bunches circulating in each ring, will require a feedback system to avoid coupled-bunch instabilities. A computer model of the storage ring, including the RF system, wake fields, synchrotron radiation loss, and the bunch-bybunch feedback system is presented. The feedback system model represents the performance of a fast phase detector front end (including system noise and imperfections), a digital filter used to generate a correction voltage, and a power amplifier and beam kicker system. The combined ring-feedback system model is used to study the feedback system performance required to sup press instabilities and to quantify the dynamics of the system. Results are presented which show the time development of coupled bunch instabilities and the damping action of the feedback system. I

D. Briggs

G. Lambertson

J. -l. Pellegrin

J. D. Fox

K. A. Thompson

L. Klaisner

P. Morton

W. Hosseini

Briggs, D

Fox, J D

Hosseini, W

Klaisner, L

Morton, P

Pellegrin, J.-L.

Thompson, K A

Lambertson, G

eScholarship - University of California

Lawrence Berkeley National LaboratoryLBL PublicationsTitleComputer Modelling of Bunch-by-Bunch Feedback for the SLAC B-Factory DesignPermalinkhttps://escholarship.org/uc/item/0xr0k4xbAuthorsBriggs, DFox, J DHosseini, Wet al.Publication Date1991-05-01eScholarship.org Powered by the California Digital LibraryUniversity of California'4 LBL-30772 Lawrence Berkeley Laboratory UNIVERSITY OF CALIFORNIA Accelerator & Fusion Research Division Presented at the IEEE Particle Accelerator Conference, San Francisco, CA, May 6-9, 1991, and to be published in the Proceedings Computer Modelling of Bunch-by-Bunch Feedback for the SLAC B-Factory Design D. Briggs, J.D. Fox, W. Hosseini, L. Klaisner, P. Morton, J.-L. Pellegrin, K.A. Thompson, and G. Lambertson May 1991 ) I ~ \ ~renee Berkeley Laboratory Library, Berkeley REFERENCE Not to be taken from this room Prepared for the U.S. Department of Energy under Contract Number DE-AC03-76SF00098 ~ ('--... ' w (): "v" _-...,J. ~· DISCLAIMER This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor the Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or the Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof or the Regents of the University of California. BECON-96 SLAC-PUB-5466 LBL-30772 Computer modelling of bunch-by-bunch feedback for the SLAC B-factory design D. Briggs, J.D. Fox, W. Hosseini, L. Klaisner, P. Morton, J.-L. Pellegrin, K.A. Thompson Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309 and G. Lambertson, Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720 * Work supported by the Director, Office of Energy Research, Office of Basic Energy Scineces, U.S. Department of Energy under Contract No. DE-AC03-76SF00098. SLAC-PUB-5466 LBL-30772 May 1991 (A) Computer modelling of bunch-by-bunch feedback for the SLAC B-factory design* D. Briggs, J. D. Fox, W. Hosseini, L. Klaisner, P. Morton, J.-L. Pellegrin, K. A. Thompson Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309 USA and G. Lambertson Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720 Abstract The SLAC B-factory design, with over 1600 high cur-rent bunches circulating in each ring, will require a feed-back system to avoid coupled-bunch instabilities. A com-puter model of the storage ring, including the RF system, wake fields, synchrotron radiation loss, and the bunch-by-bunch feedback system is presented. The feedback system model represents the performance of a fast phase detec-tor front end (including system noise and imperfections), a digital filter used to generate a correction voltage, and a power amplifier and beam kicker system. The combined ring-feedback system model is used to study the feedback system performance required to sup-press instaJ;>ilities and to quantify the dynamics of the sys-tem. Results are presented which show the time develop-ment of coupled bunch instabilities and the damping ac-tion of the feedback system. I. INTRODUCTION The large average current in the SLAC B-factory de-sign is distributed into many bunches of sufficiently small charge to minimize the beam-beam interaction and single-bunch instabilities. Although the cavity higher-order-modes (ROMs) will be strongly damped (Q < 70), there will still be significant coupling of the longitudinal and transverse motion of adjacent bunches via wakefields. Fur-thermore, the high-Q accelerating mode can also strongly couple the bunches longitudinally. The resulting instabili-ties will be controlled via wideband, bunch-by-bunch feed-back. Such a feedback system can handle disturbances to the bunch motion arising from any source, including but not limited to wakefields and injection errors. The feedback systems to control the longitudinal and transverse coupled-bunch instabilities will be similar in ar-chitecture. Since the signal detection and kicker require-ments are