Preface The industrial brushless servomotor has developed through a remarkable combination of mechanical, electrical, power electronic and microelectronic technologies, and both the operation and application of the motor rely on many interdependent factors I have tried to cover the fundamentals of the subject in a logical manner, taking a step-by-step approach, describing first the construction of the brushless machine itself and how it works, second, how the current is supplied, third, how the motor behaves when it is loaded and finally how it is rated and selected for a particular duty The book covers the important motor and load characteristics which affect the design of the control system, but does not include a detailed treatment of control techniques which are well described elsewhere The first chapter is devoted to a brief review of the brushed, permanent magnet motor This allows the early introduction to the book of some basic groundwork using what is perhaps a more familiar machine, and also allows a clearer comparison to be made with the brushless type later on Throughout I have been aware of the needs of engineers and students with no previous knowledge of how brushed or brushless motors work, and so both forms are explained from first principles Theoretical analysis is developed in relation to practical examples, and rules of thumb are suggested wherever possible Any equations for motor rating and selection are simple enough for numerical results to be found using a calculator or spreadsheet My hope is that this publication will be of xiv Preface some help to those who are already using brushless motors in servomechanisms, as well as to those who are studying the electrical and mechanical properties which are involved The practical nature of this book has been made possible by the generous supply of technical advice from the members of staff of SEM Ltd I wish to acknowledge a debt of gratitude to Paul Newall for his constant support and for the many hours of his time taken up by our discussions, and also to Van Hamlin and Omar Benzaid for their readily given advice and practical help I am also indebted to several members of staff of the University of Bristol, and wish to acknowledge here the help given by two in particular Duncan Grant suggested the basic idea for the book and followed through with advice and encouragement from start to finish I am also extremely grateful to have had the very willing help, particularly with the systematic solution of quartic equations, of Gordon Reece of the Department of Engineering Mathematics Finally, I would like to give a special thanks to Paul Prater of Lewis Berl Automation Acknowledgement The various photographs were kindly supplied by the following companies: SEM Ltd, Kangley Bridge Road, London SE 26 5AS, UK Parker Hannifin GmbH, Hauser Division, Robert-Bosch-Str 22, 77656 Offenburg, Germany List of units Unit symbol Name A oc ampere degree centigrade henry millihenry hertz joule kilogram metre millimetre ampere-turn henry per metre kilogram-square metre newton newton metre radian microradian per newton metre second millisecond metre per second newton metre per radian newton metre per radian per second tesla volt volt per radian per second watt degree centigrade per watt weber ohm H mH Hz J kg m mm A-turn H/m kg m N Nm rad #rad/Nm S ms m/s Nm/rad Nm/rad sT V V/rad s -1 W ~ Wb f~ List of symbols Symbol Definition AC B C Cp d D DC e E F G H i I lrms lS J J Jm JL Jr KE K'r L LEE m N N~ alternating current magnetic flux density compliance profile constant screw pitch damping constant direct current base of the natural logarithm electromotive force (emf) force gear ratio magnetic field intensity instantaneous current current root-mean-square current continuous stall current imaginary operator x / ~ moment of inertia motor moment of inertia load moment of inertia ratio of load to motor moments of inertia voltage constant torque constant inductance brushless motor inductance, line to line length mass number of turns number of turns on a sinusoidal winding Un/ts T #rad/Nm rn Nm/rad s -l V N A/m A A A A kg m kg m E kg m V/rad s -l Nm/rad s H in kg List of symbols