9 Introduction to superconducting magnets* · 2016. 1. 6. · Introduction to superconducting...

Mauricio Lopes – FNAL

9

Introduction to superconducting

magnets*

* From: “Superconducting Accelerator Magnets” by Paolo Ferracin, Ezio Todesco, Soren O. Prestemon and Helene Felice, January 2012

A Brief History of the

Superconductivity

US Particle Accelerator School – Austin, TX – Winter 2016 2

Heike Kamerlingh Onne

1908 – Successfully liquified helium (4.2 K)

1911 – Discovered the superconductivity

while measuring the conductivity of Mercury

as function of temperature

1913 – Nobel prize


1933 – Walther Meissner and Robert

Ochsenfeld discover perfect diamagnetic

property of supeconductors.

1935 – First theoretical works on SC by Heinz

and Fritz London

1950 – Ginzburg and Landau proposed a

macroscopic theory for SC.

Meissner effect

A Brief History of the

Superconductivity

Why using SC magnets?


�� =�

�=

�� + 2��

��

Example: Lets calculate the magnetic rigidity for a 1 TeV proton:

�� ≈1��

�≈ 3333.

Let us assume a maximum field of 1.5 T; the circumference of such machine will be:

� = 2222

� = 2�� ≈ 14�

The Tevatron was the first machine to use large scale superconductor magnets

with a 4.2 T in a 6.3 km circumference!

Critical surface


Critical surface for different SC

materials


10

100

1,000

10,000

100,000

1,000,000

0 5 10 15 20 25 30 35

Applied Field, T

Sup

erco

nduc

tor

Crit

ical

Cur

rent

Den

sity

,A/m

m²

YBCO: Tape, || Tape-plane, SuperPower (Usedin NHMFL tested Insert Coil 2007)

YBCO: Tape, |_ Tape Plane, SuperPower (Usedin NHMFL tested Insert Coil 2007)

Bi-2212: non-Ag Jc, 427 fil. round wire, Ag/SC=3(Hasegawa ASC-2000/MT17-2001)

Nb-Ti: Max @1.9 K for whole LHC NbTi strandproduction (CERN, Boutboul '07)

Nb-Ti: Nb-47wt%Ti, 1.8 K, Lee, Naus andLarbalestier UW-ASC'96

Nb3Sn: Non-Cu Jc Internal Sn OI-ST RRP 1.3mm, ASC'02/ICMC'03

Nb3Sn: Bronze route int. stab. -VAC-HP, non-(Cu+Ta) Jc, Thoener et al., Erice '96.

Nb3Sn: 1.8 K Non-Cu Jc Internal Sn OI-ST RRP1.3 mm, ASC'02/ICMC'03

Nb3Al: RQHT+2 At.% Cu, 0.4m/s (Iijima et al2002)

Bi 2223: Rolled 85 Fil. Tape (AmSC) B||, UW'6/96

Bi 2223: Rolled 85 Fil. Tape (AmSC) B|_, UW'6/96

MgB2: 4.2 K "high oxygen" film 2, Eom et al.(UW) Nature 31 May '02

MgB2: Tape - Columbus (Grasso) MEM'06

2212round wire

2223tape B|_

At 4.2 K UnlessOtherwise Stated

Nb3SnInternal Sn

Nb3Sn1.8 K

2223tape B||

Nb3SnITER

MgB2film MgB2

tape

Nb3Al:RQHT

1.9 K LHCNb-Ti

YBCO B||c

YBCO B||ab

NbTi Parameterization


( )

−=

7.1

00 1

CCC T

TBTB

where BC0 is the critical field at zero temperature (BC0 ~14.5 T)

γβα

−

−

=

7.1

0_

11),(

cccrefc

c

T

T

B

B

B

B

B

C

J

TBJ

where JC_ref is the critical current density at 4.2 K and 5 T (JC_ref ~ 3000 A/mm2);

C, α, β and γ are fitting parameters:

C ~ 31.4 T

α ~ 0.63

β ~ 1.0

γ ~ 2.3

(Lubell’s formula)

(Bottura’s formula)

Nb3Sn Parameterization


22

0

2

)(1

),(1

)(),,(

−

−=

εεεε

ccc T

T

TB

B

B

CTBJ

−

−

−=

)(77.11

)(31.01

)(1

),(

0

2

0

2

00 εεεε

cccc

c

T

TLn

T

T

T

T

B

TB

where: ( ) 2/17.1

_0 1)( εαε −= mCC

( )7.1

_0 1),( εαε −= mcc BTB

( ) 3/17.1

_00 1)( εαε −= mcc TT

and:

α = 900

ε = -0.003

Tc0_m = 18K

C0_m = 48500 AT1/2/mm2

(for Jc = 3000 A/mm2 @ 4.2 K and 12 T)

(Summer’s formula)

Strand Fabrication


Superconducting cables

• Most of the superconducting coils for particle accelerators are wound from a multi-strand cable.

• The advantages of a multi-strand cable are:

– reduction of the strand piece length;

– reduction of number of turns• easy winding;

• smaller coil inductance– less voltage required for power supply during ramp-up;

– after a quench, faster current discharge and less coil voltage.

– current redistribution in case of a defect or a quench in one strand.

• The strands are twisted to

– reduce interstrand coupling currents (see interfilament coupling currents)• Losses and field distortions

– provide more mechanical stability

• The most commonly used multi-strand cables are the Rutherford cable and the cable-in-conduit.



• Rutherford cables are fabricated by a cabling machine.

– Strands are wound on spools mounted on a rotating

drum.

– Strands are twisted around a conical mandrel into an

assembly of rolls (Turk’s head). The rolls compact the

cable and provide the final shape.



• The final shape of a Rutherford cable can be rectangular or trapezoidal.

• The cable design parameters are:

– Number of wires Nwire

– Wire diameter dwire

– Cable mid-thickness tcable

– Cable width wcable

– Pitch length pcable

– Pitch angle ψcable (tanψ cable = 2 wcable / pcable)

– Cable compaction (or packing factor) kcable

– i.e the ratio of the sum of the cross-sectional area of the strands (in the direction parallel to the cable axis) to the cross-sectional area of the cable.

• Typical cable compaction: from 88% (Tevatron) to 92.3% (HERA).

cablecablecable

wirewirecable tw

dNk

ψπ

cos4

2

=


Cable insulation

• The cable insulation must feature– Good electrical properties to withstand

high turn-to-turn voltage after a quench.

– Good mechanical properties to withstand high pressure conditions

– Porosity to allow penetration of helium (or epoxy)

– Radiation hardness

• In NbTi magnets the most common insulation is a series of overlapped layers of polyimide (kapton).

• In the LHC case:– two polyimide layers 50.8 µm thick

wrapped around the cable with a 50% overlap, with another adhesive polyimide tape 68.6 µm thick wrapped with a spacing of 2 mm.


Superconducting Magnets Design

Perfect dipole

-60

0

60

-4 0 0 4 0

+

+-

-

-60

0

60

-4 0 0 40

+

+-

-

A wall-dipole, cross-section A practical winding with flat cables

1 - Wall dipole (similar to the window frame magnet)



Perfect dipole

2 - Intersecting ellipsis

-6 0

0

6 0

-4 0 0 4 0

+

+-

-

-60

0

60

-40 0 40

+

+-

-

Intersecting ellipses A practical (?) winding with flat cables


Intersecting Cylinders


within a cylinder carrying uniform current j0, the field is perpendicular to the radial

direction and proportional to the distance to the center r:

Combining the effect of the two cylinders

Similar proof for intersecting ellipses

200 rj

Bµ

−=

{ } 0sinsin2 2211

00 =+−= θθµrr

rjBx

{ } sj

rrrj

B y 2coscos

200

221100 µθθµ

−=+−=


Perfect dipole

3 – Cos(θ) current distribution

-6 0

0

6 0

-4 0 0 4 0

+

+-

-j = j0 cos θ

θ



Perfect quadrupole


-60

0

60

-40 0 40

++

- - j = j 0 cos 2θ

θ

- -

++

Quadrupole as an ideal cos2θ

-60

0

60

-40 0 40

++

- -

- -

+

+

Quadrupole as two intersecting ellipses

Dipole design using sector coils


-6 0

0

6 0

0

+

+-

-j = j0 cos θ

θ+

+

-

-

αααα

w

r

∑∞

=

−

=

1

1

)(n

n

refn R

zCzB

n

ref

refn z

R

R

IC

−=

0

0

2πµ

0

0

0

01

cos

2

1Re

2 z

I

z

IB

θπµ

πµ −=

−= θρρ ddjI →

απµθρρ

ρθ

πµ α

α

sin2cos

22 00

1 wj

ddj

Bwr

r

−=−= ∫ ∫−

+

Multipoles of a dipole sector coil


∫ ∫∫ ∫−

+−

−

−

+−

−−=−−=α

α

α

α

ρρθθπ

µθρρ

ρθ

πµ wr

r

n

nref

wr

rn

nref

n ddinRj

ddinRj

C 11

01

0)exp(

)exp(

22

+−=r

wRjB

ref1log)2sin(

02 α

πµ

n

rwr

n

nRjB

nnnref

n −−+−=

−−−

2

)()sin(2 2210 απ

µ

for n = 2

for n > 2

+−=

wrr

jRB

ref 11

3

)3sin(2

0

3

απ

µ( )

+−=

33

40

5

11

5

)5sin(

wrr

jRB

ref απ

µ

for α=π/5 (36°) or for α=2π/5 (72°) B5=0for α=π/3 (60°) B3=0

Multi-sector dipole coil


0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

45.0

50.0

α1

α2α3

+−

+−=

wrr

jRB

ref 11

3

3sin3sin3sin 123

20

3

αααπ

µ

+−

+−=

33

123

40

5)(

11

5

5sin5sin5sin

wrr

jRB

ref αααπ

µ

=+−=+−

0)5sin()5sin()5sin(

0)3sin()3sin()3sin(

123

123

αααααα

(48°,60°,72°) or (36°,44°,64°) are some of the possible solutions

[0°-43.2°, 52.2°-67.3°] sets also B7 = 0 !

Multi-sector dipole coil


0)3sin()3sin()3sin()3sin()3sin( 12345 =+−+− ααααα




0)11sin()11sin()11sin()11sin()11sin( 12345 =+−+− ααααα(B3, B5 and B7) = 0

[0°-33.3°, 37.1°- 53.1°, 63.4°- 71.8°] sets (B3, B5, B7 , B9 and B11) = 0!

Examples


0

20

40

60

0 20 40 60x (mm)

y (m

m)

RHIC main dipole - 1995Tevatron main dipole - 1980

0

20

40

60

0 20 40 60x (mm)

y (m

m)

Two layer design


0

50

0 50

α1

α2

r

w

+−

++

+−∝

wrwrwrrB

211

)3sin(11

)3sin( 213 αα

+−

++

+−∝ 3323315 )2(

1

)(

1)5sin(

)(

11)5sin(

wrwrwrrB αα

0

20

40

60

80

0 20 40 60 80x (mm)

y (m

m)

72o

37o

0

15

30

45

60

75

90

0.0 0.1 0.2 0.3 0.4 0.5 0.6w/r (adim)

Ang

le (

degr

ees)

a1 - first branch a2 - first branch

a1 - second branch a2 - second branch

α1 α2

α2α1

Tevatron main dipole

Quadrupole design using sector coils


-5 0

0

5 0

-50 0 50

+

+

+

+

--

- -

r w

∑∞

=

−

=

1

1

)(n

n

refn R

zCzB

n

ref

refn z

R

R

IC

−=

0

0

2πµ

20

0

20

02

2cos

2

1Re

2 z

RI

z

RIB refref θ

πµ

πµ

−=

−= θρρ ddjI →

[ ]

