Transcript Document

Beam Loading and Low-level RF Control in Storage Rings
Alessandro Gallo, INFN - LNF
Lecture I
• Static Beam Loading:
– Lumped model for the generator-cavity-beam system;
– Optimal cavity tuning and coupling factor;
– Tuning Loop
• Cavity and Beam response to phase
and amplitude modulations:
– Modulation transfer functions;
– Beam transfer function;
– Pedersen model.
• Stability of the generator-cavity-beam system:
– Robinson instability;
– Robinson 2nd limit
PEP-II Low Level RF Control Loops
• Static Beam Loading
STATIC BEAM LOADING
We know from the longitudinal dynamics theory of storage rings that the motion of a single
particle is stably focused around an equilibrium position known as “synchronous phase”.
The synchronous phase corresponds to an
accelerating voltage on the particle equal
to its average single-turn loss, which
accounts for synchrotron radiation
emission and interaction with the vacuum
chamber wakefields. In synchrotrons this
also accounts for the required energy
gain/turn during ramping.

f f
1
 2 c
p p 
For storage rings with positive dilation
factor the synchronous phase is placed on
the accelerating voltage positive slope;
the contrary for negative dilation factor
rings.
The particle absorbs energy from the accelerating field but also contributes to the total
voltage. The particle induced voltage can be easily calculated treating the cavity
accelerating mode as a lumped resonant RLC resonator interacting with an impulsive
current.
THE CAVITY RLC MODEL
The beam-cavity interaction can be conveniently described by introducing the resonator
RLC model. According to this model, the cavity fundamental mode interacts with the
beam current just like a parallel RLC lumped resonator. The relations between the RLC
model parameters and the mode field integrals wr (mode angular resonant frequency),
Vc (maximum voltage gain for a particle travelling along the cavity gap for a given field
level), U (energy stored in the mode), Pd (average power dissipated on the cavity walls)
and Q (mode quality factor), are given by:
Mode field integrals
 Ez ( z ) e
Vc 
jw r
z
c
Cavity RLC model
Vc2
Rs 
2 Pd
dz
gap
Pd 
U
1
Rsurf H t2 d

2S
1
C
wr R Q
1
(E 2  H 2 )d

4V
w rU
Vc2
Q
; R/Q 
Pd
2w rU
RLC model parameters
Note: we assume that positive voltages accelerate
the beam, which is therefore modelled as a current
generator discharging the cavity.
L
1
w r2C

RQ
wr
STATIC BEAM LOADING
A particle travelling along the cavity gap is a dfunction
current generator exciting the RLC resonator. The
particle induced voltage in the cavity vq(t) is a decaying
cosine wave generated at the particle passage. The
fundamental theorem of beam loading states that the
particle “sees” only half of its induced voltage step Vq.
According to the model, the expression for vq(t) is:


vq (t )  Vq  1(t  tq )  cos w r (t  tq )  e
Vq 
w
 r (t t q )
2Q
q
 qw r R Q
C
The particle in a storage ring are gathered in bunches
that typically show a gaussian longitudinal profile. The
bunches are normally much shorter than the RF
wavelength and interact with the accelerating field
similarly to macroparticles. The voltage induced in the
cavity by a short gaussian bunch is still an exponentially
decaying cosine wave with a finite rise time (related to
the bunch duration).
STATIC BEAM LOADING
It can be easily verified that both the
amplitude and the slope of the
voltage induced in a cavity by a
single passage of a short bunch are
almost negligible with respect to the
cavity accelerating voltage. What
can not in general be neglected is the
voltage generated by the cumulative
effect of many bunch passages.
In fact, the time distance between adjacent
bunches is normally much smaller than the cavity
filling time. This is true even in single bunch
operation whenever the ring revolution period is
smaller than the cavity filling time. In this case
the voltage kicks associated to the passage of the
bunches add coherently, and the overall beam
induced voltage can grow substantially and even
exceed the generator induced voltage.
N
f
Tb
STATIC BEAM LOADING
The cumulative contribution of many bunch passages depends essentially on the RF
harmonic of the beam spectrum. Also neighbour lines play a role in case of uneven bunch
filling pattern (gap transient effect).
The static beam loading problem consists in computing:
- the overall contribution of the beam to the
accelerating voltage;
- the power needed from the RF generator to
refurnish both cavity and beam;
- the optimal values of cavity detuning and cavityto-generator coupling to minimize the power request
to the generator.

