# A Compact Triple Band Antenna Based on Multiple Split-Ring Resonators for Wireless Applications
## 3 Results and Discussion
The resonant frequencies from the analytical approach were estimated using two distinct methods:  
* Method 1 models the split-ring resonator as a resonant ring, where the effective electrical length is proportional to the physical circumference. The resonance frequency is calculated using the following expression: \[f_{SRR}=\frac{c}{\left(2.\pi.r.\sigma_{SRRn}+L_{f}\right)\sqrt{\epsilon_{re}}}\] (25)
* Method 2 treats the arms of the SRR as equivalent dipole antennas with an effective length \(L_{SRRn}\), leading to the resonance frequency \[f_{SRR}=\frac{c}{2.L_{SRRn}.\sqrt{\epsilon_{re}}}\] (26) where \[L_{SRRn}=Wn+Lf+\sum_{i=1}^{i=4}L_{ni}\] (27)  
As an illustrative example, for \(SRR_{1}\), the total effective length is given by  
\[L_{SRR1}=W_{n}+L_{f}+L_{11}+L_{12}+L_{13}+L_{14}=35.45\;\mathrm{mm}\]  
Both methods provide results that closely match those obtained from numerical simulations, validating the proposed modeling approach. The results show strong agreement between the analytical study and the numerical simulations and the equivalent circuit model, especially for the first three resonances of SRR 1, SRR 2 and SRR 3. This confirms the validity of the circuit-based model in accurately representing the behavior of the SRR. Minor deviations observed at higher frequencies (SRR 4) are due to the small size of this ring and parasitic effects. Overall, the agreement between the different methods confirms the robustness of the design and the reliability of using analytical and circuit-based techniques to predict the electromagnetic response of the proposed metamaterial structure. Figure 8 presents the surface current distribution the proposed multi-SRR-Based antenna at operating frequencies of 2.5 GHz, 3.5 GHz and 6.8 GHz, respectively. At 2.5 GHz, the current is predominantly concentrated around the largest ring (SRR 1). This confirms that this ring plays the dominant role for the resonance at the lower frequency. \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**SRR** & **Radius (mm)** & **Length** & **Equivalent Circuit** & **fr (GHz)** & **fr (GHz)** & **fr (GHz)** & **fr (GHz)** \\  & **(mm)** & **Parameters** & **Analytical** & **Analytical** & **ADS** & **CST** \\  & & & & **(Method 1)** & **(Method 2)** & **Circuit** & **Simulation** \\ \hline
1 & 11.22 & 35.45 & \begin{tabular}{c} C\({}_{0}\) = 0.21 pF, L\({}_{0}\) = 4.2 nH, \\ C\({}_{1}\) = 306 pF, L\({}_{1}\) = 1.17 nH, \\ R\({}_{1}\) = 790 \(\Omega\) \\ \end{tabular} & \begin{tabular}{c} 2.55 GHz \\ \end{tabular} & \begin{tabular}{c} 2.58 GHz \\ \end{tabular} & \begin{tabular}{c} 2.65 \\ \end{tabular} & \
\begin{tabular}{c} 2.65 \\ \end{tabular} &
\begin{tabular}{c} 2.65 \\ \end{tabular} \\ \hline
2 & 7.94 & 26.13 & \begin{tabular}{c} C\({}_{2}\) = 3.74 pF, L\({}_{2}\) = 0.52 nH, \\ R\({}_{2}\) = 500 \(\Omega\) \\ \end{tabular} & \begin{tabular}{c} 3.53 GHz \\ \end{tabular} & \begin{tabular}{c} 3.50 GHz \\ \end{tabular} & \
\begin{tabular}{c} 3.60 \\ \end{tabular} &
\begin{tabular}{c} 3.50 \\ \end{tabular} \\ \hline
3 & 5.67 & 19.70 & \begin{tabular}{c} C\({}_{3}\) = 0.70 pF, L\({}_{3}\) = 1.25 nH, \\ R\({}_{3}\) = 1 \(\Omega\) \\ \end{tabular} & \begin{tabular}{c} 4.90 GHz \\ \end{tabular} & \begin{tabular}{c} 4.80 GHz \\ \end{tabular} & \
\begin{tabular}{c} 4.84 \\ \end{tabular} &
\begin{tabular}{c} 4.83 \\ \end{tabular} \\ \hline
4 & 3.40 & 12.90 & \begin{tabular}{c} C\({}_{4}\) = 0.70 pF, L\({}_{4}\) = 0.995 nH, \\ R\({}_{4}\) = 84 \(\Omega\) \\ \end{tabular} & \begin{tabular}{c} 7.40 GHz \\ \end{tabular} & \begin{tabular}{c} 7.06 GHz \\ \end{tabular} & \
\begin{tabular}{c} 6.03 \\ \end{tabular} &
\begin{tabular}{c} 6.80 \\ \end{tabular} \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Comparison of resonant frequencies obtained from analytical study (two methods), equivalent circuit analysis and numerical simulation for the proposed multi-SRR-based antenna. Its dimensions result in an electrical length that matches the wavelength associated with 2.5 GHz.