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Improved Performance Characteristics of InductionMachines with
Non-Skewed Asymmetrical Rotor Slots
Submitted to the School of Electrical Engineering in partial
fulfillment of therequirements for the degree of Licentiate
RATHNA KUMAR SASTRY CHITROJU
Licentiate ThesisElectrical Machines and Power Electronics
School of Electrical EngineeringRoyal Institute of Technology
(KTH)
Stockholm, Sweden 2009
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TRITA-EE 2009:043ISSN 1653-5146ISBN 978-91-7415-453-5
Electrical Machines and Power ElectronicsKTH School of
Electrical Engineering
SE-100 44 StockholmSWEDEN
Akademisk avhandling som med tillstånd av Kungl Tekniska
högskolan framläg-ges till offentlig granskning för avläggande av
teknologie licentiatexamen i elek-trotekniska system tisdag den 24
November 2009 klockan 10.00 i D2 (Entreplan),Lindstedtsv 5, Kungl
Tekniska högskolan, Valhallavägen 79, Stockholm.
© Rathna Kumar Sastry Chitroju, October 2009
Tryck: Universitetsservice US AB
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iii
Abstract
Induction machines convert more than 55% of electrical energy
into var-ious other forms in industrial and domestic environments.
Improved perfor-mance, especially by reduction of losses in
induction machines hence can sig-nificantly reduce consumption of
electricity. Many design and control methodsare adopted to make
induction machines work more efficiently, however cer-tain design
compromises are inevitable, such as skewing the rotor to improvethe
magnetic noise and torque characteristics increase the cross
current lossesconsiderably in a cage rotor, degrading the
efficiency of the motor. Cross-current losses are the dominating
stray losses which are dependent on severalfactors among them are
percentage skew and the contact resistance betweenthe rotor bars
and laminations. It is shown in this thesis that implementing
adesign change which has non-skewed asymmetrical distribution of
rotor slotscan serve the same purpose as skewing i.e., reduction of
the magnetic noise,thereby avoiding the negative effects of skewing
the rotor slots especially byreducing the cross-current losses.
Two design methodologies to introduce asymmetry in rotor slots
are pro-posed and the key performance characteristics like torque
ripple, radial airgap forces are computed both numerically and
analytically. Radial forces ob-tained from the finite element
method are coupled to the analytical tool forcalculating the
magnetic noise. A spectral method to calculate and separatethe
radial forces into vibration modes and their respective frequencies
is pro-posed and validated for a standard 4-pole induction motor.
The influence ofrotor slot number, eccentricity and skew on radial
forces and magnetic noiseare studied using finite element method in
order to understand the vibrationaland acoustic behavior of the
machine, especially for identifying their sources.The validated
methods on standard motors are applied for investigating
theasymmetrical rotor slot machines.
Radial air gap forces and magnetic noise spectra are computed
for thenovel dual and sinusoidal asymmetrical rotors and compared
with the stan-dard symmetrical rotor. The results obtained showed
reduced radial forcesand magnetic noise in asymmetrical rotors,
both for the eccentric and non-eccentric cases. Based on the
results obtained some guide lines for designingasymmetrical rotor
slots are established. Magnitudes of the harmful modesof vibration
observed in the eccentric rotors, which usually occur in
reality,are considerably reduced in asymmetrical rotors showing
lower sound inten-sity levels produced by asymmetrical rotors. The
noise level from mode-2vibration in a 4-pole standard 15 kW motor
running with 25% static eccen-tricity is decreased by about 6 dB,
compared to the standard rotors. Henceimproved performance can be
achieved by removing skew which reduces crosscurrent losses and by
employing asymmetrical rotor slots same noise level canbe
maintained or can be even lowered.
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iv
Keywords: Skewing, cross current losses, radial magnetic forces,
mag-netic noise, asymmetrical rotor slots, eccentricity, stator
vibration, two-dimensionalfast fourier transform, finite element
method.
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Acknowledgments
I would like to express my deep gratitude to my supervisor Prof.
Chandur Sadaranganifor his remarkable, positive and composed
supervision. I would like to thank Dr.Heinz Lendenmann, group
manager for Machines at ABB Corporate Research, forhis
encouragement and support which was one of the reasons that
initiated my PhDstudies. I want to thank Yujing Liu, senior
principal scientist at ABB CRC for hisinspiring suggestions both at
work and social meetings; Robert. J Andersson for hishelp in
providing machine input data from the Oskar program; Tech. Lic.
Chris-ter Danielsson, my previous colleague for his support and
encouragement. Specialthanks to versatile Tech. Lic. Mats Leksell
and Dr. Juliette Soulard at our depart-ment for their admirable
hard work and inspiring academic comments on variousaspects. I
would like to thank Dr. Stefan Östlund for proof reading this
report andalso for giving me the opportunity to assist him with
course work at the department.
Special thanks to Ulf Carlsson fromMWL laboratory, KTH and
Bhavani Shankarfrom Signals and Systems department, KTH for their
suggestions during some keystages of the project.
I would like to greatly acknowledge Elforsk and Elektra
foundations for theirfinancial support and their inputs during the
yearly meetings.
Special thanks to my intimate EME colleagues; Dmitry
Svechkarenko for hishelp with Latex, many linguistic aspects and
obviously for his invitations to manysocial events; Alexander
Stening for his help with the Swedish language (esp. slang)and
culture; Henrik Grop for his ultimate Swedish humor. I thank my
office roommate Antonious Antonopoulos for his all time
co-operation and support. I alsotake this opportunity to
acknowledge all my formal and present colleagues at EMEfor
contributing their efforts to make the department proud and for
providing apeaceful working environment.
My sweet family especially my grand parents and parents are
acknowledged fortheir endless love, support and also for giving me
this opportunity to study abroad.Although Chandana, my fiance is
new in my life she is greatly acknowledged forher moral support and
understanding nature. I would like to acknowledge Kumar
v
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vi
and Rao families for giving me joyful social life and good
company in Sweden.
I thank all my well wishers namely Vijaya Chandramauli, Waqas
Arshad, KailashSrivastava, Kamesh Ganti, Kashif Khan, John Rödin,
François Besnard, SumanVodnala, priests Roger Stenzelius and
Daniella Åslund for their all time love andsupport. Last but not
least our economists Eva & Emma Petterssons,
computeradministrator Peter Lönn, lab technician Olle Bränvall,
Administrator Brigitt Hög-berg are greatly acknowledged for their
continuous help and support. I should alsomention and thank Birka
for impressing me with her funny deeds and for joiningme to those
nice walks in the woods behind the KTH campus.
Rathna ChitrojuStockholm, October 2009.
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List of Figures
1.1 Electrical energy consumption chart . . . . . . . . . . . .
. . . . . . . . 21.2 Motor distribution in Sweden . . . . . . . . .
. . . . . . . . . . . . . . . 31.3 Illustration of cross currents
in skewed and non-skewed rotor . . . . . . 51.4 Composition of
stray load losses (0.2-37 kW size motor) [1] . . . . . . . 61.5
Cross current and rotor copper losses for various skew . . . . . .
. . . . 6
2.1 Flowchart for analytical calculation procedure [2] . . . . .
. . . . . . . . 192.2 Schematic for the harmonic distribution in
induction motor . . . . . . . 202.3 Formula table for various
harmonics in induction motor . . . . . . . . . 212.4 Stator
vibration modes . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 222.5 History of noise minimization methods in electrical
motors . . . . . . . . 252.6 Schematic for noise generation in
electrical motors (IM) . . . . . . . . . 26
3.1 Simple spectral analysis on the results obtained from FEM .
. . . . . . 313.2 Illustration of 2D spectral analysis . . . . . .
. . . . . . . . . . . . . . . 333.3 Mode number and frequency
separation of radial forces . . . . . . . . . 353.4 Demonstrating
the accuracy of the spectral analysis . . . . . . . . . . . 363.5
Stator schematic showing the nodal points . . . . . . . . . . . . .
. . . . 373.6 Identification table for different modes of vibration
. . . . . . . . . . . . 373.7 Phase angle at circumferential
positions 1, 3, 5 and 7 i.e., for mode-2
identification . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 383.8 Simulation tool specification . . . . . . . .
. . . . . . . . . . . . . . . . . 38
4.1 2D FEM models for various rotor slot number Qr . . . . . . .
. . . . . . 424.2 Characteristics for various rotor slot numbers .
. . . . . . . . . . . . . . 434.3 Flux density harmonics for
various rotor slot numbers . . . . . . . . . . 444.4 Torque ripple
harmonics for various rotor slot numbers . . . . . . . . . . 464.5
Time variation of the spatial force harmonics . . . . . . . . . . .
. . . . 474.6 Frequency spectra for the radial air gap forces for
various rotor slot number 484.7 Radial air gap force spectra for
various rotor slots computed using
ASLERM . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 494.8 Spatial harmonic spectra for various eccentric
rotors . . . . . . . . . . . 554.9 Spatial harmonic variation with
time . . . . . . . . . . . . . . . . . . . . 56
vii
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viii List of Figures
4.10 Comparison of radial air gap forces computed with varying
eccentricities 574.11 FEM sliced models for non-skewed and skewed
rotors . . . . . . . . . . 604.12 Flux density in the air gap (T) .
. . . . . . . . . . . . . . . . . . . . . . 604.13 Spatial air gap
flux density spectra for the sliced models . . . . . . . . . 614.14
Torque ripple comparison for skewed and non-skewed rotors . . . . .
