Updated status of the PSB impedance model C. Zannini and G. Rumolo Thanks to: E. Benedetto, N. Biancacci, E. Métral, B. Mikulec, N. Mounet, T. Rijoff, B. Salvant, W. Weterings

Overview Introduction Present PSB impedance model – Indirect space charge Numerical form factor for ISC computation – Resistive wall – Extraction kicker – Broadband impedance of step transitions – KSW Global PSB impedance model – Comparison with tune shift measurements – Summary and future plans

Impedance calculations for  In the LHC, SPS, PS CST EM simulations are performed in the ultra-relativistic approximation (  Analytical calculation (applies only to simple structures) 3D EM simulations (CST Particle Studio) The use of 3D EM simulations for  is not straightforward at all

3D CST EM simulation for  To single out the impedance contribution the direct space charge must be removed Depend only on the source contribution due to the interaction of beam and external surroundings C. Zannini et al. Electromagnetic Simulations for Non-ultrarelativistic Beams and Application to the CERN Low Energy Machines, IPAC14, to be published

Overview Introduction Present PSB impedance model – Indirect space charge Numerical form factor for ISC computation – Resistive wall – Extraction kicker – Broadband impedance of step transitions – KSW Global PSB impedance model – Comparison with tune shift measurements – Summary and future plans

Present PSB transverse impedance model Elements included in the database: – Analytical calculation of the resistive wall impedance that takes into account the different PSB vacuum chambers weighted by the respective length and beta function. Also the iron in the magnet is taken into account Beam pipe

Present PSB transverse impedance model Elements included in the database: – Analytical calculation of the resistive wall impedance that takes into account the different PSB vacuum chambers weighted by the respective length and beta function. Also the iron in the magnet is taken into account – Extraction kicker impedance due to the ferrite loaded structure impedance due to the coupling to the external circuits (analytical calculation) KickersBeam pipe

Present PSB transverse impedance model Elements included in the database: – Analytical calculation of the resistive wall impedance that takes into account the different PSB vacuum chambers weighted by the respective length and beta function. Also the iron in the magnet is taken into account – Extraction kicker impedance due to the ferrite loaded structure impedance due to the coupling to the external circuits (analytical calculation) – Indirect space charge impedance KickersBeam pipe Indirect space charge impedance

Present PSB transverse impedance model Elements included in the database: – Analytical calculation of the resistive wall impedance that takes into account the different PSB vacuum chambers weighted by the respective length and beta function. Also the iron in the magnet is taken into account – Extraction kicker impedance due to the ferrite loaded structure impedance due to the coupling to the external circuits (analytical calculation) – Indirect space charge impedance KickersBeam pipe Indirect space charge impedance

Present PSB transverse impedance model Elements included in the database: – Analytical calculation of the resistive wall impedance that takes into account the different PSB vacuum chambers weighted by the respective length and beta function. Also the iron in the magnet is taken into account – Extraction kicker impedance due to the ferrite loaded structure impedance due to the coupling to the external circuits (analytical calculation) – Indirect space charge impedance KickersBeam pipe Indirect space charge impedance

Present PSB transverse impedance model Elements included in the database: – Analytical calculation of the resistive wall impedance that takes into account the different PSB vacuum chambers weighted by the respective length and beta function. Also the iron in the magnet is taken into account – Extraction kicker impedance due to the ferrite loaded structure impedance due to the coupling to the external circuits (analytical calculation) – Indirect space charge impedance KickersBeam pipe Indirect space charge impedance

Present PSB transverse impedance model Elements included in the database: – Analytical calculation of the resistive wall impedance that takes into account the different PSB vacuum chambers weighted by the respective length and beta function. Also the iron in the magnet is taken into account – Extraction kicker impedance due to the ferrite loaded structure impedance due to the coupling to the external circuits (analytical calculation) – Indirect space charge impedance – Broadband impedance of step transitions KickersBeam pipe Indirect space charge impedanceStep transitions

