The main synchrotron SIS100 of the Facility for Antiproton and Ion Research (FAIR) will be equipped with a bunch-by-bunch feedback system to damp longitudinal beam oscillations. In the basic layout, one three-tap finite impulse response (FIR) filter will be used for each single bunch and oscillation mode. The detected oscillations are used to generate a correction voltage in dedicated broadband radio frequency (RF) cavities. The digital filter is completely described by two parameters, the feedback gain and the passband center frequency, which have to be defined depending on the longitudinal beam dynamics. In earlier works, the performance of the closed loop control with such an FIR-filter was analyzed and compared to simulations and measurements with respect to the damping of coherent dipole and quadrupole modes, the first modes of oscillation. This contribution analyzes the influence of cavity beam loading on the closed loop performance and the choice of the feedback gain and passband center frequency to verify future high current operation at FAIR

Groß, Kerstin

Klingbeil, Harald

Lens, Dieter Etienne Mia

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Impact of Simplified Stationary Cavity Beam Loading on the Longitudinal Feedback System for SIS100

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IMPACT OF SIMPLIFIED STATIONARY CAVITY BEAM LOADING ONTHE LONGITUDINAL FEEDBACK SYSTEM FOR SIS100∗K. Gross†, TUD, TEMF, Darmstadt, GermanyH. Klingbeil, GSI and TUD, TEMF, Darmstadt, GermanyD. Lens, TUD, RMR and TEMF, Darmstadt, GermanyAbstractThe main synchrotron SIS100 of the Facility for Antipro-ton and Ion Research (FAIR) will be equipped with a bunch-by-bunch feedback system to damp longitudinal beam oscil-lations. In the basic layout, one three-tap finite impulseresponse (FIR) filter will be used for each single bunchand oscillation mode. The detected oscillations are usedto generate a correction voltage in dedicated broadband ra-dio frequency (RF) cavities. The digital filter is completelydescribed by two parameters, the feedback gain and the pass-band center frequency, which have to be defined dependingon the longitudinal beam dynamics. In earlier works [1, 2],the performance of the closed loop control with such an FIR-filter was analyzed and compared to simulations and mea-surements with respect to the damping of coherent dipoleand quadrupole modes, the first modes of oscillation.This contribution analyzes the influence of cavity beam load-ing on the closed loop performance and the choice of thefeedback gain and passband center frequency to verify futurehigh current operation at FAIR.BEAM LOADINGGbsGbsGbc−Gbcbunch phase feedbackbunch length feedbackFIRKI∫FIRKI∫BpphaseBaamplitudeavpbabpv−Figure 1: Block diagram for small modulation with thenotation of [3] (p: phase, a: amplitude, v: voltage, b: beamand Gbs , Gbc , Bp , Ba : transfer functions, see also Eqs. (1-5)).To eliminate the coupling term tanΦs , usually used inthe Pedersen model [3, 4], shown as part of Fig. 1, the gapvoltage modulations,v(x) = (1 + ãv ) sin(x + p̃v ) (1)were chosen in quadrature with the first harmonic of thebeam currentib (x) = Y I0(1 + ab ) sin(x − Φs + pb ) (2)∗ Work supported by the German Federal Ministry of Education and Re-search (BMBF) under the project 05P12RDRBF.† gross@temf.tu-darmstadt.dewhere Y I0 denotes the steady state amplitude, Y the relativebeam loading andΦs the synchronous phase angle, by settingav = ãv − tanΦs p̃v and pv = p̃v + tanΦs ãv (3)as sketched in Fig. 2.The single particle dynamics are then described by∆̈ϕ = ω2syn,0[v(Φs + ∆ϕ) − sinΦs]with the gap voltageamplitude included in the stationary synchrotron frequencyωsyn,0. This choice of the gap voltage modulations fromxx ΦsΦsp̃v , 0ãv , 0v(x)pv , 0av , 0v (x)Figure 2: Convenient choice of gap voltage modulations.Eqs. (3), in addition, reduces the cavity transfer functionsEqs. (5) for the beam current (with an equivalent meaningof the tilde) to a simpler form. Nevertheless, the followingconsiderations, other than e.g. Eq. (6), are shown forΦs = 0.The SIS100 accelerating system will be based on theSIS18 cavities with a loaded Q in the order of 10. Thusit is a reasonable assumption to consider only the fundamen-tal harmonic of the beam induced voltage.Figure 3 shows the frequency diagrams coming along withthe well-known Robinson stability diagram [3, 5] calculatedwithout the common approximations for a rigid bunch and ahigh loaded quality