Accuracy of the popular Wheeler inductance equation.
Winfield Hill
A revered formula for calculating inductance recently came under attack.
I'm referring to the Wheeler formula and a discussion on 17 Aug 1997 in
the thread "Re: Calculate large loop inductance ??". Let's explore this.
Mike <AM...@worldnet.att.net> said,
>> Hi Jim,
>> Try A squared times N squared divided by 9A + 10B = Inductance
>> in Microhenries.
>> Where A = radius of the loop
>> B = length of the loop
>> N = the number of turns
>> Hope that helps
>> Mike
and José <jose...@aol.com> (JoseSainz) replied,
> This formula is very wrong for loops whose length is not equivalent to
> its diameter. There is another that uses a current sheet (sorry left it
> at work) with NBS correction factors for the various shape factors. If
> someone is interested I can dig up the basic routine I used in a design
> based around this formula.
Yes, Jose, it's very easy to underestimate the usefulness of Wheeler's
simple equation. To see the complex world that awaits one seeking
accurate analytical inductance-calculation results, we have only to glance
at the hairy formulas in Grover's classic book "Inductance Calculations"
and read his preface comment that this book was to simplify our lives!
It was into this hairy world that Wheeler introduced a simple equation for
inductance calculations (Proc IRE, 16, p. 1398, 1966).
It's easy to see why some might be critical. But Wheeler's equation has
stood the test of time, and appears in every major work from Terman to the
ITT Reference Data for Radio Engineers to the ARRL Handbook (but not in
Grover, Giacoletto or Physics Vade Mecum). Also, most any question about
inductance calculations on sci.electronics will turn up this equation in
one of it's several forms. [Hah! I should rest my case right there!]
So one should tread very carefully in criticizing this equation.
Jose says, "This formula is very wrong for loops whose length is not
equivalent to its diameter."
This is incorrect. Sure, the length/diameter ratio is very important.
The equation can written to highlight the parameter D/L, and shows the
role Wheeler intended:
D N^2 D^2 N^2 / 40 L
L = ----------- = -------------- uH / inch
18 + 40 L/D 1 + 0.45 D/L
Not to be guilty of mere rhetoric, here's the meat: Measurements from 23
different single-layer coils with inductances ranging from 2.1 to 195uH.
Significantly, they have length/diameter ratios from 0.68 to 6.25 ( D/L =
diameter/length from 0.16 to 1.5), from 12 to 100 turns, winding pitches
from 4 to 33/inch, overall lengths from 1.5 to 10 inches, and diameters
from 0.55 to 6.3 inches. Each coil is analyzed with the Wheeler equation,
and compared to the measured inductance.
I've further rearranged the Wheeler equation to a form showing a winding
pitch parameter, p turns/inch.
D^2 p N 1
L = -------- * ----------------
40 1 + 0.45 D p / N
This makes clear that for long coils, once you pick a coil-winding pitch,
the inductance scales by N, rather than by N^2. Of course, the length
scales as well. My "winding-pitch" version of the Wheeler equation is
used in the spreadsheet below (the 1st term is divided by the 2nd).
measured Wheeler eqn. calculated
L/D turns/in inductance 1st-term 2nd- inductance errors
ratio dia length pitch turns (uH) 0.1Npr^2 term uH uH %
1+0.9rp/N
0.68 2.57 1.75 8.00 14 10.80 18.49 1.66 = 11.14 0.34 3.1%
0.80 3.00 2.40 8.00 20 22.00 36.00 1.54 = 23.38 1.38 6.3%
0.88 3.07 2.70 13.90 38 79.35 122.82 1.51 = 81.22 1.87 2.4%
1.38 1.07 1.48 8.14 12 2.11 2.80 1.33 = 2.11 -0.00 -0.1%
1.60 3.00 4.80 8.00 39 56.60 70.20 1.28 = 54.98 -1.62 -2.9%
1.64 3.05 5.00 4.00 20 15.50 18.61 1.27 = 14.60 -0.90 -5.8%
1.67 3.00 5.00 8.00 40 59.60 72.00 1.27 = 56.69 -2.91 -4.9%
1.76 3.07 5.40 13.90 75 195.00 245.64 1.26 = 195.56 0.56 0.3%
1.97 2.00 3.93 14.00 55 61.50 77.00 1.23 = 62.65 1.15 1.9%
2.38 2.10 5.00 6.00 30 16.00 19.85 1.19 = 16.69 0.69 4.3%
2.76 1.07 2.95 8.14 24 4.78 5.59 1.16 = 4.81 0.03 0.6%
2.81 1.05 2.95 16.33 48 17.30 21.60 1.16 = 18.61 1.31 7.6%
2.78 1.80 5.00 10.00 50 34.70 40.50 1.16 = 34.85 0.15 0.4%
3.13 1.60 5.00 4.00 20 4.50 5.12 1.14 = 4.48 -0.02 -0.5%
3.28 3.05 10.00 4.00 40 35.20 37.21 1.14 = 32.72 -2.48 -7.0%
3.33 3.00 10.00 8.00 80 134.00 144.00 1.14 = 126.87 -7.13 -5.3%
3.49 0.55 1.92 33.00 63 14.10 15.72 1.13 = 13.92 -0.18 -1.3%
4.76 2.10 10.00 6.00 60 35.00 39.69 1.09 = 36.26 1.26 3.6%
5.56 1.80 10.00 10.00 100 75.20 81.00 1.08 = 74.93 -0.27 -0.4%
6.25 1.60 10.00 4.00 40 9.80 10.24 1.07 = 9.55 -0.25 -2.5%
1.17 6.30 7.40 3.38 25 57.00 83.80 1.38 = 60.59 3.59 6.3%
1.20 4.25 5.09 5.50 28 45.00 69.55 1.38 = 50.56 5.56 12.3%
1.71 3.50 6.00 5.67 34 41.00 59.00 1.26 = 46.74 5.74 14.0%
The last three coils were made with edge-wound copper strip.
Taking all the wire coils as a class, the mean error is -0.02%, while the
rms error is 3.9%. The maximum errors are -7.0% to +7.6% (note that the
two coils with these extremes both have a length/diameter ratio near 3.0).
Examining scatter-plots of error vs. length/diameter, pitch, inductance,
length or diameter shows no systematic relationship. I even checked error
vs. measurement order! Sorry, no correlation is evident in any instance!
Any argument that the Wheeler equation requires diameter near length
surely isn't supported by these measurements.
