Review of the beam loading in high intensity machines

Notes for an invited talk at the ICFA Beam Dynamics Mini Workshop on Beam Loading, Feb 23-25, 1998, Tanashi, Tokyo.

I thank Dr. Yoshiharu Mori and the JHF team for this special invitation to Review Beam Loading Issues, and am honoured to accept. I shall give a tour of the "conventional wisdom" at the time of the TRIUMF KAON Factory; but my view point does not imply that other approaches are wrong.
Because there are a number of representatives of DRAL present, I also take this opportunity to acknowledge that without the encouragement of David Gray, an RF engineer and administrator at ISIS (formerly the SNS), I would not be here presenting a talk on RF-related issues 15 years later.

LCR model used in neighbourhood of cavity resonance

Model used for 80 years or more? Recent expositions in order of decreasing rigour: BUT the equivalent circuit must take into account fact that the beam sees cavity in parallel with generator impedance; and so beam does not see "bare cavity impedance", but rather the "loaded impedance". Likewise, the generator sees cavity in parallel with beam; the gap voltage V is function of beam and generator currents and so input impedance = Vg(Ig+Ib)/Ig is also modified.

Equivalent circuit model is used to find Steady State conditions

Optimum power condition is when generator sees pure resistive load and this leads to the "matched generator curve": I_b/Ig as a function of Psi, with bunch phase phi_b as parameter. Basically, I_b is in quadrature with I_g, and detuning f_res away from f_rf achieves steady state beam-loading compensation. Do not have to operate on this curve; but if not, then drive-amplifier must accept reactive power reflected from the cavity=load. Do not forget that in a machine with large beam current, the tuning range for beam-load compensation can easily exceed the frequency sweep of the RF! and must be included in the tuner specification.

Single Bunch Beam-Loading Instability

 Originally discovered by Ken Robinson 1964 [CEAL-1010 (n=0,m=1) dipole mode], this has been re-derived many times. In the naive theory, there are essentially two stability conditions:
  1.  tan(Psi)>0 [below transition energy]
  2. I_b/I_g < cos(phi_b)*2/sin(2Psi)
The first is an impedance driven or dynamical instability at the synchrotron frequency f_s. This is a "low-current" instability. The second is a "power-limited" instability at roughly d.c. This is a "high-current" instability with a threshold.
Condition (i) is identical with that for static beam-loading compensation and so there is "natural damping" of beam oscillations by the cavity impedance. [f_res > f_rf for stability below transition]
Interception of optimum generator curve with Robinson limit (ii) is inevitable at some I_b/I_g unless phi_b=0. BUT THIS IS TOO NAIVE!
The modern understanding of this instability is based on the Pedersen1 model which considers the transmission of amplitude modulation (AM) and phase modulation (PM) through the system. The model has two ingredients:
  1. geometric x-coupling because (in the presence of I_b) PM (or AM) of I_g leads to both PM and AM of the cavity total drive current I_T=I_b+I_g.
  2. dynamical x-coupling because the cavity transfer functions from I_T to V change when the cavity is detuned.
Since the cavity will be detuned (for static compensation) if there is large I_b, so (b) is a corollary of (a).
Let |phi|   =  | G_pp G_pa | |phi|
    |amp|_out | G_ap G_aa | |amp|_in
Effect (a) is modeled by a rotation matrix. Effect (b) is modeled by a frequency dependent matrix elements which can be found from either the impedance or the governing differential equation. Effect (a) is largely responsible for the "power limited" instability (ii). Effect (b) is largely responsible for the "low current" instability (i).
Examples of Robinson instability -- from beam loading multi-particle simulations in time domain (Shane TRI-DN-88-34 and HEAC 89 Tsukuba)
The Theory has been elaborated in a variety of ways. e.g. Landau damping (due to large amplitude, non-linear incoherent oscillations) allows marginal operation for small negative detuning. Landau damping is progressively stronger for higher azimuthal modes m=2,3,... but is usually inadequate for the dipole mode.

Quadrupole mode (n=0,m=2)

 The "low current" Robinson instability is naturally damped by the detuned cavity impedance; it is also partially Landau damped. However, this leaves the "high current" instability to worry about. Moreover, low-level loops often anti-damp. Also if coherent tune-shifts (due to non-cavity-impedance) are large enough, mode moves out of incoherent band and Landau lost.
Under the assumption of rigid modes, TS Wang made an analysis (plus numerical study) for Dipole-Quadrupole coupling (LANL AT6:ATN-89-1) for non-accelerating beams (phi_b=0). This was extended to acceleration (which introduces cross-coupling) by Shane1.
There are significant modifications to the instability threshold conditions, most notably at small detuning angles and large beam-load (as was originally seen in TRI-DN-88-34).
Stable Operating Conditions depend on Current Generator and Detuning.

