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This article proposes a systematic synthesis method to identify the optimum converter topology for a specified voltage conversion ratio. This gain-oriented converter synthesis is formulated as an inverse problem around the inductor flux balance equations. The proposed synthesis method consists of two processes. A set of distinct converter topologies is first synthesized from the specified voltage conversion ratio by obtaining the solutions for a set of six inverse equations in twelve unknowns. The optimal topology for any intended application can be identified based on the desired performance metrics. The complete synthesis method is illustrated with quadratic buck–boost gain as an example. In total, 25 distinct topologies are synthesized using the first process, and their feasibility is validated by PLECS simulation. In total, three case studies are reported to explain the second process. Ground-referenced output and the minimum number of switches are used as the first two selection criteria in all the case studies. Minimum peak inductor current, minimum component stress factor, and minimum voltage and current stress are the third selection criterion in Case Studies-1, 2, and 3, respectively. The steady-state operations of two optimal converters selected in Case Study-1 were verified experimentally. [ABSTRACT FROM AUTHOR]