Abstract The stability of a beam in synchrotrons with a digital longitudinal feedback system is treated. A longitudinal feedback system is required in synchrotrons to stabilize the high intensity beams against longitudinal instabilities and to damp the phase injection errors of a bunch. Damping rates of the digital longitudinal feedback system in dependence of its gain and delay are analysed

V M Zhabitsky

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DIGITAL LONGITUDINAL FEEDBACK SYSTEMS IN SYNCHROTRONS
V. M. Zhabitsky, Joint Institute for Nuclear Research, Dubna, Russia
Abstract
The stability of a beam in synchrotrons with a digital
longitudinal feedback system is treated. A longitudinal
feedback system is required in synchrotrons to stabilize the
high intensity beams against longitudinal instabilities and
to damp the phase injection errors of a bunch. Damping
rates of the digital longitudinal feedback system in depen-
dence of its gain and delay are analysed.
INTRODUCTION
Longitudinal feedback systems are necessary in syn-
chrotrons to reach the accuracy and stability required for
reproducible beam performance [1, 2]. A digital bunch-by-
bunch feedback [3] individually steers each bunch by ap-
plying electromagnetic kicks every time the bunch passes
through the kicker (DK). The kick value is in proportion
to the bunch deviation from the synchronus phase at the
beam position monitor (BPM) location. The combiner (see
Fig. 1) generates the wideband horizontal, vertical or sum
signal, which is then demodulated to base-band by the de-
tector. A stable beam rejection module removes useless
DK BPM
Beam
Stable Beam
Rejection
RF Power
Amplifier
Detector
Modulator
Combiner
Delay ADCDAC DSP
Figure 1: Block diagram of a digital feedback system
stable beam components from the signal, which is even-
tually digitized (ADC), processed (DSP), and re-converted
(DAC) to analog by the digital processor. While in trans-
verse feedback systems amplifier and kicker operate in
base-band, longitudinal feedbacks require a modulator that
translates the correction signal to the kicker operation fre-
quency. The delay line adjusts the timing of the signal to
match the bunch arrival time. Conditions for damping of
the coherent synchrotron oscillations are discussed below.
BASIC NOTIONS
The RF system accelerates particles of a bunch in syn-
chrotrons by producing a time-varying electric field in an
accelerating station. The particle’s energy growth per turn
is given by [4, 5, 6]
E[n + 1]− E[n] = qV̂RF sin φ[n] , (1)
where q is the charge of the particle, qV̂RF is the maximal
energy gain per turn and φ[n] is the phase of the RF when
the particle crosses the middle of the accelerating station
at the point sRF of the orbit in the moment t[n] at the n-th
turn (the origin of time for the RF phase is taken at zero
crossing of the RF voltage with positive slope; s is the co-
ordinate along the orbit). It is also assumed in (1) that the
electric field is the even function of (s − sRF). Details of
mode structure, geometry, transit time factor, etceteras are
ignored in (1) by including these features in V̂RF.
It will be assumed further that the particle with the en-
ergy E[n] = mc2γ[n] passes the n-th turn with the speed
cβ[n] along the orbit with the circumference C[n] so that
the angular frequency ω[n] and the revolution period are
T [n] = 2π/ω[n] = C[n]/(cβ[n]); here c is the speed of
light, m is the particle mass, and γ = 1/
√
1− β2. In ac-
cordance with these definitions the RF phase growth per
turn can be written for the particle as follows:
φ[n+1]−φ[n] =
t[n+1]∫
t[n]
ω̂RF(t)dt = ωRF[n+1] T [n+1], (2)
where ωRF[n + 1] is the average value of the angular RF
frequency ω̂RF during the turn T [n + 1] ≡ t[n + 1]− t[n].
The phase φ[n] can be kept unchanged (modulo 2π) at
the value φs when the particle returns to the same accel-
erating section after one revolution period. The phase φs
is also called the synchronous phase. In what follows, the
subscript “s” is used for synchronous quantities. The syn-
chronous phase growth per turn in accordance with (2) is
φs[n + 1]− φs[n] = ωRF[n + 1] Ts[n + 1] = 2πhRF , (3)
where hRF = (ωRF/ωs) is the RF harmonic number. Let
ρsBs be the momentum rigidity of the synchronous particle
moving along the orbit with the circumference 2πRs. Then
the synchronous particle’s energy growth pur turn is
Es[n + 1]− Es[n] = qVRF sinφs[n] = q 2πRsρsḂs . (4)
Eq. (3) and Eq. (4) are the system of difference equations
for definition of φs[n] and Es[n]. Let us assume that these
solutions have been found. Therefore the motion of an arbi-
trary particle relative to a hypothetical synchronous particle
can be examined.
