these parameters are discussed in chapter 3. 28

57 Chapter 3 Review: determination of motor parameters Common to all of the programmed current waveform methods discussed in chapter 2 is the need for accurate information about motor parameters. If the back EMF, cogging torque, current flow and rotor position are known for a motor then it is theoretically possible to eliminate pulsating torque. Problems are created if these parameters are determined inaccurately. This chapter will review existing work done on the determination of motor parameters to ensure that the experimental methods implemented are based on the most accurate motor information available. Consideration will be given to determination by calculation and by measurement. Explanation will also be given on how the variation of each property will affect the pulsating torque. The final part of the chapter will discuss the importance of sensitivity analysis and work that has been done on that subject. 3.1 Back EMF Back EMF is the induced EMF created by the interaction of rotor and stator magnetic fields when the motor is turned. It is normally measured with the phases open-circuit while the motor is being rotated by an external drive. Though measurement is reasonably straightforward, a brief discussion is provided as to the likely harmonics present, 29

58 CHAPTER 3. REVIEW: DETERMINATION OF MOTOR PARAMETERS so that the credibility of the measured results can be checked Analytical determination Generally the back EMF in a PMAC is a shape that is somewhere between a sinusoid and trapezoid [3]. Such a signal only contains odd harmonics, so most back EMF waveforms will contain only odd harmonics. Faraday s law defines that the back EMF is the derivative with respect to angle of the flux linkage. The flux linkage itself is a function of air-gap flux density, winding inductance, winding resistance, the number of slots, the number of magnets and geometrical factors. In his book, Brushless Permanent Magnet Motor Design [17], Hanselman provides a series of predicted back EMF shapes for various motor configurations. These can be used for a rough guide to check against experimental results Finite element analysis (FEA) Much work has been done in determining the back EMF using finite element analysis mostly as part of the design process for new motors. As the motor design is beyond the scope of this research these methods will not be discussed in detail. Of interest however, is an analysis done by Patterson [47] on a very similar motor to the one used for the experimental part of this research. Results suggested an error of 1% between the back EMF calcluated from FEA and the measured back EMF. Determination of back EMF using finite element analysis, while potentially accurate, usually assumes that the back EMF will be identical for each electrical revolution. Unless specifically catered for, manufacturing variation will lead to errors in the determined waveform Measurement Measurement of back EMF is generally straightforward. In their book Design of Brushless Permanent-Magnet Motors [48](p11-2) Hendershot and Miller describe it as: perhaps the simplest and most useful test which can be performed on a brushless DC motor. This statement can be extended to other types of PMAC motors. An external drive is needed to provide rotation while the open circuit voltage is measured. As the 30

59 3.2. COGGING TORQUE final goal is the speed normalised back EMF, it is also critical that an accurate measurement of velocity is obtained. This is particularly important if there is any speed fluctuation during the measurement resulting from cogging torque Effect of error on pulsating torque As long as the assumption holds that the back emf is proportional to velocity (see section 2.1.1), the shape of the back emf will remain constant. That allows the back EMF error to be divided into offset and scaling errors. In a similar way to the current errors discussed in section 3.3.2, an offset error will lead to a pulsating torque harmonic at the fundamental electrical frequency and a scaling error will lead to a pulsating harmonic at twice the electrical frequency. Temperature Mattavelli, Tubiana and Zigliotto [49] suggested that the back EMF could possibly vary with motor temperature, but not significantly. 3.2 Cogging torque The formal definition of cogging torque is: pulsating torque components generated by the interaction of the rotor magnetic flux and angular variations in the stator magnetic reluctance. [3]. For the purposes of PRCW methods, it is the component of pulsating torque that is independent of excitation current. Any current reference waveform method that seeks to remove pulsating torque must compensate for the cogging torque and so needs an accurate, time invariant description of this torque. In contrast to the back EMF, experimental determination of cogging torque is much more difficult, mainly because of the difficulty in decoupling the cogging toque from either the torque ripple or from dynamic effects resulting from the speed variation of the rotor. 31

60 CHAPTER 3. REVIEW: DETERMINATION OF MOTOR PARAMETERS Analytical determination As with back EMF, it is worth considering which harmonics will be present in the cogging torque to provide a guide to the validity of any measurements. The fundamental cogging torque harmonic will be: [6][p247] where: f cogging = N slot N pole pair (3.1) N slot = number of stator slots N pole pair = number of pole pairs Detailed analytical calculations can be done to determine the actual waveform. The cogging waveform depends on the tooth Fourier series coefficients which are a function of the magnetic field distribution around each tooth, the air gap length and the size of the slot opening between teeth [17](p210). This detailed analysis is beyond the scope of this thesis FEA As with back EMF, much work has been done on the calculation of cogging torque by FEA. It is generally done as an aid to motor design and so is beyond the scope of this research. Some research, such as that presented by Islam, Mir and Sebastian [50] and Shaotang, Namuduri and Mir [51] is relevant to this research because they use FEA to consider the effect of motor construction variation on pulsating torque Experimental Experimental methods for determining cogging torque can be classified into static, quasi-static and dynamic methods. Static Some static measurements of cogging torque are only designed to determine the absolute magnitude. For instance, a method is described by Caricchi et al. [52] where a 32

61 3.2. COGGING TORQUE measurement was taken of the force applied to a lever arm sufficient to move the rotor from one equilibrium state to the next. The static method presented by Chandler [53] allows the determination of the actual waveform. The rotor was mounted in a rotary dividing head from a milling machine. The stator was held in place by a beam with a strain gauge attached. By rotating the dividing head the force on the strain gauge could be noted. Problems were noted with this method however. For adequate sensitivity, the strain gauge was placed on a flexible beam. This flexibility made accurate measurements difficult, particularly when the motor was crossing from unstable position from one tooth to the next. Quasi-static A quasi-static method is described by Aghili, Buehler and Hollerbach [36] where the motor velocity is kept sufficiently low (1 o /s), to ensure that the inertial torque does not interfere with the measurement. Their setup is driven by a hydraulic motor with the pressure set sufficiently high to ensure that the angular speed remained constant regardless of the test motor torque. Dynamic Most motor measurement equipment is configured for measurements while the motor is rotating. As such, it is attractive to attempt cogging torque measurement while the motor is rotating. This form of measurement is discussed by Holtz and Springob [4] and by Bianchi and Bolognani [54]. As an in-line torque sensor was used, both authors pointed out the need for keeping rotor speed constant during the measurements. When measuring cogging torque dynamically, the location of the torque sensor is important. Figure 3.1 shows a free body diagram (FBD) for the motor. The two possibilities for the location of a torque sensor are: 1. In-line torque sensor which will measure the load torque in the shaft. 2. Reaction torque sensor which will measure the reaction torque on the back of the stator. 33

62 CHAPTER 3. REVIEW: DETERMINATION OF MOTOR PARAMETERS Figure 3.1: FBD for motor Figure 3.2: FBD for rotor In the case of the in-line torque sensor, if a FBD is taken of just the rotor (see figure 3.2) then the sum of the moments in the axis of rotation gives: T motor = T load + Jα + bω (3.2) On the other hand, with a reaction torque sensor (see FBD in figure 3.3) the sum of moments in the axis of rotation gives (assuming that the reaction torque sensor is suitably stiff): T motor = T reaction (3.3) 34

63 3.2. COGGING TORQUE Figure 3.3: FBD for stator Care must be taken when using an in-line torque sensor, as compensation has to be made for inertial torques due to acceleration. This issue is avoided by the use of a reaction torque sensor. Many other published works ([55], [56], [57], [50], [58] and [59]) present data for experimentally measured cogging torque but have limited information as to how these measurements were performed Effect of error on pulsating torque As the pulsating torque is the sum of the torque ripple and the cogging torque, the pulsating torque will increase by the error in the estimate of the cogging toque. Errors can be classified into: manufacturing (magnet placement, eccentricity and material property variation), and operating point dependent (temperature and torque set point dependent). This classification is important as theoretical determination will not usually account for manufacturing errors, whereas direct measurement will measure these errors. Variation that is operating point dependent will be difficult to determine by either method. Material properties Cogging torque is a function of the magnetic interaction between the permanent magnets and the steel in the stator. As such, any variation in the magnetic properties of 35

64 CHAPTER 3. REVIEW: DETERMINATION OF MOTOR PARAMETERS either of these materials will cause variation in cogging torque. Morcos, Brown and Campbell [60] suggested that poor uniformity of the magnets can lead to high cogging torque. Manufacturing tolerances Islam, Mir and Sebastian [50] report that if a magnet is misplaced from its perfect position by 1 mechanical degree (in a 6 pole 27 slot motor), then the magnitude of the cogging torque can be increased by over three times. Obviously this result will vary with motor topology but the fact that accurate manufacturing is critical is still worth noting. It is suggested in by Hartman and Lorimer [61] that other common defects are non-concentric stators and rotors (equivalent to misalignment in an axial flux motor such as that used for this research). Temperature Grcar et al. [9] claimed that the waveform of the cogging torque varies with the operating conditions (temperature), however the extent of this possible variation is not reported Variation with torque set-point Although usually defined as being independent of current flow, Gieras and Wing [6][p247] suggest that cogging torque can be affected by saturation effects associated with high current flow. Dai, Keyhani and Sebastian [62] discuss how an uneven tooth flux density can cause a current related component of the cogging torque. 3.3 Current Unlike the back EMF and cogging torque, the optimal current is calculated and so does not need to be determined. However, as current is used as the feedback variable in the controller, its measurement is critical. Any measurement errors, along with any errors in the controller itself will lead to pulsating torque. 36

65 3.4. ROTOR POSITION Measurement Current measurement is normally done by measuring the voltage across a shunt or by a hall effect device. Chapter 6 contains a detailed discussion of current sensors including justification of the sensors chosen for this research Effect of error on pulsating torque Current measurement error is normally considered as a combination of an offset and a gain error (i.e. the assumption is made that the output remains linear). Analysis of current error has been discussed in several publications [63], [64], however the most detailed analysis is reported by Chen, Namuduri and Mir [51]. Offset error The addition of an offset into the torque equation (equation 2.2) induces a torque ripple at the fundamental electrical frequency. Chen, Namuduri and Mir calculated that in the worst case, a 1% error in offset could lead to a 4% error in torque ripple. Scaling error The addition of a scaling error to the torque equation (equation 2.2) leads to a torque ripple of twice the fundamental electrical frequency. Chen, Namuduri and Mir suggested that in the worst case, a 1% scaling error between sensors in different phases could lead to a 2.3% torque ripple. 3.4 Rotor Position Measurement Normally rotor position is either measured by an encoder, where the position signal is a digital signal, or by a resolver where the signal is a sinusoidally varying analog signal. Detailed discussion of position sensors is presented in chapter 6. 37

66 CHAPTER 3. REVIEW: DETERMINATION OF MOTOR PARAMETERS Effect of error on pulsating torque Analysis by Chen, Namuduri and Mir [51] suggests that for a low resolution encoder (10 electrical degrees/ count in their setup) the induced torque ripple can be up to 5%. This emphisises the need for a high resolution encoder when implementing PRCW methods. 3.5 Sensitivity analysis Jahns and Soong [3] noted that there was minimal work published on sensitivity analysis for PRCW methods. Since then there has not been much further work done on the subject. Grcar et al. [65] stated that Since motor parameters can considerably vary under a wide range of operating conditions (temperature, saturation, load variations), a sensitivity analysis must be included as a part of the design for every particular drive. They do not however, report the results of their analysis. 3.6 Summary of methods for determination of motor parameters This chapter discussed several methods for the determination of motor properties. In most cases, to account for manufacturing errors, experimental methods are preferred to theoretical methods. Theoretical methods are important as a guide to the credibility of the experimental results. 38

