On Periodic Modes of the Generator–Motor System with an Electric Machine Amplifier

M. I. Kutarev

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

Introduction

At present, the generator–motor system with an electric machine amplifier has found wide application in complex regulated electric drives. Electric machine amplifiers are used either as exciters of generators in high-power drives, such as the main drive of heavy metalworking machine tools and rolling mills, or as generators in low-power installations, for example in the drives of light metalworking machine tools, in feed drives, etc. The electric machine amplifier with a transverse field, having comparatively small winding time constants and a large amplification factor, has become an integral part of electric machine automation, making it possible to regulate the speed of the drive motor over a wide range, to maintain it constant with high accuracy, to maintain a constant or prescribed ratio of the speeds of different parts of a mechanism, thereby ensuring the necessary technological requirements, and also to obtain optimal transient processes of the electric drive in the sense of reducing their duration and limiting surges of current, speed, etc., thus increasing the productivity of mechanisms.

Automatic control systems employing electric machine automation, as it is used and developed, are becoming the most effective in terms of ensuring the required technological process and more flexible in terms of obtaining all possible types of motion of actuating mechanisms.

With the development of complex automatic control systems, the problems of stability and quality of regulation have become, as never before, urgent. In the linear theory of automatic control, major results have been achieved. However, taking into account the fact that in real systems the characteristics of individual links are, in the general case, nonlinear, and that nonlinear elements have increasingly been introduced into automatic control systems—for example, to improve dynamic properties or to limit (cut off) current, voltage, speed, etc.—linear theory becomes powerless to explain and, still more, to give an answer satisfactory for practice concerning the motions arising in these systems. Very often, therefore, automatic control systems calculated by linear methods behave unstably during adjustment, generate oscillations, and the selected parameters have to be changed. Therefore, the problem of solving problems of nonlinear automatic control systems has become especially important and, in particular, the problem of determining periodic modes.

History of the Question and Formulation of the Problem

In the field of oscillations in electrical machines, major studies were first carried out by Academician M. V. Shuleikin, who discovered the possibility of the occurrence and existence of oscillations in electrical circuits containing no capacitors, and gave a sufficiently complete mathematical investigation of the occurrence and existence of oscillations in systems with DC commutator machines. However, by treating the problem as linear, he could not give a correct assessment of the physical essence of the oscillations arising in these systems. A number of investigators working in this field [8, 9, 10, 11, 12, and 13] solved the problem posed either by linear methods, following M. V. Shuleikin, or else did not carry out any mathematical analysis at all, confining themselves to a description of the systems in which the oscillations were observed.

Fig. 1.

Among the various systems with DC commutator machines considered by the authors of the above-mentioned literature, the most general, from the standpoint of explaining the nature of the oscillations that arise, is the generator—motor system [20]. The main feature of this system is the presence of a series excitation winding of the generator (Fig. 1), which under certain conditions creates a “negative” resistance, i.e., an element that causes the excitation of oscillations, and of the nonlinear magnetization characteristic of the electrical machine, which limits the amplitude of the oscillations. And only the consideration of this system as a nonlinear self-oscillatory one, as presented in [6] and similarly in [5], gives a correct idea of the nature of the oscillations that arise. But both in [6] and in [5] only the condition of self-excitation and existence of self-oscillations is considered, while the question of the influence of the system parameters and of the load on the motor shaft on the magnitude of the amplitude, frequency, and form of the self-oscillations remains open.

