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.Practical continuous functions for the
internal impedance of solid cylindrical conductors.
By David W. Knight
 Zint.pdf .DOI: 10.13140/RG.2.1.3865.1284  Open
document spreadsheets referred to in the discussion: Xirac.ods
, Xi_aca.ods ,
Xi_ted-ml.ods ,
Xi_aca_ml.ods ,
eff_skd.ods , Flint.ods
, Li_aca.ods ,
Li_acagt3ml.ods ,
Li_pacaml.ods ,
Zint_calcs.ods .Online references:  Optimisation
of multi-parameter empirical fitting funcs: see Data
analysis.
Practical
considerations in the calculation of Kelvin functions and elliptic
integrals, Robert S. Weaver, Oct. 2009. (electronbunker.ca/eb/) InductanceExamples.ods
.Abstract: 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 behaves incorrectly at low-frequencies. 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, which is used in conjunction with an optimised Kelvin function algorithm to give a calculation routine with a maximum error of <0.01ppM (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%. Given that the common wire-making metals have temperature coefficients of resistivity of around 0.4%/K, and an error of around 0.1% is inherent in assuming unit relative permeability for non-magnetic materials, these functions are comfortably adequate for practical circuit-modelling purposes and offer a considerable saving in machine time. The approximation methods demonstrated constitute a departure from the 'brute-force' fitting approach and have potential for application in other situations. The text provides a detailed description of the techniques used. |
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