Acceleration in presence of perturbations
Let δE[n] and δφ[n] be small deviations of energy and
phase of the particle from corresponding synchronous val-
ues at the n-th turn:
δE[n] ≡ E[n]− Es[n], δφ[n] ≡ φ[n]− φs[n] .
Let ΔVRF, ΔωRF and ΔBs be small deviations of the accel-
erating voltage, the angular RF frequency and the magnetic
WEPSB026 Proceedings of RuPAC-2010, Protvino, Russia
05 Beam Dynamics and Electromagnetic Fields
224
field from designed values. Therefore the particle’s energy
deviation from the synchronous one gained per turn in ac-
cordance with (1) and (4) is
δE[n + 1]− δE[n]
= q(VRF + ΔVRF[n]) sin(δφ[n] + φs)− qVRF sin φs . (5)
In what follows, δ represents a difference taken with re-
spect to the synchronous quantity at a given time (conse-
quently at the given turn) and Δ represents an increment
during acceleration. The phase deviation gained per turn in
accordance with (2) and (3) is given by
δφ[n + 1]− δφ[n] =
t[n+1]∫
t[n]
(ωRF(t) + ΔωRF(t))dt− 2πhRF
= ωRF[n + 1] T [n + 1]− 2πhRF + ΔφRF[n + 1] , (6)
where the phase shift due to a small modulation of the RF
frequency is
ΔφRF[n + 1] =
t[n+1]∫
t[n]
ΔωRF(t)dt . (7)
The deviation of the revolution period T [n] from Ts[n] is
determined by [4, 7] the energy deviation δE[n] and the
magnetic field error ΔBs[n]:
T [n] ≈
(
1 +
ηs δE[n]
β2s [n] Es[n]
− αs ΔBs[n]
Bs[n]
)
Ts[n] .
Here ηs = αs − (1/γ2s ) is the phase slip factor and αs is
the momentum compaction factor [7]. Therefore the phase
deviation gained per turn is
δφ[n + 1] = δφ[n]− 4π
2ν2s
qVRF cosφs
δE[n + 1]
+ ΔφRF[n + 1]− 2παshRF ΔBs[n + 1]
Bs[n + 1]
, (8)
where the synchrotron oscillation tune νs is given by [7]:
νs =
√
− hRFηs
2πβ2s Es
qVRF cosφs .
For an adiabatic process of acceleration it is possible to
neglect dependences of the coefficient before δE in Eq. (8)
on time and n.
Eq. (8) and Eq. (5) are the system of difference equations
for definition of δφ[n] and δE[n]. It corresponds to differ-
ential equations in [4] for synchrotron oscillations in pres-
ence of pertirbations and coincides with difference equa-
tions in [6, 7] in the case of ΔVRF = ΔφRF = ΔBs = 0.
The system of Eq. (8) and Eq. (5) can be solved, for exam-
ple, using the Z-transform [8] in a such way that was done
for the transverse feedback system in [9].
Synchrotron oscillations
If ΔVRF = ΔφRF = ΔBs = 0 then for small δφ lin-
earization of Eq. (5) in combination with Eq. (8) gives the
synchrotron oscillation equation as difference equations in
the matrix form:(
δφ[n + 1]
δE[n + 1]
)
= M̂ ×
(
δφ[n]
δE[n]
)
, (9)
where
M̂ ≡
⎛⎝1− 4π2ν2s − 4π2ν2sqV̂RF cosφs
qV̂RF cosφs 1
⎞⎠ .
Consequently the particle dynamics is determined by roots
zk of the characteristic equation:
det
(
zkÎ − M̂
)
= z2k − (2− 4π2ν2s )zk + 1 = 0 , (10)
where Î is the identity matrix. Eigenvalues of Eq. (10) are
z± = exp(±j2πν) , sin πν = πνs
that is well known result [5].
