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Winding Distribution Effects on Induction Motor Rotor Fault Diagnosis
8th IFAC Symposium on Fault Detection, Supervision and Safety of Technical Processes (SAFEPROCESS) August 29-31, 2012. Mexico City, Mexico
Winding Distribution Effects on Induction Motor Rotor Fault Diagnosis Pezzani C.* Bossio G.* De Angelo C.* *Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina *Grupo de Electrónica Aplicada, Facultad de Ingeniería, Universidad Nacional de Río Cuarto, X5804BYA, Río Cuarto, Córdoba, Argentina (Tel: +54-358-4676598; e-mail: [email protected]).,
Abstract: The use of stator currents harmonics as a potential tool for the diagnosis of rotor faults in induction motors is analyzed in this paper. The analysis presented is focused in those components produced by the interaction between the rotor faults and the harmonics of the spatial distribution of stator windings. These components are also compared with those resulting from the interaction between voltage harmonics and the rotor fault. A multiple coupled circuit model of the induction motor is used to evaluate the sensitivity of the proposed components for different winding configurations. Finally, experimental and simulation results are presented to validate the proposal. Keywords: Broken Bars - Fault Diagnosis - Induction Motor - WFA 1. INTRODUCTION Fault diagnosis in electrical machines, particularly in induction motors (IM), has been a very active research area in recent times. An important variety of strategies based on the measurement of machine currents has been proposed as options to the mechanical vibration analysis already accepted, for predictive maintenance of electrical motors (Bellini et al., 2008a ). Among these strategies, the motor current signature analysis (MCSA) is the most widespread. Particularly, broken bars or end-rings on the rotor produce sidebands around the fundamental component separated from it at twice the slip frequency (2sfs), where s is the slip and fs the supply frequency. The analysis of the amplitude of these sidebands has been proposed in numerous studies as an indicator of rotor faults (Bellini et al., 2001). However, the amplitude of these components also depends on the severity of the fault, and on factors such as the load level and inertia of the motorload set, among others (Bellini et al. 2001). What’s more, low-frequency oscillations in the load torque affect motor currents in a very similar way; that is, introducing sidebands around the fundamental component. Situations of this nature hinder the correct quantification of the failure and may even induce false diagnoses (Schoen and Habetler, 1995). To solve these problems, several proposals have been presented. Particularly, the analysis of current around high order harmonics has been proposed with the objective of improving rotor fault diagnosis (Schoen and Habetler, 1995). In general, these high frequency components are less sensitive to load oscillation due to the "filter" effect that inertia produces on oscillations. These high frequency components can be caused by the interaction between the asymmetry of the rotor - produced by broken bars - and either the harmonics due to supply voltage distortions or 978-3-902823-09-0/12/$20.00 © 2012 IFAC
unbalances, or the spatial harmonics due to the stator windings distribution or the rotor slots harmonics (RSH). The use of the components produced by the supply voltage distortion or unbalance to generate rotor fault indicators was proposed by Bruzzese (2008). Although these indicators are less sensitive to amplitude variations of voltage harmonics, it becomes necessary a certain degree of distortion. This can be quite easily achieved in inverter-fed motors. However, in motors fed directly from the grid, it may not occur, even less in medium voltage motors. Moreover, the use of components produced by a discrete distribution of the rotor (RSH) requires knowledge on some construction parameters such as the number of rotor bars and also requires an accurate estimate of the motor slip (Bellini et al., 2008b). The non-sinusoidal distribution of windings produces spatial harmonics on magnetomotive force (mmf), mainly at frequencies given by (6k ± 1)fs. The interaction between these spatial harmonics and rotor asymmetry produces characteristic components in the current spectrum of the IM. The components at frequencies (5-4s)fs, (5-6s)fs, (7-6s)fs and (7-8s)fs, located around the harmonics 5th and 7th, are the most significant ones (Sobczyk and Maciolek, 2003). The use of these components as indicators of the presence of broken bars has been presented in (Bossio et al. 2009) (Sobczyk and Maciolek, 2003). In (Bossio et al. 2009), it is analyzed the behaviour of the components (5-4s)fs and (7-6s)fs, for a particular distribution of windings, and for different numbers of broken bars, load inertia variations and supply voltage conditions. It was demonstrated that the magnitude of these components, besides increasing with the number of broken bars, hardly depends on the inertia of the whole system.
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This paper extends the analysis of the components (5-4s)fs and (7-6s)fs as indicators of broken bars in the IM. A multiple coupled circuit model of the IM is used (Luo et al., 1995, Bossio et al. 2004) to evaluate the sensitivity of the proposed components for different winding configurations. For this, winding functions are expressed in general by a Fourier series expansion. Through numerical simulation, using the proposed model, the amplitude of fault indicators are evaluated in terms of the harmonics of the winding distribution. The influence of voltage distortion on these indicators is also analyzed through simulation. Finally, experimental results that validate the proposal are presented.
Spatial Distribution”, and N (, z , r ) is the “Winding Modified Two-dimensional Function”. The latter can be expressed as, N (, z, r ) n , z, r 1 2 L g 1 , z , r 2
L
0 0 where
g 1 , z, r
(2)
n , z , r g 1 , z, r dzd
is the mean value of the air-gap
function inverse. 2.2. Winding distribution
2. MOTOR MODEL For the analysis of the harmonic components of the IM current, a multiple-coupled circuit model is used, proposed by Luo et al. (1995). This approach allows considering the real distribution of windings in the stator and rotor bars. Saturation effects and eddy currents are neglected in this formulation and the rotor bars are supposed to be isolated. In general, the IM consists of m stator circuits and k rotor bars. Particularly, the rotor squirrel-cage is modelled as k identical and equally spaced loops, consisting of two consecutive bars plus a current loop on one of the end-rings (Fig. 1). The end-ring current (ie) becomes zero if the ring is complete.
From (1) and (2), it is possible to determine the coupling inductances between each circuit of the IM. One way to accomplish this is by expressing the winding functions by Fourier series. This also allows analyzing separately each distribution harmonic. Below, the distribution of windings using Fourier series is widely discussed and the calculation of rotor and stator mutual inductances is presented. 2.2.1 Stator windings distribution Figure 2 shows the distribution of the stator windings corresponding to phase a for a winding with full pitch and equal number of turns in all coils. In this figure, is the pitch and q is the number of coils per pole, Nt is the number of turns per phase and per pole and P is the number of pole pairs. The stator slots pitch γ is represented by, 2 Rs ,
(3)
where Rs is the number of stator slots.
