Fractional Calculus-Based Processing for Feature Extraction in Harmonic-Polluted Fault Monitoring Systems

Abstract

Fault monitoring systems in Induction Motors (IMs) are in high demand since many production environments require yielding detection tools independent of their power supply. When IMs are inverter-fed, they become more complicated to diagnose via spectral techniques because those are susceptible to produce false positives. This paper proposes an innovative and reliable methodology to ease the monitoring and fault diagnosis of IMs. It employs fractional Gaussian windows determined from Caputo operators to stand out from spectral harmonic trajectories. This methodology was implemented and simulated to process real signals from an induction motor, in both healthy and faulty conditions. Results show that the proposed technique outperforms several traditional approaches by getting the clearest and most useful patterns for feature extraction purposes.

1. Introduction

Squirrel-cage motors are the most popular kind of Induction Motors (IMs), which are the main element in almost all industrial facilities and railway transportation systems [1]. It is because of their important advantages in terms of cost, reliability, and robustness [2]. For their operating conditions under corrosive and harsh environments, these electro-mechanical converters require special supervision and maintenance to avoid, in the most extreme case, an industrial blackout [3,4]. On intermediate cases of mechanical, thermal, electrical, and external stresses, severe electrical and mechanical faults can be caused by accelerating functional aging in parts, such as the rotor, the stator or even its bearings [5,6]. Although the prevalence of rotor fault is not the highest, the period from which a bar fails until it suffers a catastrophic failure is wide enough to assess the evolution of frequency patterns. These patterns are hard to extract because harmonic contamination in inverter-fed systems introduces frequencies that overlap fault characteristics. Thus, their detection with spectral analysis techniques is compromised [7]. Nowadays, there is an increased interest in sensorless control systems of induction motors for faulty symptoms detection and diagnosis purposes [8,9,10,11]. The broken rotor bars (BRBs) are the main failure to be analyzed by using spectral analysis [12,13]. For instance, a sensorless IM drive for broken rotors was discussed in [14]. Zolfaghari et al. proposed a BRB detector based on a wavelet feature analysis and a feed-forward multi-layer perceptron as a classifier for fault detection [15]. In the same context, Lamim et al. employed empirical demodulation and discrete wavelet transform techniques for diagnosing BRBs in IMs with extreme slip conditions [16]. More recently, Skowron et al. trained an efficient self-organized neural network for classifying electrical faults of induction motors [17]. In addition, some studies about fault-tolerant-control of electrical drivers in industrial processes and fault-diagnosis methodologies are discussed in [18].

Therefore, Motor Current Signature Analysis (MCSA) is one of the most used methods because it is non-invasive and versatile [11,19]. It facilitates the maintenance with cloud computing-based activities through the stator current acquisition. Running conditions such as voltage variations or harmonic sources introduce undesired frequency components in the stator current. This has given way to recent research activity, which is focused on the resolution improvement of joint time-frequency representations [20]. Bellini et al. proposed statistical time-domain techniques to track grid frequency and machine slip [21]. Their approach was used to tune the parameters of a zoom Fast Fourier Transform (ZFFT) algorithm, which increases the frequency resolution in detecting faulty motors. Likewise, Kowalsky et al. applied the FFT algorithm to the instantaneous active and reactive powers and the electromagnetic torque [22]. They detected early stator short-circuit and BRB faults of IMs fed by a variable frequency converter. Benbouzid et al. published an interesting comparison of signal processing techniques for identifying BRBs and bearing failures in IMs [23]. Thus, they developed a portfolio of techniques to select the proper method for a certain fault, according to their features. They also discussed windowing techniques and analyzed the side-lobe leakage effects of classical windowing using spectral analysis. In this study, the proposed method based on fractional calculus reduces those negative effects for processing features.

