A novel semianalytical methodology is used to analyze a periodic array of printed metallic closed ring elements in a multilayered dielectric structure. This approach is unique in that it is the first methodology capable in modeling structures with resonant implants and interelement dimensions well beyond the effective medium theory. In addition, it yields computational efficiency by 2 orders of magnitude over standard computational methods in computing the scattering parameters for proximity equilibration cell (PEC) closed ring multilayered (electromagnetic band gap and photonic band gap (PBG)) structures. Moreover, it provides physical insight in the implementation of metallic implants for practical applications. This methodology satisfies the Kramers-Kronig relations and causality, and therefore it allows for the development of semianalytical expressions for the composite's wave impedance, index of refraction, as well as the permittivity and permeability parameters accounting for full dispersion. For general artificial multilayered structures (PBG metamaterials) with centrosymmetric scattering matrices, the composite may be replaced by an equivalent homogeneous dispersive magneto-dielectric material and may be used for the design of integrated circuits, filters, and antennas using standard methods. Otherwise, use of the scattering matrix approach to obtain the effective parameters is valid only for semi-infinite structures. The upper band edge is determined by the host material uniquely and the bandwidth is determined by the shunt susceptance for different PEC ring inclusions.