# Smart Antenna for Application in UAVs
## 3 P-ESPAR Antenna Study
### Equivalent Circuit  
The planar antenna, from the point of view of the excitation signal source, can be described by an equivalent circuit, for example, a parallel resistor, inductor and capacitor (RLC) circuit, as shown in Figure 6.  
To determine the value of \(R\), \(L_{c}\) e \(C\) is necessary to carry out the study of the equivalent circuit, i.e., it is necessary to study the value of the current (\(\overline{I}\)), represented through:  
\[\overline{I}=\frac{V}{R+j(\omega C-\frac{1}{\omega L_{c}})} \tag{1}\]  
To maximize the antenna current module, it is necessary for the imaginary component of the previous equation to be equal to zero:  
\[(\omega C-\frac{1}{\omega L_{c}})=0\Leftrightarrow\omega=\frac{1}{\sqrt{L_{c }C}} \tag{2}\]  
After analysis of the previous equation, it was noticed that the \(L_{c}\) and C were dependent on the resonant frequency of the antenna. In the case of this antenna, the variation of the radiation lobe is dependent on the capacitances of the varicap diodes. These diodes add a capacity to the \(2C_{T}\), as shown in Figure 7[16]. Before analysis to the system of equations, it is necessary to realize that \(C_{Teff}\
eq C_{T}\), that is, \(C_{Teff}\) corresponds to the value of the effective capacity of the diodes in the circuit. This value is directly dependent on the positioning of the diodes in the patch, as represented in the following equation and Figure 8, where variables \(O\) and \(L\) are represented [16]:  
\[C_{Teff}=C_{T}\cos^{2}(\pi\theta),\;\;with\;\;\;\;\theta=\frac{O}{L} \tag{3}\]  
Figure 6: Equivalent circuit of planar antenna [16]. Figure 7: Equivalent circuit of P-ESPAR antenna [16]. In case of the equivalent circuit of the antenna, with the integration of the diodes, it is necessary to change the equivalent capacity, since capacity \(C\) is added to the capacity of the diodes. The following equation represents this change:  
\[(\omega C-\frac{1}{\omega L_{c}})=0\Leftrightarrow\omega=\frac{1}{\sqrt{L_{c}(C+ 2C_{\text{Teff}})}} \tag{4}\]  
To determine \(L\) and \(C\), the CST Microwave studio simulator was used to obtain the resonance-frequency values of the antenna without and with diodes, \(f_{0}\) and \(f_{c}\), respectively (Table 5). The capacity of the diodes considered for simulation purposes was \(C_{T}=1\) pF. The following system of equations allows \(L_{c}\) and \(C\)[16]:  
\[\left\{ \begin{array}{l}2\pi f_{0}=\frac{1}{\sqrt{L_{c}C}}\\ 2\pi f_{c}=\frac{1}{\sqrt{L_{c}(C+2C_{\text{Teff}})}}\end{array} \right. \tag{5}\]  
After obtaining the results of \(L_{c}\) and \(C\) of the RLC circuit, it was necessary to calculate the value of resistor \(R\). Thus, analysis was made to the circuit, considering that the relation between the complex amplitude of the current in the resistor (\(I_{R}\)) and the complex amplitude of the total excitation current (\(I_{exc}\)), where (\(G=\frac{1}{R}\)). \[I_{R}=G_{R}V \tag{6}\]  
\[I_{exc}=G_{T}V \tag{7}\]  
\[\frac{I_{R}}{I_{exc}}=\frac{G_{R}V}{G_{T}V}\Leftrightarrow\frac{I_{R}}{I_{exc }}=\frac{G_{R}}{G_{R}+j(\omega C-\frac{1}{\omega L_{c}})} \tag{8}\]  
Initially, it was established that reflection coefficient \(\mid\)S11\(\mid\) cannot exceed \(-10\) dB. For this reason, the ratio in dB between the complex amplitude of the current in the resistance and the complex  
Figure 8: Representation of variables \(O\) and \(L\) in the patch. \begin{table}
\begin{tabular}{c c} \hline \hline \multicolumn{2}{c}{**Two Layers of FR-4 with One Layer of Air**} \\ \hline \(f_{0}\) [GHz] & 1.388 \\ \(f_{c}\) [GHz] & 1.383 \\ \(O\) [mm] & 31.000 \\ \(C_{\text{Teff}}\) [PF] & 0.065 \\ \(L_{c}\) [nH] & 2.668 \\ C [pF] & 4.932 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Results obtained for the calculation of L and C.  
amplitude of the total current cannot be greater than \(-10\) dB, as represented in the following equation.