The thin-substrate approximation for reflection from the end of a slab-loaded parallel-plate waveguide with application to microstrip patch antennas

Integrals arising in the Wiener-Hopf solution for wide microstrip antennas are approximated in closed form for the case of an electrically thin substrate. The resulting expressions are used to obtain simple, closed-form expressions for resonant frequency and unloaded radiation
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for rectangular microstrip patch antennas which are valid when commonly used quasi-static formulas no longer hold. Numerical comparisons show significant shifts in resonant frequencies as compared with existing theories.