Re: Quarter Wavelength Frequency

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

Original poster: "Malcolm Watts" <m.j.watts-at-massey.ac.nz> 

Hi Paul,

On 23 Jul 2004, at 12:08, Tesla list wrote:

 > 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.

I note that every attempt (successful or otherwise) to formulate Fr
for a single-layer resonator incorporates the h/d ratio in one form
or another (Medhurst does for example) so this cannot be an accident.
All past attempts to do this using wirelength and h/d range from
relatively simple formulations to some that can only be described as
arcane. I suspect some of these types of formulae were concocted to
fit particular cases.

Malcolm

 > 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
 > --
 >
 >
 >