A time-frequency method for induction motor fault feature extraction using instantaneous power signals

To solve the issue that the characteristic frequencies of the broken rotor bar fault (BRBF) are always submerged by the fundamental component during the stator current spectrum analysis in light loads, this paper proposes an effective approach based on the instantaneous power signal and the local maximum synchrosqueezing transform (LMSST). By multiplying the current and voltage, the instantaneous power signal can provide more fault features and enhance fault component amplitudes. The LMSST method is used to obtain a time-frequency analysis with more concentrated energy, and more fault characteristics are extracted from the reconstructed signal, which helps to accurately identify the motor BRBF.


Introduction
Induction motor is widely used in various industries, as a power source of a system.As they frequently starting and driving heavy workloads, the temperature of the motor rises and the stress increases.Under the high temperature and the sustained stress, its rotor bar and end ring may be open-welded or broken [1].If not discovered in time, the bar and the end ring would be damaged seriously.As a result, the system would be shut down or damaged.
Motor current signature analysis (MCSA) is a widely used method.It performs spectral analysis of a stator current signal to detect the occurrence of fault characteristic frequency [2].However, the characteristic frequency components are always submerged by the fundamental components and they are difficult to be discovered on spectrum.In order to find the fault components easily, Boudraa applied the autoregressive model to process stator current signal of a broken bar motor [3].Kumar extracted the stator current envelope using the Hilbert transform and then made envelope spectrum to get the brokenbar fault characteristic frequency [4].Yu proposed a motor fault diagnosis method using local maximum synchrosqueezing transform.His experimental results showed that the reliability of diagnosis could be improved by combining the vibration signal and the time-frequency (TF) analysis method [5].
In this paper, a TF method using instantaneous power signals is proposed to detect the BRBF in lightly loaded motors with mild severity.The structure of this paper is subdivided into five parts, the first of which is dedicated to explaining the superiority of instantaneous power signals.The second part details the theory of the LMSST.In the third part, the simulation signals are employed to verify the proposed method.In the fourth part, experimental validations are provided.The last part presents the conclusion.

Instantaneous power
The instantaneous power is defined as: (1) where ua(t) is A-phase stator voltage signal of an induction motor.ia(t) is A-phase stator current.where f1 is the source power frequency of the motor, namely, f1 is the fundamental frequency.Um, Im are the maximum values of voltage and current signals respectively.φ is the initial phase angle.Substituting (2) into (1), the instantaneous power can be expressed as: As we known, if broken bar fault takes place in the motor, stator current signal usually includes these two fault characteristic frequency components [2] : (1-2s)f1 and (1+2s)f1.So A-phase stator current signal ia(t) can be described as following: where s is the slip ratio, Ibr1 and φbr1 are the amplitude and initial phase of the fault characteristic component respectively.Ibr2 and φbr2 are the amplitude and initial phase of the characteristic component, respectively.Under the circumstances, the instantaneous power p(t) can be written as m 1 From Equation ( 5), several frequency components can be found: 2f1: 2 times fundamental frequency 2sf1: fault characteristic frequency 2(1-s)f1, 2(1+s)f1: fault characteristic frequency pairs.From Equation (2), it can be seen that i(t) and u(t) have the same frequency components.According to the correlation analysis theory, Equation ( 1) is an amplitude enhancement operation for the same frequency components in both signals.Similarly, in Equation ( 5), the amplitudes of all components are enhanced by the voltage.
As shown above, compare with stator current signal, the instantaneous power signal provides much more fault features, in the same time, the amplitude of these fault features is enhanced.

The local maximum synchrosqueezing transform
where Ak(t) and φk(t) are the instantaneous amplitude (IA) and instantaneous phase (IP) of the k-th mode.
The short-time Fourier transform (STFT) of the signal s(t) is defined as where is the frequency rearrangement operator,  is the instantaneous frequency after compression.
The LMSST improves the degree of energy aggregation in the spectrogram by detecting local maxima in the frequency direction of the spectrogram, and then performing frequency point compression [6].The specific assignment rules are expressed as arg max ( , ) , [ , ], ( , ) 0 ( , ) 0, ( , ) 0 where ∆ is the discrete frequency interval. ( , ) G t is the STFT time frequency spectrogram of s(t).Since the FFT of g(t) reaches its maximum at 0, Equation ( 7) can be derived as ( ), [ ( ) , ( ) ] ( , ) 0, s(t) can be reconstructed using the following equation

Simulation validation
A simulation current signal and its spectrum is shown in Figure 1.The fundamental frequency f1 is 50 Hz, and the frequency sf1 about fault feature is 0.5 Hz.The sampling frequency is 6400 Hz and the sampling time is 2s.
   2) (13) Figure 2 shows p(t) and its spectrum.As we seen, a significant frequency component 100 Hz is highlight on the spectrum Figure 2(b).besides this, the components,1 Hz (2sf1) and 2 Hz (4sf1) and 0 Hz (dc) are displayed on the zoomed spectrum Figure 2(c).The characteristic components of the BRBF obtained from the instantaneous power signal are still easily submerged in the noise signal, so the LMSST is used to further process the signal.The LMSST contour plot of p(t) is shown in Figure 3.A distinguish TF trajectory is displayed, and it is concentrated around 100 Hz.Using Equation (11) to reconstruct the component around 100 Hz, Figure 4 gives the reconstructed signal and its envelope spectrum [7].The two characteristic components: 1 Hz and 2 Hz are obviously displayed on the envelope spectrum.
where TFR(t,ω) is the time-frequency result.α is usually set to 3. The lower Rényi entropy indicates that the TF method can produce a more energy-concentrated TF representation.In Table 1 the LMSST method has the lowest Rényi entropy, i.e., it provides the most energy-concentrated TF representation.Table 1.Rényi entropy of different TF methods.

