Low and Intermediate β Cavity - CERNMAFIA, Microwave Studio, HFSS, etc. 3D software is usually very...

Thomas Jefferson National Accelerator FacilityOperated by the Southeastern Universities Research Association for the U.S. Department of Energy

Low and Intermediate βCavity Design

A Tutorial

Jean DelayenJefferson Lab

2003 SRF Workshop

Operated by the Southeastern Universities Research Association for the U.S. Department of Energy

Thomas Jefferson National Accelerator Facility

A Few Obvious Statements

Low and medium ββ<1Particle velocity will change

The lower the velocity of the particle or cavity βThe faster the velocity of the particle will changeThe narrower the velocity range of a particular cavityThe smaller the number of cavities of that βThe more important it is that the particle achieve design velocity

Be conservative at lower βBe more aggressive at higher β



Two main types of structure geometriesTEM class (QW, HW, Spoke)TM class (elliptical)

Design issues of medium β elliptical cavities are similar to those of β=1Most of the talk will be on TEM-class cavitiesFor TM-class cavities see:

Design criteria for elliptical cavitiesPagani, Barni, Bosotti, Pierini, Ciovati, SRF 2001.

Challenges and the future of reduced beta srf cavity designSang-ho Kim, LINAC 2002.

A Few More Statements



A Word on Design Tools

TEM-class cavities are essentially 3D geometries

3D electromagnetic software is availableMAFIA, Microwave Studio, HFSS, etc.

3D software is usually very good at calculating frequenciesNot quite as good at calculating surface fields

Use caution, vary mesh sizeRemember Electromagnetism 101



Design Tradeoffs

Number of cellsVoltage gainVelocity acceptance

FrequencySizeVoltage gainRf lossesEnergy content, microphonics, rf controlAcceptance, beam quality and losses



Energy Gain Transit Time Factor - Velocity Acceptance

Assumption: constant velocity

( ) cos( )W q E z t dzω φ∆+•

-•

= +Ú

0 0cos ( ) ( )

Max ( ) cosTransit Time Factor

( )

( ) cos( ) Velocity Acceptance

Max ( ) cos

W q W T W E z dz

zE z dzc

E z dz

zE z dzc

TzE z dzc

φ β

ωβ

ωβ

βωβ

∆ ∆ ∆ Θ

Θ

+•

-•+•

-•+•

-•

+•

-•+•

-•

= =

Ê ˆÁ ˜Ë ¯

=

Ê ˆÁ ˜Ë ¯

=Ê ˆÁ ˜Ë ¯

Ú

Ú

Ú

Ú

Ú



Transit Time Factor

(a)

(b)



Velocity Acceptance for 2-Gap Structures

0 0

0 0

0

0 0

sin sin2 2

( )sin sin

2 2

x xT

x x

β βπα πβ βββ

β πα π

Ê ˆ Ê ˆÁ ˜ Á ˜Ë ¯ Ë ¯

=Ê ˆ Ê ˆÁ ˜ Á ˜Ë ¯ Ë ¯

00 2x

lβ λ=



Velocity Acceptance for 3-Gap Structures

0 0 0

0 0 0

0

0 0 0

sin cos cos3 3

( )sin cos cos

3 3

x x xT

x x x

β β βπα πα πβ β βββ

β πα πα π

È ˘Ê ˆ Ê ˆ Ê ˆ-Í ˙Á ˜ Á ˜ Á ˜Ë ¯ Ë ¯ Ë ¯Î ˚=

È ˘Ê ˆ Ê ˆ Ê ˆ-Í ˙Á ˜ Á ˜ Á ˜Ë ¯ Ë ¯ Ë ¯Î ˚

00 2x

lβ λ=



Higher-Order Effects

( )20 (2) (2)0

(2)

(2)2

0

cos ( ) ( ) sin 2 ( )

( ) ( ) ( ) /4

( ) ( ) ( ) ( )( )4

s

s

q WW q W T T T

Wk dT k T k T k k c

dkk T k k T k k T k T kT k dk

k

φ β β φ β

ω β

π

∆∆ ∆

•

È ˘= + +Î ˚

= - =

+ - -¢ ¢= - ¢¢Ú



If characteristic length <<λ (β<0.5), separate the problem in two parts:Electrostatic model of high voltage regionTransmission line

