Murphy, J B

Reusch, M F

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Particle Accelerators, 1993, Vol. 42(3 - 4), 171-179Reprints available directly from the publisherPhotocopying permitted by license only©1993 Gordon & Breach Science Publishers, S.A.Printed in the United States of America.APPLICATION OF DIFFERENTIAL ALGEBRA DATOOLS TO MAGNETIC FIELD CALCULATIONS*JAMES B. MURPHyl and MICHAEL F. REUSCH2lNational Synchrotron Light Source,Brookhaven National Laboratory, Upton, NY 119732Grumman Space Systems, 4 Independence Wa)J, Princeton, NJ 08540(Received 1 July 1992; in final form 24 June 1993)The calculation of the magnetic field for an arbitrary-geometry air-core magnet is performed by numerouscomputer codes. However, the accelerator scientist is often interested in the multipole content of the fields,which can be difficult to compute accurately and to high orders for arbitrary coil geometry. Applying thetools of differential algebra, a computer code has been devised that can compute arbitrary-order multipolesand the changes in these multipoles as a function of the coil locations for air-core magnets.KEY WORDS: Electromagnetic Field Calculations, Magnetic Fields, Magnets: Air-Core1 INTRODUCTIONCompact electron storage rings with superconducting dipole magnets are being builtas sources of soft x-rays for x-ray lithography.l In several of these designs the dipolemagnets are of the coil-dominated air-core variety. For example, the SXLS ring,under construction at BNL, contains two 1800 air-core superconducting dipoles witha nominal field of 3.87 1: Because the machine is so compact, with a circumferenceof only 8.5 meters, the dipoles must be "combined-function" magnets that contain agradient for vertical focusing and a sextupole component for chromaticity correction.The field is generated by a set of coils shown schematically in Figure 1.The magnetic field in an air-core magnet is given by the Biot-Savart Law,2(1)where J( f} is the current density in the conductor and r is the observation point.* Work performed under the auspices of the u.s. Department of Energy and funded by the DefenseAdvanced Research Projects Administration171172 J.B. MURPHY and M.R REUSCHFIGURE 1: Coil geometry for 1800 air core superconducting dipole magnetFor a given coil configuration, the field can in principle be computed to the limits ofthe computer's precision, since there is no cumbersome mesh required as in an irondominated magnet. Numerous computer codes3 - 5 exist to carry out air-core magneticfield calculations. The difficulty remains in obtaining the correct multipole coefficientsand determining the sensitivity of these to the coil locations.2 THREE-DIMENSIONAL NONISOMAGNETIC FIELDS AND MULTIPOLECOEFFICIENTSFor the complicated coil geometry given in Figure 1, the magnetic field is trulythree-dimensional, the curvature of the magnet is large (p = 60 cm) and, for an air-core magnet, the field is nonisomagnetic. The general expression for the midplanesymmetric magnetic field in curvilinear coordinates (x, y, s) is given by Brown andServranckx6 . For the present purpose it suffices to write out the expression only tosecond order in the transverse coordinates x and y,Bx(x, y, s) = AllY + A 12xy + ...1 2 1 2By(x,y,s)=A10+A11X+2!A12X +2!A30y + ...1 [ .Bs(x, y, s) = (1 + hx) AlOy + .,.(2a)(2b)(2c)DA TOOLS FOR MAGNETIC CALCULATIONS 173where Al,n(s) == anBy(s)jaxn, A30 (s) == -.A lO - A12 - hAll, Alo == dAlo/ds andh == 1/p(s).The coefficients Al,n (s) == anBy (s) / axn, are the multipole coefficients widely usedin accelerator physics; AlO == Dipole, All == Quadrupole and A 12 == Sextupole,etc. There are two other types of terms in Equation (2): nonisomagnetic terms, forexample, Alo ; and combined function feed-up terms such as hAll.Several methods to obtain the multipole coefficients have been used in the past:1. Numerical finite differencing to compute derivatives.2. Fitting of the pointwise field data by polynomials in x and/ory.3. Fast Fourier Transform (FFT) of the field on a circle surrounding the referencetrajectory.Each of these methods suffers from drawbacks for