Implications of beam filling patterns on the design of recirculating Energy Recovery Linacs
IImplications of Beam Filling Patterns on the Design of Recirculating EnergyRecovery Linacs
S. Setiniyaz ∗ and R. Apsimon † Engineering Department, Lancaster University, Lancaster, LA1 4YW, UK andCockcroft Institute, Daresbury Laboratory, Warrington, WA4 4AD, UK
P. H. Williams
Cockcroft Institute, Daresbury Laboratory, Warrington, WA4 4AD, UK (Dated: July 20, 2020)Recirculating energy recovery linacs are a promising technology for delivering high power particlebeams ( ∼ GW) while only requiring low power ( ∼ kW) RF sources. This is achieved by deceleratingthe used bunches and using the energy they deposit in the accelerating structures to accelerate newbunches. We present studies of the impact of the bunch packet filling pattern on the performance ofthe accelerating RF system. We perform RF beam loading simulations under various noise levels andbeam loading phases with different injection schemes. We also present a mathematical descriptionof the RF system during the beam loading, which can identify optimal beam filling patterns underdifferent conditions. The results of these studies have major implications for design constraints forfuture energy recovery linacs, by providing a quantitative metric for different machine designs andtopologies. I. INTRODUCTIONA. Introduction into ERLs
There is an increasing interest in Energy RecoveryLinacs worldwide due to their unique promise of com-bining the high-brightness electron beams available fromconventional linacs with the high average powers avail-able from storage rings. Applications requiring thisstep-change in capability are coming to the fore in awide variety of fields, for example high energy particlephysics colliders [1], high luminosity colliders for nuclearphysics [2], free-electron laser drivers for academic and in-dustrial purposes [3, 4], and inverse Compton scatteringsources [5, 6]. The first high average power applicationdemonstrated on an ERL was the multi-kW lasing of theJLab IR-FEL [7].Historically, an effective method to cost-optimise anelectron linac (where beam dynamics restrictions allow)is to implement recirculation [8, 9], i.e. acceleratingthe beam more than once within the same RF struc-tures. Analogously, one may implement recirculation inan ERL, accelerating and decelerating within the samestructures. This has been successfully demonstrated inthe normal-conducting Novosibirsk infrared FEL [10].There are a number of GeV scale user facilities pro-posed that are therefore based upon recirculating super-conducting ERLs [1, 11, 12], and two test facilities arecurrently attempting such a multi-turn ERL demonstra-tion [13, 14].It is thus timely to explore the implications of this rel-atively new accelerator class. Unlike a linac or storage ∗ [email protected] † [email protected] ring, there is large number of degrees of freedom in thebasic accelerator topology. For example one may choosea dogbone or racetrack layout, subsequent acceleratingpass may be transported in common or separate beamtransport, and decelerating passes may be transportedpairwise with their equivalent accelerating beam in com-mon or separate transport [15–17].In this article we explore the consequence of thesechoices on the most important aspect of an ERL-baseduser facility, the RF stability. Specifically, we considerall possible beam filling patterns in an N-pass recirculat-ing ERL and their interaction with the accelerator low-level RF control system. We show that there are optimalchoices, and note which topologies allow these optima tobe chosen.It is vital that this analysis is performed during the de-sign stage of an ERL-based facility as it fixes the pass-to-pass path length required in the recirculation transportat the scale of multiples of the fundamental RF wave-length, typically many metres, therefore any path lengthvariability built in to allow pass-to-pass RF phase vari-ation cannot correct for this macro scale requirement.Similarly, transverse phase advance manipulations thatare capable of mitigating BBU thresholds [18] would notbe effective against sub-optimal filling pattern generatedinstabilities.We first introduce beam filling and beam loading pat-terns, and describe how they affect cavity voltage. Wethen describe an analytical model of beam loading anduse this to make predictions about the system. The nextsection describes beam loading simulations while vary-ing different parameters such as the signal-to-noise ratio( S/N ) and synchronous phase. We will expand thesestudies to sequence preserving scheme in the section IVand compare all the simulations results in the section V. a r X i v : . [ phy s i c s . acc - ph ] J u l FIG. 1: Simple recirculating linac diagram.
B. Filling patterns
In this article, we note that the topology of the recir-culating ERL can impact the filling pattern or orderingof the bunches. We start with a simple recirculating ERLwith single arc on two sides as shown in Fig. 1 and dis-cuss more complex setup later on. We consider a 6-turnERL with 3 acceleration and 3 deceleration turns. In or-der to minimise cavity voltage fluctuations, we allow forspacing between injected bunches which become filled bybunches on subsequent passes. Here we elucidate the ex-act choices in which that process occurs. As an exampleFig. 