Design of a three-level Hysteresis controller for a four-leg voltage source inverter in αβo frame This paper studies Design and simulation of a three-level Hysteresis current controller for a four-leg inverter connected to a distribution system in αβo frame. It is necessary to transfer ABC frame to αβo frame in order to avoid practical issues of implementing hysteresis modulation. This paper concentrates on choosing the best switching states for the fourth leg of the Inverter to reduce low order harmonics and losses. Finally Matlab-Simulink was used for validating the proposed idea.