In a heavy-ion synchrotron the bunched beam can perform longitudinal oscillations around the synchronous particle (single bunch dipole oscillation, SBDO). If disturbances/instabilities exciting the SBDO exceed the rate of Landau damping, the beam can become unstable. Furthermore, Landau damping is accompanied by an increase of the beam emittance which may be undesired. Thus, control efforts are taken to stabilize the beam and to keep the emittance small. It is known that for a single harmonic cavity and a small bunch the SBDO oscillates with the synchrotron frequency if the oscillation amplitudes are small. For a larger bunch or a double harmonic RF systems that introduces nonlinearities, this is no longer valid. This work shows how the frequency of the SBDO can be determined in general. As a result, the SBDO can again be modeled as a harmonic oscillator with an additional damping term to account for Landau damping. This model can be used for feedback designs which is shown by means of a simple example. As the frequency of the SBDO and the damping rate depend on the size of the bunch in phase space, it is shown how this information can be obtained from the measured beam current

Adamy, Jürgen

Grieser, Jochen

Klingbeil, Harald

Lens, Dieter Etienne Mia

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A Method to Obtain the Frequency of the Longitudinal Dipole Oscillation for Modeling and Control in Synchrotrons with Single or Double Harmonic RF Systems

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A METHOD TO OBTAIN THE FREQUENCY OF THE LONGITUDINALDIPOLE OSCILLATION FOR MODELING AND CONTROL INSYNCHROTRONS WITH SINGLE OR DOUBLE HARMONIC RFSYSTEMSJ. Grieser∗ , D. Lens ∗, H. Klingbeil † , J. Adamy ∗, TU Darmstadt, GermanyAbstractIn a heavy-ion synchrotron the bunched particle beamscan perform longitudinal oscillations of several modes.These oscillations are damped by Landau damping, but canbecome unstable if driven accordingly. Furthermore, Lan-dau damping is accompanied by filamentation which in-creases the longitudinal beam emittance and thus reducesthe beam quality. To stabilize the beam and to keep theemittance low, control measures are taken. However, anycontroller design requires knowledge about the oscillationfrequency of the mode which is to be damped. For a singleharmonic RF voltage and a very small bunch, the longitu-dinal equations of motion of the particle can be linearized,and all the particles oscillate with approximately the samesynchrotron frequency which equals the frequency of thedipole oscillation. For a larger bunch, or for a doubleharmonic RF system introducing nonlinearities around thereference point, this is no longer valid. In this work, wepresent a method to obtain the frequency of the dipole os-cillation for larger bunches with a single or double har-monic RF system and show how this can be used for a statespace controller design.SINGLE AND DOUBLE HARMONIC RFVOLTAGEIn this paper a single and a double harmonic RF systemVsh(τ) = V̂1 sin(ωRF τ) , (1a)Vdh(τ) = V̂1 sin(ωRF τ) + V̂2 sin(2ωRF τ +Δϕdh)(1b)are considered, where V̂2 and Δϕdh are chosen in such away that a saddle point in τR [7] occurs, where τ denotesthe arrival time with respect to the zero crossing of the sin-gle harmonic sinusoidal component of the RF voltage. Forthe sake of simplicity only the stationary case below tran-sition energy with ϕR = ωRF τR = 0 is considered. Fig. 1shows the different RF voltages.Furthermore only low beam intensities are consideredand therefore space charge effects introducing a syn-chrotron frequency tune shift [6] are neglected.∗ {jgrieser/dlens/adamy}@rtr.tu-darmstadt.de† klingbeil@temf.tu-darmstadt.de / H.Klingbeil@gsi.de00ωRF tV(t)ϕR = 0−V̂1V̂1−π −π/2 π/2 πFigure 1: single and double harmonic gap voltage withidentical first cavity voltage amplitude V̂1STATE OF THE ARTUsing the phase space variablesϕ = ωRF τ and ΔEωR withΔE = E − ER, the longitudinal equations of motion fora single particle can be linearized for V (ϕ) = Vsh(ϕ) andshort bunches [3]. In this case the barycenter of the bunchoscillates with the frequencyfsyn,0 = fR√hqV̂1|ηR cosϕR|2πβ2RER(2)with the harmonic number h, the phase slip factor ηR =1γ2tr− 1γ2R, the angular revolution frequency ωR, the speedβ normalized to the speed of light, the total energy ER andthe charge q of the particles.In case of a realistic bunch length this linearization is nolonger valid. As a result, the