A reflex klystron is an oscillator, so it must satisfy the Barkhausen condition: loop gain exactly 1 with a total phase shift that is a multiple of 2π — option 4.

\(|A\beta|=1,\qquad \angle A\beta=2n\pi\)

Why the two halves are both necessary.

The phase condition ensures the feedback is positive: the signal returning to the cavity must arrive in step with what is already there, reinforcing it. A multiple of \(2\pi\) means exactly that. A multiple of \(\pi\) — option 2 — includes the odd multiples, which is a 180° inversion, so the returning signal would cancel rather than reinforce and the oscillation would die.

The gain condition ensures the amplitude settles. If the loop gain were less than 1, as option 1 states, each round trip would return less than was sent and the oscillation would decay to nothing. If it were greater than 1, as option 3 states, the amplitude would grow until some non-linearity stopped it — which is exactly what happens at start-up, but the steady operating condition is the point where the growth stops, and that is where the loop gain has fallen to unity.

How the reflex klystron meets these conditions physically. A single cavity does both jobs. Electrons pass through the gap and are velocity modulated; the repeller, held at a negative potential, turns them back so they pass through the same gap a second time. If they return at the right moment they deliver energy to the cavity instead of taking it.

That moment is set by the transit time in the repeller space, and the requirement is

\(T=\left(n+\dfrac{3}{4}\right)\ \text{cycles}\)

The three-quarter cycle is what makes the returning bunch arrive in the retarding phase of the gap field, and the whole cycles n are the \(2n\pi\) of the phase condition. Each value of n is a distinct mode of the tube, reached by a different repeller voltage — which is why a reflex klystron's output power and frequency vary in discrete humps as the repeller voltage is swept, and why it is tuned electronically over a small range within each mode.

Hence, the loop gain is 1 and the phase shift a multiple of 2π.