@Lol999, the devices you are considering are cheap toys, not transformers,
per se. They produce very short, not very intense arcs, that may make for a disappointing Jacob's Ladder display. They consist of a driver oscillator for the primary of a resonant fly-back type transformer whose secondary winding has thousands of turns of lightly insulated wire. That is where the "Tesla Coil" terminology, sometimes used to describe these devices, originates. By building up a strong magnetic field in the primary, using a low-voltage DC, and then rapidly collapsing that field by interrupting the current, high-frequency, high-voltage oscillations are created in the secondary. This is exactly the manner in which ignition sparks for internal combustion engines (ICEs) were generated, hence the name ignition coil sometimes is applied to such a coil, even when it is not used to fire the cylinders of ICEs. Other names, such as induction coil and medical coil have also been applied. Whatever you call them, they all depend on periodically interrupting a low-voltage, high current, magnetizing field in the primary circuit to generate from the rapidly collapsing magnetic field a high-voltage, low current, electrical field in the secondary winding.
I have seen these devices sold to schools as a means of creating short high-voltage arcs for various classroom demonstrations. The early ones used a simple magnetic reed switch, controlled by a knob on the handle, to select the interruption frequency and duration, thereby giving some control over the strength of the sparks produced.
Before CRTs vanished from television sets, to be replaced with "flat screen" LED, LCD, and plasma arrays, the high voltage that was needed to accelerate cathode ray tube electrons to high enough energy to illuminate the phosphor viewing screen was generated with similar circuits. The main difference was the CRT needed a DC power supply, and transformers can only produce AC. Some sort of high-voltage rectifier was necessary to produce a DC output, but this is not necessary to create an "arcs and sparks" Jacob's Ladder display. So your "toy" "ignition" transformers, although perhaps lacking a rectified DC output, may be adequate for your purposes of creating a very small demonstration Jacobs Ladder.
I will get around to advising you on how to keep from burning them up in a moment.
Jacob's Ladders have quite a history as special-effects props in science-fiction and horror films.
Here is a film clip from an early effort. All of these Hollywood props were built from neon sign transformers, many if not all, probably salvaged from the constant re-modeling and re-building projects Hollywood is famous for. Anyway, these are very serious transformers capable of illuminating thousands of watts of luminous tubing with hundreds of milliamperes of current at tens of thousands of volts. That's more than enough to kill you.
Most of the smaller ones, like those used to light up beer signs in bars, operate directly off 120 VAC, 60 Hz, utility power and do so continuously without overheating. Other versions use 240 VAC, 50 Hz, but are also designed for continuous operation. In the USA, the secondary voltage must be current limited by the transformer design, and for secondary voltages greater than 15,000 VAC the secondary winding must be center-tapped and grounded to the steel case of the transformer. It is wise to also connect this "ground" to the utility neutral and to a purpose-driven earth ground, but AFAIK Code doesn't require it.
Since you haven't acquired any parts yet, I would suggest you look for a small neon sign transformer on eBay or Amazon. Purchase one with at least 5000 VAC and 30 mA current capability. Using a 12 VDC wall-wart for a power supply, Google a circuit to rig up a 555 timer and a relay with a 12 VDC coil and SPST (or DPDT) 120 VAC, 10A contacts to control application of power to the primary of the transformer. Buy a short extension cord with at least 14 gauge wire and cut it in half. Wire the wall-wart 12 VDC output and the 555 circuit with the relay between the two halves of the extension cord. The half that plugs into the wall goes to normally-open contacts on the relay. The other half of the extension cord accepts a convenience plug from the NST and is wired to the other half of the normally-open contacts on the relay. Pressing a button will charge up a capacitor and turn on the 555 which will operate the relay. Place a 1N2007 rectifier across the relay coil, cathode to the wall-wart positive supply terminal, anode toward the 555. After a period of time has elapsed, the capacitor will be discharged enough to turn off the 555 which will de-actuate the relay and turn off the Jacob's Ladder. Let us know if you need any help finding a circuit for the 555 timer. Maybe I'll breadboard this after you get the NST and show you how it works. Or maybe some other "hands on" type of member will do that for you. I have all the parts in front of me, just not enough time to work on it right now.
While you are waiting around, Google to find out what voltage is required to cause an arc to jump between two needle electrodes in air. Multiply that number by at least two to account for round wire electrodes in the Jacob's Ladder. A 5000 VAC NST should not require an auxiliary ignition electrode, but a sewing needle you hold on a plastic stick should work.
If you want to prevent the toy versions from burning up, a simple push-button switch to turn it on for brief periods should suffice just as well as a timer... but you will get tired pushing the damn switch while manning the advertising display booth. I would opt for the "professional" approach of a real neon sign transformer and a delay-off timer. Good luck, and please keep us posted on your progress.
