Saturday, February 9, 2013

The Electromagnetic Torque and the Load-Angle Characteristic of a Synchronous Generator

The Electromagnetic Torque and the Load-Angle Characteristic of a Synchronous Generator

Le t us see how the power P and the electromagnetic torque Tem of a synchronous generator depend on the load angle θ (θ < 0) . To this encl, we will refer to the phasor diagram in Fig 8.

The electric power supplied by the three phases of a synchronous genera tor is given by

P = 3VI cos φ = 3E0I cos φ0= 3E0I cos (φ - θ) (15.9)

Taking the Ė0 phasor as hypotenuse, we construct a right triangle in which is a part of one of the sides. The other side of the triangle, opposite the angle 8, is

I cos φ = E0sin │θ│

On the basis of the same diagram,

E0 cos φ0 = V cos φ

and so we may express the electric power of a synchronous generator as

P=3VIcos φ =3E0V sin │θ│/ (15.10)

The electromagnetic torque developed by the interaction of the armature current with the magnetic field of the machine is con­nected to its electric power by a well-known simple relation

Tem = Prot

where the synchronous angular velocity of the rotor is

ɷrot = 2πn/60 = 2πf/p

Hence,

Tem = (3p/2πf) E0V sin │θ│/ (15.11)

Since in a large power system the voltage V and the frequency f are constant, it follows that, given a constant excitation current, the power and the electromagnetic torque of a synchronous generator depend solely on the load angle │θ│. This sinusoidal relation is known as the power/angle or torque/ angle characteristic of a synchron­ous generator (Fig 10). For power and electromagnetic torque, it only differs in scale.

The power-vs-angle and torque-vs-angle characteristics give an insight into what happens in a synchronous generator as its load is varied.

The Electromagnetic Torque and the Load-Angle Characteristic of a Synchronous Generator

The work done by the prime mover is converted to electric energy, and this is delivered to the line. As the torque developed by the prime mover increases, T > Tpm1 = Tem1 (point 1), the rotor is accelerat­ed, and the load angle │θ│increases . After several oscillations about the synchronous angular velocity the balance between the driving torque supplied by the prime mover and the opposing electromagnet­ic torque developed by the generator is restored, T pmz = T emz (point 2) at a new value of the load angle │θ│: │θ2│> │θ1│.

A synchronous genera tor will give a stable performance if │θ│ varies from 0 to π/2 . At │θ│= π/2, the electric power is

A maximum Pmax = 3E0V/ (15.12)

and the electromagnetic is likewise a maximum

Tem,max = 3pE0V/ɷ (15.13)

The range of values π /2 - │θ│ defines the margin of stability for a synchronous generator. At │θ│ > π /2 a synchronous generator tends to be unstable. Now the driving torque supplied by the prime mover exceeds the opposing electromagnetic torque of the driven synchronous generator. The excessive torque (Tpm > Tem) acce­lerates the rotor still more, thereby leading to an increase in │θ│ and a further decrease in the torque of the generator, and so on, until the generator drops out of synchronism. The margin of stability π /2 - │θ│, of a synchronous generator at an increased load can be re­stored by increasing the excitation current (point 3).