Wheeler's Formulae

John Crabtree, KC0G, Minneapolis, USA (2019 March 04):

In the March 2019 issue of Radcom, Andy Talbot, G4JNT, briefly discussed formulas for the calculation for inductance.  From a link he provided I found my way to your web site, and your "Solenoid Inductance Calculation" paper.
     On page 44 [of version 0.20] you state: "More intriguingly however, it is in the same form as the expression for Kw25 given earlier (i.e., the approximation for Nagaoka's coefficient extracted from Wheeler 1925). It therefore hints at the underlying deduction that led to Wheeler's most famous formula." I think that it is useful to go look at some of Wheeler's writings, and his oral histories to discover how he developed the 1925 formula.

1.  Derivation of the 1925 formula
The story is mostly set out in Wheeler [1982].  He started with Formula (153) in Dellinger [1918] i.e.,

     L = K 0.03948 a2 n2 / b

where L is in uH, a is the radius in metres, b is the length in metres, and K is a function of 2a/b

Wheeler was motivated to find a formula to replace the table of Nagaoka's coefficient.  Wheeler  "... analyzed 1/L for its near-linear slope and intercept in terms of shape (b/a).  I found that the asymptotic straight line gives a remarkably close approximation....".    Wheeler noted : "The straight line yielded a ratio near 9/10 for the coefficients in the denominator."   I can only assume that this was the table of Nagaoka's coefficient, and that Wheeler had in fact created an asymptotic approximation for Nagaoka's coefficient.

Things get interesting when one puts Wheeler's approximation into Dellinger's Formula (153)  Converting to inches one gets:

1.0027 a2 n2 / (9a + 10b)

So why did the 1.0027 multiplier disappear?  I can think of a couple of reasons  The first is that Wheeler simply overlooked it, i.e. made a mistake.  The second, which I think is far more likely to be the case, is that he deliberately omitted it.  Wheeler was interested in developing simple formulas for use by engineers.  (See Wheeler's oral histories (1985) and (1991)).  Omitting the multiplier 1.0027 makes the formula easier to remember and use, and does not impact Wheeler's stated accuracy.
     Another way to find out what exactly Wheeler did would be to try and track down what happened to his notebooks.  Back in 1982 Wheeler noted that they were kept in storage at the Hazeltine Technical Information Center.  Hazeltine has  been bought and sold since, and is now known as BAE Systems Sensor Systems 

2.  Other matters
My assumption is that, for whatever reason, Ramo et al went back to the original (Derringer or earlier) formula, and somehow realized that the multiplier of 1.0027 was omitted from Wheeler's formula.
     In the reference [Wheeler 1929] below, you will find discussion of Wheeler's 1928 formula.  There [one R R Batcher] gave a drivation of Wheeler's 1925 formula, starting from a slightly different place.

3.  References
H.A. Wheeler, "Simple Inductance Formulas for Radio Coils". Proc. IRE, Vol. 16, No. 10, October 1928, pp 1398-1400

[Wheeler 1929]
"Discussion on Simple Inductance Formulas for Radio Coils". Proc. IRE, Vol. 17, No. 3, March 1929, pp 580-582
Comments by R R Batcher and Wheelers's response.

[Wheeler 1982]
H.A. Wheeler, "The Early Days of Wheeler and Hazeltine Corporation - Profiles in Radio and Electronics", Hazeltine Corporation, 1982, pp 392-394, Section 10.1, Inductance Formulas 1928.

[Dellinger 1918]
J.A. Derringer et al, "Radio Instruments and Measurements". Bureau of Standards, C74 (Circular No. 74), March 1918. You can find this at: https://archive.org/details/radioinstruments00unitrich/page/n8
The relevant section starts on page 252, and refers to a table which was calculated by Nagaoka [change .../n8 to .../282 to go straight to the page].

Harold Wheeler Oral History (1985)    

Harold Wheeler Oral History (1991)

DWK (G3YNH) replies (2019 March 04):

Thank you so much for your diligence in uncovering this extra material.  I thought I had covered Wheeler's various formulae pretty well, but obviously I have missed a fair bit.

Going through what I wrote (actually in 2008, but converted to pdf in 2016), and in view of Rodger Rosenbaum's comments, I recall that a factor of about 1.00275 that arises from the inch to SI conversion accounts for my tweak of the coefficient in the denominator to 0.4502.  That it somehow appeared mysteriously without comment or attribution in Ramo et al warrants some thought.
I'm not sure we'll get to the bottom of this matter, but can I ask your permission to add an edited version of this [correspondence], to the page 'Inductance Calculation'?

John Crabtree (2019 March 05):

You are very welcome.  I think you are right - we will not get to the bottom of this.  But this information moves the story forward.

The factor 0.4502 in Ramo is interesting.   I don't see why the factor 1.00275 should account for this.  The 1.00275 accounts for the correction needed for an infinitely long coil (?)  Wheeler (1982) stated that the coefficients in his formula were very close to the ratio 9:10.   If one went back to the table in Dellinger (1918) or even Nagaoka's original work, and recalculated Wheeler's approximation in the form 1/(1+Na/10b), what would one get ?   I think that it would make some sense to look at the range for which Wheeler specified his formula.

I think you are right - Ramo et al re-derived the formula.   But we don't know where they started from. 

DWK (2019 March 05)

I think Ramo et al did what I did, which was to put the formula into a form analogous to capacitance and immediately spot that an approximation for Nagaoka's coeff. could be picked out of it by deduction - because Nagaoka's coeff. goes to 1 for infinite length / Diam.  The inch to SI translation gets lost in that operation however - it is just a small(ish) hidden error.  In Ramo et al, rearranging what they did into the form I used gives the empirical coeff in the denominator as 0.45.  It was me who tweaked it to 0.4502 to make the expression asymptotic.  I didn't bother to to try using 0.45 and multiplying the whole thing by 1.00275 because I felt that it was unjustifiable to have two empirical coefficients.  Doing the converse and multiplying Wheeler 1925 by 0.997256 anyway gives a curve that is practically identical to to the tweaked Ramo formula (W25a).