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MICROCOPY RESOLUTION TEST CHART
NATIONAL BUREAU OF STANDARDS - 1963
ORNU-P-1816
Corf650512-14
SEP 16 1965
SOME PROBLEMS IN THE DESIGN OF AV 8th ORDER POWER OPTIMIZED COIL
J. N. Luton, Jr.
Investigators have been concerned for some time with the design

of magnet coils to produce a given field with a minimum of power.
now called Fabry factors, which relate the flux density produced by
air-cored coils of various shapes to the power consumed by them. Bitter,
Gauster, Gaume, and others have extended this work of the power op-
timization of coils.
RMLEASED FOR ANNOUNCEMENT
IN NUCLEAR SCIENCE ABSTRACTS
Efforts at producing fields of desired shapes also have a long
history. For instance, Helmholtz and Maxwell, respectively, originated
and
2 and 3-coil systems for producing fields uniform in space. One
approach to the problem, described in a systematic way by Garrett,' is to
express the field as a series of zonal harmonics and then to control the
coficients of the series. For points along the coil axis, the series
RELEASED FOR ANNOUNCEMENT ·
IN NUCLEAR SCIENCE ABSTRACI'S
for the flux density takes the simple form
B = A + A2 ( )2 +43 1 579 + ! .......
where 2, the axial distance from the coil center, is less than the coil
inner radius a: A 4th order system, such as a Helmholtz pair, is
characterized by having the coefficient Ag = 0, a 6th order system by
A, = A = 0, etc. Neglecting the changes in the higher order coefficients,
the field becomes more uniform as the coil geometry is changei to make
more coefficients vanish.
"Researtile seasoned xe the rorátiAtomic Energy Commission under contract

