Analysis and design of short, iron-free dipole magnets Page: 2 of 6
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ANALYSIS AND DESIGN OF SHORT, IRON-FREE DIPOLE MAGNETS*
A. R. Harvey
Lawrence Livermore National Laboratory
P.O. Box 808, L-544
Livermore, Ca. 94550SUMMARY
Iron-free, dipole magnets are used extensively as
steering magnets to correct for the bending, induced
by extraneous -magnetic fields, of particle beams that
are being transported in vacuum. Generally, the
dipoles are long enough that the space occupied by
the end conductors is small compared to toe overall
magnet length. In a recent application, however,
this criteria did not apply, This has motivated a
reanalysis of the characteristics of a system of
sma'l aspect ratio (length/diameterl dipoles that are
spaced at relatively large anial distances. The
folliwina observations and conclusions resulted fron
this analysis:
1. The effective magnet length is a simple function
of the anal conductor lengths, their relative
orientishon, and the magnet diameter.
2. The overall magnet strength is a function of
axialy parallel conductors only.
3. ind conductors should be placed in a sinle
plan noeal to the a.is at each end of the magnet.
4, Flat wound and formed magnets lead to
cont :rahl cost editions over mornr cnvntional
'11,r) nethons.
5. >c'easng the number of turns yields more
faon1. e poer supply matching, httne fneid
anifor-ito, and more fianrahle heat dissipation.
6. !ncreasina the numtr of and upturning the end
conductors provide a more favorable flivd profile at
the ends of the magnet.
The above points are discussed in this report. Some
fab ;ation techniques which are being :leveloped in a
pretntpe mngnet are also discussed.
NOTES ON EQUATIONS Ai SYMBOLS
have used pseudo-fortran line style equations
throughout this report for ease of reproduction on
stanJard word processing equipment. Integrals and
sumations, followed by their limits, are spelled
out. Constants such as MU and Pi are phonetic
spellings of their greek counterparts. I is
universally designated as current and N as turns.
Indexed quantities are enclosed by parenthesis and
paired by the letter J. Magnetic fields are prefaced
with B and all incremental values are prefaced with
U. Directional quantities are sufficed with the
letters Y or Z. Z is universally designated to be
the beam axis. Mathematical notations, with the
above exceptions, are as defined in fortran.
Although no specific units are expressed, consistent
units are implied.
FIELD ANALYSTS
Fig. lA is a graphical representation of a
cross-section of the upper half of a classical dipole
magnet. The conductor profile is shaped such that,
* Lawrence Livermore National Laboratory is operated
Energy under Contract No. W-7405-Eng-4B.if filled with many uniform current density elements
the following relationship holds.
(1) N(THETA)'kN(O)*I'COS(THETA)
fliRE iA
CLAS&ICAl OVltf'1 I l
* I
- I n
I I l
FIGURE IS
CASIBE" APPRK MAi
) K
TIOThis current distribution produces a uniform
transverse field at any location within the magnet
bore.
(2) 004 JO*NI/(29P)
Where N1 is the integrated sum of the ampere-turns,
In practice this winding configuration is difficult
to achieve. A typical winding approximation is shown
in figure 18. The transverse field contribution at
the center bore for an infinitely long current
element would be:
(3) BY (J)=JO*CO5(THETA(J))/(2*PI*R)
The net field for all of the elements would be:
4) BY (TDTAL)=2*10*I/(PI*R)*SUM
(0=1 TO N)COS(THETA(J))
by the University of California for the Department ofThis work is performed by LLNL for the Department of Defense under DARPA (DOD) Arpa Order 3718, Amendnent
f12, monitored by NSWC under Contract No. N60921-80-WR-WO1UB.
1r
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Harvey, A.R. Analysis and design of short, iron-free dipole magnets, article, October 21, 1981; [Livermore,] California. (https://digital.library.unt.edu/ark:/67531/metadc1113627/m1/2/: accessed June 17, 2024), University of North Texas Libraries, UNT Digital Library, https://digital.library.unt.edu; crediting UNT Libraries Government Documents Department.