ON THE SELF-OSCILLATORY PROPERTIES OF A SIMPLE “GENERATOR—MOTOR” SYSTEM

L. I. Gandzha

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

The most general case of a simple “generator—motor” system is a system with a series-excited generator and motor, shown in Fig. 1. The study of this system is associated with very great difficulties if it is regarded as nonlinear. To see this, let us describe it by the corresponding equations. The starting point is the equation

\[ e_g = RI + L\frac{dI}{dt} + e_d, \tag{1} \]

which describes the transient process of the current \(I\) in the power circuit and is valid, obviously, under the following assumptions:

1) absence of armature reactions of the generator and motor; this presupposes either the presence, in the system of Fig. 1, of machines with compensating windings, or the use of internal magnetization characteristics of the machines that take account of armature reactions;

2) absence, for each of the machines, of mutual inductance between the independent-excitation winding and the armature and series-excitation windings, which means independence of the currents in the independent-excitation windings from the transient process of the current \(I\) in the power circuit, and their constancy in the absence of external actions on the circuits of the independent-excitation windings;

3) absence of oscillations of the generator speed when the load current oscillates;

4) absence of the influence of eddy currents arising in the massive parts of the machines during the transient process.

We shall adhere to these assumptions throughout the subsequent exposition. On the basis of them, we may use the usual “static” magnetization characteristics of the machines.

Taking the foregoing into account, we may regard the e.m.f. of the generator \(e_g\) as a linear function of the total magnetic flux \(\Phi_g\) of the generator, composed of the flux \(\Phi_{gn}\) of its independent and the flux \(\Phi_{gp}\) of its series excitation windings; on the basis of the generator magnetization curve, the fluxes \(\Phi_{gn}\) and \(\Phi_{gp}\) are nonlinear functions of the corresponding currents \(i_{gn}\) and \(I\), respectively, of the independent-excitation winding of the generator and of the power circuit; therefore \(e_g\) is a nonlinear function of two independent variables—the currents \(i_{gn}\) and \(I\):

\[ e_g = e_g(\Phi_g) = e_g(\Phi_{gn}, \Phi_{gp}) = e_g(i_{gn}, I). \tag{a} \]

Under the same conditions, the counter-e.m.f. of the motor \(e_d\) can be represented as a nonlinear function of the motor speed \(n\) and its

of the resultant magnetic flux \(\Phi_d\), composed of the flux \(\Phi_{d n}\) of its independent excitation winding and the flux \(\Phi_{d p}\) of its series excitation windings. Proceeding from the magnetization curve of the motor, these fluxes are nonlinear functions of the corresponding currents \(i_{d n}\) and \(I\), respectively, in the independent excitation winding of the motor and in the power circuit; therefore \(e_d\) may be represented as a function of the speed \(n\), the excitation current \(i_{d n}\), and the current \(I\):

\[ e_d=C'_{ed}\Phi_d n=C'_{ed}(\Phi_{d n}+\Phi_{d p})n=\varphi_{de}(i_{d n}, I)n. \tag{б} \]

The first term on the right-hand side of equation (1) represents the voltage drop in the active resistance of the power circuit. It also includes the voltage drop in the transition layer “brush—collector,” which is a function of many variables: the current \(I\), the direction of this current, the pressure exerted on the brushes, the rotational speed of the armature, the grade of the brushes, and so forth.

Fig. 1.

Fig. 2.

Since, on the one hand, most of these factors cannot be taken into account or expressed analytically, and, on the other hand, their influence—except for the current \(I\)—on the voltage drop at the brushes is small, in what follows we shall take into account only the dependence of the voltage drop at the brushes on the current \(I\). Then, as is known, the current–voltage characteristic of the power circuit may be represented in the form of the curve shown in Fig. 2; from what has been said it is clear that the resistance \(R\) of the power circuit is a nonlinear function of the current \(I\):

\[ R=R(I). \tag{в} \]

