Origin of Power Converters. Tsai-Fu Wu

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      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

(2.8)

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 V_o 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:

(2.9)

      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 V_i, V_o, and form a loop and one of the voltages is a dependent variable. Since we do not take output from capacitor C₁, it can be removed from the circuit, as shown in Figure 2.6f. When moving inductor L₁ and diode D₁ from the return path to the forward, we can obtain the boost converter, as shown in Figure 2.6g. With the decoding and synthesizing processes, the buck‐boost and boost converters can be evolved from the buck converter. Thus, the buck converter can be considered preliminarily the origin of power converters.

      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). Similarly, the transfer ratio of the buck‐boost converter in DCM operation can be derived as

(2.10)

by substituting the transfer ratio D shown in (2.8) with the ratio, d₁/(d₁ + d₂), of the buck converter in DCM operation. The transfer ratio of the boost converter, therefore, can be derived as

(2.11)

      Again, if

      (2.12)

and d₂ is replaced with (1 − D), the boost converter is, therefore, in CCM operation, and the transfer ratio will become the one shown in (2.9).

Based on the transfer ratio of the buck converter in DCM operation, those of the buck‐boost and boost converters can be derived correspondingly. The transfer ratios of the buck‐boost and boost converters in DCM operation can be also derived directly based on volt‐second balance principle, and they come out the same expressions as those shown in (2.10) and (2.11). This confirms that the decoding and synthesizing processes can be applied for both DCM and CCM operations. In the rest of the chapters, for simplicity, the discussion of decoding and synthesizing processes will be based on CCM operation only. From now on, the transfer ratio will be treated as a transfer code for further decoding processing.

      From the above discussion, we observed that the original PWM code is D, and the derived codes include (1 − D), 1/(1 − D), and D/(1 − D), which can be adopted as fundamental codes in decoding transfer codes. Buck converter is the origin, and the evolved converters are buck‐boost and boost converters up to this moment. In the evolution process, the evolved converters are not always directly evolved from the original converter, but they can be evolved from the evolved converters or their descendant converters, like that the boost converter is evolved from the buck‐boost converter instead of the buck converter, while the buck‐boost converter