Conical primary formula questions.

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

Original poster: "Pete Komen by way of Terry Fritz <twftesla-at-uswest-dot-net>" <pkomen-at-zianet-dot-com>

After much fiddling with numbers on an Excel spreadsheet, I think that the
formula for the inverse conical coil is off or more precisely the
measurement of H and W and R are not what they should be.  If measured on
the cone shape they yield a higher value for the cone than for a flat or
helical coil of the same number of turns.  The reduction in radius for the
conical coil indicates to me that the inductance should fall, not increase.

>From the archives:
http://www.pupman-dot-com/listarchives/1998/December/msg00023.html
Case 3: Inverse Conical Primary:


                                    / \
    --  o                          /    o
     |   o                        /    o
     |    o   N turns            /    o
           o                 Z  /    o
     h      o                  /    o   /
             o                /    o   /
     |        o              /    o   /  Angle = X
     |         o              \  o   /
    --          o               o    ------------
                        |
        |   w   |   R   |
                        |
            |<--  W  -->|
                        ^
                 Center | Line

          Z = Coil Width (hypotenuse length)
          X = Angle of Cone
          h = Z*sin(X)  Effective vertical Height
          w = Z*cos(X)  Effective horizontal Width
          W = R + w/2   Average horizontal Radius


     L1 = W^2*N^2/(9*W+10*h)  (Vertical Inductance Component)

     L2 = W^2*N^2/(8*W+11*w)  (Horizontal Inductance Component)

      L = SQRT[(L1*Sin(X))^2 + (L2*cos(X))^2]

Call this one the Widely Accepted Method (WAM).

I feel a bit presumptuous and I could well be wrong, but I have a modified
method which gives what I believe are better results.

For example:

Assume 15 turns of 1/4 inch tubing spaced 3/8 inch apart.  W (spiral) equals
H (helical) equals Z above equals 9 inches (I've even wondered about that,
it doesn't really take into account the spiral...).  Assume that the radius
of the inner wind is 5.5 inches.

Helical:  R = 5.5 inches, H = 9 inches
Spiral:   R = 10 inches, W = 9 inches
I would use those values to calculate the inductance for the spiral and the
helix; then plug those inductances into the conical formula and vary only
the angle.

For the widely accepted method (WAM), the following would be used:
Angle of elevation of cone is X.
Helical:  R = 5.5 + 9 * cosine(X) / 2,  H = 9 * sine(X)
Spiral:  R = 5.5 + 9 * cosine(X) / 2,    W = 9 * cosine(X)

These are used to calculate the inductance for the helical and spiral
components for each angle considered.  Then those inductances are plugged
into the WAM equation.

                        in microhenries
Angle radians           My Method   Inductance WAM
0     0                 125.70      125.70
15    0.261799167       122.07      130.94
30    0.523598333       111.56      131.58
45    0.7853975          95.34      120.62
60    1.047196667        75.73       99.76
75    1.308995833        57.27       73.43
90    1.570795           48.79       48.79

The key thing is the increase (over the larger R flat spiral) at 15 and 30
degrees (there is a maximum around 24.6 degrees).  I can imagine an ellipse
if this was graphed but not this bulge for low angled conical coils.

I also observe in these formulas that the inductance is (sort of)
proportional to the average radius.  So it doesn't make sense to me, that a
little angle from horizontal would increase the inductance.

My questions are these:

Where does the formula for conical primaries originate?  Has this been
checked against actual measurements?  (I would build and measure but I am
already spending enough building a coil.  Now is not the time to spend money
on test equipment).  Is there some reason that the inductance using WAM
increases for the conical primary beyond the flat spiral value while the
average radius is decreasing?

Pete Komen

Special thanks to Matt Behrend for his kind responses when I emailed him
with this question.