Modeling longitudinal bunched beam dynamics in hadron synchrotrons using scaled fourier-hermite expansions

To devise control strategies and to analyze the stability of systems with feedback, a set of few ordinary differential equations (ODEs) describing the underlying dynamics is required. It is deduced by combining two approaches not used in that context before: (I) Numerical Fourier-Hermite solutions of the Vlasov equation have been studied for over fifty years [1, 2]. The idea to expand the distribution function in Fourier series in space and Hermite functions in velocity is transferred to the dynamics of bunched beams in hadron synchrotrons in this contribution. The Hermite basis is a natural choice for plasmas with Maxwellian velocity profile as well as for particle beams with Gaussian momentum spread. The Fourier basis used for spatially nearly uniform plasmas has to be adapted to bunched beams where the beam profile is not uniform in phase. (II) This is achieved analogously to the deduction of the three term recursion relations to construct orthogonal polynomials, but applied to Fourier series with the weight function taken from the Hamiltonian. The resulting system of ODEs for the expansion coefficients of desired order - dependent on the number of functions retained - is roughly checked against macro particle tracking simulations

Impact of Simplified Stationary Cavity Beam Loading on the Longitudinal Feedback System for SIS100

The main synchrotron SIS100 of the Facility for Antiproton and Ion Research (FAIR) will be equipped with a bunch-by-bunch feedback system to damp longitudinal beam oscillations. In the basic layout, one three-tap finite impulse response (FIR) filter will be used for each single bunch and oscillation mode. The detected oscillations are used to generate a correction voltage in dedicated broadband radio frequency (RF) cavities. The digital filter is completely described by two parameters, the feedback gain and the passband center frequency, which have to be defined depending on the longitudinal beam dynamics. In earlier works, the performance of the closed loop control with such an FIR-filter was analyzed and compared to simulations and measurements with respect to the damping of coherent dipole and quadrupole modes, the first modes of oscillation. This contribution analyzes the influence of cavity beam loading on the closed loop performance and the choice of the feedback gain and passband center frequency to verify future high current operation at FAIR

A Method to Obtain the Frequency of the Longitudinal Dipole Oscillation for Modeling and Control in Synchrotrons with Single or Double Harmonic RF Systems

In a heavy-ion synchrotron the bunched beam can perform longitudinal oscillations around the synchronous particle (single bunch dipole oscillation, SBDO). If disturbances/instabilities exciting the SBDO exceed the rate of Landau damping, the beam can become unstable. Furthermore, Landau damping is accompanied by an increase of the beam emittance which may be undesired. Thus, control efforts are taken to stabilize the beam and to keep the emittance small. It is known that for a single harmonic cavity and a small bunch the SBDO oscillates with the synchrotron frequency if the oscillation amplitudes are small. For a larger bunch or a double harmonic RF systems that introduces nonlinearities, this is no longer valid. This work shows how the frequency of the SBDO can be determined in general. As a result, the SBDO can again be modeled as a harmonic oscillator with an additional damping term to account for Landau damping. This model can be used for feedback designs which is shown by means of a simple example. As the frequency of the SBDO and the damping rate depend on the size of the bunch in phase space, it is shown how this information can be obtained from the measured beam current

On the Impact of Empty Buckets on the Ferrite Cavity Control Loop Dynamics in High Intensity Hadron Synchrotrons

Due to technical reasons two of ten buckets have to stay empty in the planned SIS100 synchrotron at the GSI Helmholtzzentrum für Schwerionenforschung. The planned low level RF control systems consist of linear P and PI type controllers. These are responsible to maintain a desired phase and amplitude of the gap voltage. In addition the cavity is controlled to follow a prescribed resonance frequency ramp. In SIS100 the acceleration will be performed by ferrite cavities with comparatively small quality factors. Therefore, effects resulting from transient beam loading have to be expected. Influences due to empty buckets are analysed in the frequency domain and particle tracking simulations are carried out to estimate the effect on the overall system with particular consideration of emittance growth and particle loss

Precise verification of phase and amplitude calibration by means of a debunching experiment in SIS18

