Beam Phase Loop - CERN Accelerator School

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Transcript Beam Phase Loop - CERN Accelerator School

Beam Loading and Low-level RF Control
in Storage Rings
Alessandro Gallo, INFN - LNF
Lecture II
• Slow Servo Loops;
• Beam Phase Loops:
– Basic loop with differentiators;
– Basic loop with integrators;
– Beam phase loop and modulation transfer functions.
• Beam Loading Compensation schemes:
– Feedforward technique;
– Direct RF feedback.
• Gap Transient
• Transition Energy Crossing
• Slow servo loops
AMPLITUDE LOOP
The automatic regulation of the generator
output level can be obtained by
implementing amplitude loops. These are
feedback systems which detect and correct
variations of the level of the cavity voltage.
If the power amplifier is not fully saturated,
the regulation can be obtained by controlling
the RF level of the amplifier driving signal.
If the amplifier is saturated, the feedback
has to act directly on the high voltage that
sets the level of the saturated output power.
Referring to the reported model, the loop
transfer function can be written in the form:
Ac
H ( s)

Aref 1  H ( s)
with H ( s) 
Beam
Ain
K C ( s) G( s)
Vmod
For little cavity detuning (QLd << 1) the cavity
response to an amplitude modulated signal is a
single pole low-pass:
C ( s) 
C0
1 s 
with  
c
2QL
AMPLITUDE LOOP
In heavy beam loading regime the cavity detuning is likely to be as large as the cavity
bandwidth or even more, so that the single pole model for the cavity amplitude response
is inappropriate. The completed Pedersen model applies in this case, involving crossmodulation blocks.
Amplitude loops may have bandwidths ranging
from few Hz to 1 MHz. Sometimes it is useful to
have large residual loop gain at line frequencies to
correct spurious modulation introduced by the
power stages. The gain and bandwidth can be
tailored by properly designing the error amplifier
transfer function G(s). For instance, if the cavity
single-pole model is adequate, high loop gains in
the low frequency region are obtainable by
implementing an integrator providing a zero for
compensating the cavity pole.
G(s) 
1  sCR2
1
with CR2 
sCR1

H (s) 
Ain
A KC0
K C ( s ) G ( s )  in
Vmod
Vmod sCR1
The low frequency gain can be boosted by properly treating the error signal. However,
the coupling between the various RF servo loops induced by the large cavity detuning
may result in a global instability of the beam-cavity system. To avoid it, gain and
bandwidth of individual loops have to be reduced.
PHASE LOOP
The cavity RF phase (or the power station
RF phase) can be locked to the reference RF
clock by another dedicated servo loop. The
need for a phase loop is not strictly related
to beam loading effects but more to ensure
synchronization between different RF cavities or between RF voltage and other subsystems of the accelerator (such as injection
system, beam feedback systems, ...).
Beam
The phase is locked to the reference by measuring the relative phase deviation by means
of a phase detector and applying a continuous correction through a phase shifter. For
loop gain and bandwidth the same considerations expressed in the amplitude loop case
hold.
C ( s)
H ( s)
c  in
 ref
1  H ( s)
1  H ( s)
with H (s)  kmodkdet C ( s) G(s)
c  in if
H ( s)  C ( s) ; H ( s) 1
• Beam phase loops
Beam Phase Loop
The beam phase loops are feedback systems aimed at adding a damping (frictional)
term in the synchrotron equation for the beam center-of-mass coherent motion.
In the basic scheme the phase of the beam is detected and, after a manipulation to
introduce a 90° phase shift at the synchrotron frequency, is applied back to the cavity.
Ideally, if the cavity modulation were exactly proportional to the time derivative of the
beam phase we would get:
c  k b ; b   s2b   s2c  b   s2k b   s2b  0
making a frictional term appearing in the synchrotron equation.
Beam Phase Loop: Pure Differentiator
A pure differentiator in the Laplace s-domain has a transfer funtion of the type G(s)=s/d.
If beam loading effects can be neglected, the open loop transfer function H(s) and the
characteristic equation have the form:
H ( s)  G ( s)  B( s) 
s
d
 s2
;
2
2
s  s
G ( s)  B( s)  1  0
The closed loop transfer function will have poles at:
  
