Longitudinal coupling impedance imposed by a beam feedback in a synchrotron

Commonly, a longitudinal beam feedback processes a slowly varying signal at zero intermediate frequency (a phase offset, an amplitude departure). Often, only a portion of the data confined in a picked-up band-pass beam signal is retained (like, say, in a purely phase feedback). Sometimes, a beam feedback employs different RF bands to pick up beam data and return a correction back to the beam. All the manipulations thus involved with signal spectra result in cross-talk between various beam-current and electric-field waves propagating along the orbit, which is shown to be described by an impedance matrix with, at most, three non-trivial elements per row. It is this matrix which gives the intuitive notion that a linear feedback is seen by a beam as an artificial coupling impedance controlled from the outside from a quantitative basis. This (impedance) approach has at least two plain advantages: (i) It allows one to mount the feedbackʹs effect into the well-established theory of longitudinal coherent instabilities to use most of its inventory: beam transfer functions, threshold maps, handling of coupled-bunch motion, etc. (ii) The destabilizing effect of the beam environment, being available in standard terms of coupling impedances, is naturally taken into account since the early stages of feedback R&D.