Due to the inherent complexity of the mechanisms needed to prescribe the heave and the pitch motions of optimal flapping-foil turbines, several research groups are now investigating the potential of using unconstrained motions. The amplitude, the phase and the frequency of such passive motions are resulting from the interaction of the foil with the flow and its elastic supports, namely springs and dampers. More specifically, this work proposes an innovative semi-passive flapping-foil turbine concept with a prescribed sinusoidal heave motion and a passive pitch motion. Two-dimensional numerical simulations have been carried out at a Reynolds number of 3.9 x 106 based on the chord length with a foil having its pitch axis located at the quarter-chord point. A parametric study has been conducted by varying the value of the static moment, which involves the distance between the center of mass and the pitch axis, and the frequency of the prescribed heave motion. Different responses of the foil have been observed and one of them corresponds to an energy-extraction regime that is characterized by periodic limit-cycle oscillations of large amplitudes with a phase lag between the heave and the pitch motions ranging between 90º and 105º. A maximum efficiency of 45.4% has been reached, hence confirming the great potential of this turbine concept. This works shows that such good performance is achieved when the center of mass of the foil is located downstream of the pitch axis and when no leading-edge vortices are formed.