Indian Journal of Pure & Appl ied Physics Vol. 39. January-February 200 1 , pp. 32-35

Taylor's series method of finding magnet pole profile .

P R Sarma & R K Bhandari

Variable Energy Cyclotron Centre, Department of Atomic Energy, Block-AF, B idhan Nagar. Calcutta 700 064

Received 7 November 2000

The tield quality of magnets can be i mproved by optimizing the shape of the pole. One often uses circular proti les in quadrupoles and sextupoles where the radius of the circle is optimized for mini mizing the dominanterror harmonics. A new method of calculating the harmonic coefficients by using Taylor's series expansion technique has been adopted. It has been applied for optimizing the radi i of the circular pole profi les in quadrupoles, sextupoles and octupoles.

1 Introduction

In the beam l i nes in accelerators and in synchrotron rings, various types of magnets are required for focusing and transporting the beam. In most of the appl ications, especial l y in synchrotrons and in h igh resolution l i nes, the magnets must h ave high field quality so that the beam quality remains good after the beam passes through the magnet. The field quality of an iron magnet depends main ly on the shape of the pole face. One can achieve h igh field quality in magnets by choosing ideal pole shapes, e.g., hyperbol ic profi le of very l arge width in the case of quadrupoles. But in practical magnets one has to have a l imited pole width in order to insert the coi l s and otherwise also the s ize of a magnet cannot be arbitrari ly large. As a result of the truncation of the pole width , field errors creep in . The field due to a magnet can be described in terms of the harmonic expansion of the potential V(r,8)

gi ven by :

U(r,8) = Q"rJlcos (n8) + 113nrlncos (3n8)

+ usnr'ncos (5n8) + . . . . . .( I )

where n = I for a d ipole, n = 2 for a quadrupole, and so on. a's are the coefficients of various harmonics. The harmonics other than the first one give the errors in the field . The harmonics have rJl_ dependence and so the most dominant error harmonic is the a3n-term. The goal i n pole designing is to find a pole shape which min imizes the magnitude of one or more dominant error harmonics .

There are various ways of reducing the error terms in a magnet. For a quadrupole, Hinterberger

and others ' used a . pure hyperbol ic profi le terminated by shims at the coi l windows. Others use hyperbol ic poles truncated with straight l ines at the outer edges2• More compl icated shapes have also been used3.4. Danby and others5.(, designed and made poles consisting entirely of p lane surfaces. In h igh field qual i ty sextupoles at RIKEN7 ideal pole profi le is used up to a point and then a long radial shim is used . Someti mes people use poles which approximate the ideal pole shape with a few straight sectionsK•

A simple design, which is often employed in quadrupoles and sextupoles, uses circular poles where the pole face radius is opt imized for reducing the dominant error harmonics9- ' 2 . The numerical method9. '2 used in finding the harmonic coefficients in these designs is to c hoose a number of points (r,,8 1 ), (r2,82) etc . on the pole face and put these coordinates in the harmonic expansion of the potential (which i s taken to be constant on the pole face) . Thus one obtains a number of equations of the form:

U(r,8) = an r;"cos (n8;) + aln r;_lncos (3n8;)

+ asn r/ncos (5n8;) + . . . = I ; i = 1 ,2, . . . (2)

This set of equations is solved to determine the coefficients.

In the present work we have used a different technique in which we expand the equation of the circ le in terms of a Taylor's series. The Taylor series coefficients are then put into the various derivatives of the harmonic potential equation . Corresponding to each derivative, we obtain one equation . This set of equations is then sol ved by the matrix inversion

SARMA & BHANDARI: MAGNET POLE PROFILE 33

technique to get the harmonic coefficients. This calculation is repeated for a number of pole radii . The radius, for which the magnitude of the dominant error harmonic coefficient is minimum, is the optimum radius. Thus this method presents an alternative · method of optimizing the pole profile. This technique can also be applied to pole shapes other than the circular.

2 Method

We do the calculation In the cartesian coordinate system as it has been found that the Taylor's series expansion is easier in this system in the present case. Also, the numerical values of the derivatives come out to be smaller in the cartesian system. This is important because the error in the calculation increases if the magnitudes of the derivatives become large. Referring to Fig. I , the equation of the circle representing the pole is given by:

. . . (3)

where R is the radius. The half-aperture is taken to be I . From this we expand x in terms of a Taylor's series in y (we do not attempt to expand y in terms of x as dyldx is infinite at y = 0). Similar Taylor's series expansions are lIsed in other fields also ' �- L�. The expansion is:

x = I +( I 12 ! )XI21y2 + ( 1 /4 ! )XI4)l + ( 1 /6 !)X(h)/, +. . . . . .(4)

where

Xi II) = dllxldyll

These derivatives evaluated at (x = I , y = 0) are given by:

xi21 = ( I IR)

X(4) = ( 1 /R)r 3(xi21n

Xl!»� = ( I I R)1 1 5x(4)Xi1 l I Xl"' = ( I I R)f 28xiC'IXI2 ' + 35(xI4)f]

Xi i II) = ( 1 /R) 145x(� )x'2 1 + 2 10 X((')X(4)]

x( )2) = ( 1 /R)l 66x" I I )Xi21 + 495x(X)x(4) + 462(x(fo» 2]

Xi l4) = ( 1 /R) [9 1 x' l 2ixi21 + I OO l x(IU)XI4) + 3003x(�)X(6)]

Xl If.) = ( 1 /R)1 1 20t'4)x'2) + 1 820x( l 2)x'4)

+ 8008x( I II)X((') + 6435(x(f')f]

The odd derivatives viz., Xi i >, Xl�) , etc . do not come into picture as x is an even function .