Rs
n2Z0
PFWD
2
VFWD

2Z 0
STATIC BEAM LOADING
To solve the static beam loading problem we refer to the following circuital model:

I
1 
1 

YL  T 
 j  wC 

Vc
RL
wL 

Ig 
2VFWD

n Z0
Ib 
2q
;
Tb
8 PFWD
Rs
fb  fs
The beam current is represented by the phasor Ib at the frequency of the RF source,
while the external RF source is represented by a current generator Ig whose amplitude
is related to the power of the forward wave launched on the transmission line.
If  > 0 the beam current phasor anticipates the total cavity voltage (i.e. fs > 0), while
it is retarded in the opposite case ( < 0  fs < 0 ).
The RF sources are normally specified by the maximum amount of power deliverable
on a matched load. This is also the power actually dissipated in the system whenever
a circulator is interposed to protect and match the source. The cavity is usually tuned
near the optimal value that minimizes the request of forward RF power to the
generator.
STATIC BEAM LOADING
 0
The analysis of the beam-cavity circuital model
leads to the reported phasor diagram. The total
current exciting the cavity is:




  1 
1 
IT  I g  Ib  Vc  Y  Vc  
 j  wC 

R
w
L


 s
It may be noticed that only the imaginary part of
the total current does depend on the cavity tuning.
Real and imaginary parts of the generator current
are then given by:



 1
Re I g  Re IT  I b  Vc
 I b cosfs 
Rs



w wr  wr w
Im I g  Im IT  I b  Vc
 I b sin fs 
RQ
 


 


fs  0
fg  0
fz  0
Being the off-resonance parameter d defined as : d 
 0
fs  0
fg  0
fz  0
w wr
w  wr

2
wr w
w
the minimum amplitude of the generator current phasor is obtained when its imaginary
part is zero. The optimal cavity tuning condition is therefore given by:
2

 Vloss 
d
Ib R Q
2qb R Q

Im I g  0  Vc
  Ib sin fs   d  
sin fs    sgn  
1  
RQ
Vc
Tb Vc
V
 c 
 
STATIC BEAM LOADING
The optimal cavity tuning condition in a
storage ring with negative dilation factor 
asks for positive values of d. This means that
the cavity has to be tuned below the
frequency of the RF generator, and the
amount of detuning is proportional to the
intensity of the stored current. The sign of
the detuning is just opposite for positive
values of the dilation factor .
<0
I0 
>0
Vc 1   
Rs sin fs
Under the optimal cavity tuning condition we also have:

 1
8 PFWD
I g  Re I g  Vc
 Ib cosfs  
Rs
Rs
 

leading to the optimal coupling condition:  opt  1 
It follows immediately that the forward
power request to the external generator is:
PFWD
PFWD
R
 s
8
  1



V

I
cos
f
b
s 
 c R


s
2
Ib Rs
I RV
P
cosfs   1  b s 2 loss  1  beam
Vc
Pcav
Vc
1 Vc2 1

 I b Vc cosfs   Pcav  Pbeam
2 Rs 2
The system is perfectly matched and no RF power is wasted because of reflections.
STATIC BEAM LOADING
The beam can be also modelled as a complex admittance equal to the ratio between the
current and voltage phasors. This model is less physical (the resistive part of the beam
admittance does not enlarge the bandwidth of the cavity!) but allows more direct
computation of the reflection coefficient at the coupling port.
Ycav 

Rs
Ybeam
n2Z0
Ybeam 
Ib jf s I b
I
e
 cosfs  j b sin fs
Vc
Vc
Vc
Zero for optimal
coupling factor
Rs
Ycav
 Z0
Rs
Y
 Z 0 beam
1
d
j
Rs
RQ
Zero for optimal
cavity detuning
 1
 d

