# An experimental residual envelope analysis-based method for incipient fault detection in induction machines

Hamza Sabir, Mohammed Ouassaid, Nabil Ngote, Mohamed Benbouzid

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Hamza Sabir, Mohammed Ouassaid, Nabil Ngote, Mohamed Benbouzid. An experimental residual envelope analysis-based method for incipient fault detection in induction machines. Scientific African, 2025, 30, pp.e02965. 10.1016/j.sciaf.2025.e02965. hal-05319051

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# An experimental residual envelope analysis-based method for incipient fault detection in induction machines

Hamza SABIR<sup>a, *</sup>, Mohammed OUASSAID<sup>b ID</sup>, Nabil NGOTE<sup>c ID</sup>,
Mohamed BENBOUZID<sup>d ID</sup>

## ARTICLE INFO

Editor: DR B Gyampoh

Keywords:

Condition monitoring

Broken rotor bar

Residual envelope

Fault detection

Induction machines

## ABSTRACT

Induction motor (IM) is a vital element of any industrial manufacturing process due to its robust construction and simple maintainability. However, an incipient fault—even on a single rotor bar—can quickly become problematic and spread to adjacent bars, exacerbated by endogenous vibrations caused by rotor imbalance. Early diagnosis is therefore crucial to prevent damage and reduce the maintenance costs required to carry out repairs. Among existing techniques, spectral analysis using Hilbert transform (HT) has long been the standard for diagnosing incipient defects, but it is less effective at detecting low-amplitude failures. In this paper, a new method, RESE (Residual Electrical Signal Extraction), is proposed to optimally assist HT in detecting incipient broken bars (BRB). Through simulations and experiments, RESE has demonstrated its ability to detect defects as small as 10 mm deep, over a load range from 0 to 100% of the rated load. The results therefore confirm that RESE is a tool of choice for the early monitoring of asynchronous motors, even when they are lightly loaded.

## Introduction

Recent developments in power electronics, combined with advanced speed control systems, have led to the widespread use of variable speed motors which, after having been used exclusively for DC motors, and more recently for squirrel cage induction motors (SCIM), are now being applied to induction machines in all their configurations. A direct consequence of this technological evolution is the logic of its growing use in all sectors of industry, and in the most sophisticated sectors such as aeronautics, nuclear energy, wind power, chemicals and rail transport [1].

Nevertheless, as with any electrical machine, all three types of induction machine are vulnerable to malfunctions [1]. Such equipment can experience failures of either electrical or mechanical origin, whether at an early stage (incipient) or at an advanced stage [2,3]. In the case of incipient faults, they may be due to manufacturing or assembly faults, or to aberrant equipment operation [4, 5]. Ambient conditions can also be an aggravating factor, as humidity, vibrations and thermal variations jeopardize the proper functioning of components.

As part of Condition Monitoring (CM) and Fault Diagnosis, various approaches are used to test the behavior of induction machines

<footnote><sup>a</sup> Laboratory of Engineering Sciences & Biosciences, Faculty of Sciences and Techniques of Mohammedia, Hassan II University, Casablanca, Morocco</footnote>

<footnote><sup>b</sup> Engineering for Smart and Sustainable Systems Research Center, Mohammadia School of Engineers, Mohammed V University in Rabat, Morocco</footnote>

<footnote><sup>c</sup> ENSMR Engineering School, Rabat, Morocco</footnote>

<footnote><sup>d</sup> University of Brest, UMR CNRS 6027 IRDL, 29238 Brest, France</footnote>

<footnote>* Corresponding author.</footnote>

<footnote>E-mail addresses: hamza.sabir@fstm.ac.ma (H. SABIR), ouassaid@emi.ac.ma (M. OUASSAID), ngotenabil@gmail.com (N. NGOTE).</footnote>
(IM). The most common approaches rely on the measurement of physical quantities such as temperature, vibration, electrical signals or acoustic emissions [6,7]. But despite advances in conventional and intelligent techniques, their effectiveness remains limited for certain types of defect, particularly in the case of incipient defects [8,9].

Generally speaking, an incipient fault does not cause clearly identified symptoms, but can go unnoticed until it causes a breakdown in a very short space of time. This makes early identification of a fault, in this case through analysis of vibration signals, essential to avoid cascading effects that will damage other plant components [10,11]. Such damage can lead to unexpected production losses with considerable economic impact. In this way, the implementation of control methods aimed at early detection of incipient anomalies represents a fundamental strategic challenge, since it should improve the availability of induction machines and reduce the costs

<figure><img src="image_5.png" /><figcaption>Fig. 1. Induction machine fault appearance percentage. (a). ERPI (b). IEEE, (c). ABB.</figcaption></figure>
incurred by downtime and maintenance.

To highlight the real motivation behind the research, it should be pointed out that the problem is a crucial one for the many in-
dustries whose operations will also depend on the reliability, availability and continuity of induction machines, since an unexpected
breakdown leading to a production stoppage has a major impact on production costs, and costs considerable material damage, without
any risk to personnel or installations [6,7].

Conventional detection methods, whether based on vibration analysis, thermography or the spectral study of electrical signals, are limited in their ability to identify incipient faults, which often involve very slight variations in the measured signals that are difficult to detect. This is particularly true because these low-level faults are masked by the machine's normal operating conditions, with the result that they cannot be detected at an early stage [9].

To overcome this shortcoming, the aim of this work is to develop a new, more refined method which combines characterization by a Hilbert transform and sophisticated processing by residual signal extraction, in order to reveal the weakest modulations, signatures of precursor faults, by eliminating the main components of the fundamental-type signal. This not only helps anticipate internal machine degradation, but also significantly reduces maintenance costs, extends the service life of motors, and improves safety and industrial production in general [11].

Motor failures can be mechanical or electrical. On the one hand, eccentricity, misalignment, bearing failures, stator failure, and the rotor-stator contact friction represent the main types of mechanical damage in IMs. On the other hand, the motor is also affected by some electrical damage such as Half BRB, one or more BRBs, poor insulation, winding stator defect, earth defect, and insulation faults between two or three phases [12]. Extensive research into the various defects in IM reveals that an undetected failure of one bar can lead to an interruption in the operation of the machine and then to an unplanned suspension of the industrial production process, resulting in significant economic expenditure. IM is among the rotationally symmetrical machines with rotating magnetic field. Therefore, any malfunction alters their symmetrical characteristics. The Electric Power-Research-Institute (EPRI), the Institution of Electrical & Electronics Engineers (IEEE), and the Asea Brown Boveri (ABB) have performed statistical analyses on IM failure [13]. Fig. 1 depicts the most common IM failures, according to their statistical study.

According to failure statistics, bearing failure, rotor failure, and stator failure are the most common failures of induction machines. The purpose is to ensure that these IMs are safer and more reliable to avoid costly repairs and limit downtime costs. Early fault identification can eliminate a subsequent fault especially, on the stator windings in case the structure of the rotor rubs directly on the stator winding. For this reason, any early identification of failure in bars of the rotor has been a consistent research theme for several researchers in electrical engineering. The paper [14] proposed a new automatic way for rotor failure identification and misalignment in IMs, which uses electromotive force (EMF) in motor starting. This advanced algorithm is constructed by associating the conventional Multiple Signal Classification (MUSIC) strategies, which provides efficient discovery of the signature of rotor faults, with a new neural network. This technique permits classifying the patterns caused by the faulty part during the motor start-up. Reference [15] presented a new model for the estimation of the rotor and stator defects. First, this model is constructed in a healthy case, then faulty models of stator inter-turn short circuit faults and BRB are created. Finally, the faulty model is implemented using the Tustin transformation, and the results obtained using MCSA (Motor Current Signature Analysis) provide accurate detection of the machine defect. All of these methods are appropriate to identify one or multiple BRBs. Regarding the nascent BRB, most of the works are limited to half BRB identification. Reference [16] demonstrates how to use the MCSA and MUSIC methods in combination to find a half-BRB in a motor operating under light loads. Reference [17] suggests a novel strategy as a deterministic method for IM BRB incipient diagnosis. This method can identify and distinguish between half and one BRB. To improve the ability to identify imminent bar ruptures, the authors of reference [18] suggest a novel approach that uses the MUSIC method. This defect is a small hole about 9 mm in depth for the bars of a cage rotor, which measure 18 mm in depth. Reference [19] suggests a novel solution based on the observation of a few statistical indicators discovered through examination of the initial stator current envelope using the HT method. This proposed HT approach is tested and validated through experimentation in the electrical lab. In [20], the authors propose the spectral analysis using HT to carry out a stator current signature analysis to detect the rotor bar that starts to break in induction motors. However, the spectrum when the bar has a fracture of only 3-mm represents a fault signature with a very small amplitude. Also, for reference [21], the spectral analysis of the HT is insufficient because of the overlap between the continuous component and that caused by the defect. In addition, at no-load motor operation, the amplitudes of the spectral components associated with the defect are no longer visible. Therefore, it can not be easily distinguished. To overcome this problem, most researchers apply the zoom on the side where the defect frequency peak is located. Especially in the case of an unloaded motor.

