IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 22, NO. 3, JUNE 2012
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SIS100 Dipole Magnet Optimization and Local Toroidal Multipoles Pierre Schnizer, Bernhard Schnizer, Pavel Akishin, Anna Mierau, and Egbert Fischer
Abstract—SIS100 is the world’s second fast ramped synchrotron using superconducting magnets. The foreseen high current operation requires a sound understanding of the eld homogeneity next to a eld parametrization which allows investigating if the existing inhomogeneity introduces transverse oscillations on the particle beam. The SIS100 dipole magnets are curved so local toroidal multipoles were developed as these follow the orbit of the ion beam. The end of the magnet creates the main eld deterioration due to the coil end shape which has to be compensated by a 3D end shim. In this paper we present the Local Toroidal Multipoles next to formulae to derive these from rotating coil probe measurements. Further we outline the possible geometrical options for meeting the eld quality target. Index Terms—Author, please supply index terms/keywords for your paper. To download the IEEE Taxonomy go to http://www. ieee.org/documents/2009Taxonomy_v101.pdf.
II. THEORY A. Local Circular Toroidal Multipoles The eld of curved magnets is described with an approximate description using Local Toroidal Multipoles on dimensionless local toroidal coordinates [6]. There application on accelerator magnet elds were already presented in [4], [7], [8], but are shortly described here to facilitate the readers comprehension of the following sections. The local toroidal coordinates are constructed by: 1.) a circle with . Any position within this circle is given in Cartesian Coordinates (1)
I. INTRODUCTION
A
NY HEAT production occurring at cryogenic temperature requires expensive cooling due to the efciency factor limited by thermodynamics. Therefore the main magnets of SIS100 were optimized to balance the eld quality versus the AC losses. This process lead to: an elliptic beam aperture ; curved dipoles, not to loose and of space for the sagitta; a small yoke, to minimize the hysteresis loss while using the cheap ampere turns of the superconductor and a small coil end closely adapted to the vacuum chamber. This reduction in magnet size let to set the requested target eld quality of 600 ppm, contrary to 200 ppm typically specied for synchrotron magnets. Further the beam will ll the whole aperture due to multi-turn injection. The chosen optimization requires to fully understand the eld within the real 3D geometry (elliptic, toroidal). In a rst step elliptic multipoles were developed [1], [2], followed by toroidal multipoles [3] (now nalized in elliptic toroidal multipoles [4]). Calculations of the magnetic eld [5] showed that the main eld deviation is created in the end part of the magnet; mainly due to the high current density in the end coil. Therefore a 3D shim will be applied to the magnet end reducing the undesired sextupole.
Manuscript received September 13, 2011; accepted November 02, 2011. Date of publication November 22, 2011; date of current version May 24, 2012. P. Schnizer, A. Mierau, and E. Fischer are with GSI Helmholtzzentrum für Schwerionenforschung mbH, Darmstadt, Germany (e-mail:
[email protected]). B. Schnizer is with Technische Universität Graz, Austria. P. Akishin is with the Joint Institute of Nuclear Research, Dubna, Russia. Color versions of one or more of the gures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identier 10.1109/TASC.2011.2177049
the reference radius. 2.) this circle center is positioned with on the torus by with the larger torus radius. The curvature of the torus is described by . Approximate (to the rst order in ) toroidally uniform particular solutions of the potential equation are obtained by approximate R-separation: , [3] with the approximate scalar toroidal multipoles
(2) Corresponding (normal and skew) vector elds are given by the gradient [3]. A toroidally uniform magnetic induction is then represented as the sum of these basis functions with
(3) and
,
its coefcients.