2 shows 3 decelerating bunches followed by 3 accel-erating ones. The accelerating bunches take energy fromthe cavity, thus decreasing the cavity voltage and viceversa, therefore mixing them can minimize cavity volt-age fluctuation. The 6 bunches form what we term abunch packet. Bunch packets are repeated and fill upthe ERL as shown by the diagram in Fig. 3. As we mixbunches executing different turns into bunch packets weemphasise that “injection” only refers to the process oftransporting a bunch from the injection line to the ERLmain ring; similarly, “extraction” refers to the process ofextracting a bunch from the ring and transporting themto the beam dump. Therefore a set of injected bunchesdo not pass through the linac as one, they are alwaysmixed with bunches executing turns in the ERL ring.The “bunch number” is the order in which bunches areinjected into a bunch packet over N turns, for examplebunch 1 (or 1st bunch) is injected on turn 1, bunch 2 (or2nd bunch) is injected as bunch 1 executes turn 2 and soon. During the operation, one bunch per packet per turnis extracted and replaced by a new bunch. Usually, notall the RF cycles are filled by bunches, but one bunch islocated at the start of a block of M otherwise unoccupiedRF cycles. These M RF cycles we call the “intra-packetblock”. In a N -turn ERL, 1 bunch packet thus occupies M × N RF cycles. In the packet illustrated in Fig 2 eachintra-packet block is coloured uniquely.We can give notation of filling pattern by describingwhich bunch goes to which intra-packet block. The num-ber indicates the bunch number and its position in thevector indicates the intra-packet block number. The fill-ing pattern of Fig. 2 is a 6-element vector [1 2 3 4 5 6].Filling pattern [1 4 3 6 5 2], for example, describes fillingdepicted in Fig. 4.Here we attempt a step-by-step explanation of packet FIG. 2: Filling of recirculating linac with filling pattern[1 2 3 4 5 6]. Blue/red bunches areaccelerated/decelerated. Phase flips at 3rd turn.FIG. 3: Filling of ERL by multiple bunch packets.construction. We assume a flexible injection timing, suchthat we can insert a small delay of less than the regularpulse spacing (but still a multiple of fundamental RF)with a regular superperiod. Such capability would benovel, though not unfeasible, within the photoinjectorlaser system. Please refer to Figs. 2, 3 and 4: We startby injecting all the bunches labeled 1. In Fig. 3, we seethat we can fit 8 packets into the ERL (the number ofpackets in the ring are arbitrarily chosen), so this ac-counts for the first 8 bunches from the injector. Thiscompletes turn 1 in Fig. 2 (or Fig. 4) (which both showsonly one of the 8 packets). The ninth bunch from theinjector becomes the first bunch 2 on the second line ofFig. 2 or Fig. 4. In the case of Fig. 2 the bunch 2 isinjected in to intra-packet block number 2. In the caseof Fig. 4 the bunch 2 is injected in to intra-packet blocknumber 6. This difference between Fig. 2 and Fig. 4is accomplished using the aforementioned flexible timingfeature of the photoinjector laser by extending the timeinterval separating the 8th and 9th bunches.. The next7 injected bunches fill up the other bunch 2 spaces in theother packets. Following through, the 17th bunch fromthe injector thus becomes bunch 3 in the packet, in bothFigs. 2 and 4 this is placed in block number 3. In thisway we build up either [1 2 3 4 5 6] for Fig. 2, or [1 4 3 6 52] for Fig. 4. We call patterns constructed in this method”First-In-First-Out” (FIFO) patterns as the order of theFIG. 4: Filling of recirculating linac with filling pattern[1 4 3 6 5 2].bunches in the packet remains constant.Another way to construct filling patterns is recombi-nation using different path lengths with a fixed injec-tion time interval. In this case, the turn number of thebunches in the packet does not change. Therefore, wename it Sequence Preserving (SP) scheme. We will dis-cuss it in more details in later sections. A point we wishto emphasize for SP scheme is that because choosing be-tween these two filling patterns implies differences in thepath lengths of many RF cycles for each individual turn,this choice is a design parameter during machine con-struction.We will also use “pattern number” for brevity to indi-cate 120 filling patterns of 6-turn ERL. The pattern num-ber i is used to indicate 120 permutations of [2 3 4 5 6]and related to the filling pattern F i as F = [1 2 3 4 5 6] ,F = [1 2 3 4 6 5] ,...F = [1 6 5 4 3 2] . (1)As there are many bunch packets in a ring, without losingthe generality we can name intra-packet block of the 1stbunch as the 1st block, i.e. the 1st bunch will always bein the 1st intra-packet block. C. Cavity voltage calculation
As the bunches pass through the linacs, they are eitheraccelerated or decelerated by the RF field in the cavity.In doing so, energy is either put into or taken out of thecavity. The cavity voltage V cav is related to the storedenergy U stored as U stored = V cav ω (cid:16) RQ (cid:17) , (2)with RQ being shunt impedance of the cavity divided byits Q-factor. For an accelerating cavity, the change instored energy from a particle bunch passing through is δU stored = 2 V cav δV cav ω (cid:16) RQ (cid:17) = − q bunch V cav . (3)Therefore, the change in cavity voltage from beam load-ing is given as δV cav = − q bunch ω (cid:18) RQ (cid:19) cos ( φ ) , (4)where φ is the phase difference between the bunch andthe RF and q bunch is the bunch charge. In general,the bunches will not necessarily pass through the cavityon-crest (maximum field) or on-trough (minimum field).When dealing with RF fields, it is convenient to considerthe field as a complex number, where only the real partcan interact with the beam at any moment in time. In-deed this implies that beam loading can only change thereal component of the cavity voltage for any given phase.In order for a recirculating ERL to operate stably overtime, we require that the vector sum of the cavity voltageexperienced by each bunch in a bunch packet must equalzero, as shown Fig. 5. If this is not the case, then therewill be a net change in stored energy in the cavity eachbunch packet, reducing the overall efficiency of the ERL.For now, we will neglect the phase of the bunches andonly consider voltages as real numbers for brevity in thefollowing mathematical description. Later we will con-sider off-crest beam loading cases by replacing binarynotation with complex notation, i.e. by replace “1” and“0” by e iφ and e − iφ . We define a recirculating ERL tobe at ‘steady state’ when all intra-packet blocks in themachine are occupied. In this case, on any given turn,half the bunches in the packet pass through the cavity ataccelerating