Very high field synchrotron magnets with iron yokes

NUCLEAR

INSTRUMENTS

AND

METHODS

I06

(1973) 595-604;

©

NORTH-HOLLAND

PUBLISHING

CO.

VERY H I G H F I E L D S Y N C H R O T R O N MAGNETS W I T H IRON YOKES J. H. C O U P L A N D ,

J. S I M K I N a n d T. C. R A N D L E

Rutherford High Energy Laboratory, Chilton, Didcot, Berkshire, England Received I1 September 1972 A variety o f dipole and two q u a d r u p o l e m a g n e t s have been studied using a computer. All the m a g n e t s have iron yokes in close association with the windings and conclusions are d r a w n a b o u t the error fields in the useful aperture arising from satura-

tion in the yoke. Shaping the iron enables the s a t u r a t i o n to be controlled, and m a g n e t s have been designed with errors small e n o u g h for use in proton s y n c h r o t r o n s with fields up to 6 T.

and Br can be expressed as 1. Introduction Even though the iron yoke of a high field magnet is ( ~ ) = B° L~sin F/cos ~) + C3 ( r ) 2 (c°s 3300) + \sin saturated it can still provide a significant saving in excitation coupled with a reduction in stored energy of the magnet. The yoke is also beneficial in that it reduces \sin 500)q-CT(a)6(\sin c°s the external leakage field and screens the useful aperture from stray external fields if not fully saturated. In choosing appropriate coil dimensions and current where a is the inner radius of the winding. densities, superconducting windings are assumed in Corresponding expressions for a quadrupole magnet these calculations. with gradient G are Only iron yokes in close proximity to the coils are considered as potentially they have the greatest gains more distant unsaturated shields are discussed elseB, I_\sin 2 0 + C6 \sin 6 0 + where 1'2) - but the savings in conductor and power supplied may be at the expense of errors in the magnetic ( r ) 8 (cos 1 0 0 " ~ [r\12(cos140'-] field pattern due to iron saturation. For magnets used +Clo \sin 100J + Cl"~a) \sin 1 4 0 ) + " ' J " at a fixed excitation the field errors can usually be corrected fairly easily, but for synchrotron magnets the The criteria applied to define a good field are that field has to be of high quality over the whole accelera- within the useful aperture the magnitude of any tion cycle, from low fields corresponding to particle harmonic present should be such that ABn/Bo < 0.001 injection to the maximum fields at ejection. Computer and the maximum field error ABy/Bo must also be less studies have therefore been made to find designs where than 0.001. the field errors remain small, AB/B < 10 - 3 at all field levels. 2. Dipole magnets This paper is concerned with magnets of constant section that are very long, or more strictly infinitely 2 . 1 . W I N D I N G ON A CYLINDER WITH CYLINDRICAL IRON long in the z direction. Figs. 1 and 10 show sections of dipole magnets and figs. 12 and 13 corresponding Three types of winding are compared; a two-sector 5) quadrupoles. Most of the computations have been done coil having constant current density, fig. li, a four-layer using a version of the Berkeley program TRIM 3'4) and coil 6) with constant current density, fig. lii, and a fourfurther details about this are given in the appendix region 7) coil where the current density is constant over together with notes on the harmonic analysis, stored a region but graded to approximate to a ' cos 0 distribuenergy and expected accuracy of the results. tion ', fig. liii. In each magnet there is a small radial gap In the following sections the magnetic fields in the between the windings and the yoke, supposedly to aperture are described by circular harmonics with allow for cooling and general engineering contingency normalised coefficients Cn. Hence for a dipole magnet at the coil iron interface. The coils extend from constructed with perfect symmetry about the principal r 1 = 5 cm to r 2 = 8 cm with the iron from r 3 = 9 cm to axes, the azimuthal and radial components of field, Bo r4 = 23.8 cm. The B-H curve used for the computa-

l

() __ rF(cos 0 ) (j ( os 60)

595

596

J.H.

