On Self-Oscillations in an Electromechanical System with an Electric Machine Amplifier

L. I. Gandzha and V. A. Sevastyanov

(Presented by the scientific seminar of the Electromechanical Faculty)

In [1; 2], self-oscillations were examined in detail in an electromechanical system (Fig. 1) in which the generator has series excitation and the motor has independent excitation. It was shown there that, if the static torque applied to the motor shaft causes a current \(I_{st}<I_{cr}\), then self-oscillations arise in the system; here \(I_{cr}\) corresponds to point \(K\) (Fig. 2a) of the generator magnetization curve, at which the tangent to it is parallel to the straight line representing the voltage drop in the power circuit, which has resistance \(R\). It was also shown that, in the region \(KA\) (and \(K_1 A_1\)) of the magnetization curve, the system possesses static stability, whereas in the region \(KK_1'\) (and \(K'K_1\)) it is unstable. Since the segments between the magnetization curve and the straight line \(IR\) represent the counter-emf of the motor \(e_d\), then, proceeding from (Fig. 2a), the static mechanical characteristic of the motor in the self-oscillation regime can readily be constructed (Fig. 2b). It was further shown that, if the inductance of the power circuit of the system (Fig. 1) is neglected, then these self-oscillations acquire the character of discontinuous, or relaxation, oscillations. In this case the motor begins to accelerate, for example, from point \(A\) to point \(K\) (Fig. 2b), reaching its maximum speed at point \(K\), which is the point where the stability of the system changes. From the state corresponding to point \(K\), the system “jumps” to the state corresponding to point \(a_1\), at which the speed remains equal to \(n_{\max}\), and the current equal to \(I_{a1}\); starting from point \(a_1\), the motor brakes to point \(A_1\), and then accelerates

Fig. 1.

in the opposite direction to the point \(K_1\), from which it “jumps” to the state characterized by the point \(a\). From the point \(a\) to the point \(A\) braking takes place, and from \(A\) to \(K\)—again acceleration, but in the forward direction, etc.

Fig. 2.

These oscillations are characterized by Fig. 3; they are symmetric with respect to the time axis, so that the absolute values of the currents after the “jump” are equal:

\[ |I_a| = |I_{a1}|. \]

This symmetry is due, first, to the assumption that the inductance of the power circuit of the system is equal to zero, and, second, to the presence in the generator of only one series winding, owing to which the hysteresis loop of the generator (Fig. 2a) proves to be symmetric with respect to the origin. In [1] the possibility is shown of obtaining such oscillations in the system of Fig. 1, which are not accompanied by reversal of the motor, but represent a process of alternation of periods of motor rotation with periods of motor rest; there it was found that such oscillations can be obtained by increasing the inductance of the circuit by additionally introducing into it a coil with iron. In the article “The Influence

Fig. 3.

inductance on the form of self-oscillations in an “electromechanical system,” published in the present volume, this question is examined in greater detail. The purpose of the present work was an experimental investigation of oscillations in a \(G\)—\(D\) system in which the function of the generator is performed by an electric-machine amplifier with a transverse field. This system is shown in Fig. 4.

Fig. 4.

As follows from it, the EMA has two excitation windings: one of them—the control winding \(OU\)—is supplied from an independent power source, while the other—the positive current-feedback winding \(OOS\)—is supplied by the voltage drop across a potentiometer connected in the main-current circuit of the system. In this system, self-oscillations were studied both with the control winding \(OU\) disconnected and connected, and with it excited to various degrees.

Fig. 5.

To explain the influence of the ampere-turns \(aw_{OU}\) of the control winding on the form of the self-oscillations, let us consider an analogous system with a generator of compound excitation, shown in Fig. 5, from which the system of Fig. 4 differs only by a large value of the amplification coefficient. In Fig. 5, \(aw_n\) denotes the ampere-turns of the independent excitation winding, and \(aw_p\) those of the series winding, connected in accordance with the independent one. If the independent excitation winding is disconnected in this system, it reduces to the system with a series-excited generator shown in Fig. 1, and the mechanism of self-oscillations in it can be explained from the standpoint set forth above (Figs. 2a, b, and c and [1, 2]). If, as before, the inductance of the power circuit is neglected, then the self-oscillations in the system of Fig. 5, with the independent excitation winding disconnected, will be “discontinuous” and symmetrical, as shown in Fig. 3. If the independent winding is now connected to a power source, then, owing to the presence of constant \(aw_n \ne 0\), the magnetic system of the generator (Fig. 5) will be preliminarily saturated, and its magnetic flux at \(I = 0\) will be greater than the magnetic flux due to residual magnetism. This will lead to a displacement of the axes \(O\) and \(e_2\) (Fig. 2a) to the right, into the position shown by the dashed line. If the same notation of the axes is retained, that is, if the dependence \(e_2\) is plotted only as a function of the current \(I\) of the power circuit, then the straight line \(I \cdot R = f(I)\) must also be shifted parallel to the right, into the position shown by the dashed line, since now at \(I = 0\) also \(I \cdot R = 0\), while \(e_2 = e_{21}\) and is determined by \(aw_n\) alone. As a result, the hysteresis loop of the ge-

Fig. 6.

