Patent Number: 046631097
Section: description

DETAILED DESCRIPTION OF THE INVENTION The present invention generates stellarator fields having favorable properties (magnetic well and large rotational transform) by a simple coil system consisting only of unlinked planar non-circular coils. At large rotational transform toroidal effects on magnetic well and rotational transform are small and can be ignored. We do so herein, specializing in straight helical systems. FIG. 1 shows a typical coil configuration for a modular stellarator of the present invention. Note that only a segment of the helix is shown and that no vacuum vessel is shown. The vacuum vessel would lie within the toroidal field coils 10 and the vessel axis should be substantially parallel (and usually coaxial) to the axis traced by the centers of the coils, X(s). In an actual stellarator, X(s) is a closed curve, the vacuum vessel a torus, and the coils equally spaced along the helical axis. Coils 10 in FIG. 1 are somewhat "bean" shaped, with the cusp portion of the bean being positioned towards the center of the stellarator. For "D" shaped coils, the flat portion of the coil would be positioned towards the center of the stellarator as shown in FIG. 4. Referring to FIG. 1, the centers of the coils trace out a helix, the location of which is given in x, y, z coordinate system by: EQU X(z)=z z+r.sub.o cos kz x+r.sub.o sin kz y (1) where r.sub.o is the coil displacement from the z axis, 2/k the periodicity length, m the period, and kr.sub.o the pitch. If s defines the distance along the coil axis, X(s) defines the coil axis. Our method of solving for the magnetic field generated by these coils is an analytical expansion about the coil axis. The expansion parameter is: ##EQU2## where a.sub.c is the coil radius and r.sub.h the helical radius of curvature. The smaller this quantity is, the easier it is to get a magnetic well. Note that if this expansion parameter approaches one, coils perpendicular to the axis collide. Requiring the expansion parameter to be small is reasonable since any stellarator that would be built would satisfy the condition: coil diameter small compared to the length of a stellarator period. In a helical coordinate system .rho., .theta., s, a general vector is given by: EQU r=X(s)+n(s).rho. cos .theta.+b(s).rho. sin .theta., (3) where the relationship between .rho. and .theta. is shown in FIG. 1A. In helical symmetry the magnetic field depends only on .rho. and .theta.. Hence, the curve of each coil is given by: EQU .rho.=a.sub.c +.delta..sub.2 cos 2.theta.+.delta..sub.3 cos 3.theta.(4) where a.sub.c, .delta..sub.2, and .delta..sub.3 are constants, and at least one of .delta..sub.2 and .delta..sub.3 is nonzero. In equation 4, .delta..sub.2 determines the ellipticity of the coil and .delta..sub.3 determines the triangularity. As .delta..sub.3 increases from 0, the coils first become increasingly "D" shaped, then somewhat "bean" shaped. Equation 4 reduces to the equation for a circular coil when .delta..sub.2 =.delta.3=0. Note that the coils are perpendicular to the coil axis, which is equivalent to having the s component of the coil current vanish. From the above, rotational transform is given by: ##EQU3## where m is the number of periods. When there is a magnetic well, a.sub.c, .delta..sub.2, and .delta..sub.3 are determined by: ##EQU4## Equation 6 gives the coil triangularity required to produce a magnetic well for a given coil ellypticity, .delta..sub.2, and a given pitch, kr.sub.o, of the coil axis. Preferably, a nonzero ellipticity and triangularity are both required to get a good magnetic well from a planar coil (both .delta..sub.2 and .delta..sub.3 .noteq.0). FIG. 2 is a contour plot of the required value of (r.sub.o /a.sub.c) (.delta..sub.3 /a.sub.c) as given by equation 6. EXAMPLE Taking kr.sub.o .perspectiveto.1, we obtain from FIG. 2 a magnetic well and large transform with a reasonable coil deformation. FIG. 3 shows a coil cross-section and corresponding magnetic flux surfaces for kr.sub.o =1, .delta..sub.2 /a.sub.c =-0.3, and .delta..sub.3 /a.sub.c =-0.1. Note that the cusp portion of this "bean" shaped coil is positioned towards the center of the stellarator (center of curvature). For this configuration, calculating rotatinal transform from equation 5, .chi./m.perspectiveto.0.43. Assuming a stellarator having m=4, and aspect ratio of 4, we obtain a beta of 37% ##EQU5## For a larger stellarator having m=10 and aspect ratio of 10, we obtain a beta of 92%. Smaller helical aspect ratios require an increasingly "bean" shaped coil (i.e. increasing triangularity) to maintain a well, whereas a "D" shaped coil is sufficient at larger helical aspect ratio.