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A radiating periodically structured surface resonator is studied from the point of view of minimizing the radiation Q-factor of the fundamental mode of the resonator. Structures with a one-dimensional array of gaps are considered, where the resonant mode is formed by the capacitance of the gaps resonating with the inductance of the conducting pathways. An eigenmode study of a range of designs of identical size demonstrates that increasing the number of gaps does not lower the Q . The results show that this family of resonators has a radiation Q that is typically ~ 2 × lower than a simple patch antenna occupying the same electrical volume. The improvement in Q is accompanied by a corresponding reduction in the directivity gain of the broadside radiation, confirming the inherent bandwidth/directivity trade-off in radiators of identical size and shape. A three-period resonator is made into an antenna by driving it from underneath the surface. The fabricated antenna has dimensions of 0.40 λ × 0.26 λ, a thickness of λ/34, and a measured 10 dB return loss bandwidth of 12.4%, ~ 4.5 × wider than a simple patch antenna of the same size. These antennas can replace patch antennas in applications demanding the widest possible impedance bandwidth from a thin antenna.