Derivation of the Wheeler Inductance Equation.
Alan Fowler
There has been considerable discussion on the accuracy of
Wheeler's formula in this forum, but I have not seen any
information on the derivation of the formula. I may have missed
an earlier posting, and if so my apologies for presenting the
information again.
WARNING !! The following has been extracted from several of my
(very) old text books and is presented in good faith. I have no
way of knowing whether the information in those books is
accurate. Most authors copy well known formulae, graphs and
tables from existing text books rather than going back to the
original source. Occasional errors slip in either in the
copying or printing. As an example, with one exception, every
reference I have seen to the resistivity of metallic chromium
gives a value of 2.6 micro-ohm-cm. The exception was a series
of measurements carried at by Kaye and Laby (Melbourne
University) that produced a value of approx 13 micro-ohm-cm.
The original NBS pamphlet gave the value as 2.6, and there is a
strong possibility this was a typing error that had not been
picked up. SO BEWARE.
The following is taken from The Radiotron Designer's Handbook,
3rd Edition, 8th Impression, 1942, edited by F. Langford-Smith,
Published by The Wireless Press for Amalgamated Wireless Valve
Company pty. Ltd., Sydney, Australia. pp 141-146.
[My notes and comments are shown like this in square brackets to
distinguish them from the Handbook - AMF]
Definition of Terms:
"Current Sheet" Inductance: For the ideal case of a very long
solenoid wound with extremely thin tape having turns separated
by infinitely thin insulation we have the well known formula for
low frequency inductance which is called the "current sheet"
inductance L.
uH = micro-Henry.
L = inductance in uH of the equivalent cylindrical "current
sheet".
Lo = inductance in uH at low frequencies.
K = Nagaoka's constant.
N = Total number of turns.
a = radius of the coil, out to the centre of the wire, in
inches.
p = pitch of the winding, centre to centre of adjacent turns,
in inches.
len = pN = the length of the coil, in inches.
[NOTE I have used "len" to avoid the confusion between between
l (lower case L) and 1 (one) - AMF]
D = Wire diameter, in inches, excluding any insulating
covering,
Pi = 3.14159 ....
S = D/p
= (diameter of wire)/(winding pitch)
= wire diameter x turns per inch.
and:
x*y = x multiplied by y,
x/y = x divided by y, and
x^y = x raised to the power y.
then:
L = 0.10028(a^2 * N^2/len)K [1]
and Lo = L(1 - len * (A + B)/(Pi*a*N*K)) [2]
or Lo = L(l - p*(A + B)/(Pi*a*K)) [3]
where:
A is a function of S, and
B is a function of N.
R.D.H gives curves for the values of A, B and K. A & B are
plotted on log-linear paper. A appears to be a straight line.
From the graph:
S A
0.005 - 4.7
0.05 - 2.55
0.5 - 0.12
1.0 + 0.55
N B
1.0 0
10 0.267
100 0.327
2a/len K [RDH] K [Henney]
0.01 0.996
0.1 0.958 0.9588
1.0 0.688 0.6884
2.0 0.53 0.5255
5.0 0.32 0.3198
10 0.2 0.2033
20 0.1236
50 0.0611
100 0.035 0.0350
[The values of A, B & K [RDH] are as close as I can read from
the small sized graph. K [Henney} is taken from a table in The
Radio Engineering Handbook, K. Henney, 3rd edit, 6th impression,
McGraw-Hill 1941. p 93. - AMF]
In practical cases with turns wound close together the wire
diameter to winding pitch ratio S usually lies between 0.8 and
0.95, depending upon the thickness of the insulation. In this
range of S, A = 0.4 +/- 0.01. If the winding consists of more
than ten turns, B = 0.3 +/- 0.035. Therefore (A + B) = 0.7 with
a maximum possible error of 20%. For ratios of 2*a/len between
0.5 and 1.7, Nagaoka's constant K = 0.7 also with an accuracy no
poorer than 20%.
We can now determine the ratio of p/a or len/a*N for which lo
will differ from L by the order of 1%. The condition is that:
0.01 = p*(A + B)/(Pi*a*K) [4]
= (p/a)*(0.7/(3.14*0.7)) approximately [5]
and p/a should not exceed 0.03 [6]
Coils wound with spaced turns will show more than 1% difference
between the true low frequency inductance Lo and the "current
sheet" value L. Hence it is most important when using the
"current sheet" formula to check at the same time the order of
magnitude of the correction. The correction sometimes amounts to
15%.
As an alternative to tables and curves for the values of
Nagaokaá constant and the constants A & B, the following
formulae are suitable for slide rule computation.
A = 2.3 log 1.73 S [7]
accurate to 1% for all values of S.
B = 0.336(1 - (2.5/N) + (3.8/N^2)) [8]
accurate within 1% when N is not less than five turns. The
value from this formula is about 5% high at N = 4 and 20% high
at N = 3.
K = 1/(1 + 0.225(a/len)) [9]
accurate within 1% for all values of 2a/len less than 3.0, that
is for all solenoids whose length exceed one-third of the
diameter.
Using this value of K, the current sheet inductance is
L = 0.10028(a^2 * N^2/len)K [10]
= a^2 * N^2/(10*len(1 + 0.9(a/len))) [11]
= a^2 * N^2/(9*a + 10*len) [12]
which is the well known Wheeler's formula.
It is also accurate within 1% for all values of 2*a/len less
than 3. Wheeler's formula gives a result about 4% low when
2*a/len = 5.0
Hopefully I have copied that correctly.
Remember that these formulae apply only to single layer
solenoids.
regards, Alan.
E&OE.
,-._|\ Alan Fowler. (Alan M. Fowler FIEAust CPEng)
/ Oz \ Mail Address: PO Box 1008G, North Balwyn 3104 Vic, AUSTRALIA.
\_,--.x/ Phone: +613-9857-7128 Member, Melbourne PC User Group.
v +-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+-+
Winfield Hill
Alan Fowler, <amfo...@melbpc.org.au> said...
>
> There has been considerable discussion on the accuracy of
> Wheeler's formula in this forum, but I have not seen any
> information on the derivation of the formula.
I did post (2 Oct) a lengthy quoatation from Wheeler's original
paper, where he gives no references and simply states,
"The new formulas were derived empirically from the inductance
formulas and curves in Circular 74 of the Bureau of Standards.
The corresponding coil formulas of this circular, however,
either rely on tables or include expressions which are
inconvenient to compute. For this reason there was a need for
more convenient formulas, even with the loss of some accuracy,
for use in the laboratory."
Therefore, the material you presented is very interesting. As
one of many who doesn't have access to Langford-Smith's book,
I'm very pleased you took the time to pass this material on!