Low-level feedbacks

 The presence of low-level feedback control loops fundamentally alters the naive criteria given above. Typically there are several loops which work with demodulated signals:
  1. phase loop compares phi_b with phi_V (damps dipole mode)
  2. AVC loop compares |V| with demand
  3. Hereward loop (HEAC 1961) damps quadrupole mode.
  4. tuning loop compares phi_g with phi_V (keeps resistive load)
The effects of these loops were first considered by Pedersen1 (1975) for the PS Booster, applied to the AGS by Raka (1976), and later revisited by Shane1 (1992). Taken singly (i.e. one loop at a time), it is found that:
  1. phase (phi_b cf phi_v) feedback extends stable region (dipole mode) to -ve tuning angles: I_b/I_g < 2/tan|2Psi|
  2. amplitude (I_b cf |V|) feedback extends stable region (quadrupole mode) to -ve tuning angles
  3. the tuning loop can reverse the naive Robinson criteria, e.g. Necessary condition (K_t-2omega_s^2tau_c)*Psi < 0
BUT Neither feedback (a) or (b) changes the power-limited criterion.
Here K_t is tuner gain, tau_c=1/sigma is cavity time constant.
Several of these predictions were verified by Pedersen1 at the CERN PS Booster.
Taken in combination, the loops, produce complicated effects. BUT the essential lesson is that the independent amplitude and phase control of V that worked satisfactorily for small beam currents does not work for larger beam currents because of the geometric and dynamic x-couplings. Unstable behaviour is almost inevitable when I_b/I_g > 2 , and other means of stabilization must be found.

Large Amplitude Coherent Oscillations

The naive Robinson criteria suggest that when phi_b=0, some stable operating point (I_b/I_g,Psi) can always be found. However, this is incorrect because only small amplitude coherent oscillations were considered. Because it is a "d.c.-instability", the 2nd Robinson condition can be generalized to the large amplitude case. Just as a single particle placed outside the incoherent RF bucket will continuously drift in RF-phase, so a bunch centroid placed outside the coherent bucket will not be captured into stable oscillations. It is found [Shane2] that the RF bucket for coherent oscillations is smaller than that for incoherent motion. The incoherent bucket shape depends on phi_s. The coherent bucket shape depends on phi_s'=phi_s+Psi+phi_g and can be vanishingly small even when there is no acceleration. Explanation: the gap voltage V_T is composed of beam and generator components V_T=V_b+V_g. Whereas the incoherent bucket depends on V_T, the coherent bucket depends only on V_g (which is phase shifted). [Both P.Wilson and R. Garoby alluded to this effect before Shane2, but did not follow up the implications.]

Coupled-Bunch Modes

 Thus far, we have considered the n=0 C.B. mode, but there are a host of other modes with phase advance 2pi*n/N between N bunches. Which of these modes are excited depends on the bandwidth and shunt resistance of the fundamental cavity mode and that of other parasitic or High Order Modes (HOMs). Also very important is the frequency spacing of the revolution harmonics. There are two extremes which encompass most cases:
  1. cavity resonances narrow-band (n/b) c.f. harmonic spacing - typical in small rings and/or high Q cavities and/or low harmonic number
  2. cavity resonances wide-band (w/b) c.f. harmonic spacing - typical in large rings and/or low Q cavities and/or high harmonic number
Both will be considered.
Let impedance Z=R+jX. CB excitation is sum is over +ve and negative frequencies, but since Z(-f)=Z^*(+f) can flip-over -ve frequencies to +ve axis. Take the example of 5 bunches below transition energy and sketch the driving/damping sidebands versus frequency.
DC RF 2RF
harmonic # -4 -3 -2 -1 h +1 +2 +3 +4 2h +1 +2
| | | | | | | | | | | | | | | | | | | | | | | | | | | | |
--------------------------------------------------------------------------------
damping | | | | | | | | | | | | | upper
CB mode #0 1 2 3 4 0 1 2 3 4 0 1 2 sideband
driving | | | | | | | | | | | | lower
CB mode #0 4 3 2 1 0 4 3 2 1 0 4 3 sideband
Sideband frequencies are m*f_s, with m=1,2,3... the "within bunch" mode number. For the cavity fundamental resonance, the growth rate is proportional to the difference in the real part of the impedance evaluated at the two mode-sidebands located symmetrically on either side of the RF.
                I_dc eta f_rf F(m,n)
1/tau_n = -------------------------- Re{Z(f_+)-Z(f_-)}
          2 (f_s/f_rev) beta^2 E_s/e
tau_n is the e-folding time: exp{t/tau_n}
F(m,n) is a form factor (very) roughly equal to unity.
The growth or damping rates are caused by the asymmetry (about the fundamental RF) in the coupling impedance.
For Case (i) [n/b and/or low h] the most prominent C.B. problem is that the cavity detuning to achieve steady-state beam-load compensation (which damps the n=0) mode will excite the h-1, h-2, etc. C.B. modes. Because n/b, asymmetry strong and growth rates can be quite fast.
For Case (ii) [w/b and/or high h] there is the possibility to excite many coupled bunch modes. Because w/b, asymmetry weak and growth rates tend to be lower (unless the cavity shunt resistance is very large).
Must not forget that cavity parasitics also drive C.B. modes...
Question: What are acceptable growth rates?
Answer: depends on storage time and on the seed values. (Since it is an exponential instability, the initial rate depends on the initial value...) Typically 1-2 e-folding times is tolerable. Seed values, or the initial mis-match, depend on injection accuracy and strategy (see below on transients).
Effect of periodic transients is to cause a bunch-to-bunch shift in f_s which can promote Landau damping of C.B. modes. However, because f_s => Sqrt(V) the mechanism is negligible (unless the disturbances are large, in which case the growth rates are already enormous).

