Is it possible to test the fan control module?

[rx6180]

Hi, newbie here. I've a suspicion that the fan control module on my 1999 C200 (W202) isn't working properly, but I'm actually not sure. My car hasn't overheated, but sat in town traffic on Saturday I felt that the cooling fan wasn't cutting in when the temperature gauge reached the point where it used to do. When I got home I lifted the bonnet with the engine running and watched the fan. I saw it try to turn, then stop, so I flicked it with my fingers (risky I know) and it began to spin slowly. When I turned the engine off but turned the ignition on again, the fan stayed stopped. Since then I've tested the fan connected directly to a 12v battery (100% fine). Then I ran the engine up to temperature and beyond - on the gauge I estimate the temperature was up to more than 90C (close to but not touching the notch below the one marked 120C) but the fan did not start - I'm sure it used to do. I daren't run the engine longer at idle because it took 15 minutes or more to get there. Since then I've learned that the control module is linked to the air conditioner, so yesterday on a short journey I turned on the air con and when I got home the cooling fan was turning over slowly, but when I turned the air con off and on again with the engine running the fan stopped. My air con is low on gas and since I hardly use it I haven't bothered having it re-gassed - could this affect the normal function of the cooling fan with a hot engine? Short of swapping the control module or having the car computer diagnosed, is there any other way to confirm the correct function of the control module besides what I've already tried? How high should the temperature gauge needle rise with the engine idling before the cooling fan cuts in? Cheers.