Enhanced broken-bar fault diagnosis in asynchronous motors using SVM classification

1. Introduction

In industrial settings, preventive maintenance is critical for anticipating equipment failures and mitigating unplanned downtime. By enabling early detection of incipient faults, it supports timely interventions that reduce operational disruptions, lower maintenance costs, and enhance both productivity and safety [1, 2]. Furthermore, such proactive strategies ensure compliance with quality and regulatory standards while significantly extending the service life of key machinery [3, 4].

Asynchronous Motors are the workhorses of modern industrial systems, widely deployed in production lines, machine tools, conveyors, pumps, and ventilation systems owing to their robustness, low cost, and high starting torque [5, 6]. Their capability for variable-speed operation and near maintenance-free service further consolidates their dominance as the most prevalent motor type in industry [6-8]. Nevertheless, despite their reliability, these machines are prone to critical faults, particularly in the rotor, whose early detection remains a key challenge for predictive maintenance.

Among the most critical faults affecting Asynchronous Motors are rotor-related defects, particularly broken rotor bars, which can lead to increased vibration, torque oscillations, overheating, and eventually catastrophic failure if left undetected [9]. Due to the difficulty of directly monitoring the rotor in operation, model-based approaches have emerged as promising alternative, relying on mathematical representations to capture fault signatures in measurable stator quantities. For instance, Amrane et al. [9] proposed a fault detection and isolation scheme using nonlinear analytical redundancy, demonstrating the potential of model-driven strategies to enhance reliability and support preventive maintenance. Nevertheless, such methods like other high-gain or adaptive observers used in electromechanical systems [10] remain sensitive to model inaccuracies, unmolded dynamics (friction, saturation), and parametric uncertainties, often requiring complex compensation mechanisms. This limitation motivates the integration of high-fidelity simulation with robust, geometry-aware classification techniques, as pursued in this work.

Recent advances in fault diagnosis have leveraged sophisticated signal processing and machine learning techniques to improve detection accuracy. For instance, time frequency decomposition methods such as Variation Mode Decomposition (VMD) [11] and Enhanced Reduced-order Symptom Recognition (ERSR) [12] have shown promise in enhancing spectral clarity and reducing data dimensionality, primarily for bearing and imbalance faults. Similarly, deep learning approaches, including CNN based classifiers trained on 2D vibration images [13] or hybrid schemes combining Bessel transform and neural networks [14], have achieved high performance in gear bearing compound fault scenarios. Even in the electromagnetic domain, wavelet-CNN architectures have been explored for modulation recognition in RF signals [15], underscoring the trend toward computationally intensive, data-hungry models. Ensemble methods and stochastic resonance techniques [16] further extend the toolbox for vibration-based monitoring of rolling elements, and unsupervised clustering such as K-means for failure mode classification [17] further extend the toolbox for vibration-based monitoring of rolling elements.

However, most existing methods are validated on bearing or gear datasets [12]. They rely heavily on experimental data, which is often unavailable for rotor faults like broken bars. Moreover, their complexity may hinder interpretability and real-time deployment in industrial settings. Given that broken-bar faults predominantly exhibit electrical signatures (notably in stator current) rather than detectable vibration patterns, a diagnosis framework must prioritize physical interpretability, data efficiency, and robustness.

Recent studies have further advanced fault signature extraction and physical modeling for condition monitoring. Improved mode decomposition methods have been proposed to enhance vibration signal denoising while preserving fault-sensitive features in rotating components such as aero-engine bearings [18]. In parallel, multi-physics vibro-acoustic models enable the prediction and evaluation of electromagnetic vibration and noise in synchronous machines, providing deeper insight into how structural or electromagnetic faults manifest in measurable signals [19, 20]. While these contributions primarily address vibration or acoustic domains, they reinforce the relevance of exploiting physically measurable quantities - such as torque and speed - as robust indicators for fault diagnosis.

