Transient beam loading and rf power evaluation for future circular colliders

Ivan Karpov* and Philippe BaudrenghienCERN, Geneva 1211, Switzerland

(Received 11 February 2019; published 6 August 2019)

Interaction of the beam with the fundamental impedance of the accelerating cavities can limit theperformance of high-current accelerators. It can result in a significant variation of bunch-by-bunchparameters (bunch length, synchronous phase, etc.) if filling patterns contain gaps that are not negligiblecompared with the cavity filling time. In the present work, this limitation is analyzed using the steady-statetime-domain approach for the high-current option (Z) of the future circular electron-positron collider andfor the future circular hadron-hadron collider. Mitigation of transient beam loading by direct rf feedback isaddressed with evaluation of additional required generator power.

DOI: 10.1103/PhysRevAccelBeams.22.081002

I. INTRODUCTION

The future circular electron-positron collider (FCC-ee)[1] as a predecessor of the future circular hadron-hadroncollider (FCC-hh) [2] is considered to be built in fiveenergy stages, defined by the physics program. To keepthe same power loss budget for the synchrotron radiationin FCC-ee, the beam current will be gradually reducedfrom 1.4 A to 5.4 mA as the beam energy increases from45.6 GeV (the Z pole) to 182.5 GeV (the tt threshold),respectively. The high current colliders, the FCC-hh andthe FCC-ee Z, with parameters summarized in Table I,will have filling patterns with gaps that are not negligiblecompared with the cavity filling time. Thus, they willsuffer from transient beam loading issues which can resultin modulation of the cavity voltage and beam parameters.For example, change of the cavity voltage phase producedby a partly filled ring will cause a significant phaseerror for newly injected bunch trains. In the hadronmachine this could result in additional emittance blowupafter filamentation as well as possible particle losses.Modulation of synchronous phase can result in a collisionpoint shift and reduce luminosity due to the presence ofa finite crossing angle and a potential contribution of thehourglass effect. This limitation can be eliminated bymatching gap transients which requires the collider ringsto have identical filling patterns, rf systems, and totalbeam currents. If there is a large bunch length spread dueto modulation of the cavity voltage, for the shortest proton

bunches Landau damping can be lost as it is proportionalto the fifth power of the bunch length [3]. This makesthese bunches potentially unstable.In general, there are two methods to analyze the beam-

induced transients: in frequency domain and time domain.The former, developed by Pedersen [5], is usually called asmall-signal model. It allows one to calculate the modu-lation of cavity voltage produced by modulation of thebeam current and it was applied for the earlier set ofthe FCC-ee Z parameters in [6]. The latter method isthe tracking of the beam and the simulation of the rfsystem evolution in the time domain [7,8] which con-verges to the steady-state regime after many synchrotronperiods. Considering a machine with large circumference,high beam current, and large number of bunches, as for thecase of high-current future circular colliders, an analytical

TABLE I. The parameters of high-intensity high-luminositycolliders used for calculations in this work [1,2,4].

Parameter Units FCC-ee ZFCC-hh

(injection/physics)

Circumference, C km 97.75Harmonic number, h 130680rf frequency, frf MHz 400.79(R=Q) Ω 42.3Beam energy, E TeV 0.0456 3.3=50dc beam current, Ib;dc A 1.39 0.52Number of bunches

per beam, M16640 10400

Bunch population, Np 1011 1.7 1.0rms bunch length, σ ps 40 338=267Momentum compaction

factor, αp10−6 14.79 101.35

Total rf voltage, V tot MV 100 12=38Number of cavities, Ncav 52 24Energy loss per turn, U0 MeV 36 0=4.67

*ivan.karpov@cern.ch

Published by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.

PHYSICAL REVIEW ACCELERATORS AND BEAMS 22, 081002 (2019)

2469-9888=19=22(8)=081002(12) 081002-1 Published by the American Physical Society

model, which does not assume the small cavity voltagemodulation, may have an advantage for calculations of thebeam-induced transients.In this work, we present analytical results of beam loading

in superconducting rf cavities modeled by a lumped circuitwith a generator linked to the cavity via a circulator [9].We consider cavity voltage, beam and generator current ascomplex phasors. This approach was successfully used toderive a power saving algorithm in the large hadron collider(LHC) [10]. It allows us to get a steady-state solution forbeam and cavity parameters (beam phase, cavity voltageamplitude, and phase) for arbitrary beam currents and fillingschemes. In Sec. II we describe the steady-state time-domainapproach. For the case of constant generator current wederive the system of equations to compute modulations ofthe beam and cavity parameters in Sec. II A and then brieflydiscuss optimal parameters for which the generator power isminimized. The equations of the frequency-domain analysisare obtained from the linearized model in Sec. II A 2. Theapproach that keeps the cavity voltage amplitude constantfor an arbitrary synchronous phase is discussed in Sec. II B.Mitigation schemes of the beam and cavity parametermodulations using the direct rf feedback around the cavityare presented in Sec. III. Section IV contains the main resultsof the transient beam loading analysis for FCC-ee Z andFCC-hh. Finally, Sec. V summarizes the main results.

II. TRANSIENT BEAM LOADING

In operation of the Z machine different filling schemescan be used. To perform systematic analysis of the transientbeam loading, we introduce the filling schemes presentedin [11] and schematically shown in Fig. 1. We consider abeam containing M bunches, which can be grouped in ntrequal trains with a distance between the first bunches of theconsecutive trains ttt. This distance should be a multiple ofthe bunch spacing tbb. Each train contains a number offilled buckets Mb ≤ ttt=tbb. The beam has a regular filling,which can also contain an abort gap of length tgap. Thus,the dc beam current depends on the filling scheme Ib;dc ¼ntrMbNpe=trev, with e the elementary charge, trev therevolution period, andNp the number of particles per bunch.