 s2
s1,2 
 j s 1   s 
2 d
 2 d 
2
Provided that d <0 (negative feedback), the pole pair
has a negative real part. A damping constant ad is added,
given by:
 s2
ad 
2 d
while critical damping is achievable under the condition:
d  
s
2


 s2
s s
  s2  0
d
2
1
0
ad  s
The gain of a pure differentiator circuit grows linearly with frequency, which is not a
realistic behaviour for any physical system.
Beam Phase Loop : Real Differentiator
A real differentiator can be obtained by using
the low frequency portion of the transfer
function of a simple band-pass filter. In this
case the loop amplifier has a transfer funtion of
the type:
s d
G( s) 
1  s 1 1  s 2 
s
1
2
leading to the characteristic equation:



 
s 4  1   2 s3  1  2   s2 s 2  1   2   1 2   s2 s  1  2  s2  0
d 

By applying the Routh-Horwitz criterion, the zeros have negative real part if:
a4  0;
a3  0;
a2 
 1
1   2



2

d
s

1
1 1   2
 1




 s2
  d 1  2
 1
  0
 d
a1a4
 0;
a3
or
a1a2a3  a12a4  a32a0  0
1
1 1   2
 1




1  2
 s2
d
 1

0
 d
Beam Phase Loop : Real Differentiator
The solutions have the form:
 s  1 2
 s  1 2
1   2 1

0
2
1  2

s
d
1
1
1 1   2
 0



 d 1  2
 s2

1

1

Real integrator, to be discussed later
s
1
2
s
For real differentiators there is a limit on the maximum achievable loop gain in stable
conditions. If 2 >> 1 this limit is given by:
1
d

2 1

2 
s
1
It may be shown that in this case, even at the maximum gain, critical damping can not be
reached.
Beam phase loops based on real differentiator have been demonstrated to be effective in
cases where limited extra-damping is needed.
By tayloring the bandpass limits 1 , 2 the loop repsonse at the revolution frequency can
be reduced to avoid excitation of coupled bunch modes different from the barycentre one.
Loop gain has also to be reduced if a the beam quadrupole resonance at 2s is excited.
Beam Phase Loop : Real Integrator
To get damping from a beam phase loop the loop amplifier
must provide 90° of phase rotation at s . If a pure delay is
used the results are similar to those obtained with differentiator.
An alternative way of generating the 90° phase shift is to
implement a real integrator, i.e. a LPF with the synchrotron
frequency placed on the falling edge. The transfer function is:
G0
G(s) 
1  s 1 
leading to the characteristic equation:
1
s
s3  1 s 2   s2 s  1  s2 (1  G0 )  0
By applying the Routh-Horwitz criterion, the zeros have negative real part if:
a3  0;
a2  0;
aa
a1  0 3  0;
a2
a0  0
1  G0  0
 

