Appendix: Airgap Field Components and the Maxwell Stress

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1 Appendix: Airgap Field Components and the Maxwell Stress Figure A.l shows an idealised machine structure. The stator and rotor surfaces are smooth; the permeability of the iron is assumed to be infinite; and the windings are represented by 'current sheets' of negligible thickness on the stator and rotor surfaces. Current flows in the axial direction, perpendicular to the plane of the paper. Let the stator linear current density be (A. I) The radial and circumferential components of the magnetic field may be found by applying Ampere's circuital law to selected paths. The radial component H 1 r may be evaluated from a path such as PQRS (figure A. I), as was done in section 4.2; from eqns (4.9) to (4.11), the result is rl Hlr =- Klm COS fj g (A.2) where r 1 is the radius of the stator current sheet and g is the radial length of the airgap; it is assumed that H 1r is constant along a radial path such as PQ. Figure A.1 237

2 238 ELECTRICAL MACHINES AND DRIVES Next we show that there must be a tangential component of magnetic field. Consider a path in the airgap which links no current, such as ABCD in figure A.l. We have 0 =f Ht ds = J: Hts ds + J: Hts ds +;. TtKtm(cos 82 -cos 8t) (A.3) where AB =CD= b. Equation (A.3) shows that there must be a tangential component H 15 along BC or AD. To evaluate this component, consider the boundary conditions at the stator and rotor surfaces. Take closed contours C1 and C2 enclosing lengths l)s 1 and l)s2 of the respective surfaces; let the ends of the contours shrink to zero in such a way that one curved side is just in the iron, while the other curved side is just in the air. For the contour C 1, the current enclosed is K 1 l)s 1, and Ampere's circuital law gives Ktl)st =f Ht ds=h15 l)st (A.4) cl Thus H 15 = K I on the airgap side of the stator surface. Since we are considering the field due to stator current alone, the contour C2 encloses no current, and we have (A.S) Thus His= 0 on the airgap side of the rotor surface. If H 1r is independent ofr, and the airgap length g is small in comparison with the radius ri, it is readily shown that His varies linearly with r from a value of 0 at the rotor surface to K I at the stator surface. These results for Hir and H 15 agree with the exact solution of the field equations given by White and Woodson [ 1], subject to the condition that g ~ r 1 A similar argument holds for the rotor field H2; the component H2r is independent of r, and H25 varies from a value of 0 at the stator surface to K 2 at the rotor surface. Consider the force exerted on an element l)s of the rotor when both stator and rotor currents are present. From eqn (1.43) the tangential Maxwell stress is BB fs = _!.!_ = BrHs = Br(His + H2s) (A.6) Jlo At the rotor surface we have H 15 = 0 and H25 = K 2 If lis the axial length of the element, the tangential force is given by (A.7) The quantity k 2 l)s is just the current /)i in this portion of the rotor surface, so eqn (A.7) becomes (A.8)

3 APPENDIX: AIRGAP FIELD COMPONENTS AND THE MAXWELL STRESS 239 Equation (A.8) is equivalent to eqn ( 4.25 ), showing that the Maxwell stress calculation is equivalent to evaluating the force on a current element in a magnetic field. Reference D. C. White and H. H. Woodson, Electromechanical Energy Conversion (New York: Wiley, 1959).

4 Bibliography The following lists are not exhaustive, and are intended as suggestions only. Background reading A. J. Compton, Basic Electromagnetism and its Applications (Wokingham: Van Nostrand Reinhold, 1986). P. Hammond, Electromagnetism for Engineers, 3rd ed. (Oxford: Pergamon Press, 1986). G. Stephenson, Mathematical Methods for Science Students, 2nd ed. (London: Longman, 1973). G. Williams, An Introduction to Electrical Circuit Theory {London: Macmillan, 1973). Further reading D. A. Bradley, Power Electronics {Wokingham: Van Nostrand Reinhold, 1987). A. E. Fitzgerald, C. Kingsley, Jr. and S.D. Umans, Electric Machinery, 4th ed. (New York: McGraw-Hill, 1983). C. B. Gray, Electrical Machines and Drive Systems (Harlow: Longman, 1989). P. Hammond, Applied Electromagnetism (Oxford: Pergamon Press, 1971). J. Hindmarsh, Electrical Machines and Drives: Worked Examples, 2nd ed. {Oxford: Pergamon Press, 1982). J. Hindmarsh, Electrical Machines and their Applications, 4th ed. {Oxford: Pergamon Press, 1984). C. W. Lander,Power Electronics, 2nd ed. (New York: McGraw-Hill, 1987). J. E. Parton, S. J. T. Owen and M.S. Raven, Applied Electromagnetics, 2nd ed. (London: Macmillan, 1984). 240

