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, All,

Thanks Kurt. Anyone who has gone to the trouble of making thorough and
precise measurements will appreciate the effort involved.

The most interesting results are those of Sk-5cm, and I'll comment on
these later, but first I'll look at the other coils, which are all
within the limits of the 'new formula'.  Sk-Long has the largest h/d
and for this we find

(Sk-Long -at- b=0.545m)

    Measured     Modeled
f1: 161.45 kHz   156.4 kHz (-3.1%)
f3: 386.43 kHz   385.2 kHz (-0.3%)
f5: 562.04 kHz   546.6 kHz (-2.7%)
f7: 710.32 kHz   677.4 kHz (-4.6%)

and the new formula predicts 158.0 kHz (-2.1%).  Wheeler/Medhurst
predicts 139.2 kHz (-13.7%), which illustrates the expected shortfall
of Wh/Med at large h/d.  The problem is not that Cmed is too high but
that the low frequency inductance Ldc (Wheeler or Nagaoka) is not the
correct inductance to use.  The current profile of a large h/d second-
ary tends towards being very roughly cosinusoidal in profile, and thus
the integral of induced EMF along the coil is going to be only around
70% of the value you would get if the current where uniformly equal to
the base current all along the coil.  Therefore the effective series
inductance (Les) at the f1 resonance can be as low as 70% of the DC
value, and it is this effect which causes the error in the use of
Wheeler/Medhurst.

Les can be estimated from Ldc by using another formula obtained by
curve fitting to a large set of simulated coils. In this case there
are two formulae:

Bare coil:

  Les =  Ldc * pow( 8.0724 + 4.5129 * h/d, -0.8016) *
               pow( 0.0079 + 1.384 * b/h, 0.2623) *
               pow( 338.89 + 18.9111 * awg, 0.1493) *
               pow( 0.346 + 4.23 * h, 0.0232) *
               pow( 115.768 + 7.1 * sr, 0.0241) + 0.658311

    average error = 1.3%, peak error = 2.8%

Toroided:

   Les =  Ldc * pow( 4.6675 + 5.5509 * h/d, -0.5983) *
                pow( -0.0147 + 0.9557 * b/h, 0.131) *
                pow( 5.2204 + 0.145 * awg, 0.0703) *
                pow( 3.94 + 0.9199 * h, 0.0487) *
                pow( 5.41 + 80.798 * sr, 0.0114) *
                pow( -4.1441 + 14.3397 * td/h, 0.1537) *
                pow( 0.312 + 0.58 * tb/h, 0.088) + 0.686997

    average error = 1.6%, peak error = 4.5%

where td is the toroid outer diameter, tb is the height of the toroid
plane above the top of the winding, and the other variables are as
previously defined.

Note that at small h/d, or high elevation, or with a wide toroid, Les
can exceed Ldc, sometimes by as much as 15% - an effect caused by
current circulating around the coil-coil and coil-toroid internal
capacitances.

Neither of these two effects leading to deviation of effective
inductance from Ldc are taken account of by the Wh/Med estimate, since
Cmed assumes operation well below the self-resonance of the coil and
therefore implies a uniform coil current and Les = Ldc. Use of Les
resonating with Cmed should produce a better estimate of f1 than
using Ldc with Cmed.

Let's test this out with Sk-Long -at- f1, outdoors, b=0.545:

>From the formula above, Les = Ldc * 0.78 = 52.8 mH, so we have, using
Les and Cmed, f1 = 157.7 kHz (-2.3%), which corrects the problem with
Wh/Med.

Turning now to Sk-5cm.  As usual, it's the results that don't match
expectations that are scientifically the most interesting and
valuable.  Sk-5cm is a good example.  The results for b=0.503m are
outside the domain of the new formula since b > h/2, and also exceed
the domain of the modeling program since b > h, so I'll look only at
the b=0.158m results.

