Stability of an ion beam in synchrotrons with digital filters in the feedback loop of a transverse damper

Abstract

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¢.

Transient beam response in synchrotrons with a digital transverse feedback system

“…Let the driving kicker K located in point s D change the angle of the bunch trajectory on Δx D [n]. For the bunch states at two consecutive turns after passing the damper kicker DK located in point s K one can write [2,3]:…”

“…The transient beam response to a driving force is analyzed below in framework of the discrete transformation approach developed in [2,3] for describing transverse beam dynamics in synchrotrons with a digital TFS.…”

Beam stability in synchrotrons with digital transverse feedback systems in dependence on beam tunes

“…(10) correspond to the solutions for frequencies of the betatron motion equation of a particle in synchrotrons. If the fractional part of the tune {Q} is not close to 0 or 0.5 [5,6], then the solutions of Eq. (10) in the linear approximation with g 1 are expressed by the following formula:…”