# Multi-Band Frequency Window for Time-Frequency Fault Diagnosis of Induction Machines

^{*}

## Abstract

**:**

## 1. Introduction

- Performing the STFT with a wide range of windows with different lengths, and selecting, for each point in the TF domain, the best result obtained at that point among the complete set of windows.
- Instead of running a separate STFT for each of the windows used in the analysis, a single, multi-band window is built by stacking all the desired analysing windows in consecutive frequency bands. This approach obtains in parallel the spectrograms corresponding to several hundreds of different analysis windows with the computing cost of a single one, which makes it suitable for fast, online diagnostic systems in transient regime.

## 2. Time-Frequency Analysis of the Machine’s Current via STFT with a Multi-Band Window

#### 2.1. Spectrogram of Machine’s Current

- For each instant $\tau $, the current signal is multiplied element by element by the conjugate of a suitable time window centered at $\tau $, $h(t-\tau )$$${i}_{\tau}\left(t\right)=i\left(t\right)h{(t-\tau )}^{*}$$$${i}_{\tau}\left(t\right)=\left\{\begin{array}{cc}\hfill i\left(t\right),& \mathrm{if}t\mathrm{is}\mathrm{close}\mathrm{to}\tau \hfill \\ \hfill 0,& \mathrm{if}t\mathrm{is}\mathrm{far}\mathrm{from}\tau \hfill \end{array}\right.$$
- The Fourier transform if applied to the time-windowed signal ${i}_{\tau}\left(t\right)$, which gives the frequency content of the current signal $i\left(t\right)$ around time $\tau $$$\begin{array}{cc}{I}_{\tau}\left(\omega \right)\hfill & =\frac{1}{\sqrt{2\pi}}\int {e}^{-j\omega t}{i}_{\tau}\left(t\right)dt\hfill \\ & =\frac{1}{\sqrt{2\pi}}\int {e}^{-j\omega t}i\left(t\right)h{(t-\tau )}^{*}dt\hfill \end{array}$$
- The energy density spectrum at time $\tau $ is obtained as$${I}_{SP}(\tau ,\omega )={\left|{I}_{\tau}\left(\omega \right)\right|}^{2}=|\frac{1}{\sqrt{2\pi}}\int {e}^{-j\omega t}i\left(t\right)h{(t-\tau )}^{\ast}dt{|}^{2}$$

#### 2.2. Frequency Shifting of the Gaussian Analysing Window

## 3. Proposed Multi-Band Analysing Window

#### 3.1. Steps for Applying the Proposed Multi-Band Window

- The analysis window (15) is built:
- first, the bandwidth of diagnostic interest [0–${f}_{b}$] is established ([0–100 Hz] in this case), which gives the maximum number of elementary Gaussian windows from (14) ($Ng=200/100=2$). The current signal is low-pass filtered with a cut-off frequency equal to ${f}_{b}$. In this work, a spectral filter which zeroes all the spectrum bins with a frequency greater than ${f}_{b}$ has been used, as in Reference [45].
- second, the parameters ${\alpha}_{k}$ of each of these windows in (15) must be selected. For an actual application of the method, a high number of ${\alpha}_{k}$ parameters would be automatically generated, as was explained in Section 3, applying an expression similar to (22) in Section 4. For the simple signal (17), only two values of $\alpha $ are used, ${\alpha}_{0}=1$ (a long window) and ${\alpha}_{1}=12.6$, tailored to the chirp component according to Reference [35].

- The spectrogram of the current signal (17) is built, using the multi-band window (18) as sliding window (Figure 5). It shows two elementary spectrograms in adjacent TF regions, obtained with a single run of the STFT algorithm. The bottom one ($\alpha =1$) locates the sinusoidal component at 50 Hz but blurs the chirp component. On the contrary, the top one ($\alpha =12.6$) locates the chirp component but widens the sinusoidal component. This second Gaussian window, and the spectrogram that it generates, have been shifted to the frequency band [100–200 Hz].
- All the stacked, elementary spectrograms obtained in step 2 are shifted back to the frequency band [0–${f}_{b}$], as shown in Figure 6. This process has a negligible computational cost, just the renumbering of the frequency axis of each elementary spectrogram.
- The points with the same time-frequency coordinates in all the relocated spectrograms obtained in step 3 (Figure 6) are combined to give a single high resolution spectrogram of the TF region of diagnostic interest, in the frequency band [0–${f}_{b}$] (Figure 7). The combination process used in this work consists in selecting, for each point of this region, the minimum value obtained among all the relocated spectrograms. The final result shows with a high resolution both the sinusoidal component at 50 Hz and the chirp component.

