Induction Machine Broken Rotor
Bar 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.
A
Typeset in L TEX 2ε
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.
2.4.
2.5.
Induction Machine Broken Rotor Bar Faults
. . . . . . . . . . . .
9
2.3.1.
Causes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.3.2.
Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10
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
. . . . . . . . . . . . . . .
18
Limitations and Possible Improvement
3. Model of an Induction Machine with Broken Rotor Bars
3.1.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
3.2.
Mathematical Model
20
3.3.
3.4.
. . . . . . . . . . . . . . . . . . . . . . . . .
3.2.1.
Mathematical Model of an Induction Machine
. . . . . . .
20
3.2.2.
Mathematical Model of Broken Rotor Bars . . . . . . . . .
28
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
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
3.4.1.
Initialization . . . . . . . . . . . . . . . . . . . . . . . . . .
38
3.4.2.
Simulation Results
38
Simulations
. . . . . . . . . . . . . . . . . . . . . .
4. High-Resolution Spectral Analysis
ii
19
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
CONTENTS
5. Implementation of Prony Analysis for Induction Motor Broken Bar
Detection
53
5.1.
5.2.
5.3.
5.4.
5.5.
5.6.
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
53
5.1.1.
Study Description . . . . . . . . . . . . . . . . . . . . . . .
54
Data Acquisition and Preprocessing . . . . . . . . . . . . . . . . .
55
5.2.1.
Sampling Frequency and Window Length . . . . . . . . . .
55
5.2.2.
Data Preprocessing . . . . . . . . . . . . . . . . . . . . . .
56
Prony Estimation and Prediction
. . . . . . . . . . . . . . . . . .
58
5.3.1.
Stator Current Modulation . . . . . . . . . . . . . . . . . .
58
5.3.2.
Fault Severity Assessment
. . . . . . . . . . . . . . . . . .
63
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
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
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
A.1. Simulation Initialization
93
. . . . . . . . . . . . . . . . . . . . . . .
A.1.1. Simulation Initialization File Startsim.m
93
. . . . . . . . .
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 stator 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 demonstrated 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 complicated 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
xii
C.1. Parameters of induction machine models used for simulations. . . . .
103
D.1. Frequency estimation results by PA and DFT for Machine1 with dierent number of broken rotor bars operating under dierent load conditions, using a data window of 500 samples and a sampling frequency
of 1000Hz. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
106
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 amplitudes 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 arbitrary 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
xiv
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. M AEf req 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. M AEf req 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
5.12. M AEf req 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 motor 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 prediction with the real current signal of a 2.2kW induction machine operating in full load with 4 broken rotor bars. . . . . . . . . . . . . .
81
5.16. DFT spectrum of the same signal data used in Figure 5.14 and Figure 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 combination 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 temperature, 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 signal. 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 processing 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 mathematician, 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. Moreover, 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 detectable via the stator current spectrum; (2) broken bar rotor faults can be modeled 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 technique.
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 induction 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 synchronous 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=
where
ωrm
ωsm − ωrm
ωsm
is the rotor rotating speed and
in mechanical radians.
ωe
ωsm =
(2.1)
2
ω is the synchronous speed
P e
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 laminations 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 motor 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
frequency.
2sf
around the supply
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 components 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 significantly 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 environment. 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]
where
k = 1, 2, 3 . . .
2k
P
(2.2)
1−s
= f 2k
±s
P
(2.3)
, and [7]
fbrb
where
fbrb = (1 ± 2ks) f
= 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
(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
of total rotor bars, only
due to a nite machine-load inertia value [20].
As the fault becomes severer, higher order harmonics will then arise. It is understandable 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 frequencies 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 applications. 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
Amplitude (dB)
−20
−40
−60
−80
−100
−120
0
50
100
150
Frequency (Hz)
200
250
200
250
200
250
(a) Healthy rotor.
0
Amplitude (dB)
−20
−40
−60
−80
−100
−120
0
50
100
150
Frequency (Hz)
(b) With 1 broken rotor bars
0
Amplitude (dB)
−20
−40
−60
−80
−100
−120
0
50
100
150
Frequency (Hz)
(c) With 3 broken rotor bars.
12
2.4. Detection of Broken Rotor Bar Faults
0
Amplitude (dB)
−20
−40
−60
−80
−100
−120
0
50
100
150
Frequency (Hz)
200
250
200
250
(d) With 5 broken rotor bars.
0
Amplitude (dB)
−20
−40
−60
−80
−100
−120
0
50
100
150
Frequency (Hz)
(e) With 8 broken rotor bars.
Figure 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 .
0
25% load
50% load
75% load
Full load
Amplitude (dB)
−20
−40
−60
−80
−100
−120
40
45
50
Frequency (Hz)
55
60
Figure 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
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 rotor 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
IBB
0.5
=
I
Nb
where
IBB
and
I
are the amplitudes of the
(1 − 2s) f
(2.4)
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 asymmetry 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
0.5
, it is considered that
Nb
0.5
there are broken rotor bars; if the ratio is smaller than
, it is considered that
Nb
sideband and the fundamental frequencies is higher than
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
IBB
I
(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
nbb
IBB
=
I
Nb
where
nbb
is the number of broken rotor bars.
(2.6)
When the machine load, and
15
Chapter 2. Broken Rotor Bar Faults in Induction Machines and Non-Intrusive Methods of Detection
0
(1−2s)f sideband amplitude (dB)
−5
−10
−15
1 broken bar
2 broken bars
3 broken bars
4 broken bars
5 broken bars
6 broken bars
7 broken bars
−20
−25
−30
−35
−40
−45
−50
25%
50%
75%
100%
Load conditions (percentage of full load)
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 and
load 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
sin α
IBB
=
I
P (2π − α)
where
α
(2.7)
is the electrical angle of a contiguous group of broken rotor bars, given
by
α=
and
P
πP nbb
Nb
(2.8)
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
Prediction 1
Prediction 2
Prediction 3
(1−2s)f sideband amplitude (dB)
−5
−10
−15
−20
−25
−30
−35
−40
−45
−50
0
1
2
Figure 2.4.: The prediction curves of
3
4
5
6
Number of broken rotor bars
(1 − 2s)f
7
8
9
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
nbb =
Nb = 32.
2Nb
(2.9)
D
10 20 + P
D is the amplitude dierence between the lower sideband frequency (1 − 2s) f
the supply frequency f in decibel.
where
and
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 signicantly 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 experimental studies. Firstly, commercial induction motors can be very expensive,
whereas using simulations on the available software is economical. Secondly, laboratory experiments need to be designed and set up, which requires the assistance
of a number of laboratory sta members.
can be avoided.
By using simulation this extra work
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 operations, 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]:
vsabc
20
=
rs iabc
s
dλabc
s
+
dt
(3.1)
3.2. Mathematical Model
vrabc
=
rr iabc
r
dλabc
r
+
dt
(3.2)
and
Notations
abc abc
abc abc
λabc
s = Lss is + Lsr ir
(3.3)
abc abc
abc abc
λabc
r = Lrr ir + Lrs is
(3.4)
abc
abc
abc
vsabc , iabc
s , λ s , v r , ir
and
λabc
r
are column vectors representing the
voltages, currents, and ux linkages of each phase in either stator or rotor, where
the subscripts
abc
s
and
r
indicate stator and rotor, respectively, and the superscript
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
where
rs
and
rr

rs,r 0
0


=  0 rs,r 0 
0
0 rs,r
denote the balanced equivalent resistance in each phase of a
healthy induction machine stator and rotor, respectively.
Notations
Labc
ss
and
Labc
rr
are matrices of the self inductance of the stator and the
rotor windings, respectively, while
Labc
sr
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:

Labc
ss

Lls + Lss
Lsm
Lsm


=  Lsm
Lls + Lss
Lsm

Lsm
Lsm
Lls + Lss

Labc
rr

Llr + Lrr
Lrm
Lrm


=
Lrm
Llr + Lrr
Lrm

Lrm
Lrm
Llr + Lrr
(3.5)
(3.6)
21
Chapter 3. Model of an Induction Machine with Broken Rotor Bars
where
Lls
is the per phase stator winding leakage inductance,
rotor winding leakage inductance,
Lrr
Lss
Lrm
is the per phase
is the self inductance of the stator winding,
is the self inductance of the rotor winding,
between the stator windings and
Llr
Lsm
is the mutual inductance
is the mutual inductance between the rotor
windings.
The stator-to-rotor mutual inductances are expressed as:

Labc
sr
where

2π
cos θr
cos θr + 2π
cos
θ
−
r
3
3
T


2π
2π
= Labc
=
L
cos
θ
cos
θ
+
cos
θ
−


sr
r
r
r
rs
3
3
2π
2π
cos θr + 3
cos θr − 3
cos θr
Lsr
and
Lrs
(3.7)
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
where
Pin
is the power input to the induction machine,
converted to mechanical work on the rotor shaft,
Pmm
Pem
Ploss
is the rate of energy
is the copper loss and
represents the rate of exchange of magnetic eld energy between windings.
The electromechanical torque is dened by the
mechanical angular speed
ωrm ,
Pem
term divided by the rotor
as
Tem =
22
(3.8)
Pem
.
ωrm
(3.9)
3.2. Mathematical Model
3.2.1.2. Park's Transformation
For convenience, mathematical transformations are often used to study the rotating 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
qd0
arbitrary rotating
abc
reference frame to an
reference frame [26]. The quadrature and direct axes are
ctitious quantities of the symmetrical three-phase induction motor.
tionship between
θ
3.1, where
abc
qd0
and arbitrary
The rela-
reference frames is illustrated in Figure
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
ˆ
t
θ (t) =
ω (ζ) dζ + θ (0)
(3.10)
0
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 ),
ˆ
where the rotor angle may be expressed as
t
ωr (ζ) dζ + θr (0)
θr (t) =
(3.11)
0
θ (0) and θr (0) stand for the initial angular values of the transformation
rotor angle, respectively, at the time t = 0.
The angles
angle and
The transformation function for stator variables may be written as
fqd0 = Tqd0 (θ) fabc
(3.12)
fqd0 and fabc can be the phase voltages,
currents, or ux linkages of the machine, and Tqd0 is the qd0 transformation matrix
where the elements of the column vectors
with the form [15]


