Polygonal coil systems for magnetic fields with homogeneity of the fourth to the eighth order

Review of Scientific Instruments

Volumn 42, Issue 6, 1971, Pages 840-857

  Garrett, Milan Wayne ^a   Pissanetzky, Sergio ^b  

^a SWARTHMORE COLLEGE   (United States)

^b UNIVERSIDAD NACIONAL DE CÓRDOBA   (Argentina)

Author keywords

[No Author keywords available]

Indexed keywords

ELECTRIC COILS; ELECTRIC FIELDS, MEASUREMENT; POLYGONAL COIL SYSTEMS; RSINA;

EID: 0347156859     PISSN: 00346748     EISSN: None     Source Type: Journal    
DOI: 10.1063/1.1685243     Document Type: Article

References (37)

1. Lee-Whiting’s Eq. (39), the exponent of q has the wrong sign (see Ref. 37).
2. Ref. 2, p. 1099, with Figs. 5 and 6. See also Ref. 25.
3. Ref. 3, pp. 2565–2568 and Figs. 1–4.
4. Ref. 2, pp. 1097–1100 and Figs. 2–6. Note that by total inhomogeneity, some authors mean [formula omitted] rather than [formula omitted] (for example, see Ref. 9).
5. Ref. 2, p. 1105 and Tables VIII and IX. Table VIII describes the same family as is now listed with [formula omitted]
6. Ref. 2, Eq. (33) and Ref. 3, Table X, with curves F, [formula omitted] of Fig. 1 and system 6 of Fig. 4. The circular systems are not new; they are included for comparison.
7. Ref. 3, Table V describes the circular analogs.
8. See Ref. 3, Table VII, for the circular case.
9. Table X lists correct solutions of Lee-Whiting’s problem, in which two axial and one transverse derivative must vanish simultaneously.
10. Maxwell’s well known three loop uniform field system (see curve H, Fig. 1 of Ref. 3), and a system computed by
11. See discussion following Eq. (22).
12. We are grateful for the able assistance of C. A. Martín, who carried out much of the arduous derivation and checking (including inevitable detours) for Table IV(b).
13. See Ref. 3, Eq. (9).
14. See Ref. 2, p. 1099; Figs. 5 and 6 show [formula omitted] contours for prolate and oblate hybrids of the second and fourth orders. In a hybrid field, the nominally dominant order is partly compensated in selected regions and (inevitably) reenforced in others, by a residual error of next lower order. Singularities result, on the axis or in the midplane, and sometimes elsewhere. The low order residual is always dominant near the origin, and it impairs homogeneity for very small errors. Hybrids have legitimate uses, but must be redesigned for each specific error magnitude (say 1%) to maximize the axial limit (prolate case) at the expense of the midplane limit, or conversely in the oblate case. Also, the error contours are typically restricted even more severely in an intermediate range of polar angles.
15. To account for the negative sign, note that an increment [formula omitted] decreases the separation of filament and field point in Eq. (4), and increases it here.
16. 6 having first validated both formulas by computing [formula omitted] for square filaments in several transverse planes, both from Table IV(a) and from the relation [formula omitted].
17. One of us has fully tested these graphical procedures with circular systems when, because of slow convergence of the harmonic series, it was necessary to compute field nets by elliptic integral formulas (see also Ref. 30).
18. The complex error topologies of Ref. 8, Fig. 2 and of Ref. 11, Fig. 8, are absent. The patterns of the latter figure, which are described as “strange and beautiful,” are a result of mixing the error modes 4 and 4,4 with the intended dominant errors (6 and 6,4) in this field, which is of the prolate hybrid type (see Ref. 25). Computation from the stated geometry confirms the axial error limits, but shows that the fourth order is dominant over most of the mapped area.
19. Firester shows only [formula omitted] and does not mention the [formula omitted] plane. His Fig. 1 corresponds to [formula omitted] since [formula omitted] vanishes in the midplane.
20. Within the range of solenoid endplanes in Table [formula omitted] the ratio [formula omitted] is everywhere positive except between the zeros of [formula omitted] at [formula omitted] and of [formula omitted] at [formula omitted]
21. Ref. 3
22. this way, Garrett has computed many circular systems, for which as many as nine parameters were simultaneously adjusted (Ref. 3, pp. 2573, 2579). Suggestions are given for rapid convergence, with computing times of less than one second per case on IBM 7090, even for solutions of the 20th order. The method of Ref. 11 is ingenious but inefficient.
23. From three equations in [formula omitted] A. Sauter and F. Sauter derived a single equation like Eq. (31), for the eighth order system with four circular loops. With it they explored graphically the range of possible solutions.
24. The solenoid provides more than half of the total field. The field [formula omitted] of a general cylindrical solenoid, of which a circular or prismatic solenoid is a special case, is exactly proportional at all interior points to [formula omitted] where Ω is the solid angle subtended by the open ends.
25. This is the case whose solution was attempted in Refs. 6 and 11. But Lee-Whiting’s Eq. (39), which is analogous to Eq. (30) above, inverts the ratio of the Cartesian coefficients [formula omitted] and [formula omitted]