more stringent for the longitudinal system and for the high energy ring (HER), we shall concentrate our discussion on this case. Basic longitudinal-feedback sys-tem specifications are shown in Table 1. The proposed system implementation, its block diagram and description, and hardware tests are discussed elsewhere [1]. Table 1: Basic feedback system specifications RF freq. 476 MHz Max. mode amplitude Injection scheme ~: injection error 6t injection error 10 ps = 0.03 rad 1/5 bunch at 60 pps 0.002 100 ps * Work supported by Department of Energy contract DE-AC03-76SF00515. II. SIMULATION MODEL The simulation model (see Fig. 1) consists of a model of the feedback system electronics, combined with a model of the dynamics of the bunches in the ring. The feedback system model simulates the transfer function of the feedback system and includes: (1) the elec-tronic properties of the phase detector, mixer, low-pass fil-ter, and A/D converter, (2) input noise, gain and offset errors, bandwidth limitations, and dynamic range of the analog components, and (3) the algorithm running in the set of digital signal processors (DSP farm), that takes as input the digitized bunch phases and calculates the longi-tudinal kick. / On uch turn, tnt kinematic modeltnrorms the eltctronlc modtl or the tongltudl~l cllsplacement of each Dune.f\ (ltctronlc Model of th• phase detection, stqnal processing. •nd kicker. Initial Conditions: Pickup Electrode I) Seam Cf6"rtnt Structure 2) 'e or Injected Dunch tnctudn. lJ 8 of othtr Dunehes physical model <4) Cavity tnOdU 5) RF. ri?Q puamettrs Average t t Syn~:rv-onous Pr.ast Hlxer, low-Pass, ADC prtOICtfd at 1n1Uallutlon tnclwu: sutract 100"1 of S)""Chronous p."\.an Kinematic model or the nolst, ortstt, gJin trror longitudinal oscillations dynamic ,.,.,!lf slnusoldallnnsrrr function usumtS Ch¥it normJIIntlon Ohcrtlt Propac;atlon an<S Kick Oandw ldth limitation moc:ltllnclueltS; IJ AF fUOOamtntal ' (wltl\ptrftet comptnsatlon) 2) Jotlghtr-<>rdtr Modr Waktrttlds In Cavltln DSP (algorithm) lJ 5ync1Votron Aacllatlon ton lnc:ludn: <4} fttdOack KICk matchtd rnur • IOf\9 osc. frtQ. lot'19ltu<ttnal sahratlon,. e Olt output tlxtel·polnt AI..V oscillation FrtQUtncy prtdiCltd at lniUallzation Ktcker lncludu: output- slntHnouV128J w/ D~wldU\ llmiUUon On tach tum, lht tltctronlc modtl I rtturns to tht klntmattc modtl a ) longitudinal kick ror uch Dunch. ' Figure 1. Block diagram of the feedback simulation model. The model of the DSP farm emulates a 20-tap finite impulse response (FIR) matched filter. The coefficients ~i of the taps comprise a sinusoid with the synchrotron period of 19.3 turns, that is, ej = AvsP sin 271"--( j -1) 19.3 (1) where j E 1, ... , n•ample•, and we take n 3 ample• = 20. Given measures <Pi(k) = cp;(k)- cp, (where cp, is the synchronous phase) of the phase of bunch i on successive turns k as input, the result of the DSP algorithm is the output: Presented at the IEEE Particle Accelerator Conference, San Francisco, CA, May 6-9, 1991 y;(k) = 2: ei~p;(k- i) (2) i=l This result is clipped to an 8-bit signed integer and used to set the phase of the kicker oscillator for that bunch on that turn. The kicker model is implemented as a phase-modulated RF kicker with a nominal4 ke V maximum out-put amplitude. The measurement of the phase of a bunch is assumed independent of its charge, i.e., it is assumed that a sepa-rate measurement of bunch charge is available for normal-ization. A propagation delay of at least one turn (7.33 ps) is enforced in the feedback transfer function. In the ring simulation model, a discrete kick is given to each bunch at a single point in the ring; that is, the system is modelled as though there were a discrete change in en-ergy at a single point on each turn. This simplification is justified since the synchrotron frequency is small compared to the revolution frequency. The kick given to bunch i is comprised of several components: (1) the RF cavity volt-age ~sin 'f'i + vcav.jbk.' where ~ is the peak generator voltage, cp; is the phase of bunch i with respect to the zero crossing of the RF, and vcav.Jbk. is the RF cavity feedback needed to control beam loading in the fundamental mode, (2) the wake field voltage vwake (including both the accel-erating mode and the HOM's) accumulated in the cavity up to the present moment, (3) the synchrotron radiation loss per turn, and (4) the voltage V;Jdbk applied to bunch i by the bunch-by-bunch feedback system. Thus, the equa-tion for the total kick is ll.