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The design of elliptical cavities

The design of elliptical cavities. Gabriele Costanza. Introduction. To design a cavity we need to characterize it from an electromagnetic and mechanical point of view. Manufacturing , cleaning , testing

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The design of elliptical cavities

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  1. The design ofellipticalcavities Gabriele Costanza

  2. Introduction • To design a cavityweneedtocharacterize it from an electromagnetic and mechanicalpointofview • Manufacturing, cleaning, testing • Chemicalpolishing: Buffered ChemicalPolishing or Electrop-polishing. Removes a damagedsurfacelayer (dueto the manufacturing process) and reducesroughness. • Heat treatment: removes H from the • Rinsingwithhighpressure, ultrapure water • Design = optimizationof the shapeof the cavitywithrespectto a set of parameters • RF parameters • Mechanical parameters

  3. Introduction • The medium βcavity has 5 cells and operates in the TM01πmode. • The longitudinal E-field has a 180 phaseshift from one cell to the next so that the particlesexperiencealways an acceleratingfield. The lengthofeach cell is then:

  4. Introduction Multicell structures: • Less expensive/m !! • Fewer couplers, easierphasing….. • Advantagesofsingle cell structures: • No fieldflattness problem • Easierto damp HOMS • The input coupler transfers less power • Easiertomanufacture and clean

  5. Example: pillbox • The simplestmodelof an acceleratingcavity = pillbox • Let’sconsider a pillboxofradius a and length h. • To find the fieldsof the accelerating mode (TM010) weneedtosolve the transverse problem: • and the longitudinal problem: • The solution consists in the eigenmodes and eigenvalues. • The accelerating (fundamental) mode is the (TM010): • The dispersion relation is: • For the TM010 mode toresonate at 704.42 MHz, a=16.29 cm

  6. The design of elliptical cavities RF parameters

  7. RF parameters • With the fieldswecancalculateseveralquantities: • Stored Energy: • Power Dissipated: • part of the energystored in the cavity is dissipated on the walls • Power exchangedwith the external circuit: • Power extracted by the HOM coupler or injected by the FPC port

  8. RF parameters • Intrinsicqualityfactor Q0: • Measuresofhowquickly the energystored in the cavity is lost by dissipation in the cavitywalls. • ExternalqualityfactorQext: • Measureshowquickly the energystored in the cavity is radiatedthrough the ports . • Geometricfactor: • Measuresofthe energylost by dissipation in the cavitywallsconsidering a Rsurfof 1 ohm. • The surfaceresistanceof SC structurescan be modeledwith: • The residualresistance is almostconstantwithtemperature and is a measureof the qualityof the material. The clearner the surface, and the purer the metal, the lower is the residualresistance. • The BCS resitancegrowsveryquicklywith the frequency and decreasesexponentiallywith the temperature.

  9. RF parameters • We define the R/Q as: • Where: is a measureofhowefficient the cavityacceleratesthebeam, • a large R/Q impliesthatlittleenergy is requiredtoproduce a large acceleration, therefore the R/Q is a measure on howefficient the energyexchangebetween a mode and the beam is (beamcouplingimpedance) • R/Qdoes not depend on the material of the cavity.

  10. RF parameters • The higher the parameter: the higher the acceleratingvoltagewithrespectto the powerdissipated • Peak Fields: • Epk/Eacc , whereEpkis the peakelectricfield on the surfaceof the cavityand • Bpk/Eacc [mT/(MV/m)], whereBpk is the peakmagneticfield on the surfaceof the cavity.

  11. RF parameters • Cell to Cell CouplingKcc: • It’s a measureof the widthof a band. It’susuallycalculatedonly for the fundamental passband. • It’simportanttohave a high cell-to-cell couplingbecause: • It’seasiertoobtain a highfieldflattness, that is, field is moreevenamong cells • enhancedfrequency separation between the 4π/5 and the π modes • HOMsarebettercoupledto the outer cells and possiblyextracted by an antenna

  12. RF parameters: summary • Rf parameters summary: , theseare not the only parameters totakeintoaccount… • The end cells and the inner cells are different because the outer cells areconnectedto the beamtubes, so I considerthemseparately • Let’stake a look at geometryof the inner cell: • 6geometric parameters: • A,B = radiusesof the major ellipse • a,b = radiusesof the smallerellipse • Riris = the radiusof the iris • D = the diameter of the cell is a tuning parameter • The end cells addother 5 parameters (for symmetriccavities)

  13. The design of elliptical cavities Mechanical parameters

  14. Mechanical parameters • Assume a wallthicknessof 3.6 mm • CavityStiffness [KN/mm]: 1 KN is applied at one end, the other end is grounded. The displacement is calculated • TuningSensitivityΔf/Δz [KHz/mm]: a displacementof 1 mm is imposed at one end, the other end is grounded. The new frequencyof the π mode is calculated. 1 KN