Notwithstanding the above complications, some researchers have identified rotor faults in inverter-fed IMs running in steady-state [6,19]. Nevertheless, traditional methods based on the Fourier transform present some limitations, especially when it is a reduced number of data samples, e.g., some signal frequencies are too close and the signal-to-noise ratio (SNR) is low. To solve these limitations, many authors have used windowed FT-based methods, i.e., time-frequency analysis techniques [3]. Most common examples of these methods are the well-known Short-Time Fourier Transform (STFT) and the Windowed Fourier Transform (WFT) [24]. They have shown fair performance for representing signals in machinery fault diagnosis but requiring an inherent trade-off between frequency and time resolution, especially to cope with non-stationary signals [25]. Many works have proposed improvements to surmount the main drawbacks of these techniques, giving way to the more sophisticated MCSA methods, but they require expensive computations [3]. Nevertheless, WFT is still considered a powerful tool for several applications when the window shape is adapted or chosen properly [26]. From an engineering point of view, it is suitable for developing low-cost feature extractions on a regular basis. In several documented WFT applications, the Gaussian window has been highlighted as having the best compensation among several constraints, i.e., the Gabor transform [27,28]. An example of them is the analysis tool presented in [29] by Lima et al. for signals captured with impacts and vibrations from a mechanical manipulator. The authors employed a WFT with multiple windows, including the Gaussian one, under the Fractional Calculus (FC) perspective. It is important to mention that, in a more recent work, the Gaussian window was parameterized to modify its shape and explore its features when it is used in a WFT-based analysis [30]. Applications of FC have gained prominence in several engineering applications; however, there are no studies about their advantages in areas such as condition monitoring. Hence, this work means a novelty in this aspect, which is also a mesmerizing topic for the practical engineering community. FC is an extension of the ordinary calculus, devoted to derivatives and integrals of arbitrary orders [31]. In the last few decades, it has been used in several research works due to the advent of modern numerical computing machines, and to its features for modeling numerous processes in fields of science and engineering [31,32]. Moreover, several definitions of fractional derivatives and integrals have been proposed and implemented in the literature [31,33], e.g., the Riemann–Liouville, Grünwald–Letnikov, Caputo–Dzhrbashyan (CD) and Weyl definitions [31,34]; and more recently, the Caputo–Fabrizio (CF) [35] and Atangana–Baleanu–Caputo (ABC) [36] definitions. In addition, the Caputo fractional derivative family has shown interesting properties and advantages for practical implementations [32,37,38]. To clearly state the purpose of this work, we study the practical advantages of using some FC concepts and we do not have the intention of delving into controversial issues about formal FC definitions. However, we invite readers to consult [38,39,40,41,42] for detailed information on this interesting topic of discussion. Therefore, and for the sake of politeness, CF and ABC are hereby treated as just fractional or non-integer operators, but not as derivatives.

This work proposes an alternative identification method, found in FC and WFT, as an advantageous form of extracting patterns from time-frequency spectra for fault monitoring of operating induction motors. Particularly, the main contribution of this work is the implementation of windows achieved by applying fractional operators under the Caputo sense to the Gaussian function. They were named fractional-based Gaussian windows. Hence, three kinds of windows were obtained from the fractional Caputo kernels reported in the literature, i.e., CD, CF, and ABC. The proposed methodology is implemented and tuned, finding the proper parameter configuration by employing a synthetic signal, before applying it in real measurements. Those data are stator current signals obtained from an inverter-fed induction motor with both healthy and fault states. The latter was provoked by a rotor-bar perforation. Moreover, the resulting spectra from our implementation are compared against those reached by using traditional windows, such as the Rectangular, Hanning, and Hamming. Overall results showed that proposed windows outperform the traditional ones, evaluated quantitatively, with an $L^2$ error value at least of 1.63%. Particularly, ABC-based windows present the most detailed spectrogram, followed by CF-based windows becoming a reliable alternative for the evaluation and monitoring of fault harmonics.

The presented manuscript is organized as follows: The first section briefly describes the fundamental concepts associated with the proposed method, such as the most common induction motor faults, the fractional Gaussian windows and how they are obtained, and the Windowed Fourier Transform. Then, numerical procedures and experiments carried out are detailed in Methodology, and results are discussed in the subsequent section. Finally, the most relevant conclusions are summarized.

