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Induction Machine Broken RotorBar Diagnostics Using Prony
Analysis
by
Shuo Chen
A thesis submitted to the School of Electrical and
Electronic Engineering of the University of Adelaide
in partial fulllment of the requirements
for the degree of
Master of Engineering Science
in
Electrical Engineering
Adelaide, Australia
April, 2008
c©2008 - Shuo Chen
All rights reserved.
Typeset in LATEX2ε
Contents
Contents i
Abstract v
Statement of Originality vii
Acknowledgement ix
List of Tables xi
List of Figures xiii
Nomenclature xvii
1. Introduction 1
1.1. Induction Machine Condition Monitoring and Fault Diagnostics . 1
1.2. Motor Current Signature Analysis . . . . . . . . . . . . . . . . . . 2
1.3. Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.4. Synopsis of Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2. Broken Rotor Bar Faults in Induction Machines and Non-Intrusive
Methods of Detection 7
2.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2. The Induction Motor . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1. The Construction of Induction Motors . . . . . . . . . . . 7
2.2.2. The Operation of Induction Motors . . . . . . . . . . . . . 8
i
CONTENTS
2.3. Induction Machine Broken Rotor Bar Faults . . . . . . . . . . . . 9
2.3.1. Causes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3.2. Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.4. Detection of Broken Rotor Bar Faults . . . . . . . . . . . . . . . . 10
2.4.1. Presentation of Broken Rotor Bar Faults in Stator Current 10
2.4.2. Detection Indices . . . . . . . . . . . . . . . . . . . . . . . 14
2.4.3. Assessment of Rotor Fault Severity . . . . . . . . . . . . . 15
2.5. Limitations and Possible Improvement . . . . . . . . . . . . . . . 18
3. Model of an Induction Machine with Broken Rotor Bars 19
3.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2. Mathematical Model . . . . . . . . . . . . . . . . . . . . . . . . . 20
3.2.1. Mathematical Model of an Induction Machine . . . . . . . 20
3.2.2. Mathematical Model of Broken Rotor Bars . . . . . . . . . 28
3.3. Model in Matlab/Simulink . . . . . . . . . . . . . . . . . . . . . . 30
3.3.1. Introduction of Matlab/Simulink . . . . . . . . . . . . . . 30
3.3.2. Model Description Equations for Matlab/Simulink . . . . . 31
3.3.3. Simulink Model in Block Diagrams . . . . . . . . . . . . . 33
3.4. Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.1. Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . 38
3.4.2. Simulation Results . . . . . . . . . . . . . . . . . . . . . . 38
4. High-Resolution Spectral Analysis 41
4.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
4.2. Comparison Between Discrete Fourier Transform and Prony Analysis 42
4.2.1. Drawbacks of Discrete Fourier Transform . . . . . . . . . . 42
4.2.2. Features of Prony Analysis . . . . . . . . . . . . . . . . . . 43
4.3. The Original Prony Method . . . . . . . . . . . . . . . . . . . . . 44
4.4. Extended Least Squares Prony Method . . . . . . . . . . . . . . . 47
4.5. Iterative Prony Method . . . . . . . . . . . . . . . . . . . . . . . . 49
ii
CONTENTS
5. Implementation of Prony Analysis for Induction Motor Broken Bar
Detection 53
5.1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.1.1. Study Description . . . . . . . . . . . . . . . . . . . . . . . 54
5.2. Data Acquisition and Preprocessing . . . . . . . . . . . . . . . . . 55
5.2.1. Sampling Frequency and Window Length . . . . . . . . . . 55
5.2.2. Data Preprocessing . . . . . . . . . . . . . . . . . . . . . . 56
5.3. Prony Estimation and Prediction . . . . . . . . . . . . . . . . . . 58
5.3.1. Stator Current Modulation . . . . . . . . . . . . . . . . . . 58
5.3.2. Fault Severity Assessment . . . . . . . . . . . . . . . . . . 63
5.4. Disadvantages of DFT and Solutions by Prony Analysis . . . . . . 65
5.4.1. Impact of Data Window Length . . . . . . . . . . . . . . . 65
5.4.2. Frequency Estimation Accuracy . . . . . . . . . . . . . . . 71
5.4.3. Small Load Conditions . . . . . . . . . . . . . . . . . . . . 74
5.5. Evaluation of Prony Analysis . . . . . . . . . . . . . . . . . . . . 75
5.5.1. Impact of Data Window Length . . . . . . . . . . . . . . . 76
5.5.2. Noise Impact . . . . . . . . . . . . . . . . . . . . . . . . . 76
5.5.3. Order Selection . . . . . . . . . . . . . . . . . . . . . . . . 76
5.6. Practical Implementation Test . . . . . . . . . . . . . . . . . . . . 78
5.6.1. Experiment Setup . . . . . . . . . . . . . . . . . . . . . . . 78
5.6.2. Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6. Conclusion 83
6.1. The Broken Rotor Bar Fault . . . . . . . . . . . . . . . . . . . . . 83
6.2. The Induction Machine Model . . . . . . . . . . . . . . . . . . . . 84
6.3. The Implementation of Prony Analysis . . . . . . . . . . . . . . . 85
6.4. Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
Bibliography 87
iii
CONTENTS
A. Important Programs 93
A.1. Simulation Initialization . . . . . . . . . . . . . . . . . . . . . . . 93
A.1.1. Simulation Initialization File Startsim.m . . . . . . . . . 93
A.1.2. Machine Parameter Initialization File motor_1hp.m . . . 94
A.2. Least Squares Prony Method . . . . . . . . . . . . . . . . . . . . . 95
B. Important Equation Derivations 99
B.1. Derivation of Eq. (3.18) . . . . . . . . . . . . . . . . . . . . . . . 99
B.2. Derivation of Eq. (3.22) . . . . . . . . . . . . . . . . . . . . . . . 99
B.3. Derivation of Eq. (3.26) . . . . . . . . . . . . . . . . . . . . . . . 100
B.4. Derivation of Eq. (3.34) . . . . . . . . . . . . . . . . . . . . . . . 101
B.5. Derivation of the coecients in Eq. (4.3) . . . . . . . . . . . . . . 102
C. Parameters of Induction Machines 103
D. Prony Analysis Results 105
iv
Abstract
On-line induction machine condition monitoring techniques have been used widely
in the detection of motor broken rotor bars for decades. Research has found that
when broken bars occur in the machine rotor, the anomaly of electromagnetic eld
in the air gap will cause two sideband frequency components presenting in the sta-
tor current spectrum. Therefore, identication of these sideband frequencies can
be used as a convenient and reliable approach to broken rotor bar fault diagnosis.
Discrete Fourier Transform (DFT) is a conventional spectral analysis method used
in this application. However, the use of DFT has several limitations. The most
important one among them is the restriction of frequency resolution by window
length. Due to this limitation, the accuracy of broken rotor bar detection can
be highly aected in cases such as light machine load and limited data records.
However, Prony's method for spectral analysis has the ability of overcoming the
restriction of data window length on the frequency resolution, from which the
DFT suers. Such feature makes Prony's method a promising choice for broken
rotor bar diagnosis when the machine is operating under light or varying load,
or when only restricted data is available. In this thesis, I have demonstrated
the implementation of this technique in the induction motor broken rotor bar
detection, revealed its better performance than DFT in terms of maintaining high
resolution in frequency domain whilst using a much shorter window, and analyzed
the inuential factors to the method of Prony Analysis (PA).
In this thesis, an induction machine model that includes broken rotor bars is
developed using Matlab/Simulink and veried by comparing the experimental
and the simulated results. The Prony Analysis method for broken bar diagnosis
is implemented and tested using both simulated and measured stator current
data. Comparisons between PA and DFT results are presented, clearly indicating
improvements of broken bar diagnostics using PA.
v
vi
Statement of Originality
I hereby declare that this is an original thesis and is entirely my own work under
the guidance and advice of my supervisor Dr. Rastko Zivanovic. This work
contains no material which has been accepted for the award of any other degree
or diploma in any university or other tertiary institution and, to the best of my
knowledge and belief, contains no material previously published or written by
another person, except where due reference has been made in the text.
I give consent to this copy of my thesis, when deposited in the Adelaide University
Library, being made available for loan and photocopying, subject to the provisions
of the Copyright Act 1968.
Shuo Chen
April 2008
vii
viii
Acknowledgments
I was just trying to have a short break to take breath from writing and rewriting
a same piece of work for several months by thanking people. However, what I
had not realized was that this would never be a task any easier than writing a
thesis. It is not because I have not learned enough aecting English words to
express my appreciation, but the fact that I believe, for a man indebted, even the
most exquisite word in any language is not competent to deliver this gratefulness.
Though clumsy, I still insist on writing down the following words, with the best I
am able to put in.
Memory has carried my thought reviewing through the time from day one when
my parents saw me o in the international airport. Their images and voices keep
on emerging in my mind like that they just happened yesterday. Thank you and
forever love to my mother, Yunfeng Lei, and father Jianguo Chen. You could not
have given any more than you have done to me. Your love, support and trust is the
invaluable wealth that I have. It has carried me for the years I lived through, and
will continue to be the inexhaustible source of my encouragement and strength
for my whole life.
Of course none of my success would have been possible without the constant
guidance and support from my supervisor, Dr. Rastko Zivanovic. Rastko demon-
strated exceptional abilities to focus on the consecutions of questions and to solve
problems by tackling the keys. He was excellent in researching and tremendous
in teaching. So many times only a few words from him would turn my jumbled
mind suddenly enlightened. His brain was an inconsumable source of knowledge
and inspirations. All these are only a few of the many things that I could learn
for a lifelong time. I could not have asked for a better supervisor.
There is no separation between personal and professional life for a postgraduate
student. My ancee, Heqing Wang, has been my friend, critic, listener, assistant,
adviser, teacher and partner from the beginning and throughout the whole time
of my study of this degree. Thank you for always being there.
ix
I would also like to thank many colleagues who had generously donated their time
to help me with my study. Especially, I would like to thank Mr. Yinan Kong who
taught me a lot in mathematics and signal processing. He explained very compli-
cated and abstract concepts by using simple words and vivid guration which a
kid would understand. Without his help, I do not know if I would survive from all
the frustrations that have happened. Many thanks to Mr. Randy Supangat and
Mr. Gene S. Liew for helping in setting up experiments in the lab. Thank you
to Ms. Hui-Min Tan and Mr. Adam Burdeniuk for kindly reading my thesis and
providing valuable comments. I am also grateful to Ms. Patricia Anderson and
Mr. Benjamin Hooper from the International Student Centre of the University
of Adelaide. I did have bothered you a lot in administrative aairs and you were
always there welcoming and helpful.
x
List of Tables
5.1. Relevant parameters of induction machines used in the study. . . . . 55
5.2. Numerical PA result of the stator current of Machine 2 with various
broken rotor bar numbers operating under full load condition using a
data window of 500 samples with the sampling frequency of 1000Hz. 62
5.3. PA and DFT results of the (1± 2s) f sideband frequencies using the
minimum window lengths with 1000Hz sampling frequency for Machine
2 operating under full load condition and with various numbers of
broken rotor bars. . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.4. PA and DFT results of the (1± 2s) f sideband frequencies using the
minimum window lengths with 1000Hz sampling frequency for Machine
2 operating under 75% load condition and with various numbers of
broken rotor bars. . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
5.5. PA and DFT results of the (1± 2s) f sideband frequencies using the
minimum window lengths with 1000Hz sampling frequency for Machine
2 operating under 50% load condition and with various numbers of
broken rotor bars. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.6. PA and DFT results of the (1± 2s) f sideband frequencies using the
minimum window lengths with 1000Hz sampling frequency for Machine
2 operating under 25% load condition and with various numbers of
broken rotor bars. . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
5.7. Estimated values of the sideband frequency components by PA and
DFT for broken rotor bar detection on Machine 2, under dierent light
load conditions and with one broken rotor bar. . . . . . . . . . . . . 74
5.8. PA result of the measured stator current signal of a 2.2kW induction
motor with 4 broken rotor bars operating under full load condition,
using a data window of 200 samples with the sampling frequency of
400Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
xi
C.1. Parameters of induction machine models used for simulations. . . . . 103
D.1. Frequency estimation results by PA and DFT for Machine1 with dier-
ent number of broken rotor bars operating under dierent load condi-
tions, using a data window of 500 samples and a sampling frequency
of 1000Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
xii
List of Figures
1.1. A ow chart of a typical MCSA system for broken rotor bar diagnosis. 3
2.1. Spectra of the simulated stator current of a 5.5kW induction motor with
32 total rotor bars operating under full load condition, with respect to
the number of broken rotor bars . . . . . . . . . . . . . . . . . . . . 13
2.2. Spectra of the simulated stator current of a 5.5kW induction motor
with 1 broken rotor bar with respect to dierent load conditions. . . . 13
2.3. Amplitude of the (1− 2s) f sideband current component in dB relative
to the fundamental frequency as the number of broken bars and load
conditions are varied. . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.4. The prediction curves of (1− 2s)f current sideband frequency ampli-
tudes in dB relative to the fundamental frequency, obtained by using
three prediction equations Eq. (2.6) (2.7) and (2.9) with respect to
the number of total rotor bars Nb = 32. . . . . . . . . . . . . . . . . 17
3.1. Relationship between abc and arbitrary qd0 reference frames. . . . . 24
3.2. Equivalent circuit representation of an induction machine in the arbi-
trary qd0 reference frame. . . . . . . . . . . . . . . . . . . . . . . . 27
3.3. Block diagram of the abc− qd0 conversion module in Simulink. . . . 34
3.4. Block diagram of the unit vector calculation module in Simulink. . . 35
3.5. Block diagram of the induction motor model module in Simulink. . . 36
3.6. Block diagram of the rotor module. . . . . . . . . . . . . . . . . . . 37
3.7. Block diagram of the qd0− abc conversion module in Simulink. . . . 37
3.8. Simulated output torque curve. . . . . . . . . . . . . . . . . . . . . 39
3.9. Comparison of the simulated and measured rotor speed curves. . . . 39
xiii
LIST OF FIGURES
3.10. Comparison of the simulated and measured stator currents. . . . . . 40
5.1. The magnitude response of the equiripple bandpass lter. . . . . . . 57
5.2. Comparisons between PA estimation and prediction results with the
simulated stator currents of Machine 2 with 0, 1, 3, 5, 7 and 8 broken
rotor bars operating under full load, using a data window of 500 samples
with the sampling frequency of 1000Hz. . . . . . . . . . . . . . . . . 60
5.3. Zoomed-in views of comparisons between PA estimation and prediction
results with the simulated stator current of Machine 2 with 1 broken
rotor bar operating under full load, using a data window of 500 samples
with the sampling frequency of 1000Hz. . . . . . . . . . . . . . . . . 61
5.4. Amplitude of the (1− 2s) f sideband frequency obtained by PA with
respect to the number of broken rotor bars in Machine 1, 2 and 3
respectively. The motors are operating under full load. . . . . . . . 64
5.5. DFT spectrum of the current signal of Machine 2 with 2 broken rotor
bars operating under full load. The data window length is 5000 samples
using a sampling frequency of 1000Hz. . . . . . . . . . . . . . . . . 66
5.6. DFT spectrum of the current signal of Machine 2 with 2 broken rotor
bars operating under full load. The data window length is 1000 samples
using a sampling frequency of 1000Hz. . . . . . . . . . . . . . . . . 66
5.7. DFT spectrum of the current signal of Machine 2 with 2 broken rotor
bars operating under full load. The data window length is 500 samples
using a sampling frequency of 1000Hz. . . . . . . . . . . . . . . . . 67
5.8. DFT spectrum of the current signal of Machine 2 with 2 broken rotor
bars operating under 25% of full load. The data window length is 2000
samples using a sampling frequency of 1000Hz. . . . . . . . . . . . . 67
5.9. Plotted comparison of the minimum window length requirements of PA
and DFT for broken rotor bar detection on Machine 2, with respect to
dierent numbers of broken rotor bars and load conditions. . . . . . . 72
5.10.MAEfreq in frequency estimation by PA and DFT in respect of window
length using 1000Hz sampling frequency for simulated current data of
Machine 2 operating under full load. . . . . . . . . . . . . . . . . . 73
5.11.MAEfreq in frequency estimation by PA and DFT in respect of window
length using 1000Hz sampling frequency for simulated current data of
Machine 2 operating under 75% of full load. . . . . . . . . . . . . . 73
xiv
5.12.MAEfreq of the 6 order PA frequency estimator of broken rotor bar
sideband frequencies with respect to the window length when using
1000Hz sampling frequency for 100 runs. . . . . . . . . . . . . . . . 77
5.13. Estimation mean absolute error as a function of measurement error
standard deviation for IRLS Prony . . . . . . . . . . . . . . . . . . . 77
5.14. Spectrum of the measured stator current of a 2.2 kW induction mo-
tor with 4 broken rotor bars using DFT with a sampling frequency of
400Hz, and a data window of 4000 samples. . . . . . . . . . . . . . 80
5.15. Comparison of the current waveforms between PA estimation and pre-
diction with the real current signal of a 2.2kW induction machine op-
erating in full load with 4 broken rotor bars. . . . . . . . . . . . . . 81
5.16. DFT spectrum of the same signal data used in Figure 5.14 and Fig-
ure 5.15 but using a window of only 200 samples with the sampling
frequency of 400Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . 81
xv
xvi
Nomenclature
A/D Analogue/Digital
CT Current Transformer
DFT Discrete Fourier Transform
DSP digital signal processing
FFT Fast Fourier Transform
FIR nite-duration impulse response
GUI Graphical User Interface
IRLS Iteratively Reweighted Least Squares
LS Least Squares
MAE Mean Absolute Error
mmf magnetic motive force
PA Prony Analysis
SVD Singular Value Decomposition
xvii
xviii
Chapter 1.
Introduction
1.1. Induction Machine Condition Monitoring
and Fault Diagnostics
Induction motors are the prime movers of industry and permeate all areas of the
modern life. Generally, they are robust and reliable. However, due to the com-
bination of poor working environments, heavy duty cycles, and installation and
manufacturing factors, internal faults often occur on the rotor, stator, bearing and
accessory parts of induction machines. The most common faults that induction
machines are aicted with include broken rotor bars, stator core and winding
faults, lamination damage, air gap eccentricity and bearing failures [1].
The faults mentioned above are potential hazards to the reliability and safety
of operation, and also increase the operational costs. The broken rotor bar is a
common type of fault in induction machine. Although it does not cause motor
failure initially, broken bar faults signicantly lower the eciency and shorten
the life of an induction machine. Damages to insulation and winding structure
may be caused consequently resulting in machine breakdown eventually. Arcing
and sparking caused when induction motors operating with broken rotor bars can
be dangerous if motors are situated in mining or petroleum environments where
ammable gasses are present [2] [3].
Therefore, the early detection of faults to prevent motor failures and potential
hazards is vital and critical to industry. The prediction of incipient faults can also
help reduce the operational costs. Such advanced warning is obviously desirable
since it allows maintenance sta to schedule outages more freely, resulting in lower
down time and capitalized losses. Systematic approaches must be exploited to
1
Chapter 1. Introduction
predict incipient machine faults. The condition monitoring and fault diagnostics
for induction machines is a vast area of study. However, the key point is to
diagnose the faults by monitoring the parameters of machine operations. By
doing so the condition of electric machines is continuously evaluated throughout
their serviceable life. A typical condition monitoring system should accomplish
the following tasks as one complete diagnosis cycle [3]:
1. The transduction task (primary signal collection);
2. The data acquisition task;
3. The signal processing task;
4. The fault diagnostic task.
Depending on the type of fault, the measurement and the analysis method both
dier. Electrical quantities are the most prevalent measurements. Condition
monitoring and fault diagnostics are usually implemented by investigating the
corresponding anomalies in machine current, voltage and leakage ux. In this
thesis, the motor stator current representations of broken rotor bar faults will be
investigated and analyzed. Other methods including monitoring the core temper-
ature, bearing vibration level, and pyrolysed products have also been reported in
the literature in order to diagnose a variety of fault conditions such as insulation
defects, partial discharge and lubrication oil and bearing degradation [3].
