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

First observation at CERN of CB instability  “Damping of the Longitudinal Instability in the CERN PS “, D. Boussard and J. Gareyte, CERN, Geneva  “Study and compensation of coherent longitudinal instability in the CERN PS“, D. Boussard, J. Gareyte, D. Kohl, CERN, Geneva Compensation techniques:  Spread in f s  Landau damping 2013 LHC Injectors Upgrade programme (LIU) LS1 Compensation technique:  Coupled bunch feedback is used to compensate CB instabilities and spare cavity (C11) is used as longitudinal damper Compensation technique:  Fully digital LLRF and new dedicated damper cavity Coupled Bunch Instability at CERN 2005 J.-L. Vallet, “Amortissement des Instabilites Longitudinales de Modes Couples,” unpublished presentation (APC), Compensation techniques:  Analog FB system and C86-96 as long. damper

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

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

How do we observe CB instabilities? 1 Turn Consider a machine with only 3 circulating bunches in h=3 and have a look to the bunches profile along turns. 5

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. 6 Direct measurement of the modes by observing the constant dephasing between consecutive bunches.

3f 0 f0f0 2f 0 5f 0 4f 0 3f 0 2f 0 +ω s f 0 -ω s 6f …… as follows from Sacherer’s formula  f CB =|(qN b +μ)f 0 +mf s | 1) Stable bunches oscillation  RF frequencies 2) Stable bunches but with different density current  all the revolution harmonics f 0 3) Synchrotron oscillations  upper and lower sideband of the rev. harmonics CB instabilities in frequency

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. 8 Frequency component of CB mode from FFT (1/2) ZOOM abs(FFT)

9 f0f0 2f 0 2f 0 +f s f 0 -f s μ= 2 Frequency component of CB mode from FFT (1/2)

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

11 LLRF BEAM CB Feedback in short DETECT SYNCHROTORN FREQUENCY SIDEBANDS GIVE A KICK IN ENERGY TO EACH MODE f0f0 +f s -f s Detect and damp synchrotron frequency side-bands of f rev harmonics.

New cavity (#25) in the PS ring M. Paoluzzi Wide-band (0.4 – >5.5 MHz, V RF = 5 kV) cavity based on Finemet material No acceleration, but damping of coupled-bunch oscillations 6-cell cavity unit Accelerating gap 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. straight section 02

13 Accelerating system using wide-bandwidth Finemet cavities. They individually cover the whole frequency range without tuning and allow multi-harmonic operation. Finemet cavity at CERN LEIR PS Booster 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

f rf Preferred detection 14 Finemet vs 10MHz cavity  10MHz spare (C11): voltage up to 20 kV and tuneable from 2.8 MHz to 10 MHz. Finemet cavity 10MHz cavity Once the 10MHz cavity is tuned it has a small frequency span.  Finemet cavity: frequency span between 0.4 and 5.5 MHz with a power up to V RF = 5 kV. Kicker base-band Cover only two harmonic number Cover all oscillation mode

15 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 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. D. Perrelet

Both feedbacks together (1 harmonic) Low-pass ADC DAC Cavity return Cavity drive sin(h FB f rev t +  sin(h FB f rev t  cos(h FB f rev t +  cos(h FB f rev t  f s side- band filter ADC Wall current monitor cos((h RF -h FB )f rev t +  sin((h RF -h FB )f rev t +  f s side- band filter  Signal processing for single harmonic  Required multiple times

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

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. 18  Find the matrix M starting from forced conditions in the system  Use the Finemet cavity to excite CB modes and study the response of the system 1970: Circulant matrix formalism

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

20 Mode excitation with the Finemet cavity of 21 bunches in h = 21 With a full machine of 21 bunches the circularity of the system is respected and the matrix is diagonal. The matrix is diagonal!!!! 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

21 With a gap of 3 bunches the circularity of the system is lost. NOT diagonal!!!! 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.

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

23 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 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: Procedure in short (2/2)

24 H. Damerau Two independent mode analysis techniques SAME RESULTS

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

Low-pass ADC DAC Cavity return Cavity drive sin(h FB f rev t +  sin(h FB f rev t  cos(h FB f rev t +  cos(h FB f rev t  Amplitude Low freq. DDS Amplitude sin cos Side-band selection Excitation frequency,  f f h FB f rev ff  Excitation frequency ~ f s away from hf rev  ~ 400 Hz at 476 kHz Prototype firmware to excite coupled-bunch oscillations Excitation of coupled-bunch oscillations: set up

 All 18 modes can be excited Mode scan with 18 bunches in h = data 2015 data  Some modes can be excited very cleanly, others as a mixture; artefact? Finemet Cavity 11 h FB 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.

Excite each mode individually and measure mode spectrum  Clean observation of all possible modes Mode scan with 21 bunches in h = 21, cavity data h FB

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

Excitation amplitude scan Vary excitation amplitude and check mode spectrum ~20 ms after excitation starts:  Oscillation amplitude proportional to excitation  linear regime  Mode amplitudes comparable to excitation with spare cavity C10-11 Absolute voltage (peak to peak) in the cavity during one of the voltage scans. 600 V peak/gap We assume 66 dB attenuation from the total gap voltage to the voltage at the output of summing point (active divider).

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

32  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  First tests without and with beam successful Coupled-bunch oscillations excited as expected Each mode can be excited individually Confirms measurements with C10-11 in 2013 Summary  Future MD Complete the excitation measurements: excite multiple modes simultaneously….  Follow-up firmware development Complete filter design for synchrotron frequency side-bands Close the loop on one harmonic Outlook

Thank you for your attention!