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This paper deals with the adaptive tuning of the stator inductance in a rotor-current-based Model Reference Adaptive System observer for the sensorless control of doubly fed induction machines. At first, the effect of mismatched parameters in this observer is discussed in order to show the considerable influence of the stator inductance on the accuracy of the estimated rotor position. Then, an adaptive tuning of the stator inductance is proposed and a small signal model is deduced in order to design the tuning loop. Moreover, a theoretical sensitivity and stability analysis is performed. Finally, the performances of the proposed scheme are experimentally investigated and validated.