1.2. Motor Current Signature Analysis
The primary concerns of the topic of induction machine condition monitoring and
fault diagnostics are the mechanism and the representation of a specic fault,
and the feasible diagnostic approaches for practice. Research has found that the
machine stator current inherently reects the overall condition of an induction
machine by presenting corresponding frequency components in the current sig-
nal. Thus, fault diagnostics can be accomplished by investigating the spectrum
of stator current [4]. This has promoted the Motor Current Signature Analy-
sis (MCSA), which was systematically developed in the end of 20th century, to
become widely adopted as an eective approach to electrical machine condition
monitoring and fault diagnostics [2][5].
An MCSA system generally consists of a current probe, a signal processing box,
and a fault detection algorithm [6]. A owchart showing the information ow in
2
1.3. Motivation
Figure 1.1.: A ow chart of a typical MCSA system for broken rotor bar diagnosis.
a typical MCSA system for broken rotor bar diagnosis is illustrated in Figure 1.1.
The stator current is measured by a current transformer (CT), and then passed
through the signal processing box for spectral analysis. During the signal process-
ing procedure, signals are low-passed ltered, analogue/digital (A/D) converted
and nally transformed into the frequency domain. By investigating the frequency
components which are present in the spectrum, fault diagnosis algorithms can be
then applied to detect the faults.
The MCSA approach has several advantages. Firstly, it uses stator current as
the measurement, which can be easily monitored by tting a clip-on CT around
the supply cable, without interfering with the machine. This is a great advantage
especially when the motor is inaccessible or is located in a hazardous environment.
Secondly, it is a non-intrusive technique, which means there is no need for any
physical impairment to the motor. Thirdly, it is used in on-line monitoring systems
as it can be undertaken when the machine is still operating, without interrupting
the industrial production.
Broken rotor bars is a frequent fault in induction machines that can be diagnosed
with the MCSA approach. It has been a popular topic of research in recent decades
[7][8][9][10]. In this thesis, the cause, impact, modelling, and detection of broken
rotor bar faults are investigated. A dynamic model of induction motor with broken
rotor bars is developed. The implementation of a high-resolution technique using
motor stator current is explored and presented.
1.3. Motivation
The typical symptom of broken rotor bar faults utilized for diagnosis purpose
is the presence of two frequency components on both sides of the fundamental
frequency in the motor stator current spectrum [11]. They are called the broken
rotor bar sideband frequencies. These sideband frequency components are usually
3
Chapter 1. Introduction
very close to the fundamental frequency and have relatively small amplitudes. This
combined with low signal to noise ratio makes the task of selecting a frequency
estimation technique dicult.
Discrete Fourier Transform (DFT) is a classical technique for spectral analysis.
Fast Fourier Transform (FFT), which is a fast computation algorithm of DFT,
has been previously adopted in the implementation of MCSA [9] [12]. However,
inherent disadvantages of using DFT, such as the impact of side lobe leakage and
the limitation of frequency resolution, limit its applicability. The two broken rotor
bar sideband frequencies move along with the variation of the rotor speed, which
is inuenced by the machine load condition. The sideband frequencies can be very
close to the fundamental frequency when the load is light. This requires the data
window used for DFT to be enlarged [13]. Additionally, DFT requires the values
of the frequencies in the target signal to be constant. However, in practice the
machine load condition is usually dynamic, resulting in frequency variations in
the current signal. Sometimes only restricted data records are available. These
factors make the enlargement of data windows troublesome or virtually impossible.
Therefore, in most applications reported in the literature, where DFT is used for
the signal processing stage of MCSA, the machine load is usually xed at full load
[8].
Due to those limitations of DFT, other methods for spectral analysis become
potential options to be considered [4]. Prony Analysis (PA) is a high-resolution
spectral analysis method developed based on the original work of the French math-
ematician, Gaspard de Prony [14]. In this thesis, Prony Analysis is proposed and
exploited for signal processing to improve the broken rotor diagnosis result. This
technique is able to achieve a high frequency resolution and estimation accuracy
in the detection of broken rotor bar sideband frequencies by using very short data
acquisition windows. Such technique overcomes the drawbacks of DFT and has
high value in practical implementation in light and variable load conditions. More-
over, it is also possible to extend this method to diagnose other types of motor
faults using spectral analysis based MCSA.
A model built in Matlab/Simulink has been developed and adopted to simulate
the operations of an induction motor with broken bars in its rotor. The motivation
to use simulations is fueled by the simplicity of varying the number of broken rotor
bars and load conditions, and also for the economic benet of using simulations.
The model is benchmarked against the experimental results of a real motor, in
4
1.4. Synopsis of Thesis
order to ascertain its validity.
1.4. Synopsis of Thesis
This research addresses the hypotheses that (1) broken rotor bar faults are de-
tectable via the stator current spectrum; (2) broken bar rotor faults can be mod-
eled and their eect on the stator current can be simulated; and (3) there are
high-resolution spectral analysis techniques that can be used to estimate nearby
frequency components in the motor stator current using a shorter data window
than DFT. While formulating a plan to investigate these hypotheses, ve questions
arise and then are answered in the thesis:
• What are the indicators of broken rotor bars in the machine stator current
spectrum;
• How can the broken rotor bar fault be described in a model;
• What are the limitations of using the traditional spectral analysis method -
DFT;
• What are the improvements of using high-resolution spectral analysis;
• What are the factors that aect the performance of the high-resolution tech-
nique.
In Chapter 2 a background knowledge of broken rotor bar faults in induction
machines and their diagnosis is reviewed. Some former research on quantitative
prediction of the fault severity is presented. In Chapter 3, a model of an induc-
tion motor with broken rotor bars is described mathematically and constructed
using Matlab/Simulink. It is also benchmarked against experimental results. This
model is used to both investigate the impact of broken rotor bar faults and, to
generate a set of representative signals to implement Prony Analysis. After that,
the Prony's method is introduced and explored in Chapter 4. Chapter 5 presents
the implementation of Prony Analysis using both simulated and measured data.
The result and comparisons with DFT are also illustrated. Finally, Chapter 6
gives the conclusion.
5
Chapter 1. Introduction
6
Chapter 2.
Broken Rotor Bar Faults in
Induction Machines and
Non-Intrusive Methods of
Detection
2.1. Introduction
Broken rotor bars are a common fault in induction machine rotors. Dedicated
diagnostic techniques and systems are demanded to detect an upcoming machine
defect as early as possible. In this chapter, features and facts of this fault and
their use in diagnosis are detailed. Up to date research in fault severity prediction
is also reported.
2.2. The Induction Motor
2.2.1. The Construction of Induction Motors
The stator of an induction machine has a cylindrical annulus magnetic core which
is formed by stacking thin electrical steel laminations with uniformly spaced slots
stamped in the inner circumference. Wound poles are formed by connecting the
coils of copper or aluminum conductors. These windings carry the three-phase
supply current which induces a rotating magnetic eld in the air gap between the
stator and rotor. The terminals of the three stator phase windings can be an
either star or delta connection.
7
Chapter 2. Broken Rotor Bar Faults in Induction Machines and Non-Intrusive Methods of Detection
The rotor consists of a cylindrical laminated iron core with uniformly spaced
peripheral slots to accommodate the rotor windings and it is supported on two
bearings. There are two main types of rotors: the squirrel-cage rotor and the
wound rotor. The squirrel-cage rotor, which is the most commonly used, has two
end rings at both ends of the rotor, with axial bars running the length of the
rotor and soldered onto the rings. There is no insulation between the rotor bars
and the walls of rotor slots. It is typical that lower-resistance cast aluminum or
copper is poured in between the iron laminates, thus the rotor bars carry the vast
majority of the rotor current ow. The windings in a wound rotor are similar to
the distributed windings in the stator. The terminals of the rotor windings are
connected via slip rings.
2.2.2. The Operation of Induction Motors
The induction motor is also called the asynchronous motor. The word induc-
tion refers to the fact that the electromagnetic eld in the rotor is induced by the
stator current, and asynchronous refers to that the rotor operates below the syn-
chronous speed when motoring and above the synchronous speed when generating.
There is no current supply to the rotor. Instead, when three-phase current ows
through the stator windings, a sinusoidally distributed air gap ux is produced,
which generates rotor current. The currents owing in rotor then magnetize the
rotor to establish the revolving rotor magnetic eld. This magnetic eld interacts
with the stator magnetic eld to force the rotor to rotate into synchronization
with the stator magnetic eld.
The mechanical angular speed of the rotor is always lower than the angular speed
of the synchronous rotating stator eld in the air gap of a motor. This velocity
dierence is the so called slip speed. For an induction machine with P poles, the
ratio of slip speed to the synchronous speed in mechanical radians is called the
per unit slip, or slip, notated as s and given by
s =ωsm − ωrm
ωsm(2.1)
where ωrm is the rotor rotating speed and ωsm = 2Pωe is the synchronous speed
in mechanical radians. ωe is given as the angular speed of stator magnetic motive
force (mmf) in electrical radians per sec.
The product of slip and fundamental frequency (frequency of the excitation cur-
8
2.3. Induction Machine Broken Rotor Bar Faults
rents) f , sf , is called the slip frequency. The magnitude of the currents owing
in the rotor is determined by the magnitude of the induced rotor voltages and
the rotor circuit impedance at slip frequency [15]. Slip is always positive if the
machine is operating in motoring mode. When the motor is lightly loaded, the
rotor rotates at a very high speed so that the slip is very small. When the motor is
heavily loaded the rotor will rotate at a relatively lower speed causing an increase
in the slip.
2.3. Induction Machine Broken Rotor Bar Faults
2.3.1. Causes
Regardless of the connecting pattern, rotors are made of skewed solid metal lam-
inations which are arranged around its cylindrical surfaces. The laminated bars
and end rings can sometimes crack or break, resulting in the so called broken rotor
bar faults.
Broken rotor bars are usually caused by fatigue stresses owe to frequent start-ups.
In industry it is normal practice to start motors direct on-line. This results in
the starting current in the rotor 5 to 8 times larger than the rated current and
also creates high centrifugal loadings on the end rings of the cage [3]. When the
start-up time is relatively long and the starting is frequent, which are commonly
required in heavy duty cycles, the thermal and mechanical stresses often cause
damages to the rotor.
Faults may also occur during manufacturing process, through defective casting in
the case of die cast rotors, or poor jointing in the case of brazed or welded end
rings. Such defects cause higher resistances in certain parts of the rotor [3].
In fact, the joints between rotor bars and end rings are the critical locations where
the cracks are most likely to occur. This is because the rotor bars must provide the
braking and accelerating forces on the end ring when the motor changes speed.
Moreover, faulty bars always happen contiguously. This is due to that rotor
bars in the neighborhood of the defective bars suer a greater current ow and
overheating thermal impact, which are the primary causes of iron damage, than
the rest one.
9
Chapter 2. Broken Rotor Bar Faults in Induction Machines and Non-Intrusive Methods of Detection
2.3.2. Impact
The induction machine is a highly symmetrical system. When an induction mo-
tor is operating under three-phase balanced supply, a symmetrical and periodic
electromagnetic eld rotating at synchronous speed is generated in the air gap.
Under ideal conditions, the current, voltage and magnetic ux are symmetrically
distributed. However, defects in the machine will distort them. The anomalies of
the rotor physical structure change the rotor resistance and inductance, and then
distort the electrical and magnetic elds, resulting in a modulated stator current
carrying the presence of broken rotor bar sidebands [3].
The impact of the broken rotor bars are various. The fault is reected in the stator
current by the presence of twice slip frequency components 2sf around the supply
frequency. Such a cyclic variation in the current reacts back on the rotor will
produce a torque variation and give rise to a like-patterned speed variation [16].
The stator core vibration pattern is also altered by the change of magnetic forces
owing to the change of the air gap ux pattern, resulting in modulated frequency
components in the stator core vibration spectrum [17]. The same frequency com-
ponents as those are in stator core vibration spectrum are also observed in the
axial ux spectrum. The vibration, ux linkage, output torque and instantaneous
power signatures have all been reported to be useful for detection but none of
them shows to be more reliable or feasible than the use of stator current [18].
Defective rotors with broken bars have a number of disadvantages. They signif-
icantly lower the machine's eciency, which considerably increases the already
high electricity costs for industry. Arcing and sparking may occur during motor
operation causing unexpected accidents if motor is situated in a hazardous envi-
ronment. Fractured bars also overheat other bars in the vicinity, which degrades
the insulation and damages the windings. Additionally, if the fault deteriorates,
there are potential hazards of machine breaking down. This is observed in the
simulation when the number of broken rotor bars increases close to one third of
the total number of rotor bars.
2.4. Detection of Broken Rotor Bar Faults
2.4.1. Presentation of Broken Rotor Bar Faults in Stator
Current
This thesis presents an implementation of MCSA for broken rotor bar detection.
10
2.4. Detection of Broken Rotor Bar Faults
As previously mentioned in 2.3.2, representative frequency components occur in
the stator current spectrum of an induction machine with defective rotor bars.
They are formed by the twice slip frequency, analytically expressed as [19]
fbrb = (1± 2ks) f (2.2)
where k = 1, 2, 3 . . . , and [7]
fbrb = f
[2k
(1− sP
)± s]
(2.3)
where 2kP
= 1, 5, 7, 11, 13.
The amplitudes of these additional frequency components in the stator current are
determined by the fault severity and decrease as the equation index k increases.
Actually, when the number of broken rotor bars is much smaller than the number
of total rotor bars, only (1± 2s) f frequency components will appear in the stator
current spectrum. The (1− 2s) f component is aected by cyclic variation in the
torque at 2sf directly. The (1 + 2s) f component is caused by the speed ripple
due to a nite machine-load inertia value [20].
As the fault becomes severer, higher order harmonics will then arise. It is under-
standable as that the more rotor bars are defective, the more seriously the stator
current will be modulated. This can be observed from Figure 2.1, which shows
the stator current spectra with respect to dierent numbers of broken rotor bars.
The simulated stator current is from a 5.5kW induction motor operating under
full load condition. The same spectral lines are also observed in the spectrum of
measured data shown in Section 5.6. Because of this, the two sideband frequen-
cies at (1± 2s) f Hz are considered to be the most characteristic indicators of
broken rotor bar faults and have been widely adopted in most practical applica-
tions. They give a straight forward indication of the extent of rotor damage, and
are well known as the broken bar sidebands.
From Eq. (2.1) it is learned that the slip s is dependent on the rotor speed. Thus,
the two broken rotor bar sideband frequencies move as the rotor speed changes.
Figure 2.2 illustrates the movement of the broken rotor bar sideband frequency
components by showing the stator current spectra of an induction motor with
one fractured rotor bar operating under 100%, 75%, 50% and 25% of rated load
together. The result clearly demonstrates that for a lighter machine load, the
11
Chapter 2. Broken Rotor Bar Faults in Induction Machines and Non-Intrusive Methods of Detection
0 50 100 150 200 250−120
−100
−80
−60
−40
−20
0Am
plitude
(dB)
Frequency (Hz)
(a) Healthy rotor.
0 50 100 150 200 250−120
−100
−80
−60
−40
−20
0
Amplit
ude (dB
)
Frequency (Hz)
(b) With 1 broken rotor bars
0 50 100 150 200 250−120
−100
−80
−60
−40
−20
0
Amplit
ude (dB
)
Frequency (Hz)
(c) With 3 broken rotor bars.
12
2.4. Detection of Broken Rotor Bar Faults
0 50 100 150 200 250−120
−100
−80
−60
−40
−20
0
Amplit
ude (dB
)
Frequency (Hz)
(d) With 5 broken rotor bars.
0 50 100 150 200 250−120
−100
−80
−60
−40
−20
0
Amplit
ude (dB
)
Frequency (Hz)
(e) With 8 broken rotor bars.
Figure 2.1.: Spectra of the simulated stator current of a 5.5kW induction motorwith 32 total rotor bars operating under full load condition, withrespect to the number of broken rotor bars .
40 45 50 55 60−120
−100
−80
−60
−40
−20
0
Amplit
ude (dB
)
Frequency (Hz)
25% load50% load75% loadFull load
Figure 2.2.: Spectra of the simulated stator current of a 5.5kW induction motorwith 1 broken rotor bar with respect to dierent load conditions.
13
Chapter 2. Broken Rotor Bar Faults in Induction Machines and Non-Intrusive Methods of Detection
(1± 2s) f sideband frequencies are closer to the fundamental frequency.
2.4.2. Detection Indices
The amplitudes of broken bar sideband components indicate the existence of a ro-
tor bar fracture. However, the (1± 2s) f sideband frequencies can be still detected
even in stator current spectrum of a healthy induction motor due to unavoidable
manufacturing asymmetries and misalignment [7]. It also may be motor current
modulation produced by other events, for example pulsating loads and the natural
imbalance of the rotor structure. This can confuse the decision making and make
it dicult to judge whether or not there are broken rotor bars.
In practice, detection indices are needed in order to set a threshold for the healthy
condition of an induction machine. However, how the amplitudes of the sideband
frequency components in the stator current spectrum of a given machine in a
certain operating condition relate to the presence or absence of broken rotor bars
is not a easy decision. Such decisions require either an experienced operator or
a knowledge based system that may include all possible fault scenarios in a data
base. Bellini [5] has proposed an empirical formula to calculate the threshold
amplitude of broken rotor bar sideband frequencies from practical experience. It
is given by
IBBI
=0.5
Nb
(2.4)
where IBB and I are the amplitudes of the (1− 2s) f sideband and the fundamen-
tal frequencies in the stator current spectrum, respectively, and Nb is the number
of total rotor bars. This equation trades o the eects of the intrinsic asymme-
try and that of the rst cracked bar, and depends on the machine size and thus
the total number of rotor bars. If the ratio of the amplitudes of the (1− 2s) f
sideband and the fundamental frequencies is higher than 0.5Nb, it is considered that
there are broken rotor bars; if the ratio is smaller than 0.5Nb, it is considered that
the machine rotor is healthy.
Another detection index threshold was proposed by Kliman [7]. He claims that
if the dierence in amplitude between the (1− 2s) f current sideband frequency
component and the fundamental frequency is less than 60dB, there is probably no
fault; if the dierence is at least 54dB, there is, very likely, a cracked bar; and if
the dierence is greater than 50dB, there is probably a broken bar.
14
2.4. Detection of Broken Rotor Bar Faults
2.4.3. Assessment of Rotor Fault Severity
The knowledge of the existence of broken rotor bars sometimes is insucient in
practice. The fault severity, which means the number of broken rotor bars in this
context, is always highly desired for the purpose of decision making on equipment
maintenance. The observable relationship between the number of broken rotor
bars and the amplitudes of the sideband frequencies indicate the possibility of a
quantitative index of the fault severity. Figure 2.3 presents an example revealing
this relationship. The amplitude of the sideband frequencies in dB with reference
to the amplitude of the fundamental frequency is calculated by using the equation
IdB = 20 log
(IBBI
)(2.5)
A 5.5kW induction motor with various numbers of defective rotor bars, which are
detailed in Chapter 5, has been simulated using the model presented in Chapter 3,
to generate the data. The load eect is also taken into account in the simulations.
Figure 2.3 shows that the amplitude of the lower broken bar sideband frequency
(1− 2s) f increases along with the accretion of the number of broken rotor bars. It
can also be observed that a lighter load causes the sideband frequency of a smaller
amplitude, and conversely a heavier load produces a larger sideband amplitude.
However, the fault severity has a greater impact on the amplitude of the broken
rotor bar sideband frequency than the load condition.
There has not been any analytical formulas which link the amplitudes of the
broken rotor bar sidebands with the actual number of fractured rotor bars since
the sideband amplitude is modied by the winding, pitch and distribution factors
and the leakage inductance [7]. However, several prediction equations which give
approximate indications of rotor defects severity based on empirical relations and
experimental experience have been proposed in earlier research. There are three
important predictions as listed below.
Prediction 1
According to Bellini [5], based on the assumption of constant load, the prediction
formula is
IBBI
=nbbNb
(2.6)
where nbb is the number of broken rotor bars. When the machine load, and
15
Chapter 2. Broken Rotor Bar Faults in Induction Machines and Non-Intrusive Methods of Detection
25% 50% 75% 100%−50
−45
−40
−35
−30
−25
−20
−15
−10
−5
0
Load conditions (percentage of full load)
(1−
2s)f
sid
eban
d am
plitu
de (
dB)
1 broken bar2 broken bars3 broken bars4 broken bars5 broken bars6 broken bars7 broken bars
Figure 2.3.: Amplitude of the (1− 2s) f sideband current component in dB rela-tive to the fundamental frequency as the number of broken bars andload conditions are varied.
consequently the rotor speed, are not constant, IBB should be replaced by the
sum of the amplitudes of the two sideband frequencies (1± 2s) f .
