Carmakers use several different wiring methods to provide power to the clutch coil. In older non-computerized vehicles power flows from a fuse to the AC switch. When turned on, power flows through the switch and then to a low and high pressure switch before flowing to the compressor clutch coil. See the wiring diagram below.

In newer vehicles where the AC is controlled by the heater control head or the PCM, the wiring is quite different. See the wiring diagram below.