Broken Rotor Bar Detection in Line-Fed Induction ... - IEEE Xplore

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IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS, VOL. 44, NO. 3, MAY/JUNE 2008

Broken Rotor Bar Detection in Line-Fed Induction Machines Using Complex Wavelet Analysis of Startup Transients Fernando Briz, Senior Member, IEEE, Michael W. Degner, Senior Member, IEEE, Pablo García, Member, IEEE, and David Bragado

Abstract—Fault detection of line-connected induction machines using complex vector wavelets to analyze the transient stator currents during startup is proposed in this paper. When a machine is connected to the line, the startup transient is characterized by large stator (and rotor) currents as well as by large slips (i.e., the rotor speed is significantly smaller than the excitation frequency). The stator current of machines with damaged rotors includes large rotor speed dependent components during the startup transient. Such components, however, fade away or coincide with components not containing fault-related information (e.g., saturation-induced components) once the machine reaches steady state. Because of this, the startup transient provides an opportunity for performing diagnostics on the machine. This paper shows that the information contained in the startup transient signal can be effectively separated and detected using a complex vector wavelet transform. Index Terms—Broken rotor bar, diagnostics of electric machines, startup transient current, wavelet analysis.

I. I NTRODUCTION

D

IAGNOSTIC testing of induction machines is of tremendous importance to many applications where, on the one hand, it is necessary to prevent unexpected equipment downtime or severe equipment damage and, on the other hand, it is not feasible or desirable to interfere with the regular operation of the machine to conduct the diagnostic test [1]–[3]. Motor current signature analysis (MCSA) has been proposed as a method to detect stator faults, rotor faults, eccentricities, and bearings faults during steady-state operation [1]–[3]. MCSA exploits the fact that an imbalanced machine, when fed with a balanced three-phase voltage, produces specific components

Paper IPCSD-07-105, presented at the 2007 Industry Applications Society Annual Meeting, New Orleans, LA, September 23–27, and approved for publication in the IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS by the Electric Machines Committee of the IEEE Industry Applications Society. Manuscript submitted for review July 1, 2007 and released for publication October 22, 2007. This work was supported in part by the Research, Technological Development and Innovation Programs of the Spanish Ministry of Science and Education-ERDF under Grants MEC-04-DPI2004-00527 and MEC-ENE2007-67842-C03-01. F. Briz, P. García, and D. Bragado are with the Department of Electrical, Computer and Systems Engineering, University of Oviedo, 33204 Gijón, Spain (e-mail: [email protected]; [email protected]; dbragado@isa. uniovi.es). M. W. Degner is with Research and Advanced Engineering, Ford Motor Company, Dearborn, MI 48121-2053 USA (e-mail: [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIA.2008.921382

in the stator currents whose magnitudes and frequencies depend on the level of asymmetry and on the fault condition creating it. The startup transient of the stator current also provides an opportunity for performing machine diagnostics. When a machine is connected to the line, the startup transient is characterized by large stator (and rotor) currents as well as by large slips (i.e., the rotor speed is significantly smaller than the excitation frequency). Damaged rotors create large rotor speed dependent components in the stator current during the startup transient, which can be easily measured. These components, however, fade away or coincide with components not containing faultrelated information (e.g., saturation-induced components) once the machine reaches steady state, making their detection difficult. Rotor bar and end-ring breakage can have several causes [1], including: 1) thermal stress/cycling; 2) magnetic stress/cycling; 3) residual stresses from manufacturing; 4) dynamic stress/cycling from shaft torques, centrifugal forces, etc.; 5) mechanical stress due to loose laminations, bearing failure, etc.; and 6) environmental stresses, such as chemicals or moisture, that cause contamination and abrasion of rotor material. Detection of damaged rotor bars using startup transients requires the use of tools capable of processing the transient signals. Wavelet functions are well suited for this purpose, and a wavelet-based analysis has already been proposed for machine diagnostics using startup transient currents [4]–[10]. In these works, standard wavelet transforms (i.e., wavelet families proven useful in other applications) are used. One inconvenience of using such wavelets is that no clear criteria for selecting the wavelet function exist (discrete wavelet transform based filtering [11], Daubechies-8 wavelet [4]–[7], [10], Daubechies40 wavelet [8]), which results in an ad hoc selection. In [9], a continuous Morlet wavelet is proposed, and the analysis is based on an estimate of the torque. This requires measurement of the stator voltage as well as an estimate of the stator resistance, which significantly complicates the method’s implementation. Another inconvenience of using standard wavelet families is that, although the wavelet-based analysis can reveal differences between a healthy and a damaged machine, there is no clear relationship between the physical characteristics of the signal and the results of the wavelet-based analysis. The use of a complex wavelet transform based on the physical properties exhibited by the fault-related currents is proposed in this paper for the detection of broken rotor bars. The wavelet is designed to detect a well-defined pattern of the stator current vector during a startup transient. Since the pattern is based on the characteristics of rotor fault related components in the stator