Terman says "This formula is accurate to within 1% for l > 0.8r, i.e. if
the coil is not too short" (Radio Engineer's Handbook, p 55). The formula
appears to match the various published curves based on the NBS data.
These generally use a parameter F in
L = F d N^2.
Wheeler's equation can be comparing to these curves (and the NBS data) by
observing:
F (Wheeler) = 1 / (18 + 40 L/D)
This shows that his equation deviating from the curves below L/D = 0.4
as Terman says. For L/D = 0.1, a very short coil indeed, it's about 10%
low. However, the NBS-data curves are for a theoretical flat sheet of
current, IIRC, and even their accuracy often falls with very short coils.
Jose, when you said, "length equivalent to diameter," implying a diameter
equals length requirement, and when you said, "The Wheeler formula was not
very correct for the first turns as it is accurate only when diameter and
length approach each other," perhaps it was the L/D > 0.4 limitation
against short-coils which you had in mind, and not a diameter = length
requirement. BTW, compared to the BASIC equation you later posted, the
Wheeler formula is 0.57% low for L/D = 0.4 improving to -0.1% for L = D
and up. So if you like your program, Jose, you have to like the Wheeler
equation!
One missing parameter in the Wheeler equation is no doubt wire diameter.
For example, I have observed that coils of with identical wire size and
spacing, but varying length, all have the same error. Also note, the
three coils made with 1/4 or 3/8-inch edge-wound copper strip (e.g. with
an unusual 5:1 shape factor and 2.8:1 spacing), are 6 to 14% high.
--
Winfield Hill hi...@rowland.org _/_/_/ _/_/_/_/
The Rowland Institute for Science _/ _/ _/_/ _/
Cambridge, MA USA 02142-1297 _/_/_/_/ _/ _/ _/_/_/
_/ _/ _/ _/ _/
http://www.artofelectronics.com/ _/ _/ _/_/ _/_/_/_/
JoseSainz
Subj: Accuracy of the popular Wheeler inductance equation.
Date: 97-08-30 15:50:36 EDT
From: hi...@rowland.org (Winfield Hil)
To: t...@netcom.com, jose...@aol.com, hi...@rowland.org
See end José
Mike <AM...@worldnet.att.net> said,
0.68 2.57 1.75 8.00 14 10.80 18.49 1.66 = 11.14 0.34 3..1%
0.80 3.00 2.40 8.00 20 22.00 36.00 1.54 = 23.38 1.38 6..3%
0.88 3.07 2.70 13.90 38 79.35 122.82 1.51 = 81.22 1.87 2..4%
1.38 1.07 1.48 8.14 12 2.11 2.80 1.33 = 2.11 -0.00 -0..1%
1.60 3.00 4.80 8.00 39 56.60 70.20 1.28 = 54.98 -1.62 -2..9%
1.64 3.05 5.00 4.00 20 15.50 18.61 1.27 = 14.60 -0.90 -5..8%
1.67 3.00 5.00 8.00 40 59.60 72.00 1.27 = 56.69 -2.91 -4..9%
1.76 3.07 5.40 13.90 75 195.00 245.64 1.26 = 195.56 0.56 0..3%
1.97 2.00 3.93 14.00 55 61.50 77.00 1.23 = 62.65 1.15 1..9%
2.38 2.10 5.00 6.00 30 16.00 19.85 1.19 = 16.69 0.69 4..3%
2.76 1.07 2.95 8.14 24 4.78 5.59 1.16 = 4.81 0.03 0..6%
2.81 1.05 2.95 16.33 48 17.30 21.60 1.16 = 18.61 1.31 7..6%
2.78 1.80 5.00 10.00 50 34.70 40.50 1.16 = 34.85 0.15 0..4%
3.13 1.60 5.00 4.00 20 4.50 5.12 1.14 = 4.48 -0.02 -0..5%
3.28 3.05 10.00 4.00 40 35.20 37.21 1.14 = 32.72 -2.48 -7..0%
3.33 3.00 10.00 8.00 80 134.00 144.00 1.14 = 126.87 -7.13 -5..3%
3.49 0.55 1.92 33.00 63 14.10 15.72 1.13 = 13.92 -0.18 -1..3%
4.76 2.10 10.00 6.00 60 35.00 39.69 1.09 = 36.26 1.26 3..6%
5.56 1.80 10.00 10.00 100 75.20 81.00 1.08 = 74.93 -0.27 -0..4%
6.25 1.60 10.00 4.00 40 9.80 10.24 1.07 = 9.55 -0.25 -2..5%
1.17 6.30 7.40 3.38 25 57.00 83.80 1.38 = 60.59 3.59 6..3%
1.20 4.25 5.09 5.50 28 45.00 69.55 1.38 = 50.56 5.56 12..3%
1.71 3.50 6.00 5.67 34 41.00 59.00 1.26 = 46.74 5.74 14..0%
The last three coils were made with edge-wound copper strip.
Taking all the wire coils as a class, the mean error is -0.02%, while the
rms error is 3.9%. The maximum errors are -7.0% to +7.6% (note that the
two coils with these extremes both have a length/diameter ratio near 3.0)..
I dont dispute Wheeler's formula, as around my work we are satisfied with
a wound coil that is within 10% of the calculated value using Wheeler.
Typically solenoids we construct have at least a 1:1 Length to Diameter
ratio. More usually a 2:1 or 3:1 is common.
BTW Winfild, I noticed that you considered your calulations where lower
than meassured. Did you use the middle of the conductor to calculate the
coil diameter?
What I cautioned was dependence on that formula when calculating the
inductance of the first turns. In radio work it is common to used tapped
coils to adjust resonance. In MW work where frequencies span a 3:1 range
it is common to use 1 coil to cover the entire band in tuners. That
results in using only a few of the many turns at the top end of the band.
Any way what caused me to look for a more accurate formula, or better said
correction factors was a case where I had to match an existing meassured
coil with calculations. This coil then had to be modified. Actually it
was a quite complicated arrangement. This was a nested 3 coil arrangement
where the inner coil was shorted, similar to a slug tuner. When fitting my
calculation to the meassured values I noticed large errors using an
"uncorrected formula". The same NBS publication I was using for the
equations for mutual coupling of nested inductors also had a table of
correction factors which I incorporated in the equation I posted.
Here is a comparison of calculations. Radius is 40". Length is Turns*.5" .
