1. The status of the magnetic model of the CERN PS. A snapshot! D. Schoerling, M. Juchno July 4th, 2014 Thanks to all people involved in the continuous improvement of the PS machine for many discussions

2. Daniel Schoerling TE-MSC-MNC Introduction 2 Proton synchrotron: >50 years of operation and no end in sight! PS optics model – Field coefficients derived from beam-based measurements – No link between powering currents and WP parameters – Can model but not predict non-linear chromaticity Magnetic model (static) – Numerical analysis – Integration with the optics model – Validation with beam-based measurements Systematic and random effects

3. Daniel Schoerling TE-MSC-MNC CERN Proton Synchrotron 3 Built in 1959 Today: key element of the LHC injector system Tunnel: 200 m in diameter Main magnets: 100 (+1) combined function units – Bending – Focusing Compact but complex design – Focusing and defocusing half-unit powered with the same main coil

4. Daniel Schoerling TE-MSC-MNC CERN Proton Synchrotron 4 Built in 1959 Today: key element of the LHC injector system Tunnel: 200 m in diameter Main magnets: 100 (+1) combined function units – Bending – Focusing Compact but complex design – Focusing and defocusing half-unit powered with the same main coil

5. Daniel Schoerling TE-MSC-MNC PS main magnetic unit 5 Combined-function magnet with hyperbolic pole shape (4 types) Saturation of iron magnetization Complex geometry of coils system

6. Daniel Schoerling TE-MSC-MNC Coil system 6 Mariusz Juchno – Magnetic Model of the CERN Proton Synchrotron 7 November, 2013 Narrow circuit B B Wide circuit I 8L

7. Daniel Schoerling TE-MSC-MNC Coil system contributions 7 – 3-Current Mode – 5-Current Mode I FW I DN I FN I DW I 8L I FN =I FW I DN =I DW I 8L – Hyperbolic pole shape – Only dipolar and quadrupolar field at low field level – Iron saturation – Sextupolar and higher order components at high field level Main coil and figure-of eight loop Pole-face windings – Conductors configuration designed to produce only up to sextupolar component – Affects tune and linear chromaticity – Non-linearities at high field (iron saturation) – Un-balanced N and W circuit current generated octupolar and higher components – Working point non-linearities! – Non-linearities at high field (iron saturation)

8. Daniel Schoerling TE-MSC-MNC Effect on the yoke magnetization 8 Main coil Figure-of-eight loop I MC = 2500 AI 8L = -600 A Focusing Narrow PFWFocusing Wide PFW I FN = 100 A I FW = 100 A 1.45 T0.0 T1.45 T0.0 T 0.16 T0.0 T 0.041 T 0.0 T 0.016 T 0.0 T

9. Daniel Schoerling TE-MSC-MNC Numerical analysis 9 Quasi-static numerical analysis (OPERA) Top-down symmetry (only normal field components analysed) Magnetization curve – Wlodarski model (extrapolation) Packing factor scaling – λ 2D = 0.925 – λ 3D = 0.9424 Circuit typeCurrent range [A]Current step [A] Main coil[500, 5500]250, 500 Figure-of-eight loop[-1200, 1200]600 Pole-face windings[-200, 200]100 Circuit typeCurrent values [A] Main coil1000, 3000, 4500, 5500 Figure-of-eight loop-1200, 0, 1200 Pole-face windings-200, 0, 200 Currents set for 2D analysis Currents set for 3D analysis

10. Daniel Schoerling TE-MSC-MNC Field decomposition & circuit efficiency 10 Decomposition assumptions  Main coil contribution not affected by other circuits  All auxiliary circuits depend on the main coil and the figure-of-eight contribution (magnetization of the whole yoke)  Pole-face winding circuits depend on their own contribution (magnetization of the pole tip)  Alteration of MC contribution due to other circuits included in that circuit contribution Concept of circuit efficiency B gap g

11. Daniel Schoerling TE-MSC-MNC Efficiency functions 11 Pole-face windings affecting their own function magnitude Shift in the current space due to figure- of-eight loop contribution Main coil circuit Auxiliary circuits Figure-of-eight loop Main Coil Focusing Wide Focusing Narrow

