I. Introduction
Saddle-shaped coils are advantageous for creating ideal multipole magnetic fields in the magnet with a cylindrical bobbin when the magnet current is azimuthally distributed as [1], [2], [3], [4], [5]. From a practical implementation point of view, the saddle-shaped structure is not only a proper structure for placing such cosine-theta current distribution but it is also convenient for implementing high-field magnetic multipole especially when iron yoke cannot be used to tune multipole field shape due to a full saturation. Thanks to these advantages, saddle-shaped coils are widely used in applications such as NMR, MRI, and high-field large-scale particle accelerators, that require a homogeneous and strong magnetic field [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17].