Noise of Polyphase Electric Motors Jacek F. Gieras

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Noise of Polyphase Electric Motors Jacek F. Gieras

Preface

It has been estimated that more than 65% of the electrical energy produced in developed
countries is consumed by electric motors. Electrical motors are the most
popular machines of everyday life. They are used either as power machines providing
propulsion torque or servo motors operating in a closed loop control with
speed or position feedback. Electric motors are embedded in larger systems as
their integral part. The noise radiated by electric motors immensely affects the
overall noise of the system.
Contemporary electric motors are designed with higher magnetic flux density
in the air gap than motors manufactured a half century ago. Higher magnetic
flux density in the air gap produces higher radial magnetic forces acting on the
stator system and, consequently, higher vibration and acoustic noise. With the
increased power density of electric motors and more demanding environmental
requirements, the prediction of noise at the early stage of design of electrical
motors has become a very important issue. Not only electromagnetic, thermal,
and economic calculations, but also the level of noise and vibration must be considered,
so that the overall performance can be optimized/balanced and specific
requirements can be incorporated in the design to avoid large retrofit expenses.
However, prediction of noise is more difficult and less accurate than, for example,
torque–speed characteristics. This is because only a very small fraction of
electrical energy is converted into acoustic energy and correct estimation of some
mechanical and acoustic parameters is very difficult.
The first book [119] on calculation of noise in electrical motors was published
by Jordan in 1950. Details of harmonic field analysis including harmonic
torques, noise, and vibration in induction motors are given in the book [87] by
Heller and Hamata published in 1977. Analysis of noise in induction machines
with emphasis on its reduction is given in monograph [248] by Yang, which
was published in 1981. The most comprehensive analysis of noise and vibration
in electrical machines is contained in the book [200] by Timar, Fazekas, Kiss,
Miklas, and Yang published in 1989. It is also necessary to mention two books
on noise and vibration in induction machines published by Russian researchers:
Shubov in 1974 [187] and Astakhov, Malishev, and Ovcharenko in 1985 [10], and
a book published by Polish researcher Kwasnicki in 1998 [127]. There is no book
published so far on noise and vibration of permanent magnet (PM) synchronous
motors. The demand on these motors is nowadays in second place, after the
demand on induction motors.
As most books on noise and vibration analysis in electrical machines were
published over two decades ago, recent advances in vibro-acoustic theories and
technologies are only accessible in learned journals and have not been captured
in a single monograph. These advances include the development and application
of numerical methods of noise computation such as the finite element method
(FEM), boundary element method (BEM), and statistical energy analysis (SEA)
[43, 230] to the prediction of noise in electrical machines. With the increase in the
importance of noise analysis and synthesis in the modern approach to the design
of electrical motors, the authors have made an attempt to prepare a modern monograph
on noise calculation in induction and PM synchronous motors addressing
electromagnetic, mechanical, and vibro-acoustic issues. The noise and vibration
of switched reluctance motors have not been considered here. The authors have
devised the book as both an electrical motor noise textbook and a handbook for
electrical machine design engineers, research scientists, and graduate students.
The book can also be helpful for multidisciplinary research teams working on
noise prediction of systems with electrical motors, e.g., electrical vehicles, industrial
electromechanical drives, HVAC (heating, ventilating, air conditioning)
systems, marine propulsion systems, airborne apparatus, elevators, office equipment,
health care equipment, etc. The authors hope that this book will fill the current
gap in modern treatment of the analysis and reduction of noise in polyphase
electric motors.