<P; = - ~:;~ [~sin 'f'i + vcav.Jbk. U I + V wake + v;fdbk] 2To . - o e ; - -cp; T£ (3) where a is the momentum compaction factor, Wrf is the RF frequency, and T£ is the longitudinal radiation damp-ing time; Eo is the ring energy, To the revolution period, and Uo the synchrotron radiation loss per turn, for a par-ticle on the design orbit. In the present simulations it is assumed that the cavity feedback is perfect, so that the part of vwake due to the fundamental mode is exactly can-celled by vcav.Jbk.. More realistic models, including the cavity phase, amplitude, and tuning loops, and modifica-tion of the impedance at coupled-bunch frequencies that fall within the bandwidth of the fundamental mode (2] are under study (3]. III. SIMULATION RESULTS Parameters used in the simulations are shown in Ta-ble 2. The harmonic number of the HER is 3492, with every other bucket filled except for a 5% gap; the to-tal current is 1.48 A. The initial conditions used were cp5 = 0.2915 rad, that is, bunch 5 starts at an 0.1 rad offset from the remaining bunches i, which were started at the synchronous phase cp; = 0.1915 rad. The feedback gain in the examples described here was set so that a 5 mrad sinu-soid on the input (at the synchrotron oscillation frequency) corresponds to a 4 keV sinusoid on the kicker output. 2 0.30 ~ (/) c ell 0.26 '6 .ell .!::;.. X 0.22 Ill E -e-0.18 4-91 0 0 20 40 • ~max• Fbk OFF o ~max• Fbk ON 60 80 Bunch Number 100 6843A4 Figure 2. Plot of the maximum bunch offset reached in 3000 turns for the first 100 bunches after the gap, with and with-out feedback. (i) c Ill '6 Ill ..=.. §: -e-(i) c Ill 0.29 0.24 0.19 0.14 0.194 '6 0.192 ~ -e-0.190 (a) 0.188 ..L._ _ __::_.L__--l. __ _!_ __ ...l._ _ ____J 0 500 1 000 1500 2000 2500 3·91 Turns 6843A1 Figure 3. Plots of the longitudinal phases of (a) the injected bunch (#5) and (b) the bunch immediately following (#6), vs turn number, in the presence of feedback. Note the ex-panded vertical scale in (b). In Fig. 2, we show a plot of maximum bunch off-sets reached after 3000 turns, with and without feedback. Note that the time between injection p11lses is 1/60 second, which is about 2300 turns. In the aL:;euce of feedback, the disturbance shown would grow even larger and propagate further back in the bunch train. With the feedback system I I I I I I I J I DSP Input and Output for Bunch 5 (/) co c Ol U5 - DSPin (5) --- DSP out (5) ·--1 I I DSP Input and Output for Bunch 6 (/) co c Ol U5 3-91 400 440 1\ I \ I \ I \ I \ I I I \ I \ I 'I. Turn DSP in (6) DSP out (6) 480 6843A3 Figure 4. Plots of the input and output of the DSP model for (a) bunch #5, and (b) bunch #6, shortly after the injection of bunch #5. Table 2: Simulation parameters for HER Bunch charge 4 x 1010 Number of bunches 1658 Bunch interval Number of cavities Freq. of strongest HOM Q of HOM R/Q of HOM (per cav.) Vg a Uo 4.2 ns 20 750 MHz 70 33 n 18.5 MV 0.00241 3.52 MeV /turn turned on, the coupled bunch excitation does not extend beyond a very few bunches. Fig. 3 shows the phases of the injected bunch (#5) and the immediately following bunch, vs turn number. The envelope of the phase of the injected bunch damps linearly, reflecting the fact that the kicker saturates, and the phase 3 of the following bunch grows quickly to a maximum and then slowly damps. The excitation of subsequent bunches is strongly suppressed. Fig. 4 shows the input and output of the DSP model for bunches #5 and #6 shortly after injection. The DSP output saturates for bunch #5, but maintains the proper 90° phase lag. Such benign saturation behavior is difficult to realize with conventional analog approaches. Fig. 5 compares the amplitude of the injected bunch #5 and following bunches, first without (Fig. 5a) and then with (Fig. 5b) a 10% bunch-to-bunch coupling in the front-end electronics and a 3% coupling in the kicker. With coupling, bunch #6 suffers a greater disturbance, but still damps, while subsequent bunches suffer only slightly. Thus the system is tolerant of a reasonable amount of bunch to bunch coupling in· the analog components. (i) c (1j '6 (1j ..!::.. 0.04 (i) c (1j '6 ~ 0.04 <~ 3-91 --------/ 1000 Turns (a) -$(5) " --- (6) (b) -$(5) " --- (6) " -.-. (7) 2000 3000 6843A2 Figure 5. Plots of the phase-space error amplitude for the injected bunch (#5)and following bunch(es), (a) with no coupling, and (b) with 10% bunch-to-bunch coupling in the front-end electronics and 3% coupling in the kicker. In conclusion, our simulations indicate that the present conceptual approach to bunch-by-bunch feedback is satis-factory. Simulations to support the detailed design effort are m progress. We thank D. Boussard, J. Galayda, Q. Kerns, F. Ped-ersen, and P. Wilson for reviewing this work, and J. Dor-fan, A. Hutton, and M. Zisman for their interest and en-couragement. REFERENCES [1] D. Briggs, et.al., "Prompt Bunch-by-Bunch Syn-chrotron Oscillation Feedback via Fast Phase Measure-ment", these proceedings. [2] D. Boussard and G. Lambert, IEEE Trans. Nucl. Sci, NS-30, (1983), p. 2239. [3] P. Corradoura and F. Volker, private communica-tions. LAWRENCE BERKELEY LAB ORA TORY UNIVERSITY OF CALIFORNIA INFORMATION RESOURCES DEPARTMENT BERKELEY, CALIFORNIA 94720 :;. .l• ~..... _,.