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BRIZ et al.: BROKEN ROTOR BAR DETECTION IN LINE-FED INDUCTION MACHINES

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Fig. 1. (a) Magnitude of the stator current vector and (b) rotor speed during startup (400 V rms, 50 Hz) with the motor driving a large inertia load. TABLE I INDUCTION MOTOR PARAMETERS (fe (50 Hz)

current vector, design of the wavelet, as well as the selection of its parameters, is deterministic. II. A NALYSIS OF THE S TATOR C URRENT V ECTOR D URING S TARTUP Fig. 1 shows the stator current vector magnitude and the rotor speed of a test machine after being connected to the line. The parameters of the machine are shown in Table I. Since the transient stator current vector after connection is a nonstationary signal, conventional fast Fourier transformbased analysis is not adequate. Instead, the short time Fourier transform (STFT) can be used for these purposes [3]. Fig. 2(a) shows the spectrogram of the stator current vector (1), which was obtained using the STFT during the transient shown in Fig. 1 for the case of a healthy machine. A sampling frequency of 5 kHz with a window width of 2 048 samples was used isqds =

2 (ia + ib ej2π/3 + ic ej4π/3 ). 3

(1)

The most significant component in the spectrogram is at ωe , which corresponds to the fundamental current. Additional components of reduced magnitude (note the logarithmic scale of the color bar) can also be observed. Horizontal lines correspond to components that do not vary with the rotor speed. Horizontal lines (beside the one caused by the fundamental excitation) can be caused by saturation, nonsinusoidal line voltages, or interactions between the fundamental excitation and asymmetries intrinsic to the design of the machine (e.g., stator slotting). The component at −ωe is primarily caused by asymmetries in the stator windings, which are intrinsic to the manufacturing process, or by differences in the gains of the phase current measurements. The dc component is caused by dc offsets in the phase current measurements. All these components generally have a small magnitude compared to the

Fig. 2. Spectrogram of the stator current vector during startup with the motor driving a large inertia load, for the case of (a) a healthy machine and (b) a machine with a broken rotor bar. Color bar units: amperes.

signals of interest, but they are still visible when a logarithmic scale for the spectrogram magnitude is used. Rotor speed dependent components can also be observed in the spectrogram. Such components have previously been reported [3] and were systematically observed to be present in all healthy machines tested during the present research. A common criterion, which is applied in MCSA methods, is to use these components to detect a fault condition when their magnitude becomes greater than a preset percentage of the fundamental current magnitude (e.g., less than 50 dB smaller) [2]. Fig. 2(b) shows the spectrogram of the stator current vector during a startup transient and during steady-state operation for the case of a machine with a broken rotor bar. The rotor was modified by drilling the end ring. This had the effect of disconnecting (breaking) this rotor bar from the rotor circuit while maintaining the continuity of the end ring. Components of the stator current vector caused by a damaged rotor bar are given by (2), where ωe is the electrical frequency, s is the slip, p the number of pole pairs, and with k/p equal to 1, −5, 7, −11, 13, . . .. The most relevant component, reflecting a broken rotor bar (3), denoted as ωbrb_2 , is obtained from (2) with k/p equal to 1. It is noted that the frequency components obtained from (2) are similar to those obtained for the case when the analysis is performed on a single-phase current instead of on the stator current vector, with k/p equal to 1, 5,

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7, 11, 13, . . ., in this case [3], since only positive frequencies exist ωbrb = ωe

(1 − s) ±s k p

ωbrb_2 = ωe (1 − 2s) = −ωe + 2ωr .