The second term on the right-hand side of equation (1) represents the component of the generator e.m.f. expended in overcoming the counter-e.m.f. of self-induction induced in the power circuit. The latter, as is known, is equal to:

\[ e_L=\frac{d\Psi}{dt}=\frac{d\Psi}{dI}\cdot\frac{dI}{dt}=L\frac{dI}{dt}, \]

where \(\Psi\) is the magnitude of the flux linkage of the power contour of the system, Fig. 1, and \(\dfrac{d\Psi}{dI}=L\) is the coefficient of differential inductance, determined from the magnetization curve (Fig. 3) and, in the presence of iron, a nonlinear function of the ampere-turns. Obviously, \(L\) may be represented as a nonlinear function of three independent variables—the currents \(i_{g n}\), \(i_{d n}\), and \(I\), since the combined action of these currents determines the degree of saturation of the machines. Then

\[ L=L(i_{g n}, i_{d n}, I). \tag{г} \]

Taking into account (a), (б), (в), and (г), we obtain, on the basis of (1):

\[ e_g(i_{g n}, I)=R(I)I+L(i_{g n}, i_{d n}, I)\frac{dI}{dt}+\varphi_{de}(i_{d n}, I)n. \tag{1a} \]

To eliminate one of the variables (\(I\) or \(n\)), we shall use the equation of motion of the motor:

\[ M=M_{cm}+M_j=M_{cm}+\frac{GD^2}{375}\frac{dn}{dt}, \tag{2} \]

where the torque \(M\) of the motor is a nonlinear function of the current and of the total magnetic flux \(\Phi_d\) of the motor, and may be represented in the form

\[ M=C'_m\Phi_d I=C'_m(\Phi_{dn}+\Phi_{dn})I=\varphi_{dm}(i_{dn}, I)I, \tag{d} \]

while the static torque \(M_{cm}\) will hereafter be regarded as constant and reactive, i.e., changing its sign abruptly when the sign of the speed changes, and thus representing a nonlinear function of the speed, shown in Fig. 4. Substituting (d) into (2), we obtain:

\[ \varphi_{dm}(i_{dn}, I)I=M_{cm}(n)+\frac{GD^2}{375}\frac{dn}{dt}. \tag{2a} \]

Differentiating (1a) with respect to time and substituting into it \(dn/dt\) from (2a), we obtain a nonlinear second-order differential equation for the motor current:

\[ \frac{d^2 I}{dt^2} +\frac{1}{L(i_{zn}, i_{dn}, I)} \left\{ \frac{d[R(I)]}{dI}I +R(I) +\frac{\partial[L(i_{zn}, i_{dn}, I)]}{\partial i_{zn}}\frac{di_{zn}}{dt} +\right. \]

\[ \left. +\frac{\partial[L(i_{zn}, i_{dn}, I)]}{\partial i_{dn}}\frac{di_{dn}}{dt} +\frac{\partial[L(i_{zn}, i_{dn}, I)]}{\partial I}\frac{dI}{dt} -\frac{\partial[e_z(i_{zn}, I)]}{\partial I} \right\}\frac{dI}{dt} + \]

\[ +\frac{375}{GD^2} \frac{\varphi_{de}(i_{dn}, I)\varphi_{dm}(i_{dn}, I)} {L(i_{zn}, i_{dn}, I)}I +\frac{1}{L(i_{zn}, i_{dn}, I)} \times \]

\[ \times \left\{ \frac{\partial[\varphi_{de}(i_{dn}, I)]}{\partial i_{dn}} \frac{di_{dn}}{dt} + \frac{\partial[\varphi_{de}(i_{dn}, I)]}{\partial I} \frac{dI}{dt} \right\}n - \]

\[ -\frac{1}{L(i_{zn}, i_{dn}, I)} \frac{\partial[e_z(i_{zn}, I)]}{\partial i_{zn}} \frac{di_{zn}}{dt} - \frac{375}{GD^2} \frac{M_{cm}(n)}{L(i_{zn}, i_{dn}, I)} \varphi_{de}(i_{dn}, I)=0; \tag{3a} \]

finally, the speed \(n\) may be eliminated by substituting its value from (1a) into (3a); however, we shall not carry out this transformation, since it is of no significance for the subsequent exposition.

