Criterion Function for Broken-Bar Fault Diagnosis in Induction Motor ...

Electromagnetics, 29:220–234, 2009 Copyright © Taylor & Francis Group, LLC ISSN: 0272-6343 print/1532-527X online DOI: 10.1080/02726340902718450

Criterion Function for Broken-Bar Fault Diagnosis in Induction Motor under Load Variation Using Wavelet Transform JAWAD FAIZ,1 B. M. EBRAHIMI,1 B. AKIN,2 and B. ASAIE1 1

Center of Excellence on Applied Electromagnetic Systems, School of Electrical and Computer Engineering, University of Tehran, Tehran, Iran 2 Department of Electrical Engineering, Texas A&M University, College Station, Texas, USA Abstract In this article, a novel criterion function is introduced to diagnose the breakage in rotor bars of induction motors. This criterion function facilitates the precise diagnosis of the fault in induction motors under load variation. It uses wavelet transform to process the stator current signal in the faulty induction motors to extract the wavelet coefficients in a specific frequency band. Furthermore, spectrum analysis of the stator current around the fundamental frequency current is used to diagnose the fault non-invasively. It is shown that the amplitudes of the frequency harmonics components .1 ˙ 2ks/fs are increased due to the load increases. A time-stepping finite element method is used for modeling the faults in induction motors. In this modeling, the effects of the spatial distribution of the stator winding, nonuniform air-gap permeance, geometrical and physical characteristics of different parts of the motor, and, finally, nonlinearity of the core materials are taken into account. The proposed algorithm is applied to the stator current of a healthy and a faulty induction motor. The simulation results are obtained, and their accuracy is verified by the experimental results. Keywords induction motor, fault diagnosis, broken rotor bars, time-stepping finite element method, load variation

1. Introduction The detection of induction motor faults under steady-state and transient operating conditions has been improved over the past two decades. In the past, features, such as increase of vibration, decrease of mean torque, increase of losses, decrease of efficiency, frequency spectrum of the magnetic flux density, torque, and speed, were used to diagnose the fault (Thomson, 1999; Landy & Moore, 1987; Arkan et al., 2005). Currently, inter-turn winding failures, bearing failures, static, dynamic and mixed eccentricity, mixed fault, as well as broken rotor bars, are detected by analyzing the frequency spectrum of the induction motor current (Nandi et al., 2005). Reliable detection of the broken rotor bars requires the correct identification of the harmonic components in the current coming from Received 28 May 2008; accepted 15 November 2008. Address correspondence to Jawad Faiz, Center of Excellence on Applied Electromagnetic Systems, School of Electrical and Computer Engineering, University of Tehran, Campus No. 2, North Kargar Avenue, Tehran, Iran, 1439957131. E-mail: [email protected]