> The following is taken from The Radiotron Designer's Handbook,
> 3rd Edition, 8th Impression, 1942, edited by F. Langford-Smith,
> Published by The Wireless Press for Amalgamated Wireless Valve
> Company pty. Ltd., Sydney, Australia. pp 141-146.
[ selected massive snips to save space ]
> L = 0.10028(a^2 * N^2/len)K [1]
Right, let's see what we've got here. OK, we see [1] is just a form
of the standard inductance equation, with Nagaoka's correction K tacked
on, making it Nagaoka's equation. The 0.10028 is the permeability of
free-space, 4 pi 10^-7 Henrys per meter, divided by 39.37 inches/meter,
and an extra times pi thrown in as well (this allows using a^2 rather
than the area of the coil).
> and Lo = L(1 - len * (A + B)/(Pi*a*N*K)) [2]
>
> or Lo = L(l - p*(A + B)/(Pi*a*K)) [3]
OK, these same equations are in Welsby (p 42), introduced as corrections
to apply to coils whose turns are spaced far apart. But Welsby doesn't
give any references, and I't be nice to know where they came from.
[ snip snip ]
> In practical cases with turns wound close together the wire
> diameter to winding pitch ratio S usually lies between 0.8 and
> 0.95, depending upon the thickness of the insulation. In this
> range of S, A = 0.4 +/- 0.01. If the winding consists of more
> than ten turns, B = 0.3 +/- 0.035. Therefore (A + B) = 0.7 ...
> For ratios of 2*a/len between 0.5 and 1.7, Nagaoka's constant
> K = 0.7 also with an accuracy no poorer than 20%.
>
> We can now determine the ratio of p/a or len/a*N for which lo
> will differ from L by the order of 1%. The condition is that:
> [snip]
> and p/a should not exceed 0.03 [6]
This is interesting. Rewritten, it says the 1% valid range is
length/diameter > N/66. We can compare this to Wheeler's criteria
of > 0.4 and observe this is true for coils with more than 26 turns.
> As an alternative to tables and curves for the values of
> Nagaokaá constant and the constants A & B, the following
> formulae are suitable for slide rule computation.
>
> A = 2.3 log 1.73 S [7]
>
> B = 0.336(1 - (2.5/N) + (3.8/N^2)) [8]
These are also in Welsby' book following [2] and [3], but still
without references.
>
> K = 1/(1 + 0.225(a/len)) [9]
Whoa! Where'd this formula come from. Also, I don't get [11]
from [10] by substituting [9] unless the 0.225 is 0.920.
Hmmm....
> L = 0.10028(a^2 * N^2/len)K [10]
>
> = a^2 * N^2/(10*len(1 + 0.9(a/len))) [11]
>
> = a^2 * N^2/(9*a + 10*len) [12]
>
> which is the well known Wheeler's formula.
Alan, is there more material there which might shed some light?
--
Winfield Hill hi...@rowland.org _/_/_/ _/_/_/_/
The Rowland Institute for Science _/ _/ _/_/ _/
Cambridge, MA USA 02142-1297 _/_/_/_/ _/ _/ _/_/_/
_/ _/ _/ _/ _/
http://www.artofelectronics.com/ _/ _/ _/_/ _/_/_/_/
James P. Meyer
On 12 Oct 1997, Winfield Hill wrote:
> > K = 1/(1 + 0.225(a/len)) [9]
>
> Whoa! Where'd this formula come from.
Isn't it Nagaoka"s constant?
Jim
P.S. As soon as I hook my flatbed scanner back up, I can scan
the chapter on inductance calculations from
Langford-Smith's book and e-mail them to you.
Winfield Hill
James P. Meyer, <jim...@acpub.duke.edu> said...
>
>On 12 Oct 1997, Winfield Hill wrote:
>
>>
>
> Isn't it Nagaoka"s constant?
>
> Jim
Hmmm, in Grover's book for it's
K = 2 pi L/D [( ln 4D/L - 0.5 ) + 1/8 (L/D)^2 ( ln 4D/L + 1/8 )
- 1/64 (L/D)^4 ( etc ) + ...]
where L/D is length/dia, and that's just for short coils!!!
However I still haven't read much of Nagaoka's detailed writing,
because they seem to be in some old Japanese journals not carried
in MIT's archives. If you have any closer reference, Jim, maybe
an NBS pub or some other discussion, I'm all eyes and ears!
Alan Fowler
amfo...@melbpc.org.au (Alan Fowler) wrote:
> There has been considerable discussion on the accuracy of
>Wheeler's formula in this forum, but I have not seen any
>information on the derivation of the formula. I may have missed
>an earlier posting, and if so my apologies for presenting the
>information again.
THIS IS A REVISED and CORRECTED VERSION OF MY ORIGINAL POSTING
of 12th October at 9.46 pm. It contains two corrections, plus
additional information in answer to e-mailed questions. I have
not deleted any of the original posting in this document to save
the reader having to refer to two documents.
> WARNING !! The following has been extracted from several of my
>(very) old text books and is presented in good faith. I have no
>way of knowing whether the information in those books is
>accurate. Most authors copy well known formulae, graphs and
>tables from existing text books rather than going back to the
>original source. Occasional errors slip in either in the
>copying or printing. As an example, with one exception, every
>reference I have seen to the resistivity of metallic chromium
>gives a value of 2.6 micro-ohm-cm. The exception was a series
>of measurements carried at by Kaye and Laby (Melbourne
>University) that produced a value of approx 13 micro-ohm-cm.
>The original NBS pamphlet gave the value as 2.6, and there is a
>strong possibility this was a typing error that had not been
>picked up. SO BEWARE.
> The following is taken from The Radiotron Designer's Handbook,
>3rd Edition, 8th Impression, 1942, edited by F. Langford-Smith,
>Published by The Wireless Press for Amalgamated Wireless Valve
>Company pty. Ltd., Sydney, Australia. pp 141-146.
NOTE THAT the 4th edition does NOT show the detail of the
derivation of Wheeler's formula.
> [My notes and comments are shown like this in square brackets to
>distinguish them from the Handbook - AMF]
>Definition of Terms:
> "Current Sheet" Inductance: For the ideal case of a very long
>solenoid wound with extremely thin tape having turns separated
>by infinitely thin insulation we have the well known formula for
>low frequency inductance which is called the "current sheet"
>inductance L.
> uH = micro-Henry.
> L = inductance in uH of the equivalent cylindrical "current
>sheet".
> Lo = inductance in uH at low frequencies.
> K = Nagaoka's constant.
> N = Total number of turns.
