  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. The proposed circuit model and
permeability function are experimentally validated. Furthermore, starting from
the generalized circuit model we derive an explicit expression for the
electromagnetic field energy density in artificial magnetic media. This
expression is valid at low frequencies and in the vicinity of the resonance
also when dispersion and losses in the material are strong. The presently
obtained results for the energy density are compared with the results obtained
using different methods.

Classical Physics