Some aspects of the design of superconducting solenoids

S o m e Aspects of the Design of S u p e r c o n d u c t i n g Solenoids M.

~/[7ood

• The Clarendon Laboratory, O.~ford

Received 2 Jtme 1962

DURING the nine months since Kropshot and Arp I wrote their review article in this journal, the development of superconducting solenoids has proceeded fast. The highest field produced in a superconducting magnet has been increased from 15 kgauss, in a solenoid constructed by Kunzler 2 (~ in. inner diameter, molybdenum-rhenium at 4"2°K) to 69 kgauss, also by Kunzler 3 (¼ in. inner diameter, Nb3Sn at 1.5 ° K). The highest field in which an alloy had remained superconducting at the time of their paper was 88 kgauss (Nb3Sn). This has been raised to 180 kgauss (Nb3Sn) and, by extrapolation, a critical field in excess of 500 kgauss has been predicted for VaGa. The prophecy that 'small size, high field, superconducting magnets will undoubtedly find their place as general laboratory equipment' looks like being fulfilled fairly soon. Many magnets have been designed, and some have been built and tested in various laboratories throughout the world, and there are now commercial concerns which undertake the design and manufacture of these devices. There is a possibility that high fields may be produced by promoting very high circulating currents in cast hollow cylinders of superconducting material. At the moment, however, most practical solenoids are being made by the conventional method of winding wire into coils, and Nb3Zr is the material most commonly used. 4 The limitations of NbaZr are that it has a low critical field in comparison with Nb3Sn and V3Ga, and that it appears to require very severe cold work before its optimum high field superconducting properties emerge. The need to draw it into wire to produce these properties not only raises its cost, but results in windings which, like all wire wound coils, have space factors in the range of 60-90 per Cent, compared with the 100 per cent applicable to the cast hollow cylinders. On the other hand, by i'educing the conductor size to fine wire, and consequently reducing the magnitude of the current it carries to the order of 10 or 20 A, the problem of getting the current from room temperature into the magnet at 4.2 ° K is made fairly simple. Nb3Zr wire is also a simple material to handle, and presents no great problems in winding. It is very desirable to minimize the quantity of conductor needed to produce any required field, not only because it is expensive and cannot be obtained in long CRYOGENICS

•

SEPTEMBER

1962

lengths, and joints present difficulties, but also because it is wasteful in terms of liquid helium to use any unnecessary material. More liquid helium is required to cool down a large magnet, and evaporation losses are greater from the larger vessel in which the magnet is placed. The advantage of using a minimum weight of conductor assumes a position in the design of superconducting windings comparable with that of the minimization of the power required for producing a magnetic field in the case of resistive windings. Factors concerning the overall size, shape, and weight, the temperature of operation, the mechanical strength, and the period of time for which the field is required all bear on the design of both types of magnet, and in particular instances one or other may dominate the situation. However, it always remains desirable to minimize the quantity of wire needed on .the one hand, and the power needed on the other. In 1898 Fabry 5 deduced the relation between the axial field H produced at the centre of a solenoid, the power required W, the space factor A (conductor volume/total winding volume), the resistivity p, and the inner radius a I.

~! Pal

...(1)

where G is a geometrical factor depending on the shape of the winding and the distribution of current density within it. Shape is defined in terms of the ratios ~ and fl, where = a2/a I and fl = b/a I as shown in Figure 1. For constant current density windings

Throughout this article, practical units are used for electrical quantities, and c.g.s, units for all others. In the case of superconducting windings, where no power is involved, it is convenient to write equation (1) in terms of the current density i W = pi 2 V

where

V = 21rAb(a~-a~)

is the volume occupied by the conductor. 297

Then

H = Gi~/2zrb(a~-a~) ,q at

. . . (2)

= KiAa~

where

K =

=

cJ

v

~

4rr .