phases and half at decelerating phases. Ascavity voltage experienced by all bunches in the packetsum to zero, there is no net energy gain or loss over bunchpacket.If we neglect the phase of the bunches and only con-sider bunches passing through the cavity on-crest and on-trough, then the change in cavity voltage due to beamloading from a bunch is simply ± q bunch ω (cid:16) RQ (cid:17) cos ( φ ),from Eq. 4. Therefore, in this case, every time a bunchpasses through a linac, the cavity voltage is incrementedor decremented by a fixed amount.FIG. 5: A diagram to show the complex voltages of fourbunches in a 4-turn ERL.TABLE I: Filling patterns and associated beam loadingpatterns. filling pattern 1 2 3 4 5 6 1 4 3 6 5 2 1 4 5 2 3 6turn 1 0 0 0turn 2 0 0 0 1 1 0 1 0 0 1 1 0 1 0turn 3 0 0 0 0 1 0 1 1 0 0 1 1 0 0 1turn 4 1 0 0 0 1 0 0 1 1 0 1 0 1 0 0 1turn 5 1 1 0 0 0 1 0 0 1 0 1 1 0 0 1 0 1turn 6 1 1 1 0 0 0 1 0 1 0 0 1 1 0 0 1 1 0turn 7 0 1 1 1 0 0 0 1 1 0 0 1 0 1 0 1 1 0turn 8 0 0 1 1 1 0 0 1 1 0 1 0 0 1 1 0 1 0turn 9 0 0 0 1 1 1 0 1 0 1 1 0 0 1 1 0 0 1turn 10 1 0 0 0 1 1 1 0 0 1 1 0 1 0 1 0 0 1turn 11 1 1 0 0 0 1 1 0 0 1 0 1 1 0 0 1 0 1turn 12 1 1 1 0 0 0 1 0 0 1 1 0 1 0 1 0 0 1 D. Beam loading pattern
Let us consider a 6-turn ERL. Table I shows how thebeam loading pattern changes turn-by-turn for the fillingpatterns [1 2 3 4 5 6], [1 4 3 6 5 2], and [1 4 5 2 3 6]. If weuse “0” and “1” to denote accelerated and deceleratedbunches, respectively, we get beam loading patterns asshown in Table I. The accelerating bunches reduce thevoltage in the cavity and vise versa. Now that we havedefined the bunch filling pattern and showed how thisis associated with a unique sequence of beam loadingpatterns, we should understand how this beam loadingpattern affects the cavity voltage. Fig. 6 shows how thebeam loading pattern can be translated into a change incavity voltage.For an ERL at steady state, the definition of “block1” is arbitrary and can be one of N choices in a N -turnERL; therefore, there are ( N − N -turn ERL. A 6-turn ERL can have 120unique filling patterns. Each of these filling patterns isassociated with a unique sequence of beam loading pat-terns. Beam loading patterns changes turn by turn andare periodic over N turns, as shown in Table I.Fig. 7 shows beam loading patterns of two filling pat-terns over 6-turns. The red beam loading pattern haslarger cavity voltage fluctuation than blue one. Thisshows some filling patterns cause larger disturbances tothe cavity voltage and RF system of the ERL than oth-ers. For a 6-turn ERL, we can evaluate the RF jittersassociated with a specific beam filling pattern and usethis to identify which patterns are optimal. In Table I,the beam loading increments have been normalised to ± ± q bunch ω (cid:16) RQ (cid:17) cos ( φ ) for brevity and clarity.For the remainder of the article, we will continue to use anormalised beam loading to help the reader understandthe methodology.Once a list of all unique filling patterns is defined, wecan determine the associated sequence of beam loadingpatterns, using the method described in Table I. To de-termine the normalised change in cavity voltage, we sim-ply calculate the cumulative sum of the beam loadingsequence. We define a specific filling pattern as F i , theassociated beam loading pattern as B ( F i ) and the nor-FIG. 8: A block diagram of the modelled LLRF systemand the feedback loop.malised change in cavity voltage as δV given as δV = cusum ( B ( F i )) = k (cid:88) j =1 B j ( F i ) . (5)We can use δV to estimate the RF stability performanceof all patterns. E. Low level RF system
For the Low level RF (LLRF) system, we model thesystem as shown in Figure 8. The cavity voltage (given asI and Q components) is added to a Gaussian distributednoise (also I and Q), whose standard deviation is definedby the
S/N ; we treat this as the only source of noise inthe system, rather than including realistic noise at eachcomponent of the LLRF controller. This is then passedthrough a 16-bit analogue-to-digital converter (ADC),before a PI-control algorithm is implemented to regu-late amplitude and phase. The PI correction algorithmalso applies limits to the range of values to model thepower limits on the amplifier. The amplifier and digital-to-analogue converter (DAC) is modeled as a resonantcircuit with a bandwidth defined by the closed-loop band-width.We model LLRF system as a proportional-integral (PI)controller [19–21]. In the PI controller, the LLRF systemfirst calculates the error u voltage, which is differencebetween actual cavity voltage V measured with set-pointvoltage V set u = V measured − V set . (6)Then, two types of corrections are made, namely the pro-portional V pro and integral term corrections V int . Theproportional term correction is calculated based on thepreviously measured dV and proportional gain G p , givenas V pro = G p u. (7)The integral term correction is calculated integratingover on all the previously measured dV and integral termgain G i , given as V int = G i (cid:90) t udt = G i (cid:88) n u n δt, (8)where t is the time measurement took place. The pro-portional and integral term corrections address fast andslow changes, respectively. The set-point voltage can beconstant (static set-point) or can change over time (dy-namic set-point). A dynamic set-point can be useful inorder to improve RF stability in a recirculating ERL be-cause it prevents the LLRF system from competing withthe beam loading voltage in the cavity. If the LLRF feed-back system can adjust its set-point voltage accordingto the anticipated beam loading, then it has a “dynamicset-point” voltage. In this case, the feedback system onlyamplifies noise. If the set-point is static, LLRF systemwill treat beam loading as noise and amplify it as well. II. ANALYTICAL MODELA. Variations in cavity voltage
