IEEE Tm.nbaotio~
on Nuchvc Science, VoLNS-24, 110.3, June 1977
ELECTRONICS FOR THE LONGITUDINAL ACTIVE DAMPING SYSTEM
FOR THE CERN PS BOOSTER
B. Kriegbaum and F. Pedersen
CERN, Geneva, Switzerland
Summary and Introduction
Precisely tracking band-pass filters centred at the
sixth and seventh harmonic of the revolution frequency
are required’). Duringtheaccelerating cycle of0.6 set
the frequency changes by a factor 2.7.
The resulting
tracking problem is solved by active two-path filters,
where the centre frequency is governed by the frequency
of a pair of sinusoidal signalsinquadrature, which are
generated from the accelerating RF frequency (fifthhar-
manic) by means of a phase-locked loop and a loop-
controlled phase shifter.
The phase change caused by
the large frequency sweep (6 or 7 MHz) in conjunction
with the delay in the feedback loop (cables, etc.) is
compensated by a digital system, which computes the re-
quired phase advance from the value of the RF frequency
and controls digitally the phase shift of the two-path
filters.
A low-frequency quadrature VCO is made to
track the synchrotron frequencyorharmonics hereof from
analogue information about bending magnet field (momen-
tum) and RF voltage.
This quadrature pair ensures
tracking of single sideband filters which permit each
individual mode sideband to be examined throughout the
cycle.
A drive system can, by means of a similar VCO,
generate any desired mode sideband, and thus excite any
given mode.
Active Damping System
Filters
Fig. 1 Transfer functions and equivalent impedance
A longitudinal pick-up signal is passed through two
band-pass filters and added to the RF signal driving
the accelerating cavity (Fig. 1).
This feedback loop
representsanartificial coupling impedance, which must
have apcsitive real part for frequencies slightly above
the sixth harmonic and a negative real part below it,
to provide damping.
With a similar impedance around
the seventh harmonic added,
dampingofthe four coupled-
bunch modes n = 1 to 4 with within-bunch mode numbers
m = 1 to 3 is obtained]).
The transfer functions Hs
and H7 required to obtain this impedance depend on the
feedback path delay tc
and the pick-up to cavity travel-
ling time for the beam tb:
tb
= T0 x e/271 = 8/w,, ,
(1)
where wg is the revolution frequency.
The impedance 2
is defined by the ratio between cavity voltage and beam
current at cavity:
I
b,cav(jw> = Ib,pu(jw) em C-jut,)
V
cav(jd
= H(jw) Ib pu (jd exp (-jatc)
(2)
Z(jw)
=v
cav(jw) jxb,cav(jO)
so the required transfer function becomes
H(jw) = Z(jw) exp [jw(tc-tb)) = Z‘(jw) exp’ (j$) . (3)’
For frequencies near the sixth and seventh harmonics,
w = kwa, k = 6, 7:
$ = kwotc - k0 ,
(4)
so the phase of Hk with respect to Zk must be offset kt3
and advanced proportional to the revolution frequency.
1
;:o,+f : f
/
vJ--b: b
Zb
k.k t
z*c
\$--A,
*
1
fqkw,, , k.6.7
W.
Fig. 2 Two-path active filter with phase-shift control
The transfer functions Hs,H7 are realized by two-
path filters*),
whose centre frequencies are controlled
by the frequency of a pair of sinusoidal signals in
quadrature (90’ relative phase shift). The use of two
paths eliminates the undesired sideband at the output of
the second mixer (Fig. 2).
Only frequencies present at
the input will be present at the output, so the
system
behaves as a linear filter.
Figure 3 shows how the two-
path filter transfer function H(s) is related to G(s),
the fixed filters between the mixers.
A low-pass to
band-pass transformation takes place;
the G(s) pole-
zero cluster becomes two clusters around ?jw,. As
IG(s-jwc)l>> /G(s+jw,)I for s = jwc, the phase $ of the
output mixer quadrature pair (az, a$) can be used to
control the phase shift of the two-path filter in the
band-pass region.
The variable phase of H(s) shows up
in the pole-zero plot as a number of real zeros, whose
positions depend on @. The purpose of the zero at the
centre frequency jw, is twofold. Firstly the 180’ jump
in phase is obtained, and secondly the unequal bunch line
is suppressed by the notch in the amplitude response.
G (9
I
7
.cM
I/\,
w
w
l!k
S-pkTW
i,.rlf
40
0
0
”
-90
Hk)r+(Gts-jqk
J~+G(s+~t)c-i*)
~~~ -4;
Me a-w
I
^ -)n
4b
~f$% ~*
zeros
Fig. 3 Low-pass to band-pass transformation
The required phase advance could be obtained by
generating the input mixer quadrature pair (al, a:) from
the output pair (a2,
a?) through cables of length tc, so
(a2,
a$) would have a leading phase o = kwstc.
