A Novel Statistical Method for Spectral Analysis of A Short-Duration Signal and Its Application to Current Data for Stator Fault Diagnosis

Abstract

In this paper, a novel approach for fault detection in the stator windings of induction motors is presented. The procedure is based on spectral analysis of the current signal. However, due to the specific target application, short duration signals (0.2 s) are utilized, which results in poor spectral resolution. To address this issue, a statistical methodology is developed to minimize uncertainty in decision-making. To construct a health indicator (HI), a statistical analysis is performed to identify spectral components that are both informative and robust. For the selected fault-related frequencies, the HI was created. Using confidence intervals and statistical testing, a fault detection scheme was proposed. The method was validated on an experimental dataset, including both healthy and faulty conditions. The method has been tested on current signals with five levels of fault severity and seven load conditions. Experimental studies on a dedicated test rig demonstrated the high efficiency of the proposed approach for such specific constraints.

1. Introduction

Induction motors are widely used in industrial applications due to their robustness, efficiency, and cost-effectiveness. However, like all rotating machinery, they are subject to various electrical and mechanical faults that can compromise reliability and shorten service life. Among these, stator winding faults—particularly inter-turn short circuits (ITSCs)—are considered especially critical because they can quickly evolve into severe failures if not detected at an early stage. Therefore, accurate and timely fault diagnosis is of great importance for predictive maintenance strategies and Industry 4.0-oriented smart monitoring systems. The adoption of condition-based monitoring and predictive maintenance strategies has become a cornerstone of Industry 4.0-oriented manufacturing systems [1,2]. Modern industrial facilities increasingly rely on intelligent monitoring systems that integrate Internet of Things (IoT) sensors and advanced analytics to enable real-time fault detection and proactive maintenance scheduling [3,4]. These data-driven approaches not only reduce unplanned downtime but also optimize maintenance costs and extend equipment service life, making reliable fault detection algorithms essential for competitive industrial operations. Early and reliable ITSC detection has long been a priority, with foundational studies analyzing turn-to-turn short circuits in operating motors and during switch-off transients directly in the frequency domain [5,6].

Traditionally, motor current signature analysis (MCSA) has served as the cornerstone of fault diagnosis in induction machines, relying on the identification of characteristic frequency components in the stator current spectrum [7,8]. Over the years, various signal processing enhancements have been introduced to improve the sensitivity of MCSA, including empirical mode decomposition (EMD), wavelet transform, and principal component analysis (PCA) [9]. These methods have proven effective for rotor bar faults, eccentricities, and bearing anomalies. For stator faults, spectral strategies are particularly effective: parametric spectral estimation sharpens fault-related lines under short records and varying loads [10]. Advanced parametric spectral estimation methods have demonstrated superior frequency resolution compared to non-parametric approaches when signal duration is restricted [10]. These high-resolution techniques address the fundamental trade-off between time and frequency resolution inherent in short-duration signal analysis. Rotor slot harmonics (RSH) have emerged as particularly sensitive fault indicators for stator winding defects, with the matrix pencil method specifically designed to extract RSH components with high precision [11]. Further contributions underline frequency-domain diagnostics under short records and inverter-fed conditions, including harmonic/sideband modeling and high-resolution estimation [12,13].

Short-Time Fourier Transform (STFT) has been widely used to generate spectrograms that reveal transient fault signatures requiring time–frequency analyses [14,15]. When combined with deep learning, STFT-based approaches have achieved high accuracy in fault classification, with some studies reporting accuracies above $97\%$ in rotor bar fault detection [16]. In [17], authors show that time–frequency trajectory tracking methods can support interpretation under variable speeds. More advanced spectral estimation methods, such as minimum-norm time–frequency analysis, have demonstrated improved resolution for multiple-fault scenarios under variable operating conditions [18]. Unlike STFT, these subspace-based approaches can achieve frequency resolution that is not strictly constrained by signal duration, making them particularly suitable for the analysis of brief measurement windows encountered in rapid industrial inspection scenarios. In parallel, statistics such as spectral kurtosis have proven effective in detecting impulsive or transient fault patterns [16]. Related cyclostationary analysis can assist in the demodulation and reinforcement of sidebands in the spectrum [19,20]. Cyclostationary signal analysis offers a powerful framework for extracting periodic modulation patterns characteristic of rotating machinery faults. Vibration and current signals from faulty motors exhibit cyclostationary behavior due to the periodic interaction between stationary and rotating components [21]. Spectral correlation and cyclic coherence maps can reveal fault-related modulation sidebands even when they are masked by noise or other cyclic components in conventional power spectra. Improved cyclostationary methods combining Teager–Kaiser energy operator (TKEO) demodulation with fast spectral correlation have successfully diagnosed broken rotor bar and bearing faults under various operating conditions [22,23]. Recent surveys synthesize early-detection strategies and data-driven pipelines for induction machines, emphasizing the practical role of spectral indicators and sidebands [16,24].

Recent years have witnessed remarkable progress in the integration of artificial intelligence (AI) with motor fault diagnosis. Deep learning models leveraging time–frequency images or raw signal data have outperformed traditional approaches in terms of accuracy and generalization. For instance, convolutional neural networks (CNNs) trained on STFT-based spectral representations, as well as lightweight models such as ShuffleNetV2, have achieved nearly $99\%$ accuracy while remaining computationally efficient [16]. Ensemble methods, such as the weighted probability ensemble deep learning (WPEDL) framework, combine features from both current and vibration signals and achieve classification rates exceeding $99\%$ across multiple fault types [16]. Additionally, graph neural networks (GNNs) have been introduced for the direct analysis of raw current and vibration signals, showing promising results in capturing complex dependencies between sensor modalities [16]. Unlike traditional CNNs that process signals as independent samples, GNNs construct graph structures that explicitly model dependencies between data points, potentially improving generalization to unseen operating conditions. In parallel, recent works report AR/Prony-based spectral MCSA and ITSC-oriented indicators under inverter-fed or variable-load conditions [25]. However, deep learning approaches often require extensive labeled datasets and lack the interpretability necessary for safety-critical industrial applications. The black-box nature of neural networks makes it difficult to validate their decisions based on a physical understanding of fault mechanisms.

With respect to stator winding faults, several recent contributions have specifically addressed ITSC detection [26]. Other works have developed indicators, such as the complex current unbalance coefficient (CCUC), to discriminate between ITSC and voltage unbalance, achieving high robustness under realistic operating conditions (see related negative-sequence compensation in [27]). Inter-turn short circuits create an imbalance in the three-phase stator windings, generating negative-sequence components that are nominally absent in healthy motors under balanced supply conditions [28,29]. Negative-sequence current analysis can distinguish ITSC-induced asymmetry from supply voltage unbalance, achieving high robustness under realistic operating conditions [29]. Space-vector trajectory analysis during start-up transients can reveal ITSC signatures that are obscured during steady-state operation [30]. Envelope energy methods applied to start-up current signals have achieved high accuracy in early ITSC fault detection, with reported detection rates of 96.9% using machine learning classification [31]. The amplification of fault signatures during transient operation, when currents reach several times their rated values, makes start-up analysis particularly attractive for detecting incipient faults. The fusion of multi-sensor data and advanced deep learning architectures has further improved early-stage fault detection capabilities [32]. While these methods represent important advances, they often rely on advanced feature engineering, extensive training datasets, or access to multiple sensing modalities, which may restrict their adoption in industrial environments. By contrast, spectral techniques that target physics-informed sidebands—such as stator-fault components and rotor-slot harmonics—offer explainability and low sensor overhead [5,6,11].