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mechanical disturbances of the rotor (Thomson & Fenger, 2003). The analysis of a faulty induction motor by the time-stepping finite element (TSFE) method obviated all problems and drawbacks of analytical and traditional lumped-parameter modeling methods (Faiz & Ebrahimi, 2006, 2007; Faiz et al., 2006, 2007a). TSFE coupled state space (TSCFE-SS) has been used to predict the characteristic frequency components that are indicative of rotor bar and connector breakages in the stator current waveform, respectively (Bangura & Demerdash, 1999). In the modeling of rotor broken bars in Elkasbagy et al. (1992) and Fiser (2001), the currents of the broken bars were taken as zero; this may not be true because there are inter-bar currents. Inter-bar currents reduced the magnetic flux density asymmetry by broken bars (Walliser & Landy, 1994). This makes detection of broken bars more difficult, particularly at early stages. In Kim et al. (1997), stator current asymmetry was employed for diagnosis of the rotor broken-bar fault. It was shown in Schoen and Habettler (1995) that the level of load and occurrence of fault influences the stator current spectrum; if the stator current is visualized in the synchronous reference, an existing magnetic asymmetry influences both d and q components of the current, while the load fluctuation affects only the q component. Sideband components around the current fundamental harmonic have been suggested for detecting broken-bar fault (Fiser, 2001; Kliman et al., 1988; Thomson & Stewart, 1988; Filippetti et al., 1996). In Nandi et al. (1999), spectrum analysis of the stator current was used to detect and recognize the number of broken rotor bars. In Bellini et al. (2000), the spectrum of the field current component id in a field-oriented controlled motor was used for fault diagnosis. In Elkasbagy et al. (1992), the amplitude of the bar current and total rotor current were used for determination of broken rotor bars. In Haji and Toliyat (2001), a pattern recognition technique based on Bayes minimum error classifier was developed to detect broken rotor bar faults in induction motors at the steady state. In Kim and Parlos (2002), a model-based fault diagnosis system was developed for induction motors using a recurrent dynamic neural network for transient response prediction and multiresolution current processing for nonstationary signal feature extraction. In Bellini et al. (2002), the sum of the side band components around the supply frequency produced by electric asymmetry was used as the diagnostic index. In Zhang et al. (2003), detection method of faulty rotor bars based on wavelet ridge was presented. In Benbouzid and Kliman (2003), the spectrum of the stator current using Park’s vector approach was used to detect broken rotor bars in faulty motors. In Calıs and Cakır (2007), a fluctuation of stator current at zero crossing times (ZCT), which is independent of the motor parameters, was used to diagnose the broken bar. In Ye et al. (2003), wavelet packet decomposition of the stator current was used for the on-line noninvasive detection of broken rotor bars. In Mirafzal and Demerdash (2005), the pendulous oscillation of the rotor magnetic field orientation was implemented as a fault signature for rotor fault diagnostic purposes at a steady-state operation. In Ondel et al. (2006), features extracted from the current and voltage measurement were used to build up a pattern recognition vector to detect broken bars. In Mirafzal and Demerdash (2006), a swing-angle indicator that is based on the rotating magnetic field pendulous oscillation concept was used for diagnosis of the broken rotor bar fault. In Mohamad et al. (2006), a discrete wavelet transform (DWT) was used to extract different harmonic components of the stator current for brokenbar fault diagnosis. In Douglas and Pillay (2005), it was shown that using high-order wavelets improve the ability to detect broken rotor bars in induction motors operating under transient conditions. In Ye et al. (2006), a wavelet packet neuro-fuzzy inference was used to set the feature coefficients of mechanical faults extracted from the stator current.

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Figure 1. Algorithm for diagnosis of broken rotor bars in faulty induction motor using simulation.

This article presents a very precise method for the diagnosis of the broken rotor bars in squirrel-cage induction motors. In Section 2, the TSFE method is used for modeling broken rotor bar fault in induction motors. In this modeling, the effects of the distributed stator winding, air-gap permeance, geometrical and physical characteristics of the extract motor, and, finally, nonlinearity of the core materials are taken into account. This modeling method facilitates the access to an accurate stator current signal. In Section 3, the experimental set-up is presented. In Section 4, the fault signatures caused by the broken rotor bars in the induction motor are presented using the spectrum analysis of stator current. It is also shown that overloading the machine increases the amplitude of the fault components in the current spectrum of a faulty induction motor. A novel criterion is proposed for the diagnosis of broken rotor bars in squirrel-cage induction motors under varying load in Section 5. The criterion function is based on wavelet coefficients of stator current in a specific frequency band. This function enables one to precisely diagnose broken rotor bars at different loads. In Section 6, the simulation and experimental results obtained by wavelet transform (WT) are presented. The accuracy of the simulations is verified by experimental results. The scheme of the fault diagnosis algorithm is presented in Figure 1.

2. Modeling and Analyzing a Faulty Induction Motor with Broken Bars Using TSFE Method The finite element method (FEM) is based on the field analysis to obtain the magnetic field distribution within the motor. Then, current, torque, and speed can be determined using other specifications of the motor. If the aim is to analyze a healthy induction motor, the symmetrical structure of the motor can be used, and a part of the motor is modeled; this shortens the computation time. In a rotor with a broken-bar induction motor, the magnetic field distribution will be asymmetrical, and the whole structure of the motor must be modeled (Figure 2(a)). In this case, the analysis time is considerably longer. In this article, transient analysis of rotating machines (RMs) is used for modeling and analyzing both healthy and faulty motors with variable speed rotor and mechanical coupling. The RM program is a transient eddy current solver, extended to include the effects of rigid body (rotating) motion (Opera-2D Reference Manual, 1999–2003). The solver also provides for the use of external circuits and coupling to mechanical equations. In the modeling of the three-phase squirrel-cage induction motor with broken bars shown in Figure 2(a), three-phase sinusoidal voltages are applied to the motor terminals as inputs with the stator phase currents as the estimated outputs. In this modeling, by directly coupling the transient equations of the external circuit showing the supply and circuit, elements


