## 3 Electrically Small Planar Resonators: A Comparative Analysis and Circuit Models
### Topologies and Circuit Models of the Isolated Resonators
There is, however, another interesting aspect that differentiates the conventional SRR and the BC-SRR. The charge distribution in the SRR (depicted in Figure 4(a)) indicates that there is an electric dipole at the first resonance frequency in the orthogonal direction to the symmetry plane of the particle (such a symmetry plane is an electric wall). This means that the SRR can also be excited by means of a time varying electric field applied in the plane of the particle co-directional to the direction of the electric dipole. Hence, the first SRR resonance can be driven electrically and/or magnetically, _i.e._, the particle exhibits bianisotropy. For the BC-SRR, charges in the upper half are also images of charges in the lower half, but this charge distribution does not result in a net dielectric dipole (Figure 4(b)). Thus, the BC-SRR is non-bianisotropic. Another possible way to avoid bianisotropy while keeping a uniplanar design is by means of the topology depicted in Figure 4(c) [19]. Essentially, the circuit model of this particle, called non-bianisotropic split ring resonator (NB-SRR), is that of the conventional SRR. However, the NB-SRR, like the BC-SRR, has an inversion symmetry with regard to its center, which means that cross polarization effects are not present, _i.e._, the first resonance frequency cannot be excited through an electric field. An alternative way to obtain inversion symmetry, thus overcoming bianisotropy is by introducing additional cuts in the edge coupled SRR, as Figure 4(d) illustrates [19]. This particle, called double slit split ring resonator DS-SRR, exhibits the same inductance as the SRR, but its capacitance is four times smaller since it is the result of the series connected edge capacitances of each SRR quarter. This means  
Figure 3: SRR topology and its equivalent circuit model. that the resonance frequency is twice the resonance frequency of the SRR, and hence the electrical size of the particle cannot be made very small. Another interesting particle that combines the advantages of small electrical size and unilplanar configuration is the spiral. The two-turn spiral 2-SR can be considered to be constituted of two rings with cuts in the same position and with an electric short between both rings through cross terminals, as shown in Figure 5(a). A quasistatic analysis of this particle [20,21] reveals that the inductance is that of the SRR with identical dimensions. However, the capacitance is the edge capacitance of the whole circumference. Therefore, the resonance frequency is half the resonance frequency of the SRR. Spirals are therefore very useful in applications where small electrical size combined with a monolayer design is required. Obviously, it is possible to etch the two spiral loops at different levels, one on top of the other, connected through a metallic via [22]. Although the price to pay in this case is the use of two metal levels, this broad side coupled spiral resonator (BC-SR, see Figure 5(b)) exhibits a very small electrical size if the metal levels are separated by a very thin dielectric layer (with this design, particle diameters as small as \(\lambda\)/100 can be achieved). Another interesting electrically small planar particle is the stepped impedance resonator (SIR). It is well known that the length of a \(\lambda\)/2 resonator can be reduced by narrowing and widening the central and outer sections, respectively, of the particle [23]. If, additionally, we fold it, as depicted in Figure 5(c), the edge capacitance between the patches enhances the capacitance of the particle and hence the electrical length is made smaller. The folded SIR exhibits bianisotropy and combined with transmission lines is very interesting in order to achieve extremely small devices, as will be shown later. Figure 4: Topology and charge distribution of (**a**) conventional SRR, (**b**) broadside coupled split ring resonators (BC-SRR), (**c**) non-bianisotropic split ring resonator (NB-SRR), and (**d**) double slit split ring resonator (DS-SRR).