2π
cos θ cos θ − 2π
cos
θ
+
3
3

2
2π
Tqd0 (θ) =  sin θ sin θ − 2π
sin
θ
+

3
3
3
1
1
1
2
2
(3.13)
2
23
Chapter 3. Model of an Induction Machine with Broken Rotor Bars
Figure 3.1.: Relationship between
It should be noted that
f0
abc
and arbitrary
qd0
reference frames.
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
The arbitrary
qd0

cos θ
sin θ
1


2π
=  cos θ − 2π
sin
θ
−
1 
3
3
cos θ + 2π
sin θ + 2π
1
3
3
(3.15)
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 reference 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 rotating 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
abc
the
reference frame to the arbitrary
functions Eq.
(3.12) and Eq.
dq0
reference frame, the transformation
(3.14) are applied to the voltages, currents and
resistances in Eq. (3.1) and Eq. (3.2). This yields [15]
vsqd0
vrqd0
=
Tqd0 (θ) rs T−1
qd0
(θ) iqd0
s
qd0
d T−1
qd0 (θ) λs
+ Tqd0 (θ)
dt
(3.16)
−1
qd0
d
T
(θ
−
θ
)
λ
r
r
qd0
qd0
= Tqd0 (θ − θr ) rr T−1
+ Tqd0 (θ − θr )
qd0 (θ − θr ) ir
dt
(3.17)
Substituting Eq.
(3.13) and Eq.
(3.15) into Eq.
(3.16) and Eq.
(3.17), and
rearranging the equations, produces
dλqd0
s
dt
(3.18)
dλqd0
r
+
dt
(3.19)
qd0
qd0
vsqd0 = rqd0
s is + E s +
vrqd0
=
qd0
rqd0
r ir
+
Eqd0
r
where

Eqd0
s

0 1 0


= ω  −1 0 0  λqd0
s ,
0 0 0
ω=
dθ
,
dt

Eqd0
r

0 1 0


= (ω − ωr )  −1 0 0  λqd0
r ,
0 0 0
ωr =
d (θr )
,
dt
and

rqd0
s
The
ir

1 0 0


= rs  0 1 0  ,
0 0 1

rqd0
r

1 0 0


= rr  0 1 0  .
0 0 1
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
dλ
terms are the rate of exchange of magnetic eld between windings.
dt
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 currents in Eq. (3.3) and Eq. (3.4), yield
abc
abc abc
= Tqd0 (θ) Labc
λqd0
s
ss is + Lsr ir
abc abc
abc
= Tqd0 (θ − θr ) Labc
λqd0
rr ir + Lrs is
r
(3.20)
(3.21)
Rearrange the equations and we nally obtain











λqs
λds
λ0s
λ0qr
λ0dr
λ00r


 
 
 
 
 
=
 
 
 
 
Lls + Lm
0
0
Lm
0
0
0
Lls + Lm 0
0
Lm
0
0
0
Lls
0
0
0
0
Lm
0
0 Llr + Lm
0
0
0
Lm
0
0
L0lr + Lm 0
0
0
0
0
0
L0lr

iqs
ids
i0s
i0qr
i0dr
i00r





















(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.
parameter
Lm ,
The
which is the magnetizing inductance on the stator side, is given
by equation
3
3 Ns
3 Ns
Lm = Lss =
Lsr =
Lrr
2
2 Nr
2 Nr
where
Eq.
Ns
and
Nr
(3.23)
are the numbers of coil in stator and rotor, respectively.
(3.22) is then substituted back into Eq.
(3.18) and Eq.
the entire machine voltage equations in the arbitrary
qd0
(3.19) to form
reference frame. The
equivalent circuit representation of an induction machine in the arbitrary reference
frame is shown in Figure 3.2.
xls , x0lr
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
The torque equation can be transformed into the arbitrary
ω
qd0 reference
and
ωr .
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 arbitrary
qd0
reference frame.
27
Chapter 3. Model of an Induction Machine with Broken Rotor Bars
0 0
0 0
0 0
Pin = vas ias + vbs ibs + vcs ics + var
iar + vbr
ibr + vcr
icr
(3.24)
By applying Park's transformation to Eq. (3.24), yield
Pin =
3
0 0
0 0
0 0
vqs iqs + vds ids + 2v0s i0s + vqr
iqr + vdr
idr + 2v0r
i0r
2
(3.25)
qd0
reference
Thus, the equation for electromechanical torque in the arbitrary
frame is
Tem =
where
P
3 P ω (λds iqs − λqs ids ) + (ω − ωr ) λ0dr i0qr − λ0qr i0dr
2 ωrm
(3.26)
is reminded as the number of machine poles. The rotor mechanical and
electrical rotating speeds
ωrm
ωr
and
have the relationship
ωrm =
2
ωr
P
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 =
3P
(λds iqs − λqs ids )
22
(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 = ωb λ
where
ωb
and
x = ωb L
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


(rr + ∆rra )
0
0


r?r = 
0
(rr + ∆rrb )
0

0
0
(rr + ∆rrc )
where
∆rra , ∆rrb
and
∆rrc
(3.28)
represent rotor resistance changes in phase a, b and c,
respectively, due to broken bar faults, dened as [11]
∆rra,b,c =
where
nbb
and
Nb
3nbb
rr
Nb − 3nbb
(3.29)
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]
"
#
2
(2Ns )2
rr =
rb +
2 r e
Nb /3
Nb 2 sin α2
where
Ns
rb
and
re
represent the rotor bar and end-ring resistances, respectively, and
is the equivalent stator winding turns.
neglected,
rr
As in the assumptions, when
re
is
then simplies to
(2Ns )2
rr ≈
rb
Nb /3
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
(2Ns )2
≈
rb
Nb /3 − nbb
rr?
and the increment
∆r
is obtained as
∆r = rr? − rr =
3nbb
rr
Nb − 3nbb
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?qd0
r

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 )
√
+
3
6
(∆rrb − ∆rrc ) sin (2θr )
r12 = − 61 (2∆rra − ∆rrb − ∆rrc ) sin (2θr) +
r13 = 31 (2∆rra − ∆rrb − ∆rrc ) cos (θr ) −
√
3
6
(∆rrb − ∆rrc ) cos (2θr )
√
3
3
(∆rrb − ∆rrc ) sin (θr )
r21 = r12
r22 = 13 (∆rra + ∆rrb + ∆rrc ) − 61 (2∆rra − ∆rrb − ∆rrc ) cos (2θr )
√
+
3
6
(∆rrb − ∆rrc ) sin (2θr )
r23 = − 31 (2∆rra − ∆rrb − ∆rrc ) sin (θr ) −
√
3
3
(∆rrb − ∆rrc ) cos (θr )
r31 = 21 r13
r32 = 21 r23
r33 = 31 (∆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 equations 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 manipulated 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. Additionally, it is helpful to write integral equations with the dependent-state variables
expressed as self-referencing integral functions of independent and dependent variables [15]. Using these suggested approaches a model can be more visually comprehensive 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

Eqd0
s

0 1 0


= ω  −1 0 0  λqd0
s
0 0 0
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 referred to stator, these equations can be written as
qd0
vsqd0 = rqd0
s is +
1 dψ qd0
s
ωb dt

vr0qd0

0 1 0
ωr 
1 dψ 0qd0
 0qd0
r
0qd0 0qd0
= rr ir −
 −1 0 0  ψ r +
ωb
ωb dt
0 0 0
(3.31)
(3.32)
31
Chapter 3. Model of an Induction Machine with Broken Rotor Bars











ψqs
ψds
ψ0s
0
ψqr
0
ψdr
0
ψ0r


 
 
 
 