Despite these technological advances, important limitations persist: application to short-duration signals is often hindered by the fundamental trade-off between time and frequency resolution, and most of the aforementioned methods lack a rigorous statistical framework for decision-making. Specifically, the significance of characteristic fault frequencies is rarely tested using statistical inference, such as confidence intervals, hypothesis testing, or test power analysis. This gap becomes particularly evident when analyzing short-duration signals (e.g., 0.2 s, 10,000 samples), where poor spectral resolution introduces high uncertainty. Although some research has proposed statistic-based spectral indicators for bearing faults [33,34], systematic statistical methodologies for short-time current signals remain scarce. Recent contributions focused on start-up features, space-vector indicators, and sequence compensation highlight the domain need but typically do not attach formal confidence measures [27,30,31]. Furthermore, deep learning approaches typically do not provide probabilistic outputs or uncertainty quantification, limiting their trustworthiness in safety-critical applications [35]. This fundamental limitation undermines the development of objective detection thresholds that balance sensitivity against false alarm rates, which is essential in industrial contexts where maintenance decisions carry significant economic consequences and regulatory compliance may require statistically justified diagnostic criteria.

The present work introduces a novel statistical approach for the spectral analysis of short-duration signals and applies it to stator fault diagnosis. The proposed approach employs confidence intervals and hypothesis testing [36,37,38] to statistically evaluate informative spectral components, enabling the construction of a health indicator (HI) that captures significant amplitude variations associated with inter-turn short circuits. Unlike purely data-driven techniques, the method provides interpretable results with quantified uncertainty, thus enhancing its trustworthiness. The contribution is further validated through experimental studies on a dedicated test rig, encompassing five levels of fault severity and seven load conditions. It is tested on previously unseen data to demonstrate robustness. The novelty of the study is a statistically grounded framework for stator winding fault detection based on a spectral health indicator (HI) constructed from very short signal records (e.g., $0.2$ s). The selection of a 0.2 s time window is motivated by the requirement to monitor machines operating under dynamic load conditions. In such applications, a longer acquisition time (e.g., 1 s) would violate the stationarity assumption due to supply frequency fluctuations, leading to spectral smearing. Furthermore, the short window aligns with the capabilities of low-cost embedded sensors designed for rapid condition assessment during short duty cycles. The proposed approach operates robustly across multiple fault severities (including a single shorted turn) and a wide range of load conditions. By explicitly modeling the sampling distribution of spectral amplitude estimates, the method forms confidence bounds and hypothesis tests at physics-informed sidebands, enabling reliable decisions even when frequency resolution is coarse and spectral estimates are biased. Experimental validation on previously unseen data indicates a high probability of detection at controlled false-alarm rates. In summary, the contribution advances spectral and statistical diagnostics of ITSC and strengthens decision-making by providing a statistically validated HI tailored to very short records.

The remainder of this paper is organized as follows. Section 2 presents the proposed methodology, including the diagnostic framework and data processing strategy. Key implementation details and statistical assumptions for the spectral analysis are provided to facilitate reproducibility and industrial transfer. Section 3 describes the experimental setup and the measurement procedure used to collect current signals at different fault stages under various operating conditions. Section 4 summarizes and discusses the aggregated diagnostic results obtained for all fault types and load levels. Finally, Section 5 provides the main conclusions and outlines directions for future research.

2. Methodology

The proposed methodology aims to diagnose inter-turn short circuits in stator windings based on the statistical analysis of amplitudes at characteristic fault frequencies (FFs) extracted from short-duration current signals. The approach combines classical spectral analysis with statistical hypothesis testing to obtain a scalar health indicator, i.e., HI value, which is subsequently used for decision-making.

The proposed diagnostic approach is organized into three main stages: dataset construction, feature-based health modeling, and validation.

1.

Dataset construction.  Raw three-phase stator current signals—denoted as IsA, IsB, and IsC—were acquired under both healthy and faulty operating conditions, comprising five fault severity levels and seven load conditions. Each signal was segmented into short, fixed-length intervals to standardize the input size and augment the number of samples for analysis. The resulting segments were labeled and divided into separate datasets for subsequent processing, ensuring that identical signals did not appear in both sets. The overall dataset construction procedure is described in detail in Section 2.3.

2.

Feature-based health modeling. To construct the health indicator, it is first necessary to identify the fault-related informative frequencies. Therefore, statistical testing is employed to determine which frequencies are statistically significant. Subsequently, the health indicator and its confidence interval are computed, both of which are essential for decision-making regarding fault diagnosis. This step consists of two main stages:

(a)

Feature selection based on statistical testing. Spectral features are extracted from the current signals to characterize both healthy and faulty operating conditions. Statistical analysis is then applied to identify the most informative features for distinguishing between these states (see Section 2.4.1).

(b)

Health indicator and confidence interval construction. Using the healthy reference data and the selected informative features, the health indicator and its confidence interval are established, providing a quantitative reference for subsequent fault detection (see Section 2.4.2).

• The step-by-step procedure of the feature-based health modeling is described in detail in Section 2.4.

3.

Validation procedure. During validation, new unseen three-phase stator current signals—representing both healthy and faulty conditions with five fault-severity levels at various load settings—are employed. For each tested signal, the computed health indicator is compared against the reference confidence range. If the indicator falls within this range, the signal is classified as healthy; otherwise, it is identified as faulty. Finally, the overall diagnostic performance of the proposed method is evaluated. The step-by-step statistical testing procedure is described in detail in Section 2.5.

2.1. Spectral Representation of the Signal

Let $x\left(t\right)$, $t=0,\dots ,T-1$, be a current signal of length T. The first stage of the method involves the computation of the signal spectrum. A one-sided periodogram (non-parametric power spectral density, PSD) is used with a Hanning window $w(·)$, and the number of FFT points is set to $N=2^{\lfloor {log}_{2}T\rfloor }$. Zero padding is applied to improve frequency resolution.

The one-sided periodogram of $x\left(t\right)$ is defined as [39]:

$${\widehat{P}}_x\left(f\right)={\displaystyle \frac{\mathrm{\Delta }t}{N}}|\sum _{n=0}^{N-1}w\left(n\right)x\left(n\right)e^{-\mathrm{i}2\pi f\mathrm{\Delta }tn}{|}^2,f\in [0,\frac{fs}{2}-1],$$

(1)

with the frequency $f\left(k\right)={\displaystyle \frac{kfs}{N}}$ for $k=0,\dots ,\frac{N}{2}$, and the $\mathrm{\Delta }f={\displaystyle \frac{fs}{N}}$ and $\mathrm{\Delta }t=t_2-t_1$. The one-sided amplitude spectral density is defined as

$${\widehat{\mathrm{SD}}}_x\left(f\right)=\sqrt{{\displaystyle \frac{2P_x{\parallel w\parallel }^2}{{\left({\sum }_{n=0}^{N}w\left(n\right)\right)}^2}}},$$

(2)

where $\parallel w\parallel$ is the Euclidean norm of vector w.

2.2. Theoretical Foundations of Stator Winding Fault Detection Based on Current Spectrum Analysis

The primary cause of characteristic harmonics in the stator current during an inter-turn short circuit (ITSC) is the disturbance of the magnetic circuit’s symmetry. In a healthy state, the stator windings produce a balanced magnetomotive force (MMF), resulting in an approximately sinusoidal flux distribution in the air gap. The occurrence of an ITSC reduces the effective number of turns in the faulty phase, leading to a significant short-circuit current that generates a locally opposing magnetic field. This negative MMF acts against the main MMF of the stator winding, causing its asymmetry, which weakens and distorts the total MMF. This disturbance creates an uneven flux distribution in the air gap, and this disturbed magnetic field subsequently interacts with the rotor. This interaction modulates the stator current with frequencies intrinsically linked to the rotor’s construction, particularly the rotor slot harmonics (RSH). The severity of the fault is directly correlated with the amplitude of these induced harmonic components; a larger fault, characterized by a higher short-circuit current and greater MMF distortion, results in a more pronounced increase in their magnitudes in the current spectrum.

