MAGNETIC DESIGN
Ezio Todesco
European Organization for Nuclear Research (CERN)
Thanks to P. Ferracin and L. Rossi
CERN Accelerator School, Erice 2013
IRON and coil magnets
Iron dominated magnets
Shape of the field given by the iron
Winding give the flux
Limited to 1.8 T by iron saturation
Winding can be resistive /superconductive
Low-loss injector magnet,
F. Borgnolutti, et al, MT22 (2012)
The supercoductive option os also called superferric – warm or cold yoke
Coil dominated magnets
Shape of the field given by the conductor position
Limited by field tolerated by conductor
Iron gives second order effect (acts as a
virtual coil, field enhancement)
Fe
0
Nb-Ti
2
4
6
Nb3Sn
8
10 12 14
Operational field (T)
CERN Accelerator School, Erice 2013
HTS
16
18
20
Superferric corrector, F. Toral, et al, MT22 (2012)
Magnetic design - 2
CONTENTS
Coil lay out and field quality constraints
Field versus coil width, superconductor and filling ratio
Dipoles
Quadrupoles
Block design
Iron and persistent currents
CERN Accelerator School, Erice 2013
Magnetic design - 3
1. FIELD QUALITY CONSTRAINTS
Field given by a current line (Biot-Savart law)
y
B( z )  B y ( z )  iB x ( z )
I 0
I
1
B( z ) 
 0
2 ( z  z0 )
2z0 1  z
z0
using
40
z 0 =x 0 +iy 0
B=B y +iB x
0
-4 0
x
0
40
z=x+iy
-4 0

1
2
3
 1  t  t  t  ...   t n1
1 t
n 1
t  1 !!!
Félix Savart,
French
(June 30, 1791-March 16, 1841)
we get
I 0
B( z )  
2z0
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 z
 

n 1  z 0 

n 1
I 0

2z0
 Rref


n 1  z 0




n 1
 x  iy 


 R 
 ref 
n 1
Jean-Baptiste Biot,
French
(April 21, 1774 – February 3, 1862)
Magnetic design - 4
1. FIELD QUALITY CONSTRAINTS
Now we can compute the multipoles of a current line at z0
n 1
 z
 Rref
I 0
   



2z0 n1  z0
n 1  z 0 
Definition of multipolar expansion
I
B( z )   0
2z0

 x  iy 

B y  iB x  10 B1  (bn  ia n )


n 1
 Rref 
4


n 1



n 1
 x  iy 


 R 
 ref 
x  iy  z0
I0  1 
B1  
Re 
2
 z0 
I 010  Rref

bn  ia n  
2z0 B1  z0
4
n 1



n 1
A perfect dipole has b1=10000, and all others bn an = 0
In log scale, the slope of the multipole decay is the logarithm of (Rref/|z0|)
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Magnetic design - 5
1. FIELD QUALITY CONSTRAINTS
Perfect dipoles
Cos: a current density proportional to cos in an annulus – One
can prove it provides pure field Cable block
60
-

wedge
j = j0 cos 
+
0
-4 0
0
-
40
+
-6 0
An ideal cos
A practical winding with
one layer and wedges
[from M. N. Wilson, pg. 33]
A practical winding with
three layers and no wedges
[from M. N. Wilson, pg. 33]
Artist view of a cos magnet
[from Schmuser]
+ self supporting structure (roman arch)
+ the aperture is circular, the coil is compact
+ easy winding, lot of experience
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Magintc design – 6
1. FIELD QUALITY CONSTRAINTS
We compute the central field given by a sector dipole with
uniform current density j
I0  1 
I cos 
B1  
Re    0
2
2 z0
 z0 
I  jdd
w
-
Taking into account of current signs
j0
B1  2
2
a r w
a 

r
cos 

dd  
a
+
r
2 j0

-
+
w sin a
This simple computation is full of consequences
B1  current density (obvious)
B1  coil width w (less obvious)
B1 is independent of the aperture r (much less obvious)
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Magnetic design - 7
1. FIELD QUALITY CONSTRAINTS
A dipolar symmetry is characterized by
Up-down symmetry (with same current sign)
Left-right symmetry (with opposite sign)
w
-
r
-
+
+
Why this configuration?
Opposite sign in left-right is necessary to avoid that the field created by the
left part is canceled by the right one
In this way all multipoles except B2n+1 are canceled
 z
B ( z )  B1  B3 
R
 ref
2


  B5  z

R

 ref
4

  ...