67 Chapter 4 Theoretical comparison of reviewed methods Chapter 2 discussed the mechanisms for the creation of pulsating torque, presented a mathematical model of PMAC machines and reviewed the published methods for pulsating torque minimisation. This chapter theoretically compares the reviewed methods when applied to a generalised motor model with a variety of back EMF and cogging torque waveforms. Initially, pulsating torque minimisation is considered without cogging torque using various back EMF waveforms. Some of these variations break the assumptions imposed by some methods to determine the sensitivity of each method to those assumptions. Next, consideration is given to the ability of each method to compensate for cogging torque harmonics. 4.1 Methodology In this chapter, all calculations are done using the mathematical computer package Matlab TM. Optimal currents are calculated using one of the PRCW methods. By assuming that the current controller exactly follows the reference current, the torque produced can be determined using equation 2.3 for a given back EMF and cogging torque. 39

68 CHAPTER 4. THEORETICAL COMPARISON OF REVIEWED METHODS 4.2 Parameter and constraint variation Baseline for comparison Generally the goals of reference current waveforms methods are minimum pulsating torque using minimum current. One baseline for comparison was a pure sine wave (SIN) as that is what is most widely used. The other baseline was what is described in [20] as minimum current method (MCM). This mode demonstrates that minimum current is used if the current is the same shape as the back EMF. The amount of current used is quoted as the percentage increase required over the MCM. The pulsating torque is quoted as the RMS variation as a percentage of full scale Scope of variation The performance of the methods is primarily affected by the back EMF and cogging torque waveforms. It is also affected by whether a star connection constraint is applied. 4.3 Performance with different back EMFs (without cogging torque) Generally the shape of the back EMF in a PMAC motor varies between sinusoidal and trapezoidal [3]. The methods were compared at each of these extremes. Another variation that may occur is that the back EMFs from each phase are unbalanced. This may occur in one of two ways, a magnitude variation or a phase variation. To test the magnitude variation, phase B was scaled to an error of ± 1%, 2% and 3% of the value of the other two phases and the effect on torque ripple was noted. To test the phase variation, phase B was shifted by 1, 2 and 3 degrees and the torque ripple noted Sinusoidal back EMF As a baseline, the three methods were compared for a sinusoidal back EMF. As expected, all three determined that the best current shape would be a sinusoid and all were capable of completely removing pulsating torque. In figure 4.1 all the lines are 40

69 4.3. PERFORMANCE WITH DIFFERENT BACK EMFS (WITHOUT COGGING TORQUE) on top of one another as all methods give the same results. Table 4.1) shows that all methods have no pulsating torque and use the same amount of current. p Current Phase A A FDM TDM PLM SIN MCM Nm 5 5 Torque output electrical angle (degrees) FDM TDM PLM SIN MCM Figure 4.1: Current and torque output for sinusoidal back EMF FDM TDM PLM SIN MCM % Pulsating Torque % Current Increase Table 4.1: Performance of PRCW methods with a sinusoidal back EMF Trapezoidal back EMF - adding odd harmonics At the other end of the range of possible back EMFs is a trapezoidal back EMF. For this analysis, an approximation of a trapezoidal back EMF was achieved by adding the appropriate odd harmonics up to a certain limit. That limit was determined by the number of harmonics required to reduce the truncation error between an ideal 41

70 CHAPTER 4. THEORETICAL COMPARISON OF REVIEWED METHODS trapezoid and the approximation to a RMS variation of less than 1%. This required harmonics up to the 17th. As with the sinusoidal back EMF, each of the methods was able to eliminate pulsating torque with a small increase in current (see figure 4.2 and Table 4.2). The increase in current required for the TMD and the PLM is 2.7% and the FDM is slightly higher at 3.5% above that required for the MCM. Current Phase A A FDM TDM PLM SIN MCM Nm Torque output electrical angle (degrees) FDM TDM PLM SIN MCM Figure 4.2: Current and torque output for trapezoidal back EMF FDM TDM PLM SIN MCM % Pulsating Torque % Current Increase Table 4.2: Performance of PRCW methods with a trapezoidal back EMF 42

71 4.3. PERFORMANCE WITH DIFFERENT BACK EMFS (WITHOUT COGGING TORQUE) Variation between phases As noted in section 2.5.1, the FDM assumes that each of the back EMFs and currents are identical and spaced 120 electrical degrees apart. To test each of the methods sensitivity to violation of this assumption tests were done varying the magnitude and the phase of phase B relative to the others and the effect on pulsating torque noted. Magnitude variation It can be seen in figure 4.3 that while the TDM and the PLM were able to cope with this variation, the output from the FDM was affected. The effect on pulsating torque however is relatively minor, with a variation of 3% only creating a 0.5% RMS variation in the torque. % % Average torque variation Pulsating torque Current requirement increase 6 % Phase B scaling (%) FDM TDM PLM FDM TDM PLM FDM TDM PLM Figure 4.3: Performance of different methods if one phase has a magnitude variation Phase variation Figure 4.4 shows that as with the magnitude variation, the TMD and the PLM still can eliminate pulsating torque if the back EMF has an offset variation. The impact of 43

72 CHAPTER 4. THEORETICAL COMPARISON OF REVIEWED METHODS this change is has a much greater affect on the output of the FDM with an offset of 3 degrees creating a 6% RMS variation in the torque. % Average torque variation FDM TDM PLM % % Pulsating torque Current requirement increase Phase B offset (degrees) FDM TDM PLM FDM TDM PLM Figure 4.4: Performance of different methods if one phase has a phase variation 4.4 Comparison of cogging torque compensation To determine the capability of each method to cancel out cogging torque, they were tested with a simulated cogging torque containing only one harmonic. Figure 4.5 considers the pulsating torque output from each method for a cogging torque containing a harmonic from 1 to 40. Table 2.1 suggested that the FDM is not capable of compensating cogging torque harmonics that are not a multiple of three. Figure 4.5 clearly demonstrates that this was the case in the simulations. As the frequency to be compensated increased, the current required by the FDM increased considerably. This current increase was due to the small size of the higher back EMF harmonics. For example, consideration of table 2.1, shows that when using the FDM, a 30th harmonic can only be compensated by a combination of a 13th and 17th or 14th and 16th current and back EMF harmonics. 44

73 4.4. COMPARISON OF COGGING TORQUE COMPENSATION As the trapezoidal back EMF only contained odd harmonics, compensation of the 30th harmonic depended on the the 13th and 17th back EMF harmonics. As these were small, the matching current harmonics had to be very large. In addition, because the back EMF only had harmonics up to 17, the FDM could only compensate up to the 33rd harmonic. Both the TDM and the PLM could successfully compensate all harmonics with minimal increase in current. % pulsating torque % current increase Pulsating torque x Current requirement cogging torque harmonic FDM TDM PLM FDM TDM PLM Figure 4.5: Ability of each method to compensate 1Nm of cogging torque at different harmonics Star connection constraint The comparisons described so far were done with a star connection constraint, however the test motor is configured to allow the star constraint to be removed. This allowed a study of how the methods performed without a star constraint. Though there is no benefit from removing this constraint if the back EMF was 45

74 CHAPTER 4. THEORETICAL COMPARISON OF REVIEWED METHODS sinusoidal, figure 4.6 and Table 4.3 show that the FDM and TDM benefited from the removal of this constraint by lowering the additional current required to 0.7%. There is no change in the PLM as it always includes a star connection constraint. FDM: Star connected currents FDM: Independent currents A B C Total TDM: Star connected currents PLM: Star connected currents Position (degrees) TDM: Independent currents PLM: Independent currents Position (degrees) Figure 4.6: Comparison of currents with and without a star connection constraint 46

75 4.5. UPPER FREQUENCY LIMIT CONSTRAINT FDM TDM PLM % Current Increase (with star constraint) % Current Increase (without star constraint) Table 4.3: Performance of PRCW methods with a modified trapezoidal back EMF 4.5 Upper frequency limit constraint The analysis presented so far, suggests that the TDM is superior. It can always eliminate pulsating torque using the same or less current than the other methods. One consideration that could limit the effectiveness of the TDM is that an upper frequency limit is not prescribed for the desired current. The existence of high frequencies in the TDM current could be an argument for using the FDM. The prescribed frequency limit of the FDM limits the current controller bandwidth required. To check this, the frequency spectrum of all methods was determined for each of the comparisons described. Figure 4.7 shows the current harmonics above the specified harmonic limit for the trapezoidal back EMF. The 17th harmonic shown is the highest in the frequency domain method. Though the TDM has a 19th harmonic, its magnitude is only about 0.1% of the fundamental, so is insignificant. If all harmonics above the harmonic limit are removed from the TDM method before the torque is calculated, there is still no noticeable pulsating torque. 47

76 CHAPTER 4. THEORETICAL COMPARISON OF REVIEWED METHODS Current harmonics % of fundamental harmonic number FDM TDM PLM Figure 4.7: Ability of each method to compensate 1Nm of cogging torque at different harmonics 4.6 Summary of theoretical comparison of PRCW methods Comparisons between different PRCW methods for various back EMF waveforms and cogging torques show that regardless of the shape of the waveforms, the time domain method (TDM) is always the best method. If implementation is to be done in a rotating reference frame (such as space vector modulation) then the use of the parklike method (PLM) may be beneficial. The frequency domain method (FDM), while unlikely to produce the best results, is useful for providing insight into pulsating torque production. A possible concern that the TDM s success might require the injection of higher harmonics than the FDM (potentially beyond the bandwidth of a current controller) was checked and found to be unfounded. 48

77 Chapter 5 Pulsating torque decoupling approach to motor parameter determination Chapter 3 discussed the importance of obtaining accurate information on motor properties to ensure minimum pulsating torque when using PRCW methods. The task of parameter determination is a considerable challenge as any measurement usually involves a number of conversions or scaling factors. Previous implementations of PRCW methods have either used datasheet values for this scaling or separate calibrations of each individual scaling factor. The limited success of these implementations suggests that a more accurate method is required for parameter estimation. This chapter presents an approach to parameter determination using pulsating torque decoupling (PTD). The proposed method uses a best guess at parameter values for an initial motor trial. Any inaccuracies in these parameter values will cause pulsating torque. By decoupling this resulting pulsating torque into components related motor parameters, inaccuracies can be quantified and compensated for in future operation. Discussion begins by restating the equation responsible for torque creation in block diagram form and highlighting sources of pulsating torque. The PTD method for decoupling the pulsating torque into the parameter errors responsible for its creation 49

78 CHAPTER 5. PULSATING TORQUE DECOUPLING APPROACH TO MOTOR PARAMETER DETERMINATION is then presented. The special case where the torque sensor only measures dynamic torque is discussed. It is demonstrated that in that situation, the overall system gain is the only additional information required. A determination method for this gain is presented. 5.1 Motor model overview Figure 5.1 is a block diagram of the experimental implementation of the general motor equation as expressed in equation 2.3. To simplify this analysis, and allow the electrical transfer function to be replaced by a simple gain, it is assumed that the current controller is effective in following the reference current. The validity of this assumption is verified in section This gain for each motor phase is designated w p, where p = a, b, c. Usually, in a practical implementation, there will also be an offset (o p ) associated with the current inverter. This model assumes that the transfer function between motor torque and measured torque is only a gain (u p ). If there is a mechanical resonance in the system or if an indirect measurement is used (such as an observer) then this assumption may not hold. For correct compensation, estimates of these three parameters are required: current inverter gain estimate (wp), current inverter offset estimate (o p) and torque sensor gain estimate (u p). Though an initial estimate is usually still required, this method allows compensation for errors in this initial estimate. The dotted black line in Figure 5.1 denotes the extremities of the hardware. The gain and offset blocks shown outside this box are software compensation for the hardware gains and offsets inside. 50

