SIMULATION RESULTS OF A FEEDBACK CONTROL SYSTEM TODAMP ELECTRON CLOUD SINGLE-BUNCH TRANSVERSE

INSTABILITIES IN THE CERN SPS∗

R. Secondo† , J.-L. Vay, M. Venturini, LBNL, USA; J.D. Fox, C.H. Rivetta, SLAC, USA;W. Hofle, CERN, Switzerland

Abstract

Transverse Single-Bunch Instabilities due to ElectronCloud effect are limiting the operation at high current ofthe SPS at CERN. Recently a high-bandwidth FeedbackSystem has been proposed as a possible solution to stabi-lize the beam and is currently under study. We analyzethe dynamics of the bunch actively damped with a simplemodel of the Feedback in the macro-particle code WARP,in order to investigate the limitations of the System such asthe minimum amount of power required to maintain stabil-ity. We discuss the feedback model, report on simulationresults and present our plans for further development of thenumerical model.

INTRODUCTION

The Electron Cloud Instability (ECI) represents a lim-itation to future intensity upgrades of the LHC injectioncomplex at CERN, [1], [2]. Large and fast growingtransverse instabilities have been reported affecting high-intensity proton beams in the SPS. One of the solutions pro-posed to mitigate this effect is a high-bandwidth Feedback(FB) Control System, [3]. We used the extensive capabil-ities of the PIC simulation framework WARP to model thebunch interaction with the e-cloud together with the damp-ing action provided by a simple and ideal Feedback. Ourpurpose is to investigate the requirements of the Systemsuch as the minimum amount of power required to main-tain stability. The control action is based on a bandpassFIR filter (refer to [4] for more details on filter design). Weran several simulations using a fixed set of initial param-eters, evaluating the beam instability growth rate and ad-justing the filter gain accordingly to achieve stability, com-paring open (FB off) and closed (FB on) loop cases andanalyzing vertical instabilities and emittance growth. Aftersetting a constant filter gain value simulations have beenperformed limiting the kick signal with different saturationlevels and varying the electron density around the acceler-ator ring, as a first step to evaluate the minimum amount ofpower needed to control efficiently the transverse motionin relation to the e-cloud density. Conclusions and futuredevelopments of the numerical model are discussed in thelast section.

∗Work supported by the US-DOE under Contract DE-AC02-05CH11231, the SciDAC program ComPASS and the US-LHC Accel-erator Research Program (LARP). Used resources of NERSC and theLawrencium cluster at LBNL

† rsecondo@lbl.gov

FEEDBACK MODEL

The Feedback is composed by three elements connectedin a loop: a receiver measures and processes the signal fromthe pick-up and estimates the vertical displacement of dif-ferent areas on the bunch, a processing channel calculatesthe control signal and finally the signal is amplified and ap-plied to the bunch by a kicker, [5]. The output signal of theFB is applied to each slice of the bunch on a one-turn delaybasis at the same position along the accelerator where thebeam is sampled. The processing channel is representedby a simple FIR filter that damps the beam vertical dis-placement while limiting the bandwidth around the nomi-nal fractional tune [Qy] = 0.185 and advancing the phaseby 90 degrees at the tune frequency. The output z i(k) ofthe FIR is calculated on 5 previous measurements of thebunch vertical displacement yi(k) as

zi(k) = a1yi(k−1)+a2yi(k−2)+ ...+anyi(k−n), (1)

where i = 1, · · · , Nslices identifies the bunch slice, k is themachine turn number, n = 5 is the # of taps and the set ofcoefficients a1, a2...an defines the impulse response of thefilter. The complete FIR output with gain G can be definedas Ci(k) = G · zi(k). The receiver and kicker are idealand have no bandwidth limitation. The action of the Feed-back system can be understood in terms of the followingsimplified linearized model of bunch dynamics

y′′ + ω2y = K(ye − y) + Δp⊥ , (2)

where y is the amplitude of the vertical oscillation of a sliceand ye the transverse offset of the electron cloud baricentercorresponding to that slice; the constant K is a measureof the interaction beam-ecloud and Δp⊥ the signal of thekicker. With the Feedback on, the vertical displacement ofeach slice is forced to zero, y � 0, reducing (2) to

|Kye| � |Δp⊥ | , (3)

suggesting that the analysis of Δp⊥ will give a measure ofthe interaction between the e-cloud and the bunch.

