Beam dynamics issues for the upgrade of the LHC … · Beam dynamics issues for the upgrade of the...

M. Migliorati on behalf of PS-LIU group

Beam dynamics issues for the upgrade of the LHC injectors at CERN

LIU - LHC Injectors Upgrade Project

Outline

•  LHC Injectors Upgrade (LIU) goals

•  Beam dynamics issues –  Space charge –  Transverse mode coupling instability –  Electron cloud –  Longitudinal coupled bunch instability

•  PS coupled bunch instability

25/09/13 Beam dynamics issues for the upgrade of the LHC injectors at CERN

Pag. 2

LHC Injectors Upgrade goals


Pag. 3



Pag. 4

PSB

SPS

PS

LHC / HL-LHC

Out

put e

nerg

y

160 MeV

1.4 GeV

26 GeV

450 GeV

7 TeV

Linac2 50 MeV

Proton flux / Beam power

Present

Linac4

PSB

PS

2.0 GeV

LIU (2019)

“The LHC Injectors Upgrade should plan for delivering reliably to the LHC the beams required for reaching the goals of the HL-LHC. This includes LINAC4, the PS booster, the PS, the SPS, as well as the heavy ion chain …”

Mandate:



Pag. 5

Param. @ LHC collision

Nominal1 25 ns

2012 50 ns

HL-LHC1 25 ns

HL-LHC1 50 ns

Int/bunch 1.15E11 ~1.6E11 2.2E11 3.5E11 Bunches 2808 1380 2808 1404

Beam current [A] 0.58 0.39 1.12 0.89 εn[µm] 3.75 ~ 2.4 2.5 3.0 β*[m] 0.55 0.6 0.15 0.15

Peak Lumi [cm-2 s-1] 1 1034 7.74 1033 24 1034 25 1034

1O. Bruning, HL-LHC/LIU day, 30/03/2012 and CERN-ATS-2012-070

Goal of HL-LHC ~ 300- 250 fb-1 per year

In 2012 produced about 0.81 fb-1 per week

Mul

ti-bu

nch

Bea

ms f

or L

HC

in th

e PS

H

. Dam

erau

, S. H

anco

ck

5

Triple splitting after 2nd injection Split in four at flat top energy

26 G

eV/c

1.4

GeV

2nd in

ject

ion

The LHC25ns cycle in the PS

ĺ Each bunch from the Booster divided by 12 ĺ�6 × 3 × 2 × 2 = 72

h = 7

Eject 72 bunches

(ske

tche

d)

Inject 4+2 bunches Jtr

Low-energy BUs

h =

84 h = 21

High-energy BU

foursplitb12.dat

inj2allb.dat

LHC25(50)ns production scheme as today


Pag. 6

Production scheme: a) Double batch injection from PSB (4 + 2 bunches, 6 bunches for PS at h=7) b) Up to 4 batches of 72 bunches each transferred to the SPS (288 bunches) Transverse emittance produced in the PSB, longitudinal in the PS •  Multiturn proton injection in PSB •  RF gymnastics in PS:

-  Triple splitting -  Acceleration -  2 x Double splittings

-  (1 Double spli-ng for 50 ns)

-  Bunch rotation

h = 42

LHC25(50)ns production scheme as today


Pag. 7

25ns%LHC%type%beams%in%the%PS%

h=7%�� 6%6%b.)%%!

h=21%��%!

Triple%splitting!

Double%splitting!

Bunch%shortening!

Double%splitting!

0! 10! 20! 30! 40! 50! 60!

50!

150!

250!

300!

200!

100!

40%M

Hz%RF%Vo

ltage%[k

V]%

Time%[ms]%

h=42%b.sp.%=%50%ns%(36%b.)%%!

h=84%b.sp.%=%25%ns%(72%b.)%%!

Double%splitting!

Adiabatic%shortening!

Bunch%rotation!

��!

11%ns!

4%ns!

25ns%LHC%type%beams%in%the%PS%

h=7%�� 6%6%b.)%%!

h=21%��%!

h=42%b.sp.=50%ns%(36%b.)%%!

h=84%b.sp.=25%ns%(72%b.)%%!

Triple%splitting!

Double%splitting!

Bunch%shortening!

Double%splitting!

h = 21

h = 9 à10 à 11à12 à 13 à 14 à 7 à 21

h =

21 à

42 à

84

+4 bunches

S. Gilardoni – Space Charge 2013 h = 9

LHC 25(50)ns alternative Production (BCMS)


Pag. 8

(BCMS “Batch Compression, Merging and Splitting in PS”)

Production scheme: a) Double batch injection from PSB (4 + 4 bunches, 8 bunches for PS at h=9) b) Up to 5 batches of 48 bunches each transferred to the SPS (240 bunches) Transverse emittance produced in the PSB, longitudinal in the PS -  Multiturn proton injection in PSB -  RF gymnastics in PS:

-  Batch compression -  Bunch merging -  Triple splitting

-  Acceleration -  2 x Double splittings

(1 Double splitting for 50 ns) -  Bunch rotation

4 bunches

Beams expected @ SPS extraction in 2014-15


Pag. 9

Bunch intensity [1011 p/

b]

Transv. Emittance [1 σ norm, mm.mrad]

Bunches per batch

(@ SPS inj.)

PSB rings Harm. @PS inj.

