Re: Quarter Wavelength Frequency

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

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

Ed Phillips wrote:

 >  For lamda I use c/Fr

Ok, the free space wavelength.

 > Lundin's approximate expression for inductance and a power
 > series approximation to Medhurst's data as given in an old
 > handbook I have.

Good and fine, respectively.

 > I attempt to measure Fr by feeding the bottom of a coil,
 > isolated as far as possible from nearby objects, with a
 > low-impedance source and observing the "top voltage" with a
 > small plate remotely located and again "as far
 > as possible" from the coil.

Can't really do better than that.

 > The signal generator I'm using is a standard
 > "Hewlett Packard" circuit, since it's the only one I have
 > putting out significant voltage in the region above 100 kHz.

I don't see any problem with the calculations or measurements,
that all looks great, but the answers are a little unexpected.

The method of Fres measurement should excite 1/4 wave resonance
so your frequency calculations must be for the 1/4 wave, I
guess you are simply using 1/(2*pi*sqrt(Lundin*Cmed)), which
should be fine.

But why I wonder, does your table tend to 0.5000 rather than
0.2500?   Is the turn count doing something here, perhaps?

(There's some doubt about the mode, because you mention a
parallel resonance, but your generator would see a series
resonance at the base terminals when driving the 1/4 wave.)

Let's tabulate the velocity factor (along the wire) as
calculated by

   velocity = 4 * wire_length * Fres
            = 4 * wire_length * c/lambda

   velocity_factor = v/c = 4 * wire_length/lambda.

(the 4 because we're supposed to be measuring the 1/4 wave).

Then your table becomes:-

  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

I would expect the factor to be a greater than unity
for typical TC L/D ratios, which they are, but it should tend
down to unity, not up to 2.

I seem to recall you or somebody reported that series for
Cmed to the list a few years ago.  I'll try to reproduce
your figures.  Chances are the precise looking values
are coming about because either the Lundin or Cmed series
is losing its physical accuracy as L/D goes up and one or
other is tending to some mathematical limit value.

 > Forgot to say that "the mathematical precision" was used with
 > tongue in cheek to say the least and I agree that the value
 > quoted does "simply reveal some mathematical relation". Two
 > figure accuracy maybe, under the right conditions.

That's appreciated Ed, of course you're quite right to throw
in the extra digits of precision in order to highlight the
point.

Thanks for reporting this.  I'd like to go away and repeat the
calcs.  If you can send in that Cmed series it would save a
daunting search for 'medhurst' in the list archives !!!

And could you say what you do with the turn count - I take it
any old value for turns produces the same effect.  Maybe you
could send a copy of your program - the source code?
--
Paul Nicholson
--