Beyond signal-based methods, the diagnostic landscape has been enriched by broader strategies such as data-driven modeling [21], ensemble learning [22], and Digital Twin frameworks [23], all of which extend the capabilities of modern condition monitoring systems. While thermal analysis can provide supplementary insights [24], and feature extraction from vibration remains effective for bearing-related faults [25], their applicability to broken-bar faults is limited: thermal response is slow, vibration signatures are often negligible, and rotor inaccessibility complicates real-time digital replication. As underlined in methodological studies [26], an effective diagnosis must align with the physical nature of the fault and data availability; this requirement motivates a focused approach that exploits easily measurable electrical quantities (stator current harmonics) in combination with high-fidelity simulation.

In this context, Support Vector Machines (SVM) offer a compelling alternative: as a supervised learning algorithm, SVM constructs optimal decision boundaries by maximizing the margin between classes, thereby ensuring high generalization capability, even with limited training data [27]. This characteristic is particularly valuable for fault diagnosis in electrical machines, where experimental fault data is scarce and costly to acquire. Recent studies have successfully applied SVM to entropy-enhanced fault diagnosis; Chen et al. [28] demonstrated that composite multiscale fuzzy slope entropy (CMFSE), when paired with SVM, achieves excellent classification performance on the CWRU bearing dataset, complementary approaches, such as hybrid term extraction for structuring domain knowledge in condition monitoring [29], further support the interpretability and documentation of diagnostic systems. Moreover, SVM supports both binary classification (healthy and faulty) and multi-class extensions (one-vs-rest, one-vs-one) for fault severity assessment, making it well-suited to the progressive nature of broken-bar faults. Unlike deep learning models, SVM provides a transparent decision mechanism, allowing for physical interpretation of classification boundaries and facilitating validation against model-based expectations [30].

Building on this rationale, this paper proposes an enhanced broken-bar fault diagnosis framework for Asynchronous Motors, based on a tightly coupled simulation–SVM pipeline. First, a high-fidelity dynamic model is developed: the healthy multi-winding Asynchronous Motors is extended to emulate broken rotor bars by introducing fault specific additive resistances, enabling realistic simulation of 0, 1, 2, or 3 broken bars under varying load and speed conditions. Second, the resulting electrical signatures (stator currents, speed ripple) are processed into discriminative feature vectors, forming a labeled dataset that eliminates the dependency on scarce experimental fault data. Finally, a supervised SVM classifier is implemented in two stages: first, binary classification to distinguish healthy from faulty states, and second, multi-class classification (using one-vs-one strategy) to assess fault severity, thereby delivering not only detection but also quantification of the defect.

To achieve these objectives, the manuscript is structured as follows: Section 2 gives technical justifications and advantages associated with the methodological choices of the proposed approach, Section 3 clarifies the foundations of Support Vector Machines for fault diagnosis; Section 4 details the dynamic modeling of the Asynchronous Motors and the simulation protocol employed to generate the training dataset; Section 5 then presents the SVM implementation and evaluates its diagnostic performance in both binary (healthy vs. faulty) and multi-class (four fault modes) scenarios; finally, Section 6 synthesizes the key findings and outlines future research directions.

2. Research significance

Although electrical machine diagnosis has evolved towards Industry 4.0 [31], critical gaps persist in the technical literature. Conventional harmonic analysis techniques fail to provide a comprehensive assessment in the presence of multiple harmonics or external noise, rendering fault segregation nearly impossible [31]. Furthermore, an inherent trade-off exists between algorithm simplicity and diagnostic accuracy, leaving most techniques vulnerable to false alarms [28]. In modern approaches, while AI (such as SVMs [31,32]) offers robust automation, challenges persist regarding industrial feasibility, high computational costs, and the requirement for extensive labeled datasets [32]. Additionally, accuracy often depends on optimal sensor placement, which varies according to configuration [32].