Several new rf cavity systems have to be realized for the FAIR synchrotrons and for the upgrade of the existing GSI synchrotron SIS18 [1]. For this purpose, a completely new low-level rf (LLRF) system architecture [2] has been developed, which is now used in SIS18 operation. Closedloop control systems stabilize the amplitude and the phase of the rf gap voltages. Due to component imperfections the transmission and the detection of the actual values lead to systematic errors without countermeasures. These errors prohibit the operation of the rf systems over the whole amplitude and frequency range within the required accuracy. To compensate the inevitable errors, the target values provided by the central control system are modified by socalled calibration electronics (CEL, [3]) modules. The calibration curves can be measured without the beam, but the desired beam behaviour has to be verified by experiments. For this purpose, a debunching scenario was selected as a SIS18 beam experiment that proved to be very sensitive to inaccuracies. In this contribution the results of this experiment are presented, showing for the first time at GSI by beam observation that the accuracy requirements are met based on predefined calibration curves

A digital beam-phase control system for a heavy-ion synchrotron with a double-harmonic cavity system

For the new Facility for Antiproton and Ion Research (FAIR) at GSI Helmholtzzentrum für Schwerionenforschung GmbH (http://www.gsi.de), the heavy ion synchrotron SIS18 will be operated with a double harmonic cavity system. The second cavity, running at twice the fundamental frequency, is used to create a lengthened bucket which introduces nonlinearities to the control system. To damp longitudinal rigid dipole oscillations a digital feedback system consisting of a filter and an integrator is used. For the existing single-harmonic setup an FIR-filter is implemented which realizes a multiple bandpass filter with the first passband centre frequency close to the synchrotron frequency. Both, the feedback gain and the passband frequency of the filter depend on the actual value of the synchrotron frequency. It was shown by simulations and in an experiment that this setup can be transferred to a double-harmonic cavity system obtaining similar results for the region of stabilizing feedback parameters, if the oscillation frequency of the bunch barycenter is considered instead of the synchrotron frequency of a linearized bucket

Generation of RF Frequency and Phase References on the FAIR Site

Based on the Bunch Phase Timing System (BuTiS) local analog radio frequency reference signals (RF references) like the particle revolution frequency and their multiple harmonics will be generated. These references are used to control the phase of the accelerator cavities to altering harmonics of the bunch revolution frequency. Delay or phase shifts from the FAIR-Center to references at the BuTiS endpoints are already compensated by the BuTiS receivers. Phase shifts from the RF reference generators to LLRF electronics can be compensated by controlling the output phases of the DDS modules of the RF references. However phase shift delays of multiple harmonics at the same interconnecting electrical path are not identical at the same time. Configurable electronics manage phase calibration of the RF references to their endpoints. Calibration may depend on frequency and harmonic of the RF reference, aging as well as on thermal effects. The electrical length and impedance of interconnecting cables for phase control loops can be compensated. This is an important feature, in particular if control loops are switched between different harmonic frequencies

Modeling and Control of Longitudinal Single-Bunch Oscillations in Heavy-Ion Synchrotrons

This thesis contributes to the modeling and analysis of longitudinal radio frequency (RF) feedback systems in heavy-ion synchrotrons. Synchrotrons are ring accelerators with a constant reference orbit of the particle beam. They allow the acceleration of particles such as electrons, protons, and heavy ions to highest energies. The desired specifications for beam properties such as the quality, energy, and intensity drive the development of new accelerator components. Among other objectives, the stabilization of the beam before and during the acceleration is desirable to preserve the beam quality. The thesis deals with the modeling of longitudinal coherent oscillations of a bunched beam. The main focus is on the usability of the models for the analysis and design of digital RF feedback loops. The analysis of these models with methods from control theory leads to new insight into the possibilities of RF feedback with regard to the longitudinal beam stabilization. In particular it is shown that the nonlinearity of the beam dynamics plays a major role in the damping of coherent oscillations of higher order. An analysis of a specific RF feedback setup and the comparison with experimental data shows the practical relevance of the models