Bas

elin

e fo

r ph

ysic

s 25 ns 1.15 2.8 72 6 7

25 ns BCMS 1.15 1.4 48 8 9

Eve

ntua

lly

for

star

tup 50 ns 1.65 1.7 36 6 7

50 ns BCMS 1.6 1.2 24 8 9

Beam are produced approx. round: linear coupling in PS used to avoid Head-tail instability After LS1 it would be possible to have less round beams if needed.

Beam perfomances as today vs HL-LHC req.


Pag. 10

•  Requirement: taking into account LHC losses and blowup –  2.2×1011 p+/b in 2.3 µm (1 sigma norm.) at 25 ns @ SPS extr. –  3.6×1011 p+/b in 2.7 µm (1 sigma norm.) at 50 ns @ SPS extr.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5

Emittance (x+y)/2 [um]

Bunch Intensity [e11]

SPS 450 GeV 25 ns SPS RF pow

erSPS Longitu

dinal insabilitie

sSPS eC

loud

HL-‐LHC

2011

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0



SPS 450 GeV 50 ns

SPS Z

S, kicker heatin

gSPS Longitu

dinal insabilitie

s

HL-‐LHC2011

2012

2012

2012 BCMS

2012 BCMS

Injector chain limitations


Pag. 11

PSB: -  High intensity beams: every PSB bunch is split 12 times (to get finally 72

bunches at 25 ns spacing) -  Space-charge issue

PS: -  Space-charge issue -  TMCI instability at transition energy -  Longitudinal coupled bunch instability -  Transient beam loading in the 10, 20 and 40 MHz RF systems

SPS: -  Space-charge -  Electron cloud instability -  TMCI instabilities -  RF power (high transient beam loading) -  Longitudinal coupled bunch instability

Basic principles of the Injector upgrade


Pag. 12

To overcome some main limitations of LHC injectors: –  Space charge current limitations

•  PSB injection : Increase injection energy in the PSB from 50 to 160 MeV Linac4 (160 MeV) to replace aging Linac2 (50 MeV)

•  PS injection: Increase injection energy in the PS from 1.4 to 2 GeV

•  Upgrade the PSB, PS and SPS to make them capable to accelerate and manipulate a higher brightness beam (feedbacks, cures against electron clouds, hardware modifications to reduce impedance, improve beam instrumentations, increase power, …)

PS coupled bunch instability: theory


Pag. 13

•  The em fields trapped in machine devices, as HOMs in RF cavities, allow different bunches to influence each other.

•  Coherent oscillation modes of a beam can increase with time producing instabilities.

•  Simple physical description: coupled harmonic oscillators

PS coupled bunch instability: theory


Pag. 14

•  The em fields trapped in machine devices, as HOMs in RF cavities, allow different bunches to influence each other.

•  Coherent oscillation modes of a beam can increase with time producing instabilities.

•  Simple physical description: coupled harmonic oscillators

HOM: external driving force acting on the

spring


spring


spring

PS coupled bunch instability: observations


Pag. 15

Measurements comparing the longitudinal limitations of

LHC25ns/LHC50ns beams and the corresponding driving

impedance sources, as far as they have been identified, are

presented in the first part of the paper. Key ingredients to

achieve bunch intensities well beyond nominal intensity of

1.3 · 1011 ppb are introduced. Thereafter, the longitudinal

performance with LHC75ns and the new beam variant with

150 ns bunch spacing (LHC150ns) are reported. Finally,

ongoing and possible future upgrades of the feedback sys-

tems are discussed.

PERFORMANCE LIMTIATIONS WITH25 AND 50 NS BUNCH SPACING

Comparing longitudinal beam stability of the LHC-type

beams with 25 and 50 ns bunch spacing at extraction is mo-

tivated by the fact that both beam types are very similar

during acceleration (see Table 1). In both cases, 18 bunches

are accelerated at h = 21. The longitudinal density is the

same as well, but intensity and longitudinal emittance of

the 50 ns variant are twice smaller.

Coupled-bunch Instabilities During Acceleration

As coupled-bunch (CB) oscillations usually start after

transition crossing, mountain range data were recorded ev-

ery 70ms starting from 100ms after transition crossing,

when the final !l is reached for LHC25ns and LHC50ns

beams (the magnetic cycle for the LHC50ns beam is 1.2 s

shorter due to single-batch transfer from the PSB [3]). The

dipole motion of each bunch is extracted from the center

position of a Gaussian function fitted to each bunch of each

turn recorded. A second fit of a sinusoidal function to the

motion of the bunch center results in oscillation amplitude,

phase and frequency, the latter being the synchrotron fre-

quency. A discrete Fourier transform converts these oscil-

lation amplitudes and phases per bunch to amplitudes and

phases per mode, the mode spectrum [4]. This analysis

technique is superior to measurements in frequency domain

since the bunches only cover 6/7 of the circumference.

Spurious frev lines due to this filling pattern are removed

as only bunches are analyzed. However, the mode num-

bers with respect to the batch, nbatch do not directly cor-

respond to frev harmonics, but each mode number nbatch

results in a spectrum of frev lines, with the lines close to

7/6nbatchfrev at maximum amplitudes.

Figures 1 and 2 show the evolution of the mode spectra

for LHC25ns and LHC50ns beams during acceleration.

Though the total intensity differs by a factor two, the mode

pattern and oscillation amplitudes are very similar, suggest-

ing a scaling proportional to longitudinal density, Nb/!lrather than intensity.

Moreover, the form of the spectrum remains unchanged

during acceleration. The modes nbatch = 1, 2 and 16, 17,those close to the RF harmonic, are strongest. No individ-

ual mode, which peaks at a certain moment during acceler-

ation and then disappears again, is observed. This points to

Figure 1: LHC25ns: CB mode spectrum during accelera-

tion averaged over ten cycles for each measurement time.

