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The plane of the loop can be oriented in an arbitrary direction, but two specific geometries, known as “radial” or “tangential” coils, are most common due to ease of fabrication, characterization and data analysis. • Special geometries to measure specific harmonics are also used. USPAS, Santa Barbara, June 23-27, 2003 5 Animesh Jain, BNL Useful Background Information Commonly Used Coordinate System Y-Axis (TOP) Aperture Currents By Bθ r B Br θ θ θ Bx X-Axis (RIGHT) Users of magnetic measurements data may use a system oriented differently, often requiring suitable transformations of the measured harmonics.

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.

The field harmonics are then deduced using a knowledge of the geometry of the coil. • A coil often uses several loops of different geometries to improve the accuracy of measurements by a process of “bucking”. USPAS, Santa Barbara, June 23-27, 2003 3 Animesh Jain, BNL A Typical Rotating Coil Setup USPAS, Santa Barbara, June 23-27, 2003 4 Animesh Jain, BNL Common Coil Geometries • All geometries employ a loop of wire, with one pair of sides parallel to the magnet axis. • The plane of the loop can be oriented in an arbitrary direction, but two specific geometries, known as “radial” or “tangential” coils, are most common due to ease of fabrication, characterization and data analysis.

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