Design and Modelling of LTS Superconducting Magnets€¦ · Design and Modelling of LTS Superconducting Magnets Magnetic design Paolo Ferracin ... L. Rossi, E. Todesco, “Electromagnetic

ASC 2012 Short Course

Design and Modelling of LTS

Superconducting Magnets

Magnetic design

Paolo Ferracin ([email protected])

European Organization for Nuclear Research (CERN)

Introduction

The magnetic design is one of the first steps in the a superconducting magnet development

It starts from the requirements (from accelerator physicists, researchers, medical doctors…others)

A field “shape”

Dipole, quadrupole, etc

A field magnitude

Usually with low T superconductors from 5 to 20 T

A field homogeneity

Uniformity inside a solenoid, harmonics in a accelerator magnet

A given aperture (and volume)

Some cm diameter for accelerator magnets, much more for detectors and fusion magnets

Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 2

Introduction

Goal of this chapter How much conductor do we need to meet the requirements? And in which configuration?

Outline

Different field shape and their function

solenoid, dipole, quadrupole

How do we create a perfect field? How do we express the field and its “imperfections”? How do we design a coil to minimize field errors? Which is the maximum field we can get? Overview of different designs


References

1. P. Ferracin, E. Todesco, S. Prestemon, H. Felice, “Superconducting accelerator magnets”, US Particle Accelerator School, www.uspas.fnal.gov.

• Units 2, 5, 8, and 9 by Ezio Todesco


2. A. Jain, “Basic theory of magnets”, CERN 98-05 (1998) 1-26 3. Classes given by A. Jain at USPAS, www.uspas.fnal.gov. 4. K.-H. Mess, P. Schmuser, S. Wolff, “Superconducting accelerator magnets”,

Singapore: World Scientific, 1996. • Chapter 4

5. Martin N. Wilson, "Superconducting Magnets", 1983. • Chapter 1

6. Fred M. Asner, "High Field Superconducting Magnets", 1999. • Chapter 9

7. Classes given by A. Devred at USPAS, www.uspas.fnal.gov. 8. L. Rossi, E. Todesco, “Electromagnetic design of superconducting

quadrupoles”, Phys. Rev. ST Accel. Beams 9 (2006) 102401. 9. S. Russenschuck, “Field computation for accelerator magnets”, J. Wiley &

Sons (2010).

http://www.uspas.fnal.gov/



Field shape: dipoles

Main field components is By

Perpendicular to the axis of the magnet z

Magnetic field steers (bends) the particles in a circular orbit

Driving particles in the same accelerating structure several times Particle accelerated energy increased magnetic field increased (“synchro”) to keep the particles on the same orbit of curvature r


BveF

reBp

Field shape: quadrupoles

The force necessary to stabilize linear motion is provided by the quadrupoles

Quadrupoles provide a field which is proportional to the transverse deviation from the orbit, acting like a spring

A series of alternating quadrupole focus the beam in both planes


GyB

GxB

x

y

Field shape: solenoids

Main field components is Bz

Parallel to the axis of the magnet z

Applications in accelerators A solenoid providing a field parallel

to the beam direction (example: LHC CMS)

A series of toroidal coils to provide a circular field around the beam (example: LHC ATLAS)

The particle is bent with a curvature radius

Other applications Fusion magnets, NMR, MRI


z

Introduction


Outline





Perfect dipole field Intercepting ellipses or circles

A uniform current density in the area of two intersecting ellipses produces a pure dipolar field

The aperture is not circular Not easy to simulate with a flat cable

The same for intercepting circles Within a cylinder carrying uniform current j0, the field is perpendicular to the radial direction and proportional to the distance to the centre r:

Combining the effect of the two cylinders

Similar proof for intersecting ellipses


-60

0

60

-40 0 40

+

+-

-

M. Wilson, [5] pg. 28

2

00 rjB

0sinsin2

2211

00

rrrj

Bx

sj

rrrj

By2

coscos2

00

2211

00

Perfect dipole field Thick shell with cos current distribution

If we assume J = J0 cos where J0 [A/m2] is to the cross-section plane

Inner (outer) radius of the coils = a1 (a2)

The generated field is a pure dipole

Linear dependence on coil width

Easier to reproduce with a flat rectangular cable


)(2

1200 aa

JBy

Perfect quadrupole field

Intercepting ellipses or circles

Thick shell with cos2 current distribution

If we assume

J = J0 cos2 where J0 [A/m2] is to the cross-section plane

Inner (outer) radius of the coils = a1 (a2)

And so on…

Perfect sextupoles: cos3 or three intersecting ellipses

Perfect 2n-poles: cosn or n intersecting ellipses


-60

0

60

-40 0 40

+

+

- -

- -

+

+

1

200 ln2 a

aJ

r

BG

y

Perfect solenoid field

Let’s consider the ideal case of a infinitely long “thin-walled” solenoid, with thickness d, radius a, and current density Jθ . The field outside the solenoid is zero. The field inside the solenoid B0 is uniform, directed along the solenoid axis and given by The field inside the winding decreases linearly from B0 at a to zero at a + . The average field on the conductor element is B0/2 Twice more efficient than thick shell dipole!!!


