Re: New formula for secondary resonant frequency

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

Original poster: "by way of Terry Fritz <twftesla-at-uswest-dot-net>" <paul-at-abelian.demon.co.uk>

Kurt,

Your measured coil results and evaluation of this proposed formula
are very much appreciated.

Kurt's table:

Coil      PN-CW  PN-half  TF-big     MM
turns     725     365      1001     1700
h         1.6     0.8      0.762    1.07696 m
d         0.58    0.58     0.2606   0.1081  m
b         0.15    0.15     0.025    0.3302  m
awg       12       12        24       24

Fres,cal  91.1    152.9    147.6     279.6   kHz
Fres,exp  90.9    150.7    148.4     276.9   kHz
Diff      0.2%      1.4%    -0.5%      1.0%  cal-exp
 
Kurt Schraner <k.schraner-at-datacomm.ch> wrote:

> As obvious, there is a difference in the results, relative to
> yours,

Your figures are correct - mine wrong. Many thanks for pointing out
my error - I pasted in the frequencies and percentage errors from an
earlier iteration. I prefer your figures :)

> As a next, 4 of my own coils are compared:

[Kurt - I shrunk the table a little to avoid 72 char wrap]

> Coil     Sk-B&W    Sk-Long    Sk-12cm     Sk-20cm 
> turns     821       1950       921        979
> h        1.768      1.41       0.585      0.68
> d        0.4013     0.1633     0.1207     0.2
> b        0.7        0.5        0.2        0.5
> awg      17.162     22.053     22         22
> 
> Fres,cal  131.4     157.3      409.0     209.3  Paul's formula
> Fres,exp  119       147.7      368       202.7
> Diff      10.4%      6.5%      11.2%      3.2%  cal-exp
>  
> Fres,cal  123.1     139.6      372.9     200.0  Wheeler/Medhurst
> Diff      3.4%      -5.5%       1.3%     -1.3%  cal-exp

I concur with your calculations.
 
> It seems, my coils are yet more happy with Wheeler/Medhurst,
> however the precision of the experimental data have to be
> considered. Regarding the instruments, I believe to be quite
> precise (specifics can be supplied). The most of error probably
> stems from the spacial situation, present, when measuring the
> coils: capacitive influence of the surroundings! The B&W coil,
> i.e., was tested in my living room, which is one floor above
> ground level, and the top of my big coil only ~0.4m from the
> ceiling.

Agreed. In the case of the B&W coil, the lack of a well defined
ground surface provides an ambiguous termination of the external
E-field, so the coil is outside the domain of the formula. My
attempts to model coils at high elevation have all failed dismally,
due possibly to the poor definition of the return path for the
external field - all were measured indoors. I'm still hoping that
someone will setup a coil outdoors and high up over a good ground
plane, so that I can see if the problem persists. Meanwhile I can
only model reliably for b <= 1.0 and the formula is only regressed
against data for 0.033 < b <= 0.5.

The Sk-12cm which is a smaller coil, low down, should be quite
accurate.
However, for Sk-12cm, something is wrong: 22 awg = 0.6438 mm diam,
so 921 turns spans 593 mm, this is longer than your coil. Perhaps
you can look into this?

Your coil Sk-Long is particularly interesting, as it falls into
the large h/d category in which modeling predicts a resonant
frequency some 10% higher than Medhurst. Your measurement falls
in between the two, and I suspect that it may suffer some of the
problems similar to the Sk-B&W, in that it is quite a large coil
to be measured indoors. If it becomes possible to measure this coil
in a large room or outdoors, over a reasonable ground plane, the
result would be interesting. My prediction is that the absence
of nearby walls and ceiling will reduce the external capacitance and
raise the frequency to nearer that of the formula.
 
I just ran Sk-Long through the precision model to obtain the following
predictions for operation in an open space at b=0.5 over a ground
plane radius at least 2m:

Ldc=67.6 mH, Cdc=35.3pF
f1: 155.9 kHz (Lee=45.7mH, Les=52.6mH, Cee=17.2pF, Ces=19.9pF)
f3: 383.7 kHz
f5: 544.9 kHz

which unsurprisingly is close to the formula prediction.

I also ran Sk-Long with walls introduced at radius 2m, and a ceiling
at height 2.3m. The result is 
f1: 147.2 kHz
which is close to your measured value. Clearly, the presence of 
walls and ceiling may well be responsible for the discrepancy with the
formula.

The Wheeler/Medhurst estimates seem to do OK at predicting your
resonances. This must be accidental, since it takes no more account
of the environment around the coil than my formula does.  The large
h/d regime, in which Wheeler/Medhurst errs on the low side, may
be offset to some extent by that fact that the larger h/d coils are
inherently more sensitive to the external capacitance, so that
nearby walls and ceiling will depress the actual frequency somewhat
more than for lower h/d coils. Hopefully as further measurements
become available, this situation will be clarified.

> BTW: Would you have perhaps at hand, a version of your function
> fa = -94.6683*awg*awg*awg + 9000.55*awg*awg - 301175*awg +
> 3.64056e+6
> beeing currently a function of awg, made a function of wire
> diameter instead, like f(wd[m])? Measured values of wire diameter
> could more easyly be introduced that way.

You can use

  awg = 1 + log(7.348e-3/wd)/0.115943

and use this awg in the formula for fa above.

Regards,
--
Paul Nicholson,
Manchester, UK.
--