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.

Bunching Distance

t_a = time at which maximum retardation occur

t_b = time at which electrons have uniform velocity

t_c = time at which maximum acceleration occur

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

ΔL = V_o(t_d - t_b) ---------------------- (1)

t_c = t_b + (π/2ω),

t_b = t_a + (π/2ω),

t_a = t_b - (π/2ω)   ------- (2)

The distance of electron at ta,

Δt = V_min (t_d - t_a) = V_min(t_d – t_b + (π/2ω)) ----------- (3)

The distance of electron at ta,

Δt = V_max (t_d - t_c) = V_max(t_d – t_b – (π/2ω)) ----------- (4)

Let V_min = V_o[1 – β_iV_i/2V_o] ------------ (5)

V_max = V_o[1 + β_iV_i/2V_o] ------------ (6)

Substitute eqn 5 in eqn 3

Δt = V_o[1 + β_iV_i/2V_o] (t_d – t_b + (π/2ω))

Δt = [V_o+ β_iV_i V_o /2V_o] (t_d – t_b + (π/2ω))

Δt = V_o t_d - V_o t_b + V_o(π/2ω) - t_d β_iV_i/2 + t_b β_iV_i/2 - β_iV_i π/4ω ------ (7)

Substitute eqn 6 in eqn 4

Δt = V_o[1 + β_iV_i/2V_o] (t_d – t_b – (π/2ω))

Δt = V_o t_d – V_o t_b – V_o (π/2ω)) + t_d β_iV_i/2 – t_b β_iV_i/2 - β_iV_i π/4ω ------ (8)

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

Equating eqn 7 and eqn 8

V_o t_d - V_o t_b + V_o(π/2ω) - t_d β_iV_i/2 + t_b β_iV_i/2 - β_iV_i π/4ω =

V_o t_d – V_o t_b – V_o (π/2ω)) + t_d β_iV_i/2 – t_b β_iV_i/2 - β_iV_i π/4ω

We get,

V_o π/ω = 2β_iV_i/2 (t_d – t_b)

We have,

(β_iV_i)(t_d – t_b) = V_o π/ω

t_d – t_{b =} V_o π/ω β_iV_i  ------------------ (9)

Subsitute, eqn (9) in eqn (1)

ΔL = V_o(V_o π/ω β_iV_i  )

Output Power and Beam Loading:

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,

I_ind = β_oI₂

I₂ = 2 I_oJ₁(x)

The output power is given by the equation

P_out = (β_oI₂)²R_sh/2

R_sh is the Resistance of the catcher cavity

R_sh = V₂/ β_oI₂

Therefore, output power = (β_oI₂)²/2 x V₂/ β_oI₂

P_out = (β_oI₂) V₂/2

The efficiency of the amplifier is defined by the equation,

η = (β_oV₂J₁(x)/V_o