## 4. Experimental Validation

- The analysis window (15) is built:
- first, the bandwidth of diagnostic interest is established. In this case, from (20), the frequency band of interest is the [0–50 Hz] band. Due to the presence of higher order harmonics in the current spectrum, apart form the LSH, a wider band [0–125 Hz] has been selected, in order to better assess the strength of the LSH compared with them. In this way, with a sampling frequency of 5000 Hz, the maximum number of elementary Gaussian windows from (14) is $Ng=5000/125=40$ windows.
- second, the parameter ${\alpha}_{k}$ of each of these windows in (15) is selected. In this case, a linear range of the parameter $\alpha $ is used, covering the range [${\alpha}_{min}=1$ – ${\alpha}_{max}=700$].

The resulting multi-band window, applying (15), is$$gm\left(t\right)=\sum _{k=0}^{39}{\left(\frac{k{\alpha}_{k}}{\pi}\right)}^{1/4}{e}^{-\frac{k{\alpha}_{k}}{2}{t}^{2}}{e}^{jk2\pi 125}$$$${\alpha}_{k}=1+k\frac{700-1}{39}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}k=0,1,\dots ,39$$ - The spectrogram of the current signal of Figure 8 is built. First, the current signal is low-pass filtered, keeping only the frequency bins of its spectrum lower than 125 Hz. After, and using the multi-band window (21) as sliding window, the STFT algorithm (4) is applied, which generates the spectrogram shown in Figure 10. This spectrogram contains 40 elementary spectrograms in adjacent TF regions (Figure 10, right), obtained with 40 different Gaussian windows (Figure 10, left), at the cost of a single run of the STFT algorithm (6 seconds with the computer of Appendix C). This computation time is not compatible with real-time applications but this is not important for the diagnosis of faults that develop progressively along hours, days or months, such as rotor asymmetries, eccentricities or bearing faults.Two of the 40 individual Gaussian windows shown in Figure 10 are displayed in Figure 11, left, along with their corresponding current spectra (Figure 11, right). The two zoomed bands corresponds to the spectrogram located in the base frequency band [0–125 Hz] (Figure 11, bottom), which defines clearly the fundamental component but blurs the fault harmonics, and to the spectrogram shifted to the frequency band [1500–1625 Hz] (Figure 11, top), which defines clearly the fault harmonics but widens the fundamental component.
- All the stacked, elementary spectrograms obtained in step 2 (Figure 10, right), are relocated in the base frequency band [0–125 Hz], by renumbering their frequency axis.
- All the relocated, elementary spectrograms obtained in step 3 (Figure 10, right), are combined to give a high resolution spectrogram of the TF region of diagnostic interest (Figure 12). The combination process used in this work consists in selecting, for each point of this [0–125 Hz] region, the minimum value obtained among all the relocated spectrograms. The final result shows with a high resolution both the sinusoidal component at 50 Hz and the LSH fault component. Unlike the individual spectrograms, the optimized spectrogram clearly shows the LSH not only during the transient period but also when the steady state is reached; it is also remarkable the set of fault-related second-order components that are revealed, which helps to get a more reliable diagnostic.

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A. Motor Characteristics

## Appendix B. Current Clamp

## Appendix C. Computer Features

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**Figure 1.**Multi-band window designed in Section 3 in the time domain (real part, top, imaginary part, middle) and in the frequency domain (bottom). This window contains 20 Gaussian windows, located in 20 adjacent frequency bands of 100 Hz, spanning linearly the range [$\alpha =1$ – $\alpha =700$].

**Figure 2.**Atoms of the multi-band window designed in Section 3 in the joint time-frequency domain. This window contains 20 Gaussian windows, located in 20 adjacent frequency bands of 100 Hz, spanning linearly the range from $\alpha =1$ (bottom) up to $\alpha =700$ (top).

**Figure 4.**Multi-band window (15) in the time domain (top) and in the frequency domain (bottom). This window contains 2 Gaussian windows, in 2 adjacent frequency bands of 100 Hz, one with $\alpha =1$ and the other one with $\alpha =12.6$.