G

0
 0
0  G0  1
The case of a pure integrator G(s)=i/s can be treated by considering G0=1/i, i 0
and the system results to be unstable.
In the real integrator case the range of acceptable gain is limited, while at the limit G0=1
one of the zeros is s=0 , which means that the closed loop system has a peaking response
at low frequency. This limitation can be circumvented by AC coupling the loop (one or
more zeros at s=0 in G(s) ). With one zero we get back to the bandpass shape of G(s).
Beam Phase Loop: Integrator over the beam-to-cavity phase
Another way of obtaining AC coupling to
implement an integrator loop amplifier is to
feedback the beam-to-cavity phase. The open
loop gain H(s) in this case results to be:
H ( s )  G ( s )  B( s )  1  G ( s )
 s2
s 2   s2
so that a double zero in the origin appears.
If we consider a pure integrator transfer function G(s)=i/s we obtain the characteristic
equation:
G ( s)  B( s)  1  1  0  s 2  i s   s2  0
which is inconditionally stable for i > 0, and the damping constant ai =i/2 is
independent on s . Similar results (inconditioned stability, damping almost independent
on s) are found by if real integrator or real double integrator transfer functions are
considered:
G0
G0
real integrator G ( s) 
;
real double integrator G ( s) 
1  s / 1 
1  s / 1 1  s / 2 
All these schemes are in principle well performing but suffer of a common drawback.
The loop is AC because of the B(s)-1 transfer function, but the branch from the phase
detector PD to the phase shifter PS (including the loop amplifier) may have large DC
gain. Any DC offset or drift from the PD would saturate either the loop amplifier or the
PS, also producing unnecessary change of the cavity driving phase.
Beam Phase Loop: Integrator over the beam-to-cavity phase
To avoid DC driving of the phase shifter, an AC coupled loop amplifier has to be
implemented. The loop amplifier is a BPF acting as an integrator for frequencies located
on the folling edge of the frequency response.
The real integrator transfer function can be
recovered by using a Voltage Controlled
Oscillator (VCO) AC coupled to the loop
amplifier. The VCO can be considered as a
phase shifter with infinite dynamics,
adding a pole at s=0 (frequency to phase
conversion). The VCO pole and the AC
loop amplifier zero cancel out, so that:
s /  AC

G( s) 
; AVCO ( s)  VCO
1  s / 1 
s
H ( s )  B( s )  1G ( s ) AVCO ( s )  
s2
G0
s 2   s2 1  s / 1
In proton synchrotron the VCO may be DC driven by a proper signal to control the radial
position of the bunches even during acceleration (revolution frequency, beam energy and
radial position in a dispersive monitor are mutually proportional in this case, except in the
vicinity of the transition energy). The beam phase loop does not interfere with low
frequency regulations and provides damping of the coherent synchrotron motion.
Beam Phase Loop: DC coupled VCO with radial loop
The DC coupling to the VCO can be
restored if an additional loop correcting
the DC set of the VCO frequency is
implemented. If a “flat response” loop
amplifier G(s) would be DC coupled to
the VCO, any disturbance at the phase
detector Dn would produce a beam
frequency deviation Dfb given by:
Dfb
i B ( s )
i  s2


Dn 1  i / s B( s )  1 s 2  i s   s2
Any phase offset disturbance drives the
beam to a staedy frequency deviation.
To avoid this effect the VCO can be feedback controlled by another DC loop looking at
the radial position of the beam in a dispersive monitor (which is proportional to both the
beam energy and the revolution frequency). Once this additional loop is set up we have:
Dfb
i B ( s )
i  s2


Dn 1  i / s B( s )  1  G f B( s ) s 2  i s   s2 (1  G f )
Beam Phase Loop: DC coupled VCO with radial loop
The radial loop compresses the residual beam frequency deviation Dfb by a factor Gf .
Looking at the roots of the characteristic equation we have in this case:
s 2  i s   s2 (1  G f )  0  s1, 2
4 s2 (G f  1)
i 

1 1
2
i2





 i
 s2G f
i
that holds under the assumptions |Gf |>>1 and i2 >>s2|Gf |.
If Gf < 0, the system is stable with one large negative root strongly damping the
coherent synchrotron motion, and a second weaker root damping the slow motion
coming from low frequency disturbances. If the time constant associated to the second
root is much longer that the synchrotron period, i.e. :
i
 s2 G f

