Phased microstrip array antenna is an effective method to generate vortex electromagnetic beam containing orbital angular momentum (OAM), and a good design of the multi-mode OAM microstrip array antenna is crucial to improve the vortex performance. We consider circular microstrip antenna with coaxial feed as the element to compose the circular antenna array, which is excited with unit amplitude and equal phase delay from element to element. Two single-ring uniform circular array antennas are designed with the array elements number as eight and sixteen respectively, and a dual-ring uniform circular array antennas is designed with eight elements in inner ring and sixteen elements in outer ring. The OAM-carrying electromagnetic vortex can be generated at 15 GHz from both sing-ring and dual-ring uniform circular array antennas, and the vortex characteristics, such as radiation patterns and OAM modes, are analyzed in detail by using a full-wave electromagnetic field simulation code HFSS. According to the simulation and optimization, with the increase of elements number in the sing-ring structure, we get the better electromagnetic wave vortex performance at the cost of a larger size array antennas. In contrast, through using collaborative optimization simulation combined with inner and outer ring, the vortex properties of dual-ring structure is better than that of the single-ring structure at suitable array configurations and phase shifts.