For all these reasons, it is strongly recommended to implement the new method to overcome the limitations of the HT.

Particular attention is devoted to the rotor defect, as it has the smallest frequency band (bandwidth =2.2.s.fs), this band is all the narrower as the slip value is low, so the fault components overlap with the continuous component, which is particularly the case when the motor is running at no load or under low load and in the event of an incipient defect. Therefore, the application of a zoom of the HT spectrum remains essential to discover the small hidden fault peak around the continuous component.

For these reasons, a new CM procedure for a clear extraction of an incipient cracking defect at a rotor bar is designed to support the limitations of the HT. The main objective of this diagnostic procedure is to detect an incipient defect at its early stage without the need to apply zoom, particularly under a large-scale load. In addition, few studies have addressed the diagnosis via the cycloergodicity of electrical signals, especially in the case of incipient defect detection. This unfortunate fact is the motivation for this new work, where a low-gravity BRB (< Half BRB) is studied. Numerous classical techniques cited in the literature review subsection have demonstrated the lack of sensitivity and reliability to identify such incipient faults, which leads to wrong alarms if employed in real industrial diagnostic systems. Furthermore, the proposed strategy is sensitive to the severity of this incipient fault. This developed approach consists of extracting a residual current signal from the envelope of the stator current signal for early identification of incipient defects
happening in the rotor bar. Finally, this advanced strategy is tested and validated in various scenarios using simulation and experimentation. Briefly, the problem of incipient fault extraction is solved without loss of performance, without cutting off the power supply, and without the application of zoomed-in spectral analysis.

The most significant contributions of this study are listed as follows:

- The separation of the components of the incipient defect and fundamental ones is performed under low-rated slip.

- No other algorithm is used for zooming in on the spectrum to discover the small defect peak hidden around the fundamental component.

- Good incipient fault extraction and identification are provided against changing load conditions.

- The technique is uncomplicated to execute, non-invasive, and one stator phase current measurement is sufficient.

- The proposed strategy can be performed in the case of inaccessible machines such as the immersed pumps and offshore wind turbines, contrary to, the methods based on the signals obtained by the accelerometer sensors, where machine accessibility is required.

- The advanced strategy is a successful tool for early identification of the incipient rotor defect. Especially, in the factories producing electric motors.

The present study consists of several sections oriented towards a progressive understanding of the proposed contributions. The second section deals with the central concept of the new diagnostic procedure, first presenting the history of the methods used before introducing the innovative algorithm proposed here. The third section is devoted to the modeling of the induction machine, both in the normal state and in a rotor asymmetry fault situation. The fourth section presents and discusses the results of both numerical simulations and experimental validations carried out in the laboratory, in order to cross-assess the validity of the proposed method. Finally, the last section outlines the major conclusions drawn from the study, and describes the prospects it opens up for the preventive maintenance of electrical machines.

## Structure of the suggested new hybrid strategy

The objective of this section is to introduce some mathematical notions of the signal processing tools, which are exploited to build the new diagnosis strategy.

### Residual electrical signal extraction

In the operating system of IM, electrical perturbations and load variation result in a nonstationary behavior of the absorbed stator line current signal. The aim is to exploit this nonstationary behavior of the stator current signal to diagnose the IMs. In the present contribution, a specific tool for electrical waveform conditioning is designed and developed. Indeed, a defect that surpasses one BRB is detected directly from the stator current waveform. However, incipient BRB defects are strictly undetectable. A new electrical concept to overcome this limitation is proposed using the idea developed by MacFadden in the case of vibratory signals captured from vibration sensors [22]. The new RESE concept ensures that all the fluctuation has been averaged together to give us a good repetitive value. Thus, the RESE method permits the detection of any perturbation from the stator current waveform, therefore, the determination of any fault. The stator line current is given by the Eq. (1):

$$I_s(t) = I_{sy}(t) + I_{smec}(t) + n(t) \quad (1)$$

where $I_s(t)$, $I_{sy}(t)$, and $I_{smec}(t)$ represent the stator current signal, the fundamental component of the stator-current, and the stator current signal linked to the reason for mechanical vibration from the machine, respectively.

The stator-line-current signal related to the mechanical oscillations and the noise, respectively. The $I_s(t)$ is used for the CM of the IM, accordingly, segregation between $I_{smec}(t)$, and $I_{sy}(t)$ is highlighted. The algorithm applied to the k periods of the stator line current of period Th is given by two steps:

Firstly, synchronization is a process that allows to synchronization of K periods of stator current signal in only the Th period, consequently, all periods should be perfectly superimposed afterward the synchronization procedure. Secondly, the mean is applied to the synchronized current using Eq. (2).

$$\langle I_s \rangle_{T_h} = \lim_{K \to \infty} \frac{1}{K} \sum_{K=0}^{K-1} i_s(t + K \cdot T_h) = I_{sy}(t) \quad (2)$$

From Eq. (2), the averaged signal $\langle I_s \rangle_{T_h}$ equals exactly the $I_{sy}(t)$ signal.

A residual stator current is obtained by a basic subtraction of the initial stator current $I_s(t)$ from the averaged signal found in Eq. (2), where only the fault-related signatures remain, as is shown in equations 3.

$$I_{sres}(t) = I_{smec}(t) + n(t) \quad (3)$$

From Eq. (3), the proposed mathematical procedure leads to separating the electrical and the mechanical behavior to extract a residual current that contains only the signature of the fault.
## Hilbert transform

The stator line current signal envelope analysis is used to examine the incipient BRB failure. The envelope is the Analytical Signal's modulus (AS) [23].

The HT $\tilde{s}(t)$ of $s(t)$ is obtained through the mathematical convolution of the signal $s(t)$ and $h(t) = \frac{1}{\pi t}$. In order to understand and determine the envelope of $s(t)$ using the analytical signal AS(t), a theoretical view is developed as shown in Eq. (4), 5, and 6.

$$HT[s(t)] = \tilde{s}(t) = s(t) * h(t) \qquad (4)$$

$$\tilde{s}(t) = vp \left\{ \int_{-\infty}^{+\infty} s(\tau)h(t-\tau)d\tau \right\} = \frac{1}{\pi t} vp \left\{ \int_{-\infty}^{+\infty} \frac{s(\tau)}{t-\tau} d\tau \right\} \qquad (5)$$

$$vp \left\{ \int_{-\infty}^{\infty} s(t)h(t-\tau)d\tau \right\} = \lim_{\epsilon \to 0} \left\{ \int_{-\infty}^{t-\epsilon} s(\tau)h(t-\tau)d\tau + \int_{t+\epsilon}^{+\infty} s(\tau)h(t-\tau)d\tau \right\} \qquad (6)$$

where vp is the principal value of Cauchy.

A practical overview to understand in detail and calculate the envelope signal of $s(t)$ by means of the analytic signal AS(t) is given by Eq. (7).

$$AS(t) = s(t) + \tilde{J}\tilde{s}(t) \qquad (7)$$

It can be represented equally by:

$$AS(t) = A(t) * e^{i\varphi(t)} \qquad (8)$$

where the instantaneous envelope is defined by the $A(t)$ signal and the phase signal is denoted by $\varphi(t)$. The instantaneous phase, and modulus, expressed in terms of $s(t)$ and $\tilde{s}(t)$, are given by Eq. (9) and 10, respectively.

$$\varphi(t) = \operatorname{Arctan} \left[ \frac{\tilde{s}(t)}{s(t)} \right] = 2\pi f_0 t \qquad (9)$$

$$A(t) = \sqrt{[s^2(t) + \tilde{s}^2(t)]} \qquad (10)$$

where $f_0$ is the instantaneous frequency.

