The terms "current-controlled" and "voltage-controlled" here refer to the method of control, no the type of motor. In electric motors, the torque produced is proportional to the current -- other things being equal. The motor will also produce a back EMF voltage proportional to its velocity.
We live in a world of voltage sources, so to obtain current control, we need a measurement of the actual current, and a feedback loop that outputs the voltage necessary to obtain the desired current. This obviously adds cost and complexity, but does provide several important performance advantages:
1. The feedback loop can dynamically and temporarily apply very high voltages to get quick current changes, effectively reducing the electrical time constant for faster responses.
2. The feedback loop will automatically compensate for the generated back EMF, apply voltages above and beyond that to get the desired currents.
3. The above two factors means that you are much more directly controlling current, and therefore torque and acceleration, of the motor, providing much better dynamic physical response.
4. Performance is much less sensitive to physical changes such as increasing resistance with temperature.
5. Control of current is your best protection against over-current conditions.
The above is true for DC or AC motors, for synchronous (e.g. brushless servo) or asynchronous (induction).
Standard volts/hertz VFDs are voltage-controlled systems. Vector-control systems, with or without shaft sensors, are current-controlled (even if marketed as VFDs). Servo-control systems are virtually always current-controlled, except maybe at the very low end.
Curt Wilson
Delta Tau Data Systems