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I have recently purchased a RoboMower. Like most of the automated mowers these use electric-fence wiring to mark the borders of the property and keep a small charge on those wires so that the mower can tell where the borders are. The RoboMower I purchased is rated for 3/4 of an acre and I have 2.25 acres to mow.

As a result I was thinking of breaking my yard into multiple 3/4 or less acre zones and using an Arduino to shift which zone is active/has current in it so that the mower would follow those boundaries. The base station would be in a place where all zones overlapped slightly. Based on what day of the week it was, the Arduino would activate the proper zone, and the mower would go out and mow in that zone because it was active. I'd clearly have to schedule this so that the switch did not happen when the RoboMower was actively mowing or it would cause all sorts of issues.

In any case, my question is: What is the hardware called that I would need in order to activate one zone or the other? Is that called a digital switch or something else? Is there such a thing of one switch that would be able to switch multiple zones, or would I basically need a bunch of switches to individually control each zone?

Each zone is a loop of wire and therefore there would not be any shared wires between the zones.

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The technical term of what you're looking for is demultiplexer. Note that the demux itself will not be enough to activate the zone, it will simply allow you to choose between different zones based on its input. It is still up to you to provide any further electronics required to enable the appropriate zone.

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