NT L/Dia Wheeler NBS error%
1 0.01 4.38 7.02 37.6
2 0.01 17.30 26.89 35.7
3 0.02 38.40 57.44 33.1
4 0.03 67.37 78.04 13.7
5 0.03 103.90 129.64 19.9
6 0.04 147.69 191.83 23.0
7 0.04 198.48 251.52 21.1
8 0.05 256.00 317.30 19.3
9 0.06 320.00 389.56 17.9
10 0.06 390.24 467.54 16.5
11 0.07 466.51 551.52 15.4
12 0.08 548.57 640.37 14.3
13 0.08 636.24 734.14 13.3
14 0.09 729.30 833.17 12.5
15 0.09 827.59 935.89 11.6
16 0.10 930.91 1043.80 10.8
17 0.11 1039.10 1156.47 10.1
18 0.11 1152.00 1273.69 9.6
19 0.12 1269.45 1321.89 4.0
20 0.13 1391.30 1518.47 8.4
Here is a comparison of calculations. Radius is 40". Length is Turns*1.0" .
NT L/Dia Wheeler NBS error%
1 0.01 4.32 6.72 35.7
2 0.03 16.84 19.51 13.7
3 0.04 36.92 47.96 23.0
4 0.05 64.00 79.32 19.3
5 0.06 97.56 116.89 16.5
6 0.08 137.14 160.09 14.3
7 0.09 182.33 208.29 12.5
8 0.10 232.73 260.95 10.8
9 0.11 288.00 318.42 9.6
10 0.13 347.83 379.62 8.4
11 0.14 411.91 444.87 7.4
12 0.15 480.00 513.62 6.5
13 0.16 551.84 585.82 5.8
14 0.17 627.20 661.27 5.2
15 0.19 705.88 739.82 4.6
16 0.20 787.69 820.97 4.1
17 0.21 872.45 905.16 3.6
18 0.22 960.00 991.91 3.2
19 0.24 1050.18 1080.95 2.8
20 0.25 1142.86 1172.54 2.5
Here is a comparison of calculations. Radius is 40". Length is Turns*2.0" .
NT L/Dia Wheeler NBS error%
1 0.03 4.21 4.88 13.7
2 0.05 16.00 19.83 19.3
3 0.08 34.29 40.02 14.3
4 0.10 58.18 65.24 10.8
5 0.13 86.96 94.90 8.4
6 0.15 120.00 128.41 6.5
7 0.17 156.80 165.32 5.2
8 0.20 196.92 205.24 4.1
9 0.22 240.00 247.98 3.2
10 0.25 285.71 293.14 2.5
11 0.28 333.79 340.64 2.0
12 0.30 384.00 390.14 1.6
13 0.32 436.13 441.60 1.2
14 0.35 490.00 494.75 1.0
15 0.38 545.45 549.53 0.7
16 0.40 602.35 605.72 0.6
17 0.43 660.57 663.35 0.4
18 0.45 720.00 722.19 0.3
19 0.47 780.54 782.11 0.2
20 0.50 842.11 843.15 0.1
What the above shows is that at few turns the Wheeler formula without
corrections has large errors. The NBS formula when fitted to the existing
coils I was working on had an error of <3% for the first few turns and .1%
or so when diameter and length where 1:1. This really showed up in my
application as the turns did not have a constant pitch. At each turn the
wire "jumped" to the next winding slot. The last example was similar to
part of the coil I was working on.
Well I hope this explains more exactly my comment.
José
brian whatcott
In article <5u9tff$g...@fridge-nf0.shore.net>, hi...@rowland.org says...
>
>A revered formula for calculating inductance recently came under
attack.
>I'm referring to the Wheeler formula and a discussion on 17 Aug 1997
in
>the thread "Re: Calculate large loop inductance ??". Let's explore this.
>
OK!
>... José <jose...@aol.com> (JoseSainz) replied,
>
>> This formula is very wrong for loops whose length is not equivalent
to
>> its diameter....
>Yes, Jose, it's very easy to underestimate the usefulness of Wheeler's
>simple equation.
Yes, Winfield, and it's very easy to overestimate the accuracy of this
empirical
formula if you take many spreadsheet values where it gives reasonable
results,
and only one in the range where it starts to go badly wrong.where l/d is
much
less than 0.8 as I see from the table below.
> L/D errors
>ratio %
>
> 0.68 3.1%
> 0.80 6.3%
> 0.88 2.4%
> 1.38 -0.1%
> 1.60 -2.9%
> 1.64 -5.8%
> 1.67 -4.9%
> 1.76 0.3%
> 1.97 1.9%
> 2.38 x 4.3%
> 2.76 0.6%
> 2.81 x 7.6%
> 2.78 0.4%
> 3.13 -0.5%
> 3.28 x -7.0%
> 3.33 -5.3%
> 3.49 -1.3%
> 4.76 3.6%
> 5.56 x -0.4%
> 6.25 -2.5%
>
> 1.17 6.3%
> 1.20 x 12.3%
> 1.71 14.0%
>...
>
>Taking all the wire coils as a class, the mean error is -0.02%, while the
>rms error is 3.9%. The maximum errors are -7.0% to +7.6% (note
that the
>two coils with these extremes both have a length/diameter ratio near
3.0)....
>Any argument that the Wheeler equation requires diameter near length
>surely isn't supported by these measurements.
Ho hum....
>
>Jose, when you said, "length equivalent to diameter," implying a
diameter
>equals length requirement, and when you said, "The Wheeler formula
was not
>very correct for the first turns as it is accurate only when diameter and
>length approach each other," perhaps it was the L/D > 0.4 limitation
>against short-coils which you had in mind, and not a diameter = length
>requirement....
Bingo!
>Winfield Hill
(Why do I keep picking on you?
If *I* don't who will???)
:-)
Best wishes
Brian
John Woodgate
In article <19970831053...@ladder01.news.aol.com>, JoseSainz
<jose...@aol.com> writes
Now, I know you don't actually say that Wheeler _introduced_ the
equation _in the cited reference_, but in fact he introduced it a great
deal earlier, possibly in Prc.IRE 16.10 of October 1928 (A.D.). There is
a large chapter (Chapter 10) in 'Radio Designer's Handbook' (or
'Radiotron ...'), by F. Langford-Smith, on all aspects of calculation of
self- and mutual inducatnce, and the effects of screening, with 56
references and three more in the Supplement. This book has been out of
print since about 1960, but in May this year Mark Zenier posted here
that Old Colony Sound Labs (http://www.audioexpress.com) claim to have
the 4th Ed. on CD-ROM, although he had trouble confirming that.
