Electromagnetic forces in high field magnet coils James Caldwell

Department of Mathematics and Computer Studies, Sunderland Polytechnic, Sunderland SR 13SD. UK (Received October 1981; revised February 1982)

There has been considerable study in recent years of the mechanical effects of the electromagnetic forces in large high field magnet coils. The problem of supporting the electromagnetic forces acting on the windings has become a limiting factor in the design of such magnets. Caldwell has already obtained a mathematical model representing the stress distribution in magnet windings but a number of simplifying assumptions were made in deriving the equations. This paper checks the accuracy of this method by comparing it with homogeneous thick cylinder theory which involves the calculation of stresses by solving the Timoshenko stress equations. Thus values of the circumferential stress are compared for two different coil configurations and reasonable agreement is obtained.

Key words: electromagnetic forces, magnet windings, Timoshenko stress equations

Introduction The electromagnetic forces generating the stresses in super- conducting magnets can be large, particularly in the case of high magnetic fields. It is important that the winding should be mechanically designed to accommodate the stresses, principally the circumferential or hoop stress.

In stress calculations for high magnetic field coils, unlike most other authors, Middleton and Trowbridge’ include axial stresses and in some cases the effects of insulation. The method of including insulation, however, involves an iterative process which is further complicated when external cylinders of finite elastic modulus are used for reinforcement. Gersdorf ef a1.3 have included both insula- tion and axial stresses in the initial formulation of the problem, with the result that no iterations are required and the final equations are easy to program.

Melville and Mattocks4 have devised a computer program which enables calculations to be made of all components of the stress in uniform current density coils used for produc- ing intense magnetic fields. The method of Gersdorf et a1.3 has been adopted and is discussed with particular reference to copper wire coils, but the technique is appropriate to all types of solenoids including superconductors. The effect of the elastic properties of the insulation is taken into account and shown to be important.

Mulhall and Prothero’have indicated that this work is incomplete in two respects. The analysis of stress in a coil may conveniently be regarded as two problems: one is the analysis of stress in an equivalent homogeneous but aniso- tropic toroid; while the other is that of relating the elastic constants of the equivalent homogeneous material to those

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of the components of the coil. In the previous work shear stresses have been neglected in the stress analysis, but with- out a demonstration that this is valid. Mulhall and Prothero’ have concluded that for coils typical of those in the larger- scale applications of superconductors, many of the simplifi- cations used in earlier work are in fact justifiable.

More recently the boundary integral equation method has been used in stress analysis. Details of this work can be found in Cruse6 and relevant papers in recent COMPUMAG conferences.

In the design of magnets it is often desirable to perform preliminary calculations on a relatively large number of alternative geometries and for this purpose a simple mathe- matical model representing the stress distribution in super- conducting magnet windings has been presented by Caldwell.’ In deriving the equations various simplifying assumptions have been made. For example, it was assumed throughout the calculations that the winding can be con- sidered as a solid body. No check on the accuracy of this method was given by Caldwell’ and so this paper aims to compare results with those obtained from a more accurate calculation of the stresses by solving the Timoshenko stress equations. This is done by calculating the circumferential stress for two different coil configurations by both methods.

Homogeneous thick cylinder theory The electromagnetic body forces for an assembly of solenoids are calculated by a computer program developed by Culwick’ which is available from Brookhaven National Laboratory. This program divides each solenoid into a

Appt. Math. Modelling, 1982, Vol 6, June 157

High field magnet coils: J. Caldwell

number of uniformly spaced filamentary conductors. Each filament is then repeatedly subdivided by 4 until the sum of the fields of the subdivisions is equal to the field of the filament within a prescribed tolerance. The forces are then calculated from the equations:

R =B,J (1)

Z=B,J (2)

where J is the current density and R, Z are thd radial and axial components of force per unit volume and B,, B, are the corresponding field components.

