Coupledbunch longitudinal instability and damper in the PS

Coupled-bunch longitudinal instability and damper in the PS L. Ventura, H. Damerau, G. Sterbini Acknowledgements: S. Gilardoni, M. Haase, M. Migliorati, M. Paoluzzi, D. Perrelet. MSWG MEETING, June 19 th 2015 1

Coupled Bunch Instability at CERN LHC Injectors Upgrade programme (LIU) Compensation technique: ² Coupled bunch feedback is used to compensate CB instabilities and spare cavity (C 11) is used as longitudinal damper. Compensation technique: ² Fully digital LLRF and new dedicated damper cavity Compensation techniques: ² Spread in fs ² Landau damping 1971 2005 2013 LS 1 2015 J. -L. Vallet, “Amortissement des Instabilites Longitudinales de Modes Couples, ” unpublished presentation (APC), 2005. First observation at CERN of CB instability ² “Damping of the Longitudinal Instability in the CERN PS “, D. Boussard and J. Gareyte, CERN, Geneva ² “Study and compensation of coherent longitudinal instability in the CERN PS“, D. Boussard, J. Gareyte, D. Kohl, CERN, Geneva Compensation techniques: ² Analog FB system and C 8696 as long. damper 2

Outlook q Observe CB Instabilities in a synchrotron q Characteristics of the new hardware in the PS q Circulant matrix approach to CB mode q Finemet cavity CB modes excitation q Conclusions 3

Outlook q Observe CB Instabilities in a synchrotron q Characteristics of the new hardware in the PS q Circulant matrix approach to CB mode q Finemet cavity CB modes excitation q Conclusions 4

How do we observe CB instabilities? Consider a machine with only 3 circulating bunches in h=3 and have a look to the bunches profile along turns. 1 Turn

CB instabilities in time Hyp: the synch. frequency is the same for all bunches what changes is the phase Due to the symmetry of the system (3 bunches in h=3 circular system) the phase displacement has to be the same for all the bunches. Direct measurement of the modes by observing the constant dephasing between consecutive bunches.

CB instabilities in frequency 1) Stable bunches oscillation RF frequencies 0 6 f 0 3 f 0 2) Stable bunches but with different density current all the revolution harmonics f 0 0 f 0 2 f 0 3 f 0 4 f 0 5 f 0 3) Synchrotron oscillations upper and lower sideband of the rev. harmonics f 0 -ωs 2 f 0+ωs ……as follows from Sacherer’s formula f. CB=|(q. Nb+μ)f 0+mfs| 6 f 0

Frequency component of CB mode from FFT (1/2) We are interested in the frequency component of the CB instability because the FB system in the PS is a Frequency Domain feedback which detects synchrotron frequency sidebands indicating CB oscillations and feed them back to the beam via the damper cavity which applies to each oscillation mode a kick. abs(FFT) ZOOM 8

Frequency component of CB mode from FFT (1/2) f 0 -fs 2 f 0+fs μ= 2 9

Outlook q Observe CB Instabilities in a synchrotron q Characteristics of the new hardware in the PS q Circulant matrix approach to CB mode q Finemet cavity CB modes excitation q Conclusions 10

CB Feedback in short LLRF DETECT SYNCHROTORN FREQUENCY SIDEBANDS f 0 -fs +fs BEAM Detect and damp synchrotron frequency side-bands of frev harmonics. GIVE A KICK IN ENERGY TO EACH MODE 11

New cavity (#25) in the PS ring • Wide-band (0. 4 – >5. 5 MHz, VRF = 5 k. V) cavity based on Finemet material • No acceleration, but damping of coupled-bunch oscillations t h g i tra 0 n o i t sec 2 M. Paoluzzi 6 -cell cavity unit Accelerating gap s Power amplifiers (solid state) • Amplifiers on gaps 2, 3, 4 and 6 are operational. • First installation of transistor power amplifiers close to beam in PS.

Finemet cavity at CERN LEIR PS Booster • Accelerating system using wide-bandwidth Finemet cavities. • They individually cover the whole frequency range without tuning and allow multi-harmonic operation. • Finemet cavities with large bandwidth: covers h = 1 and h = 2 without need for tuning. • Moderate voltage per gap, many gaps → Solid state amplifiers PS Finemet cavity is used not for accelerating purpose but as a LONGITUDINAL DAMPER 13

Finemet vs 10 MHz cavity q 10 MHz spare (C 11): voltage up to 20 k. V and tuneable from 2. 8 MHz to 10 MHz. q Finemet cavity: frequency span between 0. 4 and 5. 5 MHz with a power up to VRF = 5 k. V. Finemet cavity 10 MHz cavity Cover all oscillation mode Kicker base-band frf Once the 10 MHz cavity is tuned it has a small frequency span. Cover only two harmonic number Preferred detection 14

New digital Low Level RF in the PS The LLRF for the wide-band kicker comprises two distinct loops: 1. Coupled-bunch feedback • Input: Wall current monitor signal • Output: Drive signal to cavity D. Perrelet 2. Compensate beam-loading: reduce cavity impedance at revolution frequency harmonics • Input: Gap return signal of the cavity • Output: Drive signal to cavity The board was originally designed for the PS 1 -turn delay feedback. 15

