Pattern multiplication, quadrature-phase sources

Pattern multiplication

Pattern multiplication states that the field pattern of an array of identical, similarly oriented non-isotropic elements is the product of the pattern of one element and the pattern of the array made of isotropic point sources (array factor). Total phase = element phase + array phase.

E(θ) = E_element(θ) × E_array(θ)

Two isotropic sources, equal amplitude, quadrature phase, d = λ/4

Let source 1 and source 2 lie on the x-axis, spacing d, with the current of source 2 lagging source 1 by 90° (α = −π/2). θ is measured from the axis, midpoint O as reference.

             P (far field)
            /
           / θ
  1 ●-----O-----● 2  -> axis (θ = 0°)
     <- d/2 -><- d/2->
   path difference = d cosθ

Total phase difference at P:

ψ = βd cosθ + α
βd = (2π/λ)(λ/4) = π/2 ,  α = -π/2
ψ = (π/2) cosθ - π/2 = (π/2)(cosθ - 1)

Adding the two fields:

E = E₀ e^(-jψ/2) + E₀ e^(jψ/2) = 2E₀ cos(ψ/2)
E = 2E₀ cos[(π/4)(cosθ - 1)]
Normalised: E_n = cos[(π/4)(cosθ - 1)]

Pattern values

Maximum at θ = 0° (towards the lagging source, along the axis); null at θ = 180°. The pattern is a cardioid (unidirectional end fire).

           90° (0.707)
          .--''--.
        .'        '.
 180° 0 1   2       > 0° (max = 1)
        '.        .'
          '--..--'
           270°

Physical reason: towards θ = 0°, the wave from source 1 travels λ/4 (90°) to reach source 2, which exactly cancels source 2's 90° lag, so the fields add. Towards θ = 180°, the travel delay and current lag add to 180°, so the fields cancel. If source 2 leads instead (α = +π/2), the cardioid points to θ = 180°.