For a given filling scheme, the abort gap length can becalculated as

tgap ¼trevngap

−��

ntrngap

− 1

�ttt þMbtbb

�þ tbb − trf ; ð1Þ

where ngap is the number of abort gaps, trf ¼ 1=frf , and frf isthe rf frequency. In FCC-hh,frftbb ¼ 10,Mb ¼ 80 is definedby machine protection requirements, and frfttt ¼ 980 isgiven by the injection kicker rise time. To prevent damageof accelerator components due to an asynchronous beamdump caused by erratic firing of the extraction kicker, thefilling scheme with four distributed abort gaps is consideredto be used in operation [12]. In total ntr ¼ 130 trains willcirculate in the machine.The gaps in the filling pattern generate a modulation

of the cavity voltage. Past work [9] based on the lumpedcircuit model (Fig. 2), has derived the differential equationfor transient beam loading coming from beam-to-funda-mental mode interaction:

IgðtÞ ¼VðtÞ

2ðR=QÞ�

1

QL− 2i

Δωωrf

�þ dVðtÞ

dt1

ωrfðR=QÞ

þ Ib;rfðtÞ2

; ð2Þ

where Ig is the generator current, V is the cavity voltage,Ib;rf is the rf component of the beam current, (R=Q) is theratio of the shunt impedance to the quality factor of thecavity fundamental mode expressed in circuit ohm, Δω ¼ω0 − ωrf is the cavity detuning, ω0 is the cavity resonantfrequency, and ωrf ¼ 2πfrf is the rf angular frequency. Theloaded quality factor 1=QL ¼ 1=Qext þ 1=Q0 is calculatedfrom the cavity quality factor Q0 and the coupler qualityfactor Qext ¼ Zc=ðR=QÞ defined by the line impedancetransformed at the gap by the main coupler with impedanceZc. Considering a superconducting cavity with Q0 ≫ Qext,the generator power can be written as [9]

PðtÞ ¼ 1

2ðR=QÞQextjIgðtÞj2 ≈

1

2ðR=QÞQLjIgðtÞj2: ð3Þ

Abort gap, tgap

Train spacing, ttt Bunch spacing, tbb

Number of trains, ntr

Number of bunches per train, Mb

FIG. 1. Sketch of the filling schemes used for the present study.

L R C

load

generator

circulator

V

FIG. 2. The lumped circuit model: a cavity is modeled by anLCR block, the coupler by a connected transmission line ofimpedance Zc, and the beam by a current source. Ib;rf is the rfcomponent of the beam current, Ig is the generator current, and Iris the reflected current, which is absorbed in the matched load(the generator is connected to the cavity via a circulator).

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In reality the rf power chain (amplifier, circulator, etc.) haslimited bandwidth and cannot track fast bunch-by-bunchvariation of the beam current. In the present work, the rfbeam current amplitude AbðtÞ is a stepwise function uðtÞwith a sampling time trf . The value of the stepwise functionin an mth rf bucket is uðmtrfÞ ¼ Ab;peak for m ¼ kþ n,where k ∈ ½0; tbb=trf − 1�, n is the index of the rf bucketfilled with bunches, and uðmtrfÞ ¼ 0 otherwise (see the left-hand plot in Fig. 3). Here, Ab;peak ¼ jFbjeNp=tbb is thepeak rf beam current amplitude averaged over a window oflength tbb with the complex form factor [13]

Fb ¼ 2F ½λðtÞ�ω¼ωrf

F ½λðtÞ�ω¼0

ð4Þ

obtained as the ratio of the Fourier transform of the bunchdensity distribution λ at the rf frequency to the one at dc.It depends on the particle distribution function, which isdescribed in the Appendix. This rf beam current representa-tion is equivalent to filtering of the beam spectrum with afunction centered at frf and having zeros at all other 1=tbbharmonics (see the right-hand plot in Fig. 3), which is alsodone in the LHC.Equation (2) links the cavity voltage to the beam and

generator current. If we set the generator current constant,we can calculate the beam induced modulation of the cavityvoltage. If, on the other hand, we impose a constant voltage(full compensation of the transient beam loading), theequation indicates that the generator current must followthe beam current modulation. This can lead to a significantincrease in required power, depending on parameters suchas cavity detuning and loaded quality factor. The above twostrategies are discussed in the following sections.

A. Case of constant generator current

The generator current remains constant and there is noattempt to compensate the transient beam loading. For

calculations of the beam induced transients, we assumemodulations of the cavity voltage and the beam current inthe form

VðtÞ ¼ AðtÞeiϕðtÞ; Ib;rfðtÞ ¼ AbðtÞe−iϕsþiϕbðtÞ; ð5Þ

with A the cavity voltage amplitude, and ϕ the modulationof the cavity voltage phase. The average beam stable phaseϕs (referred as the synchronous phase below) can be foundas the phase of the complex form factor Fb ¼ jFbje−iϕs

(see the Appendix). The modulation of the beam phase ϕbshould satisfy the following relation:

