B-field “Lithography”
Designing Printed Circuit B-field Coils
from the Inside Out
QuickT ime™ and a
decompressor
are needed to see thi s pi cture.
Background – Maxwell Equations
The Maxwell equations arise out of the electromagnetic force laws (Coulomb’s
law and the Biot-Savart law) by separating the two charges into force (’, J ’) and
source (, J) terms. The corresponding force (E, B) and source (D, H) fields
are connected by the constitutive relations. The fields take on a life of their own,
for example, with inclusion of Faraday’s law (tB). The displacement current
(tD) was added to explicitly conserve electric charge.
Electric
Application #2 – Guide Field
Christopher B. Crawford, Greg Porter, Yunchang Shin
University of Kentucky
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Abstract: Traditionally the design cycle for magnetic fields involves guessing at a reasonable conductor and
magnetic material configuration, using finite element analysis (FEA) software to calculate the resulting field,
modifying the configuration, and iterating to produce the desired results.
Magnetic
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Force:
We take the opposite approach of specifying the required magnetic field, imposing it as a boundary condition on
the region of interest, and then solving the Laplace equation to determine the field outside that region. The exact
conductor configuration along the boundaries is extracted from the magnetic scalar potential in a trivial manner.
Source:
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We are designing a holding field for the UCN guide of the Fundamental
Neutron Physics Beamline (FnPB) at the SNS
The last 2 metres of guide have stringent field requirements:
– Zero B-field outside the coil (can’t perturb the mu-metal shielding)
– Uniform dipole guide field inside the coil (to preserve neutron polarization)
– Smooth taper in the field from 5 G to 100 mG with small enough gradients to
transport the neutron spin adiabatically to the measurement cell
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Constitutive:
This method is being applied to design a coils for the neutron EDM experiment, and an RF waveguide in a new
design of a neutron resonant spin flipper for the n-3He experiment. Both experiments will run at the Spallation
Neutron Source (SNS) at Oak Ridge National Laboratory.
These boundary conditions come from integrating the Maxwell equations across
the boundary surface (along n) of a layer of surface charge
, current
or discontinuities in ²,¹,or ¾.
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On exterior boundaries, the longitudinal and transverse boundary conditions are
redundant, and either n£E (Dirichlet) or n¢D (Neuman) must be specified, but not
both. For interior boundaries, you have one condition for each side of the
surface, so both conditions must be specified.
The space-time symmetry of the equations is manifest using differential forms in
4-space. With the following definitions,
Our goal is to design a coil with specified B-fields in
certain regions – for example: a uniform field inside
the coil, and zero field outside the coil. An
intermediate region is left to match up the two fields
and satisfy Maxwell equation. Currents will only be
used on the boundaries, so we can use the Laplace
equation r2Ám=0 with the scalar magnetic potential.
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2) Fix the winding current I0 and plot the level surfaces of
the Ám on each boundary. The resulting traces correspond
to the positions of the winding.
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The FEA postprocessor-generate level curves can be
fed directly into a CAM milling machine, to cut out the
current traces of the magnet on a PCB board.
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Instead of cutting out thin wires, it is better to remove
thin traces between each conductor. This offers less
resistance, and a more continuous current distribution.
1) Solve for the field Ám using FEA software in the
intermediate region
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Use Neumann boundary conditions
( Ám/n = n¢B/¹) specified by
the required interior and exterior fields.
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Any complex geometry can be wound with one circuit
which bridges over from one level potential to the next
and then back.
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We use this condition because n¢B is continuous
across the boundary and does not depend on K,
which is still unknown.
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The above procedure does not account for the boundary
conditions n£H= K of the fixed interior or exterior fields.
Separate coils must be made for this side of the
boundary using the same technique.
The Maxwell equations reduce to two exact sequences (since d2=0):
Im
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Im
Ker
Im
Ker
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The gauge functions specify multiple solutions to dA=F, ie. A’=A+dÂ.
The proof is as follows: the equipotential curve Ám = In
is by definition perpendicular to rÁm.
The boundary condition n£H= K implies that K is also
perpendicular to rÁm.
Since both lie on the surface, K must flow along the
equipotential lines as stated.
The solution to dG=J and is unique up to boundary conditions with the help of
the complement dF=0.
Likewise one can fix A by specifying d*A=0 (Lorenz gauge), which determines
 up to 2Â=0 (unique with boundary conditions)
(2
- d *d*) A = *J
or
2A=
In the Lorenz gauge,
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Electric/magnetic symmetry is restored in regions without charge. In those
regions, A can be replaced by a scalar magnetic potential Ám. Note that this is a
source potential, not a force potential.
Áe is measured in volts (V), while Ám is measured in amperes (A).
There cannot be a source in the corresponding Laplace equation.
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Bt=0 on ends so
solution is axially
symmetric
equipotentials M=c
form winding traces for
current on face
n£(H=rM)
end plates connect
along inside/outside
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Top view
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3-d layout of flux return
and inner coils
beam right
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y
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Field taper designed to
Preserve adiabaticity.
The curvature field lines
are unavoidable, but can
Be minimized.
B

x
J
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beam left
To calculate the current flowing between two
equipotential lines, integrate K along a field line
between the two potentials:
3-d model of tapered coil calculated in
COMSOL. The ribons represent
current loops, placed at
equipotentials of Ám
We are designing a new resonant RF spin flipper to be
used in the upcoming n-3He experiment at the SNS,
with transverse instead of longitudinal B-fields.
Important features in this design are:
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The potentials (Á,A) exists because the source terms can be separated from the
force equation. This is due to the absence of magnetic charges. Strictly
speaking, the ratio of electric to magnetic charge qe/qm is a constant so that field
fields can be rotated in (E,H) and (D,B) to remove one source term.
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Application #1 – RF Cavity
-J .
In components:
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Does it really work?
Ker
Ker
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z
Im
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Technique for Designing Fields
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Force:
Source:
In this case the equation *d* d A =
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Can flip transversely or longitudinally polarized neutrons
No fringe fields at the end – 100% spin flip efficiency
Compact design – it can fit inside of the n-3He solenoid
Matches the driver electronics of the NPDG spin flipper
The RF cavity operates in TEM mode with conductors in
the middle of the guide. Thus the RF solution is the
same as the static B-field solution.
n-3He experimental setup:
FnPB cold
neutron guide
supermirror
bender polarizer
(transverse)
10 Gauss
solenoid
1010 steel
flux return
Magnetostatic FEA
calculation in COMSOL
NPDGamma
windings
n-3He
windings
-metal
shield
3He
Beam
Monitor
transition field
(not shown)
3He
RF spin
rotator
target /
ion chamber
red - transverse B-field lines
blue - end-cap windings
FNPB
n-3He
jmax =152 A/m
Pmax =11.3W/m2
P ~ 100 W
0 m – 100 mG
1 m – 189 mG
2 m – 460 mG
3 m – 2.4 G
4 m ~ 10 G
1.16 A
50 windings