Fig. 1. Squirrel-cage equivalent circuit. N s (φ )
2.1 Calculation of inductances To calculate self and mutual inductances of the model, Modified Winding Function Approach is used (Bossio et al. 2004), which allows considering non-uniformity, both radial and axial, on windings and on the air-gap This makes it possible to analyze the effects of winding distribution, skew and the variations in the air-gap due to either eccentricity or rotor and stator slots simultaneously. Mutual inductance between any two circuits, A and B of the motor can be expressed as, LBA r 0 r
2 0
0
L
nB , z , r N A , z , r g 1 , z , r dz d ,
Nt 2q
φ
Fig. 2 Winding distribution. Analyzing the distribution of windings in a similar way to Lipo (1996), it is possible to express winding function for the stator circuits in Fourier series as shown in (4). N s
to an arbitrary point in the air gap. g 1 , z , r is the
q=2
2π P
(1)
where 0 is air permeability, r is the mean radius of the air gap, and z are angular and axial positions corresponding
Nt 2
2 k dn k pn sin nP n odd N t n1 n
(4)
where k dn and k pn are the distribution factor and pitch factor, respectively. They are given by:
inverse of the air-gap function, n , z , r is the “Winding
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kdn sin nqP 2 q sin nP 2 n odd,
(5)
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
k pn cos nqP 2
n odd.
nr (φ ,θ r )
(6)
From (4), it can be observed that the amplitude of each harmonic corresponding to the windings distribution is given by the expression: k n 2 N t k dn k pn n
n odd.
θr −
θr +
2
αr 2
2π φ
αr
(7)
1−
As it can be seen, the amplitude of each harmonic directly depends on the winding distribution (factors kdn , k pn ). Equation (4) represents the winding function for a stator phase under uniform air-gap conditions. In the case of threephase IM, the functions of the remaining phases are obtained by shifting the previous function.
αr
α r N r ( φ ,θ r ) 2π
θr − −
(a)
αr 2π
αr
θr +
2
αr 2 2π φ
αr
(b)
Fig. 3. (a) Spatial winding distribution and (b) Winding function corresponding to a rotor loop. 3. PROPOSED STRATEGY
2.2.2 Distribution of rotor circuits Figure 3 (a) shows winding distribution for a rotor loop considering axial uniformity (without skewing). Such distribution can be expressed as, 0 nr r , 1 0
0 r r 2 r r 2 r r 2,
rotor adds higher frequency components to the current spectrum. Some of these components are the result of the interaction between rotor asymmetry and the spatial harmonics of the windings distribution. Those components at (5-4s)fs, (5-6s)fs, (7-6s)fs and (7-8s)fs frequencies are the most significant. The first two are mainly due to the 5th harmonic of the stator winding distribution, whereas the rest of them are due to the 7th harmonics (Sobczyk and Maciolek, 2003).
(8)
r r 2 2
where, r 2 Rr .
(9)
Rr is the number of rotor slots and bars.
Considering air-gap uniformity, the winding function for the rotor loop (Fig. 3 (b)) results: r 2 N r r , 1 r 2 r 2
0 r r 2 r r 2 r r 2 r r 2 2
.
(10)
2.3. Example of calculation of inductances Lsr From winding functions (4) and (8) of stator circuits and the rotor loops, it is possible to determine the self and mutual inductances of the motor. As an example, the calculation of the coupling inductance between the stator winding and any rotor loop is presented. By replacing (4) and (10) into (1), it is obtained, Lsr r
20 rL gP
2
n1
kn kbn sin nPr n odd, n
(11)
Where kbn sin nP r 2.
(12)
As it was discussed above, the rotor asymmetry adds characteristics components to the spectrum of the stator currents, which allows the detection and diagnosis of this type of faults. Apart from the sidebands at frequency (1 ± 2 s ) f s , the presence of asymmetries in the
The proposal presented in this paper focuses on the use of the components at (5-4s)fs and (7-6s)fs as a complement to traditional indicators (sidebands in (1±2s)fs) for a correct diagnosis of rotor faults. As shown in the previous section (eq. (7)), the amplitude of each winding distribution harmonic is directly dependent on the characteristics of the distribution. Table 1 shows the fifth-harmonic component of the winding distribution in percent of the fundamental component for several distributions. Among three-phase motors, the "phase belt" 2P q most commonly used is that of 60º and in some cases, the one of 120º (Alger, 1995; Cochran, 1989). On the other hand, for two-phase motors, 90º is generally used and 180º in some cases. As it can be seen from the results shown in this table, for most common distributions, the fifth-harmonic amplitude with respect to the fundamental component does not change significantly with an increase in the number of coils per pole q. Analyzing (11), it can be deduced that the amplitude of the harmonics of coupling inductances linearly depends on the harmonic components of the winding distribution. Therefore, if the fifth-harmonic component does not vary significantly when modifying the winding distribution, then it will not change in the mutual inductance. Then it is expected that the proposed indicators can be found in most motor with a rotor fault, since they directly depend on the 5th harmonic of the
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stator winding distribution. Unlike other components, those located at (5-4s)fs and (7-6s)fs frequencies, do not dependent of the voltage distortion or unbalance.
around the fundamental component, but it produces new components in the frequency spectrum (5-4s)fs and (7-6s)fs. Finally, Fig. 4 (c) shows the simulation results considering all the harmonics of the winding distribution. As it can be observed, the incorporation of all distribution harmonics does not change significantly the proposed components.
Table 1. Fifth-harmonic Component of the Winding Distribution (%) Phase Belt 2P q 60º 5.359 4.288 4.124 4.069 4.044 4.000
1 2 3 4 5
90º -8.284 -4.692 -4.288 -4.158 -4.100 -4.000
120º -20.000 -5.359 -4.533 -4.288 -4.181 -4.000
To analyze the effects of voltage harmonic distortion on the proposed components, the IM was simulated under the same conditions as the previous case but including voltage distortion. Figure 5 show the results obtained with 5% fifth harmonic and a 3% seventh harmonic.
