1. Time Domain Modelling for Crab Cavity LLRF Performance
A. C. Dexter
Cockcroft Institute (Lancaster University)
Part of this work was funded through EUCARD
It has also been funded by STFC via the Cockcroft Institute

2. Lancaster time domain modelling
Cavity modes of interest modelled as LCR circuits.
Assume large intrinsic Q (>1000) so envelope equations apply.
Allow modes to be perturbed by micro-phonics and Lorentz de-tuning.
Time step for envelope equations typically one RF period.
State determined by slowly varying I and Q component in each mode.
Beam-loading gives step changes in I and Q components of each mode.
Bunch form factor accounts for long bunches.
Voltage probe measurement is derived from the voltage in each mode with coupling factors. Expected measurement errors are added randomly.
Controller reacts to “measured” voltage with delays and possibly feedforward.
Klystron model typically includes just its output cavity. I
IQ Detector
Controller
Vector Modulator Q
cavity pick-up
Micro-phonics
Klystron
beam offset

3. Problems addressed to date
(2008) Phase and amplitude stability as function of controller for ILC crab cavity
(2009) Phase and amplitude in SPL SCRF cavities driven with 704 MHz Magnetrons
(2012) Cavity spectral noise computation for the (Jlab/Lancaster) 4 rod crab cavity in the LHC as needed for bunch lifetime calculations.
(2012) Transverse kick following quench of superconducting crab cavity
(2016) Validation of Low Order Mode active damping concept
A. C. Dexter, G. Burt, R. Apsimon, “Active lower order mode damping for the four rod LHC crab cavity”, NIMA Vol 844, pp 62-71, 2017
(2017) Modelling of acceleration in the LHC with full detuning
Bunch arrival times depend on previous kick, only average phase in acceleration cavities is controlled
(2018) Modelling of crab cavity response in the SPS with a faulty pick up
(2019) “Prediction of Beam Losses during Crab Cavity Quenches at the HL-LHC”
R. Apsimon, G. Burt, A.C. Dexter et al. in review for PRAB
In this work the time domain cavity simulation was linked with particle tracking using SixTrack

4. Example crab LLRF parameters
We have different versions of code for different machines and cavity types
Master oscillator frequency (MHz)
Bucket frequency (MHz)
Energy set point (Joules per cell
Maximum Amp Power (Watts per cell)
Maximum beam offset (mm)
1 for random offset 0 for periodic
Offset fluctuation frequency (Hz)
Initial Bunch phase retard (degrees)
Bunch phase jitter (degrees)
1 for random charge fluc 0 for periodic
Phase jitter frequency (Hz)
Bunch charge fluctuation (fraction)
1 for random charge fluc 0 for periodic
Charge fluctuation frequency (Hz)
Bunch charge (Coulombs) e-8
Bunch train length (seconds) e-6
Bunch train gap length (seconds) e-6
RF advance time (seconds) e-6
Cavity freq. shift from microphonics Hz
Vibration frequency of cavity (Hz)
Initial vibration phase (degrees, sin) :
Measurement phase error in degrees
Measurement amplitude error as fraction
Delay for control system in seconds : e-6
Control update interval in seconds : e-6
Initial gain constant for controller
Amplifier Bandwidth : e6
Measurement filter bandwidth : e6
Feed forward sum jump (~1.2e6) : e6
LHC Bunch structure was hard coded
Three input files
1st defines the cavity
2nd defines the LLRF
3rd defines the calculation
The dataset here corresponds to the hypothetical case of the 4 rod crab cavity with an 8.5 kW solid state amplifier. Results shown on next slide as illustration
Offset frequencies were chosen to assist illustrative graphs rather than being based on likely values.
Chosen large so can be seen on timescale of milli-seconds

5. LHC crab cavity simulation example
Simulations from 2012 of four rod crab cavity in LHC with full bunch train
Simulation shows the effect of a 2kHz Microphonic and a 5kHz offset fluctuation
Simulation is made assuming no measurement errors or random fluctuations
Power follows beam offset
Dips follow gaps in bunch structure
Amplitude correction depends on gain
Phase follows microphonics

6. Coding detail
Time Domain is a little brute force but is good to assist understanding
Mode I & Q (Ar &Ai) components step with 4th order Runge Kutta algorithm
Envelope equations are with respect to the master oscillator
Each bunch (ib) is followed – one turn maps can be applied
For dipole cavities solve in terms of accelerating field off centre by l/4
Centre angular frequency of mode
do j=1, modes
mode_phase = wc0(j)*bunch_time + bunch_phase_err
bunch_kick(ib) = 0.5*(Ar(j)*cos(mode_phase) - Ai(j) *sin(mode_phase)
! apply half kick before bunch changes cavity voltage
Voltage_kick(js) = 0.5*bunch_charge*wm*ROQ(j)
if(mode_order(j).eq.1) Voltage_kick(js) = Dipole_kick_factor*Voltage_kick(js)
! Increment in phase and quadrature voltage components
Ar(js) = Ar(js) - Voltage_kick(j) * cos(mode_phase)
Ai(js) = Ai(js) - Voltage_kick(j) * sin(mode_phase)
bunch_kick(ib) = bunch_kick(ib) *(Ar(js)*cos(mode_phase) - Ai(js) *sin(mode_phase)
sum_bunch_kick(ib) = sum_bunch_kick(ib) + bunch_kick(ib)
! apply second half kick after bunch changes cavity voltage
end do
Can we predict the noise floor from the measurement errors? To get dBc/Hz divide by

