# Smart Antenna for Application in UAVs
## 3 P-ESPAR Antenna Study
### Equivalent Circuit
This equation is solved to obtain resistance value (Table 6):  
\[-10=20\log\left|\frac{1}{1+j\left(\frac{\omega_{c}^{\mathrm{C}}}{G}-\frac{1}{ \omega_{c}^{\mathrm{L}}G}\right)}\right| \tag{9}\]  
To study the coupling, it is necessary to consider the dielectric loss quality factor (\(Q_{c}\)), antenna patch quality factor (\(Q_{d}\)), metal loss quality factor (\(Q_{o}\)), and the quality factor of antenna radiation (\(Q_{nd}\)), represented by the following equations. Variables a, b, W, and \(L\) are represented in Figure 9[17; 18]. \[Q_{c}=h\sqrt{\pi f\mu\sigma}\] \[\mu-Vacuum\ permeability\to 4\pi*10^{-7}\] \[\sigma-Electrical\ conductivity\ of\ copper\to 5.81*10^{7}\] \[Q_{d}=\frac{1}{\tan\delta} \tag{10}\] \[Q_{0}=\frac{Q_{c}Q_{d}}{Q_{c}+Q_{d}}\] (11) \[Q_{rad}=\frac{\ell\pi^{3}\epsilon_{0}}{32h}\frac{WL}{\sin\left( \frac{2h}{\alpha}\right)}\frac{bZ_{w}}{a}\] \[\epsilon_{0}-Permittivity\ of\ vacuum\to 8.85418782*10^{-12}\] (12) \[Z_{w}-Impedance\to 50\Omega\]  
Analyzing the quality factor of the dielectric, it is perceptible that it depends on the loss tangent (\(\tan\delta^{\prime}\)) of the same. In the case of the antenna with the FR-4 substrate, the medium was not homogeneous, that is, there were three media with different values of \(\tan\delta^{\prime}\). As such, it was necessary to calculate the value of the equivalent loss tangent (\(\tan\delta^{\prime}\)). To perform the study of the tangent of losses, it was necessary to establish the equivalent circuit for the substrate, as shown in \[-10=20\log\left|\frac{1}{1+j\left(\frac{\omega_{c}^{\mathrm{C}}}{G}-\frac{1}{ \omega_{c}^{\mathrm{L}}G}\right)}\right| \tag{9}\]0 Each substrate corresponded to a parallel RC circuit, whereby the three underlying substrates corresponded to the series of these circuits as represented. For calculating \(\tan\delta^{\prime}\), it is necessary to consider the \(W\), \(L\), \(h\), and \(a\) of each substrate. \begin{table}
\begin{tabular}{l c} \hline \hline  & **Two Layers of FR-4 with One Layer of Air** \\ \hline \(R\) [k\(\Omega\)] & 2.978 \\ \(L_{c}\) [nH] & 2.668 \\ C [pF] & 4.932 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Values obtained from the RLC equivalent circuit. Figure 9: Variation of the magnetic field in the substrate for the fundamental mode [17]. The antenna resembles a three-component system, whereby the \(\tan\delta^{\prime}\) for these circuits is given by the following equation [19; 20]:  
\[\tan\delta^{\prime}=\frac{C_{1}C_{2}\tan\delta_{3}+C_{1}C_{3}\tan\delta_{2}+C_{2 }C_{3}\tan\delta_{1}}{C_{1}C_{2}+C_{2}C_{3}+C_{1}C_{3}}\ \,\ \ \ with\ \ \ C=\varepsilon_{0} \varepsilon_{r}\frac{WL}{h} \tag{13}\]  
After calculating \(\tan\delta^{\prime}\) (Table 7), all conditions are met to calculate the quality factors and evaluate the antenna coupling. Thus, the coupling conditions are as follows [17]: \(Q_{rad}=Q_{0}\), critical coupling; \(Q_{rad}>Q_{0}\), undercoupled; and \(Q_{rad}<Q_{0}\), overcoupled (Figure 11). The used diodes were the varicap diodes of Infineon BBY53-03W. These have been selected because they were closest to the antenna capacity. The characteristics of the varicap diodes are shown in Table 8. \begin{table}
\begin{tabular}{c c c c} \hline \hline \(\tan\delta^{\prime}\) & 0.014 & \(L_{1}=L_{2}=L_{3}\) & 274.900 mm \\ \(\tan\delta_{2}\) & 0.000 & \(h_{1}=h_{3}\) & 1.575 mm \\ \(\tan\delta_{3}\) & 0.014 & \(h_{2}\) & 1.090 mm \\
4.700 & \(C_{1}=C_{3}\) & 0.725 nF & \\
1.00059 & \(C_{2}\) & 0.220 nF & \\ \(W_{1}=W_{2}=W_{3}\) & 99.820 mm & \(\tan\delta r\) & 0.00528755 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Parameters and results. Figure 11: Quality factor as a function of substrate height. Figure 10: Representation of the equivalent circuit of antenna substrates with FR-4 [18]. After analyzing the results of Table 9, it was possible to conclude that the imposed requirements were all fulfilled and that all the conditions for the construction and tests were gathered.