On Transient Processes and Oscillations in the System “Series-Wound Generator—Separately Excited Motor”

L. I. GANDZHA, Yu. I. POTEKHIN

As was shown in [1], in the system represented in Fig. 1, possessing resistance \(R\) and inductance \(L\) of the power circuit, the following stationary or established regimes are possible:

I. The motor-rest regime, when it is disconnected from the generator and is not rotating.

II. The braking regime, when the motor is connected to the generator, but by a large static torque \(M_{st}\) is brought to a complete stop.

Fig. 1

III. The regime of motor rotation at constant speed, possible in the case where the motor current, determined by the load on its shaft, is greater than the current \(I_{kr}\), which is critical for the generator (Fig. 2), and corresponds to point \(K\) of its magnetization curve, separating the region of stable (from point \(K\) toward larger currents) and unstable (from point \(K\) toward smaller currents) operation.

The same regime is also possible in the case where the motor load current, determined as

\[ I_{st}=\frac{M_{st}}{c_m}, \tag{1} \]

is less than \(I_{kr}\), but the straight line \(\Delta U=I\cdot R=\psi(I)\) passes at an angle \(\beta>\alpha\) (Fig. 2), which, as shown in [1], is the condition for stable operation of the system in Fig. 1 (here \(c_m\) is the coefficient of proportionality between torque and current at constant and rated flux \(\Phi_n\) of the motor).

Fig. 2

IV. The regime of undamped oscillations of both electrical (generator e.m.f. \(e_2\), counter-e.m.f. \(e_d\), and motor current \(I\)) and mechanical (torque \(M\) and speed \(n\) of the motor) quantities. This regime occurs in that

case, if \(\beta<\alpha\), which determines the presence of the critical point \(K\), and if \(I_{cm}<I_{kp}\).

In [1] it was shown that these oscillations are nonlinear and that, for a satisfactory interpretation of these oscillations, the system of Fig. 1 must be considered as nonlinear. The other operating conditions of the motor, which transfer it from one stationary regime to another, are nonstationary or transient. Since the transition of the motor from one stationary regime to another is accompanied by a change in the energy accumulated in the various elements of the circuit (the inductance \(L\), the kinetic energy determined by the motor speed \(n\) and its flywheel moment \(GD^2\)), and since this energy cannot change instantaneously, these transient processes proceed in time and are accompanied by a change in the motor speed. The latter may change due to the following factors:

a) a change in the load on the motor shaft; b) a change in the parameters of the system \((R, L)\); c) a change in the magnetic flux \(\Phi\) of the motor; d) a change in the voltage at the motor brushes, which can be achieved, for example, by connecting a resistance in parallel with the generator field winding or in parallel with the motor armature.

Without discussing here the transient processes associated with regulation of the motor speed and caused by the conditions listed in items b, c, and d, let us establish the possible transient processes associated with a change in the load on the motor shaft (item a). They are possible in the system of Fig. 1 in the following variants:

1. The transient process of the motor from the instant it is connected to the generator (point 1 on the oscillogram of Fig. 3) to the instant of starting from rest (point 2 in Fig. 3) is characterized by the fact that, with the motor at rest, the current increases from zero to the starting current \(I_{mp}\), at which rotation of the motor begins. This is a transient process in a static system, reducible to an inductance with iron connected to an alternating voltage, since the latter, in accordance with the magnetization curve (Fig. 2), increases as the current increases; it precedes the two other transient processes indicated in items 2 and 3.

2. The transient process of the motor from the stationary rest regime (item I) to the stationary regime of rotation at constant speed (item III), which occurs in the case when the static torque \(M_{cm}\) applied to the motor shaft determines, in the motor armature, a load current:

\[ I_{cm}=\frac{M_{cm}}{c_m}>I_{kp} \quad \text{for } \alpha>\beta. \]

It is also possible in the case when

\[ I_{cm}=\frac{M_{cm}}{c_m}<I_{kp} \quad \text{for } \alpha<\beta. \]

This transient process begins at the instant the motor starts from rest after its connection, i.e., after the end of the process indicated in item I; in this case the system of Fig. 1 must already be considered as dynamic.