7. Spectral Analysis
Bunch Lifetime depends on spectral density near betatron frequencies
Run simulation for ~0.1 seconds to get 10 Hz steps in spectrum
Spectrum shows peaks at input frequencies as expected
Beam offset frequency = kHz
Cavity microphonics = kHz
Revolution frequency = kHz
Simulation here without measurement errors
kHz + 5 kHz
22.49 kHz - 5 kHz
Gaps in the bunch structure drive large responses
Large Fourier coefficients occur at the disturbance frequencies mixed with harmonics of the revolution frequency. The level in between peaks is determined by the sampling interval. Note that unit are DB below the carrier and not dBc/Hz.

8. RF Spectral Noise and Beam Effects
Both the phase f and the offset x are oscillatory
at crab cavity
Can compute cumulative offsets of particles by combining voltage kicks with one turn maps as a function of their Betatron and Synchrotron phases. Continue for de-coherence time then assume random walk.
betatron motion
synchrotron motion
crab voltage kick
axis
Have assumed a cosine function for synchrotron phase but would like a better approximation for the LHC long bunch.
Noise at synchrotron frequency gives differential transverse kicks
Noise at betatron frequency gives differential longitudinal kicks

9. With Errors Amplitude measurement error = 0.01%
Can we predict the noise floor from the measurement errors? To get dBc/Hz divide by
Amplitude measurement error = 0.01%
Phase measurement error = degrees
Time delay = 0.5 ms

10. LHC Crab Cavity Quench Studies
Transverse kick following quench + detune of superconducting crab cavity
(results for slow undetected quench below)
Quench from
Qext= to
Qext = 30000 in
1ms
+ detune
½ bandwidth
Detune plus quench can gives about 20 degrees phase error in this case.

11. Preliminary SPS Beam-loading Data 1
Fill = + 60 bunches
+ 20 gaps
+ 60 bunches
+ 784 gaps
Bunch offset = mm
RF system had restricted power availability
IOT had quadratic power response
Probe gave low reading during a bunch train due to poor positioning
All features were easily incorporated into the time domain model.
Measurements from MD7

12. Preliminary SPS Beam-loading Data 2
Fill = + 60 bunches
+ 20 gaps
+ 60 bunches
+ 784 gaps
Bunch offset = mm
RF system had restricted power availability
IOT had quadratic power response
Probe gave low reading during a bunch train due to poor positioning
Measurements from MD7
Spikes not seen in simulation as I had not been informed that a fast analogue feed back was used to limit power but it takes hundreds of nano seconds to act ( ~ 10 bunches).

13. 4 Rod cavity LOM excitation – bunch kicks
The 400 MHz four rod crab cavity has to have an acceleration mode near 370 GHz
Accelerating modes take energy from the beam and for an unbroken bunch train give the same kick to each bunch. This is not true when the train has gaps.

14. 4 Rod cavity LOM resonant excitation
Excitation of the LOM voltage depends on the Mode frequency and has resonances at multiples of the bunch repetition frequency (40 MHz here)
Strong damping limits the voltage induced and the maximum kick.

15. LLRF scheme to damp LOM
Away from resonance use active damping to compensate gaps so that every bunch gets the same kick. Important to limit bunch growth. This kick can be set at precisely zero or alternatively at the steady state point.
VCO
PLL filter
Phase Detector
IQ Detector
Controller
Vector Modulator I Q
Beam pick-up
Crab Cavity
set point
÷ 4
x 37
Bunch
Counter
Processor
Must drive at LOM frequency.
Must be synchronised to bunches with a set fractional ratio to minimise noise at small offset frequencies.
Must have virtual response in less than 25ns hence an element of feed forward.
Each bunch rotates cavity phasor to an new location.
The controller moves the set point simultaneously (every 25 ns) so that departures form the desired phase are corrected
Multiplying bunch interval by 9⅓ means that only 3 set points are needed

16. Active Damping Applied
No power need during train, 100W needed to correct during gap

17. The End
Thanks to
Graeme Burt for managing the UK contribution to the Hi- Lumi LHC
Philippe Braudrenghien and Rama Calaga for hosting Emi Yamakawa at CERN to work on LLRF system issues for Hi- Lumi LHC and for many helpful discussions.
Robert Apsimon for linking particle tracking to time domain cavity simulations and undertaking detailed quench studies.

18. Cavity Model Input coupler
Mode damping included in mode circuit resistance
Solve in time domain with 4th order Runge Kutta one time step per RF cycle