## 2 Modeling
### Modeling  
In this section, the general formulation used to obtain the average and small-signal state-space models of the push-pull converter is developed. Considering the criteria of generality and accuracy, the model integrates all the practical non-idealities of the converter. As for the state vector, as usual, it is formed by the currents or voltages in the energy storage elements, many of them inductances or parasitic capacitances, respectively. Also, from a controller design perspective, the model outputs integrate the main converter variables that are to be controlled, i.e., output voltage and current, which are also measurable. Regarding model inputs, the average developed model includes the converter input voltage as the input variable (single input-multiple output (SIMO) model), while the small-signal model also includes the duty cycle (multiple input-multiple output (MIMO) model). Finally, for their determination, the differential equations governing the converter dynamics are defined using classical circuit theory, applying mesh analysis based on [PERSON]'s voltage and current laws. As previously mentioned, to obtain the dynamical model, the behavior of the push-pull converter will be analyzed based on the operation of its electronic switches. Thus, for the three modes of operation, cases 1-3, the equivalent circuits of Figures 2 and 3a-c, respectively, are obtained. From here, for cases 1-3, the state variables (1)-(8) are obtained. \[v_{L_{P_{1}}}(t)=L_{P_{1}}\frac{di_{L_{P_{1}}}(t)}{dt}=-i_{L_{P_{1}}}(t)\cdot R _{Lp}+i_{C_{P_{1}}}(t)\cdot R_{C_{P}}+v_{C_{P_{1}}}(t)-i_{R_{Nu}}(t)\cdot R_{ Nu} \tag{1}\]  
\[v_{L_{P_{2}}}(t)=L_{P_{2}}\frac{di_{L_{P_{2}}}(t)}{dt}=-i_{L_{P_{2}}}(t)\cdot R _{Lp}+i_{C_{P_{2}}}(t)\cdot R_{C_{P}}+v_{C_{P_{2}}}(t)+i_{R_{Nu}}(t)\cdot R_{ Nu} \tag{2}\]  
\[v_{L_{M}}(t)=L_{M}\frac{di_{L_{M}}(t)}{dt}=i_{R_{Nu}}(t)\cdot R_{Nu} \tag{3}\]  
\[v_{L_{S_{1}}}(t)=L_{S_{1}}\frac{di_{L_{S_{1}}}(t)}{dt}=-i_{L_{S_{1}}}(t)\cdot R _{L_{S}}-N\cdot i_{R_{Nu}}(t)\cdot R_{Nu}-v_{C_{S_{1}}}(t) \tag{4}\]\[v_{L_{S_{2}}}(t)=L_{S_{2}}\frac{di_{L_{S_{2}}}(t)}{dt}=-i_{L_{S_{2}}}(t)\cdot R_{L _{S}}+N\cdot i_{R_{Nu}}(t)\cdot R_{Nu}-v_{C_{S_{2}}}(t) \tag{5}\]  
\[i_{C_{P1}}(t)=C_{P1}\frac{dv_{C_{P1}}(t)}{dt}=\frac{v_{in}(t)}{R_{C_{P}}}-\frac{ v_{C_{\text{OSS}}1}(t)}{R_{C_{P}}}-\frac{v_{C_{P_{1}}}(t)}{R_{C_{P}}} \tag{6}\]  
\[i_{C_{P2}}(t)=C_{P2}\frac{dv_{C_{P2}}(t)}{dt}=\frac{v_{in}(t)}{R_{C_{P}}}-\frac {v_{C_{\text{OSS}}2}(t)}{R_{C_{P}}}-\frac{v_{C_{P_{2}}}(t)}{R_{C_{P}}} \tag{7}\]  
\[i_{C_{F}}(t)=C_{F}\frac{dv_{C_{F}}(t)}{dt}=i_{L_{F}}(t)-i_{R}(t) \tag{8}\]  
where \(N=N_{s}/N_{p}\), and \(i_{R_{Nu}}\) is given by (9):  
\[i_{R_{Nu}}(t)=i_{L_{P_{1}}}(t)-i_{L_{M}}(t)-i_{T_{1}}(t) \tag{9}\]  
By following any of Figure 3, currents \(i_{T_{1}}\) and \(i_{T_{2}}\) in the transformer are given, respectively, by (10) and (11). \[i_{T_{1}}(t)=i_{T_{2}}(t)+\Big{(}-i_{L_{S_{1}}}(t)+i_{L_{S_{2}}}(t)\Big{)}N \tag{10}\]  
\[v_{L_{P_{2}}}(t)=L_{P_{2}}\frac{di_{L_{P_{2}}}(t)}{dt}=-i_{L_{P_{2}}}(t)\cdot R _{Lp}+i_{C_{P_{2}}}(t)\cdot R_{C_{P}}+v_{C_{P_{2}}}(t)+i_{R_{Nu}}(t)\cdot R_{ Nu} \tag{2}\]0  
The converter output current (\(i_{R}\)) is calculated from the converter output voltage (\(v_{R}\)) and the load resistance (\(R\)) as (12). \[v_{L_{P_{2}}}(t)=L_{P_{2}}\frac{di_{L_{P_{2}}}(t)}{dt}=-i_{L_{P_{2}}}(t)\cdot R _{Lp}+i_{C_{P_{2}}}(t)\cdot R_{C_{P}}+v_{C_{P_{2}}}(t)+i_{R_{Nu}}(t)\cdot R_{ Nu} \tag{2}\]1  
In the same way, the rest of the state variables can be obtained, now particularized for each operating mode. Thus, for case 1, the state variables (13)-(17) are obtained.