Producing circular field harmonics inside elliptic magnet apertures with superconducting canted-cosine-theta coils

Superconducting magnets with noncircular aperture are desired for accelerators and many other high-field applications. This paper presents new methods for the analytic design of elliptic bore superconducting accelerator magnets. Part 1 of this work shares the derivation of current to field relations between a sheet current density on an elliptic cylinder and the magnetic field harmonics inside the aperture. This result is explored in the general context of elliptic bore magnet design with relevant scaling laws compared between elliptic and circular bore magnets. In part 2, this approach is applied to the specific geometry of canted-cosine-theta (CCT) accelerator magnets, enabling analytic winding design for single or mixed circular harmonics within elliptic aperture CCT magnets.

Figure 1

Constant $\eta$ and $\psi$ surfaces for the elliptic coordinates form confocal cylinders and planes.

Figure 2

The first four elliptic field harmonics for an example elliptic boundary.

Figure 3

The current densities in Table 1 are applied to produce the first four circular field harmonics inside an example elliptic aperture.

Figure 4

An example choice between a circular or elliptic bore quadrupole magnet with matched major axis. We use comparisons of this type to quantify the advantages of elliptic bore magnets.

Figure 5

Stored energy as a function of ellipticity normalized to the circular, $\epsilon =0$ case (see Fig. 4).

Figure 6

Total amp-turns as a function of ellipticity normalized to the circular, $\epsilon =0$ case (see Fig. 4).

Figure 7

The local unit tangent, radial, and binormal vectors ($\widehat{\mathit{t}}$, $\widehat{\mathit{\eta }}$, $\widehat{\mathit{b}}$) at point $\mathit{p}$ of a parametric winding path constrained to the surface of an elliptic cylinder.

Figure 8

We assume the winding path is axially periodic with pitch length $w$.

Figure 9

Single layer elliptic CCT windings producing the first four circular harmonics (from Table 3).

Figure 10

An elliptic CCT dipole magnet with alternation of tilt angle and current direction between layers to cancel the solenoidal field in the aperture.