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Modified Cos θ Coil

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Modified Cosθ Coil

A new approach in Cosθ coil design to increase the uniformity of the B field

Riccardo Schmid, CIT

A Cosθ coil with fewer turns (20 vs. 40) produces a B field with a greater gradient (greater non-uniformity).

It is possible to achieve greater uniformity by combining the fields of two coils with different gradients

mGauss

Plot of Bx(x) for Cosθ coil N=20 (black) N=40 (red) turns

cm

Prototype geometry (R=8.75cm, L=87.5cm)

The main coil consists of a 40 turn Cosθ coil with current I1.

The secondary coil consists of a 8 turn Cosθ coil with 2/5 the current in the main coil, running in the opposite direction.

mGauss

cm

+

cm

mGauss

Plot of Bx(x) for main Cosθ coil N=40 turns

Plot of Bx(x) for secondary Cosθ coil N=8 turns with current I`= -2/5 I0

The main coil consists of a 40 turn Cosθ coil with current I1.

The secondary coil consists of a 8 turn Cosθ coil with 2/5 the current in the main coil, running in the opposite direction.

mGauss

Bx(x)

Black: N=40 Cosθ coil

Red: N=40-8 Modified Cosθ coil

cm

Plot of Bx(x) for Cosθ coil N=40 in black compared tomodified Cosθcoil 40-8 turns red (main coil N=40 turns, secondary coil N=8 turns)

Prototype geometry (R=8.75cm, L=87.5cm)

The main coil consists of a 40 turn Cosθ coil with current I1.

The secondary coil consists of a 8 turn Cosθ coil with 2/5 the current in the main coil, running in the opposite direction.

cm

Bx(y)

Black: N=40 Cosθ coil

Red: N=40-8 Modified Cosθ coil

mGauss

Plot of Bx(y) for Cosθ coil N=40 in black compared tomodified Cosθcoil 40-8 turns red (main coil N=40 turns, secondary coil N=8 turns)

Prototype geometry (R=8.75cm, L=87.5cm)

The main coil consists of a 40 turn Cosθ coil with current I1.

mGauss

Bx(z)

Black: N=40 Cosθ coil

Red: N=40-8 Modified Cosθ coil

cm

Plot of Bx(z) for Cosθ coil N=40 in black compared tomodified Cosθcoil 40-8 turns red (main coil N=40 turns, secondary coil N=8 turns)

Prototype geometry (R=8.75cm, L=87.5cm)

Δ

Δ / 2

3Δ / 2

3Δ

- It is possible to implement a secondary Cosθ coil without modifying the structure of the basic coil.
- The secondary coil will have a number of turns equal to a divisor of the number of turns of the main coil by an odd number.
- No additional grooves need to be machined
- The current in the secondary coil can be implemented as current “missing” from the main. If the main coil is constructed by winding a wire multiple times, the secondary coil can be implemented by subtracting windings in some of the turns.

The Volume Averaged Gradient calculations of the B field inside the fiducial volume show a lower Average Gradient for the N=40-8 coil

<|dBx/dx|> calculated over a grid of points inside the fiducial volume for the prototype geometry

Average Gradient Magnitude for different coils:

N=20 <|dBx/dx|> = 53.988 nT/m

N=40 <|dBx/dx|> = 15.531 nT/m

N=40-8 <|dBx/dx|> = 2.184 nT/m

For N=40-8 Gradient sign changes inside the fiducial volume:

N=40-8 < dBx/dx > = .668 nT/m

(not magnitude of gradient)

Prototype geometry (R=8.75cm, L=87.5cm)

nT/m

N=20 <|dBx/dx|> = 53.988 nT/m

nT/m

N=40 <|dBx/dx|> = 15.531 nT/m

nT/m

N=40-8 <|dBx/dx|> = 2.184 nT/m

- Modified Cosθ coil N=40-8 seems to offer a smaller gradient compared to N=40 in the region of interest (fiducial volume)
- Simple implementation in existing prototypes (N=40 at Caltech)
- Increase in the current necessary to drive both coils at original B field is small (<10% increase)
To do:

- Measurement of the field on prototype
- Calculations for new geometry L/R~6.4 (current calculation L/R=10)
- Possible optimization varying secondary coil current ratio (now -2/5), number of secondary coil turns (currently 8)
- Monte Carlo simulation of geometric phase