Figure 2: LHC50ns: Development of the CB mode spec-

trum during acceleration. Same vertical scale (arbitrary

units) as Fig. 1.

a driving impedance with a relative bandwidth of at least

1.4% (the frev swing after transition crossing, which is

given by ! 1"!1" 1/"2

tr, "tr = 6.1).

Coupled-bunch Instabilities on the Flat-top

After arrival on the flat-top, the batch is normally syn-

chronized with the SPS on h = 1, before the splittings start.

For a symmetric splitting of all bunches, CB oscillations

must not be present at this point. To allow for easier anal-

ysis of CB instabilities on the flat-top, the splittings were

disabled and the beam kept at low RF voltage at h = 21 for

about 150ms until extraction. The RF voltages of 10 kV

(LHC25ns) and 20 kV (LHC50ns) correspond to the val-

ues at the start of the bunch splittings. Figure. 3 illustrates

the slowly growing oscillations during the flat-top with low

voltage at h = 21. Dipole oscillations develop especially

at the tail of the batch. The same mode analysis proce-

dure described above has been applied to the well devel-

oped CB oscillations close to extraction. The mode spec-

tra, averaged over ten cycles, are shown in Fig. 4. Again,

MOPD52 Proceedings of HB2010, Morschach, Switzerland

194 Commissioning, Operations and Performance

Measurements comparing the longitudinal limitations of

LHC25ns/LHC50ns beams and the corresponding driving

impedance sources, as far as they have been identified, are

presented in the first part of the paper. Key ingredients to

achieve bunch intensities well beyond nominal intensity of

1.3 · 1011 ppb are introduced. Thereafter, the longitudinal

performance with LHC75ns and the new beam variant with

150 ns bunch spacing (LHC150ns) are reported. Finally,

ongoing and possible future upgrades of the feedback sys-

tems are discussed.

PERFORMANCE LIMTIATIONS WITH25 AND 50 NS BUNCH SPACING

Comparing longitudinal beam stability of the LHC-type

beams with 25 and 50 ns bunch spacing at extraction is mo-

tivated by the fact that both beam types are very similar

during acceleration (see Table 1). In both cases, 18 bunches

are accelerated at h = 21. The longitudinal density is the

same as well, but intensity and longitudinal emittance of

the 50 ns variant are twice smaller.

Coupled-bunch Instabilities During Acceleration

As coupled-bunch (CB) oscillations usually start after

transition crossing, mountain range data were recorded ev-

ery 70ms starting from 100ms after transition crossing,

when the final !l is reached for LHC25ns and LHC50ns

beams (the magnetic cycle for the LHC50ns beam is 1.2 s

shorter due to single-batch transfer from the PSB [3]). The

dipole motion of each bunch is extracted from the center

position of a Gaussian function fitted to each bunch of each

turn recorded. A second fit of a sinusoidal function to the

motion of the bunch center results in oscillation amplitude,

phase and frequency, the latter being the synchrotron fre-

quency. A discrete Fourier transform converts these oscil-

lation amplitudes and phases per bunch to amplitudes and

phases per mode, the mode spectrum [4]. This analysis

technique is superior to measurements in frequency domain

since the bunches only cover 6/7 of the circumference.

Spurious frev lines due to this filling pattern are removed

as only bunches are analyzed. However, the mode num-

bers with respect to the batch, nbatch do not directly cor-

respond to frev harmonics, but each mode number nbatch

results in a spectrum of frev lines, with the lines close to

7/6nbatchfrev at maximum amplitudes.

Figures 1 and 2 show the evolution of the mode spectra

for LHC25ns and LHC50ns beams during acceleration.

Though the total intensity differs by a factor two, the mode

pattern and oscillation amplitudes are very similar, suggest-

ing a scaling proportional to longitudinal density, Nb/!lrather than intensity.

Moreover, the form of the spectrum remains unchanged

during acceleration. The modes nbatch = 1, 2 and 16, 17,those close to the RF harmonic, are strongest. No individ-

ual mode, which peaks at a certain moment during acceler-

ation and then disappears again, is observed. This points to

Figure 1: LHC25ns: CB mode spectrum during accelera-

tion averaged over ten cycles for each measurement time.

Figure 2: LHC50ns: Development of the CB mode spec-

trum during acceleration. Same vertical scale (arbitrary

units) as Fig. 1.

a driving impedance with a relative bandwidth of at least

1.4% (the frev swing after transition crossing, which is

given by ! 1"!1" 1/"2

tr, "tr = 6.1).

Coupled-bunch Instabilities on the Flat-top

After arrival on the flat-top, the batch is normally syn-

chronized with the SPS on h = 1, before the splittings start.

For a symmetric splitting of all bunches, CB oscillations

must not be present at this point. To allow for easier anal-

ysis of CB instabilities on the flat-top, the splittings were

disabled and the beam kept at low RF voltage at h = 21 for

about 150ms until extraction. The RF voltages of 10 kV

(LHC25ns) and 20 kV (LHC50ns) correspond to the val-

ues at the start of the bunch splittings. Figure. 3 illustrates

the slowly growing oscillations during the flat-top with low

voltage at h = 21. Dipole oscillations develop especially

at the tail of the batch. The same mode analysis proce-

dure described above has been applied to the well devel-

oped CB oscillations close to extraction. The mode spec-

tra, averaged over ten cycles, are shown in Fig. 4. Again,

MOPD52 Proceedings of HB2010, Morschach, Switzerland

194 Commissioning, Operations and Performance

LHC25ns: CB mode spectrum during acceleration (18 bunches, h=21) averaged over ten cycles for each measurement time.

