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This paper presents a discrete-time adaptive current controller for grid-connected pulse width modulation voltage source converters with LCL filter. The main attribute of the proposed current controller is that, in steady state, the damping of the LCL resonance does not depend on the grid characteristic since the adaptive feedback gains ensure a predefined behavior for the closed-loop current control. An overview of model reference adaptive state feedback theory is presented aiming to give the reader the required background for the adaptive current control design. The digital implementation delay is included in the model, and the stability concerning the variation of the grid parameters is analyzed in detail. Furthermore, current distortions due to the grid background voltage are rejected without using the conventional stationary resonant controllers or the synchronous proportional-plus-integral controllers. Simulation and experimental results are presented to validate the analysis and to demonstrate the good performance of the proposed controller for grid-connected converters subjected to large grid impedance variation and grid voltage disturbances.