(2) (3)

Using (2) and (3), the stator current vector can then be modeled as (4), where the first term on the right-hand side represents the current of a healthy machine and the rest of the terms represent the rotor fault-induced components isqds isqds healthy+Ibrb_2 ej(−θe +2θr )+ΣIbrb_k ej(−θe +θr (1+k/p)) . (4) The term Ibrb_2 in (4), which is obtained with k/p equal to 1, was specifically excluded from the summation (which accounts for the remaining terms k/p equal to −5, 7, −11, 13, . . .,) since it will be the focus of further analysis. Some interesting conclusions can be reached from Fig. 2(b). First, rotor speed dependent components can be readily observed during the startup transient, but they have a significantly reduced magnitude in the steady state. In addition, all of the rotor speed dependent components converge to frequencies in the steady state that are spectrally close to the fundamental excitation frequency or its harmonics, making their separation in the steady state difficult, especially if the machine is not heavily loaded. This leads to the conclusion that significantly richer information exists in the current vector during startup than exists during steady-state operation. The separation of this information from the overall stator current requires careful signal processing. The STFT, combined with a spectrogram representation of Fig. 2, is one option, but, while visually insightful, it is not easy to establish a metric that indicates the rotor condition.

method. The angle θrw is then obtained by integrating ωrw , as shown in (6) ψ(t) = hej(−θe (t)+2θr w(t)) θe (t) = ωe dt, θr w(t) = ωrw (t)dt.

(5) (6)

It should be noted that determination of the absolute value of ωrw is not critical; only the shape of ωrw during the startup transient is important. The wavelet (5) can be scaled (i.e., stretched or shrunk) to effectively sweep the expected range of startup transient time lengths and shifted to find the exact time in which the startup transient occurred. This is done by defining θrw (7) and h as a function of scale (denoted as a) and time translation (denoted as b). (Discussion about the design of h is presented in the next section). The resulting wavelet function is given by (8) θrw (a, b) = ωrw ((t − b)/a) dt (7) ψ(a, b) = h(a, b)e j(−θe +2θrw (a,b)) .

(8)

Since the wavelet definition (8) relies on an estimate of the rotor speed shape, the reliability of the method can be reduced if significant variations in the load characteristics and, consequently, in the rotor speed shape occur. It is emphasized that variations of the transient length do not prevent the method from providing reliable fault detection, as will be demonstrated in Section IV. Instead of assuming a rotor speed shape for the design of the wavelet, it is also possible to estimate the rotor speed for each start-up transient (e.g., by means of ridge detection in the spectrogram) [10]. This would have the benefit of reducing the effects due to variations of the rotor speed shape. However, the results reported in [10] show significant inaccuracies, with a significant harmonic content, in the estimated rotor speed. Its impact on the performance of the method should, therefore, be thoroughly investigated. B. Wavelet Transform

III. D ETECTION OF D AMAGED R OTOR B ARS U SING C OMPLEX V ECTOR W AVELETS A. Wavelet Design Wavelets are an efficient tool for pattern detection in transient signals. This can be done by designing a wavelet resembling the shape of the signal to be detected. Damaged rotor bars induce a component Ibrb_2 (4) in the stator current vector. A wavelet function ψ can be defined (5) that resembles this component during a startup transient. This wavelet consists of a complex exponent multiplied by a windowing function h. The complex exponent is a function of the electrical angle θe and an estimate of the rotor angle θrw . The angle θe is easily obtained by integration of ωe , which is known and constant (6). Obtaining the angle θrw is not so straightforward, since the rotor angle (or speed) is not usually available in line-connected machines. Consequently, an estimate of the rotor speed during the startup transient ωrw is used. This estimate is obtained from the −ωe + 2ωr component in the spectrogram shown in Fig. 2 by using a simple polynomial curve fit

Once ψ is defined (8), the coefficient of the wavelet transform C is obtained using (9) [12], with ‘∗’ standing for the complex conjugate 1 (9) ψ ∗ (a, b)isqds dt. C(a, b) = √ a This coefficient is a function of a (i.e., how much the base wavelet is stretched or shrunk) and b (i.e., how much the wavelet is shifted in time with respect to the signal being analyzed). Evaluation of (9) is made by changing a and b at short, regular steps, with the resulting transformation usually being referred to as a continuous wavelet transform [12]. Selecting a and b in (9) is straightforward since they are directly related to characteristics of the signal being analyzed. The limits for a are related to the minimum and maximum startup transient time lengths, and the limits for b are related to how accurately the startup transient can be detected. Discussion on the selection of these limits is presented in Section VII. It can be observed that, although the integral symbol was used in (9),


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Fig. 3. Magnitude of the stator current vector component at −ωe + 2ωr [Ibrb_2 in (4)] versus ωr for the case of a healthy machine and a machine with a broken rotor bar during the startup transients of Fig. 2.

