# Equivalent Circuits for Microwave Metamaterial Planar Components
## 4 Experimental Determination of the Metamaterial Nature of a Resonator
In this paper, owing to the possibility of having a fixed footprint of the resonator, the frequency of operation is tuned to change the internal complexity. Figure 6: Triangular resonators: (**a**) full metal patch, and (**b**) Sierpinski triangle with first-level internal complexity. The shape is used for the 20 GHz nominal frequency of operation, with the triangle edge of 6 mm, while the same shape has been scaled down to a 4 mm for a higher frequency of resonance (close to 27 GHz). The resonator size has been imposed according to the resonance frequency expected for an equilateral triangle, as a first approximation, which depends on the edge length. As demonstrated in [31], the coplanar excitation is more difficult to be understood from the spectral point of view compared with a microstrip excitation, but the frequency of resonance of the main mode can be approximatively predicted. Then, two different resonators have been studied, having edge lengths of 6 mm and 4 mm, respectively, to excite the main mode at two different frequencies, corresponding, in the present case, to values around 20 GHz and 27 GHz. The experimental characterization of the proposed configurations was performed using an on-wafer system with coplanar probes, namely ground-signal-ground (GSG) tips touching the input/output short transmission lines used to feed the resonators. The central conductor of the probe brings the signal to the central conductor of the CPW, while the lateral tips are utilized as a ground reference, touching the lateral grounds of the CPW. A vector network analyzer (VNA), i.e., a measurement system commonly used in microwave engineering to characterize a multi-port device, has been used for recording the amplitude and phase of the reflected and transmitted signals. In our case, an HP8510C VNA (from Keysight Technologies, formerly Hewlett and Packard) is linked to a [PERSON] manual probe station with micrometric control for positioning the CPW probes. A PC is connected to remotely control the instruments. A simple schematic of the experiment is shown in Figure 7, detailing the measurement procedure. An essential step in the characterization is the calibration of the measurement system, intended to utilize the hardware and software necessary for de-embedding the measurement itself. In this case, a through-reflect-line (TRL) calibration was performed using standard transmission lines with different lengths (and phase shifts) determined by the line lengths. They are standard lines, designed in agreement with the frequency range imposed for the measurement, following design criteria now well-established in the scientific literature [32]. The calibration allows the artificial movement of the measurement plane at the device input/output, de-embedding the contribution of discontinuities and cables up to the device under test (DUT). In our measurements, the reference plane is given by the position of the probes shown in Figure 7. This measurement protocol, i.e., choice of probes, calibration, and measurements once the frequency range and resolution, and the input power, are chosen, is the most reliable technique for characterizing devices on-wafer. No limitations are encountered when properly designed CPW lines are used for the TRL calibration and one of the standard lines is verified after that, used as a test to validate the calibration itself. In Figure 8, the measured frequency response is shown, with resonance frequencies \(F_{r}\) working around 20 and 27 GHz, respectively. The \(F_{r}\) values have been chosen at the design level to be coherent with current satellite communications, focusing on the K-Band within the range currently considered for uplink and downlink operations. Then, the main resonance mode has been selected according to the chosen length for the triangle edge. Since two different frequencies have been selected, despite the presence of high order modes, focus has been given to the main resonance and to its modifications induced by the internal complexity of the triangle.