PHYSICS OF OSCILLATIONS IN THE “GENERATOR WITH SERIES—MOTOR WITH SEPARATE EXCITATION” SYSTEM

L. I. GANDZHA

Introduction

Questions concerning the study of oscillations in electromechanical systems have become especially important since the principles of electric-machine automation and the theory of automatic control came into wide use in the theory and practice of electric drive [1].

Among the various electromechanical systems prone to oscillations, a special place is occupied by systems in which oscillations are caused by the nonlinearity of the magnetization characteristics of electrical machines and by the presence in them of series excitation windings, which, under certain conditions, create a “negative” resistance in the system.

These may include “generator–motor” systems in which the motor has separate excitation, while the generator has either series, mixed, or shunt excitation in the presence of a compensating winding or an additional-pole winding. When, in one of the latter two systems, the independent excitation winding of the generator is disconnected, the latter becomes a series-excited generator, and the system as a whole reduces to the first, oscillation-prone system [1;11]. Also included here may be the “generator–motor” system in which, as the generator, an electric-machine amplifier with a transverse field is used.

Fig. 1

With the control winding disconnected (independent excitation), the electric-machine amplifier can be excited by the action of the compensating winding and thus becomes a series-excited generator [2]; the presence in it of a double amplification cascade does not change the essence of the matter. It is obvious that the “generator with series—motor with separate excitation” system (Fig. 1) is, in the sense indicated above, common to all the systems listed; therefore, the study of oscillations arising in it will make it possible to elucidate the nature of oscillations arising in a number of other systems as well. These oscillations are of considerable interest also from the point of view that they may be

used for carrying out automatic reversal in the electric-drive system of mechanisms with reciprocating motion [3; 4], and also from the point of view of the general theory of electrical machines.

History of the Question

The oscillations arising in the system we are considering have been described far from fully in [5; 6; 7; 8; 9; 10]. In [5; 6; 7] only a brief description of the phenomenon is given, without deep mathematical analysis. In [8] an attempt is made at a mathematical investigation of the system, the latter being treated as linear, and the sustained oscillations arising in it as harmonic. The most detailed mathematical analysis of the system is given in the works of M. V. Shuleikin [9; 10]. However, this analysis too is insufficient, since the problem is still treated as linear.

The elementary explanation of the physical nature of the oscillations under consideration, given in the literature cited above, is basically reduced to the following statements. In Fig. 2 there are shown the magnetization characteristic of the generator with series excitation \(e_g=\varphi(I)\) and the straight line \(\Delta U=I.R=\psi(I)\), representing the voltage drop in the power circuit (Fig. 1), which has resistance \(R\).

If the magnetization characteristic at the origin is steeper than the straight line \(\Delta U=I.R\), then, as is known, the point \(A\) of intersection of the curve \(e_g=\varphi(I)\) and the straight line \(\Delta U=I.R\) will determine that emf of the generator \(e_{gA}\) to which the generator will build up when connected to the load. From the beginning of the acceleration of the motor, its speed will increase; its counter-emf \(e_d\) will also increase, since the latter, with constant and nominal magnetic flux of the motor, is determined by the expression:

\[ e_d=c_e\cdot n, \]

where \(n\) is the speed of the motor,
\(c_e\) is the coefficient of proportionality between \(n\) and \(e_d\) at constant and nominal magnetic flux \(\Phi_n\).

The current in the process of acceleration of the motor will decrease from the value \(I_A\) to the value corresponding to the load on its shaft. In this case the equality

\[ e_g=\Delta U+e_d=I.R+e_d, \tag{a} \]

must be satisfied, which is valid only when static regimes are considered, when the magnitude of the counter-emf of self-induction of the power circuit may be neglected. Hence it follows that, in the same axes, \(e_d\) is represented by the ordinate segment lying between the curve \(e_g=\varphi(I)\) and the straight line \(\Delta U=I.R=\psi(I)\), for example by the segment \(BB_1\). If the current \(I_B\) corresponds to the load applied to the motor shaft, then, when the current decreases from the value \(I_A\) and reaches the value \(I_B\) corresponding to point \(B\), the further increase in speed and decrease in current will cease, and the motor will operate in a stable steady regime. This region of stable operation lies between point \(A\) and point \(K\), determined by the tangent to the curve \(e_g=\varphi(I)\), parallel to the straight line \(\Delta U=I.R=\psi(I)\), and relation (a) is valid for it. If, however, the resisting torque applied to the motor shaft is so small that the motor current caused by it \(I_c<I_{cr}\), then, in the process of acceleration, the motor current will decrease, tending toward the value \(I_c\). As before, with decreasing current the motor speed will increase, and with it \(e_d\). Up to point \(K\) the growth of \(e_d\) will satisfy equation (a). From point \(K\), however, the segment representing \(e_d\) will begin to decrease (Fig. 2), which does not correspond to the growth of \(e_d\) and \(n\) as the current decreases. Hence it follows [5; 6] that a further increase in the motor speed is impossible, and thus

as, with a further decrease of the current to the left of point \(K\), \(e_\theta > e_2\), the current changes its direction, the generator is remagnetized up to point \(A_1\), and the motor proves to be in a braking regime. After stopping, the motor will begin to accelerate in the opposite direction, until its current again reaches the value of the critical current corresponding to the critical point \(K_1\). Beyond this point the phenomenon of remagnetization of the generator is repeated, and in the circuit of Fig. 1 undamped oscillations arise, both of electrical quantities \((e_2, e_\theta, I)\),

