Evaluation of inductance for various distributions of windings on straight ferromagnetic cores: an unusual approach

The detailed electromagnetic duality that relates small magnetic structures and small electric structures at ultralow and extremely low frequencies has recently been treated. The strict duality relationship holds provided that the structures possess similar geometry. For example, a short electric dipole antenna gapped in the middle is strongly related to the magnetic structure consisting of a short coil mounted in the middle of a rod of highly permeable material. In particular, the inductance of the latter magnetic structure can be derived directly from the capacitance between the electric dipole arms. The relationship has been thought to hold only for situations where the coil in the center of the core is very short. Here, however, we show that inductance can be evaluated even in cases where the windings are distributed beyond the center and cover relatively long portions of the ferromagnetic core. The extension of the theory described here relies on the fact that the distribution of the flux along the core is triangular-a fact derived from the similarity of the magnetic structure to a short electric dipole antenna, in which the current has a triangular profile. The triangular distribution provides a basis for evaluating the inductance of extended coils by relying on integration of the flux profile through the windings along the core. The results of the evaluation are in good agreement with laboratory measurements. The evaluation method, despite its simplicity, is based on analytical approach, unlike the many empirical methods of inductance evaluation described in the literature.