JB 00

From ideal to practical configuration

How can I reproduce thick shell with a cos distribution with a cable?

Rectangular cross-section and constant current density

First “rough” approximation Sector dipole

Better ones Single layer with wedges to reduce current towards 90

More layers with different angles


As a result, the field is not perfect anymore

How can I express in improve the “imperfect” field inside the aperture?

Introduction


Outline





Maxwell equations for magnetic field

In absence of charge and magnetized material

If (constant longitudinal field), then

Field representation Maxwell equations


0

z

B

y

B

x

BB zyx

t

EJB

000

0,,

x

B

y

B

z

B

x

B

y

B

z

BB

yxxzzy

0

z

Bz

0

y

B

x

B yx 0

x

B

y

B yx

Field representation Analytic functions

If

Maxwell gives

and therefore the function By+iBx is analytic

where Cn are complex coefficients

Advantage: we reduce the description of the field to a (simple) series of complex coefficients


0

z

Bz

0

y

B

x

Bxy

0

x

B

y

Bxy

0

0

x

f

y

f

y

f

x

f

yx

yx

1

1

),(),(

n

n

nxy iyxCyxiByxB

Cauchy-Riemann conditions

1

1

1

1

),(),(

n

n

nn

n

n

nxy iyxiABiyxCyxiByxB

What are these coefficients (or harmonics)?

For n=1 dipole

For n=2 quadrupole

Field representation Harmonics


1

1

1

1

),(),(

n

n

nn

n

n


11 iABiBB xy

yAxiAyiBxBiyxiABiBB xy 222222

[K.-H. Mess, et al, [4] pg. 50]

1B

2B

1A

2A


So, each coefficient corresponds to a “pure” multipolar field


1

1

1

1

),(),(

n

n

nn

n

n


The field harmonics are rewritten as (EU notation)

We factorize the main component (B1 for dipoles, B2 for quadrupoles)

We introduce a reference radius Rref to have dimensionless coefficients We factorize 10-4 since the deviations from ideal field are 0.01%

The coefficients bn, an are called normalized multipoles bn are the normal, an are the skew (adimensional)

1

1

1

4 )(10

n

n ref

nnxyR

iyxiabBiBB

[K.-H. Mess, et al, [4] pg. 50]


The coefficients bn, an are called normalized multipoles

bn are the normal, an are the skew (adimensional)

Reference radius is usually chosen as 2/3 of the aperture radius


-40

0

40

-40 0 40

+

+

+

+

--

- -

r

R ref

x

y

1

1

1

4 )(10

n

n ref

nnxyR

iyxiabBiBB


Important property: starting by the multipolar expansion of a current line (Biot-Savart law)


1

11

00

0

1

1

00

0

22)(

n

n

ref

n

ref

n

n

R

iyx

z

R

z

I

z

z

z

IzB

0

0

0

0

0

1

1

2)(2)(

z

zz

I

zz

IzB

-40

0

40

-40 0 40

z 0 =x 0 +iy 0

x

y

z=x+iy

B=B y +iB x

)()()( ziBzBzB xy

1

1

1

4 )(10

n

n ref

nnxyR

iyxiabBiBB

1

010

4

0

2

10

n

ref

nnz

R

Bz

Iiab


One can demonstrate that with line currents with a dipole or a quadrupole symmetry, most of the multipoles cancelled

For n=1 dipole Only b3, b5, b7, ….. are present

For n=2 quadrupole Only b6, b10, b14, ….. are present

…and so on

These multipoles are called allowed multipoles

The field quality optimization of a coil lay-out concerns only a few quantities

For a dipole, usually b3 , b5 , b7 , and possibly b9 , b11


K.-H. Mess, et al, [4]

Back to the original issue: From ideal to practical configuration

How can I reproduce thick shell with a cos distribution with a cable?

