field calculations for ring cooler
TRANSCRIPT
5 February 2002 Ring Cooler Magnet SystemS.Kahn
Page 1
Field Calculations for Ring Cooler
Steve Kahn
5 February 2002
IIT Mucool Meeting
5 February 2002 Ring Cooler Magnet SystemS.Kahn
Page 2
Ring Cooler Geometry
TOSCA finite element program models one octant of r ing.
5 February 2002 Ring Cooler Magnet SystemS.Kahn
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Approach to Field Calculations
• The smallest geometric symmetry unit is the octant. This is not the smallest magnetic symmetry unit because of the field flips.
• If we can assume that the magnet iron is linear we can separate the octant into sub-regions:
– Large Solenoid
– Wedge Dipole
– Small Solenoid
• We can calculate the contributions of these magnets and add them.
– We can evaluate how good an approximation this is.
• The alternative would be to treat the system as a whole.
– The finite element problem would have ~1000000 nodes and would take a very long time to run!
– This is not a simple topology for programs like TOSCA.
5 February 2002 Ring Cooler Magnet SystemS.Kahn
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Dipole Magnet Specifications
• The preliminary parameters of the bending magnet as provided:
– Bending Radius Rbend =52 cm
– Bending Angle Θbend = 45º
– Field Strength B = 1.46 T at reference radius
– Normalized Field Gradient:
• dBy/dx × (R/Bo) = -0.5
– Radius of aperture Raperture = 17 cm.
5.0−=B
r
dr
dB
5 February 2002 Ring Cooler Magnet SystemS.Kahn
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Sketch of Dipole Magnet
Pole face is shaped to achieve required gradient
The design of the wedge magnet is from P. Schwandt(dated 30 Jan 01).
It has been revised.
5 February 2002 Ring Cooler Magnet SystemS.Kahn
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Saturation in Dipole Magnet
• Figure shows the permeability for the vertical midplane of the magnet.
• µ<10 on inner edge of the aperture.
5 February 2002 Ring Cooler Magnet SystemS.Kahn
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By along Vertical Symmetry Plane
• Figure shows three curves:
– Ideal Field:
• 2D field from shaped iron pole and effective yoke width.
• Calculate index=0.473
– 3D Field Calculation
• Calculation using TOSCA
• Gives index=0.47
– 2D cylindrical Calculation
• Uses same pole profile, but has closed cylinder out of plane.
By Along Vertical Symmetry Plane
0
5000
10000
15000
20000
25000
30000
0 20 40 60 80 100 120
r , cm
By,
Gau
ss
3 D model
2 D cylindrical
Ideal 2D
5 February 2002 Ring Cooler Magnet SystemS.Kahn
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By Off Vertical Symmetry Plane
By on Different Planes
0
5000
10000
15000
20000
30 40 50 60 70
Radius, cmB
y,
ga
us
s
0 degrees
6 degrees
11 degrees
17 degrees
22 degrees
Angle Position index
0 0.473
5.625 0.469
11.5 0.516
17.125 0.584
22.5 0.746
Index Calculated on Difference Planes:
5 February 2002 Ring Cooler Magnet SystemS.Kahn
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Dipole Field along Reference Path
Field Along Reference Path
0
2000
4000
6000
8000
10000
12000
14000
16000
-125 -75 -25 25 75 125
Path Position, cm
By,
gau
ss
Inside Gap
Field Clamp
Figure 4: Field components for a path displaced 10 cm vertically from the
reference path
Fields 10 cm Off Axis in Dipole Magnet
-10000
-5000
0
5000
10000
15000
-125 -75 -25 25 75 125
Path Position, cm
B,
ga
uss
By
Bz
Bx
Figure 3: By along central
reference path.
5 February 2002 Ring Cooler Magnet SystemS.Kahn
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Avoiding a 3D Map
• The 2D part of a long accelerator magnet is traditionally parameterized by Fourier harmonics.
• A generalization of the traverse field that takes into account the s dependence of the field has the form:
��
�
+��
���
�++−
+��
���
�++−=
)2sin(128
161
)sin(192
5
8
3)(),,(
54
24
32
22
2
44
14
22
12
1
ϕ
ϕϕ
rds
Kdr
ds
KdrK
rds
Kdr
ds
KdzKsrB
5 February 2002 Ring Cooler Magnet SystemS.Kahn
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Transverse Harmonics as a Function of s
Dipole Component
0
2000
4000
6000
8000
10000
12000
14000
16000
0 10 20 30 40 50
Path Posit ion, cm
B0,
gau
ss
Midplane Fit
Fourier
Quadrupole Harmonic
-0.015
-0.01
-0.005
0
0.005
0.01
0 10 20 30 40 50
Path Pos it ion, cm
b1,
1/c
m
Midplane Fit
Fourier
Sextupole Component
-0.006
-0.005
-0.004
-0.003
-0.002
-0.001
0
0.001
0 10 20 30 40 50
Path Position, cm
b2,
1/c
m2
MidPlane Fit
Fourier
� ϕϕϕπ
dBBo
)2cos()(1
Calculated along the reference path
5 February 2002 Ring Cooler Magnet SystemS.Kahn
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Dipole Profile Fits
b1 harmonic
Calculation
Fit
Bo Harmonic
Fit to
tanh[k(x-χ)]-tanh[k(x+χ)]
5 February 2002 Ring Cooler Magnet SystemS.Kahn
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Preliminary Results from Fit
From Bphi From BrFit results
a lambda z0 a lambda z0
Dipole -7358.6 11.392 22.244 -7643.2 23.329 21.3950.12909 4.64E-04 3.33E-04 0.1199 3.34E-04 2.81E-04
Quadrupole52.52856.529 7.0725 23.769 48.213 6.45E+00 18.6228.05E-03 1.97E-03 2.16E-03 8.22E-03 7.06E-04 1.00E-03
Sextupole 1.0784 6.895 21.415 1.1072 7.7163 24.1296.10E-04 3.85E-03 5.06E-03 5.48E-04 3.35E-03 4.00E-03
5 February 2002 Ring Cooler Magnet SystemS.Kahn
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Dipole Field Description
• The harmonic description is currently in Muc_Geant.
