1. Unit 9 Electromagnetic design Episode II
Soren Prestemon and Paolo Ferracin
Lawrence Berkeley National Laboratory (LBNL)
Ezio Todesco
European Organization for Nuclear Research (CERN)

2. QUESTIONS
Given a material and an aperture, and a quantity of cable, what is the strongest dipole / quadrupole we can build ?
Can we have explicit equations ?
Or do we have to run codes in each case ?

3. CONTENTS 1. Dipoles: short sample field versus material and lay-out
2. Quadrupoles: short sample gradient versus material and lay-out
3. A flowchart for magnet design

4. 1. DIPOLES: FIELD VERSUS MATERIAL AND COIL THICKNESS
We recall the equations for the critical surface
Nb-Ti: linear approximation is good
with c~6.0108 [A/(T m2)] and B*c2~10 T at 4.2 K or 13 T at 1.9 K
LHC production: c~6.7108 [A/(T m2)] and B*c2~13.4 T at 1.9 K

5. 1. DIPOLES: FIELD VERSUS MATERIAL AND COIL THICKNESS
The current density in the coil is lower
Strand made of superconductor and normal conducting (copper) 
Cu-sc 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)
The filling ratio ranges from ¼ to 1/3
The critical surface for j (engineering current density) is

6. 1. DIPOLES: FIELD VERSUS MATERIAL AND COIL THICKNESS
Examples of filling ratio in dipoles (similar for quads)
Copper to superconductor ranging from 1.2 to 2.2
Extreme case of D20: 0.43
Void fraction from 11% to 18%
Insulation from 11% to 18%
Case of FNAL HFDA: 24% for insulation

7. 1. DIPOLES: FIELD VERSUS MATERIAL AND COIL THICKNESS
We characterize the coil by two parameters
: how much field in the centre is given per unit of current density
: ratio between peak field and central field
for a sector dipole or a cos ,   w
for a cos dipole, =1
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

8. 1. DIPOLES: FIELD VERSUS MATERIAL AND COIL THICKNESS
We can now compute the maximum current density that can be tolerated by the superconductor (short sample limit)
the short sample current is
and the short sample field is

9. 1. DIPOLES: FIELD VERSUS MATERIAL AND COIL THICKNESS
When we go for larger and larger coils we increase both  (central field) and  (peak field)
when c >>1 (large coil)
The interplay between these two factors constitutes the problem of the coil optimization
Note that the slope of the critical surface c defines the “large coil”
Note that for large coils the filling factor becomes not relevant

10. 1. DIPOLES: FIELD VERSUS MATERIAL AND COIL THICKNESS
Examples
The quantity lkcg is larger than 1 in the 6 analysed dipoles, and is around 5 for dipoles with large coil widths (SSC, LHC, Fresca)
This means that for these three dipoles we are rather close to the maximum field we can get with Nb-Ti

11. 1. DIPOLES: FIELD VERSUS MATERIAL AND COIL THICKNESS
Examples of g (central field per unit of current density)
Good agreement (within 4%) between 5 dipoles (no grading) and [0°-48°,60°-72°] lay-out
Five cases give 10-20% due to grading (see Unit 11)

12. 1. DIPOLES: FIELD VERSUS MATERIAL AND COIL THICKNESS
Numerical evaluation of  for different sector coils
To compute the peak field one has to compute the field everywhere in the coil, and take the maximum
One can prove that the maximum is always on the border of the coil – useful to reduce the computation time
Tevatron main dipole –
location of the peak field
RHIC main dipole –
location of the peak field
LHC main dipole –
location of the peak field

13. 1. DIPOLES: FIELD VERSUS MATERIAL AND COIL THICKNESS
Numerical evaluation of  for different sector coils
For large widths,   1
This means that for very large widths we can reach B*c2 !

14. 1. DIPOLES: FIELD VERSUS MATERIAL AND COIL THICKNESS
Numerical evaluation of  for different sector coils
For interesting widths (10 to 30 mm) is
The cos approx having =1 is not so bad for w>20 mm
The [0°-48°,60°-72°] is well fit by with a~0.045

15. 1. DIPOLES: FIELD VERSUS MATERIAL AND COIL THICKNESS
Examples of l (ratio between peak field and central field)
Agreement with the semi-analytical fit found for the [0°-48°,60°-72°] lay-out is good (within 2% in the analysed cases)

16. 1. DIPOLES: FIELD VERSUS MATERIAL AND COIL THICKNESS
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
for a [0°-48°,60°-72°] coil, a= 0=6.6310-7 [Tm/A]
for Nb-Ti c~6.0108 [A/(T m2)] and B*c2~10 T at 4.2 K or 13 T at 1.9 K
Cos model:
=210-7 [Tm/A]

17. 1. DIPOLES: FIELD VERSUS MATERIAL AND COIL THICKNESS
Evaluation of short sample field in sector lay-outs and cos model for a given aperture (r=30 mm)
Tends asymptotically to B*c2, as B*c2 w/(1+w), for w
Similar results for different position of wedges

18. 1. DIPOLES: FIELD VERSUS MATERIAL AND COIL THICKNESS
Dependence on the aperture
For very large aperture magnets, one has less field for the same coil thickness
For small apertures it tends to the cos model

19. 1. DIPOLES: FIELD VERSUS MATERIAL AND COIL THICKNESS
Case of Nb3Sn
The critical surface is not linear, it can be solved numerically
The saturation for large widths is slower (due to the shape of the surface)

20. CONTENTS 1. Dipoles: short sample field versus material and lay-out
2. Quadrupoles: short sample gradient versus material and lay-out
3. A flowchart for magnet design

21. 2. QUADRUPOLES: GRADIENT VERSUS MATERIAL AND COIL THICKNESS
The same approach can be used for a quadrupole
We define
the only difference is that now  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 independent of r !