12. Daniel Schoerling TE-MSC-MNC Validation of the model formulas 12 Cycle step Measured B tr [10 -4 T] Alone in S.C. I mc [A] Full S.C. I mc [A] Estimated |ΔB tr | [10 -4 T] Injection1013.7 ± 0.03404.9 ±0.03404.5 ±0.031 Extraction6665.84 ± 0.042667.94 ± 0.052665.89 ±0.055 Quasi-static analysis  Virgin magnetization curve  Equivalent packing factor scaling (laminations, block gaps)  No history dependent effects Pre-excitation during measurements B-train system  Peaking strips (“Marker)  Search coils History dependent effects in the machine

13. Daniel Schoerling TE-MSC-MNC 13 Cornuet D. and Sharifullin Z.Magnetic measurements on the PS magnet unit 17 with Hall probes, Technical Report AT-MA Note 92-93, CERN, Geneva, 1992 “Recent” magnetic measurements Asklöv, A.Magnetic measurement on the CERN proton synchrotron, Master’s Thesis, LITHIFM- EX-05/1463-SE, Linköpings universitet, Linköping, 2005. B. Kuiper & G. Plass, Measurements on the prototype magnet unit, PS/Int MM 59-5, Geneve 1959 Summary  Extensive measurements performed in 1959 including dynamic effects  Hall probe measurements performed in 1992 and 2005 Planned: Rotating coil measurement at DC to check also higher order multipoles What was done from our side: Comparison with simulations!

14. Daniel Schoerling TE-MSC-MNC Validation of the model formulas 14 Cycle |B mag. model – B meas. | [10 -4 T] |G mag. model – G meas. | [10 -4 T/m] DirectOpt.DirectOpt. E2217 A967664 B35312040 C80632099 LHC593220160 Cycle ε B [%] B err. [10 -4 T] ε G [%] G err. [10 -4 T/m] E ±0.20 ±4 ±0.25 ±17 A ±0.13 ±8 ±0.18 ±49 B ±0.08 ±10 ±0.15 ±71 C ±0.08 ±10 ±0.15 ±78 LHC ±0.05 ±6 ±3 ±157 Cyclep 0 [GeV/c]B avg [T]I MC [A]I 8L [A]I FN/FW [A]I DN/DW [A] E3.50.167669.20.0 A14.00.6672677.5450.439.5-45.1 B24.01.1494732.00.077.088.0 C26.01.2575413.21257.9200.799.8 LHC26.01.2575400.71452.8206.786.9 Current configurations Measurement errors Model validation (2D)

15. Daniel Schoerling TE-MSC-MNC Analysis of effective magnetic length and field integral corrections Multipolar field distribution along the beam trajectory Integration regions  Magnet ends  Junction  Block gaps 15

16. Daniel Schoerling TE-MSC-MNC Effective magnetic length corrections 16 Bending length correction  Very good agreement  Data processing differences Gradient length correction  Offset – junction correction  Processed measurement data – no contribution of the junction region  Beam-based correction Beam based Bending length correction Gradient length correction

17. Daniel Schoerling TE-MSC-MNC Effective field integral corrections 17 Sextupolar correction  Higher field region – significant 3D effects  Beam based adjustment required Sextupolar correction Octupolar correction  Low field region – linear bare machine working point  High field region – significant 3D effects  F and D not cancelled at high field Beam based

18. Daniel Schoerling TE-MSC-MNC Auxiliary coils corrections 18 Figure-of-eight loop corrections  Approximated with the bare machine field corrections Pole-face windings corrections  Difference in magnetic lengths of F and D circuits remain close to physical length difference of these circuits (8 mm)  Difference in magnetic lengths of N and W circuits up to 8 cm for octupolar component indicates that even in 3CM contributions of N and W circuits do not cancel one another completely 8L PFW

19. Daniel Schoerling TE-MSC-MNC Magnet representation in the optical model 19 “Official” optics – Magnetic parameters – beam based measurements – No link between currents and field parameters – Other elements fixed (SBEND) or unused (some MULTIPOLE elements, junction SBEND element)

