ON THE STUDY OF SELF-OSCILLATIONS IN A GENERATOR–MOTOR SYSTEM WITH SERIES-EXCITED MACHINES

I. S. AVRAAMOV

(Presented by the scientific seminar of the Electromechanical Faculty)

In modern technology, a generator–motor system with series-excited machines (Fig. 1) is used, especially in high-power drives of metal-cutting machine tools, as well as in the electric drives of aircraft units [1, 12]. The positive qualities of the system—a wide range of speed regulation, the absence of an exciter, and the extraordinary simplicity and reliability of the control circuit—have made it fairly widespread. However, the power of the machines used is limited by large losses in the regulating rheostats, and also by the possibility of oscillations arising in the motor current and speed. The occurrence of oscillations in high-power drives may lead to serious accidents.

Fig. 1.

At present, only a few works have been published in the literature concerning the generator–motor system with series-excited machines [1, 2, 3, 4, 5, 6, 12]. These works deal mainly with the static mechanical characteristics of the system. The question of studying oscillations, however, has been covered quite inadequately. In some works only the possibility of the occurrence of oscillations is indicated, without any attempt to analyze them. Only in work [7] is a mathematical analysis of oscillations given; in this case the system was considered as linear, and the oscillations as harmonic, which explains the mechanism of their occurrence in a completely unsatisfactory manner. In work [1] the first attempt was made to study the system

as nonlinear; in it an analytical method is given for investigating static mechanical characteristics, but the characteristics calculated in the region of small currents differ greatly from the experimental ones, which does not make it possible to judge correctly the processes that occur in machines at small currents.

In the present work an attempt is made to investigate the oscillations arising in the given system with allowance for nonlinear parameters, to determine the character of the oscillations, and to clarify the conditions for stable operation of this system. In addition, the study of nonlinear systems is at present an exceptionally important problem. In the article by Corresponding Member of the Academy of Sciences of the USSR Prof. L. R. Neiman [8] it is stated: “Considerably less developed, and at the same time acquiring at present exceptionally great importance, is the theory of nonlinear electric and magnetic circuits, the parameters of which depend sharply on the intensity of the processes occurring in them . . . If previously nonlinearity was taken into account in circuit theory as a complicating factor, then now the fullest possible development of the theory of nonlinear circuits acquires primary importance in electrical automation, telemechanics, electrical measuring technology, radio engineering, electric power engineering, and other fields.”

The investigation of oscillations of an electromechanical system with many nonlinearities is also of interest from the point of view of the idea of oscillatory mutual assistance among various fields of physics and technology. Since “the theory of oscillations unites physical phenomena that are identical in the form of the laws governing them” [9], investigations of oscillations in a nonlinear electromechanical system can be used in other fields of physics and technology.

In the present work an electromechanical system is investigated with nonlinear electrodynamic capacitance, nonlinear magnetization characteristics of the machines, and a nonlinear dependence of the voltage drop in the contact between the commutator and the brushes.

Static Mechanical Characteristics

The equation of the static speed characteristic can be obtained from the equation of the electrical equilibrium of the system

\[ e_g = IR + e_{dv} + L(I)\frac{dI}{dt}, \tag{1} \]

where

\[ e_g = f(I) \]
is the e.m.f. of the generator—a nonlinear function of the current, owing to saturation of the generator;

\[ IR = F(I) \]
is the voltage drop in the active resistance of the power circuit—a nonlinear function of the current, owing to the nonlinearity of the resistance in the transition layer “brush–commutator”;

\[ L(I) \]
is the total coefficient of self-induction of the system;

\[ I \]
is the current flowing in the power circuit;

\[ e_{dv} = C'_{e_{dv}}\cdot \Phi_{dv}\cdot n \]
is the counter e.m.f. of the motor, equal to the product of the coefficient \(C'_{e_{dv}}\), depending on the design data of the machine, the magnetic flux, and the motor speed.

For the static regime one should put in (1)

\[ L(I)\frac{dI}{dt}=0. \]

Substituting in (1) the value of \(e_{dv}\), setting \(L(I)\frac{dI}{dt}=0\), and solving the resulting...

