its to run a small air pump but a regular 12v supply is to little
Transformerless supplies are sometimes used in things that have no external "user" connections, like plug-in remote controlled power sockets.
They are normally used for very low power devices.
For one to work, the impedance of the "voltage dropping" components must be matched to the maximum load current, and there must be a load to always take that current, or the output voltage will rise to the same as the input voltage.
With a typical tiny bit of electronics, the permanent load can just be a small zener diode, to take the excess current.
Your 12V air compressor probably takes 5 - 10A.
That means the stabilising "load" would have to take more that 10A to prevent the voltage rising above the 12V range.
And the series impedance has to be such that 10A is passed through the rectifier to the load all the time.
The 120W shunt regulator will cost at least $10 and run very hot, needing a lot of cooling.
The total series impedance to pass 10A from 110V will be around 10 ohms. Doing that with a resistor would dissipate another 1000 Watts as heat; basically an 1KW electric heater element.
Using a capacitor in place of a resistor, you would need a 270uF 250V AC rated one.
Something like this:
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As you would be using a bridge rec, you should actually use four of those to keep it a balanced circuit, so two in parallel connected from each AC input to the bridge rec.
You would end up with something massive, weighing possibly around 10 - 20lb, costing at least a couple of hundred dollars, that wastes large amounts of power and heat, & is also lethally dangerous.
Or you just buy a proper transformer, or pay about $20 for a complete, ready built, 12V DC power unit.
In other words, "transformerless" power supplies are totally impractical for anything other than trivial loads.