# Bridgeless Buck-Boost PFC Rectifier with Positive Output Voltage
## 3 Basic Operating Principles  
For analysis convenience, some assumptions are given below:  
1. All the switches, diodes, inductor and capacitor are considered as ideal.
2. The PWM signals for \(S_{1}\), \(S_{2}\) and \(S_{3}\) are the same.
3. The value of \(C_{o}\) is large enough to render the voltage across it constant at \(V_{o}\).
4. The circuit operates in the DCM.  
According to these above assumptions, Figure 2 shows the key waveforms for the proposed PFC rectifier operating under the positive input voltage. There are six states are to be described as following.  
1. _State 1_ [\(0\leq t\leq DT_{s}\)]: As the voltage \(v_{in}\) is positive, \(S_{1}\), \(S_{2}\), \(S_{3}\), \(D_{1}\), and \(D_{3}\) are ON, whereas \(D_{2}\), \(D_{4}\), and \(D_{o}\) are OFF. During this state, the current \(i_{in}\) flows through \(S_{1}\), \(S_{2}\), \(S_{3}\), \(D_{1}\), \(D_{3}\), and \(L\), as shown in Figure 3a. At the same time, as shown in Figure 2, the voltage across \(L\) is \(v_{in}\), making \(L\) magnetized and the current \(i_{L}\) increasing linearly. Moreover, the output energy needed is offered by \(C_{o}\).
2. _State 2_ [\(DT_{s}\leq t\leq(D+\Delta)T_{s}\)]: As the input voltage is still positive, \(S_{1}\), \(S_{2}\), and \(S_{3}\) are OFF, whereas \(D_{1}\), \(D_{2}\), \(D_{3}\), \(D_{4}\) and \(D_{o}\) are ON. During this state, the current \(i_{in}\) is zero, whereas the current \(i_{L}\) continuously flows through these five diodes, as shown in Figure 3b. At the same time, as shown in Figure 2, the voltage across \(L\) is \(-V_{o}\), rendering \(L\) is demagnetized and the current \(i_{L}\) decreasing linearly. In addition, the output energy needed is provided by the energy stored in \(L\).
3. _State 3_ [\((D+\Delta)T_{s}\leq t\leq T_{s}\)]: As the input voltage still keeps positive, \(S_{1}\), \(S_{2}\), \(S_{3}\), \(D_{1}\), \(D_{2}\), \(D_{3}\), \(D_{4}\) and \(D_{o}\) are OFF, as shown in Figure 3c. Namely, there is no current flowing through \(L\), as shown in Figure 2, and hence the output energy needed is supplied from \(C_{o}\).  
Figure 2: Voltage and current of the inductor operating in the discontinuous conduction mode (DCM) under the positive input voltage.  
For the negative input voltage to be considered, there are also three states, that is, states 4, 5 and 6. The behavior in state 4 is similar to that in state 1 except that the diodes \(D_{1}\) and \(D_{3}\) conducted in state 1 is replaced by the diodes \(D_{2}\) and \(D_{4}\) conducted in state 4. Also, the behaviors in states 5 and 6 are the same as those in states 2 and 3, respectively.  
Based on the volt-second balance for the inductor \(L\), the following equation must be held:  
\[\frac{1}{L}v_{in}\cdot D\cdot T_{s}+\frac{1}{L}\cdot(-V_{o})\cdot\Delta_{1} \cdot T_{s}+\frac{1}{L}\cdot 0\cdot\Delta_{2}\cdot T_{s}=0 \tag{1}\]  
Also,  
\[D+\Delta_{1}+\Delta_{2}=1 \tag{2}\]  
By rearranging (1) according to (2), the voltage gain can be obtained:  
\[\frac{V_{o}}{v_{in}}=\frac{D}{\Delta_{1}} \tag{3}\]  
Equation (3) is only for the convertor working in the DCM. But if the converter operates in the BCM or CCM, then the value of \(\Delta_{2}\) will be zero and the voltage gain shown in (3) can be rewritten to  
\[\frac{V_{o}}{v_{in}}=\frac{D}{1-D} \tag{4}\]