... equation with respect to \(n\), we obtain finally the equation of the static speed characteristic

\[ n=\frac{e_g-IR}{c'_{dv}\Phi_{dv}} =\frac{e_{dv}}{\left(\dfrac{E}{n}\right)} =\frac{\psi(I)}{\varphi(I)} . \tag{2} \]

In the numerator, \(e_g-IR\), or \(f(I)-F(I)=\psi(I)\), is the nonlinear counter-e.m.f. characteristic of the motor. In the denominator is the quantity “volts per revolution.” The characteristic

\[ \frac{E}{n}=\varphi(I) \]

is the magnetization curve of the motor. To construct the speed characteristic, it is necessary to know the dependences \(e_g=f(I)\), \(\left(\dfrac{E}{n}\right)=\varphi(I)\), and \(IR=F(I)\), which may be obtained experimentally (Figs. 2a, 2b).

Fig. 2.

The dependence \(e_{dv}=\psi(I)\) is obtained as a result of subtracting the ordinates of the characteristic \(IR=F(I)\) from the ordinates of the characteristic \(e_g=f(I)\), taken at identical currents. In Fig. 3 the static speed characteristic \(n=P_1(I)\) is shown, calculated according to the method described and plotted point by point for the system of Fig. 1 with \(R_{rg}=\infty,\ R_{rd}=\infty\).

Since, in order to regulate the speed of the motor, it is necessary to shunt the field windings of the machines, the construction of artificial characteristics in this case is carried out in the same way, but taking into account the reduction of the generator or motor excitation currents. Consider the case when \(R_{rd}=\infty,\ R_{rg}<\infty\). Let us denote by \(\mu_g\) the ratio of the total current \(I\) (Fig. 1) to the current flowing through the generator field winding,

Fig. 3.

\[ \mu_g=\frac{I}{I_{vg}} . \]

When the resistance \(R_{r2}\) is connected in parallel with the generator field winding, the total resistance of the excitation circuit will be

\[ R_{obshch\,v2}=\frac{R_{v2}R_{r2}}{R_{v2}+R_{r2}} . \]

Then one may write

\[ IR_{obshch\,v2}=I_rR_{r2}=I_{v2}R_{v2}, \]

whence

\[ \frac{I}{I_{v2}}=\frac{R_{v2}+R_{r2}}{R_{r2}}=\psi_2 \]

or

\[ \psi_2=\frac{R_{v2}}{R_{r2}}+1 . \tag{3} \]

When the resistance \(R_{r2}\) is connected, the total resistance of the circuit, which is now equal to

\[ R_{obshch}=R_{ya2}+R_{yo}+R_{ovd}+R_{obshch\,v2}, \]

will also change.

The characteristics are constructed in the following way.

1. Having chosen the resistance of the rheostat \(R_{r2}\), from equation (3) we determine \(\psi_2\).

2. We assign arbitrary values of the current \(I\) in the power circuit and, for each of its values, determine the current flowing through the generator field winding,

\[ I_{v2}=\frac{I}{\psi_2}. \]

From the characteristic \(e_g=f(I)\), the value of the generator e.m.f. will now correspond to the current \(I_{v2}\). Since \(e_{dv}\) is determined graphically as the difference of the ordinates of the characteristics \(e_g=f(I_{v2})\) and \(IR=F(I)\), it is necessary to reconstruct the magnetization characteristic of the generator in the coordinate axes \(e_g, I\) (Fig. 4).

1. We construct the new characteristics \(IR_{obshch\,1}=F_1(I)\) and \(e_{dv}=\psi_1(I)\), where \(R_{obshch\,1}\) is the total resistance of the power circuit when the resistance \(R_{r2}\) is connected in parallel with the generator field winding.

2. We determine the values \(e_{dv}=\psi_1(I)\) and

\[ \frac{E}{n}=e(I) \]

for various currents and, by equation (2), find the points of the artificial speed characteristic.

In Fig. 4: vertical axis \(e_g\); horizontal axis \(I\); curves \(e_g=f(I_{v2})\) and \(e_g=f(I)\); marked currents \(I_{v2}\), \(I_1\); \(\psi=1.6\).

