Transcript Document

"Realistic" Ring Cooler Magnetic Fields -The Next Generation
Steve Bracker
Workshop on Ring Coolers
University of Mississippi
March 11-12, 2004
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 . . . .
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.
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
TIME in Project
Installation
Operation
An Alternate Trajectory . . .
Rigor on
"Realistic"
Vague
Speculations
Conceptual
Design
Real
Design
Fabrication
TIME in Project
Installation
Operation Suicide
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
TIME in Project
Installation
Operation Suicide
... or one can just get on with things and hope.
In building up complex magnetic-field-generating-things (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.
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.
We know how to do
finite straight line
current carriers...
For m0/4p = 1 and I = 1,
B at P = (sin q2 - sin q1) / z
(B points out of page)
P
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)
-q1 q2
Yp
Xp
(0,0,0)
(L,0,0)
Xp/Yp = tan (-q1)
q1 = atan (-Xp/Yp)
(L-Xp)/Yp = tan (q2)
q2 = atan ((L-Xp)/Yp)
Z
X
Bx = 0
By = 0
Bz = (m0/4p) I (sin q2 - sin q1) / Yp
Y
P
Changing notation in preparation
for generalizing this:
Yp -> D
By -> Br (radial component)
Bz -> Bt (tangential component)
(Xp,D,0)
-q1 q2
D
Xp
(0,0,0)
(L,0,0)
Xp/D = tan (-q1)
q1 = atan (-Xp/D)
(L-Xp)/D = tan (q2)
q2 = atan ((L-Xp)/D)
X
Bx = 0
Br = 0
Bt = (m0/4p) I (sin q2 - sin q1) / D
Z
Rotating P around the X axis in the Y->Z direction by angle t:
D = sqrt(Xp^2 + Zp^2)
t = atan (Zp/Xp)
Bx = 0
By = -Bt sin (t)
...and we
Bz = Bt cos (t)
can express this in a
coordinate system that makes sense
to a GEANT simulation.
X'
(z'3, x'3)
(z'2, x'2)
(z'1,0)
(z'2, -x'2)
(z'4, x'4)
(z'5, 0)
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'
(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.
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 (current=I)
then defines the field at every point
away from a conductor.
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.
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
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
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.
X'
Questions:
Add one more point to the
magnet description? A bit
of concavity/convexity normal
to the particle direction.
(z'2,x'2)
(z'1,0)
(0,x'6)
(z'3,x'3)
(z'4,x'4)
(z'5,0)
Z'
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
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?
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
BfieldTest1 X vs Z
150
150
100
100
50
50
0
0
X
X
BfieldTest1 X vs Z
-50
-50
-100
-100
-150
-150
-100
-50
0
Z
50
100
150
-150
-150
-100
-50
0
Z
50
100
150
Magnets azimuthally centered in
60 degree cells; injection at cell
boundary
Romulus decided he would prefer
to inject into the leading edge of
the magnet. One parameter changed
(aCpc), and . . .
BfieldTest1 X vs Z
150
150
100
100
50
50
0
X
X
BfieldTest1 X vs Z
0
-50
-50
-100
-100
-150
-150
-100
-50
0
Z
50
100
150
-150
-150
-100
-50
0
Z
50
100
150
A little Coil Design Tool (an Excel spreadsheet) helps the user compose the coil definitions.
Cell Geometry
Cell Geometry
100
100
90
90
80
80
P3
70
60
50
30
20
90
100
80
70
BfieldTest1 X vs Z
BfieldTest1 X vs Z
150
150
100
100
50
50
X
X
50
Z
Z
0
0
-50
-50
-100
-100
-150
-150
40
-100
90
100
80
70
60
50
40
30
20
0
10
-10
-20
-30
-40
-50
-60
-70
-80
-100
-90
-90
-100
30
-80
-90
20
-70
-80
0
-60
-70
10
-50
-60
-10
-40
-50
-20
-30
-40
-30
-20
-30
-40
-10
-20
-50
X
0
-10
-100
P5
P1
10
0
-60
P5
P1
P2
-80
20
10
P4
40
P2
-90
30
60
P4
40
-70
50
X
P3
70
60
-100
-50
0
Z
50
100
150
-150
-150
-100
-50
0
Z
50
100
150
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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