Properties of Current Systems with Special Reference to the IBM 7090,
ORNL-3318 (November 19, 1962).
-LEGAL NOTICE -
TM, report me prepared u wa mtovat of Oorenmeat spookord work. Heltber Who Valled
suaus, for the Countıslon, nor may person acung on behalf of the Commission:
n. Maker way miraty or representation, expreid or implied, muh respect to the accu-
rasy, completno, or whoels of the information coolalod u We report. or tal de vie
of way lafortualion, appuntu, methodo or procesi di cloned in Wo report may not infringe
printly one richua; or
B. Annei wy lladulluar wu neprct to the one of, or for dumugto resdung from ide
um. itu laformation, apparatu, authod, or proceu daclound in to report.
Au und in the above, pornou acting on behalf of the Conoscolon" facludes way en.
plo: " or contractor of the Commission, or employee of much contractor, to the extent wat
ral employee or contractor of the Comalostoa, or employee of such coalractor prepares,
di.manaus, or provides access to say Information purnit to Wo eaplojoamat or contract
vlu the conmiooloa, or No employment wild such contractor.
. In some cases, it would be desirable to have a field which 18 both
strong and uniform over a large volume, i.e., we desire a power-
optimized coil of high order. Gauster and Garrett' have derived a
method and written a computer code for finding the current density
distribution which produces the maximum field strength for a given
power under the restriction that the coil be of a certain order. The
present paper describes some work built on this base aime i at producing
a field of about 60 kg, homogeneous over a 12" sphere, with a solenoid
which consumes 7 Mw.
The coil weight was first fixed by cost considerations at approxi-
mately 7000 lbs. This corresponds to a volume factor v = (ar - 1) B
.
.
. .
.
os 100. (a = outer diameter/inner diameter; B = coil length/inner
.
. . .
.
diameter). The field homogeneity optimization procedure was performed
,.
,
with various B values and showed that a flat optimum for the Fabry
.,.
me*:,
factor G occurred at :B = 3.0. For this optimization procedure it was
to
m
i
-..
initially assumed that the coil is composed of sixty flat "pancakes"),4
of equal axial thickness and uniform current density. The resulting
:
..
ein •
.
.
ness. 5.ve ...vai
axial current density distribution for an 8th order coil is given in
Fig. 1. Figure 2 shows, for coils of several orders, curves of constant
magnitude of the flux density B, these curves being the intersections of
the. B = constant axisymmetrical surfaces with the z-r plane. The 0.1%
eve
L
-
..
W. F. Gauster and M. 'W. Garrett, Thermonuclear Div. Semiann. Progr.
Rept. Apr. 30, 1964, ORNL-3652, pp. 105-12.
.
In a two-layer pancake, & conductor spirals inwardly in one plane
to the inner surface of the coil, crosses to the adjacent plane, and
spirals in this plane back to the outer surface of the coil. Any number
of such conductors may be wound symmetrically into a pancake.
*J. N. Luton, Jr., Thermonuclear Div. Semiann. Prog. Rept. Oct. 31,
1965, ORNL-3564, pp. 121-25.
m
contours are shown, that is, inside the volume limited by the surfaces
B = 1.001 and B = 0.999, the flux density deviates less than 0.1%
from its value at the coil center. In each case the volume increases
appreciably as the order 18 increased, and the magnitude of the field
at the coil center decreases by a small amount. For example, the
field strength decreases 2.8% when the order le raised from 6 to 8,
whereas for the same step the volume enclosed by the contours of Fig.
2 increases by about 72%. For this reason, and because of certain
practical differences which later appeared, it was decided to construct
a coil which would approach as closely as possible the 8th order
current distribution of Fig. 1.
. . Based on the axial separations of the peaks and dips of the 1*
V8 z curves (Fig. 1.), an actual width of conductor was next chosen.
In order to be compatible with the cross-sectional area demanded by
the current and voltage capacities of the existing power supply, the
actual width selected was larger than the width used in the 60-pancake
coil. The coil was then reoptimized using the actual widths of pancakes.
The resulting current densities are shown normalized in Table 1 in
column "Desired 1*".
Since large tubular conductors of nonstandard size cannot be
purchased in small quantities, it is not economically, feasible to
purchase a separate conductor size for each of the desired i* values
specified in Table 1. However, many "effective" conductor sizes can be
obtained by winding a pancake from conductors of a fixed axial width but
with different radial heights. Since coils of large diameter require
several parallel water paths for sufficient cooling, the above "mixing"
can conveniently be done. In this way many different current densities
can be achieved with only a few different conductor sizes. Thus, if mis
the number of water paths and n the number of conductor sizes, then the
number N of different average current densities
• which can be provided 18 just the number of combinations of n different
things taken m at a time with repetitions of items being permitted,
that 18,
N.On + m-]
(n + m - 1)!
m! ( In + m - 1) – m)!
In our case, with seven water paths, three conductor sizes give a
sufficiently large number of current density possibilities (N = 36).
In some instances, it is desirable to adjust current density in
axial steps of one conductor width instead of steps of one two-layer
pancake. This may be done by the splicing, at the cross-over, of
conductors of differing radial thickness, still using the scheme of
"mixing" conductors described above. However, it might be accomplished
more advantageously by mixing, conductors in the type of three-layer
pancake previously described." In such a pancake, each water path
• appears in exactly two layers, the middle layer and one or the other
of the outer layers. The average current density of each of the outer
layers can therefore be independently controlled, and the current density
of the middle layer is the average of the current densities of the two
outer layers, weighted by the number of water paths in the respective
outer layers. Thus, if one outer layer has a current density of 1,
different from that of the other outer layer, in, then the current density
of the middle layer is
24+ mzizi
(2)
omy 3
1
and so can be varied between the limits of 1, and so by the proper choice
of m, and mg, the number of water paths in the respective outer layers.
21
1