More profound investigations carried out in the electric-drive laboratory of the Tomsk Polytechnic Institute and published in [20] showed that, in the given system, self-oscillations can exist only when the static torque on the motor shaft is less than a certain critical torque determined by the parameters and the form of the nonlinear magnetization characteristic of the generator; whereas, when the static torque is greater than the critical torque, the self-oscillations disappear and the system operates stably. These investigations and certain indications in the literature [7] provide a basis, after studying in detail the properties of the given self-oscillatory system and applying the means of electric-machine automation, for using these properties for contactless reversal of the drive in modern automatic-control schemes of mechanisms with reciprocating motion, since the given system (Fig. 1) cannot be applied directly to a concrete mechanism because of a number of inherent shortcomings:

1) large current surges during self-oscillations, when \(R_{\mathrm{я}} = 0\) and \(R_{\mathrm{ш}} = 0\);

2) absence of the possibility of regulating the motor speed over a wide range in steady-state operation at a static torque greater than the critical torque, while preserving self-oscillations for reversal of the motor when the load is removed;

3) regulation of the motor speed is associated with large losses of electrical energy in the active resistance \(R_{\mathrm{я}}\) or \(R_{\mathrm{ш}}\), especially in high-power machines.

These difficulties can be eliminated by using means

electrical-machine automation for controlling self-oscillations and the speed of the motor in the steady-state mode.

Consider the circuit shown in Fig. 2, which represents a widely used system for automatically maintaining the constancy of the motor speed, provided by the winding of positive current feedback \(OU_1\) and the winding of negative voltage feedback \(OU_2\). The motor speed is regulated by the control winding \(OU_3\). With a change in the magnetomotive force of the control winding \(OU_3\), the voltage of the EMA, the excitation current of the generator, the generator voltage, and consequently the motor speed change. If, in this circuit, only one positive current feedback is left connected, then we arrive at a self-oscillatory system similar to the system shown in Fig. 1, with the sole difference that instead of a generator with a series excitation winding, a generator with a parallel winding is used, connected through an amplifier to the voltage drop from the current in the power circuit.

Fig. 2.

On the other hand, such a system is a special case of operation of the system shown in Fig. 2, when the slider of the potentiometer \(RC\) is set in a position in which the current in the control winding \(OU_3\) is equal to zero, and the feedbacks are chosen so that the ampere-turns of the winding \(OU_1\) are significantly greater than the ampere-turns of the winding \(OU_2\), and the latter can be neglected. Consequently, finding and investigating periodic regimes, in particular self-oscillations, in a generator—motor system with an EMA in the presence of positive current feedback are necessary both from the standpoint of using the self-oscillations of the given system for contactless reversal of the motor, since in this case it is necessary to know the amplitude and frequency of the self-oscillations as functions of the system parameters in order to be able to calculate the reversal time and current surges, and from the standpoint of correctly choosing the parameters and the magnitude of the feedbacks to ensure the stability of the system in its usual configuration.

Equation of motion of the system and method of investigation

Let us write the equations of electrical and mechanical equilibrium of the system shown in Fig. 2, in the presence of only one positive current feedback. Taking the inductances of the excitation windings, as well as the inductances and resistances of the machine armatures, to be constant and independent of the current flowing in them, we shall have the following system of equations:

\[ e_{\mathrm{e}} = L_{\mathrm{B}} \frac{di_{\mathrm{B}}}{dt} + r_{\mathrm{B}} i_{\mathrm{B}}; \tag{1} \]

\[ e_{\mathrm{e}} = \varphi(i_{oy}); \tag{2} \]

\[ e_{\mathrm{G}} = L \frac{dI}{dt} + RI + e_{\mathrm{D}}; \tag{3} \]

\[ e_{\mathrm{G}} = f(i_{\mathrm{B}}); \tag{4} \]

\[ Ir_1 = L_{oy}\frac{di_{oy}}{dt} + r_{oy}i_{oy}; \tag{5} \]

\[ M=\frac{GD^2}{375}\frac{dn}{dt}+M_{st}; \tag{6} \]

\[ M=C_M I; \tag{7} \]

\[ e_D=C_E n, \tag{8} \]

where \(e_{\varepsilon}\) is the e.m.f. of the EMA;

\(i_{oy}\) is the excitation current of the EMA flowing in winding \(OY_1\);

\(e_G\) is the e.m.f. of the generator;

\(i_v\) is the generator excitation current;

\(I\) is the current in the power circuit;

\(n\) is the motor speed;

\(e_D\) is the e.m.f. developed by the motor when rotating in the magnetic field;

\(C_E, C_M\) are coefficients determined by the design of the motor;

\(M\) is the torque developed by the motor on the shaft;

\(M_{st}\) is the torque of the resistance forces applied to the motor shaft;

\(GD^2\) is the flywheel moment of the rotating masses, referred to the motor shaft;

\(L, L_v, L_{oy}, R, r_v, r_{oy}\) are the inductances and resistances of the power, excitation, and control circuits;

\(r_1\) is the resistance of the positive current feedback.

