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Two-Phase Auto-Piloted Synchronous Motors and Actuators

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IMPERIAL COLLEGE OF SCIENCE AND TECHNOLOGY DEPARTMENT OF ELECTRICAL ENGINEERING TWO-PHASE AUTO-PILOTED SYNCHRONOUS MOTORS AND ACTUATORS it Thesis submitted for the degree of Doctor of Philosophy of the University of London. it by Amadeu Leao Santos Rodrigues, M.Sc. (D.I.C.) July 1983 The thesis is concerned with certain aspects of the design and per- formance of drives based on variable speed synchronous motors fed from variable frequency sources, controlled by means of rotor-position sensors. This auto-piloted or self-synchronised form of control enables the machine to operate either with an adjustable load angle or an adjustable torque angle depending on whether a voltage or a current source feed is used. D.c. machine-type characteristics can thus be obtained from the system. The thesis commences with an outline of some fundamental design aspects and a summary of torque production mechanisms in electrical machines. The influence of configuration and physical size on machine performance is discussed and the advantages of the use for servo applic- ations of direct-drive as opposed to step-down transmissions are explained. Developments in permanent magnet materials have opened the way to permanent magnet motors of improved performance, and a brief review of the properties of the various materials presently available is given. A finite-difference method using magnetic scalar potential for calculating the magnetic field of permanent magnets in the presence of iron cores and macroscopic currents is described. A comparison with the load line method is made for some typical cases. Analogies between the mechanical commutator of a d.c. machine and the electronic commutator of the system studied are drawn and a theory valid for damper-less motors operating under both steady and dynamic conditions is presented. The design, manufacture and testing of a 1000 lb.ft, limited motion, rare-earth permanent-magnet-excited rotary actuator, begun in an earlier project, is described. Computer solutions for the numerical calculation of air gap flux density distribution, air gap inductance and demagnetisation current limit are given and correlated with test results. D.c. disc motors with printed circuit armatures are now widely used in servo applications. The design, construction and testing of a per- manent-magnet-excited, rotating-armature, auto-piloted synchronous disc motor is described. Two types of rotor construction were investigated. Finally, experimental results serve to supplement a theoretical com- parison between disc and equivalent drum motors. Both coil excitation and permanent magnet field excitation are considered. ACKNOWLEDGEMENTS The work described in this thesis was carried out under the supervision of Dr H.Bolton, B.Sc.