INFLUENCE OF PARAMETERS AND FEEDBACKS ON SELF-OSCILLATIONS OF AN AUTOMATIC CONTROL SYSTEM

M. I. Kutarev

(Presented by the scientific-methodological seminar of the EMF TPI)

Introduction

As the automatic control system under investigation, we consider a generator—motor system widely used in electric-drive practice, with an electric-machine amplifier as the generator exciter (Fig. 1).

The presence in this system, together with the regulating device—the rheostat $RC$ and the control winding of the EMA—$OU_3$, of positive current feedback $OU_1$ and negative voltage feedback $OU_2$ makes it possible automatically to maintain the counter-e.m.f., and, with constant excitation flux, also the speed constant when the load changes, thereby implementing the so-called principle of automatic control with respect to the disturbing action [1, 2].

Fig. 1.

One may indicate three principal internal regularities inherent in this system and determining not only the qualitative aspect of the control process, but also the overall character of its motion.

First, this system, like any automatic-control system having negative feedbacks as an inalienable factor, is prone to self-excitation of oscillations.

Second, positive current feedback determines the presence of the properties of compound systems, consisting in the fact that the series field winding, under certain conditions, creates “negative” resistance, making the equilibrium state of the system unstable and its motion oscillatory [3, 4].

Third, the system under consideration has elements with essentially nonlinear characteristics. A distinctive feature of nonlinear automatic-control systems is the possibility of the existence of periodic regimes, in the present case self-oscillations [2, 5, 6].

Naturally, the self-oscillations arising in a system intended for stabilization of a regime are a harmful, undesirable phenomenon that must be combated. On the other hand, self-oscillations of systems with compound properties can in principle be used for controlling a motor as a function of current, in particular, for contactless reversal of a motor [7, 8]. Hence follows the necessity of investigating the nonlinear automatic-control system in order to determine the parameters, the conditions of self-excitation, and the stability of the self-oscillations arising in it.

In this connection it is appropriate to quote an excerpt from the report of Doctor of Technical Sciences M. A. Aizerman at the Second All-Union Conference on the Theory of Automatic Control: “So long as the task consisted in preventing oscillations, decisive importance belonged to the problem of stability conditions as conditions under which oscillations cannot arise. But with the development of technology it became clear that a system can be controlled, adjusted, and tuned only if one knows how to control the undamped oscillations that arise, how at will to excite or suppress them, and how arbitrarily to change their parameters.”

At present, by means of special methods [5, 6, 9, 10, 11], a considerable number of problems connected with determining and investigating self-oscillations of nonlinear systems have been solved.

As for generator–motor systems using various kinds of amplifiers, including EMU, for purposes of control and regulation, no detailed investigation with such a formulation of the problem has been carried out. As a rule, the system was linearized and investigated from the standpoint of the quality of regulation and stability conditions.

The present article gives some results which, in the author’s opinion, are the most important, of an investigation of the influence of linear and nonlinear parameters, feedback gain coefficients, and the load on the motor shaft on the parameters of self-oscillations and on the conditions for their self-excitation and suppression.

Equation of Motion of the System

The system under investigation is, in the general case, described by a nonlinear differential equation of high order. Taking into account the absence of regular methods for solving such equations, a method of investigation was adopted that makes it possible to solve the posed problem most fully. This method is based on the introduction of certain simplifying assumptions that lower the order of the equation, and on the use, for its solution, of the method of qualitative integration by constructing integral curves in the phase plane. After the basic regularities and relations inherent in the system had been identified, corrections were introduced for certain parameters that had been neglected at the first stage.