20. Daniel Schoerling TE-MSC-MNC Magnet representation in the optical model 20 Modified optics – Input from new magnetic model – 3D effects correction – numerical analysis – Link between currents and coefficients – Main coil current value optimized in the field-control loop manner – Beam-based adjustment of the reference working point Only quadrupolar and sextupolar component – Unused elements were remove – Two equivalent models tested (MADX and MADX+PTC)

21. Daniel Schoerling TE-MSC-MNC Non-linear chromaticity analysis 21 Initial optimization of the main coil current Field correction adjustments based on beam measurements Final optimization of the main coil current Non-linear chromaticity analysis

22. Daniel Schoerling TE-MSC-MNC Nonlinear chromaticity (3.5 GeV/c) 22 FN FW DN DW 8L ΔQ Δξ Magnetic center offset? Feed-down Constant magnetic lengths

23. Daniel Schoerling TE-MSC-MNC Nonlinear chromaticity (14 GeV/c) 23 FN FW DN DW 8L Over-/under estimated tune offset -> sensitivity to radial position

24. Daniel Schoerling TE-MSC-MNC 14 GeV/c Transfer Matrices 24 Δ I FN Δ I FW Δ I DN Δ I DW Δ I 8L Δ Q x 0.00280.0046-0.0031-0.0027-0.0012 Δ Q y -0.0013-0.00320.00510.00410.0012 Δ ξ x 0.1122-0.01150.0770-0.01670.0008 Δ ξ y -0.07360.0077-0.10600.0223-0.0003 Matrix measured in 2008 Corresponding FN and DN tune elements  Numerical matrix – similar  Measured matrix – factor 2 difference  Numerical model idealized Chromaticity elements  Magnetic lengths Δ I FN Δ I FW Δ I DN Δ I DW Δ I 8L Δ Q x 0.00440.0045-0.0022-0.0028-0.0013 Δ Q y -0.0021-0.00280.00430.00450.0013 Δ ξ x 0.1412-0.03820.0777-0.01690.0008 Δ ξ y -0.09190.0209-0.11350.02810.0002 Reproduced with the model M cj = Δc/ΔI j  c = Q x, Q y, ξ x, ξ y  I = FN, DN, FW, DW, 8L

25. Daniel Schoerling TE-MSC-MNC Δ I FN Δ I FW Δ I DN Δ I DW Δ I 8L Δ Q x 0.00280.0049-0.0032-0.0026-0.0013 Δ Q y -0.0011-0.00300.00570.00410.0013 Δ ξ x 0.1259-0.02230.0855-0.02420.0006 Δ ξ y -0.08130.0108-0.12490.03850.0003 14 GeV/c Transfer Matrices 25 Δ I FN Δ I FW Δ I DN Δ I DW Δ I 8L Δ Q x 0.00280.0046-0.0031-0.0027-0.0012 Δ Q y -0.0013-0.00320.00510.00410.0012 Δ ξ x 0.1122-0.01150.0770-0.01670.0008 Δ ξ y -0.07360.0077-0.10600.0223-0.0003 Reproduced with the model (dp/p= -1.9x10-3) Corresponding FN and DN tune elements  Both predictions idealized  dp/p= -1.9x10-3 offset  Sensitive to radial loop (-3.67±0.35mm with respect to geometrical center) Chromaticity elements  FW and DW – sensitive to radial position  Magnetic lengths Matrix measured in 2008 M cj = Δc/ΔI j  c = Q x, Q y, ξ x, ξ y  I = FN, DN, FW, DW, 8L

26. Daniel Schoerling TE-MSC-MNC Element sensitivity to the beam radial position 26 Focusing Defocusing Quadrupole contribution ΔG [Tm -1 /A] 2008 2012 Model Matrix measurement (2008) Radial loop pickups adjustment (2009)  3.5mm deviation of the radial beam position Sensitivity measurement (2012) 2008 elements consistent with 3.5mm offset (RL 2.5mm) Magnetic model – still 2.5mm offset (RL 1.6mm) FN and DN tune elements – 3.5GeV/c cycle