LHC50ns: development of the CB mode spectrum (18 bunches h=21) during acceleration

rise time (a.u.)

rise time (a.u.)

PS coupled bunch instability: observations


Pag. 16

Mountain range density plot of a batch of 18 bunches kept at h = 21 with 20 kV (the initial condition for the splitting h = 21 → 42) along the flat top (LHC50ns).

Figure 3: Mountain range density plot of a batch of 18bunches kept at h = 21 with 20 kV (the initial condition forthe splitting h = 21 ! 42) along the flat top (LHC50ns).

1 3 5 7 9 11 13 15 17Mode number, nbatch

Mod

eam

plitu

de!a.u."


Mod

eam

plitu

de!a.u."

Figure 4: Comparison of the CB mode spectrum on theflat-top of LHC25ns (left) and LHC50ns (right) beams. Inboth cases a batch of 18 bunches is kept at low voltage onh = 21 until extraction. The vertical scale of both plots isidentical.

LHC25ns and LHC50ns beams feature very similar modespectra despite their difference in total intensity and longi-tudinal emittance, confirming the aforementioned scalingwith longitudinal density.

The strongest modes present on the flat-top are nbatch =11 and 12, very different from the dominant modes ob-served during acceleration. This suggests that the drivingimpedance changes. Indeed the configuration of the ten2.8 " 10MHz cavities is modified after the arrival on theflat-top. During acceleration, all 10MHz cavities are ac-tive, close to full voltage (20 kV per cavity) and with theirgaps open. To achieve the moderate RF voltages before thesplitting on the flat-top in a well controlled fashion, eightof ten cavities are switched off in a sequence leaving onlytwo active cavities. The unused cavities remain tuned closeto the RF harmonic but are short-circuited by a gap relay.

It was found that the residual impedance of those short-circuited, inactive cavities represents an important part ofthe impedance driving the CB instabilities on the flat-top.Two passive impedance reductions are described below.

Results from recent beam tests pushing the intensity ofthe LHC25ns and LHC50ns beams towards ultimate, in-dicate that intensities up to almost 1.9 · 1011 protons per

extracted bunch within a longitudinal emittance slightlyabove !l = 0.38 eVs can be obtained for both bunch spac-ings. In addition to the passive cures, active CB feedbackon h = 19/20, mainly damping the dominant modes n = 1,2 and 16, 17, significantly improves stability during accel-eration. Fast controlled longitudinal blow-up directly aftertransition crossing avoids a too fast growth of CB instabil-ities in the case of LHC25ns.

Detuning Unused RF Cavities

The ten ferrite-loaded cavities of the main accelerationsystem in the PS can be electrically tuned in three groups(tuning current loops are in series) from 2.8" 10MHz. Onthe flat-top, cavities were originally short-circuited only,but the tuning current was kept unchanged as if they werestill operational on h = 21 or h = 14. This was found tocontribute to the excitation of CB instabilities, hence a newtuning scheme was implemented. At the moment the lastcavity of a tuning group is programmed to zero, the group israpidly tuned to a parking frequency of 3.1MHz (h = 6.5),the lowest possible in-between two frev harmonics.

Gap Relays

Each of the ferrite-loaded cavities consists of two "/4-resonators, each of them with an acceleration gap, con-nected in parallel by two coaxial bars and the tuningloop [5]. Originally, both gaps were short-circuited, butin 1991, shortly after the significant impedance reductionby direct RF feedback [6], the second gap relay was re-moved to reduce maintenance costs. Following the analysisof the instability observations presented above, four of theten cavities have been re-equipped with a second gap re-lay. The beam induced voltage along an acceleration cycle(fixed-target SPS) measured across the left and right gapsis shown in Fig. 5. The asymmetry of the induced volt-

0.2 0.3 0.4 0.5 0.6 0.7 0.80.0

0.5

1.0

1.5

2.0

Time !s"

Vpe

ak!kV"

Right gap

0.2 0.3 0.4 0.5 0.6 0.7 0.80.0

0.5

1.0

1.5

2.0

Time !s"

Vpe

ak!kV"

Left gap

Figure 5: Beam induced voltage (cavity in straight section46) measured on left and right accelerating gaps. Black:both gap relays closed; red: left relay closed only; blue:right relay closed only.

age is different from cavity to cavity, but a reduction of thevoltage by more than a factor of two, due to the second gap

Proceedings of HB2010, Morschach, Switzerland MOPD52

Commissioning, Operations and Performance 195

Figure 3: Mountain range density plot of a batch of 18bunches kept at h = 21 with 20 kV (the initial condition forthe splitting h = 21 ! 42) along the flat top (LHC50ns).


Mod

eam

plitu

de!a.u."


Mod

eam

plitu

de!a.u."

Figure 4: Comparison of the CB mode spectrum on theflat-top of LHC25ns (left) and LHC50ns (right) beams. Inboth cases a batch of 18 bunches is kept at low voltage onh = 21 until extraction. The vertical scale of both plots isidentical.

LHC25ns and LHC50ns beams feature very similar modespectra despite their difference in total intensity and longi-tudinal emittance, confirming the aforementioned scalingwith longitudinal density.

