Generalized permeability function and field energy density in artificial magnetics

The equivalent circuit model for artificial magnetic materials based on various arrangements of split rings is generalized by taking into account losses in the substrate or matrix material. It is shown that a modification is needed to the known macroscopic permeability function in order to correctly describe these materials. Depending on the dominating loss mechanism (conductive losses in metal parts or dielectric losses in the substrate) the permeability function has different forms. Furthermore, an explicit expression is derived for the electromagnetic field energy density in the material. This expression is valid at low frequencies and in the vicinity of the resonance also when dispersion and losses in the material are strong. Microstrip antennas loaded with artificial magneto-dielectric substrates are introduced as an example of practical applications benefiting from the present results.