79 5.1. MOTOR MODEL OVERVIEW Figure 5.1: Block diagram of parameters to be determined Known parameters The PTD approach requires that some parameters are assumed to be accurate (shown in white in figure 5.1). Experiments (see chapter 7) determined that the back EMF (k θ,p ) was accurate. The reference current (i ref θ,p also known. ref ) and the torque measured (τθ ) are Block diagram simplification Pulsating torque is created when one of the parameters in Figure 5.1 is inaccurately determined (for example, w p w p ). To assist the analysis, the parameters and estimates can be replaced by other parameters which represent the estimation error. ( ) wp u p α p = wpu (5.1) p ( ) wp u p β p = (o p o p) (5.2) τ cog θ u p = u p u τ cog θ (5.3) p (5.4) With these new variables, Figure 5.1 can be modified as shown in Figure

80 CHAPTER 5. PULSATING TORQUE DECOUPLING APPROACH TO MOTOR PARAMETER DETERMINATION Figure 5.2: Block diagram of parameters to be determined (simplified) 5.2 Decoupling of pulsating torque: determination of current imbalance and cogging torque If the assumptions made are valid for the a particular motor, and if the ideal currents have been correctly calculated by one of the methods presented in chapter 4, pulsating torque will only come from an error in either: 1. cogging torque; and/or 2. an unbalance in the current, caused by an offset or gain error in the current sensors. These are shown in pink in Figures 5.1 and 5.2, and their determination will be the focus of this section. By decoupling the pulsating torque into the components created from each of these errors, it is possible to determine where the errors lie and compensate accordingly. To do this, it is important to note that the cogging torque will be independent of current input. The cogging torque is redefined as the residual resulting from a least squares minimisation matching the electro-magnetic torque to the measured torque. 52

81 5.2. DECOUPLING OF PULSATING TORQUE: DETERMINATION OF CURRENT IMBALANCE AND COGGING TORQUE General formula including scaling errors, offset errors and cogging torque Rewriting Equation 2.3 with the current offset and scaling factors as shown in Figure 5.2: where: τ meas θ = p=a,b,c (i θ,p α p + β p ) k θ,p + τ cog θ (5.5) τ meas θ = measured torque at angle θ (Nm) i θ,p = current in phase p at angle θ (A) α p = current scaling error in phase p β p = current offset error in phase p k θ,p = normalised back EMF for phase p at angle θ (V s/rad) τ meas θ = measured torque at angle θ (Nm) or in matrix notation: where: τ meas = (I ref K) α + (K) β + τ cog (5.6) τ meas = Θ 1 vector of the measured torque (Nm) I ref = Θ 3 matrix of the reference current (A) α = 3 1 vector of current scaling error β = 3 1 vector of current offset error K = Θ 3 matrix of the back EMF (V s/rad) τ cog = Θ 1 vector of the cogging torque estimate (Nm) = element-wise multiplication operator if we concatenate the matrices to let: X = ( I K 53 ) K (5.7)

82 CHAPTER 5. PULSATING TORQUE DECOUPLING APPROACH TO MOTOR PARAMETER DETERMINATION and concatenate the vectors to let: y = α (5.8) β then: τ meas = X y + τ cog (5.9) where y and τ cog are unknown. We previously defined cogging torque as the residual from from a least squares minimisation matching the electro-magnetic torque to the measured torque. The Moore- Penrose pseudo inverse is a convenient way of conducting a least squares minimisation [66]. By using this inverse and assuming that τ cog will be the residual, y can be found. where: + = pseudo-inverse operator y = X + τ meas (5.10) The residual ( τ cog ) can then be found by rearranging equation 5.9: τ cog = τ meas X y (5.11) Determination of y and τ cog over operating range The vector y is attributed to errors in the current sensor system, so regardless of speed and torque set-point it should remain constant. The cogging torque ( τ cog ) should also be independent of operating point. This method will only be valid if y and τ cog are the independent of speed and torque. One method to ensure that the same y is determined for all operating points is to combine all tests at different operating points into one long X. This matrix will have 6 columns and the number of rows will be Φ times the number of different operating points considered. Though this gives only one y for all trials, it does give a different τ cog for every trial. The validity of this method will be determined by the error between the determined residuals ( τ cog ). 54

83 5.2. DECOUPLING OF PULSATING TORQUE: DETERMINATION OF CURRENT IMBALANCE AND COGGING TORQUE Compensation for α, β and τ cog Once α, β and τ cog have been determined from an uncompensated set of measurements over the operating range, they can be used to pre-compensate I ref to cancel their effect for future operation. First I ref is split up into the current required to achieve the reference torque (I τref ) and the current required to compensate for τ cog (I cog): I ref = I τref I cog (5.12) where: I τref = Θ 3 matrix of the current required to achieve the reference torque(a) I cog = Θ 3 matrix of the current required to compensate for cogging torque (A) A PRCW method is used to ensure that: I τref K = τ ref (5.13) and: I cog K = τ cog (5.14) the reference current can be improved by compensating with the previously determined α and β I ref = I τ ref I cog 1 Θ 1 β 1 Θ 1 α (5.15) where: 1 Θ 1 = Θ 1 vector of ones 55

84 CHAPTER 5. PULSATING TORQUE DECOUPLING APPROACH TO MOTOR PARAMETER DETERMINATION If this modified reference current is used in equation 5.6: τ meas = I τ ref I cog 1 Θ 1 β T 1 Θ 1 α T K α + (K) β + τ cog (5.16) = ( I τref I cog 1 Θ 1 β T ) K + (K) β + τ cog (5.17) = ( I τref I cog ) K + τ cog (5.18) = I τref K I cog K + τ cog (5.19) But from equations 5.13 and 5.14, I τref K = τ ref and I cog K = τ cog so: τ meas = τ ref τ cog + τ cog (5.20) τ meas = τ ref (5.21) 5.3 Modifications if only dynamic torque measurement is available For reasons explained in section 6.6.2, a dynamic torque sensor was used for this research which did not measure the average component of the torque. diagram to represent this variation is shown in Figure 5.2. The block Figure 5.3: Block diagram of parameters to be determined (simplified) With measurement done in this way, equation 5.5 is modified to: 56

85 5.3. MODIFICATIONS IF ONLY DYNAMIC TORQUE MEASUREMENT IS AVAILABLE where: τ meas θ = p=a,b,c (i θ,p α p + β p ) k θ,p + τ cog θ τ average (5.22) τ average = average torque output Or in matrix form: where: τ meas = (I ref K) α + (K) β + τ cog τ average( 1 Θ 1 ) (5.23) 1 Θ 1 = Θ 1 vector of ones This equation presents τ average as an additional unknown. This can be determined however, by noting that it is the component of the torque that is independent of the effects of α, β and τ cog θ. As such it can be formulated as a gain multiplied by the ideal electromagnetic torque: τ average ( 1 Θ 1 ) = ɛ(i ref K) (5.24) where: ɛ = overall system gain independent of α, β and τ cog = 3 1 vector of ones Further analysis assumes that the overall gain of the system (ɛ) can be found. Its determination is the discussed in section 5.4 By substituting equation 5.24 into equation 5.23: τ meas = (I ref K) α + (K) β + τ cog ɛ(i ref K) (5.25) = (I ref K)( α ɛ( )) + (K) β + τ cog (5.26) Now analysis can continue as previously described, however now the vector y is: 57

86 CHAPTER 5. PULSATING TORQUE DECOUPLING APPROACH TO MOTOR PARAMETER DETERMINATION y = α ɛ( ) (5.27) β once y is determined using the least squares minimisation, a knowledge of ɛ will allow determination of alpha. 5.4 System gain determination The previous section demonstrated that if a dynamic torque sensor was used, compensation for the current imbalance and cogging torque is only possible if the overall gain of the system (ɛ) is known. An underlying assumption is that the transfer function can be approximated by a simple gain. This section will present a method for checking if the relationship can be approximated by a gain and for determining the magnitude of that gain Transfer function determination overview Transfer function determination is done by considering the response of the output to a known input. A frequency response function (H(ω)) is normally used [67](eq 7.10). where: H(ω) = B(ω) A(ω) (5.28) H(ω) = frequency response function A(ω) = frequency input function B(ω) = frequency output function Randall[67] suggests that if full control of the input is possible (as in this situation) a better estimate of the transfer function can be by multiplying numerator and denominator of H(ω) by the complex conjugate of A(ω) which gives: H 1 (ω) = G AB(ω) G AA (ω) (5.29) where: 58

87 5.4. SYSTEM GAIN DETERMINATION H 1 (ω) = alternate frequency response function G AB (ω) = cross spectrum G AA (ω) = input auto spectrum Input and Output To create the known input function requires the injection of harmonics over the frequency range. For the situation of interest, the ability of PCRW methods to cancel out unwanted harmonics can be extended to inject a range of electro-magnetic torque harmonics. By measuring the output harmonics in the measured torque signal the transfer function can be determined Transfer function Once the input and output signals are determined, Equation can be used to determine the transfer function. This can be done using the Matlab TM function tfestimate. This function uses Welch s averaged periodogram method [68]. If there are no mechanical resonances in the system, the response when plotted, should have the same magnitude over the frequency range and zero phase lag. It is possible that this analysis could be extended to situations where the transfer function is not a simple gain. As previously discussed, this could occur when mechanical resonances are present in the system or the measurement of the torque is indirect such as when using an observer. N diaye, Espanet and Miraoui [69] presented an experimental setup that could benefit from such analysis. They indirectly measured the pulsating torque using an accelerometer attached to the housing of the motor. In that situation, knowing how the accelerometer output related to motor torque would be beneficial Coherence To ensure that the estimated transfer function is accurate, the coherence was used. In his book Frequency Analysis [67], Randall states: the coherence gives a measure of the degree of linear dependence between the two signals, as a function of frequency. If there is poor linear dependence between the input and output signals then the 59

88 CHAPTER 5. PULSATING TORQUE DECOUPLING APPROACH TO MOTOR PARAMETER DETERMINATION determined transfer function is meaningless. 7.4). The coherence is calculated from the two autospectra and the cross spectrum [67](eq γ 2 (ω) = G AB (ω) 2 G AA (ω)g BB (ω) (5.30) where: G AB (ω) = cross spectrum G AA (ω) = input auto spectrum G BB (ω) = output auto spectrum A perfect linear relationship between input and output will give a coherence of one, suggesting that confidence can be placed in the determined transfer function. 5.5 Sensitivity analysis To give an insight into the accuracy required for the determination of α, β and τ cog, a sensitivity analysis was conducted. This was done using Matlab TM to calculate the expected pulsating torque as determined by Equation 5.5 when inaccurate inputs were used. This is based on work done by Poels [70]. To keep with the definition of α as the imbalance in the current gain, α a is increased while α b and α c are decreased. The factor plotted is the ratio: α a α b,c. Figure 5.4 is an overview of how much of each error is allowed to vary for the RMS pulsating torque to remain within 1%. More detailed results for various magnitudes of error are shown in Section C.1. Using figure 5.4 it can be estimated that α needs to be accurate within 5%, β within 0.2Amps and the magnitude of τ cog within 10%. 60

