IEEE Transactions on Nuclear Science, Vol. NS-32, No. 5, October 1985

MAGNETIC FIELD COMPUTATIONS OF FRINGE FIELDSBETWEEN A DIPOLE AND A QUADRUPOLE MAGNET

L. Rinolfi, CERN, CH-1211 Geneva 23, SwitzerlandK. Preis, K.H. Richter, H. Stogner, IGTE, Austria

The aim of this paper is to present theresults, of 3-D calculations, on the interferencesbetween a dipole and a quadrupole magnet in the CERNAntiproton Collector lattice.

Introduction

The uniformity of field in the ends of thegood field region of a dipole is affected by theproximity of other magnets. In the ACqL ring wheremagnets are short and of large2 aperture , this effectdeserves particular attention .

With a configuration as shown in Figure 1, itwas planned to study:- the end field of the dipole in the presence of an

unexcited quadrupole;- the end field of the quadrupole in the presence of

an unexcited dipole;- the end fields of the dipole and the quadrupole

with both at nominal current.We describe here the first case where the

quadrupole is simulated by a plate with the samepermeability Pr as the dipole iron. The angle is7.50 and the distance between the plate and the centreof the dipole face is 0.495 m (Fig. 2). For compa-rison, computations were maje with a plate parallel tothe end face of the dipole

Finite element software package

To calculate the magnetic field distributionwithin the dipole, the Finite Element Method (FEM) isused. The three-dimensional (3-D) software packagefor numerical calculations was developed at IGTE(Institute of foundations and theory of electricalengineering) at the Technical University of Graz,Austria. The software package is based oq 20-nodedhigher order isoparametric finite elements

Mathematical description

The field problem under consideration can bedescribed by the well-known Maxwell equations:

curl H= 3 (1);curl E = 0 (2);div B=O (3);B= pFH (4)

where Se is the impgsed current density, E theelectric field, H and B are the magnetic excitationand magnetic flux density, respectively. The magneticpermeability Pr is assumed to be isotropic andconstant.

In our case of stationary fields, the Maxwellqquations (1) and (2) are decoupled, which means thatE, which causes the current density 4e9 can beexpressed by integration of the equation (2):

E = -grad 4

where 4 is the electrical scalar potential.

(5)

Since the d~vergence of t is zero, the magne-tic flux density B can be described by a magneticvector potential A

B = curl A (6)Substituting t in eq. (1) and usinq equations

(4) and (6), one obtains

curl (t curl A) = Se

Fig. 1: Dipole quadrupole arrangement in the ACOL

lattice

Ipr = 531

where u =1/p is the magnetic reluctivity.Introducing vector identities, equation (7) becomes

grad u x curl A + u grad div A - AM=e (8)

Imposing the condition div A = 0, we obtainour final differential equation

grad u x curl - A = Je (9)

which has+ to be solved for the magnetic vector

potential A.

The boundary conditions for our problem are

restricted to Dirichlet and homogeneous Neumann

boundary types.

+ Using the FEM, the maqnetic vector potentialA, the current density Je and the magneticreluctivity u can be expressed by

+ n +

A = Y Ni Aii=1

+ n +

= Ni Jeii=1

I =1838A

nU = NNi A

i=1Fig. 2: Top view of dipole and tilted plate

(10)

(11 )

(12)

0018-9499/85t1000-3616$01.00( 1985 IEEE

(7)

3616

3617

for each finite element. The Ni's are the so-calledshape functions which are functions of the localcoordinates of a finite element vnd n is the numberof nodes (20 in our case). The Ails are the unknownodal values of the magnetic vector potential, the

Jei's and °J' s are the known nodal values of thecurrent density and reluctivity, respectively.

Applying Galerkins method and equations (10 to12) for solving equation (9), we obtain a volumeintegral over the domain 0 for each finite element:

f N.[grad( Y Niui) x curl( Y Ni At - Niui NiAi0

The deviation from uniformity is characterizedby ABy By

Bo Bo(16)

where Bo is the magnetic field at the centre of themagnet.

For a beam which is passing through a magnet,the relevant quantity is the integrated field throughthe magnet along the z axis:

+ c

IF I By(x,y,z) dz (17)

- T N i . ] dQ = 0 (13)

Integrating Equ. (13) by pasts and introducing theadequate boundary conditions for the finite elementsleads to a set of linear equations which can bewritten as

[k]A}= (R} (14)

for each finite element. The solving procedureapplied in the 3-D FEM software package is theconjugate gradient method preconditionVd by anincomplete Cholesky factorization (SICCG)

Finite element model of the arrangement

The geometry of the dipole is shown in Fig. 3.

The (2n-pole) components of the field are derivedfrom harmonic analysis of the integrated field IF

+00 +00 o nI = f B dz = f (B + iB )dz = a (x + iy) (18)F -00 -co y x n=u n

In the median plane Bx= 0, therefore

IF a + ax + a2x + a 3x + ...(19)

The coefficients ao, a 1, adipole, quadrupole, sextupole ... com

The variation of IF is

f B dzAIF 0 Y

0 y'F o (1J By dz)x\

2 are callednponents.

