I have a simplistic circuit and wish to add a battery backup (see image), in the event that the power supply is temporarily interrupted. I found a youtube that provides this circuit and the narrator says that the power supply voltage must be greater than the battery voltage. Unfortunately, he doesn't say by how much. Is there a simple rule/calculation for this?

For the curious, my load is a motion-detecting kitchen faucet.
With thanks.

 

Below is the LTspice simulation of the circuit.
As you can see, the battery current through D2 basically stops when the supply voltage equals 9.2V.
So the supply voltage, at a minimum, can be no lower than the battery voltage plus 0.2V.

 

WOW!!!! Thank you for taking the time to solve my problem!! Much appreciated.

 

A Single 9V-Battery may not be able to supply enough Current to operate a Solenoid-Valve,
but if all You need is "Memory-Retention" it will probably work just fine,
as long as You remember to replace the Battery once a year.

For more Current Capacity You can simply install ~4 or ~5
9-Volt Batteries in parallel, each with their own individual "D2" Diode.

The Batteries, if kept dry, will actually last ~4 to ~5 years if not used,
but if You don't change them for new ones, or at least test them once a year or so,
they are guaranteed to be stone-dead right when You need them the most.
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The diode switched battery backup circuit presented is certainly very effective within it's capabilities. And the comments in posts #2 and #4are correct.
I am not sure what sort of load the motion detecting faucet presents, although I did install one for a kitchen remodeling project I did for others a few years ago. That installation included a small mains operated power supply, I do not recall the voltage, but the supply was quite small. My impression was that the faucet would also function manually as well as in a motion sensing mode.
One interesting possibility would be to use a rechargeable battery and add a series resistor so that the supply would provide a very small float charge as long as mains power was available. At that point the battery life would be several years, at least.