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A polarization-independent metamaterial analog of electromagnetically induced transparency (EIT) at microwave frequencies for normal incidence and linearly polarized waves is experimentally and numerically demonstrated. The metamaterial consists of coupled “bright” split-ring resonators (SRRs) and “dark” spiral resonators (SRs) with virtually equal resonance frequencies. Normally incident plane waves with linear polarization strongly couple to the SRR, but are weakly interacting with the SR, regardless of the polarization state. A sharp transmission peak (i.e., the transparency window) with narrow spectral width and slow wave property is observed for the metamaterial at the resonant frequency of both, the bright SRR and the dark SR. The influence of the coupling strength between the SRR and SR on the frequency, width, magnitude, and quality factor of the metamaterial's transparency window is theoretically predicted by a two-particle model, and numerically validated using full-wave electromagnetic simulation. In addition, it is numerically demonstrated that the EIT-like metamaterial can be employed as a refractive-index-based sensor with a sensitivity of 77.25 mm/RIU, which means that the resonance wavelength of the sensor shifts 77.25 mm per unit change of refractive index of the surrounding medium.