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Figure 2. Time variations of current of a full load: (a) healthy induction motor and (b) induction motor with four broken bars.

are combined with field equations in the FEM. Also, motion equations must be combined with field equations in the FEM. Modeling of a faulty induction motor has four basic parts: A. B. C. D.

Modeling motor elements considering their physical characteristics. Modeling motor supply taking into account the electrical circuits. Modeling load of motor considering the electromechanical links. Modeling fault including broken rotor bars.

It is noted that A through C can be also used in the modeling and analysis of a healthy induction motor (Faiz & Ebrahimi, 2006, 2007; Faiz et al., 2006, 2007a; Bangura & Demerdash, 1999).

2.1.

Modeling Motor Elements

In the modeling, the spatial distribution of the stator windings, time and spatial harmonics, slots on both sides of the air gap, and nonlinear characteristics of the cores are taken into account. The stator consists of laminated M-19 sheets. There are 36 slots in the stator filled with copper and 44 bars in the rotor. As shown in Table 1, the resistance of rotor bar of the healthy and faulty motors are 39.42 and 2,500 , respectively, and the resistance of the end ring is 3.07 . In this modeling, resistance of the rotor bars is equivalent with the rotor bar and end ring resistance. The mesh for FE modeling has 19,942 elements and 9,992 nodes.

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J. Faiz et al. Table 1 Specifications of the proposed induction motor

Number of poles Number of phases Outer diameter of stator (inch) Inner diameter of stator (inch) Air gap length (inch) Core length (inch) Shaft diameter (inch) Length of rotor slot (inch) Number of stator slots

4 3 8.188 4.875 0.013 2.5 1.36 0.53 36

Number of rotor slots Width of stator slot opening (inch) Height of stator slot (inch) Height of rotor slot (inch) Power factor Rated voltage (V) Rated frequency (Hz) Rated power (hp) Skew (radian)

44 0.293 0.663 0.532 0.786 230 60 3 0.425

2.2. Modeling Motor Supply A three-phase voltage source is used for simulations and tests. A DC generator has been coupled to the motor as the load. The excitation current of the generator has been adjusted in order to regulate the output voltage. A high-power resistance box has been connected to the terminals of the generator. The resistance of this box can be selected step by step by a selector on the box. In the operating motor, a suitable position of the selector is selected and, consequently, the induction motor loaded. By regulating the output voltage of the generator inserted in the excitation current pass, the load level is regulated precisely. At no-load, the induction motor speed is close to the the synchronous speed, and by loading the motor, the speed is decreased; the required speeds, as shown in Table 1, are achieved as described above. Therefore, a speed range between 1,795 and 1,725 rpm of slip range 0.0025 to 0.04 is possible by the regulated load. Referring to Figure 2, the proper connection of the FE region to the supply needs the correct link of the stator resistance, supply, and stator windings. The two-dimensional propagation equation is r #r A D J0 ;

(1)

where J0 is the surface current density, A is the potential vector, and # is the magnetic reluctivity. The current density consists of three parts: the first part is due to the supply; the second part is the eddy current that arises from the induced electrical field, which is generated by the time-varying magnetic flux; and the third is due to the motional induced voltage. Therefore, Eq. (1) is rewritten as r #r A D

Vs `

@A C r A; @t

(2)

where Vs is the applied voltage, ` is the stack length in the z direction, and v is the conductor speed with respect to B. Using a reference frame that is assumed to be fixed against a proposed element, the relative speed v is equal to zero, and Eq. (2) is simplified as r #r A D

Vb `

@A : @t

(3)

In order to link the field and circuit equations, it is necessary to estimate the total current of every conductor. Therefore, by integrating J0 over the cross-sectional area, the