1. The transient process from the state of rest of the motor (item I) to the stationary regime of undamped oscillations (item IV) is possible for \(\alpha>\beta\) and if \(I_{cm}<I_{kp}\). It also begins at the instant the motor starts from rest after being switched on and follows after the end of the transient process indicated in item I.

2. The transient process from a stationary regime satisfying the conditions of item III to another one satisfying the same conditions. According to the character of the change in load, it may be subdivided into:

a) the case of an increase in load from \(I_{cm1}>I_{kp}\) to \(I_{cm2}>I_{cm1}\) for \(\alpha>\beta\), or from \(I_{cm1}<I_{kp}\) to \(I_{cm2}>I_{cm1}\) and \(I_{cm2}<I_{kp}\) for \(\alpha<\beta\).

In the particular case, \(I_{cm2}\) may be equal to the current \(I_m\) in the braking mode; then the steady-state mode following the transient process will satisfy the conditions of item II, and not item III;

b) the case of a decrease in load from \(I_{cm2}>I_{kp}\) to \(I_{cm1}>I_{kp}\) \((I_{cm2}>I_{cm1})\) for \(\alpha>\beta\), or from \(I_{cm2}<I_{kp}\) to \(I_{cm1}<I_{cm2}\) for \(\alpha<\beta\). In the particular case \(I_{cm2}\) may be equal to \(I_m\); then the steady-state mode preceding the transient process satisfies the conditions of item II, and not item III.

1. The transient process when the system passes from the steady-state mode of item IV (or item III) to the steady-state mode of item III (or item IV).

It may be subdivided into:

a) the case of an increase in load from \(I_{cm1}<I_{kp}\) to \(I_{cm2}>I_{kp}\) for \(\alpha>\beta\). In this case there is a transition from the steady-state mode of item IV to the same steady-state mode of item III. In the particular case \(I_{cm2}\) may be equal to \(I_m\), and then a transition is made from the conditions of item IV to the conditions of item II;

b) the case of a decrease in load from \(I_{cm2}>I_{kp}\) to \(I_{cm1}<I_{kp}\) for \(\alpha>\beta\), when the system passes from the conditions of item III to the conditions of item IV. If, in the particular case, \(I_{cm2}=I_m\), then a transition is made from the conditions of item II to the conditions of item IV.

1. The transient process from a steady-state mode satisfying the conditions of item IV to a steady-state mode satisfying the same conditions; it likewise may be subdivided into

a) the case of an increase in load from \(I_{cm1}<I_{kp}\) to \(I_{cm2}<I_{kp}\) \((I_{cm2}>I_{cm1})\) for \(\alpha>\beta\), and

b) the case of a decrease in load from \(I_{cm2}<I_{kp}\) to \(I_{cm1}<I_{kp}\) \((I_{cm2}>I_{cm1})\) for \(\alpha>\beta\).

From the standpoint of studying nonlinear oscillations and the character of their establishment, the greatest importance [3] belongs to the steady-state mode of undamped oscillations (item IV) and the transient processes indicated in items 3 and 5, which bring the system to undamped oscillations (item IV) after switching on the motor (item I) or from the steady-state mode of rotation at constant speed (item III); some qualitative features of the noted modes are considered below.

The mode of undamped oscillations, from the physical standpoint, is considered in detail in [1]; it is characterized by the fact that, while being steady-state, it consists of a series of periodically repeating transient processes continuously passing one into another. Indeed, as follows from [1] and Fig. 3, undamped speed oscillations are periodically repeating processes of motor reversal; the reversal of the motor, in turn, may be broken down into a starting process and a braking process; in the latter case, as indicated in [1], the motor passes through all stages of braking (recuperative, dynamic, plugging, etc.). Thus, here it is appropriate to speak of transient processes in the mode of steady-state (established) undamped nonlinear oscillations.

Let us further recall that the static mechanical characteristic of the motor (Fig. 4), \(n=f(I)\), in the system of Fig. 1 and in the mode of undamped oscillations accompanied by reversal of the motor, may be obtained as:

\[ n=\frac{e_2-I.R}{c_e}, \tag{2} \]

from the relation:

\[ e_2=I\cdot R+e_0=I.R+c_e\cdot n, \tag{3} \]

if the latter is applied to the part \(aAKB_1A'a_1\) of the generator hysteresis loop when the current changes during the oscillation process from positive values to negative ones, and to the part \(a_1A_1K_1BA'a\) of the hysteresis loop when the current changes from negative values to positive ones (Fig. 5).

