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Antenna

Helical Antenna Calculator (Axial Mode)

Design an axial-mode helical antenna: enter frequency, turns, helix diameter, turn spacing and wire size to get directivity, half-power and first-null beamwidths, axial ratio, input resistance and pitch angle by Kraus's formulas, with warnings outside their valid range.

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Formula

D=15 N Cλ2Sλ,HPBW=52∘CλNSλ,FNBW=115∘CλNSλ,AR=2N+12N,R≈140 Cλ  ΩD = 15\,N\,C_\lambda^{2}S_\lambda,\quad \mathrm{HPBW} = \frac{52^\circ}{C_\lambda\sqrt{N S_\lambda}},\quad \mathrm{FNBW} = \frac{115^\circ}{C_\lambda\sqrt{N S_\lambda}},\quad AR = \frac{2N+1}{2N},\quad R \approx 140\,C_\lambda\;\Omega

Reference: Kraus & Marhefka, Antennas for All Applications, 3rd ed. (2002), ch. 8; Balanis, Antenna Theory, 4th ed. (2016), §10.3.1

N— Number of turns
C_λ— Turn circumference πD in free-space wavelengths
S_λ— Turn spacing in free-space wavelengths
α— Pitch angle, tan α = S/(πD) (°)
D— Directivity (Kraus's approximation)
R— Input resistance with an axial feed (Ω)

How It Works

An axial-mode helix is a conductor wound into a coil of NN turns, each about one wavelength around, standing on a ground plane and fed against it. It radiates a single beam along its own axis, circularly polarised, over nearly an octave of bandwidth. That combination is why helices point at satellites, track GPS and feed dishes.

John Kraus built and measured the first ones in 1946 and reduced what he saw to a handful of empirical formulas. They are written in the free-space wavelength λ\lambda: the turn circumference Cλ=πD/λC_\lambda = \pi D/\lambda, the turn spacing Sλ=S/λS_\lambda = S/\lambda, and the pitch angle α\alpha, with tan⁡α=S/(πD)\tan\alpha = S/(\pi D).

D≈15 N Cλ2 SλD \approx 15\,N\,C_\lambda^{2}\,S_\lambda
HPBW≈52∘CλNSλ,FNBW≈115∘CλNSλ\mathrm{HPBW} \approx \frac{52^\circ}{C_\lambda\sqrt{N S_\lambda}}, \qquad \mathrm{FNBW} \approx \frac{115^\circ}{C_\lambda\sqrt{N S_\lambda}}
AR=2N+12N,R≈140 Cλ ΩAR = \frac{2N+1}{2N}, \qquad R \approx 140\,C_\lambda\ \Omega

Directivity grows in proportion to the number of turns, so the beamwidth narrows only as 1/N1/\sqrt{N}: doubling the turns from 10 to 20 takes a one-wavelength helix from 32.89° to 23.26°. The axial ratio tends to 1 as turns are added. The input resistance, about 140 Ω for a one-wavelength turn with an axial feed, is nearly real across the band.

Kraus's directivity is an upper estimate

King and Wong (1980) measured uniform helices of many lengths and found that Kraus's formula overestimates the gain, more so for long helices. Treat the directivity here as an optimistic bound. The antenna simulator linked from this page solves the same helix with NEC-2 over a ground plane, and its answer is the one to design to.

Validity

The formulas hold for a circumference of 0.75 to 1.33 wavelengths, a pitch angle of 12° to 14°, and at least 3 turns (Kraus and Marhefka, 3rd ed., chapter 8; Balanis, 4th ed., section 10.3.1). Outside that region the results are still shown, with a warning naming the bound that was crossed. The conductor diameter and the ground plane size do not enter the formulas.

Worked Example

Problem: Design check for a right-hand helix for a 435 MHz amateur-satellite downlink: 12 turns of 6 mm tube on a 220 mm diameter, 160 mm between turns.

Step 1 - Wavelength: λ = c/f = 299792458 / 435e6 = 689.18 mm

Step 2 - Normalise the geometry: C = π × 220 = 691.15 mm, so C/λ = 1.00286 S/λ = 160 / 689.18 = 0.23216 α = atan(160 / 691.15) = 13.03° All three are inside the axial-mode region, and N = 12 is above 3, so no warning is raised.