Present PSB transverse impedance model Elements included in the database: – Analytical calculation of the resistive wall impedance that takes into account the different PSB vacuum chambers weighted by the respective length and beta function. Also the iron in the magnet is taken into account – Extraction kicker impedance due to the ferrite loaded structure impedance due to the coupling to the external circuits (analytical calculation) – Indirect space charge impedance – Broadband impedance of step transitions – KSW magnets Extraction KickerBeam pipe Indirect space charge impedanceStep transitions KSW

Overview Introduction Present PSB impedance model – Indirect space charge Numerical form factor for ISC computation – Resistive wall – Extraction kicker – Broadband impedance of step transitions – KSW Global PSB impedance model – Comparison with tune shift measurements – Summary and future plans

Indirect space charge Analytical calculation based on the PSB aperture model (provided by O. Berrig) K. Y. Ng, Space charge impedances of beams with non-uniform transverse distributions Circular pipe Rectangular pipe Elliptic pipe [a, b, L, β x, β y, Apertype] i

PSB: indirect space charge impedance

Overview Introduction Present PSB impedance model – Indirect space charge Numerical form factor for ISC computation – Resistive wall – Extraction kicker – Broadband impedance of step transitions – KSW Global PSB impedance model – Comparison with tune shift measurements – Summary and future plans

Indirect space charge: refinement of the calculation Using numerical form factors

Indirect space charge: refinement of the calculation Using numerical form factors

PSB: indirect space charge impedance

Overview Introduction Present PSB impedance model – Indirect space charge Numerical form factor for ISC computation – Resistive wall – Extraction kicker – Broadband impedance of step transitions – KSW Global PSB impedance model – Comparison with tune shift measurements – Summary and future plans

Resistive wall impedance Wall thicknessWall (σ el )Background material Dipoles0.4 mm S/mIron (silicon steel) Quadrupoles1.5 mm S/mIron (silicon steel) Straight sections1 mm S/mVacuum Analytical calculation based on the PSB aperture model (provided by O. Berrig) Calculation performed with the TLwall code [a, b, L, β x, β y, Apertype] i

Resistive wall impedance Wall thicknessWall (σ el )Background material Dipoles0.4 mm S/mIron (silicon steel) Quadrupoles1.5 mm S/mIron (silicon steel) Straight sections1 mm S/mVacuum Analytical calculation based on the PSB aperture model (provided by O. Berrig) Calculation performed with the TLwall code [a, b, L, β x, β y, Apertype] i Vertical

Overview Introduction Present PSB impedance model – Indirect space charge Numerical form factor for ISC computation – Resistive wall – Extraction kicker – Broadband impedance of step transitions – KSW Global PSB impedance model – Comparison with tune shift measurements – Summary and future plans

is the impedance calculated using the Tsutsui formalism Constant horizontal impedance A theoretical calculation for the C-Magnet model

PSB extraction kicker: impedance due to the ferrite loaded structure Z M Vertical

PSB extraction kicker: impedance due to the coupling to the external circuits Z TEM Cables in the open-short configuration Horizontal

Overview Introduction Present PSB impedance model – Indirect space charge Numerical form factor for ISC computation – Resistive wall – Extraction kicker – Broadband impedance of step transitions – KSW Global PSB impedance model – Comparison with tune shift measurements – Summary and future plans

Broadband impedance of a step transition Based on the results of 3D EM simulations, the broadband impedance contribution due to an abrupt transition is independent of the relativistic beta. Therefore, based on the aperture model, the generalized broadband impedance of the PSB transitions has been calculated as: C. Zannini, Electromagnetic simulations of CERN accelerator components and experimental applications. PhD thesis, Lausanne, EPFL, CERN-THESIS

Broadband impedance of step transitions L Weak dependence on the relativistic beta and L

Broadband impedance of step transitions L Weak dependence on the relativistic beta and L

Overview Introduction Present PSB impedance model – Indirect space charge Numerical form factor for ISC computation – Resistive wall – Extraction kicker – Broadband impedance of step transitions – KSW Global PSB impedance model – Comparison with tune shift measurements – Summary and future plans

