An Intelligent Harmonic Synthesis Technique for Air-Gap Eccentricity Fault Diagnosis in Induction Motors

In stator current signal based IM fault detection, the characteristic frequency components f _e in Hz used to detect static and dynamic eccentricity defects [21] are given bywhere f _s is the supply frequency in Hz; R is the number of rotor slots; s is the slip; f _r is the rotor rotating speed; k = 1, 2, 3, …; p is the number of pole pairs; \(\beta = 1, \, 3, \, 5, \ldots\) is the order of the stator time harmonics; \(\alpha\) is the eccentricity order: \(\alpha = 0\) denotes static eccentricity and \(\alpha = 1\) denotes dynamic eccentricity.

The Kth order fault harmonic series f _{e_K} will bewhere the Kth order origin frequency component \(f_{o\_K}\) is

To explore the relationship among different fault harmonic series, the fault harmonic series of interest are synthesized to reveal fault characteristic features. If the first P fault harmonics (\(\beta\) = 1, 3, 5, …, 2P–1) are considered for synthesis, the harmonic frequency band corresponding to the Kth fault harmonic series will be extracted asand the bandwidth in Hz is \(2Pf_{s}\) +1.

To synthesize fault harmonic series, the frequency bands are converted to a discrete-point domain representation. The discrete-point closest to \(f_{o\_K}\) is considered as the first discrete-point in q _K , defined as D _k {1}. Suppose u _f discrete data points represent unit frequency distance in the spectrum, the bandwidth of Eq. (4) in discrete-point domain will be formulated aswhere \(\left\langle \cdot \right\rangle\) denotes the integer function. Then the harmonic frequency band of Eq. (4) in the discrete-point domain will be

The root-mean-square (RMS) operation is applied to synthesize harmonic frequency bands. Given the harmonic frequency bands of M interested fault harmonic series, the ith component in the synthesized spectrum v will be derived as

The mth harmonic (\(\beta\) = 2 m–1) of the Kth order fault harmonic series will be derived from Eq. (2),

Through the synthesis operation in Eq. (7), the mth fault harmonic over all M harmonic frequency bands is synthesized to form a new fault frequency component. Then the mth synthesized fault frequency component in the discrete-point domain will be derived in the synthesized spectrum by the following representation:

The local spectra encompassing the mth synthesized fault frequency component will bewhere d is the half bandwidth of the local spectra in the frequency domain. The mean value of this local spectrum is the fault indicator regarding the mth synthesized fault frequency component, denoted by F _m .

To illustrate the operation of the proposed HSP and LSA processes, a simulated current spectrum with three air-gap eccentricity frequency harmonic series is illustrated in Figure 1. It is normalized by deducting mean and then being divided by its standard deviation. In Figure 1, the black arrow points 50 Hz supply frequency; magenta arrows with number 1, green arrows with number 2 and yellow arrows with number 3 denote three fault harmonic series respectively. The origin frequency components \(f_{o\_K}\) are 114.1 Hz, 142.4 Hz, and 170.7 Hz, respectively. The harmonics of defect features are computed using Eq. (2) with f _s  = 50 Hz and \(\beta\) = 1, 3, 5, 7, 9. Figure 1 shows that some fault characteristic frequency components are buried in the spectrum, and it is difficult to predict which characteristic frequency components will protrude in the spectrum. The extracted frequency bands represented in Eq. (6) of three fault harmonic series, and synthesized spectrum represented in Eq. (7) with fault related local spectra are demonstrated in Figure 2. The dashed vertical lines indicate the boundaries of local spectra encompassing synthesized fault frequency components. Figures 2(a), 2(b) and 2(c) represent three frequency bands computed by Eq. (4) with staring frequencies \(f_{o\_K}\) being 114.1 Hz, 142.4 Hz, and 170.7 Hz, respectively. Figure 2(d) shows results of the synchronization of spectra in Figures 2(a), 2(b) and 2(c) using root-mean-square calculation using Eq. (7). It is seen from Figure 2(d) that the fault characteristic features can be highlighted in the spectrum after the HSP operation, which could be used for IM fault detection. In this simulation, the supply frequency f _s  = 50 Hz; \(f_{o\_K}\) is the starting point of the spectrum in Figure 2(d). Based on Eqs. (2) and (3), the fault frequencies in Figure 2(d) should appear at 50 Hz, 150 Hz, 250 Hz, 350 Hz and 450 Hz. The frequency components outside the boundaries of local spectra are usually caused by the modulation of load variation and supply frequency, and the synchronization of different harmonic series local bands.