> a = radius of the coil, out to the centre of the wire, in
>inches.
diam = 2a = diameter of the coil.
> p = pitch of the winding, centre to centre of adjacent turns,
>in inches.
> len = pN = the length of the coil, in inches.
> [NOTE I have used "len" to avoid the confusion between between
>l (lower case L) and 1 (one) - AMF]
> D = Wire diameter, in inches, excluding any insulating
>covering,
> Pi = 3.14159 ....
> S = D/p
> = (diameter of wire)/(winding pitch)
> = wire diameter x turns per inch.
>and:
> x*y = x multiplied by y,
> x/y = x divided by y, and
> x^y = x raised to the power y.
>then:
> L = 0.10028(a^2 * N^2/len)K [1]
[The factor 0.10028 is derived from the permeability of free
space, 4 * Pi * 10^-7 Henry per meter, divided by 39.37
inches/meter and multiplied by Pi to convert a^2 to an area.
(Explanation courtesy of Winfield Hill)]
> and Lo = L(1 - len * (A + B)/(Pi*a*N*K)) [2]
> or Lo = L(l - p*(A + B)/(Pi*a*K)) [3]
>where:
> A is a function of S, and
> B is a function of N.
> R.D.H gives curves for the values of A, B and K. A & B are
>plotted on log-linear paper. A appears to be a straight line.
>From the graph:
---------------------------------------------
> S A
---------------------------------------------
> 0.005 - 4.7
> 0.05 - 2.55
> 0.5 - 0.12
> 1.0 + 0.55
----------------------------------------------
> N B
----------------------------------------------
> 1.0 0
> 10 0.267
> 100 0.327
------------------------------------------------------------
> 2a/len K [RDH] K [Henney]
------------------------------------------------------------
> 0.01 0.996
> 0.1 0.958 0.9588
> 1.0 0.688 0.6884
> 2.0 0.53 0.5255
> 5.0 0.32 0.3198
> 10 0.2 0.2033
> 20 0.1236
> 50 0.0611
> 100 0.035 0.0350
> [The values of A, B & K [RDH] are as close as I can read from
>the small sized graph. K [Henney} is taken from a table in The
>Radio Engineering Handbook, Keith Henney, 3rd edit, 6th impression,
>McGraw-Hill 1941. p 93. - AMF]
> In practical cases with turns wound close together the wire
>diameter to winding pitch ratio S usually lies between 0.8 and
>0.95, depending upon the thickness of the insulation. In this
>range of S, A = 0.4 +/- 0.01. If the winding consists of more
>than ten turns, B = 0.3 +/- 0.035. Therefore (A + B) = 0.7 with
>a maximum possible error of 20%. For ratios of 2*a/len between
>0.5 and 1.7, Nagaoka's constant K = 0.7 also with an accuracy no
>poorer than 20%.
> We can now determine the ratio of p/a or len/a*N for which lo
We can now determine the ratio of p/a or len/a*N for which Lo
>will differ from L by the order of 1%. The condition is that:
> 0.01 = p*(A + B)/(Pi*a*K) [4]
> = (p/a)*(0.7/(3.14*0.7)) approximately [5]
> and p/a should not exceed 0.03 [6]
> Coils wound with spaced turns will show more than 1% difference
>between the true low frequency inductance Lo and the "current
>sheet" value L. Hence it is most important when using the
>"current sheet" formula to check at the same time the order of
>magnitude of the correction. The correction sometimes amounts to
>15%.
> As an alternative to tables and curves for the values of
>Nagaokaá constant and the constants A & B, the following
>formulae are suitable for slide rule computation.
[I haven't been able to find a reference to the derivation of
these formulae. My best guess is that the formulae have been
simplified by dropping small terms or by choosing a formula
which gave a "best fit" to within 1% over the range of
interest.]
> A = 2.3 log 1.73 S [7]
> accurate to 1% for all values of S.
> B = 0.336(1 - (2.5/N) + (3.8/N^2)) [8]
> accurate within 1% when N is not less than five turns. The
>value from this formula is about 5% high at N = 4 and 20% high
>at N = 3.
> K = 1/(1 + 0.225(a/len)) [9 wrong const]
[ The R.D.H. uses either diameter or radius in the above
formulae which makes them very hard to follow. I changed these
to use the coil radius only, to avoid errors, then in converting
"0.45(d/len)" I divided by 2 instead of multiplying. My
apologies.]
K = 1/(1 + 0.9(a/len)) [9]
> accurate within 1% for all values of 2a/len less than 3.0, that
>is for all solenoids whose length exceed one-third of the
>diameter.
> Using this value of K, the current sheet inductance is
> L = 0.10028(a^2 * N^2/len)K [10]
> = a^2 * N^2/(10*len(1 + 0.9(a/len))) [11]
> = a^2 * N^2/(9*a + 10*len) [12]
> which is the well known Wheeler's formula.
> It is also accurate within 1% for all values of 2*a/len less
>than 3. Wheeler's formula gives a result about 4% low when
>2*a/len = 5.0
Esnault-Pelterie# quotes:
K = 1/(0.9949 + 0.4572(diam/len)) [13]
accurate within 0.1% for all values of diam/len between 0.2 and
1.5.
# M.R. Esnault-Pelterie, "On the Co-efficient of Self Inductance
of Circular Cyldrical Coils", Comptes Rendus, Tome 205, No. 18
p. 762, November 3, (1937) and No. 20, p. 885, November 15,
(1937)
The following table compares the values calculated from eqn [13]
against the table in Henney referred to earlier.
2a/len Calculated Henney
0.1 0.9610 0.9588
0.2 0.9205 0.9201
0.3 0.8833 0.8838
0.4 0.8491 0.8499
0.5 0.8173 0.8181
0.6 0.7879 0.7885
0.7 0.7605 0.7609
0.8 0.7349 0.7351
0.9 0.7110 0.7110
1.0 0.6887 0.6884
1.1 0.6676 0.6673
1.2 0.6479 0.6475
1.3 0.6292 0.6290
1.4 0.6116 0.6115
1.5 0.5950 0.5950
1.6 0.5792 0.5795
1.7 0.5643 0.5649
1.8 0.5501 0.5511
1.9 0.5366 0.5379
2.0 0.5238 0.5255
[These are my calculations - AMF - 13-Oct-97]
Neither of the above expressions [9] & [13] for Nagaoka's
constant is valid for very short coils. For such cases the
first expression may be modified to:
K = 1/(1 + 0.45(diam/len) - 0.005(diam/len)^2) [14]
accurate to 2% for all values of diam/len from zero to twenty.
It is not often that solenoids shorter than one twentieth of
their diameter have to be considered.
