# Ferrites and Nanocrystalline Alloys Applied to DC-DC Converters for Renewable Energies
## 3 Core Dimensions, Core Losses, and Winding Losses
### Calculation of the Dispersion Inductance  
The magnitude of the dispersion inductance (\(L_{d}\)) in the MFT is crucial determining the control and operation range of the DC-DC converters with DAB [24].  
The dispersion inductance represents the dispersed flow of the transformer observed from an equivalent circuit. In the case of MFTs, a low dispersion inductance represents a lower dispersed flow, which results in a tendency to obtain high efficiency. The calculation of this parameter is presented and verified using the finite element method in [24] to assure high precision in the calculation. The classical equation to calculate \(L_{d}\) and a new equation presented in [24] are shown in Equations (12) and (13), respectively. Equation (13) presents a higher precision [24]. In this document, \(L_{d}\) is calculated with Equation (13).  
\[L_{d}=\mu_{0}MLT_{pri}\frac{m_{1}^{2}N_{1}^{2}}{h_{\omega}}\bigg{[}d_{iso}+ \frac{m_{1}d_{pri}+(m_{1}-1)d_{pri}+m_{2}d_{sec}+(m_{2}-1)d_{ins2}}{3}\bigg{]} \tag{12}\]  
\[\begin{split} L_{d}=\mu_{0}\frac{N_{11}^{2}}{h_{\omega}}m_{1}& [MLT_{iso}m_{1}d_{iso}+MLT_{pri}\frac{(m_{1}-1)(2m_{1}-1)}{6}d_{ins1}+MLT_{ sec}\frac{m_{1}(m_{2}-1)(2m_{2}-1)}{6m_{2}}d_{ins2}\\ &+MLT_{pri}\frac{\sin\left(\frac{2\Delta_{1}}{\Delta\delta} \right)4a\delta^{2}\left(m_{1}^{2}-1\right)+4d_{pri}(2m_{1}^{2}+1)}{24\left( \sin\frac{2\Delta_{1}}{\Delta\delta}\right)^{2}}\\ &-MLT_{pri}\frac{\kappa\delta^{2}\left(\frac{4\Delta_{1}}{\Delta \delta}\right)\left(2m_{1}^{2}+1\right)-8d_{pri}(1-m_{1}^{2})\cos\left(\frac{2 \Delta_{1}}{\Delta\delta}\right)}{24\left(\sin\frac{2\Delta_{1}}{\Delta \delta}\right)^{2}}\\ &+MLT_{sec}\frac{m_{1}}{m_{2}}\frac{\sin\left(\frac{2\Delta_{ 2}}{\Delta\delta}\right)4a\delta^{2}\left(m_{1}^{2}-1\right)+4d_{acc}\left(2m_ {1}^{2}+1\right)}{24\left(\sin\frac{2\Delta_{2}}{\Delta\delta}\right)^{2}}\\ &-MLT_{sec}\frac{m_{1}}{m_{2}}\frac{\kappa\delta^{2}\sin\left( \frac{4\Delta_{2}}{\Delta\delta}\right)\left(2m_{2}^{2}+1\right)}{24\left( \sin\frac{2\Delta_{2}}{\Delta\delta}\right)^{2}}+MLT_{sec}\frac{m_{1}}{m_{2}} \frac{8d_{sec}\left(1-m_{2}^{2}\right)\cos\left(\frac{2\Delta_{2}}{\Delta \delta}\right)}{24\left(\sin\frac{2\Delta_{2}}{\Delta\delta}\right)^{2}}\bigg{]} \end{split} \tag{13}\]  
where:  
\(\mu_{0}\)= vacuum permeability  
\(d_{ins1}\) = insulation distance between the layers of the primary  
\(d_{ins2}\) = insulation distance between the layers of the secondary  
\(m_{1}\) = number of layers in the primary  
\(m_{2}\) = number of layers in the secondary  
\(MLT_{iso}\) = mean length of the isolation distance  
\(MLT_{pri}\) = mean length turns of primary portion  
\(MLT_{sec}\) = mean length turns of secondary portion  
\begin{table}
\begin{tabular}{c c c} \hline
**MFT** & **MFT with Ferrite Core** & **MFT with Nanocrystalline Core** \\ \hline \(L_{d}\) & 4.83 W & 1.69 W \\ \(P_{\delta}\) & 2.29 W & 1.32 W \\ \(L_{tot}\) & 7.12 W & 3.01 W \\ Efficiency & 99.28\% & 99.69\% \\ \hline \end{tabular}
\end{table}
Table 5: Losses calculated of the MFTs.  
Table 6 shows the calculated values of \(L_{d}\) for the two transformers. Figure 5 show the curves of \(L_{d}\) values in a range from 600 Hz a 20 kHz for both MFTs, they were obtained with the \"Model 891, 300 kHz Bench LCR Meter-B&K Precision\".  
According to Table 6, the MFT with a nanocrystalline core has 51% lower \(L_{d}\) than the MFT with ferrites. The precise calculation of the \(L_{d}\) is essential to obtain a representative equivalent circuit of the MFT, and thus verify their operation in simulation, as presented in Section 4.