COUPLAND

tions, see fig. 2a, is based on 'Losil' modified to 4°K, cf. the experimental values of Mclnturff8), and extended

et al.

pessimistically to high fields. It is also degraded to allow 4.C

~

SECTOR

o

2"~ < 2-z m

8''~

g

2.C D c~ z

.s

Hxl00 a ~b hx10

100

'~

1.2 ~:Z:7

"

-

0.8

~

0.~ - LAYER

0.0

I I 410 610 B40 1010 12'30 ~400 ,600 MAGMETISIHGFORCE,H~AMPERETURNSPER METRE

210

Fig. 2. B - H curves. (a) ' L o s i l ' - m o d i f i e d values; (b) Bsat further reduced to 1.65 T.

10.0 TWO-SEC T O R ~ rJ

L-

-4 r~

II -41

'3

!

J

iq-

--FOUR- REGION

% ~, 6,c u

~ D

4.0

@ <

2.0

o R

-

R

E

G

I

O

N

o,

q F O U R

0,0

4-0 Bo(T)

5-0

2-0 FOUR-REGION ~Z12t0 ~ 0"0

c 1"0

" 3-O & ~ . . ~ . ~ . .

LAYER I

6.0

FOUR-klAYER 6'0

(T}

r2 | Fig. 1. Sections of magnets (one quadrant only).

~-2.0 ~E

Fig. 3. Comparison of the three types of magnet with the same yoke and coil dimensions.

VERY HIGH

FIELD

SYNCHROTRON

TABLE 1 T h e magnetic induction, just inside the iron yoke at the pole a n d median plane, for each o f the three coil types. B0 = 3 . 7 T Coil type

Magnetic induction at the pole

Magnetic induction at the median plane

(T)

(T)

2.51 2.17 2.51

1.66 1.66 1.66

Two-sector Four-layer Four-region

for voidage in the iron structure which is necessarily laminated for pulse operation. Fig. 3 shows the magnitudes of C 3 and C 5 as functions of field level for the different coil geometries. The higher order harmonics are not shown as they are hardly affected by saturation in the yoke and have a negligible effect on the field in the useful aperture r < 0.8a. The non-zero values for C3 and C5 at low fields are due to 4.0 IRO/~R(

2-0

0"0

c

2.

t

~ B o%, ~ ! O ! 3 " O

N r3=9.0cm, r& :23.8cm.

6!0

2"0 ON r3=9-0cm , r4 = 21-0cm. -~.0

MAGNETS

597

the particular angular boundaries chosen for the computation and are not significant since they can be made zero by small alteration in the coil boundaries, see the next section. From the graphs it is seen that the changes in harmonics are similar but that the four-layer winding shows least variation over the range 0 to 5T. The differences in behaviour between the three designs can be understood from the flux values at the inner boundary of the iron (see table 1), which also indicates that saturation first occurs in the polar region causing large positive changes in C a with a much smaller negative change in C5, (see also fig. 3). As the excitation is increased the return flux causes saturation in the median plane which is reflected as a negative change in C3. For all the coil systems the onset of this depends inversely with iron thickness, see fig. 4, which refers to the four layer case. Below saturation the peak flux at the inner surface of the iron can be calculated from analytic formulae which assume that the permeability p for the iron is constant. They are reasonably correct if ( p - 1 ) / ( p + 1) approximates to unity and confirm the general character of the computed results as saturation begins to take effect. In the saturated condition the computed values indicate that the return flux distributes itself nearly uniformly across the iron thickness. It has been observed by Parzen and Jellett 9) at Brookhaven that the field quality can be improved by choosing a particular thickness for the iron and from these computations one can see the cancelling effect of the changes in C 3. The cancellation is not perfect, but clearly from fig. 4 a considerable improvement in field quality for B o up to 4 T is obtained by choosing the iron thickness as 12 cm in this case.