z2
z4
4 
B( z )  B1 1  10 b3 2  b5 4  ...
 R

Rref

ref


these multipoles are called “allowed multipoles”
Remember the power law decay of multipoles with order
And that field quality specifications concern only first 10-15 multipoles
The field quality optimization of a coil lay-out concerns only a few quantities !
Usually b3 , b5 , b7 , and possibly b9 , b11
CERN Accelerator School, Erice 2013
Magnetic design - 8
1. FIELD QUALITY CONSTRAINTS
Multipoles of a sector coil
C n  2
n 1
j 0 Rref
2
a rw
a 

r
exp( in )
n
dd  
j 0 R ref
n 1
j 0 Rref

a
rw
a
r
 exp( in )d
 w
sin( 2a ) log 1  
r

for n=2 one has
B2  
and for n>2
n 1
j 0 Rref
2 sin( an) (r  w) 2n  r 2 n
Bn  

n
2n

1 n

 d
Main features of these equations
Multipoles n are proportional to sin ( n angle of the sector)
They can be made equal to zero !
Proportional to the inverse of sector distance to power n
High order multipoles are not affected by coil parts far from the centre
CERN Accelerator School, Erice 2013
Magnetic design - 9
1. FIELD QUALITY CONSTRAINTS
First allowed multipole B3 (sextupole)
2
 0 jRref
sin( 3a )  1
1 
B3 
 


3  r r  w
for a=/3 (i.e. a 60° sector coil) one has B3=0
w
-
a
+
r
-
+
Second allowed multipole B5 (decapole)
4
 0 jRref

sin( 5a )  1
1


B5 

3
3 


5 r
r  w 
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 !
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Magnetic design - 10
1. FIELD QUALITY CONSTRAINTS
Coil with two sectors
50.0
45.0
B3 
B5 
 0 jR
2
ref

 0 jR
4
ref

sin 3a 3  sin 3a 2  sin 3a 1
3
sin 5a 3  sin 5a 2  sin 5a 1
5
1 
1



r
r

w


 1
1
 3 
(r  w) 3
r
40.0
35.0
30.0
25.0



a3 a
2
20.0
a1
15.0
10.0
5.0
0.0
Note: we have to work with non-normalized multipoles, which can
be added together
Equations to set to zero B3 and B5
sin( 3a 3 )  sin( 3a 2 )  sin( 3a1 )  0

sin( 5a 3 )  sin( 5a 2 )  sin( 5a1 )  0
There is a one-parameter family of solutions, for instance
(48°,60°,72°) or (36°,44°,64°) are solutions
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Magnetic design - 11
1. FIELD QUALITY CONSTRAINTS
With one wedge one can set to zero three
multipoles (B3, B5 and B7)
What about two wedges ?
sin( 3a 5 )  sin( 3a 4 )  sin( 3a 3 )  sin( 3a 2 )  sin( 3a 1 )  0
sin( 5a 5 )  sin( 5a 4 )  sin( 5a 3 )  sin( 5a 2 )  sin( 5a 1 )  0
One wedge, b3=b5=b7=0
[0-43.2,52.2-67.3]
sin( 7a 5 )  sin( 7a 4 )  sin( 7a 3 )  sin( 7a 2 )  sin( 7a 1 )  0
sin( 9a 5 )  sin( 9a 4 )  sin( 9a 3 )  sin( 9a 2 )  sin( 9a 1 )  0
sin( 11a 5 )  sin( 11a 4 )  sin( 11a 3 )  sin( 11a 2 )  sin( 11a 1 )  0
One can set to zero five multipoles (B3, B5, B7 , B9 and B11)
~[0°-33.3°, 37.1°- 53.1°, 63.4°- 71.8°]
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Two wedges, b3=b5=b7=b9=b11=0
[0-33.3,37.1-53.1,63.4- 71.8]
Magnetic design - 12
1. FIELD QUALITY CONSTRAINTS
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 …)
Codes can estimate and optimize (e.g. ROXIE) – but never lose the
feeling of what you are doing ! (more info USPAS Unit 8)
60
y (mm)
40
20
0
0
Our case with two wedges
20
x (mm)
40
RHIC main dipole
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60
Magnetic design - 13
CONTENTS
Coil lay out and field quality constraints
Field versus coil width, superconductor and filling ratio
Dipoles
Quadrupoles
Block design
Iron and persistent currents
CERN Accelerator School, Erice 2013
Magnetic design - 14
2. DIPOLES: FIELD VERSUS MATERIAL
AND COIL THICKNESS
The coil width is the main parameter of magnet design
First decision of the magnet designer: how much superconductor ?
2500
Eng. curent density (A/mm2)
Eng. curent density (A/mm2)
B1  jw
Nb-Ti 1.9 K
2000
1500
1000
500
2500
Nb-Ti 1.9 K
2000
1500
1000
500
0
0
0
5
Field (T)
10
 High field
 Large coil  $$
 Lower current density
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15
0
5
Field (T)
10
15
 Low field
 Smaller coil
 Larger current density
Magnetic design - 15
2. DIPOLES: FIELD VERSUS MATERIAL
AND COIL THICKNESS
Aim: approximate analytical equations for magnetic design
We recall the equations for the critical surface
Nb-Ti: linear approximation is good
with s~6.0108 [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!
jsc,c ( B )  s (b  B ),
8000
Nb-Ti at 1.9 K
Nb-Ti at 4.2 K
2
jsc(A/mm )
6000
4000
2000
0
0
5
10
15
B (T)
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Magnetic design - 16
2. DIPOLES: FIELD VERSUS MATERIAL
AND COIL THICKNESS
The current density in the coil is lower
Strand made of superconductor and normal conducting (copper)
Cu/noCu is the ratio between the copper and the superconductor, usually
ranging from 1 to 2 in most cases
If the strands are assembled in rectangular cables, there are voids:
w-c is the fraction of cable occupied by strands (usually ~85%)
The cables are insulated:
c-i is the fraction of insulated cable occupied by the bare cable (~85%)
The current density flowing in the insulated cable is
reduced by a factor  (filling ratio)
   w c  c i
1
1   Cu / noCu
The filling ratio ranges from ¼ to 1/3
The critical surface for j (engineering current density) is
j c ( B)  j sc,c ( B)
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jc ( B)  s(b  B)
Magnetic design - 17
2. DIPOLES: FIELD VERSUS MATERIAL
AND COIL THICKNESS
We characterize the coil by two parameters
B  c j
B p  B   c j
c: how much field in the centre is given per unit of current density
: ratio between peak field and central field
jc ( B)  s(b  B)
B p , ss
 cs

