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�Abstract -- The diagnosis of induction machine faults is commonly carried out by means of Motor Current Signature Analysis (MCSA), i.e., by classical spectrum analysis of the input currents. Specifically in case of broken bars, the amplitude of the left sideband component of a phase current is monitored in order to sense its signature. However MCSA has some drawbacks that are still under investigation. The main concern is that an efficient frequency transformation cannot be made under speed–varying condition, since slip and speed vary and so does the left sideband frequency. In this paper, an advanced use of the Discrete Wavelet Transform (DWT) is proposed to overcome the limitation of the classical approaches based on Fourier Analysis (FA). Experimental and simulation results show the validity of the developed approach, leading to an effective diagnosis method for broken bars in induction machines.
Index Terms -- Fault diagnosis, broken rotor bars, speed transient, wavelet analysis.
I. INTRODUCTION n-line diagnosis and the early detection of faults in induction machine have attracted the attention of
researchers, since they allow reducing maintenance costs and down-time. Investigations on different failure modes in induction motors have revealed that 19% of the overall motor faults are related to the rotor part [1]. A detailed analysis of this type of fault can be found in [2]. For several applications, the motors operate continuously in time-varying conditions. In these contexts, the classical application of Fourier Analysis (FA) for processing variables fails as slip and speed vary. Thus the fault frequency components are spread over a bandwidth proportional to the speed variations. Among different solutions, high resolution frequency estimation [3] and more recently Signal Demodulation (SD) technique [4] have been developed to reduce the effect of the non periodicity on the analyzed signals. These techniques, based on FA, give high quality discrimination between healthy and faulty conditions but do
Y. Gritli and A. Chatti are with National Institute of Applied Sciences
and Technology, University of Carthage. Tunis – Tunisia. (e-mail: [email protected], [email protected] ).
C. Rossi, A. Stefani, F. Filippetti, L. Zarri are with University of Bologna, Department of Electrical Engineering -Italy. Phone: +39 051 20 93 564 Fax: +39 051 209 3588. (e-mail: [email protected], [email protected], [email protected], [email protected], [email protected] ).
not provide time domain information. This shortcoming was overcome by Wavelet Analysis, which provide greater resolution in time for high frequency components and greater resolution in frequency for low frequency components. Wavelets have a window that automatically adjusts to give the appropriate resolution observed on approximations and details signals. This method was used with different approaches for the diagnosis of rotor anomalies in induction machines such as wavelet ridge method [5] or wavelet coefficients analysis [6]. Later, approximations and details signals were investigated for reproducing the instantaneous evolution of rotor fault frequency components [7]-[8] and quantifying the fault extents [9], [10]. In the majority of the listed contributions, tracking rotor fault frequency components on multi frequency bands complicates the diagnosis process. The use of a single detail signal in [5] and more recently a single approximation signal in [10], whose level is imposed by the sampling frequency, were proposed for the same purposes. However the dependency on an appropriate choice of the sampling frequency penalizes these approaches, because the band of the fault harmonic component is not known in advance, due to the time varying operating conditions. On the other hand, the change of the sampling frequency in order to isolate rotor fault frequency components, as proposed in the most of the previous works, complicates considerably the processes of rotor fault detection based on wavelet analysis. In this paper, a simple and effective method is presented to solve the above open points in time-varying operating conditions. The technique has been first investigated for the diagnosis of stator and rotor electrical faults in Doubly Fed Induction Machine (DFIM) [11-12]. Hence, the proposed contribution is intended to be a validation of this new approach, based on DWT, applied to the detection of rotor bar breakage. This paper is organized as follows. Section II presents a diagnosis approach for fault detection based on an improved use of DWT. Simulations and experimental results for rotor broken-bar fault analysis under speed transient conditions are presented and commented in section III. Finally conclusions and perspectives are given in section IV.
