Array Factor and Array Pattern of a Phased Antenna Array

Antenna arrays are an important class of antennas which are widely used in point to point communication systems, where a very high directive beam of radiation is needed. In The Finite Dipole Antenna we saw that a dipole antenna has an omnidirectional pattern which has a very low directivity. In this problem we will see how to arrange several dipole antennas into an array to obtain a directive radiation pattern.

Problem Statement

Assume an array of antennas which consists of $$N$$ half-wavelength dipoles working at $$300\, \mathrm{MHz}$$. The dipoles are located with exactly the same orientation at positions which are given by $\mathbf{r}_n=n\, d\,\mathbf{e}_z\,, \quad n=\{0,1,2,...,N-1\}$ where $$d$$ is the distance between two adjacent dipoles. In addition, each dipole has a feed current which is given by $\tilde{I}_n=I_n e^{-\mathrm{i}n\phi_0}\,, \quad n=\{0,1,2,...,N-1\}$ where $$\phi_0$$ is a fixed parameter which is called the progressive phase.

1. Find the array factor for an array with 6 elements (N=6) with spacing of $$\lambda/2$$ ($$d=\lambda/2$$). The elements have uniform amplitude with progressive phase of zero (i.e. $$\tilde{I}_n=I_0$$).
2. Sketch the array pattern in y-z plane.
3. Calculate the first-null beamwidth (FNBW) and side lobe level (SLL) of the array pattern.
4. Sketch the overall pattern of the array while all dipoles are oriented in x-direction.