BEAM DYNAMICS EXPERIMENTS USING TRANSVERSE FEEDBACK SYSTEM
Takao IEIRI, Kazuhisa NAKAJIMA, Kaoru YOKOYA and Yoshihiko MIZUMACHI
National Laboratory for High Energy Physics
Oho-machi, Tsukuba-gun, Ibaraki-ken, 305, Japan

ABSTRACTEXCITATION OF BETATRON OSCILLATION

The transverse feedback system with a positive
feedback method can induce coherent betatron oscillation.
The damping rate of betatron oscillation wasmeasured using the method as a function of beam current
and chromaticity.
The growth rate of the head-tailinstability is estimated from the damping rate, sincethe growth rate is equal to the damping rate at thethreshold.
The transverse wake function is obtained
from the growth rate as a function of bunch length.

It is necessary to excite stable coherent betatronoscillation for the experiments of transverse beam
dynamics.
Positive feedback of the system was testd
for this purpose.
As we increased the feedback gain,
the system began to oscillate with the fractional betatron frequency which was easily read by a frequency
counter.
The amplitude of the self-oscillation has
been controlled by the gain and phase of the feedback
loop.
The phase for maximum amplitude was an antipole
to that for maximum damping rate. The threshold of theoscillation depended on beam current.
The self-oscillation has proved to have fluctuation of ± l kHz about
the betatron frequency, which is not the tune-spread.
The origin of the fluctuation is not clear. The amplitude seen in Fig. l corresponds to a peak-to-peak valueof 0. 5 mm.
An amplitude modulation has been observedwith frequency of several 10 Hz which may be related tosynchrotron motion.
The modulation appears when theamplitude is relatively large.

INTRODUCTION
The transverse feedback systeml) enhances damping
rate of coherent betatron oscillation in addition tothat of beam itself. The damping rate is controlled by
feedback gain of the system as shown in Fig. 7 of Ref.
l. When the feedback gain is relatively low, however,the damping rate depends on beam current and chromaticity.
If the system is switched to a positive feedback
loop and set in suitable gain,
stable betatronoscillation is induced without an external force.
Exponential decay of the oscillation is observed justafter the self-oscillation due to the positive feedbackis switched off. The damping rate can be measured fromthe observation.
When a chromaticity parameter is negative in theAR, the head-tail instability occurs.
The transverse
feedback system cures the head-tail instability of thelowest mode m = 0.
Transverse displacement signals
from the instability are detected through pick-up
electrodes,
amplified and fed back on deflection
electrodes with a phase shift as to counteract thebetatron motion.
The instability occurs, when thegrowth rate exceeds the damping rate.
We can estimate
the growth rate of the instability from measuring thedamping rate.
Transverse wake field is obtained fromthe growth rate, if the other parameters contributing
to the growth rate are known.
BASIC EQUATIONS

Fig. l The self-oscillation. The frequency
shows the fractional betatron frequency.

The gro~Jh rate of the head-tail instability for m
0 is givenz as follows:

A nonlinear damping effect which depends on amplitude of oscillation is introduced in order to explain
the bahavior of the self-oscillation.
Assuming thatthe motion obeys the van der Pol 1 s equation shown in
Eq. (A4) in Appendix, the amplitude of the oscillationin stationary state is(l)TI:electric charge
rest mass of electron
relativistic factor
angular betatron frequency
momentum compaction factor
chromaticity
bunch length r.m.s. valuetransverse wake function inunit azimuthal length
beam current of a bunch

XQ

=2t

fo -a effok(3)The derivation of Eq. (3) is seen in Appendix, whereaf and a ff are antidamping rate due to the feedbacksy~tem ana {he damping due to beam itself for small
amplitude, respectively, and k is a constant describing amplitude-dependent damping as seen in Eq. (A3).
The amplitude as a function of feedback gain is shownin Fig. 2.
The measured value almost agrees withthe calculated one using Eq. (3), where k is 400
12
sec- ·mm- •The transverse wake function wo makes transverse
kick propotional to a displacement.
The relation to
the coupling impedance is expressed as

MEASUREMENT OF DAMPING RATEwo (cr)(2)

A decaying oscillation is observed just after theself-oscillation is switched off.
The oscilla)ion isdetected by the bunch oscillation detector 1 •An
output signal from the detector is displayed on anoscilloscope.
Time constant of exponential decay ismeasured from the display.
The damping rate is anwhere I (Z (w)) is an imaginary part of the transverse
impedan~e.
The wake function is a function of bunch
length.

160

M0717(sec)_1r-----------------------------,
lb= 3.5±0.5mA
Vc:1.9 MV(mm)4.0--x--oalcololed ooe(!)

Imeasured value300x+

3. 0"0::J<110 200

E- z.o01

0c

Q.

+EcJ

u100

10feedback gain
Fig. 2(dB)

Amplitude of self-oscillation vs feedback gain.
The valve of 0 dB for the feedback gain corresponds to the threshold of the oscillation.

