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. 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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