...How does that even work? Now, I'm no expert, but I thought I had a pretty firm grasp on Ohm's law, and conservation of energy is one of the basic laws of physics. I mean, we're not all carrying around little perpetual-energy devices in our coat pockets, now are we? As the saying goes, "something's gotta give."
(Note: It's very probable that I am completely wrong here. This is my roundabout way of asking for someone to enlighten me further on the subject.)
Conservation of energy is the reason it works. You can't get 6 volts out of a 4.2 volt battery without exchanging something else to get it, and the exchange here is current for voltage. So we pull more amps from the battery in order to convert it to more voltage for the atomizer. Watts stay the same on both sides (minus a little bit lost to heat and such during the conversion process, but I'm going to leave that out here for simplicity's sake - almost all of these chips have an efficiency percentage on the spec sheet to describe this. A DNA40 is 92% efficient, so it loses 8%, which means the wattage on the battery side is going to be 8% higher than what the display says in order for the wattage to the atty to be accurate).
Basically, wattage has to be the same going into the chip as it is coming out of it, but voltage comes in at a set level of somewhere between 4.2 and 3.2-ish, depending on the battery level. So if we're sending 25 watts to the atomizer and we know the battery is at 4.2 volts, then to get that 25 watts (this is where Ohm's law comes in), we have to pull 5.95 amps from the battery. As the battery drains, the voltage coming in goes down, so the current draw goes up (because, for example, it takes 7.81 amps to get the same 25 watts when you only have 3.2 volts to work with). On the battery side, the chip varies its resistance in order to pull the correct amperage to get the output you want. Resistance of the atomizer only comes into play to determine how many volts to send to it. It doesn't affect anything on the pre-conversion side.
Does that make sense?
tl;dr Watts in = watts out. Which means that any time you're set over 4.2v, amps from the battery are higher than amps to the coil. That's why the spec sheets for all these high wattage devices require high drain batteries.
ETA: Steam engine's battery drain calculator has a setting for regulated devices and shows you what's going on for both sides of the equation. You can put all the variables in and see how it changes the numbers, if you want to play around with it.
It's not that any of your math is incorrect, it's just that you're only looking at half of the picture, and it's the half that's less relevant to the topic of battery life. When dealing with a regulated mod circuit, there are two different sets of Ohm's law calculations, one before you convert the output from the battery and another on the other side of the chip after it's been converted. The wattage is what links the two. You did all the calculations for the atomizer side, but the battery doesn't see any of that. It's got its own circuit, with a different set of numbers for voltage, amperage, and resistance.
Edited multiple times to try to reword more clearly. Hope it worked.
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