5 BIBLIOGRAPHY 241 G. R. Siemon and A. Straughen, Electric Machines (Reading, Mass.: Addison Wesley, 1980).

6 Answers to Problems Chapter N The machine will not work Use Ampere's circuital law and the reciprocal property of mutual inductance H; 7.76 A; H; 1790 N ll2 I:U sin 2cp A= 1.14; B = ; hysteresis loss 57 W; eddy current loss 13 W. Chapter per cent; KV 2 T= ----=- (R +Kwri dw (a) 20 A; (b) 90 rad/s; (c) 20 A; (d) 10 A; (e) 95 rad/s;- + 40w = dt Chapter l : v 1i 1 =v 2 i 2 ; Z 1 =k 2 /Z 2 ; inductanceofvaluek 2 C Rc = ; Xm = ;Re = ; Xe = ; n = 3.04; (a) V; (b) per cent; (c) 138 per cent (a) Excessive magnetising current will burn out the transformer. (b) Very low magnetising current, eddy-current loss unchanged, hysteresis loss times normal value. 242

7 ANSWERS TO PROBLEMS 243 Chapter F(O) is the displacement of the 8 axis to give equal positive and negative areas T =-fk-:n rr,-3/a 1 sin{2( w- Wr )t + 2a}. Chapter kw; 500 V; 58.3 A; 0.857; 45 kw [= _!_ fl( PTo ) 2rr J I J tan By Lenz's law, the rotor oscillations will be damped. Chapter T= mp X Iz. 1 w sxm R 2 +R 2 sxm m /, '/ Effective rotor resistance R ;;s in the first case, R; /(2 - s) in the second. Two equivalent circuits in series, with element values halved. Chapter Single-phase power factor= 0.900; three-phase power factor= Chapter Fundamental: 197 W; 5th harmonic: 38.9 W; 7th harmonic: 10.1 W. Chapter Be = Bm = tbs b = (Sr- Si)(Ti- h)/(ti - Tc); a= b + Si; k = (Ti - Tr)/(Sr- Si) J.

8 Index AC drives 193 (see also Induction motors, Inverters and Synchronous motors) adjustable-frequency operation 201 drive stability 209 induction motor drives 204 load-commutated inverter operation 203 motor losses 210 nonsinusoidal supply waveforms 209 pulsating torque 210 regeneration 20 1 reluctance motor drives 20 1 synchronous motor drives 201 vector control 207 AC generators 77, 79 (see also Synchronous generators and Induction generators) AC motors (see AC drives, Induction motors and Synchronous motors) AC/DC converters 177 (see also DC drives) active load 185 discontinuous current 186 half -con trolled bridge 18 5 inverter action 185 single-phase bridge 178 three-phase bridge 182 Accelerometer, force-balance 8 Alignment torque 19, Ill Alternator (see Synchronous generators) Ampere's circuital law 30 Analogies, electric and magnetic circuit 31 Armature reaction 57 Armature windings 50, 132 Back EMF 64 Breakdown torque 161 Brush contact loss 61 Brushes 45, 52, 55,61 Brushless DC motors 1, 227 characteristics 232, 233 delta-connected 231 sinewave 232 squarewave 228 star-connected 229 Brushless excitation 131, 234 Cage rotor 148 Capacitor, synchronous 139 Capacitor motors 167 Capacitor-start motors 1 70 Choppers 187, 221 Classical machines 1 Commutation 55,57 Commutator 45, 50 Compensating winding 57 Compensator, synchronous 139 Compound motor 69 Controlled rectifier (see AC/DC converters) Converter, ACJDC (see AC/DC converters) Core loss 61, 92, 134, 164 Coupled circuits 12, 14 Cross-field machines 1 Current density 4 Current density, linear 1 04 Current transformer 98 Damper windings 137 DC choppers 187 DC drives 174, 188 (see also AC/DC converters and DC motors) closed-loop control 189 harmonics 191 motor losses 190 motor speed control 188 power factor 191 speed reversal

