Subharmonic Instability Limits for ${\rm V}^2$ -Controlled Buck Converter With Outer Loop Closed/Open

General closed-form subharmonic oscillation conditions are obtained for V
²
-controlled buck converters. Both constant switching period and constant on-time operations are analyzed with the outer voltage loop closed or open. Once an arbitrary linear feedback is given, the associated closed-form stability condition of the converter is readily obtained. Past research results based on the describing function technique and the Floquet theory become special cases. This paper provides an alternative and easier method to analyze subharmonic instability.