20-T, 120-cm-I.R. Target Magnet with Large Axial Gaps at 3 m & 9 m
Bob Weggel, Magnet Optimization Research Engineering, LLC
12/31/2011
This report extends one entitled "20-T, 120-cm-I.R. Target Magnet with Layer-Wound Resistive
Magnet," attached as "Layer120cm3%pdf" (≡ "Layer120cm3pc.pdf") in my e-mail of 1:40 AM on
5/11/2011. That report designed 20-tesla target magnets optimized by a computer program with major
upgrades to its predictions of superconductor current density, magnet stresses and strains, and costoptimization parameters. The field-and-temperature dependence of the non-copper current density in
the Nb3Sn in the strands of the superconducting cable is as shown in Fig. 1, generated by Eq. (1):
𝑗(𝐵, 𝑇) ≅
46,630
(1 − 𝑡1.247 )2 𝑏 0.437 (1 − 𝑏)1.727
𝐵
[A/mm2].
(1)
The magnetic flux density 𝐵 is in teslas; 𝑡 and 𝑏 are, respectively, the normalized temperature 𝑇/𝑇𝑐 and
normalized magnetic flux density 𝐵⁄𝐵𝑐 (𝑇). Equation (1) resembles that of “A general scaling relation for
the critical current density in Nb3Sn,” by A Godecke, B ten Haken, H H J ten Kate and D C Larbalestier
(2006 Supercond. Sci. Technol. 19 R100 doi: 10.1088/0953-2048/19/10/R02), but gives a much closer fit
to the 𝑗(𝐵, 𝑇) data for Nb3Sn of the ITER barrel magnet tabulated on page 645 of Case Studies in
Superconducting Magnets, by Y. Iwasa. Fig. 2 plots the parameter 𝐵𝑐 (𝑇) = 20.8 − 1.27 𝑇 − 0.0234 𝑇 2
needed by Eq. (1). Data points from which to generate the curve fit of Fig. 2 came from 𝑇𝑐 = 18.2 𝐾 and
the 𝑗(𝐵|𝑇) curves of Fig. 3a-b, for which, by extrapolation, 𝑗(𝐵|𝑇) = 0 at [28.8 T, 1.8 K], [24.5 T, 4.2 K],
and [16.1 T, 10 K].
Field and Temperature Dependence of Current Density of Nb3Sn Strands
2000
2
Nb3Sn-strand non-copper current density [A/mm ]
1600
1200
1000
800
600
10 K
9K
8K
7K
6K
5K
4.2 K
500
400
300
200
5
6
7
8
9
10
11
12
13
14
15
16
Ambient field [T]
Bob Weggel 5/9/2011
Fig. 1: Non-copper current density vs. field and temperature for Nb 3Sn strands of ITER barrel magnet.
Temperature Dependence of Critical Field, Bc, of Nb3Sn
30
27
24
2
Bc = 30.8 - 1.27 T - 0.0234 T
Critical field, Bc [T]
21
18
15
12
9
6
3
0
0
2
4
6
8
10
12
14
16
Temperature, T [K]
18
Bob Weggel 5/9/2011
Fig. 2: Curve fit of data generated by Fig. 3a-b. 𝐵𝑐 (𝑇) = 20.8 − 1.27 𝑇 − 0.0234 𝑇 2 teslas.
Non-Copper Current Density in Nb3Sn Strands of ITER Cable-in-Conduit Conductors
Non-Copper Current Density in Nb3Sn Strands of ITER Cable-in-Conduit Conductors
55
2000
50
1800
45
2
Non-copper current density [A/mm ]
1600
40
½
[A/mm ]
2½
1400
1200
Current density
½
1.8 K (internal tin)
1000
4.2 K (internal tin)
800
4.2 K (ITER)
600
j = 3.23 (28.8 - B)
35
½
j = 4.10 (25.1 - B)
30
½
j = 2.18 (25.1 - B)
25
½
j = 2.10 (24.5 - B)
20
½
j = 2.34 (16.1 - B)
15
4.2 K (ITER barrel)
400
10
10 K (ITER barrel)
200
0
5
0
8
10
12
14
16
18
20
22
24
26
28
8
9
10
11
12
13
14
15
16
17
18
19
20
Ambient field [T]
Ambient field [T]
21
22
23
24
25
26
27
28
29
Bob Weggel 5/9/2011
Bob Weggel 5/9/2011
Fig. 3a-b: Curve fits of 𝑗(𝐵|𝑇) by 𝑗(𝐵)~(𝐵𝑐 − 𝐵)2 . Left: 𝑗(𝐵|𝑇). Right: √𝑗(𝐵|𝑇). For ITER barrel conductor,
extrapolation of the red curve to 𝑗 = 0 gives 𝐵𝑐 = 16.1 T at 10 K; extrapolation of the green curve gives 𝐵𝑐 = 24.5 T
at 4.2 K. For internal-tin conductor, 𝐵𝑐 = 25.1 T at 4.2 K (blue curve) and 28.8 T at 1.8 K (black curve).
