# 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  )

The difference between the exit and entrance energies must be supplied by the buncher cavity to bunch the electron beam. Thus the electron beam is energised by energy of the cavity. This phenomenon is known as beam loading. The magnitude of induced current under beam loading is given by,

Iind = βoI2
I2 = 2 IoJ1(x)

The output power is given by the equation
Pout = (βoI2)2Rsh/2

Rsh is the Resistance of the catcher cavity
Rsh = V2/ βoI2

Therefore, output power = (βoI2)2/2 x V2/ βoI2

Pout = (βoI2) V2/2

The efficiency of the amplifier is defined by the equation,

η = (βoV2J1(x)/Vo