Cylindrical Rotor Machine:-
The synchronous generator, under the assumption of constant synchronous reactance, may be considered as representable by an equivalent circuit comprising an ideal winding in which an e.m.f. Et proportional to the field excitation is developed, the winding being connected to the terminals of the machine through a resistance ra and reactance (Xl + Xa) = Xs all per phase. The principal characteristics of the synchronous generator will be obtained qualitatively from this circuit.
Generator Load Characteristics:-
Consider a synchronous generator driven at constant speed and with constant excitation. On open circuit the terminal voltage V is the same as the open circuit e.m.f. Et. Suppose a unity-power-factor load be connected to the machine. The flow of load current produces a voltage drop IZs in the synchronous impedance, and terminal voltage V is reduced. The complexor diagram for three types of load. It will be seen that the angle σ between Et and V increases with load, indicating a shift of the flux across the pole faces due to cross- magnetization. The terminal voltage is obtained from the complex summation
Algebraically this can be written
for non-reactive loads. Since normally r is small compared with X
so that the V/I curve, is nearly an ellipse with semi-axes Et and Isc. The current Isc is that which flows when the load resistance is reduced to zero. The voltage V falls to zero also and the machine is on short-circuit with V = 0 and
For a lagging load of zero power-factor, The voltage is given as before and since the resistance in normal machines is small compared with the synchronous reactance, the voltage is given approximately by
which is the straight line marked for cos φ = 0 lagging. A leading load of zero power factor. will have the voltage
another straight line for which, by reason of the direct magnetizing effect of leading currents, the voltage increases with load.
Intermediate load power factors produce voltage/current characteristics resembling those in, The voltage-drop with load (i.e. the regulation) is clearly dependent upon the power factor of the load. The short-circuit current Isc at which the load terminal voltage falls to zero may be about 150 per cent (1.5 per unit) of normal current in large modern machines.
The voltage-regulation of a synchronous generator is the voltage rise at the terminals when a given load is thrown off, the excitation and speed remaining constant. The voltage rise is clearly the numerical difference between Et and V, where V is the terminal voltage for a given load and Et is the open-circuit voltage for the same field excitation. Expressed as a fraction, the regulation is
Comparing the voltages on full load (1.0 per unit normal current) in, it will be seen that much depends on the power factor of the load. For unity and lagging power factors there is always a voltage drop with increase of load, but for a certain leading power factor the full-load regulation is zero, i.e. the terminal voltage is the same for both full and no-load conditions. At lower leading power factors the voltage rises with increase of load, and the regulation is negative. The regulation for a load current I at power factor cos φ is obtained from the equality,
from which the regulation is calculated, when both Et and V are known or found.
Generator Excitation for Constant Voltage:-
Since the e.m.f. Et is proportional to the excitation when the synchronous reactance is constant, the Eqn. 31 can be applied directly to obtain the excitation necessary to maintain constant output voltage for all loads. All unity-and lagging power-factor loads will require an increase of excitation with increase of load current, as a corollary of Low-leading-power-factor loads, on the other hand, will require the excitation to be reduced on account of the direct magnetizing effect of the zero- power-factor component. Shows typical e.m.f./current curves for a constant output voltage. The ordinates of are marked in percentage of no-load field excitation, to which the e.m.f Et exactly corresponds when saturation is neglected.
Generator Input and Output:-
For any load conditions as represented by, the output per phase is P = V I cos φ. The electrical power converted from mechanical power input is per phase
Resolving Et along I
The electrical input is thus the output plus the I2R loss, as might be expected. The prime mover must naturally supply also the friction, windage and core losses, which do not appear in the phasor diagram.
In large machines the resistance is small compared with the synchronous reactance so that θ = arc tan(xs/r) ≈ 90◦, it can be shown that
Thus the power developed by a synchronous machine with given values of Et V and Zs is proportional to sinσ: or, for small angles, to σ, and the displacement angle σ representing the change in relative position between the rotor and resultant pole- axes is proportional to the load power. The term load-, power- or torque-angle may be applied to σ.
An obvious deduction from the above is that the greater the field excitation (corresponding to Et) the greater is the output per unit angle σ: that is, the more stable will be the operation.
Salient Pole Rotor Machine:-
As discussed earlier in the behaviour of a synchronous machine on load can be determined by the use of synchronous reactance xs which is nothing but the sum of xa and xl , where xa is a fictitious reactance representing the effect of armature reaction while xl is the leakage reactance. It was also mentioned that this method of representing the
effect of armature reaction by a fictitious reactance xa was applicable more aptly only for a cylindrical rotor (non-salient pole) machine. This was so as the procedure followed therein was valid only when both the armature and main field m.m.f.’s act upon the same magnetic circuit and saturation effects are absent.
Theory of Salient-pole machines (Blondel’s Two-reaction Theory):-
That the effect of armature reaction in the case of a salient pole synchronous machine can be taken as two components – one acting along the direct axis (coinciding with the main field pole axis) and the other acting along the quadrature axis (inter-polar region or magnetic neutral axis) – and as such the mmf components of armature-reaction in a salient-pole machine cannot be considered as acting on the same magnetic circuit. Hence the effect of the armature reaction cannot be taken into account by considering only the synchronous reactance, in the case of a salient pole synchronous machine.
In fact, the direct-axis component Fad acts over a magnetic circuit identical with that of the main field system and produces a comparable effect while the quadrature-axis component Faq acts along the interpolar space, resulting in an altogether smaller effect and, in addition, a flux distribution totally different from that of Fad or the main field m.m.f. This explains why the application of cylindrical-rotor theory to salient-pole machines for predicting the performance gives results not conforming to the performance obtained from an actual test.
Blondel’s two-reaction theory considers the effects of the quadrature and direct-axis components of the armature reaction separately. Neglecting saturation, their different effects are considered by assigning to each an appropriate value of armature-reaction “reactance,” respectively xad and xaq . The effects of armature resistance and true leakage reactance ( xl ) may be treated separately, or may be added to the armature reaction coefficients on the assumption that they are the same, for either the direct-axis or quadrature-axis components of the armature current (which is almost true). Thus the combined reactance values can be expressed as :-
In a salient-pole machine, xaq, the cross- or quadrature-axis reactance is smaller than xad, the direct-axis reactance, since the flux produced by a given current component in that axis is smaller as the reluctance of the magnetic path consists mostly of the interpolar spaces.
It is essential to clearly note the difference between the quadrature- and direct-axis components Iaq, and Iad of the armature current Ia, and the reactive and active components
aa and Iar. Although both pairs are represented by phasors in phase quadrature, the former are related to the induced emf Et while the latter are referred to the terminal voltage V . These phasors are clearly indicated with reference to the phasor diagram of a (salient pole) synchronous generator supplying a lagging power factor (pf) load. We have
where σ = torque or power angle and φ = the p.f. angle of the load.
the two reactance voltage components Iaq ∗Xsq and Iad ∗ Xsd which are in quadrature with their respective components of the armature current.
The resistance drop Ia ∗ Ra is added in phase with Ia although we could take it as Iaq ∗ Ra and Iad ∗ Ra separately, which is unnecessary as
Actually it is not possible to straight-away draw this phasor diagram as the power angle σ is unknown until the two reactance voltage components Iaq ∗ xsq and Iad ∗ xsd are known. However this difficulty can be easily overcome by following the simple geometrical construction, assuming that the values for terminal voltage V , the load power factor (pf) angle φ and the two synchronous reactances xsd and xsq are known to us.