Rectangular cross-section and constant current density

First “rough” approximation Sector dipole

Better ones Single layer with wedges to reduce current towards 90

More layers with different angles


Now, I can use the multipolar expansion to optimize my “practical” cross-section

Introduction


Outline





A “good” field quality dipole Sector dipole

We compute the central field given by a sector dipole with uniform current density j

We start from Biot-Savart law and integrate

And we obtain

Multipoles n are proportional to sin (n angle of the sector) They can be made equal to zero !


rr ddjI +

+

-

-

a

w

r

n

rwr

n

nRjB

nnn

ref

n

2

)()sin(2 221

0 a

a

rr

r

a

a

sin2cos

22 00

1 wj

ddj

B

wr

r


First allowed multipole B3 (sextupole)

for a=/3 (i.e. a 60° sector coil) one has B3=0

Second allowed multipole B5 (decapole)

for a=/5 (i.e. a 36° sector coil) or for a=2/5 (i.e. a 72° sector coil)

one has B5=0

With one sector one cannot set to zero both multipoles … let us try with more sectors !


wrr

jRB

ref 11

3

)3sin(2

0

3

a

33

4

0

5

11

5

)5sin(

wrr

jRB

ref a

+

+

-

-

a

w

r


Coil with two sectors

Note: we have to work with non-normalized multipoles, which can be added together

Equations to set to zero B3 and B5

There is a one-parameter family of solutions, for instance (48°,60°,72°) or (36°,44°,64°) are solutions


wrr

jRB

ref 11

3

3sin3sin3sin 123

2

0

3

aaa

33

123

4

0

5)(

11

5

5sin5sin5sin

wrr

jRB

ref aaa

0.0

5.0

10.0

15.0

20.0

25.0

30.0

35.0

40.0

45.0

50.0

a1

a2a3

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

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

123

123

aaa

aaa


We have seen that with one wedge one can set to zero three multipoles (B3, B5 and B7)

What about two wedges ?

One can set to zero five multipoles (B3, B5, B7 , B9 and B11) ~[0°-33.3°, 37.1°- 53.1°, 63.4°- 71.8°]


0)3sin()3sin()3sin()3sin()3sin( 12345 aaaaa





One wedge, b3=b5=b7=0 [0-43.2,52.2-67.3]

Two wedges, b3=b5=b7=b9=b11=0 [0-33.3,37.1-53.1,63.4- 71.8]


Let us see two coil lay-outs of real magnets The RHIC dipole has four blocks


0

20

40

60

0 20 40 60x (mm)

y (

mm

)


Limits due to the cable geometry Finite thickness one cannot produce sectors of any width

Cables cannot be key-stoned beyond a certain angle, some wedges can be used to better follow the arch

One does not always aim at having zero multipoles There are other contributions (iron, persistent currents …)


0

20

40

60

0 20 40 60x (mm)

y (

mm

)

A “good” field quality dipole Sector quadrupole

Let’s look at the quadrupoles

First allowed multipole B6 (dodecapole)

for a=/6 (i.e. a 30° sector coil) one has B6=0

Second allowed multipole B10

for a=/10 (i.e. a 18° sector coil) or for a=/5 (i.e. a 36° sector coil)

one has B5=0

The conditions look similar to the dipole case …


44

5

0

6

11

6

)6sin(

wrr

jRB

ref a

88

8

0

10

11

10

)10sin(

wrr

jRB

ref a

-50

0

50

-50 0 50

+

+

+

+

--

- -

r w


For a sector coil with one layer, the same results of the dipole case hold with the following transformation

Angles have to be divided by two

Multipole orders have to be multiplied by two

Examples One wedge coil: ~[0°-48°, 60°-72°] sets to zero b3 and b5 in dipoles

One wedge coil: ~[0°-24°, 30°-36°] sets to zero b6 and b10 in quadrupoles

The LHC main quadrupole is

based on this lay-out: one wedge

between 24° and 30°, plus one on

the outer layer for keystoning the

coil


0

20

40

0 20 40 60 80x (mm)

y (

mm

)


Examples One wedge coil: ~[0°-12°, 18°-30°] sets to zero b6 and b10 in quadrupoles

The Tevatron main quadrupole is based on this lay-out: a 30° sector coil with

one wedge between 12° and 18° (inner layer) to zero b6 and b10 – the outer layer

can be at 30° without wedges since it does not affect much b10

One wedge coil: ~[0°-18°, 22°-32°] sets to zero b6 and b10

The RHIC main quadrupole is based on this lay-out – its is the solution with the smallest angular width of the wedge


0

20

40

0 20 40 60 80x (mm)

y (

mm

)

0

20

40

0 20 40 60 80x (mm)

y (

mm

)

Introduction


Outline





Maximum field and coil thickness Dipoles

We recall the equations for the critical surface Nb-Ti: linear approximation is good

with s~6.0108 [A/(T m2)] and B*c2~10 T at 4.2 K or 13 T at 1.9 K

This is a typical mature and very good Nb-Ti strand

Tevatron had half of it!


