Improved performance characteristics of induction machines with non-skewed symmetrical rotor slots

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Improved Performance Characteristics of Induction Machines with Non-Skewed Asymmetrical Rotor Slots

Submitted to the School of Electrical Engineering in partial fulfillment of the requirements for the degree of Licentiate

RATHNA KUMAR SASTRY CHITROJU

Licentiate Thesis

Electrical Machines and Power Electronics School of Electrical Engineering Royal Institute of Technology (KTH)

Stockholm, Sweden 2009

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ISSN 1653-5146

ISBN 978-91-7415-453-5

SE-100 44 Stockholm SWEDEN 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.

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Abstract

Induction machines convert more than 55% of electrical energy into various other forms in industrial and domestic environments. Improved performance, especially by reduction of losses in induction machines hence can significantly reduce consumption of electricity. Many design and control methods are adopted to make induction machines work more efficiently, however cer- tain design compromises are inevitable, such as skewing the rotor to improve the magnetic noise and torque characteristics increase the cross current losses considerably in a cage rotor, degrading the efficiency of the motor. Cross- current losses are the dominating stray losses which are dependent on several factors among them are percentage skew and the contact resistance between the rotor bars and laminations. It is shown in this thesis that implementing a design change which has non-skewed asymmetrical distribution of rotor slots can serve the same purpose as skewing i.e., reduction of the magnetic noise, thereby avoiding the negative effects of skewing the rotor slots especially by reducing the cross-current losses.

Two design methodologies to introduce asymmetry in rotor slots are proposed and the key performance characteristics like torque ripple, radial air gap forces are computed both numerically and analytically. Radial forces obtained from the finite element method are coupled to the analytical tool for calculating the magnetic noise. A spectral method to calculate and separate the radial forces into vibration modes and their respective frequencies is proposed and validated for a standard 4-pole induction motor. The influence of rotor slot number, eccentricity and skew on radial forces and magnetic noise are studied using finite element method in order to understand the vibrational and acoustic behavior of the machine, especially for identifying their sources.

The validated methods on standard motors are applied for investigating the asymmetrical rotor slot machines.

Radial air gap forces and magnetic noise spectra are computed for the novel dual and sinusoidal asymmetrical rotors and compared with the standard symmetrical rotor. The results obtained showed reduced radial forces and magnetic noise in asymmetrical rotors, both for the eccentric and non- eccentric cases. Based on the results obtained some guide lines for designing asymmetrical rotor slots are established. Magnitudes of the harmful modes of vibration observed in the eccentric rotors, which usually occur in reality, are considerably reduced in asymmetrical rotors showing lower sound intensity levels produced by asymmetrical rotors. The noise level from mode-2 vibration in a 4-pole standard 15 kW motor running with 25% static eccentricity is decreased by about 6 dB, compared to the standard rotors. Hence improved performance can be achieved by removing skew which reduces cross current losses and by employing asymmetrical rotor slots same noise level can be maintained or can be even lowered.

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Keywords: Skewing, cross current losses, radial magnetic forces, mag- netic noise, asymmetrical rotor slots, eccentricity, stator vibration, two-dimensional fast fourier transform, finite element method.

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Acknowledgments

I would like to express my deep gratitude to my supervisor Prof. Chandur Sadarangani for his remarkable, positive and composed supervision. I would like to thank Dr.

Heinz Lendenmann, group manager for Machines at ABB Corporate Research, for his encouragement and support which was one of the reasons that initiated my PhD studies. I want to thank Yujing Liu, senior principal scientist at ABB CRC for his inspiring suggestions both at work and social meetings; Robert. J Andersson for his help in providing machine input data from the Oskar program; Tech. Lic. Chris- ter Danielsson, my previous colleague for his support and encouragement. Special thanks to versatile Tech. Lic. Mats Leksell and Dr. Juliette Soulard at our department for their admirable hard work and inspiring academic comments on various aspects. I would like to thank Dr. Stefan Östlund for proof reading this report and also for giving me the opportunity to assist him with course work at the department.

Special thanks to Ulf Carlsson from MWL laboratory, KTH and Bhavani Shankar from Signals and Systems department, KTH for their suggestions during some key stages of the project.