synchrotron frequency spreadof the particles has to be taken into account and the rigiddipole mode does not oscillate with the linear synchrotronfrequency, but with a smaller frequency (“coherent syn-chrotron frequency”). For small displacements ϕB  π ofthe bunch barycenter a way to estimate the coherent syn-chrotron frequency is given e.g. in [1]:Ωsyn,coh = ωsyn,0√1|um|∫ ϕm2ϕm1(V (ϕ)V̂1)2dϕwith um =∫ ϕm2ϕm1(Y (ϕ) − Y (ϕm2)) dϕ; Y (ϕ) =1V̂1∫ ϕ0V (ϕ) dϕ; ϕm1,2: left/right end of the bunch. How-ever, it is not specified how the end of the bunch is to bedetermined.Another common way to obtain the synchrotron fre-quency for a larger bunch is the measurement of the beamtransfer function [5]. However, these measurements arerather intricate and the obtained values are only valid forthe very bunch size and ion type used in the experiment..WEPPP061 Proceedings of IPAC2012, New Orleans, Louisiana, USAISBN 978-3-95450-115-12846Copyrightc ○2012byIEEE–ccCreativeCommonsAttribution3.0(CCBY3.0)—ccCreativeCommonsAttribution3.0(CCBY3.0)06 Instrumentation, Controls, Feedback and Operational AspectsT05 Beam Feedback SystemsSINGLE HARMONIC OPERATIONCoherent Synchrotron Frequency for LargeBunchesIn the following, the standard deviation of the bunch indirection of ϕ will be denoted by σϕ. For different par-ticle densities, the bunch length will be defined as 4σϕ.As the bucket length equals 2π, feasible values will beσϕ ∈ [0; π/2].The nonlinear sinusoidal voltage leads to lower syn-chrotron frequencies of particles with a larger phase devia-tion ϕwhich decreases the coherent synchrotron frequency.According to [3] the oscillation period of a particle with amaximum phase coordinate ϕ∗ can be written asfsyn(ϕ∗) = fsyn,0π2K (sin(ϕ∗/2)),where K(k) denotes the complete elliptic integral of thefirst kind. The oscillation frequency of the barycenter ofthe bunch can be calculated for different particle densitieswith a moment method that was developed in [4]. For aGaussian density and the assumption that the bunch shapeis approximately ellipsoidal, the frequency for small dipoleamplitudes is derived asfsyn,coh,g(σϕ) = fsyn,0√√√√1 + 7∑n=1(−1)nn! 2nσ2nϕand for an ellipsoidal bunch with a uniform density asfsyn,coh,u(σϕ) = fsyn,0√√√√1 + 7∑n=1(−1)nn! (n+ 1)!σ2nϕ .Fig. 2 compares the two coherent frequencies with the syn-chrotron frequency for the trajectory ϕ∗ = 2σϕ. It is ap-parent that as a rule of thumb, the synchrotron frequencyat 2σϕ is a good estimate for the coherent frequency of thebarycenter, i. e.fsyn(ϕ∗ = 2σϕ) ≈ fsyn,coh,u(σϕ) ≈ fsyn,coh,g(σϕ) .0 0.1 0.2 0.3 0.4 0.50.40.60.81σϕ/πFrequencies/fsyn,0fsyn(ϕ∗ = 2σϕ)fsyn,coh,g(σϕ)fsyn,coh,u(σϕ)Figure 2: coherent frequency of the barycenter for a single-harmonic setup compared to the synchrotron frequency.Fig. 3 shows the relative error of this estimate. For abunch to bucket length ratio of up to 2σϕ/π = 0.7, theerror of this rule is less than 5% for both Gaussian and uni-form distributions. For a uniform distribution 2σϕ equalsthe maximum phase deviation of the border of the bunch.0 0.1 0.2 0.3 0.4 0.5051015GaussianUniformσϕ/πRelativeerror/%Figure 3: relative error if the coherent frequency is es-timated with the synchrotron frequency for the trajectoryϕ∗ = 2σϕDOUBLE HARMONIC OPERATIONFor obvious reasons the equations of motion can not belinearized if a double harmonic voltage is applied. As afirst approach to obtain the coherent synchrotron frequencyalso the 2σϕ-trajectory can be considered. The synchrotronfrequency of a particle moving along a certain trajectorycan be derived asfsyn(ϕ∗) = fsyn,0π sin ϕ∗2√2K(√12(1 + sin2 ϕ∗2)) (3)as shown e.g. in [3]. Macro particle simulations show thatthe synchrotron frequency of the 2σϕ-trajectory is a goodfirst estimate of the coherent synchrotron frequency but canstill be improved. This can be seen in Fig. 5 which showsthe percental error Δfsyn,coh =fsyn,coh,sim−fsyn(2σϕ)fsyn,coh,simofthis approach along with an improved estimate which ispresented below. In the simulation a matched bunch (σϕ =0.28π) was shifted by ϕB(t = 0) = 0.07π.To obtain a better estimate for the coherent synchrotronfrequency at first macro particle simulations are performedfor several bunch sizes σϕ and barycenter deviationsϕB(t = 0). The simulated coherent synchrotron frequen-cies are compared to the synchrotron frequencies along thetrajectories in phase space, and for each simulation the tra-jectory is determined whose synchrotron frequency equalsthe coherent synchrotron frequency. This yields a value ofϕ∗opt for each bunch size σϕ and phase deviationϕB which,inserted in Eq. (3), results in the coherent synchrotron fre-quency. The simulated values of ϕ∗opt for40Argon18+ ionswith a kinetic energy of Ekin = 11.4Mevu are shown inFig. 