b
1   cs
 cs
Bss 
b
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1   cs
Current density j (A/mm2)
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
Margin: you must stay at a certain distance from the critical surface
2500
(typically 80% of jss, Bss)
2000
* -B)
j=s(B
j=
s(b-B)
c2
1500
1000
Bp=cj
500
[Bp,ss,jss]
0
0
5
10
Magnetic field B (T)
15
Magnetic design - 18
2. DIPOLES: FIELD VERSUS MATERIAL
AND COIL THICKNESS
Hypothesis of 60sector coil:
B  c j
B1  
2 j 0

w sin

3
 c   c0 w
This is the easy part – with two sectors a bit more realistic
ar
 ( w, r ) ~ 1 
w
a~0.045
 [adim]
Ratio peak field/central field: empirical fit (one can make
1.3
better
TEV MB
HERA MB
SSC MB
LHC MB
MSUT
HFDA
1.2
RHIC MB
Fresca
D20
NED
1.1
1.0
0.0
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0.5
1.0
equivalent width w/r
1.5
2.0
Magnetic design - 19
2. DIPOLES: FIELD VERSUS MATERIAL
AND COIL THICKNESS
We now can write the short sample field for a sector coil as
a function of
 s
B*c2
Bss 
c
Material parameters c,
1   cs
Cable parameters 
Aperture r and coil width w
Best values: a=0.045 c0=6.6310-7 [Tm/A]
b
 ( w, r ) ~ 1 
ar
w
 c ~  c0 w
for Nb-Ti s~6.0108 [A/(T m2)] and b~10 T at 4.2 K or 13 T at 1.9 K
(see also Excel file available in material)
Bss ~
 c 0 ws
 ar 
1  1   c 0 ws
w

b
Please note: this is a handy estimate, neglecting iron, to have an idea of the trends
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Magnetic design - 20
2. DIPOLES: FIELD VERSUS MATERIAL
AND COIL THICKNESS
Evaluation of short sample field in sector lay-outs for a
different apertures
Please note that the operational field is ~80% of this value
Tends asymptotically to b~ 13 T, as b w/(1+w), for w
Example: LHC coil ~30 mm width, short sample ~10 T, operational ~8 T
Nb-Ti 1.9 K
Central field (T)
10
r=28 mm
5
r = 50 mm
r = 75 mm
0
0
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10
20
30
Coil width (mm)
40
50
Magnetic design -.21
2. DIPOLES: FIELD VERSUS MATERIAL
AND COIL THICKNESS
Case of Nb3Sn – an explicit expression
An analytical expression can be found using a hyperbolic fit
8000
6000
Nb-Ti at 1.9 K
Nb-Ti at 4.2 K
Nb3Sn at 1.9 K
2
jsc(A/mm )
b

j c ( B )  s  1
B 
4000
Nb3Sn at 4.2 K
that agrees well between 11 and 17 T 2000
with s~3.9109 [A/(T m2)]
0
0
5
10
15
20
B (T)
and b~21 T at 4.2 K, b~22 T at 1.9 K
Using this fit one can find explicit expression for the short sample
field

s c 0 w 
4b
Bss 
 1  1

2  s c 0 w

25
and the constant c  are the same as before (they depend on the layout, not on the material)
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Magnetic design -. 22
2. DIPOLES: FIELD VERSUS MATERIAL
AND COIL THICKNESS
Evaluation of short sample field in sector lay-outs for a
different apertures
s c 
4b
Tends asymptotically to b~22 T but slowly
B ss 