0

2000

4000

6000

8000

0 5 10 15B (T)

j sc(

A/m

m2)

Nb-Ti at 1.9 K

Nb-Ti at 4.2 K

),()( *

2, BBsBj ccsc


The current density in the coil is lower

Strand made of superconductor and normal conducting (copper) There are voids and insulation

The current density flowing in the insulated cable is reduced by a factor (filling ratio)

It ranges from ¼ to 1/3

The critical surface for j (engineering current density)


)()( , BjBj cscc

)()( *

2 BBsBj cc


We characterize the coil by two parameters

c: how much field in the centre is given per unit of current density

Cos dipole

Sector dipole

: ratio between peak field and central field

for a cos dipole, =1

For a sector and in general is = 1.05 - 1.15


jB c jBB cp

wj

B2

01

a

sin

2 01 w

jB


We can now compute what is the highest peak field that can be reached in the dipole in the case of a linear critical surface


)( ,

*

2, sspccccssp BBsjB

*

2,1

c

c

cssp B

s

sB

)()( *

2 BBsBj cc

0

500

1000

1500

2000

2500

0 5 10 15

Curr

ent

den

sity

j (

A/m

m2)

Magnetic field B (T)

j=s(B*c2-B)

Bp=cj[Bp,ss,jss]

*

21

c

c

ss Bs

sj

*

21

c

c

c

ss Bs

sB

We can now compute the maximum current density that can be tolerated by the superconductor (short sample limit)

and the bore short sample field (in the centre not on the conductor)

jB c jBB cp


We got an equation giving the field reachable for a dipole with a superconductor having a linear critical surface The plan: use c from sector coils and try to find an estimate for which characterize the lay-out What is (ratio between peak field and bore field)?

To compute the peak field one has to compute the field everywhere in the coil, and take the maximum


*

21

c

c

c

ss Bs

sB

0

20

40

60

80

0 20 40 60 80x (mm)

y (

mm

)0

20

40

60

80

0 20 40 60 80x (mm)

y (

mm

)

0

20

40

60

80

0 20 40 60 80x (mm)

y (

mm

)


Numerical evaluation of for different sector coils

For interesting widths (10 to 30 mm) is 1.05 - 1.15

This simply means that peak field 5-15% lager

The cos approx having =1 is not so bad for w>20 mm

Typical hyperbolic fit with a~0.045


1.0

1.1

1.2

1.3

1.4

1.5

1.6

0 20 40 60 80width (mm)

(

adim

)

1 layer 60

1 layer 48-60-72

1 layer 42.8-51.6-67

1 layer three blocks

Fit

w

arrw 1~),(


We now can write the short sample field for a sector coil as a function of

Material parameters c, B*c2

Cable parameters

Aperture r and coil width w

a=0.045 c0=6.6310-7 [Tm/A]

for Nb-Ti s~6.0108 [A/(T m2)] and B*c2~10 T at 4.2 K or 13 T at 1.9 K


*

21

c

c

c

ss Bs

sB

w

arrw 1~),(

wcc 0~

*

2

0

0

11

~ c

c

c

ss B

sww

ar

swB



0

5

10

15

0 20 40 60 80equivalent width (mm)

Bss

(T

)

r=30 mm r=60 mm

r=120 mm Cos theta

Nb-Ti at 4.2 K =0.35

Nb3Sn at 4.2 K =0.35

Maximum gradient and coil thickness Quadrupoles

The same approach can be used for a quadrupole We define

the only difference is that now c gives the gradient per unit of current density, and in Bp we multiply by r for having T and not T/m

We compute the quantities at the short sample limit for a material with a linear critical surface (as Nb-Ti)

Please note that is not any more proportional to w and not any more independent of r !


j

Gc

rG

Bp

*

2,1

c

c

c

ssp Bsr

srB

*

21

c

c

ss Bsr

sj

*

21

c

c

c

ss Bsr

sG

r

wcc 1ln0


The ratio is defined as ratio between peak field and gradient times aperture (central field is zero …)

Numerically, one finds that for large coils

Peak field is “going outside” for large widths


0

20

40

0 20 40 60 80x (mm)

y (

mm

)

0

20

40

0 20 40 60 80x (mm)

y (

mm

)

1.1

1.2

1.3

1.4

1.5

0 1 2 3w/r (adim)

(

adim

)

[0,30]

[0-24,30-36]

[0-18,22-32]


The ratio is defined as ratio between peak field and gradient times aperture (central field is zero …)

The ratio depends on w/r

A good fit is

a-1~0.04 and a1~0.11 for the [0°-24°,30°-36°] coil

A reasonable approximation is ~0=1.15 for ¼<w/r<1


r

wa

w

rarw 11 1),(

1.0

1.1

1.2

1.3

1.4

1.5

0.0 0.5 1.0 1.5 2.0aspect ratio w eq/r (adim)

[

ad

im]

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current grading


We now can write the short sample gradient for a sector coil as a function of

Material parameters s, B*c2

(linear case as Nb-Ti)

Cable parameters

Aperture r and coil width w

Relevant feature: for very large coil widths w the short sample gradient tends to zero !