3D model of the KSW magnet

Coating of the ceramic chamber

Analytical model: axially symmetric multilayer structure Assuming uniformity of the coating thickness Resistance of the coating The beam coupling impedance strongly depends on the resistance of the coating

KSW magnets Vertical R=8 Ω

Overview Introduction Present PSB impedance model – Indirect space charge Numerical form factor for ISC computation – Resistive wall – Extraction kicker – Broadband impedance of step transitions – KSW Global PSB impedance model – Comparison with tune shift measurements – Summary and future plans Other devices studied – Shielding of pumping ports – Insulated flanges

Present PSB transverse impedance model Elements included in the database: – Analytical calculation of the resistive wall impedance that takes into account the different PSB vacuum chambers weighted by the respective length and beta function. Also the iron in the magnet is taken into account – Extraction kicker impedance due to the ferrite loaded structure impedance due to the coupling to the external circuits (analytical calculation) – Indirect space charge impedance (analytical calculation) – Broadband impedance of step transitions – KSW magnets Extraction KickerBeam pipe Indirect space charge impedanceStep transitions KSW

Total horizontal driving impedance of the PSB E E Contributions of the extraction kicker due to the coupling with external circuits

Total vertical driving impedance of the PSB E E

Overview Introduction Present PSB impedance model – Indirect space charge Numerical form factor for ISC computation – Resistive wall – Extraction kicker – Broadband impedance of step transitions – KSW Global PSB impedance model – Comparison with tune shift measurements – Summary and future plans Other devices studied – Shielding of pumping ports – Insulated flanges

D. Quatraro, Collective effects for the LHC injectors: non- ultrarelativistic approaches. PhD thesis, Bologna, University of Bologna, CERN- THESIS Measurement data

Effective impedance of the PSB Measurements at different energies are consistent with a missing ~2 MΩ/m Z eff x,y [MΩ/m] E kin =160 MeVE kin =1.0 GeVE kin =1.4 GeV Indirect space charge1.50/ / /1.25 Kicker cables0.0084/00.012/ /0 Kicker ferrite-0.044/ /0.11 Steps0.53/0.63 Resistive wall0.03/ /0.07 KSW0/0.0130/ / Total (expected)2.03/ / /2.0 Total (measured )?/13.0?/4.6?/3.8

Comparison between measurements and model of the vertical coherent tune shifts at different energies

Overview Introduction Present PSB impedance model – Indirect space charge Numerical form factor for ISC computation – Resistive wall – Extraction kicker – Broadband impedance of step transitions – KSW Global PSB impedance model – Comparison with tune shift measurements – Summary and future plans Other devices studied – Shielding of pumping ports – Insulated flanges

Summary and future plans Measurements at different energies are consistent with a missing ~2 MΩ/m of the PSB impedance model Measurements of the coherent horizontal and vertical tune shift Update of the model according to new understandings and identification of significant impedance sources Using the PSB impedance model for beam dynamics studies

Additional impedance studies not discussed here Impedance model of the new H- injection region – Comparison between Inconel undulated chamber and titanium coated ceramic chamber – Pumping ports – Foil section Finemet cavities – The longitudinal impedance does not depend on the relativistic beta Impedance of insulated flanges with and without RF bypass

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Resistive wall impedance: impact of the iron Resistive wall vertical generalized impedance of the PSB

Resistive wall impedance f rel = 10 kHz K. G. Nilanga et al., Determination of complex permeability of silicon-steel for use in high frequency modelling of power transformers, IEEE TRANS. ON MAGNETICS, VOL. 44, NO. 4, APRIL A. Asner et al., The PSB main bending magnets and quadrupole lenses, Geneva, April 1970.

