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 (kgm2).
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, ve, as
where f is the system frequency in Hz.
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