SIMULATION RESULTS

The set of initial parameters used in all simulations isreported in Table 1.

Several Single-Bunch simulations were performed set-ting an initial vertical offset δy = 0.1 · σy = 0.269 mmto all the bunch slices and assuming a uniform distributionof electrons ne = 1012m−3 at each station around the ring.

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Table 1: WARP Parameters Used in the SPS Simulations

Parameter Symbol Value

beam energy Eb 26 GeVbunch population Nb 1.1 x 1011

rms bunch length σz 0.229 mrms transv. emittance εx,y 2.8, 2.8 mm·mradrms longit. emittance εz 0.0397 eV·srms momentum spread δrms 1.9 x 10−3

beta functions βx,y 33.85, 71.87 mbetatron tunes Qx,y 26.13, 26.185chromaticities Q′

x,y 0, 0RF cavity Voltage V 2 MVmom. compact. factor α 1.92 x 10−3

circumference C 6.911 km# of beam slices Nslices 64# of stations/turn Ns 20

The beam in open loop starts undergoing a growing verticalinstability for ne = 0.5 · 1012m−3. A preliminary estima-tion of the bunch interacting with the e-cloud in open loopgave a vertical instability growth rate between 1/20 - 1/50turns. With gain G = 0.1 the FIR filter is able to dampgrowth rates slower than 1/20 turns, consequently a valueof G > 0.1 is needed to efficiently damp the transversemotion of the slices. Fig. 1 shows the absolute value ofthe vertical motion of the bunch centroid in open loop (FBoff) compared with the closed loop case (FB on). Whenthe Feedback is active with G = 0.1 the damping action ofthe filter allows a strong reduction of the vertical instabil-ity and emittance growth, Fig. 2, albeit insufficient as theemittance increases by about 40% after 2000 turns. Usinga gain G = 0.2 provides an even greater control with nodetected emittance growth after 2000 turns.

0

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Abs Vertical Centroid Displacement 26 GeV - De = 1e12 e/m3

Open LoopFB on, G = 0.1FB on, G = 0.2

Figure 1: Absolute value of the bunch centroid vertical dis-placement vs turns. The centroid is calculated averagingthe vertical displacement of the bunch over 64 slices.

0

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t)⟩ /

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Relative rms Vertical Emittance Comparison

Open LoopFIR gain = 0.1FIR gain = 0.2

Figure 2: Relative vertical rms emittance growths in openloop (blue) and closed loop setting G = 0.1 (red) and G =0.2 (magenta).

Kick signal eV⋅s/m-No Saturation-G=0.2- De=1e12 e/m3

5001000

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Figure 3: Kick signal eV·sec/m, slices vs turns. Gain G =0.2. Slices close to 1 correspond to the bunch tail.

Fig. 3 shows the momentum applied to the bunch ineV·sec/m units in the case of Fig. 1 with FB on andG = 0.2 (low vertical instability). The Power SpectralDensity of the kick signal (Δp⊥ according to Eq. 3) in thiscase is reported in Fig. 4, the power over turns increasesspreading over the spectrum of frequencies but most of itis concentrated in the bandwidth within 1 GHz. One ofthe limitations of the system is given by the amplifier thatdrives the kicker. If the amplifier saturates the kicker signalcould not be sufficient to damp the instability efficiently.We ran simulations keeping the FIR gain G = 0.2 and forc-ing the kicker signal to saturate at an arbitrary value in mo-mentum units with the purpose of understanding the limitsof the kicker efficiency in controlling the transverse insta-bility. In the results shown we considered two saturationlevels at 9.58·10−6 eV·sec/m and 2.874·10−5 eV·sec/m,respectively about 10% and 30% of the maximum momen-tum measured in the case with no limitation (Fig. 3).We choose two slices as representative of the behavior ofthe tail (blue trace) or head (red trace) areas in the bunch.In Fig. 5 is reported the kick signal limited at 9.58·10−6

eV·sec/m (top) and the correspondent vertical displacement(bottom). All the slices are initially damped, slices in thetail area require more power to be controlled (as in Fig. 3)and the kick signal is saturated faster respect to the head.After 1200 turns a large growing instability shows up and

