AC resistance of a round wire

Direct current distributes itself uniformly over the cross-section: the current density is the same at the centre and at the surface. Alternating current does not behave that way. A changing current produces a changing magnetic field inside the metal, and by Faraday's law that field induces eddy currents in the same metal.

The direction of those eddy currents varies with radius. Near the axis the induced current opposes the main current and subtracts from it; near the surface it coincides and adds. The current density therefore falls from the surface towards the centre, and the core of the conductor carries little current although the copper there is identical.

Formally this is described by a diffusion equation for the current density, obtained by substituting Ohm's law into Maxwell's equations and neglecting displacement current. For a plane surface the solution is exponential:

J(x) = J₀ · exp(−x / δ)

where J₀ is the current density at the surface in A/mm², x is the distance inward from the surface in mm and δ is the skin depth in mm. The quantity δ is defined as the depth at which the current density falls by a factor of e ≈ 2.718, that is to 36.8 % of its surface value.

For a cylinder the solution is more involved: surface curvature matters, and instead of an exponential the answer involves Bessel functions of a complex argument. Their real and imaginary parts are the Kelvin functions ber and bei, and that pair underlies the calculation above.

The consequence is straightforward. Since part of the cross-section is excluded from conduction, the effective area is smaller than the geometric one, and resistance is inversely proportional to area. The ratio of total to effective area is the AC resistance factor, written Ξ after Medhurst:

Ξ = R_AC / R_DC = A / A_eff

A useful identity follows: the conducting fraction of the copper is simply 1/Ξ. If the calculator shows R_AC/R_DC = 4, then a quarter of the section is working and three quarters of the copper has been bought, wound and heated to no purpose.

The exact answer is a combination of Kelvin functions. Everything else is a way of doing without them where summing series is inconvenient or impossible. Below is what each method is and where its validity ends.

Kelvin functions: the exact solution

The factor is expressed through ber, bei and their derivatives. The expression is not an approximation: it is a direct consequence of Maxwell's equations in cylindrical geometry. The only difficulty is computational. The series converge well, but at large argument the numerator and denominator are assembled from large numbers of similar magnitude, and double precision ceases to suffice: from about q = 65 the result carries noise at the level of a few parts per million.

Knight resolves this by switching to a bridging polynomial whose coefficients are fitted so that both routes agree at the crossover to within about 0.002 ppM. The scheme has no upper frequency limit and an overall error below 0.01 ppM. It is the default here.

TED-ML: a compact substitute

Knight's 2016 closed form works in two stages. First the skin depth is modified so that at low frequency it tends to the conductor radius rather than to infinity, since otherwise the approximation breaks down near d = δ. The residual error, which has the shape of a nearly symmetric bell, is then removed by a modified Lorentzian function.

Accuracy is ±0.09 % across the whole range with no series involved: three exponentiations and one exponential. Useful where the calculation must be embedded in an environment without Bessel functions, such as a spreadsheet macro or firmware.

The Mingli-Yu asymptotic form

Work by Chinese researchers published in 2004: two polynomials in inverse powers of the argument with twelve coefficients. It yields ±0.1 ppM, but only for q ≥ 11.5, that is at sufficiently high frequency. Below that the expression is not applicable and turns negative.

The Terman formula

The handbook classic: R_AC/R_DC ≈ d/(4δ) + 1/4. The first term is the area of an annulus taken as perimeter times skin depth. The constant is usually presented as an empirical correction, although it can be derived: expanding the exact annulus expression with the curvature term gives exactly 1/4 as the leading correction. The limit is confirmed numerically, the difference being 0.250007 at q = 20000.

The formula holds for d/δ > 4. Below that it is badly in error and, more importantly, does not tend to unity at low frequency: at direct current it returns 0.25 instead of 1. The table above shows where it starts to mislead for your parameters.

Resistance is only the real part of a conductor's internal impedance. The full quantity is complex:

Z_i = R_AC + j·2πf·L_i

The imaginary part accounts for the energy of the magnetic field inside the metal itself. Some field lines close within the conductor, and the energy stored there is seen by the circuit as additional inductance.

At direct current the internal inductance equals μ/(8π), which is exactly 50 nH per metre. The remarkable feature is its independence of diameter: a fine strand and a heavy busbar contribute the same. For a 10 µH coil wound with one metre of wire that is about 0.5 % of the total inductance - negligible in rough estimates, already troublesome in precise measurement.

As frequency rises the current is displaced towards the surface, less internal field remains, and L_i falls in proportion to 1/√f. The resulting picture is characteristic: the resistive part grows, the reactive part decays, and the phase angle of the internal impedance creeps from zero towards 45°. This is classic dispersive behaviour, caused not by a resonance but by the interaction between field penetration depth and conductor thickness.

L_i is evaluated by the Flint algorithm: the same Kelvin functions in a different combination, the same error below 0.01 ppM and the same branch at q = 65.