1
s
 G f 
i
s
the beam follows adiabatically the motion induced by the slow disturbances.
The DC coupled VCO together with radial loop has been widely used in proton
synchrotron. This set-up has shown good performances in strongly damping the coherent
motion (which can not be even measured) and controlling the beam radial position.
Beam Phase Loop Analysis with Modulation Transfer Functions
We have so far analyzed beam phase loops
neglecting the modulation transfer functions,
which have to be included whenever beam
loading effects are relevant.
Referring to the complete block diagram aside,
the open loop transfer H(s) function is given by:
H ( s) 
g
G ( s ) G pp
( s )B ( s )  1
1  B ( s ) G bpp ( s )
If we consider a pure integrator loop amplifier G(s)=i/s and a synchronous phase
s=± p /2 (to account for the 2 cases sgn(h)=± 1), the characteristic equation is:
 s2  s2 s2  2s   2 1  tan2 z   s2 2Y tanz  i ss   2 1  tan2 z  Y tanz 
leading to the stability conditions:
s  p / 2 
2



0
and
Y

z

sin 2 z 


2
 z  0 and Y 
tan2 z 


s  p / 2 

 z  0 and



 z  0 and


2
sin 2 z
2
Y
tan2 z 
Y
Beam Phase Loop Analysis with Modulation Transfer Functions
The presence of the beam phase loop enlarge the Robinson 1st stability limits since also a
region with z < 0 (z > 0 for h < 0 ) becomes accessible. This is because the strong loop
damping of the coherent motion overrides the Robinson antidamping.
The Robinson 2nd limit is unaffected since it is a DC instability, and the beam phase loop
has no DC gain in the considered configuration.
However, realistic models of RF systems are much more complicated since other loops
have to be included (at least tuning and amplitude loops) and real transfer functions contain
delay terms together with roll-off frequencies which are intrinsic of the used devices.
Realistic cases can be better treated numerically. Numerical approach shows that the
presence of many loops decreses the stability region of the system, especially if the loop
bandwidths are comparable with (or larger than) the cavity half-bandwidth .
In the oversimplyfied scenario of pure integrator loop transfer functions, s=± p /2, g=0,
  i and B(s)=0, F. Pedersen has derived the following stability criterion:
Y
2
 a t t  a  p




t  a  p  p  a
where p, a, t are the integrator constants of the phase, amplitude and tuning loops.
If a too low limit on Y results, the beam loading effects have to be cured with dedicated
techniques to cancel or limit the perturbation induced by the beam signal in the RF system.
• Feedforward
Technique
Beam Loading Compensation: The Feedforward Technique
A very elegant way to compensate the beam
loading effects is the so called “feedforward”.
If a beam signal sample is injected back in
the RF driving path with a proper amplitude
and phase, it is possible to generate through
the RF power source a contribution to the
cavity voltage equal and opposite to that
induced by the beam. Ideally, looking at the
system from reference clock path, no beam
induced effects are visible.