Both feedbacks together (1 harmonic) sin(h. FB frevt + f) sin(h. FB frevt) Low-pass Cavity return ADC DAC Low-pass cos(h. FB frevt + f) Cavity drive cos(h. FB frevt) cos((h. RF-h. FB)frevt + f) Wall current monitor fs sideband filter ADC fs sideband filter sin((h. RF-h. FB)frevt + f) Signal processing for single harmonic Required multiple times

Outlook q Observe CB Instabilities in a synchrotron q Characteristics of the new hardware in the PS q Circulant matrix approach to CB mode q Finemet cavity CB modes excitation q Conclusions 17

1970: Circulant matrix formalism In 1970 for the first time was introduced the formalism that allowed to solve this set of homogeneous equations by a matrix method. The matrix solving the phase space system is a circulant matrix M and so the system stability can be studied by finding the eigenvalues and eigenvectors of the matrix M if the matrix can be put in diagonal form. Find the matrix M starting from forced conditions in the system Use the Finemet cavity to excite CB modes and study the response of 18 the system

Circulant matrix approach to CB mode The matrix M is a BLOCK CIRCULANT matrix identified by a complex circular vector with proper delay z-1 which bounds all the bunches. The first column of the matrix composed by sub-matrices which represent rotation in the phase space. Z-1 19

Mode excitation with the Finemet cavity of 21 bunches in h = 21 This plot is the result of beam measurement. We excite each mode individually, measure the mode spectrum, evaluate amplitude and phase for each one and plot a set of mode spectra amplitude in color. Mode spectra The matrix is diagonal!!!! With a full machine of 21 bunches the circularity of the system is respected and the matrix is diagonal. 20

Mode excitation with the Finemet cavity of 18 bunches in h = 21 This plot is the result of beam measurement. We excite each mode individually, measure the mode spectrum, evaluate amplitude and phase for each one and plot a set of mode spectra amplitude in color. NOT diagonal!!!! With a gap of 3 bunches the circularity of the system is lost. 21

Procedure in short (1/2) The longitudinal pickup read only the beam position xi M: BLOCK CIRCULANT MATRIX C: CIRCULANT MATRIX 22

Procedure in short (2/2) The dynamic can be re-formulated in a complex amplitude space (phasor space): To study the stability of the system we need to find eigenvalues and eigenvectors Eigenvalues D is the ratio between two consecutive turns of modes evolution: To move in the mode space, starting from the information of the centroid position we can: 23

Two independent mode analysis techniques H. Damerau SAME RESULTS 24

Outlook q Observe CB Instabilities in a synchrotron q Characteristics of the new hardware in the PS q Circulant matrix approach to CB mode q Finemet cavity CB modes excitation q Conclusions 25

Excitation of coupled-bunch oscillations: set up Prototype firmware to excite coupled-bunch oscillations sin(h. FB frevt + f) sin(h. FB frevt) Low-pass Cavity return ADC DAC Low-pass cos(h. FB frevt + f) Excitation frequency, Df cos(h. FB frevt) Amplitude Df sin h. FBfrev Low freq. DDS cos Side-band selection Amplitude Cavity drive f Excitation frequency ~ fs away from hfrev ~ 400 Hz at 476 k. Hz

Mode scan with 18 bunches in h = 21 We excite each mode individually, measure the mode spectrum, evaluate amplitude and phase for each one and plot a set of mode spectra amplitude in color. 2015 da ta 2013 da Finemet Cavity 11 ta h. FB q Some modes can be excited very cleanly, others as a mixture; artefact? h. FB q All 18 modes can be excited

Mode scan with 21 bunches in h = 21, cavity 11 Excite each mode individually and measure mode spectrum 2013 da ta h. FB Clean observation of all possible modes

Mode scan with 21 bunches in h = 21, Finemet Excitation of each mode with the prototype firmware to excite CB oscillations Upper side-band: n = nexc h. FB Lower side-band: n = 21 - nexc h. FB Every oscillation mode from n = 1… 21 can be excited on both side-bands 29

Excitation amplitude scan Vary excitation amplitude and check mode spectrum ~20 ms after excitation starts: Absolute voltage (peak to peak) in the cavity during one of the voltage scans. We assume 66 d. B attenuation from the total gap voltage to the voltage at the output of summing point (active divider). 600 V peak/gap Oscillation amplitude proportional to excitation linear regime Mode amplitudes comparable to excitation with spare cavity C 10 -11

Outlook q Observe CB Instabilities in a synchrotron q Characteristics of the new hardware in the PS q Circulant matrix approach to CB mode q Finemet cavity CB modes excitation q Conclusions 31

Summary q A CB mode analysis technique has been presented using the circulant matrices formalism. The mathematical model has been applied to the measured data to analyze the longitudinal profiles of the bunch train and to perform the mode analysis Two independently analysis technique which gives the same results q First tests without and with beam successful • • • Coupled-bunch oscillations excited as expected Each mode can be excited individually Confirms measurements with C 10 -11 in 2013 Outlook q Future MD • Complete the excitation measurements: excite multiple modes simultaneously…. q Follow-up firmware development • Complete filter design for synchrotron frequency side-bands • Close the loop on one harmonic 32

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