STUDY OF THE DYNAMIC STABILITY OF SYNCHRONOUS RELUCTANCE MOTORS ON AN ANALOG COMPUTER

E. V. Kononenko, A. F. Fink

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering)

One of the questions in the theory of synchronous reluctance motors (SRM) that has not been sufficiently developed is the study of dynamic stability under a sudden change in load. In the simplest way, this problem can be solved by the equal-area method [1]. If such an approximate estimate shows that the stability margin of an SRM considerably exceeds the required value, then there is no need to carry out stability calculations by more accurate methods. However, refinement of the calculations is necessary if there is reason to believe that a more accurate determination of dynamic stability will make it possible to improve the technical and economic characteristics of the SRM, and also to take into account a larger number of factors affecting stability.

The study of the dynamic stability of an SRM consists not only in verifying that the motor maintains synchronism under a specified disturbance of the operating mode. It is also necessary to determine the limiting permissible disturbances corresponding to the boundary of dynamic stability.

Dynamic-stability calculations are most simply carried out, for the complete system of nonlinear differential equations, on an analog computer [2].

In the general case, a disturbance of the steady-state operating mode of an SRM may have the most varied character in time. In considering the influence of motor parameters on dynamic stability, we shall restrict ourselves to electromechanical transients caused by a sudden application of load in the form of a rectangular pulse, of magnitude \(M_o\) and duration \(\Delta t\). We shall also assume that before and after the load application the SRM operates at no load.

All calculations connected with studying the influence of motor parameters on dynamic stability were performed on an EMU-10 analog computer according to the structural diagram given in [2].

The procedure adopted for finding the boundary of dynamic stability was the following [3]. For a duration \(\Delta t\) of the applied torque \(M_o\), two of its limiting values, \(M_{o1}\) and \(M_{o2}\), were found by gradually increasing the torque amplitude. These torques were to differ from one another by a prescribed amount (not more than 1 percent) and to satisfy the condition that, at the smaller value \((M_{o1})\), the SRM after removal of the load operated stably, whereas at the larger value \((M_{o2})\) it fell out of synchronism.

The investigations carried out made it possible to establish that, in order to determine the boundary of dynamic stability, for the type of disturbance under consideration it is sufficient to trace the motion of the rotor only in the first swing after load shedding. If the angle in the first swing begins to decrease, then the SRM does not fall out of synchronism and, after the transient processes have died out, operates stably, as is seen from the oscillogram shown in Fig. 1. Moreover, in the case when the motor rotor has a complete starting winding, upon crossing the boundary of dynamic stability the SRM is again pulled into synchronism after one or several “slips” of the rotor relative to the stator magnetic field (Fig. 2).

Fig. 1. Oscillogram of the motion of the SRM rotor under a sudden application of load without falling out of synchronism

Fig. 2. Oscillogram of the motion of the SRM rotor under a sudden application of load with falling out of synchronism and subsequent synchronization

In studying the influence of the SRM parameters on dynamic stability, it is most convenient to construct the stability boundaries in the form of curves

\[ \frac{M_{\partial}}{M_m} = f(\Delta t). \]

Here \(M_m\) is the maximum torque of the established synchronous operating mode. This makes it possible to estimate the influence both of the magnitude and of the duration of action of the torque \(M_{\partial}\).

Figure 3 gives the dynamic-stability boundaries of SRMs having different time constants of the starting windings. The remaining motor parameters were taken as constant and equal to:

\[ x_d = 2.33; \qquad x_q = 0.45; \qquad r = 0.049; \qquad T_j = 1 \text{ sec}; \]

\[ x_{rd} = 2.31; \qquad x_{rq} = 0.428; \qquad T_q = 0.25T_d. \]

It follows from the data presented that, with an increase in the time constants \(T_d, T_q\), the region of stable operation of the SRM expands. Therefore, in designing SRMs, the starting windings should have the smallest possible active resistance.

A large influence on the dynamic stability of the SRM is exerted by the moment of inertia of the rotating masses. As is seen from Fig. 4, with an increase in \(T_j\) the region of stable operation expands, since motors with large moments of inertia are considerably more difficult to bring out of the state of equilibrium.

The influence of the active resistance of the stator winding of an SRM on the region of dynamic stability is characterized by the curves presented in Fig. 5. From these curves it is seen that, with an increase in \(r\), the region of stable operation narrows. This can be explained by the influence of electro-

Fig. 3. Dependences of the boundaries of dynamic stability on the time constants of the rotor windings

Fig. 4. Dependences of the boundaries of dynamic stability on the inertia constant of the rotating masses

magnetic transients in the stator winding. The rate of damping of the aperiodic components of the current and flux linkages arising in the stator winding during transients is proportional to the active resistance of the stator winding. Consequently, as \(r\) increases, the electromagnetic transients exert a smaller influence. The influence of the parameter ratio

\[ \frac{x_d}{x_q} \]

on the region of dynamic stability was also determined. The studies showed that

Fig. 5. Dependences of the boundaries of dynamic stability on the active resistance of the stator winding

an increase in the ratio

\[ \frac{x_d}{x_q} \]

shifts the regions of stable operation toward a decrease.

From the curves presented (Figs. 3, 4, 5) it is evident that the boundaries of dynamic stability depend substantially on the duration of action \(\Delta t\). For short durations of action, the dynamic stability considerably exceeds the static stability. As \(\Delta t\) increases, the region of dynamic stability approaches the static one. For certain parameter ratios, the region of dynamic stability may even be smaller than the value of the static overload capacity.

If electromagnetic transients are neglected, the study of dynamic stability can be reduced to the solution of only one equation of rotor motion.

A comparison of the limits of dynamic stability calculated from the complete equations and from the equation of rotor motion is presented in Fig. 6. As follows from these curves, taking into account the transient electromagnetic processes in the stator winding increases the dynamic-stability margin by 10–15 percent.

Fig. 6. Comparison of the results of calculating the limits of dynamic stability from the complete equations (solid curve) and from the equation of rotor motion (dashed curve)

REFERENCES

1. E. V. Kononenko, A. F. Fink. Pulling a synchronous reluctance motor into synchronism. Izvestiya TPI, vol. 145, 1966.

2. E. V. Kononenko, A. F. Fink. Application of the EMU-10 computing machine to the study of transient electromechanical processes in synchronous reluctance motors. Izvestiya TPI, vol. 160, 1966.

3. I. M. Seryi, Yu. I. Yushmanov, A. A. Yanko-Trinitskii. The influence of damping moments on the dynamic stability of a synchronous motor. Izvestiya vuzov “Elektromekhanika”, 1961, No. 7.