A typical reason your controller won't turn on is because it's not being supplied enough power (if any). The potential causes of this problem are numerous. Here is how to identify and fix it:
Testing Connections
Tools and Materials
Multimeter
Soldering iron/solder
Wire stripper
How To
Use a multimeter's continuity testing feature to check certain connections (see pictures)
If there isn't any signal through a certain connection, you may need to strip that wire and replace it
If all the connections are fine, try the tests below
Testing For a Short Circuit
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A short circuit is when the power and ground connections come in contact. This could lead to very bad damage to the controller's core parts or the battery.
Tools and Materials
Multimeter
How To
Unplug everything and take out the batteries if possible
Using the multimeter's resistance testing feature(Ω), place the probes between the controller's power(+) and ground(-)
If the resistance is 0Ω or close to it, there is a short in the system. There are many ways this could've happened. They are very hard to fix, so you should check out step 8
Testing Voltage
Tools and Materials
Multimeter
Rock Candy Ps3 Controller Driver
How To
Rock Candy Wireless Ps3 Controller Driver Download
Ps3 Controller
Plug the remote in or put in the batteries
Use a multimeter's voltage testing feature(V) to check the voltage coming from the console/batteries
Compare the voltage to the ones on this table
If the voltage is less than it should be, there is something wrong with the console, and beyond the scope of this Instructable (I have no experience with fixing consoles)