2. Foundations

2.1. Induction Motor Faults

Several faults of different nature can be slightly, independently, or simultaneously produced. Three of the most common faults in Squirrel-cage IMs are [43]:

• Bearing fault due to the thermal and electrical burden, insufficient lubrication, etc.;
• Air–gap eccentricity fault caused by a significant variation in the rotor-to-stator spacing; and
• Broken Rotor Bars (BRBs) fault mainly produced by mechanical, thermal, magnetic, electrical, and environmental stresses.

Faults always disturb the correct behavior of an induction motor, which has a well-known signature in all the signals related to its operation mode. Thus, fault presence characteristics are noticed from current, mechanical, acoustic, thermal, and magnetic signals [1,2]. Therefore, the motor fault signal is commonly employed for analyzing the IM healthiness. It is known as Motor Current Signature Analysis (MCSA) and it is widely used due to its non-invasive and versatile features [19,20].

This work focused on experiments and numerical implementations of faults caused by a rotor breakage since, in its intermediate stage, it can be detected to stunt the IM accelerated aging [4]. However, frequency-related indicators are complicated to extract from an inverter-fed IM rotor because of power conversion harmonics appear in the stator current spectrum and they bury fault-related frequencies given by

$$f_{\mathrm{BRB}}=\left[\frac{k}{p}(1-s)\pm s\right]f_s.$$

(1)

These BRB frequencies $f_{\mathrm{BRB}}$ depend on the motor characteristic values $k/p=2n+1$, $\forall n\in {\mathbb{Z}}_{++}$, the per-unit motor slip s, the number of poles p, and the supply current frequency $f_s$. Fault frequencies are well-known as Left Side-band Harmonic (LSH) for the minus sign and Right Side-band Harmonic (RSH) for the plus sign [3,4,5].

2.2. Fractional Gaussian Windows

Fractional Gaussian windows are obtained by applying non-integer order operators to the Gaussian function. To do that, as a first step, it is required to find the fractional functions. Thus, the generalized Caputo fractional operator is used according to [38], defined as,

$${}_0{}^{\mathrm{C}}D_t^{\nu }f\left(t\right)={\int }_0^t{k}_{\nu }(t-\tau ){\left.{\partial }_{t}f\left(t\right)\right|}_{t=\tau }d\tau ,$$

(2)

since $\nu \in [0.0,1.0]$ is the fractional (non-integer) order, $f\left(t\right):\mathbb{R}\to \mathbb{R}$ is a real, absolutely continuous, and causal function, $\tau \in [0.0,t]$ is a dummy time variable (related with the convolution operator), ${\partial }_t\{·\}$ is the ordinary first derivative operator, and $k_{\nu }\left(t\right)$ is the kernel following the properties given in [38]. Three of the commonly used kernels for this definition are Caputo–Dzhrbashyan (CD) in Equation (3), Caputo–Fabrizio (CF) in Equation (4), and Atangana–Baleanu–Caputo (ABC) in Equation (5), which are detailed as follows:

$$\begin{array}{cc}\hfill k_{\nu }^{\mathrm{CD}}\left(t\right)& =\frac{t^{-\nu }}{\mathrm{\Gamma }(1-\nu )},\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill k_{\nu }^{\mathrm{CF}}\left(t\right)& =\frac{M\left(\nu \right)}{1-\nu }{e}^{-\frac{\nu }{1-\nu }t},\hfill \end{array}$$

(4)

$$\begin{array}{cc}\hfill k_{\nu }^{\mathrm{ABC}}\left(t\right)& =\frac{M\left(\nu \right)}{1-\nu }{E}_{\nu }\left(-\frac{\nu }{1-\nu }{t}^{\nu }\right).\hfill \end{array}$$

(5)

