# Minimization of the Decoupling Circuit (DC-Link) in Induction Motors
## 3 Results
Considering the voltage ripple values obtained in simulations using the theoretically developed minimum required capacitance values, a comparison with the initially proposed voltage ripple is carried out; Figure 13 presents this comparison. The data in the previous table were used to calculate the error percentages between what was proposed and what was obtained in simulations under the minimum capacitance equation developed, obtaining the results shown in Table 7.  
As shown in Table 7, the highest error percentage obtained using the proposed equation is 6.25%, with respect to the proposed initial value of voltage ripple. Once the correct operation of the equation has been validated, Table 8 shows a comparison between the capacitance value obtained using the proposed equation and the capacitance value required using the following existing equations in the literature [30; 31], respectively. \[C=\frac{Po}{2 fV_{dc}V} \tag{23}\]  
\[C_{DC-link}=\frac{I_{c,RMS}}{V_{Ripple}2\pi f_{sw}} \tag{24}\] \begin{table}
\begin{tabular}{c c} \hline
**Proposed Voltage Ripple** & **Capacitance Obtained Using the Proposed** \\
**(\%)** & **Equation \(\mu\)F** \\ \hline
3 & 361.2 \\
4 & 270 \\
5 & 217 \\
6 & 180.6 \\
7 & 155.8 \\
8 & 136 \\ \hline \end{tabular}
\end{table} 
Table 6: Proposed voltage ripples and obtained minimum capacitance. \begin{table}
\begin{tabular}{c c} \hline
**Measurement Number** & **Error Percentage (\%)** \\ \hline
1 & 4.6 \\
2 & 3.75 \\
3 & 4.8 \\
4 & 5.51 \\
5 & 5.4 \\
6 & 6.25 \\ \hline \end{tabular}
\end{table} 
Table 7: Percentage error between proposed voltage ripples and ripples obtained in simulation. The minimum capacitance value required by the system using the equation developed in this paper is lower by 2.6 and 1.51 times, compared with [30] and [31], respectively. \begin{table}
\begin{tabular}{c c} \hline \hline
**Equation Used** & **Capacitance Obtained** \\  & \(\mu\)F \\ \hline Proposed & 365 \\
[30] & 967 \\
[31] & 552.62 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Capacitance values using existing equations and the proposed equation, with a 3% voltage ripple. Figure 12: Voltage ripples obtained in simulation, using the capacitance with the proposed equation: (a) 3%; (b) 4%; (c) 5%; (d) 6%; (e) 7%; (f) 8%. Is important to emphasize that the developed equation can be applied in different applications only by modifying the proportionality factors. For example, for applications or scenarios where an induction motor service factor of 1.25 is required (whose specifications are described in the previous section), only the proportionality factors are recalculated, resulting in (25), (26), and (27); these factors are used in the minimum capacitance equation developed in this paper, resulting in (28), proposing a voltage ripple of 3%. \[K_{dc}=2.900 \tag{25}\]  
\[K_{4}=1.950 \tag{26}\]  
\[K_{6}=1.800 \tag{27}\]  
\[C=378\;\mu \tag{28}\]  
Finally, the process to achieve the transfer of the analysis and the developed equation is listed below:  
1. Select the application for which the induction motor will be used. Select the application for which the induction motor will be used to obtain the design specifications for the system. 2. Identify the study intervals. That is the intervals of maximum energy return to the decoupling system. 3. Calculate the proportionality factors through (6), considering the intervals established in the previous point. 4. Substitute the obtained values of proportionality factors and the design specifications of the selected application in the developed main equation of minimum required capacitance (19).