On the Stability of Limit Cycles in Resonant DC-to-DC Power Converters Victor M. Hern´ andez†‡ 1 , Ram´ on Silva‡ 2 , Hebertt Sira-Ram´ ırez‡ ‡CINVESTAV-IPN, Departamento de Ing. El´ ectrica, Secci´ on Mecatr´ onica. Avenida IPN No. 2508. Col. San P. Zacatenco. A.P. 14740. 07300 M´ exico, D.F., M´ exico. {vmhernan,hsira}@mail.cinvestav.mx, Phone:(52-55) 50 61 38 00, ext. 6311. †Universidad Aut´ onoma de Quer´ etaro. Facultad de Ingenier´ ıa. Centro Universitario Cerro de las Campanas. Quer´ etaro, M´ exico. Abstract In this note, we use the Poincar´ e map to study the sta- bility of induced limit cycles in a series resonant DC- to-DC power converter. We prove global exponential stability of the unique limit cycle. We establish a means of computing a priori the voltage supplied to the load. We also study the robustness properties of the output voltage when different values of load are used. Finally, we succeed to compute the ripple voltage. Some simu- lation results are presented to verify our findings. Keywords: Poincar´ e map, Resonant DC-to-DC Power Converters, Hybrid Systems, Dynamical Systems. 1 Introduction. Resonant converters work under the principle of switch- ing at either zero voltage or zero current. This is achieved thanks to the introduction of a LC resonant circuit connected in series, or in parallel, to the main switching device [1], [2]. Several approaches have been proposed for the analysis of resonant converters. Stud- ies based on DC aproximated considerations have been presented by Vop´ erian and Cuk [3], [4]. Oruganti and Lee [5], [6] introduced control strategies based on the state variable representation. Some controllers have been designed introducing the Lyapunov stability cri- terion (Stankovic [9]). Finally, Sira-Ram´ ırez and Silva [11] have presented a novel idea for the modelling and control design of both series and parallel resonant power converters. The main tools used are the flatness (see Fliess et al. [8]) and the hybrid properties of these 1 V.M. Hern´ andez work is supported by a scholarship from the Universidad Aut´ onoma de Quer´ etaro, M´ exico. 2 R. Silva work is supported by CONACYT, M´ exico. converters. Further, they have presented very good re- sults in both simulations and laboratory experiments. Modelling of further resonant converters using the flat- ness approach can be found in Silva [12]. However, they do not prove stability of the induced limit cycle. Further, the voltage at the load is not known a priori nor the ripple voltage. This is important because these voltages must be known before connecting the load. On the other hand, in the simulations and experiments presented in [11] it is apparent the robustness of the voltage at the load when the latter changes. However, this property is not explained. On the other hand, in Zavala [15] the Poincar´ e map has been proposed and used to analyze hybrid systems. In this paper we use the Poincar´ e map as a tool to: 1) prove that the limit cycle is globally exponentially stable, 2) determine the voltage at the load, 3) explain the robustness of the voltage at the load when this takes different values and 4) determine the ripple voltage . We restrict our study to the case of series resonant converters. This paper is organized as follows. In section 2 we present the dynamic model of the converter under study as well as some preliminary results. In section 3 we present a brief review of the Poincar´ e map thechnique. Our main results are presented in sections 4, 5 and 6. Some sim- ulation results are shown in section 7, whereas section 8 is devoted to give some concluding remarks. 2 The dynamic model of series resonant DC-to-DC power converters. This section is entirely based on the results reported in [11]. The electric circuit of a series resonant DC-to-DC power converter is presented in fig. 1. The correspond- ing dynamic model is given as: L di dt = -v - v 0 sign(i)+ E(t) (1)