Study of the Asynchronous Mode of Salient-Pole Synchronous Machines

E. V. Kononenko, T. V. Chesheva

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

This article presents the results of studies of the asynchronous mode of two salient-pole synchronous motors differing from one another by the presence of an incomplete (machine No. 1) and a complete (machine No. 2) damper winding on the rotor. The rated power of each machine is 7.5 kVA, \(U_n = 400\) V, \(2p = 6\), \(n_n = 1000\) rpm. The parameters of the machines and the experimental characteristics of the asynchronous mode \([M_c = f(s);\ I = f(s)]\) for these machines were published in [1]. Use of the results of [1] made it possible, without carrying out an experiment, to analyze the asynchronous mode of machines with different rotor asymmetry [2]. The use of a digital computer for the calculations made it possible to carry out fairly broad studies of the influence of the machine parameters and the degree of rotor asymmetry on the starting characteristics of synchronous machines. Since the calculations of the asynchronous mode (meaning the static starting characteristics, i.e., the dependences of the average asynchronous torque and stator current on slip) were performed by three methods [2]—exact, approximate, and combined—the results of these studies made it possible to evaluate the applicability and degree of accuracy of each of these methods.

Calculations by the exact method were carried out on an M20 digital computer. By the approximate and combined methods, the starting characteristics were calculated manually using values of the operational reactances \(X_d(js)\) and \(X_q(js)\), calculated from general formulas [2] on a digital computer. It should be noted that all calculations were carried out in the slip range from 0 to 1.0 for 16 values, with an increased number of points in the region of subsynchronous speed, for four values of the active resistance of the stator winding: \(r = 0;\ 0.0246;\ 0.05;\ 0.1\). Additional resistances of various values were inserted into the field-winding circuit of both machines, equal to \(3.7 R_f\) (\(R_f\) is the resistance of the field winding), \(9R_f\), \(16R_f\), \(24R_f\). The calculations were performed in relative units.

The studies carried out made it possible to conclude that separate consideration of phenomena along the longitudinal \(d\) and transverse \(q\) axes of the rotor (the approximate method) is valid only when the stator winding is superconducting (\(r = 0\)), that is, the reverse-sequence stator flux linkage is equal to zero, or when the rotor is symmetrical \([X_d(js) = X_q(js)]\). In real motors with an asymmetrical rotor, the approximate calculation method is in-

accurate. Using an approximate method, it is impossible, for example, to detect the dip in the curve of the electromagnetic torque of rotation, the pulsating torque, and the stator current at subsynchronous speed.

Studies have shown that the form of the starting characteristics of a synchronous salient-pole machine is determined mainly by the magnitude and ratio of the established values of the operational impedances \(X_d(js)\) and \(X_q(js)\). The presence of magnetic asymmetry leads to the fact that the curve of the average asynchronous torque \(M_c\), unlike in asynchronous motors, passes through zero at \(s > 0\), while at synchronous rotational speed \(M_{c0} \ne 0\). The magnitude \(M_{c0}\) depends on the degree of magnetic asymmetry, determined by the difference of the synchronous inductive reactances along the \(d\)- and \(q\)-axes, and on the active resistance of the stator winding \(r\)

\[ M_{c0}=-\frac{1}{2}\frac{r(x_d-x_q)^2}{(r^2+x_d x_q)^2}U^2 . \tag{1} \]

The considered properties of the starting characteristics have a substantial influence on motor starting, especially at the final stage—when pulling into synchronism. The use of approximate calculation methods does not make it possible to reveal these features. In Fig. 1, for comparison, the mechanical characteristics of both machines are given, calculated by exact and approximate methods. Comparing these characteristics, we see that neglecting the active resistance of the stator winding in the examples considered leads to inadmissibly large errors. This is explained primarily by the difference in the established values of the operational impedances along the rotor axes at the corresponding slips. The change in the moduli of these impedances is shown in Fig. 2. In the case when \(X_d(js)=X_q(js)\), the approximate method becomes exact. Figure 1 presents the curves \(M_c=f(s)\) for four values of \(r\). From Fig. 1 it is seen that the values of the starting torque decrease with increasing \(r\). In addition, with increasing \(r\) incre—

Fig. 1. Mechanical characteristics of synchronous motors: a) calculation by the approximate method (machine No. 1); b) calculation by the exact method (machine No. 1); c) calculation by the approximate method (machine No. 2); d) calculation by the exact method (machine No. 2)

Fig. 2. Dependences of the moduli of the steady-state values of the operational impedances on slip for various external-circuit resistances: a) for machine No. 1, b) for machine No. 2

the dip region in the range of subsynchronous speed increases, and the peak of the dip shifts into the region of small slips. The discrepancy between the characteristics \(M_c=f(s)\), calculated by the exact and approximate methods, increases with increasing \(r\).