I would like to greatly acknowledge Elforsk and Elektra foundations for their financial support and their inputs during the yearly meetings.

Special thanks to my intimate EME colleagues; Dmitry Svechkarenko for his help with Latex, many linguistic aspects and obviously for his invitations to many social 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 room mate Antonious Antonopoulos for his all time co-operation and support. I also take this opportunity to acknowledge all my formal and present colleagues at EME for contributing their efforts to make the department proud and for providing a peaceful working environment.

My sweet family especially my grand parents and parents are acknowledged for their 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 for her moral support and understanding nature. I would like to acknowledge Kumar

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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, Kailash Srivastava, Kamesh Ganti, Kashif Khan, John Rödin, François Besnard, Suman Vodnala, priests Roger Stenzelius and Daniella Åslund for their all time love and support. Last but not least our economists Eva & Emma Petterssons, computer administrator Peter Lönn, lab technician Olle Bränvall, Administrator Brigitt Hög- berg are greatly acknowledged for their continuous help and support. I should also mention and thank Birka for impressing me with her funny deeds and for joining me to those nice walks in the woods behind the KTH campus.

Rathna Chitroju

Stockholm, October 2009.

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List of Figures

1.1 Electrical energy consumption chart . . . 2

1.2 Motor distribution in Sweden . . . 3

1.3 Illustration of cross currents in skewed and non-skewed rotor . . . 5

1.4 Composition of stray load losses (0.2-37 kW size motor) [1] . . . 6

1.5 Cross current and rotor copper losses for various skew . . . 6

2.1 Flowchart for analytical calculation procedure [2] . . . 19

2.2 Schematic for the harmonic distribution in induction motor . . . 20

2.3 Formula table for various harmonics in induction motor . . . 21

2.4 Stator vibration modes . . . 22

2.5 History of noise minimization methods in electrical motors . . . 25

2.6 Schematic for noise generation in electrical motors (IM) . . . 26

3.1 Simple spectral analysis on the results obtained from FEM . . . 31

3.2 Illustration of 2D spectral analysis . . . 33

3.3 Mode number and frequency separation of radial forces . . . 35

3.4 Demonstrating the accuracy of the spectral analysis . . . 36

3.5 Stator schematic showing the nodal points . . . 37

3.6 Identification table for different modes of vibration . . . 37

3.7 Phase angle at circumferential positions 1, 3, 5 and 7 i.e., for mode-2 identification . . . 38

3.8 Simulation tool specification . . . 38

4.1 2D FEM models for various rotor slot number Qr. . . 42

4.2 Characteristics for various rotor slot numbers . . . 43

4.3 Flux density harmonics for various rotor slot numbers . . . 44

4.4 Torque ripple harmonics for various rotor slot numbers . . . 46

4.5 Time variation of the spatial force harmonics . . . 47

4.6 Frequency spectra for the radial air gap forces for various rotor slot number 48 4.7 Radial air gap force spectra for various rotor slots computed using ASLERM . . . 49

4.8 Spatial harmonic spectra for various eccentric rotors . . . 55

4.9 Spatial harmonic variation with time . . . 56 vii

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4.10 Comparison of radial air gap forces computed with varying eccentricities 57

