2. Agnieszka Chmielińska
Spiral beam screen for the FCC-hh injection kicker magnet: longitudinal and transverse impedance
Agnieszka Chmielińska
École Polytechnique Fédérale de Lausanne
CERN TE - ABT – PPE
May 14, 2019
Acknowledgements:
M. Barnes, S. Arsenyev, G. Bellotto, F. Caspers, A. Ferricker, I. Karpov, A. Lasheen, B. Popovic, B. Salvant, E. Shaposhnikova, W. Weterings, C. Zannini
CERN IT: M. Alandes Pradillo, P. Llopis Sanmillan

3. Introduction Motivation New concept
Conventional beam screen design (general features)
High voltage issues
New concept
Spiral beam screen
Longitudinal beam coupling impedance of the FCC-hh injection kicker magnet
Conventional beam screen (initial model)
Optimization of the conventional design
Impact on beam stability in the longitudinal plane
4. Transverse beam coupling impedance of the FCC-hh injection kicker magnet
Eigenmode and Wakefield simulations
Comparison of the conventional and spiral beam screen design
5. Conclusions

4. Motivation General features:
24 (NiCr) straight screen conductors inserted into grooved slots of alumina tube to shield ferrite yoke from the EM fields induced by the circulating beam:
→ reduced beam coupling impedance
Capacitive coupling to ground at the upstream end allows to preserve fast magnetic field rise time:
→ no major Eddy current loops
Overlap length determines resonances in the longitudinal impedance spectrum
→ possibility to shift resonant modes
into higher frequencies [1]:
i.e.: post LS1 MKI: Loverlap=130mm [2],
upgraded MKI8D: Loverlap=56mm [3].
Power loss in the ferrite yoke is also reduced due to the damping provided by the ferrite rings positioned around each end of the beam screen [3].
Conventional beam screen design (Catia model):

5. Motivation Magnet voltage and voltage of screen conductor (#1):
High voltage issues:
During magnetic field rise and fall, a significant voltage is induced on the screen conductors [2]:
→ possibility of HV breakdowns between adjacent screen conductors or/and between screen conductors and metallic cylinder.
Non-symmetric vacuum gap is introduced to reduce the electric field.
Maximum conductor and inter-conductor voltages (for 60 kV PFN voltage):
Beam screen cross-section:
Induced voltage:
Max. induced voltage:
Induced voltage:
Max. difference between adjacent conductors:
for

6. New concept: spiral beam screen
Each conductor has a form of spiral:
Spiral beam screen:
The coordinates of the i-th spiral screen conductor:
where and Nturn is the number
of turns in the kicker magnet aperture.
For the spiral beam screen, the distance of each screen conductor from the GND busbar changes along the z direction:
P-pitch, R-beam screen radius, a-spiral angle.
Beam screen cross-section:
Induced voltage:
For an integer number of turns along the aperture:
the induced voltage on each screen conductor is the same,
this is approximately one-half of the worst case straight screen conductor.
Spiral beam screen is expected to provide significantly improved HV performance

7. Beam screen for the FCC-hh injection kicker magnets: longitudinal impedance

8. Conventional beam screen (initial design)
Initial design: Loverlap=56 mm + non symmetric vacuum gap as in the upgraded LHC MKI8D installed in YETS 2017/2018.
Upstream end:
Downstream end: 8
Induced voltage: 8

9. Conventional beam screen (initial design)
Longitudinal impedance spectrum:
Power deposition distribution:
new
Evaluated worse case scenarios by shifting resonance peaks:
expected total power loss: 30.2 W/33.4 W/29.3 W
(smaller than for the LHC MKI8D [3]),
design is tolerant against relatively small shifts in the frequency of the resonance.
Power deposition distribution evaluated using two methods:
distributions are very similar as for the upgraded LHC MKI8D (Loverlap=56 mm), which was verified through measurements [3],
new method of calculating the power deposition distribution [4] improves the accuracy by about 10%.
Since the upgraded magnet LHC MKI8D did not limit LHC operation [5], no heating issues are expected for the FCC-hh injection kickers during operation with nominal beam parameters.
Next steps:
Investigation of possible means of beam coupling impedance improvement (longitudinal and transverse).
Improvement of the HV performance.