+−=−= ∫ ∫+

r

wRjdd

RjB

refwr

r

ref1ln2sin

42cos

28

0

02

02 α

πµ

θρρρ

θπ

µ α

Multipoles of a quadrupole sector coil


( )

+−= 44

50

6

116

)6sin(

wrr

jRB ref α

πµ

( )

+−= 88

80

10

11

10

)10sin(

wrr

jRB ref α

πµ

for α=π/6 (30°) one has B6 = 0

for α=π/10 (18°) or α=π/5 (36°) one sets B10 = 0

It follows the same philosophy of the Dipole design!

Examples


Tevatron main quadrupole

0

20

40

0 20 40 60 80x (mm)

y (m

m)

RHIC main quadrupole

0

20

40

0 20 40 60 80x (mm)

y (m

m)

~[0°-12°, 18°-30°] ~[0°-18°, 22°-32°]

0

20

40

0 20 40 60 80x (mm)

y (m

m)

LHC main quadrupole

~[0°-24°, 30°-36°]

Peak field and bore field ratio (λ)


0

20

40

60

80

0 20 40 60 80x (mm)

y (m

m)

LHC main dipole –location of the peak field

0

20

40

60

80

0 20 40 60 80x (mm)

y (m

m)

RHIC main dipole –location of the peak field

0

20

40

60

80

0 20 40 60 80x (mm)

y (m

m)

Tevatron main dipole –location of the peak field

Peak field and bore field ratio (λ)


1.0

1.1

1.2

1.3

1.4

1.5

1.6

0 20 40 60 80width (mm)

λ (

adim

)

1 layer 60

1 layer 48-60-72

1 layer 42.8-51.6-67

1 layer three blocks

1.0

1.1

1.2

1.3

1.4

1.5

1.6

0 20 40 60 80width (mm)

λ (

adim

)

1 layer 601 layer 48-60-721 layer 42.8-51.6-671 layer three blocksFit

w

arrw +1~),(λ a ~ 0.045

Examples


1.0

1.1

1.2

1.3

0.0 0.5 1.0 1.5 2.0equivalent width w/r

λ [

adim

]

TEV MB HERA MB

SSC MB RHIC MB

LHC MB Fresca

MSUT D20

HFDA NED

Operational Margin


24 %

Lorentz Forces


• A superconducting accelerator magnet has a large magnetic stored

energy

• A quench produces a resistive zone

• Current is flowing through the magnet

• The challenge of the protection is to provide a safe conversion of the

magnetic energy to heat in order to minimize

– Peak temperature (“hot spot”) and temperature gradients in the magnet

– Peak voltages

• The final goal being to avoid any magnet degradation

– High temperature => damage to the insulation or stabilizer

– Large temperature gradient => damage to the conductor due to differential thermal

expansion of materials

Joule Heating

Voltages (R and L)

Quench protection



Quench

Normal zone growth

Detection

Power supply

switched offProtection

heatersExtraction Quenchback

Trigger protection options

Current decay in the magnet

I

t

Magnetic energy

Converted to heat by Joule heating

General quench protection diagram

The faster this chain happens the safer is the magnet

Training


Magnetization


Summary


• Design and Fabrication of Superconducting Magnets belong to a different

Universe

• Although the mathematical formulation for the field generation is shared, the

design of superconducting magnets involves many other aspects:

o Thermal considerations

o Mechanical Analysis

o Fabrication techniques

o Quench Protection

o Material Science

• If one is interested to learn more about superconducting magnet, one should

attend to the Superconducting Accelerator Magnets USPAS course. The material

for that course can be found at:

http://etodesco.web.cern.ch/etodesco/uspas/uspas.html

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Unusual design examples

9 Introduction to superconducting magnets* · 2016. 1. 6. · Introduction to superconducting...

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