Non-invasive detection of air gap eccentricity in synchronous machines using current signature analysis

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Synchronous Machines Using Current Signature Analysis

by

Ilamparithi Thirumarai Chelvan

B. E., Anna University, India 2005

M.Tech., Indian Institute of Technology, Delhi, India 2007

A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY

in the Department of Electrical and Computer Engineering

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Non-Invasive detection of air gap eccentricity in synchronous

machines using current signature analysis

by

Ilamparithi Thirumarai Chelvan

B. E., Anna University, India 2005

M.Tech., Indian Institute of Technology, Delhi, India 2007

Supervisory Committee

Dr. Subhasis Nandi

Supervisor (Department of Electrical and Computer Engineering)

Dr. Ashoka K. S. Bhat

Department Member (Department of Electrical and Computer Engineering)

Dr. Andrew Rowe

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Supervisory Committee

Dr. Subhasis Nandi

Supervisor (Department of Electrical and Computer Engineering)

Dr. Ashoka K. S. Bhat

Department Member (Department of Electrical and Computer Engineering)

Dr. Andrew Rowe

Outside Member (Department of Mechanical Engineering)

ABSTRACT

Air gap eccentricity fault is one of the major faults that afflict the life and performance of rotating machines. Eccentricity fault, in the worst case, causes a stator rotor rub. Thus, a condition monitoring scheme to identify eccentricity fault at its initial stage is necessary. The most widely practised air gap monitoring schemes for synchronous machines are expensive and invasive sensors based. This work has focussed on developing an inexpensive, non-invasive, air gap monitoring technique especially for salient pole synchronous machines. Motor current signature analysis has been mostly preferred for the above mentioned purpose. By monitoring the frequency spectrum of the machine’s current, faulty condition can be isolated provided the fault specific frequency components are known beforehand.

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The research work, therefore, has developed a specific permeance function using binomial series for salient pole machines that can be used to identify eccentricity specific harmonic components in the line current spectrum. Then by performing the magneto-motive force – specific permeance analysis the characteristic frequency components have been predicted. In order to validate the prediction as well as to identify a trend in the variation of these harmonic components with changing levels of eccentricity, mathematical models of a three phase reluctance synchronous machine and a three phase salient pole synchronous machine based on modified winding function approach have been developed. The models have been made to incorporate static, dynamic and mixed eccentricity conditions of varying severity. Also time stepped finite element based models have been simulated in Maxwell-2D to verify the theoretical predictions. With the help of eccentrically cut bushings, experiments were then conducted in the laboratory to corroborate the proposed eccentricity detection scheme.

It has been observed that non-idealities such as supply time harmonics, machine constructional asymmetry, supply voltage unbalance etc. negatively impact the diagnostic technique. Consequently, a residual estimation based fault detection scheme has been implemented successfully to distinguish eccentricity fault from healthy condition. Moreover, detection logic have been put forth to discriminate the type of eccentricity and to estimate the severity of the fault.

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List of Tables

Table 2.1 Different values of the coefficients for healthy, DE, SE and ME RSM………... 47 Table 2.2 Different values of the coefficients for healthy, DE, SE and ME SPSM……… 48 Table 3.1 First few harmonic components of stator line current phasor under different conditions……… 56 Table 3.2 First few harmonic components of field current under different conditions………. 65 Table 5.1 Normalized magnitudes of triplen harmonics in stator line current space

vector of RSM under HE and SE conditions………. ……… 120 Table 5.2 Normalized magnitudes of fault specific harmonics in stator line current space vector of RSM under HE and DE conditions……….. 122 Table 5.3 Normalized magnitudes of fault specific harmonics in stator line current

space vector of RSM under HE and ME conditions………. 122 Table 5.4 Normalized magnitudes of triplen harmonics in stator line current space

vector of SPSM under HE and SE conditions……… 125 Table 5.5 Normalized magnitudes of fault specific harmonics in stator line current

space vector of SPSM under HE and DE conditions……… 125 Table 5.6 Normalized magnitudes of fault specific harmonics in stator line current

space vector of SPSM under HE and ME conditions…….... ……… 126 Table 5.7 Normalized magnitude of +3f and -9f harmonic components of line

current space vector at full load and different levels of eccentricity for the RSM………... 129 Table 5.8 Normalized magnitude of -5f and +7f harmonic components of line

current space vector at full load and different levels of eccentricity for the RSM………. 129 Table 5.9 Normalized magnitude of +0.5f and +2f harmonic components of line

current space vector at full load and 40% ME conditions for the RSM………...……… ………. 130

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Table 5.10 Normalized magnitude of 3f and 9f harmonic component of line current at full load and different conditions with 8 coefficients……… 134 Table 5.11 Normalized magnitude (in dB) of +15f harmonic component of line

current at full load and different SE conditions……… ………. 136 Table 5.12 Normalized magnitudes of triplen harmonics in stator line current space vector of inverter fed RSM under HE and SE conditions………….. 153 Table 5.13 Normalized magnitudes of fault specific harmonics in stator line current

space vector of iniverter fed RSM under HE and DE

conditions……….. 154 Table 5.14 Normalized magnitudes of fault specific harmonics in stator line current

space vector of iniverter fed RSM under HE and ME

conditions……….. 154 Table 5.15 SE severity estimation in RSM based on percentage increase in triplen harmonic components of stator current phasor at full load… ……… 159 Table 5.16 DE fault severity estimation in RSM based on percentage increase in 30 Hz and triplen harmonics of stator current phasor at full load…….. 161 Table 5.17 SE severity estimation in SPSM based on percentage increase in triplen harmonic components of stator current phasor at rated current using data after residual estimation………..……. 163 Table 5.18 DE severity estimation in SPSM based on percentage increase in 30 Hz

and triplen harmonics of stator current phasor at rated current using data before residual estimation……….…... 163 Table 5.19 SE severity estimation in SPSM based on percentage increase in 30 Hz

and triplen harmonics of stator current phasor at rated current using data before residual estimation……….…... 164 Table 5.20 SE severity estimation in SPSM based on percentage increase in SE

specific harmonic components of field current using data after residual estimation……….. 164 Table 5.21 DE severity estimation in SPSM based on percentage increase in 30 Hz

and SE specific harmonics of field current using data before residual estimation……….. 165

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Table D1 Different tap positions on the transformer to obtain balanced and unbalanced supply data sets……….. 194

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List of Figures

Figure 1.1 C1 is the centre of stator and C2 is the centre of rotor. In a healthy machine C1 and C2 coincide. C1 is the centre of rotation for dynamic eccentric (DE) condition, whereas C2 is the centre of rotation for static eccentric (SE) condition. With mixed eccentricity (ME) the centre of rotation can be anywhere between C1 and C2. ...3 Figure 2.2 Air gap of healthy salient pole machine for 𝜃𝑟 = 0. Effective d-axis air

gap length = d meters; effective q-axis air gap length = q meters. ... 28 Figure 2.3 Top: Inverse air gap; Middle: Square of air gap; Bottom: Inverse of square

of air gap. All the plots are for 𝜃𝑟 = 0. ... 29

Figure 4.1 Elementary induction machine………70

Figure 4.2 MWFA simulated, stator magnetizing inductances of RSM under different eccentric conditions; (a) Healthy condition (b) 40% SE condition (c) 40% DE condition (d) 40% ME (20% SE and 20% DE) condition ... 73

