to approximately 105 rad/sec at the time when Te equal to Tl. At this time, accelerating torque is zero and the rotor is running above synchronous speed; hence, load angle δ, and thus motor torque Te will keep on decreasing. Decrease in torque Te results in decrease in output power of the machine that causes the rotor to decelerate toward synchronous speed. Hence, due to rotor inertia, it will continue to decelerate below synchronous speed, and consequently, load angle δ begins to increase, with increase in generated torque Te. In this way, damped oscillation of machine continues and settles to a new-steady state value.
Figure 1.2 Dynamic response of motor following the change in load torque showing (a) motor torque Te, (b) rotor speed ωr, and (c) load angle, delta.
Figure 1.3 Dynamic response of motor following the change in load torque showing stator currents (a) ia and (b) ix.
Figure 1.4 d-q component of stator winding currents (a) Iq1, (b) Id1, (c) Iq2, and (d) Id2.
Due to the increase in generator prime mover input torque, increase in stator phase current can be noted in Figure 1.3, which is required to meet the increased output. The increase in stator current is associated with the increase in active output power of the generator, while maintaining its operation at constant power factor (i.e., constant reactive power). Hence, the change in q-axis component of stator current (active component of current changes from 1.8 A at 50% output power to 3.6 A at rated output, approximately) is depicted in Figure 1.4, with no change in its d-axis component of stator current (reactive component of current), of both the winding sets abc and xyz.
1.5 Stability Analysis Results
For small signal stability analysis, eigenvalue is calculated from system characteristic equation:
(1.54)
where unknown root λ is calculated (i.e., eigenvalue) with A and I as system matrix and identity matrix, respectively. For a system to be stable, all the real and/or real component of eigenvalues must be negative [15, 18–20].
State equation (1.53) and equation 8.3-45 of reference [15] are using nine and seven state variables, respectively. Hence, nine and seven eigenvalues will be obtained for six and three-phase generator, respectively. In six-phase generator, out of nine evaluated eigenvalues, three eigenvalues are complex conjugate pairs and the remaining are real. Evaluated eigenvalue of six-phase and three-phase generator is given in Tables 1.1 and 1.2, respectively. Eigenvalue was evaluated by considering the same flux level in both three- and six-phase machine. This was ensured by considering the stator voltage of three-phase machine as twice of six-phase machine [27]. Hence, value of the terminal voltage for three-phase and six-phase machine was taken as 240 and 120 V, respectively. Results are given for both machine considering the same load at 50% of rated value, at power factor 0.85 (lagging). It is worthwhile to mention here that it is a difficult to establish a correlation of eigenvalue with machine parameter [15]. It has been considered by changing a machine parameter, keeping other at nominal value and noting the variation in eigenvalue [18].
Table 1.1 Eigenvalues of six-phase synchronous generator.
| Nomenclature | Eigenvalues |
| Stator eigenvalue I | −107.8 ± j104.7 |
| Stator eigenvalue II | −19.2 ± j110.3 |
| Rotor eigenvalue | −5.1 ± j38.2 |
| Real eigenvalue | −9136.3, −703.5, −21.0 |
Table 1.2 Eigenvalues of three-phase synchronous generator.
| Nomenclature | Eigenvalues |
| Stator eigenvalue | −38.3 ± j103.2 |
| Rotor eigenvalue | −27.9 ± j50.4 |
| Real eigenvalue | −8719.7, −503.3, −10.7 |
In Equation (1.53), derivative component (i.e., with its elements with subscript p) is indicated by coefficient matrix E, with remaining terms (i.e., subscript k) of linearized machine equations are shown by the coefficient matrix F. Matrices E and F elements are defined in the Appendix.
1.5.1 Parametric Variation of Stator
During the analysis, it was assumed that the two sets of stator winding, say, abc and xyz, are identical. Hence, value of resistance and winding leakage inductance will be same (i.e.,
With increase in stator resistance, generator operation tends toward stability from both stator and rotor side due to higher magnitude of negative real component of eigenvalue, as shown in Figure 1.5a for three-phase generator and Figures 1.6a and b for six-phase generator. Real eigenvalue I and II was found to be decreased as shown in Figures 1.5b and c for three-phase generator, and in Figures