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This article presents an analytical method to derive an optimum converter topology for a specified voltage conversion ratio (G). An optimum topology for any given G can be identified after obtaining the comprehensive pool of converter topologies. This gain-oriented converter synthesis is formulated as an inverse problem around the principle of inductor flux balance. A step-by-step method is proposed to solve the inverse problem and obtain a comprehensive set of non-redundant converter topologies for any given voltage conversion ratio. Consequently, the number of possible converter topologies for a specified voltage conversion ratio is also identified. The optimum converter topology is chosen from the comprehensive set based on three selection criteria, viz., ground-referenced output, a minimum number of switches, and minimum peak inductor current. The proposed solution methodology is illustrated with quadratic boost and quadratic buck gains as case studies. A comprehensive set of non-redundant topologies is identified in both cases. The steady-state operations of all the topologies are studied with PLECS simulation. Based on the selection criteria, an optimum topology is selected in each case, and their steady-state operation is verified experimentally using lab-developed prototypes. [ABSTRACT FROM AUTHOR]