~ -6.0

2.2. SHAPING OF THE IRON SECTION 8"0

r3= 9.0cm , r4 = 18.0cm -10.0

-12.0

%

1~.o ~

z-2.0

2!o

~ , ~ o ~ I R . A 3.0

ON r3= 9"0 crn, rt, =23.8cm I I 5.o 6.0 IRON r3=9.0cm,r4 = 21"0cm. IRON r 3 = 9-0 cm, r4= 18-0 cm

~E

Fig. 4. H a r m o n i c s f r o m four-layer coil with various iron thicknesses.

There are two ways of reducing the field errors caused by iron saturation. In the first one can try and arrange that the pattern of saturation in the iron is 'cos 0 like' so as to generate a more nearly uniform perturbation to the field, albeit a negative one since the saturation is associated with a reverse azimuthal component of field intensity H o around the inner iron boundary. Such an improvement can be achieved by azimuthal variation in the thickness of the iron yoke. The effect of a smooth cut out in the outer surface of the iron is shown in fig. 5 for different depths of this cut out. This graph, which shows the maximum change in Ca and C5, or spread, taken over the excitation range 0 < B 0 < 4 . 5 T indicates that these changes can be reduced by choosing an optimum depth. Cut outs at different azimuthal angle, q~, show different proportions in their effect on Ca and C5. Thicker shields

598

J . H . COUPLAND et al.

require deeper cut outs at a different angle to produce the same effects, since s a t u r a t i o n at the pole then tends to dominate. Quite large volumes of iron have to be removed a n d the m a j o r disadvantage of the m e t h o d is TABLE 2 Iron enhancement from yokes with shaped outer boundaries. The cut outs all have ~b= 54°, c~= 18°. Depth of the cut out (cm)

Field from The enhancement from the coil B0 (coil+iron) the iron as a percentage alone of the coil's field (T) (T)

0.0 3.0 5.0

0.73 0.73 0.73

1.08 1.07 1.05

48% 46% 44%

0.0 3.0 5.0

3.40 3.40 3.40

4.74 4.60 4.43

38% 35% 30%

the associated reverse field effect, or reduction in the field e n h a n c e m e n t due to the proximity of the iron. This is illustrated in table 2 for the cut out of fig. 5 applied to a four-region winding with r~ = 5 c m , r2 = 8 cm, and thin iron r 3 = 9 cm a n d r4 = 18 cm. The second a n d more profitable alternative in these magnets is to consider shaping the inner surface of the iron yoke. It is only effective where the iron is sufficiently thick that saturation in the polar region is the pred o m i n a n t cause of field error. If the inner b o u n d a r y of the iron is shaped as shown in fig. 6, then the flux density at the surface of the iron is reduced, as are the c o n s e q u e n t changes in C 3 and C5 due to saturation. The spreads in h a r m o n i c coefficients over the range 0 < B o < 4.5T for the three coils of fig. 1 are plotted as a function of the depth of the iron cut away at the pole. These results, c o m p u t e d for an iron thickness of 14.8 cm, show substantial i m p r o v e m e n t s particularly

,~

r2 r3

I

J

]

x

(a) SHAPE OF THE CUT OUT (a) S H A P E OF THE CUT O U T AT THE P O L E

4.0 C 5 F OUR- L A Y E R ~ / / I ' ~

% 0.0

/

DEPTH OF THE CUT O U T (cm)

// / "

/

FOUR-REGION

~o -4.(

~ -'m~

u u_ © -8.0

#

~.o

1

~

4,0 e.- -'--C 5 F O U R

~C"'3 "

TWO-SECTOR FOUR-REGiON +,./

-12.{

-- - - - - - - e - - - - - - - -

/ / C 3 FOUR-LAYER /

/

C5

LAYER

TWO-SECTOR

FOU

-REG,ON

/ /

"~''f/

o.c , -

I

~3 TWO SECTO ~ ' ~ F O U R - REGION

~

1to

~.o

d,DEPTH OF THE CUT OUT (cm)

-1~.0 -2-C

(b)

SPREAD IN C 3 AND C5 OVER THE RANGE 0 TO 4.5T, AS A FUNCTION OF THE CUT OUT DEPTH, c

Fig. 5. Cut outs on the outside of the yoke.