 
=
 
 
 
 
xls + xm
0
0
xm
0
0
0
xls + xm 0
0
xm
0
0
0
xls
0
0
0
0
xm
0
0 xlr + xm
0
0
0
xm
0
0
x0lr + xm 0
0
0
0
0
0
x0lr











iqs
ids
i0s
i0qr
i0dr
i00r











(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
ψds
i0s
0
ψqr
0
ψdr
i00r
ˆ rs
= ωb
vqs +
(ψmq − ψqs ) dt
xls
ˆ rs
(ψmd − ψds ) dt
= ωb
vds +
xls
ˆ
ωb
=
{v0s − i0s rs } dt
xls
ˆ rr0
ωr 0
0
= ωb
+ ψdr + 0 ψmq − ψqr dt
ωb
xlr
ˆ rr0
ωr 0
0
0
= ωb
vdr − ψqr + 0 (ψmd − ψdr ) dt
ωb
xlr
ˆ
ωb
0
= 0
{v0r
− i00r rr0 } dt
xlr
0
vqr
ψmq = xm iqs + i0qr
(3.35)
ψmd = xm (ids + i0dr )
32
(3.34)
(3.36)
3.3. Model in Matlab/Simulink
ψqs − ψmq
xls
ψds − ψmd
ids =
xls
0
0
ψqr − ψmq
i0qr =
x0lr
ψ0 − ψ0
i0dr = dr 0 md
xlr
ψqs = xls iqs + ψmq
iqs =
ψds = xls ids + ψmd
0
ψqr
= x0lr i0qr + ψmq
0
= x0lr i0dr + ψmd
ψdr
(3.37)
where
0 ψqs ψqr
ψmq = xM
+ 0
xls
xlr
0
ψds ψdr
ψmd = xM
+ 0
xls
xlr
(3.38)
1
1
1
1
=
+
+ 0
xM
xm xls xlr
(3.39)
and
It should be mentioned that for a squirrel cage induction machine, the rotor voltages
0
0
, vdr
vqr
and
0
v0r
in the
qd0
reference frame are equal to zero [28].
The rotor motion in terms of mechanical speed is calculated by
Tem = J
where
Tem
dωrm
+ Tload + Tdamp
dt
is the electromechanical torque in Eq. (3.27),
(3.40)
Tload
is the mechanical
Tdamp is the damping torque in the direction opposite to
dωrm
the rotor rotation, and J
is the inertia torque to the accelerating torque. J
dt
2
denotes the rotor inertia in kg · m .
torque applied by load,
3.3.3. Simulink Model in Block Diagrams
In Simulink, modules of the dynamic system of an induction machine are constituted 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 conveniently traced and saved by using the Scope and the To Workspace blocks,
respectively. Some other Simulink blocks used in the model include, but not limited in, are the Clock, Sum, Gain, Mux, Integrator and Trigonometric
Function blocks.
abc − qd0 converinduction motor qd0 model
The induction motor model contains four major modules: the
sion module, the unit vector calculation module, the
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.
module has rotor angular speed as its input, and
as outputs. The angle
θr
The rotor angle calculation
sin θr , sin 2θr , cos θr
and
is calculated directly by using Eq. (3.11), and is used
as inputs to the transformation functions Eq. (3.12). The unit vectors
cos θr
are obtained by taking the sine and cosine of
stator and rotor variables in the
34
cos 2θr
qd0
sequences.
sin θr
and
θr , and are used for calculating
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
q -axis and zero sequence blocks
The d-axis block is similar to the q -axis block. The rotor
rotor speed using Eq. (3.40). The structure of the
are shown in Figure 3.5.
module block is shown in Figure 3.6.
3.3.3.4.
qd0 − abc
Conversion Module
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
This module performs the reverse operation to the
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
140
120
100
Torque (Nm)
80
60
40
20
0
−20
−40
0
0.2
0.4
0.6
0.8
1
Time (s)
1.2
1.4
1.6
1.8
2
Figure 3.8.: Simulated output torque curve.
1.5
Rotor speed in per unit
Measured data
Simulated data
1
0.5
0
0
0.2
0.4
0.6
0.8
1
Time (sec)
1.2
1.4
1.6
1.8
2
Figure 3.9.: Comparison of the simulated and measured rotor speed curves.
39
Chapter 3. Model of an Induction Machine with Broken Rotor Bars
80
Measured data
Simulated data
60
Stator current (A)
40
20
0
−20
−40
−60
−80
0
0.2
0.4
0.6
0.8
1
Time (sec)
1.2
1.4
1.6
1.8
2
(a)
20
Measured data
Simulated data
15
Stator current (A)
10
5
0
−5
−10
−15
−20
1
1.05
1.1
1.15
Time (sec)
1.2
1.25
(b) A zoomed-in view.
Figure 3.10.: Comparison of the simulated and measured stator currents.
40
1.3
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 =
where
∆f
1
1
fs
=
=
N
N Ts
T
fs is the sampling frequency, N is
window, Ts is the sampling interval and T
denotes the frequency dierence,
the number of data points within the
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 machine 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 resolution 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 fundamental 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 impossible.
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 sampled data as a linear combination of damped exponential functions. The typical
application of Prony Analysis is the parametric analysis of transient signals initiated 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 values than DFT using the same length data window.
•
Prony Analysis does not have the problem of the spectral leakage phenomenon.
•
Prony Analysis can compute the amplitudes, frequencies, phases and damping 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
N
has
complex samples
sum of
q
x[1], . . . , x[N ],
x [n]
the Prony method will t the data with a
complex exponential functions
x̂ [n] =
q
X
Ak exp [(αk + j2πfk ) (n − 1) Ts + jϕk ]
(4.1)
k=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
fk
is the sinusoidal frequency in
ϕk
is the initial phase in
sec−1 ,
Hz ,
radians
The objective is to estimate the frequencies
Ak
and phases
ϕk .
and
fk ,
damping factors
αk ,
amplitudes
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
pear in complex conjugate pairs.
Thus the
q
exp (αk + j2πfk )
must ap-
is always assumed to be even for
convenience. Then, Eq. (4.1) can be expressed in the form of
x̂ [n] =
q
X
hk zkn−1
k=1
where
hk
and
zk
are complex parameters dened as
hk = Ak exp (jϕk )
and
zk = exp [(αk + j2πfk ) Ts ]
44
(4.2)
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
X
| [n]|2
n=1
where
[n] = x [n] − x̂ [n] = x [n] −
q
X
hk zkn−1
k=1
representing the complex error between the original data samples
linear approximation
ρ
x̂ [n].
For a real signal
x [n],
x [n]
and the
minimizing the squared error
is obtained by setting the derivatives with respect to
hk
and
zk
to zero. This
yields:
∂ρ
= c1 + c2 hk = 0
∂hk
.
∂ρ
= c3 + c4 hk = 0
∂zk
(4.3)
The minimization problem is with respect to parameters
ously.
The coecients
c1,2,3,4
in Eq.
hk , zk
and
p
simultane-
(4.3) involve sums of exponentials
zk .
To
solve for the coecients , it yields
c1 c4 = c2 c3 ,
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
Y
(z − zk ) =
q
X
am z p−m
(4.4)
m=0
k=1
The linear dierence equation whose homogeneous solution is given by Eq. (4.4)
is
q
X
am x [n − m] = 0
(4.5)
m=0
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
q + 1, . . . , 2q .
The coecients
am
n=
form a linear predictive relationship among the
available samples and the relationship can be then expressed as the
q×q
Toeplitz
structure matrix equation


x [q]
x [q − 1] · · · x [1]
a1


 x [q + 1]
x [q]
· · · x [2]   a2

 .
.
.
.
..

 .
.
.
.
.
.
.
.

 .
x [2q − 1] x [2q − 2] · · · x [q]
aq
By solving the matrix equation, the
am



x [q + 1]




 x [q + 2] 
 = −

.



.
.



x [2q]
(4.6)
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
zk
αk
and the sinusoidal frequency
fk
can be determined from roots
by using the relationships
αk = ln |zk | /Ts
(4.7)
fk = tan−1 [Im {zk } /Re {zi }] /2πTs
(4.8)
and
hk
from hk
Finally, these roots are used to obtain the complex parameter
The amplitudes
relationships
46
Ak
and initial phases
ϕk
are determined
in Eq.
(4.2).
by using the
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
Step 2
am .
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
X
hk zkn−1 + [n]
(4.11)
k=1
where
[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
q
even time intervals by
2q
observations sampled at
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
[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
determined approach,
N
[n]
over data
data points, where
N > 2q ,
1 6 n 6 N.
In an over-
are utilized to compose the
linear prediction equations. Thus, the linear dierence equation Eq. (4.5) should
be modied to [14]:
q
X
am (x [n − m] + [n − m]) = 0
(4.12)
m=0
LS Prony method ignores the past noise values, then the Eq. (4.12) becomes
q
X
am x [n − m] = e [n]
(4.13)
m=0
which actually denes a forward linear prediction error equation. Thus each
am
terms a linear prediction parameter and is selected to minimize the linear prediction total squared error
PN
n=q+1
|e [n]|2 .
The minimization can be done by using
the covariance method, or alternatively, by using the SVD for ecient computation 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
H Z Z h = ZH x
(4.14)
can be yielded from Eq. (4.11). The equation components, which are the
matrix
48
Z,
the
q×1
vector
h,
and the
N ×1
data vector
x
are dened as
N ×q
4.5. Iterative Prony Method



Z=


1
z1
1
z2
···
···
1
zp
.
.
.
.
.
.
..
.
.
.
.
z1N −1 z2N −1 · · · zpN −1



,




h1


 h2 

h= . 
,
 .. 
h3


x [1]


 x [2] 

x= . 

 .. 
x [N ]
and the superscript
H
ZH Z
Hermitian matrix. The four parameters of Prony model can
forms a
q×q
means the matrix complex conjugate transposition, so that
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 suboptimum 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 developed 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
min T W (a)
(4.15)
a, where the superscript T indicates matrix transpose, and W (a)
covariance matrix for the errors [n] at each data point in the rst step of
with respect to
is the
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 equation Eq.
(4.5), when substitute the error-free data with real signal data, the
dierence equation Eq.
spanning
N
(4.12) is satised for each sample in the data window
samples. It may be expressed in the matrix notation [33]
49
Chapter 4. High-Resolution Spectral Analysis
Xa + b + D (a) = 0
(4.16)
where
aT =
h
i
a1 a2 . . . aq
q×1
is the
column vectors of dierence equation coef-
cients,
T
=
h
i
[1] [2] . . . [N ]
is the
N ×1
column vectors of error components
on each sample,
h
bT = x [q + 1]

x [q]

 x [q + 1]
X=
.