Fig. 4.

The artificial speed characteristics when a resistance is connected in parallel with the motor field winding can be constructed by the same method, but with allowance for the weakening of the magnetic flux of the motor. The construction is carried out in the following sequence.

1. Having chosen the resistance of the rheostat \(R_{r0}\), from an equation analogous to equation (3) we determine the value \(\psi_{dv}\) for the motor,

\[ \psi_{dv}=\frac{R_{v0}}{R_{r0}}+1 . \]

1. We assign arbitrary values of the current \(I\) in the power circuit and determine, for each of its values, the current flowing through the motor field winding

\[ I_{vd}=\frac{I}{\psi_d}. \]

1. From the magnetization characteristic of the motor we determine the value

\[ \left(\frac{E}{n}\right), \]

corresponding to \(I_{vd}\).

1. We construct new characteristics \(IR_{\mathrm{tot},2}=F_2(I)\) and \(e_{dv}=\psi_2(I)\), where \(R_{\mathrm{tot},2}\) is the total resistance of the power circuit when the resistance \(R_{rd}\) is connected in parallel with the motor field winding.

2. We determine the values \(e_{dv}=\psi_2(I)\) and

\[ \left(\frac{E}{n}\right)=\varphi(I_{vd}) \]

for various currents, and from equation (2) find the points of the speed characteristic.

In Fig. 3, for the machines under test, artificial characteristics are constructed for weakening the magnetic flux of the generator \(n=P_2(I)\) and of the motor \(n=P_3(I)\) and \(n=P_4(I)\). Mechanical characteristics can be obtained from speed characteristics by using the transition curve \(M=\Phi(I)\) (Fig. 5), which is constructed from the expression

\[ M=\frac{C'_M}{C'_e}\,\varphi(I)\,I. \]

Fig. 5.

The latter is obtained from

\[ M=C'_M\Phi_{dv}I, \]

taking into account that \(\Phi_{dv}\) is determined from the expression

\[ C'_{e\,dv}\Phi_{dv}=\frac{E}{n}=\varphi(I). \]

The mechanical characteristic \(n=P_1(M)\) for \(R_{rd}=\infty\) and \(R_{r2}=\infty\) is shown in Fig. 3.

A joint study of the static mechanical characteristic of the motor and of the mechanism shows that, on the sections where

\[ \frac{dM}{dn}<\frac{dM_{\mathrm{st}}}{dn}, \]

stable operation of the drive is observed [4]. It is seen from Fig. 3 that, for \(M_{\mathrm{st}}=\mathrm{const}\), this condition is satisfied on the section \(n_0-k'_1\) and on the section to the right of point \(k'_2\) of the characteristic \(n=P_1(M)\). The section \(k'_1-k'_2\), located between the two extreme values of the mechanical characteristic, corresponds to unstable operation of the drive because the condition

\[ \frac{dM}{dn}<\frac{dM_{\mathrm{st}}}{dn} \]

is not satisfied.

Thus, the points \(k'_1\) and \(k'_2\) (Fig. 3), at which \(n=n_{\max}\) and \(n=n_{\min}\), are points of change of stability. These points on the characteristic \(n=P_1(I)\) correspond to the points \(k_1\) and \(k_2\).

Oscillations in the generator--motor system with series-excited machines

For studying the dynamics of this system it is necessary to have a differential equation of motion. In all subsequent derivations, in order to simplify them, the nonlinear characteristics \(e_g=f(I)\) and \(IR=F(I)\)

are replaced by a single nonlinear characteristic \(\psi(I)=f(I)-F(I)\), shown in Fig. 2a. In this connection, equation (1), taking into account the expression \(e_{\partial v}=\varphi(I)n\) and the characteristic \(\psi(I)\), is rewritten as follows:

\[ \psi(I)-\varphi(I)n-L(I)\frac{dI}{dt}=0. \tag{1a} \]