**Figure 5.**Spectrogram of the current signal (17) obtained in step 2 with the multi-band window (right). The Gaussian component of the window with $\alpha =1$ (bottom, left) locates the sinusoidal component at 50 Hz (bottom, right) but fails to resolve the chirp component. On the contrary, the Gaussian component with $\alpha =12.6$ (top, left) locates the chirp component but widens the sinusoidal component (top, right). The Gaussian window that is frequency shifted (left) generates a spectrogram that is also frequency shifted (right).

**Figure 6.**Relocation in the frequency axis of the elementary spectrograms, so that all of them span the same frequency band [0–${f}_{b}$].

**Figure 7.**High resolution spectrogram of the current signal (17) —for each point of the time frequency (TF) region of interest, the minimum value obtained among all the stacked spectrograms of Figure 5, right, is selected. The final result shows with a high resolution both the sinusoidal component at 50 Hz and the chirp component.

**Figure 8.**Start-up current of the motor of Appendix A with a broken bar fault.

**Figure 9.**Multi-band window (21) in the time domain (real part,

**top**, imaginary part,

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

**bottom**). This window contains 40 Gaussian windows, located in 40 adjacent frequency bands of 125 Hz, with values of $\alpha $ ranging from ${\alpha}_{min}=1$ to ${\alpha}_{max}=700$.

**Figure 11.**Zoom of Figure 10 showing two of the individual Gaussian windows contained in the multi-band window of Figure 10 (

**left**) and the corresponding spectrograms generated with them (

**right**). The two zoomed bands corresponds to the spectrogram located in base frequency band [0–125 Hz] (

**bottom**), which defines clearly the fundamental component but blurs the fault harmonics and to the spectrogram shifted to the frequency band [1500–1625 Hz] (

**top**), which defines clearly the fault harmonics but widens the fundamental component.

**Figure 12.**High resolution spectrogram of the current of Figure 8—for each point of the TF region of interest, the minimum value obtained among all the stacked spectrograms of Figure 10, right, is selected. The final result shows with a high resolution both the sinusoidal component at 50 Hz and the lower side band harmonic (LSH) fault component.

Type of Fault | Fault Harmonics Frequency | |
---|---|---|

Shorted coils | ${f}_{1}\left(k\pm n\frac{1-s}{p}\right)$ | $k=1,3,5\dots $ $n=1,2,3,\dots $ |

Rotor asymmetries | ${f}_{1}\left((1-s)\frac{k}{p}\pm s\right)$ | $\frac{k}{p}=1,3,5\dots $ |

Mixed eccentricity | $|{f}_{1}\pm k{f}_{r}|$ | $k=1,2,3\dots $ |

Bearing (outer race) | $\frac{{N}_{b}}{2}{f}_{r}\left[1-\frac{{D}_{b}\mathrm{cos}\left(\beta \right)}{{D}_{c}}\right]$ | |

Bearing (inner race) | $\frac{{N}_{b}}{2}{f}_{r}\left[1+\frac{{D}_{b}\mathrm{cos}\left(\beta \right)}{{D}_{c}}\right]$ | |

Bearing (balls) | $\frac{{D}_{c}{f}_{r}}{2{D}_{b}}\left[1-{\left(\frac{{D}_{b}\mathrm{cos}\left(\beta \right)}{{D}_{c}}\right)}^{2}\right]$ |

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## Share and Cite

**MDPI and ACS Style**

Burriel-Valencia, J.; Puche-Panadero, R.; Martinez-Roman, J.; Sapena-Baño, A.; Riera-Guasp, M.; Pineda-Sánchez, M. Multi-Band Frequency Window for Time-Frequency Fault Diagnosis of Induction Machines. *Energies* **2019**, *12*, 3361.
https://doi.org/10.3390/en12173361

**AMA Style**

Burriel-Valencia J, Puche-Panadero R, Martinez-Roman J, Sapena-Baño A, Riera-Guasp M, Pineda-Sánchez M. Multi-Band Frequency Window for Time-Frequency Fault Diagnosis of Induction Machines. *Energies*. 2019; 12(17):3361.
https://doi.org/10.3390/en12173361

**Chicago/Turabian Style**

Burriel-Valencia, Jordi, Ruben Puche-Panadero, Javier Martinez-Roman, Angel Sapena-Baño, Martin Riera-Guasp, and Manuel Pineda-Sánchez. 2019. "Multi-Band Frequency Window for Time-Frequency Fault Diagnosis of Induction Machines" *Energies* 12, no. 17: 3361.
https://doi.org/10.3390/en12173361