Dipolar motion approximation
Let us consider the bunch as a statistical collection of
many particles [2], each particle oscillating around the syn-
chronous phase. Let f(φ, E) dφ dE be the probability that
the RF phase is between φ and φ + dφ and the particles
energy is between E and E +dE when the particle crosses
the middle of the RF cavity at the n-th turn with phase φ[n]
and energy E[n]. The first order moment (centre of charge)
of φ[n] at the n-th turn is
〈φ[n]〉 ≡
∫∫
φ f(φ, E) dφ dE ,
∫∫
f(φ, E) dφ dE = 1,
where the integration covers the entire bunch. Conse-
quently with respect to synchronous phase and energy one
can write:
〈δφ[n]〉 =
∫∫
(φ− φs[n]) f(φ, E) dφ dE ; (11a)
〈δE[n]〉 =
∫∫
(E − Es[n]) f(φ, E) dφ dE . (11b)
Let us now assume a stationary distribution so that f(φ, E)
does not depend on t and n. Applying (11) to (8) and (5)
for small δφ one can obtain:
〈δφ[n + 1]〉 = 〈δφ[n]〉 − 4π
2ν2s
qVRF cosφs
〈δE[n + 1]〉
+ 〈ΔφRF[n + 1]〉 − 2πhRF
〈
ΔBs[n + 1]
Bs[n + 1]
〉
; (12a)
〈δE[n + 1]〉 = 〈δE[n]〉+ qVRF〈δφ[n]〉 cos φs
+q 〈ΔVRF[n]〉 sinφs + q 〈ΔVRF[n] δφ[n]〉 cosφs . (12b)
The above equations are identical to Eq. (8) and Eq. (5)
describing the motion of each particle that cannot be in-
dividually observed by the instrumentation. On the other
hand the motion of the centre of charge of the bunch, also
called dipolar motion, can easily be monitored.
LONGITUDINAL FEEDBACK SYSTEMS
Let a pick-up (see Fig. 1) measure 〈δφ[n]〉 for the bunch
in place close to the accelerating section. The signal from
the BPM can be used for the correction of the RF parame-
ters via the modulator in the feedback loop when the same
bunch crosses the accelerating section after one turn.
Proceedings of RuPAC-2010, Protvino, Russia WEPSB026
05 Beam Dynamics and Electromagnetic Fields
225
RF voltage modulation. Feedback is achieved pass-
ing the beam phase information through the loop to modify
the amplitude at the RF cavity. Let the kick ΔVRF[n] be in
proportion to 〈δφ[n− 1]〉 but ΔφRF = ΔBs = 0 :
ΔVRF[n] = −gφ qVRF4π2ν2s
u[n− 1] 〈δφ[n− 1]〉 . (13)
Here u[n] is the Heaviside step function and gφ is the gain
of the feedback loop. Consequently Eqs. (12) can be writ-
ten in the matrix form that coinsides with (9) but
M̂ ≡
⎛⎜⎜⎝ 1− 4π
2ν2s + ĝ −
4π2ν2s (1 − ĝ)
qVRF cosφs(
1− ĝ
4π2ν2s
)
qVRF cosφs 1− ĝ
⎞⎟⎟⎠ ,
where ĝ ≡ gφ tan φs. The particle dynamics is determined
by roots zk of the characteristic equation:
z2k − (2− 4π2ν2s )zk + 1− gφ tanφs = 0 . (14)
Eigenvalues of Eq. (14) for a small gφ are
z± = exp
(
−gφ
2
tan φs
)
exp(±j2πν) ,
cos 2πν = (1− 2π2ν2s ) exp
(gφ
2
tanφs
)
.
The damping regime corresponds to gφ tanφs > 0. It can
be obtained during the Bs ramp only (tan φs = 0). It
should be noted that the bunch returns to the synchronous
phase as quickly as possible without coherent oscillations
(ν = 0 ⇒ critically damped oscillator) for
(gφ)opt ≡ g∗φ = 4π2ν2s (1 − π2ν2s )/ tanφs .
RF phase modulation. Let the phase shift due to
a small modulation of the RF frequency in accordance
with (7) is chosen in proportion to 〈δφ[n]〉 but deviations
ΔVRF = ΔBs = 0 :
〈ΔφRF[n + 1]〉 = −gφ〈δφ[n]〉 , (15)
where gφ is the gain of the feedback loop. Consequently
Eqs. (12) can be written in the matrix form that coincides
with (9) but
M̂ ≡
⎛⎝1− 4π2ν2s − gφ − 4π2ν2sqV̂RF cosφs
qV̂RF cosφs 1
⎞⎠ .