Fig. 7.

Fig. 8.

1 second

generator and the static mechanical characteristic of the motor will be represented as shown in Figs. 6a and 6b.

It is clear from it that both the absolute values of the currents \(+I_a\) and \(-I_{a1}\) after the “jumps,” and of the currents \(+I_{\mathrm{cr}}\) and \(-I_{\mathrm{cr}}\), as well as the absolute values of the maximum speeds \(+n_{\max}\) and \(-n_{\max}\) in adjacent half-periods, will be unequal; and the “discontinuous” oscillations in this case are represented by Fig. 7.

Fig. 9.

Fig. 10.

The times of adjacent half-periods will obviously also be unequal, since attaining a greater maximum speed will require a greater time. Similar oscillations will also be obtained in the system of Fig. 4. The presence of inductance in the power circuit rules out the possibility of “discontinuous” oscillations in the system, and in reality the oscillations will differ from those shown in Figs. 3 and 7. Below are given oscillograms of the current \(I\), speed \(n\), and voltage \(U\) in the system of Fig. 4. The oscillogram of Fig. 8 corres-

Fig. 11.

Visible labels and annotations in the figure: \(n\), \(u\), \(i\); \(1620\ \mathrm{rpm}\), \(300\ \mathrm{V}\), \(15.6\ \mathrm{A}\), \(4.32\ \mathrm{A}\), \(54\ \mathrm{V}\), \(20\ \mathrm{rpm}\); \(t\), sec; \(1\) second.

Fig. 13.

Visible labels and annotations in the figure: \(n\), \(u\), \(i\); \(200\ \mathrm{V}\), \(1520\ \mathrm{rpm}\), \(3.7\ \mathrm{A}\); \(t\), sec; \(1\) second.

Fig. 12.

Visible labels and annotations in the figure: \(e_2\); \(i=1\).

corresponds to the experiment with the control winding disconnected, when the system of Fig. 4 is reduced to the system of Fig. 1. In the “discontinuous” interpretation they are qualitatively characterized by Fig. 3. The oscillograms of Figs. 9 and 10 were taken with the control winding excited to such an extent that reversal of the motor is still observed. At the same time, the oscillogram of Fig. 10 corresponds to a larger \(aw_{oy}\), which brings the axis \(e_2\) still closer to the critical point \(K\) (Fig. 6). However, the points of unstable equilibrium \(B_1\) and of change of stability \(K_1'\) have not yet disappeared; this makes possible remagnetization of the EMA toward negative values of \(I\) and \(e_2\), up to the point \(a_1\), and the appearance of a negative motor speed, although a small one. These oscillograms, in the “discontinuous interpretation,” are described by Figs. 6 and 7. The oscillogram of Fig. 11 characterizes the establishment of self-oscillations after the motor is connected to the EMA. It shows that if the first half-period after connection of the motor corresponds to positive values of \(I\) and \(e_2\), then in the odd half-periods the maximum values of the current, voltage, and motor speed are greater than the maximum values of the same quantities in the even half-periods. If \(aw_{oy}\) is increased to such an extent that the axis \(e_2\) approaches the point \(K\) still more closely, and the branch \(Ka_1\) of the hysteresis loop does not intersect the straight line \(IR=f(I)\), then the points \(B_1\), \(K_1'\), and \(A_1'\) disappear, and the system will not be able to pass beyond the point \(B_1\) into the region of stable operation at negative currents. The only region of stable operation is the branch \(KA\) of the hysteresis loop, toward which the system will tend. Therefore, in this case reversal of the motor is impossible; however, oscillations of the speed will occur, since the point \(K\) is still located to the right of the axis \(e_2\) (Fig. 12), and since the system still retains a segment of the magnetization curve between the point \(K\) and the axis \(e_2\), characterizing unstable operation of the system.

Oscillations of this kind are represented by the oscillogram of Fig. 13. Finally, by a further increase of \(aw_{oy}\) one can shift the point \(K\) to the left of the axis \(e_2\), and then the oscillations disappear, since the system of Fig. 4 during operation will be characterized only by the stable region corresponding to the portion of the magnetization characteristic located to the right of the point \(K\).

The foregoing shows that, by changing the ampere-turns \(aw_{oy}\) or \(aw_{н}\) in the system of Fig. 5, one can substantially change the form of the self-oscillations.

In carrying out the present work, experiments were performed with the following machines:

EMA type-12A; \(0.5\) kW; \(230\) V; \(2.2\) A; motor: \(0.6\) kW; \(170\) V; \(3.5\) A; \(1500\) rpm.

LITERATURE

1. Gandzha L. I. Physics of oscillations in the system “Generator with series excitation—motor with independent excitation.” Proceedings of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov, vol. 76, 1953.

2. Gandzha L. I. and Potekhin Yu. I. On transient processes and oscillations in the system “Generator with series excitation—motor with independent excitation.” Proceedings of the Tomsk Order of the Red Banner of Labor Polytechnic Institute named after S. M. Kirov, vol. 76, 1953.