>
>It's easy to see why some might be critical. But Wheeler's equation has
>stood the test of time, and appears in every major work from Terman to the
>ITT Reference Data for Radio Engineers to the ARRL Handbook (but not in
>Grover, Giacoletto or Physics Vade Mecum). Also, most any question about
>inductance calculations on sci.electronics will turn up this equation in
>one of it's several forms. [Hah! I should rest my case right there!]
True, but it has been copied from one to the other, not necessarily
_verified_ as you have done.
>
>So one should tread very carefully in criticizing this equation.
>
>Jose says, "This formula is very wrong for loops whose length is not
>equivalent to its diameter."
>
>This is incorrect. Sure, the length/diameter ratio is very important.
>The equation can written to highlight the parameter D/L, and shows the
>role Wheeler intended:
>
> D N^2 D^2 N^2 / 40 L
> L = ----------- = -------------- uH / inch
> 18 + 40 L/D 1 + 0.45 D/L
>
Mind your L's (;-).
[lotsofgooddatawithregretsnip]
For VERY short coils, I think my 1.6N^2 uH/metre of perimeter (1.6 can
go up to 2 for single-turn coils of several metres dimension in
buildings in practice) is a good guide. Curiously, this is around twice
as much as the Wheeler formula gives if you blatantly misuse it by
setting length = 0. However, Langford-Smith quotes a 'Wheeler's formula
for short coils' (equation 16) which shows the inductance tending
logarithmically to infinity as the length and cross-sectional area of
the coil tend to zero! So perhaps it is not so curious after all!
--
Regards, John Woodgate, Elector of Rayleigh. Phone +44 (0)1268 747839
Fax +44 (0)1268 777124. OOO - Own Opinions Only. You can fool some of
the people all of the time, but you can't please some of the people
any of the time.
Winfield Hill
brian whatcott, <in...@intellisys.net> said...
>
> hi...@rowland.org says...
>>
>> A revered formula for calculating inductance recently came under
>>
>>... José <jose...@aol.com> (JoseSainz) replied,
>>
>>> This formula is very wrong for loops whose length is not equivalent
>>
>> Yes, Jose, it's very easy to underestimate the usefulness of Wheeler's
>> simple equation.
> Yes, Winfield, and it's very easy to overestimate the accuracy of this
> empirical formula if you take many spreadsheet values where it gives
> reasonable results, and only one in the range where it starts to go
> badly wrong.where l/d is much less than 0.8 as I see from the table
> below.
>> L/D errors
>> ratio %
>>
>> 0.68 3.1%
>> 0.80 6.3% <- wh: Is this the value you're pinpointing? I'd
>> 0.88 2.4% like to point out that in this region, Wheeler
>> 1.38 -0.1% still matches very closely the NBS curves.
>> 1.60 -2.9%
>> 1.64 -5.8% <----
>> 1.67 -4.9% <----|
>> 1.76 0.3% |
>> 1.97 1.9% |
>> 2.38 x 4.3% <----|
>> 2.76 0.6% |<==== Brian, tell me what you make of all
>> 2.81 x 7.6% <----| these errors? Note they are positive
>> 2.78 0.4% | as well as negative.
>> 3.13 -0.5% |
>> 3.28 x -7.0% <----|
>> 3.33 -5.3% <----
>> 3.49 -1.3%
>> 4.76 3.6%
>> 5.56 x -0.4%
>> 6.25 -2.5%
>> ...
>>
>> Taking all the wire coils as a class, the mean error is -0.02%, while
>> the rms error is 3.9%. The maximum errors are -7.0% to +7.6% (note
>> that the two coils with these extremes both have a length/diameter
>> ratio near 3.0)....
>> Any argument that the Wheeler equation requires diameter near length
>> surely isn't supported by these measurements.
>
> Ho hum....
>
>> Jose, when you said, "length equivalent to diameter," implying a
>> diameter equals length requirement, and when you said, "The Wheeler
>> formula was not very correct for the first turns as it is accurate
>> only when diameter and length approach each other," perhaps it was
>> the L/D > 0.4 limitation against short-coils which you had in mind,
>> and not a diameter = length requirement....
>
> Bingo!
>
> (Why do I keep picking on you?
>
> If *I* don't who will???)
Hah! Thanks, Brian, I needed that. I sometime tend to over-dramatize,
and am often looking for an excuse. Frankly, I couldn't tell if Jose
was really picking on Wheeler or not.
But when Jose posted his BASIC subroutine, which was a (presumably
accurate) form of the NBS sheet formulations, as an alternate to Wheeler,
I had to comment. This is because over this range we're discussing,
Wheeler very closely matches BNS and therefore the program isn't a
significant replacement at all!
He has posted some data in support of his argument, which I will study.
You implied that my data shows Wheeler falling apart below L/D = 1,
which isn't apparent to me, but at any rate, this can't be any more so
than saying the NBS curves are falling apart. In practise, I think both
are pretty useful to L/D below say 0.3, and below that all the simple
formulas get into trouble.
OK, Brian, take your best shot!
Winfield Hill
JoseSainz, <jose...@aol.com> said...
>
>Subj: Accuracy of the popular Wheeler inductance equation.
>Date: 97-08-30 15:50:36 EDT
>From: hi...@rowland.org (Winfield Hil)
-> A revered formula for calculating inductance recently came under attack.
-> I'm referring to the Wheeler formula and a discussion on 17 Aug 1997 in
-> the thread "Re: Calculate large loop inductance ??". Let's explore this.
->
-> JoseSainz, <jose...@aol.com> said..
->> This formula is very wrong for loops whose length is not equivalent to
->> its diameter. ...
->
-> This is incorrect. Sure, the length/diameter ratio is very important.
-> The equation can written to highlight the parameter D/L, and shows the
-> role Wheeler intended:
->
-> D N^2 D^2 N^2 / 40 L
-> L = ----------- = -------------- uH / inch
-> 18 + 40 L/D 1 + 0.45 D/L
->
-> Not to be guilty of mere rhetoric, here's the meat: Measurements from 23
-> different single-layer coils with inductances ranging from 2.1 to 195uH.
-> Significantly, they have length/diameter ratios from 0.68 to 6.25 ( D/L =
-> diameter/length from 0.16 to 1.5), from 12 to 100 turns, winding pitches
-> from 4 to 33/inch, overall lengths from 1.5 to 10 inches, and diameters
-> from 0.55 to 6.3 inches. Each coil is analyzed with the Wheeler equation,
-> and compared to the measured inductance.
->
-> I've further rearranged the Wheeler equation to a form showing a winding
-> pitch parameter, p turns/inch.
