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Exploring the Characteristics of the Cos(θ) Coil in Magnetic Field Applications

This paper from the University of Kentucky delves into the detailed analysis of the cos(θ) coil in the context of magnetic scalar potential and field equations. It elucidates the boundary conditions, flux lines, and flow sheets pertinent to this coil design, discussing its practical implications in various topologies and configurations. The study comprehensively covers different types of cos(θ) coils, including solenoids and rounded variations, and examines their role in dipole moments and neutron beam manipulation. The research also addresses optimization techniques and the significance of symmetric coil properties for diverse geometries, offering insights into field cancellation and application versatility.

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Exploring the Characteristics of the Cos(θ) Coil in Magnetic Field Applications

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  1. The cos-theta coil re-re-visited Christopher Crawford, University of Kentucky DNP Fall Meeting, Newport News, VA 2013-10-26

  2. Magnetic scalar potential • Field Equationsfield potential • Boundary conditionsfield potential B.C.’s:Flux lines bounded by charge Flux lines continuous Flow sheets continuous (equipotentials) Flow sheets bounded by current

  3. What is a cos θ coil ? U = -x = - ρ cos φ CYLINDER SPHERE Symmetryz φ Wire pos. φi θi Const. surf. ρ0 r0 Moment Topology infinite bound U = -z = - r cos θ Cos θ coil U=-z Solenoid Cos φ coil

  4. Single cos θ coil

  5. Single solenoid

  6. Single cos φ coil

  7. Arbitrary angle

  8. Flux containment • Three limits of boundary conditions in the return yoke: • μ = ∞(ferromagnetic) • Magnetization currents • High static shielding factor • THIN! • Image currents: automatic self-compensation (approximate) (exact) • μ = -1(superconductor) • Super- currents • Infinite dynamic shielding factor • Need space between the coil and shield • μ = 1(wires) • Conductor currents • No external field distortion • Must calculate wire positions! • Three topologies: • sphere • Separate shells • cylinder • Common surface • Restores z-symmetry • torus • No flux return

  9. Double cos θ coil • Dipole Moments • μinner = - μouter • I = ΔA/A H0

  10. Double cos φ coil • n3He Spin Rotator • (TEM RF mode): • Reverses either longitudinal OR transverse polarizedneutrons • No fringe fieldsin the neutronbeam • Self-shielding –no eddy currentsin Aluminum enclosure

  11. Discretization of cos φ coil • Standard winding:one wire at centerof each slice • Optimization:– nominal dipole m=1– vanishing higher moments m=3,5,7,… • Nonlinear equations:– solved iteratively by Newton-Rhapson method for φi– up to m=15 (15 wires) or m=27 (30 wires)

  12. Discretization of cos φ coil • Equally spaced wiresFourier cosine series • Optimization:– nominal dipole m=1– vanishing higher moments m=3,5,7,… • Linear equations:– unitary matrix– can null N-1 odd moments using N wire pairs– can tune individual currents in situ– use as shim coils for series cos θ winding

  13. Rounded cos φ coil

  14. Rounded cos θ coil B0

  15. Conclusion • The Cos θ coil can be classified according tosymmetry, moment, and topology • Can use double layers for exact field-cancellation • Use the scalar potential, one can apply the properties ofa symmetric cos θ coil to any geometry

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