# Quaternion Signal Analysis for Detection of Broken Rotor Fault Degrees in Induction Motors
###### Abstract
The works presented above classify BRB using methods that involve a certain level of complexity and require significant computational resources. Utilizing quaternions could simplify this process, as they provide a unique mathematical framework because they are an extension of complex numbers, consisting of a real part and a three-dimensional vector of imaginary components [15]. This distinctive configuration has spurred interest across various fields, revealing quaternions' versatility and potential in advanced applications. For example, in image processing, quaternions have been shown to facilitate image enhancement and manipulation in ways that surpass conventional methods [16], offering new data representation and transformation possibilities. In robotic motion tracking, as detailed in [17], quaternions provide a sophisticated framework for accurately monitoring and controlling robot movement, delivering the precision essential for executing complex tasks. In control systems, quaternions are equally valuable, enabling the design of robust control mechanisms with high accuracy and reliability [18]. The versatility of quaternions in various domains highlights their impact on advancing technological innovation. In spectral analysis, quaternions have been effectively combined with the Fourier Transform, as demonstrated in [19], to perform comprehensive spectral analysis of three-phase currents in induction motors, achieving a bearing fault classification accuracy of approximately 94.2%. Likewise, in [20], researchers developed the Quaternion Empirical Wavelet Transform (QEWT) based on the Quaternion Fourier Transform to diagnose outer ring, inner ring, and rolling bearing faults, illustrating quaternions' capability in high-precision fault detection. We have demonstrated the versatility of quaternions in previous works as [21, 22], where we employed Quaternion Signal Analysis (QSA) to detect multiple faults and their combinations in induction motors. This methodology involves rotating quaternion signals and performing statistical analyses, calculating mean, variance, kurtosis, skewness, standard deviation, root mean square, and shape factor. Subsequently, machine learning techniques are applied to classify various fault conditions and combinations. This approach highlights the effectiveness of QSA in capturing and classifying complex fault patterns. Similarly, the QSA was used in [23] to diagnose short-circuit damage in transformer windings demonstrating that this method is not exclusive to classifying faults in induction motors. QSA has shown high accuracy in fault classification through simple mathematical operations in the time domain, even using four statistical values from three-phase currents as in [24], where four levels of inter-turn short circuit faults per phase and healthy states were classified. On the other hand, regression methods utilize statistical processes to model relationships between variables, providing simplicity and low computational costs, especially with polynomial regressions. Furthermore, techniques like Fourier regressions can improve the effectiveness of stepped models. Studies have introduced methodologies for modeling predictive wear functions capable of quantifying BRB faults providing accurate forecasts and optimal performance as in [25]. In this, the authors present regression-based methods to evaluate BRB degradation using curve-fitting techniques such as linear, polynomial, and least squares regression with a power law. The method utilizes fault sideband parameters from the current spectrum, machine rating, and load torques of 50%, 75%, and 100% to gauge BRB severity levels from one to six, providing a structured model for fault severity assessment. As BRB faults in induction motors develop gradually, establishing an effective model to predict future behavior with straightforward calculations becomes crucial. This approach not only helps prevent severe damage but also ensures timely maintenance, safeguarding your equipment and investment.