As a result of the FFT transformation of the stator current signal, an ITSC fault manifests as an increase in the amplitudes of several characteristic frequency groups of components. The first group comprises the basic fault harmonics in the low-frequency range, which are a direct consequence of the MMF disturbance and depend on the rotor speed or slip. These components are calculated using the following formula:

$$f_{sh}^{\left(1\right)}(k,m)=f_s\left(k{\displaystyle \frac{(1-s)}{p_p}}+m\right),$$

(3)

where $f_s$—supply voltage frequency, $p_p$—number of pairs of poles, s—slip, $k=1,2,3,\dots$, $m\in {\mathbb{Z}}_{\mathrm{odd}}$, where m belongs to the odd integer values.

The second, and particularly sensitive, group consists of the rotor slot harmonics (RSH) in the medium-frequency range, which are related to the physical construction of the rotor. The amplification of RSH during ITSC occurs because the stator asymmetry created by the shorted turns distorts the air-gap magnetomotive force (MMF), causing uneven flux distribution. This disturbed field interacts with the rotor’s physical structure (rotor slot spacing), thereby modulating the stator current with frequencies intrinsically linked to rotor geometry. Unlike low-frequency fault harmonics, RSH are less affected by slip variations at varying loads, making them more robust indicators. While these harmonics exist under normal conditions with small amplitudes, a stator fault significantly amplifies them. The general formula for these components is

$$f_{sh}^{\left(2\right)}(k,m)=f_s\left(k{N}_r{\displaystyle \frac{(1-s)}{p_p}}+m\right),$$

(4)

where $N_r$—number of rotor bars.

The most diagnostically valuable is typically the Lower Rotor Slot Harmonic (LRSH) given by

$$f_{sh}^{\left(2\right)}(1,1)=f_s\left(N_r{\displaystyle \frac{(1-s)}{p_p}}-1\right).$$

(5)

The third group includes supply harmonics, calculated as

$$s{f}_n=n{f}_s,$$

(6)

which are multiples of the fundamental frequency; however, these are less specific indicators compared to the first two groups.

It is essential to consider practical aspects, such as load dependence, as the frequency position of $f_{sh}$ components varies with slip. Under very light loads, they may approach supply-related harmonics, complicating detection. Furthermore, low-amplitude fault harmonics in the early stages can be masked by noise in conventional FFT analysis. For a measured rotor speed of n, the slip is determined as

$$s={\displaystyle \frac{n_s-n}{n_s}}.$$

(7)

The synchronous speed $n_s$ can be calculated using the following formula:

$$n_s={\displaystyle \frac{60·f_s}{p_p}},$$

(8)

where $p_p$ denotes the number of pole pairs.

In the presented approach, seven fault-related frequencies, denoted as $F{F}_l\in \mathrm{FF}$, are investigated. They consist of the third supply harmonic, i.e., $s{f}_3$, and the rotor slot harmonics for $k=1$ and $m\in \{-5,-3,-1,1,3,5\}$. The resulting vector of potential fault frequencies $\mathrm{FF}$ considered in this study is defined as follows:

$$\mathrm{FF}:=[F{F}_1,F{F}_2,\dots ,F{F}_7]=[s{f}_3,f_{sh}^{\left(2\right)}(1,-5),f_{sh}^{\left(2\right)}(1,-3),f_{sh}^{\left(2\right)}(1,-1),f_{sh}^{\left(2\right)}(1,1),f_{sh}^{\left(2\right)}(1,3),f_{sh}^{\left(2\right)}(1,5)].$$

(9)

Each empirical frequency estimate ${\widehat{FF}}_l$ is obtained as

$${\widehat{FF}}_l=arg\underset{f\in [F{F}_l-2\mathrm{\Delta }f,F{F}_l+2\mathrm{\Delta }f]}{max}{\widehat{\mathrm{SD}}}_x\left(f\right),$$

(10)

where ${\widehat{\mathrm{SD}}}_x\left(f\right)$ denotes the estimated spectral density of the signal.

The selection of these components was determined experimentally by comparing healthy and faulty operating conditions. The chosen spectral components exhibit the largest differences in spectral amplitudes, making them the most informative for fault characterization.

The graphical illustration of $\widehat{F{F}_l}$ detection for different fault stages, i.e., healthy signal (upper panel), fault 1 (middle panel), and fault 5 (bottom panel), is presented in Figure 1. The ${\widehat{FF}}_l$ is marked with a black dashed line, with the region of local maximum exploration, i.e., $[F{F}_l-2\mathrm{\Delta }f,F{F}_l+2\mathrm{\Delta }f]$ marked with a shaded gray area.

The exemplary theoretical values of $F{F}_l$ used in this study (for load 6) for the analyzed motor, with the number of rotor slots equal to $N_r=26$, a measured rotor speed of $n=1407\mathrm{rpm}$, and a current supply of $f_s=50$ Hz are listed in

Table 1. Characteristic fault frequencies used in the proposed method (load 6).

${\mathit{FF}}_{\mathit{l}}$	RSH	Value [Hz]
$F{F}_1$	$s{f}_3=3·50$	150.00
$F{F}_2$	$f_{sh}^{\left(2\right)}(1,-5)=50\left(26·{\displaystyle \frac{1-0.062}{2}}-5\right)$	359.7
$F{F}_3$	$f_{sh}^{\left(2\right)}(1,-3)=50\left(26·{\displaystyle \frac{1-0.062}{2}}-3\right)$	459.7
$F{F}_4$	$f_{sh}^{\left(2\right)}(1,-1)=50\left(26·{\displaystyle \frac{1-0.062}{2}}-1\right)$	559.7
$F{F}_5$	$f_{sh}^{\left(2\right)}(1,1)=50\left(26·{\displaystyle \frac{1-0.062}{2}}+1\right)$	659.7
$F{F}_6$	$f_{sh}^{\left(2\right)}(1,3)=50\left(26·{\displaystyle \frac{1-0.062}{2}}+3\right)$	759.7
$F{F}_7$	$f_{sh}^{\left(2\right)}(1,5)=50\left(26·{\displaystyle \frac{1-0.062}{2}}+5\right)$	859.7

.

Using the formula for RSH, see Equation (4), and for the formula for the slip and synchronous speed, see Equations (7) and (8); one can note that

$$1-s=1-{\displaystyle \frac{n_s-n}{n_s}}={\displaystyle \frac{n}{n_s}}.$$

(11)

Substituting into the fault frequency model from Equation (4) gives

$$f_{sh}^{\left(2\right)}(k,m)=f_s\left({\displaystyle \frac{k{N}_r}{P_p}}{\displaystyle \frac{n}{n_s}}+m\right).$$

(12)

Using $n_s={\displaystyle \frac{60f_s}{P_p}}$ (hence ${\displaystyle \frac{1}{n_s}}={\displaystyle \frac{P_p}{60f_s}}$), the expression simplifies to a linear form

$$f_{sh}^{\left(2\right)}(k,m)={\displaystyle \frac{k{N}_r}{60}}n+f_{s}m.$$

(13)

This linear dependence yields a direct sensitivity of the predicted fault frequency to rotor-speed estimation errors. For the parameters $N_r=26$, $f_s=50\mathrm{Hz}$, $k=1$, and $m=1$, the expression becomes

$$f_{sh}^{\left(2\right)}(1,1)={\displaystyle \frac{26}{60}}n+50.$$

(14)

A rotor-speed estimation error $\delta n$ produces a frequency prediction error

$$\delta f\approx {\displaystyle \frac{d{f}_{sh}^{\left(2\right)}(1,1)}{dn}}\delta n={\displaystyle \frac{26}{60}}\delta n.$$

(15)

To ensure that the true spectral peak remains inside the search window of $\pm 2\mathrm{\Delta }f$, the condition

$$\left|\delta f\right|\le 2\mathrm{\Delta }f$$

(16)

must hold, which yields the admissible speed error bound

$$\left|\delta n\right|\le {\displaystyle \frac{2\mathrm{\Delta }f}{26/60}}\approx {\displaystyle \frac{2\mathrm{\Delta }f}{0.4333}}\mathrm{rpm}.$$

(17)

Summarizing, for the $\mathrm{\Delta }f\approx 6\mathrm{Hz}$, small speed/load variations within $\left|\delta n\right|\le {\displaystyle \frac{2·6}{0.4333}}\approx 27\mathrm{rpm}$ will not affect the effectiveness of the proposed method, as the spectral peak of empirical frequency estimate ${\widehat{FF}}_l$ in Equation (10) will be properly assigned.

