ON QUASILINEAR AND QUASI-RELAXATION SELF-OSCILLATIONS IN AN ELECTROMECHANICAL SYSTEM

V. M. Rikkonen

(Presented by the Scientific-Methodological Seminar of the Electromechanical Faculty)

In connection with the spread of complex schemes of electric-machine automation, questions concerning the study of oscillatory phenomena and the stability of systems with respect to the occurrence of oscillatory phenomena—and, in particular, in the generator–motor system—are at present acquiring ever greater importance.

As is known, the analysis of complex automatic-control schemes is carried out by the methods and various stability criteria of automatic-control theory. Oscillatory phenomena, or periodic regimes of control systems, are determined by the nonlinearities existing in the system; the great variety of periodic motions arising in systems and observed in practice is explained both by the number and type of these nonlinear characteristics and by their location in the structural scheme of the system.

When the phenomenon of oscillations in control systems is regarded as an undesirable and complicating factor, one often confines oneself to finding the frequency of possible oscillations and to stability criteria for the system, without touching upon the physical picture or the nature of the given phenomenon.

If one turns to the theory of electrical machines and the theory of electric drive, then transient processes in the G–M system, taking into account various nonlinear characteristics, have been and still remain at present unresolved scientific problems. In this connection, full member of the Academy of Sciences of the Armenian SSR A. G. Iosif’yan wrote [1]: “As is known, the theoretical foundations for the study of transient processes are based on methods of analyzing the behavior of systems that require the solution of nonlinear differential equations describing nonstationary processes in electric circuits, and differential equations describing processes connected with the motion of inertial masses.

To facilitate methods of analysis, one usually neglects the nonlinear parameters of current circuits, excludes from consideration the phenomenon of saturation, eddy-current losses, hysteresis..., and also excludes the variable velocity of moving masses. Linearization of equations is the result of extraordinary mathematical difficulties arising in solving these equations in general form.”

Meanwhile, quantities of such practical importance as the duration of the transient regime, the amplitude of oscillations, the number of limiting cycles of oscillations, the analysis of losses during these oscillations, the phenomena of self-excitation and self-oscillation, depend very strongly on the nonlinear parameters of the electric circuits and on the mechanical masses moving with acceleration. Consequently, the main attention of researchers should be directed to the field of analysis of transient processes with allowance for nonlinear parameters.”

The problem of solving transient processes in systems with allowance for the principal nonlinearities arises in various fields of engineering, for example, in electric drive, theoretical electrical engineering, and so on. In the theory of electric machines, to solve nonlinear problems, methods of successive intervals, various methods of linearizing nonlinear characteristics, and numerical and graphical-analytical methods are applied.

However, there exist special methods of the theory of nonlinear oscillations, developed mainly by Soviet physicists—academicians Mandelstam, Papalexi, and Andronov—with the aid of which it is possible to explain a number of essentially important properties of the system mentioned by A. G. Iosifyan.

The generator–motor system is the principal part of an electric-machine automatic-control circuit, and therefore the questions of the occurrence of free oscillations in this system and the physics of the phenomenon of self-oscillations in it are highly topical.

It is quite natural, in solving the problems that arise, to make use of the indications of the theory of nonlinear oscillations, in which the problems of finding the limiting cycles of self-oscillations and the phenomenon of self-excitation of the system are solved, and the physical picture of the given phenomenon is revealed.

Since in the theory of electric machines and in the theory of electric drive self-oscillations have scarcely been studied, it is simpler to begin the investigation by considering the simplest system, for which we shall take a separately excited series generator–motor system (Fig. 1).

Fig. 1.

Here, AD is the prime mover, the energy source;
G—OPV is the separately excited series generator;
\(L(I)\) is a variable or constant inductance;
D—OVD is the separately excited motor.

In the theory of oscillations, self-oscillatory systems have a structural diagram of the form shown in Fig. 2.