Prediction 2
The second quantitative fault evaluation equation is proposed by Hargis [16] as
IBBI
=sinα
P (2π − α)(2.7)
where α is the electrical angle of a contiguous group of broken rotor bars, given
by
α =πPnbbNb
(2.8)
and P is the number of machine poles. This method makes the assumptions of
constant rotor speed and that nbb Nb.
Prediction 3
Thomson [2] proposed a modied version of Hargis's prediction equation Eq. (2.7),
given as
16
2.4. Detection of Broken Rotor Bar Faults
0 1 2 3 4 5 6 7 8 9−50
−45
−40
−35
−30
−25
−20
−15
−10
−5
0
(1−
2s)f
sid
eban
d am
plitu
de (
dB)
Number of broken rotor bars
Prediction 1Prediction 2Prediction 3
Figure 2.4.: The prediction curves of (1− 2s)f current sideband frequency ampli-tudes in dB relative to the fundamental frequency, obtained by usingthree prediction equations Eq. (2.6) (2.7) and (2.9) with respect tothe number of total rotor bars Nb = 32.
nbb =2Nb
10D20 + P
(2.9)
whereD is the amplitude dierence between the lower sideband frequency (1− 2s) f
and the supply frequency f in decibel.
To study the property of these prediction equations, Figure 2.4 reveals the trend of
the amplitude changes of the lower broken bar sideband (1− 2s) f with respect to
fault severity, indicated by the above three prediction equations. The machine is
a 5.5kW three-phase induction machine with 32 rotor bars. Each of the predicting
curves clearly shows that the sideband amplitude increases with the number of
broken bars when this number is less than half of the total rotor bar number in
one rotor phase. However, the curve of Prediction 2 falls down after the number
of broken rotor bars is greater than 4. This is due to the assumption made in
Prediction 2 that the number of defective rotor bars should be much smaller than
the total rotor bar number. The curve of Prediction 3 has an improved tendency
compared with that of Prediction 2. It still shows increasing sideband amplitude
with respect to the number of broken rotor bars even when more than half of the
total rotor bars in one rotor phase fail. Thus, unlike Prediction 2, Prediction 3 does
not constrain nbb. The curve of Prediction 1 shows a similar trend to Prediction
17
Chapter 2. Broken Rotor Bar Faults in Induction Machines and Non-Intrusive Methods of Detection
3 but with a higher amplitude. This dierence decreases as the number of broken
rotor bars increases.
2.5. Limitations and Possible Improvement
Despite the advantages of MCSA, its dependability and accuracy are aected sig-
nicantly by external factors, for example, load condition, because of the inherent
drawbacks of DFT. Failures of detection of rotor faults using MCSA have been
reported in the literature [9] and [16], due to the randomly uctuating load.
It is also a major problem in light load conditions, as the two broken bar sideband
frequencies are very close to the fundamental frequency, making it dicult to
detect them. Further details of the drawbacks of DFT is described in the Section
4.2 of Chapter 4.
18
Chapter 3.
Model of an Induction Machine
with Broken Rotor Bars
In order to investigate the impact of broken rotor bars and to generate a set of
representative signals to test the spectral analysis methods, a dynamic induction
machine model has been developed. Simulations have been used for analysis in
this research due to the benets of model based studies the overall nancial
and manpower cost for simulation is signicantly less than that needed for ex-
perimental studies. Firstly, commercial induction motors can be very expensive,
whereas using simulations on the available software is economical. Secondly, lab-
oratory experiments need to be designed and set up, which requires the assistance
of a number of laboratory sta members. By using simulation this extra work
can be avoided. Thirdly, physically breaking the rotor bars is not a easy task.
If a comprehensive study is desired, a number of identical rotors are required.
Holes are usually drilled in a dierent number of the rotor bars to construct the
dierent degrees of broken rotor bar fault. In addition, for each case that to be
studied, the motor has to be opened and the rotor has to be manually installed.
If broken rotor bar faults in motors of dierent power is to be studied, the work
needs to be repeated for each motor, and the high power motors can be physically
really huge. In contrast, using a machine model simulations provides the benets
of great exibility in changing machine parameters, the number of broken rotor
bars, and load conditions. In the end of this chapter, limited laboratory results
are used for verication.
3.1. Introduction
A reliable model is essential for accurate simulation and fault prediction. The
19
Chapter 3. Model of an Induction Machine with Broken Rotor Bars
model should be realistic, yet general. It must be able to incorporate all of the
important dynamic characteristics, during both transient and steady-state oper-
ations, and be able to simulate the operation of both healthy induction machines
and those with defective rotors. The desired simulated stator current needs to
be able to reect on the impact of broken rotor bar faults and their inuencing
factors. All machine parameters should be accessible for variations in values. The
model also should be simple to understand and easy to manipulate.
There are many dynamic induction machine models that have been well developed
via dierent approaches [21] [22] [23]. A mathematical model based on the Coupled
Circuit Approach is introduced in this chapter. After the construction of a general
induction machine model, the next step is to specically model the broken rotor
bars. This is accomplished by unbalancing the rotor resistance, which is described
in 3.2.2 in detail.
Matlab/Simulink is a powerful tool for modeling and implementing simulations
and is simple to use [15]. In this research, the machine model has been constructed
with programs coded in Matlab/Simulink. Simulation results are used for the
study of broken rotor bar detection using Prony Analysis. The simulation results
are also presented and compared with experimental results in this chapter to
validate the model.
3.2. Mathematical Model
3.2.1. Mathematical Model of an Induction Machine
3.2.1.1. Machine Model in Traditional abc Frame of Reference
Usually the Coupled Circuit Approach is used to describe the electromagnetic
relationships of induction machines with wound rotors. Induction motors with
squirrel-cage rotors can be considered equivalent to the wound rotor motors in
terms of equivalent rotor resistance [24]. The stator and rotor circuits of an
induction machine are magnetically coupled. Using the Coupled Circuit Approach
and matrix notation, an idealized induction machine may be presented in terms
of the rst-order dierential equations of the voltages in the motor natural abc
reference frame as [25]:
vabcs = rs i
abcs +
dλabcsdt
(3.1)
20
3.2. Mathematical Model
vabcr = rri
abcr +
dλabcrdt
(3.2)
and
λabcs = Labcss iabcs + Labc
sr iabcr (3.3)
λabcr = Labcrr iabcr + Labc
rs iabcs (3.4)
Notations vabcs , iabcs , λabcs , vabcr , iabcr and λabcr are column vectors representing the
voltages, currents, and ux linkages of each phase in either stator or rotor, where
the subscripts s and r indicate stator and rotor, respectively, and the superscript
abc denotes the three phases. In an ideal induction machine, the resistance in
each stator or rotor phase is assumed to be equal. Thus, notations rs and rr
are diagonal matrices with one phase equivalent resistance of the stator or rotor,
whichever the subscript indicates, as the non-zero elements, given as
rs,r =
rs,r 0 0
0 rs,r 0
0 0 rs,r
where rs and rr denote the balanced equivalent resistance in each phase of a
healthy induction machine stator and rotor, respectively.
Notations Labcss and Labc
rr are matrices of the self inductance of the stator and the
rotor windings, respectively, while Labcsr and Labc
rs are matrices of the stator-to-rotor
and rotor-to-stator mutual inductances, respectively.
The submatrices of the stator-to-stator and rotor-to-rotor winding inductances
are formed as follows:
Labcss =
Lls + Lss Lsm Lsm
Lsm Lls + Lss Lsm
Lsm Lsm Lls + Lss
(3.5)
Labcrr =
Llr + Lrr Lrm Lrm
Lrm Llr + Lrr Lrm
Lrm Lrm Llr + Lrr
(3.6)
21
Chapter 3. Model of an Induction Machine with Broken Rotor Bars
where Lls is the per phase stator winding leakage inductance, Llr is the per phase
rotor winding leakage inductance, Lss is the self inductance of the stator winding,
Lrr is the self inductance of the rotor winding, Lsm is the mutual inductance
between the stator windings and Lrm is the mutual inductance between the rotor
windings.
The stator-to-rotor mutual inductances are expressed as:
Labcsr =
[Labcrs
]T= Lsr
cos θr cos(θr + 2π
3
)cos(θr − 2π
3
)cos(θr − 2π
3
)cos θr cos
(θr + 2π
3
)cos(θr + 2π
3
)cos(θr − 2π
3
)cos θr
(3.7)
where Lsr and Lrs are the peak values of the stator-to-rotor and rotor-to-stator
mutual inductance, respectively, θr is the electrical angle between the a-phase axes
of the stator and the rotor, namely the rotor angle, and the superscript T denotes
the transpose of the matrix.
Above equations together show that the stator and rotor voltage equations are
coupled to one another through the mutual inductance terms, which are a function
of rotor angle [15]. Thus, the coupled terms interact and vary with the rotor
position and time.
For a complete model, a torque equation is also needed. The torque equation is
deduced by applying energy conservation, which is given by Eq. (3.8) in the case
of an induction machine.
Pin = Pem + Ploss + Pmm (3.8)
where Pin is the power input to the induction machine, Pem is the rate of energy
converted to mechanical work on the rotor shaft, Ploss is the copper loss and
Pmm represents the rate of exchange of magnetic eld energy between windings.
The electromechanical torque is dened by the Pem term divided by the rotor
mechanical angular speed ωrm, as
Tem =Pemωrm
. (3.9)
22
3.2. Mathematical Model
3.2.1.2. Park's Transformation
For convenience, mathematical transformations are often used to study the rotat-
ing electric machinery. This is because the coecients of the voltage dierential
equations are time-varying except when the machine is at standstill.
Park's transformation transforms variables from the abc reference frame to an
arbitrary rotating qd0 reference frame [26]. The quadrature and direct axes are
ctitious quantities of the symmetrical three-phase induction motor. The rela-
tionship between abc and arbitrary qd0 reference frames is illustrated in Figure
3.1, where θ is the stator transformation angle, which is the angle between the
q-axis of the arbitrary reference frame that rotates at an angular speed of ω in the
direction of the rotor rotation and the a-axis of the stationary stator winding. It
can be calculated by
θ (t) =
ˆ t
0
ω (ζ) dζ + θ (0) (3.10)
where ζ is the dummy variable of integration. ωr is the electrical angular speed of
rotor rotation in radian per second. It is easy to observe that the transformation
angle for rotor parameters is (θ − θr), where the rotor angle may be expressed as
θr (t) =
ˆ t
0
ωr (ζ) dζ + θr (0) (3.11)
The angles θ (0) and θr (0) stand for the initial angular values of the transformation
angle and rotor angle, respectively, at the time t = 0.
The transformation function for stator variables may be written as
fqd0 = Tqd0 (θ) fabc (3.12)
where the elements of the column vectors fqd0 and fabc can be the phase voltages,
currents, or ux linkages of the machine, andTqd0 is the qd0 transformation matrix
with the form [15]
Tqd0 (θ) =2
3
cos θ cos(θ − 2π
3
)cos(θ + 2π
3
)sin θ sin
(θ − 2π
3
)sin(θ + 2π
3
)12
12
12
(3.13)
23
Chapter 3. Model of an Induction Machine with Broken Rotor Bars
Figure 3.1.: Relationship between abc and arbitrary qd0 reference frames.
It should be noted that f0 represents a scaled version of the zero sequence terms
from symmetrical components.
The inverse of the transformation equation Eq. (3.12) may be expressed as
fabc = Tqd0 (θ)−1 fqd0 (3.14)
where
Tqd0 (θ)−1 =
cos θ sin θ 1
cos(θ − 2π
3
)sin(θ − 2π
3
)1
cos(θ + 2π
3
)sin(θ + 2π
3
)1
(3.15)
The arbitrary qd0 reference frame can be chosen to rotate at a designated speed
in the same direction as the rotor rotation to simplify the model. In practice,
two often used reference frames for the analysis of induction machine in dierent
scenarios are the stationary and the synchronous reference frames. The rotor ref-
erence frame rotating at the same speed as the rotor is used infrequently. With
arbitrary rotating reference frames, it is convenient to convert to any reference
frame as desired. This can be easily accomplished by setting the reference rotat-
ing speed ω equal to either zero, the synchronous speed, or the rotor speed, for
stationary, synchronous or rotor reference frame applications [15].
24
3.2. Mathematical Model
3.2.1.3. Machine Model in Arbitrary dq0 Frame of Reference
In the next step, to transform the machine voltage and ux linkage equations in
the abc reference frame to the arbitrary dq0 reference frame, the transformation
functions Eq. (3.12) and Eq. (3.14) are applied to the voltages, currents and
resistances in Eq. (3.1) and Eq. (3.2). This yields [15]
vqd0s = Tqd0 (θ) rsT−1qd0 (θ) iqd0s + Tqd0 (θ)
d[T−1qd0 (θ)λqd0s
]dt
(3.16)
vqd0r = Tqd0 (θ − θr) rrT−1qd0 (θ − θr) iqd0r + Tqd0 (θ − θr)
d[T−1qd0 (θ − θr)λqd0r
]dt
(3.17)
Substituting Eq. (3.13) and Eq. (3.15) into Eq. (3.16) and Eq. (3.17), and
rearranging the equations, produces
vqd0s = rqd0s iqd0s + Eqd0s +
dλqd0s
dt(3.18)
vqd0r = rqd0r iqd0r + Eqd0r +
dλqd0r
dt(3.19)
where
Eqd0s = ω
0 1 0
−1 0 0
0 0 0
λqd0s , Eqd0r = (ω − ωr)
0 1 0
−1 0 0
0 0 0
λqd0r ,
ω =dθ
dt, ωr =
d (θr)
dt,
and
rqd0s = rs
1 0 0
0 1 0
0 0 1
, rqd0r = rr
1 0 0
0 1 0
0 0 1
.The ir terms are the voltages produce copper losses, the E terms represent the
speed voltages which determine the rate of energy converted to mechanical work,
25
Chapter 3. Model of an Induction Machine with Broken Rotor Bars
and the dλdt
terms are the rate of exchange of magnetic eld between windings.
The details of the derivation of the Eq. (3.18) and Eq. (3.19) can be found in
Appendix B.1.
By applying the Park's transformation to the ux linkages, inductances and cur-
rents in Eq. (3.3) and Eq. (3.4), yield
λqd0s = Tqd0 (θ)(Labcss iabcs + Labc
sr iabcr)
(3.20)
λqd0r = Tqd0 (θ − θr)(Labcrr iabcr + Labc
rs iabcs)
(3.21)
Rearrange the equations and we nally obtain
λqs
λds
λ0s
λ′qr
λ′drλ′0r
=
Lls + Lm 0 0 Lm 0 0
0 Lls + Lm 0 0 Lm 0
0 0 Lls 0 0 0
Lm 0 0 L′lr + Lm 0 0
0 Lm 0 0 L′lr + Lm 0
0 0 0 0 0 L′lr
iqs
ids
i0s
i′qr
i′dri′0r
(3.22)
The derivation of Eq. (3.22) is presented in detail in Appendix B.2. The primed
rotor quantities in the equation denote values referred to the stator side. The
parameter Lm, which is the magnetizing inductance on the stator side, is given
by equation
Lm =3
2Lss =
3
2
Ns
Nr
Lsr =3
2
Ns
Nr
Lrr (3.23)
where Ns and Nr are the numbers of coil in stator and rotor, respectively.
Eq. (3.22) is then substituted back into Eq. (3.18) and Eq. (3.19) to form
the entire machine voltage equations in the arbitrary qd0 reference frame. The
equivalent circuit representation of an induction machine in the arbitrary reference
frame is shown in Figure 3.2. xls, x′lr and xm denote the stator leakage reactance,
the referred rotor leakage reactance, and the stator magnetizing reactance in ohms.
Eqs, Eqr, Eds and Edr are speed voltages dependent on the speed terms ω and ωr.
The torque equation can be transformed into the arbitrary qd0 reference frame in
a similar manner. The power conservation equation Eq. (3.8) can be extended to
26
3.2. Mathematical Model
(a) q-axis
(b) d-axis
(c) zero-sequence
Figure 3.2.: Equivalent circuit representation of an induction machine in the ar-bitrary qd0 reference frame.
27
Chapter 3. Model of an Induction Machine with Broken Rotor Bars
Pin = vasias + vbsibs + vcsics + v′ari′ar + v′bri
′br + v′cri
′cr (3.24)
By applying Park's transformation to Eq. (3.24), yield
Pin =3
2
(vqsiqs + vdsids + 2v0si0s + v′qri
′qr + v′dri
′dr + 2v′0ri
′0r
)(3.25)
Thus, the equation for electromechanical torque in the arbitrary qd0 reference
frame is
Tem =3
2
P
ωrm
[ω (λdsiqs − λqsids) + (ω − ωr)
(λ′dri
′qr − λ′qri′dr
)](3.26)
where P is reminded as the number of machine poles. The rotor mechanical and
electrical rotating speeds ωrm and ωr have the relationship
ωrm =2
Pωr
The derivation of Eq. (3.26) can be found in Appendix (B.3). The torque equation
can also be expressed by using the ux linkage relationship in Eq. (3.22), that is
Tem =3
2
P
2(λdsiqs − λqsids) (3.27)
Moreover, machine parameters are always determined in terms of the ux linkage
per second, ψ, and the reactance, x, instead of λ and L in experiments. These
quantities have the following relationship:
ψ = ωbλ and x = ωbL
where ωb is the base value of the angular frequency calculated by
ωb = 2πf.
3.2.2. Mathematical Model of Broken Rotor Bars
Having constructed a general model for induction machines, the next key task is
to model the broken rotor bars. An induction machine is a highly symmetrical
28
3.2. Mathematical Model
electromagnetic system. Any fault will induce a certain degree of asymmetry.
Broken bars in induction machines can cause asymmetry in the resistances of
rotor phases, which results in asymmetry of the rotating electromagnetic eld in
the air gap between the machine stator and rotor. In turn this will eventually
induce frequency harmonics in the stator current. Therefore, in the mathematical
model, an additional resistance is added into each of the rotor phases to simulate
broken rotor bar faults [24] [21]. The rotor resistance matrix rr in Eq. (3.2) should
be modied accordingly as
r?r =
(rr + ∆rra) 0 0
0 (rr + ∆rrb) 0
0 0 (rr + ∆rrc)
(3.28)
where ∆rra, ∆rrb and ∆rrc represent rotor resistance changes in phase a, b and c,
respectively, due to broken bar faults, dened as [11]
∆rra,b,c =3nbb
Nb − 3nbbrr (3.29)
where nbb and Nb are reminded as the number of broken and the total rotor bars,
respectively.
The function of rotor resistance change ∆rra,b,c due to rotor defects is derived
based on the assumption that the broken bars are contiguous, neither the end
ring resistance nor the magnetizing current are taken account. The rotor phase
equivalent resistance of a healthy induction motor is given as [11]
rr =(2Ns)
2
Nb/3
[rb +
2
Nb
(2 sin α
2
)2 re]
where rb and re represent the rotor bar and end-ring resistances, respectively, and
Ns is the equivalent stator winding turns. As in the assumptions, when re is
neglected, rr then simplies to
rr ≈(2Ns)
2
Nb/3rb
Then, the resistance of one phase rotor with nbb contiguous broken rotor bars
becomes
29
Chapter 3. Model of an Induction Machine with Broken Rotor Bars
r?r ≈(2Ns)
2
Nb/3− nbbrb
and the increment ∆r is obtained as
∆r = r?r − rr =3nbb
Nb − 3nbbrr
Next, substitute the modied rotor resistance for the original rotor resistance
matrix in Eq. (3.2), and then apply the previously described method steps of
transforming quantities from the abc to qd0 reference frame, yielding [21]
4r?qd0r =
r11 r12 r13
r21 r22 r23
r31 r32 r33
(3.30)
where the elements of the matrix are
r11 = 13
(∆rra + ∆rrb + ∆rrc) + 16
(2∆rra −∆rrb −∆rrc) cos (2θr)
+√
36
(∆rrb −∆rrc) sin (2θr)
r12 = −16
(2∆rra −∆rrb −∆rrc) sin (2θr) +√
36
(∆rrb −∆rrc) cos (2θr)
r13 = 13
(2∆rra −∆rrb −∆rrc) cos (θr)−√
33
(∆rrb −∆rrc) sin (θr)
r21 = r12
r22 = 13
(∆rra + ∆rrb + ∆rrc)− 16
(2∆rra −∆rrb −∆rrc) cos (2θr)
+√
36
(∆rrb −∆rrc) sin (2θr)
r23 = −13
(2∆rra −∆rrb −∆rrc) sin (θr)−√
33
(∆rrb −∆rrc) cos (θr)
r31 = 12r13
r32 = 12r23
r33 = 13
(∆rra + ∆rrb + ∆rrc)
3.3. Model in Matlab/Simulink
3.3.1. Introduction of Matlab/Simulink
Simulink is an extended software package of Matlab that can be used to model,
simulate and analyze dynamic systems. It provides a graphical modeling interface
30
3.3. Model in Matlab/Simulink
facilitated by programming [15]. To set up a dynamic model for a complex system
in Simulink, the mathematical description of the system is required. These equa-
tions need to be adjusted for the implementation in Simulink. Then, a dynamic
system simulation can be completed by using the Simulink model editor to create
block diagrams, and then commanding Simulink to run the system model for a
specied start and stop time. A Simulink block diagram model can be manipu-
lated graphically to depict the time-dependent mathematical relationships of the
system among the system inputs, states, and outputs.