 
I f  Ib  It  I d
The RF generator current in the
model can be split in 2 contributions,
coming from drive and feedback
signals. If the feedback is properly
set, the current If and Ib cancel out,
so that the total current It exciting the
cavity is equal to the current Id
generated by the drive signal alone.
Beam Loading Compensation: The Feedforward Technique
It has to be pointed out that the “feedforward” can not change the static beam loading
effects. The power delivered by the generator is just the same, and so it is the beam
induced voltage and the need for a cavity detuning proportional to the stored current.
What the feedforward does change is the dynamics of the beam loading, leading to a
substantial simplification of the Pedersen representation of the system.
p
g
G pp
(s )
g
G pa
(s )
Drive
g
Gap
(s )
a
g
Gaa
(s )
p
 G bpp (s )
 G bpa (s )
Feedforward
a
b
 Gap
(s )
b
 Gaa
(s )
Cavity +
p
+
B(s )
Beam
p
G bpp (s )
G bpa (s )
b
Gap
(s )
+
a
b
Gaa
(s )
1
tan s
a
The feedback signal
and the beam have
the same modulation
transfer functions to
the cavity voltage
(the 2 phasors have
the same phase).
So the feedback
cancel out the beam
to cavity modulation
terms in the Pedersen
model.
“Plain” modulation functions (no vector projection)
Beam Loading Compensation: The Feedforward Technique
Feedforward technique has been successfully implemented in some “hystorical” proton
synchrotron machines such as the CERN PS and ISIS.
However, in spite to the great elegance and conceptual simplicity, the compensation
scheme is quite critical for a number of reasons, namely:
•
non linearity and drifts of the characteristics of the RF power amplifier as well as any
other element along the chain will reduce the degree of beam signal cancellation;
• the frequency change during the acceleration process in proton synchrotrons asks for a
frequency independent compensation, not easy to be obtained;
• frequency response of the power amplifier and of the other elements in the chain may
lead to imperfect cancellation over a wide span of the beam modulation tranfer
functions;
• overall delay of the feedforward path must be small, or exactly equal to 1 turn to avoid
excitation of non-barycentric coupled bunch modes.
Some of these difficulties may be overcome with different compensation techniques, such
as the “direct RF feedback”.
• Direct RF
Feedback
Beam Loading Compensation: The Direct RF Feedback
In the “direct RF feedback” configuration a sample of the cavity voltage
is re-injected back and added to the
RF drive. The effect of this loop is
that of reducing the cavity impedance
as seen by the beam by a factor equal
to the open loop gain.
The cavity voltage is related to the
beam current and to the RF drive
signal by:
Z L ( s)
H ( s) / F ( s)
 Vd ( s )
1  H (s)
1  H ( s)
 Z L (s)
H (s)  K (s)
F ( s)
  1 RL
Vc ( s )   I b ( s )
with
In the limit of large loop gain (H0>>1) the cavity equivalent impedance and the cavity
voltage are given by:
Z (s)
Z ( s) V ( s)
Vd ( s )
Z L' ( s )  L ;
Vc ( s )   I b L  d

H0
H0
F0 H 0  F0
Beam Loading Compensation: The Direct RF Feedback
In the limit of large loop gain (H0  )
the direct RF feedback is equivalent to the
feedforward technique: the beam induced
voltage is cancelled and the cavity voltage
is entirely due to the RF drive signal.
Obviously, the gain can not be infinite but
it is actually limited by the total delay tt of
the loop path. The physical delay tp (the
total length of the connection) and the
group delay tg (the derivative of the phase
response of the bandwidth limited devices
such as the RF power source) contribute
both to tt (typically  some 100 ns) .
H 0 e j tt
A realistic expression for open loop gain has H(j) is:
H ( j ) 
1  j D  BW
In order to maximize H0 is necessary to “trim” the delay with the loop phase shifter to the
condition rtt=2np (r=cavity resonant frequency). This condition has to be maintained
also when the cavity is detuned to match the static beam loading. Under this condition,
being M the design loop phase margin, the maximum allowed gain H0 is given by:
p / 2  M
H0 
 BW t t
Beam Loading Compensation: The Direct RF Feedback
Depending on the total delay tt and cavity bandwidth BW the equivalent cavity impedance
is reduced and deformed as shown. Even thought it is still not zero, the reduction may be
sufficient to weaken the beam loading effects to a tolerable level.
t t  300 ns
 BW  2p  10 kHz
t t  300 ns
 BW  2p  10 kHz
Similarily to the case of the feedforward technique, the direct RF feedback does not affect
the static beam loading aspects. Again, the effect is to compensate the dynamics of the
beam loading, in terms of modification of the modulation transfer functions.
Beam Loading Compensation: The Direct RF Feedback
The direct and cross modulation transfer functions are in this case given by:
1  Z L' s  j  Z L' s  j 
1  Z L' s  j  Z L' s  j 
G pp ( s)  Gaa ( s)   '
 '
 '
; Gap ( s)  G pa ( s)   '

