In general, spectral magnitude and fault sidebands are used to estimate the severity of BRB faults, but it is prone to generating false spectral components. As false frequency components do not inversely affect time series data, the proposed adaptive EMD technique will be undertaken in the time domain.

The relationship between two time series signals can be estimated by correlation coefficients of two stationary signals, but not for non-stationary signals whose statistical quantities are time varying. Fault related features from IMs could be non-stationery [23]. Different from general EMD processing, in the proposed adaptive EMD technique, the sliding window is determined adaptively by the estimation of the instantaneous period of the IMF; this provides the intrinsic relationship between the related signatures. In addition, the value of correlation is used to select the fault represented frequencies. Once the correlation value goes below to a certain threshold, the process will stop in calculating next IMF [20]. In this technique, only fault represented IMFs (which also have higher correlation coefficient) are calculated. Hence, it can process signals with any length. Figure 1 shows the block diagram of proposed adaptive EMD technique.

Due to the rotor asymmetry and imbalance in cage winding, there is a backward rotating field at slip frequency analogous to the forward rotating rotor [1]. As a result, current and induced electromagnetic signal are found at different slip frequencies. In addition, due to rotor oscillation, there exits an upper sideband current component in the stator winding [14]. Considering all these effects, sideband frequencies f s b due to broken rotor bars will be

f s b = f 1 ( 1 ± 2 s ) (1)

where f 1 is the power line frequency in Hz and s is the slip. As the EMD is based on the calculation of the envelope of maxima and minima of the input signal [18], fault related envelope can be independent of the f 1 . The line current I L is modeled as

I L = I A sin ( ω s t + α ) (2)

where I A is the amplitude of the current, and α is the phase angle of the sinusoidal line current.

Consider the sideband components with phase angle, then Equation (2) becomes

I L = I A sin ( ω s t + α ) + I l 1 sin ( ( 1 − 2 s ) ω s t + α l 1 )               + I r 1 sin ( ( 1 + 2 s ) ω s t + α r 1 ) + I l 2 sin ( ( 1 − 4 s ) ω s t + α l 2 )               + I r 2 sin ( ( 1 + 4 s ) ω s t + α r 2 ) + I l 3 sin ( ( 1 − 6 s ) ω s t + α l 3 ) + ⋯ = sin ( ω s t ) ( I L cos ( α ) + ∑ k I l k cos ( 2 k s ω s t − α l k ) + I r k cos ( 2 k s ω s t + α r k ) )       + cos ( ω s t ) [ I L sin ( α ) − ∑ k I l k sin ( 2 k s ω s t − α l k ) + I r k sin ( 2 k s ω s t + α r k ) ]

= X ( t ) sin ( ω s t ) + Y ( t ) cos ( ω s t ) = X 2 ( t ) + Y 2 ( t ) sin ( ω s t + tan − 1 ( X ( t ) Y ( t ) ) ) = I L M sin ( ω s t + β ( t ) ) (3)

where I l k and α r k are the current and phase of the left sideband; I r k and α l k are related to right sideband quantities. I L M is the mean value of the current amplitude. Equation (3) infers that the fault related envelope can be separable from the line frequency component, which can provide a better understanding of the fault features.

In the EMD process, the signal is decomposed into a set of IMFs. An IMF is a function that satisfies the following two requirements:

1) In the whole data set, the difference between the number of extrema and the number of zero-crossings points is either 0 or 1.

2) The mean value of the envelope due to the local maxima and local minima is zero.

Variable amplitude and frequency are represented as functions of time in the IMF. Spline interpolation can be used to determine the extreme points and form the negative and positive envelopes. The mean value of these two envelopes is subtracted from the original signal and the residual is investigated to see whether it satisfies these two criteria for IMF or not. After calculating each IMF, the correlation will be checked. If the correlation is above some threshold (usually determined by trials and errors) then the next IMF is computed.

On the other hand, from our systematic investigation, it is observed that the dot product between two IMFs is not zero. That means that EMD cannot strictly decompose IMF orthogonally. As a result, we cannot predict that fault frequency is particularly strict in one IMF, which can also be verified by correlation analysis as discussed in Section 3.