A suggestion for modeling induction machines in Matlab/Simulink from both [15]
and [27] is that integral equations are preferable than dierential equations. Addi-
tionally, it is helpful to write integral equations with the dependent-state variables
expressed as self-referencing integral functions of independent and dependent vari-
ables [15]. Using these suggested approaches a model can be more visually com-
prehensive and have less chances of errors.
3.3.2. Model Description Equations for Matlab/Simulink
This section describes a modular Simulink model of an induction machine built
according to the mathematical description in 3.2. The eect of broken rotor bars
is considered and applied to the described model.
When use the stationary reference frame, the stator speed voltage terms
Eqd0s = ω
0 1 0
−1 0 0
0 0 0
λqd0s
in Eq. (3.18) will be eliminated. Eq. (3.18), (3.19) and (3.22) are often expressed
in terms of ux linkage per second and reactance, as these are the parameters
which are usually measured in experiment. With the rotor parameter values re-
ferred to stator, these equations can be written as
vqd0s = rqd0s iqd0s +1
ωb
dψqd0s
dt(3.31)
v′qd0r = r′qd0r i′qd0r − ωrωb
0 1 0
−1 0 0
0 0 0
ψ′qd0r +1
ωb
dψ′qd0r
dt(3.32)
31
Chapter 3. Model of an Induction Machine with Broken Rotor Bars
ψqs
ψds
ψ0s
ψ′qr
ψ′drψ′0r
=
xls + xm 0 0 xm 0 0
0 xls + xm 0 0 xm 0
0 0 xls 0 0 0
xm 0 0 x′lr + xm 0 0
0 xm 0 0 x′lr + xm 0
0 0 0 0 0 x′lr
iqs
ids
i0s
i′qr
i′dri′0r
(3.33)
As stated, in models built in Simulink, integral equations are used rather than
dierential equations. The model description equations Eq.(3.31), (3.32), and
(3.33) can then be rearranged as follows for simulation [15]. A detailed derivation
is discussed in Appendix B.4.
ψqs = ωb
ˆ vqs +
rsxls
(ψmq − ψqs)dt
ψds = ωb
ˆ vds +
rsxls
(ψmd − ψds)dt (3.34)
i0s =ωbxls
ˆv0s − i0srs dt
ψ′qr = ωb
ˆ v′qr +
ωrωbψ′dr +
r′rx′lr
(ψmq − ψ′qr
)dt
ψ′dr = ωb
ˆ v′dr −
ωrωbψ′qr +
r′rx′lr
(ψmd − ψ′dr)dt (3.35)
i′0r =ωbx′lr
ˆv′0r − i′0rr′r dt
ψmq = xm(iqs + i′qr
)ψmd = xm (ids + i′dr)
(3.36)
32
3.3. Model in Matlab/Simulink
ψqs = xlsiqs + ψmq iqs =ψqs − ψmq
xls
ψds = xlsids + ψmd ids =ψds − ψmd
xls
ψ′qr = x′lri′qr + ψmq i′qr =
ψ′qr − ψ′mqx′lr
ψ′dr = x′lri′dr + ψmd i′dr =
ψ′dr − ψ′mdx′lr
(3.37)
where
ψmq = xM
(ψqsxls
+ψ′qrx′lr
)ψmd = xM
(ψdsxls
+ψ′drx′lr
) (3.38)
and
1
xM=
1
xm+
1
xls+
1
x′lr(3.39)
It should be mentioned that for a squirrel cage induction machine, the rotor volt-
ages v′qr, v′dr and v
′0r in the qd0 reference frame are equal to zero [28].
The rotor motion in terms of mechanical speed is calculated by
Tem = Jdωrmdt
+ Tload + Tdamp (3.40)
where Tem is the electromechanical torque in Eq. (3.27), Tload is the mechanical
torque applied by load, Tdamp is the damping torque in the direction opposite to
the rotor rotation, and J dωrm
dtis the inertia torque to the accelerating torque. J
denotes the rotor inertia in kg ·m2.
3.3.3. Simulink Model in Block Diagrams
In Simulink, modules of the dynamic system of an induction machine are consti-
tuted of Function Blocks. Each function block implements one of the equations
33
Chapter 3. Model of an Induction Machine with Broken Rotor Bars
Figure 3.3.: Block diagram of the abc− qd0 conversion module in Simulink.
in 3.3.2. Shared variables are transferred between blocks. Any variable can be con-
veniently traced and saved by using the Scope and the To Workspace blocks,
respectively. Some other Simulink blocks used in the model include, but not lim-
ited in, are the Clock, Sum, Gain, Mux, Integrator and Trigonometric
Function blocks.
The induction motor model contains four major modules: the abc − qd0 conver-
sion module, the unit vector calculation module, the induction motor qd0 model
module, and the qd0− abc conversion module. Each module is explained in detail
as below.
3.3.3.1. abc− qd0 Conversion Module
This block converts variables from the abc reference frame to the qd0 reference
frame by applying the Park's transformation function Eq. (3.12). In this model
the induction machine is connected to a three-phase balanced voltage supply.
Thus the stator phase voltages are transformed. Figure 3.3 shows the Simulink
representation of this module.
3.3.3.2. Unit Vector Calculation Module
Figure 3.4 provides an insight view of this block. The rotor angle calculation
module has rotor angular speed as its input, and sin θr, sin 2θr, cos θr and cos 2θr
as outputs. The angle θr is calculated directly by using Eq. (3.11), and is used
as inputs to the transformation functions Eq. (3.12). The unit vectors sin θr and
cos θr are obtained by taking the sine and cosine of θr, and are used for calculating
stator and rotor variables in the qd0 sequences.
34
3.3. Model in Matlab/Simulink
Figure 3.4.: Block diagram of the unit vector calculation module in Simulink.
This block is also used to set the rotor starting position by assigning the initial
rotor angle, if needed.
3.3.3.3. Induction Motor Model Module
This module is the core component of the induction machine model. It contains
four subsystem blocks: the q, d and 0 sequence modules and a rotor module,
coupling with one another. In each of the qd0 sequences modules, currents and
ux linkages are calculated. Eq. (3.34) to (3.37) are properly organized in each
module, so that in each state integral is a function of only other state variables
and model inputs. The rotor block calculates the rotor output torque and the
rotor speed using Eq. (3.40). The structure of the q-axis and zero sequence blocks
are shown in Figure 3.5. The d-axis block is similar to the q-axis block. The rotor
module block is shown in Figure 3.6.
3.3.3.4. qd0− abc Conversion Module
This module performs the reverse operation to the abc − qd0 module for current
variables using function (3.14). Stator currents in the qd0 reference frame are
taken as inputs and then transformed to the three-phase currents in normal abc
reference frame. It is from this module the stator current is collected for the
analysis for broken rotor bar detection. Figure 3.7 presents the inside of this
block.
35
Chapter 3. Model of an Induction Machine with Broken Rotor Bars
(a) q-axis block.
(b) Zero sequence block.
Figure 3.5.: Block diagram of the induction motor model module in Simulink.
36
3.3. Model in Matlab/Simulink
Figure 3.6.: Block diagram of the rotor module.
Figure 3.7.: Block diagram of the qd0− abc conversion module in Simulink.
37
Chapter 3. Model of an Induction Machine with Broken Rotor Bars
3.4. Simulations
3.4.1. Initialization
To simulate an induction motor in Simulink, the Simulink model needs to be
initialized previously to assign values to all parameters. Simulation conditions
must also be set up. Both of these tasks can be done by using the M-Files scripts
in Matlab. An example is provided in Appendix A.1. The inputs to the induction
motor model are the three-phase supply voltage and the load torque. The outputs
are the three-phase currents, the resulting electromechanical torque, and the rotor
rotating speed. Both the number of broken rotor bars and the machine load can
be easily changed to any desired values.
3.4.2. Simulation Results
Based on the described induction motor model and machine parameters that are
measured from a real three-phase, 4-pole, 5.5kW induction motor, simulations in
Matlab/Simulink have been implemented to obtain the stator current, rotor speed
and output torque. In order to validate this model, laboratory tests on the real
motor described above have also been conducted. The rotor speed and stator
current are measured and compared with the simulated data. Figure 3.8 presents
the output torque curve obtained from the simulation. The simulated rotor speed
is plotted together with the measured signal in Figure 3.9. It can be observed that
there is a considerable agreement between the two speed curves. The simulated
stator current is plotted with a small phase shift in respect to the measured data
for comparison in Figure 3.10. The two stator current plots also match with each
other well. Dierences in magnitude in the transient state exist but both the
simulated and measured currents enter the steady state at the same time. Also,
zoomed in view of the comparison in steady state is presented as only the current
in steady state is related to the methods of broken rotor bar diagnostics in this
thesis. The spectra of the simulated stator current of the induction machine with
broken rotor bars have been presented previously referring to Chapter 2.
38
3.4. Simulations
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
−40
−20
0
20
40
60
80
100
120
140
Torqu
e (Nm
)
Time (s)
Figure 3.8.: Simulated output torque curve.
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
0.5
1
1.5
Rotor
spee
d in p
er un
it
Time (sec)
Measured dataSimulated data
Figure 3.9.: Comparison of the simulated and measured rotor speed curves.
39
Chapter 3. Model of an Induction Machine with Broken Rotor Bars
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−80
−60
−40
−20
0
20
40
60
80
Stator
curre
nt (A)
Time (sec)
Measured dataSimulated data
(a)
1 1.05 1.1 1.15 1.2 1.25 1.3−20
−15
−10
−5
0
5
10
15
20
Stator
curre
nt (A)
Time (sec)
Measured dataSimulated data
(b) A zoomed-in view.
Figure 3.10.: Comparison of the simulated and measured stator currents.
40
Chapter 4.
High-Resolution Spectral Analysis
4.1. Introduction
Discrete Fourier Transform (DFT), which is the discrete form of Fourier analysis,
is the most commonly adopted tool for signal processing in frequency domain.
In practice, Fast Fourier Transform (FFT) is usually employed as an ecient
algorithm to compute the DFT . However, there are several inherent drawbacks
of DFT, which limit the condition of its application. Trade-os have to be made
accordingly.
Extensive research in the last a few decades has led to a great development of
modern digital spectral estimation techniques [14] [29] [30]. Their advantages
such as higher frequency resolution and increased signal detectability have shown a
promising potential of improvement in the induction machine condition monitoring
application. Originally invented by French Mathematician Gaspard de Prony in
1795, a high-resolution spectral analysis technique called Prony Analysis (PA)
now has been largely extended for dealing with data corrupted with noise. Its
most useful feature for motor condition monitoring applications is that it can
maintain high resolution in frequency domain whilst using short data windows.
This advantage overcomes the problems from which DFT suers. In this chapter,
the Prony Analysis and its extensions are introduced in detail.
41
Chapter 4. High-Resolution Spectral Analysis
4.2. Comparison Between Discrete Fourier
Transform and Prony Analysis
4.2.1. Drawbacks of Discrete Fourier Transform
DFT is a well-known and widely employed spectral analysis technique in motor
condition monitoring. It is computationally ecient and easy to achieve. However,
detection results to the desired level of precision are still hard to obtain due to its
several inherent drawbacks.
The most prominent performance limitation of DFT is the achievement of high
frequency resolution. The frequency resolution is dened as the minimum dif-
ference in hertz between two frequency components which allows to resolve two
distinct peaks in the spectrum. The frequency resolution of DFT is determined
by the length of data window. It is calculated roughly as the reciprocal to the
time duration over which sampled data is available [13], given by the equation
∆f =fsN
=1
NTs=
1
T
where ∆f denotes the frequency dierence, fs is the sampling frequency, N is
the number of data points within the window, Ts is the sampling interval and T
represents the total sampling time. Thus, to obtain a higher frequency resolution,
a longer data window is required.
Another major disadvantage of DFT is that the impact of side wiggles (Gibbs
oscillations) [13]. The implicit windowing process when using DFT causes side lobe
leakage in spectral domain [31], obscuring and distorting other spectral response
in its vicinity. This especially depresses the dierentiation of the broken rotor bar
sideband frequencies as it is the case that two small frequency peaks present closely
to a large peak. There are also other limitations such as that the time domain
noise in the signal is distributed uniformly by DFT in the frequency domain,
which limits the certainty of computing frequency width, magnitude, and phase
[32] and, it is also known that DFT may cause spurious spectral components in
the spectrum, which will confuse the detection on desired frequency components.
Therefore, trade-os among leakage suppression, resolution and stability are hard
to be fullled when using DFT.
These limitations of DFT can be particularly troublesome in real induction ma-
chine condition monitoring situations. Short data records are usually required
because of the instability of machine load condition, which causes time-varying
42
4.2. Comparison Between Discrete Fourier Transform and Prony Analysis
broken rotor bar sideband frequencies. However, on the other hand, a high reso-
lution is required to observe the two broken rotor bar sideband frequencies when
the machine is operating with light load as they can be very close to the funda-
mental frequency. This means longer data acquisition time in the case of using
DFT. Moreover, in practice, sometimes only restricted data records are available.
This also makes enlarging the data window to obtain a high resolution impos-
sible. Thus, the detection of broken rotor bars using DFT can be dicult and
unauthentic.
4.2.2. Features of Prony Analysis
Prony Analysis is a linear prediction method for modelling a set of uniformly sam-
pled data as a linear combination of damped exponential functions. The typical
application of Prony Analysis is the parametric analysis of transient signals initi-
ated by disturbances in electrical circuits [33]. It is also widely used in biomedical
science [34], environmental engineering [32], radar [35], sonar [36], geophysical
sensing and speech processing [37]. It has the following key features.
• Prony Analysis is parametric whereas DFT is non-parametric.
• Prony Analysis needs uniformly sampled signal data.
• Prony Analysis ts the signal data to a model represented as a sum of
damped exponential functions.
Main advantages of Prony Analysis above DFT may be briey summarized that
• Prony Analysis is able to work with signicantly shorter data windows to
maintain a high frequency resolution, compared to DFT.
• Prony Analysis generally has a higher accuracy in estimating frequency val-
ues than DFT using the same length data window.
• Prony Analysis does not have the problem of the spectral leakage phe-
nomenon.
• Prony Analysis can compute the amplitudes, frequencies, phases and damp-
ing factors of the tted signal whereas DFT can not determine the damping
factors.
43
Chapter 4. High-Resolution Spectral Analysis
4.3. The Original Prony Method
The original Prony method seeks to t a deterministic exponential model to
equally spaced data points. It was discussed in detail by Marple [14] and Therrien
[29]. Here will give a brief review of this technique. Assuming signal data x [n]
has N complex samples x[1], . . . , x[N ], the Prony method will t the data with a
sum of q complex exponential functions
x [n] =
q∑k=1
Ak exp [(αk + j2πfk) (n− 1)Ts + jϕk] (4.1)
for n = 1, 2, . . . , N and k = 1, 2, . . . , q, where
Ak is the amplitude of the complex exponential,
αk is the damping coecient in sec−1,
fk is the sinusoidal frequency in Hz, and
ϕk is the initial phase in radians
The objective is to estimate the frequencies fk, damping factors αk, amplitudes
Ak and phases ϕk. If these function coecients are determined correctly, then
the plot of the estimation of the signal within the data window, and that of the
prediction of the future signal after the data window, should t the original signal
with a high degree of accuracy.
Since only real signals are considered, the signal poles exp (αk + j2πfk) must ap-
pear in complex conjugate pairs. Thus the q is always assumed to be even for
convenience. Then, Eq. (4.1) can be expressed in the form of
x [n] =
q∑k=1
hkzn−1k (4.2)
where hk and zk are complex parameters dened as
hk = Ak exp (jϕk)
and
zk = exp [(αk + j2πfk)Ts]
44
4.3. The Original Prony Method
The tting of a designated signal is usually accomplished by minimizing the total
squared error over the N data values [13]
ρ =N∑n=1
|ε [n]|2
where
ε [n] = x [n]− x [n] = x [n]−q∑
k=1
hkzn−1k
representing the complex error between the original data samples x [n] and the
linear approximation x [n]. For a real signal x [n], minimizing the squared error
ρ is obtained by setting the derivatives with respect to hk and zk to zero. This
yields:
∂ρ
∂hk= c1 + c2hk = 0
∂ρ
∂zk= c3 + c4hk = 0
. (4.3)
The minimization problem is with respect to parameters hk, zk and p simultane-
ously. The coecients c1,2,3,4 in Eq. (4.3) involve sums of exponentials zk. To
solve for the coecients , it yields
c1c4 = c2c3,
which turns out the minimization to be a dicult nonlinear problem. Derivations
of these coecients can be found in Appendix B.5. In this case, no analytic
solution is available.
Prony's method addresses this problem by determining the zk elements separately
and then considering Eq. (4.2) as a set of linear simultaneous equations to solve for
hk. The key of Prony method is in the fact that to see the Eq. (4.2) as the solution
to a homogeneous linear dierence equation with constant coecients. These
coecients are identied by computing the eigenvectors of a suitably calculated
covariance matrix [14]. A polynomial can be formed accordingly with roots zk
45
Chapter 4. High-Resolution Spectral Analysis
φ (z) =
q∏k=1
(z − zk) =
q∑m=0
amzp−m (4.4)
The linear dierence equation whose homogeneous solution is given by Eq. (4.4)
is
q∑m=0
amx [n−m] = 0 (4.5)
with complex coecients am such that a0 = 1.
The original Prony method assumes that the number of available data samples
is equal to the unknown parameters, so the dierence equation is valid for n =
q + 1, . . . , 2q. The coecients am form a linear predictive relationship among the
available samples and the relationship can be then expressed as the q× q Toeplitzstructure matrix equation
x [q] x [q − 1] · · · x [1]
x [q + 1] x [q] · · · x [2]...
.... . .
...
x [2q − 1] x [2q − 2] · · · x [q]
a1
a2
...
aq
= −
x [q + 1]
x [q + 2]...
x [2q]
(4.6)
By solving the matrix equation, the am coecients, which are the function of zk,
can be determined.
Next, the roots of Eq. (4.4) can be determined by polynomial factoring, and the
damping factor αk and the sinusoidal frequency fk can be determined from roots
zk by using the relationships
αk = ln |zk| /Ts (4.7)
and
fk = tan−1 [Im zk /Re zi] /2πTs (4.8)
Finally, these roots are used to obtain the complex parameter hk in Eq. (4.2).
The amplitudes Ak and initial phases ϕk are determined from hk by using the
relationships
46
4.4. Extended Least Squares Prony Method
Ak = |hk| (4.9)
and
ϕk = tan−1 [Im hk /Re hk] (4.10)
To sum up, the Prony method consists of three steps [14]:
Step 1 Determine the linear prediction parameters that t the observed data.
This step is undertaken by solving Eq. (4.6) for the coecients am.
Step 2 Find roots of the characteristic polynomial formed from the linear pre-
diction coecients and determine the estimates of the damping factor and
frequency of each of the exponential terms. This step consists of polynomial
factoring Eq. (4.4) and solving Eq. (4.7) and Eq. (4.8).
Step 3 Solve the original set of linear equation to yield the estimates of the
exponential amplitude and sinusoidal initial phase. This step is to solve the
original matrix equation Eq. (4.2), where the matrix of the time-indexed z
elements has a Vandermonde structure.
