I'm using 10 servos. I know I can't power them with my arduino, so I want to use AA batteries to power them. How many AA batteries do I need then and how long will they last?
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The answer is somewhere between one and one billion AA batteries. You first need to know how much current the servos will consume. That, of course, depends on what the servos are, what they are driving, and how often and how far they move.
You need to ensure that you are providing the right voltage for your servos. You certainly don't want to be driving that number of servos direct from the Arduino's 5V pin, so you will want to run them direct from the AA batteries themselves. So check the voltage that the servos need, and provide that voltage in batteries. 6V is a common voltage for servos, and so 4xAA batteries (4x1.5 = 6) would provide the right voltage.
Would they provide enough current? Possibly, possibly not. If you can run 2 servos off 4x AA batteries but adding the 3rd makes it unreliable with brown-outs then you obviously need more. So you would need to add another group of 4 AA batteries in parallel with the previous group. That arrangement is known as "4S2P" or "4 batteries in series, and 2 chains in parallel". The voltage stays at 6V (4x1.5) but the current capacity is doubled.
Add as many chains in parallel as you need - the more you add the longer the batteries will last.
Once you have all your 10 servos connected and running off enough batteries you may be able to predict the length of time the batteries will last. Batteries have a capacity measured in milli-amp-hours, or mAh. This is the amount of current they can provide over a single hour before the battery is considered flat. So if you can then measure the average current the whole setup is using then you can calculate how many hours it would take to rack up that "mAh" limit.
Say you use 2000mAh batteries (typical for AA batteries). That is 2A for one hour. Or, if you draw less, say 1A, then it would be for 2 hours. Of course, this is only a "rule of thumb" and different current draws will cause the battery to discharge in different ways giving slightly different results. But it's good enough for a rough estimate.
So say your system draws on average 3A, and you have 4 chains of 2000mA batteries in parallel. That's a total of 2000x4=10,000mAh (or 10Ah to change the order of magnitude of the units). At 3A from a 10Ah battery bank would be 10Ah / 3A = 3.3333 or around three-and-a-half hours of operation.