1 .5

1 .0 iii' a: :;) ... a: 0.5 w � � I&. -I '" 0.0 % I&. 0 I/) t: z -0.5 ::I Z :::. > -1.0

-1 .5 '--_---' __ --'-_..;.....-'-__ ...L..._--' 0.0 0.5 1 .0 1 .5 2.0 2.5

X (IN UNITS OF HALF-APERTURE)

Fig. I - Circular pole profile of a magnet; R is the radius of the circle in units of the half-aperture of the magnet

The potential equation in the cartesian system is given by:

V(x,y) = Re[an(x+iy)lI + a�n(x+i.Yrll + a5n(x+iy)'I1 + . . . 1 = r. an V(n.x,y) 1 . . . (5)

The various derivatives of V(n,x,y) with respect to y are:

VI I ) (n) = n (}2) (n) = n[x(21 - V( I ) (n- I )] lJ3) (n) = n[2lJ') (n- I ).t2) + (}2) (n- I )]

lJ2m)(n) = n[.t2m) + 2m- I C2 V(2) (n_ l )xI2m.21 + . . . _ lJ2m- l ) (n- I )] ·V(2m+I )(n) = n[x(2m)Cl lJ l ) + (2mm' I )C3VO) (n- I )XI2m'2 ) + . . . + lJ2m)(n_ I )]

and so on. We now write down the fol lowing set of equations :

an + a3n + asn + a7n + . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . = I all(}2) (n) + a3nlJ2) (3n) + a5nlJ2) (5n)

+ a7nlJ2) (7n) + . . . . = 0

a1lV(4) (n) + a3nlJ41 (3n) + a5nU4) (5n)

+ a7nU4) (7n) + . . . . = 0 . . . (6)

The above equations are · solved by the matrix inversion technique to get the coefficients an, a3n, a'n

34 INDIAN J PURE & APPL PHYS, VOL 39, JANUARY-FEBRUARY 200 1

etc . This is done repeatedly by varying the radius R. The optimum radius is that value of R for which the magnitude of a,n is minimum or zero.

...... W " ::l � n:: w 0. < I LL .J < :r u. o (J) !: z ::l Z ..... (J) J -o < " :E ::l :E i= 0. o

1.25 ,.......---------...,

0.95 '--........ ......I.-�-.I..-.l....-........ --'

o 2 4 6 8 10 12 14

NO. OF COEFACIENTS

Fig. 2 - Calculated optimum radius for a quadrupole as a function of the number of coefficients used. The dotted l ine is for R = 1 . 1 44

3 Results and Discussion

The new method has been applied for optimizing the circu lar pole profiles of quadrupole, sextupole and octupole magnets. It has not been applied to a dipole as it is not realistic to have a dipole of circu lar pole profi le. It is obvious that the determination of optimum R becomes more and more accurate as we include more and more number of coefficients in Eq . (6). Fig. 2 shows the calculated opti mum radius for a quadrupole as a ftlncti on of the number of coefficients used. It is clear that the optimum value settles down to a value

of 1 . 1 44. This is in agreement with the value 1 . 1 45 obtained with the earlier numerical method . The optimum radi i for the sextupole and the octupole come out to be 0.55 and 0.35 times the half­aperture. These values are also c lose to the values obtained earlierl 2 .

The method can be extended to include other pole shapes also for minimizing higher harmonics. An example of a simple extension of the circu lar profi le is the fol lowing:

(Py2 + (x- I -R)2 = R2 . . . (7)

where d is a constant. For d< I the pole becomes elongated along the y-axis and so the pole width increases. One can then optimize R and d to

minimize both a.," and a5n' Here the formu lation remains the same, only Xl21 is modified to Xl21 = diR. This has been used to reduce I al ii I for a quadrupole to below I .x 1 0" whereas a circular profi le gi ves

I a lii I = 2.4 x 1 0-·'. The optimum values of R and d in this case are 0.90 1 and 0.8 1 respective ly .

Both the present technique and the earl ier numerical technique sol ve a number of l inear simultaneous equations for the evaluation of the harmonic coefficients. The earlier technique depends on the evaluation of the potential at a number of points chosen on the selected profi le. The separation between the points has to be chosen systematical ly, otherwise one gets inconsistent results . In the present method no such problem is encountered.

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