.
.

x [N − 1]

aq aq−1

 0 aq
D=
..
 ..
.
 .
0 ···
x [q + 2] . . . x [N ]
i
is the
x [q − 1]
x [q]
···
···
x [1]
x [2]
.
.
.
..
.
.
.
.
(N − q)×1 column vectors of data,






is the
(N − q)×q data matrix, and
x [N − 2] · · · x [N − q]
· · · a1
aq−1 · · ·
..
.
0
..
.
aq
1
a1
···
aq−1

··· 0

··· 0 

. 
..
..
.
.
.
. 
· · · a1 1
0
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
is expressed by rearranging Eq. (4.16), yielding
= −D (a)+ (Xa + b)
(4.17)
where + indicates the matrix pseudoinversion. Thus, the optimal estimates of
am
the dierence coecients
correspond to the minimum of the error norm
T .
This is the nonlinear problem addressed by Eq. (4.15) and can be formulated in
terms of IRLS, that is,
a
minimizes
n
o
min (Xa + b)T W (a) (Xa + b)
a
where
h
i−1
W (a) = D (a)D (a)T
is a real symmetric positive denite weighting
matrix. The least squares solution is then returned to the linear system
0
with covariance matrix proportional to
Eq. (4.16).
50
(4.18)
W (a),
Xa + b =
subject to the relation given by
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 described 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 spectrum, 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 usually 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
28
Machine 2
5.5
4
32
Machine 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 machine power on the broken rotor bar sideband frequencies.
The relevant
parameters of the induction machine models utilized for study in the chapter 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 discrete 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
of samples
N
Lw ,
sampling frequency
fs
and the number
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
Hz. In the digital domain,
2
there is no way to distinguish these aliasing frequencies from the frequencies that
actually lie between 0 and
fs
Hz. Therefore, these aliasing frequencies need to be
2
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 procedures 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 attenuate the noise and undesired frequency components and to separate the broken
rotor bar sideband frequency components in the spectrum. By doing so, the performance 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
20
Magnitude (dB)
0
−20
−40
−60
−80
−100
0
20
40
60
Frequency (Hz)
80
100
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 rooting of high-order polynomials. Both of them are computational intensive operations. 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 demand 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 current 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 currents 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 prediction 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
20
20
Current Signal
Prony Estimation
10
Magnitude (A)
Magnitude (A)
10
Current Signal
Prony Prediction
0
−10
0
−10
−20
0
0.1
0.2
0.3
Time (sec)
0.4
−20
0.5
0.5
0.6
0.7
0.8
Time (sec)
0.9
1
(a) PA estimation of healthy stator cur-(b) PA prediction for the period into the
rent.
future.
20
20
Current Signal
Prony Estimation
10
Magnitude (A)
Magnitude (A)
10
0
−10
−20
0
Current Signal
Prony Prediction
0
−10
0.1
0.2
0.3
Time (sec)
0.4
−20
0.5
0.5
0.6
0.7
0.8
Time (sec)
0.9
1
(c) PA estimation of 1 broken rotor bar sta-(d) PA prediction for the period into the
tor current.
future.
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
20
20
Current Signal
Prony Estimation
10
Magnitude (A)
Magnitude (A)
10
0
−10
−20
0
Current Signal
Prony Prediction
0
−10
0.1
0.2
0.3
Time (sec)
0.4
0.5
−20
0.5
0.6
0.7
0.8
Time (sec)
0.9
1
(e) PA estimation of 3 broken rotor bars(f ) PA prediction for the period into the
stator current.
future.
59
Chapter 5. Implementation of Prony Analysis for Induction Motor Broken Bar Detection
20
20
Current Signal
Prony Estimation
10
Magnitude (A)
Magnitude (A)
10
0
−10
−20
0
Current Signal
Prony Prediction
0
−10
0.1
0.2
0.3
Time (sec)
0.4
−20
0.5
0.5
0.6
0.7
0.8
Time (sec)
0.9
1
(g) PA estimation of 5 broken rotor bars(h) PA prediction for the period into the
stator current.
future.
20
20
Current Signal
Prony Estimation
10
Magnitude (A)
Magnitude (A)
10
0
−10
−20
0
Current Signal
Prony Prediction
0
−10
0.1
0.2
0.3
Time (sec)
0.4
0.5
−20
0.5
0.6
0.7
0.8
Time (sec)
0.9
1
(i) PA estimation of 7 broken rotor bars(j) PA prediction for the period into the
stator current.
future.
20
20
Current Signal
Prony Estimation
10
Magnitude (A)
Magnitude (A)
10
0
−10
−20
0
Current Signal
Prony Prediction
0
−10
0.1
0.2
0.3
Time (sec)
0.4
0.5
−20
0.5
0.6
0.7
0.8
Time (sec)
0.9
1
(k) PA estimation of 8 broken rotor bars(l) PA prediction for the period into the
stator current.
future.
Figure 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. Prony Estimation and Prediction
20
20
Current Signal
Prony Estimation
10
Magnitude (A)
Magnitude (A)
10
Current Signal
Prony Prediction
0
−10
0
−10
−20
0.05
0.1
Time (sec)
0.15
−20
0.55
0.6
Time (sec)
0.65
(a) Zoomed-in view of PA estimation of 1(b) Zoomed-in view of PA prediction for
broken rotor bar stator current.
the period into the future.
Figure 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.
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
PN
M AEf itting =
where
[n]
| [n] |
N
n=1
(5.2)
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 frequency 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
PA
Fundame
load condition using a data window of 500 samples with the sampling frequency of 1000Hz.
True Value
(1 + 2s) f
N/A
(Hz)
56.2802
56.0568
N/A
(Hz)
43.1537
43.4589
43.7195
43.9432
N/A
(Hz)
1.7278
1.1840
0.8468
0.6129
0.3946
0.1854
N/A
59.0370
58.2446
57.6647
57.2150
56.8487
56.5450
56.2852
56.0410
N/A
0.2866
0.2828
0.2643
0.2085
0.1669
0.1346
0.0941
0.0482
N/A
50.0001
50.0000
49.9999
49.9999
50.0000
50.0000
50.0000
50.0000
50.0000
7.2384
7.0388
6.9352
6.8689
6.8239
6.8105
6.7990
6.7937
6.7916
0.0024
0.0039
0.0027
0.0029
0.0058
0.0045
0.0058
0.0052
0.0012
0.0100
0.0164
0.0093
0.0124
0.0309
0.0232
0.0260
0.0219
0.0208
Amplitude
43.9432
56.5407
42.7867
2.1860
(1 − 2s) f (1 + 2s) f (1 − 2s) f
Estimation Prediction
Table 5.2.: Numerical PA result of the stator current of Machine 2 with various broken rotor bar numbers operating under full
Number
of
0
43.7198
56.8470
42.3341
2.7316
M EAf itting M EAf itting
1
43.4593
57.2125
41.7558
Amplitude
2
43.1530
57.6672
40.9622
-ntal
3
42.7875
58.2415
Amplitude
4
42.3328
59.0334
Frequency
5
41.7585
(Hz)
6
40.9666
broken
7
bars
8
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 frequencies 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
−5
−10
Magnitude (A)
−15
−20
−25
−30
−35
−40
PA estimation values
Prediction 1
Prediction 2
Prediction 3
−45
−50
0
1
2
3
4
5
Number of broken rotor bars
6
7
8
(a) Machie 1: 2.2kW, 4 poles and 28 total rotor bars.
0
−5
−10
Magnitude (A)
−15
−20
−25
−30
−35
−40
PA estimation values
Prediction 1
Prediction 2
Prediction 3
−45
−50
0
1
2
3
4
5
6
Number of broken rotor bars
7
8
9
(b) Machine 2: 5.5kW, 4 poles and 32 total rotor bars.
0
−5
−10
Magnitude (A)
−15
−20
−25
−30
−35
−40
PA estimation values
Prediction 1
Prediction 2
Prediction 3
−45
−50
0
2
4
6
8
Number of broken rotor bars
10
12
14
(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 with
respect to the number of broken rotor bars in Machine 1, 2 and 3
respectively. The motors are operating under full load.
64
5.4. 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
Fundamental Frequency: 50Hz
(1−2s)f sideband frequency: 43.8Hz
−20
Amplitude (dB)
−30
(1+2s)f sideband frequency: 56.2Hz
−40
−50
−60
−70
−80
−90
−100
0
10
20
30
40
50
60
Frequency (Hz)
70
80
90
100
Figure 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.
0
Fundamental Frequency: 50Hz
−10
(1−2s)f sideband frequency: 44Hz
−20
Amplitude (dB)
−30
(1+2s)f sideband frequency: 56Hz
−40
−50
−60
−70
−80
−90
−100
0
10
20
30
40
50
60
Frequency (Hz)
70
80
90
100
Figure 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
5.4. Disadvantages of DFT and Solutions by Prony Analysis
0
−10
−20
Amplitude (dB)
−30
−40
−50
−60
−70
−80
−90
−100
0
10
20
30
40
50
60
Frequency (Hz)
70
80
90
100
Figure 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.
0
−10
−20
Amplitude (dB)
−30
−40
−50
−60
−70
−80
−90
−100
0
10
20
30
40
50
60
Frequency (Hz)
70
80
90
100
Figure 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
Chapter 5. Implementation of Prony Analysis for Induction Motor Broken Bar Detection
UE =
where
fest
|fest − ftrue | × 100
%
ftrue
denotes the estimated frequency value whilst
(5.3)
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 reveal 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 required 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 the
minimum window lengths with 1000Hz sampling frequency for Machine
2 operating under full load condition and with various numbers of
broken rotor bars.
Minimum
Full load
(1 − 2s)f
(1 + 2s)f
sideband
component (Hz)
sideband
dow
component (Hz)
win-
length
(Number
of
samples)
Number
of broken
rotor bars
True
Value
DFT
PA
True
Value
DFT
PA
DFT
PA
1
43.9431 44.0000 43.9653 56.0569 56.0000 56.0568
1000
40
2
43.7197 43.7500 43.7129 56.2803 56.2500 56.1944
800
40
3
43.4598 43.3300 43.4606 56.5402 56.6700 56.5695
900
40
4
43.1556 43.0800 43.1550 56.8444 56.9200 56.8299
1300
40
5
42.7849 42.8600 42.7812 57.2151 57.1400 57.2452
900
40
6
42.3318 42.2200 42.3331 57.6682 57.7800 57.6651
900
40
7
41.7562 41.8200 41.7576 58.2438 58.1800 58.2510
1100
40
8
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 the
