Dispersion relation of split-ring resonators magnetically coupled to a TEM wave

An electromagnetic wave dispersion equation in a metamaterial consisting of an array of split-ring resonators (SRRs) is derived for the determination of stop bands and passbands, showing the kind of wave which propagates, whether it is forward or backward. This derivation takes into account the magnetic coupling between the adjacent SRRs to consider propagation of magneto inductive waves (MI). The mathematical formulation is based on a coupled circuit model of transmission lines, that is, by using lumped-element circuits to model the SRR array.