Traveling-Wave Tube (TWT) synchronism condition In a TWT, the phase velocity of the axial field on the slow-wave structure is kept how, relative to the electron beam velocity, for effective energy transfer?

Introduction / Context:
A Traveling-Wave Tube (TWT) achieves amplification by continuous interaction between an electron beam and an RF wave slowed down by a slow-wave structure (e.g., helix). Proper synchronism between beam velocity and wave phase velocity is essential for net transfer of kinetic energy from electrons to the RF field.

Given Data / Assumptions:

• Electron beam with average velocity v_e.
• Axial RF wave with phase velocity v_p along the slow-wave structure.
• Goal: set relation between v_p and v_e for gain.

Concept / Approach:
For sustained energy transfer, electrons must “overtake” the RF phase such that they see a retarding electric field on average, giving up kinetic energy to the wave. This requires v_e to be slightly greater than v_p, which equivalently means the phase velocity is kept slightly less than the electron velocity. Exact tuning is adjusted via beam voltage and circuit parameters to maintain synchronism over the device length.

Step-by-Step Solution:

Verification / Alternative check:
Small-signal TWT theory shows growth when the beam line intersects the circuit dispersion just above the wave's phase velocity, yielding a negative beam power change and positive RF power growth.

Why Other Options Are Wrong:

• Equal to v_e: narrow interaction; practical operation prefers a slight margin for consistent gain.
• Slightly more than v_e: electrons would gain energy from the wave.
• Equal to c: impossible in a slow-wave helix and not conducive to interaction.

Common Pitfalls:
Confusing group and phase velocities; assuming maximum gain at exact equality rather than slightly less v_p than v_e.

Final Answer:
slightly less than velocity of electron