(Eng.), Ph.D., Reader in Electrical Machines at the Department of Electrical Engineering, Imperial College of Science and Tech- nology, London. For the invaluable help, advice, constant encouragement and stimulating discussions during the course of this work, the author would like to express his extreme gratitude to his supervisor. The bulk of the manufacturing work of the prototypes and test rigs was carried out in the Electrical Engineering Workshop at Imperial College, under the supervision of Mr. R. Moore to whom, along with his staff, the author is grateful. Thanks are also due to Mr. R.B. Owen for his practical assistance and advice during the work. Considerable assistance was received from Dr. D. Pedder and Mr N. Mallinson on problems associated with the electronic portions. The author is deeply indebted to the "Universidade Nova de Lisboa" for all the facilities provided and to the "Instituto Nacional de Investigacao Cientffica" for generous financial support received during his studies in England, without which this work would not have been possible. Thanks are finally due to many colleagues and friends at Imperial College who have helped to make the research enjoyable. CONTENTS Page ABSTRACT ii ACKNOWLEDGEMENTS iv CONTENTS v INTRODUCTION 1 CHAPTER 1: SOME FUNDAMENTAL ASPECTS OF ELECTRICAL MACHINES 7 1.1 - GENERALITIES 7 1.2 - ELECTROMAGNETIC TORQUE IN INDUCTIVELY COUPLED CIRCUITS 7 1.2.1 - Particular cases 9 1.3 - MAGNETIC AND ELECTROMAGNETIC MACHINES 14 1.3.1 - Electromagnetic machines 14 1.3.2 - Magnetic machines 15 1.4 - ELECTRICAL MACHINES TOPOLOGY 18 1.5 - TORQUE EXPRESSION FOR A TWO-PHASE MULTI-POLE SYNCHRONOUS MOTOR 20 1.6 - POLE PROFILE FOR A SINUSOIDAL AIR GAP FLUX DENSITY DISTRIBUTION 30 1.7 - SPECIFIC TANGENTIAL FORCE DEVELOPED BY AN ELECTROMAGNETIC MACHINE 32 1.8 - VARIATION OF STATOR INSIDE RADIUS WITH STATOR OUTSIDE RADIUS 34 vi Page 1.9 - DYNAMIC FACTORS OF A MOTOR 36 1.9.1 - Optimal gear ratio 37 1.9.2 - Variation of dynamic factors with motor proportions 39 1.9.3 - Comparison between a direct and an optimal gearing drive 41 1.10 - HYDRAULIC MOTOR VERSUS ELECTRIC TORQUE MOTOR 45 1.11 - COMPARISON BETWEEN ARCUATE AND CYLINDRICAL CONFIGURATIONS 48 1.12 - COMPARISON BETWEEN DISC AND CYLINDRICAL CONFIGURATIONS 50 1.13 - EQUIVALENT CIRCUIT AND PHASOR DIAGRAM OF A SYNCHRONOUS MOTOR 55 1.14 - INDUCTANCE CALCULATION 57 1.14.1 - Expressions for leakage inductance 58 1.14.2- Air gap inductance of q coils linked 60 1.15 - STATOR FLUX DENSITY ON LOAD 67 CHAPTER 2: PERMANENT MAGNET FIELD ANALYSIS 73 2.1 - GENERALITIES 73 2.2 - FUNDAMENTAL PARAMETERS OF A PERMANENT MAGNET MATERIAL 74 2.2.1 - Intrinsic magnetisation or polarisation 75 2.2.2 - The various permeabilities 79 2.2.3 - The Energy Product 81 2.2.4 - The shape factor 84 2.3 - CLASSIFICATION OF PERMANENT MAGNET MATERIALS 84 2.4 - MAGNET ARRANGEMENT IN ROTATING ELECTRICAL MACHINES 88 2.5 - THE CALCULATION OF A PERMANENT MAGNET UNDER STATIC CONDITIONS 91 2.5.1 - Working point for the minimum volume of the magnet material 94 vn 2.5.2 - Leakage factor formulae for specific permanent magnet configurations 2.6 - THE CALCULATION OF A PERMANENT MAGNET UNDER DYNAMIC CONDITIONS 2.6.1 - Recoil constant energy contours 2.7 - COMPUTATION OF THE FIELD DISTRIBUTION IN A PERMANENT MAGNET MACHINE 2.7.1 - Modelling the magnetic field problem in permanent magnet devices 2.7.2 - Linear representation of demagnetisation characteristics 2.7.3 - Derivation of finite-difference equations for magneti scalar potential in polar coordinates 2.7.4 - Field solution at no-load 2.7.5 - Field solution on load 2.7.6 - Flux plot 2.7.7 - Derivation of finite-difference equations for magneti scalar potential in cylindrical coordinates 2.8 - COMPARISON OF LOAD LINE AND MESH ANALYSIS METHOD 2.8.1 - Calculation of the air gap flux density using the load-line method 2.8.2 - Calculation of the air gap flux density using the mesh analysis method 2.9 - MANUAL SOLUTION FOR THE MAXIMUM PERMISSIBLE STATOR CURRENT CALCULATION CHAPTER 3: AUTO-PILOTED SYNCHRONOUS MOTOR THEORY 3.1 - GENERALITIES 3.2 - MECHANICAL COMMUTATOR REPLACEMENT 3.3 - TRANSISTOR-SWITCHED TWO-PHASE SYNCHRONOUS MOTOR PRINCIPLE viii Page 3.4 - SINE WAVE OPERATION 157 3.5 - TYPES OF ROTOR POSITION SENSORS 159 3.6 - OVERALL SYNCHRO-RESOLVER SYSTEM 166 3.6.1 - Influence of the ratio w /m r c in synchro-resolver system 174 3.7 - ANALYSIS OF THE SYNCHRONOUS MOTOR WITH ROTOR POSITION CONTROL 177 3.7.1 - Steady-state theory of a synchronous motor operating at variable-frequency and fixed load angle 180 3.7.2 - Steady-state theory of a synchronous motor operating at variable-frequency and fixed torque angle 193 3.7.3 - Operating chart for a synchronous motor at variable frequency 203 3.8 - MOTOR RESPONSE TO A STEP INPUT SIGNAL 204 CHAPTER 4: LIMITED MOTION, ROTARY BRUSHLESS ACTUATOR 208 4.1 - GENERALITIES 208 4.2 - DESIGN OF ACTUATOR STATOR 209 4.3 - WINDING ARRANGEMENT 213 4.3.1 - Winding factor 214 4.4 - LEAKAGE INDUCTANCE CALCULATION 215 4.4.1 - Slot leakage inductance 215 4.4.2 - End-winding leakage inductance 218 4.5 - DESIGN OF ACTUATOR ROTOR 219 4.6 - AIR GAP FLUX DENSITY DISTRIBUTION ON NO LOAD 221 4.6.1 - Flux set up per tooth pitch by the B.P.R.E. magnet 222 4.7 - NORMAL FORCE SEEN BY THE SHAFT 225 4.8 - AIR GAP INDUCTANCE CALCULATION 228 4.9 - STATOR CURRENT LIMITATION 231 ix 4.9.1 - Current limit for starting saturation of the magnetic circuit 4.9.2 - Current limit for starting demagnetisation of the permanent magnets 4.10 - FUNDAMENTAL ELECTROMAGNETIC TORQUE 4.11 - PREDICTED RIPPLE TORQUE 4.12 - TEST RESULTS 4.12.1 - Static torque tests 4.12.2 - Stator winding inductance measurement 4.12.3 - Air gap flux density measurement 4.12.4 - Temperature-rise test 4.12.5 - Rotor inertia measurement 4.12.6 - Viscous damping coefficient measurement 4.13 - DESCRIPTION OF THE COMPLETE ACTUATOR SYSTEM 4.13.1 - Synchro-resolver characteristics 4.13.2 - Machine requirements for electronic supply matching 4.13.3 - Actuator dynamic behaviour 4.14 - POSITION CONTROLLER 4.14.1 - System response to a step input 4.14.2 - Effect of tachometer feedback 4.14.3 - System response with variable input 4.15 - NOVEL TACHOMETER CHAPTER 5: TWO-PHASE AUTO-PILOTED DISC SYNCHRONOUS MOTOR 5.1 - GENERALITIES 5.2 - MACHINE CONFIGURATION 5.2.1 - Wire-wound rotor construction 5.2.2 - Printed rotor layout 5.3 - AIR GAP FLUX DENSITY CALCULATION Page 5.4 - ARMATURE REACTION AND INDUCTANCE CALCULATION 300 5.5 - BACK E.M.F. AND TORQUE CONSTANT 301 5.5.1 - Torque constant evaluation 305 5.6 - TWO-PHASE INVERTER CIRCUITRY 306 5.7 - MACHINE OPERATION FROM A SINUSOIDAL SUPPLY 309 5.8 - TEST RIG FOR OBTAINING SPEED CHARACTERISTICS 313 5.8.1 - The speedometer 314 5.8.2 - The eddy-current brake 315 5.9 - TEST RESULTS 317 5.9.1 - Air gap flux density measurement 317
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