English

Computer Modelling of Bunch-by-Bunch Feedback for the SLAC B-Factory Design

Lawrence Berkeley National LaboratoryLBL PublicationsTitleComputer Modelling of Bunch-by-Bunch Feedback for the SLAC B-Factory DesignPermalinkhttps://escholarship.org/uc/item/12h113p2AuthorsBriggs, DFox, J DHosseini, Wet al.Publication Date1991-05-01Copyright InformationThis work is made available under the terms of a Creative Commons Attribution License, available at https://creativecommons.org/licenses/by/4.0/eScholarship.org Powered by the California Digital LibraryUniversity of California'4 LBL-30772 Lawrence Berkeley Laboratory UNIVERSITY OF CALIFORNIA Accelerator & Fusion Research Division Presented at the IEEE Particle Accelerator Conference, San Francisco, CA, May 6-9, 1991, and to be published in the Proceedings Computer Modelling of Bunch-by-Bunch Feedback for the SLAC B-Factory Design D. Briggs, J.D. Fox, W. Hosseini, L. Klaisner, P. Morton, J.-L. Pellegrin, K.A. Thompson, and G. Lambertson May 1991 ) I ~ \ ~renee Berkeley Laboratory Library, Berkeley REFERENCE Not to be taken from this room Prepared for the U.S. Department of Energy under Contract Number DE-AC03-76SF00098 ~ ('--... ' w (): "v" _-...,J. ~· DISCLAIMER This document was prepared as an account of work sponsored by the United States Government. While this document is believed to contain correct information, neither the United States Government nor any agency thereof, nor the Regents of the University of California, nor any of their employees, makes any warranty, express or implied, or assumes any legal responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by its trade name, trademark, manufacturer, or otherwise, does not necessarily constitute or imply its endorsement, recommendation, or favoring by the United States Government or any agency thereof, or the Regents of the University of California. The views and opinions of authors expressed herein do not necessarily state or reflect those of the United States Government or any agency thereof or the Regents of the University of California. BECON-96 SLAC-PUB-5466 LBL-30772 Computer modelling of bunch-by-bunch feedback for the SLAC B-factory design D. Briggs, J.D. Fox, W. Hosseini, L. Klaisner, P. Morton, J.-L. Pellegrin, K.A. Thompson Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309 and G. Lambertson, Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720 * Work supported by the Director, Office of Energy Research, Office of Basic Energy Scineces, U.S. Department of Energy under Contract No. DE-AC03-76SF00098. SLAC-PUB-5466 LBL-30772 May 1991 (A) Computer modelling of bunch-by-bunch feedback for the SLAC B-factory design* D. Briggs, J. D. Fox, W. Hosseini, L. Klaisner, P. Morton, J.-L. Pellegrin, K. A. Thompson Stanford Linear Accelerator Center, Stanford University, Stanford, CA 94309 USA and G. Lambertson Lawrence Berkeley Laboratory, University of California, Berkeley, CA 94720 Abstract The SLAC B-factory design, with over 1600 high cur-rent bunches circulating in each ring, will require a feed-back system to avoid coupled-bunch instabilities. A com-puter model of the storage ring, including the RF system, wake fields, synchrotron radiation loss, and the bunch-by-bunch feedback system is presented. The feedback system model represents the performance of a fast phase detec-tor front end (including system noise and imperfections), a digital filter used to generate a correction voltage, and a power amplifier and beam kicker system. The combined ring-feedback system model is used to study the feedback system performance required to sup-press instaJ;>ilities and to quantify the dynamics of the sys-tem. Results