Large growth rates

If growth rates are comparable with synchrotron frequency, then it becomes very difficult to design a multi-bunch feedback-damper system. Instead, the problem must be cured at the root-cause.
E.g. use RF feedback (direct or 1-tune-delay) to reduce impedance to a level that dampers can be used.
E.g. damp cavity HOMs with selective coupling into resistive loads; this reduces peak growth rate rate, but not integrated growth if frequency sweeps through HOM -- but usually you "win out".

Transient Loading Effects

When I_b changes rapidly there may be rapid variations of voltage V. This is a broad and important issue and includes a number of effects which are usually divided into 2 groups:
  1. Injection/Extraction transients
  2. Periodic transients.
Moreover, both groups must be considered from two stand-points:
  1. detrimental effects on the beam
  2. high-power requirements to compensate voltage transients.
These issues were first discussed by Kerns & Flood in 1965. Both groups (i)+(ii) cause AM and PM of the RF voltage. However, AM is usually unimportant as it only changes the bunch shape (height and length) by a small amount because these scale as Sqrt(V). Hence, we shall concentrate on the PM effect.
Should not forget that the effects (i)+(ii) may need to be summed together. Further, when transient effects are considered for storage rings and "RF feedback" is used, the transient RF power may even dominate over the static power requirements. Though we shall not dwell on HOMs, it must be remembered that parasitics make a contribution to the transient and should be damped. Perry Wilson (1974?)

Injection/Extraction Transients

During transfer of bunched beams between two rings, the sudden arrival or exit of beam current causes a disturbance to the RF voltage in both the receiving and giving machines. Sequential transfers of batches from a small to a larger ring (to fill the latter) will cause successive transients that not only affect the new beam but also perturb the accumulated beam from previous transfers. [There is a similar effect from successive partial extractions from one ring to multiple target stations, e.g. LANL PSR]
The sudden change in I_b will cause AM and PM of the RF voltage responsible for capture and acceptance. These modulations can cause emittance growth, and/or produce large "seeds" for coupled bunch modes. To prevent this, ideally, the variation of V must be damped on a time-scale short c.f. synchrotron period.
Unfortunately, exact calculation of the transients (and the power required to compensate them) is complicated by several factors: (a) low-level amplitude and phase loops will try and restore the voltage to nominal values (even if there is no RF feedback) -- but are usually slow (b) the cavity detuning will attempt to follow the arrival of each new batch. So, not only must one consider the cavity response/filling time but also that the cavity represents a non-matched load to the generator. [Can do anticipatory detuning for single batch injection into a small ring -- but cannot do this for multi-batch filling of a large collector ring.]
However, Boussard1,2 gave formulae to estimate the peak power requirements for a variety of simplified cases. Further, computer programs exist for calculation of the phase and amplitude excursions during injection/extraction (for rigid dipole motion) including the detuning variation. N.B. If a strong RF feedback is applied to the cavity, then the impedance becomes locally flat and the natural Robinson damping of n=0 modes ceases; consequently, a beam-phase-loop is essential.
To estimate emittance increase, one may perform multi-particle tracking: e.g. beam loading options in LONG1D (Shane4) and later ESME (FNAL). [N.B. can make analytic estimates, but a 2nd order perturbation theory is required.]

