Studies of beam dynamics in relativistic klystron two-beam accelerators Page: 33 of 229
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accelerating gradient with longitudinal position including beam loading effects,
G,(z) = Go exp (-az) - Ior [1 - exp (-az)] . (2.3)
The power flowing in the structure at any point is then given by PS(z) = G2/(2ar). At
the end of the structure (z=L), the gradient has the finite value
GL = Go exp (-T) - Ior 1 - exp (-T
and the remaining power (PSL) flows into the matched load. The fraction of the input
power deposited into the load is given by
CIS = PSL/Po = (GL/Go)2 [exp (-T) - X (1 - exp 2
with x = Ior /Go.
The power delivered to the beam during its interaction with the structure is Pb = IoV,
where V beam-loaded voltage gain
V(z) = dz'GS(z').
Performing this integration we obtain the power absorbed along the structure length
Pb(z) = (IoGoL + IorL) (1 - exp (-az) - IorL,
and the net absorbed power at z=L is
[(1 x 1 - ex; (-T))-x
[Iep()PbL = IoGoL 1+ X) T
The net efficiency of extraction is then
/b = PbLPo = 2X (1 + X) (1 - exp (-T))-2XT. (2.4)
Additionally, by averaging the beam-loaded gradient (2.3) over the length of the structure
we obtain
G = Go [(1+ X) - T ()L
This last quantity in brackets is the inverse of the peak field enhancement factor which
relates the peak on-axis fields in the structure to the average accelerating gradient, and15
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Lidia, Steven M. Studies of beam dynamics in relativistic klystron two-beam accelerators, thesis or dissertation, November 1, 1999; California. (https://digital.library.unt.edu/ark:/67531/metadc724108/m1/33/: accessed April 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.