 1  1

2  s c

20
Central field (T)
Nb3Sn 1.9 K
15
Nb-Ti 1.9 K
10
r=28 mm
5
r = 50 mm
r = 75 mm
0
0
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10
20
30
Coil width (mm)
40
50
Magnetic design -. 23
2. DIPOLES: FIELD VERSUS MATERIAL
AND COIL THICKNESS
Summary
Current density j (A/mm2)
2500
2000
j=s(B*c2-B)
Nb-Ti is limited at 10 T
[B ,j ]
B = j
Nb3Sn allows to go towards 15 T
Approaching the limits of each material implies
very large coil and lower current densities – not so effective
Operational current densities are typically ranging between 300 and
600 A/mm2
1500
1000
500
p
0
0
15
5
10
Magnetic field B (T)
15
600
11T
HFD
10
LHC
SSC
5
RHIC
MSUT D20
LD1
FRESCA2
FRESCA
HERA
Tevatron
10
20
RHIC
30
40
50
60
Equivalent coil width (mm)
70
Operational bore field versus coil width
(80% of short sample at 1.9 K taken for models)
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LHC
LD1
D20
FRESCA2
HERA
200
Nb-Ti
Nb3Sn
Nb3Sn (in construction)
100
80
HD2
MSUT
SSC
Tevatron
Fresca
300
Nb-Ti
Nb3Sn
Nb3Sn (in construction)
11 T
500
400
0
0
HFD
current density jo (A/mm2)
HD2
Bore field (T)
p,ss ss
c
0
0
10
20
30
40
50
60
Equivalent coil width (mm)
70
80
Operational overall current density versus coil width
(80% of short sample at 1.9 K taken for models)
Unit 9: Electromagnetic design episode II – 9.24
2. QUADRUPOLES: GRADIENT VERSUS MATERIAL
AND COIL THICKNESS
Nb-Ti case, =0.3
Gss 
See appendix


 c 0 ln 1 
w
s
r
r
w

 w
1   a1  1  a1 r c 0 ln 1  s
w
r
r


b
350
Nb-Ti 1.9 K
Gradient (T/m)
300
250
200
150
100
r=28 mm
r = 50 mm
r = 75 mm
50
0
0
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10
30
20
Coil width (mm)
40
50
Magnetic design -. 25
2. QUADRUPOLES: GRADIENT VERSUS MATERIAL
AND COIL THICKNESS
Nb3Sn case, k=0.33
About 50% larger gradient for the same aperture

4b
Gss 
 1  1

2  rs c

r=28 mm
500


 c   c 0 ln 1 
r = 50 mm
400
r = 75 mm
Central field (T)
s c 
Nb3Sn 1.9 K
w

r
300
 ( w, r ) ~ a 1
200
r
w
 1  a1
w
r
100
0
0
10
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20
30
Coil width (mm)
40
50
Magnetic design -. 26
CONTENTS
Coil lay out and field quality constraints
Field versus coil width, superconductor and filling ratio
Dipoles
Quadrupoles
Iron and persistent currents
Block design
CERN Accelerator School, Erice 2013
Magnetic design - 27
3. IRON YOKE – WHAT THICKNESS
Iron is mainly used to avoid leaks of flux outside the
magnet
A rough estimate of the iron thickness necessary
The iron cannot withstand more than 2 T
R1
R2
RI
Shielding condition for dipoles:
rB ~ t iron Bsat
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
tiron ~
rB 28 * 8.3

~ 100 mm
Bsat
2
Shielding condition for quadrupoles:
CERN Accelerator School, Erice 2013
r 2G
~ t iron B sat
2
Magnetic design -. 28
3. IRON YOKE – IMAGE METHOD
Positive side effect: increase the main field for a fixed current
Examples of several built dipoles
Smallest: LHC  16%
Largest: RHIC  55%
Lower impact on short sample (a few percent for LHC)
B1iron   1 (r  w)r