Figure 4.3 MWFA simulated, stator magnetizing inductances of SPSM under

different eccentric conditions; (a) Healthy condition (b) 50% SE condition (c) 50% DE condition (d) 50% ME (25% SE and 25% DE) condition. .... 73 Figure 4.4 MWFA simulated, stator mutual inductances of RSM under different

eccentric conditions; (a) Healthy condition (b) 40% SE condition (c) 40% DE condition (d) 40% ME (20% SE and 20% DE) condition ... 74 Figure 4.5 MWFA simulated, stator mutual inductances of SPSM under different

eccentric conditions; (a) Healthy condition (b) 50% SE condition (c) 50% DE condition (d) 50% ME (25% SE and 25% DE) condition ... 75 Figure 4.6 MWFA simulated, stator rotor mutual inductance of RSM under different

eccentric conditions; (a) Healthy condition (b) 40% SE condition (c) 40% DE condition (d) 40% ME (20% SE and 20% DE) condition ... 76 Figure 4.7 MWFA simulated, stator rotor mutual inductance of SPSM under different

eccentric conditions; (a) Healthy condition (b) 50% SE condition (c) 50% DE condition (d) 50% ME (25% SE and 25% DE) condition ... 77

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Figure 4.8 MWFA simulated, stator field winding mutual inductances of SPSM under different eccentric conditions; (a) Healthy condition (b) 50% SE condition (c) 50% DE condition (d) 50% ME (25% SE and 25% DE) condition ... 78 Figure 4.9 MWFA simulated, rotor loop magnetizing inductances of SPSM under

different eccentric conditions; (a) Healthy condition (b) 50% SE condition (c) 50% DE condition (d) 50% ME (25% SE and 25% DE) condition ... 79 Figure 4.10 MWFA simulated, rotor loop mutual inductances of SPSM under different

eccentric conditions; (a) Healthy condition (b) 50% SE condition (c) 50% DE condition (d) 50% ME (25% SE and 25% DE) condition ... 81 Figure 4.11 MWFA simulated, rotor loop field mutual inductances of SPSM under

different eccentric conditions; (a) Healthy condition (b) 50% SE condition (c) 50% DE condition (d) 50% ME (25% SE and 25% DE) condition ... 82 Figure 4.12 Flowchart showing steps involved in obtaining inductances using

Magnetostatic solver of FE simulation package MAXWELL -2D ... 84 Figure 4.13 FE simulation models of (a) RSM and (b) SPSM built in MAXWELL –

2D. ... 85 Figure 4.14 FE simulated, stator rotor mutual inductance of RSM under different

eccentric conditions; (a) Healthy condition (b) 40% SE condition (c) 40% DE condition (d) 40% ME (20% SE and 20% DE) condition ... 86 Figure 4.15 MWFA simulated, stator rotor mutual inductance of SPSM under different

eccentric conditions; (a) Healthy condition (b) 50% SE condition (c) 50% DE condition (d) 50% ME (25% SE and 25% DE) condition ... 87 Figure 4.16 MWFA simulated, time domain signal of line current (left) and complex

frequency spectrum (right) of line current space vector of RSM at full load. (a) HE condition (b) 40% SE condition (c) 40% DE condition (d) 40% ME (20% SE, 20% DE) condition... 94 Figure 4.17 MWFA simulated, time domain signal of line current (left) and complex

frequency spectrum (right) of line current space vector of SPSM at full load. (a) HE condition (b) 34% SE condition (c) 34% DE condition (d) 50% ME (33% SE, 17% DE) condition... 95

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Figure 4.18 MWFA simulated, time domain signal (left) and frequency spectrum (right) of field current of SPSM at full load. (a) HE condition (b) 34% SE condition (c) 34% DE condition (d) 50% ME (33% SE, 17% DE)

condition. ... 96

Figure 4.19 Flowchart showing steps involved in setting up TSFE simulation model. 98 Figure 4.20 TSFE simulated, time domain signal of line current (left) and complex frequency spectrum (right) of line current space vector of RSM at full load. (a) HE condition (b) 40% SE condition (c) 40% DE condition (d) 40% ME (20% SE, 20% DE) condition... 100

Figure 4.21 TSFE simulated, time domain signal of line current (left) and complex frequency spectrum (right) of line current space vector of SPSM at full load. (a) HE condition (b) 34% SE condition (c) 34% DE condition (d) 50% ME (33% SE, 17% DE) condition... 101

Figure 4.22 TSFE simulated, time domain signal (left) and frequency spectrum (right) of field current of SPSM at full load. (a) HE condition (b) 34% SE condition (c) 34% DE condition (d) 50% ME (33% SE, 17% DE) condition. ... 102

Figure 5.1 1.5 hp, 3 phase, RSM in its original sheet metal enclosure ... 107

Figure 5.2 2 kW, 3 phase, SPSM in its original sheet metal enclosure ... 107

Figure 5.3 1.5 hp, 3 phase, RSM in new custom built enclosure... 108

Figure 5.4 2 kW, 3 phase, SPSM in the new rugged enclosure ... 108

Figure 5.5 40% SE bushings for RSM. Bushing for driving end (left); bushing for non-driving end (right). All dimensions are in mm... 109

Figure 5.6 40% DE bushings for RSM. Bushing for driving end (left); bushing for non-driving end (right). All dimensions are in mm... 110

Figure 5.7 50% SE bushing for SPSM. All dimensions are in mm. ... 110

Figure 5.8 50% DE bushing for SPSM. All dimensions are in mm... 111

Figure 5.9 40% SE sleeves housing the bearings of the RSM corresponding to driving end (6205ZZ) (left) and non-driving end (6203ZZ) (right). ... 112

Figure 5.10 40% DE sleeves inside the new bearings of the RSM corresponding to driving end (R18Z) (left) and non-driving end (6904Z) (right). ... 112

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Figure 5.11 A bearing of SPSM (6208RSR) (left); the bearing with both 50% SE and 50% DE bushings (right). ... 113 Figure 5.12 Modified end plate of the RSM with provision to accommodate the SE

bushing. ... 113 Figure 5.13 End plate of the RSM with non-driving end bearing and corresponding SE

bushing. ... 114 Figure 5.14 Experimental set up showing the RSM driving a separately excited DC

generator along with the data acquisition system... 116 Figure 5.15 Experimental set up showing the SPSM driving a separately excited DC

generator along with the data acquisition system... 117 Figure 5.16 Experimental complex FFT of line current space vector of RSM at full

load. (a) Healthy condition; (b) 20% SE condition; (c) 40% SE condition; (d) 60% SE condition. ... 119 Figure 5.17 Experimental complex FFT of line current space vector of RSM at full

load. (a) Healthy condition; (b) 40% DE condition; (c) 40% ME (20% SE, 20% DE) condition. ... 121 Figure 5.18 Experimental complex FFT of line current space vector of SPSM at full

load and rated excitation. (a) Healthy condition; (b) 33.34% SE condition; (c) 33.34% DE condition; (d) 50% ME (33.33% SE, 16.67% DE) condition. ... 124 Figure 5.19 Experimental normalized magnitude of +3f component’s residual estimate

of RSM with various degrees of SE at different load conditions. ... 130 Figure 5.20 Experimental normalized magnitude of +7f component’s residual estimate

of RSM with various degrees of DE at different load conditions. ... 130 Figure 5.21 Experimental normalized magnitude of +2f harmonic component’s

residual estimate of RSM under healthy and 40% ME condition with different load settings. ... 131 Figure 5.22 Experimental, normalized magnitude of 3rd harmonic current’s residual

estimate of RSM under healthy (with supply unbalance) and SE conditions with different load settings computed using 8 coefficients and 9 data sets. ... 131