Taking into consideration that

\[ \Phi_{\partial v}=\frac{\varphi(I)}{C'_{e\partial v}};\quad M=C'_M\Phi_{\partial v}I;\quad M_{st}=C'_M\Phi_{\partial v}I_{st}(I); \]

\[ I_{st}(I)=\frac{M_{st}C'_{e\partial v}}{C'_M\varphi(I)} \tag{4} \]

and considering the case where \(M_{st}=\mathrm{const}\), on the basis of the drive equation of motion

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

we obtain:

\[ I=\frac{GD^2 C'_{e\partial v}}{375\,C'_M\varphi(I)}\frac{dn}{dt}+I_{st}(I). \tag{5} \]

The differential equation for the current is obtained if equation (1a) is divided by \(\varphi(I)\) (in order to get rid of the variable coefficient at \(n\)), differentiated, and the derivative \(\frac{dn}{dt}\) is replaced from expression (5). Finally, the expression for the current will be as follows:

\[ L(I)\frac{d^2I}{dt^2} +\left[ L'(I)\frac{dI}{dt} +\frac{\varphi'(I)\psi(I)}{\varphi(I)} -\frac{\varphi'(I)}{\varphi(I)}L(I)\frac{dI}{dt} -\psi'(I) \right]\frac{dI}{dt} \]
\[ +\left[I-I_{st}(I)\right] \frac{375\,C'_M[\varphi(I)]^2}{GD^2\,C'_{e\partial v}} =0. \tag{6} \]

In equation (6), \(L(I)\)—the inductance of the armatures and of the excitation windings of the generator and motor—is a nonlinear function of the current owing to saturation of the magnetic system of the electric machines. The dependence \(L(I)\) is given in Fig. 6.

Fig. 6.

The coefficient at \(\frac{dI}{dt}\) in equation (6) is a dissipative term consisting of the nonlinear functions \(L(I)\), \(\varphi(I)\), \(\psi(I)\), and their derivatives. The coefficient at \([I-I_{st}(I)]\) is the reciprocal electrodynamic capacitance

\[ \frac{375\,C'_M[\varphi(I)]^2}{GD^2\,C'_{e\partial v}} = \frac{1}{C_{dyn}(I)} = \left[\frac{1}{\mathrm{farad}}\right]. \]

The dependence of the nonlinear electrodynamic capacitance \(C_{\mathrm{dyn}}(I)\) is shown in Fig. 6. In view of what has been said, equation (6) is a differential equation of motion of a system with nonlinear electrodynamic capacitance, inductance, and a dissipative term.

Investigation of the Mechanical Transient Process

The equation of the mechanical transient process, in which only the mechanical inertia of the drive is taken into account, is obtained from equation (6), putting \(L(I)=0\):

\[ \left[ \frac{\varphi'(I)\psi(I)}{\varphi(I)}-\psi'(I) \right]\frac{dI}{dt} + \left[I-I_{\mathrm{st}}(I)\right] \frac{375\,C'_M[\varphi(I)]^2}{GD^2 c'_{e\,\mathrm{dv}}} =0. \tag{7} \]

Let us write (7) with respect to \(\dfrac{dI}{dt}\), taking (2) into account, and investigate the equation obtained

\[ \frac{dI}{dt} = \frac{375\,C'_M[\varphi(I)]^2\,[I-I_{\mathrm{st}}(I)]} {GD^2 c'_{e\,\mathrm{dv}}\,[\psi'(I)-\varphi'(I)n]}. \tag{8} \]

According to equation (4), \(I_{\mathrm{st}}(I)\), for \(M_{\mathrm{st}}=\mathrm{const}\), assumes different values depending on the current \(I\) (Fig. 5). Under the condition \(M=M_{\mathrm{st}}\)

\[ c'_{e\,\mathrm{dv}}\Phi_{\mathrm{dv}}I = c'_{e\,\mathrm{dv}}\Phi_{\mathrm{dv}}I_{\mathrm{st}}(I), \]

whence it follows that \(I=I_{\mathrm{st}}(I)\). Consequently, by equation (8), \(\dfrac{dI}{dt}=0\). If the magnitude \(M_{\mathrm{st}}\) causes the system to operate on the sections of stable operation of the speed characteristic \((n_0-k_1\) and \(k_2-a)\) (Fig. 7), then the motor will operate with constant speed and current

\[ \left(\frac{dI}{dt}=0\right). \]

Fig. 7.

