There is a system in your compressor control that detects belt slippage as well as other items like coolant overheat,low pressure cut off, freeze up protection, engagement delay etc. This system resets after cycling the key. It sounds to me like you may have a worn or loose belt that is causing the speed sensor on the back of the compressor to detect a potential lock-up and then turn the compressor off. Another possibility is a bad shock on the belt tensioner which allows the belt to shudder. You can often hear this in the car as a rattle when you first accelerate. I hope this helps.
Good Luck