# Analytical Selection of Leakage Inductance for Single-Phase Dual Active Bridge Converters
## 4 Design of High-Frequency Transformer
### Core Losses  
For a single-phase transformer design, core loss calculation typically involves the application of the Steinmetz equation. However, given the non-sinusoidal nature of the input voltage, the Improved Generalized Steinmetz Equation [24] is preferred for a more accurate estimation of the core losses. This equation yields the core losses (\(P_{c}\)) per volume formulated as follows:  
\[P_{c}=2^{\alpha+\beta}K_{i}f^{\alpha}B_{m}^{\beta} \tag{34}\]  
where \(\alpha\), \(\beta\), and \(K_{i}\) represent the Steinmetz coefficients.  
The required leakage inductance for the proposed approach is significantly less than that of the SPS-based design approach. As a result, the size of the transformer needed for the converter is considerably reduced. This size reduction is further highlighted through a normalized comparison, as depicted in Figure 9.  
The designed transformer to validate the proposed methodology has its leakage inductance determined using Equation (30) for the laboratory prototype of the converter. The specifications for this prototype are provided in Section 5. The cross-sectional area (\(A_{c}\)) and core volume (\(V_{c}\)) are measured at 356.76 mm\({}^{2}\) and 94,310.6 mm\({}^{3}\), respectively. For identical converter specifications, the SPS-based method necessitates a transformer with 321.43 \(\mu\)H of leakage inductance, which is more than twice the size compared to the proposed method. Opting for the same material, the most suitabl  
Figure 9: Normalized comparison of high-frequency transformer.  
is the 9478117002 E-type core. A comprehensive comparative analysis between the two transformers, including considerations of volume, weight, and core losses, is detailed in Table 3.  
One significant aspect to consider is the heat generation in the single-phase transformer. Heat generation primarily arises from core losses and copper losses. In the proposed method, core losses are reduced due to the smaller size of the required core. Copper losses occur in the transformer windings due to the resistance of the transformer windings and are proportional to the square of the current flowing through the windings.  
When comparing the heat generated in transformers for different approaches, the conventional SPS-based approach typically results in higher core and copper losses due to the need for larger transformers and higher current flows. This leads to more heat generation, affecting the transformer's efficiency and thermal management. Conversely, the proposed method requires a smaller transformer because the required leakage inductance is smaller, which is made possible by the reduced leakage inductance required with the developed power controller-based modulation method for the same active power requirement. Additionally, the current flowing through the transformer is up to 51% less with the same load requirement using the proposed modulation method. This reduction in current substantially decreases copper losses, and the smaller core size reduces core losses. By minimizing both core and copper losses and reducing the current flow, the proposed method generates less heat, thus enhancing the overall efficiency and reliability of the transformer. This approach is crucial for better thermal performance and a longer operational lifespan of the transformer.