# Uniform Linear Antenna Array Beamsteering Based on Phase-Locked Loops
## 2 Antenna Basics
### Array Theory  
There are six design parameters in an antenna array. They are the layout/geometrical configuration, and number of elements, spacing of elements, excitation amplitude, excitation phase, and element type/radiation pattern. In this paper, we are only interested in the number of elements and the relative phase of each element. The other parameters remain fixed as a linear array with each element spaced a distance of \((\lambda/2)\) apart and every element being a balun-fed dipole [1], all with the same excitation amplitude. To understand the theory of arrays, we first start with the special case of a two-element array with equal amplitude excitation [2]. The electric field at a point in the far-field from the two elements is the sum of the electric fields of each individual element. If we assume that there is negligible coupling between the elements, then we can consider each element independently (Figure 1). For two identical dipole elements separated by distance d as shown in Figure 1, their resultant electric fields at P is derived as shown from Equations (1)-(7).  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameter** & **Desired Range** & **Value Single Element** \\ \hline Frequency (GHz) & 2.4 & 2.4 \\ \hline Return loss (dB) & 10 & 10 \\ \hline Gain (dB) & 10 & 2.1 \\ \hline
3 dB beamwidth (\({}^{\circ}\)) & 21 & Omni-directional \\ \hline
10 dB bandwidth (MHz) & 100 & 100 \\ \hline \hline \end{tabular}
\end{table}
Table 1: Design parameters of ULA at 2.4 GHz.  
Here, \(A_{1}\), \(A_{2}\) are the normalised field magnitudes at distances \(r_{1},r_{2}\) and polarization vectors \(\rho_{1},\rho_{2}\), respectively. \(k\) is the phase constant given by \(2\pi/\lambda\), \(\beta\) is the phase difference between the two elements. \(\theta_{1},\phi_{1}\), and \(\theta_{2},\phi_{2}\) are the positions in azimuth and elevation, respectively, with reference to the array centre. Both elements are identical with the same excitation amplitude; therefore, \(A_{1}=A_{2}\), they are both physically aligned, so the same polarization, therefore \(\rho_{1}=\rho_{2}\). If we make the assumption that point P from Figure 1 is sufficiently far away, then we can set \(\theta_{1}=\theta_{1}=\theta\) and \(\phi_{1}=\phi_{2}=\phi\). This means that both \(E_{1}\) and \(E_{2}\) can be written as below in Equation (3):  
\[E_{1}=E_{2}=\frac{A_{1}(\theta_{1},\phi_{1})}{r_{1}}e^{-j(kr_{1}-\frac{\theta} {2})}\rho \tag{3}\]  
As mentioned, the resultant field is the summation of the fields from each element. The final simplified resultant far-field is given by Equation (7).  
\[E=E_{1}+E_{2} \tag{4}\]  
\[E=\rho\frac{A_{1}(\theta,\phi)}{r}\left[e^{-j(k(r-\frac{\theta}{2}\cos\theta) -\frac{\theta}{2})}+e^{-j(k(r+\frac{\theta}{2}\cos\theta)+\frac{\theta}{2})}\right] \tag{5}\]  
This can be simplified by moving the common factor of \(e^{-jkr}\),  
\[E=\rho\frac{A_{1}(\theta,\phi)}{r}e^{-jkr}\left[e^{j(\frac{jk}{2}\cos\theta+ \frac{\theta}{2})}+e^{-j(k\frac{\theta}{2}\cos\theta+\frac{\theta}{2})}\right] \tag{6}\]  
and further simplified by using [PERSON]'s formula  
\[=\rho\frac{A_{1}(\theta,\phi)}{r}e^{-jkr}\left[2\cos\left(\frac{kd\cos\theta+ \beta}{2}\right)\right] \tag{7}\]