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Figure 3. On-line current monitoring system.

current will be iD

“

Vb `

@A : @t

(4)

This current passes through the supply, resistance Rext , and inductanceLext . Therefore, Vs D Rext i C Lext

di C E; dt

(5)

where E is the induced voltage in the FE region. Summing up the equations, the following general equation can be obtained: 2 3 J0 @ 6 7 ŒP ŒA E i C ŒA E i T D 4 0 5 : (6) @t Vs By solving Eq. (6), the magnetic potential distribution within the motor and stator phase current are obtained. Figure 3(a) and 3(b) show the time variations of the stator current of the healthy and faulty (four broken rotor bars) induction motors from start-up to steady-state mode. 2.3.

Modeling the Load

In modeling the motor, one has to consider the interaction of the drive elements and magnetic forces. These, in turn, influence the magnetic field within the motor. Therefore, it is necessary to simulate the rotation of the driving parts of the motor. 2.4.

Modeling Fault

In the modeling and fault diagnosis of broken bars (Fiser, 2001), the bar current is taken to be equal to zero. It is noted that a zero current in a particular bar increases the currents of the adjacent bars considerably (Landy & Moore, 1987). This implies a considerable asymmetry in the rotor circuit and, consequently, asymmetry in the field produced by the rotor currents. Even a broken bar can have a nonzero current, depending on the

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type of construction and the way aluminum is used in manufacturing. In fact, suitable current paths exist between the bars of the squirrel-cage rotor; for example, current can enter the bar where it is connected to the end ring and return through the rotor core. Additional current paths are created because of injection of the high-pressure molten aluminum. At the time of injecting high-pressure molten aluminum into the rotor core, the molten aluminum penetrates between the sheets, which can generate a conducting path between two adjacent bars. Therefore, in the modeling, the broken-bar current is considered nonzero, but the resistance of the broken bar is taken to be high. Figure 2(a) shows the stator phase current of a healthy motor from start-up to the steady-state mode. Figure 2(b) shows the corresponding current for a motor with four broken rotor bars. Comparison of the marks shown in Figures 2(a) and 2(b) indicates that in the transient mode, the amplitude of the stator current of the faulty motor is increased, such that the amplitude of the second peak increases from 54 A in the healthy motor to 62 A in the faulty motor. Also, comparison of the steady-state mode of the healthy and faulty motors shows that the fault causes distortion of the stator current waveform, such that the mean current of the stator current decreases from 9.12 A in the healthy motor to 8.74 A in the motor with four broken bars.

3. Experimental Set-Up A number of identical induction motors are artificially modified to test broken rotor bar and eccentricity faults. A conventional test bed is used in order to validate the proposed method to detect stator current fault signature components of the three-phase induction machine. The on-line current monitoring system has been shown in Figure 3. The tested motor is a three-phase, four-pole, 230-V induction motor with 36 stator slots and 44 rotor slots. The rotor material is aluminum, and skewing is 0.425 rad. Table 1 summarizes the specifications of the proposed motor. The tests have been carried out on both a healthy motor and a motor with bored bars. The resistances of the healthy and faulted rotor bars are 39.42 and 2,500 , respectively. Voltages and currents of the motor connected to a DC generator have been sampled by a data acquisition system. The 3-hp induction motor is loaded by a DC generator at different load levels. The quantities have been measured for four rotor bars and different loads (0, 33, 66, 100, and 133% rated load). A signal conditioning board including voltage and current sensors is designed and connected to data acquisition board through voltage amplifiers to scale the magnitude and low-pass filters (LPFs). In order to obtain raw current voltage data, a 1.25-MS/s 12-bit resolution data acquisition card is used for off-line tests. A 16-bit A/D at 256 kHz SR760 FFT spectrum analyzer (Sunnyvale, California, USA) is used to monitor the real time current and voltage spectrums.

4. Spectrum Analysis of Stator Current under Load Variation The particular pattern frequency (1 ˙ 2ks/fs (k is an integer number, s is slip, and fs is supply frequency) used for broken-bar fault diagnosis relates to the slip of induction motor, which, in turn, closely correlates with load variation. Although determination of broken rotor bar occurrence and recognition of its number has been presented in (Faiz et al., 2007b), the effects of the load variation on the fault diagnosis procedure have not been considered. Therefore, the investigation of load variation on the fault diagnosis procedure is necessary.


