## 2 Electrically Small Resonators for Metamaterial and Microwave Circuit Design  
As we have mentioned before, in semilumped resonators the inductance and capacitance can be associated to a certain element of the resonator topology (in certain cases, a single element can provide the inductive and capacitive behavior). To clarify this aspect, let us consider the implementation of a transmission line with a shunt connecting the series resonator to the ground (Figure 1(a)). From the transmission line theory, it is well known that such resonators can be implemented by means of a \(\lambda\)/4 open stub, as depicted in Figure 1(b) [2]. However, the shunt-connected resonator can be alternatively realized by means of a stepped impedance shunt stub (SISS), as shown in Figure 1(c) [11]. The narrow section of the stub is essentially a series inductance, whereas the wide section behaves as a grounded patch capacitance. Both designs of Figure 1(b,c) provide the same resonance frequency, as derived from the transmission zero present in the transmission coefficient (Figure 1(d)). Such a transmission of zero is caused by the short at the position of the resonator. However, it can be appreciated that the length of the SISS is significantly smaller than the length of the transmission line stub, as corresponds to a semilumped component. The considered SISS is an open resonator, since it is driven by the current generated by the line. Let us now consider electrically small closed resonators, which can be electrically or magnetically driven. It is obvious that if a \(\lambda\)/2 transmission line is closed in a ring configuration, as shown in Figure 2(a), this resonator can be excited by means of an axial time varying magnetic field, or even by means of an electric field in the plane of the particle. Since the electrical length of the ring is \(\lambda\)/2 at resonance, it follows that its diameter is \(\lambda\)/2\(\pi\) at this frequency, _i.e._, relatively small compared to the wavelength. However, this ring resonator is a distributed element since it operates under the same principle than the unfolded counterpart. In order to reduce the size of a split ring, one possible solution is to add an inner ring with the gap on the opposite side, as depicted in Figure 2(b) [12]. By virtue of the mutual coupling between both rings, the fundamental resonance of the resulting particle is lower than the fundamental resonance of any of the individual rings, and thus the particle is electrically smaller than a single ring resonator [13]. The particle of Figure 2(b) is called split ring resonator (SRR), and, as long as its diameter can be made much smaller than the wavelength, this particle is useful for the implementation of effective media with controllable electromagnetic properties, also called metamaterials. Specifically, since, above SRR resonance, the magnetic field generated by the induced currents in the SRR is opposite to the incident magnetic field, this particle is useful for the synthesis of negative effective permeability media. Figure 2(c) shows the first three-dimensional metamaterial structure that was conceived to exhibit backward (or left-handed) wave propagation [14]. The necessary negative effective permeability and permittivity to achieve left-handedness was due to the presence of SRRs and metallic posts, respectively. Through these two examples, it is clear that electrically small resonators have a high potentiality in both microwave engineering and metamaterial synthesis. Nevertheless, microwave circuits and antennas based on metamaterial concepts and implemented by means of electrically small resonators like the SRR have been reported. Therefore, the analysis and comparison of such electrically small resonators is of highest interest and is the purpose of the next section. Figure 1: (**a**) Transmission line loaded with a grounded resonator. (**b**) Implementation of the resonator by means of a \(\lambda\)/4 open stub. (**c**) Implementation of the resonator by means of a stepped impedance shunt stub (SISS).