# Equivalent Circuits for Microwave Metamaterial Planar Components
## 3 Metamaterial Resonators  
Two main advantages are recognized in the scientific literature about metamaterial resonators: (1) a sharp response at the resonance frequency, contributing to a higher quality factor, and (2) a reduction of the electrical size, using a physical size smaller than the wavelength of the incident wave [13]. In our case, an additional advantage is given by the fixed footprint, and the internal complexity has the task of changing the operation frequency. The last characteristic makes the Sierpinski triangles suitable for realizing arrays with the fine-tuning of bandwidth and frequency of operation, combining different geometries within a purposely designed array. In our study, we propose the utilization of equilateral triangles because they can be easily integrated into a more complicated array, but this is a possibility and not a limitation. The general configuration in Figure 2b can be used for modeling circuits with \(Z_{2}\) describing a resonant circuit. Alternatively, we can include a resonant circuit within two ordinary transmission line sections to bring the signal to the resonator. Using \(Z_{2}\) as a resonator, we can have, for a band-stop configuration, a series of \(L_{r}\) and \(C_{r}\) fed by a TL with characteristic impedance \(Z_{0}\) leading to the classical equations:  
\[\omega_{r}=\frac{1}{\sqrt{L_{r}C_{r}}}\;Z_{0}=\sqrt{\frac{L_{r}}{C_{r}}} \tag{5}\]  
From the above equations, we can obtain:  
\[L_{r}=\frac{Z_{0}}{\omega_{r}}\;C_{r}=\frac{1}{Z_{0}\omega_{r}} \tag{6}\]  
So, from the experimental determination of the characteristic impedance (or its definition by design) and the frequency of resonance \(f_{r}=\frac{\omega_{r}}{2\pi}\), we obtain the values of the equivalent lumped components for the resonator. The measurement of the quality factor leads to the additional element, i.e., the resistor, which describes the losses of the resonator. For a series resonator:  
\[Q=\frac{\omega_{r}L_{r}}{R_{r}}=\frac{1}{\omega_{r}C_{r}R_{r}}=\frac{\omega_{ r}}{\Delta\omega_{r}} \tag{7}\]  
where \(R_{r}\) is the resonator resistance, and \(Q\) is the quality factor obtained measuring the frequency of resonance and the linewidth \(\Delta\omega_{r}\) of the resonator, from which we can get the resistance. We can also write, in general, that \(L_{r}=L_{r0}\mu_{r}\) and \(C_{r}=C_{r0}\varepsilon_{r}\), where the suffix \"0\" is valid for the in-vacuum equivalent lumped component, and the dielectric constant and magnetic permeability as a multiplication factor are included when an actual medium with its properties is considered. The above concepts are essential for a complete numerical determination of the lumped equivalent circuital components because both inductance and capacitance can be regarded as proportional to the magnetic permeability \(\mu_{r}\) and dielectric permittivity \(\varepsilon_{r}\), respectively. On the other hand, when the circuital analysis aims to determine a negative sign for both quantities, i.e., \(\mu_{r}\) and \(\varepsilon_{r}\), the proportionality is enough because \(L_{r0}\) and \(C_{r0}\) always have positive values. Nevertheless, a resonator is characterized by vanishing the reactive contribution at the resonance frequency, maintaining only the resistive part, i.e., the real part of the impedance. For this reason, it is not fully correct to determine a possible metamaterial response at resonance with this approach because, in this case, \(Im\{Z_{2}\}=0\), and the initial equations must be modified to obtain reasonable values accounting for a possible reactive contribution. Since TLs feed a resonator, we can consider the entire device, including a cascaded configuration where the resonator is placed in the middle of the structure with two TLs at the input/output ports.