## 5 Parametric Study
The effect of a resonating conductor (gold, platinum, tantalum) indicated that the response of \(S_{21}\) was the same as copper, whereas nickel used as a patch showed the same response with lower magnitude. Since gold, platinum, and tantalum are very expensive metals, annealed copper was used as a resonating conductor for the proposed design. Figure 5: The \(S_{21}\) plot for (**a**) changing of split gap of SRR, (**b**) changing of split gap of Coptic cross resonator, (**c**) varying the width of the Coptic cross resonator, and (**d**) effect of different conductors used in resonator. **Effect of substrate material on \(S_{21}\) response:**  
Three different substrate materials were used for substrate selection. The three substrates are FR-4 (lossy and loss-free), Rogers RT-5880 (lossy), and Rogers RT-6002 (lossy). All substrate properties are different. In the first analysis, the dielectric constant, dielectric loss, and thickness (mm) of FR-4 (lossy) are 4.3, 0.025, and 1.6, respectively. Figure 6a shows that this material has demonstrated quadruple resonance frequency with the magnitudes of \(-29.45\), \(-25.44\), \(-19.05\), and \(-24.45\) dB; on the other hand, FR-4 (loss-free) material exhibits the same resonance with the magnitudes of \(-39.35\), \(-44.55\), \(-38.94\), and \(-49.02\) dB. The dielectric constant, dielectric loss, and thickness (mm) of Rogers RT-5880 (lossy) and RT-6002 (lossy) are 2.2, 0.0009, and 1.575 and 2.94, 0.0012 and 1.524, respectively. As shown in Figure 6a, both substrate materials exhibit triple resonance frequency with good magnitude. Compared to the FR-4 substrate, the resonance frequency of both Rogers materials moves to a higher value. However, the performance of the FR-4 substrate material is better than the other substrate materials. Therefore, the FR-4 substrate material was selected solely because it was consistent with the purpose of this study. Figure 6: The S21 plot for (**a**) effect of substrate material, (**b**) effect of substrate length, and (**c**) the effect of different boundary conditions. **Effect of substrate length on \(S_{21}\) response:**  
The design starts with an initial selection of basic dimensions and considers the same general subwavelength criteria as \(\lambda/10\). The \(S_{21}\) plot for different substrate lengths is exhibited in Figure 6b. Since the substrate length and resonance frequency are inversely proportional to each other, we examined the performance of four different unit cell substrate length, such as 14, 13, 12, and 11 mm, in order to obtain the lowest resonance frequency in the lower dimension as the higher EMR. For the effect of substrate length on \(S_{21}\) response, it was found that first and second resonance frequency shifted to higher value but third and fourth resonance frequency decreased slightly. The unit cell with four different substrate lengths exhibits the same resonance frequency. Finally, the proposed metamaterial resonator with a substrate length of 11 mm on each side is selected to achieve quadruple resonance frequency covering for S-, C-, X-, and Ku-band. **Effect of different boundary condition on \(S_{21}\) response:**  
The aim of this study is to elucidate the effects of different boundary conditions on modified Coptic cross shape-based metamaterial in the frequency range of 1 to 16 GHz. Five different boundary conditions, perfect electric conductor (PEC)- perfect magnetic conductor (PMC), PMC-PEC, PEC-PEC, PMC-PMC, and Unit Cell, are used to describe the metamaterial performance. The proposed metamaterial is used to demonstrate that the use of different boundary conditions can lead to completely different electromagnetic responses, as shown in Figure 6c. In the first study, PEC-PMC (E-field in \(x\)-axis and H-field in \(y\)-axis) boundary conditions are used in modified Coptic cross shape-based metamaterial; there are quadruple resonance frequencies in a given frequency range, representing 2.02, 6.985, 9.985, and 14.425 GHz.