A General Method to Study Multiple Discontinuous Conduction Modes in DC–DC Converters With One Transistor and Its Application to the Versatile Buck–Boost Converter

Abstract

The discontinuous conduction mode (DCM) is usually studied in single-diode and single-inductor converters, where only one DCM exists. However, multiple DCMs can appear in multidiode and multi-inductor topologies and the methodology to identify and characterize these multiple modes is not evident. In this article, a general method to study multiple DCMs is presented. The first step of the method consists in finding out the number <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> , which is the number of diodes conducting current passing exclusively through inductors when the transistor turns <sc xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">off</small> . For a given <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> value, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$2^{n}$</tex-math></inline-formula> possible conduction modes are expected: 1 continuous mode and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$2^{n}-1$</tex-math></inline-formula> DCMs. The second step is to create an <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -dimensional space called “ <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -space.” In the <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$k$</tex-math></inline-formula> -space, the converter operation describes a straight line when the load changes. This straight line called “converter trajectory” passes through different <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> -dimensional enclosures. Each one of these enclosures represents a different conduction mode. The third step is to determine the borders between conduction modes which are subspaces of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$(n-1)$</tex-math></inline-formula> dimensions. This method must be followed for both control strategies (i.e., open- and closed-loop controls). The proposed method is applied to the versatile buck–boost converter. Experimental results verify the theoretical analysis for all the identified conduction modes.