Low-Frequency Characterization of Switched dc-dc Converters

Techniques are developed for the approximate representation of switched dc-dc converters by time-averaged models. Simple analytical expressions in terms of the circuit components are derived for the characteristic transient and frequency responses of averaged power-stage models for use in designing and understanding the behavior of the actual switched power stages. High-order systems can be analyzed by the averaging technique without a commensurate increase in complexity. Two functional blocks are necessary to construct a switched converter: the switch controller, which is relatively well understood, and the power stage. When concreteness is necessary, a particular pulse- width modulator is chosen for the switch controller and is thoroughly analyzed. The output of representative power stages (buck, boost, and buck-boost) is a complicated nonlinear function of the switch controller and source input, and since conventional methods of nonlinear analysis are shown to be intractable or uninterpretable, attention is focused on the challenge of obtaining useful design equations. The difficulty encountered in the nonlinear analysis of switched power stages is successfully surmounted by the semiheuristic development of a continuous power-stage model. Since the characteristic response times of state variables in the switched power stage are invariably large with respect to the switching period, discontinuous forcing functions in the equivalent circuits are averaged over a time interval comparable with the switching period without appreciably affecting the nature of the response. Consequently, the averaged model is limited to response times greater than the averaging interval. Equivalent circuits and analytic expressions for the transient and frequency response of each power-stage type are then derived from the averaged models. A linearized control-input transfer function, obtained for small amplitude variations of the averaged control, reveals a dependence of effective circuit component values upon the switch duty ratio, and the possible existence of a positive real zero. The unusual behavior predicted above is confirmed by an analog computer simulation of both the switched and averaged power stages. It is also shown experimentally that closed-loop stability of the switched power stage is adequately predicted by the averaged model. The averaging technique is thus a powerful analytical tool for exposing inherent characteristics of switched circuits.