## 4 Equivalent Circuit and Design Analysis  
Numerous approaches can be taken for remodeling the equivalent circuit. The lumped equivalent circuit approach is one of the approaches that consider the microwave elements, including the inductance, resistance, capacitance, and conductance [38]. The metallic conductor is addressed with the inductor property when the equivalent circuit model is developed. The capacitive impact is represented by the split gap. Every SRR serves as a combination of inductance L and capacitance C. The split ring acts as a resonator, resonating at a specific frequency. The proposed structure is based on double SRR with some modifications. The length and spacing of the ring resonator may be adjusted to modify the resonance frequency. \[f=\frac{1}{2\pi\sqrt{LC}} \tag{11}\]  
The total inductance L and capacitance C of the proposed unit cell can be achieved from Equations (12) and (13), respectively [39]. \[L=0.01\times\mu_{0}\left\{\frac{2(d+g+h)^{2}}{(2w+g+h)^{2}}+\frac{\sqrt{\left(2 w+g+h\right)^{2}+l^{2}}}{(d+g+h)}\right\}t \tag{12}\]  
\[C=\epsilon_{0}\left[\frac{2w+g+h}{2\pi(d+h)^{2}}\ln\left\{\frac{2(d+g+h)}{(a- l)}\right\}\right]t \tag{13}\]  
where \(\mu_{0}=4\pi\times 10^{-7}\) H/m, \(\mu_{0}=8.854\times 10^{-12}\) F/m, \(h\) = thickness of the substrate, \(d\) = split distance, \(l\) = resonator length, \(w\) = resonator width, \(t\) = thickness of the copper resonator, and g = split gap of the ring. The equivalent circuit of the unit cell and the S21 response from the CST and ADS simulator are illustrated in Figure 6. Table 3 represents the values of the circuit components. The inductances L1 and L2 with the collaboration of capacitances C1 and C2 indicate the model of the equivalent circuit of the outer SRR of the design. At the same time, inductances L3 and L4 and capacitances C3 and C4 represent the inner SRR as an equivalent circuit. The outer and inner SRR relate to two metallic stripes, which are represented with the LC combination of L7, C7, and L8, C8, respectively. The inductances and capacitances of the Double C shape are indicated by the components of L5, L6, C5, and C6. This part of the circuit is related to the inner SRR, whereas component C9 is made by the split gap. The simulation software Advanced Design System (ADS) was used to determine the values of these circuit components; the acquired values are characterized in Table 3. These values are considered from the unit cell resonances of S21. It can be seen in Figure 6b that both results show slight variations, as the ADS simulator was used to provide the estimated structure of the lumped elements with manual fitting and lower boundary values, whereas the CST simulator used the finite integration method with a computing mesh algorithm. After \begin{table}
\begin{tabular}{c c c c c c} Structure & Design 1 & Design 2 & Design 3 & Design 4 & Proposed design \\ \hline Resonance & 3.53, 10.29, & 3.41, 9.19, 12.9 GHz & 3.41, 8.72, 9.25, & 3.43, 8.77, 9.25, & 2.4, 5.6, 8.93, 11.5 GHz \\ frequencies & 13.42 GHz & & 12.87 GHz & 12.85 GHz & 2.4, 5.6, 8.93, 11.5 GHz \\  & \(-40.79\), \(-47.37\), & \(-42.49\), \(-44.98\), & \(-41.38\), \(-35.89\), & \(-39.9\), \(-35.11\), & \(-32.27\), \(-38.28\), \\ Magnitude in dB & \(-34.61\), & \(-38.36\) & \(-35.68\), \(-40.44\) & \(-39.9\), \(-32.43\) & \(-25.99\), \(-44.7\) \\ Bandwidth Span & 3.25-38, 9.38-11.1, & 3.1-3.6, 8.3-9.8, & 3.18-36, 8.27-8.88, & 3.19-3.6, 8.28-8.87, & 2.3–24.9, 5.27-5.87, \\ (under \(-10\) dB) & 13.2-13.8 & 12.5-13.36 & 9.0–9.7, 12.55-13.19 & 9.0–9.7, 12.5-13.15 & 8.81–8.98, 10.61–12.26 \\ \hline \end{tabular}
\end{table} 
Table 2: Proposed unit cell parameters values. studying the unit cell design and equivalent circuit model, the responsible components that trigger the resonance at a certain frequency are identified. The L1, L3, C1, C3, L8, and C8 are responsible for the resonance of 2.4 GHz, whereas the L3, C3, L7, and C7 represent the resonance of 5.6 GHz. The resonance of 8.93 GHz is responsible for the components of C9, L5, and C5.