B. Local Elliptic Toroidal Multipoles The coordinates are quite new. They are obtained by rotating off center elliptic coordinates , by an angle :
A segment of a torus with elliptic cross section is given by , with the elliptic aspect , , and the eccentricity . ratio
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Apart from a factor equation is:
IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 22, NO. 3, JUNE 2012
the potential
with , the multipole components of the magnetic eld, and the sensitivity of a radial coil probe by
(11)
Approximate multipoles are solutions accurate to the rst order in
(4)
with L the length of the coil, and the other and inner radius of the coil [9], [10]. A toroidally uniform eld varies over the coil rotational axis. The coil’s rotation axis is in the equatorial plane of the torus segment, which is the reference volume in the gap of the curved magnet. Now the uniform toroidal eld has to be integrated over the coils surface and then solved and sorted for the coefcients and . This yields [4], [8]
(5) with
(12) (6)
The corresponding toroidal components of the magnetic induction are obtained by gradients. These are rather involved in elliptic toroidal coordinates. We prefer to dene them in the Cartesian coordinates
(7)
and
are given below. These differ only by
(13) with the horizontal offset of the coil probe from the torus center line. So it sufces to compute only one matrix, say . is then . Each matrix element of and comprises at most 4 terms. The conversion matrix can be written in the following form
(8) The real, the imaginary part respectively of (4) and , (5) are separately used to replace , respectively by polynomials in , . Taking into account the factor leads to the nal result:
:
(14)
The matrix consists of four submatrices whose magnitude depend on the ratio . Only the rst matrix U is independent of the coil probes offset from the coordinate system. All matrices dependent on d are zero if , except for . and are given by
(9) and a corresponding result for the imaginary part. The are given in (1). The connection between the Fourier components of the rotating coil voltage and the expansions coefcients , can then be done analogously to the case of the circular toroidal multipoles [4].
(15) respectively. All following matrices (denoted with a large ) are lower triangular matrices with all elements on the diagonal equal to zero, thus is given by
(16)
III. APPLICATION: ROTATING COIL IN A CURVED MAGNET The output voltage of a rotating coil is given for
The matrix
(10)
is given by
(17)
SCHNIZER et al.: SIS100 DIPOLE MAGNET OPTIMIZATION AND LOCAL TOROIDAL MULTIPOLES
For machines with an aspect ratio as found for SIS100 or SIS300 the matrix is in the order of 100 ppm. It can be neglected except for the main multipole. The values of the matrix get of similar size as the values for . The values are given for the offset , a typical upper displacement limit found when measuring accelerator magnets. Also when measuring straight magnets special methods are applied to obtain the offset d from the measured data set [9]. Thus the authors believe that the artifacts can be minimized by similar adequate procedures [16].
TABLE I PARAMETERS FOR DIFFERENT MACHINES
with
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given by
B. Choosing the Coil Length (18)
describes the smear out of one toroidal multipole over the whole spectrum measured by the rotating coil depending on the curvature of the torus (described by and the coil geometry). The matrix is the only one, which does not depend on the torus curvature ratio . Its non zero elements are given by
A long coil will result in larger sensitivity (see (11)) and thus in better measurement performance. On the other hand the matrix depends on the coil length (see (18)). The magnitude of the coefcients depends on as for , with the component being orders of magnitude larger than the second term of . Its total contribution to the matrix is given by (see (20)). can be related to by
(21) (19) This term is equivalent to the formula derived for recalculating plane circular multipoles. This term is the equivalent as found, when a coil probe is translated by a distance in a cylindrical eld [11]. For the discussion below the matrix is dened
describes a translation of a plane coordinate system. Plane circular multipoles have been used for describing rotating coil measurements for many different machines; and it was shown that the parameter can be deduced from the measurement data. Therefore it can be desirable to keep the contributions of and of of similar magnitude. The factor varies across the matrix, but the largest terms in the matrix are in the side diagonal , if . Therefore the coil length should be in the range of
(20) (22)
only using , setting the second term of It is obtained from to zero. The matrices and can be inverted numerically. A. Magnitude of the Terms
C. Compensating Systems
The formulae given above were evaluated for the following different machines: the Large Hadron Collider (LHC) at CERN [12], SIS100 [13], [14] and SIS300 at GSI, and NICA [15] at JINR (see Table I). The parameters given in Table I were used to calculate the coefcients of the matrices. Accelerators require a eld description with an accuracy of 1 unit and roughly 0.1 unit for the eld homogeneity (1 unit equals 100 ppm). Therefore any contribution less than 1 ppm can be ignored. For LHC is very small and thus the correction of all matrices are very small (less than 1 ppm) except for the matrix , where the values close to the diagonal get to a size of 2000 ppm. This value may seem to exceed the target value for the eld description; but the higher order multipoles are in the order of 100 ppm; thus the effective artifact will be safely below the target value of 10 ppm.