Fig. 2

and of mechanical quantities \((M, n)\). In [7] the phenomenon of “overturning” of the electromotive forces in the region between the origin of coordinates and the critical point \(K\) is explained by the inertia of the motor armature, which has reached a high speed, as a result of which \(e_\theta\) becomes greater than \(e_2\), the current changes its direction to the opposite one, and an oscillatory regime sets in.

Critique of views on the nature of the oscillations under consideration

Let us summarize the basic propositions arrived at by the authors of the literature cited above:

1. If the load current is less than the critical current \(I_{kp}\) (Fig. 2), then, upon passing beyond the critical point \(K\) toward the origin of coordinates, a further increase in speed—and consequently also in the counter-e.m.f. of the motor—is impossible. In this case, beyond point \(K\), \(e_\theta > e_2\); an “overturning” of the electromotive force of the generator occurs, the current changes direction to the opposite one, and an oscillatory regime sets in [5].

2. Starting from point \(K\), the motor continues to rotate by inertia, as a result of which \(e_\theta\) becomes greater than \(e_2\), and the phenomenon of motor oscillations sets in [7].

3. Since the motor has independent excitation, the dependence \(e_\theta = \varphi_1(I)\) will, at various rotational speeds, represent a line parallel to the axis of abscissae [6].

4. The asymmetric character of the curves \(I = f(t)\) and \(n = f_1(t)\) (Fig. 3) is explained by the influence of the self-induction of the power circuit [6].

5. Stable operation of the circuit of Fig. 1 is impossible [7].

6. The limiting points determining the amplitude of the current and the e.m.f. of the generator during the process of oscillation are the points \(A\) and \(A_1\), which are the excitation thresholds of the generator.

These propositions prove to be not entirely sufficient, and sometimes even incorrect, for satisfactorily explaining the physical picture of the onset of oscillations.

Let us consider the first two propositions in more detail.

If, under a small motor load, its current decreases during acceleration, then, regardless of whether it becomes less than the critical current or not, the motor speed will increase. Therefore, the assertion that beyond the critical point an increase in speed is impossible is incorrect; if the motor current has become less than the critical current and continues to decrease further, the motor speed, and with it also \(e_0\), will increase, although not so intensively, since, when supplied by a generator with series excitation, the latter, as the current decreases, will develop a lower voltage and thereby, to a certain extent, compensate for the increase in motor speed. What has been said is fully confirmed by the oscillogram (Fig. 3), taken at no load and with the motor brushes set on the neutral; it shows a continuous increase in speed, reaching its maximum at a current equal to zero; this statement is also confirmed by analysis of the equation of motion, which, for the case of motor acceleration, has the form:

Fig. 3

\[ M - M_{cm} = \frac{GD^2}{375}\,\frac{dn}{dt}; \]

bearing in mind that, with constant and nominal magnetic flux of the separately excited motor,

\[ M = C'_m \Phi_n I = C_m I, \]

we obtain:

\[ I - \frac{M_{cm}}{C_m} = I - I_{cm} = \frac{GD^2}{375C_m}\,\frac{dn}{dt}. \tag{6} \]

In these equations:

\[ \begin{aligned} M &\text{— the torque developed by the motor on the shaft,}\\ M_{cm} &\text{— the static torque referred to the motor shaft,}\\ GD^2 &\text{— the flywheel moment referred to the motor shaft,}\\ \Phi_n &\text{— the nominal magnetic flux of the motor,}\\ C_m &\text{— the coefficient of proportionality between current and torque}\\ &\quad\text{at nominal and constant magnetic flux,}\\ I_{cm} &\text{— the static current caused by the load } M_{cm} \text{ applied to the motor shaft.} \end{aligned} \]

Since at the maximum speed of the motor its acceleration is equal to zero, on the basis of (6) we obtain that, for

\[ n=n_{\max}, \]

\[ I=I_{st}, \]

whence it follows that at no load, when \(M_{st}=0\) and \(I_{st}=0\), the speed reaches its maximum value at current \(I=0\). If, however, the current \(I_{st}\ne 0\) and is less than the current \(I_{cr}\), then the speed reaches its maximum value at

\[ I=I_{st} \]

in the region between the origin of coordinates and the critical point \(K\). It follows from this that, in the cases considered, the speed inevitably increases if the motor load current assumes values smaller than \(I_{cr}\) (Figs. 2 and 3).

