G3YNH info: Refs for Inductors and Transformers.

Notes and References for Chapter 3.

[1] "RF Coils, Helical Resonators and Voltage Magnification by Coherent Spatial Modes", K L and J F Corum, Microwave Review, Sept 2001 p36-45.
[1a] "Class Notes: Tesla Coils and the Failure of Lumped-Element Circuit Theory", Kenneth and James Corum.
[1b] "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. http://www.nedyn.com/TeslaIntlSymp2006.pdf .

[2] Radio Designer's Handbook, Ed. Fritz Langford-Smith. 4th edition. 4th impression (with addenda), Iliffe Publ. 1957 [A later reprint exists (1967) ISBN 0 7506 36351]
Chapter 10: Calculation of inductance. Wheeler's formula: section 10.1(iii), p432.
Chapter 11: Design of radio Frequency Inductors. Section 11.2 (ii), p451: Medhurst's formula. Section 11.5: Short-Wave Coils [uses SWG wire sizes, but SWG diameters in mils (1 mil=0.001"=25.4μm) are given in section 38.19, p1409.].
[Erratum: In chapter 36, Design of FM receivers, by E Watkinson: Rosa's formula for the inductance of a wire has been transcribed incorrectly, and the subsequent example on p1287 is incorrect.].

[4] Fields and Waves in Communication Electronics, Simon Ramo, John R.Whinnery, Theodore Van Duzer, 3rd edition. Publ. John Wiley & Sons Inc. 1994. ISBN 0-471-58551-3.
The symbol

means "equal by definition".
3.16 (p149-153): Penetration of Electromagnetic Fields into a Good Conductor (skin effect).
3.17 (p153-155): Internal impedance of a plane conductor.
4.4 (p180-182): Skin effect in practical conductors.
4.5 (p182-186: Impedance of round wires.
9.8 (p476-478): The idealized helix and other slow-wave structures.
9.9 (p479-481): Surface Guiding.

[4a] Fields and Waves in Communication Electronics, Simon Ramo, John R.Whinnery, Theodore Van Duzer, Publ. John Wiley & Sons Inc. 1965. Library of congress cat. card no. 65-19477.
4.12 (p249-254): Penetration of Electromagnetic Fields into a Good Conductor (skin effect). 5.16 (p291-293): Current distribution in a wire of circular cross section.
5.17 (p294): Internal impedance of a round wire.
5.19 (p298-301): Impedance of a coated conductor.
5.20 (p301-303): Impedance of a thin-walled tubular conductor.
5.24 (p309-311): External inductance of a circular loop.
5.25 (p311-313): Inductance of practical coils.
6.04 (p330-334): Imperfect Dielectrics and Conductors (Kramers-Krönig relations).
6.13 (p358-361): Refractive index.
8.17 (p467-470): The idealized helix and other slow-wave structures.

[5] "A Handbook Formula for the Inductance of a Single-Layer Circular Coil" R. Lundin Proc. IEEE, vol. 73, no. 9, pp. 1428-1429, Sep. 1985.
Formula for current sheet inductance accurate to < 3 parts in 10⁶.

A calculation comparing Lundin's formula against Lorenz's exact expression has been performed by Rodger Rosenbaum, see [Lundin_err.gif]. The claimed accuracy of 3ppm is bettered by a comfortable margin and, for almost any purpose, Lundin's formula can be considered to be exact.

[6] "Simple Inductance Formulas for Radio Coils", Harold A Wheeler, Proc. Inst. of Radio Eng., 1928, Vol 16 pt 10, pp1398-1400.

Wheeler's 1928 formula for a single-layer solenoid is given in its original form as:
L = a² N² / (9a + 10b) [microHenries] , b > 0.8a
Where b is the coil length in inches, and a is the radius in inches. Factoring b from the denominator gives:
L = 10^-6 a² N² / [ b (10 + 9a/b)] [Henrys]
To convert this formula to SI units, we will use the symbols r = radius, D = 2r = diameter,

= solenoid length. The quantity a/b is dimensionless, and so we can immediately substitute in the denominator:
L = 10^-6 a² N² / [ b (10 + 9r/

)] = 10^-6 a² N² / [ b (10 + 4.5 D/

)]
Factoring 10 from the denominator gives:
L = 10^-7 N² ( a² / b ) / (1 + 0.45 D/

) [Henrys]
Now recall that Nagaoka's coefficient → 1 when the coil becomes very long and thin, i.e., when D/

→ 0. Hence, according to the asymptotic behaviour, we can extract an approximation for Nagaoka's coefficient as:
kW28 = 1 / (1 + 0.45 D/

) ,

> 0.4D
This is the identification used in the text; but as has been pointed out by Rodger Rosenbaum*, Wheeler's formula is not an asymptotic approximation. If we assume it to be so, a small discrepancy arises between the inch and the metric forms of the complete inductance formula. If we express the error as a proportion, p, we have:
L = μ0 N² π r² kW28 /

= 10^-7 p N² a² kW28 / b
substituting μ0 = 4π × 10^-7 gives:
4π² r² /

= p a² / b
Now, since 1" = 25.4mm**, then:
r = 25.4 × 10^-3 a and

= 25.4 × 10^-3 b
Using these substitutions:
4π² (25.4 × 10^-3 a)² / (25.4 × 10^-3 b) = p a² / b
i.e.:
p = 4π² × 25.4 × 10^-3 = 1.002 751 807
Hence Wheeler's formula comes out at 1/1.002751807 = 0.997255744 of the long coil asymptotic value for the current sheet formula (i.e., 0.274% low). This choice however, makes it better than the formula given in section 3-6b in the region around

/D=0.7, but inferior for longer coils (see Rodger's error plot). These subtleties do not affect the use of either formula in the preferred application; i.e., as a simple approximation for use with scientific hand calculators.