These kernels have different features about continuity and memory, i.e., CF is singular and ABC is non-local [38,44,45]. Furthermore, $\mathrm{\Gamma }(·)$ is the well-known Gamma function, $M\left(\nu \right)$ is an “unknown” normalization function, commonly assumed as $M\left(\nu \right)=1.0$ [46], and $E_{\nu }\left(t\right)$ is the one-parameter Mittag–Leffler function. Subsequently, recall the classical expression for the Gaussian function,

$$G_{\mu ,\sigma }\left(t\right)=\frac{1}{\sqrt{2\pi }\sigma }{e}^{-\frac{{(t-\mu )}^2}{2{\sigma }^2}},$$

(6)

where $G_{\mu ,\sigma }\left(t\right):\mathbb{R}\to \mathbb{R}$ with $\mu$ and $\sigma$ as the mean and standard deviation, respectively, and t is the independent variable related either with time or space. Moreover, it is required to calculate the first ordinary derivative of the Gaussian function, as

$${\partial }_t{G}_{\mu ,\sigma }\left(t\right)=\frac{(\mu -t)}{{\sigma }^2}{G}_{\mu ,\sigma }\left(t\right),$$

(7)

to obtain fractional operators by utilizing the Caputo definition from Equation (2). Therefore, fractional forms of the Gaussian function, using the aforementioned kernels, are found as follows:

$$\begin{array}{cc}\hfill {}_0{}^{\mathrm{CD}}D_t^{\nu }{G}_{\mu ,\sigma }\left(t\right)& =\frac{1}{\sqrt{2\pi }{\sigma }^3\mathrm{\Gamma }(1-\nu )}{\int }_0^t\frac{\mu -\tau }{{(t-\tau )}^{\nu }}{e}^{-\frac{{(\tau -\mu )}^2}{2{\sigma }^2}}d\tau ,\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill {}_0{}^{\mathrm{CF}}D_t^{\nu }{G}_{\mu ,\sigma }\left(t\right)& =\frac{1}{\sqrt{2\pi }{\sigma }^3}{\int }_0^t\frac{\mu -\tau }{1-\nu }{e}^{-\nu \frac{t-\tau }{1-\nu }-\frac{{(\tau -\mu )}^2}{2{\sigma }^2}}d\tau ,\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {}_0{}^{\mathrm{ABC}}D_t^{\nu }{G}_{\mu ,\sigma }\left(t\right)& =\frac{1}{\sqrt{2\pi }{\sigma }^3}{\int }_0^t\frac{\mu -\tau }{1-\nu }·E_{\nu }\left(-\nu \frac{{(t-\tau )}^{\nu }}{1-\nu }\right)e^{-\frac{{(\tau -\mu )}^2}{2{\sigma }^2}}d\tau .\hfill \end{array}$$

(10)

It is noteworthy that only Equation (8) may be called the Caputo fractional derivative of the Gaussian function [39,41]. These formulae can be written in a closed-form, but, for the sake of brevity, we invite the reader to perform this enriching experience. More information about these can be consulted in [37]. Thus, expressions in Equations (8)–(10) give way to multiple windows (patterns) based on fractional forms of the Gaussian function, by evaluating them in a defined time series and assuming values for $\mu$, $\sigma$, and $\nu$.

2.3. Windowed Fourier Transform

Windowed Fourier transform (WFT) is a natural generalization of Short-Time Fourier Transform (STFT) where different window functions $w\left(t\right)$ are employed [24,30,47]. Consider a real and causal signal $s\left(t\right):{\mathbb{R}}_{+}\to \mathbb{R}$ in time t (or space) and a finite energy window $w\left(t\right)$, then the WFT of $s\left(t\right)$ is obtained as

$$\widehat{s}(t,f)={\int }_0^{\infty }s\left(\tau \right)w(t-\tau )e^{-\mathrm{i}2\pi f\tau }d\tau ,$$

(11)

since f is the frequency variable.

