Always remember that safety is an overriding concern in the primary area of a Tesla Coil. That is where there is enough Voltage with enough current capability to electrocute you if you do not keep electrical lab safety in the front of your mind. You get exactly one chance to be safe - no scond chances in many cases!
I am making the assumption that you are trained and qualified in electrical safety. Your own safety is up to you. I am not responsible for your safety.
Primary Capacitor Considerations
The primary capacitor bank and the primary power supply have an interaction that limits the maximum practical capacitance value for a given primary supply. See my previous write up on "Primary Power Supply Considerations" for details.
For a single NST (12-15 kV, 30 mA short circuit current) a good design goal for the primary capacitor bank value is on the order of 2000 to 3000 pF (2-3 nF) max with a working Voltage of at least 30 kV DC (the actual peak Voltage would normally run 20 kV buy transients push the peak higher). If you use two NST's properly phased and paralleled, you can double that capacitance. It is a good idea to have a capacitor protective spark gap set for 30 kV directly in parallel with the capacitor - that is nominally about 0.50 inch spacing for half inch diameter spheres. Make sure the capacitor protection spark gap is not in line of sight from the main spark gap because when the main spark gap fires it produces UV light which can trigger another spark gap even though it is otherwise below its normal breakdown Voltage.
The capacitor bank is one of the more challenging components to do right for a Tesla Coil. The discharge pulse currents are in the 100's of Amps peak. Ordinary metalized film capacitors won't stand up to the pulse currents. And the capacitor bank has to be non-inductive or that is going to limit your peak current and spark output. Non-inductive construction means the individual plates are stacked and connected In parallel as opposed to one long rolled up capacitor. There are non-inductive type film caps sold that will work. These are typically polypropylene "extended foil" construction. That means that the metal part of the capacitor structure internal to the part sticks out each end and then was welded or soldered together.
Capacitors are usually specified in micro-farads, nano-farads, or pico-farads.
Micro is 10^-6, Nano is 10^-9, Pico is 10^-12
Or 1000 pF = 1 nF; 1000 nF = 1 uF
Building Your Own Capacitor
If you build your own caps you have to design for a higher Voltage than the nominal working Voltage requirement. Think of the capacitor as many individual plate capacitors in parallel - a single fault in any of the dielectrics will lead to failure of the whole capacitor. To account for both reliability and inconsistent material with spot weaknesses you should design for a dielectric that can nominally withstand around 2 times the normal working Voltage to take into account the material variations and spot weaknesses across the material. Also unless the capacitor is vacuum impregnated with a high quality insulator you have to account for the fact that high Voltages will cause break down along the surface of the film between conductors - especially in the presence of carbon, moisture or dust. If you don't have the capability to do a high quality vacuum impregnation of the assembled capacitor you should allow a total of 2 inches of surface leakage length between the metal capacitor plates (conductor edge over the insulator edge and back around to the next plate.)
Theoretically, for a well made parallel plate capacitor with all the air spaces pressed out, the capacitance per plate (per dielectric layer in stacked plates) is:
C = K x 0.2248 x A/D in pico farads
K is the relative dielectric constant for the dielectric material
A is the plate area of the overlapping metal plates in square inches
D is the dielectric thickness
Note that because capacitance is a function of the overlapping metal plate area and you have to leave surface creepage spacing around the edges of the metal plates - If you build your own capacitors it is best to go with the biggest practical plate size. When I was in high school and built my first NST powered TC I used a single piece of plate glass about 3 feet by 3 feet with aluminum foil epoxied to each side of the glass with contacts bonded to the foil. I left a 1 inch space all around the foil back from the edge. Eventually even though the glass was 1/4 inch thick it punctured in one spot. Ordinary window glass has voids (bubbles) and specks of mineral impurities that act as weaknesses. I was able to cut the foil away from the defect and apply an epoxy patch to make it work again. I would now recommend now that you use one sheet of plexiglass that is at least 1/8 inch thick if you wanted to go cheap on the capacitor - It is easier to cut and work with than glass and I believe has less defects. Polycarbonate (Lexan) is even better because it won't splinter like plexiglass when cut and drilled. You can tuck the big capacitor plate up under the primary and secondary coils. Don't forget to use adequate air-space or insulation in mounting the capacitor. Wood is not to be trusted as an insulator at high Voltages.
Dielectric Properties for Tesla Coil Capacitor Construction
Material. Volts/mil Kr Loss Tan Cost.