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Figure 5.23 Experimental normalized magnitude of 3rd harmonic current’s residual estimate of RSM under healthy (with supply unbalance) and SE conditions with different load settings computed using 8 coefficients and 9 data sets. 0 Hz was used for determining coefficients. ... 132 Figure 5.24 Experimental normalized magnitude of 3rd harmonic current’s residual

estimate of RSM under healthy (with supply unbalance) and SE conditions with different load settings computed using 14 coefficients and 27 datasets. ... 133 Figure 5.25 Experimental normalized magnitude of 7th harmonic current’s residual

estimate of RSM under healthy (with supply unbalance) and DE conditions with different load settings computed using 8 coefficients and 9 datasets. ... 134 Figure 5.26 Experimental normalized magnitude of 2nd harmonic current’s residual

estimate of RSM under healthy (with supply unbalance) and 40 ME condition with different load settings computed using 8 coefficients and 9 datasets. ... 135 Figure 5.27 Variation of residual estimated fault specific frequency components of the

stator line current space phasor under different eccentric conditions at full load; (a) +9f (SE), (b) +7f (DE), (c) +2f (ME). ... 138 Figure 5.28 Effect of load variation on the fault specific frequency components in the

stator line current space vector under healthy and different eccentric conditions; (a) +9f (SE), (b) +7f (DE), (c) +2f (ME). ... 139 Figure 5.29 Variation of residual estimated fault specific frequency components of the

field current under different eccentric conditions at full load; (a) 4f (SE) (b) 6f (DE) (c) 3.5f (ME). ... 140 Figure 5.30 Effect of load variation on the fault specific frequency components in field

current under healthy and different eccentric conditions; (a) 4f (SE), (b) 6f (DE), (c) 3.5f (ME) ... 142 Figure 5.31 Variation of -180 Hz component’s residual estimate of the stator line

current space phasor under different SE conditions at full load; (a) 0.9 lag, (b) UPF, (c) 0.9 lead. ... 144

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Figure 5.32 Variation of 420 Hz component’s residual estimate of the stator line current space phasor under different DE conditions at full load; (a) 0.9 lag, (b) UPF, (c) 0.9 lead. ... 145 Figure 5.33 Variation of 90 Hz component’s residual estimate of the stator line current

space phasor under different ME conditions at full load; (a) 0.9 lag, (b) UPF, (c) 0.9 lead... 146 Figure 5.34 Effect of load variation on the fault specific frequency components in

stator space current phasor under healthy and different SE conditions at different power factor; (a) 0.9 lag; (b) UPF; (c) 0.9 lead. ... 148 Figure 5.35 Effect of load variation on the fault specific frequency components in

stator space current phasor under healthy and different DE conditions at different power factor; (a) 0.9 lag; (b) UPF; (c) 0.9 lead. ... 149 Figure 5.36 Effect of load variation on the fault specific frequency components in

stator space current phasor under healthy and different ME conditions at different power factor; (a) 0.9 lag; (b) UPF; (c) 0.9 lead. ... 151 Figure 5.37 Effect of load variation on the fault specific frequency component in field

current under healthy and different ME conditions at 0.9 lead power factor. ... 152 Figure 5.38 Variation of fault specific frequency component’s residual estimate of

inverter fed RSM’s stator line current space phasor under different eccentricity conditions at different loads; (a) +540 Hz variation under SE condition, (b) -660 Hz variation under DE condition, (c) +90 Hz variation under ME condition. ... 156 Figure 5.39 Complex frequency spectrum of (a) utility fed RSM and (b) inverter fed

RSM under no load healthy condition………..157 Figure 5.40 Variation of average value of percentage increase in negative sequence

third harmonic component and positive sequence ninth harmonic component present in the stator line current phasor of RSM under different load conditions. ... 160 Figure 5.41 Variation of average value of difference in the percentage increase of 30

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sequence ninth harmonic component present in the stator line current phasor of RSM under different load conditions. ... 161 Figure A.1 Rotor lamination of the RSM showing the rotor bar arrangement. All

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List of Abbreviations

DAQ Data Acquisition

DE Dynamic Eccentricity

DSP Digital Signal Processor

EMF Electro Motive Force

FE Finite Element

FFT Fast Fourier Transform

FTI Fault Type Indicator

HE Healthy condition

IE Inclined Eccentricity

MCSA Motor Current Signature Analysis

ME Mixed Eccentricity

MMF Magneto Motive Force

MWFA Modified Winding Function Approach

PMSM Permanent Magnet Synchronous Machine

PSD Power Spectral Density

RSM Reluctance Synchronous Machine

SE Static Eccentricity

SPSM Salient Pole Synchronous Machine TSFE Time Stepping Finite Element

UMP Unbalanced Magnetic Pull

UPF Unity Power Factor

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List of Symbols

A Fourier series coefficient of inverse of air gap

B Air gap flux density in T

D Variable related to dynamic eccentricity

F Magneto-motive force (MMF) in A-t

G Coefficient of components of inverse air gap function

𝑰� Current phasor in A

𝐼𝑒𝑠𝑡

�� Estimated fault current signature

𝐈𝐦 Array of measured current phasors 𝐼�� 𝐼𝑚1 , �� etc 𝑚2

Ires Residual estimate of fault current signature

J Moment of inertia in kg-m2

L Inductance in H

M Coefficient of components of MMF

Na, Nb Winding functions

P Specific permeance in Wb/At-m2

R Resistance in Ω

S Variable related to static eccentricity

T Torque in Nm

𝑽� Voltage phasor in V

𝐕𝐦 Matrix of measured voltage phasors �� 𝑉𝑉1+𝑚1, �� etc 1−𝑚1

W Energy in J

as Absolute value of static eccentricity in m

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a1 Normalized value of static eccentricity with respect to air-gap

a2 Normalized value of dynamic eccentricity with respect to air-gap

f Supply frequency in Hz

fde Harmonics of DE machine in Hz

fhe Harmonics of healthy machine in Hz

fl Low frequency sideband component in Hz

fme Harmonics of ME machine in Hz

fr Rotor frequency in Hz

fse Harmonics of SE machine in Hz

g Air gap in m

g-1 Inverse of air gap in m-1 𝑘� Complex coefficients

𝐊 Array of complex coefficients 𝑘��,𝑘₀ �� etc ₁

l Stack length of motor in m

n Order of space harmonics

p Fundamental pole pair number

r Mean radius of motor in m

s Slip

𝜑 Angular reference in stator in rad 𝜑′ Angular reference in rotor in rad 𝜃𝑑 Rotor pole width in rad

𝜃𝑟 Rotor position in rad

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ω Angular supply frequency in rad/s

µ0 Permeability of air in H/m

λ Flux linkage in Wb-t

ν Number of data sets used for residual estimation < > Average value

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Acknowledgments

I would like to express my sincere thanks and gratitude to my supervisor, Dr. Subhasis Nandi, for his time, effort and guidance throughout this program. I would also like to thank him for providing me with financial support and adequate research facilities. I would like to thank my supervisory committee members, Dr. A. K. S. Bhat for being a constant source of inspiration, and Dr. A. Rowe for his time and encouragement.

