# Quaternion Signal Analysis for Detection of Broken Rotor Fault Degrees in Induction Motors
## 4 Conclusions  
Few studies have developed methodologies to quantify gradual levels of faults, such as BRB faults, in induction motors. The method presented here accurately quantifies BRB failure severity using Quaternion Signal Analysis (QSA), polynomial regression, and Fourier series modeling. This approach provides incremental records of motor behavior, enabling precise estimation and timely prevention of severe faults in the time domain.  
The mean RMSE results indicate consistent performance across regression orders, window sample sizes, and modeling techniques. Polynomial modeling, in particular, demonstrates that changes in function order and window size have minimal impact on method effectiveness. This robustness allows for low-order and low-sample implementations, which are computationally efficient as they rely solely on fundamental addition and multiplication operations, unlike Fourier series, which require sine and cosine harmonics.  
Future work will test this methodology by expanding the number of BRB conditions to detect partial BRBs and validating it with breaks in the junction between the bar and the side rings. In addition, we are interested in applying our development to assess wear levels in various rotating machine components, such as bearings and gearboxes, to validate their performance across diverse failure sources.  
The QSA method was designed to adapt to different load conditions in induction motors; however, this adaptability needs to be validated. Consequently, we will conduct an analysis to examine how the ability to detect fault signatures depends on various load levels in the induction motor.  
Conceptualization, J.L.C.-H. and M.A.I.-M.; methodology, J.L.C.-H. and D.L.A.-O.; software, J.L.C.-H.; validation, D.L.A.-O. and R.C.-S.; formal analysis, J.L.C.-H. and M.A.I.-M.; investigation, J.L.C.-H.; resources, M.A.I.-M.; data curation, D.L.A.-O.; writing--original draft preparation, J.L.C.-H.; writing--review and editing, D.L.A.-O. and M.A.I.-M.; visualization, J.L.C.-H. and R.C.-S.; supervision, R.C.-S.; funding acquisition, M.A.I.-M. All authors have read and agreed to the published version of the manuscript.  
The APC was funded by the Electronic Engineering Department of Universidad de Guanajuato under POA 2024.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Window**} & \multirow{2}{*}{**2**} & \multicolumn{3}{c}{**Order**} \\  & & **2** & **3** & **4** & **5** & **6** \\ \cline{2-6}
**Samples** & _Mean (min, max)_ & _Mean (min, max)_ & _Mean (min, max)_ & _Mean (min, max)_ & _Mean (min, max)_ \\ \hline
4000 & 0.303 (0.16, 0.66) & 0.391 (0.16, 1.28) & 0.739 (0.18, 3.06) & 1.051 (0.17, 4.48) & 2.727 (0.07, 19.73) \\ \hline
6000 & 0.268 (0.19, 0.53) & 0.379 (0.17, 1.34) & 0.702 (0.16, 3.11) & 1.396 (0.18, 6.00) & 2.663 (0.12, 36.69) \\ \hline
8000 & 0.273 (0.16, 0.69) & 0.414 (0.15, 2.45) & 0.978 (0.23, 7.26) & 1.765 (0.14, 12.63) & 3.684 (0.12, 68.57) \\ \hline
10,000 & 0.293 (0.17, 0.70) & 0.643 (0.16, 3.60) & 1.227 (0.18, 4.69) & 3.107 (0.21, 26.46) & 6.125 (0.11, 81.33) \\ \hline
12,000 & 0.308 (0.18, 0.74) & 0.524 (0.17, 4.67) & 0.951 (0.18, 8.13) & 2.321 (0.15, 26.73) & 4.252 (0.10, 80.28) \\ \hline
14,000 & 0.301 (0.17, 0.64) & 0.579 (0.14, 3.13) & 0.998 (0.16, 4.89) & 2.728 (0.17, 20.43) & 7.288 (0.09, 162.18) \\ \hline
20,000 & 0.325 (0.18, 0.70) & 0.634 (0.15, 3.37) & 0.921 (0.15, 5.55) & 1.972 (0.15, 21.71) & 3.653 (0.09, 29.71) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Mean, minimum, and maximum RMSE in each order from two to six using Fourier series modeling.