2.3. Dataset Construction

Both healthy and faulty three-phase stator current signals were used to build the database for analysis, including different load levels. Each raw signal was divided into fixed-length windows (in the presented analysis, $0.2$ s) to ensure that uniform data frames were generated from the input data. Based on this segmentation, the obtained signals were organized into datasets. Three datasets were prepared from healthy signals, i.e., ${\mathcal{A}}_0$, ${\mathcal{A}}_1$, and ${\mathcal{A}}_2$, and two from faulty ones, i.e., ${\mathcal{B}}_1$ and ${\mathcal{B}}_2$, with all datasets containing an equal number of segments. The desired feature of the datasets is their independence (the datasets come from the same machine but from different experiments run at different times). Spectra are computed using fixed parameters and reveal stable, i.e., spectral estimates and their confidence intervals do not vary systematically across adjacent windows. The resulting datasets are summarized in

Table 2. Division of healthy and faulty datasets into equinumerous subsets used in statistical testing.

	Healthy Dataset $\mathcal{A}$	Faulty Dataset $\mathcal{B}$
Dataset for $\widehat{FFCI}$ and $\widehat{HICI}$ construction	$x_i^{*}\in {\mathcal{A}}_0$,
Datasets for feature selection	$x_i\in {\mathcal{A}}_1$	$y_i\in {\mathcal{B}}_1$
Datasets for validation procedure	$s_i\in {\mathcal{A}}_2$	$u_i\in {\mathcal{B}}_2$
for $i=[1,\dots ,M]$ in each dataset

.

This division ensures a strict separation between datasets of equal size used for different purposes, thereby facilitating a reliable evaluation of diagnostic performance. In the present analysis, each subset contains $M=100$ segments of $0.2$ s. The algorithm of the dataset construction (for each current phase, fault type, and load) is presented in Algorithm 1.

Algorithm 1: Dataset construction (for each current phase, fault type, and load)

Data: Current-signal datasets: healthy $\mathcal{A}$ and faulty $\mathcal{B}$ Result: Datasets ${\mathcal{A}}_0,{\mathcal{A}}_1,{\mathcal{A}}_2\in \mathcal{A}$ and ${\mathcal{B}}_1,{\mathcal{B}}_2\in \mathcal{B}$ of healthy and faulty data Dataset construction: 1. Collect raw current signals under healthy ($\mathcal{A}$) and faulty ($\mathcal{B}$) conditions. 2. Divide each raw signal into fixed-length windows to ensure uniform input size. 3. Label the obtained segments and organize them into disjoint datasets. 4. Select M segments for each subset (${\mathcal{A}}_0$, ${\mathcal{A}}_1$, ${\mathcal{B}}_1$, ${\mathcal{A}}_2$, ${\mathcal{B}}_2$). Dataset for $\widehat{FFCI}$ and $\widehat{HICI}$ construction: $x_i^{*}\in {\mathcal{A}}_0$ (reference healthy), Datasets for feature selection: $x_i\in {\mathcal{A}}_1$ (healthy), $y_i\in {\mathcal{B}}_1$ (faulty), Datasets used for validation procedure: $s_i\in {\mathcal{A}}_2$ (healthy), $u_i\in {\mathcal{B}}_2$ (faulty). 5. Save the prepared datasets with corresponding labels and segment indices for further diagnostic analysis.

In the proposed methodology, the construction of the health indicator involves an internal statistical testing procedure, namely a feature selection approach, which is independent of the final validation stage of the entire diagnostic framework. This separation is required since the algorithm for selecting informative fault frequencies ($F{F}_l$) relies on an internally defined statistical test, which is explicitly described in the next section.

2.4. Feature-Based Health Modeling

The feature-based health modeling stage employs three datasets: two healthy datasets, ${\mathcal{A}}_0$ and ${\mathcal{A}}_1$, and one faulty dataset, ${\mathcal{B}}_1$. The signals $x_i^{*}\in {\mathcal{A}}_0$ are used to establish the confidence intervals ${\widehat{FFCI}}_l$ for the spectral amplitudes at each characteristic frequency ${\widehat{FF}}_l$, as well as the confidence interval for the health indicator $\widehat{HICI}$. The signals $x_i\in {\mathcal{A}}_1$ and $y_i\in {\mathcal{B}}_1$ are then used to estimate the empirical size (Type I error) and empirical power of the statistical test defined below, enabling the selection of informative fault frequencies. The two main stages include:

• the selection of informative ${\widehat{FF}}_l$ features, described in detail in Section 2.4.1;
• the construction of the confidence interval for the decision-maker health indicator discussed in Section 2.4.2.

2.4.1. Feature Selection Based on Statistical Testing

For each fault frequency $F{F}_l$ from healthy signals $x_i^{*}\in {\mathcal{A}}_0$, where $l=1,2,\dots ,7$, and $i=1,\dots ,M$, a confidence interval ${\widehat{FFCI}}_l$ is estimated as follows:

$${\widehat{FFCI}}_l=\left[q_{\alpha /2}\left({\widehat{\mathrm{SD}}}_{x_1^{*}}\left({\widehat{FF}}_l\right),\dots ,{\widehat{\mathrm{SD}}}_{x_M^{*}}\left({\widehat{FF}}_l\right)\right),q_{1-\alpha /2}\left({\widehat{\mathrm{SD}}}_{x_1^{*}}\left({\widehat{FF}}_l\right),\dots ,{\widehat{\mathrm{SD}}}_{x_M^{*}}({\widehat{FF}}_l\right)\right],x_i^{*}\in {\mathcal{A}}_0,$$

(18)

where $q_{\alpha /2}(·)$ is the empirical quantile of order $\alpha /2$ with $\alpha =0.01$. The confidence intervals ${\widehat{FFCI}}_l$ are used to assess which of the $\widehat{FF}$ are informative. For this purpose, a statistical test was defined that involved the verification of two criteria. For a signal z, the statistic ${\mathcal{T}}_z$ and the test ${\phi }_z$ are defined as follows:

$${\mathcal{T}}_z\left(l\right):={\widehat{\mathrm{SD}}}_z\left({\widehat{FF}}_l\right),{\phi }_z\left(l\right)=\left\{\begin{array}{cc}1,\hfill & {\mathcal{T}}_z\left(l\right)\notin {\widehat{FFCI}}_l,\hfill \\ 0,\hfill & {\mathcal{T}}_z\left(l\right)\in {\widehat{FFCI}}_l,\hfill \end{array}\right.$$

(19)

with the following testing hypotheses:

$$H_{0,l}:{\mathcal{T}}_z\left(l\right)\in {\widehat{FFCI}}_l\left(\mathrm{healthy}\right),H_{1,l}:{\mathcal{T}}_z\left(l\right)\notin {\widehat{FFCI}}_l\left(\mathrm{faulty}\right).$$

(20)

For healthy signals $x_i\in {\mathcal{A}}_1$ and faulty signals $y_i\in {\mathcal{B}}_1$, where $i=1,\dots ,M$ the empirical size and power of the test [37,38] denoted ${\phi }_z$ at $F{F}_l$ respectively are defined as follows:

$$\mathrm{Criteria}1:{\widehat{\kappa }}_{{\phi }_x}\left(l\right)={\displaystyle \frac{1}{M}}\sum _{i=1}^M{\phi }_{x_i}\left(l\right),\mathrm{Criteria}2:{\widehat{\pi }}_{{\phi }_y}\left(l\right)={\displaystyle \frac{1}{M}}\sum _{i=1}^M{\phi }_{y_i}\left(l\right),l=1,\dots ,7.$$

(21)

The selected informative set of fault frequency indexes is defined as follows:

$${\mathcal{L}}_{FF}:=\left\{l\in \{1,\dots ,7\}:{\widehat{\kappa }}_{{\phi }_x}\left(l\right)\le 0.01\wedge {\widehat{\pi }}_{{\phi }_y}\left(l\right)\ge 0.99\right\}.$$

(22)

This means that the informative frequencies are those for which, under healthy conditions, the corresponding amplitudes fall within the confidence interval ${\widehat{FFCI}}_l$ with a Type I error probability not exceeding $1\%$, and under faulty conditions, they fall outside this interval with a confidence level of $99\%$.

