Battery to battery charging - Ah vs Wh

Assuming you have these two batteries:

Battery A: 12V, 15 Ah = 180 Wh
Battery B: 5V, 20 Ah = 100 Wh

Can you use battery A to fully charge battery B?

Battery A has less capacity (with respect to charge) compared to battery B, but battery A has more stored energy at 180 Wh.

It seems like there are two ways to think about the transfer (in an ideal case):

Ah: After charging, battery A is left with 0 Ah and battery B is 3/4 charged with 15 Ah.

Wh: After charging, battery A is left with 80 Wh and battery B is fully charged with 100 Wh.

This seems like a situation where, using a water analogy, you have a smaller amount of water (in battery A) trying to fill a larger vessel (battery B). The smaller amount of water is at a higher pressure (and therefore greater energy), but that doesn't make up for the fact that there isn't enough water.

Is this correct? Or is stored energy (i.e. Wh) what really matters and electrons are somehow pulled from the environment to make up for the deficit in stored charge (Ah)?

• Welcome to EE.SE! An excellent question and well worded. Are you considering only linear regulated charging or also allowing DC-to-DC switching converters between the two batteries? – user2943160 Jul 8 '16 at 21:10
• I'm interested in whether or not fully charging the battery is possible, so any method that works is acceptable. – sparky Jul 11 '16 at 22:55

You can fully charge battery B if you use a dc-dc converter (switching regulator) to drop the 12V to 5V. In this case the WH is the number to look at. You will have to take into account the fact that the batteries are not perfect and so you lose a bit of power in charge/discharge and the efficiency of the switching regulator (typicaly 90%+).

Think compressed air not water.

In a further attempt to clarify this say I have a fully charged 12V battery of 100AH made up of 6 cells, how many AH can I get from this at 6V? If I split my battery into 2 6V batteries each of 3 cells and 100AH and then connect them in parallel I end up with a 6V 200Ah battery.

When it comes to charging one battery from the other, it's the charge content that would matter and not the power. Your water analogy is correct.

Charge content of the battery(Q)= Ixt

where I=current t=time for which the current can be supplied at rated voltage

So, charge content of battery A = 3/4 times Charge content of battery B

Therefore, assuming ideal transfer of energy and no energy wastage in any form, battery A would end up with 0 charge and battery B would end up 3/4th charged.