will conduct, and the voltage across L1 is −Vo, as illustrated in Figure 2.5b. Thus, based on the volt‐second balance principle, we can have the following equation:
Figure 2.5 (a) The buck converter, (b) inductor voltage VL1 and current iL1, and (c) those in DCM operation.
where D is the duty ratio of switch S1 and Ts is the switching period. Since inductor current iL1 never drops to zero, this operation mode is called CCM. From (2.1), we can derive the input‐to‐output voltage transfer ratio below:
(2.2)
If inductor current iL1 drops to zero before turns on switch S1 again, as shown in Figure 2.5c, the operation mode is called DCM. Again, based on the volt‐second balance principle, we can have the following equation:
where d1 is the duty ratio of switch S1, d2 is the duty ratio of diode D1, and (1 − d1 − d2)Ts is the dead time. From (2.3), we have the following input‐to‐output voltage transfer ratio under DCM operation:
If
(2.5)
the operation will be CCM and d1 = D. Thus, the CCM can be considered a special case of DCM. No matter what mode of operation, the configuration of buck converter keeps unchanged. For simplicity and considering most of converters operated in CCM, we will first discuss the evolution of converters based on the voltage transfer ratio under CCM operation.
2.2.2 CCM Operation
A buck converter and its voltage transfer ratio under CCM operation are shown in Figure 2.6a. When taking the output from capacitor C1, we can have the following voltage transfer ratio:
(2.6)
If taking the output from capacitor C2, we will have a transfer ratio of
(2.7)
The voltage transfer ratio of a buck‐boost converter is
and it can be decoded into the form shown in Figure 2.6b, in which the forward‐path gain is D and the feedback‐path gain is unity. The gain D can be synthesized with a buck converter for its single‐ended characteristic and meeting the requirement of a forward path in a control system with forward and feedback paths. The unity‐gain feedback is synthesized by feeding back the output voltage Vo to the input, and there is no power flow from this feedback path to the output. Thus, the overall converter configuration depicted in Figure 2.6c can synthesize the control block diagram shown in Figure 2.6b satisfactorily and correctly. Redrawing the circuit shown in Figure 2.6c yields the buck‐boost converter, as shown in Figure 2.6d, which is in the form that people are familiar with.
Figure 2.6 Decoding, synthesizing, and evolution of buck‐boost and boost converters from the buck converter.
Similarly, we can take the output from capacitor C2 of the buck‐boost converter depicted in Figure 2.6d, and we have the following voltage transfer ratio or code:
which is the input–output voltage transfer ratio or code of the boost converter in CCM operation. Redrawing the circuit of Figure 2.6d yields the one shown in Figure 2.6e, in which voltages Vi, Vo, and
2.2.3 DCM Operation
The above decoding process is based on CCM operation of the converters. In order to confirm the decoding process is also working for DCM operation, the input–output voltage transfer ratios of the converters in DCM operation are discussed as follows.
For the buck converter shown in Figure 2.6a and operated in DCM, the input–output voltage transfer ratio was derived and expressed in (2.4).