Polycarbonate. 380 3 0.0012 Moderate
Plexiglass. 400 4 0.001 Low
Phenolic. ~350 5 < 0.03 Moderate
Polypropylene. 500 2.3 0.0002 Low
PET 400 3 0.002 Moderate
UHMW PE 900 2.3 0.0002 Moderate
Mica composite 450 4-5 Excellent High
FR4 bare PCB ~300 4.5 0.02 High
Window Glass ~200 ~7 Good Moderate
Teflon (PTFE) 1500 2.1 0.001 Very high
Formica laminate 450 ~4.7 0.04 Low
ABS plastic 400 ~3 <0.02 Moderate
Acetal Plastic. 500 3.7 0.005 Moderate
(1) Where multiple published values exist - I used lowest one for breakdown.
(2) All assume moisture free and clean dielectric.
(3) All are for between planar surfaces (not point breakdown or surface creep).
(4) Technical glass much higher Volts/mil but window glass has voids and inclusions.
(5) Phenolic is wood/paper/cellulose fill - lowest cost option.
(6) FR4 V/mil for aged board material, through the board.
(7) 1 mil is 0.001 inch
The typical extended foil, high peak current capacitors available from distributors like Mouser Electronics are only available in working Voltages up to a few kV. That means you have to stack them in series. You must use identical caps in the bank otherwise when the caps discharge they will not share the Voltage equally. And if they are operating near their rated limit and a single cap fails shorted it will over-stress the lot and cause cascading multiple failures. You also need to use high value resistors in parallel with each step in the capacitor ladder to help spread the stress. Note that connected in series with equal capacitance the net capacitance is C/n where C is the value per cap and n is the number of series stages. If you stack identical capacitors in parallel the net capacitance is C x N where C is the capacitance per cap and N is the number of capacitors in parallel. If necessary to make the required value from the caps you can get, you can place identical capacitors in parallel per rung of each step in the series ladder.
For current capacity reasons it is better to parallel several (identical) capacitors of the operating Voltage rating you need to get the capacitance you need than to series larger value but lower operating Voltage rated caps to get the Voltage rating you need. But if you have no other options, it can work to series the caps in a ladder - But you must allow a very generous margin on the Voltage ratings to account for non- equal sharing of the Voltage.
Adding equal resistors across each rung in the ladder of the series capacitors helps balance the Voltage stress. Generally, like in the case of bleeder resistors; you don't want to burn more than 10% of your supply current drain in the resistors but need to burn at least 5% of the supply current drain to have a practical effect. Resistors have both Voltage and power ratings. The power dissipated in the resistor is:
P = (V^2) / R in Watts, Volts (RMS), and Ohms.
The power and Voltage rating will be specified by the manufacturer. Unlike capacitors, series or paralleling resistors to get the required Voltage or power rating is not so fussy as long as the individual resistors do not exceeded their ratings. The only other consideration for resistors is that if you are at or above 1/2 the power rating you need to physically space out the resistors a bit away from each other so they don't contribute to heating each other.
Primary Wiring Considerations
The primary discharge path from and through the primary capacitor, main spark gap or interrupter, primary coil, and back to the primary capacitor has amazingly high peak currents in the hundreds of Amps - by intention. While that won't burn out ordinary gauge wire because of the low duty cycle, the self inductance of the wiring will limit the peak currents and hence your spark length. You reduce the self-inductance of this wiring by using either wide stripes or tubes of wiring materials and keeping the path length short. I prefer to use 1/4 inch wide copper braid. If you don't have that you can use the outer conductor of RG-58 (50 Ohm RF coax) which is normally a copper braid. You can strip the braid out of the RF cable or use the cable as is (connect to the center conductor too). When I use bare braid I cover it with heat shrink tubing and shrink it in place and then double up on the insulation by pulling it through vinyl plumbing type tubing.
The path between the primary power transformer secondary and the capacitor bank only carries the relatively low charge currents so wiring gauge is not a consideration here. However, you must use wiring that is rated for the Voltage.
Standoffs, Insulators, and Ground Plane in the Primary Area
When dealing with high Voltage wiring, you generally either wire directly to the components which are spaced above the mounting plane or use stand-off insulators as tie points. Bear in mind that NST outputs are generally center-tapped to case ground - which must be tied to the mains supply safety ground (green wire in the USA). So each side of the NST is at 1/2 of its output Voltage. For an NST you need at least an inch spacing of the output above the mounting plane - be sure to account for the screw clearance inside the insulator. You can use commercial ceramic or porcelain,stand-off spacers, or build your own spacers out of phenolic rod or polycarbonate.
I advise using a piece of sheet aluminum or aluminum flashing that goes beneath the NST and the primary wiring that is well grounded to the frame or safety ground and the NST case. The reason is three fold:
Safety - Do not underestimate the possibility of a fire in the primary area.
Electrical safety - Better a wire short out to safety ground than have an object electrified.
RF Grounding - Helps reduce undesired RF interference signal radiation
Happy but safe coiling!