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A new class of pulse-frequency modulated systems is presented in this paper. These systems referred to here as have many advantages over previously used schemes such as integral PFM. Most significant advantages are improved stability and simpler physical implementation of the modulator. The major part of this paper is concerned with the study of sustained oscillations using a specially developed quasi-describing function. One important feature of these kinds of PFM systems is that they often present a limit annulus and not a limit cycle, a feature which is common in most nonlinear discrete feedback systems. A few examples with experimental verification are presented and the limitations of the method are discussed.