2.4.2. Health Indicator and Confidence Interval Construction

For a given ${\mathcal{L}}_{FF}$, the spectrum-based health indicator, HI, for signal z is defined as the sum of spectrum amplitudes at the informative ${\widehat{FF}}_l$, i.e.:

$${\widehat{HI}}_z=\sum _{l\in {\mathcal{L}}_{FF}}{\widehat{\mathrm{SD}}}_z\left(F{F}_l\right).$$

(23)

For healthy signals $x_i^{*}\in {\mathcal{A}}_0$, $i=1,\dots ,M$, a confidence interval of the indicator $\widehat{HI}$ is estimated as follows:

$$\widehat{HICI}=\left[q_{\alpha /2}({\widehat{HI}}_{x_1^{*}},\dots ,{\widehat{HI}}_{x_M^{*}}),q_{1-\alpha /2}({\widehat{HI}}_{x_1^{*}},\dots ,{\widehat{HI}}_{x_M^{*}})\right],x_i^{*}\in {\mathcal{A}}_0,\alpha =0.01.$$

(24)

The $\widehat{HICI}$ serves as a quantitative reference for subsequent fault detection and decision-making. It reflects the deviation of the current signal from the healthy reference state, enabling the assessment of the machine’s operational health and facilitating the identification of emerging faults with statistical confidence.

The algorithm for feature-based health modeling is summarized in Algorithm 2.

Algorithm 2: Construction of the health indicator confidence interval (for each current phase, fault type, and load)

Data: Current-signal datasets: healthy ${\mathcal{A}}_0,{\mathcal{A}}_1$ and faulty ${\mathcal{B}}_1$ Result: The confidence interval $\widehat{HICI}$ of healthy dataset Feature-based health modeling: 1. Import signals from ${\mathcal{A}}_0$, ${\mathcal{A}}_1$, and ${\mathcal{B}}_1$. Feature selection 2. For each $x_i^{*}\in {\mathcal{A}}_0$, $x_i\in {\mathcal{A}}_1$, and $y_i\in {\mathcal{B}}_1$, compute the spectrum and estimate ${\widehat{FF}}_l$ using Equation (10). 3. Determine confidence intervals ${\widehat{FFCI}}_l$ from dataset ${\mathcal{A}}_0$, using Equation (18). 4. Compute empirical size and test power: ${\widehat{\kappa }}_{{\phi }_x}\left(l\right)$ and ${\widehat{\pi }}_{{\phi }_y}\left(l\right)$, using Equation (21). 5. Select informative ${\widehat{FF}}_l$, i.e., ${\mathcal{L}}_{FF}$ according to Equation (22). Confidence interval construction 6. Compute the health indicator ${\widehat{HI}}_{x_i^{*}}$ from signals $x_i^{*}\in {\mathcal{A}}_0$ using the selected $\widehat{FF}$ Equation (23). 7. Establish the indicator confidence interval $\widehat{HICI}$ from healthy dataset ${\mathcal{A}}_0$ using Equation (24).

2.5. Validation Procedure

For a signal z with the HI-based statistic ${\widehat{HI}}_z$, the test ${\varphi }_z\left(HI\right)$ is defined as follows:

$${\varphi }_z\left(\mathrm{HI}\right)=\left\{\begin{array}{cc}1,\hfill & {\widehat{HI}}_z\notin \widehat{HICI},\hfill \\ 0,\hfill & {\widehat{HI}}_z\in \widehat{HICI},\hfill \end{array}\right.$$

(25)

with the following testing hypotheses:

$$H_0:{\widehat{HI}}_z\in \widehat{HICI}\left(\mathrm{healthy}\right)\mathrm{vs}.H_1:{\widehat{HI}}_z\notin \widehat{HICI}\left(\mathrm{faulty}\right).$$

(26)

For the signals $s_i\in {\mathcal{A}}_2$ (healthy) and $u_i\in {\mathcal{B}}_2$ (faulty), where $i=1,\dots ,M$, the indicator values ${\widehat{HI}}_{s_i}$ and ${\widehat{HI}}_{u_i}$ are computed using Equation (23), and each signal is classified according to the HI-based decision rule defined in Equation (25). The count of correct decisions (CCD) is then defined as

$${\widehat{\mathrm{CCD}}}_H=\sum _{s_i\in {\mathcal{A}}_2}{\varphi }_{s_i}\left(\mathrm{HI}\right)\left(\mathrm{healthy}\right),{\widehat{\mathrm{CCD}}}_F=\sum _{u_i\in {\mathcal{B}}_2}{\varphi }_{u_i}\left(\mathrm{HI}\right)\left(\mathrm{faulty}\right),$$

(27)

where ${\varphi }_z\left(\mathrm{HI}\right)$ is the decision function defined in Equation (25), returning 1 for faulty signals and 0 for healthy ones.

The diagnostic performance is evaluated using the fault detection rates (FDRs):

$${\mathrm{FDR}}_{\mathrm{H}}={\displaystyle \frac{{\widehat{\mathrm{CCD}}}_H}{M}}\left(\mathrm{healthy}\right),{\mathrm{FDR}}_{\mathrm{F}}={\displaystyle \frac{{\widehat{\mathrm{CCD}}}_F}{M}}\left(\mathrm{faulty}\right).$$

(28)

A higher ${\mathrm{FDR}}_{\mathrm{F}}$ value (closer to 1) indicates superior fault detection performance; ${\mathrm{FDR}}_{\mathrm{F}}=1$ means that 100% of faulty signals are correctly identified as faulty. Conversely, a lower ${\mathrm{FDR}}_{\mathrm{H}}$ value (closer to 0) reflects better acceptance of healthy signals; ${\mathrm{FDR}}_{\mathrm{H}}=0$ implies that none of the healthy signals were incorrectly classified as faulty, i.e., 100% of healthy signals were correctly identified as healthy. The desired diagnostic property is to achieve ${\mathrm{FDR}}_{\mathrm{H}}$ close to 0 and ${\mathrm{FDR}}_{\mathrm{F}}$ close to 1. The algorithm of the validation procedure is summarized in Algorithm 3.