This paper addresses these critical gaps through an innovative hybrid method combining sensor curve analysis to determine the optimal speed as a function of torque, with SVMs, to enhance broken-bar fault diagnosis in Asynchronous Motors. The following table summarizes the technical justifications and advantages associated with the methodological choices of the proposed approach.

Based on the methodological foundations established in Table 1, the results enhance existing knowledge by offering a more reliable diagnostic framework for Industry 4.0. In contrast to methods listed in the review [31], which often require sophisticated hardware or are susceptible to load variations, the proposed approach aims to optimize the accuracy-cost trade-off. To better contextualize the contribution of this article, the following table presents the positioning of the proposed method relative to the state of the art.

Thus, this synthesis establishes a foundation for more precise predictive maintenance, meeting the reliability and adaptability requirements of industrial electrical machine systems [31, 32].

3. Dynamic model of the asynchronous motors

To enable data-driven fault diagnosis, a high-fidelity simulation framework is developed, grounded in a multi-winding representation of the Asynchronous Motors. This approach ensures physical realism while providing full control over fault severity essential for generating a labeled dataset adapted to supervised learning [33-35].

While classical modeling approaches (the fifth-order state-space representation in the $\left(d,q\right)$ frame provide a compact description of global machine dynamics [5, 6, 8, 9], they lack the granularity needed to capture localized rotor faults such as broken bars. To address this limitation and enable fault-specific simulation, a multi-winding equivalent circuit model is adopted, which explicitly represents individual rotor bars and end ring segments. This representation, detailed in the following subsection, forms the foundation for realistic emulation of progressive bar breakage and subsequent dataset generation for SVM-based diagnosis [33-35].

3.1. Healthy multi-winding asynchronous motors model

To model broken-bar faults, a multi-winding equivalent circuit is adopted, enabling explicit representation of each rotor bar and end-ring [33, 34]. Fig. 1 illustrates how this approach resolves the localized electromagnetic perturbations due to fractures.

Fig. 1Rotor configuration: a) intact, b) with broken bar

a) Healthy rotor

b) Faulty rotor with broken bar

Conventional dynamic modeling of Asynchronous Motors relies on the Park transformation, which reduces the three-phase system to an equivalent two-phase $\left(d,q\right)$ representation. While this yields a compact fifth-order state-space model – widely used for control and global performance analysis [30, 31] – it inherently averages rotor behavior and thus cannot resolve localized electromagnetic asymmetries, such as those induced by broken rotor bars:

1

$\left\{\begin{array}{l}\begin{array}{l}\frac{d}{dt} i_{ds}=\left(\frac{1}{\sigma L_s}\right)\left(-R_{sm}{i}_{ds}+ \sigma L_s{\omega }_s{i}_{qs}+\frac{L_m}{L_r{T}_r} {\varnothing }_{dr}+ \frac{L_m P}{L_r} {\varnothing }_{qr}{\omega }_r+V_{ds}\right),\\ \frac{d}{dt} i_{qs}=\left(\frac{1}{\sigma L_s}\right)\left(-R_{sm}{i}_{qs}- \sigma L_s{\omega }_s{i}_{ds}+\frac{L_m}{L_r{T}_r} {\varnothing }_{qr}+ \frac{L_m P}{L_r} {\varnothing }_{dr}{\omega }_r+V_{qs}\right),\\ \frac{d}{dt} {\varnothing }_{dr}=\frac{L_m}{T_r}{i}_{ds}-\frac{1}{T_r} {\varnothing }_{dr}+\left({\omega }_s-P {\omega }_r\right){\varnothing }_{qr},\end{array}\\ \frac{d}{dt} {\varnothing }_{qr}=\frac{L_m}{T_r}{i}_{qs}-\frac{1}{T_r} {\varnothing }_{qr}+\left({\omega }_s-P {\omega }_r\right){\varnothing }_{dr},\\ \frac{d}{dt} {\omega }_r=\left(\frac{P L_m}{J T_r}\right)\left(i_{qs}{\varnothing }_{dr}- i_{ds}{\varnothing }_{qr}\right)+ \frac{1}{J} \left(- {C_r-K}_f{\omega }_r\right),\end{array}\right.$

where, $R_{sm}= R_s-\frac{{L_m}^2}{L_s{T}_r}$; $T_r= \frac{L_r}{R_r}$ and $\sigma =1-\frac{{L_m}^2}{L_s{L}_r}$.