Step 3 - Directivity (Kraus): D = 15 × 12 × 1.00286² × 0.23216 = 42.03, which is 16.24 dBi

Step 4 - Beamwidths: C_λ √(N S_λ) = 1.00286 × √(12 × 0.23216) = 1.6739 HPBW = 52° / 1.6739 = 31.07° FNBW = 115° / 1.6739 = 68.70°

Step 5 - Polarisation and feed: AR = (2 × 12 + 1) / (2 × 12) = 1.042 R = 140 × 1.00286 = 140.4 Ω

Step 6 - Mechanical length: L = N × S = 12 × 160 = 1920 mm

Kraus's 16.24 dBi is an upper estimate. Measured gain is lower, so check the design in the antenna simulator before cutting the tube, and match the 140.4 Ω feed to 50 Ω.

Practical Tips

  • ✓Match the feed by flattening the first quarter turn into a strip that runs close to the ground plane, or with a quarter-wave transformer of about 84 Ω, the geometric mean of 140 and 50 Ω.
  • ✓Make the ground plane at least about three-quarters of a wavelength across. A cup-shaped reflector a little larger than the helix improves the front-to-back ratio.
  • ✓Choose the turns from the gain you need, then check the beamwidth: each doubling of N adds about 3 dB and narrows the beam by a factor of √2.
  • ✓Use a non-conducting former such as PVC or PTFE and keep metal fixings off the axis, where the field is strongest.

Common Mistakes

  • ✗Taking Kraus's directivity as the gain you will measure. King and Wong's measurements show the formula overestimates, and the error grows with the number of turns, so budget the link on a solved or measured figure.
  • ✗Winding the helix with the wrong sense. A right-hand helix receives right-hand circular polarisation, and a reflector such as a dish reverses the sense, so a dish feed is wound the opposite way to a direct-radiating helix.
  • ✗Measuring the diameter to the outside of the tube. Kraus's circumference is taken on the conductor's centre line, so use the mean diameter, the outside diameter minus one conductor diameter.
  • ✗Feeding the 140 Ω helix straight from 50 Ω coax. The mismatch alone gives a VSWR of 2.8, so add a matching section first.

Frequently Asked Questions

A helix whose turns are each about one wavelength around. It radiates a single beam along its axis with nearly circular polarisation. The normal mode, with turns much smaller than a wavelength, radiates broadside instead and is a different antenna.
It is Kraus's empirical formula, which is known to overestimate. King and Wong (1980) measured helices of many lengths and found lower gain than Kraus predicts, increasingly so for long helices. Use it for a first design, then solve the helix in the antenna simulator linked on this page.
Kraus's formulas were fitted to helices with a pitch angle between 12 and 14 degrees, a circumference between 0.75 and 1.33 wavelengths and at least 3 turns. Outside that region the formulas are extrapolated, so the results are shown with a warning rather than hidden.
About 140 times the circumference in wavelengths, in ohms, with an axial feed: roughly 140 ohms for a one-wavelength turn. It is nearly resistive across the band, which makes a fixed matching section practical.
Not in Kraus's formulas, which depend only on the circumference, the spacing and the number of turns. Thickness matters for strength, losses and the simulator's thin-wire model, which is why the calculator carries it to the antenna simulator.

Methodology & References

References

  • Antennas for All Applications, 3rd ed. — John D. Kraus & Ronald J. Marhefka (2002), ch. 8 — Helical antennas, axial mode
  • Antenna Theory: Analysis and Design, 4th ed. — Constantine A. Balanis (2016), §10.3.1 — Helical antenna, axial-mode design equations
  • Characteristics of 1 to 8 wavelength uniform helical antennas — H. E. King & J. L. Wong, IEEE Trans. Antennas Propag., AP-28(2), pp. 291–296 (1980) — Kraus's gain formula overestimates measured gain

Reproduces Kraus's equations exactly, including the ten-turn, one-wavelength example; the doubling-N identity is exact. For a solved pattern and impedance, open the antenna simulator from this page.

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