tesla coil - Calculate Capacitance in inductors

I have an air core inductor, and I need to know the capacitance for a tesla coil design- "Secondary Coil Capacitance", not the Top Load capacitance, though. I have already figured out the inductance from this question, but I need to know the capacitance also, and because my inductor is not standard (the wire is spaced by fishing line, and not just next to each other), the formulas don't apply from here.

How can I calculate the capacitance of an air core inductor, here are my parameters:

Height = 46.5 Inches

Diameter= 12 Inches

Turns = 1066 turns

Wire Gauge= 24AWG

Spaced out with = Fishing line the same size as the wire

""

Answer

Air core inductors can have small capacitance. You can get an idea how small by considering the capacitance of two wires parallel to each other of length \$\pi D\$ (where \$D\$ is the coil diameter). That's like the turn to turn capacitance in the coil, and it's a small number. Of course, the total capacitance will be more than just the series combination of these turn to turn capacitances since the windings are coupled and will add in kind of a series/parallel way, but still a small number. That might make it hard to just measure by resonance. Probe capacitance of ~15pF or 20pF could be close to the coil capacitance.

Medhurst did a lot of work during the 1940's on inter-winding capacitance of air cored single layer solenoids. An equation that is an extension of Medhursts work is:

\$C_{\ell }\$ = \$\frac{4 \ell \epsilon _o \epsilon _{\text{rx}} \left(\frac{1}{2} k_C \left(\frac{\epsilon _{\text{ri}}}{\epsilon _{\text{rx}}}+1\right)+1\right)}{\pi \text{Cos}^2 \psi }\$

where
\$k_C\$ = \$\frac{0.106 D^2}{\ell ^2}+\frac{0.717439 D}{\ell }+0.933048 \left(\frac{D}{\ell }\right)^{3/2}\$

and \$D\$ is coil diameter, \$ \ell\$ is the coil length (or height), \$\psi \$ = \$\tan ^{-1}\left(\frac{p}{\pi D}\right)\$, \$p\$ is the pitch or turn spacing, \$\epsilon _o\$ = 8.854pF/m, \$\epsilon _{\text{ri}}\$ is relative permittivity of the coil form (2.56 for polystyrene, for example), and \$\epsilon _{\text{rx}}\$ is relative permittivity of space external to the coil (probably just air). For your case of \$D\$ = 0.3048m, \$ \ell\$ = 1.1811m, and \$p\$ = 1.08712mm, \$C_ \ell\$ ~ 21pF.

Reference for this is "The self-resonance and self-capacitance of solenoid coils" by David Knight. The equation is 5.3 on p 25, but there is a lot of detail of its development.

You might also be interested in the work of Gary Johnson who has written a book about Tesla coils. Here is a sample "Tesla Coil Impedance", and the rest is on his web page.