In abnormal operation, the alternator may be subjected to transient conditions which cannot be explained from steady-state theory. These transients may occur from- (i) Switching (ii) Sudden changes of load, (iii) Sudden short circuit (either line-to-line or line-to-neutral or from a symmetrical short circuit).
These short circuits may develop severe mechanical stresses on the armature coils and their end turns or large torques, which may damage the alternator or its prime mover. The analysis of a synchronous machine under conditions of such transients is useful in predicting the possible conditions that may result from abnormal operation resulting from short circuits. However, here the discussions will be limited to symmetrical short circuit (short circuit on all the output terminals of a 3-phase alternator).
Effects of short circuit currents can be conveniently studied by using the concept of constant flux linkage, which may be stated as below:
It is impossible to change the flux linkages in a closed electrical circuit that contains no resistance or capacitance (i.e. in a purely inductive circuit).
ADVERTISEMENTS:
In case of a synchronous machine, the armature and field windings can be assumed almost purely inductive because they do not contain any capacitance and their resistances are almost negligible in comparison to their inductive reactances.
So the flux linkages in the armature circuit and the field circuit cannot be changed suddenly by the application of short circuit to the armature winding. For maintaining these flux linkages constant, large changes of current may take place in both of the windings when the short circuit occurs in order to keep their respective flux linkages constant.
Reactance of Synchronous Machines:
The current flowing in the armature of a synchronous generator when its terminals are short circuited is similar to that flowing when a sinusoidal voltage is suddenly applied to an R-L series circuit. However, there is one important difference, that is, in case of an R-L series circuit, reactance X (ωL) is a constant quantity where as in case of the synchronous generator the reactance is not a constant one but is a function of time.
In practice three discrete values are assigned and thus we have three reactances-direct-axis subtransient reactance symbolised as X”d, direct-axis transient reactance symbolised as X’d and direct-axis synchronous reactance symbolized as Xd.
ADVERTISEMENTS:
In short circuit studies only the above direct-axis reactances are involved. This can be justified as follows:
Short-circuit currents, in general, feed through lines or transformers where the inductive reactance is quite large in comparison to resistance, making the power factor approach zero as currents go lagging and when the armature current is in quadrature (lagging) with the excitation voltage (or no-load voltage) the entire armature mmf acts directly upon the magnetic paths through the salient poles and all of the armature mmf is directly opposing the mmf of the salient pole field windings. Now these direct-axis reactances of the synchronous machine will be briefly described.
Under normal no-load condition of operation, there is no mmf due to armature reaction. When a sudden three-phase short circuit occurs at the terminals of a synchronous generator, the current in the armature circuit increases suddenly to a large value (the symmetrical short circuit current is limited only by the leakage reactance of the machine) and since the resistance of the circuit is negligible as compared to its reactance, the current is highly lagging and the power factor is approximately zero.
A sudden increase in armature current is accompanied by armature reaction. Since the air gap flux (main flux) cannot change instantaneously (concept of constant flux-linkage), to counter the demagnetization of the armature short circuit current (armature reaction acts on the direct axis in a direction to oppose the main excitation) currents appear in the field winding as well as in damper windings in a direction to help the main flux.
ADVERTISEMENTS:
This happens due to the transformer action at the instant of short circuit. Thus during the initial part of the short circuit, the equivalent circuit appears as shown in Fig. 4.4 (a)—field winding reactance Xf, damper winding reactance Xdw appear in parallel with Xa.
The resistance/reactance ratio of damper windings is higher than that of the field winding, and the current in the damper windings falls to zero after three or four cycles, when the main flux assumes the value it would have had if there has been no damper windings. The effect of damper windings is, therefore, to increase the initial value of short circuit current, and to maintain a slightly increased current for three or four cycles after the short circuit.
As the damper winding currents are first to die out, Xdw effectively becomes open circuited and at a later stage Xf becomes open circuited. The machine reactance thus changes from the parallel combination of Xa, Xf and Xdw during the initial period of the short circuit to Xa and Xf in parallel [Fig. 4.4 (b)] in the middle period of short circuit, and finally to xa in steady-state [fig. 4.4 (c)].
