## 2 Large-Signal Model of Q1S AC-DC Converter
 \tag{3}\]  
By putting three phase shifts' relationships of the ITPS in [16] into (3), the variation rate of the average transformer current within a half switching cycle is  
\[L_{leak}\frac{\mathrm{d}\langle i_{p}\rangle_{T_{S}}}{\mathrm{d}t}=\frac{2}{T_ {S}}\left( \begin{array}{cc}\int_{0}^{D_{\varphi}\frac{T_{S}}{2}}\left(\frac{ v_{dc1}}{L_{leak}}\right)\mathrm{d}t+\int_{0}^{D_{1}\frac{T_{S}}{2}}\left(\frac{- nv_{dc2}}{L_{leak}}\right)\mathrm{d}t\\ +\int_{0}^{(1-D_{1}-D_{\varphi})\frac{T}{2}}\left(\frac{v_{dc1}-nv_{dc2}}{L_{ leak}}\right)\mathrm{d}t\end{array} \right)\equiv 0 \tag{4}\]  
From Figures 1 and 2a, the state equation of \(v_{dc1}\) during \(t\in[t_{0},t_{3})\) is expressed as  
\[C_{1}\frac{\mathrm{d}v_{dc1}}{dt}= \begin{cases}i_{g}-i_{p},&t\in[t_{0},t_{1}) \\ i_{g}-i_{p},&t\in[t_{1},t_{2})\\ i_{g}-0,&t\in[t_{2},t_{3})\end{cases}  \tag{5}\]  
The average value of (5) within a half switching cycle is  
\[C_{1}\frac{\mathrm{d}v_{dc1}}{\mathrm{d}t}=i_{g}-\frac{2}{T_{S}}\left(\int_{t _{0}}^{t_{1}}i_{p}\mathrm{d}t+\int_{t_{1}}^{t_{2}}i_{p}\mathrm{d}t\right)=i_{ g}-\langle i_{dc1}\rangle_{T_{S}} \tag{6}\]  
The state equation of \(i_{g}\) in the positive half switching cycle is acquired from Figure 1:  
\[L_{g}\frac{\mathrm{d}i_{g}}{\mathrm{d}t}+i_{g}r_{L}=u_{g}-v_{dc1},\quad t\in[t _{0},t_{3}) \tag{7}\]  
\(r_{L}\) in (7) is called the filter parasitic resistor of the filter inductor. On the basis of (4), (6), and (7), the large-signal model of the converter under the modulation in Figure 2b is  
\[\frac{\mathrm{d}}{\mathrm{d}t} \begin{bmatrix}\langle i_{p}\rangle_{T_{S}}\\ v_{dc1}\\ i_{g}\end{bmatrix} = \begin{bmatrix}0&0\\ 1/C_{1}&0\\ -r_{L}&1/L_{g}\end{bmatrix}\begin{bmatrix}i_{g}\\ u_{g}\end{bmatrix} + \begin{bmatrix}0&0\\ -1/C_{1}&0\\ 0&-1/L_{g}\end{bmatrix}\begin{bmatrix}\langle i_{dc1}\rangle_{T_{S}}\\ v_{dc1}\end{bmatrix}  \tag{8}\]  
Similarly, the large-signal model in the negative half grid cycle is derived as  
\[\frac{\mathrm{d}}{\mathrm{d}t} \begin{bmatrix}\langle i_{p}\rangle_{T_{S}}\\ -v_{dc1}\\ i_{g}\end{bmatrix} = \begin{bmatrix}0&0\\ 1/C_{1}&0\\ -r_{L}&1/L_{g}\end{bmatrix}\begin{bmatrix}i_{g}\\ u_{g}\end{bmatrix} - \begin{bmatrix}0&0\\ -1/C_{1}&0\\ 0&-1/L_{g}\end{bmatrix}\begin{bmatrix}\langle i_{dc1}\rangle_{T_{S}}\\ v_{dc1}\end{bmatrix}  \tag{9}\]  
In order to characterize (8) and (9) in a unified form, define the following variables. \[v_{dc1}\_{u}= \begin{cases}v_{dc1}&Q_{1}\&Q_{4}=1\\ -v_{dc1}&Q_{2}\&Q_{3}=1\\ \end{cases}  \tag{10}\]From (8)-(10), the large-signal model of the Q1S AC-DC converter in the whole grid cycle is  
\[\frac{\mathrm{d}}{\mathrm{d}t} \begin{bmatrix}\langle i_{p}\rangle_{T_{S}}\\ v_{dc1,u}\\ i_{g}\end{bmatrix} = \begin{bmatrix}0&0\\ 1/C_{1}&0\\ -r_{L}&1/L_{g}\end{bmatrix}\begin{bmatrix}i_{g}\\ u_{g}\end{bmatrix} + \begin{bmatrix}0&0\\ -1/C_{1}&0\\ 0&-1/L_{g}\end{bmatrix}\begin{bmatrix}i_{g1}\\ u_{gL}\end{bmatrix}  \tag{11}\]  
From (11), it is obvious that the differential value of the average transformer current in a switching cycle is zero. This feature is very interesting, because under the ITPS, \(L_{t\_{t\_{t\_{t\_{t\}t\}t\}}}\) does not cause the transient process of the transformer current, and in consequence the total order number of the system decreases. It means even the step change of the three phase shift occurs, the transformer current will reach its final value immediately, and there is no transient process; therefore, the Q1S AC-DC converter under the ITPS is a linear system in essence. From (11), the transfer functions from \(u_{g}\) to \(i_{g}\) and from \(i_{av1\_u}\) to \(i_{g}\) can be expressed as  
\[\left\{ \begin{aligned} & G_{u_{g}}z_{i_{g}}(s)=\frac{i_{g}(s)}{u_{g}(s)}= \frac{s\mathrm{C}_{1}}{L_{g}\mathrm{C}_{1}s^{2}+r_{L}\mathrm{C}_{1}s+1}\\ & G_{i_{av2}i_{g}}(s)=\frac{i_{g}(s)}{i_{av1\_u}s}=\frac{1}{L_{g}\mathrm{ C}_{1}s^{2}+r_{L}\mathrm{C}_{1}s+1}\end{aligned} \right.