Idealized Coil Cross Sections with Minimized Conductor Area for High Field Dipoles

In the design of superconducting accelerator magnets, the shape of the coil cross section is mainly driven by the minimization of the conductor volume, constrained by requirements on the central magnetic field and its homogeneity. Such optimizations commonly assume either a Block or Cosine Theta coil type, which is then filled with (predetermined) rectangular or key-stoned Rutherford cables. By optimizing the positions, angles, and the number of turns, the field quality requirements and cost minimization are achieved. However, this leaves to wonder what the optimal coil geometry looks like, when such practical constraints are not present. Although such a coil cross section has always been presented as the intersection between two ellipses, this method results in a noncircular aperture and is thus not fully representative of a realistic coil. This paper introduces a method in which organically shaped (nongraded) dipole coil layouts are optimized without any assumptions on the conductor. The resulting layouts are presented as a function of overall current density, aperture size, and required magnetic field (inside the aperture). The layouts presented should be viewed as an ultimate limit of what can be achieved, for comparison with real coil layouts, and as an initial guide for finding an optimal cross section for a realistic magnet.