Time Period of Oscillation

Time Period of Oscillation

The synchronous machines are associated with free oscillations having some natural time period. There may be swinging of machine due to variations in load or due to any other reason. If the time period of these oscillations coincide with natural time period of the machine then there may be higher amplitude of oscillations which may pull the synchronous machine out of synchronism.

Let us derive the expression for time period for these oscillations.

Let     P_SY = Synchronizing power per phase

T_SY = Synchronizing torque per phase, E = Induced e.m.f. per phase

Z_s = Synchronous Impedance, N_s = Synchronous speed in r.p.m. 

n_s = Synchronous speed in r.p.s.,

P = No. of poles

ɑ = Angle of swing by an alternator in electrical radian

The synchronizing power developed is given by,

P_SY = ɑ E² / Z_S = E² / Z_S per elecrical radian per phase

We have, 1 electrical radian = P/2 × mechanical radian

For mechanical radian displacement,

The period of undamped free oscillations is given by,

t = 2π √ J/T

Here, J = Moment of Inertia in kg-m²

Substituting value of T obtained in above expression,

Here kVA is full load kVA rating of an alternator.

If synchronous speed is expressed in rpm, then time t in seconds is given by

Review Question

1. Derive the expression for the time period of oscillations in a synchronous machine.