The influence of the parameters of the damper winding on the starting characteristics of a synchronous machine can be traced from Fig. 3. Shown here are the dependences \(M_c=f(s)\), \(I=f(s)\), and \(M_{\text{pl}}=f(s)\) (pulsating torque) for both machines, in the external circuit of which a resistance equal to \(24R_f\) is connected. The parameters of the damper winding along the transverse axis in the machine with a complete damper winding are almost half those of the machine with an incomplete damper winding, and the difference in the steady-state values of the operational impedances \(X_d(js)\) and \(X_q(js)\) has decreased considerably, as is seen from Figs. 2 and 3. This circumstance led to a decrease in the magnitude of the dip in the range of subsynchronous speed on the mechanical characteristic (at \(r=0\)) and to a reduction of the pulsating torque during starting. This circumstance had almost no effect on the magnitude of the stator current (the starting current of the machine with a complete damper winding increased only slightly).

The influence of additional resistance connected into the field-winding circuit on the starting of synchronous motors can be traced from Fig. 4. The greater the resistance of the stator winding, the more noticeable is the influence of the additional external resistance in the region of small slips, and the less it affects starting \([M_c=f(s)]\). It has also been established that the greater the asymmetry of the rotor, the larger the value of the minimum additional resistance that must be connected into the external circuit so that the ma-

Fig. 3. Starting characteristics of machines in whose external circuit a resistance equal to \(24R_f\) is connected:
a) curve 1 (machine No. 2); c) curve 2 (machine No. 1). The parameters of the machines along the longitudinal axis are identical. Solid lines — \(r = 0.1\); dashed lines — \(r = 0\)

Fig. 4. Starting characteristics of synchronous machines as a function of the additional resistance connected into the field-winding circuit

the machine could pass through the region of subsynchronous speed and reach full synchronous speed. Increasing the field resistance reduces the dip in the mechanical and current characteristics and leads to a decrease in the pulsating torque in the region of small slips, which is an important positive factor.

A comparison of the effective values of the stator current also indicates a discrepancy between the curves \(I = f(s)\) calculated by the exact and approximate methods; this discrepancy is especially pronounced in the region of subsynchronous speed (Fig. 5). The magnitude of the dip, which cannot be detected by an approximate calculation when \(r\) is quite appreciable, depends mainly on the value of the active resistance of the stator winding: as \(r\) increases, the dip in the current curve is smoothed out, and the starting current decreases.

From the curves (Figs. 5 and 6) it is seen that the amplitude of the pulsating torque and of the stator current at synchronous speed and during rotation practically does not depend on the magnitude of the active resistance of the stator winding;

Fig. 5. Curves of variation of the stator current for machine No. 1; calculation by the approximate method (dashed lines), calculation by the exact method (solid lines)

Fig. 6. Curves of variation of the amplitude of the pulsating torque for a motor with an incomplete damper winding (machine No. 1) as a function of slip, calculated by the exact method for different \(r\)

The value of \(M_{\mathrm{pl}}\) reaches a significant magnitude depending on the machine parameters and the degree of rotor asymmetry.

For the same synchronous machines, the static starting characteristics were calculated by the combined method [2]. The calculation results are summarized in Table 1 for machine No. 1—with an incomplete damper winding and an active resistance of the stator winding equal to 0.0246. Analysis of the calculation results showed that it is inadvisable to use the combined method for calculating the static starting characteristics of synchronous machines. To make this conclusion convincing, the table gives the values of the average asynchronous torque calculated for the corresponding slips by the exact and approximate methods. Since the characteristics \(I=f(s)\), calculated by the combined method, are analogous to those calculated by the approximate method, our conclusion about the inadvisability of using the combined method applies mainly only to the torque characteristics of synchronous machines.

Table 1

References

1. V. J. Chalmers, A. K. Agarwal. Importance of direct and quadrature-axis damper impedances in starting performance of salient-pole motors, Proc. IEE, vol. 113, No. 4, 1966.

2. E. V. Kononenko, T. V. Chesheva. On methods for calculating the starting characteristics of synchronous salient-pole machines (present collection).