STUDIES OF ENERGY RECOVERY LINACS AT ... - CASA

FIG. 7.6: Generic layout for a feedback system in an ERL.
170
where M p
34 is the matrix element that transforms an angular kick from the cavity
HOM to a vertical displacement at the downstream pickup and td is the feedback
delay time. For td = 0, the feedback is a bunch-by-bunch system in the sense that
each bunch generates its own error signal and then is corrected by the kicker using
that signal. For the situation where td = 0, a bunch generates an error signal which
is only applied to the n th trailing bunch, for example.
The displacement on the second pass at the cavity due only to the effects of
the feedback is written as
yF B(t ′ ) = gM k 34yp(t ′ ) = gM p
34M k 34
′ V (t − Tr − td)
pb(c/e)
(7.3)
where M k 34 is the matrix element that transforms an angular kick from the feedback
kicker to a vertical displacement at the cavity. Finally, the net displacement on the
second pass is given by the sum of Eq. (7.1) and Eq. (7.3)
y2(t ′ ) =
1
M34V (t
pb(c/e)
′ − Tr) + gM p
34M k 34V (t ′ − Tr − td)
(7.4)
With this new expression for the bunch displacement, the threshold current can
be derived following the same steps outlined in Section 4.4 by replacing Eq. (4.33)