The strongest modes present on the flat-top are nbatch =11 and 12, very different from the dominant modes ob-served during acceleration. This suggests that the drivingimpedance changes. Indeed the configuration of the ten2.8 " 10MHz cavities is modified after the arrival on theflat-top. During acceleration, all 10MHz cavities are ac-tive, close to full voltage (20 kV per cavity) and with theirgaps open. To achieve the moderate RF voltages before thesplitting on the flat-top in a well controlled fashion, eightof ten cavities are switched off in a sequence leaving onlytwo active cavities. The unused cavities remain tuned closeto the RF harmonic but are short-circuited by a gap relay.

It was found that the residual impedance of those short-circuited, inactive cavities represents an important part ofthe impedance driving the CB instabilities on the flat-top.Two passive impedance reductions are described below.

Results from recent beam tests pushing the intensity ofthe LHC25ns and LHC50ns beams towards ultimate, in-dicate that intensities up to almost 1.9 · 1011 protons per

extracted bunch within a longitudinal emittance slightlyabove !l = 0.38 eVs can be obtained for both bunch spac-ings. In addition to the passive cures, active CB feedbackon h = 19/20, mainly damping the dominant modes n = 1,2 and 16, 17, significantly improves stability during accel-eration. Fast controlled longitudinal blow-up directly aftertransition crossing avoids a too fast growth of CB instabil-ities in the case of LHC25ns.

Detuning Unused RF Cavities

The ten ferrite-loaded cavities of the main accelerationsystem in the PS can be electrically tuned in three groups(tuning current loops are in series) from 2.8" 10MHz. Onthe flat-top, cavities were originally short-circuited only,but the tuning current was kept unchanged as if they werestill operational on h = 21 or h = 14. This was found tocontribute to the excitation of CB instabilities, hence a newtuning scheme was implemented. At the moment the lastcavity of a tuning group is programmed to zero, the group israpidly tuned to a parking frequency of 3.1MHz (h = 6.5),the lowest possible in-between two frev harmonics.

Gap Relays

Each of the ferrite-loaded cavities consists of two "/4-resonators, each of them with an acceleration gap, con-nected in parallel by two coaxial bars and the tuningloop [5]. Originally, both gaps were short-circuited, butin 1991, shortly after the significant impedance reductionby direct RF feedback [6], the second gap relay was re-moved to reduce maintenance costs. Following the analysisof the instability observations presented above, four of theten cavities have been re-equipped with a second gap re-lay. The beam induced voltage along an acceleration cycle(fixed-target SPS) measured across the left and right gapsis shown in Fig. 5. The asymmetry of the induced volt-

0.2 0.3 0.4 0.5 0.6 0.7 0.80.0

0.5

1.0

1.5

2.0

Time !s"

Vpe

ak!kV"

Right gap

0.2 0.3 0.4 0.5 0.6 0.7 0.80.0

0.5

1.0

1.5

2.0

Time !s"V

peak

!kV"

Left gap

Figure 5: Beam induced voltage (cavity in straight section46) measured on left and right accelerating gaps. Black:both gap relays closed; red: left relay closed only; blue:right relay closed only.

age is different from cavity to cavity, but a reduction of thevoltage by more than a factor of two, due to the second gap

Proceedings of HB2010, Morschach, Switzerland MOPD52

Commissioning, Operations and Performance 195

Comparison of the CB mode spectrum on the flat-top of LHC25ns (left) and LHC50ns (right) beams. In both cases a batch of 18 bunches is kept at low voltage on h = 21 until extraction. The vertical scale of both plots is identical. The mode 0, the rigid motion of the whole train (all bunches in phase), is damped by the RF system.

PS coupled bunch instability: study


Pag. 17

•  The main sources of longitudinal coupled bunch instability in the PS are thought to be the 10 MHz RF cavities.

•  The coupling impedance of the cavities is not a simple resonator due to the feedback loop and the power amplifier.

SOURCES OF IMPEDANCEObvious sources of longitudinal impedance in the PS are

the RF cavities. Only the 10 and 200MHz RF systems werenot short-circuited. The frequency of the 200MHz RF sys-tem is too high to explain the dependence on bunch lengthand the excitation of dipole modes. Therefore, the 10MHzRF system [2] is the most probable impedance source. Asimplified model of the system is sketched in Fig. 6.

Figure 6: Simplified model of the 10MHz RF system, in-cluding ferrite loaded cavities in the PS and power am-plifier with a tuned grid resonator and the fast feedbackaround the amplifier.

According to this model, the impedance seen by thebeam can be written as

Zc =dV

dIB=

ZGgP gG

ZGgP gG! + (RG + Z)/(RGZ), (4)

where ZG(") is the impedance of the grid resonator, gP

and gG are the effective trans-conductances of driver andfinal amplifier, ! represents attenuation and delay of thefeedback loop. The parameters of the model have beenmatched to reproduce the measured open and closed looptransfer functions of six of the ten cavities. The result forRe{Zc} is shown in Fig. 7.

!15 !10 !5 0 5 10 15 20k " #!#0

0

0.5

1

1.5

2

Re"Z#$k

$%,all1

0cavities

negativefrequencies

positivefrequencies

Figure 7: Real part of the total impedance of the 10MHzcavities at the revolution harmonics.

As the impedance Zc covers several revolution harmon-ics around h"0, the narrow-band approximation is not ap-plicable for the estimation of the growth rates of the CBoscillations. For dipole modes, the growth rates 1/# =

Im{(! ! "s0)/"s0} are given by the eigenvalues, $k ofthe system [3, 4]

$kjk "!!

l!="!