4.4. Extended Least Squares Prony Method
It should be noticed that in the original Prony's method there is no noise model.
This means that the actual noise present in the data will be approximated entirely
by complex exponentials, leaving an un-modeled residual energy which manifests
itself as parameter estimation errors [38]. It is because of this, the performance
of the original Prony method is unstable if there is noise in the signal data.
However, in practice, acquired signal data is always embedded in noise. The Eq.
(4.2) should be modied as the following form for noise corrupted signals [39].
x [n] =
q∑k=1
hkzn−1k + ε [n] (4.11)
where ε [n] is known as the exponential approximation error and noise which is
assumed to be Gaussian distributed and white. If the noise present in the signal
is not white then standard ltering methods can be used to whiten the signal so
that this model applies, too.
47
Chapter 4. High-Resolution Spectral Analysis
The classical Prony method models a sequence of 2q observations sampled at
even time intervals by q exponential functions at the most. In practice, there
are usually more data points than the minimum number of samples needed to t
a model of order q. To deal with practical situations, appropriate least squares
procedures and Singular Value Decomposition (SVD) [40] are employed in the rst
and third steps of the original Prony method, and this is called the extended Least
Squares (LS) Prony method [41]. The goal of the algorithm is to minimize the
total squared error over all sampled data with respect to the complex parameters
and the number of exponents.
As mentioned in Section 4.3 that the minimization of the norm of the exponential
error ε [n] is a dicult nonlinear problem. The extended LS Prony method employs
a suboptimum solution that predicts the linear prediction approximation error
e [n] instead of the exponential error ε [n] over data 1 6 n 6 N . In an over-
determined approach, N data points, where N > 2q, are utilized to compose the
linear prediction equations. Thus, the linear dierence equation Eq. (4.5) should
be modied to [14]:
q∑m=0
am (x [n−m] + ε [n−m]) = 0 (4.12)
LS Prony method ignores the past noise values, then the Eq. (4.12) becomes
q∑m=0
amx [n−m] = e [n] (4.13)
which actually denes a forward linear prediction error equation. Thus each am
terms a linear prediction parameter and is selected to minimize the linear predic-
tion total squared error∑N
n=q+1 |e [n]|2. The minimization can be done by using
the covariance method, or alternatively, by using the SVD for ecient computa-
tion of the pseudoinverse and projection matrices [29] [40].
The third step of the original Prony method also switches to a linear least square
procedure. The complex-valued q × q matrix normal equation
[ZHZ
]h =
[ZHx
](4.14)
can be yielded from Eq. (4.11). The equation components, which are the N × qmatrix Z, the q × 1 vector h, and the N × 1 data vector x are dened as
48
4.5. Iterative Prony Method
Z =
1 1 · · · 1
z1 z2 · · · zp...
.... . .
...
zN−11 zN−1
2 · · · zN−1p
, h =
h1
h2
...
h3
, x =
x [1]
x [2]...
x [N ]
and the superscript H means the matrix complex conjugate transposition, so that
ZHZ forms a q × q Hermitian matrix. The four parameters of Prony model can
be determined in the same way by using Eq. (4.7) - (4.10).
4.5. Iterative Prony Method
As mentioned that the original Prony method does not perform well when there is
addictive noise present in the signal data. The LS Prony can deal with practical
situations but it is still inconsistent in providing unbiased parameter estimates as
the number of sampled points increases [42] [43]. It is considered as a subopti-
mum solution as this approach does not make a separate estimation of the noise
process, but ts the exponentials to any noise present in the data [14]. However,
this can be improved signicantly by the iterative Prony method.
The Iteratively Reweighted Least Squares (IRLS) Prony method has been devel-
oped for the identication of the resonant-grounded system parameters based on
fault records of a power system [33]. The major improvement to the LS Prony
method is that it solve the weighted least squares problem
minεTW (a)ε
(4.15)
with respect to a, where the superscript T indicates matrix transpose, andW (a)
is the covariance matrix for the errors ε [n] at each data point in the rst step of
Prony method. It iteratively minimizes the total squared error, so that to lter
out noise more eciently. Accurate parameter identication has been achieved as
a result.
We start the IRLS Prony method by the model given by Eq. (4.11), noting that
the measurement errors are assumed to be independent and normally distributed.
Taking into account that the error-free signal satises exactly the dierence equa-
tion Eq. (4.5), when substitute the error-free data with real signal data, the
dierence equation Eq. (4.12) is satised for each sample in the data window
spanning N samples. It may be expressed in the matrix notation [33]
49
Chapter 4. High-Resolution Spectral Analysis
Xa + b + D (a)ε = 0 (4.16)
where
aT =[a1 a2 . . . aq
]is the q × 1 column vectors of dierence equation coef-
cients,
εT =[ε [1] ε [2] . . . ε [N ]
]is the N × 1 column vectors of error components
on each sample,
bT =[x [q + 1] x [q + 2] . . . x [N ]
]is the (N − q)×1 column vectors of data,
X =
x [q] x [q − 1] · · · x [1]
x [q + 1] x [q] · · · x [2]...
.... . .
...
x [N − 1] x [N − 2] · · · x [N − q]
is the (N − q)×q data matrix, and
D =
aq aq−1 · · · a1 1 0 · · · 0
0 aq aq−1 · · · a1 1 · · · 0...
. . . . . . . . . · · · . . . . . ....
0 · · · 0 aq aq−1 · · · a1 1
is the (N − q) × N coecients
matrix for errors.
In order to minimize the sum of squared error over the available data, the error ε
is expressed by rearranging Eq. (4.16), yielding
ε = −D (a)+ (Xa + b) (4.17)
where + indicates the matrix pseudoinversion. Thus, the optimal estimates of
the dierence coecients am correspond to the minimum of the error norm εTε.
This is the nonlinear problem addressed by Eq. (4.15) and can be formulated in
terms of IRLS, that is, a minimizes
mina
(Xa + b)T W (a) (Xa + b)
(4.18)
where W (a) =[D (a)D (a)T
]−1
is a real symmetric positive denite weighting
matrix. The least squares solution is then returned to the linear system Xa + b =
0 with covariance matrix proportional to W (a), subject to the relation given by
Eq. (4.16).
50
4.5. Iterative Prony Method
One aspect of the IRLS method that needs to be addressed to attention is that
because the elements of the weighting matrix W (a) depend on the unknown
parameters, it is essential to apply an iterative scheme using the estimates obtained
at the previous iteration. Thus, results of the LS Prony method are used here as
the initial values of the iteration process. Then an iteratively reweighting process
is the next step based on error residue criteria and the iteration count [33]. In
Matlab, the computation algorithm can be achieve by using the function
lscov.
51
Chapter 4. High-Resolution Spectral Analysis
52
Chapter 5.
Implementation of Prony Analysis
for Induction Motor Broken Bar
Detection
5.1. Introduction
In this chapter the implementation of Prony Analysis (PA) for induction motor
broken rotor bar diagnostics is described, demonstrated and discussed. There are
three major parts of interest for study. Firstly, the eect of broken rotor bar
fault on motor stator current spectrum will be illustrated with comparisons of the
results between Prony Analysis and Discrete Fourier Transform (DFT). Secondly,
the Prony Analysis will be evaluated in terms of accuracy and limitations. In the
end, the verication of Prony Analysis by using measured stator current data will
be presented.
Simulation data of induction motor stator currents obtained from the model de-
scribed in Chapter 3 is used for the rst two studies and real data measured from
laboratory-based experiments is used for the verication. All data analysis and
gures are undertaken and plotted in Matlab.
This chapter sought to address the following aims:
• To demonstrate the implementation of Prony Analysis for induction motor
broken rotor bar detection.
• To study how the number of broken rotor bars and the machine load aect
the detection process.
53
Chapter 5. Implementation of Prony Analysis for Induction Motor Broken Bar Detection
• To investigate how the data processing algorithms perform in the presence
of noise and load variation using data windows with dierent lengths.
• To compare Prony Analysis and DFT.
• To understand the limitations of Prony Analysis.
5.1.1. Study Description
In order to examine the aspects listed above, a number of simulation cases have
been designed. The model parameters for simulation are varied in a systematic
manner. Simulated induction motor stator current signals are analyzed using both
Prony Analysis and DFT for comparison. The method of Prony Analysis employed
in this chapter refers to IRLS Prony in all of the applications, and the model
order is chosen as six, unless otherwise indicated. Inuencing factors to both the
value and the amplitude of broken rotor bar sideband frequencies are investigated.
Studies in Section 5.3 uses simulated data corrupted with noise obtained by adding
normally (Gaussian) distributed random signals to the machine input voltages.
The noise level in the simulation signals is determined as approximated -80dB
to the supply frequency, unless otherwise advised. This value is chosen since the
noise level presented in measured signals from laboratory experiments is observed
to be always between -80dB to -90dB.
Designs of the simulation cases are described as below.
5.1.1.1. Eect of Broken Rotor Bars on Stator Current
In the study of the eect of broken rotor bar faults on motor stator current spec-
trum, the number of fracture rotor bars, load conditions and the parameters of the
machine are varied in a systematic manner. Prony Analysis results are presented
and compared with DFT results.
Fault Severity A severer fault means a bigger number of broken rotor bars. The
motor breaks down when this number exceeds a certain limit, which is usu-
ally close to one third of the total number of rotor bars. In a squirrel-cage
rotor, one third of the rotor bars together is considered equivalent to one
rotor phase of a wound rotor.
Load Conditions The machine load is varied from full-load to non-load, giving
a range of the movement of two broken rotor bar sideband frequencies from
54
5.2. Data Acquisition and Preprocessing
Table 5.1.: Relevant parameters of induction machines used in the study.
Machine number Rated power (kW) Number of poles Number of rotor bars
Machine 1 2.2 4 28Machine 2 5.5 4 32Machine 3 35 8 52
around 7 Hz apart from the fundamental frequency to only a few decimal
hertz. Small load conditions, which refers to conditions that the machine
load is less than 5% of the rated load in this chapter, are also investigated.
Comparisons of data window length and frequency estimation accuracy are
addressed between Prony Analysis and DFT.
Machine Parameters Simulations of various induction motors are utilized for
generalizing of the machine model and for studying the impact of the ma-
chine power on the broken rotor bar sideband frequencies. The relevant
parameters of the induction machine models utilized for study in the chap-
ter are shown in Table 5.1 whilst full parameters are listed in Appendix
C.
5.1.1.2. Evaluation of Prony Analysis
The evaluation of Prony Analysis is conducted by introducing inuencing factors
and examining their impact on frequency estimation accuracy. The factors which
are focused on for discussion are the data window length and the signal noise
level. Section 5.5 will give more insights of Prony Analysis to help understand its
advantages and limitations.
5.2. Data Acquisition and Preprocessing
5.2.1. Sampling Frequency and Window Length
Current signals obtained in practice are analog. They need to be sampled for
digital signal processing (DSP) applications. The sampling frequency denes the
number of samples per second sampled from a continuous signal to make a dis-
crete signal. The data window length may be described as the number of data
points sampled in a period of time with a determined sampling frequency. The
55
Chapter 5. Implementation of Prony Analysis for Induction Motor Broken Bar Detection
relationship between window length Lw, sampling frequency fs and the number
of samples N is shown in equation
Lw =N
fs(5.1)
The basic requirement for sampling frequency is the Nyquist Sampling Theorem,
which denes that the lowest sampling frequency should be at least twice as high as
the highest frequency components of the signal. Because of the requirement of the
DSP techniques, the naturally analog stator current signals must be sampled as
discrete data points. Any analog frequencies greater than the Nyquist frequency,
which refers to the frequency component at half the sampling frequency, after
sampling, will alias with frequencies between 0 and fs
2Hz. In the digital domain,
there is no way to distinguish these aliasing frequencies from the frequencies that
actually lie between 0 and fs
2Hz. Therefore, these aliasing frequencies need to be
removed from the analog signal before sampling by an A/D converter.
5.2.2. Data Preprocessing
5.2.2.1. Preltering
Filtering in the frequency domain is a usually employed prior to the DSP proce-
dures to gain improved results. For all DFT applications in this chapter, signal
data is processed by a Hanning window before applying the FFT algorithm, in
order to decrease the spectral leakage eect and to shape the signal spectrum.
The objective of ltering the signal prior to applying Prony Analysis is to atten-
uate the noise and undesired frequency components and to separate the broken
rotor bar sideband frequency components in the spectrum. By doing so, the per-
formance of Prony Analysis can be improved signicantly [44] [38] [45].
Therefore, a bandpass nite-duration impulse response (FIR) lter is designed
and employed to process signals before applying Prony Analysis. A bandpass
lter will pass all frequency components of a signal within a designated frequency
range, namely the pass band, and to reject all other frequency components of
a signal outside this range. Thus, the use of a bandpass lter in the frequency
domain eliminates all other frequency components which are not, or less related
to induction machine broken rotor bar diagnostics, leaving only the fundamental
and the two sideband frequencies. The number of signal poles in the ltered stator
56
5.2. Data Acquisition and Preprocessing
0 20 40 60 80 100−100
−80
−60
−40
−20
0
20
Frequency (Hz)
Mag
nitu
de (
dB)
Figure 5.1.: The magnitude response of the equiripple bandpass lter.
current is then known as three. The order of the Prony Analysis algorithm can
be chosen as six, as a consequence.
Thank to the powerful function and the Graphical User Interface (GUI) design
modules of Matlab, lter designing is easy to accomplished by using the Filter
Design Toolbox [46]. In this research, an FIR equiripple bandpass lter has been
employed. The specication of this FIR lter is as following:
lower stopband edge: 37 Hz, attenuation: 60 dB
lower passband edge: 40 Hz, ripple: 1 dB
upper passband edge: 60 Hz, ripple: 1 dB
upper stopband edge: 63 Hz, attenuation: 60 dB
The bandwidth of this lter is 20Hz centered at the 50Hz fundamental frequency.
This is decided by the range of the movement of the two broken bar sideband
frequencies. The magnitude response of the lter is plotted in Figure 5.1.
5.2.2.2. Removing The Constant Oset
In real recorded data, there is such a concern that a DC component may be caused
in the signal by the electronic devices that used in the test. The DC component
can result in signicant errors in the Prony Analysis results. The bias introduced
components will toward a zero frequency. Therefore, data needs to be corrected
57
Chapter 5. Implementation of Prony Analysis for Induction Motor Broken Bar Detection
before sent for Prony Analysis. The preprocess includes removing any linear trend
(detrending) and the signal mean [32] [47].
5.2.2.3. Downsampling
Prony Analysis involves the solution of over-determined linear equations and root-
ing of high-order polynomials. Both of them are computational intensive opera-
tions. The iterative algorithm is the recurrence of these processes. It is because
of so, the amount of data used in the algorithm can be a great concern in practice
as a large number of data will increase the complicity of the equations and de-
mand a huge computational eort and a long computational time. Downsampling
the data signal can eectively reduce the number of data points and the eort
of computation. Therefore, a downsampling process is sometimes desired for the
benet of computational eciency and the performance of Prony Analysis, when
the original sampling frequency is high. To avoid aliasing, anti-aliasing low pass
lter should be implemented before downsampling.
5.3. Prony Estimation and Prediction
5.3.1. Stator Current Modulation
While the number of broken rotor bars increases, the anomaly of the ux linkage
within the motor aggravates consequently, which will cause a higher degree of cur-
rent distortion. This phenomenon is observed from the stator current waveform.
Here, Machine 2 is simulated with a supply of 50Hz three-phase voltage with rated
amplitude and loaded with rated load. The number of broken rotor bars is varied
from zero, which indicates the healthy status of the induction motor, up to 8,
which is close to one third of the number of total rotor bars.
Selected estimation and prediction results of Prony Analysis on faulty stator cur-
rents are plotted in the left and right sides of Figure 5.2, respectively, together with
the simulated current signals for comparison. The estimation refers to estimating
the signal data within the data window used for the PA algorithm, whilst the pre-
diction refers to predicting the future signal data after the window. The prediction
waveform is obtained by plotting the signal data with parameters gained from the
estimation procedure. The window length used for Prony Analysis is 500 samples
58
5.3. Prony Estimation and Prediction
0 0.1 0.2 0.3 0.4 0.5−20
−10
0
10
20
Mag
nitu
de (
A)
Time (sec)
Current SignalProny Estimation
(a) PA estimation of healthy stator cur-rent.
0.5 0.6 0.7 0.8 0.9 1−20
−10
0
10
20
Mag
nitu
de (
A)
Time (sec)
Current SignalProny Prediction
(b) PA prediction for the period into thefuture.
0 0.1 0.2 0.3 0.4 0.5−20
−10
0
10
20
Mag
nitu
de (
A)
Time (sec)
Current SignalProny Estimation
(c) PA estimation of 1 broken rotor bar sta-tor current.
0.5 0.6 0.7 0.8 0.9 1−20
−10
0
10
20
Mag
nitu
de (
A)
Time (sec)
Current SignalProny Prediction
(d) PA prediction for the period into thefuture.
with the sampling frequency of 1000Hz. Zoomed-in views of the estimation and
prediction results are presented in Figure 5.3 for the one broken rotor bar case.
The numerical result of Prony Analysis for Figure 5.2 is displayed in Table 5.2.
A complete result of the same format but for four dierent load conditions (full,
75%, 50% and 25% load) is presented in Appendix D. As a linear prediction,
Prony method estimates the information of frequency, amplitude, damping factor
and phase within a designated signal and tries to t a model to the signal. Only
0 0.1 0.2 0.3 0.4 0.5−20
−10
0
10
20
Mag
nitu
de (
A)
Time (sec)
Current SignalProny Estimation
(e) PA estimation of 3 broken rotor barsstator current.
0.5 0.6 0.7 0.8 0.9 1−20
−10
0
10
20
Mag
nitu
de (
A)
Time (sec)
Current SignalProny Prediction
(f) PA prediction for the period into thefuture.
59
Chapter 5. Implementation of Prony Analysis for Induction Motor Broken Bar Detection
0 0.1 0.2 0.3 0.4 0.5−20
−10
0
10
20
Mag
nitu
de (
A)
Time (sec)
Current SignalProny Estimation
(g) PA estimation of 5 broken rotor barsstator current.
0.5 0.6 0.7 0.8 0.9 1−20
−10
0
10
20
Mag
nitu
de (
A)
Time (sec)
Current SignalProny Prediction
(h) PA prediction for the period into thefuture.
0 0.1 0.2 0.3 0.4 0.5−20
−10
0
10
20
Mag
nitu
de (
A)
Time (sec)
Current SignalProny Estimation
(i) PA estimation of 7 broken rotor barsstator current.
0.5 0.6 0.7 0.8 0.9 1−20
−10
0
10
20M
agni
tude
(A
)
Time (sec)
Current SignalProny Prediction
(j) PA prediction for the period into thefuture.
0 0.1 0.2 0.3 0.4 0.5−20
−10
0
10
20
Mag
nitu
de (
A)
Time (sec)
Current SignalProny Estimation
(k) PA estimation of 8 broken rotor barsstator current.
0.5 0.6 0.7 0.8 0.9 1−20
−10
0
10
20
Mag
nitu
de (
A)
Time (sec)
Current SignalProny Prediction
(l) PA prediction for the period into thefuture.
Figure 5.2.: Comparisons between PA estimation and prediction results with thesimulated stator currents of Machine 2 with 0, 1, 3, 5, 7 and 8 bro-ken rotor bars operating under full load, using a data window of 500samples with the sampling frequency of 1000Hz.
60
5.3. Prony Estimation and Prediction
0.05 0.1 0.15−20
−10
0
10
20
Mag
nitu
de (
A)
Time (sec)
Current SignalProny Estimation
(a) Zoomed-in view of PA estimation of 1broken rotor bar stator current.
0.55 0.6 0.65−20
−10
0
10
20
Mag
nitu
de (
A)
Time (sec)
Current SignalProny Prediction
(b) Zoomed-in view of PA prediction forthe period into the future.
Figure 5.3.: Zoomed-in views of comparisons between PA estimation and predic-tion results with the simulated stator current of Machine 2 with 1broken rotor bar operating under full load, using a data window of500 samples with the sampling frequency of 1000Hz.
frequency and amplitude are the parameters of interest to induction machine
broken rotor bar diagnostics. The damping factor may be used associated with
the amplitude to eliminate feigned results in the case of using a model order higher
than the number of actual signal poles.
