V.Kashikhin , Fermilab, P.O. 500, Batavia, IL, 60510Low- Field Very Large Hardron Collider Magnet Parameters:
Magnet Field :
Injection - 0.1 Tesla
Maximum - 2.0 Tesla
Magnet : Double C Gradient Transmission Line Magnet
Core : Low Carbon Steel, Warm
Winding: One Turn Superconducting, NbTi,Transmission Line Type
Current : 0.3 100 kA
Air Gap: Double 20 mm
Gradient: 3 4 %/cmGood Field Quality Region ( 10-4 )
Injection - 18 mm diameter
Maximum - 10 mm diameter
Outer Dimensions :
Resistive Model and Measurement Stand for the Magnet Field Quality Tests Field - up to 2 Tesla
Width: 250 mm
Height: 600 mm
Length (prototype): 60 m
Winding - Copper ,Water Cooled, 10 Turns
Length - 1 meter
Iron Core Double C, Removable, Easy Reassemble with new Laminations
Laminations Low-Carbon Steel, Laser cutted
Type of Field Measurements:
Flat Coil moved in horizontal
Direction, Integral Field
Number of Turns - 3
Length - 760 mm
Width - 1.5 mm
Moving - X,Y +/- 25.4 mm
Accuracy - several microns
Test of the Field Value in the Gap:
Hall Probe Teslameter , 10-5
Power Supply :
Current - 0 - 10 kA
Transmission Line Magnet Test Stand
Flat Coil Field Measurement Stand
Iron Core Laminations
Reduce in the pole center saturation effects using crenelated laminations
( now tested crenelations 0.43 mm in height, 20 mm width)
each 10th lamination is crenelated
EPAC 98 , "Pole Profile Optimization of VLHC Transmission Line Magnet" , G.W.Foster, V.Kashikhin, V.Kashikhin Jr. see Appendix
AppendixPole Profile Optimization of VLHC Transmisssion Line Magnet
G.W.Foster, V.Kashikhin, V.Kashikhin - Jr
Fermilab, P.O. Box 500, Batavia, IL 60510EPAC 98
A pole profile optimization method has been developed for the accelerator magnets. It is based on the analytical solution of reverse Coushys problem combined with the numerical solution of nonlinear Poissons equation. This method was used for the pole profile optimization of VLHC transmission line magnet . It was received the good field quality in magnet aperture for the fields up to 2 Tesla.
One of the more complicated problem in design of magnets for accelerators is to form needed magnetic fields with very high accuracy at low pole width and correspondingly weights of magnets. In common case, it is the nonlinear reverse problem of magnetostatic and can not be solved by analytic methods. There were suggested various methods [2 4] of solution this problem but, in practice, usually use numerical solution of Poissons equation with step by step correction of pole profile. It is labor intensive procedure, based mostly on the experience of designer. For linear problem, when the saturation effects of iron core do not influence on the field distribution, was designed number of analytic methods [4-6]. In this work described the method, which combine the accuracy of analytic approach for the reverse problem solution with efficiency of numerical methods. The optimization process was used for the pole profile design of the Very Large Hardron Collider (VLHC) prototype magnet . At the first step with help to the analytic transformations of Taylors series, using field harmonics as initial data, was received the pole profile, which was optimized during next analytic-numeric iterative solutions. "Crenelation" technique  gave possibility to receive magnetic field in range from 0.1 Tesla up to 2 Tesla with quality 2*10-4 .
Analytic solution of the Reverse Magnetostatic Problem
Usually for accelerator magnets are known from the beam dynamics analysis the demands to the field harmonic content and the aperture dimensions. Magnetic field in the middle plane of magnet may be expressed
where am - field harmonics, xo - normalization parameter.
If the magnetic permeability of ferromagnetic poles is infinite and the areas with currents are far away, the field in magnet air gap is defined by the form of pole profiles. Scalar potential of this field satisfies Laplaces equation
and in middle plane is known the field distribution or the scalar potential derivative
It is well-known Coushys problem for the upper half plane. Solution may be received on the base of the scalar potential analytic continuation by Taylors series
Pole surface in this case coincides with equipotential on which U(x,y) = const. For magnet with the symmetry relatively Y axis the scalar potential on the pole surface may be defined trough the field in the center Up = - By(0,0) *d , where d is the magnet air gap . So, for the defining pole profile coordinates, at known Up and Byo, it is need to solve equation (1) relatively unknown Yp coordinates of the pole profile. It seems rational to use for it the method of analytic transformations of Taylors series, widely used in a high precision calculations of astronomy. Express Yp as the series
Substituting (2) in (1) and extracting from the first term of (1) Yp,then have
Replacing Byo and Yp by their series receive the equation for the bn coefficients definition
The bn will calculate by sequential iterations, when on the first iteration all bn=0 for the right part equation. Substituting known first field harmonics and scalar potential Up on pole surface it may be found the more accurate bn values. This process are repeated until the following condition is satisfied
wheree accuracy of bn calculation, i number of iteration, N number of terms
used for calculations.
This method was proposed and sucsessfully tested in  for the dipole magnet with 5 cm air gap.
Pole Profiles Optimizer
The pole profiles optimizer is created as the additional modules for the base software package POISSON and consists of the following parts:
harmonic analysis module
pole profile regeneration module
For the data input is used original POISSON preprocessor. Harmonic analysis module is based on the algorithm of field interpolation in the magnet middle plane by polynomial:
where are polynomial coefficients (harmonics) defined by Newtons interpolation algorithm :
where , - divided differences degree () and ; ;
; ; .
