Improved prescription for winding an electromagnet
IImproved prescription for winding an electromagnet
Christopher B. Crawford and Joseph P. Straley
Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506-0055, USA (Dated: July 4, 2020)We describe an improvement on the magnetic scalar potential approach to the design of anelectromagnet, which incorporates the need to wind the coil as a helix. Any magnetic field that canbe described by a magnetic scalar potential is produced with high fidelity within a Target region; allfields are confined within a larger Return. The helical winding only affects the field in the Return.
PACS numbers: 07.55.Db, 41.20.Gz, 03.50.De
I. USING THE MAGNETIC SCALARPOTENTIAL TO DESIGN A COIL
In practical uses of magnetism it is sometimes desirableto be able to create a very well characterized magneticfield; for example, a field that is uniform within a definedregion (hereafter, the Target), and with a surroundingregion (the Return) which confines the field, so that thereis no magnetic disturbance outside the device. Ref [1]has described an algorithm to do this that makes use ofthe magnetic scalar potential to determine the surfacecurrent densities on the interfaces.Here is how the algorithm works: the specified mag-netic field inside the Target is represented by a scalarpotential such that (cid:126)H target ( (cid:126)r ) = − (cid:126) ∇ Φ target . A currentfield (cid:126)K ( (cid:126)r ) at the surface of the Target is constructed sothat only the normal component of (cid:126)H is present just out-side the target. It can be shown that this surface currentdensity flows along the equipotentials of Φ target ; havingthe current between any two equipotentials be equal tothe potential difference ensures that the tangential com-ponents of the field are terminated. This establishes thephysical interpretation of magnetic scalar potential asa “source potential” in analogy with charge being thesource of electric flux lines.The Return envelopes the Target, excepting possiblyparts of the Target where there is no normal field. Thenormal component of the field is already specified at theinterface between the Target and the Return, and it isrequired to be zero at the exterior surface. Then themagnetic scalar potential Φ return defined in the Returnis determined by Neumann boundary conditions. Thecurrent field on both the inner and outer surfaces of theReturn is determined by this scalar potential in the sameway as above.The result is a complete description how to constructsurface current distributions that exactly produce withinthe Target any field configuration that is consistent withMaxwell’s equations. Choosing equal increments of thescalar potentials divides the surfaces into ribbons whichcan be turned into physical wires, all carrying the samecurrent.For the cases that the Return is one region that com-pletely envelopes the Target, a simplification of this al-gorithm is possible: as described, there are two current sheets right next together at the interface between theTarget and the Return, and these can be merged. Thenthe current sheet on the surface is determined by the gra-dient of the difference between the two scalar potentials,and flows along contours of constant value of this differ-ence. In the continuum limit (i.e. very fine wires) thesurface current density is (cid:126)K ( (cid:126)r ) = − ˆ n × (cid:126) ∇ δ Φ , (1)where the stream function [2–4] δ Φ = Φ return − Φ target is the difference between the scalar potentials on the twosides of the surface and ˆ n is the outward-directed normal.A small source of dissatisfaction with this algorithm forconstructing designed-field coils is that what it prescribesis the current that should flow in closed loops around thesurface of the Target, and on the surfaces of the Return.The effect of breaking each loop and connecting it to itsneighbors to make a series circuit introduces the equiva-lent of a single wire lying transverse to the loops; its ef-fect can be nearly canceled by running the return currentwire right on top of the interconnections, but this leavesa magnetic dipole line source. Since the perfect field en-visioned by the algorithm is otherwise only spoiled byexponentially small corrections due to wire discreteness,this would appear to be the largest design error in theconstructed field.Here we will describe a modification of the algorithmwhich removes this defect, allowing the construction of amore nearly perfect coil. In the next section we will workthough a problem is that exactly solvable, but not quitetrivial; later we will generalize this to arbitrary geometry. II. SPHERICAL ELECTROMAGNET