I would like to thank the University of Victoria for awarding me graduate fellowship and travel grants.

I would like to express my special thanks to Mr. Rodney Katz, for his suggestions and preparation of the experimental setup, Mr. Mike Milroy for fabricating the experimental machine, Mr. Robert Fichtner for providing me with equipment, and Mr. Kevin Jones for providing support with software maintenance. My thanks also to Ms. Vicky Smith, Ms. Moneca Bracken, Ms. Lynne Barrett, Ms. Janice Closson, Ms. Monique Frappier, Mr. Dan Mai, Mr. Paul Fedrigo, Mr. Brent Sirna, Ms. Lynn Palmer, Mr. Erik Laxdal and Mr. Steve Campbell for helping me during my stay here.

I would like to convey my gratitude to Dr. Akshay K. Rathore and Mr. Sriram J. Reddy for helping me settle down and for providing me with invaluable suggestions. I also express my thanks to Dr. Mohamed Almardy, Dr. Xiaodong Li, Mr. Dhaval Shah, Mr. Clive Antoine, Mr. Qing Wu, Mr. Jonathan Weaver, Mr. Yimian Du, Mr. Shantanav Bhowmick, Mr. Adrian Engel, Mr. Daniel Figueira Sandoval, and Mr. H. Nagendrappa, for providing an excellent ambience in the work place.

I would like to thank Dr. P. Jothirmayanantham for being a wonderful companion, Mr. Jerome Etwaroo, Mr. Derek Swallow, Ms. Valerie Burgess and Mr. Marshall Burgess for making my stay in Victoria memorable. My whole hearted thanks to all my friends for wishing me success.

Finally, my utmost thanks to my beloved family members for sacrificing their desires for my betterment.

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Introduction

Electric machine forms the backbone of any modern day industry. Electric machines play a key role in electricity generation, whether it is from a conventional source or a non conventional source, in transporting materials, in compressing fluids and in commuting people [1]. By its very nature, an electric machine requires materials that not only have excellent electrical and magnetic properties but also extremely good mechanical strength. In addition, there is a necessity to withstand high temperatures that are generated within a machine. Thus, a machine typically demands the best of all the worlds [1]. To make matters worse, a machine may be subjected to different operational conditions such as frequent starting, occasional overloading, unbalanced supply voltage, voltage surges, dirty environment etc. [1].

Unfortunately, nothing is perfect and electric machines are no exceptions. Consequently, every electric machine is bound to fail. When such a vital cog fails rather unexpectedly, the consequences can be potentially disastrous. Thus, the need to monitor the condition of a machine to detect a lurking danger at its very inception is undeniable. Moreover, the benefits that can be reaped by implementing a condition monitoring scheme are tremendous. The advantages such as minimum downtime, optimum maintenance schedule, fore warning of imminent failures, etc. save huge costs that would otherwise be incurred [1], [2].

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Based on the components that lead to the failure of a machine, faults are classified into different categories. Literature survey reveals that there are four major faults that afflict the life and performance of electric machines. They are bearing faults, stator inter turn fault, rotor faults and eccentricity faults [3-5]. Eccentricity fault, being the focus of this work, has been delineated below.

1.1 Eccentricity

In an ideal machine, the axis of the stator and that of the rotor coincide resulting in a perfectly symmetrical distribution of air gap around the periphery of the rotor. When the two axes do not coincide, the perfect symmetry is lost and the condition is known as eccentricity. Some degree of eccentricity is unavoidable and is inherently present even in a newly manufactured machine due to manufacturing tolerances. Eccentricity is considered as a fault if the air gap asymmetry exceeds 10% of the nominal air gap length [3], [6].

1.1.1 Types of eccentricity

Eccentricity is predominantly classified into two types: static eccentricity (SE) and dynamic eccentricity (DE) [6-8]. SE ensues whenever the rotor rotating about its own axis is displaced from the geometric axis of the stator [9]. In case of SE the point of minimum air gap length remains fixed in space. Oval stator core, bearing wear and improper assembling of rotor at the commissioning stage are some of the most common reasons for the occurrence of SE. DE occurs when the rotor, whose axis is displaced from that of the stator, is rotating about the stator’s geometric axis. Under such a

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circumstance, the point of minimum air gap length rotates along with the rotor. DE is primarily caused by bent shaft, unbalanced magnetic pull, bearing wear, or mechanical resonance.

Usually both SE and DE tend to exist simultaneously leading to what is known as mixed eccentricity (ME) [10], [11]. Under such a situation the axis of rotation of the rotor is in between its own geometric axis and the geometric axis of the stator. In ME condition the position of minimal air gap length rotates with the rotor and also the magnitude of minimal air gap length changes with position. Figure 1.1 gives a pictorial description of healthy and eccentric conditions using the cross sectional view of a salient pole machine. In addition to the above mentioned, one more type of eccentricity is reported in the literature. It is known as inclined eccentricity (IE) [9], [12]. In case of IE the shaft is tilted and therefore the air gap is asymmetrical along the length of the axis.

Figure 1.1 C1 is the centre of stator and C2 is the centre of rotor. In a healthy machine C1 and C2 coincide. C1 is the centre of rotation for dynamic eccentric (DE) condition, whereas C2 is the centre of rotation for static eccentric (SE) condition. With mixed eccentricity (ME) the centre of rotation can be anywhere between C1 and C2. d-axis implies polar axis; q-axis implies inter polar q-axis.

d-axis

q-axis

d-axis

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1.1.2 Need for eccentricity monitoring

Eccentricity fault, while not as common in occurrence as bearing fault, stator turns fault or rotor winding fault, is still a major reason for motor failure and is often reported in literature [3-5]. Also, on many occasions bearing faults manifest themselves as eccentricity faults. Increased vibration, increased noise levels, higher electromagnetic stress, increased unbalanced magnetic pull (UMP), increased bearing wear are some of the major consequences of eccentricity. These may in turn increase the level of eccentricity with time. In addition, high levels of eccentricity can damage bearings. In the worst case eccentricity can result in stator rotor rub thereby causing a permanent damage to the motor [6-8]. Hence, it is vital to monitor eccentricity.

1.2 Eccentricity Detection Techniques

The study of eccentricity in induction motors has been the subject of interest for several decades now. In many of the classical research publications, however, the focus was on investigating the effect of eccentricity and expressing it in terms of air gap magnetic field [7]. Later on, with the idea of condition monitoring gaining importance, several techniques to detect eccentricity were attempted and have been reported in literature. Measuring acoustic noise, mounting search coils in stator slots to measure air gap flux, winding coil around the shaft to detect axial flux signal were some of the methods that were proposed [7], [13]. The aforementioned techniques were either invasive in nature or unreliable. Therefore, newer methods were explored. Motor casing vibration spectrum analysis and motor current signature analysis (MCSA) have been the two most preferred methods for eccentricity detection in induction motors because of

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their advantages such as being non-invasive in nature, ease of implementation, higher reliability etc.