For short solenoids Wheeler's formula then becomes:
L = a^2 * N^2/[9 - (a/5*len)]*a + 10*len] [15]
Also accurate to 2% for all values of 2a/len from zero to
twenty. The error approaches +2% when diam/len = 2.0 to 3.5 and
at diam/len = 20. The error approached -2% in the range 10 to
12.
> Hopefully I have copied all that correctly.
> Remember that these formulae apply only to single layer
>solenoids.
I think we will be lucky to get any better results unless we can
get hold of a copy of the original Bureau of Standards Circular
74 (1924) [Which makes it just 3 years older than me]
I had access to a copy when I was working on the design of
aerial coupling units, but that was 40 years ago. It was
regarded as a relatively modern paper back then.
I'm surprised that somebody hasn't used the exact values in
Circular 74 to write a computer program to do all the
calculations for us. Anyone know of one?
Winfield Hill
Alan Fowler, <amfo...@melbpc.org.au> said...
>
>amfo...@melbpc.org.au (Alan Fowler) wrote:
>> The following is taken from The Radiotron Designer's Handbook,
>> 3rd Edition, 8th Impression, 1942, edited by F. Langford-Smith,
>> Published by The Wireless Press for Amalgamated Wireless Valve
>> Company pty. Ltd., Sydney, Australia. pp 141-146.
>
> NOTE THAT the 4th edition does NOT show the detail of the
> derivation of Wheeler's formula.
Very interesting. [snip]
>> L = 0.10028(a^2 * N^2/len)K [1]
>>
>> and Lo = L(1 - len * (A + B)/(Pi*a*N*K)) [2]
>
>> or Lo = L(l - p*(A + B)/(Pi*a*K)) [3]
>
> [I haven't been able to find a reference to the derivation of
> these formulae. My best guess is that the formulae have been
> simplified by dropping small terms or by choosing a formula
> which gave a "best fit" to within 1% over the range of
> interest.]
>
>> A = 2.3 log 1.73 S [7]
>>
>> B = 0.336(1 - (2.5/N) + (3.8/N^2)) [8]
>
> K = 1/(1 + 0.9(a/len)) [9] corrected
>
>> L = 0.10028(a^2 * N^2/len)K [10]
>
>> = a^2 * N^2/(10*len(1 + 0.9(a/len))) [11]
>
>> = a^2 * N^2/(9*a + 10*len) [12]
>
>> which is the well known Wheeler's formula.
>
> Esnault-Pelterie# quotes:
>
> K = 1/(0.9949 + 0.4572(diam/len)) [13]
>
> accurate within 0.1% for all values of diam/len between 0.2 and 1.5.
>
> # M.R. Esnault-Pelterie, "On the Co-efficient of Self Inductance
> of Circular Cyldrical Coils", Comptes Rendus, Tome 205, No. 18
> p. 762, November 3, (1937) and No. 20, p. 885, November 15, (1937)
Aha! Very very interesting! And yet another old hard-to-get
publication! We'll see just how good the MIT archives are anyway,
and find out just exactly how Esnault-Pelterie came up with [13].
Also, since they published this in 1937, nine years after Wheeler's
paper, I wonder how they refer to his work? Were they trying to
establish a basis for it (which Weeler avoided doing in his very
short paper), or were they trying to improve upon it?
> The following table compares the values calculated from eqn [13]
> against the table in Henney referred to earlier.
> calculated
> 2a/len Calculated Henney Grover error
> 0.1 0.9610 0.9588 0.958807 +0.23 %
> 0.5 0.8173 0.8181 0.818136 -0.10
> 1.0 0.6887 0.6884 0.688423 +0.04
> 1.5 0.5950 0.5950 0.595039 0.00
> 2.0 0.5238 0.5255 0.525510 -0.33
>
> [These are my calculations - AMF - 13-Oct-97]
[abbreviated, Grover values and calc. error added] I'm curious
about the Henney reference you quote. The values seem the same
as in Grover (tables 36 and 37), which I think are the same as
circular 74.
> Neither of the above expressions [9] & [13] for Nagaoka's
> constant is valid for very short coils. For such cases the
> first expression may be modified to:
>
> K = 1/(1 + 0.45(diam/len) - 0.005(diam/len)^2) [14]
>
> accurate to 2% for all values of diam/len from zero to twenty.
> It is not often that solenoids shorter than one twentieth of
> their diameter have to be considered.
>
> For short solenoids Wheeler's formula then becomes:
>
> L = a^2 * N^2/[9 - (a/5*len)]*a + 10*len] [15]
>
> Also accurate to 2% for all values of 2a/len from zero to
> twenty. The error approaches +2% when diam/len = 2.0 to 3.5 and
> at diam/len = 20. The error approached -2% in the range 10 to 12.
Are the above two formulas also from Esnault-Pelterie, or were they
in R.D.H. instead (and attributed to Esnault-Pelterie)? I see that
formula [14] is also in Welsby's book, but unattributed. He uses
it, rather than Wheeler's formula, as the basic formula to use.
As far as I can tell, none of these simple formulas come from any
simple mathematical manipulation of the long series formulas usually
resulting from theory.
> I think we will be lucky to get any better results unless we can
> get hold of a copy of the original Bureau of Standards Circular
> 74 (1924) [Which makes it just 3 years older than me]
I have a copy (it's at work now). After zeroxing much of it at the
Boston Public Library (neither MIT nor Harvard had it), our librarian
was able to buy a used copy from Powells, I think. There were three
copies available at the time. It's a complete thick book, as you'll
remember. Yes, it's possible the formulas are in there. I also
have at work a collection of about 30 papers from that era to scour
for these old formulas, if your Esnault-Pelterie source doesn't clear
things up!
> I'm surprised that somebody hasn't used the exact values in
> Circular 74 to write a computer program to do all the
> calculations for us. Anyone know of one?
Actually, the BASIC program Jose posted on 24 Aug, used values
matching Henney, as you referred to, which appear to be a 4-digit
version of the Grover tables 36 and 37, which I think are in fact
the BS tables. So Jose's program fills that bill, at least to
3 digit accuracy. I'll post a copy of it.
But as you pointed out, all this is for a single-layer solenoid,
and in fact, since they're current-sheet formulas, they may have
substantial error when the coils are made with thick wire or tubing
or have considerable spacing between turns.
Grover addresses this with a correction formula, which he credits
to the theories of Rosa, and which uses two parameters G and H,
given by tables 38 and 39 in his book. And there's the correction
formulas with A and B, you give above (which is also in Welsby).
I remember seeing more formulas in some of my papers.