minimum window lengths with 1000Hz sampling frequency for Machine
2 operating under 75% load condition and with various numbers of
broken rotor bars.
Minimum
75% of
full load
(1 − 2s)f
(1 + 2s)f
sideband
component (Hz)
sideband
dow
component (Hz)
(Number
win-
length
of
samples)
Number
of broken
rotor bars
True
Value
DFT
PA
True
Value
DFT
PA
DFT
PA
1
45.6169 45.7100 45.7327 54.3831 54.2900 54.3361
1400
120
2
45.4564 45.4500 45.4251 54.5436 54.5500 54.4233
1100
100
3
45.2692 45.3800 45.1949 54.7308 54.6200 54.5766
1300
100
4
45.0567 45.0000 45.0953 54.9433 55.0000 55.0240
1000
100
5
44.7970 44.6700 44.8032 55.2030 55.3300 55.2007
1500
100
6
44.4837 44.5500 44.3839 55.5163 55.4500 55.4848
1100
80
7
44.0983 44.1700 44.1251 55.9017 55.8300 55.9781
1200
40
8
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 the
minimum window lengths with 1000Hz sampling frequency for Machine
2 operating under 50% load condition and with various numbers of
broken rotor bars.
Minimum
50% of
full load
(1 − 2s)f
(1 + 2s)f
sideband
component (Hz)
sideband
dow
component (Hz)
win-
length
(Number
of
samples)
Number
of broken
rotor bars
True
Value
DFT
PA
True
Value
DFT
PA
DFT
PA
1
47.1621 47.1400 47.2649 52.8379 52.8600 52.9784
1400
260
2
47.0576 46.9200 46.9146 52.9424 53.0800 52.9900
1300
280
3
46.9376 46.9200 46.9644 53.0624 53.0800 53.1240
1300
280
4
46.7941 46.6700 46.8485 53.2059 53.3300 53.2321
1200
240
5
46.6348 46.6700 46.5908 53.3652 53.3300 53.3897
1200
220
6
46.4431 46.3600 46.5076 53.5569 53.6400 53.4716
1100
210
7
46.2033 46.1500 46.1479 53.7967 53.8500 53.8513
1300
200
8
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 the
minimum window lengths with 1000Hz sampling frequency for Machine
2 operating under 25% load condition and with various numbers of
broken rotor bars.
Minimum
25% of
full load
(1 − 2s)f
(1 + 2s)f
sideband
component (Hz)
sideband
dow
component (Hz)
(Number
win-
length
of
samples)
Number
of broken
rotor bars
70
True
Value
DFT
PA
True
Value
DFT
PA
DFT
PA
1
48.6136 48.5200 48.5329 51.3864 51.4800 51.3717
2700
500
2
48.5609 48.4600 48.5544 51.4391 51.5400 51.5580
2600
460
3
48.5008 48.4000 48.4907 51.4992 51.6000 51.4398
2500
300
4
48.4356 48.3300 48.5897 51.5644 51.6700 51.5490
2400
250
5
48.3477 48.2600 48.2545 51.6523 51.7400 51.6249
2300
250
6
48.2655 48.1800 48.2685 51.7345 51.8200 51.7513
2200
230
7
48.1305 48.1000 48.1462 51.8695 51.9000 51.8691
2100
230
8
47.9873 47.8900 48.0157 52.0127 52.1100 51.9144
1900
220
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 window 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 minimum 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 accuracy 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
Window length (number of samples)
3000
2500
DFT
2000
PA
1500
1000
rs
0 a
1 b
2 or
3 rot
4
5 ken
6
7 bro
8
f
ro
be
um
500
0
Full load
75%
50%
Load condition (percentage of full
25%
load)
N
Figure 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.
PNt
M AEf req =
where
Nt
U Ent
100Nt
nt =1
(5.4)
is the number of independent trials.
Figure 5.10 and Figure 5.11 shows the accuracy of frequency estimation in terms of
M AEf req
of using both Prony Analysis and DFT. The
M AEf req
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
M AEf req
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
0
10
MAE of PA result for (1−2s)f sideband
MAE of PA result for (1+2s)f sideband
MAE of DFT result for (1−2s)f sideband
MAE of DFT result for (1+2s)f sideband
Frequency estimator error (MAE)
−1
10
DFT for (1−2s)f sideband
−2
10
−3
10
DFT for (1+2s)f sideband
PA for (1−2s)f sideband
−4
10
PA for (1+2s)f sideband
500
Figure 5.10.:
M AEf req
1000
1500
2000
2500
Window length (number of samples)
3000
3500
in frequency estimation by PA and DFT in respect of win-
dow length using 1000Hz sampling frequency for simulated current
data of Machine 2 operating under full load.
0
10
MAE of PA result for (1−2s)f sideband
MAE of PA result for (1+2s)f sideband
MAE of DFT result for (1−2s)f sideband
MAE of DFT result for (1+2s)f sideband
Frequency estimator error (MAE)
−1
10
DFT for (1−2s)f sideband
−2
10
−3
10
PA for (1−2s)f sideband
−4
10
500
Figure 5.11.:
DFT for (1+2s)f sideband
M AEf req
1000
PA for (1+2s)f sideband
1500
2000
2500
Window length (number of samples)
3000
3500
in frequency estimation by PA and DFT in respect of win-
dow length using 1000Hz sampling frequency for simulated current
data 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 and
DFT for broken rotor bar detection on Machine 2, under dierent
light load conditions and with one broken rotor bar.
Minimum
Load
condition
(1 − 2s)f
(1 + 2s)f
sideband
component (Hz)
sideband
dow
component (Hz)
win-
length
(Number
of
samples)
(% of full
True
load)
value
DFT
PA
True
value
DFT
PA
DFT
PA
100
20%
48.8954 49.0000 48.5888 51.1046 51.0000 51.6599
3000
15%
49.1742 49.0700 49.1661 50.8258 50.9300 51.6534
4000
200
10%
49.4525 49.500
6000
1000
49.3222 50.5475 50.5000 50.4061
5%
49.7277 49.7500 49.4312 50.2723 50.2500 50.1761
12000
2000
1%
49.9459 49.95
35000
2000
49.7089 50.0541 50.0500 50.8899
Analysis is capable of estimating the sideband frequency components more accurately than DFT. It is also observed that the
M AEf req
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 broken 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 sideband 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 frequencies. 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, respectively, due to the extremely light load.
The frequency dierence between the
nearest high peak is only 0.05Hz. These would make the two frequency components 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
M AEf req
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
M AEf req
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
M AEf req
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
M AEf itting
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
10
(1−2s)f sideband
(1+2s)f sideband
Frequency estimator error (MAE)
−1
10
−2
10
−3
10
−4
10
0
Figure 5.12.:
50
100
M AEf req
150
200
250
300
350
Window length (number of samples)
400
450
500
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.
0
10
−1
Fitting error (MAE)
10
−2
10
−3
10
−4
10
0
0.02
0.04
0.06
0.08
Measurement error (standard deviation)
0.1
0.12
Figure 5.13.: Estimation mean absolute error as a function of measurement error
standard 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. The
part of signal is 0.0014 and the
M AEf itting
of the estimation
M AEf itting 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
−10
−20
Amplitude (dB)
−30
43.3 Hz
56.7 Hz
−40
−50
−60
−70
−80
−90
−100
0
20
40
60
80
100
120
Frequency (Hz)
140
160
180
200
Figure 5.14.: Spectrum of the measured stator current of a 2.2 kW induction motor
with 4 broken rotor bars using DFT with a sampling frequency of
400Hz, and a data window of 4000 samples.
Table 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.
PA
True
value
sideband
Fundamental frequency
(1 + 2s) f
80
sideband
Amplitude
Value
Amplitude
(Hz)
(dB)
(Hz)
(dB)
43.3333
43.3019
-22.7483
44.0000
-23.8800
50.0000
49.9919
0
50.0000
0
56.6667
56.6883
-30.6257
56.0000
-31.4000
(Hz)
(1 − 2s) f
Value
DFT
5.6. Practical Implementation Test
10
Current Signal
Prony Estimation
8
6
Magnitude (A)
4
2
0
−2
−4
−6
−8
−10
0
0.05
0.1
0.15
0.2
0.25
Time (sec)
0.3
0.35
0.4
0.45
0.5
(a) Sampled real data and PA estimation for the period of sampling
10
Current Signal
Prony Estimation
8
6
Magnitude (A)
4
2
0
−2
−4
−6
−8
−10
0.5
0.55
0.6
0.65
0.7
0.75
Time (sec)
0.8
0.85
0.9
0.95
1
(b) Sampled real data and PA prediction for the period into future.
Figure 5.15.: Comparison of the current waveforms between PA estimation and
prediction with the real current signal of a 2.2kW induction machine
operating in full load with 4 broken rotor bars.
0
−10
−20
Amplitude (dB)
44 Hz
−30
56 Hz
−40
−50
−60
−70
−80
0
20
40
60
80
100
120
Frequency (Hz)
140
160
180
200
Figure 5.16.: DFT spectrum of the same signal data used in Figure 5.14 and Figure
5.15 but using a window of only 200 samples with the sampling
frequency 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 condition. When the machine is operating under full load, these two sideband frequencies 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 modulation 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 amplitude 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 parameters 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 detection using both simulated and measured data. The result is also compared with
DFT result.
Data pre-conditioning such as ltering, downsampling and constant oset removing needed to be conducted prior to running the Prony Analysis algorithm. Especially, 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 correctly. 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 detection 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 implemented 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
Bibliography
[1] O. V. Thorsen and M. Dalva, A survey of faults on induction motors in
oshore oil industry, petrochemical industry, gas terminals and oil reneries,
IEEE Trans. Ind. App., vol. 31, pp. 11861196, Sept./Oct. 1995.
[2] W. T. Thomson and D. Rankin, Case histories of rotor winding fault diagnosis in induction motors, in
Proc. 2nd Int. Conf. Condition Monitoring,
pp. 798819, University of Swansea, Mar. 1987.
[3] P. J. Tavner and J. Penman,
Condition Monitoring of Electrical Machines.
