Firing HV spark:
http://www.instructables.com/id/Start-Fires-With-Electricity-DIY-Arc-Lighter/

I actually looked around and found that instrusctables too. He should not be playing around with mains and HV, as it looks like he did not mention ANY precautions to take and common risks, other than just saying that it is dangerous. And his DC power supply is an incredibly basic and terrible design. It does not even have any snubber network!

I also found this website that says that they make HV arc lighters that acheive very high temperatures but will not fatally electrocute you: https://www.arclighter.com/impulse-faq/. So that means it should be possible to get arcs with relatively low currents.

And lastly, what if you started a high frequency high voltage arc, but then very quickly (a few uS) switched over and applied a medium-high dc voltage? If that is possible, could you then supply much more current as the ionized air would be very conductive? I have a 60V 2A DC-DC boost converter that would be able to supply some power. So is this possible?

 

Both lighters are similar, they are not DC HV, but high frequency AC HV. Such (HF) voltage is absolutely not dangerous for human, you even can not feel it. You can only thermally burn your skin in spark, like in flame.

You can discharge 60V in gap after previous heating electrodes by spark to 3500 Celsius degrees.
Edit:
How small is resistance of gap in process of 1mA 10KV (10W) spark? It is 10,000/0.001=10,000,000Ohm=10MOhm.

 

I've decided to revisit this project. I want to have a working arc lighter for when I next go camping, in a month or two. But I also want it to be DIY, for customization and other reasons.

For the battery, 18650s seem to make sense here. I'll make sure to get a proper BMS. I am thinking 3s to get higher voltages. A recommendation for 3 pieces 2Ah+ 18650s with 10A+ discharge would be great. The ones I have found are too expensive (shipping or just total cost), or BS (9000mAh cells). I have found some good BMSs that are reasonably priced, and a good holder for the cells.

There are plenty of circuits out there. I want to design/use one that is fairly efficient (70%+). The goal is 10-15kV at 3-4mA. That means up to 60 watts in. I want a lot of power in the arc, for easy ignition. I also want to ensure it will not deliver more current than that, for safety reasons. So one idea I am considering is some sort of adjustable VCO that controls and H-bridge powering the primary. The issue here is that it must deliver fairly constant power on the output. An impedance of 2.5 ohms means 5A and 63W at 12.6V (100% voltage), but 3.6A and 32.4W at 9V (cutoff). So it needs to adjust somehow. A VCO, decreasing inductive reactance with voltage to increase current and acheive constant power, seems like a good approach.

I do not want to make a resonance circuit because the inductance will be too hard to predict, and may change.

 

I will be getting this PWM board to quickly test out the inductance of the coil, and to find out the turn ratio. Then I will determine the appropriate frequency range of the VCO. I may also go with a center tapped coil to only need two n-fets for the bridge. But this means a larger turn ratio, and it may be too large. It also means less inductance. But it may still be beneficial. I will have to do the tests to find out.

 

Expendable butane lighters are reasonable and taking a couple of each outdoors trip they should last a couple of years except if you intend to begin many flames day by day. I don't see a less complex less expensive option.

 

If you can find an older lighter that uses flint and lighter fluid, replacement flints were available and still are. Instead of lighter fluid you can use gasoline, the unleaded kind, which is all you can get in the US these days. Just don't carry it in your pocket. At $3 per gallon, and given that my old ZIPPO lighter would fill up with half a spoonful of gas, the fuel cost will be very small indeed. A secondary benefit is that you can have a bit of gasoline on hand to aid in igniting stubborn kindling. I am talking about a few drops, before the fearful folks go into a safety panic. One drop of gas will help slightly damp paper ignite when a match alone will not do it. This is an old survival trick. No need to haul around a heavy transformer just to ignite a campfire for cooking and staying warm.

 

If you can find an older lighter that uses flint and lighter fluid, replacement flints were available and still are. Instead of lighter fluid you can use gasoline, the unleaded kind, which is all you can get in the US these days. Just don't carry it in your pocket. At $3 per gallon, and given that my old ZIPPO lighter would fill up with half a spoonful of gas, the fuel cost will be very small indeed. A secondary benefit is that you can have a bit of gasoline on hand to aid in igniting stubborn kindling. I am talking about a few drops, before the fearful folks go into a safety panic. One drop of gas will help slightly damp paper ignite when a match alone will not do it. This is an old survival trick. No need to haul around a heavy transformer just to ignite a campfire for cooking and staying warm.

I already have a lot of the stuff, and I want to make an arc lighter for other reasons too. The transformer I have is about 1 in*1 in*1 in. It can generate a 5mm arc with the right circuitry. I actually made an oscillator with a shunt, comparator, and mosfet that worked fairly well. The one issue is that it keeps killing the mosfet.