Disconnect each air conditioner in your system from its power supply, confirming that the system is safe to repair using a voltage tester. Locate the external outdoor condenser and isolate it from the power supply using the breaker switch, usually found mounted away from the condensing unit.
Locate the main access panel to the system. Each air conditioning unit will have its own panel, but the main access panel can be identified through the safety warning symbols displayed on its box. Unscrew the cover to the access panel from its plate and lift away, revealing the internal wiring.
Find the damaged contactor and remove any wires attached it, making a note of the color and mounting position for each of the wires for later use. Check the condition of the contact points at the end of the wires, removing any rust or corrosion using wire wool. Unscrew each of the connection points on the mounting plate of the damaged contactor and it will come away cleanly, leaving points for the new contactor to be attached to.
Place and hold the new contactor against the mount. Test to see if the old points line up with the mounting plate on the new unit, screwing the contactor as tight as possible onto the unit. If the old points do not line up, you should use self-threading screws rather than drilling new pilot holes.
Reconnect the wires to the new contactor, consulting your notes taken earlier. If the connectors on the wires fit loosely to the new contactor, use a pair of pliers to crush the ends of the connector into a tighter fit. Do not place the cover back onto the unit yet, but instead reconnect the power supplies to each air conditioner, leaving the outdoor condenser disconnected.
Using the thermostat controls for the air conditioning system set the internal climate to a cooler temperature. Go back to the contactor and check that its mechanism has been pulled in, confirming that the system is operational again. Replace the cover to the main access panel and switch the outdoor condenser back on to complete the repair process.