The energy source in a self-oscillatory system must exist in one form or another, since in every real oscillatory system there is dissipation of the energy of the oscillatory con-

For the established periodic motions to be maintained, it is necessary to replenish the energy reserve in the circuit. The role of feedback consists in acting on the “valve,” which opens access of energy into the system.

Such a system, isolated from external actions (autonomous), has the property of self-regulation of the inflow of energy, and all the properties of free motions in the system will be explained only by the characteristics of the system itself.

Fig. 2.

The system \(G—D\) adopted by us (Fig. 1) can be represented by a structural diagram analogous to Fig. 2 if one singles out the oscillatory element, consisting of the inductance due to the flux linkages over the entire circuit \(G—D\), and the electrodynamic capacitance of the rotating motor—\(C_0\). In the general case, both the inductance and the capacitance of such an oscillatory element will depend on the “intensity” of the process, i.e., on the magnitude of the flowing current and of the small masses moving with acceleration, and consequently the oscillatory circuit will be nonlinear.

The nonlinearity of the generator characteristic, owing to saturation of the magnetic system, will appear in the structural diagram as a nonlinear element playing the role of a limiter of the growth of oscillations. Thus we obtain Fig. 3, analogous to Fig. 2.

Fig. 3.

The role of the valve is played by the series-excitation winding of the generator, through which the current of the circuit \(G—D\) flows and which at the same time performs the function of positive current feedback.

In electric-machine automatic-control systems there occur more complicated circuits in which there are feedbacks with respect to current, voltage, and feedbacks with respect to speed, supplied to the control windings of the generator or of the electric-machine amplifier. In addition to these excitation windings, there may be independent-excitation windings or setting windings (Fig. 4).

If from the circuit of Fig. 4 one removes the IEW—the independent-excitation winding—and the SW—the shunt winding, then, for a definite current gain and positive feedback coefficient, we obtain the circuit we have adopted (Fig. 1).

The ampere-turns of the DNF and DSF alter the self-oscillatory properties of the system, but in principle the method of investigation and the physics of the phenomenon must remain the same.

Fig. 4.

For the G—D circuit, the voltage-balance equation will be

\[ \frac{d\psi}{dt}+IR+2\Delta U_{\mathrm{sh}}+E_0=E_a(I), \tag{1} \]

where
\(R\)—the total active resistance of the circuit;
\(2\Delta U_{\mathrm{sh}}\)—the voltage drop at the brush contacts on the generator and motor;
\(E_0\)—the counter-e.m.f. of the motor;
\(E_a(I)\)—the internal characteristic of the generator.

We shall regard the principal nonlinearity in the G—D circuit as the nonlinearity \(E_a(I)\); the remaining coefficients of equation (1), such as the electrodynamic capacitance and inductance, may be taken as constant under certain conditions. In fact, when a large linear inductance is included in the circuit, the flux linkage in the G—D circuit is represented by a weakly nonlinear curve \(\psi(I)\), which may be replaced by a rectilinear characteristic.

For the nonlinearity of the curve \(E_a(I)\) we shall use an approximation by a third-degree polynomial, taking into account the characteristic nonlinearity of magnetization,

\[ E_a(I)=\alpha' I-\beta' I^3 . \tag{2} \]

This approximation is often adopted in the theory of nonlinear oscillations for characteristics with saturation and makes it possible, “by means of simple mathematical calculations, to obtain results that are readily surveyable and at the same time, from the qualitative point of view, agree sufficiently closely with experimental data” [2].

