# Reflex Klystron Oscillator Theory

Reflex Klystron Oscillator:

The Reflex Klystron Oscillator is a low power microwave oscillator generating power in the range of 10 -500 MW. The output frequency range of the oscillator is in the range of 1 to 25 GHz. The efficiency of Reflex Klystron is about 20 – 30%. This type of oscillator is mainly used in microwave laboratories for microwave measurements and as local oscillators in commercial and military radars. The Reflex Klystron Oscillator theory is explained below.

Reflex klystron is a single cavity klystron that can be used as low power microwave oscillator. Due to the DC voltage in the cavity, a Radio Frequency (RF) noise is generated. The electron beam injected from the cathode is velocity modulated by the cavity gap voltage. Electrons which encounter the positive half cycle of the RF field will get accelerated. Those electrons which encounter zero RF field will pass with unchanged velocity (constant velocity). Electrons which encounter negative half cycles will be retarded. Some of the accelerated electrons enter the ‘repeller space’ with greater velocities and these electrons are retarded by the retarding field in the repeller space. Since the repeller space is at negative potential, the electrons entering the repeller space are repelled back. These electrons give up Kinetic Energy into the field on their return journey. The electrons are finally collected by walls of the cavity by keeping unity gain and phase-shift of 2π sustained oscillations are obtained.

Velocity Modulation:

The uniform velocity with which the electrons enter the cavity is given by,
Vo = √(2eVo/m)

Where, e – charge of electron
m – mass of electron.
Substitute values of e and m.

Vo = 0.596 x 106 √Vo m/s ----------------- 1

The velocity with which the electrons leave the cavity is given by,

V(t1) = Vo[1 + β1V1/Vo Sin(ωt1- θg/2)]

The electric field inside the cavity is given by,

E = [Vr + Vo + V1Sin(ωt1)]/L
where, Vr – repeller voltage

Vo – output voltage/DC gain beam voltage.
L – Length of drift space

If V1Sin(ωt1) is less than Vr + Vo, then

E = (Vr + Vo)/L --------------------- (3)

The force of electron is given by
F = -eE

Substituting for E

We have, F = -e((Vr + Vo)/L) ------------------------ (4)
F = ma = m(d2z/dt2) ----------------- (5)

Equating eq (4) and (5)

m(d2z/dt2) = -e((Vr + Vo)/L)
d2z/dt2 = e((Vr + Vo)/Lm)
dz/dt = -e((Vr + Vo)/Lm)(t-t1) + K1

The above equation represents the velocity change taking place inside the cavity. The velocity change is obtained as a ratio of change in distance to change in time where z is the distance.

Output Power and Efficiency:

For the electron beam to generate maximum amount of energy, the returning electron beam should pass the cavity gap with less retardation. Thus, for maximum energy transfer the DC transit angle is given by,

Θ01 = ω(t2 – t1) = ωTo
Θ01 = 2Nπ

Where, N = the no: of modes
N = n-¼

Therefore Θ01 = 2π(n-¼)
= 2πn – π/2 --------------------- (1)

The magnitude of current in the cavity is given by,

I2 = 2Ioβi J1(x’) -------------------- (2)

The DC power supplied is given by,

Pdc = VoIo ---------------------- (3)

The AC power supplied is given by,

Pac = V1I2/2

Substituting for I2 from eq(2)
Pac = V1Io βi J1(x’) -------------------- (4)

Let, the bunching parameter of the reflex klystron is given by
x’ = βi V1θo’/2Vo

Therefore, V1 = 2Vo x’/ βiθo

Substituting for θo’ from eq (1)
V1 = 2Vo x’/ βi(2πn – π/2) ------------------------- (5)
Substitute eq (5) in (4)

Pac = 2Vo x’ Io J1(x’)/(2πn – π/2) -------------------- (6)

The efficiency of the klystron is given by,
η = Pac/ Pdc

Substituting values from eq (6) and (3)
η =[2Vo x’ Io J1(x’)/(2πn – π/2)] x 1/VoIo
η =[2 x’ J1(x’)/(2πn – π/2)]
For a given beam voltage, Vo
The repeller voltage, Vr is related to number of modes as,

Vo/(Vr + Vo)2 = e/m (2πn – π/2)2/8ω2L2

The admittance of the klystron is the ratio of induced current to induced voltage.
Ie, Ye = i2/V2
Where, i2 = 2IoβiJ1(x’)e-jθo’

V2 = I2Rsh
= 2 x’Vo e-jπ/2/ βi(2πn – π/2)

Substitute I2 and V2,

Ye = (2Ioβi J1(x’)e-jθo’) (βi(2πn – π/2)) / 2 x’Vo e-jπ/2
= Ioβi2 J1(x’)e-jθo’(2πn – π/2) / x’Vo e-jπ/2

Hence the admittance of the Reflex Klystron is obtained as:

Ye = Ioβi2 J1(x’)ej(π/2 - θo’) (2πn – π/2) / x’Vo