It is presented in this article, a simplified theory of the active lift turbine which has been the subject of several patent[4, 5, 11]. A simplified theory is proposed to extend the Betz limit of the yield on vertical axis wind turbine. This work can be extended either on wind driven or marine current turbine. Based on kinetic energy calculation , that theory demonstrates that the radial force acting on the blade can be used to extend the maximum recoverable power, mainly by transforming a linear motion into a rotating motion. The geometry of this new type of vertical axis turbine is described. Finally the driving power is calculated through the tangential power coefficients. Consequences on the velocity variation generation and induced vibrations are exposed. Two parts is treated in this article : The calculation of power coefficient and the calculation natural modes of vibration