You can find beautiful explanation about this matter in "POWER SYSTEM DYNAMICS Stability and Control" reference book as below.
Figure (4.8) shows three characteristic states that correspond to three different stages of rotor screening. Immediately after the fault, the current induced in both the rotor field and damper windings forces the armature reaction flux completely out of the rotor to keep the rotor flux linkages constant, Figure 4.8a, and the generator is said to be in the subtransient state.
Figure 4.8 The path of the armature flux in: (a) the subtransient state (screening effect of the damper winding and the field winding); (b) the transient state (screening effect of the field winding only); (c) the steady state. In all three cases the rotor is shown to be in the same position but the actual rotor position corresponding to the three states will be separated by a number of rotations.
As energy is dissipated in the resistance of the rotor windings, the currents maintaining constant rotor flux linkages decay with time allowing flux to enter the windings. As the rotor damper winding resistance is the largest, the damper current is the first to decay, allowing the armature flux to enter the rotor pole face. However, it is still forced out of the field winding itself, Figure 4.8b, and the generator is said to be in the transient state. The field current then decays with time to its steady-state value allowing the armature reaction flux eventually to enter the whole rotor and assume the minimum reluctance path. This steady state is illustrated in Figure 4.8c and corresponds to the flux path shown in the top diagram of Figure 4.5b.
It is convenient to analyse the dynamics of the generator separately when it is in the subtransient, transient and steady states. This is accomplished by assigning a different equivalent circuit to the generator when it is in each of the above states, but in order to do this it is first necessary to consider the generator reactances in each of the characteristic states.
The inductance of a winding is defined as the ratio of the flux linkages to the current producing the flux. Thus, a low-reluctance path results in a large flux and a large inductance (or reactance) and vice versa. Normally a flux path will consist of a number of parts each of which has a different reluctance. In such circumstances it is convenient to assign a reactance to each part of the flux path when the equivalent reactance is made up of the individual path reactances. In combining the individual reactances it must be remembered that parallel flux paths correspond to a series connection of reactances while series paths correspond to a parallel connection of reactances. This is illustrated in Figure 4.9 for a simple iron-cored coil with an air gap in the iron circuit. Here the total coil flux Ф consists of the leakage flux Фl and the core flux Фc. The reactance of the leakage flux path is X l
, while the reactance of the core flux path has two components.
The first component corresponds to the flux path across the air gap X ag
and the second component corresponds to the flux path through the iron core X Fe
. In the equivalent circuit X l is connected in series with the parallel combination X ag
and X Fe
. As the reluctance of a flux path in iron is very small compared with that in air X Fe
>> X ag
and the total reactance of the winding is dominated by those parts of the path that are in air: the air gap and the flux leakage path. Consequently X ≈ X ag + X l . These principles are applied to the synchronous generator in Figure 4.10 for each of the three characteristic states. It also introduces a number of different reactances, each of which corresponds to a particular flux path:
The reactances of the damper winding and the field winding are also proportional to the flux path around the winding. Thus X D
and X f
are proportional to the actual damper and field winding reactance respectively.
In Figure 4.10 the armature reactances in each of the characteristic states are combined to give the following equivalent reactances:
In the subtransient state the flux path is almost entirely in air and so the reluctance of this path is very high. In contrast, the flux path in the steady state is mainly through iron with the reluctance being dominated by the length of the air gap. Consequently, the reactances, in increasing magnitude, are X "d < X 'd < Xd. In the case of large synchronous generators X 'd is about twice as large as X "d, while X d is about 10 times as large as X "d.
As previously discussed, following a fault the generator becomes a dynamic source that has both a time-changing synchronous reactance X(t) and an internal voltage E(t). Dividing the generator response into the three characteristic states with associated constant reactances makes it easier to analyse the generator dynamics. Rather than considering one generator model with time-changing reactances and internal voltages, it is convenient to consider the three states separately using conventional AC circuit analysis. This is illustrated in Figure 4.11. The rms value of the AC component of the armature current IAC
(t) is shown in Figure 4.11a. This AC component was previously shown in the top diagram of Figure 4.7b. The continuously changing synchronous reactance X, shown in Figure 4.11b, can be calculated by dividing the open-circuit emf E by the armature current IAC(t). In each of the three characteristic states, the generator will be represented by a constant emf behind a constant reactance X "d , X 'd , X d, respectively. Dividing the emf by the appropriate reactance will give the subtransient, transient and steady-state currents.
POWER SYSTEM DYNAMICS
Stability and Control
Warsaw University of Technology, Poland
Janusz W. Bialek
The University of Edinburgh, UK
James R. Bumby
Durham University, UK