Design of spiral and multiple split-ring resonators for the realization of miniaturized metamaterial samples

We present the design of miniaturized resonant inclusions to be employed in the practical realization of metamaterial samples with anomalous values of the real part of the permeability. Such inclusions, in fact, can be employed in the design of both mu-negative (MNG) materials and artificial magnetodielectrics (with negative and high-positive values of the real part of the permeability, respectively). The inclusions here considered are the multiple split-ring resonators (MSRRs), that represent a straightforward extension of the commonly used split-ring resonators (SRRs), and the spiral resonators (SRs), that enable a greater miniaturization rate. Some physical insights on the resonance mechanism and on the inherent saturation of the resonant frequency when increasing the number of the rings of the MSRRs and the number of the turns of the SRs are given in the paper. New and accurate analytical design formulas, based on a quasi-static model, for both MSRRs and SRs are derived and tested through a proper comparison with the existing formulas and full-wave numerical results. Both MSRRs and SRs are shown to be useful to reduce the electrical dimensions of the resonant inclusions when synthesizing artificial metamaterials.