The GMD Method for Inductance Calculation Applied to Conductors with Skin Effect

The GMD method (geometric mean distance) to calculate inductance offers undoubted advantages over other methods. But so far it seemed to be limited to the case where the current is uniformly distributed over the cross section of the conductor, i.e. to DC (direct current). In this paper, the definition of the GMD is extended to include cases of nonuniform distribution observed at higher frequencies as the result of skin effect. An exact relation between the GMD and the internal inductance per unit length for infinitely long conductors of circularly symmetric cross section is derived. It enables much simpler derivations of Maxwell’s analytical expressions for the GMD of circular and annular disks than were known before. Its salient application, however, is the derivation of exact expressions for the GMD of infinitely long round wires and tubular conductors with skin effect. These expressions are then used to verify the consistency of the extended definition of the GMD. Further, approximate formulae for the GMD of round wires with skin effect based on elementary functions are discussed. Total inductances calculated with the help of the derived formulae for the GMD with and without skin effect are compared to measurement results from the literature. For conductors of square cross section, an analytical approximation for the GMD with skin effect based on elementary functions is presented. It is shown that it allows to calculate the total inductance of such conductors for frequencies from DC up to 25 GHz to a precision of better than 1 %.