The assertion that beyond the critical point the motor rotates by inertia is also incorrect; although the motor current has become less than the critical value, the motor has not been disconnected from the generator and, consequently, has not been left to free run by inertia; the law governing the change in its speed continues to depend on the law governing the change in the generator current and e.m.f.

Nor is the assertion that beyond the critical point \(e_d>e_g\) quite correct. In fact, it follows from Fig. 2 that in the vicinity of point \(K\), on the contrary, the e.m.f. of the generator is greater than the counter-e.m.f. of the motor, and only near the origin of coordinates can the inequality \(e_d>e_g\) hold.

The third proposition is plainly erroneous. At different motor currents \(e_d\) cannot remain constant, since, in accordance with the mechanical characteristic of the motor, as the current changes its speed also changes, and with it \(e_d\) as well.

The fourth proposition is not quite sufficient. The asymmetry of the curves \(I=f(t)\) and \(n=f_1(t)\) (Fig. 3) should be explained not simply by the influence of the inductance of the power circuit, but by the influence of the inductance with iron; moreover, the shape of the curves is also determined by the fact that during a half-period the motor is in starting and braking modes, for which, as is known, the current and speed curves have different shapes. In addition, the influence of the flywheel moment of the drive and of the magnitude of the load on its shaft is certainly manifested here.

The fifth proposition is not quite correct: unstable operation of the motor occurs at loads smaller than the critical current \(I_{cr}\). At currents greater than \(I_{cr}\), the system operates quite stably.

Finally, the last proposition is incorrect. The points \(A\) or \(A_1\) can be reached by the generator only in the process of its excitation when switched on to a stationary motor. In the process of remagnetizing it in the direction from point \(A\) to point \(A_1\), the sign of \(e_g\) changes to the opposite. In this case the rotating motor is in a plugging mode; its counter-e.m.f. is added to \(e_g\), and the current peak during remagnetization of the generator, determined as

\[ I_{\max}=\frac{e_g+e_d}{R}, \]

will be greater than the current \(I_{A1}\), determined for a stationary motor as

\[ I_{A1}=\frac{e_g}{R} \]

(Fig. 2).

In addition to what has been said, all the propositions listed do not answer sufficiently satisfactorily the question: how and why does remagnetization of the generator occur? This question can be answered by considering the system of Fig. 1 only as a nonlinear one and by analyzing its stability.

Differential Equation of Drive Motion

The electromechanical transient process in the system shown in Fig. 1 may be described by the following equations:

\[ e_z=I\cdot R+L\frac{dI}{dt}+e_a, \tag{1} \]

\[ M=M_{st}+M_j=M_{st}+\frac{GD^2}{375}\frac{dn}{dt}, \tag{2} \]

where

\[ M_j \]

is the dynamic torque,

\[ L \]

is the self-inductance coefficient of the power circuit, assumed hereafter to be constant.

Restricting the discussion to no-load operation of the motor \((M_{st}=0)\) and assuming a constant and nominal magnetic flux of the motor,

\[ M=C_m\cdot I, \]

we obtain from (2):

\[ I=\frac{GD^2}{375\,C_m}\frac{dn}{dt}. \tag{3} \]

Under the same condition we have

\[ e_a=C_e\cdot n. \tag{4} \]

Next, in (1), \(e_z\) is a nonlinear function of the current:

\[ e_z=\varphi(I). \]

Substituting (4) and (5) into (1) and differentiating, we obtain:

\[ \varphi'(I)\frac{dI}{dt} = R\frac{dI}{dt} + L\frac{d^2 I}{dt^2} + C_e\frac{dn}{dt}, \tag{6} \]

and substituting \(\dfrac{dn}{dt}\) from (3) into (6):

\[ \varphi'(I)\frac{dI}{dt} = R\frac{dI}{dt} + L\frac{d^2 I}{dt^2} + \frac{375 C_m\cdot C_e}{GD^2}\cdot I, \]

or

\[ L\frac{d^2 I}{dt^2} + \left[ R-\varphi'(I) \right] \frac{dI}{dt} + \frac{375 C_m\cdot C_e}{GD^2}\cdot I = 0. \tag{7} \]

The nonlinear second-order differential equation obtained describes the transient process of the motor current. In it, \(\varphi'(I)\) is the tangent of the angle \(\alpha\) of inclination between the tangent to the curve \(e_z=\varphi(I)\) and the axis of abscissas at the point of this curve under consideration (Fig. 2). As is seen from Fig. 2, for

\[ I>I_{cr} \]

\[ \alpha<\beta \]

and

\[ \tg\alpha<\tg\beta, \]

while for

\[ I<I_{cr} \]

\[ \alpha>\beta \]

and

\[ \tg\alpha>\tg\beta. \]

It follows from this that in (7) the coefficient of the first derivative proves to be negative for \(I<I_{kr}\) and positive for \(I>I_{kr}\).

