A/C compressor clutch
The low pressure switch is 'between' (electrically) the A/C control unit and the clutch. The low pressure switch is just a simple switch (open or closed).
The low pressure sensor is normally closed when there is adequate pressure in the system (enough refrigerant) and temperature warm enough - about 50F or so. If the system is low on refrigerant or the ambient temperature is too low, the pressure will drop and the switch opens, ending current to the clutch. When the refrigerant quantity is adequate and the ambient temperature is 'moderate' (like 60-70) the switch will cycle on and off. This is due to the compressor pulling the low pressure down. Then the pressure builds due to the evaporator (the coil inside the car) sending more hot refrigerant back to the low side. Then the clutch comes on again when the pressure is high enough.
So, if the clutch doesn't come on at all, the switch is probably simply open (I don't think the analyzer can 'see' beyond the low pressure switch as it 'sees' the signal going to the low pressure switch).
Three possibilities; low refrigerant, really cool ambient, or bad switch. I've had two bad switches on GM cars. They're easy to replace on newer cars as there's a schraeder valve that the low pressure switch screws into so refrigerant doesn't leak out when the switch is unscrewed. Make sure what kind of switch your year car has before unscrewing the switch or all the refrigerant may leak out.
An easy test is to remove the lead from the low pressure switch and short the two contacts in the lead with a paper clip. If the compressor runs, the refrigerant is likely low (unless it's freezing outside). If the compressor doesn't run, it's not one of the 3 possibilities.
Don't let the compressor run more than a few seconds; if the refrigerant is low, running the compressor will damage it.
Hope this helps.