Taking \(M_s=0\), we obtain d’Alembert’s equation in the form

\[ j\frac{d\omega}{dt}=C_m I . \tag{3} \]

Having accepted the stated assumptions and differentiating equation (1) with respect to time, taking expression (3) into account, we obtain:

\[ L\frac{d^2 I}{dt^2}+\left(R-\alpha'+3\beta'I^2\right)\frac{dI}{dt}+ \frac{C'_e C_M}{j}\,I=0. \tag{4} \]

In analyzing electromechanical transient processes, the current in the system and the rotational speed of the motor must primarily be considered. Since \(I=\dfrac{j}{C_M}\cdot\dfrac{d\omega}{dt}\) (3), (4) takes the form

\[ L\frac{d^2\omega}{dt^2} +\left[R-\alpha'+\beta_1\left(\frac{d\omega}{dt}\right)^2\right]\frac{d\omega}{dt} +K\cdot\omega=0, \tag{5} \]

where

\[ \beta_1=\beta'\left(\frac{j}{C_M}\right)^2,\qquad K=\frac{C'_e C_M}{j}. \]

In equation (5) one may distinguish a linear oscillatory link, to which the circular frequency of the circuit is inherent,

\[ h_0=\sqrt{\frac{C'_e C_M}{jL}}. \tag{6} \]

The expression \(\dfrac{j}{C'_e C_M}=C_0\) plays the role of an electrodynamic capacitance, whose physical meaning consists in the accumulation of kinetic energy by the rotating drive.

Since \(C_M=K_1\Phi_0\) and \(C'_e=K_2\Phi_0\), where \(\Phi_0\) is the excitation flux of the motor, the frequency of the linear circuit is

\[ h_0\equiv\Phi_0 \quad \text{and} \quad C_0=\frac{1}{\Phi_0^2}. \tag{7} \]

To analyze the obtained equation (5), it is necessary to express it in relative units. This necessity is explained not so much by the conveniences and simplifications that appear when relative units are used, as by the fact that the theory of nonlinear oscillations widely uses the method of comparing various coefficients in equations. It is clear that such a comparison can be made only for quantities having the same dimension or expressed in dimensionless form.

In a system of relative units, as is known, all quantities must be referred to one link, the nominal quantities of which are taken as the base quantities. In the present case it is convenient to take as the base values the nominal data of the motor under investigation,

\[ E_n=I_n R_n=C'_e\omega_n. \]

To express the inductance, the flux linkage over the entire \(\Gamma—D\) circuit must be referred to the armature winding, whose nominal flux linkage \(\Phi_{d.n.}W_{a.d}\) we shall take as the base.

In relative units

\[ e_a(i)=\alpha i-\beta i^3;\qquad \psi(i)=l_0 i. \]

Then equation (4) will be written as:

\[ \xi l_0 \frac{d^2 i}{dt^2}+(r-\alpha+3\beta i^2)\frac{di}{dt} +\frac{C_M C_e'}{J R_H}\,i=0, \tag{8} \]

where

\[ \xi=\frac{W_{ya0}\Phi_{dH}}{E_H} =\frac{W_{yad}\Phi_{dH}\cdot \pi a}{p\cdot \omega_H \cdot W_{yad}\Phi_{dH}} =\frac{\pi a}{p\cdot \omega_H}\ [\mathrm{sec}]. \tag{9} \]

If now the natural frequency of the linear circuit is taken as the unit of time, \(\tau=h_0t\), then in dimensionless units we obtain

\[ \frac{d i^2}{d\tau^2} +\frac{1}{\xi l_0 h_0}(r-\alpha+3\beta i^2)\frac{di}{d\tau} +i=0. \tag{10} \]

For equation (5), respectively, we obtain

\[ \frac{d^2\omega}{d\tau^2} +\frac{1}{\xi l_0 h_0} \left[ r-\alpha+\frac{\beta}{(\xi l_0 h_0)^2} \left(\frac{d\omega}{d\tau}\right)^2 \right]\frac{d\omega}{d\tau} +\omega=0. \tag{11} \]

The quantity \(\xi l_0 h_0\) may be called the wave resistance of the circuit in dimensionless units. Indeed,

\[ \xi l_0 h_0 =\xi l_0\sqrt{\frac{C_M C_e'}{J \xi l_0 R_H}} =\sqrt{\frac{\xi l_0}{C_0R_H}} =\frac{1}{R_H}\sqrt{\frac{L_0}{C_0}} =\rho. \tag{12} \]

For the conservative oscillatory circuit obtained from (11), when the second term is equal to 0, we have:

\[ \frac{1}{2}\left(\frac{d\omega}{d\tau}\right)^2+\frac{1}{2}\omega^2=C. \tag{13} \]

This equality expresses the law of conservation of energy for an ideal conservative circuit.

