The expressions for vector potential of a single helical conductor have been derived from the definitive integral form, treating a single helical current as a current distribution on the surface of a infinitely thin cylinder due to the delta function.

The self-inductance for an infinitely long helical conductor and the mutual inductance between two coaxial helical conductors are investigated over the whole range from solenoids with an infinitesimal pitch length to straight conductors with an infinite pitch length. The principal terms, apart from the logarithmically divergent term due to the infinite length of the self-and mutual inductances, are rigorously obtained, evaluating the double integrals of Neumann's formula, using analytical expressions for the vector potential of a single helical thin conductor. The relations between the conventional approximate and the rigorously obtained expressions for the self-and mutual inductances of solenoids are also compared. This analytical method is applied for calculations of the self-inductance of a twisted bifilar lead and for the current distribution of a twisted superconducting 6 around 1 strand cable with insulated strands, using the cancellation of the logarithmically divergent term. As a result, it is shown that the analytical method for the inductance calculation for infinitely long helical conductors is useful, by obtaining results consistent with the magnetic energy calculation.

An analytical expression for the mutual inductance for long coaxial helical conductors or solenoids is derived on the basis of Neumann’s formula for the whole range from 0 to of the pitch length, including the cases of the mutual inductance of long concentric closely wound helical solenoids and that of long parallel thin conductors as two extreme cases. In addition, an approximate expression for the self-inductance for a long helical round conductor is obtained.

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