2  Z L  j 
2 j  Z L  j 
Z L  j  
Z L  j  
where the impedance is that reduced by the feedback. Typical plots of the module of the
direct and cross modulation transfer functions are:
Beam Loading Compensation: The Direct RF Feedback
Since the cavity voltage almost entirely come from the RF drive signal, it is easy to show
that the projected modulation transfer functions from the generator are almost equal to the
unprojected ones, while those related to the beam are (to the first order) negligible:
d
d
G dpp ( s )  Gaa
( s )  G pp ( s ); G dpa ( s )  Gap
( s )  G pa ( s )
b
G bpp ( s )  Gaa
( s )  0;
b
G bpa ( s )  Gap
(s)  0
• The direct RF feedback drastically reduces the beam induced modulation on the cavity
voltage, while the system seen by the RF drive path appears much more broadband, with
flattened modulation transfer functions;
• All the coherent effects driven by the cavity impedance are drastically weakened, since
the equivalent cavity impedance as seen by the beam is reduced;
• The implementation and the set up of the feedback hardware are reasonably simple, and
mantaining the optimal compensation in different operational conditions is not critic;
• The direct RF feedback is presently the most used scheme to compensate the beam
loading dynamic effects, originally proposed for hadron stotrage rings and now widely
used also for lepton machines.
• Gap Transient
GAP TRANSIENT
Many storage rings are operated with a gap in the bunch filling pattern. This is quite
common in e- rings to avoid ion trapping, while in synchrotron light sources experiments
may require particular beam temporal structure.
In presence of a gap a head and a tail of the bunch train can be identified. The longrange wakes sampled by each bunch depend on the bunch position along the train.
Limiting our attention to the beam interaction with a cavity accelerating mode we can
immediately conclude that different bunches along the train experience different kicks
from the beam induced voltage. This generates a spread of the parasitic losses along the
train and, as consequence, a spread of the synchronous phases of the bunches.
The effects of the gap in the beam can be deduced from time or frequency domain
approach. In frequency domain in the no gap case, the cavity is excited with a frequency
comb with line spacing given by the reciprocal of bunch time spacing.
Only
the
RF
line
significantly interacts with
the cavity, while the other
lines are responsible of the
transient voltage across
the bunch that we have
neglected so far.
GAP TRANSIENT
As soon as a gap in the bunch filling pattern appears, the beam
spectrum becomes much more populated, with line spacing equal
to the ring revolution frequency 0 (much lower than the bunch
repetition frequency). Beside the RF line, many other lines
interact with the cavity impedance generating a non-harmonic
voltage VNH(t), i.e. a voltage which has only the revolution
periodicity. This non-harmonic term is synchronous across any
given bunch, but not across different bunches in the train.
 