If we consider the effects of beam loading and noise,the cavity voltage, V cav , can be expressed as: V cav = V + V b + V n , (9)where V is the steady state cavity voltage, which we willassume to be time-independent, V b is the voltage contri-bution due to beam loading, and V n is the voltage con-tribution due to all noise sources in the system. We shallassume that noise originates from the electronics in thelow-level RF system (LLRF), which in turn introducesnoise to the cavity voltage. How the noise propagatesthrough the RF system depends on the behaviour of theLLRF system as well as the beam loading patterns, butthe noise voltage in the cavity can be defined as σ V n = α RF | V | S/N , (10)where
S/N is the voltage signal to noise ratio and α RF is a constant of proportionality, which depends on theparameters of the system. From Eq. 9, we can obtain anexpression for the cavity voltage squared: V cav = V + V b + V n + 2 V V b + 2 V V n + 2 V b V n . (11)We shall assume that V b and V n are independent variablesand that V is constant, therefore, from Eq. 9 and 11, weobtain expressions for the mean and standard deviationof the cavity voltage. (cid:104) V cav (cid:105) = V + (cid:104) V b (cid:105) + (cid:104) V n (cid:105) σ V cav = (cid:112) (cid:104) V cav (cid:105) − (cid:104) V cav (cid:105) (12)FIG. 9: The RMS fluctuation of the normalized beamloading pattern of 6-turn ERL.If V b and V n have zero mean, then Eq. 12 produces theexpected result that (cid:104) V cav (cid:105) = V . Because noise andbeamloading is independent, (cid:104) V b V n (cid:105) = (cid:104) V b (cid:105)(cid:104) V n (cid:105) . (13)Therefore, σ V cav = (cid:113) σ V b + σ V n . (14)From Eqs. 10 and 14, we can express the noise on thecavity voltage as σ V cav = (cid:115) σ V b + α RF V ( S/N ) . (15)The σ V b is pattern specific, and depends on topology ofthe ERL as well as the expected beam jitters. The volt-age fluctuation due to the beam loading and given by σ V b = σ V pattern δV (16)where σ V pattern is RMS fluctuation of the normalizedbeam loading pattern over all turns of the machine. The σ V pattern for all 120 patterns is shown in Fig. 9 for a 6-turnERL, where we have assumed a FIFO schemes, where theorder of the bunch packet does not change turn by turn.One can see that σ V pattern varies by approximately a fac-tor of 2 depending on the choice of filling pattern. B. Variations in amplifier power
From [22], the cavity voltage can be determined froman envelope equation ˙ V cav ω + (cid:20) ω + ω Q L ω + j ω − ω ωω (cid:21) V cav = j ˙ V amp + ω V amp ωQ e . (17)Where ω is the resonant frequency of the cavity, ω is theamplifier drive frequency, Q L and Q e are the loaded andexternal Q-factors respectively and P amp is the forwardpower from the amplifier expressed as a voltage as V amp = (cid:115) (cid:18) RQ (cid:19) Q e P amp . (18)If we assume that the cavity is driven at the resonantfrequency and that the cavity is at steady state, thenfrom Eq. 17, we obtain V cav = 2 Q L Q e V amp , (19)thus P amp = Q e (cid:16) RQ (cid:17) Q L | V cav | . (20)From Eqs. 11 and 20, we obtain (cid:104) P amp (cid:105) = Q e (cid:16) RQ (cid:17) Q L (cid:2) V + (cid:104) V β (cid:105) + (cid:104) V n (cid:105) +2 V (cid:104) V β (cid:105) + 2 V (cid:104) V n (cid:105) + 2 (cid:104) V β (cid:105)(cid:104) V n (cid:105) ] . (21)Note that for the beam loading terms, we now use V β rather than V b . This is because the LLRF feedback algo-rithm determines the power required to maintain a stablecavity voltage. If we implement a static set point algo-rithm, then V β = V b , if a dynamic set point algorithm isused then V β = δV b , which is an error residual when sub-tracting the expected beam loading voltage from the realvalue. This error residual depends on pattern number,LLRF algorithm, gains and other factors.We should note that for the amplifier power, the noisehas a simpler relationship to the signal to noise ratio thanthe noise observed on the cavity voltage (Eq. 10) becausethe noise on the amplifier is the measured noise amplifiedby the proportional gain of the LLRF, so σ V n = G p S/N V . (22)If we assume that V β and V n are independent and zeromean, then Eq. 21 can be simplified as: (cid:104) P amp (cid:105) = Q e V (cid:16) RQ (cid:17) Q L (cid:34)(cid:32) G p ( S/N ) (cid:33) + σ V β V (cid:35) . (23)By a similar method, we can also determine the standarddeviation on the amplifier power as σ P amp ≈ Q e V (cid:16) RQ (cid:17) Q L (cid:115) G p ( S/N ) + 2 G p ( S/N ) + ∆∆ = (cid:104) V β (cid:105) − (cid:104) V β (cid:105) V + 4 (cid:104) V β (cid:105) V . (24)For low signal to noise ratios, the first terms dominates,whereas for high signal to noise ratios, we encounter anoise floor due to either beam loading (static set-point)or a residual error (dynamic set-point); this noise floorwill be pattern dependent. For the first term, note thatit is independent of beam loading pattern and therefore,for lower signal to noise ratios, we expect σ P amp to beindependent of beam loading pattern. III. BEAM LOADING SIMULATION
The cavity voltage fluctuation can be simulated by sim-ulating beam loading and its interaction with RF sys-tem [22]. In this work we have extended beam loadingtype to accelerating and decelerating. In acceleratingmode, voltage changes due to the beam loading is sub-tracted from cavity voltage and vise versa.
A. Static and dynamic set-points
Before running simulations, it is important to deter-mine the set-point voltage of LLRF system. As we men-tioned earlier, there are two types of set-point voltages:dynamic and static set-points. During the beam loading,the cavity voltage fluctuates but the net beam loading ofa packet is zero and voltage will return to nominal volt-age. So, there is no need for LLRF correction for beamloading. The dynamic set-point is designed to excludebeam loading correction. In static set-point, however,the LLRF system treats beam loading as noise, tries tocorrect to the oscillatory beam loading, and thus becomesunstable. Therefore, the dynamic set-point is better thanstatic set-point as it creates less cavity voltage fluctua-tion and requires much less amplifier power. This is alsoconfirmed by simulations shown in Fig. 10.
