# Equivalent Circuits for Microwave Metamaterial Planar Components
## 4 Experimental Determination of the Metamaterial Nature of a Resonator
Figure 9: \(\,Im(Z_{1})\) and \(\,Im(Z_{2}^{\prime})\) equivalent impedances for the C0 and C1 triangles working around 20 GHz. C0 always exhibits \(\,Im(Z_{1})<0\) and \(\,Im(Z_{2}^{\prime})>0\) around the frequency of resonance. For C1, the two quantities are negative on the right side of the resonance frequency, and there is a transition below the resonance. because of the mirrored arrangement. It might be stressed that a symmetric configuration was always the best solution for all the studied resonators from the electrical performance point of view, mainly because a CPW favors a symmetric coupling considering the specific geometry of the feeding line [31; 32; 33]. The layout of the hexagonal resonators implemented by the C0 and C1 configurations is shown in Figure 11, while in Figure 12, the Transmission Parameter S\({}_{21}\) is plotted. The results in Figure 12 are not easy to understand, as only the C1 configuration is critically matched for both frequency ranges, exhibiting a deep notch. For the C0 triangles, the resonance is unexpectedly strongly depressed around 20 GHz, while additional modes around 21.5 GHz are excited for both structures based on the C0 and C1 resonators. This result is possible when exciting resonating modes with an external coupling. A modal analysis can help determine the position of the maximum of the fundamental mode, even if, for practical applications, a simple configuration is encouraged and the principal mode cannot always be excited with the chosen feeding system. At 27 GHz, the resonance of C1 is well represented, but C0 has a much lower intensity. The above findings prove that a general criterion to match the single resonator and the coupled ones is not straightforward. Fixing some geometrical parameters, like the distance between coupled resonators and their distance to the central conductor of the CPW feeding line or the length of the input/output feeding lines, cannot contribute to shifting the operative frequency and simultaneously preserving the exact electrical matching when increasing the internal complexity. Despite the wideband response of the CPW alone, the coupling between CPW and resonators and the mutual coupling among the resonators needs further refinement. The distance between the individual triangles in the C0 configurations should be optimized, even if C0 performs  
Figure 11: Triangular resonators are arranged in a hexagonal geometry by coupling three of them and mirroring the structure with respect to the central conductor of the CPW feeding line. The figure is for the 20 GHz resonators, and the 27 GHz configuration has been scaled down, as in the case of the individual triangles. Figure 12: Resonance frequency response for the 20 and 27 GHz hexagonal structures. well as a single resonator. At the same time, C1 is suitable for single and coupled structures, exhibiting a good response in both cases. The results for C0 and C1 hexagons are plotted in Figures 13 and 14. The previous figures show that, despite showing different levels of electrical matching, the C0-based structures can be reasonably considered metamaterial-like around the resonance frequency. In contrast, C1 structures are better matched, but a different comment is necessary due to the response of the equivalent \(Im(Z_{2}^{\prime})\). In this case, a configuration with C1 arranged in a hexagon with a near-to-zero value has been obtained for both frequencies at resonance while \(Im(Z_{1})<0\) in the same range. Then, the resonance frequency can delimit two regions close to the resonance \(F_{r}\): (1) for \(F<F_{r}\), the structure shows a metamaterial response with \(Im(Z_{2}^{\prime})>0\) and \(Im(Z_{1})<0\), and (2) for \(F>F_{r}\), and the two imaginary values are negative, with evidence for a purely capacitive response.