Taking into account the adopted assumptions, the equation of the reduced system was derived in the following form:

\[ T_v B \frac{d^2 n}{dt^2} + (T_v + B)\frac{dn}{dt} + n = \frac{K}{R_v C_e}\, \varphi\!\left[ i_z+ \left( \frac{r_1}{r_{oy_1}} - R_o K_1 \right) \left( \frac{C_e B}{R}\frac{dn}{dt} + I_c \right) - K_1 C_e n \right] - \frac{R}{C_e} I_c, \tag{1} \]

where:

\[ T_v=\frac{L_v}{R_v} \]
— the time constant of the generator excitation circuit, sec;

\[ L_v,\ R_v \]
— the inductance, assumed constant, and the resistance of this circuit;

\[ B=\frac{G D^2 R}{C_e C_m \cdot 375} \]
— the electromechanical time constant of the motor, sec;

\[ R \]
— the total resistance of the power circuit;

\[ G D^2 \]
— the flywheel moment of the rotating masses, referred to the motor shaft;

\[ C_e,\ C_m \]
— constant coefficients determined by the motor design;

\[ n \]
— motor speed, rpm;

\[ i_z \]
— the constant current in the setting winding \(OU_z\);

\(K\) — a coefficient characterizing the angle of inclination of the generator magnetization characteristic;

\(r_1\) — the resistance of the power circuit from which the voltage drop is taken in order to provide positive current feedback;

\(r_{oy_1}\) — the resistance of winding \(OY_1\);

\(R_{\theta}\) — the resistance of the motor armature;

\[ K_1=\frac{R_{n_1}}{R_n r_{oy_2}+R_{n_1}(R_n-R_{n_1})} \]

— a coefficient characterizing the negative voltage feedback;

\(R_n\) and \(R_{n_1}\) — the resistance of the potentiometer and of that part of it from which the voltage is supplied to winding \(OY_2\);

\(r_{oy_2}\) — the resistance of control winding \(OY_2\);

\(I_c\) — the static motor current characterizing the load.

In what follows we shall regard the load as constant, independent of the speed (Fig. 2), but dependent on its sign. According to the load equation,

\[ I_c=I_{c_1}\operatorname{sign} n \tag{2} \]

we have \(I_c=I_{c_1}\) for \(n>0\) and \(I_c=-I_{c_1}\) for \(n<0\).

The nonlinear function appearing on the right-hand side of the equation represents the dependence of the EMU e.m.f. on the resultant current of the control windings, when their numbers of turns are the same, and is characterized by the graph in Fig. 3.

The investigation of the motions of the system according to equation (1) is conveniently carried out in two stages: first consider the properties of the system in the absence of negative voltage feedback, and then take its influence into account for each of the particular cases analyzed at the first stage.

Self-oscillations of the system for \(K_1=0\) and \(i_3=0\). This particular case has been considered in sufficient detail in the author’s article, published in the present volume of the collection Izvestiya TPI, “On periodic regimes of a generator–motor system with an electromechanical amplifier.” Since in this case the nonlinear term of equation (1) depends only on

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

the integral curves in the phase plane can be constructed by Liénard’s method [6] using the isocline of horizontal tangents. In the cited work, by approximating the EMU magnetization curve according to [12] by a third-degree polynomial, an analytic expression was obtained for the isocline of horizontal tangents, and, together with this, the possibility of studying the influence of the parameters.

Fig. 2.

Taking into account that a third-degree polynomial gives a considerable error in the approximation, in the present work we shall show an exact graphical method for constructing the isocline of horizontal tangents, making it possible to reveal more deeply the physics of the self-excitation of self-oscillations and to take hysteresis into account. On the other hand, according to the equation of mechanical equilibrium of the system,

\[ I-I_c=\frac{C_e B}{K}\frac{dn}{dt}. \tag{3} \]

the phase plane can be considered directly in the axes of the variables \(I\) and \(n\), which completely characterize the motor as an object of automatic control.