Oscillogram No. 2

Visible labels: \(U=f_3(t)\); \(I=f_2(t)\); \(n=f_1(t)\); \(t=0,\ I_{\mathrm{initial}}=0\); \(I_{\mathrm{tr}}\); \(1\ \mathrm{sec}\); \(I_{\max}\); \(t\); \(M_1=0.65\,\alpha_{\mathrm{nom}}\), \(M_n=14.3\,\mathrm{ohm}/\mathrm{min}\), \(M_1=2.7\,\alpha_{\mathrm{nom}}\).

Fig. 3

Visible labels: \(n\); \(I\); \(a_1\); \(q\); \(A\); \(A_1'\); \(B_0\); \(B\); \(K\); \(K_1\); \(+n_{\max}\); \(-n_{\max}\); \(n_1\); \(n_2\); \(-I_d\); \(I_3\); \(-I_{\mathrm{cr}}\); \(+I_2\); \(+I_{\mathrm{cr}}\); \(I_1\); \(+I_d\); \(a\).

Fig. 4

Visible labels: \(e_g\); \(I\); \(-I_a\); \(-I_1\); \(+I_{\mathrm{cr}}\); \(I\); \(I_1\); \(I_a\); \(a_1\); \(A_1'\); \(A_1\); \(B_0\); \(B\); \(K_e\); \(K\); \(A\); \(N\); \(a\); \(e_{g1}\); \(e_{g2}\); \(e_g\); \(e_{g3}\); \(I_R\).

Fig. 5

The dynamic system under consideration, in transient processes, is described by the equations:

\[ e_z=\varphi(I)=I.R+L\frac{dI}{dt}+e_0, \tag{4} \]

\[ M=c_m I-M_{cm}+\frac{GD^2}{375}\cdot\frac{dn}{dt}. \tag{5} \]

Restricting ourselves to consideration of the no-load case \((M_{cm}=0)\), from (5) we obtain:

\[ I=\frac{GD^2}{375\,c_m}\cdot\frac{dn}{dt}; \tag{6} \]

whence

\[ n=\frac{375\,c_m}{GD^2}\int I\,dt. \tag{7} \]

Bearing in mind that

\[ e_0=c_e\cdot n, \tag{8} \]

substituting (7) and (8) into (4) and differentiating, we finally obtain:

\[ L\frac{d^2I}{dt^2}+\left[R-\varphi'(I)\right]\frac{dI}{dt} +\frac{375c_m c_e}{GD^2}\cdot I=0. \tag{9} \]

Substituting (6) into (4), we obtain:

\[ L\frac{d^2n}{dt^2}+R\frac{dn}{dt} +\frac{375c_e c_m}{GD^2}\cdot n =\frac{375c_m}{GD^2}\,\varphi(I). \tag{10} \]

Let us assume that the inductance of the system is so small that it may be neglected; then, for \(L=0\), from (9) and (10) we shall have first-order equations of the degenerate system describing the transient process of the motor current and speed:

\[ \left[R-\varphi'(I)\right]\frac{dI}{dt} +\frac{375\,c_m c_e}{GD^2}\cdot I=0, \tag{11} \]

\[ \frac{GD^2R}{375\,c_m}\cdot\frac{dn}{dt} +c_e n=\varphi(I)=I.R+c_e\cdot n. \tag{12} \]

Next, from (12) we have:

\[ \frac{dn}{dt}=\frac{375c_m}{GD^2R}\left[\varphi(I)-c_e n\right], \tag{13} \]

or, taking (6) into account:

\[ \left. \begin{aligned} I&=\frac{GD^2}{375\,c_m}\frac{dn}{dt} =\frac{1}{R}\left[\varphi(I)-c_e n\right],\\ I.R&=\left[\varphi(I)-c_e n\right], \end{aligned} \right\} \tag{14} \]

and also

\[ \frac{d^2n}{dt^2} =\frac{375^2c_m^2c_e}{(GD^2)^2}\cdot \frac{I}{\left[\varphi'(I)-R\right]}. \tag{15} \]