are presented which show the time develop-ment of coupled bunch instabilities and the damping ac-tion of the feedback system. I. INTRODUCTION The large average current in the SLAC B-factory de-sign is distributed into many bunches of sufficiently small charge to minimize the beam-beam interaction and single-bunch instabilities. Although the cavity higher-order-modes (ROMs) will be strongly damped (Q < 70), there will still be significant coupling of the longitudinal and transverse motion of adjacent bunches via wakefields. Fur-thermore, the high-Q accelerating mode can also strongly couple the bunches longitudinally. The resulting instabili-ties will be controlled via wideband, bunch-by-bunch feed-back. Such a feedback system can handle disturbances to the bunch motion arising from any source, including but not limited to wakefields and injection errors. The feedback systems to control the longitudinal and transverse coupled-bunch instabilities will be similar in ar-chitecture. Since the signal detection and kicker require-ments are more stringent for the longitudinal system and for the high energy ring (HER), we shall concentrate our discussion on this case. Basic longitudinal-feedback sys-tem specifications are shown in Table 1. The proposed system implementation, its block diagram and description, and hardware tests are discussed elsewhere [1]. Table 1: Basic feedback system specifications RF freq. 476 MHz Max. mode amplitude Injection scheme ~: injection error 6t injection error 10 ps = 0.03 rad 1/5 bunch at 60 pps 0.002 100 ps * Work supported by Department of Energy contract DE-AC03-76SF00515. II. SIMULATION MODEL The simulation model (see Fig. 1) consists of a model of the feedback system electronics, combined with a model of the dynamics of the bunches in the ring. The feedback system model simulates the transfer function of the feedback system and includes: (1) the elec-tronic properties of the phase detector, mixer, low-pass fil-ter, and A/D converter, (2) input noise, gain and offset errors, bandwidth limitations, and dynamic range of the analog components, and (3) the algorithm running in the set of digital signal processors (DSP farm), that takes as input the digitized bunch phases and calculates the longi-tudinal kick. / On uch turn, tnt kinematic modeltnrorms the eltctronlc modtl or the tongltudl~l cllsplacement of each Dune.f\ (ltctronlc Model of th• phase detection, stqnal processing. •nd kicker. Initial Conditions: Pickup Electrode I) Seam Cf6"rtnt Structure 2) 'e or Injected Dunch tnctudn. lJ 8 of othtr Dunehes physical model <4) Cavity tnOdU 5) RF. ri?Q puamettrs Average t t Syn~:rv-onous Pr.ast Hlxer, low-Pass, ADC prtOICtfd at 1n1Uallutlon tnclwu: sutract 100"1 of S)""Chronous p."\.an Kinematic model or the nolst, ortstt, gJin trror longitudinal oscillations dynamic ,.,.,!lf slnusoldallnnsrrr function usumtS Ch¥it normJIIntlon Ohcrtlt Propac;atlon an<S Kick Oandw ldth limitation moc:ltllnclueltS; IJ AF fUOOamtntal ' (wltl\ptrftet comptnsatlon) 2) Jotlghtr-<>rdtr Modr Waktrttlds In Cavltln DSP (algorithm) lJ 5ync1Votron Aacllatlon ton lnc:ludn: <4} fttdOack KICk matchtd rnur • IOf\9 osc. frtQ. lot'19ltu<ttnal sahratlon,. e Olt output tlxtel·polnt AI..V oscillation FrtQUtncy prtdiCltd at lniUallzation Ktcker lncludu: output- slntHnouV128J w/ D~wldU\ llmiUUon On tach tum, lht tltctronlc modtl I rtturns to tht klntmattc modtl a ) longitudinal kick ror uch Dunch. ' Figure 1. Block diagram of the feedback simulation model. The model of the DSP farm emulates a 20-tap finite impulse response (FIR) matched filter. The coefficients ~i of the taps comprise a sinusoid with the synchrotron period of 19.3 turns, that