⚡ electrical engineering

Low-Complexity Recursive NARX for Discriminating Incipient Stator Inter-Turn Short Circuits from Load Variations in Three-Phase Induction Motors

This paper presents a low-complexity recursive NARX model with static-gain adaptation that effectively discriminates incipient stator inter-turn short circuits from load variations in three-phase induction motors by detecting deviations in recursive coefficients, achieving high accuracy and fast response suitable for embedded condition monitoring without requiring complex computations or classifiers.

Original authors: Márcio Falcão Santos Barroso, Lane Maria Rabelo, Ana Paula Lima dos Santos

Published 2026-10-15

📖 7 min read🧠 Deep dive

CC BY 4.0

Original authors: Márcio Falcão Santos Barroso, Lane Maria Rabelo, Ana Paula Lima dos Santos

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the industrial world, the three-phase induction motor is the silent workhorse. Found in everything from factory pumps to ventilation fans, these machines keep production lines moving. However, like any mechanical system, they are prone to wear and tear. One of the most dangerous faults is a short circuit between the tiny copper windings inside the motor's stator, the stationary part that creates the magnetic field. When just a few turns of wire touch each other, a small, localized current begins to circulate. If left unchecked, this can overheat the insulation, leading to a catastrophic failure that stops the entire machine. The challenge for engineers is not just to find a broken motor, but to spot the problem while it is still small and manageable. This is difficult because the electrical signals that reveal a tiny fault also change when the motor simply works harder or carries a heavier load. Distinguishing a developing defect from a normal shift in workload has long been a hurdle for reliable, automated monitoring.

A team of researchers at the Federal University of São João del-Rei in Brazil has developed a new way to solve this specific problem. They created a simple, low-cost digital tool that can tell the difference between a motor that is just working hard and one that has a tiny, early-stage short circuit. Instead of trying to build a perfect, high-fidelity digital copy of the motor that mimics every physical detail, the researchers took a different approach. They built a very small, simplified mathematical model that acts as a sensitive detector. This model is designed to stay fixed in its structure, meaning it does not change its shape or complexity. Instead, when the motor's behavior shifts due to a fault, the numbers inside the model are forced to change to keep up. By watching how these numbers drift, the system can identify a fault even when the motor is undergoing significant changes in load.

To test this idea, the researchers set up a laboratory experiment using a standard three-phase induction motor. They carefully created controlled faults by shorting out a small number of turns in the motor's windings. In one scenario, they shorted about 1.5% of the turns in a single phase, and in another, they shorted about 3%. These are very small defects, the kind that are difficult to detect before they cause major damage. To make the test realistic, they also subjected the motor to a large, normal change in workload, increasing the electrical current by nearly 49%. The goal was to see if their system could ignore this massive load change while still sounding an alarm for the tiny short circuits.

The system they built relies on a method called recursive parameter estimation. In plain terms, this means the system updates its understanding of the motor's behavior one moment at a time, using only the most recent voltage and current readings. It does not need to store vast amounts of historical data or perform complex calculations that require powerful computers. The model uses just four adjustable numbers to describe the relationship between the voltage going into the motor and the current coming out. When the motor is healthy, these numbers stay relatively steady. When a fault occurs, the physical properties of the motor change, and the system must adjust these four numbers to match the new reality. The researchers found that by freezing the structure of the model and only allowing these numbers to move, they could clearly see the difference between a fault and a load change.

A crucial part of their success was a technique called static-gain adaptation. Without this feature, the system would mistake a heavy load for a fault because both cause the electrical signals to change. The adaptation acts like a volume knob that automatically adjusts the model's sensitivity to the current operating level. This allows the system to ignore the normal, expected shifts in performance caused by the load, while remaining highly sensitive to the specific, irregular changes caused by a short circuit. When the researchers tested their system without this adaptation, it failed to detect the smallest faults and struggled to distinguish the larger ones from normal load variations. With the adaptation in place, the system successfully separated the fault signals from the load signals.

