The trinary switch has two internal switches as you know. One controls the compressor clutch, and one controls cooling fan for the condenser.
The A/C system has to have at least 30psi in the system for anything to work (compressor protection). Thats the first switch. Under 30psi the switch is open, no clutch engagement. 30 to 380psi the switch is Normally Closed (active), above 381psi, the switch opens again to protect the compressor. So if the system is full, this half of the Trinary will be NC with the key off, engine off.
At the same time the above is going on, the second switch is Normally open till the system reaches 220 psi (normal operating pressure). Once the system reaches 220 psi the second switch turns on the fan relay though the ground side. This switch stays closed (active) as long as the pressure stays above 220.
The idea behind the Normally Closed at 220psi is to keep from spiking the alternator when the A/C turns on. It spreads the load out a bit. The clutch pulls a lot of power along with the cabin cooling fan, you don't need the condensor fan kicking in at the same time, so there is a few seconds delay while the system builds pressure before the fan kicks on.
If you wire the load side of the relay from a battery post, the fan will run anytime the system is at 220 or above. This will keep the A/C system from going into high pressure on hot days once you turn off the engine to park, the fan will continue to run till it has time to cool the system down.
You can add a third manual override switch to the fan relay. Just use the same ground side as the trinary does and run it parallel to the Trinary.
Recheck your switch with the system running, if the fan doesn't run, follow the wires back to the relay, then the trinary. You should find the problem pretty quick. Just remember you could be looking for a ground and not battery voltage, it depends on how someone has wired it.
There could also be a forth switch involved, a adjustable fan control. It would be wire to the fan relay as well. Its used to turn the fan on and off at a desired temperature, but the Trinary will override it when the system calls for it. http://www.hotrodders.com/forum/attachment.php?attachmentid=14367
The link above shows how it must be wired for it work correctly.
Never run a second switch in line with the Trinary or you could damage the compressor. You can run parallel switches to control the fan, but don't mess around with the clutch side of the Trinary.