is, ej = AvsP sin 271"--( j -1) 19.3 (1) where j E 1, ... , n•ample•, and we take n 3 ample• = 20. Given measures <Pi(k) = cp;(k)- cp, (where cp, is the synchronous phase) of the phase of bunch i on successive turns k as input, the result of the DSP algorithm is the output: Presented at the IEEE Particle Accelerator Conference, San Francisco, CA, May 6-9, 1991 y;(k) = 2: ei~p;(k- i) (2) i=l This result is clipped to an 8-bit signed integer and used to set the phase of the kicker oscillator for that bunch on that turn. The kicker model is implemented as a phase-modulated RF kicker with a nominal4 ke V maximum out-put amplitude. The measurement of the phase of a bunch is assumed independent of its charge, i.e., it is assumed that a sepa-rate measurement of bunch charge is available for normal-ization. A propagation delay of at least one turn (7.33 ps) is enforced in the feedback transfer function. In the ring simulation model, a discrete kick is given to each bunch at a single point in the ring; that is, the system is modelled as though there were a discrete change in en-ergy at a single point on each turn. This simplification is justified since the synchrotron frequency is small compared to the revolution frequency. The kick given to bunch i is comprised of several components: (1) the RF cavity volt-age ~sin 'f'i + vcav.jbk.' where ~ is the peak generator voltage, cp; is the phase of bunch i with respect to the zero crossing of the RF, and vcav.Jbk. is the RF cavity feedback needed to control beam loading in the fundamental mode, (2) the wake field voltage vwake (including both the accel-erating mode and the HOM's) accumulated in the cavity up to the present moment, (3) the synchrotron radiation loss per turn, and (4) the voltage V;Jdbk applied to bunch i by the bunch-by-bunch feedback system. Thus, the equa-tion for the total kick is ll.<P; = - ~:;~ [~sin 'f'i + vcav.Jbk. U I + V wake + v;fdbk] 2To . - o e ; - -cp; T£ (3) where a is the momentum compaction factor, Wrf is the RF frequency, and T£ is the longitudinal radiation damp-ing time; Eo is the ring energy, To the revolution period, and Uo the synchrotron radiation loss per turn, for a par-ticle on the design orbit. In the present simulations it is assumed that the cavity feedback is perfect, so that the part of vwake due to the fundamental mode is exactly can-celled by vcav.Jbk.. More realistic models, including the cavity phase, amplitude, and tuning loops, and modifica-tion of the impedance at coupled-bunch frequencies that fall within the bandwidth of the fundamental mode (2] are under study (3]. III. SIMULATION RESULTS Parameters used in the simulations are shown in Ta-ble 2. The harmonic number of the HER is 3492, with every other bucket filled except for a 5% gap; the to-tal current is 1.48 A. The initial conditions used were cp5 = 0.2915 rad, that is, bunch 5 starts at an 0.1 rad offset from the remaining bunches i, which were started at the synchronous phase cp; = 0.1915 rad. The feedback gain in the examples described here was set so that a 5 mrad sinu-soid on the input (at the synchrotron oscillation frequency) corresponds to a 4 keV sinusoid on the kicker output. 2 0.30 ~ (/) c ell 0.26 '6 .ell .!::;.. X 0.22 Ill E -e-0.18 4-91 0 0 20 40 • ~max• Fbk OFF o ~max• Fbk ON 60 80 Bunch Number 100 6843A4 Figure 2. Plot of the maximum bunch offset reached in 3000 turns for the first 100 bunches after the gap, with and with-out feedback. (i) c Ill '6 Ill ..=.. §: -e-(i) c Ill 0.29 0.24 0.19 0.14 0.194 '6 0.192 ~ -e-0.190 (a) 0.188 ..L._ _ __::_.L__--l. __ _!