Agenda:

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

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: compute the impedance for a model as close as possible to the real one, compute the impedance for any value of β (i.e. in PS we have β=0.91 at injection). update our machine models in HEADTAIL simulations.

3. General kicker analysis 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). (a) Tsutsui’s model Ferrite t b Vacuum a PEC (b) Flat chamber model

4. Tsutsui-Wang’s method Method description: A field matching method is applied: Divide geometry in ferrite (F) and vacuum (V) subdomains. Solve Helmholtz equation in F + boundaries Solve Helmholtz equation in vacuum splitting the inner field in Evacuum=Esource+Eresidual. The residual field can be expressed in terms of waveguides modes (HOMogeneus Helmholtz equation in vacuum). Hom.Helmholtz  Eferrite F V + Hom.Helmholtz  Eresidual “free space+plates”  Esource Approximation: the source field is approximated as being in free space limited by two vertical parallel plates. Avantage: 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: Set matching condition for Ez, Hz, Ex, Dy at the ferrite-vacuum boundary. 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. Impedance calculation: Basically integrating Eresidual along the paths shown in the pictures (X cross = path; green spot = Beam position). x x x x x ZyQuadrupolar ZxQuadrupolar ZyDipolar ZxDipolar Zlongitudinal Technical Note: Direct and indirect SC effects have been directly separated at the beginning splitting the vacuum field as sum of Evacuum=Esource+Eresidual. In N.Mounet-E.Metral method this is done at the end, separating the impedance contributions. Beta and models:

6. Wang-Tsutsui Impedances Relativistic β starts to be significantly different from 1 in PSB and PS at injection. We choose three values of β in Wang-Tsutsui impedance calculation: 0.85, 0.9, 0.99999 x x x LHC PS SPS LinacPSB

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

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

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

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

11. Wang-Tsutsui Impedances

12. Benchmarks 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 → ∞ a 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.

13. 1- Tsutsui-Wang Vs Mounet-Metral 1- Theory Vs Theory ferrite Good agreement between the two theories! Ferrite Model a Re(Z) increase with β Im(Z) decreasewith β Longitudinal impedance for N.Mounet-E.Metral model and N.Wang-H.Tsutsui one.

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

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

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

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

18. 2- Tsutsui-Wang Vs CST A 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 given by code artefacts dued to ports setup. β=1 β=0.95

19. Conclusions 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. 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.

20. Bibliography • "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)

21. Tsutsui-Wang’s method: detailed description We assume a longitudinal dependency given by: F V Since we can express the field in sum of TE and TM modes (TEM not supported) we get: 2 pairs of Helmholtz equation per region. 4 Unknowns 4 Unknowns Vacuum F Top-Bottom/ Left-Right simmetry and lateral PECs reduce 3 unknowns per equation. V 1 Unknown 1 Unknown

22. 4 Unknowns 4 Unknowns F V Left-Right simmetry, lateral and covering PECs reduce 3 unknowns per equation. Ferrite 1 Unknown 1 Unknown We end up with 4 unknowns, 2 from vacuum + 2 from ferrite slabs. The last layer that separate vacuum from ferrite gives 4 equations. Homogeneus system, has only the trivial solution: no source, no field. 2 1

23. We plug in a source beam distribution travelling along the center of the kicker. We get a “driven” Helmholtz equation. F V Source The solution is the sum :homogenus case (waveguide modes) +particular solution (source field). V The source field is calculated assuming to be in free space and adding metal plats

24. Matching procedure 2 New inhomogeneus system leading to non trivial solution. 1 Beam • This analysis: • can be followed for any value of beta; • allows easy impedance calculations.