(b) S P R E A D S OF T H E

OF C3 A N D Cs CUT OUT D E P T H ,

AS d

A FUNCTION

Fig. 6. Cut outs on the inside of the yoke.

VERY

HIGH

FIELD SYNCHROTRON

for the two-sector and four-region coils which as already seen in fig. 3 are m o r e prone to saturation in the polar region. Care has to be taken in not choosing t o o great a depth with this simple shape as it increases the spread in C5; this is also sensitive to the angle c¢ at which the cut out starts for m o r e c o m p l e x shapes. In contrast with external cut outs, shaping the inner surface of the iron strongly affects the harmonic coefficients at all levels of excitation. Adjustments have therefore to be m a d e to the coil boundaries to offset these changes. However, they are small and additive with little or no iterative effect on the choice of cut out despite the non linear nature of the problem. The magnitude o f the harmonic offsets which have to be supplied by the coil assuming it to be surrounded by a circular iron boundary are plotted in fig. 7 as a function o f the depth d.

599

MAGNETS

jo=17000Acm -2 equivalent to 6 . 4 x 1 0 5 ampere turns. Field from iron 1.45 T. Stored energy, 1.5 x 105 J m -~ . 4 layers of equal thickness, 01 = 81.49 °, 02 = 63.08 °, 03 = 54.60 ° and 04 = 28.21 o, for the windings rl = 5.0 cm, r2=8.0cm, for the iron r 3 = 9 . 0 c m , r4=23.8cm. Polar cut out on the inside of the iron, d = 1.4 c m with = 30 °. This is the basic design of a magnet n o w being

5,0 4"0

IRON

~-3"0 o m 2.0 1.0 X

2.3. EXAMPLES To illustrate the power of these design arguments, the c o m p u t e d results are given for a design of a 0 to 4.8 T magnet in fig. 8, and for a 0 to 6.0 T magnet in fig. 9, where the field harmonics and field errors are shown plotted as a function of the central field B o. Further details o f the magnets are:

0"0

I

I

Jo. CURRENTDENSITY IN THE WINDINGS(kAcm-2 )

(a) SECTION THROUGH THE MAGNET

(b) EXCITATION CURVE

2'0 ~r o 1.0

Magnet A : B o = 4.8 T, av. current density in the w i n d o w area,

¢0

0"0

o

2!o Bo(T)

24.0

-1.0

d2B/B ° x (c) d-~-/ , AT-~.-= 0.8, y = 0 . O , AS A FUNCTION OF Bo

%

160 I'Cl

N a_ o u

12.0

o 0"0 UE ~1,0

g ~z

8.0

c~

4.0

1"0

2'3B°(T) 3"0 ~

c5

I

J

-T4 7 OF C n

(d) M A G N I T U D E

D z

ON THE 0"0

0.5

,.'0

1!5

d, DEPTHOFTHE

2'.o CUT

z'.5

OUT (crn)

~o l-0[ ~ I~

-8.0

E5

1"0 I

0 I0

1.0

2;o

~ Bo(T)

x

3;o

e.---.O

AXIS ~io

ONTHE AXIS

(,z) AB/ao~ AT rG: 0 . 8 , AS A FUNCTION OF Bo Fig. 7. Harmonic offsets produced by cut outs at the pole of the yoke.

Fig. 8. Magnet A.

600

J.H.

COUPLAND

et al.

c o n s t r u c t e d at the Rutherford L a b o r a t o r y and named AC4. 6'C

M a y n e t B." 3.0 2-0 1.0

°'°

×

~

'

;

~ ¢

1'~ ,'4 ,L

Jo, CURRENT DENSITY IN THE ¢IH4D!NGS (k Acre-2

(a) SECTION THROUGH THE MAGNET

(b) EXCITATION CURVE

In some circumstances the magnitude of a given harmonic must be very small, for example in slow extraction in addition to limits on C3 the C5 term has to be particularly small t°) and the simple cut out o f fig. 6 may not be adequate, in this case the easiest solution is probably a small increase in the inner radius of the iron.

