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Based on the mathematical model of the current source inverter (CSI) in the stationary alphabeta frame, this paper proposes advanced stationary frame voltage control methods for CSI with linear and nonlinear loads to achieve zero steady-state error respectively. The voltage control strategy based on a special resonant controller for CSI with linear loads is firstly analyzed in detail, on the basis of which, the voltage control method for CSI with nonlinear loads is proposed and the principle of it is investigated. Because the alphabeta components of CSI output voltage are decoupled in the stationary frame, they can be controlled independently and the controllers can be designed easily. Meanwhile, these voltage control methods don't require accurate circuit parameters, so they can also improve the control system's robustness. At last, all these analyses and the effectiveness of the proposed control methods are firstly validated by simulation, and then are verified further in a 5 kVA laboratory prototype.