You'll recall that with my assortment of 20 single-layer test coils
(post of 30 Aug), although most were accurate, I saw up to +/- 7%
error from Wheeler = from Nagaoka. Hmmm, maybe someday I'll redo
those measurements with more care, and more instruments, and apply
them to more of the coil formulas we've been uncovering.
Winfield Hill
Winfield Hill, <hi...@rowland.org> said...
>
> Alan Fowler, <amfo...@melbpc.org.au> said...
>
>> I'm surprised that somebody hasn't used the exact values in
>> Circular 74 to write a computer program to do all the
>> calculations for us. Anyone know of one?
>
> Actually, the BASIC program Jose posted on 24 Aug, used values
> matching Henney, as you referred to, which appear to be a 4-digit
> version of the Grover tables 36 and 37, which I think are in fact
> the BS tables. So Jose's program fills that bill, at least to
> 3 digit accuracy. I'll post a copy of it.
OK, Here's Jose's BASIC subroutine. Note, I don't agree with his
comments about Wheeler!!!
--- Win
The following is a Basic file for calculating inductance of coils.
The first is the familiar Wheeler. The second is the one based on a
current sheet with correction factors from an NBS publication. On a
design I participated in we had a tapped coil with a tap at each turn.
The Wheeler formula was not very correct for the first turns as it is
accurate only when diameter and length approach each other. The 2nd
formula tracked my meassured data pretty well when I accounted for
shielding of the room.
This is valid when only considering inductance.
José Sainz
-----------
CoilEquation: ' formula for inductance of coil
'RAD= "mean" radius, NT=number of turns, CWL=coil winding length
IF InductAccuracy = 0 THEN
INDUCT = (RAD ^ 2 * NT ^ 2) / (9 * RAD + 10 * CWL)
ELSE
DL = 2 * RAD / CWL
IF DL = 0 THEN K = 1
IF DL <= .05 THEN K = 1 - .0209 * (DL - 0) / .05: GOTO IndQuit
IF DL <= .1 THEN K = .9791 - .0203 * (DL - .05) * 20: GOTO IndQuit
IF DL <= .15 THEN K = .9588 - .0197 * (DL - .1) * 20: GOTO IndQuit
IF DL <= .2 THEN K = .9391 - .019 * (DL - .15) * 20: GOTO IndQuit
IF DL <= .25 THEN K = .9201 - .0185 * (DL - .2) * 20: GOTO IndQuit
IF DL <= .3 THEN K = .9016 - .0178 * (DL - .25) * 20: GOTO IndQuit
IF DL <= .35 THEN K = .8838 - .0173 * (DL - .3) * 20: GOTO IndQuit
IF DL <= .4 THEN K = .8665 - .0167 * (DL - .35) * 20: GOTO IndQuit
IF DL <= .45 THEN K = .8499 - .0162 * (DL - .4) * 20: GOTO IndQuit
IF DL <= .5 THEN K = .8337 - .0156 * (DL - .45) * 20: GOTO IndQuit
IF DL <= .55 THEN K = .8181 - .015 * (DL - .5) * 20: GOTO IndQuit
IF DL <= .6 THEN K = .8031 - .0146 * (DL - .55) * 20: GOTO IndQuit
IF DL <= .65 THEN K = .7885 - .014 * (DL - .6) * 20: GOTO IndQuit
IF DL <= .7 THEN K = .7745 - .0136 * (DL - .65) * 20: GOTO IndQuit
IF DL <= .75 THEN K = .7609 - .0131 * (DL - .7) * 20: GOTO IndQuit
IF DL <= .8 THEN K = .7478 - .0127 * (DL - .75) * 20: GOTO IndQuit
IF DL <= .85 THEN K = .7351 - .0123 * (DL - .8) * 20: GOTO IndQuit
IF DL <= .9 THEN K = .7228 - .0118 * (DL - .85) * 20: GOTO IndQuit
IF DL <= .95 THEN K = .711 - .0115 * (DL - .9) * 20: GOTO IndQuit
IF DL <= 1 THEN K = .6995 - .0111 * (DL - .95) * 20: GOTO IndQuit
IF DL <= 1.05 THEN K = .6884 - .0107 * (DL - 1) * 20: GOTO IndQuit
IF DL <= 1.1 THEN K = .6777 - .0104 * (DL - 1.05) * 20: GOTO IndQuit
IF DL <= 1.15 THEN K = .6673 - .01 * (DL - 1.1) * 20: GOTO IndQuit
IF DL <= 1.2 THEN K = .6573 - .0098 * (DL - 1.15) * 20: GOTO IndQuit
IF DL <= 1.25 THEN K = .6475 - .0094 * (DL - 1.2) * 20: GOTO IndQuit
IF DL <= 1.3 THEN K = .6381 - .0091 * (DL - 1.25) * 20: GOTO IndQuit
IF DL <= 1.35 THEN K = .629 - .0089 * (DL - 1.3) * 20: GOTO IndQuit
IF DL <= 1.4 THEN K = .6201 - .0086 * (DL - 1.35) * 20: GOTO IndQuit
IF DL <= 1.45 THEN K = .6115 - .0084 * (DL - 1.4) * 20: GOTO IndQuit
IF DL <= 1.5 THEN K = .6031 - .0081 * (DL - 1.45) * 20: GOTO IndQuit
IF DL <= 1.55 THEN K = .595 - .0079 * (DL - 1.5) * 20: GOTO IndQuit
IF DL <= 1.6 THEN K = .5871 - .0076 * (DL - 1.55) * 20: GOTO IndQuit
IF DL <= 1.65 THEN K = .5795 - .0074 * (DL - 1.6) * 20: GOTO IndQuit
IF DL <= 1.7 THEN K = .5721 - .0072 * (DL - 1.65) * 20: GOTO IndQuit
IF DL <= 1.75 THEN K = .5649 - .007 * (DL - 1.7) * 20: GOTO IndQuit
IF DL <= 1.8 THEN K = .5579 - .0068 * (DL - 1.75) * 20: GOTO IndQuit
IF DL <= 1.85 THEN K = .5511 - .0067 * (DL - 1.8) * 20: GOTO IndQuit
IF DL <= 1.9 THEN K = .5444 - .0065 * (DL - 1.85) * 20: GOTO IndQuit
IF DL <= 1.95 THEN K = .5379 - .0063 * (DL - 1.9) * 20: GOTO IndQuit
IF DL <= 2 THEN K = .5316 - .0061 * (DL - 1.95) * 20: GOTO IndQuit
IF DL <= 2.1 THEN K = .5255 - .0118 * (DL - 2) * 10: GOTO IndQuit
IF DL <= 2.2 THEN K = .5137 - .0112 * (DL - 2.1) * 10: GOTO IndQuit
IF DL <= 2.3 THEN K = .5025 - .0107 * (DL - 2.2) * 10: GOTO IndQuit
