LINEAR APPROXIMATION OF SELF-OSCILLATIONS IN AN ELECTROMECHANICAL SYSTEM

L. I. GANDZHA

(Presented by the scientific seminar of the Electromechanical Faculty)

In [1; 2], “discontinuous” oscillations in a nonlinear electromechanical system consisting of a separately excited generator and an independently excited motor (Fig. 1) connected to this generator were examined in detail. It was shown there that an idea of “discontinuous” oscillations in this system can be obtained as a result of idealizing this system, if the inductance of the power circuit is neglected. It was shown that the mechanical characteristic of the motor \(n=f(I)\) of this self-oscillating system, in the “discontinuous” interpretation, gives an idea of the limiting cycle of the self-oscillations. It is also indicated there

Fig. 1.

that taking the inductance into account excludes the possibility of “discontinuous” oscillations, and the self-oscillations of the current and speed of the system on the phase plane are characterized by limit cycles

\[ \frac{dI}{dt}=f_1(I) \quad \text{and} \quad \frac{dn}{dt}=f_2(n) \]

of a considerably more complex form.

The self-oscillating system of Fig. 1 considered in [1; 2], in the absence of load on the motor shaft \((M_{\mathrm{st}}=0)\), is described by the nonlinear differential equation of the second order

\[ L\frac{d^2 I}{dt^2}+\left[R-\varphi'(I)\right]\frac{dI}{dt} +\frac{375c_e\cdot c_M}{GD^2}\cdot I=0, \tag{1} \]

obtained as a result of the simultaneous solution of the equations

\[ e_2=I\cdot R+L\frac{dI}{dt}+e_0, \tag{2} \]

\[ M-M_j=\frac{GD^2}{375}\frac{dn}{dt}, \tag{3} \]

\[ M=c_M\cdot I, \tag{4} \]

\[ e_0=c_e\cdot n, \tag{5} \]

\[ e_0=\varphi(I), \tag{6} \]

where:

\(e_2, e_0, I\) are the generator e.m.f., the counter-e.m.f. and the motor current;

\(M, n, GD^2\) are the torque developed by the motor on the shaft, its speed, and the flywheel moment;

\(L, R\) are the coefficient of self-inductance and the resistance of the power circuit;

\(c_M, c_e\) are the proportionality coefficients between the torque and the motor current and between its counter-e.m.f. and speed at the nominal magnetic flux.

\(\varphi'(I)\) is the derivative of the generator e.m.f. with respect to current, determined from the magnetization curve of the generator.

The difficulty of studying the self-oscillations of this or a similar system is due to a number of reasons, of which we shall indicate the principal ones.

1. It does not appear possible to express the nonlinear dependence \(e_2=\varphi(I)\) analytically, all the more so since such a dependence must approximate the hysteresis loop of the generator, along which its remagnetization takes place in the process of self-oscillations.

2. The coefficient of self-inductance \(L\) depends on the current, which introduces an additional nonlinearity into the system and complicates its analysis.

3. In a number of cases the magnetic flux of the motor does not remain constant and equal to the nominal value, but changes with the change in the motor current (for example, when the motor has a series field winding, or when the motor brushes are shifted from the neutral). Therefore the coefficients \(c_M\) and \(c_e\) in (1) become variable and dependent on the current, which introduces still another additional nonlinearity into the system.

The problem of studying self-oscillations is considerably simplified if, as a first approximation, one sets \(L=\mathrm{const}\) and \(\Phi=\mathrm{const}\) and applies a piecewise-linear approximation to the magnetization curve of the generator (Fig. 2).

