9V NiCad Battery back-up/Charger circuit needed

[tranerek]

I need a simple, CHEAP circuit to sense input power between
0 and 22VAC and if input power is less than 10VAC switch over to the battery. The circuit also needs to keep the battery charged when input power is available.

 

[BrianG]

You don't actually need to "switch over" to the battery when the input drops to 10V as there are various circuits which can give you what you need with a few diodes, resistors and perhaps a transistor or regulator chip.
However, before you can design the circuit you need to define the acceptable range for the "9V" output and the load current, what kind of battery you are using and what its actual on-charge voltage needs to be.