4.11 FEM sliced models for non-skewed and skewed rotors . . . 60

4.12 Flux density in the air gap (T) . . . 60

4.13 Spatial air gap flux density spectra for the sliced models . . . 61

4.14 Torque ripple comparison for skewed and non-skewed rotors . . . 61

4.15 Analytical results for skewed and non-skewed rotor models . . . 62

4.16 Sound intensity for varying static eccentricity . . . 64

5.1 2D FEM models for dual rotors . . . 68

5.2 Rotor slot types I & II . . . 69

5.3 Combination of rotor(1) in Type I and rotor(1) in Type II . . . 69

5.4 Rotor slot dimensions . . . 71

5.5 2D FEM models for sinusoidal rotors . . . 71

6.1 Instantaneous torque for the dual rotors . . . 74

6.2 Space harmonics for dual slot rotors at one instant of time . . . 76

6.3 Torque ripple comparison for the dual rotors . . . 78

6.4 Spatial spectra variation with time for the dual rotors . . . 79

6.5 Radial force spectrum for dual rotors . . . 80

6.6 Sound intensity for dual rotors . . . 81

6.7 Electromagnetic torque for sinusoidal rotors . . . 82

6.8 Space spectrum at one instant of time for sinusoidal rotors . . . 84

6.9 Spatial spectrum variation with time for sinusoidal rotors . . . 85

6.10 Radial force spectrum for sinusoidal rotors . . . 88

6.11 Sound intensity for sinusoidal rotors . . . 89

6.12 Rotor designs with sinusoidal modulated slots . . . 90

6.13 Characteristics for 36-28 sinusoidal asymmetrical rotors . . . 91

6.14 Asymmetrical versus standard rotors – Radial force spectra . . . 93

6.15 Sound intensity for asymmetrical rotors . . . 95

7.1 Induction motor efficiency categorization tables . . . 107

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Introduction

Induction motors are one of the widely used motors due to their robust construction and the ability to control their speed – thanks to the latest developments in power electronics which made this control possible. Applications of induction motors range from many industry sectors, including power industry, food & beverage, metal processing, 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 and vibrations are major goals for the motor manufacturers. Motor manufacturers today, have shifted their focus away from efficiency class 2 (EFF2¹) motors to higher efficiency 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’s electricity consumption of induction motors accounts for approximately 55–65% of the industrial electricity consumption, see Fig. 1.1. Hence even a smallest improvement in motor efficiency (or reduction in losses), can significantly reduce the energy consumed globally or in a single installation. The smaller motors, which are of major interest in this thesis, generally have efficiency values around 70–90%, and loss reductions 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, the emerging 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 improvement of induction motors. Not only on the motor side but there is an equal opportunity for energy savings by looking at the whole system using variable speed drives. Its

1see Appendix 1

2Information Management Systems

1

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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 motors being sold each year do not meet the efficiency standards. Replacing motors with a higher efficiency (NEMA premium or EFF1) will save approximately 3 to 8 percent in electricity per year and this replacement can have a dramatical impact on the induction motor market over the next 3 years [3].

Electric motors account for about 65 percent of the total electricity consumption in the industrial sector and 38 percent in the service sector of Sweden. The so-called asynchronous motor, is the most common motor type and accounts for 90 percent of 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, compressors and for air conditioning in apartments [4]. The motor distribution in Sweden is as shown in Fig. 1.2.

1.2 Background of the thesis

The focus of this work, as the title suggests, is mainly on improving the performance of induction motors, in particular the reduction of losses. Such improvements can be achieved either by modifying the control or by introducing some changes in the machine design. The initial focus has been on machine design improvements where traditionally compromises have to be made e.g., introducing rotor skew reduces the noise and vibration level but increases the stray losses due to cross-currents between 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 eliminate these stray losses is to keep the rotor slots straight and investigate new methods to tackle 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 radial forces produced in the air gap [5]. These forces can be characterized as rotating waves with different mode³ numbers having different distribution around the air gap circumference. These waves propagate at various frequencies around the air gap, acting both on stator and the rotor [7]. Radial forces which lead to deformations such as the 4-node (mode-2) vibrations in 4-pole motors produce significant magnetic noise [8]. Existing techniques to reduce magnetic forces and noise are to use an optimum stator and rotor slot combinations [9] and by skewing the rotor slots [10].

A design modification, which allows the rotor slots to be non-skewed i.e., keeping them straight to avoid cross current losses and the use of asymmetrical rotor slots to keep magnetic noise at a desirable level will be the main topic of this thesis. A design methodology to introduce various asymmetries in the rotor slots and methods to analyze these motors will be presented in the thesis. The focus will be on 4-pole induction motors for which some novel rotor configurations with asymmetrical slots will be investigated. The analytical source code, ASLERM⁴has been used for the radial magnetic force and noise calculations in the standard motors. Finite element method (FEM) analysis is adopted to obtain accurate results for the new rotor designs. Prior to this the FEM results are validated against the analytical results on the standard motors.