It can be observed from the comparison gures that the more defective rotor
bars there are, the more severely the stator current is modulated. From the
less distorted current waveforms caused by a few broken rotor bars to the highly
distorted current waveforms caused by a number of broken rotor bars, it is shown
explicitly that both the estimates and the predictions t the data within and after
the window perfectly. The Mean Absolute Error (MAE) presented in Table 5.2 is
calculated as the mean of the absolute error over the whole length of the plotted
data and given as
MAEfitting =
∑Nn=1 | ε [n] |N
(5.2)
where ε [n] is the error calculated on each sampled data point and N is the total
number of the data points. All fundamental and sideband frequency components
have been estimated accurately, shown in the table comparing with the true fre-
quency values. The true values of the two sideband frequencies are calculated by
rstly averaging the rotor speed and then using equation (1± 2s) f . The result
is sucient to prove the credibility of Prony Analysis as it does not only exactly
model the available data but also well predicts the future data.
61
Chapter 5. Implementation of Prony Analysis for Induction Motor Broken Bar Detection
Table5.2.:
Numerical
PA
resultof
thestator
curren
tof
Mach
ine2with
variousbroken
rotorbar
numbers
operatin
gunder
full
loadcon
dition
usin
gadata
window
of500
samples
with
thesam
plingfreq
uency
of1000H
z.
TrueValu
ePA
Number
ofbrokenbars
(1−
2s)f
(Hz)
(1+
2s)f
(Hz)
(1−
2s)f
(Hz)
Amplitu
de
(1+
2s)f
(Hz)
Amplitu
deFundam
e-ntal
Freq
uency
Amplitu
deEstim
ationMEAfittin
g
Pred
ictionMEAfittin
g
0N/A
N/A
N/A
N/A
N/A
N/A
50.00006.7916
0.00120.0208
143.9432
56.056843.9432
0.185456.0410
0.048250.0000
6.79370.0052
0.02192
43.719856.2802
43.71950.3946
56.28520.0941
50.00006.7990
0.00580.0260
343.4593
56.540743.4589
0.612956.5450
0.134650.0000
6.81050.0045
0.02324
43.153056.8470
43.15370.8468
56.84870.1669
50.00006.8239
0.00580.0309
542.7875
57.212542.7867
1.184057.2150
0.208549.9999
6.86890.0029
0.01246
42.332857.6672
42.33411.7278
57.66470.2643
49.99996.9352
0.00270.0093
741.7585
58.241541.7558
2.186058.2446
0.282850.0000
7.03880.0039
0.01648
40.966659.0334
40.96222.7316
59.03700.2866
50.00017.2384
0.00240.0100
62
5.3. Prony Estimation and Prediction
5.3.2. Fault Severity Assessment
Though there are higher order harmonics of the broken rotor bar sideband fre-
quencies presenting in the stator current spectrum, as shown in Section 2.4, the
(1 − 2s)f Hz and (1 + 2s)f Hz sideband frequency components are the most
characteristic indicators of broken rotor bar faults. The amplitudes of these two
sideband frequencies are subject mainly to the the number of broken rotor bars
whilst the values of them are subject mainly to load conditions.
However, there has not been a precise mathematical denition that can determine
the exact number of broken rotor bars using the amplitudes of these sideband
frequencies. Predictive formulas introduced in Chapter 2 indicate an approximate
degree of the fault severity. This works together with empirical judgment to make
reasonable predictions.
Machine 1 to 3 are simulated under full load separately. The amplitude of the left
broken rotor bar sideband frequency (1− 2s) f is plotted in Figure 5.4 in terms
of dB with respect to the number of broken rotor bars for an intuitionistic view.
The three prediction equations given in Chapter 2 are also drawn together.
It is observed that for Machine 1 and 2, the amplitude of the (1− 2s) f sideband
frequency obtained by Prony Analysis can be predict well by Prediction 1 when
the number of broken rotor bars are less than four, which is approximate half of
the number of total rotor bars in one rotor phase. When the number of broken
rotor bars exceeds four, the Prediction 1 underestimates the sideband amplitude
within 4dB, and thus overestimates the number of broken rotor bars when given
an amplitude value. It is also observed in Figure 5.4(a) and (b) that the amplitude
curve of the (1− 2s) f sideband obtained by Prony Analysis is always approxi-
mate 5dB above the Prediction 3 curve, regardless the number of broken bars. A
corrector may be employed in these cases to give a more accurate prediction of the
number of broken rotor bars. For the result of a higher power Machine 3 shown
in Figure 5.4(c), the Prediction 1 overestimates the amplitude of the (1− 2s) f
sideband than the Prony Analysis result when the number of broken rotor bars is
less than 6, and overestimates it when the number of broken rotor bars increases
further. However, the dierence between the Prediction 1 and the Prony Analysis
result is always within 4dB. Nevertheless, in practice the broken rotor bar faults
is desired to be detected in an early stage. Machines allowed to operate with a
large number of broken bars are very rare.
63
Chapter 5. Implementation of Prony Analysis for Induction Motor Broken Bar Detection
0 1 2 3 4 5 6 7 8−50
−45
−40
−35
−30
−25
−20
−15
−10
−5
0
Mag
nitu
de (
A)
Number of broken rotor bars
PA estimation valuesPrediction 1Prediction 2Prediction 3
(a) Machie 1: 2.2kW, 4 poles and 28 total rotor bars.
0 1 2 3 4 5 6 7 8 9−50
−45
−40
−35
−30
−25
−20
−15
−10
−5
0
Mag
nitu
de (
A)
Number of broken rotor bars
PA estimation valuesPrediction 1Prediction 2Prediction 3
(b) Machine 2: 5.5kW, 4 poles and 32 total rotor bars.
0 2 4 6 8 10 12 14−50
−45
−40
−35
−30
−25
−20
−15
−10
−5
0
Mag
nitu
de (
A)
Number of broken rotor bars
PA estimation valuesPrediction 1Prediction 2Prediction 3
(c) Machine 3: 35kW, 8 poles and 52 total rotor bars.
Figure 5.4.: Amplitude of the (1− 2s) f sideband frequency obtained by PA withrespect to the number of broken rotor bars in Machine 1, 2 and 3respectively. The motors are operating under full load.
64
5.4. Disadvantages of DFT and Solutions by Prony Analysis
5.4. Disadvantages of DFT and Solutions by
Prony Analysis
5.4.1. Impact of Data Window Length
It is known that the frequency resolution of DFT, which indicates the capability
of distinguishing neighboring frequency components, lies solely on the length of
sampling time, or the data window length with a given sampling frequency. It
is because of this, the frequency resolution is a major problem when using DFT,
especially in the application of induction machine broken rotor bar diagnostics
where due to restrictions that the window length can not be enlarged as desired.
This disadvantage is demonstrated as follows. Figure 5.5 to Figure 5.8 present
examples of the stator current spectra obtained by DFT using windows of dierent
lengths. Machine 2 is simulated under full load. Two broken rotor bars are chosen
just for demonstration. The sampling frequency is 1000Hz and the signal data is
processed through a Hanning window before applying FFT.
In Figure 5.5, a window of 5000 data points is used, which requires a sampling time
of 5s and provides a frequency resolution of 0.2Hz. The two sideband frequencies
are observed distinctly in the spectrum, noticing the true values of the lower and
higher sideband frequency components are calculated as 43.7198Hz and 56.2802Hz,
respectively, and the frequency values given by DFT are 43.8000Hz and 56.2000Hz.
If a shorter data window, for example that of 1000 data points is used, the two
frequencies are still visible but with quite a low denition, as shown in Figure 5.6.
However, Figure 5.7 shows when the window size is reduced to 500 samples, DFT
fails to distinguish the two broken rotor bar sideband frequencies.
This disadvantage can be even worse as that if the machine load is lighter, the data
window required for DFT becomes much longer. Figure 5.8 shows the spectral
result of using a window of 2000 data points and a same sampling frequency for the
same machine as above but operating under 25% of full load. It can be seen the
two sideband frequency components have merged into the fundamental frequency
already and are not able to be observed.
For the convenience of comparison, minimum window lengths are dened as the
threshold of the window length requirement for both Prony Analysis and DFT.
It is the shortest data window required in order to provide a sucient degree of
accuracy in frequency estimation. Here, this criterion of accuracy is dened as
the unitary frequency error, given by
65
Chapter 5. Implementation of Prony Analysis for Induction Motor Broken Bar Detection
0 10 20 30 40 50 60 70 80 90 100−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Am
plitu
de (
dB)
Frequency (Hz)
Fundamental Frequency: 50Hz
(1+2s)f sideband frequency: 56.2Hz
(1−2s)f sideband frequency: 43.8Hz
Figure 5.5.: DFT spectrum of the current signal of Machine 2 with 2 broken ro-tor bars operating under full load. The data window length is 5000samples using a sampling frequency of 1000Hz.
0 10 20 30 40 50 60 70 80 90 100−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Am
plitu
de (
dB)
Frequency (Hz)
Fundamental Frequency: 50Hz
(1+2s)f sideband frequency: 56Hz
(1−2s)f sideband frequency: 44Hz
Figure 5.6.: DFT spectrum of the current signal of Machine 2 with 2 broken ro-tor bars operating under full load. The data window length is 1000samples using a sampling frequency of 1000Hz.
66
5.4. Disadvantages of DFT and Solutions by Prony Analysis
0 10 20 30 40 50 60 70 80 90 100−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Am
plitu
de (
dB)
Frequency (Hz)
Figure 5.7.: DFT spectrum of the current signal of Machine 2 with 2 broken rotorbars operating under full load. The data window length is 500 samplesusing a sampling frequency of 1000Hz.
0 10 20 30 40 50 60 70 80 90 100−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Am
plitu
de (
dB)
Frequency (Hz)
Figure 5.8.: DFT spectrum of the current signal of Machine 2 with 2 broken rotorbars operating under 25% of full load. The data window length is2000 samples using a sampling frequency of 1000Hz.
67
Chapter 5. Implementation of Prony Analysis for Induction Motor Broken Bar Detection
UE =|fest − ftrue| × 100
ftrue% (5.3)
where fest denotes the estimated frequency value whilst ftrue indicates the true
frequency value calculated from motor slip. The minimum or threshold window
length is then dened as the shortest window needed for either Prony Analysis or
DFT to maintain the unitary error UE of frequency estimation within 0.3%. It
also should be kept in mind that the exact value of each threshold is dependent
on its particular condition. It may vary when condition changes.
As an example, a number of simulations have been conducted and the results re-
veal that for Machine 2 with two broken rotor bars operating under full load, when
using a sampling frequency of 1000Hz, DFT requires a minimum data window with
the length of 800 data points (frequency resolution: 1.25Hz). This requirement
is elevated to a 2600 points window (frequency resolution: 0.38Hz) when there is
25% of full load and up to a 6000 points window (frequency resolution: 0.1Hz)
when the load is only 10% of full load. True values of both the lower and higher
broken bar sideband frequencies calculated from motor slip are 48.5624Hz and
51.4376Hz, and 49.4525Hz and 50.5475Hz in the 25% and 10% load conditions,
respectively. If a lighter load condition applies, an even longer sampling time is re-
quired consequently. These limitations make the accuracy of the broken rotor bar
detection in induction machine suer considerably from machine load variations
or lack of data records.
However, there is no such restrictions for Prony Analysis. To generalize the ability
of maintaining a necessary frequency resolution for both Prony Analysis and DFT
with respect to the data window length, Table 5.3 to Table 5.6 list the true and
estimated values of sideband frequency components obtained by using windows of
the minimum lengths for broken rotor bar detection on Machine 2. The number
of broken rotor bars and the load condition are varied. The sampling frequency
used in these simulations is 1000Hz. The stator currents used for both approaches
are the same one in each case.
In Table 5.3, the result shows that when using even a window size as small as only
40 samples, Prony Analysis is still able to estimate the values of the sideband
frequency components, whereas in the case of using DFT, windows which are
more than 20 times longer are required. Similar result is also observed for other
load conditions. Besides that the load condition aects considerably the values
of sideband frequencies, it is noticed from the table that the number of broken
68
5.4. Disadvantages of DFT and Solutions by Prony Analysis
Table 5.3.: PA and DFT results of the (1± 2s) f sideband frequencies using theminimum window lengths with 1000Hz sampling frequency for Machine2 operating under full load condition and with various numbers ofbroken rotor bars.
Full load(1−2s)f sidebandcomponent (Hz)
(1+2s)f sidebandcomponent (Hz)
Minimum win-dow length(Number ofsamples)
Numberof brokenrotor bars
TrueValue
DFT PATrueValue
DFT PA DFT PA
1 43.9431 44.0000 43.9653 56.0569 56.0000 56.0568 1000 402 43.7197 43.7500 43.7129 56.2803 56.2500 56.1944 800 403 43.4598 43.3300 43.4606 56.5402 56.6700 56.5695 900 404 43.1556 43.0800 43.1550 56.8444 56.9200 56.8299 1300 405 42.7849 42.8600 42.7812 57.2151 57.1400 57.2452 900 406 42.3318 42.2200 42.3331 57.6682 57.7800 57.6651 900 407 41.7562 41.8200 41.7576 58.2438 58.1800 58.2510 1100 408 40.9606 41.0000 40.9664 59.0394 59.0000 59.0160 1000 40
Table 5.4.: PA and DFT results of the (1± 2s) f sideband frequencies using theminimum window lengths with 1000Hz sampling frequency for Machine2 operating under 75% load condition and with various numbers ofbroken rotor bars.
75% offull load
(1−2s)f sidebandcomponent (Hz)
(1+2s)f sidebandcomponent (Hz)
Minimum win-dow length(Number ofsamples)
Numberof brokenrotor bars
TrueValue
DFT PATrueValue
DFT PA DFT PA
1 45.6169 45.7100 45.7327 54.3831 54.2900 54.3361 1400 1202 45.4564 45.4500 45.4251 54.5436 54.5500 54.4233 1100 1003 45.2692 45.3800 45.1949 54.7308 54.6200 54.5766 1300 1004 45.0567 45.0000 45.0953 54.9433 55.0000 55.0240 1000 1005 44.7970 44.6700 44.8032 55.2030 55.3300 55.2007 1500 1006 44.4837 44.5500 44.3839 55.5163 55.4500 55.4848 1100 807 44.0983 44.1700 44.1251 55.9017 55.8300 55.9781 1200 408 43.5844 43.6400 43.5918 56.4156 56.3600 56.4245 1100 40
69
Chapter 5. Implementation of Prony Analysis for Induction Motor Broken Bar Detection
Table 5.5.: PA and DFT results of the (1± 2s) f sideband frequencies using theminimum window lengths with 1000Hz sampling frequency for Machine2 operating under 50% load condition and with various numbers ofbroken rotor bars.
50% offull load
(1−2s)f sidebandcomponent (Hz)
(1+2s)f sidebandcomponent (Hz)
Minimum win-dow length(Number ofsamples)
Numberof brokenrotor bars
TrueValue
DFT PATrueValue
DFT PA DFT PA
1 47.1621 47.1400 47.2649 52.8379 52.8600 52.9784 1400 2602 47.0576 46.9200 46.9146 52.9424 53.0800 52.9900 1300 2803 46.9376 46.9200 46.9644 53.0624 53.0800 53.1240 1300 2804 46.7941 46.6700 46.8485 53.2059 53.3300 53.2321 1200 2405 46.6348 46.6700 46.5908 53.3652 53.3300 53.3897 1200 2206 46.4431 46.3600 46.5076 53.5569 53.6400 53.4716 1100 2107 46.2033 46.1500 46.1479 53.7967 53.8500 53.8513 1300 2008 45.8893 46.0000 45.8860 54.1107 54.0000 54.1576 1000 120
Table 5.6.: PA and DFT results of the (1± 2s) f sideband frequencies using theminimum window lengths with 1000Hz sampling frequency for Machine2 operating under 25% load condition and with various numbers ofbroken rotor bars.
25% offull load
(1−2s)f sidebandcomponent (Hz)
(1+2s)f sidebandcomponent (Hz)
Minimum win-dow length(Number ofsamples)
Numberof brokenrotor bars
TrueValue
DFT PATrueValue
DFT PA DFT PA
1 48.6136 48.5200 48.5329 51.3864 51.4800 51.3717 2700 5002 48.5609 48.4600 48.5544 51.4391 51.5400 51.5580 2600 4603 48.5008 48.4000 48.4907 51.4992 51.6000 51.4398 2500 3004 48.4356 48.3300 48.5897 51.5644 51.6700 51.5490 2400 2505 48.3477 48.2600 48.2545 51.6523 51.7400 51.6249 2300 2506 48.2655 48.1800 48.2685 51.7345 51.8200 51.7513 2200 2307 48.1305 48.1000 48.1462 51.8695 51.9000 51.8691 2100 2308 47.9873 47.8900 48.0157 52.0127 52.1100 51.9144 1900 220
70
5.4. Disadvantages of DFT and Solutions by Prony Analysis
rotor bars also has a small inuence on them. This is because the increase of
resistance on rotor due to fractured rotor bars can be treated as a small increase
of load. When the number of broken rotor bars rises, the two sideband frequencies
are slightly more apart from the fundamental frequency. It also can be observed
that the minimum window length requirement for Prony Analysis decreases as
the number of broken rotor bars increases. This is because the amplitudes of the
sideband frequencies are higher when the fault is severer. This makes it easier for
Prony Analysis to estimate their values.
Table 5.3 to Table 5.6 are plotted together in the three-dimensional Figure 5.9
to illuminate the impact of load and broken rotor bar numbers on the minimum
length requirements of data windows for both Prony Analysis and DFT. It should
also be noticed that the data acquisition time is in direct proportion to data win-
dow length when the sampling frequency is determined. Thus, it is observed that
in all circumstances those have been taken into account, Prony Analysis needs
a considerably shorter data window (or data acquisition time) for distinguishing
the two sideband frequency components compared to DFT. It also can be seen
that when the machine load decreases, which means the two broken bar sideband
frequency components will move close to the fundamental frequency, the mini-
mum window length required for DFT increases dramatically, whereas that for
Prony Analysis only goes up sightly. The trends of the minimum window length
requirements of the two methods illustrate that if higher frequency resolutions are
needed, the data window (or data acquisition time) for DFT has to be enlarged (or
lengthened) signicantly to observe close frequencies, whereas the Prony Analysis
only needs slightly longer data windows (or data acquisition time).
5.4.2. Frequency Estimation Accuracy
In practice, more precise estimates of the values of the broken rotor bar sideband
frequencies facilitate decisions on the existence of broken rotor bars to be more
creditable. The accuracy of the frequency estimation by DFT depends on the
frequency resolution, which is solely determined by the sampling time. The accu-
racy of the frequency estimation by Prony Analysis is also aected by the window
length, but in a dierent manner. In this section, they are compared in terms of
the unitary Mean Absolute Error of the frequency estimation, which is calculated
as the mean of the unitized absolute error of frequency estimates for a number of
independent runs. The equation is given as
71
Chapter 5. Implementation of Prony Analysis for Induction Motor Broken Bar Detection
25%50%75%Full load
0123456780
500
1000
1500
2000
2500
3000
Numbe
r of b
roke
n roto
r bar
s
Load condition (percentage of full load)
Win
dow
leng
th (
num
ber
of s
ampl
es)
DFT
PA
Figure 5.9.: Plotted comparison of the minimum window length requirements ofPA and DFT for broken rotor bar detection on Machine 2, with respectto dierent numbers of broken rotor bars and load conditions.
MAEfreq =
∑Nt
nt=1 UEnt
100Nt
(5.4)
where Nt is the number of independent trials.
Figure 5.10 and Figure 5.11 shows the accuracy of frequency estimation in terms of
MAEfreq of using both Prony Analysis and DFT. The MAEfreq is calculated for
100 independent trials with each window size, varying from 500 samples to 3500
samples using 1000Hz sampling frequency. To construct the independent trials for
testing the two spectral analysis methods, 10 independent runs of the simulation
with random selection of errors for 40s are executed. A dierent segment of data is
used for each trial. The numerical results of using the two methods with a window
size of 500 samples can be found in Appendix D. The model has been used in
these simulations is Machine 2 with broken rotor bar number varying from 1 to 8
and load varying as 100%, 75%, 50% and 25% of the rated load. The amplitude
of the sideband frequencies is in dB with reference to the fundamental frequency.