B. Simulation parameters
The simulation parameters are shown in Table II. Wesimulated 6-turn ERL, so there are 6 bunches in thepacket. The bunch charge was set high to increase theeffect of the beam loading and to allow us to explore thebehaviour of the RF system under extreme conditions.The circumference is set to 360 m, so number of RF cy-cles in the ring would be 1200 for a 1 GHz RF frequency. (a)(b)
FIG. 10: Comparison of static and dynamic set-pointsfor filling pattern [1 2 3 4 5 6] when
S/N = 7 . × .(a) cavity voltage and (b) amplifier power as function oftime.We set 1 intra-packet block is 10 RF cycles, so 20 packetsfill up the ring. New bunches replaced old bunches, untiltotal of 96 turns are tracked, which is about 121 µ s timeduration. We scanned through all the 120 filling patternsof 6-turn ERL. C. Simulation results
1. Comparison of optimal and non-optimal patterns
Firstly, we have looked at the effect of beam loadingpattern on the cavity voltage and amplifier power. Asshow in Fig. 11, the simulation results are shown for anoptimal filling pattern [1 4 3 6 5 2] indicated by blueline and a non-optimal pattern [1 2 3 4 5 6] indicated byred line. The optimal pattern is better, because it cre-ates much smaller cavity voltage fluctuations as shown insub-figures (a) and (c) and requires less amplifier poweras shown in sub-figures (b) and (d). The sub-figures (a)and (b) are simulation results when
S/N = 7 . × andTABLE II: Simulation parameters. Machine parameters value bunch charge q bunch µ snumber of turns tracked 96tracking time duration 121 µ s Cavity parameters cavity voltage ( V ) 18.7 MVR/Q 400RF frequency 1 GHz LLRF parameters latency 1 µ sdigital sampling rate 40 MHzclosed-loop bandwidth 2.5 MHzproportional controller gain G p G i (c) and (d) are results when S/N = 7 . × . Increas-ing the S/N reduced cavity voltage fluctuation slightlyand amplifier power significantly. Simulation results con-firmed that certain patterns are better from the perspec-tive of cavity voltage jitters, RF stability, and power re-quirements.
2. Noise scan
We observed the cavity voltage jitters and amplifierpower is reduced when
S/N is increased. To investigatenoise dependence, we have performed simulations withfilling patterns [1 4 3 6 5 2] and [1 2 3 4 5 6] by varying
S/N . The results are shown in Fig. 12 for (a) σ V cav , (b) σ P amp , and (c) average P amp .In Fig. 12 (a), we see that the σ V cav is more sensitiveto the filling pattern than S/N . In other words, σ V cav is dominated by filling pattern. σ V cav reaches patternspecific limit σ V b around 10 , so S/N needs to largerthan 10 to minimize cavity voltage jitters.In Fig. 12 (b) and (c), we see σ P amp and average P amp are sensitive to noise than filling pattern. To minimizepower consumption P amp around to 11.15 kW, the S/N has to be larger than 10 . Two patterns has similar am-plifier power fluctuations σ P amp up to S/N = 10 . Be-yond this point, σ P amp reach filling pattern specific floors.The analytical model underestimates P amp as shown inFig. 12 (b) at high noise. As the noise increase, the am-plifier starts to have saturation. In this case, the propor-tional term can’t provide sufficient power. As the powershortage build up, the integral term will start to makecorrection and add power the cavity. The simulation canmodel the proper PI controller and have integral term.But the analytical doesn’t have the integral term and thus can’t include the power from integral term. Thiswill cause analytical model to fail at very high noise lev-els and accounts for the difference between the analyticmodel and simulation.The typical S/N range for a real LLRF system isaround 10 − . In the figures, we cover a very widerange of S/N , including values which far exceed the real-istic range of values. The reason for this is to allow us toexplore the behaviour of the RF and LLRF system in thelimit of ultra-low noise, which allows us to study featuresthat are not visible at realisable values of
S/N , such asthe pattern-dependent noise floor in Fig. 12 (c).
3. Cavity voltage
The cavity voltages jitters σ V cav of all 120 filling pat-terns are shown in Fig. 13. We see that σ V cav is dif-ferent when different set-points are used. The dynamicset-point is better because it gives smaller cavity voltagejitters. The filling patterns No. 60 (pattern [1 4 3 6 5 2])and 61 (pattern [1 4 5 2 3 6]) are optimum for both set-points. There are other patterns [1 4 2 5 3 6], [1 4 2 5 6 3],[1 4 3 6 2 5], [1 4 5 2 6 3], [1 4 6 3 2 5], and [1 4 6 3 5 2]are optimal only for dynamic set-point. This indicatesthat depending on the set-point type, the Figure Of Merit(FOM) to estimate σ V cav is different. For static set-point,the FOM can be given as σ V cav = σ V turns = (cid:118)(cid:117)(cid:117)(cid:116) N t i = N t (cid:88) i =1 ( ¯ V i ) , (25)with ¯ V i being the average voltage of i th turn, and N t being number of turns. In this case, we averaging voltageover one turn and get ¯ V i first, then calculating the RMSof these N t turns. As shown in Fig. 13 (a), the FOMroughly overlaps with simulation. Although, the FOMdoesn’t predict jitters exactly, but it can find optimalpattern quickly without simulations. For dynamic set-point, the FOM is Eq. 15. The theoretical predictionmatches simulation results exactly for S/N = 1 × asshown in Fig. 13 (b).We see the dynamic set-point give smaller jitters. Thepatterns [1 4 3 6 5 2] and [1 4 5 2 3 6] (pattern number60 and 61) are optimal in both set-points. Optimal pat-tern has 2 −
4. Amplifier power results
The required average amplifier powers P amp for dif-ferent patterns and different S/N are given in Fig. 14.We see that the average P amp is reduced from 28 kWto 11.13 kW, when the S/N increased from 7 . × to7 . × t . When S/N reduced further, the P amp is re-duced to minimum of 11.147 kW, which is the resistive (a) S/N = 7 . × (b) S/N = 7 . × (c) S/N = 7 . × (d) S/N = 7 . × FIG. 11: Comparison of patterns [1 4 3 6 5 2] and [1 2 3 4 5 6] with dynamic set-point at different
S/N . (a) and (c)cavity voltage. (b) and (d) amplifier power. (a) and (b) are the results when
S/N = 7 . × . (c) and (d) are theresults when S/N = 7 . × .power loss. This shows that ERLs can be operated withvery low power, when S/N is sufficiently high.
