How a helix works
An axial-mode helix is a coil of wire about one wavelength in circumference, mounted in front of a flat reflector. It radiates a circularly polarised beam along its axis, which makes it ideal for satellites: the signal's polarisation rotates as a satellite tumbles and moves across the sky, and a circularly polarised antenna doesn't suffer the fading that a linear antenna would. Helices are popular for the QO-100 geostationary satellite's 2.4 GHz uplink, for 23cm and 13cm ATV, and as feeds for dishes.
Gain ≈ 10.8 + 10 log₁₀(C² N S ÷ λ³) dBi (Kraus)
Beamwidth ≈ 52 λ1.5 ÷ (C √(N S)) degrees Impedance ≈ 140 × C ÷ λ Ω
These are John Kraus's classic design equations. Later measurements showed that the gain formula is optimistic, by around 2 to 4 dB for longer helices, so treat the lower figure shown as a realistic expectation. Gain rises with the number of turns, but more slowly than the formula suggests beyond about 15 turns.
Polarisation
A helix wound like a normal screw thread (clockwise as it goes away from you, viewed from the reflector end) gives right-hand circular polarisation, and the opposite winding gives left-hand. Most satellites use right-hand, but check, as the wrong sense loses 20 dB or more. If the helix feeds a dish, the reflection reverses the polarisation, so wind it the opposite way to the polarisation you want from the dish.
Matching to 50 Ω
The feed impedance of about 140 Ω needs matching to 50 Ω coax. The usual method is to flatten and bend the first quarter turn of the helix closer to the reflector, or solder a thin copper strip to it, which forms a short transmission-line transformer. Adjust it while watching the SWR on a NanoVNA. Connect the helix to the centre pin of an N-type socket on the reflector, and keep the first turn at the right height above the reflector, about a quarter of the turn spacing to start with.
Questions
What size helix for QO-100?
For 2,400 MHz, a 1 λ circumference is a coil about 39.8 mm in diameter with turns spaced about 31 mm apart. Five to seven turns is enough to feed a dish, and more turns for use on its own.
How many turns should a helix have?
Gain rises with turns. Three to six turns makes a good dish feed, and 10 to 20 turns a stand-alone antenna with about 12 to 15 dBi.
Which way should I wind a helix?
Like a normal screw thread, clockwise moving away from the reflector, for right-hand circular polarisation. Reverse it for a dish feed if you want right-hand from the dish.
What is the feed impedance of a helix?
About 140 Ω for a 1 λ circumference. Match it with a flattened first quarter turn or a strip transformer.