. . 614.15 Analytical results for skewed and non-skewed rotor
models . . . . . . . 624.16 Sound intensity for varying static
eccentricity . . . . . . . . . . . . . . . 64
5.1 2D FEM models for dual rotors . . . . . . . . . . . . . . .
. . . . . . . . 685.2 Rotor slot types I & II . . . . . . . . .
. . . . . . . . . . . . . . . . . . . 695.3 Combination of rotor(1)
in Type I and rotor(1) in Type II . . . . . . . . 695.4 Rotor slot
dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
715.5 2D FEM models for sinusoidal rotors . . . . . . . . . . . . .
. . . . . . . 71
6.1 Instantaneous torque for the dual rotors . . . . . . . . . .
. . . . . . . . 746.2 Space harmonics for dual slot rotors at one
instant of time . . . . . . . . 766.3 Torque ripple comparison for
the dual rotors . . . . . . . . . . . . . . . 786.4 Spatial spectra
variation with time for the dual rotors . . . . . . . . . . 796.5
Radial force spectrum for dual rotors . . . . . . . . . . . . . . .
. . . . . 806.6 Sound intensity for dual rotors . . . . . . . . . .
. . . . . . . . . . . . . 816.7 Electromagnetic torque for
sinusoidal rotors . . . . . . . . . . . . . . . . 826.8 Space
spectrum at one instant of time for sinusoidal rotors . . . . . . .
846.9 Spatial spectrum variation with time for sinusoidal rotors .
. . . . . . . 856.10 Radial force spectrum for sinusoidal rotors .
. . . . . . . . . . . . . . . . 886.11 Sound intensity for
sinusoidal rotors . . . . . . . . . . . . . . . . . . . . 896.12
Rotor designs with sinusoidal modulated slots . . . . . . . . . . .
. . . . 906.13 Characteristics for 36-28 sinusoidal asymmetrical
rotors . . . . . . . . . 916.14 Asymmetrical versus standard rotors
– Radial force spectra . . . . . . . 936.15 Sound intensity for
asymmetrical rotors . . . . . . . . . . . . . . . . . . 95
7.1 Induction motor efficiency categorization tables . . . . . .
. . . . . . . . 107
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Contents
List of Figures vii
Contents ix
1 Introduction 11.1 Significance of improved efficiency in
induction motors . . . . . . . . 11.2 Background of the thesis . .
. . . . . . . . . . . . . . . . . . . . . . . 2
1.2.1 Skewing in sinusoidal fed induction motors . . . . . . . .
. . 41.2.2 Additional losses and magnetic noise in inverter fed
induction
motors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . 51.3 Motivation and goal of the thesis . . . . . . . . . . . .
. . . . . . . . 7
1.3.1 Proposed design solution – Non-skewed asymmetrical rotor .
81.3.2 Challenges in the new design solution . . . . . . . . . . .
. . 8
1.4 Literature survey . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . 91.4.1 Concept of asymmetrical rotor slots . . . .
. . . . . . . . . . 91.4.2 Calculation of magnetic forces and noise
– Methods other
than the classical analytical approach . . . . . . . . . . . . .
101.5 Scientific contribution . . . . . . . . . . . . . . . . . . .
. . . . . . . 101.6 Organization of the thesis . . . . . . . . . .
. . . . . . . . . . . . . . 111.7 Publications . . . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . 12
2 Radial magnetic forces and magnetic noise in induction motors–
Methods of analysis 152.1 Analytical approach for calculating
radial force waves . . . . . . . . 17
2.1.1 Classical theory for calculating radial magnetic forces .
. . . 182.2 Rotor bar induction currents approach . . . . . . . . .
. . . . . . . . 232.3 FEM analysis with spectral methods . . . . .
. . . . . . . . . . . . . 232.4 Magnetic equivalent circuit
approach . . . . . . . . . . . . . . . . . . 242.5 Combined
classical and finite element method approach . . . . . . . 242.6
Magnetic noise production and its modeling for induction motors . .
25
2.6.1 An overview of magnetic noise in induction motors . . . .
. . 25
3 Finite element method approach for radial forces computation
29
ix
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x CONTENTS
3.1 Spectral analysis for radial forces computation . . . . . .
. . . . . . 293.1.1 1D Spectral analysis . . . . . . . . . . . . .
. . . . . . . . . . 293.1.2 2D Spectral analysis . . . . . . . . .
. . . . . . . . . . . . . . 323.1.3 Spectral accuracy . . . . . . .
. . . . . . . . . . . . . . . . . . 353.1.4 Phase difference method
. . . . . . . . . . . . . . . . . . . . . 36
3.2 Simulation tool specification . . . . . . . . . . . . . . .
. . . . . . . . 39
4 Calculations and validations on standard 4-pole induction
motors 414.1 Influence of rotor slot number on the radial magnetic
force spectrum 414.2 Effect of eccentricity on radial magnetic
force spectrum . . . . . . . 544.3 Influence of skew on radial
magnetic force spectrum - Using skewed
FEM model . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 594.4 Noise computation and comparisons . . . . . . . . .
. . . . . . . . . 634.5 Validation results and conclusions . . . .
. . . . . . . . . . . . . . . . 63
5 Methods to introduce asymmetry in rotors 675.1 Asymmetry in
rotating structures . . . . . . . . . . . . . . . . . . . . 675.2
Dual slot rotors . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . 67
5.2.1 Slot combinations where rotor and stator slots are not
equal . 685.2.2 Slot combinations where rotor and stator slots are
equal . . . 68
5.3 Progressive sinusoidal rotors . . . . . . . . . . . . . . .
. . . . . . . . 705.4 Combined dual and progressive sinusoidal
rotors . . . . . . . . . . . 715.5 Manufacturing of asymmetrical
rotor slots . . . . . . . . . . . . . . . 71
6 Asymmetrical rotor slots versus standard rotors – Results
anddiscussion 736.1 Comparison of asymmetrical slot rotors with the
standard symmetric
rotors . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 736.2 Comparison of eccentric asymmetrical slot rotors
with eccentric stan-
dard rotors . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 926.3 Some guidelines for the design of novel
asymmetrical rotors for in-
duction rotors . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . 94
7 Conclusions and future work 977.1 Conclusions . . . . . . . .
. . . . . . . . . . . . . . . . . . . . . . . . 977.2 Future work .
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
98
Bibliography 99
Glossary 103
List of Symbols 105
Appendix 1 107
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Chapter 1
Introduction
Induction motors are one of the widely used motors due to their
robust constructionand the ability to control their speed – thanks
to the latest developments in powerelectronics which made this
control possible. Applications of induction motorsrange from many
industry sectors, including power industry, food & beverage,
metalprocessing, textiles and utilities to domestic appliances.
1.1 Significance of improved efficiency in induction motors
Improved performance such as reduction of losses (= higher
efficiency), noise andvibrations are major goals for the motor
manufacturers. Motor manufacturers to-day, have shifted their focus
away from efficiency class 2 (EFF21) motors to higherefficiency
motors in order to significantly reduce the consumption of
electricity.IMS2 research shows that the movement towards higher
efficiency motors will af-fect the worldwide ac induction motor
market over the next few years. Today’selectricity consumption of
induction motors accounts for approximately 55–65% ofthe industrial
electricity consumption, see Fig. 1.1. Hence even a smallest
improve-ment in motor efficiency (or reduction in losses), can
significantly reduce the energyconsumed globally or in a single
installation. The smaller motors, which are of ma-jor interest in
this thesis, generally have efficiency values around 70–90%, and
lossreductions required to achieve efficiency class 1 (EFF1) can be
up to 40%. How-ever, induction motors are already very efficient
and it is a very mature technology.Though the scope of improving
the efficiency of these motors seems to be bleak, theemerging
calculation methods, computational capacity, manufacturing
techniques,material advancements have increased the scope for
improvement in these motors.Thus a constant effort is continuously
being made for the performance improvementof induction motors. Not
only on the motor side but there is an equal opportunityfor energy
savings by looking at the whole system using variable speed drives.
Its
1see Appendix 12Information Management Systems
1
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2 CHAPTER 1. INTRODUCTION
Process heating
6-10 %
Electrochem
2-4 %
Household
4-6 %
Line losses
6-10 %
Others
2-5 %
Lighting
8-10 %
Other motors
2-5%
Arc furnace
12-16 %
Motor systems
55-65 %
Air compressors
9%
Material
processing
12%
Material
handling
7%
Fans
8%
Pumps
14%
Refrigiration
4%
Figure 1.1: Electrical energy consumption chart
worthwhile to note that, three-quarters of the world’s
industrial ac induction mo-tors being sold each year do not meet
the efficiency standards. Replacing motorswith a higher efficiency
(NEMA premium or EFF1) will save approximately 3 to 8percent in
electricity per year and this replacement can have a dramatical
impacton the induction motor market over the next 3 years [3].
Electric motors account for about 65 percent of the total
electricity consumptionin the industrial sector and 38 percent in
the service sector of Sweden. The so-calledasynchronous motor, is
the most common motor type and accounts for 90 percentof
electricity consumption of all electric motors in the power range
of 0.75–375 kW.These electric motors are mainly used within
industry in fans, pumps, compressorsand for air conditioning in
apartments [4]. The motor distribution in Sweden is asshown in Fig.
1.2.
