in DCM operation can be derived as
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
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 is evolved from the buck converter.
2.2.4 Inverse Operation
By exchanging the roles of the active and passive switches in the converters shown in Figure 2.6, we have the inverse converters, as shown in Figure 2.7, and their corresponding transfer ratios can be derived as 1/D, (1 − D), and (1 − D)/D, which are the reciprocals of D, 1/(1 − D), and D/(1 − D), respectively. These codes provide more choices for decoding transfer codes. Illustrations of decoding the transfer codes of PWM converters in terms of the fundamental codes discussed previously will be presented in later chapters.
Figure 2.7 (a) Inverse buck, (b) inverse boost, and (c) inverse buck‐boost with the transfer ratios of 1/D, (1 − D), and (1 − D)/D, respectively.
Typically, the inverse converters do not operate independently since its output sink will transfer power back to the input source in unidirection. They usually work with other regular converters to control power flow between input and output, which can achieve higher step‐up or step‐down power conversion. The regular and the inverse buck, buck‐boost, and boost converters are considered the fundamental converters since in the decoding process, their transfer codes will be used frequently and they are with second‐order filters only.
2.3 Duality
In mathematical optimization theory, duality means that optimization problems may be viewed from either of two perspectives, the primal problem or the dual problem. Similarly, in circuit theory, duality means that viewing from either of the two variables, voltage and current, can yield the same output expression for the two dual circuits. Typical examples of two RLC networks are shown in Figure 2.8, in which the series RLC network is driven by a voltage source, while the parallel RLC network is driven by a current source. The circuits shown in Figure 2.8a and b are dual. It can be observed that when replacing the voltage source with a current source and the components in series with the ones in parallel, we can obtain a dual circuit of the other and we can determine the branch currents in parallel from the voltages across the branches in series. The dual networks shown in Figure 2.8 have one‐to‐one correspondence.
Conventionally, with the duality theory, the boost converter can be derived from the buck converter. The buck converter is first represented in the form shown in Figure 2.9a, in which the output inductor–capacitor filter is simplified to a current sink. Then, replacing the voltage source Vi with a current one Ii, the switch S1 in series with a parallel one, the diode D1 in parallel with a series one, and the output current sink Io with a voltage sink Vo can yield the boost converter shown in Figure 2.9b. They have one‐to‐one correspondence. Since the converters are represented in voltage/current sources and voltage/current sinks, they are called topologically dual converters. When comparing the topological converters with the component converters shown in Figure 2.10, one can reveal that they do not have one‐to‐one correspondence and, in fact, they are not dual converters. In Figure 2.10a, capacitor C1 realizes the voltage source Vi, while in Figure 2.10b, capacitor C1 and inductor L1 together realize the current source Ii. Similarly, the output current sink Io is realized with capacitor C2 and inductor L1 in Figure 2.10a. In the buck and boost converters, the voltage source or sink is realized with a single capacitor. Why is a current source or sink realized with a capacitor and an inductor rather than a single inductor? Is it because there is no current source or sink, or even the configurations of the buck and boost converters are incorrect? The answers to these questions will be left for later discussions in Chapter 11.
Figure 2.8 Dual RLC networks (a)
