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G. for normal quadrupole) that the polarity be changed when viewed from the opposite end. This amounts to an extra change in sign of all the harmonics in such cases. USPAS, Santa Barbara, June 23-27, 2003 15 Animesh Jain, BNL Symmetries & Allowed Harmonics • Accelerator magnets often have definite symmetries (or antisymmetries) in the current distribution. These symmetries lead to only certain harmonics being allowed. • The coordinate transformations derived for harmonics may be used to identify the allowed terms.

For normal quadrupole) that the polarity be changed when viewed from the opposite end. This amounts to an extra change in sign of all the harmonics in such cases. USPAS, Santa Barbara, June 23-27, 2003 15 Animesh Jain, BNL Symmetries & Allowed Harmonics • Accelerator magnets often have definite symmetries (or antisymmetries) in the current distribution. These symmetries lead to only certain harmonics being allowed. • The coordinate transformations derived for harmonics may be used to identify the allowed terms.

Rref ∞ X ∞  z′ ′ ′ (Bn + iAn ) ∑ R n = n0  ref z0 = x0+ iy 0 = r0. exp (i ξ) ∞   x0 + iy0  ( k − n0 )!   ( Bn′ + iAn′ ) = ∑ ( Bk + iAk )     k =n  ( n − n0 )! ( k − n )!  Rref  ∞ k −n n − n0      z0  R  ref k −n k − n0      z ′   Rref  k −n     n − n0 n0 = 0: US ; n ≥ n0 n0 = 1: European All higher harmonics contribute to a given harmonic in the displaced frame. This is referred to as FEED DOWN of harmonics USPAS, Santa Barbara, June 23-27, 2003 9 Animesh Jain, BNL Centering of Harmonics • If X-Y is the measurement frame, and X’-Y’ is the magnet frame, then one can compute the “centered” harmonics, provided the offsets (x0, y0) are known.

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BNL - Superconducting Accelerator Magnets [presentation slides] by A. Jain


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