Polynomial coefficients are used with the pole profile regeneration. Described algorithm ensures an exact evaluation of polynomial coefficients up to 15th harmonics (practically were used 6 first harmonics). Regeneration pole profile module is based on the analytical approach previously described. On each iteration is produced the improvement of the polynomial coefficients and the new pole profile synthesis. On the convergence of iterativ? process highly influence the amount of taken into account harmonics. Synthesis of a minimal pole width (with artificially cuted off corners) and the convergence of process is limited by 5-6 harmonics taken into account during optimization process. High harmonics defines the form of the pole ends and their influence extremely connected with the coners saturation effects. For the saturated poles ends, which close to straight angle the convergence is worse. Number of harmonics also influence on the convergence velocity. Most optimal is the algorithm to allow the first harmonic vary until it reach required value, then to allow first and second vary until each of them become the required value and so on, adding each cycle one harmonic until all harmonics will satisfy the needed accuracy.
Pole profile optimization of VLHC transmission line magnet
The optimizer was successfully used for the VLHC transmission line magnet pole profile optimization. It was need to generate the pole of minimal width which provide the following parameters: field gradient -0.04 %/cm in circular good-field aperture with the diameter 2cm at B0.1T and 1cm at B2T. Required field quality . Magnet cross-ection is shown on Fig.1.
During the first calculations was generated the pole profile of a minimal width, which ensures required field quality at 0.1T in 2cm diameter area. The good field aperture at high fields decreased up to 0.6 cm. It is explained by the large saturation of inner pole corner in relation to outer corner. This effect causes the gradient change up to 3.98%/cm. After optimization were found more optimal parameters of inner and outer pole corners:
inner corner radius 2.38cm
outer corner radius 0.52cm
Due to this the gradient at high field was stabilized at 4.006%/cm.Good field aperture are showm on Fig 2 (low field) , on Fig. 3 (high field).
Fig. 2 Good-field aperture for low field level
Fig. 3 Good-field aperture for high field level
However the good-field aperture less than required value at high field. The reason of it was too large value sextupole at high field (at low field ). Such large sextupole is connected with larger saturation of the pole corners on a comparison with the pole center. Usual solution of this problem is to increase the pole width to get 1cm of good-field aperture at high field. But that however disagree with the accepted concept of minimization pole width in the whole range of field variations, since low field good aperture correspondingly increases more than required value.
Crenelation technique for field quality improving
One of the ways to receive the good field quality in all range of field variations is the crenelation technique. This technique was proposed by G.W.Foster in . Since the magnet pole is assembled from laminated steel there is a possibility to reduce common steel «density» by removing a part of material each -th lamination - to crenelate it. The steel packing factor in crenelated area will be correspondingly decreased
where - total number of laminations , - number of crenelated laminations .
For described dipole magnet it is necessary to crenelate central area of the pole tip to increase saturation in this part of the pole up to a level of pole corners saturation. Such pole of equal saturation will ensure more stable field distribution in the whole range of fields, because each point of steel will lie on about the same point of curve. During realization this optimizing technique was found strong influence of crenelation profile at high field levels. Accordingly for the crenelated surface optimization was used the algorithm similar used for the pole tip generation. Also a large importance has the right choice of packing factor for crenelated area. At high for compensation of pole corners saturation is necessary to increase height of crenelation surface. But it hampers smoothing of field distribution in magnet gap as the influence of pole surface ~ , where is adistance between gap axes and pole surface. At low crenelation surface has small height and influences almost as basic pole surface, but in such strong compensated system is probably deterioration of field quality at middle field levels because of «bad» flux redistribution between crenelated and non-crenelated areas.
For ARMCO type steel used in tentative calculations were most optimal and crenelation height in the center of pole. During optimization was generated the basic pole profile atfield 1.1?, where the effects of saturation begin to influence on the field quality. Then was generated the crenelated area at high field and checked field quality in the range 0-2T. The best results were obtained for the generation of the crenelated surface at 1.7T field. Figures 4, 5, 6 show the good field aperture at various field levels.
Fig. 4 Good-field aperture for low field level
Fig. 5 Good-field aperture for middle field level
Fig. 6 Good-field aperture for high field level
 G.W.Foster. Magnetic Calculations for the PIPETRON Transmission Line Magnet. Proceedings of Snowmass 96, January 1997.
 K.A. Halbach. Program for inversion system analysis and its application to the design of magnets. Proceedings of the Second International Conference on Magnet Technology, Oxford, 1967, p. 47 59.
 A.G. Armstrong, et. al. Automated optimization of magnet design using the boundary integral method. IEEE Transactions of Magnetics, MAG-18, No.2, 1982, p.620 623.
 S.C. Snowdon Magnet profile design. IEEE Transactions on Nuclear Science, Vol.5-18, No.3, 1971, p.848 852.
 V.Kashikhin, E.Lamzin. Formation of Homogeneous Magnetic Fields in Dipole Magnets of Particle Accelerators. Journal of Technical Physics, Vol. 58, N 4,1988,p.728736.
 V. Kashikhin. Synthesis of Electromagnet Pole Configuration on the Base of Solution Coushys Problem for Laplaces Equation. Preprint/NIIEFA,E-0707, 1885, p.15.
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