(20)

For computational convenience, the integrals are takenfrom the centre of the magnet to infinity.

From the above quantities we define an effec-tive length for the magnet:

1 XQ =- f B (x, y, z) dz (21)eff B o y

and AQeff

where to is

= eff fo (22)

Fig.3: Model of dipole for numeric fieldcalculations

The final 3-D structure consists of 18 layerswith 5184 finite elements. The total number ofunknowns (nodal values of components of the magneticvector potential) is 58500.

1

2QD f= B (x=0, y=0, z) dzy

Using equation (19) we have

aO

Bo

(23)

(24)

The end field contribution begins with a 1 (nosymmetry) or a2 (with symmetry). The fundamental termao is,by definition, included in the effective length.

Dipole end fields

The dipole has a magnetic field B which variesaccross the aperture due to end fields and otherperturbations.

In order to respect boundary conditions in thedipole, there is only the vertical component By inthe median plane x = 0, y = 0. We expand By as

follows

B = B (1 + b x + b 2+ ...) (15)y b1 b2

The odd terms bn vanish when there is symmetry.

All variations of these quantities will beconsidered within the good field region of the dipole,x = ± 186 mm.

Sextupole term due to end field only

The dipole studied here is a combination ofwindow-frame and H magnet types. The permeability isassumed to be uniform and equal to Pr= 531.The current density is constant: coil 1 has2.469 x 10 6 A/m2 and coil 2 has 2.872 x 10 6 A/mn2.

From Parzen's formula 7, we get

b 7 x 10- 3m2

1

3618

Dipole with tilted plate simulating a quadrupole

In this case, there is no symmetry and allterms of equation (15) can be present.

From the results obtained by the model descri-bed above, we plot the curve My/Bo versus x

(Figure 4).

-14

Fig. 4: Magnetic field in the median plane

Within the limited accuracy of this curve

(due to limitations from the mesh), the field B in themedian plane x=O, y=O, does not depart from Bo up tox = ± 14 cm. The results obtained for the integratedfield IF (Figure 5) are very accurate. The effect ofthe tilted plate shows up in the slope of the curve.

Curve fitting for magnetic field, integratedfield and effective length give the values summarizedin Table 1.

Quar.(m- )

Sext.(m 2)

Oct. DecaLp .

(m-3) (m- )

AB /B 0 0 0 1 .5yo0At /o 9.1 10 -0.08 2.0 58.2

eF/F o9.1 10- -0.09 2.0 76.4Al/F 9.1

Table 1: Multipole components from the studiedconfinurrt inn

AIcI,fs

1o-3

Fig. 5: Integrated field in the median plane

Next, we consider the effective PE

Aieff/Zo plotted in Figure 6.

Fig. 6: Effective length in the median plane

Table 2 gives a summary of the dipole charac-teristics pertubed by the vicinity of a tilted platesimulating a quadrupole.

Conclusion

X The results of the computation show errors

which are very small and while it is necessary to cor-

rect them, this is within the range of correctiontechniques by shimming used to linearise the field.One obvious improvement might be a full three dimen-sion model of an excited neighbouring magnet.

Acknowledgements

The authors wish to thank E. Jones for hissupport and H.H. Umstattter for useful discussions.They also are very grateful to W.M. Rucker andJ. Franz at IGTE for helpful collaboration.

engthReferences

[1] ACOL magnets for the design report, L. Rinolfi,PS/AA/AC 16, August 1983.

[2 Design study of an antiproton collector for the

e antiproton accumulator, edited by E.J.N. Wilson,CERN 83-10, October 1983.

[3] End field of a dipole in presence of a quadrupolearrangement as in the ACOL lattice, L. Rinolfi,H.H. Umstatter (CERN), K. Preis, H. St6gner (IGTEGraz), PS/AA/Note 85-3, January 1985.

[4 1 Application of higher order finite elements in

3-D magnetic field calculations, H. St6gner,K. Preis, J. Appl. Phys. Vol 53 (1982),p. 8411-8413.

[5 1 The finite element method, O.C. Zienkiewicz,McGraw-Hill Book Cie, London 1977.

[6 1 The sifted incomplete Cholesky factorization,T.A. Manteuffel, report No. 9 snd No. 78-8226

Sandia Laboratoris, Albuquerque N.M. (1978).[7 ] Magnetic fields for transporting charged beams,

G. Parzen, BNL 50536, January 1976.

At the At the edge of qoodaB= 0.945 m Units Theore- centre field region

tical ofg = 0.066 m values dipole

x = 0 x- 18 cm x= + 18 cm

B (T) 1.600 1.637 1.639 1.639y

AB /B 0 1.2 x 0-3 1.2 x 10- 3iTo

I fJ 8 dz T.m 1.56104 1.69122 1.68958 1.69218F sy

AIf/IF0 0 -0.97 x 10- 3 0.57 x 10- 3

teff (half dipole) m 0.97565 1.03312 1.03086 1.03245

At / Q 0 -2.2 x 10- 3 -0.65 x

eff o

(teff - B)/g 1.30 1.32