Q5Microwave Theory And Techniques

Question

Q.5. Derive the equation of velocity modulation for a Two-Cavity Klystron amplifier.

Answer 1

Setup: in a two-cavity klystron amplifier, electrons are accelerated from the cathode by a DC voltage V0, acquiring a uniform initial velocity v0 = √(2eV0/m), and then pass through the gap of the first (buncher/input) cavity, which is excited by the small RF input signal to be amplified, producing a small time-varying gap voltage V1sin(ωt) superimposed on the DC beam voltage (with V1 ≪ V0).

Velocity modulation in the buncher gap: an electron transiting the narrow gap (transit time assumed short enough that the RF field is approximately constant during transit — the usual thin-gap approximation) gains or loses kinetic energy depending on the instantaneous gap voltage at the moment of transit. An electron entering the gap at time t0 experiences an effective gap voltage V1sin(ωt0), so by energy conservation, its exit kinetic energy is:

\frac{1}{2} m v (t_{0})^{2} = e V_{0} + e V_{1} sin (ω t_{0})

Solving for v(t0) and factoring out the DC term:

v (t_{0}) = \frac{2 e}{m} [V_{0} + V_{1} sin (ω t_{0})] = v_{0} 1 + \frac{V _{1}}{V _{0}} sin (ω t_{0})

Small-signal (linearizing) approximation: since V1 ≪ V0 (a small-signal input), the square root is expanded using the binomial approximation √(1+x) ≈ 1+x/2 for small x:

v (t_{0}) \approx v_{0} [1 + \frac{V _{1}}{2 V _{0}} sin (ω t_{0})]

This is the standard velocity-modulation equation: the exit velocity of each electron is now a sinusoidal function of its entry time t0, oscillating about the unperturbed velocity v0 with a small fractional modulation depth V1/2V0. Electrons entering during the positive half-cycle of the gap voltage leave slightly faster than v0; those entering during the negative half-cycle leave slightly slower.

From velocity modulation to bunching: this velocity-modulated beam then enters a field-free drift space (drift tube) toward the output (catcher) cavity. Because faster electrons (which entered slightly later, near the positive peak of the RF cycle) gradually catch up with slower electrons that entered slightly earlier, the initially velocity-modulated but density-uniform beam progressively develops periodic density bunches as it travels down the drift space — this conversion of velocity modulation into density (current) modulation via differential transit time is the essential 'bunching' mechanism of the klystron, and the resulting bunched beam current, when it passes through the output cavity gap, induces a strongly amplified RF voltage there, delivering the amplified output signal — completing the basic klystron amplification mechanism from the velocity-modulation equation derived above.

Q5Microwave Theory And Techniques

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