x v i i P Psp R Rth RthT"m RLL p F SI s T TL rs rsoac Trms t t1 V v x s # Oo Oss Opk Oav Omin 0p O" Te Tm power speed-sensitive loss resistance thermal resistance motor rating coefficient brushless motor resistance, line to line profile distribution factor radius international system of units Laplace operator s - a + j w torque load torque continuous stall torque continuous rated torque required torque time duty operation time duty cycle period circuit input voltage velocity linear displacement energy magnetic flux stator angle of sinewave motor conductors permeability temperature ambient temperature steady-state winding temperature peak, winding ripple temperature above O0 average, winding temperature above O0 minimum, winding ripple temperature above O0 angular displacement angle of load rotation real part of Laplace operator s electrical time constant of motor mechanical time constant of motor W W ~ ~ ms/W m S- l Nm Nm Nm Nm Nm s s s V m/s m J Wb rad H/m ~ ~ ~ ~ ~ oC rad or ~ rad s s xviii TM Tth Tw 02 6dm 6OL / Wc Listof symbols mechanical time constant of motor and load thermal time constant of motor thermal time constant of motor winding angular velocity motor velocity load velocity constant velocity of motor constant velocity of load s S S rad/s rad/s rad/s rad/s rad/s CHAPTER I BRUSHED D C MOTORS I.I Introduction Industrial brushless servomotors can be divided into two main types One operates in a similar way to the three-phase synchronous motor and the other is a relatively simple development of the brushed DC motor Both types of brushless motor have the same sort of construction and have an identical physical appearance Both have many characteristics similar to those of a permanent magnet brushed DC motor, and both are operated from a source of direct current A review of the features of the permanent magnet brushed motor is therefore a convenient first step in the approach to the brushless type In this first chapter, the relationships between the supply voltage, current, speed and torque of the brushed motor are developed from fundamental electromagnetic principles Attention is also given to the factors controlling the steady-state speed of the unloaded motor The later part of the chapter is devoted to the question of DC motor rating Only the basic ideas are covered at this stage, in preparation for the more detailed treatment in Chapter The power losses which lead to motor temperature rise are identified, and the main factors affecting the final steady-state Industrial Brushless Servomoters 1.2 temperature are explained for both continuous and intermittent operations of the motor The scope of this chapter is confined to cases where the losses during periods of speed change are insignificant in Comparison to those generated during the periods of constant motor speed 1.2 Operational principles Motor construction Figure 1.1 shows the essential parts of a rudimentary permanent magnet DC motor Two conductors are connected in series to form a winding with one turn The winding has a depth ! and width 2r metres and is mounted between the poles of a permanent magnet The winding is free to rotate about the dotted axis and its ends are connected to a DC source through sliding contacts to form a circuit carrying current I A The main diagram is drawn for the moment when the conductors are passing the centre of the poles The contacts allow the direction of current in the winding to reverse as it moves through the vertical position, ensuring that the direction of flow through the conductors is always the same relative to the direction of the magnetic field In other words, it does not matter in the diagram which side of the winding is to the left or right when we look at how torque is produced Torque production The torque produced by the motor in Figure 1.1 is the result of the interaction between the magnetic field and the currentcarrying conductors The force acting on each conductor is shown as F Some simple magnetic principles are involved in the evaluation of the torque Brushed DC motors + v Rotatingcontacts S S S / T / $ / S S S y s , 2r i I Conductor .