## 2 Voltage-Current Characteristic  
Figure 1 shows the typical topology of an LCL-DAB converter, consisting of input and output parts, two H bridges, an LCL resonant tank, and a high-frequency transformer. In the LCL-DAB converter, each H bridge is composed of four switches, used for DC-AC and AC-DC conversion. The LCL resonant tank is the key component for power transmission, including two filter inductors, \(L_{1}\) and \(L_{2}\), and a filter capacitor, \(C_{1}\). Compared to a line-frequency transformer, a high-frequency transformer can decrease the volume of the converter to increase the power density, and the turns ratio of it is \(N\):1. To make the structure symmetric, \(L_{1}=L_{2}\) by default. The LCL tank complicates the time-domain analysis of the converter characteristics, so an effective solution is to adopt the phasor method. The LCL-DAB converter can be equivalent to a simplified model, as shown in Figure 2. The voltages at both sides of the LCL resonant tank are three-level square waves,  
so they can be seen as the superposition of infinite sinusoidal voltages. The voltages \(u_{\rm P}\) and \(u_{\rm S}\) at each frequency excite the currents \(i_{\rm P}\) and \(i_{\rm S}\) at the same frequency. Taking forward power flow as an example, Figure 3 shows the main operating waveforms with the DPS modulation scheme, which shows that the duty ratios of all gate signals are 50%. The inner phase shift time \(d_{1}T\) is the phase shift time between \(S_{1}\) and \(S_{4}\), or \(Q_{1}\) and \(Q_{4}\), corresponding to the time interval of zero voltage time of \(u_{\rm P}\) and \(u_{\rm S}\). The outer phase shift time \(dT\) is defined as the time difference between the symmetry axis of \(u_{\rm P}\) and \(u_{\rm S}\), and its polarity determines the direction of power flow. Since the time origin setting does not affect the conclusion of the following analysis, the origin of the time axis is set to the midpoint of the positive level of \(u_{\rm S}\) for convenience of derivation. Therefore, the time-domain expressions of \(u_{\rm P}\) and \(u_{\rm S}\) are  
\[u_{\rm P}(t)=\left\{ \begin{array}{ll}0&(\frac{1-4d}{4}T<t<\frac{1-4d+2d_{1}}{ 4}T)\\ -U_{\rm in}&(\frac{1-4d+2d_{1}}{4}T<t<\frac{3-4d-2d_{1}}{4}T)\\ 0&(\frac{3-4d-2d_{1}}{4}T<t<\frac{3-4d+2d_{1}}{4}T)\\ U_{\rm in}&(\frac{3-4d+2d_{1}}{4}T<t<\frac{5-4d-2d_{1}}{4}T)\\ 0&(\frac{5-4d-2d_{1}}{4}T<t<\frac{5-4d}{4}T)\end{array} \right. \tag{1}\]  
\[u_{\rm S}(t)=\left\{ \begin{array}{ll}0&(\frac{1}{4}T<t<\frac{1+2d_{1}}{4}T) \\ -NU_{\rm o}&(\frac{1+2d_{1}}{4}T<t<\frac{3-2d_{1}}{4}T)\\ 0&(\frac{3-2d_{1}}{4}T<t<\frac{3+2d_{1}}{4}T)\\ NU_{\rm o}&(\frac{3+2d_{1}}{4}T<t<\frac{5-2d_{1}}{4}T)\\ 0&(\frac{5-2d_{1}}{4}T<t<\frac{5-2d_{1}}{4}T)\end{array} \right. \tag{2}\]  
Figure 1: Topology of the LCL-DAB DC-DC converters. Figure 2: Equivalent model of the LCL-DAB converter. For a periodic signal \(f(x)\), it can be expanded into the form shown in (3), i.e., Fourier series. \[f(x)=\frac{1}{2}a_{0}+\sum\limits_{n=1}^{\infty}[a_{n}\cos(n\omega x)+b_{n}\sin(n \omega x)] \tag{3}\]  
\[\left\{ \begin{array}{l}a_{n}=\frac{2}{T}\int\limits_{T}f(x)\cos(n\omega x)dx\\ b_{n}=\frac{2}{T}\int\limits_{T}f(x)\sin(n\omega x)dx\end{array} \right. \tag{4}\]  
where \(T\) is the period of \(f(x)\) and \(\omega=2\pi/T\). Since the voltages and currents of the LCL resonant tank are periodic signals, the Fourier series expressions of them can be derived to (5) by substituting (1) and (2) into (4), respectively. Then, the phasor expression at each frequency can be expressed as (6). \[\left\{ \begin{array}{l}u_{\text{P}}(t)=\sum\limits_{n=1,3,5,\ldots}(-1)^{ \frac{n-1}{2}}\frac{4U_{\text{in}}}{n\pi}\cos(n\pi d_{1})\cos(n\omega_{s}t+2n \pi d)\\ u_{\text{S}}(t)=\sum\limits_{n=1,3,5,\ldots}(-1)^{\frac{n-1}{2}}\frac{4 NU_{ \text{in}}}{n\pi}\cos(n\pi d_{1})\cos(n\omega_{s}t)\end{array} \right. \tag{5}\]  
\[\left\{