1.2.1

E

ccentricity

detection in induction motors

In [7], the authors opted for a unified monitoring system based on stator line current and stator frame vibration signals. Characteristic fault frequencies in both line current spectrum and vibration spectrum were arrived at using magneto motive force (MMF) – specific permeance analysis. Experiments were carried out in the laboratory to confirm the existence of eccentricity specific frequency components. The magnitudes of the fault frequency components were found to increase with increase in eccentricity. Finally, on-site tests were conducted to test the effectiveness of the developed eccentricity detection scheme. With SE, the characteristic frequency component present in the line current showed a large increase, whereas the one in the vibration spectrum did not show much deviation leading to ambiguity. Also, the positioning of accelerometer was found to affect the sensitivity of the signal acquired - a major limitation of vibration based monitoring scheme.

Normally, both SE and DE occur simultaneously. Thus, ME specific frequency components in the air gap field, stator line current and casing vibration spectrum have been identified in [8] by following the MMF-specific permeance analysis. The low frequency side band components that emanate due to ME were studied as a function of different combinations of SE and DE levels. The effect of loading on these sideband components was also investigated. It has been reported that loading affected the magnitudes of the side band components considerably in both the line current spectrum

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and casing vibration spectrum. By using both line current and vibration components, the authors were able to detect the dominant fault type in case of a ME condition. However, the severity of the faults could not be found.

To predict the severity of eccentricity fault, an attempt has been made in [6] using a time stepping finite element (TSFE) model of the motor. As a first step, only SE was considered. The results obtained from experiments and the finite element (FE) simulations were compared. The TSFE model was found to predict accurately the magnitude of the harmonic components present in the line current spectrum under healthy condition. However, the predicted magnitudes had considerable error with the introduction of SE. The effects of supply harmonics that were neglected in the FE simulation could have resulted in the difference. The Achilles heel of the TSFE model, however, is its computational time.

Around the same time when TSFE models were used for identifying the severity of eccentricity faults, a coupled circuit model of cage induction motor was developed to study the effect of air gap eccentricity [13]. The advantages of a model based diagnostic system for studying eccentricity faults are several. Effects of non-sinusoidal supply as encountered in inverter fed drives can be simulated. Moreover, thresholds for various faulty conditions can be decided based on the simulations. Another major advantage of the model based system is its ability to discriminate a fault from a false positive condition. As the accuracy of the model plays a vital role in such a diagnostic scheme, the authors of [13] have derived the model using winding function approach (WFA). In this technique, the actual geometry and winding layout is used for calculating the parameters of the model. The model was used for simulating both the transient and steady state

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behaviour of the motor under healthy as well as SE condition. While simulating the SE condition, the authors have made use of the WFA leading to some inaccuracies in the mutual inductance computation.

To overcome the drawback of [13], the authors of [10] and [11] have made use of the modified winding function approach (MWFA) to model eccentricity in cage induction motors. As a result, the mutual inductances between stator and rotor were computed with higher accuracy. The inductance profiles were verified using a FE model of the motor. Also, the MWFA based model of the motor was used to simulate the motor under SE, DE and ME conditions. The simulation results showed the presence of different eccentricity related frequency components in the line current spectrum. Experiments were conducted to validate the simulation results. MWFA based model was found to be much faster than FE models, though the accuracy of the former was not as good as the latter.

Meanwhile in [14], it was established that eccentricity specific high frequency components given by (1.1) are generated in the line current only if the rotor bar number and the pole pair number of a cage induction motor conform to certain relationship. 𝑓ℎ = �(𝑘𝑅 ± 𝑛𝑑)1−𝑠_𝑝 ± 𝜗� 𝑓 (1.1) where R is the rotor bar number, f is the stator supply frequency in Hz, k is any integer, nd

is the order of eccentricity; nd is 0 for SE and 1, 2, 3… for DE, s is the slip, ϑ is the order

of time harmonic of supply frequency.

However, the low frequency components corresponding to ME condition remained unaffected by the pole pair – rotor bar combination. Simulation results, obtained using MWFA based model, and experimental results corroborated the theoretical expectation,

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implying that high frequency component based eccentricity detection must be used with caution.

Rotor bar number independent detection of eccentricity in induction motors then became the focus of the researchers. In [15], the authors have been able to detect eccentricity without the rotor bar information. The terminal voltage signals of the motor were recorded when the motor was turned-off. A short time Fourier transform (STFT) was performed on the first few cycles of the captured line voltage and the eccentricity specific frequency components present in the voltage spectrum were monitored. The only disadvantage of the proposed diagnostic technique is that it requires the motor to be turned-off for the purpose of eccentricity detection.

In another work, eccentricity fault of inverter fed induction motor has been diagnosed, independent of rotor bar number, at standstill [16]. The authors have made use of the inverter to produce a pulsating magnetic field of desired voltage and frequency at different angular positions. The currents in the stator windings were captured. Using the equivalent circuit model of the motor, the variation in the equivalent impedance was computed. With eccentricity, the equivalent leakage inductance was found to increase and was used as the fault indicating parameter. Experimental results obtained on a test set up have been employed to verify the method. The major advantages of the method include parameter independent detection, minimal influence of loading, minimum hardware requirement and ease of testing. However, the method cannot identify the type of eccentricity fault as the testing is done at standstill conditions.

It is clear from the literature review that lot of work has been reported on eccentricity detection in induction motors. However, the same cannot be said with respect

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to synchronous machines. In fact, until the turn of the century synchronous motors were not the prime subject of interest [17], [18]. With the greater emphasis on energy efficiency and with the advancements in the fields of power electronics and digital control, along with the reduced cost of rare-earth permanent magnets, modern day industrial drives are increasingly using synchronous machines. Consequently, researchers all around the world have shifted their focus to fault diagnosis of synchronous machines in the last decade.

1.2.2 Eccentricity detection in synchronous machines

One of the methods to detect air-gap eccentricity in hydroelectric generators using optical sensors has been mentioned in [2]. In this scheme, two collimated narrow beams of light are projected at a known angle across the air-gap. A pair of retro-reflective strips mounted on the opposing surface reflects the light back to the source. By measuring the time intervals between the reflected pulses the air-gap length can be computed.

Another method used to measure the air-gap in salient pole synchronous machine has been briefly described in [19]. In this method, a capacitor is formed between stator bore and rotor pole by placing a partly conductive plate on the stator bore. A high frequency voltage is applied to the plate and the current flowing into the plate is measured. The capacitance value obtained depends upon the air gap between the plate and the rotor pole. Usually samples are taken every few milli-seconds thus measuring the air-gap at various instants as the rotor rotates. Also, several capacitive sensors will be placed around the periphery of the stator bore to map the air gap with respect to rotor position. The present top of the line commercially available technique to detect

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eccentricity in salient-pole synchronous machines also relies on air gap capacitive sensor, where multiple passive sensors are glued to the stator laminations [20].

A different method of monitoring air gap in hydro generators using inductive sensors has been reported in [21]. The voltages induced in the coils that are mounted on the periphery of the stator stack have been used to identify the fault. The method has been found capable of detecting absolute values of air gap eccentricity. In [22], different methods to diagnose SE in a synchronous machine have been discussed. The possibility of using axial shaft voltages as an effective fault indicator has been explored. However, the results were not conclusive enough to detect SE.

All these techniques are intrusive methods of detecting air gap eccentricity and therefore, none of them is the ideal choice. A non-invasive scheme is far more desirable because of its convenience as a process need not be disrupted while the condition of the machine is being monitored. Moreover, a cost effective non-invasive technique in combination with a sensor based scheme will increase reliability. Thus, the pursuit of a non-invasive detection scheme for synchronous machines led to further developments.

