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## Introduction

In 1999, J. B. Pendry proposed a way to construct o 'left-handed' medium (i.e., negative refractive index) based on a split ring resonator (SRR) structure [1]. This was the beginning to develop materials with properties not present in nature, the so-called metamaterials. The SRR consist of a pair of concentric metallic rings, etched on a dielectric substrate, with slits etched on opposite sides.

## Equivalent circuit

### Split ring resonator (SRR)

The electrical performance of a SRR can be well-reproduced by a LC circuit [2]. The conventional SRR behaves as a resonant magnetic dipole that can be excited by an axial magnetic field. If we define C0=2πr0Cpul as the total capacitance between the two rings, where Cpul is the per-unit-length capacitance of a CPS line [3], the resulting series capacitance CS of its equivalent circuit is given by two capacitors of C0/2 in series, that is CS=C0/4. On the other hand, the series inductance can be approximated by that of a single ring of width w and radius the average between the two rings r0. Consequently, the resonant frequency of a SRR is given by
 $f_{0,SRR}=\frac{1}{2\pi \sqrt{L_SC_S}}$ (Eq. 1)

### Complementary split ring resonator (CSRR)

It is also possible to synthesize the electrical counterpart of the SRR, by etching its negative image on a substrate (i.e., complementary SRR, or CSRR). The CSRR  behaves  as  an  electric dipole that can be excited by an axial electric field [4]. In this case, the capacitance CC can be approximated by that of a slot of radius r0 and width w. On the other hand, if we define the total inductance  L0=2πr0Lpul, where Lpul is the per-unit-length inductance of a CPW transmission line [3], the resulting inductance LC of the equivalent circuit is given as LC=L0/4. Therefore, the resonant frequency of the CSRR can be obtained as
 $f_{0,CSRR}=\frac{1}{2\pi \sqrt{L_CC_C}}$ (Eq. 2)

 Fig. 1: Split ring resonator (SRR) and complementary split ring resonator (CSRR), and its corresponding equivalent circuits.

## References

[1] J. B. Pendry, A. J. Holden, D. J. Robbins, W. J. Stewart, "Magnetism from conductors and enhanced nonlinear phenomena," IEEE Trans. Microwave Theory Tech., vol. 47, no. 11, Nov. 1999.
[2] R. Marqués, F. Mesa, J. Martel, F. Medina, "Comparative analysis of edge- and broadside-coupled split ring resonators for metamaterial design - Theory and experiments," IEEE Trans. Antennas Propag., vol. 51, no. 10, Oct. 2003.
[3] I. Bahl, P. Bhartia, Microwave solid state circuit design. Wiley, 2nd ed., 2003.
[4] J. D. Baena, J. Bonache, F. Martín, R. Marqués Sillero, F. Falcone, T. Lopetegi, M. A. G. Laso, J. García-García, I. Gil, M. F. Portillo, M. Sorolla,, "Equivalent-circuit models for split-ring resonators and complementary split-ring resonators coupled to planar transmission lines," IEEE Trans. Microw. Theory Tech., vol 53, no. 4, Apr. 2005.