As is seen from the equations presented, the system under study has two nonlinear elements in the form of the magnetization characteristics of the EMA and of the generator, represented by equations (2) and (4).

Taking into account that nonlinear differential equations have no general methods of solution, and that the equation describing the state of the system under study with respect to any variable, as is seen from equations (1)—(8), will have order higher than the second, self-oscillations can most simply be found and investigated by approximate methods developed for nonlinear automatic-control systems of arbitrary order by L. S. Goldfarb [16] with the use of the frequency stability criterion, or by E. P. Popov [17] with the use of the Mikhailov stability criterion, and based on the harmonic-balance method of N. M. Krylov and N. N. Bogolyubov [14, 15]. The application of these methods is based on the assumption that the form of the self-oscillations is close to harmonic and remains so under sufficiently broad variations of the system parameters. Not yet knowing what form the self-oscillations have in the system under study, we are forced to abandon these methods and choose a more refined method that makes it possible to find and investigate all types of motion possible in this system.

Taking into account the smallness of some terms in the system of equations (1)—(8), it is possible to reduce the equation of motion of the system to a second-order equation and, using the phase plane for the investigation, to solve the posed nonlinear problem in its entirety, as A. A. Andronov did [1, 21], i.e., not only to find the existence of self-oscillations in the system and determine their stability, but also to find, as functions of the system parameters, of the load on the motor shaft, and of the type of nonlinear characteristics, the conditions for self-excitation and the form of the self-oscillations and, finally, to investigate the stability of the equilibrium state and of the steady-state motion of the system.

We shall dwell on the latter method of investigation, for which purpose we adopt the following simplifications:

1) we neglect the inductances of the power circuit, the short-circuit circuit, and the control winding of the EMA, since in machines of normal design

they are tens of times smaller than the inductance of the generator’s shunt field winding;

2) we neglect hysteresis, armature reaction, eddy currents, and the mutual inductance of the machine windings, solving the problem for the time being in a first approximation.

Operation of the system for \(M_{st}=0\)

Taking into account the assumptions adopted and solving the system of equations (1)—(8) with respect to \(n\), after simple transformations we obtain the equation of motion of the system:

\[ \frac{d^{2}n}{dt^{2}}+ \left\{ \frac{1}{B}+ \frac{f'(i_{B})}{f(i_{B})T_{B}R_{B}I} \left[RI_{B}i_{B}-f(i_{B})\varphi(i_{oy})\right] \right\} \frac{dn}{dt} + \frac{f'(i_{B})i_{B}}{f(i_{B})T_{B}B}\,n=0, \tag{9} \]

where

\[ B=\frac{GD^{2}R}{C_{E}C_{M}\cdot 375} \]

is the electromechanical time constant of the motor, sec;

\[ T_{B}=\frac{L_{B}}{r_{B}} \]

is the electromagnetic time constant of the generator field winding, sec.

The differential equation obtained, taking into account two nonlinearities, does not lend itself to investigation, since its coefficients contain other variables and their nonlinear functions, which depend in a complicated way on \(n\). Taking this circumstance into account, it is necessary to make the assumption that one of the machines operates on the rectilinear portion of the magnetization characteristic. The only way to obtain the equation of motion of the system with respect to \(n\) such that its coefficients depend only on \(n\), \(\frac{dn}{dt}\), or on their nonlinear functions is to linearize the generator magnetization curve, which in this case is expressed by the equality:

\[ e_{r}=Ki_{B}, \tag{10} \]

where \(K\) is a constant coefficient.