-::::: ;~.:: ~:~ View A j Figure 1.1 Principle of the permanent magnet brushed DC motor Magnetic flux r The amount of magneticflux in a magnetic field tells us how much magnetism is present By itself, it does not give the strength of the field The flux may be represented by lines drawn between the poles of the magnet and in the old British system the unit of flux was, in fact, the line In the SI system Industrial Brushless Servomoters 1.2 the unit is the weber, denoted by Wb, where one weber is equivalent to 10 lines in the old system Magnetic flux density B As its name suggests, the term magneticflux density describes the concentration of the magnetic field The SI unit of magnetic flux density is the tesla, denoted by T, where a tesla is equal to one weber per square metre The force on a c o n d u c t o r When a conductor of length l, carrying a current/, is placed in a magnetic field of uniform flux density B, it is found that the conductor is acted on by a force which is at right angles to both the field and the conductor The force is greatest when the conductor and field are also at right angles, as in Figure 1.1 In this case, the force is given by f = BlI (N) The unit of force is the newton, denoted as N The direction of F can be found by the 'left-hand motor rule' This states that the thumb of the left hand points in the direction of the force, if the first finger of the hand is pointed in the direction of the field and the second finger in the direction of the current Torque Force F acts on each conductor of the winding shown in Figure 1.1 The torque produced at each conductor is T = Fr (Nm) The unit of torque is the newton metre, denoted as Nm The radius of action of F around the axis falls as the winding moves away from the horizontal position, reducing the torque In the figure, the winding lies in a plane between the centres of the fiat poles of the magnet, where B is greatest With such a pole shape the flux will be less dense at other winding positions, reducing the torque still further Motor rating and selection 171 period may be neglected in the calculation of the rms torque The Soac curve for M05 shows the nominal torque continuously available at 2025 rpm to be 4.1 Nm, and so the maximum torque available at the safety margin is / - 3.5 Nm This figure is above the torque required and so motor M05 would not overheat, whatever the actual values of torque constant and winding resistance within the 4-10% band Table 5.4 shows the results when the above calculations are made for the next two smaller motors in Table 5.2, using speed-sensitive losses of 30 and 25 W respectively The table shows a good safety margin for the rating of the motor initially selected on the basis of its rating coefficient The margin for M04 is too small, and M03 is unsuitable even in the nominal case Table 5.4 Motor torques and ratings for Example 5.5 M05 M04 i Nominal Tsoac Required Trms Tsoac/Trms M03 i,i 4.1 3.3 1.24 3.4 3.1 1.10 ,, 2.7 2.9 0.93 The ball screw drive In Chapter the pitch of the ball screw was optimized for any drive motor, and the rating of the motor used in Example 4.3 was not considered We can now check if the motor used in the example was the most suitable for the given load, on the basis of the motor rating Example 5.6 Select a correctly rated motor f o r the load o f E x a m p l e 4.3 The load constants are x = 0.025 m = 0.120 s 172 IndustrialBrushless Servomoters t' - F 0.120 q = s IO00N - m 5.0kg I (_~]_]]] Jm t ~ 5.5 rn J, I I ;~; 031 O)m i 20 60 120 t ms Figure 5.9 Motor torque and velocity for Example 5.6 Figure 5.9 shows the motor torque and velocity profiles The motor should satisfy Rth'rm < o.8~ mx262 where - 0.511 + x/(1 + A2)] and Motor rating and selection 173 Inserting the numerical values gives A - 885, and 62- 15.4 The motor rating coefficient is found to be RthT"m < 3.4 • 10 -3 Table 5.2 gives M03 as the initial choice Following the method used in Example 4.3, using C p - 18 and assuming a screw inertia of J s w - 0.00003 kgm 2, gives the optimum pitch of a ball screw driven by M03 as dA 7.2 mm The load moves 25 mm as the screw and motor shaft turn through 25/7.2 = 3.47 revolutions in 0.12 s The motor shaft rotates through 0m = 3.47 • 27r = 