1.O

% 8"5

×

¢0

dc

!0 o

?

~L~

E 1.,2

o as m[~

%1 #= -2< d2Blm (c) ~ / D o ,

If the effects o f iron saturation are avoided by increasing the separation between coil and yoke for example, then an increase of r 3 from 9 to 11 cm with r4 unchanged gives an acceptable field quality. The changes in field enhancement and stored energy are given in table 3.

x AT ~-= 0 . 8 , y=0.0, AS A FUNCTION OF Bo

,.oF ~

0"0

I'01

(d)

2'01

Bo (T) 3i0

B o = 6.0 T, av. current density in the window area, J0 = 1 4 5 0 0 A c m - 2 equivalent to 9 . 4 x 105 ampere turns. Field from iron 1.6 T. Stored enelgy 3.0x 105j m - 1.6 layers o f equal thickness, 0 t = 81.14 °, 02 = 67.04 °, 03 = 57.16 °, 04 = 49.44 °, 05 = 42.60 °, 06 = 20.85 °, for the windings r~ = 5 . 0 c m , r 2 = 10.0cm, for the iron r3=ll.0cm, r4 = 3 2 . 0 c m . Polar cut out on the inside o f the iron d = 0.32 cm with ~ = 13.6 °.

~

C

4"0

5 6"0

The effect of B - H curve on field quality in magnet A can be examined by computing with a modified B - H curve, fig. 2b, having a reduced level for saturation, B s a t = 1.65 T. Using this curve the same saturation effects occur at correspondingly lower values of the central field B o, but no change in magnitude is seen in the variation o f harmonics or field quality.

MAGNITUDE OF C n

Y AX IS

% o

r£ o

1

1!0 O

o

2[0 Bo (T} 3"0 O

(z) ' ~ % o ,AT 7~r =0'8'AS

a

~

~ 4.0

X

5"0

A FUNCTION

2.4.

ONA ×THE IS

OTHER DIPOLE MAGNETS

It is of interest to optimise and c o m p a r e the computed field quality of other designs with lectangular windings, which are more conventional and perhaps easier to wind. Three examples are presented here. The first, as illustrated in fig. 10i, has a non-circular useful

OF Bo

F i g . 9. M a g n e t B.

TABLE 3 C h a n g e s in field e n h a n c e m e n t

M a g n e t w i t h r a = 9.0 c m , r4 = 2 3 . 8 c m Stored energy (J/m) Iron enhancement

(T)

and stored energy.

M a g n e t w i t h r3 = 11.0 c m , r4 = 23.8 c m Stored energy (J/m) Iron enhancement

B0(T)

4.6

1.38 x l0 s

1.40

1 . 4 8 x 105

1.18

1.4

1.22 x 104

0.50

1.31 x 104

0.40

(T)

VERY HIGH

FIELD

SYNCHROTRON

aperture, a simple coil configuration and a cylindrical iron yoke. The reduced vertical aperture may be no handicap as synchrotrons usually require more horizontal aperture than vertical for the beam, and the extra vertical clearance between the coils and iron may be of practical help in arranging for the end windings of

IRON

10~0¢m

(i)

