I am currently working on a project involving multiple servos (6 or 7) and wanted to power them with my converted computer power supply (5V 22A). I am using an arduino uno and would like to power that with a 9V wall wart. I assume that i can do this as long as the ground between the Arduino and computer power supply are connected? Also, should I add components to protect the Arduino and if so, what?
I think you are correct: you can use two separate supplies as long as their grounds are connected. This is in fact a good idea if you're going to be powering "high current" loads like servos.
If you're powering the Arduino with a regular 9V wall wart I don't think you need any special protection for it.
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The Ruggeduino: compatible with Arduino UNO, 24V operation, all I/O's fused and protected