10. Conventional beam screen: optimization
Longitudinal impedance spectrum:
There is still possibility to improve conventional design.
Reduction of the overlap length:
best result for Loverlap=44mm.
This technique is limited due to the shift of the low frequency mode (~4 MHz) into higher frequencies.
Cumulative power loss:
Total power loss vs. Loverlap:

11. Spiral beam screen: generalized formula
Induced voltage:
Straight conductors: = =
Very good agreement between simulations and analytical predictions.

12. Spiral beam screen (optimized design)
Additional options:
Integer number of turns in the aperture
Symmetric vacuum gap (2mm width)
Ungraded lengths of screen conductors
Reduced overlap length
Possibility to connect all spirals together
Future option to use switch and minimize beam coupling impedance once machine is filled

13. Spiral beam screen (optimized design):
Longitudinal impedance spectrum:
Cumulative power loss:
Induced voltage:
No heating issues are expected
for the FCC-hh injection kickers with the spiral beam screen.
Induced voltage:

14. Impact on longitudinal beam stability (I)
Threshold for the coupled bunch instability in the longitudinal plane:
[6]
Contribution to the threshold at ~1000MHz:
18 kickers x 200 Ω / 8 MΩ ≅ 0.05%
Conclusion: Kicker magnets are very unlikely to drive coupled bunch instability in the longitudinal plane.

15. Impact on longitudinal beam stability (II)
Contribution to the total impedance budget for the loss of Landau damping.
Im(Z||/n):
Total impedance budget: 200 mΩ (assuming margin of a factor of 2 in comparison with LHC) [7].
Effective impedance calculation:
Conclusions:
For 18 kicker magnets with the conventional beam screen, the contribution to the total impedance budget is between 2.5% to 3%.
In the worst case, for the spiral with 3 turns, the contribution is between 5.4% to 5.7%.
The contribution of the FCC injection kicker magnets to the total impedance budget is not critical.
Without optimization, for 120 m long injection system (as initially assumed), the contribution would be 16% to 18%.

16. Beam screen for the FCC-hh injection kicker magnets: transverse impedance

17. Transverse impedance: Eigenmode simulations
Theoretical background (see [8]).
Computation method:
Wakefield simulations are necessary to determine resonant frequencies of the dipolar and the quadrupolar modes (5 simulations) (i.e. 10 modes). Note: typical simulation time for the kicker magnet (simplified model) on Linux HPC is 5 days!
For each mode determined in Wakefield we run a simulation in the Eigenmode Solver over a very narrow frequency range (10 simulations). The output typically consists of several eigenmodes (i.e. 5, hence 5*10=50 modes to be analysed).
Note: typical simulation time is 1 day.
Python post-processing is necessary to determine transverse modes.
Python post-processing is necessary to analyze f(x)=R/Q|| (x), calcualte the fit and derivatives.
Dipolar and quadrupolar impedance is calculated using the resonator model.

18. Results: conventional beam screen
Higher impedance predictions from Eigenmode simulations can be explained by not fully decayed wake potential in Wakefield Solver simulations (wakelength=500 m).
Modes at approximately: 20 MHz, 60.5 MHz, 100 MHz.
The resonant frequency is related to the length of the screen conductors (lambda/4 resonance?).
 f=3e8/(2*sqrt(9.9)*(( )/1000))=22.9MHz ?
Schematic drawing:
Mode at 20 MHz
Mode at 60.5 MHz

