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Low and Intermediate βCavity Design
A Tutorial
Jean DelayenJefferson Lab
2003 SRF Workshop
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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
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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
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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
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Design Tradeoffs
Number of cellsVoltage gainVelocity acceptance
FrequencySizeVoltage gainRf lossesEnergy content, microphonics, rf controlAcceptance, beam quality and losses
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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
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Transit Time Factor
(a)
(b)
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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=
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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=
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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
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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
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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
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A Simple Model: Loaded Quarter-wavelength Resonant Line
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 ¯
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Center conductor voltage
Center conductor current
Line impedance
A Simple Model: Loaded Quarter-wavelength Resonant Line
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 ¯
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Loading capacitance
Peak magnetic field
A Simple Model: Loaded Quarter-wavelength Resonant Line
( ) ( )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 ˛
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Power dissipation (ignore losses in the shorting plate)
Energy content
A Simple Model: Loaded Quarter-wavelength Resonant Line
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
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Geometrical factor
Shunt impedance R/Q
A Simple Model: Loaded Quarter-wavelength Resonant Line
( )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
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A Simple Model: Loaded Quarter-wavelength Resonant Line
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A Simple Model: Loaded Quarter-wavelength Resonant Line
MKS units, lines of constant normalized loading capacitance Γ/λε0
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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
- + =
+ + =
= -
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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
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Profile of Constant Surface Magnetic Field
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Profile of Constant Surface Magnetic Field
MKS units, lines of constant normalized loading capacitance Γ/λε0
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Some Real Geometries (λ/4)
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Some Real Geometries (λ/4)
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Some Real Geometries (λ/2)
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Some Real Geometries (λ/2)
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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
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