/sci/ - Science & Math

>>6527744
bump 4 interest

en.wikipedia.org/wiki/Traveling-wave_tube

velocity modulation?

>>6527897
good copy / paste, but it in no way answers OPs question.

What is the name of the effect that causes the amplification in a TWT?

>>6527904
>velocity modulation?
that's about as close as anything I've found, thank you anon.

it seems the velocity of the charge in conjunction with the velocity of the wave interact.

the whole thing reminds me of aeroelastic flutter, but with EM instead.

Chu & Jackson have the following for the TWT anylysis
2.1 General Field and Wave Equations.

If the fields are circularly symmetric about the axis and assumed to vary with <span class="math">e^{j\omega t \,- \, \gamma\, z}[/spoiler]

The TE wave is described by:
(1)
<span class="math"> \gamma E_\phi + j\omega\mu \, H_r = 0 [/spoiler]
<span class="math">\frac {1}{r} \frac {\partial}{\partial_r}\, (rE_\phi) + j\omega\mu \, H_r = 0[/spoiler]
<span class="math">\frac {\partial H_z}{\partial_r} + \gamma H_r + j\omega\epsilon \, E_\phi = -J_\phi [/spoiler]

The TM wave is described by:
(2)
<span class="math">\gamma H_\phi - j\omega\epsilon \, E_r = J_r [/spoiler]
<span class="math">\frac {1}{r} \, \frac {\partial}{\partial_r} \, (rH_\phi) - j\omega\epsilon \, E_z = J_z[/spoiler]
<span class="math">\frac {\partial E_z}{\partial_r} + \gamma E_r - j\omega\epsilon \, H_\phi = 0 [/spoiler]

Where:
z is helix axis
r is helix radius
<span class="math">\phi[/spoiler] is angle of helix (thread pitch angle)
<span class="math"> \gamma = \alpha + j\beta [/spoiler] is the propagation constant along the z axis
<span class="math">E_z , E_r , E_\phi [/spoiler] are the electric field components
<span class="math">H_z , H_r , H_\phi [/spoiler] are the magnetic field components
<span class="math">J_z , J_r , J_\phi [/spoiler] are the components of the vector current density

>>6528347
(cont)
The grouping of field components into TE and TM waves is for mathematical convenience only. All six components are required to satisfy the boundary conditions on the helix. From the field equations the following inhomogeneous wave equations for <span class="math"> H_z \: and \; E_z\:[/spoiler] can be deduced.
(3)
<span class="math">\frac{1}{r}\,\frac {\partial}{\partial_r} \,(r\frac {\partial H_z}{\partial_r})+(\gamma^2 + k^2)\,H_z \; = \; -\frac {1}{r}\,\frac {\partial}{\partial_r}\,(r\:j_\phi)[/spoiler]

(4)
<span class="math">\frac {1}{r}\,\frac{\partial}{\partial_r}\,(r\frac{\partial E_z}{\partial_r})+(\gamma^2 + k^2)\,E_z \; = \; -\frac {(\gamma^2 + k^2)}{j\omega\epsilon} \, J_z \: + \: \frac {\gamma}{j\omega\epsilon}\,\frac{1}{r}\,\frac{\partial}{\partial_r}\,(rJ_r)[/spoiler]

where <span class="math">k^2\,=\,\omega^2 \mu\epsilon [/spoiler]

2.2 TE wave within the electron beam.

Since the electrons are assumed to have no transverse motion, the a-c current density J has only one component namely <span class="math">J_z [/spoiler]. Thus equation (3) for <span class="math">H_z [/spoiler] reduces to the form of a homogeneous wave equation:
(5)
<span class="math">\frac{1}{r}\,\frac{d}{d_r}\,(r\frac{dH_z}{d_r})+(\gamma^2 + k^2)\,H_z \; = 0 [/spoiler]

The other components of the TE wave are given by:
<span class="math">E_\phi \, = \, \frac{j\omega\mu}{\gamma^2 \,+ \,k^2}\,\frac{dH_z}{d_r} [/spoiler] and <span class="math">H_r \,= \, \frac{-\gamma}{\gamma^2 \,+\,k^2}\,\frac{dH_z}{d_r}[/spoiler]

>>6528351
(cont)
The solution of equation (5) involves Bessel's functions. It is convenient to write the solution in the form:
(6)
<span class="math">H_z \,=\,\{ A_1 I_0 \,(pr) \, + \, A_2 K_0 \, (pr) \} \, e^{j\omega t\, - \, \gamma z} [/spoiler]

where:
(7)
<span class="math">p^2 \, = \,-(\gamma^2 \, + \, k^2) [/spoiler]

and <span class="math">I_0(x) , K_0(x) [/spoiler] are modified Bessel's functions of the zeroth order, related to the more familiar Bessel's and Hankel's funcitons through the equations:
(8)
<span class="math">I_\mu(x) \,= \, j^{-\mu} \, j_\mu (jx) [/spoiler]
<span class="math">K_\mu(x) \,= \, \frac{\pi}{2} \, j^{\mu +1} \,H_\mu^{(1)} \, (jx) [/spoiler]

