"Realistic" Ring Cooler Magnetic Fields --

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"Realistic" Ring Cooler Magnetic Fields -- The
Next Generation
Steve Bracker Workshop on Ring Coolers University
of Mississippi March 11-12, 2004
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Don sprang this talk on me without much advance
warning. Usually I prepare slides for talks on
the plane bus on the way to the conference, so I
was relaxed until someone brought to my attention
that I was already here. Trouble... Hence I
thought it best to rummage around in the archives
for a talk I could dust off and revise a bit.
Happily I found an old position paper I had put
together for Bill Clinton a number of years ago
just in case he asked . . . .
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On The Meaning of "Sex" "Realistic"
Has come to mean "better than a box field
and maybe more or less satisfies Maxwell's
Equations".
At some point in the design process, we must mean
more 1. The field is that one field of all
Maxwellian fields generated by some
well-specified apparatus (e.g. coil set). 2.
Engineers assure us that such an apparatus can
be built to sufficient precision, and operated
without causing region-wide power blackouts. 3.
Simulations assure us that the apparatus will
still perform when constructed and maintained to
feasible precision and stability.
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Paraphrasing one not-so-great man . . .
"Whenever I hear 'realistic' and 'field' in the
same sentence, I reach for my revolver."
An Idealized Evolution of Rigorous Realism
Through Time
Rigor on "Realistic"
Vague Speculations
Conceptual Design
Real Design
Fabrication
Installation
Operation
TIME in Project
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An Alternate Trajectory . . .
Rigor on "Realistic"
Vague Speculations
Conceptual Design
Real Design
Fabrication
Installation
Operation
Suicide
TIME in Project
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Projects often slowly slide from Conceptual
Design to Real Design without much fanfare. Great
Peril that the need for rigorous realism in the
design of essential components will be realized
very late, so that great strain will be placed on
critical manpower late in the design process....
Overwhelming but essential effort to achieve
rigorous realism
Area of Maximum Peril
Rigor on "Realistic"
Vague Speculations
Conceptual Design
Real Design
Fabrication
Installation
Operation
Suicide
TIME in Project
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... or one can just get on with things and hope.
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In building up complex magnetic-field-generating-t
hings (MFGTs), there are only a few
computationally interesting sub-things, among
them 1. differential straight-line current
elements (Biot-Savart integration, slow) 2.
finite-length straight-line current segments
(simple analytic expression) 3. cylindrical
current sheets (e.g. BSHEET)
If magnets for cooling rings can be represented
as aggregates of 2 and 3, then there is some
hope of generating a truly realistic field map
before our sun leaves the main sequence.
If magnets for cooling rings can be approximated
as aggregates of these sub-things, then there is
some hope of generating approximations to a
realistic field that are sufficient to test
whether cooling rings cool, and how sensitive
their performance is to details of the magnetic
field map.
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A good magnetic field generator should include
the ability to alter MFGTs in ways not too
dissimilar to those alterations inevitable due to
imperfect construction, operational
instability, and inexact field simulation methods.
To "demonstrate cooling" (before you build and
operate a cooler) you have to show that adequate
cooling takes place not in "one ring to rule them
all" but a whole ensemble of rings which span the
phase space of "rings you may end up with" when
all the vicissitudes of design, construction,
installation and operation are accounted for.
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For m0/4p 1 and I 1,
We know how to do finite straight line current
carriers...
P
B at P (sin q2 - sin q1) / z
(B points out of page)
q1
q2
z
Y
P
For a straight current segment of length L lying
on the X axis from (0,0,0) to (L,0,0), and an
observation point P in the XY plane at
(Xp,Yp,0), carrying current I in the X direction
(Xp,Yp,0)
q2
-q1
Yp
Xp
X
(L,0,0)
(0,0,0)
Bx 0 By 0 Bz (m0/4p) I (sin q2 - sin q1)
/ Yp
Xp/Yp tan (-q1) q1 atan (-Xp/Yp) (L-Xp)/Yp
tan (q2) q2 atan ((L-Xp)/Yp)
Z
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Y
P
Changing notation in preparation for generalizing
this Yp -gt D By -gt Br (radial component)
Bz -gt Bt (tangential component)
(Xp,D,0)
q2
-q1
D
Xp
X
(L,0,0)
(0,0,0)
Bx 0 Br 0 Bt (m0/4p) I (sin q2 - sin q1)
/ D
Xp/D tan (-q1) q1 atan (-Xp/D) (L-Xp)/D tan
(q2) q2 atan ((L-Xp)/D)
Z
Rotating P around the X axis in the Y-gtZ
direction by angle t D sqrt(Xp2 Zp2) t
atan (Zp/Xp) Bx 0 By -Bt sin (t) Bz
Bt cos (t)
...and we can express this in a coordinate system
that makes sense to a GEANT simulation.
