# Ferrites and Nanocrystalline Alloys Applied to DC-DC Converters for Renewable Energies
## 3 Core Dimensions, Core Losses, and Winding Losses
### Geometric Dimensions  
The geometric dimensions of the core obtained with each material can be used to estimate the final power density in the MFT. The calculation of the core dimensions is obtained based on the input power, turns ratio, MFT frequency, design flux density, and the minimum required efficiency. These parameters are reflected in the transversal section area of the MFT (\(A_{c}\)). This area confines the magnetic flux of both windings, its precise calculation represents a tendency to obtain an MFT with high performance, favorable temperature conditions, high power density, and high efficiency for a specific frequency. In addition, from this result, the amount of material required in the core for the initial design power and frequency is obtained [16].  
\[A_{c}=\frac{\left(P_{in}+\frac{p_{in}}{\eta_{1}}\right)\cdot 10^{4}}{k_{f}k_{u} \int\!fW_{a}B_{ac}} \tag{1}\]  
where \(P_{in}\) is the input power in the MFT, \(\eta_{1}\) is the minimum efficiency required to obtain, \(K_{f}\) is the waveform factor (4.44 for sine waves, 4.0 for square waves), \(K_{u}\) is the window utilization factor (0.4), \(J\) is the current density, \(f\) is the frequency, \(B_{ac}\) is the design flux density, and \(W_{a}\) the window area.  
For \(B_{ac}\), two conditions must be met. The first condition is that \(B_{ac}<B_{max}\) (\(B_{max}\), maximum flux density of the material). The second condition is that given a \(B_{ac}\) and frequency the specific power losses (w/kg) are obtained, which must meet the following condition.  
\[\frac{P_{fe1}}{kg}\cdot(Weight)<1\%Pin \tag{2}\]  
where P\({}_{fe1}\) are the specific power losses calculated and weight are the kg obtained from the MFT after Equation (1). From the initial condition of efficiency greater than 98%, due to this, a maximum of 1% is designated for losses in the core and 1% for losses in the winding.  
Figure 3 shows the dimensions of the nanocrystalline alloy and ferrite cores.  
The nanocrystalline core obtained is 73.26% smaller than the ferrite core. The reason for this is the different value the flux density of each material.  
The cores are toroidal type because they are readily available for both materials, and also because its lower dispersed flux associated with the geometry compared to the CC or EE core types. Higher flux density means higher power density in the MFT [19]. The calculation of the number of turns in the MFT of the primary winding (\(N_{p}\)) and secondary winding (\(N_{s}\)) is carried out employing Equations (3) and (4), respectively. Table 3 presents a  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Material 1** & **Material 2** \\ \hline Material & Ferrite (3C90) & Nanocrystalline (Vitroperm 500 F) \\ Core & Block core & Laminate (0.02 mm) \\ Maximum flux density & 0.35 T & 1.2 T \\ Permeability & 2000–4000 & 15,000–150,000 \\ Cost & Low & High \\ \hline \hline \end{tabular}
\end{table}
Table 2: Characteristics of ferrites and nanocrystalline alloys for MFT design.  
comparison of the winding volume (\(V_{winding}\)), core volume (\(V_{core}\)), core weight (\(W_{core}\)), and total weight (\(W_{MFT}\)) of each MFT designed.  
\[N_{p}=\frac{V_{in}\cdot 10^{4}}{k_{f}B_{ac}fA_{c}} \tag{3}\]  
\[N_{s}=\frac{N_{p}\cdot V_{0}}{V_{in}} \tag{4}\]  
where \(V_{in}\) is the input voltage, and \(V_{o}\) is the input voltage.  
The winding volume of the nanocrystalline MFT is 46.93% lower than the ferrite MFT. The total calculated volume of the MFT with a nanocrystalline core is 56.29% lower than the MFT with a ferrite core.