# Equivalent Circuits for Microwave Metamaterial Planar Components
## 5 Conclusions  
This paper developed a simple approach to experimentally derive the equivalent circuit of planar microwave components, namely microwave resonators fed by CPW lines, to determine the ordinary or metamaterial response of the device. The method is based on \(Z\)-parameters analysis and specifically on the sign of \(Im(Z_{2}^{\prime})\) and \(Im(Z_{1})\), i.e., the two branches of the T-model for a transmission line section, including the resonator and the  
Figure 14: \(Im(Z)\) for the 27 GHz hexagonal resonators. Figure 13: \(Im(Z)\) for the 20 GHz hexagonal resonators. feeding line, respectively. The T-model has been used and related to the experimental data to derive the possible metamaterial response of the investigated structures, analyzing the sign (positive or negative) of the impedances defined along the branches of the T-network. In particular, the studied configurations were triangular planar resonators coupled with a CPW line and symmetrically arranged to the central conductor of the CPW. According to the proposed method, the RH (ordinary) or LH (metamaterial) nature of the device was investigated, concluding that the possible LH response of the equivalent circuit confirms the device's metamaterial nature, i.e., a series capacitor followed by a shunt inductor (LH structure) opposed to the ordinary series inductor with a shunt capacitor (RH configuration). Two triangular patches, designed with sizes fulfilling the requirement of 20 GHz and 27 GHz operation, have been compared with Sierpinski triangles with the same edge length of the entire patch, and with the first level of complexity, classically considered a fractal shape. Moreover, the same building blocks have been used to manufacture hexagonal structures, coupling three triangles and mirroring them with respect to the CPW line. The results obtained from the analysis of the inferred \(Z\)-parameters gave evidence for an equivalent circuit not always related to a metamaterial behavior. The single resonators and the coupled ones yield different responses despite the presumed metamaterial nature due to the fractal design for C1 and the ordinary one for C0. As a result, the complexity of the structure must include the analysis of the feeding system and the boundaries of the resonators, involving parasitic contributions not easily obtainable with ab initio calculations or de-embedded with a calibrated measurement, and is strictly related to the technology and to the necessity of having a feeding system. Then, the measured C0 configuration, i.e., the simple patch, appeared to be more clearly related to the metamaterial response despite the non-fractal shape. At the same time, the C1 structures (first-level complexity of the Sierpinski triangle) could be even better coupled from an electrical point of view but do not necessarily exhibit a metamaterial response. This result is particularly evident in the coupled resonators, where the boundary conditions are less critical with respect to the dominant area occupied by the entire resonating structure, thus behaving as a simple resonator, with \(Im(Z_{2}^{\prime})=0\). The above findings show that even a metamaterial-inspired resonating structure could behave as an ordinary material or vice-versa, depending on the coupling mechanisms and additional equivalent lumped components to be considered. In conclusion, the approach proposed in this contribution aims to determine experimentally the equivalent dielectric constant and magnetic permeability of a metamaterial-inspired structure or possibly for configurations not necessarily considered from the beginning as metamaterial resonators. Still, they can be recognized as left-handed (LH) structures if considered as a whole. This is particularly important when not only single resonators are designed, but also arrays are considered, typically accompanied by complicated coupling solutions and feeding networks.