To predict the maximum stress (at the inner radius) in each solenoid, the computer program uses Eq.
(5.35b) on p. 124 of Solenoid Magnet Design by Montgomery & Weggel or, equivalently, Eq. (3.77b) on
p. 101 of Case Studies in Superconducting Magnets. The solenoid is of inner radius a1, outer radius a2,
current density j, bore field B1, and external field B2, and is of isotropic material of Poisson’s ratio ν = 0.3.
The predicted stress is:
𝜎𝑚𝑎𝑥 = 𝑗
(7𝑎12 + 14𝑎1 𝑎2 + 85𝑎22 )(𝐵1 + 𝐵2 ) + 14(𝑎12 𝐵1 + 𝑎22 𝐵2 )
.
120(𝑎1 + 𝑎2 )
Fig. 4 presents the results. Note that for a radially-thin solenoid (radius ratio α ≡ a2/a1 ≈ 1), the peak
stress is σmax = j1 a1 <B>, where <B> = (B1+B2)/2, the average field in the windings. In a solenoid of radius
ratio α = 1.6 and field ratio β = 0 (appropriate for the most-upstream superconducting coil of a 20-T
target magnet), the normalized stress σ* ≡ σmax / (j1 a1 B1) is 0.85. For a solenoid of radius ratio α = 2.8
and field ratio β = 0.7 (appropriate for a pancake-wound resistive magnet of O.R. = 50 cm and I.R. = 17.5
cm) the normalized stress σ* is 2.85.
Normalized Maximum Hoop Stress *  max./(B1 j1 a1)
1.0
1.0
1.5
2.0
2.5
3.0
3.5
0.9
0.8
Field ratio   B2/B1
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
1.0
1.2
1.4
1.6
1.8
2.0
2.2
2.4
2.6
2.8
3.0
Radius ratio   O.D./I.D.
Bob Weggel 5/8/2011
Fig. 4: Normalized maximum hoop stress σ* ≡ σmax./(B1 j1 a1) vs. radius ratio α ≡ O.R./I.R. & field ratio β ≡ B2/B1.
Fig. 5a-d, generated by a finite-element-method program, confirms that the magnet-optimization
program does indeed generate designs in which the peak strain in every coil is very nearly the ~0.4%
that should be acceptable for all the magnet materials: copper and stainless steel for the resistive coils,
and Nb3Sn, copper stabilizer and Incoloy 908 or other conduit material for the superconducting coils.
Fig. 5a-d: Hoop strain εhoop (color & contour lines) of 20-T target magnet with layer-wound resistive coils and
upstream superconducting (SC) coils of 120-cm inner radius. Total stored energy = 2.88 GJ. Field homogeneity =
3.3% peak-to-peak. Resistive magnet has five nested two-layer coils of MgO-insulated hollow conductor, graded
from 23.8 mm square Japanese-Hadron-Facility hollow conductor (inner coil) to ~35 mm square (outer two coils).
Supporting each coil is a tube or wrap of 600 MPa design stress and 200 GPa Young's modulus. Field contribution =
5.3 T at 12 MW; ∆Tmax = 70 oC with water flow of 59 liters/sec (∆P = 40 atm, 4 hydraulic paths per layer). The mostupstream SC coil has an O.R. of 193 cm, a length of 3.3 meters, a weight of 130 metric tons, and is 9% Nb3Sn, 51%
steel, 32% (Cu+He) & 8% insulation. The endmost coil shown is 1.4% SC, 4.5% steel, 75% (Cu+He) & 19% insulation.
a) Entire magnet upstream of 9 m. b) Resistive magnet. c) Upstream SC coils. d) Downstream SC coils.
To accommodate support structure and the many vessel walls, radiation shields and vacuum spaces of
cryostats, the magnet design above needs much larger axial gaps between its cryostats, each of which
should be not much longer than 6 meters if the vessel containing its shielding is not to sag excessively
with only cantilever support. Fig. 6 shows the cross section of coils in a system with an intercoil axial gap
of 50 cm near z = 3 m and 35 cm near z = 9 m. Fig. 7 shows that its on-axis field profile matches very
closely the desired one.