Minor Remarks on Compensation

There will be some (small) Robinson damping of bunch oscillations by proper detuning (unless there is strong RF feedback).
Though anticipatory detuning can reduce required reactive current, it is better still to have a fast tuner.
The main effect of the incremental changes in I_b is to introduce a current component in quadrature with I_g. A simple and expedient measure (Griffin IEEE Nuc. Sci. Vol. NS-22 No.3, 1975 pp 1910-1913) is to have a fast loop inject (via the amplifier) a compensating quadrature component I_g' based on comparison of Phase[Ig] with Phase[Vg]. In the FNAL Main Ring, an amplifier b/w 5MHz and loop response 300 ns was used. This is a "half-way house solution" that is a precursor of RF Feedback.
Because the beam induced voltages can be comparable with that due to the generator, it may not be appropriate to use small amplitude AM transfer functions to calculate the transients. Order of magnitude estimates can be based on: V_beam(t)=Delta I_b R_shunt {1-exp(-sigma*t)}

Periodic Transient Calculations

Whenever the RF buckets are not uniformly filled, there will be beam Fourier components at multiples of the revolution frequency in addition to the strong beam components at harmonics of the RF. If there is impedance at any of these frequencies (e.g. cavity fundamental and HOMs), then voltage components will be developed that may perturb the RF buckets. If bunches are not individually matched/tailored to their buckets, then there is the possibility for emittance growth, and for seeding of C.B. instabilities. Typically perturbations of few % are acceptable, unless buckets are very full [do not forget space-charge].
The effect is worrisome when: Often, the effects can be estimated by small-amplitude modulation theory because the revolution harmonics are "small" It is trivial to write a computer program for calculating the harmonic content of beams with various filling patterns. One such code "SPECTRA" (Shane5) allows for a hierarchy of bunches, batches, periods, super-periods, etc. as occurs in large machines.

Voltage perturbation (simple estimates)

Modulation at revolution harmonics, n, of RF=h*f_rev
delta-V(n)     Z(h+n)I_b(h+n)
---------   =   -------------
  V(h)              Z(h)I_g(h)
For case of "Empty buckets", when summed over all harmonics:
delta-V      [empty] I_b R_shunt
------- = h --------------
V_0          [full] V_0 Q
The formulae of Boussard1 may be used to estimate the peak power required to compensate these transients.
Though, as discussed below, these perturbations can be canceled by feed-back and feed-forward techniques, Boussard3 has suggested in 1991 a more "mature view": try to live with them un-compensated. This view is echoed by Pedersen2 for PEP II. If one can match periodic phase-transients between machines at extraction/injection then their "nuisance value" is essentially eliminated. [Controversial?]
Periodic beam loading results in a modulation (PM and AM) of the accelerating voltage. There is a steady-state adjustment of the synchronous phase of each bunch so that each receives the correct energy gain despite the periodical perturbation of the RF bucket centres. In other words the, the bunches are no longer equidistant.
Let |phi|  =  | G_pp G_pa | |phi|
    |amp|_v   | G_ap G_aa | |amp|_b
where G_** are transfer functions for modulations of Laplace frequency "s" about the carrier RF. To find the effect of "p" revolution harmonic at frequency (h+p)f_rev, substitute s=j*2Pi*p*f_rev with j=Sqrt(-1).
[N.B. care needed with transfer func when "s" comparable with sigma.]
We know amp_b(s) from beam current spectrum, and must find equilibrium values of phi_b, phi_v & amp_v. In absence of perturbation: (V/2)I_b sin(Phi_b) = V I_dc sin(Phi_s) is equilibrium power condition. With periodic transient:
[(1+G_aa)amp_b +G_ap*phi_b] sin(Phi_b) + [(1+G_pp)phi_b +G_pa*amp_b] cos(Phi_b) =0.
If Phi_b=0 (usually true at transfer for hadron machines)
              - G_pa
phi_b = ------- amp_b and phi_v = -phi_b
           1+G_pp
Better gap-transient matching between machines can be achieved by adjusting parameters (R,Q,V,Psi) so that the same equilibrium conditions are achieved in both machines -- a small amount of reactive power may be required. [N.B. we assume HOMs are damped so that fundamental resonance dominates.]
Boussard3 also gives time-domain formulae for calculation of equilibrium bunch phases along the batch. If RF feedback is used (say to avoid Robinson instability), one runs into a problem: the feedback tries to reduce all the equilibrium phases to zero (and cancels all our good intentions). Pedersen2 suggested to phase modulate the RF demand (along the batch) to avoid the problem -- but the electronics is fearsome. E.g. "RF Feedback Simulation Results for PEP II", R.Tighe PAC'95.
[N.B. the equilibrium bunch phases do not change when double-comb, 1-turn-delay RF feedback is applied.]
[N.B. this "live with it" philosophy is probably not applicable to a Collector Ring which receives several batches with long dwell times during which there are different filling patterns.]

