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I have a circuit that can use a 9V battery for power, but also want to alternatively connect to a power supply (usually higher that 9V) without taking out the battery.

Is it OK to just connect the power supply parallel with the battery? Or add a diode between the power supply and battery? Will it still use the battery power even it is always connect to power supply?

I have an idea that to add a JFET between the battery and the circuit, and connect the gate to power supply. So that when there is power supply, it with cut off the battery. Is it necessary?

Given that the power supply can be a few volt higher than the battery, such as 12V. The circuit has used zener diode to regulate voltage, so the supply voltage is not a problem.

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No, you can not safely connect a 12V power supply in parallel with a 9V battery.

The easiest way to isolate the two supplies is with two diodes. The anodes of the diodes go to your two sources. The cathodes are connected together and then to the load. Use Schottky diodes if you want to minimize the voltage drop.

There are more sophisticated schemes and this site is full of them.

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