D. Property of optimal patterns
In Fig. 15, we compared cavity voltage of optimal andnon-optimal patterns, indicated by blue and red lines re-spectively. In sub-figure (a), voltage of optimal pattern[1 4 3 6 5 2] fluctuates less than ± .
024 MV range of18.7 MV, while non-optimal pattern [1 2 3 4 5 6] has3 times larger fluctuation. We see similar 3-up-3-downand up-down fluctuations as in Fig. 7, but here we have20 bunch packets, so these fluctuations are repeated 20times in each turn. Revolution times is about 1.2 µ s, soevery 1.2 µ s turn changes.The optimum filling patterns [1 4 3 6 5 2] and[1 4 5 2 3 6] (pattern number 60 and 61) and their as-sociated beam loading patterns are given in Table I. Weobserve their two consecutive bits are in either up-down(10) or down-up (01) pairs. Such combinations limit cu-mulative sum of beam loading pattern to a range of [ −
1, 1], and thus minimizes jitters. We also see 1 pair flips(“1” and “0” switch positions) per turn. The change from“0” to “1” (acceleration to deceleration) happens in 3rdto 4th turn transition and the change from “1” to “0”is the new bunch replacing the extracted bunch. There-fore, in optimal patterns, consecutive pairs are made upby bunches that are 3 turns apart like [1 4], [2 5], and[3 6].Patterns [1 4 2 5 3 6], [1 4 2 5 6 3], [1 4 3 6 2 5],[1 4 5 2 6 3], [1 4 6 3 2 5], and [1 4 6 3 5 2] also haveabove motioned properties of optimal patterns. However,they are only optimal for dynamic set-point and not forstatic set-point. Therefore, these 6 patterns are DynamicSet-Point Optimal (DSPO) patterns, while [1 4 3 6 5 2]and [1 4 5 2 3 6] are All Set-Point Optimal (ASPO) pat-terns. Of course, a ASPO pattern is a DSPO patternby definition. The difference between the ASPO pattern[1 4 3 6 5 2] and DSPO pattern [1 4 3 6 2 5] is shownin Fig. 15. Both patterns have same fluctuation range,but the turn average of the DSPO is larger in the 1st,4th, and 7th turns. So, σ V turn of pattern DSPO is larger,which makes it non-optimal for static set-points accord-0 (a)(b)(c) FIG. 12: RMS cavity voltage (a), average amplifierpower (b), and RMS amplifier power (c) as function of
S/N for patterns [1 4 3 6 5 2] and [1 2 3 4 5 6].ing to Eq. 25.
E. Off-crest beam loading
So far, we have studied the effects of beam loading foron-crest phases. In applications such as FELs, bunches (a)(b)
FIG. 13: Simulated σ V cav of 120 patterns with (a) staticand (b) dynamic set-points compared to prediction.The S/N was set to 1 × to turn off the noise.must be compressed during acceleration to achieve highpeak current, then stretched and energy compressed ondeceleration to eliminate adiabatic energy spread growth.Beams must therefore pass through the RF system offcrest [7, 23]. In recirculating ERLs, we want to minimizethe net beam loading of a packet, so the in-phase (I) andquadrature phase (Q) components of the beam loadingof a packet should sum to approximately zero, i.e. thevector sum of the voltage changes sums to zero for thebunch packet. By doing so, the amplitude and phase ofthe cavity voltage changes minimally after a packet. Thisimplies that the phase and amplitude perturbations frombeam loading cancel out over a bunch packet, as shown inFig. 16. Here, by “mirror turns” we meant turns that hassame energy but the bunch phase is offset by π radians.In 6-turn ERLs, turn 1 and 6, 2 and 5, and 3 and 4 aremirror turns. Mirror bunches have same energy and off-set angles as shown in Fig. 16, so their vector sum is zero.In Fig. 16, φ is the off phase angle of 1st and 6th turns; φ is the off phase angle of 2nd and 5th turns; φ is theoff phase angle of 3rd and 4th turns.1FIG. 14: Average amplifier power P amp of 6-turn ERLpatterns at different S/N .
1. Phase angle jitters
We have estimated off-crest cavity voltage phase fluc-tuation for 120 patterns of the 6-turn ERL and resultsare given in Fig. 17. The
S/N was set to 10 to turn offthe noise. We have simulated two sets of off-set angles φ , , = 20 ◦ , − ◦ , ◦ and φ , , = 20 ◦ , − ◦ , − . ◦ . Wesee that: (1) phase jitters is pattern dependent; (2) phasejitters is off-phase angle dependent; (3) in the worst casescenario, the RMS cavity phase jitters is less than 0 . ◦ ,even at fairly large off-set angles. (4) the jitters in theon-crest case is negligible.For the two ASPO patterns (pattern number 60 and61), the first off-set angles φ , , = 20 ◦ , − ◦ , ◦ hassmaller jitters of 0 . ◦ . The σ φ cav pattern is approx-imately up-side down of σ V cav , as can be seen fromFigs 17 and 18 (a). This is more obvious for φ , , =20 ◦ , − ◦ , − . ◦ angle sets. This indicates if a patternhas larger amplitude jitters, then it tends to have smallerphase jitters, and visa versa.
2. Cavity voltage and amplifier power jitters
We have also estimated cavity voltage and amplifierpower jitters and results are given in Fig. 18. The differ-ence in on- and off-crest cases are insignificant. The av-erage amplifier power is the same as on-crest case, whichis about 11.15 kW for all filling patterns.