27. Daniel Schoerling TE-MSC-MNC Nonlinear chromaticity (26 GeV/c) 27 FN FW DN DW 8L Similar observations-3.63±0.14mm offset

28. Daniel Schoerling TE-MSC-MNC Nonlinear chromaticity (2 GeV) 28 FN FW DN DW 8L Linear coupling Unbalanced PFW Strong nonlinearities From study on PFW correction during injection [measurement: A. Huschauer] Non-linear chromaticity within 18% No octu-/decapolar correction

29. Daniel Schoerling TE-MSC-MNC Defocusing 2 GeV Transfer Matrices 29 Δ I FN Δ I FW Δ I DN Δ I DW Δ I 8L Δ Q x 0.02230.0231-0.0115-0.0145-0.0064 Δ Q y -0.0108-0.01420.02170.02280.0066 Δ ξ x 0.7438-0.21020.4209-0.08790.0059 Δ ξ y -0.48690.1169-0.60260.15380.0012 Δ Q’’ x 110.3123-167.4144-47.557081.40529.4781 Δ Q’’ y -86.9241102.123992.1193-95.9876-3.1709 Δ I FN Δ I FW Δ I DN Δ I DW Δ I 8L Δ Q x 0.02190.0213-0.0116-0.0142-0.0061 Δ Q y -0.0094-0.01310.02300.02070.0061 Δ ξ x 0.8569-0.08900.4671-0.1353-0.0196 Δ ξ y -0.43960.0380-0.54220.17460.0212 Δ Q’’ x 207.91304.6475-218.1390-105.730217.2275 Δ Q’’ y -115.96331.983936.2483-9.29809.8931 Matrix calculated with the new model Matrix measured in 2012 Sextupole Contribution ΔS/I [Tm -2 /A] Focusing Tune elements  No significant offset Linear chromaticity elements  Discrepancies for FW and DW elements Non-linear chromaticity elements  Significant inconsistencies

30. Daniel Schoerling TE-MSC-MNC 2 GeV Linearization 30 Initial WPMeas. pred.Model pred Meas.ModelMeas.ModelMeas.Model. QxQx 6.105 6.1026.1066.116.105 QyQy 6.205 6.2076.2056.2016.205 ξxξx 0.720.690.670.710.670.72 ξyξy -1.03-0.98-0.81-0.74-1.55-1.03 Q’’ x 2105184534182920-2530 Q’’ y -874-1007-1537-20377561129 Horizontal Linearization – target: Q x ’’ = 0 Initial WPMeas. pred.Model pred Meas.ModelMeas.ModelMeas.Model. QxQx 6.105 6.1116.1026.1116.105 QyQy 6.205 6.1996.2056.2026.205 ξxξx 0.720.690.740.63 0.72 ξyξy -1.03-0.98-2.24-1.59-1.22-1.03 Q’’ x 21051845-569-7679561187 Q’’ y -874-100712061487-2400 Vertical Linearization – target: Qy’’ = 0 Minimization of non-linear chromaticity  Measurement matrix prediction ineffective  Numerical matrix prediction significantly reduces non-linear chromaticity Other test cases  Linear chromaticity – FW & DW elements  Discrepancies close to 3CM  Initial matching validity

31. Daniel Schoerling TE-MSC-MNC 31 Summary Part I A detailed magneto-static model for almost all combinations of currents was developed. By linking this magnetic model to the optics model it become possible to:  Reconstruct the working point transfer matrices for any energy.  Predict for the first time in the history of the PS the higher-order chromaticity function among other working point parameters.  Analyze the transfer matrix sensitivity to the radial beam position. No means of predicting resonances Further reading: M. Juchno, Magnetic Model of the CERN Proton Synchrotron, PhD thesis, EPFL, 2013

32. Daniel Schoerling TE-MSC-MNC 32 Why doing even more? See H. Damerau et al., TUXA02, IPAC’12, New Orleans Higher brightness/intensity beams are required for the LHC to achieve its high luminosity objective  Consolidating and upgrading PSB, PS, SPS and using the newly built LINAC4  PS’ injection energy will be increased from 1.4 to 2 GeV to reduce space charged induced tune shift Working point control (under good control) Resonance compensation scheme required Upgrade program for hardware in the PS machine Much more activities outside our group…

