Axial Electric Field and Wave Modes in TWT

Axial Electric Field in TWT:

It defines how the electric field changes with convection current.

Let ‘L’ be the inductance, ‘C’ be the capacitance, ‘I’ be the alternating current, ‘V’ be the alternating voltage and ‘i’ be the convection current. The current flowing is defined by the Equation,

∂I/ ∂z = -C∂V/ ∂t - ∂i/ ∂z -------------------------- (1)

Let,

∂/ ∂z = -γ and ∂/ ∂t = jω

Therefore equation (1) becomes,

-γ I = - jωCV + γi ---------------------------- (2)

The voltage across the electric field is defined as,

∂V/ ∂z = -L∂I/ ∂t

-γV = - jωLI

I = γV/ jωL --------------------------- (3)

Sub eq (3) in (1)

-γ²V/ jωL = -jωCV + γi

γ²V = -jωL(-jωCV + γi)

= -CVω²L - jωL γi

When the convection current, i = 0

γ²V = -CVω²L

γ² = -Cω²L

γ =√(-Cω²L)

γ = jω√(LC)

This equation represents propagation constant γ for an ideal transmission line.

ie, γ₀ = jω√(LC)

γ₀² = -ω²LC ---------------------------- (4)

When the convection current is not zero,

γ²V = -ω²LCV – γijωL

γ²V + ω²LCV = -γijωL

V(γ² + ω²LC) = -γijωL

V = -γijωL/(γ² + ω²LC)

Sub in eqn (4)

V = -γωLij/(γ² - γ₀²) ------------------------- (5)

Characteristic impedance,

Z_o = √(L/C)

Multiplying both sides by γ₀

γ₀ Z_o = γ₀√(L/C)

Sub for γ₀ from eq (4)

γ₀ Z_o = jω√(LC). √(L/C)

= jωL

Sub the value of γ₀ Z_o in eq (5)

V = -iγ γ₀ Z₀ /(γ² - γ₀²)

The electric field is defined as,

E_z = -∇. V

E_z = -∂V/ ∂z  (we have, ∂/ ∂z = -γ)

E_z = γV

Sub. for V.

Therefore, E_z = -γ² γ₀ Z₀ i/(γ² - γ₀²)

This equation is called circuit equation as it determines the electric field from the convection current, i.

Wave Modes in TWT:

The solution for propagation constant ‘γ’ in the electronic and circuit equations provides four distinct solutions. These four solutions represent four modes of propagation in a travelling wave tube. The values of four propagation constants are given by,

γ₁ = -β_e c √(3/2) + jβ_e [1 + (c/2)]

γ₂ = β_e c √(3/2) + jβ_e [1 + (c/2)]

γ₃ = jβ_e [1 - c]

γ₄ = -jβ_e [1 – c³/4]

The wave corresponding to γ₁ is a forward wave and its amplitude grows exponentially with distance. The wave corresponding to γ₂ is a forward wave whose amplitude decays exponentially. The wave corresponding to γ₃ is a forward wave whose amplitude remains constant. The wave corresponding to γ₄ is a backward wave whose amplitude remains constant. The growing wave propagates at a velocity slightly lower than electron velocity and energy flows from electron beam to travelling wave. The decaying wave propagates in the same manner as that of the growing wave but the energy flows from travelling wave to electron beam. The constant amplitude wave travels with a velocity higher than electron velocity and no energy transfer occurs. The backward wave propagates in negative z direction with a velocity higher than electron velocity.

Gain Characteristics:

The gain of the travelling wave tube is defined as,

A_p = 10 log |output voltage/input voltage|²

A_p = -9.54 + 47.3 NC dB

Where, N is length and C is gain parameter.