i (t )  e   I k e jk  0t  s  ; VTot (t )  Vc cos(t )  VNH (t )
k 0


jk ( 0t  s ) 
VNH (t )  e   I k Z (k r ) e

 k h

The voltage VNH(t) kicks the various bunches by different amounts and has different slope
across them. This have noticeable implications.
Since the total voltage VTot(t) has to kick all the bunches by the same amount Vloss (the
particle energy loss per turn), the harmonic part of the cavity voltage Vc(t) has to
compensate the kick spread due to VNH(t). Each bunch founds its energy equilibrium
position at some particular phase n that changes from bunch to bunch.
GAP TRANSIENT
The spread of the voltage kicks associated to VNH(t)
increases with the average current and with the gap
width, and it is converted in a spread of the bunch
synchronous phases n, through the local slope of the
harmonic voltage Vc(t). The smaller is the slope, the
larger is the synchronous phase spread. The effect is
enhanced in systems implementing Landau harmonic
cavities, where the voltage across the bunch is kept at
zero-slope to produce non-linear bunch lengthening.
The synchronous phase spread due to gap
transient can be computed analytically (in
frequency or time domain approach) or
numerically, on the base of macro-particle
tracking codes. Numerical solutions are selfconsistent because follow the bunches in
finding their equilibrium position. Numerical
solutions also show that the bunch phase
equilibrium distribution is almost linear in most
cases, even in presence of Landau cavities.
GAP TRANSIENT
Analytical computation of the synchronous phase distribution
in frequency domain starts from the unperturbed beam
spectrum and proceeds in iterative way. The spectrum of a
bunch train of Nb bunches spaced by mTRF (h=harmonic
number, m=any integer divisor of h) is given by:
On the base of this spectrum the total cavity voltage VTot(t)
can be calculated, and the bunch equilibrium phases are
obtained from the solutions of VTot(nTRF+n)=Vloss. In most
cases the distribution is almost linear so that we can write:
Under this assumption the spectrum of the beam
can be re-computed accordingly to:
and the calculation can be repeated more
precisely on the base of the new spectrum.
Corrections of the spectrum due to the time
displacement of the bunches from the original
common position s can be relevant.
I k'  I
Ik  I
sin p mkNb h 
Nb sin p mk h 
DTn  DT0  nDT
sin kNb 2  2p m h   r DT 
Nb sin k 2  2p m h   r DT 
GAP TRANSIENT
Analytical computation of the synchronous phase distribution can be also approached in
time domain. In this case it is convenient to represent the total voltage VTot(t) as an
amplitude and phase modulated sine-wave in the form: VTot (t )  A(t )  cost   (t )
with A (t) and (t) periodic with the revolution period.
Since the beam shows a “static” amplitude modulation at the revolution frequency, A (t)
and (t) can be derived by making use of the modulation transfer functions from beam
amplitude to cavity amplitude and phase:
b
Gaa


 ; Gb
Y   coss s   2 tan  z sin s  coss 

s  2 s   1  tan  z
2
2
2

ap



Y  sin s s   2 tan  z coss  sin s 

s 2  2 s   2 1  tan 2  z

If the revolution frequency is much larger than the cavity
bandwidth ( / fr  0), the transfer functions are integrators:
Y sin s
s
s
which for |s|p/2 give no amplitude modulation and linear
phase modulation with maximum phase deviation Dmax:
b
Gaa

