As M.B.Doc points out the inside controller grounds the outside controller to tell the outside controller (compressor control relay)to switch on the compressor.
This signal passes through the low/high pressure switch which will be a closed circuit if the system has pressure.
This is the first point for testing. You should test the voltage at the switch while running. If it is close to zero volts you should then lift one terminal off the low pressure switch. One side of the circuit should go to battery voltage. That is the side that goes to the compressor control relay. If one side of the switch is high and the other low then the switch is open and low refrigerant charge is the answer.
That was the simple part of testing. It allows you to decide whether you have a climate control problem (inside the car) or a compressor control problem (under the hood - for engine protection).
The problem can be very frustrating. I am going to paste a post that I placed to a Tech Only forum to give an example of how much trouble this can be to diagnose. It is actually a Diesel but the considerations are similar.
Here it is:
Well I have been here before, thought I had seen it all, but I found a unique model in a OM606 E300 Diesel 1995.
The car came in with the complaint that the A/C wasn't working. Some other history: the car was just sold by a local car rental place that wholesales many european cars. The new customer is tolerant but eager. We are working for the car rental place.
The first trip we find nothing wrong except for the low speed aux fan fuse and a slightly low system. We repair and three days later its back. After two days of driving it and no problems we decide to replace the pushbutton control unit (we have to do something as we know it will be back otherwise - we stock the PBC so it will sit in this car for a while). We also remove the compressor control relay and find that its circuit board has had the relay pins resoldered (well I guess I can't do that), I thought about disconnecting the full throttle switch thinking it may stick (I've seen that). When I got to the switch it was disconnected. I seem to be following in someones footsteps.
The car leaves does great for four days, back again. No problem, we slide that one year only compresor control relay in there and off she goes. Only a day this time.
Of course we have monitored the compressor speed signal and the engine speed signal, found no faults in the A/C self diagnostics. This time we decide to get mean with it and we disassemble the relay connector enough to back probe it and we make a chart of the pin characteristics as seen on our scope. The car of course is working all the time. The only time we saw the car not working we were able to verify the proper signal at the low/high pressure switch. So we know the problem is in the compressor control relay ciruitry.
We also know that the problem never occurs to a running system. It always occurs as a will not engage problem never a disconnect problem. Based on this we tried disconnecting the comp speed signal. The engine started and after a delay the compressor engaged for a little over a second. We know this is not the way it happens. So we disconnected the engine speed signal. Ah Ha! this does keep the compressor from starting off, it never peeped.
So we decide to investigate the speed signal. I knew from earlier diesels that the comp relay doesn't look at an inductive AC speed signal; that the signal is converted to a pulsed DC signal by the EGR controller I though on the earlier diesels. Turns out that the ISC (Idle speed control) relay does that job on this guy. The signal goes to the tach the A/C comp relay and is used by the ISC. Well this little fuel pump relay sized device has self diagnostics. It has a possibilty for 5 different codes; pretty simple.
On reading the codes we got a code 2 (crankshaft sensor)and code 3 (engine coolant temp). Oh Boy now we're getting some where. But where? The code 3 reset but nothing we could do would reset the #2 code. The whole time the A/C and the tach and the square wave 8v signal is going fine. Does anyone have any experience with the ISC self-diagnostics? I don't know which parts to shot gun next? The ISC controller is high on the list with the non removalble code. The crank sensor is up there too, maybe that little box knows something.
Anyway I figured I would write this all down so it might jar something loose. Even restating it I don't know what next except some more parts replacement.
BTW on the first trip we also reduced the compressor shoe gap to 0.020" from 0.030". We have since verified that no compressor signal comes from the relay when it is failing.
Owner 24 bay BSC
Bosch Master, ASE master L1
26 years MB technician