Finally, from (11) we obtain:

\[ \frac{dI}{dt} =\frac{375\,c_e c_m}{GD^2}\cdot \frac{I}{\left[\varphi'(I)-R\right]}. \tag{16} \]

Suppose that the motor, while in braking mode (point \(A\) in Figs. 4 and 5), begins to move without deviating from the static mechanical characteristic; such motion can be effected if the motor is loaded with a large \(M_{st}\) and brought to a stop (point \(A\), Figs. 4 and 5), and then the magnitude \(M_{st}\) is gradually and slowly reduced; then the motor will accelerate from point \(A\) to point \(K\) with such small accelerations that its actual speed in the transient regime will be practically equal to the speed read at the same current from the mechanical characteristic.

Bearing in mind that, when the current changes from positive values to negative ones, the system can move only along the branches \(aAKB_1A_1'a_1\) of the hysteresis loop (Fig. 4) and of the mechanical characteristic (Fig. 5), whereas when the current changes from negative values to positive ones it can move only along the branches \(a_1A_1KBA'a\), we see that, on the basis of (12), the speed \(n\) is a single-valued function of the current. Furthermore, when the system moves from point \(A\) to point \(K\), the speed increases, as follows from Figs. 4 and 5, and also from (12) and (13), since \(\dfrac{dn}{dt}\) in the interval from \(I_A\) to \(I_{kr}\) remains positive. Since in the same interval \(\dfrac{d^2 n}{dt^2}\), on the basis of (15), is a negative quantity, the integral curve \(n=f_1(t)\) is turned with its convexity toward the side opposite the time axis; \(\dfrac{dn}{dt}\) on this basis, and in accordance with (13), decreases, and therefore, according to (14), the current also decreases. Moreover, from (13) and (14) it follows that \(\dfrac{dn}{dt}\), and consequently also the current \(I\), are multivalued functions of the speed: indeed, to the same, for example, values \(n_1\) and \(e_{d1}\) (Figs. 4 and 5) there correspond different values \(I_1\), \(I_2\), and \(I_3\) for the current; from (15), however, it follows that \(\dfrac{d^2 n}{dt^2}\) in this case is a single-valued function of the current.

The fact that for \(n=f_1(t)\) its first derivative \(\dfrac{dn}{dt}\), proportional to the current, is determined as a function of the current not single-valuedly, whereas \(\dfrac{d^2 n}{dt^2}\) is single-valued, means that the system under consideration admits continuous periodic solutions for \(n=f_1(t)\) and only “discontinuous” periodic solutions for its derivative \(\dfrac{dn}{dt}\), and consequently also for the current \(I=f_2(t)\) [3].

When the system reaches the speed \(n_{\max}\), corresponding to point \(K\) (Figs. 4 and 5), for which

\[ \varphi'(I)=R \]

the derivative \(\dfrac{dI}{dt}\), on the basis of (16), becomes infinite; this indicates that at point \(K\) there is a violation of the holomorphism of the function \(\dfrac{dI}{dt}=F(I)\); the latter becomes nonanalytic and ceases to satisfy the theorem on the existence and uniqueness of the solution for \(I=f_2(t)\) [3, 4]. Since at point \(K\) the equations given above do not describe the behavior of the system, an additional condition is necessary that would justify the presence of current oscillations in the system (these os—

…must be due to the fact that the occurrence in the system of continuous oscillations of \(n\) entails oscillations of \(\dfrac{dn}{dt}\), and consequently of the current, since

\[ I=\frac{GD^2}{375c_{\mathrm{m}}}\,\frac{dn}{dt}\,). \]

This condition reduces to the fact that the kinetic energy accumulated in the system at the maximum speed \(n_{\max}\), corresponding to point \(K\) and equal to

\[ \frac{I\omega_{\max}^{2}}{2}=\frac{GD^{2}\cdot n_{\max}^{2}}{3583} \]

cannot change instantaneously (otherwise an infinitely large power would be required of the system for this).