In [17] and [18], dynamic air gap eccentricity in salient pole synchronous machine has been modeled based on MWFA. It has been identified that 17th harmonic component of stator current spectrum is a good indicator of DE fault. However, the reason behind the increase in magnitude of various harmonics has not been explained. Also, the model did not include the damper bars on the rotor. In a similar work reported in [23], the accuracy of the model built has been improved by incorporating the variation of MMF between two slots. However, the authors have accounted the variation of air gap length only along d-axis while neglecting the air gap variation along q-axis.

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Stray magnetic field of a synchronous machine has been used to identify rotor faults in [24]. A FE model of the machine has been built to compute the stray magnetic field around the machine. Applying classical frequency analysis, the fault specific low frequency components present in the stray magnetic field were established. Experiments on a 30 kVA, 4 pole, synchronous generator were then carried out to validate the predicted results. Though the method could identify the presence of rotor fault, it could not distinguish between rotor winding (field winding short circuit) fault and eccentricity fault. Also, in order to measure the stray magnetic field, highly expensive fluxgate magnetometers of high sensitivity have to be employed. Finally, the reason behind the presence of different fault specific harmonic components has not been explored.

An experimental study to detect bearing fault and eccentricity fault in permanent magnet synchronous machine (PMSM) has been reported in [25]. Experiments were performed on two 380 V, 6000 rpm, 6 pole, vector controlled PMSM drives to study individually the effect of each fault. In order to detect faults, harmonics of the stator currents in dq0 reference frame also had to be monitored as results obtained from stator current signatures in abc reference frame alone were inconclusive. Further, a combination of different harmonic components has been suggested rather than monitoring any one frequency component. In [26] and [27], several advanced signal analysis techniques such as continuous wavelet transforms, Wigner Ville distribution, Zero-Atlas-Marks distribution etc. have been resorted to diagnose eccentricity by monitoring torque change. Still, it was not possible to identify the severity of individual type of eccentricity.

Rotor faults in turbo-generators have been studied in [28] using transient FE analysis. The response of a two pole turbo-generator to DE fault has been analysed in

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both time and frequency domain. Frequency domain signatures in the no-load voltage spectrum of the generator have been found to identify the existence of rotor fault, though fault discrimination could not be achieved. In another work, a 3-dimensional FE model of a 26.7 kVA, 4 pole, 50 Hz turbo-generator has been used to study the effects of SE, DE, ME and IE [29]. Harmonic contents of two 90° shifted magnetic flux density signals have been used to identify and differentiate the fault.

Various rotor faults of a PMSM have been implemented in an experimental setup and the possibility of using the current and voltage signals to detect the air gap eccentricity has been investigated in [30]. In this paper, the frequency of rotor speed and its integral multiples in the line current and voltage were taken as fault frequencies. Though DE affected the fault frequency components considerably, the results obtained for SE did not show appreciable variations from healthy condition.

In [31], an improved air gap model of both polar and inter-polar regions of a salient pole synchronous machine has been presented to accommodate eccentricity. SE and DE in synchronous generator have then been modeled using MWFA. The no-load emf and current spectra have been analyzed to study the effect of SE and DE individually. With SE the spectral pattern obtained for each phase was found to be different. Therefore, the authors suggested the use of space vector analysis. However, the effect of ME was not studied in this work. In another approach no-load electromotive force (EMF) space vector loci has been used to detect SE and DE in synchronous generators [32]. The amplitude of no-load EMF space vector was found to increase with eccentricity. While SE resulted in oval shaped EMF space vector loci, DE did not affect the shape. Thus, the

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shape of the loci has been used to distinguish the type of eccentricity. But there was no information regarding the effect of ME on the EMF space vector loci.

In [33], a TSFE model of a PMSM has been used to study the effect of SE, while in [34], [35] the effects of SE, DE and ME have all been studied using a TSFE model. The authors have introduced a new index for eccentricity fault detection in PMSM [34-36]. However, as all types of eccentricities affected the same frequency components, the authors had to resort to Neural Network based classifier to detect the type of eccentricity. Even then only SE or only DE fault could be successfully identified. Thus, in case of ME the detection scheme failed to solve its purpose. Moreover, in predicting the line current harmonics the authors have ignored the rotor space angle and eccentricity dependence of the air gap flux density coefficients when deriving equation (27) from (26) in [34].

Air gap eccentricity in PMSM has been detected using d-axis inductance in [37]. In this work, the d-axis inductance has been found to decrease with increase in eccentricity due to change in the degree of magnetic saturation. The inverter driving the motor has been used to perform a test for eccentricity at standstill when the motor is turned-off. Being a standstill test, the method is free from load oscillations. However, the fault indicating parameter has been found influenced by the motor design and the temperature of the permanent magnet.

In [38], a synchronous machine with eccentricity has been modeled to study its performance under three phase fault and after fault removal. The variation of inductance profiles with eccentricity has been simulated in [38] and has been validated experimentally in [39]. The details of modeling the inverse air gap function under eccentricity, however, have not been discussed in these works.

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In [40] and [41], FE based simulation has been used along with the standard Fourier series method to model the air gap variation of a salient pole generator. With the more accurate specific permeance function so obtained, the effects of different types of eccentricities in a synchronous generator have been studied. The authors have been able to diagnose eccentricity by analyzing harmonic components in the voltage spectrum of a machine operating at no-load. However, the effect of stator MMF has been neglected in these works. Also, the permeance function described in [41] is based on cylindrical rotor machines.

In [42], a theory to identify SE in synchronous generators has been developed. Rotor field current and no-load voltage signals have been used for the purpose of fault detection. SE has been found to introduce a double fundamental frequency ripple in the rotor current. Harmonics introduced in the no-load voltage spectrum due to SE have been found to depend upon the winding structure. Simulations based on modified winding function approach have been performed to confirm the theoretical prediction. Experimental results reported in [43] have been found to validate the theory. With the addition DE, a sideband component around the fundamental frequency was found to appear. However, the theory behind the appearance of the sideband component has not been mentioned. Moreover, as in [40] and [41], the effect of stator MMF has not been included.

In [44], the authors have extended the work done in [23] by accommodating the effect of saturation in the MWFA model. Also, the actual shape of rotor poles has been considered while modeling the air gap function along d-axis. The results obtained from the MWFA based model have been compared with FE results and were found matching

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closely. Though the accuracy of the model has been improved, the variation of air gap along q-axis was not accounted. Further, the reason behind the presence of various harmonics in the current spectrum has not been explained.

Based on the literature review it is clear that the reported works either do not present complete mathematical modeling of the motor or they do not give a detailed analysis for the results obtained. Also, no breakthrough results have been obtained to detect ME and IE conditions in synchronous machines. Thus, based on the present situation it has become clear that eccentricity detection research in synchronous machines is still at its initial stage.

1.3 Motivation

Synchronous motors can be broadly classified into three categories: (i) machines with dc field excitation of which there are two types namely salient pole and round rotor synchronous machines (ii) Reluctance synchronous motors (RSM) with no dc field winding and (iii) Permanent magnet synchronous machines (PMSM). Machines belonging to category (i) are usually the largest ones and can be several hundreds, even few thousands of megawatts [45]. They are the ones used for power generation. The large synchronous motor drives (tens of megawatts) also belong to this category.

