In this paper, we analyze low-profile circularly polarized (CP) antennas comprising a crossed-dipole radiator on finite artificial magnetic conductor (AMC) surfaces. The crossed dipole is fed by a pair of vacant-quarter printed rings to produce CP radiation. The AMC structure consists of a lattice of square metal plates on a grounded dielectric substrate with connecting pins between the patches and the ground plane. In this paper, we focus on the excitation of surface waves propagating on the finite-sized AMC surface, which generates extra resonances and CP radiations for the radiation structures. We predict the surface-wave resonances using a cavity model. In this model, the finite-sized AMC structure is considered as a waveguide resonator. We verify the predicted results computationally using the finite element method-based full-wave electromagnetic solver Ansoft high-frequency structure simulator. The results show that these extra resonances and corresponding CP radiations can be used to broaden the impedance matching and axial ratio bandwidths of the antennas, respectively.