# A Compact Triple Band Antenna Based on Multiple Split-Ring Resonators for Wireless Applications
## 2 Construction of the Proposed Multi-SRR-Based Antenna
### Modeling and Analysis of the Equivalent Circuit for SRR-Based Antenna  
Figure 4 illustrates the proposed equivalent circuit of an SRRs-loaded antenna. The antenna is modeled using a series inductor (\(\mathbf{L_{0}}\)) and capacitor (\(\mathbf{C_{0}}\)), which represent the quasi-static capacitive and inductive behavior of the antenna input. Together, these elements form the foundation of the antenna's low-frequency equivalent circuit model. Each split-ring resonator is represented by a parallel RLC circuit, leading to a cascade of four parallel RLC branches. When the split-ring resonator functions as a radiating antenna element, its equivalent circuit is altered due to near-field interactions with the feed line. These interactions introduce a parallel resonant (tank) circuit behavior, resulting in maximum impedance and establishing a balanced current distribution across its arms. In the equivalent circuit, the resistance (R) primarily represents the radiation resistance, responsible for radiating power. Figure 3: Evolution of the proposed multi-SRR-loaded antenna. (**a**) Reference antenna. (**b**) Front and side views with 1 SRR (Antenna 1), (**c**) with 2 SRRs (Antenna 2), (**d**) with 3 SRRs (Antenna 3) and (**e**) with 4 SRRs (Antenna 4). In this section, a detailed modeling and analysis of the equivalent circuit for the proposed SRR-based antenna is presented, beginning with the analysis of a parallel RLC circuit. The total admittance \(Y\) of a parallel RLC circuit [22; 23; 24] is the sum of the individual admittances of the resistor, inductor and capacitor, \(Y=Y_{R}+Y_{L}+Y_{C}\), and is expressed as  
\[Y=\frac{1}{R}+j\omega C-\frac{j}{\omega L}=\frac{1}{R}+j\bigg{(}\omega C-\frac {1}{\omega L}\bigg{)} \tag{6}\]  
where \(\omega\) is the operating angular frequency defined as \(\omega=2\pi\)f, and \(\omega_{0}\) is the resonant angular frequency, given by Equation (4). At this frequency, the inductive and capacitive effects cancel out, satisfying the condition  
\[\omega_{0}C-\frac{1}{\omega_{0}L}=0 \tag{7}\]  
The quality factor \(Q\) for the parallel RLC is given by  
\[Q=\omega_{0}\text{RC} \tag{8}\]  
Figure 4: Proposed multi-SRR-based antenna equivalent circuit. At resonance, \(\omega_{0}=\omega\), and the circuit exhibits purely resistive behavior. By substituting Equations (7) and (8) into Equation (6), the total admittance at resonance simplifies as follows:  
\[Y=\frac{1}{R}\bigg{[}1+jQ\bigg{(}\frac{\omega}{\omega_{0}}-\frac{\omega_{0}}{ \omega}\bigg{)}\bigg{]} \tag{9}\]  
According to Equation (9), the impedance \(Z\) of a parallel \(RLC\) circuit can be expressed as [22; 23; 24]  
\[Z_{RLC}(\omega)=\frac{1}{Y}=\frac{R}{1+jQ\bigg{(}\frac{\omega}{\omega_{0}}- \frac{\omega_{0}}{\omega}\bigg{)}} \tag{10}\]  
Since Figure 4 contains four cascaded parallel \(RLC\) circuits, the total impedance of the equivalent circuit can be expressed as the sum of the individual resonant circuits. The impedance \(Z_{RLC}(\omega)\) is given by [22; 23; 24]  
\[Z_{RLC}(\omega)=\sum_{n=1}^{4}\frac{R_{n}}{1+jQ_{n}\left(\frac{\omega}{\omega_ {n}}-\frac{\omega_{n}}{\omega}\right)} \tag{11}\]  
where \(n\) represents the number of resonant modes needed to accurately synthesize the antenna's input impedance. \(R_{n}\), \(L_{n}\), \(Q_{n}\) and \(C_{n}\) represent the resistance, inductance, quality factor and capacitance, respectively, describing each resonant mode. \(\omega_{n}\) is the resonant angular frequency of the nth resonant mode, given by \(\omega_{n}=\frac{1}{\sqrt{\frac{L_{n}C_{n}}{\omega}}}\). The quality factor is defined as \(Q_{n}=\omega_{n}R_{n}C_{n}\). These four cascaded parallel RLC circuits are connected in series with the antenna input impedance \(Z_{feed}(\omega)\), which accounts for the effects of \(L_{0}\) and \(C_{0}\). In this model, \(L_{0}\) represents the antenna input inductance, while \(C_{0}\) corresponds to the total low-frequency (quasi-static) capacitance seen at the antenna input.