Now, using the aforementioned fractional Gaussian formulae as windows, the fractional WFT of $s\left(t\right)$ is written as

$${\widehat{s}}_{\nu }(t,f)={\int }_0^{\infty }s\left(\tau \right){}_0{}^{\mathrm{C}}D_t^{\nu }{G}_{\mu ,\sigma }(t-\tau )e^{-2\mathrm{i}\pi f\tau }d\tau ,$$

(12)

where ${}^{\mathrm{C}}D_t^{\nu }{G}_{\mu ,\sigma }\left(t\right)$ $\forall \mathrm{C}\in \{\mathrm{CD},\mathrm{CF},\mathrm{ABC}\}$ from Equations (8)–(10). When $\nu \to 0$, any fractional Gaussian window becomes the classical Gaussian window; then, Equation (12) gives the well-known Gabor transform.

3. Methodology

Experimental data used in this work were obtained through the test bench shown in Figure 1. It comprises a three-phase induction motor Siemens^® Erlangen™ 1LA7083-4AA10 (Erlangen, Bavaria, Germany), star-connected, with two pole pairs, 28 bars, 1395 rpm, rated-current of 1.9 A and 0.75 kW (Figure 1a). This motor was fed by a voltage inverter Allen Bradley^® (Milwaukee, WI, USA), model PowerFlex 40, set for an operating frequency of 50 Hz through its integral keypad, and with a supplying voltage of 230/400, a rated current of 6 A, a nominal power from 0.4 to 7.5 kW, and a switching frequency of 4 kHz (Figure 1b). The IM start-up was programmed using a voltage-frequency (V-f) linear control within an open-loop strategy, configured in such a way that it reaches the steady-state set frequency in 10 s. The load torque was provided by an electromagnetic brake LN^® SE2662-5R that, for this case study, is set to apply 50% of the full-load level (Figure 1c). The stator current was acquired by a Hall Effect current transducer by LEM (Milwaukee, WI, USA). A National Instruments NI cDAQ-9174 base platform (Austin, TX, USA) with an NI 9215 acquisition module was used for data acquisition with a sampling frequency of 50 kHz and a sampling time of 15 s, 10 s at transient regime, and 5 s at steady-state. The motor was tested first under healthy conditions (Health). Fault conditions were produced by drilling a complete hole in one of the 18 mm radial-length rotor bars (BRB fault). Further details about the bar-breakage drill can be found in [7].

Resulting spectra were analyzed both qualitatively and quantitatively. For the quantitative analysis, the $L^2$-error presented in [48] was implemented as a detectability metric and defined by

$$e_{\mathfrak{f}}\left(\widehat{s}\right)\stackrel{▵}{=}\frac{\left|\right|{\widehat{s}}_{\mathfrak{f}}-\widehat{s}{\left|\right|}^2}{\left|\right|\widehat{s}{\left|\right|}^2},$$

(13)

where $\widehat{s}(t,f)$ and ${\widehat{s}}_{\mathfrak{f}}(t,f)$ are the current signal spectra from the induction motor in healthy and faulty conditions, respectively. These spectra were obtained by employing the different methodologies studied in this work. Hence, experiments were carried out on two-sequential stages that are described as follows. The first stage consisted in preparing analysis methodologies by using concepts described above, i.e., WFT in Equation (12) via fractional Gaussian windows from Equations (8)–(10). A synthetic electrical current signal $s\left(t\right)$ was implemented to emulate the time-frequency behavior of a real induction motor with one broken rotor bar. This signal was defined as a combination of three linear Chirp signals, with frequencies $f_e+\lambda \Delta f$, $\lambda \in \{-1,0,1\}$, and the well-known Additive White Gaussian Noise (AWGN) signal $\eta \left(t\right)$, as shown

$$s\left(t\right)=\sum _{\lambda =-1}^1{A}_{\lambda }sin\left(2\pi \left[f\right(t)+\lambda \Delta f]t+\varphi \right)+\sigma \eta \left(t\right),$$