Advanced Diagnosis of Broken Bar Fault in Induction Machines by Using Discrete Wavelet
Transform under Time-Varying Condition Y. Gritli*,**, C. Rossi**, Member, IEEE, L. Zarri**, Member, IEEE,
F. Filippetti**, Member, IEEE, A. Chatti*, D. Casadei**, Senior Member, IEEE, and A. Stefani**
O
2011 IEEE International Electric Machines & Drives Conference (IEMDC)
978-1-4577-0061-3/11/$26.00 ©2011 IEEE 424
II. FAULT FREQUENCY TRACKING – THE PROPOSED APPROACH
A. Rotor fault frequency propagation Like any other rotating electrical machine, the squirrel cage induction motor is subjected to both electromagnetic and mechanical forces symmetrically repartitioned. In healthy conditions only the fundamental frequency f exists in stator currents. If the rotor part is damaged, the rotor symmetry of the machine is lost and a reverse rotating magnetic field related to an inverse sequence component of the rotor currents, at frequency –sf (s denotes the slip), appears. This inverse sequence is reflected on the stator side and produces the frequency component (1−2s)f. Consequently a torque ripple and a speed ripple are generated at frequency 2sf, which modulates the rotating magnetic flux [2]. This modulation produces two current components, i.e., an additional left-side component at (1−4s)f and a right side component at (1+2s)f. Following this interaction process, the frequency content of the stator current shows a series of fault components at the following frequencies:
fbrb=(1±2ks)f (with k=1,2,3,…). (1) The propagation principle of this chain of frequency components is illustrated in Fig. 1.
Freq
uenc
y Pro
paga
tion
f
1/2sf
Time Propagation
1/f
Time-Frequencies Propagation
1/(1-2s)f
(1-2s)f2sf
±3sf
±sf
±1/sf*
Mechanicalspeed
periods
Rotorfreq.
Mechanicalspeedfreq.
Statorperiods
Statorfreq.
(1+2s) f
±1/3sf*
1/(1+2s)f
Rotorperiods
Fig. 1. Time-frequency propagation of rotor fault components: the sign (–) in time domain corresponds to inverse current sequence components. Under speed–varying condition, the sideband components expressed by (1), whose amplitudes have to be monitored for diagnostic purposes, are spread in frequency. It turns out that the direct application of the classical FA to machine stator currents is not effective. A simple solution for an efficient diagnosis of rotor broken bars for induction machines, under speed-varying conditions, is presented in the next section. B. Rotor fault frequency tracking under time-varying operating conditions The main feature of wavelet analysis is its high multiresolution analysis capability. Wavelet analysis is a
signal decomposition, using successive combinations of approximation and detail signals. The procedure for two-level decomposition is shown in Fig. 2, where (LPF) and (HPF) are respectively a low pass filter and a high pass filter. By using Nyquist sampling criterion, the signal is then down sampled by factor 2. The coefficients a1 and d1 represent respectively approximation and detail signals at first level of decomposition. The procedure is repeated until the original signal is decomposed to a pre-defined decomposition level.
2
2
H.P.F
L.P.F
2
2
H.P.F
L.P.F
d1(n)
a1(n)
a2(n)
d2(n)i(n)
1st Leveldecomposition
2nd leveldecomposition
Fig. 2 Wavelet Transform using a filter banks
With the well known dyadic down sampling procedure, frequency bands of the jth level decomposition are related to the sampling frequency fs. Hence, these bands given by [0, 2-
(J+1)fs] or [2-(J+1)fs , 2-Jfs] cannot be changed unless a new acquisition with different sampling frequency is made [13]. This fact complicates any fault detection based on DWT, particularly in speed-varying condition. In this paper, an efficient solution to overcome this limitation is proposed. Hereafter the sampling frequency fs is assumed equal to 3.2kHz, and a nine level decomposition (J=9) is chosen to cover the frequency bands in which the fault component is tracked. With regard to the type of mother wavelet, a 10th order Daubechies family is chosen, although other families (Dmeyer and Coiflet) also allow a clear detection of the phenomenon. Under healthy conditions, only the fundamental component f exists in the stator currents. If the rotor part is damaged, the first fault frequency component that occurs in the stator currents is (1–2s)f (Fig. 1).