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The frequency spectrum of a motor that has a four-broken-bar rotor and is driven at various load levels are presented in Figures 4 through 7. Comparison of Figures 4 and 5 indicates that the amplitude of harmonic components .1 2s/fs increases from 70 dB in the no-load case to 64 dB in the 33% of rated load case. Also, amplitude of harmonic components .1 C 2s/fs increases from 75 dB in the no-load case to 65 dB in the 33% of rated load case. Figure 6 shows a similar spectrum for 66% of rated load. A 33% increase of the load compared with Figure 5 increases the amplitude of the harmonic components .1 2s/fs to 55 dB and .1 C 2s/fs to 57 dB. Finally, at the rated load, the corresponding amplitudes rise to 47 dB and 55 dB, respectively. Referring to Figures 4 through 7, the higher loads increase the amplitude of harmonic components .1 ˙ 2s/fs . An interesting point in Figures 4 through 7 is that a varying load not only affects the amplitude of the harmonic components but also influences the frequency of the harmonic components, such that the difference between the frequency of

Figure 4. Normalized line current spectra of a faulty no-load induction motor with four broken bars: (Left) simulated and (Right) experimental.

Figure 5. Normalized line current spectra of a faulty induction motor with four broken bars under 33% of rated load: (Left) simulated and (Right) experimental.



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J. Faiz et al. Table 2 Amplitude of harmonic components in faulty induction with broken bar on different load in dB Healthy motor

Percentage of rated load 0% 33% 66% 100% 133%

.1

2s/fs 76 67 60 54 50

.1

2s/fs 75 63 55 50 49

.1 C 2s/fs 77 62 53 52 51

Faulty motor with four broken bars .1 C 4s/fs 80 66 56 55 52

.1

4s/fs 72 60 52 45 43

.1

2s/fs 68 50 41 38 33

.1 C 2s/fs 75 53 43 40 38

.1 C 4s/fs 73 61 54 52 49

harmonic components due to the broken bars increases compared with the fundamental frequency. The reason is that a higher load decreases the speed and slip of the motor; therefore, the slip and difference of frequencies .1 ˙ 2s/fs , in respect to the fundamental frequency, rises. Table 2 summarizes the amplitude of harmonic components .1 ˙ 2ks/fs for k D 1, 2 of healthy and faulty motors with four broken bars between 0% and 133% of rated load with a 33% step size.

5. Wavelet Transform The waveforms associated with fast electromagnetic transients are typically nonperiodic signals that contain both high-frequency oscillations and localized impulses superimposed on the power frequency and its harmonics. These characteristics present a problem for the traditional discrete Fourier transform (DFT), because its use assumes a periodic signal. As electrical machine disturbances are subject to transient and nonperiodic components, the DFT alone can be an inadequate technique for signal analysis. If a signal is altered in a localized time instant, the entire frequency spectrum can be affected. To reduce the effect of nonperiodic signals on the DFT, the short-time Fourier transform (STFT) is used. It assumes a local periodicity within a continuously translated time window. The WT is a powerful signal processing tool used in power systems. The WT, like the STFT, allows time localization of different frequency components of a given signal; however, with one important difference—the STFT uses a fixed width windowing function. As a result, both frequency and time resolution of the resulting transform will be fixed, but in the case of the WT, the analyzing functions, which are called wavelets, will adjust their time-widths to their frequencies in such a way that higher frequency wavelets will be very narrow and those with lower frequency will be broader. Therefore, in contrast to the STFT, the WT can isolate the transient components in the upper frequency isolated in a shorter part of a power frequency cycle. The ability of the WT to focus on short time intervals for high-frequency components and long intervals for low-frequency components improves the analysis of the signals with localized impulses and oscillations. Figure 8 illustrates the implementation procedure of a DWT, in which S is the original signal and the low-pass and high-pass filters are designated by LPF and HPF, respectively. At the first stage, an original signal is divided into two halves of the frequency bandwidth and sent to both the HPF and LPF. Then the output of the LPF is further cut into half of the frequency bandwidth and sent to the second stage; this procedure is repeated until the signal is decomposed to a pre-defined level. If the original signal is being sampled at fs Hz, the highest frequency that the signal could contain, from Nyquist’s theorem, would be fs =2 Hz. This frequency would be seen