nonisomagnetic combined-function bending magnets.Numerical finite differencing yields unacceptable results for all but the lowest ordermultipole coefficients.Typical problems encountered with polynomial fitting of data are as follows:How many data points are required?Over what range in x or y should the fit be applied? And,What is the maximum order of the polynomial to be used?Finally, the FFT method, which is best suited for straight multipole elements such asquadrupoles or sextupoles, runs into trouble with the nonisomagnetic and combined-function feed-up terms.3 APPLICATION OF DA TOOLS TO FIELD COMPUTATIONSWhat is really desired is to be able to obtain a Taylor series expansion of equation(1) in x, y and s directly. This is precisely what the DA software of Berz7- 8 can do.In this way the multipole coefficients, the nonisomagnetic terms, and the combined-function feed-up terms are obtained without any of the problems associated with theabove methods. We have written a computer code to perform these magnetic fieldcalculations and added DA capabilities to be able to compute the desired multipolesthroughout the magnet [Al,n (s)] and the changes in these multipoles due to positionalerrors in the coil placement9 .The coils in the SXLS air-core superconducting magnet are constructed out of arcsor straight sections of rectangular cross section. For a magnet constructed of arcs ofrectangular cross section with a uniform current density Ji , the vertical field in themidplane is given as a single numerical integral over the angular extent of the coil,and the radial and vertical integrals in Equation (1) can be carried out analytically:174 J.B. MURPHY and M.E REUSCH3-D View Top Viewyzh.I, w·~_I---&- -6- X-+..:-----f--+--~ Xr .IzFIGURE 2: Geometry of arc of rectangular cross sectioncoils 1¢2 [ (vi 2 2 rI... 2 2 )By(x) = J.lo L Ji d¢ yIn r - rx cos '/' + y + x +:c - cos¢ (3)41T i=1 ¢1 X Xx rl...l ( Jr2 - 2rx cos ¢ + y2 + x 2 + y)--Cos~ n2 Jr2 - 2rx cos ¢ + y2 + x 2 - Y( ) ]r=ri+W,Y=Yi+h• rI... -1 y(r - x COS ¢)-x SIn ~tanx sin ¢Jr2 - 2rx cos ¢ + y2 + x2 r=ri ,Y=Yiwhere ri and Yi are the lower left hand coordinates of the coil location, Wi and hi arethe width and height of the coils, respectively (see Figure 2). For a midplane symmetricmagnet the above sum need only be done over the top plane coils and the result issimply doubled. A simpler expression, where all the integrals can be done analytically,for a straight section of rectangular cross section is given in Reference 3.From the point of view of the DA tools, the above expression is simply an analyticfunction of several variables:(4)The DA tools provide for arbitrary-order differentiation with respect to any of thevariables (¢, Xi, ri, Yi). It must be noted that the derivatives are not computed asnumerical finite difference quotients, which would give very poor accuracy.DA TOOLS FOR MAGNETIC CALCULATIONS4 MULTIPOLE COEFFICIENTS175The DA package can compute the Taylor coefficients, a;:By (s), defined as follows,where the a;:By (s) are evaluated on the reference orbit and x is the deviation from thereference orbit in the radial direction. The Taylor coefficients are simply the multipolecoefficients widely used in accelerator physics codes. The nonisomagnetic terms inequation (2) are also directly computed as a,,;,a;:By(s) for arbitrary m and n. Fora midplane symmetric magnet the other two field components, B x and B s ' can beobtained from the a,,;,a;:By (s) or an expression such as equation (3) can be used withthe DA tools to produce these multipole components.5 COIL POSITION TOLERANCESThe DA tools can also be used to compute the effects of coil positioning errors onthe various multipole by producing a Taylor series of jj with respect to the position ofeach of the coils, ri and Yi, which serve as limits to the integrals in Equation (3). Sucha tool is very valuable to set limits on the coil positioning errors during constructionof the magnet. Table 1 shows some of the possible coefficients.6 APPLICATION TO AIR-CORE SUPERCONDUCTING DIPOLE MAGNETSThe DA-equipped version of the Biot-Savart was used to layout the set