MAGNET WITH A N O N - C I R C U L A R AND A CYLINDRICAL IRON YOKE

B

~ IRON

APERTURE

601

MAGNETS

a finite length magnet. A new computer programme developed by Newman et al. 11) is used in designing the magnet. The normalised coefficients of the field harmonics C 3 and C5 are reduced to less than 0.001 first by adjustment of the size and position of the current blocks and then, with the aid of an integer optimisation procedure, a limited number of conductors are removed leaving spaces in the windings. This magnet then shows the same kind of variation in field quality with central field, B 0, as for the four-layer coil and in the same way it is possible to correct the errors by a cut out in the polar region of the yoke (see fig. 10i). In this example there are some high order harmonics remaining, particularly C7 = 0.002, but it is thought that these are not an inherent limitation but a consequence of the limited nature of the integer optimisation. The second example is the Umst~itter design 12) which as proposed (see fig. 10ii), was thought to be capable of maintaining a uniform field in the aperture up to B 0 ----Bsa t ( c o s ~ ) - 1 , whilst limiting the flux entering the iron to B,a t. It does pose practical difficulties in that the required current density in the corners A and A ' has to be increased to 2jo with (1 - c o t 2 c0j o in the corners B and B'. Although the iron has adequate cross section for the return flux, the differential reluctance of the iron path causes the flux lines to crowd to the corners A and A'. This leads to the poor performance shown in fig. 11 where there is a rapid deterioration in harmonic coefficients for B0 > 2.5 T.

3-of

2.0

(ii)

UMSTATTER

MAGNET

1.0

0.0 5"0 o -1-0

-2.0

!~ON

-3.0

4-0

0cm +30 4 (iii)

WINDOW

FRAME

5"0

MAGNET

Fig. 10. O t h e r dipole magnets.

6-0

Fig. 11. H a r m o n i c coefficients o f the Umst/itter m a g n e t (~t = 45°).

602

J . H . COUPLAND et al.

The third example, shown in fig. 10iii, is the traditional ' w i n d o w f r a m e ' magnet which shows large changes in field quality above 2.0 T. Saturation o f the poles always dominates however thick the iron on the median plane. W h e n the pole saturates C3 is positive and is associated with a tangential field B~ as the flux for preference tries to cut the top and b o t t o m corners o f the yoke, cf. the median plane in the Umst/itter case. N o method o f correcting these error fields is found to be sufficiently effective. The coil may be extended into the yoke so that it becomes an ' H ' magnet, an air gap introduced between the top o f the coil and the yoke and the pole face dished. Whilst these modifications give some improvement, the changes in harmonics with field level, B 0, are still an order o f magnitude greater than acceptable. If a ' window f r a m e ' or ' H ' magnet is still preferred for reasons of easier construction, then it would have to have the winding divided into sections ~3) with separate excitation according to field level, or be fitted with auxiliary pole face windings.

3. Quadrupole magnets 3.1. SECTOR COILS WITH CYLINDRICAL YOKES The field from two-sector quadrupole windings analogous to fig. li is c o m p u t e d at increasing excitation for three different iron thicknesses, r 3 = 9 cm and r, = 13, 18 and 23.8 cm respectively. Only the coefficient C6 is shown in fig. 12 as a function of the field

gradient; changes in the other coefficients are negligible and it is assumed for this discussion that this behaviour is typical of quadrupoles corresponding to all three geometries of fig. 1. The requirement for use as focussing elements in a separated-function high field synchrotron is for gradients variable in strength up to 100 T m -~ and constant over the useful aperture to within 0.001. This criterion is satisfied out to 80% o f the coil aperture for all three iron thicknesses, and in the case of 5 cm thickness the maximum gradient error at r = 0.8a is 0.06%. Although further shaping o f the yoke to improve the field is not necessary, it is found that for the thick iron, the residual change in C6 can be reduced by removing iron on the inner b o u n d a r y in the polar regions. At low gradients the iron enhancement is 22% whilst at 100 T m -1 the iron provides 18% of the total field gradient. The total stored energy for r 4 = 23.8 cm is 1.4× 105 J m -L. 3.2. PANOFSKY QUADRUPOLE This type 14) of quadrupole might be expected to show better behaviour with respect to iron saturation than the window frame and Umst/itter designs since the awkward corner now becomes a pole in the square aperture version. The c o m p u t e d values bear this out