1.2 Background of the thesis
The focus of this work, as the title suggests, is mainly on
improving the performanceof induction motors, in particular the
reduction of losses. Such improvements canbe achieved either by
modifying the control or by introducing some changes in themachine
design. The initial focus has been on machine design improvements
wheretraditionally compromises have to be made e.g., introducing
rotor skew reduces thenoise and vibration level but increases the
stray losses due to cross-currents be-tween the rotor conductors
and laminations, degrading the efficiency of the motor.Moreover,
the emf induced in a skewed rotor is comparatively lower than in a
non-skewed rotor resulting in a reduced output torque. A simple
solution to eliminatethese stray losses is to keep the rotor slots
straight and investigate new methods totackle the increased noise
level.
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1.2. BACKGROUND OF THE THESIS 3
Others
0.8%
High efficiency
3-phase motors
2.2 %
DC motors
2.1 %1-phase induction
motors
2.3%
3-phase induction
motors 92.6%
3-phase induction motors
1-phase induction motors
Others
High efficiency 3-phase motors
DC motors
Figure 1.2: Motor distribution in Sweden
Major contribution of magnetic noise in induction motors are
from the radialforces produced in the air gap [5]. These forces can
be characterized as rotatingwaves with different mode3 numbers
having different distribution around the airgap circumference.
These waves propagate at various frequencies around the airgap,
acting both on stator and the rotor [7]. Radial forces which lead
to deforma-tions such as the 4-node (mode-2) vibrations in 4-pole
motors produce significantmagnetic noise [8]. Existing techniques
to reduce magnetic forces and noise are touse an optimum stator and
rotor slot combinations [9] and by skewing the rotorslots [10].
A design modification, which allows the rotor slots to be
non-skewed i.e., keepingthem straight to avoid cross current losses
and the use of asymmetrical rotor slotsto keep magnetic noise at a
desirable level will be the main topic of this thesis. Adesign
methodology to introduce various asymmetries in the rotor slots and
meth-ods to analyze these motors will be presented in the thesis.
The focus will be on4-pole induction motors for which some novel
rotor configurations with asymmetri-cal slots will be investigated.
The analytical source code, ASLERM4 has been usedfor the radial
magnetic force and noise calculations in the standard motors.
Finiteelement method (FEM) analysis is adopted to obtain accurate
results for the newrotor designs. Prior to this the FEM results are
validated against the analyticalresults on the standard motors.
3The term ‘mode’ denotes the number of complete cycles of a
physical quantity along theperiphery of the stator [6]
4ASLERM – Analytical program for calculating radial forces and
magnetic noise in inductionmotors (coded in FORTRAN language)
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4 CHAPTER 1. INTRODUCTION
The work presented in this thesis focuses on investigating the
effects of intro-ducing asymmetrical rotor slots on the performance
of induction motors. Spatialflux density spectra, magnetic forces,
vibration modes and magnetic noise of induc-tion motors are studied
through analytical and FEM simulations. A novel methodfor magnetic
force calculations from FEM results data is introduced to study
theasymmetrical rotor slot designs.
The work presented in this thesis is purely theoretical. For a
complete pictureof the novel concepts experimental investigation
will be required to verify the the-oretical investigations in this
thesis. Furthermore the starting performance mustbe investigated
both theoretically and experimentally. The main challenge is
todesign a rotor configuration having minimum rotor bar losses with
lower noise leveland which also has good starting characteristics.
Ultimately, the disadvantages ofskewing the rotor should be
eliminated and thus an improved rotor design can beobtained. This
is proposed as future work.
1.2.1 Skewing in sinusoidal fed induction motorsSpace and time
harmonic fields associated with the parasitic effects in induction
mo-tors are produced due to the slotting, phase unbalance, rotor
eccentricity, magneticsaturation, and magnetostrictive expansion of
the core laminations. The contribu-tion of the above sources can be
seen in the form of radial air gap forces in themotor. These
harmonic fields induce emf, which due to the short circuited
endrings circulate currents in the rotor windings (bars). These
harmonic currents inthe rotor interact with the harmonic fields
from stator to develop harmonic torques,vibrations and magnetic
noise. Skewing is the technique used since a few decadesin
induction motors where the rotor or the stator slots are twisted to
get a moreuniform torque, less noise, and better voltage waveform.
The most common andeffective method is to skew the rotor slots by
one stator slot pitch [10].
By introducing skew, the voltage induced by the flux is
displaced longitudinallyalong the rotor bar resulting in increased
reactance of the rotor bar [5]. The ax-ial variation of the flux is
subsequently increased and hence additional losses
areconsequential. The change in flux along the rotor axis induces a
voltage and hencecurrents will flow through the bars via the
laminations. This basic phenomena ofcross currents in the rotor
bars can be explained using the diagram shown in Fig.1.3, where φ1
and φ2 are the fluxes enclosed, say in loop 1 and 2, respectively.
Therotor is skewed by one stator slot pitch denoted by τs. It can
be seen from Fig.1.3 that ideally, φ1 6= φ2 and φ1 = φ2, when the
rotor is skewed and non-skewed,respectively. Hence, due to the
axial variation of the slot harmonic flux enclosedby the rotor
bars, high frequency currents will flow in the rotor bars and the
lami-nations. These currents are called inter-bar or cross
currents.
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1.2. BACKGROUND OF THE THESIS 5
Stator Stator
Skewed rotor Non-skewed rotor
s
2
1 1
2
s
Figure 1.3: Illustration of cross currents in skewed and
non-skewed rotor
In induction motors the cross current losses are considered
under stray losses,which by definition refers to the additional
losses that occur in the machine overthe normal losses that are
considered in usual induction motor performance calcu-lations. The
composition of stray load losses for low voltage induction motors
isshown in Fig. 1.4. As seen, the cross current losses can be a
significant part ofthe overall stray load losses if the rotor is
skewed. The cross current losses varydepending on the amount of
skew, inter-bar resistance (cross resistance) and thestator/rotor
slot ratio [1].
Some pre-calculations were carried out using an analytical
program called EDDY5on a 4-pole, 36/28 slot combination, 15 kW
motor. The cross current losses for var-ious skew is shown in Fig.
1.5, where ‘ρ’ is the contact resistivity between the rotorbar and
lamination. It is observed that stray load losses decrease to zero
when therotor skew is removed.
In general the positive and negative effects of removing skew
can be listed asshown in Table 1.1
1.2.2 Additional losses and magnetic noise in inverter
fedinduction motors
The output voltage and output current of an inverter are
non-sinusoidal and containhigher time harmonics which are generated
due to switching of solid state devices.
5EDDY – An in-house analytical program for the calculation of
losses in induction motors(coded in FORTRAN language)
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6 CHAPTER 1. INTRODUCTION
Pulsation losses
17%
Surface losses
38%
Circulating
losses
11%
Leakage flux
losses
3%
Cross current
losses
31%
Figure 1.4: Composition of stray load losses (0.2-37 kW size
motor) [1]
0
50
100
150
200
250
300
350
400
450
-5 -4 -3 -2log( ) [ -m]
Pow
er l
oss
es [
W]
25% skew
50% skew
75% skew
100% skew
150% skew
Figure 1.5: Cross current and rotor copper losses for various
skew
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1.3. MOTIVATION AND GOAL OF THE THESIS 7
Table 1.1: Positive and negative effects of removing skew
Positive effects
Reduced cross currents.Simplified construction of the
rotor.Improved casting of the rotor.Improved power factor and thus
reduced copper losses.Reduced fan size due to the reduction in
losses.
Negative effectsIncreased magnetic noise level.Higher amplitude
of harmonic magnetic fields.Increased synchronous torques during
start-up.
The losses in electrical machines with non-sinusoidal current in
the stator windingincrease as compared to an equivalent machine
with sinusoidal stator current. Themagnetic noise of an induction
motor fed from a pulse width modulation (PWM) in-verter with
switching frequencies upto 7 kHz can increase by about 7 to 15
dB(A).For higher switching frequencies between 7–16 kHz the
increase in the magneticnoise is lower, usually 2 to 7 dB(A) [11].
The radial magnetic force spectrum for aninverter fed induction
motor is richer and the chance of matching the exciting
fre-quencies with the natural frequencies of the stator system is
increased. It is a wellknown fact that a right choice of slot
combination is the key for a quiet operationof a sinusoidal fed
motor, however, it is proven that noise produced from a PWM
orinverter fed motors are invariant of the slot combinations [1].
Conventional designtechniques such as skew does not work
effectively for inverter fed machines due tothe noise produced from
the slot ripple at a large range of frequencies.
The technique proposed in this thesis i.e., the modulation of
rotor slots providesthe possibility to eliminate or cancel the
harmful force and noise components gen-erated from either the motor
or the converter as long as an appropriate modulationtechnique is
employed. The slot ripple produced by equi-slot pitched slots can
beremoved by employing irregular slot pitch.
It is worthwhile to note that magnetostrictive forces (not
considered in thiswork) will have a significant effect on the
magnetic noise in inverter fed inductionmotors [1].
1.3 Motivation and goal of the thesis
As described in Section 1.2.1, the increase of stray losses in a
skewed rotor is mainlydue to the cross currents flowing from the
conductor into the laminations. Theselosses are strongly related to
the contact resistance between the rotor bar and thelaminations
which depends on several factors including the casting process.