21.8 rad, and the motor speed during the fiat-topped section of the velocity profile of Figure 5.9 is ! Ca.) m m tp[1 0.5(pl -k-P2)] giving ! = 273 rad/s - 2607 rpm The inertia of mass m reflected to the motor side of the screw is m d / r (Section 4.5) The motor torque required for acceleration or deceleration of the motor, screw and load masses in Figure 5.9 is md d T- (Jm + Jsw + 47r2 )~ ]a;m giving TI - 2.55 Nm and T2 - - N m The work done in supplying the load force over a distance of one screw pitch is F d - 27rTo where To is the output torque of the motor (assuming negligible 174 Industrial Brushless Servomoters 5.5 losses at the screw) The motor output torque required for the supply of the load force is therefore Fd T0= The Soac diagram for M03 is shown in Figure 5.2 The speedsensitive loss at 2600 rpm is found to be Psp ~ 30 W The total motor torque required for the supply of the load force is therefore Fd Psp T - ~ ~+-~ = 1.26Nm The torques required during the three sections of the period of motion are T~+T3 = 3.81 Nm T3 = 1.26 Nm T2 + T3 = 0.42 Nm The rms value of the three torques is Trm~ = 1.74 Nm The maximum, nominal torque is Tsoac = 2.65 Nm The maximum safe torque in the worst case would be Tsoac/1.17 = 2.26 Nm Table 5.5 shows the results when the calculations are made for M02 using a speed-sensitive loss of 25 W The motor is seen to be unsuitable, even on a nominal basis M03 has a more than adequate safety margin, which is reflected by the amount by which its rating coefficient falls within the limit set by the loading conditions of Example 4.3 M05, the original motor chosen at random (for example 4.3) would clearly be much too large Table 5.5 Motor torques and ratings for Example 5.4 M03 M02 2.65 1.74 1.52 1.28 1.38 0.93 i, Nominal Tsoac Required Trms Tsoac/Trms Motor rating and selection 5.5 175 Precautions The temperature of 150~ is the absolute maximum for any part of a brushless servomotor Overheating of the motor or its environment is an obvious risk if the safety margin used in selecting the motor is too small On the other hand, an overcautious approach can result in an unnecessarily large and expensive motor The final choice is often influenced by several factors, and no figure can be given for a safety margin which suits all circumstances Throughout this chapter we have allowed for 10% variation of the stator resistance and torque constant, between individual motors of the same type We have catered for this variation by allowing a margin of 17% between the required rms torque and the rated torque of the motor It is in fact possible for the torque constant to fall by 10% as the temperature rises, mainly due to the fall in the magnetic intensity of the permanent magnets The same allowance for the stator resistance is, however, very generous The normal variation is much smaller, being dependent, for example, on very small differences between the nominal and actual diameter of the wire used in the manufacture of the winding The conclusion is that the margin of 17% between the required and rated torque should be adequate, assuming that there is no other reason why the motor should be derated In practice there may be several thermal effects to be taken into account, apart from the effect of the tolerance allowed in the values of the motor constants For example, an excessive amount of heat transmitted directly from a hot motor may cause distortion of the frame of a precision mechanism such as a ball screw It is also important that the overall design of any system should allow for sufficient space between the hot surface of the motor and any heat-sensitive equipment, or allow for the accommodation of a larger motor which can 176 Industrial Brushless Servomoters 5.5 run at a lower temperature Care is also required in the way the motor is fitted to the frame of the installation As already mentioned in Section 5.2, the thermal resistance of a motor is likely to be higher than the published value when it is thermally isolated from the flame Thermal isolation can be deliberate, or it can be the accidental result of the insertion of electrical insulation material at all points of metal to metal