– I have calculated harmonics of Bφ(s) and Br(s) at positions along a reference path through the dipole magnet at 7 different radii using the TOSCA program.
– I have fit the previous formula (2 transparencies ago) parameterizing K(s) as:
• Kn(s) = a[tanh((z-zo)/λ)–tanh((z+zo)/λ)] for each n
• The parameters for K(s) come from a combined fit of Br and Bφ
– The fits for the Dipole and Sextupole components look reasonably good. The fit to the Quadrupole component is not that stable.
• Note the difference in the Br and Bφ for zo for the quadrupole fit.
– Since the harmonics are power series r the field at large radius will be less reliable. This can be seen in the field for r>14 cm.
5 February 2002 Ring Cooler Magnet SystemS.Kahn
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Tosca Model for Wedge Dipole
End of Return Iron for Large Solenoid
10 cm gap between dipole and shield15 cm endplate
95 cm radius
Coils in this Model are assumed to be
superconducting
5 February 2002 Ring Cooler Magnet SystemS.Kahn
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Long Solenoid Magnet
coil
Thick Flux Retur n
Vanadium Permadur End Plate
Field Clamp
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Fields in Long Solenoid
Solenoid Fields
0
5000
10000
15000
20000
25000
30000
35000
40000
45000
50000
0 100 200 300 400 500
Axial Position, cm
B,
ga
uss
Bz
Br_5_cm
Br_10_cm
5 February 2002 Ring Cooler Magnet SystemS.Kahn
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Comments on Solenoid Field
• The end plate effectively separates the solenoid field from the dipole for the case with small aperture coils.
– This still has to be shown for the large coil case.
• Radial field are present only in vicinity of end plate.
– Extend approximately one coil diameter (as expected).
5 February 2002 Ring Cooler Magnet SystemS.Kahn
Page 19
Short Solenoid Magnet
5 February 2002 Ring Cooler Magnet SystemS.Kahn
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Short Solenoid Fields
Bz in Field Flip Solenoid
-60000
-40000
-20000
0
20000
40000
60000
-75 -25 25 75
Ax ial Position, cm
Bz,
ga
uss
Bz(0 cm)
Bz(10 cm)
Bz(20 cm)
Br in Field Flip Solenoid
-20000
-10000
0
10000
20000
30000
40000
50000
-100 -50 0 50 100
Axial Position, cmB
r, g
au
ss
Br(10 cm)
Br(20 cm)
5 February 2002 Ring Cooler Magnet SystemS.Kahn
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Ideal Short Solenoid Field Flip Magnet
• Top figure shows Valerie’s design for the short field flip solenoid.
• The lower figure shows the field and dispersion for that magnet.
• This form of the field assumes that there is a Neumann condition at the end of the magnet.
– The field will continue ~2 T forever.
Bz on axis
5 February 2002 Ring Cooler Magnet SystemS.Kahn
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Short Solenoid Fields From Opera2D
Field with iron covering ends
Field with aperture for muons
This forces field to be sort of parallel to axis
How critical is it to have this field plateau?
5 February 2002 Ring Cooler Magnet SystemS.Kahn
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Field Maps
• The TOSCA program can generate 3D field grids.– Wedge Dipole
• Opera2D is used to generated the following grids with 1 cm×1 cm segmentation:– Large Solenoid– Tail of Dipole in Large Solenoid– Tail of Dipole in Small Solenoid– Small Solenoid
• The gufld (or guefld) subroutine in GEANT will have to keep track of which octant it is in, which grid it should use and what signs to apply to the field for each point in space.
5 February 2002 Ring Cooler Magnet SystemS.Kahn
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Field in Geant
5 February 2002 Ring Cooler Magnet SystemS.Kahn
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Tracking Muons Around the Ring
• Tracking to test the field.
– Set all absorber material to vacuum.
– Remove all RF.
– Cooling ring now is just a storage ring.
• Current State:
– A so-called on momentum muon can get 2 revolutions around the ring.
– This just requires tuning the overall dipole field with a small scale factor.
– There is a very small transverse displacement necessary since the reference path is not exactly circular because of the fringe field.
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This is the only graph I could find. The on momentum track shows very little Larmor motion.
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Current Status
• I am beginning to look at tracking muons through the ring with the RF. I have large losses at this point.
– This is merely debugging a new system. I need to put more work here.
• I have placed all of the code and field maps for this realistic field representation into the MUC_GEANT CVS.
– They are available for others to use.
– There are still some details that need to be fixed.