## 3 Simulation Results
### Simulation Results in a Circuit with FOC at Rated Motor Shaft Rotation Frequency
But here a problem arises related to the fact that the presence of filters in the compensator leads to a change in the phase shift between the voltage and current of the corresponding phases. Therefore, with this approach, it is quite difficult to determine the convergence time. It is proposed to determine the convergence time when the amplitudes of the spatial vectors of the stator phase currents coincide in the case of powering the drive from a sinusoidal voltage system and using the drive circuit with a compensator. The amplitude of the stator phase current spatial vector can be determined as follows [29]:  
\[\mathrm{I_{s}}=\sqrt{\mathrm{I_{s\alpha}^{2}+\mathrm{I_{s\beta}^{2}}}} \tag{21}\]  
where \(\mathrm{I_{s\alpha}}\) is the projection of the stator phase current spatial vector onto the \(\alpha\) axis and \(\mathrm{I_{s\beta}}\) is the projection of the stator phase current spatial vector onto the \(\beta\) axis. Projections of the stator phase current spatial vector onto the \(\alpha\) and \(\beta\) axes are determined as follows [29]:  
\[\left\{ \begin{matrix}\mathrm{I_{s\alpha}}=\frac{2}{3}\cdot\left(\mathrm{I_{sa }}-\frac{1}{2}\cdot\left(\mathrm{I_{sb}}+\mathrm{I_{sc}}\right)\right)\\ \mathrm{I_{s\beta}}=\frac{1}{\sqrt{3}}\cdot\left(\mathrm{I_{sb}}-\mathrm{I_{sc }}\right)\end{matrix} \right. \tag{22}\]  
where \(\mathrm{I_{sa}}\), \(\mathrm{I_{sb}}\), and \(\mathrm{I_{sc}}\) are the phase currents of the induction motor stator. In accordance with Equations (21) and (22), time diagrams of the amplitudes of the spatial vectors of the stator phase currents were built for the case when the drive is powered from a sinusoidal voltage system and the drive circuit with a compensator. As a result of the simulation, the values of the motor shaft rotation frequency were also obtained for all cases \(\mathrm{n_{r}}\). All the results described above are presented in Table 3.  
The analysis of the results shown in Table 3 was carried out according to four criteria (in order of the criterion importance):  
1. The largest value of efficiency. 2. The smallest value of the torque ripple factor. 3. The shortest convergence time for a stable mode. 4. The shortest convergence time during motor acceleration. The analysis of the results presented in Table 4 showed that the drive circuit with FOC in the presence of a compensator with elliptic filter with adapted cut-off frequency has the highest efficiency value, the lowest values of the torque ripple factor, convergence time in stable mode, and convergence time during motor acceleration. At JSC \"Ukrzaliznytsia\", 14 DS-3 series electric locomotives are operated at the Kyiv-Pasazhyrskyi locomotive depot. According to post-repair test data from this depot in 2024, the torque ripple coefficient for the traction drives of these locomotives without compensators ranged from 27.5% to 30.5%. As a result of the simulation, this parameter was calculated as 29.3%, which confirms the high accuracy of the obtained result. Simulation Results in a Circuit with FOC at a Steady Motor Rotation Frequency, Which Is Lower than the Nominal Rotation Frequency  
The performance of the traction drive system with FOC at the nominal shaft speed of the motor was investigated for five distinct cases: with a Butterworth filter in the reactive power compensator, with a direct Chebyshev filter in the reactive power compensator, with  
\begin{table}
\begin{tabular}{l c c c c c c c c c c} \hline \hline \multicolumn{1}{c}{**Parameter**} & \multicolumn{1}{c}{**\(\mathrm{n_{r}}\)**} & \multicolumn{1}{c}{**\an inverse Chebyshev filter in the reactive power compensator, with a Bessel filter in the reactive power compensator, with an elliptic filter in the reactive power compensator. For circuits with compensators, the investigations were conducted at a fixed and adapted filter cut-off frequency to the motor shaft rotation frequency. The simulation was carried out for the following conditions:  
* PWM frequency \(\mathrm{f_{PWM}}=20\cdot\mathrm{f_{nom}}=1116\) Hz.