180º -20.000 8.284 5.359 4.692 4.424 4.000
In Fig. 5 (a), it can be seen that the components around the fundamental do not changed due to voltage distortion. Around the harmonics, as it is proposed by Bruzzese (2008), new components at (5+2s)fs and (7-2s)fs frequencies appear even for a sinusoidal distribution of windings. Figure 5 (b) shows the results incorporating the fifth harmonic in the winding distribution; the components at (5-4s)fs and (7-6s)fs frequencies mentioned above can also be observed. Comparing figures 5 (a) and 5 (b), it can be deduced that the fifth harmonic component does not produce significant changes on the components at (5+2s)fs and (7-2s)fs frequencies. Similarly, comparing Fig. 4 (b) and 5 (b), it can be seen that the voltage distortion hardly affects the components at (5-4s)fs and (7-6s)fs frequencies. Figure 5 (c) shows the results obtained considering all the harmonics of the winding distribution. As in the previous case, it can be observed that including all harmonics do not significantly affect the proposed components.
4. SIMULATION ANALYSIS Through using the model proposed in the previous sections, the IM was simulated with rotor asymmetry. The breakage of one or more bars or a section of the end-ring was included in the model as presented in (Toliyat and Lipo 1995; Bossio et al. 2005) The data and parameters of the motor used are listed in Table 2 in the Appendix. Figure 4 shows the frequency spectra of current for three broken bars and different winding distributions. For the three cases shown, a balanced sinusoidal supply voltage and a 100% load are applied.
44
46
48 50 52 Frequency (Hz)
54
56
44
46
0.2 Amplitude (A)
0.2 Amplitude (A)
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
0.16 0.12 0.08 0.04 0
Figure 6 shows the amplitude of the sidebands around the fundamental frequency and the current harmonics, as a function of the amplitude of the winding distribution fifth harmonic for three broken bars. These results correspond to the motor fed by sinusoidal voltage and a 100% load. It can be observed that the components at (1±2s)fs frequency do not significantly vary with the fifth-harmonic component of the winding distribution whereas the components at (5-4s)fs and (7-6s)fs frequencies do almost linearly.
Amplitude (A)
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
Amplitude (A)
Amplitude (A)
Figure 4 (a) shows the frequency spectrum around the fundamental component as well as around the fifth and seventh harmonics, for sinusoidal distribution case. In this figure the sidebands at (1±2s)fs, typical for rotor faults, are observed. Because the 5th harmonic of the winding distribution was not included in the model, components at (5 - 4s)fs and (7-6s)fs does not appear in the spectrum. Figure 4 (b) shows the result obtained by incorporating the fifthharmonic component to the winding distribution. This component hardly affects the amplitude of the sidebands
48 50 52 Frequency (Hz)
54
56
s
0.12 0.08 (7−6s)f
s
0.04 0
230
250
270 290 310 Frequency (Hz)
330
350
230
250
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
44
46
48 50 52 Frequency (Hz)
54
56
0.2
(5−4s)f
0.16
Amplitude (A)
q
270 290 310 Frequency (Hz)
330
350
(5−4s)fs
0.16 0.12 0.08
(7−6s)f
s
0.04 0 230
250
270 290 310 Frequency (Hz)
330
350
(a) (b) (c) Fig. 4. Amplitude of the sidebands at fundamental frequency and current harmonics for three broken bars: (a) with sinusoidal winding distribution, (b) fifth-harmonic in the winding distribution, (c) real winding distribution. (Simulation).
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44
46
48 50 52 Frequency (Hz)
54
56
0.2
0.2
0.16
0.16
0.12 0.08
(5+2s)fs
0.04
(7−2s)fs
0
Amplitude (A)
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
44
46
(5−4s)fs
48 50 52 Frequency (Hz)
56
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
7f
s
(7−2s)fs
(5+2s)fs
(7−6s)fs
0.04
0.16
250
270 290 310 Frequency (Hz)
330
46
48 50 52 Frequency (Hz)
54
7f
5fs
(5−4s)fs
56
s
0.12 0.08
(7−2s)fs (7−6s)fs
(‘5+2s)f
s
0.04 0
0 230
44
0.2
5fs
0.12 0.08
54
Amplitude (A)
Amplitude (A)
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
Amplitude (A)
Amplitude (A)
Amplitude (A)
SAFEPROCESS 2012 August 29-31, 2012. Mexico City, Mexico
350
230
250
270 290 310 Frequency (Hz)
330
230
350
250
270 290 310 Frequency (Hz)
330
350
(a) (b) (c) Fig. 5. Amplitude of the sidebands at fundamental frequency and the current harmonics for three broken bars and voltage distortion: (a) with sinusoidal winding distribution, (b) fifth-harmonic in winding distribution, (c) real winding distribution. (Simulation)
Amplitude (A)
1.2
0.8 0.6
(1+2s)fs
0.4
(1−2s)f
s
(5−4s)f
s
0.2 0 0
44
46
(7−6s)fs 0.2
0.4 0.6 5th Harmonic (pu)
0.8
1
Amplitude (A)
Amplitude (pu)
1
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
Fig. 6. Amplitude of the sidebands at the fundamental frequency and current harmonics, as a function of the 5th harmonic of the winding distribution for three broken bars. (Simulation).
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
44
48 50 52 Frequency (Hz)
46
54
56
(b)
Fig.7. Frequency spectrum around the fundamental component for: (a) healthy motor, (b) three broken bars.
Amplitude (A)
0.2
Figure 7 shows the frequency spectrum around the fundamental current for healthy motor and for the motor with three broken bars, both cases with constant nominal load. Figure 7 (b) shows the sidebands at 2sfs, due to the presence of broken bars.
5f
0.16
7fs
s
0.12 0.08 0.04 0 230
250
0.2
Amplitude (A)
The components (5-4s)fs and (7-6s)fs around the 5 th and 7th harmonic for these tests are shown in Fig. 8. As it can be observed in Fig. 8 (a) for the case of a healthy motor, components around the 5 th and 7th harmonics are not present. On the other hand, Fig. 8 (b) shows the components at (5-4s)fs and (7-6s)fs frequencies produced by rotor faults. From the same figures, it can be observed that these components appear even when the motor is fed with voltages with low distortion (THD <2%). 5 th and 7 th current harmonics are in this case, 1.4% and 1% of the fundamental current, respectively.