Algorithm 3: Validation procedure (for each current phase, fault type and load)

Data: Current-signal datasets: healthy ${\mathcal{A}}_2\in \mathcal{A}$ and faulty ${\mathcal{B}}_2\in \mathcal{B}$;         The confidence interval health indicator of healthy dataset: $\widehat{HICI}$ Result: Classification of the tested signal as healthy or faulty Validation procedure: 1. Import signals from ${\mathcal{A}}_2$ and ${\mathcal{B}}_2$. 2. For each $s_i\in {\mathcal{A}}_2$, $u_i\in {\mathcal{B}}_2$, compute the spectrum and estimate informative ${\widehat{FF}}_l,l\in {\mathcal{L}}_{FF}$ using Equation (10). 3. For each signal z, compute $\widehat{HI}\left(z\right)$ using Equation (23). 4. Apply the HI-based decision rule Equation (25): If ${\widehat{HI}}_z\in \widehat{HICI}$: classify as healthy. If ${\widehat{HI}}_z\notin \widehat{HICI}$: classify as faulty. 5. Calculate ${\mathrm{FDR}}_{\mathrm{H}}$ and ${\mathrm{FDR}}_{\mathrm{F}}$ using Equation (28) and report the overall diagnostic performance.

Figure 2 depicts the schematic block flowchart of the proposed methodology, covering dataset construction (Stage 1), feature extraction and selection (Stage 2.1), health-indicator and confidence intervals calculation (Stage 2.2), and the final decision and validation procedure (Stage 3). The figure provides a compact overview that is referenced throughout the section.

3. Experiments

Experimental research aimed at detecting and analyzing inter-turn short circuits in the stator winding of an induction motor was conducted on a specially designed laboratory stand. This chapter details the stand’s configuration, the measuring components used, and the adopted methodology for fault modelling.

3.1. Research Stand and Experimental Methodology

The research stand consisted of two fundamental subsystems: a drive system with the motor under test and a data acquisition and measurement system (see Figure 3). The main component was an Sh90L-4 squirrel-cage induction motor (Cantoni Group, Cieszyn, Poland) with a rated power of 1.5 kW, which served as the test object. This motor was powered via a frequency converter operating in scalar control mode (U/f = const) with an open speed loop. The output frequency range was set from 10 to 50 Hz to cover typical industrial variable-speed drive scenarios, from light-load (10 Hz) to rated-speed (50 Hz @ 1500 rpm) operation.

The load torque was generated in a controlled manner by a second drive unit, which utilized a Permanent Magnet Synchronous Motor (PMSM). This configuration, with the PMSM acting as a load motor, allowed for the precise application of a variable mechanical torque to the shaft of the motor under test, enabling the observation of its behaviour across different operating points.

3.2. Modelling Stator Winding Faults

A crucial aspect of the stand was the physical capability to simulate inter-turn short circuits. For this purpose, the tested Sh90L-4 motor was specially modified. A series of taps were brought out from the original stator winding, corresponding to a specific number of turns in each phase (in the range from 0 to 10 turns). These taps were connected to an external terminal board, as illustrated in Figure 4.

A short circuit between selected turns was realized by directly connecting (shorting) the corresponding terminals on this board (Figure 5). It is important to emphasize that no additional current-limiting resistance was introduced into the short-circuit loop. This approach enabled the replication of conditions similar to a real, sudden fault, characterized by significant currents flowing through the shorted loop.

Following this approach, we explicitly define five fault-severity levels by the number of shorted turns ($N_{sh}$), which we use as an unambiguous measure of severity (independent of operating-point-dependent short-circuit current). The levels are summarized in

Table 3. Definition of fault-severity levels by the number of shorted turns.

Level	${\mathit{N}}_{\mathbf{sh}}$
Fault 1	1
Fault 2	2
Fault 3	3
Fault 4	4
Fault 5	5

.

From a diagnostic perspective, fault 1 is the least severe in terms of operational disruption (the motor can still run), but it is the most challenging to detect due to the weak signal signature. However, early detection at this stage is critical because the high circulating current in the shorted loop creates a localized hotspot. Without intervention, this leads to rapid insulation degradation, propagating the fault to more turns and eventually causing a catastrophic phase-to-ground or phase-to-phase failure.

3.3. Data Acquisition System

The measurement of current signals was fundamental to the diagnostic process. For recording the three-phase stator currents, three LEM LA 25-NP current transducers (LEM International SA, Meyrin, Switzerland) were used. The main metrological parameters of these transducers are summarized in

Table 4. Key parameters of LEM LA 25-NP.

Parameter	Value
Measurement range [A]	$\pm 36$
Bandwidth [kHz]	[DC, 150]
Power supply voltage [V]	$\pm 15$
Accuracy [%]	$\pm 0.5$
Ambient operating temperature [°C]	$[-40,+85]$

.

The voltage signals from the outputs of the LEM transducers were fed to a high-quality National Instruments NI PXIe-4492 data acquisition card, (National Instruments NI, Austin, TX, USA) mounted in a PXI system. This card, with key parameters presented in

Table 5. Key parameters of NI PXIe 4492.

Parameter	Value
A/D converter (ADC) resolution	24
Sample rates range	100 S/s to 204.8 kS/s
Analog Input Coupling	AC/DC
High-Pass Filter Cutoff Frequency	0.5 Hz

, ensured signal recording with very high resolution.

The entire measurement process, including card control, data collection, preliminary processing, and visualization, was managed by a proprietary application developed in the LabVIEW environment.

4. Results

In this section, the analyzed signals are presented. The signals correspond to different fault stages, including the healthy condition and the most severe fault (fault 5). The datasets $\mathcal{A}$ and $\mathcal{B}$ are independent (the datasets come from the same machine but from different experiments run at different times). Spectra are computed using a Hann-windowed periodogram with fixed parameters, and stability is verified by confirming that spectral estimates and their confidence intervals do not vary systematically across adjacent windows. The measurements were conducted under different levels of load conditions (from level 0 to 6), with the load gradually increasing over time. The recorded signals (with each of the three phase currents overlapping) are shown in Figure 6 with the zoomed in view presented in Figure 7. There is a visible increase in the signal amplitude for fault 2–fault 5, corresponding to the increasing load over time for load levels greater than 2. The load profiles estimated from the signals are illustrated in Figure 8, with the zoomed in view presented in Figure 9. Estimated load profiles exhibit slight variations between fault categories, resulting from fault-induced changes in current amplitude and waveform distortion. For higher fault severities (faults 2–5), phase divergence becomes more pronounced due to increased stator current asymmetry. The load level was estimated from the RMS value of the stator current.

4.1. Dataset Construction for Analyzed Current Data

For each load level, 5 s current signals were extracted for further analysis (highlighted in the green shaded areas in Figure 6, Figure 7 and Figure 8). The sampling frequency of the measurements was 50 kHz. Considering the requirements of the intended industrial application, the maximum signal length was limited to 0.2 s (10,000 samples). In the spectral analysis, the FFT length was set to 8192 points, which resulted in a frequency resolution of approximately 6 Hz. Each 5 s signal (for every load and current phase) was further divided into 0.2 s segments to create equal-sized datasets: ${\mathcal{A}}_0$, ${\mathcal{A}}_1$, and ${\mathcal{A}}_2$ for healthy signals, and ${\mathcal{B}}_1$ and ${\mathcal{B}}_2$ for faulty signals. Each dataset contained 100 segments that were used for statistical testing. The indices of the selected signals in each dataset were determined using a pseudo-random number generator that produces a deterministic sequence of numbers that appear random. For example, set ${\mathcal{A}}_0$ contains indices starting with $[73,229,92,220,224,180,\dots ]$, whereas set ${\mathcal{A}}_1$ starts with $[74,93,96,44,211,153,\dots ]$. This segmentation procedure ensures a balanced and statistically representative dataset for evaluating the proposed method.

4.2. Partial Results—Feature Selection Based on Statistical Testing

In this section, the results of the proposed methodology are presented for the data corresponding to the maximum load applied during the experiment (load 6). As discussed in Section 2.2, fault detection under very light loads can be challenging. Therefore, load 6 was selected as the most representative operating condition for demonstrating the analysis results. The following subsections present the obtained health indicators, confidence intervals, and diagnostic performance measures.