AðtÞ cos ½ϕs − ϕbðtÞ þ ϕðtÞ� ¼ Vcav cosϕs; ð6Þ

where Vcav ¼ V tot=Ncav is the average cavity voltage, V totis the total rf voltage, and Ncav is the number of rf cavities.Substituting Eq. (5) in Eq. (2) and separating real andimaginary parts, we get the following system of equations:

dAðtÞdt

¼ −AðtÞτ

þ ðR=QÞωrf ×

�Ig;c cos½ϕL − ϕðtÞ�

−AbðtÞ cos½ϕs − ϕbðtÞ þ ϕðtÞ�

2

�; ð7Þ

dϕðtÞdt

¼ Δωþ ðR=QÞωrf

AðtÞ ×

�Ig;c sin½ϕL − ϕðtÞ�

þ AbðtÞ sin½ϕs − ϕbðtÞ þ ϕðtÞ�2

�; ð8Þ

with the cavity filling time τ ¼ 2QL=ωrf . The constantgenerator current amplitude Ig;c, and the loading angle ϕL

that is the phase of the generator current with respect to theaverage cavity voltage can be calculated by averagingEq. (2) over one turn

FIG. 3. Left: an example of the amplitude of the rf component of the beam current within one turn for FCC-hh filling scheme with asingle abort gap. Right: corresponding amplitude of the beam spectrum (blue) and the amplitude of the filter function (red).

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Ig;ceiϕL ¼ 1

trev

Ztþtrev

tIgðuÞdu

¼ 1

trev

Ztþtrev

t

�VðuÞ

2ðR=QÞ�

1

QL− 2i

Δωωrf

�

þ dVðuÞdu

1

ωrfðR=QÞ þIb;rfðuÞ

2

�du

¼ Vcav

2ðR=QÞ�

1

QL− 2i

Δωωrf

�þ hIb;rfi

2; ð9Þ

where we use the fact that in the steady-state situation thecavity voltage is a periodic function Vðtþ trevÞ ¼ VðtÞ,and the average rf harmonic of the beam current is

hIb;rfi ¼1

trev

Ztþtrev

tIb;rfðuÞdu ¼ jFbjIb;dce−iϕs : ð10Þ

Unfortunately Eqs. (6), (7), (8) can not be solvedanalytically, but the results of numerical calculations willbe presented in Sec. IV.

1. Optimal parameters

The generator power is constant for the case of constantgenerator current and can be expressed using Eqs. (3), (9),(10) as

Pg;c ¼1

2ðR=QÞQLjIg;cj2

¼ 1

2ðR=QÞQL

��Vcav

2ðR=QÞQLþ jFbjIb;dc cosϕs

2

�2

þ�

Vcav

ðR=QÞΔωωrf

þ jFbjIb;dc sinϕs

2

�2�: ð11Þ

Setting the partial derivative of the average generator powerwith respect to Δω to zero we get the optimal cavitydetuning

Δωopt ¼ −ωrfjFbjIb;dcðR=QÞ sinϕs

2Vcav: ð12Þ

The optimal loaded quality factor that minimizes Pg;c forΔω ¼ Δωopt is

QL;opt ¼Vcav

ðR=QÞjFbjIb;dc cosϕs; ð13Þ

and the optimal generator power is

Pg;opt ¼VcavjFbjIb;dc cosϕs

2: ð14Þ

For the optimal parameters, however, the beam is unstablebecause the second Robinson limit [14] is reached, whichreads as

Y < −2 sinϕs

sinð2ϕzÞ: ð15Þ

When no loop around the cavity is present, with Y ¼jFbjIb;dcðR=QÞQL=Vcav the relative beam loading [15], andthe detuning angle ϕz is defined as tanϕz ¼ 2QLΔω=ωrf .In this case Y ¼ 1= cosϕs and ϕz ¼ −ϕs, so Eq. (15) is notsatisfied.

2. Linearized model (frequency domain calculations)

To linearize Eqs. (6)–(8) we define the normalizedmodulations of the beam current amplitude ab and thecavity voltage amplitude aV

AbðtÞ ¼ jFbjIb;dc½1þ abðtÞ�; AðtÞ ¼ Vcav½1þ aVðtÞ�ð16Þ

and assume jaV j ≪ 1, jϕj ≪ 1 and jϕbj ≪ 1, while jabjcould be 100% modulated. The linearized system ofequations is

daVðtÞdt

¼ −�1

τþ Δωopt

tanϕs

�aVðtÞ þ ðΔωopt − ΔωÞϕðtÞ

þ abðtÞΔωopt

tanϕs; ð17Þ

dϕðtÞdt

¼�Δω −

Δωopt

tan2ϕs

�aVðtÞ −

�1

τ−Δωopt

tanϕs

�ϕðtÞ

− abðtÞΔωopt; ð18Þ

ϕbðtÞ ¼ ϕðtÞ − aVðtÞtanϕs

: ð19Þ

Then applying a Laplace transformation we obtain thetransfer functions from the beam current amplitude to thecavity voltage amplitude, the cavity voltage phase, andthe beam phase:

aVab

¼ −Δωopt

DðsÞ�Δωopt

sin2ϕs− Δω −

�sþ 1

τ

�cotϕs

�; ð20Þ

ϕ

ab¼ −

Δωopt

DðsÞ��

Δωopt

sin2ϕs− Δω

�cotϕs þ

�sþ 1

τ

��; ð21Þ

ϕb

ab¼ −

Δωopt

DðsÞsin2ϕs

�sþ 1

τ

�; ð22Þ

where

DðsÞ ¼�sþ 1

τ

�2

− Δω�Δωopt

sin2ϕs− Δω

�; ð23Þ

and a is the Laplace image of the variable a. Equations (20)and (21) are identical to Eqs. (13) and (14) in Ref. [16] with

