Analytical Calculation of Mutual Inductance Between Two Coaxial Thick Coils With Rectangular Cross Section Using Modified Bessel and Hypergeometric Functions

Building on our previous work, we propose a novel series expansion for mutual inductance of two coaxial thick cylindrical coils with rectangular cross-section, where the outer radius of one coil is smaller than or equal to the inner radius of the other. The obtained expansion contains hypergeometric functions and converges absolutely for all aforementioned configurations. Unlike previous semi-analytical methods that are mostly based on reducing the six-fold integral to simpler expressions, followed by numerical integration, our approach is the first fully analytical calculation method for radially separated coils. We tackle the numerical challenges associated with evaluating the hypergeometric functions and implement them in highly performant C code. We find that our approach is in excellent agreement with state-of-the-art semi-analytical approaches, matching their accuracy (14 to 15 significant digits) and surpassing previously achieved performance. Accuracy is reduced to 8 significant digits only when the outer radius of the primary coil is the same as the inner radius of the secondary coil. Noting the algebraic convergence in this regime, we estimate the remainder of the sum, improving accuracy by 1 to 2 orders of magnitude.