Experimental validation
The test rig consists of a three-phase induction motor and a DC generator which simulates the load torque [9].Main parameters of the motor are as following: 0.75 kW, 220 V/3.02 A, 4 poles, 34 rotor bars, power supply frequency is 60 Hz (f1), rated speed is 1715 rpm, rated torque is 4.1 Nm.In order to mimic broken bar fault, holes were drilled on the rotor.Stator current and voltage signals are used to verify the above method with one rotor bar broken at 25% of the rated load (1 Nm).The sampling frequency is 6400 Hz.

The instantaneous power signal spectrum analysis
Figure 6 shows the instantaneous power signal and its spectrum.The frequency components: 1 Hz (2sf1) and 2 Hz (4sf1) are highlighted in Figure 6(c).By calculation, the slip is 0.83% and the rotation frequency fr is 29.75 Hz.It's shown by Figure 6(b) that as the rotor bar damaged in early stage, the frequency components 29.73 Hz (fr), 59.46 Hz (2fr), 90.27 Hz (2f1-fr) and 149.73 Hz (2f1+fr) appears.These frequency components can be treated as signatures of the rotor bar damaged.

LMSST analysis
The instantaneous power signal is processed again using the LMSST method.Figure 7 shows the contour plot of the LMSST.And energy at harmonics: 2f1,4f1, 6 f1, 8 f1, etc. are dominated, while the energy at other frequencies is very low, which cannot be shown in the contour plot.By extracting the ridges of the contour plot, we obtain a series of IF trajectories shown in Figure 8.These IF trajectories are located around 2f1,4f1, 6 f1, 8 f1, 12 f1, respectively.9.The components 1 Hz (2sf1) and 2 Hz (4sf1) are very obvious in the envelope spectrum, and the frequency 29.73 Hz (fr) is also here.Reconstructing the trajectory of the 6f1 component, Figure 10 shows the results.The amplitude of reconstructed signal is obviously modulated.And more feature frequencies of the BRBF are extracted such as 1 Hz (2sf1), 3 Hz (6sf1) and 4 Hz (8sf1), also the frequency 29.73 Hz (fr).In contrast, only the components 1 Hz (2sf1) and 2 Hz (4sf1) can be extracted from the envelope of the original instantaneous power signal.

Conclusion
In this paper, the instantaneous power signal of the induction motor is combined with the LMSST to diagnose the BRBF.By providing more fault features and larger fault component amplitudes, the instantaneous power signal solves the issue that the BRBF characteristic frequencies are easily

Figure 1 (Figure 1 .
Figure 1.Current signal and its spectrum.To build the instantaneous power signal p(t), the corresponding voltage signal is denoted as 1 220cos(2 )

Figure 2 .
Figure 2. Instantaneous power signal and its spectrum.The characteristic components of the BRBF obtained from the instantaneous power signal are still easily submerged in the noise signal, so the LMSST is used to further process the signal.The LMSST contour plot of p(t) is shown in Figure3.A distinguish TF trajectory is displayed, and it is concentrated around 100 Hz.Using Equation (11) to reconstruct the component around 100 Hz, Figure4gives the reconstructed signal and its envelope spectrum[7].The two characteristic components: 1 Hz and 2 Hz are obviously displayed on the envelope spectrum.

Figure 6 .
Figure 6.Instantaneous power signal with one broken bar and its spectrum.

Figure 7 .
Figure 7. LMSST result Figure 8. IF trajectories Taking the trajectory of the 2f1 component to reconstruct which has the highest energy, the reconstructed signal and its envelope spectrum are shown in Figure 9.The components 1 Hz (2sf1) and 2 Hz (4sf1) are very obvious in the envelope spectrum, and the frequency 29.73 Hz (fr) is also here.Reconstructing the trajectory of the 6f1 component, Figure10shows the results.The amplitude of reconstructed signal is obviously modulated.And more feature frequencies of the BRBF are extracted such as 1 Hz (2sf1), 3 Hz (6sf1) and 4 Hz (8sf1), also the frequency 29.73 Hz (fr).In contrast, only the components 1 Hz (2sf1) and 2 Hz (4sf1) can be extracted from the envelope of the original instantaneous power signal.

Figure 9 .Figure 10 .
Figure 9. Reconstructed signal of the 120 Hz ridge and its envelope spectrum.

Table 1
[8]ts the Rényi entropy values of the above time-frequency transform.The Rényi entropy is defined as[8]