Helical Antenna — Axial Mode, Kraus Formulas & Circular Polarisation

Helical Antenna — Axial Mode

The helical antenna is a wire wound in a helix and fed at one end against a ground plane. Depending on the circumference-to-wavelength ratio it can operate in two distinct modes. The axial mode (end-fire) is the most useful for gain antennas — it produces a circularly polarised beam along the helix axis and is widely used in satellite ground stations, CubeSats, and amateur radio.

Normal Mode vs Axial Mode

Normal mode: circumference C ≪ λ; radiation broadside (perpendicular to axis); linearly polarised; low gain. Used as a compact antenna when space is limited.
Axial mode: circumference C ≈ λ; radiation along the axis; circularly polarised; high gain. This is the practical gain antenna.

Axial-Mode Condition

The helix operates in axial mode when:

\(0.75 < \dfrac{C}{\lambda} < 1.33 \quad \text{where } C = \pi D\)

The optimal pitch angle (angle of the helix wire relative to the horizontal plane) is approximately α ≈ 14°, which corresponds to a turn spacing S ≈ λ/4 when C ≈ λ.

Gain and Beamwidth (Kraus Formulas)

\(G \approx 15\,N\!\left(\frac{C}{\lambda}\right)^{\!2}\!\frac{S}{\lambda} \quad \text{(linear, not dB)}\)

\(\text{HPBW} \approx \dfrac{52°}{\dfrac{C}{\lambda}\sqrt{N\dfrac{S}{\lambda}}}\)

where N is the number of turns. Gain increases with N — a 10-turn helix at C = λ, S = λ/4 gives G ≈ 15×10×1×0.25 = 37.5 (≈ 15.7 dBi).

Input Impedance

The input impedance of the axial-mode helix is nearly resistive and approximated by:

\(Z_{\rm in} \approx 140\,\dfrac{C}{\lambda}\;\;[\Omega]\)

At the optimum C/λ = 1, Zin ≈ 140 Ω. A quarter-wave stripline transformer between the helix feed and the 50 Ω coax connector matches the impedance: the transformer impedance is Z_t = √(140 × 50) ≈ 84 Ω.

Axial-mode helix: N turns of diameter D, turn spacing S, fed against a ground plane.

Circular Polarisation

The axial-mode helix produces circular polarisation naturally. The sense (RHCP or LHCP) is determined by the winding direction:

Right-hand winding (thumb along axis, fingers curl in direction of travel) → RHCP
Left-hand winding → LHCP

For satellite reception (LEO downlinks, GPS), RHCP is standard. The axial ratio of a well-designed axial-mode helix is typically better than 1 dB (very close to perfect circular polarisation) over the 3 dB beamwidth.

Design Procedure

Choose target frequency f → compute λ = c/f
Set D = λ/π (so C = λ) and S = λ/4 for optimal pitch angle
Choose N for desired gain: G ≈ 15N×1×0.25 = 3.75N → e.g. N = 10 gives G ≈ 37.5 (15.7 dBi)
Compute Zin ≈ 140 Ω; design λ/4 transformer to 50 Ω (Z_t ≈ 84 Ω stripline)
Ground plane diameter should be at least 0.75λ to 1λ

🔧 Related Calculators

Helical Antenna Calculator → Dipole Antenna → Array Factor Plotter →