By A. Jain

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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.

The harmonics allow computation of field everywhere in the aperture (within a circle of convergence) using only a few numbers. • These coefficients obviously depend on the choice of origin and orientation of the coordinate system. Measured harmonics, therefore, often need to be centered and rotated. USPAS, Santa Barbara, June 23-27, 2003 6 Animesh Jain, BNL Centering of Harmonics: Definitions Y Y' x x' y' x0 0 O' r O z0 = x0+ iy 0 = r0. exp (i ξ) ξ P y z' = x' + iy' z = x + iy =(x'+x0) + i (y'+y0) X' y0 USPAS, Santa Barbara, June 23-27, 2003 X 7 X’-Y’ is a coordinate system displaced with respect to the X-Y frame by x0 along X-axis and by y0 along the Y-axis.

The integrator drift, however, can be a problem, and needs correction. 14 Animesh Jain, BNL Signal from a Tangential Coil Y Br ∆ θ+ ∆ / 2 Coil Φ(t ) = NL ∫ B r ( Rc , θ) Rc dθ θ− ∆ / 2 Rc θ Coil Support Flux through the coil at time t is: ω N = No. of turns L = Length ∆ = Opening angle δ = angle at (t = 0) ω = angular velocity θ = ωt + δ (angle at t) USPAS, Santa Barbara, June 23-27, 2003 X n 2 NLRref Rc n∆ =∑ sin × n 2 n =1 Rref [Bn sin(nωt + nδ) + An cos(nωt + nδ)] ∞ The periodic variation of flux is described by a Fourier series, whose coefficients are related to the Normal and Skew harmonics, and geometric parameters of the coil.