Exemplary current signals for healthy and faulty conditions are presented in Figure 10. Each subplot contains six different signals, although some of them overlap and are therefore not fully visible. A slight phase shift between the signals can be observed when comparing healthy and faulty cases. There is also a small difference in the maximum amplitudes among the current types (IsA, IsB, and IsC), but it is difficult to distinguish visually.

In Figure 11, the signal spectra are presented. The 0.2 s signal duration yields a frequency resolution of approximately 6 Hz (with 8192-point FFT), which constrains the separation of closely spaced harmonics. Despite this limitation, the proposed statistical framework successfully identifies informative fault frequencies.

The fault frequencies, denoted as ${\widehat{FF}}_l$, are identified as local maxima within the theoretical fault frequency range and a window of $\pm 2$ samples (according to Equation (10)). This range is indicated by the gray shaded area.

As shown in Figure 11, certain fault frequencies, such as $F{F}_4$ (559.7 Hz) and $F{F}_5$ (659.7 Hz), exhibit noticeable variations in spectral amplitude as the damage level increases. To verify this observation, the amplitudes of the spectrum at $F{F}_l$ are analyzed. The corresponding ${\widehat{FF}}_l$ values are summarized in

Table 6. Exemplary empirical values of $F{F}_l$ for IsA, fault 5, and load 6.

${\widehat{\mathit{FF}}}_{\mathit{l}}$	Value [Hz]
${\widehat{FF}}_1$	152.6
${\widehat{FF}}_2$	347.9
${\widehat{FF}}_3$	463.9
${\widehat{FF}}_4$	561.5
${\widehat{FF}}_5$	659.2
${\widehat{FF}}_6$	762.9
${\widehat{FF}}_7$	860.6

.

Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17, Figure 18, Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24 and Figure 25 present the results of testing the ${\widehat{FF}}_l$ amplitudes for each fault stage and healthy data.

Spectrum amplitudes at the fault frequencies ${\widehat{FF}}_l$, obtained from the healthy dataset ${\mathcal{A}}_0$ (shown as blue dots), are used to construct the $99\%$ confidence intervals ${\widehat{FFCI}}_l$ (blue shaded areas) and to perform the corresponding hypothesis testing. Figure 12, Figure 13, Figure 14, Figure 15, Figure 16, Figure 17 and Figure 18 present the testing results for the healthy data ${\widehat{FF}}_l\in {\mathcal{A}}_1$ (red dots). Each figure includes results for the three current phases—IsA, IsB, and IsC—displayed in separate subplots, with the titles indicating the empirical size ${\widehat{\kappa }}_{{\phi }_x}$ of the test. The desired outcome is ${\widehat{\kappa }}_{{\phi }_x}\le 0.01$. As observed, four frequencies $l\in \mathcal{L}$ do not meet this criterion, with the following results: ${\widehat{\kappa }}_{{\phi }_x\left(2\right)}=0.04$ (IsB, IsC), ${\widehat{\kappa }}_{{\phi }_x\left(3\right)}=0.02$ (IsA), ${\widehat{\kappa }}_{{\phi }_x\left(4\right)}=0.02$ (IsA), and ${\widehat{\kappa }}_{{\phi }_x\left(7\right)}=0.02$ (IsB). Consequently, $F{F}_2$ for IsB and IsC, $F{F}_3$ for IsA, $F{F}_4$ for IsA, and $F{F}_7$ for IsB are excluded from the set of informative fault frequencies, as they do not satisfy the expected result of Criterion 1.

Figure 19, Figure 20, Figure 21, Figure 22, Figure 23, Figure 24 and Figure 25 present the results of testing the faulty data ${\widehat{FF}}_l\in {\mathcal{B}}_1$ (red dots). Each figure shows the results for individual current phases—IsA, IsB, and IsC—in separate subplots, with titles indicating the test power ${\widehat{\pi }}_{{\phi }_y}$, for which the desired outcome is ${\widehat{\pi }}_{{\phi }_y}\ge 0.99$.

For ${\widehat{FF}}_5$, the value of ${\widehat{\pi }}_{{\phi }_y}=1$ is achieved for all fault stages and current phases. In contrast, for ${\widehat{FF}}_7$, none of the analyzed cases reaches the expected value. Consequently, $F{F}_7$ is excluded from the set of informative fault frequencies, as it does not satisfy the expected result of Criterion 2.

Among all analyzed frequencies, $F{F}_4$ and $F{F}_5$ most consistently meet the testing criteria, indicating a strong sensitivity of their spectral components to fault-related variations in the current signal.

The summarized testing results are presented in Figure 26 and Figure 27. Figure 26 presents the aggregate results for the healthy-case test (test on healthy data) across loads and fault frequency indexes for all currents. Figure 27 presents the aggregate results for the faulty-case test (test on faulty data) across loads and fault frequency indexes for all currents.

In

Table 7. Informative set ${\mathcal{L}}_{FF}$ of fault frequency indexes for data with different loads (datasets A_1 and B_1).

	Current A	Current B	Current C
Load 0	5	4, 5	5
Load 1	none	4	4
Load 2	4	4	4, 5
Load 3	4	4, 5	none
Load 4	4, 5	none	5
Load 5	4	5	5
Load 6	5	4, 5	5

, the summarized results of feature selection based on statistical testing are presented for all load levels and current phases.

Since, for some analyzed cases, it was not possible to fully satisfy the statistical test assumptions and extract an informative set ${\mathcal{L}}_{FF}$ of fault frequency indexes (as even a minor deviation in the test power or the empirical size exceeding, e.g., 0.005 leads to rejection), to ensure consistency and robustness, the informative set ${\mathcal{L}}_{FF}$ of fault frequency indexes identified at maximum load (load 6) for each current separately is adopted across all load levels. This choice is justified by the superior statistical separability between healthy and faulty conditions observed at higher loads.

Therefore, without a significant loss of generality, the set of fault frequency indexes identified as informative for load 6 is adopted for all load levels, namely, for IsA—${\mathcal{L}}_{FF}=5$, for IsB—${\mathcal{L}}_{FF}=\{4,5\}$, and for IsC—${\mathcal{L}}_{FF}=5$.

4.3. Partial Results—Health Indicator and Confidence Interval Construction

The informative fault frequencies (IsA—$F{F}_5$, IsB—$F{F}_4$, $F{F}_5$, and IsC—$F{F}_5$) are used to calculate the $\widehat{HI}$ values. The 99% confidence intervals $\widehat{HICI}$, computed based on the healthy dataset ${\mathcal{A}}_0$ for each current phase, are represented by the blue shaded areas and are used in the decision-making process.

Figure 28 presents the results of testing the healthy dataset ${\mathcal{A}}_2$ (red dots), with the corresponding fault detection rate ${\widehat{FDR}}_H$ values indicated in the subplot titles. The expected outcome is ${\widehat{FDR}}_H=0$, meaning that all healthy data samples are correctly classified as healthy. Each figure shows the results for the individual current phases—IsA, IsB, and IsC—in separate subplots.

As observed, the ${\widehat{FDR}}_H$ values for all current phases remain close to zero, confirming the high reliability of the proposed approach in correctly identifying healthy operating conditions.

Figure 29 presents the results of testing the faulty dataset ${\mathcal{B}}_2$ (red dots), with the corresponding fault detection rate ${\widehat{FDR}}_F$ values indicated in the subplot titles. The expected outcome is ${\widehat{FDR}}_F=1$, which means that all $\widehat{HI}$ values corresponding to faulty data are correctly classified as faulty. Each figure shows the results for the individual current phases—IsA, IsB, and IsC—in separate subplots.

As observed, the ${\widehat{FDR}}_F$ values for all current phases are equal to 1, confirming the full detection of faulty conditions by the proposed method.