9 INDEX 245 torque reversal 190 DC generators 61 (see also DC machines) permanent-magnet 62 separately excited 62 shunt 62 tachogenerator 62, 72 DC machines 45 (see also DC generators and DC motors) armature reaction 57 armature windings 50 brush contact loss 61 commutation 57 commutator 45, 50 compensating winding 57 constants 54, 56 construction 52 core loss 61 cross-field 1 disc type 73 efficiency 61 elementary 45 energy conversion 58 field winding 55 fundamental principles 46 general equations 54 generated voltage 46, 49, 54 goodness factor 66 heteropolar 46 homopolar 8, 45 interpoles 55 linear approximation 56 losses 58, 60 magnetic forces 57 magnetisation curve 56 Maxwell stress 57 moving-coil 72 multi-pole 54 open-circuit characteristic 56 permanent-magnet 48, 62, 66 printed armature 73 reactance voltage 55 rotationalloss 61 sign convention 59 slotted armature 57 special machines 72 torque 50, 54 DC motors 64 (see also Brushless DC motors, DC drives and DC machines) back EMF 64 compound 69 disc type 72 ideal characteristics 65 linear 73 moving-coil 72 permanent-magnet 66 printed armature 73 separately excited 65 series 68 shunt 66 speed control 66, 67, 188 starting 71 torque/speed characteristics 67-9 Delta connection 82 Depth of penetration 28 Direct axis 143 Direct-axis reactance 143 Direct-on-line (DOL) starting 71, 162 Disc motor 72 Distributed windings 103 (see also Sinusoidally distributed windings) Eddy current loss 25 Electrical angle 122 Electrodynamic wattmeter 22 Electromagnet 18 Electromagnetic forces 4, 14 energy methods 20 Maxwell stress 15 on a conductor 5 on iron parts 15,21 Electromagnetic induction 2, 10 calculation 13 Faraday's law 12 inductance 13, 14 moving conductor 2 Electromechanical energy conversion 6 Energy, magnetic field 20 Energy and inductance 22 Energy conversion 6, 58 Energy methods 20 Equivalent circuit induction machine 152 synchronous machine 133 transformer 89, 93 Excitation voltage 133 Excitation winding Exciter 130 Faraday Faraday disc machine 8 Faraday's law 12 Ferromagnetic materials 23

10 246 Field winding 55, 130 Flux, definition 10 Flux cutting rule 4 Flux linkage 10 Force, electromagnetic (see Electromagnetic forces) Fractional slip 151 Friction and windage loss 61 Fringing 35 Generated voltage AC machines 45,77 DC machines 46, 49,54 Generator, elementary 7, 7 8 Generators (see DC generators, Induction generators and Synchronous generators) Goodness factor 66 Gyrator 100 Harmonics space 127 time 191, 209 Homopolar machines 8, 45 Hysteresis 24 Hysteresis loss 24 Ideal transformer 84 Impedance transformation 86 Induced electric field 3 Induced EMF (see Electromagnetic induction) Inductance 11 energy 22 induced EMF 13 leakage 87 mutual 11 self 11 Induction, electromagnetic (see Electromagnetic induction) Induction generators 148, 160 Induction machines 148 (see also Induction generators and Induction motors) braking 160, 172 construction 148 core loss 164 dynamic braking 172 electromagnetic action 150 equivalent circuit 152, 156 leakage reactance 15 6 losses 164,210 magnetising current 1 56 INDEX magnetising reactance 156 multi-pole 1 57 power relationships 1 58 rotationalloss 164 rotor efficiency 15 9 slip 15 1 slip frequency 151 torque 152, 159 torque/speed characteristic 159, 161 Induction motors 148, 160, 161 (see also AC drives and Induction machines) applications, single-phase 170 breakdown torque 161 cage-rotor 148 capacitor 167 capacitor-start I 70 characteristics 162 efficiency 162 Kramer system 165 plugging 161 pole-amplitude modulation 166 pole-change windings 166 power factor 162 rotor resistance, effect of 161 shaded-pole 169 single-phase 167 slipring 148, 165 speed control 164, 204 split-phase 169 starting 162 wound rotor 148, 165 Instrument transformers 97 Interpoles 55 Inverters 1, 194 (see also AC drives) autosequentially commutated inverter 199 current-regulated PWM inverter 197 current-source inverter 199 sinusoidal PWM inverter 197 six-step inverter 195 three-phase bridge inverter 194 voltage-source inverter 194 Kramer system 165 Laminations 27 Leakage 35 Leakage flux 88 Leakage inductance 88 Leakage reactance 92, 133, 15 6

