Maths and Theory

I have tried hard to avoid everything harder that a square root! The simplest way to do the calculations is with one of the various dedicated programs available for download on the web or use a spreadsheet then you can easily produce tables giving a range of values.

Inductance and coil proportions

The inductance L of a solenoid coil is given by the formula:

L = N*N*R*R / (9*R + 10*H)

Where L the inductance is in uH, N is the number of turns, H is the height of the coil and R is the radius in inches. It is possible to show that the maximum inductance Lmax for a given length of wire is obtained when h=0.9R.

Such a fat squat coil would not be practical the voltage difference between turns would be very high and it would probably breakdown. So we compromise. I was interested to know how the inductance fell as the shape of coil changed. I used a spread sheet to work out the inductance of a given piece of wire for a variety of values of the radius. The radius of the coil and the length of wire gives you the number of turns, the number of turns multiplied by the wire diameter gives you the height of the coil.

As the expected the maximum inductance Lmax occurred when H = 0.9R. I then plotted the induction of the coil as a percentage of Lmax against the ratio of H to R. The shape of graph was the same for a variety of wire lengths and thicknesses.

Y axis is inductance as % of Lmax, X axis is H / R

The significance of all this is that we want coils which have the least resistance for a given induction to maximise Q. It makes sense therefore to avoid the tall skinny coils and go for a bigger radius.

Maximum Q

Wire resistance is not the only factor we also want to minimise the self capacitance of the coil - when this is taken into account it turns out that for maximum Q the height should equal the coil diameter.

Flat Spirals

The formula for the inductance of a flat spiral is:

L = R * R * N * N / (8 * R + 11 * W)

where R is the average radius, W is the difference between the outermost radius and the innermost radius, N is the number of turns. L is inductance in microhenries, uH. R and W are in inches.

Conical Forms

These are usually calculated by taking the average radius and the height and using the equation for a solenoid form. But a very shallow slope may be better approximated by ignoring the height and using the flat spiral equation - must investigate this sometime.

RLC Circuits

An equivalent circuit to a tesla secondary coil

The simplest way to treat the tesla coil is as an RLC circuit.Where R is the ac resistance of the coil and other losses in the system, L is the inductance of the coil and C is the sum of the self capacitance of the coil and the capacitance of the top load. Of course this is a gross oversimplification - it ignores for example the capacitance between turns, but it is sufficient to give us reasonable approximations.

The resonance frequency of a LC circuit is given by:

f = 1/(2pi SQROOT(L*C))

We could in theory calculate the output voltage as:

Vout = SQROOT(L/C) * Vin / R

in practice V out is hard to measure and R the sum of the losses is hard to measure.

But while from the point of view of the generator the tesla secondary is a series RLC circuit. From the point of view of the output terminal the coil is a parallel tuned RLC circuit.

The output impedence is given by:

Zout = L / (C*R)