Traveling-Wave Tube (TWT): The apparent axial phase velocity of the RF electric field along a helical slow-wave structure is approximately equal to which of the following?

Introduction / Context:
In a helical TWT, the RF circuit is a slow-wave structure. The wave follows the helix and therefore has a reduced axial phase velocity compared with free space. Matching this reduced axial phase velocity to the electron beam velocity enables efficient energy exchange and high gain over wide bandwidth.

Given Data / Assumptions:

• Helix pitch p is the axial advance per turn, and circumference C is approximately 2πr.
• The RF wave travels close to c along the wire path itself.
• We are interested in the axial projection of that motion.

Concept / Approach:

Because the wave travels around the helix, only the axial component of its velocity contributes to the phase advance seen along the TWT axis. Geometrically, the axial component is the full path velocity multiplied by p/C, giving v_phase,axial ≈ c * (p / C). Designers select helix radius and pitch to achieve the desired synchronism with the beam for the operating band.

Step-by-Step Solution:

Verification / Alternative check:

Dispersion curves for helical slow-wave structures show axial phase velocity well below c and approximately proportional to the pitch to circumference ratio for small frequency ranges, validating the geometric argument.

Why Other Options Are Wrong:

• The speed of light c would not allow synchronism with the comparatively slower electron beam.
• c * (circumference / pitch) implies a superluminal axial velocity, which is unphysical for the phase projection.
• The product c * pitch * circumference has incorrect units and no physical meaning for velocity.
• Dividing c by the number of helix turns per wavelength is not the correct geometrical projection.

Common Pitfalls:

Confusing axial phase velocity with group velocity or beam velocity, and neglecting that dispersion makes v_phase frequency dependent. The simple geometry gives intuition, while detailed design uses full dispersion models.

Final Answer:

c * (helix pitch / helix circumference)