General Remark on Transients

As noted by Kerns & Flood, the voltage variations will be small if the cavity stored energy is large and the shunt resistance is small; so that small R/Q is preferred. Despite the benefits, there is a cost in terms of power requirement. "Cavities of large stored energy U will have relatively large power losses in the walls. Second the necessity of tuning the cavity will mean the tuner power will increase as the cavity stored energy: U_tuner=2*(Delta-f/f_RF)*U"
If cavity b/w small enough (i.e. high Q and/or low harmonic number) and/or sufficiently detuned, then no periodic transients.
If very low Q, then cavity response instant and transients "filled in" provided amplifier has sufficient b/w and power. But disadvantage that low Q often associated with low R_shunt.

Dual Harmonic RF:

see my other talk!

Summary

Beam loading => => More Powerful Control Methods Are Required ! 

Compensation

Use of Terms
Feedback : correction signal from V_gap
Feedforward: correction signal from I_b There is NO "time-ordering" implied by these names.

Feed-forward

Low-level

Beam signal from pick-up is filtered to get RF component I_b(t) (the full vector with no-demodulation) and added to LOW-LEVEL drive chain and hence injected on the input of the RF amplifier (Boussard2). The steady-state beam-load compensation is not changed. The amplitude and phase loops now act on I_g (as if no I_b vector) and the geometric x-coupling between loops is reduced or removed. Original analysis by Boussard: CERN SPS/ARF/DB/gs/78-16. Tutorial by Shane: chapter 7.5 of 1996 USPAS course. The signal corresponding to I_b does not need to be synthesized accurately, just well enough to reduce the couplings.
Advantages: The Robinson "power limited" instability threshold is raised by an order of magnitude. Used in CERN PS with factor 8-10 improvement. Injection transients at the RF are also much reduced. Cheap!
Disadvantages: Difficult to apply this scheme while frequency is varying (Pedersen4). For the PS, coarse correction all along cycle + fine corrections at critical points (Boussard2). No effect on periodic transients. Limited by b/w of low-level RF, which is usually small. May reduce effective gain of tuning loop.

High-level

The idea is to inject an image of I_b directly into the cavity so as to cancel the beam. The beam RF current from a PU is amplified and injected into the cavity with opposite phase by means of a SEPARATE RF power amplifier (Pedersen4). Steady state conditions ARE changed, and the cavity is not detuned.
One-Turn Delay
If the pick-up to cavity delay is adjusted to exactly one turn (T_rev), beam-loading compensation can be achieved not only at RF but also (h+n)f_rev. The result is a rapidly changing beam-impedance, ideally zero at n*f_rev, but twice as large at intermediate frequencies (n+1/2)f_rev where there are no beam current components (Boussard2). With a 1-turn delay and perfect cancellation, any voltage disturbance only lasts T_rev which is short c.f. T_s. The method is simple to implement at fixed frequency, as in the CERN ISR (Schnell, IEEE Nuc. Sci. 1977) and practical with digital technology in a swept frequency system. Expensive but powerful solution:  But care needed in CB computations because of phase-rotations due to the filter & delay.
N.B. A simple feed-forward scheme was used at RAL SNS (Ian Gardner, 1985) for compensation of beam loading at the fundamental and multiples of the RF.
Feedforward can also be viewed as a means to reduce the effective impedance of the cavity as seen by the beam. At the RF, the beam induced voltage is zero for perfect correction (Boussard2).

Amplifier Feedback (RF feedback or Fast feedback)

Essentially the idea is to reduce the cavity impedance by a classic feedback. The scheme was discussed by Kerns & Flood (FNAL) in 1965 and later by M. Lee at SLAC, but was first implemented at CERN ISR, then at AA and PS Booster and elsewhere. The "State of the Art" is described by Pedersen2.
The gap-voltage signal (i.e. whole vector with no demodulation) is fed back to a high-power summing point before the final power amplifier. Suppose the f/b path has gain A. The feedback open loop gain is then H=A*R_shunt, and the closed loop transfer function becomes: Z' = Z/(1+H) and the effective shunt resistance and quality factor are reduced by the same factor (1+H) [but R/Q does not change]. The demand current at the summing point is increased by the factor (1+H) I_d=I_g[1+H] but the generator current (inside the loop) is unchanged and so the cavity detuning for steady-state beam-load compensation is not altered. Also the quiescent power requirements are unchanged. However, the low-level phase and amplitude control loops now act on I_d which can be much larger than I_b if the gain is large enough. Hence, as it appears from outside the loop, the geometric cross-coupling is eliminated and the system stability is restored. Robinson limit/threshold raised by a factor [1+H].