  15. Mechanical parameters • PressureSensitivity [Hz/mbar]: vibrations coming from varioussources cause the detuningof the cavity. The major contributor is the variation of the helium pressure. In this simulation a uniform pressureof 1 mbar is appliedto the external boundary. The frequencyshift is calculated. Bothendsaregrounded

  16. Mechanical parameters • Lorentz DetuningCoefficient [Hz/(MV/m) 2]: The Lorentz Detuning Coefficient is defined as • The frequency detuning is caused by the EM pressure on the cavitywalls. The pressure is • Bothendsaregrounded

  17. The design of elliptical cavities Design

  18. Design • The radiusof the iris is a verypowerfulvariableto trim the RF parameters • All the other parameters have a ”second order” influcence • Toomany parameters to design an entirecavity all at once • Design flow: • All the cells are designed with COMSOL. I wrote a codetoexploreonesectionof the parameter space at a time. The codelaunches COMSOL tosimulate the structure,tunes the cell to 704 MHz and calculates the RF parameters. The mechanical simulations areperformedonly on the full cavity. • Thereare 5 RF parameters, the optimal choice is not obvious! (tradeoffs) RF Parameter calculation & selectionof the best geometry RF Parameter calculation & selectionof the best geometry Inner cell cavity end cell

  19. Parameter trends • All the parmeters areconnectedbetweeneachother and it’s not clearwhat the ”best solution” is • For example: Kcc Peak Fields Riris R/Q G

  20. More on parameter trends Highpeakfieldscan limit the maximum achievable gradient - A ”tall” minor ellipseleadsto a lowerelectricpeakfield (αincreases). - A ”large” major ellipseleadsto a lowermagneticpeakfield - B has littleinfluence on the RF properties. - The same appliesto the outer cells butit’shardertoachieve the same performancedueto the beamtube

  21. The code • The optimizingcode…

  22. The code • The optimizingcode…

  23. The design of elliptical cavities Results

  24. 63_2+31 63+2 57_2+20 largerdomeellipse=>higherKcc Found in ”Medium βEllipticalCavity – CyromoduleTechnologyDemonstrator”. S. Molloy Canweusehigher gradients?

  25. Courtesyof Paolo Pierini, HPSL Workshop Results Lower beta => lower R/Q => SmallerRiris SPL CDR II 4.5 cm Riristoincrease The R/Q but a lower beta LeadstohigherKcc

  26. Results 63+2

  27. Results 63+2

  28. Results 63+2 • The cavitiestendtohavebetterperformances for β>βg

  29. Results 63+2

  30. Results The cavities must be tunedtoobtain a highfieldflattness

  31. Results 63+2

  32. Results 57_2+20

  33. Results 57_2+20

  34. Results 57_2+20

  35. Results

  36. Results 57_2+20

  37. Results 63_2+31

  38. Results 63_2+31

  39. Results 63_2+31

  40. Results 63_2+31

  41. Results 63_2+31

  42. The design of elliptical cavities Bonus Section(ifyou’re not toobored….) SLUT, TACK

  43. Results: HOM 1pole list All HOMswiththeir R/Q’sare calculatedupto 3 GHz. Studyof the HOMsstarted 2.111337 GHz Two modes closeto6f0 : f0 = 352.21 MHz 2.11135 GHz Does this mode reallyexist?

  44. On the numberof cells per cavity βg The lowe the numberof cells, the higher the maximum Eacc. The maximum is not obtained at the geometric beta The higher the numberof cells, the lower the energy / velocityacceptancebut 4 cell cavitiesleadtolonger accelerator & more€

  45. On the numberof cells per cavity Cryostat FillingFactor = Cryostatacceleratingefficiency = βg =0.69 Is a higherβg better? 6 cavitiesper cryo βg =0.67 5cavities per cryo 4 cavities per cryo βg =0.65 2 m 1 m 10 cm 15 cm

  46. On the numberof cells per cavity βg • Higherβg => widerenergy/velocityacceptance, higherinjectionenergy => morespokes. Aretheymoreefficient / less expensivethanellipticalcavities? • If not it’spossibletouse ”few” βg = 0.65 ell. cavities (lowerinjectionenergy) and morehighβcavitieswhicharemoreefficientthanβg = 0.67 cavities • Lowerβg => lowerperformances (butit’spossibletofind a goodcompromise). Cavities for βg<1 have a smallervolume, for the same frequency, w.r.t βg=1 cavities, and lowerEaccbecauseof the reducedlength => higherpeakfields

  47. Simulations ofstiffenedcavities 63_2+31

  48. Someresults 63+2 2 63

  49. Someresults 57_2+20 57_2 20

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