## 3 Circuit Models
### Differential Microstrip Lines Loaded with CSRRs  
Differential microstrip lines exhibit two fundamental modes, the even and the odd mode. Hence, the symmetry plane is an electrical wall for the odd mode and a magnetic wall for the even mode. This means that if a CSRR is symmetrically etched in the ground plane (_i.e._, with its symmetry plane aligned to that of the line), as Figure 5a illustrates, the even mode is expected to be suppressed in the vicinity of the fundamental resonance, whereas the line is expected to be transparent for the odd mode. The reason is that the CSRR exhibits a magnetic wall at the fundamental resonance frequency. The circuit model of  
Figure 4: Illustration of symmetry rupture by square-shaped inclusions in the approach based on pairs of resonators, where a CPW is symmetrically loaded with a pair of SRRs. Note that the symmetry planes of the SRRs are not required to be parallel to the longitudinal \(x\)-direction, preventing mixed coupling. this structure is depicted in Figure 5b [36]. The element \(L\) models the inductance of the lines, \(C\) accounts for the electric coupling between the lines and the resonator (and the line capacitance as well), and the CSRR is modeled by the parallel resonant tank, \(L_{e}\)-\(C_{c}\). Finally, \(C_{m}\) and \(L_{m}\) model the mutual capacitance and inductance between the coupled lines, respectively. The circuit model of Figure 5b explains that the differential signals are insensitive to the presence of the CSRRs, while these resonators prevent the transmission of the common mode at the fundamental resonance frequency of the CSRRs. The simplified equivalent circuit model under common-mode excitation is depicted in Figure 6a, whereas that for the odd mode is depicted in Figure 6b. For the odd mode, the resonator is short circuited to the ground, and the resulting model is the one of a conventional transmission line. For the even mode, we obtain the same circuit as the one of a CSRR-loaded line [37], but with modified parameters. Thus, a similar stop band behavior for the common mode is obtained. To validate the model, we have considered a CSRR unit cell (see the inset in Figure 7) with narrow and closely spaced apart slot rings (\(c=0.2\) mm and \(d=0.2\) mm). These values are close to the limit of the available technology and guarantee that the particle is electrically small. With these values, the model of the CSRR reported in [37], and the per-unit length capacitance of the coupled lines (common mode), we have estimated the side of the CSRR in order to obtain a transmission zero frequency at \(f_{z}=1.4\) GHz (obviously, optimization has been required since the model reported in [37] is valid under conditions not exactly fulfilled in the structure under study). The simulation of the common-mode insertion loss is depicted in Figure 7. The circuit simulation of the structure with the parameters extracted according to the procedure reported in [38] is also depicted in Figure 7, and good agreement with the electromagnetic simulation is noticeable. In order to enhance the rejection bandwidth, an order-three structure with tightly  
Figure 5: Topology (**a**) and circuit model (**b**) (elemental cell) of a differential microstrip line loaded with a CSRR. Figure 6: Circuit models for the even mode (**a**) and odd mode (**b**). coupled CSRRs has been implemented [36]. The bandwidth is closely related to the level of inter-resonator coupling. Therefore, we have separated 0.15 mm the CSRR in order to broaden the bandwidth as much as possible. The simulation of the resulting structure is shown in Figure 8, where for comparison purposes we have also included the circuit simulation. The simulated circuit is that resulting by cascading the elemental cells, but with the addition of coupling capacitances between adjacent resonators. The coupling capacitance has been considered to be an adjustable parameter, and we have found that the capacitance that provides a better fit is \(C_{R}=0.11\) pF.