But the current, in this case, can change instantaneously, “by a jump,” since we have set \(L=0\) and since the change in current is now not associated with a change in the energy of the magnetic field, which under this condition is equal to:

\[ \frac{L\cdot I^{2}}{2}=0. \]

Therefore, as soon as, under the admitted idealization, the system reaches point \(K\), the current in it will instantaneously change from the value \(I_{kp}\), corresponding to \(n_{\max}\) (Fig. 5), to the value \(I_{a_1}\), corresponding to the same speed, and the system will “jump” from the state determined by point \(K\) to the state determined by point \(a_1\), where the equations given above will again determine the behavior of the system. Thereafter the system will proceed, in accordance with what was said above, along the section \(a_1A_1K_1\) (Figs. 4 and 5) until, at point \(K_1\), a new “jump” occurs from point \(K_1\) to point \(a'\), etc.

Fig. 6

Thus, under the idealization admitted above, “discontinuous” self-oscillations arise in the system, which are also called relaxation oscillations [3; 5] (Fig. 6).

Since in “discontinuous” self-oscillations the current changes by “jumps” from the values \(+I_{kp}\) and \(-I_{kp}\), respectively, to the values \(-I_{a_1}\) and \(+I_a\), at constant values \(+n_{\max}\) and \(-n_{\max}\) of the speed (Fig. 5), the motor cannot operate on the sections \(KB_1A'_1a_1\) and \(K_1BA'a\) of the mechanical characteristic.

The process of its operation in self-oscillations in the \(n\)—\(I\) axes is characterized by the closed curve \(aAKa_1A_1K_1a\); since

\[ I=\frac{GD^2}{375c_{\mathrm{m}}}\,\frac{dn}{dt}=F_1\!\left(\frac{dn}{dt}\right), \]

this closed curve in the axes \(n\)—\(F_1\!\left(\dfrac{dn}{dt}\right)\) gives an idea of the limiting cycle of speed self-oscillations in the phase plane.

Since in a real system \(L \ne 0\), the current cannot change instantaneously, since this is associated with a change in the energy accumulated in the magnetic field. Therefore, during self-oscillations the current reaches its maximum values after a certain time different from zero, during which the speed has time to decrease somewhat. This is illustrated by oscillograms No. 2 (Fig. 3) and No. 1 (Fig. 7); both oscillograms show self-oscillations recorded with identical parameters of the circuit of Fig. 1, but with different initial conditions. For oscillogram No. 2 the initial conditions were chosen as:

\[ \text{at} \qquad t = 0 \qquad I_{\mathrm{init}} = 0 \]

and the oscillogram itself characterizes the process of establishing self-oscillations when the motor is connected to the generator.

Since, after obtaining oscillogram No. 2, the maximum current during oscillations was determined, when taking oscillogram No. 1 other initial conditions were adopted:

\[ \text{at} \qquad t = 0 \qquad I_{\mathrm{init}} = 32a > I_{\max}. \]

Since the current \(I_{\max}\) occurs in the motor plugging regime, in order to obtain a current \(I_{\mathrm{init}} > I_{\max}\), the motor under test, while connected to the generator, was forcibly rotated through a belt drive by an auxiliary motor in the plugging regime at a speed ensuring the condition \(I_{\mathrm{init}} > I_{\max}\).

To obtain identical conditions with respect to the flywheel moment in both experiments, when oscillogram No. 2 was recorded the auxiliary motor remained coupled by means of the belt drive to the motor under test, but was disconnected from the supply network.

Processing the oscillograms made it possible to obtain phase portraits of the self-oscillations both for current (Fig. 8) and for speed (Fig. 9). The process of establishing self-oscillations is represented here by phase trajectories 1 and 2, the numerical designations of which correspond to the numbers of the indicated oscillograms. It follows from Figs. 8 and 9 that the limiting cycles shown (bold closed curves), and consequently also the periodic nonlinear motions, are stable.

