Practical continuous functions for the internal impedance of solid cylindrical conductors

Methods for calculating the internal impedance of round wires are investigated. 'Exact' calculation using Kelvin Bessel functions runs into difficulties at radio frequencies due to rounding errors in computer floating-point arithmetic. Specialist techniques (such as the use of high-precision BCD arithmetic) could be used to circumvent this problem; but for general modelling, the use of approximations is common practice. The traditional 'thick-conductor approximation' for AC resistance is inaccurate and has an incorrect lower limit. An improved derivation with allowance for surface curvature gives rise to an asymptotic form that converges with the Bessel calculation for only moderately large arguments. This is modified to give a bridging polynomial, that is used in conjunction with an optimised Kelvin function algorithm to give a calculation routine with a maximum error of < 0.01 ppM (assuming double-precision arithmetic) and no upper frequency limit. A bridging polynomial is also used for the internal inductance case and gives the same overall accuracy. Inaccuracy and range restrictions can also be avoided without the need for complicated computer programs. A generalised method for producing continuous doubly asymptotically-correct approximations (ACAs) accurate to within a few percent is demonstrated. Suitable choice of ACA allows further correction using modified Lorentzian (ML) functions, leading to a family of compact formulae. The best of these for AC resistance (Rac-TEDML) is accurate to within ±0.09%; and the best for internal inductance (Li-PACAML) is accurate to within ±0.02%.