The particle dynamics is determined by roots zk of the
characteristic equation:
z2k − (2 − gφ − 4π2ν2s )zk + 1− gφ = 0 . (16)
Eigenvalues of Eq. (16) for a small gφ are
z± = exp
(
−gφ
2
)
exp(±j2πν), sin πν = πνs exp
(gφ
4
)
.
Therefore the damping regime corresponds to gφ > 0 and
(gφ)opt ≡ g∗φ = 4πνs − 4π2ν2s .
Feedback loop with delay. In practice there is a delay
between the measurement of 〈δφ[n]〉 and the RF kick. For
example, one can write for (15):
〈ΔφRF[n + 1]〉 = −gφu[n− q̂] 〈δφ[n− q̂]〉 , (17)
where q̂ is the number of turns in the feedback loop de-
lay. Applying the bilateral Z-transform [8] in Eqs. (12)
and Eq. (17) it is not difficult to obtain the characteristic
equation for calculation of eigenvalues:
z2k − (2 − gφz−q̂k − 4π2ν2s )zk + 1− gφz−q̂k = 0 . (18)
φ
g
0 0.02 0.04 0.06 0.08 0.1 0.12
dτ
 / 
re
v
T
0
0.01
0.02
0.03
0.04
0.05
Delay:
= 0 q
= 5 q
= 10 q
Figure 2: Dependences of maximal damping rates on gφ
Dependences of maximal damping rates Trev/τd =
MAX |(ln |zk|)| on feedback gains gφ are shown in Fig. 2
for νs = 0.005. It should be emphasised that g∗φ values
decrease with the growth of q̂ values.
ACKNOWLEDGEMENTS
The author would like to thank T. Linnecar, A. Blas and
P. Baudrenghien (CERN) for valuable discussions.
REFERENCES
[1] W. Schnell. “Equivalent Circuit Analysis of Phase-Lock
Beam Control Systems”. CERN 68–27, ISR Division,
CERN, Geneva, 17 July 1968.
[2] P. Baudrenghien. “Low level RF systems for synchrotrons”.
CAS – CERN Accelerator School: Radio Frequency Engi-
neering, Seeheim, 8-16 May 2000, pp. 146 – 209. CERN
2005-003, Geneva, 2005.
[3] M. Lonza. “Multi-bunch feedback system”. CAS – CERN
Accelerator School: Digital Signal Processing, 31 May – 9
June 2007, Sigtuna, Sweden, pp. 285 – 330. CERN–2008–
003, Geneva, 2008.
[4] H. Bruck. “Accélérateurs Circulaires De Particules”. Insti-
tute National des Sciences et Techniques Nucléares Saclay.
Presses Universitares de France, 1966.
[5] I. M. Kapchinsky. “Theory of Resonance Linear Accelera-
tors”. M.: Atomizdat, 1966.
[6] G. Dôme. “Theory of RF acceleration and RF noise”. CAS
– CERN Accelerator School: Antiprotons for Colliding-
beam Facilities, 11–21 October 1983, Geneva, Switzerland,
pp. 215 – 260. CERN–84–15, Geneva, 1984.
[7] Alexander Wu Chao (Editor) and Maury Tinger (Editor).
“Handbook of Accelerator Physics and Engineering”. World
Scientific, 1999.
[8] Alan V. Oppenheim and Ronald W. Schafer, with John R.
Buck. “Discrete-Time Signal Processing”. Prentice-Hall,
2nd edition, 1999.
[9] V. M. Zhabitsky, I. L. Korenev, L. A. Yudin. “Transverse
Feedback System with Digital Filter”. Proceedings of the
1993 Particle Accelerator Conference, 17–20 May 1993,
Washington, USA, pp. 2543–2545. IEEE, 1993.
WEPSB026 Proceedings of RuPAC-2010, Protvino, Russia
05 Beam Dynamics and Electromagnetic Fields
226


DIGITAL LONGITUDINAL FEEDBACK SYSTEMS IN SYNCHROTRONS

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DIGITAL LONGITUDINAL FEEDBACK SYSTEMS IN SYNCHROTRONS

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