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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 3
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 4
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).
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 5
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 6
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 7
z
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 8
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 9
-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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 10
)(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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 11
-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!!!
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 12
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 13
As a result, the field is not perfect anymore
How can I express in improve the “imperfect” field inside the aperture?
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 14
Maxwell equations for magnetic field
In absence of charge and magnetized material
If (constant longitudinal field), then
Field representation Maxwell equations
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 15
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 16
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 17
1
1
1
1
),(),(
n
n
nn
n
n
nxy iyxiABiyxCyxiByxB
11 iABiBB xy
yAxiAyiBxBiyxiABiBB xy 222222
[K.-H. Mess, et al, [4] pg. 50]
1B
2B
1A
2A
Field representation Harmonics
So, each coefficient corresponds to a “pure” multipolar field
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 18
1
1
1
1
),(),(
n
n
nn
n
n
nxy iyxiABiyxCyxiByxB
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]
Field representation Harmonics
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 19
-40
0
40
-40 0 40
+
+
+
+
--
- -
r
R ref
x
y
1
1
1
4 )(10
n
n ref
nnxyR
iyxiabBiBB
Field representation Harmonics
Important property: starting by the multipolar expansion of a current line (Biot-Savart law)
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 20
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
Field representation Harmonics
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 21
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 22
Now, I can use the multipolar expansion to optimize my “practical” cross-section
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 23
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 !
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 24
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
A “good” field quality dipole Sector dipole
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 !
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 25
wrr
jRB
ref 11
3
)3sin(2
0
3
a
33
4
0
5
11
5
)5sin(
wrr
jRB
ref a
+
+
-
-
a
w
r
A “good” field quality dipole Sector dipole
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 26
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
A “good” field quality dipole Sector dipole
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°]
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 27
0)3sin()3sin()3sin()3sin()3sin( 12345 aaaaa
0)5sin()5sin()5sin()5sin()5sin( 12345 aaaaa
0)7sin()7sin()7sin()7sin()7sin( 12345 aaaaa
0)9sin()9sin()9sin()9sin()9sin( 12345 aaaaa
0)11sin()11sin()11sin()11sin()11sin( 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]
A “good” field quality dipole Sector dipole
Let us see two coil lay-outs of real magnets The RHIC dipole has four blocks
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 28
0
20
40
60
0 20 40 60x (mm)
y (
mm
)
A “good” field quality dipole Sector dipole
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 …)
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 29
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 …
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 30
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
A “good” field quality dipole Sector quadrupole
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 31
0
20
40
0 20 40 60 80x (mm)
y (
mm
)
A “good” field quality dipole Sector quadrupole
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 32
0
20
40
0 20 40 60 80x (mm)
y (
mm
)
0
20
40
0 20 40 60 80x (mm)
y (
mm
)
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 33
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!
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 34
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
Maximum field and coil thickness Dipoles
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)
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 35
)()( , BjBj cscc
)()( *
2 BBsBj cc
Maximum field and coil thickness Dipoles
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 36
jB c jBB cp
wj
B2
01
a
sin
2 01 w
jB
Maximum field and coil thickness Dipoles
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 37
)( ,
*
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
Maximum field and coil thickness Dipoles
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 38
*
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
)
Maximum field and coil thickness Dipoles
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 39
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~),(
Maximum field and coil thickness Dipoles
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 40
*
21
c
c
c
ss Bs
sB
w
arrw 1~),(
wcc 0~
*
2
0
0
11
~ c
c
c
ss B
sww
ar
swB
Maximum field and coil thickness Dipoles
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 41
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 !
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 42
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
Maximum gradient and coil thickness Quadrupoles
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 43
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]
Maximum gradient and coil thickness Quadrupoles
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 44
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)
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ad
im]
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current grading
Maximum gradient and coil thickness Quadrupoles
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 !
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 45
*
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
Maximum gradient and coil thickness Quadrupoles
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% !!