$A(t)$ and $\varphi(t)$ give a physical representation of how the energy contained in the signal $s(t)$ changes over time.

## Motor current signature analysis

One of the most popular techniques for CM of the IM in the industrial sector is MSCA. The stator current signal can be examined by the MCSA, which provides early BRB detection. The MCSA is a fundamental signal processing tool for signature extraction that involves FFT analysis. The FFT algorithm is applied to analyze the current signal waveforms. A Hanning window will be used in convolution with these current signals [24]. The coefficients of the Hanning function of length $N_L$ are calculated by Eqs. (11), 12, and 13.

$$w(n) = w_0 \left( \frac{N_L}{N} \left( n - \frac{N}{2} \right) \right) \qquad (11)$$

$$w_0(x) = \begin{cases} \frac{1}{2} \left( 1 + \cos \left( \frac{2\pi x}{N_L} \right) \right) = \cos \left( \frac{\pi x}{N_L} \right), & |x| \le \frac{N_L}{2} \\ 0, & |x| > \frac{N_L}{2} \end{cases} \qquad (12)$$

$$w(n) = \frac{1}{2} \left( 1 - \cos \left( \frac{2\pi n}{N} \right) \right), \quad 0 \le n \le N \qquad (13)$$

where $N + 1$ is the number of samples, and the integer $N$ can be an odd or even number.

The fault is identified directly from the spectrum by extracting the frequency components with the highest magnitudes. Each type of defect is mathematically associated with a predefined frequency component. Therefore, the spectrum of the analytical signal modulus presents peaks with specific frequencies of the BRB fault. These frequencies of defect $f_d$ are theoretically expressed by Eq. (14) [25].

$$f_d = 2kfs_s \qquad (14)$$

where $f_s$ is the power supply frequency, $k$ is an integer, and $s$ is the slip.
## The new advanced approach

To overcome the insufficient accuracy and sensitivity of conventional MCSA and HT methods to detect an incipient BRB defect. The new RESE method consist on synchronization to align the envelope with characteristic events, in order to better identify faults, because it's a matter of subtracting the synchronized envelope from the original envelope. This method allows an efficient separation between the stator current associated with the mechanical structure and those associated with the electrical part. Thus, it is noticed that only the signal of the mechanical phenomena is conserved in the residual envelope $I_{res}(t)$. Finally, it's an exceptional and attractive property that permits an electrical diagnosis-linked indicator to detect incipient faults. The main steps and the flowchart of the advanced processing algorithm are illustrated in Fig. 2.

## Modeling of machines

### Modeling of the healthy machine

The d and q components of the voltage can be expressed as given in the system of Eq. (15).

$$ 
\left\{
\begin{aligned}
V_{qs} &= R_s I_{qs} + \frac{d}{dt} \Phi_{qs} + \omega_x \Phi_{ds} \\
V_{ds} &= R_s I_{ds} + \frac{d}{dt} \Phi_{ds} - \omega_x \Phi_{qs} \\
V_{qr} &= R_r I_{qr} + \frac{d}{dt} \Phi_{qr} + (\omega_x - \omega) \Phi_{dr} \\
V_{dr} &= R_r I_{dr} + \frac{d}{dt} \Phi_{dr} - (\omega_x - \omega) \Phi_{qr}
\end{aligned}
\right.
\qquad (15) $$

The components d and q of the flux can be given as a function of the currents by the system of Eq. (16).

$$ 
\left\{
\begin{aligned}
\Phi_{qs} &= L_s i_{qs} + L_m i_{qr} \\
\Phi_{ds} &= L_s i_{ds} + L_m i_{dr} \\
\Phi_{qr} &= L_r i_{qr} + L_m i_{qs} \\
\Phi_{dr} &= L_r i_{dr} + L_m i_{ds}
\end{aligned}
\right.
\qquad (16) $$

<figure><img src="image_6.png" /><figcaption>Fig. 2. Structure and flowchart of the proposed algorithm.</figcaption></figure>
$L_s = l_s - m_s$: Cyclic self-inductance of the stator.

$L_r = l_r - m_r$: Cyclic self-inductance of the rotor.

$L_m = \frac{3}{2}M_{sr}$: Mutual cyclic inductance stator-rotor.

$\omega = p \cdot \Omega$: Electrical pulsation of the rotor.

with the magnetic leaks are totalized at the stator ($L_f = L_s - L_m$ and $L_r - L_m = 0$) the flux equations are expressed in Eq. (17).

$$ \left\{ \begin{aligned} \Phi_{qs} &= (L_f + L_m) i_{qs} + L_m i_{qr} \\ \Phi_{ds} &= (L_f + L_m) i_{ds} + L_m i_{dr} \\ \Phi_{qr} &= L_m (i_{qr} + i_{qs}) \\ \Phi_{dr} &= L_m (i_{dr} + i_{ds}) \end{aligned} \right. \qquad (17) $$

The rotor frame is selected as a reference. The components $d$ and $q$ of the voltage are obtained from Eqs. (18) and 19.

$$ U_{dqs} = R_s I_d + \frac{d}{dt}\Phi_{dqs} + \omega_x P(\pi/2)\Phi_{dqs} \qquad (18) $$

$$ U_{dqr} = 0 = R_r I_{dqr} + \frac{d}{dt}\Phi_{dqr} + (\omega_x - \omega)P(\pi/2)\Phi_{dqr} \qquad (19) $$

where

$$ P(\theta) = \begin{bmatrix} \cos(\theta) & -\sin(\theta) \\ \sin(\theta) & \cos(\theta) \end{bmatrix} \qquad (20) $$

Fig. 3 shows the equivalent electrical diagram of the asynchronous machine in a dynamic regime, with the totalized leakage at the stator [26].

To access the rotor and design the complete machine model, the mechanical Eq. (21) must be considered:

$$ J \frac{d}{dt} \Omega = T_{em} - T_r - f_v \Omega \qquad (21) $$

where $\Omega = \dot{\theta} = \frac{w}{p} T_{em}$, and $T_r$ are the electromagnetics and the resistive torques, respectively. $J, p$, and $f_v$ are the inertia coefficient, the number of pole pairs, and the load viscous friction coefficient, respectively.

The electromagnetic torque in terms of flux and currents can be expressed as shown in Eq. (22).

$$ T_{em} = p \cdot (\Phi_{qs} i_{ds} - \Phi_{ds} i_{qs}) \qquad (22) $$

The substitution of Eq. (22) in (21) gives:

$$ \frac{d}{dt}\omega = \frac{p^2}{J}(\Phi_{dr} i_{qs} - \Phi_{qr} i_{ds}) - \frac{p}{J}T_r - \frac{f_v}{J}\omega \qquad (23) $$

Considering the dynamics of rotor fluxes, stator currents and electromechanical dynamics, the representation of the state of the so-called healthy machine is given by the system of Eqs. (24) [27,28].

$$ \left\{ \begin{aligned} \dot{\mathbf{x}} &= f(\mathbf{x}) + g \mathbf{u} \\ \mathbf{y} &= h(\mathbf{x}) \end{aligned} \right. \qquad (24) $$

<figure><img src="image_7.png" /><figcaption>Fig. 3. Model of the healthy machine in the Park reference frame linked to the rotor.</figcaption></figure>
where:

$\underline{x} = [i_{ds} \ i_{qs} \ \Phi_{dr} \ \Phi_{qr} \ \omega \ \theta]^{T}$ is the state vector.

$\underline{u} = [u_{ds} \ u_{qs} \ T_r]^{T}$ is the system input.

