Re: Inductance of a conical coil

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

Original poster: "Barton B. Anderson by way of Terry Fritz <teslalist-at-qwest-dot-net>" <classictesla-at-netzero-dot-com>

Hi Antonio,

Yes, I remember the discussion. There was no solution posted. I believe we 
did agree the inverse conical equation popular was grossly in error. 
Following that discussion I took a look at it and found the error jumped 
all over the place with the angle.

Here's the basic popular equation:

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 (versus horizontal plane)
          h = Z*sin(X)  Effective vertical Height
          w = Z*cos(X)  Effective horizontal Width
          W = R + w/2   Average horizontal Radius


     L1 = W2*N2/(9*W+10*h)  (Vertical Inductance Component)

     L2 = W2*N2/(8*W+11*w)  (Horizontal Inductance Component)

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

This formula weights the horz and vert Wheeler equations, but is poor in 
accuracy. I've added my own "weighting" to this of which the final L above 
is multiplied by a factor based on the angle.

I then have:
L = [SQRT[(L1*Sin(X))2 + (L2*cos(X))2] ] * my factor

First I find the cosine and sine of the angle in radians and denote them as
"sina" and "cosa". My factor is then: ((cosa^2+sina^2)/SQRT(cosa+sina))*2

When applied to L as above, it results fairly decent, but of course cannot 
pull out Fantc accuracy for conical coils.

Example: (inches)
Angle = 20
Inside Diam = 11.375
Outer Diam = 30.223
Outer Top Height = 3.43
Wire Diam = 0.375
Turns = 13.6

Equation Outputs (uH):
Cone Eq.    w/factor       Fantc
113.93      100.64         101.63

Note, Vert L1(vert) = 156.4 and L2(Horz) = 107.05.

For reference: at 20 degree, sina = 0.939693, cosa = 0.342020. My factor 
then = 0.883292708 and this is what was applied to Cone Eq. of 113.93 to 
arrive at 100.64.

That's how I worked around the error. Of course, we could just use Fantc 
and similar programs to find a better L, but for a 
quick-pop-in-a-spreadsheet equation, I've simply applied this factor.

Take care,
Bart

Tesla list wrote:

>Original poster: "Antonio Carlos M. de Queiroz by way of Terry Fritz 
><teslalist-at-qwest-dot-net>" <acmq-at-compuland-dot-com.br>
>
>Hi:
>
>I was reviewing the archives looking for a good approximate
>formula for the inductance of a conical coil. There was a
>discussion years ago, but didn't come to a conclusion. That
>formula that makes an average between the Wheeler approximations
>for solenoids and flat coils is very poor. The Wheeler formula
>for flat coils is also poor.
>What would be the best formulas now?
>
>By the way, I have added mutual inductance calculation to my
>Teslasim design/simulation program.
>http://www.coe.ufrj.br/~acmq/programs
>
>Antonio Carlos M. de Queiroz
>
>
>
>