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the substitution ϕB ¼ π=2þ ϕs as we use the electronmachine convention. For a detuning slightly larger thanoptimal [Eq. (12)] Δωm ¼ Δωopt=sin2ϕs, which is calledthe “magic” detuning [17], and simple first order responsescan be obtained from Eqs. (20)–(22)

aVab

¼ −Δωoptτ cotϕs

1þ τs;

ϕ

ab¼ −

Δωoptτ

1þ τs;

ϕb

ab¼ Δωmτ

1þ τs: ð24Þ

This magic detuning is used in the following calculations,because for the optimal detuning and optimal loadedquality factor the beam is unstable as was discussed above.This results in an increase of power consumption by afactor 1þ 1=ð2 tanϕsÞ2, which is about 1.025 for theFCC-ee parameters.If the cavity filling time is significantly longer than the

beam gaps, the modulation of cavity voltage (amplitudeand phase) and beam phase will be almost linear in thebeam gaps. We propose to use the following equations toestimate the peak-to-peak beam phase modulation:

maxϕb −minϕb ¼ jΔωmtgapj; ð25Þ

and peak-to-peak phase modulation of the cavity voltage

maxϕ −minϕ ¼ jΔωopttgapj; ð26Þ

which can be obtained from transfer functions in Eq. (24)assuming a step excitation due to the abort gap. The latterequation was originally proposed in Ref. [18] for the caseof optimum cavity detuning.

B. Case of constant cavity voltage amplitude

The shift in collision point (z-vertex) due to beam phasemodulation can be mitigated by a proper match of fillingpatterns, rf systems, and total beam currents. However,amplitude modulation of the cavity voltage can still affectbunch-by-bunch parameters (bunch length, longitudinalemittance, synchrotron frequency, etc.) and beam stability.Reference [10] proposes to keep constant the amplitude ofthe cavity voltage while allowing its phase modulation.This approach is currently used in the LHCwhere the stablephase is very close to π=2 (electron machine convention).The generator current amplitude is constant, while its phaseis modulated. In the more general case (ϕs ≠ π=2) a similarscheme would require a significant amplitude modulationof the generator current as will be shown in this section.If the cavity voltage amplitude remains constant, the

beam phase modulation ϕb is at all time equal to the phasemodulation of the cavity voltage ϕ [see Eq. (6)]. Assumingthat the generator current is modulated in amplitude Ag andphase ϕg in the form IgðtÞ ¼ AgðtÞeiϕgðtÞ, cavity voltage is

VðtÞ ¼ VcaveiϕðtÞ, and the rf component of the beamcurrent Ib;rfðtÞ ¼ AbðtÞe−iϕsþiϕbðtÞ ¼ AbðtÞe−iϕsþiϕðtÞ, werewrite Eq. (2),

AgðtÞeiϕgðtÞ−iϕðtÞ

¼ Vcav

2ðR=QÞQLþAbðtÞcosϕs

2

− i

�ΔωVcav

ωrfðR=QÞ−Vcav

ωrfðR=QÞdϕðtÞdt

þAbðtÞsinϕs

2

�: ð27Þ

From Eq. (3) the instantaneous generator power in thiscase is

PðtÞ ¼ 1

2ðR=QÞQLjIgðtÞj2

¼ 1

2ðR=QÞQL

�Vcav

2ðR=QÞQLþ AbðtÞ cosϕs

2

�2

þ 1

2ðR=QÞQL

�ΔωVcav

ωrfðR=QÞ −Vcav

ωrfðR=QÞdϕðtÞdt

þ AbðtÞ sinϕs

2

�2

: ð28Þ

It can be minimized if the term in the second set of bracketsequals zero. Using Eq. (12) the required phase modulationof the cavity voltage is

ϕðtÞ ¼ −Δωopt

jFbjIb;dc

Zt

t0

AbðuÞduþ Δωðt − t0Þ þ ϕðt0Þ

ð29Þ

where ϕðt0Þ is an integration constant. In the steady-statesituation ϕðtþ trevÞ ¼ ϕðtÞ and the beam current ampli-tude averaged over a turn is jFbjIb;dc [see Eq. (10)]resulting in the detuning being optimal Δω ¼ Δωopt,similar to [10]. Thus the right-hand side of Eq. (27)becomes real and requires the generator current phase tobe the same as the phase modulation of the cavity voltage(exp ½iϕgðtÞ − iϕðtÞ� ¼ 1). Also the generator currentamplitude should follow the beam current modulation.Finally, the peak generator power can be minimized for

the loaded quality factor

QL;fd ¼Vcav

ðR=QÞAb;peak cosϕs; ð30Þ

and is given as

Pg;peak ¼VcavAb;peak cosϕs

2¼ Pg;opt

Ab;peak

jFbjIb;dc¼ Pg;opt

�htrfMtbb

�: ð31Þ

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For the maximum bunch spacing available in operation, theratio of the peak beam current amplitude to the average rfbeam current is around one because bunches are moreuniformly distributed in the ring. For the smallest bunchspacing (tbb ¼ trf ), h=M ≫ 1, and more generator power isrequired compared to the case without transient beamloading compensation.This scheme is very attractive for hadron colliders

(ϕs ≈ π=2) where the real valued term on the right-handside of Eq. (27) can be reduced significantly by choosinga large QL resulting in low power consumption. In leptoncolliders where ϕs differs from π=2, keeping voltageamplitude constant results in excessive required power.One must make a trade-off: accept “some” modulation ofthe cavity voltage within the available rf power budget.

