Terahertz vortex beam generators have potential applications in optical micro-manipulation, terahertz communications and many other fields. A broadband vortex beam generator in a terahertz frequency range is proposed based on the metasurface of double-split resonant rings’ array. The designed structure consists of two layers, i.e., the top layer, which is a metasurface of double-split resonant rings, and the bottom layer, which is the dielectric layer of polymide. The numerical simulation of the cell structure array is performed by using the CST microwave studio. In order to obtain the best performance, the structure parameters of metasurface are continuously optimized and a set of optimal geometric parameters is finally determined. The simulation results show that the circularly polarized incident light can be converted into corresponding cross-polarized transmitted light. By rotating the metal resonant ring on the top layer, the cross-polarized transmitted light can be controlled to have the same amplitude and correspondingly different phases. The relationship between the phase change and the angle of rotation conforms to the P-B phase principle. These cell structures are arranged according to a specific order and can form the vortex phase plates for generating the vortex beams with different topological charges. Taking the topological charge numbers 1 and 2 for example, two kinds of vortex phase plates are designed. The characteristics of the circularly cross-polarized vortex beams generated by a circularly polarized wave perpendicularly incident on the vortex phase plates are numerically analyzed. The results show that the ideal vortex beams with different topological charge numbers are generated. The characteristics of vortex beams appear to be consistent with those theoretical results. Moreover, the vortex beams can be generated in a frequency range from 1.39 THz to 1.91 THz. The operating bandwidth is much wider than the previously obtained result of the transmission terahertz vortex phase plates. The transmission is higher than 20%, and the maximum value of transmission can reach 24%, which is close to the theoretical limit value of the single-layered transmission-type metasurface. This work provides a reference for generating the terahertz vortex beams based on metasurface. It is expected to possess a practical application in generating the device of terahertz vortex beam.