To explain the method of graphical construction of the isocline of horizontal tangents on the phase plane with axes \(n, I\), let us first consider equation (1) for \(I_c=0\). Substituting from equation (3)

\[ \frac{dn}{dt}=\frac{R}{C_e B}I \]

into equation (1) and eliminating time, we obtain, for

\[ \frac{dI}{dt}=\theta=0, \]

the equation of the isocline of horizontal tangents:

\[ n=a[\varphi(bI)-cI], \tag{4} \]

where

\[ a=\frac{K}{R_v C_e};\qquad b=\frac{r_1}{r_{oy_1}};\qquad c=\frac{R R_v}{K}\left(1+\frac{T_v}{B}\right). \]

From equation (4) it is clear that the isocline of horizontal tangents can be constructed when \(\varphi(bI)\) is specified graphically.

In the second quadrant (Fig. 4), neglecting hysteresis, the nonlinear function \(\varphi(bI)\) is plotted in the axes \(e_{\mathrm{emf}}\) and \(i_{oy}\), and in the first quadrant—in the axes \(e_{\mathrm{emf}}\) and \(I\), where the value \(b=b_1\) corresponds to the curve \(\varphi_1(I)\), \(b=b_2\) corresponds to \(\varphi_2(I)\), etc. Now drawing the straight line \(c_1 I\) and multiplying, according to equation (4), the difference of the ordinates enclosed between this straight line and, for example, the function \(\varphi_1(I)\) in Fig. 4, we find the isocline 8 corresponding to the given parameters as

\[ n=f(a_1,b_1,c_1,I). \]

Fig. 3.

It is not difficult to see that, given an exact expression for the nonlinear characteristic, this method makes it possible quite simply and clearly to investigate the influence of the parameters on the shape of the isocline of horizontal tangents, and hence also of the phase portrait characterizing the motion of the system under study. Starting from Lienard’s method of construction [6], it can be said at once that isocline 9, for \(a=a_1\), \(b=b_1\), and \(c=c_1\), corresponds to a phase portrait characterizing a self-oscillatory character of motion with a smaller value of the amplitude of the current and speed than isocline 8 for the values of the coefficients \(a=a_1\), \(b=b_1\), and \(c=c_1\).

Isocline 10, in contrast to the first two, corresponds to a stable equilibrium position and a damped character of motion in the system. Leaving, for example, the coefficients \(b\) and \(c\) constant and changing only \(a\), one can thereby change to a considerable extent the form of the isocline and, consequently, the form of the self-oscillations. It is clear that the smaller the coefficient \(a\), the closer in form the self-oscillations are to harmonic ones, and, conversely, the larger \(a\) is, the closer the form of the oscillation is to relaxation oscillations. The phase portraits and oscillograms of the processes corresponding to various combinations of parameters for \(I_c=0\) were presented in the author’s above-mentioned work; therefore there is no need to repeat them, especially since the physics of the process and the form of the phase portraits corresponding to this particular case remain the same.

However, when there is a load on the motor shaft and the phase plane in the variables \(n\) and \(I\) is considered, it becomes possible to obtain several distinct portraits, making it possible to reveal more deeply the physics of the self-oscillations.

From equation (1), taking into account equalities (2) and (3), we find the equations of the isoclines of horizontal tangents:

\[ n=\frac{K}{R_{\mathrm{v}}C_e}\left[\varphi\left(\frac{r_1}{r_{\mathrm{oy}_1}}I\right)-\left(\frac{T_{\mathrm{v}}}{B}+1\right)\frac{RR_{\mathrm{v}}}{K}I\right] +\frac{T_{\mathrm{v}}R}{BC_e}I_c; \tag{5} \]

\[ n=\frac{K}{R_{\mathrm{v}}C_e}\left[\varphi\left(\frac{r_1}{r_{\mathrm{oy}_1}}I\right)-\left(\frac{T_{\mathrm{v}}}{B}+1\right)\frac{RR_{\mathrm{v}}}{K}I\right] -\frac{T_{\mathrm{v}}R}{BC_e}I_c. \tag{6} \]

It follows from these equations that, for \(I_c\ne0\), there exist two isoclines of horizontal tangents having the form of the isocline for \(I_c=0\) and shifted relative to it by the amount \(\pm\dfrac{T_{\mathrm{v}}R}{BC_e}I_c\).

