**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)

-γ

^{2}V/ jωL = -jωCV + γi
γ

^{2}V = -jωL(-jωCV + γi)
= -CVω

^{2}L - jωL γi
When the convection current, i = 0

γ

^{2}V = -CVω^{2}L
γ

^{2}= -Cω^{2}L
γ =√(-Cω

^{2}L)
γ = jω√(LC)

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

ie, γ

_{0}= jω√(LC)
γ

_{0}^{2}= -ω^{2}LC ---------------------------- (4)
When the convection current is not zero,

γ

^{2}V = -ω^{2}LCV – γijωL
γ

^{2}V + ω^{2}LCV = -γijωL
V(γ

^{2}+ ω^{2}LC) = -γijωL
V = -γijωL/(γ

^{2}+ ω^{2}LC)
Sub in eqn (4)

V = -γωLij/(γ

^{2}- γ_{0}^{2}) ------------------------- (5)
Characteristic impedance,

Z

_{o}= √(L/C)
Multiplying both sides by γ

_{0}
γ

_{0}Z_{o}= γ_{0}√(L/C)
Sub for γ

_{0}from eq (4)
γ

_{0}Z_{o}= jω√(LC). √(L/C)
= jωL

Sub the value of γ

_{0}Z_{o}in eq (5)
V = -iγ γ

_{0}Z_{0}/(γ^{2}- γ_{0}^{2})
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}= -γ^{2}γ_{0}Z_{0}i/(γ^{2}- γ_{0}^{2})
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,

γ

_{1}= -β_{e}c √(3/2) + jβ_{e}[1 + (c/2)]
γ

_{2}= β_{e}c √(3/2) + jβ_{e}[1 + (c/2)]
γ

_{3}= jβ_{e}[1 - c]
γ

_{4}= -jβ_{e}[1 – c^{3}/4]
The wave corresponding to γ

_{1}is a forward wave and its amplitude grows exponentially with distance. The wave corresponding to γ_{2}is a forward wave whose amplitude decays exponentially. The wave corresponding to γ_{3}is a forward wave whose amplitude remains constant. The wave corresponding to γ_{4}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.

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