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

2. Magnetic scalar potential
B.C.’s: Flux lines bounded by charge Flux lines continuous
Flow sheets continuous (equipotentials) Flow sheets bounded by current
Field Equations field potential
Boundary conditions field potential
Geometrical Gauss -> Ampere’s law
U interpretation as boundary currents
Statement in terms of boundary conditions
Technique for calculating coils

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

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 polarized neutrons
No fringe fields in the neutron beam
Self-shielding – no eddy currents in Aluminum enclosure

11. Discretization of cos φ coil
Standard winding: one wire at center of 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 wires Fourier 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 to symmetry, moment, and topology
Can use double layers for exact field-cancellation
Use the scalar potential, one can apply the properties of a symmetric cos θ coil to any geometry