The best analytical formulae for the self-inductance of rectangular coils of circular cross section available in the literature were derived from formulae for the partial inductance of straight wires, which, in turn, are based on the well-known formula for the mutual inductance of parallel current filaments, and on the exact value of the geometric mean distance (GMD) for integrating the mutual inductance formula over the cross section of the wire. But in this way, only one term of the mutual inductance formula is integrated, whereas it contains also other terms. In the formulae found in the literature, these other terms are either completely neglected, or their integral is only coarsely approximated. We prove that these other terms can be accurately integrated by using the arithmetic mean distance (AMD) and the arithmetic mean square distance (AMSD) of the wire cross section. We present general formulae for the partial and mutual inductance of straight wires of any cross section and for any frequency based on the use of the GMD, AMD, and AMSD. Since partial inductance of single wires cannot be measured, the errors of the analytical approximations are computed with the help of exact computations of the six-dimensional integral defining induction. These are obtained by means of a coordinate transformation that reduces the six-dimensional integral to a three-dimensional one, which is then solved numerically. We give examples of an application of our analytical formulae to the calculation of the inductance of short-circuited two-wire lines. The new formulae show a substantial improvement in accuracy for short wires.

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Abstract. Let G = P U(1, d) be the group of holomorphic isometries of complex hyperbolic space H d C . The latter is a Kähler manifold with constant negative holomorphic sectional curvature. We call a finitely generated discrete group Γ = g 1 , . . . , gn ⊂ G a marked classical Schottky group of rank n if there is a fundamental polyhedron for G whose sides are equidistant hypersurfaces which are disjoint and not asymptotic, and for which g 1 , . . . , gn are side-pairing transformations. We consider smooth families of such groups Γt = g 1,t , . . . , gn,t with g j,t depending smoothly (C 1 ) on t whose fundamental polyhedra also vary smoothly. The groups Γt are all algebraically isomorphic to the free group in n generators, i.e. there are canonical isomorphisms φt : Γ 0 → Γt. We shall construct a homeomorphism Ψt ofwhich is equivariant with respect to these groups:which is quasiconformal on ∂H d C with respect to the Heisenberg metric, and which is symplectic in the interior. As a corollary, the limit sets of such Schottky groups of equal rank are quasiconformally equivalent to each other.The main tool for the construction is a time-dependent Hamiltonian vector field used to define a diffeomorphism, mapping D 0 onto Dt, where Dt is a fundamental domain of Γt. In two steps, this is extended equivariantly toThe method yields similar results for real hyperbolic space, while the analog for the other rank-one symmetric spaces of noncompact type cannot hold.

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 %.