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Antennas · Elevation pattern & takeoff angle

Antenna pattern over ground

How height and ground shape the vertical pattern of a dipole or a vertical: the takeoff angle of the main lobe, its −3 dB elevation range and the nulls, for up to three heights at once. A far-field reflection model, not NEC.

What this model leaves out

This is a far-field reflection model: the direct wave plus one wave reflected from flat, uniform ground, using the plane-wave Fresnel coefficients. It is not NEC. It ignores:

  • The ground (surface) wave. Verticals over real ground show a null right at 0°; in reality a ground wave exists there, but it dies off quickly with distance at HF.
  • Near-field ground loss under the antenna. That sets a vertical's efficiency (how good the radials are) and a low dipole's loss. Patterns here are shapes, in dB relative to each trace's own peak, not gain.
  • Coupling between antenna and ground at low heights. The current is assumed to stay sinusoidal. Below about 0.1 λ this is poor, so heights under 0.1 λ are flagged and under 0.02 λ are refused.
  • Terrain, nearby objects, feed-line radiation and wire thickness.

For exact work use a NEC-based program: EZNEC, 4nec2 or MMANA-GAL. For low antennas over real ground, check that the program's ground model covers the ground under the antenna, not only the far-field reflection.

How it works

The ground is described by its complex relative permittivity ε = εr − jσ/(2πfε0) ≈ εr − j60σλ. At elevation ψ the reflected wave is the incident wave times
Γh = (sinψ − √(ε − cos²ψ)) / (sinψ + √(ε − cos²ψ)) for horizontal polarisation and
Γv = (ε sinψ − √(ε − cos²ψ)) / (ε sinψ + √(ε − cos²ψ)) for vertical polarisation.

A perfect conductor gives Γh = −1 and Γv = +1, the classic image antenna. Real ground gives Γv = −1 at the horizon and a minimum near the pseudo-Brewster angle: that's why verticals over poor ground lose their lowest angles and over sea water keep them. A horizontal dipole at height h over perfect ground has a broadside pattern ∝ |sin(2πh/λ · sinψ)|, so its first lobe peaks where sinψ = λ/(4h): 30° at λ/2, straight up at λ/4.

The pattern is the λ/2 dipole's free-space pattern cos(π/2·cosθ)/sinθ times the ground factor, worked out separately for the horizontally and vertically polarised parts of the field. The ground-mounted λ/4 vertical is integrated along its sinusoidal current. Plots are in dB relative to each trace's own maximum (0 dB = the marked dot), on a linear dB scale down to −30 dB.

  • Reflection coefficients: ITU-R Recommendation P.2040-4 (09/2025), §2.2.1.1, eq. 31a (TE) and 31b (TM), with medium 1 = air and incidence angle 90° − ψ.
  • Complex permittivity: ITU-R P.527-6 (09/2021) eq. 1b and 3a (ε″ = σ/(2πfε0) = 17.98σ/fGHz); P.2040-4 eq. 9–12.
  • Ground constants: ITU-R P.527-6, Attachment to Annex 1, Fig. 24 (curves A–E, values below ~30 MHz): sea water εr 70, σ 5 S/m; fresh water 80, 0.003; wet ground 30, 0.01; medium dry 15, 0.001; very dry 3, 0.0001. Read off a log-scale graph. Above about 30 MHz the land and fresh-water conductivities in that figure rise; use Custom if that matters.
  • Half-wave dipole pattern and the image method over a conducting plane: N. K. Nikolova, McMaster University, Lecture 9, eq. 9.43 (λ/2 dipole), 9.61 (vertical element over ground), 9.77 (horizontal element over ground); the same results are in C. A. Balanis, Antenna Theory, §4.6 and §4.7 “Ground Effects”.
  • Not modelled: ground wave, near-field ground loss, ground coupling at low height. See the box above.