# Exploring Carbon-Fiber UAV Structures as Communication Antennas for Adaptive Relay Applications
## 2 Materials and Methods
### Impedance Matching  
The impedance matching procedure, whereby the antennas are tuned to match the impedance of the transmission line, thereby minimizing reflections, is shown in the following. This optimization enhances the antenna's efficiency, resulting in both an extended operational range and increased longevity. Essentially, the process involves shifting the operating point to the center of the Smith Chart, Figure 5, which corresponds to a pure resistance of 50 \(\Omega\). Impedance matching may be achieved by incorporating a segment of cable in series and/or an open or short-circuited parallel stub (i.e., a segment of cable assumed to be lossless) into the circuitry [19; 20; 21]. Alternatively, matching can also be realized by using lumped elements such as capacitors, inductors, and resistors [22]. For the purposes of this report, the only available option is to implement a parallel open or shorted stub. The complex reflection coefficient extracted from the VNA at our central frequency (1.2 GHz) is given by Equations (4) and (5):  
\[\text{Antenna A}:\ z_{L}=\frac{33.1-11.86j}{50}=0.662-0.2372j \tag{4}\]  
\[\text{Antenna B}:\ z_{L}=\frac{35.7-10.12j}{50}=0.714-0.2024j \tag{5}\]  
These values are plotted on the Smith Chart (see Figure 5). We then proceed to design a matching network by adding a series cable segment along with a parallel open st  
Figure 5: Impedance matching on the Smith Chart. The colored markers on the Smith Chart correspond to distinct stages of the impedance-matching procedure, specifically, the dark green marker denotes the antenna's initial normalized impedance; the light green marker indicates the impedance locus after insertion of the series transmission-line section; and the yellow marker represents the locus following implementation of the parallel open-circuit stub. Initially, a cable series was introduced. The antenna's starting point (depicted in dark green) moves to a second point on the Smith Chart (depicted in light green) in a clockwise direction. The length of the cable is calculated as follows:  
\[2\theta_{\mathrm{s}}=360^{\circ}-180^{\circ}-109^{\circ}=113^{\circ} \tag{6}\]  
\[\theta=\frac{l}{\lambda}360^{\circ} \tag{7}\]  
Thus, the required cable length is:  
\[l_{s}=x\lambda\frac{\theta_{s}}{360^{\circ}} \tag{8}\]  
where \(x\) is the velocity factor, indicating the speed at which the signal propagates through the cable. In this example, an RG58C/U cable is used, featuring a velocity factor of 66%. Taking these considerations into account, the calculated length of the series cable is \(l_{s}\) = 0.0259 m.  
The next step in the impedance matching procedure is to add an open stub in parallel. This corresponds to moving from the light green point to the center (black point) of the Smith Chart. The length of the parallel stub is determined by:  
\[b_{p}=0.7\;and\;\theta_{p}=tan^{-1}\big{(}b_{p}\big{)}=34.99^{\circ} \tag{9}\]  
\[l_{p}=x\lambda\frac{\theta_{s}}{360^{\circ}}=0.0161\mathrm{m}=1.61\;\mathrm{cm} \tag{10}\]  
By applying the same reasoning for Antenna B, we obtain \(l_{s}\) = 0.0268 m and \(l_{p}\) = 0.0142 m.  
In these matching-network formulas, both \(\theta_{s}\) and \(\theta_{p}\) are electrical lengths--that is, the phase shifts (in degrees) that a wave experiences as it travels along those pieces of transmission line \(\theta_{s}\) corrects the real part of the load, while \(\theta_{p}\) provides the exact reactance needed to cancel out any remaining imaginary component. The parameter \(b_{p}\) represents the normalized susceptance of the parallel open stub--essentially a measure of how much the stub contributes to the imaginary part of the admittance. It is used to determine how long the stub must be (in electrical degrees) to provide the necessary reactive compensation, and its value is translated into the angle \(\theta_{p}\) using the arctangent function highlighted in Equation (9). Figure 6: Zoomed view of impedance matching on the Smith Chart.