89 5.6. SUMMARY OF PTD APPROACH TO MOTOR PARAMETER DETERMINATION Figure 5.4: α, β, τ cog variation leading to 1% RMS τ meas 5.6 Summary of PTD approach to motor parameter determination This chapter has presented a new pulsating torque decoupling (PTD) approach to the determination of the motor parameters responsible for the creation of pulsating torque. It was shown that determination is possible from the pulsating torque itself by defining the cogging torque as the residual resulting from a least squares minimisation matching the electromagnetic torque to the measured torque. Some additional information is required about the system but it was demonstrated that this can be limited to knowledge of only the overall system gain. A theoretical sensitivity analysis was presented to provide a measure of the accuracy required for the determination of α, β, and τ cog. This PTD approach still requires a carefully designed experimental setup to achieve 61

90 CHAPTER 5. PULSATING TORQUE DECOUPLING APPROACH TO MOTOR PARAMETER DETERMINATION the required accuracy. The following chapter discusses how such a setup was implemented for this research. 62

91 Chapter 6 Experimental setup Section 2.8 highlighted problems in past experimental implementations of PRCW methods. This chapter will explain the Charles Darwin University experimental setup and describe how it has overcome or minimised the effect of these problems. Following a discussion of the motor tested and a definition of the pulsating torque target, past PRCW implementation problems are reviewed. The operating range of the motor was defined by the mechanical design so this is presented first. With the operating range defined, the design of the current controller and data-logging are then presented. Each of these designs included both hardware and software. 6.1 Motor description The test motor used was an axial flux motor originally developed as a high efficiency, high torque, direct-drive push-bike motor with a rated power of 200W, a rated torque of 14Nm and a rated voltage of 24V. It was particularly appropriate for this research as it has a non-sinusoidal back EMF and a relatively high cogging torque. If such a motor could be controlled to have low pulsating torque, it would broaden the range of motors able to produce smooth torque and considerably reduce the constraints on motor designers. 63

92 CHAPTER 6. EXPERIMENTAL SETUP 6.2 Review of past implementation problems Section 2.8 discussed past experimental implementations and highlighted the following problems: 1. the determination of motor parameters to a suitable accuracy; 2. the availability and location of a suitable torque sensor; 3. the provision of a smooth load to the motor; 4. the presence of mechanical resonances in the experimental setup. Determination of motor parameters was discussed in Chapters 3 and 5, the other three items had to be addressed by: mechanical design, design of the current controller and design of the data acquisition system. 6.3 Mechanical design Design decisions Several design decisions were made to overcome past implementation problems. 1. the motor and brake would be mounted on the same shaft and bearings; 2. a reaction torque sensor would be used; 3. all measurements would be taken at a motor velocity range well below any system natural frequencies; 4. the load would be applied using an eddy current brake. Use of the same bearings Wu and Chapman [22] discussed that the flexible coupling in their setup caused system dynamics that were difficult to model and compensate for. The best way to avoid a flexible coupling was to mount the motor and load on the same shaft. 64

93 6.3. MECHANICAL DESIGN Use of a reaction torque sensor Mounting the motor and load on the same shaft required the use of a reaction torque sensor rather than an in-line sensor. As discussed in section 3.2.3, using a reaction torque sensor also simplified the dynamic measurement of cogging torque. The company Transducer Techniques [71], also suggests that the use of a reaction torque sensor is preferred because it avoids the complexity involved in acquiring signals from a rotating shaft. Avoidance of natural frequencies By ensuring that measurements were not affected by mechanical resonances, a linear relationship could be assumed between motor torque and measured torque. As this was desirable, it was decided that the range of motor velocities would be limited by the first natural frequency of the test setup. For this reason, one of the mechanical design goals was to make the first natural frequency as high as possible. The determination of natural frequencies is known as modal analysis. Choice of an eddy current brake As discussed in 2.9.2, for load application, previous PRCW method researchers have either used: DC motors, hysteresis brakes, eddy current brakes, hydraulic motors or friction brakes. As the primary goal of this research was pulsating torque minimisation, those methods with any potential to have their own torque variation (DC motors, hysteresis brakes and friction brakes) were excluded. Hydraulic motors were too complex and expensive. Eddy current brakes were the only option remaining. The disadvantage of eddy current brakes is their inability to provide zero speed or high frequency loads. As this research was to focus on steady state operation, the high frequency issue was not a concern. It was decided if zero speed trials were required, a mechanical means of locking the rotor could be implemented. The proportionality of torque to speed was seen as an advantage in that it would allow self speed regulation, something that Wu and Chapman [22] had difficulty with when using their hysteresis brake. The use of an eddy current brake would require that an additional motor was able 65

94 CHAPTER 6. EXPERIMENTAL SETUP to be connected to measure the back EMF. If an eddy current brake was to become a reality, a design had to be found that would allow a reasonable range of torque to be applied over the speed range specified by the first natural frequency. The brake also had to be adjustable to allow testing at a range of speeds and torques Design order These decisions placed restrictions on the system that would define the operating range of the motor. The ability to achieve a high first natural frequency would define the range of motor velocities which would in turn, define the torque available from the eddy-current brake. As such the design needed to be done in a specific order. Rotor axle Design modification of the rotor and stator was beyond the scope of this research and as such, design started by considering these components. Mounting the motor and brake on one set of bearings required the rotor to be on the opposite side of the stator to the bearings (see Figure 6.1). The dimensions of the stator defined the length and diameter of the shaft supporting the rotor. This shaft was the starting point for modal analysis (detailed analysis B.2). Even with the thickest, shortest axle possible, simulations showed the first natural frequency was at about 700Hz. Without modification of the rotor or stator, this defined the first natural frequency in the system. 66

95 6.3. MECHANICAL DESIGN Figure 6.1: Stylised motor assembly cross section - full detail see A.2.2 Upper frequency limit (bandwidth) With the first natural frequency defined, the upper frequency limit could be determined to ensure measurements were unaffected by resonance. In their book Vibration Testing: Theory and Practice [72](p166), McConnell and Varoto state that for a mass, spring, damper system such as this test setup, the assumption of a linear sensor gain is only valid at frequencies well below the natural frequency. They suggest if that by only measuring frequencies at least five times lower than the natural frequency of the system, the linearity error is less than 5%. For this reason an upper frequency limit of 140Hz was chosen. Upper harmonic limit To find the upper angular velocity limit from the upper frequency limit required the harmonic limit. Hung and Ding [16] suggested that for the FDM, the upper limit of the torque harmonics is the sum of the upper limits of the back EMF and current harmonics. The harmonic limit of the current is usually set as the same as the harmonic limit of the back EMF. Section 4.5 showed that for the TDM and PLM, if the back EMF has the same or less harmonics than a trapezoid, the current limit can be set as 67

96 CHAPTER 6. EXPERIMENTAL SETUP the same as that for the FDM. It follows that the harmonic limit of the torque is twice that of the back EMF. Back EMF measurements on the test motor (see section 7.4.2) showed that the significant back EMF harmonics were up to the 72nd (9th harmonic in electrical degrees). This indicated the 144th torque harmonic as the upper harmonic limit. Upper angular velocity limit Frequencies produced by the motor are the product of the motor velocity and the motor harmonics. Therefore, the upper angular velocity limit is the upper frequency limit divided by the harmonic limit. upper frequency limit upper angular velocity limit = upper harmonic limit = (6.1) (6.2) = 0.97Hz (6.3) So, to avoid mechanical resonance problems, the maximum angular velocity should be around 1Hz. It was decided that measurements should be taken over a range of angular velocities. The minimum velocity of this range would limit the ability of the eddy-current brake to provide a range of reference torques. In order to provide a reasonable velocity range, it was decided that the minimum of the velocity range must be at least as low as 0.5Hz. Eddy-current disk With the angular velocity range and first natural frequency defined, the eddy current disk and its axle could be designed. Detailed eddy current design is presented in section B.1. Because of uncertainties in the design process, the expected load torque was a range. The highest achievable torque to provide braking over the defined motor velocity range was between 3Nm and 9Nm. Though this would only use a portion of the rated torque of the motor, the benefits of the completely smooth torque output of an eddy current brake justified this limitation. 68

97 6.4. CURRENT CONTROL DESIGN Mechanical design of housing, bearings and base With the design of the axle and eddy-current brake finalised, the housing, bearings and base could be designed. Further modal analysis ensured that there were no natural frequencies below the previously defined 700Hz. Tapered roller bearings were chosen for maximum stiffness and contactless seals were used to minimise friction. A final assembly drawing is provided in Section A.2.2 and a full set of part drawings in Section A.2.3. Encoder Bearing Housing Eddy Current Brake Stator Rotor Piezoelectric sensors Figure 6.2: CDU Experimental Setup (Magnet portion of the eddy current brake has been removed for clarity) 6.4 Current control design As PRCW methods rely on accurate following of the reference current, accurate current control is critical. Current controller design was influenced by the decision to use a full bridge topology and included: sensor selection, hardware design and software design. 69

98 CHAPTER 6. EXPERIMENTAL SETUP Design decisions Full bridge motor topology To ensure the greatest flexibility of reference current waveforms, it was desired that the star connection constraint be able to be imposed or removed. For this reason a full bridge drive topology [17](Figure 8-5) was chosen. This has been previously implemented by Aghili, Buehler and Hollerbach [36], N diaye, Espanet and Miraoui [69] and Mieno and Shinohara [73]. Independent current control loops allow inclusion of the star constraint in software by ensuring the reference currents always sum to zero. Figure 6.3 shows a Simulink TM block diagram of this topology and figure 6.4 shows the star connected topology. S1 S2 S5 S6 S9 S10 Supply voltage S3 S4 S7 S8 S11 S12 Figure 6.3: Full bridge drive topology S1 S2 S3 Supply voltage S4 S5 S6 Figure 6.4: Star connected drive topology 70

99 6.4. CURRENT CONTROL DESIGN Current sensor selection PRCW methods require a position sensor, to determine the reference current and a current sensor to ensure the actual current matches the reference current. Current sensor The current sensor measurement range is defined by the rated torque of the motor and the magnitude of the back EMF. Using any of the PRCW methods for the maximum possible torque of 9N m required a current measurement range of about 50A. As the accuracy desired of the system is a few percent, the current sensor over the measurement range should considerably lower than this. Most current sensors either measure the voltage across a shunt or use a hall effect device. The hall effect devices such as the range manufactured by LEM TM are usually the most convenient as the signal is already isolated and amplified to a range compatible with the input of an analog to digital converter (ADC). In his 2004 masters thesis [74], Camilleri states the LEM current sensors work as an excellent traction drive current sensor. He goes on to note that: The main disadvantage of the LEM system is cost. As cost would only be an issue in mass production, for this experimental setup, a hall effect device from LEM TM was chosen. Both open-loop and closed-loop sensors are available. Closed-loop sensors are more accurate. The sensor chosen was a closed-loop type (LEM LTS-25 NP) which had an accuracy over the measuring range of 0.7%. One potential issue with this sensor was the susceptibility of the analog voltage output to noise. To minimise this problem, the current sensors were mounted on a separate board with a separate power supply mounted away from the current inverter. The design of the sensor board is shown in section A.3.4. Position sensor PRCW methods require a much higher resolution encoder than traditional current control methods as they are effectively seeking to control much higher harmonics. The resolution of the sensor was defined from the need to control up the the 144th harmonic. For adequate control of a frequency range, the sampling frequency should be 71

