Analytical modeling of antisymmetric split-ring resonators coupled with transmission line

Coupled-mode theory is applied to obtain an analytic form of scattering parameters for a class of transmission line metamaterials with antisymmetric split-rings. The same structure is modeled with equivalent circuit, which includes electric and magnetic coupling with the line and inter-resonator coupling. Modified even/odd analysis is used to obtain scattering parameters from the equivalent circuit. These two methods are shown to be equivalent in a narrow band, and their constants related. The obtained results are compared with full-wave simulations and measurements, and it is shown that both methods give accurate approximation in one octave frequency band. The derived analytic expressions are suitable for study of resonant phenomena, with potential practical applications for filters, phase shifters, delay lines, and sensors.