Within the limits of small changes in \(I\), system (7) may be regarded as linear; since the signs of the coefficients of a second-order differential equation are stability criteria, we conclude that for \(I>I_{kr}\) the system under consideration is stable, and for \(I<I_{kr}\) it is unstable.

If we set \(L=0\), then (7) degenerates into

\[ \left[ R-\varphi'(I)\right]\frac{dI}{dt}+\frac{375 C_m \cdot C_e}{GD^2}\cdot I=0, \tag{8} \]

which describes the mechanical transient process. It follows from it that the stability or instability of the system is determined only by the form of the nonlinear characteristic \(e_2=\varphi(I)\).

In order subsequently to judge the nature of the transient processes in the system, let us consider the stability conditions for the operation of its individual links: the motor and the generator.

Mechanical characteristic and stability of motor operation

The static mechanical characteristic of the motor can be obtained from expression (a):

\[ e_2=I\cdot R+e_0, \]

or

\[ \varphi(I)=I\cdot R+C_e\cdot n=I\cdot R+C'_e\cdot \Phi_n \cdot n, \]

whence

\[ n=\frac{e_2-I\cdot R}{C_e}=\frac{\varphi(I)-I\cdot R}{C'_e\cdot \Phi_n}. \tag{9} \]

It should be borne in mind that in the oscillatory regime the generator is remagnetized not along the curve \(e_2=\varphi(I)\) shown in Fig. 2, but along the hysteresis loop (Fig. 5); in this case, when the current changes from \(+I_{\max}\) to \(-I_{\max}\), the generator is remagnetized along the curve \(a\, b\, a_1\), and when the current changes from \(-I_{\max}\) to \(+I_{\max}\)—along the curve \(a_1\, b_1\, a\). Therefore the segments enclosed between the curve \(e_2=\varphi(I)\) and the straight line \(\Delta U=IR\), and measuring on the drawing scale the back e.m.f. of the motor, must be measured in the region marked with vertical hatching when the current changes from \(+I_{\max}\) to \(-I_{\max}\), and in the region marked with oblique hatching when the current changes from \(-I_{\max}\) to \(+I_{\max}\). The magnitudes of these segments, expressed in volts, for

Fig. 4

at different currents are the values of the numerator of the right-hand side of (9). Taking this into account, the mechanical characteristic of the motor shown in Fig. 4 can be constructed. From a comparison of Figs. 4 and 5 it follows that, when the current changes from \(+I_{\max}\) to \(-I_{\max}\), the speeds on the mechanical characteristic should be read in the region marked by vertical hatching (Fig. 4), and when the current changes from \(-I_{\max}\) to \(+I_{\max}\), in the region marked by oblique hatching. It also follows from Figs. 4 and 5 that, in contrast to Fig. 2, we obtain not two but four critical points \(K, K_1, K'_1\), and \(K'\), which determine the stability thresholds of the system, since for them in (7) and (8)

\[ [R-\varphi'(I)]=0. \]

Further, it follows from Fig. 4 that the portions of the mechanical characteristic \(aK, aK', a_1K_1\), and \(a_1K'_1\) are descending and characterize stable operation of the drive at constant static torque \(M_{st}\), while the portions \(KK_1\) and \(K'K'_1\) are rising and characterize unstable operation of the drive at constant static torque. Indeed, let \(M_{st}\) remain constant, and let \(M\) and \(n\) be the values of the motor torque and speed in the established equilibrium state preceding the disturbance of equilibrium. Let, further, a disturbance arise in the system and transient deviations of torque \(\Delta M\) and speed \(\Delta n\) take place. Then the new values of torque and speed in the transient process will be equal to

\[ M+\Delta M, \]

\[ n+\Delta n \]

and the equation of motion will take the form:

\[ (M+\Delta M)-M_{st}=\frac{GD^2}{375}\frac{d(n+\Delta n)}{dt}. \tag{10} \]

Since in the steady-state regime preceding the disturbance, \(M=M_{st}\), and the speed \(n\) is a constant quantity, from (10) we obtain:

\[ \Delta M=\frac{GD^2}{375}\frac{d(\Delta n)}{dt}. \tag{11} \]

