0195136047.pdf

586 ROTATING MACHINES

and the armature current,

I ̄a=

E ̄f−V ̄t

jXs

=

Ef cosδ−Vt+jEf sinδ

jXs

(13.3.7)

whereXsis the synchronous reactance per phase. Since

I ̄a∗=Ef cosδ−Vt−jEf sinδ

−jXs

=

Efsinδ

Xs

+j

Efcosδ−Vt

Xs

(13.3.8)

therefore,

P=

VtEfsinδ

Xs

(13.3.9)

and

Q=

VtEf cosδ−Vt^2

Xs

(13.3.10)

Equations (13.3.9) and (13.3.10) hold for a cylindrical-rotor synchronous generator with negligible

armature resistance. To obtain the total power for a three-phase generator, Equations (13.3.9) and

(13.3.10) should be multiplied by 3 when the voltages are line to neutral. If the line-to-line

magnitudes are used for the voltages, however, these equations give the total three-phase power.

The power-angle or torque-angle characteristic of a cylindrical-rotor synchronous machine

is shown in Figure 13.3.3, neglecting the effect of armature resistance. The maximum real power

output per phase of the generator for a given terminal voltage and a given excitation voltage is

Pmax=

VtEf

Xs

(13.3.11)

Any further increase in the prime-mover input to the generator causes the real power output to

decrease. The excess power goes into accelerating the generator, thereby increasing its speed

and causing it to pull out of synchronism. Hence, thesteady-state stability limitis reached when

δ=π/2. For normal steady operating conditions, the power angle or torque angle is well below

90°. The maximum torque orpull-out torqueper phase that a round-rotor synchronous motor can

develop for agradually appliedload is

−δ δ

Real power or torque

Pull-out torque

as a generator

Generator

0

Pull-out torque

as a motor

Motor

π

−π

−π/2 π/2

Figure 13.3.3Steady-state power-angle

or torque-angle characteristic of a cylin-

drical-rotor synchronous machine (with

negligible armature resistance).