The 12th Symposium on Accelerator Science and Technology, Wako, Japan 1999

A Study on a High Performance Dipole Magnet for an Accelerator
Shiro NINOMIYA, Kenji SATO, Tadahiro OHISHIand Shinji AN 0
Research Center for Nuclear Physics (RCNP), Osaka University,
Mihogaoka 10-1, lbaraki, Osaka 567-0047, Japan
Abstract

where y-direction is a direction of the dipole field and zdirection is that of the beam. Since a magnetic field toA new theoretical method by using three dimensional the y-direction, By, should be symmetrical with respect
Laplace's equation for determination of the pole shape of to the medium plain of the magnet, therefore,
a dipole magnet was proposed. The calculation showssome distinctive features, one of which is a shim structure.
A new magnetic field measurement apparatus using
moving search coils was also developed. A preliminary
experimental result shows that the apparatus is useful The scalar potential '1/J should be solution of three dimensional Laplace's equation, i.e., 1':.'1/J = 0, and then we canfor the H-type magnet.
obtain1 Introduction
In resent years, high energy beam generated by asynchrotron-type accelerator have been required not only
for pure physics but also for technology and medical care.
Under this condition, a smaller synchrotron-type acceler(3)ator have been favored mainly for cost performance and,
therefore, accelerator elements, such as a dipole magnet,
a quadrupole magnet and an rf cavity should be smaller. We assumed fl(x, z) as
In the case of a small dipole magnet, the edge effectfl(x, z) = -Bog(x)G(z),(4)
of the magnet to an expectable dipole field comes notto ignore. Therefore, two demensional Laplace's equation and two deminsional numerical computation are not where Bo is a constant. In order to realize almost conenough to determine the pole shape of the small magnet stant By within certain regions, the functions g( x) andwholly. In fact, the edge structure of the magnet pole G( z) are adopted as
was generally determined by an empirical way mainly
through the magnetic field measurements, which comes1( ) [ erf (x+xo)
g(x) =- - - erf (x-xo)]-- ,that no guiding principle for the small dipole magnet was
2erf ~~x~x<Yxfound out. Three dimensional numerical computation is(5)seen to be not so useful by now. One idea to determine
more theoretically the pole structure of the small dipole andmagnet is using analytical solution of three dimensional
1( ) [ erf (z+zo)
Laplace's equation, which was proposed by Langenbeck
G(z)=- - -erf (z-zo)]-- ,2erf ~~z~z
in a quadrupole magnet case[l].
<Yz(6)

2 Three dimensional Laplace's equation

where xo and zo are measures of the pole width and thepole length, respectively, ~x and ~z are mesures of thestray magnetic field extension and error function, er f( x),is represented as

The pole shape of a dipole magnet is regarded as asurface with constant scalar potential, when the permeability of the matter of which the magnet pole consists
is high enough. Though this condition is not satisfied for
2-t2
a real magnet, the surface with constant scalar potential(7)
erf(x) = J1T
e dt.
may be useful at the start point of the determination of
the pole shape. After calculation, successional magnetic
Figure 1 shows a typical shape of the function g( x)field measurements with changing a part of pole shapeorG(z). Roughly speaking, the magnetic field By related
may make us find a rather good pole shape.
tog(x)and G(z) through eq. (2) is almost constant at
The scalar potential, '1/J, is assumed aslxl < x 0 and lzl < z0 , decreases smoothly at lxl ,. . , x 0 and
00lzl "'"'zo and almost zero at lxl > xo and lzl > zo.

Jor'1/J(x, y, z) =L ym fm(x, z).

(1)

m=O-310-The 12th Symposium on Accelerator Science and Technology, Wako, Japan 1999

was found out from the limited calculations. Near thecenter of the magnet, i.e., for smaller \Z\, Y ~ 1, i.e.,
minimum gap, within certain range of \XI in this case
\XI ~ 6, which corresponds to \xo\. The value of Y
increases rather sharply without the range of \X\. Forlarger \Z\, Y is also constant but is bigger than that forsmaller \ZI with small \X\. Near the \XI correspondingto \xo I, as \X\ increases \Y I firstly decreases slightly andthen increases sharply. In other words, cross-sectionalview of the pole has two projections at the pole end,
which may coresspond to the presence of the shim structure, which is well known to be one of the effective methods for a wider good magnetic field region. For N = 1,
qualitative tendency is very similar with that for N =2.

