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 (  t B). The displacement current (  t D) was added to explicitly conserve electric charge. B-field “Lithography” Designing Printed Circuit B-field Coils from the Inside Out 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. 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. 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. Christopher B. Crawford, Greg Porter, Yunchang Shin University of Kentucky +  ElectricMagnetic Constitutive: 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. The space-time symmetry of the equations is manifest using differential forms in 4-space. With the following definitions, The Maxwell equations reduce to two exact sequences (since d 2 =0): The gauge functions specify multiple solutions to dA=F, ie. A’=A+dÂ. 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   Â=0 (unique with boundary conditions) In this case the equation *d* d A = (   - d *d*) A = *J or   A= -J. In components: In the Lorenz gauge, Im Ker Im Ker Im KerIm Force: Source:                                                                    B t =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 3-d layout of flux return and inner coils     x z Top view B J beam left beam right  y   1.16 A 50 windings 0 m – 100 mG 1 m – 189 mG 2 m – 460 mG 3 m – 2.4 G 4 m ~ 10 G j max =152 A/m P max =11.3W/m 2 P ~ 100 W  - metal shield 1010 steel flux return 10 Gauss solenoid RF spin rotator 3 He target / ion chamber supermirror bender polarizer (transverse) FnPB cold neutron guide 3 He Beam Monitor transition field (not shown) FNPBn- 3 He red - transverse B-field lines blue - end-cap windings 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: –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- 3 He 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- 3 He experimental setup: NPDGamma windings n- 3 He windings Magnetostatic FEA calculation in COMSOL Force: Source: Technique for Designing Fields Application #1 – RF Cavity Application #2 – Guide Field 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 q e /q m is a constant so that field fields can be rotated in (E,H) and (D,B) to remove one source term. Electric/magnetic symmetry is restored in regions without charge. In those regions, A can be replaced by a scalar magnetic potential Á m. Note that this is a source potential, not a force potential. Á e is measured in volts (V), while Á m is measured in amperes (A). There cannot be a source in the corresponding Laplace equation. 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. Does it really work? 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 r 2 Á m =0 with the scalar magnetic potential. 1) Solve for the field Á m using FEA software in the intermediate region Use Neumann boundary conditions (  Á m /  n = n¢B/¹) specified by the required interior and exterior fields. We use this condition because n¢B is continuous across the boundary and does not depend on K, which is still unknown. 2) Fix the winding current I 0 and plot the level surfaces of the Á m on each boundary. The resulting traces correspond to the positions of the winding. 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. 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. Any complex geometry can be wound with one circuit which bridges over from one level potential to the next and then back. 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 proof is as follows: the equipotential curve Á m = I n is by definition perpendicular to rÁ m. The boundary condition n£H= K implies that K is also perpendicular to rÁ m. Since both lie on the surface, K must flow along the equipotential lines as stated. To calculate the current flowing between two equipotential lines, integrate K along a field line between the two potentials: 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 Field taper designed to Preserve adiabaticity. The curvature field lines are unavoidable, but can Be minimized. 3-d model of tapered coil calculated in COMSOL. The ribons represent current loops, placed at equipotentials of Á m