## 3 Modeling of the LCL-DAB Converter with DPS Modulation Scheme  
Due to the switching action of the LCL-DAB converter, a severe nonlinear is presented in the system, which brings great difficulty to the system design process. Therefore, the small-signal analysis method is adopted to build the mathematical model of the LCL-DAB converter. The variables are approximated to the 0 and 1 st order, which can avoid complex calculations and ensure high accuracy. With DPS modulation scheme, the switching functions of H1 and H2 are defined respectively as  
\[s_{1}(t)= \begin{cases}1,&\text{S}_{1}\text{ and }\text{S}_{4}\text{ are turned on}\\ 0,&\text{S}_{1}\text{ and }\text{S}_{3},\text{ or }\text{S}_{2}\text{ and }\text{S}_{4}\text{ are turned on}\\ -1,&\text{S}_{2}\text{ and }\text{S}_{3}\text{ are turned on}\end{cases}  \tag{10}\]  
\[s_{2}(t)= \begin{cases}1,&\text{Q}_{1}\text{ and }\text{Q}_{4}\text{ are turned on}\\ 0,&\text{Q}_{1}\text{ and }\text{Q}_{3},\text{ or }\text{Q}_{2}\text{ and }\text{Q}_{4}\text{ are turned on}\\ -1,&\text{Q}_{2}\text{ and }\text{Q}_{3}\text{ are turned on}\end{cases}  \tag{11}\]  
Therefore, based on the above analysis, the time-domain expression of \(u_{\text{P}}\) and \(u_{\text{S}}\) can be rewritten to  
\[u_{\text{P}}(t)=\text{s}_{1}(t)\cdot U_{\text{in}} \tag{12}\]  
\[u_{\text{S}}(t)=\text{s}_{2}(t)\cdot NU_{\text{o}} \tag{13}\]  
Since \(s_{1}(t)\) and \(s_{2}(t)\) are symmetrical in each half period, the DC terms of them are zero, and the real and imaginary parts of the 1 st coefficient of their Fourier series expression are  
\[\left\{ \begin{array}{l}\langle\text{s}_{1}\rangle_{\text{1R}}=\frac{2}{\pi }\cos(\pi d_{1})\cos(2\pi d)\\ \langle\text{s}_{1}\rangle_{\text{1I}}=\frac{2}{\pi}\cos(\pi d_{1})\sin(2\pi d) \end{array} \right. \tag{14}\]  
\[\left\{ \begin{array}{l}\langle\text{s}_{2}\rangle_{\text{1R}}=\frac{2\cos( \pi d_{1})}{\pi}\\ \langle\text{s}_{2}\rangle_{\text{1I}}=0\end{array} \right. \tag{15}\]  
Assuming that the current of inductors and the voltage of capacitors are state variables, so the continuous-time large-signal average equations are  
\[\left\{ \begin{array}{l}L\frac{\text{d}ip(t)}{\text{d}t}=s_{1}(t)u_{\text{in }}(t)-u_{\text{C1}}(t)\\ L\frac{\text{d}is(t)}{\text{d}t}=-Ns_{2}(t)u_{\text{o}}(t)+u_{\text{C1}}(t)\\ C_{\text{dc2}}\frac{\text{d}u_{\text{C}_{dc2}}(t)}{\text{d}t}=Ns_{2}(t)i_{\text{S }}(t)-\frac{u_{\text{o}}(t)}{R}\\ C_{1}\frac{u_{\text{C}_{dc1}}(t)}{\text{d}t}=ip(t)-i_{\text{S}}(t)\\ u_{\text{C}_{dc2}}(t)=u_{\text{o}}(t)\end{array} \right. \tag{16}\]  
It can be analyzed that all the DC terms of \(i_{\text{P}}\), \(i_{\text{S}}\) and \(u_{\text{C1}}\) are equal to zero, and the ac terms of \(u_{\text{o}}\) are zero. Then, a new matrix equation can be derived as  
\[\dot{\mathbf{X}}_{1}=\mathbf{A}_{1}\cdot\mathbf{X}_{1}+\mathbf{B}_{1}\cdot[ \langle u_{\text{in}}\rangle_{0}] \tag{17}\]  
where  
\[\mathbf{X}_{1}=\big{[} \begin{array}{cc}\langle i_{\text{P}}\rangle_{\text{1R }}&\langle i_{\text{P}}\rangle_{\text{1I}}&\langle i_{\text{S}}\rangle_{\text{1R }}&\langle i_{\text{S}}\rangle_{\text{1I}}&\langle u_{\text{C1}}\rangle_{\text {1R}}&\langle u_{\text{C1}}\rangle_{\text{1I}}&\langle u_{\text{o}}\rangle_{0} \end{array} \big{]}^{\text{T}} \tag{18}\]\[\mathbf{A}_{1}= \begin{bmatrix}0&\omega&0&0&\frac{-1}{L}&0&0\\ -\omega&0&0&0&0&\frac{-1}{L}&0\\ 0&0&0&\omega&\frac{1}{L}&0&\frac{-2N\cos(\pi d_{1})}{\pi L}\\ 0&0&-\omega&0&0&\frac{1}{L}&0\\ \frac{1}{\mathrm{C}_{1}}&0&\frac{-1}{\mathrm{C}_{1}}&0&0&\omega&0\\ 0&\frac{1}{\mathrm{C}_{1}}&0&\frac{-1}{\mathrm{C}_{1}}&-\omega&0&0\\ 0&0&\frac{4N\cos(\pi d_{1})}{\pi\mathrm{C}_{42}}&0&0&0&\frac{-1}{\hbar\mathrm{ C}_{42}}\end{bmatrix}  \tag{19}\]  
\[\mathbf{B}_{1}= \begin{bmatrix}\frac{2\cos(\pi d_{1})\cos(2\pi d)}{\pi L}&\frac {2\cos(\pi d_{1})\sin(2\pi d)}{\pi L}&0&0&0&0&0\end{bmatrix} ^{\mathrm{T}} \tag{20}\]  
For the design and analysis of the system controller, the input-output small-signal model needs to be built, which is linear.