IF DL <= 2.4 THEN K = .4918 - .0102 * (DL - 2.3) * 10: GOTO IndQuit
IF DL <= 2.5 THEN K = .4816 - .0097 * (DL - 2.4) * 10: GOTO IndQuit
IF DL <= 2.6 THEN K = .4719 - .0093 * (DL - 2.5) * 10: GOTO IndQuit
IF DL <= 2.7 THEN K = .4626 - .0089 * (DL - 2.6) * 10: GOTO IndQuit
IF DL <= 2.8 THEN K = .4537 - .0085 * (DL - 2.7) * 10: GOTO IndQuit
IF DL <= 2.9 THEN K = .4452 - .0082 * (DL - 2.8) * 10: GOTO IndQuit
IF DL <= 3 THEN K = .437 - .0078 * (DL - 2.9) * 10: GOTO IndQuit
IF DL <= 3.1 THEN K = .4292 - .0075 * (DL - 3) * 10: GOTO IndQuit
IF DL <= 3.2 THEN K = .4217 - .0072 * (DL - 3.1) * 10: GOTO IndQuit
IF DL <= 3.3 THEN K = .4145 - .007 * (DL - 3.2) * 10: GOTO IndQuit
IF DL <= 3.4 THEN K = .4075 - .0067 * (DL - 3.3) * 10: GOTO IndQuit
IF DL <= 3.5 THEN K = .4008 - .0064 * (DL - 3.4) * 10: GOTO IndQuit
IF DL <= 3.6 THEN K = .3944 - .0062 * (DL - 3.5) * 10: GOTO IndQuit
IF DL <= 3.7 THEN K = .3882 - .006 * (DL - 3.6) * 10: GOTO IndQuit
IF DL <= 3.8 THEN K = .3822 - .0058 * (DL - 3.7) * 10: GOTO IndQuit
IF DL <= 3.9 THEN K = .3764 - .0056 * (DL - 3.8) * 10: GOTO IndQuit
IF DL <= 4 THEN K = .3708 - .0054 * (DL - 3.9) * 10: GOTO IndQuit
IF DL <= 4.1 THEN K = .3654 - .0052 * (DL - 4) * 10: GOTO IndQuit
IF DL <= 4.2 THEN K = .3602 - .0051 * (DL - 4.1) * 10: GOTO IndQuit
IF DL <= 4.3 THEN K = .3551 - .0049 * (DL - 4.2) * 10: GOTO IndQuit
IF DL <= 4.4 THEN K = .3502 - .0047 * (DL - 4.3) * 10: GOTO IndQuit
IF DL <= 4.5 THEN K = .3455 - .0046 * (DL - 4.4) * 10: GOTO IndQuit
IF DL <= 4.6 THEN K = .3409 - .0045 * (DL - 4.5) * 10: GOTO IndQuit
IF DL <= 4.7 THEN K = .3364 - .0043 * (DL - 4.6) * 10: GOTO IndQuit
IF DL <= 4.8 THEN K = .3321 - .0042 * (DL - 4.7) * 10: GOTO IndQuit
IF DL <= 4.9 THEN K = .3279 - .0041 * (DL - 4.8) * 10: GOTO IndQuit
IF DL <= 5 THEN K = .3238 - .004 * (DL - 4.9) * 10: GOTO IndQuit
IF DL <= 5.2 THEN K = .3198 - .0076 * (DL - 5) * 5: GOTO IndQuit
IF DL <= 5.4 THEN K = .3122 - .0072 * (DL - 5.2) * 5: GOTO IndQuit
IF DL <= 5.6 THEN K = .305 - .0069 * (DL - 5.4) * 5: GOTO IndQuit
IF DL <= 5.8 THEN K = .2981 - .0065 * (DL - 5.6) * 5: GOTO IndQuit
IF DL <= 6 THEN K = .2916 - .0062 * (DL - 5.8) * 5: GOTO IndQuit
IF DL <= 6.2 THEN K = .2854 - .0059 * (DL - 6) * 5: GOTO IndQuit
IF DL <= 6.4 THEN K = .2795 - .0056 * (DL - 6.2) * 5: GOTO IndQuit
IF DL <= 6.6 THEN K = .2739 - .0054 * (DL - 6.4) * 5: GOTO IndQuit
IF DL <= 6.8 THEN K = .2685 - .0052 * (DL - 6.6) * 5: GOTO IndQuit
IF DL <= 7 THEN K = .2633 - .0049 * (DL - 6.8) * 5: GOTO IndQuit
IF DL <= 7.2 THEN K = .2584 - .0047 * (DL - 7) * 5: GOTO IndQuit
IF DL <= 7.4 THEN K = .2537 - .0045 * (DL - 7.2) * 5: GOTO IndQuit
IF DL <= 7.6 THEN K = .2491 - .0043 * (DL - 7.4) * 5: GOTO IndQuit
IF DL <= 7.8 THEN K = .2448 - .0042 * (DL - 7.6) * 5: GOTO IndQuit
IF DL <= 8 THEN K = .2406 - .004 * (DL - 7.8) * 5: GOTO IndQuit
IF DL <= 8.5 THEN K = .2366 - .0094 * (DL - 8) * 5: GOTO IndQuit
IF DL <= 9 THEN K = .2272 - .0086 * (DL - 8.5) * 2: GOTO IndQuit
IF DL <= 9.5 THEN K = .2185 - .0079 * (DL - 9) * 2: GOTO IndQuit
IF DL <= 10 THEN K = .2106 - .0073 * (DL - 9.5) * 2: GOTO IndQuit
IF DL <= 11 THEN K = .2033 - .0133 * (DL - 10): GOTO IndQuit
IF DL <= 12 THEN K = .1903 - .0113 * (DL - 11): GOTO IndQuit
IF DL <= 13 THEN K = .179 - .0098 * (DL - 12): GOTO IndQuit
IF DL <= 14 THEN K = .1692 - .0087 * (DL - 13): GOTO IndQuit
IF DL <= 15 THEN K = .1605 - .0078 * (DL - 14): GOTO IndQuit
IF DL <= 16 THEN K = .1527 - .007 * (DL - 15): GOTO IndQuit
IF DL <= 17 THEN K = .1457 - .0063 * (DL - 16): GOTO IndQuit
IF DL <= 18 THEN K = .1394 - .0058 * (DL - 17): GOTO IndQuit
IF DL <= 19 THEN K = .1336 - .0052 * (DL - 18): GOTO IndQuit
IF DL <= 20 THEN K = .1284 - .0048 * (DL - 19): GOTO IndQuit
IF DL <= 22 THEN K = .1236 - .0085 * (DL - 20) / 2: GOTO IndQuit
IF DL <= 24 THEN K = .1151 - .0073 * (DL - 22) / 2: GOTO IndQuit
IF DL <= 26 THEN K = .1078 - .0063 * (DL - 24) / 2: GOTO IndQuit
IF DL <= 28 THEN K = .1015 - .0056 * (DL - 26) / 2: GOTO IndQuit
IF DL <= 30 THEN K = .0959 - .0049 * (DL - 28) / 2: GOTO IndQuit
IF DL <= 35 THEN K = .091 - .0102 * (DL - 30) / 2: GOTO IndQuit
IF DL <= 40 THEN K = .0808 - .008 * (DL - 35) / 2: GOTO IndQuit
IF DL <= 45 THEN K = .0728 - .0064 * (DL - 40) / 5: GOTO IndQuit
IF DL <= 50 THEN K = .0064 - .0053 * (DL - 45) / 5: GOTO IndQuit
IF DL <= 60 THEN K = .0611 - .0043 * (DL - 50) / 10: GOTO IndQuit
IF DL <= 70 THEN K = .0528 - .0061 * (DL - 60) / 10: GOTO IndQuit
IF DL <= 80 THEN K = .0467 - .0048 * (DL - 70) / 10: GOTO IndQuit
IF DL <= 90 THEN K = .0419 - .0038 * (DL - 80) / 10: GOTO IndQuit
IF DL <= 100 THEN K = .0381 - .0031 * (DL - 90) / 10: GOTO IndQuit
IF DL > 100 THEN K = .035: CWL = RAD / 50
IndQuit:
INDUCT = .03948 * 2.54 * (RAD ^ 2 * NT ^ 2) * K / CWL
END IF
RETURN
Winfield Hill
Alan Fowler, <amfo...@melbpc.org.au> said...