/3,og,

~ + ~/(/32 + ec2)

and is a function of ~ and/3 only. The volume occupied by the conductor may be written as

... (3)

v = Caa where

C = 2rr/3(~2 - I)

and is also a function of 0~and/3 only. i~

20

'1 O~= 02/01

I?

js = b/a~

l Figure 1

Now to minimize the quantity of wire required, it is necessary first to make A and i as large, and a I as small as possible. These values substituted in equation (2) give the minimum value of K which will yield the required field H. Values of ~ and./3 must then be chosen which give the minimum value of C compatible with this minimum value of K. This can be done quite simply by reference to the curves of constants C and K plotted against ~ and/3, as in Figure 2. The two equations

to their limits. High coolant velocities can keep )~ large; experiments are contracted into small volumes to keep a, small; p can be reduced far below the figure for copper at room temperatures by cooling the conductors in liquid nitrogen, hydrogen, or helium, and by using ultra-pure metals with low magnetoresistance, like sodium and aluminium; G can be increased to the calculable optima set for various coils, shapes, and current density distributions. In the case of H = K ) t i a , , definite limitations apply to )t, i, and at, and only K can be increased indefinitely, subject to considerations of the size and weight of the winding, the liquid helium required to cool it, and the availability and cost of the conductors. It is interesting to note that of the two geometrical factors, G always has an optimum value, and this is usually with quite small values of ~ and r , whereas K always increases for increasing 0~and fl values. The fact that the space factor Ais outside the square root sign in the relationship for superconducting windings, makes it particularly important to use tightly packed windings with a minimum of insulation. The variation of ,~ with different thicknesses of insulation of wires ranging from 0.127 mm to 0.508 mm (0.005 in. to 0-020 in.) in diameter is given in Table 1. The maximum value of i is set by the properties of the conductor material, and by the margin which must be allowed between the usual operating value, and the value at which the conductor reverts to the normal resistive state. The relationship Hoc at is misleading in that it suggests that H should be increased by increasing av If at is increased, and the shape is maintained, i.e. K remains constant, the volume of the winding will increase in proportion with a 3, and this is clearly a very uneconomical way of raising the field, a I must be kept to a minimum, and K increased to give the required field.

¢ 20 100 200 400 600 10 SO 150 300 500

n = Gx/[(W~)/(pa,)] for resistive windings, and

Isi LI

\

H = KAia t

for superconducting coils, contain the important parameters with which the magnet designer can play, and their comparison is interesting. In the case of the first of these two, there are known definite limitations on the possible values of G, A, a~, and p, and only Whas less well defined limits, set by conditions of heat transfer, mechanical stresses, and, more often, the capacity of the power supply and the cooling installation. The desirability of keeping the power requirement low is indicated by the way in which other parameters are driven 298

i" "

•

2.0 1.5

1

2

3

4

5

Figure 2. C and K factor curvesfor magnet coils with uniform current density and of rectangular cross-section. C is shown by broken lines, and K is shown by unbroken lines CRYOGENICS

• SEPTEMBER

1962

Table 1. Theoretical M a x i m u m Values o f the Space F a c t o r

for Various Wires Sizes and Thicknesses of Insulation.

Values in Actual Windings are Usually a Few per Cent Lower

Insulation thickness on radias (ram)

Wire diameter (mm)