this equation actually operates with sampled signals, with the integration being transformed into a summation in the practical implementation. C. Selection of the Windowing Function The windowing function h in (5) is required so that the wavelet ψ has a finite length and a smooth transition, since the complex exponential term in (5) has a constant magnitude equal to one. Fig. 3 shows the magnitude of the stator current vector −ωe + 2ωr component versus ωr for the case of a healthy machine and for the case of a machine with a broken rotor bar. They were obtained from signal processing of the spectrograms in Fig. 2(a) and (b), respectively. While the differences between the two signals are noticeable, both signals increase significantly for values of ωr near ωe (50 Hz). This is due to the fact that the frequency being tracked (3) approaches the fundamental frequency of the current, which has a much larger magnitude and is difficult to separate from the desired signal. A distortion is also observed for values of ωr near 0, which is caused by the spectral closeness of the −ωe component. It follows from the previous discussion that the windowing function h should vary as a function of the rotor speed. More specifically, h should be equal to zero for its beginning and ending values (i.e., for values of ωrw near 0 and ωe ). This is schematically shown in Fig. 4. Given the rotor speed shown in Fig. 4(a), a suitable windowing function could be of the form shown in Fig. 4(b).The window has two transition regions (t1 − t2 and t3 − t4 ) of variable magnitude and a mid-region (t2 − t3 ) of constant magnitude. The transition regions are defined by (10) and (11), with their design being similar to a hanning window 1 − cos(α12 ) , with 2 (t − t1 )π and t1 < t < t2 = (t2 − t1 ) 1 + cos(α34 ) , with = 2 (t − t3 )π and t3 < t < t4 . = (t4 − t3 )

h12 = α12 h34 α34

(10)

(11)

This window will be used for most of the experimental results presented in this paper. It is noted, however, that a variety of windowing functions can be used, each providing similar results. This will be discussed in more detail in Section V.

Fig. 4. Construction of the complex vector wavelet ψ. (a) Estimated rotor speed. (b) Windowing function h. (c) Real and imaginary components of ψ. (d) Phase angle of ψ.

The real and imaginary components of the resulting wavelet function (5) are shown in Fig. 4(c), with its magnitude and phase in Fig. 4(b) and (d), respectively. IV. E XPERIMENTAL R ESULTS The proposed method was tested using the machine in Table I. Fig. 5(a) and (b) show the estimated rotor speed ωrw and the windowing function h defined by (10) and (11) for different scales of a = 0.2 and a = 2, which are the limits over which the wavelet transform (9) was evaluated, and for a = 1. Fig. 5(c) shows the magnitude of the windowed stator current for these values of a. Fig. 6(a) shows the magnitude of the C coefficient of the proposed wavelet transform for the case of a healthy machine. Fig. 6(b) shows the coefficient of the wavelet transform for the case of a machine with a broken rotor bar. The estimated start of the transient was assigned a time t = 0; therefore, the b parameter of the wavelet transform (i.e., the time translation of the wavelet function ψ) was evaluated in a range around this value (−0.2 to 0.3 s in the figure). The a parameter was varied within a range of 0.2 to 2 times the base function. The region of Fig. 6(b) that shows the most significant differences with respect to Fig. 6(a) is zoomed. The values of a and b that define this region are of greatest interest for this particular startup of a machine with a damaged rotor bar. Some interesting conclusions can be reached by comparing Fig. 6(a) and (b).

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Fig. 7. Magnitude of the wavelet transform coefficient |C| during a startup transient, as a function of a (scale), (b = −0.04 s), for the case of a healthy machine and a machine with a broken rotor bar.

Fig. 8. Magnitude of the stator current vector and of the windowed stator current vector during startup transient. Fig. 5. (a) Estimated rotor speed, (b) windowing function, and (c) magnitude of windowed stator current vector resulting from applying the windowing functions to the stator current vector magnitude of Fig. 1. In all cases, b = 0.

Fig. 6. Magnitude of the wavelet transform coefficient |C| during the startup transient shown in Fig. 5 as a function of a and b (relative to the estimated start of the transient). (a) Healthy machine. (b) Machine with a broken rotor bar.

