## 3 Electrically Small Planar Resonators: A Comparative Analysis and Circuit Models
### Transmission Lines Loaded with Electrically Small Resonators
The circuit models depicted in Figures 11 and 12 also apply for microstrip lines loaded with complementary spirals or with other complementary closed particles. Concerning the trade-off between particle size and bandwidth, the same comments in reference to SRR-loaded lines also apply (see [38] for more details and for numerical calculations). The main relevant limitation of SRR- and CSRR-loaded lines is the narrow bandwidth, which is related to the limited coupling between the line and the resonator. In band pass structures, moderate and wide bands have been achieved by merging the backward and the forward wave transmission bands [41]. However, the validity of the models of the SRR- and CSRR-loaded lines in the forward wave transmission band is restricted to only a small portion of the band [42]. The DS-SRR (or DS-CSRR) has been considered in order to implement moderate or even wideband stop band structures, but this particle is electrically larger than the SRR (or CSRR), and therefore the accuracy of the circuit model to properly describe lines loaded with such particles is questionable. An alternative approach to implement wide band structures is the combined use of OSRRs and OCSRR in transmission lines. Let us consider a CPW loaded with an OSRR (Figure 13(a)). The circuit model of this structure is not merely a series resonator between the input and output ports of the CPW. As discussed in [43], the  
\[L_{p}\text{'}=2L_{p}+\frac{L}{2} \tag{5}\]0 **(a)** Typical topology of a CSRR loaded microstrip line. (**b)** Equivalent circuit model of the unit cell. \[L_{p}\text{'}=2L_{p}+\frac{L}{2} \tag{5}\]1 **(a)** Typical topology of a CSRR- and gap-loaded microstrip line. (**b)** Equivalent circuit model of the unit cell. (**c)** Transformed circuit model. line introduces some phase shift that must be taken into account for an accurate modeling. Such phase shift can be accounted for by cascading phase shifting lines to the series resonator, \(L_{s}\)-\(C_{s}\), modeling the OSRR. In practice, such shifting lines can be modeled by means of a lumped capacitance \(C\) and inductance \(L\), and the circuit model is that shown in Figure 13(b), which in turn can be simplified to the model depicted in Figure 13(c). For OCSRR loaded lines (Figure 14), a similar phenomenology applies. That is, the structure cannot be simply modeled by means of a shunt connected parallel resonator. A certain phase shift appears, and the equivalent circuit model is that depicted in Figure 14(b,c). In the models of Figures 13 and 14, \(L\) and \(C\) must be considered parasitic elements. The frequency response of CPW transmission lines loaded with either OSRRs or OCSRRs reveals that the structures exhibit a band pass behavior with very wide transmission bands. As will be shown in the next section, by cascading OSRR and OCSRR-loaded CPWs, it is possible to implement transmission lines with composite backward and forward characteristics, as well as wide band filters. As compared to other structures based on closed rings, the main advantage of these OSRR and OCSRR based lines, is that the equivalent circuit of the structures models their behavior to a very good approximation within the whole transmission band. \[L_{p}\text{'}=2L_{p}+\frac{L}{2} \tag{5}\]2 Layout (**a**), circuit model (**b**) and simplified circuit model (**c**) of a CPW transmission line loaded with a pair of open complementary split ring resonators, where \(C\)’\({}_{p}=2C_{p}+2C\) and \(L\)’\({}_{p}=L\)/2. The backside strips (in dark grey) connecting the different ground plane regions are necessary to prevent the slot mode of the CPW and the second resonance of the open complementary split ring resonators. \[L_{p}\text{'}=2L_{p}+\frac{L}{2} \tag{5}\]3 Layout (**a**), circuit model (**b**) and simplified circuit model (**c**) of a CPW transmission line loaded with a series connected open split ring resonator (\(L\)’\({}_{s}=L_{s}+2L\)). To end this section, we will consider the SIRs coupled to transmission lines.