# Analysis of AC DC Four-Switch Boost-Buck Battery Charger Converter for EV Applications
## 3 Converter Analysis
### State-Space Averaging and Small-Signal Model Analysis in Boost Mode  
As mentioned, the converter operates in CCM, and the switching cycle of the converter consists of two states. The duty cycle \(d_{1}\) represents the fraction of time during which the switch \(Q_{1}\) is on and \((1-d_{1})\) is representing the time of the switch \(Q_{1}\) is off. The state variables of the converter in the boost mode are the output capacitor voltage \(v_{o}(t)\), intermediate capacitor voltage \(v_{c}(t)\), inductor current \(i_{L1}(t)\), and second inductor current \(i_{L2}(t)\) as depicted in Figure 3a. Under small-ripple approximation, assuming \(v_{g}(t)\) is positive and follows a rectified sine waveform, an analysis of the different converter switching states yields average state-space equations that describe the operation in the boost mode. These equations capture the relations between the state variables, input voltage, output voltage, and control parameters, which can be expressed as  
\[ \begin{array}{lll}\frac{d\dot{t}_{11}(t)}{dt}&=&\frac{v_{g}(t)+v_{c}(t)\;(d_ {1}(t)-1)}{L_{1}},\\ \frac{d\dot{t}_{2}(t)}{dt}&=&\frac{v_{g}(t)-v_{o}(t)+d_{1}(t)v_{c}(t)}{L_{2}}, \\ \frac{dv_{c}(t)}{dt}&=&\frac{(1-d_{1}(t))\;(d_{1}(t)-d_{1}(t)\;i_{L2}(t)}{L_{ 2}},\\ \frac{dv_{c}(t)}{dt}&=&\frac{i_{L2}(t)}{\zeta_{o}}-\frac{v_{o}(t)}{R_{o}\zeta_{ o}},\end{array}  \tag{1}\]  
where \(R_{o}\) is the output resistance calculated from the output power \(P_{o}\) and average output voltage \(V_{o}\). Equations in (1) are inherently nonlinear due to the presence of multiplication between time-varying variables. Therefore, the linearization of the equations is necessary to facilitate the analysis. The steady-state values of the converter variables are obtained by equating the differential equations to zero. Additionally, it is assumed that the variations in \(v_{g}(t)\) occur at a much slower rate compared to the converter dynamics, and the converter bandwidth is at least ten times lower than the switching frequency. Consequently, the converter operates in proximity to an equilibrium point, allowing it to be represented linearly. The converter dynamics around the equilibrium point can be represented as  
\[ \begin{array}{l}v_{g}(t)\ =\ V_{g},\\ d_{1}(t)\ =\ D_{1}\ +\ \Delta d_{1}(t),\\ i_{L1}(t)\ =\ I_{L1}\ +\ \Delta i_{L1}(t),\\ i_{L2}(t)\ =\ I_{L2}\ +\ \Delta i_{L2}(t),\\ v_{c}(t)\ =\ V_{C}\ +\ \Delta v_{c}(t),\\ v_{o}(t)\ =\ V_{o}\ +\ \Delta v_{o}(t),\end{array}  \tag{2}\]  
Figure 4: Operation principle for (**a**) boost mode; (**b**) boost and buck mode. where variables \(V_{g},D_{1},I_{L1},I_{L2},V_{C},V_{o}\) express the dc equilibrium point or the steady-state values and the other terms \(\Delta d_{1}(t),\Delta i_{L1}(t),\Delta i_{L2}(t),\Delta v_{c}(t),\Delta v_{o}(t)\) represent small variation around the steady-state values. The obtained equilibrium points (steady-state values) are the following:  
\[ \begin{array}{ll}I_{L1}&=\frac{D_{1}V_{g}}{R_{o}(D_{1}-1)^{2}}\;,\\ I_{L2}&=\;-\frac{V_{g}}{R_{o}(D_{1}-1)},\\ V_{C}&=\;V_{o},\\ V_{o}&=\;\frac{V_{g}}{1-D_{1}}.\end{array}  \tag{3}\]  
In this converter, the control objectives involve regulating the average voltage output \(V_{o}\) and reshaping the line current through control of the input current \(i_{g}(t)\). To achieve these control objectives, two transfer functions need to be calculated: \(G_{i_{g}d_{ho}}(s)\), which represents the relationship between the duty cycle \(d_{1}(s)\) and the rectified input current \(|i_{g}(s)|\), and \(G_{d_{ho}}(s)\), which represents the relationship between the duty cycle \(d_{1}(s)\) and the output voltage \(vo(s)\). By calculating the transfer functions \(G_{i_{g}d_{ho}}(s)\) and \(G_{vd_{ho}}(s)\), the converter's control system can be designed and analyzed using classical control techniques. These two transfer functions are expressed as follows:  
\[ \begin{array}{ll}G_{i_{g}d_{ho}}(s)=\frac{\Delta i_{g}(s)}{\Delta i_{g}(s)}, \\ G_{vd_{ho}}(s)=\frac{\Delta V_{o}(s)}{\Delta i_{g}(s)},\end{array}