This paper describes the equivalent-circuit model of a metamaterial composed of conducting spheres and wires. This model involves electromagnetic coupling between the conductors, with retardation. The lumped-parameter equivalent circuit, which imports retardation to the electromagnetic coupling, is developed in this paper from Maxwell's equation. Using the equivalent-circuit model, we clarify the relationship between the retardation and radiation loss; we theoretically demonstrate that the electromagnetic retardation in the near-field represents the radiation loss of the meta-atom in the far-field. Furthermore, this paper focuses on the retarded electromagnetic coupling between two meta-atoms; we estimate the changes in the resonant frequencies and the losses due to the distance between the two coupled meta-atoms. It is established that the dependence characteristics are significantly affected by electromagnetic retardation.

An equivalent-circuit model is an effective tool for the analysis and design of metamaterials. This paper describes a systematic and theoretical method for the circuit modeling of meta-atoms. We focus on the structures of wired metallic spheres and propose a method for deriving a sophisticated equivalent circuit that has the same topology as the wires using the partial element equivalent circuit (PEEC) method. Our model contains the effect of external electromagnetic coupling: excitation by an external field modeled by voltage sources and radiation modeled by the radiation resistances for each mode. The equivalent-circuit model provides the characteristics of meta-atoms such as the resonant frequencies and the resonant modes induced by the current distribution in the wires by an external excitation. Although the model is obtained by a very coarse discretization, it provides a good agreement with an electromagnetic simulation.