## 3 Electrically Small Planar Resonators: A Comparative Analysis and Circuit Models
### Transmission Lines Loaded with Electrically Small Resonators
The bandwidth can be inferred from the dispersion relation, given by:  
\[\cos\beta\text{'}=1+\frac{Z_{s}(\omega)}{Z_{p}(\omega)} \tag{6}\]  
with \(Z_{s}\) and \(Z_{p}\) being the series and shunt impedance, respectively, of the \(\pi\)-model of the considered line (for a T-model, the same expression is also valid). The characteristic impedance of a periodic structure whose unit cell can be described by a \(\pi\)-model is:  
\[Z_{B}(\omega)=\sqrt{\frac{Z_{s}(\omega)Z_{p}(\omega)/2}{1+\frac{Z_{s}(\omega) }{2Z_{p}(\omega)}}} \tag{7}\]  
Figure 8: (**a**) Typical topology of a SRR loaded CPW. (**b**) Equivalent circuit model of the unit cell. The magnetic wall concept has been applied, so that the model is that of the one half of the structure. Figure 9: (**a**) Typical topology of a SRR- and strip-loaded CPW. (**b**) Equivalent circuit model of the unit cell. (**c**) Transformed circuit model. and for a T-model, it takes the form:  
\[Z_{B}(\omega)=\sqrt{Z_{s}(\omega)[Z_{s}(\omega)+2Z_{p}(\omega)]} \tag{8}\]  
It has been mentioned before that the electrical size of the SRR can be decreased through different strategies. However, as discussed in [38], reducing the particle size has the effect of narrowing the bandwidth. Therefore, a trade-off between size and bandwidth when choosing the loading particles must be taken into account. This important aspect is illustrated in Figure 10, where the transmission coefficient of a CPW transmission line loaded with different resonant elements is depicted. If, for instance, a notch filter with much localized frequency is required, the use of a 2-SR spiral or a BC-2 SR, rather than a SRR, is recommended. However, for applications of strip- and resonator-loaded CPW as bandpass filters, SRRs are necessary if the required bandwidth is moderate. Let us now consider microstrip lines loaded with CSRRs. The typical topology and circuit model (unit cell) are depicted in Figure 11[25]. The line is described by the inductance \(L\) and by the capacitance \(C\), which models also the electric coupling between the line and the CSRR, described by the inductance \(L_{c}\) and the capacitance \(C_{c}\). The structure exhibits a stop band behavior that has been interpreted as being due to the negative effective permittivity of the structure in the vicinity of CSRR resonance. A left handed transmission band can be generated by merely cutting gaps in the microstrip line, above the positions of the CSRRs [26]. In this case the structure is a passband and can be modeled according to the circuit depicted in Figure 12(b), where the gap is modeled by a \(\pi\)-model consisting of the series capacitance \(C_{s}\) plus the fringing capacitances \(C_{f}\). Such a model can be transformed to the model depicted in Figure 12(c), with the following transformations [39]:  
Figure 10: Layouts and transmission coefficients of several CPWs loaded with pairs of SRRs, BC-SRRs and 2-SRs (**a**) and with shunt strips (**b**). Relevant dimensions are: Ring width \(c=0.2\) mm, distance between the rings \(d=1\) mm, external radius \(r_{ext}=5.4\)mm for the SRRs; \(c=0.961\) mm, \(d=0.19\) mm, \(r_{ext}=3.36\) mm for the SRs; and \(c=2.12\) mm, \(d=0.2\) mm, \(r_{ext}=3.17\) mm for the BCSSRs. For the CPW the central strip width is \(W_{C}=10\) mm, the width of the slots is \(G=1.59\) mm and the length \(D=12.2\) mm. Shunt strip width \(w_{S}=0.2\) mm. The considered substrate characteristics are: dielectric constant \(\varepsilon_{r}=10.2\) and thickness \(h=2.54\) mm. From [38]; copyright © 2010, John Wiley & Sons; reprinted with permission. \[2C_{g}=2C_{s}+C_{par} \tag{9}\] \[C=\frac{C_{par}\left(2C_{s}+C_{par}\right)}{C_{s}} \tag{10}\]  
where \(C_{par}=C_{f}+C_{L}\). Like for the SRR-loaded CPW, CSRR loaded lines can be analyzed through their lumped element equivalent circuit, and expressions (6) and (8). The parameter extraction technique for these CSRR-loaded lines has been published in [40].