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The Self-Resonance and Self-Capacitance of Solenoid Coils: Applicable Theory, Models and Calculation Methods

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The Self-Resonance and Self-Capacitance of Solenoid Coils: Applicable Theory, Models and Calculation Methods 1 The self-resonance and self-capacitance of solenoid coils: applicable theory, models and calculation methods. By David W Knight1 Version2 1.00, 4th May 2016. DOI: 10.13140/RG.2.1.1472.0887 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. Keywords: circuit modelling, solenoid coil, self-capacitance, self-resonance, helical waveguide, sheet helix, sheath helix. Acknowledgements I would like to thank Prof. Tuck Choy (M0TCC) for detailed reading and discussion of this work, and for providing reference materials and corrections. I would also like to thank Bob Weaver for discussions, corrections, inductance calculation programs, and reference materials; Alex Pettit (KK4VB) for the long coil measurements discussed in the text; Dr Duncan Cadd (G0UTY) for the donation of a large PTFE-cored test coil; and Dr Richard Craven for discussions and for introducing me to the literature on folded helical resonators. 1 Ottery St Mary, Devon, UK. website: http://g3ynh.info/ http://orcid.org/0000-0003-0499-3938 2 Please check the author's website for updates and supplementary material. http://g3ynh.info/zdocs/magnetics/appendix/self-res.html © D. W. Knight, 2009 - 2016. 2 Contents Abstract ..............................................................................................................................................1 Acknowledgements ...........................................................................................................................1 1.0 The inductor modelling problem ................................................................................................3 1.1 Scattering experiment .................................................................................................................8 1.2 Conductor length vs. SRF .........................................................................................................12 1.3 Single-conductor transmission-line ..........................................................................................14 1.4 Normal-mode antennas and Tesla coils ....................................................................................18 1.5 Helical resonators .....................................................................................................................20 1.6 Helical waveguide models overview .........................................................................................21 1.7 Velocity-factor for a finite free helix ........................................................................................23 Simple fringe-field correction ....................................................................................................27 Decline of velocity factor with frequency .................................................................................29 1.8 Velocity factor for an infinite free helix ....................................................................................30 Kandoian & Sichak ....................................................................................................................34 2.0 Self-capacitance .......................................................................................................................39 2.1 Propagation of uncertainty from SRF to self-capacitance ........................................................39 2.2 Self-capacitance derived from the conductor-length ................................................................41 3. Medhurst's formula ...................................................................................................................43 Repeat of Medhurst's data analysis ...........................................................................................44 4. Coil-former dielectric ...............................................................................................................45 5. Empirically-corrected formula for self-capacitance .................................................................50 6. Inter-turn capacitance ...............................................................................................................53 7. Comparison of self-capacitance formulae ................................................................................55 8. Tubular coil formers .................................................................................................................60 9. Axial induced electric-field capacitance ..................................................................................62 Correction for internal dielectric ...............................................................................................65 10. Velocity limiting .......................................................................................................................70 11. Free coil SRF calculation ..........................................................................................................77 Helical-line surge resistance ......................................................................................................79 Uniform current surge resistance ...............................................................................................80 Nominal VF - short coil limiting case .......................................................................................81 Nominal VF - Iterative solution ................................................................................................83 Pitch-angle correction ................................................................................................................88 Long-coil correction ..................................................................................................................89 Coil former dielectric .................................................................................................................91 Model properties, limitations and overall calculation procedure ..............................................95 12. Pseudo self-capacitance ............................................................................................................98 13. Discussion and summary .........................................................................................................101 Glossary .........................................................................................................................................103 Revision history ..............................................................................................................................105 3 1.0 The inductor modelling problem Of the circuit theory elements: resistance, capacitance and inductance; the latter is the least amenable to realisation in practical devices. The reason is that the lumped-component theory depends on the assumption that every physical dimension is negligible in comparison to the operating wavelength. In a wound inductor, the difficulty lies with the length of the piece of wire inside it. Although that wire might be coiled-up in a small volume, its full length nevertheless dictates the inductor's high-frequency behaviour. When a solenoid coil is operated in the regime in which wire length is negligible in comparison to wavelength, its partial inductance3 can be calculated with good accuracy from magnetic considerations alone. A straightforward basis for so doing is the hypothetical current-sheet inductor that, with corrections for realistic wire, allows the inductance of coils of small pitch-angle to be determined from physical parameters to an accuracy of around one part-per-thousand4. Magnetic theory must, of course, be respected in the asymptotic behaviour of any hypothesis that purports to describe inductors at high frequencies, and this has been a weakness in some of the more influential studies. At audio and low-radio frequencies, the situation is complicated by the onset of the skin effect, and its companion, the proximity effect. The internal impedance of isolated wires is
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