Themain research question investigated in this work is to look for
a substantial reductionof cross-current losses keeping the noise
level to a possible minimum. This can be
-
8 CHAPTER 1. INTRODUCTION
achieved by simply removing skew and by employing various slot
combinations.However, it is important to note that the skew removal
causes additional parasiticeffects such as an increase in noise
level and undesirable dips in the starting torquecharacteristics
[12]. To minimize these effects the rotor design can be modifiedand
this could be achieved by introducing asymmetry in rotor slots
[13], [14], [15].The presented work deals with the design
modifications in the rotor slots (pitch)to minimize the magnetic
noise (radial forces), where both analytical and finiteelement
calculation methods are employed to evaluate the motor
performance.
1.3.1 Proposed design solution – Non-skewed
asymmetricalrotor
The increased magnetic noise (radial forces) due to the
parasitic effects caused byeliminating skew can be minimized by
adopting the following design changes in therotor:
• choosing rotor slot number, Qr with special regard to magnetic
noise
• introducing asymmetrical rotor slot configurations
• increased air gap of the motor
• optimizing the stator core (resonance frequency etc)
Research on choosing the right slot combinations for induction
motors has beenreported widely since the late 40’s [9]. The
existing slot combinations are theend-result of such optimization.
The option of increased air gap is obviously nota desirable design
change, the reason being that smaller the air gap the largeris the
torque produced by any motor. Asymmetrical rotor is the best
possibledesign change of interest which needs rigorous
investigation via simulations andexperiments. Optimizing the stator
core is not the solution for eliminating the noiseat the source
level, but is a method to withstand the effect after its
production,which is not an effective method. Thus design of
non-skewed asymmetric inductionrotors is proposed and investigated
in this thesis.
1.3.2 Challenges in the new design solutionThe following
challenges have to be met for the above mentioned design
solution:
1. To design a high performance non-skewed rotor, either for
symmetrical ornon-symmetrical rotor, a detailed information about
the dominating noise(radial forces) component characteristics
associated with various rotor de-signs, is necessary. A realistic
and accurate method has to be developed forthis purpose.
-
1.4. LITERATURE SURVEY 9
2. It is important to identify the various harmful vibration
modes of the sta-tor yoke which can lead to mechanical
deformations. A method to calculateradial forces and noise level
using the data obtained from FEM calculationsis needed. FEM has
been chosen as it has an added advantage to the accu-racy that
permeance variations can be considered by just modifying the
rotorgeometry.
3. Asymmetry can be introduced in many ways and a quantified
method tointroduce asymmetry has to be developed.
4. As mentioned in Section 1.2, the slot combinations that do
not lead to 4-node radial forces (which are considered to be the
most harmful for the sizeof motors considered here) in the air gap
however can lead to synchronoustorques during the start. Thus
starting properties have to be thoroughlyinvestigated.
5. Ultimately, in order to cut down the manufacturing lines a
common rotordesign for 4-6-8 pole motors could be designed.
It is important for the reader to note that the work presented
here covers onlythe first three challenges.
1.4 Literature survey
In this section a brief description of the previous work done
related to the modulatedor asymmetrical slots and various methods
to analyze the radial magnetic forces ininduction motors are
presented.
1.4.1 Concept of asymmetrical rotor slotsVery few findings were
found in the literature on asymmetrical slot configurations,some of
the important findings are summarized below.
Some guidelines for reducing magnetic noise using asymmetrical
rotor slots havebeen reported in [14], where a method to analyze
the torque and voltage equationsof an induction motor with an
unequal slot pitch cage rotor is developed and theelectromagnetic
force wave characteristics with various non-uniform slot pitches
areclarified. One of the important conclusion from this work is
that the unbalance inthe current and torque harmonics are
increased, with the increase in the asymmetrybut the fundamental
component remains almost unaffected [14].
Toshiba corporation proposed the inequality slot pitched rotor
to make the en-ergy dissipation distributed for a wide frequency
range. This idea was proved bypower spectrum and auto-correlation
functions of information theory, and tested
-
10 CHAPTER 1. INTRODUCTION
on real production motors [16]. It is suggested by the research
work done at Kan-gawa Institute of Technology that the effective or
the only possible way to reducemagnetic noise in induction motors
driven by inverter is to employ non-uniformpitched slots [17].
Effects of asymmetrical rotor slot pitch is studied and reportedby
A. Tenhunen et.al, the report shows improved force spectrum when
the rotorslotting is modified by introducing asymmetrical slot
pitch [13].
1.4.2 Calculation of magnetic forces and noise – Methods
otherthan the classical analytical approach
To evaluate spatial and time harmonics a two-dimensional
approach is needed tostudy the radial magnetic forces. It is
evident that magnetic forces form the keycharacteristic in
predicting the magnetic noise in induction motors. The relevantwork
contributed to such 2D approaches in the past is briefly described
in thissection.
V. Ostovic and G. Boman proposed a method for computation of
space andtime harmonics of radial air gap force in induction
motors. The radial air gap forcein two dimensions (space and time)
is computed by using the Magnetic EquivalentCircuit (MEC) method.
The results obtained show clearly the influence of rotorskewing,
rotor static and dynamic eccentricity on the force spectrum [8].
Similarly,a quantitative method to analyze the time and space
harmonics in the magneticfield distribution, using FEM with the
time stepping method considering the rotormovement is reported by
H. Mikami et.al. This method was claimed to be time sav-ing and the
results obtained match well with the measurements [18]. L.
Vandeveldehas proposed a method using magnetic equivalent circuit
(MEC). In this methodthe various effects which determine the modal
contents of radial force are takeninto account: the spectrum of
applied voltage (for inverter supply), the windings,slotting,
eccentricity and saturation of iron. Magnetic equivalent circuit
developedin frequency-order domain which is coupled to an
electrical circuit, is used for theanalysis. These calculations
were experimentally verified [19]. Pedro Vincent .et.alhave
published a method to compute electromagnetic force distribution
along theair gap using the results obtained from FEM. The computed
radial air gap forcesis further analyzed using double Fourier
series one in space and the other in time.This method consists of
FEM simulations of the magnetic field in an electrical ma-chine and
characterization of the forces into rotating waves of different
wavelengths(modes) and frequencies [20].
1.5 Scientific contribution
A list of contributions from this work is presented in this
section.
• Spectral analysis is employed for extracting the radial air
gap force distri-bution in terms of frequency and vibration mode
number. Two dimensional
-
1.6. ORGANIZATION OF THE THESIS 11
Fast Fourier Transform (FFT) is applied on the space-time data
obtained fromFEM simulations. This analysis is the key to identify
the dominating modesand the corresponding frequencies of the radial
air gap forces and magneticnoise.
• Spectral analysis usually demands bulk amount of data (here
from FEM) toobtain the accurate results, which is time consuming
and tiresome, hence asimpler method called phase difference method
is adopted for the extractionof vibration modes and the
corresponding frequency spectrum. This methodis similar to the
method that is used in real time vibration testing of
inductionmotors.
• The rotor skew is modeled by using the so called slice model
in FEM forcomparisons with the non-skewed rotor.
• Two methods to introduce asymmetry are proposed, namely,
sinusoidal asym-metry and dual asymmetry. Some useful guidelines
are formulated for design-ing these asymmetrical rotors.
• The flux density values obtained from the finite element model
are coupledto the analytical tool to calculate the magnetic noise
produced by the motor.
1.6 Organization of the thesis
The outline of the chapters in this thesis are described as
follows:
Chapter 2: The classical theory behind the computation of radial
magnetic forcesand noise in the conventional induction motor is
described in this chapter. Anoverview of magnetic noise, its causes
and sources in induction motors is presented.A brief explanation of
the space harmonics in induction motors and their calcula-tion
using analytical formulas is presented. Finally, various methods to
calculateradial air gap forces in induction motors are described
and the advantages of thechosen analytical method for analysis is
explained.
Chapter 3: This chapter describes the finite element method
approach in com-puting the radial forces and magnetic noise, taking
eccentricity into account. Thepost-processing issues of the finite
element result data such as spectral analysis in1D and 2D, spectral
accuracy, the new phase difference method, combined FEMand
analytical method are described. An optimal method using FEM which
isimplemented in this work for analyzing the asymmetrical rotor
slots is finally de-scribed in this chapter.
Chapter 4: Calculations and validations on a 4-pole, 15 kW
standard motor withand without eccentricity is presented in detail
in this chapter. Comparisons aremade against the analytical and FEM
simulations. The influence of skew using
-
12 CHAPTER 1. INTRODUCTION
FEM based sliced model is studied and explained in a separate
section. Optimizingthe rotor slot number for the same stator is
also presented. This chapter concludesthat the proposed method of
analysis in Chapter 3 can be applied for studying thenew
asymmetrical rotor design proposed in Chapter 5.
Chapter 5: Methods to introduce asymmetry in rotors such as, the
dual slot ro-tors, slot combinations where rotor and stator slots
are equal, progressive sinusoidalrotors and combined dual
sinusoidal rotors is presented in this chapter. The designrules and
the predictions behind the design rules are explained. Proposals of
somedesigns are discussed and the simulations performed on some
selected designs aredescribed in Chapter 6
Chapter 6: This chapter consists of results and comparisons of
standard rotorsagainst asymmetrical rotors. The identification of
dominant forces and noise com-ponents for various proposed designs
are explained and their performance is ana-lyzed. Some guidelines
to design asymmetrical rotor slots are also given.
Resultcomparisons for the novel asymmetrical slot rotors and
standard rotors with andwithout eccentricity are presented.
Chapter 7: This chapter is dedicated to the conclusions from the
results obtained.Future work is suggested for a complete
performance analysis of the proposed in-duction rotors.