However,
for tC =
1.2 Wsecitwould require 4cables, each approxi-
mately 250 m long, so this solutionwas abandoned. Instead
1695
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a digital phase control in steps of 90
'
is used.
A 45'
error in phase is acceptable as it only reduces the
damping effect by a
For 90' steps, the required
(a~, a$) has a simple relation
to
(al, a:) (Fig. 4). A
simpler realization is, furthermore, obtained by commu-
tating the low-frequency inputs (y, y*) to the output
mixers and leaving (az, a$) fixed, (as, a$) = (al, a?).
4 Fig.
L
4
“1
I
ai
‘“1
0,v l o;Y=
o,Y=a;(-Y)
a,(-Yl+ o;(-YY
q-v’) l o;v
Two-path filter with digital phase control
The two bits controlling the four-position switch
are generated by a counter,
which repetitively counts
the accelerating RF frequency during a fixed count time,
which is set to match the required phase advance rate
(Fig. 5).
The preset value of the counter priortoeach
counting period gives the required phase offset k9. The
counter overflows each time the required phase advance
exceeds 2n.
The count result is storedina Z-bit buffer
register while the next counting takes place.
a offset (4bits, 2205/step)
Fig. 5 Digital phase advance control
The quadrature pairs are generated by the quadra-
ture revolution frequency generator2'3) (Fig. 6).
The
frequency of a VCO divided by k is phase-locked to the
accelerating frequency divided by h = 5. As the fre-
quency of the two inputs to the phase discriminator are
equal fk/k = fRR/h,
the output frequency becomes fk =
= kfR-f/h.
The VCO (varicap-inductor) gives asinusoidal
output of high spectral purity and low distortion. The
amplitude is kept constant by an AVC loop. The VCOout-
put is passed through a voltage-controlled phase shifter
(varicaps) to produce the quadrature component.
The
phase of the two outputs are compared in a phase dis-
criminator,
which controls the phase shifter to give
90' phase shift independent of frequency. A relative
vector error of less than 1% has been achieved (AA <
-t 0.1 dB, &$ < 006), so a suppression better than 40 dB
of
the
undesired sideband in the two-path filters can
be obtained.
ra.&k’
Divider
Fig. Quadrature
revolution
frequency generator
A pick-up AVC*) keeps the peak'value of the bunch
signal at a fixed level to avoid saturation of the in-
put mixers in the filters (Fig. 7). The outputs of the
filters are added to the RF signal driving the acceler-
ating cavity.
Although the impedance of the cavity at
the sixth and seventh harmonics is 40 to 50 dB below
its resonant impedance (fifthharmonic), sufficientvolt-
age (So-100 V) can be obtained to get damping rates
several times the highest growth rates’).
The maximum
gain is determined by noise and maximum avajlable volt-
age, as the noise must not saturate'the system. The q
fact that the accelerating cavity was used for feedback
has greatly reduced the cost, as only low-level elec-
tronics has been built.
The cavity has also influenced
the choice of harmonics').
The phase offset of -90°,
caused by the capacitive cavity impedance at the feed-
back frequencies,
is compensated by the phase-advance
control previously described.
5;
L
Fig. 7 Active damping system (per ring)
Mode Analysis System
Sixmodesidebands (Fig. 1) are treated together in
each active damping filter. Single sideband filters2v3)
with narrow bandwidth permit the examination of these
individually (Fig. 8).
This is important for growth
and dampin
P
rate measurements
with
or without the
damping on ).
b,=elqt
t.ZDkHz IF-
”
’ Mixers Lowpass
Mixers
S!$j&
* Y8 2
-; D
%
r
.
t
IFI
( Lc=$Lin
&kc+,
q i 2n20k+mws
tit+
2r2Dk
XI
f
T
* )
YI +
I
Y w
*
Dut-
;xy )w f-g--y
I
t
w
%
I
Fig. 8 Single sideband filters, mode analyser
The mixing processes are similar to the two-path
filters except that the output mixers are fed by a low-
frequency quadrature pair,
whose frequency is related to
to the synchrotron frequency as follows:
fL = 20 kHz +
+ mf,. The desired sideband will then appear at the
output as a fixed frequency, 20 kHz. The lower sideband
comes
out
as 20 kHz in the sum output and the upper side-
band as 20 kHz in the difference output. Fixed 20 kHz
band-pass filters then eliminate the undesiredsidebands.
1696
The amplit:lde can be sbserved at the output of a detec-
tor ,
which can operate
in either iinear
Jr
logarithmic
node.
Sideband amplitudes from approximately 30-95 dE
below the main RF line can be observed.
