ON LIMIT-CYCLES AND THE DESCRIBING FUNCTION-METHOD IN PERIODICALLY SWITCHED CIRCUITS

This paper begins with an examination of existence, uniqueness, and stability of limit cycles in periodically switched circuits. The motivation comes from the field of power electronics where switched circuit models composed of passive elements, independent sources, and ideal switches are studied. The paper then studies the describing function method for computation of limit cycles in these switched circuits. Typical power circuit models have nonlinear elements with characteristics that do not satisfy a Lipschitz continuity condition. As a result of these nonsmooth characteristics, previously developed justifications for the describing function method are not applicable. The present paper develops a justification for the describing function method that relies on the incrementally passive characteristics of the network elements comprising typical power electronic circuit models. This justification holds for nonsmooth circuit nonlinearities, and takes the form of a set of asymptotically convergent bounds on the errors incurred with the describing function method. In particular, the developed bounds become arbitrarily tight as the number of harmonics included in the analysis increases.