Abstract: Hexagonal modular multilevel converter (MMC) and nonagonal MMC are two emerging direct AC-AC converters with a circular topology. Owing to their advantages in device count, they can be used as an alternative topology of cascaded H-bridge multilevel matrix converter, and have broad development prospects in the fields of offshore wind power fractional frequency transmission system. However, these two topologies are bridge arm multiplexing structures, which may result in excessive stress of a single bridge arm; therefore, their advantages over M3C in the number of devices will be weakened, the control degrees of freedom will be reduced, and hence, traditional stress optimization methods are not applicable. To further enhance their advantages in the number of devices, in this paper, hexagonal MMC and nonagonal MMC are unified in circular MMC topologies, the mathematical model of bridge arm current in circular MMC is established, and the influence of system parameters on current stress of circular MMC bridge arm is explored. It is proved that the phase angle difference between ports of nonagonal MMC is the key parameter affecting current stress of circular MMC bridge arm, and the comprehensive current cycle component stress of circular MMC bridge arm has a periodic relationship with the phase angle difference between the ports. By adjusting the phase angle difference between the ports of nonagonal MMC with the same frequency, the comprehensive current cycle component stress of bridge arm can be reduced by as great as 29.25% according to theoretical analysis. Finally, the correctness of the conclusions of this paper is verified by building a simulation model on the RT-LAB simulation experimental platform.