Zc(k#)k# jk!

" !

"!

dg(r)dr

J1(k#r)J1(kr)dr .

(5)The synchrotron frequency in the center of the bunch isgiven by "s0 and J1 is the Bessel function of the first order;k = n + lM , k# = n + l#M ; the eigenvector (Fouriertransform of the current perturbation) is jk.Solving the system Eq. (5) for its eigenvalues $k results

in the growth rates shown in Table 1. Due to the symme-

Table 1: CB growth rates from eigenvalues of Eq. (5).Mode number n = 1 n = 2 n = 3Growth rate, 1/# 2.5 s"1 3.0 s"1 1.0 s"1

try of Eq. (5), the modes n = 6, 5, 4 have the same, butnegative growth rates as modes n = 1, 2, 3 (Table 1). Themeasured growth rates are slightly higher. A delayed oc-currence of the n = 3 mode, when other CB modes arealready well developed, may explain the large scatteringmeasured (compare Fig. 3).

CONCLUSIONLongitudinal CB growth rates and mode spectra mea-

surements in the CERN PS indicate that the main 10MHzRF system represents the most probable impedance sourcefor excitation. Due to the strong feedback, the impedanceof the cavities covers 4 harmonics at each side of the RFharmonic. The shape of these impedance lines predicts an = 2 mode being stronger than n = 1. This has alsobeen observed with beam. Methods to damp the CB os-cillations with a wide-band feedback or by optimizing thecavity impedance with the one-turn-delay feedback are cur-rently studied.Work supported by EU Design Study DIRACsecondary-

Beams (contract 515873).

REFERENCES[1] F. Pedersen, F. Sacherer, “Theory and Performance of the

Longitudinal Active Damping System for the CERN PSBooster”, PAC’77, IEEE Trans. Nucl. Sci., Vol. NS-28, 1977,pp. 1396-1398

[2] D. Grier, “The PS 10 MHz Cavity and Power Amplifier”,PS/RF Note 2002-073, CERN, Geneva, Switzerland, 2002

[3] J. L. Laclare, “Bunched Beam Coherent Instabilities”, CERN87-03, CAS on General Acc. Phys., Oxford, U. K., 1985, pp.264-326

[4] E. Shaposhnikova, “Analysis of Coupled Bunch InstabilitySpectra”, CERN-SL-99-040 HRF, CERN, Geneva, Switzer-land, 1999

FRPMN069 Proceedings of PAC07, Albuquerque, New Mexico, USA

05 Beam Dynamics and Electromagnetic Fields

4182

D04 Instabilities - Processes, Impedances, Countermeasures

1-4244-0917-9/07/$25.00

c�2007 IEEE

7 8 9 10 11 12 13 14f (MHz)

0

200

400

600

800

1000

1200

1400

1600

Re[Z

] (O

hm)



Pag. 18

•  A simulation code, deriving from an older code that was developed to study the coupled bunch instabilities in DAΦNE, has been adapted to the PS machine.

•  The main difference with respect to the DAΦNE code is the presence of the frequency domain longitudinal feedback instead of the time domain FB.

KK

DA!NE TECHNICAL NOTEINFN - LNF, Accelerator Division

Frascati, June 23, 1993

Note: G-1 9

A TIME DOMAIN SIMULATION CODE OF THE LONGITUDINALMULTIBUNCH INSTABILITIES

M. Bassetti, A. Ghigo, M. Migliorati, L. Palumbo, M. Serio

1 . INTRODUCTION

The analytical study of the longitudinal dynamics of a beam interactingwith an RF cavity is generally performed only in the case of small oscillationsof equispaced equal bunches around their synchronous phase[1]. Furthermorea complete analytical treatment of the dynamics in the presence of a bunch-by-bunch feedback system to control longitudinal coupled bunch instabilitieshas not yet developed.

The purpose of this note is to describe the main features of a simulationcode that executes a tracking of the longitudinal oscillations of the bunchesfor DA!NE, with the aim of including the main phenomena affecting the beamdynamics (i.e. the bunch-by-bunch feedback, the effect of the HOMs, thesynchrotron radiation).

2 . THE SIMULATION CODE

The code can simulate different starting conditions. In order to study thetransient beam loading effects on the HOMs that may be dominant in highintensity accelerators, such as DA!NE, we have simulated the injection of abunch, assuming all the others already stored in the ring.

We model each bunch as a single particle of given charge. Under thiscondition it is possible to simulate only the "rigid" oscillations that, however,are the most dangerous for the beam stability and, at the moment, the onlyones that will be cured with the longitudinal feedback system.

Basically the core of the algorithm can be divided into three main parts:

1) propagation around the ring2) feedback effect3) beam-cavity interaction.

In Fig.1 we show a simplified flow chart of the code. The input data havebeen divided into three files: one for machine and cavity parameters, anotherfor all the bunches, and the last one for the longitudinal feedback system.



Pag. 19

•  According to measurements in 2007, the growth rates range from about 1 s-1 to 5 s-1, with a weak dependence on the longitudinal emittance.

•  The unstable modes are n=1, 2, 3

Figure 2: Growth rate measurements of a CB mode (n = 2,5, m = 1, 2, . . .); fs ! 0.2 kHz after transition. Greentrace: peak detected beam signal; yellow and blue traces:down-converted base-band signals around the 5th harmonicof !0; horizontal scale: 100ms per division.

