Green cognitive radios are promising for high energy efficiency (EE) in the future of wireless communications. Spectrum sensing refers to an energy-consuming procedure that allows cognitive users to independently identify unused radio spectrum segments and prevent interference to primary users and it should be minimized due to resource limitations. In this paper, we present a wireless multiple-access channel function to establish the primary users' presence and the mean EE optimization problem of the cognitive radio systems with mathematical structure computation purposes. EE as the average throughput to the average energy consumption ratio is used to measure the network's performance subject to detection constraint. More specifically, since secondary users are generally battery-powered devices, saving on energy is crucial. Our aim is to decrease the energy consumption of secondary users since while maximizing e the total EE while and preserving the accurate sensing by maximizing the sensing time detection subject to secondary user power constraints and minimum data rate. Based on this, to address the non-convexity optimization problem, one can consider energy-efficient power allocation based on an iterative method. Simulation results show the optimal ity of our framework when combined by the multiple-access channel computation based scheme. Green cognitive radio should consider the trade off against its complexity and maximum available EE metric. However, one can observe from the simulation results that the improved EE presented in this work yields much higher than other ones with the same detection performance in comparison.