1.7 Publications
The work presented in this monograph has resulted in two
international and oneresidential conference publications listed
below:
1. Design and Analysis of Asymmetrical Rotor for Induction
MotorsR. Chitroju and C. Sadarangani.Proceedings of the
International Conference of Electrical Machines
(ICEM’08),Vilamoura, Portugal, September 2008.
2. Phase Shift Method for Radial Magnetic Force Analysis in
Induction Motorswith Non-Skewed Asymmetrical Rotor SlotsR. Chitroju
and C. Sadarangani.Proceedings of the International Conference of
Electrical Machines and Drives(IEMDC’09), Miami, Florida, May
2009.
3. Noise Minimization Method for Induction Motors Using
Asymmetrical RotorSlots
-
1.7. PUBLICATIONS 13
R. Chitroju and C. Sadarangani.Residential Conference on Noise
and Vibration: Emerging MethodsKeble College, Oxford, UK, April
2009.
-
Chapter 2
Radial magnetic forces andmagnetic noise in induction motors–
Methods of analysis
This chapter deals with the theoretical description of
analytical and FEM methodsfor computing magnetic forces and noise
in induction motors. The fact that remov-ing skew increases
magnetic noise makes it important to understand and identifythe
sources of radial forces leading to magnetic noise and develop
methods to reducethem. Some interesting findings on different
existing methods for force computationin induction motors are
presented in this chapter and the methods which are usedin this
thesis are justified. Finally, a brief theory about the noise
production, itssignificance and calculations are described.
Modeling of flux density, magnetic forces as traveling waves in
induc-tion motors
A traveling wave can be defined as a physical disturbance that
travels throughspace and time, with transfer of energy. For
example, a mechanical wave is a wavethat travels in a medium due to
the restoring forces it produces upon deformation.If all the parts
constituting a medium were rigidly fixed (like the stator of a
induc-tion motor), then they would all vibrate as one, with no
delay in the transmission ofthe vibration. On the other hand, if
all the parts were free, then there would not beany transmission of
the vibration. Wave physics reveals that in any traveling wavethe
phase of a vibration i.e., its position within the vibration cycle
is different forneighboring points in space because the vibration
reaches these points at differenttimes [21].
The flux density waves produced by the interaction of stator mmf
waves andthe permeance waves can also be treated as traveling waves
as they rotate along
15
-
16CHAPTER 2. RADIAL MAGNETIC FORCES AND MAGNETIC NOISE IN
INDUCTION MOTORS – METHODS OF ANALYSIS
the circumference of the air gap with finite wavelengths1. Such
a traveling wavemathematically can be described by Eq. 2.1, where k
represents the wavelengthnumber, B is the amplitude (half the
peak-to-peak value) of the flux density, ψ isphase angle at time
zero w.r.t the reference position, ωkt is the temporal
angularfrequency. At a particular position θ in space, b is the
flux density periodic functionwith temporal frequency given by fk =
ωkt2π
b (θ, t) = B · cos (kθ − ωktt+ ψ) (2.1)Thus the spatial angular
speed of the wave along the circumference becomes
wks = ωktk . The wavelength (k) of the traveling wave can be
written as a spatialharmonic related to fundamental i.e., k = µp,
where µ is the harmonic order num-ber and p is the pole pair
number.
The speed of the space harmonic can be written as
ωµs = vµ · ω1s = vµ · ω1p
(2.2)
where vµ is the velocity with which the harmonic rotates.
Rewriting Eq. 2.1,the flux density as a traveling wave can be
expressed as
bµ (θ, t) = Bµ · cos µ(pθ − vµω1t+ ψµ
µ
)(2.3)
Force density waves in the air gap are produced due to the
interaction of any twoflux density waves, which mathematically can
be expressed as their simple product.Considering two such flux
density traveling waves as written in Eq. 2.4 and Eq.2.5 for mth
and nth space harmonics, respectively.
bm (θ, t) = Bm · cos m(pθ − vmω1t+ ψm
m
)(2.4)
bn (θ, t) = Bn · cos n(pθ − vnω1t+ ψn
n
)(2.5)
The product of these two waves can be simplified using
appropriate trigonomet-ric relations into a sum of positive
rotating force wave and a negative rotating forcewave, both in
general can be written as in Eq. 2.6 and its wave characteristics
aretabulated in Table 2.1.
σr (θ, t) = σ̂r · cos r(pθ − vrω1t+ ψr
r
)(2.6)
1the distance between two sequential crests (or troughs).
Generally in machine domain it isrepresented as the pole-pair
number of the harmonic.
-
2.1. ANALYTICAL APPROACH FOR CALCULATING RADIAL FORCEWAVES
17
Table 2.1: Wave parameters for the force wave produced due to
interaction of twoflux density waves
Positive wave Negative wave
σr+ = 12BmBn σr− =12BmBn
r+ = m+ n r− = m− nθ+ = θm+θn2 r− =
θm−θn2
vr+ =mvm+nvnr+ vr− =mvm−nvn
r−
fr+ =mvm + nvn fr− =mvm − nvn
Consequently, the traveling force wave results in vibrations
which can also betreated as series of sinusoidal waves with phase
displacement. The radial displace-ment can be written as in Eq.
2.7.
Yr (θ, t) = Ŷr · cos r(pθ − vrω1t+ ψr
r
)(2.7)
As a result of the radial forces and vibrations acoustic noise
is generated ac-cording to the principles of wave mechanics. The
sound intensity produced can becalculated by using an appropriate
radiation model [22]. The description of suchmodels is out of scope
of this thesis and hence it will be mentioned that the spher-ical
radiation model is used for noise computation in this thesis and
the noise levelin decibels (dB) is calculated by using Eq. 2.8
Lr = 20 log(
9.05× 104fr√N2relYr
)(2.8)
where, fr is the vibration frequency and Nrel is the relative
radiated (noise)power.
2.1 Analytical approach for calculating radial force waves
Based on the traveling wave theory explained in the previous
section, differentanalytical methods were developed and were widely
reported in the literature [7], [5],[23], [24]. The accuracy of
such methods depends on the extent of physical detailsincluded in
the calculation method e.g. consideration of the geometry,
materialproperties, effects of saturation and the eccentricities.
Though most of the methodsuse the concept of traveling waves for
radial force calculations in common, the basiccomputation of mmf
and air gap permeance differs. The modeling of the air gapis
necessarily a key for the accuracy of any such method. Some of
these methodsare described briefly in this section with regard to
the flux density and radial forcewave calculations in the air gap
of induction motors.
-
18CHAPTER 2. RADIAL MAGNETIC FORCES AND MAGNETIC NOISE IN
INDUCTION MOTORS – METHODS OF ANALYSIS
2.1.1 Classical theory for calculating radial magnetic forcesThe
procedure for calculating radial magnetic forces and further noise
is shown inthe flow chart, Fig. 2.1. The flux density waves are the
result of the interaction ofmmf waves from the stator windings,
also called as the current loading and the airgap permeance waves.
The product of the fundamental mmf and the constant termof
permeance gives the fundamental air gap flux density which induce
secondarycurrents, and produce torque. A detailed summary of the
harmonics generation ininduction motor and the equations for
calculations are shown in Fig. 2.2 and Fig.2.3, respectively. This
harmonic division is well understood and has been widelydocumented
in literature [7], [5], [22]. The most common permeance wave
modelimplemented is described in [25], [22], [26], [23]. However,
the permeance wavesdue to slotting, eccentricity and saturation are
evaluated separately using Fouriercomponents and then are combined
together. By doing so, in this method some ofthe unknown effects
could have been excluded.
Consideration of skew
Skewing of induction rotors is one of the main topic in this
thesis work, hence abrief description of the equations considered
in the analytical calculations are givenbelow.
Skewing is usually defined in terms of stator slot pitch, which
is also called theskew pitch. For example for a motor with 36/28
slot configuration, the rotor slotsare usually skewed by one stator
slot pitch which implies that the skew pitch is one.
In the classical approach the skew angle γskewis calculated from
the skew pitchSP and is given by Eq. 2.9 where, Qs is the number of
stator slots.
γskew =2πQs
SP (2.9)
Skewing the rotor slots aids in the production of the damping
fields which canbe calculated using Eq. 2.10. Damping fields are
the fields induced by the rotorresidual fields into the air gap
which damp the main stator fields. These fieldsinduce various
harmonics in the rotor.
Baν = χνBν (2.10)
where, Bν is the undamped stator field and the damping factor χν
is given byEq. 2.11
χν =
√1− ξ2skewξ2T
2 (1 + σg)− (ξskew.ξT )2(1 + σg)2 (1 + (βν/sν))
(2.11)
-
2.1. ANALYTICAL APPROACH FOR CALCULATING RADIAL FORCEWAVES
19
Electrical loading Air gap permeance
Induced airgap flux density
Radial force wave
Deformation of lamination packet
Deformation at the core surface
Sound pressure level
Relative radiation
efficiency
)(sin),( txAtxa )(cos),( txtxo
)(cos),(vvvv
tvxBtxb
)(cosˆ),(rrrr
txrtx
)(cosˆ),(rrrr
txrYtxY
)(cosˆ),(rrrr
txrYtxY
dBNYfLrelrrr
)ˆ1005.9(log20 4
relN
Figure 2.1: Flowchart for analytical calculation procedure
[2]
where, sν is the harmonic slip, βν is the specific rotor
resistance and ξskew is theskew factor given by Eq. 2.12, σg is the
rotor stray flux, ξT is an empirical slottingcorrection factor.