100 CHAPTER 6. EXPERIMENTAL SETUP 5 to 10 times higher than the closed loop bandwidth [75](p357). Control of harmonics is somewhat analogous to control of frequencies. Instead of a requirement for the sampling frequency to be 10 times faster than the controlled frequency, the position information needs to be available 10 times faster. For the test motor, this specified that a sensor of at least 1440 pulses per revolution was required. The decision to mount the entire test setup on one set of bearings required the use of a hollow shaft encoder. To keep the system stiff enough, an encoder for a 30mm diameter axle was required. Modal analysis showed that a shaft of that diameter would keep the first natural frequency above 700Hz. The sensor chosen was a 12-bit (4096 states), gray code, absolute encoder (PCA ANHM-30RR-MAA1/4096). Using an absolute encoder had the added benefit of avoiding the need for an initialisation routine and the use of gray code ensured a more robust digital signal. 6.5 Hardware selection and design In addition to sensors, a digital signal processor (DSP) was required to implement the control algorithms and a current inverter was required to create the desired current from the supply voltage. DSP selection The most accurate current control would come from having the fastest possible control loop. This meant that the DSP needed to be as fast as possible. The decision to use a full bridge drive topology required three independent current control loops running simultaneously. The chosen DSP needed to have at least six pairs of pulse width modulation (PWM) outputs and three ADC inputs. A Texas Instruments TM DSP was chosen (TMS320F2812) mounted on a Spectrum Digital TM ezdspf2812 board. This decision was made as CDU staff had prior experience with other TMS products. It was also an attractive option as Mathworks TM had recently released a Simulink TM toolbox for this chip. This allowed the programming of the DSP to be done in the block diagram language of Simulink TM. 72

101 6.5. HARDWARE SELECTION AND DESIGN Current inverter Consideration of figures 6.3 and 6.4 shows that the use of the full bridge drive topology required 12 switches instead of the six required for a star connected topology. This requirement for twelve switches, excluded the use of a commercially available module as off the shelf modules are only designed with six switching components. The other requirements of the current inverter were that it had to be designed for the rated bus voltage (24V ) and the phase currents (50A). The switching components chosen were MOSFETs from International Rectifier TM (IRF2907). This decision was based on their current and voltage ratings, their low drain-source on resistance (R DS(on) ), their low cost and availability. These were driven by IR2110 gate drivers from International Rectifier TM. A photo of the current inverter is shown in appendix A Current control software The main challenge for the control software was to run the current control loop as fast as possible while still achieving suitable accuracy. The final controller is shown in Figure PWMbias W1 W2 W3 C28x PWM C28x ADC I_ref Current reference generator error 23 Kp 327 Ki Vy = Vu * 2^ 5 Qy = Qu >> 5 Ey = Eu P_shift Vy = Vu * 2^ 12 Qy = Qu >> 12 Ey = Eu I_shift W1 W2 W3 PWM_hi C28x PWM PWM_lo Ifb Memory Figure 6.5: Simulink TM model of current controller 73

102 CHAPTER 6. EXPERIMENTAL SETUP Current feedback The current feedback is the raw data from the ADC. As the DSP has a 12-bit ADC, the feedback current is represented as a number from Current reference generator The current reference generator uses position data from the encoder as an input to a lookup table for each phase. To speed up real time calculations and allow direct comparison with the feedback values, the lookup tables were converted to ADC values (1-4096) off-line. Proportional - integral (P I) controller The pink blocks in Figure 6.5 represent the implementation of the P I controller. This was chosen as an appropriate controller on the assumption that the motor was a firstorder system. Derivative (D) control is normally added if additional bandwidth is desired, however it can lead to instability if noise is present in the system [75](p161). As it was anticipated that the bandwidth of the controller would be adequate, a P I controller was implemented instead of a P ID controller. When implementing an integral controller, the error needs to be multiplied by the sampling time. In this implementation, to minimise calculation time, this was done by bit shifting. The actual multiplication done by the bit shift will not be exactly the same as multiplication by the sample time. This was compensated for by adjusting the value of the integral gain (K i ). Additional bit shifting was also used in both the proportional and integral sections to avoid quantisation issues resulting from fixed point arithmetic. PWM outputs The output of the controller was sent to the standard PWM blocks provided by the Simulink TM link library to the DSP. 74

103 6.6. DATA ACQUISITION 6.6 Data acquisition A data acquisition system was required to determine the effectiveness of the proposed methods. It was required to be able to measure the currents and motor torque and log them in a suitable form Design decisions Data acquisition was based on a Labview TM platform due to software availability. To minimise data storage size, a position based logging system was chosen. This meant that the torque and the currents were logged each time there was an change in encoder position. For every revolution, there would be 4096 measurements of each variable. This greatly simplified later order analysis Data acquisition hardware Torque sensor In their paper Measurement of torque ripple in PM brushless motors[35], Sun et al. suggested that a torque sensor with a range large enough to measure the average torque would have inadequate resolution to measure the pulsating torque. The best way to achieve adequate resolution for pulsating torque measurement was to use a sensor that only measures the dynamic torque. That way none of the range is used up measuring the zero frequency (DC) portion. The chosen setup needed a dynamic reaction torque sensor capable of withstanding the maximum torque that could be applied by the eddy current brake (9Nm) and of measuring the range of the pulsating torque (expected to be ±2N m). The sensor chosen needed a bandwidth capable of measuring the pulsating torque and of analysing the resonant frequencies in the system. Although the frequency range of measurement to avoid resonant frequencies was 140Hz, if possible, the bandwidth of the sensor should be substantially higher to allow analysis of the resonant frequencies and allow the potential for future testing of the motor at higher angular velocities. As described in section 3.2.3, to allow the simplification of the dynamic measurement of cogging torque, the reaction torque sensor must also be suitably stiff. 75

104 CHAPTER 6. EXPERIMENTAL SETUP The chosen configuration of a reaction torque sensor required the housing to be supported by four force sensors (see Figure 6.2). To ensure adequate sensor resolution, piezoelectric sensors (PCB 208C01) were chosen. These are dynamic sensors, so their range was chosen to suit only the expected magnitude of the pulsating torque. Their upper bandwidth is 36kHz and their stiffness is 1kN/µm. These sensors were interfaced to the data acquisition card by a line powered signal conditioner (PCB 482A16). Data acquisition card The data acquisition card needed at least 12 digital inputs for the encoder and 7 analog inputs (3 currents and 4 forces). To ensure compatibility with Labview TM, which is made by National Instruments TM, a National Instruments TM data acquisition card was chosen (NI PCI-6259). Interface PCB A dedicated PCB was required to route the signals from the torque sensor signal conditioner, the encoder and the current sensor board to the data acquisition card. The decision to log data when an encoder pulse changed was difficult to implement with the gray-code encoder. The least significant (highest frequency) bit in a gray-code encoder only changes every second state. If this were used, the sampling frequency would be half the encoder frequency. To overcome this, a series or exclusive or (XOR) gates were implemented in hardware to convert the gray-code to binary. A photo of the interface PCB is shown in appendix cha:appendixa Data acquisition software The data acquisition software implemented was implemented in Labview TM using a series of virtual instruments. Data was acquired in a batch which was then saved to a text file. Figure 6.6 shows a screen shot with current, torque and velocity logged. 76

105 6.7. EXPERIMENTAL SETUP SUMMARY Figure 6.6: Screen shot of Labview TM virtual instrument 6.7 Experimental setup summary This chapter discussed how the CDU experimental setup was realised, including component selection and design. This created a platform to overcome problems faced by previous researchers and was suitable for experimental comparison of PRCW methods (chapter 2) and methods of parameter estimation (chapters 3 and 5). The results of this testing are presented in the next chapter (chapter 7). 77

106 CHAPTER 6. EXPERIMENTAL SETUP 78

107 Chapter 7 Results Chapters 2 and 4 discussed PRCW methods and chapters 3 and 5 considered the parameter determination so critical for their successful implementation. This chapter will present an experimental comparison of PRCW methods and methods of parameter determination. To begin with, the assumptions made when developing the motor model discussed in section are validated, and the performance of the current controller is checked. Motor parameters are estimated using analytical methods outlined in chapter 3 and then determined using experimental methods from that chapter. Results are then determined using the PTD method proposed in chapter 5 and an experimental comparison is shown between each of the published PRCW methods. Finally a sensitivity analysis is provided for the effect of motor parameter variation on pulsating torque. Though all analysis relating to these results has been done in mechanical degrees, presentation is in electrical degrees (or electrical harmonics) for clarity. 7.1 Validity of assumptions Section presented the three assumptions made when using PRCW methods: 1. Back EMF is proportional to angular velocity; 2. Torque produced is proportional to phase currents, and 79

108 CHAPTER 7. RESULTS 3. Infinite DC bus voltage Back EMF proportional to angular velocity The back EMF was measured using an DC motor coupled to the test motor by two v-belt pulleys and a large o-ring. This was the simplest way to transfer torque in a smooth way with an appropriate gear ratio to allow the drive motor to operate at a higher speed than the low speed of the test motor. This setup is shown in Figure 7.1 Figure 7.1: External drive for measurement of back EMF Figure 7.2 shows the raw back EMF taken at different angular velocities. Close inspection suggests that the shape of the waveform varies. This however is due to angular velocity variations resulting from the cogging torque. Figure 7.3 shows the average normalised torque. Figure 7.4 shows that following normalisation, the error between the back EMFs from different angular velocities was less than 0.2%, validating the assumption that back EMF is proportional to angular velocity is valid. 80

109 7.1. VALIDITY OF ASSUMPTIONS Raw back EMF data (phase A) back EMF (V) Hz 0.6Hz 0.7Hz 0.8Hz 0.9Hz 1.0Hz encoder position Figure 7.2: Raw back EMF over the angular velocity range Mean normalised back EMF 0.15 normalised back EMF (V.s/rad) phase A phase B phase C encoder position Figure 7.3: Mean normalised back EMF 81

110 CHAPTER 7. RESULTS % error % error over speed range Phase A Hz 0.6Hz 0.7Hz 0.8Hz 0.9Hz 1.0Hz % error % error over speed range Phase B Hz 0.6Hz 0.7Hz 0.8Hz 0.9Hz 1.0Hz % error % error over speed range Phase C encoder position 0.5Hz 0.6Hz 0.7Hz 0.8Hz 0.9Hz 1.0Hz Figure 7.4: Percentage error of normalised back EMF over the angular velocity range 82

111 7.1. VALIDITY OF ASSUMPTIONS size of output harmonics y = 0.82*x data 1 linear size of input harmonics x residual size of input harmonics Figure 7.5: Proportionality of torque to current Torque proportional to current Aghili, Buehler and Hollerbach [36] present a method for checking the proportionality of current to torque at three torque harmonics. For this research, the proportionality was checked over the entire range by injecting all harmonics as described in Section Figure 7.5 shows that when the current magnitude is increased five times the torque output remains proportional Infinite DC bus voltage The assumption that the DC bus voltage is infinite is valid as long as the PWM signal from the current control loop is not saturated. This situation will usually arise when the motor is running at high angular velocities and the differential between the bus voltage and the back EMF is small. As all of the testing has been done as low angular velocities, this was not an issue. To ensure that the PWM signal was not saturated, the signal was checked during each measurement. The pulse width did not exceed 50% for any trial, so the system was not close to saturating. Hence it was valid to make 83

112 CHAPTER 7. RESULTS 40 PLANT: open loop curve fit From: u1 To: y1 30 Magnitude (db) Phase (deg) experimental approx (no delay) approx (1T delay) approx (2T delay) approx (3T delay) Frequency (rad/sec) Figure 7.6: Bode plot of Plant the infinte DC bus voltage assumption for motor modelling. 7.2 Current controller Accurate current control is critical for PRCW methods. To do this required an accurate model of the system System characterisation Section explained the choice of a P I controller. This choice was based on the assumption that the system was first-order. To determine the transfer function of the windings, voltage waveforms were induced into the system at different frequencies and the resulting current logged. Figure 7.6 shows the results of this testing along with a fitted first-order system. Fitting was done using the Matlab TM tfestimate function which is based on Welch s averaged periodogram method [68]. Analysis of the phase indicated that there was a delay in the system. Fitting suggested that this was 3 sample times. 84

