# Analysis of Non-Minimum Phase System for AC/DC Battery
## 3 Converter Analysis
### State-Space Averaging and Small-Signal Model Analysis in Boost Mode  
In CCM, the switching cycle has two states. The duty cycle \(d_{1}\) is the fraction of time in which the controlled switch is on. The state-space averaging model is a weighted sum of the state equations in each switching state. The weights are the corresponding fractions of time. The state variables of the converter are the input inductor current \(i_{L1}(t)\), second inductor current \(i_{L2}(t)\), intermediate capacitor voltage \(v_{c}(t)\) and finally the output capacitor voltage  
Figure 2: Simplified model for different mode: (a) Boost mode. (b) Buck mode. \(v_{o}(t)\) as shown in Figure 2a. Assume \(v_{g}(t)\) is positive as a rectified sine waveform. After analyzing different converter switching states, average state-space equations are obtained to represent the converter operation in boost mode as follows:  
\[ \begin{array}{rcl}\frac{di_{L1}(t)}{dt}&=&\frac{v_{g}(t)-\ v_{o}(t)+d_{1}(t)v_{ c}(t)}{L_{1}},\\ \frac{di_{L2}(t)}{dt}&=&\frac{v_{o}(t)\ -\ v_{c}(t)}{L_{2}},\\ \frac{dv_{c}(t)}{dt}&=&\frac{i_{L2}(t)-d_{1}(t)i_{L1}(t)}{\text{C}},\\ \frac{dv_{o}(t)}{dt}&=&\frac{i_{L1}(t)\ -\ i_{L2}(t)}{\text{C}_{o}}-\frac{v_{o}(t) }{R_{o}\text{C}_{o}},\end{array}  \tag{1}\]  
where \(v_{g}(t)\) is the input voltage, \(v_{o}(t)\) is the output voltage, \(d_{1}(t)\) is the duty cycle for the boost mode, \(v_{c}(t)\) is the voltage across the intermediate capacitor \(C\), \(i_{L1}(t)\) is the current through the inductor \(L_{1}\), \(i_{L2}\) is the current in the inductor \(L_{2}\) and \(R_{o}\) is the output resistance calculated from the output power \(P_{o}\) and average output voltage \(V_{o}\). The above equations in (1) are nonlinear due to the multiplication of time-varying quantities. Most of the ac circuit analysis such as Laplace transform and Bode plot are not suitable for nonlinear systems. So, linearization is required for Equation (1). On top of that, linearization is a linear approximation of a nonlinear system that is valid in a small region around an operating point. All the variables in (1) are averaged without switching ripple. The steady-state values of the converter variables are obtained from (1) after equating to zero the different differential equations, and the variations in \(v_{g}(t)\) are assumed to be much slower than the converter dynamics, so the converter always works close to an equilibrium point where it can be represented as follows:  
\[ \begin{array}{rcl}V_{C}&=&V_{o},\\ D_{1}&=&1-\frac{V_{g}}{V_{o}},\\ I_{L1}&=&\frac{V_{o}^{2}}{R_{o}V_{g}},\\ I_{L2}&=&-\frac{-\ V_{o}^{2}+\ V_{g}V_{o}}{R_{o}V_{g}}.\end{array}  \tag{2}\]  
Then, the converter variables can be decomposed as:  
\[ \begin{array}{rcl}v_{g}(t)&=&V_{g},\\ d_{1}(t)&=&D_{1}\ +\ \Delta d_{1}(t),\\ i_{L1}(t)&=&I_{L1}\ +\ \Delta i_{L1}(t),\\ i_{L2}(t)&=&I_{L2}\ +\ \Delta i_{L2}(t),\\ v_{c}(t)&=&V_{c}\ +\ \Delta v_{c}(t),\\ v_{o}(t)&=&V_{o}\ +\ \Delta v_{o}(t),\end{array}  \tag{3}\]  
where variables \(V_{g},D_{1},I_{L1},I_{L2},V_{C},V_{o}\) express the dc equilibrium point and the other terms \(\Delta d_{1}(t),\Delta i_{L1}(t),\Delta i_{L2}(t),\Delta v_{c}(t),\Delta v_{o} (t)\) represents small variation around it. In this converter, the voltage output \(V_{o}\) should be controlled for regulating its average, and the input current \(i_{L1}(t)\) is controlled to reshape the line current. Thus, two transfer functions must be calculated: the duty cycle to input inductor current \(Gi_{L1}d\_u(s)\) and the duty cycle to output voltage \(Gvd\_u(s)\). Linearizing after substituting (3) in (1) and transform differential equations to the Laplace domain allow to obtain:\[ \begin{array}{rcl}G{l_{L1}}{d_{-}}u(s)&=&\frac{{{A_{1}}\left({{s^{3}}+{A_{2}}\,{s ^{2}}+{A_{3}}\,{s}+{A_{4}}}\right)}}{{Den_{-}}u(s)},\\ Gvd_{-}u(s)&=&\frac{{{A_{v}}^{2}\,{V_{g}}\left({{R_{o}}\left({{A_{v}}\,{C\,L_{2} }\,{s^{2}}+1}\right)-{A_{v}}^{2}\,{s}\left({{L_{1}}-{L_{2}}\left({\frac{1}{{{A_ {v}}}}-1}\right)}\right)}\right)}}{{Den_{-}}u(s)},\end{array}  \tag{4}\]  
where,  
\[