I spotted the very interesting, classical paper,

- Formula for the inductance of a helix made with wire of any section, by Chester Snow, Scientific papers of the Bureau of Standards, No. 537 (Part of Vol. 21), submitted February 25, 1926, published November 10, 1926.

The original paper is available online at the NIST Virtual Library. This publication for the first time takes care also of the real, helical structure of a coil inductance, providing a higher precision than the former methods for inductance calculation, e. g., by approximating the real coils through current sheets with correction values, or by summation of self and mutual inductances of wire rings and circular filaments. Calculation examples are given in the paper for wires of round and rectangular section.

The formulas in the paper are rather complex, and their lengthy derivation intractable (for me). For sure they were very tedious to compute at their time of publishing. And the paper tries by judicious neglect of terms of higher order to keep the complexity manageable. Back then the word "computer" was already known, but it meant staff teams who were trained to calculate formulas mostly by hand. Integrals (e. g., function B1(k) in the paper) were solved by a mechanical graphing device named integraph, an ingenious precision machine long since forgotten.

The beauty of the formulas by Chester Snow is in the possibility to arrive at a very precise result for the inductance of a wire coil, considering its helicity and wire diameters. Today modern PCs can calculate complex formulas like the ones by Snow easily, and there is no real need anymore to use any of the imprecise "handbook formulas" for coil inductance, which have been published in the meantime.

The date 25 February 2016 is the 90th anniversary of the submission date from Snow's paper.
To make his inductance calculation formulas easily usable in today's practice,
and to have them accessible for comparison with new developments,
i coded them in the language Lua.
The program code tries
to follow the formulas exactly (while fixing obvious typos),
and to reference the original formulas wherever possible.
The integraph method for calculation of function `B1(k)`

is replaced by Gauss-Legendre integration,
using constants calculated by this
Lua program.
In addition i have tried to recalculate a few constants
to a higher precision than given in the paper.
But comparison between the original and the new values shows,
that in the paper their precision was already chosen to be
just high enough not to influence the inductance result notably,
and short enough for efficient calculation by hand.

Here now is the fresh Lua program, written as a library:

To see how it works, just "`require()`

" it from a Lua prompt and run the `test()`

program,
or remove the comment dashes in the line "`-- t.test()`

",
and run the code directly by the Lua interpreter from the command line.

I would like to recommend also the webpage Numerical Methods for Inductance Calculation by Robert Weaver, who discusses Snow's helical inductance calculation method from the above paper in detail, and who also provides a fresh calculation method, thriving for utmost precision.

Have fun!

This page first put online 22 February 2016.