With cooling, beam phase space density increases, which makes the beam motion intrinsically unstable. To suppress instabilities, dampers are required. With a progress of digital technology, digital dampers are getting to be more and more preferable. Conversion of an analog signal into digital one is described by a linear operator with explicit time dependence. Thus, the analog-digital converter (ADC) does not preserve a signal frequency. Instead, a monochromatic input signal is transformed into a mixture of all possible frequencies, combining the input one with multiples of the sampling frequency. Stability analysis has to include a cross-talk between all these combined frequencies. In this paper, we are analyzing a problem of stability for beam transverse microwave oscillations in a presence of digital damper; the impedance and the space charge are taken into account. The developed formalism is applied for antiproton beam in the Recycler Ring (RR) at Fermilab.

Transverse beam instability of a coasting beam with a digital damper is examined. Threshold of instability is calculated in specific cases with Landau damping taken into account. The results are applied to the Fermilab Recycler Ring. Some improvement of existing RR damper is proposed.

We present a short summary of use of longitudinal and transverse dampers in the Tevatron Run II operation (2001-2006).

This book presents the latest results from high energy physics laboratories. The topics discussed include: Cosmology, Heavy Ions, Electroweak, Heavy Flavour Physics and CP Violation/Rare Decays, QCD and Beyond the Standard Model, Planck Scale Physics, Accelerator and Non-Accelerator Physics and Instrumentation.

Fermilab Booster has two transverse dampers which independently suppress beam instabilities in the horizontal and vertical planes. A suppression of the common mode signal is achieved by digital notch filter which is based on subtracting beam positions for two consecutive turns. Such system operates well if the orbit position changes sufficiently slow. Unfortunately it is not the case for FNAL Booster where the entire accelerating cycle consists of about 20000 turns, and successful transition crossing requires the orbit drifts up to about 10 [mu]m/turn, resulting in excessive power, power amplifier saturation and loss of stability. To suppress this effect we suggest an improvement of the digital filter which can take into account fast orbit changes by using bunch positions of a few previous turns. To take into account the orbit change up toN-th order polynomial in time the system requires (N + 3) turns of "prehistory". In the case of sufficiently small gain the damping rate and the optimal digital filter coefficients are obtained analytically. Numerical simulations verify analytical theory for the small gain and predict a system performance with gain increase.

The second edition of Flight Stability and Automatic Control presents an organized introduction to the useful and relevant topics necessary for a flight stability and controls course. Not only is this text presented at the appropriate mathematical level, it also features standard terminology and nomenclature, along with expanded coverage of classical control theory, autopilot designs, and modern control theory. Through the use of extensive examples, problems, and historical notes, author Robert Nelson develops a concise and vital text for aircraft flight stability and control or flight dynamics courses.