Realistic%20Model%20of%20the%20Solenoid%20Magnetic%20Field

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Realistic Model of the Solenoid Magnetic Field

• Paul S Miyagawa, Steve Snow
• University of Manchester

• Objectives
• Closed-loop model
• Field calculation corrections
• Helical model
• Realistic model
• Results
• Conclusions further work

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Objectives

• A useful test of the Standard Model would be
  measurement of W mass with uncertainty of 25 MeV
  per lepton type per experiment.
• Momentum scale will be dominant uncertainty in W
  mass measurement.
• Momentum accuracy depends primarily on alignment
  and B-field.
• Solenoid field mapping team targetting an
  accuracy of 0.05 on sagitta to ensure that
  B-field measurement is not the limiting factor on
  momentum accuracy.
• Mapping machine will take measurements of the
  solenoid B-field.
• Simulations of the machine performance have been
  based on a simple model of the B-field.
• Want to test the performance with a more
  realistic model.
• Realistic model seeks to improve on several
  aspects
• Calculating the field from a curved current path
• Modelling the actual current path
• Modelling the shape of the solenoid

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Closed-Loop Model

• Previously, the solenoid was modelled as a series
  of closed circular loops of infinitely thin wires
  evenly spaced in z.
• For calculating the field, each loop approximated
  as 800 straight-line segments, and Biot-Savart
  law applied to each segment.
• Added in field due to magnetised iron (4 of
  total field).
• Model is symmetric in ? and even in z.

current 7600 A Nominal current at 2 T
number of loops 1154 Number of electrical turns
half z-length 2645.5 mm Includes shrinkage due to cold and excitation
radius 1246 mm Average of internal and external radii
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Field calculation corrections (1)

• We approximate the true current path, which is
  helical, with straight-line segments whose end
  points lie on the true path.
• This leads to two types of error
• Inscribed polygon error
• If a circle is replaced by an inscribed polygon,
  the average radius of the polygon is less than
  the radius of the circle.
• This effect can be corrected by increasing the
  radius by 2/3 of S where S is the sagitta between
  the arc and the chord.
• Line segment integration error
• The natural approximation is to use the midpoint
  of the segment to calculate r between the segment
  and the measurement point.
• This leads to an error because r is different for
  each point on the line. This error is more
  significant than the inscribed polygon error.
• A correction factor can be added to the magnitude
  of the field

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Field calculation corrections (2)
z 0, r 0
z 1, r 0
z 0, r 0.8
z 1, r 0.8

100 steps most likely sufficient, but we use 200
to be safe.
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Helical Model

• First step towards a realistic model is to
  replace the series of closed loops with a helical
  coil.
• The current starts at (0,R,hz), winds in
  anti-clockwise direction, terminates at
  (0,R,-hz). Note that this does not form a closed
  path.
• The overall dimensions of the coil are the same
  as for the closed-loop model.
• Model is neither symmetric in ? nor even in z.
• Main differences with closed-loop model are at
  the ends of the coil due to difficulty in lining
  up the ends.

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Realistic Model

• The ATLAS solenoid consists primarily of four
  main coil sections.
• The last loop from each section is welded to the
  first loop of the next section to form a single
  coil.
• At end A, an extra cable is welded to the last
  loop, and the two cables are routed up through
  the services chimney.
• At end C, an extra cable is welded to the last
  loop. The two cables form the return coil, which
  is routed along the cryostat surface to end A and
  up into the chimney.
• Cables in the chimney are magnetically shielded.
• The realistic model models the current through
  these main components.

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Main Coil Sections

• From end A to end C, the four coil sections are
  labelled H-A, F-A, F-C and H-C.
• Each section is modelled as a shorter version of
  the helical coil with its own length and number
  of loops.
• The loops at either end of each section are
  treated in the welds, so are not included in the
  main sections.

Section Length (mm) Number of loops
H-A 1315.0 286
F-A 1321.0 288
F-C 1325.0 288
H-C 1326.5 288
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Internal Welds

• Physically, the last loop from one section lies
  parallel to the first loop of the next section.
  These two loops are welded together along their
  entire length so that electrically they are one
  loop.
• Two models of these welds can be used
• The physical representation uses two loops
  running in parallel. The current gradually
  transfers from the first loop to the second.
• The electrical representation uses a single loop
  of double pitch. This loop carries the full
  current over its entire length.

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End Welds

• At each end of the solenoid, the extra cable is
  welded to the main cable over a 30-cm portion.
  The current starts entirely in the main cable,
  and 20 is transferred to the extra cable by the
  end of this portion.
• The two cables follow the curvature of the
  solenoid over 4.5, but are electrically isolated
  from each other. They carry the current in the
  ratio 8020.
• The two cables then curve away from the solenoid
  until they are tangent to the chimney angle
  (11.25 from vertical). The radius of curvature
  is 165 mm. As the cables are still electrically
  isolated, they carry the current in the same
  8020 ratio.

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Return Coil

• The return coil consists of the two cables from
  the end C weld running flat along the cryostat
  surface. The extra cable rests on top of the main
  cable.
• The cables are electrically isolated, so they
  carry the current in ratio 8020.
• In the realistic model, the cables connect to the
  cables from the end A weld so as to form a closed
  current path for the entire solenoid.

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Shape Deformations

• The dimensions given are for a warm coil in
  quiescent conditions.
• The real coil will be deformed
• Shrinkage due to cool down. This is modelled as
  an overall scale reduction of 0.41.
• Bending due to field excitation. This is modelled
  as the radius r being a parabolic function of z.
  At the centre, ?r 0.90 mm at the coil ends, ?r
  0.38 mm, ?z 1.09 mm.

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Results (1)
14
Conclusions
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Further Work