~radiation

inverse of the time constant.
The damping rate observed is much faster than theThe horizontal
radiation damping rate of 23 (sec) - 1 .
damping rate is measured as a function of the chro= i; (1, p/p) and beammaticity parameter defined by !:lv
current as shown in Fig. 3 and 4:" Thl damping rate isroughly propotional to chromaticity and beam current
when bunch length is constant (o = 1.1 em).
These phenomena cannot be explained by the damping
mechanism due to momentum-dependent tune-spread produced by sextupoles, because each particle has sameAmean momentum during a synchrotron oscillation.
head-tail damping effect is considered in order to exThe head-tailplain the result of the measurement.
effect is damping if i; > 0 and growth when i; < 0 as
The damping and growth rates
expected from Eq. (l).
are equal when only the sign of chromaticity is changed
under the condition that other parameters are same. An
effective damping rate a ff. is the sum of those ofradiation damping a 0 and h~aa-tail damping a 1 ,damping rateo~,--'--'---'--'--~--L-'---'--'---!--:---'---'·10.5

0chromaticity
Fig. 3Damping rate vs chromaticity. The solidline is calculated valve using eq. (4).

(sec)-,r---------------- ------------------------- s_4_o_''-'--,
2005 :+1.5±0.5
Vc =1. 9 MV

I01I

6. 100

EIcJuI(4)damping rate

The value of a1 is obtained from the horizontal growthrate of the head-tail instability described in the next
The value of a ff is shown
chapter using the Eq. (1).
in solid line in Fig. 3 and 4 to be compafed with themeasured one. These two values roughly agree with each
When beam current and chromaticity are relaother.
tively high, other second-order damping mechanisms
should be considered.

00105

beam current (rnA)

Fig. 4Damping rate vs beam current. The solidline is calculated valve using eq. (4).

Bunch length is inverse to a square root oftheory.
Ascavity voltage, before bunch lengthening starts.
cavity voltage is increased or bunch length shortens,
the slope seen in Fig. 5 becomes steeper, which shows
an increase of the wake function defined in Eq. (2) .
The wake function is shown in Fig. 6 as a function ofbunch length fogether with a calculated value reported
He calculated the transverse wakeChin 3 •by Y.
function of bellows and cavities installed in the AR.
The measured value involves total contribution fromWe should take count of aaccelerator components.
contribution from other vacuum components than thebellows and the cavities at short bunch length of 1 em.

HEAD-TAIL INSTABILITY
An experiment was carried out at the 2.5 GeV injec. tion. When vertical chromaticity is negative, a bunch
begins to oscillate in vertical direction at threshold
The oscillation grows up and causes beam
current.
Beam current is limited by the instability.
loss.
Maximum beam current as the threshold value has beenmeasured as a function of feedback gain of the system.
The relation between the feedback gain of the systemand the vertical damping rate has been measured underThen,the same condition using the injection kickers.
we obtain the relation between the growth rate and thethreshold current, since the growth rate is equal tothe damping rate at the threshold.
The result is shown in Fig. 5. The growth rate is
propotional to threshold current as is predicted by theCONCLUSION
Damping and exciting of betatron oscillation are
surveyed in the AR using the transverse feedback sys-161

other vacuum components than the bellows and thecavities should be taken into account in order toexplain the difference between the measured andthe calculated values.

(sec) 1 r-------------------------------------s-•o_7_2_1~---o--- Vc =2.5 MV

REFERENCES

Vc = 2.0 MV

1000

1.

- • - Vc =1.0 MV-·-.t~.--Vc = 0. 5 MV

2.
.r.

3.

3:

0a, 500

T. Ieiri, Y. Mizumachi and J. -1. Pellegrin, "TheTransverse Feedback System for the TRISTAN AR",contributed to this conference.
A.W. Chao, "Coherent Instabilities of a Relativistic Bunched Beam", SLAC-PUB-2946, June 1982 .
Y. Chin, "Transverse Impedance of Bellows in theAR", (in Japanese) TN-84-009, 1984.
Appendix0u~

An equation of motion without the transverse feedback is'QJ>

d 2xdxCftZ + 2a.eff dt +a.eff:v:
wo:Fig. 5

Vertical growth rate vs threshold current
with a parameter of cavity voltage.
Vertical chromaticity is s
9.6.

(vwo)2_(A1)x - 0effective damping rate
fractional part of the tuneangular revolution frequency

We take count of the feedback effect.
Negative feedback increases damping and positive feedback produces
antidamping. Then, the equation isy840721

(A2)

00::

The + sign in the bracket in Eq. (A2) is damping andthe - sign is antidamping.
Assuming that the damping
rate depends on an amplitude due to nonlinear fields,
we putl<(...... 10000Irti

10c

(A3)

Xc

where a.f and a. ff are damping rates for small amplitude and~ is a Eon~tant coefficiency. The equation ofmotion for positive feedback is::::J<1.>-"'Cl~TX
15000(A4)

<1.>
Ul

.....

<1.>

The Eq. (A4) is known as the van der Pol's equation.
An amplitude in stationary state is obtained from Eq.
(AS) .

>

UlcCl
.....
.....

{(a. fo - a. ) - kx 2 #effo
dt· d xone revolution
J00Fig. 6

tern.

1.0

2.0

bunch length(]' rms (em)We put x = xosin(vwot),
Eq. (A4) is much larger
quation (AS) becomes0(AS)since the oscillating term inthan the damping term. The e-Transverse wake function vs bunch length.
The solid line shows calculated value forbellows and cavities.

0The following is remarked.

From the above equation we obtain

1) The damping rate of a single bunch depends on beam
current and chromaticity.
The effective damping
rate is considered, which is the sum of radiation
damping and head-tail damping rates.
2) The self-oscillation of betatron is carried out by
positive feedback of the system. The oscillation
has the amplitude determined by Eq. (3).
This
technique is used for the measurement of the effective damping rate.
3) The transverse wake function of the AR is measured
as a function of bunch length from the growth rateof the head-tail instability. A contribution fromXQ

1622(A6)