The RSMs, on the other hand, have been built only up to a few kilowatts. A RSM has many innate advantages namely rugged construction, no rotor copper loss and zero speed regulation [46-49]. Moreover, the major drawback of RSM, poor power factor, has been considerably improved by incorporating suitable design modifications, such as multiple flux barriers, axially laminated anisotropic rotor instead of regular induction

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motor rotors with cut-outs in q-axis, and control algorithms [49-52]. Consequently, RSM is found to offer an efficient solution for applications that require high dynamic performance (e.g. machining) and wide constant power mode of operation (e.g. spindle drives) [53], [54]. In addition, the saliency of the rotor structure in RSM has been identified as a blessing in sensor less rotor position estimation [55], [56]. Thus, the obvious implication is the increased use of RSM in future.

PMSMs are the most popular ones for drives up to several tens of kilowatts as they offer excellent power density, power factor, and lower size than equivalent induction motor. They are now very popular in automotive applications [57], [58]. Thus, each type of synchronous motor offers its own advantages and as a result has its own applications, implying that different approaches of fault diagnosis may have to be taken for different types of synchronous motors.

For this thesis work it has been decided to work first on RSM with damper bars. Once the fault signatures for this motor are clearly understood, fault diagnosis of conventional salient pole synchronous machine will be easier. This is because, only a little have been reported pertaining to modeling salient-pole synchronous machines for eccentric conditions and almost all of them make use of only the field winding in their models. Moreover, most of the work has neglected the air gap variation along q-axis while modeling the specific permeance function. The reason attributed to the approximation is the complexity involved in accommodating eccentricity in salient pole machines using the standard Fourier series method. Consequently, the specific permeance expressions so derived do not reflect the actual variation and have higher inaccuracies especially at larger eccentricity levels. Also, there is no clear formulation of theory

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behind most of the results reported. In addition, fault detection in salient-pole synchronous machines for ME, the most realistic form of eccentricity [3], has not yet been done in an unambiguous manner. MWFA, which has been successfully used to model all types of eccentricities in induction motors, has been used sparingly with synchronous machines. As seen in case of induction motors, MWFA based approach, unlike FE, is much quicker and can be used to avoid slotting and saturation effects that might mask the fault signatures [14], [59]. All these have generated enough curiosity to propose the following research objectives.

1.4 Objectives

The objectives are summarized as follows:

1. To use MWFA to generate a coupled magnetic circuit model for different (SE, DE and ME) eccentricity faults initially in RSM and later in salient-pole synchronous machine (SPSM).

2. To use MCSA on the data obtained from the coupled magnetic circuit model to unambiguously detect the type of eccentricity and the severity of the fault, initially in RSM and later in SPSM. Vibration analysis of simulated flux density may also be carried out if MCSA does not show enough promise. Transient approaches like checking the terminal voltages of the motor at switch-off etc. may also be explored.

3. To simulate the synchronous machine for different types of eccentricities using commercial FE package Maxwell and to compare with the results obtained from the coupled circuit model. This is to (a) check the inductance profiles generated

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by MWFA and (b) fine tune the model based signatures obtained using MWFA and coupled magnetic circuit analysis.

4. To implement the proposed eccentricity diagnostic scheme on an experimental motor to validate the method.

1.5 Significance

As mentioned earlier, synchronous motor drives are increasingly used in many industrial drives and are predicted to increase in the years to come. Almost all modern day drives using synchronous motors are controlled using DSP or microprocessor based controllers and already have motor’s current and voltage information. Therefore, adding fault diagnostic feature to such drives requires no additional hardware and can be implemented easily by modifying merely the software. So including this feature will not result in significant increase in the cost of the existing drive system. On the other hand, condition monitoring feature will help increase the reliability of the drive and thus, improve the marketability of the product.

1.6 Thesis Outline

In Chapter 2, the need for deriving an expression that can represent the inverse air gap of a salient pole machine under any eccentricity condition will be elucidated. The drawbacks of the methods reported in the literature will be made clear and a new method will be proposed that can address the problem. Detailed derivations of the inverse air gap function corresponding to each type of eccentricity fault will be shown. In Chapter 3, the derived inverse air gap functions will be made use of in the MMF-specific permeance

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analysis. Various harmonic components present under different eccentric conditions in the line current spectrum of both RSM and SPSM will be derived. Finally, the harmonic components present in the field current of the SPSM will be obtained.

The first half of Chapter 4 will deal with MWFA based inductance computation of RSM and SPSM under healthy as well as eccentric conditions. FE simulations carried out in Maxwell – 2D, commercial FE software, to validate the inductance plots will be shown. In the second half of the chapter, a state space model of RSM and SPSM built in Matlab will be presented. Simulation results obtained from the coupled circuit model will be reported. Simulations performed using TSFE model of the motors and the results generated will be documented. Finally, a comparison of MWFA based model and FE model will be done.

Chapter 5 will begin with a detailed description of the experimental setup, followed by analysis of experimental results. The effects of non-ideal conditions will be discussed. A method based on residual estimation will be used to minimize the effects of non-ideal conditions. Eccentricity detection based on stator current signatures for RSM and SPSM and field current signature based eccentricity detection in case of SPSM will be reported. Finally, in Chapter 6, conclusions of the research work will be summarized by highlighting the contributions made. Also, the future scope of the research work will be outlined.

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Chapter 2 Inverse Air Gap Modeling

For eccentricity fault diagnosis, inverse air gap function plays a vital role and therefore gains special significance. In order to appreciate the need for inverse air gap function, it is necessary to understand its influence on eccentricity faults detection through motor current signature analysis (MCSA), a non-invasive detection scheme.

2.1 Motor Current Signature Analysis (MCSA)

MCSA is a popular frequency domain approach to detect various faults in electric machines. The basic operating principle of MCSA is based on monitoring the frequency spectrum of the machine’s current. The current spectrum of a machine contains certain frequency components under healthy condition. However, with the occurrence of a fault, the harmonic spectrum of the machine’s current gets modified. Depending upon the type of fault and the winding configuration, new harmonic components emanate and/or the existing components get modulated. Thus, by monitoring the frequency spectrum of the machine current, the occurrence of a fault can be identified. Figure 2.1 shows a very basic condition monitoring system based on MCSA.

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AC AC AC AC Motor Data acquisition system Current signals Voltage signals Voltage probe2 Voltage probe1 FFT PC

Figure 2.1 Condition monitoring system based on MCSA

2.2 Need for Inverse Air Gap Function

Though the operating principle of MCSA is simple, in a practical machine, slotting and supply voltage time harmonics injects many harmonics in the line current. Thus, due to the presence of numerous frequency components in the current spectrum of a machine, the process of comparing the current spectra at different instants to diagnose faults can become overwhelming. The tediousness of the process can be minimized considerably provided the frequency components associated with any faulty condition are known beforehand. These fault specific frequency components are known as characteristic frequencies. Once the characteristic frequencies are known, it is sufficient enough to monitor the variation of only these components in the frequency spectrum to achieve desired fault diagnosis.

From the literature it is clear that fault specific frequency components can be easily obtained by performing MMF – specific permeance analysis. When multiplied with

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MMF, the specific permeance results in an expression of air gap flux density that can very easily predict the harmonics in the machine line current spectrum. Thus, the specific permeance expression plays a very vital role in this method.The inverse air gap function directly determines the specific permeance of a machine. Therefore, its formulation becomes critical.