->
-> D^2 p N 1
-> L = -------- * ----------------
-> 40 1 + 0.45 D p / N
->
-> [ snip results ]
->
-> Taking all the wire coils as a class, the mean error is -0.02%, while the
-> rms error is 3.9%. The maximum errors are -7.0% to +7.6% (note that the
-> two coils with these extremes both have a length/diameter ratio near 3.0).
-> Examining scatter-plots of error vs. length/diameter, pitch, inductance,
-> length or diameter shows no systematic relationship. I even checked error
-> vs. measurement order! Sorry, no correlation is evident in any instance!
->
-> Any argument that the Wheeler equation requires diameter near length
-> surely isn't supported by these measurements. [snip]
->
-> Wheeler's equation can be comparing to these curves (and the NBS data) by
-> observing:
->
-> F (Wheeler) = 1 / (18 + 40 L/D)
->
-> [snip]
->
-> Jose, when you said, "length equivalent to diameter," implying a diameter
-> equals length requirement, and when you said, "The Wheeler formula was not
-> very correct for the first turns as it is accurate only when diameter and
-> length approach each other," perhaps it was the L/D > 0.4 limitation
-> against short-coils which you had in mind, and not a diameter = length
-> requirement. BTW, compared to the BASIC equation you later posted, the
-> Wheeler formula is 0.57% low for L/D = 0.4 improving to -0.1% for L = D
-> and up. So if you like your program, Jose, you have to like the Wheeler
-> equation!
->
-> One missing parameter in the Wheeler equation is no doubt wire diameter.
-> For example, I have observed that coils of with identical wire size and
-> spacing, but varying length, all have the same error. [snip]
> I don't dispute Wheeler's formula, as around my work we are satisfied with
> a wound coil that is within 10% of the calculated value using Wheeler.
> Typically solenoids we construct have at least a 1:1 Length to Diameter
> ratio. More usually a 2:1 or 3:1 is common.
Jose, you miss my point, which is that the Wheeler formula is basically just
the same as the sheet formulas for all the pratical ranges we're discussing.
In fact, to set aside its usefulness even below L/D = 0.4 just because its
values deviate from the NBS sheet curves by -0.6% really begs a more
significant issue, which is that BOTH of these methods are really over-
simplifications (e.g. ignoring coil height) and in my (limited) experience
don't seem to fare all that well with real coils (e.g. the 3.9% rms error
I observed was really the error from the NBS formulation, since within this
range Wheeler matches closely). So if in the "well-behaved" operating
region, the NBS curves are off by 4% rms, 7% max, why quibble over a
systematic -0.6% at one extreme?
Furthermore, in my experience with very short coils, as L/D is reduced to
0.1 the measured inductance fall well under the NBS-calculated values by 20
to 30% or even more, so the -10% Wheeler answer in this region more closely
approximates a real coil. I'll have more on that later, and also look
forward to examining your data below.
> BTW Winfield, I noticed that you considered your calulations where lower
> than meassured. Did you use the middle of the conductor to calculate the
> coil diameter?
Yes, I measured from the middle of the conductor. For the 20 wire coils,
which had L/D from 0.68 to 6.25, the Wheeler calculations where NOT lower
than measssured; the mean error was -0.02%. Also, the largest deviations
were balanced at -7.0% and +7.6%.
> What I cautioned was dependence on that formula when calculating the
> inductance of the first turns. In radio work it is common to used tapped
> coils to adjust resonance. In MW work where frequencies span a 3:1 range
> it is common to use 1 coil to cover the entire band in tuners. That
> results in using only a few of the many turns at the top end of the band.
I agree with this caution, but do take issue with your stronger statement.
> Any way what caused me to look for a more accurate formula, or better said
> correction factors was a case where I had to match an existing meassured
> coil with calculations. ...
I would also like to find some better formulas for really short coils. I'll
sign off here and read the rest of your posting where you discuss this, and
examine the data later.
I do hope we're not wasting too much time on this! My wife wants to get me
away from the computer, and go play tennis. Meanwhile she started cleaning
out her dress closet. Oh! I should have said "cleaning up"!
James P. Meyer
On 31 Aug 1997, Winfield Hill wrote:
> > There is a large chapter (Chapter 10) in 'Radio Designer's Handbook'
> > (or 'Radiotron ...'), by F. Langford-Smith, on all aspects of
>
> Yes, another book I had access to as a kid in college in the early 60's.
> Too bad I didn't appreciate them at the time and buy my own copies.
> Anybody want to sell one?
I felt like a kid at Christmas when I found one at a local used
book store several years ago. The owner had no idea what it was so I got
it for 10 bucks. AoE, first edition, on the other hand was $28.15. Now
if I could just find an early edition of the Editor's and Engineer's
audio handbook, 1955 or there abouts, my life would be complete.
Jim
Winfield Hill
John Woodgate, <j...@jmwa.demon.co.uk> said...
>
> hi...@rowland.org (Winfield Hill) said,
>> It was into this hairy world that Wheeler introduced a simple
>> equation for inductance calculations (Proc IRE, 16, p. 1398, 1966).
>
> equation _in the cited reference_, but in fact he introduced it a
> great deal earlier, possibly in Prc.IRE 16.10 of October 1928 (A.D.).
Oops, I'm sorry, John. Yes, the 16th volume of the Proc IRE, which I
cited is actually dated October 1928, not 1966. Hah! A.D. indeed.
> There is a large chapter (Chapter 10) in 'Radio Designer's Handbook'
> (or 'Radiotron ...'), by F. Langford-Smith, on all aspects of
> calculation of self- and mutual inductance, and the effects of
> screening, with 56 references and three more in the Supplement.
> This book has been out of print since about 1960, but in May this
> year Mark Zenier posted here that Old Colony Sound Labs
> (http://www.audioexpress.com) claim to have the 4th Ed. on CD-ROM,
> although he had trouble confirming that.
Yes, another book I had access to as a kid in college in the early 60's.
Too bad I didn't appreciate them at the time and buy my own copies.
Anybody want to sell one?
>> The equation can written to highlight the parameter D/L, and shows
>> the role Wheeler intended:
>>
>> D N^2 D^2 N^2 / 40 L
>> L = ----------- = -------------- uH / inch
>> 18 + 40 L/D 1 + 0.45 D/L
>>
>
> Mind your L's (;-).
Hmm, many books use l (lower-case L), which looks too much like 1 (one).
Maybe a better character is z, the length in the z-axis! Then one
can use d, which I prefer to D anyway, except that D and d are often
used for coil dia and wire dia, respectively. But then "a" can be used
for wire radius, as physicists like. They also use R for coil radius.
