# Comparison of Hysteresis Based PWM Schemes \(\Delta\Sigma\)-PWM and Direct Torque Control
## 4 Simulation Results
### Stationary Behavior of Both PWM Schemes  
Figure 6 shows the stationary operation of the studied PMSM when using \(\Delta\Sigma\)-PWM. Figure 6a presents the phase current, whereas Figure 6c shows the transformed currents in \(\alpha/\beta\) as well as \(d/q\) coordinates. The behavior of the field-orientated control approach can be noted. The nominal operating point is reached with a \(q\) current of \(1.5\cdot i_{0}\) and a \(d\) current of \(0\,\mathrm{A}\). All shown quantities have a certain oscillation around the reference point. Figure 6b compares the differences between the reference and actual values of the torque. Given a stationary operation point, the requested value is achieved. Nevertheless, the torque is subject to some small superimposed quantization noise in a range of \(\pm 1.79\,\%\). By increasing \(N_{1}\), the hysteresis width becomes larger. The number of switching events decreases and the magnitude of torque oscillation increases. Because of the high frequent bit stream, no equation can be derived which forecasts the absolute torque ripple. The flux components in \(\alpha/\beta\) coordinates are presented in Figure 6d. This diagram shows a phase shift of the stator flux space vector with respect to the \(d\) axis. According to (9) and (10), the field-orientated current \(i_{sq}\) causes a \(q\) component in flux space vector of \(9.77\,\%\) to the nominal flux. Due to the approach of field-orientated control, which is specified by the cascades of \(\Delta\Sigma\)-PWM, a phase shift of \(5.58^{\circ}\) occurs. The undesired \(q\) component influences the magnitude of the flux space vector. The current controller in field coordinates uses \(d\) current for controlling flux space vector only. Thus, the effect of \(i_{sq}\) on the flux space vector is not measured and does not result in a phase shift in the waveforms from Figure 6d. The results from the DTC simulation are presented in Figure 7, which is structured as Figure 6. \begin{table}
\begin{tabular}{c c c} \hline \hline \multicolumn{3}{c}{**Control parameters of \(\Delta\Sigma\)-PWM**} \\ \hline Hysterisis element & \(N_{1}=58\) & \(N_{2}=78\) \\ \hline Current controller & \(V_{iq}=2\,\mathrm{V}\,/\,\mathrm{A}\) & \(T_{iq}=0.000\,74\,\mathrm{s}\) & \(V_{id}=2\,\mathrm{V}\,/\,\mathrm{A}\) & \(T_{id}=0.000\,74\,\mathrm{s}\) \\ \hline Speed controller & \(V_{\omega}=2\,\mathrm{As}\) & \(T_{\omega}=0.03\,\mathrm{s}\) \\ \hline \multicolumn{3}{c}{**Control parameters of DTC**} \\ \hline Flux controller & \(\Delta\Psi_{+}=0.43\,\mathrm{m}\mathrm{W}\mathrm{b}\) & \(\Delta\Psi_{-}=-0.43\,\mathrm{m}\mathrm{W}\mathrm{b}\) \\ \hline Torque controller & \(\Delta T_{+}=0.03\,\mathrm{N}\mathrm{m}\) & \(\Delta T_{0}=-0.03\,\mathrm{N}\mathrm{m}\) & \(\Delta T_{-}=-0.09\,\mathrm{N}\mathrm{m}\) \\ \hline Speed controller & \(V_{\omega}=226.39\,\mathrm{As}\) & \(T_{\omega}=0.04\,\mathrm{s}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Control parameters of \(\Delta\Sigma\)-PWM and DTC. In contrast to \(\Delta\Sigma\)-PWM, all physical quantities shown are distorted by a higher proportion of harmonics. The torque, for example, shows an absolute ripple of \(\pm 2.0\,\%\) related to the nominal load. Thus, the torque is always within the defined tolerance. Furthermore, \(i_{sd}\) is permanently negative, resulting in a phase shift of \(3.8^{\circ}\) of the current with respect to the \(d\) axis. The reason for the negative \(d\) current is based on the field forming property of \(i_{sq}\). According to (9) and (10), the magnitude of the flux space vector is increased by \(0.4\,\%\). The magnitude with this offset is now closer to the upper boundary of the flux controller (\(\Delta\Psi_{+}=0.5\,\%\) of nominal flux) than to the lower boundary. To compensate for this unwanted amplification, the control enforces a negative current in the \(d\) axis. Figure 6: Stationary behavior of simulation results at nominal point for \(\Delta\Sigma\)-PWM. Figure 8 compares the frequency spectra of the phase currents. The electrical fundamental frequency is 55.33 Hz.