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

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

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.

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.

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

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.

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

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

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

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

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

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.

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)

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.

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)

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

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

pgsstststst

1pe

6

pm

3

7

pQgsg 1

ps

15 4 2,3

Rotor Residual Field, Rotor field 9

stv

ev

mv

gv p s

pQgrg 2

Residual

fields from

eccentricity

Residual

fields from

saturation

Residual

fields from

fundamental

pQgQgrsg 21

Arm

atu

re R

eact

ion o

f

resp

ecti

ve

pri

mar

y f

ield

fro

m

stat

or

Win

din

g h

arm

on

ics

for

resp

arm

ature

rea

ctio

n

strgvQg2 erg vQg2 mrg vQg2

12 13

11

10

MMF

fundamental wave

pQgrg 2 srg Qg2

)(),( txBtxb

)(),(vvvv

tvxBtxb

Cross section

changes at the end

Rotor

Air

gap

Stator

Supp

ly

Figure 2.2: Schematic for the harmonic distribution in induction motor

2.1. ANALYTICAL APPROACH FOR CALCULATING RADIAL FORCEWAVES 21

Ty

pe o

f fi

eld

R

ef

Ord

er n

um

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Freq

uen

cy

rela

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/2

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Co

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tal

field

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tor f

ield

prod

uced

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m t

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urren

t

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da

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perm

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nce

1.0

pB

Win

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arm

on

ics (

wit

h

inte

gra

l slo

t w

ind

ing

in

sta

tor)

21

(61)

vp

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1

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IN

Sq

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Am

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ro

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al

to

the c

urren

t 0

.00

50.0

8p

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Sta

tor s

lot

ha

rm

on

ics

3p

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vs

1v

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f1.

1.

v

e

IN

Sq

p

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v

Am

pli

tud

e p

ro

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al

to

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40.1

pB

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ics

du

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oth

sta

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oto

r s

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vr

s2

1s

pQg

ff

r

Nv

11

21

21

1v

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Ng

pB

CF

SC

FR

B

Eccen

tric

ity

fie

lds

61

ev

p

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ff

(1);

0N

ee

f

fs

f

p

1 2ve

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B0,

(1)

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fs

p

, sta

tic,

dy

na

mic

0.1

0.3

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dep

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ds o

n %

ag

e

of

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ity

Sa

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73

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p3

vm

Nf

fvm

mp

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0.1

5p

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Resu

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m t

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am

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field

s

8res

vv

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da

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field

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gr

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12

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ad

dep

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lott

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fun

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tal

cu

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BC

FR

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m

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21

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Ne

ff

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30

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~2

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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