III. COMPENSATION OF TRANSIENTBEAM LOADING

Partial compensation of the beam induced voltagemodulation can easily be done using direct rf feedbackaround the cavity and amplifier (Fig. 4). This has been usedin most high-intensity accelerators since the 1980s [19].In addition to regulation of the cavity field in the presenceof amplifier drifts and rf noise, it counteracts the longi-tudinal coupled-bunch instability driven by the fundamen-tal cavity impedance Zb [8,16,20]. The direct rf feedbackreduces the cavity fundamental mode impedance seen bythe beam such that the impedance of the closed loop is

ZclðωÞ ¼ZbðωÞ

1þGZbðωÞe−iτdωþiϕadj; ð32Þ

where G is the feedback gain, τd is the overall loop delay,and ϕadj is the phase adjustment required to correctly set thefeedback negative at the detuned cavity resonant frequency.The flat response is achieved and the feedback loopstability is guaranteed for 1=G ¼ 2ðR=QÞωrfτd [19]. Inthe presence of the direct rf feedback the generator currentin Eq. (2) can be written as

IgðtÞ ¼ Ig;ceiϕL þ G½Vrefðt − τdÞ − Vðt − τdÞ�; ð33Þwhere VrefðtÞ is a reference signal whose shape depends onthe compensation schemes discussed below. The constantoffset Ig;ceiϕL in Eq. (33) guarantees the average cavity

voltage amplitude to be equal Vcav for the finite value of thefeedback gain G.

A. Partial compensation

The cavity voltage amplitude A0ðtÞ and phase ϕ0ðtÞ arecalculated for constant generator current, magic detuningΔω ¼ Δωm, and optimal loaded quality factorQL ¼ QL;opt.The reference signal is then

VrefðtÞ ¼ Vcav þ ð1 − kÞ½A0ðtÞeiϕ0ðtÞ − Vcav�; ð34Þwhere k is the compensation factor. For k ¼ 0, there is nocompensation and no extra power is required for thegenerator during operation while the beam stability ismaintained thanks to the reduction of cavity impedanceEq. (32). For k ¼ 1 there is full compensation of the transientbeam loading, which will come at a large cost in peakgenerator power.

B. Constant cavity voltage amplitude (LHC fulldetuning scheme)

As was discussed in Sec. II B, the aim is to keep thecavity voltage amplitude constant and minimize the peakgenerator power Δω ¼ Δωopt and QL ¼ QL;fd. Then thereference signal can be obtained from Eq. (29)

Vref;fdðtÞ

¼Vcav exp

�−iΔωopt

Zt

t0

AbðuÞ− jFbjIb;dcjFbjIb;dc

duþ iϕðt0Þ�;

ð35Þ

where ϕðt0Þ is adjusted to have a zero phase modulationaveraged over a turn. As mentioned above this is a verygood scheme for hadron colliders with ϕs ≈ π=2.

C. Constant cavity voltage amplitude and phase(LHC half-detuning scheme)

In proton machines with ϕs ≈ π=2 the half-detuningscheme was proposed in [21] that keeps the cavity voltageconstant in amplitude and phase. Power consumption isminimum and the same in beam and no-beam segments forthis scheme if the following cavity detuning and loadedquality factor are used:

Δω1=2 ¼ −ωrfAb;peakðR=QÞ

4Vcav; QL;1=2 ¼

2Vcav

ðR=QÞAb;peak;

ð36Þrespectively. In this case the reference signal is constant

Vref;1=2ðtÞ ¼ Vcav: ð37Þ

The LHC was operated in this mode until 2014.FIG. 4. The direct rf feedback model.

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IV. RESULTS AND COMPARISONS

This section presents the results of transient beamloading calculations for high-current future circular col-liders. It is an extended version of the study presented in[22]. For the parameters of the FCC-ee Z with a variety ofdifferent filling schemes we present systematic compar-isons of time-domain and frequency-domain approaches.Then, evaluation of the generator power during the rampand transients at flat top for FCC-hh are discussed.

A. FCC-ee Z machine

We assume a constant generator current for evaluation ofthe beam induced transients (Sec. II A). To solve

Eqs. (6)–(8), the Euler method is used with total calculationtime longer than the cavity filling time (τ ≈ 37 μs), suffi-cient to get the steady-state solution. In the frequency-domain approach the Fourier image of the beam currentmodulation is multiplied by transfer functions [Eq. (24)with substitution s ¼ iω] and then the inverse Fouriertransform is used to obtain amplitude and phase modu-lations of the cavity voltage as well as the beam phasemodulation. An example of a reasonable agreement oftime-domain and frequency-domain calculations is shownin Fig 5. There is a strong modulation due to the abort gapand a finer structure due to the gaps between trains.To study modulation of the beam and the cavity

parameters systematically, the scan for different train

FIG. 5. Comparison of the time-domain [Eqs. (6)–(8)] and the frequency-domain approaches [Eqs. (24)] for the Z machine for magicdetuning Δωm ¼ Δωopt=sin2ϕs ¼ 13.1 kHz and the cavity filling time τ ¼ 37 μs. The figure shows the amplitude modulation of thecavity voltage (left) and the phase modulation of the cavity voltage (right) within one turn.

FIG. 6. Dependence of the peak-to-peak values for the beam phase modulation (left) and the phase modulation of the cavity voltage(right) on the train spacing for a bunch spacing of 15 ns. The results from the analytic formulas are given by Eq. (25) (left) and byEq. (26) (right).

