## 2 Unit Cell Array Analysis with FEM Approach  
The proposed unit cell structure with different array sizes is shown in Figure 1. Here, the red-colored SRRs indicate radiating patch and the yellow color indicates polyimide substrate material. The radiating patch contains two types of SRRs, and the complete unit cell's dimensions are given in Table 1. Not only is unit cell analysis insufficient for evaluating performance, but array analysis is also required to assess the effectiveness of the suggested structure. The CST tool was used to simulate the various sizes of arrays, such as 1 \(\times\) 1, 1 \(\times\) 2 and 2 \(\times\) 2, as shown in Figure 1, and the simulation setup for the unit cell in CST tool is given in Figure 2. The numerical simulations were done using CSTMW Studio, which uses FEM as a method for problem solving that needs accurate as well as quick responses. The CST time domain solver has a great benefit, as the scaling of resource requirements is done linearly with the number of mesh nodes. As a result, big radiating structures such as arrays can be handled using this CST tool. The problem can be solved in three different stages: unit cell analysis, array analysis and result analysis. By utilizing the FD solver in the CSTMW Studio software, the unit cell analogy and metamaterial array structure were numerically performed. Since it can calculate all the ports at once, the FD solver is a powerful solver with exceptional computation effectiveness for simulating multi-port systems such as array structures. On the other hand, the ports were arranged along both positive and negative z-axes using a TEM wave with no electric or magnetic fields along the propagating wave. While doing the simulation in CST studio first, the frequency range is chosen, and then in the simulation setup, ideal magnetic and electric conditions are set towards X- and Y-directions for determining S-parameters, in order to calculate the effective parameters of the medium, like epsilon, mu and n.  
When the S\({}_{11}\) magnitude is minimum, then the magnitude of S\({}_{21}\) at that resonant frequency is maximum. From Figure 3a, we can conclude that for the proposed unit cell, \begin{table}
\begin{tabular}{c c} \hline
**Variable** & **Dimension (mm)** \\ \hline L & 10 \\ w & 10 \\ r\({}_{\text{int}}\) & 3.5 \\ r\({}_{\text{out}}\) & 4.3 \\ g & 0.4 \\ p & 0.8 \\ p\({}_{1}\) & 1.09 \\ d & 0.3 \\ d\({}_{1}\) & 0.5 \\ x\({}_{1}\) & 4.8 \\ x\({}_{2}\) & 3.2 \\ x\({}_{3}\) & 1.6 \\ x\({}_{4}\) & 0.8 \\ \hline \end{tabular}
\end{table} 
Table 1: Dimensions of the unit cell. Figure 1: Geometrical arrangement for the (**a**) unit cell, (**b**) 1 \(\times\) 2 and (**c**) 2 \(\times\) 2 array cells. Figure 2: Simulation set up for the proposed unit cell structure. the \(S_{11}\) parameter resonates at 9.84 GHz and 11.46 GHz frequencies with magnitudes \(-31.68\) dB and \(-23.13\) dB, respectively, and \(S_{21}\) for the proposed unit cell, resonated at 8 GHz, 8.435 GHz and 11.11 GHz frequencies with magnitudes of \(-33.20\) dB, \(-37.13\) dB and \(-25.89\) dB, respectively. Similarly, for the \(1\times 2\) sized array, the \(S_{11}\) parameter resonates at 9.82 GHz and 11.47 GHz frequencies with magnitudes of \(-28.83\) dB and \(-22.62\) dB, respectively, and the \(S_{12}\) for the proposed unit cell was resonated at 8 GHz, 8.44 GHz and 11.10 GHz frequencies, with magnitudes of \(-33.20\) dB, \(-37.59\) and \(-26.31\) dB, respectively. Finally, for the \(2\times 2\) sized array, the \(S_{11}\) parameter resonates at 9.85 GHz and 11.45 GHz frequencies with magnitudes of \(-30.23\) dB and \(-24.18\) dB, respectively, and the \(S_{12}\) for the proposed unit cell resonated at 8 GHz, 8.44 GHz and 11.09 GHz frequencies with magnitudes of \(-32.56\) dB, \(-37.25\) and \(-25.77\) dB, respectively. We can thus conclude that the return loss parameter (\(S_{11}\)) and transmission loss (\(S_{12}\)) of various array topologies was exactly the same, with just a slight variation in the resonance frequency and its magnitude.