In what follows we analyze equation (11) by qualitative methods, by constructing integral curves on the phase plane according to the theory \([2,3]\).

According to equality (13), the energy of the conservative circuit remains constant and, consequently, the motion in this case is represented by a circle whose radius vector is

\[ r=\sqrt{2W}, \tag{14} \]

where \(2W\) is the doubled energy of the conservative circuit.

On the phase plane, along the abscissa axis we lay off the values of \(\omega\), and along the ordinate axis, the values of \(\dfrac{d\omega}{d\tau}\).

To find the magnitude of the current in the system of relative units, we have the equality

\[ i=\frac{\omega}{\rho}. \tag{15} \]

Indeed,

\[ \omega=\frac{d\omega}{d\tau} =\frac{C_M I_H}{h_0J\omega_H}\,i =\sqrt{\frac{C_M C_e'\xi l_0 R_H}{J R_H^2}}\,i =\frac{1}{R_H}\sqrt{\frac{L_0}{C_0}}\,i =\rho\cdot i. \]

Thus, the solutions of equation (11) will be integral curves in the phase plane, giving a representation of the motion in the system; the motor rotational speed and the current in the system that are of interest to us, taking into account equality (15), will be found as the projections of the representative point onto the coordinate axes.

Regime of Establishment of Quasilinear Self-Oscillations

We rewrite equation (11) in the following form:

\[ \frac{d^{2}\omega}{dt^{2}}+\frac{1}{\rho}\left[r-\alpha+\frac{\beta}{\rho^{2}}\left(\frac{d\omega}{dt}\right)^{2}\right]\frac{d\omega}{dt}+\omega=0. \tag{16} \]

The construction of integral curves for equation (16) can be carried out either by the method of isoclines or by Liénard’s method [2,4].

In the present case the dissipative term of the equation is a nonlinear function of the derivative of the variable \(\omega\); therefore Liénard’s method, which greatly simplifies the construction of integral curves, can be applied [3,4].

For this method it is necessary to construct the isocline of horizontal tangents, according to which the integral curves themselves are then constructed. If we set \(x=\omega\), \(y=\dfrac{d\omega}{d\tau}\), then from (16) we obtain the system of equations

\[ \begin{gathered} \frac{dy}{d\tau}+\frac{1}{\rho}\left[r-\alpha+\frac{\beta}{\rho^{2}}y^{2}\right]y+x=0,\\[6pt] \frac{dx}{d\tau}=y. \end{gathered} \tag{17} \]

We obtain the equation of the integral curves in the phase plane if we eliminate time by dividing the first equation by the second:

\[ \frac{dy}{dx}= \frac{-\dfrac{1}{\rho}\left[r-\alpha+\dfrac{\beta}{\rho^{2}}y^{2}\right]y-x}{y}. \tag{18} \]

We obtain the equation of the isocline of horizontal tangents from the condition \(\dfrac{dy}{dx}=0\), which gives us

\[ x=\frac{1}{\rho}(\alpha-r)y-\frac{\beta}{\rho^{3}}y^{3}. \tag{19} \]

This is the equation of a cubic parabola. Knowing the coefficients \(\alpha\), \(r\), \(\beta\), and \(\rho\), we construct, in relative units, from equation (19) the isocline of horizontal tangents in the phase plane.