TABLE I FREQUENCY BANDS AT EACH LEVEL
Approximations «aj »
Frequency bands (Hz)
Details « dj »
Frequency bands (Hz)
a9 : [0 – 3.125] d9 : [3.125 – 6.25] a8 : [0 – 6.25] d8 : [6.25 – 12.5] a7 : [0 – 12.5] d7 : [12.5 – 25] a6 : [0 – 25] d6 : [25 – 50] a5 : [0 – 50] d5 : [50 – 100] a4 : [0 – 100] d4 : [100 – 200] a3 : [0 – 200] d3 : [200 – 400] a2 : [0 – 400] d2 : [400 – 800] a1 : [0 – 800] d1 : [800 – 1600]
In time varying conditions the magnitudes of this fault
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component cannot be detected through a classical frequency analysis since it is spread in a wide frequency range. A simple processing of the stator phase current allows shifting the fault component (1-2sf) to a prefixed frequency band. In such a way, all the information related to the fault is isolated and confined in a single frequency band. More in details, a frequency sliding fsl is applied to the stator phase current at each time slice as shown in (2), so that the harmonic component of interest is moved to a frequency band in which the fault component will be tracked. Then the real part of the shifted signal is analyzed by means of DWT.
2( ) R e ( ) s li s aj f tI t t es l
π−= ⎡ ⎤⎣ ⎦
(2)
The choice of fsl is made by taking into account the speed range of the considered transient. The evaluation of fsl is made in such a way that the contribution of the fault frequency component, for the whole considered slip range, is shifted into one of the intervals [0, 2-(J+1)fs] or [2-(J+1)fs , 2-J fs] and consequently the DWT is applied to analyze only the frequency band of interest. In this way, the detection and quantification get easier.
aJ dJ dJ-1
[0 : fsam/2J+1] [fsam/2J+1 : fsam/2J] [fsam/2J : fsam/2J-1] [fsam/4 : fsam/2]
d1
Frequency
Fig. 3. The DWT filtering process
As illustrated in Fig. 3, the DWT analysis divides the signal into logarithmically–spaced frequency bands. From this bandwidth segmentation, it is possible to realize that the approximation aJ and detail dJ have the same and the smallest frequency band width equal to fsam/2J+1. Moreover, approximation aJ is less subjected to the overlapping effect than dJ due to its proximity to 0Hz [14]. Finally, approximation a9 was chosen for extracting the contribution of the fault component (1-2s)f. Therefore the choice of fsl is made to shift these fault components inside the frequency interval [0, 3.125] Hz corresponding to approximation a9 (see Table I). The proposed approach was applied to the stator current with fsl=45.5Hz to isolate the contribution of the (1-2s)f component and to allow a great precision in filtering the effects of the fundamental component and of other harmonics around the fault component of interest. With regard to the type of mother wavelet, the use of low order Daubechies mother wavelet (db10) has provided satisfactory results. Several types of mother wavelets (Daubechies, Coiflet, and Symlet) were tested in the analysis. Regardless of its different properties, the qualitative analysis of the results showed that no significant advantages appeared. Once the state of the machine has been qualitatively diagnosed, a quantitative evaluation of the fault
degree is necessary. For this purpose a dynamic multiresolution mean power indicator mPaj at different resolution levels j was introduced as a diagnostic index to quantify the extent of the fault. This index is defined as shown in (3):
2
1
1( ) ( )n
j s a jn
m P a i a nn
Δ
=
=Δ ∑
(3)
where aj(n) is the approximation signal of interest (a9 in our case), Δn denotes the number of samples and j the decomposition level. The fault indicator is periodically calculated over time, every 400 samples using a window of 6400 samples as depicted in Fig. 4 where δn is a set of 400 samples and Δn a set of 6400 samples.
''aj''
sign
al
Time
Samples
nΔ
nδTIN=1
TIN=2TIN=3
''aj''
sign
alse
quen
ces
Fig. 4 Principle of time interval calculation, TIN: Time Interval Number.
These values were regulated experimentally to reduce variations that can lead to false alarms in healthy operating conditions of the motor. When the fault occurs, the energy distribution of the signal is changed in the resolution levels related to the characteristic frequency bands of the default. Hence, the energy excess localized in the approximation is considered as an anomaly indication in case of rotor broken bar.