The equations relating the stress components and strains for axially symmetric systems when shear is neglected are:3

dr r

d*z dz +Z(ryz)=o

(3)

(4)

Ee, = a’, - vae - VU, (5)

EeO = - vu, f ue - vu, (6) Ee, = - vu, - vag + a, (7)

where a,. Q and o, are the radial, circumferential and axial stresses respectively and e,, ee and e, the corresponding strains. E is Young’s modulus and v is Poisson’s ratio for the material.

By introducing:

du U

er =G ee =- (8) r

where u is the radial displacement, it is possible to solve equations (3)-(7) and thus find an expression for the circumferential stress oe. This has been done by Middleton and Trowbridge’but, since they only outline the method without including a detailed derivation of the equations, this has been included below.

On eliminating u from equation (8) we obtain:

From equations (5) and (6) we have:

E(e, - ee) = (1 + ~)(a, - a~) (10)

On differentiating equation (6) with respect to r we obtain:

E dee do,. doe ---z-v -+--vdol dr dr dr dr

and using this equation with equation (9) gives:

(11)

dee Er---pE(el-ee)=r

doe do, do, --g-V-g-VT

From equations (3) and (10) we have:

(1 + war - *e) =;(er _ ee) r

doe do, do, =-- v--v- dr dr dr

=-(l+v)(?+R)

and hence:

(12)

On integrating equation (12) with respect to r we obtain:

a,g + CJ, = (Del - VU& + va,(r, Z) - (1 + V) s

R dr

rl

where oer and a,, are the circumferential and axial stresses at the inner coil radius r = rl for some z plane. The radial stresses at the inner (r = rr) and outer radii (r = r2) are taken as zero in this case.

Using the notation:

r s r”R dr = I,

the sum of the circumferential and radial stresses becomes:

(00 + Ur)=(Uel-VUzl)+ W-PI0

wherep = 1 + v.

(13)

Rewriting equation (3) as:

da, *r -ae=-rR-r-

dr

and adding to equation (13) yields:

2a,.=(ue,-vo,3+vu,-pIo-rR-r?

which, on rearranging and multiplying by r, gives:

$ (r2a,) = (a01 - vu,,) + vra, - pIor - r2R

Integration with respect to r yields:

r’a, = $(aet - vu,r)(r’ - r:)

+vfruzdr-p]rIodr-I2

rl f-1

The third term on the right-hand-side can be integrated by parts to give:

r

I rI,(r) dr =G I, - $I,

rl

and hence the radial stress is given by:

-~(ue’-vaz’)(l-~)-;l,--:ijl-~)I, Or -

r +v

r2 s ra, dr

Note that if the axial stress a, is constant, then:

r V

-

r2 s

6 ro,&=T I-- ( 1 r2

r1

158 Appl. Math. Modelling, 1982, Vol 6, June

High field magnet coils: J. Caldwell

and therefore:

Clerarly the axial stress u, is obtained by direct integration of equation (4) and uel is obtained from equation (14) by inserting the condition that u,. = 0 at r = r2.

Hence the sequence of computation is as follows:

(1) a, from equation (4) (2) uer from equation (14) with r = r2 (3) ur from equation (14) (4) (se from equation (13) (5) e,, ee, e, from equations (5), (6) and (7) (6) u from equation (8).

From equation (13) the circumferential stress is given by:

Ue=Ue(r,)-uU,(r,)-U,+vU,-(l+U)

r2 X

s R dr (1%

rl and this expression, together with the other stress and strain components, is then evaluated using a computer program.

Comparison of methods and conclusions

By making a number of simplifying assumptions Caldwell’ obtained the following three equations for the circumferen- tial stress:

J{BI(r~+rlr2-2r~+B2(2r~-r~r2-r~)I 00 =

6r h (r&-J (16)

WI + B2)(4 - rT> Ue =

4r h (r2lrd (17)

00 = &WI + B2Wl + r2> (18)

where B, = II and B2 = B(r2). In order to assess the accuracy of these equations values

of the circumferential stress oe (which is the dominant one) have been calculated for each of the equations (16), (17) and (18) and compared with the values obtained by the more rigorous method (equation 15) for two different coil configurations.