11 INDEX 247 Lenz's law 12 Linear current density 104 Linear machines DC 73 induction 170 reluctance 146 synchronous 145 Linearity, magnetic 11, 3 5 Load angle 136 Lorentz equation 2 Losses (see also Core loss and Rotational loss) DC machine 60 eddy current 25, 92 hysteresis 24, 92 induction machine 164,210 synchronous machine 133 transformer 92 Loudspeaker, moving-coil 8 Magnet, permanent (see Permanent magnets) Magnetic circuit 28 analogies 31 calculations 32 diagram 32 parallel paths 36 Magnetic field energy 20 Magnetic flux 10 Magnetic fringing 35 Magnetic leakage 35 Magnetic linearity 11, 35 Magnetic materials 23 Magnetic potential difference 31 Magnetic saturation 24, 34, 56, 63,65 Magnetisation curve 23, 56 Magnetising current 91, 134, 15 6 Magnetising reactance 92, 134, 156 Magnetomotive force 31, 104 Maxwellstress 15,57,110,237 Meter, moving-coil 8 Motional induction 2 Motional induction formula 4 Motor, elementary 7 Motors (see Brushless DC motors, DC motors, Induction motors, Stepper motors, Switched reluctance motors and Synchronous motors) Moving-coil DC motor 72 loudspeaker 8 meter 8 Multi pole machines 54, 121, 136, 157 Mutual inductance 11 Negative phase sequence 86 Non-classical machines 1 Oersted 1 Open-circuit characteristic 56 Open-circuit test 9 5 Penetration depth 28 Permanent magnets 24, 38 Alnico 38, 40 coercivity 39 demagnetisation characteristic 39 energy product 41 ferrite 38, 40 load line 39 neodymium-iron-boron 38, 39 rare-earth 38 recoil 40 remanence 3 9 samarium-cobalt 38, 40 Permeability 24 Permeance 38 Phase sequence 83 Plugging 161 Pole-amplitude modulation 166 Pole-change motors 166 Polyphase systems 79 Pony motor 142 Positive phase sequence 83 Potential difference, magnetic 31 Power factor correction 139 Pull-out torque 137,223 Quadrature axis 143 Quadrature-axis reactance 143 Reactance voltage 55 Rectifier (see AC/DC converters) Regulation 96 Reluctance 32 Reluctance motors 145 Reluctance torque 143 Reversal of rotating field 116 Rotary amplifiers 1 Rotating magnetic field 113 definition 113 induced voltage 116 multi-pole 121 reversal of direction 116