If both static and dynamic eccentricities occur simultaneously, the following characteristic frequency components could also be excited [22], given by

The related harmonic series can be expressed as

The local spectra of the first Q harmonics in Eq. (12) will be transformed to the discrete-point domain, and then synthesized into one spectrum in RMS form using Eq. (7). The mean value of the synthesized spectrum is considered as a fault indicator denoted by F _d1. The fault indicator derived from Eq. (13) will be denoted by F _d2.

To examine the effectiveness of the proposed HS technique, a series of tests will be conducted for IM air-gap eccentricity fault diagnosis using stator current signals. This work focuses on both static and dynamic eccentricity fault detection. The parameters of the proposed HS technique used in the following tests are given in Table 1. To thoroughly analyze fault information, the first 20 fault harmonic series (i.e., M = 5) are synthesized and the first eight harmonics (P = 8) in the synthesized spectrum are utilized to generate fault indicators F _m , m = 1, 2, …, 8. The frequency band [0, 8000] Hz is used for analysis. To analyze more informative fault features in harmonic series in Eqs. (12) and (13), the first ten harmonics are used to derive fault indicators F _d1 and F _d2. To capture the fault features and exclude the interference nearby, the half bandwidth of the local spectra is selected as 2 Hz.

To derive F _m for eccentricity fault detection, the coefficient \(\alpha\) = 0 in Eq. (3) is set for static eccentricity detection and \(\alpha\) = 1 for dynamic eccentricity analysis. To evaluate the effectiveness of the HS technique, the power spectral density (PSD) is used for comparison. To implement PSD, the mean values of ten local bands are used as fault indicators, whose half bandwidth is set as 2 Hz. Eight of ten center frequencies of the local bands are calculated using Eq. (1) with k = 1 and \(\beta\) = 1, 2, …, 8. The other two center frequencies are computed using \(f^{\prime}_{e1}\) with k = 1 and \(f^{\prime}_{e2}\) with k = 2. In addition, a variant of the HS technique is employed for comparison, in which the center characteristic frequencies in the synthesized local spectra rather than the mean values are utilized as fault indicators, and is denoted by CHS. The frequency components in Eq. (9) with m = 1, 2, …, 8, and the two center frequencies of the synthesized local spectra derived from Eqs. (12) and (13), respectively, are used as fault indicators of the CHS method. The Hilbert-Huang transform (HHT) based fault indicators [13] will be used for comparison. Since this work focuses on the analysis of the current signal based fault indicators, only the current signal related fault indicators provided in Ref. [13] will be used for comparison. The HHT is applied to current signal analysis to generate fault indicators. The mean values of four local bands centered at \(f^{\prime}_{e1}\) with k = 1, 2, and \(f^{\prime}_{e2}\) with k = 2, 3, and averaged Hilbert marginal spectrum in local bands centered at \(f^{\prime}_{e1}\) with k = 1, and \(f^{\prime}_{e2}\) with k = 2, in the first two intrinsic mode functions (IMF), and averaged instantaneous amplitudes of the first two IMFs are used as fault indicators. The half bandwidth of these local bands in HHT is set as 2 Hz.

The HHT, CHS and the proposed HS technique generate fault indicators from the PSD spectrum. To examine if the fault indicators could be used for different classifiers, both the support vector machine (SVM) [23] and linear discriminant analysis (LDA) [24] are utilized to test the accuracy of different fault detection techniques. A series of tests have been conducted for this verification; however, the test results corresponding to five load conditions (i.e., 0, 20%, 50%, 70% and 100% load levels) will be used for demonstration. In data preparation, 200 data sets are collected for each IM condition (healthy, static air-gap eccentricity fault, and dynamic air-gap eccentricity fault) in each of five load conditions. The sampling frequency is f _p  = 20000 Hz and the time span of each data set is 3 s. The fault indicators of PSD, HHT, CHS and the proposed HS are extracted from these totally 3000 data sets.