The design of the magnet of Fig. 6 optimized the winding depth of every coil, but not the endplane
locations of the superconducting coils downstream of z = 3 meters. The magnets of Figs. 8 through 15
adjust all magnet lengths and intercoil gaps--other than those at z = 3 meters and 9 meters--to minimize
a weighted sum of resistive-magnet power consumption, superconductor cost, and accuracy of on-axis
field profile. For Figs. 8 & 9 the respective axial gaps are 60 cm and 45. For Figs. 10 & 11, the gaps are 80
cm and 60 cm. For Figs. 12-5, the gaps are 100 cm and 80 cm. Note that even with axial gaps this large,
the on-axis field profile approximates quite closely the desired profile.
Fig. 6a-b: Resistive & upstream 12 SC coils of Magnet IDS120i3m with intercoil axial gap of 50 cm flanking z = 3 m
and 35 cm at z = 9 m. Left: Coil cross sections. Right: Field magnitude (color & contours) and direction (arrows).
On-Axis Field Profile of Target Magnet "IDS120i"
20
Total field
Desired field
SC magnet
On-axis field [T]
10
Resistive magnet
5
2
Field error 
3
2
5x10 (B) /B
1
0.5
-450
-300
-150 -75
0
150
300
450
600
750
Distance along axis [cm]
900
1050
1200
Bob Weggel
1350
1500
12/28/2011
Fig. 7: On-axis field profiles of superconducting, resistive and combined magnet of Fig. 6.
Fig. 8a-b: Resistive & upstream 12 SC coils of Magnet IDS120i3m' with axial gaps of 60 cm & 45 cm at 3 m & 9 m;
all coil ends optimized. Left: Coil cross sections. Right: Field magnitude (color & contours) & direction (arrows).
On-Axis FIeld Profile of Target Magnet IDS120i3m
20
Total field
15
Desired field
SC magnet
10
Cu magnet
On-axis field [T]
5
2
1.5
1
.5
Field error 
4
2
10 (B) /B
.2
0.1
-450
-300
-150
0
150
300
450
600
750
900
1050
1200
1350
1500
Distance along axis [cm]
Fig. 9: On-axis field profiles of superconducting, resistive and combined magnet of Fig. 8.
Fig. 10a-b: Resistive & upstream 12 SC coils of Magnet IDS120i3m` with axial gaps of 80 cm & 60 cm at 3 m & 9 m;
coil ends optimized. Left: Coil cross sections. Right: Field magnitude (color & contours) & direction (arrows).
On-Axis FIeld Profile of Target Magnet IDS120i3m`
20
Total field
15
Desired field
SC magnet
10
Cu magnet
On-axis field [T]
5
2
1.5
1
.5
Field error 
4
2
10 (B) /B
.2
0.1
-450
-300
-150
0
150
300
450
600
750
900
1050
1200
1350
1500
Distance along axis [cm]
Fig. 11: On-axis field profiles of superconducting, resistive and combined magnet of Fig.10.
Fig. 12a-b: Resistive & upstream 12 SC coils of Magnet IDS120i3m~; axial gaps of 100 cm & 80 cm at 3 m & 9 m;
coil ends optimized. Left: Coil cross sections. Right: Field magnitude (color & contours) & direction (arrows).
On-Axis FIeld Profile of Target Magnet IDS120i3m~
20
Total field
15
Desired field
SC magnet
10
Cu magnet
On-axis field [T]
5
2
1.5
Field error 
4
2
10 (B) /B
1
.5
.2
0.1
-450
-300
-150
0
150
300
450
600
750
900
1050
1200
1350
1500
Distance along axis [cm]
Fig. 13: On-axis field profiles of superconducting, resistive and combined magnet of Fig.12.
Fig. 14: Field magnitude (color & contours) & direction (arrows) & coil cross-sections of resistive & upstream 12 SC
coils (SC #2 has zero cross section) of Magnet IDS120i3m#; axial gaps of 100 cm & 80 cm at z= 3 ms & 9 m; coil
ends optimized; outer diameters of SC coils #3 & #4 limited, to improve fabricability.
On-Axis FIeld Profile of Target Magnet IDS120i3m#
20
Total field
15
Desired field
SC magnet
10
Cu magnet
On-axis field [T]
5
2
1.5
1
.5
Field error 
3
2
5x10 (B) /B
.2
0.1
-450
-300
-150
0
150
300
450
600
750
900
1050
1200
1350
Distance along axis [cm]
Fig. 15: On-axis field profiles of superconducting, resistive and combined magnet of Fig.14.
1500