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### Natural Frequency Electric Motor Equation and Calculator

**Electric Motors, Generators and Drives**

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The undamped natural frequency of oscillation of a electric motor in a synchronous machine connected to an infinite system is:

Where:

f_{n} = natural frequency in cycles per minute

f = Frequency of motor output (Hz)

n = synchronous speed in revolutions per minute

P_{r} = synchronizing torque coefficient

W = weight of all rotating parts in pounds

k = radius of gyration of rotating parts in feet

When a pulsating torque is applied to its shaft, the synchronous machine rotor will oscillate about its average angular position in the rotating magnetic field produced by the currents in the stator. As a result of this oscillation, a pulsating torque will be developed at the air gap, a component of which is proportional to the angular displacement of the rotor from its average position. The proportionality factor is the synchronizing torque coefficient, Pr. It is expressed in kilowatts, at synchronous speed, per electrical radian.

Factors Influencing Pr

The value of Pr, for a given machine, is dependent upon:

- the voltage and frequency of the power system,
- the magnitude of the applied load,
- the operating power factor,
- the power system impedance, and
- the frequency of the torque pulsations. It is recommended that, unless other conditions are specified, the value of Pr submitted be that corresponding to operation at rated voltage, frequency, load, and power factor, with negligible system impedance and a pulsation frequency, in cycles per minute, equal to the rpm for synchronous motors and equal to one-half the rpm for synchronous generators.

Reference: NEMA MG 1-2009

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