# Compact Broadband Antenna with Vicsek Fractal Slots for WLAN and WiMAX Applications
## 2 Antenna Geometry Design
### Design Evolution Procedure  
Figure 2 presents the evolutionary stages of the investigated antenna. The antenna evolution process starts with a rectangular microstrip antenna excited by a feed line. Thereafter, four triangular parts are cut in the four angles of the rectangular radiator in order to make an octagonal radiator. Subsequently, the octagonal patch is loaded with the first and the second iterations of Vicsek fractal slots to achieve the final geometry. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Parameters**} & **Dimensions** & \multirow{2}{*}{**Parameters**} & **Dimensions** \\  & **(mm)** & & **(mm)** \\ \hline \(W_{s}\) & 50 & \(W\) & 13 \\ \hline \(L_{s}\) & 50 & \(L\) & 10 \\ \hline \(W_{f}\) & 3.5 & \(a\) & 1.3 \\ \hline \(L_{f}\) & 15 & \(g\) & 2 \\ \hline \(L_{g}\) & 15 & \(d\) & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Dimensions of the suggested antenna. Figure 1: The suggested antenna structure with notations: (**a**) top view and (**b**) back view. The first step in the design of the suggested antenna is displayed in Figure 2a. A rectangular microstrip antenna of length \(L_{p}\) and width \(W_{p}\) was taken as a reference antenna (simple microstrip patch). The dimensions of the rectangular radiator were determined using the theory of transmission lines using the following Equations (1)-(4) [23]:  
The width of a microstrip antenna can be calculated by:  
\[W=\frac{c}{2f_{r}}\sqrt{\frac{2}{\varepsilon_{r}+1}} \tag{1}\]  
where \(c\) and \(\varepsilon_{r}\) are the speed of light (\(3\times 10^{8}\) m/s) and relative permittivity of the medium, respectively, and \(f_{r}\) represents the resonant frequency at which the antenna is designed. The effective dielectric constant of the substrate is yielded by [23]:  
\[\varepsilon_{reff}=\frac{\varepsilon_{r}+1}{2}+\frac{\varepsilon_{r}-1}{2} \times\left[1+12\frac{h}{W}\right]^{-\frac{1}{2}} \tag{2}\]  
where \(h\) is the substrate thickness. Finally, the length of the rectangular radiator is given by [24]:  
\[L=\frac{c}{2f_{r}\sqrt{\varepsilon_{eff}}}-2\Delta L \tag{3}\]  
In Equation (3), \(\Delta L\) corresponds to the increase in length of the antenna due to fringing effects and is given by [23]:  
\[\Delta L=0.412\times h\times\frac{\left(\varepsilon_{reff}+0.3\right)+\left( \frac{W}{h}+0.264\right)}{\left(\varepsilon_{reff}-0.258\right)+\left(\frac{ W}{h}+0.8\right)} \tag{4}\]  
The corresponding simulated S\({}_{11}\) of the different steps of the design process of the investigated antenna is depicted in Figure 3. The simple microstrip patch antenna generates  
Figure 2: Evolution process of the suggested antenna: (**a**) simple microstrip patch (STEP I), (**b**) truncated patch (STEP II), (**c**) slotted patch (STEP III), (**d**) suggested patch (STEP IV). one resonant frequency at 7.4 GHz, and it provides a bandwidth of 1.5 GHz. The truncated patch antenna structure is obtained by cutting four triangular shapes in the four angles of the radiating element to obtain an octagonal patch antenna as depicted in Figure 2b. It is noticed from Figure 4 that the truncated patch antenna provides a broadband of 3.91 GHz from 2.29 to 6.2 GHz with one resonant frequency at 3.8 GHz. In the third step, the first iteration of Vicsek fractal slots is incorporated in the radiating element (slotted patch) as presented in Figure 2c. The construction of Vicsek fractal slots is based on square rotated by \(\pi/4\) and decomposed into nine squares.