4.By introducing a smoothing function, ϕ∗opt can be ex-pressed in terms of σϕ:ϕ∗opt ≈ −0.4287σ2ϕπ+ 2.1898σϕ . (4)This improves the percental error of the estimate signifi-cantly, as can be seen in Fig. 5. Similar results are obtainedfor different ion types and beam energies..Proceedings of IPAC2012, New Orleans, Louisiana, USA WEPPP06106 Instrumentation, Controls, Feedback and Operational AspectsT05 Beam Feedback SystemsISBN 978-3-95450-115-12847 Copyrightc ○2012byIEEE–ccCreativeCommonsAttribution3.0(CCBY3.0)—ccCreativeCommonsAttribution3.0(CCBY3.0)σϕ/πϕB/πϕ∗ opt/π *1*20.50.150.250.350.50.150.250.20.40.6Figure 4: simulated values of ϕ∗opt for different bunch sizesσϕ and barycenter deviationsϕB . *1: For unrealistic smallbunches with a large barycenter deviation, ϕ∗ depends onϕB . *2: For realistic bunch sizes ϕ∗ depends only on σϕΔf syn,coh[%]σϕ/πΔfsyn,eff,2σϕΔfsyn,eff,ϕ∗0.05 0.1 0.15 0.2 0.25 0.3-5051015Figure 5: percental deviation of synchrotron frequency ofthe ϕ∗ = 2σϕ-trajectory and ϕ∗opt according to smoothingfunction Eq. (4).CONTROLLER DESIGNTo damp the dipole oscillation a state space controller isdesigned based on the model of a harmonic oscillator withthe coherent synchrotron frequency. The two variables arethe bunch barycenter phase ϕB and the energy deviationΔEB . The actuating variable is the phase shift ϕgap of theaccelerating voltage e.g. for a double harmonic systemVdh(ϕ) = V̂1 sin(ϕ− ϕgap)+ V̂2 sin(2ϕ− 2ϕgap +Δϕdh) . (5)In the SIS 18 at GSI the beam current is measured fromwhich ϕB − ϕgap is obtained [2]. The cavities are syn-chronized to direct digital synthesis (DDS) modules, whichmeans that in order to calculate ϕB a second DDS has tobe run to obtain the phase shift ϕgap of the RF voltage.As the energy deviation ΔEB can not be measured, an ob-server is needed. The controller performance is improved ifϕB −ϕgap is fed back which smoothens the progression ofthe bunch barycenter. An additional integrator is added asa countermeasure to a remaining offset. The closed controlloop is shown in Fig. 6.dynamics ofbunch barycenteradditionalintegratorobserverϕBcontroleffort ϕgapusspstate spacecontroller(ϕB − ϕgapΔEB)− −Figure 6: closed control loop: bunch barycenter(ϕB , ΔEB)T , control effort of state space controllerussp = −kT · (ϕB − ϕgap, ΔEB)TThe controller is optimized for robustness against para-meter uncertainties and keeps the beam emittance to a mini-mum, as shown in Fig. 7. The simulation parameters wereσϕ = 0.28π, ϕR = 0, double harmonic RF system withV̂1 = 16000V, 40Argon18+ ions with a kinetic energy ofEkin = 11.4MeVu , ϕB(t = 0) = 0.1π.ϕB−0.2π00.2πemittance100%103%105%107%200 800 1400 2000number of turns in synchrotronFigure 7: phase deviation of bunch barycenter and beamemittance growth in the open loop and in the closed loopcase.REFERENCES[1] O. Boine-Frankenheim, T. Shukla, “Space Charge Effects inBunches for Different RF Wave Forms”, Phys. Rev. SpecialTopics - Accelerators and Beams 8 (2005)[2] H. Klingbeil et al., “A Digital Beam-Phase Control Systemfor Heavy-Ion Synchrotrons”, IEEE Trans. on Nucl. Sc. 54(6)(2007)[3] S. Y. Lee, Accelerator Physics, (Singapore: World ScientificPublishing Co. Pte. Ltd., 2004)[4] D. Lens, Modeling and Control of Longitudinal Single-BunchOscillations in Heavy-Ion Synchrotrons, Dissertation, TUDarmstadt (2012)[5] K. Y. Ng, Physics of Intensity Dependent Beam Instabilities,(Singapore: World Scientific Publishing Co. Pte. Ltd., 2006)[6] K. Y. Ng, “Comments on Landau Damping due toSynchrotron Frequency Spread”, FERMILAB-FN-0762-AD(2005)[7] Throughout this paper the index R indicates the reference par-ticle...WEPPP061 Proceedings of IPAC2012, New Orleans, Louisiana, USAISBN 978-3-95450-115-12848Copyrightc ○2012byIEEE–ccCreativeCommonsAttribution3.0(CCBY3.0)—ccCreativeCommonsAttribution3.0(CCBY3.0)06 Instrumentation, Controls, Feedback and Operational AspectsT05 Beam Feedback Systems

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A Method to Obtain the Frequency of the Longitudinal Dipole Oscillation for Modeling and Control in Synchrotrons with Single or Double Harmonic RF Systems

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