The results showed that the nonlinear model they chose was significantly better than a traditional linear model at tracking the motor's behavior. In their tests, the nonlinear approach reduced the error in predicting the motor's output by about 22% compared to the linear version. More importantly, the system could clearly distinguish between the healthy motor, the motor with a 1.5% short, and the motor with a 3% short, even while the load was changing by nearly half. The researchers observed that different numbers in their model reacted in different ways. One specific number, related to how the input voltage affects the current after a short delay, proved to be the most reliable indicator of a fault. Another number was very sensitive to the load changes, which helped the system understand the context of the operation.

The system operates at a speed of 1,000 samples per second, which is fast enough to catch these early signs of trouble. However, the time it takes to raise an alarm is not limited by how fast the computer can calculate. Instead, the delay is about 1.2 seconds, which is the time required for the system to gather enough evidence to be sure that the change in the numbers is a real fault and not just random noise or a temporary fluctuation. This delay is acceptable for a predictive maintenance system, which aims to warn operators before a failure occurs, rather than to instantly trip a circuit breaker like a traditional safety device.

The study also highlighted that the system needs to monitor all three phases of the motor's electricity supply. When a fault occurs in one phase, the numbers in the model for that specific phase show a clear deviation. The phases that are not affected show much smaller changes. This means that in a real-world installation, the system would run three parallel monitors, one for each phase, and flag the one that is showing the most significant drift. This approach does not require knowing in advance which phase might fail.

The researchers emphasize that their method is designed for embedded systems, meaning it could run on the small, inexpensive microcontrollers already found in many industrial drives and protection relays. It does not require the heavy computational power needed for complex machine learning algorithms or the massive data storage of digital twin simulations. The entire process involves simple arithmetic operations that can be performed with very little memory. This makes it a practical candidate for retrofitting existing machinery, allowing older motors to be monitored with modern, intelligent diagnostics without replacing the hardware.

While the tests were conducted on a single motor in a controlled laboratory setting, the findings offer a promising path forward for industrial reliability. The study demonstrates that a simple, interpretable model, when combined with smart adaptation to changing conditions, can outperform more complex approaches in the specific task of separating early faults from normal operational changes. The work suggests that the future of motor monitoring may not lie in increasingly complex artificial intelligence, but in refining simple, physics-based tools that are robust, fast, and easy to implement in the real world. The researchers plan to validate their approach on motors powered by variable-speed drives and under a wider range of fault conditions, but the initial results provide a strong foundation for a new, low-cost strategy to keep industrial machinery running safely.

Technical Summary: Low-Complexity Recursive NARX for Discriminating Incipient Stator Inter-Turn Short Circuits

Problem Statement
The early detection of stator inter-turn short circuits (ITSCs) in three-phase induction motors (TIMs) presents a significant diagnostic challenge. While incipient faults (involving only a few turns) can lead to severe insulation degradation and catastrophic failure, they produce subtle changes in terminal variables. A primary difficulty is distinguishing these fault-induced dynamics from legitimate, large-scale variations in operating points, such as load changes. Industrial detectors must remain robust to normal load fluctuations (e.g., a ~48.5% current variation) to avoid false alarms, yet remain sensitive enough to detect faults as small as 1.5% of a stator phase. Existing high-fidelity models or machine learning approaches often lack the physical interpretability, low computational burden, or explicit mechanism to decouple load effects from fault signatures required for embedded implementation.

Methodology
The study proposes a low-complexity, model-based strategy using a Recursive Nonlinear Autoregressive with Exogenous Inputs (NARX) model. The approach diverges from traditional "digital twin" objectives; the identified model is not designed to minimize free-run simulation error but to act as a dynamic-change detector.