_ __ ...l._ _ ____J 0 500 1 000 1500 2000 2500 3·91 Turns 6843A1 Figure 3. Plots of the longitudinal phases of (a) the injected bunch (#5) and (b) the bunch immediately following (#6), vs turn number, in the presence of feedback. Note the ex-panded vertical scale in (b). In Fig. 2, we show a plot of maximum bunch off-sets reached after 3000 turns, with and without feedback. Note that the time between injection p11lses is 1/60 second, which is about 2300 turns. In the aL:;euce of feedback, the disturbance shown would grow even larger and propagate further back in the bunch train. With the feedback system I I I I I I I J I DSP Input and Output for Bunch 5 (/) co c Ol U5 - DSPin (5) --- DSP out (5) ·--1 I I DSP Input and Output for Bunch 6 (/) co c Ol U5 3-91 400 440 1\ I \ I \ I \ I \ I I I \ I \ I 'I. Turn DSP in (6) DSP out (6) 480 6843A3 Figure 4. Plots of the input and output of the DSP model for (a) bunch #5, and (b) bunch #6, shortly after the injection of bunch #5. Table 2: Simulation parameters for HER Bunch charge 4 x 1010 Number of bunches 1658 Bunch interval Number of cavities Freq. of strongest HOM Q of HOM R/Q of HOM (per cav.) Vg a Uo 4.2 ns 20 750 MHz 70 33 n 18.5 MV 0.00241 3.52 MeV /turn turned on, the coupled bunch excitation does not extend beyond a very few bunches. Fig. 3 shows the phases of the injected bunch (#5) and the immediately following bunch, vs turn number. The envelope of the phase of the injected bunch damps linearly, reflecting the fact that the kicker saturates, and the phase 3 of the following bunch grows quickly to a maximum and then slowly damps. The excitation of subsequent bunches is strongly suppressed. Fig. 4 shows the input and output of the DSP model for bunches #5 and #6 shortly after injection. The DSP output saturates for bunch #5, but maintains the proper 90° phase lag. Such benign saturation behavior is difficult to realize with conventional analog approaches. Fig. 5 compares the amplitude of the injected bunch #5 and following bunches, first without (Fig. 5a) and then with (Fig. 5b) a 10% bunch-to-bunch coupling in the front-end electronics and a 3% coupling in the kicker. With coupling, bunch #6 suffers a greater disturbance, but still damps, while subsequent bunches suffer only slightly. Thus the system is tolerant of a reasonable amount of bunch to bunch coupling in· the analog components. (i) c (1j '6 (1j ..!::.. 0.04 (i) c (1j '6 ~ 0.04 <~ 3-91 --------/ 1000 Turns (a) -$(5) " --- (6) (b) -$(5) " --- (6) " -.-. (7) 2000 3000 6843A2 Figure 5. Plots of the phase-space error amplitude for the injected bunch (#5)and following bunch(es), (a) with no coupling, and (b) with 10% bunch-to-bunch coupling in the front-end electronics and 3% coupling in the kicker. In conclusion, our simulations indicate that the present conceptual approach to bunch-by-bunch feedback is satis-factory. Simulations to support the detailed design effort are m progress. We thank D. Boussard, J. Galayda, Q. Kerns, F. Ped-ersen, and P. Wilson for reviewing this work, and J. Dor-fan, A. Hutton, and M. Zisman for their interest and en-couragement. REFERENCES [1] D. Briggs, et.al., "Prompt Bunch-by-Bunch Syn-chrotron Oscillation Feedback via Fast Phase Measure-ment", these proceedings. [2] D. Boussard and G. Lambert, IEEE Trans. Nucl. Sci, NS-30, (1983), p. 2239. [3] P. Corradoura and F. Volker, private communica-tions. LAWRENCE BERKELEY LAB ORA TORY UNIVERSITY OF CALIFORNIA INFORMATION RESOURCES DEPARTMENT BERKELEY, CALIFORNIA 94720 :;. .l• ~..... _,.

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(A) Computer modelling of bunch-by-bunch feedback for the SLAC B-factory design*

(A) Computer modelling of bunch-by-bunch feedback for the SLAC B-factory design*

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