Recently, experimental results have demonstrated that perpendicular magnetic tunnel junction (p-MTJ) with the antiferromagnetic(AFM)/ferromagnetic (FM)/oxide structure can achieve field-free spin-orbit torque (SOT) switching since the AFM metal strip can not only generate the SOT, but also provide an exchange bias (HEX), making it suitable for practical applications. However, owing to that the HEX is weak, such field-free SOT switching is incomplete, thus resulting in severe switching reliability. In addition, a large SOT switching current (ISOT) is also required, leading to high switching energy dissipation. In this paper, to address these issues, the voltage-controlled magnetic anisotropy (VCMA) is introduced to assist the SOT switching, and such novel switching method is referred as voltage-gated SOT (VGSOT). First, we develop a physics-based compact model for the three-terminal VGSOT-MTJ device, which includes three modules, i.e., the electrical module, the tunnel magnetoresistance module and the dynamic switching module. Then, the impact of the VCMA effect on the field-free SOT switching is investigated by solving a modified Landau-Lifshitz-Gilbert (LLG) equation with consideration of the VCMA, SOT and HEX. Simulation results show that thanks to the introduction of the VCMA effect, the critical ISOT can be reduced greatly, and the incomplete field-free SOT switching can be completed. With further analysis, we obtain a special switching condition, under which complete SOT field-free switching can be achieved with a shortest path and ultra-low power. Moreover, a novel write pulse scheme is proposed to achieve high speed and reliability.