3The term ‘mode’ denotes the number of complete cycles of a physical quantity along the periphery of the stator [6]

4ASLERM – Analytical program for calculating radial forces and magnetic noise in induction motors (coded in FORTRAN language)

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The work presented in this thesis focuses on investigating the effects of introducing asymmetrical rotor slots on the performance of induction motors. Spatial flux density spectra, magnetic forces, vibration modes and magnetic noise of induction motors are studied through analytical and FEM simulations. A novel method for magnetic force calculations from FEM results data is introduced to study the asymmetrical rotor slot designs.

The work presented in this thesis is purely theoretical. For a complete picture of the novel concepts experimental investigation will be required to verify the theoretical investigations in this thesis. Furthermore the starting performance must be investigated both theoretically and experimentally. The main challenge is to design a rotor configuration having minimum rotor bar losses with lower noise level and which also has good starting characteristics. Ultimately, the disadvantages of skewing the rotor should be eliminated and thus an improved rotor design can be obtained. This is proposed as future work.

1.2.1 Skewing in sinusoidal fed induction motors

Space and time harmonic fields associated with the parasitic effects in induction motors are produced due to the slotting, phase unbalance, rotor eccentricity, magnetic saturation, and magnetostrictive expansion of the core laminations. The contribution of the above sources can be seen in the form of radial air gap forces in the motor. These harmonic fields induce emf, which due to the short circuited end rings circulate currents in the rotor windings (bars). These harmonic currents in the rotor interact with the harmonic fields from stator to develop harmonic torques, vibrations and magnetic noise. Skewing is the technique used since a few decades in induction motors where the rotor or the stator slots are twisted to get a more uniform torque, less noise, and better voltage waveform. The most common and effective method is to skew the rotor slots by one stator slot pitch [10].

By introducing skew, the voltage induced by the flux is displaced longitudinally along the rotor bar resulting in increased reactance of the rotor bar [5]. The axial variation of the flux is subsequently increased and hence additional losses are consequential. The change in flux along the rotor axis induces a voltage and hence currents will flow through the bars via the laminations. This basic phenomena of cross currents in the rotor bars can be explained using the diagram shown in Fig.

1.3, where φ1and φ2are the fluxes enclosed, say in loop 1 and 2, respectively. The rotor 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 enclosed by the rotor bars, high frequency currents will flow in the rotor bars and the laminations. 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

Ws

I2

I1

I2

Ws

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 over the normal losses that are considered in usual induction motor performance calculations. The composition of stray load losses for low voltage induction motors is shown in Fig. 1.4. As seen, the cross current losses can be a significant part of the overall stray load losses if the rotor is skewed. The cross current losses vary depending on the amount of skew, inter-bar resistance (cross resistance) and the stator/rotor slot ratio [1].

Some pre-calculations were carried out using an analytical program called EDDY⁵ on 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 rotor bar and lamination. It is observed that stray load losses decrease to zero when the rotor skew is removed.

In general the positive and negative effects of removing skew can be listed as shown in Table 1.1

1.2.2 Additional losses and magnetic noise in inverter fed induction motors

The output voltage and output current of an inverter are non-sinusoidal and contain higher 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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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 -2

log(U) [ȍ-m]

Power losses [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 effects

Increased 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 winding increase as compared to an equivalent machine with sinusoidal stator current. The magnetic noise of an induction motor fed from a pulse width modulation (PWM) inverter 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 magnetic noise is lower, usually 2 to 7 dB(A) [11]. The radial magnetic force spectrum for an inverter fed induction motor is richer and the chance of matching the exciting frequencies with the natural frequencies of the stator system is increased. It is a well known fact that a right choice of slot combination is the key for a quiet operation of a sinusoidal fed motor, however, it is proven that noise produced from a PWM or inverter fed motors are invariant of the slot combinations [1]. Conventional design techniques such as skew does not work effectively for inverter fed machines due to the 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 provides the possibility to eliminate or cancel the harmful force and noise components generated from either the motor or the converter as long as an appropriate modulation technique is employed. The slot ripple produced by equi-slot pitched slots can be removed by employing irregular slot pitch.

