# Comparison of Hysteresis Based PWM Schemes \(\Delta\Sigma\)-PWM and Direct Torque Control
## 5 Conclusions  
In this paper, the differences between \(\Delta\Sigma\)-PWM and DTC were analyzed to benchmark both schemes. A simulation model was used to compare several quantities with a focus on the behavior of the switching frequency. Using an equal maximum switching frequency of 30 kHz for both schemes, the simulation results were discussed. The aim of this paper was in particular to show the differences in the instantaneous switching frequency over angular positions. Because of the fixed sequence, \(\Delta\Sigma\)-PWM distributed the switching events more evenly among all inverter half bridges. DTC, in contrast, often used only one inverter leg causing the instantaneous switching frequency to be more widespread (see also Table 4). Whereas the \(\Delta\Sigma\)-PWM had its minimum switching frequency near the angles of the maximum phase voltages, here the DTC switched more often. Figure 14: Torque ripple and average switching frequency at various operating points. Figure 13: Average switching frequency at various operating points. All compared quantities from Section 4 are summarized in Table 4. For comparison, the ripples of torque, voltage and flux and the switching frequencies are listed. Due to the lower average switching frequency, all ripples with the DTC are higher. The additional simulations show that, DTC has a lower torque ripple than \(\Delta\Sigma\)-PWM at the same average switching frequency. However, because of the lower variation of the instantaneous switching frequency, a higher average switching frequency can be achieved with the \(\Delta\Sigma\)-PWM on the same inverter than with the DTC. Since the ripple is inverse to the switching frequency, a lower ripple is possible with the \(\Delta\Sigma\)-PWM than with the DTC with the same hardware. According to this a higher control performance can be achieved with the \(\Delta\Sigma\)-PWM than with the DTC. Conceptualization, M.T. ; Investigation, K.K. ; Supervision, W.S. All authors have read and agreed to the published version of the manuscript. This publication was funded by the Publication Fund of the Technische Universitat Braunschweig. Not applicable. The data presented in this study are available on request from the corresponding author. The authors declare no conflict of interest. The following abbreviations are used in this manuscript: \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Scheme**} & \multicolumn{3}{c}{**Ripple**} & \multicolumn{3}{c}{**Switching Frequency**} \\  & **Torque** & **Voltage** & **Flux** & **Max** & **Min** & **Average** \\ \hline \(\Delta\Sigma\)-PWM & 1.79 \% & 3 \% & 0.37 \% & 30.37 kHz & 18.8 kHz & 25.6 kHz \\ \hline DTC & 2 \% & 4.4 \% & 1 \% & 30.5 kHz & 660 Hz & 10.1 kHz \\ \hline \hline \end{tabular}
\end{table} 
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