113 7.2. CURRENT CONTROLLER 45 Simulated plant step resonse current (A) time (sec) x 10 3 Figure 7.7: Simulated step response of system PI controller determination Controller design was done based on the system model determined. For a first order system with a delay, a standard way of designing a P I controller is to use the Zeigler- Nichols reaction curve method [75](p167). This is based on the gain of the system and the ratio of the delay to the 63% rise time. The delay was already determined. The 63% rise time was found by simulating a step response to the determined transfer function in Matlab TM. This is shown in Figure 7.7. K o = 46.3 (7.1) τ o = (7.2) ν o = (7.3) where: K o = system gain τ o = delay time ν o = 63% rise time 85

114 CHAPTER 7. RESULTS For a P I controller [75](p168): K p = 0.9ν o K o τ o (7.4) = 0.72 (7.5) and T i = 3τ o (7.6) = (7.7) where: K p = controller gain T i = integral time constant Implementation into the fixed-point DSP controller, as presented in Figure 6.5, required some scaling: K DSP p = K p 2 5 (7.8) = 23 (7.9) and K DSP i = K p T i 2 12 t s (7.10) = 327 (7.11) Verification To demonstrate the accuracy of the controller, the reference and feedback currents were compared. Figure 7.8 shows that the feedback current closely follows the reference current. 86

115 7.2. CURRENT CONTROLLER syncronisation check phase A I ref I fb 10 current (A) encoder position Figure 7.8: Current error 87

116 CHAPTER 7. RESULTS 7.3 Eddy current brake testing Section explained that due to uncertainties in material properties and magnetic flux density, the strength of the eddy current brake at the slowest measurement speed (0.5Hz) could only be determined to be between 3Nm and 9Nm. Once the current controller was functioning, the brake could be tested. It was found that the maximum torque that could be applied at 0.5Hz was 5Nm. Though this meant that only about a third of the rated torque (14Nm) of the motor would be used, measurements could still be taken over a reasonable range (1 5Nm). 7.4 Parameter determination As mentioned in Chapter 3 the critical motor parameters for PRCW methods are: back EMF, cogging torque and current flow. To ensure that these are determined correctly also required a knowledge of the gain and offset of the current inverter and the gain of the torque sensor. This section will discuss parameter determination by datasheet values and traditional methods. The next section will discuss parameter determination using the PTD method presented in Chapter Scaling factors The scaling factors were initially determined using datasheet values. Power supply gain The current reference is converted to ADC values off-line to maximise controller speed. The power supply gain therefore, relates current in ADC values to current in amps. It is more intuitive to initially analyse the gain starting at the current sensor and then take the inverse. The current sensor (LEM LTS25NP) converts the phase current to a voltage: DSP: V olts Amp = (7.12) A voltage divider is then used to ensure that the signal is suitable for the 3.3V 88

117 7.4. PARAMETER DETERMINATION This signal is then fed into the ADC: V olts V olt = 2 3 (7.13) Therefore the total gain is: ADC counts V olt = (7.14) V olts Amp V olts ADC counts V olt V olt ADC counts Amp = The gain from ADC counts to current in amps is the inverse: (7.15) = 22.8 (7.16) Amps = (7.17) ADC count = w p (7.18) where: w p = estimated current inverter gain for phase p Power supply offset Figure 5.1 showed that in addition to a power supply gain, there was also an offset. In this experimental setup this was due to 0 Amps coming out of the current sensor as 2.5 V olts. From the previous section it can be determined that 2.5 V olts is equivalent to 2276 ADC counts, so: o p = 2276 ADC counts (7.19) where: o p = estimated current inverter offset for phase p 89

118 CHAPTER 7. RESULTS Torque sensor gain From the torque sensor calibration certificate, the output of the piezoelectric sensor (PCB 208C01) is: V olts N = (7.20) This is acting on a moment arm on the housing base of 120mm so: V olts Nm = 0.109/0.12 (7.21) = (7.22) So by taking the inverse: Nm volt = 1.10 (7.23) = u p (7.24) where: u p = estimated torque transducer gain for phase p Back EMF Section showed that back EMF of the test motor is proportional to angular velocity. For PRCW methods, in particular the FDM, the harmonic limit of the back EMF is also important. Back EMF harmonic analysis Figure 7.9 shows that the dominant harmonics in the back EMF of the test motor are the odd harmonics. Based on Section 3.1.1, that result was expected for a back EMF between a sine wave and a trapezoid. The magnitude of the harmonics decreases rapidly. The FDM requires a harmonic limit to be chosen. To determine this harmonic limit, consideration was given to the 90

119 7.4. PARAMETER DETERMINATION Harmonic content of back EMF phase A phase B phase C relative magnitude electrical harmonics Figure 7.9: Harmonic content of back EMF Error from truncation of back EMF phase A phase B phase C % error harmonic limit Figure 7.10: Error from truncation of back EMF 91

120 CHAPTER 7. RESULTS 0.6 Measured Cogging Torque 0.4 Torque sensor output voltage (V) Encoder position φ Figure 7.11: Dynamically measured cogging torque percentage error induced by truncating the back emf at different harmonics. These results are shown in figure After the inclusion of the 9th harmonic, the error remaining was about 0.2% which is smaller than the accuracy of the current sensor (0.7%). For this reason, control was attempted up to the 9th electrical hamonic (72nd mechanical harmonic) Cogging torque Section 3.2 discussed that cogging torque can be measured either statically or dynamically. The decision to use a reaction torque sensor, allowed dynamic measurements without the need to compensate for inertial forces. piezoelectric sensors are suitably stiff (see section 6.6.2). This assumption holds as the Measurements were taken at the same time the back EMF was measured, while the test motor was being rotated by another motor. Figure 7.11 shows the average resulting cogging torque in the time domain. Section explained that the dominant cogging torque harmonic would be defined by the number of stator slots divided by the number of pole pairs. For the test motor this will be: 48 8 or the 6th electrical harmonic (48th mechanical harmonic). 92

121 7.5. PARAMETER DETERMINATION - PTD METHOD 1 Measured cogging torque harmoincs relative magnitude electrical harmonics Figure 7.12: Measured cogging torque harmonics Figure 7.12 shows the cogging torque harmonics. As expected, the dominant harmonics are multiples of the 6th electrical harmonic. There are also smaller harmonics at multiples of the 2nd. The error associated with the cogging torque determination at different speeds was 2.5%. This is shown in Figure Parameter determination - PTD method Scaling factors Overall system transfer function As only the dynamic torque was measured, the magnitude of the overall system transfer function needed to be determined. This determination follows analysis done by Grassens[76]. Section 5.4 presented a method for calculating the overall system transfer function in the frequency domain. Initially datasheet values were used to determine the gain. Harmonics were induced into the electro-magnetic torque and the response of the measured torque recorded. The result of these measurements would determine any 93

122 CHAPTER 7. RESULTS Percentage error of the averaged Cogging Torque 0.5 Hz 0.6 Hz 0.7 Hz 0.8 Hz 0.9 Hz 1.0 Hz % Error Encoder position φ Figure 7.13: Dynamically measured cogging torque error 94

123 7.5. PARAMETER DETERMINATION - PTD METHOD 10 H = T p /T em Magnitude Phase Coherence Frequency Figure 7.14: System transfer function estimation error in the datasheet values. Figure 7.14 shows the results of this method over the operating range. It should be noted that these are mechanical harmonics. To demonstrate the linear dependence of the input and output signals, the coherence was determined. The bottom portion of Figure 7.14 shows that the coherence is very close to 1 over the harmonic range indicating a strong linear dependence. Though the system response is generally a straight line, there are some obvious peaks. These are caused by pulsating torque harmonics due to current imbalance or cogging torque (i.e. a lot more torque comes out of the system than was induced into the system). To overcome this issue, all harmonics that were multiples of 8 (electrical harmonics) were removed. The resulting response is shown in Figure The important point to note from Figure 7.15 is that the magnitude is relatively constant over the harmonic range and the phase lag is close to zero. This confirms that there are no resonant frequencies in the operating range and that the transfer function can be approximated by a gain for further analysis. 95

124 CHAPTER 7. RESULTS 1.5 H selected & Least Squares Magnitude Phase Coherence Frequency Figure 7.15: System transfer function estimation (no multiple of 8 harmonics) 96

125 7.5. PARAMETER DETERMINATION - PTD METHOD Overall system scaling Figure 7.15 showed that the magnitude of the overall system (ɛ) was about 0.8. Consideration of how the accuracy of this parameter estimation effects the pulsating torque is considered in figure This value could now be used to determine the current imbalance and cogging torque as in equation Current imbalance and cogging torque Once the overall system gain was determined, current imbalance and cogging torque could be found by decoupling the pulsating torque as described in Section 5.2. To do this required an initial series of measurements to be taken over the operating range ( Hz, 1 5Nm) without compensation. The pulsating torque from these measurements was decoupled to determine the cause. Figure 7.16 demonstrates how the torque was decoupled for one set-point. The top part of the figure is the measured torque. In the second part of the figure, the torque has been split up into the electromagnetic torque X y and the residual (designated here as z). This residual should be the cogging torque, however figure 7.12 demonstrated that the cogging torque only contained harmonics that were multiples of the 2nd electrical harmonic. For this reason, these harmonics were taken from z and assumed to be the cogging torque. This left a component (described in the figure as rest ) for which the source was unknown. This decopling process can also be shown as harmonics (figure 7.17). Section discussed that the vector y, which describes the current imbalance, could be forced to be the same over the entire operating range. τ cog however, would be determined independently for each set-point. A measure of the validity of the PTD method is the variation of the τ cog determined for each set-point. Figure 7.18 shows τ cog determined for all set-points. Figure 7.19 shows the percentage error of each of these determined waveforms away from the average. 97

126 CHAPTER 7. RESULTS Decoupling of pulsating torque (time domain) torque (Nm) torque (Nm) torque (Nm) encoder position Xy z Tc rest Figure 7.16: Decoupling the pulsating torque (time domain) - ( rest is the remaining pulsating torque for which the source is unknown) Decoupling of pulsating torque (frequency domain) torque (Nm) torque (Nm) torque (Nm) harmonic number Xy z Tc rest Figure 7.17: Decoupling the pulsating torque (frequency domain) - ( rest is the remaining pulsating torque for which the source is unknown) 98

127 7.5. PARAMETER DETERMINATION - PTD METHOD 0.6 Determined cogging torque for different set points (Nm) encoder position Figure 7.18: Determined cogging torque for different set-points Cogging torque error for different setpoints % error encoder position Figure 7.19: Cogging torque error for different set-points 99

128 CHAPTER 7. RESULTS Verification of determination of current imbalance To ensure that the decoupling of the pulsating torque was working effectively, a set of trails were taken with an induced current offset error and another set of trials with an induced scaling error. Table 7.1 shows the determination errors. The values calculated were generally a good estimate of the induced value, and the third column shows the error. In some cases, this error was reasonably high (24% for β c ), however as the fourth column shows, the resulting pulsating torque expected if the incorrect estimate was used was always less than 0.5%. Induced value Calculated value Error Resulting pulsating torque if estimate used (%RMS) α a = α a = % 0.54% α b = α b = % 0.27% α c = α c = % 0.54% β a = 1.0 β a = % 0.002% β b = 0.5 β b = % 0.028% β c = 0.5 β c = % 0.439% Table 7.1: Induced scaling and offset errors and the found compensating values 7.6 Comparison of PRCW methods Pulsating torque comparison Once the current errors and estimate of the cogging torque had been determined, it was possible to compare the different PRCW methods using both traditional and PTD compensation for these parameters. Figure 7.20 shows a summary of the results. The red circles on the left show the uncalibrated results and the green crosses on the right are the results when the PTD method was used to create calibrated results. Each of the data points in figure 7.20 is the mean of a series of experiments taken 100

