# Comparison of Hysteresis Based PWM Schemes \(\Delta\Sigma\)-PWM and Direct Torque Control
## 4 Simulation Results
### Comparison of Instantaneous Switching Frequencies  
Hysteresis based PWM schemes are characterized by a variable switching frequency. The literature distinguishes between instantaneous and average switching frequency. According to [36], the instantaneous switching frequency \(f_{sw}\) can be calculated by the time difference between a first (\(t_{on1}\)) and its following rising edge (\(t_{on2}\)), with  
\[f_{sw}=\frac{1}{t_{on2}-t_{on1}}. \tag{17}\]  
Due to the hysteresis elements, the instantaneous switching frequency can vary during an electrical cycle. Thus, a maximum and minimum switching frequency of the inverter can be measured. Furthermore, [36] defines an average switching frequency \(f_{m}\) for capturing those oscillations by  
\[f_{m}=\frac{N_{ris}}{\Delta T}. \tag{18}\]  
In (18) the number of rising edges \(N_{ris}\) of one inverter half bridge are divided by a previously defined period \(\Delta T\). Within this paper, the time period \(\Delta T\) is set to the duration of one electric rotation. Additionally, the average switching frequency of all inverter half bridges \(f_{\overline{m}}\) can be calculated by  
\[f_{\overline{m}}=\frac{1}{3}\cdot(f_{m1}+f_{m2}+f_{m3}). \tag{19}\]  
Given (17), Figure 11 shows the instantaneous switching frequencies for \(\Delta\Sigma\)-PWM and DTC as a function of the electric angle. Figure 11a,b presents the filtered phase voltages, where, in addition, also the angle boundaries of \(\Delta\Sigma\)-PWM and the sector boundaries of DTC are marked.  
Figure 11c shows the instantaneous switching frequency for \(\Delta\Sigma\)-PWM. All three half bridges of the inverter operate with a switching frequency in a range between 18.8 kHz and 30.7 kHz at any angle of electric rotation. The instantaneous switching frequency reaches its minimum if the phase voltage reaches its maximum or minimum. During this moment, the control passes an angle boundary and the reference voltage space vector is aligned with a voltage vector of the inverter hexagon. Thus, the reference voltage space vector can only be synthesized by the corresponding switching state. This switching state is kept as long as possible without violating the switching sequence given in (16). As a result, the number of switching operations is minimized and the instantaneous switching frequency is reduced [6]. \(\Delta\Sigma\)-PWM evenly uses all inverter half bridges at any angle of electric rotation.  
Figure 11d presents the instantaneous switching frequency for DTC. In this case, huge oscillations ranging from 660 Hz to 30.5 kHz are observed. DTC seems to prioritize certain switching states periodically. The instantaneous switching frequency always has its maximum when the corresponding phase voltage reaches its minimum or maximum (see Figure 11b). At this moment, the voltage space vector crosses a sector boundary. During such a transition, the switching state which should decrease the magnitude of the flux space vector accelerates the flux space vector more than changing its magnitude (see Section 4.3). Hence the flux controller changes its state infrequently. To keep the torque constant, the active voltage vector must often be interrupted by a zero vector. To do so, only one inverter half bridge has to change its state. The switching frequency of this half bridge will increase whereas the switching frequency of both other half bridges remains low. The flux controller increases its activity towards the midpoint of the sector. Consequently, active voltage vectors are changed more often and all inverter half bridges operate with an approximately same instantaneous switching frequency.  
The average switching frequency is 10.1 kHz for DTC and 25.6 kHz for \(\Delta\Sigma\)-PWM. The flux ripple discussed in Sections 4.1-4.3 is directly related to the switching frequency. Since DTC switches less frequent than \(\Delta\Sigma\)-PWM, the harmonics in physical values are higher.  
Figure 11: Phase voltage and instantaneous switching frequency.