F. Bunch charge jitter
Bunch charge modulations for a recirculating ERL in-troduces a unique source of noise that is unlike othersources we have considered thus far in this article. Anerror on bunch charge persists over all turns in the ERLbefore the beam is dumped. As a result, the noise spec-trum from charge modulation is significantly narrower (a)(b)
FIG. 15: Comparison of V cav and turn average of V cav of different patterns. (a) ASPO and non-optimalpattern. (b) DSPO and non-optimal pattern.FIG. 16: Definition of off-set angels in off-crestbeamloading.than the white noise we have assumed for other noisesources. For the 6-turn ERL we consider in this pa-per, the effective noise spectrum for the bunch chargejitter is peaked at approximately 140 kHz, and thereforeit is within the closed-loop bandwidth of 2.5 MHz forthe LLRF controller. For small bunch charge errors, theLLRF system is easily able to correct the error, whereas2FIG. 17: Cavity voltage phase jitters of off-crest beamloading for 120 patterns for 6-turn ERL.for larger values, it will struggle and the charge jitterbecomes the dominant noise source.We performed beam loading simulations to investigateeffect of bunch charge jitter on the cavity voltage andamplifier power. The jitter was assumed to be Gaussian.RMS bunch charge jitters with 2% and 12% were sim-ulated. Simulations were carried out for 120 filling pat-terns with the S/N = 7100, bunches launched on crest,and both set-points. The results are given in Fig. 19 forRMS cavity voltages in sub-figures (a) and (d), for aver-age amplifier powers in (b) and (e), and RMS amplifierjitters in (c) and (f). The sub-figures (a), (b), and (c) areresults for dynamic set-points and (d), (e), and (f) are forstatic set-points. We see charge jitters does not increasecavity voltage jitters for both static and dynamic set-points, even when σ q = 12%. We see the filling patternand other noises are dominant over charge jitter noise. G. Energy modulation
It is possible that disturbances, such as charge jitter,beam loading, or other noise or jitter sources, may resultin an energy modulation on the accelerating or deceler-ating beam. The stored energy in the cavity is given inthe Eq. 2. Therefore, the change in energy of the cav-ity when a beam passes through is equal to minus theenergy change of the particle bunch as it passes throughthe cavity ( q bunch V cav e jφ ), where φ is the RF phase atwhich the bunch passes through the cavity: δU stored = ( V cav + δV ) − V cav ω (cid:16) RQ (cid:17) = q bunch V cav e jφ (26)Usually, Eq. 26 is simplified to a linear approximationby assuming that the change in cavity voltage is smallcompared to the cavity voltage, in which case, we obtain δV = q bunch ω (cid:16) RQ (cid:17) e jφ , which is independent of the cavity (a)(b) FIG. 18: Cavity voltage fluctuation (a) and amplifierpower amplitude fluctuation (b) of off-crest beamloading for 120 patterns for 6-turn ERL. The S/N is10 .voltage, and small modulations on the cavity voltage donot lead to an energy modulation on the bunches. How-ever, if we don’t approximate Eq. 26, we get that thechange in cavity voltage due to beam loading is: δV = − V cav (cid:34) (cid:115) − q bunch ωV cav (cid:18) RQ (cid:19) e jφ (cid:35) ≈ q bunch ω (cid:18) RQ (cid:19) e jφ (cid:20) q bunch ω V cav (cid:18) RQ (cid:19) e jφ + · · · (cid:21) (27)The second term in Eq. 27 does result in an energymodulation, and in fact it is the dominant term for caus-ing an energy modulation. If we use the values fromTable II, we find that the second term in Eq. 27 is ap-proximately 0.06% of the magnitude of the first term.Therefore, the resultant energy modulation caused bybeam loading in our hypothetical recirculating ERL isnegligible, hence the energy modulation due to effectssuch as charge jitter will be even smaller and for mostscenarios it can be neglected. However, if we operate3 (a) Dynamic set-point (b) Dynamic set-point (c) Dynamic set-point(d) Static set-point (e) Static set-point (f) Static set-point FIG. 19: Bunch charge jitter simulation results with RMS bunch jitters of 0, 2%, and 12%. (a), (b), and (c) areresults with dynamic set-point; (d), (e), and (f) are results with static set-point.at very high frequency ( ∼ THz), very high bunch charge(which would exceed the threshold current for an ERL),or the cavity operates at very low voltages ( < kV) thenthe higher order terms in Eq. 27 become significant. Thiswould also mean that the machine is operating in a non-linear regime, which would not be beneficial. IV. SEQUENCE PRESERVING SCHEME
For a recirculating linac to be an ERL, there has to bean extra path length to delay the bunch by 180 ◦ phaseto switch from accelerating mode to decelerating mode.By adjusting the delay length or by implementing moresophisticated arcs, topologies, and injection scheme, onecan manipulate the bunch order or bunch spacing. Theextra path length can be in the form of a longer arc [24]or a chicane [25]. By introducing this additional pathlength, the topology changes from the “0” topology ofFig. 1 to the “8” topology of Fig. 20. More complicatedtopologies can be achieved by setting all the arcs to dif-ferent lengths [11, 16, 26].Here we discuss “8” topology as an example to showthat it can maintain an ‘up-down-up-down’ ([1 0 1 0 1 0])beam loading pattern for all turns; which is preferable forcavity voltage and RF stability. It is achieved by utilisingan injection and delay scheme shown the Fig. 21. Sucha scheme preserves { } bunch-turn numbersequence and [1 0 1 0 1 0] beam loading pattern. Bunch- turn number sequence { } indicates the firstbunch of bunch packet is at 4th turn, the second bunchis at 1st turn and so on. In SP schemes the new bunch isinjected to the head of the packet and the bunch 3 of theearlier packet is delayed to join subsequent packet. Inthe previously described FIFO scheme, the new bunch isinjected to the position of the dumped bunch and thusthe bunch-turn number sequence changes turn-by-turn.Of course, one can maintain ‘up-down-up-down’ pat-terns with more complicated topologies as well. The pre-sented SP pattern is suitable for both simple or com-plicated topologies as it can maintain the favoured ‘up-down-up-down’ beam loading pattern and there is nodifference from the RF system perspective. For this SPscheme, the cavity voltage fluctuates within ± . L for delay.The length of delay can be given as4FIG. 20: Topology with extra arc length for phase flipand/or delay.FIG. 21: Topology with an extra arc 6 length topreserve { } bunch-turn number sequence. (a)Depiction of two bunch packets before entering the arcs.