*

21

c

c

c

ss Bsr

sG

r

wa

w

rarw 11 1~),(

r

wrw cc 1ln),( 0

*

2

011

0

*

2

1ln11

1ln

1c

c

c

c

c

c

ss B

sr

wr

r

wa

w

ra

sr

w

Bsr

sG


Evaluation of short sample gradient in several sector lay-outs for a given aperture (r=30 mm)

No point in making coils larger than 30 mm !

Max gradient is 300 T/m and not 13/0.03=433 T/m !! We lose 30% !!


0

100

200

300

400

0 10 20 30 40 50 60 70 80 90 sector width w (mm)

Gss (

T/m

)

[0,30]

[0-24,30-36]

[0-18,22-32]

G*=B

*c2 /r

-30%


Gss as a function of aperture (maximum achievable)

Gain of Nb3Sn is ~50% in gradient for the same aperture (at 35 mm)

Gain of Nb3Sn is ~70% in aperture for the same gradient (at 200 T/m)


0

100

200

300

400

500

600

0 10 20 30 40 50 60 70 80 90 100

Gra

die

nt

[T /

m]

Aperture radius r [mm]

Nb3Sn at 1.9 K

Nb-Ti at 1.9 K

=0.35


Gss as a function of aperture (maximum achievable and with different coil thicknesses)


0

50

100

150

200

250

300

50 75 100 125 150 175 200

Gra

die

nt

[T /

m]

Aperture [mm]

Nb3Sn at 1.9 K

w/r=0.25

w/r=0.5

w/r=1

w/r=2

Operational margin

Magnets have to work at a given distance from the critical surface, i.e. they are never operated at short sample conditions

At short sample, any small perturbation quenches the magnet

One usually operates at a fraction of the loadline which ranges from 60% to 90%

This fraction translates into a temperature margin


Loadline with 20% operational margin Operational margin and temperature margin

Operational margin


0

100

200

300

400

500

600

0 10 20 30 40 50 60 70 80 90 100

Gra

die

nt

[T /

m]

Aperture radius r [mm]

Nb3Sn at 1.9 K

Nb-Ti at 1.9 K

=0.35

0

100

200

300

400

500

600

0 20 40 60 80 100 120 140 160 180 200

Oper

atio

nal

gra

die

nt

[T /

m]

Aperture [mm]

80% of Nb3Sn at 1.9 K

80% of Nb-Ti at 1.9 K

Iron yoke

An iron yoke usually surrounds the collared coil – it has several functions

Keep the return magnetic flux close to the coils, thus avoiding fringe fields

In some cases the iron is partially or totally contributing to the mechanical structure

RHIC magnets: no collars, plastic spacers, iron holds the Lorentz forces

LHC dipole: very thick collars, iron give little contribution

Considerably enhance the field for a given current density The increase is relevant (10-30%), getting higher for thin coils

This allows using lower currents, easing the protection

Increase the short sample field The increase is small (a few percent) for “large” coils, but can be considerable for small widths

This action is effective when we are far from reaching the asymptotic limit of B*

c2


Iron yoke

A rough estimate of the iron thickness necessary to avoid fields outside the magnet

The iron cannot withstand more than 2 T

Shielding condition for dipoles:

i.e., the iron thickness times 2 T is equal to the central field times the magnet aperture – One assumes that all the field lines in the aperture go through the iron (and not for instance through the collars)

Example: in the LHC main dipole the iron thickness is 150 mm

Shielding condition for quadrupoles:


satironBtrB ~

satironBtGr

~2

2

mm 100~2

3.8*28~

sat

ironB

rBt

Iron yoke

The iron yoke contribution can be estimated analytically for simple geometries

Circular, non-saturated iron: image currents method

Iron effect is equivalent to add to each current line a second one

at a distance

with current

Limit of the approximation: iron

is not saturated (less than 2 T)


rr

2

' IR

II1

1'

r

r 'R I

Iron yoke

Impact of the iron yoke on short sample field Large effect (25%) on RHIC dipoles (thin coil and collars)

