Transverse feedback system with digital filter

Abstract

Main theoretical results for a transverse feedback system with a digital IIR-filter (infinite duration impulse response filter) and FIR-filter (finite duration impulse response filter) are described. The Z-transform method is used to solve the problem of the beam dynamics in the accelerator with a digital feedback. The analytical solution for the damping time and for the eigen frequencies are obtained and the system stability analyzed.

Nonlinear damping of injection oscillations

“…The order of these corrections is given by small parameter g. The amplitude and phase are the functions of a n . Hence, for their derivatives we can write: da n dn = gA 1 घa n ङ + g 2 A 2 घa n ङ + : : : ; (6) dऐ n dn = ऌ + gऄ 1 घa n ङ + g 2 ऄ 2 घa n ङ + : : : : (7) Functions ऎ m as the periodical functions of ऐ n can be expanded into the Fourier series: ऎ m घa n ; ऐ n ङ = ऍ m0 घa n ङ + 1 X k=2 ëऍ mk घa n ङ coskऐ n + ae mk घa n ङ sin kऐ n ë ;…”

“…For the left-hand side of (3) we expand all values into the Taylor series taking into account (6) and (7). The first approximation of these expansions is:…”