# Equivalent Circuits for Microwave Metamaterial Planar Components
## 4 Experimental Determination of the Metamaterial Nature of a Resonator
First, the increase in the internal complexity of the triangle contributes to a shift in the resonance frequency. Unbalanced configurations are obtained if no attention is paid to balancing the RH and LH contributions of the structure, which would not be ideal and would always be affected by parasitic contributions. For this reason, close to the resonance frequency, a gap is opened in the dispersion relation [10]. It is also true that when the wavelength is much larger than the discontinuities encountered during the propagation, the structure can easily be considered balanced. During the passage from RH to LH behavior, a TL section will be characterized by a dispersion relation passing from negative to positive values of the excited wavevector [10]. Then, it would be natural to expect that at resonance, a resonating structure could pass from an ordinary response to a metamaterial one, with a change in the sign for the dielectric constant and the magnetic permeability. Since we are focused on resonator configurations, this response should be concerned with a narrow-band behavior limited to the resonance frequency and its boundaries. Using the approach of the equivalent circuit proposed in this contribution, the impedance values have been plotted around the resonance frequency for different configurations and frequencies of operation, comparing C0 and C1. Figure 9 plots the imaginary parts of \(Z_{1}\) and \(Z_{2}\)' for the 20 GHz resonator in the C0 and C1 configurations. The \(Z\)-values have been obtained from the \(S\)-parameters using literature formulas, as specified before in the text. At resonance and surrounding frequencies,  
Figure 8: The frequency response for the triangles C0 and C1 for the two configurations: (**a**) 20 GHz and (**b**) 27 GHz. Figure 7: Schematic of the measurement system, composed of a VNA and a probe station remotely controlled by a PC. The device under test (DUT) is connected at the end of the cables by means of GSG coplanar probes, and the calibration is valid from the ends of the tips. C0 always shows a monotone response of the two quantities, which do not change when the sign passes through the frequency of resonance at 20.16 GHz, while, in the case of C1, the right side of the resonance peak exhibits negative values of both \(\,Im(Z_{1})\) and \(\,Im(Z_{2}^{\prime})\) up to frequencies of a few hundred MHz from the resonance frequency. A possible interpretation of this behavior returns to the generally expected response in Figure 3, i.e., around the resonance frequency, the C0 configuration, despite nominally not being a fractal configuration, always exhibits a metamaterial response because \(\,Im(Z_{2}^{\prime})>0\) and \(\,Im(Z_{1})<0\) over the entire resonance frequency band. The C1 configuration exhibits a more complicated response because we can distinguish three frequency ranges: (1) before the resonance, we have \(\,Im(Z_{2}^{\prime})>0\) and \(\,Im(Z_{1})<0\), nominally corresponding to a metamaterial structure, (2) just after the resonance, both quantities are negative, thus giving evidence for a dominant capacitive contribution on both branches of the equivalent T-circuit, and finally, (3) the response is inverted, with \(\,Im(Z_{2}^{\prime})<0\) and \(\,Im(Z_{1})>0\) at the end of the resonance curve, corresponding to the response of an ordinary TL. Figure 10 shows a result similar to the previous one, with a metamaterial response for C0 and mainly a capacitive contribution for C1 around the resonance frequency. A more complicated configuration has been studied, including three coupled resonators with the C0 and C1 configurations, also, in this case, mirrored to the central conductor of the CPW feeding line and for the same frequency ranges, i.e., close to 20 GHz and 27 GHz. The complete structure of the coupled resonators looks like a hexagon  
Figure 10: \(\,Im(Z_{1})\) and \(\,Im(Z_{2}^{\prime})\) frequency response for the C0 and C1 triangles working close to 27 GHz.