The growth rate 1/" of the CB oscillation is computedfrom a linear fit to the logarithmic amplitude envelope ofthe measured signal in the region of exponential growth;saturation is thus not taken into account. The measuredgrowth rates versus longitudinal bunch emittance are plot-ted in Fig. 3. The emittance is adjusted with a controlledlongitudinal blow-up at flat-bottom. Only a very weak de-

1.6 1.8 2 2.2 2.4 2.60

2

4

6

8n ! h " 5 ! 2, n ! 5

1.6 1.8 2 2.2 2.4 2.60

2

4

6

8n ! h " 6 ! 1, n ! 6

1.6 1.8 2 2.2 2.4 2.60

2

4

6

8n ! h " 3 ! 4, n ! 3

1.6 1.8 2 2.2 2.4 2.60

2

4

6

8n ! h " 4 ! 3, n ! 4

1.6 1.8 2 2.2 2.4 2.60

2

4

6

8n ! h " 1 ! 6, n ! 1

1.6 1.8 2 2.2 2.4 2.60

2

4

6

8n ! h " 2 ! 5, n ! 2

Growthrate,1

!#"1!s#

Longitudinal emittance, !l "eVs#Figure 3: Growth rates 1/" versus longitudinal emittance#l, measured at the harmonics 1 to 6 of !0.

pendence of the growth rates on the longitudinal bunchemittance is observed. This indicates an impedance sourcein the area with flat dependence of the growth rate on bunch

length (for m = 1: "bunch ! 0.5/fr). The large scatter ofthe measurements can be related to small intensity varia-tions of each individual cycle. The n = 3 and 4modes startto develop 200ms later in the cycle and have larger scattersince the bunches already oscillate on the other modes.

MODE ANALYSISThe dipole mode CB spectrum at a certain moment in

the cycle can be found from mountain range measure-ments spanning a few periods of the synchrotron oscillation(Fig. 4). In this case, the center positions of the bunches

Time$

Figure 4: Mountain range density plots (490ms after tran-sition, see Fig. 2) of all seven bunches during 15ms (fs !210Hz). The horizontal span of each frame is 160 ns; thedistance between two bunches (reduced) is 304 ns.

versus time is computed from the center position of a Gaus-sian fit to each bunch in each frame. A nearly sinusoidalmotion of the bunch centers with time is observed. A fit ofa sinusoidal function to the motion of the bunch centers

$̂k sin(!st + %k) =M!1!

n=0

!̂n sin"

!st +2&nk

M+ "n

#

(2)results in an oscillation amplitude $̂k (k: bunch index), anoscillation phase %k and a frequency, corresponding to fs.The n = 0 . . . 6 mode amplitudes !̂n and phases "n arecomputed applying a DFT according to

!̂ne!i!n =1M

M!1!

k=0

$̂ke!i!k · e2"ikn/M . (3)

Fig. 5 illustrates the CB mode spectrum obtained fromthe mountain range data shown above. This mode spectrum

0 1 2 3 4 5 6Mode number, n

Figure 5: Dipole CB mode amplitudes !̂n extracted fromthe mountain range measurement shown in Fig. 4.

is very reproducible. The n = 2 oscillation is excited firstand has the maximum amplitude.

Proceedings of PAC07, Albuquerque, New Mexico, USA FRPMN069


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c�2007 IEEE


4181

SOURCES OF IMPEDANCEObvious sources of longitudinal impedance in the PS are

the RF cavities. Only the 10 and 200MHz RF systems werenot short-circuited. The frequency of the 200MHz RF sys-tem is too high to explain the dependence on bunch lengthand the excitation of dipole modes. Therefore, the 10MHzRF system [2] is the most probable impedance source. Asimplified model of the system is sketched in Fig. 6.

Figure 6: Simplified model of the 10MHz RF system, in-cluding ferrite loaded cavities in the PS and power am-plifier with a tuned grid resonator and the fast feedbackaround the amplifier.

According to this model, the impedance seen by thebeam can be written as

Zc =dV

dIB=

ZGgP gG

ZGgP gG! + (RG + Z)/(RGZ), (4)

where ZG(") is the impedance of the grid resonator, gP

and gG are the effective trans-conductances of driver andfinal amplifier, ! represents attenuation and delay of thefeedback loop. The parameters of the model have beenmatched to reproduce the measured open and closed looptransfer functions of six of the ten cavities. The result forRe{Zc} is shown in Fig. 7.

!15 !10 !5 0 5 10 15 20k " #!#0

0

0.5

1

1.5

2

Re"Z#$k

$%,all1

0cavities

negativefrequencies

positivefrequencies

Figure 7: Real part of the total impedance of the 10MHzcavities at the revolution harmonics.

As the impedance Zc covers several revolution harmon-ics around h"0, the narrow-band approximation is not ap-plicable for the estimation of the growth rates of the CBoscillations. For dipole modes, the growth rates 1/# =

Im{(! ! "s0)/"s0} are given by the eigenvalues, $k ofthe system [3, 4]

$kjk "!!

l!="!

Zc(k#)k# jk!

" !

"!

dg(r)dr

J1(k#r)J1(kr)dr .

(5)The synchrotron frequency in the center of the bunch isgiven by "s0 and J1 is the Bessel function of the first order;k = n + lM , k# = n + l#M ; the eigenvector (Fouriertransform of the current perturbation) is jk.Solving the system Eq. (5) for its eigenvalues $k results

in the growth rates shown in Table 1. Due to the symme-

Table 1: CB growth rates from eigenvalues of Eq. (5).Mode number n = 1 n = 2 n = 3Growth rate, 1/# 2.5 s"1 3.0 s"1 1.0 s"1

try of Eq. (5), the modes n = 6, 5, 4 have the same, butnegative growth rates as modes n = 1, 2, 3 (Table 1). Themeasured growth rates are slightly higher. A delayed oc-currence of the n = 3 mode, when other CB modes arealready well developed, may explain the large scatteringmeasured (compare Fig. 3).

