Arbitrary-shaped single-layer coil self-inductance using shape functions

Coils of various shapes are used in various applications such as sensing and magnetic energy storage. Successful practical application of coils requires knowledge about their self-inductance, which is one of their fundamental properties. In the past, approximation analyses have been performed on coils of various shapes by means of theoretical analysis methods and finite element methods. In this paper, we propose a theory for determining the self-inductances of arbitrary-shaped single-layer coils: this theory is based on the novel concept of adopting a shape function for a coil of a particular shape to determine the coil´s self-inductance. We used the double Fourier transform method to derive equations for determining the self-conductance of coils of various shapes. Then, to determine the validity of the proposed theory, we experimentally measured the self-inductances of coils and compared the measured self-inductance values with those calculated theoretically using the proposed theory of shape functions. The comparison results showed that the calculated and measured values were in good agreement, which confirms the validity of our proposed theory. Therefore, we conclude that by using shape functions, it is possible to perform a unified and highly detailed theoretical analysis of magnetic fields in arbitrary-shaped coils and determine their self inductances.