129 7.6. COMPARISON OF PRCW METHODS over the entire torque and velocity range. Figures 7.21 and 7.22 show the original baseline and the best result as achieved by the compensated time domain method. All of the other detailed results are presented in appendix C. Also in appendix C is a harmonic analysis of each method showing which harmonics were removed during calibration. As discussed in section 2.8.1, to numerically compare results, the pulsating torque will be defined as the ratio of RMS pulsating torque to maximum torque. RMS of pulsating torque (Nm) Comparison of methods for pulsating torqe minimisation uncalibrated calibrated 1 0 SIN FDM FDM* TDM TDM* PLM SIN FDM FDM* TDM TDM* PLM Figure 7.20: Comparison of PRCW methods (* refers to the use of the star connection constraint) Baseline - sine wave As expected, the sine wave shaped current used as a baseline has the most pulsating torque ( 8 9%). The use of calibration did not have a significant effect. Uncalibrated PRCW methods For the uncalibrated PRCW methods, datasheet values were used for the system gains and independently determined cogging torque data was used. This approach, as proposed by previous researchers, reduced the pulsating torque to about ( 3 4%). 101

130 CHAPTER 7. RESULTS RMS of pulsating torque: SIN RMS of pulsating torque (Nm) torque setpoint (Nm) speed (Hz) Figure 7.21: Uncalibrated sin wave method There was minimal difference between each of the methods with the only obvious variation an increase in pulsating torque when the additional star connection constraint was applied. PTD calibration Each of the methods was then tested over the operating range using the current imbalance and cogging torque determined from the PTD parameter determination. As with the uncalibrated results, the difference between PRCW methods was minimal. The TDM did have slightly lower pulsating torque as suggested in simulations (section 4.6). Far more significant was the improvement possible for all methods by the use of PTD calibration. Using this method to compensate for current imbalance and cogging torque, the pulsating torque was reduced to about ( 1%) Current use comparison In addition to considering the pulsating torque, it is worth considering the current waveforms, which are shown in Figure It can be seen that each PRCW method produces a distinctly different waveform. 102

131 7.6. COMPARISON OF PRCW METHODS RMS of pulsating torque: TDM RMS of pulsating torque (Nm) torque setpoint (Nm) speed (Hz) Figure 7.22: TDM - without star connection constraint 103

132 CHAPTER 7. RESULTS current (A) Phase A current for different methods SIN FDM FDM* TDM TDM* PLM encoder position Figure 7.23: Comparison of PRCW current waveforms (* refers to the use of the star connection constraint) Figure 7.24 compares the current usage increase of each method to the current used for a sinusoidal waveform. The percentage increase is much larger at the smaller set points as the PRCW methods are using current to compensate for the cogging torque regardless of set point. Table 7.2 compares the increase in average RMS current used by each method over the entire operating range. SIN FDM TDM PLM % Current Increase (without star constraint) % Current Increase (with star constraint) Table 7.2: Average total RMS current usage increase over operating range 7.7 Sensitivity analysis Jahns and Soong[3] stated that: motor parameter sensitivity of these algorithms has received little attention in the literature to date. To ensure a thorough analysis, the estimations of α, β, τ cog and ɛ were varied. 104

133 7.7. SENSITIVITY ANALYSIS RMS current (A) Total RMS current increase for different methods SIN FDM FDM* TDM TDM* PLM torque setpoint (Nm) Figure 7.24: RMS current increase from PRCW methods (* refers to the use of the star connection constraint) 105

134 CHAPTER 7. RESULTS rms error due to change in α a Measurements Fit on measurement 1 4 Fit on measurement 6 9 Theoretical rms error 8 9 rms error α a Figure 7.25: Sensitivity of pulsating torque to a α error This analysis is based on work done by Poels[70]. The sensitivity of pulsating torque to α, β and τ cog was discussed in Section 5.5. The experimental results have been plotted against these results and show a close match. Two points are of particular interest; 1. For all the analysis, even when the parameters are accurately determined, the pulsating torque always remains at least 1%. 2. The sensitivity of the overall system gain is low. Variation of up to 20% in the estimate will only produce a small increase in puslating torque. 7.8 Summary of results This chapter validated the assumptions made when developing both the PRCW methods and the PTD method for parameter determination. The parameters determined using traditional methods and the parameters determined using the PTD method were presented. Finally, a comparison was made between different PRCW methods 106

135 7.8. SUMMARY OF RESULTS rms error due to change in β a rms error Measurements Fit on measurement 1 5 Fit on measurement 7 11 Theoretical rms error β a Figure 7.26: Sensitivity of pulsating torque to a β error rms error due to change in τ cog rms error Measurements Fit on measurement 1 8 Fit on measurement Theoretical rms error factor of τ cog Figure 7.27: Sensitivity of pulsating torque to a τ cog error 107

136 CHAPTER 7. RESULTS 1.4 rms error due to change in ε rms error ε Figure 7.28: Sensitivity of pulsating torque to a ɛ error 108

137 7.8. SUMMARY OF RESULTS for each method of parameter determination. The significance of these results will be discussed in Chapter

138 CHAPTER 7. RESULTS 110

139 Chapter 8 Discussion As stated in the introduction, the goal of this research was to compare published PRCW methods with a well designed experimental setup and determine if any of these methods could provide a smooth torque output. To achieve a smooth torque output using PRCW methods requires: a good experimental setup, an accurate estimate of motor parameters and the choice of the most appropriate methods. The discussion will consider each of these issues. A summary of this analysis is provided in Chapter 9 - Conclusions. 8.1 Experimental setup Several parts of the experimental setup proved to be critical, the most important being: 1. the ability of the current control loop to follow the reference current; 2. an easily modelled relationship between electro-magnetic torque and measured torque; and 3. the ability to apply a smooth torque with a regulated speed Current control loop For PRCW methods, the sole focus of the motor controller is current control. Information is used from the position and current sensors to ensure that the actual current 111

140 CHAPTER 8. DISCUSSION follows the reference current. Without accurate tracking, it is impossible for any of the proposed methods to work. The controller implemented for this research was a very high speed PI controller (loop speed 100kHz) which was tuned using the Zeigler-Nichols reaction curve tuning rules [75](p167). This tuning method was chosen after it was determined that the system was a first order system with a time delay. The accuracy of the current is clearly dependant on the accuracy of the current feedback Transfer function between electo-magnetic torque and measured torque The control system is judged by the measured torque but it is the electro-magnetic torque that is controlled. Without a thorough understanding of how electro-magnetic torque affects the measured torque, it will be impossible to minimise pulsating torque. For example, cogging torque should be compensated by creating an equal and opposite torque ripple in the electro-magnetic torque. Without understanding the relationship between the measured cogging torque and the induced electro-magnetic torque ripple it is impossible to make them equal and opposite. The ideal scenario is if this transfer function is a simple gain. Difficulties are created when mechanical resonances or inertia forces create a more complex transfer function that is not so easily modelled. In this research, potential problems were avoided by careful design of the test setup to avoid mechanical resonances and by the use of a reaction torque sensor to avoid the measurement of inertia forces. By injecting a spectrum of induced harmonics in the electromagnetic torque over the range of interest, it was demonstrated that the response magnitude of the measured torque remained constant and that the phase lag was minimal (see figure 7.15) Smooth load application The final critical element of a good setup for testing PRCW methods is a means of applying a smooth torque. Any pulsating torque from the load has a detrimental effect on the measurement of actual motor torque. In addition, as PRCW methods 112

141 8.2. PARAMETER ESTIMATION are torque controlled methods, the load also needs to be able to accurately regulate speed to ensure that the measurements take place in a steady state environment. For this research, an eddy current brake was chosen to ensure a completely smooth load and an inherently self-regulated speed. 8.2 Parameter estimation The biggest challenge for the implementation of PRCW methods is the accurate determination of motor parameters. Initial examination of the published methods suggests that only the back EMF and cogging torque are required. In addition, it is important to determine any current offset or scaling errors. Once the back EMF has been determined, the most accurate way of determining the cogging torque and current measurement errors is to use the PTD method developed in chapter 5. This is done by defining the cogging torque as the residual after a least squares approximation is done between the electro-magnetic torque and the measured torque at all torque set-points Effect of determination uncertainty Figure 7.19 suggested an error of 8% in the determintion of the cogging torque. The sensitivity analysis (figure 7.27) showed that this uncertaintly could lead to a pulsating torque of 1.5%. Table 7.1 showed the error in determination of the offset and scaling errors is only a few percent for α but was up to 24% for β (0.12A). Consideration of the sensitivity analysis however (figure 7.26), suggested that this uncertainty would lead to a pulsating torque of less than 1% Potential parameter estimation in mass production One of the stated advantages of PRCW methods was that they are able to reduce restrictions on the design and manufacturing processes. If the main goal is to reduce restrictions on the design process, for instance to allow the motor designer to focus on achieving maximum average torque, parameter estimation can be done once for that design. 113

142 CHAPTER 8. DISCUSSION Problems arise however if the goal was to reduce the restrictions on the manufacturing process. This would usually be done by relaxing tolerances on either physical dimensions or material properties which would lead to variations in motor parameters between individual motors. To overcome this problem, as part of the final testing of each motor, a brief test run would be required to allow determination of the unique motor parameters. 8.3 Comparison of PRCW methods Pulsating torque Theoretical comparison When compared theoretically, using a sinusoidal and a trapezoidal waveform with a star constraint imposed, all methods were capable of eliminating pulsating torque. The FDM required slightly more current to do this than the TDM and the PLM. If any imbalance between phases was present, the FDM could no longer completely eliminate pulsating torque. When the star constraint was removed, the current required by the FDM and the TDM reduced, however as the PLM has an inherent star constraint, the current it used was unchanged. When the ability of each method to compensate for cogging torque was checked, it was demonstrated that the FDM was only able to do this for cogging torque harmonics that were multiples of three. In addition, it was shown that as the cogging torque harmonics became higher, the current required for the FDM to compensate increased dramatically. Experimental comparison A summary of the experimental comparison was presented in figure The baseline for experimental comparison was using a sine wave current shape with current scaling and offset values taken from datasheets. The RMS pulsating torque for this method was 8-9%. Published PRCW methods were compared to this baseline using the same current 114

143 8.3. COMPARISON OF PRCW METHODS parameters and a cogging torque waveform determined using separate dynamic measurements. There was no significant difference between different methods and all were able to reduce the pulsating torque to 3-4% RMS. The difference between different PRCW methods may have been more pronounced if the back EMF or the cogging torque contravened the additional assumptions required for the FDM, which would have reduced its effectiveness. The final comparison was to use the PTD method developed in this thesis to calibrate the current scaling and offset and to estimate the cogging torque waveform. Using this calibration, all PRCW methods were capable of achieving around 1% RMS pulsating torque. As with the uncalibrated results there was still not a large difference between different PRCW methods, however the TDM was slightly better at minimising pulsating torque. These results suggest that if PRCW methods used with motor and controller parameters accurately determined from the PTD method, they are worthy of consideration for the control of motors requiring a smooth torque output as defined in section Remaining pulsating torque To determine the source of the remaining pulsating torque, it is worth considering the reconstructed torque which is found by multiplying the feedback current by the back EMF. Figure 8.1 shows the RMS reconstructed pulsating torque for the calibrated time domain method trial. Over the operating range, the RMS pulsating torque is about 1% which is the same size as the measured pulsating torque (presented in figure 7.22). This indicates that the remaining pulsating torque could be attributed to inaccuracies in the current controller. If this were the case, to reduce the pulsating torque further would require a more accurate current controller. 115

