Hello
I'm currently using 4 1.5V AA batteries in series to supply power to my Arduino Leonardo.
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The combined voltage is about 6V not accounting losses, is that ok for my Arduino? My load is approx. 618mWH/day.
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I connected an LED from "+" to 5V and "-" to Ground, but as soon as I turned on the Arduino, it burnt out. I'm not sure the cause? Is it related to my battery power supply?
Thanks
Cookie 
6v (or more for fully charged batteries is too high - check out the spec sheet for the processor). As far as the LED goes, an LED has approximately 2v drop across it (depends on the color) - needs to have the current limited to prevent it from burning out. In your case, you put it across the 6v supply from the description. What you need to do is limit the current to the LED so you take 6v (supply) - 2v (drop across LED) = 4 volts. Most LED's don't want more than 20 ma (0.020A) or so. Using ohms law, you take 4v/0.020 and get 200 ohms. 220 ohms is a common standard value (330 is another common one that would work fine). You wire the battery positive to the resistor to the LED (observe polarity) and the LED connects to ground.
Hi Mikey,
I mean I used the 6V (4 AA batteries) as the Arduino power supply. I connected the batteries to the power jack.
It should have a power regulator right?
Yes, but you are in a grey area - most regulators need 1.5-2v drop across them to work well, so to use a regulator, you need 7v input or so. If you look at the technical details for the Uno R3 on the Adafruit page HERE you will see that the input voltage is listed as "7-12 volts". The Arduino also has provisions for feeding 5v directly in, but you are over that with the batteries which add up to 6v (or more for nice fresh batteries). You can use rechargable batteries which have a lower voltage NiCd is somewhere around 1.2 per cell. NiMH are also in the 1.2 volt range and are generally good batteries to work with (more info HERE ) 4 of those gives somewhere around 4.8v. I'm not sure just how far down the 5v Arduino will go and remain stable - you need to research that.
Figure 29-2 in the 32U4 datasheet shows 4.5V min for 16 MHz.
There you go - looks like 4.5 is the bottom end.
You were not clear in the initial post when you said "it burnt out" - I assumed you meant the LED (and my comments about needing a resistor are still valid). Did you mean the LED burned out or the Arduino (which would probably be the regulator trying to source enough current for the LED) burned out??? If you had the batteries plugged into the jack on the board, then you were going through the on-board regulator.