Lower-level RF feedback

 An alternative implementation is to take the feedback signal and sum before the pre-amplifier. [BUT the summing point must manipulate the full vector, and not be so "low down" that demodulated signals are used.] Though this is cheaper, the delay and bandwidth limitation makes this variant very unattractive.
The "amplifier feedback" automatically generates the correct compensating signal, which is another way of saying that it keeps the controlled parameter constant (Boussard2). Because part of the gap voltage is due to the beam, any beam disturbances (e.g. injection transient loading) will be automatically compensated. However, the cost is a large power requirement because the power-tube must, in principle, be able to supply the complete I_b in quadrature with I_g.
Gain-delay limitation
Feedback inevitably involves a delay (say 100 ns for high-level or 10 us for low-level f/b) and this introduces the possibility of a HF instability if the product of gain (A) and delay (T) is too large. This limitation was first recognised by Kerns & Flood, and rederived many times (Boussard, Rees1, Shane6 etc.)
A*T < (Q/R)/(2*f_rf) is the absolute limit when (f_rf*T<1/2)
A*T*(-1)^m < (Q/R)/(2*f_rf)/m is the absolute limit when (f_rf*TRevolution harmonic limitation In order to give fast suppression of transients, the RF feedback must be quite wide band. Consequently, the cavity resonance is flattened and broadened by feedback. Though the impedance is reduced at the RF, it will be boosted for some revolution harmonics. The key to avoiding periodic transients and C.B. modes is to make the feedback gain A so large that the boosted harmonics are pushed so far away from the fundamental resonance that even after "boosting" the impedance there is negligible. This has been discussed by Rees1, Boussard, Shane7. E.g. an approximate criterion to avoid boosting the (h+n) revolution harmonic is: (1+H)>(2Q*n/h)^2
where H=A*R_shunt, Q is quality factor before f/b and h*f_rev=f_rf. However, the required gain might not be achievable due to the gain limitation! N.B. The value of "n" at which the impedance (before feedback) becomes negligible is proportional to h(R/Q) and can be calculated from the allowable C.B. growth rate and R_shunt. For cavities which are narrowband c.f. the revolution harmonic spacing, it is sufficient to take n=1; this is typical in small booster-type machines with low harmonic number.
Advantages of Amplifier Feedback
  1. Most powerful compensation scheme known.
  2. Reduction of cavity Z over large b/w
  3. Large impedance reduction factors 10-100.
  4. Very fast response (can be less than revolution period)
  5. Suppresses injection transients.
  6. Suppresses periodic transients and C.B. modes if enough gain at revolution harmonics.
  7. Many successful examples of operation worldwide
  8. No need for critical adjustments
(Even novices can set this up in a couple of days e.g. TRI-DN-92-K198 -- though R&D leading up to this, for amplifiers and combiners, may take 2 years.)
Disadvantages
  1. High Power combiner needed (not cheap)
  2. Problems for frequency swing (due to delay) but can program phase correction using digital technology.
  3. May boost the impedance at revolution harmonics due to flattening of the resonance (important in large rings with large h). (iv) If A(f) not pure real, then Z' becomes asymmetric about f_res which may promote C.B. instability.
  4. In a wideband design must damp HOMs because of the progressive phase change due to the delay.
  5. Gain-delay limitation.
Can always consider 1 turn delay feedback to reduce problems (iii)-(v).

One-turn delay feedback

Boussard4,5 (SPS, 1983) Long delay will severely limit the bandwidth of RF feedback. In order to avoid this problem, note that we only need a large gain in the vicinity of the revolution harmonics. Outside the bands, gain can be small or zero. If the delay is made exactly one turn, then open loop phase is zero at revolution harmonics. =>
Place comb-filter with 1-turn delay in feedback path from cavity gap to amplifier input. Comb filter has poles (i.e. large gain) at revolution harmonics, n*f_rev. When the loop is closed, the system response is like a notch filter with (almost) zero gain at n*f_rev. Let w=2Pi*n*f_rev
Comb filter & delay = G/[exp{j*w}-K], G,K constants
                               [exp(jw) -K]
Closed loop Z' = Z -----------------  with Z=cavity impedance
                     {[exp(jw)-K] +Z*G}
@ RF+nf_rev Z'= Z [1-K]/[1-K+ZG] approx = zero
@ RF+(n+.5)f_rev Z'= Z [1+K]/[1+K-ZG] approx = 2*Z if (1+K)=2ZG but no beam components at (n+.5)f_rev, so no problem.
By making K close to unity, the RF feedback approaches the theoretical performance of the feedforward correction, but with the inherent advantage of a closed loop system: no critical adjustments.
The reduced impedance will suppress periodic transients and C.B. modes. Can damp several azimuthal modes, m=1,2. But range of damped CB modes n, is limited:
Double peaked comb filter
The limitations to C.B. mode damping in the Boussard design can be overcome by the use of a more sophisticated filter in the f/b path (Pedersen2). It is the impedance at the synchrotron sidebands which is important for coupled bunch. Simple comb filter has a pole at each revolution harmonic. Double comb has poles at each synchrotron sideband and a zero at revolution harmonic. Hence closed loop response will have:
  1. double notches (zeros) at sidebands of every revolution harmonic
  2. the regular cavity impedance Z at revolution harmonics
  3. 2*Z at RF+(n+.5)f_rev.
Therefore CB modes are damped. BUT there is no impact on periodic transient beam loading! However, remember, that if the bunch-equilibrium-phases are matched between rings, then the periodic transient effect is not an issue.
One difficulty with the scheme is that the periodic transients are not compensated -- but they will be seen by a DIRECT fast-feedback (short delay) if this is used in addition to the 1-turn-delay system. Hence the power tube/klystron will saturate/limit while trying to suppress the periodic transients unless there is a preprogrammed variation of the demand vector V that follows the equilibrium phase of each individual bunch (e.g. PEP II)