Between 4% and 10% for most of the others

(both Nb-Ti and Nb3Sn)


0

20

40

60

80

0 20 40 60 80x (mm)

y (

mm

)

RHIC main dipole and yoke

0

20

40

60

80

100

0 20 40 60 80 100x (mm)

y (

mm

)

D20 and yoke

0

5

10

15

20

25

30

0 10 20 30 40 50 60equivalent width w (mm)

Gai

n i

n B

ss [

%]

TEV MB HERA MB

SSC MB RHIC MB

LHC MB Fresca

MSUT D20

HFDA NED

Iron yoke

Similar approach can be used in quadrupoles Large effect on RHIC quadrupoles (thin coil and collars)

Between 2% and 5% for most of the others

The effect is smaller than in dipoles since the

contribution to B2 is smaller than to B1


0

20

40

0 20 40 60 80 100x (mm)

y (

mm

)

0

20

40

0 20 40 60 80x (mm)

y (

mm

)

0

5

10

15

20

25

0.0 0.2 0.4 0.6 0.8 1.0 1.2weq/r (adim)

Gss

in

crea

se d

ue

to i

ron

[%

] ISR MQ TEV MQ

HERA MQ SSC MQ

RHIC MQ RHIC MQY

LHC MQY LHC MQ

LHC MQM LHC MQXA

LHC MQXB

Conclusions


Outline





Appendix I: A review of dipole lay-outs



RHIC MB Main dipole of the RHIC

296 magnets built in 04/94 – 01/96

Nb-Ti, 4.2 K

weq~9 mm ~0.23

1 layer, 4 blocks

no grading

0

20

40

60

80

100

0 20 40 60 80 100x (mm)

y (

mm

)

0

2

4

6

8

10

12

0 20 40 60 80w (mm)

B (

T)

sector [0-48,60-72]

No iron

With iron

B*

c2

2. Magnetic design Design and modeling of LTS superconducting magnets, October 7, 2012 58

Unit 11: Electromagnetic design episode III – 11.59


Tevatron MB Main dipole of the Tevatron

774 magnets built in 1980

0

2

4

6

8

10

12

0 20 40 60 80w (mm)

B (

T)

sector [0-48,60-72]

No iron

With iron

B*

c2

0

20

40

60

80

100

0 20 40 60 80 100x (mm)

y (

mm

)

Nb-Ti, 4.2 K

weq~14 mm ~0.23

2 layer, 2 blocks

no grading

2. Magnetic design Design and modeling of LTS superconducting magnets, October 7, 2012


HERA MB Main dipole of the HERA

416 magnets built in 1985/87

Nb-Ti, 4.2 K

weq~19 mm ~0.26

2 layer, 4 blocks

no grading

0

2

4

6

8

10

12

0 20 40 60 80w (mm)

B (

T)

sector [0-48,60-72]

No iron

With iron

B*

c2

0

20

40

60

80

100

0 20 40 60 80 100x (mm)

y (

mm

)



SSC MB Main dipole of the ill-fated SSC

18 prototypes built in 1990-5

Nb-Ti, 4.2 K

weq~22 mm ~0.30

4 layer, 6 blocks

30% grading

0

2

4

6

8

10

12

0 20 40 60 80w (mm)

B (

T)

sector [0-48,60-72]

No iron

With iron

B*

c2

0

20

40

60

80

100

0 20 40 60 80 100x (mm)

y (

mm

)



HFDA dipole Nb3Sn model built at FNAL

6 models built in 2000-2005

Nb3Sn, 4.2 K

jc~2000 to 2500 A/mm2 at 12 T, 4.2 K (different strands)

weq~23 mm ~0.29

2 layers, 6 blocks

no grading

0

20

40

60

80

100

0 20 40 60 80 100x (mm)

y (

mm

)

0

5

10

15

20

25

0 20 40 60 80w (mm)

B (

T)

sector [0-48,60-72]No ironWith iron

B*

c2



LHC MB Main dipole of the LHC

1276 magnets built in 2001-06

Nb-Ti, 1.9 K

weq~27 mm ~0.29

2 layers, 6 blocks

23% grading

0

20

40

60

80

100

0 20 40 60 80 100x (mm)

y (

mm

)

0

2

4

6

8

10

12

14

0 20 40 60 80w (mm)

B (

T)

sector [0-48,60-72]

No iron

With iron

B*

c2



FRESCA Dipole for cable test station at CERN

1 magnet built in 2001

Nb-Ti, 1.9 K

weq~30 mm ~0.29

2 layers, 7 blocks

24% grading

0

2

4

6

8

10

12

14

0 20 40 60 80w (mm)

B (

T)

sector [0-48,60-72]

No iron

With iron

B*

c2

0

20

40

60

80

100

0 20 40 60 80 100x (mm)

y (

mm

)


6. A REVIEW OF DIPOLE LAY-OUTS

MSUT dipole Nb3Sn model built at Twente U.