(b) Green bunch at 3 rd turn gets delayed. Bunches attheir 6 th turn are extracted. (c) Green bunch at 3rdturn is delayed and joined pink packet. When thepacket passes injection point, all the bunch numbers areincremented by 1. (d) A new bunch is injected into pinkpacket. New circulation starts with (a) again.∆ L = nL packet + mL block + λ RF . (28)with n = 0 , , , ... , m = 0 , , , ... , L packet being thelength occupied by a bunch packet, L block being thelength occupied by a intra-packet block, and λ RF beingthe wave length of RF cycle. When m = n = 0, the bunchflips phase but remains in the same packet; which is thecase of the simple recirculating FIFO scheme describedin earlier sections. The beam line layout described in [25]can be an example of this. When m, n (cid:54) = 0, the bunchesdon’t only flip phase, but also move to later blocks andpackets.Note that sequence { } indicates the turnnumber of bunches and should not be confused with fill-ing pattern [1 5 2 6 3 4], which describes filling order. Angal-Kalinin et al. , proposed [11] a similar SP schemeas { } for the purpose of separating low en-ergy bunches to minimize Beam-Breakup (BBU) insta-bility [27]. BBU is a major limiting factor for the ERLbeam current [28] and we will investigate it further in afuture study. V. COMPARISON OF SIMULATION RESULTS
Simulations were performed for SP with on- and off-crest beam loadings and static and dynamic set-points.The results are overlaid for comparison and given inFig. 22 and Fig. 23. The
S/N was set to 7 × toobserve the behavior of the system with moderate noise.Fig. 22 shows results with on-crest beam loadings only.Fig. 23 shows results with dynamic set-point only. A. Comparison of dynamic and static set-points ofFIFO and SP
In the sub-figure (a) of Fig. 22, we see SP can haveslightly lower cavity voltage jitter σ V cav than FIFO. Thedifference in σ V cav between of different patterns of SP arenot as significant as FIFO. SP is insensitive to set-pointsregardless of patterns, wile for FIFO is only insensitiveat optimal filling patterns (pattern number 60 and 61).In the sub-figure (b) of Fig. 22, we see the phase jittersare noise dominated and remained low at around 10 − degrees. This shows at S/N of 7 × , the phase jitters isnegligible for all injection schemes, set-points, and fillingpatterns.In the sub-figure (c) of Fig. 22, we see injectionschemes, filling patterns, and set-points all can affectthe average beam power. Firstly, we see SP requiresminimum power regardless of set-points and filling pat-terns. Secondly, when FIFO is combined with the dy-namic set-point, the average power is minimized as well.Thirdly, When FIFO is with static set-point, the fillingpattern becomes the most important factor in determin-ing the average power. When the pattern is optimal,the power 14.9 kW is very close to minimum power of11.3 kW. If one combines FIFO with static set-point andthe worst filling pattern, the average power can be as highas 333 kW, which is 30 times of minimum. All these areimportant factors to consider and optimize when design-ing ERLs to minimize power consumption.In the sub-figure (d) of Fig. 22, we see σ P amp has similarshape as average P amp . It is because σ P amp is determi-nant factor for P amp . At dynamic set-point, the σ P amp is very small at about 2 kW for all patterns and injec-tion schemes, which is consistent with our earlier results.On the other hand, for the static set-point σ P amp canrange from 10 −
270 kW, depending on the filling patternand injection schemes. SP with static set-point is signifi-cantly better than FIFO with static set-point, except forthe optimal patterns of FIFO.5 (a) (b)(c) (d)
FIG. 22: Comparison of SP and FIFO at dynamic set-point with on-crest beam loadings: (a) cavity voltage jitters;(b) cavity phase jitters; (c) average amplifier power; and (d) amplifier power jitters.Over all, dynamic set-points is better than static asit causes less jitters and requires less power. When set-point is static, the optimal patterns can lower jitters andpower to near the minimum. SP is more stable thanFIFO, even when it is with static set-point.
B. Comparison of on- and off-crest
In the sub-figure (a) of Fig. 23, we see off-crest beamloading lowers cavity jitters slightly, which could be dueto the fact that at off-crest phases electron bunchestake/deposit less energy from/to the cavity than on-crest.In the sub-figure (b), we phase jitters increased more than1 order of magnitude for off-crest cases. Therefore, off-crest beam loading causes increase in the phase jitters,but the jitters after the increase is still small at 0 . − . VI. CONCLUSION
We studied recirculating ERL beam loading instabil-ities of different filling patterns under various noises,phases, and injection schemes by combining analyticalmodel with simulations. Simulation results agreed withanalytical predictions with some minor differences at veryhigh or very low noises, possibly due to the non-linearityof the system. These studies give us useful insight toERL beam loading with different filling patterns, LLRFsystems, and injection schemes.We found filling patterns, S/N, and LLRF set-pointsare important for maintaining stable cavity voltage andlowering consumed RF power. We identified optimal fill-ing patterns for 6-turn ERL, but our methodology can beapplied for finding optimal patterns of other multi-turn6 (a) (b)(c) (d)
FIG. 23: Comparison of SP and FIFO at dynamic set-points when beam loading is at on-crest: (a) cavity voltagejitters; (b) cavity phase jitters; (c) average amplifier power; and (d) amplifier power jitters.ERLs as well. Optimal filling patterns lower cavity volt-age jitters and amplifier power significantly. Our studiesshow that ERL LLRF requires dynamic set-point volt-age. The cavity voltage is more sensitive to the fillingpatterns than noise. The amplifier power jitters is moresensitive to noise than filling patterns. For our setup pa-rameters, when
S/N is increased to 7 × or more, theaverage amplifier power can be reduced to minimum ofaround 11 kW. Lowering noise is critical for lowering theamplifier power. The effect of charge jitters and off-crestbeam loading on the cavity voltage and amplifier powerare negligible. The off-crest beam loading increased thecavity phase jitters by one order of magnitude, but jittersare still small at around 0 . − . ACKNOWLEDGMENTS
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