Plane wave expansion method was employed to study the complete photonic band gap (PBG) of two-dimensional air annular photonic crystals (PCs). Comparison of the complete PBG of the air annular photonic crystal and the commonly used PCs (square-and triangular-lattice dielectric rods in air and square-and triangular-lattice air holes in dielectric background) reveals that a larger complete PBG can be obtained for the air annular PCs. Furthermore, the complete PBG is observed for the air annular PCs even if the dielectric contrast is low, while it is difficult to achieve a complete PBG for the commonly used PCs. For example, for the square and triangular lattice air-hole PCs, the complete PBG is observed when the refractive index of the dielectric materials is arger than 2.8 and 3.2, respectively. But the complete PBG can be observed when the refractive index is arger than 2 for the air annular PCs.