factor of the cavity. To be able to copewith long bunches (σ = 33◦ as standard deviation at thebottom of Fig. 3) the beam dynamics were taken from [5],replacing the beam transfer functions for short bunches (heree.g. σ = 6◦) [6]:Bp (s) =ω2syn,0s2 + ω2syn,0and Ba (s) ∼−2ω2syn,0s2 + 4ω2syn,0(4)The white curve indicates the optimum (concerning powerconsumption) detuning ΦL = 0 (generator current and gapvoltage in phase) for the loading angle ΦL and stationarybeam loading compensation. The diagrams were calculatedfor Q = 10 and a synchrotron tune of 10−3 but are validfor all cavities that react on modulations of the beam cur-rent within one period of a synchrotron oscillation. This,however, primarily depends on the real part of the dominantpole of the cavity impedance with the resonant frequencyω0, which is the half cavity bandwidth α =ω02Qfor Q ≥ 125th International Particle Accelerator Conference IPAC2014, Dresden, Germany JACoW PublishingISBN: 978-3-95450-132-8 doi:10.18429/JACoW-IPAC2014-TUPRI073TUPRI0731736ContentfromthisworkmaybeusedunderthetermsoftheCCBY3.0licence(©2014).Anydistributionofthisworkmustmaintainattributiontotheauthor(s),titleofthework,publisher,andDOI.06 Instrumentation, Controls, Feedback & Operational AspectsT05 Beam Feedback Systems024Y0 π4024ΦZY0 π4ΦZ0 π4ΦZFigure 3: Damping, coherent dipole and quadrupole fre-quency (left to right) for short (top) and long (bottom)bunches; with from ωsyn,0 to 2ωsyn,0 for thequadrupole frequency and from 0 to ωsyn,0 otherwise.and ω02Q(1 −√1 − 4Q2)for Q ≤ 12. To support the follow-ing simplification, a more detailed evaluation regarding thedependence of the extended Robinson stability diagrams(Fig. 3) on the synchrotron tune and the loaded Q-factorwas done based on the transfer functions given in the Ap-pendix. Supposing that the cavity reacts on changes in thebeam current virtually immediately, the transfer functionscan be taken as quasi-static, with Gs = 1 and Gc = 0 astransfer functions for the total current, and hence for thebeam currentG̃bs = Y cosΦZ (sinΦZ cosΦs − cosΦZ sinΦs ) (5a)G̃bc = Y cosΦZ (cosΦZ cosΦs + sinΦZ sinΦs ) (5b)Gbs = Y sinΦZ cosΦZ/ cosΦs (5c)Gbc = Y cos2ΦZ/ cosΦs (5d)is obtained, where ΦZ is the cavity detuning angle. Thisgives the high current limit from the stability considerationsby Robinson [3]Y =2 cosΦssin(2ΦZ )which is due to the loss of phase focusing as the phase anglebetween the generator induced voltage ΦZ − ΦL and thesynchronous phase Φs is 90◦= Φs + ΦZ − ΦL with thesteady state condition [3]tanΦL =Y cosΦs − tanΦZY sinΦs + 1to keep the total gap voltage independent of Y andΦZ . Thusa single particle arriving too late in the cavity is no longeraccelerated with respect to the synchronous particle.From pb = Bp (s)pv and pv = Gbs pb using Eqs. (4) and(5) the coherent synchrotron frequency with beam loading,corresponding to the dipole frequency shift along the whitecurve in Fig. 3, is found to be [7]:ω2syn,bl = ω2syn,0(cosΦs − Y12sin(2ΦZ ))(6)It is shown in the following that Eq. (6) is important to tunethe passband center frequency of the bunch phase feedback.LONGITUDINAL FEEDBACK SYSTEMThe closed loop bunch-by-bunch feedback system basi-cally consists of• broadband measurement of the beam current• de-multiplexing of the beam signal• one DSP-system per bunch with– IF pre-processing of the pulsed signals– phase / amplitude detection(actuating variables, see Eq. (2))– one digital FIR-filter for each oscillation mode– post-processing of the filter output(control variables, see Eqs. (1) and (3))• multiplexing of the control signals• two broadband kicker cavitiesThe scope of the longitudinal feedback system is to reducedilution in longitudinal phase space due to filamentation.Different single bunch modes are damped, in the first in-stance dipolar (m = 1) and quadrupolar (m = 2) oscillations,whereas all coupled bunch modes (0 ≤ n ≤ h − 1) are cov-ered by the bunch-by-bunch processing. The modularity andscalability of the digital control system allow considerableflexibility regarding the oscillation modes and frequencies.Fig. 4 (top) shows the basic stability diagrams of the FIR-filters (feedback gain and passband center frequency) usedto damp undesired beam phase [1] and bunch length [2]oscillations. The large stability regions indicate robustnessagainst parameter uncertainties. The dynamics of the digitalFIR-filter withk =fsample2 fpassare implemented as:y(t) = −14x(t) +12x(t − kTsample) −14x(t − 2kTsample) (7)It is followed by an integrator with the gain KI as shown inFig. 