The equation of the separating curve [3], which separates the increment and decrement regions of the phase plane, is obtained from the condition

\[ 2\varepsilon_q(y)=\frac{1}{\rho}\left(r-\alpha+\frac{\beta}{\rho^{2}}y^{2}\right)=0, \tag{20} \]

whence the separating straight lines obtain the expression

\[ y=\pm \rho \sqrt{\frac{\alpha-r}{\beta}} . \tag{21} \]

In Fig. 5 the isocline of horizontal tangents according to (19) is constructed; with its help, integral curves have been constructed on the phase plane, giving a complete qualitative and quantitative picture of the behavior of the system.

Fig. 5.

To obtain the rotation speed of the motor and the current in the system as functions of time, which are of interest to us, we carry out a point-by-point construction according to formula [5]

\[ \Delta t_i=\int_{x_i}^{x_{i+1}}\frac{dx}{y} = \frac{x_{i+1}-x_i}{y_{i+1}-y_i} \ln\frac{y_{i+1}}{y_i}. \tag{22} \]

It is evident from Fig. 5 that, for small values of the coefficient \(\dfrac{\alpha-r}{\beta}\)

in formula (19) the isocline of horizontal tangents deviates little from the ordinate axis. This can be achieved by increasing the active resistance of the circuit \(G-D\), or else by a large value of the wave resistance of the circuit. In this case one obtains the so-called “soft” self-excitation of the system, without any external disturbing actions.

In the initial stage of self-oscillation buildup, at small amplitudes in the incremental region, energy accumulates in the oscillatory circuit. As the energy in the system increases, the radius vector at first increases only slightly; then the spiral unfolds more rapidly—the inflow of energy into the circuit reaches a maximum; subsequently the increase of the radius vector slows down, and the representative point approaches the limit cycle, which is characterized by the fact that over one period of oscillation the energy of the circuit remains constant. The constancy of the energy in the established periodic motion in the system is explained by the fact that the representative point passes twice per period through the incremental region and twice through the decremental region, in which the radius vector decreases, since the energy of the circuit is dissipated in the latter region.

Naturally, in established motion over the course of a period the balance must be maintained between the energy entering the circuit and the energy lost by the circuit.

The form of the self-oscillations in this case differs little from harmonic motion, since the isocline of horizontal tangents is located close to the ordinate axis; in the present case the self-oscillations are called quasilinear.

With an increase of the coefficient \(\dfrac{\alpha-r}{\rho}\), the isocline of horizontal tangents moves farther away from the ordinate axis; this means larger amounts of energy entering in the incremental region; in this case the system must penetrate more deeply into the decremental region of the phase plane in order that the energy balance in the self-oscillations be maintained. The form of the motion deviates more significantly from harmonic motion and, consequently, appreciable higher-order harmonic components appear in the current and velocity curves.

It is of interest to determine the amount of energy that enters the circuit and is then returned by the oscillatory circuit during self-oscillations. In equation (16) the dissipative term can be denoted as follows:

\[ -\frac{1}{\rho}\left[r-\alpha+\frac{\beta}{\rho^2}\left(\frac{d\omega}{dt}\right)^2\right]\frac{d\omega}{dt} = 2\varepsilon\varphi_1(\omega)\dot{\omega} = F_1(\omega). \tag{23} \]

If equation (13) expresses the constancy of the energy of the conservative circuit, then under self-oscillations, when on average over a period the constancy of the energy of the oscillatory circuit is likewise observed, the work of the “external” force applied to this circuit over the period must be equal to 0.

This is written in the following way:

\[ W=\int_{0}^{T}2\varepsilon\varphi_1(\omega)\dot{\omega}^{\,2}\,d\tau = \Phi\,2\varepsilon\varphi_1(\omega)\dot{\omega}\,d\omega = 0, \tag{24} \]

which means that the integral over the closed contour of the dissipative term must be equal to 0.