Since <span class="math">H_z[/spoiler] is continuous and finite at r=0, the constant <span class="math">A_2 [/spoiler] in equation (6) must be identically zero. The solutions for the components of the TE wave within the electron beam are:
(9)
<span class="math">H_z \,= \, A_1 I_0 \,(pr)e^{j\omega \,t \,- \gamma z} [/spoiler]
<span class="math">H_r \,= \, A_1 \frac{\gamma}{p} I_1 \,(pr)e^{j\omega \,t \,- \gamma z}[/spoiler]
<span class="math">E_\phi \,= \, -A_1 \,\frac {j\omega \mu}{p} \, I_1 \,(pr)e^{j\omega \,t \,- \gamma z} [/spoiler]

>>6528363
this is all from
>FIELD THEORY
OF TRAVELING-WAVE TUBES
L. J. CHU AND D. JACKSON

In this report, only the helix type of traveling-wave tube will be considered.
It consists of a cylindrical helical coil which, in the absence of an electron beam, is
capable of supporting a wave along the axis of the helix with a phase velocity substantially
less than the light velocity. When an electron beam is shot through the helix,
the electrons are accelerated or decelerated by the field of the wave, especially the
longitudinal electric field. As a result, the electrons will be bunched. The bunched
beam travels substantially with the initial velocity of electrons which is usually
different from the phase velocity of the wave. Because of the bunching action, there
will be, in time, more electrons decelerated than those accelerated over any cross
section of the helix or vice versa. As a result, there will be a net transfer of
energy from the electron beam to the wave or from the wave to the beam. The bunching
of the electrons produces an alternating space charge force or field which modifies
the field structure of the wave and consequently its phase velocity. The average
energy of the electron beam must change as it moves along on account of the energy
transfer. The process is continuous and a rigorous solution to the problem is probably
impossible. The procedure of analysis is therefore to find the modes of propagation
which can have exponential variation along the tube in the presence of the electron
beam. We are interested in those modes which will either disappear or degenerate into
the dominant mode when the beam is removed. By studying the properties of these modes
and combining them properly, we hope to present a picture of some of the physical
aspects of the helix-type traveling-wave tube.

>>6528363
(cont)
2.3 Dynamics of the Electron Beam.

It will be seen from Eq. (4) that a knowledge of <span class="math">J_z[/spoiler] as a function of <span class="math">E_z[/spoiler] is necessary in order to obtain the solution of the TM wave within the electron beam. To find such a relationship, the behavior of the electrons under the action of electric and magnetic fields must be considered.

As was indicated earlier, the motion of the electrons is assumed to be confined to the axial direction. In practice this assumption is very nearly realized by means of the focusing action of a strong d-c magnetic field applied parallel to the helix axis. It is also assumed that the a-c components of charge, current, electron velocity vary exponentially with the same propagation constant as the wave traveling in the helix, while the average electron velocity is substantially constant over a finite section of the helix. This last assumption depends on the tacit supposition that the phenomena can be described by a small-signal analysis.

The notation used will be the following:
<span class="math">\rho[/spoiler] = a-c component of charge density
<span class="math">\rho_0[/spoiler] = average value of charge density
v = a-c component of electron velocity
<span class="math">v_o[/spoiler] = average value of electron velocity
<span class="math">J_z[/spoiler]= a-c component of current density
<span class="math">J_0[/spoiler]= average value of current density
<span class="math">\frac {e}{m}[/spoiler]= ratio of charge to mass of electron

Thus:
<span class="math">\vec{J} \, = \,\vec{i_r} J_r \,+ \, \vec{i_\phi} J_\phi \,+ \, \vec {i_z} (J_z \,+ \, J_0)[/spoiler]
<span class="math">\rho_T \, =\, \rho_0 \, + \, \rho[/spoiler]
where the <span class="math">\vec {i}[/spoiler] are unit vectors in cylindrical coordinates.

Continuity of charge demands that:

(10)
<span class="math">\nabla \cdot \, \vec {J} \,+ \, \frac {\partial \rho_T}{\partial_t} \, = \,0 [/spoiler]

>>6528430
(cont)
since
<span class="math">J_\phi \, = \, J_r \, = \, 0 [/spoiler]
and
<span class="math"> \frac {\partial J_0}{\partial_z} \, = \, \frac {\partial \rho_0}{\partial_t} \, = \, 0 [/spoiler]
the continuity equation becomes:
<span class="math"> \frac {\partial J_z}{\partial_z} \, = \, - \frac {\partial \, \rho}{\partial t} [/spoiler]
or

(11) <span class="math"> J_z \, = \, \frac {j\omega}{\gamma} \, \rho [/spoiler]

The force equation for non-relativistic motion is:
(12)
<span class="math">\vec{F} \, = \, q \; |\vec{E} \, + \, \vec{v} \cdot \vec{B}|[/spoiler]

Since the velocity of the electrons is a small fraction of the velocity of light, the force due to the magnetic field can be neglected in comparison to the force due to the electric field, so that:
(13)
<span class="math"> \frac {d}{dt} \, (v_0 \, + \, v) \, = \, - \frac {e}{m} \, E_z [/spoiler]

Now
(14)
<span class="math"> \frac {d}{dt} \, (v_0 \, + \,v) \, = \; \frac {dv}{dt} \, = \; \frac {\partial v}{\partial t} \, + \: v_0 \, \frac {\partial v}{\partial z} \, = \; v_0 \, (j \frac {\omega}{v_0} \, - \gamma )v [/spoiler]

Then eq (13) can be written as:
(15)
<span class="math"> v \, = \, \frac { (- \frac {e}{m} ) \, E_z}{v_o \, (j \, \frac {\omega}{v_0} \, - \, \gamma )} [/spoiler]

>>6527744
nobody?

>>6528715
nope