contact with the flame The maximum rms torque available in either case will be lower than the figure given by the Soac curve, which is plotted experimentally when the motor is bolted directly to a typical flame If possible, a heatsink should be fitted whenever the flame has to be electrically or thermally isolated from the motor Assuming that the motor has been selected correctly on the basis of its ability to supply the required torque, and that all other factors have been taken into account, any overheating which does occur is normally the result of accidental misuse Computer controlled duty cycles are very easy to change, and a motor which has been correctly rated for a particular torque profile is likely to overheat if the profile is made more demanding It is, of course, sometimes difficult to judge the range of the future demands of an application Where space permits, the fitting of a blower will help an existing motor to cope with a moderate increase in demand References Miller, T J E (1989) Brushless Permanent-Magnet and Reluctance Motor Drives Clarendon Press Hendershot, J R Jr and Miller, T J E (1994) Design of Permanent-Magnet Motors Magna Physics Publishing and Clarendon Press Dote, Y (1990) Brushless Servomotors: Fundamentals and Applications Clarendon Press Electro-craft Corp (1980) DC Motors, speed controls, servo systems Armstrong, R W Jr (1998) Load to Motor Inertia Mismatch: Unveiling the Truth Drives and Controls Conference, Session 6, 17-22 Newall, P (1998a) Models of the Motor and Load: Analytical Solutions Paper DD0369, SEM Ltd Newall, P (1998b) Motor and Load Mechanical Resonance Paper DD326, SEM Ltd Newall, P (1998c) Models of the Motor and Load." Numerical Solutions Paper DD0370, SEM Ltd Newall, P (1996) Brushless Motor Thermal Models Paper DDO230, SEM Ltd Index Absolute encoder 75 AC tacho, output 72, 89 Adhesives l Air gap flux density 30, 37, 45, 60, 66, 72 Alnico 61 Alternating current: rectangular 5, 28, 36, 80 sinusoidal 37 Ambient temperature 20 Ampere-conductor, rotating 51 Ampere-turn 54 Amplifier: servo 99 tuning parameters 118 Amplitude modulation 72 Analogue: models 145 signals 96 Angular velocity Applications, motors and sensors 95 Armature reaction 59, 72 Audible noise 30 Backdrive 133 Back e.m.f trapezoidal 43 Backlash: ball and lead screw 133 gears 124 Ball screw transmission 133 backdrive 133 compliance 145 heat distortion 175 inertia 135 optimization 134 resolution 133 Bearings: ball screw 145 damping 141, 145 seal 105, 147 Belt and pulley transmission 130 damping 141 inertia 132 optimization 131 Bipolar junction transistor 76 BJT 76, 79 Blowers 150, 176 Brakes 66, 133 Brushed motor, s e e Permanent-magnet brushed DC motor Brushes inspection 24 Brushless motor 28 air gap flux density 30, 37, 45, 66, 72 armature reaction 59 back e.m.f 36, 38, 39, 46, 69 beating and seal 105, 147 circulating current 41 cogging torque 66 coil 37 commutation 69, 85, 87, 95, 101 construction 29 continuous torque, current 53 cooling 29 current regulation and control 83, 84, 118 damping 141, 144 demagnetization 58, 59, 60, 62 drive 69 dynamic equation 104 eddy current heating, rotor and stator 147 effective resistance 53 electrical equation 104 180 Index electrical time constant 105 electromechanical equation 108 emergency brake 66 equivalent circuit 103 fractional slotting 66 frequency response 109, 140 frictional loss 105, 147 heatsinks 150, 176 hub 30, 45, 147 i2R power loss 29, l l9, 125, 141,147, 149 inductance 97, 99, 102 intermittent rating 153 laminations 30, 147 magnets and materials 37, 45, 61 mechanical time constant 107 misuse 176 motor temperature 160 permanent-magnet rotors 29 pole number 51 power range 64 rating 146 rating coefficient 162 safe speed and torque 148 servomotor 30, 37 shaft compliance 137 single-phase 37 sinusoidal (sinewave) 37 slots, skewing 30 SOAC curves 148 speed and position sensors 71 speed limit 63, 162 speed-sensitive loss 149 squarewave 37 stall torque and current 52 stator core 30 thermal characteristics 63 thermal models 159 thermal resistance 53 thermal time constant 153 torque constant 34, 43, 51 torque loss 155 torque production 34, 38, 48 torque