1. Please note that the bunch phase feedback works withboth inputs pb − p̃v and pb − pv if Φs changes adiabaticallyas constant offsets are suppressed by the FIR-filter (7). Fig-ure 4 verifies the strong dependence of the passband centerfrequency fpass of the bunch phase feedback (left columns)on the relative beam loading Y according to Eq. (6). Thepredicted optimum for a short bunch marked with a white¨x¨ on the k-axis in the left column of Fig. 4 is remarkablyclose to the optimum k found from the eigenvalue analysisof the full order closed loop system.In addition to the references [1, 2] the analysis includesthe coupling of the two feedback loops. The slower (dueto more filter taps) bunch phase feedback dominates thedynamics, resulting in the feignedly extended optimum ofthe bunch length feedback, with respect to reference [2]. Theparameters of the other loop are each set to the value markedwith a white ¨o¨ for respective scenario (Y , σ). For thescaling of the KI -axes please refer to [1] and [2, Eq. (24)].5th International Particle Accelerator Conference IPAC2014, Dresden, Germany JACoW PublishingISBN: 978-3-95450-132-8 doi:10.18429/JACoW-IPAC2014-TUPRI07306 Instrumentation, Controls, Feedback & Operational AspectsT05 Beam Feedback SystemsTUPRI0731737ContentfromthisworkmaybeusedunderthetermsoftheCCBY3.0licence(©2014).Anydistributionofthisworkmustmaintainattributiontotheauthor(s),titleofthework,publisher,andDOI.0KI0KIfsample2·2 fsyn,0fsample2· fsyn,0fsample2· 23fsyn,00kKIfsample2·2 fsyn,0fsample2· fsyn,0fsample2· 23fsyn,0kfsample2·2 fsyn,0fsample2· fsyn,0fsample2· 23fsyn,0kfsample2·2 fsyn,0fsample2· fsyn,0fsample2· 23fsyn,0kFigure 4: Impact of (relative) beam loading, Y ∈ {0,0.75,1.5} (top to bottom), on the closed loop performance for dampingof dipole (left) and quadrupole (right) oscillations for short and long bunches, respectively.CONCLUSIONIt was shown that the simple estimation of the coherentsynchrotron frequency from Eq. (6) is sufficient for the tun-ing of the longitudinal feedback system when beam load-ing becomes relevant during future high current operationat SIS100. This result may be limited to heavy ion syn-chrotrons as the coherent synchrotron frequency decreaseswith beam loading only if the loaded quality factor of thecavity is small compared to the inverse of the synchrotrontune times the harmonic number.ACKNOWLEDGMENTThe authors would like to thank Prof. J. Adamy for hisconfidence and support.APPENDIXTo find the scope of application of Eq. (6) it is importantto use the full 4th order cavity transfer functionsGs (s) =2α(s3+(2α−ωtZ )s2+2ω (αtZ+ω)s+2αω2 (t2Z+1))s4+4αs3+4(αωtZ+α2+ω2)s2+8αω (αtZ+ω)s+4α2ω2 (t2Z+1)Gc (s) =2αs(tZ s2+(2αtZ+ω)s+2ωtZ (αtZ+ω))s4+4αs3+4(αωtZ+α2+ω2)s2+8αω(αtZ+ω)s+4α2ω2 (t2Z+1)with the angular RF frequency ω and the abbreviationtZ := tanΦZ =ω20−ω22αω.Usually 1st or 2nd order transfer function are appliedwhen dealing with beam loading due to high Q-factors [3–5].REFERENCES[1] H. Klingbeil et al., “A digital beam-phase control system forheavy-ion synchrotrons”, IEEE Trans. Nucl. Sci. 54, 2604(2007).[2] D. Lens, H. Klingbeil, “Stability of longitudinal bunch lengthfeedback for heavy-ion synchrotron”, Phys. Rev. ST Ac-cel. Beams 16, 032801 (2013).[3] F. Pedersen, “Beam loading effects in the CERN PS booster”,IEEE Trans. Nucl. Sci. 22, 1906 (1975).[4] D. Boussard, “Design of a ring RF system”, CERN AcceleratorSchool, CERN-SL-91-02-RFS-REV (1991).[5] K.W. Robinson, “Radiofrequency acceleration II”, CEA(MIT-Harvard)-11 (1956).[6] K. Groß, D. Lens, “Modeling longitudinal bunched beam dy-namics in hadron synchrotrons using scaled Fourier-Hermiteexpansions”, Proc. of IPAC’13, WEPEA010, Shanghai, China.[7] S.R. Koscielniak, “Coherent and incoherent bucket for a beamloaded RF system”, Part. Accel. 48, TRI-PP-94-8 (1994).5th International Particle Accelerator Conference IPAC2014, Dresden, Germany JACoW PublishingISBN: 978-3-95450-132-8 doi:10.18429/JACoW-IPAC2014-TUPRI073TUPRI0731738ContentfromthisworkmaybeusedunderthetermsoftheCCBY3.0licence(©2014).Anydistributionofthisworkmustmaintainattributiontotheauthor(s),titleofthework,publisher,andDOI.06 Instrumentation, Controls, Feedback & Operational AspectsT05 Beam Feedback Systems

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Impact of Simplified Stationary Cavity Beam Loading on the Longitudinal Feedback System for SIS100

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