Numerical Methods for
Inductance Calculation

Introduction

In 2008 I was building a tuned loop antenna for one of my homebrew receivers, and needed to find an accurate inductance formula for a flat spiral loop of large diameter. I was unable to find anything suitable over the internet, and started digging through some old books. F.W. Grover's book Inductance Calculations, Working Formulas and Tables [2] contains a wealth of information. However, while this book remains a standard reference to this day, it was published in 1946 when electronic calculators and computers were generally unavailable. As a result, most of the complex formulae were put into table form in order to make hand calculation easier. I wanted to use a spreadsheet to do the calculations, and so I began to research the literature to find the original formulae from which these tables were derived. Some of them were given explicitly in Grover's book, while some others required a lot more digging. Many of the underlying formulae were series expansions of elliptic integral formulae or other similarly intractable formulae.

The purpose of this webpage is to discuss some of the methods available to calculate inductance using the power of the computer to its best advantage. The emphasis here will be on numerical methods rather than the underlying physics of inductance. However, there are ample references given to the original sources. Examples are given in Javascript which may be run from any modern web browser, and BASIC which may be used directly as macros in Open Office Calc, and can be converted with little effort to Visual Basic to work with MS Excel. These routines may then be used either directly to calculate inductance, or to provide a very accurate set of data with which to develop simpler fitting functions (some examples of which will be given).

Units

The units used in most of the old reference materials are not SI (Système International d'Unités), as this system didn't exist at the time most of the papers were published. In addition, if you track the units used in the calculations, you will note that the resulting inductance is in units of length. As time progressed, the constant µ_o, the permeability of free space, began to be included in the formulae (typically µ_o/4π). Since µ_o is equal to 4π×10^–7 Henries/meter, this factor converts the length units into inductance units, Henries, so that the dimensions are now consistent.

Many of the early papers used centimeters and microhenries in their formulae (there was an implied conversion from length to inductance), and conversion factors were all rolled into one numeric constant, often something like 0.002. Initially, I opted to stay with the original system of units so that the referenced formulae were not altered and thus easier to trace back to their source, and also because I was concerned about introducing inadvertent errors into the formulae during conversion. However, that decision made the work more difficult to follow, and so the pages are now in the process of revision to include SI equivalents of the original formulae. All newly added pages will be in SI units, and the existing pages will be revised and updated as time permits. Please refer to the Updates page to check the current status.

Also, please note that wherever the constant µ_o appears, it may be replaced with µ, which is the absolute permeability of the medium surrounding the inductor (i.e., the core material), or with µ_oµ_r, where µ_r is the relative permeability of the medium surrounding the inductor.

Acknowledgements

I would like to express my thanks to David Knight G3YNH, whose website and personal correspondence has been invaluable to this work, and to Rodger Rosenbaum for providing me access to a treasury of historical scientific papers.

Part 1 - Elliptic Integral Formulae

1a - Lorenz & Nagaoka, Current Sheets
1b - Maxwell and Coaxial Circular Conductors
1c - Multi-Layer Coils
1d - Special Coils - The Mystery Crystal Set Coil

Part 2 - Inductance Calculation Refinements

2a - Rosa's Round Wire Corrections
2b - Circular Filaments and Helical Corrections
2c - The Helical Formula, and Coils of large Pitch Method

Part 3 - Empirical Methods

3a - Solvers and Optimizers
3b - Wheeler's Continuous Inductance Formula

Part 4 - Geometric Mean Distance

4a - Derivation and Application