Let us note that for \(I_c=0\), according to equation (1), in the equilibrium position of the system \(n=0\), and, taking equation (3) into account, \(I=0\).

Consequently, all the isoclines (5), including the isoclines of horizontal tangents, converge on the phase plane at a “singular point” lying at the origin of coordinates.

In the case \(I_c\ne0\), in the equilibrium position, according to equation (3), \(I=I_c\), and from equation (1) we obtain:

\[ n=n_c=\frac{K}{R_{\mathrm{v}}C_e}\varphi\left(\frac{r_1}{r_{\mathrm{oy}_1}}I_c\right) -\frac{R}{C_e}I_c. \tag{7} \]

Fig. 4.

The last equality is the equation of the static mechanical characteristic of the system.

Substituting \(I=I_c\) into equation (5) and \(I=-I_c\) into equation (6), we obtain \(n=n_c\) and \(n=-n_c\). This means that the “singular point” lies at the intersection of the mechanical static characteristic (curve 3 in Fig. 5) and the isocline of horizontal tangents corresponding to the given parameter value. Hence it is clear that the construction of the phase portrait by Lienard’s method must be carried out in the plane \(n>0\) relative to the “singular point” \(O_1\) for \(I=I_c\) and \(n=n_c\), and in the plane \(n<0\) relative to \(O'_1\) for \(I=-I_c\) and \(n=-n_c\), according to the isoclines of horizontal tangents corresponding to these points, in this case curves \(1\) and \(1'\) in Fig. 5. This phase portrait is characterized by curve \(a\). The same figure also gives phase portraits corresponding to the most characteristic types of motion of the system with an unchanged value of the parameters and with different magnitudes of the load.

In order not to clutter the drawing, the isoclines corresponding to another value of \(I_c\) have been omitted, and only the arrangement of the “singular points” and the corresponding phase portraits is shown. Comparing curve \(a\) with curve \(d\), constructed from isocline 2 at \(I_c=0\), we see that, in the presence of a load on the motor shaft, the maximum current deviation during self-oscillations is smaller, while the velocities are larger. When the sign of the velocity changes, curve \(a\) undergoes a sharp change, characterizing a sharp increase in the current and, consequently, in the derivative of the velocity.

For \(I_{c2}>I_{c1}\) we obtain the “singular points” \(O_2\) and \(O_2'\) and the phase portrait \(b\), characterizing a self-oscillatory regime of the system with an even sharper jump in current when the sign of the velocity changes.

Fig. 5.

In addition, this regime is characterized by the fact that when the current changes from point 1 to point 2, i.e., until the current \(I\) reaches the value \(I_{c2}\), the motor is motionless. With a further increase in the current, the self-oscillations may become one-sided, without a change in the sign of the motor velocity. This regime is characterized by curve \(v\), constructed with respect to \(O_3\). It should be noted that, for a certain value of the parameters, there exists in the system a certain critical load current \(I_k\), at which the isocline of horizontal tangents has a maximum (point \(K\)). For \(I_c<I_k\) the equilibrium position (the “singular point”) is unstable, and the regime of the system is self-oscillatory. For \(I_c>I_k\), the “singular point,” for example \(O_4\), lies entirely in the decremental region, giving rise to a stable equilibrium position and, consequently, to steady motion at \(I=I_c\) and \(n=n_c\). The last two regimes may occur both in the first and in the third quadrant of the phase plane, depending on the sign of the disturbance applied to the system. In the electric-drive laboratory of TPI, not only the regimes obtained have been confirmed experimentally, but also the influence of the load on the parameters of the self-oscillations. Here we shall confine ourselves to stating the final results, namely that a constant load, causing a complex form of self-oscillations, decreases the current amplitude and increases the velocity amplitude and the period of the self-oscillations.

