# A Compact Triple Band Antenna Based on Multiple Split-Ring Resonators for Wireless Applications
## 2 Construction of the Proposed Multi-SRR-Based Antenna  
### Split-Ring Resonator Unit Cell Design  
The structure of the split-ring resonator unit cell (SRR) investigated in this paper is shown in Figure 1a,b. It consists of a single octagonal ring with a gap. This SRR was made on a low-cost, lossy FR4 substrate whose unit cell has a thickness of 0.035 mm. The substrate features a dielectric constant of 4.3 and a loss tangent of 0.025. The SRR cell is placed on a substrate area of 23 \(\times\) 23 mm\({}^{2}\) and was simulated using CST Microwave Studio. To investigate the properties of the metamaterial, a waveguide simulation is performed by applying electric (E) and magnetic (H) boundary conditions along the Y and Z directions to emulate the walls of a virtual waveguide. A plane wave is aligned parallel to the SRR surface in the X direction. The specific design parameters of the unit cell are Gap = 1.98 mm, R1 = 11.22 mm, R2 = 10.11 mm and Wd = 7.74 mm. The resonant frequency of the SRR can be calculated as follows [16]:  
\[f_{SRR}=\frac{c}{2\pi r\sqrt{\epsilon_{eff}}} \tag{1}\]where \(r\) is the radius of the ring, and \(\epsilon_{eff}\) is the effective dielectric constant and can be calculated as  
\[\epsilon_{eff}=\frac{\epsilon_{r}+1}{2} \tag{2}\]  
where \(\epsilon_{r}\) represents the relative permittivity of the FR4 substrate. The equivalent circuit of the resonator is shown in Figure 1c. When a time-varying magnetic field H interacts with a split-ring resonator, it induces a circulating current I along the resonator's structure. This electromagnetic response can be effectively modeled using a series RLC circuit, where the resonant frequency is primarily governed by the following:  
* Inductance (L): Represents the self-inductance of the SRR loop, which is related to the size and shape of the ring.
* Capacitance (C): Corresponds to the gap of the SRR.
* Resistance (R): Represents the total ohmic losses in the metallic structure, as well as dielectric losses in the substrate material.  
This series RLC circuit models the SRR as a passive resonant scatterer (Figure 2c), responding to incident electromagnetic waves through current-driven excitation. At resonance, the incident magnetic field induces a strong circulating current within the SRR loop, enabling energy storage and scattering. The total impedance is  
\[Z_{series}=R+j\bigg{(}\omega L-\frac{1}{\omega.C}\bigg{)} \tag{3}\]  
At resonance:  
\[\omega_{0}=\frac{1}{\sqrt{\text{LC}}} \tag{4}\]  
Figure 2a shows the simulated reflection coefficient (\(S_{11}\)) and transmission coefficient (\(S_{21}\)) of the unit cell. The transmission coefficient (\(S_{21}\)) reaches its maximum at 2.7 GHz, which agrees well with the theoretical value of 2.6 GHz calculated from Equation (1). This agreement confirms that the split-ring resonator (SRR) operates effectively at the desired resonant frequency, demonstrating its suitability as a radiating element for antenna applications. Figure 2b shows the effective permeability (\(\mu\)) of the unit cell, extracted from the real and imaginary components of \(S_{11}\) and \(S_{21}\) using CST Microwave Studio simulations [21].  
\[\mu_{eff}=\pm\sqrt{\frac{\left(1+S_{11}\right)^{2}-S_{21}^{2}}{\left(1-S_{11} \right)^{2}-S_{21}^{2}}} \tag{5}\]  
Equation (5) [21] was used to calculate the permeability values. It is obvious that the octagonal SRR exhibits negative permeability values in the frequency range of 2.5 GHz to 3 GHz.  
Figure 1: Split-ring resonator unit cell. (**a**) Front view. (**b**) Boundary settings. (**c**) Equivalent circuit.