Antenna Array Calculator

Analyze a uniform linear array (ULA) of isotropic elements. Compute main beam direction, required phase shift, beamwidths, sidelobe level, and detect grating lobes. Based on classical antenna theory (Balanis, Kraus).

Normalized to wavelength (d/λ).
0° = end‑fire, 90° = broadside.
Compute array factor at this angle (normalized, dB).
8‑element broadside (d=λ/2, α=0°)
8‑element end‑fire (d=λ/4, α=-90°)
Scan to 30° (d=λ/2, N=16)
Grating lobes (d=0.8λ, α=0°)
Privacy first: All calculations run locally in your browser – no data sent to servers.

Uniform Linear Array (ULA) Theory

A uniform linear array consists of N identical antenna elements equally spaced along a line, with equal amplitudes and a constant phase shift α between adjacent elements. The array factor (AF) for isotropic elements is given by:

AF(θ) = sin(N ψ / 2) / (N sin(ψ / 2))

where ψ = k d cosθ + α, k = 2π/λ, θ is measured from the array axis (0° = end‑fire, 90° = broadside).

The main beam occurs when ψ = 0, i.e., cosθmax = – α λ / (2π d). For a desired beam direction θ0, the required phase shift is α = – (2π d / λ) cosθ0.

Key Parameters & Approximations (for large N)

  • Half‑Power Beamwidth (HPBW): Broadside case (θ=90°) ≈ 0.886 λ/(N d) [rad]; End‑fire case (θ=0°) ≈ 2√(0.886 λ/(N d)) [rad]. Our calculator uses numerical root‑finding for accuracy and reports total beamwidth for end‑fire.
  • First‑Null Beamwidth (FNBW): Broadside ≈ 2 λ/(N d) [rad]; End‑fire ≈ 2√(2λ/(N d)) [rad]. We compute exact nulls from ψ = ±2π/N, accounting for visibility, and report total width for end‑fire.
  • Sidelobe Level (SLL): For uniform excitation, the first sidelobe is at –13.26 dB relative to the main lobe (valid for N ≥ 3).
  • Grating Lobes: Occur when |ψ| = 2πm (m≠0) for some θ in visible region. Condition: d/λ ≥ 1/(1+|cosθ|). Typically d/λ > 0.5 may cause grating lobes when scanning.
Example: An 8‑element array with d = λ/2, α = 0° gives a broadside beam at 90°. HPBW ≈ 0.886/(8×0.5) = 0.2215 rad = 12.7°. FNBW ≈ 2/(8×0.5)=0.5 rad = 28.6°. Sidelobe = –13.26 dB. Verified by our calculator.

Practical Applications

  • Phased Array Radar: Electronic beam steering for weather surveillance, air traffic control.
  • 5G/6G Beamforming: Massive MIMO base stations use ULAs to direct signals to users.
  • Radio Astronomy: Interferometer arrays synthesize large apertures.
  • Sonar: Underwater acoustic arrays.

Derivation & Assumptions

The array factor formula assumes isotropic radiators, no mutual coupling, and far‑field observation. The pattern is periodic in ψ; the visible region corresponds to θ ∈ [0°,180°] → ψ ∈ [−kd+α, kd+α]. Grating lobes appear when the periodicity creates additional main lobes within the visible region.

The beamwidth approximations above are derived from the sinc‑pattern and are accurate within a few percent for N ≥ 5. Our calculator finds HPBW numerically by solving |AF(θ)|² = 0.5 (relative to peak). For end‑fire arrays, the beamwidth is reported as the total width (twice the distance from peak to half‑power point on the visible side), following standard antenna definitions.

Step‑by‑Step Calculation

  1. Input N, d/λ, and either α or θ₀.
  2. Compute main beam direction from ψ = 0 condition.
  3. Compute HPBW by searching around the peak for half‑power points (numerical method). For end‑fire, the result is doubled to obtain total beamwidth.
  4. Compute FNBW from exact null positions ψ = ±2π/N, considering whether each null lies in the visible region. For end‑fire with only one visible null, the width is taken as twice the angular distance from peak to that null.
  5. Sidelobe level is taken as -13.26 dB for N ≥ 3; for N=2 it is not applicable.
  6. Grating lobe check: determine if any integer m ≠ 0 yields ψ = 2πm within the visible range and with |cosθ| ≤ 1. The range of m is dynamically computed based on kd and α.
  7. Optional: compute AF at a user‑specified angle (dB).

Reference Table (Uniform Array, d/λ = 0.5, broadside)

N HPBW (deg) FNBW (deg) SLL (dB)
4 25.5 57.3 -13.3
8 12.7 28.7 -13.3
16 6.4 14.3 -13.3
32 3.2 7.2 -13.3
Case Study: 5G Base Station Beam Steering

A 5G gNB uses a 16‑element array with d = λ/2 (at 3.5 GHz, λ ≈ 85.7 mm, spacing ≈ 42.9 mm). To serve a user at 30° from broadside (θ = 60° from axis). Using our calculator in “Given desired beam direction” mode with θ₀ = 60°, N=16, d/λ=0.5, we get α ≈ –(2π×0.5)×cos60° = –π×0.5 = –90°. The HPBW ≈ 6.4°, which provides good directivity. The calculator also confirms no grating lobes because d/λ=0.5 ≤ 1/(1+|cos60°|)=1/(1.5)=0.666. This information helps the beamforming software set the phase shifters correctly.

Common Misconceptions

  • “Beamwidth is constant” – No, it depends on scan angle; end‑fire beam is much wider.
  • “d/λ = 0.5 always avoids grating lobes” – Only for broadside; when scanning, grating lobes may appear even with d/λ < 0.5.
  • “Array factor alone gives total pattern” – In reality, element pattern multiplies the AF, modifying sidelobes and beamwidth.

Applications Across Domains

  • Radar: Phased array antennas for target tracking.
  • Wireless Communications: Smart antennas, MIMO.
  • Radio Astronomy: Interferometric imaging.
  • Acoustic Arrays: Underwater sonar, microphone arrays.

Based on authoritative antenna theory – This tool implements formulas from “Antenna Theory: Analysis and Design” by Constantine A. Balanis (4th ed., Wiley) and “Antennas” by John D. Kraus. All numerical methods follow standard practices. Last revised March 2026 by the GetZenQuery RF engineering team.

Frequently Asked Questions

θ = 0° corresponds to the direction along the array axis (end‑fire). θ = 90° is perpendicular to the array (broadside).

We start from the peak angle and step away until the normalized power drops to 0.5. A bisection search ensures accuracy better than 0.1°. For end‑fire, the resulting single‑side width is doubled to give total beamwidth.

When element spacing is large enough (typically d/λ > 0.5 for broadside), additional main lobes appear in visible space. This can cause ambiguity in direction finding and should be avoided in most designs.

No, it assumes isotropic elements. For real antennas (dipoles, patches), the total pattern is AF multiplied by the element factor. Sidelobes may be further suppressed.

Negative α means the phase lags progressively along the array (element 1 has phase 0, element 2 has –α, etc.). This steers the beam toward the end‑fire direction (θ decreasing from 90°).

This calculator assumes uniform amplitude (all elements equal). For tapered arrays (e.g., Dolph‑Chebyshev), sidelobes are lower but beamwidth wider – not covered here.
References: Balanis, C.A. “Antenna Theory: Analysis and Design” (4th ed.); Kraus, J.D. “Antennas” (2nd ed.); IEEE Std 145‑2013 “IEEE Standard for Definitions of Terms for Antennas”.