Concept: High Voltage Power Supply

Clearly you need to do more "research" on exactly what and how a Tesla Coil works. Google is your friend here. There are scores of Tesla sites available that will help you. This is probably not one of them because the theory and design of Tesla coils is a rather specialized hobby. It can also be quite dangerous if you don't know what your are doing.

Your "oscillator --> step up transformer --> voltage multiplier" paradigm has nothing to do with Tesla coils. Tesla coils are resonant transformers and do not use "half wave multipliers" or anything involving rectification. The "turns ratio" in a Tesla coil has virtually nothing to do with the output voltage.

Your paradigm does describe how a Cockroft-Walton high voltage generator works. Until I retired last year, I operated a tandem linear particle accelerator that was provided with up to 1.7 MV of terminal potential from a Cockroft-Walton high voltage power supply. It uses a pair of tetrode vacuum tubes in a push-pull 47 kHz oscillator circuit with about 8 KV plate voltage to excite an "air-core" transformer mounted inside a steel pressure vessel pressurized with 125 psig sulfurhexaflouride gas, The transformer steps the 47 KHz primary excitation up to about 50 kV. This is capacitively coupled to the Cockroft-Walton stack of series-connected rectifiers and capacitors... some five hundred of them, IIRC, driven and charged in parallel by the capacitive coupling plates, called "dees" in this device because of their shape. The entire assembly is about five feet long by three feet in diameter, possible only because of the high-voltage insulating dielectric properties of the SF6 gas.

I am not saying you need one point seven million volts to "experiment" with ion propulsion, but you do need a steady source of high-voltage direct current that is current-limited. The Cockroft-Walton design can easily provide what you need using commonly available components. Depending on the size of the energy storage capacitors, the short-circuit current can be limited to relatively small values... perhaps not entirely safe, but certainly manageable with due care. You do need circuitry to shut down the oscillator in case of a current fault (such as arcing) and a "bleeder" chain that can be switched in to discharge the energy storage capacitors.

Or you can pursue your interest in Tesla coils. AFAIK this will lead to no practical results, but it can be fun for awhile.
Arc and sparks and corona discharges are fun to watch as long as no one and no thing gets damaged. Unfortunately this is not always the case if you have the full faith and credit of a large utility backing things up. Be careful!

 

Those are typical components for a voltage multiplier. Diodes with 15 to 20 kV ratings would be preferred. These are made by soldering a stack of 1 kV chips together and packaging the stack as a single high-voltage diode. Forward voltage drop is about 0.6 V per diode in the stack, sixteen diodes typical per stack. You can connect two stacks in series and pot them in epoxy for even higher reverse voltage capability with forward voltage drops in the 10 to 15 V range. This is how the diodes in my accelerator are made. The diode stacks are commercial products.

Another source for high-voltage capacitors used to be TV "door knob" capacitors used in the high voltage section of tube-type television sets. These are hard to find now, but there is lots of surplus available for exorbitant prices. Typical ratings are in the 15 to 20 kV range with capacitance of 350 pF to 600 pF or so,

Glassman High Voltage has been manufacturing high-voltage, low current, laboratory power supplies for decades, using a transistorized power oscillator to drive a "fly-back" transformer (which is a type of Tesla coil) whose high voltage secondary drives a voltage multiplier. Everything is "air insulated" up to about 100 kV with the multiplier mounted inside a plastic enclosure. Very robust, very reliable design. An operating manual for their earlier models can be found here as a PDF file.

Tesla coils depend on the fast time-rate-of-change of magnetic flux coupling energy from the primary to a resonant secondary. The 'turns ratio" does not directly account for the voltage multiplication. A spark-commutated Tesla coil is most common, usually driven by a current-limited neon sign transformer. However, as the Glassman design illustrates, a solid state power oscillator works well, as do vacuum tube designs, All depends on how much power you need and whether you want continuous output or pulses.

Battery powered Tesla coils would avoid the problem of having to filter high-frequency currents back-feeding into the mains, which is a typical problem with spark-commutated coils driven from high-voltage transformers. However, this should not be a problem with a solid-state or vacuum tube oscillator design using either sinusoidal or "square wave" pulsed excitation and adequate power supply filtering.

Why do you want battery operation?

 

As mentioned by Hop you could use a NST (Neon Sign Transformer). They plug into the mains and can provide the high output voltage needed to jump the spark gap. They are self limited with respect to current due to high magnetic leakage so have a sort of built in safety mechanism.

With regards to using a transformer in reverse are you talking about a mains transformer? If so you are only going to get mains out. I personally wouldn't do this, the transformer is not designed to work this way.

Adam

 

Well, the main idea is to have fun and learn something. That looks like a safe enough sized coil to start with. If you want to learn a bit more about how Tesla coils work, here is a PDF file that explains it without overwhelming you with math. It takes a very hands-on approach to how one 2nd year undergraduate physics student constructed a Tesla coil for a science exhibit in 2012. The "paper" is written as if were to be submitted to a peer-reviewed scientific journal, with plenty of citations throughout the text, lots of professional looking color pictures, charts and drawings, and a bibliography at the end with hot-links. All in all, a delight to read and not at all like most of the stuff you see online from the Tesla hobby community.

In my youth, back in the 1950s, I was introduced to Nikola Tesla's works through accidentally reading about him in a book borrowed from the library. Up until that time my electrical gods included people like Thomas Alva Edison, Michael Faraday, and (although I didn't have the math to understand his work) James Clerk Maxwell. I knew these people were important to 20th century progress in electrical technology, but no one even mentioned Tesla. It was only many years later, in no small amount thanks to the Internet, that Tesla has finally been acknowledged for his contributions to electrical engineering. Turns out he earned (and spent) a fortune pursuing his ideas about wireless transmission of power, finally dying penniless on January 7, 1943 in a dingy hotel room in New York City.

If you get tired of Tesla coils, the Philo T. Farnsworth Fusor is a good thing to learn about, and perhaps to build.