56
(a)
48 50 52 Frequency (Hz)
6. EXPERIMENTAL VALIDATION To validate this analysis, experimental results were obtained with the same motor and two different rotors: a healthy one and other with three broken bars. The technical data of the motor used as well as the data related to the winding distribution are shown in Table 2 in the Appendix.
54
330
350
(a)
5fs
0.16 0.12
270 290 310 Frequency (Hz)
7fs
(5−4s)fs (7−6s)fs
0.08 0.04 0 230
250
270 290 310 Frequency (Hz)
330
350
(b)
Fig.8. Frequency spectrum around 5th and 7th harmonics. Healthy motor, (b) motor with 3 broken bars.
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To evaluate the sensibility of the analyzed components to voltage distortion, the analysis of the current for the motor with three broken bars and distorted voltage was carried out (4% 5th and 1.3% 7th). The obtained results are shown in Fig. 9. From this figure, it can be observed that there are no significant changes in the components at (5-4s)fs and (7-6s)fs frequencies with respect to the result with lower harmonic distortion, previously shown in Fig. 8 (b). Amplitude (A)
0.2
0.12
7f
5fs
0.16
s
(7−6s)fs
(5−4s)fs
0.08 0.04 0 230
250
270 290 310 Frequency (Hz)
330
350
Fig. 9. Frequency spectrum for three broken bars and harmonic distortion (4% of the 5th harmonic and 1.3% of the 7th). 7. CONCLUSIONS In this paper, the behaviour of the components at (5-4s)fs and (7-6s)fs frequencies was analyzed, as indicators of broken rotor bars in induction motors. It was demonstrated that these components, which depend on the distribution of the stator windings and on the asymmetry caused by the fault, are present in most motors with broken bars. This is mainly because for most commonly used configurations of windings, the 5th harmonic components of the spatial distribution of windings practically remain unchanged. It was also demonstrated that these components are not so sensible to distortions in the supply voltages. These features allow the use of these components as rotor fault symptoms, complementing in this way the traditional diagnosis based on the sidebands around the fundamental current. If supply voltage have some degree of distortion, then it is possible to use the indicators proposed in (Bruzzese, 2008) to support the diagnosis.
An Extension of the Modified Winding Function Approach. IEEE Trans. on Energy Conversion, Vol. 19, No. 1, pp. 144-150. Bossio G., De Angelo, C., Solsona, J., García, G.O. and Valla, M.I. (2005). Effects of Rotor Bar and End-Ring Faults Over the Signals of a Position Estimation Strategy for Induction Motors. IEEE Transaction on Industry Applications, Vol. 41, Nº4, pp. 1005-1012. Bossio, G.R., De Angelo, C.H., Pezzani, C.M., Bossio, J.M., and Garcia, G.O. (2009). Evaluation of Harmonic Current Sidebands for Broken Bar Diagnosis in Induction Motors. IEEE Symposium for Electrical Machines, Power Electronics & Drives 2009 (SDEMPED 2009), Cargese, France, p. TF0020. Bruzzese, C. (2008). Analysis and Application of Particular Current Signatures (Symptoms) for Cage Monitoring in Nonsinusoidally Fed Motors With High Rejection to Drive Load, Inertia, and Frequency Variations. IEEE Trans. on Industrial Electronics, vol. 55, pp. 4137-4155. Cochran P. (1989), Polyphase Induction Motors, Analysis: Design, and Application, CRC Press. Lipo T. (1996), Introduction to AC Machine Design, Wisconsin Power Electronics Research Center, University of Wisconsin, Vol. 1. Luo X., Liao, Y., Toliyat, H., El-Antably, A. and Lipo, T.A. (1995). Multiple Coupled Circuit Modeling of Induction Machines. IEEE Transactions on Industry Applications, Vol. 31, Nº 2, pp. 311-318. Schoen, R.R. and Habetler, T.G. (1995). Effects of timevarying loads on rotor fault detection in induction machines. IEEE Trans. on Industry Applications, Vol. 31, pp. 900-906. Sobczyk, T.J. and Maciolek, W. (2003) Diagnostics of rotorcage faults supported by effects due to higher MMF harmonics. IEEE Power Tech Conference Proceedings, Bologna, Vol.2. Toliyat H.A. and Lipo, T.A. (1995). Transient Analysis of Cage Induction Machines Under Stator, Rotor Bar and End Ring Faults. IEEE Transactions on Energy Conversion, Vol. 10, Nº 2, pp. 241-247.