A Simple Model: Loaded Quarter-wavelength Resonant Line



Basic Electrostaticsa: concentric spheresb: sphere in cylinderc: sphere between 2 planesd: coaxial cylinderse: cylinder between 2 planes

Vp : Voltage on center conductorOuter conductor at groundEp: Peak field on center conductor




Capacitance per unit length

Inductance per unit length

0 0

0 0

2 21ln ln

Cbr

πε πε

ρ

= =Ê ˆ Ê ˆÁ ˜ Á ˜Ë ¯ Ë ¯

0 0

0 0

1ln ln2 2

bLr

µ µπ π ρ

Ê ˆ Ê ˆ= =Á ˜ Á ˜Ë ¯ Ë ¯



Center conductor voltage

Center conductor current

Line impedance


02( ) sinV z V zπλ

Ê ˆ= Á ˜Ë ¯

02( ) cosI z I zπλ

Ê ˆ= Á ˜Ë ¯

0 00

0 0 0

1ln , 3772

VZI

µη ηπ ρ ε

ΩÊ ˆ

= = =Á ˜Ë ¯



Loading capacitance

Peak magnetic field


( ) ( )0 0

2cotan cotan2( )

ln / ln /

zz

b r b

π π ζλλε λε

ρΓ

Ê ˆ Ê ˆÁ ˜ Á ˜Ë ¯ Ë ¯

= =

00

m, A/m1ln sin m, T

2300 cm, G

: Voltage across loading capacitance

9 mT at 1 MV/m

p

p

HV

c Bb

B

VB

ηπρ ζ

ρ

Ï ¸ Ï ¸Ê ˆÔ Ô Ô ÔÊ ˆ= Ì ˝ Ì ˝Á ˜Á ˜ Ë ¯Ë ¯Ô Ô Ô ÔÓ ˛ Ó ˛



Power dissipation (ignore losses in the shorting plate)

Energy content


2 02 2

20

2 22

1 sin1 1/8ln sin

2

sp

s

RP Vb

RP E

ζ πζρλ πππ η ρ ζ

β λη

++=

µ

( )2 0

20

2 2 30

1 sin18 ln 1/ sin

2

pU V

U E

ζ πζπε πλ πρ ζ

ε β λ

+=

µ



Geometrical factor

Shunt impedance R/Q


( )00

ln 1/2

1 1/sbG QR

G

ρπ η

λ ρη β

= =+

µ

222

0

0

2

sinln32 211 1/ sin

shs

sh s

bRR

R R

π ζρηπ λ ρ ζ πζ

πη β

=+ +

µ

( )24 /pV P

( )2

02

sin16 2ln 1/ 1 sin

sh

sh

RQ

RQ

π ζη ρ

π ζ πζπ

η

=+

µ




MKS units, lines of constant normalized loading capacitance Γ/λε0



More Complicated Center Conductor Geometries

2 2

2

2 2

2

1 0ln 41 0ln 4

( )( ) ( )/

d v d dv vd d dd i d di id d d

i zz C zdi dz

ρ πζ ρ ρ ζ ζ

ρ πζ ρ ρ ζ ζ

Γ

- + =

+ + =

= -



Constant logarithmic derivative of line capacitanceGood model for linear taper

Constant surface magnetic field

exp( / )01 d 1 ( )

d

z drC r z bC z d b

Ê ˆ= - = Á ˜Ë ¯

( ) ( )i z r zµ

22 2

2 2

1 4 0ln( / )

d r dr rdz r b r dz

πλ

Ê ˆ- + =Á ˜Ë ¯

More Complicated Center Conductor Geometries



Profile of Constant Surface Magnetic Field



Profile of Constant Surface Magnetic Field

MKS units, lines of constant normalized loading capacitance Γ/λε0



Some Real Geometries (λ/4)



Some Real Geometries (λ/2)



Parting Words

In the last 30+ years, the development of low and medium βsuperconducting cavities has been one of the richest and most imaginative area of srfThe field has been in perpetual evolution and progressNew geometries are constantly being developedThe final word has not been said

The parameter, tradeoff, and option space available to the designer is large

The design process is not, and probably will never be, reduced to a few simple rules or recipesThere will always be ample opportunities for imagination, originality, and common sense

Low and Intermediate β Cavity - CERNMAFIA, Microwave Studio, HFSS, etc. 3D software is usually very...

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