The 12th Symposium on Accelerator Science and Technology, Wako, Japan 1999

Robinson Instability under Cavity Voltage Feed-backObservation at the SPring-8 Storage Ring
T. Nakamura, T. Takashima and T. Ohshima
SPring-S
1-1-1 Kouto, Mikazuki-cho, Sayo-gun, Hyogo 679-519S, Japan

1 Introduction
The observation of the Robinson instability undercavity voltage feed-back loop was performed at theSPring-S storage ring and the results were comparedwith the analysis in [1]. First, we describe the application of the theory of the Robinson instability undercavity feed-back loops based on the reference [?] to The
SPring-S storage ring and show the results of the calculation. Then we show the results of the observation andthe comparison with the calculated results.

2 Calculation for the SPring-8 Storage Ring

The static part of phasers of cavity voltages areshown in Fig. 1 where Zb = -2Io and \i = Z(wRF )zb =- 2IoRL cos 'lj;e''l/J are the static part of the beam
current[2] and the beam induced voltage,respectively,
and 10 is the stored current. We do not show the suffix
0 which was used to tag static part of parameters in [1]and we use ¢ instead of ¢c in [1].
The resonant frequency Wr is controlled by the cavity
tuner control system to keep !::,. 'lj; and Vc = IVc I is controlled by the amplitude feed-back loop and ¢ should
fulfill the equation Vc cos¢ = Uo to compensate the energy loss of the beam. Now we have Vb, Vc, ¢and l::,.'lj;.

The parameters of the SPring-S storage ring andsymbols used in this paper are shown in Table 1. TheSPring-S storage ring has three RF stations(B,C,D).
Each station has one klystron, eight single-cell RF cavities and two feed-back loops, one for amplitude and one
for phase of sum of voltages of cavities.

Im

Table 1SPring-S Parameters
Electron Energy
Eo
RF Frequency
fRF
hHarmonic Number
Radiation Loss/TurnUoAcceleration Voltage/Turn
Vc
Max. Stored Current
Io
Synchrotron Frequency
fs
Number of RF stations
Ns
Number of Cavities/StationCavity Parameters
Shunt Impedance
Rc
Unloaded Q
Qc(3Coupling Factor
l::,.'lj;
Detuning Phase Angle

7.975
50S.5S
2436
S.91
12100
"' 1.5
3GeV
MHz-ib = -2loMVMVrnAkHz

u 0 = Vc cos<!>
Fig. 1 Phaser Diagram of the Static Voltages in RFCavity. The positive direction of the angles iscounter-clockward.
Glancing Fig. 1, we haves
3400002-5

MDanddeg.

tan'lj;2.1 Static Voltages in Cavity
We have to know the impedance of the cavity bysolving the phaser diagram of the static voltages in acavity. Every cavity in each station has the same parameters and the sum of these cavity voltage is controlled by
two feed-back loops, one for amplitude and one for phase.
Thus, we treat them as one cavity whose impedance isS times of a actual single cavity.
A cavity has a resonator type impedance of the form

where Wr is the resonant frequency and Q L = Q c j (1 +(3),RL = Rc/ (1 +(3). Wr is dynamically controlled by a tuner
control system and (3 is a coupling factor shown in Table
1.

=2IoRL
1tan!::,. 'lj; - - - A - - - sin (¢ - !::,. 'lj;) (3)cosu'lj; Vc

where we used Wbl = IZzbl = 2IoRLcos'lj;. The resonant
frequency Wr is obtained through tan 'lj; and its definition
Eq. (1).

2.2 Feed-Back Controller
Each feed-back loop has one feed-back controllers
which consists of a filter circuit and an amplifier of gain
Go. The filter circuit is shown in Fig. 2 and the firststage of the circuit is a high-pass filter and the second
is a low-pass filter. The matrix element of the filter for
one feed-back loop isZt =1iwCL(ZH + R) + 1'

ZH =RH
1 + iwCHRH(4)

where R = lkf2 and the other parameters are shown inTable 2.

-242-The 12th Symposium on Accelerator Science and Technology, Wako, Japan 1999

Q)

Fig. 2 The filter circuit Z f in a feed- back controller ofthe SPring-S storage ring. The values of RH,
CH and CL are listed in Table 2.
Table 2Filter Parameters. The raw k is for amplitude
feed- back loop and the raw ¢ is for phase.
Filter A
CH = Op,F
CL = 24.7p,FRH/k~l

at Station
DB
101010101010
1.1 1.1 1.1
1.1 1.1 1.1
1.1 1.1 1.1
101010101010
1.1 1.1 1.1cset
A-1k¢
kA-2

¢kA-3

¢kA-4¢"g 0.1.<;::;

10.01

Filter B
CH = 10nF
CL = 4.7p,FRH/kfl
at StationDBset
B-1 k 10 10 10¢ 10 10 1010 10B-2 k 3¢ 10 10 10103
B-3 k 3¢ 10 10 1010Frequency I HzFig. 4 Loop gain of the feed-back loop without beam
current for the parameters listed in Table 2c

The calculated growth rates are shown in Fig. 5 andFig. 6 for parameters in Table 2.

100-At the experiment, we used two set of CH andCL;Filter A and Filter B, and used several set of valueof RH for six feed-back loops;A-1,2,3,4 for Filter A andB-1,2,3 for Filter B, as shown in Table 2.
The frequency response of the feed- back loop at zero
beam current is shown in Fig. 3 and Fig. 4 for Filter A
and Filter B with several values of RH. Filter A has narrower frequency range compared with Filter B becauseof the large value of C£.

.;

rJl..._-''..._0.....t1.l~-100.s~-200~

0

1-<tJ...

,.;~-300-400,.;

,.;

,.;

,.;

--......

