Historical research guide · 1146 · 2015

Magnetic Design of Superconducting Magnets

E. Todesco

In one minute

A superconductor is not one number. Ezio Todesco of CERN, teaching magnet designers at the CERN Accelerator School, sets out the working form of that fact: a conductor tolerates a combination of current density and magnetic field, at a fixed temperature, and the boundary of what it tolerates is a surface in that space. For niobium-titanium the usable stretch of that boundary is close to a straight line, and Todesco gives its two numbers — a slope of about five hundred amps per square millimetre gained for each tesla the field is lowered, and an intercept near thirteen tesla at 1.9 kelvin, ten tesla at 4.2 kelvin. For niobium-tin the line does not work; the boundary curves the other way, and he fits it with a shifted hyperbola instead. The point of the lecture is what happens when that surface meets the magnet's own loadline: where they cross is the magnet's limit.

Connection

This is the sheet that puts absolute units under chapter 11's superconductors. Everywhere else on the site the three limits of a superconductor are drawn in reduced units, as a shape; here are kelvin, tesla and amps per square millimetre for two named conductors, from the engineer who designs the magnets.

From the source

A superconductor tolerates a given combination of current density and magnetic field, at a fixed temperature, over an approximately triangular zone of the current-density-versus-field plane — the critical surface. It is the intersection of that surface with the magnet's own loadline, not the surface alone, that sets what field a magnet reaches.Section 3.1, opening sentence; Section 3.3 and Fig. 14
For niobium-titanium the critical surface is well approximated over a few tesla by a straight line: the superconductor current density falls as the field rises, with a slope of about 5.0 × 10⁷ amps per square metre per tesla — that is, lowering the field by one tesla buys about 500 amps per square millimetre.Section 3.1, Eq. (32) and the paragraph following it
Absolute anchors for niobium-titanium: about 2000 amps per square millimetre at 1.9 kelvin and 9 tesla, giving a field intercept near 13 tesla; and about 2000 amps per square millimetre at 4.2 kelvin and 6 tesla, giving an intercept near 10 tesla. Cooling from 4.2 to 1.9 kelvin shifts the whole critical surface by roughly 3 tesla.Section 3.1, paragraph following Eq. (32)
Niobium-tin cannot be fitted with the same straight line, because its critical surface carries a non-negligible positive curvature. Todesco uses a shifted hyperbola instead, which corresponds to a pinning force linear in the field, and reports it works well over a large range of fields.Section 3.1, Eqs. (33) and (34)
Absolute anchors for niobium-tin: with the hyperbolic fit at 4.2 kelvin, about 2700 amps per square millimetre at 12 tesla and 1450 amps per square millimetre at 15 tesla, which Todesco says are close to the best achievable values; the field parameter is about 22 tesla at 4.2 kelvin and about 24 tesla at 1.9 kelvin.Section 3.1, paragraph following Eq. (34)
The current density a coil actually carries is lower than the superconductor's own, because an insulated coil is only a fraction of superconductor — Todesco gives that fraction as usually between 0.25 and 0.40, and the overall current density is the superconductor value multiplied by it.Section 3.1, Eqs. (35) and (36)

The way in

https://arxiv.org/abs/1501.07149

Historical source-access record. This guide links to the original; it does not reproduce the paper text.

WHAT WAS READ. The author's PDF was downloaded from arXiv on 2026-09-12 and read as text; the file is 24 pages, sha256 593f34785e9a01d380a608051e120cdb75c961232236514f3790d22db6351fbe. The licence was checked on the item's own arXiv record, not inferred from the volume: the page for 1501.07149 declares Creative Commons Attribution 3.0, and so does the page for the parent proceedings 1502.02950. TEXT. The abstract and Section 3.1, the section this site needs, are carried below under that licence; the rest of the lecture — sector-coil multipoles, peak field, iron, block coils — is not reproduced and is at the source. The PDF sets its equations as typeset mathematics and the extraction flattened them, so the two critical-surface fits are written here in words and plain notation rather than reproduced as display equations, and Figure 11 is described rather than reproduced.

How to cite it

E. Todesco (2015) Magnetic Design of Superconducting Magnets arXiv:1501.07149

Where it sits in the curriculum

Gravity control and superconductors