Since \(M\) is a function of \(n\) (Fig. 4), then, considering small values \(\Delta M\) and \(\Delta n\), one may assume that within these values \(M\) varies as a function of \(n\) according to a linear law. Then

\[ \Delta M=\frac{dM}{dn}\cdot \Delta n, \tag{12} \]

and substituting (12) into (11), we obtain:

\[ \frac{dM}{dn}\cdot \Delta n=-\frac{GD^2}{375}\frac{d(\Delta n)}{dt}, \]

whence

\[ \frac{d(\Delta n)}{\Delta n} = -\frac{375}{GD^2}\cdot \frac{dM}{dn}\cdot dt \]

or

\[ \ln(\Delta n)-\ln C = -\frac{375}{GD^2}\cdot \frac{dM}{dn}\cdot t, \]

where
\(C\) is the constant of integration. Further:

\[ \Delta n=C\cdot e^{-\frac{375}{GD^2}\frac{dM}{dn}\cdot t}, \]

The constant \(C\) is determined from the initial conditions:

\[ \text{for } t=0,\qquad \Delta n=C=\Delta n_{\text{init}}. \]

Then

\[ \Delta n=\Delta n_{\text{init}}e^{-\frac{375}{GD^2}\frac{dM}{dn}\cdot t}, \tag{13} \]

from which it follows that, in order for the deviation \(\Delta n\) of the speed from the equilibrium state to decay with time, the condition

\[ \frac{dM}{dn}<0 \]

is necessary.

As is evident from Fig. 4, this condition is satisfied by the descending portions of the mechanical characteristic, which, in this way, ensure stable motion of the drive in the case under consideration. The same result can be obtained if one considers the case in which \(M_{\text{st}}\) is a function of speed and increases as the speed increases [12].

Stability of generator operation

As is known, the generator is excited up to the voltage corresponding to point \(A\), the intersection of its magnetization characteristic with the straight line of voltage drop in the generator armature circuit and in the load circuit (Fig. 6). Thus, point \(A\) is the equilibrium point of the generator connected to a static load, and for it

\[ e_{gA}=I_A R. \tag{14} \]

If a disturbance of equilibrium has arisen in the system, then the system will be described by the equation:

\[ e_g=iR+L\frac{dI}{dt}. \tag{15} \]

If the current changes by a small amount \(\Delta I\), then the corresponding increments of the generator e.m.f. and of the voltage drop will be \(\Delta e_2\) and \(\Delta(I\cdot R)\). Within the limits of small quantities \(\Delta I\), we may assume that \(\Delta e_2\) varies according to a linear law; then (Fig. 6)

\[ \begin{aligned} \Delta e_2 &= \frac{de_2}{dI}\cdot \Delta I,\\ \Delta(I\cdot R) &= \frac{d(I\cdot R)}{dI}\cdot \Delta I \end{aligned} \tag{16} \]

Fig. 6

It is further evident that

\[ \begin{aligned} e_2 &= e_{2A}+\Delta e_2,\\ I\cdot R &= I_A\cdot R+\Delta(I\cdot R), \end{aligned} \tag{17} \]

and, substituting (16) and (17) into (15), we obtain:

\[ \left[ e_{2A}+\frac{de_2}{dI}\cdot \Delta I \right] - \left[ I_A R+\frac{d(I\cdot R)}{dI}\cdot \Delta I \right] = \]

\[ = L\frac{dI}{dt} = L\frac{d(I_A+\Delta I)}{dt} = L\frac{d(\Delta I)}{dt}, \]

or, taking (14) into account:

\[ \frac{1}{L} \left[ \frac{de_2}{dI} - \frac{d(I\cdot R)}{dI} \right]\cdot \Delta I = \frac{d(\Delta I)}{dt}, \]

whence

\[ \frac{1}{L} \left[ \frac{de_2}{dI} - \frac{d(I\cdot R)}{dI} \right]dt = \frac{d(\Delta I)}{\Delta I}, \]

or

\[ \ln(\Delta I) = \frac{1}{L} \left[ \frac{de_2}{dI} - \frac{d(I\cdot R)}{dI} \right]\cdot t + \ln K, \]

where \(K\) is the constant of integration, determined from the initial conditions: at \(t=0\), \(\Delta I=\Delta I_{\text{init}}=K\).

Finally,

\[ \Delta I = \Delta I_{\text{init}}\cdot e^{ \frac{1}{L} \left[ \frac{de_2}{dI} - \frac{d(IR)}{dI} \right]\cdot t }. \tag{18} \]

Since \(L\) is a positive quantity, it is obvious that the condition for the damping of \(\Delta I\) as \(t\) tends to infinity is that the angle of inclination of the straight line \(I R\) be greater than the angle of inclination of the curve \(e_2=\varphi(I)\).

If, for the equilibrium point of the system under consideration, this condition is satisfied, as in the case of Fig. 6, then the equilibrium is stable, and conversely.