Data Generation: A physics-based symmetric $dq$ model, parameterized with standard test data, was used to generate persistently exciting healthy data. This "white-box" reference ensured the identification process started with physically consistent dynamics.
Model Structure: A deliberately low-order, second-order polynomial NARX surrogate was selected. The structure was frozen after batch identification, consisting of four deterministic terms:
$y(k) = \theta_1 u(k-1) + \theta_2 y(k-2) + \theta_3 u(k-2) + \theta_4 u(k-1)y(k-1)$
where $u$ is phase voltage and $y$ is phase current.
Recursive Estimation: An online Recursive Least Squares (RLS) algorithm with a forgetting factor ( $\lambda$ ) tracks the four coefficients ( $\theta_1$ to $\theta_4$ ) sample-by-sample. The implementation avoids matrix inversion by utilizing rank-one updates, requiring only a $4 \times 4$ covariance matrix and scalar operations per sample.
Static-Gain Adaptation: An auxiliary static characteristic adapts the model gain to the operating point. This mechanism is critical for absorbing benign load-dependent gain changes, preventing them from being misinterpreted as dynamic fault signatures in the recursive coefficients.
Local Interpretation: The coefficients are interpreted via local linearization. $\theta_2$ relates to effective poles (denominator), $\theta_1$ and $\theta_3$ to effective zeros (numerator), and $\theta_4$ to operating-point dependence. Faults manifest as drift in these coefficients rather than changes in the model structure.

Key Contributions
The paper outlines eight specific contributions:

A transparent identification pathway from physics-based models to a minimal diagnostic surrogate.
A comparative analysis showing the nonlinear NARX model achieves ~22% lower validation error than a corresponding linear model.
A local pole/zero interpretation of recursive coefficient drift, linking parameter changes to effective motor dynamics.
Explicit ablation demonstrating that static-gain adaptation is essential for fault/load discrimination.
Experimental validation against 1.5% and 3% ITSCs versus a 48.5% load-current variation.
Analysis of the forgetting factor trade-off between sensitivity, variance, and detection delay.
Phase-wise analysis confirming that all three phases must be monitored, as the fault signature is strongest in the affected phase.
An industrial implementation assessment highlighting the method's suitability for embedded systems (no FFT, no trained classifiers, no matrix inversion).

Experimental Results
Experiments were conducted on a 3 cv, 220 V, 60 Hz induction motor test bench with controlled ITSCs (3 and 6 shorted turns) and load transitions.

Discrimination: The recursive coefficients successfully distinguished incipient faults from load variations. Specifically, the delayed-input numerator coefficient ( $\theta_3$ ) provided the clearest discrimination, showing significant displacement for faults but minimal movement during load changes (after gain adaptation).
Coefficient Roles: $\theta_2$ (pole-related) was highly sensitive to both faults and load changes, making it less specific. $\theta_4$ was primarily responsive to operating-point changes, serving as contextual information.
Performance: The nonlinear model outperformed the linear counterpart. The static-gain adaptation was proven critical; without it, the system lost sensitivity to the 1.5% fault and struggled to distinguish the 3% fault from load changes.
Latency: The observed detection delay was approximately 1.2 seconds (1,200 samples at 1 kHz). The authors clarify this is an "evidence accumulation" latency required to statistically separate the fault from noise, not a computational execution delay.
Computational Load: The estimator operates sample-by-sample with $O(p^2)$ complexity ( $p=4$ ), requiring no matrix inversion or windowing, making it feasible for implementation on standard industrial microcontrollers.

Significance and Claims
The paper positions this work as a practical, industrial solution for predictive maintenance rather than a high-fidelity simulation tool. Its significance lies in:

Separation of Concerns: It explicitly separates the goal of simulation (high fidelity) from diagnosis (sensitivity to dynamic change), using a frozen, low-order structure to force fault dynamics into coefficient drift.
Embedded Feasibility: The method eliminates computationally expensive operations (FFT, matrix inversion, neural inference) typically found in modern fault diagnosis, relying instead on simple scalar recursions.
Robustness: By combining a nonlinear structure with static-gain adaptation, the method achieves robustness against large load variations while maintaining sensitivity to incipient faults that conventional protection relays might miss.
Modest Scope: The authors acknowledge limitations, noting that the current validation is limited to grid-fed laboratory motors with seeded faults. They state that future work is required to benchmark execution times on target hardware and validate performance under inverter-fed (PWM) operation and diverse supply disturbances. The method is presented as a complementary layer to conventional protection, targeting the incipient stage of degradation.

Technical Summary: Low-Complexity Recursive NARX for Discriminating Incipient Stator Inter-Turn Short Circuits

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