## 5 Electromagnetic Fields and Surface Current Analysis
### Equivalent Circuit Model of the Proposed Metamaterial Unit Cell  
An equivalent circuit was implemented for the proposed metamaterial unit cell to authenticate the results. The unit cell comprises two SRRs and two epsilon-shaped metal strips which contain both capacitive and inductive elements. Each metal strip functioned as an inductor, while every split gap was a capacitor. Consequently, the whole unit cell can be presented by an LC resonance circuit. If L and C represent the inductance and capacitance of the inductor and capacitor, respectively, then the resonance frequency (n) for the LC circuit can be determined using the Equation (11) [32]:  
\[\text{n}=\frac{1}{2\pi\left(\text{LC}\right)^{1/2}} \tag{11}\]  
The capacitance of the capacitor formed by the slits can be calculated using Equation (12):  
\[\text{C}=\in_{\text{r}}\in_{0}\frac{\left(\text{F}\right)\text{A}}{\text{d}} \tag{12}\]  
Figure 10: Surface current at (a) 2.38 GHz, (b) 4.55 GHz and (c) 9.42 GHz.  
Figure 9: Magnetic field distribution at (a) 2.38 GHz, (b) 4.55 GHz and (c) 9.42 GHz.  
where \(\in_{0}\) and \(\in_{r}\) are the permittivity in free space and the relative permittivity, respectively, while A is the cross-sectional area of the conducting strip and d is the split gap:  
\[\frac{\mathrm{L(nH)}}{1}=2\times 10^{-4}\left[\ln\left(\frac{1}{\mathrm{w}+ \mathrm{t}}\right)+1.193+0.02235\left(\frac{\mathrm{w}+\mathrm{t}}{1}\right) \right]\mathrm{k_{g}} \tag{13}\]  
Here, \(\mathrm{k_{g}}\) is the correction factor, w is the width, \(\mathrm{l}\) is the length and \(\mathrm{t}\) is the thickness of the strip.  
Figure 11 represents a predictable equivalent circuit for the proposed metamaterial unit cell. In the equivalent circuit \(\mathrm{L_{1}}\), \(\mathrm{L_{2}}\), \(\mathrm{L_{3}}\), \(\mathrm{L_{4}}\), \(\mathrm{L_{5}}\), \(\mathrm{L_{6}}\) and \(\mathrm{L_{7}}\) depict the inductors while \(\mathrm{C_{1}}\) to \(\mathrm{C_{9}}\) represent the capacitors. The equivalent circuit for the proposed unit cell was performed using the advanced design system (ADS) software. The first SRR is represented by the inductors \(\mathrm{L_{1}}\), \(\mathrm{L_{2}}\) and capacitors \(\mathrm{C_{1}}\), \(\mathrm{C_{2}}\), respectively, which belong to the first resonance frequency of 2.375 GHz. Similarly, \(\mathrm{L_{3}}\), \(\mathrm{L_{4}}\) and \(\mathrm{C_{3}}\), \(\mathrm{C_{4}}\) are used for the second SRR, contributing to the second frequency of 4.52 GHz. Meanwhile, the two epsilon-shaped resonators are replaced by the inductors \(\mathrm{L_{5}}\) to \(\mathrm{L_{7}}\) and the capacitors \(\mathrm{C_{5}}\) to \(\mathrm{C_{7}}\), which are responsible for the third resonance frequency of 9.35 GHz, where \(\mathrm{C_{8}}\) and \(\mathrm{C_{9}}\) are the coupling capacitors. The values of all inductors and capacitors are tabulated in Table 3. The two transmission coefficients (\(\mathrm{S_{21}}\)) for the unit cell extracted from CST and ADS are presented in the Figure 12. Based on Figure 12, the 3 resonance frequencies that were attained by ADS simulation are 2.375 GHz, 4.52 GHz and 9.35, whereas the CST-simulated frequencies were 2.38 GHz, 4.55 GHz and 9.24 GHz, indicating very negligible disparity.