Bearing in mind that, during oscillations, the remagnetization of the generator from point \(a\) to point \(a_1\) takes place along the hysteresis loop (Fig. 5) in the direction indicated by the arrows, we are convinced that the points \(A\), \(A_1\), \(A'\), and \(A'_1\) are points of stable equilibrium, while the points \(B\) and \(B_1\) are points of unstable equilibrium of the generator. The points \(K\), \(K'\), \(K_1\), and \(K'_1\), however, are points of change of stability, since at these points the angles of inclination of the straight line \(I R\) and of the branches of the loop \(e_2=\varphi(I)\) are equal. In accordance with (18), the deviation \(\Delta I\) appearing in the system does not damp with increasing \(t\), but increases on the segments of the loop \(K K_1\) and \(K'_1 K'\).

If we further suppose that the system operates, for example, in the regime corresponding to point \(B\), and that the current changes by \(\Delta I\) toward positive current values, then it will move away from the position of unstable equilibrium in the direction toward point \(a\). Conversely, in order for the system to move away from the same position of unstable equilibrium toward point \(a_1\), the current must acquire an increment \(\Delta I\) toward negative values. Thus, the necessary condition for remagnetization of the generator from point \(a\) to point \(a_1\) is that the current, while decreasing from \(+I_{\max}\), necessarily pass beyond the value \(I_{B_1}\), corresponding to the unstable equilibrium state of the generator at point \(B_1\).

Influence of Inductance on the Character of Oscillations

Let us now return to the investigation of the system under consideration as a whole. Since, in accordance with (1), the current in the power circuit of the system of Fig. 1 is determined by the inductance \(L\) of the circuit, the essential question is: how does the magnitude of the inductance affect the character of the oscillations that arise? Suppose (Fig. 4) that, in the process of oscillations, the current at a given instant of time, while remaining positive, decreases; then the generator is demagnetized along the upper part \(a A K b\) of the hysteresis loop. The sign of the counter-e.m.f. of self-induction will then be such that, according to Lenz’s law, it will oppose the decrease of the current; as the current decreases, \(e_2\) will also decrease, its value at \(I=0\) being determined by the residual magnetism of the machine. Further, as the current decreases, the motor counter-e.m.f. increases with the rise of speed, which, as was indicated above, in no-load operation of the motor reaches its maximum value at \(I=0\).

The relation of the quantities

\[ e_2=\varphi(I),\quad e_L=L\frac{dI}{dt},\quad e_{\theta}=C_e\cdot n\quad \text{and}\quad I R \]

in a particular case may be such that the current passes through its zero value and begins to increase toward negative values. Since in this case \(e_L\) does not change its sign, it will oppose the increase of the current in the region of negative currents, and, because of this, depending on the magnitude of \(L\), the current may either pass or not pass beyond the value

\[ I=I_{B_1}, \]

corresponding to point \(B_1\) of the unstable equilibrium state of the generator. Let us consider both of these cases.

1. The Case of Small Inductance

For a small value of \(L\), the current passes beyond the value \(I = I_{B_1}\), and the generator moves away from the point \(B_1\) of unstable equilibrium, tending toward the region \(K_1 A'_1 a_1\), which characterizes its stable operation. At the same time \(I\) and \(e_2\) continue to increase in the direction of negative values. Since negative current is a braking current for the motor, and \(e_2\) and \(e_0\) are then directed in the same sense, the motor is in the plugging regime and is braked intensively. Its current in the plugging regime reaches the value \(-I_{\max}\) and decreases during braking. As the current increases from zero to \(-I_{\max}\), the generator is remagnetized along the branch \(\delta B_1 K_1 A'_1 a_1\); as the current decreases from \(-I_{\max}\), it begins to demagnetize along the branch \(a_1 A_1 K'_1 \delta_1\) of the hysteresis loop. Starting from the point \(A_1\) (Figs. 4, 5), the motor reverses, thus passing from the plugging regime into the motoring regime. After the current passes through zero and subsequently increases into the region of positive values, the process of remagnetization of the generator and reversal of the motor is repeated, as a result of which periodic oscillations arise, represented by the oscillogram in Fig. 3; the latter was taken at the smallest value of \(L\), corresponding to the self-inductance of the armatures of the machines. In Fig. 7 the hysteresis loop is shown to scale

Fig. 7

of the tested generator, corresponding to the oscillogram of Fig. 3. From consideration of Figs. 4 and 5, the following possible operating regimes of the motor in the system under consideration follow:

a) throughout the entire region \(AK\delta\) of the generator hysteresis loop, the motor operates in the motoring acceleration regime during forward rotation in the case where \(M_{st}=0\); under this condition the speed reaches its maximum value at \(I=0\);

b) if \(M_{st}\ne 0\), then in the region \(AK_2\) of the hysteresis loop the motor operates in the motoring acceleration regime during forward rotation, and at

\[ I = I_{st} = \frac{M_{st}}{C_m} \]

the motor speed reaches the value \(n_{\max}\). In the region

the segment \(ab\) of the hysteresis loop, the motor operates in the motoring mode under braking with forward rotation;

c) at point \(b\) of the hysteresis loop, the motor operates in the mode of ideal no-load operation with forward rotation;

