## 3 Analysis and Comparison with Fundamental Bandwidth Limit  
Figure 5 presents the theoretical maximum bandwidth of the patch antenna according to the permittivity of the substrates. The bandwidth of the patch antenna can be calculated using the following equation [6,8]:  
\[BW=\frac{VSWR-1}{Q\sqrt{VSWR}} \tag{1}\]  
Figure 4: Radiation pattern of the proposed antenna: (**a**) _zx_-plane; (**b**) _zy_-plane. Figure 3: Reflection coefficient of the proposed antenna. where \(Q\) is the quality factor of the antenna. When the surface wave power of the patch antenna is much smaller than the radiated power, the expression for the bandwidth of the patch antenna can be approximated as follows [6; 8]:  
\[BW=\frac{16}{3\sqrt{2}}\frac{c_{1}p}{e_{r}}\bigg{(}\frac{1}{e_{r}}\bigg{)}\bigg{(} \frac{h}{\lambda_{0}}\bigg{)}\bigg{(}\frac{W}{L}\bigg{)} \tag{2}\]  
where \(e_{r}\) is the radiation efficiency of the patch antenna, and \(h\) is the total thickness of the substrate. \(p\) means the ratio of the power radiated by the patch to that radiated by an equivalent Hertzian dipole of the same moment. This \(p\) factor can be approximately 1.0 when the patch antenna has a small size [8]. \(W\) and \(L\) are the width and length of the conductive resonator of the patch antenna, and \(c_{1}\) is expressed as follows [6; 8]:  
\[c_{1}=1-\frac{1}{n^{2}}+\frac{2/5}{n^{4}} \tag{3}\]  
where \(n\) is \(\sqrt{\varepsilon\mu}\), and \(\varepsilon\) and \(\mu\) are the permittivity and the permeability of the antenna substrates (the upper and lower substrate layers in the proposed antenna), respectively. According to Equations (1) and (2), in order to achieve the theoretical maximum bandwidth of the patch antenna, the quality factor of the system should be minimized. To do so, the area of the conductive radiator and the thickness of the substrate should be enlarged. In addition, the permittivity of the substrate in the patch should be minimized. The results in Figure 3 are then compared with the theoretical bandwidth calculated using Equation (2). The solid and dashed lines represent the theoretical maximum bandwidth of the patch antenna for which the thicknesses of the substrates are \(h=0.05\lambda\) and \(h=0.13\lambda\), respectively. In this result, \(p\) and \(e_{r}\) are assumed to be 1 and 0.85, respectively, and the ratio of \(W\) and \(L\) is 1. The maximum bandwidth of the dual patch (dotted and dash-dotted lines) is assumed to be twice that of the single patch (solid and dashed lines), considering their two resonators. Therefore, the theoretical maximum bandwidth of the dual patch with the quasi-air substrate (\(e_{r}=1.8\)) is calculated to be 26%. In fact, the bandwidth of the simulation model for the proposed dual-patch antenna has a bandwidth of 21%. In addition, the fundamental bandwidth limits of the antennas with the RF-35 and FR-4 substrates are 15.5% and 13.3%, respectively. These results demonstrate that by using the proposed antenna structure, broadband characteristics close to the fundamental bandwidth limit of the patch antenna can be achieved, and at the same time, the durability of the antenna is improved for various applications. Figure 5: Fundamental bandwidth limit of the patch antenna according to the dielectric constant. Figure 6 represents the fundamental bandwidth limit of the dual-patch antenna (\(h=0.13\lambda\)) according to the antenna's efficiency [6,8]. As can be seen in this figure, when there is no reduction in bandwidth due to the antenna's efficiency (\(e_{r}=1\)), the fractional bandwidth of the antenna with the quasi-air substrate is 30.8%. On the other hand, when the antenna's efficiency is reduced to 0.6, the bandwidth is also reduced to 18.5%. Table 2 shows comparisons of the antenna's characteristics with other stacked coupled-fed antennas. As can be seen in this table, the antennas in [5,13] with air foam substrates have a broad bandwidth while having a relatively low profile. In addition, an antenna in [11] achieved broader bandwidth through a shorted patch and an L-shaped probe structure.