First, the coefficient of the wavelet transform for the case of a healthy machine, shown in Fig. 6(a), has a rather constant value (uniform color), independent of the values for a and b. This means that the wavelet transform effectively eliminates components of the stator current vector different from the one modeled by the wavelet function ψ (5). Second, the coefficient of the wavelet transform for the case of a machine with a broken rotor bar has large values for well-defined values of a and b, which reflects that a good correlation exists between the wavelet ψ and the signal for these values of a and b. Fig. 7 shows the value of C, as a function of a, from Fig. 6 for b = −0.04 s, for the cases of a healthy machine and a machine with a broken rotor bar. It can be observed that the maximum value of C is obtained for a = 0.86 (i.e., the startup transient was shorter than the wavelet base function). It can also be noted from Fig. 7 that a metric as simple as the peak value of |C| can effectively be used to indicate the condition of the machine.To assess the robustness of the method against variations in the startup transient characteristics, the analysis was repeated with the machine driving a significantly smaller inertia. Fig. 8 shows the magnitude of the stator current vector during the startup as well as the windowed current. Fig. 9(a) and (b) show the magnitude of the coefficient C for the case of a healthy machine and of a machine with a broken rotor bar, respectively. Regardless of the significant change in the startup transient’s length, the wavelet transform in Fig. 9(b) clearly displays the presence of a fault. Fig. 10 shows the values of C, as a function of a for b = 0.06 s, both for the healthy m achine and for the machine with a broken rotor bar. It is finally noted that the application of the method requires a minimum transient length. Such limitation has often been reported for diagnostics methods using startup transients [4]–[7]. Though this issue has not yet been thoroughly investigated for the proposed method, it is expected, from the preliminary results, that transient lengths in the range of one second would be enough to allow reliable fault detection.


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Fig. 11. (a) Estimated rotor speed. (b) Different windowing functions h.

Fig. 9. Magnitude of the wavelet transform coefficient |C| during the startup transient shown in Fig. 8, as a function of a and b (relative to the estimated start of the transient). (a) Healthy machine. (b) Machine with a broken rotor bar. Fig. 12. Magnitude of the wavelet transform coefficient |C| during a startup transient, as a function of a (scale) for the case of a healthy machine and a machine with a broken rotor bar with three different windowing functions (b).

Fig. 10. Magnitude of the wavelet transform coefficient |C| during a startup transient, as a function of a (scale), (b = −0.04 s), for the case of a healthy machine and a machine with a broken rotor bar.

V. W INDOWING F UNCTION D ESIGN The basic criteria for the design of the windowing function was given in Section III-B, with the most important requirement being that the wavelet function (5) be equal to zero for values of ωrw near zero and ωe . In addition, smooth functions (i.e., without sudden variations) are preferred, as they are less sensitive to noise present in the sampled currents. Provided that these two conditions are met, the proposed method was observed to be highly insensitive to the design of the windowing function. To illustrate this, the different windowing functions shown in

Fig. 11 were tested for their suitability. The experimental results obtained with the windowing function used in Section IV [windowing function #1 in Fig. 11(b)] will be used as a reference. To ease the comparison, all the windows in Fig. 11(b) were normalized to have the same energy (area underneath their curves). Windowing function #2 in Fig. 11(b) was deliberately designed to violate the rules stated above. It was created using (10) and (11), the same as windowing function #1, but with the intervals t1 − t2 and t3 − t4 incorrectly set, which results in the magnitude of h not being zero for values of ωrw near zero and ωe [Fig. 11(a)]. This results in the coefficient C [Fig. 12(a)] showing a tremendous amount of distortion compared to that obtained with window #1 (Fig. 7) and makes fault detection unreliable. Windowing function #3 in Fig. 11(b) corresponds to a hanning window. Fig. 12(b) shows the resulting coefficient C, as a function of the scale a. It can be observed that the differences between the healthy and faulty machines are readily

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Fig. 14. Spectrogram of the phase-A current during a startup with the motor driving a large inertia load, for the case of machine with a broken rotor bar [same transient as Fig. 2(b)].

Fig. 13. (a) Definition of the windowing function h versus the rotor speed, (b) and (c) magnitude of the wavelet transform coefficient |C| during the startup transients shown in Figs. 1 and 8, respectively, as a function of a (scale), for the case of a healthy machine and a machine with a broken rotor bar, with ωr1 = 1 Hz, ωr2 = 5 Hz, ωr3 = 40 Hz, ωr4 = 45 Hz, respectively.