B1
  1 RI2
4
RI/r (adim)
+15%
+20%
3
+40%
2
+50%
Forbidden zone
+25%
TEV MB
SSC MB
LHC MB
MSUT
HFDA
+30%
HERA MB
RHIC MB
Fresca
D20
NED
1
0.0
0.5
1.0
equivalent width w/r
1.5
2.0
Iron saturation in RHIC magnet [R. Gupta]
For high field magnet iron gets saturated – mirror approximation not
valid, nonlinear effect –computed with FEM (Opera, Ansys, ROXIE)
CERN Accelerator School, Erice 2013
Magnetic design -. 29
3: PERSISTENT CURRENTS
The filaments get magnetized during a field change
Since they are superconductive, current flow forever  persistent
B
B
a
a
B
-
b
+ B
+
B
-
+
-
+
B
Magnetization for ramping field according to Bean model
Persistent current measured vs computed in Tevatron dipoles From P. Bauer et al, FNAL TD-02-040 (2004)
These currents have a large impact at injection on field quality
Effect proportional to filament size
One can decide to correct with wedges at injection and have residual
at high field or viceversa (depends on the magnet function)
CERN Accelerator School, Erice 2013
Magnetic design -. 30
CONTENTS
Coil lay out and field quality constraints
Field versus coil width, superconductor and filling ratio
Dipoles
Quadrupoles
Iron and persistent currents
Block design
CERN Accelerator School, Erice 2013
Magnetic design -. 31
4. OTHER DESIGNS: BLOCK
Block coil (HD2, HD3, Fresca2)
Cable is not keystoned, perpendicular to the midplane
Ends are wound in the easy side, but must be flared to make space
for aperture (bend in the hard direction)
Internal structure to support the coil needed
HD2 design: 3D sketch of the coil (left) and magnet cross section (right)
[from P. Ferracin et al, MT19, IEEE Trans. Appl. Supercond. 16 378 (2006)]
CERN Accelerator School, Erice 2013
Magnetic design -. 32
4. OTHER DESIGNS: BLOCK
Block coil – HD2 & HD3
100
y (mm)
80
60
40
20
0
0
20
40
60
x (mm)
80
100
B * c2
B (T)
Two layers, two blocks
Enough parameters to have a good field quality
Ratio peak field/central field not so bad:
1.05 instead of 1.02 as for a cos with the same
quantity of cable
Ratio central field/current density is 12%
25
less than a cos with the same quantity of
20
cable: less effective than cos theta
15
Short sample field is around 5% less
10
than what could be obtained by a cos
5
with the same quantity of cable
0
Reached 87% of short sample
Elegant, but mechanical support is an issue
CERN Accelerator School, Erice 2013
sector [0-48,60-72]
No iron
With iron
0
20
40
60
w (mm)
80
Magnetic design -. 33
100
100
80
80
60
60
y (mm)
y (mm)
4: BLOCK VS COS THETA
40
40
20
20
0
0
0
20
40
60
x (mm)
80
100
Cos theta coil in Tevatron dipole
Square vs circle: Vitruvian man, Leonardo
CERN Accelerator School, Erice 2013
0
20
40
60
x (mm)
80
100
Block coil in HD2/3
Square vs circle: Bologna city centre
Magnetic design -. 34
CONCLUSIONS
Main parameter to choose for a magnet design
Current density and coil width
Field quality can be solved with azimuthal layout (some wedges)
Looks complicate, but it is not
Dipole: field propto coil width and current density
Quadrupole: gradient propto ln(1+w/r) and current density
In both cases, adding more and more coil is not worth – asymptotic
limit – important to know where to stop
Other factors: protection, mechanics
Most magnets work with a current density around 500 A/mm2
Cos theta is the workhorse of accelerator magnets
Block design is interesting but needs more experience
CERN Accelerator School, Erice 2013
Magnetic design -..35
REFERENCES
General magnet design
R. Wilson “Superconducting magnets”, Oxford press
P. Schmuser, K. Mess, S. Wolff “Superconducting accelerator
magnets”, World Scientific
USPAS 2012 course H. Felice, P. Ferracin, S. Prestemon, E. Todesco
www.cern.ch/ezio.todesco/uspas/uspas.html
Field vs coil width
L. Rossi, E. Todesco, `Electromagnetic design of superconducting
quadrupoles', Phys. Rev. STAB 9 102401 (2006).
L. Rossi, E. Todesco, `Electromagnetic design of superconducting
dipoles based on sector coils', Phys. Rev. STAB 10 112401 (2007).
Codes
Roxie Ansys Opera
CERN Accelerator School, Erice 2013
Magnetic design -..36
APPENDIX
Quadrupole equations
A gallery of coil lay outs
CERN Accelerator School, Erice 2013
Magnetic design -..37
5. QUADRUPOLES: GRADIENT VERSUS MATERIAL
AND COIL THICKNESS
The same approach can be used for a quadrupole
We define
G
Bp

j
rG
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)
B p , ss
r cs

b
1  r cs
c 
 cs
Gss 
b
1  r cs
s
jss 
b
1  r cs
Please note that  is not any more proportional to w and not any
more independent of r !
w

 c   c 0 ln 1 
c0
=6.6310-7 [Tm/A
CERN Accelerator School, Erice 2013


r
] also in this case, by chance as in the dipole
Magnetic design -. 38
5. QUADRUPOLES: GRADIENT VERSUS MATERIAL
AND COIL THICKNESS
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
y (mm)
40
 ( w, r ) ~ a 1
a1=0.11
r
w
 1  a1
w
r
20
1.5
0
20
40
x (mm)
60
RHIC main quadrupole
40
ISR MQ
SSC MQ
RHIC MQ
LHC MQM
LHC MQXA
1.4
80
 [adim]
0
y (mm)
a-1=0.45
1.3
TEV MQ
LEP I MQC
RHIC MQY
LHC MQY
HERA MQ
LEP II MQC
LHC MQ
LHC MQXB
1.2
1.1
20
current grading
1.0
0
0.0
0
20
40
x (mm)
60
80
0.5
1.0
1.5
aspect ratio w eq/r (adim)
2.0
LHC main quadrupole
CERN Accelerator School, Erice 2013
Magnetic design -. 39
5. QUADRUPOLES: GRADIENT VERSUS MATERIAL
AND COIL THICKNESS
We now can write the short sample gradient for a sector
coil as a function of
 s
Material parameters s, b
(linear case as Nb-Ti)
Cable parameters 
Aperture r and coil width w
Gss 


