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This paper investigates dynamic balancing of flying capacitor multilevel (FCML) converters with coupled inductors. Coupled inductors help to reduce the ripple current, accelerate transient response, and balance the flying capacitors of FCML converters at steady-state. However, coupled inductors also change the dynamic balancing properties compared to uncoupled inductors, and these principles must be understood for robust design. As an extension of a previously developed feedback mechanism for understanding the steady-state behaviors of coupled inductors in FCML converters, this paper derives models of coupled inductor FCML converters in dynamic operation, revealing several key insights: (i) the multi-resonant behavior of large-order FCML converters and their dependence on the initial conditions, (ii) how power dissipation relates to balancing speed, and (iii) the relation between multiphase and multilevel FCML balancing. The insights uncovered by this paper can provide useful guidelines for designing multi-phase self-balanced FCML converters with coupled inductors. 

This article investigates the modeling, analysis, and design methods for passively balancing flying capacitor multilevel (FCML) converters using coupled inductors. Coupled inductors synergize with FCML converters by reducing inductor current ripple, reducing switch stress, and, as proven in this article, by providing flying capacitor voltage balancing. This enables FCML topologies to be scaled well to larger systems. This article proves that coupled inductors can solve the unbalancing problem in many FCML converters. Moreover, tools are developed to thoroughly explain and quantify coupled inductor balancing, allowing general design guidelines to be offered for robust coupled inductor FCML converters. Finally, this article derives the limitations of coupled inductor balancing with respect to the number of phases, levels, and the required coupling ratio. The key principles of coupled inductor FCML balancing in steady state are demonstrated with a systematic theoretical framework and extensive experimental and simulation results.