This paper presents an improved method using inverse hyperbolic functions to calculate the mutual inductance and self-inductance of coaxial circular coils with rectangular cross section and uniform current density in air. The method is simple, fast, and easy to use, which only entails numerical integration with one variable. The method is derived by using inverse hyperbolic functions but can be extended to using logarithmic functions. An algorithm and a leading part checking method to resolve singularities are presented, and their theoretical foundations are fully analyzed. The results of several testing cases calculated by the proposed method were compared with those in the literature, and they were in excellent agreement.Index Terms-Circular coils, inductance, inverse hyperbolic functions, numerical stability. 1051-8223

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