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spacings is performed for a fixed bunch spacing tbb ¼15 ns and the maximum total number of bunches M ¼16640 (Fig. 6). Examples of filling scheme used forcalculations are listed in Table II. For each train spacingthe peak-to-peak value of the beam phase modulation(the left-hand plot) and the phase modulation of the cavityvoltage (the right-hand plot) are calculated. In general,the peak-to-peak values of phase modulations increasewith larger train spacing because of longer abort gap.This comes from the fact that for larger ttt, less bunchtrains can circulate in the machine and it is more difficultto distribute them uniformly. For example, the abort gapwill be the largest if the ring is filled with a single train.The results obtained in frequency domain are slightlylarger than in the time domain. The difference is due tononlinearities, which are accurately treated in time-domain calculations, while in the frequency-domaincalculations they are neglected. The analytical formulas[Eqs. (25), (26)] agree very well with the results of thetime-domain calculations (see the dashed black linesand the blue solid lines in Fig. 6, correspondingly). Fortttfrf > 100, the peak-to-peak beam phase modulation islarger than 70 ps, which corresponds to about 60 ps peak-to-peak phase modulation of the cavity voltage [thedifference is the factor 1= sin2 ϕs, see Eqs. (25) and(26)]. However, in operation, the shift of collision pointcan be minimized by matching abort gap transients.The dependence of the peak-to-peak phase modulationof the cavity voltage on the train spacing for differentbunch spacings is shown in Fig. 7. In general, a largerbunch spacing leads to smaller transients because of moreuniform filling that results in a shorter abort gap.The comparison of amplitude modulations of the cavity

voltage with the direct rf feedback (τd ¼ 700 ns, similar tothe loop delay in the LHC [20]) for k ¼ 0.3 and k ¼ 0 [seeEq. (34)] is shown in Fig. 8. For partial compensation(k ¼ 0.3) one can see about 15% reduction of the peak-to-peak value with about 10% increase of the instantaneousgenerator power (Fig. 9). Further optimizations withdifferent filling schemes and parameters of the direct rffeedback can be performed to minimize transient beamloading in the FCC-ee Z. Note that the 10% increase inrequired power appears during the abort gap and the firstfew bunch trains (duration related to the rf feedbackdelay). It can thus be clamped (and reduced) with a tinyeffect on the beam.

B. Results for FCC-hh

The loss of Landau damping (LLD) in the longitudinalplane [3] is one of the limiting factors which defines theparameters of the FCC-hh rf system [2,23,24]. The pro-posed ramp lasts for 20 min and contains a linear part[region (ii) in Fig. 10] with 10% of parabolic parts at bothends [regions (i) and (iii) in Fig. 10]. The full longitudinalemittance and the filling factor qp—the ratio of themaximum energy offset in the bunch to the bucket half-height—define the rf voltage for a given momentumprogram. The former increases with square root depend-ence on the beam energy (from 3.1 to 13 eVs) in order to bebelow the LLD threshold during the ramp. The latter is

FIG. 7. Dependence of the peak-to-peak phase modulation ofthe cavity voltage from Eq. (26) on the train spacings for differentbunch spacings.

TABLE II. Examples of filling schemes used in Fig. 6.

Train spacing(rf buckets) Mb ntr M tgap (μs)

120 15 1082 16230 2.185240 30 541 16230 2.260360 46 360 16560 2.920480 61 270 16470 2.995

FIG. 8. Comparison of amplitude modulation of the cavityvoltage with the direct rf feedback for different compensationfactor k.

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adjusted with linear and constant dependence on time tohave the proper cavity voltage during the cycle which doesnot exceed 2 MV per cavity (see Fig. 10) [4].The corresponding voltage program and synchronous

phase as a function of time are shown in Fig. 11. Thesynchronous phase deviates from π=2 significantly duringthe ramp which means that the optimum detuning schemecould be an advantage in this case in terms of powerconsumption (Fig. 12). The synchronous phase differs fromthe phase of the synchronous particle as discussed in theAppendix. At the injection energy the half-detuningscheme will be used with QL ¼ 2.45 × 104 resulting inabout 60 kW required peak generator power. To maintaincavity voltage amplitude and phase we assume the sametotal loop delay of the direct rf feedback τd ¼ 700 ns as inthe LHC. After machine filling is done, the LHC fulldetuning scheme can be switched on and used for the initialstage of the ramp where the synchronous phase is close

to π=2. In this region the quality factor increases linearly upto the value defined by Eq. (13). Then we propose to use theconstant generator current scheme until the end of the ramp(no beam loading compensation) as it requires less gen-erator power compared with the LHC full detuning scheme(see Fig. 12).Comparing the results with the constant generator

current at flat top for a single abort gap (8.65 μs) andfour abort gaps (4 × 2.87 μs), the beam phase modulationis smaller for more uniformly distributed bunches, asexpected (Fig. 13, left). It is less than 40 ps which is verysmall compared to the 1 ns long FCC-hh bunches. Thepeak-to-peak cavity voltage amplitude modulations forboth cases are smaller than 4% (Fig. 13, right) and producebunch length modulations that are negligible comparedwith the natural spread between bunches (about 5% in

FIG. 10. The energy program and evolution of the filling factorduring the ramp in the FCC-hh: (i) the parabolic, (ii) linear, and(iii) parabolic parts of the energy program.

FIG. 11. Evolution of the cavity voltage, synchronous phase ofthe beam (red solid line), and the phase of the synchronousparticle (red dotted line) during the ramp for the energy programand the filling factor shown in Fig. 10.

FIG. 12. Evolution of the generator power and the loadedquality factor during the ramp for the energy program and thefilling factor shown in Fig. 10.

FIG. 9. Instantaneous generator power with the direct rf feed-back for calculations shown in Fig. 8.