Winding an infinite solenoid as a helix also introducesan axial current, but this does not affect the field insideit. This can be readily generalized to any azimuthallysymmetric object. First consider the case that the fieldin the Target is zero, but there is with an axial currenton the surface of the Target. Let the symmetry axis be z ,and the shape of the object be ρ ( z ). Current conservationrequires that the current through any cross section be thesame; then from outside this is indistinguishable from along straight wire along the axis (this wire will have to a r X i v : . [ phy s i c s . c l a ss - ph ] J a n actually exist beyond the object); the added field out-side will be the corresponding field, which has only a ˆ φ component. This adds no field component normal to thesurface, and the parallel component is exactly canceledby the surface current. The field inside is unaffected.Adding the axially symmetric field inside the Target, wefind the corresponding field in the Return by the algo-rithm described above, and add the currents and fieldsjust constructed to find a consistent set of fields producedby wires that wind around the surfaces.For the case of a sphere of radius P with only anaxial current, the surface current density is K axial = − ˆ θI/ (2 πP sin θ ) (in spherical coordinates) and the fieldoutside is (cid:126)H = ˆ φI/ πr sin θ . We can represent this as amagnetic scalar potential (cid:126)H = − (cid:126) ∇ Φ axial , whereΦ axial = − φI/ π (2)This is a multiply valued function; we can make it singlevalued by introducing a branch surface interrupting anypath that wraps around the sphere or the wire extension.It has the same discontinuity I everywhere across thebranch surface, which can be taken to be the half-plane φ = 0 for r > P – but note that the gradient of Φ axial (the magnetic field) is continuous everywhere except atthe poles θ = 0 and θ = π .Now consider how the magnetic potential constructiondescribes the same sphere when it has a uniform mag-netic field inside (but no axial current), and all magneticflux is enclosed within a Return of radius Q . Inside, themagnetic potential isΦ target = − zH = − H r cos θ. (3)The potential in the Return isΦ return = H P ( r + Q / r ) cos θQ − P . (4)At the boundary the normal component of the mag-netic field is continuous, so that − ∂∂r Φ target = 0. Thesurface current density that matches up the tangentialmagnetic fields is (cid:126)K inner = ˆ φ H Q Q − P sin θ ≡ ˆ φS inner sin θ (5)on interface between the Target and the Return, and (cid:126)K outer = − ˆ φ H P Q − P sin θ ≡ − ˆ φS outer sin θ (6)on the outer surface of the Return. If we slice up thesphere surface at equal intervals of the magnetic poten-tial, or equivalently at equal intervals of the coordinate z (call this interval D ), each of the resulting rings onthe surface of the Target is carrying the same current (cid:126)I = ˆ φDS inner . The rings have the same extent in the co-ordinate z but varying width on the surface of the sphere. FIG. 1: An illustration of the helical winding for the spher-ical electromagnet. An actual coil would use a much smallercurrent and far more turns
Now we make this set of rings helical, so that after oneturn the bottom of this ring is higher (measured alongthe z axis) by the amount D . This adds an axial cur-rent (cid:126)I helix = ˆ zDS inner in the form of the surface currentdensity K axial . This modifies the potential in the Re-turn Φ return (4) by adding an axial potential of the formΦ axial (2) given above. As explained above, this has noeffect inside the sphere.The interesting point is that in moving up one turn, thescalar potential difference Φ return − Φ target decreases by DS , and in going around one turn, the potential outsidedue to the axial current Φ axial increases by DS . Thisshows that the condition Φ return + Φ axial − Φ target = constant generates a helical winding for a spherical elec-tromagnet that completely and efficiently covers the sur-face. The same construction applies to the outer surfaceof the return with the omission of Φ target ; observe thatall of the current that was moving along the z axis on theTarget will return along the outer surface in agreementwith both Ampere’s law and current conservation.Including the effects of the axial currents, the magneticpotential in the Return is given by˜Φ return = H P ( r + Q / r ) cos( θ ) Q − P − I π φ (7)The potential difference across the surface of the targetand across the outer surface of the return determines thesurface current