1. Update on transient beam loading at injection
Ivan Karpov and Philippe Baudrenghien
Acknowledgments:
Helga Timko and John Molendijk
LHC/HL-LHC Beam Dynamics WG meeting

2. One turn delay feedback
Recap on the progress
Circulator
𝐼 g , generator current
𝐼 b,rf , rf component of the beam current
Generator
rf cavity
𝐼 r , Reflected current
delay
Load
𝑉, cavity voltage – +
Digital rf feedback Σ Σ
𝑉 ref , reference voltage + +
𝜖= 𝑉 ref −𝑉, error signal +
Analog rf feedback Σ +
One turn delay feedback

3. One turn delay feedback
Recap on the progress
Circulator
𝐼 g , generator current
𝐼 b,rf , rf component of the beam current
Generator
rf cavity
𝐼 r , Reflected current
delay
Load
𝑉, cavity voltage – +
Digital rf feedback Σ Σ
𝑉 ref , reference voltage + +
𝜖= 𝑉 ref −𝑉, error signal +
Analog rf feedback Σ +
One turn delay feedback
BLonD and stand-alone implementations of detailed model of direct rf feedback (analog and digital) agree well

4. One turn delay feedback
Recap on the progress
Circulator
𝐼 g , generator current
𝐼 b,rf , rf component of the beam current
Generator
rf cavity
𝐼 r , Reflected current
delay
Load
𝑉, cavity voltage – +
Digital rf feedback Σ Σ
𝑉 ref , reference voltage + +
𝜖= 𝑉 ref −𝑉, error signal +
Analog rf feedback Σ +
One turn delay feedback
Simulations with simplified model of one-turn delay feedback (OTFB) show large power transients at injection
→ This needs to be verified with the accurate model of OTFB including low pass filter (LPF)

5. OTFB model with LPF AC coupling
Error signal
Sampling interval, 25 ns
𝑦 𝑛 =𝑦 𝑛−1 1− Δ𝑡 𝜏 AC +𝑥 𝑛 −𝑥 𝑛−1
AC coupling
𝑛 rev = 𝑡 rev /Δ𝑡
𝑦 𝑛 = 𝑎 OTFB 𝑦 𝑛− 𝑛 rev +𝐾 1− 𝑎 OTFB 𝑥(𝑛 − 𝑛 OTFB )
OTFB
the number of taps of finite impulse response (FIR) filter
𝑦 𝑛 = 𝑖=0 𝑁 tap 𝑏 𝑖 𝑥 𝑛−𝑖
LPF
𝑛 OTFB = 𝑛 rev − 𝑁 tap −1 2 − 𝑛 delay
Total delay in OTFB branch
AC coupling
In the LHC 𝑎 OTFB = , 𝐾=10, 𝜏 AC =110 𝜇s, 𝑁 tap =63
→ How 𝑛 delay affects power transients?

6. Power transients with full OTFB model
Injection of 1000 Gaussian bunches with 𝜏 4𝜎 =1.5 ns ( 𝐹 𝑏 =0.64) and 𝑁 𝑝 =2.3× ;
rf cavities are pre-detuned with Δ𝜔=2𝜋Δ𝑓
𝑃 theory = 𝑉 cav 𝐼 b,rf 8
→ Power overshoot is either during transients or in steady-state
→ Optimum delay is about 42 samples (1050 ns) with 𝑃 max =1.2 𝑃 theory

7. Cavity-beam-generator model
Tuner
Circulator
𝐼 g , generator current
𝐼 b,rf , rf component of the beam current
Generator
rf cavity
𝐼 r , Reflected current
delay
Load
𝑉, cavity voltage – +
Digital rf feedback Σ Σ
𝑉 ref , reference voltage + +
𝜖= 𝑉 ref −𝑉, error signal
Analog rf feedback + Σ +
One turn delay feedback
→ Tuner is important for comparisons with measurements

8. Tuner model (Preliminary)
Adaptation rate
Δ𝑓 𝑓 0 new = Δ𝑓 𝑓 0 old − 𝜇 2 Im 𝑉 𝐼 𝑔 min +Im 𝑉 𝐼 𝑔 max 𝑉 cav 2
Tuning algorithm*
Cavity voltage and generator current values are down sampled using Cascaded integrator–comb (CIC) filter
𝑦 𝑛 =2𝑦 𝑛−1 −𝑦 𝑛− 𝑥 𝑛 −2𝑥 𝑛−8 +𝑥 𝑛−16
Minimum and maximum values are calculated over one turn
*P. Baudrenghien, The Tuning Algorithm of the LHC 400 MHz Superconducting Cavities, CERN-AB , 2007

9. Tuner model. Initial tests
Injection of 12 bunches after 100 turns, then 144 bunches at turn 500,
after 250 turns tuner and beam phase adaptation are on,
Initially cavity is on tune, 𝜏 4𝜎 =1 ns and 𝑁 𝑝 =1.2× 10 11
With 12 bunches Δ𝑓=0.37 Δ 𝑓 theory
With bunches Δ𝑓=0.96 Δ 𝑓 theory

10. Comparison with measurements in steady-state
E = 6.5 TeV, 2244 bunches, 𝑁 𝑝 =1.2× 10 11
Simulations:
Δ𝑓=0.935 Δ 𝑓 theory , 𝑛 delay =46, 𝜏 4𝜎 =1 ns
*T. Mastoridis, P. Baudrenghien and J. Molendijk, PRAB 20, (2017)

11. Comparison with measurements in steady-state
E = 6.5 TeV, 2244 bunches, 𝑁 𝑝 =1.2× 10 11
Simulations:
Δ𝑓=0.935 Δ 𝑓 theory , 𝑛 delay =46, 𝜏 4𝜎 =1 ns
*T. Mastoridis, P. Baudrenghien and J. Molendijk, PRAB 20, (2017)

12. Comparison with measurements in steady-state
E = 6.5 TeV, 2244 bunches, 𝑁 𝑝 =1.2× 10 11
Simulations:
Δ𝑓=0.953 Δ 𝑓 theory , 𝑛 delay =42, 𝜏 4𝜎 =1 ns
*T. Mastoridis, P. Baudrenghien and J. Molendijk, PRAB 20, (2017)

13. Comparison with measurements in steady-state
E = 6.5 TeV, 2244 bunches, 𝑁 𝑝 =1.2× 10 11
Simulations:
Δ𝑓=0.953 Δ 𝑓 theory , 𝑛 delay =42, 𝜏 4𝜎 =1 ns
*T. Mastoridis, P. Baudrenghien and J. Molendijk, PRAB 20, (2017)

14. Conclusions
Detailed model of OTFB in the LHC was implemented in the time-domain beam-cavity-generator interaction equations.
The power transients depend significantly on the delay in the OTFB branch.
Preliminary results with implemented tuner show overall good agreement with steady-state measurements but it depends on the delay in the OTFB branch.
Next steps:
Comparison with MD data and BLonD model

15. Thank you for your attention!