-- ---//

A-20Stored Current I AFig. 5 Growth rate vs stored current for filter A. The
growth rate includes radiation damping.
The growth rate is positive above 0.6 A with theparameter B-3.
With the parameter A-3, the gain of amplitude feedback loop at the synchrotron frequency is larger thenA-1 and growth rate is larger than A-1. On the other
hand, with the parameter A-4, the gain of phase feedback loop at the synchrotron frequency is larger thenA-1 and growth rate is smaller than A-1.

-20-40oiJQ)"0..._ -60Q)
rJl'\~..c -80~•• A...... ~. R =1.1k.Q

-100

""'•

H'·'·

-120

3 Simple Model

10Frequency I Hz
Fig. 3 Phase shift of the feed-back loop without beam
current for parameters listed in Table 2.

2.3 Growth Rate and Frequency Shift
We use the eigenvalue equation Eq.() and Eq.() in[1] to obtain the growth rates and the frequency shiftsof the instability. At solving these equations, we haveto care that C, R, G 0 and Zt in them have frequency
dependence and we used an iteration scheme to solve it.

We show a simple model to explain the effect of thefeed-back loop on synchrotron motion of a beam.
We assume that Q value of a cavity is lower enoughso that the cavity voltage respond immediately to thechange of driving current such as ib or i 9.
Assume the beam executes synchrotron motion. If
the beam has a shift of the timing T < 0 as shown inFig. 7, this delay produces shift of the beam current,b.~b· This produces shift of the cavity voltage, b. Vb.
The feed-back loops try to compensate this shift; theamplitude feed-back loop produces b. V9 k which reduceenergy gain of beam and the phase feed- back loop pro--243-

The 12th Symposium on Accelerator Science and Technology, Wako, Japan 1999

200by a phase detector.
The observed frequency response of the synchrotron
motion is shown in Fig. 8. As the stored current increases, the peak height become higher and the resonance width become narrower which means damping
time is longer because instability growth rate cancel theradiation damping rate. Above stored current 0.6 A; thelarge amplitude of the synchrotron motion was observed
without external excitation.
With A-1, which is the nominal parameter of thestorage ring, the growth rate is far below zero and thesynchrotron sideband at RF frequency is less than -60dBof main peak.

100,-...

......
"'......

0Q)

-100

'-'
......

~

0::

-200
..c:.......
~

0

1-<

0-300
-400

Stored Current I A
Fig. 6 Growth rate vs stored current for filter B. With
B-3, Growth rate exceeds 0 at stored current
more than rv 0.6A.
duces ~ V9 ,p which increase energy gain of beam. If thephase delay of the feed-back loop is 0, the total energy
gain of the beam during the energy shift 8 < 0 in synchrotron motion of the bunch or that during 8 > 0 is
0. In actual case, the feed-back loops have a phase delay as shown in thus total energy gain of the beam by~ Vgk during 8 < 0, is negative and the total energy gain
during 8 > 0 is positive hence the amplitude feed-backloop excite the synchrotron oscillation. On the other
hand, ~V9 ,p increase the energy gain during 8 < 0 anddecrease the energy gain during 8 > 0, which means thephase feed-back loop damps the synchrotron oscillation.
Imh<ol-ac~

lt'~/ ::4ilVgk

ilVb///.-.-- .--.-- .-- LlVgk/'\..

_ilVg<l>ReFig. 7 Phaser diagram of the small shift of the cavity
voltage. iii, is produced by small timing delayof the beam for T < 0. V9k and Vg,p are voltage
produced by the amplitude and the phase feedback loop to iii,. The static part of voltages are
the same as Fig. 1.
From [1], the amplitude of synchrotron oscillation ofthe beam, I<PI, is Qs = !TEWS rv 20 times larger than themodulation of the cavity voltage 1¢1 or lkl which drives<f). The loop gain of the feed-back should be smaller than
1/Qs at the worst case that phase delay is 90 deg. But
in case of Filter B, the gain at fs is still higher than
1/Qs and may become unstable.
4 Observation

The experiment was performed with the parameterB-3. The synchrotron motion of the beam was excited bya phase modulation on the RF acceleration voltage andthe phase oscillation of the beam current was observed

Response of Synchrotron Motion
,-...

0.002

0.2'"0~P>--<"'s~0e o.150.0015Q)"'::l='

J.-;::

0.001

0.1s'E.
.....
S"
0.
~,-...

,_0\Q)"' 0.05..c:p.,0.0005

C<l0;J>

0~

~
0~

'-'

1450

1500
1550 1600
1650
frequency (Hz)Fig. 8 Observed frequency response of the amplitudeof phase oscillation of the beam with Filter B3. The data for 60mA is 2 magnitude larger
than the data at 16mA and 40mA. The damping time/rate of the synchrotron motion fromthe width of the data are 5.5ms/182s- 1 for
16mA, 12ms/83s- 1 for 40mA and the beam
was unstable at 60mA. These values are the expected from the theory as shown in Fig. 6. The
peak height is also proportional to the dampingtime.
5 Conclusion

The effect of the cavity voltage feed- back loop on theRobinson instability was analyzed and growth rate andsynchrotron frequency shift is obtained. It shows that,in actual machine which has slower synchrotron oscillation frequency and large beam loading like the SPring8 storage ring, the frequency response of the feed-backloop must be slower enough to suppress gain at the synchrotron oscillation frequency to get stable operation athigh current.
The author thanks Dr. N. Kumagai, SPring-8, whosuggested the existence of the effect of the feed- back loopon the synchrotron oscillation of the beam and encouraged us to search parameters for stable operation.
References[1] T. Nakamura, "Robinson Instability under Cavity
Voltage Feed-back", This proceedings.
[2] P. Wilson, SLAC-PUB-2884(1991),SLAC.

-244-