## 3 Antenna Design and Analysis
### Initial DR Dimensions and Parametric Study  
Initial values of the DRA dimensions are calculated using dual-band formulas given in [10]. While calculating the dimensions of the proposed dual-band DRA, all the equations given in [10] for dual band DRAs were used. Only the equations for kx1 and kx2 for TE\({}^{y}\)\({}_{111}\) and TE\({}^{y}\)\({}_{211}\) and ky1 and ky2 for TE\({}^{x}\)\({}_{111}\) and TE\({}^{x}\)\({}_{121}\) modes were modified because the excited higher order modes in the presented DRA are TE\({}^{y}\)\({}_{111}\) and TE\({}^{y}\)\({}_{211}\), TE\({}^{x}\)\({}_{111}\) and TE\({}^{x}\)\({}_{121}\) as compared to the higher order mode excited in [10] which is TE\({}^{y}\)\({}_{113}\). The modified values are given in Equations (1) and (2). \[Kx1=\frac{\pi}{a},\ Kx2=\frac{2\pi}{a}\ and\ Kz1=\frac{\pi}{2h},\ Kz2=\frac{ \pi}{2h} \tag{1}\]  
\[Ky1=\frac{\pi}{b},\ Ky2=\frac{2\pi}{b}\ and\ Kz1=\frac{\pi}{2h},\ Kz2=\frac{ \pi}{2h} \tag{2}\]  
where kx, ky, and kz are the wave numbers in the x, y, and z direction, respectively. The design formulas given in [10] are applicable to a rectangular shaped DRA only. Therefore, based on these formulas, the dimensions of the dual-band rectangular DRA were obtained. The DRA was simulated and corresponding S-parameters are shown in Figure 2a. S-parameters from port 2 are not shown as they are similar to port 1. It is clear from this figure that there are three resonances at 1.68 GHz, 2.33 GHz, and 2.54 GHz. The first resonance relates to TE111 mode with poor matching while the second resonance is due to some unknown higher order mode at 2.33 GHz with low simulated gain (3.8 dBi). The third resonance is because of the TE121 mode. In order to improve matching and shift the resonance frequencies to the required bands of interest, the size of the DRA needs to be changed. As the modes excited are TE\({}^{x}\) and TE\({}^{y}\), the resonance frequency of the excited modes is sensitive to the dimension of the DRA along x and y-axis. Also, to maintain symmetry between the ports along with change in size in the x and y-axis, the rectangular shape was modified to an L-shaped DRA. Doing so, the size of the DRA has been modified in the x and y-axis (dimension a and b) which in turn changes the wave numbers of kx and ky, thus shifting the lower resonance frequency towards the higher band while merging the two resonances at the upper band to give a wide band. The S-parameters of the L-shaped DRA are shown in Figure 2b. As evident from the figure, the resonance at the lowest frequency has shifted to 1.8 GHz while at the upper frequency band, the two resonances have merged to obtain a wideband and matching has been improved. However, isolation between the ports at the bands of interest is poor. To improve port isolation, cylindrical air-gap and metal strips have been incorporated in the L-shaped DRA and the results are shown in Figure 2c. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Value (mm)** & **Parameter** & **Value (mm)** & **Parameter** & **Value (mm)** \\ \hline \(a\) & 20 & b & 25 & \(h\) & 22 \\ _sl_ & 20 & sw & 6 & _pl_ & 23 \\ _pw_ & 3 & s1 & 5.25 & _s2_ & 3 \\ _s3_ & 4.5 & mw & 3 & _rad_ & 6 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Presented design optimized dimensions. A parametric study on cylindrical air-gap radius _rad_ and length of metallic strips _sl_ is performed to evaluate the performance of the design. Figure 3 shows the effect of the cylindrical air-gap radius _rad_ on reflection coefficient of port 1 (S\({}_{11}\)). As both ports 1 and 2 are symmetrical, only the results of port 1 are shown in the parametric study. As can be observed from the figure that the radius of the cylindrical air-gap has a little effect on reflection coefficient. A minor shift in frequency can be observed in Figure 3 at both frequency bands.