(Sk-5cm -at- b=0.185m)

    Measured     Modeled            New formula
f1: 919.45 kHz   1027 kHz (+11.7%)  1058 kHz (+15.1%)

The new formula agrees (within its limits) with the model,  which is
to be expected,  but the model is very high in its prediction of this
coil.  I have a number of other results on small coils which exhibit a
similar large positive error.  I don't yet know if this is a defect in
the model - a problem which shows up only at small radius perhaps (*),
or whether it is due to some stray capacitance or other influence
affecting the measurements more noticeably on small coils. 
Consequently I'd very much like to understand what's going on here
with Sk-5cm. Comparing the low frequency parameters:

     Measured    Modeled
Ldc:  5.203mH    5.19 mH
Rdc: 21.7 ohms  20.88 ohms (-3.8%)

gives a reasonable match, the higher actual resistance might be caused
by a slight stretch of the wire during winding, so no cause for
concern there.  The difficulty of measuring f1 on a small coil is 
apparent from the fact that the frequency went up slightly on lowering
the elevation. With an effective shunt capacitance of around 5.6pF, it
only takes 1pF of stray capacitance applied to the top of the coil to
produce nearly 10% shift in f1.  This possibility could be confirmed
by looking at f3 and f5.  If stray external C is to blame, then f3
should be predicted about 5% high and f5 should be at about +3%. Then

     Modeled   - StrayC =  Estimate
f3:  2524 kHz  -  5%    =  2398 kHz. 
f5:  4423 kHz  -  3%    =  4290 kHz.

To reach these frequencies you would have to excite the coil with the
3rd harmonic of the generator's square wave.

Kurt wrote:
> Regarding harmonics: the small coils showed only none to very
> weak signal, so I wasn't able to determine more.

> The ground floor surface was humid but not wet, consisting of
> cobbles (in german: "Kopfsteinpflaster").
 
> I've noticed a problem in reliably measuring the resonant
> frequencies of the coils: I've experienced an interaction of my
> "signal generator - feed line - coil" resonance system, in the
> way, that more capacity between generator and coil changes the
> Fres to lower values

These three comments are interesting. Taken together they suggest that
there may be a problem with the grounding arrangements. The
sensitivity of f1 to feedline capacitance should be slight, and
padding to a low impedance at the coil base ought to take care of it.
In your case you seem to be having quite some trouble with it.
I usually recommend the use of a tin foil ground plane for the coil to
resonate against, which provides a well defined ground, and the feed
can be brought in beneath the foil. Without this, the return path for
the coil current is via a circuitous high resistance path through the
earth and back to your signal generator. I wonder if the coil is
exhibiting a very low Q factor in its test position? It should be
getting on for 200, maybe more, but if it's down around 50 or less
there will be problems, and the f1 will be reduced below the predicted
value.

It ought to be possible to get reliable measurements on this small
coil indoors, with the coil at low elevation over a tinfoil ground-
plane. A coax feed with a shunt resistor connected directly between
the coil base and groundplane should give reliable measurements. At
10dBm the output of your generator is quite low and the risk is that
probe capacitance will depress f1. You might prefer to arrange the
scope to measure base current, with the aid of a series resistance or
CT, and tune for maximum base current instead. If you're interested in
investigating this discrepancy further, I think that would be the way
to go.

> BTW: I'm happy to see, how the things now have reversed, and with
> the exception of the small coil, Pauls predictions look more
> close to the measurements :o)).

Well I'm very confident about the Wh/Med shortfall prominent at high
h/d and the explanation I've given for it in terms of the effective
inductance being different from the DC value. Yours and a number of
other results help to confirm this picture. I've given formulae for
the effective inductance and f1 which appear to give better accuracy,
at least for coils with a diameter greater than around 10cm.
It remains to be seen whether the underlying model is faulty for coils
narrower than this, and if so why. As always, experiment decides the
matter, and further measurements on small diameter coils will be most
welcome.

(*) For eg, the model does not account for the presence of the coil-
former dielectric. It may become significant at small radius.

Perhaps we can continue discussions off-list, or on tssp, as the
credulity and frequency of crank postings on this list has exceeded my
tolerance threshold, and I wish to disconnect.

Cheers,
--
Paul Nicholson,
Manchester, UK.
--