It is observed that although the MAEfreq of DFT decreases when the data win-
dow is enlarged, in general, much better estimation results obtained by Prony
Analysis using a window of the same length is achieved. With a range of dierent
window lengths taken into account, the comparison result highlights that Prony
72
5.4. Disadvantages of DFT and Solutions by Prony Analysis
500 1000 1500 2000 2500 3000 3500
10−4
10−3
10−2
10−1
100
Window length (number of samples)
Fre
quen
cy e
stim
ator
err
or (
MA
E)
MAE of PA result for (1−2s)f sidebandMAE of PA result for (1+2s)f sidebandMAE of DFT result for (1−2s)f sidebandMAE of DFT result for (1+2s)f sideband
DFT for (1−2s)f sideband
DFT for (1+2s)f sideband
PA for (1−2s)f sideband
PA for (1+2s)f sideband
Figure 5.10.: MAEfreq in frequency estimation by PA and DFT in respect of win-dow length using 1000Hz sampling frequency for simulated currentdata of Machine 2 operating under full load.
500 1000 1500 2000 2500 3000 3500
10−4
10−3
10−2
10−1
100
Window length (number of samples)
Fre
quen
cy e
stim
ator
err
or (
MA
E)
MAE of PA result for (1−2s)f sidebandMAE of PA result for (1+2s)f sidebandMAE of DFT result for (1−2s)f sidebandMAE of DFT result for (1+2s)f sideband
DFT for (1−2s)f sideband
DFT for (1+2s)f sideband
PA for (1−2s)f sideband
PA for (1+2s)f sideband
Figure 5.11.: MAEfreq in frequency estimation by PA and DFT in respect of win-dow length using 1000Hz sampling frequency for simulated currentdata of Machine 2 operating under 75% of full load.
73
Chapter 5. Implementation of Prony Analysis for Induction Motor Broken Bar Detection
Table 5.7.: Estimated values of the sideband frequency components by PA andDFT for broken rotor bar detection on Machine 2, under dierentlight load conditions and with one broken rotor bar.
Loadcondition
(1−2s)f sidebandcomponent (Hz)
(1+2s)f sidebandcomponent (Hz)
Minimum win-dow length(Number ofsamples)
(% of fullload)
Truevalue
DFT PATruevalue
DFT PA DFT PA
20% 48.8954 49.0000 48.5888 51.1046 51.0000 51.6599 3000 10015% 49.1742 49.0700 49.1661 50.8258 50.9300 51.6534 4000 20010% 49.4525 49.500 49.3222 50.5475 50.5000 50.4061 6000 10005% 49.7277 49.7500 49.4312 50.2723 50.2500 50.1761 12000 20001% 49.9459 49.95 49.7089 50.0541 50.0500 50.8899 35000 2000
Analysis is capable of estimating the sideband frequency components more ac-
curately than DFT. It is also observed that the MAEfreq remains in almost the
same level for Prony Analysis regardless of the window length, though there is an
approximate minimum window length required for it to be able to function, as
described previously in 5.4.1.
5.4.3. Small Load Conditions
The window length can be critical for on-line diagnosis of induction machine bro-
ken rotor bars. The eect of small or changing load is desired to be eliminated
as much as possible. Therefore the requirement of long data windows for DFT
makes it very dicult to detect the sideband frequencies in such conditions.
The general impact of load condition on the movement of broken rotor bar side-
band frequencies in the stator current spectrum has been described in Chapter
2. Apart from that, another interesting fact is that what the smallest machine
load is, for Prony Analysis to be able to accurately estimate the two sideband fre-
quencies. Table 5.7 displays the true and estimated values of sideband frequency
components obtained by Prony Analysis for Machine 2 in regard to conditions
that the load is less that 25% of the full load. The sampling frequency is 1000Hz.
Table 5.7 shows that the Prony Analysis has successfully estimated the two broken
bar sideband frequencies for Machine 2 with one broken rotor bar in conditions
that the load is as light as only 1% of full load. This is an extreme case as it is
required to distinguish considerably small peak in the vicinity of a much higher
74
5.5. Evaluation of Prony Analysis
frequency peak. The amplitude of the two sideband frequencies in the 1% load
case, are -65.33dB and -71.77dB lower than the fundamental frequency, respec-
tively, due to the extremely light load. The frequency dierence between the
nearest high peak is only 0.05Hz. These would make the two frequency compo-
nents almost enshrouded in the noise oor or the side lobe of the fundamental
frequency due to spectral leakage. With the same load condition, the one broken
rotor bar case can be considered as the most dicult one. Thus, it is reasonable
to state that if more than one broken bar exist in the rotor, they also can be
detected when the machine is operating with the same small load. To achieve the
similar resolution, DFT requires much longer data windows as presented, which
is obviously inconvenient and dicult in practice.
Since the small value of motor slip when the load is light, more sideband harmonics
will also appear very close to the fundamental frequency if the number of broken
rotor bars is more. This increases the complexity of the computation for Prony
Analysis. Therefore, a lter of narrower passband is used to eliminate those
interfering frequency components. Here, the lter used for small load conditions
has the frequency passband of f ± 3Hz, passing only the fundamental frequency
and the two sideband frequencies.
5.5. Evaluation of Prony Analysis
The purposes of this section are to try to characterize the inuential factors and
to have a better understanding of how to adjust them to make Prony Analysis
work better. Though the original Prony method was invented about more than
200 years ago, it has not been practically used until the recent decades after the
theory of modern spectral estimation. A number of modied versions [33][48][43]
have been proposed to overcome its problems of inconstancy and sensitivity when
analyzing noise corrupted signals. The IRLS Prony method [33] is a signicant
improvement of the Prony algorithm based on LS Prony and has shown a strong
ability of dealing with noisy signals in practice. It is therefore the method chosen
to be utilized in this research.
However, the success of Prony tting and frequency estimation is subject to a
number of inuencing factors. Only experiential conclusions have been made so
far on how these factors aect. Signal noise level, amplitude of each frequency
component contained in the signal, window length, sampling frequency, algorithm
order choice, the number of samples taken into the estimation process or even the
75
Chapter 5. Implementation of Prony Analysis for Induction Motor Broken Bar Detection
choice of data segment all aect the estimation result. This is both reported in
[47] and observed by the author.
5.5.1. Impact of Data Window Length
It has been shown in 5.4.2 that if the length of data window for Prony Analysis is
more than 500 samples with 1000Hz sampling frequency, even if the window length
is enlarged further, the accuracy of frequency estimation remains in a similar level.
However, if a considerably small data window is used, the performance of Prony
Analysis is eected greatly by the length of the window.
Figure 5.12 displays the MAEfreq of the higher and lower sideband frequency
estimations with respect to the length of the data window less than 500 samples
using 1000Hz sampling frequency. The data used is the stator current signal of
Machine 1 operating with 2 broken rotor bars with full load. The MAEfreq is
calculated using Eq. (5.4) for 100 independent trials. It is observed that if only a
little data is available the estimator error can be much higher. However, after the
amount of data samples used in the Prony algorithm achieving a certain number,
the frequency estimator error falls down and then remains almost steady even the
data window length continues to increase. The length of the data window at this
turning point of the MAEfreq curve is the minimum window length requirement
which was previously mentioned in 5.4.1.
5.5.2. Noise Impact
Noise presented in the signal can deteriorate the performance of Prony Analysis.
The MAEfitting of Prony estimation calculated using Eq. (5.2) is displayed in
Figure 5.13. The signal is sampled with a sampling frequency of 1000Hz and the
data window is 500 samples. The standard deviation of the measurement error is
simulated by using the random number generator. It is increased from 0.005 to
0.105 with a step of 0.01. The results clearly show the degree of accuracy that
the IRLS Prony is able to perform in modeling the original signal. A higher noise
level decreases the accuracy of estimate.
5.5.3. Order Selection
The selection of model order can be a critical and tough task as it directly aects
the performance of Prony Analysis [32] [47]. Some selecting approaches have been
76
5.5. Evaluation of Prony Analysis
0 50 100 150 200 250 300 350 400 450 500
10−4
10−3
10−2
10−1
100
Window length (number of samples)
Fre
quen
cy e
stim
ator
err
or (
MA
E)
(1−2s)f sideband(1+2s)f sideband
Figure 5.12.: MAEfreq of the 6 order PA frequency estimator of broken rotor barsideband frequencies with respect to the window length when using1000Hz sampling frequency for 100 runs.
0 0.02 0.04 0.06 0.08 0.1 0.12
10−4
10−3
10−2
10−1
100
Measurement error (standard deviation)
Fitt
ing
erro
r (M
AE
)
Figure 5.13.: Estimation mean absolute error as a function of measurement errorstandard deviation for IRLS Prony
77
Chapter 5. Implementation of Prony Analysis for Induction Motor Broken Bar Detection
proposed but they are complicated and may not work in all situations [14] [37].
Normally the number of sinusoids in a signal is unknown. If the order chosen
is smaller than the number of sinusoids present, great error can occur and cause
the tting to fail. Generally, increasing the model order can be useful to improve
the tting estimation result [14]. However, the algorithm may suddenly fail at
some points as the order increases [47]. The computational diculty rises due
to high order polynomials and longer computational time is caused accordingly.
Moreover, feigned poles will be resulted from the calculation, which confuse with
the actual ones. Thus, roots inspection is necessary to eliminate fake signal poles.
This can be accomplished by discarding the results with very small or even zero
amplitudes, or very high damping coecients.
As there is no straightforward method of computing the order for an arbitrary
system, an initial estimate in needed. An empirical rule is to start with one third
of the number of data points in the window, and then increase the order until the
original signal is well tted [49].
However, prior knowledge of the number of signal poles can greatly reduce the
eort on selecting a proper order. Fortunately, it is the case in induction machine
broken rotor bar diagnosis applications. The number of sinusoids is known as
three (the fundamental frequency and two sideband frequencies). Moreover, for
the IRLS Prony method, the iteration process depresses the impact of the order
chosen. Thus, an order of 6 can be chosen in most of the situations as there are
only three frequency components in the bandpass ltered current signal. Only in
some of the previous small load conditions, the algorithm fails with an order of 6.
If this is the case, simply choosing another order or applying the one third rule
will most likely solve the problem.
5.6. Practical Implementation Test
5.6.1. Experiment Setup
In order to verify the Prony Analysis method for practical uses, measured data
from laboratory experiments is used in this section. A commercial 2.2kW, 50Hz,
4 poles induction machine, which has a standard cast aluminum squirrel cage
rotor with 32 rotor slots, is used in the test. A separately-excited DC generator
is coupled via a belt as load, and is loaded by using a variable resistance bank.
78
5.6. Practical Implementation Test
The broken rotor bar fault is constructed by cutting holes through the rotor bars
at the joints with the end ring using a ne milling cutter.
The stator current is sensed by a Hall-Eect clamp probe, passed through an
anti-aliasing lter, which is an 8th order Butterworth lter, and nally sampled
by an A/D converter with a sampling frequency of 400Hz. This gives a Nyquist
frequency of 200Hz. The sampling time is 20s, which gives the length of the data
window is 8000 samples. A custom written LabVIEW data acquisition system is
used for data acquisition [50].
The captured data was then analyzed using both Prony Analysis and DFT in
Matlab. The data is passed through a Hanning window before applying DFT,
and is passed through an FIR bandpass lter before applying Prony Analysis.
5.6.2. Test Results
Examples are presented in this section for demonstration and verication of the
Prony Analysis. Four bars are cut in the rotor and the induction motor is managed
to operate with full load. The spectrum of the measured stator current using DFT
is presented in Figure 5.14. The two sideband frequency components at 43.30Hz
and 56.70Hz and their harmonics can also be observed. The whole data is used
for DFT, giving the frequency resolution determined as 0.05Hz.
In this implementation a remarkably short data window of only 200 samples is used
for Prony Analysis. The order of the algorithm is chosen as 6 since the number of
frequency components is known. The Prony Analysis result is displayed in Figure
5.15 and Table 5.8. The rotor speed is measured using a digital photo tachometer
to calculate the slip and the true value of sideband frequencies.
The plotted estimation and prediction results in Figure 5.15 both demonstrate
excellent matches with the real data waveforms. TheMAEfitting of the estimation
part of signal is 0.0014 and theMAEfitting of the prediction part of signal is 0.0127.
The numerical result in Table 5.8 shows a high resolution achieved by Prony
Analysis, while the DFT result presents big errors in determining the frequency
values. The spectral estimation result obtained by DFT using 200 data samples
is also plotted in Figure 5.16 to compare against the result of Prony Analysis.
The comparison between the Prony Analysis and DFT results clearly shows the
superiority of the Prony Analysis.
79
Chapter 5. Implementation of Prony Analysis for Induction Motor Broken Bar Detection
0 20 40 60 80 100 120 140 160 180 200−100
−90
−80
−70
−60
−50
−40
−30
−20
−10
0
Frequency (Hz)
Am
plitu
de (
dB)
43.3 Hz56.7 Hz
Figure 5.14.: Spectrum of the measured stator current of a 2.2 kW induction motorwith 4 broken rotor bars using DFT with a sampling frequency of400Hz, and a data window of 4000 samples.
Table 5.8.: PA result of the measured stator current signal of a 2.2kW inductionmotor with 4 broken rotor bars operating under full load condition,using a data window of 200 samples with the sampling frequency of400Hz.
PA DFTTruevalue(Hz)
Value(Hz)
Amplitude(dB)
Value(Hz)
Amplitude(dB)
(1− 2s) f sideband 43.3333 43.3019 -22.7483 44.0000 -23.8800Fundamental frequency 50.0000 49.9919 0 50.0000 0
(1 + 2s) f sideband 56.6667 56.6883 -30.6257 56.0000 -31.4000
80
5.6. Practical Implementation Test
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5−10
−8
−6
−4
−2
0
2
4
6
8
10
Magni
tude (A
)
Time (sec)
Current SignalProny Estimation
(a) Sampled real data and PA estimation for the period of sampling
0.5 0.55 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1−10
−8
−6
−4
−2
0
2
4
6
8
10
Magni
tude (A
)
Time (sec)
Current SignalProny Estimation
(b) Sampled real data and PA prediction for the period into future.
Figure 5.15.: Comparison of the current waveforms between PA estimation andprediction with the real current signal of a 2.2kW induction machineoperating in full load with 4 broken rotor bars.
0 20 40 60 80 100 120 140 160 180 200
−80
−70
−60
−50
−40
−30
−20
−10
0
Frequency (Hz)
Am
plitu
de (
dB)
44 Hz
56 Hz
Figure 5.16.: DFT spectrum of the same signal data used in Figure 5.14 and Figure5.15 but using a window of only 200 samples with the samplingfrequency of 400Hz.
81
Chapter 5. Implementation of Prony Analysis for Induction Motor Broken Bar Detection
82
Chapter 6.
Conclusion
The data window limitation in induction machine condition monitoring and fault
diagnostics is a critical and real challenge in practice. Traditional spectral analysis
approach such as DFT suers from this due to its inherent drawbacks and to other
objective causes, for instance, the light load condition and the load variation.
Additionally, research in diagnosing the broken rotor bars of induction machines
can be dicult and expensive, because of the eort needed to make this fault in
an induction machine manually and the high cost of induction motors.
This thesis has shown the implementation of the Prony Analysis in the diagnostics
of broken rotor bars in an induction motor. A model is used to simulate the
operation of an induction motor with broken rotor bar faults and to generate
data for tests. Laboratory measurement is used to validate the Prony Analysis
approach. In the thesis, the nature of induction motor broken rotor bar faults is
studied, a model-based diagnosis approach is developed, a high-resolution spectral
analysis technique that overcomes the disadvantages of the DFT is applied and
veried, and the nature of this technique is investigated. This chapter gives an
overview of these achievements.
6.1. The Broken Rotor Bar Fault
Broken rotor bar faults are reected in the stator current of an induction machine
as the presence of specic frequency components. The most consistent and widely
adopted ones among these characteristic frequency components are the two broken
rotor bar sideband frequencies, (1± 2s) f .
Generally, the values of these sideband frequencies are sensitive to the load condi-
tion. When the machine is operating under full load, these two sideband frequen-
cies are 2sf Hz away from the fundamental frequency. This frequency dierence is
83
Chapter 6. Conclusion
always within 10Hz. When the load becomes light, these two sideband frequencies
move towards the fundamental frequency, due to the decrease of the slip. They
can be very close to the fundamental frequency until disappeared since the slip
equals to zero if the load is zero. This has been addressed in Section 2.4. Thus,
the data windows need to be enlarged to gain higher resolution in the frequency
domain for DFT. This trades o the requirement of short data window due to
load variation. This has been shown in Section 5.4.
The fault severity has a major eect on the amplitude and a minor eect on
the value of the sideband frequency components. The amplitude of the sideband
frequencies rises when the number of broken rotor bars increases. This has led
to predictively quantitative evaluations of the number of broken rotor bars using
prediction equations introduced in Section 2.4. Three predictions of the amplitude
of the (1− 2s) f sideband are compared in Section 5.3. The result shows that
the Prediction 1 performs the best in predicting the amplitude of the (1− 2s) f
sideband when the number of broken rotor bars is less than half of the number
of total rotor bars in on rotor phase. It underestimates this amplitude within
4dB if the number of broken rotor bars increases further. The result also shows
the value in quantitative evaluation of Prediction 3. It gives a constant small
underestimated amplitude value regardless the number of broken rotor bars.
Additionally, broken rotor bar faults distort the motor stator current. The mod-
ulation is severer if more defective rotor bars exist, as revealed in Section 5.3. A
series of sideband harmonics will also arise when the number of broken rotor bars
increases, described in Section 2.4. The induction motor normally breaks down
when this number goes up close to the number of total rotor bars in one rotor
phase.
6.2. The Induction Machine Model
The broken rotor bar fault of induction motors can be simulated by increasing
the resistance of the rotor phase where the fault occurs. The model is built in the
arbitrary qd0 reference frame and is achieved in Matlab/Simulink. To validate
the model, measured data from laboratory experiments is used for comparison.
This model successfully simulates the dynamic and steady operating state of an
induction motor with or without broken rotor bar faults. The broken rotor bar
sideband frequency components well present in the stator spectrum. Their ampli-
tude and values change according to the change of fault severity and the machine
84
6.3. The Implementation of Prony Analysis
load. Result in Section 2.4 and Section 3.4 has shown the validity of this model.
With this model, the load condition and the number of broken rotor bars can be
easily changed, which provides great convenience for study. All machine param-
eters are also accessible. This gives the advantage of study the broken rotor bar
fault on dierent machines.
6.3. The Implementation of Prony Analysis
The Prony Analysis is implemented for induction motor broken rotor bars detec-
tion using both simulated and measured data. The result is also compared with
DFT result.
Data pre-conditioning such as ltering, downsampling and constant oset remov-
ing needed to be conducted prior to running the Prony Analysis algorithm. Es-
pecially, preltering the signal signicantly improves the analysis result.
The Prony Analysis is not only able to well estimate the data within the data
window, but also predict the future data with high precision, as shown in Section
5.3. In Section 5.4, compared with DFT, the Prony Analysis has demonstrated
great advantages in terms of using much shorter data windows to satisfy the
same frequency resolution requirement, and gaining a higher accuracy of frequency
estimation using the same length windows. This gives the Prony Analysis the
ability to detect the broken rotor bar sideband frequency components in light and
varying load conditions. Result shown in Section 5.4 proofs the minimum window
length required by DFT can be 6 to 20 times longer than Prony Analysis in light
load conditions.
The Prony Analysis algorithm needs a minimum length window to function cor-
rectly. If this minimum window length is not achieved, Prony Analysis may fail
to produce accurate result. However, the accuracy of frequency estimation us-
ing Prony Analysis shows great independence on the data window length after
the minimum length requirement has been met. This result is shown together in
Section 5.4 and Section 5.5.
The noise level aects the accuracy of Prony Analysis. A higher noise level will
result in a lower estimate accuracy. Thus, in practice, decreasing the noise level by
lowpass or bandpass ltering can signicantly improve the performance of Prony
Analysis.
85
Chapter 6. Conclusion
The order of the Prony algorithm is always chosen as twice the number of frequency
components present in the signal. Thus, previous knowledge of the number of
signal poles is desired. Besides, since the Prony Analysis is a high computational
cost algorithm, the order is preferred to be as small as possible.
