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Grover's 'Inductance Calculations' Supplementary information and errata

Version 1.00. 4th June 2012 David Knight* & Rodger Rosenbaum

David Knight* & Rodger Rosenbaum *Ottery St Mary, Devon, England. http://www.g3ynh.info/zdocs/ Inductance Calculations: Working Formulas and Tables F W Grover, 1946 and 1973. Dover Phoenix Edition 2004. ISBN: 0 486 49577 9. 2009 reprint ISBN13: 9780486474403 Dr Frederick W Grover's monograph for engineers and scientists engaged in the accurate calculation of self and mutual inductance. The book is based on the work carried out by E B Rosa and F W Grover during their distinguished careers at the American National Bureau of Standards during the first half of the 20th Century. Much of the information is given in tabular form; as befits the calculation methods used at the time, but source material is fully referenced, and most of the generating functions are presented and explained. Contents: 1. General principles. 2. Methods of calculating inductances. 3. Geometric Mean Distances. 4. Construction of and method of using the collection of working formulas. Part 1: Circuits whose elements are straight filaments. 5. Parallel elements of equal length. 6. Mutual inductance of unequal parallel filaments. 7. Mutual inductance of filaments inclined at an angle to each other. 8. Circuits composed of combinations of straight wires. 9. Mutual inductances of equal parallel polygons of wire. 10. Inductance of single-layer coils on rectangular winding forms. Part 2: Coils and other circuits composed of circular elements. 11. Mutual inductance of coaxial circular filaments. 12. Mutual inductance of coaxial circular coils. 13. Self-inductance of circular coils of rectangular cross section. 14. Mutual inductance of a solenoid and a coaxial circular filament. 15. Mutual inductance of coaxial single-layer coils. 16. Single-layer coils on cylindrical winding forms. 17. Special types of single-layer coil. 18. Mutual inductance of circular elements with parallel axes. 19. Mutual inductance of circular filaments whose axes are inclined to one another. 20. Mutual inductance of solenoids with inclined axes and solenoids and circular coils with inclined axes. 21. Circuit elements of larger cross sections with parallel axes. 22. Auxiliary tables of functions which appear frequently in inductance formulas. 23. Formulas for calculation of the magnetic force between coils. 24. High frequency formulas. Errata and supplementary data Errors and their corrections are shown below in the form of scanned pages marked in pencil. Most of these were discovered through the careful work of Rodger Rosenbaum:

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Pages 19 - 22

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A complete recalculation of tables 1, 2, and 3 has been given by Rodger Rosenbaum 1 . Most of the errors are small except for the one shown above, which should be marked on any working copy.

Page 34 The above contains a simple arithmetic error: 0.002 x 304.8 x 3.6889 = 0.7162

Chapter 15: Pages 122 - 141

A discussion of the accuracy of Grover's tabular calculation method is given by Engel and Rohe 2 . Claims made in that paper relating to inaccuracies in Grover's method may however be flawed due to the assumption that the FEEFC calculations carried out were equivalent to actual measurements. Rodger Rosenbaum investigated this matter in April 2009, and apart from changes in the last digit of some tabulated values, no errors were found in Grover's work.

1 GMDs of equal parallel rectangles. Re-calculation of Grover's table 1, 2, and 3. Rodger Rosenbaum. 2009.

Available from g3ynh.info. 2 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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Pages 143 - 147

Formula 119 on page 143, is Coffin's formula truncated, which is in turn an extended version of the Rayleigh-Niven formula 3 . The second term in the series should be: . . . + (β²/8)[ln(4/β) + 1/4] - . . . Consideration was given to the possibility that Grover might be correcting errors in the 1911 document, but a comparison against Lundin's formula 4 indicated that ¼ was the best choice for the term in question. Rodger Rosenbaum has since compared the formula against Lorenz's exact expression, and confirms this conclusion as shown below:

3 Formulas and Tables for the calculation of mutual and self induction. E B Rosa and F W Grover. 3rd edition

1916 with 1948 corrections BS Sci. 169 (available from g3ynh.info). See pages 116 - 117, formulas 69 and 71. 4 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.

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Note however, Grover's comment above: "For values of β as large as ¼ three terms will suffice for an accuracy better than 1 part in 1000". As is confirmed by Rodger's calculation, even with the error, this statement is correct. Rodger comments: "When you look at the plots, it is clear that ¼ is the right value. The error as β gets small descends right down to the round-off error of the arithmetic for a value of ¼, but not for 1/8. The fact that the error is better than 1 part in 1000, as Grover says, even with 1/8, suggests that he did some calculations with the 1/8 value, and noticed the error was 1 part in 1000, failing to notice that with ¼ the error would be much smaller." Rodger has also calculated the exact values of Nagaoka's coefficient K as they should appear in tables 36 and 37 (p144-147), (see below). Some minor differences occur, and the tables can be amended using the data provided. The exact formula for Nagaoka's coefficient is:

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where K[x] and E[x] are elliptic integrals of the first and second kind respectively; and z=2a/b

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Pages 149 - 150

A recalculation of Rosa's mutual inductance correction factor H (Table 39, p150) to 10 decimal places, and a discussion of the GMD method, is given by Bob Weaver 5 . Grover's table has no errors greater than 1 in the last digit, but the more accurate information may be preferred. Bob's article also provides information on the coding of calculation routines. A one-line continuous formula which gives H with a maximum absolute error of better than ±0.000 000 02 is given by David Knight 6 .

Page 151 A comma has been inserted instead of a decimal point: p = 30.5510

5 Investigation of Rosa's round-wire mutual inductance correction formula. Robert Weaver, 2008. Available from

g3ynh.info. 6 Rosa's mutual inductance correction for the round-wire solenoid. David Knight, 2010. available from

g3ynh.info.

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Page 156

Arithmetic error: -0.0023 x 7051 = -16.2173 7051 - 16.2 = 7034.8

Page 161

On page 161, the last line in formula 134 reads: a=2.451b But this has to be 2a=2.451b The number "2" is missing. For confirmation, see line 1 of formula 134: 2a/b=2.451

(error found by Thomas Heckel, Feb. 2011)

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Page 164

Here the error can be seen by looking back at formula 135 (p163); Grover used the diameter (12 inches) instead of the radius (6 inches). The above should read:

inches) instead of the radius (6 inches). The above sho ΔL = 0.004 π (20)(6 × 2.54)(-0.2453) = -0.9396 µH,

and the inductance of the helix is:

and the inductance of the helix is: L = 70.71 - (-0.9396) = 71.65 µH.

David Knight & Rodger Rosenbaum. April 2009. Updated March 2011, June 2012 (converted to pdf).