Fig. 9 Quadrature VCO, 0 .?-40 k;iz
The low-frequency quadrature pair is generated by
a quadrature VCO ‘) (Fig. 9). A fixed frequency quadra-
ture pair is mixed with a sinusoidal signal from a VCO.
A quadrature pair of variable frequency is produced by
the difference mixing product.
The sum frequency is
suppressed by low-pass filters. A frequency discrimi-
nator stabilizes the output frequency and linearizes the
voltagr-to-frequency transfer function.
An AVC loop
keeps the output amplitude constant. The quadratureVC0
is
controlled by the synchrotron frequency programmer2)
to tiscillate at 20 kHz + mfs (ai&. 10).
A voltage pro-
portional to the synchrotron
frequency is produced as a
non-linear filnction of IB (bending magnet current LX mo-
mentum) multiplied by the square root of the cavity
voltage.
Higher harmonics can be selected by a switch,
and a voltage equivalent to 20 kHz is added.
Non-l~neor
function
. 10 Synchrotron frequency programmer
Fig
Ihe complete mode analysis system (Fig. 11) pro-
vides parallel observation of all four coupled-bunch
modes n = 1 to 4 with switch selectionof the within-bunch
number m= 1 to 3. Octupole modes (m= 4) have been ob-
served on an experimental basis.
A multiplexing system
permits the use of the system
with any of the four Booster
rings.
The second quadrature VCO oscillates at mf, and
is uses in the drive system described below.
Ig (momentum)
VRF(CQV1ty volt )
-Ciuad Rev Freq Generator
lode Outputs
Fig. 11
Node analysis system
Drive System, RF inock-oat
TJ take full advantage of the mode analysis system,
it is convenient to be a51e to excite a single node.
This
is lone by driving
it at its corresponding frequency
kc., >
t Tms.
This sideband is generated by adding a low-
frequency (mws) quadrature pair to the inpu:s of theout-
put mixers of the active damping filters (Fig. l.?).Upper
or lower sidebands are selected by the sign cf the cl-c’:
phase.
c, =c
Jmwst
a,,=ew
I
,~,,i(mw,t~~) a~m=ei(qt-~)
b, z,lwtt
I
Fig. 12 Drive system
Fig. 13 Restoring drive fre-
quency from IF.
RF knock-out consists ofmeasuringamplitudeandphase
responseofthe beaminthe neighbourhood of a mode sideband
resonance.
The mode analysis and drive systems are used
to do this. To obtain the Ihase,
the 20 kbz IF (inter-
mediate frequency) of the mode analyser has to be nixed
back to the driving frequency (Fig. 13).
Originally,
a voltage ramp was applied to the quad-
rature VCO to sxeep across the desired frequency range.
This method was abandoned because the constant drive level
was too high when the frequency was inside the band of
incoherent synchrotron frequencies, where the amplitude
response is strcng,
and too low when outside, where the
amplitude response is xeak. Also the bands;idth of the
mode analyser (= 700 Hz) was too wide compared with the
desired frequency resolution, .If,=vm=2C-40 Hz, so
the signal-to-noise ratic was lower than necessary.
In the second and more successful scheme the quad-
rature VCO was phase-locked to an HP network analyser,
which was used to measure phaseandamplitude response.
The frequency was changed manually and the drive level
readjusted for each frequency.
A Tektronix DPO (digital
processing oscilloscope) was used to collect thedata and
convert them to a stability diagram ‘) .
Averaging over
several cycles reduced the signal-to-noise ratio.
The
typical drive levels used correspond to 0.1 RF degrees
centre-of-mass motion for the dipoie mode.
For fixed frequency drive the frequency resolution
is kfr = l/t,xc,
where text is the excitation time. Beam
dynamics considerations4) show that the maximum drive
level has to be scaled as A,,, 0: (Of,)‘.
I3y taking into
acccunt that a bad signal-to-noise ratio has to be im-
provedby averaging, the
total measurement time scales ‘1s
tr%
a Cfdet/(&fr)6,
where hfdet is the detector bandwidth,
/Lfdet 1 ;ifr.
A proper choice of drive level and excita-
tiontime (resolution) is therefore crucialforagood result.
Acknowledgements
The authors want to thank F. Sacberer for his
valuable comments and steady interest in the project,
and X.R. Reich for his support and encouragement.
P. Asboe-Hansen has made valuable contributions to the
development of the mode analyser and quadrature revolu-
tion frequency generator.
References
1) F. Pedersen and F. Sacherer, Theory and performance
of the longitudinal active damping system for the
CERN PS Booster, these proceedings.
2) B. Kriegbaum and F. Pedersen (unpublished).
33 P. Asboe-Hansen, Lnt. Rep. CERX/PS/OP 76-6 (1976).
$1
H.G. Hereward, CERN 65-X (1965).
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