Figure 3 shows the experiment setup used in this test. The tested IMs are 3-phase, 1/3 hp motors made by Marathon Electric. Its speed is controlled by a speed controller (VFD-B) with output frequency 0.1–400 Hz. A magnetic clutch system (PHC-50) is used as the dynamometer to provide external loading, with torque ranging from 1 to 40 N·m. A gearbox (Boston Gear 800) is used to adjust the speed ratio of the dynamometer. An encoder (NSN-1024-2 M-F) is used to measure the shaft speed. Phase current signals are measured by the use of current sensors (LTS 6-NP). A data acquisition board (Quanser Q4) is used for signal measurement. Motor specifications are given in Table 2. The static air-gap eccentricity is implemented by introducing a 0.025 inch horizontal travel of the bearing housing of end bells in the IM. The dynamic air-gap eccentricity is implemented by intentionally bending rotor in the center 0.0127 cm to 0.0254 cm, while making sure the rotor does not bind the stator in the IM [25]. Based on the motor specification, the introduced air-gap eccentricity faults are considered to be at the early stage of the IM fault.

Static air-gap eccentricity fault detection is considered as a binary classification problem; the data sets collected from a healthy IM and an IM with the dynamic air-gap eccentricity defect belong to one class; the data sets collected from the IM with the static air-gap eccentricity defect belong to the other class. The averaged successful rate of ten runs three-fold cross validation in terms of static air-gap eccentricity fault diagnosis using SVM and LDA for classification is summarized in Tables 3 and 4, respectively. It is seen from Tables 3 and 4 that the proposed HS technique has the highest averaged successful rates in both training and testing. The proposed HS technique outperforms PSD under all five load levels, because it could explore more fault information and enhance fault features using HSP. The HS outperforms the CHS in all five scenarios, because the mean value of the local band is a more representative feature for fault detection. HS is superior to the HHT in that more effective fault features are extracted and employed for fault detection.

The score plots of SVM and LDA with respect to the HS static eccentricity fault indicators for static eccentricity diagnosis are illustrated in Figure 4. Totally 600 samples are shown in each plot. The scores of SVM classification and LDA classification are normalized over [–1, 1]. In each axis (x-axis or y-axis), –1 and 1 denote two different classes respectively. The closer the sample is to the –1 or 1, the more probably the sample belongs to the class denoted by –1 or 1. The zero-vertical line and zero-horizontal line are the separating hyper-planes to differentiate two classes using SVM and LDA respectively. From Figures 4(a)–4(d), 4(g), and 4(h), it is seen that the two classes could be separated using either SVM or LDA under 0, 20% and 70% load levels. It is seen in Figures 4(e) and 4(f) that there is one sample misclassification in both training and testing in 50% load state. The samples in Figures 4(i) and 4(j) are a bit scattered because heavy load condition weakens fault features, however, the samples could still be correctly classified mostly in 100% load level. Therefore, the proposed HS technique is a useful tool to detect IM static air-gap eccentricity fault in different load conditions.

For dynamic air-gap eccentricity fault diagnosis, the data sets corresponding to healthy IM condition and static air-gap eccentricity condition will belong to one class; the data sets corresponding to dynamic air-gap eccentricity condition belong to the other class. The averaged successful rates of ten runs three-fold cross validation in terms of dynamic air-gap eccentricity fault diagnosis using SVM and LDA for classification is given in Tables 5 and 6, respectively. From Tables 5 and 6, it is seen that the proposed HS technique has the highest accuracy in both training and testing. HS technique outperforms PSD in all five scenarios because of the extensive fault information exploration and fault features enhancement. The proposed HS is superior to the CHS in all five cases because of its effective local fault information analysis. HS outperforms HHT because more effective fault features are extracted for fault analysis.

The score plots of SVM and LDA with respect to the HS dynamic eccentricity fault indicators for dynamic eccentricity diagnosis are shown in Figure 5. From Figures 5(a)–5(d), it is seen that the two classes could be separated using either SVM or LDA under 0% and 20% load levels. It can be seen from Figure 5(f) that there is one sample misclassification in testing by LDA in 50% load condition. The samples in Figures 5(g)–5(j) have larger variance due to interference of the heavy load condition; however, most samples could still be correctly classified in 70% and 100% load conditions. It demonstrates that the proposed HS technique is a useful tool to detect IM dynamic air-gap eccentricity fault in different load conditions.