## 3 Electrically Small Planar Resonators: A Comparative Analysis and Circuit Models
### Topologies and Circuit Models of the Isolated Resonators
Figure 5: Topology of the (**a**) two-turn spiral resonator (2-SR), (**b**) BC-SR, (**c**) folded stepped impedance resonator (SIR) and (**d**) open split ring resonator (OSRR) (including the circuit model). The previous particles are closed particles that can be magnetically and, in some cases, electrically driven. Open particles can also be of interest in applications where the resonator must be excited by means of a voltage or current source. Inspired by the topology of the SRR of Figure 3, the open split ring resonator (OSRR) was presented in 2004 [24]. This particle can be constructed by means of two concentric rings with cuts in the same position. By elongating the rings, shaping them as a hook (Figure 5(d)), we create the two required terminals for particle excitation. The incident current is transferred from one ring to the other through the distributed edge capacitance between the two rings. According to that, it is clear that the resonance frequency of this series resonator is identical to the resonance frequency of the two-turn spiral resonator, 2-SR. Therefore OSRRs are very interesting for applications where small series (open) resonators implemented in one metal layer are required. Indeed, the OSRR, rather than an open SRR, is an open 2-SR. However, we will preserve the original term assigned to this particle in [24]. Concerning open SIRs, the SISS reported in Section 2 is indeed an open version of the SIR. It is an open series resonator shunt connected to the host microstrip line. Open SIRs can also be implemented in CPW structures by etching this particle in the back substrate side, with the wide strip sections below the central strip and ground planes. This enhances the electric coupling between the line and the SIR and hence very small resonance frequencies can be achieved with small particles. Therefore, electrically driven open SIRs can be made electrically very small. Let us now consider the application of duality to some of the previous particles. If the topology of the SRR is removed from a metallic film, the resulting particle (Figure 6) can be considered the negative image, or the complementary counterpart, of the SRR. For this reason, this particle has been called complementary split ring resonator (CSRR) [25]. It can be easily demonstrated by applying the Babinet principle, that the CSRR can be excited by means of an axial time varying electric field or by means of a magnetic field applied to the plane of the particle [26]. Under ideal conditions, _i.e._, a perfectly conducting and zero-thickness metallic screen, it can be demonstrated that the resonance frequency of the CSRR is identical to the resonance frequency of the SRR (provided identical dimensions and substrate are considered). Indeed, the capacitance of the CSRR, \(C_{c}\), is the capacitance of a disk of radius \(r_{o}\)-\(c/2\) surrounded by a ground plane at a distance \(c\) of its edge. The inductance is given by the parallel connection of the two inductances of the metallic strips connecting the inner and outer metallic regions of the CSRR. These inductive values are given by \(L_{o}\)/2, where \(L_{o}=2\pi rL_{pu}\), with \(L_{pu}\) being the per unit length inductance of the CPWs connecting the inner disk to ground. Figure 6: Complementary split ring resonator (CSRR) topology and its equivalent circuit model. Geometrical parameters of the CSRR are identical to those of Figure 3 referred to the SRR. Complementarity can also be applied to the NB-SRR, DS-SRR, 2-SR, SIR and OSRR. Let us consider in certain detail the complementary counterparts of the SIR and OSRR, since they are of special interest for microwave and metamaterial-based circuit design. The complementary version of the open split ring resonator was called open complementary split ring resonator (OCSRR) [27]. The OCSRR is the complementary (negative) image of the OSRR, but it can also be considered to be the open version of the CSRR. However, the OCSRR must be considered with some caution, since the metallic terminals of the OSRR disappear by applying duality.