Taking this into account, equation (9) is transformed into the following:

\[ \frac{d^{2}n}{dt^{2}}+ \left[ \frac{1}{B}+\frac{1}{T_{B}}-\frac{K\varphi(i_{oy})}{T_{B}R_{B}I} \right]\frac{dn}{dt} + \frac{1}{BT_{B}}\,n=0. \tag{11} \]

Solving equations (5)—(7) with respect to \(i_{oy}\) under the assumptions adopted, we have:

\[ i_{oy}=\frac{r_{1}GD^{2}}{r_{oy}C_{M}\cdot 375}\,\frac{dn}{dt}. \tag{12} \]

To investigate the system on the phase plane it is necessary to choose a suitable approximation of the magnetization curve of the EMA. In this case a convenient form of approximation is a complete cubic polynomial, which most simply expresses the single-valued odd magnetization characteristic of the machine [19] in the form:

\[ \varphi(i_{oy})=\alpha i_{oy}-\beta i_{oy}^{3}. \tag{13} \]

Substituting (13) into (11) and taking (12) into account, we obtain:

\[ \frac{d^{2}n}{dt^{2}}+ \left[ \frac{1}{B}+\frac{1}{T_{B}} -\frac{\alpha Kr_{1}}{T_{B}KR_{B}r_{oy}} + \frac{\beta Kr_{1}^{3}\left(\dfrac{GD^{2}}{C_{M}\,375}\right)^{2}}{T_{B}R_{B}r_{oy}^{3}} \left(\frac{dn}{dt}\right)^{2} \right]\frac{dn}{dt} + \frac{1}{T_{B}B}\,n=0. \]

Introducing the notation

\[ a=\frac{1}{T_B}+\frac{1}{B},\quad b=\frac{\alpha K r_1}{T_B R r_B r_{oy}},\quad c=\frac{\beta K r_1^3\left(\dfrac{GD^2}{C_M\cdot 375}\right)^2}{T_B R r_B^3 r_{oy}},\quad \omega_0^2=\frac{1}{T_B B}, \]

we obtain the equation of motion of the system in the simpler form:

\[ \frac{d^2 n}{dt^2}+\left[(a-b)+c\left(\frac{dn}{dt}\right)^2\right]\frac{dn}{dt}+\omega_0^2 n=0. \tag{15} \]

In this equation the coefficients \(a, b, c\), and \(\omega_0\) are constants determined by the parameters of the system; \(\omega_0\) has the physical meaning of the natural frequency of oscillation of the system.

The phase plane of the system described by equation (15) is the plane with axes \(n\) and \(\dfrac{dn}{dt}\). This is the most suitable choice of coordinates for the given system, since for \(M_{cm}=0\) the phase plane is the plane of the motor speed \(n\) and, on the corresponding scale, the current of the main circuit \(I\). The dissipative term of equation (15) is a function only of \(\dfrac{dn}{dt}\). In this case the phase portrait may be constructed from the isocline of horizontal tangents by Liénard’s method [18].

The state of the system described by equation (15) is represented at any instant of time by the position of a representative point on the phase plane. As the system moves, this point moves over the phase plane, tracing on it a continuous curve, called an integral curve of the system. From the theory of oscillations it is known that if the representative point describes a closed trajectory, called the limit cycle of self-oscillations, then a periodic process with constant frequency and amplitude exists in the system.

On the basis of the methods set forth in detail in the literature [1, 18], we derive the equation of the isoclines, i.e., the locus of points of the phase plane at which the integral curves have tangents with angular coefficient

\[ \Theta=\frac{d\left(\dfrac{dn}{dt}\right)}{dn}. \]

From equation (15) we have:

\[ n=\frac{(b-a)\dfrac{dn}{dt}-c\left(\dfrac{dn}{dt}\right)^3-\Theta\dfrac{dn}{dt}}{\omega_0^2}. \tag{16} \]

For \(\Theta=\infty\) we obtain the isocline of vertical tangents, in this case the axis \(n\).

