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This circuit is for the cabin light on my airplane. The light is independent of the master switch (equivalent of the iginition switch on a car). You want to be able to use the light of unloading bags, etc. but if you forget to turn it off, it will run the battery down.
So I set up the circuit with the master switch feeding the left side on the circuit, but with constant power to the relay coil. So when the master switch is turned off, the capacitor charge feeds the transistor until it bleeds down. My problem is getting that bleed down time long enough...
Here is a circuit similar to one I am using for another application. The simulation shows a delay of ~360s (6min).
The problem is that if I installed it in my Cessna, I could still run the battery dead because the switch is a slide switch. If the switch gets left in the on position, it will kill the battery. You would have to replace the switch with a momentary spring-return push-button to eliminate the dead battery possibility.
Resistor R4 is there to provide some significant current to keep the switch contacts clean, and provides the discharge path for the timing network. R2 limits the initial inrush current. The delay time is created by C1 discharging to the threshold voltage of M1 (~3V) from one diode drop below the battery voltage, about 2(R1+R4)C1.
Note the traces in the simulation. Green V(sw) shows when the switch is closed. Red V(g) shows M1's gate voltage. Lt. Blue V(d) is M1's drain. Dk. Blue V(load) is the voltage at the dome lamp. Purple is the current through the relay coil.
Note that you do not need to build the stuff in the dashed box. V2 is the human flipping the switch S1. R3 is the existing dome lamp(s). V1 is the battery.
M1 is almost any modern NFet which has an Vds >30V (50V if your airplane is 28V) and an Ron<250mΩ.
btw, for the ones who will notice, there is no, nor does there need to be, a snubber diode across the relay coil.