ξskew =sin(ν γskew2
)
ν(γskew
2) (2.12)
Skew also has the influence on the production of rotor residual
fields in cagewinding produced by the rotor bar currents.
Iring,ν =1
Λνξskewξν(1 + kν)
Bν√2
(2.13)
-
20CHAPTER 2. RADIAL MAGNETIC FORCES AND MAGNETIC NOISE IN
INDUCTION MOTORS – METHODS OF ANALYSIS
Unsymmetrical current system
Symmetrical and sinus format
current system
Discontinous winding distribution
mmf fundamental wave
Permeance variations
mmf harmonics
Constant permeance
End
harmonics
Eccentricity
harmonics
Saturation
harmonics
Harmonics
from stator
and Rotor
slots
Stator slot
harmonics
Stator
fundamental
field
Winding
harmonics
Primary fundamental, Stator field
End
harmonics
Eccentricity
harmonics
Saturation
harmonics
Harmonics
from stator
and rotor
slots
Rotor slot
harmonics
Rotor
fundamental
field
Winding
harmonics
Residual
fields from
end regions
Residual field
harmonics
from stator and
rotor slots
Residual
fields from
the windings
Supply current harmonics
8
Eccentricity
of the rotorSaturation in the
magnetic circuit
Slot opening in the
stator and rotor
Mean air gap
length
Winding
harmonics
ps
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Rotor Residual Field, Rotor field 9
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Residual
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Residual
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Residual
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Figure 2.2: Schematic for the harmonic distribution in induction
motor
-
2.1. ANALYTICAL APPROACH FOR CALCULATING RADIAL FORCEWAVES
21
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tud
e,
vB
9v
Qg
r2
s
pQg
ff
r
v1
2
Ro
tor f
un
da
men
tal
cu
rren
t lo
ad
ing
-resid
ua
l
field
10
pQ
gr
2s
pQg
ff
r
N1
12
11
z
e
BI
Lo
ad
dep
en
den
t a
mp
litu
de
21
1.0
gB
Ro
tor s
lott
ing
fie
ld f
ro
m
fun
da
men
tal
cu
rren
t 1
1p
Qg
r2
s
pQg
ff
r
N1
12
21
.N
gp
BC
FR
B
Resid
ua
l fi
eld
fro
m
eccen
tric
ity
1
21
2p
Qg
r
Ner
Ne
ff
s
pQg
ff
11
21
1 2e
p
e
p
BB
e, rela
tiv
e e
ccen
tric
ity
2 2
1 1
1,0.
050.
15
1,0.
100.
30
g g
pB
pB
Resid
ua
l fi
eld
fro
m
sa
tura
tio
n1
3p
Qg
r3
2s
pQg
ff
r
Ne
13
23
mm
p
mp
BB
~2
10
.4g
B
Figure 2.3: Formula table for various harmonics in induction
motor
-
22CHAPTER 2. RADIAL MAGNETIC FORCES AND MAGNETIC NOISE IN
INDUCTION MOTORS – METHODS OF ANALYSIS
(a) mode-0 (b) mode-1 (c) mode-2 (d) mode-3
Figure 2.4: Stator vibration modes
The rotor fields induced in a cage winding is the sum of the
stator windingfields and the slotting fields. The rotor circuit has
to be taken into considerationto determine the relation between the
bar currents and the stator flux fields. Therotor ring currents can
be calculated using Eq. 2.13 [22], where Λν is the averageperemance
of the air gap, Bν is the stator flux density harmonic, kν is the
rotorgeometrical leakage co-efficient for the νth harmonic. The
skew angle is also con-sidered during the calculation of the
vibration amplitude and the noise level forthe rotor laminations,
especially for the vibration order r = 1 also called
Ruttelsoscillations. Different vibration patterns are shown in Fig.
2.4 and the vibrationamplitude is given by the equation Eq.
2.14
Yr =πlfeRa
Cblηξskewσ̂r(2.14)
where, Cbl is the bending constant of the shaft, Ra radius of
the rotor lami-nations, η is the resonance factor, lfe is the
effective length of the iron, ξskew isthe skew factor of the rotor
given by Eq. 2.12. In the case of skew the torsionaloscillations
also have to be considered in addition to the bending
oscillations.
Weh effect:
According to the theory discussed in Section 2.1.1, the mode
numbers are ob-tained from the sum or difference of the pole pair
numbers (wavelengths) of anytwo flux density waves. It was believed
until late 80’s that the major cause for theradial magnetic forces
were the contributions from smaller mode numbers. How-ever, Weh in
1986 showed the fact that pole pair numbers greater than the stator
orrotor teeth number have significant influence on the stator yoke
deformation owingto the anisotropy caused by the teeth [27]. This
is also called sub-wave generation.Note that the Weh effect is
considered in the analytical calculations performed inthis thesis
work.
-
2.2. ROTOR BAR INDUCTION CURRENTS APPROACH 23
2.2 Rotor bar induction currents approach
A method to analyze the rotor bar currents induced by the mmf
wave from thestator using voltage and torque equations, for
induction motors with unequal slotpitches is developed by Mineyo
et.al. The obtained results are used to find the mmfand air gap
flux density distribution, and to investigate the modulation
methodsthat can suppress the sources of noise and vibrations in
unequal slot pitched rotors[14].
2.3 FEM analysis with spectral methods
For a valid distribution of the forces acting radially between
the stator and therotor, it is necessary to determine the space and
time characteristics of the instan-taneous values of magnetic
induction in the air gap. In the numerical approachair gap can be
accurately modeled using the finite element method as the
physicalgeometry, material properties and non-linearity can be
accurately defined. Further,the Maxwell’s force tensors and the
flux density products are obtained directly bylooking at the
results in the air gap. Both flux density components Bn and
Btconsist of sum of all the space harmonics which means both force
densities (normaland tangential components) consist of products of
two random flux density waves.The radial force components are
responsible for the unnecessary forces and the tan-gential forces
are responsible for the useful torque. Thus obtained radial forces
canbe expressed as a series of Fourier components in order to
identify the key forcecomponents causing magnetic noise.
Only the static eccentricity can be considered as the existing
FEM tool couldn’thandle the dynamic eccentricity problem. The
influence of skew can also be studiedseparately by using a quasi-3D
FEM model. A detailed method of analysis of thisapproach is
presented in Chapter 3 as this method is used for the calculations
oftorque ripple, in this thesis work.
The radial force densities as functions of space and time can
further be ana-lyzed using spectral methods to characterize these
waves into mode numbers andfrequencies. Recalling the fact from the
first paragraph of this chapter, the basicrule of wave mechanics
(italicized text) the phase of the space harmonics have tobe
considered in the spectral analysis.
A similar method to the above has been reported in [28]. The
radial electromagnetic force distribution is developed into a
double Fourier series in space andtime. The Fourier coefficients
are obtained by integrating the quantities from timestep to time
step. Simulations are performed with a frequency resolution of 1
Hzso as to study the stress waves for each spatial mode. This
approach is chosen forthe analysis of radial forces and noise in
induction motors which is discussed more
-
24CHAPTER 2. RADIAL MAGNETIC FORCES AND MAGNETIC NOISE IN
INDUCTION MOTORS – METHODS OF ANALYSIS
comprehensively in Chapter 3.
2.4 Magnetic equivalent circuit approach
In the MEC method of calculation the motor is divided into Qr
rotor and Qs statorsectors and an equivalent circuit is built for
each sector and are combined [26].All the parameters of the model
such as scalar magnetic potential, tooth and yokefluxes, magnetic
reluctances, mmf sources are resolved into different frequenciesand
orders which can be written as shown in Eq. 2.15
Xi(t) =∑
k
<{Xkexp
(j · [2πfkt− rk 2π
Qi])}
(2.15)
where, < is the magnetic reluctance, Xi is the quantity for
ith sector, f , k arethe frequency and order of the component, Q is
the slot number of the rotor or thestator under consideration. As
explained in Section 2.1.1, according to the rotatingfield theory,
the air gap flux density as a function of time t and angular
position θcan be written as a series of traveling waves as shown in
Eq. 2.16.
B (θ, t) =∑
k
<{Bkexp
(j · [2πfbt− kbθ]
)}(2.16)
MEC method has many advantages for solving problems including
simultane-ous time-spatial analysis of electro mechanical systems,
such as speed, accuracyand possibility to include almost all the
physical phenomenon of interest. A briefdescription about this
method is also given in [29].
2.5 Combined classical and finite element method approach
A combined, analytical and FEM approach for the computation of
flux densityand radial forces has been published by Stefan Toader
in 1992 [30]. The analyticalmethod used was similar to the
classical method where the permeance waves aremultiplied by mmf
waves. By extracting the flux density values from the FEMresult at
two time instants, with the help of the analytical expressions it
is shownthat it could be possible to identify the origin and
production of force waves moreaccurately. Despite of the advantages
of using FEM, this method can only beused for one particular motor
owing to its tedious process of computation. It canbecome extremely
tedious process if this method has to be performed on
severalmachines. This method can only be used to understand the
complex phenomenarelated to the harmonics in one particular motor
at a time. However, a convenientmethod could be developed by
performing rigorous analysis on numerous machines.