Eliminating from (1a) the current \(I\) by substituting its value from (2a), we finally obtain the nonlinear second-order differential equation for the motor speed:

\[ \begin{aligned} \frac{d^{2}n}{dt^{2}} &+\left\{ \frac{R(I)}{L(i_{gn}, i_{dn}, I)} +\frac{375}{GD^{2}}\,M'_{st}(n) -\frac{1}{\varphi_{dm}(i_{dn}, I)} \frac{\partial[\varphi_{dm}(i_{dn}, I)]}{\partial i_{dn}} \frac{di_{dn}}{dt}\right.\\ &\left.\qquad -\frac{1}{\varphi_{dm}(i_{dn}, I)} \frac{\partial[\varphi_{dm}(i_{dn}, I)]}{\partial I} \frac{dI}{dt} \right\}\frac{dn}{dt}\\ &+\frac{375}{GD^{2}}\left\{ \frac{R(I)}{L(i_{gn}, i_{dn}, I)} -\frac{1}{\varphi_{dm}(i_{dn}, I)} \frac{\partial[\varphi_{dm}(i_{dn}, I)]}{\partial i_{dn}} \frac{di_{dn}}{dt}\right.\\ &\left.\qquad -\frac{1}{\varphi_{dm}(i_{dn}, I)} \frac{\partial[\varphi_{dm}(i_{dn}, I)]}{\partial I} \frac{dI}{dt} \right\}M_{st}(n)\\ &+\frac{375}{GD^{2}} \frac{\varphi_{dm}(i_{dn}, I)\varphi_{de}(i_{dn}, I)} {L(i_{gn}, i_{dn}, I)}\,n -\frac{375}{GD^{2}} \frac{\varphi_{dm}(i_{dn}, I)} {L(i_{gn}, i_{dn}, I)}\,e_{2}(i_{gn}, I)=0. \end{aligned} \tag{3b} \]

It follows from equations (3a) and (3b) that, under the assumptions mentioned above, the system of Fig. 1 is represented as a dynamical system with three degrees of freedom, since it is described by three independent variables—the currents \(i_{gn}\), \(i_{dn}\), and \(I\).

The investigation of these equations in general form presents insurmountable difficulties. It is therefore natural to begin the study of the system of Fig. 1 with an investigation of the simpler particular cases that follow from it.

Let us adopt the following assumptions:

1. We shall assume that no external actions are applied to the circuits of the separately excited windings of the machines. Then \(i_{gn}=\mathrm{const}\) and \(i_{dn}=\mathrm{const}\), \(\dfrac{di_{gn}}{dt}=0\) and \(\dfrac{di_{dn}}{dt}=0\), and the corresponding terms of equations (3a) and (3b) containing these derivatives become zero; \(i_{gn}\) and \(i_{dn}\) in the corresponding functions entering equations (3a) and (3b) may be regarded as parameters, which leads to the dependence of these functions on only one variable, \(I\). Then the corresponding partial derivatives become ordinary derivatives with respect to \(I\).

2. Let \(M_{st}(n)=0\), i.e., let us consider transient processes in the system of Fig. 1 in the absence of load on the motor shaft; in this case the terms of equations (3a) and (3b) containing \(M_{st}(n)\) and \(M'_{st}(n)\) become zero.

3. Let us further put \(L(i_{gn}, i_{dn}, I)=\mathrm{const}\) and \(R(I)=\mathrm{const}\); then the terms of equations (3a) and (3b) containing derivatives of \(L(i_{gn}, i_{dn}, I)\) and \(R(I)\) also become zero.

Applying what has been said to (3a) and (3b), we obtain the simplified equations describing the system of Fig. 1:

\[ L\frac{d^{2}I}{dt^{2}} +\left[\varphi'_{de}(i_{gn}, I)n+R-e'_{2}(i_{gn}, I)\right]\frac{dI}{dt} +\frac{375}{GD^{2}}\varphi_{de}(i_{dn}, I)\varphi_{dm}(i_{dn}, I)I=0; \tag{4a} \]

\[ L\frac{d^{2}n}{dt^{2}} +\left[ R-L\frac{\varphi'_{dm}(i_{dn}, I)} {\varphi_{dm}(i_{dn}, I)}\frac{dI}{dt} \right]\frac{dn}{dt} +\frac{375}{GD^{2}}\varphi_{dm}(i_{dn}, I)\varphi_{de}(i_{dn}, I)n -\frac{375}{GD^{2}}\varphi_{dm}(i_{dn}, I)e_{2}(i_{gn}, I)=0^{*}. \tag{4b} \]

\[ \text{*} \]

After what has been stated above, the corresponding functions in equations (4a) and (4b) (for example, \(\varphi_{de}(i_{dn}, I)\)) are regarded as functions of one variable \(I\) and of the constant parameters \(i_{g}\) and \(i_{dn}\), which in the present case indicate the presence in the system of Fig. 1 of independent excitation windings.

