Magnetic Design of Superconducting Magnets

E. Todesco (CERN), Magnetic Design of Superconducting Magnets, CERN Yellow Report CERN-2014-005, pages 269 to 292, a lecture to the CAS-CERN Accelerator School Superconductivity for Accelerators, Erice, Italy, 24 April to 4 May 2013, edited by R. Bailey. Preprint at arxiv.org/abs/1501.07149. Carried here under CC BY 3.0.

Abstract

In this paper we discuss the main principles of magnetic design for superconducting magnets (dipoles and quadrupoles) for particle accelerators. We give approximated equations that govern the relation between the field/gradient, the current density, the type of superconductor (Nb−Ti or Nb₃Sn), the thickness of the coil, and the fraction of stabilizer. We also state the main principle controlling the field quality optimization, and discuss the role of iron. A few examples are given to show the application of the equations and their validity limits.

Keywords: magnets for accelerators, superconducting magnets, magnet design.

3.1 Critical surfaces

Superconductors are able to tolerate a given combination of current density and field, in an approximately triangular zone of the (B, j) space, at a fixed temperature. For Nb−Ti, a good approximation is the linear function

j_sc(B) = s (b − B) (32)

where b and s are two free parameters that fit the critical surface. The fit is accurate over a few tesla (see Fig. 11). Typically, the slope s is about 5.0 × 10⁷ A·m⁻²·T⁻¹, i.e. lowering the field by 1 T gains 500 A·mm⁻². At 1.9 K and 9 T, typical values are about 2000 A·mm⁻², thus yielding b about 13 T. At 4.2 K and 6 T, typical values are as well around 2000 A·mm⁻², thus yielding b about 10 T, i.e., the whole critical surface at 1.9 K is shifted left by about 3 T.

For Nb₃Sn, the linear approximation is not valid since there is a non-negligible positive curvature (see Fig. 11). A good approximation that provides an analytical solution when intersecting with the magnet loadline is a shifted hyperbola:

j_sc(B) = s (b/B − 1) (33)

This is an empirical fit corresponding to a pinning force linear in the field,

F(B) = j_sc(B) · B = s (b − B) (34)

and works very well over a large range of fields (see Fig. 11). Again, s and b are two free parameters, with the same dimensions as in the previous case (a current density per tesla and a field, respectively). Using s about 3.15 × 10⁹ A·m⁻²·T⁻¹ and b about 22 T at 4.2 K, we obtain a current density of 2700 A·mm⁻² at 12 T and of 1450 A·mm⁻² at 15 T, which are currently close to the best achievable values. Extrapolation at 1.9 K can be achieved with s about 3.3 × 10⁹ A·m⁻²·T⁻¹ and b about 24 T. As before, these are indicative values, and one can work out the best parameters for each critical surface and range of interest.

The insulated coil contains only a fraction, κ, of superconductor, with κ usually in the range 0.25–0.40. So the equation for the overall current density (including stabilizer, voids, and insulation) is

j(B) = κ s (b − B) for Nb−Ti (35)

and

j(B) = κ s (b/B − 1) for Nb₃Sn. (36)

(Figure 11 of the lecture, not reproduced here, plots the four curves this section describes: the Nb−Ti critical surface at 4.2 K and at 1.9 K, and the Nb₃Sn critical surface at 4.2 K and at 1.9 K, with current density on the vertical axis from 0 to 3500 A·mm⁻² and field on the horizontal axis from 0 to 20 T. The Nb₃Sn hyperbolic fits lie so close to the standard Kramer fit that Todesco notes they are barely distinguishable.)