
456 Microwave Engineering
In Equations 12.84 and 12.85,
υ
0
is the dc electron velocity,
υ
e
is the magni-
tude of the velocity uctuation in the velocity-modulated electron beam,
ω
e
is the angular frequency of velocity uctuation and
θ
e
is the phase angle of
the velocity uctuation.
Substitution of Equation 12.85 in Equation 12.83 gives
m
υ
e
ω
e
sin (
ω
e
t +
θ
e
) = e E
1
sin(
ω
t +
β
p
z) (12.86)
There will be interaction between the electric eld and the electron beam,
provided the velocity of the modulated electron beam and the dc electron
velocity are approximately the same, that is,
υ
≈
υ
0
. Thus, the distance trav-
elled by the electrons is
z =
υ
0