(14)

where $A_{\lambda }$ is the normalized amplitude per component $\lambda$, $\Delta f$ is the frequency spacing, $\varphi$ is the phase, and $\sigma$ is the standard deviation of the noise. It is directly related to the signal-to-noise rate (SNR). In addition, the temporal frequency variation $f\left(t\right)$ was determined by

$$f\left(t\right)=\left\{\begin{array}{cc}{f}_0+m_{f}t,\hfill & 0\le t\le t_{ss},\hfill \\ f_{ss},\hfill & t>t_{ss},\hfill \end{array}\right.$$

(15)

since $f_0$ is the fundamental frequency, $m_f$ is the frequency slope for the linear Chirp behavior, and $f_{ss}$ is the stationary frequency due to the electrical supply. Additionally, the time series t was defined as $t=\{0,t_s,2t_s,\cdots ,t_{ss},t_{ss}+t_s,\cdots ,t_f\}$, where $t_s$ is the sampling period and $t_f$ is the last time sample. Likewise, $t_{ss}$ indicates when IM enters the stationary state.

Table 1. Values used for the synthetic signal $s\left(t\right)$ in (14).

Parameter (Unit)	Value	Parameter (Unit)	Value
$A_0$ (A/A)	1.0	$m_f$ (s${}^{-2}$)	5.0
$A_{-1},A_1$ (A/A)	0.3	$t_f$ (s)	15
$f_0$ (Hz)	0.0	$t_s$ (ms)	6.4
$f_{ss}$ (Hz)	50	$t_{ss}$ (s)	10
$\Delta f$ (Hz)	5.0	SNR (dB)	25

presents all the values employed for implementing the synthetic signal in (14).

Subsequently, the synthetic signal was processed using several cases of study to find the proper parameters for the proposed methodologies, i.e., window length $n_{sc}$, fractional-order $\nu$, and standard deviation $\sigma$, as presented in

Table 2. Values considered for the proposed methodologies.

Parameter	Set of Values
$n_{sc}$	80, 100, 120, 150, 180, and 220
$\sigma$	0.1, 0.2, 0.3, ⋯, 1.0
$\nu$	0.01, 0.02, 0.03, ⋯, 0.90

. To assess the fractional Gaussian forms, the time series t was defined between $\mu -3\sigma$ and $\mu +3\sigma$, with $n_{sc}$ equally spaced samples. A sampling number of 4096 was employed for implementing the discrete-time Fourier transform.

Lastly, the fractional methodologies were implemented for real signals, which were obtained from the aforementioned experimental setup. In this work, two current signals were studied as illustrative examples related to the motor operating under two fault conditions, i.e., healthy and one broken rotor bar. Those signals were preprocessed via the following methodology: remove bias and initial noise; rescale in time to obtain a signal composed of 10 s and 5 s of transient and stationary states, respectively; decimate in cascade with factors of 5, 4, 4, and 4; filter using an 8th order low-pass Chebyshev filter; and rescale the signal between 0 and 1. Resulting signals were of 2344 samples with 156.25 Hz of sampling frequency. Additionally, three commonly used windows, such as the rectangular, Hanning, and Hamming, were implemented for comparison purposes.

4. Results and Discussion

Figure 2 presents an illustrative example of the fractional Gaussian patterns from Equations (8)–(10) with $\mu =5.0$, $\sigma =\sqrt{2}/4$, and $\nu =0.0,0.17,\cdots ,1.0$. These curves begin at the original function when $\nu =0.0$ and end at its ordinary first derivative when $\nu =1.0$. However, patterns based on Caputo–Dzhrbashyan (CD) kernel present the fastest convergence in Figure 2a, followed by the ones of Atangana–Baleanu–Caputo (ABC) in Figure 2c. In addition, shapes achieved from Caputo–Fabrizio (CF) in Figure 2b have central symmetry around $\nu =0.5$, compared to the left and right symmetries of Figure 2a,c, respectively. Thus, it is inferable that ABC-based patterns have more details when $\nu =0.0$ and, conversely, CD-based ones when $\nu =1.0$.