Y coss
;
Dmax 
b
Gap

1 R
IbTr
2QV
GAP TRANSIENT: Conclusions
• A gap in the filling pattern generates a distribution of bunch equilibrium phase n;
• The deviation from a common position respect to an external RF clock has potential
drawbacks. Optimal synchronization with synchronous feedback systems is affected. In
multibunch colliders the interaction point IP is displaced from bunch to bunch, unless
the gap transient effects of the 2 beams are perfectly matched;
• The gap transient effect is generated by all the long range wakefields of the ring. The
cavity accelerating mode is certainly the most important source, but cavity HOMs
together with any vacuum chamber trapped mode have to be taken into consideration;
• The gap transient effect can be hardly compensated by external active feedback system.
This is because the generator-cavity system is quite narrowband and it can’t provide
compensation over the required bandwidth. This is especially true for the amplitude
modulation part, because the RF power saturation do not allow cancelling the
perturbation. For the phase modulation part a partial compensation is possible by
overmodulating the generator through a high gain RF loop. However, large loop gain
and bandwidth are not easy to obtain (limitations come from group delay of the system)
and may interfere with others feedback loops;
• In extreme cases (such as gap transient with Landau cavities) the bunch dynamics is
also affected. The slope of the voltage across the bunch varies along the train, and
bunches show different synchrotron frequencies. This cause a bunch-to-bunch Landau
effect which may possibly stabilize the coupled-bunch dynamics.
• Transion Energy
Crossing
TRANSITION ENERGY CROSSING
We have so far considered machines having positive or negative dilation factor h. As
matter of fact, in protons or heavy ions synchrotron the energy is raised by smoothly
increasing the bending and focusing magnetic fields (dB/dt > 0 during a given time).
Because of the synchronous phase
stability principle, if dB/dt is small
enough, the bunch is accelerated by the
RF field and its energy follows
adiabatically the B-field increase
executing small synchrotron oscillations.
h
Df f
1
 2  ac
Dp p g
While energy and g factor increase, the
ring may cross the point where:
1
h  2  a c  0  g  g t  1/ a c
g
The beam energy Et corresponding to gt
is called “transition energy”. Dilation
factor h changes sign across gt.
Near transition energy the ring becomes isochronous, i.e. particles with different momenta
have the same revolution frequency. The synchrotron frequency gets to zero and the
synchronous phase stability principle is violated. The acceleration process is not adiabatic
anymore, and the time duration of the transition crossing is called the “non-adiabatic time”.
TRANSITION ENERGY CROSSING
Single and multi-particle dynamics near transition is peculiar. Beam quality degradation
and beam loss may occur for a number of reason, namely:
• As transition is approached, the RF bucket elongates in the momentum direction and
shrinks in the phase direction. If ring momentum acceptance is exceeded, the particle is
lost. Momentum spread grows, and bunch length reduces so that space charge effects
are enhanced;
• Short bunches at very low h values are likely to undergo m-wave instability;
• A rapid RF phase jump from s to s is necessary at transition. However particles
arriving late at transition get less energy kick and accumulate more delay and eventually
never reach transition and get lost.
Several schemes have been proposed and tested to limit the beam quality and intensity
losses across the transition. They consist in lattice gymnastics or RF gymnastics, or
eventually both.
Concerning the lattice, a sudden change of the gt value (gt jump scheme) obtained by
pulsing some quadrupole magnets near transition has been demonstrated to speed-up the
crossing process resulting in an increased efficiency of the beam transport through
transition.
It has to be noticed that lattices with negative momentum compaction ac value (sometimes
indicated as imaginary gt lattices) get rid of transition crossing, being h always positive.
Here we are more interested in the RF gymnastics for transition crossing.
TRANSITION ENERGY CROSSING
A detailed microscopic analysis of the beam behaviour near transition reveals that most of
the problems encountered in transition crossing come from the fact that during the nonadiabatic time the RF longitudinal focusing is not needed. Revolution frequency is almost
momentum independent, so particles in the bunch head receive insufficient acceleration
(due to the RF voltage slope) and the contrary for bunch tail particles, and this energy
compensation lack accumulates from turn to turn.
Independently on their phase error, bunch particles all need the same voltage kick during
non-adiabatic time to follow the ring energy ramp, while RF slope should be better reduced
to zero. Then, we may distinguish three different strategies to optimize the RF set-up across
the transition energy:
• Synchronous phase jump of 2s. This is
the minimal necessary condition to have
stable motion on both transition sides.
However, the phase jump is synchronized
with the transition crossing of the average
particle in the bunch, while individual
particles will cross transition in different
times because of their momentum
deviation. Some of the beam can be
transported through transition, but with
poor efficiency.
g
gt
Tc
t0
Tc
t
TRANSITION ENERGY CROSSING:
The “duck-under” scheme
• At beginning of the non adiabatic time the RF phase is shifted from s to 0 and the
amplitude decreased to the merely needed kick to follow energy ramping. The bunch
center sits on the RF crest and the effects coming from the RF slope are minimized.
However, the typical bunch length is tens of RF degrees, and mis-acceleration second
order effects may still spoil beam quality. Once the non-adiabatic time is over the RF
phase is finally jumped to s. This scheme is called the “duck-under”.
g
gt
 E0 f rev g t4
Tc  Trev 
 4p hg Vc coss

t0
Tc
Tc
t
13




TRANSITION ENERGY CROSSING:
The “slide-under” scheme
• This scheme is based on the same principle as the “duck-under” but the accelerating
voltage during the non-adiabatic time is flattened by adding an harmonic
contribution. In this way the kick for the off-time particles in the bunch is corrected
up to the 2nd order. Once the transition is completed the RF voltage is shifted to
accelerate the bunch on the negative slope. This scheme is called the “slide-under”.
g
gt
 E0 f rev g t4
Tc  Trev 
 4p hg Vc coss

t0
Tc
Tc
t
13



