# Equation of Motion of a Rotating Machine

The kinetic energy of a rotating mass, such as a large synchronous generator, is and the angular momentum is

where *v* is the synchronous speed of the rotor (rad/s) and I is the moment of inertia (kgm^{2}).

The inertia constant *(H)* is defined as the stored energy at synchronous speed divided by the rating of the machine G expressed in volt-amperes (VA). Hence

and the stored kinetic energy is

The angular velocity can be expressed in electrical degrees per second, v_{e}, as

where f is the system frequency in Hz.

Then

with

The inertia constants (*H*) of generating sets tend to similar values. The inertia constant for steam or gas turbo-generator units decreases from 10Ws/VA (generator and turbine together) for machines up to 30 MVA to values in the order of 4 Ws/VA for large machines, the value decreasing as the rating increases. For salient-pole hydro-electric units, *H* depends on the number of poles; for machines operating in the speed range 200-400 r.p.m. the value increases from about 2 Ws/VA at 10 MVA rating to 3.5 Ws/VA at 60MVA. A mean value for synchronous motors is 2 Ws/VA. Large wind turbines have an inertia constant in the range 2-5 Ws/VA.

The net accelerating torque on the rotor of a machine is

where AP *=* net power corresponding to AT,

In equation (8.2) a reduction in electrical power output results in an increase in d. Sometimes, the mechanical power input is assumed to be constant and the equation of motion becomes

or