6.4. Future Work
The Prony Analysis implementation in this thesis for broken rotor bars detec-
tion in an induction motor has shown a great advantage over DFT in terms of
using shorter data windows and achieving higher frequency estimation accuracy.
It is therefore to think promisingly that this high-resolution approach can be im-
plemented for the detection of other kinds of machine faults by signal spectral
analysis, where trade-os on the data window length have to be made.
86
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92
Appendix A.
Important Programs
Models, simulations and data analysis were undertaken using the MATLAB R2006b.
Appendix A contains important programs coded in the MATLAB environment.
A.1. Simulation Initialization
Matlab scripts presented in this section are modied from reference [15].
A.1.1. Simulation Initialization File Startsim.m
% This file sets up the motor parameters, initial conditions, and
% mechanical loading in the MATLAB workspace for simulation.
% Load three-phase induction motor parameters
motor_1hp % load motor parameters from
% motor_1hp.m
% Initialize to start from standstill with machine unexcited
Psiqso = 0; % stator q-axis total flux linkage
Psipqro = 0; % rotor q-axis total flux linkage
Psidso = 0; % stator d-axis total flux linkage
Psipdro = 0; % rotor d-axis total flux linkage
wrbywbo = 0; % pu rotor speed
tstop = 10;
93
Appendix A. Important Programs
% program time and output arrays of repeating sequence signal
% for Tmech
tmech_time = [0 0.5 1 2 3 tstop]; % base torque operation
%tmech_value = [0 0 0 0 0 0]*Tb;
%tmech_value = [-.25 -.25 -.25 -.25 -.25 -.25]*Tb;
%tmech_value = [-.5 -.5 -.5 -.5 -.5 -.5]*Tb;
%tmech_value = [-.75 -.75 -.75 -.75 -.75 -.75]*Tb;
tmech_value = [-1 -1 -1 -1 -1 -1]*Tb;
disp('Set up for running im.m.);
disp('Perform simulation then type return for plots');
A.1.2. Machine Parameter Initialization File
motor_1hp.m
% Parameters of a 1ph induction motor.
%%----------rated & based values------------------%%
P = 4; % number of poles
frated = 50; % rated frequency in Hz
Vrated = 200; % rated line to line voltage in V
Prated = 746; % rated output power in W
wb = 2*pi*frated; % base electrical frequency
we = wb;
wbm = 2*wb/P; % base mechanical speed
Sb = Prated; % rating output power in VA
Vm = Vrated*sqrt(2/3); % magnitude of phase voltage
Vb = Vm; % base voltage
Ib = (2*Sb)/(3*Vb); % base current
Zb = Vb/Ib; % base impedance in ohms
Tb = Sb/wbm; % base torque
%%---------------------------------------------%%
Tfactor = (3*P)/(4*wb); % factor for torque expression
N = 28; % total rotor bar number
n = 1; % broken bar number
94
A.2. Least Squares Prony Method
a0 = 0; % initial rotor angle
%%----------machine parameters------------------%%
rs = 3.35; % stator winding resistance in
% ohms
xls = 6.94e-3*wb; % stator leakage reactance in
% ohms
xplr = xls; % rotor leakage reactance
xm = 163.73e-3*wb; % stator magnetizing reactance
rpr = 1.99; % referred rotor wdg resistance
% in ohms
dr = (rpr*n)/(N/3-n); % broken bar effect
xM = 1/(1/xm + 1/xls + 1/xplr);
J = 0.15; % rotor inertia in kg·m2H = J*wbm*wbm/(2*Sb); % rotor inertia constant in sec
Domega = 0; % rotor damping coefficient
A.2. Least Squares Prony Method
The Least Square Prony algorithm is modied from [51] in Matlab. Parameters
of the algorithm can vary from occasion to occasion.
%%----------Least Square Prony Method----------%%
%% Use of SVD
%%----------Parameters set-up------------------%%
close
global order N Z input_signal
%downsample(); % downsampling if needed
fs=; % sampling frequency
T=1/fs; % sampling interval
t=[0:T:]; % sampling time
input_signal = ias; % input signal
order = ; % model order
N = length(input_signal); % number of data samples
%%------step 1: Compose equations and solve the complex
95
Appendix A. Important Programs
%% coefficients a[m]-------%%
X = []; % compose the X matrix
m = N - order;
step = order;
for d = 1:order
for j = 1:m
X(d,j) = input_signal(step-1 + j);
end
step = step-1;
end X=X';
z = [];
for l = 1:(N-order)
z(l,1) = -input_signal(l+order);
end
theta = pinv(X)*z; % evaluation of the
% overdetermined linear
% simultaneous equations
%%------step 2: Solve the complex roots z-------%%
LPM = [1 theta']; % a0=1
rootz = roots(LPM); % roots of LPM - the homogeneous
% linear constant-coefficient
% difference equation
Freq = imag(log(rootz))/(2*pi*T);
damping_factor = log(abs(rootz))/T;
%%------step 3: Solve the complex coefficients h-------%%
Z=[]; % compose the Z matrix
for k = 1:N
for m = 1:order
Z(k,m) = rootz(m)^(k-1);
end
end
V=[];
for n = 1:N
V(n)=input_signal(n);
96
A.2. Least Squares Prony Method
end
V=V';
H = pinv(Z)*V;
phase_rad = angle(H);
amplitude = abs(H);
result=[Freq damping_factor phase_rad amplitude];
disp(' Freq; damping_factor; phase_rad; amplitude');
disp(result)
%%------------Draw the Prony estimation----------%%
y=0;
for n=1:2:order;
y=y+2*amplitude(n)*exp(damping_factor(n)*t).*
cos(2*pi*Freq(n)*t+phase_rad(n));
end
plot(t,input_signal,'k')
ylabel('Magnitude (A)')
xlabel('Time (s)')
hold
plot(t,y,':kx')
legend('Current Signal','Prony Estimation');
xlim([0 length(y)]);
97
Appendix A. Important Programs
98
Appendix B.
Important Equation Derivations
B.1. Derivation of Eq. (3.18)
vqd0s = Tqd0 (θ) rsT−1qd0 (θ) iqd0s + Tqd0 (θ)
d[T−1qd0(θ)λqd0
s ]dt
The following time-derivative term may be expressed as [15]
d[T−1qd0(θ)λqd0
s ]dt
=d[T−1
qd0(θ)]dt
· λqd0s + T−1qd0 (θ) · d(λqd0
s )dt
=
− sin θ cos θ 0
− sin(θ − 2π
3
)cos(θ − 2π
3
)0
− sin(θ + 2π
3
)cos(θ + 2π
3
)0
dθdt· λqd0s + T−1
qd0 (θ) · d(λqd0s )dt
Substituting this back to Eq. (3.16), obtain Eq. (3.18). The procedure of getting
(3.19) is similar.
B.2. Derivation of Eq. (3.22)
λqd0s = Tqd0 (θ)(Labcss iabcs + Labc
sr iabcr)
By applying the Park's transformation to the above ux linkage equations, obtain
λqd0s = Tqd0 (θ) Labcss T−1
qd0 (θ) iqd0s + Tqd0 (θ) Labcsr T−1
qd0 (θ − θr) iqd0r
= 23
cos θ cos(θ − 2π
3
)cos(θ + 2π
3
)sin θ sin
(θ − 2π
3
)sin(θ + 2π
3
)12
12
12
Lls + Lss Lsm Lsm
Lsm Lls + Lss Lsm
Lsm Lsm Lls + Lss
×
cos θ sin θ 1
cos(θ − 2π
3
)sin(θ − 2π
3
)1
cos(θ + 2π
3
)sin(θ + 2π
3
)1
iqs
ids
i0s
99
Appendix B. Important Equation Derivations
+23Lsr
cos θ cos(θ − 2π
3
)cos(θ + 2π
3
)sin θ sin
(θ − 2π
3
)sin(θ + 2π
3
)12
12
12
cos θr cos
(θr + 2π
3
)cos(θr − 2π
3
)cos(θr − 2π
3
)cos θr cos
(θr + 2π
3
)cos(θr + 2π
3
)cos(θr − 2π
3
)cos θr
×
cos (θ − θr) sin (θ − θr) 1
cos(θ − θr − 2π
3
)sin(θ − θr − 2π
3
)1
cos(θ − θr + 2π
3
)sin(θ − θr + 2π
3
)1
iqr
idr
i0r
= 23
32
(Lls + Lss − Lsm) 0 0
0 32
(Lls + Lss − Lsm) 0
0 0 32
(Lls + Lss) + 3Lsm
iqs
ids
i0s
+Lsr
32
0 0
0 32
0
0 0 0
iqr
idr
i0r
Because of Lsm = Lss cos 2π
3= −1
2Lss, the above equation can be written as Lls + 3
2Lss 0 0
0 Lls + 32Lss 0
0 0 Lls
iqd0s +
32Lsr 0 0
0 32Lsr 0
0 0 0
iqd0r
Similarly, we can obtain the rotor ux linkages in the qd0 reference frame
λqd0r =
32Lsr 0 0
0 32Lsr 0
0 0 0
iqd0s +
Llr + 32Lrr 0 0
0 Llr + 32Lrr 0
0 0 Llr
iqd0r
Then Eq. (3.22) is obtained by expressing the above two equations together
compactly.
B.3. Derivation of Eq. (3.26)
When substitute the voltages in the qd0 form power equation
Pin = 32
(vqsiqs + vdsids + 2v0si0s + v′qri
′qr + v′dri
′dr + 2v′0ri
′0r
)we yield
Pin = 32
[rsi
2qs + ωλdsiqs + d
dt(λqsiqs)
+rsi2ds − ωλqsids + d
dt(λdsids)
100
B.4. Derivation of Eq. (3.34)
+2rsi20s + 2 d
dt(λ0si0s)
+r′ri′2qr + (ω − ωr)λ′dri′qr + d
dt
(λ′qri
′qr
)+r′ri
′2dr − (ω − ωr)λ′qri′dr + d
dt(λ′dri
′dr)
+ 2r′ri′20r + 2 d
dt(λ′0ri
′0r)]
= 32
(rsi
2qs + rsi
2ds + 2rsi
20s + r′ri
′2qr + r′ri
′2dr + 2r′ri
′20r
)+3
2
[ω (λdsiqs − λqsids) + (ω − ωr)
(λ′dri
′qr − λ′qri′dr
)]+3
2ddt
(λqsiqs + λdsids + 2λ0si0s + λ′qri
′qr + λ′dri
′dr + 2λ′0ri
′0r
)The i2r terms are the copper losses, the idλ
dtterms are the rate of the exchange
of magnetic eld energy between windings, and the ωλi terms represent the rate
of energy converted to mechanical work. The electromechanical torque developed
by the machine is given by the sum of ωλi terms divided by the mechanical speed
ωrm = 2Pωr, yield
Tem = 32Pωr
[ω (λdsiqs − λqsids) + (ω − ωr)
(λ′dri
′qr − λ′qri′dr
)]
B.4. Derivation of Eq. (3.34)
From Eq. (3.31), obtain
vqs = 1ωb
d(ψqs)
dt+ rsiqs
Thus,
ψqs = ωb´
(vqs − rsiqs) dt
From Eq. (3.33), obtain
ψqs = (xls + xm) iqs + xmi′qr and ψmq = xm
(iqs + i′qr
)Thus,
iqs = ψqs−ψmq
xls
Finally, obtain
ψqs = ωb´
vqs + rsxls
(ψmq − ψqs)dt
The rotor ux linkages can be obtained using the similar procedure.
101
Appendix B. Important Equation Derivations
B.5. Derivation of the coecients in Eq. (4.3)
ρ =∑N
n=1 |ε [n]|2
=∑N
n=1
(x [n]−
∑qk=1 hkz
n−1k
)2=∑N
n=1 x [n]2 − 2∑N
n=1 x [n]∑q
k=1 hkzn−1k +
∑Nn=1
(∑qk=1 hkz
n−1k
)2;
∂ρ∂hk
= ∂∂hk
[−2∑N
n=1 x [n](h1z
n−11 + . . .+ hkz
n−1k + . . .+ hqz
n−1q
)+∑N
n=1
(h1z
n−11 + . . .+ hkz
n−1k + . . .+ hqz
n−1q
)2]= −2
∑Nn=1 x [n]
∑qk=1 z
n−1k +2
∑Nn=1
(h1z
n−11 + . . .+ hkz
n−1k + . . .+ hqz
n−1q
)∑qk=1 z
n−1k
= −2∑N
n=1
∑qk=1
x [n] zn−1
k − zn−1k
(∑qi=1 hiz
n−1i
)∂ρ∂zk
= ∂∂zk
[−2∑N
n=1 x [n](h1z
n−11 + . . .+ hkz
n−1k + . . .+ hqz
n−1q
)+∑N
n=1
(h1z
n−11 + . . .+ hkz
n−1k + . . .+ hqz
n−1q
)2]= −2 (n− 1)
∑Nn=1 x [n]
∑qk=1 hkz
n−2k
+2∑N
n=1
(∑qk=1 hkz
n−1k
)(n− 1)
(h1z
n−21 + . . .+ hkz
n−2k + . . .+ hqz
n−2q
)= −2 (n− 1)
∑Nn=1
∑qk=1
x [n]hkz
n−2k − hkzn−2
k
(∑qi=1 hiz
n−1i
)Set ∂ρ
∂hk= 0 and ∂ρ
∂zk= 0, yield:∑N
n=1
∑qk=1
x [n] zn−1
k − zn−1k
(∑qi=1 hiz
n−1i
)= 0, and∑N
n=1
∑qk=1 (n− 1)
x [n] zn−2
k − zn−2k
(∑qi=1 hiz
n−1i
)= 0.
Thus,
c1 =∑N
n=1
∑qk=1 x [n] zn−1
k ;
c2 =∑N
n=1
∑qk=1 z
n−1k
(∑qi=1 z
n−1i
);
c3 =∑N
n=1
∑qk=1 (n− 1)x [n] zn−2
k ;
c4 =∑N
n=1
∑qk=1 (n− 1) zn−2
k
(∑qi=1 z
n−1i
).
102
Appendix C.
Parameters of Induction Machines
Table C.1.: Parameters of induction machine models used for simulations.
Machine 1 Machine 2 Machine 3
Output power(kW)
2.2 5.5 35
Rated frequency(Hz)
50 50 50
Rated voltage(V)
220 415 460
Poles 4 4 8Number of rotor
bars28 32 52
Stator windingresistance (Ω)
0.435 1.003 0.187
Stator leakagereactance (Ω)
1.554 2.57 0.502
Rotor leakagereactance (Ω)
1.554 2.57 0.502
Stator magnetizingreactance (Ω)
26.13 44.307 13.08
Referred rotorwinding resistance
(Ω)1.016 1.4735 0.228
103
Appendix C. Parameters of Induction Machines
104
Appendix D.
Prony Analysis Results
105
Appendix D. Prony Analysis Results
TableD.1.:
Freq
uency
estimation
results
byPAandDFTfor
Mach
ine1
with
dieren
tnumberofbroken
rotorbars
operatin
gunder
dieren
tload
condition
s,usin
gadata
window
of500
samples
andasam
plingfreq
uency
of1000H
z.
Full
Load
TrueValu
eDFT
PA
Number
ofbrokenbars
(1−
2s)f
(Hz)
(1+
2s)f
(Hz)
(1−
2s)f
(Hz)
Amplitu
de
(dB)
(1+
2s)f
(Hz)
Amplitu
de
(dB)
(1−
2s)f
(Hz)
Amplitu
de
(dB)
(1+
2s)f
(Hz)
Amplitu
de
(dB)
143.9432
56.056844.0000
-32.360056.0000
-44.410043.9432
-31.279956.0410
-42.98122
43.719856.2802
44.0000-25.7700
56.0000-38.3300
43.7195-24.7258
56.2852-37.1771
343.4593
56.540744.0000
-21.830056.0000
-35.110043.4589
-20.915856.5450
-34.08274
43.153056.8470
44.0000-19.2200
56.0000-33.2600
43.1537-18.1365
56.8487-32.2430
542.7875
57.212542.0000
-16.330058.0000
-31.380042.7867
-15.270757.2150
-30.35566
42.332857.6672
42.0000-13.2000
58.0000-29.5000
42.3341-12.0713
57.6647-28.3792
741.7585
58.241542.0000
-10.910058.0000
-28.710041.7558
-10.157058.2446
-27.92048
40.966659.0334
40.0000-10.0800
60.0000-29.6500
40.9622-8.4658
59.0370-28.0473
75%Load
TrueValu
eDFT
PA
Number
ofbrokenbars
(1−
2s)f
(Hz)
(1+
2s)f
(Hz)
(1−
2s)f
(Hz)
Amplitu
de
(dB)
(1+
2s)f
(Hz)
Amplitu
de
(dB)
(1−
2s)f
(Hz)
Amplitu
de
(dB)
(1+
2s)f
(Hz)
Amplitu
de
(dB)
145.6170
54.3830Fail
Fail
Fail
Fail
45.6164-33.1123
54.3805-40.9856
245.4560
54.5440Fail
Fail
Fail
Fail
45.4569-26.4389
54.5445-35.0059
345.2694
54.7306Fail
Fail
Fail
Fail
45.2715-22.4082
54.7325-31.7183
445.0564
54.9436Fail
Fail
Fail
Fail
45.0535-19.4734
54.9459-29.5699
544.7988
55.201244.0000
-17.820056.0000
-28.800044.7975
-16.903555.2036
-27.89276
44.484155.5159
44.0000-14.7300
56.0000-26.7200
44.4863-14.0924
55.5152-26.1252
744.0981
55.901944.0000
-12.120056.0000
-25.410044.0984
-11.122255.9030
-24.39258
43.584056.4160
44.0000-10.1800
56.0000-24.9200
43.5898-9.1925
56.4117-23.9287
106
50%
Load
TrueValue
DFT
PA
Number
ofbroken
bars
(1−
2s)f
(Hz)
(1+
2s)f
(Hz)
(1−
2s)f
(Hz)
Amplitude
(dB)
(1+
2s)f
(Hz)
Amplitude
(dB)
(1−
2s)f
(Hz)
Amplitude
(dB)
(1+
2s)f
(Hz)
Amplitude
(dB)
147.1622
52.8378
Fail
Fail
Fail
Fail
47.1628
-36.4707
52.8462
-40.4929
247.0590
52.9410
Fail
Fail
Fail
Fail
47.0556
-30.0562
52.9414
-34.8171
346.9409
53.0591
Fail
Fail
Fail
Fail
46.9382
-25.7854
53.0614
-30.9764
446.7990
53.2010
Fail
Fail
Fail
Fail
46.7984
-22.6637
53.2017
-28.5780
546.6380
53.3620
Fail
Fail
Fail
Fail
46.6317
-19.9379
53.3683
-26.5194
646.4375
53.5625
Fail
Fail
Fail
Fail
46.4356
-17.3182
53.5678
-24.7541
746.1967
53.8033
Fail
Fail
Fail
Fail
46.1946
-14.5572
53.8069
-22.9079
845.8868
54.1132
Fail
Fail
Fail
Fail
45.8847
-11.7460
54.0792
-22.7825
25%
Load
TrueValue
DFT
PA
Number
ofbroken
bars
(1−
2s)f
(Hz)
(1+
2s)f
(Hz)
(1−
2s)f
(Hz)
Amplitude
(dB)
(1+
2s)f
(Hz)
Amplitude
(dB)
(1−
2s)f
(Hz)
Amplitude
(dB)
(1+
2s)f
(Hz)
Amplitude
(dB)
148.6128
51.3872
Fail
Fail
Fail
Fail
48.6885
-43.8687
51.2795
-45.8390
248.5624
51.4376
Fail
Fail
Fail
Fail
48.5101
-39.1548
51.4898
-41.3964
348.5005
51.4995
Fail
Fail
Fail
Fail
48.4519
-34.2296
51.4848
-36.4170
448.4335
51.5665
Fail
Fail
Fail
Fail
48.4322
-30.7713
51.5702
-32.6599
548.3528
51.6472
Fail
Fail
Fail
Fail
48.3539
-28.0812
51.6434
-30.2020
648.2635
51.7365
Fail
Fail
Fail
Fail
48.2575
-25.4226
51.7414
-27.9801
748.1286
51.8714
Fail
Fail
Fail
Fail
48.1384
-22.7975
51.8603
-25.8499
847.9869
52.0131
Fail
Fail
Fail
Fail
47.9911
-20.1440
52.0073
-23.8403
107
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