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OK, Herb, you get first crack at these:
Pressure balanced mixer valves somehow compensate for a
reduction in water pressure on one side (either cold or hot)
by reducing the flow on the other side to keep the shower
temperature constant. This works for large swings in pressure,
and also for pressure rises (after the flush tank filler shuts
off, for instance) with no perceptible hysteresis. There are
no inputs to the device other than the hot and cold water
supplies.
How do pressure balanced mixer valves work? Electrical analogs
gratefully accepted.