Re: Quarter Wavelength Frequency

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

Original poster: Ed Phillips <evp-at-pacbell-dot-net> 

"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.)"

	You're correct about how I defined "wavelength".  I don't know why the
wavelength ratio approaches 1/2, but I've seen that mentioned in the
literature somewhere.  Is that really 1/2, or ~1/2?  I don't have any
idea.  While the inductance calculation was run for a fixed winding
pitch (winding factor and wire size), but Medhurst data is stated to be
independent of the winding factor in one of the earlier references I
have read and to which I no longer have access. I'll trying to find the
measurements for the four coils I measured.  Three of them had an inside
diameter of about 3.17" and were about 15" long, with wire size from #26
to #30 (I have a lot of the latter and it seems to wrok fine).  The
fourth coil has an ID of 5.25" and a length of 18", and is wound with
litz wire and turns spaced with cotton string.  All seemed to check
quite closely, somewhat to my original surprise.  I don't begin to have
the ambition to make a bunch of coils of different geometries and
winding pitches, but just to suggest that the calculations are a useful
guide for coil tuning.

	I'm not sure what mode is being excited when a self-resonant coil with
no external capacitance loading is fed from one end, but the frequency
observed corresonds to the measured frequency of the same secondaries
excited with a normal primary and loosely (spelling?) coupled.

	All of this was done many years ago and I'll have to dig for the data.

"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."

	Hadn't thought about this at all so no useful comments.  When I have a
chance I'll go over stuff on helical antennas.  "REFERENCE DATA FOR
RADIO ENGINEERS" by FT&T has quite a bit on them but I've never paid
much attention.  I have always thought of an unloaded TC as being
equivalent to an extremely short helical antenna and tried to calculate
the radiation resistance once.  It turns out to be nil which probably
explains why our coils don't create more of a ruckus than they do.

	The program is a few lines of QuickBasic code and I'll send the text
listing later.  In order to call it forth I have to shut down this Mac
and restart it in a different mode, something I don't want to bother to
do right now.  Here is the listing for the inductance calculation:

"Calculation of inductance by Lundin's approximation to Nagaoka's
constant.
[Letter to Proceedings of the IEEE, Volume 75, Number 9, September 1985
pp 1428 =1429]

FOR A SOLENOID OF DIMENSIONS:
DIAMETER  (INCHES) = D
LENGTH  (INCHES) = LE
NUMBER OF TURNS = N

CALCULATE
X=D/LE
X2=X^2

A(X)=(1+.383901*X+.017108*X^2)/(1+.258952*X)
B(X)=(.093842*X+.002029*X^2-.000801*X^3)

IF X = > 1
K = (.6366198#/X)*((LOG(4*X)-.5)*FNA(1/X2)+FNB(1/X2))
INDUCTANCE =.0250688*D*X*N^2*K    MICROHENRIES

IF X < = 1
K=FNA(X2)-.42441318#*X
IND=.0250688*D*X*N^2*K    MICROHENRIES

  I can't find the original letter, so the stuff above is a rewrite of
the expressions in the Basic program I wrote at the time; hope I didn't
make any mistakes.  "Just in case" here are the original Basic
statements:

INPUT "DIAMETER, LENGTH, (INCHES) AND NUMBER OF TURNS"; D,L,N
DEF FNA(X)=(1+.383901*X+.017108*X^2)/(1+.258952*X)

DEF FNB(X)=(.093842*X+.002029*X^2-.000801*X^3)
X=D/L
X2=X^2
IF X<1 THEN LT1
K=(.6366198#/X)*((LOG(4*X)-.5)*FNA(1/X2)+FNB(1/X2))

LT1:
K=FNA(X2)-.42441318#*X
IND=.0250688*D*X*N^2*K    ' INDUCTANCE IN MICROHENRIES"

	Wow but this is long but may of interest to someone besides Paul or I'd
try to send it direct.  Criticisms and corrections and rebuttals
welcome.

Ed