Using these patterns on WFT and the synthetic signal from Equation (14) with data in Table 1, we conducted a parametric study varying $n_{sc}$, $\sigma$, and $\nu$ according to Table 2. It was noticed that resulting spectrograms slightly change when $\sigma$ is modified, as expected because fractional patterns were obtained via a time series dependent on $\sigma$. Nonetheless, it was found that, when $\sigma$ equals 0.3 for all the fractional Gaussian windows, they achieve the most detailed spectra. Likewise, $\nu$ equals 0.1 was the best selection. It is observed that these windows have qualitatively similar behaviors when their parameters are modified. However, the window length $n_{sc}$ was recognized as the most sensitive parameter. Figure 3 shows the spectrogram achieved from the synthetic signal via CD-based Gaussian windows, for example, using $n_{sc}$ equal to 100, 125, 150, and 200. When the signal has one linear frequency modulation, the optimal window length is $n_{sc}^{*}=100$ in the time-frequency sense and by employing the expression reported in [49]. Figure 3a displays a spectrogram with some disturbances between both sides of frequency components around the main one. This undesired time-frequency resolution is also observed when $n_{sc}$ is vastly increased, as Figure 3c exhibits for $n_{sc}=200$. Hence, spectra are improved when $n_{sc}$ is about 150 samples (Figure 3b).

Subsequently, experimental current measurements from an induction motor with health and BRB fault condition are studied via the proposed methodology. Previously obtained values for parameters are used as starting points in a fine-tuning procedure to improve spectra. Figure 4 shows the best spectrograms found from both healthy (left) and faulty (center) signals, and the difference spectrum of them (right). Each row corresponds to a fractional Gaussian window implemented in WFT. With $n_{sc}=150$ samples, the most detailed spectra are reached when $\nu$ is 0.03, 0.015, and 0.1 for windows-based on CD, CF, and ABC, respectively. Similar to curves analyzed in Figure 2, $\nu$ values for the three kinds of fractional windows preserve the same window shapes and improve the performance of the traditional bell shape of Gaussian function. Moreover, $\sigma$ value for both CD and ABC-based windows was chosen 0.3, and for CF-based window was 0.2. It was noticed that the CF one is more sensible to $\sigma$ variations than the others, which reveals a strong dependence between weight distribution and variance. This behavior makes sense because either this fractional kernel (Caputo–Fabrizio) and Gaussian function have explicit exponential functions. Therefore, any variation in the kernel argument directly affects the function. Unfortunately, it ought not to occur in derivatives, but that matter is beyond our scope. In the qualitative point of view, spectra achieved present defined patterns for fault frequencies, i.e., RSH and LSH (cf. Section 2.1). The best spectrum is obtained when ABC-based Gaussian window is used in WFT analysis (see Figure 4c). Nevertheless, the CF-based window could be considered a good-enough alternative due to its closeness to the ABC-based window results—particularly because these windows have closed expressions that simplify their implementation in specific purpose devices [37].

In the same fashion, those current signals were studied using WFT and employing traditional windows, i.e., Rectangular, Hanning and Hamming windows with 150 samples. Figure 5 plots spectra resulting from said implementation. On the one hand, in Figure 5a, one notices that the rectangular window yields a spectrum where it is hard to distinguish between a healthy and faulty operating condition. On the other hand, in Figure 5b,c, Hanning and Hamming windows show better time-frequency results, but they lack a crystal-clear distinction between the fundamental signal and the faulty signals of the broken rotor bar condition. Problematic trajectories and blurred zones are indicated in Figure 5b. When employing Hamming window, however, the fault can be associated with harmonic curves visualized in the resulting spectrum (see Figure 5c).

Finally,

Table 3. Summary of results obtained from all the methodologies studied in this work.