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Figure 8. Implementation of DWT.

at the output of the high-frequency filter, which is the first detail. Thus, the band of frequencies between fs =2 and fs =4 would be captured in detail 1; similarly, the band of frequencies between fs =4 and fs =8 would be captured in detail 2, and so on. The sampling frequency in this article is taken 960 Hz, and Table 3 shows the frequency levels of the wavelet function coefficients. Daubechies-8 wavelet is used in this article as a mother wavelet. The efficiency of Daubechies wavelets based on the accurate reconstruction of power system transient signals, as described in (Faiz et al., 2007b), and the suitability of Daubechies-8 for the analysis of power system transients from the family of Daubechies wavelets, as described in (Angrisani et al., 1996), is the basis for choosing Daubechies-8. Moreover, according to (Douglas & Pillay, 2005), use of high-order wavelets such as Daubechies-8 can improve the precision of diagnosis of the broken rotor bars. As previously discussed, rotor broken bars generate side band components around the fundamental frequency. On the other hand, referring to Table 3 indicates that the wavelet coefficient in D4 consists of side band components around the fundamental frequency. Therefore, in this article, a wavelet coefficient in D4 has been used for broken-bar fault diagnosis.

6. Diagnosis of Broken Rotor Bar in Induction Motor under Load Variation Using Criterion Function It is clear from the analysis described in Section 4 that the broken rotor bars produce special harmonic components with a certain frequency pattern in the stator current. Furthermore, it was shown that the frequencies of these harmonics are around the fundamental component. Now, considering the classification of the presented frequency in Table 3, the wavelet coefficients in D4 is chosen for fault diagnosis. In Figures 9(a) and 9(b), the fluctuation in the absolute value of the wavelet coefficient in D4 for the

Table 3 Frequency levels of wavelet coefficients Wavelet analysis

Frequency components (Hz)

D1 D2 D3 D4 A5

480-240 240-120 120-60 60-30 30-0

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Figure 9. Fluctuation in the absolute value of the wavelet coefficient in D4 for the full-load induction motor: (a) healthy, (b) faulty with one broken bar, (c) faulty with four broken bars, and (d) experimental result for faulty with four broken bars.

healthy full-load motor and faulty full-load motor with one broken rotor bar is presented, respectively. Comparison of the fluctuation in the absolute value of the wavelet coefficient in D4 for the healthy motor and the motor with one broken rotor bar confirms the increase of its value due to the fault occurrence. The average fluctuation in the absolute value of the wavelet coefficient in D4 increases from 0.0782 in the full-load healthy motor to 0.1375 in a full-load motor with one broken rotor bar. By increasing the number of broken rotor bars to four, this average value raises to 0.5469, which is shown in Figure 9(c). This change relates to the harmonic components in the stator current, which causes oscillation in D4 and absolute value of its amplitude. Referring to Figures 9(a) through 9(c), the average fluctuation of the absolute value of the wavelet coefficient in D4 is increased by raising the number of broken bars to four. Table 4 indicates the average fluctuation in the absolute value of the wavelet coefficient in D4 and the average value of the current amplitude. Meanwhile, according to Table 4, the average value of the current amplitude is reduced due to the broken rotor bars, such that the average value of the current decreases from 9.87 A in the healthy motor to 8.97 A in the motor with one broken bar. By increasing the number of broken bars to four, this value decreases to 8.74. As shown in Table 4, the average fluctuation in the absolute value of the wavelet coefficient in D4 is a convenient criterion for diagnosis of the broken bars. According to the above-mentioned Table 4 Criterion function for different loads of faulty induction motor with broken bars Number of broken bars

Percentage of rated load

Mean current (A)

