# Uniform Linear Antenna Array Beamsteering Based on Phase-Locked Loops
## 2 Antenna Basics
### Uniform Linear Array with \(N\) Elements  
The form in Equation (7) can be seen as two different attributes. The left-hand side is related to the individual element and is called the _Element Factor_ and the right is related to the array and is called the _Array Factor_.  
\[Total\ Field=Element\ factor(E.F)\times Array\ factor(A.F). \tag{8}\]  
Figure 1: A 2-element linear phased antenna array.  
Hence, the total field E can be expressed as the product of field due to single element and array factor. This is termed as pattern multiplication. This method is applicable to any array consisting of identical elements. Therefore, the overall pattern can be controlled by controlling the \(E.F\) or \(A.F\) or both.  
\[A.F=2\cos\left(\frac{kd\cos\theta+\beta}{2}\right) \tag{9}\]  
\(A.F\) depends on the number of element (N), where N = 2, in the above Equation (9), distance between the elements \(d\), and the amplitude and phase excitations of the array elements \(\beta\). Having derived Equation (9) for the case of a two-element array, this can now be generalised to an N-element array. The derivation shows that the field strength in the far-field is given by the \(E.F\) times the \(A.F\) as from Equation (8). The \(A.F\) for an N-element ULA is derived as follows:  
\[A.F =1+e^{j(kd\cos\theta+\beta)}+e^{j(2 kd\cos\theta+\beta)}+\ldots\] \[\ldots+e^{j(N-1)(kd\cos\theta+\beta)}.\] \[=\sum_{n=1}^{N}e^{j(n-1)(kd\cos\theta+\beta)}=\sum_{n=1}^{N}e^{j(n -1)\psi} \tag{10}\]  
where \(\psi=kd\cos\theta+\beta\). This is sufficient for calculating the A.F. and is the basis of the MATLAB code (source code is available [6]) used to plot the 2-element array in Figure 2.  
However, a greater understanding of how the number of elements and progressive phase affect the A.F. can be gained by further simplifying Equation (10). Multiplying both sides by \(e^{j\psi}\) results in Equation (11) [2]:  
\[A.F=e^{j\left(\frac{N-1}{2}\right)\psi}\left[\frac{\sin\left(\frac{N\psi}{2} \right)}{\sin\left(\frac{\psi}{2}\right)}\right] \tag{11}\]  
For small values of \(\psi\), this can be approximated to  
\[A.F=\left[\frac{\sin\left(\frac{N\psi}{2}\right)}{\sin\left(\frac{\psi}{2} \right)}\right]\approxeq\left[\frac{\sin\left(\frac{N\psi}{2}\right)}{\left( \frac{\psi}{2}\right)}\right] \tag{12}\]  
The peaks and nulls can be found by equating the A.F to 0 in (12). \(\sin(N\psi/2)=0\) when \((N\pi d\cos\theta+\beta)/2\lambda=n\pi\). Peaks occur when \(n\
eq N,2N,3N,\ldots\) and nulls occur when \(n=1,2,3,\ldots\). [2]. The total number of nulls and peaks is dependent on the separation distance \(d\) and progressive phase \(\beta\). The role of \(\beta\) in beam steering is discussed in Section 2.3.  
When A.F. = 0, Equation (12) can be solved for \(\theta\) (Equation (13)) and used to calculate the positions of peaks and nulls in the radiation pattern.  
\[\theta_{n}=\arccos\left[\frac{\lambda}{2\pi d}\left(-\beta\pm\frac{2n\pi}{N} \right)\right] \tag{13}\]  
Depending on the direction of maximum radiation, we can classify the array into a broadside array (radiation maximum is perpendicular to the array length) and an end-fire array (radiation maximum is along the array length) [2].