Within the limits \(\pm I_{\max}\) of the established self-oscillations (Figs. 3, 7), the hysteresis loop of the generator under test was then recorded; it is represented by solid lines in Fig. 10. It was found that, by its extreme points \(a\) and \(a_1\), the hysteresis loop lay on the straight line of voltage drop \(IR\); this means that, at the instant when the current reaches its maximum value, the motor speed and its counter-e.m.f. become equal to zero. This is confirmed by the dynamic mechanical characteristic (the solid curve in Fig. 11), which is the result of processing one period of the established self-oscillations from the oscillograms of Figs. 3 and 7; from the oscillograms, paired values of currents and speeds corresponding to definite instants of time were taken and plotted on the \(n\)--\(I\) axes (Fig. 11). In the same Fig. 11, for comparison, the static mechanical characteristic, calculated on the basis of Fig. 10 according to the formula, is shown by a dashed line:

\[ n = \frac{e_d}{c_e}. \]

Since, when operating on the static mechanical characteristic, the current, according to what was stated above, changes by a jump at an unchanged speed \(n_{\max}\), its maximum value \(I'_{\max}\) in the static mechanical characteristic is greater than \(I_{\max}\) in the dynamic mechanical characteristic. In practice, the value of the current \(I'_{\max}\) was determined by pro-

Oscillogram M1.

Visible curve labels: \(U=I_n^2(t)\), \(I=I_2^0(t)\), \(M_3^0(t)\); \(I_{\max}\), \(I_{\text{ssk}}\); \(t=0,\ I_{\max}=33^\circ\); \(M_1=0.65M_n,\ M_n=\dots,\ M_3=2\dots\) [[unclear: remaining handwritten formula]].

Fig. 7

Fig. 9

Axes: horizontal \(n\); vertical \(\dfrac{dn}{dt}\). Curves marked \(1\), \(2\), \(3\).

Fig. 8

Axes: horizontal \(I\); vertical \(\dfrac{dI}{dt}\). Curves marked \(1\), \(2\), \(3\).

Fig. 10

continuing the branch \(Ba\) while removing the hysteresis loop until the segment \(-e_{\mathrm{dyn\,max}}\), equal in absolute value to \(+e_{\mathrm{dyn\,max}}\), has fitted between the continuation of the branch \(Ba\) (dashed line) and the straight line \(I \cdot R\); this condition for determining \(I'_{\max}\) means constancy of the speed \(n_{\max}\) when the current changes abruptly from the value \(-I_{\mathrm{cr}}\) to the value \(+I'_{\max}\).

Oscillogram No. 2 (Fig. 3) indicates an interesting feature of the transient processes. During the first half-period after the motor is switched on to the generator, magnetization of the generator proceeds along the initial magnetization curve \(cba\) (Fig. 10). At the beginning of the transient process \(I=0\), and the voltage is equal to the e.m.f. due to the residual magnetic flux. Therefore, at the beginning of the first half-period (Fig. 3) the phase current lags behind the voltage. By the end of the first half-period the generator, being demagnetized along the branch \(aKc\) (Fig. 10), reaches a state in which the current again becomes zero, and the voltage becomes equal to the e.m.f. due to the residual magnetic flux; therefore, at the end of the first half-period the voltage

Fig. 11

lags in phase behind the current. In the subsequent half-periods, when the generator is in the mode of cyclic reversal of magnetization, the voltage lags behind the current both at the beginning and at the end of the half-periods. At the instants of time corresponding to the maximum values of current and voltage, the oscillogram shows coincidence in phase of current and voltage, which corresponds to points \(a\) and \(a_1\) in Fig. 10, for which the current and voltage simultaneously attain their maximum values. Thus, the hysteresis of the generator causes a variable phase shift between current and voltage during one half-period.

The experiments whose results are presented in the present work were carried out in the electric-drive laboratory of the Tomsk Polytechnic Institute on machines with the following characteristics:

Generator: 8 hp, 220 V, 1300 rpm, excitation—series.

Motor: 3.7 hp, 110 V, 31 A, 1180 rpm, excitation—separate.

REFERENCES

1. L. I. Gandzha. Physics of oscillations in the system “series generator—separately excited motor” (published in the present volume of the Proceedings of TPI).
2. A. A. Andronov and S. E. Khaikin. Theory of Oscillations, Part 1. ONTI, NKTP, 1937.
3. V. I. Smirnov. A Course of Higher Mathematics for Engineers and Physicists, Vol. 3. Gostekhizdat, 1933.
4. Edited by Doctor of Technical Sciences K. M. Polivanov. Physical Foundations of Electrical Engineering. Gosenergoizdat, 1950.