YI

IRON

X

y , ocm.b 9.0 cm 4-E

j

SECTION I~ON r 3:9,0cm, r 4 =23'8crn ;RON r 3 ~ 9"0crn r6 ~ 18.0crn

3< o

1-0

G c~ 2.0

0"0

20 f

40 I

GRADIENT (Tin-1 ) 60 80 I

[

100 I

% ~-zo 1.0 < X 0.0

,@

A

A

2~

40 60 GRADtE

I

I

I

I

100

IRON r 3 =9"0cm, r4 = 14"0cm

-1.0

Fig. 12. Harmonic coefficient, C6, of a two-sector quadrupole (rl = 5.0 cm, r2 = 8.0 cm) with different iron thicknesses,

c~ D -2-0 z

e._

~ = 1 2 . 0 c r n 15.0crn

-3.0

Fig. 13. Panofsky quadrupole; harmonic coefficient, C~, as a function of the gradient for different iron thicknesses.

VERY

HIGH

FIELD

SYNCHROTRON

and values of C6 for two thicknesses of iron are plotted in fig. 13 again as a function of field gradient. Both magnets satisfy the 0.1% criterion on gradient at r < 0.8a up to a maximum of 100 T m -1. The results of fig. 13 are also interesting in that they show that saturation on the equator is the more important and that this dominance is increased for the thicker yoke! Cutting away iron from the corners to make the outer boundary more circular increases the degree of saturation at the pole and hence reduces the errors in field gradient. A disadvantage of this square construction when using field dependent superconductors in the windings is that there is a rise of field (x/2 times) at the conductor corresponding to the useless corner region of the available aperture.

4. Conclusion The computed results in general show that for magnets where the iron is closely associated with the winding so that it makes a significant contribution to the useful field, those based on circular sections give the best field when the iron saturates, and of those considered the four-layer is the best. It also uses the least amount of conductor. Though saturation resulting in field errors occurs at the equator and the pole, a balance can be achieved by shaping to give remarkably good fields over the range 0 to 6 T for dipoles and gradients 0 to 100 T m -1 for quadrupole magnets. The most efficient design will have the least iron removed from a section that is thick enough to carry the return flux calculated as if distributed unifotmly through it. Internal cut outs for correction of field errors are preferred as these preserve the field enhancement fcom the iron. The likely accuracy and other comments and details concerning the method of computing are discussed in the appendix. It is hoped that with present developments in accurately wound superconducting magnets it will be possible to test these predictions experimentally in the fairly near future.

Appendix Notes on the computation

The computer program T R I M 3'4) uses a finite difference method on a non-uniform triangular mesh to obtain a numerical solution of the two dimensional equation for the axial component of vector potential, A, which in effect is a scalar for this application since Ax = Ay = 0 for magnets with jx = i x = O. Curl (#-1 curl A) = /~oJ, where the quantity/~-1 is a non-linear function of the

603

MAGNETS TABLE 4 Comparison with analytic calculation.

H a r m o n i c coefficient C . calculated analytically f o r / t = 10

1

Computed values obtained from TRIM, #= 10

1.0000

3 5 7

1.0012

-- 1.79 × 10 _8 0 . 0 0 × 10 _8 O.Ol × IO- z 0,00× lO -3 0 . 0 0 × lO -3 - - 0 . 9 6 x lO -3 2.08 × lO -3

9 II 13 15

-- 1.85 × 0.20x --O. l O × --0.03 × O.lOx -- 1.12 x 1.81 ×

10 -3 10 - a 10 -3 lO 3 10 -3 10-3 10 3

local field or derivative of A. Only one quadrant of the magnet is represented, the rest being implied either by zero vector potential or reflecting mesh boundaries. The maximum number of mesh points available when using the IBM 360/195 is 10 000, and for effective use it is found sufficient to terminate the problem at a distance of about 3 times the size of the magnet. For this degree of truncation, the difference between writing zero vector potential on this boundary, making it reflecting, i.e. zero gradient, or fitting to an asymptotic r -z dipole field is negligible in the region of interest. The results are found to be relatively insensitive to the shape and size of the triangles in the air and coil regions, but for the iron, the triangles have to be as small as possible for realistic results at high fields. The harmonic content of the field in the aperture of the magnet is obtained by a standard least squares fit of the computed values of A at the irregular mesh points to the general analytic expression, =

cos O + ½ C 3

+kC5

cos 3 0 +

]

cos 5 0 + . . . .