Let us consider particular cases following from equations (4a) and (4b).

1) Let the system of Fig. 1 have no independent excitation winding of the generator and let \(i_{gn}=0\); then from (4a) and (4b) we obtain equations describing the behavior of the “generator--motor” system with a series generator and a compound-excited motor:

\[ L\frac{d^2 I}{dt^2}+\left[\varphi'_{de}(i_{dn}, I)n+R-e'_g(I)\right]\frac{dI}{dt}+ \]

\[ +\frac{375}{GD^2}\varphi_{de}(i_{dn}, I)\varphi_{dm}(i_{dn}, I)I=0; \tag{5a} \]

\[ L\frac{d^2 n}{dt^2}+\left[R-L\frac{\varphi'_{dm}(i_{dn}, I)}{\varphi_{dm}(i_{dn}, I)}\frac{dI}{dt}\right]\frac{dn}{dt}+ \]

\[ +\frac{375}{GD^2}\varphi_{dm}(i_{dn}, I)\varphi_{de}(i_{dn}, I)n -\frac{375}{GD^2}\varphi_{dm}(i_{dn}, I)e_g(I)=0. \tag{5b} \]

2) Let now the generator have no series excitation winding; then \(e_g=\mathrm{const}\) does not depend on \(I\) and \(e'_g(I)=0\); in this case, from (4a) and (4b) we obtain equations describing the behavior of the “generator--motor” system with an independently excited generator and a compound-excited motor:

\[ L\frac{d^2 I}{dt^2}+\left[\varphi'_{de}(i_{dn}, I)n+R\right]\frac{dI}{dt} +\frac{375}{GD^2}\varphi_{de}(i_{dn}, I)\varphi_{dm}(i_{dn}, I)I=0; \tag{6a} \]

\[ L\frac{d^2 n}{dt^2}+\left[R-L\frac{\varphi'_{dm}(i_{dn}, I)}{\varphi_{dm}(i_{dn}, I)}\frac{dI}{dt}\right]\frac{dn}{dt} +\frac{375}{GD^2}\varphi_{dm}(i_{dn}, I)\varphi_{de}(i_{dn}, I)n- \]

\[ -\frac{375}{GD^2}\varphi_{dm}(i_{dn}, I)e_g(i_{gn})=0. \tag{6b} \]

3) Let the motor have no independent excitation winding, while the generator has compound excitation. Then \(i_{dn}=0\), and from (4a) and (4b) we obtain the equations:

\[ L\frac{d^2 I}{dt^2}+\left[\varphi'_{de}(I)n+R-e'_g(i_{gn}, I)\right]\frac{dI}{dt}+ \]

\[ +\frac{375}{GD^2}\varphi_{de}(I)\varphi_{dm}(I)=0; \tag{7a} \]

and

\[ L\frac{d^2 n}{dt^2}+\left[R-L\frac{\varphi'_{dm}(I)}{\varphi_{dm}(I)}\frac{dI}{dt}\right]\frac{dn}{dt} +\frac{375}{GD^2}\varphi_{dm}(I)\varphi_{de}(I)n- \]

\[ -\frac{375}{GD^2}\varphi_{dm}(I)e_g(i_{gn}, I)=0, \tag{7b} \]

describing the “generator--motor” system with a compound-excited generator and a series-excited motor.