The reactance represented by the machine in the initial period of the short circuit i.e. [Xl + {1/(1/Xa) + (1/Xf)+ (1/Xdw)}] is called the subtransient reactance (X”d)of machine; while the reactance effective after the damper winding currents have died out, i.e., [Xl + {1/(1/Xa)+ (1/Xf)}] is called the transient reactance X’d of the machine.
ADVERTISEMENTS:
Of course, the reactance under steady-state short circuit conditions is the synchronous reactance of the machine. Obviously X”d < X’d < Xd. The machine thus offers a time-varying reactance which changes from X”d to X’d and finally to Xd.
Analysis of Short-Circuit Oscillogram:
When a short circuit occurs across the terminals of a synchronous generator the initial short circuit current is limited by the sub-transient reactance for a few cycles, later on it is controlled by the transient reactance. Finally the short circuit current settles down to the steady- state short circuit value limited by the synchronous reactance of the machine.
At least two phases will have asymmetry in the short circuit waves, the extent to which asymmetry occurs depending upon the value of the phase voltage at the instant of short circuit. Those phases in which the voltage is near its maximum value at the instant of short circuit will have least asymmetry while those where the voltage is at or near its zero will have a greater or lesser degree of asymmetry, the maximum asymmetry being equal to the otherwise short circuit current.
ADVERTISEMENTS:
The current can be split up into unidirectional or asymmetrical component, and an alternating component of supply frequency. The dc component represents the displacement of the ac wave from the zero axis and is therefore, sometimes known as dc offset current.
If dc component is subtracted from the short circuit current wave, we will be left with the ac symmetrical component as shown in Fig. 4.6. The characteristics of short circuit currents of a 3-phase synchronous generator are shown in Fig. 4.5. These curves indicate that the short circuit occurred at the instant when the voltage of phase R was maximum (assuming that the machine was not delivering any current to the load prior to fault).
The transient reactances can be obtained from any one of the three oscillograms by graphical methods. In case of using Fig. 4.5 (b) or 4.5 (c), it is necessary to eliminate the dc components and plot the envelopes of ac components only.
Figure 4.6 illustrates the short circuit current for one phase with dc component eliminated.
The short circuit period can be divided into three periods-initial sub-transient period, lasting only for the first few cycles, during which the current decrement is very rapid, the middle transient period, covering a relatively longer time, during which the current decrement is moderate; and finally the steady-state period.
In accordance with the foregoing theory of changing reactance we have –
Sub-transient current (maximum) excluding dc component,
I”max = Oc = Eg max/X”d …(4.4)
Transient current (maximum) excluding dc component,
I’max = Ob = Eg max/X’d …(4.5)
Steady-state current (maximum).
Imax = Oa = Eg max/Xd …(4.6)
where Eg max is the maximum voltage from one terminal to neutral on no load.
Note that over any one cycle the current is approximately sinusoidal. So for using rms or effective values, it is quite reasonable to divide the maximum values of Fig. 4.6 by √2. The new curve so obtained is shown in Fig. 4.7 and is known as a decrement curve.
Thus using rms values, the initial rms currents corresponding to the above mentioned three periods are as follows:
Sub-transient current (rms) excluding dc component,
I” = Eg/X”d …(4.7)
Transient current (rms) excluding dc component,
I’ = Eg/X’d …(4.8)
And steady-state current (rms),
I = Eg/Xd …(4.9)
Where,
the rms voltage from one terminal to neutral.
The sub-transient and transient decay periods have time constants T” and T’ respectively. By determining these time constants it is possible to express the instantaneous current by –
If dc component is to be included, then we find the effective value of armature short circuit current, which is given as –
The momentary short circuit which a breaker should be capable of carrying is given by Eq. (4.15) but according to AIEE Switchgear Committee, the multiplying factor should be taken as 1.6 instead of √3. This multiplying factor can be further reduced to 1.5 for 5 kV breaker or less.