But then some, like Terman, use "a" for _mean_ winding diameter. Pender
uses b for length, while Terman uses L_0 and "l" (lower-case L). Sigh.
> For VERY short coils, I think my 1.6 N^2 uH/metre of perimeter (1.6
> can go up to 2 for single-turn coils of several metres dimension in
> buildings in practice) is a good guide. Curiously, this is around twice
> as much as the Wheeler formula gives if you blatantly misuse it by
> setting length = 0. However, Langford-Smith quotes a 'Wheeler's formula
> for short coils' (equation 16) which shows the inductance tending
> logarithmically to infinity as the length and cross-sectional area of
> the coil tend to zero! So perhaps it is not so curious after all!
Very interesting.
JoseSainz
>Subject: Re: Accuracy of the popular Wheeler inductance equation.
>From: "James P. Meyer" <jim...@acpub.duke.edu>
>Date: Sun, 31 Aug 1997 20:08:44 -0400
>Message-id:
><Pine.SOL.3.91.970831...@godzilla1.acpub.duke.edu>
>
>On 31 Aug 1997, Winfield Hill wrote:
>
>> > (or 'Radiotron ...'), by F. Langford-Smith, on all aspects of
>>
>> Too bad I didn't appreciate them at the time and buy my own copies.
>> Anybody want to sell one?
>
>book store several years ago. The owner had no idea what it was so I got
>it for 10 bucks. AoE, first edition, on the other hand was $28.15. Now
>if I could just find an early edition of the Editor's and Engineer's
>audio handbook, 1955 or there abouts, my life would be complete.
>
> Jim
I totally agree about the old tomes. I graduated college in '79. So I
heard from the old hands about all these wonderful books filled with
knowledge. Some fortunately were passed on to others at my work. Some
other engineers obtained Taiwanese reprints. Unfortunately in my
generation all these books have long been out of print. The NBS formula's
I mentioned before were from a gentleman who before coming to my company
worked for the NBS, the book was his copy, deeded to another engineer my
age when he retired.
José
Fred E. Davis
On Sun, 31 Aug 1997 20:08:44 -0400, "James P. Meyer"
<jim...@acpub.duke.edu> wrote:
Concerning 'Radiotron Designer's Handbook' by F. Langford-Smith, ...
>On 31 Aug 1997, Winfield Hill wrote:
>> Yes, another book I had access to as a kid in college in the early 60's.
>> Too bad I didn't appreciate them at the time and buy my own copies.
>> Anybody want to sell one?
Heck no! I'm keeping mine!
> I felt like a kid at Christmas when I found one at a local used
>book store several years ago.
I bumped into a fellow (I won't mention his name; you'll see why!) who was
throwing away a copy! He felt it was old and out of date. I gladly gave it
a new home, and it has a place of respect on my reference shelf where it
will stay.
Winfield Hill
James P. Meyer, <jim...@acpub.duke.edu> said...
>
>On 31 Aug 1997, Winfield Hill wrote:
>
>> > (or 'Radiotron ...'), by F. Langford-Smith, on all aspects of
>>
>> Too bad I didn't appreciate them at the time and buy my own copies.
>> Anybody want to sell one?
>
> it for 10 bucks. AoE, first edition, on the other hand was $28.15.
Hmmm, at that ratio, I should be able to find one for about $20.
However, I'd pay more than that. How about a copy of AoE in trade,
anybody? Sort of transistors, ICs, etc. for tubes!
brian whatcott
In article <5ucb56$o...@fridge-nf0.shore.net>, hi...@rowland.org says...
>....
> You implied that my data shows Wheeler falling apart below L/D = 1,
> which isn't apparent to me, but at any rate, this can't be any more so
> than saying the NBS curves are falling apart. In practise, I think both
> are pretty useful to L/D below say 0.3, and below that all the simple
> formulas get into trouble.
>
> OK, Brian, take your best shot!
>
>--
>Winfield Hill
You want simple, and you want L/D <0.3
OK How about Nagaoka?
Inductance of Single Layer RF Coils (Nagaoka)
L = k(pi.d.T)^2 / 10^3.ell
L = inductance microhenries
d = diam of coil in cm.
ell = length of coil in cm.
T = total number of turns
k = factor depending on ratio d/ell see table below:
d/ell k
0.1 0.959
0.2 0.920
0.4 0.850
0.8 0.735
1.0 0.688
2.0 0.526
4.0 0.365
5.0 0.320
Notice d/ell not L/D
(From Radio and Television Engineers' Reference Book, Hawker&Pannett
3rd Ed. (1960) Could swap for AofE? )
Regards
brian whatcott <in...@intellisys.net> Altus OK
John Woodgate
In article <5udd5a$m...@snews1.zippo.com>, brian whatcott
<in...@intellisys.net> writes
Nagaoka WAS the inductance calculation expert at NBS.
Winfield Hill
brian whatcott at in...@intellisys.net says...
>
>In article <5ucb56$o...@fridge-nf0.shore.net>, hi...@rowland.org says...
>>....
>> You implied that my data shows Wheeler falling apart below L/D = 1,
>> which isn't apparent to me, but at any rate, this can't be any more
>> so than saying the NBS curves are falling apart. In practise,
>> I think both are pretty useful to L/D below say 0.3, and below
>> that all the simple formulas get into trouble.
>>
>> OK, Brian, take your best shot!
>
>
> OK How about Nagaoka?
>
> Inductance of Single Layer RF Coils (Nagaoka)
>
> L = k(pi.d.T)^2 / 10^3.ell
>
> L = inductance microhenries
> d = diam of coil in cm.
> T = total number of turns
> k = factor depending on ratio d/ell see table below:
>
> d/ell k
> 0.1 0.959
> 0.2 0.920
> 0.4 0.850
> 0.8 0.735
> 1.0 0.688
> 2.0 0.526
> 4.0 0.365
> 5.0 0.320
Hey! I sure wouldn't call a cumbersome lookup table simple!
It makes any playing with parameters to design a coil really painfull!
Anyway, I don't understand why Nagaoka used (pi d N)^2, which has an
extra (pi d) in it, which must then be included as an extra (1/pi*d)
within the k. Perhaps it was due to the force of convention at the
time. Consider this form instead, which is taken from the classic
series formula for a current sheet, which Grover credits to Rayleigh
and Niven in 1881 (I've modified the formulas a bit),
L = 2 pi d Kx N^2 nH/cm and,
K(x) = (ln x - 1/2) + 2/x^2 (ln x + 1/8) - 4/x^4 (ln x -2/3) + ...
where d = coil diameter, N = turns, and K(x) is a parameter dependant
on d/l (ok, ell), using x = 4d/b = 8r/b, where b is the length of the
coil (b instead of ell, how's that?).