The table is computed from the exact solution. It has a feature worth noticing: the values repeat along the diagonals. Ø0.5 mm at 1 MHz and Ø5 mm at 10 kHz both give 2.166. The reason is that the factor depends on the single combination d·√f, so the table is essentially one-dimensional and the universal curve above exhausts every possible case.

The calculation describes an isolated conductor: the return path and neighbouring turns are assumed far enough away that their magnetic field does not distort the current distribution. The condition holds for a length of wire in free space and for a coil with a large pitch, where the pitch-to-diameter ratio exceeds ten.

Inside a tight winding it does not. The field of neighbouring turns redistributes the current within the conductor, adding proximity-effect losses that in a multilayer winding frequently exceed the contribution of the conductor's own skin effect. That calculation follows Dowell's model and lies outside this tool: for windings there is the litz winding calculator.

Also excluded: non-sinusoidal current waveforms, which must be resolved into harmonics with the dissipation summed over each; magnetic conductor materials; tubular and bimetallic conductors; surface coatings. Rectangular cross-sections follow different expressions.

1. Knight D. W. Practical continuous functions for the internal impedance of solid cylindrical conductors. Version 2.08.1, 27 April 2016. DOI: 10.13140/RG.2.1.3865.1284. - Primary source: the Xirac and Flint algorithms, the TED-ML and Li-PACAML closed forms, and the analysis of thick-conductor approximations.
2. Mingli W., Yu F. Numerical calculations of internal impedance of solid and tubular cylindrical conductors under large parameters. IEE Proceedings - Generation, Transmission and Distribution, 2004, vol. 151, no. 1, pp. 67-72. - Asymptotic polynomials for large arguments (Northern Jiaotong University, Beijing).
3. Weaver R. S. Practical considerations in the calculation of Kelvin Bessel functions and complete elliptic integrals K and E. 2009. - Recurrence relations for the ber and bei series and their derivatives.
4. Abramowitz M., Stegun I. A. Handbook of Mathematical Functions. National Bureau of Standards, 1964, sections 9.9-9.10. - Definitions of the Kelvin functions as power series.
5. Terman F. E. Radio Engineers' Handbook. McGraw-Hill, 1943. - The classic d/(4δ) + 1/4 approximation.
6. Wheeler H. A. Formulas for Skin Effect. Proceedings of the IRE, 1942, vol. 30, no. 9, pp. 412-424. - Series expansion of the thick-conductor approximation.
7. Medhurst R. G. H. F. Resistance and Self-Capacitance of Single-Layer Solenoids. Wireless Engineer, February 1947, pp. 35-43. - Origin of the notation Ξ for the AC resistance factor.
8. Ramo S., Whinnery J. R., Van Duzer T. Fields and Waves in Communication Electronics. Wiley. - Standard text; derivation of the skin effect from Maxwell's equations.
9. Payne A. Skin Effect, Proximity Effect and the Resistance of Circular and Rectangular Conductors. 2016. - Review of the formulas and their limits of validity.
10. IEC 60228:2023 Conductors of insulated cables. - Standard resistivity values and the temperature model.
11. IEC 60287-1-1:2023 Electric cables. Calculation of the current rating. Part 1-1. Edition 3.0, May 2023. - Current standard for cable AC resistance, with skin and proximity effect factors.
12. Dowell P. L. Effects of eddy currents in transformer windings. Proceedings of the IEE, 1966, vol. 113, no. 8, pp. 1387-1394. - Winding loss model, outside the scope of this tool.
13. Sullivan C. R. Optimal Choice for Number of Strands in a Litz-Wire Transformer Winding. IEEE Transactions on Power Electronics, 1999, vol. 14, no. 2, pp. 283-291. - Choice of strand diameter and count for litz wire.
14. Sullivan C. R., Zhang R. Y. Simplified Design Method for Litz Wire. IEEE APEC, 2014. - Simplified design procedure for litz constructions.
15. Peng C., Chen G., Wang B., Song J. Semi-Analytical AC Resistance Prediction Model for Litz Wire Winding in High-Frequency Transformer. IEEE Transactions on Power Electronics, 2023, vol. 38, no. 10, pp. 12730-12742. DOI: 10.1109/TPEL.2023.3293483. - Recent semi-analytical model for litz windings (Shanghai Electric, Shanghai).
16. Chen T., Zhao Z., Shen Z., Jia H., Ji J. Litz-Wire Winding Loss Calculation Method for Optimal Design of High-Frequency Transformers. IEEE Journal of Emerging and Selected Topics in Power Electronics, 2024, vol. 12, no. 2, pp. 2027-2040. DOI: 10.1109/JESTPE.2024.3361924. - Loss calculation aimed at design optimisation (Hebei University of Technology, Tianjin; Southeast University, Nanjing).
17. Li X., Li Y., Wan Z. Semianalytical Calculation Model for Winding Loss of Litz Wire in High-Frequency Transformer. IEEE Transactions on Power Electronics, 2024, vol. 39, no. 12, pp. 15481-15492. DOI: 10.1109/TPEL.2024.3431911. - Refined litz loss model (Hebei University of Technology, Tianjin).