-
2.6. MAGNETIC NOISE PRODUCTION AND ITS MODELING FORINDUCTION
MOTORS 25
1910 1925-30 1940 1957-67 1970 1980 2000 - 2007
• Oversized electrical machines
• Negligible aerodynamic noise
• Bearings-well understood technology
• Focus on electromagnetic noise
• major interest for induction machines
Finding optimum slot
numbers
Optimum slot combinations
for different power rating
Improved exploitation of active and
passive parts, weight decreased and
noise increased
Machines from factories move
to domestic applications, 30
kW motor in 10 years gave 10
dB increase in noise
Decrease in electromagnetic noise
achieved. Focus on aerodynamic origin,
ventillation noise
Solid state devices - increased
time harmonics hence
increased noise levels
Design of outer frame
at the recievers end
Vibro-acoustic theories;
Finite element method
Boundary element method
Statistical energy analysis
Figure 2.5: History of noise minimization methods in electrical
motors
2.6 Magnetic noise production and its modeling forinduction
motors
A brief history related to magnetic noise in electrical machines
is depicted in Fig.2.5. The time line shown in the figure is self
explanatory showing the improvementsachieved in the area of noise
minimization in electrical motors. In general predict-ing noise is
difficult and is not very accurate specially at the design stage of
anyelectrical machine, one of the reasons being that only a small
portion of energy isconverted to noise and also the estimation of
mechanical and acoustic parametersare very difficult. However, the
noise and vibration level requirements are increas-ingly stringent,
since reduction of noise and vibrations is equally important to
thephysical health, machine economics and productivity [7].
2.6.1 An overview of magnetic noise in induction motors
A detailed schematic for noise production in induction motors is
shown in Fig. 2.6.It can be identified that the major sources of
magnetic noise are the exciting forceharmonics resulting from
various harmonics in the flux density wave produced inthe air gap.
Reducing noise at source level should be one of the main
objectivesand also an effective method to handle the problem.
-
26CHAPTER 2. RADIAL MAGNETIC FORCES AND MAGNETIC NOISE IN
INDUCTION MOTORS – METHODS OF ANALYSISConnected with
auxilary
functions
of electrical machines
Aero
dyn
am
ic
Ele
ctr
om
agn
eti
c
Ex
cit
ing
fo
rce h
arm
on
ics
Slo
t h
arm
on
ics
Sa
tura
tion
harm
on
ics
Su
pp
ly w
ith
ha
rm
on
ic c
on
ten
t
Vib
ra
tion
Vib
ra
tin
g
ca
pab
ilit
ySou
nd
rad
iati
ng
ca
pab
ilit
yV
ibra
tion
tra
nsm
itti
ng
& S
ou
nd
rad
iati
on
Sou
nd
prop
ag
ati
on
Sou
nd
sen
sin
g
Str
uctu
re-b
orn
e
sou
nd
Sou
nd
at
hu
man
s
Ex
cit
ing f
orces,
Dir
ectl
y
con
necte
d w
ith
EM
con
versio
n
No
ise
No
ise s
ou
rce
En
vir
on
men
t
Sen
sor
Pa
ssiv
e s
yste
m
pa
ra
mete
rs
Active system
parameters
Mech
an
ica
l
Asym
metr
y i
n t
he s
tato
r &
ro
tor w
ind
ing
Eccen
tric
assem
bly
of
ro
tor &
sta
tor
Dyn
am
ic e
ccen
tric
ity
of
ro
tor
Bearin
g e
ffect,
Ben
t sh
aft
en
d,
Jou
rn
al
ava
lity
, L
oose a
ssem
bly
,
Fan
bla
de d
esig
n
Ven
t secti
on
s i
n r
oto
r&
sta
tor
Air
-bo
rn
e
sou
nd
Figure 2.6: Schematic for noise generation in electrical motors
(IM)
-
2.6. MAGNETIC NOISE PRODUCTION AND ITS MODELING FORINDUCTION
MOTORS 27
Significance of magnetic noise
As shown in Fig. 1.1, motor systems consume more than 55% of
electricalenergy and hence are one of the main source of noise in
industrial environments.There is a growing concern for silent
workplaces because of the increased awarenessof the negative
effects of noise on people exposed to noise for long periods of
time.Even today, after the delivery of any motor for an industrial
or domestic usage,the customers usually still need to take many
measures to protect the environmentfrom the acoustic noise1. Some
of such measures are listed below.
• Isolation of the source through the appropriate placement,
incorporation orvibration damping through the use of metal or air
springs.
• Reduction at the source or along the way through enclosures by
silencers onthe exhaust pipe, or by lowering the speed of cutting
tools or fans.
• Replacement or modification of machines - for example, drive
belts instead ofgears, or power tools instead of pneumatic
tools.
• Use of quieter material - for example, rubber-lined
containers, conveyors andvibrators. Active noise reduction in
certain vulnerable conditions.
• Preventive maintenance - when the machine is worn out the
noise level canincrease.
Actually, such post purchase measures are frustrating and are
not the solutionpreferred by the customers. Also the solution can
be expensive and additional costsfor modifying the thermal design
of the motor can rise, especially if the motor ishoused in an extra
sound proof frame. The best way is always to reduce the noiseat the
source level in the motor design stage. Today improved sound and
vibrationtechnologies are required for a large proportion of the
products supplied by manyindustries.
Of all the input power supplied to a motor a part of it is
dissipated as heatanother part is consumed in the ventilating
system and the third or the smallestpart is lost in sound. However,
this kind of power loss can be very significant if themotor noise
goes beyond the standard level.
Just to give an idea of importance of the noise reduction in
electrical machinestoday, one can say that, in some cases a high
noise level is reason enough to mod-ify completely a product line.
There are four factors which are accelerating thedevelopments of
silent electrical machines [31].
1source:"Ljud och oljud - för utbildning och praltisk
bullerbekämpning", Prevent 2004, "Eu-ropeiska arbetsmiljöveckan
2005", faktablad 58 från Europeiska arbetsmiljöbyrån
-
28CHAPTER 2. RADIAL MAGNETIC FORCES AND MAGNETIC NOISE IN
INDUCTION MOTORS – METHODS OF ANALYSIS
1. Increased usage of machines in domestic environments which
concerns thesociety in general.
2. Demands imposed by standardization entities related to the
electrical machinenoise reduction.
3. Demands from customers that often go beyond the standards,
searching forelectrical machines with features that meet their
special requirements.
4. Besides all the above, silent motor means a good quality
motor, this is astrong motivation to build a quieter machine.
It is important to note that of all the noise sources magnetic
noise is the secondhighest source in 2 and 4 pole machines, after
the ventilation noise.
-
Chapter 3
Finite element method approachfor radial forces computation
Radial forces computation using air gap flux densities obtained
as functions of spaceθ and time t from the finite element method is
presented in this chapter. Further,the radial force analyses i.e.,
extracting the Fourier components of the magneticforces that can be
used to identify the vibration modes that lead to magnetic noiseare
discussed. A simpler method called phase difference method is
presented as analternative to the 2D spectral analysis. This
chapter concludes with an optimummethod to analyze asymmetrical
rotor slot motors proposed later in this work.
3.1 Spectral analysis for radial forces computation
FEM approach has many advantages over the classical analytical
models describedin the previous chapter. Details such as the
physical geometry, saturation of theiron, rotor eccentricities can
be easily modeled, which allows an accurate calculationof the
permeance provided an accurate mesh discretisation is adopted. It
also offersthe possibility to quickly modify the design geometry in
order to study variousnew non-symmetrical rotor design changes.
Though the method of analysis is notstraight forward, some extra
post-processing and application of spectral analysiscan be used to
produce promising results. A step-by-step post-processing of
thefinite element results is presented in the following
sections.
3.1.1 1D Spectral analysis
From the rotating field and traveling wave theories we know that
flux density in theair gap which is the interaction of stator and
rotor mmf waves and the permeancewaves, vary both in space and
time. Thus the instantaneous flux density b can beexpressed as a
function of space x and time t as written in Eq. 3.1.
29
-
30CHAPTER 3. FINITE ELEMENT METHOD APPROACH FOR RADIAL
FORCES COMPUTATION
bν (x, t) = Bν cos (νx− ωνt− ψν) (3.1)where ν is the space
harmonic number, Bν is the peak value of the flux density
space harmonic, ων is the angular frequency and ψν is the phase
angle of the spaceharmonic.
The air gap forces can be calculated by using Maxwell’s tensor
given by Eq.3.2 for the radial force component and Eq. 3.3 for the
tangential force component(torque producing), where, bn and bt are
the normal and tangential flux densitycomponents, respectively.
σr (x, t) =1
2µ0
[bn (x, t)2 − bt (x, t)2
](3.2)
σt (x, t) =1µ0
[bn(x, t) · bt (x, t)] (3.3)
The radial force σr calculated from Maxwell’s tensor can also be
represented asa traveling wave given by Eq. 3.4, where σ̂r is the
peak value of the radial force.Note that the variable ν
representing the pole pair number in Eq. 3.1 transformsto variable
r, which is twice the pole pair number (also called the mode
number).The tangential component has negligible contribution to the
total forces and noiseproduction, hence can be ignored in this
analysis.
σr (x, t) = σ̂r cos (rx− ωrt− ψr) (3.4)
By using simple spectral analysis like FFT in 1D an elementary
force spectrain space and time can be obtained separately. As an
illustration the flux densitydistribution around the air gap
(space) of a motor simulated in FEM is chosen asshown in Fig. 3.1a
and the time variation of flux density at a point in space is
cho-sen as shown in Fig. 3.1b. The corresponding radial forces1
obtained by applyingMaxwell’s tensor are shown in Fig. 3.1c and
Fig. 3.1d, respectively. In order tostudy the Fourier harmonic
components of the radial air gap forces these resultsshould be
separated into spatial and temporal spectra as shown in Fig. 3.1e
andFig. 3.1f, respectively.
