In recent years, a great interest has been observed in the development and use of new materials in microwave technology. Particularly, a special interest has been observed in the use of superconducting materials in microwave integrated circuits, this is due to their main characteristics. In this paper, the complex resonant frequency problem of a superconductor patch over Ground Plane with Rectangular Aperture is formulated in terms of an integral equation, the kernel of which is the dyadic Green’s function. Galerkin’s procedure is used in the resolution of the electric field integral equation. The surface impedance of the superconductor film is modeled using the two fluids model of Gorter and Casimir. Numerical results concerning the effect of the thickness of the superconductor patch on the characteristics of the antenna are presented.