1. Agenda: General kickers analysis Wang-Tsutsui method for computing impedances Benchmarks Conclusions Bibliography Acknowledgments: E.Métral, M.Migliorati, A.Mostacci, N.Mounet, B.Salvant, H.Tsutsui, V.Vaccaro, N.Wang, C.Zannini. Kickers analysis and benchmark N.Biancacci

2. General kicker analysis Kickers are one of the most important contributors to the global value of impedance in accelerator rings. Constant studies are carried on at CERN in order to correctly evaluate their impedance contribution and, in case, reduce it. In this direction we want to: 1.compute the impedance for a model as close as possible to the real one, 2.compute the impedance for any value of β (i.e. in PS we have β=0.91 at injection). 3.update our machine models in HEADTAIL simulations.

3. The inner C-shape magnet has been modeled in many different ways. Mainly we’ll consider Tsutsui’s model (case a) comparing it with a flat geometry model studied by N.Mounet-E.Metràl (case b). General kicker analysis Vacuum Ferrite PEC t b a (a) Tsutsui’s model (b) Flat chamber model

4. Tsutsui-Wang’s method F Method description: V A field matching method is applied: 1.Divide geometry in ferrite (F) and vacuum (V) subdomains. 2.Solve Helmholtz equation in F + boundaries 3.Solve Helmholtz equation in vacuum splitting the inner field in E vacuum =E source +E residual. The residual field can be expressed in terms of waveguides modes (HOMogeneus Helmholtz equation in vacuum). Hom.Helmholtz  E ferrite + Hom.Helmholtz  E residual “free space+plates”  E source Approximation: the source field is approximated as being in free space limited by two vertical parallel plates. Avantages: 1) the impedance will be computed only using the homogeneus solution, directly separating direct SC due to the beam itself, and indirect SC due to horizontal image currents. 2) Avails the following Fourier development for matching on ferrite-vacuum layer.

5. Tsutsui-Wang’s method Method description: 4.Set matching condition for Ez, Hz, Ex, Dy at the ferrite-vacuum boundary. 5.The system coming out from matching procedure is a 4x4 system solvable symbolically. Some symmetry consideration around source field leads to further semplifications in the final unknowns expression. 6.Impedance calculation: Basically integrating E residual along the paths shown in the pictures (X cross = path; green spot = Beam position). xxx ZlongZdrivZdet beta=1H.Tsutsui B.Salvant beta<=1N.Wang N.Biancacci Beta and models: Zlongitudinal ZxDipolar ZxQuadrupolar x x ZyDipolar ZyQuadrupolar Technical Note: Direct and indirect SC effects have been directly separated at the beginning splitting the vacuum field as sum of E vacuum =E source +Er esidual. In N.Mounet-E.Metral method this is done at the end, separating the impedance contributions.

6. Wang-Tsutsui Impedances PSSPS LHC Linac PSB machineKinetic energy (Extraction) β at extraction LINAC50 MeV0.314 PSB1.4 GeV0.916 PS26 GeV0.9993 SPS450GeV0.999998 LHC3.5 TeV0.999999991 We choose three values of β in Wang-Tsutsui impedance calculation: 0.85, 0.9, 0.99999 x x x Relativistic β starts to be significantly different from 1 in PSB and PS at injection.

7. β=0.99999 β=0.9 β=0.85 Wang-Tsutsui Impedances

8. β=0.99999 β=0.9 β=0.85 Wang-Tsutsui Impedances

9. β=0.99999 β=0.9 β=0.85 Wang-Tsutsui Impedances

10. β=0.99999 β=0.9 β=0.85 Wang-Tsutsui Impedances

12. 2- Tsutsui-Wang Vs CST The same structure is implemented in CST. Beta less than one simulations should agree with N.Wang theory. Tsutsui β=1 already benchmarked in the past. 1- Tsutsui-Wang Vs Mounet-Metral N.Mounet and E.Metràl developed the analysis for a two infinite parallel multilayer flat chamber, for any β. Taking Tsutsui–Wang's theory in the limit a → ∞ we should have a convergence between these two models. a Benchmarks a → ∞

13. 1- Theory Vs Theory Good agreement between the two theories! Longitudinal impedance for N.Mounet-E.Metral model and N.Wang-H.Tsutsui one. Ferrite Model 1- Tsutsui-Wang Vs Mounet-Metral a ferrite

14. 1- Tsutsui-Wang Vs Mounet-Metral a ferrite Im(Z) decrease with β ! !

15. From theory, the imaginary part of transverse propagation constant in ferrite becomes negative (damping modes). -1/Ky~2cm < t = 6cm One more check... 2 layers 1 layer Ky ( f ) 1- Tsutsui-Wang Vs Mounet-Metral Eliminating PECs and extending ferrite to infinity we expect the beam “doesn't see” the boundaries from ~10MHz. ≈ f >10MHz

16. 1- Tsutsui-Wang Vs Mounet-Metral Graphite Im(Z) decrease with β

17. 1- Tsutsui-Wang Vs Mounet-Metral Graphite Im(Z) decrease with β

18. A first rough model for MKP was studied in CST and compared with Wang’s impedances. The real part of Zlong shows a good agreement for different values of β. On the contray the imaginary part shows a strong discrepancy probably dued to code artefacts in the ports setup and SC effects inclusion. β=1 β=0.95 2- Tsutsui-Wang Vs CST

19. 2- Benchmarking ● N.Wang’s formulas were benchmarked with Tsutsui’s ones in the limit β  1 with success. ● N.Wang formulas were benchmarked with N,Mounet-E.Metral flat chamber showing basically a good agreement. Simulations for ferrite and graphite were performed. ● N.Wang formulas were benchmarked also with CST code without success. Probably a problem in the ports setup. 1- Tsutsui-Wang model ● Tsutsui-Wang model for kicker was studied in dedail understanding procedure and main assumptions ● Longitudinal, dipolar impedance was derived implementing N.Wang new formulas for β<=1. Also the quadrupolar component has been derived. ● Imaginary part of the impedance is mainly decreasing for high frequencies (above 1GHz), the real part is instead increasing. Conclusions

20.  "Coupling impedance and collective effects in the RCS ring of the China spallation neutron source" N. Wang, PhD thesis  "Longitudinal wakefields and impedance in the CSNS/RCS" N. Wang, Q. Qin, EPAC 2008  "Transverse Coupling Impedance of a Simplified Ferrite Kicker Magnet Model", H. Tsutsui  "Some Simplified Models of Ferrite Kicker Magnet for Calculation of Longitudinal Coupling Impedance", H. Tsutsui, CERN-SL-2000-004-AP, 2000  Impedances of an Infinitely Long and Axisymmetric Multilayer Beam Pipe: Matrix Formalism and Multimode Analysis / Mounet, N (EPFL, Lausanne) ; Metral, E (CERN) Bibliography