# Minimization of the Decoupling Circuit (DC-Link) in Induction Motors
## 2 Energy Flow Analysis of the Study Case
### Case Study Selection
#### 2.1.2 Speed Variation at Constant Load  
In this operation profile, the speed variation at a constant load is performed, which consists of modifying the speed of the electric motor while maintaining a constant load. This is done to observe the behavior of the energy under frequency and voltage variations.  
Figure 4: Inverter current, varying torque: (a) at 30% of rated torque; (b) at 60% of non-minimum torque; (c) at full load; (d) at 50% of rated torque; and (e) at 10% of rated torque.  
Figure 3: Torque variation steps: (a) 30% of rated torque; (b) 60% of rated torque; (c) at full load; (d) 50% of rated torque; and (e) 10% of rated torque.  
To perform this operation profile, different voltage and frequency levels are applied to the electric motor, trying to maintain a constant load. In this way, the behavior and performance of the motor under different operating conditions can be evaluated. Figure 5 shows the inverter current obtained at different test speeds at constant load, which allows observation of the variation of the current as a function of the speed of the electric motor.  
To perform the speed variation at constant load, previously defined speed variation steps are used. These steps are shown in Figure 6 and represent the different frequency and voltage levels applied to the electric motor used as load.  
Considering the information shown in Figures 5 and 6, it can be observed that in this operating profile, the highest current return from the inverter to the decoupling capacitor occurs when varying from the idle state to the rated speed. Realizing a comparison between the critical point obtained in the first scenario (torque variation) and the critical point obtained in this second operation profile, the critical point that occurs when the speed of the electric motor is varied from the idle state to reach 100% of the nominal speed in the shortest possible time, under the conditions established in the work, is selected as a case study, from which the mathematical equation of minimum capacitance developed in this article is based.  
To develop the required minimum capacitance equation, it is necessary to identify the critical intervals of the returned power, in the previously selected case study. Subsequently, simulations were performed with the specifications shown in Table 2, to obtain the signal patterns of the inverter and the DC-link in these critical intervals.  
Figure 5: Inverter current, constant load at variable speed. With speed steps from: (a) 0% to 100%; (b) 100% to 80%; (c) 80% to 35%; (d) 35% to 90%.  
Figure 6: Speed variation ranges from: (a) 0% to 100%; (b) 100% to 80%; (c) 80% to 35%; (d) 35% to 90%.  
The waveform pattern of the inverter current and capacitor voltage (VC) waveforms are shown in Figure 7, zooming in on Figure 8.  
From Figure 8, it can be seen that the pattern is directly affected by the switching signals. However, it was detected that a certain pattern is followed every six intervals. For this reason, the signal is sectioned to deduce the capacitance in the largest ripple presented (worst case), Figure 9, Table 3 defines the intervals. Observing that in sections four, five, and six is when the steepest positive slopes are presented, this indicates that these are the moments in which more energy is returned.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameter** & **Value** \\ \hline \multicolumn{2}{c}{DC–AC Converter and Modulation} \\ Inverter & 33-phase H-bridge inverter \\ Modulation & SPWM \\ DC source inverter & 320 V \\ Modulating signal frequency & 60 Hz \\ Carrier signal frequency & 3300 Hz \\ Modulation index & 0.9 \\ \multicolumn{2}{c}{DC-link} \\ Capacitor & 200 \(\mu\) \\ ESR & 0.20 \(\Omega\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Design specifications used for the analysis.  
Figure 8: Inverter current zooming in on the critical case.  
Figure 7: Capacitor voltage waveform pattern (**a**); inverter current (**b**) in the critical case.