[^1]

*Abstract*--- Dipole magnets with fields beyond 16T will require
superconducting coils that are at least 40 mm thick, an applied
pres-stress around 150 MPa and a protection scheme for stored energy in
the range of 1-2 MJ/m. The coil size will have a direct impact on the
overall magnet cost and the stored energy will raise new questions on
protection. To reduce coil size and minimize risk, the coil may have to
be graded. Grading is achieved by splitting the coil into several layers
with current densities that match the short sample field in each layer.
Grading, especially at high fields, can be effective; however it will
also significantly raise the stress. In this paper we report on the
results of a study on the coil size and field relation to that of the
stress and stored energy. We then extend the results to graded coils and
attempt to address high stress issues and ways to reduce it.

*Index Terms*---Grading, Superconducting magnets, Stress, Stored energy.

# INTRODUCTION

[W]{.smallcaps}hen the LHC pushed NbTi conductor closer to its
high-field limit, the 56 mm bore dipoles reach 9.7 T at 1.9 K, and a
stored energy of 334 kJ/m (per bore) \[1\]. Replacing the cable in that
magnet with identical size Nb~3~Sn conductor (31.3 mm of overall coil
thickness) would raise the field to 15.3 T, the stored energy to 900
kJ/m, and the Lorentz stress from 88 MPa to 220 MPa. If we farther grade
the coil (20.3 mm layer 1 and 11.0 mm layer 2) the central field will
increase to 16.2 T, the Lorentz stress in layers 1 and 2 will increase
to 202 MPa and 377 MPa respectively, and the stored energy will increase
to 1.14 MJ/m. (we assume a Nb~3~Sn wire carrying 3000 A/mm^2^ at 12 T,
4.2 K in the non-copper with a 50% reduced engineering current density).

This comparison between NbTi and Nb~3~Sn conductors points out both the
promise and the challenge for high field magnets. Accelerator magnets
can not be treated as a "one-of-a-kind" magnet; their cost must be at a
minimum and the reliability high. Pushing the limits on high field
Nb~3~Sn magnets requires the best superconductor, a reasonably small
size magnet, and a compact structure. In this paper we address the
relations between field, coil size, bore diameter, stress and stored
energy, and we then extend them to graded coils pointing out areas where
we presently meet the challenge and areas that will require further R&D.

# Dipole model

## Model Description

To study the limits of superconducting dipole magnets, we formulated a
simple but realistic model based on several assumptions: 1) the bore is
round and the coil is a thick cylinder (Fig 1); 2) the engineering
current density is equal to *J~e~cos* and the field is therefore a pure
dipole; 3) the field magnitude in the bore and along the conductor inner
surface is identical; 4) the short-sample current density in the
superconductor *J~ss~* and the engineering current density *J~e-ss~*
(obtained by averaging *J~ss~* over copper, insulation, and voids) are a
function of field *B* and temperature *T*; 5) the coil is initially not
graded (section II) followed by graded coils (section III); 6) there is
no ferromagnetic material nearby.

![](media/image1.png){width="1.738888888888889in"
height="1.9013888888888888in"}

Fig. 1. A pure dipole with a *J~e~cos* current density.

We summarized our previous work on none-graded coils \[2\]. A simple
relation exists between the central dipole field *B*~0~, the engineering
current density *J~e~*, and the coil thickness *w* \[3\]:

$B_{0} = \frac{\mu_{0}J_{e}}{2}w$.

If we extend the engineering current density *J~e\ ~*up to the
short-sample current density limit *J~e-ss~*, the field will reach its
corresponding short-sample value *B~ss~*. By applying Summer's empirical
short-sample relation \[4\] to the above equation, we obtain

$w_{\text{ss}}(B_{\text{ss}},T) = \frac{2B_{\text{ss}}}{\mu_{0}J_{e - \text{ss}}(B_{\text{ss}},T)}$,

where the coil thickness is expressed as a function of short-sample
field and temperature only. We note and emphasize that the relation is
*independent* of bore diameter.

The parametric analysis assumes the best commercially available
superconductor (Nb~3~Sn with 3000 A/mm^2^ at 12 T and 4.2 K) and a
Rutherford cable with 50 % non-copper, a 12 % void fraction and an 11 %
insulation fraction (Fig. 2).

![](media/image2.png){width="3.245833333333333in"
height="2.6430555555555557in"}

Fig. 2. Short-sample *J~ss~* and engineering current density *J~e-ss~*
of Nb~3~Sn superconductor.

## Parametric dependencies

### Coil Thickness

The coil thickness *w~ss~* at short-sample is plotted in Fig. 3 for two
different operating temperatures. We note that, at 1.9 K, a 7 mm thick
coil is sufficient to generate a 10 T field, and a 100 mm thick coil
will be needed for a 19 T field. Operating at 1.9 K requires
significantly less conductor than at 4.2 K, an advantage that becomes
even more significant as the field approaches 20 T.

![](media/image3.png){width="3.245833333333333in" height="2.575in"}

Fig. 3. Coil thickness of Nb~3~Sn dipole magnets at short-sample.

