An approach to 3D magnetic field calculation using numerical and differential algebra methods
Motivated by the need for new means for specification and determination of 3D fields that are produced by electromagnetic lens elements in the region interior to coil windings and seeking to obtain techniques that will be convenient for accurate conductor placement and dynamical study of particle motion, we have conveniently gene the representation of a 2D magnetic field to 3D. We have shown that the 3 dimensioal magnetic field components of a multipole magnet in the curl-fire divergence-fire region near the axis r=0 can be derived from one dimensional functions A{sub n}(z) and their derivatives (part 1). In the region interior to coil windings of accelerator magnets the three spatial components of magnet fields can be expressed in terms of harmonic components'' proportional to functions sin (n{theta}) or cos (n{theta}) of the azimuthal angle. The r,z dependence of any such component can then be expressed in terms of powers of r times functions A{sub n}(z) and their derivatives. For twodimensional configurations B{sub z} of course is identically zero, the derivatives of A{sub n}(z) vanish, and the harmonic components of the transverse field then acquire a simple proportionality B{sub r,n} {proportional to} r{sup n-1} sin (n{theta}),B{sub {theta},n} {proportional to} r{sup n-1} cos (n{theta}), whereas in a 3-D configuration the more complex nature of the field gives rise to additional so-called psuedomultipole'' components as judged by additional powers of r required in the development of the field. Computation of the 3-D magnetic field arising at a sequence of field points, as a direct result of a specified current configuration or coil geometry, can be calculated explicitly through use of the Biot-Savart law and from such data the coefficients can then be derived for a general development of the type indicated above. We indicate, discuss, and illustrate two means by which this development may be performed.
- Research Organization:
- Lawrence Berkeley National Lab. (LBNL), Berkeley, CA (United States)
- Sponsoring Organization:
- USDOE; USDOE, Washington, DC (United States)
- DOE Contract Number:
- AC03-76SF00098
- OSTI ID:
- 7252409
- Report Number(s):
- LBL-32624; SC-MAG-395; ON: DE93001590
- Country of Publication:
- United States
- Language:
- English
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Related Subjects
MAGNETIC FIELDS
THREE-DIMENSIONAL CALCULATIONS
SUPERCONDUCTING SUPER COLLIDER
ALGEBRA
BEAM TRANSPORT
DIFFERENTIAL CALCULUS
MAGNET COILS
MULTIPOLES
NUMERICAL SOLUTION
ELECTRIC COILS
ELECTRICAL EQUIPMENT
EQUIPMENT
MATHEMATICS
STORAGE RINGS
430200* - Particle Accelerators- Beam Dynamics
Field Calculations
& Ion Optics