Characteristic are two-sided self-oscillations with stopping of the motor and one-sided self-oscillations with respect to \(n\), with and without a change in the sign of the current.

Case of hard self-excitation of self-oscillations

Up to now, the no-load characteristic has been assumed rectilinear at the origin of coordinates, and therefore, if the equilibrium-position point was unstable, then self-oscillations were excited in the system under an arbitrarily small disturbance. Such a regime is called a soft regime of self-excitation. In reality, at the origin of coordinates the characteristic has considerable curvature (curve 1 in Fig. 3), and therefore, for a certain combination of parameters it may turn out that

Fig. 6.

the straight line \(c_1 I\) has two points of intersection, \(A\) and \(B\), with the curve \(\Phi_2(I)\) (Fig. 4), and the isocline of horizontal tangents has the form of curve 1 (Fig. 6,a). In this case the “singular point” lies in the decremental region (the zone hatched between the points \(B\) and \(B_1\)), characterizing a stable state of the system. However, if such a disturbance is applied to the system that the point representing the system goes beyond the closed curve 3, called the unstable limit cycle of self-oscillations [5], then it will subsequently begin to move along an unwinding spiral, winding from within onto the stable limit cycle of self-oscillations (curve 2). Such a regime of excitation of self-oscillations is called hard and is the most characteristic one for systems in which the nonlinearity is due to the magnetization characteristic of an electric machine.

Using the graphical method of constructing the isocline of horizontal tangents, let us present the construction of the limit cycle of self-oscillations with allowance for the presence of hysteresis in the nonlinear characteristic (Fig. 6,b). For a qualitative assessment of the influence of hysteresis on the parameters of the established self-oscillations, the hysteresis loop has been taken considerably wider than the actual one.

Let the point representing the motion of the system (Fig. 6,b), under an external action, be at point \(A\) of the phase plane. Then, according to curve 2 (Fig. 3), when the current changes from \(+I_{\max}\) to \(-I_{\max}\), the integral curve is constructed with respect to isocline 2 (Fig. 6,b), and when the current changes from \(-I_{\max}\) to \(+I_{\max}\), it is constructed with respect to isocline 3.

Comparing the obtained phase portrait (curve 4) with the limit cycle (curve 5) constructed with respect to isocline 1 (without taking hysteresis into account), we see that even an intentionally widened hysteresis loop does not exert a substantial influence on the parameters of the established self-oscillations. Owing to a somewhat greater increase of the maximum deviation of the velocity as compared with the current, the phase portrait is somewhat stretched along the \(n\)-axis, determining a small increase in the period of oscil-

tions. However, taking into account that, for the actual width of the hysteresis loop, these changes fall within the limits of the errors allowed both in measuring the parameters and in constructing the phase portrait, they may be neglected.

Self-oscillations of the system for \(K_1=0,\ i_3\ne 0\) and \(I_c=0\)

Taking into account the notation of equality (4), from equation (1) we find the equation of the isocline

\[ n=a\{\varphi[b(I+I_3)]\}-cI, \tag{8} \]

where \(I_3=i_3\frac{1}{b}\) is a certain current of the main circuit equivalent to the constant control current in \(OU_3\) with the same number of turns in \(OU_1\) and \(OU_3\).

In Fig. 7,a are shown plots of the nonlinear function for \(I_3=0\), \(I_3=I_3'\), and \(I_3=I_3''>I_3'\). Carrying out, according to equation (8), graphical constructions analogous to those in Fig. 4, we obtain isocline 4 (Fig. 7,b), corresponding to curve 1 (Fig. 7,a), and isocline 6, corresponding to curve 2. In the equilibrium position we have: \(I=0\) and \(n=n_3=a\varphi(i_3)=a\varphi_1(I_3)\). This means that the “singular points” on the phase plane for \(I_3=I_3'\) are point \(O_1'\), and for \(I_3=I_3''\)—point \(O_2'\).

Fig. 7.