In processing, the Hilbert transform is applied to each of the IMFs such that

S ( t ) = ℜ { ∑ j = 1 n a j ( t ) exp [ i ∫ w j ( t ) d t ] }

where a j ( t ) and w j ( t ) are the respective instantaneous amplitudes and instantaneous frequencies of the jth IMF, and n is the total number of IMF generated by EMD.

The residual is a monotonic function:

S ( t ) = ℜ { ∑ j = 1 n − 1 a j ( t ) exp [ i ∫ w j ( t ) d t ] } (5)

To calculate the IMF, the mean value x ¯ ( t ) of the envelope of input signal x ( t ) is computed as x ¯ = E v max ( t ) + E v min ( t ) 2 , where E v max ( t ) and E v min ( t ) are maximum and minimum values of the envelope. Then

h 1 = x ( t ) − x ¯ ( t ) = I L M ( t ) sin ( ω s t + β ( t ) ) − E v max ( t ) + E v min ( t ) 2 (6)

If h 1 ( t ) doesn’t satisfy the criteria for IMF, then x ( t ) is replaced by h 1 ( t ) and the same process continues until it satisfies the criteria then,

I M F 1 = h 1 ( t ) = I L M ( t ) sin ( ω s t + β ( t ) ) − E v max ( t ) + E v min ( t ) 2 (7)

Then the input signal for the next IMF is calculated as,

x 2 ( t ) = x ( t ) − I M F 1 = I L M ( t ) sin ( ω s t + β ( t ) ) − [ I L M ( t ) sin ( ω s t + β ( t ) ) − E v max ( t ) + E v min ( t ) 2 ] = I L M ( t ) sin ( ω s t + β ( t ) ) − [ I L M ( t ) sin ( ω s t + β ( t ) ) − E v max ( t ) + E v min ( t ) 2 ] = E v max ( t ) + E v min ( t ) 2 (8)

It is noted that applying MCSA in the IMF is highly dependent on the bandwidth of the DAQ system. The proof of the use of multiple IMFs to detect the BRB faults and operating points will be discussed in the following section.

The experimental setup used in this test is shown in Figure 2. The tested motor (model 056T34F5301 from Marathon Electric) has 34 rotor bars and 24 stator slots. The speed of these IMs is controlled by a variable frequency drive (VFD) (from Delta Electronics Inc.). In this experiment, 50 Hz and 60 Hz supply frequencies are used, switching at 15 kHz. The IM output shaft is coupled to a gearbox (from Boston Gear) to adjust the speed ratio of the dynamometer. A magnetic particle clutch (PHC-50 from Placid Industries) is used as a dynamometer to provide variable load via an external DC power source. Current signals are collected by the use of a wireless smart current sensor DAQ system, developed in our research team [24], which can measure 2 - 3 phase current signals. The vibration signals corrected by using the 3D vibration sensor are used to estimate the motor shaft rotating speed, which is then used to compute the slip [24].

The current signals are collected when they have reached their rated load levels. The speeds and slip ( f r , slip) at full load in 50 Hz and 60 Hz are 2910 RPM (48.5 Hz, 3%) and 3498 RPM (58.3 Hz, 2.83%), respectively. Four load conditions are tested: coupled shaft with minimum load (33% of full load), medium load (67% of full load), and full load (100% of full load) controlled by adjusting the magnetic clutch current. Decoupled condition corresponds to zero load state (0% of full load) when the motor is disconnected with the gearbox.

The effectiveness of the proposed adaptive EMD technique is evaluated under a series of tests corresponding to different load and speed operating conditions.

Five 1/3 HP IMs are tested in this work: two healthy IMs, and three faulted IMs with one, two, and three simulated BRBs, respectively. Figure 3 shows an example of an IM with two simulated BRBs.

At first, EMD is applied to the stator current. Figure 4(a) illustrates the first three IMFs of healthy motor and Figure 4(b) shows the IMFs of faulty motor using current signals at full load state. A comparison of Figure 4(a) and Figure 4(b) indicates that BRB can increase the oscillation of the IMF₂ amplitude. In this case, only the first three IMFs are calculated, as after these the correlation coefficients are almost negligible.