144 CHAPTER 8. DISCUSSION Error of the Current Sensors and Control Loop RMS error of I fb Torque [Nm] Speed [Hz] Figure 8.1: Pulsating torque due to current controller error Current usage Figure 7.24 and Table 7.2 demonstrate that the TDM always used the least amount of current. The inclusion of a star constraint slightly increased the required current for both the FDM and the TDM. If a star connection constraint is imposed, the current required from the TDM and PLM was very similar Preferred method In practice, the use of the TDM is recommended because of its superior results and ease of implementation. If implementation is to be done in a rotating reference frame (such as space vector modulation) then the use of the PLM may be beneficial. Though the FDM provides some useful insights into the production of pulsating torque there seems no practical situation where it would be the preferred method. 116

145 Chapter 9 Conclusion The goal of this research was to compare different programmed reference current waveform (PRCW) methods and determine if any would be suitable to control a permanent magnet alternating current (PMAC) motor in an application requiring a smooth torque output. Three commonly reference PRCW methods were considered: the frequency domain method (FDM) where the analysis is done in the frequency domain, the time domain method (TDM) where the analysis is done in the time domain and the park-like method (PLM) where the analysis is done in a rotating reference frame. To compare methods, the pulsating torque was defined as the percentage ratio of RMS pulsating torque to maximum torque. The theoretical analysis presented in Chapter 4 suggested that if the three phases were balanced and if the cogging torque contained only harmonics that were multiples of three then all three methods would give similar results. If these assumptions were violated, the TDM and the PLM remained effective, however the FDM method became less effective. It was also demonstrated was that the removal of the star constraint (sum of currents in all phases equal to zero) slightly reduced the amount of current required. To compare the results experimentally, an setup was designed, addressing issues highlighted by previous PRCW researchers. Resonant frequencies in the operating range were avoided, dynamic reaction torque sensors were used and an eddy current brake was used to ensure a smooth load was applied. 117

146 CHAPTER 9. CONCLUSION When the three methods were compared experimentally, the results supported the theoretical analysis. Initially, the PRCW methods were compared using datasheet values for the current scaling and offset and a cogging torque waveform determined off-line. Using these values, all PRCW methods were capable of reducing the cogging torque to 3 4% from the baseline of a pure sine wave which had a pulsating torque of about 8 9%. The difference in pulsating torque between methods was minimal with the TDM producing slightly better results. The TDM method also used slightly less current. To further reduce the pulsating torque, a more accurate estimation was required for both the current offset and scaling and the cogging toque waveform. This greater accuracy was achieved by using the pulsating torque decoupling (PTD) method developed in this thesis (chapter 5). In this method, parameter errors could be determined from the pulsating torque. This was done by redefining the cogging torque as the residual from a least squares minimisation between the electromagnetic torque and the measured torque. Using this PTD method, the pulsating torque for all methods could be reduced to about 1%. Once again, the difference between methods was minimal with teh TDM producing slightly better results. A sensitivity analysis, both theoretical and experimental showed that to remain below 1% pulsting torque, the current scaling needed to be accurate to within 2-3%, the current offsset to within 0.1A and the estimate of the cogging torque within 5% (see 7.7). Within these ranges, all measurements showed about a 1% pulsating torque, even when the parameter estimation was highly accruate. In an attempt to find the source of this error, the torque was reconstructed from the meausred feedback current and the back EMF. This showed an error over the measurement range of about 1%, suggesting that the source of the remaining pulsating torque may be the inability of the current controller to follow the reference torque. This research has demonstrated that if motor and controller parameters are accurately estimated, PRCW methods can be used to to control motors requiring a smooth torque output. To adequately determine these parameters, the PTD method 118

147 9.1. FURTHER WORK was developed and implemented. It is envisaged that in a high volume production environment, the PTD method could be used to determine motor and controller parameters. This could be done either once for a given motor design and then applied to all individual motors or if there was substantial manufacturing variation, the methods could be used as part of the final motor testing, from a brief test run. 9.1 Further work This work highlighted several areas worth of further work: 1. Current controller accuracy 2. Varying operating conditions 3. Adaptive control Current controller accuracy It was discussed in section that much of the remaining pulsating torque could be attributed to inaccurate following of the reference current. Though the current controller was designed and tested to a high accuracy, once the pulsating torque due to parameter estimation inaccuracies had been removed, the small amount of error in the current controller was the biggest remaining factor creating pulsating torque. As such, the development of a higher accuracy controller using more accurate current sensors is the next logical step to reducing pulsating torque further Operating conditions This study was conducted under carefully controlled conditions. It is possible that these conditions may not be so favourable if: 1. the motor was operated in a speed range with natural frequencies present; 2. dynamic torque set-points were used, or 3. motor parameters varied due to temperature or other effects. 119

148 CHAPTER 9. CONCLUSION Those situations would most likely have a negative impact on the ability of PRCW methods to minimise pulsating torque. The ability of the methods proposed by this research to deal with these variations should be analysed Adaptive control It was mentioned in the introduction that adaptive control is most beneficial if the operating conditions are varying. If this is the case, as discussed above, adaptive control or an alternative such as iterative learning control would be worth considering. Of particular interest is that this research has determined ways to find and compensate for errors in the estimation of motor parameters by measuring pulsating torque. If this pulsating torque could be determined on-line using an observer then the estimation of motor parameters could be updated while the motor is operating. 120

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159 Appendix A Hardware design A.1 Hardware overview Motor: Single sided, 16 pole, three-phase, axial flux motor (Patent pending, Fasco Pty Ltd) Rated voltage: 24V, Rated current: 30A, Rated torque: 8.5Nm Piezoelectric torque sensors: PCB Piezotronics - 208C01 Encoder: PCA - ANHM30HSMAA1/04096 Bearings: SKF 40x80 tapered roller bearings DSP: Texas Instruments TMS320F2812; programmed with Simulink TM C2800 embedded coder Data logging: National instruments data acquisition cards (PCI-6259, PCI-6602) interfaced with LabVIEW. 131

160 APPENDIX A. HARDWARE DESIGN A.2 A.2.1 Mechanical drawings Overview Bearing Housing Encoder Stator Eddy Current Brake Rotor Piezoelectric sensors Figure A.1: CDU motor test assembly A.2.2 Assembly drawing 132

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162 APPENDIX A. HARDWARE DESIGN A.2.3 Part drawings 1. Welding drawing of base 2. Machining drawing of base 3. Piezoelectric sensor spacer 4. Housing 5. Axle 6. Rotor disk 7. Eddy current disk 8. Seal 9. Nut spacer 134

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172 APPENDIX A. HARDWARE DESIGN A.3 Electrical Design A.3.1 Overview Figure A.2 shows an overview of the electrical components designed and built for this research. The following figures show a close up of each component. Figure A.2: Overview of electrical components 144

173 A.3. ELECTRICAL DESIGN A.3.2 Current inverter Figure A.3: Overview of current inverter A.3.3 Gate drive and MOSFET module Figure A.4: Closeup of gate drive and MOSFET module 145

174 APPENDIX A. HARDWARE DESIGN A.3.4 Current sensor board Figure A.5: Current sensor board with LEM LTS25NP A.3.5 Labview data acquisition board Figure A.6: Labview interface board 146

175 A.3. ELECTRICAL DESIGN A.3.6 DSP interface board Figure A.7: DSP interface board 147

176 148 APPENDIX A. HARDWARE DESIGN

177 Appendix B Additional Calculations B.1 Design and sizing of eddy current brake The eddy current brake was designed to provide a reasonable load at the minimum of the planned motor velocity range (0.5Hz). Eddy current brake design was done using the model presented by Schieber [77]. Due to space requirements, electromagnets were not possible. Instead, two rings of permanent magnets were designed with a mechanism to change the phase of these rings. The field strength would be maximum when the rings were aligned to opposite poles and minimum when they were aligned to like poles. The final assembled brake is shown in Figure B

178 APPENDIX B. ADDITIONAL CALCULATIONS Figure B.1: Assembled eddy current brake 150

179 B.1. DESIGN AND SIZING OF EDDY CURRENT BRAKE B.1.1 Schieber s model Schieber s model is for a rotating system and is valid for low speed only. It was chosen as the CDU eddy current brake would be low speed. Though it is for electro-magnets, it only uses the field strength so it should be valid for permanent magnets as well. Where: T = 1 2 σδωπr2 m 2 B 2 z ( (r/a) 2 ) 1 (1 (m/a) 2 ) 2 (B.1) σ = electrical conductivity of the rotating disk δ = disk thickness ω = angular velocity of the rotating disk π = constant coefficient r = radius of the magnet m = distance of disk axis to pole face centre a = disk radius B z = z component of magnetic flux density, z-axis is the direction of the centre of the electromagnetic pole. Electrical conductivity The material chosen for the disk was 2024 grade aluminium due to its good machinability and availability. Data [78] for the electrical conductivity of 2024 grade aluminium varied between 1.734e 7 Siemens/m and 2.764e 7 Siemens/m. Sheet thickness and radius The disk thickness and radius was chosen by carefully balancing the torque requirement of the disk with the modal analysis. A width of 5mm was chosen. The disk drawing in Section A.2.3 shows that the disk was tapered for maximum stiffness where it was not running through the magnets. 151

180 APPENDIX B. ADDITIONAL CALCULATIONS Electo-magnet radius To maximise magnet coverage of the disk, rectangular magnets were chosen. This required a small modification to be made to the model as it was for circular magnets. This issue was resolved by determining the equivalent magnet radius to provide the same magnet area as the rectangular magnets chosen. lmagnet w magnet r equivalent = π = π = 0.022m (B.2) Distance of disk axis to pole face centre This was defined by the magnet size and disk radius. Magnetic flux density The strongest magnets available with an appropriate geometry were N48 designation. The magnet supplier advised that for the proposed air gap we could expect and air gap flux density between 0.8T and 1.1T. Summary of variable values σ = 1.734e 7 Siemens/m e 7 Siemens/m. δ = 0.005m ω = πrad/sec π = π r = 0.022m m = 0.075m a = 0.1m B z = 0.8T - 1.1T 152

181 B.1. DESIGN AND SIZING OF EDDY CURRENT BRAKE Use of multiple magnets This model is for only one set of magnets. The assumption was made that by adding multiple poles the torque applied would also be multiplied by the same factor. The maximum number of magnets able to fit in the space available was ten. B.1.2 Results The uncertainly involved with the electrical conductivity and the flux density meant that the expected torque could only be provided as a range. Figure B.2 shows that for the range of flux densities and electrical conductivities expected, the applied torque ranged from 3Nm to 9Nm. 2.8 x Torque possible from eddy current brake electrical conductivity (Siemens/m) flux density (T) 5 6 Figure B.2: Expected torque from eddy-current brake 153

182 APPENDIX B. ADDITIONAL CALCULATIONS B.2 Modal analysis of experimental setup The CDU motor and load setup has been analyzed and redesigned using a finite element analysis package 1 to ensure that all resonant frequencies are higher than the expected range of fluctuating torque harmonics. Design involved a careful balance of weight and stiffness to ensure that each part had no natural frequencies below the target of 700Hz. Figure B.3 shows a representation of the first natural frequency for the entire assembly. Figure B.3: Results of modal analysis 1 ProEngineer - Mechanica TM 154