## 2 Antenna Design
### The Design of Unit Cell and Proposed Series-Fed Antenna
The chosen patch length \(L_{p}\) of 13.7 mm achieves an optimal trade-off, resulting in both a low reflection coefficient and a good axial ratio, signifying a well-optimized antenna for circular polarization. Figure 5a shows how increased truncation shifts the resonant frequency lower and alters the reflection coefficient. Figure 5b shows the axial ratio's sensitivity to \(a\), with the optimal circular polarization observed at \(a=1.39\) mm. The chosen truncated corner length \(a\) of 1.39 mm is an optimal trade-off for achieving robust resonance and high-quality circular polarization. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Parameter** & **Value** & **Parameter** & **Value** \\ \hline \(L_{sub}\) & 25 & \(L_{1}\) & 1.6 \\ \(W_{sub}\) & 40 & \(L_{2}\) & 2.45 \\ \(L_{p}\) & 13.7 & \(L_{3}\) & 1.6 \\ \(a\) & 1.39 & \(L_{4}\) & 5.65 \\ \(b\) & 0.57 & \(W_{1}\) & 1.09 \\ \(W_{TL}\) & 0.4 & & \\ \hline broadside & 0 & & broadside & 13.7 \\ \hline beam-tilted & 1.2 & & beam-tilted \\ \(\theta=15^{\circ}\) & & & \(\theta=15^{\circ}\) \\ \hline beam-tilted & 2.1 & & beam-tilted \\ \(\theta=30^{\circ}\) & & & \(\theta=30^{\circ}\) \\ \hline beam-tilted & 2.9 & & beam-tilted \\ \(\theta=45^{\circ}\) & & & \(\theta=45^{\circ}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Design parameters of the proposed antennas (unit: millimeters). Figure 4: Parameter study of the unit cell for varying \(L_{p}:\) (**a**) reflection coefficient, (**b**) AR. Figure 3: Simulated surface current distributions of the patch of the unit cell with period \(T\) at 5.84 GHz: (**a**) \(t=0\), (**b**) \(t=T/4\), (**c**) \(t=T/2\), and (**d**) \(t=3T/4\). Figure 2: Simulated results of the patch of unit cell: (**a**) reflection coefficient, (**b**) AR, (**c**) RHCP realized gain in \(\phi=90^{\circ}\) plane, and (**d**) RHCP realized gain in \(\phi=0^{\circ}\) plane. \[AF_{Linear}=\sum_{n=1}^{N}e^{i(n-1)(kd\cos\theta+\beta)} \tag{1}\]  
\[ArrayPattern=EP\times AF \tag{2}\]  
Equation (1) represents the array factor of a linear array (\(AF_{Linear}\)), where \(kdcos(\theta)+\beta\) signifies the phase difference of each element, with \(N\) being the element number of the linear array, \(k\) the wave number, \(d\) the distance between elements, \(\theta\) the angle with respect to the array axis, and \(\beta\) the progressive phase shift applied to the elements [30]. \(N\) is 12, and the distance between antenna elements is kept at 25 mm \((0.487\lambda_{0}\), where \(\lambda_{0}\) is the free space wavelength at 5.84 GHz) to prevent grating lobes and maintain a compact array. At 5.84 GHz, to achieve broadside radiation and beam tilting angles of 15\({}^{\circ}\), 30\({}^{\circ}\), and 45\({}^{\circ}\) with respect to the array axis, the \(\beta\) values are calculated in ascending order of beam tilting angles as 0\({}^{\circ}\), \(-45.35^{\circ}\), \(-87.61^{\circ}\), and \(-123.89^{\circ}\). Equation (2) represents the radiation pattern formula for an array antenna, where the radiation pattern of the array is calculated as the product of the Element Pattern (EP) of a single antenna and the array factor. Figure 6 shows the normalized radiation pattern of the proposed linear array series-fed array antenna, calculated by multiplying the array factor from Equation (1) with the element pattern from Figure 2c. As the phase difference from 0\({}^{\circ}\) increases, the beam tilting angle of the normalized radiation pattern also increases, reaching maximum magnitude at 15\({}^{\circ}\), 30\({}^{\circ}\), and 45\({}^{\circ}\). Figure 5: Parameter study of the unit cell for varying \(a\): (**a**) reflection coefficient, (**b**) AR. Figure 6: Calculated and normalized radiation pattern of the linear array antenna for different \(\beta\) at 5.84 GHz. Figure 7 shows simulated results for different \(W_{2}\) in the U-shaped bend transmission line. Figure 7a shows the phase of the transmission coefficient of the unit cell according to the change in \(W_{2}\).