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This paper presents a strategy for the analytic determination of the natural voltage balancing dynamics of flying capacitor converters. The approach substitutes Double Fourier series representations of the PWM switching signals into a nonlinear dynamic circuit model of the converter. The result reduces to a linearised state space model that can be readily solved, with the Fourier solution coefficients defining the state space matrix terms. The solution can be readily developed for converters of any level, and allows rapid analytical investigation of the dynamic (and static) balancing behaviour over a wide range of conditions. Furthermore, the approach allows powerful strategies such as root locus to be used to investigate the converter's performance as a function of changes in parameters such as modulation index and load. The analysis approach has been fully verified by comparing it against experimental results on a low voltage prototype converter.