As is evident, the value of the force is determined by the curve of horizontal tangents, which is a function only of \(\omega\); to find the integral (24) this “force” must be represented as a function of \(\omega\). This can be done with the aid of the limit cycle, which establishes the relation between \(\dot{\omega}\) and \(\omega\). Figure 6 shows the dependence \(F(\omega)\) constructed in this way.

Fig. 6.

The area enclosed by this curve is the energy in the scale of the relative units adopted by us; the contour traversed as the representative point moves along the limit cycle in the clockwise direction bounds an area equal to the energy entering the system when the incremental region is traversed (Fig. 6)—\(+\); the contour traversed counterclockwise bounds an area giving the value of the energy lost by the contour in the decremental region—\(-\).

Naturally, in the established regime of self-oscillations the areas with \(+\) and \(-\) must be equal. In Fig. 6 both the qualitative and the quantitative aspect of the process of self-oscillations in our system is clearly visible.

For considerable coefficients

\[ 2\varepsilon=\frac{\alpha-r}{\rho} \]

quasi-relaxation self-oscillations arise in the system, acquiring the characteristic form of purely relaxation oscillations as \(\varepsilon \to \infty\).

For convenience in analyzing equation (16), we perform linear transformations of the variables by the substitution

\[ x=\sqrt{\frac{\beta}{\rho^{3}\,2\varepsilon}} . \]

Thus, from (16) we obtain the expression

\[ \ddot{x}-2\varepsilon\left[1-x^{2}\right]\dot{x}+x=0 . \tag{25} \]

The equation of the isocline of horizontal tangents is

\[ x_{1}=2\varepsilon(1-y^{2})\,y . \tag{26} \]

In Fig. 7 an isocline is constructed for \(2\varepsilon=4.61\), and from it the limit cycle is constructed.

Because \(2\varepsilon \gg 1\), the motion in the system is sharply decremental or sharply incremental in all regions of the phase plane except the field of directions near the isocline of horizontal tangents. The limit cycle is reached within the first quarter of a period for any initial conditions. Here we do not see the former phenomenon of self-buildup; the oscillatory link seems to lose its oscillatory properties.

From the point \(P_0\) the representative point moves almost by a jump downward to its intersection with the isocline of horizontal tangents at the point \(P_1\). Then the motion proceeds along the lower branch of the isocline of horizontal tangents; in the vicinity of the point \(P_2\) the representative point will go almost vertically upward to its intersection with the upper branch of the isocline.

In Fig. 7 the curves of interest to us, \(\omega=f(t)\) and \(I=\varphi(t)\), are shown; they were constructed with the aid of the limit cycle. As can be seen, the form of the curves deviates considerably from harmonic form and assumes the characteristic sawtooth shape inherent in relaxation oscillations.

Such self-oscillations are called quasi-relaxation oscillations.

Fig. 7.

Conclusions

By the methods of the theory of nonlinear oscillations it is possible to solve a number of nonlinear problems arising in the practice of electric drives.

In the simplest system \(G\)—\(D\), periodic motions of the free system, or self-oscillations, may be of two types depending on the magnitude of the coefficient \(\dfrac{\alpha-r}{\rho}\) in the nonlinear equation; for one and the same accelerating voltage, either quasilinear or quasi-relaxation self-oscillations may arise in the system; in the steady-state regime it is possible to establish the energetics of the self-oscillations.

References

1. A. G. Iosifyan. Scientific Problems of Domestic Electromachine Building, Elektrichestvo, No. 7, 1955.
2. A. A. Andronov, S. E. Khaikin. Theory of Oscillations, ONTI, 1937.
3. K. F. Teodorchik. Self-Oscillatory Systems, 3rd ed., 1952.
4. J. Stoker. Nonlinear Oscillations in Mechanical and Electrical Systems, IIL, Moscow, 1953.
5. Edited by Doctor of Technical Sciences Polivanov. Physical Foundations of Electrical Engineering, Gosenergoizdat, IIL, 1950.