Rs I b
I
cosfs  jRs 
 b sin fs 
Vc
 R Q Vc

 1
 d

Rs I b
I
cosfs  jRs 
 b sin fs 
Vc
 R Q Vc


STATIC BEAM LOADING
For a given required cavity voltage Vc, the optimal values of the cavity detuning d and
coupling coefficient  are both dependent on the instantaneous stored current Ib. The
accelerating cavities are equipped with tuners, that are devices allowing small and
continuous cavity profile deformation to real time control the resonant frequency position.
Sometimes cavities are also equipped with
variable input couplers but they are never
operated in regime of continuous 
adjustment. In general the coupling
coefficient is adjusted just once to match the
maximum expected beam current. At lower
currents the system is partially mismatched,
but the available RF power is nevertheless
sufficient to feed the cavity and the beam.
It turns out that the cavity tuning d and the
generator power PFWD have to follow the
actual current value in order to keep the
accelerating voltage constant preserving
Ib Ibmax
good matching conditions.
These tasks are normally accomplished by dedicated slow feedback systems, namely the
“tuning loop” and the “amplitude loop”.
TUNING LOOP
The tuning loop restores automatically and
continuously the cavity resonant frequency
to compensate the beam reactive admittance.
The loop controls the RF phase between the
cavity voltage and the forward wave from
the generator. Phase drifts are corrected by
producing mechanical deformations of the
cavity profile by means of dedicated devices
(plungers, squeezers, ...). In fact the loop
controls the phase of the transfer function:
Beam
1
2
 Rs

1   Z0
Vc
2n
n 2 Ycav  Ybeam 

 2n


1
R
R I
VFWD
Z0  2
1  s Ycav  Ybeam  1  jQLd  L b e jf s

Vc
n Ycav  Ybeam 
Q
R I
QL  0

QLd  L b sin fs
1 
V
Vc
with
  c   arctan
Rs
RL I b
VFWD
R

1
cosfs
L
1 
Vc
By controlling the set point of the variable phase shifter the phase of the transfer function
can be locked to 0 or to any other value f0.
TUNING LOOP
The best cavity tuning is obtained by locking the loop to f0 = 0. In this case we have:
RL I b
sin fs
R I
I RQ
Vc
 arctan
 0  QLd  L b sin fs  0  d   b
sin fs  0
RL I b
V
V
c
c
1
cosfs
Vc
QLd 
For dynamics considerations, sometimes it is useful to set a phase f0 slightly different
from zero. In this case we have:
RL I b
sin fs
 R I

R I
Vc
 arctan
 f0  QLd   L b sin fs  tanf0 1  L b cosfs 
R I
Vc
Vc


1  L b cosfs
Vc
The reflection at the
  1 RL I b

cosfs
input coupler port is not
0  j tanf0 
  1 Vc
minimized in this case,

with 0 
1 j tanf0 
 RL I b

and the overall efficiency
1 
cosfs 
Vc
of the system is reduced.


QLd 
Tuning loops are generally very slow since they involve mechanical movimentation
through the action of motors. Typical bandwidth of such systems are of the order of 1 Hz.
Tuning systems are also necessary to stabilize the cavity resonance against thermal drifts.
• Cavity and Beam
response to phase and
amplitude modulations
CAVITY RESPONSE to AM and PM SIGNALS
Servo-loops and feedback loops apply AM and PM modulation to the cavity. If the cavity
is detuned the modulating signals are filtered and mixed by the cavity frequency response:
vi (t )  Ai 1  ai (t )coswt 
wr  w
vi (t )  Ai coswt  fi (t )
vo (t )  Ao coswt  fo  fo (t )
Ltransform

x(t )

xˆ ( s)
vi (t )  Ai 1  ai (t )coswt 
vi (t )  Ai coswt  fi (t )
vo (t )  Ao 1  ao (t )coswt 
G( s) 
aˆo ( s) fˆo ( s)
1


aˆi ( s) fˆi ( s) 1  s / 
wr  w
with  
wr
2QL
vo (t )  Ao 1  ao, a (t ) coswt  fo  fo, a t 




vo (t )  Ao 1  ao, p (t ) cos wt  fo  fo, p t 
fˆo, p ( s)
aˆo, p ( s)
fˆo,a ( s)
Gaa ( s) 
; G pp ( s) 
; Gap ( s) 
; G pa ( s) 
aˆi ( s)
aˆi ( s)
fˆi ( s)
fˆi ( s)
aˆo,a ( s)
It may be demonstrated that direct and cross modulation transfer functions are given by:
1  Z s  jw  Z s  jw 
1  Z s  jw  Z s  jw 
G pp ( s)  Gaa ( s)  

;
G
(
s
)