$\underline{y} = [i_{ds} \ i_{qs} \ \omega]^{T}$ is the system output.

$$f(\underline{x}) = \begin{bmatrix} -\frac{R_s + R_r}{L_f} \cdot i_{d_s} + \omega \cdot i_{q_s} + \frac{R_r}{L_f \cdot L_m} \cdot \varnothing_{d_r} + \frac{\omega}{L_f} \cdot \varnothing_{q_r} \\ -\frac{R_s + R_r}{L_f} \cdot i_{q_s} - \omega \cdot i_{d_s} + \frac{R_r}{L_f \cdot L_m} \cdot \varnothing_{q_r} - \frac{\omega}{L_f} \cdot \varnothing_{d_r} \\ R_r \cdot i_{d_s} - \frac{R_r}{L_m} \cdot \varnothing_{d_r} \\ R_r \cdot i_{q_s} - \frac{R_r}{L_m} \cdot \varnothing_{q_r} \\ \frac{p^2}{J} \cdot (i_{q_s} \cdot \varnothing_{d_r} - i_{d_s} \cdot \varnothing_{q_r}) - \frac{f_v}{J} \cdot \omega \\ \omega \end{bmatrix} \qquad (25)$$

$$g = \begin{bmatrix} \frac{1}{L_f} & 0 & 0 \\ 0 & \frac{1}{L_f} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & \frac{p}{J} \\ 0 & 0 & 0 \end{bmatrix} \qquad (26)$$

$$h(\underline{x}) = \begin{bmatrix} i_{d_s} \\ i_{q_s} \\ \omega \end{bmatrix} \qquad (27)$$

## Modeling of the faulty machine

In the context of rotor faults diagnosis, the development of a model is mainly motivated by the possibility to simulate these faults. In addition to their complexity, the disadvantage of these models is that they require a thorough knowledge of the electrical parameters of the machine. A rotor bar failure causes a nonstationary field in the rotor, directed according to the angle $\theta$ of the faulty bar. An additional parameter $\eta_0$ is naturally introduced to quantify the rotor defect. The rotor, in the two-phase reference frame, therefore has a third winding short-circuited due to the squirrel cage, through which a fictitious fault current $i_0$ flows and whose number of fictitious turns is proportional to the fault rate.

To account for this field anomaly, this winding must necessarily have the same direction as the faulted bar. Therefore, one fault parameter is introduced to allow the detection of broken bars:

The defect ratio $\eta_0$ is equal to the ratio of the number of coils in defect divided by the total number of turns. This parameter quantifies the unbalance and can give the number of broken bars [27].

Since the number of rotor turns is fictitious, for a $N_b$ bar rotor, a rotor turn is a mesh consisting of two bars shorted by two ring portions, then the total number of rotor turns is equal to the number of bars in the rotor. A fictitious phase is therefore made up of $\frac{N_b}{3}$ bars. For $N_{bb}$ broken bars on a phase, the expression for the fault ratio $\eta_0$ is given by Eq. (28) [27,29]:

$$\eta_0 = \frac{3 \cdot N_{bb}}{N_b} \qquad (28)$$

The voltage and flux equations of the faulty coil expressed in the two-phase reference frame of axis d and q related to the rotor are expressed in the system of Eq. (29).
$$ egin{cases} 0 = \eta_0 \cdot R_r \cdot i_0 + \frac{d}{dt} \Phi_0 \\ \Phi_0 = \frac{2}{3} \eta_0^2 \cdot L_r \cdot i_0 + \sqrt{\frac{2}{3}} \eta_0 \cdot R_r \cdot L_r \cdot [\cos(\theta_0) \sin(\theta_0)] \left( \dot{i}_{dqs} + \dot{i}_{dqr} \right) \end{cases} \quad (29) $$

where

$$ \eta_0 = \frac{\text{Number of fictive turns in defect}}{\text{Total number of fictive turns on one phase}} \quad (30) $$

The current $i_0$ in the winding representing the fault is therefore the origin of the magnetic field stationary regarding the rotor. This magnetic field is the origin of the flux $\Phi_0$. The projection of $i_0$ and $\Phi_0$ on to the d and q Park axes, gives the stationary vectors as shown in Eq. (31) and 31.

$$ \dot{i}_{dq0} = \left[ \begin{matrix} \cos(\theta_0) \\ \sin(\theta_0) \end{matrix} \right] \dot{i}_0 \quad (31) $$

$$ \Phi_{dq0} = \left[ \begin{matrix} \cos(\theta_0) \\ \sin(\theta_0) \end{matrix} \right] \cdot \Phi_0 \quad (32) $$

Relation (29) becomes a relation between stationary vectors regarding the rotor. Thus, in the rotor frame, the set of matrix equations at the stator, rotor bar, or rotor winding is given by Eqs. (33), 34 and 35 [27].

$$ \left\{ \begin{aligned} u_{dqs} &= R_s \cdot \dot{i}_{dqs} + \frac{d\Phi_{dqs}}{dt} + \omega_x P(\pi/2) \Phi_{dqs} \\ \Phi_{dqs} &= L_f \cdot \dot{i}_{dqs} + L_m \cdot \left( \dot{i}_{dqs} + \dot{i}_{dqr} + \sqrt{\frac{2}{3}} \cdot \eta_0 \cdot \dot{i}_{dq0} \right) \end{aligned} \right. \quad (33) $$

$$ \left\{ \begin{aligned} 0 &= R_r \cdot \dot{i}_{dqr} + \frac{d\Phi_{dqr}}{dt} \\ \Phi_{dqr} &= L_m \cdot \left( \dot{i}_{dqs} + \dot{i}_{dqr} + \sqrt{\frac{2}{3}} \cdot \eta_0 \cdot \dot{i}_{dq0} \right) \end{aligned} \right. \quad (34) $$

$$ \left\{ \begin{aligned} 0 &= \eta_0 \cdot R_r \cdot \dot{i}_{dq0} + \frac{d\Phi_{dq0}}{dt} \\ \Phi_{dq0} &= \sqrt{\frac{2}{3}} \cdot \eta_0 \cdot L_m \cdot Q(\theta_0) \cdot \left( \dot{i}_{dqs} + \dot{i}_{dqr} + \sqrt{\frac{2}{3}} \cdot \eta_0 \cdot \dot{i}_{dq0} \right) \end{aligned} \right. \quad (35) $$

where:

$$ Q(\theta_0) = \begin{bmatrix} \cos^2(\theta_0) & \cos(\theta_0)\sin(\theta_0) \\ \cos(\theta_0)\sin(\theta_0) & \sin^2(\theta_0) \end{bmatrix} \quad (36) $$

$$ P(\theta) = \begin{bmatrix} \cos(\theta) \cos(\theta + \frac{\pi}{2}) \\ \sin(\theta) \sin(\theta + \frac{\pi}{2}) \end{bmatrix} \quad (37) $$

The flux equations of the asynchronous machine in rotor fault [27], by analogy with the equivalent diagram reduced to the primary, become:

$$ \left\{ \begin{aligned} \Phi_{dqs} &= \Phi_{dqf} + \Phi_{dqm} = L_f \cdot \dot{i}_{dqs} + L_m \cdot \left( \dot{i}_{dqs} + \dot{i}_{dqr} - \tilde{\dot{i}}_{dq0} \right) \\ \Phi_{dqr} &= \Phi_{dqm} = L_m \cdot \left( \dot{i}_{dqs} + \dot{i}_{dqr} + \tilde{\dot{i}}_{dq0} \right) \\ \tilde{\Phi}_{dq0} &= \eta_0 \cdot Q(\theta_0) \cdot \Phi_{dqm} \end{aligned} \right. \quad (38) $$

where:
$$ \tilde{i}_{dq0} = -\sqrt{\frac{2}{3}} \cdot \eta_0 \cdot \dot{i}_{dq0} \qquad (39) $$

$$ \tilde{\Phi}_{dq0} = \sqrt{\frac{2}{3}} \cdot \Phi_{dqm} \qquad (40) $$

$$ \tilde{i}_{dq0} = \frac{2}{3} \cdot \frac{\eta_0}{R_r} \cdot Q(\theta_0) \cdot \frac{d\Phi_{dqm}}{dt} = R_{def}^{-1} \cdot \frac{d\Phi_{dqm}}{dt} \qquad (41) $$

The equivalent diagram of the induction machine with a rotor winding fault in the Park reference frame of Park linked to the rotor is represented by the Fig. 4 [28,29].