rating 52 transfer function 113 trapezoidal 43 voltage constant 36, 43, 51 windage loss 147 winding 37 winding connections, delta and wye 41 winding temperature 153 Brushless signals Brushless signals resolver 72 92 tachometer 72 90 Chopped current 83 Circulating current 41 Clockwise rotation 93 Closed-loop control 98 Coercivity: hard magnet 59 intrinsic 59 Cogging torque 66, 96 Coil 37 Commutation brushed brushless 69, 85, 86 limit 63 sensors 71 three-phase 85 Commutator Comparator 99 Compliance 137 ball screw 145 motor shaft 137 Cooling, forced 17 blowers 150, 176 Conductor: force on rotating ampcrc- 51 Conjugate pole ll0 Continuous stall current and torque 52, 148 Control: closed-loop 98 current 83, 118 four-quadrant 100 integrated system 96 machine tools 96 position 100 servo 23, 100 unidirectional 100 Core 16, 30 Current: chopped 83 circulating 41 continuous stall 52, 149 control 83, 118 demagnetizing 60 eddy 16 no-load overload 60 Index regulation 84 ripple 25 r.m.s 18 Damping: bearing 141, 145 damping constant (viscous) 16 eddy current 138, 141 friction 138, 141 i2R 141 viscous 16, 141 DC tacho 72 Degrees, mechanical and electrical 93, 95 Delta connection 41 Demagnetization 24, 59, 60, 62 current 60 recovery 58, 59 Digital encoder 73 Diode, freewheeling (flywheel) 82 Disc: brake 66 encoder 73 Drive 25, 69, 85 closed-loop 99 digital 73, 96 Duty cycle 14 computer controlled 176 Dynamic equation 104 Eddy current 16, 30, 147 Effective resistance, sinusoidal motor 53 Efficiency, load velocity profile 123,128 Electrical degrees 93 Electrical equation 104 Electrical time constant 105 Electromechanical equation 108 Electromotive force direction E.m.f End rings 32 Encoder: absolute 75 digital 73 incremental 73 optical 75 Sincos 75 sine/cosine 75 Energy 122 Equation: dynamic 104 181 electrical 104 electromechanical 108 steady-state 102 Error signal 100 Exterior rotor 37 Fan motor 37, 95 Faraday 34 Feedback 99 Ferrite magnet 23 Field, s e e Magnetic field Field strength, s e e Magnetic intensity Flux, s e e Magnetic flux Force: on conductor direction electromotive (e.m.f.) magnetomotive (m.m.f.) 54 radial 31 shear 31 Forced cooling 17 blowers 150, 176 Form factor 19, 151 Fractional slotting 66 Freewheeling (flywheel) diode 82 Frequency: PWM 83, 147 resolver 72 resonant 138, 140, 143, 145 response 109 Friction: beating seal 105, 147 load 141 power loss 16 Fringing 38, 45 Full-bridge inverter 83, 85 Functional transform 112 Gate turn-off thyristor 77 Geared transmission: backlash 124 inertia 120 losses 130 optimization 124 planetary 124 reducer 124 Glass-fibre tape 32 GTO 77 Half-bridge inverter 80, 82 Hall-effect sensor 71, 87 Hall plate 71 1182 Index Hard magnet 58 coercivity 59 demagnetization 59 intrinsic flux density 58 materials 61 permeability 60 recoil permeability 59 remanence 59 temperature effects 61 Heating, brushless motor: rotor 147 stator 29, 147 Heatsink 150, 176 High inertia, motors and loads 145 High load force 133 Hub 30, 45, 147 Hysteresis, magnetic flux density 56 hard magnet 58 IGBT 76, 79 Incremental encoder 73, 92 signal ripple 96 Incremental motion 97, 124, 147, 159 motor selection 160 Inductance: brushed motor 101 brushless motor 97, 99, 102 Induction motor 95 Inertia, s e e Moment of Inertia match 124, 143 Inertial load 124 Inspection, brushes 24 Insulation: electrical 176 thermal 150, 176 Insulated gate, bipolar junction transistor 76 Intensity, magnetic 55 Intermittent operation 14, 18, 146 motor rating and torque 17, 153 motor test 157 rule of thumb 22, 155 Intrinsic coercivity 59 Inverter 29, 70 motor torque control 83 single-phase, full-bridge 82 three-phase, full-bridge 85, 86 single-phase, half-bridge 80 i2R loss 16, 21, 29, 63, 119, 121, 126, 129, 141, 143, 149, 154 minimization 119, 129 Iron loss 16, 102, 147 Joule 122 Laminations: rotor hub 30, 147 stator core 30 Laplace transformation 111 Lead screw transmission 133 Limit on speed, brushless motor 63, 162 Line-to-line resistance 52 Linear translation 119 Load force 133 Load friction 141 Load torque 127 Load velocity profile 120 optimum 122, 128 parabolic 128 profile constant 123 trapezoidal 120 Losses, s e e Power losses Machine