The main multipole of an accelerator magnet is typically 1000 to 10000 times larger than the higher order multipoles. Therefore one reduces the sensitivity to the main component subtracting two coil rotating on a common support (e.g. [9], [10]). is typically reduced by a factor 500 to 1000 for a carefully fabricated coil array (thus ). This setup increases the measurement accuracy of the higher order multipoles, as the coil probe system is more immune to mechanical imperfections of the rotation, and less accurate electronics are required. The second term of (18) contains the ratio , which would get large for . The multipole , measured with the compensated system is not used in the eld calculation, but it is taken from the measurement with the “absolute coil probe”. Therefore of the absolute system is to be used in the ratio. (The same procedure is applied when measuring a straight magnet using circular cylindrical multipoles.)
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IEEE TRANSACTIONS ON APPLIED SUPERCONDUCTIVITY, VOL. 22, NO. 3, JUNE 2012
Fig. 1. First allowed multipoles versus coil height. The harmonics shown are total harmonics for the magnet. Original position circle 10 mm up triangle triangle up 40 mm up square 100 mm up hexagon. down 20 mm up
Fig. 2. Three-dimensional end model.
Fig. 3. Aperture widening to increase the eld quality.
IV. FIELD QUALITY The eld quality of the ends of the SIS100 dipoles was considered as improvable. The eld of a conventional magnet can only be considered as iron dominated for that part, where the yoke closes the ux loop. This is the central part, especially at a eld of roughly 0.25 T as foreseen for the SIS100 dipole at injection. The coil end itself, however, is to be seen as a magnet together with its image currents. The magnet is designed with a minimal coil height minimizing the AC losses [17]. A larger coil head does increase the eld quality, but not signicantly (see Fig. 1). Thus the coil height was not changed. Calculating the multipoles versus one could see that a large positive sextupole is added in the end. So the yoke end should be changed to create the counter sextupole so that the integral would be diminished. To minimize the calculation time, an end model was made with its length determined on an appropriate full 3D model, showing that a 200 mm iron yoke length was appropriate (see Fig. 2). A natural approach is to add a sextupole like shape to the pole prole [18], which allows to diminish the sextupole. An other one is widening the aperture end keeping the coil in place (see Fig. 3). The sextupole can be diminished next to the decapole (see Fig. 4). This can be taken further even changing the sign of the sextupole but with an increase of the decapole
Fig. 4. Deformation eld, original data circles, end widening V1 squares. (a) . (b) . gles down, end widening V2
trian-
(see Fig. 5); thus countermeasures for the higher order multipoles have to be considered. The superimposed parabola was chosen as it can be machined on an insert iteratively until the desired eld quality is reached. V. CONCLUSION Local Toroidal coordinates allow describing the eld in a dipole following the particle curvature. These create a “feed up” effect. For SIS100 only the quadrupole component is affected
SCHNIZER et al.: SIS100 DIPOLE MAGNET OPTIMIZATION AND LOCAL TOROIDAL MULTIPOLES
Fig. 5. Deformation eld V2 original data different yoke ends.
circles, diamonds
a study of
within the required accuracy, if the coordinate systems match perfectly. The local toroidal coordinates can be obtained from a measurement coil of reasonable length as dened by (22). The magnet end is the main source for the total eld distortion. Different forms of ideal shape were already presented. Its nal form will be optimized in an iterative process measuring the eld using the method presented above. REFERENCES [1] P. Schnizer, B. Schnizer, P. Akishin, and E. Fischer, “Magnetic eld analysis for superferric accelerator magnets using elliptic multipoles and its advantages,” IEEE Trans. Appl. Supercon., vol. 18, no. 2, pp. 1605–1608, June 2008. [2] P. Schnizer, B. Schnizer, P. Akishin, and E. Fischer, “Theory and application of plane elliptic multipoles for static magnetic elds,” Nucl. Instr. Meth. A, vol. 607, no. 3, pp. 505–516, 2009. [3] P. Schnizer, B. Schnizer, P. Akishin, and E. Fischer, “Plane elliptic or toroidal multipole expansions for static elds. applications within the gap of straight and curved accelerator magnets,” Int. J. Comp. Mathematics in Electrical Engineering (COMPEL), vol. 28, no. 4, 2009.
