# Design and Control of a Battery Charger/Discharger Based on the Flyback Topology
## 5 Design Procedure and Application Example
### Selection of the HFT  
The second step is to select the HFT. Figure 5 shows the effect of \(L_{m}\) and \(n\) values on both \(\delta l_{m}\) and \(F_{sw}\) magnitudes. Such a figure was constructed using expressions (16), (17), (33) and (44) with the parameters in Table 1; the figure shows that reducing \(L_{m}\) increases, exponentially, the current ripple; while the switching frequency depends on both the ripple and inductance magnitudes. Decreasing the turn ratio \(n\) affects, in a similar way, both the current ripple and switching frequency. Thus, decreasing at the same time both the inductance and turn-ratio is not a suitable option; instead, increasing both \(L_{m}\) and \(n\) enables to select small ripples and low switching frequencies, but the HFT becomes bulkier, heavier, and more expensive. The previous figure is useful to design a custom HFT since such an analysis enables to fine-tune both the turn-ratio and magnetizing inductance. However, construct an HFT is not a trivial task since specialized equipment is needed, e.g., a frequency analyzer. Therefore, using a commercial HFT is a simpler alternative; for example, Table 2 reports the characteristics of four commercial HFT designed for dc/dc applications. The turn-ratio of those HFT can be used to extract sections of Figure 5 to evaluate the current ripple and switching frequency imposed by those transformers. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameter** & **symbol** & **Value** \\ \hline Battery voltage & \(v_{b}\) & 12 [V] \\ Reference value for the bus voltage & \(v_{r}\) & 48 [V] \\ Maximum ripple of the DC bus voltage & \(\delta v_{dc}/v_{dc}\) & 0.5 [\%] \\ Maximum perturbation of the DC bus voltage & \(\Delta v_{dc}/v_{dc}\) & \(\pm\)3.5 [\%] \\ Settling time of the DC bus voltage & \(t_{s}\) & 1 [ms] \\ Maximum switching frequency & \(F_{sw,max}\) & 30 [kHz] \\ Maximum ripple of the magnetizing current & \(\delta l_{m}\) & 5 [A] \\ Maximum dc current & \(i_{dc}\) & \(\pm\)1 [A] \\ Maximum instantaneous current perturbation & \(\Delta i_{dc}\) & \(\pm\)2 [A] \\ Maximum dc current derivative & max\(\left(\frac{dif_{w}}{dt}\right)\) & \(\pm\)50 [A/ms] \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Parameters for the application example. Figure 6 shows the effect of \(L_{m}\) on both \(\delta_{im}\) and \(F_{sw}\) for the turn-ratio values of the HFT reported in Table 2, where a wide range of turn-ratios are available; i.e., from \(n=1.4\) to \(n=12\). The figure shows that XFRMS (\(n=1.4\)) and Vitec (\(n=5.4\)) HFT are able to provide both \(\delta_{im}<5\) [A] and \(F_{sw}<30\) [kHz], operating near the limit of the feasibility zone due to the \(L_{m}\) and \(L_{k}\) values of those HFT, which are reported in Table 2. However, the Pulse (\(n=12\)) HFT will provide a switching frequency under 20 [kHz], which is not desirable since that frequency is in the range of the audible noise. A similar, but much worst condition is achiever with the Nascent (\(n=8\)) HFT, which will provide a switching frequency under 5 [kHz]. Another important analysis needed to define the HFT concerns the range of duty cycle available for the SMC: the average duty cycle of the flyback converter must not be near the saturation, otherwise the duty cycle could be saturated when the SMC acts to compensate a perturbation; therefore, it is desirable to have a duty cycle margin for the SMC operation. In this example, a margin of 30% is selected, thus the average duty cycle must be between 30% and 70%; such a value can be modified depending on the load profile. The average duty cycle achieved with the HFT of Table 2 was calculated from expressions (6), (15), and (16), and it is summarized in Figure 7. This analysis shows that XFRMS (\(n=1.4\)) and Pulse (\(n=12\)) transformers require an average duty cycle of 74% and 25%, respectively, thus those HFT are out of the desired duty cycle range.