33. Daniel Schoerling TE-MSC-MNC 33 Methodology

34. Daniel Schoerling TE-MSC-MNC 34 Structural analysis The simulation and measurement [1] of the deformation of the magnet are similar The magnetic field is used to derive the normal and skew components of the magnetic fields in Taylor series The effect on the optics of the machine were calculated with MAD-X and PTC The effect of the deformation is especially visible for 26 GeV/c, because F  B 2 The mechanical deformations cannot explain the resonances at low energy [1] M. Buzio, M. Tortrat, Deformation of the PS reference magnet U101 during operation: geometrical survey and impact on B-train magnetic field measurements, April 2010 QxQx QyQy xx yy 14 GeV/c, normal components (negligible difference between magnetic and structural) 6.20586.30320.20230.6837 14 GeV/c, normal & skew com.6.20586.30320.20220.6839 26 GeV/c, no deformation, only normal components6.26866.22190.03810.4770 26 GeV/c, with deformation, only normal components6.26476.21790.11960.3961 26 GeV/c, with deformation, normal & skew components6.26476.21790.16460.3506 UY

35. Daniel Schoerling TE-MSC-MNC 35 Vacuum chamber influence Dipole [T] Quadrupole [T/m] Sextupole [T/m 2 ] Octupole [T/m 3 ] Permeability Weld 1.0000 1.23645.23272.7668-36.3321 Permeability Weld 1.0030 1.23645.23272.7483-36.5692 Difference 1.56  10 -05 5.67  10 -05 0.01850.2371 PS spare vacuum chambers stored in building 169 Permeability measurements with Dr. Foerster Magnetoscop 1.069 Pre-measurements have shown that the permeability is very small Calibration with a relative permeability of 1.0037 Largest measured relative permeability was on a long vacuum chamber with around 1.002 Limitations Usually thick samples required but permeability is very small and therefore, the influence on the magnetic field is also small. Introduced welding seams (in red)

36. Daniel Schoerling TE-MSC-MNC 36 Material uncertainty Shuffling was performed  Reduction of the spread per yoke  Minimization of uncertainties in the magnetic field Epstein-frame measurement of electrical steel  2-5% anisotropy in steel  Good correlation to split-coil measurements Fit with Wlodarski’s model for measured magnet (limited improvement)

37. Daniel Schoerling TE-MSC-MNC 37 Geometrical measurement Unit 17 was measured with a laser tracker with 19.05 mm offset to the plane Fitting by rotating and translating the measurement data to nominal surface was applied (normally offset by 19.05 mm) Standard deviation from this nominal surface was calculated D. Schoerling, Analysis of PS main magnet geometrical measurements, Unit 17, EDMS 1336186, 2013

38. Daniel Schoerling TE-MSC-MNC 38 2D magneto-static simulations 2D calculation including Gaussian distribution of the position of the coils and the shape of the iron with up to 22 DOFs per magnet (OPERA) 1000 models per magnet type and current level have to be calculated (<1 d with advanced and additional licenses, before 10 d) Performed for momentum of 2.14 GeV/c, 2.78 GeV/c, 14 GeV/c, 26 GeV/c Coils can be displaced, no rotation: Main coils (2 x 4 DOFs),  = 3 mm F8 (2 x 4 DOFs),  = 1 mm PFW (2 x 2 DOFs),  = 0.7 mm Iron is displaced in y-direction,  = 0.02/3 mm 2.14 GeV/c Reference radius r = 10 mm

39. Daniel Schoerling TE-MSC-MNC 39 3D magneto-static simulations Time consuming Monte-Carlo study performed. New features were implemented together with Vectorfields (deforming of mesh). Each block was shifted, the pole face and coils were altered to simulate the effect on the magnetic field

