# Quaternion Signal Analysis for Detection of Broken Rotor Fault Degrees in Induction Motors
###### Abstract  
Fault detection in induction motors is essential for maintaining the reliability of industrial operations. In practical applications, induction motors experience gradual wear on critical components, such as rotor bars, affecting their performance. This paper introduces a new methodology for modeling predictive wear functions related to rotor faults in induction motors, providing accurate forecasts and optimal performance through Quaternion Signal Analysis in the time domain. Our approach accurately detects wear in broken rotor bars and anticipates their degradation over time. The methodology involves coupling four vibration signals from the motor, representing them as quaternion coefficients, and calculating their rotational attributes to derive a statistical mean. We employ polynomial and Fourier regression techniques to construct a predictive wear function. We assess its accuracy through root mean square error (RMSE) analysis, which improves with increased sample size and regression complexity. Our findings indicate that polynomial regression, particularly at the second degree, achieves superior RMSE results compared to Fourier regression, even within limited sample windows. This approach offers a robust framework for early fault detection and wear prediction in induction motors, supporting enhanced maintenance strategies. Article  
Academic Editors: [PERSON], [PERSON], [PERSON] and [PERSON]. rotor bar wear across states ranging from healthy to BRB. In wavelet algorithms, the authors of [8] successfully classified multiple BRB faults by extracting statistical features directly from raw signals using advanced techniques such as Matching Pursuit (MP) and Discrete Wavelet Transform (DWT). Moreover, several studies leverage machine learning methods like k-Nearest Neighbor (KNN) and support vector machines (SVMs) for BRB classification. For instance, the approach in [9] integrates Discrete Wavelet Transform (DWT) with these machine learning algorithms to achieve robust multi-class BRB classifications. Recently, there has been a surge of interest in utilizing machine learning and deep learning techniques for fault detection in induction motors. This trend reflects these approaches' proven accuracy and efficiency in identifying and diagnosing faults, marking significant progress in predictive maintenance [6]. Specifically, for BRB fault detection using time-domain signals, studies such as [10] have explored various machine learning algorithms--including fuzzy ARTMAP networks, SVM, KNN, and Multilayer Perceptron (MLP) networks--without requiring prior domain knowledge. This adaptability underscores the capability of machine learning methods to capture complex patterns related to BRB faults. Additionally, ref. [11] presents an innovative algorithm that classifies BRB conditions by applying statistical metrics derived from motor vibration signals, such as kurtosis, skewness, and variance. These metrics are then processed through classifiers like Naive Bayes, MLP, SVM, and Optimum-Path Forest, further demonstrating the efficacy of machine learning in accurate BRB fault classification. In [12], homogeneity and kurtosis metrics were employed to classify motor conditions, from healthy states to the presence of two BRBs, using a backpropagation artificial neural network. This approach demonstrates the power of neural networks in identifying complex data patterns and accurately diagnosing motor faults. Similarly, ref. [13] leveraged stator current signals to obtain comprehensive two-dimensional insights into BRB severity across time and frequency domains, calculating entropy at various load levels to assess fault impact. Conversely, in [14], the authors proposed a degradation model for BRB faults by applying principal component analysis (PCA). Using raw torque, stator current, and speed signals, they estimated the remaining useful life of the BRBs, highlighting PCA's utility in modeling the degradation process.