### Bore Diameter and Cost

Since the field depends on the coil thickness and not on the bore
diameter, we may claim that a dipole with a zero bore diameter has the
same field as a coil with any bore diameter, as long as the coil
thickness is constant. This gives us the opportunity to separate the
cost of the field from the cost of the bore.

The cost of the coil is proportional to its area, which is given by
*w^2^ + 2wR~1~* (where *w* is the coil thickness, and *R~1~* is the bore
radius). We associate the first term with the area of a no bore coil,
and the second term with an additional area representing the
contribution of a bore. Accordingly, the cost of the field is
proportional to *w^2^*, but the additional cost of the bore is linearly
proportional to the bore diameter. For example at high fields (18 T),
doubling the bore diameter from 25 mm to 50 mm increases the amount of
conductor by only 26 %. We conclude that at very high fields, the effect
of the bore diameter on the overall cost of the conductor is minor
(grading will be addressed in the following section).

### Lorentz Stress

The azimuthal Lorentz stress in a *cos* dipole is the integrated
azimuthal Lorentz force with respect to (no shear) \[5\]. Expressed as

$\sigma_{\theta} = \frac{\mu_{0}J_{e}^{2}}{2}\frac{r}{2}(R_{2} - \frac{R_{1}^{3}}{3r^{2}} - \frac{2}{3}r)\text{cos}^{2}\theta$,

the stress exhibits a maximum along the coil mid-plane ( *= 0*), at a
radius nearly two thirds of the coil thickness (setting /*r = 0* and
solving the cubic relation in *r*).

If we apply the short-sample field, coil thickness *w~ss~*, and a given
bore diameter Ф to the above equation, we arrive at the maximum
short-sample stress ~-*max-ss*~ (Fig. 4).

![](media/image4.png){width="3.245833333333333in"
height="2.557638888888889in"}

Fig. 4. Maximum azimuthal Lorentz stress as a function of the
short-sample field at 1.9 K, and the bore diameter in mm.

The zero bore solution is a monotonic increasing function of the field
and is at the minimum for any bore diameter at that field. Surprisingly,
for certain conditions the maximum stress decreases as the short-sample
field increases. As the field increases, so do coil thickness *w~ss~*
and Lorentz force; however, the rate of increased coil thickness is
greater than the corresponding Lorentz force, thereby reducing the
stress. At high fields the coil thickness dominates and the stress
asymptotically approaches that for a zero bore solution, equal to
*~-max-ss~* = (3*B~ss~^2^*)/(8~0~).

### Stored Energy

The stored energy *E* of a dipole increases quadratically with field
*B*, bore radius *R~1~*, and coil thickness *w*:

$E = \frac{{\pi B}^{2}}{3\mu_{0}}\frac{w^{2}}{2}\left\lbrack 1 + 6\left( \frac{R_{1}}{w} \right)^{2} + 4\left( \frac{R_{1}}{w} \right) \right\rbrack$.

Fig. 5 is a log plot of the short-sample stored energy *E~ss~* for a
number of different bore diameters, including a zero bore.

![](media/image5.png){width="3.245833333333333in"
height="2.5319444444444446in"}

Fig. 5. Short-sample stored energy in dipoles at 1.9 K for various bore
diameters.

As for the stress, we can associate the energy of a zero bore diameter
with the term outside the square brackets in the formula above, and the
terms within the bracket as an additional bore contribution. At high
fields, when the coil thickness *w* increases, the additional
contribution of bore diameter to the stored energy becomes less
effective and the stored energy asymptotically approaches that of a zero
bore.

If we focus on the curves representing a bore diameter closer to the LHC
dipole (50-60 mm), we notice that at 15 T the stored energy is close to
1 MJ/m) reaching 7 MJ/m at 20 T.

# Graded Coils

## Model Description

Grading the coil can effectively reduce overall size while the field
remains the same. Grading takes advantage of the drop in field within
the coil in order to raise the current density in several discrete outer
layers, thus matching that drop. That way the superconductor can reach
its short-sample simultaneously throughout the cross-section, thereby
reducing the overall size.

### Coil Thickness

Grading significantly impacts the coil thickness especially at high
field: for example, at 16 T a two layer graded coil reduces its
thickness by 23%. That number increases to 32% at 18 T and 42% at 20 T
(Fig. 6). Increasing the number of graded coils from 2 to 4 reduces the
overall coil thickness by an additional 10%. Fig. 7-9 show the relation
between coil thickness, current density and field at the conductor as a
function of the dipole field of a 50 mm bore magnet.

### Bore Diameter

Whereas in a non-graded magnet the coil thickness does not depends on
the bore diameter that is no longer true for graded coils. The overall
coil thickness of a graded coil increases by 5% for an increase of bore
diameter from a 30 mm to 50 mm.

![](media/image6.png){width="3.5097222222222224in"
height="2.821527777777778in"} Fig. 6. Reduction in coil thickness of a 2
layer and a 4 layer graded magnet at short-sample (1.9 K).