12 248 INDEX space and time phasors 119 three-phase winding 11 5 two-phase winding 114 voltage-current relationship 118 Rotational loss 61, 164 Salient poles 102, 142 Saturation 24, 34, 56, 63, 65 Schrage motor 1 Scott transformer connection 99 Self-inductance 11 Semiconductor devices 175 gate turn-off thyristor (GTO) 176, 188 insulated gate bipolar transistor (IGBT) 177,188 power MOSFET 177, 188 power transistor 176,188,194, 219 rectifier diode 175,178,182 thyristor(scr) 175,180,184, 199 Separately excited generators 62 Separately excited motors 65 Series motors 68 Shaded-pole motors 169 Short-circuit test 95 Shunt generators 62 Shunt motors 66 Sign convention 13, 59 Silicon steel 24 Sinusoidally distributed fields 102, 1 OS (see also Rotating magnetic field) combination 107 space phasor 108 vector representation 108 Sinusoidally distributed windings 1 OS current density 105, 106 rotating magnetic field 113 torque 110 Skin depth 28 Skin effect 28 Slip 151 Slip frequency 15 1 Slip ring induction motors 148, 165 Sliprings 45, 130, 148 Slotted armature 57 Soft start 71, 162 Space harmonics 12 7 Space phasors definition 1 09 relationship with time phasors 119 Speed control (see AC drives, DC drives, DC motors, Induction motors, Stepper motors and Switched reluctance motors) Split-phase motors 169 Squirrel-cage rotor (see Cage rotor) Star connection 82 Star-delta starting 162 Star -delta transformation 82 Starting DC motors 71 induction motors 162 synchronous motors 142 Steinmetz law 24 Stepper motors 1, 131,213 bipolar drive 220 chopper drive 221 closed-loop control 226 drive circuits 218 hybrid 216 multi-stack 214 multi-step operation 222 open-loop control 224 permanent-magnet 213 position control 225 pull-out torque 223 resonance 223 single stack 214 slewing 223 speed control 225 stattjstop rate 223 static torque 222 torque production 216, 218 unipolar drive 219 variable-reluctance 214 velocity profile 225 Stress, Maxwell 15, 57,110,237 Stress vector 16 Superconducting machine DC 9 synchronous 131 Switched reluctance motors 1, 131, 214,234 comparison with other drives 236 controllers 235 torque/speed characteristics 235 Synchronous capacitor 139 Synchronous compensator 139 Synchronous generators 130, 138 Synchronous machines 130 armature winding 132 brushless excitation 130 characteristics 135, 139

13 INDEX 249 damper windings 137 direct axis reactance 143 equivalent circuit 133 excitation voltage 133 excitation winding 130 exciter 130 leakage reactance 133 load angle 136 losses 133 magnetising current 134 magnetising reactance 134 multi-pole 136 phasor diagrams 132, 144 quadrature-axis reactance 143 reluctance torque 143 salient-pole 102, 142 superconducting 130 synchronous reactance 135 synchronous speed 136 synchronous torque 136 torque/load-angle characteristic 136, 143, 144 Synchronous motors (see also AC drives and Synchronous machines) linear reluctance 146 linear synchronous 145 power factor correction 13 9 pull-out torque 137 reluctance motors 145 starting 142 V -curves 141 Synchronous reactance 135 Synchronous speed 136 Tachogenerators 62, 72 Three-phase systems 80 Time harmonics 191, 209 Torque alignment 19, 111 breakdown 161 calculation 110 DC machine 50, 54 induction machine 152, 159 pull-out 137,223 reluctance 143 synchronous 136 Transformation impedance 86 ratio 85 star-delta 82 Transformer induction 10 Transformers 83 approximate equivalent circuit 93 auto-transformers 97, 162 construction 84 core loss 91 current transformers 98 determination of parameters 94 efficiency 96 equivalent circuit 89, 93 ideal 84 impedance transformation 86 instrument transformers 97 leakage inductance 88 leakage reactance 91 magnetising current 91 magnetising reactance 92 open-circuit test 95 phasor diagram 91 primary 84 ratio of transformation 85 regulation 96 Scott connection 99 secondary 85 short-circuit test 95 tests 94 variable transformers 97 voltage equation 91 voltage transformers 97 Two-phase system 79 Universal motors 70 V-curves 141 Variable-speed drives (see AC drives, Brushless DC motors, DC drives and Switched reluctance motors) Vernier motor 214 Voltage (see Electromagnetic induction and Generated voltage) Voltage transformer 97 Wattmeter, electrodynamic 22 Windage and friction loss 61 Windings (see also Sinusoidally distributed windings) AC machine 123 DC machine 50 Wound-rotor induction motors 148, 165

Contents. About the Authors. Abbreviations and Symbols

Contents. About the Authors. Abbreviations and Symbols About the Authors Preface Abbreviations and Symbols xi xiii xv 1 Principal Laws and Methods in Electrical Machine Design 1 1.1 Electromagnetic Principles 1 1.2 Numerical Solution 9 1.3 The Most Common