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A new behavior of the field dependence of microwave surface resistance (Rs), which was observed on a microstrip resonator and may be caused by the edge of the center strip, is reported in this paper for epitaxial high-Tc superconducting (HTSC) thin films. The exhibited behavior is that Rs remains almost unchanged below a certain rf magnetic field Hrf, and then increases abruptly at this field, after which it increases in proportion to Hrf. To explain the behavior, the morphology of the microstrip resonator was examined by atomic force microscopy (AFM), which showed that the edge of the resonator was damaged in some regions because of the acid etching. If the damaged edge is considered as a weakened granular superconductor, the observed Rs behavior could be explained well in terms of the high-frequency critical state model. This implies that the edge condition should be considered in studying the field dependence of Rs when the planar resonator technique is used.