## 4 Results
### Steady-State Performance  
With the aim to evaluate the operating performance generated by the proposed sensorless scheme at the steady-state, experiments were carried out under the circumstances that the speed command was set as 50 rpm and 500 rpm, as exhibited in Figure 8a,b, respectively. In these two cases, the torque command was given as 6 N\(\cdot\)m. It can be observed that both speed and torque can remain steady around the given value as expected in the two cases, and the scheme yields a sinusoidal stator current with THD values of about 14.22% and 11.46%, respectively.  
Furthermore, experimental waveforms considering the NPV balance are displayed in Figure 9. Certainly, both phase-A current and the rotor position angle are severely distorted, and the NPV increases to approximately 200 V before the control strategy is introduced. In particular, the motor speed produces large oscillations so that the whole system is out of control. Comparatively, speed can remain steady around 500 rpm, and the phase-A current changes into a nearly sinusoidal waveform once the NPV balance control is involved. In the meantime, the NPV returns quickly to 0 V within 200 ms, which is in accordance with the theoretical analysis. Overall, the proposed sensorless scheme gives an advantageous performance in terms of the neutral-point voltage balance.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Items** & **Specifications** \\ \hline Rated power & 2.2 kW \\ Rated current & 5 A \\ Rated voltage & 380 V \\ Rated torque & 14 N\(\cdot\)m \\ Direct axis inductance & 24 mH \\ Quadrature axis inductance & 36 mH \\ Stator resistance & 5.25 \(\Omega\) \\ Rated speed & 1500 rpm \\ Moment of inertia & 0.001 kg\(\cdot\)m2 \\ Pole pairs & 2 \\ Permanent magnet flux linkage & 0.8 Wb \\ \hline \hline \end{tabular}
\end{table}
Table 2: Motor parameters.  
Figure 7: Experimental setup.  
Next, to clearly analyze the position-tracking performance of the proposed scheme, the speed tracking error and position tracking error are introduced by  
\[\left\{\begin{array}{l}e_{Nr}=\left(N_{r}{}^{real}-N_{r}\right)/N_{r}{}^{ref} \times 100\%\\ e_{\theta}=\left(\theta^{real}-\theta\right)/2\pi\times 100\%\end{array}\right. \tag{18}\]  
where \(N_{r}{}^{real}\) and \(\theta^{real}\) represent the actual rotor speed and the actual position angle, respectively.  
Specifically, the position-tracking performance is tested under the two cases in Figure 8, as presented in Figures 10 and 11. It is worthwhile highlighting that the estimated value of the rotor speed follows the actual value accurately, not only under the condition that the speed command is set at 50 rpm but also when it is set at 500 rpm. In detail, the maximum speed tracking error at 50 rpm is within 25%; meanwhile, the maximum speed tracking error is within 10% at 500 rpm. On the other hand, the position-tracking error \(e_{\theta}\) of both operating conditions is no more than 5%, indicating that the proposed sensorless scheme satisfactorily reflects the position-tracking performance.  
Figure 8: Experimental results in a steady-state with a 6 N-m torque command: (**a**) 50 rpm speed command; (**b**) 500 rpm speed command.  
Figure 9: Experimental results of the NPV balance control.