# Power Factor Correction Application Based on Independent Double-Boost Interleaved Converter (IDBIC)
## 2 Converter Topology Analysis
### State-Space Modeling of the Converter  
Observing the switching states presented in Figure 3, the state-space analysis of the proposed converter has been made [29]. Thus, the expressions that describe the behavior of the converter at CCM were first derived for the operation at a duty cycle smaller than 0.5. Figure 3: Switching stages for the proposed PFC converter. Thus, from relation (1), the state-space system for the S1 switching configuration is characterized by the matrices:  
\[A_{S1}= \begin{bmatrix}0&0&0&0\\ 0&0&0&\dfrac{-1}{L_{2}}\\ 0&0&\dfrac{-1}{RC_{1}}&\dfrac{-1}{RC_{1}}\\ 0&\dfrac{1}{C_{2}}&\dfrac{-1}{RC_{2}}&\dfrac{-1}{RC_{2}}\end{bmatrix} ;\;B_{S1}= \begin{bmatrix}\dfrac{1}{L_{1}}\\ \dfrac{1}{L_{2}}\\ 0\\ 0\end{bmatrix}  \tag{6}\]  
The representation of relation (2) under state matrices will yield:  
\[A_{S2}= \begin{bmatrix}0&0&\dfrac{-1}{L_{1}}&0\\ 0&0&0&\dfrac{-1}{L_{2}}\\ \dfrac{-1}{C_{1}}&0&\dfrac{1}{RC_{1}}&\dfrac{1}{RC_{1}}\\ 0&\dfrac{1}{C_{2}}&\dfrac{-1}{RC_{2}}&\dfrac{-1}{RC_{2}}\end{bmatrix} ;\;B_{S2}= \begin{bmatrix}\dfrac{1}{L_{1}}\\ 0\\ 0\\ 0\end{bmatrix}  \tag{7}\]  
For the expressions in (3) and (4), corelated with the switching states S5 and S3, the state-space matrices will be:  
\[A_{S5}= \begin{bmatrix}0&0&\dfrac{-1}{L_{1}}&0\\ 0&0&0&0\\ \dfrac{1}{C_{1}}&0&\dfrac{-1}{RC_{1}}&\dfrac{-1}{RC_{1}}\\ 0&0&\dfrac{-1}{RC_{2}}&\dfrac{-1}{RC_{2}}\end{bmatrix} ;\;B_{S5}= \begin{bmatrix} \dfrac{1}{L_{1}}\\ \dfrac{1}{L_{2}}\\ 0\\ 0\end{bmatrix}  \tag{8}\]  
and  
\[A_{S3}= \begin{bmatrix}0&0&\dfrac{-1}{L_{1}}&0\\ 0&0&0&\dfrac{-1}{L_{2}}\\ \dfrac{1}{C_{1}}&0&\dfrac{-1}{RC_{1}}&\dfrac{-1}{RC_{1}}\\ 0&\dfrac{-1}{C_{2}}&\dfrac{1}{RC_{2}}&\dfrac{1}{RC_{2}}\end{bmatrix} ;\;B_{S3}= \begin{bmatrix}0\\ \dfrac{1}{L_{2}}\\ 0\\ 0\end{bmatrix}  \tag{9}\]  
while  
\[C_{S1}=C_{S2}=C_{S5}=C_{S3}= \begin{bmatrix}1&0&0&0\\ 0&1&0&0\\ 0&0&1&0\\ 0&0&0&1\end{bmatrix}  \tag{10}\]  
Considering the representation from Figure 2b and the duty cycle as \(d\), the state matrices averaged over one switching period will result the following:  
\[ \begin{array}{l}A_{av}=A_{S1}\;d+A_{S2}(0.5-d)+A_{S5}\;d+A_{S3}(0.5-d)\\ B_{av}=B_{S1}\;d+B_{S2}(0.5-d)+B_{S5}\;d+B_{S3}(0.5-d)\\ C_{av}=C_{S1}\;d+C_{S2}(0.5-d)+C_{S5}\;d+C_{S3}(0.5-d)\end{array}  \tag{11}\]To obtain the expressions depicting the continuous, linear behavior of the converter, the small signal analysis is implied to the linearized model. Small variations of the input variables, \(\widetilde{v}_{rec}\) and duty cycle \(\widetilde{d}\), around the quiescent operating point of the converter will result in small variations of the output variables \(\widetilde{i}_{L1}\), \(\widetilde{i}_{L2}\), \(\widetilde{v}_{C1}\) and \(\widetilde{v}_{C2}\). \[\left\{ \begin{array}{l}\dot{\widetilde{x}}=A_{av}\,\widetilde{x}+B_{av}\, \widetilde{v}_{rec}+E_{av}\,\widetilde{d}\\ \widetilde{y}=C_{av}\,\widetilde{x}\end{array} \right. \tag{12}\]  
where \(E_{av}=[(A_{S1}+A_{S5})-(A_{S2}+A_{S3})]X+[(B_{S1}+B_{S5})-(B_{S2}+B_{S3})]V_{ rec}\). After specific algebraic operation, the Laplace domain solutions of the state vectors will determine the output to input transfer functions. Thus, the solution for the output to input voltage transfer function will be:  
\[\frac{\widetilde{Y}}{\widetilde{V}_{rec}}=C_{av}\;(sI-A_{av})^{-1}.B_{av} \tag{13}\]  
and for the output to duty cycle variation:  
\[\frac{\widetilde{Y}}{\widetilde{D}}=C_{av}\;(sI-A_{av})^{-1}.E_{av} \tag{14}\]  
The steady-state values of the state variables can be determined as:  
\[Y=-C_{av}\;(A_{av})^{-1}.B_{av}\;V_{rec} \tag{15}\]  
Concerning the operation at a duty cycle greater than 0.5, the switching states from Figure 2a are partially similar with the ones described by (1) and (3). The only particular switching state for this mode of operation is S4, which is characterized by the equations:  
\[\left\{