It's a very safe power supply -- I assume the 2.2k was intended to be a pull-up, but with the unfortunately placed(?) 1N4007, it'll never turn on.
The freq pot should have a certain minimum resistance in series with it (maybe 1/10th to 1/100th its total resistance), and a maximum in parallel -- this ensures that the 555 never tries to eat itself making too high of a frequency, nor gets stuck in a fixed state if the pot goes open (0Hz isn't much AC).
Not as dangerous in this case, since the 555 is limited by the 10k to pin
7, and is only making a ramp, but too high or low a switching frequency will still cause problems for the output half.
Likewise, the PWM control shouldn't go all the way to 2/3*Vcc, but a bit shy of that. So the output doesn't accidentally get stuck on 100% duty. Changing the resistor below the pot to 15k would be about right.
You can also replace the 555 with the other half of the '393, e.g.:
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As mentioned, the output stuff is missing a pull up, so at worst, the LM393 sets the output high all the time (PWM control at negative side), and the output just kind of casually drifts up due to leakage (primarily of the top 1N4007). As the MOSFET goes into conduction, it starts to cook, magic smoke releases, and your supply ends up shorted (if it's beefy enough to clear the short, the magic smoke will transition into magic fireworks before everything settles down).
Most of the time, it'll just be switching terribly, because (assuming the
2.2k is wired in) the pull-up capacity is severely lacking to drive a MOSFET gate at any kind of frequency.
IRF740 is also a poor choice for 12V 5A. If the flyback voltage is low, an IRFZ34N would be passable, but more likely, something like "20N25" would be better (i.e., ~20A, 250V, N channel). You'll still want a heatsink, but it won't be as bad as the IRF740 (which just isn't worth using on 12V).
The TVS diodes are also rather restrictive, and not used correctly. The most likely result (once it's switching, that is) is a cooked schottky diode (it's only rated 40V, and will avalanche at 50-60V for a few pulses before failing shorted).
Here's a more complete example, using a BJT output:
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Instead of an FBT, an ignition coil is of course suitable. In that case, I would recommend a 200-600V transistor (BJT or MOSFET), and instead of UF4007 and the RC damping network, a TVS diode (e.g., P6KExxxA, SMDJxxxA) rated for about half the Vceo/Vdss of the transistor. So, 100-300V diode,
200-600V transistor. As long as the diode doesn't cook, you can pretty much crank as much power from this as you like.
This isn't variable frequency, because the flyback is pretty standard (it's either intended for 15.7kHz from a TV, or 30-100kHz from a monitor). Ignition coils are also fairly standard, given there are a few types to choose from (old fashioned points, modern transistor driven, CDI). You could pick a couple capacitor options for each class of component and leave it at that (frequency isn't as important as PWM, anyway).
Tim