>
>> [The values of A, B & K [RDH] are as close as I can read from
>> the small sized graph. K [Henney} is taken from a table in The
>> Radio Engineering Handbook, Keith Henney, 3rd edit, 6th impression,
>> McGraw-Hill 1941. p 93. - AMF]
I haven't seen this book. Can you tell us anything more about its
usefulness. Recommended?
> # M.R. Esnault-Pelterie, "On the Co-efficient of Self Inductance
> of Circular Cyldrical Coils", Comptes Rendus, Tome 205, No. 18
> p. 762, November 3, (1937) and No. 20, p. 885, November 15, (1937)
Ah, Comptes Rendus, the famous French philosophical journal. OK, we
can get this from the MIT library - no doubt the mother-lode is here!
> I think we will be lucky to get any better results unless we can
> get hold of a copy of the original Bureau of Standards Circular
> 74 (1924) [Which makes it just 3 years older than me]
Having read through this document several times in August, I don't
think we'll find the history for Wheeler's equation there. However,
if I can find the time at work, I'll recheck (or maybe remember to
bring it home!). Rather, the answer must surely like with Wheeler
(who isn't telling) or perhaps from the writings in Comptes Rendus.
Alan Fowler
hi...@rowland.org (Winfield Hill) wrote:
>Alan Fowler, <amfo...@melbpc.org.au> said...
>>
>>> [The values of A, B & K [RDH] are as close as I can read from
>>> the small sized graph. K [Henney} is taken from a table in The
>>> Radio Engineering Handbook, Keith Henney, 3rd edit, 6th impression,
>>> McGraw-Hill 1941. p 93. - AMF]
> I haven't seen this book. Can you tell us anything more about its
> usefulness. Recommended?
Not recommended. If someone gave you a copy you may want to
keep it. I keep mine more for sentimental reasons. It was one
of the first Technical books I bought. It cost me A$1.00 (=
US$2.00 at the time) about 1946. I was studying first year
university at the time, and earning A$6.02 per week so even a
dollar was expensive.
He makes a few comments you may find of interest.
"Bank Winding" is one result of the attempts to devise a
multilayer coil with relatively low internal capacity. The
turns are wound in the order shown in the following
cross-section.
6 9 12 15 18 21 24 27
5 4 8 11 14 17 20 23 26
1 2 3 7 10 13 16 19 22 25
Henney gives a series of formulae for inductance from
Circular 74, including a straight round wire, a pair of straight
round wires, a square of round wire, a rectangle of round wire,
a grounded horizontal wire, a circular ring of circular
cross-section, a single layer solenoid, and a range of
multi-layer coils. I would say at a guess that the formulae are
a direct copy from Circ. 74. Incidentally the "Inductance"
chapter was written by Gomer L. Davies B.S., Engineer,
Washington Institute of Techology, Washington D.C.
One more comment, from the chapter on "Electrical
Measurements" written by R.F. Field (really!) and John H.
Miller. They say "The self and mutual inductance air cored
coils ..... can be calculated from their dimension
with an accuracy of better than 2 parts in 1000,000."
Unfortunately, they do not say how.
>
>> # M.R. Esnault-Pelterie, "On the Co-efficient of Self Inductance
>> of Circular Cyldrical Coils", Comptes Rendus, Tome 205, No. 18
>> p. 762, November 3, (1937) and No. 20, p. 885, November 15, (1937)
> Ah, Comptes Rendus, the famous French philosophical journal. OK, we
> can get this from the MIT library - no doubt the mother-lode is here!
>> I think we will be lucky to get any better results unless we can
>> get hold of a copy of the original Bureau of Standards Circular
>> 74 (1924) [Which makes it just 3 years older than me]
> Having read through this document several times in August, I don't
> think we'll find the history for Wheeler's equation there. However,
> if I can find the time at work, I'll recheck (or maybe remember to
> bring it home!). Rather, the answer must surely like with Wheeler
> (who isn't telling) or perhaps from the writings in Comptes Rendus.
I have a feeling that you, or someone else, said that some
of the formulae in Circ. 74 were in the form of series. Is this
correct?
regards, Alan.
Winfield Hill
Alan Fowler, <amfo...@melbpc.org.au> said...
>>>> the small sized graph. K [Henney} is taken from a table in The
>>>> Radio Engineering Handbook, Keith Henney, 3rd edit, 6th impression,
>>>> McGraw-Hill 1941. p 93. - AMF]
>
>> usefulness. Recommended?
>
> Not recommended. If someone gave you a copy you may want to
> keep it. I keep mine more for sentimental reasons. It was one
> of the first Technical books I bought. It cost me A$1.00 (=
> US$2.00 at the time) about 1946. I was studying first year
> university at the time, and earning A$6.02 per week so even a
> dollar was expensive.