tool control 96 Machining, 'two and a half axis' 68 Magnet, see Permanent magnet Magnetic circuits: coercivity 59 hard magnet 58 intrinsic coercivity 59 permeability 55 recoil permeability 59 relative permeability 55 remanence 59 Magnetic field 54 Magnetic flux cutting 8, 33 fringing 38, 45 linkage 33 radial rotating 36 Magnetic flux density hysteresis 56 irregularity 30 Magnetic intensity 55 Magnetic strength, s e e Magnetic intensity Magnetomotive force 54 Measurement, shaft position and speed 89, 92 Mechanical degrees 92 Mechanical resonance 137 inertia match 143 Mechanical time constant 107 Index Metal oxide, semiconductor field-effect transistor 76 Minimization: cogging torque 67 i2R loss 119, 129 magnet cost 60 Misuse, brushless motor 176 M.m.f 54 Models: analogue, thermal 145, 159 Modulation: amplitude 72 pulse-width 83 Moment of inertia 104 ball screw 135 belt and pulley 133 gear reducer 120 high, motors and loads 145 reflected 125 MOSFET 76, 77, 79 Motor: brushless 28 induction 95 permanent magnet, brushed DC synchronous 1, 36, 37, 46 Neodymium-iron-boron 61 Newton, Newton-metre NIB 61 No-load current No-load speed 7, 9,12 Open-loop 23 motor-load characteristics 101 Operational transform 112 Optical sensor, encoder 73, 75 Optimization: ball screw and lead screw 134 belt and pulley 131 gear reducer 124 load velocity profile 122 Oscillatory response 111, 137, 141 Overload current 60 Overshoot, motor speed 111, 116 Parabolic velocity profile 128 Permanent magnet: alnico 61 characteristics 56 cost 61 ferrite 23 hard 58 materials 61 183 neodymium-iron-boron 61 NIB 61 rare-earth 61 rotor mounting 31 samarium cobalt 61 shape 37, 45 spacing 66 Permanent-magnet brushed DC motor alternating back e.m.f armature reaction 59 brush inspection 24 commutation 5, 101 commutation limit 63 construction 2, damping constant 16 eddy currents 16 ferrite 23 forced cooling 17 inductance 101 intermittent operation 14, 17 12R loss 15, 18, 21 iron losses 16 motor temperature 17, 19 no-load speed 7, 9, 12 power losses 15 rating 14 servomotor 23 Slunits 9, 16 SOAC curves 17 speed regulation constant 12 speed-torque characteristics 12 stall current 13 thermal characteristics 17 thermal resistance 20 thermal time constant 21 torque constant torque loss 16 voltage constant windage loss 16 winding temperature 20, 22, 26 Permeability 55 hard magnet 59 recoil 59 relative 55 Pick-and-place 130 Pitch: screw thread 135 stator, slot 67 Pole, magnetic number 51 spacing 66 1114 Index Pole, s-plane 113 conjugate 110 resonant frequency 144 Position: control 98, 100 measurement 87, 92 Power 10 Power electronics 75 inverter 80 semiconductor switches 76 Power losses 15 beating 15 beating seal 15 eddy current 16 frictional 16 i-'R 16 iron 16 speed-sensitive 16 Power range, brushless motors 64 Power transmission: ball screw, lead screw 133 belt and pulley 130 gear reducer 120 Profile, s e e Load velocity profile Pulse-width modulation 83 PWM 83 switching frequency 83, 147 Quadrants: flux density 58 torque-speed 100 Radial flux 5, 30 Radial force 31 Rare-earth 61 Rating: brushed motor 14 brushless motor 146 coefficient 162 continuous 17, 151 criterion 161 incremental motion 159 intermittent torque 17, 153 rule of thumb 22, 155 SOAC 17, 148 steady-state 151 torque 148, 149 Ratio, gear reducer: optimization 124 rule of thumb 127 Reaction, armature 59, 72 Recoil permeability 59 Reducer 124 power loss 130 Reflected inertia 125 Regulation: constant 12 current 84 speed 100 Relative permeability 55 Remanence 59 Resistance: line-to-line 52 sinusoidal motor 53 temperature coefficient of 149 thermal 20, 153 tolerance 147, 175 winding 10, 52 Resolution 136 Resolver 72 Resonance, torsional 137 inertia match 143 Resonant frequency 138, 143, 145 tests 140, 144 Response 109 frequency 109 oscillatory I 11, 137, 141 steady-state 110 transient 109 unstable 110 Ripple: current 25 incremental encoder signal 96 motor torque 35 winding temperature 21, 153 R.m.s current, torque 18, 166 Root-mean-square 18 Rotating ampere-conductors 51 Rotating flux 36, 51 Rotation, clockwise 93 Rotor, brushless motor 30 eddy currents 147 exterior 37 heating 147 hub 30, 45, 147 laminations 