It is not difficult to see that if point \(k_1\) (in Fig. 7,a), which determines, according to equation (8), the maximum value of the difference between the ordinates of the curve \(\varphi_1(I+I_3')\) and the straight line \(cI\), lies in the plane \(I>0\), then the corresponding point \(K_1'\) on the phase plane also lies at \(I>0\). In this case point \(O_1'\) [9] is a point of unstable equilibrium, determining the existence of self-oscillations in the system (curve 5 in Fig. 7,b). If \(I_3''>I_3'\), and point \(k_2\) and the corresponding \(k_2'\) lie in the plane \(I<0\), then point \(O_2'\) becomes a point of stable equilibrium, determining the motion of the system for \(I=0\) and \(n=n_3''=a\varphi(I_3'')\). The phase portrait in this case has the form of curve 7 in Fig. 7,b. For some value \(I_3'<I_3'''<I_3''\), self-oscillations may

will turn from two-sided, with a change in sign of the speed and current (curve 5 in Fig. 7,b), into one-sided with respect to speed, but, as on curve 5, with equal current amplitudes in the positive and negative half-periods.

Self-oscillation of the system for \(K_1=0,\ i_z\ne 0,\ I_c\ne 0\)

For this case the isocline equations have the form:

\[ n=a\{\varphi[b(I+I_z)]-cI\}+c_1 I_c, \tag{9} \]

\[ n=a\{\varphi[b(I+I_z)]-cI\}-c_1 I_c, \tag{10} \]

where

\[ c_1=\frac{T_b R}{BC_e}. \]

The method for constructing these isoclines remains analogous to the case \(I_c\ne 0\) and \(i_z=0\), with the sole difference that, instead of the initial isocline described by equation (4), one of the isoclines according to equation (8) is taken.

The equilibrium condition of the system is still found from equations (3) and (1) and is characterized by: \(I=I_c\) and

\[ n=n_c=\frac{K}{R_\omega C_e}\,\varphi\left[\frac{r_1}{r_{oy_1}}(I_c+I_z)\right]-\frac{R}{C_e}I_c. \tag{11} \]

Upon substituting \(I=I_c\), equation (9) becomes the equation of the static mechanical characteristic (11). This means that the “singular points” lie at the intersection of the static mechanical characteristic (curve 2 in Fig. 8,a), constructed according to equation (11), and the isocline of horizontal tangents corresponding to the given values of \(I_z\) and \(I_c\). Let isocline 4 in Fig. 7,b be the initial one for equations (9) and (10). Constructing it without changes in Fig. 8,a (curve 1), we see that for \(I_z=I'_z\) and \(I_c=0\) the “singular point” lies at the intersection of the isocline with curve 2 and lies on the \(n\)-axis at \(I=0\).

Fig. 8.

If a load is applied to the motor shaft such that \(I_c=I_{c1}\), then for \(n>0\) the phase trajectory is constructed with respect to point \(O_1\) along isocline 3, shifted relative to the initial one by \(+c_1 I_{c1}\), while for \(n<0\) it is constructed with respect to \(O'_1\) along isocline 5, shifted relative to the initial one to the left by the same amount. Comparing the phase portraits—curve 5 in Fig. 7,b for \(I_z=I'_z\) and \(I_c=0\), and curve 6 in Fig. 8,a for \(I_z=I'_z\) and \(I_c=I_{c1}\)—we see that in the latter case the self-oscillations have become one-sided.

relative velocity with the current amplitude in the positive half-period, greater than in the negative one. As the load is increased, when \(I_{c_2}>I_k\), the “singular point” \(O_2\) becomes a point of stable equilibrium. In this case the phase trajectory has the form of a twisting spiral (curve 7 in Fig. 8,a), characterizing an oscillatory damped transient process and, after its completion, the established motion of the system at \(I_{c_2}=\mathrm{const}\) and \(n_{c_2}=\mathrm{const}\). Figure 8,b gives oscillograms confirming the results obtained.