distinguishable but with reduced values of C compared to windowing function #1 (Fig. 7). This can be explained by the fact that windowing function #3 weighs the mid portion of the signal more heavily, where the −ωe + 2ωr component of the stator current vector has a reduced magnitude (Fig. 3). Two consecutive hanning windows can also be used [windowing function #4 in Fig. 11(b)]. This windowing function more closely resembles the shape of the stator current vector −ωe + 2ωr component’s magnitude when a fault in the rotor exists (Fig. 3) than window #3, providing a greater weighting to the portion of the signal that reflects a fault in the rotor. Fig. 12(c) shows the results provided by this window. All the windowing functions presented so far were designed as a function of time [see (10) and (11)]. It should be noted that the time intervals t1 − t2 and t3 − t4 were actually chosen according to the shape of the rotor speed ωrw . This suggests the idea of designing the windowing function directly as a function of the rotor speed, instead of time. This is schematically shown in Fig. 13(a). Two transition regions, ωr1 − ωr2 and ωr3 − ωr4 , are defined with a mid-region ωr2 − ωr3 of constant magnitude. Similarly to (10) and (11), the transition regions are defined by h12 =

1 − cos(γ12 ) (ωrw − ωr1 )π , with γ12 = 2 (ωr2 − ωr1 ) and

h34

ωr1 < ωrw < ωr2

(12)

1 + cos(γ34 ) (ωrw − ωr3 )π , with γ34 = = 2 (ωr4 − ωr3 ) and

ωr3 < ωrw < ωr4 . (13)

Fig. 13(b) shows the results of the wavelet transform using (12) and (13), normalized to have the same energy as the earlier

windowing functions. The results are similar to those obtained with windowing function #1. In general, defining the windowing function as a function of rotor speed, (12) and (13), would be preferred, since it explicitly establishes the limits for the different regions as a function of the rotor speed shape ωrw , which has been shown to be the key quantity for a correct design. It can also be concluded from the previous analysis that the precise design of the windowing function is not critical for the accuracy of the method if simple, well-defined requirements are met. VI. M ODIFICATION OF THE M ETHOD FOR USE W ITH A S INGLE -P HASE C URRENT M EASUREMENT The wavelet-based analysis described so far was performed on a transient stator current vector (1). It is also possible to use this method, with relatively minor modifications, on a single-phase current. Fig. 14 shows the spectrogram of the phase-A current during the same startup transient of Fig. 2(b). Components in the stator phase current caused by a damaged rotor bar are readily seen, which are now modeled by (14) [3], with k/p equal to 1, 5, 7, 11, 13, . . .. The biggest difference with respect to the case of the stator current vector is that only positive frequencies are shown (or, more precisely, negative frequencies will mirror positive frequencies). The most relevant component, reflecting a broken rotor bar (15), is obtained from (14) with k/p equal to 1 (1 − s) ±s (14) ωbrb = ωe k p ωbrb_2 = |ωe (1 − 2s)| = | − ωe + 2ωr | .

(15)

To account for the fact that the frequencies are now only positive, the wavelet (8) can be modified to (16) with the discussion on the design of the windowing function presented in the previous sections still being valid ψ(a, b) = h(a, b) cos (−θe + 2θrw (a, b)) .

(16)

The most significant differences that can be observed between the spectrograms shown in Fig. 2(b) (current vector) and Fig. 14 (phase current) occur at the very beginning of the transient. For the current vector case, the frequency of the component being tracked (3) is spectrally close to the component at −ωe , which has a relatively reduced magnitude. On the other


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Fig. 15. Magnitude of the wavelet transform coefficient |C| for a single-phase current measurement during a startup transient, as a function of a (scale), for the case of a healthy machine and a machine with a broken rotor bar, for two different startup transient lengths. Subplot (a) corresponds to the startup transient shown in Fig. 1(a). Subplot (b) corresponds to the startup transient shown in Fig. 8.

hand, for the single-phase current case, the component being tracked (15) is spectrally close to the component at ωe (i.e., the fundamental current), which has a much larger magnitude. To minimize the interference caused by the ωe component at the beginning of the transient, the windowing function needs to be redesigned, increasing both ωr1 and ωr2 from (12), as shown in Fig. 13(a) [or increasing t1 and t2 from (10) and Fig. 4(b)]. This reduces the portion of the transient signal being effectively integrated by the wavelet transform (8) and results in decreased sensitivity. Fig. 15 shows the magnitude of the C coefficient from the processing of a single-phase current. Slight differences with respect to the case of using the stator current vector can be observed. It can be concluded from these results that the proposed method is also applicable when measuring and processing only a single-phase current. This reduces the hardware requirements but results in a slight decrease in the performance of the method. VII. I MPLEMENTATION OF THE M ETHOD The proposed method requires the measurement of three current sensors for the implementation described in Section III. The use of two current sensors is adequate for these purposes, however, and is the more typical implementation for calculating the stator current vector. A single-current sensor is only needed for the implementation described in Section VI. The method can be implemented for continuous monitoring or as a part of periodic maintenance. It is equally valid for both delta- and wye-connected machines and for machines with open as well as closed rotor slots. For the experiments presented in this work, conventional Hall effects current sensors and 12-bit A/D converters were used. Fig. 16 shows the flowchart of the method. For the wavelet design, the rotor speed ωrw during a startup transient is estimated and stored. As already mentioned, the spectrogram of the startup current of a healthy machine was used to estimate the rotor speed for this work. The windowing function h is then