The denominator of equation (8) can also assume different values depending on the current. Thus, for \(I>I_{k_2}\) or \(I<I_{k_1}\),

\[ [\psi'(I)-\varphi'(I)n]<0; \]

for \(I=I_{k_1}\) or \(I=I_{k_2}\), when the system has reached the stability boundary,

\[ [\psi'(I)-\varphi'(I)n]=0 \]

and for \(I_{k_1}<I<I_{k_2}\)

\[ [\psi'(I)-\varphi'(I)n]>0. \]

These conclusions were drawn on the basis of an analysis of the characteristics (Figs. 2a, 2b, and 3) and are proved below. Table 1 shows the results of processing the experimental and calculated characteristics given in Figs. 2a, 2b, and 3. From Table 1 it is seen that near the values of the currents \(I_{k_2}=25\div 30\,a\) and \(I_{k_1}=6\div 8a\) the sign of \(\left|\psi'(I)-\varphi'(I)n\right|\) changes from minus to plus, passing through zero, and conversely. That \(\left|\psi'(I)-\varphi'(I)n\right|=0\) when \(I=I_{k_1}\) and \(I=I_{k_2}\) is also evident from the following. Differentiating (1a) with respect to the current, taking into account that \(L(I)=0\), we obtain

\[ \psi'(I)-n'\varphi(I)-\varphi'(I)n=0; \]

Table 1

taking into account that at the critical points \(k_1\) and \(k_2\), when \(I=I_{k_1}\) or \(I=I_{k_2}\), the speed on the speed characteristic has its maximum or minimum value, when \(n'=0\), we shall have \(\psi'(I_{k_2})-\varphi'(I_{k_2})n=0\). Thus, at a current equal to \(I_{k_1}\) or \(I_{k_2}\), the denominator of equation (8) becomes zero, and the first derivative of the current with respect to time becomes infinite. This indicates that the current at the points of the speed characteristic \(k_2\) or \(k_1\) must instantaneously change its value. Since we have put \(L(I)=0\) and the change in current is now not connected with a change in the energy of the magnetic field, the system will reach the new state corresponding to point \(b\) or \(a\) (Fig. 7) as a result of a “jump,” during which the current \(I\) can change instantaneously, while the speed remains unchanged. Let us trace the possible motions in the system under the condition \(M'_{k_1}<M_{st}<M'_{k_2}\), when the possible equilibrium state at \(M=M_{st}\) is unstable. When the switch \(P\) (Fig. 1) is closed, the motor is connected to the generator and the process of excitation of the machines begins, accompanied by a surge of current with the motor armature at rest. At a certain value of the torque the motor will begin to accelerate. During the acceleration process \(I\) decreases, and the motor tends toward the equilibrium state in which \(M=M_{st}\) and \(I=I_{st}\). As soon as the motor speed reaches the value \(n_{\max}\), corresponding to point \(k_2\), the equation of motion (8) loses its analyticity, since

\[ \frac{dI}{dt}=\infty . \]

The subsequent motion is determined by the jump condition: the current changes instantaneously, in a jump, while the motor speed cannot change instantaneously, since its change is connected with a change in the kinetic energy of the drive,

\[ J\frac{\omega_{\max}^{2}}{2}=\frac{GD^{2}\pi^{2}}{7200\,g}\,n_{\max}^{2}. \]

Otherwise, a system of infinite power would be required. Therefore the speed before the jump, corresponding to point \(k_2\), must be equal to the speed after the jump, i.e., to the speed at point \(b\) of the speed characteristic (Fig. 7), or

\[ n_{k_2}=n_b . \]