To faithfully reproduce bar fracture effects, the healthy machine model is extended with two physically meaningful parameters that enable precise fault localization and severity control:

– The angular position ${\theta }_0$ of the broken bar (relative to the stator reference frame).

– A severity factor $k_f$ defined as the ratio of faulty to healthy inter-turn equivalents - allowing graded emulation from incipient cracks to complete bar rupture.

Under the assumption of an intact rotor (i.e., all bars healthy), the reduced multi-winding model simplifies to the form presented in [33-35]:

2

$\left[\begin{array}{ccccc}{L}_{sc}& 0& -\frac{1}{2} N_r{M}_{sr}& 0& 0\\ 0& L_{sc}& 0& \frac{1}{2} N_r{M}_{sr}& 0\\ -\frac{3}{2} N_r{M}_{sr}& 0& L_{rc}& 0& 0\\ 0& \frac{3}{2} N_r{M}_{sr}& 0& L_{rc}& 0\\ 0& 0& 0& \frac{1}{2} N_r{M}_{sr}& L_e\end{array}\right] \frac{d}{dt}\left[\begin{array}{c}{I}_{ds}\\ I_{qs}\\ I_{dr}\\ I_{qr}\\ I_e\end{array}\right]$
$\begin{array}{c} =⌈\begin{array}{c}{V}_{ds}\\ V_{qs}\\ 0\\ 0\\ 0\end{array}⌉-\left[\begin{array}{ccccc}{R}_s& {-L}_{sc} {\omega }_r& 0& -\frac{1}{2} N_r{M}_{sr}{\omega }_r& 0\\ L_{sc} {\omega }_r& R_s& \frac{1}{2} N_r{M}_{sr}{\omega }_r& 0& 0\\ 0& 0& R_r& 0& 0\\ 0& 0& 0& R_r& 0\\ 0& 0& 0& 0& R_e\end{array}\right] \left[\begin{array}{c}{I}_{ds}\\ I_{qs}\\ I_{dr}\\ I_{qr}\\ I_e\end{array}\right]\end{array}.$

In Eq. (2), the key electromagnetic parameters are defined as follows: $M_{sr}$ is mutual inductance between stator phase a and a rotor mesh:

3

$M_{sr}=\frac{4}{\pi }\frac{{\mu }_0}{p^2 e} N_s R\sin \left(\frac{\alpha }{2}\right),$

where: $\alpha =2 \pi /N_r$ is the angular pitch between adjacent rotor bars, ${\mu }_0=$4$\pi$×10^-7 H/m is the permeability of free space, and $e$ – air gap thickness (m), $N_{s }$ – number of stator slots, $R$ – rotor radius (m), $N_r$ – number of rotor bars, $L_{sr}$ – self-inductance of stator phase:

4

$L_{sc}=\frac{6}{\pi }\frac{{\mu }_0}{p^2 e} {N_s}^2 l R+ L_{sf},$

where: $l$ – active rotor length (m), $L_{sf }$ – stator leakage inductance ($H$), $L_{rc }$ – self-inductance of a rotor mesh:

5

$L_{rc}=L_{rp}-M_{rr}+2\frac{L_e}{N_r}+2 L_b \left(1- \cos \alpha \right),$

with:

6

$L_{rp}=\frac{{2 \pi \mu }_0}{ e} \frac{\left(N_r-1\right)}{{N_r}^2 }l R , M_{rr}=-\frac{{2 \pi \mu }_0}{ e} \frac{1}{{N_r}^2 }l R.$

3.3. Simulation setup and dataset generation

Broken rotor bars constitute one of the most prevalent mechanical faults in Asynchronous Motors. To capture the associated electrical signatures and assess their impact on machine performance, a high-fidelity numerical simulation is carried out. Following established practice [30, 31], a fully severed bar is modeled by increasing its resistance by a factor of 11, an approximation that reflects the near-open-circuit condition while preserving numerical stability.

