The transient beam response to an externally applied impulse force in synchrotrons with a digital transverse feedback system is studied. Experimental data from the LHC on damping of coherent transverse oscillations excited by the discrete-time unit impulse are analysed. Good agreement on the measured and theoretically predicted decrements has been obtained. A method of feedback fine tuning, based on measurements of the bunch response to the harmonic excitation impulse, is discussed.
The stability of an ion beam in synchrotrons with digitalˇlters in the feedback loop of a transverse damper is treated. A transverse feedback system (TFS) is required in synchrotrons to stabilize the high-intensity ion beams against transverse instabilities and to damp the beam injection errors. The TFS damper kicker (DK) corrects the transverse momentum of a bunch in proportion to its displacement from the closed orbit at the location of the beam position monitor (BPM). The digital signal processing unit in the feedback loop between BPM and DK ensures a condition to achieve optimal damping. Damping rates of the feedback systems with digitalˇlters are analyzed in comparison with those in an ideal feedback system. •¨¢μ¤ÖÉ¸Ö •¥ §Ê²ÓÉ ÉÒ É¥μ•¥É¨Î¥¸±μ£μ¨¸¸²¥¤μ¢ ´¨Ö ʸÉμ°Î¨¢μ¸É¨¨μ´´μ£μ ¶Êα ¢¸¨´Ì•μÉ•μ-Ì ¢ § ¢¨¸¨³μ¸É¨μÉ ¶ • ³¥É•μ¢ ͨ˕μ¢ÒÌ Ë¨²ÓÉ•μ¢ ¢ Í¥ ¶¨μ¡• É´μ°¸¢Ö §¨¸¨¸É¥³ ¶μ¤ ¢²¥´¨Ö (') ±μ£¥•¥´É´ÒÌ ¶μ ¶¥•¥Î´ÒÌ ±μ²¥¡ ´¨° ¶Êα. ' ¶•¨³¥´ÖÕÉ¸Ö ¢¸¨´Ì•μÉ•μ´ Ì¸Í¥²ÓÕ¸μ-§¤ ´¨Ö ʸ²μ¢¨°¤²Ö ¶•¥¤μÉ¢• Ð¥´¨Ö • §¢¨É¨Ö ±μ£¥•¥´É´ÒÌ ¶μ ¶¥•¥Î´ÒÌ´¥Ê¸Éμ°Î¨¢μ¸É¥° ¶Êα , É ±¦¥ ¤²Ö ¤¥³ ¶Ë¨•μ¢ ´¨Ö μ¸É ÉμδÒÌ ±μ²¥¡ ´¨°Î ¸É¨Í ¶μ¸²¥¨´¦¥±Í¨¨. ' μ¡¥¸ ¶¥Î¨¢ ÕÉ ±μ••¥±Í¨Õ ¶μ ¶¥•¥Î´μ£μ¨³ ¶Ê²Ó¸ ¸£Ê¸É±μ¢´ ± ¦¤μ³ μ¡μ•μÉ¥¸ ¶μ³μÐÓÕ ¤¥Ë²¥±Éμ• ¸ÊÎ¥Éμ³ ¤ ´´ÒÌ μ¸³¥Ð¥´¨¨Í¥´É• ÉÖ¦¥¸É¨ ¶Êα ,¨ §³¥•¥´´ÒÌ ¤ ÉΨ±μ³ ¶μ²μ¦¥´¨Ö. "²Ö ¤μ¸É¨¦¥´¨Ö μ ¶É¨³ ²Ó´ÒÌ Ê¸²μ¢¨°¤¥³ ¶Ë¨•μ¢ ´¨Ö ¶μ ¶¥•¥Î´ÒÌ ±μ²¥¡ ´¨°¸£Ê¸É±μ¢¨¸ ¶μ²Ó §ÊÕÉ¸Ö Í¨Ë•μ¢Ò¥ ³¥Éμ¤Ò μ¡• ¡μɱ¨¸¨£´ ²μ¢ ¢ Í¥ ¶¨μ¡• É´μ°¸¢Ö §¨³¥¦¤Ê ¤ ÉΨ±μ³ ¶μ²μ¦¥´¨Ö¨¤¥Ë²¥±Éμ•μ³. •¨¢μ¤ÖÉ¸Ö ¤ ´´Ò¥ μ § ¢¨¸¨³μ¸ÉÖÌ É¥³ ¶ ¶μ¤ ¢²¥´¨Ö ±μ£¥•¥´É´ÒÌ ±μ²¥¡ ´¨°¤²Ö • §²¨Î´ÒÌ ¶-• ³¥É•μ¢ ͨ˕μ¢ÒÌ Ë¨²ÓÉ•μ¢.
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.
The problem of transverse emittance blow-up from beam injection errors in synchrotrons with nonlinear feedback systems is considered. The relative emittance growth is calculated for linear and nonlinear feedback transfer functions. Effects of an increase of the damping decrement of the beam coherent oscillations and of a decrease of the coherent transverse amplitude spread of different bunches in case of the damper with a positive cubic term in the feedback transfer function are discussed.
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