Study On Border Collision And Bifurcation Of Piecewise Smooth Systems In DC-DC Switching Converter

Switching converters are the main technology of the switched power supply, almostof the industrial and civil switched power supply consists of the switching converters.Recently, it is the most important fundamentals for second power transformation anduse of high quality and high e?ciency. In the other hand, switching converters exhibit awealth of nonlinear phenomena, including bifurcations, coexisting attractors, and chaos,etc. The prime source of nonlinearity presented in the circuits is the nonlinear components(e.g., the power diodes and MOSFET) and control methods (e.g.,pulse-width modulation).Lately, the academic training of engineers has traditionally emphasized linear methods(e.g., Fourier and Laplace transforms and transfer functions). By their very nature, linearmethods alone give little insight into nonlinear phenomena and are of limited value inpredicting them. Using nonlinear dynamics to investigate the nonlinear phenomena of theswitching converters is importantly and realistically economical and applied significancefor directing engineers to improve the stability of the actual systems.The large number of the other switching converters are made of the direct current-direct current converters (DC-DC switching converters). There has a direct function ofguidance for the other topological structure to study the DC-DC switching converters.Nonlinear phenomena in DC-DC switching converters have been investigated extensivelyin recent years. A series of research methods have been formed, for example, one andtwo dimension discrete-time maps have been gotten with using numerical iterate, maxi-mal Lyapunov exponent has been used to estimate the periodic orbits or chaotic orbitsin bifurcation process and the stability of the system has been analyzed based on theeigenvalue of the Jacobian matrix , etc. But DC-DC switching converters are a piecewisesmooth system and have a specially nonlinear characteristic, and the piecewise smoothsystems are one of the main reason for producing bifurcation and chaos. So studying oneshas more actual significance. And few studies in nonlinear analysis of the DC-DC switch-ing converters have been preformed on no-isolated switching converters (Buck, Boost,Buck-Boost and C′uk converter), because there is not a universal variable like inductancecurrent of buck converter to analyze numerically in isolated switching converters. More-over, the studies of border collisions and bifurcation in isolated switching converters arevery poor. Symbolic sequence is used to analyze border collision and bifurcation phe-nomena in switching converters initially. Two dimension coarse-grained method do not analyze the more operation modes in switching converters. Symbolic sequence method ofswitching block and block switching sequence must be combine the periodic number of thesystem by observing the waveform of the converters to distinguish standard bifurcationsand border collision bifurcation. Therefor, this paper will bring forward new concepts andwill investigate the border collision and bifurcation phenomena in no-isolated and isolatedDC-DC switching converters based on nonlinear dynamics from beginning the piecewisesmooth systems.The main fruits are included:(1)A class of piecewise smooth maps with two border and three zones is derived to de-scribe the dynamics of a current-programmed buck converter operating in a discontinuousmode. The existence and stability condition of the fixed point are analyzed concretelyand how the operation modes of the converter charge in border collision point is analyzedtoo.(2)Total ampere-turns of the primary and secondary windings of a transformer are takenas one of variables to analyze the bifurcation behaviors and the border collision phe-nomenon on voltage mode controlled ?yback converter. A class of piecewise smooth mapswith five border and six zones is derived as the switching mode state equations in termsof total ampere-turns. Based on these equations, the border collision phenomena andbifurcations behaviors are analyzed and that the main reason of the border collision inDC-DC converter is the conversion of operation modes is presented.(3)Voltage mode controlled ?yback converter with a transformer are investigated exten-sively due to happening quasi-periodicity and frequency locking phenomena easily in highfrequency switching converters. Using stroboscopic sampling, the hopf bifurcation dia-gram with the control period of high frequency as a parameter is obtained. Hopf bifurca-tion in terms of fixed point equation with the calculation of eigenvalues at losing stabilityof equilibrium point in current continuous mode is studied. The correct expression ofnational period with the transformer is presented and the relationship between nationalperiod and control period at bifurcation point is explained. According to bifurcation dia-gram, the evolvement of quasi-periodicity and frequency locking phenomena is analyzedand rotation number is used to distinguish those phenomena. The embodiments of theFarey principle, devil's staircase and the expression of rotation number of quasi-periodicityare investigated as well.(4)Using the characteristic of switching converters and logical arithmetic method of com-puter that change decimal system from binary system, the new main switching block is putforward and instead of the operation mode in time sequence analysis. How the switching mode change is clear using this main switching block. Based on main switching block andusing the coarse-grained symbolic sequence method, the second switching block and theduplicate symbolic sequence are put forward to analyze the border collision phenomenonand bifurcation behavior. Based on the duplicate symbolic sequence, weight complexityis defined and utilized to analyze border collision and bifurcation phenomenon qualita-tively, so the across-the-aboard symbolic sequence and complexity analytical method areput forward to analyze the border collision and bifurcation phenomenon in switchingconverters.