Analytical theory of an unloaded rectangular microstrip patch

A graphical method for determining the size of a resonant rectangular microstrip patch for a specified frequency is given. Using an expression obtained from the reflection coefficient of a transverse electromagnetic (TEM) wave in a semi-infinite microstrip patch, the resonance condition of a given mode for a patch of finite size is derived in a manner analogous to that of a rectangular, waveguide cavity. Radiation is shown to be in the form of both surface waves and space waves and is dependent on the angles of incidence for waves impinging onto the edges of the patch. By varying the aspect ratio, it is also possible to modify the
Q
factor of a resonant path. Analytical expressions for the resonant frequency and the
Q
factor are then derived under the assumption of a very thin substrate slab, encountered frequently in the application. Design curves in the form of the "Argand diagram" are then presented for microstrip patches with fixed aspect ratios.