Log-periodic antenna calculator (LPDA)

About

The log-periodic dipole array (LPDA) is a wideband antenna that holds its performance steady across a wide frequency range. It is built from a number of dipoles of different lengths, spaced along a boom according to a logarithmic rule.

τ — scaling factor, σ — relative spacing, α — apex angle; fed from the narrow end

Log-periodic antenna layout: τ = 0.85, σ = 0.155

Standard bands Enter the frequencies manuallyUHF (430–450 MHz)GSM (800–960 MHz)LTE (1700–2200 MHz)WiFi/LTE (2300–2700 MHz) Lowest frequency (f₁) MHz Highest frequency (fₙ) MHz Feedpoint impedance (Z₀) 50 Ω75 Ω Ω Scaling factor (τ) - Relative spacing (σ) optimum = 0.156 - Diameter of the largest element mm Boom diameter mm Boom shape Round Square Units Millimetres (mm)Centimetres (cm)Metres (m)

Notes on the calculation:

• Standard ranges: pick one from the list of common frequency ranges, or enter your own.
• Frequency range: the lower frequency must be below the upper one. The wider the range, the more elements you need.
• Input impedance: the standard 50 Ω and 75 Ω are supported.
• Scale factor τ (tau): the ratio between adjacent element sizes. Range: 0.8–0.98.
  • Smaller τ gives a wider band but more elements
  • Larger τ gives a narrower band but fewer elements
• Relative spacing σ (sigma): the spacing between elements. Range: 0.03–0.3.
  • The optimum is worked out automatically: σ = 0.243τ − 0.051
  • You can adjust it by hand to suit what you want
• Diameters: the diameter of the largest element and of the boom, in millimetres.
• Boom section: round or square.
• Units: results can be shown in millimetres, centimetres or metres.

What sets a log-periodic apart:

• Bandwidth: the frequency coverage can reach 10:1 and beyond
• Steady performance: gain and pattern stay practically unchanged across the whole range
• Typical gain: usually 6–12 dBi, depending on the number of elements
• Unidirectional: it has a clear main lobe
• Build complexity: it needs accurate construction and careful adjustment

Where it is used:

• Television antennas covering many channels
• Amateur multiband antennas
• Test equipment and monitoring
• Military and special-purpose communications

📐 The formulas

The calculator uses formulas recovered by reverse-engineering reference designs:

Main parameters

1. Number of elements:

n = ⌈ln(fₙ/f₁) / ln(1/τ)⌉ + 6

where the extra constant 6 gives the right element count for a practical build.

2. Apex angle:

α = arctg(σ/2) × (180/π) × 2

3. Wavelength at the lowest frequency:

λ₁ = c / f₁ [mm]

Element dimensions

4. Length of the largest element:

L₁ = (λ₁/4) × 0.996

5. Lengths of the remaining elements (a geometric progression):

Lᵢ₊₁ = Lᵢ × τ

Spacing between elements

6. The first spacing:

S₁ = σ × λ₁

7. The remaining spacings (a geometric progression):

Sᵢ₊₁ = Sᵢ × τ

Further parameters

8. Characteristic impedance of the feeder line:

Zf = 120 × ln(2D/d) / π

9. Gap in the feeder line:

G = d × 2.4

10. Distance from the first element to the shorting stub:

w = L₁ × 0.5

11. Approximate gain:

G = 6.5 + 1.5 × log₁₀(n) [dBi]

Symbols:

• f₁, fₙ — the lower and upper frequencies [MHz]
• τ — the scale factor (period factor)
• σ — the relative spacing
• c = 300,000,000 m/s — the speed of light
• d — element diameter [mm]
• D — boom diameter [mm]
• n — number of elements

Practical advice:

1. For the widest bandwidth: use τ = 0.85–0.88 with the matching optimum σ
2. For more gain: use τ = 0.90–0.95 (narrower band, more gain)
3. Construction: the best results come from a dielectric boom
4. Feed: connect at the smallest (first) element
5. Adjustment: use an antenna analyser across the whole range
6. Accuracy: the calculator agrees with professional designs to within 2 %

© 2026 UR3PKI · CyberDev.Space · Content licensed under CC BY-NC-SA 4.0.

How to cite this calculator

UR3PKI. «Log-periodic antenna calculator (LPDA)». CyberDev.Space. https://cyberdev.space/en/radio/calculators/antennas/logoperiodic (licence CC BY-NC-SA 4.0).

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