III. BROKEN BAR DETECTION AND QUANTIFICATION
In order to test the effectiveness of the rotor faults diagnostic procedures a transient model of the faulty squirrel cage induction machine is required. The model developed in [15] is proper for this aim. In that model, the following geometry of the rotor is considered: the N bars and the 2N end ring segments constitute a network with N+1 loops with N independent mesh currents and a circulating current in one of the end rings. The electrical parameter values of the N independent mesh loops are identical under healthy operating conditions but obviously differ if a break occurs. When this approach is adopted, it is necessary to compute the mutual inductance coefficients, which are dependent on the rotor position. This model is referred to a reference frame rotating with the rotor, therefore rotational voltages are only induced by stator currents. The assumed stator and rotor m.m.f. distributions allow an easy determination of the stator-stator, stator-rotor and rotor-rotor mutual inductances.
426
Fig. 5. Stator Windings and rotor loops of the model
By using the following assumptions: • infinite iron permeability • smooth air gap • quadrature-phase symmetrical stator windings with sinusoidal distribution of the air gap flux density • a rotor symmetrical cage, whose bars are insulated from the iron core
it is possible to obtain a set of first order voltage differential equations: two equations for the transformed stator windings, N equations for the rotor bar loops, and an extra equation corresponding to one of the end ring loops. Anyway the model does not loose in generality when rotor faults are considered. For more details, the model, implemented in Matlab Simulink®, is completely developed in [15]. In order to validate the results obtained in simulations a test bed was realized. A 7.5 kW, 2 poles pair induction machine was used (see Appendix). Two rotors are available, one healthy and one with a drilled rotor bar. The induction machine is coupled to a 9kW separately excited DC machine controlled in speed. This allows to reproduce the simulated speed transients. Photos of the experimental test bed are shown in Fig. 6. Machine currents and speed are sampled at 3.2 kHz with a time duration of 10 seconds. A smaller number of points could be used without affecting the performances of the proposed procedure.
Fig. 6. Test bed photo (left). The faulty rotor with one drilled rotor bar (right).
The induction motor has been initially simulated in a healthy conditions during a prefixed transient from 1496 rpm to 1460 rpm, corresponding to slip ranges from s=0.002 to s=0.026. Another simulation, for the same speed transient
was made with the machine operating with one rotor broken bar. Simulation results under healthy and faulty conditions are reported in Fig. 7. As explained in section III, the frequency band of interest for tracking the (1-2s)f fault component contribution is the approximation signal a9. For the sake of clarity only the a9 signal will be reported. In healthy condition, the 9th approximation signal a9, resulting from the wavelet decomposition of the stator phase current, does not show any kind of variation (Fig. 7-e). This indicates the absence of the fault component (1-2s)f, leading to diagnose the healthy condition of the motor under speed-varying condition.
However, in faulty condition (one rotor broken bar) approximation a9 shows significant variation in magnitude observed in Fig.7-f. During the first time period (t=0s to t=2s), under a constant speed of 1496 rpm, the 9th approximation signal, representative of the (1-2s)f frequency component, do not show any important variations. This is due to the fact that at low load level (under 4% of the nominal torque) the fault frequencies are practically superimposed to the fundamental and magnitudes are not detectable. During deceleration (around t=2s), a9 shows particular magnitude escalation proportional to the abrupt deceleration until reaching a quasi steady-state magnitude (under 66% of the nominal torque). More in detail, the oscillations observed in the signal a9, with quasi-constant amplitude, follow a characteristic pattern that fits the evolution in frequency of the fault component (1-2s)f, during the speed transient. The experimental results presented in Fig. 8 corroborate simulations although the magnitude evolutions, in some tests, are even bigger than those registered in simulation. The 9th approximation signal obtained from the experimental results shows the sensitivity and the effectiveness of this particular signal a9 to reproduce the contribution of the frequency component (1-2s)f under rotor broken bar condition. As introduced in the previous section, a fault indicator based on a cyclic mean power calculation of the approximation a9, resulting from the wavelet decomposition applied to one stator current, has been proposed to evaluate quantitatively the degree of the fault extent. The cyclic evolution of the fault indicator mPa9 issued from simulations (Fig. 7), under healthy and rotor broken bar conditions are depicted in Fig. 9. In healthy condition, and under large range of speed variations, mPa9 does not show any significant change. Consequently, the indicator values for the healthy motor are considered as a baseline to set the threshold for discriminating healthy from faulty conditions. Under rotor broken bar condition the calculated mPa9 indicator shows a noteworthy increase. The large energy deviation observed in faulty condition proves the effectiveness of the proposed approach, since the transient speed motor operation does not disturb the fault assessment, in comparison with the healthy case.