Fig. 16. Flowchart showing implementation of the method.

designed as discussed in Section V and stored for later use. The stored ωrw and h are then used for the wavelet-based analysis each time that a startup occurs. The peak value of |C| has been used in the analysis presented to this point to indicate the condition of the machine. It is noted, however, that this implies the establishment of a threshold that differentiates a healthy from a faulty machine, which cannot be easily done when no previous data from the machine of interest exist. It is recommended that Cpeak be measured when a machine is first installed or is known to be healthy; after that, increments or changes in the value of Cpeak would indicate a deterioration of the rotor. Tracking the changes in Cpeak could also be used for diagnostics in machines already installed and for which the condition of the rotor is unknown. The experimental results shown in Figs. 6 and 9 used a wide range of values for a and b to better show the results of the method. However, to reduce the computational and time requirements of the method, the range of evaluation for a and b parameters can be significantly reduced by simple analysis of the startup transient current vector magnitude (Figs. 1 and 8). The start of the transient can typically be accurately established, which suggests that b could be restricted to a single value. However, it was found convenient in practice to allow a narrow range of evaluation for b. This allows for compensation of effects like those caused by the nonideal (nonrepetitive) behavior of the breakers, the variation of the instant in time relative to the period of the phase voltages in which the breakers were ordered to close, as well as for inaccuracies in the rotor speed ωrw . For the experiments presented in this paper, the time translation parameter b was changed in steps of 5 ms, with a range of variation in the order of ±0.1 s relative to the theoretical start of the transient. As for the selection of the range of a, line-connected machines often show repetitive startup transients, allowing for narrow bounds to a, which would be specific for each application. For the case of machines showing significant variations in

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the startup transient length, simple threshold-based analysis of the startup current can be used to dynamically adapt the limits for a for each startup transient. For the experiments presented in this paper, a was changed in steps of 0.02. The calculation of the C coefficient for each value of a and b, including scaling of h and ωrw , took ≈0.06 s for a standard computer. In the top-right corner of Figs. 6(b) and 9(b), the wavelet transform coefficient is shown with a and b restricted to a reduced range of values, which were automatically obtained by the analysis of startup transient current described above. VIII. C ONCLUSION Broken rotor bar detection in line-connected induction machines using complex wavelets to analyze the stator currents during startup transients was presented in this paper. These wavelets allow for the accurate detection of fault-related components in the stator current vector that indicate damaged rotor bars. In comparison with standard wavelet-based methods, the wavelet design is made following a well-defined procedure. Because of this, limits for the scale and time translation parameters of the transform are easily established, reducing the computational requirements of the transformation. Experimental results confirm the effectiveness of the method in detecting damaged rotor bars. ACKNOWLEDGMENT The authors wish to acknowledge the support and motivation provided by the University of Oviedo, Spain, and Ford Motor Company, USA. R EFERENCES [1] S. Nandi and H. A. Toliyat, “Condition monitoring and fault diagnosis of electrical machines—A review,” IEEE Trans. Energy Convers., vol. 20, no. 4, pp. 719–729, Dec. 2005. [2] W. T. Thomson and M. Fenger, “Current signature analysis to detect induction motor faults,” IEEE Ind. Appl. Mag., vol. 7, no. 4, pp. 26–34, Jul./Aug. 2001. [3] M. E. H. Benbouzid, “A review of induction motors signature analysis as a medium for faults detection,” IEEE Trans. Ind. Electron., vol. 47, no. 5, pp. 984–993, Oct. 2000. [4] H. Douglas, P. Pillay, and A. K. Ziarani, “A new algorithm for transient motor current signature analysis using wavelets,” IEEE Trans. Ind. Appl., vol. 40, no. 5, pp. 1361–1368, Sep. 2004. [5] H. Douglas, P. Pillay, and A. K. Ziarani, “Detection of broken rotor bars in induction motors using wavelet analysis,” in Proc. IEEE IEMDC, Jun. 2003, vol. 2, pp. 923–928. [6] H. Douglas and P. Pillay, “The impact of wavelet selection on transient motor current signature analysis,” in Proc. IEEE Int. Conf. IEMDC, 2005, pp. 80–85. [7] H. Douglas, P. Pillay, and A. K. Ziarani, “Broken rotor bar detection in induction machines with transient operating speeds,” IEEE Trans. Energy Convers., vol. 20, no. 1, pp. 135–141, Mar. 2005. [8] J. A. Antonino-Daviu, M. Riera-Guasp, J. R. Folch, and M. P. M. Palomares, “Validation of a new method for the diagnosis of rotor bar failures via wavelet transform in industrial induction machines,” IEEE Trans. Ind. Appl., vol. 42, no. 4, pp. 990–996, Jul./Aug. 2006. [9] F. Niu and J. Huang, “Rotor broken bars fault diagnosis for induction machines based on the wavelet ridge energy spectrum,” in Proc. 8th ICEMS, Sep. 2005, vol. 3, pp. 2274–2277. [10] R. Supangat, N. Ertugrul, W. L. Soong, D. A. Gray, C. Hansen, and J. Grieger, “Detection of broken rotor bars in induction motor using starting-current analysis and effects of loading,” Proc. Inst. Electr. Eng.—Electr. Power Appl., vol. 153, no. 6, pp. 848–855, Nov. 2006.