d) in the region \(bd\), regenerative braking of the motor takes place during forward running, since \(e_b\) opposes \(e_g\) and \(e_b > e_g\);

e) at point \(d\) of the hysteresis loop, the motor operates in the mode of dynamic braking during forward running, since the generator is not excited;

f) in the region \(dB_1 k_1 A'_1 a_1 A_1\), the motor is in the plugging mode during forward running. In this case, on the branch \(K' b B_1 A'_1 a_1\) of the mechanical characteristic (Fig. 4) the motor does not operate; indeed, for small \(L\) one may neglect the influence of inductance and assume that the current changes from \(I = I_{kp}\) to \(I = -I_{\max}\) practically instantaneously; at the same time the speed cannot change instantaneously, since a change in speed is associated with a change in the kinetic energy accumulated by the flywheel masses of the drive and equal, at \(n_{\max}\), to

\[ \frac{J \omega^2_{\max}}{2} = \frac{G D^2 (2\pi n_{\max})^2}{4g \cdot 60^2} = \frac{G D^2 \cdot 2\pi \cdot 2\pi n^2_{\max}}{4g \cdot 60 \cdot 60} = \frac{G D^2}{3583}\, n^2_{\max}. \]

Therefore the motor reaches point \(a_1\) in the plugging mode practically at unchanged speed;

g) in the region \(A_1 K'_1 b_1\), the motor mode is the same as in item a, but with reverse rotation of the motor;

h) for \(M_{st} \ne 0\), in the region \(A_1 K'_1 \Gamma_1\), the motor mode is analogous to that described in item b, but with reverse rotation of the motor;

i) at point \(b_1\), the motor mode is the same as in item c, but with reverse rotation;

j) in the region \(b_1 d_1\), it is the same as in item d, with reverse rotation of the motor;

k) at point \(d_1\), it is the same as in item e, with reverse rotation of the motor;

l) in the region \(d_1 B K' A' a A\), it is the same as in item f, with reverse rotation of the motor; analogously to what was said in item f, the motor does not operate on the branch \(K'_1 b_1 B K' A' a\) of the mechanical characteristic (Fig. 4), reaching the state corresponding to point \(a\) practically at unchanged speed.

The indicated boundaries of the various modes are approximate, since they were considered with reference to the static mechanical characteristic of Fig. 4, without taking into account the electromagnetic and electromechanical inertia of the system.

2. The case of large inductance

For a large value of \(L\), it may turn out that the current does not reach the value \(I_{B1}\) (Fig. 5). In this case remagnetization of the generator up to the state determined by point \(a_1\) will be impossible, since the generator tends to move away from the unstable equilibrium state determined by point \(B_1\) into the region \(K A a\) of its stable operation.

Thus, the operating mode of the generator will be determined not by the former hysteresis loop (Fig. 5), but by a new one, shown in Fig. 9.

What has been said is confirmed by the oscillogram of Fig. 8, taken with a large additional inductance connected into the power circuit of the circuit of Fig. 1, and with all other conditions the same as when obtaining the oscillogram of Fig. 3. The oscillogram of Fig. 8 corresponds to the hysteresis loop obtained for the generator tested and shown in Fig. 9, on which, for comparison, the corresponding portions of the hysteresis loop shown in Fig. 7 and obtained for the case of small inductiv-

...inductance. As can be seen in Figs. 8 and 9, in the course of the oscillations the absolute values of the maximum currents in the positive and negative regions turn out to be unequal. The same is also observed for small inductance (Fig. 3), but there the difference between the absolute values of the maximum currents is considerably smaller than in the case of large inductance. Further, from

Fig. 8

Fig. 8 it follows that, for the given large value of \(L\), the phenomenon of reversal of the motor does not occur. When the current decreases from \(+I_{\max}\) to zero, the speed of the motor increases to \(n=n_{\max}\) at \(I=0\) and no load of the motor, which corresponds to what was said earlier. In this case the demagnetization of the generator proceeds along the branch \(a\delta\) of the hysteresis loop, and the motor in this

Fig. 9

case is in the motoring acceleration mode with forward rotation. At point \(\delta\) there is ideal no-load operation of the motor with forward rotation; on the segment \(\delta 0\) the motor is in the mode of regenerative braking with forward rotation, since \(e_2\) is opposed to \(e_\delta\), and at point \(0\) it is in the mode of dynamic braking with forward rotation. Further, on the segment \(0a_1\) there is a plugging mode with forward rotation, since \(e_2\) coincides with \(e\delta\); because, with further braking, the speed of the motor decreases, the motor current, having reached the value \(-I_{\max}\), also decreases, and the generator begins to demagnetize along the segment \(aA_1\) of the hysteresis loop. Point \(A_1\) is a point of stable equilibrium of the generator; however, it will be passed through, since at \(I=-I_{A_1}\) the motor in the braking mode continues to reduce its speed, and with it the braking current will also decrease. This corresponds to the segment \(A_1\delta_1\) of the loop; near point \(\delta_1\) on the segment \(A_1\delta_1\) the generator enters the unstable part of the segment and tends to move away from point \(A_1\) toward point \(a\). At point \(\delta_1\), as well as on the segment \(\delta_1 0_1\), there is an interesting case when \(I=0\) (point \(\delta_1\)) or assumes positive values (segment \(\delta 0\)) under