Contents

1 Generation and Radiation of Noise in Electrical Machines 1
1.1 Vibration, sound, and noise . . . . . . . . . . . . . . . . . . . . . 1
1.2 Sound waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.3 Sources of noise in electrical machines . . . . . . . . . . . . . . . 5
1.3.1 Electromagnetic sources of noise . . . . . . . . . . . . . . 5
1.3.2 Mechanical sources of noise . . . . . . . . . . . . . . . . 7
1.3.3 Aerodynamic noise . . . . . . . . . . . . . . . . . . . . . 7
1.4 Energy conversion process . . . . . . . . . . . . . . . . . . . . . 7
1.5 Noise limits and measurement procedures for electrical
machines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
1.6 Deterministic and statistical methods of noise prediction . . . . . 13
1.7 Economical aspects . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.8 Accuracy of noise prediction . . . . . . . . . . . . . . . . . . . . 18
2 Magnetic Fields and Radial Forces in Polyphase Motors Fed with
Sinusoidal Currents 21
2.1 Construction of induction motors . . . . . . . . . . . . . . . . . . 21
2.2 Construction of permanent magnet synchronous brushless
motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 A.C. stator windings . . . . . . . . . . . . . . . . . . . . . . . . 25
2.4 Stator winding MMF . . . . . . . . . . . . . . . . . . . . . . . . 28
2.4.1 Single-phase stator winding . . . . . . . . . . . . . . . . 28
2.4.2 Three-phase stator winding . . . . . . . . . . . . . . . . . 32
2.4.3 Polyphase stator winding . . . . . . . . . . . . . . . . . . 33
2.5 Rotor magnetic field . . . . . . . . . . . . . . . . . . . . . . . . 36
2.6 Calculation of air gap magnetic field . . . . . . . . . . . . . . . . 37
2.6.1 Effect of slots . . . . . . . . . . . . . . . . . . . . . . . . 37
2.6.2 Effect of eccentricity . . . . . . . . . . . . . . . . . . . . 40
2.6.4 Effect of rotor saliency . . . . . . . . . . . . . . . . . . . 44
2.7 Radial forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.7.1 Production of radial magnetic forces . . . . . . . . . . . . 45
2.7.2 Amplitude of magnetic pressure . . . . . . . . . . . . . . 48
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2.7.3 Deformation of the stator core . . . . . . . . . . . . . . . 49
2.7.4 Frequencies and orders of magnetic pressure . . . . . . . 50
2.7.5 Radial forces in synchronous machines with
slotted stator . . . . . . . . . . . . . . . . . . . . . . . . 51
2.7.6 Frequencies of vibration and noise . . . . . . . . . . . . . 54
2.8 Other sources of electromagnetic vibration and noise . . . . . . . 58
2.8.1 Unbalanced line voltage . . . . . . . . . . . . . . . . . . 58
2.8.2 Magnetostriction . . . . . . . . . . . . . . . . . . . . . . 58
2.8.3 Thermal stress analogy . . . . . . . . . . . . . . . . . . . 62
2.8.4 FEM model . . . . . . . . . . . . . . . . . . . . . . . . . 62
3 Inverter-Fed Motors 65
3.1 Generation of higher time harmonics . . . . . . . . . . . . . . . . 65
3.2 Analysis of radial forces for nonsinusoidal currents . . . . . . . . 66
3.2.1 Stator and rotor magnetic flux density . . . . . . . . . . . 67
3.2.2 Stator harmonics of the same number . . . . . . . . . . . 68
3.2.3 Interaction of stator and rotor harmonics . . . . . . . . . . 69
3.2.4 Rotor harmonics of the same number . . . . . . . . . . . 70
3.2.5 Frequencies and orders of magnetic pressure for
nonsinusoidal currents . . . . . . . . . . . . . . . . . . . 70
3.2.6 Interaction of stator harmonics of different numbers . . . . 70
3.2.7 Interaction of switching frequency and higher time
harmonics . . . . . . . . . . . . . . . . . . . . . . . . . . 71
3.2.8 Interaction of permeance and magnetomotive force
(MMF) harmonics . . . . . . . . . . . . . . . . . . . . . 71
3.2.9 Rectifier harmonics . . . . . . . . . . . . . . . . . . . . . 71
3.3 Higher time harmonic torques in induction machines . . . . . . . 71
3.3.1 Asynchronous torques . . . . . . . . . . . . . . . . . . . 71