The equivalent diagram can be represented by considering the equivalent resistance $R_{r_{eq}}$ and that by putting in parallel the resistance $R_r$ and the fault resistance $R_{def}$ as shown in Eq. (42).

$$ R_{r_{eq}}^{-1} = R_r^{-1} + R_{def}^{-1} = R_r^{-1} + \frac{2}{3} \cdot \eta_0 \cdot R_r^{-1} \cdot Q(\theta_0) \qquad (42) $$

The expression of the equivalent matrix is obtained by the inverse using Eq. (43).

$$ R_{r_{eq}} = R_r + R_{def} = R_r - \frac{\alpha}{1 - \alpha} \cdot Q(\theta_0) \cdot R_r \qquad (43) $$

where:

$$ \alpha = \frac{2}{3} \eta_0 \qquad (44) $$

$$ R_r = R_r \cdot I_2 = R_r \cdot \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \qquad (45) $$

$$ R_{def} = -\frac{\alpha}{1-\alpha} \cdot R_r \cdot \begin{bmatrix} \cos^2(\theta_0) & \cos(\theta_0)\sin(\theta_0) \\ \cos(\theta_0)\sin(\theta_0) & \sin^2(\theta_0) \end{bmatrix} \qquad (46) $$

The electrical equations of both induction machines in rotor fault remain the same, in a reference frame linked to the rotor, except that the resistance $R_r$ is replaced by the equivalent resistance $R_{r_{eq}}$ as shown in Eq. (47).

$$ R_{r_{eq}} = R_r \cdot \begin{bmatrix} 1 & 0 \\ 0 & 1 \end{bmatrix} \pm R_{def} \qquad (47) $$

In general, the equivalent rotor resistance $R_{r_{eq}}$ is given by the Eq. (48).

$$ R_{r_{eq}} = R_r \pm R_{def} \qquad (48) $$

An equivalent schematic is obtained, in Fig. 5, with the series connection of the resistor $R_r$ and the fault matrix $R_{def}$ [27,28].

This model offers the possibility to simulate either a broken bar fault in SCIM and a short circuit fault between the rotor winding turns in WRIM. The broken bar fault is simulated by increasing the rotor resistance value by simply adding a fault resistor $R_{def}$ to the rotor resistance, in this case, the formula (48) becomes:

$$ R_{r_{eq}} = R_r + R_{def} \qquad (49) $$

The simulated global system is obtained by introducing the electromechanical equation of the machine. Thus, the induction machine with a bar break fault or circuit fault between the rotor winding turns can be described by the system of differential Eqs. (50) [27, 29].

<figure><img src="image_8.png" /><figcaption>Fig. 4. Rotor defect model of the induction machine in the Park frame.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>Fig. 5. Final rotor fault model of the machine.</figcaption></figure>

$$ \left\{ \begin{array}{l} \dot{\underline{x}} = f(\underline{x}) + g \underline{u} \\ \underline{y} = h(\underline{x}) \end{array} \right. \qquad (50) $$

where:

$\underline{x} = [i_{ds} \ i_{qs} \ \Phi_{dr} \ \Phi_{qr} \ \omega \ \theta]^{T}$ is the state vector.

$\underline{u} = [u_{ds} \ u_{qs} \ T_{r}]^{T}$ is the system input.

$\underline{y} = [i_{ds} \ i_{qs} \ \omega]^{T}$ is the system output.

$$ f(\underline{x}) = \begin{bmatrix} -A + B \\ [R_{eq}] \cdot i_{dq_1} - [R_{eq}] \cdot L_m^{-1} \cdot \Phi_{dq_1} \\ \frac{p^2}{J} \cdot (i_{q_1} \cdot \Phi_{d_1} - i_{d_1} \cdot \Phi_{q_1}) - \frac{f_v}{J} \cdot \omega \\ \omega \end{bmatrix} \qquad (51) $$

where:

$$ A = \left( (R_s \cdot I_2 + [R_{eq}]) \cdot L_f^{-1} + \omega \cdot P \left(\frac{\pi}{2}\right) \right) \cdot i_{dq_1} \qquad (52) $$

<figure><img src="image_10.png" /><figcaption>Fig. 6. Model of the SCIM in the MATLAB®Simulink environment.</figcaption></figure>
$$B = \left( [R_{eq}] \cdot L_m^{-1} + \omega \cdot P \left(\frac{\pi}{2}\right) \right) \cdot L_f^{-1} \cdot \Phi_{dq_r} \quad (53)$$

$$g = \begin{bmatrix} \frac{1}{L_f} & 0 & 0 \\ 0 & \frac{1}{L_f} & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & \frac{p}{J} \\ 0 & 0 & 0 \end{bmatrix} \quad (54)$$

$$h(x) = \begin{bmatrix} i_{ds} \\ i_{qs} \\ \omega \end{bmatrix} \quad (55)$$

## Simulation and experimental results

### Simulation platform

The operation of the SCIM is first simulated in the healthy case and then in the case of an incipient rotor bar crack. It is important to note that the defect of a small crack in the rotor bar generally increases the equivalent resistance of a rotor winding. Fig. 6 shows a visual Simulink implementation used for the test of the machine with incipient BRB fault.

The block diagram adopted for the implementation and the analysis of a nascent BRB in IM, represented in Fig. 7, is implemented and performed on the Matlab/Simulink platform. The IM specifications are given in Table 1.

The incipient failure of the BRB is created by a slight decrease in the value of the overall rotor resistance. At a sampling rate of 3200 Hz, the extracted stator line current signal is sampled. As a result, the simulation includes a global number of samples $N_T = 50,048$ samples ($N = 64$ samples per 50 Hz).

### Experimental test

The block diagram of the test performed to investigate an incipient BRB fault is depicted in Fig. 8. The three-phase IM involved in the experimental bench, shown in Fig. 9, has similar characteristics to the simulated motor (Table 2). This motor is connected to the electrical grid of 230V/400 V and 50 Hz. The incipient defect has been created. The stator current signal is measured through one current sensor and captured with a numerical oscilloscope. The time of acquisition is of 10 s, with a sample rate $F_s = 25$ kHz. Finally, the value of the rotation speed is captured by an optical sensor. The fault was caused by a 3 mm deep hole on one of the rotor blades at the cage boundary. The motor, supplied at 230/400 V - 50 Hz in a temperature-controlled environment, was tested under various loads from 0 % to 100 %, with voltage imbalances reduced to a minimum.

### Performances of HT/MCSA and the proposed technique

The first objective of the simulation and experiment design is to measure the motor stator current at full load, half load and no load for the healthy case and the incipient fault condition.

As a second step, the stator current envelope is extracted using HT in situations of healthy and incipient rotor defect. The superposition of the stator and stator envelope current waveforms under full load, half load, and no load is shown in Figs. 10 and 11. These figures seem to include information regarding the incipient rotor defect. Consequently, the spectrum analysis must be used to identify the defect's signature. Afterward, the stator line current envelope is then assigned to the MCSA method to extract an incipient

<figure><img src="image_11.png" /><figcaption>Fig. 7. Schematic representation of the diagnostic process.</figcaption></figure>
<table><caption>**Table 1**
Specification of The Implemented IM.</caption><thead><tr><td>Machine Parameters</td><td>Values</td></tr></thead><tbody><tr><td>Rated output power</td><td>P = 3 KW</td></tr><tr><td>Number of pole pairs</td><td>N<sub>P</sub> = 2</td></tr><tr><td>Rated Stator Phase Voltage</td><td>U<sub>s</sub> = 400 V</td></tr><tr><td>Supply Frequency</td><td>f<sub>s</sub> = 50 Hz</td></tr><tr><td>Rotor Inductance</td><td>L<sub>r</sub> = 0.011 H</td></tr><tr><td>Stator Inductance</td><td>L<sub>s</sub> = 0.011 H</td></tr><tr><td>Mutual inductance</td><td>L<sub>m</sub> = 0.2395H</td></tr><tr><td>Rotor Resistance</td><td>R<sub>r</sub> = 4 Ω</td></tr><tr><td>Stator Resistance</td><td>R<sub>s</sub> = 1.9 Ω</td></tr><tr><td>Machine Inertia</td><td>J = 0.1282 Kg.m2</td></tr></tbody></table>

<figure><img src="image_12.png" /><figcaption>**Fig. 8.** Schematic representation of the experimental implementation.</figcaption></figure>

<figure><img src="image_13.png" /><figcaption>**Fig. 9.** Experimental setup for defect diagnosis (a) Healthy rotor (b) Creation of an incipient BRB failure using a drilling machine (c), Incipient BRB fault (d).</figcaption></figure>
<table><caption>**Table 2**
Comparison of simulated and theoretical defect signatures.</caption><thead><tr><th>Load level</th><th>Speed and signature fault values</th><th>Defective case</th><th>Healthy case</th></tr></thead><tbody><tr><td>No-load IM (0 %)</td><td>Simulated speed (rpm)<br>Theoretical f<sub>d</sub> frequency (Hz)<br>Simulated f<sub>d</sub> frequency (Hz)</td><td>1461.5<br>2.567<br>2.581</td><td>1462.6<br>No upper sideband<br>No upper sideband</td></tr><tr><td>Half-load IM (50 %)</td><td>Simulated speed (rpm)<br>Theoretical f<sub>d</sub> frequency (Hz)<br>Simulated f<sub>d</sub> frequency (Hz)</td><td>1404<br>6.4<br>6.452</td><td>1407.3<br>No upper sideband<br>No upper sideband</td></tr><tr><td>Full-load IM (100 %)</td><td>Simulated speed (rpm)<br>Theoretical f<sub>d</sub> frequency (Hz)<br>Simulated f<sub>d</sub> frequency (Hz)</td><td>1326.5<br>11.567<br>11.54</td><td>1332.5<br>No upper sideband<br>No upper sideband</td></tr></tbody></table>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>

