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I have a 1.5 amp-hour, 12 V battery and I have a 10 amp-hour 12 V battery. I know the voltage will increase to 24 V when they are put in series, but I don't know what goes on amp-hour wise.

Does the new amp-hour rating take the amp hour of the low amp hour rating, does it take the 10 amp hours or is it like an average?

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It is bad practice to connect batteries in series when they don't have the same capacity. The battery with the smaller capacity will be empty before the larger one, resulting in a lower voltage for the smaller battery. At that point things will start to get interesting as the larger battery will start to charge the smaller one through the connected circuit and with reversed voltage. The cell is not designed for being reversed and charged and bad things may happen a.o.: leaking acid and exploding. Neither of these situations are desirable. This is also the reason why most manuals of battery operated devices urge to replace all batteries at the same time.

batteries in series

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    \$\begingroup\$ Interesting. Now, there's certainly a margin of error in the capacity, right? So, even with two theoretically identical batteries, in practice one will always have a slightly smaller capacity. I'm curious to know: what exactly goes on that prevents bad things from happening in this case, i.e. when the difference in capacity (and hence in voltage while both batteries are approaching EOL but one is closer) is small enough? \$\endgroup\$ – matteo Oct 28 '18 at 16:58
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When in series, the smaller battery will deplete before the bigger one. When a battery reaches its last bits of energy stored, its voltage drops rapidly.

Therefore, after using up more than 1.5Ah, the first battery's voltage will drop significantly, while the other will keep supplying current at 12V, since it still has 8.5Ah left.

You can't let that smaller battery operate in these conditions. This means that you would effectively have a 24V 1.5Ah battery.

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spot on , and one of the main problem comes in when we are charging those batteries and as they are connected in series so the battery with the minimum capacity charges first and its resistance increases so as its voltage and if we are using smart chargers to charge them it will feel that the batteries are charged and will stop charging. and we will be left with the uncharged big battery...

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