Figure 5 shows the correlation between the signal and IMFs. It is seen that the first IMF, or IMF₁, has the highest correlation value, as it has higher energy

frequency components. IMF₂ mainly includes RBR related freqeuency harmonics, while the remaining correlation values after IMF₂ are very small. It is important to note that although BRB sideband components exits in all the IMFs, the correlation coefficients decrease with the increase of the IMF order, up to negligible. In our proposed adaptive EMD technique, the IMF₁ will be used to detect the rotor bar faults and IMF₂ is applied to diagnose the fault conditions.

Different from the previous theoretical analysis in the literature where BRB feature is extracted from spectral magnitude of IMFs without considering fundamental line frequencies, the proposed adaptive EMD technique considers both the component of fundamental line frequency and the fault related sidebands to predict rotor bar fault.

Figure 6 shows the processing results of healthy and faulty motors at no load and full load condition, respectively. It is seen that spectra of stator current signals cannot detect rotor bar faults at no load or low load conditions. In our proposed adaptive EMD technique, the Hilbert transform is performed for the first

IMF before doing spectral analysis. It is also noted that a new frequency pattern of ( 1 ± 2 s ) f s is added to the fundamental frequency domain in the electromagnetic force profile. Torque and speed fluctuations will result in some extra sideband components. It means that if the motor speed is fixed, the sidebands may disappear, which is the reason sometimes sideband components may not be recognized in one or two sides.

In general, MCSA-based BRB fault detection uses the spectral magnitude of these sideband frequencies to determine the severity of the faults. However, a major problem for such an approach is that speed/load oscillation can generate some extra frequency components in the vicinity of fault characteristic frequency and its harmonics, which will result in false alarms in online IM health condition monitoring. The proposed adaptive EMD technique applies a new approach to distinguish these false alarms and real IM faults. Compared to low speed load oscillation, the variation of BRB frequencies are much higher in MCSA spectrum. From this investigation, it can be found that from no load to full load conditions, the BRB frequency component varies from 60.5 Hz to 63.39 Hz with a difference of 2.89 Hz; however, the literature indicates that frequency variation is about 0.035 Hz only, which is almost negligible. Therefore, the proposed adaptive EMD technique can distinguish false alarms to improve IM monitoring accuracy.

To further investigate the severity of the fault under different operating conditions, Figure 7(a) shows the fluctuation of IMF₂ at 3600 RPM for 1 and 3 BRBs at the full load condition. It is seen that the magnitude of IMF₂ with 3 BRBs is greater than the magnitude of 1 BRB. Figure 7(b) illustrates the results of IMF₂ for a motor with 3 BRBs at two respective speeds of 3600 RPM and 3000 RPM, at full load condition.

Figure 8 presents the results of IMF₂ at medium load state (i.e., 67% of full load) for different fault and speed conditions. It is seen that the magnitude of IMF₂ at 3600 RPM is much higher than that at 3000 RPM. Therefore, the distinction for different motor speed is clear, even though the distinction for different number of BRBs becomes less clear in this case.

Figure 9 and Figure 10 illustrate the impact of small load and no load conditions (33% versus 0%) on different fault states and different operating conditions. The amplitude of IMF₂ changes with the variation of load and fault severity. In this work it is found that the amplitude of the IMF₂ increases with the increase of load and with the increase in number of BRB.

From the above result analysis, it is seen that in a 34-bar rotor, if one bar is broken it influences 1/34 times of the total operation. But if 3 bars are broken it will affect 3/34 times of the total operation. Increasing the number of BRBs will result in asymmetry and saturation in the breakage area. The adjacent bars of broken bars will draw more current in this case. As a result, local flux density and

fluctuation increase. Similar phenomena can be observed in load increment. Increasing load level causes additional current in breakage area, which also causes saturation and fluctuation. In addition, motor load increment will increase the slip, the difference between synchronous speed and rotor speed; it also increases the IMF amplitude. But with the increase of speed, slip decreases. This slip reduction will lead to amplitude reduction of IMF. In this work the analysis is undertaken in the time domain whereas conventional EMD-based BRB analysis is performed in the frequency domain. In general, spectral analysis cannot provide accurate results in no load or light load condition (Figure 9 and Figure 10).