Nuclear Instruments and Methods in Physics Research A 391 (1997) 64-68 cm . . mm

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Id ELSWIER

NUCLEAR INSTRUMENTS

& METHODS IN PHYSICS RESEARCH

Section A

Longitudinal coupling impedance imposed by a beam feedback in a synchrotron

Sergei Ivanov”

Institute for High Energy Physics, Protvino, Moscow Region, 142284, Russia

Abstract Commonly, a longitudinal beam feedback processes a slowly varying signal at zero intermediate frequency (a phase offset,

an amplitude departure). Often, only a portion of the data confined in a picked-up band-pass beam signal is retained (like, say, in a purely phase feedback). Sometimes, a beam feedback employs different RF bands to pick ulj beam data and return a correction back to the beam. All the manipulations thus involved with signal spectra result in cross-talk between various beam-current and electric-field waves propagating along the orbit, which is shown to be described by an impedance matrix with, at most, three non-trivial elements per row. It is this matrix which gives the intuitive notion that a linear feedback is seen by a beam as an artificial coupling impedance controlled from the outside from a quantitative basis. This (impedance) approach has at least two plain advantages: (i) It allows one to mount the feedback’s effect into the well-established theory of longitudinal coherent instabilities to use most of its inventory: beam transfer functions, threshold maps, handling of coupled-bunch motion, etc. (ii) The destabilizing effect of the beam environment, being available in standard terms of coupling impedances, is naturally taken into account since the early stages of feedback R&D.

Keywords: Longitudinal dynamics; Synchrotron; Beam feedback; Impedance; Bunched beam; Instability

1. Introduction

This paper is rather a qualitative presentation. It should not be treated as a comprehensive description of the subject. A full write-up is available [I]; its brief excerpts are published [2].

The impedance approach, when a longitudinal coupled- bunch (CB) beam feedback (FB) is considered, has been applied extensively to tailor out feasible technical contours of the two band-pass FBs near the RF foreseen in IHEP’s UNK Project - a DC-coupled RF FB around a final PA [3], and an AC-coupled CB beam FB [4].

A few papers should be mentioned as a prerequisite of this study.

First, [5] yields a theory of a two-path amplitude-phase FB versus beam transfer functions (BTFs) Gfa, G:p, G”,, and Gt, operating in amplitude-phase parameter space. It provides an adequate tool to treat a narrow-band FB whose bandwidth AcYIJ’~~’ is much smaller than the rotation frequency wO. Such a circuit readily handles an in-phase azimuthal CB mode IZ = 0 of beam multipole oscillations. Still, far less obvious is its effect on the higher azimuthal

* Corresponding author. Tel.: +7 95 217 58 49; fax: +7 95 217 58 49; e-mail: ivanov_s@mx.ihep.su

CB modes n # 0, 2nnlM being the bunch-to-bunch phase shift, and M the number of bunches per beam.

Second, in [6] a now standard approach to bunched- beam coherent instabilities is pioneered. Its major con- stituent is a matrix plane-wave BTF (‘beam admittance’) Y,,,(n) that relates linearly various electric-field and beam-current waves propagating along the orbit:

E,(a) eik*-iat 3 (via matrix Y,,,(a)) *J,,(Q) eik’a-inr.

(1)

Here 6 is the azimuth in the co-rotating frame, integer k is the wave number, t is time and 0 is the frequency of the Fourier transform seen as a side-band w = km, + 0 in the lab-frame. This plane-wave representation provides a natural basis to treat azimuthal CB modes n. The whole essential inventory of a conventional instability theory - effect of Landau damping, bunch form-factors, within-

bunch multipole oscillations, threshold map technique, etc. -is thus worked out in terms of these very Y,,,(a).

An attempt to merge the FB and beam instability studies is undertaken in [7] which gives amplitude-phase BTFs Gz of, say, [5] through plane-wave BTFs Y,,.(@. However, on adopting such a standpoint, only an in-phase azimuthal CB mode n =0 and a beam FB with Aw’~’ -=z co,, are readily involved in the analysis. Here, one moves right in

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S. Ivanov / Nucl. Instr. and Meth. in Phys. Res. A 391 (1997) 64-68 65

the opposite direction. The basic plane-wave BTFs Y,,,(R) are retained throughout. It is the open-loop beam PB circuit itself whose effect is written in terms of an artificial coupling impedance controlled from the outside.

