## 2 Operation Principle of the Conventional DCM-CRM Cuk PFC
The discharge time \(t_{\text{off1}}\) of \(i_{L1}\) can be derived from the volt-second balance of the input inductor \(L_{1}\). \[t_{\text{off 1}}=\frac{V_{M}|\text{sin}(\omega t)|}{(V_{\rm C1}-V_{M}| \text{sin}(\omega t)|)}t_{\text{on}} \tag{6}\]  
Figure 2: Key operation waveforms. Thus, based on (5) and (6), the input current \(i_{in}(t)\) is represented by the average current \(i_{L1}(t)\) flowing through the input inductor during the switching cycle. Consequently, the input current can be given as  
Figure 2.0  
where \(T_{s}\) is the switching period and \(t_{off1}\) is the time for the input inductor current to drop to zero. According to (2), the peak current \(i_{L2\_pk}(t)\) of the output inductor \(L_{2}\) can be expressed as  
Figure 2.1  
The average output inductor current is equivalent to the output current; hence, the output current \(I_{\rm o}\) can be expressed as  
Figure 2.2  
where \(t_{off2}\) is the discharge time for the output inductor current to drop to zero. Based on the volt-second balance of inductor \(L_{2}\), the \(L_{2}\) discharge time \(t_{off2}\) formula can be expressed as  
Figure 2.3  
As shown in Figure 2, the output inductor operates in CRM mode, and from Equation (9), the peak current \(i_{L2\_pk}(t)\) of the inductor \(L_{2}\) can also be expressed as  
Figure 2.4  
Substituting Equation (10) into Equation (9) yields an expression for the conduction time \(t_{on}\) of the switch as  
Figure 2.5  
Neglecting the voltage ripple of the intermediate capacitor, Equations (8) and (11) indicate that, when the converter is delivering energy to the load, the conduction time \(t_{on}\) of \(S_{1}\) can also be expressed as  
Figure 2.6  
When the switch is turned off, by associating Equations (10) and (13), the turn-off time \(T_{off}\) of switch \(S_{1}\) is obtained as  
Figure 2.7  
From (5), it can be seen that the discharge time of the inductor \(L_{1}\) is the longest when \(i_{L1\_pk}(t)\) reaches the maximum at \(\mid\sin(\omega t)\mid=1\) in a half-line cycle, and the longest discharge time \(t_{\rm offmax}\) can be expressed as  
Figure 2.8Inductor \(L_{1}\) can be operated in DCM mode for the entire frequency cycle by making it operate in DCM mode at the peak input voltage. The conditions for the inductor \(L_{1}\) to operate in DCM mode can be expressed as  
Figure 2.9  
By associating (15) and (16), the condition under which the inductor \(L_{1}\) works in DCM can be obtained as  
\[\frac{di_{L1}}{dt}=\frac{v_{\text{Rec}}}{L_{1}} \tag{1}\]0  
Without considering the converter losses, it can be obtained based on the conservation of input power and output power:  
\[\frac{di_{L1}}{dt}=\frac{v_{\text{Rec}}}{L_{1}} \tag{1}\]1  
According to (7) and (12), Equation (18) can be rewritten as  
\[\frac{di_{L1}}{dt}=\frac{v_{\text{Rec}}}{L_{1}} \tag{1}\]2  
where \(K=L_{2}/L_{1}\). Based on Equation (7), the expression of PF can be expressed as  
\[\frac{di_{L1}}{dt}=\frac{v_{\text{Rec}}}{L_{1}} \tag{1}\]3  
According to (19), the magnitude of \(V_{\text{C1}}\) can be computed using an iterative method once the load parameters are established. Figure 3 illustrates the variations in the PF and the intermediate capacitor voltage as functions of the effective input voltage \(V_{\text{in\_RMS}}\) for different ratios of \(K\). From Figure 3, it is evident that, for a constant \(K\), \(V_{\text{C1}}\) increases with the increase in \(V_{\text{in\_RMS}}\). Concurrently, within the range of 90 Vac to 240 Vac, the PF experiences a slight decrease as \(V_{\text{in\_RMS}}\) rises. Additionally, for a given \(V_{\text{in\_RMS}}\), \(V_{\text{C1}}\) rises with an increase in the ratio \(K\), while the PF diminishes as \(K\) decreases. \[\frac{di_{L1}}{dt}=\frac{v_{\text{Rec}}}{L_{1}} \tag{1}\]4 The relation curves of \(V_{\text{in\_RMS}}\) and PF, \(V_{\text{C1}}\) with different ratios \(K\).