I think I understand what you’re trying to get at.
A 12 volt pump will not run off of 110 volt and a 110 volt pump will not run off 12 v. 110 is alternating current or ac. 12 v is direct current or dc. Now, if you have a 110 volt pump you can run it off of batteries but only if you change the dc power (batteries) to ac power. You do this using a power inverter. It converts dc to ac so you can run that 110 volt pump off of batteries. You aren’t running it off of dc though because you’re converting it to ac power just like you would have in your house.
Not sure if this is a good example but here goes. Imagine you have a bunch of solar panels. You have to store that power in battery banks for use at night or cloudy days. You have to use a power inverter to convert that battery power to ac so you’re able to power regular household appliances. If you took a 12 volt pump and hooked it up to those same batteries right on the terminals that pump would run. Now, if you took that same 12 volt pump and, plugged it into an outlet of your house, it’s not going to turn on because that power inverter converted the dc power to ac.
It’s not as simple to run a 12 volt pump off of 110 volt. You mentioned a transformer. You’re correct in that one will convert ac power to dc but there is so much more to it than converting dc to ac. It’s pretty simple doing it for things like thermostats or low voltage 12 v lighting but it’s not so easy when powering devices that require a lot of amps. Sure, it can be done but it’s not feasible and not as simple as going from dc to ac.
I’m not that good at explaining things so hopefully you kind of get what I’m saying. I’m sure Alex @Infinity can make it easier to understand and he knows more than I do about electrical so he can correct me if I’m explaining things incorrectly.
Anyways, we own in a split level so I was able to tie into the rectangle duct of the hvac quite easily because the duct work in between the basement ceiling and upstairs floor inside is located about 5 foot up the garage wall. I made up a rectangle damper door and installed it on a damper motor. The damper motor is controlled by a second thermostat that’s next to the regular thermostat. My wife can set it on whatever she wants and the house stays right on it. When it calls for heat the damper motor opens the damper door and the blower blows into the house ducts. When it’s satisfied the damper opens and allows the blower to blow hot air in the garage through vent that I cut into the side of the rectangle duct. Anyways, I originally was just using a 24 vdc transformer that I purchased at a big box store. I burned up two of them within a few weeks. That’s what got me doing a little reading about transformers and found out I needed one with a higher amp rating. I don’t really know anything about it but I do remember coming across a lot of electronics guys having to build circuit boards to convert ac to dc. Maybe it was more of a want rather than need. Although I’m sure they have them I also don’t remember seeing convenient ways of converting ac to dc as easily as doing the opposite with an inverter.
