Reader's Digest version:
A VFD is a 3 part power conversion system.
1) It rectifies the AC to DC, called the "Converter Section". This section uses the incoming AC power as the "raw material" for creating a completely new power source for the motor. So from that standpoint, anything other than raw power needs by the motor is isolated from the AC line, i.e. power factor, terminal voltage etc.. The only limit is that the Converter cannot create voltage that is not present on the AC line terminals. It rectifies to the peak voltage value of the sine wave (as opposed to the RMS value) but for the most part, that's somewhat irrelevant because when the motor uses it, it uses RMS anyway, so the net effect is for all intents, the same or less as the incoming.
2) The "DC Bus" section smooths and conditions the DC with inductors and/or capacitors to remove the ripple resulting from rectification. It also usually supplies the control power for the electronics, although that varies from one mfr to another. Some have a separate SMPS to do that.
3) It recreates a pseudo-sine wave AC output of variable voltage and variable frequency in the "Inverter Section" by firing high speed transistors in a PWM pattern. To provide a constant level of output torque from the motor, the VFD maintains a constant ratio of RMS voltage an frequency, referred to as the V/Hz ratio. This can be tweaked to get specific performance out of the motor, but in general the motor's operation is dependent upon this being maintained and controlled.
So when your input voltage drops to 80% of normal, the VFD can no longer provide 100% voltage at full speed. If by chance your commanded speed was no more than 80% of full speed, then no problem. But if you commanded more than 80% speed when the line dropped to 80%, the VFD was no longer able to maintain the proper V/Hz ratio and the motor lost torque. When that happens, slip increases and the motor draws more current, possibly to the point of overloading.
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