For \(\Theta=0\) we obtain the isocline of horizontal tangents

\[ n=\frac{(b-a)\dfrac{dn}{dt}-c\left(\dfrac{dn}{dt}\right)^3}{\omega_0^2}. \tag{17} \]

To construct the isocline of horizontal tangents on the phase plane, and then to construct from it the phase portrait of the system, we investigate equation (17).

1. Let us find the points of intersection of the isocline of horizontal tangents with the axis \(\dfrac{dn}{dt}\), setting \(n=0\):

\[ c\left(\frac{dn}{dt}\right)^3-(b-a)\frac{dn}{dt}=0; \tag{18} \]

\[ \left(\frac{dn}{dt}\right)_1 = 0; \tag{19} \]

\[ \left(\frac{dn}{dt}\right)_{2,3} = \pm \sqrt{\frac{b-a}{c}}. \tag{20} \]

1. Let us find the extremal values \(n_{\max}\) on the isocline of horizontal tangents, by taking the derivative \(\frac{dn}{dt}\) from equation (17) and setting it equal to zero. We then obtain:

\[ \frac{dn}{dt} = \pm \sqrt{\frac{b-a}{3c}} \tag{21} \]

at \(n_{\max}\), and substituting (21) into (17), we find:

\[ n_{\max}=\frac{2}{3\omega_0^2}(b-a)\sqrt{\frac{b-a}{3c}}. \tag{22} \]

Thus, the phase plane is divided into two regions:

1. The region where
   \[ \frac{dn}{dt}<\left|\sqrt{\frac{b-a}{c}}\right|. \]

In this case the damping coefficient (the dissipative term) of equation (15) will be negative. The motion of the system is of a growing character, i.e., incremental, accompanied by an accumulation of energy in the system.

Fig. 3.

Fig. 4.

1. The region where
   \[ \frac{dn}{dt}>\left|\sqrt{\frac{b-a}{c}}\right|. \]

In this case the damping coefficient is positive. The motion of the system, accompanied by dissipation of energy, decays, i.e., it is decremental.

The value
\[ \frac{dn}{dt}=\left|\sqrt{\frac{b-a}{c}}\right| \]
is the boundary separating the incremental, unshaded in Fig. 3,a, and the decremental, shaded in the same figure, regions. In this case the damping coeffi-

is equal to zero; the system degenerates into a conservative one, having a harmonic character of motion with frequency \(\omega_0\) and with amplitude determined by the energy stored in the system.

Let us consider the influence of the system parameters on the character of the motion:

a) Let

\[ b_1>a_1;\qquad c=c_1;\qquad n_{\max c}=n_{\max c_1}. \]

This case corresponds to the phase portrait shown in Fig. 3 and representing a limit cycle of self-oscillations. Unfolding the phase portrait in time, we obtain a law of variation of speed and current similar to Fig. 3,b, where the oscillogram of the process corresponding to the given ratio of parameters is presented.

b) Let

\[ b_2>a_2;\qquad b_2-a_2<b_1-a_1; \]

\[ c_2<c_1;\qquad n_{\max c_2}<n_{\max c_1}. \]

This case corresponds to the phase portrait in Fig. 4,a and to the corresponding oscillogram of the process in Fig. 4,b.

c) Let

\[ b_3=a_3;\qquad b_3-a_3=0; \]

\[ c_3<c_2;\qquad n_{\max c_3}=0. \]

In this case the incremental region is absent (Fig. 5,a). The motion in the system will decay, as shown in Fig. 5,b.

Fig. 5.

Consequently, by changing the parameters of the system so that the difference \(b-a\) decreases, we can change the form of the self-oscillations from a form similar to relaxation oscillations (Fig. 3) to a form of almost harmonic oscillations (Fig. 4). Having made this difference equal to zero, we obtain an oscillatory decaying type of motion (Fig. 5). From the phase portraits we see that the equilibrium state of the system, under the given approximation of the magnetization curve and under the adopted assumptions, is unstable for \(b-a>0\), determining a soft mode of self-excitation of oscillations, and stable for \(b-a<0\).

