Steady-State Indeterminacy in Lossless Switched-Mode Power Converters

Abstract

In this article, it is shown for the first time that lossless switched-mode power converters may not possess a unique steady-state solution. Rather, they can exhibit an inherent <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">indeterminacy</i> in their steady-state behavior, and a unique solution is only found when losses are included in the analysis. Even more interestingly, it is shown that lossless converters of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">odd order</i> never possess a unique steady-state solution. Indeterminate converters can be intrinsically sensitive to parasitic resistances and other nonideal effects, a phenomenon that practically manifests itself in the form of undesirable voltages or circulating currents possessing, in general, both dc and ac components. This article first sets a general mathematical framework for a systematic assessment of steady-state indeterminacy. This is subsequently linked to the geometry of the state vector periodic motion in the converter's <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</i> -dimensional state space. On the basis of the developed theory, odd-order converters are shown to always be affected by the indeterminacy issue. Besides inherent theoretical importance in the power electronics field, the results of this article provide a deeper justification for the observed behavior of several converter topologies of significant practical relevance, the most notable being multiphase and multilevel dc–dc converters, examples of which are discussed as case studies.