This (impedance) approach to CB beam FB is a fre- quency-domain technique. Still, it does not necessarily imply a narrow-band FB whose Am’“’ is much smaller than the bunch-to-bunch recurring frequency 1Llw,. Indeed, as Awcm’ extends beyond Mw,,/2, the beam FB fails to distinguish between CB modes n, #n,. Such a CB beam FB eventually turns into a ‘bunch-by-bunch’ beam FB. Of course, its characteristic equation becomes more complex, but it does account for an adverse effect of finite response times of a real pick-up, an acting device, electronics, etc. There is hardly any other way to treat this problem rigorously.

This paper puts forward an approach most suitable to study a longitudinal CB beam F’B with ALO’“‘> coo. Such a circuit is a system of a large multi-bunch synchrotron with a low rotation frequency w,+ Usually, the beam is injected into such a ring by trains to fill the orbit partially. Hence, the FB’s response time should be <2~7/w, so as to handle individual bunch trains. From these the danger of destabili- zation of higher off-resonance CB modes n at large FB gains arises. Moreover, a proton synchrotron usually accelerates long bunches. Thus, the effect of the beam FB on the dipole, quadrupole, sextupole, etc. within bunch modes should be properly estimated as well.

2. A beam feedback

Unfortunately, to recover dynamical consequences of switching on a beam FB, one has to intervene in RF- engineering affairs. A block diagram of a CB beam FB that

processes a slowly varying signal is shown in Fig. 1. There, H”.“‘(Sw) are two low-pass base-band transfer functions. The others are band-pass transfer functions: front-end electronics’ admittance S(w); current-to-current gain K(o) through a final PA; T(w) from the external drive current to the AD gap voltage; and -T’, W’(w) from the beam current to the AD or PU gap voltages, respectively. Phase (p’ of the up-mixing carrier is set equal to 4 of the low-level drive that would have provided an in-phase contribution of the AD to the net accelerating field. Phase 4 of the down-mixing carrier is adjusted w.r.t. 4’ so as to settle transit-time effects due to finite the PU-AD distance.

In principle, Fig. 1 shows just the data flow through a beam FB.

On the one hand, it may well be taken as a guideline for a straightforward technical realization. In this case, one gets a two-path FB circuit relying on a filter method [S] with down and up frequency conversion. It operates in an in-phase-quadrature signal parameter space, and is hence inherently linear at any input. This scheme is adopted in

[3,41. On the other hand, in practice, to study widespread

beam FBs operating in polar amplitude-phase parameter space one has to employ a linear small-signal decomposi- tion near the FB’s set-point. In this case, Fig. 1 would represent an equivalent local Cartesian in-phase-quadra- ture FB.

Thus, Fig. 1 shows quite a general scheme that includes many other particular options widely used in practical applications:

Distinct RF bands that may be employed to pick up beam data (*Km,,), to feed corrections back to the beam (ih’o,), and for RF acceleration itself (Fhw,). Here, arbitrary harmonic numbers are implied: i#h’ and h, h’#h.

Fig. 1. Beam FB layout.

II. COHERENT OSCILLATIONS

66 S. Ivanov I Nucl. Instr. and Meth. in Phys. Res. A 391 (1997) 64-68

2. Any path transfer functions H”‘(w)#H’“‘(w), includ- ing, say, H”‘(w)=0 (amplitude control off =a purely ‘phase’ FB). Identical path transfer functions Z#“‘(w)=H’“‘(w). Ex- cept for a factor l/2, this trivial case stands for a band-pass FB which does not employ a frequency conversion at all. Provided the latter is still im- plemented, the perfectly balanced (due to HcC)(w)= H’“‘(w)) frequency up and down mixing would hide the frequency conversion inside the electronics. Setting &=h’ =h and joining a PU and an AD into a single device AC, an accelerating cavity, as shown schematically in Fig. 2, would result in an RF FB around a final PA. This circuit can be viewed as a degenerate (singular) CB beam FB. Due to an extra FB loop closed it can become self-excited without a beam. The primary purpose of this RF FB is stabilization of the accelerating voltage amplitude and phase against any (periodic with 27r//w,,) external interference. Natu- rally, the major source of the latter is the bunched beam (with unequal bunches) itself. Hence, as a side effect through BTF, one gets (i) transient beam loading compensation, and (ii) damping of beam instabilities at the high-Q RF cavity fundamental mode. This DC- coupled RF FB is now a standard option for a large synchrotron [9].