*

** Strictly 1" = 25.400051 mm, for the U.S. inch in 1928. See Kaye and Laby, 12th edition, 1959, p2-3.

[7] "Inductance Formulas for Circular and Square Coils" Harold A Wheeler, Proc. IEEE (Letters), Vol 70, No 12, Dec 1982, p1449-1450.

[8] "Accurate Single-Layer-Solenoid Inductance Calculations", Hank Meyer W6GGV, QST Technical Correspondence, April 1992, p76-77.
"Corrections to Accurate Single-Layer Solenoid Inductance Calculations", Hank Meyer. QST July 1992, p73.
Complains of inaccuracy of Wheeler's formula, but fails to appreciate that Wheeler's formula is a current-sheet approximation. Confuses coil-former diameter with coil diameter. Fits Nagaoka's coefficient (taken from Langford-Smith's graph) to simple functions, but these functions are seriously inaccurate in the region of interest. The long-coil formula is far less accurate than Wheeler's formula, and the short-coil formula is less accurate than truncating the Rayleigh-Niven formula to a single term. Equation 4, for Rosa's mutual inductance correction is only accurate to 20.6% and does not return 0 when N=1. Computer programs developed from this article should be avoided.

Additional errata: April: Eq.5 is not Nagaoka's coefficient, it is formula (119) from [Grover 1946] p143. This formula is a truncated version of Coffin's formula with a transcription error in the second term. The second term should read ...+ (B²/8)[ln(4/B) + 1/4]-... (see erratum of [Grover 1946]. Meyer adds his own transcription error also: the last term of Eq. 5 should be (5B⁶/1024){...}. Eq. 6 is not Nagaoka's coefficient, it is the Webster-Havelock formula [Grover 1946] p 121). The accuracy of Eq. 5 without corrections is 0.66% and Eq.6 is 0.06%, not one part in 10⁵ as stated.
April: Ref 3 should read pp1-33.
July p73: The statement that inductance is independent of frequency is incorrect.

[9]

[10] "Stray Capacitances of Single-Layer Solenoid Air-Core Inductors", G. Grandi, M K Kazimierczuk, A Massarini, U Reggiani. IEEE Transactions on Industry Applications, Vol 35, No. 5, Sept/Oct 1999, p1162-1168.
Available from: Classic Tesla at time of writing.
www.classictesla.com/download/ia99.pdf .

[11] "Lumped Parameter Models for Single- and Multiple-Layer Inductors", A Massarini, M K Kazimierczuk, G Grandi. Proc PESC '96, June 1996, p295-301.
Available from: Classic Tesla at time of writing.
www.classictesla.com/download/pesc96.pdf .

[12] Sort Ferrites: Properties and Applications. E C Snelling. 2nd ed. Butterworth. 1988. ISBN 0-408-02760-6.
Winding self-capacitance. p330-335.

[13] "Measurement of Self-Capacitance for windings on High-Permeability Ferrite Cores". V Yurshevich, S Lomov and J Jankovskis. Measurement Science Review, Vol 1, No. 1, 2001. p219-222.
www.measurement.sk/Papers3/Yur.pdf .

[14] "Numerical calculations of internal impedance of solid and tubular cylindrical conductors under large parameters" W. Mingli and F. Yu (Northern Jiaotong University, School of Electrical Engineering, Beijing, China). IEE Proceedings, Generation, Transmission and Distribution. January 2004, Vol 151, Issue 1, p. 67-72.

[15] "Practical Model and Calculation of AC resistance of Long Solenoids". E. Fraga, C Prados, and D.-X Chen. IEEE Transactions on Magnetics, Vol 34, No. 1. Jan 1998.

[18] "Theory of the Beam-Type Traveling-Wave Tube". J R Pierce. Proc. IRE. Feb. 1947. p111-123. See Appendix B, p121-123, "Propagation of a wave along a helix", which gives Schelkunoff's derivation of propagation parameters for the Ollendorf sheath-helix.

[19] "Coaxial Line with Helical Inner Conductor". W Sichak. Proc. IRE. Aug. 1954. p1315-1319. Correction Feb. 1955, p148.

[20] Radio Frequency Transistors, Norm Dye and Helge Granberg. Motorola inc. / Butterworth Heinemann, Newton MA. 1993. ISBN 0-7506-9059-3
Line-length resonance: p142.

[27] "A Comparison of Single-Layer Coaxial Coil Mutual Inductance Calculations Using Finite-Element and Tabulated Methods", Thomas G Engel and Stacy N Rohe. IEEE Transactions on Magnetics, Vol 42, No. 9, Sept. 2006. p2159-2163.
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Ch.3 Contents