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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 ofellipticalcavities

Gabriele Costanza


Introduction
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


Introduction1
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:


Introduction2
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


Example pillbox
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



Rf parameters
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


Rf parameters1
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.


Rf parameters2
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.


Rf parameters3
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.


Rf parameters4
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


Rf parameters summary
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)


The design of elliptical cavities1

The design of elliptical cavities

Mechanical parameters


Mechanical parameters
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


Mechanical parameters1
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


Mechanical parameters2
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



Design
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


Parameter trends
Parameter trends

  • All the parmeters areconnectedbetweeneachother and it’s not clearwhat the ”best solution” is

  • For example:

Kcc

Peak Fields

Riris

R/Q

G


More on parameter trends
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


The code
The code

  • The optimizingcode…


The code1
The code

  • The optimizingcode…



63_2+31

63+2

57_2+20

largerdomeellipse=>higherKcc

Found in ”Medium βEllipticalCavity – CyromoduleTechnologyDemonstrator”. S. Molloy

Canweusehigher gradients?


Results

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


Results1
Results

63+2


Results2
Results

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Results3
Results

63+2

  • The cavitiestendtohavebetterperformances for β>βg


Results4
Results

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Results5
Results

The cavities must be tunedtoobtain a highfieldflattness


Results6
Results

63+2


Results7
Results

57_2+20


Results8
Results

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Results9
Results

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Results11
Results

57_2+20


Results12
Results

63_2+31


Results13
Results

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Results14
Results

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Results15
Results

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Results16
Results

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The design of elliptical cavities4

The design of elliptical cavities

Bonus Section(ifyou’re not toobored….)

SLUT, TACK


Results hom 1pole list
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?


On the number of cells per cavity
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€


On the number of cells per cavity1
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


On the number of cells per cavity2
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


Simulations of stiffened cavities
Simulations ofstiffenedcavities

63_2+31


Some results
Someresults

63+2

2

63


Some results1
Someresults

57_2+20

57_2

20


Some results2
Someresults

63_2+31

63_2

31


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