Paper  Title  Page 

WEPPP061  A Method to Obtain the Frequency of the Longitudinal Dipole Oscillation for Modeling and Control in Synchrotrons with Single or Double Harmonic RF Systems  2846 


Funding: This work was partly funded by GSI Helmholtzzentrum für Schwerionenforschung GmbH In a heavyion 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 For a larger bunch or a double harmonic RF systems that introduces nonlinearities**, this is no longer valid.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. * F. Pedersen and F. Sacherer, IEEE Transactions on Nuclear Science, 24:1296–1398, 1977 ** A. Hofmann and S. Myers, Proc. of the 11th International Conference on High Energy Acceleration, 1980 

THPPC076  Comparison of LLRF Control Approaches for High Intensity Hadron Synchrotrons: Design and Performance  3464 


Funding: Federal Ministry of Education and Research A usual and effective way to realize closedloop controllers is to use cascaded SISO feedback and to rely on some kind of linear PID structure with parameters adjusted manually in simulations or experiments. Such a control may not reach optimal performance if the system is coupled or nonlinear. Regarding intense beams, longitudinal beam loading can be compensated by detuning. But the coupling between phase and amplitude (or I and Q component) highly depends on the tuning, that is on the resonant frequency of the cavity. It is derived that cavity and beam dynamics thus show bilinear nature, i.e. belong to a well investigated class of nonlinear systems with appropriate control strategies available*. Different controller designs are compared in terms of performance but also design transparency, the need of previous knowledge like the expected magnitude of beam loading and adaptability to different conditions, e.g. during acceleration or if applied to the full range of ion species as at GSI. The performance evaluation is based on macroparticle tracking simulations. In particular avail and limits of an optimal (quadratic cost) MIMO controller for bilinear systems are shown**. * H.K. Khalil: Nonlinear Systems, 3rd Edition, PrenticeHall, 2002 ** Z. Aganović, Z. Gajić: Linear Optimal Control of Bilinear Systems, SpringerVerlag, 1995. 