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the LHC). This means that the FCC-hh can be operated atthe flat top with magic detuningΔωm ¼ −4.7 kHz, optimalloaded quality factor QL;opt ¼ 3 × 105, and about 102 kWinput power keeping the generator drive constant.We also realized that for the cases of larger phase

modulations of the cavity voltage (above 20°) the systemof Eqs. (6)–(8) becomes unstable. This indicates that theRobinson stability criterion could be different for the caseof large transients. However, the beam remains stable evenfor the optimal parameters in the presence of the direct rffeedback (see for example [25]).

V. CONCLUSIONS

In the present work the beam loading in high-currentfuture circular colliders has been analyzed. We have shownthat the equations of frequency-domain analysis can beobtained by linearization of the more accurate time-domainequations that are valid for any values of the beam currentmodulations.For the assumed regular filling schemes, the main con-

tribution to the phase and amplitude modulations of thecavity voltage comes from the abort gap. For the FCC-ee Zthe resulting peak-to-peak value of the bunch-by-bunchphase modulation is larger than 70 ps for an abort gaplonger than 2 μs. The larger train spacings lead to strongermodulation of cavity and beam parameters, which can bereduced by using filling schemes with larger bunch spacings.The direct rf feedback can mitigate the transient beamloading at the cost of additional generator power. For theoverall loop delay τd ¼ 700 ns, about 10% increase of thepeak generator power is sufficient to reduce the peak-to-peakamplitude modulation of the cavity voltage by 15%.In the FCC-hh the magic detuning without transient beam

loading compensation should be used during the ramp.Transient beam loading compensation schemes demandvery large peak power because the synchronous phasediffers significantly from π=2. Without compensation, themaximum generator power during acceleration is about

420 kW and at flat top it is slightly above 100 kW. Withfour distributed abort gaps the peak-to-peak values of thebeam phase modulation and the cavity voltage amplitudemodulation are less than 40 ps and 2%, respectively. Thismeans that the FCC-hh can be operated without transientbeam loading compensation at the collision energy.This paper has derived power requirements based on

the beam-cavity-generator interaction at the fundamentalmode, described in Eq. (2). The rf component of the beamcurrent is considered as an external “perturbation”. Furtherstudies should focus on development of a model includingboth longitudinal dynamics (the action of cavity voltageon the beam) and hardware limitations (such as nonlinearityof the generator, limited bandwidth of the power parts,misalignments of the low level rf and tune settings, rf noise,etc.), similar to the studies done for PEPII [8] and the LHC[26]. A possible impact of mismatched transients onluminosity depending on the collision scheme of bothFCC-ee and FCC-hh needs to be also addressed.

ACKNOWLEDGMENTS

We thank Elena Shaposhnikova, Dmitry Teytelman,Rama Calaga, Andrew Butterworth, Alessandro Gallo,and Olivier Brunner for useful discussions and comments.

APPENDIX: LINE DENSITY CALCULATION

For the case of a single rf system with a sinusoidal rfwave, the Hamiltonian is

Hðϵ;φÞ ¼ hη2β2Es

ϵ2 þ eV tot

2πWðφÞ: ðA1Þ

Here ϵ ¼ Ep − E energy offset of the particle with respectto the energy of the synchronous particle E, h is theharmonic number, η ¼ αp − 1=γ2 is the slip factor, and V tot

is the total rf voltage. The normalized rf potential W isdefined as

FIG. 13. Modulations of the beam phase (left) and of the cavity voltage amplitude (right) at flat top in FCC-hh for different abort gapconfigurations and constant generator current.

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WðφÞ ¼ sinϕs0 − sinφ − ðϕs0 − φÞ cosϕs0; ðA2Þwhere the phase of the synchronous particle (electronmachine convention) is

ϕs0 ¼ arccos

�U0 þ ΔEeV tot

�ðA3Þ

so that at each turn the rf cavities give a momentum kick,which compensates the energy loss due to synchrotronradiation U0 and produces acceleration with the energygradient ΔE. In the present work we assume the binomialparticle distribution as a function of Hamiltonian

ψ ½Hðϵ;φÞ� ¼ ψ0½Hmax −Hðϵ;φÞ�μ; ðA4Þwhere ψ0 is the normalization constant, and Hmax ¼eV totWmax=ð2πÞ is the Hamiltonian of the particle withthe maximum energy offset, andWmax is the correspondingvalue of the rf potential. In the LHC μ ≈ 2 and it is assumedto be μ ¼ 1.5 for proton bunches in the FCC-hh becausecontinuous emittance blow-up is applied during the rampand at flat top resulting in flatter bunches. For μ ≫ 1 thedistribution becomes Gaussian, which is the case forelectron bunches in the FCC-ee. After integration over ϵone gets the line density [27]

λðφÞ ¼ λ0½Wmax −WðφÞ�μþ12 ðA5Þ

with λ0 the normalization constant. Finally, the complexform factor can be obtained using Eq. (4). For short FCC-eebunches jFbj ≈ 2 exp ½−ðωrfσÞ2=2� and ϕs ≈ ϕs0, while forlong FCC-hh bunches the bunch shape asymmetry issignificant, so that the synchronous phase of the bunchdiffers from one of the synchronous particle (see Fig. 11).

[1] A. Abada et al., FCC-ee: The Lepton Collider, Eur. Phys. J.Spec. Top. 228, 261 (2019).

[2] A. Abada et al., FCC-hh: The Hadron Collider, Eur. Phys.J. Spec. Top. 228, 755 (2019).

[3] F. J. Sacherer, A longitudinal stability criterion for bunchedbeams, IEEE Trans. Nucl. Sci. 20, 825 (1973).