Letchworth, England: Research Studies Press Ltd., 1987.
[4] M. E. H. Benbouzid, A review of induction motors signature analysis as a
medium for faults detection,
IEEE Trans. Ind. Electron.,
vol. 47, pp. 984
993, Oct. 2000.
[5] A. Bellini, F. Filippetti, G. Franceschini, C. Tassoni, R. Passaglia, M. Saottini, G. Tontini, M. Giovannini, and A. Rossi, Enel's experience with on-line
diagnosis of large induction motors cage failures, in
Conf. Rec. 2000 IEEE
Ind. App. Conf., vol. 1 of 2000, (Rome, Italy), pp. 492498, 2000.
[6] J. M. B. Siau, A. L. Gra, W. L. Soong, and N. Ertugrul, Broken bar
detection in induction motors using current and ux spectral analysis, in
AUPEC'03, (Christchurch, New Zealand), Oct. 2003.
[7] G. B. Kliman, R. A. Koegl, J. Stein, and R. D. Endicott, Noninvasive detection of broken rotor bars in operating induction motors,
Energy Conversion, vol. 3, pp. 873879, Dec. 1988.
IEEE Trans.
[8] D. R. Rankin, The industrial application of phase current analysis to detect
rotor winding faults in squirrel cage induction motors,
Journal, vol. 9, pp. 7784, 1995.
Power Engineering
iss. 2.
[9] W. T. Thomson and M. Fenger, Current signature analysis to detect induction motor faults,
IEEE In. App. Magazine, pp. 2634, July/Aug. 2001.
87
BIBLIOGRAPHY
[10] R. Zivanovic and S. Chen, Fault diagnostics of induction machines using
prony analysis, in
Proc. IEEE Int. Conf. Power Tech., no. 496, (Lausanne,
Switzerland), July 2007.
[11] F. Filippetti, M. Martelli, G. Franceschini, and C. Tassoni, Development of
expert system knowledge base to on-line diagnosis of rotor electrical faults of
induction motors, in
Conf. Rec. IEEE IAS Annu. Meeting, (Houston, TX),
pp. 9299, Oct. 1992.
[12] S. A. S. A. Kazzaz and G. K. Singh, Experimental investigations on induction machine condition monitoring and fault diagnosis using digital signal
processing techniques,
Electric Power Systems Research,
vol. 65, pp. 197
221, Jun. 2003.
[13] R. W. Hamming,
[14] S. L. Marple,
Digital Filters.
Englewood Clis, NJ: Prentice Hall, 1983.
Digital Spectral Analysis With Applications.
Englewood Clis,
NJ: Prentice Hall, 1987.
Dynamic Simulation of Electric Machinery Using MATLAB/SIMULINK. New Jersey: Prentice Hall, 1998.
[15] C.
M.
Ong,
[16] C. Hargis, B. G. Gaydon, and K. Kamash, The detection of rotor defects in
induction motors, in
Proc. IEEE Conf. Electrical Machines ²C Design and
Application, no. 213, pp. 216220, 1982.
[17] W. T. Thomson, R. A. Leonard, A. J. Milne, and J. Penman, Failure identication of oshore induction motor systems using on-line condition monitoring, in
Proc. 4st National Conf. Reliability, (Birmingham, UK), pp. 2C/3/1
2C/3/11, 1983.
[18] I. Ahmed, R. Supangat, J. Grieger, N. Ertugrul, and W. L. Soong, A baseline
study for on-line condition monitoring of induction machines, in
AUPEC'04,
(Brisbane, Australia), Sept. 2004 2004.
[19] F. Filippetti, G. Franceschini, C. Tassoni, and P. Vas, Ai techniques in
induction machines diagnosis including the speed ripple eect,
Ind. App., vol. 34, pp. 98108, Jan./Feb. 1998.
IEEE Trans.
[20] A. Bellini, G. Franceschini, C. Tassoni, and A. Toscani, Assessment of induction machines rotor fault severity by dierent approaches, in
32th IEEE Annu. Conf., 6-10 Nov. 2005.
IECON'05,
[21] P. M. Santos, M. B. R. Correa, C. B. Jacobina, E. R. C. da Silva, A. M. N.
Lima, G. Didiery, H. Raziky, and T. Lubiny, A simplied induction machine
88
BIBLIOGRAPHY
model to study rotor broken bar eects and for detection, in
IEEE, pp. 17, Jun. 2006.
PESC'06, 37th
[22] A. L. Orille, G. M. A. Sowilam, and J. A. Valencia, A new simulation of
symmetrical three phase induction motor under transformations of park,
Computers & Industrial Engineering, vol. 37, pp. 359362, 1999.
[23] J. F. Bangura and N. A. Demerdash, Diagnosis and characterization of eects
of broken bars and connectors in squirrel-cage induction motors by a timestepping coupled nite element state space modeling approach,
Energy Conversion, vol. 14, pp. 11671176, Dec. 1999.
IEEE Trans.
[24] A. Bellini, F. Filippetti, G. Franceschini, C. Tassoni, and G. B. Kliman,
Quantitative evaluation of induction motor broken bars by means of electrical signature analysis,
IEEE Trans. Ind. App.,
vol. 37, pp. 12481255,
Sept./Oct. 2001.
[25] P. C. Krause, O. Wasynczuk, and S. D. Sudho,
ery.
Analysis of Electric Machin-
New York: IEEE Press, 1995.
[26] H. A. Toliyat,
DSP-Based Electromechanical Motion Control.
CRC Press,
2004.
[27] B. Ozpineci and L. M. Tolbert, Simulink implementation of induction machine model - a modular approach, in
IEEE. Int. Conf. Electric Machines
and Drives, IEMDC'03, vol. 2, pp. 728734, Jun. 2003.
[28] S. R. Shaw and S. B. Leeb, Identication of induction motor parameters from
transient stator current measurements,
IEEE Trans. Ind. Electron., vol. 46,
pp. 139149, Feb. 1999.
[29] C. W. Therrien,
Discrete Random Signals and Statistical Signal Processing.
Englewood Clis, NJ: Prentice Hall, 1992.
[30] S. M. Kay,
Modern Spectral Estimation: Theory and Application.
Englewood
Clis, NJ: Prentice Hall, 1988.
[31] S. L. Marple, A tutorial overview of modern spectral estimation, in
Conf. Acoustics, Speech and Signal Processing (ICASSP),
Int.
vol. 4, pp. 2152
2157, 23-26, May 1989.
[32] S. Singh, Application of prony analysis to characterize pulsed corona reactor measurements, Master's thesis, Dept. Elec. Comp. Eng., University of
Wyoming, Aug. 2003.
89
BIBLIOGRAPHY
[33] R.
Zivanovic,
P.
Schegner,
O.
Seifert,
and
G.
Pilz,
Identication
of
the resonant-grounded system parameters by evaluating fault measurement
records,
IEEE Trans. Power Delivery, vol. 19, pp. 10851090, July 2004.
[34] H. P. Sava and J. T. E. McDonnell, Modied forward-backward overdetermined prony method and its application in modeling heart sounds,
IEEE Trans. Vision, Image and Signal Processing,
vol. 142, pp. 375380,
Dec. 1995.
[35] C. W. Chuang and D. L. Moatt, Natural resonances of radar targets via
prony's method and target discrimination,
IEEE Trans. Aerospace and Elec-
tronic Systems, vol. AES-12, pp. 583589, Sept. 1976.
[36] R. Walker, A. Ashley, and P. Kavanagh, Noise normalization of broadband
sonar data in bearing space, in
IEEE Int. Conf. on Acoustics, Speech and
Signal Processing, ICASSP'83, vol. 8, pp. 375378, Apr. 1983.
[37] M. Wax and T. Kailath, Detection of signals by information theoretic criteria,
IEEE Trans. Acoustics Speech and Signal Processing,
vol. 33, pp. 387
392, Apr. 1985.
[38] T. H. Friddell, J. A. Ritcey, and D. Haynor, Data pre-conditioning for improved performance of the prony method, pp. 360364, 1989. Maple Press.
[39] S. Chen and R. Zivanovic, A novel high-resolution technique for induction
machine broken bar detection, in
AUPEC'07, no. 05-09, (Perth, Australia),
9-13, Dec. 2007.
[40] P. Schegner, G. Pilz, and C. Wallner, Analysis of noisy voltage signal with
a high resolution of frequency for the closing of transmission lines, in
14th Power Systems Computation Conference (PSCC),
Proc.
(Sevilla, Spain), 24-
28, Jun. 2002.
[41] R. N. McDonough and W. H. Huggins, Best least-squares representation of
signals by exponentials,
IEEE Trans. Autom. Control, vol. AC-13, pp. 408
412, Aug. 1968.
[42] S. P. Robinson and P. M. Harris, Modelling acoustic signals in the calibration of underwater electroacoustic transducers in reverberant laboratory
tanks, tech. rep., National Physical Laboratory, Teddington, Middlesex,
March 1999. NPL Report CMAM 029.
[43] M. R. Osborne and G. K. Smyth, A modied prony algorithm for exponential
tting,
90
SIAM J. Sci. Comput., vol. 16, pp. 119138, 1995.
BIBLIOGRAPHY
[44] R. Zivanovic and P. Schegner, Pre-ltering improves prony analysis of disturbance records, in
Proc. 8th Int. Conf. Developments in Power System
Protection, (Amsterdam, The Netherlands), pp. 780783, 5-8 April 2004.
[45] R. Kumaresan and Y. Feng, Fir preltering improves prony's method,
Trans. Signal Processing, vol. 39, pp. 736741, Mar. 1991.
[46] The MathWorks, Inc., 3 Apple Hill Drive, Natick, MA,
User's Guide, 3.4 ed., Mar. 2006.
IEEE
Filter Design Toolbox
Cognitive Eects on The Neurophysiology And Biomechanics
of Stroke Recovery. PhD thesis, Div. Eng. & App. Sci., Harvard University,
[47] S. G. Diamond,
Cambridge, Massachusetts, Jan. 2004.
[48] C. W. Therrien and C. H. Velasco, An iterative prony method for arma
signal modeling,
IEEE Trans. Signal Processing, vol. 43, pp. 358361, Jan.
1995.
[49] D. J. Trudnowski, Order reduction of large-scale linear oscillatory system
modes,
IEEE Trans. Power System, vol. 9, pp. 451458, Feb. 1994.
[50] N. Ertugrul,
tories.
LabVIEW for electric circuits, machines, drives, and labora-
National Instruments virtual instrumentation series, Upper Saddle
River, NJ: Prentice Hall, 2002.
[51] E. W. Tan, Power system dynamic security assignment via prony analysis,
Master's thesis, Dept. Inf. Tech. & Elec. Eng., University of Queensland, Oct.
2003.
91
BIBLIOGRAPHY
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;
we = wb;
wbm = 2*wb/P;
Sb = Prated;
Vm = Vrated*sqrt(2/3);
Vb = Vm;
Ib = (2*Sb)/(3*Vb);
Zb = Vb/Ib;
Tb = Sb/wbm;
% base electrical frequency
%
%
%
%
%
%
%
base mechanical speed
rating output power in VA
magnitude of phase voltage
base voltage
base current
base impedance in ohms
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·m2
H = 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)
qd0
vsqd0 = Tqd0 (θ) rs T−1
qd0 (θ) is + Tqd0 (θ)
qd0
d[T−1
qd0 (θ)λs ]
dt
The following time-derivative term may be expressed as [15]
qd0
d[T−1
qd0 (θ)λs ]
dt
=