[11] J. M. Aller, T. G. Habetler, R. G. Harley, R. M. Tallam, and S. B. Lee, “Sensorless speed measurement of AC machines using analytic wavelet transform,” IEEE Trans. Ind. Appl., vol. 38, no. 5, pp. 1344–1350, Sep./Oct. 2002. [12] Wavelet Toolbox User’s Guide, The MathWorks, Inc., Natick, MA, 2007.

Fernando Briz (A’96–M’99–SM’06) received the M.S. and the Ph.D. degrees from the University of Oviedo, Gijón, Spain, in 1990 and 1996, respectively. From June 1996 to March 1997, he was a Visiting Researcher at the University of Wisconsin, Madison. He is currently an Associate Professor in the Department of Electrical, Computer and Systems Engineering, University of Oviedo. His topics of interests include control systems, high-performance ac drives control, sensorless control, diagnostics, and digital signal processing. Dr. Briz received the 2005 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS Third Place Prize Paper Award and was the recipient of two IEEE Industry Applications Society Conference prize paper awards in 1997, 2003, and 2007, respectively.

Michael W. Degner (S’95–A’98–M’99–SM’05) received the B.S., M.S., and Ph.D. degrees in mechanical engineering from the University of Wisconsin, Madison, in 1991, 1993, and 1998, respectively. In 1998, he joined the Ford Research Laboratory, Dearborn, MI, working on the application of electric machines and power electronics in the automotive industry. He is currently the Manager of the Electric Machine Drive Systems Department of the Hybrid Electric Vehicle and Fuel Cell Vehicle Laboratory in Research and Advanced Engineering, Ford Motor Company, where he is responsible for the development of electric machines, power electronics, and their control systems for hybrid and fuel-cell vehicle applications. His interests include control systems, machine drives, electric machines, power electronics, and mechatronics. Dr. Degner received the 2005 IEEE TRANSACTIONS ON INDUSTRY APPLICATIONS Third Place Prize Paper Award and has been the recipient of several IEEE Industry Applications Society Conference paper awards.

Pablo García (S’02–M’06) received the M.S. and the Ph.D. degrees in electrical engineering and control from the University of Oviedo, Gijón, Spain, in 2001 and 2006, respectively. From 2002 to 2006, he was the recipient of a fellowship of the Personnel Research Training Program funded by the Spanish Ministry of Education. In 2004, he was a Visiting Scholar at the Wisconsin Electric Machines and Power Electronics Consortium (WEMPEC), University of Madison, Wisconsin. He is currently with the Department of Electrical, Computer and Systems Engineering, University of Oviedo. His research interests include sensorless control of ac machines, diagnostics of inductions motors, signal processing, and neural networks.

David Bragado was born in Gijón, Spain, in 1978. He received the B.Sc. and the M.Sc. degrees in computer science from the University of Oviedo, Gijón, Spain, in 2001 and 2005, respectively. He is currently working toward the Ph.D. degree in the Department of Electrical Engineering, University of Oviedo. His current research is focused on condition monitoring of electrical machines, including lineconnected as well as fed from soft-starters and inverters.