[4] I. Karpov and E. Shaposhnikova, in The fifth AnnualMeeting of the Future Circular Collider Study, Brussels(2019), https://indico.cern.ch/event/727555/contributions/3439881/attachments/1868115/3073133/Beam_dyn_and_RF_IK_ES.pdf.

[5] F. Pedersen, Beam loading effects in the CERN PS booster,IEEE Trans. Nucl. Sci. 22, 1906 (1975).

[6] D. Teytelman, The Third Annual Meeting of the FutureCircular Collider Study, Berlin (2017), https://indico.cern.ch/event/556692/contributions/2590414/attachments/1468181/2271216/3WE16C.pdf.

[7] J. M. Byrd, S. De Santis, J. Jacob, and V. Serriere, Transientbeam loading effects in harmonic rf systems for light sources,Phys. Rev. ST Accel. Beams 5, 092001 (2002).

[8] C. Rivetta, T. Mastorides, J. D. Fox, D. Teytelman, andD. Van Winkle, Modeling and simulation of longitudinal

dynamics for Low Energy Ring–High Energy Ring at thePositron-Electron Project, Phys. Rev. STAccel. Beams 10,022801 (2007).

[9] J. Tückmantel, CERN Report No. CERN-ATS-Note-2011-002 TECH, 2011, https://cds.cern.ch/record/1323893/files/CERN-ATS-Note-2011-002%20TECH.pdf.

[10] T. Mastoridis, P. Baudrenghien, and J. Molendijk, Cavityvoltage phase modulation to reduce the high-luminosityLarge Hadron Collider rf power requirements, Phys. Rev.Accel. Beams 20, 101003 (2017).

[11] I. Karpov, R. Calaga, and E. Shaposhnikova, High ordermode power loss evaluation in future circular electron-positron collider cavities, Phys. Rev. Accel. Beams 21,071001 (2018).

[12] E. Renner, M. J. Barnes, W. Bartmann, F. Burkart, E.Carlier, L. Ducimetiere, B. Goddard, T. Kramer, A.Lechner, N. Magnin, V. Senaj, J. Uythoven, P. V. Trappen,and C. Wiesner, in Proceedings of the 9th InternationalParticle Accelerator Conference, Vancouver, British Co-lumbia, Canada (2018), pp. 10–13, http://ipac2018.vrws.de/papers/tupaf058.pdf.

[13] P. F. Tavares, A. Andersson, A. Hansson, and J. Breunlin,Equilibrium bunch density distribution with passive har-monic cavities in a storage ring, Phys. Rev. ST Accel.Beams 17, 064401 (2014).

[14] K. W. Robinson, Report No. CEAL-1010, 1964, https://www.osti.gov/biblio/4075988.

[15] Y is the ratio of the beam induced voltage with cavity ontune (and no generator current) to the desired Vcav [14].

[16] F. Pedersen, CERN Report No. CERN-PS-92-59-RF,1992, https://cds.cern.ch/record/244817.

[17] D. Boussard, in Handbook of Accelerator Physics andEngineering, 2nd ed. (World Scientific, Singapore, 2013),p. 131, https://www.worldscientific.com/doi/pdf/10.1142/8543.

[18] D. Boussard, in Proceedings of 1991 IEEE Particle Accel-erator Conference (APS Beam Physics) (1991), p. 2447,https://cds.cern.ch/record/221476/files/cer-000134053.pdf.

[19] D. Boussard, Control of cavities with high beam loading,IEEE Trans. Nucl. Sci. 32, 1852 (1985).

[20] P. Baudrenghien and T. Mastoridis, Fundamental cavityimpedance and longitudinal coupled-bunch instabilities atthe High Luminosity Large Hadron Collider, Phys. Rev.Accel. Beams 20, 011004 (2017).

[21] D. Boussard, CERN Report No. CERN-SL-91-16-RFS,1991, https://cds.cern.ch/record/221153.

[22] I. Karpov and P. Baudrenghien, in Proceedings of 61stICFA Advanced Beam Dynamics Workshop (HB 2018),Daejeon, Korea (JACoW Publishing, Geneva, 2018),pp. 266–271, https://doi.org/10.18429/JACoW-HB2018-WEP2PO003.

[23] E. Shaposhnikova, in The Third Annual Meeting of theFuture Circular Collider Study, Berlin (2017), https://indico.cern.ch/event/556692/contributions/2484257/attachments/1467380/2269248/FCChh_RF_Berlin.pdf.

[24] E. Shaposhnikova and I. Karpov, The Fourth AnnualMeeting of the Future Circular Collider Study, Amsterdam(2018), https://indico.cern.ch/event/656491/contributions/2930759/attachments/1631764/2602214/FCC_Week_2018_Shaposhnikova_Karpov.pdf.

TRANSIENT BEAM LOADING AND rf POWER … PHYS. REV. ACCEL. BEAMS 22, 081002 (2019)

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[25] A. Mosnier, F. Orsini, and B. Phung, in Proceedings ofthe 6th European Particle Accelerator Conference(EPAC’98), Stockholm, Sweden (1998), http://accelconf.web.cern.ch/Accelconf/e98/PAPERS/WEP28G.PDF.

[26] T. Mastorides, C. Rivetta, J. D. Fox, D. V. Winkle, and P.Baudrenghien, RF system models for the CERN Large

Hadron Collider with application to longitudinal dynamics,Phys. Rev. ST Accel. Beams 13, 102801 (2010).

[27] T. Linnecar and E. N. Shaposhnikova, CERN ReportNo. SL-Note-92-44-RFS, 1992, https://cds.cern.ch/record/703261.

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