density. Slicing each boundary surfacealong lines of constant potential difference in incrementsof I turns each one into a ribbon that winds around thesurface, connecting up with the next ribbon. Explicitly,the potential differences are δ ˜Φ inner = 32 H P Q Q − P cos θ − I π φ (8)at the interface between the Target and the Return, and δ ˜Φ outer = − H Q P Q − P cos θ + I π φ (9)on the exterior surface of the Return. The tangential(along the surface) gradient of the potential differencescorresponds to a surface current density (Eq. 1) thattwists about the sphere in such a way as to match thetangential component of the field inside with the tangen-tial field outside according to Ampere’s law. Beyond theTarget, there are a pair of wires along the axis, carryingthe current I to the outer surface of the Return. In con-structing the spherical magnet, the field inside and thecurrent I are independent parameters, which determinethe width of the windings (or the spacing of the equiv-alent wires). Once the potentials have been chosen, thepath of the windings is determined by the condition ofconstant potential difference.The combined potential ˜Φ return actually has a non-physical discontinuity DS at each crossing of the branchcut, which must be built into numerical solutions, butthe result is equivalent to the multi-valued Φ axial poten-tial with no discontinuities as describe above, for whichthe entire winding is traced out by a single equipotentialcontour on both the inside and outside of the Return. III. THE GENERAL CASE
The construction described above will work with lit-tle modification for any system with an axis of rotation;the only difference is that the construction of Φ return willentail solving the Laplace problem with Neumann bound-ary conditions in a more complicated geometry. However,it can also be generalized to any Target region of arbi-trary shape and physically allowable magnetic field. Thefirst step is to follow the basic algorithm [1] to learn howto produce a designed field, by finding the scalar poten-tials Φ target and Φ return . Equal intervals of δ Φ describeribbon loops on the surfaces which carry equal current.Though this doesn’t yet specify how to make a coil, it ap-proximates its form. The points (the four dots in Fig. 2)where δ Φ is maximal and minimal on each surface arethe places where the coil on a surface must start andend. For fields of dipolar character in a suitable Targetand Return, there will be just one maximal and minimalpoint on each surface.Now we need to choose a “wiring diagram”: Jumperwires connecting the minimal point on the Target to themaximal point on the Return, and vice versa for the otherpair of extrema, with the current supply interpolatedin one of these paths via a twisted pair. At this pointwe also decide how large a current I to use relative to the size of the field H to be constructed (which deter-mines the width of the ribbons, just as in the case of asolenoid). Assuming that the Return wraps around theTarget, choose a branch surface whose boundary edgeincludes the wires connecting the Target to the Return,and traverses across the surfaces of the Target and theReturn so that the surface interrupts any path throughthe Return that encloses the Target. Treating this as aspecial kind of boundary, the Return is now singly con-nected.As in the case of the spherical electromagnet, we nowconstruct a second scalar potential Φ axial in the Returnthat has has vanishing normal derivative at the surface ofthe Target and the Return, and a discontinuity of magni-tude I across the branch surface. This is not a standardNeumann boundary condition, though perfectly well de-fined. a) a)b) b) c)d)d) FIG. 2: An illustration of generalized axial windings, usedto connect all equipotential contours of the Target and Re-turn fields into a continuous helical winding. a) axial currentsflow from the maximum to minimum of Φ return − Φ target onthe inner surface of the Return and back from maximum tominimum of Φ return on the outer surface. b) two Jumpersconnect the axial currents to make a complete circuit. Thecorresponding toroidal winding field is confined to the Returnso it does not disturb the inner or outer specified fields. c)the entire coil is energized from a twisted pair inserted intoone Jumper on the outer surface. d) the two Jumpers canbe extended outside the Return to form a filament circuitwith calculable potential Φ wire with the same singularity asΦ axial , so that the latter can be solved from the differenceusing standard Neumann boundary conditions. (Here’s a way to turn it