Flux linkage in helical windings and application to pick-up coils

The paper presents an exact theory for the calculation of the flux linkage of multi-turn coils. The helical nature of the winding and the effects due to its complex geometry, including the leads, are taken into account. The parametric description of a surface bounded by the winding is considered, which leads to the determination of the analytical expression of the flux. The theory is applied to pick-up coil magnetic sensors which, within such general setting, can be fully characterized. As particular cases, some widely used pick-up coil types are discussed. A precise definition of the magnetic section and of the magnetic axis are given, supported by their analytical evaluation. From the developed formulas design criteria directly follow. Finally, the practical applicability of the theory is widened by considering a non-ideal geometry of the winding, specifically to analyze the common effect of turns tilting.