Coil 1 was used in the prototype superconducting motor developed at international Research and Development Co. Ltd, and later tested by driving a cooling water pump at Fawley Power Station and had the following parameters:

Inner radius rl = 1.11 m Outer radius r2 = 1.29 m Winding length 26 = 0.55 m Total Ampere turns NI=3x10dA

For equations (16), (17) and (1 S), the values of Br and B2 were obtained from the BNL program.2 The circumferen- tial stresses obtained including those from equation (15) are plotted in Figure 1.

The second coil magnet was for the 7.0 Wb/m’bubble chamber at the Rutherford High Energy Laboratory. The solution of equation (15) for this coil is given by Middle- ton and Trowbridge’ for a particular value of z and the variation of B,(r) with radius is also plotted. Values of B,

15”

1 Colt lnslde radius

Coil outside, mdius

1.11 1.14 1.17 1.20 r,(m)

1.23 1.26 1.2

Figure 1 Stresses for Coil 1 using equations (15)-(18)

8-

6-

4-

2-

O- 0.9

I I I 1 f-

Co11 inslde Coil outside I

radius radtus

I I I I

,

I I

I

(18)

1.2 1.5 1.U r,(m)

Figure 2 Stresses for Coil 2 using equations (15)-(18)

and B2 were obtained from this plot. Coil 2 is a Helmholtz pair with the following parameters:

Inner radius 0.95 m Outer radius 1.70 m Total winding length (including gap) 2.30 m Length of gap between coils 0.15 m Ampere turns (per coil) 1.02x 10’A

The circumferential stresses for this coil obtained by using equations (15), (16), (17) and (18) are plotted in Figure 2.

Appl. Math. Modelling, 1982, Vol 6, June 159

High field magnet coils: J. Caldwell

On comparing the results we fiid that in both cases the discrepancy between equations (16) and (15), though significant, is not large and that for both coils equation (17) gives the highest value of stress. At r = rI, where the peak stress occurs, equations (16) and (15) are in close agreement for Coil 1 but the agreement is not particularly good for Coil 2.

It appears from the limited comparison made above that the thick/thin cylinder approach of equation (16) gives results close enough to the more complex theory of equa- tion (15) to justify its use for preliminary calculations of coil stresses, particularly as the applicability of solid cylinder theory at all has not been conclusively demon- strated and any results must be considered as approximate. In addition, the simple expression given in equation (18) seems to give a good first approximation to the peak stress for Coil 1. The intermediate approximation of equation (17) is apparently of little value except that when used it seems to err on the high side which is useful information. None of the equations (16), (17) or (18) can, however, be a complete

substitute for the full theory, if only because they give no information about the radial and axial stresses.

The comparison has been carried out for two particular coil geometries which have been used in practice. However, comparisons made for other theoretical coil geometries provide similar agreement and this effectively gives us more confidence in the simplified formulae.

References 1 Caldwell, J. Appl. Mat/l. Mod. 1982,6, 61 2 Middleton, A. J. and Trowbridge, C. W. Prod. 2tjd IN. Conk

Magrzet Teclzrtol., Oxford, 1967, p. 140 3 Gersdorf, R. et al. Rev. Sci. Itlstrurn. 1965, 36, 1100 4 Melville, D. and Matticks, P. G. J. P!I)Q. D. 1972, 5, 1745 5 Mulhall, B. E. and Prothero, D. H. J. PhJU. D. 1973,6, 1973 6 Cruse, T. A.J. Comput. Struct. 1973, 3, 509 7 Culwick, B. B. Brookhaven National Lab. Rep. No. HH05-0,

BBC-l, 1965 (unpublished) 8 Timoshenko, S. and Goodier, J. N. ‘Theory of elasticity’,

McGraw-Hill, London, 1951

160 Appl. Math. Modelling, 1982, Vol 6, June