Operation of the System for \(M_{st}\ne 0\)

Let a static torque in the form of dry friction be applied to the motor shaft, produced by a brake, which depends only on the sign of the speed and does not depend on its magnitude (Fig. 6), i.e.

\[ M_{st}=M_{st_1}\operatorname{sign} n, \tag{23} \]

and, with constant excitation current of the motor, taking into account equation (7),

\[ I_{st}=I_{st_1}\operatorname{sign} n. \tag{24} \]

Taking into account the preceding idealizations, let us solve the system of equations (1)—(8) with respect to \(n\). Then

\[ \frac{d^2 n}{dt^2} +\left\{ \frac{1}{B}+\frac{1}{T_B} -\varphi'\!\left(I_{st}\frac{r_1}{r_{oy}}\right) \frac{K}{T_B R r_B} +I_{st}\frac{K r_1 \beta C_E B}{T_B R^2 r_B r_{oy}} \left(\frac{dn}{dt}\right) +\right. \]

\[ \left. +\frac{K r_1^3 \beta C_E B^2}{T_B R^3 r_B r_{oy}^3} \left(\frac{dn}{dt}\right)^2 \right\} \frac{dn}{dt} +\frac{1}{T_B B}n = \frac{1}{T_B B}n_{st}, \tag{25} \]

where

\[ \varphi\!\left(I_{cm}\frac{r_1}{r_{oy}}\right) = \alpha\left(\frac{r_1}{r_{oy}}\right)I_{cm} - \beta\left(\frac{r_1}{r_{oy}}\right)^3 I_{cm}^{3}; \tag{26} \]

\[ \varphi'\!\left(I_{cm}\frac{r_1}{r_{oy}}\right) = \alpha\left(\frac{r_1}{r_{oy}}\right) - 3\beta\left(\frac{r_1}{r_{oy}}\right)^3 I_{cm}^{2} \tag{27} \]

is the value of the nonlinear function and its derivative at \(i_{oy}=I_{cm}\dfrac{r_1}{r_{oy}}\) (Fig. 7), and

\[ n_{cm}=\frac{R}{C_E}\left( \frac{K\varphi\!\left(I_{cm}\dfrac{r_1}{r_{oy}}\right)}{Rr_B} - I_{cm} \right) \tag{28} \]

is the equation of motion of the system in the steady state.

Fig. 6.  Fig. 7.

Indeed, let us consider the steady-state process of the system for \(I=I_{cm}\) and \(n=n_{cm}\). Taking into account that in this case all derivatives in equations (1)—(8) become zero, we have:

\[ Ki_B=IR+C_En, \tag{29} \]

\[ \varphi\!\left(I\frac{r_1}{r_{oy}}\right)=i_Br_B, \tag{30} \]

whence

\[ n=\frac{R}{C_E}\left( \frac{K\varphi\!\left(I\dfrac{r_1}{r_{oy}}\right)}{Rr_B} - I \right) \tag{31} \]

or, upon substituting \(I=I_{cm}\) and \(n=n_{cm}\), we obtain equation (28), and the equation of approximation and its derivative in the form (26) and (27).

Transferring the origin of coordinates on the phase plane to \(\pm n_{cm}\) and taking into account that \(I_{cm}>0\) for \(n>0\) and \(I_{cm}<0\) for \(n<0\), we obtain two equations of motion of the system:

\[ \frac{d^2 n_1}{dt^2} + \left[ (a-b)+c\frac{dn_1}{dt} + g\left(\frac{dn_1}{dt}\right)^2 \right]\frac{dn_1}{dt} + \frac{1}{T_BB}n_1=0 \tag{32} \]

for \(n>0\), where

\[ n_1=n-n_{cm}, \tag{33} \]

\[ \frac{d^2 n_2}{dt^2} + \left[ (a-b)-c\frac{dn_2}{dt} + g\left(\frac{dn_2}{dt}\right)^2 \right]\frac{dn_2}{dt} + \frac{1}{T_BB}n_2=0 \tag{34} \]

for \(n<0\), where

\[ n_2=-(n-n_{cm}). \tag{35} \]