REFERENCES Alger P.L. (1995). Induction Machines. Their behavior and uses. Gordon and Breach Science Publishers, Switzerland. Bellini, A., Filippetti; F., Franceschini, G., Tassoni, C., and Kliman, G.B. (2001). Quantitative evaluation of induction motor broken bars by means of electrical signature analysis. IEEE Trans. on Industry Applications, Vol. 37, No.5, pp. 1248-1255. Bellini, A., Filippetti, F., Tassoni, C., and Capolino, G. (2008a). Advances in Diagnostic Techniques for Induction Machines. IEEE Trans. on Industrial Electronics. Vol. 55, pp. 4109-4126. Bellini, A., Yazidi, A., Filippetti, F., Rossi, C., and Capolino, G.A. (2008b). High Frequency Resolution Techniques for Rotor Fault Detection of Induction Machines. IEEE Transactions on Industrial Electronics, vol. 55, pp. 42004209. Bossio G.R., De Angelo, C.H., Solsona, J., García, G.O. and Valla, M.I. (2004). A 2D- Model of the Induction Motor: 1381
Appendix A: TECHNICAL DATA AND PARAMETERS OF THE IM Table 2. IM technical data and parameters Induction Motor Rated Power 5.5 kW Rated Voltage 380 V Rated Frequency (fs) 50 Hz Rated Current 11.1 A Rated speed 1470 rpm Power Factor 0.85 Stator Winding 18 turns per coil, 2 coil per group, 4 group per phase, series connection, step 1:10:12; Stator Slots 48 Rotor Bars 40 Inertia 0.02 Kgm2
Winding Distribution Effects on Induction Motor Rotor Fault Diagnosis Pezzani C.* Bossio G.* De Angelo C.* *Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Argentina *Grupo de Electrónica Aplicada, Facultad de Ingeniería, Universidad Nacional de Río Cuarto, X5804BYA, Río Cuarto, Córdoba, Argentina (Tel: +54-358-4676598; e-mail: [email protected]).,
Abstract: The use of stator currents harmonics as a potential tool for the diagnosis of rotor faults in induction motors is analyzed in this paper. The analysis presented is focused in those components produced by the interaction between the rotor faults and the harmonics of the spatial distribution of stator windings. These components are also compared with those resulting from the interaction between voltage harmonics and the rotor fault. A multiple coupled circuit model of the induction motor is used to evaluate the sensitivity of the proposed components for different winding configurations. Finally, experimental and simulation results are presented to validate the proposal. Keywords: Broken Bars - Fault Diagnosis - Induction Motor - WFA 1. INTRODUCTION Fault diagnosis in electrical machines, particularly in induction motors (IM), has been a very active research area in recent times. An important variety of strategies based on the measurement of machine currents has been proposed as options to the mechanical vibration analysis already accepted, for predictive maintenance of electrical motors (Bellini et al., 2008a ). Among these strategies, the motor current signature analysis (MCSA) is the most widespread. Particularly, broken bars or end-rings on the rotor produce sidebands around the fundamental component separated from it at twice the slip frequency (2sfs), where s is the slip and fs the supply frequency. The analysis of the amplitude of these sidebands has been proposed in numerous studies as an indicator of rotor faults (Bellini et al., 2001). However, the amplitude of these components also depends on the severity of the fault, and on factors such as the load level and inertia of the motorload set, among others (Bellini et al. 2001). What’s more, low-frequency oscillations in the load torque affect motor currents in a very similar way; that is, introducing sidebands around the fundamental component. Situations of this nature hinder the correct quantification of the failure and may even induce false diagnoses (Schoen and Habetler, 1995). To solve these problems, several proposals have been presented. Particularly, the analysis of current around high order harmonics has been proposed with the objective of improving rotor fault diagnosis (Schoen and Habetler, 1995). In general, these high frequency components are less sensitive to load oscillation due to the "filter" effect that inertia produces on oscillations. These high frequency components can be caused by the interaction between the asymmetry of the rotor - produced by broken bars - and either the harmonics due to supply voltage distortions or 978-3-902823-09-0/12/$20.00 © 2012 IFAC
unbalances, or the spatial harmonics due to the stator windings distribution or the rotor slots harmonics (RSH). The use of the components produced by the supply voltage distortion or unbalance to generate rotor fault indicators was proposed by Bruzzese (2008). Although these indicators are less sensitive to amplitude variations of voltage harmonics, it becomes necessary a certain degree of distortion. This can be quite easily achieved in inverter-fed motors. However, in motors fed directly from the grid, it may not occur, even less in medium voltage motors. Moreover, the use of components produced by a discrete distribution of the rotor (RSH) requires knowledge on some construction parameters such as the number of rotor bars and also requires an accurate estimate of the motor slip (Bellini et al., 2008b). The non-sinusoidal distribution of windings produces spatial harmonics on magnetomotive force (mmf), mainly at frequencies given by (6k ± 1)fs. The interaction between these spatial harmonics and rotor asymmetry produces characteristic components in the current spectrum of the IM. The components at frequencies (5-4s)fs, (5-6s)fs, (7-6s)fs and (7-8s)fs, located around the harmonics 5th and 7th, are the most significant ones (Sobczyk and Maciolek, 2003). The use of these components as indicators of the presence of broken bars has been presented in (Bossio et al. 2009) (Sobczyk and Maciolek, 2003). In (Bossio et al. 2009), it is analyzed the behaviour of the components (5-4s)fs and (7-6s)fs, for a particular distribution of windings, and for different numbers of broken bars, load inertia variations and supply voltage conditions. It was demonstrated that the magnitude of these components, besides increasing with the number of broken bars, hardly depends on the inertia of the whole system.
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This paper extends the analysis of the components (5-4s)fs and (7-6s)fs as indicators of broken bars in the IM. A multiple coupled circuit model of the IM is used (Luo et al., 1995, Bossio et al. 2004) to evaluate the sensitivity of the proposed components for different winding configurations. For this, winding functions are expressed in general by a Fourier series expansion. Through numerical simulation, using the proposed model, the amplitude of fault indicators are evaluated in terms of the harmonics of the winding distribution. The influence of voltage distortion on these indicators is also analyzed through simulation. Finally, experimental results that validate the proposal are presented.
Spatial Distribution”, and N (, z , r ) is the “Winding Modified Two-dimensional Function”. The latter can be expressed as, N (, z, r ) n , z, r 1 2 L g 1 , z , r 2
L
0 0 where
g 1 , z, r
(2)
n , z , r g 1 , z, r dzd
is the mean value of the air-gap
function inverse. 2.2. Winding distribution
2. MOTOR MODEL For the analysis of the harmonic components of the IM current, a multiple-coupled circuit model is used, proposed by Luo et al. (1995). This approach allows considering the real distribution of windings in the stator and rotor bars. Saturation effects and eddy currents are neglected in this formulation and the rotor bars are supposed to be isolated. In general, the IM consists of m stator circuits and k rotor bars. Particularly, the rotor squirrel-cage is modelled as k identical and equally spaced loops, consisting of two consecutive bars plus a current loop on one of the end-rings (Fig. 1). The end-ring current (ie) becomes zero if the ring is complete.
From (1) and (2), it is possible to determine the coupling inductances between each circuit of the IM. One way to accomplish this is by expressing the winding functions by Fourier series. This also allows analyzing separately each distribution harmonic. Below, the distribution of windings using Fourier series is widely discussed and the calculation of rotor and stator mutual inductances is presented. 2.2.1 Stator windings distribution Figure 2 shows the distribution of the stator windings corresponding to phase a for a winding with full pitch and equal number of turns in all coils. In this figure, is the pitch and q is the number of coils per pole, Nt is the number of turns per phase and per pole and P is the number of pole pairs. The stator slots pitch γ is represented by, 2 Rs ,
(3)
where Rs is the number of stator slots.
