# Quad-Band Uniformly Spaced Array Antenna Using Diverse Patch and Fractal Antennas
## 2 Materials and Methods
### Patch and Fractal Element Antennas  
The quad-band array antenna embodies four types of microstrip patch element antennas, which resonate at different frequency bands. Figure 3 depicts a set of different kinds of patch antennas for the element antennas. The approach to designing the element antennas is to use shape variants of patch antennas and Sierpinski fractal antenna designs. The dimensions of the element antennas are \(w_{s}\) = 9.8 mm, \(l_{s}\) = 20.2 mm, \(w_{b1}\) = 18 mm, \(w_{b2}\) = 20.5 mm, \(w_{b}\) = 9 mm, \(l_{s1}\) = 5.1 mm, \(l_{s2}\) = 2.5 mm, \(l_{s3}\) = 1.3 mm, \(l_{s4}\) = 4.8 mm, \(l_{s5}\) = 2.4 mm, and \(w_{b1}\) = 12 mm. It has been reported that a large variance of resonant frequencies can be obtained by altering the shapes of microstrip patch element antennas [17]. The length \(l\) of the square patch antenna with width \(w\) can be defined as [26]  
\[l\approx\frac{c}{2f_{0}\sqrt{\epsilon_{eff}}}-2(0.412t)\frac{(\epsilon_{r}+0.3 )(w/t+0.264)}{(\epsilon_{r}-0.258)(w/t+0.8)} \tag{1}\]where \(c\) is the speed of light in a vacuum, \(t\) is the thickness of the substrate, and \(\varepsilon_{eff}\) is the effective dielectric constant, which is 1.75. The side length \(s\) of the Sierpinski gasket antenna is determined from resonant frequencies [23] at  
\[s\approx\left\{ \begin{array}{c}\frac{1}{\sqrt{3}}\big{[}0.3069+0.68\rho\cdot x \big{]}\frac{c}{f_{0}}\big{(}\zeta-1\big{)}^{n}-\frac{t}{\sqrt{\varepsilon_{r} }}\ \ \text{for}\ n=0\\ \frac{0.52}{\sqrt{3}}\frac{c}{f_{0}}\big{(}\zeta-1\big{)}^{n}-\frac{t}{\sqrt {\varepsilon_{r}}}\ \ \text{for}\ n>0\end{array} \right. \tag{2}\]  
where \(\left\{ \begin{array}{c}x=0\ \text{for}\ k=1\\ x=1\ \text{for}\ k>1\end{array} \right.\) and \(\rho=\zeta-0.23075\), where \(h\) is the height of the antenna, \(k\) is the iteration number, the band number \(n\) is zero, and \(\zeta\) is the scale factor of the classical Sierpinski gasket, which is typically 0.5. The results of this study are obtained using the full-wave simulation tool HFSS. The graphs of the reflection coefficients are provided in Figure 4. The reflection coefficients obtained using High Frequency Structure Simulator (HFSS) are provided in Figure 4. The center frequencies of the bow tie, the fat bow tie and the square patch antennas are simulated as 2.8 GHz, 4.4 GHz, and 5.1 GHz, respectively. The ratio of different resonant frequencies between bow tie and square patch antennas can be derived as \(f_{1}/f_{2}=0.549\). The resonant frequency of the fat bow tie antenna is located between these, and can be tuned by changing a parameter of \(w_{b}\). It is worth noting that a larger width of \(w_{s}\), compared to the one for \(w_{b}\), is used for bow tie antennas to tune the resonant frequencies. The ratio of resonant frequencies between bow tie fractal and square patch antennas increases to \(f_{1}/f_{2}=0.486\). Table 1 summarizes the types of element antennas, resonant frequencies, and relevant wireless communication services. Broader options for element antennas are presented, including patch elements with Sierpinski-gasket fractal antennas. The requirement for multiple frequency bands is met using bow tie and fat bow tie antennas with Sierpinski gaskets. Two different iterations of 2 and 3 are used for bow tie fractal antennas, which provide resonant frequencies at 2.6 GHz and 2.47 GHz. To the best knowledge of the authors, a fat bow tie Sierpinski fractal antenna is presented here for the first time, and further miniaturization of around 15% is achieved with the proposed antenna, compared to the typical fat bow tie antenna. The four designs that resonate at the center frequencies in accordance with the wireless communication services are selected, namely square, fat bow tie, fat bow tie fractal, and bow tie fractal patch antennas. Figure 1: Schematic description of quad-band array antenna: (**a**) top view of radiating elements, including feed network on bottom layer; (**b**) exploded view of the antenna.