G
(
s
)


ap
pa
2  Z  jw 
Z  jw  
2 j  Z  jw 
Z  jw  
CAVITY RESPONSE to AM and PM SIGNALS
Being the cavity impedance in the Laplace domain expressed by:
2 s
wd
w RF  w r
Z ( s)  RL 2
with
w

w


w


tan
f
;
tan
f


Q
d


r
z
z
L
2

s  2 s  w r2
where fz fy is the phase of the cavity impedance at the generator frequency w, one gets:
G pp ( s )  Gaa ( s ) 

 s   2 1  tan 2 f z


s 2  2 s   2 1  tan 2 f z

; Gap ( s )  G pa ( s )  
 tan f z s

s 2  2 s   2 1  tan 2 f z

The general form of the modulation transfer functions features 2 poles (possibly a
complex conjugate pair) and 1 zero, and degenerate to a single pole LPF response if the
cavity is perfectly tuned (cross modulation terms vanish in this case).
CAVITY RESPONSE to AM and PM SIGNALS
Since the total current IT is the vector difference of the two currents Ig and Ib, the final
modulation transfer function from Ig and Ib to the cavity voltage Vc are given by:
 0
 0
g
Gaa
 ...
sin f g  fT ;
g
Gap
 ...
G bpp  G pp
Ib
I
cosfb  fT   G pa b sin fb  fT ;
IT
IT
b
Gaa
 ...
G bpa  G pa
Ib
I
cosfb  fT   G pp b sin fb  fT ;
IT
IT
b
Gap
 ...
g
G pa
 G pa
Ig
IT
Ig
IT
cosf g  fT   G pa
Ig
cosf g  fT   G pp
Ig
IT
IT
similar expressions
sin f g  fT ;
g
G pp
 G pp
CAVITY RESPONSE to AM and PM SIGNALS
The complete expressions for the modulation transfer functions can be worked out with
some algebra and result to be:
g
G pp

g
G pa

G bpp



 1  Y cosfs s   2 1  tan 2 f z  Y cosfs  tan f z sin fs 

s 2  2 s   2 1  tan 2 f z

 0
 tan f z  Y sin fs s   2Y sin fs  tan f z cosfs 


s 2  2 s   2 1  tan 2 f z


Y   cosfs s   2 tan f z sin fs  cosfs 


s 2  2 s   2 1  tan 2 f z


Y  sin fs s   2 tan f z cosfs  sin fs 
G bpa

Y
I b RL
q R
sgn(fs )
g
g
g
g
 2 b L ; Gaa
 G pp
; Gap
 G pa
;
1
Vc
Tb Vc
sgn( )

s 2  2 s   2 1  tan 2 f z

 0
The beam, with the uncommon exception of bunch length comparable with the RF
wavelength, can only exhibit phase modulation (the amplitude of the beam spectrum line
at the generator frequency is fixed and equal to 2qb/Tb).
Beam Transfer Function
Let’s consider the synchrotron motion fb(t) of a particle in a storage ring where the
accelerating voltage is not modulated (i.e. fc(t)=fclock , being fb(t) measured respect to the
same fclock). The synchrotron equation has the form:
fb  2 fb  w sfb  0
with w s  w RF
 Vc sin fs
2  2h E / e
where  accounts for frictional terms which may damp or amplify the oscillations. If the
cavity phase fc(t) is modulated the synchrotron equation becomes (neglecting  ):
L transform
fb  w s2fb  w s2fc


s
fb ( s)  w s2fb ( s)  w s2fc ( s)
2

fb ( s)
w s2
B( s) 
 2
fc ( s) s  w s2
The “Beam transfer function” B(s) measures the response of the beam to a cavity phase
modulation in the Laplace s-domain. The response is one-to-one at dc (the beam follows
the slow phase motion of the cavity) and peaks at the synchrotron frequency (infinitely if
no damping is provided).
The function B(s) represents the forward block in the active feedback systems aimed at
generating some damping term  to stabilize the beam.
The beam phase also depends on the cavity voltage amplitude, according to:
cosfb  
Vloss
V
1 dVc
  sin fb dfb   loss
dV

d
f

c
b
Vc
tanfb  Vc
Vc2
CAVITY RESPONSE to AM and PM SIGNALS:
PEDERSEN MODEL
The whole generator-cavity-beam system can be graphically represented in a diagram
called Pedersen Model. The modulation transfer functions vary with the stored current and
definitely couple the servo-loops and the beam loops implemented around the system.
Generator
p
Cavity +
g
G pp
(s )
p
+
B(s)
Beam
p
G bpp (s )
g
G pa
(s )
G bpa (s )
g
Gap
(s )
b
Gap
(s )
a
a
g
Gaa
(s )
+
a
b
Gaa
(s )
1
tan fs
• Stability of the
generator-cavity-beam
system
Stability of the System without Loops
Even if no AM and PM are applied from the generator (no loops acting on the system),
still there is an intrinsic loop from cavity to beam and backward. The stability of the
system can be derived from the characteristic equation:


1
B( s) G bpp ( s) 
G bpa ( s)  1  0
tan fs


The zeros of this equation must have negative real part to avoid system instability. If the
equation is put in polynomial form a4 s 4  a3s3  a2 s 2  a1s  a0  0 we must have:
a4  0;
a3  0;
a2 
a1a4
 0;
a3
a1a2a3  a12a4  a32a0  0
These conditions are fulfilled if the two following relations are satisfied:
sgn   tanf z   0
and
sgn  
Y sin 2f z 
1
2 sin fs
These are the Robinson instability limits and are predicted in the identical form by the
theory of the longitudinal coupled bunched instabilities applied to the beam barycentre
motion ( CB mode “0”, all bunches in phase).
Stability of the System without Loops
The 1st Robinson condition states that the beam is
stable at positive  if the upper synchrotron
sideband samples a larger cavity resistive
impedance with respect to the lower synchrotron
sideband (the opposite for negative ).
 >0 => unstable
 <0 => stable
 >0 => stable
 <0 => unstable
w  ws
w  ws
w  ws
w  ws
The damping constant R is given by:
 R  sgn   
w s Ib
Z r w  w s   Z r w  w s 
4Vc sin fs
The condition R > 0 (beam stability) is immediately satisfied if sgn   tanfz   0
This is not a primiry issue since the beam loading detunes the cavity in the direction where
Robinson damping is generated.
The Robinson 2nd limit can be also derived from the bunched beam coherent instabilities
theory. The coherent frequency of the barycentre motion of the bunches wsc is given by:
w sc  w si
 


Ib
1
1
Z i w  w sc  Z i w  w sc
2 Vc sin fs

and the Robinson 2nd limit is reached when wsc gets to zero, i.e. :
1
I b Zi w 
0
Vc sin fs

Y sin 2f z 
1 
2 sin fs
sgn  
Y sin 2f z 
1
2 sin fs
Stability of the System without Loops
The Robinson 2nd limit has also another important interpretation. From the phasor diagram
the following general relations can be derived:




1 Ib
1 sin f g  f z
Y

cosf z IT
cosf z sin f g  fs
Robinson
2nd limit


 0
fs  0


1 sin f g  f z sin 2f z 
 1
cosf z sin f g  fs 2 sin fs 


sin f g  fs  sin fs 
sin f g  f z sin f z 
 1

f g  f z  fs
The portion of the cavity voltage induced by the external generator is the phasor Vg=ZLIg .
The phase of Vg relative to Ig is fz . At the 2nd Robinson limit the absolute phase of Vg is
fgfz  fs which means that the beam is on the crest of Vg . For the barycentre coherent
motion only Vg provides longitudinal focusing since the beam induced voltage simply
follows the bunches and does not contribute to the restoring force. At the Robinson 2nd limit
there is no residual focusing for the coherent synchrotron oscillation and the beam is
suddenly lost.
Stability of the System without Loops
For an optimally tuned cavity the 2nd Robinson limit appears at the threshold:
tan f z  sgn   Y sin fs 
 Yth  sgn 
2 sin fs 
sin 2f z 
Yth 
1  tan 2 f z  1  Yth2 sin 2 fs 
 sgn    sin fs 

Yth
tanf z 
1
V
 c
cosfs Vloss
In the optimal tuning case, the Robinson
2nd limit in terms of current gives:
Overvoltage factor
I th Rs
1

1   Vc cosfs

Ith  I  opt
1 
1 
which means that the limit is reached beyond the current value I-opt matching the input
coupling. However, getting close to the 2nd Robinson limit the coherent frequency of mode
0 can be very low. The motion of the bunches gets very sensitive to the low frequency
noise in the RF systems, and some beam feedback systems designed for the nominal
synchrotron frequency can work improperly. Feed-forward and direct RF feedback
systems can be implemented to avoid such effect.
It can be demonstrated that the 2nd Robinson limit can be avoided also by moving from the
optimal tuning condition adding some extra tuning f0 to the system. The coherent
frequency of mode 0 does not get to zero in this case but tends lowering to an asymptotic
non-zero value. However, this cure has a cost in terms of RF power reflected from the
cavity coupler and require therefore oversized RF power sources.