Let us replace the magnetization curve of the generator by a broken line consisting of \(m\) straight-line segments; then the dependence (6) of the generator e.m.f. on the current for some \(i\)-th segment (Fig. 2) will be expressed by the equation

\[ e_{2i}=\varphi_i(I_i)=e_{2i0}+\varphi'_i(I_i)I_i \quad (i=1,2,\ldots,m), \tag{7} \]

valid within the given segment, where \(\varphi'_i(I)=\mathrm{const}^{1}\). Solving now simultaneously (2), (3), (4), (5), and (7), we obtain:

\[ L\frac{d^2 I_i}{dt^2} +\left[R-\varphi'_i(I_i)\right]\frac{dI_i}{dt} +\frac{375c_e\cdot c_M}{GD^2}I_i=0. \tag{8} \]

On the basis of the adopted approximation for the magnetization curve and from (8), we shall determine what the character of the transient process of the current will be and in what way the limiting cycle under self-oscillations will be approximated.

\(^{1}\) The adopted notation \(I_i\) characterizes the instantaneous value of the current within the limits of the \(i\)-th segment.

A necessary condition for the occurrence of self-oscillations is the presence of a critical point \(K\), where the stability of the system changes, on the magnetization curve (Fig. 2), which is found by drawing a tangent to the curve \(e_2=\varphi(I)\), parallel to the straight line \(I\cdot R=\psi(I)\). In turn, the presence of the point \(K\) leads to the conditions:

\[ \varphi'(I)>\psi'(I)=R \quad \text{for } I<I_{kp}; \tag{9} \]

\[ \varphi'(I)<\psi'(I)=R \quad \text{for } I>I_{kp}; \tag{10} \]

\[ \varphi'(I)=\psi'(I)=R \quad \text{for } I=I_{kp}. \tag{11} \]

Fig. 2.

The solution of the linear differential equation (8), as is known, has the form

\[ I_i=A_{1i}e^{p_{1i}t}+A_{2i}e^{p_{2i}t}, \tag{12} \]

where:

\(A_{1i}\) and \(A_{2i}\) are constants of integration, which must be determined from the initial conditions for the \(i\)-th segment under consideration (for \(t=0\), \(I_i=I_{i1}\)).

\(p_{1i}\) and \(p_{2i}\) are the roots of the characteristic equation, which, on the basis of (8), can be written in the form:

\[ Lp_i^2+\left[\,R-\varphi'_i(I_i)\,\right]p_i+\frac{375c_e\cdot c_m}{GD^2}=0. \tag{13} \]

The roots \(p_{1i}\) and \(p_{2i}\) in general form are determined as

\[ p_{(1,2)i}= \frac{ -\left[R-\varphi'_i(I_i)\right]\pm \sqrt{\left[R-\varphi'_i(I_i)\right]^2-4L\frac{375c_e\cdot c_m}{GD^2}} }{2L}. \tag{14} \]

Analyzing (9), (10), (11), (12), and (14), we arrive at the following possible cases.

1st case. \(\varphi'_i(I_i)\) is so large that the conditions are satisfied:

\[ \varphi'_i(I_i)>R, \tag{15} \]

\[ [R-\varphi'_i(I_i)]^2>4L\,\frac{375c_e\cdot c_m}{GD^2}. \tag{16} \]

Condition (15) determines the minus sign of the second term in (8), which means that the system of Fig. 1 has “negative” resistance. Condition (16), in turn, determines the roots \(p_{1i}\) and \(p_{2i}\) from (14) as real, positive, and distinct; accordingly, the transient process of the current according to (12), within the limits of the considered portion of the magnetization curve, is defined as aperiodically increasing.

2nd case. Condition (15) holds, and

\[ [R-\varphi'_i(I_i)]^2=4L\,\frac{375c_m\cdot c_e}{GD^2}. \tag{17} \]

In this case the system still has “negative” resistance, but the roots from (14) are real, positive, and equal \((p_{1i}=p_{2i}=p_i)\). The transient process of the current is still aperiodically increasing, but it is now described not by (12), but by the equation

\[ I_i=A_{1i}e^{p_{1i}t}+A_{2i}e^{p_i t}\cdot t. \tag{18} \]

3rd case. Condition (15) is satisfied, and

\[ [R-\varphi'_i(I_i)]^2<4L\,\frac{375c_e c_m}{GD^2}. \tag{19} \]

The system still has “negative” resistance; the roots \(p_{1i}\) and \(p_{2i}\) from (14) are obtained as conjugate complex numbers with positive real parts. The transient process in this case will be oscillatory increasing and is described by equation (12), which by familiar methods can easily be transformed into trigonometric form.