3.3.2 Pulsating torques . . . . . . . . . . . . . . . . . . . . . . 72
3.4 Higher time harmonic torques in permanent magnet (PM)
brushless machines . . . . . . . . . . . . . . . . . . . . . . . . . 73
3.5 Influence of the switching frequency of an inverter . . . . . . . . 73
3.6 Noise reduction of inverter-fed motors . . . . . . . . . . . . . . . 75
4 Torque Pulsations 77
4.1 Analytical methods of instantaneous torque calculation . . . . . . 77
4.2 Numerical methods of instantaneous torque calculation . . . . . . 78
4.3 Electromagnetic torque components . . . . . . . . . . . . . . . . 79
4.4 Sources of torque pulsations . . . . . . . . . . . . . . . . . . . . 80
4.5 Higher harmonic torques of induction motors . . . . . . . . . . . 80
4.6 Cogging torque in permanent magnet (PM) brushless motors . . . 81
4.6.1 Air gap magnetic flux density . . . . . . . . . . . . . . . 82
4.6.2 Calculation of cogging torque . . . . . . . . . . . . . . . 84
4.6.3 Simplified cogging torque equation . . . . . . . . . . . . 87
4.6.4 Influence of eccentricity . . . . . . . . . . . . . . . . . . 88
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4.6.5 Calculation and comparison with measurements . . . . . . 92
4.7 Torque ripple due to distortion of EMF and current waveforms
in permanent magnet (PM) brushless motors . . . . . . . . . . . . 94
4.8 Tangential forces vs. radial forces . . . . . . . . . . . . . . . . . 99
4.9 Minimization of torque ripple in PM brushless motors . . . . . . . 102
4.9.1 Slotless windings . . . . . . . . . . . . . . . . . . . . . . 102
4.9.2 Skewing stator slots . . . . . . . . . . . . . . . . . . . . 103
4.9.3 Shaping stator slots . . . . . . . . . . . . . . . . . . . . . 103
4.9.4 Selection of the number of stator slots . . . . . . . . . . . 104
4.9.5 Shaping PMs . . . . . . . . . . . . . . . . . . . . . . . . 104
4.9.6 Skewing PMs . . . . . . . . . . . . . . . . . . . . . . . . 104
4.9.7 Shifting PM segments . . . . . . . . . . . . . . . . . . . 104
4.9.8 Selection of PM width . . . . . . . . . . . . . . . . . . . 104
4.9.9 Magnetization of PMs . . . . . . . . . . . . . . . . . . . 105
4.9.10 Creating magnetic circuit asymmetry . . . . . . . . . . . 105
5 Stator System Vibration Analysis 107
5.1 Forced vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
5.2 Simplified calculation of natural frequencies of the
stator system . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.3 Improved analytical method of calculation of
natural frequencies . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.3.1 Natural frequency of the stator core . . . . . . . . . . . . 112
5.3.2 Natural frequency of a frame with end bells . . . . . . . . 114
5.3.3 Natural frequency of a stator core–frame system . . . . . 115
5.3.4 Effect of the stator winding and teeth . . . . . . . . . . . 116
5.3.5 Analytical calculation of natural frequencies for a stator
core-winding-frame system . . . . . . . . . . . . . . . . 117
5.4 Numerical verification . . . . . . . . . . . . . . . . . . . . . . . 120
5.4.1 FEM modeling . . . . . . . . . . . . . . . . . . . . . . . 120
5.4.2 Comparison of analytical calculations with the FEM . . . 122
6 Acoustic Calculations 127
6.1 Sound radiation efficiency . . . . . . . . . . . . . . . . . . . . . 127
6.2 Plane radiator . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6.2.1 Infinite plates . . . . . . . . . . . . . . . . . . . . . . . . 130
6.2.2 Finite plates in bending motion . . . . . . . . . . . . . . 133
6.3 Infinitely long cylindrical radiator . . . . . . . . . . . . . . . . . 141
6.4 Finite length cylindrical radiator . . . . . . . . . . . . . . . . . . 145
6.4.1 Acoustically thin shells . . . . . . . . . . . . . . . . . . . 147
6.4.2 Acoustically thick shells . . . . . . . . . . . . . . . . . . 149
6.4.3 Modal radiation efficiencies of acoustically
thick shells . . . . . . . . . . . . . . . . . . . . . . . . . 154