It is worthwhile to note that magnetostrictive forces (not considered in this work) will have a significant effect on the magnetic noise in inverter fed induction motors [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 mainly due to the cross currents flowing from the conductor into the laminations. These losses are strongly related to the contact resistance between the rotor bar and the laminations which depends on several factors including the casting process. The main research question investigated in this work is to look for a substantial reduction of cross-current losses keeping the noise level to a possible minimum. This can be

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achieved by simply removing skew and by employing various slot combinations.

However, it is important to note that the skew removal causes additional parasitic effects such as an increase in noise level and undesirable dips in the starting torque characteristics [12]. To minimize these effects the rotor design can be modified and 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 finite element calculation methods are employed to evaluate the motor performance.

1.3.1 Proposed design solution – Non-skewed asymmetrical rotor

The increased magnetic noise (radial forces) due to the parasitic effects caused by eliminating skew can be minimized by adopting the following design changes in the rotor:

• choosing rotor slot number, Qrwith 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 been reported widely since the late 40’s [9]. The existing slot combinations are the end-result of such optimization. The option of increased air gap is obviously not a desirable design change, the reason being that smaller the air gap the larger is the torque produced by any motor. Asymmetrical rotor is the best possible design change of interest which needs rigorous investigation via simulations and experiments. Optimizing the stator core is not the solution for eliminating the noise at 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 induction rotors is proposed and investigated in this thesis.

1.3.2 Challenges in the new design solution

The following challenges have to be met for the above mentioned design solution:

1. To design a high performance non-skewed rotor, either for symmetrical or non-symmetrical rotor, a detailed information about the dominating noise (radial forces) component characteristics associated with various rotor designs, is necessary. A realistic and accurate method has to be developed for this purpose.

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1.4. LITERATURE SURVEY 9

2. It is important to identify the various harmful vibration modes of the stator yoke which can lead to mechanical deformations. A method to calculate radial forces and noise level using the data obtained from FEM calculations is needed. FEM has been chosen as it has an added advantage to the accuracy that permeance variations can be considered by just modifying the rotor geometry.

3. Asymmetry can be introduced in many ways and a quantified method to introduce 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 size of motors considered here) in the air gap however can lead to synchronous torques during the start. Thus starting properties have to be thoroughly investigated.

5. Ultimately, in order to cut down the manufacturing lines a common rotor design for 4-6-8 pole motors could be designed.

It is important for the reader to note that the work presented here covers only the first three challenges.

1.4 Literature survey

In this section a brief description of the previous work done related to the modulated or asymmetrical slots and various methods to analyze the radial magnetic forces in induction motors are presented.

1.4.1 Concept of asymmetrical rotor slots

Very 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 have been reported in [14], where a method to analyze the torque and voltage equations of an induction motor with an unequal slot pitch cage rotor is developed and the electromagnetic force wave characteristics with various non-uniform slot pitches are clarified. One of the important conclusion from this work is that the unbalance in the current and torque harmonics are increased, with the increase in the asymmetry but the fundamental component remains almost unaffected [14].

Toshiba corporation proposed the inequality slot pitched rotor to make the energy dissipation distributed for a wide frequency range. This idea was proved by power spectrum and auto-correlation functions of information theory, and tested

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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 reduce magnetic noise in induction motors driven by inverter is to employ non-uniform pitched slots [17]. Effects of asymmetrical rotor slot pitch is studied and reported by A. Tenhunen et.al, the report shows improved force spectrum when the rotor slotting is modified by introducing asymmetrical slot pitch [13].

1.4.2 Calculation of magnetic forces and noise – Methods other than the classical analytical approach

To evaluate spatial and time harmonics a two-dimensional approach is needed to study the radial magnetic forces. It is evident that magnetic forces form the key characteristic in predicting the magnetic noise in induction motors. The relevant work contributed to such 2D approaches in the past is briefly described in this section.

