In this paper a simplified model based on the exact discrete-time map of a buck switching power converter with proportional control, which captures all its dynamics, allows deriving a closed-form stability condition for predicting fast-scale instability boundary. This condition analytically demonstrates the validity of the recently proposed ripple-based index to predict fast-scale period-doubling, hitherto based on an a priori hypothesis and simulation validation, thereby demonstrating the use of the ripple index as a design-oriented tool. The equivalence of the ripple index to the condition derived from the discrete-time map endorses its use as a means to characterize the complete design space against fast-scale instabilities.

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