Impedance matrix - an unified approach to longitudinal coupled-bunch feedbacks in a synchrotron

Characteristic equation of coupled-bunch motion of beam governed by a feedback (FB) is given to find FB´s stabilizing effect against coherent instabilities or, say, injection error damping rates. Quite a general FB´s schematics is involved: (i) it has two paths, the in-phase and quadrature (or amplitude and phase in a small-signal approach), with unequal gains; (ii) may employ distinct RF-bands to pick-up beam data and feed correction back to the beam. To account for cross-talk between various field and beam current harmonics inflicted by frequency down- and upmixing, an impedance matrix (with, at most, three non-trivial elements per row) is introduced as a natural concept to gain insight into `FB and beam´ dynamics. The important class of FBs to counteract heavy beam loading of accelerating cavities is included into analysis as a particular case