# Fault Diagnosis of Induction Machines in a Transient Regime Using Current Sensors with an Optimized Slepian Window

^{*}

## Abstract

**:**

## 1. Introduction

- The Slepians are the band-limited functions that are the most concentrated in a fixed time interval in the ${L}^{2}$-norm [59]. Therefore, they can be considered as the optimal window for TF analysis of non-stationary currents [60], because they can highlight the energy content of the current signal in the joint time-frequency domain with the highest possible resolution among all the almost time- and band-limited windows, including the truncated Gaussian window.
- Alternatively, the Slepians can be considered as the time-limited functions that are the most concentrated in a fixed frequency interval in the ${L}^{2}$-norm. That is, for a given bandwidth, they are the shortest possible windows that can be used for generating the current spectrograms, which allows the reduction of the time needed to build such spectrograms.

## 2. The Slepian Functions for Fault Diagnosis of Rotating Electrical Machines in the Transient Regime

#### 2.1. Theoretical Introduction to the Slepian Functions

#### 2.2. Energy of the Slepian Windows in a Time Interval

#### 2.3. Energy of the Slepian Windows in a Frequency Interval

#### 2.4. Energy of the Slepian Windows in the Joint TF Domain

#### 2.5. Comparison between the Slepian Window and the Gaussian Window

#### 2.6. Proposed Method for the Choice of the Parameters of the Slepian Window

## 3. STFT of the Start-Up Current of a Simulated IM Using the Slepian Window

#### 3.1. Evolution of the LSH during the Start-Up Transient of an IM

#### 3.2. Choice of the Parameters of the Slepian Window

#### 3.3. Detection of the LSH Fault Component with the Slepian Window

## 4. Experimental Validation on a High-Power, High-Voltage IM

#### 4.1. Choice of the Parameters of the Slepian Window for the Tested IM

#### 4.2. Application of the Slepian Window to the Fault Diagnosis of the Tested IM

## 5. Cost-Effective IM Fault Diagnosis Using the Truncated Slepian Window

- Reducing the length of the FFT to the time duration ${T}_{W}$ of the Slepian window in Equation (35), much smaller than the length of the current signal ${T}_{s}$; that is, using a truncated Slepian window with a length equal to ${T}_{W}$, instead of the length of the current signal. This is equivalent to setting $\Delta F=1/{T}_{W}$ in Equation (37).
- Increasing the time shift of the window in successive FFTs to a value of $1/{B}_{W}$, where ${B}_{W}$ is the frequency bandwidth of the Slepian window in Equation (35), much longer than the time step between consecutive samples of the current, $1/{F}_{sampling}$. That is, setting $\Delta T=1/{B}_{W}$ in Equation (37).

#### 5.1. Comparison between the Spectrograms Generated with the Truncated Gaussian Window and with the Truncated Slepian Window

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A Simulated IM

## Appendix B Industrial IM

## Appendix C Computer Features

## References

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**Figure 2.**Choice of the parameters of the Slepian window so that the aspect ratio of its Heisenberg box coincides with the slope of the trajectory of the related fault component in the time-frequency (TF) plane.

**Figure 3.**Time evolution of the left sideband harmonic (LSH) (

**top**), of the motor speed (

**middle**) and of the motor slip (

**bottom**) during the start-up transient of the simulated induction machine (IM) given in Appendix A. The vertical line corresponds to the time when the slip $s=0.5$ is reached.

**Figure 6.**Slepian window (${B}_{W}=20.85$ Hz, ${T}_{W}=383.7$ ms) optimized for the maximum overlap with the LSH trajectory in the time domain (

**top**) and in the frequency domain (

**bottom**).

**Figure 7.**Slepian window (${B}_{W}=20.85$ Hz, ${T}_{W}=383.7$ ms) optimized for representing the LSH, as a 2D view (

**top**) and as a 3D view (

**bottom**) in the time-frequency plane. The white line marks the trajectory of the LSH in this plane.

**Figure 8.**Time-frequency-amplitude pattern generated by the LSH obtained with the optimized Slepian window (${B}_{W}=20.85$ Hz, ${T}_{W}=383.7$ ms), as a 2D view (

**top**) and as a 3D view (

**bottom**).

**Figure 9.**(

**a**) Entropy of the time-frequency analysis of the LSH using the Slepian window, as a function of the parameter ${B}_{W}/{T}_{W}$; (

**b**) zoomed area of the entropy in the interval close to the optimum value of ${B}_{W}/{T}_{W}$. The vertical line corresponds to the minimum entropy value, which coincides with the criteria of maximum overlapping between the Slepian window and the LSH, as proposed in this paper.

**Figure 10.**Rotor of the high-power, high-voltage IM given in Appendix B (

**left**) and the detail of the broken rotor bar (

**right**), used in the experimental validation of the proposed method.

**Figure 11.**Stator current during the start-up transient of the high-power, high-voltage IM given in Appendix B with a broken bar fault.