Self-oscillation of the system in the presence of negative voltage feedback

Since the nonlinear term of equation (1) in this case depends on the independent variable \(n\) (for \(K_1\ne0\)) and on its derivative \(\dfrac{dn}{dt}\), Liénard’s method for constructing the phase portrait of the system becomes unsuitable. This circumstance makes it necessary to fill the entire phase plane with isoclines [5]. For example, for an isocline with angular coefficient of the tangents to the integral curves equal to \(\theta\), from equation (1), taking into account equation (3), for the phase plane in the axes of the variables \(n\) and \(I\) we have:

\[ \theta=\frac{dI}{dn}= \frac{ \dfrac{K}{K_{\delta}C'_e}\,\varphi \left[ i_3+\left(\dfrac{r_1}{r_{0'1}}-R_0K_1\right)I-C_eK_1n \right] -\left(\dfrac{T_v}{B}+1\right)\dfrac{R}{C'_e}I +\dfrac{T_vR}{BC'_e}I_c-n }{ \dfrac{T_v}{B}\left(\dfrac{R}{C'_e}\right)^2(I-I_c) }. \tag{12} \]

To obtain the equation of an isocline it is necessary, at \(\theta=\mathrm{const}\), to solve equation (12) with respect to \(I\) or \(n\). Taking into account that the order of the isocline equation both with respect to \(I\) and to \(n\) is determined by the order of the equation approximating the magnetization curve, and that, to simplify the computational operations, it should be as low as possible, we choose, as the analytical expression of the nonlinear function, one of the equalities given in [13], which for our case has the following form:

\[ e_{\mathrm{EMU}}=\frac{i_p}{A_1+B_1i_p}, \tag{13} \]

where \(A_1\) and \(B_1\) are constant coefficients determined from the graph of the nonlinear function, and \(i_p=i_1+i_3-i_2\) is the resultant current of the control windings when they have an equal number of turns.

This expression, while approximating the magnetization curve with sufficient accuracy and defining an equation of second-order isoclines, has only the drawback that it has no inflection point, i.e., it is not symmetric with respect to the origin of coordinates (as a complete cubic is) when \(i_p\) changes from \(i_{p\max}\) to \(-i_{p\max}\). However, this equality can be used successfully if, in constructing the isoclines, the condition of positivity of the argument of the nonlinear function is strictly observed; on the phase plane (Fig. 9,a) this is expressed by a certain separating straight line \(l\):

\[ n=\frac{ i_3+\left(\dfrac{r_1}{r_{0'1}}-R_0K_1\right)I }{ C_eK_1 }, \tag{14} \]

passing through the first and third quadrants. Filling with isoclines the region of the phase plane located above this straight line in the [[unclear: word beginning “na-”]]

of the direction of traversal of the phase plane counterclockwise, and, taking into account that the isoclines for equal \(\theta\) are symmetric with respect to the “singular points,” one can, without resorting to repeated calculations for \(-i_p\), fill in with them the remaining part of the phase plane.

Fig. 9.

Figure 9 gives, by means of this method, the construction of phase portraits representing the most characteristic types of motion of the system for various combinations of parameters and of load on the motor shaft. Under each phase portrait an oscillogram is given, confirming the presence in the system of the obtained form of motion.

If one takes into account that, for the selected scales, harmonic oscillations of \(n\) and \(I\) correspond in the phase plane to a limit cycle in the form of a circle, and that the scales for all cases in Fig. 9 have been chosen

