Properties of a transverse damping system, calculated by a simple matrix formalism

In a synchrotron, proton beams with injection steering errors perform coherent betatron oscillations, possibly of large amplitude. The oscillations may be damped by using a system of a beam position monitor and a variable, fast kicker combined in a feedback loop to form a `transverse damper´. The system of ring, beam and damper can be modeled by iteration of a matrix mapping once per turn. This paper reports the calculation of damping rates, and coherent tune shifts by analytic solution of the recursions. Two cases are treated: (i) kick proportional to beam displacement; and (ii) `bang-bang´ damping in which, above a certain threshold, the kick depends only on the sign (+/-) of the displacement. We demonstrate (under certain conditions) that the `bang-bang´ scheme provides a linear damping of the amplitude and no tune shift, and (for the same peak power) is faster than the conventional proportional damper which produces an exponential damping with time