![](media/image7.png){width="3.5083333333333333in"
height="2.421527777777778in"}

Fig. 7. Incremental layer thickness of a 4 layer graded coil 50 mm bore
magnet (1.9 K).

![](media/image8.png){width="3.5083333333333333in"
height="2.5659722222222223in"}

Fig. 8. Engineering current density of a 4 layer graded 50 mm bore
magnet (1.9 K).

![](media/image9.png){width="3.453472222222222in"
height="2.4895833333333335in"}

Fig. 9. Field in the conductor of each layer of a 4 layer graded magnet
(50 mm bore, 1.9 K).

### Lorentz Stress

Grading reduces the coil thickness at the expense of an increase in
conductor stress. At 18 T (1.9 K) grading will reduce the coil thickness
from 72 mm to 49 mm, but, in a 56 mm bore diameter, the maximum stress
will rise from 170 MPa to an unacceptable level of 380 MPa (Fig. 10,11).
Since reducing the coil thickness has a major impact on reducing the
magnet cost, we are left with an important R&D issue on how to bring
down the stresses. Reducing the stress through stress management should
be confined to the second layer of a two layer graded magnet thereby
maintaining overall high engineering current density.

![](media/image10.png){width="3.5097222222222224in"
height="2.7618055555555556in"}

Fig. 10. Grading an 18 T dipole reduces the coil thickness but raises
the stress (56 mm bore, *T* = 1.9 K).

The formulas for the mid-plane Lorentz stress in a two layers graded
magnet with current densities *J~e1~* and *J~e2\ ~*are written below,

$\begin{array}{r}
\sigma_{\theta 1} = \frac{\mu_{0}J_{e1}}{2}\frac{r}{2}\lbrack J_{e1}(R_{2} - \frac{R_{1}^{3}}{3r^{2}} - \frac{2}{3}r) + J_{e2}(R_{3} - R_{2})\rbrack \\
\sigma_{\theta 2} = \frac{\mu_{0}J_{e2}}{2}\frac{r}{2}\lbrack J_{e2}(R_{3} - \frac{R_{2}^{3}}{3r^{2}} - \frac{2}{3}r) + J_{e1}\frac{R_{2}^{3} - R_{1}^{3}}{3r^{2}}\rbrack
\end{array}$,

![](media/image11.png){width="3.5097222222222224in"
height="2.7618055555555556in"}

Fig. 11. Maximum stress in a graded (2 layers) and none graded magnet
(50 mm bore, 1.9 K).

### Stored Energy

Grading has only a small affect on the stored energy. At 18 T the stored
energy of a 50 mm bore magnet is 1.9 MJ/m a decrease of 10 % from the
2.17 MJ/m none graded magnet (Fig. 5).

# conclusion

The following conclusions, regarding minimum coil size, coil stress, and
magnet stored energy, can be drawn: pushing dipole fields to 18 T at
1.9K in a 50 mm bore would require a 72 mm thick none-graded coil, the
Lorentz stress would be 170 MPa and the stored energy 1.8 MJ/m.

Grading the same coil into two layers will reduce the overall coil
thickness to 49 mm, reduce the Lorentz stress in layer one (31 mm thick)
to 160 MPa, increase the Lorentz stress in layer 2 (18 mm thick) to 380
MPa and reduce the stored energy to 1.9 MJ/m.

At very high fields, the effect of bore size on conductor amount, peak
stress, and the stored energy is small.

Grading the coils reduces the overall thickness at the expense of higher
stress. Assuming we are at coil stress limit (150-200 MPa), new designs
will have to be developed that intercept the Lorentz forces. However we
need to be aware that stress management reduces not just the stress, but
the coils current density as well. Stress management of the outer layer
only is a good way to keep the current density high and reduce the
stress.

# References

1.  O. Bruning, et al., "LHC Design Report v.1: the LHC Main Ring",
    CERN-2004-003-V1, 2004.

2.  S. Caspi and P. Ferracin "Limits of Nb~3~Sn Accelerator Magnets",
    *PAC 05*, <http://snsapp1.sns.ornl.gov/pls/pac05/toc.htm>, paper
    TOAA, 2005.

3.  R. Meuser, "Magnetic field for a thick cos-n winding", Engineering
    Note M5254, Lawrence Berkeley National Lab., October 1978

4.  L.T. Summers, et al., *IEEE Trans. Magnetics*. 27 (1991) 2041.

5.  R. Meuser, "Stress in thick cos-n winding", Engineering Note M5256,
    Lawrence Berkeley National Lab., November 1978.

[^1]: ^^Manuscript received September 15, 2005. This work was supported
    under contract DE-AC02-05CH11231 by the Director, Office of Energy
    Research, Office of High Energy Physics, U.S. Depart­ment of Energy.

    S. Caspi, P. Ferracin, and S. Gourlay are with Lawrence Berkeley
    National Laboratory, Berkeley, CA 94720 (phone: 001 510 486 7244;
    fax: 001 510 486 5310; e-mail:
    [s_caspi@lbl.gov](mailto:pferracin@lbl.gov)).