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.
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Acknowledgments: E.Métral, A.Mostacci, N.Mounet,M.Migliorati, B.Salvant, H.Tsutsui, N.Wang, C.Zannini.
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.
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
(b) Flat chamber model
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).
“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.
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).
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:
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
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 → ∞
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- Theory Vs Theory
Good agreement between the two theories!
Re(Z) increase with β
Im(Z) decreasewith β
Longitudinal impedance for N.Mounet-E.Metral model and N.Wang-H.Tsutsui one.
Im(Z) decrease with β
Re(Z) decrease with β
From theory, the imaginary part of transverse propagation constants becomes infact negative (damping modes). -1/Ky~2cm < t = 6cm
Ky ( f )
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.
1- Tsutsui-Wang model
We assume a longitudinal dependency given by:
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.
Top-Bottom/ Left-Right simmetry and lateral PECs reduce 3 unknowns per equation.
Left-Right simmetry, lateral and covering PECs reduce 3 unknowns per equation.
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.
We plug in a source beam distribution travelling along the center of the kicker. We get a “driven” Helmholtz equation.
The solution is the sum :homogenus case (waveguide modes) +particular solution (source field).
The source field is calculated assuming to be in free space and adding metal plats
New inhomogeneus system leading to non trivial solution.