>
> He makes a few comments you may find of interest.
>
> "Bank Winding" is one result of the attempts to devise a
> multilayer coil with relatively low internal capacity. The
> turns are wound in the order shown in the following
> cross-section.
>
> 6 9 12 15 18 21 24 27
> 5 4 8 11 14 17 20 23 26
> 1 2 3 7 10 13 16 19 22 25
That's interesting. Yes, Terman has a bank winding drawing as well,
(fig 53 b) but with only two layers. You can see the difficulty
this scheme has in getting from 6 to 7, 9 to 10, etc., especially
with fat wire. One can suffer the resulting lump, not turn after
turn at the same spot... OK, rotate the lump back at bit at each
turn, etc.
Eyeing this schem, and realizing the separation of turns 3 to 6 from
7 to 9, etc is important to reduce capacity, I came up with my sort
of combo layer-winding, bank-winding, universal-winding scheme. Do
you remember this drawing?
|<----- 1.0" ----->| cross-section views
.-. .-. .-. .-.
| || || || |
'-' '-' '-' '-' bundle wire order
top \ o o o o o o o o 4 5 4 5
mid \ o \ o \ o \ o \ 3 3
bottom o o o o o o o o 1 2 1 2
The idea is to separate the banks, making room for the top-to-bottom
wire, and reducing capacitance and proximity effects as well. Then
to get the inductance back up to our goal, a small bunch of 5 turns
is used in each bank.
I have expanded upon this scheme for a new coil, which has achieved
a Q of over 1000. Watch for more about this in an upcoming thread
"The Great Litz-Coil Shootout."
> One more comment, from the chapter on "Electrical
> Measurements" written by R.F. Field (really!) and John H.
> Miller. They say "The self and mutual inductance air cored
> coils ..... can be calculated from their dimension
> with an accuracy of better than 2 parts in 1000,000."
> Unfortunately, they do not say how.
Hah! They get that from all the infinite-series formlas and 6-place
tables in the B.S. publications, plus Grover's book etc. However,
I suspect this simply means one can accurately calculate with the
tables, etc, a 5 or 6 figure number matching the long current-sheet
formulas calculations! As far as real world experimental results...
Grover does some example calculations showing great accuracy. For
example, when discussing the mutual inductance of loosly-coupled coils,
he points out )page 135) that one of the applicable formulas (108) has
nearly equal terms which are subtracted from each other. He improves
the situation by deriving a new formula using differences (112) and
calculates M = 1.0862 uH. Then he says this formula is accurate,
since the "true value of the mutual inductance in this case, as found
by the most accurate absolute formulas (here he references a 1912 B.S.
paper) is about M = 1.0865 uH.
So we see that he's checking the results to an amazing 0.0003 uH on
paper!! Can you imagine these things have ever been checked in the
real world to 0.01% = 0.3nH?
>>> # M.R. Esnault-Pelterie, "On the Co-efficient of Self Inductance
>>> of Circular Cyldrical Coils", Comptes Rendus, Tome 205, No. 18
>>> p. 762, November 3, (1937) and No. 20, p. 885, November 15, (1937)
>
>> Ah, Comptes Rendus, the famous French philosophical journal. OK, we
>> can get this from the MIT library - no doubt the mother-lode is here!
>
>>> I think we will be lucky to get any better results unless we can
>>> get hold of a copy of the original Bureau of Standards Circular
>>> 74 (1924) [Which makes it just 3 years older than me]
>
>> Having read through this document several times in August, I don't
>> think we'll find the history for Wheeler's equation there. However,
>> if I can find the time at work, I'll recheck (or maybe remember to
>> bring it home!). Rather, the answer must surely like with Wheeler
>> (who isn't telling) or perhaps from the writings in Comptes Rendus.
>
> I have a feeling that you, or someone else, said that some
> of the formulae in Circ. 74 were in the form of series. Is this
> correct?
Yes. Well at least many of the formulas referenced there are originally
infinite series. However, many of them are quite useful with only a few
terms.
I have the 1937 Comptes Rendus paper in my hands now, and after a quick
scan, I can say that he does seem to have done some original work.
However, the papers were apparently two lectures at the Academie des
Sciences, and the printed version has ZERO references.
One interesting note, the session begins with the Academie president's
"memoires et communications" in which he says the dissapearance of
Amelia Earhardt and the importance of transatlantic communications,
blah blah and the study of the inductor is very important for
communications, and blah blah Air France ... [I'll have to ask.]
Back to Esnault-Pelterie's paper.
He starts off talking about the formula with the famous K (you know the
one with the K which includes everything, maybe even the kitchen sink!),
and he writes his first equation for L in the limit,
L_lim = (pi D N)^2 / B
and then says, well
L = K L_lim
and that K is just a dimensionless coefficient that Nagaoka and Sakuri
have calculated using elliptical functions taken from the Lorentz
formula.
Then he says, let's assume 1/K = 1 + lambda, and he makes reference to
the table created by the "Japanese wizards" for K, which is a function
of D and B (B is the coil's length), and says he noticed that lambda
can almost be called D/B times another constant. So one can write,
L = L_lim / ( 1 + X D/B )
and then he says, without any introduction,
"In taking X = 0.452, the values given by this formula follow
those of Nagoaka and Sakurai with a precision of 3%, and, for
extension to infinity, you have, L_real = L_lim."
He says this is as good as the wizards result, "des savants japonais".
[ Note, here he's really saying K = 1 / 1 + 0.452 D/B ]
Then he immediately says, suppose we have instead
k = K = 1 / ( a + b D/B )
and a = 0.9949 and b = 0.4572, and then he has a table of results
showing that the error is then half a thousandth. This is all just
pulled out of a hat (very similar to Wheeler's article with his
equation and numbers), and with no mention whatsoever of Wheeler,
who published his equation 9 years earlier, and was certainly
VERY well known by that time. Even in France.
Then he says,
"This is a remarkable result, because in practise, I have found
the use of this formula preferable in every respect to the use
of a table or a diagram."
Indeed. Finally he writes,
pi^2 D^2 N^2
L = ------------ a = 0.9949
a + b D / B b = 0.4572
which as we see from the Radiotron Designer's Handbook exercise, is
just Wheeler with a slight adjustment in the values.
So in conclusion, we have to say that M. Robert Esnault-Pelterie
performed an empirical match to the B.S. Nagaoka formula and tables,
just as Wheeler said he did 9 years earlier (likely even using his
formula as a starting point, no?), not exactly a derivation, per se.