4) In the absence of independent excitation windings both in the motor and in the generator, \(i_{gn}=0\) and \(i_{dn}=0\), and from (4a) and (4b) we have equations for the “generator--motor” system with series-excited machines:

\[ L\frac{d^2 I}{dt^2}+\left[\varphi'_{de}(I)n+R-e'_g(I)\right]\frac{dI}{dt} +\frac{375}{GD^2}\varphi_{de}(I)\varphi_{dm}(I)I=0; \tag{8a} \]

\[ L\frac{d^2 n}{dt^2}+\left[R-L\frac{\varphi'_{dm}(I)}{\varphi_{dm}(I)}\frac{dI}{dt}\right]\frac{dn}{dt} +\frac{375}{GD^2}\varphi_{dm}(I)\varphi_{de}(I)n- \]

\[ -\frac{375}{GD^2}\varphi_{dm}(I)e_g(I)=0. \tag{8b} \]

5) In the absence of an independent field winding in the motor and a series winding in the generator, from (4a) and (4b) we obtain equations for a system consisting of an independently excited generator and a series-excited motor:

\[ L\frac{d^2 I}{dt^2}+ \left[\varphi'_{de}(I)n+R\right]\frac{dI}{dt} +\frac{375}{GD^2}\varphi_{de}(I)\varphi_{dm}(I)I=0; \tag{9a} \]

\[ L\frac{d^2 n}{dt^2}+ \left[ R-L\frac{\varphi'_{dm}(I)}{\varphi_{dm}(I)}\frac{dI}{dt} \right]\frac{dn}{dt} +\frac{375}{GD^2}\varphi_{dm}(I)\varphi_{de}(I)n - \frac{375}{GD^2}\varphi_{dm}(I)e_z(i_{gn})=0. \tag{9b} \]

6) Let, further, the motor have no series field winding, while the generator has compound excitation; let \(i_{dn}=i_{d\mathrm{nom}}\); then \(\Phi_d=0\), \(\Phi_{dn}=\Phi_{d\mathrm{nom}}\), and from (6) we obtain:

\[ e_d=\varphi_{de}(i_{dn})n=c_e n, \tag{e} \]

whence it follows that

\[ \varphi_{de}(i_{dn})=c_e=\mathrm{const} \quad\text{and}\quad \varphi'_{de}(i_{dn})=\frac{dc_e}{dI}=0. \]

In exactly the same way, from (d) it follows that

\[ M=\varphi_{dm}(i_{dn}, I)I=\varphi_{dm}(i_{dn})I=c_m I \tag{zh} \]

and, consequently,

\[ \varphi_{dm}(i_{dn})=c_m=\mathrm{const} \quad\text{and}\quad \varphi'_{dm}(i_{dn})=\frac{dc_m}{dI}=0; \]

then, using (e) and (zh), from (4a) and (4b) we obtain equations for the system “compound generator—independently excited motor”:

\[ L\frac{d^2 I}{dt^2}+ \left[R-e'_z(i_{gn}, I)\right]\frac{dI}{dt} +\frac{375}{GD^2}c_e c_m I=0; \tag{10a} \]

\[ L\frac{d^2 n}{dt^2} +R\frac{dn}{dt} +\frac{375}{GD^2}c_m c_e n -\frac{375}{GD^2}c_m e_z(i_{gn}, I)=0. \tag{10b} \]

1. Under the same conditions as in item 6, but in the absence of an independent field winding in the generator (\(i_{gn}=0\)), from (4a) and (4b) we obtain equations for the system “series generator—independently excited motor”:

\[ L\frac{d^2 I}{dt^2} +\left[R-e'_z(I)\right]\frac{dI}{dt} +\frac{375c_m c_e}{GD^2}I=0; \tag{11a} \]

\[ L\frac{d^2 n}{dt^2} +R\frac{dn}{dt} +\frac{375c_m c_e}{GD^2}n -\frac{375c_m}{GD^2}e_z(I)=0. \tag{11b} \]

1. Finally, for an independently excited generator and motor, from (4a) and (4b) we have the equations:

\[ L\frac{d^2 I}{dt^2} +R\frac{dI}{dt} +\frac{375c_m c_e}{GD^2}I=0; \tag{12a} \]

\[ L\frac{d^2 n}{dt^2} +R\frac{dn}{dt} +\frac{375c_m c_e}{GD^2}n -\frac{375c_m}{GD^2}e_z(i_{gn})=0. \tag{12b} \]

from which the well-known equations can easily be obtained:

\[ \frac{d^{2}I}{dt^{2}}+\frac{1}{T}\frac{dI}{dt}+\frac{1}{TB}I=0; \]

\[ \frac{d^{2}n}{dt^{2}}+\frac{1}{T}\frac{dn}{dt}+\frac{1}{TB}n-\frac{1}{TB}\frac{e_z}{c_e}=0. \]

The various variants of the simple “generator–motor” system shown in Fig. 1, described by equations (4a)—(12b), are, for \(i_{zn}=\mathrm{const}\) and \(i_{sn}=\mathrm{const}\), dynamical systems with one degree of freedom. The simplest of them, described by equations (12a), (12b), is a linear system, and therefore self-oscillations are impossible in it.