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[4] P. Schnizer, B. Schnizer, P. Akishin, and E. Fischer, “Toroidal circular and elliptic multipole expansions within the gap of curved accelerator magnets,” in 14th Int. IGTE} Symposium, , Austria, September 2010, Graz: Institut für Grundlagen und Theorie der Elektrotechnik, Technische Universität Graz. [5] E. Fischer, P. Schnizer, A. Mierau, S. Wilfert, A. Bleile, P. Shcherbakov, and C. Schroeder, “Design and test status of the fast ramped superconducting SIS100 dipole magnet for FAIR,” IEEE T. Appl. Supercon, vol. 21, no. 3, pp. 1844–1848, June 2011. [6] W. D. D’haeseleer, W. N. G. Hitchon, J. D. Callen, and J.-L. Shohet, Flux Coordinates and Magnetic Field Structure. : Springer, 1990. [7] P. Schnizer, B. Schnizer, P. Akishin, and E. Fischer, “Theoretical eld analysis for superferric accelerator magnets using plane elliptic or toroidal multipoles and its advantages,” in The 11th European Particle Accelerator Conference, June 2008, pp. 1773–1775. [8] P. Schnizer, E. Fischer, H. Kiesewetter, A. Mierau, and B. Schnizer, “Field measurements on curved superconducting magnets,” IEEE T. on Appl. Supercon., vol. 21, no. 3, pp. 1799–1803, June 2011. [9] A. K. Jain, “Harmonic coils,” in CAS Magnetic Measurement and Alignment, S. Turner, Ed. : CERN, August 1998, pp. 175–217. [10] P. Schnizer, “Measuring System Qualication for LHC Arc Quadrupole Magnets,” Ph.D. dissertation, TU Graz, , 2002. [11] A. K. Jain, “Basic theory of magnets,” in CAS Magnetic Measurement and Alignment, S. Turner, Ed. : CERN, August 1998, pp. 1–21. [12] O. Brüning, P. Collier, P. Lebrun, S. Myers, R. Ostojic, J. Poole, and P. Proudlock, LHC Design Report. Geneva: CERN, 2004. [13] E. Fischer, P. Schnizer, A. Akishin, R. Kurnyshov, A. Mierau, B. Schnizer, S. Y. Shim, and P. Sherbakov, “Superconducting SIS100 prototype magnets design, test results and nal design issues,” IEEE T. Appl. Supercon., vol. 20, no. 3, pp. 218–221, June 2010. [14] FAIR—Facility for Antiprotons and Ion Research, Technical Design Report Synchrotron SIS100, December 2008. [15] H. G. Khodzhibagiyan et al., “Superconducting magnets for the NICA accelerator complex in Dubna,” IEEE T. Appl. Supercon, vol. 21, no. 3, pp. 1795–1798, June 2011. [16] L. Bottura, M. Buzio, G. Deferne, P. Schnizer, P. Sievers, and N. Smirnov, “Magnetic measurement and alignment of main LHC dipoles and associated correctors,” IEEE T. Appl. Supercon, vol. 12, no. 1, pp. 1659–1662, March 2002. [17] E. Fischer, R. Kurnishov, and P. Shcherbakov, “Finite element calculations on detailed 3D models for the superferric main magnets of the FAIR SIS100 synchrotron,” Cryogenics, vol. 47, pp. 583–594, 2007. [18] E. Fischer et al., “Design and operation parameters of the superconducting main magnets for the SIS100 accelerator of fair,” in Proceedings of the IPAC’11, San Sebastián, Spain, September 2011, pp. 2451–2453.
