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Time Domain Modelling for Crab Cavity LLRF Performance

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Presentation on theme: "Time Domain Modelling for Crab Cavity LLRF Performance"— Presentation transcript:

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


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