Antenna arrays and two-element expression

An antenna array is an arrangement of two or more similar antennas (elements), placed with definite spacing and fed with definite current amplitude and phase, so that their fields add in wanted directions and cancel in others. Arrays give high directivity, narrow beams and beam steering that a single element cannot give.

Array of two isotropic point sources

Let two isotropic sources lie on the x-axis with spacing d, equal amplitude E₀, and the current of source 2 leading source 1 by phase α. Point P is in the far field at angle θ from the array axis. The midpoint O is the phase reference.

                    P (far field)
                   /
                  / θ
   1 ●-----------O-----------● 2 ---> axis
      <--- d/2 ---><--- d/2 --->
 extra path from 1 compared to 2 = d cosθ

Total phase difference between the two waves at P:

ψ = βd cosθ + α ,   β = 2π/λ

With O as reference, source 1 contributes a phase −ψ/2 and source 2 a phase +ψ/2:

E = E₀ e^(-jψ/2) + E₀ e^(jψ/2)
  = E₀ [2 cos(ψ/2)]
  = 2E₀ cos(ψ/2)

E = 2E₀ cos[(βd cosθ + α)/2], normalised E_n = cos(ψ/2).

Special cases

Main points:

• The pattern depends only on d/λ and α, not on E₀.
• Maximum occurs where ψ = 0, i.e. cosθₘ = −α/(βd).
• If the sources are not isotropic, the total pattern = element pattern × this array factor (pattern multiplication).