Transcranial magnetic stimulation (TMS) is a popular method for the non-invasive stimulation of neurons in the human brain. It has become a standard instrument in experimental brain research and approved for a range of diagnostic and therapeutic applications in neurology and psychiatry. For major depressive disorder, TMS offers effective treatment with lower side effects than other therapies. Depression treatment is currently either performed with focal figure-of-eight coils or with rather distributed H1-type coils. In conventional protocols, the latter appears to even show superior performance. Still, their use is rather limited due to the complexity of the H coil. The H coil furthermore includes various segments that complicate manufacturing and drive the cost. In this paper, we present an electromagnetic equivalent for the H1 coil, which we call surface-projected, H1sp, through vector projection and Huygens’ and Love’s equivalence principle. The latter in principle allows to generate the electromagnetic field distribution inside a closed equivalence volume, e.g., a sphere, exclusively with currents on any closed surface around the volume. We aim at an anatomy-independent equivalent, i.e., the coil should generate the same induced electric field conditions as the original coil for any anatomy inside this equivalence volume, which matching in even a high number of sample anatomies could not fulfill. As the highest current utilization and efficiencies are achieved when all coil winding elements are as close to the target—i.e., the brain—as possible, we derived an equivalent entirely residing on a spherical shell on top of the head. In contrast to other coil design or optimization approaches, the procedure does not require any ad-hoc steps or heuristics but is an explicit forward Hilbert-space vector projection or base change. The resulting equivalent coil generates by design the same field conditions for any head anatomy inside but omits many complexity-, loss-, and leakage-flux-generating features of the initial H1 coil, such as the vertical coil winding segments or hard-to-manufacture sharp turns. For the same induced electric field magnitude and spatial profile, the equivalent H coil requires < 65% of the magnetic field energy and < 55% of the wire length. The comparably simple winding pattern promises the use of notable thicker wire than the initial H1 coil design with several relatively sharp corners.
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