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The national power provider here is not great. There is often a 10 - 30 V difference between phases.

Will the voltage exiting a 3-phase variable frequency drive be the same as the input voltage, or does the magic voodoo in there have any balancing effect?

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On the input side of a VFD is a rectifier, which will maintain a relatively steady DC voltage on a DC link bus. On the output side of a VFD is some type of inverter that will draw from the DC link bus and feed the variable frequency voltage & current to the motor being driven. Because the output is decoupled from the input by the DC link bus, the voltage level and phase-to-phase imbalances on the input will not affect the output voltage.

Here is a reference for how a VFD works.

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