MEASUREMENT OF HIGHER MULTIPOLE COMPONENTS IN QUADRUPOLE MAGNET
A. Ando, K. Rosano, M. Kibayashi, and M. Tsukihara*
Research Center for Nuclear Physics, Osa.ka University, Ibaraki, Osaka 567*Sumitomo Heavy Industries, Ltd.,
Accelerator Business Center, Technical Department,
Niihama, Ellime 792

Abstract

Principle of measurment

Higher multipole components of a quadrupole magnet were directly measured by the fast Fourier transformation of the signal froma rotating coil, and compared with the results of polynomial fitting
for the field mapping data in the medium plane by a Hall generator.

The magnetic field inside the magnet is considered as two dimensional, for the uniformity along the magnet axis ( z- direcrion ).
The field is described by the z-component of vector potential, Az.
Expanding Az as00

Quadrupole magnet
The full scale model of the quadrupole magnet was designed
and constructed for the design study of the multipurpose storage ring(MSR) in RCNP 1 . The parameters of this model are given in Table 1.
The core material is the silicon steel S-23 of the Japanese Industrial
Standard. Each core was stamped out from the 0.5 mm thick sheet.
Also the 0.35 mm thick sheet was used to adjust the stucked corelength. The pole face contour is similar to that of the main quadrupole
magnet of the KEK 12-GeV proton synchrotron 2 , except that ourmagnet has the symmetry between the horizontal direction and thevertical one.

Table 1. Parameters of quadrupole magnet
Maximum field
Core length ( Lc)
Bore radius (rB)Maximum current
ResistanceInductance

2.0
0.4
50
770
3533- LAnrncos(nO + Xn),n=lwhere the cylindrical coodinates (r, 0, z) are chosen, and An and Xn
are the amplitude and the phase of the 2n-th multi pole field, we obtain
the pickup signal from the rotating coil as V = :Z::::;"=l Vn whereVn = wM Lnan An sin( nO + Xn)·
Here (} = wt, w is the angular frequency of the rotation, and a., Jv[and L are the radius, turns and the length of the rotating coil, respectively.
The differential coeffient of the field at the magnet center isthen given by

T/mmmmA
mrlft HryField measuring system
The block diagram of the field measuring system is shown inFig.l. The pickup coil is rotated at 30 Hz by the AC 100 V synchronous motor. The signal from the coil is sampled and converted
to the digital datum at every external clock. The rod mounted at
the rotating axis makes the trigger signal at every passage throughthe photo-interrupter. At first, the control computer, PC-9801VX,
gives the digitizer the number of the data to be taken, then the startcommand. The digitizer makes the digital datum train by A/D conversion from the first trigger signal after the start command. Theanalog signal between -5 V and +5 V is converted to the integer from0 to 4095. The typical frequency of the external clock is 4096 Hz.

d(n-l)B(n-l)!Vn

I dx(n-l)Y I

wMLon

We can easily obtain the strength of Vn by the spectral analysis of
the signal V. In this metod, the main term Am, for example A2
in a quadrupole magnet, does not disturb the higher order signals
Vn( n > m) as far as the rotation is stable.
Data reduction
An example of the spectra obtained by the fast Fourier transformation (FFT) is shown in Fig.2. The spectra are unfortunately
not the sum of the harmonics of the rotation frequency, since the total sampling time is not exactly the multiple of the rotation period.
In such a case, the spectral shape is given byRotating coil

The short and the long rotating coil are used for the pickup.
The parameters of these coils are given in Table 2.

Table 2. Parameters of rotation coil

Radius (a)Length (L)Turns (M)

mmmm

Short

Long37.55
101241.53

1000
Fig. 1 Block diagram of measuring system.

-160-

sin X

Table 3. Multipole components in effective lengthx-·

Wn- WTXI'2L3(xl0-2m-1)(A)

where Wn is the angular frequency of each spectrum, and T is thetotal sampling time.

30025020015010050

In this measurement we took typically 4096 data in T = 1 sec.
We replaced V2 by the J2Vn where Vn was the read-out strength
when the signal is directly inputed to the digital volt meter. For
farther analysis, we are preparing a fitting program to obtain the true
amplitude from the descrete spectra.

L4Ls

L6

(xl0- 1m- 2)(xlO m- 3)(xl04m-4)

1.28

7.10

2.28

8.86

5.93
7.46
7.0
6.817.01

9.16
7.84
8.11
7.19
8.21
10.1

G/I (T/rn/A)

The dipole component is explained by the setting error of therotation coil. The r.m.s. of the distance between the magnet axis andthe rotaion one is estimated as 0.25 mm.
0.0315=dBy/dx, and the efThe strength of the field gradient, G
fective length, L 0 , are plotted in Fig.3 and Fig.4 for some levels of theexcitation current, I. G /I in Fig.3 agrees well with the calculatedvalue, 3.116 x 10- 2 , from Table 1. From Fig.4, the effective lengthis approximated asr~ .---------1--------._0.0310

0.03050100

200300I (A)Fig. 3 Excitation curve of the field gradient.

Higher multipoles
We define the normalized strength of the higher multi pole component as

L 9 (rnrn)

460,-----,-~~-.~---,------,-----,------,Ln=Jd(n-l)BY dz/dz(n-1)

JGdz..-- ""'

1················+···················1······'\:·"\.··············+·················+······················+················ j"\.

(m-(n- 2)).

