# Analytical Selection of Leakage Inductance for Single-Phase Dual Active Bridge Converters
## 2 Steady State Analysis  
A single-phase DAB DC/DC converter incorporates a high-frequency transformer, which is essential for power flow from source to load. It employs two H-bridges to manage the conversion from the input DC-link voltage (\(V_{in}\)) to the output voltage (\(V_{o}\)). Figure 1 illustrates the schematic layout of the DAB converter, illustrating its key components and the interconnections between the H-bridges and the transformer. The leakage inductance of the transformer winding is denoted by \(L_{e}\) and the combined parasitic resistance of the transformer winding and the on-state resistance of the semiconductor switches is represented by \(R_{e}\), while \(v_{pr}(t)\) and \(v_{sr}(t)\) represent the input and output voltages of the high-frequency transformer. The variables used in this section are listed in Table 1.  
The converter utilizes three control variables to manage the active and circulating powers: the inner phase shift of the primary H-bridge (\(d_{1}\)), the inner phase shift of the secondary H-bridge (\(d_{2}\)), and the phase shift between the voltages of the two H-bridges (\(d_{3}\)). Figure 2 illustrates the primary and secondary voltage waveforms along with the control variables \(d_{1}\), \(d_{2}\), and \(d_{3}\). The range of these control parameters is limited to \(0\leq d_{1}\leq 1\), \(0\leq d_{2}\leq 1\), and \(-1\leq d_{3}\leq 1\) ensuring precise control over power flow in the system. By choosing appropriate values for these control variables, the power flow within the converter can be effectively controlled. The transformer windings voltage can be expressed as [15]  
\[v_{pr}(t)=\sum_{n=1,3,5,..}^{\infty}\frac{4V_{in}}{n\pi}\cos\left(n\pi\frac{d_{1} }{2}\right)\sin(n\omega_{s}t), \tag{1}\] \begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & **Description** \\ \hline \(V_{in}\) and \(V_{o}\) & Input and output voltage \\ \(f_{s}\) & Switching frequency \\ \(L_{e}\) & Leakage inductance \\ \(R_{e}\) & Sum of switches on-resistance \\  & and transformer winding resistance \\ \(v_{pr}(t)\) and \(v_{sr}(t)\) & Primary and secondary side voltage of transformer \\ \(i_{lc}(t)\) & Inductor current \\ \(d_{1}\) and \(d_{2}\) & Primary and secondary side H-Bridge’s inner phase shift \\ \(d_{3}\) & Phase shift between H-bridge voltages \\ \(\varphi\) & Total phase shift between transformer voltages \\ \(\omega_{s}(t)\) & Angular frequency of voltages and current \\ \(v_{pr1}(t)\), \(v_{pr3}(t)\), and \(v_{pr5}(t)\) & 1 st, 3 rd, and 5 th harmonics of \(v_{pr}(t)\) \\ \(v_{sr1}(t)\) & 1 st harmonics of \(v_{pr}(t)\) \\ \(i_{lc1}(t)\), \(i_{lc3}(t)\), and \(i_{lc5}(t)\) & 1 st, 3 rd, and 5 th harmonics of \(i_{lc}(t)\) \\ \(P_{1}(t)\) and \(Q_{1}(t)\) & First harmonic active and reactive powers \\ \(v_{pd}(t)\) and \(v_{pq}(t)\) & Primary first harmonic voltages in \(dq\) reference frame \\ \(v_{sd}(t)\) and \(v_{sq}(t)\) & Secondary first harmonic voltages in \(dq\) reference frame \\ \(l_{lat}(t)\) and \(i_{lap}(t)\) & First harmonic inductor current in \(dq\) reference frame \\ \(T_{s}\) & Sampling time \\ \(V_{n\!rf}\) & Reference voltage \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Nomenclature. Figure 2: Primary and secondary H-bridge voltage waveforms and phase shifts. \[v_{sr}(t)=\sum_{n=1,3,5,..}^{\infty}\frac{4V_{o}}{n\pi}\cos\left(n\pi\frac{d_{2}}{ 2}\right)\sin(n\omega_{s}t-n\varphi), \tag{2}\]  
\[\varphi=\left(d_{3}+\frac{d_{2}-d_{1}}{2}\right)\pi, \tag{3}\]  
where \(\varphi\) represents the total phase shift between the two voltages and \(\omega_{s}\) denotes the angular frequency. The current flowing through the inductor can be expressed as  
\[i_{le}(t)=\sum_{n=1,3,5,..}^{\infty}\frac{4V_{in}(t)}{n\pi Z}\cos(n\pi\frac{d_{ 1}}{2})\sin(n\omega_{s}t)-\frac{4V_{o}(t)}{n\pi Z}\cos(n\pi\frac{d_{2}}{2}) \sin(n\omega_{s}t-n\varphi) \tag{4}\]  
where \(Z=\sqrt{{R_{e}}^{2}+\left(n\omega_{s}{L_{e}}\right)^{2}}\).