# High Frequency Transformers for Solid-State Transformer Applications
## 2 Modulation of the Converter System
### Transformer Modeling
\[f\big{(}A_{p}\big{)}=a_{0}\ A_{p}{}^{2}-a_{1}\ A_{p}{}^{\frac{7}{4}}+a_{2}\] (3) where the parameters \(a_{0}\), \(a_{1}\) and \(a_{2}\) are given by Equations (4), (5) and (6) respectively [30]. \[a_{0}=\frac{k_{c}\ K_{c}\ f^{\alpha}\ B_{max}^{\beta}}{\rho_{w}\ V_{w}\ k_{u}}\] (4) \[a_{1}=\frac{h_{c}\ k_{a}\ \Delta T}{\rho_{w}\ k_{w}\ k_{u}}\] (5) \[a_{2}=\left(\frac{\sum VA}{K_{V}\ f\ B_{sat}\ k_{f}\ k_{u}}\right)^{2}\] (6) Based on the determined analytical value, \(A_{p}\), a magnetic core with an equal or greater value should be chosen. 5. Calculate the Number of Turns: The number of turns of each winding can be described by Equation (7). \[N=\frac{V_{rms}}{K_{v}\ B_{max}\ A_{m}\ f}\] (7) where the parameter \(A_{m}\) represents the effective cross-sectional area of the core, which is given by the product between the physical cross-sectional area of the core (\(A_{c}\)) and the staking factor (\(k_{f}\)) [17;30]. 6. Conductor Sizing: For the dimensioning of the conductors to be used it is necessary to determine the current density (\(J_{o}\)). To determine this parameter, it is necessary to compare the value of the optimum flux density (\(B_{opt}\)) and the value of the saturation flux density (\(B_{sat}\)). Based on [30]:
7. If \(B_{opt}<B_{sat}\), the value of \(J_{o}\) is given by Equation (8). \[J_{o}=K_{t}\;\sqrt{\frac{\Delta T}{2k_{u}}}\;\frac{1}{\sqrt[8]{A_{p}}} \tag{8}\]
* If \(B_{opt}>B_{sat}\), the value of \(J_{o}\) is given by Equation (9). Figure 3:0 (9)  
For parameter \(A_{p}\), the value of the previously chosen magnetic core should be used. Where the parameters \(V_{c}\), \(V_{w}\) represent the volume of the core and the volume of the windings respectively, these parameters can be calculated from Equations (10) and (11), respectively. Where \(l_{c}\) is the average length of the magnetic part of the core, \(MLT\) is the average length of a coil, and \(W_{a}\) is the area of the magnetic core window. Figure 3:1  
Figure 3:2  
The \(l_{c}\) and \(MLT\) length can be obtained using Equations (12) and (13) respectively, where dimensions \(a\), \(b\), \(d\), \(f\) and \(u\) are obtained from the core dimensions, as in Figure 3:3  
Figure 3:4  
Figure 3:5  
To minimize the skin effect and the proximity effect, the choice of the Litz wire should consider the previously determined current density. However, the sizing of the internal structure of the Litz wire is highly complex. In [20; 31], simplified methods for sizing Litz wire are presented. In this study, we used the data provided by the manufacturers, namely the relationship between section/number of single conductors and the type of construction. * Estimation of Leakage Inductance: The estimation of leakage inductance can be obtained through [PERSON]'s model, but this model was designed to calculate the leakage inductance in a transformer composed of foil conductor windings. Considering that the model under study uses Litz wire windings, it will first be necessary to perform the equivalence between Litz wire and foil conductors and subsequently estimate the leakage inductance value through [PERSON]'s model. The thickness of a sheet conductor, \(d_{eq}\), may be given by Equation (14), where \(d\) represents the diameter of each conductor that makes up the Litz wire. For the height of the foil winding, it will be considered that it occupies the entire height of the window, a situation that will allow simplification of the calculation. The equivalent section can be given by Equation (15), where \(h_{w}\) and \(d_{w}\) represent, respectively, the height and width of the winding composed of Litz conductors. Thus, the number of sheet conductors, m, will be given by Equation (16), where \(N\) represents the number of turns of each winding, and \(n\) represents the number of single conductors that make up the Litz wire [32]. Figure 3:6  
Figure 3:7Figure 3:8  
Using [PERSON]'s model, the estimate of the leakage inductance, referenced to the primary, is given by Equation (17).