Realizations of double negative (DNG) metamaterials using lumped element representations of lossless and lossy transmission lines are reported. Double negative media are artificial materials in which the permittivity and permeability are both negative. Analytical and numerical results for a variety of transmission line network representations explain how they can exhibit low-pass, high-pass, and band-pass dispersion characteristics when a DNG medium is produced in a specified frequency region of interest. Detailed dispersion properties of these various configurations are given. Comparisons with previous formulations are provided. It is shown conclusively that the DNG metamaterials occur only when the overall equivalent transmission line per-unit-length impedance is capacitive and the overall equivalent transmission line per-unit-length admittance is inductive. A convenient categorization of the possible DNG medium designs in transmission line constructs or more complex 2-D or 3-D inclusion based structures thus is provided.