Fig. 1. Squirrel-cage equivalent circuit. N s (φ )
2.1 Calculation of inductances To calculate self and mutual inductances of the model, Modified Winding Function Approach is used (Bossio et al. 2004), which allows considering non-uniformity, both radial and axial, on windings and on the air-gap This makes it possible to analyze the effects of winding distribution, skew and the variations in the air-gap due to either eccentricity or rotor and stator slots simultaneously. Mutual inductance between any two circuits, A and B of the motor can be expressed as, LBA r 0 r
2 0
0
L
nB , z , r N A , z , r g 1 , z , r dz d ,
Nt 2q
φ
Fig. 2 Winding distribution. Analyzing the distribution of windings in a similar way to Lipo (1996), it is possible to express winding function for the stator circuits in Fourier series as shown in (4). N s
to an arbitrary point in the air gap. g 1 , z , r is the
q=2
2π P
(1)
where 0 is air permeability, r is the mean radius of the air gap, and z are angular and axial positions corresponding
Nt 2
2 k dn k pn sin nP n odd N t n1 n
(4)
where k dn and k pn are the distribution factor and pitch factor, respectively. They are given by:
inverse of the air-gap function, n , z , r is the “Winding
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kdn sin nqP 2 q sin nP 2 n odd,
(5)
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k pn cos nqP 2
n odd.
nr (φ ,θ r )
(6)
From (4), it can be observed that the amplitude of each harmonic corresponding to the windings distribution is given by the expression: k n 2 N t k dn k pn n
n odd.
θr −
θr +
2
αr 2
2π φ
αr
(7)
1−
As it can be seen, the amplitude of each harmonic directly depends on the winding distribution (factors kdn , k pn ). Equation (4) represents the winding function for a stator phase under uniform air-gap conditions. In the case of threephase IM, the functions of the remaining phases are obtained by shifting the previous function.
αr
α r N r ( φ ,θ r ) 2π
θr − −
(a)
αr 2π
αr
θr +
2
αr 2 2π φ
αr
(b)
Fig. 3. (a) Spatial winding distribution and (b) Winding function corresponding to a rotor loop. 3. PROPOSED STRATEGY
2.2.2 Distribution of rotor circuits Figure 3 (a) shows winding distribution for a rotor loop considering axial uniformity (without skewing). Such distribution can be expressed as, 0 nr r , 1 0
0 r r 2 r r 2 r r 2,
rotor adds higher frequency components to the current spectrum. Some of these components are the result of the interaction between rotor asymmetry and the spatial harmonics of the windings distribution. Those components at (5-4s)fs, (5-6s)fs, (7-6s)fs and (7-8s)fs frequencies are the most significant. The first two are mainly due to the 5th harmonic of the stator winding distribution, whereas the rest of them are due to the 7th harmonics (Sobczyk and Maciolek, 2003).
(8)
r r 2 2
where, r 2 Rr .
(9)
Rr is the number of rotor slots and bars.
Considering air-gap uniformity, the winding function for the rotor loop (Fig. 3 (b)) results: r 2 N r r , 1 r 2 r 2
0 r r 2 r r 2 r r 2 r r 2 2
.
(10)
2.3. Example of calculation of inductances Lsr From winding functions (4) and (8) of stator circuits and the rotor loops, it is possible to determine the self and mutual inductances of the motor. As an example, the calculation of the coupling inductance between the stator winding and any rotor loop is presented. By replacing (4) and (10) into (1), it is obtained, Lsr r
20 rL gP
2
n1
kn kbn sin nPr n odd, n
(11)
Where kbn sin nP r 2.
(12)
As it was discussed above, the rotor asymmetry adds characteristics components to the spectrum of the stator currents, which allows the detection and diagnosis of this type of faults. Apart from the sidebands at frequency (1 ± 2 s ) f s , the presence of asymmetries in the
The proposal presented in this paper focuses on the use of the components at (5-4s)fs and (7-6s)fs as a complement to traditional indicators (sidebands in (1±2s)fs) for a correct diagnosis of rotor faults. As shown in the previous section (eq. (7)), the amplitude of each winding distribution harmonic is directly dependent on the characteristics of the distribution. Table 1 shows the fifth-harmonic component of the winding distribution in percent of the fundamental component for several distributions. Among three-phase motors, the "phase belt" 2P q most commonly used is that of 60º and in some cases, the one of 120º (Alger, 1995; Cochran, 1989). On the other hand, for two-phase motors, 90º is generally used and 180º in some cases. As it can be seen from the results shown in this table, for most common distributions, the fifth-harmonic amplitude with respect to the fundamental component does not change significantly with an increase in the number of coils per pole q. Analyzing (11), it can be deduced that the amplitude of the harmonics of coupling inductances linearly depends on the harmonic components of the winding distribution. Therefore, if the fifth-harmonic component does not vary significantly when modifying the winding distribution, then it will not change in the mutual inductance. Then it is expected that the proposed indicators can be found in most motor with a rotor fault, since they directly depend on the 5th harmonic of the
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stator winding distribution. Unlike other components, those located at (5-4s)fs and (7-6s)fs frequencies, do not dependent of the voltage distortion or unbalance.
around the fundamental component, but it produces new components in the frequency spectrum (5-4s)fs and (7-6s)fs. Finally, Fig. 4 (c) shows the simulation results considering all the harmonics of the winding distribution. As it can be observed, the incorporation of all distribution harmonics does not change significantly the proposed components.
Table 1. Fifth-harmonic Component of the Winding Distribution (%) Phase Belt 2P q 60º 5.359 4.288 4.124 4.069 4.044 4.000
1 2 3 4 5
90º -8.284 -4.692 -4.288 -4.158 -4.100 -4.000
120º -20.000 -5.359 -4.533 -4.288 -4.181 -4.000
To analyze the effects of voltage harmonic distortion on the proposed components, the IM was simulated under the same conditions as the previous case but including voltage distortion. Figure 5 show the results obtained with 5% fifth harmonic and a 3% seventh harmonic.