This may not look so very simple, but it lets us examine what's going
on. For example, we can use it to find out just how much error all
these current-sheet formulas have for real wires!
For a single turn, Grover takes only the first term (page 143), which
slightly under-determines a long sheet, but is fine for one turn:
L = 2 pi d [ln x - 0.50] N^2 (119a)
This can be compared to the formula for a single turn of round wire:
L = 2 pi d [ln x - 1.75] N^2 (119b)
where x = 4 d/dw, where dw is the diameter of the wire. [Don't ask
me exactly how Grover got that 1.75 term! Anyway, if we assume a
sheet length equal to the wire diameter, b = dw, as is customary,
the ratio [ln x - 1.75]/[ln x - 0.50] tells us about how much the
sheet formulas over-estimates a true _wire_ coil's inductance in
the limit of one turn:
error
wire-size vs dia: error with *0.833
d/dw d = 10" d = 1m ln x ratio error correction
---- -------- ----- ---- ----- ------ -------
10 Grosbeak 4.382 0.678 -32.2% -18.6%
40 0.4" 3/0 " 5.275 0.738 -26.2% -11.4% -|
100 0.1" 10 3/0 6.685 0.798 -20.2% -4.2% -| practical
1000 .01" 30 18 8.987 0.853 -14.7% +2.4% -| wire-size
10000 1mil 50 38 11.289 0.884 -11.6% +6.1% -| range
10^5 10^-4 - 58 13.592 0.905 -9.5% +8.6%
10^6 10^-5 - - 15.894 0.919 -8.1% +10.3%
->infinity - - - 1.00 0.0% +20.0%
In the last column we see that a -16.7% correction (*0.833) improves
the accuracy to under about +/- 6% for typical real-world wire sizes.
In other words, for a single turn, the correct inductance is 16.7%
less than the sheet formulas say.
Hah! Note that for the 1.0" dia wire, I've selected a power-industry
steel-reinforced aluminum size, called "Grosbeak." Try bending that
into a 10" circle!
This 15 to 20% discrepancy between the commonly-used sheet formulas
and tables is the reason I'm so skeptical of the criticism of
Wheeler's formula, especially when based upon comparisons to the
former. My own measurements of very short coils have shown the
correction above is a step in the right direction. As it happens,
Wheeler's formula goes half way there already! That's not half bad!
If I have an error in my thinking here, I'd sure appreciate someone
pointing it out.
> (From Radio and Television Engineers' Reference Book, Hawker&Pannett
> 3rd Ed. (1960) Could swap for AofE? )
I love those old books. Sure, I'll swap with you, if you're willing
to part with it!
--
Winfield Hill hi...@rowland.org
Rowland Institute for Science
Cambridge, MA 02142
JoseSainz
>Subject: Re: Accuracy of the popular Wheeler inductance equation.
>From: hi...@rowland.org (Winfield Hill)
>Date: 2 Sep 1997 01:54:21 GMT
>Message-id: <5ufrkd$9...@fridge-nf0.shore.net>
>
>
>JoseSainz, <jose...@aol.com> said...
>>
>> ... Unfortunately in my generation all these books have long been
>> out of print. The NBS formula's I mentioned before were from a
>> gentleman who before coming to my company worked for the NBS,
>> the book was his copy, deeded to another engineer my age when
>> he retired.
The publication title is Radio Instruments and Measurements - Circular No.
74 published 1937, with the correction table appearing at the rear of the
book. Sorry I'm still at home for the holiday & found the reference in the
header of the full program I was using.
> I've never seen the actual NBS publications, although it's likely
> the MIT library has them. José, were you aware that all the parameters
> in your program are in tables 36 and 37 of Grover's book? Carried out
> to an extra 2 digits (silly people), and credited to Nagaoka, who also
> published most of his papers in Japanese journals c. 1909 - 11.
> However, Nagaoka did publish some B. of S. Sci. Papers, as Grover calls
> them.
I will look for a copy of Grover in the work library (as well as some of
the other books mentioned in this thread).
> As I mentioned to Brian Whatcott, I like this form of the classic sheet
> equations (which I re-arranged from Grover):
>
> L = 2 pi d K_x N^2 nH/cm and,
>
> K_x = (ln x - 1/2) + 2/x^2 (ln x + 1/8) - 4/x^4 (ln x -2/3) + ...
>
> where x = 4 d / b, and b is the length. By inspection, we can see
> that K_x is related to the K in Grover's tables, and your BASIC
> subroutine as follows:
>
> K = ( 2 b / pi d ) K_x.
>
> Since the ratio b / d is the input parameter for the tables, that works
> out fine. Using the series, which Grover attributes to Rayleigh and
> Niven in 1881, to calculate a few values from the table seems to get
> the same answer, to 5 or 6 places.
>
> Read my posting to Brian. In that I show that all these sheet equations
> fail to correctly predict the inductance of true single-turn wire coils,
> overshooting by about 17% on average. I think this more closely matches
> some of my experimental observations, but not your own as I understand it.
> Ironically, Wheeler comes closer than the sheet equations.
> I'd appreciate your comments.
I will resurrect my data at work and publish the compararitve table
tomorrow from meassurements on a large Litz coil.
I am working on the effect of radial shielding as to how it affects coil
inductance. Hopefully I can post more information tomorrow after checking
my equations.
José
Winfield Hill
JoseSainz, <jose...@aol.com> said...
>
> ... Unfortunately in my generation all these books have long been
> out of print. The NBS formula's I mentioned before were from a
> gentleman who before coming to my company worked for the NBS,
> the book was his copy, deeded to another engineer my age when
> he retired.
I've never seen the actual NBS publications, although it's likely
the MIT library has them. José, were you aware that all the parameters
in your program are in tables 36 and 37 of Grover's book? Carried out
to an extra 2 digits (silly people), and credited to Nagaoka, who also
published most of his papers in Japanese journals c. 1909 - 11.
However, Nagaoka did publish some B. of S. Sci. Papers, as Grover calls
them.
As I mentioned to Brian Whatcott, I like this form of the classic sheet
equations (which I re-arranged from Grover):
L = 2 pi d K_x N^2 nH/cm and,
K_x = (ln x - 1/2) + 2/x^2 (ln x + 1/8) - 4/x^4 (ln x -2/3) + ...
where x = 4 d / b, and b is the length. By inspection, we can see
that K_x is related to the K in Grover's tables, and your BASIC
subroutine as follows:
K = ( 2 b / pi d ) K_x.
