With the increasing demand for potential applications of vortex beams in various fields of modern optics, the generation of optical vortex beams has attracted great interest. Based on a flat plate made of an electro-optical crystal, a method to generate optical vortex beams assisted by the Pockels effect is proposed. This method allows an orbital-angular-momentum-tunable range of $ \pm 2\hbar$ with a finite phase-modulated region. We simulate the propagation of optical beams transmitted from the flat plate and investigate the orbital-angular-momentum-mode spectra of the transmitted optical beams. The mode spectra accord well with the simulation results. The proposed method will be applied to fields where tunable optical vortex beams are required, such as optical communication and optical manipulation.