The transfer function for the proposed feedback system for the LHC can be described as a low-pass filter with a half-bandwidth of 730 kHz. Its response is thus described by the function
where 
kHz.
A is adjusted so that
the maximum gain of all the multibunch modes will cancel a rise time
of 10 ms. A particle receives a kick approximately two turns after
the pickup sees it, at approximately the same point in the ring
[11]. To get damping, I set the pickup to kicker
distance to be 127.25 betatron oscillations (at the zero-current
betatron frequency). See [1] for a discussion of how
feedback is modeled.
Figure 5 shows the effect of the feedback on the multibunch growth rates.
Figure 5: Growth rates of multibunch modes at injection, with feedback.
It can be seen that many of the m=0 modes that had large growth rates in Fig. 3 have been eliminated. However, Fig. 6 demonstrates that many of the m=0 modes have not been eliminated.
Figure 6: Growth rates of m=0 multibunch modes at injection, with feedback.
This is because the bandwidth of the feedback is very low, and thus its damping is over two orders of magnitude smaller than the peak damping rate for some of the m=0 multibunch modes. In fact, Fig. 6 indicates that the insufficiently damped modes are not only from cavity modes, but also from the resistive wall impedance.
The feedback damped only 49% of the highest peak shown in
Fig. 6. Thus, since in the high-frequency tail of the
feedback response (9), the real part of the transfer
function is proportional to 
, increasing the
bandwidth by a little over a factor of 1.4 should be sufficient to
damp all of the modes. It turns out, for reasons possibly related to
mode coupling, that it is necessary to increase the bandwidth by a
factor of two to damp all the m=0 modes. Once this is done, the
growth rates look like those shown in Fig. 7.
Figure 7: Growth rates of multibunch modes at injection,
with feedback having a half bandwidth of 1460 kHz. All m=0 modes are damped.
