# Analytical Selection of Leakage Inductance for Single-Phase Dual Active Bridge Converters
## 3 Reactive Power Command Generation and Calculation of Leakage Inductance  
### Generation of Reactive Power Command  
The total active power can be expressed as the sum of odd harmonics [16] via  
\[ \begin{array}{ll}P_{T}&=\sum\limits_{n=1,3,5..}^{\infty}P_{n}\\ &=\sum\limits_{n=1,3,5..}^{\infty}\frac{8V_{in}V_{o}}{n^{3}\pi^{2}\omega_{s} L_{e}}\cos\bigg{(}n\pi\frac{d_{1}}{2}\bigg{)}\cos\bigg{(}n\pi\frac{d_{2}}{2}\bigg{)}\sin(n \varphi)\end{array}  \tag{21}\]  
Figure 4: Control block diagram. Figure 5 displays curves representing the first, third, and fifth harmonics' active power components, along with the total active power. These power components are normalized using \(P_{N}=V_{in}{}^{*}V_{o}/(8f_{s}{}^{*}L_{e})\), with \(f_{s}\) denoting the switching frequency. It can be noticed that the graph illustrating the total power obtained through the conventional piecewise linear approach (\(P_{T}\)) peaks at a maximum value of 1. Particularly noteworthy is the first harmonic's active power component (\(P_{1}\)), which, among all the odd harmonics (\(P_{1}\), \(P_{3}\), and \(P_{5}\)), exhibits the highest amplitude, closely resembling the overall power obtained through the conventional piecewise linear method. The variables used in this section are listed in Table 2.  
The first harmonics' active and reactive powers can be expressed as [17]  
\[ \begin{bmatrix}P_{1}(k)\\ Q_{1}(k)\end{bmatrix} =\frac{1}{\omega_{s}L_{e}} \begin{bmatrix}v_{p1}(k)v_{s1}(k) \sin(\varphi)\\ v_{p1}(k)(v_{p1}(k)-v_{s1}(k)\cos(\varphi))\end{bmatrix}  \tag{22}\]  
where \(v_{p1}(k)\) and \(v_{s1}(k)\) represent the root mean square (rms) voltages of the first harmonic components of the primary and secondary side voltages, respectively. The expressions for the active (\(i_{leP1}\)) and reactive (\(i_{leQ1}\)) components of the inductor current can be expressed as  
\[ \begin{bmatrix}i_{leP1}(k)\\ i_{leQ1}(k)\end{bmatrix} =\frac{1}{\omega_{s}L_{e}} \begin{bmatrix}v_{s1}(k)\sin( \varphi)\\ v_{p1}(k)-v_{s1}(k)\cos(\varphi)\end{bmatrix}  \tag{23}\]  
Using Equations (22) and (23), the rms inductor current (\(i_{leRMS}(k)\)) can be expressed as  
\[i_{leRMS}(k)=\frac{1}{\omega_{s}L_{e}}\sqrt[4]{v_{s1}}^{4}(k)+4v_{s1}{}^{2}(k) \omega_{s}L_{e}\cdot Q_{1}(k)-4\omega_{s}{}^{2}L_{e}{}^{2}\cdot P_{1}{}^{2}(k) \tag{24}\] \begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & \multicolumn{1}{c}{**Description**} \\ \hline \(P_{T}\) & Total active power \\ \(v_{p1}(t)\) & First harmonic component of the primary side rms voltage \\ \(v_{p2}(t)\) & First harmonic component of the secondary side rms voltage \\ \(i_{leP1}(t)\) & Active component of the first harmonic inductor current \\ \(i_{leQ1}(t)\) & Reactive component of the first harmonic inductor current \\ \(i_{leRMS}(t)\) & Inductor RMS current \\ \(I_{max}\) & Maximum Leakage Inductance \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Nomenclature. Figure 5: Active power curves for different harmonics components. It is evident from Equation (24) that through the appropriate selection of \(v_{s1}(k)\) and \(Q_{1}(k)\), the current through the inductor can be minimized for any load. The criterion for achieving the minimum rms current through the inductor at any load is given by  
\[Q_{1}(k)=0 \tag{25}\]  
\[v_{s1}(k)=\sqrt{2\omega_{s}L_{e}\cdot P_{1}(k)} \tag{26}\]  
Therefore, the phase shift \(d_{2}\) can be computed using Equation (26) via  
\[d_{2}(k)=\frac{2}{\pi}\cos^{-1}\left(\sqrt{\frac{\omega_{s}L_{e}\cdot P_{1}(k )}{4{V_{o}}^{2}}}\right) \tag{27}\]  
Figure 6 shows the reactive power range with active power demand, indicating that even at the same load, the amount of circulating power can be different based on the settings of the control variables. It becomes evident from the graph that precise adjustment of all phase shifts can lead to the elimination of the circulating power. To consistently achieve the zero reactive power condition across all ranges of active power demands, the leakage inductance must not exceed \(L_{emax}\).