180º -20.000 8.284 5.359 4.692 4.424 4.000
In Fig. 5 (a), it can be seen that the components around the fundamental do not changed due to voltage distortion. Around the harmonics, as it is proposed by Bruzzese (2008), new components at (5+2s)fs and (7-2s)fs frequencies appear even for a sinusoidal distribution of windings. Figure 5 (b) shows the results incorporating the fifth harmonic in the winding distribution; the components at (5-4s)fs and (7-6s)fs frequencies mentioned above can also be observed. Comparing figures 5 (a) and 5 (b), it can be deduced that the fifth harmonic component does not produce significant changes on the components at (5+2s)fs and (7-2s)fs frequencies. Similarly, comparing Fig. 4 (b) and 5 (b), it can be seen that the voltage distortion hardly affects the components at (5-4s)fs and (7-6s)fs frequencies. Figure 5 (c) shows the results obtained considering all the harmonics of the winding distribution. As in the previous case, it can be observed that including all harmonics do not significantly affect the proposed components.
4. SIMULATION ANALYSIS Through using the model proposed in the previous sections, the IM was simulated with rotor asymmetry. The breakage of one or more bars or a section of the end-ring was included in the model as presented in (Toliyat and Lipo 1995; Bossio et al. 2005) The data and parameters of the motor used are listed in Table 2 in the Appendix. Figure 4 shows the frequency spectra of current for three broken bars and different winding distributions. For the three cases shown, a balanced sinusoidal supply voltage and a 100% load are applied.
44
46
48 50 52 Frequency (Hz)
54
56
44
46
0.2 Amplitude (A)
0.2 Amplitude (A)
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
0.16 0.12 0.08 0.04 0
Figure 6 shows the amplitude of the sidebands around the fundamental frequency and the current harmonics, as a function of the amplitude of the winding distribution fifth harmonic for three broken bars. These results correspond to the motor fed by sinusoidal voltage and a 100% load. It can be observed that the components at (1±2s)fs frequency do not significantly vary with the fifth-harmonic component of the winding distribution whereas the components at (5-4s)fs and (7-6s)fs frequencies do almost linearly.
Amplitude (A)
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
Amplitude (A)
Amplitude (A)
Figure 4 (a) shows the frequency spectrum around the fundamental component as well as around the fifth and seventh harmonics, for sinusoidal distribution case. In this figure the sidebands at (1±2s)fs, typical for rotor faults, are observed. Because the 5th harmonic of the winding distribution was not included in the model, components at (5 - 4s)fs and (7-6s)fs does not appear in the spectrum. Figure 4 (b) shows the result obtained by incorporating the fifthharmonic component to the winding distribution. This component hardly affects the amplitude of the sidebands
48 50 52 Frequency (Hz)
54
56
s
0.12 0.08 (7−6s)f
s
0.04 0
230
250
270 290 310 Frequency (Hz)
330
350
230
250
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
44
46
48 50 52 Frequency (Hz)
54
56
0.2
(5−4s)f
0.16
Amplitude (A)
q
270 290 310 Frequency (Hz)
330
350
(5−4s)fs
0.16 0.12 0.08
(7−6s)f
s
0.04 0 230
250
270 290 310 Frequency (Hz)
330
350
(a) (b) (c) Fig. 4. Amplitude of the sidebands at fundamental frequency and current harmonics for three broken bars: (a) with sinusoidal winding distribution, (b) fifth-harmonic in the winding distribution, (c) real winding distribution. (Simulation).
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44
46
48 50 52 Frequency (Hz)
54
56
0.2
0.2
0.16
0.16
0.12 0.08
(5+2s)fs
0.04
(7−2s)fs
0
Amplitude (A)
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
44
46
(5−4s)fs
48 50 52 Frequency (Hz)
56
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
7f
s
(7−2s)fs
(5+2s)fs
(7−6s)fs
0.04
0.16
250
270 290 310 Frequency (Hz)
330
46
48 50 52 Frequency (Hz)
54
7f
5fs
(5−4s)fs
56
s
0.12 0.08
(7−2s)fs (7−6s)fs
(‘5+2s)f
s
0.04 0
0 230
44
0.2
5fs
0.12 0.08
54
Amplitude (A)
Amplitude (A)
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
Amplitude (A)
Amplitude (A)
Amplitude (A)
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350
230
250
270 290 310 Frequency (Hz)
330
230
350
250
270 290 310 Frequency (Hz)
330
350
(a) (b) (c) Fig. 5. Amplitude of the sidebands at fundamental frequency and the current harmonics for three broken bars and voltage distortion: (a) with sinusoidal winding distribution, (b) fifth-harmonic in winding distribution, (c) real winding distribution. (Simulation)
Amplitude (A)
1.2
0.8 0.6
(1+2s)fs
0.4
(1−2s)f
s
(5−4s)f
s
0.2 0 0
44
46
(7−6s)fs 0.2
0.4 0.6 5th Harmonic (pu)
0.8
1
Amplitude (A)
Amplitude (pu)
1
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
Fig. 6. Amplitude of the sidebands at the fundamental frequency and current harmonics, as a function of the 5th harmonic of the winding distribution for three broken bars. (Simulation).
0.35 0.3 0.25 0.2 0.15 0.1 0.05 0
44
48 50 52 Frequency (Hz)
46
54
56
(b)
Fig.7. Frequency spectrum around the fundamental component for: (a) healthy motor, (b) three broken bars.
Amplitude (A)
0.2
Figure 7 shows the frequency spectrum around the fundamental current for healthy motor and for the motor with three broken bars, both cases with constant nominal load. Figure 7 (b) shows the sidebands at 2sfs, due to the presence of broken bars.
5f
0.16
7fs
s
0.12 0.08 0.04 0 230
250
0.2
Amplitude (A)
The components (5-4s)fs and (7-6s)fs around the 5 th and 7th harmonic for these tests are shown in Fig. 8. As it can be observed in Fig. 8 (a) for the case of a healthy motor, components around the 5 th and 7th harmonics are not present. On the other hand, Fig. 8 (b) shows the components at (5-4s)fs and (7-6s)fs frequencies produced by rotor faults. From the same figures, it can be observed that these components appear even when the motor is fed with voltages with low distortion (THD <2%). 5 th and 7 th current harmonics are in this case, 1.4% and 1% of the fundamental current, respectively.