4th case. Inequality (15) becomes an equality

\[ \varphi'_i(I_i)=R. \tag{20} \]

The system, in accordance with (8), has no resistance; the roots from (14) are obtained as imaginary conjugates, and the transient process will be described by equation (12), which for this case, in trigonometric form, represents undamped harmonic oscillations of the current.

5th case. The following conditions hold:

\[ R>\varphi'_i(I_i), \tag{21} \]

\[ [R-\varphi'_i(I_i)]^2<4L\,\frac{375c_e c_m}{GD^2}. \tag{22} \]

Under these conditions the system will have positive resistance, and the roots will be conjugate complex numbers with negative real parts. The transient process is still described by (12), which in trigonometric form represents damped oscillations.

6th case. Condition (21) is satisfied, and

\[ [R-\varphi'_i(I_i)]^2=4L\,\frac{375\cdot c_e\cdot c_m}{GD^2}. \tag{23} \]

The resistance of the system remains positive, while the roots are real, negative, and equal. As in the second case, this transient process is a limiting one between damped oscillatory and damped aperiodic, and is described by (18).

7th case. (21) is satisfied and

\[ [R-\varphi'(I_i)]^2 > 4L \frac{375\, c_e \cdot C_M}{G D^2}. \tag{24} \]

With positive resistance of the system, the roots obtained are real, negative, and different, which characterizes an aperiodic damped transient process described by (12).

Fig. 3.

For definiteness, let us consider the increase of current from zero to \(+I_{\max}\) during the positive half-period of self-oscillations. As was clarified in [1,2] and according to Fig. 3, when the current increases the process of reversal of magnetization of the generator will proceed along the lower branch of the hysteresis loop, and when it decreases—along the upper branch, in the directions indicated by the arrows.

Let us divide the lower branch into 7 sections in such a way that for each of them the conditions corresponding to the cases considered above are satisfied, and, in accordance with this, number the sections with the digits \(1, 2, \ldots, 7\). Comparing Fig. 4, which represents the curve of the transient process \(I=F(t)\) over one half-period of self-oscillation, with Fig. 3, and dividing the latter into sections of the same name \(1, 2, \ldots, 7\) within the range of current variation from 0 to \(I_{\max}\) by points corresponding to the current values \(I_{1-2}, I_{2-3}, \ldots, I_{6-7}, I_{\max}\) in Fig. 3, we may say that this part of the curve from \(I=0\) to \(I=I_{\max}\) can be constructed in the following sequence:

on section 1—as the curve of an aperiodic increasing process;
on section 2—as the curve of a transient process that is limiting between an aperiodic increasing and an oscillatory increasing process;
on section 3—as the curve of an oscillatory increasing process;
on section 4—the same transient process with undamped oscillations;
on section 5—the same oscillatory damped process;

on segment 6—as that of the boundary between oscillatory decaying and aperiodic decaying;

on segment 7—as the curve of an aperiodic decaying transient process.

The same may be said when considering the decrease of the current from \(I_{\max}\) to 0, when the remagnetization of the generator takes place along the upper branch of the hysteresis loop, which may be divided into the corresponding sections \(1', 2' \ldots 7'\) (Fig. 3), and also when considering the changes of the current from 0 to \(-I_{\max}\) and from \(-I_{\max}\) to 0.

Fig. 4.

Let us now consider the transient process in the phase plane in a rectangular coordinate system

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

When the current changes from 0 to \(I_{1-2}\), when an increasing aperiodic regime occurs, the representative point in the phase plane will move along a segment of a parabolic curve [3, 4] with an equilibrium state of the type of an unstable node at the origin (Fig. 5a).

