# Investigation of piecewise linear power electronic systems

This paper is divided into four chapters. Chapter 1 covers the most important ideas in dynamical systems (in an informal way) and explains the necessity for such models.

The goal of Chapter 2 is to sketch the mathematical background of the thesis. This starts with the basic definitions of differential equations, the stability of the orbits and special solutions such as equilibrium points and periodic orbits. Special forms of these concepts are shown in the case of linear, nonlinear and piecewise linear equations. The main part of this chapter is a mathematical introduction to dynamical systems and the techniques for investigating the structure of their phase space (Poincaré maps and bifurcation diagrams). The chapter ends with a brief description of chaos in dynamical systems.

Chapter 3 is a demonstration of the mathematical tools shown in Chapter \ref{ch:math}. After creating the mathematical model for the DC/DC buck converter, I show the existence and uniqueness of a periodic orbit of the open loop controlled converter. Using this a result I prove that periodic solutions of the closed loop controlled converter also exist. For the sake of simplicity I only investigate continuous conduction mode, as there are only two alternating structures when the converter operates in continuous conduction mode (CCM). At the end of the chapter a method (using the auxiliary state vector) is given to examine the stability of periodic orbits. This alternative method determines the Jacobian of the Poincaré map without calculating its derivatives.

The main part of my thesis is the examination of the peak current controlled DC/DC flyback converter. Again, there are more possible structures of the flyback converter, I chose the parameters in a way that only two of them switch to each other. Allowing the converter to enter a third region of the state space would only result in a more complicated model and longer simulation runs. I investigate it analytically using Wolfram Mathematica, model it in MATLAB/Simulink and carry out measurement on the converter that I built in the laboratory. I used MATLAB/Simulink to generate the control algorithm of the circuit, this algorithm was implemented on DSP. The results are compared in Chapter 4