6.4.4 Modal averaged radiation efficiency . . . . . . . . . . . . 157
6.4.5 Validity of using an infinite length model . . . . . . . . . 161
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6.4.6 Effects of boundary conditions on the radiation
efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . 164
6.5 Calculations of sound power level . . . . . . . . . . . . . . . . . 166
6.5.1 Sound power radiated from a stator . . . . . . . . . . . . 167
6.5.2 Total sound power of an induction motor . . . . . . . . . 168
6.5.3 Permanent magnet synchronous motors . . . . . . . . . . 171
7 Noise and Vibration of Mechanical and Aerodynamic Origin 175
7.1 Mechanical noise due to shaft and rotor irregularities . . . . . . . 175
7.2 Bearing noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.2.1 Rolling bearings . . . . . . . . . . . . . . . . . . . . . . 176
7.2.2 Sleeve bearings . . . . . . . . . . . . . . . . . . . . . . . 180
7.3 Noise due to toothed gear trains . . . . . . . . . . . . . . . . . . 180
7.4 Aerodynamic noise . . . . . . . . . . . . . . . . . . . . . . . . . 181
7.5 Mechanical noise generated by the load . . . . . . . . . . . . . . 184
8 Acoustic and Vibration Instrumentation 187
8.1 Measuring system and transducers . . . . . . . . . . . . . . . . . 187
8.2 Measurement of sound pressure . . . . . . . . . . . . . . . . . . 189
8.2.1 Choice of microphones . . . . . . . . . . . . . . . . . . . 189
8.2.2 The sound pressure sensor–condenser microphone . . . . 189
8.2.3 Sound level meter . . . . . . . . . . . . . . . . . . . . . . 193
8.2.4 Acoustic calibrator . . . . . . . . . . . . . . . . . . . . . 196
8.2.5 Level recorder . . . . . . . . . . . . . . . . . . . . . . . 197
8.3 Acoustic measurement procedure . . . . . . . . . . . . . . . . . . 197
8.3.1 Effect of the operator on measurement results . . . . . . . 197
8.3.2 Measurement position . . . . . . . . . . . . . . . . . . . 198
8.3.3 Standing waves . . . . . . . . . . . . . . . . . . . . . . . 198
8.3.4 Measurements of ambient sound pressure levels . . . . . . 198
8.3.5 Corrections for background sound during source
measurements . . . . . . . . . . . . . . . . . . . . . . . . 199
8.3.6 Polar plots . . . . . . . . . . . . . . . . . . . . . . . . . 200
8.4 Vibration measurements . . . . . . . . . . . . . . . . . . . . . . 200
8.4.1 Theory of operation of vibration-measuring
transducer . . . . . . . . . . . . . . . . . . . . . . . . . . 201
8.4.2 Characteristics of piezoelectric accelerometers . . . . . . 207
8.4.3 Other vibration transducers . . . . . . . . . . . . . . . . . 211
8.5 Frequency analyzers . . . . . . . . . . . . . . . . . . . . . . . . 213
8.6 Sound power and sound pressure . . . . . . . . . . . . . . . . . . 214
8.7 Indirect methods of sound power measurement . . . . . . . . . . 215
8.7.1 Determination of sound power in an anechoic/
semianechoic room . . . . . . . . . . . . . . . . . . . . . 215
8.7.2 Reverberation room . . . . . . . . . . . . . . . . . . . . . 216
8.7.3 Juxtaposition principle using a reference
sound source . . . . . . . . . . . . . . . . . . . . . . . . 217
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8.8 Direct method of sound power measurement — sound
intensity technique . . . . . . . . . . . . . . . . . . . . . . . . . 217
8.8.1 Historical perspective . . . . . . . . . . . . . . . . . . . . 217
8.8.2 Theoretical background . . . . . . . . . . . . . . . . . . 217
8.8.3 Sound intensity probe . . . . . . . . . . . . . . . . . . . 219
8.8.4 External noise suppression . . . . . . . . . . . . . . . . . 221
8.8.5 Error considerations . . . . . . . . . . . . . . . . . . . . 221
8.8.6 Dynamic capability and pressure-intensity index . . . . . 223
8.9 Standard for testing acoustic performance of rotating
electrical machines . . . . . . . . . . . . . . . . . . . . . . . . . 224
8.9.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . 224