The same will occur when the current changes from \(I_{1-2}\) to \(I_{2-3}\) under a regime that is the boundary between aperiodic increasing and oscillatory increasing transient regimes; the representative point will move along segment 2 of a phase trajectory of parabolic type with an unstable node at the origin (Fig. 5b). When the current changes from \(I_{2-3}\) to \(I_{3-4}\), when the transient process is oscillatory increasing, the representative point will move along segment 3 of a phase trajectory of the type of an unwinding spiral with an equilibrium state of the type of an unstable focus at the origin (Fig. 5c).

The change of current from \(I_{3-4}\) to \(I_{4-5}\), in the case of an undamped oscillatory regime, will correspond in the phase plane to segment 4 of an ellipse with its center at the origin (Fig. 5d).

In the oscillatory decaying regime, when the current varies within segment 5 from \(I_{4-5}\) to \(I_{5-6}\), in the phase plane we obtain a segment

Fig. 5.

5 of a phase trajectory of the type of a twisting spiral with a stable focus at the origin (Fig. 5d).

Next, to segment 6 of the change in current within the limits from \(I_{5-6}\) to \(I_{6-7}\), with a regime on the phase plane that is the boundary between damped oscillatory and aperiodic regimes, there will correspond segment 6 of a parabolic phase trajectory with a stable node at the origin (Fig. 5e).

Finally, to segment 7, within whose limits the current changes from \(I_{6-7}\) to \(I_{\max}\), with an aperiodic damped regime on the phase plane, there also corresponds segment 7 of a parabolic phase trajectory with a stable node at the origin (Fig. 5g). The segments \(1, 2, \ldots, 7\) thus obtained on the phase plane together constitute a part of the limit cycle of self-oscillation corresponding to the first quadrant. Extending the foregoing to the case of a decrease of the current from \(+I_{\max}\) to 0 and of its change in the negative half-period from 0 to \(-I_{\max}\) and from \(-I_{\max}\) to 0, one can obtain the entire limit cycle (Fig. 5z). Thus, under a piecewise-linear approximation of the magnetization curve, the limit cycle is approximated by segments of parabolas, spirals, and ellipses, which are phase trajectories of possible linear systems of the second order, with a successive change of the equilibrium-state points from an unstable node to an unstable focus, a center, a stable focus, a stable node, and then in the reverse order for the half-period of oscillation under consideration. Parts of these phase trajectories can be constructed from the equations given in [3, 4]. Proceeding from the fact that segment 1 of the limit cycle (Fig. 5) is part of a parabolic phase trajectory with an unstable node and is, therefore, ascending, the following essential conclusion may be drawn: in a self-oscillatory system in which an aperiodic increasing regime is possible (or a regime that is the boundary between an aperiodic increasing and an oscillatory increasing regime), the point of the limit cycle corresponding to the maximum value of

\[ \frac{dI}{dt}, \]

is displaced from the axis

\[ \frac{dI}{dt} \]

in the direction of motion of the representative point; accordingly, in the curve of the transient self-oscillatory process, the inflection point lying on the part of the curve corresponding to the change of \(I\) from 0 to \(I_{\max}\) and from 0 to \(-I_{\max}\) cannot be located on the time axis, and is displaced from it toward the corresponding extreme value of the current.

LITERATURE

1. Gandzha L. I. Physics of oscillations in the system “generator with series motor with independent excitation,” Izvestiya Tomskogo ordena Trudovogo Krasnogo Znameni politekhnicheskogo instituta im. S. M. Kirova, vol. 75, 1953.

2. Gandzha L. I., Potekhin Yu. I. On transient processes and oscillations in the system “generator with series motor with independent excitation,” Izvestiya Tomskogo ordena Trudovogo Krasnogo Znameni politekhnicheskogo instituta im. S. M. Kirova, vol. 76, 1953.

3. Andronov A. A., Khaikin S. E. Theory of Oscillations. ONTI NKTP USSR, 1937.

4. Edited by Polivanov K. M. Physical Foundations of Electrical Engineering. Gosenergoizdat, 1950.