Mean distortion in D4

Criterion function

0 1 4 4 4 4 4

100% 100% 0% 33% 66% 100% 133%

9.87 8.97 9.54 8.92 8.81 8.74 8.57

0.0782 0.1375 0.0923 0.3220 0.4044 0.5469 0.5674

0.792% 1.533% 0.97% 3.61% 4.59% 6.25% 6.62%


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facts, the following criterion function is proposed for diagnosis of the broken bars: Criterion function D

Average of fluctuations for (|D4|) ; Mean current

(7)

where the fluctuation in the absolute value of the wavelet coefficients in D4 is expressed in per-unit with respect to the average amplitude of the corresponding currents. Therefore, the value of proposed criterion function increases by occurring the breaking in the bars of the rotor and has ascending trend. According to Table 4, the value of the criterion function for a healthy motor is 0.792%, and for the motor with one broken rotor bar, it is 1.533%. A noticeable difference between these two values makes the proposed index able to diagnose a faulty motor from a healthy one. Meanwhile, by increasing the number of broken rotor bars from one to four, this index increases to 6.25%. A considerable difference between these two values makes the introduced criterion function able to determine the number of broken bars. Figures 10(a) and 10(b) present the fluctuation in the absolute value of the wavelet coefficient in D4 for the motor with four broken bars, running at no load and 33% of rated load, respectively. The fluctuation in the absolute value of the wavelet coefficient in D4 for a faulty motor with 0% and 33% of the rated load have been compared and shows that the oscillation of wavelet coefficients in D4 increases. The reason for this increase is the rise of the amplitude of harmonic components, .1˙2Ks/fs , in the stator current. In addition, the average of fluctuations in absolute value of the wavelet coefficients in D4 increases from 0.0923 in the no-load faulty motor to 0.3220 in a faulty motor with 33% of the rated load. By increasing the load to 66% of the rated load, this average value increases to 0.4044, which has been shown in Figure 9(c). This phenomenon is due to the increase of the harmonic components in the stator current, which causes oscillation in D4 and absolute value of its amplitude. Referring to Figure 9(d), an increase is seen in the fluctuations in the average of the absolute value of the wavelet coefficients in D4 by increasing the load to 100% of the rated load. This fluctuation average in the absolute value of the coefficients increases to 0.5469 for 100% of the rated load. Table 4 indicates that the average value of the current amplitude reduces due to the increase in load, such that the current decreases from 9.54 A in the no-load motor to 8.92 A in the motor with 33% of the rated load. By increasing the load to 66%, this value is decreased to 8.81 A. The trend of this reduction continues by increasing the load. Finally, for a motor with 100% of the rated load, the current reduces to 8.74 A, and with 133% of the rated load, the current value reduces to 8.57 A. Therefore, the value of

Figure 10. Simulation of wavelet coefficient absolute in D4: (a) 0% of rated load, (b) 33% of rated load, and (c) 66% of rated load.

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the criterion function increases by increasing the load. According to Table 4, the value of the criterion function for the no-load faulty motor is 0.97% and for the faulty motor with 33% of the rated load, is 3.61%. A noticeable difference between these two values makes the proposed criterion function able to diagnose a faulty motor in different loads. It is observed that the criterion function increases from 4.59% in 66% of the rated load to 6.25% at the rated load. Finally, the criterion function increases to 6.62% for 133% of the rated load. As shown in Table 4, the introduced criterion function is a suitable index for diagnosis of the broken bars in a faulty induction motor under different loads. Figure 9(d) shows the experimental results for four rotor broken bars, which are in good agreement with the simulation results shown in Figure 9(c).

7. Conclusions In this article a novel criterion function was introduced to diagnose the breakage in rotor bars of the faulty induction motors. This criterion function helps to accurately diagnose the fault in induction motors under load variation. The broken rotor bars are also detected using spectrum analysis of stator current. It is experimentally proven that an increase of the load is followed by an increase in harmonic component amplitude of a faulty induction motor. A TSFE method was used to model the faulted induction motor. In this modeling, the effects of the spatial distribution of stator winding, nonuniform air-gap permeance, geometrical and physical characteristics of different parts of the motor, and the nonlinearity of the core materials were taken into account. The proposed algorithm was applied to the stator current of a healthy and faulty induction motor. The accuracy of the simulations was finally verified by the experimental results.

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