 c 0 ln 1 
Gss 
 ( w, r ) ~ a 1
c
1  r cs
b
r
w
 1  a1
w
r
w
s
r
r
w

 w
1   a1  1  a1 r c 0 ln 1  s
w
r
r




 c ( w, r )   c 0 ln 1 
w

r
b
Relevant feature: for very large coil widths w the short sample
gradient tends to zero !
CERN Accelerator School, Erice 2013
Magnetic design -. 40
APPENDIX
Quadrupole equations
A gallery of coil lay outs
CERN Accelerator School, Erice 2013
Magnetic design -..41
6. A REVIEW OF DIPOLE LAY-OUTS
RHIC MB
Main dipole of the RHIC
296 magnets built in 04/94 – 01/96
100
12
60
B * c2
10
40
B (T)
y (mm)
80
Nb-Ti, 4.2 K
weq~9 mm ~0.23
1 layer, 4 blocks
no grading
20
8
6
sector [0-48,60-72]
No iron
With iron
4
2
0
0
0
20
40
60
x (mm)
CERN Accelerator School, Erice 2013
80
100
0
20
40
60
w (mm)
80
Unit 11: Electromagnetic design episode III – 11.42
6. A REVIEW OF DIPOLE LAY-OUTS
Tevatron MB
Main dipole of the Tevatron
774 magnets built in 1980
100
12
60
B * c2
10
40
B (T)
y (mm)
80
Nb-Ti, 4.2 K
weq~14 mm ~0.23
2 layer, 2 blocks
no grading
20
8
6
sector [0-48,60-72]
No iron
With iron
4
2
0
0
0
20
CERN Accelerator School, Erice 2013
40
60
x (mm)
80
100
0
20
40
60
w (mm)
80
Unit 11: Electromagnetic design episode III – 11.43
6. A REVIEW OF DIPOLE LAY-OUTS
HERA MB
Main dipole of the HERA
416 magnets built in 1985/87
100
12
60
B * c2
10
40
B (T)
y (mm)
80
Nb-Ti, 4.2 K
weq~19 mm ~0.26
2 layer, 4 blocks
no grading
8
6
sector [0-48,60-72]
No iron
With iron
4
20
2
0
0
0
20
40
60
x (mm)
CERN Accelerator School, Erice 2013
80
100
0
20
40
60
w (mm)
80
Unit 11: Electromagnetic design episode III – 11.44
6. A REVIEW OF DIPOLE LAY-OUTS
SSC MB
Main dipole of the ill-fated SSC
18 prototypes built in 1990-5
100
12
60
B * c2
10
40
B (T)
y (mm)
80
Nb-Ti, 4.2 K
weq~22 mm ~0.30
4 layer, 6 blocks
30% grading
20
8
6
sector [0-48,60-72]
No iron
With iron
4
2
0
0
0
20
40
60
x (mm)
CERN Accelerator School, Erice 2013
80
100
0
20
40
60
w (mm)
80
Unit 11: Electromagnetic design episode III – 11.45
6. A REVIEW OF DIPOLE LAY-OUTS
HFDA dipole
Nb3Sn model built at FNAL
6 models built in 2000-2005
100
25
60
B * c2
20
40
B (T)
y (mm)
80
Nb3Sn, 4.2 K
jc~2000 A/mm2 at 12 T, 4.2 K
weq~23 mm
~0.29
2 layers, 6 blocks
no grading
20
15
10
sector [0-48,60-72]
No iron
With iron
5
0
0
0
20
40
60
x (mm)
CERN Accelerator School, Erice 2013
80
100
0
20
40
60
w (mm)
80
Unit 11: Electromagnetic design episode III – 11.46
6. A REVIEW OF DIPOLE LAY-OUTS
LHC MB
Main dipole of the LHC
1276 magnets built in 2001-06
100
60
40
B (T)
y (mm)
80
Nb-Ti, 1.9 K
weq~27 mm ~0.29
2 layers, 6 blocks
23% grading
20
0
0
20
40
60
x (mm)
CERN Accelerator School, Erice 2013
80
100
B * c2
14
12
10
8
6
4
2
0
sector [0-48,60-72]
No iron
With iron
0
20
40
60
w (mm)
80
Unit 11: Electromagnetic design episode III – 11.47
6. A REVIEW OF DIPOLE LAY-OUTS
FRESCA
60
40
B (T)
y (mm)
Dipole for cable test station at CERN
1 magnet built in 2001
Nb-Ti, 1.9 K
weq~30 mm ~0.29
100
2 layers, 7 blocks
24% grading
80
20
0
0
20
40
60
x (mm)
CERN Accelerator School, Erice 2013
80
100
B * c2
14
12
10
8
6
4
2
0
sector [0-48,60-72]
No iron
With iron
0
20
40
60
w (mm)
80
Unit 11: Electromagnetic design episode III – 11.48
6. A REVIEW OF DIPOLE LAY-OUTS
MSUT dipole
cilinder
yoke
collar
Nb3Sn model built at Twente U.
1 model built in 1995
windings
wedge
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
100
80
25
60
B * c2
20
40
B (T)
y (mm)
insert
20
15
10
sector [0-48,60-72]
No iron
With iron
5
0
0
0
20
40
60
x (mm)