1 model built in 1995

Nb3Sn, 4.2 K

jc~1100 A/mm2 at 12 T, 4.2 K

weq~35 mm ~0.33

2 layers, 5 blocks

65% grading

0

20

40

60

80

100

0 20 40 60 80 100x (mm)

y (

mm

)

0

5

10

15

20

25

0 20 40 60 80w (mm)

B (

T)


B*

c2



D20 dipole Nb3Sn model built at LBNL (USA)

1 model built in ???

Nb3Sn, 4.2 K

jc~1100 A/mm2 at 12 T, 4.2 K

weq~45 mm ~0.48

4 layers, 13 blocks

65% grading

0

20

40

60

80

100

0 20 40 60 80 100x (mm)

y (

mm

)

0

5

10

15

20

25

0 20 40 60 80w (mm)

B (

T)


B*

c2



HD2 Nb3Sn model being built in LBNL

1 model to be built in 2008 Nb3Sn, 4.2 K

jc~2500 A/mm2 at 12 T, 4.2 K

weq~46 mm ~0.35

2 layers, racetrack, no grading

0

20

40

60

80

100

0 20 40 60 80 100x (mm)

y (

mm

)

0

5

10

15

20

25

0 20 40 60 80w (mm)

B (

T)


B*

c2


Appendix II: A review of quadrupole lay-outs



RHIC MQX Quadrupole in the IR regions of the RHIC

79 magnets built in July 1993/ December 1997

Nb-Ti, 4.2 K

w/r~0.18 ~0.27

1 layer, 3 blocks, no grading

0

20

40

60

0 20 40 60 80 100 120x (mm)

y (

mm

)

0

50

100

150

200

0.0 0.5 1.0 1.5 2.0w eq /r (adim)

Gss

(T

/m)

sector [0-24,30-36]

No iron

With iron

B*

c2 /r



RHIC MQ Main quadrupole of the RHIC

380 magnets built in June 1994 – October 1995

Nb-Ti, 4.2 K

w/r~0.25 ~0.23

1 layer, 2 blocks, no grading

0

20

40

60

0 20 40 60 80 100 120x (mm)

y (

mm

)

0

50

100

150

200

250

300

0.0 0.5 1.0 1.5 2.0w eq /r (adim)

Gss

(T

/m)

sector [0-24,30-36]

No iron

With iron

B*

c2 /r



LEP II MQC Interaction region quadrupole of the LEP II


Nb-Ti, 4.2 K, no iron

w/r~0.27 ~0.31

1 layers, 2 blocks, no grading

0

20

40

60

0 20 40 60 80 100 120x (mm)

y (

mm

)

0

20

40

60

80

100

120

140

0.0 0.5 1.0 1.5 2.0w eq /r (adim)

Gss

(T

/m)

sector [0-24,30-36]

No iron

B*

c2 /r



LEP I MQC Interaction region quadrupole of the LEP I

8 magnets built in ~1987-89

Nb-Ti, 4.2 K, no iron

w/r~0.29 ~0.33


0

20

40

60

0 20 40 60 80 100 120x (mm)

y (

mm

)

0

20

40

60

80

100

0.0 0.5 1.0 1.5 2.0w eq /r (adim)

Gss

(T

/m)

sector [0-24,30-36]

No iron

B*

c2 /r



Tevatron MQ Main quadrupole of the Tevatron

216 magnets built in ~1980

Nb-Ti, 4.2 K

w/r~0.35 ~0.250


0

20

40

60

0 20 40 60 80 100 120x (mm)

y (

mm

)

0

50

100

150

200

250

0.0 0.5 1.0 1.5 2.0w eq /r (adim)

Gss

(T

/m)

sector [0-24,30-36]

No iron

With iron

B*

c2 /r



HERA MQ Main quadrupole of the HERA

Nb-Ti, 1.9 K

w/r~0.52 ~0.27

2 layers, 3 blocks, grading 10%

0

100

200

300

0.0 0.5 1.0 1.5 2.0w eq /r (adim)

Gss

(T

/m)

sector [0-24,30-36]

No iron

With iron

B*

c2 /r

0

20

40

60

0 20 40 60 80 100 120x (mm)

y (

mm

)



LHC MQM Low- gradient quadrupole in the IR regions of the LHC


Nb-Ti, 1.9 K (and 4.2 K)

w/r~0.61 ~0.26


0

20

40

60

0 20 40 60 80 100 120x (mm)

y (

mm

)