d[T−1
qd0 (θ)]
dt
−1
· λqd0
s + Tqd0 (θ) ·
d(λqd0
s )
dt

− sin θ
cos θ
0
d(λqd0

 dθ
s )
qd0
−1
2π
=  − sin θ − 2π
·
λ
+
T
(θ)
·
cos
θ
−
0
 dt
s
qd0
dt
3
3
− sin θ + 2π
cos θ + 2π
0
3
3
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)
abc
abc abc
λqd0
= Tqd0 (θ) Labc
s
ss is + Lsr ir
By applying the Park's transformation to the above ux linkage equations, obtain
−1
qd0
abc −1
qd0
λqd0
= Tqd0 (θ) Labc
s
ss Tqd0 (θ) is + Tqd0 (θ) Lsr Tqd0 (θ − θr ) ir



cos θ cos θ − 2π
cos θ + 2π
Lls + Lss
Lsm
Lsm
3
3



2π
= 23  sin θ sin θ − 2π
sin
θ
+
Lls + Lss
Lsm
  Lsm

3
3
1
1
1
Lsm
Lsm
Lls + Lss
2
2
2



cos θ
sin θ
1
iqs



2π
×  cos θ − 2π
sin
θ
−
1   ids 
3
3
cos θ + 2π
sin θ + 2π
1
i0s
3
3
99
Appendix B. Important Equation Derivations

2π
cos
θ
+
cos θ cos θ − 2π
3
3


2π
+ 23 Lsr  sin θ sin θ − 2π
sin
θ
+

3
3

1
2
1
2
1
2


2π
cos θr
cos θr + 2π
cos
θ
−
r
3
3


2π
cos
θ
cos
θ
+
 cos θr − 2π

r
r
3
3
cos θr − 2π
cos θr
cos θr + 2π
3
3



cos (θ − θr )
sin (θ − θr )
1
iqr



×  cos θ − θr − 2π
sin θ − θr − 2π
1   idr 
3
3
cos θ − θr + 2π
sin θ − θr + 2π
1
i0r
3
3
 3


(L
+
L
−
L
)
0
0
i
ls
ss
sm
qs
2



3
= 23 
(Lls + Lss − Lsm )
0
0
  ids 
2
3
0
0
(Lls + Lss ) + 3Lsm
i0s
2

 3

0 0
iqr
2



3
+Lsr  0 2 0   idr 
0 0 0
i0r
= − 12 Lss , the above equation can
Lsm = Lss cos 2π
3
 3



Lls + 23 Lss
0
0
L
0
0
sr
2

 qd0 

3
3
i
+
L
0
L
0  iqd0
0
L
+
 0
 s

ls
r
2 ss
2 sr
0
0
Lls
0
0
0
Because of
Similarly, we can obtain the rotor ux linkages in the