After the jump of the current to point \(b\), the subsequent motion will proceed continuously and will be described by equation (8). The motion will pass—

move along the stable segment of the characteristic in the direction of increasing current, since \(M<M_{st}\) and \(I<I_{st}(I)\), until the system reaches point \(k_1\). At point \(k_1\), \(\dfrac{dI}{dt}\) again tends to infinity, which causes the subsequent current jump to the value \(I_a\). The condition for the jump in this case will be

\[ n_{k_1}=n_a . \]

After the jump the system will again tend toward the equilibrium state, repeating the motion along the segment of the characteristic \(a-k_2\), etc. The alternation of the motions described, accompanied by current jumps, leads to the occurrence in the system of “discontinuous” self-oscillations for the current and continuous self-oscillations for the speed. The closed curve \(b, k_1, a, k_2\) constructed in Fig. 7 is the dynamic speed characteristic under the adopted idealization. Fig. 8 presents an oscillo-

Fig. 8.

gram of the transient process of speed and current for the circuit of Fig. 1 with \(R_{pc}=\infty\) and \(R_{pd}=\infty\). In Fig. 7 (curve 2) the same transient process is shown in speed—current axes, obtained by processing the indicated oscillogram. As can be seen from comparison of curves 1 and 2, the calculated characteristic 1 satisfactorily reflects the quality of the transient process. The quantitative discrepancy of the characteristics presented is due to the idealization that was used in the investigation.

Electromechanical Transient Process

The investigation of the electromechanical transient process is carried out according to equation (6), in which \(L(I)\), \(\varphi(I)\), \(\psi(I)\), and \(C_{\mathrm{dyn}}(I)\) are nonlinear functions, while \(M_{st}=\mathrm{const}\).

Integration of differential equation (6) by analytical methods proves impossible. In view of the indicated circumstances, methods of qualitative integration may be applied to the study of this dynamic system; among them, the most suitable for the given system is the isocline method [10, 11]. This method makes it possible to construct a phase portrait for the current. However, taking into account all the indicated nonlinearities leads to a very complicated isocline equation. Therefore we solve equation (6) taking into account only the essential nonlinearities \(\psi(I)\) and \(C_{\mathrm{dyn}}(I)\), and take the characteristics \(L(I)\) and \(\varphi(I)\) to be linear:

\[ L(I)=\mathrm{const}, \]

\[ \varphi(I)=k_{\partial}I. \]

We approximate the nonlinear characteristic \(\psi(I)\) by a second-degree polynomial

\[ \psi(I)=\alpha I-\beta I^2, \tag{9} \]

where \(\alpha\) and \(\beta\) are constant coefficients whose values are determined by the characteristic. The numerical values of \(\alpha\) and \(\beta\) are as follows:

\[ \alpha=2.65, \]

\[ \beta=0.02. \]

Substituting (9) into (6), making the change of variables

\[ \frac{dI}{dt}=y;\qquad \frac{d^2 I}{dt^2}=\frac{dy}{dt}=\frac{dy}{dI}\,y=\Theta y \]

and solving the resulting equation with respect to \(y\), we obtain the equation of the isoclines:

\[ y=I\left[\left(-\frac{\Theta}{2}+\frac{\beta I}{2L}\right)\pm \right. \]

\[ \left. \pm\sqrt{\left(-\frac{\Theta}{2}+\frac{\beta I}{2L}\right)^2+ \frac{k^2 d_0 I^2 \cdot 375 \cdot C'_M}{GD^2\cdot C'_{e\,dv}} -\frac{k d_0\cdot 375}{GD^2\cdot L}\,M_{st}}\right]. \tag{10} \]

Assigning various values to the current \(I\) and to \(\Theta=\dfrac{dy}{dI}\), we construct on the phase plane a family of isoclines (Fig. 9).