19. Results: conventional beam screen
Quadrupolar modes found only in the Wakefield simulations.

20. Results: spiral beam screen
Very good agreement between Eigenmode and Wakefield simulations with fully decayed wake potential.
Modes at approximately: 105 MHz, 150 MHz, 193 MHz, 237 MHz.
Example: mode at 193 MHz f
x[m]
Mode at 193 MHz:

21. Results: spiral beam screen
Spiral beam screen cancels out quadrupolar component of transverse beam coupling impedance.
NOTE: Code to analyze the transverse impedance for other devices is accessible on CERN gitlab:

22. Remark: transverse impedance budget
From recent studies, 18 kicker magnets with a conventional or a spiral beam screen design will not fit into the transverse impedance budget. In particular, 2 magnets with a 3 turn spiral would fit.
For the spiral design, the frequency of the resonant modes can be tuned, several options can be considered: i.e: 
magnets with a different integer number of turns around the aperture 
magnets with the same integer number of turns around the aperture, but with more turns outside the aperture
magnets with a different length of the beam screen 
perhaps magnets with a non-integer number of turns
9 detuned pairs of kicker magnet will fit into impedance budget.
There is no possibility of tuning the frequency of the conventional beam screen design without a destructive impact on the longitudinal impedance .

23. Next step
Longitudinal beam coupling impedance measurements of the prototype alumina tube with spiral screen conductors applied using copper paint.

24. Conclusions
Conventional beam screen (initial design) will not result in heating issues for the FCC injection kicker magnets.
The longitudinal impedance of the conventional design can be further optimized → possible means of improvement for the LHC injection kickers.
New concept of the spiral beam screen has been developed:
significantly improved high voltage performance
longitudinal impedance is very well understood → new formula proposed
flexibility in terms of different design options (number of turns, connection of spirals, etc.) allowing to achieve low longitudinal beam coupling impedance and proper power deposition distribution that will not cause heating issues.
no quadrupolar component of transverse beam coupling impedance (symmetric vacuum gap)
possibility to tune resonant frequencies of the transverse modes
good field homogeneity (Opera simulations)
fast field rise and fall times (recent PSpice simulations)
Longitudinal and spiral design are not expected to impact longitudinal beam stability in FCC.
Good agreement between the new Eigenmode method and Wakefield simulations for transverse impedance. The spiral design with possible tweaks will fit into FCC transverse impedance budget.

25. References
H. Day, "Beam Coupling Impedance Reduction Techniques of CERN Kickers and Collimators", Ph.D. thesis, School of Physics and Astronomy, The University of Manchester, Manchester, United Kingdom, June 2013, CERN-THESIS (available online), p. 203.
M. J. Barnes et al., "Reduction of surface flashover of the beam screen of the LHC injection kickers", in Proc. of IPAC'13, Shangai, China, May 2013, paper MOPWA032, pp
V. Vlachodimitropoulos, M.J. Barnes, A. Chmielińska, "Preliminary results from validation measurements of the longitudinal power deposition model for the LHC injection kicker magnet", in Proc. IPAC'18, Vancouver, Canada, May 2018, paper WEPMK005, pp
A. Chmielińska, "FCC-hh kicker magnet design, longitudinal impedance and heating aspects", CERN, Impedance meeting, 15 June 2018,
M. J. Barnes et al., "Operational Experience of a prototype LHC injection kicker magnet with a low SEY coating and redistributed power deposition", presented at IPAC19, Melbourne, Australia, May 2019, paper THPRB072.
E. Shaposhnikova, "Longitudinal stability of the LHC beam in the SPS“, SL-Note HRF, CERN, Geneva, Switzerland, August 2001, p.4.
M. Benedikt et al., "Future Circular Collider Study: The Hadron Collider (FCC-hh)", CERN, Geneva, Switzerland, Rep. ACC , December 2018, p. 14.
S. Arsenyev, "A method for computing driving and detuning beam coupling impedances of an asymmetric cavity using eigenmode simulations",

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