STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA

with Eq. (7.4). It follows that
KF B ≡
etoωk(R/Q)M eff
2(c/e)pb
where M eff takes the place of M ∗ in Eq. (4.37) and is given by
171
(7.5)
M eff = M34 + gM p
34M k 34e iωtd (7.6)
After explicitly performing the sum, a modified dispersion relation is found
1
Io
= KF Be iΩTr
ξ sin(ωto)
1 − 2ξ cos(ωto) + ξ2
(7.7)
where ξ is given by Eq. (4.42). After assuming a perturbative solution and expanding
in Ω, the threshold current is found to be
where
2Vb
Ith = −
(7.8)
k(R/Q)QLM F B
M F B = M34e ωTr
2Q L sin(ωTr) + gM p
34M k 34e ω(Tr+t d )
2Q L sin(ω(Tr + td)) (7.9)
As expected, with no feedback g = 0, the threshold current derived in Section 4.2
(and using an alternative method in Section 4.4) is recovered.
For a physically viable solution, the threshold current must be a positive quan-
tity. This condition requires that M F B < 0 for Eq. (7.8) to be valid. Equation (7.8)
is based on a perturbative treatment of the problem. Thus, for the perturbative solu-
tions to be valid, the gain, g, must be less than |M34/M p
34M 34
k | for sin(ω(Tr +td)) > 0
and greater than |M34/M p
34M 34
k | for sin(ω(Tr + td)) < 0.
When these conditions are not met, the system is said to be in a pseudo-stable
regime, where the negative threshold current implies beam stability. As discussed