Effects of the Leakage Flux in the Inhomogeneity of Magnetic Field in a Sectorial
Electromagnet
Yasuhiko Okuma
Research Center for Nuclear Physics, Osaka University, Suita, Osaka.
The difference between magnetic field strengths in the center and in a measuring
point in a sector type of a slim electromagnet with_~rdinary iron pole piece& had beenmeasured under the degree of measuring precision 10in such a way as to not cause theeddy-current effect and magnetic after-effect. Important sources producing the inhomogeneities of magnetic fields in the pole pieces with a sharp-cornered profile are thesaturation and hysteresis effects in the central ray, and the saturation and magnetic
asymmetry of the geometry of pole pieces in the radial direction. The saturation effectis the superimposition of the internal flux and leaka~e flux from the edges to centerin the pole pieces. 'rhe hysteresis effect is the difference between leakage fluxes in
the magnetization and demagnetization processes because of the saturation in the polepieces. The magnetic asymmetry of the geometry of pole pieces is the difference between
leakage fluxes in pole-piece sides with the small and large curvature radii. The B-constant design of pole pieces has excellent effects in avoiding the inhomogeneities ofmagnetic fields caused by the above sources, although the small hysteresis effect due
to the residual magnetization in the pole pieces remains.
The magnet has a cugvature radius 1000
mm, deflecti~n angle 45 and pole-piece width 240 mm.
The thickness of thB main air
gap is 39.899=0.004 rom. The magnet has ther-~~~~r-~~----r--=~----~"~'''~·~·-~~-..~.~.
u.u--- ...1.uve ,.ne&biuty!~e;~c~~~~:~~i~~~s~ 6 ~!r;f:~e~P:~~~~a~:~hinfrom the yoke. The power suppl;r 5has a curr-.
.
ent stability better than 5xl0 • The mag£~~~e0~· ;~~= f~~~efr~~~.magnet~c propernet has two types of pole pieces with thePP
sharp-cornered (SCOF) and B-constant (B-const) profiles, which are made of a commercial grade of two kinds of low carbon and forged iron plates in table l, The differential probe consisting of the fixed and searchHall element_~r~~es has been used to detect the small differences of field strengths
less than 10 •The analog system can plot the magnetic field difference between thefixed and search probes versus the position of search probe.
Excitation curves --- Excitation curves in all pole pieces
in table l sf~it into two different curves SCOF and B-conExt:JtatronCUIW
st in fig.l.
Curves do not depend on the pole-piece iron
•SCOF1&315• SCOF 2, but on the pole-piece profile. The leakage factor in the"'B-const 1• B-ccnst 2
pole pieces SCOF is derived from .the splitting of the excSCOF
itation curve into two curves, ~S the pole pieces B~const
are assumed to have no leakage.
A leakage factor in thepole pieces SCOF is klif3 in the region ofthe critical fie10
ld strength B =10.3 kG.
Magnetic field distributions ---.Fig.2 shows the distribut1ons of the magnet1c field difference on the distance along the central ray R with a parameter of the field strength in the SCOF3 (a) and B-const2 (b), The field distributions along the central ray in the pole pieces SCOF change
LNilageofactot!n SCCFk.-Jfore"r~with the !ie·ld strength in dependence on the magnetization-demagnetization process because of the. hysteresis effect.
The field distributions along the centrai ray in the polepieces SCOF .and B-const .are summarized in figs.4(a) and(b), respectively,.
Fig.3 shows the distributions of the magnetic fieldI<'ig.2. 1'1agnetization curves.
inhomogeneity on the distant along the radial direction
with two parameters of the field strength and field homo- Critical field drengths Bandthe leakage.factor k ingenizers in the SCOfl, SCOF2 and B-constl. The The field
the region of B in the polestrength along the radial direction in the pole pieces
pieces SCOF are given ·in theSCOF shows the asymmetrical distri.butions de~'nding onfigure.
the field strength ·to the pole-piece center.
Magnetization curves --- Fig•4 shows the dependences of the magnetic field difference
on the field strength along the central ray which are related to the magnetization-demagnetization process •. Fig.5 gives the dependence of the magnetic field difference on
the field strength .along the radial direction.
2)
The field difference is represented by a eq.:--SCXF3--

1a1_,s"'COrist2--lbl

•I

..••

...

··1--------!.--"--'---["...

,...!.-,:)_

Fig.2. Nagnetic field distributions along the central ray in the sc;JF.~ (a) and B~const2(b), ,immbers on curves denote the order of field settino; ana also represent field
Jrtrength corresponding to numbers in fig.4. Distributions are summarized in fig.4.

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'"

C•1·SXF3,:::::,... .,...... ___--n_.,....
...,.. .., ,"'_....

n~~g;4.