identical, then from the deformation of the form of the phase portrait one may judge the influence of one or another factor on the form of the self-oscillations in time, and from the dimensions of the limit cycles—the amplitude. Since quantities proportional to acceleration are plotted along the ordinate axis of the phase plane, and the speed itself along the abscissa axis, the deformation of the phase portrait makes it possible to judge the change in the period of the self-oscillations. Comparing, for example, the phase portraits (Fig. 9, a and b) constructed for different values of \(K_1\), we see that a certain increase in the current amplitude and a more substantial decrease in the speed amplitude appreciably deform the phase portrait, stretching it along the \(I\) axis and thereby causing an increase in the frequency of the self-oscillations. The form of the self-oscillations remains close to harmonic. When the resistance \(r_1\) is decreased, the equilibrium position may become stable (Fig. 9, v), characterizing an oscillatory damped nature of the motion. In the presence of a setting winding, the equilibrium-position points shift to the right along the \(n\) axis, determining two-sided self-oscillations asymmetric with respect to \(n\) (Fig. 9, g) or one-sided self-oscillations with respect to \(n\) (Fig. 9, d) for the same value of the current amplitude in the positive and negative half-periods. With a further increase in \(i_z\), the equilibrium position (Fig. 9, e) becomes stable, and the character of the system motion is oscillatory damped. In the presence of a load on the motor shaft, for example \(I'_c\), and with \(i_z = 0\), the self-oscillations again become symmetric with respect to \(n\) and \(I\) (Fig. 9, zh), but with a characteristic abrupt change in the current when the sign of the speed changes, determining a complex form of self-oscillations. In the case \(I''_c > I'_c\) (Fig. 9, z) the phase portrait characterizes two-sided self-oscillations with stopping of the motor. With a further increase in the load (Fig. 9, i) the equilibrium position of the system becomes stable, determining its motion at \(I = I_c\) and \(n = n_c\). And, finally, in the presence of current in the setting winding and of a load on the motor shaft (Fig. 9, k), the self-oscillations of the system become asymmetric both with respect to \(n\) and to \(I\), causing, in the general case, current and speed amplitudes in the positive half-period greater than in the negative one. Figure 9, k shows a phase portrait characterizing the most interesting self-oscillatory regime of the system, in which the motor performs one-sided impact-like motions.

In conclusion, we note that under all possible conditions for the existence of self-oscillations, their frequency increases when the gain coefficient of the negative voltage feedback is increased and decreases when the load on the motor shaft is increased.

Conclusion

Taking into account that all the theoretical arguments presented in the work are fairly well confirmed qualitatively by the experimental data, one may draw the general conclusion that the adopted method for investigating the motions of the system on the phase plane is correct and sufficiently effective.

For the quantitative determination of the parameters of self-oscillations, at present there exist only approximate methods [2, 11], which give a satisfactory result in the case of self-oscillations close to harmonic. The error introduced into the calculations by neglecting small parameters for systems with machines of normal design is small and lies within the accuracy of measurement of the “basic” parameters and of graphical constructions [9].

It follows from this that the chosen method is also suitable for an approximate quantitative estimate of the parameters of self-oscillations in the case of their complex form, caused, for example, by the presence of a constant load on the motor shaft.

Thus, the approach to analyzing the behavior of a nonlinear automatic-control system under various combinations of its parameters and feedbacks from the standpoint of the theory of nonlinear oscillations makes it possible not only to explain the physical nature of the type of motion obtained as a result of experiment, including self-oscillations, but also to predict the possibility of their existence without a preliminary experiment. Thus, as a result of the investigation, self-oscillations have been found and explained which are close in form to harmonic and relaxation oscillations, asymmetric with respect to velocity or asymmetric with respect to both independent variables, for various combinations of the current in the setting winding of the EMU and the load on the motor shaft, and, finally, self-oscillations with stopping of the motor.

The most interesting and valuable part of the work is the investigation of the influence of a constant load on the motor shaft on the character of the system’s motion; as a result, the property of the system has been determined whereby it performs self-oscillations under a load smaller than a certain critical value and operates with a constant motor speed under a larger load, and also realizes impulse modes of motor operation.

In this connection, the results of the investigation are of undoubted interest both for the theory of automated electric drive and for its practice, for example, for providing motor control directly as a function of the load without the use of relay-contactor equipment.

References

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