0

## Bunching Process in Microwave

• Bunching Process in Microwave:

Once the electrons leave the buncher cavity, they drift with a velocity along the space between two cavities. The effect of velocity modulation produces, bunching of electron beam or current modulation. The electrons that pass the buncher cavity with zero voltage travel with unchanged velocity and become the bunching centre. Electrons that pass the bunching cavity during positive half cycles of microwave input become faster and electrons that pass during the negative half cycle become slower.
ta = time at which maximum retardation occur
tb = time at which electrons have uniform velocity
tc = time at which maximum acceleration occur

Bunching centre is the point at which electron density is maximum. The distance to the bunching centre,

ΔL = Vo(td - tb) ---------------------- (1)
tc = tb + (π/2ω),
tb = ta + (π/2ω),
ta = tb - (π/2ω)   ------- (2)

The distance of electron at ta,
Δt = Vmin (td - ta) = Vmin(td – tb + (π/2ω)) ----------- (3)

The distance of electron at ta,
Δt = Vmax (td - tc) = Vmax(td – tb – (π/2ω)) ----------- (4)

Let Vmin = Vo[1 – βiVi/2Vo] ------------ (5)
Vmax = Vo[1 + βiVi/2Vo] ------------ (6)

Substitute eqn 5 in eqn 3

Δt = Vo[1 + βiVi/2Vo] (td – tb + (π/2ω))
Δt = [Vo+ βiVi Vo /2Vo] (td – tb + (π/2ω))
Δt = Vo td - Vo tb + Vo(π/2ω) - td βiVi/2 + tb βiVi/2 - βiVi π/4ω ------ (7)

Substitute eqn 6 in eqn 4

Δt = Vo[1 + βiVi/2Vo] (td – tb – (π/2ω))
Δt = Vo td – Vo tb – Vo (π/2ω)) + td βiVi/2 – tb βiVi/2 - βiVi π/4ω ------ (8)

The necessary condition at which electrons meet at a distance ΔL is,

Equating eqn 7 and eqn 8

Vo td - Vo tb + Vo(π/2ω) - td βiVi/2 + tb βiVi/2 - βiVi π/4ω =
Vo td – Vo tb – Vo (π/2ω)) + td βiVi/2 – tb βiVi/2 - βiVi π/4ω

We get,

Vo π/ω = 2βiVi/2 (td – tb)

We have,

iVi)(td – tb) = Vo π/ω
td – tb = Vo π/ω βiVi  ------------------ (9)

Subsitute, eqn (9) in eqn (1)

ΔL = Vo(Vo π/ω βiVi  )