TX to Ae
Losses and Q

Inductor self-resonance and self-capacitance

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)
velocity factor comparison
click to enlarge in a new window

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

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.

Experiment & measurement
Coil resonance experiments

Inductor resonance and self-resonance experiments .
Photographs and practical details of some coil resonance experiments.

Solenoid self-resonance measurements by Alex Pettit, KK4VB

Self-capacitance of toroidal inductors.

Capacitor standardisation using a reference inductor.
Parallel resonance indicated using a neon strobotron tube

Evaluation and optimisation of current transformer bridges.

References & further reading

System of Transmission of Electrical Energy. N Tesla, 1900, US Pat. No. 645576.

On the construction of Tesla Transformers: Period of oscillation and self-inductance of the coil [Zur construction von Teslatransformatoren. Schwingungsdauer und Selbstinduction von Drahtspulen)], P Drude, Annalen der Physik 1902, 314(10 & 11), p293-339, 590-610.
English translation by David Knight and Robert Weaver. arXiv:1605.04196 .

The Calibration of Wave-Meters for Radio-Telegraphy, G W O Howe.
Proc. Phys. Soc. London. Vol 24, issue 1, 1st Dec. 1911, p251-259.

Systematic study on self-capacity of coils for radio use. I Yamamoto. JIEEJ, 45 (1925) No. 440, p259-277.

Die Grundlagen der Hochfrequenztechnik, Franz Ollendorff. Section 54, Pages 79-87,
Das dynamische Feld der mehrwindigen Spule [The dynamic field of multi-turn coils].

Remarks on the self-capacitance of coils [Remarques sur la capacité propre des bobines], P Cassou and J Cayrel, Comptes Rendus Académie des Sciences, Vol 198 (1934), p1305-1308.
English translation by Kate Knight and David Knight.

H F Resistance and Self-C of Single-Layer Solenoids. R G Medhurst, 1947.

Travelling Wave Tubes. J R Pierce. BSTJ* 1950: 29(1) p1-59, 29(2) p189-250, 29(3) p390-460, 29(4) p608-671.

EM Wave Propagation on Helical Conductors, S. Sensiper, MIT Research Lab Electronics Tech Report No. 194, May 1951.

Some wave properties of helical conductors. J H Bryant. Electrical Communication 31(1), 1954. p50-56.

Coaxial Line with Helical Inner Conductor. W Sichak. Proc. IRE. Aug. 1954. p1315-1319. Correction Feb. 1955, p148. Reprinted in Electrical Communication, 32(1), March 1955. p62-67.

Wide frequency-range tuned helical antennas and circuits. A G Kandoian, W Sichak. Electrical Communication 30(4), 1953. p294-299. Correction: 31(1), 1954, p49.
Considerations Regarding the Kandoian & Sichak Formula, Andrés Esteban de la Plaza. (accessed 28th May 2021).

Self-resonant frequencies of air-core single-layer solenoid coils calculated by simple method, Caio Marcelo de Miranda, Sergio Francisco Pichorim, Elec. Eng. 97(1) p57-64, Mar. 2014.
Self-resonant frequencies, standing waves , and impedance behaviour of air-core solenoidal coil. Miranda & Pichorim 2015. DOI 10.1109/ICEAA.2015.729736.
Self-Resonant Frequencies. Miranda and Pichorim 2015.
These papers are available from researchgate.net.

Helical Antennas, Ref Data for Radio Eng., 4th edn. 1956, ITT corp., p682-687.

Helix Waveguide. S P Morgan, J A Young. BSTJ*. 1956: 35(6), p1347-1384.

Folded-over helical resonator. P Vizmuller, US Pat. # 4422058, 1983. [see also Vizmuller's book 'Filters with Helical and Folded Helical Resonators' (1987) ISBN 0-89006-244-7 (not available online)].

Fields and Waves in Communication Electronics, Simon Ramo, John R.Whinnery, Theodore Van Duzer, 3rd edn. Publ. John Wiley & Sons Inc. 1994. ISBN 0-471-58551-3.
9.8: The idealized helix and other slow-wave structures. 9.9: Surface Guiding.

Stray Capacitances of Single-Layer Solenoid Air-Core Inductors, G. Grandi, M K Kazimierczuk, A Massarini, U Reggiani. IEEE Trans. on Industry Apps., Vol 35, No. 5, Sept/Oct 1999, p1162-1168.

Filters and an Oscillator Using a New Solenoid Model, Randy Rhea, Applied Microwave & Wireless, Nov 2000, p30-42.

RF Inductor Modelling for the 21st century. Leslie Green. EDN, Sept. 2001.

RF Coils, Helical Resonators and Voltage Magnification by Coherent Spatial Modes. K L and J F Corum, Microwave Review, Sept 2001 p36-45.
Class Notes: Tesla Coils and the Failure of Lumped-Element Circuit Theory, Kenneth and James Corum.
(now seems to be available again from: ece.k-state.edu/people/faculty/gjohnson/ )
Multiple Resonances in RF Coils and the Failure of Lumped Inductance Models. K L Corum, P V Pesavento, J F Corum. 6th International Tesla Symposium 2006.

Solid State Tesla Coil. Gary L Johnson, 2001.

Practical continuous funcs for internal Z of solid cyl. conductors. D W Knight 2010

A study of secondary winding designs for the two-coil Tesla transformer. R M Craven, PhD 2014.

Paul Drude's Prediction of Nonreciprocal Mutual Inductance for Tesla Transformers, Bart McGuyer, 2014, PLoS ONE. 9(12):e115397. doi: 10.1371/journal.pone.0115397.
Bart McGuyer's Tech Notes: Tesla coils, inductor transmission line models, and more. Note 11 is an extension of the work in the preceding refererence

Electrical oscillations in antennas and inductance coils. J M Miller, BBS 14, 1918, p677-696.

* Unfortunately, Bell System Technical Journal articles are no longer free from Alcatel-Lucent. The articles are now available via IEEE Xplore (subscription required). Links to BSTJ articles can be found atArchive.org, but the files appear to be stubs and cannot be opened by personae non gratae.

TX to Ae
Losses and Q

© D W Knight 2008 - 2021.