## 3 Simulation Results  
The winding function model described in Section 2 was used for the transient and steady-state analysis of the motor's performance parameters under healthy and broken bar conditions. The broken bars were simulated by increasing the value of corresponding entries in the resistance matrix. Figure 3b shows the speed-torque dynamic response under healthy and faulty conditions. The zoomed window shows the shift of the operating point at the load application with a slight increase in the slip. The speed and torque ripples due to broken bars are visible in Figure 3a,b.  
Figure 3: (**a**) The speed dynamic response under healthy and faulty cases. (**b**) The speed–torque dynamic response under the healthy and broken bar cases with shifting of the operating point upon the application of a load in the zoomed window.  
Figure 2: Flowchart for better frequency resolution and development of contour plot.  
These ripples were because of the fault-based induced voltage on the stator side. The frequency of those harmonics depended upon the slip, as discussed previously. During the transient interval, the slip changed dramatically from one to the nominal value, generating a nonstationary signal in the current. This transient interval was so narrow that any specific pattern was hardly visible with rated voltage conditions because of the low-frequency resolution. This phenomenon is described in Figure 4. The motor's simulated transient current at 10%, 15%, 25%, and the rated voltage is presented. The envelope is highlighted by plotting the absolute value of the analytical function given below in (19) and (20). In all subfigures, the \(x\)-axis represents the time in seconds while the y-axis represents the amplitude in amperes. Moreover, as the constant air-gap is considered in the simulation model, the non-stationary signal contains only broken bar based pattern.  
\[i_{A}(t)=i(t)+j\,\hat{i}(t) \tag{19}\]  
where \(i(t)\) is the current, \(j\) is the imaginary unit representing the complex number and \(\hat{i}(t)\) is its Hilbert-transformed signal. This can be achieved by the convolution of the signal with \(1/\pi t\), as shown by the following equation:  
\[H(t)=\frac{1}{\pi}\int_{-\infty}^{\infty}\biggl{(}\frac{f(x)}{t-x}\biggr{)}dx \tag{20}\]  
All these steps can be achieved in Matlab by using the simple command ab (hilbert([t])). The envelope contains all harmonics other than the fundamental component. For a steady state signal, the Hilbert filter is the right choice for attenuation of the fundamental component. However, in the transient regime, the phase shift of all frequency components poses a problem.  
Hence, the nonstationary current signatures were recovered by attenuating the fundamental component using the IIR band-stop filter, as shown in Figure 4 (middle and last  
Figure 4: The motor’s simulated startup currents at different voltage levels (**top row**), the recovered nonstationary signal (**middle row**) and the nonstationary signal at 10% of the rated voltage representing the changing frequency pattern (**bottom**).  
row). With a decrease in the applied voltage, the fault-based nonstationary signal increased in length and became more legible. The time-frequency analysis of those non-stationary signals at rated voltage and 10% of the rated voltage is shown in Figure 5. With reduced voltage and the corresponding increase in the signal length, the V-shaped pattern became visible, being hardly present in the current at the rated voltage. Moreover, the supply frequency component (50 Hz) was attenuated using the proposed filter. The fault pattern became more legible by using filter and contour plots together. The contour plot shows the areas with a 95% confidence interval. With the rated voltage, the total power was confined in the region until 0.5 s without any pattern (Figure 5b), while with extended time, the pattern became visible as in Figure 5d. In the magnitude scalogram, the color bar represents the amplitude of the specific frequency component in amperes.