## 2 Antenna Design
### Single Antenna Design Evolution Steps  
Figure 2a depicts a fractal structure used to design a sole antenna element. A spherical patch of radius A is utilized as the basis structure in step 1 (0 th iteration), and its dimensions are determined using Equation (1). \[a_{e}=\frac{1.8412\times\mathrm{V_{o}}}{2\times 3.14\times\mathrm{f_{r}}\times \sqrt{\varepsilon_{r}}} \tag{1}\]  
where \(\mathrm{V_{o}}\) is the free space speed of light; \(\varepsilon_{r}\) is the dielectric constant of material. Step 2 involves scaling the base structure by a factor of one-third (first iteration), resulting in 5 small circles with radius A1. One central circle is then taken out of the 5 small circles, leaving 4 circles in total. Iteration 2 (step 2) is achieved by repeating the previous iteration's steps on the remaining 4 circles, which entails dividing the 4 circles once more into 5 small circles of radius (A2), removing one center circle from each of the 5 small  
Figure 2: (**a**) Basic element design evolution process. (**b**) Scattering parameters of single antenna. circles, and leaving 16 circles in total. In the case of step 3 (iteration 3), the remaining 16 circles are then further divided into 5 smaller circles of radius (A3), and once again the one center circle is eliminated from each of the 5 smaller circles, leaving a total of 64 circles. Table 1 shows the method used to calculate the radiuses A1, A2, and A3. Due to fabrication limitations and the intricate structure, additional iterations are not possible. Once the patch structure has been created, then, using Equation (2), the size of the microstrip feedline is determined. The suggested single antenna element's dimensions are listed in Table 2.  
\[\text{F}_{\text{w}}\text{=}\frac{7.48\text{H}}{\text{e}^{(\text{Z}_{0}\times \frac{\sqrt{(\text{r}_{\text{r}}+14)}}{87})}}-1.25\text{t} \tag{2}\]  
The S-parameters are analyzed in order to understand more about the antenna's performance. Figure 2b displays the single-antenna iteration stages and scattering parameter results and Table 3 displays the findings of various iteration studies regarding the effect of their findings. The graphic makes it abundantly evident that an increase in the number of iterations will result in an increase in both the bandwidth and the bands. For the 0 th iteration, the proposed antenna employs a patch with a circular form as depicted in step 1 of Figure 2a. The circular geometry of the antenna facilitates the presence of numerous resonant modes, each associated with a distinct frequency band. In the context of the initial iteration, the antenna demonstrates resonance at three distinct frequencies, specifically 9 GHz, 18.8 GHz, and 26.4 GHz, as shown in Table 3. Furthermore, when a single slot (1 st iteration) is applied to the patch, as depicted in step 2 of Figure 2a, the antenna exhibits resonance at three distinct multiband frequencies: 9.3 GHz, 21.7 GHz, and 26.2 GHz. The resonance behavior of the antenna changes due to slot loading on the surface of the radiating patch. This may be due to the possible changes in surface current distributions, as a result of which, the electric and magnetic field distribution alters leading to changes in the resonant behavior of the antenna with the generation of additional resonances. The additional resonances in conjunction with main resonance of the initial patch results in a broader bandwidth in comparison to the initial iteration. For the 2 nd iteration, the resonance occurs at four different frequencies below a level of \(-10\) dB S11. As a result, the iteration is carried out for the next step (iteration 3), which results in an antenna that resonates for five frequencies while increasing bandwidth. The conclusion that can be drawn from this is that the antenna configuration for the third iteration gives balanced outcomes in the form of resonating frequencies and bandwidth.