V. Ostovic and G. Boman proposed a method for computation of space and time harmonics of radial air gap force in induction motors. The radial air gap force in two dimensions (space and time) is computed by using the Magnetic Equivalent Circuit (MEC) method. The results obtained show clearly the influence of rotor skewing, rotor static and dynamic eccentricity on the force spectrum [8]. Similarly, a quantitative method to analyze the time and space harmonics in the magnetic field distribution, using FEM with the time stepping method considering the rotor movement 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. Vandevelde has proposed a method using magnetic equivalent circuit (MEC). In this method the various effects which determine the modal contents of radial force are taken into account: the spectrum of applied voltage (for inverter supply), the windings, slotting, eccentricity and saturation of iron. Magnetic equivalent circuit developed in frequency-order domain which is coupled to an electrical circuit, is used for the analysis. These calculations were experimentally verified [19]. Pedro Vincent .et.al have published a method to compute electromagnetic force distribution along the air gap using the results obtained from FEM. The computed radial air gap forces is 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 machine 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 distribution in terms of frequency and vibration mode number. Two dimensional

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1.6. ORGANIZATION OF THE THESIS 11

Fast Fourier Transform (FFT) is applied on the space-time data obtained from FEM simulations. This analysis is the key to identify the dominating modes and the corresponding frequencies of the radial air gap forces and magnetic noise.

• Spectral analysis usually demands bulk amount of data (here from FEM) to obtain the accurate results, which is time consuming and tiresome, hence a simpler method called phase difference method is adopted for the extraction of vibration modes and the corresponding frequency spectrum. This method is similar to the method that is used in real time vibration testing of induction motors.

• The rotor skew is modeled by using the so called slice model in FEM for comparisons with the non-skewed rotor.

• Two methods to introduce asymmetry are proposed, namely, sinusoidal asymmetry and dual asymmetry. Some useful guidelines are formulated for designing these asymmetrical rotors.

• The flux density values obtained from the finite element model are coupled to 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 forces and noise in the conventional induction motor is described in this chapter. An overview of magnetic noise, its causes and sources in induction motors is presented.

A brief explanation of the space harmonics in induction motors and their calculation using analytical formulas is presented. Finally, various methods to calculate radial air gap forces in induction motors are described and the advantages of the chosen 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. The post-processing issues of the finite element result data such as spectral analysis in 1D and 2D, spectral accuracy, the new phase difference method, combined FEM and analytical method are described. An optimal method using FEM which is implemented in this work for analyzing the asymmetrical rotor slots is finally described in this chapter.

Chapter 4: Calculations and validations on a 4-pole, 15 kW standard motor with and without eccentricity is presented in detail in this chapter. Comparisons are made against the analytical and FEM simulations. The influence of skew using

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FEM based sliced model is studied and explained in a separate section. Optimizing the rotor slot number for the same stator is also presented. This chapter concludes that the proposed method of analysis in Chapter 3 can be applied for studying the new 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 sinusoidal rotors and combined dual sinusoidal rotors is presented in this chapter. The design rules and the predictions behind the design rules are explained. Proposals of some designs are discussed and the simulations performed on some selected designs are described in Chapter 6

Chapter 6: This chapter consists of results and comparisons of standard rotors against asymmetrical rotors. The identification of dominant forces and noise components for various proposed designs are explained and their performance is analyzed. Some guidelines to design asymmetrical rotor slots are also given. Result comparisons for the novel asymmetrical slot rotors and standard rotors with and without 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 induction rotors.

1.7 Publications

The work presented in this monograph has resulted in two international and one residential conference publications listed below:

1. Design and Analysis of Asymmetrical Rotor for Induction Motors R. 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 Motors with Non-Skewed Asymmetrical Rotor Slots

R. 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 Rotor Slots

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1.7. PUBLICATIONS 13

R. Chitroju and C. Sadarangani.

Residential Conference on Noise and Vibration: Emerging Methods Keble College, Oxford, UK, April 2009.

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Chapter 2

Radial magnetic forces and

magnetic noise in induction motors – Methods of analysis

This chapter deals with the theoretical description of analytical and FEM methods for computing magnetic forces and noise in induction motors. The fact that remov- ing skew increases magnetic noise makes it important to understand and identify the sources of radial forces leading to magnetic noise and develop methods to reduce them. Some interesting findings on different existing methods for force computation in induction motors are presented in this chapter and the methods which are used in this thesis are justified. Finally, a brief theory about the noise production, its significance 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 through space and time, with transfer of energy. For example, a mechanical wave is a wave that 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 induction motor), then they would all vibrate as one, with no delay in the transmission of the vibration. On the other hand, if all the parts were free, then there would not be any transmission of the vibration. Wave physics reveals that in any traveling wave the phase of a vibration i.e., its position within the vibration cycle is different for neighboring points in space because the vibration reaches these points at different times [21].