40. Daniel Schoerling TE-MSC-MNC 40 Resonance compensation (R. Wasef) Magnetic & alignment (  = 200  m) errors are essential for space charge studies because at low energy (bare machine) they are the main cause of resonance excitation, and cause therefore losses and emittance growth PS is implemented in MAD with ideal lattice In MAD the main magnets are divided in 4 half units 2D & 2F  400 elements Magnetic errors (Systematic & Gaussian distribution ,  ) can be implemented for each element in the lattice up to the normal & skew octupolar component. For each half unit one set of multipolar field errors is created, i.e., 400 numbers per multipolar field error have to be generated F F F F D D D D Half unit

41. Daniel Schoerling TE-MSC-MNC 41 Resonance compensation (R. Wasef) In the 80’s several compensation schemes using normal and skew sextupoles in the PS (sections 2, 52, 14, 64) were applied: Y. Baconnier, Tune shifts and stop bands at injection in the CERN proton synchrotron, CERN/PS 87-89 (PSR), 1987 The air-cooled sextupole magnets have been installed in the winter shutdown in sections 2, 52, 14, 72 (instead of 64) A compensation scheme for each of the resonances 2Q x +Q y =19 and 3Q y =19 was implemented, using the new locations and the magnetic field error distribution Compensating both resonances requires larger skew sextupole fields, which cannot be generated with the currently installed magnet-power supply installation 601 602 C

42. Daniel Schoerling TE-MSC-MNC 42 Resonance compensation (A. Huschauer) Scan Direction Compensated resonance 2q x +q y =1

43. Daniel Schoerling TE-MSC-MNC 43 Resonance compensation (A. Huschauer) Scan Direction Compensated resonance 3q y =1

44. Daniel Schoerling TE-MSC-MNC 44 Summary Part II Deforming the magnet due to magnetic forces is a systematic effect that has a large impact on the field distribution at high field and only a negligible influence at low field. Estimating the permeability of the beam pipe and calculating the influence on the field distribution, it could be shown that this systematic effect is negligible. The influence of anisotropy in the steel of the magnets is negligible. The random effects were investigated by performing Monte Carlo simulations with 2D and 3D finite element models. The 2D simulations showed that skew components can be neglected and the standard deviation is small. The 3D simulations showed larger skew components but also a small standard deviation. Therefore, the field distribution variation from magnet to magnet is expected to be small in PS. The presented data will be used to enhance the resonance compensation scheme after re-start of the CERN injector complex and beam-based measurements will be performed. Further reading: D. Schoerling, Prediction of the field distribution in CERN-PS magnets, TUPRO107, IPAC 2014.

45. Daniel Schoerling TE-MSC-MNC 45 Conclusion and outlook Very precise systematic and random model of the PS magnet available! What is next?  Magnetic measurements with rotating coils  More geometrical measurements of magnets to understand better the mechanical errors  More 3D simulations (cross-check with other mesh, update with other mechanical errors, improving the precession by setting the potential manually, etc.)  Measurements in the machine. What could be in the far-future?  Hysteresis effects  Eddy current effects including vacuum chamber effects * *B. Auchmann, Compensation of Eddy-Current Effects in PS Vacuum Chambers by Pole-Face Windings, 2007, EDMS #973216

46. Daniel Schoerling TE-MSC-MNC Appendix: 14 GeV/c Transfer Matrices 46 Reproduced with the model Matrix predicted in 1974 Corresponding FN and DN tune elements  Both predictions idealized Chromaticity elements  FW and DW - magnetic lengths M cj = Δc/ΔI j  c = Q x, Q y, ξ x, ξ y  I = FN, DN, FW, DW, 8L Δ I FN Δ I FW Δ I DN Δ I DW Δ I 8L Δ Q x 0.00440.0045-0.0022-0.0028-0.0013 Δ Q y -0.0021-0.00280.00430.00450.0013 Δ ξ x 0.1412-0.03820.0777-0.01690.0008 Δ ξ y -0.09190.0209-0.11350.02810.0002 Δ I FN Δ I FW Δ I DN Δ I DW Δ I 8L Δ Q x 0.00460.0047-0.0025-0.0031-0.0018 Δ Q y -0.0025-0.00320.00460.00480.0019 Δ ξ x 0.1279-0.02220.0744-0.01440.0000 Δ ξ y -0.08730.0130-0.10620.02190.0000