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PSD of the kick signal - 26 GeV - De = 1e12 e/m3 FIR gain=0.2

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Figure 4: Power Spectral Density of the kicker signal, fre-quencies vs turns, dB units.

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Tail slice Head slice

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Tail slice Head slice

Figure 5: Absolute value of the kick signal limited at9.58·10−6 eV·sec/m (top) and vertical displacement (bot-tom) for a slice in the tail (blue) and in the head (red).

a similar effect occurs in the head a few turns later. If thekick signal is limited at 2.874·10−5 eV·sec/m, Fig. 6, allslices are damped in this range of turns and the kick signaldo not saturate. In Fig. 7 we keep a saturation thresholdof 2.874·10−5 eV·sec/m but we set a density of electronsne = 2 ·1012m−3. In this case the maximum signal able tobe processed by the amplifier and kicker is not enough tocontrol the instability, when the power stage saturates theslices are affected by large transverse oscillations.

CONCLUSION

Large and fast growing transverse instabilities in high-intensity proton beams represent a limitation for futureupgrades of the SPS operation at high current. A high-bandwidth Feedback Control System could be a potentialsolution to this problem. We have started to investigateSystem requirements and limitations using WARP simu-lation framework, starting from a simplified model of theFeedback based on a FIR filter. Single-Bunch simulationswith gain G = 0.2 and e-cloud density ne = 1012m−3

show a well damped vertical motion of the bunch. Howeverthe control action fails and a growing instability shows upin the case of a limitation on the kicker signal of 9.58·10−6

momentum. In addition setting a larger electron den-

0

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Kick signal with saturation at 2.874⋅10-5 eV⋅s/m - G = 0.2

Tail slice Head slice

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vert

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]

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Vertical displacement [m], Head and Tail - G = 0.2

Tail slice Head slice

Figure 6: Absolute value of the kick signal limited at2.874·10−5 eV·sec/m (top) and vertical displacement (bot-tom) for a slice in the tail (blue) and in the head (red) of thebunch. The uniform electron density is ne = 1 · 1012m−3.

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Kick signal with saturation at 2.874⋅10-5 eV⋅s/m - G = 0.2 - De = 2⋅1012

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Vertical displacement [m], Head and Tail - G = 0.2 - De = 2⋅1012

Tail slice Head slice

Figure 7: Absolute value of the kick signal limited at2.874·10−5 eV·sec/m (top) and vertical displacement (bot-tom) for a slice in the tail (blue) and in the head (red) of thebunch. The uniform electron density is ne = 2 · 1012m−3.

sity value increases the beam-ecloud interaction and morepower is needed to damp the beam efficiently. We are cur-rently improving the feedback model including more real-istic models for the receiver, power amplifier and kicker.We plan to continue to study the bunch dynamics in closedand open loop for different electron densities. We still needto investigate more accurately if the effects seen in the sim-ulation are caused by the beam physics or resulting fromaspects of the numeric simulation.

REFERENCES

[1] G. Arduini et al., CERN-2005-001, (2005) 31-47

[2] G. Rumolo et al., PRL 100 (2008) 144801

[3] J. D. Fox et al., Proc. of IPAC 2010, Kyoto, Japan.

[4] R. Secondo et al., Proc. of ECLOUD 2010, Cornell Univer-sity, NY, USA.

[5] J.-L. Vay et al., Simulation of E-Cloud Driven Instability andits Attenuation Using a Feedback System in the CERN SPS,Proc. of IPAC 2010, Kyoto, Japan.

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