<figure><img src="image_16.png" /><figcaption>**Fig. 10.** Simulated Stator line current and stator-current envelope waveforms for the healthy and incipient BRB defect cases.</figcaption></figure>

<figure><img src="image_17.png" /><figcaption>**Fig. 11.** Experimental Stator line current and stator current envelope waveforms for the healthy and incipient BRB defect cases.</figcaption></figure>

BRB fault. In particular, when the fault severity is less than half-BRB. Figs. 12 and 13 show the stator current envelope spectra at three load levels. The outcomes of both the simulation and the experiment show that the spectra are nearly identical. The 0 Hz frequency component, which corresponds to the DC component, is the most visible component in the spectrum of the stator current envelope. Effectively, no other signature is observable. As a result, an incipient BRB can not be directly extracted by this technique. Therefore, using a zoom is still necessary to discover the small hidden fault peak around the DC component or eliminate the DC component before implementing the MCSA, although for both solutions, the frequency resolution is never assured. Especially, in the case of an incipient BRB fault

From the previous findings, it is necessary to implement the conditioning of the stator line current envelope signal to overcome the limitations of the early diagnosis by the complete elimination of the DC component (electrical phenomena) to achieve a good spectral resolution for the detection of incipient BRB defect without the use of spectrum zoom. For these reasons, the novel RESE algorithm is established to extract the residual current envelope from the stator line current envelope. Figs. 14 and 15 depict the envelope waveforms appropriate for all load conditions. The observable variation in the residual envelopes provides prompt information on the
<figure><img src="image_18.png" /><figcaption>a. Unloaded IM -Simulation Results-</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>b. Half-load IM -Simulation Results-</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>c. Full-load IM -Simulation Results-</figcaption></figure>

<figure><img src="image_21.png" /><figcaption>a. Unloaded IM -Experimental Results-</figcaption></figure>

<figure><img src="image_22.png" /><figcaption>b. Half-load IM -Experimental Results-</figcaption></figure>

<figure><img src="image_23.png" /><figcaption>c. Full-load IM -Experimental Results-</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>a. Unloaded IM -Simulation Results-</figcaption></figure>

<figure><img src="image_25.png" /><figcaption>b. Half-load IM -Simulation Results-</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>c. Full-load IM -Simulation Results-</figcaption></figure>

<figure><img src="image_27.png" /><figcaption>a. Unloaded IM -Experimental Results-</figcaption></figure>

<figure><img src="image_28.png" /><figcaption>b. Half-load IM -Experimental Results-</figcaption></figure>

<figure><img src="image_29.png" /><figcaption>c. Full-load IM -Experimental Results-</figcaption></figure>

<caption>Fig. 15. Comparison of the residual envelope waveforms for the healthy and incipient BRB defect cases – Experimental Results-.</caption>
<figure><img src="image_30.png" /><figcaption>Fig. 16. Comparison of the spectrums of the residual current envelope for the healthy and incipient BRB defect cases - Simulation Results.</figcaption></figure>

<figure><img src="image_31.png" /><figcaption>Fig. 17. Comparison of the spectrums of the residual current envelope for the healthy and incipient BRB defect cases – Experimental Results.</figcaption></figure>

existence of the fault, according to the simulated and experimental findings. The temporal analysis of the residual envelope can provide a rich basis for extracting information on the presence of an incipient BRB fault. However, this temporal analysis remains insufficient for an accurate identification and localization of this incipient BRB failure. For effective fault localization, the residual-current envelope signals are processed using the MCSA method. In Figs. 16 and 17, the obtained spectra are depicted. These figures show that as the machine load increases, the magnitude of the fault signature in the residual current envelope spectrum also increases.

## Performances of the proposed approach

To prove the effectiveness of the proposed method and the impact of the fault when the IM operates at three different speeds. The steady-state mechanical velocity waveforms for all scenarios are illustrated in Fig. 18. This figure illustrates that an incipient BRB fault causes minor vibrations, whose amplitude may change further with velocity. As a result, the degree of this incipient BRB fault's fluctuation in rotational speed is directly proportionate.

Finally, these vibrations generate a worsening of the BRB as soon as the fault is increased. Afterward, the values of the mean measured speed, in the healthy and the defective cases for three load levels, as well as the theoretical signature, the experimental and simulation signature already presented in the residual envelope spectrums, are given in Tables 2 and 3.

The experimental and simulated residual envelope signature values agree with the theoretical fault signature values obtained from (14). As a result, it is easy to differentiate between faulty and healthy cases using the residual envelope's spectrum. The MCSA

<figure><img src="image_32.png" /><figcaption>Fig. 18. Comparison of the speed rotation for the healthy case and defective cases of incipient BRB defects.</figcaption></figure>
<table><caption>**Table 3**
Comparison of experimental and theoretical defect signatures.</caption><thead><tr><th>Load level</th><th>Speed and signature fault values</th><th>Defective case</th><th>Healthy case</th></tr></thead><tbody><tr><td rowspan="3">No-load IM (0 %)</td><td>Speed (rpm)</td><td>1467</td><td>1469</td></tr><tr><td>Theoretical f<sub>d</sub><br>frequency (Hz)</td><td>2.2</td><td>–</td></tr><tr><td>Experimental fd<br>frequency (Hz)</td><td>2.25</td><td>–</td></tr><tr><td rowspan="3">Half-load IM (50 %)</td><td>Speed (rpm)</td><td>1407</td><td>1417</td></tr><tr><td>Theoretical f<sub>d</sub><br>frequency (Hz)</td><td>6.2</td><td>–</td></tr><tr><td>Experimental fd<br>frequency (Hz)</td><td>6.25</td><td>–</td></tr><tr><td rowspan="3">Full-load IM (100 %)</td><td>Speed (rpm)</td><td>1328</td><td>1349.8</td></tr><tr><td>Theoretical f<sub>d</sub><br>frequency (Hz)</td><td>11.467</td><td>–</td></tr><tr><td>Experimental fd<br>frequency (Hz)</td><td>11.5</td><td>–</td></tr></tbody></table>

<table><caption>**Table 4**
Quantitative comparison of detection performance between HT/MCSA and RESE methods according to Motor load.</caption><thead><tr><th>Motor Load</th><th>Method</th><th>Detected Frequency (Hz)</th><th>Fault Amplitude</th><th>SNR Approx. (dB)</th><th>Detection Capability</th></tr></thead><tbody><tr><td rowspan="2">0 %</td><td>HT/MCSA</td><td>Not discernible</td><td>Approximately same as healthy case</td><td>&lt; 5 dB</td><td>Failed</td></tr><tr><td>RESE</td><td>8.25</td><td>747.9</td><td>&gt; 10 dB</td><td>Successful</td></tr><tr><td rowspan="2">50 %</td><td>HT/MCSA</td><td>Slight visible offset</td><td>low</td><td>≈ 6 dB</td><td>Uncertain</td></tr><tr><td>RESE</td><td>8.25</td><td>9616</td><td>&gt; 15 dB</td><td>Successful</td></tr><tr><td rowspan="2">100 %</td><td>HT/MCSA</td><td>Visible frequency</td><td>moderate</td><td>≈ 10 dB</td><td>Acceptable</td></tr><tr><td>RESE</td><td>11.5</td><td>2.85 × 10<sup>4</sup></td><td>&gt; 20 dB</td><td>Excellent</td></tr></tbody></table>

approach can therefore be used to extract an incipient BRB fault from the residual current of the stator current envelope.

