Features of end-fire array

An end fire array is a linear array whose maximum radiation is along the array axis. It is obtained by feeding equal-amplitude elements with a progressive phase lag equal to their spacing in radians (α = −βd), so the fields add only in the direction of the array line.

  ● → ● → ● → ● → ●  ====> main beam
  phase: 0  -βd -2βd -3βd -4βd

Features

1. Direction of maximum: θ = 0° (or 180°) along the axis; ψ = βd(cosθ − 1) = 0 at θ = 0°.
2. Phase condition: α = −βd; for d = λ/4, α = −90° between neighbours.
3. Pattern: usually unidirectional (e.g. two elements at λ/4 give a cardioid); a pencil-shaped beam for many elements.
4. Beamwidth: null-to-null beamwidth ≈ 2√(2λ/(nd)) = 2√(2λ/L), wider in angle than broadside for the same length.
5. Directivity: D ≈ 4L/λ (= 4nd/λ), about twice that of a broadside array of the same length.
6. Hansen–Woodyard condition: extra phase δ = π/n (α = −(βd + π/n)) narrows the beam and raises directivity to about 7.2L/λ (1.8 times ordinary end fire).
7. Spacing: kept small (≈ λ/4, < λ/2) to avoid a back lobe and grating lobes.
8. Applications: Yagi-Uda TV antennas, axial-mode helix, VHF/UHF point-to-point links.