## 3 Electrically Small Planar Resonators: A Comparative Analysis and Circuit Models
### Transmission Lines Loaded with Electrically Small Resonators  
In microwave applications, rather than isolated, the resonators considered in the previous sub-section are coupled (or connected) to host transmission lines (microstrip lines, CPWs, _etc._). Despite the fact that such electrically small particles can also be present in metallic waveguides, the main interest of this paper is planar technology. For this reason, the present section is devoted only to the study of planar transmission lines loaded with semilumped resonators. Let us start with the equivalent circuit models of transmission lines loaded with closed resonators. Typically, the SRR (or the 2-SR, NB-SRR, BC-SRR and DS-SRR) are useful particles to load a CPW transmission line. They can also be used by coupling them to a microstrip transmission line, but the corresponding circuit models are not so accurate. The typical topology of a CPW loaded with pairs of SRRs, as well as the lumped element equivalent circuit model of the unit cell, are depicted in Figure 8[34,35]. The SRRs are etched in the back substrate side, below the slots, so the particles can be excited by the time varying magnetic field generated by the current flowing through the line. The circuit model of the unit cell must take into account the resonator and the line, as well as the coupling between the resonator and the line and, eventually, inter-resonator coupling. In many applications, coupling of the inter-resonator is not required (_i.e._, the resonators are separated enough) and can be neglected. This is the case considered here. Nevertheless, the analysis of transmission lines loaded with tightly coupled resonators has been carried out in [36]. In the circuit model of Figure 8, \(L\) and \(C\) are the line inductance and capacitance, _L_\({}_{s}\) and _C_\({}_{s}\) are the inductance and capacitance of the SRR (or the considered coupled particle), and \(M\) is the mutual coupling between the line and the SRRs. The structure of Figure 8 exhibits a stop band behavior that has been interpreted as consequence of the negative effective permeability of the structure (caused by the presence of the SRRs) above resonance, and by the highly positive effective permeability below SRR resonance. To implement a backward wave (or left handed) CPW transmission line, inductive strips between the central strip and the ground planes, above the positions of the SRRs, have been introduced (Figure 9(a)) [34,35]. With the presence of the shunt connected strips, the structure switches its frequency response to a bandpass type response, where wave propagation in the first allowed (narrow) band is backward (typically this structure exhibits a forward wave transmission band at higher frequencies). In this case, the circuit model is that depicted in Figure 9(b), where _L_\({}_{p}\) accounts for the pair of shunt connected strips. The circuit model depicted in Figure 9(b) can be transformed to the circuit depicted in Figure 9(c) [35]. The following transformations apply:  
\[L_{s}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\, \,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,  
\[L\text{'}=\left(2+\frac{L}{2L_{p}}\right)\frac{L}{2}-L_{s}\text{'} \tag{4}\]  
\[L_{p}\text{'}=2L_{p}+\frac{L}{2} \tag{5}\]  
The circuit model of Figure 9(c) is useful for design purposes since a parameter extractor has been developed [37], namely, from the electromagnetic simulation (or measurement) of the frequency response of a SRR-loaded CPW, we can easily infer the element values of the circuit model. The technique is very useful but it is out of the scope of this paper. Moreover, from the equivalent circuit model of the unit cell, we can easily predict the bandwidth of the allowed bands, as well as the characteristic impedance of the propagating structures in those bands.