We determine the initial conditions from equation (1). We set \(n=0\) (\(e_{dv}=0\)) and \(I=I_{tr}=I_{st}(I)\); then

\[ \left(\frac{dI}{dt}\right)_{tr\,og} = \frac{e_{g\,tr}-I_{tr}R}{L}. \]

We determine \(e_{g\,tr}\) from the characteristic \(e_g=f(I)\) (Fig. 2a). Using the known method [11], we construct the integral curve of the transient process on the phase plane (Fig. 9). The presence of a closed integral curve (limit cycle), of an energy source that does not possess oscillatory properties (an asynchronous motor), and also of a nonlinear oscillation limiter (the nonlinear characteristic of the generator) makes it possible to conclude that the oscillations arising in the system are self-oscillations. The limit cycle of the self-oscillations makes it possible to determine directly the amplitude of the oscillations, whose period is determined when constructing the transient process in time. The expansion of the phase portrait in time is carried out by formula [11]

\[ \Delta t_i=\int_{I_i}^{I_{i+1}}\frac{dI}{y} = \frac{I_{i+1}-I_i}{y_{i+1}-y_i}\cdot \ln\frac{y_{i+1}}{y_i}. \]

The transient process \(I(t)\) is shown in Fig. 9.

The phase portrait for the motor speed in the given system is difficult to obtain because the differential equation

motion for the speed is a function of three variables: speed, derivative of speed, and current. Therefore, to study the transient process \(n(t)\), we proceed as follows. From equation (1), after substituting \(e_g=f(I)\) and \(e_{dv}=\varepsilon(I)\,n\), we have:

\[ n_{\text{dyn}}= \frac{f(I)-IR(I)-L(I)\dfrac{dI}{dt}}{\varepsilon(I)}. \tag{11} \]

In the equation of the isoclines, \(L(I)=\text{const}\). For greater accuracy, in (11) we take \(L(I)\) to be a nonlinear function, shown in Fig. 6. For specified current values, \(L(I)\) is determined from Fig. 6, \(\dfrac{dI}{dt}\)—from the phase portrait (Fig. 9), \(f(I)\) and \(IR(I)\)—from Fig. 2a, and \(\varepsilon(I)\)—from Fig. 2b.

Fig. 9.

The dynamic speed characteristic \(n_{\text{dyn}}(I)\) is constructed according to (11) in Fig. 10. Having the characteristic \(I(t)\) and \(n_{\text{dyn}}(I)\), we construct the characteristic of the transient process \(n(t)\), which is shown in Fig. 11.

The investigations carried out above allow the following conclusions to be drawn.

1. The oscillations arising in the generator–motor system with series-excited machines (an electromechanical system with nonlinear electrodynamic capacitance) are self-oscillations, and the system itself is self-oscillatory.

1. The occurrence of oscillations takes place as a result of the system passing from the stable region of the machine characteristics into the unstable one, and back again; moreover, for a generator–motor system with series-excited machines, in the absence of resistances shunting the excitation windings of the machines, the current and the speed vary in the region of positive values. In this case the oscillations occur without remagnetization of the generator and reversal of the motor.

Fig. 10.

In the figure: vertical axis \(n\), rev/min; horizontal axis \(\alpha, I\); curve \(n(\alpha)\).

Fig. 11.

In the figure: vertical axis \(n\), rev/min; horizontal axis \(t\); curve \(n(t)\); \(\alpha, I\).

1. Analysis of discontinuous oscillations in the system makes it possible to determine, in a first approximation, the influence of the load \((M_{st})\) on the oscillations. Oscillations arise in the system when \(M_{k1}<M_{st}<M_{k2}\). In all other cases, stable operation of the system is observed, with constant \(n\), \(I\), and \(M\), which are determined by the load and by the mechanical characteristic of the system.

1. The study of the mechanical transient process when \(L(I)=0\), without resorting to integration of the nonlinear differential equation (7), gives an idea of the nature of the oscillations in the generator–motor system with series-excited machines. For a more accurate quantitative and qualitative study of the oscillations in this system, it is necessary to investigate the electromechanical transient process.

2. The given method for studying the electromechanical transient regime satisfactorily reflects both the qualitative and the quantitative aspects of the processes occurring in the system.

3. With the aid of this method one can construct the dynamic speed and mechanical characteristics of the system.

4. With the aid of this method it becomes possible to study in greater detail the influence of nonlinear parameters and of the load on self-oscillations, and to reveal the physics of self-oscillations.

The author expresses gratitude to Docent L. I. Gandzha for a number of suggestions that were taken into account in preparing the article for publication.

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