Feature	Fractional Gaussian Windows			Traditional Windows
	CD	CF	ABC	Rectangular	Hanning	Hamming
Mean ${}^a$	5.0	5.0	5.0	–	–	–
Standard Deviation ${}^a$	0.3	0.2	0.3	–	–	–
Non-integer order ${}^a$	0.03	0.015	0.1	–	–	–
Main-lobe width ${}^b$	21.5	2.15	21.5	11.7	18.6	16.6
Side-lobe Attenuation ${}^c$	−51.3	−58.9	−58.3	−13.3	−31.5	−42.6
Leakage factor ${}^d$	0.00	0.00	0.00	9.37	0.05	0.04
$L^2$–metric ${}^d$	1.624	1.631	1.633	0.615	1.358	1.220

With ${}^a$ dimensionless, ${}^b$ in $\pi \times {10}^{-3}$ rad/sample, ${}^c$ in dB, and ${}^d$ in %. CD, CF, and ABC refer to the Caputo–Dzhrbashyan, Caputo–Fabrizio, and Atangana–Baleanu–Caputo operators, respectively.

presents several features from results obtained by implementing fractional-based and traditional windows. These features consist of the main-lobe width, the relative side-lobe attenuation, the leakage factor of each window frequency response, and the $L^2$-metric value as a quantitative measurement for the fault detectability. In addition, it shows the best parameter values found for each fractional Gaussian window, i.e., mean, standard deviation, and fractional order. It is observed that proposed windows provide a main-lobe width larger than those obtained via traditional ones. Such a fact means that these windows prioritize the fundamental frequency in a great manner, by reducing the presence of undesired components. The leakage factor approaches zero when any fractional window is employed, contrary to the values obtained from traditional windows. A leakage factor equals zero means a minimum quantity of spurious components in the spectrum. A counterexample is observed in the spectrogram achieved with the rectangular window (Figure 5a), which has an associated leakage factor of 9.37% (Table 3). Moreover, one of the most critical features is the relative side-lobe attenuation, which is directly associated with the $L^2$-metric or the fault detectability. Therefore, fractional Gaussian windows show greater detectability measurements than the traditional ones, as expected from the qualitative analysis, whereas their corresponding side-lobe attenuation factors are significantly lower. Furthermore, it is remarked that the ABC-based Gaussian window has the most crystal-clear spectrogram, i.e., the highest detectability factor, where the fault detection in an induction motor with BRB condition is qualitatively easy to recognize.

5. Conclusions

This work proposed an alternative methodology, which is founded in Fractional Calculus (FC) concepts and Windowed Fourier Transform (WFT), for monitoring and diagnosis of induction motors in startup and steady-state. This method is boosted up thanks to fractional operators in the Caputo sense that were applied to the Gaussian function to obtain several patterns used as windows. Particularly, three kinds of fractional-based windows were achieved from definitions reported in the literature, i.e., Caputo–Dzhrbashyan (CD), Caputo–Fabrizio (CF), and Atangana–Baleanu–Caputo (ABC). These windows were implemented in software and tested through a synthetic signal, which emulates the fed current signal of an induction motor with one broken rotor bar (BRB) fault. Then, proposed methodologies were tuned and implemented in two real (and noisy) current signals with healthy and BRB faulty conditions, for illustrative purposes. Parameters such as window length $n_{sc}$, standard deviation $\sigma$, and fractional order $\nu$ were selected. From the achieved spectra, it was noticed that ABC-based Gaussian windows present the most detailed spectrogram with $n_{sc}=150$, $\sigma =0.3$ and $\nu =0.1$, followed by CF-based windows, which render a feasible methodology for a low-cost computing implementation. Moreover, three traditional windows, such as Rectangular, Hanning and Hamming windows, were implemented for comparative purposes. It was noticed that fractional-based Gaussian windows outperformed the traditional ones via visual inspection and quantitative evaluation of resulting spectra. This quantitative evaluation was done employing the $L^2$-error as a fault detectability metric to compare spectra from healthy and faulty signals. Detectability values associated with fractional Gaussian windows were about 20% greater than the best value from traditional ones, i.e., Hanning window. Hence, the proposed methodology improves the time-frequency analysis of the induction motor with, at least, a BRB fault condition, whereas the highest performance was reached when the ABC-based form is employed (cf. Table 3).