CERN Accelerator School, Erice 2013
80
100
0
20
40
60
w (mm)
80
Unit 11: Electromagnetic design episode III – 11.49
6. A REVIEW OF DIPOLE LAY-OUTS
D20 dipole
Nb3Sn model built at LBNL (USA)
1 model built in ???
100
25
60
B * c2
20
40
B (T)
y (mm)
80
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
20
15
10
sector [0-48,60-72]
No iron
With iron
5
0
0
0
20
40
60
x (mm)
CERN Accelerator School, Erice 2013
80
100
0
20
40
60
w (mm)
80
Unit 11: Electromagnetic design episode III – 11.50
6. A REVIEW OF DIPOLE LAY-OUTS
HD2/3
Nb3Sn model being built in LBNL
2 models to be built in 2008/2013
100
25
60
B * c2
20
40
B (T)
y (mm)
80
Nb3Sn, 4.2 K
jc~2500 A/mm2 at 12 T, 4.2 K
weq~46 mm ~0.35
2 layers, racetrack, no grading
20
15
10
sector [0-48,60-72]
No iron
With iron
5
0
0
0
20
40
60
x (mm)
CERN Accelerator School, Erice 2013
80
100
0
20
40
60
w (mm)
80
Unit 11: Electromagnetic design episode III – 11.51
6. A REVIEW OF DIPOLE LAY-OUTS
Fresca2 dipole
Nb3Sn test station founded by UE
cable built in 2004-2006
Operational field 13 T
To be tested in 2014
CERN Accelerator School, Erice 2013
Nb3Sn, 4.2 K
jc~2500 A/mm2 at 12 T, 4.2 K
weq~80 mm ~0.31
Block coil 4 layers
Unit 11: Electromagnetic design episode III – 11.52
6. A REVIEW OF QUADRUPOLES 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
200
B * c2 /r
150
40
Gss (T/m)
y (mm)
60
20
100
sector [0-24,30-36]
No iron
With iron
50
0
0
0
20
CERN Accelerator School, Erice 2013
40
60
x (mm)
80
100
120
0.0
0.5
1.0
1.5
w eq /r (adim)
2.0
Unit 11: Electromagnetic design episode III – 11.53
6. A REVIEW OF QUADRUPOLES 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
60
300
B * c2 /r
40
Gss (T/m)
y (mm)
250
20
200
150
sector [0-24,30-36]
No iron
With iron
100
50
0
0
0
20
CERN Accelerator School, Erice 2013
40
60
x (mm)
80
100
120
0.0
0.5
1.0
1.5
w eq /r (adim)
2.0
Unit 11: Electromagnetic design episode III – 11.54
6. A REVIEW OF QUADRUPOLES 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
60
140
B * c2 /r
40
Gss (T/m)
y (mm)
120
20
0
100
80
60
40
sector [0-24,30-36]
20
No iron
0
0
20
CERN Accelerator School, Erice 2013
40
60
x (mm)
80
100
120
0.0
0.5
1.0
1.5
w eq /r (adim)
2.0
Unit 11: Electromagnetic design episode III – 11.55
6. A REVIEW OF QUADRUPOLES LAY-OUTS
ISR MQX
100
y (mm)
IR region quadrupole of the ISR
8 magnets built in ~1977-79
Nb-Ti, 4.2 K
w/r~0.28 ~0.35
1 layer, 3 blocks, no grading
B * c2 /r
Gss (T/m)
80
60
40
sector [0-24,30-36]
No iron
With iron
20
0
0.0
0.5
1.0
1.5
w eq /r (adim)
2.0
50
0
0
CERN Accelerator School, Erice 2013
50
100
x (mm)
150
Unit 11: Electromagnetic design episode III – 11.56
6. A REVIEW OF QUADRUPOLES 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
100
B * c2 /r
80
40
Gss (T/m)
y (mm)
60
20
60
40
sector [0-24,30-36]
20
0
No iron
0
0
20
CERN Accelerator School, Erice 2013
40
60
x (mm)
80
100
120
0.0
0.5
1.0
1.5
w eq /r (adim)
2.0
Unit 11: Electromagnetic design episode III – 11.57
6. A REVIEW OF QUADRUPOLES 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
B * c2 /r
250
200
40
Gss (T/m)
y (mm)
60
20
150
100
sector [0-24,30-36]
No iron
With iron
50
0
0
0
20
CERN Accelerator School, Erice 2013
40
60
x (mm)
80
100
120
0.0
0.5
1.0
1.5
w eq /r (adim)
2.0
Unit 11: Electromagnetic design episode III – 11.58
6. A REVIEW OF QUADRUPOLES LAY-OUTS
HERA MQ
Main quadrupole of the HERA
60
300
40
200
Gss (T/m)
y (mm)
Nb-Ti, 1.9 K
w/r~0.52 ~0.27
2 layers, 3 blocks, grading 10%
20
0
B * c2 /r
sector [0-24,30-36]
