It's an essay :thumbdo:
I do agree it can be a sensor that switches the fan on, but here is my story:
This is for my e46 323i - 2000. The car occasionally overheated. All sorts of components were replaced (water pump, thermostat, etc.).
I told the guy (3rd party Mechanic), Listen it is the fan, but every time he listens to it, the fan comes on. I had the fan replaced by BMW Leo Haese Hatfield :Yuck: and left the car there for the day to pick it up in the morning.
Upon arrival I had to wait. After 30 minutes, I became impatient (how's that for being patient :rollsmile

and enquired about this delay. They reckon that the battery needed charging first. Ok fine. I paid the fine and walked out of the dealership to get to the car - that fan was pumping full blast. "that's not normal" I thought, but then again it was a new fan.
I drove of with this windmill of a car to my place of work (about 6km). At every traffic light and stop street I could hear this monster fan pumping air like you cannot believe. I parked the car in the parking lot and left, with that fan still about to take off :argh:
I worked for a while and decided to return to the car to check if everything is OK. At that very moment the alarm made a funny sound and everything was just dead.
Clearly the battery was drained.
Called them and they jump started me. I left for the DS and they confirmed that the incorrect fan was installed into my car and a new one was on order. Later that afternoon a new fan arrived and all was fine since then.
************
The fan works with SCR's (Silicone Controlled Rectifiers), a transistor like device that "fires" coils in succession to control the rotation of the fan exactly. I was pretty amazed when I opened it, to see the technology used.
That electronic circuitry was situated in the middle of the fan (Siemens fan). The reworked fan had the circuitry moved out of the way (it looks like a hard drive) into a sealed unit, with a harness going to the middle that connects the coils. I hope this information helps
Here is the invoice of the Incorrect fan part:
I would really show BMW this invoice and twist their arm for a better price.
Goodluck.