Asymptotic formula for the resonant frequencies of a circular microstrip antenna

The resonant frequencies of both the axial and nonaxial symmetric modes of a circular microstrip antenna or resonator are solved asymptotically. The technique used is matched asymptotic expansions which is widely used in fluid mechanics but little used in electromagnetism. In this technique, the region around the open resonator is divided into the exterior region, the edge region, and the interior region. Leading order solution is derived in each region and matched to each other asymptotically. This provides an asymptotic eigenequation which can be solved approximately to give an asymptotic formula. The asymptotic formula agrees well with previous numerical methods, especially when the ratio of the dielectric substrate thickness to disk radius is d/a→0.