Modeling Metamaterial Transmission Lines Loaded With Pairs of Coupled Split-Ring Resonators

A lumped-element equivalent circuit model of the unit cell of metamaterial transmission lines loaded with pairs of coupled split-ring resonators (SRRs) is presented. It is assumed that the dominant coupling mechanism between the SRRs forming the pair is magnetic, and that the distance between SRRs of adjacent cells is high enough to neglect such additional inter-resonator coupling. SRRs are oriented with their symmetry plane orthogonal to the line axis. Under these conditions, the line-to-SRR coupling is also magnetic, the electric coupling being negligible. The presented model accounts for the rupture of symmetry that can be caused, for instance, by asymmetric dielectric loading of the SRRs. Thus, the analysis is carried out on a general model where the SRRs of the pair have different inductance and capacitance. Then, different cases are studied, in particular a line with identical SRRs, and a line with different SRRs, but with the same resonance frequency. It is shown that coupling between SRRs tends to far or split the resonance frequencies of the loaded lines (transmission zeros), except for the symmetric case, where only one resonance (different to the one of uncoupled SRRs) appears. The model is validated by comparing circuit simulations using extracted parameters with electromagnetic simulations and experimental data.