Cathode-Follower

Another variant of the idea to reduce the cavity impedance to the beam is "cathode follower" whose big advantage is that no external feedback network is required to achieve a feedback ratio of beta=1. The "cathode follower" scheme was adopted for the LANL PSR and is described by Hardek1; the scheme was also considered at BNL by Puglisi1.
Comments:

C.B. modes

The direct amplifier feedback has a weak positive effect on C.B. modes The single comb delayed feedback has a good positive effect on C.B. The double comb delayed feedback has a strong suppression of C.B. modes.

Beam Loading Compensation -miscellany

Both feedback and feedforward will: But the effectiveness is not uniform, and needs careful study.
The beam current modulations whose effects are to be suppressed must be within the current capability (and b/w) of the generator when feedback or feedforward is used.
Feed forward offers, at least in principle, the possibility of complete beam-loading cancellation in contrast to feedback which brings about only a reduction of beam loading (Gamp1). However, in practise the beam signal has to be correctly delayed/phased and amplified before the "true image" is injected into the cavity, and errors typically limit the image fraction to 90% of the beam.
The "feedback" is generally preferred (Garoby) over the "feedforward" because: it is easier to set up (on the bench) whereas the open-loop aspect of feed forward implies lengthy adjustment with beam. Further, over the long term, while feedback is stable against drift of component values, feedforward must be manually corrected.

Frequency Swing

Except for machines with fixed RF, delay-feedforward and delay-feedback (TRI-DN-89-K82) can only be envisaged with the help of modern digital signal processing technology.
For booster-type machines with a large frequency sweep, and that use direct RF feedback, centre frequency of amplifiers should either track the RF, or required bandwidth of amplifiers should include width of cavity fundamental resonance and also the sweep range.
Large, fast frequency swings as may occur in booster-type synchrotrons can be difficult for the tuner to follow accurately if the cavities are too narrow band; and so very high Q might not be worth pursuing in this case.

Bandwidth of 1-turn-delay

Though the comb filter used in long-delay feedback and/or feedforward can reject all h revolution harmonics, it may not be essential for the amplifier bandwidth to extend over such a large range. E.g. in a large ring with b batches separated by b kicker gaps, the strong revolution harmonics are at the RF and h(+/-)b, and so a bandwidth of 2b would be adequate.
If, however, the cavity resonance width is equal to the RF, then a wideband amplifier would be used to suppress all the C.B. modes.

