2. Coupled bunch instabilities in the CERN PS
Speaker: L. Ventura
Acknowledgments: H. Damerau, S. Gilardoni, M. Migliorati, G. Sterbini.
LIU-PS Student Day
21 April 2015

3. Outlook My PhD work COUPLED-BUNCH INSTABILITIES METHOD AND TOOLS
WHAT are they?
WHY are we concerned about it?
WHERE are CB instabilities observed in the PS?
HOW are they compensated?
My PhD
work
METHOD AND TOOLS
CIRCULANT MATRIX
ALGORITHM to analyze longitudinal profiles of the bunch train and to perform the mode analysis of the system.

4. What are CB instabilities...?
Interaction of charged particles/cavity  wakefields act back on the beam and produce growth of oscillations.
If growth rate stronger than the natural damping  oscillation gets unstable.
Wakefields are proportional to the bunch charge instabilities are current dependent.
Cross-talk of bunches which are coupled with the machine impedance
Two pendulums of equal length (BUNCHES) and connected by a soft spring (IMPEDANCE), comprise a 2-degree-of-freedom system, with two natural modes of vibration.
Coupled motion in which energy and amplitude are traded back and forth between the two pendulums.

5. LIU-PS PROJECT New LHC beam planned for the LIU project
Double the intensity: from 1.3 · 1011 p/b  2.6 · 1011 p/b (72 bunches in h = 84)
Increased voltage in the cavity induced by the beam
Instabilities are current dependent and they will become
more threatening with new beam parameters

6. The LHC25ns cycle in the PS pre-LS1
Longitudinal dipolar Coupled-Bunch Instability
gtr
Eject 72 bunches
Inject 4+2 bunches
h = 7
h = 21
h = 84
Split in four at flat top energy
Triple splitting after 2nd injection

7. Feedback System
The PS can presently work with a narrowband FB system by using the spare 10 MHz RF cavity (power up to 20 kV, it is tuneable from 2.8 MHz to 10.1 MHz, h = [6...21]) as a longitudinal kicker. Due to its limited bandwidth, the FB can address only two adjacent modes.
During the long shutdown (LS1) in , a new dedicated larger band- width cavity has been installed in the straight section 2 of the PS: the Finemet© cavity. With a voltage up to 5 kV and bandwidth from MHz, the cavity can provide the correcting RF voltage to address all the oscillation modes in h = 21.
FINEMET©
Digital feedback electronics
D.Perrelet
WCM
(beam)
FB system
Drive signal to cavity
Kicker cavity
Beam
‘PS Low Level electronics and tests’, H. Damerau , Finemet review, 9-10 September, CERN, Geneva

8. Circulant matrix Accepted at IPAC 2015
A circulant matrix is formed by a vector c by cyclically permuting the entries.
The filled PS train has a circulant symmetry, so the evolution in the longitudinal phase space of the machine can be described using a block circulant matrix.
A bunch oscillating with synchrotron oscillation is rotating in the phase space.
Δp/p
Evolution of nb bunches in the normalized longitudinal phase space described with a stationary block circulant matrix where each block represents a rotation matrix. x

9. If M known, the stability of the system can be investigated by the eigenvalues of the M if the matrix can be put in diagonal form.
Example: For the simple case of nb=2, M will be a 4x4 matrix. m1 m2
BLOCK CIRCULANT MATRIX
COMPLEX CIRCULANT MATRIX
Find C and derive M by transforming each element of C in a pure rotation block of M
Derive the M matrix starting from the measurement data from the longitudinal PU.

10. Longitudinal PU read only bunch position xi
From time space to phasor space
Δp/p
All bunches in the system has the same Qs Φ x
Longitudinal PU read only bunch position xi
The dynamic can be re-formulated in a complex amplitude space (phasor space):

11. From phasor to mode space
Properties of circulant matrices
Eigenvectors
Eigenvalues
D is the ratio between two consecutive turns

12. Recap…..
Fit
DFT
With M we obtain a linear evolution of the machine.

13. Measured data analysis
1) Loading Trev & beam signal data and synchronization
Settings for the acquisition:
CH1: WCM95 signal
CH2: Trev signal
Sampling frequency: 400 MHz (h≈841, ns sampling)
Synchronize using CH1
One external trigger from CTRV
Perform the gating
We find the zero crossing of the Trev (we choose arbitrarily the rising edge) and reconstruct one revolution period.
Trev
Beam signal synchronized

14. 2) Locate for each bunch centre of mass xi
Sampling frequency of 400 MHz in h≈841 -> ≈841 points to reconstruct one revolution period …..but we want to gate in h=21 so we normalized in respect to harmonic 21 by doing a resampling in h=21*500.
Normalize each Trev between 0 and 1
Trev

15. 3) Centre of mass evolution xi
We find the center of mass using a fast algorithm which identifies the centroid of each bunch with a weighted average for each turn in 160 ms .
Longitudinal oscillation of the centroid ~ 1ns well below the limit of 2.5 ns of the sampling.

16. 4) Fit on the centroid evolution
Δ=Delta between consecutive windows=100 turns
W=Window width=2000 turns
Number of window=700
Turns W
Fit performed using a moving window which covers about one synchrotron oscillation.
Where Xi is the phasor representing the complex amplitude of the bunch.

17. 5) Oscillation mode Wn
Allows to compute amplitude Ai and phase Φi for each oscillation mode from the ai and ϕi of each bunch.

18. Conclusions
Coupled-bunch instabilities cause one of the most relevant limitations for the high-brightness beams in the LHC injectors, in particular in the CERN Proton Synchrotron, the injector where this instability is more critical.
In the frame of the LIU project the beam intensity will be increased from 1.3 · 1011 p/b to 2.6 · 1011 p/b, this making CB instabilities even more problematic.
During LS1, a new dedicated larger band-width cavity has been installed, the Finemet© cavity, which can provide the correcting RF voltage to address all the oscillation modes in h = 21.
A CB mode analysis technique has been presented by using the circulant matrices formalism. The mathematical model has been explained and applied to the measured data to analyze the longitudinal profiles of the bunch train and to perform the mode analysis.
Accepted at IPAC 2015

19. Next Step FB system PS ring
The development of a Simulink© model of the machine and the feedback is ongoing as an auxiliary tool in view of the 2015 commissioning of longitudinal feedback in the PS.
The digital card of the low level RF is available but the firmware for the signal processing has still to be finalized.
FB system
PS ring
Non-linear aspects to be included in the FB system:
the limited maximum voltage;
the error induced by the 14-bit quantization of the ADCs and DACs;
the noise of the wall current monitor;
……….

20. THANK YOU FOR YOUR ATTENTION!