30, 147 shaft 30, 137 s-plane 110 Safe speed and torque 17, 148 Samarium cobalt 61 Selection, brushless motor 146, 160, 164, 168 criterion 161 Index Semiconductor switches 76 Sensor 71 AC tacho 72 applications 95 DC tacho 72 Hall-effect 71 optical 73 resoiver 72 shaft speed and position 72 Sincos, sine/cosine 75 Servo control 23, 100 Servo system 100 Servomotor 23, 30, 37 Shaft, brushless motor 30 compliance 137 direction of rotation 93 speed and position measurement 89, 92 Shear force 31 SI units 9, 16 Sincos, sine/cosine encoder 75, 96 Single-phase inverter 80, 82 Single-phase motor 37 Sinusoidal distribution: air gap flux 45 winding 45, 72 Sinusoidal (sinewave) motor 37, 45, 64 applications, sensors 96 back e.m.f 46 commutation 92 effective resistance 53 flux density waveform 45 magnet shape 45 stall torque and current 53 thermal resistance 52 torque 52 torque constant 51, 53 torque production 48 voltage constant 52 winding distribution 45 Skewing 30, 38, 66 Slot pitch 67 Slots 30 Slotting, fractional 66 SOAC curve 17, 25, 148 Speed: control, s e e Servo control limit, brushless motor 63, 162 measurement 24, 89, 92 no-load 7, 9, 12 overshoot 111, 116 regulation, regulator 100 185 stability 110, 118, 139 steady-state 10 voltage limited 63, 162 Speed-sensitive loss 15, 148 Speed and torque, safe values 17, 148 Squarewave motor, s e e Trapezoidal motor Stability, motor speed 110, 118, 139 Stall torque and current 53, 147 Star (wye) connection 41 Stator, brushless motor 29 core 30 eddy currents 30, 147 heating 29, 147 laminations 30 slots 30 tolerance, resistance 147, 175 Steady-state: current 10 equation 102 rating 151 response 110 speed 10 temperature 20, 147 torque 11 Step input 103 Switching devices, s e e Semiconductor switches Switching frequency 83 Synchronous motor 1, 36, 37, 46 Tachometer 24 AC, DC 72 Temperature: ambient 20 coefficient of resistance 149 motor maximum 18, 149, 175 steady-state 20, 147 winding 20, 21, 153 winding ripple 21, 153 Tesla Thermal: characteristics 63 isolation 150, 176 resistance 20, 153 time constant 20, 154 Three-phase: commutation 85 connections 41 inverter 83, 85 Thyristor 76 GTO 77 186 Index Time constant: electrical 105 mechanical 107 thermal 21, 154 Tolerance 147, 167, 175 Torque cogging 66, 96 continuous stall 53, 147 control 83 intermittent 153 irregularity 45 load 10, 127, 130 loss 16, 150 rating 146, 149 r.m.s 18, 166, 175 sinusoidal motor 48 steady-state 11 trapezoidal (squarewave) motor 38 Torque and speed, safe values 17, 148 Torque constant 7, 43, 51, 53 tolerance 147, 175 Torsional resonance 137 inertia match 143 Transfer function 113 Transform: functional 112 Laplace I 11 operational 112 Trapezoidal back e.m.f 40, 69 Trapezoidal (squarewave) motor 37, 41,64 applications, sensors 95 back e.m.f 39, 69 circulating current 41 commutation 87, 89 flux density waveform 37 magnet shape 37 stall torque and current 53 thermal resistance 52 torque 52 torque constant 43 torque irregularity 45 torque production 38, 43 voltage constant 43 Transient response 109 Translation 119 Trapezoidal velocity profile 120 Two-and-a-half-axis machining 68 Unidirectional control 100 Unstable response 110 Velocity profile, s e e Load velocity profile Viscous damping 16, 105 damping constant 16 Voltage constant 9, 43, 52 Voltage, speed limiting 63, 162 Weber Windage 16, 105, 147 Winding 2, 5, 34, 37, 41 i2R loss 15, 21, 147 maximum temperature 17, 27, 154 temperature ripple 21, 155 thermal time constant 21, 155 Winding resistance 10, 53 tolerance 147, 175 Worst case 175 Wye (star) connection 41 ... temperature above O0 average, winding temperature above O0 minimum, winding ripple temperature above O0 angular displacement angle of load rotation real part of Laplace operator s electrical time... physical appearance Both have many characteristics similar to those of a permanent magnet brushed DC motor, and both are operated from a source of direct current A review of the features of the permanent... practice would cause a reduction in WNL Figure 1.8 has been drawn for a fixed value of supply voltage For any particular motor, a family of linear speedtorque characteristics can be drawn for a