## 3 Proposed Wideband, Planar, Bow-Tie Antenna  
The final antenna design and its dimensions are shown in Figure 7. The proposed bow-tie antenna is entirely planar--there are no multilayered components, and the antenna lays in one plane (the XZ plane). The dimensions are optimized using a genetic algorithm (GA) optimization. Since the radiation pattern of the antenna is bi-directional, meaning that there is a negligible difference between the frontwards and backwards realized gains, the cost function is simplified to include \(S_{11}\) only to maximize the \(-10\) dB bandwidth as follows:  
\[\text{Cost Function}=\sum_{f_{l}=1.0}^{f_{l}=3.0}\left(S_{i}-(-15)\right) \left(\text{dB}\right), \tag{1}\]  
where \(S_{i}\) is the \(S_{11}\) (dB) value at various frequencies between \(f_{l}\) (1.0 GHz) and \(f_{l}\) (3.0 GHz) [18]. The GA uses the flare angle and radii of each constituent bow tie as the variables, contrary to the designs in Figure 5, where the radius of each bow tie is the same. These variables are constrained to ensure that the geometry always comprises three distinct bow ties. The nomenclature of the three different bow ties follows the pattern established in Section 2, where each name corresponds to the flare angle of the constituent bow tie (rounded to the nearest whole number). The longest bow tie (bow-tie-86) has a radius of 41.1 mm and  
Figure 6: Simulated \(S_{11}\) for the triple bow ties with the same radius, without (Figure 5a) and with circular top loading (Figure 5b), and for the proposed wideband, planar, bow-tie antenna (Figure 7), and the simulated input impedance of the proposed wideband, planar, bow-tie antenna. Figure 7: Dimensions and geometry of the proposed wideband, planar, bow-tie antenna. an angle of 86.3\({}^{\circ}\) and the circular top loading has the same radius. The middle bow tie (bow-tie-130) has the largest radius, 42.8 mm, and an angle of 129.8\({}^{\circ}\). Finally, the shortest bow tie (bow-tie-166) has a radius of 32.9 mm and the largest angle, 165.5\({}^{\circ}\). Since the radius of bow-tie-86 is slightly decreased to 41.1 mm, compared to that of bow-tie-80 in Section 2, the lowest frequency in the \(-\)10 dB bandwidth is slightly shifted to the right, as shown in Figure 6. However, a bandwidth increase of 14.8% on the upper end, compared to that for the design in Figure 5b, is shown. This is because of the increased angle (166\({}^{\circ}\)) paired with the decreased radius (32.9 mm) of bow-tie-166, compared to bow-tie-160 in Section 2. Consequently, the superposition of these bow ties with varying flare angles and radii paired with circular top loading generates the oscillation of \(R_{in}\) at around 50 \(\Omega\), and \(X_{in}\) at around 0 \(\Omega\), as depicted in Figure 6. This leads to a wider impedance bandwidth of 80.2% (1.26-2.97 GHz), compared to that of the initial design shown in Figure 5a, valued at 57.9% (1.24-2.25 GHz). The proposed antenna is studied at four evenly spaced frequencies within the \(-\)10 dB bandwidth of the presented antenna--1.26 GHz (the lowest frequency), 1.84 GHz (one-third of the \(-\)10 dB bandwidth), 2.41 GHz (two-thirds of the \(-\)10 dB bandwidth), and 2.97 GHz (the highest frequency)--to investigate the contribution of each bow tie to the wider bandwidth, shown in Figure 8a--d, respectively. Some important features of the strong currents are indicated by the black circles and the black arrows next to them, to show the direction of the strong currents. These indicators are only shown on the right side of each antenna since the antenna is symmetrical across the z-axis. The blue and red circles represent the first and second nulls, respectively, where the currents traveling in different directions cancel each other out. The double-sided, blue arrows between the first null and the feed indicate the distance between them.