This study includes a quantitative comparison of the detection performance of the classic HT/MCSA method in the noise configuration and that of our new RESE method according to the signal-to-noise ratio (SNR), with the support of a summary, in Table 4, enabling to visualize the results, grouping together the main measured indicators passed by the engine load. To calculate SNR:

$$SNR_{db} = 10. \log_{10} \left( \frac{A_{defect}^2}{A_{noise}^2} \right) \qquad (56)$$

SNR: Signal-to-Noise Ratio, expressed in decibels (dB)

$A_{defect}$: signal amplitude in the presence of the fault

$A_{noise}$: noise amplitude

According to the results provided in this table, the RESE method is superior to the HT/MCSA method in terms of both sensitivity and robustness, particularly in difficult, low-load configurations for which the HT/MCSA method does not provide detection.

An overview of the different approaches under investigation, including comparisons and limitations, is presented in Table 5. Table 6 compares the proposed strategy with other strategies developed in the published research literature since 2016.

It's also worth noting that the RESE topology has several real effects compared with the conventional method. For, unlike

<table><caption>**Table 5**
Comparison of the techniques used in this study.</caption><thead><tr><th>Method</th><th>Attributes and incipient fault diagnosis performances</th></tr></thead><tbody><tr><td>Classical HT/ MCSA</td><td><ul><li>Analysis of the stator current envelope does not provide clear information about the presence of an incipient BRB fault.</li><li>Frequency resolution is not remarkable.</li><li>Extraction of the incipient BRB defect is impossible.</li><li>Separation of the incipient BRB fault and noise is difficult</li></ul></td></tr><tr><td>RESE</td><td><ul><li>Temporal analysis of the envelope of the residual current provides information on the presence of the fault</li><li>Does not allow for to anticipation of the type and level of a small defect</li></ul></td></tr><tr><td>Proposed approach</td><td><ul><li>Non-intrusive, straightforward implementation and single-phase measurement are sufficient.</li><li>Unlike techniques based on the analysis of the signal acquired by the accelerometer, which require direct access to the machine, this technique can be applied to offshore wind turbine motors and generators that are inaccessible.</li><li>The frequency signature of the defect is more apparent, especially when the motor is not loaded.</li><li>Gives information regarding the existence of an incipient BRB fault.</li><li>It is an effective tool for the early identification of the IMs fault in the industries that produce electric motors.</li></ul></td></tr></tbody></table>
<caption>**Table 6**</caption>

Performance comparison between the suggested approach and other approaches presented in the literature.

<table><thead><tr><th>Year (Reference)</th><th>Technique/ Approach</th><th>Defect severity</th><th>Load</th><th>Detection Threshold</th><th>Computational Complexity</th><th>Limitations</th></tr></thead><tbody><tr><td>2016 [17]</td><td>MCSA and MUSIC</td><td>Half-BRB</td><td>low load</td><td>~10 %</td><td>High</td><td>Poor accuracy under low load and sensitive to noise.</td></tr><tr><td>2017 [18]</td><td>Estimation of statistical parameters using HT.</td><td>Half-BRB</td><td>0 % and 100 %</td><td>~10 %</td><td>Moderate</td><td>Not robust under variable loads and affected by signal noise.</td></tr><tr><td>2017 [20]</td><td>Homogeneity</td><td>Half-BRB</td><td>25 %</td><td>~10 %</td><td>Low</td><td>Lack of precision in early stage detection.</td></tr><tr><td>2018 [30]</td><td>MCSA and MUSIC</td><td>Half-BRB</td><td>5 %, 10 %, 20 %, 30 %, 50 % and 75 %</td><td>5–10 %</td><td>High</td><td>Complex implementation and less effective under low load.</td></tr><tr><td>2019 [31]</td><td>- MCSA is used for fault extraction.<br/>- Conventional Neural Networks (CNN) is used for classification</td><td>Incipient BRB</td><td>50 % and 75 %</td><td>5–10 %</td><td>High</td><td>Requires labeled datasets, retraining and ineffective under low load.</td></tr><tr><td>2020 [32]</td><td>- Short Time Fourier Transform (STFT) is employed to transform the measured signals to images.<br/>- A Convolutional Neural Networks (CNN) is used as features classifier</td><td>Incipient BRB</td><td>-</td><td>5–10 %</td><td>High</td><td>High computational burden, requires image processing and large data.</td></tr><tr><td>2021 [33]</td><td>- The combined ETSA/MCSA method is applied to the Park current components for fault extraction.<br/>- Fuzzy logic system is used for classification of incipient BRB faults.</td><td>Three levels of Incipient BRB</td><td>0 %</td><td>~5 %</td><td>Moderate</td><td>Needs expert-defined fuzzy rules and moderate sensitivity.</td></tr><tr><td>2022 [34]</td><td>- The combined RESE/DWT method is used for fault extraction<br/>- Fuzzy logic system is used for incipient rotor fault classification</td><td>Incipient BRB</td><td>0 %, 50 % and 100 %</td><td>~3–5 %</td><td>Moderate</td><td>Requires tuning of decomposition levels and fuzzy system.</td></tr><tr><td>2022 [35]</td><td>A combination of MCSA and DC injection braking approaches is used.</td><td>Incipient BRB</td><td>0 %</td><td>~5 %</td><td>Moderate</td><td>Not suitable for continuous online monitoring.</td></tr><tr><td>2023 [36]</td><td>HT and Particle Swarm Optimization (PSO)</td><td>Half-BRB</td><td>25 %</td><td>~5–10 %</td><td>High</td><td>Optimization-based; high computational time and parameter sensitivity.</td></tr><tr><td>Proposed work</td><td>The new HT/RESE approach.</td><td>Incipient BRB</td><td>0 %, 50 % and 100 %</td><td>≤ 2% under no-load</td><td>Moderate</td><td>Currently offline; needs extension to other machine types.</td></tr></tbody></table>

conventional approaches which often rely on computationally-intensive spectral processing, RESE synthesizes a subtraction between residual envelopes, and thus considerably simplifies processing, and reduces resource requirements. However, this simplification does not imply any loss of sensitivity: on the contrary, it makes it possible to detect weak defects in constrained environments without using intrusive instrumentation. What's more, the method is highly robust to load variation, as well as to the noise of the industrial environment, which is a real plus for a reliable and repeatable industrial application. Finally, RESE is an original, lightweight and effective solution to meet the new needs of industrial systems.

## Conclusion

In this paper, a novel method based on residual stator current envelope analysis for incipient fault detection of induction motors is verified and validated by simulation and experimental tests. Five main steps recap the newly proposed method:

- Extract the stator line current from one stator phase for three load levels i.e. 0 %, 50 %, and 100 %.

- Determine the stator current envelope using HT.

- Demonstrate the weakness of traditional HT using the MCSA method for early detection of an incipient defect of 3-mm crack depth of a rotor bar, even for three load levels, namely 0 %, 50 %, and 100 %.

- Extract the residual envelope using the RESE algorithm.

- Validate simulation and experimentation results via the comparison with the theory.

The simulation and experimental results confirm that the residual envelope analysis allows a low-severity fault identification (Less severe than half the BRB). Finally, this method can be implemented in factories producing IM and the manufacturing chain as an effective way to diagnose an incipient rotor BRB defect.

Finally, this work demonstrates an effective method for detecting early defects smaller than half a broken rotor bar (<, ½ BRB),
even at low loads. The sensitivity represents a significant technical advance, enabling early detection, actionable diagnostics, and proactive maintenance that allows for rapid intervention before operational efficiency declines and costly motor failure occurs.

## Credit author statement

The author declares that the research presented in this article is the result of their own work. All listed authors have made significant contributions to the design, analysis, and writing of the article. There is no conflict of interest related to this publication. Furthermore, the author confirms that all results have been obtained honestly, and all sources of information used are properly cited.

## Declaration of competing interest

The authors of the manuscript, declare that we have no conflict of interest.

## References

[1] M.E.H. Benbouzid, Signal Processing for Fault Detection and Diagnosis in Electric Machines and Systems, IET, London, 2020, p. 284. ISBN 978-1-83953-025-8.

[2] C. Morales-perez, J. Rangel-magdaleno, H. Peregrina-barreto, et al., Incipient broken rotor bar detection in induction motors using vibration signals and the orthogonal matching pursuit algorithm, IEEE Trans. Instrum. Meas. 67 (9) (2018) 2058-2068.

[3] X. Ying, Performance evaluation and thermal fields analysis of induction motor with broken rotor bars located at different relative positions, IEEE Trans. Magn. 46 (5) (2010) 1243-1250.