Magnetization curves along the central ray. Arrows and numbers on curves denotethe order of field setting.
~B=B-u(R or r)=(H or r/d)•s1nP•(a/~)
SCOF2seeP<=(R or r/d)·~·~H
where B and B(R or r) are air-gap field strengths at R or r=O and R or r, respectively,p. and )L are the permeability of the vacuum
and relative permeability of the pole-pieceiron, respectively, d is the half air-gap
thickness at R or r=O, H is the magnetic'field, 8 is the inclination of a magnetic
induction line in the pole piece and B ·irri8l •is the component of the distance along thetI'pole-piece face which is averaged_over theihalf whole of the distance R or r. The depe~. • !
i •ndence of the field difference on the fieldstrength is interpreted to be the B-H curve
0r magnetization curve in the pole pieces,Fig.3. Magnetic field distributions along
The maGnetic field in the pole pieces
the radial direction in the SCOFl, SCOF2
SCOF exhibits the saturation effects dependand B-constl. Letters (e-a) on curves 'indi- ing on the distance, field strength and polecate the order of field .setting and also-piece iron. F1g.4 shows the saturations
represent field strengths corresponding to depending on the-distance Rand field strenletters in fig.5. Solid and dotted curves
gth in the SCOF3 (a), and on the pole-piece
denote distributions in magnets with and· iron and field 'strength in the SCOFl andVlithout the field homogenizers.
SCOF2 (b). Fig.5 gives the saturations depending on the -distance r and field strengthin the SCOFl (a), and on the pole-piece ironand field strength in· the SCOFl and SCOF2(b). Both po·le-piEu;e'sides with ·the small
and large cu~vat~~e radii ;show a perfectly
same saturat1on •c: .. awn
The hystered·s effect is the pnenomenon
ll:roaw., ...:IWIZ
which is related to the· field-setting proce=t:.·.
dure of· the magnetization-de magnetization ·and is observed !!ijdz~ the degree of measuring
precision ~0 •Different pa:;ts along
0 I2l·I_.,..,'' 'the central ray·in the'pole _piec.es SCOF-aft"'~·never go through the same fysteresis loopoecause of the saturation in the pole pieces
F1g.5. magnetization ~urv_es _a~ong the_ radi- ,, Fig.4 showsthe· hysteresises depending olial direction. Curves 1n pos1t1v7 and nega- the field-setting speed dB/dt and distance R
tive distances represent those 1n. polein the SCOf3 (a) and on the pole-piece ironpi'!lce sides with the _large and small curv- _in the SCOFl and SCOF2 ('b).
.
at,ure ra.di~, respec~1vely, The W is a main.\. field distribution before ·the saturaair-gap th1ckness d1stance (W=40 mm) •.tion resu·lting. from the leakage· 'flux in theSC0>3 i.s shoVIn with a bold solid curve in fig.3,3J l'he asymmetry o·f .th'!l. fiel<i di'.stribution. to the pole-piec.e center along the radial direction in the. pole pieces SCOF· will·
be due. to the difference- betVIeen lea:ka5e fluxes in pol·e-piece s.ides Vlith the small ·andlarge c\trvature radii, which re-sults from the. ma.:.netic asy·umetry· of the gi!ometz:y· ofpole pieces along the radial direction.
.
··B-constant design of pole pieces - - 'fhe 'B-constant design of pole pieces has 'Prominent
effects ·.i.n avoiding many phenomena caused in the pole pieces ·SCQF,.., ..
Hysteresis loop SCOfl and SC0~'2 in the pole pieces SCOF fa:J.l into approximately straight lines with the, small cycling 'behaviour .d-constl and B-const2 in the pole pieces
B-const, respectively, in fig.4(b), 'fhe fact means that .,tl;le~·pole pie<;lils.~const _exhibitl
neithe: ~he satu:a~io_n or b~ysteresis effects and also :·pe_rme'aoilities iri' tlie pole pieces
have f1n1te, pol(1,tJ,ye and constant values over the whole working field region,·
Magnetization ·curves in the po.re· pieces 3-const depend on the disj;ance: r, show .noany dependence on the field strength an4 have a perfectly same behaviour in both polepiece sides with the. small and large curvature radii in fig.5(c),..

I

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1) Y.:ikum_a_, K.Yagi, T.Kumabe and K.Natsuda, Nucl. Instr. and ;1eth, l!2.'.· (1972) 317.
2) Y.Okuma, K.Yagi, I.Kumabe and K.Natsuda, Nucl. Instr·, and Heth, ~ (19'/9) 27,
3) Y.Okuma, K.tagi, LKumabe and K.Hatsuda, present in elsewhere.
2 and 3) ·y,()kurna, LYae;i, Ll:umabe and K.l1atsuda, to .be published in the Intermag Con!.
Pros., Boston, (1·980).
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Acknowledgement: 1'he autnor wisnes to thank Prol'.I.Kumabe, Prof.K.Yagi and late Dr.Ki
i•latsuda, and a.lso Prof .M._Kon<lo for their supportl and encourage,nents throughout the .work•.

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