Re: Inductance of a conical coil

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

Original poster: "Antonio Carlos M. de Queiroz by way of Terry Fritz <teslalist-at-qwest-dot-net>" <acmq-at-compuland-dot-com.br>

Tesla list wrote:

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

 >           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

This reduces to 2/sqrt(cosa+sina)

 > 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.

The factor is then 1/sqrt(cosa+sina)
(sina and cosa are interchanged in your text)

 > 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.

I have run some tests. The correction factor 1/sqrt(cos(a)+sin(a))
works well. It has a maximum value of 0.841 for a 45 degrees coil.
I notice, however, that the sqrt factor reduces the factor excessively
when the angle is close to 0 or to 90 degrees.
I propose then this correction factor: 1-k*sin(2*a)
k is a constant chosen so the calculation is "exact" for some particular
geometry. For example, consider a coil with:
minimum radius=0.1 m
winding length=0.1 m
angle varying between 0 and 90 degrees with the horizontal.
100 turns, 1 mm wire.
My formula requires a factor k=0.1466 to match Fantc exactly at a=45
degrees. The correction factor is then (1-0.1466*sin(2*a))
A comparison (sizes in meters, L in mH):
a     rmin  rmax    h       Fantc  cone   Bart  Antonio
0     0.1   0.2     0       3.78   3.85   3.85  3.85
22.5  0.1   0.1924  0.0383  3.64   4.03   3.52  3.61
45    0.1   0.1707  0.0707  3.26   3.82   3.21  3.26
67.5  0.1   0.1383  0.0924  2.70   3.02   2.64  2.71
90    0.1   0.1     0.1     2.07   2.08   2.08  2.08
For your coil (rmin=0.1445 m, rmax=0.3838 m, h=0.0871 m, n=13.6), I
obtain L=0.103 mH (Fantc: 0.102; Your correction: 0.101).

Note that there is still a problem with Wheeler's formula for flat
coils, that doen't match Fantc well. By the way, a convenient expression
for it (identical to the original) is:
Lflat=((rmin+rmax)*N)^2/(60*rmax-28*rmin) uH
Multiply by 100e-6/2.54 to use sizes in meters and get the result in H.

Antonio Carlos M. de Queiroz