## 3 Evolution  
Although the final structural design is achieved after numerous iterations through an optimization process, for clarity, we have summarized the evolution process in three simple steps. To achieve the final design, variations in the size and shape of the structural components are made at each step. The step-wise summary of the activity is given below.  
Step 1: Initially, as depicted in Figure 3a, a simple structure is created with a circular patch, which is fed by a feedline. The rectangular-shaped ground plate is aligned at the level just below the circular disk of the feed. The structure generates wideband characteristics; however, the S\({}_{11}\) is poor, and the bandwidth is much larger than we require.  
Initially, the radius 'a' of the circular patch, which can directly impact the resonant frequency, can be calculated using the formula below [29].  
\[a=\frac{\mathrm{F}}{\sqrt{1+\frac{2\mathrm{h}}{\pi.\varepsilon_{r}.F}\left[ln \left(\frac{\mathrm{F}\pi}{2\mathrm{h}}\right)+1.7726\right]}}\]  
where \(\mathrm{F}=\frac{8.791\times 10^{6}}{f_{r}\sqrt{\varepsilon_{r}}}\), h is the substrate height and \(\varepsilon_{r}\) is the substrate's dielectric constant.  
Due to fringing effects, the dimensions of the circular patch are modified, and the effective radius (a\({}_{\mathrm{eff}}\)) of the circular patch can be calculated using the equation below.  
\[a_{eff}=a\sqrt{1+\mathrm{q}},\ \mathrm{where}\ q=\frac{2\mathrm{h}}{\pi. \varepsilon_{r}.a}\left[ln\left(\frac{\mathrm{a.\tau}}{2\mathrm{h}}\right)+1. 7726\right]\]  
Figure 2: Reflection coefficient (S\({}_{11}\)) vs. frequency for single-element antenna.  
Step 2: In this step, two alterations are made. Firstly, we place two protruding stubs on the extreme edges of the ground plane. This helps in increasing the effective length of the radiating surface, and it also reduces the overall size of the design. Secondly, in the feed-plane, two slits are etched out on both sides of the feedline, as displayed in Figure 3b. With these changes, the antenna starts resonating at the desired frequency.  
Step 3: To further improve upon the return loss, a half-disk-shaped parasitic structure is placed on top of the circular patch, as shown in Figure 3c.  
A parasitic element is a conductive element that acts like a passive radiator and that can be used to improve the antenna's parameters, such as the bandwidth and the reflection coefficients. In ref. [30], an improvement in the bandwidth and the stable radiation pattern is suggested by the use of a parasitic element. Also, ref. [31] suggests an improvement in both the bandwidth and impedance matching. In the proposed structure, the addition of a parasitic patch has resulted in a better reflection coefficient, which is depicted in Figure 3c. Moreover, the introduction of a parasitic structure forms guided E and H fields between the radiating and the parasitic patch due to the creation of a coupling effect in the gap, as illustrated in Figure 4. This leads to an enhancement of characteristics of the design, such as the reflection coefficients (due to better impedance matching). The guided E and H fields in the coupling gap can be represented by an equivalent circuit where L and C are placed in a parallel combination [32; 33]. The distribution of the E field between the radiating circular patch and the parasitic structure is represented in Figure 5.  
Figure 3: Step-wise antenna design.