115 December 2011 Holger Witte Brookhaven National Laboratory Advanced Accelerator Group Elliptical...

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1 15 December 2011 Holger Witte Brookhaven National Laboratory Advanced Accelerator Group Elliptical Dipole

Transcript of 115 December 2011 Holger Witte Brookhaven National Laboratory Advanced Accelerator Group Elliptical...

Page 1: 115 December 2011 Holger Witte Brookhaven National Laboratory Advanced Accelerator Group Elliptical Dipole.

115 December 2011

Holger WitteBrookhaven National Laboratory

Advanced Accelerator Group

Elliptical Dipole

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Motivation• Bending magnets in

muon collider: – exposed to decay

particles – a few kW/m– from short lived

muons

• Distribution is highly anisotropic – large peak at the

midplane (Mokhov)

• One suggestion: open midplane dipoles– Issue: filed quality

Nikolai Mokhov, in “Brief Overview of the Collider Ring Magnets Mini-Workshop, Telluride 2011.

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Task

Inside pipe width = 5 cmInside pipe height = 2 cm

From: Suggested shield & cos theta dipole dimensions R. B. Palmer, 5/26/11

Tungsten liner

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Methodology developed for Integrable Optics Lattice (FNAL)

• Task: generate certain vector potential

• Singularities• Difficult to

approximate with multipole fields

• Ideally non-circular aperture – 2 cm horizontal, 4 cm vertical

B22

)()(),(

gf

tyxU

c

ycxycxc

ycxycx

2

22222

2222

acos2

1)(

acosh1)(

2

2

g

f

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• Vector potential at point P due to current I (in z-direction):

• Magnetic field:

Vector Potential of Single Line Current

P

I

x

y

a

RIrAz ln

2),( 0

Rr

a

y

AB

x

AB z

xz

y

,

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• Required: desired Az and coil bore

• A~I, therefore:

• P2:

• Generally:

Methodology

I1 P1

1111 zAIA

P2

2121 zAIA

),...,,(),...,,( 21112111 znzzn AAAIAAA

A11=VP @ P1 for unit current I1

A21=VP @ P2 for unit current I1

A12=VP @ P1 for unit current I2

Beam Aperture

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• Same is true for multiple currents and positions P

• Formalism:

• Linear equation system: Ax=b

Methodology: Formalism

P1P2 P3 P3 P4

I1

I3I3

I4I2

zn

z

z

nnnnn

n

n

A

A

A

I

I

I

AAA

AAA

AAA

2

1

2

1

21

22221

11211

A · x = b

A11, A12, ... are known (can be calculated – unit current Im, calculate Az at Pn)

b: also known (this is the vector potential we want)

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Example: Quadrupole

Current

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Rectangular Shape

Conductor Reference Az

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From 2D to 3D

Vector addition• Power each current strand individually – Very inefficient, clumsy – Not very elegant

• Known current distribution

• Helical coil: vector addition of two currents, which always intersect at the correct angle

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• Easy if functional relationship is known (i.e. cos theta)

• Here:– (x,y) position known

need to determine z• dz=dI

From 2D to 3D

dzzn

in

0

In+1 In

In-1ds

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Quadrupole

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Quadrupole

Calculated for two coils

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Task

Inside pipe width = 5 cmInside pipe height = 2 cm

From: Suggested shield & cos theta dipole dimensions R. B. Palmer, 5/26/11

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Concept: Elliptical Helical Coil

x (m)

y (m

)

Task: Find 2D current distribution which generates (almost) pure dipole field

Calculate this for a set of positions on ellipse

A-axis: 9.1 cm /2B-axis: 13.77 cm /2

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Answer: Current Distribution

Normalized current density vs. azimuthal angle

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Implementation: Elliptical Helical Coil

40 turns

Spacing: 20 mm(= length about 0.8 m + “coil ends”)

Single double layer

Current in strand: 10 kA(=400 kA turns)

Average current density: 10 kA/(20mmx1 mm)=500A/mm2

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Field Harmonics

Normalized to Dipole field of 1T

Evaluated for radius of 25 mm

Well behaved: small sextupole component at coil entrance and exit

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Field along z

z (m)

B (

T)

10 kA = 1.1T

All unwanted field components point symmetric to the origin should disappear (e.g. Bz)for 4-layer arrangement

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Other Geometries?

• Well-known: intersecting ellipses produce dipole field

• Worse performance– Field quality– Peak field on wire

• Less flexible• Coil end problem?• Geometry problem

– Approximation with blocks

• Stresses?

J+ J-

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Additional Slides

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• Introduce tune shift to prevent instabilities– Introduces

Landau damping• One option for

high intensity machines

• Key: Non-linear block– Length 3 m

Integrable Optics

13 m

Nonlinear Lens Block

10 cm

5.26F F

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Required Vector Potential

• Singularities• Difficult to

approximate with multipole fields

• Ideally non-circular aperture – 2 cm horizontal, 4 cm vertical

B22

)()(),(

gf

tyxU

c

ycxycxc

ycxycx

2

22222

2222

acos2

1)(

acosh1)(

2

2

g

f

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Integrable Optics - Field

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Quadrupole

Gauging

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Gauging• Circular coil: constant

current in longitudinal direction will cause a uniform vector potential A0 within this circle

• Az(x,y)=A1(x,y)+A0

• N.b.:

• Ergo: changes vector potential but not field

• Allows to shift current

y

AB

x

AB z

xz

y

,

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Gauging for elliptical coils

• For elliptical coils (or other shapes): some modest variation of Az

• Example: quadrupole• Correction per current

strand: 2kA• Field: 0.3 mT

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• Required: desired vector potential– Defined by application

• Required: beam aperture– Defined by application– (Real coil will be slightly

larger)

Methodology

Az

Beam Aperturex

y

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• Define point P1 on desired cross-section (known Az)

• Define current I1

(for example on coil cross-section)

• Az can be calculated from

Methodology (cont.)

I1 P1

a

RIrAz ln

2),( 0