School of Electronic and Information Engineering South China University of TechnologyNonlinear Phenomena in Power Electronic Circuitsn Chen Yanfengeeyfchenscut.edu.cn2012.2.13 Contentsv1. Nonlinear Study on PENonlinearity the default property of power electronics circuitsClassic example of nonlinear studyRecent study from a bifurcation perspectiveCircuits have been studied in detail,Literature reviewCurrent state of findingsBasic problems in analysisWhat models to use?Analysis techniques (Averaging, Discrete-time map)Summary of observationsPossible engineering applicationsSummary of nonlinear study on PE 21、Nonlinear Study on PENature of operationTime varying different systems at different timesANDNonlinear the time durations are related nonlinearly with the output function3Nonlinearity the default property of power electronics circuitsPower electronics engineers/researchers are dealing with nonlinear problemsMuch of power electronics is about identifying nonlinear phenomena and “taming” them to do useful applications Classic examples: Averaging (R. David Middlebrook, Richard Bass) Discrete-time modeling (Harry Owen, Fred Lee) Stability analysis (George Verghese) Phase-plane analysis/control (Fred Lee, Bill Burns) Series approximation (R. Tymerski)4Classic example of nonlinear studyBoost converter with current-mode controlIrefiLIrefiLd 0.5Simple analysis reveals a change of stability status at a critical duty cycle of 0.5.The circuit is actually stable beyond the critical point, though operates with a longer period. This period- doubling phenomenon was observed long ago.5Recent study from a bifurcation perspectiveIrefiLT/CR = 0.125T/CR = 0.625samplediLsampledIrefWith the help of computers, we can study the phenomenon in more detail.Bifurcation diagrams (panoramic view of stability status)We can plot bifurcation diagrams for different sets of parametersSampled variable at steady state versus parameter, e.g., iL(nT) vs. Irefnormal period-1 operationbifurcation point6Circuits whose bifurcation behaviours have been studied in detailVoltage-mode and current-mode controlled simple buck and boost converters, and many others7ViniLiLvoPeriod-doubling and chaos in voltage-mode controlled buck converterbifurcation diagramchaotic attractorHamill et al. (1990)8Period-doubling and chaos in current-mode controlled boost converterbifurcation diagramIrefiL(nT)9Hopf Bifurcation in parallel boost convertersnormal period-1limit cycle of long periodquasi-periodic orbitbifurcation diagram10Literature reviewHamill and Jefferies 1988 - First analysis of bifurcation and chaotic dynamics in a first order PWM voltage-mode controlled converter. Deane and Hamill 1990 - Analysis of bifurcation in first order and second order PWM buck converters.Hamill et al. 1992 - Derivation of an iterative map to analyze bifurcation in a buck converter in continuous mode.Deane 1992 - First report on chaotic behaviour in a current-controlled boost converter.Tse 1994 - Derivation an iterative map to demonstrate period-doubling cascades in a boost converter in discontinuous mode. 11Literature review (cont.)Chakrabarty et al. 1995 / Fossas and Olivar 1996 - Further study of chaos in a PWM buck converter. Poddar et al. 1995 / Batlle et al. 1996 - Control of chaos in dc/dc converters. Tse and Chan 1995 - Study of bifurcation and chaos in a fourth order current-controlled Cuk converter. Chan and Tse 1996 - Study of various routes to chaos in a current-controlled boost converter. Banerjee et al. 1997 - Examination of bifurcation and chaos in the light of “border collision bifurcation”.Tse 1997 Analysis of autonomous converters using averaged models. 12Di Bernardo et al. 1998 - Study of various sampling and their applications in the identification of bifurcation and chaos.Chan and Tse 1998 Proof of period-doubling in discontinuous converters using Schwarzian derivativesDi Bernardo et al. 1998 - Analysis of the non-smooth dynamics (such as grazing, skipping and sliding) of dc/dc converters. El Aroudi et al. 1999 - Identification of quasi-periodicity and chaos in a boost converter. Mazumder, Nayfeh and Borojevich 1999 - Fast- and slows-scale instabilities.Iu and Tse 2000 - Study of bifurcation in parallel convertersLiterature review (cont.)13Review paperIEEE Proceedings Vol. 90 No. 5 pp. 768781 May 2002Literature review (cont.)14Two types of bifurcation* seen in power electronicsStandard bifurcations (found in other systems as well) Period-doubling Hopf (Neimark-Sacker) Saddle-nodeBorder collision (characteristic of power electronics) Abrupt change of stability status due to saturationCurrent state of findings *Bifurcation refers to sudden change of qualitative behaviour of a dynamical system when a certain parameter is varied.15Standard bifurcations Buck converters (voltage-mode) period-doubling