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X'
A magnet made up from a small ensemble of
finite straight-line current carriers. To avoid
unseemly charge buildups, I suppose we
should make them all carry the same current,
though we might turn individual segments on
and off for sensitivity studies.
(z'3, x'3)
(z'4, x'4)
(z'2, x'2)
(z'1,0)
(z'5, 0)
Z'
(z'2, -x'2)
(z'4, -x'4)
(z'3, -x'3)
Eight parameters (z'1, x'2, z'2, x'3, z'3, x'4,
z'4, z'5) define the shape of the magnet.
Corresponding points in the right half
by symmetry. 8 straight current segments per
magnet.
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Looking down toward -Y
Looking sideways toward X
X
Y
Rcpc
Ycpc
Acpc
Z
Z
Three more parameters Rcpc, Acpc, Ycpc define
the entire 12-coil assembly. In total there are
11 parameters defining the field shape. One more
(currentI) then defines the field at every
point away from a conductor.
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Four current "models"
1. A non-Maxwellian "box-field" which has
constant B (0,By,0) between the vertical pairs
of coils and (0,0,0) outside it.
2. A Maxwell-compliant single-turn-per-magnet
field, computed from the 8 straight-line segments
per magnet.
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3. A Maxwell-compliant multiple-turn-per-magnet
field stacked in Y, so that each turn has exactly
the same shape and size. Requires two more
parameters number of turns in each stack
turn-to-turn Y separation
dStack
nStack 4
4. A Maxwell-compliant multiple-turn-per-magnet
field with layers of coil stacks. Requires two
more parameters layer separation in X-Z plane
number of layers in X-Z plane.
dLayer
nLayer 3
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Eight parameters to describe geometry of one
(outermost) coil
x'1, x'2, z'2, x'3, z'3, x'4, z'4, x'5
xCoil1, xCoil2, zCoil2, xCoil3, zCoil3,
xCoil4, zCoil4, xCoil5
Four parameters to describe stacking and layering
nStack, dStack, nLayer, dLayer
Three parameters to describe distribution of coil
assemblies around ring
rCpc, aCpc, yCpc
One parameter to set the magnet current
magCurrent
16-parameter field
In array MagnetParam
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call Bfield (magnetParam, position, model,
field) magnetParam 16 input reals describing
magnet configuration position 3 input reals
specifying (x,y,z) where field is to be
found model 1 - box field 2 -
single-coil per magnet 3 - vertical stacks of
coils in each magnet 4 - horizontal layers and
vertical stacks of coils in each magnet field
3 output reals returning (Bx, By, Bz)
Progressively implemented one model at a time.
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X'
Questions
Add one more point to the magnet description? A
bit of concavity/convexity normal to the particle
direction.
(z'3,x'3)
(0,x'6)
(z'4,x'4)
(z'2,x'2)
(z'5,0)
(z'1,0)
Z'
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Add one more parameter to the magnet
configuration? nCells The number of pairs of
magnets distributed (uniform angular
spacing) around the ring.
nCells 6
nCells 8
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Discussion Points
1. This field should explore most of the
physically interesting field-shape issues for
rings of this type. It is sufficiently general to
allow us to test the effects of small
perturbations to field shape on cooling
performance. True?
2. It is possibly practical to call the field
generator directly from GEANT, without the use of
a secondary field grid. How many points per
second must the generator be capable of to permit
this without incurring unacceptable slowdown? How
odious is the care and feeding of a secondary
field? 3. Ring designs of this kind have
appeared so far with 4 and 6 "cells". Are still
different numbers of cells contemplated? Should
the ring design be generalized (1 more parameter)
to allow for N cells? 4. No field map should be
believed without a "second opinion". Is
there anyone out there who would be willing to
undertake a second implementation of exactly this
field in a manner that yields Bx,By,Bz directly?
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Ensure that the rotation transforms are working
correctly using the box field. Generate a toroid
of particle positions (left) and see where the
field is non-zero (right).
Particle Positions Examined
Particle positions with Nonzero Field
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Romulus decided he would prefer to inject into
the leading edge of the magnet. One parameter
changed (aCpc), and . . .
Magnets azimuthally centered in 60 degree cells
injection at cell boundary
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A little Coil Design Tool (an Excel spreadsheet)
helps the user compose the coil definitions.
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Next Steps -- 1. Rework parameter definitions
as needed document. 2. Produce the
single-coil-per-magnet Maxwellian field. 3.
Check 2. by checking with Maxwell, testing simple
symmetric cases, etc. 4. Generate grid field
(format-compatible with FindFieldAnywhere) if and
only if speed dictates. Compare interpolation in
grid to primary field. 5. Add stacking and
layering if simulation results suggest it's
useful.
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