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In this paper, we perform an analysis of anomalous extraordinary transmission (ET) through hole arrays in a rectangular lattice. This phenomenon is a peak of high or even total transmission when the polarization is parallel to the short periodicity of the hole array, and it only happens when the perforated plate is backed by a dielectric slab. The analysis is done using equivalent circuit concepts, and the performance is evaluated for several combinations of dielectric slabs on both faces. Two types of anomalous ET are shown, depending on the dielectric slab parameters. The phenomenon is also analyzed in terms of grounded slab diffraction modes and following the transversal resonance method as an open cavity resonance. Finally, an extension to rectangular waveguides is done, recognizing the similarities, but also emphasizing the intrinsic differences, related with the different modal distribution in the free-space and waveguide models. These results can help to fully understand anomalous ET with simple circuit concepts and to deepen the physical understanding of the problem by observing the same phenomenon from different points of view. Applications in filters, both in free-space and waveguide architectures, as well as in microscopy, are foreseen.