Re: Quarter Wavelength Frequency

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

Original poster: "Paul Nicholson" <paul-at-abelian.demon.co.uk> 

Ed Phillips recently posted a table relating wire length to free
space wavelength for unloaded coils.   I added a column for
velocity factor = 4 * wire_length/lambda, and we got:-

 >   L/D   length of wire/lambda     velocity_factor
 >   0.5   0.228                      0.912
 >   1.0   0.298                      1.192
 >   1.5   0.343                      1.372
 >   2.0   0.374                      1.496
 >   3.0   0.413                      1.652
 >   4.0   0.435                      1.740
 >   5.0   0.449                      1.796
 >   7.0   0.466                      1.864
 >   10    0.478                      1.912
 >   100   0.49998                    1.99992
 >   1000  0.50000                    2.00000

where L/D = axial_length/diameter = h/d, lambda is an abbreviation
for 'free space wavelength'.

Ed's figures are based on Ldc (via Lundin) and Medhurst C from
a series.

The observations are that:-

a) Ed's calculations tend to velocity factor 2.0 as h/d tends to
infinity.  We might have expected unity here on the basis that
the coil's becoming more stretched out like a straight wire.

b) The actual figure that Ed's calcs tend to is an exact 2.0000...
which probably indicates a mathematical limiting value.

In order to see where these observations stand, I ran through my
database of about a dozen accurately measured coils.  The following
table reports the measured frequencies and wire lengths, and the
velocity factor calculated from them:-

System       Fres    c/fres/4  wire      h/d   vfactor
sk38b50    221.3kHz    338.9m  417.4m   1.15   1.23
pn1        150.7kHz    497.7m  659.9m   1.36   1.33
pn2         92.0kHz    815.2m 1321.0m   2.84   1.62
tfltr      148.4kHz    505.4m  818.7m   2.92   1.62
sk20b49    217.2kHz    345.3m  607.9m   3.26   1.76
mwa1-4hd0  224.0kHz    334.8m  582.5m   4.00   1.74
mm3         61.9kHz   1211.6m 2077.9m   4.65   1.71
sk12b49    405.1kHz    185.1m  340.4m   4.83   1.84
tfsm1      358.8kHz    209.0m  398.8m   6.15   1.91
mm4        237.0kHz    316.5m  572.0m   6.78   1.81
sk5b503    979.7kHz     76.6m  149.9m   8.04   1.96
sk16b50    152.3kHz    492.4m  999.5m   8.71   2.03
mm1        455.5kHz    164.7m  347.3m   8.92   2.11
mm2        276.9kHz    270.9m  577.1m   9.97   2.13

(The above are all bare coils, ie no toploads or top probes or
anything to perturb the frequency.  c = 300e6).

Bearing in mind that these are measured values, we do seem to have
the real coils tending to a high velocity factor as h/d increases.

Is there anybody out there with a coil with h/d > 10 ???  If so,
we want your measurements!

The interesting thing is that these coils all have a variety of
turns and pitches, yet they all land within a narrow range of
one another when h/d is plotted against velocity factor.

This implies that we can get a good estimate for Fres by simply
taking the free space quarterwave frequency for the straight wire
and then multiplying by the corresponding velocity factor for
the given h/d.

In other words,

   Fres = Phillips(h/d) * 75e3/wire_length  (kHz)

where Phillips(A) is a function interpolated from the right hand
two columns of the above table and the wire_length is in metres.

I think this is a very interesting observation by Ed and, along with
the fact that the velocity factor increases well beyond unity, ought
to be telling us something quite general about coil resonance.

Thanks, Ed, for bringing up this neat little observation.  It hints
at a fairly simply stated mathematical relation between the overall
coil geometry and the Fres.  A very nice result.

I'll now go away and test the this 'Phillips function' against a
large database of a few thousand simulated coils to try to pin
down a semi-empirical formula for it.
--
Paul Nicholson
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