The self-resonance and self-capacitance of solenoid coils
By David W Knight

Self-resonance and self-capacitance of solenoids . DOI: 10.13140/RG.2.1.1472.0887 + Open Document spreadsheets and coil data F61-32T.ods , Medhurst.ods , CL_theor_test.ods , 18T_scat_Howe.ods , CL_axial-field.ods . Helical_vf.ods . Maxi Spring Air Core Inductors. Coilcraft Document 185-1, 2004 (accessed 5th Jan. 2016)

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Abstract:
The data on which Medhurst's semi-empirical self-capacitance formula is based are re-analysed in a way that takes the permittivity of the coil-former into account. The updated formula is compared
with theories attributing self-capacitance to the capacitance between adjacent turns, and also with transmission-line theories. The inter-turn capacitance approach is found to have no predictive power. Transmission-line behaviour is corroborated by measurements using an induction loop and a receiving antenna, and by visualising the electric field using a gas discharge tube. In-circuit solenoid self-capacitance determinations show long-coil asymptotic behaviour corresponding to a wave propagating along the helical conductor with a phase-velocity governed by the local refractive index (i.e., v = c if the medium is air). This is consistent with measurements of transformer phase error vs. frequency, which indicate a constant time delay. These observations are at odds with the fact that a long solenoid in free space will exhibit helical propagation with a frequency-dependent phase velocity > c. The implication is that unmodified helical-waveguide theories are not appropriate for the prediction of self-capacitance, but they remain applicable in principle to open-circuit systems, such as Tesla coils, helical resonators and loaded vertical antennas, despite poor agreement with actual measurements. A semi-empirical method is given for predicting the first self-resonance frequencies of free coils by treating the coil as a helical transmission-line terminated by its own axial-field and fringe-field capacitances.

This article is also available from ResearchGate.net
https://www.researchgate.net/publication/301824613

Please also see Additional comments and further work.

Some suggestions for modelling the dielectric effects of tubular coil formers are given by Prof. Tuck Choy:
On the effects of dielectric or permeable formers on the inductance and self-capacitance of solenoids. Tuck Choy. 2015.

The scientific data analysis methods used in this study are explained in a separate article:
Scientific Data Analysis. D W Knight. 2009.