The next system [item 7 and equations (11a), (11b)] is already nonlinear, but the simplest one, since equation (11a) contains only one second nonlinear term, whose nonlinearity is determined by the form of the magnetization characteristic \(e_z=e_z(I)\). Owing to the latter, this system allows the possibility of the occurrence of self-oscillations considered in [1, 2, 3, 4]. As was indicated in the cited literature, these self-oscillations, at small inductances, may be accompanied by reversal of the motor, while at large \(L\) they may turn out to be one-sided (without change of sign) with respect to the motor speed.

The transition from the system of item 7 to the system of item 6 does not introduce superfluous nonlinearities and is accompanied by a transition of self-oscillations with motor reversal into one-sided self-oscillations; at large ampere-turns of the independent excitation winding of the generator these self-oscillations may disappear altogether. This case is considered in [5].

Adding to the motor in the system of item 7 a series excitation winding, we obtain the system described in item 1. This system is more complex, since it already contains two nonlinearities: in the second and third terms of equation (5a). As was indicated in [6], in such a system self-oscillations with motor reversal and with unequal half-periods arise, since in one half-period the magnetic flux characterizing the third term in (5a) reaches its greatest value, and in the other its least.

The system described in item 4 is also self-oscillatory, since in (8a) there is a nonlinearity in \(\frac{dI}{dt}\), inherent in the magnetization characteristic of the generator. The third term in (8a) is also nonlinear, and this system also contains two nonlinearities. Some self-oscillatory properties of this system are described by us in the article “On self-oscillations in a generator–motor system with series-excitation machines,” published in the present collection. The system described in item 3, being a variety of the system mentioned in item 4, is also self-oscillatory to the extent that the independent excitation winding of the generator retains the nonlinearity of the magnetization characteristic of the generator and, consequently, the nonlinearity of the second term in (7a).

The system of Fig. 1 also retains self-oscillatory properties, since the second term in (4a) is nonlinear on account of the magnetization characteristic of the generator; at the same time it is also a system with two nonlinearities.

As for the systems described in items 2 and 5, no self-oscillations are observed in them, since in them \(e_z=\mathrm{const}\) and in (6a) and (9a) the nonlinearity of the second term is not determined by the magnetization characteristic of the generator.

In the case of variable inductance the system of Fig. 1 will be a second-order system with one degree of freedom, containing three nonlinearities. Thus:

1) the system of Fig. 1 is the most general case of a simple self-oscillatory “generator—motor” system with one degree of freedom, under the assumptions set forth above;

2) the study of self-oscillations in such a complex system should expediently begin with the study of the simplest system mentioned in Sec. 7, gradually complicating the problem by taking account of additional nonlinearities.

Literature

1. Gandzha L. I., Physics of oscillations in the “generator with series—motor with independent excitation” system, Izv. TPI, vol. 76, 1954.

2. Gandzha L. I. and Potekhin Yu. I., On transient processes and oscillations in the “generator with series—motor with independent excitation” system, Izv. TPI, vol. 76, 1954.

3. Gandzha L. I., Linear approximation of self-oscillations in an electromechanical system, Izv. TPI, vol. 82, 1956.

4. Gandzha L. I., The influence of inductance on the form of self-oscillations in an electromechanical system, Izv. TPI, vol. 82, 1956.

5. Gandzha L. I. and Sevastyanov V. A., On self-oscillations in an electromechanical system with an electric-machine amplifier, Izv. TPI, vol. 82, 1956.

6. Gandzha L. I., The self-reversing drive as an element of electric-machine automation, dissertation, Tomsk, 1948.