__The V-Curve of a Synchronous Generator __

A valuable property of a synchronous generator operating into a large power system is that its reactive current can be controlled by varying its excitation current. For a better understanding of the matter, let us turn to the phase phasor diagram of a synchronous generator (see Fig 8) and analyse it from this point of view (Fig 11) . If the power developed by a synchronous generator, *P* = ɷ_{rot}*T*_{em}, and the voltage across the system busbars, *V*, are constant, then the products of the terms in Eq. (15.10)

*I* cos φ = *I*_{act} = const (15.14)

*E*_{0} sin │θ│ = ɷ Ψ_{0 }sin │θ│ =const

are independent of the excitation current. However, a change in the excitation current brings about a change in the associated flux linkage with the stator phase winding, Ψ̇_{0} , and in the emf *Ė*_{0} induced by this flux linkage in the phase winding.

It follows from the equation of- electric state of a stator phase, Eq. (15.8), that this can happen only if there is a proportionate change in the phase current

*İ*= *İ*_{act} + *İ*_{reac}

or, to be more exact, in the reactive current, *İ*_{reac} .

When the excitation current is less than some limiting value,

*I*_{f} < *I*_{f,Ifm} (*P*), the current in a capacitive reactive component, *I*_{reae},_{c} . When the excitation current is greater than some limiting value, *I*_{f} > *I*_{f},_{1fm}, the current in a synchronous generator contains an inductive reactive component, *I*_{reae,L} (Fig. 11). The phase angle respectively is φ < 0 and φ > 0 . In consequence, underexcitation makes the reactive power of a synchronous generator capacitive in its effect,

Q_{C} = 3*VI*_{reac},_{c}

and overexcitation makes it inductive in its effect

Q* _{L}* = 3

*VI*

_{reac,L }

If a synchronous generator operates into a large power system with *V* = const, its equivalent circuit may be drawn up as a parallel combination of two current sources, one being a source of active current which depends on the torque developed by the prime mover, *I*_{act} = *f* (*T*_{pm}), and the other being a source of reactive current which depends on the torque developed by the prime mover and the excitation current, *I*_{reac} = *f* (*I*_{f}, *T*_{pm}).

In the special case of a zero prime-mover torque, *T*_{pm} = 0, the phase equivalent circuit of a synchronous generator operating into a large system will contain no active current source, and the reactive current will be solely dependent on the excitation current,

*I*_{reac} = *f* (*I*_{f})·

The plots of the current of a synchronous machine operating into a large power system with *V* = const, as a function of the excitation (field) current, *I* = *f *(*I*_{f}), at a constant prime-mover torque; *T*_{pm} = const, are called the *V*-curves of the machine (Fig. 12) . At some low value of excitation current the angle │θ│(see Fig 10) may exceed π/2 in value, and the stability of the machine may be upset. The greater the active power of a synchronous generator, the greater the excitation current at hich the machine may lose its stability. In Fig 12 the boundary of stable operation for a synchronous generator is shown by a dashed line .

The torque developed by the prime mover is zero, *T*_{pm} = 0, then, on neglecting all forms of loss, we may take it that the current in a synchronous generator is purely reactive (see Fig 12, *P* = 0) :

*İ* (*I*_{f}) = *İ*_{reac} (*I*_{f}) = (*Ė* - *V̇*)/*j* = (*j*ɷΨ̇_{0} (*I*_{f}) - *V̇*)/*j* (15.15)

Now the generator current is a linear function of the excitation current. The function *I* = *f* (*I*_{f}) ceases to be linear only at high values of excitation current owing to the saturation of the magnetic circuit of the machine.