John Crabtree, KC0G,
(2019 March 04):
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.
page 44 [of version 0.20] you state: "More intriguingly however, it is in the same
form as the expression for Kw25
(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
his oral histories to discover how he developed the 1925 formula.
1. Derivation of the 1925 formula
story is mostly set out in Wheeler . He
Formula (153) in Dellinger  i.e.,
L = K 0.03948 a2
L is in uH, a is the radius in metres, b is the length in metres, and K
is a function of 2a/b
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
Formula (153) Converting to inches one gets:
/ (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
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
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
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
Wheeler, "Simple Inductance Formulas for Radio Coils". Proc. IRE, Vol.
16, No. 10, October 1928, pp 1398-1400
"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, "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.
Derringer et al, "Radio Instruments and Measurements". Bureau of
Standards, C74 (Circular No. 74), March 1918.
can find this at:
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].
Wheeler Oral History (1985)
Wheeler Oral History (1991)
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.
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.
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'?
(2019 March 05):
are very welcome. I think you are right - we will not
get to the bottom of this. But this information moves 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
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.
think you are right - Ramo et al re-derived the formula.
But we don't know where they started from.
(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).