It takes a threshold amount of current in a Tesla coil arc to be 'suitably' bright. This may involve subtleties such as the depth of ionization of atmospheric gas molecules. When lightening strikes, the electromotive force (emf or voltage) has become high enough to rip valence electrons out of the nitrogen & oxygen gas molecules in the atmosphere from 'there to there' so to speak. These electrons constitute current flow. A neutral atom of nitrogen has 7 protons (+charge) in the nucleus and 7 electrons (-charge) orbiting the nucleus (oxygen atom has +8 and -8). But gases are diatomic meaning molecules of gas have two atoms each, N2, O2 etc. So a neutral molecule of nitrogen has 14 electrons and protrons, neutral oxygen has 16 protrons and electrons.
When one or more electrons are ripped from a molecule it has an imbalance in (+) and (-) charges and therefore becomes a 4th state of matter - plasma. Plasma, ionized molecules plus free electrons, in an electric field will accelerate in opposite directions. Since a nitrogen ion is very much heavier (~51,200 times) than an electron, the electrons do most of the moving. The free electron will accelerate to high speed before crashing into another molecule of gas and knocking out more electrons and maybe be captured by that molecule. When an electron is captured by an ion it gives up accumulated kinetic energy which energy is seen as Tesla Coil arc light.
The usual first choice of power for TC builders is a neon sign transformer (NST). They are rated in kilovolts and milliamperes such as 12 kV and 30 mA (caution; LETHAL shock hazard!). Unlike normal transformers, sign transformers have special cores built to provide a flux leakage path (air-gap) in parallel with the secondary high voltage winding's iron core. As the NST secondary load current is increased, a portion of the primary coil flux finds it easier to jump the air gap thus limiting the power to the secondary coil. A NST secondary can be continuously shorted without harming the transformer. Shorting the secondary of a powered up ordinary transformer will invoke the proverbial smoke test!
Power is the product of voltage times current. The TC builders goal is to produce a 'long impressive looking' arc. Since a bright arc requires a certain minimum current, the input power required to make TC arcs is more or less proportional to the desired arc length for given coil efficiency. I'm going to take a stab at it here; it takes about 40-50 watts TC input per inch of TC arc length. So a 12 kV 30 ma transformer may yield 7.5 to 9 inch arcs. A variety of neon sign transformers are available up to 60 mA. Though I have not tried this; identical transformers can be used in parallel to increase power input. Also see chart of spark length vs voltage in CRC Handbook of Physics & Chemistry 61th Edition page E-55, the correlation is not linear. NST output voltage selection is based on adequate voltage to ignite an arc in the TC's primary spark gap; usually 9kV to 15kV. A 9 kV 30 mA NST will be same size (weight) and power as a 18 kV 15 mA NST.
Need for maximum power transfer invokes a really neat guiding principle. Maximum power is transfered when the source impedance is equal to the load impedance. For the Tesla primary circuit this means matching the NST output impedance with the primary circuit capacitor impedance. To first order, the NST output impedance (Z)=Volts/Amps. For 12,000 volts at 0.030 amps the output impedance is 4E+5 (400,000) ohms. To get maximum energy out of the transformer secondary into the primary capacitor as stored charge requires the capacitive reactance to be equal to 4E+5 (12000/0.030) ohms. The capacitor impedance formula is: Z=1/(2*π*f*C). Solving for the capacitance, (C)=1/(2*π*f*Z) where freq=60 Hz in America (50 Hz in Europe). The capacitance should be 0.0066 uF. The capacitor voltage rating needs to be high enough to withstand voltage produced at resonance in the primary that could be up to double the open circuit NST RMS output (34 kV). I'm going to conduct a test to check impedance linearity of a NST; this will determine if the sweet spot may be slightly less than or greater than the predicted optimum capacitance.
The TC secondary is a single layer coil wound on an insulating cylindrical form. The coil can be wound to within 1/2" to 3/4" inch of the end to permit mounting end baffles for structurally support; Use nylon screws. The coil should be close wound with formex or formvar insulated magnet wire. I spray the finished coil with glossy polyurethane (available in aerosol cans). Spinning the coil in a lathe or coil machine until the polyurethane gets tacky is helpful preventing 'runs'. Wire from B&S-23 to B&S-30 gage works.