The simulation considers progressive fault severity by successively introducing failures in three adjacent bars ($k=$0, $k=$1, $k=$2, $k=$ 3). Under a constant load torque $C_r=$ 3.5 N/m applied at $t=$ 0.3 s, the fault scenarios are triggered at:

– $t=$ 0.8 s: one broken bar.

– $t=$ 1.2 s: two broken bars.

– $t=$ 1.6 s: three broken bars.

For the three-bar fault case, the resulting equivalent resistances are: $S_1=S_4=$6.1455×10^-5 Ω, $S_2=S_3=$ 3.1819×10^-5 Ω.

Fig. 2Learning curve of the rotor current (A) for a healthy and faulty asynchronous motors

The dynamic responses in Figs. 2-5 reveal clear electromechanical deviations induced by progressive bar breakage and show that:

1) The motor starts without load and during the initial transient (≈ 0.25 s), stator and rotor currents exhibit dynamic behavior before settling into steady-state sinusoidal waveforms centered around zero. The electromagnetic torque stabilizes at 0 N·m, while the mechanical speed reaches its no-load nominal value of 314 rad/s.

2) Under loaded operation, two distinct behaviors are observed:

– Healthy case: A slight reduction in speed occurs, while the torque rises to match the applied load (3.5 N·m). Stator and rotor currents remain sinusoidal and unchanged in amplitude.

– Broken-bar case: As the number of broken bars increases (1 at $t=$ 0.8 s, 2 at $t=$ 1.2 s, 3 at $t=$ 1.6 s), the following trends are observed:

a) Increased over currents in rotor bars adjacent to the fractures.

b) Growing amplitudes of speed and torque ripples.

c) Higher amplitudes in stator currents.

These effects are most pronounced in the three-bar fault scenario.

Fig. 3Learning curve of the stator current (A) for a healthy and faulty asynchronous motors

Fig. 4Electromagnetic torque (N·m) learning curve for an asynchronous motors in normal and fault conditions

Fig. 5Learning curve of the mechanical speed (rad/s) for a healthy and faulty asynchronous motors

4. Support vector machine for broken-bar fault diagnosis

Support Vector Machines (SVM) provide a powerful framework for supervised classification, particularly well-suited to fault diagnosis where labeled datasets are limited but physically meaningful. As illustrated in Fig. 6, SVM constructs an optimal hyperplane that maximizes the margin between classes, enhancing generalization and robustness to noise. For broken-bar faults in Asynchronous Motors, the two SVM strategies is implemented: binary classification (healthy and faulty) and multi-class classification (0, 1, 2, or 3 broken bars).

Fig. 6Maximum-margin hyperplane separating healthy (class 0) and faulty (class 1) states

Fig. 7Multiclass separation for broken-bar severity levels (0 to 3 bars)

a) Input space

b) Feature space

c) Decision boundaries

4.1. Mathematical foundations of SVM

4.1.1. Linear decision function

For a feature vector $x\in {\mathbb{R}}^n$, the linear classifier is defined as:

13

$h\left(x\right)=\mathrm{ }{\omega }^{T}x+b,$

where: $\omega \in {\mathbb{R}}^n$ is the weight vector and $b\in \mathbb{R}$ the bias term. The predicted class is:

14

$\widehat{y}=\mathrm{ }\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\left(h\left(x\right)\right).$

