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Re: Conical primary formula questions.



Original poster: "Bert Hickman by way of Terry Fritz <twftesla-at-uswest-dot-net>" <bert.hickman-at-aquila-dot-net>

Pete,

You make a compelling argument... and I plead guilty as the source. :^)

The formula was originally developed as a hybrid of the flat and helical
formulas. It was initially checked against a couple of conical primaries
that others had constructed and reported on in that timeframe (it was
originally developed back in 1996 or 1997). I had a similar problem - no
actual conical primary to compare it against. The hybrid formula matched
Wheeler at 0 and 90 degrees, and appeared to be close enough for the two
samples reviewed (5-10%) for coil developers at that time. It provided a
"quick and dirty" closed-form estimate for coil designers since none had
existed for conical primaries before. 

We now have more accurate and better estimating "tools" and models (most
notably Mark and Paul's programs) and we are striving for much greater
precision. Since these newer modeling programs actually compute the
inductance, it should be possible to develop a more accurate closed form
estimate for the conical primary. And, now that these tools are
available (and verified experimentally), it would be interesting to see
if your revised formula indeed provides a better estimate. Will look at
this evening... 

-- Bert --
-- 
Bert Hickman
Stoneridge Engineering
Email:    bert.hickman-at-aquila-dot-net
Web Site: http://www.teslamania-dot-com

Tesla list wrote:
> 
> 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.

-- 
Bert Hickman
Stoneridge Engineering
Email:    bert.hickman-at-aquila-dot-net
Web Site: http://www.teslamania-dot-com