Longitudinal coupling impedance

A counterpart of Eq. (1) which closes a conventional theory of coherent instabilities is the relationship

J,(o) eiks-inr -(via linear Maxwell equations)

jE,,(fi)_eik’+iot. (2)

Commonly, it is put down in terms of a longitudinal coupling impedance Z,,(w). Since [lo], it proves to be an extremely convenient interface between beam dynamics and electrodynamics of the beam passive environment, and is defined by

xx AC

W BTF --

Fig. 2. Transformation from a CB beam FE to an RF FB.

Here ‘ - ’ is a conventional sign yielding Re Z,,(w) 20 for a passive structure, 27rR, is the circumference of the orbit, and w = kc+, + D is the frequency in lab-frame (a pair of indices in Z,,. is reserved for later use).

The impedance concept of Eq. (3) involves a contribu- tion of the beam surroundings smoothed over a turn. Accounting for the lumped nature of vacuum chamber components would have resulted in a matrix Z,,,(w). Fortunately, a slowly (101 -=x w,) perturbed beam looks like a periodic ‘comb’ filter, and the beam harmonic behaves as J,(k’w,, + 0)-J,(a)&,,,. Hence, retaining diag(Z,,(w)) would suffice.

It should be noted that the impedance matrix due to beam FBs (with a frequency conversion inside) has nothing to do with the lumped effects. It accounts for a smoothed contribution of the beam active environment as well.

Frequency down and up mixing inside a beam FB circuit is sketched in Fig. 3. To account for manipulations thus involved with signal spectra, one has to extend the definition of Eq. (3) by introducing an impedance matrix with, at most, three non-trivial elements per row (labels A, B and C correspond to those in Fig. 3):

+zc-,._,(kw, + a)J,_,._,(a)l,) (4)

Here, positive wave-numbers k-h’ >O are taken for definiteness. The region k= - h’<O is arrived at through the impedance reflection property Z,, * (w) = Z _-k,_k,(-~*)*. Term A with Re Z,,(w)?0 represents the acting device treated as a passive structure. Terms B and C describe the active FB circuit itself. They may have any sign of Re Z!$(w), and are responsible for damping of coherent motion. Term C vanishes given H’“‘(w)=H’“‘(o),

i.e. for a perfect balance of up and down frequency mixing. Terms A and B merge whenever h(PU)= h’(AD). The latter is always the case in RF FBs around a final PA.

In principle, Eq. (4) could have been established a priori referring to: (i) the linearity of the Maxwell equations, and (ii) the linearity of the ideal (the bandwidth of H’“.“‘(w)

being do.+, < (6, h’)q,) procedure of down and up mixing of frequencies w.r.t. a slow amplitude-modulated signal. Of course, in [1,2] this result is obtained directly, and the matrix elements of Eq. (4) are available in terms of FB section transfer functions and set-point parameters (their lengthy expressions are not shown here).

Nevertheless, putting aside a distinction between deduc- tive and inductive ways to write down Eq. (4), formally it is nothing but just a more general linear relation between .Ik and Ek, to be inserted into a conventional theory of bunched-beam longitudinal instabilities.

S. Ivanov I Nucl. Instr. and Meth. in Phys. Res. A 391 (1997) 64-68

Fig. 3. Conversion of spectra inside a CB beam FB.

4. Characteristic equation

The most transparent results are obtained for a narrow- band FB and a relatively short bunch half-length AtYo,

beam mode (n. m) with k = n + MZ and 0 =mQ, (4 /w,, is a small-amplitude longitudinal tune) one gets the following characteristic equation of a beam governed by a CB beam FB:

w,, 5 Aw’+-’ i< MwJ2, (Aw’m~lwo)A~o -K G-. (5) 1 + CJ,(~,(,n)Y,.,.(~) + {$‘(Q,Y&!(L!n,, = 0. (6)

Let us label azimuthal CB modes with n = 0, 1, 2 ,..., M - 1, and within-bunch modes with m= 1, 2, 3,... (dipole,

Here, C = 0: l(hV, sin pJ, V, is the accelerating voltage, lp,

quadrnpole, sextnpole, etc). At a side-band w =kw, + fi of is the stable phase angle (4pS > 0 below transition), J, is the beam current averaged over the orbit, and Yk,,(0)=

Fig. 4. Resonant frequency lines of a narrow-band FB.