**Figure 12.**Slepian window (${B}_{W}=11.55$ Hz, ${T}_{W}=692.8$ ms), optimized for detecting the LSH during the start-up of the high-power, high-voltage IM given in Appendix B, represented in the time (

**top**) and in the frequency (

**bottom**) domains.

**Figure 13.**Heisenberg’s box of the atom of the Slepian window (${B}_{W}=11.55$ Hz, ${T}_{W}=692.8$ ms), optimized for detecting the LSH during the start-up transient of the high-power, high-voltage IM given in Appendix B. The white line marks the estimated trajectory of the LSH in the time-frequency plane.

**Figure 14.**Spectrogram of the stator current computed with the proposed Slepian window, optimized for detecting the LSH during the start-up of the high-power, high-voltage IM given in Appendix B, with a broken bar (

**top**) and in healthy conditions (

**bottom**).

**Figure 15.**Amplitude of the LSH due to the broken rotor bar during the start-up of a healthy and faulty machine extracted from Figure 14. The average value of the LSH of the healthy machine (blue line) is $-56.36$ dB and of the faulty machine (red line) is −41.67 dB.

**Figure 16.**TF distribution of the stator current of the faulty machine presented in Section 4, using the full length TF analysis with a Slepian window (154.65 s, 186608 kB) (

**top**) and using the proposed reduced length TF analysis with the truncated Slepian window (0.59 s, 59 kB) (

**bottom**).

**Figure 17.**Reduced spectrogram of the high-power, high-voltage faulty machine given in Appendix B with a broken bar during the start-up transient using the truncated Slepian window (

**top**) and using the truncated Gaussian window (

**bottom**).

**Table 1.**Comparison of the parameters of the STFT of the current signal using the traditional full-length analysis and the proposed reduced length TF analysis, where ${T}_{s}$ is the length of the current signal, ${F}_{sampling}$ is the sampling frequency and ${T}_{W}$ and ${B}_{W}$ are the parameters of the Slepian window obtained from Equation (27).

Full-Length TF Analysis | Reduced Length TF Analysis | |
---|---|---|

Window duration (s) | ${T}_{s}$ | ${T}_{W}=8/{B}_{W}$ |

Shift step (s) | $1/{F}_{sampling}$ | $1/{B}_{W}$ |

FFT length (samples) | ${T}_{s}\xb7{F}_{sampling}$ | ${T}_{W}\xb7{F}_{sampling}$ |

Number of FFTs | ${T}_{s}\xb7{F}_{sampling}$ | ${T}_{s}\xb7{B}_{W}$ |

**Table 2.**Comparison of the parameters of the STFT of the current signal using the full-length and the proposed reduced length TF analysis, applied to the example presented in Section 4, where ${T}_{s}$ is the length of the current signal, ${F}_{sampling}$ is the sampling frequency and ${T}_{W}$ and ${B}_{W}$ are the parameters of the Slepian window obtained from Equation (27).

${\mathit{T}}_{\mathit{s}}=8.2$ s , ${\mathit{F}}_{\mathit{sampling}}=6.4\phantom{\rule{4.pt}{0ex}}\mathbf{kHz}$, $\mathit{T}=0.6928\phantom{\rule{4.pt}{0ex}}\mathbf{s}$ and $\mathit{B}=11.55\phantom{\rule{4.pt}{0ex}}\mathbf{Hz}$ | ||
---|---|---|

Full-Length TF Analysis | Reduced Length TF Analysis | |

Window’s length (s) | $8.2$ | $0.6928$ |

Shift step (s) | $1.56\times {10}^{-4}$ | $0.087$ |

FFT length (samples) | 52,480 | 4434 |

Number of FFTs | 52,480 | 95 |

Time needed for computing the spectrogram (s) | $154.65$ | $0.59$ |

Memory needed for computing the spectrogram (kB) | 186,608 | 59 |

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**MDPI and ACS Style**

Burriel-Valencia, J.; Puche-Panadero, R.; Martinez-Roman, J.; Sapena-Bano, A.; Pineda-Sanchez, M.
Fault Diagnosis of Induction Machines in a Transient Regime Using Current Sensors with an Optimized Slepian Window. *Sensors* **2018**, *18*, 146.
https://doi.org/10.3390/s18010146

**AMA Style**

Burriel-Valencia J, Puche-Panadero R, Martinez-Roman J, Sapena-Bano A, Pineda-Sanchez M.
Fault Diagnosis of Induction Machines in a Transient Regime Using Current Sensors with an Optimized Slepian Window. *Sensors*. 2018; 18(1):146.
https://doi.org/10.3390/s18010146

**Chicago/Turabian Style**

Burriel-Valencia, Jordi, Ruben Puche-Panadero, Javier Martinez-Roman, Angel Sapena-Bano, and Manuel Pineda-Sanchez.
2018. "Fault Diagnosis of Induction Machines in a Transient Regime Using Current Sensors with an Optimized Slepian Window" *Sensors* 18, no. 1: 146.
https://doi.org/10.3390/s18010146