Re: Inductance calculations

["Tesla list" <tesla-at-pupman-dot-com>]

Original poster: "Antonio Carlos M. de Queiroz by way of Terry Fritz <teslalist-at-qwest-dot-net>" <acmq-at-compuland-dot-com.br>

Tesla list wrote:

 > Original poster: "Barton B. Anderson by way of Terry Fritz 
<teslalist-at-qwest-dot-net>" <classictesla-at-netzero-dot-com>

Bart,

 > Tonight I tore down and stretched out the coil. Yes, it is tubing (I needed
 > the coil to hold shape). When you mentioned the smaller wire size, I
 > decided to stretch it out to at least the maximum of my meter leads. The
 > results are impressive! Once again, nature tells the tale.
 >
 > Dimensions: r1,r2=.485775,  h=2.1336 at turn 5, wd=.009525): (L=uH,
 > units=meters).
 >
 > height    turns    meas.    fantc    Wheeler    Lundin  | inca.h    Snow
 > ------    -----    -----    -----    -------    ------    ------    -----
 > 2.1336      5       17      9.02     9.06       9.08    | 18.15     18.73
 > 1.7051      4       13      6.92     6.97       6.97    | 14.26     14.65
 > 1.27635     3       10      4.86     4.89       4.89    | 10.41     10.63
 > 0.84785     2       6       2.87     2.90       2.89    |  6.63      6.70
 > 0.4191      1       3       1.08     1.09       1.09    |  3.02      3.00

So, old Maxwell was right. The values from Inca are with the
approximation
of considering the loops closed (as it draws). With true loops the
values
get closer to Snow's formula (but the calculation is slow). The closed
rings approximation works very well, even for 1 turn.

 > Note:
 > With the XT's 1uH minimum, I measured 2uH of lead inductance (already
 > accounted for in numbers above). The leads inductance did toggle between 1
 > and 2uH, but settled on 2uH. Thus, I feel that inca and Snow are likely
 > calculating the inductance better than I can measure.

I am now after an "exact" formula, at least geometrically, with
Maxwell's
method. The integration that Inca does numerically appears to be
solvable.
It's just a question of finding a closed expression for the mutual
inductance between two filamental turns of a conical spiral. The exact
expression, taking into account the curvature of the wire, is probably
known too. I found a book and some papers that possibly have the problem
worked out.

 > I think you have a winner with Inca! This coil was spaced out far enough
 > that the conductor size appears small relative to the distance. And with
 > that thought, if we were capable of decreasing the conductor size
 > extremely, we should be able to achieve the same inductances with less 
spacing.

Surely, but the difference is small. Ex: my secondary coil with 28.2
mH was wound with 0.2 mm wire (the default in Inca). If I decrease
the with diameter Inca gives: (Wheeler gives 28.21 mH)
0.2 mm  -> 28.27 mH
0.1 mm  -> 28.30 mH
0.05 mm -> 28.35 mH
For the primary circuit in the Fantc example, that started this thread,
(Wheeler gives 37.96 uH), Inca gives, for different wire diameters:
20 mm -> 39.32 uH
10 mm -> 41.40 uH
  5 mm -> 43.55 uH
  2 mm -> 46.43 uH
  1 mm -> 48.60 uH
.1 mm -> 55.84 uH
Wheeler's formula appears to work well for closely would spirals. With
5 turns in 20 cm, the maximum wire diameter is 4 cm (Inca -> 37.54 uH).
I saw somewhere (Grover?) a correction for empty space between windings.
Maybe this is what is missing in Fantc, Acmi, and Mandk. Maxwell's
method
takes this into account automatically.

 > Obviously, these geometry's are impractical to coiling, but excellent for
 > the test. I think I've learned a little something today. Very cool!

Thank you for the experiments.

Antonio Carlos M. de Queiroz