No iron
With iron
100
0
0
20
CERN Accelerator School, Erice 2013
40
60
x (mm)
80
100
120
0.0
0.5
1.0
1.5
w eq /r (adim)
2.0
Unit 11: Electromagnetic design episode III – 11.59
6. A REVIEW OF QUADRUPOLES 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
B * c2 /r
500
400
40
Gss (T/m)
y (mm)
60
20
300
200
sector [0-24,30-36]
No iron
With iron
100
0
0
0
20
CERN Accelerator School, Erice 2013
40
60
x (mm)
80
100
120
0.0
0.5
1.0
1.5
w eq /r (adim)
2.0
Unit 11: Electromagnetic design episode III – 11.60
6. A REVIEW OF QUADRUPOLES LAY-OUTS
LHC MQY
60
300
40
200
Gss (T/m)
y (mm)
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%
20
0
B * c2 /r
sector [0-24,30-36]
No iron
With iron
100
0
0
20
CERN Accelerator School, Erice 2013
40
60
x (mm)
80
100
120
0.0
0.5
1.0
1.5
w eq /r (adim)
2.0
Unit 11: Electromagnetic design episode III – 11.61
6. A REVIEW OF QUADRUPOLES LAY-OUTS
LHC MQXB
60
400
40
300
Gss (T/m)
y (mm)
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%
20
B * c2 /r
200
sector [0-24,30-36]
No iron
With iron
100
0
0
0
20
CERN Accelerator School, Erice 2013
40
60
x (mm)
80
100
120
0.0
0.5
1.0
1.5
w eq /r (adim)
2.0
Unit 11: Electromagnetic design episode III – 11.62
6. A REVIEW OF QUADRUPOLES 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
B * c2 /r
500
400
40
Gss (T/m)
y (mm)
60
20
300
200
sector [0-24,30-36]
No iron
With iron
100
0
0
0
20
CERN Accelerator School, Erice 2013
40
60
x (mm)
80
100
120
0.0
0.5
1.0
1.5
w eq /r (adim)
2.0
Unit 11: Electromagnetic design episode III – 11.63
6. A REVIEW OF QUADRUPOLES 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
B * c2 /r
500
400
40
Gss (T/m)
y (mm)
60
20
300
200
sector [0-24,30-36]
No iron
With iron
100
0
0
0
20
CERN Accelerator School, Erice 2013
40
60
x (mm)
80
100
120
0.0
0.5
1.0
1.5
w eq /r (adim)
2.0
Unit 11: Electromagnetic design episode III – 11.64
6. A REVIEW OF QUADRUPOLES LAY-OUTS
LHC MQXA
60
400
40
300
Gss (T/m)
y (mm)
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%
20
B * c2 /r
200
sector [0-24,30-36]
No iron
With iron
100
0
0
0
20
CERN Accelerator School, Erice 2013
40
60
x (mm)
80
100
120
0.0
0.5
1.0
1.5
w eq /r (adim)
2.0
Unit 11: Electromagnetic design episode III – 11.65
6. A REVIEW OF QUADRUPOLES LAY-OUTS
LHC MQXC
Nb-Ti option for the LHC upgrade
LHC dipole cable, graded coil
2 short models built in 2011-3
w/r~0.5 ~0.33 2 layers, 4 blocks
y (mm)
60
40
20
0
0
20
CERN Accelerator School, Erice 2013
40
60
x (mm)
80
100
120
Unit 11: Electromagnetic design episode III – 11.66
6. A REVIEW OF QUADRUPOLES LAY-OUTS
LARP TQ/LQ
90 mm aperture Nb3Sn option for the
LHC upgrade (IR triplet)
~5 short model tested in 2005-2010
Two structures: collars (TQC) and shell (TQS)
3 3.4-m-long magnets tested in 2010-13
w/r~0.5 ~0.33 2 layers, 3 blocks
CERN Accelerator School, Erice 2013
Unit 11: Electromagnetic design episode III – 11.67
6. A REVIEW OF QUADRUPOLES LAY-OUTS
LARP HQ
120 mm aperture Nb3Sn option for the
LHC upgrade (IR triplet)
2 short model tested in 2011/2013
w/r~0.5 ~0.33 2 layers, 4 blocks
y (mm)
60
40
20
0
0
20
CERN Accelerator School, Erice 2013
40
60
x (mm)
80
100
120
Unit 11: Electromagnetic design episode III – 11.68
6. A REVIEW OF QUADRUPOLES LAY-OUTS
MQXF
150 mm aperture Nb3Sn option for the
LHC upgrade (IR triplet)
first short model tested in 2014
w/r~0.5 ~0.33 2 layers, 4 blocks
CERN Accelerator School, Erice 2013
Unit 11: Electromagnetic design episode III – 11.69