Since the ratio b / d is the input parameter for the tables, that works
out fine. Using the series, which Grover attributes to Rayleigh and
Niven in 1881, to calculate a few values from the table seems to get
the same answer, to 5 or 6 places.
Read my posting to Brian. In that I show that all these sheet equations
fail to correctly predict the inductance of true single-turn wire coils,
overshooting by about 17% on average. I think this more closely matches
some of my experimental observations, but not your own as I understand it.
Ironically, Wheeler comes closer than the sheet equations.
I'd appreciate your comments.
--
Winfield Hill
JoseSainz, <jose...@aol.com> said...
> I don't dispute Wheeler's formula, as around my work we are satisfied with
> a wound coil that is within 10% of the calculated value using Wheeler.
> Typically solenoids we construct have at least a 1:1 Length to Diameter
> ratio. More usually a 2:1 or 3:1 is common. [snip]
Not to damm with faint praise, let's be clear that Wheeler is not simply
a 10% grade formula. Used for L/D > 0.4, as suggested, it's basically
good to 0.5% or better, and about 0.1% over the range of most high-Q coils.
> What I cautioned was dependence on that formula when calculating the
> inductance of the first turns. [snip interesting radio-work discussion]
> When fitting my calculation to the measured values I noticed large
> errors using an "uncorrected formula". ...
>
> Here is a comparison of calculations.
> Wheeler
> NT L/Dia Wheeler NBS error%
> 1 0.00625 4.38 7.02 37.6 Radius is 40".
> 2 0.0125 17.30 26.89 35.7 Length is Turns*0.5"
> 3 0.01875 38.40 57.44 33.1
> 4 0.0250 67.37 78.04 13.7
> 5 0.03125 103.90 129.64 19.9
> 10 0.0625 390.24 467.54 16.5
> 15 0.0938 827.59 935.89 11.6
> 20 0.125 1391.30 1518.47 8.4
> 10 0.125 347.83 379.62 8.4 Radius is 40".
> 20 0.250 1142.86 1172.54 2.5 Length is Turns*1.0"
> 10 0.250 285.71 293.14 2.5 Radius is 40".
> 15 0.375 545.45 549.53 0.7 Length is Turns*2.0"
> 20 0.500 842.11 843.15 0.1
[ snip identical calculated errors for the same Length/Dia ratios ]
> What the above shows is that at few turns the Wheeler formula without
> corrections has large errors. The NBS formula when fitted to the existing
> coils I was working on had an error of <3% for the first few turns ...
I was disappointed, when examining your table, to see you had simply shown
the Wheeler - vs - Nagaoka deviations, without measurements. Many of us
have long since plotted Wheeler on one of our factor "F" graphs (i.e.
L = F d N^2) to see the deviation below L/D < 0.4, growing to 10% at 1/10.
Although for my small-diameter coils these measurements aren't easy, my
own experience has been that for L/D less than say 0.2, all the sheet-based
formulas have increasing errors. I remember at the time repeating some
of my short-coil measurements for accuracy, but will re-measure them, along
with some new ones my technician made last week. Nonetheless, if it is
still available, it would be gratifying to see some of your data. E.g.,
did the Nagaoka values for ONE turn really agree to 3% or better?
I'm very interested in more details of your measurements.
> [ The NBS formula ... had an error ] of 0.1% or so when diameter and
> length were 1:1.
Not to belabor the point, but as does Wheeler's simple equation.
> Well I hope this explains more exactly my comment.
José, thanks for your detailed story. Could you tell us what kind of
radio receivers / antennas these are, and more about them?
Winfield Hill
JoseSainz, <jose...@aol.com> said...
>
> hi...@rowland.org (Winfield Hill) said,
>>
>>JoseSainz, <jose...@aol.com> said...
>>>
>>> ... Unfortunately in my generation all these books have long been
>>> out of print. The NBS formula's I mentioned before were from a
>>> gentleman who before coming to my company worked for the NBS,
>>> the book was his copy, deeded to another engineer my age when
>>> he retired.
>
> The publication title is Radio Instruments and Measurements - Circular
> No. 74 published 1937, with the correction table appearing at the rear
> of the book. Sorry I'm still at home for the holiday & found the
> reference in the header of the full program I was using.
>
>> I've never seen the actual NBS publications, although it's likely
>> the MIT library has them. José, were you aware that all the parameters
>> in your program are in tables 36 and 37 of Grover's book? Carried out
>> to an extra 2 digits (silly people), and credited to Nagaoka, who also
>> published most of his papers in Japanese journals c. 1909 - 11.
>> However, Nagaoka did publish some B. of S. Sci. Papers, as Grover calls
>> them.
>
> I will look for a copy of Grover in the work library (as well as some of
> the other books mentioned in this thread).
>
>> As I mentioned to Brian Whatcott, I like this form of the classic sheet
>> equations (which I re-arranged from Grover):
>>
>> L = 2 pi d K_x N^2 nH/cm and,
>>
>> K_x = (ln x - 1/2) + 2/x^2 (ln x + 1/8) - 4/x^4 (ln x -2/3) + ...
>>
>> where x = 4 d / b, and b is the length. By inspection, we can see
>> that K_x is related to the K in Grover's tables, and your BASIC
>> subroutine as follows:
>>
>> K = ( 2 b / pi d ) K_x.
>>
>> Since the ratio b / d is the input parameter for the tables, that works
>> out fine. Using the series, which Grover attributes to Rayleigh and
>> Niven in 1881, to calculate a few values from the table seems to get
>> the same answer, to 5 or 6 places.
I should have said, valid only for dia/length > 2.
I took this from formulas 118 and 119 in Frederick W. Grover's book
"Inductance Calculation" Van Nostrand (1946) and Dover (1962), by
re-arranging them into a more simple form, better suited to internet
ASCII typing. For a very short coil, a single term of the series
suffices and Grover says three terms retain a 0.1% accuracy for
b/d < 1/4 which is x > 16. For coils as long as dia = length, b/d = 1,
the error has grown to -1.6% and cannot be fixed with any number of
terms. Clearly very useful for short coils, the series is better
replaced by Wheeler's formula or Nagaoka's tables (or your program)
for longer ones.
> I will resurrect my data at work and publish the compararitve table
> tomorrow from meassurements on a large Litz coil.
We do look forward to that!