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 46
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%
Maximum gradient and coil thickness Quadrupoles
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)
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 47
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
Maximum gradient and coil thickness Quadrupoles
Gss as a function of aperture (maximum achievable and with different coil thicknesses)
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 48
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 49
Loadline with 20% operational margin Operational margin and temperature margin
Operational margin
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 50
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 51
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:
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 52
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)
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 53
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)
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 54
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 55
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
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 56
Appendix I: A review of dipole lay-outs
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 57
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
Appendix I: A review of dipole lay-outs
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
Appendix I: A review of dipole lay-outs
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
)
2. Magnetic design Design and modeling of LTS superconducting magnets, October 7, 2012 60
Appendix I: A review of dipole lay-outs
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
)
2. Magnetic design Design and modeling of LTS superconducting magnets, October 7, 2012 61
Appendix I: A review of dipole lay-outs
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
2. Magnetic design Design and modeling of LTS superconducting magnets, October 7, 2012 62
Appendix I: A review of dipole lay-outs
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
2. Magnetic design Design and modeling of LTS superconducting magnets, October 7, 2012 63
Appendix I: A review of dipole lay-outs
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
)
2. Magnetic design Design and modeling of LTS superconducting magnets, October 7, 2012 64
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)
sector [0-48,60-72]No ironWith iron
B*
c2
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 65
6. A REVIEW OF DIPOLE LAY-OUTS
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)
sector [0-48,60-72]No ironWith iron
B*
c2
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 66
6. A REVIEW OF DIPOLE LAY-OUTS
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)
sector [0-48,60-72]No ironWith iron
B*
c2
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 67
Appendix II: A review of quadrupole lay-outs
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 68
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 69
Appendix II: A review of quadrupole lay-outs
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 70
Appendix II: A review of quadrupole lay-outs
LEP II MQC Interaction region quadrupole of the LEP II
8 magnets built in 1991-3
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 71
Appendix II: A review of quadrupole lay-outs
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
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
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 72
Appendix II: A review of quadrupole lay-outs
Tevatron MQ Main quadrupole of the Tevatron
216 magnets built in ~1980
Nb-Ti, 4.2 K
w/r~0.35 ~0.250
2 layers, 3 blocks, no grading
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 73
Appendix II: A review of quadrupole lay-outs
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
)
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 74
Appendix II: A review of quadrupole lay-outs
LHC MQM Low- gradient quadrupole in the IR regions of the LHC
98 magnets built in 2001-2006
Nb-Ti, 1.9 K (and 4.2 K)
w/r~0.61 ~0.26
2 layers, 4 blocks, no grading
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 75
Appendix II: A review of quadrupole lay-outs
LHC MQY Large aperture quadrupole in the IR regions of the LHC
30 magnets built in 2001-2006
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 76
Appendix II: A review of quadrupole lay-outs
LHC MQXB Large aperture quadrupole in the LHC IR
8 magnets built in 2001-2006
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 77
Appendix II: A review of quadrupole lay-outs
SSC MQ Main quadrupole of the ill-fated SSC
Nb-Ti, 1.9 K
w/r~0.92 ~0.27
2 layers, 4 blocks, no grading
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
)
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 78
Appendix II: A review of quadrupole lay-outs
LHC MQ Main quadrupole of the LHC
400 magnets built in 2001-2006
Nb-Ti, 1.9 K
w/r~1.0 ~0.250
2 layers, 4 blocks, no grading
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 79
Appendix II: A review of quadrupole lay-outs
LHC MQXA Large aperture quadrupole in the LHC IR
18 magnets built in 2001-2006
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 80
Appendix II: A review of quadrupole lay-outs
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)
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 81
Appendix II: A review of quadrupole lay-outs
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)
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 82
Appendix III: Numerical tools
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 83
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 84
Appendix III: Numerical tools
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 85
Appendix III: Numerical tools
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
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 86
Appendix III: Numerical tools
The ROXIE code
Quench propagation in an LHC main dipole
Splices in the LHC main dipole Interconnection between LHC main dipoles
Design and modeling of LTS superconducting magnets, October 7, 2012 2. Magnetic design 87