[Inductance calculation method for transcranial magnetic stimulation figure-8 coils Accounting for spatial mutual inductance].

Transcranial magnetic stimulation (TMS) is widely used in the treatment of neuropsychiatric disorders, and the stimulation coil constitutes a critical component of TMS devices. Existing inductance calculations for figure-8 coils suffer from three notable shortcomings: they are mostly limited to single-layer geometries, they still rely on elliptic integrals, and they systematically overlook spatial mutual coupling among windings. To address these issues, we propose a high-precision analytical method that accommodates both flat spiral and multi-layer stacked coil configurations. Instead of adopting a conventional lumped-equivalent model, the proposed approach decoupled the total inductance into four independent contributions: self-inductance, coaxial layer-to-layer mutual inductance, coplanar mutual inductance, and non-coplanar mutual inductance. The self-inductance term was computed through a modified magnetic circuit procedure. The coaxial mutual inductance was obtained from parametric curves that incorporated the wire cross-section, built upon the geometric mean distance theory. The coplanar and non-coplanar mutual inductances were derived by combining the Maxwell-Wien series expansion with geometric perturbation theory. The overall coil inductance was then determined by analyzing the circuit connection. Finite element simulations and experimental measurements verified the accuracy and robustness of the method at both the individual-component level and the system level, laying a theoretical foundation for the optimal design of high-performance TMS coils.