# Ferrites and Nanocrystalline Alloys Applied to DC-DC Converters for Renewable Energies
## 3 Core Dimensions, Core Losses, and Winding Losses
### Core Losses and Winding Losses  
At medium frequencies, high permeability is always desirable. According to Table 2, nanocrystalline alloys have a higher flux density and higher permeability. Table 4 shows the power density in kW/L for the flux density of the MFTs designed for each material.  
Core losses and winding losses of both MFTs are calculated using the method presented in [16].  
Winding losses:  
\[L_{\omega}=L_{p}+L_{s} \tag{5}\]  
\[L_{p}=\left(l_{in}^{2}\right)\cdot R_{1} \tag{6}\]  
\[L_{s}=\left(l_{o}^{2}\right)\cdot R_{2} \tag{7}\]  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**MFT** & **Flux Density** & **Power Density (kW/L)** \\ \hline Ferrite core & 0.2 T & 16.13 \\ Nanocrystalline core & 0.8 T & 36.91 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Power density in kW/L of the MFTs.  
Figure 3: Geometric dimensions of: (**left**): nanocrystalline core, (**right**): ferrite core.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **MFT with Ferrite Core** & **MFT with Nanocrystallyne Core** \\ \hline \(V_{core}\) & 22.03 cm\({}^{3}\) & 5.89 cm\({}^{3}\) \\ \(V_{winding}\) & 39.95 cm\({}^{3}\) & 21.20 cm\({}^{3}\) \\ \(V_{total}\) & 61.98 cm\({}^{3}\) & 27.09 cm\({}^{3}\) \\ \(W_{core}\) & 100 yrs & 32.9 yrs \\ \(W_{MFT}\) & 238 yrs & 102 yrs \\ \hline \hline \end{tabular}
\end{table}
Table 3: Volume and weigh of both MFTs.  
where \(L_{\omega}\) are the winding losses in watts, \(L_{p}\) losses in the primary winding, \(L_{s}\) losses in the secondary winding, \(I_{in}\) primary winding current, \(I_{o}\) secondary winding current, \(R_{1}\) primary winding resistance, and \(R_{2}\) secondary winding resistance.  
\(R_{1}\) y \(R_{2}\) are calculated with Equations (8) and (9), respectively.  
\[R_{1} =\big{(}MLT_{1}\cdot N_{p}\cdot\mu\Omega/cm_{1}\big{)}\cdot 10^{-6} \tag{8}\] \[R_{2} =\big{(}MLT_{2}\cdot N_{s}\cdot\mu\Omega/cm_{2}\big{)}\cdot 10^{-6} \tag{9}\]  
where \(MLT_{1}\) and \(MLT_{2}\) are the mean lengths of the primary and secondary windings, respectively; \(\mu\Omega/cm_{1}\) and \(\mu\Omega/cm_{2}\) are the resistances per centimeter of the primary and secondary winding conductors, respectively.  
The MFT with ferrite core features an \(R_{1}\) = 0.0300 \(\Omega\), \(R_{2}\) = 0.1222 \(\Omega\), \(L_{p}\) = 2.52 W, \(L_{s}\) = 2.31 W, and an \(L_{\omega}\) = 4.83 W. The MFT with nanocrystalline core has an \(R_{1}\) = 0.0114 \(\Omega\), \(R_{2}\) = 0.0465 \(\Omega\), \(L_{p}\) = 0.88 W, \(L_{s}\) = 0.81 W, and \(L_{\omega}\) = 1.69 W.  
Core losses:  
Figure 4 shows core losses (W/kg) versus flux density for the nanocrystalline material used in this paper (Vitroperm 500F), for a frequency of 20 kHz [23]. In [23], a more detailed characterization of these nanocrystalline materials is presented for a wider range of frequencies.  
The core losses of the MFT are calculated by Equation (10).  
\[P_{fe}=(P_{fe1})\cdot W_{fe} \tag{10}\]  
where \(P_{fe1}\) are the material losses (W/kg) and \(W_{fe}\) the core weight (kg). In this case for nanocrystalline alloys, \(P_{fe1}\) is 40 W/kg for 20 kHz), and \(W_{fe}\) = 0.033 kg, this specified for 1 kVA. It can be notice that \(P_{fe}\) is directly proportional to the frequency and the flux density.  
These results are subsequently verified with experimental testing in Section 5. Table 5 shows the core losses (\(P_{fe}\)), the winding losses (\(L_{\omega}\)), and the total losses (\(L_{tot}\)) of both MFTs at 20 kHz and 1 kVA.  
The MFT with nanocrystalline core has 65% fewer winding losses than the ferrite MFT. Regarding core losses, the nanocrystalline MFT has almost 42% fewer losses than the ferrite MFT. Finally, the total losses of nanocrystalline MFT are 3.01 W for an efficiency of 99.69%. In the case of the ferrite, MFT total losses are 7.12 W, and efficiency is 99.28%. The total losses and efficiency obtained in these designs are experimentally verified in Section 5.  
Figure 4: Core losses versus flux density for nanocrystalline materials (20 kHz).