3
L
2 sr

qd0
be written as
reference frame


Llr + 23 Lrr
0
0
0
0



3
L
0  iqd0
0
Llr + 32 Lrr 0  iqd0
s +
r
2 sr
0
0
0
0
Llr

λqd0
=
r
0
0
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
Pin =
3
2
2
rs iqs + ωλds iqs +
+rs i2ds − ωλqs ids +
100
form power equation
0 0
0 0
0 0
vqs iqs + vds ids + 2v0s i0s + vqr
iqr + vdr
idr + 2v0r
i0r
3
2
we yield
Pin =
qd0
d
dt
d
dt
(λqs iqs )
(λds ids )
B.4. Derivation of Eq. (3.34)
+2rs i20s + 2 dtd (λ0s i0s )
=
0 0
+rr0 i02
qr + (ω − ωr ) λdr iqr +
d
dt
λ0qr i0qr
0 0
+rr0 i02
dr − (ω − ωr ) λqr idr +
d
0 0
+ 2rr0 i02
0r + 2 dt (λ0r i0r )
d
dt
(λ0dr i0dr )
0 02
0 02
rs i2qs + rs i2ds + 2rs i20s + rr0 i02
qr + rr idr + 2rr i0r
+ 23 ω (λds iqs − λqs ids ) + (ω − ωr ) λ0dr i0qr − λ0qr i0dr
3
2
+ 32 dtd λqs iqs + λds ids + 2λ0s i0s + λ0qr i0qr + λ0dr i0dr + 2λ00r i00r
The
i2 r
terms are the copper losses, the
i dλ
dt
terms 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
ωrm =
2
ω , yield
P r
Tem =
3 P
2 ωr
ωλi terms divided by the mechanical speed
ω (λds iqs − λqs ids ) + (ω − ωr ) λ0dr i0qr − λ0qr i0dr
B.4. Derivation of Eq. (3.34)
From Eq. (3.31), obtain
vqs =
1 d(ψqs )
ωb dt
+ rs iqs
Thus,
ψqs = ωb
´
(vqs − rs iqs ) dt
From Eq. (3.33), obtain
ψqs = (xls + xm ) iqs + xm i0qr
and
ψmq = xm iqs + i0qr
Thus,
iqs =
ψqs −ψmq
xls
Finally, obtain
ψqs = ωb
´n
vqs +
rs
xls
o
(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=
PN
n=1
n−1 2
h
z
k
k
k=1
n=1
PN
PN
P
P
Pq
2
n−1 2
= n=1 x [n] − 2 n=1 x [n] qk=1 hk zkn−1 + N
h
z
;
k
k
n=1
k=1
h P
n−1
= ∂h∂ k −2 N
+ . . . + hk zkn−1 + . . . + hq zqn−1
n=1 x [n] h1 z1
i
P
n−1
n−1
n−1 2
+ N
h
z
+
.
.
.
+
h
z
+
.
.
.
+
h
z
1 1
k k
q q
n=1
Pq
PN
Pq
P
n−1
n−1
n−1
n−1
h
z
+
.
.
.
+
h
z
+
.
.
.
+
h
z
= −2 n=1 x [n] k=1 zkn−1 +2 N
1
k
q
1
q
k
k=1 zk
n=1
Pq
P
Pq n−1
n−1
− zkn−1
= −2 N
i=1 hi zi
n=1
k=1 x [n] zk
h P
n−1
n−1
n−1
= ∂z∂k −2 N
x
[n]
h
z
+
.
.
.
+
h
z
+
.
.
.
+
h
z
1
k
q
1
q
k
n=1
i
2
P
n−1
+ . . . + hk zkn−1 + . . . + hq zqn−1
+ N
n=1 h1 z1
Pq
P
n−2
= −2 (n − 1) N
k=1 hk zk
n=1 x [n]
Pq
P
n−1
(n − 1) h1 z1n−2 + . . . + hk zkn−2 + . . . + hq zqn−2
+2 N
k=1 hk zk
n=1
Pq Pq
P
n−1
n−2
− hk zkn−2
= −2 (n − 1) N
k=1 x [n] hk zk
i=1 hi zi
n=1
=
∂ρ
∂hk
∂ρ
∂zk
| [n]|2
PN
x [n] −
Pq
∂ρ
= 0 and ∂z
= 0, yield:
k
PN Pq Pq
n−1
n−1
− zkn−1
= 0, and
n=1
k=1 x [n] zk
i=1 hi zi
Pq
PN Pq
n−2
n−1
− zkn−2
= 0.
k=1 (n − 1) x [n] zk
i=1 hi zi
n=1
Set
∂ρ
∂hk
Thus,
PN Pq
x [n] zkn−1 ;
P
Pq
Pq
n−1
n−1
;
c2 = N
n=1
k=1 zk
i=1 zi
P
Pq
n−2
c3 = N
;
n=1
k=1 (n − 1) x [n] zk
P
Pq
Pq
n−2
n−1
c4 = N
.
n=1
k=1 (n − 1) zk
i=1 zi
c1 =
102
n=1
k=1
Appendix C.
Parameters of Induction Machines
Table C.1.: Parameters of induction machine models used for simulations.
Output power
(kW)
Rated frequency
(Hz)
Rated voltage
(V)
Poles
Number of rotor
bars
Stator winding
resistance (Ω)
Stator leakage
reactance (Ω)
Rotor leakage
reactance (Ω)
Stator magnetizing
reactance (Ω)
Machine 1
Machine 2
Machine 3
2.2
5.5
35
50
50
50
220
415
460
4
4
8
28
32
52
0.435
1.003
0.187
1.554
2.57
0.502
1.554
2.57
0.502
26.13
44.307
13.08
1.016
1.4735
0.228
Referred rotor
winding resistance
(Ω)
103
Appendix C. Parameters of Induction Machines
104
Appendix D.
Prony Analysis Results
105
Appendix D. Prony Analysis Results
True Value
DFT
PA
dierent load conditions, using a data window of 500 samples and a sampling frequency of 1000Hz.
Table D.1.: Frequency estimation results by PA and DFT for Machine1 with dierent number of broken rotor bars operating under
Full
Load
Number
-44.4100
43.4589
43.7195
43.9432
-15.2707
-18.1365
-20.9158
-24.7258
-31.2799
58.2446
57.6647
57.2150
56.8487
56.5450
56.2852
56.0410
-27.9204
-28.3792
-30.3556
-32.2430
-34.0827
-37.1771
-42.9812
(dB)
-38.3300
43.1537
-12.0713
Amplitude
56.0000
-35.1100
42.7867
-10.1570
(Hz)
56.0000
-33.2600
42.3341
(1 + 2s) f
-32.3600
56.0000
-31.3800
41.7558
(dB)
(Hz)
-25.7700
56.0000
-29.5000
Amplitude
44.0000
-21.8300
58.0000
-28.7100
(Hz)
(Hz)
44.0000
-19.2200
58.0000
(1 − 2s) f
56.0568
44.0000
-16.3300
58.0000
(dB)
(Hz)
56.2802
44.0000
-13.2000
Amplitude
43.9432
56.5407
42.0000
-10.9100
(Hz)
43.7198
56.8470
42.0000
-28.0473
(1 + 2s) f
1
43.4593
57.2125
42.0000
59.0370
(dB)
2
43.1530
57.6672
-8.4658
Amplitude
3
42.7875
58.2415
40.9622
(1 − 2s) f (1 + 2s) f (1 − 2s) f
4
42.3328
-29.6500
of
5
41.7585
60.0000
broken
6
-10.0800
bars
7
40.0000
PA
59.0334
DFT
40.9666
True Value
8
75%
Load
Number
Fail
Fail
45.0535
45.2715
45.4569
45.6164
-16.9035
-19.4734
-22.4082
-26.4389
-33.1123
55.2036
54.9459
54.7325
54.5445
54.3805
-27.8927
-29.5699
-31.7183
-35.0059
-40.9856
(dB)
Fail
Fail
44.7975
Amplitude
Fail
Fail
(Hz)
Fail
Fail
-28.8000
(1 + 2s) f
(Hz)
Fail
Fail
(dB)
Fail
Fail
56.0000
Amplitude
(Hz)
Fail
Fail
-26.1252
(Hz)
54.3830
Fail
-17.8200
55.5152
-24.3925
(1 − 2s) f
(Hz)
54.5440
Fail
-14.0924
55.9030
-23.9287
(dB)
45.6170
54.7306
44.0000
44.4863
-11.1222
56.4117
Amplitude
45.4560
54.9436
-26.7200
44.0984
-9.1925
(Hz)
1
45.2694
55.2012
56.0000
-25.4100
43.5898
(1 + 2s) f
2
45.0564
-14.7300
56.0000
-24.9200
(dB)
3
44.7988
44.0000
-12.1200
56.0000
Amplitude
4
55.5159
44.0000
-10.1800
(1 − 2s) f (1 + 2s) f (1 − 2s) f
5
44.4841
55.9019
44.0000
of
6
44.0981
56.4160
broken
7
43.5840
bars
8
106
107
50%
46.9409
46.7990
46.6380
46.4375
46.1967
45.8868
3
4
5
6
7
8
48.6128
48.5624
48.5005
48.4335
48.3528
48.2635
48.1286
47.9869
1
2
3
4
5
6
7
8
52.0131
51.8714
51.7365
51.6472
51.5665
51.4995
51.4376
51.3872
Fail
Fail
Fail
Fail
Fail
Fail
Fail
Fail
(Hz)
(Hz)
(Hz)
broken
bars
(1 − 2s) f (1 + 2s) f (1 − 2s) f
Fail
Fail
Fail
Fail
Fail
Fail
Fail
Fail
(Hz)
of
Number
Load
54.1132
53.8033
53.5625
53.3620
53.2010
53.0591
52.9410
52.8378
(Hz)
True Value
47.0590
2
25%
47.1622
(Hz)
(1 − 2s) f (1 + 2s) f (1 − 2s) f
True Value
1
bars
broken
of
Number
Load
Fail
Fail
Fail
Fail
Fail
Fail
Fail
Fail
(dB)
Fail
Fail
Fail
Fail
Fail
Fail
Fail
Fail
(Hz)
(1 + 2s) f
Fail
Fail
Fail
Fail
Fail
Fail
Fail
Fail
(Hz)
(1 + 2s) f
DFT
Amplitude
Fail
Fail
Fail
Fail
Fail
Fail
Fail
Fail
(dB)
Amplitude
DFT
Fail
Fail
Fail
Fail
Fail
Fail
Fail
Fail
(dB)
Amplitude
Fail
Fail
Fail
Fail
Fail
Fail
Fail
Fail
(dB)
Amplitude
47.9911
48.1384
48.2575
48.3539
48.4322
48.4519
48.5101
48.6885
(Hz)
(1 − 2s) f
45.8847
46.1946
46.4356
46.6317
46.7984
46.9382
47.0556
47.1628
(Hz)
(1 − 2s) f
-20.1440
-22.7975
-25.4226
-28.0812
-30.7713
-34.2296
-39.1548
-43.8687
(dB)
Amplitude
52.0073
51.8603
51.7414
51.6434
51.5702
51.4848
51.4898
51.2795
(Hz)
(1 + 2s) f
54.0792
53.8069
53.5678
53.3683
53.2017
53.0614
52.9414
52.8462
(Hz)
(1 + 2s) f
PA
-11.7460
-14.5572
-17.3182
-19.9379
-22.6637
-25.7854
-30.0562
-36.4707
(dB)
Amplitude
PA
-23.8403
-25.8499
-27.9801
-30.2020
-32.6599
-36.4170
-41.3964
-45.8390
(dB)
Amplitude
-22.7825
-22.9079
-24.7541
-26.5194
-28.5780
-30.9764
-34.8171
-40.4929
(dB)
Amplitude