II. COHERENT OSCILLATIONS

68 S. Ivanov I Nucl. Instr. and Meth. in Phys. Res. A 391 (1997) 64-68

-200 -250 -150 -50 50 150 250

Re(O, Ohm

A 0 200

Re(<), Ohm

Fig. 5. Residual instability driving impedances of ACs. Fig. 6. Stabilizing effect of CB beam FB.

Y::!(n) is a plane-wave BTF. For reference, a bunch without incoherent tune spread responds at O-mQ, as

YLT!(fi) = im2F~~!/(Qz - (mf@), (7)

where Fkkym’ is the bunch form-factor; Fkk!m)-+kk’lj,,,, when the bunch half-length A$ -+ 0.

In Eq. (6), &(O) and &‘“‘(fi) are the instability driving (reduced) impedances made of coupling impedances, Eq. (4), and their negative-frequency counterparts,

Z’m,

lp-Q(L?) = k;,ki_-h,+&:w, + 0)

k : Z cm) ~;,+,+&q, + fin)

+ (-1)” k:

+ (2 terms: kt + k;, h’ + - h’, h -+ _

(8)

Ii).

(9)

Here, R,,, and k:, are the wave numbers from a set {n+MZ}, I is an integer, that occur inside the FB band- width which is sketched in Fig. 4. Terms with a factor

(-1)” in Eq. (9) are responsible for a discrimination between m =odd and m =even imposed by FBs with H’“‘(w)#H’“‘(w); &‘“‘(fi) may be scaled ~1 /JO by elec- tronics.

Characteristic Eq. (6) is reduced to a form nearly amenable to a conventional threshold map technique. The second part of Eq. (5) has allowed us to employ BTFs Y,,,,,(O) and Y,,.(n), both taken at central frequencies h’w, of an AD, and ,&I, of a PU. Notice the possibility of the BTF Y,,.(O) sign reversal for long bunches and E#h’.

The impedance treatment of a longitudinal FB was employed at length in, e.g. [3,4] to derive technical contours of the systems for the IHEP’s UNK project. As an example, Fig. 5 shows the stabilizing effect of the RF FB around a final PA w.r.t. CB multipole instabilities excited by the AC fundamental mode. Fig. 6 shows further stabilization provided by a beam FB near the RF. Both figures are plotted as conventional threshold diagrams in the complex (Z)-plane. The parameter of the instability driving the impedance godograph is the CB mode index n incremented counter-clockwise. Curve A is the threshold curve for dipole oscillations m = 1, curve B is for quad- rupole oscillations m = 2. The stable region lies to the left of the threshold curves.

Figs. 5 and 6 along with the other technical details which can be found in [l-4] allow one to draw the conclusion that impedance treatment is a straightforward practical tool to get a deep and detailed insight into ‘beam & feedback’ system dynamics.

References

[l] S. Ivanov, IHEP Report 96-08 (1996). [2] S. Ivanov, Proc. 1995 Particle Accelerator Conf., Dallas,

USA, Vol. 5 (1995) p. 2949.

[3] S. Ivanov, IHEP Report 94-43 (1994). [4] S. Ivanov and A. Malovitsky, IHEP Report 96-7 (1996).

[5] F. Pedemen, IEEE Trans. Nucl. Sci. NS-22 (1975) 1906.

[6] A. Lebedev, Atom. Energy 25(2) (1968) 100.

[7] E. Shaposhnikova, CERN SL/94-19 (RF) (1994).

[8] F. Pedersen, CERN/PS/90-49 (AR) (1990). [9] D. Boussard, CERN/SPS/85-31 (ARF) (1985).

[lo] A. Lebedev and E. Zhilkov, Nncl. Ins& and Meth. 45 (1966)

238.