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As I know it, a VSD alters the frequency to change the speed of a motor. I also understand that since the back emf of the rotor is directly proportional to the speed of the rotor, the slower the motor runs the higher the overall current there is going to be. Which explains why current is higher at starting of the motor, or at stand still altogether.

However, I've heard that the VSD can limit the current at starting of the motor whilst the motor accelerates from low speed to high speed. How is this possible? How does the VSD control the current of the motor whilst the motor is running slower than usual?

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VSD stands for variable speed (DC or AC ) drive. A VFD is a DC controlled Variable Frequency AC Drive.

Modern VFD's are highly programmable to sense RPM, current and control V/f with PWM. There may be a hundred control options or parameters or a simple user interface.

Current is a result of dependent on controlled acceleration, external load factor and magnetic voltage-dependent excitation currents.

With no load, current may be limited by a constant acceleration rate of increasing f while keeping V/f constant. With a load, it depends how it changes from start to finish, so each application may have custom start-stop profiles.

Thus the acceleration rate can be constant to max RPM or controlled by load or limited by either. The reciprocal function of braking is similar.

DC motors with full voltage applied can draw up to 10x the rated max current and power with the start surge or full stop with 0V. Doing this often can lead to thermal issues.

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I also understand that since the back emf of the rotor is directly proportional to the speed of the rotor, the slower the motor runs the higher the overall current there is going to be.

The flaw in your understanding here is that this related to a FIXED frequency being applied to the motor, where the stator frequency is much higher than the rotor frequency. A VFD is altering the frequency WITH the voltage and maintaining the same ratio of V/Hz as the motor was designed for, so it can make the motor produce rated torque the entire time. In a fixed frequency environment, that lack of back emf causes the motor to pull high REACTIVE current, but without doing useful work. The VFD maintains the motor's ability to be performing useful work (i.e. acceleration) without having to deal with reactive current.

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In any motor, the current drawn is proportional to the torque demaned by the mechanical load. When the drive is in the process of accelerating a motor (at a predefined rate), and senses that the drawn current is exceeding the programmed limit, it diverges from the programmed acceleration profile and reduces the acceleration just enough to keep the current within bounds. Reducing the acceleration reduces the torque required to overcome inertia.

How this is accomplished in practice depends on which kind of variable speed drive you have: A VSD driving a plain brushed motor will just reduce the rate of change of voltage, but a VFD using field-oriented control may be more complex.

the slower the motor runs the higher the overall current there is going to be.

This is generally not true, as a VSD won't output a constant voltage: Voltage is what you vary to control speed (in a DC motor), with AC motors you have to control voltage and frequency. To control the torque, you can vary the current, but of course you cannot control both voltage and current (or speed and torque, respectively) simultaneously.

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