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Arduino Nano is a great small board. It comes with a USB port so you can easily program it.
60mA * 300 = 18,000mA = 18Amp @ 5vdc (90 watts)
Batteries are rated in mAH (milli amp hours), so a 18,000mAH battery would last one hour with all the LEDs at full brightness.
The easiest thing for something on this scale would be a USB Battery Pack. The only issue would be that it's only rated for about 4.8A output. So you might need to get multiple to handle the load. The other option would be going with some Sealed Lead Acid batteries.
What are you trying to build? Seems like a lot of LED's to be portable
I want to set it and forget it - leave it all set up and have it run on its own.
Seems like it is the solar/battery math that I need help with. If I get the math right it will work - or tell me that it'll cost too much to be worth the project. Without the math I'd just be slapping parts together and doing some trial and error.
I've read that a 10,000mah battery can power the cam for a day. So I was thinking (guessing) a battery bank with double that capacity would be about right - something like this which also allows for 4 amp and pass through charging, which seems to be a good thing.
The solar panel would need to be sized to reliably put 10,000 mah back into the battery per day. (Although I suppose a battery with pass through charging can possibly run the cam and charge the battery at the same time - thus not using battery capacity when the sun is out - but better to plan for cloudy days) From the little I know, that is a function of the panel size and efficiency and my geographic location - on average.
Plus the charging cable that comes with the pro controller. This combination, with the switch starting at 100% and plugging it in when it hit about 60% lasted me just under 13 hours. It doesn't charge the switch fast, took about 2 hours to get from 60 back up to 100.