Generalized Permeability Function and Field Energy Density in Artificial   Magnetics Using the Equivalent Circuit Method
  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. 
Pekka Ikonen and Sergei Tretyakov