# Minimization of the Decoupling Circuit (DC-Link) in Induction Motors
## 2 Energy Flow Analysis of the Study Case
### Development of a Mathematical Model  
For the analysis and mathematical development of the proposed capacitance equation, the analysis of the currents presented in Figure 10 are used as a starting point. The development of the calculation of the minimum capacitance required is addressed in Equations (1)-(19), which start from [PERSON]'s current law, taking as reference the incoming and outgoing currents in the decoupling circuit, as well as the operating mode of the load [39; 40]. Subsequently, the data pertaining to the critical case of operation, which is characterized by the highest energy return to the decoupling circuit, is addressed. This analysis neglects the effects of heating and degradation of the DC-link. Using [PERSON]'s law of nodes, obtaining:  
\[I_{dc}=i_{C}+i_{inv} \tag{1}\]  
where \(I_{DC}\) is the DC-DC converter current; \(i_{C}\) is the capacitor current and \(i_{inv}\) is the inverter current. Knowing that the capacitor current is equal to the capacitance multiplied by the derivative of the voltage across the capacitor concerning time (2). Substituting (2) in (1) we obtain (3):  
\[i_{c}=C\frac{dVc}{dt} \tag{2}\]  
Figure 10: Identification of study currents. \begin{table}
\begin{tabular}{c c} \hline
**Section** & **Interval** \\ \hline
1 & \(t_{0}\leq t\leq t_{1}\) \\
2 & \(t_{1}\leq t\leq t_{2}\) \\
3 & \(t_{2}\leq t\leq t_{3}\) \\
4 & \(t_{3}\leq t\leq t_{4}\) \\
5 & \(t_{4}\leq t\leq t_{5}\) \\
6 & \(t_{5}\leq t\leq t_{6}\) \\ \hline \end{tabular}
\end{table} 
Table 3: Distribution of sections and intervals in the critical case. Figure 9: Maximum energy return intervals. \[I_{dc}=C\frac{dVc}{dt}+i_{inv} \tag{3}\]  
where \(C\) is the decoupling capacitance; \(\frac{dVc}{dt}\) is the ratio of voltage change with respect to time. The capacitance \(C\) is solved from (3) and (4) assuming the derivatives as an increase in voltage value, \(C\) is obtained with (5). \[C=\frac{I_{dc}-i_{inv}(\Delta t)}{\Delta V_{c}} \tag{4}\]  
\[C=\frac{I_{dc}-i_{inv}(t)}{\Delta V_{c}} \tag{5}\]  
Subsequently, the ratio of the nominal power of the system with the power of the case study is determined, in which the critical point of returned energy is carried out because under this circumstance the capacitor must be able to handle the maximum variation. The above relation is performed employing proportionality factors, obtaining (6). \[K=\frac{P_{mxreq}}{P_{nom}} \tag{6}\]  
where \(K\) is the general proportionality factor; \(P_{nom}\) s is the nominal power of the system and \(P_{mxreq}\) is the worst-case power (case study). Deducting \(P_{mxreq}\) from (6) (7) and (8) are obtained:  
\[P_{mxreq}=KP_{nom} \tag{7}\]  
\[P_{mxreq}=I_{mxreq}V_{dc} \tag{8}\]  
where \(I_{mxreq}\) is the worst case current (study case) and \(V_{DC}\) is the DC-DC converter voltage. Equaling (7) and (8) we obtain (9). From this, the maximum current required in the worst case is obtained according to (10). \[KP_{nom}=I_{mxreq}V_{dc} \tag{9}\]  
\[I_{mxreq}=\frac{KP_{nom}}{V_{dc}} \tag{10}\]  
Considering the intervals discussed at the end of the previous section where the case study was selected. It is solved \(i_{inv}\) of (11), considering the study times, obtaining (12):  
Figure 9:0  
Figure 9:1  
However, as presented in Figure 9 the current \(I_{mxreq5}\) of the intervals \(t_{4}\) and \(t_{5}\) are zero. Considering this solution (12) is obtained (13). Figure 9:2  
To obtain the inverter current in terms of power, (10) is substituted into (13), and (14) is obtained:  
Figure 9:3  
The proportionality factors \(K_{4}\) and \(K_{6}\) have a negative value derived from the direction of the current, which is returned from the load to the source. Subsequently, it is sought to obtain a relationship between the intervals used in (14), concerning the inverter switching frequency used. This is because when varying said frequency, the duration of the intervals used in the capacitance equation developed is modified. Through the analysis of the intervals in which the critical case occurs under the conditions established in the work.