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I am having trouble figuring out what kind of time delayed relay I need for this circuit or if anything will work even. It is used to keep a dome light on for a few seconds in a vehicle after the switch is opened (which signifies the door being closed).

My dilemma is that when the switch is closed it could trigger the relay somehow but if the relay is then putting out power on the same line isn't it continuously triggering itself? (I am familiar with relays but not time delay relays)

Circuit

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You don't want to use an actual time delay relay, as they are quite expensive ($30 or more).

This circuit should do what you want using the ubiquitous 555 IC in a monostable mode:

enter image description here

SW1 is the existing car door switch. I show a resistor to +12v for the purpose of demonstrating this circuit, but in reality the dome switch should already be putting out +12v when closed.

When closed, the switch feeds 12v into the 555 (which doesn't trigger it, since the 555 triggers on a signal going to ground). The switch is also fed into the OR gate, which should operate the dome light via the transistor output stage and small relay. By using a relay, you won't have to worry about messing up the wiring in your car, just connect it where the switch used to be.

When the door is closed, the falling edge of the door switch triggers the 555, which operates the 555, and keeps the dome light on via the OR gate.

Currently the timer is set for five seconds. You can modify the value with this calculator.

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