The coefficients of equations (32) and (34) are constant quantities and are equal to:

\[ a=\frac{1}{T_B}+\frac{1}{B};\quad b=\frac{K\varphi'\!\left(I_{cm}\frac{r_1}{cm\,r_{oy}}\right)}{T_B R r_B};\quad c=\frac{I_{cm}K r_1 C_E B}{T_B R r_B r_{oy}};\quad g=\frac{K r_1^3 C_E^2 B^2 \beta}{T_B R^3 r_B r_{oy}^3}. \]

Taking this into account, the equations of the isoclines of horizontal tangents are written as:

\[ n_1= \frac{ (b-a)\dfrac{dn_1}{dt} - c\left(\dfrac{dn_1}{dt}\right)^2 - g\left(\dfrac{dn_1}{dt}\right)^3 }{\omega_0^2} \tag{36} \]

for \(n>0\),

\[ n_2= \frac{ (b-a)\dfrac{dn_2}{dt} + c\left(\dfrac{dn_2}{dt}\right)^2 - g\left(\dfrac{dn_2}{dt}\right)^3 }{\omega_0^2} \tag{37} \]

for \(n<0\).

Consider the motion of the system when the parameters of the system remain constant, while \(I_{cm}\) changes in magnitude.

a) Let \(I_{cm}=I_{cm_1}\).

Knowing the value \(I_{cm_1}\), we find the corresponding \(i_{oy_1}\) from equation (5). Then, from the magnetization characteristic of the EMA (Fig. 7), we find \(\varphi(i_{oy_1})\) and \(\varphi'(i_{oy_1})\), and from formula (28), \(n_{cm_1}\). Determining the difference \((b_1-a)\), the coefficients \(c_1\), \(g_1\), and laying off on the phase plane, to the left and to the right of the origin, segments equal to \(\pm n_{cm_1}\), we construct, from equations (36) and (37), the isoclines of horizontal tangents. The phase portrait (Fig. 8, a):

Fig. 8.

Fig. 9.

is constructed, for \(n>0\), along the isocline of horizontal tangents constructed with respect to the axis \(\dfrac{dn_1}{dt}\), and for \(n<0\) along the isocline constructed with respect to the axis \(\dfrac{dn_2}{dt}\). At \(n=0\) the integral curves are matched. The corresponding oscillogram is given in Fig. 8,b.

b) Let \(I_{st}=I_{st_2}\); \(I_{st_2}>I_{st_1}\); \(n_{st_2}>n_{st_1}\); \(c_2>c_1\); \((b_2-a)<(b_1-a)\). In this case we obtain two phase portraits (Fig. 9,a). The point representing the system moves along one of them depending on the direction in which the system received its initial deviation. The motion of the system is one-sided, without reversal of the motor. The oscillogram of the process is given in Fig. 9,b.

Fig. 10.

c) Let \(I_{st}=I_{st_3}\); \(I_{st_3}>I_{st_2}\); \(n_{st_3}>n_{st_2}\); \((b_3-a)=0\).

In this case, for any deviation, the self-oscillations die out; the system operates stably at \(+I_{st}, +n_{st}\) or at \(-I_{st}, -n_{st}\), depending on the initial deviation of the system. The phase portrait and oscillogram are given in Fig. 10.

It is seen from Fig. 8 that the curve of speed changes smoothly with time, while the current and the derivative of the speed undergo a sharp change when the speed passes through the zero value. This character of the motion is explained by the fact that the equation of mechanical equilibrium of the system, for \(n<0\) and \(I_{st}<0\),

\[ I_{(1)}=I_{d(1)}-I_{st}; \tag{38} \]

for \(n>0\) and \(I_{st}>0\)

\[ I_{(2)}=I_{d(2)}+I_{st}, \tag{39} \]

where \(I_{(1)}\) and \(I_{(2)}\) are the motor currents at the instants of time corresponding to positions 1 and 2 in Fig. 8, which are sufficiently close to the position at which \(n=0\);

