transient beam loading at injection - cern
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Transient beam loading at injection
Ivan Karpov and Philippe Baudrenghien
Power requirements at injection
The full-detuning scheme has no advantage during machine filling (previous meeting)
β The half-detuning scheme needs to be used
Required peak power in steady-state situation πHD =πcav απΌb,rf
8, but is it
the same during injection process?
β Evaluation of power including details of LLRF system is necessary
Peak beam rf current
2
Cavity-beam-generator model developed for FCC
3
rf cavity
Load
Circulator
Generator
LLRF Ξ£
β
+
πΌb,rf, rf component of
the beam current
πref, reference voltage
π, cavity voltage
πΌg, generator current
πΌr, Reflected current
π π‘ , πΌb,rf π‘ , πΌg π‘ , πΌr π‘ are time-varying complex phasors rotating with angular rf frequency πrf
ππ π‘
ππ‘= βπ π‘
1
πβ πΞπ + πrf π /π πΌπ π‘ β
πΌb,rf π‘
2
*J. TΓΌckmantel, Cavity-Beam-Transmitter Interaction Formula Collection with Derivation, CERN-ATS-Note-2011-002, 2011
For given πΌg π‘ and πΌb,rf π‘ the cavity voltage can be found from*
Cavity filling time π = 2πL/πrf, cavity detuning Ξπ = πr β πrf, π /π = 45 Ξ©
β How do we get πΌb,rf π‘ and πΌg π‘ ?
π = πref β π, error signal
rf component of the beam current
4
The rf power chain (amplifier, circulator, etc.) has limited bandwidth
For power transient calculations, we are interested dynamics of the system for the first few turns after injection
β πΌb,rf π‘ can be replaced by a stepwise function π(π‘) with sampling rate 1/π‘bb = 40 MHz (π‘bb - bunch spacing), so
πΌb,rf π‘ = βπ απΌb,rf π(π‘)
β Synchrotron motion can be neglected
Peak rf current απΌb,rf =ππpπΉb
π‘bbBunch form factor πΉπ = 2πβ
πrf2 π2
2 πp - number of particles per bunch
Fourier transform
Generator current as output of LLRF module
5
Delay, πdelay Gain, G
OTFB
AC coupling AC coupling
πΌg π‘ π π‘Ξ£+
+
First simplified model (analog direct rf feedback): πΌg π‘ = πΊ π π‘ β πdelay = πΊπ(π‘ β πdelay)
Correction signal Error signal
π π‘
The direct feedback gain is defined by the loop stability πΊ = 2 π /π πrfπdelayβ1
for πdelay = 650 ns
For the finite gain cavity voltage will be lower than πref
It improves longitudinal multi-bunch stability
Generator current as output of LLRF module
6
Delay, πdelay Gain, G
OTFB
AC coupling AC coupling
πΌg π‘ π π‘Ξ£+
+
Model for analog and digital direct rf feedback:ππΌg π‘
ππ‘=πΌg π‘
πdπd+πΊ
πdπ π‘ β πdelay + πΊ
ππ π‘ β πdelay
ππ‘
Correction signal Error signal
π π‘
In the LHC πd = 10, πd β2
πrev=
π‘rev
π, for the revolution period π‘rev β 88.9 ΞΌs
Frequency dependent gain
ππrev
1
πd
1/πd1/πdπd
With digital rf feedback error in cavity voltage can be reduced
Generator current as output of LLRF module
7
Delay, πdelay Gain, G
OTFB
AC coupling AC coupling
πΌg π‘ π π‘Ξ£+
+
Model for one-turn delay feedback:
Correction signal Error signal
π π‘
In the LHC πOTFB =15
16, πΎ = 10, πAC = 100 ΞΌs.
OTFB reduces transient beam loading and improves longitudinal multi-bunch stability
Frequency dependent gain
ππrev
1
πd
1/πd1/πdπd
π¦ π‘ = πOTFBπ¦ π‘ β π‘rev + πΎ 1 β πOTFB π₯(π‘ β π‘rev + πdelay)
Removes DC offset
from the signal
Model AC coupling: π¦ π‘
ππ‘= β
π¦ π‘
πAC+
ππ₯ π‘
ππ‘
π₯
π¦
π₯π¦
Results: analog DFB only (1/2)
8
Injection of 3 Γ 48 bunches with πΉπ = 1 and ππ = 2.3 Γ 1011; rf cavities are pre-detuned with Ξπ = 2πΞπ
πcav απΌb,rf8
β The requested power is below steady-state limit, but what happens with cavity voltage?
π π‘ =1
2π /π πL πΌg π‘
2
*J. TΓΌckmantel, Cavity-Beam-Transmitter Interaction Formula Collection with Derivation, CERN-ATS-Note-2011-002, 2011
Generator power*
Results: analog DFB only (2/2)
9
Injection of 3 Γ 48 bunches with πΉπ = 1 and ππ = 2.3 Γ 1011; rf cavities are pre-detuned with Ξπ = 2πΞπ
πcav
As expected for the finite gain, the voltage is lower than it is requested
β This explains lower power consumption
Results: analog + digital DFB (1/2)
10
Injection of 3 Γ 48 bunches with πΉπ = 1 and ππ = 2.3 Γ 1011; rf cavities are pre-detuned with Ξπ = 2πΞπ
There is a small overshoot in power after injection
Results: analog + digital DFB (2/2)
11
Injection of 3 Γ 48 bunches with πΉπ = 1 and ππ = 2.3 Γ 1011; rf cavities are pre-detuned with Ξπ = 2πΞπ
Some modulation of the cavity voltage amplitude and more significant modulation of the cavity voltage phase
Results: analog + digital DFB + OTFB (1/2)
12
Injection of 3 Γ 48 bunches with πΉπ = 1 and ππ = 2.3 Γ 1011; rf cavities are pre-detuned with Ξπ = 2πΞπ
There is a difference between first and the second turn after injection
Significant overshoot due to action of OTFB
First turn
Second turn
Results: analog + digital DFB + OTFB (2/2)
13
Injection of 3 Γ 48 bunches with πΉπ = 1 and ππ = 2.3 Γ 1011; rf cavities are pre-detuned with Ξπ = 2πΞπ
First turn
Second turn
Better compensation of the cavity voltage at the second turn by OTFB costs significantly more power
Conclusions
β’ Detailed model of LLRF in the LHC was implemented in the time-domain beam-cavity-generator interaction equations.
β’ Preliminary results show that one turn delay feedback can cause problems during injection process resulting in large power transients. Possible solution would be reduction of OTFB gain during machine filling.
β’ Next steps:
β’ Comparison with MD data and BLonD model
14
Benchmarks
15
Expected impulse response constant of OTFB
πOTFB =π‘rev
1 β πOTFBβ 1.5 ms
Long term evolution
16
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18
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