# Numerical Methods forInductance Calculation

### Geometric Mean Distance — Its Derivation and Application in Inductance Calculations

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###### Synopsis

This article begins with an informal introduction to the use of Geometric Mean Distance (GMD) in inductance calculations, and why it is important. This is followed by brief discussion of the logarithm function. Because it pervades the subject of GMD calculation, it is important to have a good working knowledge of various logarithmic identities and integrals of formulae involving logarithms. The issue of integrating across logarithmic anomalies (i.e., log(0) ) is also treated.

The self GMD is derived in complete detail for some very simple shapes, in order to demonstrate the general analytical method of calculation. This is followed by a discussion of the use of numerical methods—The Monte Carlo Method, in particular—for the GMD calculation of more difficult shapes. Detailed examples are given, which result in empirical formulae for the self GMD of a triangular area, the mutual GMD of elliptical areas and the self GMD of elliptical loci.

The use of GMD in inductance calculations is often accompanied by the proviso, that the "diameter of the conductor should be considerably smaller than the diameter of the winding." However, there is almost nothing available in the literature giving quantitive information about the limits of accuracy of the GMD method. This is treated for the cases of self inductance of a circular loop, and the mutual inductance of two parallel circular loops. Empirical functions are developed, which give the estimated error due to the approximation of the GMD method. These error formulae may be used to calculate correction factors. An example of this is given for a multi-turn solenoid coil.

The final part of the article deals with the calculation of the self GMD of groups of objects. In particular, closed form formulae are developed for the self GMD of linear arrays of circular conductors and linear arrays of thin strips of conductors. These are further developed into a new closed form formula for Rosa's round wire inductance corrections, and formulae for the inductance of short coils which account for conductor shape.

The complete article follows:

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Numerical Methods 3b
Numerical Methods Introduction