ds
qs
ei
rii
427
The corresponding results, issued from experimental tests (Fig. 8) under healthy conditions and under the drilled rotor bar are depicted in Fig. 10. Experimental results corroborate the simulation ones, thus proving a clear quantitative broken bar detection.
IV. CONCLUSION A diagnosis procedure for detection and quantification of rotor broken bar in squirrel cage motor, in speed-varying condition is proposed here. The presented approach is based on an improved use of the DWT by a simple pre-processing of the stator phase current. Once the state of the machine is qualitatively diagnosed, a cyclic time-frequency fault indicator was introduced to quantify the fault extent. It is obvious that the method is particularly advantageous even when the fault component of interest is very close to the fundamental one or in case of other faults that would produce their appearance of a
peculiar fault frequency. The proposed approach allows, with high precision, the extraction of the contribution of each frequency component separately. Simulation and experimental results demonstrate the effectiveness of this new approach for the detection of rotor broken bar in squirrel cage motor, under time varying conditions. Eventually, the technique can be extended to the investigation of mixed mechanical or/and electrical fault conditions.
V. APPENDIX Rated data Value Rated Power kW 7.5Rated stator voltage V 380Rated frequency Hz 50Rated speed rpm 1440Pole pairs 2Number of rotor bars 28
Fig. 7. Simulation results: DWT analysis of stator phase currents underhealthy and rotor broken bar conditions: (a, c and e) and (b, d and f) are theresults corresponding to healthy and faulty rotor case respectively.
Fig. 8. Experimental results: DWT analysis of stator phase currents underhealthy and rotor broken bar conditions: (a, c and e) and (b, d and f) are theresults corresponding to healthy and faulty rotor case respectively.
0 2 4 6 8 10
1460
1480
1500
Sp
eed
(rp
m)
- a -
0 2 4 6 8 10
-20
0
20
- c -
Cu
rren
t (A
)
0 2 4 6 8 10
-20
0
20
- d -
Cu
rren
t (A
)
0 2 4 6 8 10
1460
1480
1500
- b -
Sp
eed
(rp
m)
0 2 4 6 8 10
-1
-0.5
0
0.5
1
Time (s)
a9- f -
0 2 4 6 8 10
-1
-0.5
0
0.5
1
- e -
Time (s)
a9
0 2 4 6 8 10
1460
1480
1500
Sp
eed
(rp
m)
- a -
0 2 4 6 8 10
-20
0
20
- c -
Cu
rren
t (A
)
0 2 4 6 8 10
-20
0
20
- d -
Cu
rren
t (A
)
0 2 4 6 8 10
1460
1480
1500
- b -
Sp
eed
(rp
m)
0 2 4 6 8 10
-1
-0.5
0
0.5
1
Time (s)
a9
- f -
0 2 4 6 8 10
-1
-0.5
0
0.5
1
- e -
Time (s)
a9
Fig. 9. Simulation results : Cyclic values of the fault indicator mPa9, resulting from the 9th wavelet decomposition level of the signals Isl under healthy and rotor bar broken (red and blue), for speed-varying condition.
Fig. 10. Experimental results : Cyclic values of the fault indicator mPa9, resulting from the 9th wavelet decomposition level of the signals Isl under healthy and rotor bar broken (red and blue), for speed-varying condition.
0 10 20 30 40 50 60 70 800
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
TIN
mP
a9
Healthy Broken bar
0 10 20 30 40 50 60 70 800
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
0.5
TIN
mP
a9
Healthy Broken bar
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