4.4. Validation Procedure for Analyzed Current Data

Figure 30 presents the summarized results of the validation procedure for all load conditions, current phases, and fault categories, using the testing datasets ${\mathcal{A}}_2$ and ${\mathcal{B}}_2$. As observed, the fault detection effectiveness for the faulty datasets reaches 100% across all analyzed cases. For the healthy datasets, the effectiveness remains nearly 100%, with ${\widehat{FDR}}_H$ values close to zero, confirming the high reliability of the proposed diagnostic methodology.

The data from the same test rig, acquired during a separate experimental session at load 0 (the most challenging case for diagnosis), are used for testing (10 signals of 0.2 s). The exemplary signals of currents A, B, and C for healthy and faulty data are presented in Figure 31. The corresponding spectra are presented in Figure 32. These separate experiments ensure that the thermal conditions, background noise, and mechanical transients are uncorrelated with the training data. The proposed statistical method was applied to this new dataset without re-training the baseline parameters. The results of health indicator testing are presented in Figure 33, Figure 34 and Figure 35.

The summarized results obtained for this independent validation scenario are presented in Figure 36 (tested healthy data) and Figure 37 (tested faulty data). This additional validation confirms that the high accuracy reported is not a result of data leakage but demonstrates the method’s capability to generalize across different operating cycles.

4.5. Influence of the Significance Level on the HI Outcome and Detection Performance

In the proposed methodology, the significance level $\alpha$ governs the strictness of the decision rule through the width of the two-sided acceptance band defined by the $(1-\alpha )$ confidence interval (CI). Decreasing $\alpha$ widens the CI (i.e., produces a more conservative acceptance band), which reduces the likelihood of spurious detections on healthy signals. Conversely, increasing $\alpha$ leads to narrower CIs and therefore a tighter acceptance band, enhancing sensitivity to deviations while potentially increasing the false-alarm rate in healthy data.

This behavior is consistent with the nominal size of a two-sided test: for a $(1-\alpha )$ confidence band, approximately $\alpha$ of healthy observations are expected to fall outside the band, with about $\alpha /2$ in each tail. As a result, a larger $\alpha$ naturally tends to increase the false discovery rate computed on healthy data, denoted as ${\mathrm{FDR}}_H$.

Figure 38 empirically confirms this relationship by showing an almost linear trend: ${\mathrm{FDR}}_H$ increases with $\alpha$ over the considered range. To evaluate whether this effect is consistent across operating conditions, Figure 39 summarizes ${\mathrm{FDR}}_H$ for all tested loads and currents. The results indicate that the increase in ${\mathrm{FDR}}_H$ with $\alpha$ is systematic across the full set of conditions, which further supports using a small $\alpha$ when false-alarm control on healthy signals is prioritized.

At the same time, selecting a too small $\alpha$ may reduce statistical power (increase Type II errors), potentially delaying fault detection. Importantly, this trade-off does not manifest as a loss of fault detection effectiveness in our experiments. Figure 40 aggregates ${\mathrm{FDR}}_F$ (computed on faulty data) across all loads and currents and shows no degradation of performance for the tested values of $\alpha$.

Therefore, while $\alpha$ primarily affects ${\mathrm{FDR}}_H$ through CI tightening/loosening, the fault-related metric ${\mathrm{FDR}}_F$ remains stable, indicating no loss in efficiency.

Based on these observations, we adopt $\alpha =0.01$ as a deliberate compromise that emphasizes robustness against false alarms while preserving fault detection performance across the full range of operating conditions.

The average processing time per 0.2 s window for the entire online pipeline (three phases, and all fault/healthy states) was 13.43 ms, i.e., about $6.7\%$ of the 200 ms window duration, leaving a $93.3$ timing margin. This confirms real-time feasibility of the online stage.

5. Conclusions

A feature-based methodology for fault detection in induction motor systems has been presented, utilizing three-phase stator current signals. In the proposed approach, frequency-domain analysis was combined with statistical testing to identify informative fault-related spectral components, construct confidence intervals, and develop a quantitative health indicator HI along with its confidence range HICI. Most existing MCSA/envelope-based approaches rely on very similar frequency-domain indicators (e.g., amplitudes at fault-related sidebands). The difference lies in how the final decision is made from these features. In standard MCSA/envelope practice, fault detection typically depends on selecting a fixed threshold, which is often chosen heuristically, tuned separately for different operating conditions, or requires expert interpretation. As a result, the decision boundary may be difficult to reproduce, and the achieved false-alarm rate is not explicitly controlled, especially when load or current conditions vary.

The proposed methodology replaces heuristic thresholding with an automated, data-driven statistical test. Empirical confidence bands are estimated from healthy data offline, and online detection reduces to verifying whether the indicator falls outside the $(1-\alpha )$ acceptance band. This yields explicit and interpretable control of the nominal false-alarm rate via $\alpha$ and provides a consistent, automatically constructed decision boundary without manual threshold selection.

It was demonstrated that the informative fault frequencies—$F{F}_5$ for IsA, $F{F}_4$ and $F{F}_5$ for IsB, and $F{F}_5$ for IsC—exhibit the highest sensitivity to fault-induced variations in the current spectrum. Through validation under multiple load conditions, it was confirmed that using data from load 6 as a reference provides robust generalization and high diagnostic accuracy across all operating conditions. For the faulty datasets, the fault detection rate ${\widehat{FDR}}_F$ was found to reach $100\%$ in all analyzed cases, whereas for the healthy datasets, ${\widehat{FDR}}_H$ values remained close to zero, indicating the absence of false alarms. These results demonstrate that the proposed health indicator and statistical testing framework offer high reliability, well-defined decision boundaries, and robustness to load variations.

Although the experimental validation was conducted on an Sh90L-4 motor, the proposed method is adaptable to induction motors with different structural parameters. Since the fault-related frequencies (such as RSH) are determined analytically based on the number of rotor bars ($N_r$) and pole pairs ($p_p$), the algorithm can be reconfigured for any motor by updating these parameters. The statistical main of the method operates independently of the specific frequency location, making it applicable to a wide range of AC machinery, provided that the characteristic frequencies do not overlap significantly with the fundamental supply frequency or its low-order harmonics.

Obtaining a guaranteed “perfectly healthy” baseline is a common challenge in industrial condition monitoring. The standard procedure is to perform the healthy state acquisition (dataset ${\mathcal{A}}_0$) immediately after motor installation or maintenance. This ensures that the baseline reflects the optimal machine state. In a bad scenario, when the system is installed on a motor that is already in operation (with unknown health status), the current state is adopted as the baseline ${\mathcal{A}}_0$. The proposed method functions as an anomaly detector. It detects statistically significant deviations from the reference state, rather than measuring absolute health against a theoretical ideal. A key advantage of our statistical approach is its adaptability. If the “baseline” motor already exhibits slight wear or noise, the variance in the spectral components in the dataset ${\mathcal{A}}_0$ will be naturally higher. Consequently, the calculated confidence interval automatically widens. This mechanism naturally desensitizes the health indicator to the pre-existing condition, ensuring that the system detects only significant future degradation (trend changes) rather than triggering false positives based on the initial state.

The proposed methodology is particularly tailored for two specific industrial scenarios where traditional MCSA is often inapplicable due to signal duration requirements. The first scenario involves automated manufacturing and robotics (e.g., pick-and-place operations), where motors operate under short duty cycles with rapid speed changes. In such regimes, acquiring a continuous 1 s steady-state signal is unfeasible, whereas the proposed method effectively utilizes brief 0.2 s stable windows for reliable diagnosis. The second scenario concerns Edge AI and IoT monitoring systems implemented on low-cost microcontrollers. The capability to achieve high diagnostic accuracy using short data buffers minimizes memory usage and computational latency, enabling cost-effective, real-time condition monitoring on embedded platforms.

In conclusion, the developed methodology can be regarded as an interpretable and statistically grounded framework for fault detection in electric machines. Future work will focus on extending the method to enable fault classification and validating the approach in real industrial environments with time-varying load profiles.