56
(a)
48 50 52 Frequency (Hz)
6. EXPERIMENTAL VALIDATION To validate this analysis, experimental results were obtained with the same motor and two different rotors: a healthy one and other with three broken bars. The technical data of the motor used as well as the data related to the winding distribution are shown in Table 2 in the Appendix.
54
330
350
(a)
5fs
0.16 0.12
270 290 310 Frequency (Hz)
7fs
(5−4s)fs (7−6s)fs
0.08 0.04 0 230
250
270 290 310 Frequency (Hz)
330
350
(b)
Fig.8. Frequency spectrum around 5th and 7th harmonics. Healthy motor, (b) motor with 3 broken bars.
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To evaluate the sensibility of the analyzed components to voltage distortion, the analysis of the current for the motor with three broken bars and distorted voltage was carried out (4% 5th and 1.3% 7th). The obtained results are shown in Fig. 9. From this figure, it can be observed that there are no significant changes in the components at (5-4s)fs and (7-6s)fs frequencies with respect to the result with lower harmonic distortion, previously shown in Fig. 8 (b). Amplitude (A)
0.2
0.12
7f
5fs
0.16
s
(7−6s)fs
(5−4s)fs
0.08 0.04 0 230
250
270 290 310 Frequency (Hz)
330
350
Fig. 9. Frequency spectrum for three broken bars and harmonic distortion (4% of the 5th harmonic and 1.3% of the 7th). 7. CONCLUSIONS In this paper, the behaviour of the components at (5-4s)fs and (7-6s)fs frequencies was analyzed, as indicators of broken rotor bars in induction motors. It was demonstrated that these components, which depend on the distribution of the stator windings and on the asymmetry caused by the fault, are present in most motors with broken bars. This is mainly because for most commonly used configurations of windings, the 5th harmonic components of the spatial distribution of windings practically remain unchanged. It was also demonstrated that these components are not so sensible to distortions in the supply voltages. These features allow the use of these components as rotor fault symptoms, complementing in this way the traditional diagnosis based on the sidebands around the fundamental current. If supply voltage have some degree of distortion, then it is possible to use the indicators proposed in (Bruzzese, 2008) to support the diagnosis.
An Extension of the Modified Winding Function Approach. IEEE Trans. on Energy Conversion, Vol. 19, No. 1, pp. 144-150. Bossio G., De Angelo, C., Solsona, J., García, G.O. and Valla, M.I. (2005). Effects of Rotor Bar and End-Ring Faults Over the Signals of a Position Estimation Strategy for Induction Motors. IEEE Transaction on Industry Applications, Vol. 41, Nº4, pp. 1005-1012. Bossio, G.R., De Angelo, C.H., Pezzani, C.M., Bossio, J.M., and Garcia, G.O. (2009). Evaluation of Harmonic Current Sidebands for Broken Bar Diagnosis in Induction Motors. IEEE Symposium for Electrical Machines, Power Electronics & Drives 2009 (SDEMPED 2009), Cargese, France, p. TF0020. Bruzzese, C. (2008). Analysis and Application of Particular Current Signatures (Symptoms) for Cage Monitoring in Nonsinusoidally Fed Motors With High Rejection to Drive Load, Inertia, and Frequency Variations. IEEE Trans. on Industrial Electronics, vol. 55, pp. 4137-4155. Cochran P. (1989), Polyphase Induction Motors, Analysis: Design, and Application, CRC Press. Lipo T. (1996), Introduction to AC Machine Design, Wisconsin Power Electronics Research Center, University of Wisconsin, Vol. 1. Luo X., Liao, Y., Toliyat, H., El-Antably, A. and Lipo, T.A. (1995). Multiple Coupled Circuit Modeling of Induction Machines. IEEE Transactions on Industry Applications, Vol. 31, Nº 2, pp. 311-318. Schoen, R.R. and Habetler, T.G. (1995). Effects of timevarying loads on rotor fault detection in induction machines. IEEE Trans. on Industry Applications, Vol. 31, pp. 900-906. Sobczyk, T.J. and Maciolek, W. (2003) Diagnostics of rotorcage faults supported by effects due to higher MMF harmonics. IEEE Power Tech Conference Proceedings, Bologna, Vol.2. Toliyat H.A. and Lipo, T.A. (1995). Transient Analysis of Cage Induction Machines Under Stator, Rotor Bar and End Ring Faults. IEEE Transactions on Energy Conversion, Vol. 10, Nº 2, pp. 241-247.
REFERENCES Alger P.L. (1995). Induction Machines. Their behavior and uses. Gordon and Breach Science Publishers, Switzerland. Bellini, A., Filippetti; F., Franceschini, G., Tassoni, C., and Kliman, G.B. (2001). Quantitative evaluation of induction motor broken bars by means of electrical signature analysis. IEEE Trans. on Industry Applications, Vol. 37, No.5, pp. 1248-1255. Bellini, A., Filippetti, F., Tassoni, C., and Capolino, G. (2008a). Advances in Diagnostic Techniques for Induction Machines. IEEE Trans. on Industrial Electronics. Vol. 55, pp. 4109-4126. Bellini, A., Yazidi, A., Filippetti, F., Rossi, C., and Capolino, G.A. (2008b). High Frequency Resolution Techniques for Rotor Fault Detection of Induction Machines. IEEE Transactions on Industrial Electronics, vol. 55, pp. 42004209. Bossio G.R., De Angelo, C.H., Solsona, J., García, G.O. and Valla, M.I. (2004). A 2D- Model of the Induction Motor: 1381
Appendix A: TECHNICAL DATA AND PARAMETERS OF THE IM Table 2. IM technical data and parameters Induction Motor Rated Power 5.5 kW Rated Voltage 380 V Rated Frequency (fs) 50 Hz Rated Current 11.1 A Rated speed 1470 rpm Power Factor 0.85 Stator Winding 18 turns per coil, 2 coil per group, 4 group per phase, series connection, step 1:10:12; Stator Slots 48 Rotor Bars 40 Inertia 0.02 Kgm2