Secondary coil resonant frequency will be affected coil turns, coil diameter, and winding length. Coil resonance from 100-400 kHz is functional. Included in next paragraph is an Equation for inductance from Inductance Calculations P-153 by Frederick W. Grover ISBN: 0-87664-557-0. Grover has a variable(r)=(diam/len) from a lookup table. I generated an algorithm that closely approximates the (F) value that is based on r=(D/L). This is need to complete the inductance computation. Grover uses a lookup chart for (F) given (r).
Inductance: (uH)=0.00254*F*D*N2 Where: (F) is Table Lookup or algorithm; (D) is the mean coil winding diameter (IN); and N is the number of turns (usually 800-3000). My (F) algorithm is: (F)=9.785*(D/L)*(1-(D/L)) 1/3 for use in the inductance Equation. The coil is wound with ~963 turns of 25 gage (0.0181",0.0195" insulated) wire. The wound length is 18.8 inches (1066 ft of wire) that weighs 1.06 Lbs. The coil resistance at ambient temperature is about 33.7Ω and resonates at 353 kHz. The secondary self-capacitance is about 7.17 pF and the HV terminal adds another 3.0 pF. The inductance computes to 20,271 uH. See TC_Coil_Equations for sizing of various TC components. The lead at the bottom end of the coil needs to be brought out to connect to ground; a suitably long ¼-20 screw through the bottom baffle centerline can be used to support the secondary coil and ground the winding. A wing-nut works nice; permits easy removal of the secondary for safe transport to next group of interested fans.
The Primary Coil design needs to consider adequate turn-to-turn spacing to prevent arc-over between turns. It is generally a good idea to wind the primary with bare heavy gage house-wiring copper or small copper tubing. The coil should be wound with an extra turn or two to permit adjusting inductance as a way to tune the Tesla Coil for optimum operation. The primary coil shape can be cylindrical or spiral. Some P-Coils are wound as conical spirals. The P-Coil also needs a value of inductance to match secondary resonant frequency when coupled with the selected Primary Capacitor. The physical size of the P-Coil and number of turns determines inductance.
The simplest (SG) for even lower power Spark Gap Tesla Coils should be made from a hard material like Heliarc Tungsten electrode material. The electrode face should be smooth ball shaped (hemispherical). A simple trick to help cool the electrodes is to embed them in a copper bar heat sink maybe ¼-3/8" diameter that is supported in an insulated bracket.
Some experimenters install a water cooling loop in the copper substrate that supports the tungsten electrode tip to enhance cooling. The electrodes can be mounted in an insulated tube that has fan forced air flow. The air flow may have two benefits; aid cooling the electrodes and sweep away remnants of ionized air in the gap. Hot electrodes break over into an arc at lower voltage than cool electrodes. Also, it takes a short period of time for air ionized during the last spark-gap spark to vanish. These ions can cause premature break-over (at lower capacitor voltage) thus affecting the TC output arcs.
A set of regularly spaced tungsten electrode tips can be mounted on an insulated disk that is motorized to provide a spinning set of spark gap electrodes. An absolute requirement for this to be successful is the motor must run synchronously with the AC power line in order to present a fixed spark gap distance at 'gap spark-time'. A hysteresis synchronous motor can be purchased (expen$ive) OR an inexpensive shaded pole induction motor can be modified to operate synchronously. There are a great number of informative SGTC references on the web; one I found recommends to remove the squirrel cage armature and machine 2 or 4 flats symmetrically on the OD to make 'virtual' poles to appear. I have not tried this but believe it will work. Another method I thought of, but untried to date, is to spot face the armature for rare earth magnet implants to be epoxied in at symmetric locations 2 or 4 places based on 2-pole or 4-pole motor type. A two pole motor runs just under 3600 RPM and a 4-pole runs just under 1800 RPM. With flats or magnets installed, the speed should be 3600 RPM or 1800 RPM. Be sure the magnets are installed with correct polarity to produce the desired fixed magnetic pole polarity for the modified armature. The squirrel cage rotor should still be self starting by induction and 'pop into sync' when getting near armature synchronous electrical (field) speed.