The aim of this experiment was to design a basic air cored resonant transformer known as a tesla coil, rated at over 300kV output. I would like to thank Jeff Messer from Zap Tesla Coil for supplying us all parts required for the construction of our transformer.
The function of this air cored resonant transformer is not in any way similar to any other transformer, and it is in fact best described as a 
velocity inhibited slow-wave helical transmission line resonator.
Circuit values can be calculated using the Air cored resonant transformer calculator
The first step is to choose an hv transformer to supply the required power for your system. I used a 450 Watt type at 15 kV output.
23 x 0.15 µF in series will give us 6.5 nF, quite close to the calculated 6.37 nF. The peak voltage in each capacitor will be 15 kV * 1.4/23 = 922 V. We choose 2 kV capacitors to be on the safe side.
Next we choose the operating frequency of our resonant coil. For my coil, I choose 240 kHz. From this value we get the quarter wavelength which gives the length of secondary coil wire. Using the calculator we get the length of 1025.26 feet or 312.5 metres. I used AWG24 which results in a coil of 18 turns per cm and a total length of 19".
There are several ways to wind up a secondary coil of 312.5 metres length of coil, but not all coils will give good results. The following formulas are used to derive the best diameter and aspect ratio for the secondary coil which mainly depends on the power P, in my case 450 Watts:
Aspect ratio height/diameter A= 13.8*P-0.197 = 4.14
Recommended secondary coil diameter D(inch)= 0.323*P0.4 = 3.7"
Of course I had to choose the nearest plastic pipe diameter available which was 4.5", and 4.5"*4.14 gives a length of 18.6" so I cut the pipe to 19".
We then put in reasonable dimensions for the primary pancake coil, until we get the correct inductance value.
The distributed capacitance of the secondary is equal to Cl
where Cl = (0.1 A + 0.32)*D ... A = secondary coil aspect ratio, D = diameter in cm
Cl = (0.1*4.2 + 0.32) * 11.5) = 8.51 pF.
The inductance of the secondary coil is 18500 µH, so the required capacitance for resonance at 240 kHz is approximately 24 pF. The external (top load) capacitance is equal to 24-8.5 pF = 15 pF. We have chosen an off the shelf sphere of radius 0.3 feet which gives 10 pF.
Assuming a conversion efficiency of about 85%, which is a reasonable value for a well designed resonator, the secondary peak voltage at resonance is given by:
Vmax = Vcap * (Eff * Lsec/Lprim)0.5
Vmax = 15 kV * 1.4 * (0.85*18500/69)0.5
Vmax= 317 kV
It is quite interesting to note that the turns ratio of the primary & secondary coils do not have any effect on the voltage gain. Also, we can have a good approximation of the actual voltage generated from the fact that air breakdown occurs at 30kV/cm. Thus:
Field at surface of our sphere = Voltage / Radius of Sphere
So Minimum Voltage for streamer to generate around the sphere > Breakdown electric field * radius of sphere
Voltage > 30kV/cm * 10.16cm
Voltage must be > 304 kV.