22. 2. QUADRUPOLES: GRADIENT VERSUS MATERIAL AND COIL THICKNESS
Please note that  is not any more proportional to w and independent of r !
Actual values fit well with the constant for [0°-24°,30°-36°] lay-out
0=6.6310-7 [Tm/A], as for the dipoles …
[0°-30°] lay-out

23. 2. 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
RHIC main quadrupole
LHC main quadrupole

24. 2. QUADRUPOLES: GRADIENT VERSUS MATERIAL AND COIL THICKNESS
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

25. 2. QUADRUPOLES: GRADIENT VERSUS MATERIAL AND COIL THICKNESS
Comparison for the ratio l between the fit for the [0°-24°,30°-36°] coil and actual values

26. 2. QUADRUPOLES: GRADIENT VERSUS MATERIAL AND COIL THICKNESS
We now can write the short sample gradient for a sector coil as a function of
Material parameters c, 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 !

27. 2. QUADRUPOLES: GRADIENT VERSUS MATERIAL AND COIL THICKNESS
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% !!
-30%

28. 2. QUADRUPOLES: GRADIENT VERSUS MATERIAL AND COIL THICKNESS
Dependence of of short sample gradient on the aperture
Large aperture quadrupoles go closer to G*=B*c2/r
Very small aperture quadrupoles do not exploit the sc !!
Large aperture need smaller ratio w/r
For r= mm, no need of having w>r

29. 2. QUADRUPOLES: GRADIENT VERSUS MATERIAL AND COIL THICKNESS
Case of Nb3Sn – in this case one can solve the equations numerically
Gain is ~50% in gradient for the same aperture (at 35 mm)
Gain is ~70% in aperture for the same gradient (at 200 T/m)

30. CONTENTS 1. Dipoles: short sample field versus material and lay-out
2. Quadrupoles: short sample gradient versus material and lay-out
3. A flowchart for magnet design

31. 3. A FLOWCHART FOR MAGNET DESIGN - DIPOLES
Having an aperture
The technology gives the maximal field that can be reached
Nb-Ti: 7-8 T at 4.2 K, 11 T at 1.9 K (80% of B*c2)
Nb3Sn: 17-20 T ?
Having an aperture and a field
One can evaluate the thickness of the coil needed to get the field using the equations for a sector coil
Cost optimization – higher fields costs more and more $$$ (or euro)

32. 3. A FLOWCHART FOR MAGNET DESIGN - DIPOLES
Having aperture, field and cable thickness
Surface necessary to get that field can be estimated (i.e. number of turns) or
Wedges are put to optimize field quality
Sometimes the cable is given, but not the field
Having the cable one can make a quick estimation of the short sample field that can be obtained with 1/2/3 … layers
Then the actual lay-out with wedges is made
At first order one aims at zero multipoles – pure field
Successive optimizations
Effect of iron (Unit 11) and persistent currents (Unit 15) is included – fine tuning of cross-section
Balance of optimization between low and high field
Once built, an additional correction is usually needed (Unit 20)
Models are not enough precise (in absolute) but work well in relative

33. 4. CONCLUSIONS
We wrote equations for computing the short sample field or gradient
This gives the dependence on the aperture, coil thickness, filling ration, material
Episode III
Can we make better ?
What about iron ?
Other lay-outs

34. REFERENCES Field quality constraints Electromagnetic design
M. N. Wilson, Ch. 1
P. Schmuser, Ch. 4
A. Asner, Ch. 9
Classes given by A. Devred at USPAS
Electromagnetic design
S. Caspi, P. Ferracin, “Limits of Nb3Sn accelerator magnets“, Particle Accelerator Conference (2005)
L. Rossi, E. Todesco, Electromagnetic design of superconducting quadrupoles , Phys. Rev. ST Accel. Beams 9 (2006)
Classes given by R. Gupta at USPAS 2006, Unit 3,4,5,6
S. Russenschuck, et al, CERN (1999) and
Numerical methods to calculate magnetic field
S. Russenschuck, “Electromagnetic design of superconducting accelerator magnets”, CERN (2004)

35. ACKNOWLEDGEMENTS
B. Auchmann, L. Bottura, A. Den Ouden, A. Devred, P. Ferracin, V. Kashikin, A. McInturff, T. Nakamoto, N. Schwerg, S. Russenschuck, T. Taylor, C. Vollinger, S. Zlobin, for kindly providing magnet designs … and perhaps others I forgot
S. Caspi, L. Rossi for discussing magnet design, grading, and other interesting subjects …
No frogs have been harmed during the preparation of these slides

36. APPENDIX A AN EXPLICIT EXPRESSION FOR NB3SN
Case of Nb3Sn – an explicit expression
An analytical expression can be found using a hyperbolic fit
that agrees well between 11 and 17 T
with c~4.0109 [A/(T m2)]
and b~21 T at 4.2 K, b~23 T at 1.9 K
Using this fit one can find explicit expression for the short sample field
and the constant   are the same as before (they depend on the lay-out, not on the material)