# A Low-Cost CPW-Fed Multiband Frequency Reconfigurable Antenna for Wireless Applications
## 2 Step by Step Design Guidelines and Approach  
### Antenna Geometry  
The corresponding geometrical dimensions of the proposed multiband reconfigurable antenna is displayed in Figure 1. This antenna utilizes a low-cost FR4-Epoxy substrate with loss tangent \(\delta\) = 0.02, dielectric constant \(\varepsilon_{r}\) = 4.4, and thickness of 1.6 mm. The overall dimensions of the antenna are \(40\times 60\) mm\({}^{2}\). The antenna design modeling is carried out in a commercially available FEM (Finite element method)-based simulator, ANSYS HFSS. Moreover, we have also validated our results on time domain-based simulator CST (FDTD) before fabrication and measurement.  
To achieve improved impedance matching and acceptable gain, the shape of the main radiating patch resembles the cedar tree and the rectangular slots are placed over an equilateral triangle. The antenna is designed in such a way to resonate on different frequency bands, using four PIN diodes, named 2, 3, 4, and 5, with a lumped element boundary in a reserved slot of 1 mm. The width of the CPW feedline is 1 mm to excite the antenna with the characteristic impedance of 50 \(\Omega\). This value of CPW feedline is evaluated by using Equations (1)-(6) [22].  
\[k^{\prime}=\sqrt{1-k^{2}} \tag{1}\]  
\[k_{1}=\frac{S_{c}}{S_{c}+2W} \tag{2}\]  
\[k_{2}=\frac{\sinh\frac{\pi a}{2h}}{\sinh\frac{\pi b}{2h}} \tag{3}\]  
\[\frac{K(k)}{K^{\prime}(k)}=\left\{\begin{array}{l}\frac{\pi}{\ln\left[\frac{ 2\left(1+\sqrt{k^{\prime}}\right)}{1-\sqrt{k^{\prime}}}\right]}\\ \frac{\ln\left[\frac{2\left(1+\sqrt{k^{\prime}}\right)}{1-\sqrt{k^{\prime}}} \right]}{\pi}\\ \frac{\ln\left[\frac{2\left(1+\sqrt{k^{\prime}}\right)}{1-\sqrt{k^{\prime}}} \right]}{\pi}\\ \end{array}\right.\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\The parameters used in these equations are described as follows: \(S_{c}\) is the width of the central conductor, \(Zo_{cp}\) is the characteristic impedance of the CPW line, \(W\) is the separation between the central conductor and ground plane, \(\varepsilon_{r}\) is the relative dielectric constant of the substrate, \(\varepsilon_{\textit{eff}}\) is an effective dielectric constant, \(h\) is the thickness of the substrate, \(K_{1}\), and \(K_{2}\) can be calculated by utilizing \(S_{c}\), \(W\), \(h\), \(a\), and \(b\) as defined before. \(K\)(\(k\)) is defined as the integral of first order with argument \(k\) or complementary argument \(k^{\prime}\).  
Table 1 displays the optimized overall dimensions of the antenna. Transmission line theory is used to calculate the effective length for a specific resonating frequency using Equation (7) taken from [23].  
\[L_{r}=\frac{c}{4f_{r}\sqrt{\frac{\varepsilon_{r}+1}{2}+\frac{\varepsilon_{r} -1}{2}\Big{(}1+\frac{12h}{W}\Big{)}^{-0.5}}} \tag{7}\]  
where \(h\) is the thickness of the substrate, \(c\) is the speed of light, \(L_{r}\) is the length of the resonant frequency, and \(\varepsilon_{r}\) is the relative permittivity of the substrate.