0

100

200

300

400

500

0.0 0.5 1.0 1.5 2.0w eq /r (adim)

Gss

(T

/m)

sector [0-24,30-36]

No iron

With iron

B*

c2 /r



LHC MQY Large aperture quadrupole in the IR regions of the LHC


Nb-Ti, 4.2 K

w/r~0.79 ~0.34

4 layers, 5 blocks, special grading 43%

0

20

40

60

0 20 40 60 80 100 120x (mm)

y (

mm

)

0

100

200

300

0.0 0.5 1.0 1.5 2.0w eq /r (adim)

Gss

(T

/m)

sector [0-24,30-36]

No iron

With iron

B*

c2 /r



LHC MQXB Large aperture quadrupole in the LHC IR


Nb-Ti, 1.9 K

w/r~0.89 ~0.33

2 layers, 4 blocks, grading 24%

0

20

40

60

0 20 40 60 80 100 120x (mm)

y (

mm

)

0

100

200

300

400

0.0 0.5 1.0 1.5 2.0w eq /r (adim)

Gss

(T

/m)

sector [0-24,30-36]

No iron

With iron

B*

c2 /r



SSC MQ Main quadrupole of the ill-fated SSC

Nb-Ti, 1.9 K

w/r~0.92 ~0.27


0

100

200

300

400

500

0.0 0.5 1.0 1.5 2.0w eq /r (adim)

Gss

(T

/m)

sector [0-24,30-36]

No iron

With iron

B*

c2 /r

0

20

40

60

0 20 40 60 80 100 120x (mm)

y (

mm

)



LHC MQ Main quadrupole of the LHC


Nb-Ti, 1.9 K

w/r~1.0 ~0.250


0

20

40

60

0 20 40 60 80 100 120x (mm)

y (

mm

)

0

100

200

300

400

500

0.0 0.5 1.0 1.5 2.0w eq /r (adim)

Gss

(T

/m)

sector [0-24,30-36]

No iron

With iron

B*

c2 /r



LHC MQXA Large aperture quadrupole in the LHC IR


Nb-Ti, 1.9 K

w/r~1.08 ~0.34

4 layers, 6 blocks, special grading 10%

0

20

40

60

0 20 40 60 80 100 120x (mm)

y (

mm

)

0

100

200

300

400

0.0 0.5 1.0 1.5 2.0w eq /r (adim)

Gss

(T

/m)

sector [0-24,30-36]

No iron

With iron

B*

c2 /r



LHC MQXC Nb-Ti option for the LHC upgrade

LHC dipole cable, graded coil

1-m-long model built in 2011-2 to be

tested in 2012

w/r~0.5 ~0.33 2 layers, 4 blocks

0

20

40

60

0 20 40 60 80 100 120

y (

mm

)

x (mm)



LARP HQ 120 mm aperture Nb3Sn option for the

LHC upgrade (IR triplet)

1-m-long model tested in 2011, more

to come plus a 3.4-m-long

w/r~0.5 ~0.33 2 layers, 4 blocks

0

20

40

60

0 20 40 60 80 100 120

y (

mm

)

x (mm)


Appendix III: Numerical tools



Main critical aspects for computational tools Electromagnetic:

2D, 3D

Coil ends

Persistent currents

Iron saturation

Mechanical: 2D, 3D

Coupling with magnetic model to estimate impact of Loretz forces

Interfaces between different components

Quench analysis, stability and protection: Thermal properties of materials

Hypothesis on cooling

Multiphysics codes: Can couple magnetic-mechanic-thermal



Computational tools are needed for the design: Electromagnetic: Roxie, Poisson, Opera, (Ansys), Cast3m …

Mechanical: Ansys, Cast3m, Abaqus, …

Quench analysis, stability and protection: Roxie, …

Multiphysics codes: Comsol, …

Stress due to electromagnetic forces

in Tevatron dipole, through ANSYS



The ROXIE code Project led by S. Russenschuck, developed at CERN, specific on superconducting accelerator magnet design

Easy input file with coil geometry, iron geometry, and coil ends

Several routines for optimization

Evaluation of field quality, including iron saturation (BEM-FEM methods) and persistent currents

Evaluation of quench



The ROXIE code

Quench propagation in an LHC main dipole

Splices in the LHC main dipole Interconnection between LHC main dipoles


Documents

Design and Modelling of LTS Superconducting Magnets€¦ · Design and Modelling of LTS Superconducting Magnets Magnetic design Paolo Ferracin ... L. Rossi, E. Todesco, “Electromagnetic