# 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
Together, these elements influence both the impedance matching and the overall resonant behavior of the proposed antenna. \(Z_{feed}(\omega)\) is expressed as  
\[Z_{feed}(\omega)=j\omega L_{0}+\frac{1}{j\omega.C_{0}} \tag{12}\]  
Equation (12) can be simplified as follows [22; 23; 24]:  
\[Z_{feed}(\omega)=j\left(\omega L_{0}-\frac{1}{\omega.C_{0}}\right)=j\frac{ \omega^{2}L_{0}C_{0}-1}{\omega.C_{0}} \tag{13}\]  
The final input impedance \(Z_{antenna}(\omega)\) of the proposed antenna is the sum of the antenna input impedance, given in Equation (13), and the equivalent impedance of the \(RLC\) network, given in Equation (11), \(Z_{antenna}(\omega)=Z_{feed}(\omega)+Z_{RLC}(\omega)\). It is expressed as  
\[Y=\frac{1}{R}+j\omega C-\frac{j}{\omega L}=\frac{1}{R}+j\bigg{(}\omega C-\frac {1}{\omega L}\bigg{)} \tag{6}\]0  
The quasi-static input capacitance \(C_{0}\) can be determined as follows [22; 23; 24]:  
\[Y=\frac{1}{R}+j\omega C-\frac{j}{\omega L}=\frac{1}{R}+j\bigg{(}\omega C-\frac {1}{\omega L}\bigg{)} \tag{6}\]1  
where \(\Delta\omega\) is the harmonic frequency step, defined as \(\Delta\omega=2\pi\Delta_{f}\). Based on the value of \(C_{0}\), the inductance \(L_{0}^{(0)}\) can be estimated by identifying the lowest frequency at which the imaginary part of the input impedance crosses zero. The inductance \(L_{0}^{(0)}\) models feed-in effects and higher-order electromagnetic phenomena and can be expressed as [22; 23; 24]  
\[Y=\frac{1}{R}+j\omega C-\frac{j}{\omega L}=\frac{1}{R}+j\bigg{(}\omega C-\frac {1}{\omega L}\bigg{)} \tag{6}\]2  
where \(\omega_{0}^{(0)}\) is the lowest frequency for which \(Im=\left\{Z_{in}\left(\omega_{0}^{(0)}\right)\right\}=0\). At each estimated resonant frequency \(\omega_{0}^{(0)}\), we take the real part of the impedance [22; 23; 24]:  
\[Y=\frac{1}{R}+j\omega C-\frac{j}{\omega L}=\frac{1}{R}+j\bigg{(}\omega C-\frac {1}{\omega L}\bigg{)} \tag{6}\]3  
Since the imaginary part is nearly zero, the input impedance is primarily resistive. We can estimate the quality factor \(Q_{n}^{(0)}\) as follows [22; 23; 24]:  
\[Y=\frac{1}{R}+j\omega C-\frac{j}{\omega L}=\frac{1}{R}+j\bigg{(}\omega C-\frac {1}{\omega L}\bigg{)} \tag{6}\]4  
Each parallel RLC branch consists of \(R_{i}\) (resistance), \(L_{i}\) (inductance) and \(C_{i}\) (capacitance). Each stage (for \(i=1\), 2, 3, 4) corresponds to one of the resonance peaks in the input impedance curve of the antenna. The inductance for each resonance branch is given by [22; 23; 24]:  
\[Y=\frac{1}{R}+j\omega C-\frac{j}{\omega L}=\frac{1}{R}+j\bigg{(}\omega C-\frac {1}{\omega L}\bigg{)} \tag{6}\]5  
where \(f_{ri}\) is the resonant frequency of the \(i\)-th mode. Based on Equation (19), we can calculate \(C_{i}\) as follows [24]:  
\[Y=\frac{1}{R}+j\omega C-\frac{j}{\omega L}=\frac{1}{R}+j\bigg{(}\omega C-\frac {1}{\omega L}\bigg{)} \tag{6}\]6  
At each resonance, the \(Q_{i}\) value provides an estimate of how narrow or wide the bandwidth is relative to the center frequency, and it is given by the following expression [24]:  
\[Y=\frac{1}{R}+j\omega C-\frac{j}{\omega L}=\frac{1}{R}+j\bigg{(}\omega C-\frac {1}{\omega L}\bigg{)} \tag{6}\]7  
where \(f_{u}\) and \(f_{l}\) are the upper and lower frequency of the bandwidth around \(f_{ri}\). As a result, the fractional bandwidth (\(FBW\)) is also determined by [24]:  
\[Y=\frac{1}{R}+j\omega C-\frac{j}{\omega L}=\frac{1}{R}+j\bigg{(}\omega C-\frac {1}{\omega L}\bigg{)} \tag{6}\]8  
The frequency bandwidth around each resonance can be estimated using \(R_{i}\) and \(C_{i}\), as follows [24]:  
\[Y=\frac{1}{R}+j\omega C-\frac{j}{\omega L}=\frac{1}{R}+j\bigg{(}\omega C-\frac {1}{\omega L}\bigg{)} \tag{6}\]9  
Based on Equations (19) and (20), the angular resonant frequency can be calculated using this equation:  
\[\omega_{0}C-\frac{1}{\omega_{0}L}=0 \tag{7}\]0