ONE MORE STEP TOWARDS PROVING CHAOS IN CONVERTERS Gerard Olivar, Enric Fossas and Carles Batlle DGDSA, Technical University of Catalonia, Av. Victor Balaguer s/n, E-08800, Vilanova i la Geltru, Spain e-mail: gerard@mat.upc.es, fossas@ioc.upc.es, carles@mat.upc.es WWW: http://www-mat.upc.es/dgdsa/ Abstract—Evidence of chaotic behavior of electri- cal circuits can be divided into three categories [1]: a) laboratory experiments, b) computer simulations and c) mathematical proofs. As the author of [1] wrote, mathematical proofs for the chaotic nature of satisfactory model systems were rare in 1987, being the field of electrical circuits no exception. In his opinion, completely convincing evidence for chaotic behavior is only given by a), b) and c) together. In this communication, basic geometrical features will be described for a discontinuous system modelling a dc-dc buck converter under voltage control. The prob- lem we are concerned with in this paper is to prove the existence of an attractor with an embedded horseshoe mechanism and discussing how it is created. The lo- cal analysis performed in [2] about the existence of strange attractors in the system is now complemented in a more geometrical way 1 . I. I NTRODUCTION Unfortunately, nowadays, mathematical proofs of chaotic behavior in electrical circuits are still rare. This also applies to dc-dc converters. Laboratory ex- periments and computer simulations are abundant in the literature regarding the dc-dc voltage-controlled buck converter ([3],[4],[5],[6]), but there are very few mathematical works about it ([2] is a remarkable ex- ception). In [2] a deep analysis of the non-smooth dynamics of this converter was established, including local analysis of grazing and sliding phenomena. This was used to explain the spiral bifurcation structure of some and periodic orbits [5]. The experimental basis of the present study is a DC-DC buck converter whose output voltage is con- trolled by a PWM with natural sampling and constant frequency, working in continuous conduction mode. This circuit is one of the simplest but most useful power converters, a chopper circuit that converts a dc input to a dc output at a lower voltage (many switched This work was partially supported by CICYT under Grant TAP97-0969-C03-01 mode power supplies employ circuits closely related to it). The two circuit topologies are described by two systems of linear differential equations 2 if if (1) where , , and is a two-valued function. It is worth noting that in each topology the system is linear, and analytical solutions can easily be com- puted. Solutions for system (1) can be obtained by joining trajectories of both topologies at the switching instants. A. The Stroboscopic Map Keeping the notation of [3],[4], the parameters of the circuit are: , , and , the resistance, the capac- itance and the inductance of the circuit; and , the lower and upper voltages of the ramp and , its period; , the gain of the amplifier; , the reference voltage, and , the input voltage. The values for the parameters which will be assumed in this paper are like in [3],[4]: , , , , , , , and . The switches are assumed to be ideal. The converter we are going to study works as follows: the voltage of the capaci- tor is applied to the positive pole of the amplifier with gain , and the reference voltage , to the nega- tive pole. The output voltage, which we will call the control voltage , is thus Then, both and , the voltage of the ramp, are applied to the comparator, and every time the output difference changes its sign, the position of the switch is commuted in such a way that it is open when the control voltage exceeds the ramp voltage and it is Throughout the paper, stands for derivation with respect to time.