Figure 1: A typical shape of the functions g( x) or G( z).
The equation to represent the pole shape of a dipole
magnet iswhere dis a gap length at the center of the dipole magnet,
which corresponds to a minimum gap.
Since eq.(8) has infinite terms at the right hand, it
is very difficult to calculate strictly. Therefore, we ignored the sign "lim" at the right hand of the eq.(8) andcalculated it for N =1 and 2.
Figure 3 shows a typical calculation of the pole shapefor N =2 represented by using normalized variable, i.e.,

where G = d/2. Even though the evaluation of the caly

2.2

1.8

1.2
X-4-2

2

4

Figure 2: A calculated pole structure of the dipole mag= 2.8.
net for N = 2, 6Z = 0.5, !J~ = 1.316 and!J:

3 A magnetic field measurement apparatus

We prepared an H-type dipole magnet with widely
adjustable edge pieces and constructed a new magnetic
field measurement apparatus for the magnet. In orderto ensure conceptual adequency of eq. (8), we need tomeasure so-called BL product with a very wide range ofx. We selected a moving search coil method, which is
generally used for magnetic field measurements of a large
amount of magnets at once. Using the rectangular searchcoil, the BL product is obtained by(BL) = - 1-NewJVdt(9)

where Nc is the turn number of the search coil, w is awidth of the coil and V is a voltage appearing between
both ends of the coil wire, respectively.
Schematic drawing of the magnetic field measurement
apparatus is shown in figure 3. Four search coils(750mmx 20mm), which are put on the medium plain of themagnet, are separated into two pieces at the center. Eachpiece is drawn out straightly toward each edge of the magnet along the beam orbit by using a long guide( ~3.5m)
and a wire and finally it is completely shifted to the freespace without any magnetic field. Separation of the moving search coils is a rather adequate method to measure
the magnetic field of the H-type dipole magnet with relatively small radius of curvature. The coil guide wasdesigned to move along the x-axis by ± llOmm. Since
the apparatus is prepared for a conceptual test, it wascarried oi1t by hand to move the search coils and the coilguide. Cost performance of this measurement apparatus
is, then, rather good due to its simple structure. In order
to measure the magnetic Held of a large amount of themagnet, automatical moving of the search coils and thecoil guide will be easily carried out without changing themain part of the apparatus design.
The turn number of the coils was firstly set to be ~700. Since the square measure of each coil was a little bit
different, we adjusted to add the correction coil empilically. Finally the difference comes to less than 2 x 10- 4 .

culations up to larger N value should be needed for detailed discussions, qualitative features of the pole shape-311-

4 Preliminary results

The 12th Symposium on Accelerator Science and Technology, Wako, Japan 1999

Figure 3: A schemati c drawing of the magnetic field measure ment apparatu s.
Some preliminar y tests were carried out. Figure 4
shows an example of an observed V signal as a functionof time. After 2.5 seconds from the rising point of V,
the value of V come to zero, which ensure that all of
the search coils are drawn out to no magnetic field space.
Observed four peaks come from the change of the drawing
Tu, ~ IOQ t/1!

:

1 .A CCII:;;:·:: ' :jwas found that new magnetic field measurem ent apparatus works as designed except for some trivial experimen tal problems, all of which are easily solved in principle.
Detailed analyses of the obtained data are under way.
The edge piece of the magnet with the shape based on
the analytical solution of the three dimension al Laplace'sequation have been made and the magnetic field measurements are now under way.

o: I. u v

. . . : . . . . : . . . • : . . . • : . . . . : . . • . : . • . • : . . . . : . . . • :·. . • . o:-IOmV

Referenc es.)7.7.~~. :-!7.l:~. 7.~. ,-; .~ ~.7.7~. ..~ ~-~.7.~. .~f::'. . ;,.
•. .•....• •. . .. . :•••••.•.•.... •...•.•..•..•.
. ..

[1] B. Langenbec k, IEEE Transcatio n on Magnet 24 (1988)
1369.

'··.·~····~····~···~····~····~····~····: .... :....
.
.
... ... .. .
..

Figure 4: An example of an obserbed voltage signal asa function of time.
speed of the search coils because we move it by hand.
Nevertheless, the reappeara nce of the measured magnetic
field was found out to be within 10- 4 , which is goodenough to judge the justificatio n of the concept discussed
above.
Magnetic field measurem ents of the dipole magnet
without any changes of the pole edge was carried out. It-312-