# Equivalent Circuits for Microwave Metamaterial Planar Components
## 4 Experimental Determination of the Metamaterial Nature of a Resonator  
The scientific literature has characterized several resonator structures, claiming their metamaterial nature. Most efforts have been dedicated to the so-called split ring resonator (SRR) [13] and recent publications have been focused on a lossless network composed of input/output lines and the resonator itself [14]. On the other hand, independently of the loss mechanisms and their inclusion in the model, an experimental approach can help in revealing the intrinsic nature of the resonator considered with all the details, including losses and feeding network. In this section, we prefer to use the definition of metamaterial-inspired resonators, checking the possible metamaterial response according to the simple method developed in the past paragraphs. A similar approach has been utilized in [15] without entering the details of the analytical derivation. A complete theoretical treatment has been proposed in [16; 17; 18; 19; 20; 21; 22; 23] to cite a few of the available papers in the literature, with the ambition to evaluate analytically all the possible contributions to be included, assisted by software simulation and experimental findings to validate the model. The chapter contribution in [22] perfectly summarizes the specific advantages of antennas suitable for forward and backward propagation and multi-band response, while in [23; 24], an explicit  
Figure 5: A Simplified approach to determine the potential metamaterial response of the resonator fed by two transmission lines. In (**a**), the central network of the entire structure is reproduced with all the components, while in (**b**), the simplified T-model with \({Z_{2}}^{\prime}\) substituting the intermediate network is proposed. derivation of the dielectric constant and magnetic permeability is also given, using the \(S\)-parameters. In [24], the original approach proposed and developed to obtain the material parameters using free wave propagation is utilized [25; 26; 27; 28; 29]. In this paper, and specifically in this section, we shall present experimental plots related to the quantities defined at the end of the previous paragraph, giving evidence for the frequencies where it is reasonable to assume a metamaterial response. The advantage of the proposed phenomenological approach is that it utilizes \(Z\)-parameters as experimental findings of the studied structure. This method gives immediate feedback about the possible metamaterial nature of the device without introducing a very detailed description. In fact, a deeper analytical derivation could fail because of the impossibility of modeling exactly the lumped elements belonging to a CRLH configuration, including parasitic components, as a function of frequency. In [14], we analyzed the electrical response of two resonators based on (1) a fractal-inspired geometry with the shape of Sierpinski triangles and (2) simple U-shapes. Intuitively, a Sierpinski geometry is expected to show a metamaterial behavior, but we have preliminarily seen that the U-shape is also favored when looking for a metamaterial structure. One consequence of the metamaterial design studied in the literature is the recognition that this characteristic is fulfilled only in a narrow bandwidth, i.e., to have a negative value for permeability and permittivity or other properties like forward and backward wave onset. The device we shall analyze is a triangular resonator made by a full metal patch coupled to a line embedded in a coplanar waveguide (CPW) configuration compared with the same structure hosting a Sierpinski triangle with a first-level internal complexity, which can be considered as a fractal geometry, often viewed as a relative of the metamaterial geometries. Analogously to [14], we define them as C0 and C1, respectively. The two configurations are shown in Figure 6. The Sierpinski geometry has been studied for decades from a mathematical point of view, and it has been proposed especially in high-frequency antennas for multi-band purposes [30].