## 4 Experimental Validation and Discussion
### Experimental Results and Discussion  
It has been observed that the characteristic frequency 2 _sf_ (where \(s\) is the slip and \(f\) is the supply frequency) is the most dominating fault component and easily obvious owing to its higher power in comparison to other fault frequency components corresponding to MVSA. When the BRB fault occurs, a torque ripple and a speed oscillation will be generated at frequency 2 _sf_, which is modulated on the rotation frequency components. Hence, this characteristic frequency will be extracted and analyzed using faulty motor with 1 BRB in the following experimental analysis for determining the robustness of proposed methodology. As shown in Figure 10a,b, the STFT-based CMS and FSC yield approximate rough results, and they provide a long carrier frequency band modulated by the fault frequency. It was difficult to find the specific carrier frequency bandwidth modulated by fault frequency content. However, the CMS based on CWT presented a relatively accurate and narrow carrier frequency bandwidth modulated by fault frequency as shown in Figure 10c. Therefore, the spectrum obtained from CMS-CWT has the ability to clearly identify and present the approximate range of carrier frequency components modulated by the cyclic frequency related to BRB faults. The precise wavelet analysis can gather the energy of  
Figure 8: Experimental setup. Figure 9: The faulty rotor simulated with one bar breakage. information related to BRB faults, so that it can satisfy the requirement of cyclic frequency and carrier frequency location. The slips of the IM with 1 BRB fault under different loads and their fault frequency components calculated by CMS, FSC, and CMS-CWT methods according to the characteristic frequency 2 _sf_ were presented in Table 3. Based on the analysis results, all the three cyclostationary analysis methods can accurately extract fault modulation components because of higher cycle frequency resolution. Meanwhile, the fault frequency 2 _sf_ is indeed effective to confirm the presence of BRB failure in IMs. Diagnosis of BRB fault signature using CMS and FSC vibration spectrums under 0%, 20%, 40%, and 80% load conditions are shown in Figures 11 and 12. It can be seen that CMS and FSC based on STFT received similar results. In a comparison with the experimental results issued from CMS-CWT as shown in Figure 13, the analysis results obtained from CMS and FSC indeed present additional harmonic components with big amplitudes or interference components that affecting the extraction of main characteristic frequency 2 _sf_, especially under the low load states such as 0%, 20%, and 40% \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Load**} & **Slip (s)** & \multirow{2}{*}{**Characteristic Frequency \(\alpha\) = 2_sf_ (Hz)} & **CMS** & **FSC** & **CMS-CWT** \\ \cline{2-4}  & **1 BRB** & & **1 BRB** & **1 BRB** \\ \hline
0\% & 0.002 & 0.20 & 0.20 & 0.20 & 0.20 \\
20\% & 0.010 & 1.00 & 0.95 & 0.95 & 0.95 \\
40\% & 0.018 & 1.80 & 1.75 & 1.75 & 1.75 \\
80\% & 0.036 & 3.60 & 3.65 & 3.65 & 3.65 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The slips and characteristic frequencies under different loads. Figure 10: Carrier frequency identification capability of (**a**) CMS, (**b**) FSC, and (**c**) CMS-CWT for the one broken bar case under 40% load. load conditions. However, this unimportant interference components were not visible in CMS-CWT vibration spectrum as presented in Figure 13. On the one hand, it demonstrated that the CMS-CWT is able to get better detectability in diagnosing the presence of BRB fault from light loads to high loads. On the other hand, this method exhibited an excellent performance and a high sensitivity to BRB fault information because of its high frequency resolution. More importantly, the CMS-CWT further improved the detection precision. Figure 11: Vibration spectrum of the healthy induction motor and faulty motor with one broken rotor bar (BRB) analyzed by CMS under **(a)** 0% load, **(b)** 20% load, **(c)** 40% load, and **(d)** 80% load.