[4] V. Mishra, P. Gupta, Wind energy generation system using wound rotor induction machine, in: Innovations in Power and Advanced Computing Technologies (i-PACT), IEEE, 2019, pp. 1-7.

[5] Hassan H. ELDEEB, Haisen ZHAO, Osama A. MOHAMMED, Detection of TTF in induction motor vector drives for EV applications via Ostu's-based DDWE, IEEE Trans. Transp. Electrif. 7 (1) (2020) 114-132.

[6] Anmol AGGRAWAL, Ibrahim Mahmood ALLAFI, Elias G. STRANGAS, et al., Off-line detection of static eccentricity of PMSM robust to machine operating temperature and rotor position misalignment using incremental inductance approach, IEEE Trans. Transp. Electrif. 7 (1) (2020) 161-169.

[7] Meidi SUN, Hui WANG, Ping LIU, et al., A novel data-driven mechanical fault diagnosis method for induction motors using stator current signals, IEEE Trans. Transp. Electrif. (2022).

[8] Wangjie LANG, Yihua HU, Chao GONG, et al., Artificial intelligence-based technique for fault detection and diagnosis of EV motors: a review, IEEE Trans. Transp. Electrif. 8 (1) (2021) 384-406.

[9] Seungdeog CHOI, Moinul Shahidul HAQUE, Md Tawhid Bin TAREK, et al., Fault diagnosis techniques for permanent magnet AC machine and drives—a review of current state of the art, IEEE Trans. Transp. Electrif. 4 (2) (2018) 444-463.

[10] Z. Huo, Y. Zhang, P. Francq, L. Shu, J. Huang, Incipient fault diagnosis of roller bearing using optimized wavelet transform based multi-speed vibration signatures, IEEE Access 5 (2017) 19442-19456.

[11] H. Darong, K. Lanyan, M. Bo, Z. Ling, S. Guoxi, A new incipient fault diagnosis method combining improved RLS and LMD algorithm for rolling bearings with strong background noise, IEEE Access 6 (2018) 26001-26010.

[12] Khaled LAADJAL, Hugo RP ANTUNES, Mohamed SAHRAOUI, et al., On-line diagnosis and discrimination of stator faults in six-phase induction motors, based on voltage symmetrical components, IEEE Trans. Transp. Electrif. (2022).

[13] Anurag Choudhary, Deepam Goyal, Sudha Letha Shimi, et al., Condition monitoring and fault diagnosis of induction motors: a review, Arch. Comput. Methods Eng. 26 (4) (2019) 1221-1238.

[14] I. Zamudio-Ramirez, J.A. Ramirez-Núñez, J. Antonino-Daví, R.A. Osornio-Rios, A. Quijano-Lopez, H. Razik, R.J. Romero-Troncoso, Automatic diagnosis of electromechanical faults in induction motors based on the transient analysis of the stray flux via MUSIC methods, IEEE Trans. Ind. Appl. (2020).

[15] J. Tang, Y. Yang, J. Chen, R. Qiu, Z. Liu, Characteristics analysis and measurement of inverter-fed induction motors for stator and rotor fault detection, Energies 13 (1) (2020) 101.

[16] J. Tang, J. Chen, K. Dong, Y. Yang, H. Lv, Z. Liu, Modeling and evaluation of stator and rotor faults for induction motors, Energies 13 (1) (2020) 133.

[17] M.M. Rahman, M.N. Uddin, Online unbalanced rotor fault detection of an IM drive based on both time and frequency domain analyses, IEEE Trans. Ind. Appl. 53 (4) (2017) 4087-4096.

[18] M. Abd-el-Malek, A.K. Abdelsalam, O.E. Hassan, Induction motor broken rotor bar fault location detection through envelope analysis of start-up current using hilbert transform, Mech. Syst. Signal Process. 93 (2017) 332-350.

[19] G.D. Baranov, E.G. Nepomuceno, M.A. Vaganov, V.Y. Ostrovskii, D.N. Butusov, New spectral markers for broken bars diagnostics in induction motors, Machines 8 (1) (2020) 6.

[20] R.A. Lizarraga-Morales, C. Rodriguez-Donate, E. Cabal-Yepez, M. LopezRamirez, L.M. Ledesma-Carrillo, E.R. Ferrucho-Alvarez, Novel FPGA-based methodology for early broken rotor bar detection and classification through homogeneity estimation, IEEE Trans. Instrum. Meas. (99) (2017) 1-10. PP.

[21] I. MARTIN-DIAZ, D. MORINIGO-SOTELO, O. DUQUE-PEREZ, et al., Analysis of various inverters feeding induction motors with incipient rotor fault using high-resolution spectral analysis, Electr. Power Syst. Res. 152 (2017) 18-26.

[22] P.D. MCFADDEN, A revised*model for the extraction of periodic waveforms by time domain*averaging, Mech. Syst. Signal Process. 1 (1) (1987) 83-95.

[23] B. Asad, \*T. Vaimann, A. Belahcen, A. Kallaste, Broken rotor bar fault diagnostic of inverter fed induction motor using FFT, \*Hilbert and\*Park's vector approach, in: 2018 XIII International\*Conference on Electrical Machines (ICEM), IEEE, 2018, pp. 2352-2358.

[24] Prajoy PODDER, Tanvir Zaman \*KHAN, Mamdudul Haque KHAN, et al., Comparative performance\*analysis of hamming, hanning and blackman window, Int. \*J. Comput. Appl. 96 (18) (2014).

[25] I. Ouachtouk, S. El Hani, S. Guedira, K. Dahi, H. Mediouni, Broken rotor bar fault\*detection based\*on stator current envelopes analysis in squirrel cage induction\*machine, in: 2017 IEEE International Electric Machines and Drives Conference, 2017, pp. 1-6.

[26] S. Bachir, Thèse de doctorat, Laboratoire d'Automatique et d'Informatique Industrielle de Poitiers, Université de Poitiers, Poitiers, France, 2002.

[27] S. Bachir, S. Tnani, J.C. Trigeassou, G. Champenois, Diagnosis by parameter estimation of stator and rotor faults occurring in induction machines, IEEE Trans. Ind. Electron. 53 (3) (2006) 963-973.

[28] \* S. Bachir, S. Tnani, G. Champenois, J.C. Trigeassou, Diagnosis of induction machines by parameter estimation, \* , in: R. Husson (Ed.), Control Methods for Electrical Machines, ISTE Ltd., and John Wiley & Sons, 2009, pp. 249-273. \*.

[29] Zoltán Németh, Miklós Kuzmann, State space modeling theory of induction machines, Pollack Period. 15 (1) (2020) 124-135.

[30] G. Singh, V.N.A. Naikan, Detection of half broken rotor bar fault in VFD driven induction motor drive using motor square current MUSIC analysis, Mech. Syst. \*Signal Process. 110 (2018) 333-348.

[31] J.E. Garcia-Bracamonte, J.M. Ramirez-Cortes, J. De Jesus Rangel-Magdaleno, et al., An approach on MCSA-based fault detection using independent component analysis and neural networks, IEEE Trans. Instrum. Meas. 68 (5) (2019) 1353-1361.

[32] M. Valtierra-Rodriguez, J.R. Rivera-Guillen, J.A. Basurto-Hurtado, J.J. De-Santiago-Perez, D. Granados-Lieberman, J.P. Amezquita-Sanchez, Convolutional neural network and motor current signature analysis during the transient state for detection of broken rotor bars in induction motors, Sensors 20 (13) (2020) 3721.
[33] H. Sabir, M. Ouassaid, N. Ngote, M. Benbouzid, A novel experimental method to detect early rotor faults in induction machines, Int. J. Energy Convers. 9 (5) (2021) 191–202.

[34] H. Sabir, M. Ouassaid, N. Ngote, An experimental method for diagnostic of incipient broken rotor bar fault in induction machines, Heliyon 8 (3) (2022).

[35] D.G. Jerkan, D. Reljić, I. Todorović, I. Isakov, V. Porobić, D. Dujić, Detection of broken rotor bars in a cage induction machine using DC injection braking, IEEE Access 10 (2022) 49585–49598.

[36] M. Valtierra-Rodríguez, J.R. Rivera-Guillen, J.J. De Santiago-Perez, G.I. Perez-Soto, J.P. Amezquita-Sanchez, Expert system based on autoencoders for detection of broken rotor bars in induction motors employing start-up and steady-state regimes, Machines 11 (2) (2023) 156.