Charge lower voltage, higher mAh with higher voltage, lower mAh battery

Question

So let's hypothetically say we have a 12V 500mAh battery and we want to charge a 4V 2000mAh battery with it, using a (hypothetical) 100% efficiency converter.

One could say that, as the charge is constant in a circuit, one can best case scenario, charge the 4V battery to 25% of its capacity.

However, if one looks at it in energy terms, the first battery could provide, let's say 500mA for one hour @ 12V, this would be converted (with 100% efficiency) to 1500mA @ 4V, so it would (provided the 4V battery could handle the current) charge the 4V battery to 75% of its capacity.

At least one of these two statements is clearly false. I would like to know which one and why. Or if both then what is the real answer.

Answer 1

Ignoring charging losses (and I'll get to that), the first statement is wrong and the second statement is correct.

Converter efficiency is measured in terms of power, not current. So, for instance, a 12-volt input at 1 amp would give a 4-volt output at 3 amps, and that's why the 75% figure is correct - it is 3 times greater than the first.

With that said, even if you have a perfect converter, batteries do not charge with perfect efficiency. At the very least, they start getting hot towards the end of the charge cycle. Even when charged, a battery's capacity has to be measured by a discharge cycle, so charging efficiency is actually efficiency through a charge/discharge cycle. Wikipedia suggests a charge efficiency for li-ion batteries of 80-90%. A standard 16-hour @ 0.1C charge cycle for NiCads produced a maximum efficiency of 62%.

Answer 2

Assuming everything is magically efficient, both statements are wrong.

Your problem is that you will lose energy as the battery is charged. This is usually converted to heat, so your second statement is wrong because even if the voltage conversions are perfectly efficient, the battery will burn energy in the charging process.

Worse, some battery chemistry require voltages higher nominal output to charge - NiMH for example, output ~1.2V, but charge at 1.4-1.6V.

Statement 1 is wrong because of your perfectly efficient clause. In a very bad circuit, you could charge a 4V battery with a 12V battery like that. You wouldn't, you'd be wasting massive amounts of energy and your 4V battery might explode. (4V sounds like lithium to me. Do you really want to know what happens if you charge a lithium cell at 12V? No, I don't want to either...)

So the true answer is "Somewhere in the middle."