The flux density waves produced by the interaction of stator mmf waves and the permeance waves can also be treated as traveling waves as they rotate along

15

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the circumference of the air gap with finite wavelengths¹. Such a traveling wave mathematically can be described by Eq. 2.1, where k represents the wavelength number, B is the amplitude (half the peak-to-peak value) of the flux density, ψ is phase angle at time zero w.r.t the reference position, ωkt is the temporal angular frequency. At a particular position θ in space, b is the flux density periodic function with temporal frequency given by fk= ^ω_2π^kt

b (θ, t) = B · cos (kθ − ωktt + ψ) (2.1) Thus the spatial angular speed of the wave along the circumference becomes wks = ^ω_k^kt. The wavelength (k) of the traveling wave can be written as a spatial harmonic 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· ω1

p (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 two flux 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 m^th and n^thspace 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 force wave, both in general can be written as in Eq. 2.6 and its wave characteristics are tabulated in Table 2.1.

σr(θ, t) = ˆσr· cos r

pθ − vrω1t +ψr

r

(2.6)

1the distance between two sequential crests (or troughs). Generally in machine domain it is represented as the pole-pair number of the harmonic.

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2.1. ANALYTICAL APPROACH FOR CALCULATING RADIAL FORCE

WAVES 17

Table 2.1: Wave parameters for the force wave produced due to interaction of two flux density waves

Positive wave Negative wave σr+ = ¹₂BmBn σr− = ¹₂BmBn

r+ = m + n r− = m − n

θ+ = ^θ^m^+θ₂ ⁿ r− = ^θ^m^−θ₂ ⁿ vr+=^mv^m_r^+nvⁿ

+ vr− =^mv^m_r^−nvⁿ

−

fr+ =mvm+ nvn f_r− =mv_m− nv_n

Consequently, the traveling force wave results in vibrations which can also be treated as series of sinusoidal waves with phase displacement. The radial displacement can be written as in Eq. 2.7.

Yr(θ, t) = ˆYr· 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 be calculated by using an appropriate radiation model [22]. The description of such models 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 level in decibels (dB) is calculated by using Eq. 2.8

Lr= 20 log

9.05 × 10⁴fr

q N_rel² Yr

(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, different analytical 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 details included in the calculation method e.g. consideration of the geometry, material properties, effects of saturation and the eccentricities. Though most of the methods use the concept of traveling waves for radial force calculations in common, the basic computation of mmf and air gap permeance differs. The modeling of the air gap is necessarily a key for the accuracy of any such method. Some of these methods are described briefly in this section with regard to the flux density and radial force wave calculations in the air gap of induction motors.

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2.1.1 Classical theory for calculating radial magnetic forces

The procedure for calculating radial magnetic forces and further noise is shown in the flow chart, Fig. 2.1. The flux density waves are the result of the interaction of mmf waves from the stator windings, also called as the current loading and the air gap permeance waves. The product of the fundamental mmf and the constant term of permeance gives the fundamental air gap flux density which induce secondary currents, and produce torque. A detailed summary of the harmonics generation in induction 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 widely documented in literature [7], [5], [22]. The most common permeance wave model implemented is described in [25], [22], [26], [23]. However, the permeance waves due to slotting, eccentricity and saturation are evaluated separately using Fourier components and then are combined together. By doing so, in this method some of the unknown effects could have been excluded.

Consideration of skew

Skewing of induction rotors is one of the main topic in this thesis work, hence a brief description of the equations considered in the analytical calculations are given below.

Skewing is usually defined in terms of stator slot pitch, which is also called the skew pitch. For example for a motor with 36/28 slot configuration, the rotor slots are 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 pitch SP and is given by Eq. 2.9 where, Qsis the number of stator slots.

