##
**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

*C*

_{0}=2π

*r*

_{0}

*C*

_{pul}as the total capacitance between the two rings, where

*C*

_{pul}is the per-unit-length capacitance of a CPS line [3], the resulting series capacitance

*C*

_{S}of its equivalent circuit is given by two capacitors of

*C*

_{0}/2 in series, that is

*C*

_{S}=C

_{0}/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

*r*

_{0}. Consequently, the resonant frequency of a SRR is given by

(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 *C*

_{C}can be approximated by that of a slot of radius

*r*

_{0}and width

*w*. On the other hand, if we define the total inductance

*L*

_{0}=2π

*r*

_{0}

*L*

_{pul}, where

*L*

_{pul}is the per-unit-length inductance of a CPW transmission line [3], the resulting inductance

*L*

_{C}of the equivalent circuit is given as

*L*

_{C}=

*L*

_{0}/4. Therefore, the resonant frequency of the CSRR can be obtained as

(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.