8.9.2 Acoustic tests on an induction motor . . . . . . . . . . . . 226
9 Numerical Analysis 231
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
9.2 FEM model for radial magnetic pressure . . . . . . . . . . . . . . 232
9.2.1 Induction motor . . . . . . . . . . . . . . . . . . . . . . . 233
9.2.2 Permanent magnet synchronous motor . . . . . . . . . . . 237
9.3 FEM for structural modeling . . . . . . . . . . . . . . . . . . . . 239
9.4 BEM for acoustic radiation . . . . . . . . . . . . . . . . . . . . . 243
9.4.1 Governing equation and boundary conditions . . . . . . . 243
9.4.2 FEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
9.4.3 BEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
9.4.4 Radiating sphere . . . . . . . . . . . . . . . . . . . . . . 248
9.4.5 Application to the prediction of radiated acoustic power
from an inverter-fed induction motor . . . . . . . . . . . . 250
9.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
10 Statistical Energy Analysis 257
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
10.2 Power flow between linearly coupled oscillators . . . . . . . . . . 259
10.2.1 Two coupled oscillators . . . . . . . . . . . . . . . . . . 259
10.2.2 Three series coupled oscillators . . . . . . . . . . . . . . 261
10.2.3 Energy exchange between groups of oscillators . . . . . . 264
10.3 Coupled multimodal systems . . . . . . . . . . . . . . . . . . . . 267
10.3.1 General SEA equations . . . . . . . . . . . . . . . . . . . 267
10.3.2 SEA model establishment . . . . . . . . . . . . . . . . . 269
10.3.3 SEA parameters . . . . . . . . . . . . . . . . . . . . . . 272
10.3.4 Limitations of SEA . . . . . . . . . . . . . . . . . . . . . 278
10.4 Experimental SEA . . . . . . . . . . . . . . . . . . . . . . . . . 281
10.4.1 General theory . . . . . . . . . . . . . . . . . . . . . . . 282
10.4.2 Recent developments . . . . . . . . . . . . . . . . . . . . 285
10.5 Application to electrical motors . . . . . . . . . . . . . . . . . . . 290
10.5.1 Subsystems of a motor structure . . . . . . . . . . . . . . 291
10.5.2 Internal and coupling loss factors . . . . . . . . . . . . . 292
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10.5.3 Input power to the stator . . . . . . . . . . . . . . . . . . 293
10.5.4 Sound power radiated from the motor structure . . . . . . 295
11 Noise Control 301
11.1 Mounting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
11.1.1 Foundation . . . . . . . . . . . . . . . . . . . . . . . . . 301
11.1.2 Principles of vibration and shock isolation . . . . . . . . . 303
11.1.3 Vibration limits . . . . . . . . . . . . . . . . . . . . . . . 306
11.1.4 Shaft alignment . . . . . . . . . . . . . . . . . . . . . . . 306
11.2 Standard methods of noise reduction . . . . . . . . . . . . . . . . 307
11.3 Active noise and vibration control . . . . . . . . . . . . . . . . . 311
11.3.1 Principles of active noise control . . . . . . . . . . . . . . 311
11.3.2 Induction motor acoustic noise reduction . . . . . . . . . 313
11.3.3 Active vibration isolation . . . . . . . . . . . . . . . . . . 315
AppendixA Basics of Acoustics 319
A.1 Sound field variables and wave equations . . . . . . . . . . . . . 319
A.2 Sound radiation from a point source . . . . . . . . . . . . . . . . 321
A.3 Decibel levels and their calculations . . . . . . . . . . . . . . . . 323
A.4 Spectrum analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 325
AppendixB Permeance of Nonuniform Air Gap 327
B.1 Permeance calculation . . . . . . . . . . . . . . . . . . . . . . . 327
B.2 Eccentricity effect . . . . . . . . . . . . . . . . . . . . . . . . . . 328
AppendixC Magnetic Saturation 333
AppendixD Basics of Vibration 337
D.1 A mass–spring–damper oscillator . . . . . . . . . . . . . . . . . . 337
D.2 Lumped parameter systems . . . . . . . . . . . . . . . . . . . . . 339
D.3 Continuous systems . . . . . . . . . . . . . . . . . . . . . . . . . 342
Symbols and Abbreviations 347