γskew= 2π

QsSP (2.9)

Skewing the rotor slots aids in the production of the damping fields which can be calculated using Eq. 2.10. Damping fields are the fields induced by the rotor residual fields into the air gap which damp the main stator fields. These fields induce various harmonics in the rotor.

Baν = χνBν (2.10)

where, Bν is the undamped stator field and the damping factor χν is given by Eq. 2.11

χν = s

1 − ξ_skew² ξ_T²2 (1 + σg) − (ξskew.ξT)²

(1 + σg)²(1 + (βν/sν)) (2.11)

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2.1. ANALYTICAL APPROACH FOR CALCULATING RADIAL FORCE

WAVES 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 )

,

( _P _P

P P P Z \

¦

^A ^x ^t

t x

a ( , ) cos( _O _O)

O /O O Z \

/

/ ^x^t ^o

¦

^x ^t

) (

cos )

,

( v v v

v xt B vx t

b Z \

) (

ˆ cos ) ,

( r r r

r xt V rx Z t \

V

) (

ˆ cos ) ,

( r r r

r xt Y rx t

Y Z \

) (

ˆ cos ) ,

( r r r

r xt Y rx t

Y Z \

dB N Y f L

rel r r

r 20log(9.05u10⁴ ˆ )

Nrel

Figure 2.1: Flowchart for analytical calculation procedure [2]

where, sνis the harmonic slip, βν is the specific rotor resistance and ξskewis the skew factor given by Eq. 2.12, σg is the rotor stray flux, ξT is an empirical slotting correction factor.

ξskew= sin ν^γ^skew₂

ν ^γ^skew₂ (2.12)

Skew also has the influence on the production of rotor residual fields in cage winding produced by the rotor bar currents.

Iring,ν = 1 Λν

ξ_skewξ_ν (1 + kν)

B_ν

√2 (2.13)

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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 rotor

Saturation in the magnetic circuit

Slot opening in the stator and rotor

Mean air gap length

Winding harmonics

P p Ps

p g s_st _st _st

st O

Q Qe pr1

6

m 3p

Q

7

p Q g _s

g ₁

Q Q p Q P_s

5 4 1 2,3

Rotor Residual Field, Rotor field 9

st

O v O ve O vm

g

O v gQ p O p O P^s

r

g ₂

O

Residual fields from eccentricity

Residual fields from

saturation

Residual fields from fundamental p

Q g Q g

r s

g ₁ ₂

Q

Armature Reaction of respective primary field from stator

Winding harmonics for resp armature reaction

st r

g g₂Q v

O Og g₂Qrve Og g₂Qrvm

12 13

11

10 MMF

fundamental wave

p Q g _r

g ₂

O Og g₂QrPs

) (

) ,

( _O _O _O

O xt B OxZt\

b

) (

) ,

( v v v

v xt B vx t

b Z \

Cross section changes at the end

RotorAirgapStatorSupply

Figure 2.2: Schematic for the harmonic distribution in induction motor

Improved performance characteristics of induction machines with non-skewed symmetrical rotor slots

Improved Performance Characteristics of Induction Machines with Non-Skewed Asymmetrical Rotor Slots

RATHNA KUMAR SASTRY CHITROJU

Licentiate Thesis

Electrical Machines and Power Electronics School of Electrical Engineering Royal Institute of Technology (KTH)

Stockholm, Sweden 2009

Acknowledgments

List of Figures

Contents

Chapter 1

Introduction

1.1 Significance of improved efficiency in induction motors

1.2 Background of the thesis

1.2.1 Skewing in sinusoidal fed induction motors

1.2.2 Additional losses and magnetic noise in inverter fed induction motors

1.3 Motivation and goal of the thesis

1.3.1 Proposed design solution – Non-skewed asymmetrical rotor

1.3.2 Challenges in the new design solution

1.4 Literature survey

1.4.1 Concept of asymmetrical rotor slots

1.4.2 Calculation of magnetic forces and noise – Methods other than the classical analytical approach

1.5 Scientific contribution

1.6 Organization of the thesis

1.7 Publications

Chapter 2

Radial magnetic forces and

magnetic noise in induction motors – Methods of analysis

2.1 Analytical approach for calculating radial force waves

2.1.1 Classical theory for calculating radial magnetic forces

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