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What are the effects for a 50Hz motor running on 60Hz?

Frequency directly represents no of rotation of rotor for that machine, the higher the frequency the more the rotation of the rotor, i.e. the rpm of the motor there by result in bearing wear and tear, if the motor run at no load, voltage will be very high than the rated value which may result in insulation failure for the motor winding.
Hz
The rated operating voltage must be the same, the motor power will reduced by 50/60 Hz. so if the motor working at the full load, load must be reduced by the same ratio, to keep the motor temperature within the rated limit.

Torque/speed curve is on blower flux. You must reduce load or increase voltage, being careful of insulation voltage level. If motor is fed by variable frequency drive, speed reference must be changed to ensure correct V/Hz control (open loop control).

Take previous considerations and even pay attention to speed/RPM changes. It will respect the ratio between 120 times frequency (f) and number of poles (n), or, RPM=(120*f)/n. Not considering mechanical losses it will be 3000 RPM for a 2 poles induction motor operating at 50Hz and 3600 RPM at 60Hz, as an example. It can overcharge the bearings and generate heat and you need also to analyze how it will affect the load attached to your motor (is it a pump, a fan?).

While talking about a three-phase motor, and you vary frequency and maintain steady V/Hz. By changing only the frequency, motor will run at 20% higher speed at 60 Hz compared to the speed at 50 Hz.

At higher speed, the motor will have better cooling by its internal fan. That is one good effect. But the motor will have worse power factor which is not good effect. Besides, magnetic system of the motor will not get into saturation at 60 Hz.

On the contrary, motor designed for 60 Hz can be magnetically overloaded if running on 50 Hz. That is kind of overload and causes more current draw. More current means more heat in the motor and could lead to failure.

Next you should consider all effects dependent on drive. Those effects are many. The motor will run 20% faster.., BUT DON'T think of that to be as simple as PROMOTION of the motor. More speed means more energy. Energy cannot be created or destroyed, but it can be transformed.

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