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## Velocity Modulation in Klystron Amplifier

• Klystron Amplifiers:

The 2 cavity klystron amplifier is widely used microwave amplifier, operated by the principles of voltage and current modulators.

Basic Operations:

A high velocity electron beam produced by the accelerating anode is passed through a buncher cavity, drift space, catcher cavity and finally collected by the collector terminals. The electrons injected from the cathode is accelerated by applying a DC voltage ‘Vg’. They arrive at the first cavity that is the buncher cavity or input cavity with uniform velocity. At the buncher cavity these electrons encounter signal voltage or gap voltage. The electrons that pass through the zeros of the gap voltage pass with unchanged velocity. The electrons that pass through positive half cycles of the gap voltage undergo acceleration in velocity. The electrons that pass through negative half cycles of the gap voltage undergo retardation in velocity. (As a result of these the electrons get bunched together as they travelled through the drift space). The variation in electron velocity in drift space is called velocity modulation. (The buncher cavity velocity modulates the electron beam). This electron beam induces a RF current in this field is opposite to the input cavity. Thus the kinetic energy is transferred from the electrons to the field capture cavity. The second cavity is called capture cavity since it captures energy from the bunch electron beam. The electrons emerging from the capture cavity are collected by the collector terminal.

Velocity Modulation in Klystron Amplifier:

The velocity of electrons before entering the buncher cavity is given by,
Vo = 2ev0/m

Where m is the mass of the electron
e is the charge of electron
v0 is the cathode potential

On substituting the values of e and m, the equation reduces to

Vo = 0.596 x 106 (v0)   m/sec ------------- 1

When the microwave signal is applied to the input terminal, the gap voltage is given by

Vs = V1sinωt ----------------- 2

V1 is the amplitude of the signal.
The average transit time through the gap at a distance ‘d’.

Τ = d/Vo = t1 – t0 -------------- 3

Where t0 is the line at which beam reaches the buncher cavity. t1 is the time at which the beam leaves the buncher cavity.

The average transit angle, θg = ω(d/Vo) = ω(t–  t0) ------------ 4

The average microwave voltage in the buncher cavity is

= V1/T ω (cos ωt0 – cos ωt1) ----------------- 5

From eq (4)
ωd/V0 = ω(t1 – t0)

ωt1 = ω(d/Vo + t0) ------------- 6

Subsituting eq (6) in eq (5)

<Vs> = V1/Tω [cosωt0 – cos(ωd/Vo + ωt0)] ------------- 7

Let ωt0 + ωd/2Vo = ωt0 + θg/2 = A
ωd/2Vo = θg/2 = B
A + B = ωt0 + θg/2 + θg/2 = ωt0 + θg
A – B = ωt0 + θg/2 - θg/2 = ωt0

A + B = ωt0 + θg  ,  A – B = ωt0   ----------------- 8

Substitute eq (8) in eq (7)

<Vs> = V1/Tω [cos(A - B) – cos(A + B)]
= 2V1/Tω.  sin A sin B
=  2V1/Tω.  sin (ωt0 + θg  ) sin (θg/2)

Let ωT = θg

Therefore, <Vs> = 2V1/ θg.  sin (ωt0 + θg  ) sin (θg/2)
<Vs> =  V1 βi sin (ωt0 + θg/2)

βi = sin (θg/2)/g/2)