CONCLUSIONLongitudinal CB growth rates and mode spectra mea-

surements in the CERN PS indicate that the main 10MHzRF system represents the most probable impedance sourcefor excitation. Due to the strong feedback, the impedanceof the cavities covers 4 harmonics at each side of the RFharmonic. The shape of these impedance lines predicts an = 2 mode being stronger than n = 1. This has alsobeen observed with beam. Methods to damp the CB os-cillations with a wide-band feedback or by optimizing thecavity impedance with the one-turn-delay feedback are cur-rently studied.Work supported by EU Design Study DIRACsecondary-

Beams (contract 515873).

REFERENCES[1] F. Pedersen, F. Sacherer, “Theory and Performance of the

Longitudinal Active Damping System for the CERN PSBooster”, PAC’77, IEEE Trans. Nucl. Sci., Vol. NS-28, 1977,pp. 1396-1398

[2] D. Grier, “The PS 10 MHz Cavity and Power Amplifier”,PS/RF Note 2002-073, CERN, Geneva, Switzerland, 2002

[3] J. L. Laclare, “Bunched Beam Coherent Instabilities”, CERN87-03, CAS on General Acc. Phys., Oxford, U. K., 1985, pp.264-326

[4] E. Shaposhnikova, “Analysis of Coupled Bunch InstabilitySpectra”, CERN-SL-99-040 HRF, CERN, Geneva, Switzer-land, 1999

FRPMN069 Proceedings of PAC07, Albuquerque, New Mexico, USA


4182


1-4244-0917-9/07/$25.00

c�2007 IEEE

Simula'on’s results Mode Number 1 2 3

Growth Rate 1.84 s-1 2.14 s-1 0.52 s-1

PS coupled bunch instability: cures


Pag. 20

RF phase of each bunch in a given ;me window

Reconstruc;on of amplitude and phase oscilla;on for each bunch Mode analysis

Select the mode to damp: aμ, θμ

Amplitude (propor;onal to aμ) and phase of the energy kick

Same kick amplitude for all the bunches (but different phase)

= aµ sin ωst +2πNb

µb+θµ!

"#

$

%&

µ=0

Nb−1

∑Ab sin ωst +θb( )

ΔVfb = −gωs

dφbdt

= −gaµ cos ωst +2πNb

µb+θµ#

$%

&

'(

φb =

φb +ωs

VRFg φb +ωs

2φb = 0

equation of motion of a bunch in presence of the feedback

α fb = −12ωs

VRFg = − 1

2ωRFη

ωsT0 E0 / e( )g

FB damping rate

PS coupled bunch instability: cures


Pag. 21

•  Up to the present intensities, only the first two dominant oscillation modes are damped by a dedicated feedback (FB) system.

•  However it will become insufficient for the beam parameters planned within the framework of the LIU upgrade of CERN’s injector chain.

•  The goal is to have a stable beam with 3e11 ppb at flat top after the splitting.

•  A new FB will therefore be installed, covering all possible modes (the instability develops above transition energy).

•  It will use a new wide-band kicker cavity based on Finemet technology driven by a digital signal processing chain.

•  In order to study the effect of the new FB on the longitudinal coupled bunch instability, we will use the time domain simulation code, with the inclusion of a frequency domain FB.

PS coupled bunch instability: conclusions …


Pag. 22

•  The simulation code to study the longitudinal coupled bunch instability has been benchmarked with the PS parameters.

•  The code has been validated under several conditions with the PS data.

•  A frequency domain longitudinal FB system has been implemented and tested.

•  The 10 MHz cavities impedance has been demonstrated to be the main source of the instability.

… and future work

•  Full simulations using PS-LIU beam parameters with longitudinal FB. •  Find the maximum power required by the FB with the PS-LIU beam

parameters.

Expected performance


Pag. 23

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5



SPS 450 GeV 25 ns

SPS RF pow

erLongitu

dinal insabilitie

s

PS RF power

Longitu

dinal insabilitie

s

HL-‐LHC

0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

4.0

0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0Em

ittance (x+y)/2 [um]


SPS 450 GeV 50 ns

SPS Longitu

dinal insabilitie

s

PSLongitu

dinal insabilitie

s

SPS TM

CI limit

HL-‐LHC

2.3e11, 2.4 µm (!) at SPS extraction

2.8e11, 2.6 µm at SPS extraction

Conclusions


Pag. 24

•  For the HL-LHC, the requirements placed on the LHC machine and the injector complex are very challenging. The LHC will need to make dramatic improvements in current, peak luminosity and efficiency. The injector complex will need to provide 25 ns beams with twice the present intensity in the present emittance, and 50 ns beams with a factor of 2.5 higher intensity and a brightness increase of 50%.

•  Upgrade of LHC beams in injectors requires to improve understanding of current limits, improve machine modeling, …

•  The 50 ns beams are more challenging with respect to the 25 ns that are within reach. In any case it is however necessary that: –  All planned upgrades must be fully effective –  There is the need to “stretch” loss/blowup levels in injectors

•  Overall, the HL-LHC requirements are not totally out of reach, although there are still several unknowns and risks.

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