The spatial spectrum obtained by choosing one frozen time
instant of the fluxdensity along the air gap can be written in the
form of Fourier series given by Eq.3.5, where tn is the frozen time
instant. Similarly, the temporal spectrum obtainedby calculating
the time variation of flux density at a fixed point in space can
bewritten as in Eq. 3.6, where xn is the arbitrary point chosen in
the air gap.
1The term ‘radial force’ mentioned in the thesis represents the
force per unit area acting inradial direction expressed in N/m2
-
3.1. SPECTRAL ANALYSIS FOR RADIAL FORCES COMPUTATION 31
0 1 2 3 4 5 6
−1
−0.5
0
0.5
1
Space co-ordinate in the air gap (radians)
Flu
xden
sity
(T)
(a) Flux density distribution around the air gapat one instant
of time
0 0.005 0.01 0.015 0.02−1.5
−1
−0.5
0
0.5
1
1.5
Time(s)
Flu
xden
sity
(T)
(b) Variation of flux density at a fixed point
0 1 2 3 4 5 60
1
2
3
4
5
6
x 105
Space co-ordinate in the air gap (radians)
Rad
ialai
rga
pfo
rce
(N/m
2)
(c) Maxwell’s tensor radial force in space
0 0.005 0.01 0.015 0.020
1
2
3
4
5
6
x 105
Time(s)
Rad
ialai
rga
pfo
rce
(N/m
2)
(d) Maxwell’s tensor radial force with time
0 1 2 3 4 5 6 7 8 9 101112131415161718192010
0
102
104
106
Mode number
Rad
ialair
gap
forc
e(N
/m
2)
(e) Radial force spatial spectrum
0 1000 2000 3000 4000
103
104
105
106
Frequency (Hz)
Radia
lai
rga
pfo
rce
(N/m
2)
(f) Radial force temporal spectrum
Figure 3.1: Simple spectral analysis on the results obtained
from FEM
-
32CHAPTER 3. FINITE ELEMENT METHOD APPROACH FOR RADIAL
FORCES COMPUTATION
σr (x)tn =∑r
σ̂r,tn cos (rx− ψr,tn) (3.5)
σrxn (t) =∑ωr
σ̂r,xn cos (ωrt− ψr,xn) (3.6)
This method is a traditional approach, also explained in [22]
which gives lim-ited information regarding the radial magnetic
forces. As seen two separate spectraare needed to analyze the
radial forces, where the spatial spectrum gives the ordernumbers
(mode) and temporal spectrum gives the speed of the radial force
compo-nents (frequency). It is in fact very difficult to separate
and categorize the radialforces into mode numbers and their
corresponding frequencies from this approach.It should also be
recalled from the previous chapter that the radial force wave
shiftsits position with time and unfortunately in this analysis the
spatial spectra can onlybe obtained at one frozen time instant
which is insufficient to analyze the radialforces.
Thus an analysis in two-dimensions is required to separate the
space and timeharmonics of the radial force waves i.e., into mode
numbers and frequencies. Notethat time harmonics here are the
harmonics due to the stator mmf and the sec-ondary currents from
the rotor which induce additional harmonics in the air gap.To
consider these effects time-stepping analysis is adopted, where the
rotor rota-tion is also taken into consideration and the flux
density variation with time alongthe spatial points in the air gap
are studied. The data obtained needs to be an-alyzed using 2D
Fourier analysis to separate the force waves into series of
Fouriercomponents in space and time, as explained in the next
section.
3.1.2 2D Spectral analysisA detailed analysis of radial forces
from the flux density waves obtained from finiteelement data can be
performed by using 2D spectral analysis. By employing acustom 2D
Fourier analysis [32] a series of Fourier components can be
obtained asexpressed by Eq. 3.7 and Eq. 3.8. Thus a detailed
spectra of forces with frequenciesand mode numbers can be
obtained.
σ̂ (r, ω) =M−1∑t=0
N−1∑
θ=0σ (θ, t) e2πj( rθN +ωtM ) (3.7)
σ (θ, t) = 1NM
N−1∑r=0
M−1∑ω=0
σ̂ (r, ω) e−2πj( rθN +ωtM ) (3.8)
As an illustration, a simple flux density wave rotating in the
air gap which canbe expressed by Eq. 3.9 is considered, where, n is
the time harmonic number, f is
-
3.1. SPECTRAL ANALYSIS FOR RADIAL FORCES COMPUTATION 33
(a) Fundamental flux density wave in the air gap
0 100 200 300 400 500 600 700 8000
0.2
0.4
0.6
0.8
1
Frequency(Hz)
Am
plitu
de(
T)
mode-1
(b) Flux density spectrum showing the fundamen-tal
(c) Flux density harmonics in the air gap
0 100 200 300 400 500 600 700 8000
0.2
0.4
0.6
0.8
1
Frequency(Hz)
Am
plitu
de(
T)
mode-0
mode-1
mode-2
mode-3
mode-4
(d) Harmonic spectrum
(e) Flux density from FEM
0 1000 2000 3000 400010
−3
10−2
10−1
100
Frequency(Hz)
Am
plitu
de(
T)
(f) Harmonic spectrum
Figure 3.2: Illustration of 2D spectral analysis
-
34CHAPTER 3. FINITE ELEMENT METHOD APPROACH FOR RADIAL
FORCES COMPUTATION
the fundamental frequency, r is the space harmonic mode number,
t and θ are thetime and space variables, respectively.
b (θ, t) = B cos (2πnft− rθ + ψ) (3.9)A graphical representation
of such a wave in two dimensions i.e., at f = 50 Hz
for the fundamental n = 1 and the first spatial harmonic r = 1
can be shown as inFig. 3.2a. This wave can be imagined as the ideal
flux density wave in the air gapwithout any harmonics. Performing
2D Fourier analysis on such a wave results ina harmonic spectrum as
shown in Fig. 3.2b, where 50 Hz frequency component ofthe first
space harmonic is clearly visible.
Further, to the fundamental wave shown in Fig. 3.2a but now with
r = 0, fewmore harmonic components are added as shown in Eq.
3.10
B (θ, t) = 1.00× cos (2π × 1× 50t+ 0× θ) + 0.80× cos (2π × 3×
50t+ 1× θ)+0.50× cos (2π × 6× 50t+ 1× θ) + 0.60× cos (2π × 5× 50t+
2× θ)+0.90× cos (2π × 7× 50t+ 3× θ) + 0.45× cos (2π × 7× 50t+ 2×
θ)
+0.3× cos (2π × 11× 50t+ 4× θ)(3.10)
This rotating wave in two-dimensions is shown in Fig. 3.2c and
the correspond-ing 2D harmonic spectrum is plotted in Fig. 3.2d. As
seen from the figure all thecomponents are computed with accurate
magnitudes without any deviation. Thesimulated rotating wave is a
pure wave without any presence of white noise2. Itis interesting to
see the flux density data obtained from FEM, extracted in the
airgap of the motor as a function of space and time shown in Fig.
3.2e. The ripplesin the wave are due to the presence of higher
harmonics obviously, but a significantnumerical noise from the
finite element mesh is also seen which can actually distortthe
amplitude estimation of the harmonic spectra.
As seen from Fig. 3.2f, the 2D spectrum obtained consists of
several componentsfrom which the precise spatial harmonics are
quite difficult to identify. However, allthe spatial components
always are not of interest, the spectra could be separatedinto mode
numbers and their corresponding frequencies as shown in Fig. 3.3a
andFig. 3.3b, respectively; provided a careful consideration of
phase of the spatialharmonics is taken. A similar approach has been
published by Boman and Ostovic[8] to calculate radial forces
leading to magnetic noise by using the MEC method.The time
variation of each individual mode can be seen in Fig. 3.3a which
givesa qualitative picture of the dominating radial force modes.
Further, analyzing the
2White noise contains an equal amount of all frequencies, the
same as white light; whichresults from the noise signal on the time
domain waveform uncorrelated from sample-to-sample.
-
3.1. SPECTRAL ANALYSIS FOR RADIAL FORCES COMPUTATION 35
0 1 2 3 4 0
1000
2000
100
102
104
106
Time samplesMode number(r)
Rad
ialair
gap
forc
es(N
/m2)
(a) Spatial harmonic variation with time
0 500 1000 1500 2000 2500 3000
102
103
104
105
106
Frequency (Hz)
Rad
ialair
gap
forc
es(N
/m2)
mode-0
mode-1
mode-2
mode-3
mode-4
(b) Frequency spectrum of the radial forces
Figure 3.3: Mode number and frequency separation of radial
forces
complex time variation of each mode a detailed temporal spectrum
is obtained asshown in Fig. 3.3b.
3.1.3 Spectral accuracySpectral accuracy depends on many factors
such as the sampling rate, length ofthe signal, presence of white
noise etc. A more comprehensive description of suchfactors is given
in [22]. It is worth noting that the spectrum shown in Fig. 3.1f
(orFig. 3.4a) is the result of a simple FFT in Matlab which chooses
the length of thesignal as the length of the FFT without any padded
zeros. In order to increase theaccuracy of the spectrum an n-point
DFT is chosen so that length of the transformis powers of two. If
the length of signal is less than n, the output is padded
withtrailing zeros to length n. It is kn