2.3 Drawbacks of Existing Inverse Air Gap Models

Until now, to perform the MMF-specific permeance analysis, the Fourier series representation of the inverse air gap has been the widely used form. However, the Fourier series model has major limitations. To understand the shortcomings of the Fourier series based model, the MMF – specific permeance analysis of an eccentric induction motor with smooth stator and smooth rotor is delineated below.

2.3.1 Fourier series model of inverse air gap

The air gap of a healthy induction motor is constant, assuming there are no slots. Let it be denoted by 𝑔₀. Therefore, its inverse is also a constant given by 1 𝑔� . With a ₀ static eccentricity (SE) of 𝑎_𝑠, the air gap of the motor is modified to (2.1);

𝑔𝑠𝑒(𝜑) = 𝑔0− 𝑎𝑠cos 𝜑 (2.1)

where 𝜑 is the angular reference in stator co-ordinates. The inverse air gap of the SE machine is then given by

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𝑔𝑠𝑒−1(𝜑) =_𝑔₀_−𝑎1_𝑠_{𝑐𝑜𝑠 𝜑}= _𝑔1₀∙_1−𝑎₁1_{𝑐𝑜𝑠 𝜑}; 𝑎1 =_𝑔𝑎₀𝑠; (2.2) The inverse air gap given above is usually represented in Fourier series form as shown in (2.3). 𝑔𝑠𝑒−1(𝜑) = 𝐴0 + ∑∞𝑘=1,2,3…𝐴𝑘𝑐𝑜𝑠(𝑘𝜑) 𝐴0= _𝑔₀_�1−(𝑎1 ₁₎2; 𝐴𝑘 = 2 𝑔0�1−(𝑎1)2∙ � 1−�1−(𝑎1)2 𝑎1 � 𝑘 . (2.3)

The specific permeance function under SE condition is then obtained by multiplying the inverse air gap function with the permeability of free space as shown in (2.4).

𝑃𝑠𝑒 = 𝜇0𝑔𝑠𝑒−1(𝜑) (2.4)

For a 3 phase integral slot winding, the stator MMF produced by a balanced three phase sinusoidal supply is given by

𝐹𝑠 = ∑∞𝑛=1,5,7,11,13…𝑀𝑛𝑠𝑐𝑜𝑠[𝑛𝑝𝜑 ± 𝜔𝑡] (2.5)

where 𝑀_𝑛𝑠 are the coefficients of components of stator MMF; 𝑛 = 1,5,7,11,13,… is the order of space harmonics; 𝑝 is the fundamental pole pair number; 𝜔 is the stator angular supply frequency.

When the stator MMF (2.5) interacts with the SE specific permeance function (2.4), it results in a flux density in the air gap with components, which with respect to stator frame of reference can be represented as

𝐵𝑠1 = 𝜇0𝐴0𝑀𝑛𝑠𝑐𝑜𝑠[𝑛𝑝𝜑 ± 𝜔𝑡] + 𝜇0𝐴𝑘𝑀𝑛𝑠[𝑐𝑜𝑠(𝑘𝜑) × 𝑐𝑜𝑠{𝑛𝑝𝜑 ± 𝜔𝑡}] . (2.6) The above flux density induces currents in the rotor. The currents circulating in the rotor result in a rotor MMF.

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where 𝜑′ is the angular reference in rotor in rad and is given as 𝜑′ = 𝜑 − 𝜃_𝑟 ; 𝜃_𝑟 is the rotor position in rad.

The rotor MMF interacting with the SE specific permeance function produces another flux density in the air gap, whose components with respect to rotor can be given by

𝐵𝑟1 = 𝜇0𝐴0𝑀𝑙𝑟𝑐𝑜𝑠 �𝑙𝜑′+ �𝑙(1−𝑠)_𝑝 ± 1� 𝜔𝑡 − 𝜃1� + 𝜇0𝐴𝑘𝑀𝑙𝑟𝑐𝑜𝑠{𝑘(𝜑′+ 𝜃𝑟)} × 𝑐𝑜𝑠 �𝑙𝜑′_{+ �}𝑙(1−𝑠)

𝑝 ± 1� 𝜔𝑡 − 𝜃1� (2.8) With respect to stator frame, the above flux density components can be re-written as 𝐵𝑠2 = 𝜇0𝐴0𝑀𝑙𝑟𝑐𝑜𝑠 �𝑙(𝜑 − 𝜃𝑟) + �𝑙(1−𝑠)_𝑝 ± 1� 𝜔𝑡 − 𝜃1� + 𝜇0𝐴𝑘𝑀𝑙𝑟𝑐𝑜𝑠{𝑘(𝜑)} × 𝑐𝑜𝑠 �𝑙(𝜑 − 𝜃_𝑟) + �𝑙(1−𝑠)

𝑝 ± 1� 𝜔𝑡 − 𝜃1� (2.9) The flux density can induce current in the stator windings provided the pole pair numbers of (2.9) match that of (2.5). The frequencies of the induced currents can be directly obtained from (2.9) since the coefficients 𝐴₀ and 𝐴_𝑘 are constants. Though the Fourier series based inverse air gap model works well for SE condition, its limitation becomes obvious with ME as shown below.

The air gap of the induction motor with absolute SE of 𝑎_𝑠 m and absolute DE of 𝑎𝑑 m is

𝑔𝑚𝑒(𝜑, 𝜃𝑟) = 𝑔0− 𝑎𝑠cos 𝜑 − 𝑎𝑑cos(𝜑 − 𝜃𝑟). (2.10)

The inverse of the air gap is represented as

𝑔𝑚𝑒−1(𝜑, 𝜃𝑟) =_𝑔₀_−𝑎_𝑠_{𝑐𝑜𝑠 𝜑−𝑎}1_𝑑_{𝑐𝑜𝑠(𝜑−𝜃}_𝑟₎ =_𝑔1₀∙_1−𝑎₁_{𝑐𝑜𝑠 𝜑−𝑎}1₂_{𝑐𝑜𝑠(𝜑−𝜃}_𝑟₎;𝑎1 = _𝑔𝑎𝑠₀; 𝑎2 = 𝑎_𝑔𝑑₀; (2.11)

The above equation when represented in Fourier series form is given by (2.12).

Non-invasive detection of air gap eccentricity in synchronous machines using current signature analysis

Synchronous Machines Using Current Signature Analysis

Non-Invasive detection of air gap eccentricity in synchronous

machines using current signature analysis

Supervisory Committee

Supervisory Committee

ABSTRACT

Table of Contents

List of Tables

List of Figures

List of Abbreviations

List of Symbols

Acknowledgments

Introduction

1.1 Eccentricity

1.1.1 Types of eccentricity

1.1.2 Need for eccentricity monitoring

1.2 Eccentricity Detection Techniques

E

detection in induction motors

1.2.2 Eccentricity detection in synchronous machines

1.3 Motivation

1.4 Objectives

1.5 Significance

1.6 Thesis Outline

Chapter 2

Inverse Air Gap Modeling

2.1 Motor Current Signature Analysis (MCSA)

2.2 Need for Inverse Air Gap Function

2.3 Drawbacks of Existing Inverse Air Gap Models

2.3.1 Fourier series model of inverse air gap