in the case of the dipole magnet, or the corresponding expression with terms in 20, 60, 100 etc. for the quadrupole. Table 4 shows a comparison for a 6 layer coil between coefficients calculated analytically and computed on T R I M both with/z = 10; using a typical B - H curve the results again agree very well with calculations at low field or/~ = ~ . With this experience it is thought that the field values in the aperture are accurate to about 2 x 10 -4. The computations assume that B and H are parallel but this is an old and unresolved question about the

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behaviour of iron that has been raised again recently by Blewett'5). Is there a purely scalar relationship between B and H when the iron is saturated, or do the directions depend on the local field from the coil, the field from the rest of the iron and its past history? At fields well below saturation the magnetisation of a domain will be parallel to one of its directions of easy magnetisation. Near saturation the directions of magnetisation of the domains are rotated reversibly to be parallel to the applied field. With regard to these computations it is thought that if the field in the aperture as computed shows little change of shape with field level then there is likely to be little error from making the scalar assumption in this case. Further evidence can also be obtained using the alternative method previously mentioned ~t) to compute the problem. With the iron divided into separate blocks this program first calculates the field due to all current elements and uses this to determine the initial magnetisation of each iron element in magnitude and direction. The field pattern is then recalculated and the final solution obtained after several iterations. It is re-assuring to discover that the final field agrees very closely with the TRIM calculation, and that during the iterations when the direction of magnetisation changes there is only a small effect on the field in the aperture. For magnets of fig. 1 there is a maximum difference of 5 x 10 - 4 in C 3 corresponding to a change in direction of magnetisation of 10°. The stored energy is found by numerical integration of the flux 45 linked by the windings,

45(J) = ff¢oi~ Ar dr dO,

taken over the windings, if the turns are of uniform density, whence the stored energy E - - .~ 45(j) dj, wherej is the cur rent density.

References 1) j. p. Blewett, Proc. 1968 Summer Study Superconducting devices and accelerators, Part Ili (Brookhaven National Laboratory, 1968)p. 1042. 2) j. H. Coupland, Proc. 3rd Intern. Conf. Magnet technology (DESY, Hamburg, 1970) in press; or Rutherford Laboratory Report RPP/A 88 (1971). a) A. M. Winslow, Proc. 1st Intern. Syrup. Magnet technology (Stanford, 1965)p. 170. 4) N. J. Diserens, Rutherford Laboratory Report RHEL/R 171, H,M.S.O. (1969). 5) j. H. Coupland, Nucl. Instr. and Meth. 78 (1970) 181. 6) A. Asner and C. lselin, Proc. 2nd Intern. Conf. Magnet technology (Oxford, 1967) p. 32. 7) R. A. Beth, 1. c. ref. 1, p. 843. 8) A. D. Mclnturff and J. Claus, Brookhaven National Laboratory Report AADD-162 (1970). 9) G. Parzen and K. Jellett, IEEE Trans. Nucl. Sci. NS-18, no. 3 (1971) 646. 10) E. Courant and V. Baconnier, Panel Discussions Proc. 8th Intern. Conf. High energy accelerators (CERN, Geneva, 1971) p. 105. 11) M. J. Newman, C. W. Trowbridge and L. Turner, Rutherford Laboratory Report RHEL/R 244, in preparation. 12) K. Leeb and H. H. Umstatter, CERN Report 66-7 (PSMD) (I 966). az) G. Parzen and K. Jellett, Brookhaven National Laboratory Reports AADD 178, 180 and 182 (1971). 14) L. N. Hand and W. K. H. Panofsky, Rev. Sci. Instr. 30 (1959) 927. 15) j. p. Blewett, in discussion, l.c. ref. 10, p. 50.