-5.
.
Two three-layer pancakes were used to achieve the eighth-order solenoid of
Table 1,. one as described above and one with constant current density.
The field contours for the eighth-order "actual case" of Table 1
are shown in Fig. 3. The contours of the actual case are quite com-
parable to those of the ideal case, except for the 0.01% contours,
which are drastically distorted. The addition of an extra turn and a
129 amp shunt to pancake number 2 changes the contours to those shown
in Fig. 4, which in every case are nearly exactly those calculated
for the current density distribution of Fig. 1. The final arrangement
of shunts will not be determined until after the coil has been constructed,
in the hopes that manufacturing tolerances may also be compensated.
FIGURE LIST
Caption
Page
3
Photo No.
ORNL-DWG-64-11802
:
Eighth
:
.
.
·
ORNL-DWG-55-4381
Fig. 1. Current Density Distribution for Ideal
.Eighth-Order Solenoid with 60 Pancakes
Fig. 2. IBI Contours with 0.1% Errors, for Power
Optimized Coils of Order 2,4,6, and 8. Ideal Gases:
js = 100, B Chosen to Maximize the Fabry Factor.
ORAL-DWG-64-11806
: Fig. 3. Constant B Contours for the Actual Eighth-
Order Solenoid.
Fig. 4. Constant IBI Contours for the Actual 8th
Order Solenoid with Shunt.
ORNL-DWG-65-4382
TABLE LIST
.
.
.
Table 1. Normalized Current Densities for an Eighth-
Order. Solenoid with Conductor Width of 1.02 in.
.
.
wa
.
.
-.-
.-.
-
.-
.-.-.-
-
I. -
a..-
n........
.......
.
.
Table 1. Normalized Current Densities for an
Eighth-Order Solenoid with Conductor Width
of 1.02 in.
1*. 1/max
Pancake
Layer
No.
Desired 1*
Actual 1*
0.511
0.519
0.833
0.814
º
0.488
0.496
º
0.715
0.870
0.715
0.899
0.999
1.0
1.0
1.0
0.870
0.898
0.698
0.715
0.543
0.544
O 418
0.409
0.323
0.307
The coil is symmetrical, about its midplane, which lies between
layers 1 and i'.
... ..
ORNL-DWG 64-4802
END OF COIL
MIDPLANE
OLLI I
2

4
6
8
10 12 14 16 18
PANCAKE NUMBER (AXIAL DISTANCE
20
22
24
26
28
30
,
figurel
ORNL-DWG 65-4384
2nd ORDER,
B = 2.7
8th ORDER, B = 3.0
♡ 70.999
0.82
.999
<1.001
4th ORDER, B= 2.9
p*= /,
0.999.
6th ORDER, B = 3.0
0.999)
0.999
O
0.2
0.8
1.0
0.4 0.6
z* = ?lo,
1B1 Contours with 0.1% Errors, for Power Optimized Coils of
Order 2, 4, 6, and 8. Ideal Cases; v= 100, B Chosen to
Maximize the Fabry Factor.

figure 2
an
ORNL-DWG 64- 41806
L
0; = 6.75 in. (COMPUTATIONAL INNER RADIUS)
4.01
m oo= 6.5 in. (PHYSICAL INNER RADIUS) 0.99
- iBi > Bo
-1B1 < B.
0.999)
11.001
.0001
(0.999
1.01
H0.9999
r (in.)
3 -

0.9999;
1.001
T 1.0001
1.000
1/0.99
0.9999
1.00040
10.999
10
z
(in.)
figure 3
I
.
:
CO.
$
9
.
ORNL-DWG 65-4382
Z169,=6.75-in. (COMPUTATIONAL INNER RADIUS)
1.01
0.99
0.9991
1.001
0.gggg
r (in.)
10.9999
21-
1.001
1B > Bo-
---Bl< B.
Ng = 34.55 (CASE SA!! 1.0001
ΓACTUAL Δα's
SYMMETRICAL CROSSOVERS
_SHUNT OF 147 amp, 37v (0.48 kg)
10.9999 10.999

3
4
5
z (in.)
6
7
8
9
1 2
TP.G. NO. 2-
10
Constant Bl Contours for the Actual 8 th Order Solenoid.
figure 4
TW
*
:
w
END
.
.
.
H4
the
i
DATE FILMED
11/ 19 /65






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