transient beam loading at injection - cern
TRANSCRIPT
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Transient beam loading at injection
Ivan Karpov and Philippe Baudrenghien
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Power requirements at injection
The full-detuning scheme has no advantage during machine filling (previous meeting)
โ The half-detuning scheme needs to be used
Required peak power in steady-state situation ๐HD =๐cav แ๐ผb,rf
8, but is it
the same during injection process?
โ Evaluation of power including details of LLRF system is necessary
Peak beam rf current
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Cavity-beam-generator model developed for FCC
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rf cavity
Load
Circulator
Generator
LLRF ฮฃ
โ
+
๐ผb,rf, rf component of
the beam current
๐ref, reference voltage
๐, cavity voltage
๐ผg, generator current
๐ผr, Reflected current
๐ ๐ก , ๐ผb,rf ๐ก , ๐ผg ๐ก , ๐ผr ๐ก are time-varying complex phasors rotating with angular rf frequency ๐rf
๐๐ ๐ก
๐๐ก= โ๐ ๐ก
1
๐โ ๐ฮ๐ + ๐rf ๐ /๐ ๐ผ๐ ๐ก โ
๐ผb,rf ๐ก
2
*J. Tรผckmantel, Cavity-Beam-Transmitter Interaction Formula Collection with Derivation, CERN-ATS-Note-2011-002, 2011
For given ๐ผg ๐ก and ๐ผb,rf ๐ก the cavity voltage can be found from*
Cavity filling time ๐ = 2๐L/๐rf, cavity detuning ฮ๐ = ๐r โ ๐rf, ๐ /๐ = 45 ฮฉ
โ How do we get ๐ผb,rf ๐ก and ๐ผg ๐ก ?
๐ = ๐ref โ ๐, error signal
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rf component of the beam current
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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
โ ๐ผb,rf ๐ก can be replaced by a stepwise function ๐(๐ก) with sampling rate 1/๐กbb = 40 MHz (๐กbb - bunch spacing), so
๐ผb,rf ๐ก = โ๐ แ๐ผb,rf ๐(๐ก)
โ Synchrotron motion can be neglected
Peak rf current แ๐ผb,rf =๐๐p๐นb
๐กbbBunch form factor ๐น๐ = 2๐โ
๐rf2 ๐2
2 ๐p - number of particles per bunch
Fourier transform
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Generator current as output of LLRF module
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Delay, ๐delay Gain, G
OTFB
AC coupling AC coupling
๐ผg ๐ก ๐ ๐กฮฃ+
+
First simplified model (analog direct rf feedback): ๐ผg ๐ก = ๐บ ๐ ๐ก โ ๐delay = ๐บ๐(๐ก โ ๐delay)
Correction signal Error signal
๐ ๐ก
The direct feedback gain is defined by the loop stability ๐บ = 2 ๐ /๐ ๐rf๐delayโ1
for ๐delay = 650 ns
For the finite gain cavity voltage will be lower than ๐ref
It improves longitudinal multi-bunch stability
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Generator current as output of LLRF module
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Delay, ๐delay Gain, G
OTFB
AC coupling AC coupling
๐ผg ๐ก ๐ ๐กฮฃ+
+
Model for analog and digital direct rf feedback:๐๐ผg ๐ก
๐๐ก=๐ผg ๐ก
๐d๐d+๐บ
๐d๐ ๐ก โ ๐delay + ๐บ
๐๐ ๐ก โ ๐delay
๐๐ก
Correction signal Error signal
๐ ๐ก
In the LHC ๐d = 10, ๐d โ2
๐rev=
๐กrev
๐, for the revolution period ๐กrev โ 88.9 ฮผs
Frequency dependent gain
๐๐rev
1
๐d
1/๐d1/๐d๐d
With digital rf feedback error in cavity voltage can be reduced
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Generator current as output of LLRF module
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Delay, ๐delay Gain, G
OTFB
AC coupling AC coupling
๐ผg ๐ก ๐ ๐กฮฃ+
+
Model for one-turn delay feedback:
Correction signal Error signal
๐ ๐ก
In the LHC ๐OTFB =15
16, ๐พ = 10, ๐AC = 100 ฮผs.
OTFB reduces transient beam loading and improves longitudinal multi-bunch stability
Frequency dependent gain
๐๐rev
1
๐d
1/๐d1/๐d๐d
๐ฆ ๐ก = ๐OTFB๐ฆ ๐ก โ ๐กrev + ๐พ 1 โ ๐OTFB ๐ฅ(๐ก โ ๐กrev + ๐delay)
Removes DC offset
from the signal
Model AC coupling: ๐ฆ ๐ก
๐๐ก= โ
๐ฆ ๐ก
๐AC+
๐๐ฅ ๐ก
๐๐ก
๐ฅ
๐ฆ
๐ฅ๐ฆ
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Results: analog DFB only (1/2)
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Injection of 3 ร 48 bunches with ๐น๐ = 1 and ๐๐ = 2.3 ร 1011; rf cavities are pre-detuned with ฮ๐ = 2๐ฮ๐
๐cav แ๐ผb,rf8
โ The requested power is below steady-state limit, but what happens with cavity voltage?
๐ ๐ก =1
2๐ /๐ ๐L ๐ผg ๐ก
2
*J. Tรผckmantel, Cavity-Beam-Transmitter Interaction Formula Collection with Derivation, CERN-ATS-Note-2011-002, 2011
Generator power*
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Results: analog DFB only (2/2)
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Injection of 3 ร 48 bunches with ๐น๐ = 1 and ๐๐ = 2.3 ร 1011; rf cavities are pre-detuned with ฮ๐ = 2๐ฮ๐
๐cav
As expected for the finite gain, the voltage is lower than it is requested
โ This explains lower power consumption
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Results: analog + digital DFB (1/2)
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Injection of 3 ร 48 bunches with ๐น๐ = 1 and ๐๐ = 2.3 ร 1011; rf cavities are pre-detuned with ฮ๐ = 2๐ฮ๐
There is a small overshoot in power after injection
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Results: analog + digital DFB (2/2)
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Injection of 3 ร 48 bunches with ๐น๐ = 1 and ๐๐ = 2.3 ร 1011; rf cavities are pre-detuned with ฮ๐ = 2๐ฮ๐
Some modulation of the cavity voltage amplitude and more significant modulation of the cavity voltage phase
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Results: analog + digital DFB + OTFB (1/2)
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Injection of 3 ร 48 bunches with ๐น๐ = 1 and ๐๐ = 2.3 ร 1011; rf cavities are pre-detuned with ฮ๐ = 2๐ฮ๐
There is a difference between first and the second turn after injection
Significant overshoot due to action of OTFB
First turn
Second turn
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Results: analog + digital DFB + OTFB (2/2)
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Injection of 3 ร 48 bunches with ๐น๐ = 1 and ๐๐ = 2.3 ร 1011; rf cavities are pre-detuned with ฮ๐ = 2๐ฮ๐
First turn
Second turn
Better compensation of the cavity voltage at the second turn by OTFB costs significantly more power
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Conclusions
โข Detailed model of LLRF in the LHC was implemented in the time-domain beam-cavity-generator interaction equations.
โข 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.
โข Next steps:
โข Comparison with MD data and BLonD model
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Benchmarks
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Expected impulse response constant of OTFB
๐OTFB =๐กrev
1 โ ๐OTFBโ 1.5 ms
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Long term evolution
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