## 2 Proposed Circuit Topology and Operation Analysis
### Boost-Typed Power-Factor Corrector  
As shown in Figure 1, the boost converter, which is composed of \(L_{PFC}\), \(S_{2}\), \(D_{S1}\), and \(C_{bus}\), performs the function of PFC. Figure 5 shows the PFC inductor current \(i_{LPFC}\) when the PFC circuit is designed to operate at discontinuous conduction mode (DCM). Since the rising slope of \(i_{LPFC}\) is proportional to the line voltage \(v_{in}\), the envelope of \(i_{LPFC}\) would be sinusoidal. The alternating current (ac)/dc converter was supplied from the ac line voltage.  
\[v_{in}(t)=V_{m}\,sin(2\pi\,f_{L}\,t) \tag{8}\]  
where \(f_{L}\) and \(V_{m}\) are the frequency and amplitude of the line voltage source, respectively. In practice, \(f_{L}\) is much lower than the switching frequency, \(f_{s}\), of the active switches. It is reasonable to consider the rectified input voltage as a constant over a high-frequency cycle. Figure 6 shows \(i_{LPFC}\) in one  
Figure 4: Voltage and current waveforms of the resonant circuit (\(f_{s}=f_{r}\)).  
Figure 3: Voltage gain versus frequency ratio.  
high-frequency cycle where \(t_{on}\) is the conduction time of switch \(S_{2}\) and \(t_{f}\) represents the fall time of \(i_{LPFC}\). The duty ratio \(D\) is defined as:  
\[D=\frac{t_{on}}{T_{s}} \tag{9}\]  
where \(T_{s}\) is the period of the switching frequency. At DCM operation, the peak value of the inductor current can be expressed as:  
\[i_{PK}(t)=\frac{|V_{m}\sin(2\pi\,f_{L}\,t)|\cdot t_{on}}{L_{PFC}}=\frac{(V_{ cbus}-V_{m}\sin(2\pi\,f_{L}\,t))\cdot t_{f}(t)}{L_{PFC}} \tag{10}\]  
From Equation (10), relationship between \(t_{on}\) and \(t_{f}\) can be described as Equation (11).  
\[t_{f}(t)=\frac{V_{m}\sin(2\pi\,f_{L}\,t)}{V_{cbus}-V_{m}\sin(2\pi\,f_{L}\,t)}t _{on} \tag{11}\]  
The average value of \(i_{LPFC}\) over a high-frequency cycle is given, and can be expressed as [23]:  
\[\overline{i_{LPFC}}(t)=\frac{\left(t_{on}+t_{f}(t)\right)\cdot i_{PK}(t)}{2T_ {s}}=\frac{D^{2}V_{m}|sin(2\pi f_{L}t)|}{2L_{PFC}f_{s}}\cdot\frac{1}{1-\frac{1 }{k}\cdot|sin(2\pi f_{L}t)|} \tag{12}\]  
where the index \(k\) is defined as:  
\[k\equiv\frac{V_{cbus}}{V_{m}} \tag{13}\]  
The input current is equal to the average value of the PFC inductor current, and can be expressed as:  
\[i_{in}(t)=\frac{D^{2}V_{m}\,sin(2\pi f_{L}t)}{2L_{PFC}f_{s}}\cdot\frac{1}{1- \frac{1}{k}\cdot|sin(2\pi f_{L}t)|} \tag{14}\]  
Figure 5: Schematic waveform of a power-factor corrector (PFC) inductor current.  
Figure 6: PFC inductor current in one high-frequency cycle.  
The input power can be obtained by taking the average of the product of the input voltage and current over one line-frequency cycle. From Equations (8) and (14), the input power can be derived as [24]:  
\[P_{in}=\frac{1}{\pi}\int_{0}^{\pi}V_{m}\,sin(2\pi f_{L}\,t)\cdot i_{in}(t)d(2\pi f _{L}t)=\frac{D^{2}V_{m}{}^{2}}{2L_{PFC}f_{s}}\cdot y \tag{15}\]  
where \(y\) is expressed as:  
\[y=\int_{0}^{\pi}\frac{sin^{2}\,\theta}{1-\frac{1}{k}sin\,\theta}\,d\theta= \frac{k^{3}}{\sqrt{k^{2}-1}}\bigg{(}1+\frac{2}{\pi}sin^{-1}\frac{1}{k}\bigg{)} -k^{2}-\frac{2}{\pi}\cdot k \tag{16}\]  
From Equations (15) and (16), the inductance of \(L_{PFC}\) can be calculated by using Equation (17).  
\[L_{PFC}=\frac{\eta D^{2}V_{m}{}^{2}}{2P_{o}f_{s}}\cdot y \tag{17}\]  
where \(P_{o}\) and \(\eta\) represent the output power and circuit efficiency, respectively.