of coils shownin Figure 1 for the superconducting dipole magnet for the SXLS storage ring at BNL.Figure 3 displays the calculated multipole components through the decapole termfor half of the symmetric magnet, the center of the magnet is taken as s = o. Nearthe ends of the magnets (s ~ 1 meter) where the field is highly nonisomagneticthe DA-equipped version of the Biot-Savart law is particularly adept at extractingthe multipole coefficients as evidenced by the smooth, continuous behavior shownin Figure 3.TABLE 1: Effects of Coil Positions on MultipolesDerivativeaTl By(s)aYl a~By(s)aYl a~By(s)Tolerance CoefficientsChange of dipole field with rlChange of quadrupole with YlChange of octupole with Yl176 J.B. MURPHY and M.E REUSCH4.0.......-.... 2.5CdenQ).....,'-'aen1.0-0.5 0 0.5 1.0 1.5 2.0Arc length, s (meters)o.......-....E -0.5Ci3~ -1.0.....,'-""~en -1.5-2.0-2.5 L.--...L..---L---L---L.----L....--I----L---&----Io 0.5 1.0 1.5 2.0Arc length, s (meters)FIGURE 3: Multiple coefficients B(x,s)=2: Bn(s) x n In!, vs. arc length for half of a symmetric 1800magnet (center of magnet is at 8=0, p=0.6037 m).DA TOOLS FOR MAGNETIC CALCULATIONS 17760 ~~-...----r-----,--r-r---r-....,4020orE -20-40-60 0 0.5 1.0 1.5 2.0Arc length, s (meters)o500-100-300 L..-J.-----"-----'-----...---'- ---""__0.5 1.0 1.5 2.0Arc length, s (meters)....-...(of)~ 300Ci5Q)~ 100(of)mFIGURE 3: (Continued)178~ 0ox::: -2~E(0-4enQ)oof--oJ~~ -6ro-8J.B. MURPHY and M.E REUSCHo 0.5 1.0 1.5 2.0Arc length, s (meters)FIGURE 3: (Continued)In Table 2 the multipole coefficients at the center of the magnet are given as wellas the changes in these coefficients due to a lO-mil (.254-mm) deviation of the coilsfrom their ideal locations. The deviations were taken in a direction to maximize thechange to each of the multipoles independently and, in this case, to maintain midplanesymmetry.TABLE 2: Ideal Multipole Coefficients and Tolerances at the Magnet CenterMultipole Ideal Ali(O) ~A I i Coil PositionA IO [T] 3.87 ± 0.009All [Tim] -1.05 ±0.14A l2 [T/m2] -9.1 ±3.4A l3 [T/m3] -5.1 ± 100A l4 [T/m4] 595.1 ± 6,000DA TOOLS FOR MAGNETIC CALCULATIONS7 CONCLUSIONS179The numerical computation of magnetic field multipoles and the effects of coil posi-tion tolerances for arbitrary geometry air-core magnets is greatly facilitated by the useof differential algebra software tools. We have developed a computer code to carry outthis analysis for air core magnets composed of straight sections and arcs of rectangularcross section. The resulting code has been applied to the design of a superconductingdipole magnets for a compact storage ring.ACKNOWLEDGEMENTSThe authors are grateful to Etienne Forest of the Center for Beam Physics at LBLfor introducing us to the DA tools. We thank Sushil Sharma for help in preparing theinput data files.REFERENCES1. J.B. Murphy, X-Ray Lithography Sources: A Review, in Proceedings of the 1989, IEEE ParticleAccelerator Conference, p. 757.2. J.D. Jackson, Classical Electrodynamics, 2nd ed. (Wiley, New York, 1975).3. A. Schleich & A. Seggesseman, "Calculation of the Electromagnetic Forces Acting on the Coils ofAnElectromagnet", in Proceedings of the 3rd International Conference on Magnet Technology MT-3, p.198 (1971).4. S.J. Sackett, EFFI - "A Code for Calculating the Electromagnetic Field Force, and Inductance inCoil Systems ofArbitrary Geometry," Lawrence Livermore National Laboratory report UCRL-52402(1978).5. TOSCA, Vector Fields Ltd, Oxford, England.6. K. Brown and R.\Z Servranckx, "First and Second Order Charged Particle Optics," Standard LinearAccelerator Center report, SLAC PUB 3381 (1984).7. M. Berz, "Differential Algebraic Descriptions ofBeam Dynamics to very High Orders," Part. Accel. 24,(1989), pp. 109-124.8. The LBL version of Berz's DA package was used to perform all the DA-related calculations.9. M.R Reusch, E. Forest and J.B. Murphy, "Panicle Tracking and Map Analysis for Compact StorageRings," in Proceedings of the 1991 IEEE Particle Accelerator Conference, p. 1651.

Application of differential algebra da tools to magnetic field calculations

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