4.1.2. Maximum-margin hyperplane

The geometric margin for sample $i$is:

15

${\gamma }_i=\mathrm{ }{y}_i \frac{{\omega }^T{x}_i+b }{‖\omega ‖},$

and the optimization problem becomes:

16

$\underset{\omega ,b}{\min }\frac{1}{2} {‖\omega ‖}^2 \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{ }\mathrm{t}\mathrm{o}\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }{y}_i\left({\omega }^T{x}_i+b\right)\ge 1, {\forall }_i.$

Fig. 8Canonical hyperplanes ωTxi+b±1 and maximum margin γ=2/ω

4.1.3. Dual formulation

Introducing Lagrange multipliers ${\alpha }_i\ge 0$, the Lagrangian is:

17

$\mathcal{L}\left(\omega ,b,\alpha \right)=\frac{1}{2}{‖\omega ‖}^2- \sum _{i=1}^N{\alpha }_i\left[y_i\left({\omega }^T{x}_i+b\right)-1\right].$

Setting derivatives to zero yields:

18

$\frac{\partial \mathcal{L}\left(\omega ,b,\alpha \right)}{\partial \omega }=0\mathrm{ }\Rightarrow \omega =\sum _{i=1}^N{\alpha }_i{y}_i{x}_i,$

19

$\frac{\partial \mathcal{L}\left(\omega ,b,\alpha \right)}{\partial b}=0\mathrm{ }\Rightarrow {\sum }_{i=1}^N{\alpha }_i{y}_i=0.$

Substituting into $\mathcal{L}$ gives the dual problem:

20

$\underset{\alpha }{\max }\sum _{i=1}^N{\alpha }_i-\mathrm{ }\frac{1}{2} \sum _{i=1}^N\sum _{j=1}^N{\alpha }_i{{\alpha }_{j}y}_i{y}_j{x_i}^T{x}_j,$

subject to ${\alpha }_i\ge 0$ and ${\sum }_i{\alpha }_i{y}_i=0$.

4.1.4. Kernel trick for nonlinear separation

For non-separable data, the kernel function $k\left(x_i,x_j\right)={\mathrm{\Phi }\left(x_i\right)}^T\mathrm{\Phi }\left(x_j\right)$ maps inputs to a higher-dimensional space. The decision function becomes:

21

$h\left(x\right)=\mathrm{ }\sum _{i=1}^N{\alpha }_i{y}_i k\left(x_i,x_j\right)+b.$

The Radial Basis Function (RBF) kernel is adopted, as it has proven highly effective for motor fault signatures in prior studies [27, 36]:

22

$k\left(x_i,x_j\right)=\exp \left(-\gamma {‖x_i-x_j‖}^2\right),$

where $\gamma >0$ controls similarity decay.

Fig. 9SVM-based diagnostic architecture for broken-bar fault detection and severity assessment

5. Fault detection and classification in asynchronous motors using SVM

5.1. Case 01: binary SVM for two-class health state classification in asynchronous motors

The validation of the proposed algorithm was performed using datasets generated from the dynamic mathematical model of the Asynchronous Motors described in Section 2. A binary support vector machine (SVM) classifier was employed to discriminate between healthy and faulty operating conditions.

Fig. 10 illustrates the training phase of the binary SVM, where samples from each class are represented by distinct markers and colors. Fig. 11, in contrast, presents the classification outcomes obtained during the testing phase.

Fig. 10Visualization of binary SVM classification training results

Fig. 11Visualization of binary SVM classification results

Both figures correspond to a dataset extracted over the time interval [1:600] and ($t\in$[0 s, 0.6 s]), under a modified rotor resistance condition ($R_r=$0.001 Ω), simulating fault scenarios.

From Fig. 10, the following observations can be drawn regarding the training stage:

1) A clear separation between the two classes is achieved by the optimal hyperplane.