-~...............;.........................;........ ·····"-·""···················"·"···"7············-L
455~----~----~----~----~~~~~--~

We can only find Sn, n = 2 + 4k as the significant components atthe center of the magnet. This suggests that the magnet is almost
completry symmetric inside the radius of the rotation coil ( 37.55 mm). The variation of S6 due to the excitation current is shown in Fig.5.
For the integral strength, i.e., the data of the long coil, we cannot distinguish Vn ( n 2:: 7 ) from the noise level. Table 3 summarizes
Ln ( n :::; 6 ) for the various currents.

100

200300I (A)Fig. 4 Effective length versus excitation current.

-LNG COIL··········SORT COIL

~ 10-lk----+----------~--------~---------+----------~---------r--------~--------~

E.....

~ 10-2 b---~----------~----------~---------r----------+----------t----------t---------~s< to-3 b-~~~-------+---------4--------~----------r---------+---------t-------~

2610

14Harmonics
Fig. 2 Typical spectra of FFT.

-161-

From these results, the effect of the fringing field becomes important for I > 200 A or G ~ 6.0 T /m. Sn in the fringing region
are measured by changing the location of the short coil from the edgeof the magnet core. The results are shown in Fig.6 where "d < 0" means "inside the magnet". We measured these unfortunately at
I == 200 A because of the cooling problems for the larger excitation
current. We cannot distinguish any spectrum at n = 3, 6, 7 and 9.
The 16-th mulipole component (n == 8 ) seems to be mainly
caused by the return flux of the fringing field. A field clamp or an end
plate can completely compensate this component though this effectin L 9 is negligibly small. To reduce dodecapole component ( n == 6), cutting the pole face seems better than attaching some end plates,
since this component is enhanced at d ~ -rB/2. For the decapolecomponent ( n == 5 ) at I == 200 A, the integral of the fringing regionis estimated as about
2 x 0.1(mV) x 75(mm) = 1.5 x 10- 6 Volt· m,which corresponds to 2.1 x 10 2 T/m3 • On the other hand, the long
70m- 3 as shown in Table 3, that is,coil measurement gives £5=

I iB< >idz
4= 70 x 3.116 x 10- 2 x 200 x 0.4575 ~ 2.0 x 10 2 T/m3.

The strength of the octupole component in Table 3 would be explained by .the similar calculation as above, although this component
is not seen in the small coil data. Consequently we can completely
reduce the higher error field ( n f. 2 + 4k ) by a suitable "pole-facecutting".

Table 4. Comparison of higher multipoles in ,the effective lengthRot. coilG(T/m)
L3(1o- 2m -l)
L4(m- 2 )L5(m- 3 )L6(104m-4)L7(1o- 5 m- 5 )Hallr:=; 50 mm

3.12

2.74-3.83

70.1
8.21

203-8.32-1.42
49.0
-9.88

Reference
1. A. Ando et a/., Proc. 6-th Symposium on Accelerator Scienceand Technology, 1987, p.310.
2. T. Kasuga et al., Proc. Conf. on Computation of Magnetic
Field, 1976, p.177.

We also measured the magnetic field at I == 88 A with a Hall
generator by mapping in the medium plane, and found the multipole
components by polynomial fitting. In the fitting, we used an iterative
method, in which once we found some polynomials, at the next step,
we added the best term to minimize the chi-square. We only show
the results for the effective length in Table 4, because at the center ofthe magnet, higher multipoles are too small to be distinguished which'term is more meaningful. Table 4 also compares with the data of theKEK 12-Gev PS 3, which are obtained by a simple polynomial fitting
including terms upto n == 14 for the field differences given by a twin

3. A. Ando et al., Proc. 9-th Int. Conf. High Energy Ace., 1974,p.610.

Short Coil

200 A

0.4 r - r - - - . - - - - - - . - - - . , - - - - - . - - - - , - ,coil.

•...•

0.3
10.------,---- --;-------,---- --,------,----- -.
6·····················k················· ·-······-········:··+···················· ·-······--·········+······················=-- ---+---,/

Yto,/.....

-..........
--~---.

//

0.0v•

/""'

,.....,...........-

v--~:r-·----.....-·/-----------------------t-----·-··-------·-··---

V5V8/10,...~~ 0.1

6~----~~---+----~------+-----~----~;--·- ...

> 0.2!

·······················t··············································t······················· ·······················t················-·····=~=J._~==J.

1.28
9.2
0.33
-70.2
4.89
6.12

2.74

In the polynomial fitting, the strength of multipole components
depend on the fitting procedure, that is, how many polynomial terms
are included and how wide the fitting reagion is. The resultant coefficients couple with each other, and may lead us to misread in particular
for multipoles which do not ordinary exist by the symmetry. Rotating
coil method is usually available for a small region because of the magnet aperture, although higher multipoles become more important as
a radius is larger. But we can correctly know multipole fields without
any interference among them. The only limit is the noise-to-signal ratio. In view of beam dynamics in a synchrotron or a storage ring, theimportant multipoles are usually those of n :=:; 10. As to these field,
we can know their strengths ( also phases ) with enough accuracy bythis rotating coil method.

Comparison with another measurments

4KEK-PS
r:=;8omm

r :=:; 80 mm.......- '.....

...........,____ _../'I'_..///... --~----,' "-:a!.'----------·-·····•···------------·/,; - - - - - -A- - - - - - ...... - - - - -2~---~!---~--=~==~~~=--~--------------------··i······------------------ ----------------------+-------------··················--·····------+------~;
0 o~----~-----1~o~o----~----~2~o~o----~------3~o0I (A)Fig. 5 Dodecapole at the magnet center versus excitation current.

-162--o. 1 ~-~5~0------~2~5~----o~-----2~5~----~5~0----~7~5~

d (distance from the edge, mm)
Fig. 6 Change of multipoles in the fringing region.