DC-DC converters are some of the most commonly used circuits in power electronics where there is the need of obtaining a regulated dc output at a desired value. State-space averaging approach is one of the most widely adopted modeling approaches for the analysis and design of such converters. It can be directly applied in a standard Laplace transform domain or frequency domain for facilitating design of the control loops. But it is found that this approach fails to predict the stability of the cycle-to-cycle fast-scale dynamics of the switched converters, and can only predict the slow-scale behaviour and the instabilities resulting in slow-scale oscillations. It is necessary to estimate the range of the external parameters that will ensure the stable period-1 operation without the onset of subharmonic oscillations. It is also necessary to get adequate knowledge of the dynamics beyond the point of instability in the power electronic circuits. The method of sampled-data modeling to obtain the discrete-time maps was developed to overcome this problem. But there exist very few circuit configurations (like, for example, the current mode controlled DC-DC converters, and the current programmed H-bridge inverter) for which the map can be obtained in closed form. We show that in a voltage mode controlled DC-DC converter, if the switching is governed by pulsewidth modulation of the first kind (PWM-1) which is more common in digital implementation, an explicit form of the stroboscopic map can be obtained. The discrete-time state-space of such a system is divided into five regions, each with a different functional form. We then analyse the bifurcation behavior using the explicit map, and demonstrate the different types of bifurcations that may occur in this system as a fixed point moves from one region to another. This includes the very interesting case of a direct transition from periodicity to quasiperiodicity through the route of border collision bifurcation. Mode-locking Keywords- DC-DC converters, pulsewidth modulation, state space averaging, sampled data model, slow– and fast-scale instability, bifurcation, bordercollision bifiucation, subharmonic oscillation, quasiperiodicity, bifurcation control, saltation matrix, and monodromy matrix.
Thesis Submitted To The Indian Institute Of Technology, Kharagpur For Award Of The Degree Of Doctor Of Philosophy By Somnath Maity Department Of Electrical Engineering Indian Institute Of Technology Kharagpur May, 2008 Supervisor ,Prof. Soumitro Banerjee AND Prof. Tapas K. Bhattacharya