into a standard Neumannboundary value problem, as illustrated in Fig 2: choose awire circuit that carries the chosen current, such that itincorporates the two Jumpers, connected by wires whichlie outside the Return and inside the Target. The Biot-Savart integral for the magnetic field of this circuit corre-sponds to a scalar potential Φ wire ( (cid:126)r ) = I Ω / π where Ω isthe solid angle subtended by the circuit viewed from thepoint (cid:126)r , and has a branch surface which can be chosento coincide with that of Φ axial . The difference betweenΦ axial and Φ wire is a solution to Laplace’s problem in-side the Return, and with normal derivative which is thedifference between those of the two potentials. The for-mer is specified by the field in the Target and the latteris calculable, so this is a standard Neumann boundaryvalue problem in a singly connected region.)As above we construct Φ chiral = Φ return +Φ axial in theReturn. The difference Φ chiral − Φ target is constant alonga path that slices the surface of the Target into one longribbon that defines the appropriate winding; similarly,Φ chiral is constant along a path on the outer surface ofthe return that defines the winding there.The wires connecting the Target to the outer surface ofthe Return should join the extrema of the relevant poten-tial differences, but the positioning of these has alreadybeen determined earlier in the algorithm. We claim thatthe wire positions continue to be appropriate: the wiresthemselves are singular points of the fields where Φ chiral takes on different values when the wire is approachedalong different paths. Then the “perturbation” due tothe tangential fields will automatically be absorbed inthe determination of the surface windings.The axial potential Φ axial produces an additional mag-netic ”winding field”. In the specific and general casesdescribed above, this was a toroidal field circling aboutthe Jumpers. The boundary conditions were constructedto confine the winding field completely inside the Return,so that it didn’t perturb either the Target or externalfields.The algorithm describes how to make a perfect elec- tromagnet in the limit of a continuous surface currentdistribution when I →
0. The effect of replacing thiswith discrete windings is exponentially localized near thesurfaces, with a healing length whose scale is set by thewinding spacing [1]. This is largest near the wire connec-tion points where the current density vanishes when thesurfaces are smooth (this can be readily seen in the case ofthe spherical magnet), and has the consequence that thelargest discreteness error occurs near the poles [6]. Wepropose that the Target and Return be deformed nearthe wire attachments, to make them somewhat conicalabout the wire (the Target becomes a lemon, and the Re-turn has the shape of an apple). The linear (rather thanquadratic) variation of the potential near the connectionpoint implies that the width of the spiralling ribbon willnot grow so much in that limit, thus decreasing the heal-ing length and making the field in the Target closer tothe design field.In summary, this paper provides a general method toconvert the series of disjoint equipotential contour loopsdescribed in [1], each carrying current I , into a contin-uous helical winding through use of an auxiliary axialpotential Φ axial representing the current I flowing fromthe lowest to the highest equipotential along each surfaceof the Return. This method is very general in the sensethat the choice of wiring circuit, current, and equipoten-tial constants classifies all possible helical windings whichapproximate the specified Target and external field witha discrete wire or trace winding. IV. ACKNOWLEDGMENTS
This work is supported in part by the U.S. Depart-ment of Energy, Office of Nuclear Physics under contractsDE-SC0008107 and DE-SC0014622, and by the NationalScience Foundation under award number PHY-0855584. [1] C. B. Crawford, arXiv:2012.00800.[2] HA Haus and JR Melcher, Electromagnetic Fields andEnergy, (Prentice-Hall, Englewood Cliffs, 1989).[3] ID Mayergoyz and G Bedrosian, “On Calculation of 3-DEddy Currents in Conducting and Magnetic Shells”, IEEETrans Magnetics 31, 1319 (1995).[4] R. A. Lemdiasov ”RF Coils for MRI” (ed. J. T. Vaughanand J. R. Griffiths; Wiley, 2012) Chapter 26, pp. 327-338..[5] P. L. Walstrom, Dipole-magnet field models based on a conformal map, Physical Review Special Topics - Acceler-ators and Beams 15 (10) 2401 (2012)[6] N. Nouri and B. Plaster, “Comparison of magnetic fielduniformities for discretized and finite-sized standard cos- θθ