2) The training data are correctly assigned to two distinct classes:

– Healthy machine (red dots): These samples are predominantly concentrated in the early interval ($t\in$[0 s, 0.2 s]), with a noticeable decline in density beyond $t=$ 0.3 s.

– Faulty machine (green dots): These samples cover the entire interval ($t\in$ [0 s, 0.6 s]), reflecting the superposition of three distinct fault conditions (1, 2, and 3 broken rotor bars), all simulated within this time window.

During the classification phase (Fig. 11), the performance of the trained SVM is assessed, yielding the following results:

1) The SVM constructs a robust optimal separating hyperplane that effectively distinguishes healthy samples from faulty ones.

2) Classification accuracy is confirmed by the distinct temporal clustering of predicted labels:

– Healthy machine (pink dots): Predicted healthy states are mainly localized in ($t\in$ [0 s, 0.3 s]), with no faulty (blue) predictions co-occurring in this interval-validating the discriminating capacity of the hyperplane.

– Faulty machine (blue dots): Predicted faulty states dominate the interval ($t\in$ [0.3 s, 0.6 s]), with no healthy (pink) predictions present further confirming the classifier’s fidelity and decision boundary robustness.

This temporal partitioning of predictions aligns closely with the physical behavior of the machine under varying fault severities and confirms the effectiveness of the binary SVM for early fault detection in Asynchronous Motors.

6. Conclusions

In summary, the proposed framework enables accurate multi-class diagnosis of broken rotor bar faults in Asynchronous Motors, using only physically measurable signals namely torque and speed combined with a two-stage SVM classifier: first binary (healthy vs. faulty), then multi-class (four severity levels). The results demonstrate that minimal instrumentation specifically, two standard laboratory-grade sensors, a torque transducer and an incremental encoder is sufficient to achieve high-fidelity health monitoring, without requiring deep learning or complex signal preprocessing.

This work establishes the steady-state torque-speed characteristic $Te\left(\Omega \right)$as a highly separable and physically measurable feature for multi-class diagnosis of rotor bar faults in Asynchronous Motors. Using a validated dynamic model, a binary SVM is first deployed to detect the presence of any fault (healthy and defective), yielding Accuracy: 93.8 %. Subsequently, a dedicated multi-class SVM is trained to discriminate among four fault-severity levels (healthy, and 1 to 3 broken bars) and achieves Accuracy: 93.9 %, Recall: 93.8 %, and F1-score: 93.8 %, confirming robust and balanced classification performance.

The discriminative power of $Te\left(\Omega \right)$ stems from its physical interpretability: broken rotor bars perturb the air-gap MMF, inducing slip-modulated torque harmonics whose spectral energy concentrates in a low-dimensional region of the $Te$, $\Omega$ plane structurally stable across load and speed variations, and inherently robust to PWM ripple and transient disturbances. This geometric separability enables even a standard RBF-SVM (hyper parameters $C=$ 10, $\gamma =$ 0.1 tuned via coarse grid search) to achieve high-margin classification without deep architectures. Crucially, the approach exhibits strong data efficiency: reducing the training set size from 600 to 400 samples induced only a 5.0 percentage-point drop in accuracy (from 93.9 % to 88.9 %), confirming robustness to limited data, a key asset for embedded deployment. Moreover, $Te$ and $\Omega$ are estimable in real time using signals natively available in vector-controlled drives (stator currents $isd$, $isq$ and rotor position from encoder or observer), requiring no additional sensors. Nevertheless, the approach entails two practical limitations that warrant explicit acknowledgment. The primary limitation – dependence on quasi-steady-state operation – is mitigated by the fact that industrial drives (pumps, compressors, conveyors). Future work will focus on experimental validation under real noise conditions and online estimation of $Te\left(\Omega \right)$ for incipient fault detection and investigation of semi-supervised learning strategies to further reduce dependency on large labeled datasets while preserving classification margins.