Compensation Strategy

Try and design the cavities from the beginning to have low R/Q (which depends basically on geometry of the RF structure and loading materials such as ferrite, etc.). Try to have few cavities, but each with large gap voltages. Of course, both these "principles" run quickly into the problem of large power requirements, and must be supplemented by other means.
To first order, there are two possible beam loading compensation strategies; and the choice depends on the ratio of the revolution frequency spacing to the bandwidth of the cavity fundamental resonance. The extremes of a small Booster ring and a large Diver ring (at the TRIUMF KAON Factory) encompass these two strategies.
Booster Ring [2Q/h > 1]
[few revolution harmonics in the cavity fundamental b/w]
  1. Use direct amplifier feedback to raise the Robinson threshold well above the beam current, and to decouple the low-level control loops (phase and amplitude, etc).
  2. Examine the impedance at h(+/-)1 and h(+/-)2 revolution harmonics and decide whether it is appropriate to push the feedback gain to the delay limit.
  3. Damp higher order modes.
  4. Install dedicated active damper to combat observed C.B. modes.
Driver Ring [2Q/h < 1]
[many revolution harmonics in the cavity fundamental b/w] Sequential injection of several batches implies the beam Fourier spectrum will change several times. After the first transfer there will be strong revolution harmonics giving periodic transients. Later, there will be stronger harmonics at h(+/-)b where "b" is the number of batches transferred.
  1. Use direct amplifier feedback to reduce the fundamental impedance and so raise the Robinson threshold.
  2. Push the gain to the delay limit to suppress revolution harmonics h(+/-)1, h(+/-)2, etc.
  3. Supplement with 1-turn-delay feedback to suppress periodic transients and reduce C.B. modes.
  4. Damp higher order modes.
  5. Install wideband active damper to combat residual C.B. modes.
  6. If measures 2 & 3 inadequate to suppress transients, then add 1-turn-delay feed forward.
[N.B. when designing the active damper, do not forget that the cavity detuning varies during the acceleration cycle, and it is quite possible to have the n=h-1 C.B. mode driven by the full cavity shunt resistance if the cavity is detuned so that f_res coincides with (h+1)f_rev.]
These recommendations are outlined in greater detail in Chapter 4 "Accelerating System" of the "KAON Factory Study Accelerator Design Report" (1990) and build on suggestions of Pedersen3.

Non-linear Effect

Resonant frequency depends on RF amplitude (Effect seen by C. Friedrichs and G Hulsey 1992) (Effect analyzed at SSC LEB: Hulsey, Petrov, Yakolev, 1992) Ferrite permeability depends on amplitude of E,B

Novel Ideas

Accelerating Cavity coupled with an energy storage cavity (T. Shintake KEK preprint 92-191)

Bibliography

  1. Boussard1: "RF power estimated for a hadron collider", SPS/ARF/DB/gw/Note/84-9
  2. Boussard2: "Control of Cavities with High Beam-Loading", CERN SPS/85-31 (ARF) or IEEE Nuc. Sci. 1985, Vancouver PAC.
  3. Boussard3: "RF Power Requirements for a High Intensity Proton Collider", CERN SL/91-16 (RFS)
  4. Boussard4: "Reduction of the apparent impedance of w/b accelerating cavities" IEEE Trans. Nuc. Sci. 1983 pp 2239-2241.
  5. Boussard5: "First tests of the RF feedback system with beam", CERN SPS/ARF/DB/gs/82-83
  6. Gamp1: "Servo Control of RF cavities under beam loading", CERN 92-03.
  7. Garoby: "Beam Loading in RF cavities", Frontiers of Particle Beams, Intensity Limitations, Springer-Verlag Lecture Notes in Physics 400, 1992.
  8. Kerns & Flood: "Stabilization of accelerating voltage under high intensity beam loading", IEEE Nuc. Sci. 1965, pp 58-64.
  9. Shane1: "Analytic Criteria for Stability of Beam Loaded Radio-Frequency Systems", Particle Accelerators, Vol. 48, No.3 (1994) pp 135-168.
  10. Shane2: "Coherent and Incoherent Bucket for a Beam Loaded RF System", TRI-DN-93-K239
  11. Shane3: "Injection Transient Compensation in the Collector", TRI-DN-91-K166
  12. Shane4: "LONG1D User's Guide", TRI-DN-97-12
  13. Shane5: "Calculation of Seed Values for C.B. Instability", TRI-DN-90-K161
  14. Shane6: TRI-DN-97-2 and TRI-DN-97-3R Shane7: TRI-DN-88-29
  15. G.Rees: "RF system design for control of heavy beam loading in circular accelerators"
  16. Hardek1: "A low-impedance 2.8 MHz pulsed bunching system for LANL PSR", Proc. 16th Power Modulator Symposium, Virginia, June 1984.
  17. Pedersen1: "Beam loading effects in the CERN PS Booster", IEEE Trans. Nuc. Sci. Vol NS-22 No.3, June 1975, pp 1906-1090.
  18. Pedersen2: "RF Cavity Feedback", CERN/PS 92-59 (RF)
  19. Pedersen3: "Beam Loading Aspects of TRIUMF KAON Factory RF Systems", TRI-DN-85-15
  20. Pedersen4: "A novel RF cavity tuning feedback scheme for heavy beam loading", IEEE Trans. Nuc. Sci. 1985 PAC, Vancouver.
  21. Puglisi1: "A cathode follower amplifier", IEEE Trans Nuc. Sci, Vol NS-30, No.4, 1983, pp 3408-3410.
  22. USPAS course: ``Radio Frequency Systems for Accelerators'' offered at the San Diego campus of the University of California in January 1996.
  23. Wilson1: Single Pass Collider Memos CN-43 and CN-74.