\(I_{d(1)}\) and \(I_{d(2)}\) are the dynamic motor currents, proportional to the derivative of the speed at the points corresponding to the same instants of time, i.e.,

\[ I_{d(1)}=\frac{GD^2}{C_M\cdot 375}\cdot \frac{dn_{(1)}}{dt}; \]

\[ I_{d(2)}=\frac{GD^2}{C_M\cdot 375}\cdot \frac{dn_{(2)}}{dt}. \]

Let us consider two cases.

a) Suppose that the circuit has a large inductance, which does not permit sharp jumps of the current, and the flywheel moment of the rotating masses reduced to the motor shaft is negligibly small; then, at the indicated instants of time, one may take \(I_{(1)}\simeq I_{(2)}\), and from equations (38) and (39) determine:

\[ I_{d(1)}\simeq I_{d(2)}+2I_{st}, \tag{40} \]

i.e., the current curve varies smoothly with time, while the derivative of the speed undergoes a jump. With the reverse combination of these parameters one may take \(I_{\partial(1)} \approx I_{\partial(2)}\); then from equations (38) and (39) we have:

\[ I_{(1)} \approx I_{(2)} - 2I_{cm}, \]

i.e., the motor current undergoes a jump of magnitude \(\sim 2I_{cm}\). In our case the flywheel moment is predominant, and therefore the current, as the speed passes through zero, undergoes a sharper change than the derivative of the speed, which is confirmed by the oscillogram (Fig. 8,b).

Thus, when a static torque is present on the motor shaft, the system, depending on its magnitude, is either in a self-oscillatory regime, in which the amplitude of the current and speed decreases with increasing load, or in a regime of steady motion at \(I = I_{cm}\) and \(n = n_{cm}\). The state of rest of the system is absent in this case. Hence there follows the existence of a certain critical value of the load current \(I_{kp}\). For \(I_{cm} < I_{kp}\) self-oscillations exist, while for \(I_{cm} > I_{kp}\) the system enters a regime of steady motion.

The latter property is a prerequisite for using the self-oscillations of this system for contactless reversal of the drive of mechanisms with reciprocating motion, where during the working stroke \(I_{cm} > I_{kp}\), and when the load is relieved \(I_{cm} < I_{kp}\). In the latter case the system enters a self-oscillatory regime, the motor reverses, and if at this time a load is again applied to the motor shaft for which \(I_{cm} > I_{kp}\), or some action is introduced that affects the change in the system parameters so that self-oscillations are not excited, the motor will rotate steadily in the reverse direction, bringing the mechanism back to its initial position, etc.

Conclusions

1. The adopted method of investigating the system under study in the phase plane makes it possible:

a) to explain quite satisfactorily the nature of the oscillations arising in the system;

b) to determine the influence of the parameters and of the load on the motor shaft on the type and form of the motions;

c) to determine the stability of self-oscillations by the presence of a stable limit cycle, the conditions for self-excitation, and also the stability of the state of rest and of the steady-motion state of the system;

d) to determine the amplitude of the current and speed as a function of the system parameters by projecting the phase portrait onto the corresponding axis, as well as the frequency of the self-oscillations by unfolding this portrait in time.

1. By varying the magnitude of the static torque, one can obtain self-oscillations of different form, deviating from a harmonic form the more strongly the greater the applied torque; one can also obtain steady motion of the system at constant motor speed for \(M_{cm} > M_{kp}\), which is most important when using self-oscillations for contactless reversal of the motor.

2. By varying the system parameters at \(M_{cm} = 0\), one can change the form of the self-oscillations from a form close to relaxation oscillations to one close to harmonic.

3. The investigations carried out make it possible to judge the motions arising in an ordinary automatic-control system, depicted

shown in Fig. 2, in the case where the potentiometer PC is set to a position for which \(i_{oy3}=0\), and when the control winding \(OU_2\) is disconnected, or when its ampere-turns are neglected, when they are significantly smaller than the ampere-turns of the control winding \(OU_1\).

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