1. Update on SPS BPM impedance B. Salvant for the 2008 impedance team

2. Reference: http://cdsweb.cern.ch/record/1208424/files/CERN-ATS-2009-046.pdf 106 BPH and 96 BPV in the SPS Low frequency imaginary impedance for a BPH : Zlong/n ~ 1 mΩ, Zx=- 0.1 kΩ/m and Zy=2 kΩ/m. Separate simulations for a BPV: Zlong/n~0.5mΩ, Zy= 0.1 kΩ/m and Zx= 0.2 kΩ/m.

3. Modelled structure for the BPH All materials are perfect conductors (except the ceramic spacers) In fact: - Electrodes + pipe should be in Stainless Steel - Casing should be in Anticorodal Ultrarelativistic beams Rs=|V|^2/Plosses

4. BPH longitudinal (with material losses) fres (GHz) Q Rs (“center”) in Ohm Rs/Q In Ohm 0.26526317666.7 0.8854822750.6 0.8924933740.8 1.05277330133.9 1.06277335914.6 1.5986728041.2 1.6136861127416.4 1.85989655676.2

5. BPH vertical (with material losses) fres (GHz) Q Rs (“center”) in Ohm Rs (y=2mm) in Ohm Abs(DeltaRs(y=2)) /Deltay in Ohm/m Abs(DeltaRs(y=2)) /Deltay/Q in Ohm/m 0.5284650522611856.1 0.5324830946419.6 1.2061276021102868.1 1.25876401049106.4 1.2716880524713.6 1.41415140683417222.6 1.61368611274113393272747.7 1.66512230673340227.3 1.859896556755901155412.9

6. BPH horizontal (with material losses) fresQ Rs (“center”) Rs (x=2mm) Abs(DeltaRs) /Deltay Abs(DeltaRs) /Deltay/Q 0.88548227528235117.3 0.89249337437820064.1 1.59867280481553798.0 1.61368611274112103200546.7 1.83390571921094512.1 1.859896556755441128512.6 Quite insignificant change of transverse displacement on shunt impedance

7. BPV

8. From PhD thesis…

9. BPV (with material losses) fresQ Rs (“center”) In Ohm Rs(center) /Q In Ohm Rs (y=2mm) In Ohm Abs(DeltaRs(y=2)) /Deltay In Ohm/m Abs(DeltaRs(y=2)) /Deltay/Q In Ohm/m Rs (x=2mm) In Ohm Abs(DeltaRs) /Deltay In Ohm/m Abs(DeltaRs) /Deltay/Q In Ohm/m 0.293239900.49315716.690170.1 0.29824115956.615951420.615962401.0 0.70452500.000 17828215.8 0.71554800.000 736116.6 1.0696549451.494311461.894816432.5 1.09266711881.8118516922.5119116682.5 1.1856102410.4254653310.723530335.0 1.215624260.043849913.6256801.1 1.57074500.000 0100.0 1.57672400.000 210871.5 1.61077300.000 0340.0 1.66168600.000 15310.8 1.871287300.00480.00540.0 1.910243000.000 01480.1 1.960199315030.8150925841.3150114390.7 Longitudinalvertical horizontal

10. Effect on the beam  Headtail simulations Simulations done at the time with small longitudinal emittance (0.15 eVs) All BPMs are lumped in one location accounting for the respective beta functions. Linear longitudinal restoring force No direct space charge

11. BPMs alone (vertical plane)

12. BPMs alone (horizontal plane)

13. Summary Effect of impedance of BPH and BPV is small on transverse single bunch dynamics Longitudinal modes : – Rs~1 to 10 k Ω – R/Q~ 5 to 20 Ω – Larger for BPHs than BPVs – Frequencies above 1 GHz Transverse modes: – Modes at 0.5 GHz (vertical for BPH) and 0.7 GHz (horizontal for BPV) – Rs~1 to 20 kΩ/m – R/Q~ 5 to 50 Ω/m

14. However… fres (GHz) Q Rs (“center”) in Ohm Rs/Q In Ohm 0.26526317666.7 0.8854822750.6 0.8924933740.8 1.05277330133.9 1.06277335914.6 1.5986728041.2 1.6136861127416.4 1.85989655676.2 ?

15. 15 Effect of matching the impedance at electrodes coaxial ports in Particle Studio simulations (BPH) Electrode coaxial port Modes are damped by the “perfect matching layer” at the coaxial port Short bunch (1 cm rms) SPS bunch (20 cm rms) Importance to match the BPM electrodes!

17. BPH fresQ Rs (“center”) Rs(center) /Q Rs (y=2mm) Abs(DeltaRs(y=2)) /Deltay Abs(DeltaRs(y=2)) /Deltay/Q Rs (x=2mm) Abs(DeltaRs) /Deltay Abs(DeltaRs) /Deltay/Q 0.260260770.377100.0783541.4 0.26526317666.717672561.017662340.9 0.52846500.0522611856.1000.0 0.53248300.0946419.6000.0 0.8854822750.627711892.528235117.3 0.8924933740.837715943.237820064.1 1.034179800.01470653.9000.0 1.05277330133.9301512941.730145170.7 1.06277335914.6359415592.03591670.1 1.206127600.021102868.1000.0 1.25876400.01049106.4000.0 1.27168800.0524713.6000.0 1.414151400.0683417222.6000.0 1.5986728041.280924043.681553798.0 1.6136861127416.4113393272747.7112103200546.7 1.63898200.0418441.9000.0 1.665122300.0673340227.3000.0 1.801302400.01154581.8000.0 1.833905710.1711590.2921094512.1 1.85989655676.255901155412.955441128512.6 Longitudinalvertical horizontal

18. Longitudinal Frequency (GHz) Rs (Ohm) (PEC) Q (PEC) R/Q (Ohm) 1.041.6 10 4 31505 1.614.6 10 4 269017 1.812.5 10 4 33577 Time domain (CST PS) Frequency domain eigenmode solver at the peaks (CST MWS)

19. Dipolar vertical impedance (PEC) Frequency Rs (Ohm) (PEC) at y=1mm Rs (Ohm) (PEC) at y=4 mm Q (PEC) Rs/Q (Ohm/m) 0.53 1.03 1.2 1.92

20. Longitudinal (with losses) Frequency Rs (Ohm) (PEC) Rs (Ohm) (with losses) Q (PEC) Q (with losses) R/Q (Ohm) 1.048.2 10 4 1.96 10 4 308074027 1.613788826206100.14 1.814.9 10 5 1.2 10 5 3380820145 Electrodes + pipe  Stainless Steel Casing  Anticorodal