PSB extraction kicker: impedance due to the coupling to the external circuits Z TEM Cables in the open-short configuration Horizontal

Theoretical estimation The C-Magnet model can support a TEM propagation Expectation The TEM mode plays a role when the penetration depth in the ferrite becomes comparable to the magnetic circuit length (below few hundred MHz). CST Particle Studio simulations Peak due to the TEM propagation C-magnet: driving horizontal impedance

is the impedance calculated using the Tsutsui formalism Constant horizontal impedance A theoretical calculation for the C-Magnet model

The theoretical predictions and simulations show a very good agreement Confirmation of the new theory CST Particle Studio is found to be a reliable tool to simulate the impedance of ferrite loaded components C-Magnet: Comparing theoretical model and 3-D simulations

Indirect space charge Analytical calculation based on the PSB aperture model (provided by O. Berrig) K. Y. Ng, Space charge impedances of beams with non-uniform transverse distributions Circular pipe Rectangular pipe Elliptic pipe

Definition of impedance Longitudinal component of the electric field in (x, y) induced by a source charge placed in (x 0, y 0 ) Depend only on the source contribution due to the interaction of beam and accelerator components EM simulator uses the total fields

Overview Introduction – PSB impedance model Updated PSB impedance model – Indirect space charge Numerical form factor for ISC computation – KSW kickers – Longitudinal impedance model – Wake model Other devices studied – Vacuum chamber of the new H- region – Finemet cavities – Shielding of pumping ports – Insulated flanges – Foil section Future plans

Inconel undulated chamber Inconel thickness: mm Vertical full aperture: 63 mm Inconel conductivity = Inconel undulated chamber Vertical full aperture: 63 mm Titanium thickness: 100 μm Titanium coated Ceramic (Al2O3) chamber (no corrugation) Alternative solution

Analytical calculation (no corrugation): comparison between Inconel and Ceramic chamber Theoretical calculation made with the TLwall code

Inconel undulated chamber: CST Particle Studio simulation The impedance contribution of the corrugation seems to be negligible

Overview Introduction – PSB impedance model Updated PSB impedance model – Indirect space charge Numerical form factor for ISC computation – KSW kickers – Longitudinal impedance model – Wake model Other devices studied – Vacuum chamber of the new H- region – Finemet cavities – Shielding of pumping ports – Insulated flanges – Foil section Future plans

Finemet cavities The impedance does not depend on the relativistic beta S. Persichelli et al., Finemet cavities impedance studies, CERN-ACC-NOTE , Ps and PSB cell cavity

Comparison between measurements and model of the vertical coherent tune shift at E kin =160 MeV 10 % 20 % 30 % 40 % 50 % 60 % 70 % 80 % 90 % 100 %

Comparison between measurements and model of the vertical coherent tune shift at E kin =160 MeV 10 % 20 % 30 % 40 % 50 % 60 % 70 % 80 % 90 % 100 %

Comparison between measurements and model of the vertical coherent tune shift at E kin =1.0 GeV 10 % 20 % 30 % 40 % 50 % 60 % 70 % 80 % 90 % 100 %

Comparison between measurements and model of the vertical coherent tune shift at E kin =1.0 GeV 10 % 20 % 30 % 40 % 50 % 60 % 70 % 80 % 90 % 100 %

Comparison between measurements and model of the vertical coherent tune shift at E kin =1.4 GeV 10 % 20 % 30 % 40 % 50 % 60 % 70 % 80 % 90 % 100 %

Comparison between measurements and model of the vertical coherent tune shift at E kin =1.4 GeV 10 % 20 % 30 % 40 % 50 % 60 % 70 % 80 % 90 % 100 %

Effect of the ceramic chamber on the KSW longitudinal impedance Titanium coating The shielding (coated ceramic chamber) strongly reduces the longitudinal impedance

Effect of the ceramic chamber on the KSW transverse impedance The coating shifts the impedance spectrum to lower frequencies

Effect of the ceramic chamber on the KSW transverse impedance The frequency shift is proportional to the coating resistance

Effect of the ceramic chamber on the KSW transverse impedance Real part of the transverse impedance frfr

Effect of the ceramic chamber on the KSW transverse impedance