with the presence in the closed circuit (Fig. 1) of currents \(e_z\), \(e_\omega\), and \(e_L\) directed concordantly toward negative values. This phenomenon is probably due to the circumstance that the change of current in the process of oscillations was considered in this work on the “static” hysteresis loop of the generator, whereas, strictly speaking, the investigation should be carried out on the “dynamic” hysteresis loop. Returning to Figs. 8 and 9, we see that the motor counter-connection regime corresponding to the branch \(\partial a_1 A_1 b_1\) of the hysteresis loop is characterized by the fact that, owing to the dependence of \(e_z\) on \(I\) and to the influence of the electromagnetic and electromechanical

Fig. 10

inertias of the system, it is not accompanied by reversal of the motor, but passes into the ideal no-load condition corresponding to point \(b_1\); the speed in this case proves to be less than \(n_{\text{ideal}}\) of the ideal no-load condition corresponding to point \(b\) of the loop. For the segment \(b_1\partial_1\) the motor regime proves to be motor braking with forward rotation; in this case, for the generator, the current proves to be opposite with respect to \(e_z\). At point \(\partial_1\) there is motor braking with \(e_z = 0\). This same regime continues until the motor comes to a complete stop, but with \(e_z\) increasing into the region of positive values (segment \(\partial_1 e\)). Finally, at point \(e\) we have an electromagnetic-brake regime, and on the segment \(eab\)—a motor regime of motor acceleration with increasing (segment \(ea\)) and decreasing (segment \(ab\)) \(e_z\).

Fig. 11

When a somewhat smaller inductance was inserted into the power circuit of the scheme of Fig. 1, with all other conditions unchanged (including \(R\)), the oscillogram shown in Fig. 10 was obtained, characterized by the presence of two extreme current values during a half-period.

The oscillogram of Fig. 11 was obtained when an even larger inductance was inserted into the power circuit of the scheme than in the experiment corresponding to Fig. 8. As is seen from Fig. 11, increasing the inductance causes a break in the speed curve, which retains the same sign; the current, in the process of oscillations, likewise does not leave the region of positive values. Conversely, with an inductance smaller than in the experiment corresponding

Fig. 8, but greater than in the experiment corresponding to Fig. 10, a considerable discontinuity is observed in the speed curve when the motor is reversed (Fig. 12).

All the oscillograms presented above were obtained on machines with the following characteristics: generator 8 hp, 220 V, 32.5 A, 1300 rpm; excitation—series; motor 3.7 hp, 110 V, 31 A, 1180 rpm; excitation—separate.

Fig. 12

When testing the circuit with the same generator, but with a motor of 3.25 hp, 220 V, 13.5 A, 1375 rpm, and with a small inductance connected into the power circuit, the oscillogram shown in Fig. 13 was obtained. Unlike all those presented above, in it the speed curve is continuous, while the current curve undergoes a discontinuity when passing through the zero value.

Fig. 13

Conclusions

1. The investigation carried out satisfactorily explains the mechanism of oscillations in the system “generator with series excitation—motor with separate excitation.”

2. It has been shown that the physics of these oscillations and their various forms can be correctly explained only if the system is regarded as nonlinear and is investigated by the methods of stability theory.

3. It has been shown that oscillations arise as a result of the system’s transition from a region of stable operation to a region of unstable operation, and conversely.

4. It has been shown that the inclusion in the system of an additional and variable inductance with iron changes, to a considerable degree, the form of the nonlinear oscillations.

LITERATURE

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3. Govorkov V. A. The works of Acad. M. V. Shuleikin on electrical machines. Elektrichestvo, No. 5, 1952.

1. L. I. Gandzha. Self-reversing electric-drive systems. Izvestiya TPI, vol. 72, 1952.

2. E. Arnold. Direct-Current Dynamo Machine, 1, 1909.

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5. N. N. Rukavishnikov. Collection of Problems on Direct- and Alternating-Current Electrical Machines. Kubuch, 1933.

6. M. V. Shuleikin. On natural oscillations in the circuit of a series motor and generator. Izvestiya of the St. Petersburg Polytechnic Institute, XI, issue 2, 1909.

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8. V. K. Popov. Application of Electric Motors in Industry, part II, issue 1. Leningrad, 1935.

9. V. K. Popov. Fundamentals of Electric Drive. Gosenergoizdat, 1945.