## 2 Materials and Methods
### Equivalent Circuit Model Analysis  
As has been widely documented in the literature, physical considerations based on an equivalent circuit model (ECM) prove highly beneficial in determining effective design guidelines for FSS filters. In this specific case, the proposed configuration comprises two lossless layers etched on opposing sides of a dielectric substrate, whose thickness (measuring 0.003\(\lambda_{L}\)) is sufficiently small that it can be disregarded in the ECM retrieval. The schematic representation of a typical FSS surrounded by air has previously been illustrated in Figure 2a. Firstly, the trend of the overall double-layer FSS impedance \(Z_{FSS}\), obtained from full-wave simulations and normal incidence, was shown and examined to understand \begin{table}
\begin{tabular}{c c} \hline \hline
**w (mm)** & **Q** \\ \hline
0.2 & 23.9 \\
0.3 & 22.7 \\
0.4 & 21.8 \\
0.5 & 21.7 \\
0.6 & 21.2 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Values of Q for varying strip widths. the corresponding circuital behavior. The real and imaginary parts of \(Z_{FSS}\) are illustrated in Figure 3. In the frequency interval from 1 GHz to 2.5 GHz, the FSS unit cell could be recognized as an LC series, which constitutes the first branch of the ECM and can be expressed as in Equation (5):  
\[Z_{FSS,1}=j\omega L_{FSS,1}+\frac{1}{j\omega C_{FSS,1}} \tag{5}\]  
At this point, to retrieve the values of \(L_{FSS,1}\) and \(C_{FSS,1}\) from the graph reported in Figure 4a, we can adopt the following procedure. By computing the derivative with respect to \(\omega\) of the expression in Equation (5) and evaluating it at \(\omega_{res,1}\) (i.e., the first series resonance frequency, approximately 1.7 GHz, Figure 3), the value of \(L_{FSS,1}\) can be obtained:  
\[L_{FSS,1}=\frac{1}{2}\left\{\frac{\partial}{\partial\omega}[\mathcal{S}m(Z_{ FSS,1})]\right\}\,\,\text{at}\,\,\omega=\omega_{res,1} \tag{6}\]  
Figure 4: (**a**) Schematic diagram of the FSS unit cell equivalent circuit model. (**b**) Comparison between \(\mathcal{S}m(Z_{FSS})\) obtained from the full wave simulations and \(\mathcal{S}m(Z_{FSS})\) reconstructed via ECM. Figure 3: Real and imaginary parts of \(Z_{FSS}\) obtained from the full wave simulations under normal incidence. Consequently, the value of \(C_{FSS,1}\) is established from the resonance condition:  
\[C_{FSS,1}=\frac{1}{\omega_{res,1}^{2}\cdot L_{FSS,1}} \tag{7}\]  
From an operative point of view, the frequency span around the series resonance constitutes a band stop region within the L-band that can also be conveniently exploited as a ground plane for future rasorber configurations. By moving upward within the frequency range, it can be deduced that the retrieved LC series is to be positioned in parallel with a capacitance indicated as \(C_{FSS,2}\) and with a resistance \(R_{FSS,2}\). The value of \(R_{FSS,2}\) can be directly evaluated at the parallel resonant point, as in the following expression:  
\[R_{FSS,2}=\underset{f}{\text{max}}\mathcal{R}e(Z_{FSS}) \tag{8}\]  
It should be noticed that this resistive value is not related in its absolute value to the losses in the copper strip constituting the FSS unit-cell or in the dielectric substrate. Instead, its role is to describe the parallel resonance at 3.68 GHz, whose Q-factor is not infinite due to the realistic material employed in the numerical simulation.