0"508

0-254

0-204

0"127

0"025 0"019 0-013

0-70 0-79 0.82

0.63 0-69 0"75

0"59 0'65 0-71

0-46 0"54 0.63

0-006

~ 0'86

0"83

0.81

0-76

0-000

0"91

0-91

0"91

0'91

By way of example we may take the case of a magnet which has a clear inner radius of 1.5 cm and which is required to produce 60 kgauss. If the former on which the coil is w o u n d is 1 m m thick, a I = 1-6 cm. If we use Nb3Zr wire 0.25 m m (0.01 in.) in diameter with 0.013 m m (0.0005 in.) o f insulation, ~ 2 0.72. If we assume we can run the coil at 95 per cent o f the critical current density specified by the suppliers, i = 3 . 6 x 104 A/cm 2. Then H = 3.6 x 104 x 0.72 x 1.6 x K = 60 kgauss, and K = 1.45. F r o m Figure 2 we can see that the values o f ~ and/3 which correspond to K = 1.45 and which give a minimum value o f C = 67, are ~ = 2-74 and/3 = 1-64. This means that the outer diameter o f the coil will be 8.76 cm and the length 5-25 cm. The volume of material composing the wire, after equation (I), will be 67 x 0.72 x 1.63 = 197 cm 3. This corresponds to 3,900 m o f wire weighing 1.57 kg. The curves o f Figure 2 can also be used in the reverse direction. If we have a 1,000 m length of the same wire, and wish to wind a general purpose solenoid, we might consider winding it on a 1 cm inner diameter tube. Then a I = 6 mm, C = 326, K = 3.1, and thus H = 48 kgauss. If, however, we want a clear inner diameter of 5 cm, a~ = 2.6 cm, C = 4, K = 0.25, and H falls to 17 kgauss. W h e n the magnetic field required is specified not as a m a x i m u m at the geometrical centre but as an average over a central volume, with a m a x i m u m allowable variation, the problem o f deciding on the best shape for the winding is more complicated. However, the fact that no flow o f coolant is required past the conductors makes it simpler to construct a superconducting winding than a resistive one in the shapes suggested by calculations for producing high homogeneity fields. The simplest way o f calculating the field at any point near the centre o f a solenoid was suggested by Garrett 6 who expressed the field as a power series using Legendre polynomials expanded a b o u t the origin. Garrett's work has been extended by Montgomery. 7 F o r a symmetrical system with its origin at the geometrical centre o f the coil, there will be no odd terms in the expansion o f the series, and the problem o f producing a h o m o g e n e o u s field is to CRYOGENICS

• SEPTEMBER 1962

add or subtract such windings as will cancel out as m a n y o f the even terms as is wished. M o n t g o m e r y has published tables o f the coefficients for terms o f the expansion up to the 8th term, for coils o f various shapes, and further tables which assist in choosing dimensions for compensating windings. The extra quantity of wire required depends on the shape o f the initial basic coil, and on the degree o f homogeneity required. In general it is possible to make substantial improvements in homogeneity for only small percentage increases in wire. To investigate some o f the problems involved in winding and operating superconducting solenoids, we have constructed the solenoid shown in Figure 3. This coil has 9,500 turns, wound from a single length of Nb3Zr wire 0.25 m m in diameter coated with 0-02 mm of nylon, giving a space factor of 0.65. The clear inner diameter is 1.85 cm, the outer diameter is 6 cm, and the length is 3.9 cm. The manufacturer's specification of the wire states that it will remain superconducting at 4.2°K when carrying 19 A in a transverse field of 60 kgauss. Nineteen amperes in this solenoid would produce 47 kgauss, and so far it has been tested to just over 40 kgauss. We are in the process of a number of investigations with this magnet and

Figure 3. SolenoM wotmd w#h Nb3Zr wire 0"25 mm in diameter with nylon insulation. Clear inside diameter, 1"85 cm; outside diameter o f winding, 6 cm; length o f winding, 3.9 cm

299

others which are being constructed, and we shall report our findings at a later date. The author is grateful t o Dr. N. Kurti for the help and advice he has given on many occasions. REFERENCES 1. KROPSnOT, R. H., and ARP, V. Cryogenics 2, 1 (1961) 2. KUNZLER,J. E., BUEHLER,E., Hsu, F. S. L., MATTHIAS,B. T., and WAHL, C. J. appl. Phys. 32, 325 (1961) 3. KUNZLER,J. E. Proc. 1st Internat. Conf. on High Magnetic Fields (Massachusetts Institute of Technology, November, 1961)

300

4. BERLINCOURT,T. G., HAKE, R. R., and LESLm,,D. H. Phys. Rev. Lett. 6, 671 (1961); KUNZLER,J. E., et al. Bull. Amer. phys. Soc. 6, 298 (1961) 5. FAaRY, C. l~clair, dlect. 17, 133 (1898) 6. GARRm, M. W. J. appl. Phys. 22, 9 (1951) 7. MONTGOMERY,D. B., and TERRELL, J. Rep. Nat. Mag. Lab., Massachusetts Institute of Technology AFOSR-I525

Note added in proof. Some aspects of this subject and other related problems have been treated by R. W. Boom and R. S. Livingston in Proc. Inst. Radio Engrs N. Y. 50 (No. 3) (March 1962)

CRYOGENICS - SEPTEMBER 1962